t) YE ul t NHS AKE len iy Oa SORES PA oh : a CANS ENN yi ‘ MA a ti SEAN Mh STAN AN 15) Ras UAH NEARER ES ao a COE aa pe y a YY ye sae s } wi i NN bat AGS i vey me AS tS aN Tie Non eee ii a ‘ a Re is x as x st) ee SEE PH Beg a ea Ne Gueieran f Bt Renee tiie i beet A i CEU HESS ior Rae y, Pip oaers centri diate . LD Hi ’ Gen ip Dice. fa se is a Dt IEAM Rise arse (ees a i Ea ei us a Pe inane Pui es ye Cornell University Library | -Sasmuen i Report upon t — 3 192 PROFESSIONAL PAPERS OF THE CORPS OF ENGINEERS OF THE UNITED STATES ARMY. PUBLISHED BY AUTHORITY OF THE SECRETARY OF WAR. HEADQUARTERS CORPS OF ENGINEERS, 1882. PROFESSIONAL PAPERS OF THE CORPS OF ENGINEERS, U. 8S. ARMY. No. 24. REPORT UPON THE PRIMARY TRIANGULATION OF THE UNITED STATES LAKE SURVEY. BY LIEUT.-COL. GC. B. COMSTOCK, CORPS OF ENGINEERS, BREVET BRIGADIER-GENERAL, U. S. A., ALWED BY THE ASSISTANTS ON THE SURVEY. WASHINGTON: GOVERNMENT PRINTING OFFICE. 4, 1882. lit LETTER OF "TRANSMITTAL. LAKE-SURVEY OFFICE, Detroit, Mich., August 12, 1882. General H. G. WRIGHT, Chief of Engineers, U. S. A.: GENERAL: I have the honor to submit herewith the final report of the Lake Survey. The original observations have not beén given in detail from the fear of making the report too bulky. But the note-books containing the original observations, as well as the reductions, will be filed in your office. It had been hoped to express all distauces in the report in terms of the international metre to be adopted by the Bureau International des Poids et Mesures, but the delay in the preparation of that metre has made it impracticable, and accordingly all distances are given in English feet. The value of the Lake-Survey metre, 21876, is probably known in terms of what the international metre will be within one or two thousandths of a millimeter, and this value is given in an appendix to the report. R1876 has been sent to the Bureau International, des Poids et Mesures for an accurate determination of its length. When this is known, the report contains the data for expressing all its distances in terms of the metre. The results for the difference of longitude of Detroit and Cambridge were not available in time for insertion in the body of the report. They are given in an appendix, and show that no modification of the longitude of Detroit with reference to Washington, as given in the report, is necessary. In the preparation of the report, while all the assistants have been engaged in the work, I have been especially indebted to Mr. R. S. Woodward, who has aided in preparing the manuscript and has attended to the proof-reading. The following chapters of the report were prepared under my supervision entirely or mainly by the persons named: Chapter I by First Lieutenant P. M. Price, U. S. Engineers. Chapter VIII by Assistant Engineer E. 8S. Wheeler. Chapters XVI to XX, and Chapters XXIV and XXV by Assistant. Engineer R. S. Woodward. ~ Chapter XXII by Assistant Engineer L. L. Wheeler. I wish to call your attention to the valuable services of the following assistants, who have remained upon the work till its close. E. S. Wheeler, employed in measurement of bases and comparisons of standards. A. R. Flint, O. B. Wheeler, R. S. Woodward, and J. H. Darling, employed in primary trian- gulation. T. Russell and T. W. Wright, employed in computation. To these gentlemen, and to those assistants, whether civilians or officers of Engineers, who have previously been employed on the Survey while it has been under ny charge, [ wish to tender my thanks for their efficient aid and for their cordial co-operation in all attempts to make the work precise and reliable. Very respectfully, your obedient servant, C. B. Comstock, * LTieutenant-Colonel of Engineers, Brevet Brigadier-General, U. 8. A. Vv TABLE OF CONTENTS. Part I.—HISTORY. CuaPTER I.—HISTORICAL ACCOUNT OF THE SURVEY OF THE NORTHERN AND NORTHWESTERN LAKES, MAY, 1841, TO JULY 1, 1881. Antroductory, OL s2ccsisscsdsig destin ks alejsinG sewiowin csv weseiscisee as cease Seen sa cles sap cee Weisiane Heoesle ce eeec ey vem see The Survey under the charge of Captain W. G. Williams, 184145, § 2.....-...--.2. 222222 eee eee eee eee eee Methods of survey, 3. Progress of field-work, 3. The Survey under the charge of Lieutenant-Colonel James Kearney, 184551, § 3.......... 22-200. eee ne eee eee Methods of survey, 5. Progress of field-work, 5. The Survey under the charge of Captain J. N. Macomb, 185156, § 4.-..--.. .. 2222 eee ee cee een cece eee - Distribution of published charts, 6. Methods of field-work, 6. Progress of field-work, 8. The Survey under the charge of Lieutenant-Colonel James Kearney, 1856-57, § 5..---. 052. .---2. eee ee eee eee The Survey under the charge of Captain George G. Meade, 1857-61, § 6-2... 02-0. eee eee ce eee ee eee eee Methods of survey, 10. Progress of field-work, 12. The Survey under the charge of Colonel J. D. Graham, 1861-64, § 7 ...... 2-22. cee ee eee ce eee cece eens / Office work, 15. Methods of survey, 15. Progress of field-work, 16. The Survey under the charge of General W. F. Raynolds, 186470, § 8...-2. 2-22. eee 8 ne cee ee ce ee eee Methods of survey, 17. Progress of field-work, 19. The Survey under the charge of General C. B. Comstock, and under the temporary charge of Captain H. M. Adams: 1620; 60, close: OF Surveys 0) Oi. 6s esis see etme sine seesinise eed cece dD ecemei-oe sls Geinbedieecagye sales OPT C Sr WOT ME acces ssn canbe. ct mies ie Ne ase wie BGS cle iS aIziaS: De paIS. Sse im cepa Simla aia aS iwuiey a isiaiessie ane Mea exes eisai. aaa alee nis weialsuasieciale Methods of fidld-WoOrk’.. 22 ...<0 . opposite 110 Herculis, 6th column, read -+0.14 for —0.14. under OBSERVATION-EQUATIONS, Ist equation, read -+A0@ for Ad, under OBSERVATION-EQUATIONS, 5th equation from last, read -+A@ for —d4. under OBSERVATION-EQUATIONS, 10th equation from last, read r=—0.01 for c=-+0,01. under OBSERVATION-EQUATIONS, last equation, read --4.74a. 3d line, 7th column, read +0.20 for —0.20. y 28th line, lst column, read 7 Herculis. 2d line, 10th column, read 048.8 for 05*.8. under OBSERVATION-EQUATIONS, 5th equation from last, read +A0 tor —Aé. under OBSERVATION-EQUATIONS, 3d equation from last, read —0.06 for + 0.06. under NORMAL Equations, 2d equation, 3d term, and 3d equation, 2d term, read —2.44 for +2.44. heading to 6th column, read C(e-+Ai + ab’n). . opposite « Aquilae, 5th column, read +0.14 for —0,14. under OBSERVATION-EQUATIONS, second equation from last, under JWeight, read 0.06 for 0.006, under RESULTs, read Ad?=—08.005 for A@=-+ 08,005. 4th line, 6th colunm, read 08*.000 for 688.000. 6th line from bottom, 8th column, read 03°.389 for 05*.389. 2d line, 7th column, read 418.000 for 418.010. 15th line of §4, read determined with Needle No.2 for that are marked with a star. Ist column, 4th item, read Saint Louis, Mich., for Saint Louis, Mo. the following items in last column should be aligned to read as follows: Place. Observer. Kenosha From a survey under Captain Cuyler. Benona G. A. Marr. Seven miles south of Sheboygan J. P. Mayer. Two miles south of Manitowoc J. R. Mayer. 1st column, 8th line from bottom, read Port Clinton for Point Clinton. under LAKE ONTARIO, 3d column, 2d line from bottom of table, read 77° 43’ for 78° 43/. PA RE T3 HISTORY. ‘ CHAPTER I. HISTORICAL ACCOUNT OF THE SURVEY OF THE NORTHERN AND NORTH- WESTERN LAKES—MAY, 1841, TO JULY 1, 1881. INTRODUCTORY. §1. The Lake Survey was begun in 1841 under an appropriation of $15,000, made in May of that year. At this time the country bordering on the lower lakes was already pretty well settled and works for the improvement or formation of harbors had been commenced at most of the im- portant points on Lakes Erie and Ontario. The upper-lake region was but thinly settled, and there were no good harbors on Lake Huron, and but one, the harbor of Chicago, on Lake Michigan. Settlers were, however, pouring in rapidly, and there was even then a large and constantly increas- ing commerce between the lake ports, especially from Buffalo to Detroit and Chicago. -Communi- cation with Lake Superior could only be had by portage around the Sault Ste. Marie, but the great mineral wealth of the Lake Superior country was attracting attention, and a survey for a ship- canal had been made in 1840 by officers of the Topographical Engineers. The lake commerce was carried on under many difficulties, which caused much loss of life and property each year. There were no charts of the lakes except the admiralty charts compiled from the surveys of Captain H. W. Bayfield, of the Royal Navy (English), and these were not in general use by the mas- ters of American vessels. These charts were the results of rapid reconnaissances, and although they showed the coast-lines with an accuracy which is remarkable considering the rough methods of surveying employed, they were of little value as hydrographical charts of the American coast, because they showed the depths of water in comparatively few places, and but a small number of the many reefs and shoals which are found along the lake shores. There were few light-houses and beacons to indicate the positions of dangers to navigation, and, in the absence of charts, pilots were obliged to rely upon their own knowledge, which was fre- quently only acquired by the vessel’s grounding on a shoal or striking a hidden rock. The navigation of the lakes is attended with peculiar dangers, because, while violent gales are frequent and the storms rival those of the ocean itself, a vessel is never more than a few hours’ run from the shore, and cannot, as is generally the case at sea, drift before the wind until the storm is over, but in a long-continued gale must be thrown upon the shore, unless a port or harbor of refuge can be entered. In 1841, a vessel leaving Chicago found no harbor or shelter in storms until the Manitou or Beaver Islands were reached, and after passing the Straits of Mackinac, it was again exposed without refuge on Lake Huron, except in the vicinity of Presqw Isle, until the head of the Saint Clair River was reached. In sailing from Chicago to Buffalo, the greatest difficulties were encountered in the vicinity of the Straits of Mackinac and in the west end of Lake Erie, on account of the many islands, shoals, and reefs found in those localities; and at the mouth of the Saint Clair River, at which no improvments had been made in 1841, and where the channels were not only ILS 2 HISTORICAL ACCOUNT [Cuar. I, circuitous and narrow, but so shoal, that vessels in low-water seasons frequently were compelled to have their cargoes taken over the bars in lighters. It was therefore with the double object of furnishing reliable charts to lake vessels, and of determining from the surveys the works of improvement which were necessary to the prosperity of the lake commerce, that Congress in 1841 directed a survey of the lakes, and that annual appro- priations, with the single exception of the year 1847, have since been made for carrying on the Survey. Some idea of the magnitude of the work may be had from the following dimensions: The American shore-line of the Great Lakes and their connecting rivers, if measured in steps of 25 miles, is about 3,000 miles, but if the indentations of the shore and the outlines of the islands be included, the developed shore-line is about 4,700 miles in length. Along rivers, and where a lake is narrow, it is necessary for navigation that both shores be mapped. This increases the length of the shore-line to be surveyed between Saint Regis, N. Y., and Duluth, Minn., to about 6,000 miles.* During the first ten years of the Survey, whilst a general geodetic survey of the entire chain of lakes was contemplated for the future, the actual operations were mainly confined to surveys of special localities where improvements were called for or where the navigation was difficult; and where the surveys were more extended they were little more than reconnaissauces. This course was made necessary because the appropriations were inadequate to the purchase of the finer instru- ments and the support of the larger force necessary for more extensive and more exact surveys, and also because of the pressing need of improvements at particular localities, for which prelimi- nary surveys. were essential. The Survey from the beginning has been conducted under the War Department, at first by the Chief of Topographical Engineers, and, after the consolidation of that corps with the Corps of Engineers in 1863, by the Chief of Engineers. An officer of engineers has always been in immediate charge of the Survey, and during the earlier years the assistants were almost exclusively engineer officers, but as the scope of the Survey was enlarged, and more assistants were required than could be spared from among the officers of engineers, civilian assistants were employed, many of whom have served for a long number of years on the Survey. In the following account of the progress of the Survey, the narrative will be divided into eight sections, corresponding to the changes in the officers in charge of the Survey. The following tables are appended : Catalogue of published charts. Statement of the annual appropriations. Officers in charge of the Survey. Officers who have served as assistants on the Survey. - . Civil assistants employed on the Survey. . Showing the annual issue of charts. So Rw E THE SURVEY UNDER THE CHARGE OF CAPT. W. G. WILLIAMS. May, 1841 To 1845. § 2. Colonel J. J. Abert, Chief of Topographical Engineers, in a letter of instructions, dated May 17,1841, directed Captain Williams to take charge of the Survey of the Northern and North- western Lakes. Captain Williams was then general superintendent of harbor improvements on Lake Erie, with an office 1t Buffalo, N. Y. He still retained the charge of these harbor works, and the office of the Lake Survey was therefore first established at Buffalo. Captain Williams remained in charge of the Survey until the fall of 1845. His annual reports to the Chief.of Topographical Engineers, and the reports of the latter to the Secretary of War, furnish the data for the following account of the Survey under his administration. The reports are very meager as respects the details of the methods of surveying and of the instruments employed. They only furnish a brief statement of the field of operations, and are mainly devoted to showing the capabilities for improvement of the different localities, and the necessity of such improvements in the interests of commerce. There are, however, in the Lake-Survey office, a few of the field note- * From memoranda respecting the Lake Survey by General C. B. Comstock, published in the Report of the Chief of Engineers for 1875. § 2.) OF THE UNITED STATES LAKE SURVEY. 3 books used by Captain Williams and his assistants, and from them it appears that the general method of making the surveys was as follows: 6 METHODS OF SURVEY. The astronomical observations for time and latitude were made with a sextant, the differences of longitude being determined apparently by the chronometric method. Azimuths were determined by observations on Polaris or the sun. : Base-lines for the triangulation were generally measured with three well-seasoned wooden rods, each about 10 feet long. These were either laid upon stakes previously leveled, or had grooves cut along their under sides so that they could be laid upon a tightly-stretched rope supported upon stakes driven at suitable intervals. When the stretched rope was used, the base was measured in sections of about 500 feet each. The Mackinac base was measured in 1844 with an iron-bar appara- tus devised by Lieutenant Gunnison. There were four bars, each 10 feet long, 14 inches wide, and 3 inch thick, having a thermometer placed at the middle, with its bulb sunk into the bar. During the measurement each of these bars rested on five rollers, which were mounted on a carriage of the best mahogany wood a few inches shorter than the iron bar and stiffened by iron plates. The rollers at the ends of the carriage had adjusting screws, by which a longitudinal or a cross motion could be given to the bar. Each carriage was supported by two tripods, with adjustable heads for leveling. When measuring, the bars were made level by means of a 12-inch striding level. Contact was made by bringing the rear end of the bar so that it just touched a hair sus. pended from the forward end of the preceding bar, the hair being made vertical by an attached ~plumb-bob swung in water. Minor bases were measured with the ordinary surveyor’s chain. The angles of the triangulation were measured with theodolites reading to 5 or 10 seconds by verniers. Generally, the pointings were made in succession at the stations around the horizon, each angle being read twice. By some observers the telescope was reversed, and the setting of the horizontal limb changed for the second set of readings. The triangulation stations consisted of a single center-post of the requisite height, stiffened by braces from the ground. To support the platform for the observer a pyramidal frame-work, composed of four corner-posts and the necessary braces, was built around the center-post, but the platform and its supports were entirely discon- nected from the center-post. The corner-posts of the pyramid were extended until they met above the center-post, and a few feet of the upper portion of the pyramid was boarded up and white- washed, and thus served as a target. There is no record of these stations having been permanently marked. Heights were determined by barometrical measurements or leveling with a Y-level. The shore-line was surveyed either with a compass and chain or with a theodolite and chain, points of the shore being determined by chained off-sets at right angles to the compass or theodo- lite courses. The principal features of the topography in the immediate vicinity of the shore were located in the same way, or by intersections from two or more stations, and the details were sketched in. But little topographical work, however, was done, beyond the rough sketching of the general character of the shores. For the hydrographical work the soundings were taken either from a row-boat or a steamer, the boats running on known courses, or between buoys and stations on the shore, and the soundings being taken at regular intervals of time. The soundings with row-boats, or the in-shore hydrog- raphy, were usually made by the parties called shore-parties, who surveyed the shore-line in con- nection with the latter work. The need of a steamer in making soundings over large areas, and for moving and supplying the various parties, was early felt, and in his report for 1842 Captain Williams submitted an estimate for the building of an iron steamer. The steamer was not finished until the spring of 1844, and in the mean time, as the surveys went on, permanent points and stations were located, to be afterwards used in the hydrographic work. The steamer was at first called the Abert, but her name was subsequently changed to the Surveyor. PROGRESS OF THE FIELD-WORK. The season for field operations was usually about five months, from May to October, the remaining seven months being spent in the office in making the reductions, computations, and plottings of the previous season’s work. 4 HISTORICAL ACCOUNT [Cnap. I, ¥ The following officers of the Corps of Topographical Engineers served at various times as assistants to Captain Williams on the Lake Survey, viz: Captain Howard Stansbury, Lieutenants Joseph E. Johnston, J. N. Macomb, J. H. Simpson, W. H. Warner, I. C. Woodruff, J. W. Gun- nison, J. D. Webster, J. W. Abert, and W. B. Franklin. The instructions to Captain Williams required him to establish a point of commencement for the Survey on the north extremity of the southern cape of the entrance to Green Bay, and also to make surveys of the places of difficult navigation in the vicinity of the Straits of Macki- nae. The point of commencement was selected not only on account of the importance to naviga- tion of a survey of the entrance to Green Bay, but also because it was regarded as a favorable point, from which to extend a system of triangulation to the Beaver and Manitou Islands and thence to the east shore of Lake Michigan. All this region of country was heavily timbered, and a great deal of labor was required to clear lines for the measurement of bases and for the trian- gulation. During the summer of 1841 a detailed topographical survey of the island of Mackinac was completed, reconnaissance surveys were made in the northern part of Lake Michigan, and a site for a base-line near the entrance to Green Bay was selected and partly cleared. A better view of the work of the Survey can perhaps be presented by tracing separately the progress made at the several localities than by giving a detailed narrative of the operations of each successive year. ; 1. Green Bay.—During the seasons of. 1842 and 1843 Lieutenant Simpson finished clearing and measured the base-line at the entrance, and located and built triangulation stations on both shores of the bay and on the islands at its mouth. He also read the angles at many of these stations. The survey of the shore-line on.the southeast side of the bay was completed from the town of Green Bay to and around the Porte des Morts entrance. In 1845 Captain Williams and Lieutenant Gunnison, on the steamer Abert, were engaged in triangulation and hydrographic work on Green Bay. s 2. Straits of Mackinac.—For this survey a base-line about 6 miles long was selected on the south shore of the straits, the west end being near McGulpin’s Point and the line running back of Old Fort Mackinac. The line having been cleared in previous years, the base was measured by Lieutenant Gunnison in the summer and fall of 1844 with the apparatus previously described. In 1842 and 1843 Lieutenant Woodruff was at work about the straits. He was engaged in both trian- gulation and shore-line work, and also assisted Captain Williams in astronomical observations. The shore-line was surveyed down as far as the entrance to Grand Traverse Bay, and about thirty miles of the northern shore of the straits was also surveyed. 3. Lake Michigan.—In 1842 and 1843 Lieutenants Gunnison and Webster ran.a line along the western shore from Chicago northward until they connected with Lieutenant Simpson’s survey of the shore-line of Green Bay. In 1843 Lieutenant Gunnison also made a survey of Grand River on the east side of the lake. The shore-line in the vicinity of Saint Joseph was surveyed by Lieutenant Franklin in 1844. These surveys were made with especial reference to harbor improve- ments, for which plans and estimates were made by the same officers. In 1843 and 1844 Lieutenant Macomb, assisted in the first year by Lieutenant Abert, nade surveys in the vicinity of Grand Traverse Bay and the Manitou Islands, connecting the latter by triangulation with the main shore. 4, Lake Saint Clair.—In 1842 Lieutenants Macomb and Warner made a survey cf the delta of the Saint Clair River with a view to plans for the improvement of the channels. 5. Lake Erie-—In 1844 Captain Williams, assisted by Lieutenants Simpson and Woodruff, on the steamer Abert, was engaged in surveying the harbors on this lake. In 1845 the survey of the west end of Lake Erie, embracing the area west of a line from Sandusky to Pointe Pelée, was commenced by Lieutenants Macomb and Woodruff. A base-line for the trigonometric survey of the islands was measured on South Bass Island, and considerable triangulation and shore-line work was effected during the season. In his annual report for 1845, Colonel Abert states that all the lake harbors, except those upon Lake Superior, have been surveyed, and that he is prepared, if authorized by Congress, to compile and publish a portfolio of them. §3.] OF THE UNITED STATES LAKE SURVEY. 5 THE SURVEY UNDER THE CHARGE OF LIEUT.-COL. JAMES KEARNEY. 1845 TO APRIL 9, 1851. § 8. Lieutenant-Colonel Kearney assumed charge of the Survey in the fall of 1845. During his administration the office of the Survey was removed to Detroit, where it has since remained. The data for an account of the Survey while under his charge are the brief summaries of opera- tions contained in the annual reports of the Chief of Topographical Engineers for 1846, 1848, and 1850, the annual report of Lieutenant-Colonel Kearney for 1849, and the incomplete set of note- books in the office records. There is no report for 1847. METHODS OF SURVEY. The methods of making the surveys were the same as those employed under Captain Williams, except that the triangulation seems to have been more carefully executed, and a greater number of readings was taken in the measurement of angles. In his report for 1849 Colonel Kearney urges very strongly upon the bureau the necessity for the purchase of improved instruments suitable for a geodetic survey, and asks for an appropriation of $10,000 for that purpose. He states that the Survey was at that time, and had from the begin- ning, been totally destitute of any astronomical and geodetic instruments, and nearly destitute of any topographical or hydrographical instruments that were fit to be taken into the field. PROGRESS OF THE FIELD-WORK. On account of the demand for officers of the Topographical Engineers for service with the armies in Mexico, there were but three officers, besides the superintendent, on the Survey in 1846, 1847, and 1848, viz, Lieutenants Macomb, Woodruff, and Gunnison in 1846, and Lieutenants Macomb, Gunnison, and Scammmon in 1847 and 1848. The operations for these years were restricted, with the exception of the examination of a few points on Lake Ontario and a reconnaissance of Lake Champlain made in 1846, to the completion of the survey of the west end of— 1. Lake Hrie——In this work the triangulation was mostly done by Lieutenant Macomb, and the topography and hydrography by Lieutenants Woodruff, Gunnison, and Scammon, assisted by Messrs. R. W. Burgess, J. F. Peter, and J. H. Forster. The survey of the west end was finished in 1848, and the drawings were forwarded to the bureau at Washington for compilation and engrav- ing in 1849. A chart of the whole of Lake Erie on a scale of 1: 400,000, which had been compiled in the Lake-Survey office, was also forwarded to Washington in the same year. The engraving of these two charts, and of one of Kelley’s and the Bass Islands on a larger scale, was completed, and their systematic distribution to navigators was begun in 1852. 2. Straits of Mackinac.—Colonel Kearney was directed to resume the survey of the straits in 1849. One triangulation party on the steamer, now called the Surveyor, under Lieutenant Macomb, and five topographical and hydrographical parties, under Lieutenant Scammon and Messrs. Hough- ton, Hearding, Burgess, and Potter, were sent into the field in the spring. Lieutenant Macomb was occupied during the season in reconnoitering for stations for the primary triangulation, clear- ing lines of sight, building stations, and preparing the base-line formerly used by Captain Williams for remeasurement. He also assisted the shore-parties in surveying shoals and reefs distant from the land. The three shore-parties first mentioned, under the direction of Lieutenant Scammon, made a survey based upon a secondary and tertiary triangulation of the shore-line and the adjacent islands from Point Saint Ignace on the north shore to the Chenaux group of islands. The other two parties, under Mr. Burgess, surveyed Bois Blane and Round Islands and the adjacent waters. 3. Miscellaneous.—In 1850 the appropriation for the Lake Survey was not made until the 28th of September, too late to allow the survey of the Straits of Mackinac to be resumed that season. The only field-work accomplished, therefore, was a survey of the Sandusky River up to Fremont, Ohio, and a survey of the harbor of Port Clinton, Ohio, both of which were made late in the fall, in obedience to orders from the Topographical Bureau. 6 HISTORICAL ACCOUNT [Cnap. I, THE SURVEY UNDER THE CHARGE OF CAPT. J. N. MACOMB. APRIL 9, 1851, TO SEPTEMBER, 1856. § 4. Captain Macomb assumed charge of the Lake Survey on the 9th of April, 1851, and in this year the Lake Survey proper may be said to have been begun, as nearly all the localities sur- veyed in previous years have since been resurveyed with greater accuracy than was possible with the means available when the original surveys were made. Captain Macomb’s letter-book is among the records of the office. It contains copies of official letters to the Topographical Bureau, the monthly reports of the progress of the work, and the annual reports of Captain Macomb. These furnish the materials for the following account of the survey under his administration. Larger appropriations being granted by Congress, Captain Macomb was enabled to procure better instruments, to introduce improved methods, and to prosecute the work more systematically than had been possible in the earlier years of the Survey. He was also able to employ a greater nuinber of assistants, who, starting on the Survey in subordinate positions, were, as they acquired experience in its different branches, promoted to more responsible positions, until they finally became chiefs of parties. DISTRIBUTION OF THE PUBLISHED CHARTS. The engraving of the three Lake Erie charts prepared under the direction of Lieutenant- Colonel Kearney having been finished in 1852, their systematic distribution to vessels was begun in the same year. The regulations then adopted for the issue of the charts are still in force, and provide that the charts shall be furnished gratuitously to any American or Canadian vessel navigating the lakes, on the presentation of a certificate of a collector of customs stating the names of the owners, place of registration, tonnage, and certifying that the vessel’s papers are in full force. The charts were issued at the Detroit office and at an agency establisbed in Buffalo. A record is kept at both places, and duplicate charts are issued only on satisfactory proof that the originals were lost or destroyed unavoidably, or by an accident, for which no blame attaches to the owners. A table giving a list of the published charts, the scales on which they are drawn, the dates of publication, the officers under whose direction the surveys were made, the draughtsmen, and the engravers, when known, is appended, and therefore no further reference to the publication of the charts will be made. I1t should, however, be stated that of late years the practice has been to have the charts photolithographed immediately on their completion by the draughtsman, and to issue these photolithographs whilst the plates are being engraved. METHODS OF FIELD-WORK. There were two general classes of parties for,the field-work; the steamer-party for the primary triangulation and off-shore hydrography, and the shore-parties for the topographical and in-shore hydrographical work. Captain Macomb took personal charge of the first party, which usually consisted of two assist- ants and the necessary crew for the Surveyor, and, in addition to the triangulation and off-shore work, made frequent inspections of the shore-parties, furnished them with supplies, and occasion- ally moved them from camp to camp. Astronomical work.—In 1852 a field astronomical transit and a Wiirdemann zenith-telescope were procured for the Survey, and during the remainder of Captain Macomb’s superintendence determinations of longitude were made by the method of lunar culminations, and latitude determi- nations were made by Captain Talcott’s method of observing with the zenith-telescope the ditfter- ences of meridian zenith distances of pairs of stars culminating near the zenith and at nearly equal distances on opposite sides of it. The first precise astronomical work done by the Lake Survey was the determination of the latitude of Detroit in the spring of 1852, the observers being Captain Macomb and Lieutenant Raynolds. Azimuth determinations were made by observations on Polaris or other close circumpolar stars at their elongations. These were made with all the accuracy * ’ $4.) OF THE UNITED STATES LAKE SURVEY. aL required for mapping purposes, but not with that precision and attention to the elimination of instrumental errors which is necessary in the most accurate geodetic work; and this remark, in fact, applies to most of the theodolite work done previous to 1870. Base-lines.—A_ great advance was made in the accuracy of the triangulation work by the pur- chase, in 1852, of a Bache-Wtirdemann compensating base-apparatus. This was constructed for the Survey under the direction of Captain Thomas J. Lee, of the Topographical Engineers, and was in all essential particulars similar to that previously made by Mr. William Wiirdemann for the Coast Survey, of which a description, with plates, is given in the Coast-Survey Report for 1854. The modifications introduced in the Lake-Survey apparatus for the purpose of making it lighter and smaller consisted mainly in the reduction of the length of the tubes from 20 feet to 15 feet. It is a contact apparatus, the measurement of a base being made with two tubes, each made up of a bar of iron and a bar of brass firmly joined together at one end, and carrying at the other a com- pensating lever so constructed that the tube shall have nearly the same length at all temperatures. The apparatus was accompanied by a 15-foot standard brass bar, whose length and coefficient of expansion were determined by Wiirdemann. It was first used on the Mackinac base in 1852, this being the only primary base measured during Captain Macomb’s charge of the Survey. Primary triangulation.—For this work a 10-inch Gambey repeating theodolite, reading by two verniers to 5 seconds, was generally used, although some of the work was done with another 10-inch instrument made by Wiirdemann, the limb of which was originally the vertical limb of the first- mentioned instrument. Several of the smaller Wiirdemann and Gambey theodolites were also occasionally used. The 10-inch Gambey was a very good instrument, and remained in constant use on the Survey until it was taken to the Mississippi River in 1876, and has since then been used in the secondary triangulation of that river. All the angles of the main triangulation seem to have been read by the principle of repetition, but there seems to have been no systematic dis- tribution of the readings on different parts of the limb in order to eliminate periodjc and accidental errors of graduation. The stations were built as already described, and were placed near the shore in order to be made available in locating the steamer-soundings. The longest lines did not exceed 20 miles, and were usually much shorter. Off-shore hydrography.—The general method of running the lines of off-shore soundings was to anchor a buoy at each extremity of the line in water sufficiently deep for the steamer to pass round it, aud to take the soundings at regular intervals of time as the steamer passed over the line between the buoys, running at a speed of about 4 miles an hour. The position of the steamer at the time of taking a sounding was also frequently determined by reading sextant angles between stations or other objects on shore. The positions of shoals and reefs were indicated by placing upon them ‘tripod stations, which were located by intersections from the stations on shore. These tripods were made of such a height as to stand two-thirds out of water when placed in position, and were secured by piling stones on a platform built between the legs of the tripod above the water. They not only answered the purpose for which they were mainly intended, but also served in the absence of the buoys, which of late years have been placed in such positions by the Light-house Establishment as beacons to warn navigators of the positions of dangerous places, and were much appreciated by the masters of vessels. On several occasions, before proceeding to his field of operations in the Straits of Mackinac, Captain Macomb went to the west end of Lake Hrie to place these tripods on shoals already known, and to locate and mark others which had been discovered since the original surveys. As the operations of the Survey were enlarged, it became evident that more than one ° steamer was required for the vigorous and economical prosecution of the work, and accordingly Captain Macomb inserted an item for the building of an iron steamer in his estimates for 1854. This side-wheel steamer, having a length of 143 feet, and a beam of 21 feet, was completed and turned over to the Survey in July, 1856. She was called the Search, a name appropriate to one of her most important uses, that of seeking out and exposing hidden dangers. Topography and in-shore hydrography.—Kach shore-party consisted of a chief of party, three or four assistants, and the requisite number of chainmen, leadsmen, and boatmen to furnish the necessary assistance to the topographers, and crews for three or four six-oared cutters. They had a complete camp equipage, and established their camps on shore, and after surveying for six or seven miles on either side of a camp, moved to a new location. Two to four such parties took the 8 HISTORICAL ACCOUNT [Cnar. 1, field each season. They extended a secondary or tertiary triangulation, developed from bases measured with wooden rods or chains, over their field of work. Frequent observations for azimuth and variation of the compass were made, on Polaris usually. The shore-line and the important features of the topography were determined with the theodolite and chain. The shores of the upper lakes being generally either densely wooded or marshy, the topography back from the shore was as a rule simply sketched, detailed surveys being made only where there were settlements or towns. The in-shore hydrography usually covered the area from the shore to the 3- or 4-fathom curve, but included the development of all shoals or dangerous places within several miles of the shore. The lines of soundings were run between buoys anchored at convenient points and sounding Stations on the shore, the soundings being taken at regular intervals of time. PROGRESS OF THE FIELD-WORK. The portions of the lakes surveyed under the direction of Captain Macomb were the Straits ‘of Mackinac and the approaches thereto for 30 or 40 miles on either side of the island of Mackinac, part of the north end of Lake Michigan, including the Beaver Island group, and the whole of the Saint Mary’s River. The survey of Saginaw Bay was commenced the last season he was in charge. Local surveys of a few harbors on Lake Superior were made by a party under Lieutenant Raynolds in 1855. lL. Straits of Mackinac.—The base-line for the triangulation of the straits was the one selected by Captain Williams in 1842, and approximately measured by Lieutenant Gunnison in 1844. The eastern section of the original line, which was about 6 miles long, was found, however, to lie through very unfavorable ground, principally swamps, and it was decided to reduce the length of the base to about 4 miles. Captain T. J. Lee came on from Washington with the new apparatus, and made the measurement in September and October, 1852. He was assisted in this work by Captain Macomb and his’officers and assistants. Comparisons of the tubes with the standard bar were made before and after the measurement. As a test of the accuracy of the measurement, a length of 1,500 feet was remeasured in a contrary direction. The ends of the base were marked by fine lines drawn upon silver 10-cent pieces, set in the upper surfaces of stone posts sunk 2 feet under the surface of the ground. Reference stones were also set at short distances from the end marks. The detailed report of Captain Lee on the measurement of this base is printed as Appendix G of the Report of the Chief of Topographical Engineers for 1854. The triangulation and off-shore hydrography were carried on by Captain Macomb and his party on the Surveyor during the seasons of 1851, 1852, and 1853, both being completed in the latter year, with the exception of a small amount of hydrographical work which was done in 1854. Many hitherto unknown shoals and reefs were discovered. An astronomical station was established upon the northwest end of Round Island, opposite Mackinac, at which observations for latitude and azimuth were made by Captain Macomb and Lieutenant Raynolds in 1853. The longitude of the station was determined in September, 1854, by a series of observations of lunar culminations made by Lieutenant Raynolds. The meridian of this station was the principal meridian for all the surveys of Captain Macomb in that portion of the lakes. In 1851 Lieutenant Scammon continued his survey of the north shore from the point at which his work terminated in 1849 to Point Detour at the mouth of the Saint Mary’s River. His field of operations included the group of islands called the Chenaux, lying in the indentations of the shore, and separated from it and from each other by narrow but generally deep and navigable channels, some of which afford excellent harbors. In 1852 he surveyed that part of the north shore lying between Pointe la Barbe and Pointe Epoufette. The remaining portions of this shore from Pointe Epoutette about 10 miles westward, and from Pointe la Barbe to Point Saint Ignace, were surveyed by Lieutenant Rose in 1853. The south shore and Mackinac and Round Islands were surveyed by Lieutenant Raynolds’ party, that part from Old Fort Mackinac to Hammond’s Bay being done in 1851, the survey of the islands and of a short distance west of Old Fort Mackinac being made the following year. In 1853, Lieutenant Raynolds, assisted by Lieu- tenant Rose and Messrs. G. W. Lamson and H. Gillman, extended the survey westward to Wau- goshance light-house, and then south nearly to Little Traverse Bay. 2. Saint Mary’s River—In view of the near completion of the ship-canal around the Sault Ste. Marie, the act making appropriations for the Survey for 1853 contained a proviso requiring § 4] OF THE UNITED STATES LAKE SURVEY. a the immediate examination and survey of obstacles to navigation in the Saint Mary’s River. This duty was assigned to Captain Scammon, who, assisted by Lieutenant Mendell and a large party, made surveys in that year of the East Neebish Rapids and of the expansion of the river called Lake George, and submitted plans for the improvement of navigation at both places. Later in the season, Captain Scammon surveyed the entrance to the river by the Detour Passage. Early in 1854, Captain Scammon proceeded to the East Neebish Rapids to mark out the channel by a system of landmarks on the shores, and having accomplished this satisfactorily, resumed the survey of the river, and at the close of the season had completed the data for a continuous chart of the river from Lake Huron to the Sault. This survey depended upon a carefully executed secondary triangulation, with several bases measured with wooden rods. In 1855 the survey of the river was continued from the Sault to Point Iroquois at the east end of Lake Superior by Lieutenant G. W. Rose. The ship-canal was opened to navigation in this year. 3. Lake Michigan—North end and Beaver Island group.—Having finished the triangulation and off-shore hydrography necessary for the completion of the chart of the Straits of Mackinac early in 1854, Captain Macomb, assisted by Lieutenant Rose, devoted the remainder of the season of 1854, and the entire season of 1855, in which he was assisted by Mr. J. A. Potter, fo the extension of the triangulation to the Beaver Island group and along the north shore of the lake, and to off- shore hydrographical work in the same locality. The shore-party work in the north section of the Beaver Island Group was carried on by Lieutenant Raynolds in 1854, until September, when he was detached to make astronomical observations on Round Island, and his party, having beeu placed under the charge of Mr. G. W. Lamson, was transferred to the north shore. In 1855, the surveys of this section were completed by two shore parties under the charge of Messrs. W. H. Hearding and G. W. Lamson, the survey of the north shore being extended to a point about 5 miles east of the Monistique River. 4, Lake Superior.—On account of the opening of the Satilt Canal in 1855, it became important that surveys of the principal harbors of Lake Superior should be made, and accordingly in that year a party under Lieutenant Raynolds proceeded to Ontonagon with orders to survey that harbor and as many of the harbors to the east of it as was possible during the season, the surveys to include the shore for six or seven miles on either side of each harbor. Lieutenant Raynolds com- pleted the surveys of Ontonagon, Eagle River, Eagle Harbor, and Agate Harbor, meeting many difficulties at the three latter places, owing to the abrupt and rocky shores, heavily wooded, which necessitated the placing of the triangulation stations inland and the cutting of lines of sight. He also determined the latitudes of Ontonagon and Eagle River, and the longitude of the latter by lunar culminations. 5. Miscellaneous surveys.——In order to publish a chart of the head of Green Bay it was found that further surveys were needed to render those made under Captain Williams available. There- fore, in July and August, 1852, Captain Macomb and Lieutenant Gunnison, with the steamer Sur- veyor, were engaged in that work. Observations for latitude and azimuth were made on Green Island by Captain Macomb and Lieutenant Gunnison, respectively, after which several triangula- ion stations were built, and while Captain Macomb read the angles at these stations, Lieutenant Gunnison made soundings in the bay. In August of the following year, Lieutenant Raynolds was sent to Green Bay to finish some topographical and other work uecessary for the chart, which was published that year. In September, 1855, Mr. G. W. Lamson surveyed the boundary lines between the United States lands and the private claims on the island of Mackinac. In the fall of 1855, Captain Macomb on the Surveyor made a reconnaissance with a view to connecting the triangulation of the Straits of Mackinac with that of the Saint Mary’s Rfver, and with the contemplated Lake Superior system. He found at the head of Lake George, on the Saint Mary’s River, a height which commanded a view of the heights near Point Iroquois and of those on the north coast of Lake Huron. He caused stations to be built on this height and on one of those at Point Iroquois. During this reconnaissance the Surveyor passed through the new canal at the Sault, being the first government vessel to make the passage. A third point on the north coast of Lake Huron was selected by Mr. G. W. Lamson, and a station was built upon it in the spring of 1856. These stations, however, were never occupied, the connections desired being eventually made by a different system. 2LS8 10 HISTORICAL ACCOUNT (cua. I, 6. Saginae Bay.—On account of the rapidly increasing commerce of the Saginaw region, and in compliance with the urgent requests for surveys and charts of that section, If was decided to place the whole force of the Survey there for the season of 1856. On the 3d of May, Captain Macomb received orders assigning him to duty in New Mexico on being relieved from the command of the Lake Survey by Lieutenant-Colonel Kearney, who, however, did not reach Detroit until the latter part of September. Lieutenants Raynolds and Rose having also been transferred to other duty early in the spring, Messrs. W. H. Hearding, G. W. Lamson, and D. F. Henry were assigned to the charge of three shore-parties. The Surveyor was not in commission that season, being laid up for extensive repairs, which her long service had rendered necessary. The new steamer Search was, however, available in July, and the party ou her, under command of Captain Meade, who joined the Survey in June, was engaged in making a general reconnaissance for the triangulation of the bay, in the building of stations, and in hydrographical work. Captain Macomb was with this party fora time. Mr. Hearding made a minute survey of the mouth of the Saginaw River and of the bar in front of it, with reference to improving the entrance, and then carried the shore-line survey north to the Opinkawning River. The shore-line, from six miles above the mouth of the Sable River southward to include Tawas Harbor, was surveyed by Mr. Lamson’s party. Lieutenant ©. N. Turnbull, having joined the Survey in June, was assigned to the charge of the party, until then under Mr. Henry. Lieutenant Turnbull made observations for latitude on Charity Islands. This party made surveys of the Charity Islands and of the east shore front Oak Point to Sand Point, where they selected and cleared a site for a base-line. THE SURVEY UNDER CHARGE OF LIEUT.-COL, JAMES KEARNEY. SEPTEMBER, 1856, 1o MAy 20, 1857. § &. The only field-work done during the short period Colonel Kearney was for the second time in charge of the Survey, was a resurvey of the Saiut Clair Flats, made by Captain Meade after his return from Saginaw Bay in the fall of 1856. This resurvey was based on three stations whose relative positions had been determined by Lieutenant J. N. Macomb in 1842, The usual office-work was carried on during the winter. In the spring, failing health required that Colonel Kearney should relinquish the command of the Survey, and he was relieved by Captain George G. Meade, who, a few years later, was the com- mander of the Army of the Potomac. THE SURVEY UNDER THE CHARGE OF CAPT. GEORGE G. MEADE. May, 1857, ro SEPTEMBER 1, 1861. * § G. The principal work accomplished by Captain Meade was the survey of the whole of Lake Huron, including the completion of that of Saginaw Bay, the entire force of the Survey being engaged on this duty during the years 1857, 1858, and 1859. In 1860 the survey of the northeast end of Lake Michigan was extended southward to include the Fox and Manitou Islands and Grand and Little Traverse Bays, and the data were thus obtained for a much-needed chart of a dangerous part of the lake passed over by the vessels sailing between the Straits of Mackinac and Chicago. Local surveys of a few harbors on Lake Superior were made in 1859, and in 1861 the general sur- vey of the lake was bégun at its western end. ® METHODS OF SURVEY. The general methods of survey employed by Captain Meade were similar to those followed by Captain Macomb. The nature of the field of operations required a combination of triangulation and astronomical work for the determination of the positions of points on the shores of Lake Huron, and made some change necessary in the method of executing the off-shore hydrography. Larger appropriations permitted a considerable expansion of the scope of the Survey, the intro- duction of more accurate methods in obtaining longitudes, and the commencement of a series of magnetic, water-level, and meteorological observations at many points on the lakes. The methods §§ 5, 6.) OF THE UNITED STATES LAKE SURVEY. 11 and instruments employed in these observations will be noticed under their appropriate headings. The method of executing the Of-shore hydrography on Lake Huron has since been used on all the other lakes, and consists in running lines of soundings, made from a steamer, from the outer limit of the in-shore hydrog- raphy to a distance of eight or ten miles into the lake. The lines are about a mile apart, and are run by the steamer’s compass, their direction being perpendicular to the general line of the coast. The position of the steamer at the time a sounding is taken is located by simultaneous theodolite pointings of two observers on shore, and is checked when practicable by sextant angles read on the steamer. The general character of the bottom of a lake is determined by running lines of soundings entirely across the lake at intervals of ten or fifteen miles. Detroit observatory.—The use of a suitable lot on Washington avenue, near Grand River avenue, having been offered gratuitously to the Survey by J: ohn Hull, esq., a wooden building was erected on it in the spring of 1857 and fitted up as a field-observatory for botlf astronomical and magnetic obser- vations. The astronomical work, under Captain Meade’s direction, was mainly done by Lieutenants C. N. Turnbull and O. M. Poe, and Mr. James Carr, and the magnetic work by Lieutenant William Proctor Smith. Early in 1858 a new astronomical transit (No. 15) and a new zenith-telescope (No. 12), both made by Wiirdemann, the telescope of each having a focal length of 32 inches, and an aperture of 23 inches, were purchased for the Survey, and at the same time a break-circuit -Sidereal clock (No. 184), a chronograph with spring governor, and four sidereal chronometers, all made by Bond & Sons, of Boston, were received. In the following spring another break-circuit sidereal clock (No. 256), asecond chronograph, and four more sidereal chronometers, all by the same makers, were added to the list of instruments. A favorable opportunity for determining the longitude of the Detroit observatory by connecting through the magnetic telegraph with an observ- atory whose longitude was well established did not occur until the winter of 185859, when it was decided to connect with the observatory of the Western Reserve College, at Hudson, Ohio, the longitude of which from the Cambridge observatory had been determined in 1849. The uninter- rupted use of the wire between Detroit and Hudson after 9 o’clock at night was offered free of charge by Anson Stager, esq., the general superintendent of the Western Union Telegraph Com- pany. The observations were made in January and February, 1859, the Detroit observer being Lieutenant Turnbull, who used a chronograph, and Professor C. A. Young, of the Western Reserve College, observing at Hudson, and using a Morse register for recording his observations. The telegraphic connections were so made that the transits of stars at either meridian were recorded at both stations with their respective local times, and the same stars being used by both observers, each star gave from each record a determination of the longitude, and the mean of these two deter- minations was free from wave-time. At the conclusion of the observations at Hudson, Professor Young visited Detroit for the determination of the relative personal equation of himself and Lieu- tenant Turnbull. The latitude of the Detroit observatory was determined by Lieutenant Turnbull by seven nights’ observations with the zenith-tclescope in April and May, 1859. In May, 1860, the difference of longitude between the Detroit observatory and that of the University of Michi- gan at, Ann Arbor, was telegraphically determined by Lieutenant Poe, at Detroit, and Professor James C. Watson, at Ann Arbor, by the exchange on two nights of arbitrary signals, the observ- atory at Ann Arbor not being provided with the necessary apparatus for the exchange of star- signals. The results not being entirely satisfactory, a second connection with Ann Arbor was made in April and May, 1861. On this occasion both observatories had chronographs, and the method of operations was similar to that employed between Detroit and Hudson. Professor Briin- now, of the University of Michigan, observed at Ann Arbor, and Lieutenant Poe and Mr. James Carr at Detroit. Subsequently Professor Briinnow connected Ann Arbor with the Hamilton Col- lege observatory at Clinton, N. Y., the observer at the latter place being Professor C. H. F. Peters. The difference of longitude between the Hamilton College and Cambridge observatories having been previously determined, a second connection between Detroit and Cambridge was thus made. Magnetic observations.—Previous to 1858 the magnetic observations had been limited to the determination by the ordinary compass of magnetic declinations at those places where observations ‘for azimuth were made by the shore or triangulation parties. In that year a portable declinometer, with detached theodolite, for the determination of the magnetic declination and horizontal intensity, 12 HISTORICAL ACCOUNT [Cuar. J, made by Jonex, of London, a Barrow dip-circle, and a Fox dip-circle, were imported, and the making of magnetic observations was assigned to Lieutenant W. P. Smith in addition to his other duties. At various times during’the seasons of 1858, 59, ’60, Lieutenant Smith made determinations of the magnetic elements at Detroit, Cambridge, Mass. (where he had been sent to compare his instru- ments with those of the Cambridge observatory, aud to be instructed hy Professor Bond in the use of them), Toronto, and at twenty-seven points on the lakes, of which three were on Lake Ontario, four on Lake Erie, six on Lake Michigan, three on Lake Superior, ten on Lake Huron, and one on the Straits of Mackinac. Tables giving the results of these observations are published in the Lake-Survey reports for 1859 and 1860. Water-level and meteorological observations.—Up to the time of Captain Meade’s assuming charge of the Survey, readings for water-level were taken on temporary gauges at the localities where surveys were being carried on, and the soundings were reduced to a certain stage of water, which was usually either the mean level during ‘the period of the survey, or the mean level during a particular season. With a view to establishing a uniform plane of reference for the soundings, as well as deciding numerous interesting questions in regard to the fluctuations of the water-level of the lakes, including the question of tides, Captain Meade, in his annual report for 1857, recom- meuded that simultaneous water-level readings, accompanied by complete meteorological observa- tions, should be made over the entire lake region. This recommendation was approved by the bureau, and the instruments, including four self-registering tide-gauges, were ordered, but were not received at Detroit until August, 1858, too late to be distributed to the different stations that season. Early in the spring of 1859, Captain Meade himself distributed and set up these instru- ments at Sacket’s Harbor, Charlotte, and Fort Niagara, on Lake Ontario; Buffalo, Erie, Cleveland, and Monroe piers, on Lake Erie; Forestville, Pointe aux Barques, Tawas, Thunder Bay Island, and Presqw’ Isle, on Lake Huron; Grand Haven, Michigan City, and Milwaukee, on Lake Michigan; Head of Saint Mary’s River, Marquette, Ontonagon, and Superior City, on Lake Superior. The instruments furnished each station were a water-gauge, barometer, psychrometer, thermometer, rain-gauge, and wind-gauge. Competent observers were employed to make daily or more frequent observations, the records being sent to the office at Detroit at the end of each month. These records were reduced and tabulated in the office, and detailed tables of the results are published in the annual reports for 1860 and 1861. In 1860, Mr. J. M. Bigelow was placed in charge of the meteorological division of the Survey. His reports discussing the results of the observations are published with the above-mentioned tables. The report for 1861 also contains a discussion of the tides and seiches (irregular oscillations) on Lakes Michigan and Superior, by Mr. O. N. Chaffee. PROGRESS OF THE FIELD-WORK. 1. Lake Huron—Sand Point base-line.—The Saginaw Bay triangulation depends upon a base about 4 miles long, méasured in October, 1857, upon the sand-spit extending into the bay from its easternshore. The Bache-Wiirdemann apparatus was used. Mr. a Wiirdemaun came on from Washington to correct a slight want of compensation in tube No. 2, and at the same time thoroughly cleaned and adjusted the whole apparatus. The peeaduuenent was made by Captain Meade, assisted by Lieutenant Turnbull and Messrs. Potter, Carr, and Casgrain. A length of’ 960 feet was remeasured as a test of the accuracy of the work. The ends of the line and two intermediate points were marked by stone posts. The latitude and longitude of the west end were determined by Lieutenant Turnbull, the former by observatious with the zenitl.-telescope, the latter by lunar culminations. Triangulation and astronomical work.—In 1857 Captain Meade on the Search finished selecting the sites and building the stations for the Saginaw Bay triangulation. He had also intended to read the angles, but finding that the duties of general supervision would prevent his accomplishing both the triangulation and off-shore hydrography, he decided to devote himself to the latter, and to assign Lieutenant Poe with a party on the Surveyor to the triangulation work. Lieutenant Poe used the 10-inch repeating Gambey theodolite the greater part of the time. He read the angles from the Charity Islands to the head of the bay in 1857. In order to see over the line from Pointe aux Barques to Pointe au Sable, 27 miles long, much higher stations than had as yet been used on §6.] OF THE UNITED STATES LAKE SURVEY. 13 the survey were required. In 1858 Lieutenant Poe built a station with a 100-foot center-post on Pointe aux Barques, and one with an 82-foot center-post on Pointe au Sable. Before occupying them be was taken seriously ill, and had to withdraw from field duty for the remainder of the season. Mr. James Carr relieved him and tinished the Saginaw Bay triangulation. It was found impracticable to extend the triangulation beyond the line Point aux Barques— Pointe au Sable on account of the character of the shores of Lake Huron. From Pointe aux Barques to the head of Saint Clair River the coast is comparatively straight and flat, and at that time was densely wooded, with a few scattering settlements on the immediate shores of the lake. To have carried a system of primary triangulation along this coast, it would have been necessary to cut out almost every line of sight through the heavy timber, and would have involved an expenditure of time and money entirely unwarranted by the appropriations mmade for the Survey. It was therefore decided to carry along the shore in connection with the shore- party work such a minor triangulation as the nature of the ground permitted, and to check this by ‘determining the positions of certain points by astronomical observations; or, in other words, to establish a series of astronomical bases, the latitudes of the ends of these thases being determined by observations with the zenith-telescope, and the differences of longitude between them and the principal meridian at Sand Point being obtained by the repeated seuustens of chronometers from one station to the other on the Lake- Survey steamers. In the longitude work, after the receipt of the second clock and chronograph, a clock and chronograph were used at each of the stations between which the chronometers were to be trans- ferred. The errors and rates of the clocks were carefully determined by as frequent observations as possible with an astronomical transit, and the errors of the chronometers on local time were determined by comparisons with the clocks. From four to twelve chronometers were used, and eight to twelve transfers were made between each pair of stations. Astronomical stations were established at Forestville, Sanilac, and Fort Gratiot, and -their latitudes and differences of longi- tude between each other and Sand Point were determined during the seasons of 1858 and 1859 by Lieutenant Turnbull and Mr. James Carr. Early in the latter year the difference of longitude between Fort Gratiot and Detroit was determined in the same manner, and the Saginaw Bay tri- angulation and longitudes dependent upon it were thus connected with the Detroit meridian as determined by telegrapb. A small portion of the south end of Lake Huron was covered by a trian. gulation extending from the head of the Saint Clair River to the line Sanilac—Cape Ipperwash, the angles being read by Lieutenant J. L. Kirby Smith in 1860. Similar considerations prevented the extension of the Saginaw Bay triangulation to the north- ward, and it was therefore connected with the triangulation of the Straits of Mackinac by a system of latitudes and azimuths, the stations being at Sturgeon Point, Thunder Bay Island, Presqw Isle, and Sand Bluff, a few miles east of Hammond’s Bay. From the observed latitudes and azi- muths at these stations their longitudes and distances apart were computed. The observations at the two first-mentioned stations were made by Lieutenant Turnbull in 1858. As a cheek upon the work, the difference of longitude between Thunder Bay Island and Sand Point was determined directly by the exchange of chronometers. The latitude and azimuth observations at Presqu’ Isle and Sand Bluff were made in 1859 by Lieutenant Poe. After completing these observations, Lieutenant Poe, with a party on the schooner Coquette, fixed the positions of points on the north shore of Lake Huron by extending the triangulation of the Straits of Mackinac eastward to the line Presqu’ Isle-Great Duck Island. In order to incor- porate into the chart of Lake Huron Bayfield’s surveys of such portions of the Canada coast as were not included in the work of the Lake Survey, it was desirable to determine the latitudes and longitudes of several points on that coast, and Goderich, which was in telegraphic connection with Detroit, and Cove Island, at the entrance to the Georgian Bay, were selected as being the inost suitable points. Lieutenant Poe, assisted hy Lieutenant Beckham, and Mr. Carr, assisted by Mr. Austin, were assigned to this duty in 1860. Lieutenant Poe, at Goderich, and Mr. Carr, at Detroit, determined the longitude of Goderich by telegraphic signals. Mr. Carr then observed for latitude at Cape Ipperwash, and afterwards occupied the Cove Island station, and chronometers were trans- ferred between it and Goderich. Lieutenant Poe then moved to Mackinac, and chronometers were transferred between the station on that island and Cove Island. A comparison was thus made Pa 14 HISTORICAL ACCOUNT [Cnap. I, between the longitude of the Mackinac station, as determined by the astronomical and triangula- tion work on the west shore of Lake Huron, and that by the astronomical work on the east shore, and the results wert found to agree very well. Lieutenant Poe and Mr. Carr finished the season’s work by determining the telegraphic difference of longitude between Detroit and Grand Haven. Mr, Carr, who observed at Grand Haven, also determined its latitude. Offshore hydrography.—The off-shore hydrography for a distance of 15 miles between Saginaw Bay and Thunder Bay; and from Thunder Bay around the northwest end of the lake to Cockburn’s Island, was done by a party on the Surveyor, under the command of Mr. J. A. Potter, in 1859. That of the rest of the lake, including Saginaw Bay, was done by the party on the Search, under the command of Captain Meade, during the seasons of 1857, °58, 59. In 1860 Lieutenant W. P. Smith on the Search ran the lines of soundings across the lake, and made observations of the temperature of the water at different depths. Topography and in-shore hydrography. —All of this work was done by parties under the charge of Messrs. Hearding, G. W. Lamson, and H. C. Penny in 1857 and 1858, and of Messrs. Hearding and Penny in 1859, in which year it was completed. Mr. Hearding’s party continued their survey of the Saginaw River, made in 1856 eastward to Quannakisse Bayou, surveyed the coast from Oak Point on the east shore of Saginaw Bay to Pointe aux Barques, and trom this point south to the head of the Saint Clair River, and made surveys of Cockburn and Drummond Islands, connecting with Captain Scammon’s survey of the Saint Mary’s River. Mr. G. W. Lamson’s party extended the survey of Tawas Harbor, made in 1856, south to the Opinkawning River, and surveyed Thunder Bay, covering it with a triangulation depending upon a base two miles in length, measured with wooden rods. Mr. Penny’s party surveyed the east shore of Saginaw Bay from Sand Point to Quannakisse Bayou, the west shore of Lake Huron from near the mouth of the Sable River to the south side of Thunder Bay, and from the north side of Thunder Bay until a connection was made with the survey of the straits at Hammond’s Bay, and the Canadian coast from the head of Saint Clair River to Cape Ipperwash. Lieutenant Poe had charge of this party during the early part of 1857, until he was assigned to triangulation duty. 2. Lake Micuican.—Nearly the entire force of the Survey was sent to the northeast end of Lake Michigan in the spring of 1860. Primary triangulation. —Lieutenant J. L. Kirby Smith, with a party on the schooner Coquette, located and built the stations required to extend the Mackinac triangulation from the line Hat Island - Pointe anx Chénes over the field of the present survey. He also, with the assistance of Mr. Carr, read the angles at all the stations except those in Grand Traverse Bay. These were occupied the following year by Mr, O. N. Chaffee. It had been intended to counect this triangula- tion with the meridian of Detroit by transferring chronometers between Pointe aux Becs Scies and Grand Haven, 6f which the longitude was determined with this object, but the exigencies of the service prevented the carrying out of the project that season. Off-shore hydrography.—Mr, J. A. Potter, with the steamer Surveyor, did the off-shore hydrog- raphy of the main coast, including the Traverse Bays and around the Manitou Islands. In September, Captain Meade brought the Search from Lake Huron and did part of the sounding around the Fox Islands. Mr. Potter did the rest of it with the Surveyor in 1861. Topography and in-shore hydrography.—Three shore-parties were engaged in this work. That of Mr. Hearding commenced its survey where the work under Captain Macomb ended, near Middle Village, and carried it south to Deep Water Point, on the east arm of Grand Traverse Bay. Mr. Penny’s party surveyed from Traverse City to Pointe aux Bees Scies. Mr. Henry’s party, after finishing the Manitou and Fox Islands, surveyed the peninsula which separates the two arms of Grand Traverse Bay, from Deep Water Point to Traverse City. ! 3. LAKE SUPERIOR.—In 1859, a party under Mr. G. W. Lamson was sent to Lake Superior and made surveys of the harbor of Marquette and of Grand Island and its approaches. The off. shore hydrography about Grand Island was done by Mr. Potter on the Surveyor in 1861. A gen- eral reconnaissance of Lake Superior was made by Captain Meade on the Search in 1859. The general survey of Lake Superior was commenced in the spring of 1861, the parties being sent to the extreme west end of thelake. Just previous to the season for commencing field operations the breaking out of the rebellion caused the withdrawal from the Survey of all the officers engaged $7.) OF THE UNITED STATES LAKE SURVEY. mS upon it as assistants, and no officers but the superintendent were employed upon the Survey from that time until the close of the war. Primary triangulation.—This was assigned to Mr. Henry, who had charge of a party on the schooner Coquette. He located, cleared, and prepared the ground for the preliminary measure- ment of a base-line about four miles long on Minnesota Point, south of Duluth, and also built and occupied six stations for the development of the triangulation from the base to the eastward. Mr. Hearding and Mr. Casgrain made the preliminary measurement of the base with wooden rods, duplicating the work as a check upon its accuracy. Off-shore hydrography.—Mr. Carr, on the Search, did the off-shore work on both shores of the head of the lake along a total distance of 63 miles. > Topography and in-shore hydrography.— My. Hearding’s party surveyed the Saint Louis River from the head of navigation at the town of Fond du Lac to its mouth at the Bay of Saint Louis, and then made surveys of this bay and of Superior and Allouez Bays. These bays are separated from the lake by the narrow sand-spits called Minnesota and Wisconsin Points. Mr. Penny’s party, connecting with Mr. Hearding’s survey, extended it along both shores of the lake, the total length of shore-line surveyed being 63 miles. 4, MISCELLANEOUS.—Various local surveys were made from year to year by the different assistants, in obedience to orders from the Topographical Bureau. Many of these were for light- house purposes. Surveys were made nearly every year at the Saint Clair Flats and at the Lake George Flats of the Saint Mary’s River, for the purpose of determining the effects of the improve- ments which were being carried on by Captain A. W. Whipple. A detailed survey of Maumee Bay and River, as far up as Toledo (of which a chart was published), was made by Messrs. Hearding, Penny, and Potter in the fall of 1857, after their return from Lake Huron. Before the parties returned from the field in,1861, Captain Meade was relieved from the charge of the Lake Survey and ordered to duty with the armies in the field. He turned over the command of the Survey to Lieutenant-Colonel J. D. Graham, of the Topographical Engineers, on the 31st of August, 1861. THE SURVEY UNDER THE CHARGE OF COL. J. D. GRAHAM, CORPS OF ENGIN EERS. SEPTEMBER 1, 1861, TO APRIL 15, 1864. § 7. When detailed as superintendent of the Lake Survey, Colonel Graham was in charge of the harbor improvements on all the lakes from Champlain to Superior, and had his office at Chicago. On removing to Detroit, he still retained the charge of the harbor works, and was in addition assigned to duty as engineer of the tenth and eleventh light-house districts, embracing all the lakes except Champlain. Colonel Graham was in charge of the Survey during three winter and two summer or field seasons. OFFICE WORK. During the winter seasons the work of the office was carried on in the usual manner. Much attention was given by Colonel Graham to the reduction, tabulation, and discussion of the water- level and meteorological observations, which were continued at the stations established by Cap- tain Meade, and at others which had been previously established by Colonel Graham in connection with his harbor works. In the annual reports for 1861, 1862, and 1863, the meteorological data are discussed by Mr. Bigelow, and the fluctuations of the water-level by Colonel Graham, who had made these a subject of study for several years. Colonel Graham, before taking charge of the Survey had, in his annual report for 1860, presented a memoir, accompamed by a record of ex- tended observations at Chicago, demonstrating the existence and character of a semi-diurnal lunar tidal wave in Lake Michigan. In the report for 1863, Colonel Graham revises his memoir and announces as the final result of his observations that the height of the mean semi-diurnal lunar tidal wave at Chicago is 0.142 foot. METHODS OF SURVEY. Colonel Graham made but two changes in the methods of field-work. He introduced the stadia for horizontal and vertical measurements in the topographical work, to which more care was 16 HISTORICAL AGCOUNT [Crrav. I, given than formerly, although the use of the stadia did not become general among the shore-parties until some years later. The second change was the substitution of the method by powder-flashes for the chronometric method in determining differences of longitude where telegraphic facilities did not exist. At each of the stations whose difference of longitude it was desired to obtain, an astronomical party with a field astronomical transit and clock or chronometer determined the local time by observing the meridian transits of stars. The observations for time were made in two series with an interval of an hour or wore between them. During the interval, a series of signals one minute apart was made, either at one of the stations or by a third party from a height visible from both astronomical stations, by flashing small charges of gunpowder. These signals were noted by the observer at each station, and the difference of the local time thus noted was the difference of longitude of the two stations. These flashes could be seen on a clear night over distances of 50 to 60 miles with the naked eye, and with the aid of a telescope over distances of 100 miles. PROGRESS OF THE FIELD-WORK. The field seasons during which Colonel Graham was in charge were those of 1862 and 1863. He had intended to continue the survey of Lake Superior, commenced in 1861 by Captain Meade, but the appropriation in 1862 became available at so late a date that the parties could not take the field until the latter part of August, when. it would have been unadvisable to send them to so distant a field. The survey of Green. Bay was therefore commenced in 1862 and continued in 1863. In the latter year a party was also sent to Lake Superior to make local surveys. Before commencing work on Green Bay in 1862, Mr. O. N. Chaffee’s party on the Surveyor and the shore-party of Mr, J. R. Mayer were occupied for some weeks, the first in surveying several shoals about the ox Islands which had not been discovered when the original surveys were made, and the second in making a detailed topographical survey of SouthgFox Island, using the stadia. 1. GREEN Bay—Astronomical work.—In 1862, an astronomical party under Mr. James Carr made latitude observations with the zenith-telescope at Rock Island, Green Island, and Fort Howard. In 1863 two parties, one under Mr. James Carr, and one for a short time under Mr. *Austin and then under Mr. O. B. Wheeler, took the field for the purpose of determining latitudes with the zenith-telescope and longitudes by powder-flash signals. In determining longitudes the station at Fort Howard was taken as the primary meridian, with the intention of subsequently connecting it telegraphically with Detroit through Chicago. These two parties determined the longitudes of Rock and Green islands, and the latitudes and longitudes of Oconto on the west coast of the bay, and of stations on Washington Island, on the bluff at the entrance to Big Bay de Noquette, on South Manitou Island, and on the southeast end of Beaver Island. ' Triangulation.—In both seasons Mr. Henry was in charge of this branch of the work, having a. party on the schooner Coquette. He was mainly engaged in making reconnaissances for a base-line ‘and for the system of triangulation, in building stations and clearing lines of sight. Some of the stations were occupied, and a preliminary measurement of the base-line, which had been established on Chambers’ Island, was made with the wooden-rod apparatus in 1863. Off-shore hydrography.—This was done in both seasons by parties on the Search and Surveyor, in charge respectively of Mr. Hearding and Mr. O. N. Chaffee. They completed the off-shore work of that part of the bay north of the Sister Islands, and in Lake Michigan carried it from Bayley’s Harbor, on the east coast of the peninsula which separates Green Bay from the lake, northward to Point Detour, the northern point of the entrance to Green Bay. Much of the time of the steamers was occupied in the building of primary triangulation stations, and in moving and supplying the shore-parties. e Topography and in-shore hydrography.—This work was done in 1862 by two parties under Messrs. Penny and Mayer, and in 1863 by three parties under Messrs. Penny, Gillman, and Mayer. In the latter year, Mr. O. N. Chaffee’s steamer-party ran the shore-line of Washington and Rock islands and the northern part of Little Bay de Noquette, including the secondary triangulation of the whole of this bay. In the two seasons, surveys were made of the west coast of Green Bay from a point six miles south of Cedar River to the north to include both shores of Little Bay de No- quette, of the peninsula between Green Bay and Lake Michigan from Egg Harbor on the former §8.] OF THE UNITED STATES LAKE SURVEY. 17 to Bayley’s Harbor on the lake, of Detroit Island, and of Plum and Pilot islands in the Porte des Morts entrance to Green Bay. 2. LAKE SUPERIOR.—In the spring of 1863, a party under Mr. J. U. Mueller, assisted by Mr. E. S. Wheeler, was sent to survey Portage Entry on Keweenaw Bay. They surveyed the shore of the bay for a distance of four miles on either side of the mouth of Portage River, Portage River from its mouth to its head, including Portage Lake, the lower portions of Sturgeon, Pike, and Pil- grim rivers, and Torch River from its mouth in Portage Lake to its source in Torch Lake. This survey embraced the positions of the cities of Houghton and Hancock, and of the principal copper- mining establishments in their vicinity. On the 15th of April, 1864, Colonel Graham was relieved of the charge of the Lake Survey, and of his harbor improvements and light-house duties, by Colonel W. F. Raynolds, Additional Aide- de-Camp and Major of Engineers. Colonel Raynolds was relieved of the charge of the harbors on Lakes Michigan and Erie in October, 1864, of other harbor-works in 1866, and of his light-house duties in 1870. s THE SURVEY UNDER THE CHARGE OF BVT. BRIG. GEN. W. F. RAYNOLDS, LIEU- TENANT-COLONEL OF ENGINEERS. APRIL 15, 1864, TO May 12, 1870. § &. The survey of Lake Superior was the main work of the Lake Survey during the six field- "seasons that General Raynolds had charge. At the close of the season of 1869 but three islands of the Apostle Group remained to be surveyed in order to complete the topographical work on the American shores of the lake. A little hydrography and the greater part of the primary triangula- tion were unfinished. Besides the Lake Superior work, the survey of Green Bay was completed, that of Lake Michigan was extended south to Two Rivers on the west shore and to Little Pointe au Sable on the east shore, the whole of Saint Clair River, and a large part of Lake Saint Clair were surveyed, and many special surveys of harbors or of localities at which works of improvement were’ in progress were made in different seasons under orders from the Engineer Department. In Octo- ber, 1865, a third steamer, a screw propeller, length 122 feet and beam 18 feet, was purchased from the Navy Department. She was named the Ada. This steamer was builf on the Clyde for a blockade runner and was captured by the Navy. Extensive alterations and additions in the way of upper works and cabin accommodations were necessary to make her suitable for use on the Survey. The office work and the publication and issue of charts were continued in the usual manner. METHODS OF SURVEY. The operations of the Survey were continued under General Raynolds on the same general plan that had been pursued by his immediate predecessors. Triangulation and astronomical work.—Where it was practicable, the surveys depended upon a system of primary triangulation with carefully measured bases. Along the shore of Lake Superior, from Grand Island to White Fish Point, and on both shores of Lake Michigan south of the entrance to Green Bay on the west and of Pointe aux Becs Scies on the east shore, where a primary triangu- lation was impracticable except at a great cost for stations and for clearing lines of sight, the posi- tions of points were determined either by observations for longitude and latitude or by the method of latitudes and azimuths. At most of the primary triangulation stations latitude and longitude observations were made. Latitudes were determined in all cases by several nights’ work with the zenith-telescope. Longitudes were determined by several nights’ work either by the method of telegraphic star-signals, in which each observer registers the time of passage of a star over his meridian on all the chronographs employed in the work, or, when the telegraph was not available, by the method of powder‘flash signals previously described, The personal equation was not in all cases determined or eliminated with precision. The two primary bases, on Keweenaw Point, Lake Superior, and Chambers’ Island, Green Bay, were measured with the Bache-Wiirdemann Sapaiatis: In June, 1865, agates were set in the ends of the 15-foot brass standard bar, changing its length. “To determine its new length five brass yards were made for the Survey by Wiirdemann, who 3L S$ 18 HISTORICAL ACCOUNT [Cnar. I, assigned their lengths and marked the same upon them. The yards were received at Detroit in March, 1867. Several series of comparisons of the standard bar with these yards were made at different times by Colonel Farquhar and Mr. Henry for the purpose of determining the length and coefticient of expansion of the standard bar. Comparisons of the tubes of the base-apparatus with the standard bar were made before and after measuring a base. The triangulation of Lake Superior was much retarded by the lack of suitable instruments. In 1864 the 10-inch Gambey theodolite was the only one in the possession of the Survey fit for primary work. In September, 1865, Gen- eral Raynolds made a requisition for three large theodolites. These were ordered by the Engineer Department from Oertling & Sons, of Berlin, but were not received at Detroit until the spring of 1869. They were used on Lake Superior during that season. They had 20-inch horizontal limbs, read by three microscopes to single seconds. Two of them turned out to be very poor, and the third was not a good instrument, all having large accidental errors of graduation. In 1867 the triangu- lation work was done with three instruments borrowed from the U. 8. Coast Survey. These were a 24-inch theodolite by Troughton, reading to single seconds by three microscopes; a 14-inch Briin- ner repeating theodolite, reading by two verniers to five seconds, and a 12-inch Gambey repeating theodolite, reading by two verniers to five seconds. In the primary work a large number of meas- urements, as a general rule, were made on each angle, but there was no uniformity as to this number, and the readings were not taken with the system which is necessary for the elimination of errors. An improvement in the method of building stations was adopted in 1864 by changing the center- post from a single piece to a tripod, the legs of which were firmly braced together and had a suffi- cient inclination to give the structure great stability. The tripod has the advantage of enabling the target and instrument either to be centered over the geodetic point or their reductions to this point to be determined, and has since been exclusively used for primary stations. The geodetic point was marked by a hole drilled in the top of a stone post sunk below the surface of the ground, or in the natural rock, where that was found. A brass frustum of a cone with a cross cut on its top surface was sometimes leaded into the top of the marking-stoné. During the season of 1865 Assistant Engineers O. B. Wheeler and 8. W. Robinson introduced on the Survey a method of sending messages by means of flashes of sunlight from a mirror, nade short or long to correspond with the Morse telegraphic code. This method of telegraphing has since been frequently employed. It was especially useful to the parties engaged in the primary triangulation of Lake Superior, messages having been successfully sent in that work over lines 50 to 90 miles long. Topography and hydrography.—In regard to the topographical work, General Raynolds, in his report for 1866, states that— The character of the country in which the surveys are being prosecuted forbids that attention to the details of topography which would otherwise be desirable. It is the exception to find anything but a dense forest, in which it is impossible to make an accurate survey without opening every foot of the lines of sight. No sketching can be done that is reliable. Parties within easy hearing distance cannot see each other. And last, though by no means least, during the summer season, which is the only one in which work can be done at all, the forests are so full of venomous insects that it is next to impossible for an instrument to be used. The stadia has been found most available for over- coming these difficulties. The hydrographic work was done in the manner hitherto in use on the Survey. The steamer- parties, in addition to doing the off-shore hydrography, were generally charged with the building of stations, with the moving and supplying of other parties, and sometimes with the reading of angles. Water-level and meteorological observations were continued at many of the stations established by Captain Meade. Mr. Bigelow continued in charge of the reduction and discussion of these observations until the 1st of January, 1867, when Mr. Henry was placed in charge of this depart- ment. Extended tables and reports, containing information in regard to the meteorology of the lakes, form appendices to the annual reports of General Raynolds. Outflow of the lakes——In 1867 the investigation of the subject of the supply of water in the chain of lakes was taken up. This duty was assigned to Mr. D. F. Henry, who was directed to carefully gauge the rivers forming the connecting links of the chain. Observations were made during the seasons of 1867-68-69. In 1867, a party under Mr. A. RB. Flint was engaged in this work on the Saint Clair and Saint Lawrence rivers, and another party under Mr. Lewis Foote . § 8.) OF THE UNITED STATES LAKE SURVEY. 19 worked on the Saint Mary’s and Niagara Rivers. The method followed during this season was generally that used by Humphreys and Abbot in their work on the Mississippi River. The bases, however, were much longer than those used on the Mississippi River, being from 700 to 1,100 feet in length, instead of 200 feet. The signals between the ends of the base were sent by ialeeraph. Several other departures from the method of using double floats, recommended in the Report on the Physics and Hydraulics of the Mississippi River, were made. In 1868, the Saint Lawrence, Niagara, and Saint Clair Rivers were gauged by parties under the immediate charge of Messrs. Flint, Foote, and David Wallace, respectively. In this season meters were used for obtaining the velocity of the currents. These were of two or three different kinds. Some were full propeller- wheels of four blades, some of two blades, and one, devised by Mr. Henry, was constructed of a set of Robinson anemometer cups set in a suitable frame, and was used where the current was so sluggish that a propeller-wheel would not move. The record was made by electrical apparatus, the circuit being broken at each revolution af the wheel. An escapement was attached to the armature of a relay coil, and a record of the revolutions of the wheel kept by a decimal train of wheel-work. The coefficient of velocity in these meters is variable for different velocities, and it is difficult to obtain it with accuracy. The friction of the parts in all those, except the anemometer cups, used in this work caused considerable uncertainty as to the accurate determination of this coefficient. In 1869 the work was continued on the Niagara and Saint Clair Rivers, under the immediate charge of Messrs. Foote and Wallace. Both meters and double floats were used for the object of testing and comparing the two methods. The detailed reports of Mr. Henry form appen- dices to the Lake-Survey Reports for 1868, 1869, and 1870. In these reports Mr. Henry differs from and criticises the methods and conclusions of Generals Humphreys and Abbot, as given in their report on the Mississippi River. A criticism of Mr. Henry’s methods of observation and reduc- tion, as well as of his conclusions, by General H. L. Abbot, is published in the Report of the Chief of Engineers for 1870. ‘ PROGRESS OF THE FIELD-WoRK. 1. RIVER AND LAKE SAINT CLAIR.—In 1867, before the navigation was open on the Upper Lakes, three large parties, on the steamers Search, Surveyor, and Ada, commanded respectively by Lieutenants James Mercur and B. D. Greene, and Mr. O. N. Chaffee, made a complete survey of the Saint Clair River from Lake Huron to the head of the delta, and of the south channel into Lake Saint Clair. The survey was completed early inJune. In 1868 the parties, having returned from Lake Superior in September, were immediately reorganized for the purpose of making a survey of Lake Saint Clair. Colonel Farquhar and Lieutenant Gregory, in charge of large parties on the Ada and Search, respectively, made surveys of the flats and of the eastern shore, and the two shore parties of Messrs. Mayer aud Molitor surveyed the west shore. Lieutenant B. D. Greene relieved Colonel Farquhar of the charge of the Ada soon after the parties took the field. About 40 miles of the east shore and all the main triangulation of the lake were yet to be done when the parties were withdrawn from the field, owing to the lateness of the season. 2. GREEN Bay AND LAKE MicHIiGAN—Base-line.—The base on Chambers’ Island for the Green Bay triangulation, which had been measured with wooden rods in 1863, was grubbed and graded during the summer of 1864, and its measurement with the Bache-Wiirdemann apparatus was commenced by Professor C. A. Young and Mr. Henry in September of the same year. When about one-third of the line had been measured, Professor Young’s duties at his college required him to leave, and the measurement was completed by Mr. Henry, assisted by Messrs. Robinson and Le Baron. The base was about 34 miles in length. The ends and four intermediate points were each marked by cuts on a frustum of a brass pyramid leaded into the top of a stone 5 feet long and 6 inches square on top, sunk just below the surface of the ground. One hundred and fifty-one tubes were remeasured as a test of the accuracy of the work. The Chambers’ Island light-house, built in 1868, was placed directly over the north end of the base, and its subsequent use as a base of verification for the triangulation extending down from Lake Superior was thus prevented. Triangulation —Mr. O. N. Chaffee had charge of the triangulation work on Green Bay during the seasons of 1864 and 1865, his party being on the schooner Coquette in the first season, until she was wrecked in the fall, and on the Surveyor in the second season, In 1864 the stations north 20 HISTORICAL ACCOUNT (Cua. 1, of Chambers’ Island, including those on Big Bay de Noquette, were occupied, and the stations south of Chambers’ Island were occupied in 1865. During the summer of 1864, Mr. Henry, on the Surveyor, was engaged in reconnoitering for the best method of making a connection between the triangulation extending from the Mackinac base to the Beaver, Fox, and Manitou Islands, and the Green Bay system. Having found that stations on Burnt Bluff at the south end of Big Bay de Noquette, and on Rock Island in the entrance to Green Bay, could be seen fronr the South Fox and North Manitou Islands, stations were built at these four points. The angles at the two latter points were read by Messrs. O. B. Wheeler and G. E. Swinscoe. Astronomical work.—In 1864 the longitude of Fort Howard, at the head of Green Bay, was determined. General Raynolds and Mr. 8. W. Robinson observed at Detroit, and Professor C. A. Young at Fort Howard, having a clock and chronograph at each station. Intermediate obser- vations were made at Chicago by Mr. O. B. Wheeler, who also determined the latitude of his station, and at Ann Arbor by Professor J.C. Watson, The star transits at each of the four stations were recorded on both chronographs. In 1865, the telegraph line having been extended from Fort Howard to Escanaba and Marquette, the differences of longitude between Marquette and Escanaba, and Escanaba and Fort Howard, were determined. The first of these differences was determined by O. B. Wheeler and 8. W. Robinson, each station being provided with a clock and chronograph, and the observers exchanging stations to eliminate personal equation. For the difference between Escanaba and Fort Howard, Mr. O. B. Wheeler and Professor C. A. Young were the observers, both having clocks and chronographs, and exchanging stations to eliminate personal equation. Mr. S, W. Robinson occupied an intermediate station at Menomonee, his observations being recorded on the chronographs at the ends of the line. Latitude observations were made at. Escanaba by Messrs. Wheeler and Robinson, and at Menomonee by the latter. After completing these obser- vations, Mr. Wheeler determined the latitude of the triangulation station at Fishdam at the head of Big Bay de Noquette, Mr. Robinson of the stations at Death’s Door Bluff on the east side of Green Bay and of Cedar River on the west side of the bay, and Professor Young of the Station at Boyer’s Bluff on Washington Island. Off-shore hydrography.—The off-shore work of that part of Green Bay south of the Sister Islands was done by parties on the Search in 1864 and on the Surveyor in 1865, under the charge, respect- ively, of Messrs. A.C. Lamson and O. N. Chaffee, Mr. Lamson’s party doing that part about Chambers’ {sland only. Topography and in-shore hydrography.—In 1864 the islands in the entrance to Green Bay, between Washington Island and Point Detour and both shores of Big Bay de Noquette, were sur- veyed by parties under the charge of Messrs. J. R. Mayer and A. F. Chaffee. Mr. A. Molitor’s party assisted in grading the base-line, and made a survey of Chambers’ Island and of about 15 wiles of the west shore of the bay between Cedar and Menomonee Rivers. In 1865 Mr. A. C. Lam- son’s and Mr. A. F. Chaffee’s parties, respectively, surveyed the east and west shores from Egg Harbor and Menomonee to the head of the bay, and Mr. O. N. Chaffee’s party made a survey of the head of the bay and the lower part of Fox River. These surveys completed the shore-line work of Green Bay. 3. LAKE MicHican.—In 1864, Mr. W. T. Casgrain’s shore-party took up the survey of the north shore of Lake Michigan at the point about 5 miles east of the Monistique River, where the survey under Captain Macomb in 1855 had terminated, and extended it southward until it joined Mr. Mayer’s work near Point Detour. The off-shore soundings along this section were made in the same year by the party on the Search, under the charge for a short time of Mr. Hearding, and, after his resignation, of Mr. A. C. Lamson. In 1866, the survey of the east shore of Lake Michigan was extended from Pointe aux Bees Scies to Little Pointe au Sable by Mr. A. F. Chaffee’s party, while that of the west shore was carried from Bayley’s Harbor to Two Rivers by Mr. Henry Gill- man’s party. The off-shore work was completed between the same limits on the east shore and was carried to about 5 miles south of Kewaunee on the west shore by a party on the Ada, under command of Mr. O. N. Chaffee, until illness compelled him to leave the field, after which Mr. A. RB. Flint had charge of the party. Five lines of soundings across the lake were also run by the steamer. The method of latitudes and azimuths was adopted for fixing the positions of points along the shores of thelake, In the fall of 1866, Lieutenant M. R. Brown and Mr. G. Y. Wisner determined §8.] OF THE UNITED STATES LAKE SURVEY. 21 ‘ e the latitudes of Kewaunee, Whitefish Point, and Sheboygan, Mr. 8. W. Robinson those of Cana Island, Clay Banks, and Rawley’s Point on the west shore, and Mr. O. B. Wheeler those of Big and Little Pointes au Sable on the east shore. The azimuths at these stations were not observed until 1871 and 1872. 4. LAKE SUPERIOR— Base-line.—After an extended reconnaissance in the summer of 1865, Mr. D. F. Henry selected and marked out a base-line for the primary triangulation of Lake Superior ov the west shore of Keweenaw Bay south of Portage Entry. The line was about 55 miles long, very level, and so located that the triangulation could be developed from it with good angles. It was, however, through a dense forest, and a great deal of labor was required to clear it and remove the stumps and roots from the ground. Inthe fall of 1866, about 8,000 feet of the line having been prepared, the measurement with the Bache-Wiirdemann apparatus was commenced by Mr. D. F. Henry, assisted by Messrs. E. S. Wheeler and David Wallace, but only two hundred and eighty- four tubes were measured, the ground being so soft and shaky from the recent grading and the heavy rains which fell during September as to make the measurement entirely unreliable. In August, 1867, the shore-parties of Lieutenant B. D. Greene and Messrs. A. C. Lamson and H. Gill- man were detached from other work and sent to the base-line to finish grading and preparing it for measurement, and on September 7 the measyrement was commenced at the South Base station and continued to the North Base station, the work being finished on the 25th of October. The ends of the line were marked by the intersection of two fine lines cut on a frustum of a brass pyr- amid leaded into the top of a stone 5 feet long. Five intermediate points were warked in the same way. A station was built near the center of the line and the measurement was checked by trian- gulation. Mr. Henry was in charge of the work, and was assisted by Messrs. E. S. Wheeler and D. Wallace. Triangulation and astronomical work.—Atter 1865 the triangulation and astronomical work on Lake Superior were done in connection with each other by the same parties. The only astronomical work of 1864 was the determination of the latitudes of Copper Harbor aud Portage Entry by Mr. O. B. Wheeler. In 1865 Messrs. O. B. Wheeler and 8. W. Robinson determined the latitudes of Mount Houghton, on Keweenaw Point, and Marquette, and their difference of longitude by powder- flashes, and also the differences of longitude between Marquette, Escanaba, and Fort Howard by telegraph. No triangulation work was done in 1864, but in 1865 Mr. Henry, with a party on the Search, commenced the triangulation of Keweenaw Bay, after selecting the base-line. He built stations on both sides of the bay, and also the Wheal Kate station on Keweenaw Point, about 15 miles northwest of the base. The angles of the secondary triangulation of the head of the bay south of the base-line were read. A general reconnaissance of the central portion of the lake was made, which resulted in fixing approximately the points of the main triangulation. From the base the system expands through very favorable triangles until the first large triangle, Wheal Kate- Vulean—Huron Mountains, is obtained. Station Vulcan is near the northern extremity of Keweenaw Point, the line Vulcan-Wheal Kate being in round numbers 48 miles long. From this line the triangle Vulean— Wheal Kate—Isle Royale is reached, and then comes Vulcan-—Isle Royale-Saint Ignace. From the line Vulcan—Saint Ignace, 93 miles long, the largest triangle of the system, Vulcan—Saint Ignace—Tip Top is obtained; the line Vulean—Tip Top being 101 miles long, and Saint Ignace-Tip Top 92 miles long. In 1866 the building of stations and moving of parties were done by Messrs. Henry and A. C. Lamson, who were in command of the steamers Search and Surveyor, respectively. Three triangulation and astronomical parties were in the field, under Lieutenant M. R. Brown-and Messrs. O. B. Wheeler and S. W. Robinson, Lieutenant Brown having general charge of the work. On account of the lack of triangulation instruments the operations were confined almost exclusively to astronomical observations. Stations Vulcan, Wheal Kate, Huron Mountains, Isle Royale, Saint Ignace, and Northeast were occupied, their latitudes being determined with the zenith telescope, and their differences of longitude by powder-flashes. Station Northeast was built in 1866 on a hill near Tip Top, but on lower ground than the latter, and was abandoned when the Tip Top station was built in 1867. In the latter part of the season of 1866, Lieutenant J. F. Gregory was in command of the Search and read the angles of the triangulation of Keweenaw Bay from South Base as far north as the line Traverse Island—Huron Island. In 1867 Brevet Lieutenant-Colonel F. U. Farquhar, Captain of Engineers, was in command of the Search 22 HISTORICAL ACCOUNT [Cuar. I, e and had charge of the reconnaissance for triangulation and the building of stations, and also exercised a general supervision over all the work on Lake Superior in the absence of General Ray- nolds. The triangulation and astronomical work were under the immediate direction of Lieutenant James Mercur until August 15, and of Lieutenant Gregory after that date. With the instruments borrowed from the Coast Survey, the angles of the triangle Vulcan — Tip Top - Saint Ignace, were read by Lieutenant Merceur and Messrs. O. B. Wheeler and G. Y. Wisner, respectively. After Lieu- tenant Gregory relieved Lieutenant Mercur the three parties were transferred to the south shore and determined latitudes, and differences of longitude by powder-flashing, of points between Marquette and the Sault Ste. Marie. Marquette, Grand Island, Pointe au Sable, Whitefish Poiut, Tahquamenon Island, Point Troquois, Sault Ste. Marie, and two points between Pointe au Sable and Whitefish Point were occupied. The triangulation of Keweenaw Bay was completed in this season by Mr, O. N. Chaffee. The only triangulation-work of 1868, besides the selection of the sites for and the building of stations at Farquhar’s Knob, Porcupine Mountains, and East Sawteeth by Colonel Farquhar, was a secondary triangulation of Whitefish Bay made by Mr. O. N. Chaffee. It depended upon a sec- ondary base measured with wooden rods near Waiska Bay. In 1869 Brevet Major J. A. Smith, Cap- tain of Engineers, had general supervision of the work on Lake Superior. Lieutenant Gregory was in command of the Search, and was charged with the duty of reading the angles of the secondary triangulation between Isle Royale and the Canadian shore north of it, and also with moving and supplying the six triangulation and astronomical parties employed in reading the angles of the main triangulation. Lieutenant Gregory was transferred to the command of the Ada on the Ist of October, and Lieutenant W. R. Livermore, having reported for duty on the Survey, was assigned to the Search. These six parties were divided into two sections, one to work west from the line Vul. can -Saint Ignace, and the other to work east of that line. The first section was under the direction of Lieutenant E. H. Ruffner, and consisted of his own party and those of Messrs. G. Y. Wisner and E. 8. Wheeler. They occupied, respectively, the stations Vulcan, Isle Royale, and Wheal Kate, and after reading the angles of this triangle Mr. Wisner occupied the secondary station, West Isle Royale. Lieutenant Ruffner then moved to Farquhar’s Knob and Mr. Wisner returned to Isle Royale, and the angles of the triangle Wheal Kate - Farquhar’s Knob-—Isle Royale were meas- ured. Mr. Wisner then occupied Porcupine Mountains, but on account of the lateness of the sea- son the readings for the triangle Farquhar’s Knob- Wheal Kate - Porcupine Mountains were not completed. The latitude of Farquhar’s Knob was determined. The second section was under the direction of Mr. O. B. Wheeler, and consisted of his own party and those of Messrs. G. A. Marr and A. R. Flint. They occupied respectively the stations Michipicoten, Saint Ignace, and Tip Top. In connection with the observers at Vulcan and Isle Royale they measured the angles of the tri- angles Vulcan - Isle Royale- Saint Ignace, Vulcan - Saint Ignace - Tip Top, and Vulean- Tip Top - Michipicoten. Paugon, Gargantua, and Mamainse were afterwards occupied and the readings of the angles at them completed. The latitudes of all these stations except Paugon and Mamainse were determined and their differences of longitude were obtained by powder-flashes. The first section used the three new Oertling theodolites, and the second had two of the Coast-Survey instru- ments and the 10-inch Gambey repeating theodolite belonging to the Lake Survey. Off-shore hydrography.—The off-shore hydrography was all done by the steamer-parties, who also moved and supplied the shore and triangulation parties, built stations, and frequently assisted both in the triangulation and topographical work, the chiefs of the steamer-parties usually exer- cising a general supervision over the other parties. It will not be attempted to describe the details of their work for each season, nor to note al] the temporary changes that occurred in the officers or assistants in command of them, but the account of their operations will be confined to a brief statement of the general field in which they worked and of the officers in charge of them. In 1864 Mr. W. H. Hearding, in charge of the Search, did the off-shore hydrography in the vicinity of Copper Harbor and of Portage Entry, and also marked Stannard’s Rock with a tripod station to serve as a beacon for navigators. In 1865 the off- shore hydrography around Keweenaw Point was done by the party on the Search under the com- mand of Mr. D. F. Henry. In 1866 the Search and Surveyor were both engaged on Lake Superior, the former under the command of Mr. D, F, Henry and the latter of Mr. A.C. Lamson. The party § 8.] OF THE UNITED STATES LAKE SURVEY. 23 on the Search sounded out some detached shoals east of Keweenaw Bay, made a minute hydro- graphical survey of Stannard’s Rock, and ran several lines of soundings across the lake. Mr. Lam- son did the off-shore hydrography of Keweenaw ‘Bay and of the coast east of it as far as Laughing Fish Point. In 1867 Colonel Farquhar on the Search and Mr. O. N. Chaffee on the Ada finished the off-shore work from Grand Island to the head of the Saint Mary’s River. Lieutenant Gregory on the Surveyor had general supervision of the work on Isle Royale until August 15, when he was relieved by Lieutenant B.D.Greene. The Surveyor party surveyed Passage and Gull Islands east of Isle Royale, and finished about one-half of the off-shore soundings around Isle Royale. In 1868 Colonel Farquhar, in command of the Ada, made the off-shore soundings along the north coast trom Isle Royale to the head of the lake, and along the south shore from Eagle River to Montreal River. Lieutenant Gregory on the Search finished the hydrographical surveys around Isle Royale, and Mr. O. N. Chaffee on the Surveyor made a hydrographical survey of Whitefish Bay. At the close of this season the only portion of the off-shore hydrography remaining unfinished was that of the south shore from Montreal River to Brulé River, including the Apostle Islands group. This work was completed in 1869 by the party on the Ada, which also assisted the shore-parties in the second- ary triangulation and topography of the Apostle Islands. The Ada was commanded at times by Major J. A. Smith and Lieutenant J. C. Mallery until October 1, when Lieutenang Gregory took charge of her. Previous to that time Lieutenant Gregory was in charge of the Search, and in addition to his duties in connection with the triangulation made general soundings across the lake. Shore-line work.—In 1864 Mr. Henry Gillman’s shore-party made a survey of Copper Harbor, finished the survey of Torch Lake, aud surveyed the shore of Keweenaw Bay from Portage Entry to Pequaquawaming Point. In 1865 Mr. Gillman filled up the gap between the surveys of Eagle River and Ontonagon, while the parties of Messrs. J. R. Mayer and A. Molitor filled up that between Copper Harbor and Portage Entry, making a careful topographical survey of a large part of the mineral region of Keweenaw Point. In 1866 the two latter parties surveyed the shore be- tween Keweenaw Bay and Marquette. In 1867 Mr. Gillman filled up the gap between the surveys of Marquette and Grand Island, while Messrs. Mayer and A. Molitor surveyed the shore from Grand Island to the head of Saint Mary’s River. In this year, parties under Lieutenant B. D. Greene and Mr. A. C. Lamson were at work on the south and north shores, respectively, of Isle Royale until the 23d of August, when, as has been stated, the parties were transferred to the Keweenaw base. In 1868 two parties, under Lieutenants B. D. Greene and J. C. Mallery, completed the survey of Isle Royale. A party under Lieutenant J.E. Griffith madea survey of the south shore from Ontonagon” to about four nftles west of the Montreal River, and was then tramsferred to the north shore, of which it surveyed about nineteen miles. The transfer was made after Lieutenant Griffith had left the field on account of ill health, and Mr. Gillman had taken charge of his party. The rest of the north shore from Pigeon River to the head of the lake was surveyed by parties under Lieutenant W.E. Rogers and Mr. Mayer. In 1869 the only portion of the survey of the American shore remaining unfinished, that between Bad River and Brulé River, including the Apostle Islands, was completed, with the exception of three small islands of the Apostle group, by the parties of Messrs. A. C. Lamson and J. R. Mayer. 5. MISCELLANEOUS.—In 1868 the large theodolites which had been ordered from Berlin not having yet arrived, the three astronomical and triangulation parties which had been organized for work on Lake Superior were assigned to the duty of determining the latitudes and longitudes of points along the shores of Lakes Ontario and Erie, between Ogdensburg and Detroit. Lieutenant E. H. Ruffner had general charge of this work, the observers being himself, Mr. O. B. Wheeler, and Mr. G. Y. Wisner. They commenced work at the three eastern stations, after finishing which the two observers farthest east swung around the one on the west, and so on. The method of star- signals was used in longitude-work, the middle observer recording his observations on the chrono- graphs at the ends of the line. The points occupied were Ogdensburg, Watertown, Oswego, Roch- ester, Buffalo, Dunkirk, in New York; Erie, in Pennsylvania ; Ashtabula, Cleveland, Sandusky, Toledo, in Ohio ; Monroe and Detroit, in Michigan. General Raynolds was relieved of the charge of the Lake Survey by Brevet-Brigadier General C. B. Comstock, Major of Engineers, on the 12th of May, 1870, 24 HISTORICAL ACCOUNT (Crar. 1, THE SURVEY UNDER THE CHARGE OF BYT. BRIG. GEN. C. B. COMSTOCK, MAJOR OF ENGINEERS. May 12, 1870, To CLOSE OF SURVEY. 9 9 AND UNDER THE TEMPORARY CHARGE OF CAPT. H. M. ADAMS, CORPS OF ENGI. NEERS, DURING THE ABSENCE IN EUROPE OF GENERAL COMSTOCK, From Aveust 14 To NOVEMBER 20, 1874, AND FROM MAy 24, 1877, TO JUNE 25, 1878. § 9. Since General Comstock assumed charge, the Survey of the Northern and Northwestern Lakes has been completed, and a continuous chain of triangulation, depending upon eight care- fully measured bases, has been extended from Saint Ignace Island, on the north shore of Lake Superior, to Parkersburg in Southern Lllinois, a distance of 10°, and from Duluth, Minn., via Chicago, to the east end of Lake Ontario, a distance along its axis of 1,300 miles. This triangula- tion was incidental to the survey of the lakes, but was measured with the greatest precision, in order that it might be of value in a more accurate determination of the form and dimensions of the earth. The description of the survey since 1870 will be arranged under the heads of (1) office work, (2) methods of field-work, (3) field-work of the Survey by years. As most of the subjects under these heads, except those relating to topography and hydrography, will be treated of at length in the other chapters of this report, the descriptions here given will be brief and general in their character. OFFICE WORK. 1. General work of the office—The office work comprises the reduction and plotting of the field- work of the several parties, the drawing of the final charts, the correspondence, money and property accounts, the issuing of the published charts, the examinations of instruments, and the investigations of the various scientific subjects connected with the Survey. Mr. J. H. Southall was the chief clerk in charge of the money accounts and of the general correspondence of the office from July 25, 1870, to July 31, 1878, when he resigned and Mr. N.S. Fisher was appointed in his place. Mr. J. Lohman has been the property clerk since February 1, 1872, and his duties have included the issuing of charts and the keeping of the registers of note-books, charts, and field sketches, reports and computations, instruments, &c. . 2. Computations and plotting of field-work.—The computations incident to the Survey are made under the direct supervision of General Comstock, and are submitted for his approval before being registered and adopted for use. All computations not self-checking are duplicated. As a general rule, each astronomical observer, on returning to the office, is required to make the first reduction of his own work, and each chief of a triangulation party reduces and tabulates in the “record of triangulation” the angles read by himself. These two classes of observers, in connection with a small force of computers (of whom Messrs. O. B. Wheeler, T. W. Wright, T. Russell, and C. H. Kummell have been longest employed), who usually remain in the office during the entire year, make the computations for the adjustment of the primary and secondary triangulation, and for the geodetic positions of the points of triangulation, and also prepare the data required by the draughts- men in the projection and drawing of the final charts. Of the shore-parties, the chief and one assistant are usually retained in the office. They compute their triangulation, the co-ordinates of the stations, and of the points located with the chain and theodolite, and plot both the topography and in-shore hydrography on a scale of 1: 10,000 on sheets called “ detail sheets.” These sheets are first divided by fine lines into squares of 1,000 meters on a side, and then the principal stations are plotted by their co-ordinates and checked by the lengths of the sides of triangles. The work which has been plotted in the field on the “ field sheets” is then plotted upon the detail sheets being adjusted to the positions of the principal stations. The soundings, after being corrected for error of lead-line, and reduced to a plane of reference, are plotted by interpolation between sound- ing stations and buoys. ~The off-shore hydrography is plotted on a scale of 1: 60,000. Each detail sheet contains a list of the note-books and field sheets from which it is plotted. §9.] OF THE UNITED STATES LAKE SURVEY. 25 3. Final charts.—The final charts are compiled and drawn by draughtsmen employed especially for that purpose. The data for the projections and the co-ordinates of all points fixed by the tri- angulations, primary, secondary, and tertiary, are furnished from the office computations. The details of the topography and hydrography are filled in from the detail sheets. The final charts, when completed and verified, are forwarded to the Engineer Department at Washington, where they are at once photolithoeraphall and afterwards engraved. The system of publication adopted since 1870 has been to publish a general chart of each lake on a seale of 1: 400,000, and to divide the shore-line of each lake into convenient sections, and publish a separate chart of each section on a scale of 1: 80,000. These are called coast charts. The charts of the rivers and a few special localities are on larger scales. The sailing lines are laid down on all charts. A list of the author- ities, a water-table showing the mean level and fluctuations of the water for certain periods, a table of magnetic variations, a table of light-houses, a list of sailing directions, and a statement of the dangers to be avoided are printed on each chart. 4, Reduction of water-level and meteorological observations.—The reduction of water-level and meteorological observations was in charge of Mr. O. B. Wheeler from March, 1871, to July 1, 1878, and of Mr, A. R. Flint after that date. Their reports are published in the annual reports. The Signal Service having established meteorological stations at or near many of the places occupied by the Lake Survey, and the Chief Signal Officer having directed that copies of the observations made at those places should be furnished the Lake Survey, the meteorological observations by the Survey were discontinued in January, 1872, at all stations but Port Austin and Monroe, Mich., and Sacket’s Harbor, New York. The observations at these stations were also discontinued in 1876. Water-level observations have been made continuously since 1870 at Milwaukee, Cleveland, Erie, and Charlotte ; since 1873 at the above places and also at Sault Ste. Marie, Marquette, Escanaba, Port Austin, Detroit, and Sacket’s Harbor; and at various times for\a year or more at Superior City, Duluth, Monroe, Buffalo, and Fort Magara: The mathod of observation at the several stations is as follows: A convenient fixed point, called the zero of gauge, is chosen. The distance of the surface of water from this zero is measured with a rod graduated to hundredths of a foot. These measurements are taken three times a day, at 7 a.m., 1p. m., and 7 p. m., local time. The zero of gauge at each station is connected with at least two permanent bench-marks, and levels are run each season to see if any change has taken place in the position of the zero. As an additional precaution, a check-point is established at each sta- tion, from which readings are taken twice a month in the same way as from the zero of the gauge. This serves to detect any change in the position of the zero, and, if any change should be found by the leveling, the check-point readings will show when the change occurred. In 1876 a re-reduction of the Lake-Survey observations since 1859 was made, and a series of observations at several other stations having been furnished to the Survey, a set of annual water-level curves, showing: the height of water referred to an established plane for each month of the years, from 1859 to 1876, on each of the lakes, except for Lake Superior, for which the dates are 1870-1876, was drawn and published in the annual report for that year. These curves have been continued and published each year since. A full description of the planes of reference and bench-marks is also given in the report for 1876. 5. Tides and seiches on the lakes.—In 1871 General Comstock commenced the examination of the subject of the tides and seiches on Lakes Michigan and Superior. For Lake Michigan, the records of a self-registering tide-gauge at Milwaukee for several years were available. The heights of the water for solar hours were read off and tabulated, for the entire lunations of which there was a record from 1867 to 1871, inclusive, and for lunar hours for the complete lunations in 1867. The examination of the solar hourly mean heights showed that there was a solar semi-diurnal tide of about four-hundredths of a foot, the tide following the sun’s upper transit being considerably the larger. This inequality was explained by the known existence of a lake breeze at Milwaukee during the summer months, a comparison of the solar diurnal curve for April and November, when the lake breeze should be weak, and that for July and August, when the lake breeze should be strongest, with that of the whole season, showing that for the former months the inequality nearly disappears, while for the latter it is cousiderably increased. The examination of the lunar hourly 4LS 26 HISTORICAL ACCOUNT [Cuar. I, mean heights showed a lunar semi-diurnal tide of eight-hundredths of a foot. The details of the examination and the theory of these tides form Appendix A of the annual report for 1872. An examination of the seiches, or irregular oscillations of the lakes, led General Comstock to conclude that they resulted from atmospheric disturbance, or, more definitely, barometric oscilla- tions and their accompanying winds over some part of the lake. The details of his examination and theory are given in Appendix B of the above report. In 1872 a tolerably complete record of the water-level at Duluth during three lunations was obtained from a self-registering tide-gauge, and examined for evidences of solar and lunar tides at that point. The result showed a lunar semi-diurnal tide of 0.14 foot. The same value was found for the solar tide following the sun’s upper transit, while that following the lower transit was very small, almost disappearing. The record did not extend over a safficient length of time to show whether this was to be attributed to the effect of a lake breeze, as was the case at Milwaukce. The fact that the same value, 0.14 foot, was found for both solar and lunar tides, while the solar tide, theoretically, should be much smaller, and that the solar tide following the lower transit nearly disappears, suggested some cause whose period is a solar day, acting to increase the solar tide following the upper transit, and to diminish the other. A land and lake breeze would be such a cause, but the data were not sufficient for the discussion of the question. 6. Standards of length—In the winter of 1870 the question of standards of length, their coefficients of expansion, and those of the tubes of the base-apparatus was taken up by General : Comstock, and a large amount of work has since been expended at various times in comparisons of the standards with each other and with the base-tubes, and in determining their coefficients of expansion. The basement of the Lake-Survey office was fitted up as a comparing-room and used until the winter of 1876. The Repsold base-apparatus having arrived in November, 1876, aud a very elaborate series of comparisons being required in connection with the determination of its constauts, a more complete comparing-room was built on the first floor of the office. Comparisons have been made there almost without interruption since the fall of 1877. The greater part of this work has been done under the immediate direction of General Comstock by Mr. E. 8. Wheeler, aided at different times by one or more of the other assistants. Previous to 1870, the standards of length used on the Survey were (1) the 15-foot brass bar, having a cross-section of 1.1 inches by 0.33 inch, made by Wiirdemann to accompany the base-apparatus. Its length and expansion were given by Wiirdemann. After the measurement of the Chambers Island base, agates were set in its ends, changing its length. Its new length was determined at the Lake-Survey office by comparing it with (2) five brass yards made for the purpose in 1867 by Wiirdemann. These yards had a cross-section 0.98 inch by 0.37 inch, and their lengths were assigned by Wiirdemann and marked upon them. On examining these yards in 1870-71, considerable discrepancies were found in their relative lengths as given above, aud their end surfaces were also found to be so irregular that no final and satisfactory value of the 15-foot bar could be obtained from them, and they were discarded. (3) General Comsiock then procured five new standard yards. They were constructed in the Office of Weights and Measures in Washington, under the direction of Professor J. E. Hilgard. They are all similar brass bars 1 inch deep by 0.6 inch wide, and 34.7 inches long, having at each end an axial cylinder 0.4 inch in diameter and 0.6 inch long. In the ends of these cylinders agates are held by the brass, which is burnished down on them. Their lengths were determined by com- parisons made by General Comstock with the “Transfer yards A and B” in the Office of Weights and Measures. (4) Early in 1875 were received two steel yards made for the Survey by Troughton & Simms, of London, the lengths and expansions of which had been determined by Colonel A. R. Clarke, of the Royal Engineers, by comparisons with the Ordnance-Survey Standard. These yards are called “Clarke yards A and B.” They are bar. of steel 0.73 inch deep and 0.50 inch thick. The ends are cylinders 0.35 inch in length and 0.25 inch in diameter, with agates set in their ends. Each yard is inclosed in an iron box, so arranged that the yard remains in the box during com- parisons. Each box has niches for four thermometers. The lengths of the brass yards and of the 15-foot brass bar were redetermined by comparisons with the Clarke yards, and these values have since been used. (5) For determining the values of the micrometer-screws of the contact-level comparators, used in the comparisons, a standard inch, graduated to tenths, and the last tenth to hundredths, has been used. This standard inch was made by Troughton & Simms, and the values § 9] OF THE UNITED STATES LAKE SURVEY. . 27 of each of its tenths and hundredths were determined by Colonel Clarke by comparisons with the Ordnance-Survey standard foot. (6) There are three standards which accompany the Repsold base-apparatus. One is a steel metre called “ R. 1876,” whose length and expansion were deter- mined at the Imperial Bureau of Weights and Measures, Berlin. The second is a metre made of two bars, zine and steel, and called the “metallic thermometer metre”; its constants have not yet (May, 1880) been determined. The third is a decimeter whose values were also determined at the above bureau. These three standards arrived in 1878. 7. Thermometers.—The lengths of the standard bars and yards and of the tubes of the base- apparatus depend upon their temperatures, and as the corrections to the thermometers then in use were unknown, the determination of these corrections was undertaken in the winter of 1870. In- vestigations and experiments relating to this subject have been carried on from time to time to the present date with the object of determining with the greatest precision the values of the thermom- eters used as standards, and of deducing from these the corrections to be applied to the indications of the working thermometers used directly to give the temperatures of the standards and tubes. The working thermometers are compared with the standard by completely immersing them and the standard in a glass vessel containing water, whose temperature is varied through the desired range, and reading the indications of both by means of a micrometer attached to the telescope of a small theodolite. The water in the vessel is kept in motion by small revolving paddles in order to secure a uniformity of temperature in it. The comparisons and tests of the freezing and boiling pcints and calibration of the standards, made in the Lake-Survey Office, have been made by Mr. Thomas Russell under the immediate direction of General Comstock. In 1870 a standard Wiirdemann thermometer was procured and used until the receipt in 1872 of a standard Troughton & Simms thermometer, whose corrections had been determined by a series of comparisons with the standard thermometer at the Kew Observatory in England. In 1875 six- teen thermometers specially adapted for use with the Clarke yards were received. Eight of these thermometers had been compared by Colonel Clarke with the Ordnance-Survey standard. Soon afterwards five excellent thermometers of Casella’s make were received. The corrections of these thermometers were determined at Kew Observatory, for both horizontal and vertical positions, in January, 1875. Two Baudin standard thermometers were procured from Paris in 1876, but the bulb of one of them was broken in transportation, and that of the other was cracked while being cali- brated. A new bulb was blown on one of them by James Green, of New York, and it was after- wards calibrated and its freezing and boiling points determined at the Lake-Survey Office. As the expansions of glass and mercury depend upon both the first and second powers of the temperature, a mercurial thermometer which is correct at its fixed points, 32° Fahr. and 212° Fahr., may be found to bein error by several tenths of a degree at 100° Fabr. when compared with an air-thermometer. As such errors affect the precise determinations of the lengths and coefficients of expansion of the standards of measure, it was deemed advisable to have one of the standard thermometers compared with an air-thermometer, and Casella 21472 was accordingly sent to Professor H. Ste. Claire Deville, Paris, in 1876, for this purpose. Professor Deville, however, found it impossible to compare it with an air-thermometer, but he made a careful study of it ahd reported that it had no errors of calibra- tion exceeding 0°.1, and that it differed from a tested standard of great perfection by only 09.02 and 0°.03 at 61° Fahr. and 91° Fahr. The same thermometer was sent in the autumn of 1879 to Professor H. A. Rowland, of Johus Hopkins University, Baltimore, to be compared with an air-thermometer there. A direct comparison with the air-thermometer was not obtained, but Professor Rowland compared it very carefully on different days, in both horizontal and vertical positions, with two fine Baudin standards, which had been previously compared with an air-thermometer, and fur- nished a system of corrections by which to reduce its indications to those of a “perfect-gas thermom- eter.” This thermometer is now the standard Lake-Survey thermometer, and a system of correc- tions depending upon it has been adopted for the other thermometers. 8. Report on European surveys.—In 1875 and 1876 numerous manuscripts, books, and maps relating to European surveys were, at the request of General Comstock, procured by the Engineer Department through the Department of State and forwarded to General Comstock for examination and report. They were carefully looked over and abstracts and translations of the more important papers and notes on the various maps were made, mainly by Captain H. M. Adams, Lieutenant P. M. 28 HISTORICAL ACCOUNT [Cmar. 1, Price, and Mr. F. W. Lehnartz. These papers, with general remarks upon them and a discussion of the character and cost of similar surveys for the United States, by General Comstock, were published as a supplement to the Lake-Survey report for 1876. METHODS OF FIELD-WORK. 1. Astronomical work. (a) Longitudes.—The instruments that have been used for time deter- minations are two Troughton & Simms astronomical transits, one of 43 inches focal length and 3 inches aperture, the second of 29 inches focal length and 24 inches aperture; two Wtirdemann transits, Nos. 1 and 15, each having a focal length of 31 inches and an aperture of 24 inches; a combined transit and zenith-telescope made by Pistor & Martins, Berlin, dated 1849, having a focal length of 24 inches and an aperture of 24 inches; a combined instrument made for the Lake Survey by the same makers, and received in Detroit in 1874, having a focal length of 24 inches; and a transit made for the Survey by Buff & Berger, of Boston, having a focal length of 39 inches, a magnifying power with the diagonal eye-piece of 87, and an aperture of 3 inches. The last instrument was received in the spring of 1876, and since that time has been used constantly as the observatory instrument at Detroit. The two Wiirdemann transits have been used more than the others. The break-circuit sidereal clock No. 184, by Bond & Sons, was used in the observatory until the spring of 1875, when it was replaced by No. 256. Two of Bond & Sons’ chronographs and several mean and sidereal chronometers by different makers have been used. In 1874, two break-circuit sidereal chronometers were procured for use by the field observers. Some longitude work by powder-flashing was done on Lake Superior in 1870-71, but with that exception the method of telegraphic signals has been.employed exclusively. Tor the determination of the longi- tude of primary triangulation points the field observer has used a chronograph, and in 1871 a clock, but since then a break-circuit chronometer. For these determinations, four complete nights’ work has been required, and the reductions have been made by the method of least squares. For the determination of the positions of points in aid of State surveys, two full nights’ work is required. The field observer uses a chronometer, and makes his observations by the eye-and-ear method, and these are reduced by the method of high and low stars. A clock and chronograph are used for all the observations made at Detroit. Time stars are selected from the American Ephemeris and from the German Catalogue of ‘539 Sternen,” and cireumpolar stars, which must be over 75° declination, are taken from one of the above or from the ‘General Bericht der Europiische Grad- messung” of 1870. The following is the programme for one night’s work, by which a complete time determination is made before and after the exchange of signals: Level readings. Circumpolar star (reversed on). Level readings. Five or more time stars. Level readings. Reversal of transit. Level readings. Five or more time stars. Level readings. Circumpolar star (reversed on). Level readings. Exchange of signals. A second time determination following the same programme as above. Level readings are also taken between the time stars when the interval is sufficient. When both observers have chronographs, signals are exchanged by automatic clock or chronometer beats by sending alternately from each station for 1™ 20° or for 2™ 20° until two sets of signals have been sent from and received at each station. When the field observer has no chronograph, the signals are sent (1) from Detroit by switching in the clock for seven minutes, the field observer td a § 9.) OF THE UNITED STATES LAKE SURVEY. 29 noting the coincidences of the armature breaks and the beats of a mean solar chronometer, which is compared before and after the exchange of signals with the sidereal chronometer used in the time observations, these comparisons being also made by coincidences; (2) from the field station by automatic breaks, if the observer has a break-circuit chronometer, otherwise by hand-breaks, coincident with the beats of his chronometer, usually by sending for 15 seconds, then omitting for 15 seconds, then sending, and so on for a period of two minutes. The personal equation of the observers is determined on two nights before going into the field, and on two nights after returning from it, by the field observer setting up his instrument near that of the Detroit observer, and both making complete time determinations entirely independent of each other. The difference of the determined times, corrected for the few feet of distance between the instruments, gives the personal equation, the clock and chronometer being compared by the exchange of signals, in precisely the same manner as when one of the observers is in the field. (b) Latitudes.—All latitude determinations are made by Captain Talcott’s method of opposite and nearly equal meridian zenith distances. Some of the observations have been made with the combined zenith-telescope and transit of Pistor & Martins, but most of them have been made with the several Wiirdemann zenith-telescopes belonging to the Survey. The telescopes of the latter instruments have a focal length of 32 inches and an aperture of 23 inches, except that of No. 19, which has an aperture of 3 inches. For the determination of the latitudes of primary triangulation points, four nights’ work is required, and for points in aid of State surveys, two nights. From twenty to thirty pairs of well determined stars must be observed each night. In 1871~72 a “ Cat- alogue of the Mean Declinations of 981 Stars between 12 hours and 26 hours of Right Ascension and 30° and 60° of North Declination for January 1, 1875,” was prepared by Professor T. H. Safford under the direction of General Comstock, and printed at the Government Printing Office, Wash- ington, 1873. Stars for latitude work have been selected mainly from this catalogue since its publication, and all the important latitude work of previous years has been recomputed with the star-places as given by it. (ce) Azimuths.—Azimuths are observed with the primary triaugulation instruments. Previous to 1875, three nights’ observations were required for the determination of primary azimuths, the method followed being that given in the Coast-Survey Report for 1866. In 1875 General Comstock issued instructions changing this method, and since then observations for primary azimuths have been made on five nights. The horizontal limb of the instrument is so placed that on each night the reading to the reference mark will exceed that of the preceding night by one-fifth of the dis- tance between the microscopes, this reading remaining constant for the night. To avoid the assumption of perfect stability of instrument from first pointing to mark to last pointing to star, or of a change in azimuth strictly proportional to the time, the following is the programme for one star for one night, the star being near elongation : Pointing to mark. Pointing to star. Level readings. Pointing to star. Pointing to mark. Reverse the telescope, keeping the same pivot in the same wye. Pointing to mark. Pointing to star. Level readings. Pointing to star. Pointing to mark. Pointing to mark. Pointing to star. Level readings. Pointing to star. Pointing to mark. 30 HISTORICAL ACCOUNT ° Crap. I, Reverse the telescope as above. Pointing to mark. Pointing to star. : Level readings. Pointing to star. Pointing to mark. As many closely circumpolar stars are observed on each night as is practicable. For secondary azimuths two such stars are observed on two nights, and the above programme is followed except that on the completion of each set of readings—to mark, star, star, mark—the horizontal limb is advanced through an angle equal to one-fourth the distance between the microscopes or verniers, and that on the second night the first reading to the mark is greater than that on the first night by one-eighth of this distance. The error of the chronometer is obtained by observing the meridian transits of well-determined stars. 2. Base-lines—The bases measured previous to 1877, viz, at Minnesota Point and Keweenaw Point, Lake Superior; Fond du Lac, Wis.; Sandy Creek, Lake Ontario; and Buffalo, N. Y.; were measured with the Bache-Wiirdemann compensating apparatus made for the Survey in 1852. In the fall of 1876, a new base-apparatus, made for the Survey under the special instructions of Gen- eral Comstock, by Repsold & Sons, of Hamburg, was received at Detroit, and with it the bases near Chicago and Olney, Tll., and Sandusky, Ohio, have been measured. The accuracy of the measurement of all these lines has been tested both by duplicating the measurement of a part or the whole of the line, and also by dividing it into two or more sections and checking the lengths of these sections by a triangulation. .Maji€., By Coma@tocksccsesecaullocesicstsnicgcs aie dayiess cincraidaree «.| E. Molitor. Detroit: Rivetsssccsccesessssaassc 1-40, 000 | 1876 |...... OO) sisies eearcs ze anestemance B.. Molitor ican cocssaniewasens LAKE ERIE. Take Bie: c2cccccccsccsssceascases 1=400;.000: 18526 lo ov cepetie’s va gheoneGs deesee ee John Lambert .........--.---- West end Lake Erie .......-...-- 1-120, 000 | 1852 | Lieut. Col. James Kearney ...|...--- OO sex seeesorsissseceseees W. Smith. Kelley’s and Bass Islands.......-. 1-50, 000 | 1852 |...... GO) cinciscreieltesajtece'sise eigaiecatees. OOrsssicnces tebe sceloccicce W.H. Dougal and J.V.N. Throop Buffalo Harbor ....-.....-...----- 1-30, 000 | 1857 | Capt. W. G. Williams and |.....-...2..-.22.--2-00.2 02. eee: W. iH. Dougal. Capt. I. C. Woodruff. Maumee Bay ........-.--..-+-2+-- 1-30, 000 | 1858 | Capt. G. G. Meade......-..-.. J.H. Foster, W. H. Hearding, Do. and P. C. Rabaut. Sandusky Bay ..-..-...-...-.-..-. 1-20, 000 | 1874 We Us MOWER. 2 ocean nent ceen Coast chart No. 2, Lake Erie. .-... 1-80, 000 | 1879 Max Franke .....--........... Coast chart No. 3, Lake Erie. .-.-.- 1-80, 000 | 1879 A. de Witzleben ee Coast chart No. 4, Lake Erie 1-80, 000 | 1879 Max Franke.......-...-.--.6- Coast chart No. 6, Lake Erie = 1-80, 000 | 1879 |.. E. Molitor ...............---.. Coast chart No. 5, Lake Erie..-.-. 1-80, 000 | 1880 A. de Witzleben...-.......... Coast chart No. 7, Lake Hrie....-.. 1-80, 000 | 1881 Ey. Molitor i255 cccescccaweccc Coast chart No. 1, Lake Erie...... 1-80, 000 | 1880 Max Franke.-......-.... “gy Wak Hrie@sccscices ngasevedsssexese 1-400, 000 | 1880 J Ue MUeWers cacccceekesce ace NIAGARA RIVER. Niagara Falls......-...--...--.+++ 1-10, 000 | 1876 C. Donovan sciesscds secdeg seca LAKE ONTARIO. Coast chart No. 1, Lake Ontario..}; 1-80, 000 | 1877 A. de Witzleben Lake Ontario: sce x. -taewsercaxs eos 1-400, 000 | 1877 J.U. Mueller........22...22.. Coast chart No. 2, Lake Ontario -. 1-80, 000 | 1878 W. A. Wansleben ....-....... Coast chart No. 3, Lake Ontario-..| 1-80, 000 | 1878 Max Franke .......--....2.... Coast chart No. 4, Lake Ontario -.| 1-80, 000 | 1878 ; Coast chart No. 5, Lake Ontario..; 1-80, 000 ; 1878 SAINT LAWRENCE RIVER. Saint Lawrence River, No. 1-30, 000 | 1874 BAO MAN cawssoc sens vaan sens Saint Lawrence River, No. 1-30, 000 | 1875 Bi Bis, BAGH OR: vise atieincs Soina caw Saint Lawrence River, No. 3...--.-. 1-30, 000 | 1875 A. de Witzleben.............. Saint Lawrence River, No. 1-30, 000 | 1876 F. A. Fisher........ Saint Lawrence River, No. 1-30, 000 | 1876 W. A. Wansleben ............ Saint Lawrence River, No. 1-30, 000 | 1876 Max Franke ...............02. §§ 11, 12.] OF THE UNITED STATES LAKE SURVEY. 43 § LL. Statement of the appropriations made for the Survey of the Northern and Northwestern Lakes since its commencement. TABLE II. — Act of Con- Act of Con- Appropriation made for— gress ap- | Amount. Appropriation made for— gress ap-| Amount. proved— ‘ proved— Hydrographical Survey of Northern and | Mar. 3, 1841 $15,000 00 || Surveys of the Northern and Northwest- Northwestern Lakes. ern Lakes. May 18,1842] 20,000 00 Mar. 1,1843| 30,000 00 DO jeistersteicrcios diaciarkisie esa read Maatastel ce ., Tune 17, 1844 | 20, 000 00 Do iseceamrcnccaaa nan anrnctaanosen eee Mar. 3,1845} 20, 000 00 DG ce ssaiesontihe wanted soneticin: casete Aug. 8,1846} 25,000 00 WO Gigdensietioracanemeenned MaDeneaeones Aug. 12,1848] 25, 000 00 DO 26 cwsseaashioden sates saeseesesess Mar. 3,1849} 10,000 00 | TDG eevedanelnetaiecneawanscy aiesenesies Sept. 28,1850] 25,000 00 | IDO sc sehan loved eta vacd esa eee Mar. 3,1851! 25,000 00 | 1 | Surveys of the Northern and Northwestern | Aug. 30,1852] 25, 000 00 Lakes. : Doe eeedeeeedyceeeese ass aawae Mar. 3,1853| 50,000 00 | DG See eeeecctee cae teen ace atten .....| Aug. 5,1854| 50,000 00 D0 Sess Mesi cette ce ag Mar. 3,1855| 50,000 00 | Dig aided Anns enue need eh Ang. 30,1856 | 50,000 00 Do ..--..220-- 2222-2 eeee eee es eeee--| Mar. 3,1857} 50,000 00 | § Surveys of the Northern and Northwest- June 12, 1858} 75, 000 00 ern Lakes and Mississippi River. June 21, 1860| 75,000 00 ern Lakes. Mar. 2,1861| 75,000 00 DDO disie dieistatnratdi vais siege ncitcictoaiainnsey Do... July 5,1862| 105, 000 00 | WDO sites scicccnacesescsteecavsnese Feb. 9, 1863 | $106, 879 00 July 2,1864| 100,000 00 Feb. 28, 1865| 125, 000 00 June 12,1866] 50,000 00 Mar. 2,1867| 77,500 00 Mar. 2,1868] 77,500 00 Tuly 20,1868} 75,000 00 Mar. 3,1869| 100,000 00 July 15,1870 | 100,000 00 Mar. 3,1871! 175,000 00 .| June 10, 1872 | 175,000 00 ..| Mar. 3, 1873] 175,000 00 .| Tune 23, 1874 | 175, 000 00 Mar. 3,1875| 150,000 00 July 31,1876] 100,000 00 Mar. 3,1877| 119,500 00 June 20,1878] 99,000 00 Mar. 3,1859/ 75,000 00 || Surveys of the Northern and Northwest- | Mar. 3, 1879 85, 000 00 June 16,1880) 40,000 00 .| Mar. 3,1881] 18, 000 00 * Appropriation $155,000, with provision that there shall not be over fifty per cent. expended during the fiscal year ending June 30, 1868, and the residue thereof shall not be expended till otherwise ordered. Restriction subsequently withdrawn by resolution No. 17, March 2, 1868. + Part of appropriation applied to Survey of Mississippi River, as provided by the act. About $16,000 was so expended. + Appropriation $110,000, of which $25,000 should be expended in continuing Survey of Mississippi River. sales of steamers and applied to the Lake Survey, as provided in the act. § One-half the amount applied to Survey of Mississippi River as provided in the act. The $9,500 was received from || An additional amount not exceeding $8,000 was reappropriated from the unexpended balance of the appropriation of June 16, 1880, § 12. Officers in charge of the Survey of the Northern and Northwestern Lakes. TABLE ITI. Name and rank. Aasenictl Relieved. Capt. W. G. Williams, Topographical Engineers ..--.....-+ seeeee sees eee eeeee cee ener e ee ee reece eee eee May 17, 1841 | Fall of 1845. Lieut. Col. James Kearney, Topographical Engineers Fall of 1845 | Apr. 9, 1851. Capt. J. N. Macomb, Topographical Engineers ...-..---.--.---------- Apr. 9,1851 | Sept., 1856. Lieut. Col. James Kearney, Topographical Engineers Sept., 1856 | May 20, 1857. Capt. George G. Meade, Topographical Engineers........---..---++--+-2+ 22 220eeer erect eet ccc May 20,1857 | Sept. 1, 1861. * Col. James D. Graham, Corps of Engineers ...--.---- +--+ +--+ -++-222 eee ee een ee eer tenes cette ees cece eee ee Sept. 1,1861| Apr. 15, 1864. Lieut. Col. W. F. Raynolds, Corps of Engineers, brevet brigadier-general, U.S. A .--.....--.--------2-------- Apr. 15, 1864 | May 12, 1870. || Maj. Cyrus B. Comstock, Corps of Engineers, brevet brigadier-general, U.S. A......-.-- vate eee ee eee e ences May 12, 1870 | Closeofsurvey. : 7 Aug. 14, 1874 | Nov. 20, 1874. + Capt. Henry M. Adams, Corps of Engineers in temporary charge.......---------+---++++--++eeeteeee reer eee i May 24, 1877 | June 25, 1878. * Col. James D. Graham became lieutenant-colonel in the Corps of Engineers when the Corps of Topographical Engineers was consoli- dated with the Corps of Engineers March 3, 1863, and was promoted to be colonel of engineers on June 1, 1863. tIn April, 1864, General Raynolds was a major of engineers, with the rank of colonel and additional aide-de-camp. He was promoted to be lieutenant-colonel of engineers March 7, 1867. || Maj. C. B. Comstock was promoted to the rank of lieutenant-colonel of engineers July 19, 1881. +Captain Adams was in charge of the Survey during the absence in, Europe of General Comstock. 44 HISTORICAL ACCOUNT § 13. Officers who have served as assistants on the Lake Survey. LCnar. I, [Those who served previous to March 3, 1863, were, at the time of their service, ofticers of the Corps of Topographical Engineers. Those serving since March 3, 1863, have been officers of the Corps of Engineers.) TABLE IV. Name. Rank and remarks. Joined. Relieved. Howard Stansbury......-..-. Captain scp occa b soo centeeecasces wadel aqacibunlas diaietis nape taeseeuie ses gemewiace 1841 1842 Joseph E. Johnston. - -| First lieutenant and brevet captain 1841 1842 James H. Simpson .......---- First lieutenants oeses cen oeiccvees Yeas neves sass se emens eevee ge ea gd pees ebteeediees 1841 1845 I. Carle Woodruff ....-....--- Second lieutenant ; first lieutenant March 31, 1842 ..........-.22..---- 2202 ee eee eee 1841 1847 John N. Macomb .......----- First lieutenant; placed in charge of the Survey April 9, 1851; captain August 1842 | Sept. 1856 4, 1851. William H. Warner.........- First lieutenant s.2..22 02-0 aes gates sdcatns Sethe tate Saami erereeen ea eetRee hs eee 1842 1843 : : : ; 1842 1849 Jobn W. Gunnison .......... Second lieutenant; first lieutenant May 9, 1846 ............2-- 2-2. e eee eee ee eee : 1851 | Apr. 1853 J.D. Webster. ........-.2.... BAGULSDaN tis. ts ceaccae Sreeaaserteaita duce sees wsdeoms Pasaahetarmeee deena eaawadada 1842 1843 James W. Abert ....-.-..---- Brevet second lieutenant 1843 1844 Williarn B. Franklin Brevet second lieutenant 1843 1845 E. Parker Scammon.......... First lieutenant; captain March 3, 1853 1847 | Nov. 1854 William F. Raynolds. ........ Second lieutenant; first lieutenant March 3, 1853............------ 22.2022. 0 2 eee 1851 | Apr. 1856 George H. Mendell........... Breve second, Menten an teen jai. eiseince sate wilt cinleies niesay Sis nies twiaiDese Laveaadd Hees He Eee: Dec 1852 1854 George W, Rose... .005200.0. Brevet second lieutenant; second lieutenant March, 1855.............---.-+-2--++- Nov. 1852 | Mar. 1856 George G. Meade..-.........- Captain; appointed superintendent of the Survey May 20, 1857 .| June, 1856/ Sept. 1, 1861 Charles N. Turnbull ......-.-. Second lieutenant)... eos csccesaeut adits GeekGemmenShereseteaseetauas aoeceiamciees June, 1856 1859 Orlando M. Poe.... Second lieutenant; first lientenant July 1, 1860........-..-.-2-2-2-. 222.2 ee eee eee Oct. 1,1856| May 1, 1861 William Proctor Smith...... Brevet second lieutenant.......--...-... 222-2 sees cece ee ee eee eee eee eee eee Nov. 12, 1857 | Feb. 14,1861 J.L. Kirby Smith............ Second lieutenants « cicccsecccs cesar ccutnmsoternsceased seeuanseeaees acmccinscecca Nov. 12,1859] June 13, 1861 Robert F. Beckham.......... Brevet second lieutenant .-...-...-.--.. 2-02 -2-02 2 eee eee eee eee cece eee eee ee Oct. 1, 1859 ‘May 3, 1861 Micah R. Brown.......-...- First lieutenant Oct. 1,1865| Mar. 12, 1867 James F. Gregory......-..--. First lieutenant July 1,1866! Nov. 19,1869 Francis U. Farquhar......... Captain; brevet lieutenant-colonel, U.S. A Mar. 4,1867| Nov. 9, 1868 James Mercur............... Second lieutenant; first lieutenant March 7, 1867.....-.......2-..22.2-0-.--- divine Fe Oct. 1, 1866] Aug. 23, 1867 Benjamin D. Greene ....... | Second lieutenant; first lieutenant March 7, 1867...........-..-..20222-2022000000- Oct. 1,1866|May 5, 1869 Ernest H. Ruffner............ Pirst Memben anit sigs ectosu’esacnsns en nels has nigererhes dered otter erertante alse magni Aug. 1867| Nov. 30,1870 Jobn C. Mallery.........-.... Hirst Membenant .22.3.00-0s1e sieresuccdscceessyadaes ceayireacamen se teiis wet nasa oeuiex vee -|Aug. 1867] Oct. 1, 1869 Joseph E. Griffith............ Second listitenan ts sces. sccsewsasedsaceaey nbs ade co Suauaasusaune ae meunoiancmncaee Aug.17,1867| May 5, 1869 William E. Rogers... ....... Second lieutenant ...--. Sept. 1867/ Aug. 6, 1869 Lewis M. Haupt ............- Second Heuben awl vcs0c vecsmsian sank conewns teases Ranimnnene koeyeaee REAR Eons Aug. 9,1867| Jan. 31,1869 Jared A. Smith .............. Tain BRAT AOR UB Aida cmaecaeanay acs Patines sRede vanseumeyiense agen June 5,1869/ Apr. 1, 1871 William R. Livermore........ First lieutenant ; captain January 22, 1870 Sept. 24,1869 | May 26, 1874 A. Nisbet Lee. ....-.......... First lieutenant; captain July 11, 1871.....-2.....0 0202220 fee eee eee eee eee May 1,1870| Feb. 28, 1874 John H. Weeden.....-.....-. Eivet lieutenant. ccerencrcctenwsssd saga teaaucs Uhamaes enecn nang May, 1870/June 7,1872 Edward Maguire ............ Hirst MeutenaMnt i.2c2 seine, tsieseceion® seieec/ho Woae MoGeRwm aa aee seers sanaela edi bree May 6,1871| May 12,1875 Charles F. Powell.... .-| First lieutenant ....-....-.-...----.-- : May 6,1871| Mar. 18,1879 Daniel W. Lockwood ......- First lieutenant; captain June 30, 1879 July 10,1872} Feb. 14,1880 Thomas N. Bailey..........- Second lieutenant .........-..----2 2-222 esse cece cee cece cece econ eee cee eeeeees June 1,1874| Aug. 23, 1876 Henry M. Adams ...... Ratios First lieutenant; captain September 2, 1874; in charge of the Survey during | June 8, 1874! Dec. 31, 1878 absence in Europe of General Comstock from August 14, 1874, to November 20, 1874, and from May 24, 1877, to June 25, 1878. PHI NG PIG ocak cacsamene First lieutenant. ....-..--.--.2- 2-20 .eeeee eee e ee eee pumane messed wed awn Yea cas May 13,1875! Dec. 7, 1880 § LA. Civil assistants employed on the Survey. [For the years previous to 1870 this list is necessarily incomplete, the office records not containing the data for making a complete list.) TABLE V. Name. Years of Occupation. : Pp service. Remarks. R. W. Burgess J. F. Peter J.H. Forster... J. Houghton, jr Years not known, but were on the Survey during part of Mr. Burgess’ term. Duration of service unknown. §§ 13, 14.] OF THE UNITED STATES LAKE SURVEY. 45 § M4. Civil assistants employed on the Survey—Continued. {For the years previous to 1870 this list is necessarily incomplete, the office records not containing the data for making a complete list. ] TABLE V—Continued. Years of Name. Occupation. BeEVIC: Remarks. J. H. Hearding..........-.--..0-0 +--+ Assistant engineer..........-.-.--..-. 1849-1850 J. A. Potter 1849-1861 W. 4H. Hearding -| 1851-1864 F. W. Wilmotte 1851-——- | Duration of service not known Henry Gillman 1851-1869 G. W. Lamson 1851-1860 TD: Beghein -< .2 se -ce2c. coseceaxessees ieee 0: ceexd sceeceeeeeedd Seaseeceesss 1853-1859 Si COlOs sxzenzvewss veseceoenesseenees|avguay WO dé vanaiasicnaininvencieee ea: 1853-—. “Do. J. U. Mueller 1854-1880 H.C. Penny.-...---+----- 2-22-22 sees 1855-1866 Re BROW 125 dances acaeateiaieesiecss 1854——_ Do. DLE. HO 89.20 cc ceisdeaceis netics kee 1854-1871 Gy Pinney» 0c neneve .| 1855-—— Do. O. N. Chaffee .| 1855-1869 | Was away one year. E. B. Wright .| 1854-1861 | Field seasons only. Fe. Wallace). acisiscacciececcgccteneccindes 1855-—— | Duration of service not known G. Wallace........-. -02 seseee eee eee 1855——_ Do. Dis CARB aic.sccnipoctes ee te seceeseaees 1857-1863 W. 'T. Casgrain 1857-1865 A.C. Lamson .- 1856-1878 J. E. Goodell 1857-—_ Do. Ts Clagne sssnevxececcses nea necsiewen cine Assistant and computer ...-.....-.--- 1857-1863 Assistant engineer...........-----..-- 1858-1864 1859-1863 1859-—— 1859-1864 1859-1864 -| 1859 -——_ J. M. Bigelow ......-------------------- 1860-1867 Henry Clague......-------------------- 1860-1870 A. F. Chaffee .........-2--+---2-2e eee eee 1862-1871 J.R. Mayer ....-.----------- 220-22 +20: 1862-1876 J. Paul Mayer .....-.-------+-++---++-+ 1862-1871 E.S. Wheeler ......-...----------+----- 1863-1882 G. A. Marr ..... 1863-1877 A.B. Flint ....... 1863-1882 F. M. Towar ....- 1863-1878 A. W. Unthank 1863-1864 Lewis Foote .-..-..-----------+--+----- 1863-1874 A. Molitor ........-..-seseee-ceeeee eee 1863-1869 S. W. Robinson 1863-1866 | Also in 1873, field season only JaH. Booth: ccc. ccineenerseeacesa sees Recorder ....-.-.----------+ eee eee eee 1863-1865 OLB. Wheéelat ...3 46.654+ 41.15% | 339.98 hk + 341. 25k 22. 59 10 63.89 | 384.48h-+ 384.50k | 45.25h + 43.28% | 339,23 h+ 341.22k | — 23,21 | es | || 33 ae | Date. 33 Y. B. Difference. 2 s et oo ze | ! z | | | 1874. ° | ; | May 18 52.43 412.53 + 412.03k 70.65h+72.70k | 341.88 h + 339.33 k | — 22. 60 : ie] 52.49 | 411. 18h -4+413.23% | 71.38h4+72.58% | 339.8544 340.70k| 23.13 | 18 52. 80 407.25h+4415.93k 72.00h+72,02k | 335.25 h + 343.91 kb 24, 22 19 52.82 411.40 h + 411. 88k ; 72.70k+71.08k 338.70 h + 340. 80k 23. 97 | 19 52.90 | 410.82h4411.95k 71.25h + 72.48% | 339.57 h + 839. 52 k 24. 29 19 52.95 | 407.92h+414.10k. 71.72h+71.80k 336.20A + 342.30% 24. 76 | 19 53.25 | 410.92h + 410.50k | 72.15h+ 69.80% | 338.77A + 340. 70k 23. 99 21 53.33 | 409.78 + 410.48k | 71.15h+ 70.15% | 338.63 h + 340.33 k 24. 40 June 10 63.41 | 386.15h + 386.18k | 48.05h+49.73k |} 338.10h + 336.45 k 27. 91 11 63.15 | 384.18 h + 387.42k | 49.25h+48.83k | 334.93 Ah + 338, 59 k 28. 73 11 63.19 | 385.72h+ 386.35k | 48.98hk+49.40k | 336.74h + 336. 95 k 28. 60 12 62.65 | 388.05h + 386.68k | 50.42k + 50.30k |) 337.63h + 336. 38 k 28. 34 | 12 62.73 | 388.38h + 386.08 | 51.083k+49.98k | 337.35h + 336.10 & | 28.79 | 12 62.94 | 386.45h + 386.50% | 50.35h+48.68k | 336.10h + 337. 82k 28, 41 | 12 62.98 | 386.85 + 385.20k | 48.80h +49.88k% | 338.05h + 335.32 k | 28. 86 13 61.77 | 388.18h + 388.75k | 52.08h+52.25k] 336.10h + 336. 50 k | 29. 46 13 61.81 | 387.37 h + 388. 88% | 52.30h+52.55k |) 335.07h + 336.33 k& | 30. 42 13 62.09 | 387.45h+ 388.55k | 48.45h+ 54.43] 339.00K + 334.12k | — 29.05 | By treating these measurements in the usual manner we get— or, at the temperature of 62°— Yes — A = 22.65-10.652 (t — 62°) Ys5 — B= 28,494 0.495 (t — 62°) Yo5= A + 22.65 + 0,13 Yes = B+ 28.49 + 0.16 §3.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 5a The probable error of a single comparison in the case of the bar -4, being + 0.44, and in the case of the bar B, + 0.55. The length of Y;; at 62° is 0.99999960 yard, consequently, at 62°— A = 0.99997695 + .00000013 B=0.99997111 +4 .00000016 A. R. CLARKE, Lieutenant-Colonel, R. FE. The value of Y;; with reference to the standard yard of England is given in Colonel Clarke’s Comparisons of Standards of Length, p. 162. It will be noticed that no probable error is assigned to the value of Y;; in terms of the standard of England, which amounts to assuming that the standard is zsac4saa0 yard longer than Y;; and virtually makes Y;; the standard. The following are Colonel Clarke’s results : Length of A at 62° F. in yards, A=0°.99997695 4 0.00000013 Length of B at 62° F. in yards, B=07.99997111+ 0.00000016 Expansion for 1° F. of A in yards, =0°.0000058740 + 0.0000000195 Expansion for 1° F. of B in yards, =07.0000058103 + 0.0000000197 In inches, the above values become— Length of A at 62° F., A4=35",999170 4- 0.0000047 Length of B at 62° F., B=35'".998960 + 0.0000058 Expansion for 1° F. of A= 0'7.00021146 + 0.00000070 Expansion for 1° F. of B= 0.00020917 + 0.00000071 STANDARD INCH. § 3. As accurate values of the micrometer-screws used in comparisons of lengths are indis- pensable, a standard inch was obtained from Mr. Simms, of London, for such determinations. It is divided into tenths, and the last tenth (from 0”.9 to 1:°.0) is divided into hundredths. Each of the tenths and each of the hundredths had its value determined by comparison with the Ordnance- Survey standard foot, by Colonel Clarke. On this inch depend the values of the micrometer-screws used in all comparisons. The following letter gives Colonel Clarke’s determinations. THE STANDARD INCH. VALUES OF THE SPACES DETERMINED BY LIEvUT.-CoL. A. R. CLarke, R. E. ORDNANCE-SURVEY OFFICE, Southampton, June 19, 1875. My Dear Srp: I send you herewith the determinations of the spaces on your standard inch. I hope youreceived the inch safely. Believe me yours, very truly, A. R. CLARKE. General CoMSsTOCK. . : «yard Values of the spaces on the standard inch. Tenths ; (anit = 1,000,000 (0.1) =yhy F+ 2.78 40.14 (0.2) = 735 F+7.01 40.13 (0.3) = 745 F4 1.06 40.10 (0.4) =745 F—0.47 4 0.12 (0.5) = 785 F—2.81 - 0.08 (0.6) =1$5 F—5.04 + 0.09 (0.7) = 35 F—9.86 + 0.12 (0.8) = 785 F—6.69 4.0.12 (0.9) = 735 F—0.59 + 0.18 (0.10) = 74%, F+ 4.40 4 0.13 Where F is the length at 62° of the Ordnance standard foot, viz: F=}t Y—0.49 Y being the true length of a yard, or te F= 75 Y—0.04 54 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. II, The values of the 10 spaces of +45 inch, composing (9.10), are as follows, counting from 9 towards 10: Space Ist=45 (9.10) + 3.64 4 0.08 ; Space 2d =y5 (9.10) — 1.44 + 0.06 Space 3d =75 (9.10) —1.59 + 0.10 Space 4th =; (9.10) + 1.50 0.06 Space 5th= yy (9.10) + 0.52 + 0.10 Space 6th = 5 (9.10) + 0.36 4 0.10 Space 7th = +5 (9.10) —1.31 + 0.09 Space 8th =-Jy (9.10) + 1.06 + 0.10 Space 9th = jp (9.10) — 0.22 + 0.08 Space 10th = yy (9.10) — 2.53 + 0.06 N. B.—The probable error of the space (9.10) is this: (9.10) =yh5F + 4.99 + 0.13 Total number of observations, 1,032. A. R. CLARKE, Lieutenant-Colonel, R. E. NoreE.—In the comparisons referred to in this chapter Colonel Clarke’s values of the spaces on the standard inch have been used. Subsequently to his determinations, a good deal of material accumulated in the determination of the values of micrometer-screws and. in direct comparisons in the Lake-Survey office. This work being completed, and being quite extended, was reduced. It consisted in comparing different spaces on the inch with a common space on a comparator-screw, the comparator-screw being used to move a slide bearing the inch, under a fixed microscope. This gave relative values of the tenths of the inch in terms of the whole inch. A similar process applied to the hundredths of an inch between 0'.90 and 1'*.00 gave the relative values of these hundredths in terms of this tenth. As the Lake-Survey determinations were in great number, it was decided to utilize them by combining them with equal weight with those of Colonel Clarke, and the result- ing means have been used as the values for the parts of the inch whenever the inch has been used to determine values of graduations on the 15-feet brass bar, or on any part of the Repsold base- apparatus. The following table gives the values of the tenths and of certain hundredths of an inch as found by Colonel Clarke; those found in the Lake-Survey office; their differences; and their weighted means. For the hundredths of an inch, as there were two independent determinations, double weight was given to the results of the Lake-Survey office. Values of spaces on the standard inch. ‘ ‘ ‘ Spacesenstond- Clarke's val: "Values of | mizus’ Lake: | meaavalves paces. Survey values.| of spaces. in. in. in. in. in. in. 0. 00 to 0. 10 0. 100100 0. 100096 + 0.000004 0. 100098 0. 10 to 0. 20 0. 100152 0. 100166 — 0.000014 0. 100159 0. 20 to 0. 30 0. 099786 0. 099783 + 0. 000003 0. 099785 0. 30 to 0. 40 0, 099945 0, 099958 — 0.000013 0. 099951 0. 40 to 0. 50 0. 099916 0. 099888 + 0.000028 0. 099902 0. 50 to 0. 60 0. 099920 0. 099906 + 0.000014 0. 099913 0. 60 to 0. 70 0. 099826 0. 099871 — 0. 000045 0. 099849 0. 70 to 0. 80 0. 109114 0. 100124 — 0. 000010 0. 100119 0. 80 to 0. 90 0. 100219 0. 100191 + 0. 000028 0. 100205 0. 90 to 1. 00 0. 100179 0. 100176 + 0.000003 0. 100177 0. 00 to 1, 00 1. 000157 1. 000159 — 0. 000002 1. 000158 0. 90 to 0. 91 0. 010149 0. 010120 + 0.000029 0. 010130 0. 91 to 0. 92 0. 009966 0. 009996 — 0.000030 ‘0. 009986 0. 92 to 0. 93 0. 009961 0. 009970 — 0.000009 0. 009967 0. 93 to 0. 94 0. 010072 | 0, 010080 — 0.000008 0, 010077 0. 94 to 0. 95 0. 010037 | 0. 010000 + 0.000037 0. 010018 0. 95 to 0. 96 0.010031 | 0. 010090 — 0.000059 0. 010070 0. 96 to 0. 97 0. 009971 0. 009944 + 0. 000027 0. 009953 0. 97 to 0. 98 0. 010056 0, 010052 + 0.000004 0. 010053 0. 98 to 0. 99 0. 010010 0. 009964 + 0. 000046 0. 009979 0. 99 to 1. 00 | 0. 009927 0. 009961 — 0.000034 6. 009950 0. 90 to 1. 00 | 0. 100180 0. 100177 + 0.000003 0. 100178 §§ 4, 5.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 55 The probable errors of the Lake-Survey work have not been computed. Colonel Clarke’s prob- able errors are small, and may be safely used as small enough in computing the probable error of the space 0.80 to 0.99, which enters the determination of R 1876 in terms of Clarke yard A. Space 0.80 to 0.99 = space 0.80 to 1.00 minus space 0'".99 to 1.00 = 07,200382 + 0'".0000065 — 0.009950 + 0,0000023 = 0.190432 + 0'".0000069 = 4™™,83692 + 0™™.00018 FIFTEEN-FEET STANDARD BAR. - § 4. This bar is a brass bar about 15 feet in length, its cross-section being 1.1 by 0”.33, In each end an agate plate has been inserted in a dovetail, and is held in position by burnishing the brass down upon it. The outer face of the agate plate coincides with the end of the bar. On one broad face and near one end there is stamped “U.S. standard, 15 feet 0.0018 inches at 62° Fah. From yards Nos. 41, 42, 43, 44,45. Expansion for 1° Fah. 0.0017 of an inch.” The length of this bar is the distance between the middle points of the agate planes, when, the stamped face being vertical and the inscription right side up, that top edge which is farthest from an observer reading the inscription is a straight line. This 15-feet bar appears to have been received by the Lake Survey in 1852 with the Bache- Wiirdemann base-apparatus. The agate planes were not put in the ends of the bar till the winter of 1865-66. They of course gave a new length to the bar. Nothing is known of the method by which the expansion of the bar for 1° F., which is stamped on it, was obtained. It is only an approximation to the true value. The length of the bar stamped on it, of course, is now incorrect. In reference to the term “U.S. standard,” it may be remarked that at one time the United States Treasury Department adopted the distance at 62° F. between the 27th and 63d inch, on a brass 82-inch scale, by Troughton and Simms, belonging to the Coast Survey, as a United States standard yard. Subsequently it was found that this so-called United States standard differed by about one-thousandth of an inch from the standard yard of England. By law of 28th July, 1866, Congress, in legalizing the metrical system, adopted in round numbers the ratios which exist between the English and French standards of length, thus assuming that the yard, foot, and inch of the United States are the sane as those of Great Britain. Hence, whenever these units are used in this chapter the standards to which they refer are those of Great Britain. The length of the 15-feet brass bar was derived from comparison with the five brass yards L. 8., Nos. 6, 7, 8, 9, 10, placed end to end in a right line, and inclosed in a long narrow box, to prevent rapid changes of temperature, the 15-feet bar being in the same box beside and within 2 inches of the yards. FIVE BRASS YARDS. § &. These yards, used in determining the length of the 15-feet bar, were constructed for the Lake Survey at the Office of Weights and Measures in Washington, in 1871, under the direction of Mr. J. E. Hilgard. They are all similar brass bars, and are marked, respectively, L. 8., No. 6, % 8, 9, 10. : "These yards are prisms one inch wide by six-tenths of an inch thick and thirty four and seven- tenths inches long, having at each end an axial cylinder four-tenths of an inch in d ameter and six- tenths of an inch long. In the ends of these cylinders agates are held by the br. ‘s, which is bur- nished down‘on them. The end surfaces of the agates are ground to a radius of 4 inches, the center of the sphere being in the axis of the bar. The distance between the middle points of these agate disks is taken as the length of the yard. These points are by construction in the prolongations of the axes of the end cylinders. The values of these yards aud their relative expansions have been determined by comparisons with each other and with the Clarke yards A and B at various times and at widely-differing tem- peratures. 56 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Crap. II, COMPARATORS. § 6. The comparators used are contact-level comparators, one by Wiirdemann and one by Stackpole. In each, a micrometer-screw moves in its own direction, a slide carrying the supports of a level having horizontal trunnions, which turn on the points of two steel screws; a vertical arm projecting from the under side of the level is in contact with one end of a sliding piece of steel, which at its other end abuts against the standard undergoing comparison, and moves parallel to the micrometer-screw on grooved rollers. One end of the level is the heavier by an ounce or two, so that by the vertical arm underneath it the sliding piece is pressed with constant pressure against the end of the standard under com- parison. Any slight motions of this sliding piece can be read on the level by reading its bubble; larger motions up to 1 inch are read by moving the slide which carries the level by turning the micrometer-screw till the bubble is near the middle of the level, and then reading the level, the micrometer-head, and an inch scale on the slide. The value of one division of the micrometer-head is approximately 0.0001, and of the level about 0.00001 for the Wiirdemann comparator. One turn of the Wiirdemann screw is approximately 0i".02 and of the Stackpole screw approximately 0i.01. . In comparisons of two end-measure yards, the two comparators are mounted on a stout timber, so that their sliding pieces shall be horizontal with axes in the same straight line, the outer ends of the sliding pieces being nearly a yard apart. The centers of the ends of the yards are then brought alternately between these sliding pieces, and the micrometer-heads are turned till the bubble is in the middle; both levels and micrometers are then read. The same being done for the: second yard, the difference of readings gives the difference of length. The Stackpole comparator is kept as nearly at a constant reading as possible, thus throwing the measurement on the Wiir- demann comparator, which has the best level. For comparisons of ‘15-feet bars the same process is used, save that the comparators are then mounted on stable stone posts entirely disconnected with the floor on which the observer stands. As in expansion-experiments quantities of nearly 0.1 have to be measured, it is necessary to know the absolute values of one turn of the micrometer-screws and their periodic errors, with precision. These have been determined for the parts of the screws used, and the resulting values have been used in reductions. The absolute values of one turn of the micrometer-screws were determined by mounting the standard inch on the comparator-slide and observing its successive divisions with a microscope, as the micrometer-screw, which was read at each observation, moved them through the field. The Wiirdemann comparator should be used only in the vicinity of or between the scale- readings, 0.1 and 0.2. At 0.11 its indications are to be multiplied by 1.004 to reduce them to inches. At this point the value of one revolution increases with increased reading at the rate of zg4y,5 part of its value per turn. Periodic error between 0.1 and 0.2 scale-reading has a maximum effect on either side of the mean of 0.00003 or a total range of 0°.00006. The periodic error was first determined by Bessel’s method by Lieutenant Lockwood, in 1874, and a later redetermination gave nearly the same value. The readings of the Stackpole comparator at 0.47 are to be multiplied by 1.003 to reduce them to inches. The value f one turn diminishes at the rate of 73455 part of its value per turn as the readings increase. ~ The periodi error of the screw amounts to 0.000019 on each side of the mean, or has a total range of 0'°.000038. This screw is mainly used between the scale-readings 0'.4 and 0.5. THERMOMETERS. § 7. The thermometers on which the results given herewith depend are the following: Ist. Standard No. 230, made by Troughton & Simms, 1871. 2d. Casella standards Nos, 21472, 21473, 21474, 21475, 21476. 3d. Four thermometers, marked A), A2, A;, As, made by Simms, to accompany Clarke yard A. . \ §§ 6, 7.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 57 4th. Four thermometers, marked B,, B,, B;, B,, made by Simms, to accompany Clarke yard B. Standard 230 has a bulb 0.88 long and 0.22 diameter. It is graduated from 20° to 220° F., in degrees, one degree having a length of 0,067. In the Casella thermometers, the length of the bulb is 1.0 inch and its external diameter is 0.31. The stem is graduated from 21° to 129° F., into half-degrees, and a degree is 0.12 in length. . In the A- and B-thermometers, the stem is bent at right angles to the bulb, which is 0.6 long and 0.2 in diameter. Itis graduated to degrees, from 22° F. to 110°, the length of one degree being 0.055. The Casella thermometers having greater sensibility, and one of them, No. 21472, having been carefully studied by Professor H. Ste. Claire Deville, it is taken as the standard. Professor Deville states that it has no errors of calibration exceeding 0°.1; that at 16° C. (= 60°.8 F.) it differed from a tested standard of great perfection by but 0°.02 and but 0°.03 at 33° C. (= 919.4 F.) No. 230 was compared with a standard at Kew Observatory in January, 1872, and the follow- ing corrections were found to the scale-readings, the stem being vertical: Scale-readings, 32° 42° 52° 62° 72° 82° 92° 212° Corrections, —0°.0 — 0°.0 — 0°.1 —0°.0 + 09.1 + 0°.2 +0°.3 +4+0°.1 In September, 1873, its freezing-point was redetermined; the correction to 32° was then — 0°.16. On November 17, 1873, its boiling-point was tested. The correction to 212° was + 0°.07, and on November 18 the correction to 32° was + 0.02. December 19, 1873, it was..---...---..--- aise seine s Joule Seba au es Homereene Soeeles sees mes — 0°. 06 January: 6), 18704, ib Was wise: ccistaccinteericets Satiaieas cewg ieee ewe dee eeace exe ceccnes =00,07 March 8, 1875,at Was» oxissesieies ewes pace sciee sting geesa taieieieie ies se See dieigieiee epee eeueeee — 0°, 24 April 2, 1877, it was ........-. aioe se retecseiecexs — 0%. 31 Regnault’s methods were carefully followed 4 in all deboumiiations: ‘8 numerous determinations were made at each date. The stem was always vertical. As this thermometer is sometimes used in a horizontal position, its error at 32° was deter- mined in that position at the same time; —0°.08 should be added to the vertical correction at 32° to give the correction at 32° with stem horizontal. The following tables give the corrections, determined at the Kew Observatory in January, 1875, for the five Casella thermometers, the stems being horizontal and vertical, respectively, and also the subsequent determinations of the freezing-points made at Detroit: CASELLA THERMOMETERS, FAHRENHEIT. (HORIZONTAL.) Corrections to be applied to scale-readings. | Date. sea atace 21472, | 21473. | 21474. | 21475. | 21476. | | ° c ! ° | ° ° ° Kew, January, 1875 ......--. fo g0 | — 00 | eee. eee os Bal Lae Soe aN eae Digitale eee I la Meseemucee ee 401 |) 401 DOs sac guscceaseracemees ee) —0.0 | +01 |) +01 | 401 | +02 Wo waineraseeacneeees ae eee Joga. | etal +01) 401 Do adanidaccascsceaieees | 45 saga | appa tase +01 | +01 DOzj-02sosesessse8 are | 50 | 401] $01 | +01 +01 | 401 Doi .cacekewtestameceems 55 —0.0 ; +01 401 | +01 | +02 DO steed eum sence! | 60 —00 | +01 ! ¢o1 | +01 | +01 ; DOweteneeectaswesasece 65 —00 ) +01 | +01 | +041 | +01 ; DOseswsias ici veserssee 70 —0.0 | +01) 401°] +01 } +01 RL Biettadeesiatiaees Sece, | oe hee] aos | eae: | ame | ewe DOcrcikenscseeaceneees | 80 —0.0 | +01 ) +01 | +01 | +01 DOSS trees see 8 | —00! +01 | +01] —00 | +01 Daren eeeaasacenees | o | —oo | +01] +01) —o0 | —o0 Dovoicstseneviekccseuek "95 oo | +01) +01) 401] +01 : D6 ecescasecieeseen ! 100 00 | +01 | +01 | +01 | +01 Detroit, March, 1875 ........ [ 32 | 40.06) 40.13) 40.14! 40.15] + 0.18 ' Detroit, December 23,1875... 32 «| —.0.01, 4 0.08} 4 0.06| +0.07| + 0.08 | Meteor Deeemrbersn Abie x! 32 — 0.04 eee re ne weet 58 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cxap. II, CASELLA THERMOMETERS, FAHRENHEIT (VERTICAL). Corrections to be applied to scale-readings. | 4 { Date. reais. 21472. 21473. 21474, 21475. | 21476. | A | oO | fe} o ° oO a | 2} +01 | +02 | +02 | +02] +02 | 35 | +401 1 402) +02) +02 | +02 | 40 | +01) +02) +02 | +02 | +02 | 4 | +02] +02 | +02 | -02 | +02 ! 50 +02 | +02) 402) +02 7 +02 55 | +01 | +02 | $02 | +02) +02 | 60 | +01 '° +02 !' +02 | 402 | +02 65 | +01) + 0.2 40.2 | +02 | +02 7 | $01) +02 402) +02) +02 7% | 401 / £02! +02) +02 | +02 | 8 | +01) +02 +02; 402 | 40.2 85 +01 | 402 402 ' +01 +402 90 +01) 402 402 ) 401 +01 95 + 0.1 +02 ° 402: 40.2 + 0.2 wo | +01 | +02 +02 | +02) 40.2 . : 105 +01 | 402 402 ) +01 ' 401 110 +01 / 402 +402 ° 402 +402 32 +010} +018 +4 0.18, + 0.20) + 0.16 | Detzalh sina 18, 1877... 32 | <<) OOD oeeaite iste ween sease Boe ect eric ciao The following table gives the corrections to be applied to the mean of the four A-thermometers, and to the mean of the four B-thermometers, to reduce their mean readings to Ordnance-Survey standard 3241, a thermometer which has been very thoroughly studied by Lieutenant-Colonel Clarke, who has given his methods and results in his Comparisons of Standards of Length. Colonel Clarke states that the A- and B-thermometers were compared with 3241 with the greatest care. The table contains, besides Colonel Clarke’s results, the results of later freezing-point deter minations at the Lake-Survey office. TROUGHTON & SIMMS 4A, de, 43, 44, Bi, Bo, Bs, By (HORIZONTAL). Corrections to be applied to scale-readings. Mean of} Mean of Date. Aj, A2,| Bi, Ba, Az, Aa.| Bs, Ba. ° ° ° 32 —0. 30 —0. 28 32 —0. 28 —0. 24 52 —0. 22 —0. 18 55 —0.18 —0.11 , a7) 0.17) —0.13 | 62 —0. 22 —0.17 . 93 —0.17 —0.13 32 —0. 37 —0. 34 32) 040, 0.38 The comparisons which fix the length of the 15-feet brass bar and its expansion depend almost entirely on the five Casella thermometers, and on the A- and B-thermometers. As the corrections of the A- and B-thermometers depend on Ordnance-Survey standard 3241, while those of the five Casella thermometers depend on a Kew standard, a comparison between the A- and B-thermometers and the Casella thermometers becomes of interest. The freezing-point determinations, of course, give the absolute errors of the 32° line. Comparisons were made between the standard 230 and the A- and B-thermometers and between 230 and 21472 in January, 1876, at 59°.5 F. This work gives us an indirect comparison between the A- and B-thermometers and 21472, § 8.) STANDARDS DEPENDENT ON THE ENGLISH YARD. 59 - The results are as follows: At 59°.5, the stems of the thermometers being horizontal, the mean of the readings of A), d2, a3, Ay, having had Colonel Clarke’s corrections, and also the corrections for subsequent change of freezing-point applied to it, and 21472 having had the Kew correction and also the correction for subsequent change of freezing-point applied to it, 21472 then read 0°.06 F. greater than the mean of Aj, Az, As, Ay. Similarly, 21472 corrected reads 0°.02 F. greater than the corrected mean of B,, B, Bs, By, at 599.5. As the Kew corrections are only given to the nearest tenth of a degree, this agreement, taken in connection with Professor Sainte Claire Deville’s comparisons with another standard which gave no greater discrepancies, indicates that Casella No. 21472 is very accurately constructed, and that, so far as a mercurial thermometer is concerned, the probable error of its corrected indications is but a few hundredths of a degree. (21472 was also compared with the other Casella thermometers at 599.5. The greatest discrepancy between the results and the Kew corrections was 0°.04.) * COMPARISON OF YARDS NUMBERS 6, 7, 8, 9, 10, WITH EACH OTHER. § 8, For comparisons of the Lake-Survey yards Nos. 6 to 10 with each other and with Clarke yards 4A and SB, the cylindrical ends of the brass yards rested in wyes, with a pressure of three or four ounces, nearly all their weight being carried by supporting spiral springs attached at one- fourth and three-fourths of the length of the yard. The comparators were firmly attached to strong timbers. The two yards under comparison were inclosed in a box with glass windows in its cover through which to read thermometers, and with slits with sliding covers through which the sus- pending wires passed, by whose aid, without opening the box, the yards under comparison were alternately placed in the wyes, which brought the centers of their ends in line with the axes of the sliding pieces of the comparators, and gave central contact for the end of the yard and the end of the sliding piece. The Clarke yards A and B remained in their cases, and were suspended and handled in the same way, the cases resting against adjustable stops, when exactly in the right position with reference to the comparators. Their cases were slightly inclined sidewise, so that these yards should slip so as to always take the same position in the case. Nearly all the com- parisons were made in the cellar of the Lake-Survey office, where the daily range of the air-tem- perature, caused mainly by the presence of the observers, rarely exceeded 2° or 3° F. Readings were usually taken at about 94 a.m. and 4 p. m., the room at other times being kept carefully closed when the temperature of the cellar was below 40° F. Therise of temperature in the box containing the yards under comparison, from the presence of the observers, during the time (about ten minutes) required for two or three comparisons, was usually about 0°.3 F., as indicated by the thermometers under the wrapping of three thicknesses of flannel which incased the brass yards. For higher temperatures the effect of the presence of the observers was less. If the temperatures of the brass yards had risen by the same amount as the temperatures of ‘their thermometers, namely, 0°.3 F., the lengths of the yards would have increased by 0'".00010. It is safe to say that they did not increase in length by more than one-half this amount, or 0.00005, which must have been nearly the same for both; so that there is no probability that, during the ten or fifteen minutes of comparison of two brass yards, the difference of their lengths changed by so much as 0'",00002 on an average. As the yards were systematically alternated in position with reference to the observer, this error was eliminated in the result. From the insignificant effect that this small rise in temperature would have on the difference in length of two similar brass bars, all comparisons made on two brass yards are used, until the thermometer has risen by 0°.3 F. In early comparisons, at lew temperatures, as many as five were obtained, occupying thirty minutes. Of these, only the first two or three have been used, as the thermometers rose 0°.6 or 0°.7 during the thirty minutes, and the temperatures of the two similar yards may not have changed equally. In later work but two comparisons were usually made, each occupying from six to ten minutes, at 9:30 a. m. and 3 p. m. In reducing the comparisons, those at the same period, and, therefore, at about the same tem- perature, have been combined so as to give a single equation of condition, to which a weight is assigned equal to the number of days of comparisons. For this purpose, the mean of the observed *See Note A, Chapter IT, § 15. 60 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. 0, differences of length is taken, and the mean of the first readings of the thermometers in each set of comparisons, as the thermometers after the first reading are slightly affected by the presence of the observers, while the temperature of the yard has probably not sensibly changed. The form of an equation of condition for yards Nos. 6 and 7 will be— (6—7) = (6 —7)g20— (62° —1°) (€;—é7) where 6 and 7 are the lengths of the two yards, ¢©° the observed temperature, and ¢ and e; the - expansions of 6 and 7 for 1° F. Or, placing (6—7)=n=observed difference of lengths ; (6—7).=x=difference of lengths at 62° ; és—é€;—=y=difference of expansions for 1° F.; 62°—1°=b ;yW ? the equation of condition takes, for 6—7, the form— n—x+by=0. For 6—8, 6—9, &c., x and y would receive one, two, &c., primes. The first column of the following table gives the dates of comparisons, the second the number of sets of comparisons, the third the number of days of comparisons, or the weights, the fourth the total range in the observed values of n, the fifth, sixth, seventh, and eighth the equations of con- dition, and the ninth their residuals. The unit is one-millionth of an inch. The values of b show how far below 62° F. the mean temperature of each group of comparisons was. 21s | | | 3 He. | ag | | ° o Date. 23 25 é no x by . 0 v | eo lB 5 a i? 2 ia |2 | 8 | | |e po | : | | °° | 1874—Jan. 14 to Feb. 12...... | 18 9} 84} — 1333 —«£ + 23.84 y o| + 21) 37-41 June 18,19........ ... 8 92; — 13389 —@ 4+ Lty =o0|/ — 8 | 60-61 | 1875—Dec. Tol .-eeveeeee 12) 6 2d — 1368) —@ + 20.10 y =o] —18 41-43 Nov. 29 to Jan. 5, 1876. 24 1d | 122) + 1 | — af | + 21.43 y! =0 0} 43-39 | 1876—Aug.7to19.........-., 10) 4] 108) + 89! —a@ | — 0.28 y/ =0 0| 61-63 | 1874—Jan. 19 to Feb. 19...... | 1: 7) 8) +1472) — a + 23.08 y” =o] + 3{ 39-40 . June 23 to 30 ..,....-. 8 3| 106) + 1509! — a — 219y" =0}] + : 63-65 | 1875—Nov. 22 to 26.......... | g! 4+ 141} 4+ 1476) — 2 + 18.47 y" =0 44-43 1874—Jan. 22 to Feb. 21...--- oa) 4! 63! —1596) — a + 21, 82 =o] +3 5 39-41 bul Ded: 5 seh trett aches | 8) 2) 88) — 1532) — ol” — 19ty” ; =O} +12 63-64 1875—Dec. 14 to 21 .......... 122 4866) «109; — 1624, — a + 23,27 yi!” | =0} —21° 40-86 1874—Jan. 27 to Feb. 27...... / 10 4) 110; + 2830' ~— (a — 2) + 23.50" —y) . =0] + 8-38-40 ° July 7 to 15..-........ ' og’ 3] 79) + 2830! — av — 2m) = Bos Gt si seo! | 63-64 Jan. 29 to Mar.2 -..... i 9) 4 SL} — 234 | — (a — a) + 22.63 (y’” — y) = 0} + 15) 39-40! July 16,17 ...--.....-. 8 2) or) — 227; — (a ~ a — 263" —y) » =0] —12 64-65, Feb. 4 to Mar.7 9 4) 96| — 3057 = (a — ae) | 2B. 87 (yy!) o| +14 37-39. 3 i Solving the equations of condition by the method of least squares, and deducing the values of the unknowns and their probable errors, the following values result: (6 — 7T)g0 = — 1329 + 12 és — @; = + 1.02 + 0.57 (6— 8)go-=+ 89413 €s — &s = — 1.01 + 0.66 (6 — 9)o2° = + 1503 + 11 6 —@=+1.47 + 0.56 Pong! aed 13 €s — Co= + 2.36 + 0.62 In determining these values no comparisons of No, 8 with another yard made prior to August 7, 1875, are used, as that yard had previously changed its length. Five days’ comparisons, March 1, 1874, at 39° F., gave 6 — 8 = — 0.00010 Two days’ comparisons, June 1, 1874, at 619.2 F., gave 6 — 8 = — 0", 00013 Two days’ comparisons, July 20, 1875, at 649° VF., gave 6 — 8 = + 0, 00020 § 9.) STANDARDS DEPENDENT ON THE ENGLISH YARD. 61 In the interval between the last two comparisons, which show an evident change of length, as their difference far exceeds any possible error of comparison, the five yards placed end to end had been compared with the 15-feet brass bar. During these comparisons the yards had been suspended by spiral springs, which allowed the end-cylinders of the yards to rest in their guiding-wyes with but a few ounces weight, and so left them with great ease of motion in the direction of their lengths. The idea at once suggests itself that, in bringing the ends of the yards in contact, the shock of contact due to the considerable mass of the yards had been sufficient to force in one of the agates at the ends of the yard No. 8 where the fitting had not been firm before, thus shortening the yard. These agates, in constructing the yards, were simply pressed into a cavity cut for them and the brass at the ends of the cylinders was burnished down around the agates. On August 2, 1875, I heated the ends of yard No. 8 to about 150° F., and then holding the yard vertical pressed the lower agate into a piece of soft wood with a pressure of 20 or 30 pounds. Then the agate and end of the cylinder were rotated under the same pressure in a hole a little larger than the agate, the upper end of the hole having been reamed out so as to bear on the brass immediately around the agate and press it against the agate. Finally, the brass was burnished down upon the agate with a hand-burnisher. Both agates of No. 8 were treated in the same way. ; Two days’ comparisons, on August 7 and 9, 1875, gave, at 62°,6 — 8 = + 0.00012, which would indicate a lengthening of No. 8 by 0.00008 since July 20, 1875. But in the previous comparisons the comparing-room was visited three or four times during the day, which makes it uncertain whether the temperatures of the two yards were precisely the same. As the burnishing could hardly have lengthened the yard, it seems possible that the small apparent change in length subsequent to July 20, 1875, was due simply to temperature errors in the previous comparisons, and that the length of yard No. 8 was not sensibly affected by the burnishing process; or, in other words, that its agates are now stable. The comparisons of the other yards at different dates give no indications of change in lengths. The following are the results of comparisons of Nos. 6 and 8 at different dates: 1. Nov. 30, 1871. Temp., 55°.5 F., 1 day’s comparisons,6—8= 0. 00000 2. March, 1874. Temp., 39° F., 5 days’ comparisons, 6 — 8 = — 0%. 00010 3. June, 1874. Temp., 62° F., 2 days’ comparisons, 6 — 8 = — 0", 00014 4, July 20, 1875. Temp., 64° F., 2 days’ comparisons, 6 — 8 = + 0.00019 Aug. 2, 1875. Both agates of No. 8 burnished down. 5. Aug. 9, 1875. Temp., 62° F., 2 days’ comparisons, 6 — 8 = + 0, 00012 6. Aug. 9, 1875. Temp., 63° F., 2 days’ comparisons, 6 — 8 = + 0. 00009 7. Dec. 1876. Temp., 41° F., 14 days’ comparisons, 6 — 8 = + 0. 00011 The comparisons of November 30, 1871, were not made under temperature conditions which could make precision certain, and their discrepancy with the second comparisons does not make it sure that yard 8 had changed length in the interval. It certainly changed length largely between June, 1874, and July, 1875, and may have changed very slightly while being burnished, although the difference of lengths on July 20, 1875, and August 9, 1875, while larger than the probable is not larger than the possible errors in the comparisons made on only two days. COMPARISON OF BRASS YARD NUMBER 6 WITH CLARKE YARDS A AND B. § 9. Having given the lengths and expansions of the other brass yards relatively to No. 6, it remains to explain how the absolute length and expansion of No. 6 were obtained from the Clarke yards A and B. No. 6 was compared with A and B, and No. 7 with A. The yard A or B remained in its iron case, and, with its thermometers in their places, was put with the brass yard in the wooden box in which the brass yards had been compared with each other. The larger part of its weight was carried by suspending-springs, and its outer iron case rested against stops, adjustable so that the centers of its end surfaces should be in line with the axes of the sliding pieces or quills of the comparators. The case was slightly inclined, so that the yard within, under its own weight, should 62 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. II, always assume the same position with reference to the case, and so with reference to the compara- tors. The brass yard was also suspended, resting its end cylinders in their supporting wyes. The two yards were alternately placed between the comparators, and comparator-readings taken on both ends. In the comparisons of No. 6 with A and B, at temperatures near or below 40° F., the three comparisons making a set required about twenty minutes. In this time the thermometer beneath the flannel wrapping of the brass yard usually rose about 0°.3 F., while those inside the cases of the yards 4 and B only rose about 0°.05 F. In comparisons near 62° F, the rise in thermometers was about one-half the above amounts. Iience, as the thermometer-temperatures of No. 6 and A or B change very unequally, and as their expansions are very different, only the first comparisons and the first temperatures, on entering the comparing-room after an absence of several hours, were used in the reductions. No comparisons were used in which the thermometer in contact with the brass yard and under three thicknesses of flannel differed by more than 0°.2 F. from the mean of the four thermometers inside the iron case of the Clarke yard; and as the presence of the observers for the two to four minutes before the thermometers were read may have slightly affected the thermometer with the brass yard, while its effect on those with the steel yards was insensible, the thermometers with the latter yards have been alone used in fixing the temperature of comparisons. No comparisons were used where the thermometers with the Clarke yards changed by more than 0°.4 F. between morning and afternoon comparisons (9:30 a. m. and 4 p. m.). No. 6 and A were compared with each other at three different periods, at temperatures varying from 37°.6 to 62°.8. No. 7 and A were compared at about 33°. As the length and expansion of No. 7, with reference to No. 6, are known with great precision, the comparisons of 7 with A were reduced to those of 6 with A, giving a fourth group. Each of these four temperature-groups of comparisons gave an equation of condition of the form (6—A)1o =(6—A)g2e— (62°—1°) (€s—€,) in which ¢° is the observed temperature, e, and es the expansions of No. 6 and 4 for 1° F., and (6—A)g0, and (es—e,), the unknowns, provided the temperatures were correctly measured. But in some thermometers, which agree with an air-thermometer at 32° and 212° F., and which have no sensible errors of construction, it is known, in consequence of the dilatations of glass and mercury depending on both the first and second powers of the temperature, that at other readings they may differ sensibly from an air-thermometer. An attempt has been made to have one of the Lake-Survey thermometers compared with an air-thermometer, but as yet no results have been received.* In the comparisons with each other of the brass yards Nos. 6, 7, 8, 9, 10, as their rela- tive expansions are very small, no sensible error is introduced into their relative lengths by an error of 0°.1 or 0°.2 F. in measuring their common temperature. But in determining the relative lengths of A and No. 6, an error of 0°.2 F., in determining their common temperature, would intro- duce an error of 0.00002, a quantity larger than the probable error of comparison. If such a systematic correction should be needed to make the Lake-Survey thermometers agree with an air-thermometer, it may be taken with sufficient accuracy between 32° and 62° F., as pro- portional to the excess of temperature over 32°, and its form will be 4 (t0—320°), The equation of condition from a group of comparisons will then take the form (6—A)o= (6 —A)ox°— § 62°— [124 (t2—32) 4] } (e¢—e,) which may be written —n+a—(a—b4) y=0 in which n is the mean difference of lengths of the two yards derived from a group of comparisons at nearly the same temperatures, of which ¢° is the mean, « is the unknown difference of length at 62°, and y the unknown relative expansion. The weight of each equation is equal to the number of its comparisons. ‘ * See note A, Chapter II, § 15. §9.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 63 The following table gives the date of comparisons, the number of days of comparisons, the number of comparisons, the temperature-range during the comparisons, the equations of condition, and their residuals, 4 being neglected. The unit is the millionth of an inch. Comparisons of No. 6 with A. i ao ' ° og 3 - pa lue] Sg 4 Date. Se| 38 me Equations of condition. z z9 | ae gS B 5 52 oO ) z |48| a | 1874—December 24 to January 4, 1875 | 6 6 | 42° to37° —2847—a- (22.25— 7.75 A) y=0 | — 4 1875—August 20 to 23....---.22-.226+- | 8] 11; 68°to 60° | + 405—x-+( 0.09-29.91 A) y=0 | — 8 October 2 to 6............-.-.--- 5 9 | 54° to 55° = — ~588-a-+-( 6.65—23.35 A) y=0 +13 January 18 to 23 (No.7 and A). 6 10 338° to 34° ~—3820—a2-+ (28.89— 1.11 A)y=0 — 1 The following table gives similar data for comparisons of No. 6 and Clarke yard B: Comparisons of No. 6 with B. ao 7 | 3 (32, 4 : % aD ae | a oO a . Date. 23 s a | pk Equations of condition. 2 2 = BER) & : a as a 6 | | 1875—January 6t015...-...02e.s02---| 6, 8 | 36°t038° | —3473—a/4+(27.72— 2.98 A) y/=0 | +1 October 7 to 8...-...-..--------- 2 4 | 55°to 54° | — 414—2/+( 6.81—23.19 A) y’/=0 | + 4 November 8 to9 ...-...--------- 6 8 | 48°t047° | —1508—a/+(14.25—15.75 A) y’=0 | — 3 In this table .c’ and y” have replaced the x and y of the preceding table, as yard B has replaced yard A. : Solving the above equations by least squares, the following values result : xv =(6—A)go=+426 — 4408 446 v'=(6—B)go=+578 — 4385 444 yes —@, =+146.94—146.94 440,82 yl =e; —@; =+146.17—-146.17 440.21 the unit being a millionth of an inch, and the probable errors being derived from the equations of condition when 4 is taken as zero. If the value of « above be subtracted from that of x’, we have A —B=152, the terms in 4 being neglected, as 4 is very small and its coefficients nearly equal. Referring to Colonel Clarke’s values of A and B, (§ 2), we find A—B=211, differing from the value just found by 59. Both Colonel Clarke’s and the Lake-Survey determination of this differ- ence are indirect; that is, they are each obtained by comparisons with a third yard. Still the differ- ence is larger than would be supposed from the probable errors of the comparisons, and it is hoped hereafter to make a direct determination of its value.* If equal weights were given to the two determinations of the difference, the two results would differ from their mean by but ysats00 Of a yard, a quantity that is not large in such work. Subtracting y’ from y we have €,—€,=+0.77 Colonel Clarke’s value (§ 2) is €p—€4 = — 2.33 * See note B, Chapter II, § 20. 64. STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuap. 11, Were the mean of these two values to be adopted, it would require a change in the values for the expansions of A and B, found by Colonel Clarke, of 745 part. Adding now the value of (6—A)g:0 to the length of A at 62° F., given by Colonel Clarke (§ 2), there results the length of No. 6 at 629° F.: No. 6=35'",999596 + 8—0.004408 4 Adding the value of (6—B),°, above, to the length of B at 62° I’, given by Colonel Clarke (§ 2), there results the length of No. 6 at 62° F.: No. 6=35",999538 + 7— 0.004385 4 the probable errors having, in both cases, been obtained by neglecting 4 as insignificant. These values differ by 0.000058, a quantity larger than the probable errors of the separate values would indicate. The mean of the two values must be taken, and the probable error of the mean must be derived from its differences from the two values. There results, then, for the length of No. 6 at 62°, from the data at present (April, 1877) avail- able No. 6=35'".999567 + 0.000020—0.004396 4 the probable error being obtained by supposing 4=0, and being ygotao0 part of the length. From § 2, ¢:, the expansion of yard A for 1° F. is 0".00021146. T'rom § 9, eg — és, is 0'".00014694 (1 — 4), and adding, the value és = 0'".0003584 — 0.00014694 4 results. Again, deriving e; from es + (¢ — és) we have és = 0'".00035534 — 0.00014617 4 Taking the mean of these two values, derived from yards A and B, respectively, we have 5 = 0'".0003569 — 0.00014656 4 + 0'.00000103. Subtracting from the length of No. 6 its excesses over Nos. 7, 8, 9, 10 (§ 8), there result the following lengths of these yards at 62° F.: No. 7= 36. 000896 + 0%. 000023 — 0. 004396 4 No. 8 = 35, 999478 + 0. 000024 — 0. 004396 4 No. 9 = 35, 998064 + 0, 000023 — 0. 004396 4 No. 10 = 36. 001115 + 0. 000024 — 0.004396 4 Summing these values and deducing the probable error, we have Yds. (64 74+8+49-+4 10) =179,999120 + 0.000103 — 0.02198 4 If the relative expansions, € — €7, 5 — €a, 5 — 3, and & — eyo, be subtracted in succession from the value of e, derived from the Clarke yards, we should have values for the absolute expansions of those yards. Summing those values, we have a value for the expansion of the five yards when placed in contact, end to end, their axes being in the same right line. That value is Crar4eeoei0 = 0".0017807 + 0™,0000053 — 0.0007328 4 COMPARISONS OF FIVE LAKE-SURVEY YARDS (NOS. 6 TO 10), PLACED END TO END, WITH 15-FEET BRASS BAR. § 10. In these comparisons the two comparators were mounted on stone posts, about 15 feet apart, their sliding pieces being in the same right line. The bar and yards, in the comparisons prior to June, 1875, were placed side by side in a closed wooden box, parallel to each other, and 2 inches apart. The15-feet bar rested on rollers 18 inches apart, and was provided with side guide-screws to limit its side motion and to aid by slight pressure in its alignment. By raising or lowering its rollers, and by moving its guide-screws, the bar, while remaining free to move, was made straight ; § 10. ] STANDARDS DEPENDENT ON THE ENGLISH YARD. 65 its straightness being tested with a silver wire strained by a known weight, the wire being verti- cally over one of the upper edges of the bar. The side of the 15-feet bar was brought into the vertical plane of the wire to within 0.01, and the top of the bar made parallel to the wire after the latter was correctéd for its computed sag within the same error. The 5 yards placed end to end were each supported at 4 and # of their lengths by spiral springs, so adjusted as to carry nearly all the weight of the yards, leaving only three or four ounces to be supported by the wyes in which the end cylinders of the yards rested. The spiral springs supporting the end yards were so inclined that these yards each pressed toward the central yard with a force of about one pound, thus securing contact between the ends of the yards. The axes of the end cylinders of the yards were brought into the same right line by adjusting the wyes in which they rested. To bring them into the proper position, with reference to a vertical plane, a small piece of a semi-cylinder of. the same diameter as the end cylinders of the yards had a point in its axis marked on its diametral plane surface. This semi-cylinder was placed in the wyes in succession, and each of the latter was moved sidewise till the point was directly under a stretched fine silver wire, 16 feet long, which vibrated just above the point. o To bring them into the proper position with reference to a horizontal plane, a level was fastened to the upper surface of one of the yards which at once gave the means of determining the difference of height of two wyes a yard apart, and of adjusting them to the proper height by means of their adjusting-screws. : As where two yards met there were two wyes very near each other, they were adjusted with reference to each other by a smaller level. Prior to July 1, 1875, the box containing the bar and yards, parallel to each other and about 2 inches apart, was mounted on two trestles of the base- apparatus, which gave lateral motion sufficient to bring the 15-feet bar and the yards alternately between the comparators. Subsequent to that date the bar and yards were mounted on a T-shaped iron beam about an inch apart, the whole being inclosed in one of the long boxes of the expansion-apparatus, the necessary lateral motion for comparisons being obtained by running the car of that: apparatus sidewise. The first comparisons of 15-feet bar with Lake-Survey yards Nos. 6, 7, 8, 9, 10, were made ina room on the first floor of the Lake-Survey office, on 17 days, between February 26, 1872, and June 24, 1873. The daily temperature-range in this room is large, though less than that of the external air. As the cross-sections of the yards are 1 by 0.6, while that of the 15-feet bar is 1.1 by 0,33, the cross-section of the yards is two-thirds greater than that of the bar. In changing temperatures that of the yards will always lag behind the temperature of the bar, so that the temperature-changes should be kept far below those of the ordinary diurnal ones. For this reason these comparisons were not considered satisfactory. They were given in the Lake- Survey Report for 1874; the difference at 62° F., bar — yards, being found 0.00839. A comparing-room was subsequently established in the cellar of the Lake-Survey office, where. the daily range in the air-temperature outside the closed box containing the yards and bar was only 2° or 3° F., this being partly caused by the presence of the observers. Fourteen days’ comparisons, between December 10 and December 30, 1873, at a mean tempera- ture of 42°.81 F., gave bar — yards = 0'4.00829. ‘Three days’ comparisons, between July 21 and 24, 1874, gave at a mean temperature of 64°.7 F., bar — yards = 0'".00837. Between June 1, 1874, and July 20, 1875, yard No. 8,as compared with No. 6, appeared to have shortened by 0.00033, there having been two days’ comparisons at the first and two at the last of those dates, the first giving at 619.2 F., 6 —8 = — 0*.00018, and the second giving at 63°.9 F., 6—8 = + 0.00020. In both sets of comparisons the room was visited too frequently (three or four times a day) for the best work, but it is improbable that either of these values is erroneous by so much as 0'".00005. The first value agrees with a value determined by five days’ comparisons on March 1, 1874. It is uncertain at what time the change in length of yard No. 8 occurred. It may have been before March, 1874, or after June, 1874, and it therefore throws uncertainty on all comparisons of the 15-feet bar with the five yards prior to August 2, 1875. 9LS 66 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Caap. II, On that date strong pressure was applied to the end agates of No. 8, the brass around them was burnished down, and No. 8 was recompared with No. 6; the resulting difference (6 — 8). = + 0'",000089 has already been given in § 7. The 15-feet bar and the five yards were again compared on August 12, 13, 14, 1875, at a tem- perature of 63° F., and on 26 days between February 28, 1876, and April 7, 1876, at temperatures varying from 34°.7 to 38°.5 F. From these two sets of comparisons the difference at 62° F, (bar — yards) has been found and the relative expansion of the bar and the five yards. Each comparison gave an equation of condition of the form— Dye +(t°—62°) e—n=0 in which Dg» is the excess of length of bar over that of the five yards in contact in a straight line; t° is the corrected temperature of the comparison; e is the excess of expansion of bar over yards for 1° F., and v is the observed difference of length of bar and 5 yards corrected for periodic error of comparator-screw. ‘ The following are the equations of condition, with dates and residuals: Date. Equations of condition. Residuals. 1875. in. in. Aug. 12 | Dee+ 0.8¢—0, 00872=0 .. 20.22... 0.cceetceeeeeeceeecccesceeeeeeeees ee + 0.00012 13'| Det 0,960; 00876=0 xc.ceceas.oascaviasieneusas veveeendes seeweaeecdan oceete + 9 TE.) Dest 0:56 0,00905=0 sic deancsec's cayecis aici eemeasueidy sone ewoaxesoseeeesen _ 21 1876. Feb. 28 | De2—26. 0e—0. 008830 «22... cece ee cece ee eee ccc eee ccc ceceeescecceeeeceeee _ 12 29' | Des=25.'8e=0, 0087820 si vacuacetuasyavinds enegwaticnnducccwaeseccncectnececeds — 7 Mari. 1: Dee—26i08:+0500878=0 eo casetcastiassence scene tedyetapetelsieeee seedy seas = 2 2 | Deo—26. 4e—0. 008710 .. 22. e cece eee cece cece e nee ee ccc enesceeceeneeeees SS 1 3 | De2—26. 6e—0. 00872=0 «2.2.22. e eee ccc eee ce ence ee ec ence ec eccceceeeeeesees _ 2 4 | Doro—26. 5¢—0. 00888=0 «2.2... ec cece cece ce eect eeeec cece neecnesncseenes ~ 18 6: | Dea —25:'Se— 0.008900 soe vie cisise,nies suieae eatmenous va saaswetceusbe sea cccteses ceed - 19 Td Diea=-28 860, 008910 sc ccccwaacciee sanneunce ciate dads aciececcee eSwaueeswsaxa ccc - 19 9 | Dez—24. le—0. 00880==0 «22.2... eee eee cece cece nee ec eeceecnecceceeeees eesti pe 8 10 | De2—24. 0e—0. O0BGB=0 . 2. eee cece ee ence ceneeeceeccceccececceecs + 9 TD:\| Daa 254388=0:/00855 20 | wires sa ceie's ue eveaeedetade de kaene eaceetassemeue dws + 16 16: Deo=2554¢ = 0.008150 se saw cinyctveiecnic caavis sh estan seinmmpaeceancsae enews exe sec + 10 18 | Der—26. 4e—0. 00866=0 .. 22... ee cece cece cee cece cece ce cenenseeecceees + 4 21 | Deo—27. 8¢—0. 00865=0 . 22... cece van ee cvceceececenscccccccsaceccesce + 5 22:'| Das—27.:2e—-0,.008590 au osazecceckwe aint ves ixie tlie uc an aeidabeicicauecwacoaaca ae + u 23 | Dez—26. 8e—0. 008710 ..-- 2... cece cen c cere ccs ca cee eenceeeececcouceceeces = 1 24 | Dez—26. 3e—0. 00858=0 2.02... c ee cece ee cece cece eeccceteeeece et ei + 2 25) Des—25; Be 0008650 oo sence deine wa Gee eevingnaeeninecbimacaeesceuecs vies + 6 27 | De2z—25. 4e—0. 00860=0 ......22..2 22.2 ee tabase we cet desea ctaa ed aemececeuaat + 1 30) | Doe 26: 0420,.00868 S01 a secs ys ces seeds eens eeation eeeeeeesutiod cocened ene + 3 Apr. 4 | Der—25, 0e-—0. 00864=0 1.2.2... cece ec cc ececceceee cece ceeeececeaacceeees + 7 5 | De2z—24. 3e—0. 008740 «1. ee eee eee ee cee cece cece ee cewesceccessccceeenas os 3 6 | Deo—24. le—0, 00876=0 2.2... eee eee ccc c eee sce ceeceeecececencceencces 4 7% | Dea 28, Ce—0,00876 20) «..s.:sis aise vercsdea essiasaeciseeicine neehtacecsamciuzamoce = 4 From February 28 to March 10, 1876, inclusive, the yards were next the observer; from March 15 to April 7 the bar was next the observer. Solving these equations of condition by least squares, we have: bar longer than the five yards, at 62°, or : : Deo= + 07.008843 + 0'2.00004 and expansion of 15-feet bar for 1° F. greater than that for five yards, by 0'".0000052 + 0'.0000017 In § 8 the sum of the expansions of the five brass yards, as derived from their comparisons with Clarke yards A and B, and each other, and from Colonel Clarke’s values of the expansions of A and B, was given— €64.74849410== 0'2-0017807 + 0'",0000053 — 0.0007328 4 § 11.) STANDARDS DEPENDENT ON THE ENGLISH YARD. 67 Adding to this the value of excess of expansion of bar over five yards just given, we obtain a value for the expansion of the 15-feet bar for 1° F. derived from that of the Clarke yards; it is— 0.001786 + 0”.0000056 — 0.0007328 4 The value derived from the direct expansion-experiments given in § 11 is— 0.001795 + 0.0000016 When it is remembered that the first value is affected by the errors of comparisons of seven different yards, its agreement with the direct value may be considered satisfactory. Adding to the sum of the lengths of the yards Nos. 6, 7, 8, 9, 10, given in § 8, namely, 64748494 10 =179.999120 + 0.000103 — 0.02198 4, the value— Dee = + 0.008843 + 0.000040 there results for length of 15-feet brass bar at 62°— bare, = 180'7.00796 + 0.000111 — 0.02198 4 It has previously been stated that 14 days’ comparisons of 15-feet bar with the five yards in December, 1873, at a mean temperature of 429.81 gave: bar — yards = 0.00829. The 24 days’ comparisons, beginning February 28, 1876, gave at 36°.2: bar—yards = 0.00871, giving, when reduced to the same temperature, a relative increase in length of bar of 0.00045. Of this increase, 0.00023 is due to the shortening of brass yard No. 8 between June, 1874, and August, 1875. See § 8. The rest, 0'".00022, may be accounted for by supposing the shortening of No. 8 to have partly oceurred before June, 1874, by supposing all the yards to have shortened slightly, or by supposing that the brass 15-feet bar has lengthened. Butas it amounts to but g3 590 part of the length of the bar, it may possibly be due to errors of comparisons. In discussing the expansion of the 15-feet brass bar (§ 11), it will be seen that the compari- sons of the brass bar with the tron bar packed in ice also indicate an increase in length in the brass bar after it had been heated from 32° to about 100°, which temperature it probably reached, of about the same amount. If the modulus of elasticity of cast brass be taken as 6450 kilograms per square millimeter, and its breaking strength as 12 kilograms, the force needed to stretch it by z$8o5 of its length would be about 4.4 kilograms per square millimeter, or more than one-third of its ultimate strength. But heating a brass bar from 32° Fahr. to 100° Fahr. stretches it by this amount (z>$8o0 of its length), and the question arises whether this heating may not produce a temporary or permanent change of length: No positive conclusions can be drawn, as the apparent change is so small that it may possibly be due to errors of comparison, its proportional amount little exceeding that of the errors in good comparisons of two yards @ trait. EXPANSION OF 15-FEET BRASS BAR FOR 1° FAHRENHEIT. § LL. A determination of the expansion for 1° F. of the standard 15-feet brass bar was made in the winter of 187071. Two methods were adopted. In the first the comparators being firmly fixed in the tops of stone piers, their stability was assumed. On consecutive days the bar was brought between the comparators, the room and bar on the first day being at the natural tempera- ture, and on the next the room being heated by a stove, so that its temperature and that of the bar were from 25° to 45° F. higher. The distance between the middle points of the comparators being taken as unchanged, the difference of the comparator-readings for hot bar and cold bar, allowance being made for the expansion of half of each comparator, gave the change of length of the bar. Temperatures of the bar were given by six thermometers. The differences in lengths of the bar, divided by the corresponding differences of temperatures, gave the expansion for 1° F, 68 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Cuap. II, § 12, The second method was to clamp five yards together, end to end, thus forming a com- pound bar about 15 feet long, to surround this compound bar with broken ice, thus making its temperature and length constant, and then to compare the 15-feet bar with it, both at the natural temperature of the room and with that temperature raised about 30° F. The value obtained for the expansion of the 15-feet brass bar by both methods was 0.00174. (See Lake-Survey Report for 1871.) The objection to the first method is that the perfect stability of the piers during changing temperatures is uncertain, and to both methods that the temperature of a brass bar like the stand- ard cannot be determined from mercurial thermometers in contact with it with any such accuracy as 0°.1 F. or 0°.2 F. unless the temperature of the bar and its surroundings has been kept within a few tenths of a degree of a stationary temperature for several hours. At the high temperatures this could not be effected with means then at command. When the air-temperature rises through many degrees to a maximum, the temperature of the bar lags behind that shown by the thermome- ters, so that the indicated temperature-change is too great, giving too small an expansion. Subse- quent work on this bar shows that at the high temperatures (70° to 80° F.), when the bar had its maximum length, its temperature was about 1° below that of the thermometers with it. Later experience threw doubt on the accuracy of the value, and a few trials in July, 1874, of a method slightly differing from the first, made it nearly certain that the above value was some- what too small. This new method was to place the 15-feet bar between the comparators, the tem- perature of the comparing-room being steady at about 64° F., to let it remain at least 16 hours, so that its thermometers might give its true temperature, then to read the comparators, and imme- diately to remove it to another room, when, for 4 hours, it was packed in ice. At the end of this time it was, while still packed in ice, placed between the comparators, which were again read. This process assumed the stability of the comparators for 4 hours of pretty steady temperature. Results were obtained on five days for the expansion of the 15-feet bar for 1° F., which varied from 0.001781 to 0i.001809, their mean being 0°.0017959. § 13. It being then pretty certain that the value found for the expansion of this bar in 1871 was not sufficiently precise, I decided to attempt the careful redetermination of this important constant, and to use the method described by Captain A. R. Clarke, Royal Engineers, in Compar- isons of Standards of Length, making such modifications as would adapt it to the inconvenient cellar in which we bad to work. The important point in Captain Clarke’s method is, that he secures steady temperatures for each of two bars for several hours by placing them in closed boxes, whose sides are copper tanks, through which water of any desired constant temperature runs. The method adopted, differing somewhat from Captain Clarke’s, was to use-a second bar con- stantly packed in ice, and to compare the 15-feet brass bar at temperatures of 32° and 90° F. with this second bar. For the second bar, an iron bar, whose cross-section was 1™.1 in depth by 0'".33 in thickness, had milled cylindrical steel pins firmly screwed into its ends; the outer ends of these cylinders were planes, very nearly at right angles with the axis of the bar, and at such a distance apart as to give the two bars the same length at about 91° F. This iron bar was mounted on ten adjustable rollers, in a semi-cylinder of quarter-inch boiler- iron, which was 15 feet long and 4 inches in diameter. This semi-cylinder was packed with pounded ice so that the bar should be entirely covered, one upper edge of the bar being tested after the ice was in to see that it did not deviate from aright line by more than one-hundredth of an inch. When the ice was well rounded over the bar, fifty pounds were required. It was replaced when the weight had diminished by about twelve pounds. The semi-cylinder was supported at one-fourth and three-fourths of its length. To test its stiff. ness, five pounds were hung at one end, giving a deflection of but 0.007. Accordingly it was assumed that the varying ice-load produced no injurious flexure of the bar due to change of form of the supporting semi-cylinder. As the points of support for the iron bar were but 19 inches apart, and the greatest ice-load §§ 12, 13.] STANDARDS: DEPENDENT ON THE ENGLISH YARD. 69 on top of the bar between two points of support could scarcely exceed half a pound, the length of the axis of the bar could not be sensibly changed by variations in this load. For cold-comparisons, the brass bar was also packed in ice, while resting on its supporting T- shaped iron bar, which was stiff like the semi-cylinder already described. For comparisons of brass bar hot with iron bar in ice, the brass bar was placed in a long, narrow, closed box, its ends only projecting very slightly from the ends of the box. The box had a cloth-lined wooden cover, and its sides were formed by two long metal tanks, each 5 inches in depth by 2.7 in thickness, the faces next the bar being of copper. In the interior of each tank was a 2-inch iron pipe running along the bottom of the tank, and then returning along the top,; through this pipe the hot water flowed. The brass bar was supported on adjustable rollers, ontop of a stout T-shaped iron bar, nearly 15 feet long, lying between the tanks. The two bars in their boxes, parallel to each other and about 3 feet apart, rested on a very stable, heavy truck, whose smooth lateral motion on iron rails brought, alternately, the centers of the end of one bar or the other between the comparators, mounted on their stone posts. The bars were adjustable both horizontally and vertically, so that when once adjusted each could at once be brought into proper position with reference to the comparators, by motion of the truck alone, whose displacement was limited by adjustable stops, and a comparison of the two bars could be made within five minutes. Hot and cold water were run into a large cask at such rates as to give a pretty steady tem- perature when well stirred ; from this, with a slight head, the water flowed through flexible tubes into the iron pipes contained in the tanks. In the first experiments, although the temperature of the water in the mixing-cask could be kept at a temperature varying from 99° to 101° F. by properly adjusting the flow into it of hot and cold water, yet the flow through the pipes in the tanks was not steady, the slit-opening of the stop- cock being clogged by the impurities in the water. The form of this opening was changed and the water passed through a cloth strainer. Afterward there was no difficulty in keeping the tempera- ture of the hot-box at a temperature of about 90° F., without a greater variation for the middle thermometer than 0°.9 in eight hours. The thermometer near the end where the water entered and left the tank was usually 0°.2 or 0°.3 greater than at the corresponding place at the other end. At first some vapor from the wood of the hot-box condensed on the under side of the glass windows in the cover through which the thermometers beside the brass bar in the hot-box were read, making the reading uncertain. Chloride of lime was used to absorb this vapor until the boxes were dry, although it is objectionable on account of danger of rust. The temperature of the cellar being about 35° F. below that of the hot bar, which was about 90° F., and the ends of the brass bar being exposed for comparison, the temperature of the ends of this hot bar was somewhat affected by the cold air outside. The interior of the hot-box near its ends was stuffed with cotton, but a thermometer within 3 inches of the ends of the bar still stood at from 2° to 3° below those at the middle. While these thermometers doubtless gave a temperature lower than that of the brass they touched, it was not deemed safe to neglect so large a difference. A stove was brought into the cellar, and during hot- comparisons the temperature of the room was raised to about 85°. The temperature of the water as it left the tanks was lowered to 90° F., giving about 89° for the temperature of the bar. After this, the thermometer near the end of the bar gave a temperature but about 0°.5 lower than that near the middle. The thermometers used were the five Casella standards Nos. 21472 to 21476, already described. They can be read to 0°.02 F. The mean temperature of the 15-feet bar during hot-comparisons was determined as follows : The early experiments with thermometers in various positions had shown that thermometers dis- tributed along the bar agreed within two or three tenths of a degree until they approached within 18 inches of the ends of the bar, when they began to fall, standing at the ends of the bar 2° or 3° lower. By heating the air in the cellar to 85° this fall was reduced to about 09.5, After this, if 70 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Caar. II, the thermometer-readings on the bar are plotted as ordinates, the distances along the bar being abscissas, the curve will be nearly straight to within 12 or 15 inches of the ends of the bar. In the comparisons used the thermometers were placed on each side of the middle of the bar at 224 and 72 inches from it, and a fifth at 3 inches from one end. Means of the readings of each ther- mometer were used for each comparison to plot the temperature-curve (whose general form was already known) for that comparison. The mean ordinate of this curve was taken as the temper- ature of the bar for this comparison. The brass 15-feet bar and the iron bar were compared with each other, both being packed in ige, on ten days between April 21 and May 1, 1875, as shown in the following table: Date. a bar — brass bar, oth in melting ice. | : wm. April 91/1875 sassainsces cise seve ccinsstasuae wes Sede emerge acess sesame teens 0. 038801 | Aptil 99: 1875 sc Jo ciacsacecesnarmactuced cena ied be heaedaee asek wee eee eu NEbERY 0. 038991 | April 28: 1876: .s.cneccaciaesedaccacatstabadneearacohe teem ommaarseteaa tegaatte 0. 038794 April 24, 1875 secisaccsencascnaee sepeceameece pace seneqeiesdcaeen sede sense res @ 038794 1 April 26/1876 oecesser aes eeee wadse tes adage vase egeeNeee ated anes sangacen tease é 0. 038880 AeprillOG 1875 tage naivie dash Ai cowase odes apes aeqcemtwndanieebiseennsseh taaenes 0. 038803 Rep rilO8 S18 78 occa de av ceassmedsmrcwted eaoeasthee tose gene darezast creme meeeenes 0. 038873 A prili99, 1875 cc cocacsace esac s peas ve ce -o ose eeeaeeks teeadt sagan way Gteoeels 0. 088954 APTI BO IST Se oniasss scenes Seees eebwtrenasnins aoe Sela amiisaisinneelr sedate soatoeere 0. 038955 Miiy T8716 cicdivotecte easyer datesine sesuede ace seeneduecnananrmaceen yereeaveeedces 0. 038966 MG@OR asesczcn dieses cdnesnehuastemenneseoeebwes scedadaneuues cgekaese ses 0. 038881 + 0. 000017 The resulting excess of length of the iron bar is 0.038881 + 0.000017 at 32°, the probable error being derived from the discrepancies between the daily results and their mean just given. After the hot-comparisons had been made the two bars, both packed on ice, were again com- pared with each other on four days from June 21 to June 25, 1875, as shown in the following table: , Iron bar— brass bar, Date: both in melting ice. : ¥ MN. JH: 21, 1876 wosencian sochanceszccs des sesiaoyiaevbs aueeciheveaweweeeces ens 0. 038595 sie Wa ABTS ee see a ect een ene aie ta rot eceraate hoee adtelies shee at 0. 088711 S AUTT To) oo] vr NR RIE HE eRe POG AES POPE RP 0. 038535 Flin SO TT a ct cts net es ana eet acer ION ats pense rege 0. 038664 Moai esha oe nace sateen eden sods tee eee ae Meets 0. 038626 + 0. 000026 The resulting excess of length of the iron bar was 0.038626 + 0'.000026, differing from the previous value by 0.000255. This difference is larger than the probable errors would lead one to expect. But it is only zs¢5s part of the length of the bar, a quantity which, in view of the many difficulties in such work, is too small to make it certain that the bar changed length during the hot- comparisons. As such a change is possible, however, equal weights have been given to the values obtained for excess of length of iron bar in ice over brass bar in ice, before and after hot.compari- sons, and their mean 0.038754 + 0.000086 has been adopted as the most probable value of their difference in length at 32° F. The comparisons of the brass 15-feet standard bar with the five brass yards, Nos. 6 to 10, given in § 10, also indicate a lengthening of the brass 15-feet standard at about this time and by about the same amount, but, as already stated, the apparent discrepancy scarcely exceeds the possible errors in good work. The comparisons between the iron bar packed in ice and the 15-feet standard brass bar at a temperature of about 90° were 19 in number on ten days, between May 26 and June 18, 1875, but only those on or after June 10 were used in obtaining the final value of the expansion. They were 12 in number, and on reducing them to a common temperature the residuals gave a probable error § 14.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 71 for their mean of + 0.000003. The resulting expansion of brass bar for 1° F. from each hot-com- parison is given below under the heading e. It has previously been stated that the early hot-comparisons were not considered satisfactory because, at first, the flow of hot water was not steady or the temperature of the air in the cellar was not near enough to that of the exposed ends of the brass bar. For these reasons no hot-com- parisons prior to June 10, 1875, were used in adopting a final value for the expansion, but the values resulting from the previous comparisons are given in the following table to show how little effect those causes of error produced. The table gives the expansion resulting from each comparison of brass bar, hot, with iron bar in ice. Date. sa ad o 1875. ° in. May 26-0 osce sinids chesnicgs vectusoresiuscaacats 59. 20 0. 0018088 = Mibiy 285 sce xicarsuiqudsev. dee waaed eid dbennoesadewe 59, 92 ‘17964. 59. 24 17946 59. 25 17948 59. 33 17993 61. 33 17997 65. 22 17975 56. 96 17939 56. 96 17923 56. 70 17947 57. 68 17914 57. 51 17930 57. 73 17946 57. 26 17962 57. 41 17967 CNRG 16 scnuwssnes iemmoewewaccsnkee semanas aoe 57. 07 17960 PONCE coro 2:5 sa ainwenaiecamiewenwere sauce cedtes 56. 59 17984 D006 18 «canommowsensn deyne cemawieneseenseeey 57. 05 17996 SUNG) LBs icaceiccah aa semnes ceed Hee seieee a serie et 57. 10 17984 The results of the hot-comparisons on and subsequent to June 10, 12 in all, were combined with the results of cold-comparisons, and the mean value of the resulting expansions was taken as the most probable value. That value is, mean expansion of 15-feet standard brass bar between 32° and 899.15 F. for 1° F., é = 0.001795 +. 0.0000016. This probable error is obtained by dividing the probable error in the determination of the change of length of the brass bar, namely, Vv (.000086)? + (.000030)2, by 579.15, and assumes that there is no constant error in the determination of this difference of temperature. As the temperature of 32° was obtained directly from melting ice, there is no thermometer- error init. But at the high temperatures, whose mean was 899.15 F., there may be a small correc- tion necessary to reduce the standard thermometer to an air-thermometer. Suppose it of the form (t0—32°)4. The corresponding correction to the expansion will be, closely enough, de = 0.001795 4 so that we have for the mean expansion of the 15-feet brass bar between 32° and 899.15 F., € = 0'°.001795 (1 — 4) + 0.0000016 § LA. In May, 1876, an additional determination of the expansion of the 15-feet brass bar for 1° F. was made, using a temperature-range from 32° to 65° F. The end surfaces of the steel pins in the ends of the iron bar having become slightly rusty, they were polished off, thus making the iron bar shorter than in the preceding expansion-work. The following table gives the results of the comparisons between the iron 15-feet bar, always packed in ice, and the brass 15-feet bar, either packed in ice or heated to about 629°, 72 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. I, The same method and the same care was used as in the determinations previously made. The column J—B gives the observed excess of length of iron over brass bar: 3 g Be oo 2 oS 5 | tA §5 Fs ej Date. gs SE I-B avD ag s BSP as 28 Sag om | ge Zaye 4 ; fa 1876. OF. oF, in. in May 6 ...... 2-2-2 eee eee eee eee cee ees 22. 00 32. 00 +0. 034373 —0. 000067 May 9 1.22. eee n cece eee e cece ee cence nen eee nee 32. 00 65.18 —0. 024697 —0. 000256 May 10 .........--.- SMES eRe ces eee 32. 00 32. 00 +0. 034298 +0. 000008 May U1 ..conosivcsesies samen sieeetrsses isieaes 32. 00 61. 60 —0. 018595 +0. 000036 MSY 1B; 12 Meas wecuelecsenitenieeceseeraaases 32. 00 61. 04 —0. 017501 —0. 000058 May 13) Spy scscedcscine cewicesericteanimesd 32. 00 61. 06 —0. 017499 —0. 000096 May: 15; i3'ipst sacassncsasmeaesszecce meas ae 32. 00 32. 00 +0. 034316 +0. 000010 ! May 15, 4). M....-.--eeceee cece eee ee cone 32. 00 32. 00 +0. 034282 —0. 000024 May 16, 3.30 p. m.......200-0.e2eeeee eee eee 32. 00 61.22} —0.017945! +0.000065 May 16; 4p: I ss-ccics covececscwsesseecnces 32. 00 61. 00 —0. 017561 +0. 000073 May 16, 8.20 p.m ..--.-- es eee eee eee eee eee 32. 00 61. 45 —0. 018541 +0. 000250 May 17 sccciscccnc satasessccen Sis ealalen aaa 32. 00 32. 00 +0. 034394 +0. 000088 MAY. 18: 6:c:0 so nine coiacvian Setes savem soon weaensers 32. 00 32. 00 +0. 034245 —0, 000061 Each observation gives an equation of condition of weight 1. Solving these by least squares, we have for mean expansion of 15-feet brass bar from 32° to 62° F. for 19 F. e = 0'2,001786 + 0'.0000015 — 0.001786 4 This value is identical with that derived from the Clarke yards, through the brass yards, § 9, except that its probable error is much less. The discrepancy, amounting to 737 part of the whole value, between this and the value 0",001795, obtained for the mean expansion between 32° and 899.15, may be accounted for in three ways: 1st. It may be attributed solely to errors of observation, and the work would still be good, as the discrepancy is small. 2d. The mean expansion of the 15-feet bar may be greater between 32° and 89° than between 32° and 62°. Working on small specimens, Fizeau has found for the expansion of a unit’s length of brass for ¢ centigrade degrees— (1034 x 10-8) t+ (76 x 10-”) ? the expansion depending on both ¢ and @. 3d. If thermometer 21472 should have a very small systematic position-correction at 62° to reduce it to an air-thermometer, and one of + 0°.3 at 89°, the two expansions would agree. The length of the 15-feet brass bar at 62° has already been given. As it was compared with the brass yards Nos. 6, 7, 8, 9, 10, at 62° on three days, as these yards were compared with each other and the Clarke yards at temperatures between 54° and 63°, and as the Clarke yards were compared with the Ordnance-Survey standard between 62° and 64°, any slight errors in the values of the relative or absolute expansions can have little effect on the resulting length of the 15-feet bar at 62°, But as the Keweenaw, Sandy Creek, and Buffalo bases depend on the length of this bar packed in melting ice, the question arises how its length at 32° should be derived from the data already given. 1st. Using the expansions of the Clarke yards given by Colonel Clarke, their lengths at or near 32° can be derived from their comparisons with the Ordnance-Survey standard at tempera- tures between 51° and 64° F. The Clarke yards, the brass yards, Nos. 6,7, 8,9, 10, and the 15-feet brass bar have been intercompared at temperatures between 33° and 44° F, Hence the length of the 15-feet brass bar at 32° F. can be obtained without using any expan- sions, except those of the Clarke yards, over a range sufficient to introduce much error. 2d. The length of the 15-feet brass bar at 32° may be derived from the length adopted at 62° by using the mean expansion from 32° to 899.15 F. already given, namely, 0.001795. § 15.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 73 dd. The length of the 15-feet brass bar at 32° I’. may be derived {rom its adopted length at 62° by using the mean expansion from 32° to 62° I. for 1° already given, namely, 0'°.001786. In view of the probability that the expansion of the brass bar increases with the temperature, and of the fact that the expansion derived from the Clarke yards agrees with that obtained directly from expansion-experiments between 32° and 62° for the length of the 15-feet brass bar at 32° the pale obtained by the first method, namely: 15 feet brass bar at 32° = 179'".95438 + 0.000120 — 0.02198 4 The probable error is derived from the probable error in the length of the bar at 62°, and from the probable error in the value of the expansion from 32° to 62°, directly determined. The saine value would result from the adopted length at 62° , it is deemed advisable to adopt, for the present, , namely, 180.00796 + 0.000111 — 0.02198 4 if the mean expansion between 32° and 629, directly obtained, namely, 0.001786, were used. If me 1 mean ee berween 32° and ot ee obtained, had been used it would have made the 1 > Length of 15-feet brass bar at 62° F. Length of 15-feet brass bar at 32° F. The results obtained for the 15-feet brass bar up to May, 1877, may now be given: = 180.00796 + 0™.000111 —0.02198 4 = 179,95438 + 0.000120 + 0.03160 4 Mean expansion between 32° and 89° for 1° F. = Mean expansion between 32° and 62° for 1° F.= Nort A, MAy, 1880. 0.001795 + 0.0000016 — 0.001795 4 0'2.001786 + 0.0000015 — 0.001786 4 § 15. Since the preceding was written, Casella thermometer 21472 has been compared, under the supervision of Professor H. A. Rowland, with two Baudin thermometers, Nos. 6163 and 6165, belonging to the Johns Hopkins University, of Baltimore. These comparisons were made at the temperatures 45°, 62°, 75°, and 90° F., and comparisons at all these temperatures, as well as freezing-point determinations, were made on each of four days, the thermometers being vertical. " * Thermometers 6163 and 6165 have been carefully and repeatedly compared, when vertical, with an air-thermometer. A description of them and of the method of comparing them with an air-ther- mometer, as well as the final corrections needed to reduce their readings when vertical to a perfect- gas thermometer, may be found in Professor Rowland’s paper on ‘Nhe Mechanical Equivalent of Heat in the Proceedings of the American Academy of Arts and Sciences for 1879. The following tables give for each day’s determinations the corrections derived through 6163 and 6165 needed to reduce the readings of 21472 to those of a perfect-gas thermometer, and the means of the results. They are derived from the inclosures in a letter to me from Professor Row- land, of March 8, 1880, and from Mr. W. W. Jacques, of January 30, 1880. 6163 is the best thermometer, double weight has been attributed to the results derived from it in obtaining the mean corrections. CASELLA THERMOMETER 21472 FAHRENHEIT. (Venricat.) In the reduction, as TABLE i.—Corrections to scale-readings to reduce to «a perfect-gas thermometer, as derived from Baudin 6163, vertical. zi a | | Be | Date. Pas 32°. 455 2°, | 15°. 0°. ‘ oF 3 | i | 1879. ° ° ° ° ° IN Gi Biss ae ne eee inaese St seselleeu ae —0.11) —0.06) —0.05) —0.09 ' NOUS 6 e secineearot en eecepe oem eet ciseiece 2d .. ~0.07) --010} 0.10; 0.09} --0.10 IN OWS Utes eaer 3d. —0.06] 0.10) 0.08} -—0.09| —0.05 Nov. 10, ....2.cececeseeeeeeeeeeeeres 4th —0.04! —0.04| 0.05] 0.05] —0. 04 Means ecmano ce vscamece en tensesee neces) Sees —0.06| —0.09 | 0.07 | —0.07 | © —0.07 74 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. II, TABLE 2.—Corrections to seale-readings to reduce to « perfect-gas thermometer, as derived from Baudin 6165, vertical. | 3 ! \ a i Ko Date. 8 320, 45°, 62°, 75°. 90°. og ° a 25s Sy pe te 1879 | ° ° ° o | CNM Bante tien basasetonereueen lates Danie onus Ge —o1| 0.09 0.05) ~0.06' WAN Gy Gieone eset asucches etre eeaees OA wissen 0.08} 0.13 0.11 0.04 , * NOM: 7 ocasewsteageasss eoneaesast eden BU acto cin Astod cians —0.12 —0.12 =08, 0.0n- PNGbu 10 TPs stncieweucies vexews¥e bas xian GOW ci |eniscacces —0.06 | 0.06 =0.05 ; 40.04 | | PiMiGa niece seach iets epeate hay leek tens J ovoccesfeeeseeeees | —0.09| —0.10 —0.07 | 00.02 t TABLE 3.—Mean corrections to scale-readings to reduce to a perfect-gas thermometer, Baudin 6163 . being given double weight. November 4-11, 1879. e Temperatures. Total corrections. b-corrections. oO ° Oo 32 —0. 06 0. 60 45 —0. 08 —0. 02 62 —0. 08 —0, 02 75 —0. 07 —0. 01 90 —0. 06 0. 00 If we call the freezing-point correction of a thermometer the a-correction, and the correction at any temperature, which remains when the freezing-point correction is subtracted from the total correction at that temperature, the b-correction, we shall have for the b-corrections of 21472, vertical, to reduce to a perfect-gas thermometer, the last column of the above Table 3. For small changes in the freezing-point correction, the total correction at any time may be found by determining the freezing-point correction at the time and adding to it the b-correction. The values of the a-corrections and of the b-corrections will differ as the thermometer is horizontal or vertical. 'Fhese corrections are for 21472 vertical, but as it has been habitually used in com- parisons of standards in a position nearly or quite horizontal, it is important to know its corrections in that position to a perfect-gas thermometer. Subsequently to the comparisons of 21472, vertical, it was compared in a horizontal position with 6163 in a vertical position, but at the higher temperatures results discrepant by half a degree were obtained. On examination, a small air-bubble was found in the bulb. It was removed and more satisfactory results were obtained, although discrepancies amounting to nearly 0.°1 F. still occurred. Professor Rowland says, ‘For the backward motion of the mercury its weight seems necessary. Hence, I never use a thermometer in the horizontal position.” After the removal of the air-bubble, 21472, horizontal, was again compared, on March 16 and 18, 1880, with 6163, vertical, by Professor Rowland. These comparisons gave the absolute errors of 21472 when in a horizontal position at certain temperatures given in the following table: TABLE 4.—Absolute corrections to Casella 21472, horizontal, derived from Baudin 6163, vertical. . 21472, vertical. 21472, horizontal. | Dates. | 329 F. | 32° F. 52°F, | 2° BF, | 75° FB. | 90° F. | ° | ° ° ° ° | ° f March 16, 1880 .----.----+--++++++2-++- —0. 09 | —0.14 —0.19 —0.18 —0, 20 0.17, March 18,1880 ...------2ee2e0-2e 20002: 0.10; 0.16, = —0.19 —0.19; 0.18) —0.13 | MGaneenensescsasecacne access custo 0.09 = 0.15) 0.19 018, —019! 0.15, Horizontal minus vertical reading ....|...........----- FOE |raen eevee -+0. 07 | +0.09 | +0. 06 |. = eats GE So ae Na ae § 16.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 15 The mean horizontal-corrections are given in the third line. Between November 4-11, 1879, and March 16-18, 1880, the freezing-point correction of 21472, vertical, had increased by —0°.03. Adding this quantity to each of the total vertical-corrections given in Table 3, the results are the absolute corrections on March 16-18, 1880, to 21472, vertical. The differences at each temperature between these quantities and the means in the above table are the changes in the absolute correc- tions when the thermometer is changed from a vertical to a horizontal position. Those changes or differences are given in the last line of Table 4. There is another method of determining the change of reading of 21472, when it is changed from a vertical to a horizontal position, which is more direct and which avoids the errors of com- parisons. This is to read the thermometer when vertical, and then, rapidly making it horizontal, to read it again before the slow temperature-change can have produced any appreciable effect. This method was used by Professor Rowland and also at the Lake-Survey oflice. Professor Row- land inserted the bulb of 21472 in a bottle, the stem projecting through the cork, and then read the thermometer many times while the bottle was rapidly changed from an upright to a horizontal or inverted position at temperatures of 72°, 90°, 96°, 979, and 100° F. In reduction he assumed . that the differences were proportional to the length of the column of mercury. At the Lake-Survey office the thermometer was immersed in water in a large tank with glass sides, the temperatures of the room being kept within 5° of the temperatures of the water. Keeping the bulb in the same place, the thermometer was read many times alternately in a vertical and a horizontal position, the intervals between readings being only one or two minutes. The resuits agreed well. Their means and those obtained by Professor Rowland are given in the following table: TABLE 5.—Horizontal minus vertical readings of Casella 21472 at different temperatures. | 32°F, | 45° F. | 62° F. | 75°F. | 90° F. j ° o | ° ° o Rowland, March 10-18, 1880 ccescuuss ccssewenes vessevans ann 0. 03 0. 04 0. 07 0. 09 0.11 Lake Survey, March 31-April 1, 1880 ......---.-.----2..--- 0.04, 0.04 0. 05 0. 08 0. 08 | | The results agree well with each other and do not differ very widely from those given in the last line of Table 4. As the second method is the more exact, the values obtained by Professor Rowland and given in Table 5 are adopted. ‘The corrections to 21472 in a horizontal position to reduce it to a perfect-gas thermometer can now be obtained for November 4-11, 1879, by combining the absolute vertical-corrections found by Professor Rowland and previously given in Table 3 with the corrections just adopted to reduce horizontal to vertical readings. In the following table the total corrections thus obtained are first given and then the part of each which is independent of the freezing-point correction. TABLE 6.—Corrections to reduce readings of Casella 21472, horizontal, to a perfect-gas thermometer, November 4-11, 1879. g29F. | 40°F. | 62°F. | 75°F. | 90°F, | ° ° ° ° ° EGE dicted ek ids onan aa nicimagwibeamescn ere —0. 09 —0.13 —0.15 —0.16 —0.17 Partials co. scouted yeawanieadee sae sae Casas 0. 00 —0. 04 —0. 06 —0. 07 —0. 08 § 16. From the numerous comparisons of Casella 21472, 21473, 21474, 21475, and 21476 with each other and with a calibrated Baudin thermometer, given in the Report of the Chief of Engineers for 1879, Appendix LL, it is known that the calibration errors of all are very small, and hence interpolation may be used for any intermediate temperature. The corrections to a perfect-gas thermometer for other thermometers can now be derived from comparisons with Casella 21472. In April, 1880, the differences between the readings of Casella thermometers 21474, 21475, and 21476, and Baudin 6131, as they were horizontal or vertical, were determined at 32°, 56°, 74°, and 76 STAND AND BASE-APPARATUS. (Cirap. IE, ARDS OF LENGTH, BASES, 4°, and of Troughton & Simms 230 at 90°, by reading them several times at the Lake-Survey ottice while in a large tank of water, whose temperature was kept very nearly stationary and at the temperature of the surrounding air, first when vertical and immediately after when horizontal, the interval of tiine being only a minute or two, the bulb remaining in the same place. The assump- tion that the differences were proportional to the length of the column of mercury above the bulb satisfied these observations well, and with that assumption the differences in the following table have been computed, those for DTt72 being derived alone from Professor Rowland’s determinations, in which 21472 was alternately vertical with bulb up and vertical with bulb down. TABLE 7.—JSifferences between readings of certain thermometers as they are horizontal or vertical. {Horizontal reading always greater. ] Casella, | | Temp. | i 2S) DGS. 230; : Temp. | Bandin 6131. 21472 24 | 21475. 21476. | ! 3 eee ey, fe | ' Bi OE F F | #F F hs. | Cc ; J | i | ° i ° ' ° | ° Oo ° } fe) °° 32 0. 03 0. 04 0, 05 | 0. 03 0. 08 \ 0 0. 000 ‘ 42 , 0. 04 0. 05 | 0. 06 | 0. 04 0.09 \ 5 0. 009 52 0. 06 0. 05 0. 06 0, 05 0.11 i 10 0. 018 62 0.07 0.06 0.07 0.06 012 4 16 0.027 | 72 0. 09 0. 07 0. 07 | 0. 07 0. 14 ! 20 ; 0. 036 ‘ 82 | 0.10 0.07, 0.08 | 0,08 015 | 25 0. 045 92 0.11 0. 08 0. 08 0.09 0.16 30 0. 054 | | 35 0. 063 An attempt was made to determine the same differences for Geissler thermometers 1, 2, 3, and 4, which are used with the Repsold base-tubes. The resulting differences were no larger than their probable errors, and hence are neglected. Between March 29 and April 1, 1880, Casella 21472, was compared, first with Casella 21474, 21475, 21476, Troughton & Simms 230, aad Baudin 6131; and, second, with Geissler 1, 2, 3, and 4. The Bandin and Geissler pienaonisiers have not inerarature een degeriiatl The bulb of standard Baudin 6131 was cracked January 3, 1879, and another bulb was made by James Green, of New York. It is graduated to fifths of centigrade degrees, from —19°.0 (. to +1019.0 C. One degree centigrade is 0.133 in length. The length of its bulb is 1.5 and diameter 0.3, The greatest calibration correction is 0°.05 C. The Geissler thermometers 1, 2, 3, and 4 are nearly alike, and consist of a bulb and an almost capillary stem. A glass cylinder ingldsee this delicate stem and carries fastened to it a graduated porcelain scale against which the capillary tube rests. They are graduated to fifths of a degree Fahrenheit from +30° F. to +120° F. One degree F.=0'".065, The bulb is 0.56 in length and 0.17 in diameter. Comparisons with other thermometers indicate that their calibration errors are not large. In comparing all the above thermometers with 21472, they were placed vertically in a large tank of water at about one inch from its glass wall, and were habitually read with a horizontal telescope, whose micrometer was used for the Casella thermometers. Two groups of comparisons were made, only 21472 and the Geisslers entering the second group. The water was kept well stirred by a paddle-wheel and the temperature of the air near the tank was kept habitually within 3° or 4° F. of that of the water. In each set of comparisons 21472 was first read, then the other thermometers, and last 21472. The greatest difference in the readings of 21472 in a set was in one case 0°.04. The mean of its readings was used. Six sets of readings, each occupying about an hour, were taken at each temperature, except in the case of the Geissler thermometers at 75° and at 90°, for which four sets were deemed enough. The greatest ranges in the temperature of the water during comparisons at the different temperatures were, at 45° F., 0°.02; at 62° F., 02.30; at 75° F., 09.05; at 6° F., 69.08, The thermometers were very near eat other, the denice § 16.] STANDARDS DEPENDENT ON THE ENGLISH YARD. T7 between the extreme ones not exceeding 6 inches. The following tables give the mean readings of these thermometers, the readings near 32° F. being freezing-point determinations : TABLE 8.—Mean readings in water. [Thermometers vertical. ] Casella. T. & S. 280. | Baudin 6131. Date. 21472. 21474, 21475. 21476. F. F. a, F. F. C. 1880. oO ° o oO fe} Oo March 16, 18, 27, 31* ... 32. 08 32. 00 32. 00 | 32. 04 32. 48 0. 356 March 29 .........-... 45. 31 45. 21 45, 25 45, 27 45.91 7. 648 March 29 ...-. 62. 48 » 62.44 62.45 | 62. 46 62. 80 17. 100 March 30 .... tia 74. 66 74, 59 74. 61 74, 61 74.79 23. 780 March 30 .-. -.....--. 90. 37 90. 31 90. 36 90. 37 90. 44 32. 450 * Casella 21472 is the mean of Professor Rowland's readings in melting ice March 16 and 18, and those made at Detroit March 27, all ver- tical. The other Casellas were observed March 27; T. & S. 230, March 31; and Baudin 6131, March 27 and 31, all vertical. TABLE 9.—Mean readings in water. | Thermometers vertical. ] Casella. Geissler. Date. 21472. 1, 2. 3. 4, F. F. F. F. Fy 1880. oO ° ° ° ° March 20* ......- preeeee 32. 08 32. 48 32. 40 81.46 32. 32 March 29 .....-...-.--.. 45. 36 45, 80 45. 60 44.72 45. 60 Mareh 29) scsi ces exewes * 62.18 62. 86 62. 49 61. 66 62. 53 APTI] ccsesivewessssiecs 75. 27 76. 08 75. 70 74.77 75. 58 Mareh 30): csacis somicnwaces 90. 40 91. 34 90. 90 90. 00 90. 80. * Casella 21472 in melting ice same as in Table 8. The others in melting ice on March 20. If in the preceding tables the freezing-point correction for each thermometer be subtracted from all the mean readings for that thermometer, a new table, having the same form, will be obtained. If now, in this new table, the changed reading for each thermometer at each tempera- ture be subtracted from the corresponding changed reading of 21472, these differences will be the differences needed to correct the other thermometers so as to make them agree with 21472. If to each of these corrections be added the correction at the same temperature needed to reduce 21472 to a perfect-gas thermometer, and already given, there will result the b-corrections needed to reduce the readings of the other thermometers, vertical, to a perfect-gas thermometer. The following tables give these corrections, repeating those for 21472. TABLE 10.—b-corrections to scale readings, vertical, to reduce to a perfect-gas thermometer. Casella. Geissler. ; Temp. | - 21472, 21474. 21475, 21476. 1 2, 3. 4. | °o ° O° ° oO ° ° ° fe} | 32 0. 00 0. 00 0.00 0. 00 0. 00 0. 00 0. 00 0. 00 45 —0, 02 0. 00 —0, 0F —0. 02 —0. 06 +0. 06 0. 00 —0. 02 62 —0. 02 —0. 06 0. 07 —0. 04 —0.30 —0.01 —0. 12 —0. 13 75 —0.01 —0. 02 —0. 04 0. 00 =O G2 —0.12 —0. 13 —0, 08 90 0. 00 ~-0, 02 —0.07 —0. 04 —0. 54 —0.18 —0. 23 —0, 16 78 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS, — [Crap. I, TABLE 11.—b-corrections to scale-readings of Baudin 6131, vertical, to reduce to a perfect-gas ther- mometer. i | ‘ Thermometer readings. | b-corrections. | s 2 °C. °C, 0.36 0. 000 i 7.65 +0. 047 17.10 +0, 134 | 23.78 +0. 226 | 32. 45 1 +0, 289 These are the b-corrections needed when the thermometers are vertical. The excesses of the readings of these thermometers, horizontal, over those when vertical have already been given in Table 7. Subtracting from each of those excesses for each thermometer the special excess at freezing-point for that thermometer, a series of remainders will be obtained which are the parts of the corrections due to the horizontal position, which are independent of the change of freezing- point produced by the horizontal position. Adding to these changed excesses the corresponding b-corrections for vertical thermometers, given in Table 10, there results the following table of b-corrections to reduce the readings of the thermometers, horizontal, to the reading of a perfect-gas thermometer. The corrections for 21473 have been derived from comparisons with 21472, given in the Lake-Survey Report, in the Report of the Chief of Engineers for 1879. TABLE 12.—b-corrections to scale-readings, horizontal, to reduce to a perfect-gas thermometer. | Casella. | Geissler. | e | = | ; i 21472 | 21473. | 21474, 21475. | 21476. il; 2. { 3. | 4. | ° | fo} o oO | o ° fe} oO | oO | oO 32, | 0. 00 | 0. 00 0.00 | 0. 00 ; 0. 00 0. 00 | 0. 00 0. 00 0. 00 | 45 ; —0. 04 —0. 04 —0.01:' —0.05' —0.03 —0. 06 ; +0. 06 0.00 | —0.02 | 62 | —0. 06 —0. 01 —0. 08 —0.09 —0. 07 —0. 30 —0. 01 —0.12 0.13 | 745) —0.07 —0. 05 —0. 05 | —0. 06 —0. 04 0.42: -0.12, -—0.13 —0.08 —0. 08 —0.10 —0.11) —0.54' —0.18 | —0.16 2 Ss | = S ro] —0.06 | —0. 22 | i i | , | TABLE 13.—b-corrections to scale-readings of Baudin 6131, horizontal, to reduce to a perfect-gas ther mometer. Thermometer-readings. b-corrections. °C. °C, 0.36 0. 000 7. 66 +0. 038 17.13 | +0. 103 23. 82 +0, 183 ¢ 32.51 | 4-0. 231 § 17. The most important of these corrections to the Casella thermometers are those at 62° F and at 90° F., horizontal, since these thermometers were all used near these temperatures in deter- mining the expansion of the 15-feet bar given in§ 14. In obtaining those temperatures the mean b-correction for all of them was taken from the Kew comparisons given in § 7. Those mean b-cor- rections, horizontal, were + 09.00 at 62° F. and — 0°.04 at 90° F. The mean b-corrections, hori- zontal, now found necessary to reduce to a pertect-gas thermometer, may be derived from Table 12, STANDARDS DEPENDENT ON THE ENGLISH YARD. 79 §§ 17, 18.] and are — 0°.06 at 62° F. and — 0°.09 at 90° F. Hence at these temperatures the corrected tem- peratures used in § 14 need a further correction of —0°.06 and — 0°.05, respectively, to reduce to a perfect-gas thermometer. The thermometers 4), A2, A3, Ay, belonging to the Clarke yard A, were compared with 21472, horizontal, in January and May, 1879, two sets of comparisons being made, whose greatest dis- crepancy in means was 0°.02. Colonel Clarke’s comparisons of these thermometers with the Ord- nance-Survey standard, given in § 7, show the relation of the mean of the B’s to the mean of the A’s. The means of the comparisons of the A’s with 21472, first corrected for error of freezing-poiut and then corrected by the b-quantities needed to reduce 21472, horizontal, to a perfect-gas ther- mometer, given in Table 12, give the following as the b-corrections needed to reduce the mean of the A’s with stems horizental to a perfect-gas thermometer. In a third column are given the b-cor- rections derived from Colonel Clarke’s work for these thermometers by subtracting the freezing- point correction from the correction at each temperature. In a fourth and fifth column the same quantities are given for the mean of the B-thermometers. : TABLE 14.—b-corrections to scale-readings of “yard” thermometers, horizontal, to reduce to « perfect- gas thermometer. Mean of A1, A2, As, Aa. Mean of Bi, Bo, Bs, Ba. Temp. care beorrections. COCKS! poprrections. [Cok Clarkesre F. F. F. FR. F. oO ° oO oO oO 32 0.00 0.00 0.00 0.00 37 OOO oat roa ee nh ne sell Rete oa ace ct 42 ETE dt ee Ne Oe PE ae aa acter ee eee ae 47 ct ON04> “sal dmedaeaeane Cah amamen aan aeeimemereceesan nee 52 +0. 05 4-0. 07 4011 | +0.13 55 +0.07 +0.11 +0.11 +0.15 37 +0. 08 +0.12 +0.09 +0.18 62 +0. 09 +0. 07 +-0.11 +0. 09 67 AIOSHO, *| i eeeaeresenasnandans cial taendecec mame se eeacaanmeenat 72 A007 |s2 een a Cusco secede cele et eb onnes Sy 7 WDAOD I ew ance ce aagtcs su cee iceman oul Aa cat arene toad 82 4202405 | lireree axicrereoaks [aed ees sete dle ama tte ee 87 SETH OS sd bach Ne ht telat al Atcha folelah fal oh tact ce 90 +0.11 +0.12 40.12 40.13 On comparing the b-corrections of these thermometers, required to reduce their readings to a perfect-gas thermometer, with those derived from Colonel Clarke’s corrections given in § 7, it is seen that the greatest differences are 0°.04 at 55° F. and 57° F. As the length of 1° on the dA’s and B’s is but 0.055, this quantity scarcely exceeds the probable errors of the comparisons, while at the important temperatures, 62° F. and 90° F., the differences are less than their probable errors. The important conclusion then follows, that when the d’s and B’s have their freezing-point correc- tions applied, and the Ordnance-Survey Standard 3241 has Clarke’s corrections applied, they then agree quite well with a perfect-gas thermometer. § 18. Having given the b-corrections for the various thermometers both when vertical and when horizontal, it remains to give the freezing-point or a-corrections, horizontal, as determined at different dates. The values of the a-corrections for intermediate dates may be determined by inter- polation. The a-corrections, vertical, at 32° when not directly observed, may be obtained from the observed a-corrections, horizontal, by adding with a negative sign the quantities opposite 32° in Table 7, remembering that the Geissler thermometers have sensibly the same readings whether horizontal or vertical. Such computed a-corrections are marked with an asterisk. 80 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. The following tables give the a-corrections for different dates : TABLE 15.—«-corrections to scale-readings of thermometers, horizontal. | Casella. ; ' Date. Place. 21472. 21473. 21474. | 21475. 21476. | | ; t i } F. \ ° January, 1875 ..-....- Kew Observatory, England 0.0 March, 1875.22.22... | Debroibscnsessassckemer cece +0. 06 \ December, 1875 . ....-- Seca iss co syacccanaetiecjas ieee ars —0.01 | September, 1876....... | Paris, Ste. Claire Deville .... 0. 00 December, 1876 .....-.. | Detroit s.ecescedveseocsews —0. 04 May 1879) aseaeseis —0.11 | Novembor, 1879 .....-. —0. 09% . March 16, 18, 1880 ...../.....- —0.12*;. March 27, 1880 .......- —0. 10* Geissler. | Date. Place. 1. | 2. 3. 4. F. | F. B, F. i | oO ° ° °o | Speraiaeep ALB TT semcenee setae De tROI aes oi ee wceeebvnavens 0.22) -013] +0.66] 0.18 / SAMUMALY 18719 2 cterscs civsate sien ceselbecicins CO. cenvidincemesisectacen act —0.55t, —0.40 +0. 50 —0. 36 + January, 1880.---..--.--+---ee+-/eeeee- 0 wavs au Sea seiiecem teak —0.50} —0.41 +0.50) —0.28) { Mar@h; 1680. ss ccicanscinacsisismceccelseaees OOiadiccedicedicu discern —0. 50 —0.40) 40.51 —0. 36 | AMD TLD: 1SBO ew acenyoreecewbeyente ereews ose ote ch ete ence ae oh dene USO OBA aie ahancdl caaiase case t Scale loose August, 1878. + Scale of No. 2 taken out to remove moisture. Refastened in a different position April 8, 1880. “Yard” thermometers. aa Mean of Mean of ! Date. Place. Ai, A», As, Aa.| Bi, Be, Bo, Ba. | | F. Fr. | Oo 3 °o | Ayn, May, 1874 serene vevcnswnn England, Clarke ..--.----0-..---0--+-+ —0. 29 —0. 26 | Mareh, 1875 scccececeec esos es ces DOtreit execs nicstismewaccins Savane d =0,37 —0. 34 December, 1875 » secs vcncceese| eee see 00) .2s56 Secceoeewed siecewes —0. 40 ~-0. 38 MAy, 1899 scesecsti te peetacnsis —0, 44 —0. 42 March, 1880 SOLS, | seem zictecis crete core i Bandin 6131. Date. Place. oe it Cc fo} May 15,1879 .--...22.----- —0. 252 February 13, 14, 1880 —0. 386 March 20, 1880 .--.......-. —0. 392 March 27, 1880 ............ —0. 356" March 31, 1880 ........---. —0. 356* {Cnar. ll, There is no difference in the horizontal and vertical readings of 6131 at freezing-point. The details of the work on which this note is based may be found in the Lake-Survey reports embraced in the Reports of the Chief of Engineers for the years 1879 and 1880. § 19.) STANDARDS DEPENDENT ON THE ENGLISH YARD. 81 § 19. The determinations of the expansions of the 15-feet brass bar, given in § 11, depend on the Kew corrections to the Casella thermometers, given in §7. As the corrections needed to reduce the readings of these thermometers when horizontal, as in the expansion work, to the readings of a perfect gas thermometer differ from the Kew corrections, a modification of those values of the expansions of the 15-feet bar for 1°°'F. results. Two sets of determinations of expansions were made, the first from temperatures at 32° F. and near 62° F.; the second at 32° F. and near 89° F, In the first set the higher mean temperature after the Kew corrections were applied was 619.79 F., giving a temperature-range of 29°.79. In the second the higher mean temperature after the Kew corrections were applied was 89°.15, giving a range of 579.15 F. As previously stated, the mean of the five Casella thermometers at 62° after the Kew corrections have been applied needs a further correction to reduce it to the reading of a perfect-gas thermometer of —0°.06, while at 90° it needs a correction of —0°.05. The mean expansion between 32° and 62°, given in § 14, namely, 0.001786, needs then a correction of + eae part, which makes it 0.001790. The mean expansion between 32° and 89° I’, given as 0.001795 in § 14, needs a correction of oe fares part, which makes it 0.001797. It has been stated in § 10 that the expansion of the 15-feet brass bar, derived from the comparisons by which its length was deduced from the Clarke yards A and B and from the expansions obtained for those yards by Colonel Clarke, was 0.001786 for 1° F. From the direct expansion determinations between 32° F. and 62° F., after the temperatures have been corrected to agree with those of a perfect gas thermometer, the resulting value of the mean expan- sion is, as has just been seen, 0'".001790. It is very difficult to assign relative weights to these two values, which indeed differ but slightly; hence their mean will be taken, giving 0.001788 for mean expansion of 15-feet brass bar between 32° F. and 62° F. for 1° F. This change in the adopted value of the mean expansion changes the difference of lengths of the brass bar at 32° and 62° I, given in § 14, increasing it by 0.00006, and the lengths to be adopted for the 15-feet bar at 32° and 62° must conform to this new difference. The length at 62° is least affected by expansion errors, since inter-comparisons of Ordnance-Survey standard yard Y;5, the Clarke yards, the brass yards, and the 15-feet bar have been made in the vicinity of 62°, so that if those comparisons alone were used, little error would arise from the errors in the different relative expansions. Besides, when the length and expansion of one bar are derived from those of another by comparisons, and subsequent direct work gives another expansion, the resulting change in the expansion does not change the difference of length of the bars at the mean temperature of comparisons. In determining the length of the 15-feet bar from yard A there were many pairs of bars to be compared. The mean of the mean temperatures did not differ widely from 50°. If the mean temperature of all these comparisons had been 50°, about one-third of the change would belong to the length at 62° and two-thirds to that at 32°. For these reasons, one-third of the change 0'",00006, or + 0.00002, will be attributed to the length of the 15-feet bar at 62° given in § 14, and two-thirds, or — 0.00004, to its length at 32°. There result then, finally, retaining the probable errors unchanged— 15-FEET BRASS BAR AT 62° F.=180'".00798 + 0.00011 15-FEET BRASS BAR AT 32° F, =179!".95434 + 0.00012 MEAN EXPANSION BETWEEN 32° AND 62° F. FoR 19°= _ 0'".001788 + 0.0000015 MEAN EXPANSION BETWEEN 32° AND 89° F. FoR 1°= _ 0',001797 + 0'.0000016 The Minnesota Point, Keweenaw, and Fond du Lac Bases and the connecting triangulation have already been computed with the lengths of the 15-feet brass bar at 32° and 62° F., given in §14. As the changes in the length of the 15-feet bar at 32° F. and 62° F. amount to only gsotooa and sgsdu00 part, respectively, quantities that are very small in comparison with the probable errors in these bases arising from other sources, a recomputation of the work is unnecessary. For the Buffalo and Sandy Creek Bases the new values will be used. Fizeau (Comptes Rendus LX VIII, p. 1125) gives for length of a brass bar at 1° C., =U, (1+1781 x 10-¢+ 98 x 10-"'’) 11Ls 82 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. II, where J, ix its length at 0° ©. This expression gives for mean expansion of 15-feet brass bar for 1° F., between 32° and 62° F., 0.001797, and between 32° and 89° F., 0.001812. The change in the mean expansion of brass as the temperature increases, found from our comparisons, is then somewhat less than that found by Fizeau. If the rate of expansion of the bar varies with the tem- perature, the length of the bar would be best expressed in the form (1) L=l, (1 at+,3°) and this should have been the form given to the equations of condition derived from the compari- sons before solving them by the method of least squares. But as the two sets of expansion deter- winations have already been solved by least squares, and as the dependence of the rate of expan- sion on the temperature, if real, is very slight, the re-solution of the equations of condition would not give results of sufficiently increased value to justify the work. Since the value of /, in (1) and the Jength of the bar at 62° F. are known, while its length at 89° F. can be computed from the mean expansion from 32° F. to 89° F., if these values be substituted with their probable errors in (1), (t—32) being substituted for t, two equations will result, from which the values of @ and (3, with their probable errors, can be obtained. The resulting values are a== (9878 +: 20.2) 10-" &=(1852+ 451)10-" and (1) may be now written, 15-feet brass bar at ¢°? F. —=179i",95434 {1+ [(98784 20.2) (£32) + (1.852 + 0.45) (¢—32)*] 10} =179".95434 + (00017776 £ 0.0000036) (t—32) + (0.0000006666 -£ 0.0000000812) (t—32)? NotE B.—May, 1880. § 20. In § 9 it is stated that no direct comparisons had been made at the time it was written between the Clarke yards A and B, either by Colonel Clarke or at the Lake-Survey office; but between May 24 and June 7, 1879, thirty sets of comparisons were made in the Lake-Survey office on thirteen different days. The comparisons were made as usual with the Wiirdemann and Stack- pole contact-level comparators. From one to three comparisons were made at each visit to the comparing-room, and the mean of the determinations of the set or visit was taken as a single result. Of the thirty such results obtained, none were used in which the mean temperatures of the two yards as indicated by the four thermometers which accompany each yard differed by more than 0°.1 F. This rejected two results. Of the thirty results, there were eight in which the differ- ence of temperature exceeded 0°.08 F., but the mean of the eight results was identical with the mean of the other twenty-two, or of the twenty-eight which were used. The mean temperature of A minus the mean temperature of B for the twenty-eight results used was+ 0°.03. As the thermom- eters can only be read to 0°.1, this quantity does not establish any difference of temperature, and in reduction it has been assumed that the yards were of the same temperature. The adjustment of the yards and comparators was changed four times, a nearly equal number of results being obtained after each adjustment. The following table gives in the successive columns the dates of comparisons, the corrected mean temperatures of yard A, the corrected mean temperatures of yard B, the differences, A—B, in inches of the Wiirdemann comparator, and the residuals. Wiirdemann inches are reduced to English inches by multiplying by 1.004. § 20.] STANDARDS DEPENDENT ON THE ENGLISH YARD. 83 Comparisons of Clarke yards A and B. Corrected Niet tem- erature. i Date. 4—B, Wirde-| Witrdemanin 7 inches. Yard A. Yard B. oF. oF, - in, in. 50. 82 50. 81 +0. 00019 —0. 00001 51.15 51. 08 0. 00016 +0. 00002 53.11 53. 04 0. 00020 —0. 00002 52, 89 52. 99 0. 00011 +0. 00007 53. 61 53. 64 0. 00013° +0. 00005 53. 79 53. 69 0. 00018 0. 00000 54. 04 53. 96 0. 00020 —0. 00002 53. 26 53. 26 0. 00018 0. 00000 52. 00 51. 95 0. 00022 —0. 00004 53. 53 53. 56 0. 00020 —0. 00002 | 54,14 54. 09 0. 00016 +0. 00002 54. 62 54, 56 0. 00018 0. 00000 55, 23 55. 22 0. 00017 +0. 00001 55. 75 55. 85 0.60019 —0. 00001 JUNG DD. sscinsiaccieneencs te 56. 34 56. 30 0. 00017 +0. 00001 Ms PUNE? «is ccecesee ya ciecaicce 56. 64 56. 64 0. 00019 —0. 00001 57. 36 57. 36 0. 00018 0. 00000 57. 64 57. 64 0. 00016 +0. 00002 56, 14 56. 04 0. 00019 —0. 00001 55. 03 55. 05 0. 00017 +0. 00001 55,11 55. 05 0. 00020 —0. 00002 60. 93 60. 84 0. 00018 0. 00000 61. 23 61. 24 0. 00018 0. 00000 61. 81 61.79 0. 00017 +0. 00001 62.13 62. 03 0. 00022 —0. 00004 62.15 62.11 0. 00022 —0. 00004 62. 43 62, 39 0. 00020 —0. 00002 62. 63 62. 61 +0. 00022 —0. 00004 56. 27 56. 24 +0.-000183 |...--.--------- At 56°.25 F., A — B= + 0.000183 + 0.000003 Wiirdemann inches == + 0.000184 + 0.000003 English inches. The relative expansion of A and Bis very small and has been neglected in computing the probable error. If it had been considered, the probable error would have been slightly less. The mean temperature of all the comparisons used is 56°.25, and at this temperature Yard A — yard B=0'",000184 + 0.000003 Using the lengths and expansions determined by Colonel Clarke and given in § 2, there results Yard A — Yard B=0",000197 at 56°.25 F, The agreement is very satisfactory, so close indeed as to make any change in Colonel Clarke’s value in consequence of the new comparisons unnecessary. 84. STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Cnap. III, CHAPTER. LLL KEWEENAW BASE. LOCATION—MARKINGS—MEASUREMENT. § 1. Keweenaw Base is approximately parallel to the west shore of Keweenaw Bay, Lake Superior, its north end being about 100 feet from the shore and its south end about 4,000 feet from it. The length of the base is 28,992 feet approximately, or 54 miles. Its height above the bay varies from 25 to 80 feet, being 68 feet at North Base and 70 feet at South Base, the slope of the ‘ground being gentle, so that the inclination of the base-tubes reached 4° in but a single case. The latitude and longitude of its north end are respectively 46° 57’ and 88° 27’. Nearly the whole length of the base is through a forest, and the surface soil is decomposed sandstone mixed with vegetable matter. § 2. The following markings of the base have been made: At South Base the end-point of the base is a cross on a piece of brass leaded into the top of a stone post 6 feet long, the top of the post being level with the ground. On the prolongation of the base southward, precisely one yard from the end of the base, is a cross on a piece of brass, set in the top of a stone post 23 feet long, the top of this post being 6 feet under ground. At right angles to the end of the base, and about 100 feet east of it, a 6-foot stone post rises 1 foot above the ground. There is a similar post about 100 feet west of the end of the base, its top being broken off. At the end of the ninety-fourth tube (1,410 feet) from south end of base, and on the base line, there is a cross on brass in the top of a stone post 23 feet long, the top of the post being 18 inches under ground. At the middle of the base (end of nine hundred and fifty-second tube) there is a cross on brass in the top of a stone post, the top of the post being 2 feet under ground. The north end of the base is marked by the intersection of two lines on the end of a copper bolt 10 inches long leaded into the solid sandstone rock, which is about 4 feet below the surface of the ground. Further details are given in the Description of Stations, Chapter XIV, A. § B. This base was originally measured in 1867, but as General W. I’. Raynolds, at that time in charge of the Lake Survey, had little confidence in the precision of the work, on account of the muddy condition of the ground during the measurement, giving instability to the measuring tubes, it was decided to remeasure it. This was done in the summer and fill of 1873 by Assistant Engineer E. 8S. Wheeler, aided by Assistant Engineers C. F. Burton and C. Pratt. Although the ground was tolerably firm during the second measurement, greater stability for the trestles was obtained by making them rest on stout wooden stakes driven to the surface of the ground. Prior to the second measurement, the base-line had been converted into a rarely used road, and in places grass had sprung up. As stakes were used the partial soil was not removed. The base-apparatus used is a compensating one by Wiirdemann, a copy of that of the Coast Survey, and is described in the Lake-Survey Report for 1868, the original being described in the Coast-Survey Report for 1854. The measuring part consists of tubes 1 and 2, each about 15 feet long. In previous bases of the Lake Survey, the measurement had been made without tents over the tubes. In this, Assistant Engineer Wheeler was directed to make the measurement under movable tents, so as to avoid, at least in part, the large daily fluctuations in the length of the base-tubes arising from unequal temperatures of the compensating bars. §§1-7.] KEWEENAW BASE. 85 § 4. Comparisons of the tubes with the standard bar were made from July 16 to August 12, 1873; measurements from August 13 to September 7, 1873; comparisons from September 8 to Sep- tember 11, 1873; measurements from September 12 to September’ 25, 1873; comparisons from Sep- tember 26 to October 5, 1873; remeasurements from October 6 to Odtsber 8, 1873; comparisons from October 9 to October 15, 1873. The first distance of ninety-four tubes was measured four times at the beginning of the meas- urement, and three times after the base had been finished, to ascertain the error in measurement. The greatest number of tubes measured in one day was one hundred and thirty on September 5, in 8" 43", giving a speed of 224 feet per hour. Measuring was done on thirty-two days, averaging seventy-eight tubes per day. DESCRIPTION OF APPARATUS—COMPARISONS WITH STANDARD BAR. § 3. A measuring tube of the Bache-Wiirdemann base-apparatus consists essentially of two parallel bars of brass and iron, each 168 inches long, rigidly connected at the rear ends by an iron cross-bar, the front ends being free to move longitudinally with reference to each other. The brass bar, 1.10 in height and 0.37 thick, is directly below the iron bar and 0.3 distant from it. The iron bar is 1.40 in height and 0.27 in thickness. Both are inclosed in a tin case. To the front end of the brass bar a vertical compensating lever is hinged, having on its rear side a knife-edge which is pressed by a spring against a steel plate on the front end of the iron bar, while, higher up, another kuife-edge on the front face of the lever has constantly pressed against it a sliding rod, whose forward end, armed with an agate plane, projects from the front end of the tin case and forms one measuring end of the tube. The object of this lever is to make the distance between the end-points of the tubes during changing temperatures constant so long as the two bars have the same temperature. At the rear end of the tube a vertical arc for measuring inclinations is fixed to the lower bar, and is read by a vernier with a level on its arm, called the sector-level, the hori- zontal axis of the vernier being perpendicular to the plane of the bars. The vernier arm carries a small cylindrical surface, the axis of which coincides when in adjustment with that of the vernier at the instant the bubble of a small contact-level moved by the arm bearing the cylindrical surface plays. A rod with parallel motion passing near the axis of the cylindrical surface abuts against this surface, while its rear end projects out of the rear end of the tube and is covered with an agate knife-edge, which forms the rear measuring end of the tube. § 6. In measurement, the base-line was divided into sections 500 feet long, by stakes carefully placed by a theodolite in the vertical plane passing through the ends of tke base-line. Two tubes were used. ‘The end-points of each tube are small agates carried by sliding rods. The forward agate was brought into the vertical plane of the base-line by means of the tri-rectangular motion with which the trestles supporting the tubes are provided, the plane being given by a theodolite set up over one of the section-stakes, and directed toward the next, the tube being intermediate. The rear agate of this tube was then brought, by means of a slow-motion screw, into contact with the front agate of the rear tube till the bubble of the contact-level could be read. Then the con- tact-level, the sector-level, the sector-arc, and the thermometers inside of the tube were read. The rear tube, with its tent, was then carried forward in front of the other tube, placed in line, and the work continued. When, as at night, it became necessary to stop work, a theodolite was set up about 10 feet from the front end of the forward tube and approximately in a vertical plane through the end of its agate, at right angles to the base-line. By means of this theodolite, and the one used for alignment, a mark was made directly under the agate on the head of a copper tack in a stout peg, the top of which was driven below the surface of the ground. The stability of this mark during the cessation of work was checked by other similar marks placed in a line perpendicular to the base-line. The record was kept in duplicate by two recorders, and all readings except those of the contact-level were in duplicate. § 7. In an apparatus like the Bache-Wiirdemann, composed of many different parts, and liable to change length at each of the places of contact, frequent comparisons with a standard are neces- sary. The standard used was the 15-feet brass bar of the Lake Survey. To determine what the lengths of the tubes were during the measurement, the comparisons were made under circumstances as nearly the same as those of measurement as was possible. The temperature of the standard 15- RG STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS, — [Cuar. III, feet bar was fixed by being packed in broken ice, which was constantly renewed. The compari- sons were under tents, and were continued from 8 a. m. to 6 p. m., so as to include the same diurnal fluctuations of temperature as took place during measurement. The method of comparison was as follows: Two contact-level comparators, reading to one hundtred-thousandth of an inch, one made by Wiirdemann and one by Stackpole, were mounted on heavy stone piers a little more than 15 feet apart. The masonry supports of these piers rested on rock, and the piers were boxed nearly to their tops to protect them from unequal changes of tem- perature. A tube resting on its trestles was placed between the comparators, the standard bar was placed on the tube and carefully aligned, and then, by motions of the trestles, one or the other was brought between the comparators, the Stackpole being kept at 4 fixed reading and the Wiirde- mann comparator being read. The stone piers proved to be very steady. In reducing, the ther- mometers were corrected for their errors and then their curves plotted. The readings on the standard bar and tubes were plotted as ordinates with times for abscissas. On inspecting the curves of comparisons of tubes with bar in ice when the temperature was above 65° F., slight depressions are seen in the tube-curve, arising from the cooling effect of the bar during the five or ten minutes it was on the-tube. In extreme cases, these depressions amount to 0.00040. As the bar in ice was placed on the tube but once in two hours, the effect is elimi- nated by neglecting these depressions in the curves. The differences between the tube- and bar- curves give the corresponding differences of length, and hence the length of the tube from the computed length of the bar at the observed temperature. The values of the comparator-screws have been carefully determined. EFFECT OF TEMPERATURE-CHANGE ON APPARATUS. § 8. Ina compensated base-apparatus, like the Bache-Wiirdemann belonging to the Lake-Sur- vey, the length of a tube may change from a change in temperature in two ways: 1. Its length may change from the unequal expansion of the brass and iron bars when their changing temperatures are constantly equal to each other, this change of length arising from incor- rect proportions in the compensating lever. This change in length from imperfect compensation is called expansion. It can be reduced to almost zero, and has been made very small in the Lake- Survey apparatus. 2, In changing temperatures the bars may not gain or lose heat alike, and the length of the tube may change from the difference in their temperatures. This is the serious difficulty in any apparatus composed of two bars, whether in the form of a compensating apparatus, like the Colby and the Bache-Wiirdemann, or in the form of a metallic thermometer like the apparatus of Borda, of Bessel, and of Repsold. The problem of keeping the two bars at the same temperature is not well solved in the compensating apparatus of the Lake Survey, and any such apparatus has the great disadvantage of giving exaggerated changes of length for any difference in temperature between the two bars. Thus a compensated arrangement of brass and iron bars, whose expansions for 1° are b and i, will change length for a difference of 1° in temperature of its two bars, by i i, or by about three times the change of length of the iron bar for 1° of temperature. § 9. Some results from comparisons will be of interest as showing the amount of such changes in the Lake-Survey base-apparatus. On August 5, 1873, during comparisons of tube 1 in fwll sunshine, the temperature of the interior of the tube rose from 60° F. at 6 a. m. to 98° F. at5 p.m. The tube reached its maximum length at 9a, m. and then decreased so as to be 0.00470 shorter at 10 p.m. The day was one of excessive range in temperature. Such change of length approaches zero when the rate of temperature-change during the day becomes very small. At the Buffalo base, in 1875, the mean length of this tube for the eleven days of comparisons was, at 1 p. m., 0.00230 greater than at 8 a. m., the comparisons being under tents. The variation for tube 2 was less. But as a part of this change might be due to expansion of tube from imperfect compensation, while the two bars have equal but varying tem- peratures, that question has been re-examined. Two attempts had been made in the office at the determinations of the expansions of the tubes, and the comparisons at the base-lines also gave data ; §§ 8-12.] KEWEENAW BASE. 87 but the quantity to be measured is so small, and is so masked by larger errors, that its determina- tion is difficult. A combination of the various values gave for both tubes an expansion less than 6.00001 for 1° F., a quantity less than the probable error of determination. Hence, in the com- putation of the base, the expansions for both tubes have been taken as zero. As the mean tempera- tures of the tubes during comparisons and during measurement of the Keweenaw Base did not differ by more than 1° I., the uncertainty in the precise values of the expansion does not affect their lengths by more than 0.00001, a quantity which may be neglected. § 10. The fact that the tubes change length during the day being known, and it also being known that their proper expansion (both b.irs having the same temperatures) is very small, it follows that the change in length given above must be due to difference of temperature of the two bars. When tube 1, on August 5, 1873, changed length during the day by 0.00470, taking one-third of this quantity as giving the change of the length ofthe iron bar relatively to the brass bar, we find that the temperatures of the iron bar relatively to the brass bar must have changed by about 1°.3 F. during the day, and that on an average comparison-day at the Buffalo Base the relative tempera- ture of the iron and brass bars must have varied by 0°.7 F. § LI. The variations in length of tubes 1 and 2 during comparisons uncer tents are well shown graphically by the curves in Plate XX VII, in which the abscissas give the hours of the day and the ordinates the corresponding excess of the length of the tube above that of the 15-feet brass bar in melting ice. The curves for tubes 1 and 2 are each the means of daily comparison-curves for twenty-three days of comparisons at the Sandy Creek and Buffalo Bases. From these curves it will be seen, that, for an average day during comparisons at those bases, tube 1 changed length between 8 a. m. and 5 p. m. by 0.00216 and tube 2 by 0.00154, these changes being due almost entirely to difference of temperatures of the brass and iron bars in each tube. In a preliminary computation of the Keweenaw Base in 1874, the lengths adopted for the tubes during measurement were derived from comparisons of the tubes with the bar in melting ice, the tubes being under a tent as in measurements, and the comparisons being repeated at short inter. vals during the working-day. The differences of lengths of bar and tube were plotted for each day as ordinates, the hours of the day being the abscissas, so that for each day of comparison a curve resembling those already given was obtained. The mean ordinate for the day was computed, giving the tube’s mean length for that day. The mean of such mean lengths for the periods of comparisons gave the length of the tube in terms of the bar in ice, which was used in computation. If the mean daily temperature-curve during comparisons had precisely the same form and dimensions as during measurement, thus giving the same mean curve for the daily changes of length of tube during comparisons and measurement, no error would be introduced into the resulting length of the whole base, although these mean values would not give the same length in remeasure. ment for a part of the base, for which the temperature-curves on the two days of measurement differed considerably. In fact, the mean temperature-range between 8 a.m. and 12 m. on the days of comparison was 69.1 F., while on the days of measurement it was 13°.2 F., a difference too great not to need a correction if any method can be found for making it. § 12. The numerous comparisons of the base-tubes with the 15-feet brass bar in ice at Sandy Creek and Buffalo Bases added largely to our knowledge of the conduct of the tubes during rapidly changing temperatures, and showed what had previously been noticed, that the mean daily length of a tube depended on the temperature-range during the morning. After some trials it was found that if the mean daily differences of length of tube and bar in ice were plotted as ordinates, the temperature-changes from 8 a. m. to 12 m. being the abscissas, the resulting curve was nearly a straight line for the Keweenaw, Sandy Creek, and Buffalo Bases, and that for each base the incli- nation of the line was nearly the same. That is, if we represent by / the difference of lengths of 15-feet bar in ice and tube, when its iron and brass bars have the same temperature; by x the change in the mean difference between 8 a. m. and 5 p. m., due to a change of temperature of 1° TF’, between 8 a. m. and 12 m.; by a the change of temperature from 8 a. m. to 12 m. on any day when the mean difference of length between 8 a. m. and 5 p. m. of bar in ice and tube isd; we shall have ‘ i+ac—a lis constant so long as no permanent change in the length of the tube takes place. 88 STANDARDS OF LENGTH, BASES, AND BASHK-APPARATUS., (Crap. III, As already stated, the expansions of the tube, both bars being always at like temperatures, is so sinall and uncertain as to be neglected. Ifit had a definite value it would also be necessary to fix a temperature for the normal length. The above equation expresses in effect that the mean length of a tube between 8 a. m. and 5 p.m. on any day is equal to its normal length plus the temperature-change from 8 to 12 a.m. mul- tiplied by a constaut. This equation is entirely empirical, and its constants might be somewhat changed by using other hours than 8 and 12 for determining the temmperature-range. Each day of comparisons continued from 8 a. m. to 5 p.m. of a tube under tent, with the bar in ice, on Kewee- naw, Sandy Creek, or Buffalo Bases, gives an equation of condition of the above form. The value of .r will be the saine in all. The value of J will be the same so long as there is no evidence of a permanent change in its value. Such change occurs between the measurement of different bases, and sometimes during the measurement of the same base. § 83. This liability to change in length is a serious evil in the Bache-Wiirdemann apparatus. There are in it thirteen joints or points of contact, at any one of which change in contact by wear ov by change of adjustment may change the length of the tube. As, in consequence of jars and expansions, the screws on which these joints depend frequently get loose, changes in length are unavoidable, and they are large enough to seriously diminish the accuracy of the work, unless much time is spent in comparisons, which would otherwise be unnecessary. In any base-apparatus it is very much to be desired that the points which fix the length of the apparatus should be rigidly connected with each other; or, if there are joints, that these should be very few in number and practically unchangeable. In one of the tubes of the Lake-Survey base-apparatus the length of the tube can be changed 0.003 inch by simply tightening or loosening the screw which forms the axis of rotation of the compensating lever; the tightening twisting the original plane of rotation. § 14. From the comparisons at Keweenaw, Sandy Creek, and Buffalo Bases, twenty-eight equations of condition of the form ltav=ad for each tube have been obtained. These equations contain for tube 1 a single value of I fer Keweenaw Base, and three values for each of the other bases, 7 being the same for all; while for tube 2 a single value for | was used for each base, # being the same for all. For the Keweenaw Base there was no evidence of change of length of tube 1 during measure- ment. Tube 2 changed length, but the comparisons of September 30 and October 1, 1873, were not included in the least-square work, as the temperature-range was small. On the Sandy Creek and Buffalo Bases, while there was no evidence of change in length of tube 2 during the measure- ment of the base, there was evidence that tube 1 on both bases changed length between the com- parisons at beginning and middle of measurement, and between the middle and end comparisons, necessitating the assumption of different lengths at each of the three periods of comparison at each base. The following table gives the dates of comparisons, the weights, the equations of condition, and the residuals. The coefficient of i in the equations is the temperature-increase from 8 to 12 wu. m. on each day, and the absolute terms are the mean excess of length of tube over that of bar in ice between 8 a.in, and 5 p. mm. of each comparison-day. TUBE 1. KEWEENAW BASE. Maxi # * | | Date. demi peratnre Weight. Equations. u. % oF, AUGUSH4; 1873-2522 sec eose 80.0 0.5 +14. 6x—0. 01916=0 —0. 00016 August 5, 1873 .-- -....22.-- 98.0 0.5 L+17.4@--0.01948=0 | 0.00010 | August 6,1873 .....2022.022- 72.8 1 l— 1Lvx—0.01734=0 | 0.00087 | August 7, 1873 ..2.2.2eee2e- 65.0 1 h+ 8.0¢—0.01743=0 | —0. 00001 | August 8, 1873 .......------- 66. 0 1 b+ &12-0.01796=0 | +0.00015 | October 2, 1873 ..--------.--- 39.3 1 h+ 1.8a—0.01680=0 | +0. 00046 | §§ 13, 14.] KEWEENAW BASE. TUBE 1.—Continued. SANDY CREEK BASE. Date. t oenrnen Weight. Equations. vO o August 13, 1874 .... 2.2.2.2. 74.9 1 lot 4.4 2—0. 01904=0 +0. 00061 August 14, 1874 ............. 83.0 1 2+10. 6 e—0. 02064=0 —0. 00014 August 15, 1874 2.2.2... 2... 77,1 1 I2+14. 0 2-0. 02068=0 +0. 00028 August 17, 1874 _............ 85. 6 1 lo+13. 4 2—0. 02159=0 —0. 00071 August 19, 1874 ....-........ 76.5 1 le+10. 9 x—0. 02059=0 —0. 00005 September 15, 1874 .......... 85.0 1 13-12. 7 a—0. 01984=0 —0. 00034 September 16, 1874 ._. “ 63.1 1 3+ 2.3 2—0. 01799=0 +0. 00010 September 17, 1874 .......... 69.0 1 ig+ 6.3 a—0. 01839=0 +0. 00024 October 17, 1874............. 61.4 aL; 411.7 e—0. 019310 —0. 00045 October 19, 1874............. 43.4 1 yt 1.4 a—0. 01769=0 —0. 00023 October 22, 1874 53. 3 a 4415. 0 a—0. 01912 =0 +0. 00018 ' October 23, 1874 x 60.3 1 14+ 22. 9 a—0. 01990=0 +0. 00049 BUFFALO BASE. 1 August 31,1875 ...2.......-. 82. 0 1 is+17. 6 x—0. 02932=0 +0. 00034 September 1, 1875 -.......-.. ! 85. 3 1 Is+14. 2 a—0. 02952=0 —0. 00032 September 2,1875 ..... 82.4 1 s+ 6.5 2—0. 02823=0 —0. 00008 September 3,1875 . 84.0 1 ls+ 8.8 x—0. 02839=0 +0. 00007 September 22, 1875 50. 6 1 ls+ 7.7 «—0. 02778=0 +0. 00009 September 23,1875 .. ...... 57.5 1 ls+15. 4 x—0. 02901=0 —0. 00009 October 11, 1875............. 44.4 1; U+ 4.8 a—0. 02662=0 —0. 00006 October 12, 1875....... ..... | 40.0 | 1 + 5.5 x—0. 02621—0 +0. 00045 October 13, 1875 : 49. 9 j 1 t7+-12. 0 a—0. 02797=0 —0. 00043 October 14, 1875 61.1 1 l7+-18. 7 x—0. 02842=0 +0. 00004 TUBE 2. KEWEENAW BASE. July 17, 1873 .. a6 / 76.6 0.5 4+ 4.8 2—0. 04360—0 +0. 00001 July 18, 1873 ............-.-- 60. 0 1 li— 4.0 2-0. 04254=0 +0. 00021 Duly 21,1878 accweesew cs vosicies { 90. 3 0.5 U}422. 3 a—0. 04473=0 +0. 00057 July 22, 1873 .....-..2-22.0-. 94.2 0.5 11416. 3 x—0. 04433—0 +0. 00039 July 23, 1873. .-......--.---- | 79. 2 1. i+ 5.8 x—0. 04438=0 —0. 00068 SANDY CREEK BASE. August 13, 1874 ....... -.--. 75.9 | 1 UVe+ 4.0 2—0. 04438=0 +0. 00007 August 14, 1874 -.-....2222.. 84. 0 1 V+10. 8 x—0. 04515=0 — 0. 00004 August 15, 1874 ..........-. 76.1 1 U2+11. 3 a—0. 04535=0 —0. 00020 August 17, 1874 ........- eens 85.1 : Ie { V2+12. 0 a—0. 04579=0 —0. 00057 August 19, 1874 ..........-.. 7.9 | 1 Ve+11. 0 2—0. 04560=0 —0. 00047 September 15, 1874 .....-.... 84.2 | ab U24+10. 1 s—0. 04537=0 —0. 00033 September 16, 1874 .......... 64. 6 | 1 Ve+ 2.3 x—0. 04372=0 +0. 00056 September 17, 1874 A 69. 8 | 1 Ve4+ 6.4 2—0. 04422=0 +0. 00046 October 17, 1874 ...----..--.- 62.5 1 U2412. 2 x—0. 04509=0 +40. 00015 October 19, 1874.....-....---. 45.0 1 Ve4+ 2.0 2—0. 04418=0 +0. 00007 October 22, 1874 ......-...--- 55. 5 1 V2+14. 6 a—0. 04520=0 +0. 00027 October 23, 1874 .......-..--- 61. 2 | 1 U2+22. 6 x—0. 04619=0 +40. 00006 BUFFALO BASE. August 28, 1875 ....-.-....- 86.0 1 V3+16. 3 «—0. 04755=0 +0. 00001 August 31, 1875......-..--.-. 84.5 1 V3+19. 5 c—0. 04750=0 +0. 00037 September 1, 1875 .........- 87.3 1 V3+13, 6 —0. 04760=0 —0. 00030 September 2, 1875 ........-.. 84.4 1 Us+ 8.2 a—0. 04653=0 +0. 00024 September 3, 1875 .....-..... 86.1 1 U3+10. 0 a—0. 04636=0 +0. 00059 September 22, 1875 .......... 51.9 1 Us+ 7.4 x—0. 04673=0 —0. 00008 September 23, 1875 .....-..-. 58.7 1 V3+15. 5 a—0. 04742=0 +0. 00006 October 11, 1875 43. 4 1 UVz+ 6.6 x—0. 04712=0 —0. 00050 October 12, 1875 39.5 a V3+ 6.0 2—0.04651=0 +0. 00005 October 13, 1875 ...........-- 48.9 1 V3+11. 5 w—0. 04741=0 —0. 00032 October 14, 1875 ......-...--. 60.5 1 V3 +18. 0 2—0. 04793=0 —0. 00021 12LS8S 89 90 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. III, § 15. Solving for each tube, by least squares, the twenty-eight equations of condition resulting from twenty-eight days’ comparisons, the following values of | and # are found: TUBE 1. Keweenaw Base, 1,—= 0.01701 + 0.00012. August 4- October 2, 1873. y= 0.01905 + 0.00015. August 13-19, 1874. Sandy Creek Base, < 1,—0",01777 + 0.00016. September 15-17, 1874. 4g —=0i.01727 + 0.00017. October 17 - 23, 1874. 1; = 0.02726 + 0.00017. August 31 — September 3, 1875. Buffalo Base, Ig = 0'".02682 + 0.00021. September 22-23, 1875. 1, = 0.02591 + 0.00016. October 11-14, 1875. For tube 1, 2 = 0.000136 + 0.000009. TUBE 2. Keweenaw Base, = 1’; = 0'".043135 + 0.00014. July 17-23, 1873. Sandy Creek Base, UV’, = 0.04406 + 0.00011. August 13- October 23, 1874. Butfalo Base, V,;—= 0.04598 + 0'".00012. August 28- October 14, 1875. For tube 2, x= 0.000097 + 0.000008. The diagrams in Plates XXVIII and XXIX show how closely these values of J and x represent the results of comparisons for each tube and each base. Abscissas are the changes in temperature of tube between 8 and 12 a. m. on a comparison-day, and the ordinates of the points inclosed by a circle with a date are the excesses of the mean tube-length on that comparison-day over that of the 15-feet bar in melting ice. Where tube 1 changed length during measurement, these ordinates have been corrected so as to free them from the effect of that change and to make them give only the temperature effect. The straight line is plotted from the first values for each base of the tube’s normal length and from the values of .r just given. The distances of the points from the right line are due to the errors of observation and theory, and for the apparatus in question must be consid- ered as reasonably small. LENGTHS OF TUBES FOR ANY DAY—LENGTHS USED IN COMPUTATION. § #6. The method of determining the value of a tube’s length to be used in the computation of any day’s measurement can now be explained. The value of the excess t of the normal length of tube 1 over bar in ice for Keweenaw Base, already given, namely, 0.01701, was used for the whole base. For each full day of measurement (from 8 a. m. to 5 p. m.) the temperature-change a, from 8 a. m. to 12 m., was taken from the ther- mometer-readings inside the tube, and this, with the proper value of x previously given, enabled us to compute 1+ ax, the excess of the mean length of the tube for that day over the bar in ice. The mean length multiplied by the number of times that tube was used during the day gave the length measured by it. When measurements were made on any day during a fraction of the period from 8 a. m. to 5 p.m. the mean length of the tube during the fraction of a day would usually differ from the mean length for the whole day, and hence the computed value, !4 ar, would need a correction to give the mean length of the tube while being used. To obtain this correction, the mean curve of § 11, giving the average daily variations of the tube’s length at Sandy Creek and Buffalo Bases, was used. The difference between the mean ordinate of this curve for the whole day and for the fraction of the day during which measurements were made, was applied to the computed value 1 + ax as a correction, and the result was used to obtain the mean length for the fraction of a day. The same method was followed in computing the mean length of tube 2 for each day or fraction of a day of measurement. But the normal length of this tube changed permanently during the measurement. On September 11, 1873, one end of the tube fell. It was caught by hands so that the blow on a man’s shoulders below it was not very severe; hence it sustained no violent jar. It §§ 15-19.] KEWEENAW BASE. 91 was at once taken to cainp and recompared. The not very satisfactory comparisons indicated no change of length since those before the measurement of the base was begun. But comparisons after the measurement of the base, on September 30 and October 1, 1873, showed that the tube then had changed length. It must then be assumed that this change took place between September 11 and September 30, 1873. For the portion of the base measured prior to September 11, the value of I’, is adopted for tube 2, which results from the least-square reductions of the comparisons for the three bases, which includes for this tube and this base the comparisons of July 17, 18, 21, 22, and 23, 1873. The value has already been given as l/; = 0.043135. The comparisons of September 30 and October 1, 1873, of tube 2 with 15-feet bar in ice, give the equations of condition September 30, 1873, 14+4.2a—0".04031=0 Weight=1 October 1, 1873, 1+ 4.0.0 — 0".04030=0 Weight=1 Using the value of x previously given for this tube, these equations give another value for 1’, to be used in the remeasurements subsequent to October 1. That value is 1 = 0.03985. For the interval between September 11 and September 30, the mean of the two values, namely, 0.04150, is used, the change in the length of tube 2 being supposed proportional to the number of tubes meas- ured. Collecting these values we have for lengths of tubes to be used in computation of the portion of the base measured on any day, Tube 1 for whole base and for remeasurements = 15-feet bar in melting ice + 0.01701 + 0.00012 + az. Tube 2 from 1st to 1028th tube = 15-feet bar in melting ice + 0.04314 + 0.00014 + ax. Tube 2 from 1029th tube to 1933d tube = 15-feet bar in melting ice + 0.04150 + ax. Tube 2 for remeasurements after measurement of base = 15-feet bar in melting ice + 0.03985 + ax. COMPUTATION OF LENGTH OF BASE. § 17. In computing the length of the base the mean length of each tube was found, as already explained, for each day of measurement, in the form: Tube’s mean length for the day —15-feet bar in melting ice + [+ az. When the measurement was only during a fraction of a day the quantity av was corrected, as has been explained in §16. Multiplying the obtained length by the number of times the tube was used during the day, we have the sum of the lengths of the tubes_for one day, and in the same way for all days of measurement. These quantities are summed. In closing on the permanent marks on the base line, several small distances less than a tube’s length were carefully measured with a seale, and their sum is added to the above length. §18. There are also several instrumental corrections to be applied to this value: 1. Correction for inclination of tube to horizontal plane. 2. Correction for reading of contact-level. 3. Correction for error in sector-level adjustment. 4. Correction for error in cylindrical surface. The first of these corrections needs no explanation. The second arises from the fact that the tube is at its definite length when the bubble is at the middle of the contact-level, and changes by very minute known quantities when the bubble is not exactly in the middle. The third error arises from the fact that the sector-level is not stable in adjustment. When found in error in adjustment, the erroris assumed to be proportional to the number of tubes measured since the last adjustment. It gives a correction to the grade-angle and to the tube-length, both of which are always small. The fourth error arises from a slight defect in the cylindrical surface, by which the tube was slightly shorter at 0° inclination than at other inclinations. The sum of these corrections applied to the sum of the lengths of tubes gives the length of base, as shown by the measurement. § 49. A portion of the Keweenaw Base equal to ninety-four tube-lengths (about 1,410 feet) was rerheasured seven times, in order to form an idea of the accuracy of the measurement. The 92 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cmap. III, seventh remeasurement is not included, as it is almost certain from the notes that an error of either 1° or 2° was made at one point in reading the inclination of the tube. The mean of the other six measurements is taken for the length of this part of the base. The following table gives the number of the measure, its date, the maximum and minimum tube-temperature during the measurement, and the excess of the separate measures over their mean: z Temperature. 2 Date. Excess. & Maximum. | Minimum. ° 2 in. 1 | August 16,18, 19,1873 ......- 82. 2 66. 6 +0. 075 2) August 20, 1873 .......-..--. 711 66. 0 —0. 008 3 | August 21, 22,1873 .......--- 81.8 62.7 —0. 003 4 | August 23, 1873 .........---- 65. 6 54.5 —0. 041 5 | October 6,7, 1873 .....--..-- 58. 5 38. 6 —0. 045 6 | October 7, 8, 1873 .....--.---- 71.2 40.0 +0. 022 Deducing from these residuals the probable error of one measurement, it is found 0.030, or about solos: It will be noticed that while on some days the whole distance was measured, for other measure- ments two or even three days were required. As this involved the difficult estimate of the mean tube-length for parts of days by the approximate method already described, it seems probable that the errors in these measurements are larger than in the continuous measurement of the base when there were relatively fewer fractional days of measurement. It will further be observed that during these measurements the temperatures varied from 38°.6 F. to 82°.2 F., and that while during the measurement of August 20 there was a change of tem- perature of but 5°.1 F., in that of October 7 and 8 there was a change of 319.2 F. These great irregularities in temperature-condition test severely the method of reduction adopted. § 20. Taking the mean value of all corrections (1+ «wx included) for this portion of the base, and adding it to the corrections for the remainder of the base, we have Keweenaw Base = 1933 (15-feet bar in melting ice)+[1]+[2]+[3]+[4]+[5]+[6], where [ ] is the sign of summation for the whole base, and 1, 2, 3, and 4 are the corrections specified by those numbers in § 18, while 5 is the value of /+ ax for any one tube, and 6 is a fractional tube length measured with a scale. [1] + [2] + [8] + [4] + [5] + [6] = — 55.116 — 6.034 — 07,101 + 0.108 + 60.366 + 56.206 = + G1'429. .:. Keweenaw Base 1933 (bar in melting ice) + 61.429, as resulting directly from measurement. This value needs some corrections, given below, and is to be reduced to the sea-level. PROBABLE ERRORS OF CORRECTIONS AND PROBABLE ERROR OF LENGTH OF BASE. § 28. An attempt will now be made to estimate the probable errors of these corrections, and the probable error of the final value for length of base. The sums of the corrections of the second, third, and fourth classes are so small for the whole base that their probable errors are neglected. § 22. The first question is as to the accuracy of the values of tube-lengths used in the com- putations. The probable errors in the values of J for tubes 1 and 2 as derived by the least-square reductions are given in § 15 as 4. 0".00012 and + 0.00014, respectively. The corresponding prob- able errors in the values of x are + 0'°.000009 and + 0".000008. As the average temperature-range betwecn 8 and 12 a. m. on days of measurement was 13°.2, the average probable errors in the values of ar would be + 0".00012 and + 0°.00010 for the tubes 1 and 2. Hence the probable error in the value of d for tube 1 would be approximately v(0'.00012) 4- (0.00012)? = + 0.00017 , and for tube 2 also + 0.00017. Multiplying the first of these by 967, the number of times tube 1 was used, and the second by 966, the number of times tube 2 was used, there result 0.164 and 0°.164. The square root of the sam of their squares gives for the probable error in the whole base from this cause + 0".232, §§ 20-24.] KEWEENAW BASE. 93 In taking for tube 2 the constant probable error + 0'".00017 multiplied by the number of tubes, the resulting probable error is theoretically overestimated, since it assumes that the actual errors for all tubes have the same sign. But this is partly compensated for by the greater probable error of the few final comparisons. § 23. Alignment was obtained after finding that the mark at the middle base was very nearly on the line through the marks at the ends of the base by setting up a theodolite at the middle of one of the halves of the base very nearly on the line, then reading by several repetitions the angle (very nearly 180°) between the ends of this half and computing the distance the stake must be moved to bring it on the line. Each of these smaller parts was then bisected in the same way, and the process continued till stakes had been set on the line 500 feet apart. By this method it is im- probable that any of those lines could have deviated from the base-line by more than one minute as the instrument used read to 10 seconds, and the probable deviation is much less. As an error of one minute would only introduce an error into that part of the base of sgag4oao7, and as the prob- able error is much less, it is insignificant in comparison with other errors and may be neglected . or, what is the same thing, these stakes 500 teet apart may be assumed to be precisely on the line’ § 24. To place a tube on the line the base-line transit-instrument, with a more powerful tele- scope than that of the theodolite, was set up on one of the stakes and pointed at the preceding and the front end of the tube placed in the vertical plane of the telescope. When the front tube got within 150 or 200 feet of the transit the latter was moved to the next 500-feet stake. The dis- tances from transit to tube varied then from 200 to, 700 feet. The length of the base gives fifty- seven of these 500-feet intervals, and so many times will the distance of the tubes from the transit vary from 700 to 200 feet. The maximum error that could be committed in putting the agate at the end of the tube on the line at the distance of 700 feet may be taken as 0.3, and at the distance of 200 feet as 0.05. From the ordinary law of error, when the maximum errors in fifty-seven obser- vations are respectively 0.3 and 0.05, we have the probable errors of one such observation re- spectively 0.085 and 0'°.014. These values give for the probable deviation from the base-line of a tube at 700 feet from the transit 0.085 /2 10s and at the distance of 200 feet = 0.014 V2 5a, 180 sin 17° In order to estimate the sum of the corrections to the length of the base arising from these deviations, suppose the whole number of tubes which make up the base divided into ten classes of 193.3 tubes each, the first class including the 193.3 tubes which have the smallest probable devi- ation, the tenth class including the 193.3 tubes having the largest probable deviation. Assume 23” as the probable deviation for all tubes of the first class (which is somewhat too small) and 138” (which is somewhat too large) as the probable deviation of all tubes of the tenth class, and the deviations of the intermediate classes of 193.3 tubes to increase for each class by 138" —23"" 9 Assuming the ordinary law of error and considering the first class of 193.3 tubes, we can compute the number of tubes of this class included in each of its ten sections. The first section of the first class is to include the number of tubes for which it is probable the deviation will be less than 23’ x0.4; the second will include the number of tubes for which the deviation should be between _ 23" x 0.4 and 23” x 0.8; the third will give the number of tubes whose deviations lie between 23” x 0.8 and 23x 1.2, and so on, the interval for each remaining section being 0.4, so that the Jast will in- clude the tubes whose deviations lie between 23” x 3.6 and 23’ x4.0. Assuming that the mean of the extreme deviations in each section may be taken as the deviation for all tubes in that section, the number of tubes in the section multiplied by the length of a tube multiplied by 2 sin? } « (where a is the mean probable deviation for this section) will be, when taken with a negative sign, the correction to the length of the base due to the deviations in this section. Doing the same for =12'.8 94 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. III, each section in this class and summing the results, we have the correction to the length of the base arising from the probable deviation of the 193.3 tubes in this first class. Doing the same for the second class of 193.3 tubes, for which the probable error is 35/’.8, and for the other 8 classes, and summing all results, we have —0".069 as the correction to the measured base arising from angular deviation of the tubes from the vertical surface through the base-line. The probable error in this correction is so Small in comparison with other probable errors that it may be neglected. § 85, The inclination of the tubes to the horizon is read to seconds by means of the sector. As the errors in reading would only give exceedingly small corrections, whose signs would on the whole be as often positive as negative, thus being eliminated in the result, they may be neglected. The sector, however, is liable. to get out of adjustment; and this adjustment was therefore exam- ined and the correction determined every few days during the remeasurement, eighteen times in all tor each tube. The maximum correction found was 115’. The error in the adopted value for any inclination will therefore be made up of two parts: first, the error in the determination of the cor- rection; secondly, the error in the assumption that between two such determinations the correction changed uniformly. The correction was determined by bringing the two agates at the ends of a tube into the same horizontal plane by means of a leveling instrument close at hand and then reading the sector. Ke- peated trials gave a range to the values of the correction of less than 20’, and as several deter- winations were made each time the correction was determined, if 5’ be taken as the probable error of the correction it is certainly large enough. If the mean value of the correction as derived from its consecutive determinations were used, the resulting correction to the base would be nearly the same as if the correction was supposed to change uniformly. Taking 5’ as the probable error of the correction for any tube of a step, the error in the horizontal projection of this tube would be 0.000151 if its inclination be taken at 2°. This value, 2°, is more than double the average inclination, and is large enough to give a correction to the horizontal projection greater than the average value. There were thirty-six determinations of the correction, hence we may consider the line as divided into steps whose average length was fifty-three tubes. If the inclination of all the tubes in one of these steps had had the same sign, the corrections to the horizontal projection of these tube-lengths arising from the actual error in determining the index-error would have had the same sign. Assuming this, and in order to overestimate rather than underestimate its effect on the length of the base, assuming that a tube whose. inclination is - 2° will give the average correction, there results for the sum of the errors in the horizontal projec- tious of the fifty-three tubes in this step 53 (0.000151)=0".008. If the inclinations of all the tubes in each step had the same sigu, since the errors in determining the index-error are independent in different steps, there would result for sum of errors in horizontal projections of tubes for whole base of thirty-six steps, 0.008 ¥36—0".048. But instead of the inclivations of the tubes all having the same sign, the two ends of the base differed in elevation by but 2 feet, so that the sum of the ascents was practically equal to that of the descents. Since the error in the horizontal projection of a tube changes sign when the inclination changes sign, it follows that in fact the positive and negative values of the correction nearly cancel each other when sumined, and give a sum that is insignificant and may be neglected in comparison with the 0.04 obtained above, when all inclina- tions in a set were supposed to have the same sign. Error also exists in the assumption that between two determinations of index-error of the sector its change was proportional to the time. There is no way of knowing precisely the amount of error in this assumption, but as the correction was sometimes constant for several days, once for ten days, and as the maximum change between any two determinations was 115”, its effect will certainly tainly be overestimated if a probable error of 10” be attributed to the inclination of any tube from this cause. The actual error in the horizontal projections of tubes is not, as in the preceding case, constant for all the tubes in a set. The sum of errors in projection would therefore be less 10>? .. : a a than (=z) times the 0".048 there estimated, all tube-inclinations in a set being supposed of the. same sign. But since there are approximately as many positive as negative inclinations, and hence §§ 25-28.] KEWEENAW BASE. 95 as many positive as negative corrections, the sum of the corrections to the projections of the tubes 2 will for the whole base be a quantity so small in comparison with (3) 0.048 that it may be neglected. Tt is then concluded that no error needing consideration arises from errors in the adopted incli- nations of tubes. § 26. When at any time, from the approach of night or from bad weather, the measurement was suspended, the terminal point of the measurement was marked by a cut in the head of a cop- per tack driven into the head of a stake, this head being below the surface of the ground as described in § 6. In thus referring the end of the tube to the ground with a good theodolite, experience shows that an error greater than 0'°.03 is impossible. A similar error may arise in placing the rear end of the rear tube over the mark on the copper tack when the measurement is resumed. Assistant Engineer E. 8S. Wheeler is of opinion that the greatest error did not exceed 0.02. There were forty-four references of tube to ground or from ground to tube; and assuming the ordinary law of error to hold, and that the maximum error in such references was 0'".03, we have for the probable error in one reference + 0.009, and the probable error in the length of the base arising from this cause + 0.06, § 27. There is an error entering into every measurement of one tube-length, arising from the pressure of contact between consecutive tubes. This pressure, when reduced to the lowest amount consistent with certainly overcoming the friction of the sliding rod, was still four ounces. Now, supposing a standing tube to be in the right position, when the next tube is brought into contact with it, the pressure between the two forces back the rear or standing tube by a certain ainount, and, of course, the front tube is moved back in contact by the same amount. When the rear tube is taken away, the front tube is relieved from the pressure of contact and moves to the rear by the same amount, and is then the standing tube, but is at a distance in rear of the position it would have had if the contact-pressure had been zero, equal to twice the disturbance produced by the contact- pressure. The amount of this disturbance will vary with the precision with which the movable parts of the trestles fit each other, and with the height to which the tube is raised by the vertical motion of the trestles. From nearly a hundred determinations of the amount of displacement, made by Assistant Engineer E. S. Wheeler and Assistant Engineer Burton in 1872 and 1873, the following mean values have been obtained for disturbance by pressure of contact, that pressure being about four ounces, the trestle being either run up to its full height or run clear down: Trestle up, 0.00034; trestle down, 0.00013. During the remeasurement of the Keweenaw Base the trestles were kept as low as possible, the average being estimated at one-fifth their total range, and hence giving 0.00017 as the average displacement. Doubling and multiplying by 1932 the number of tubes less one, there results a correction to the length of the base of —0.657. Taking into consideration the varying conditions, as cleanness of the movable parts of the trestles, firmness of ground-support, and the difficulty of measuring with precision movements so small as 0.00017, its probable error may easily be 0.00003 or 0*,00004. Taking the latter, we have for the total probable error, 2 x 1932 x 0'",00004.= + 0.155 ; and for the correction to the measurement of the base, —0™.657 + 0.155. § 28. It has already been stated that the length of tube 2 changed between September 11, 1873, and September 30, the value of 1,’ changing from 0'".043135 to 0'".03985. In the computation of the length of this part of the base it has been assumed that the rate of change with reference to the number of tubes measured was constant. The maximum error resulting from this assumption would occur if the change had been instantaneous and had happened on September 11 or on Sep- tember 30. Its amount then would be 490 x0™.001642=0".804. The change probably occurred from the gradual loosening of some of the screws of the apparatus, but in view of the uncertainty that the change was uniformly distributed over the whole period, and of the fact that the probable error of tube 2 from the 1029th tube to the 1933d tube has been taken in § 22 at the same value as prior to tube 1029 (a value undoubtedly too small, though data do not exist for its determination), it is thought that the probable error from these causes may reach one-fourth of 0.804, and 0,201 is accordingly taken. 96 STANDARDS OF LENGTH, BASES, ETO. [Cuar. ILL, §§ 29-33, RESUME. § 29. The probable errors in the length of the base resulting from the various causes are then as follows : From probable errors in lengths of tubes (§ 22) ..---.-.----- -e0- 22 ee 2 eee eee cee eee eee 0,232 From probable error in inclinations (§ 25).----...--- +... eee eee eee cee ee ee eee eee +0in,000 From probable error in reference to and from the ground (§ 26)..--.-.----------------- -L0i7.059 From probable error in contact displacement (§ 27)..--....---------------------------- -0i".155 From probable error in change of length of tube 2 (§ 28)...---..----- 2+. ------ eee eee +0".201 Taking the square root of the sum of the square of these quantities, we have for the probable error in the length of the base 0.349 (error in length of 15-feet standard bar not included). § 30. In § 19 the probable error in one measurement of ninety-four tubes deduced from the discrepancies of six measurements is given as 0'",03. These discrepancies arise from errors in adopted lengths of tubes due either to temperature or to permanent changes in length, from errors in inclination, in alignment, and in references to the ground. They would give for the probable error in the length of the whole base due to these causes + 0'".03 = + 0.136. This does not include the errors in the value of / nor in the estimate of contact displacement, but if these were added the probable error in the length of the base would be less than that derived above. § BI. The mean height above sea level of Keweenaw Base is 662 + 1 feet, giving a correction to reduce to sea level of — 11'".022 + 0'.017. In §§ 24 and 27 the estimated corrections to length of base for errors in alignment and for contact displacement are given as — 0.069 and — 0'".657, respectively. Applying these corrections to the value given in § 20, namely, 1933 (15-feet bar at 32° F.) + 61.420, Keweenaw Base reduced to sea level = 1933 (15-feet bar in melting ice) + 49.68 + 0,349. § $8. Taking for the length of the 15-feet bar in melting ice the value given in Chapter II, § 14, namely, 179.95438 + 0.00012, we have for the length of the KEWEENAW BASE REDUCED TO SEA LEVEL==347901'".50 + 0'.419, § BB. As a check on the parts of the base into which it was divided near its middle, the angles of two triangles having a common vertex, and the parts of the base for opposite sides, were read, excepting those at the middle of the base. Computing the north half of the base from the whole base, the computed length was 1.3 less than the measured length, while the computed south half was 1.3 greater than the measured length. As the angles of this small triangulation were not well measured, the agreement is as good as was to be expecteél. An approximate reduction of the 1867 measurement of the Keweenaw Base in Note-book S. 288, gives 347,904.154 American inches for the length of the base not reduced to the sea-level. The corresponding value derived from the remeasurement as follows from the data given above is 347,912.52 British inches. Of this correction to the older value, namely, + 8.37 inches, + 12.18 inches arises from the change in value for length of the 15-feet brass bar at 62°. It was formerly not very accurately known in terms of the uncertain American yard (which has now properly been dropped) as 180.00166 Ameri- can mches at 62°, (See Lake-Survey Report, 1868.) It is now accurately known in terms of the British standard yard of the Ordnance Survey, its length at 62° being given in Chapter II, § 11, as 180.00796 British inches. The rest of the correction to the older result, namely, — 3.81 inches, arises from differences in the two measurements, and in the methods of reduction. Cuap. IV, §§ 1-4.] MINNESOTA POINT BASE. 97 CHAPTER IV. MINNESOTA POINT BASE. LOCATION—MARKINGS—MEASUREMENTS. § 1. Minnesota Point base-line is on a sand-spit which separates the western end of Lake Superior from Superior Bay. This spit is narrow, its greatest width not exceeding 700 feet, and its highest points, which are sand-hills, are not more than 15 feet above the lake. The length of the base-line is, approximately, 19,871 feet. The north end of the base is in latitude 46° 45’ and in longitude 92° 05’. The ends of the base are marked by crosses on brass plugs inserted in heavy stones, whose tops are 2 feet below the surface of the ground. Near each end of the base three stones similar to the marking-stone, and 15 feet from it, were set, one in the prolongation of the base-line, and one on either side at right-angles to the base-line, in order to check any disturbance of the principal stones. The dimensions of the stones are 3% by 1* by 1%. The ground is 2 feet above the lake at North Base and 33 feet at South Base. The base was measured under the direction of Captain (afterward General) G. G. Meade, with wooden rods, as a secondary base, in 1861. It was measured in 1870 as a primary base by General C. B. Comstock, aided by Lieutenant J. H. Weeden and Assistant Engineer E. 8. Wheeler, with the Bache-Wiirdemann base-apparatus of the Lake-Survey. § 2. Comparisons of the measuring-tubes with the 15-feet standard bar extended from July 22 to August 9 before the measurement of the base, and from August 31 to September 2, 1870, after the measurement of the base. The measurement was made between August 13 and August 31, 1870. The number of tubes measured was 1,325, each being approximately 15 feet in length. The greatest speed of measurement was on August 27, 1870, when 200 tubes were measured in 84 12™, giving a speed of 366 feet per hour. This great speed was only possible from the fact that the soil was of clean sand, on which the trestles that support the tubes could be rapidly placed, and were immediately stable. METHOD OF MEASUREMENT—DETERMINATION OF TUBE-LENGTHS. § 8. A description of the method of using the Bache-Wiirdemann base-apparatus has been given in Chapter III. Accordingly, it will only be necessary here to refer to points in which its use at the Minnesota Point Base differed from that at the Keweenaw Base. 1. In the measurement of the Minnesota Point Base the feet of the trestles rested usually on the soil (sand) without the intervention of supporting stakes. 2. The tubes in ineasurement had no tent over them to protect them from the sun. 3. The pegs over which the theodolite was set to give alignment to the tubes were about 1,000 feet apart. The experience gained on this and the Fond du Lac Base led to modifications of methods for the Keweenaw Base. s § 4. The comparisons of the tubes with the standard 15-feet bar, to determine the lengths which they actually had during the days of measurement, were made in a house at Superior City. The methods were similar to those used at the Keweenaw Base, described in Chapter III, save that the standard bar was in air instead of being packed in ice, and had its temperature determined by mercurial thermometers lying in its box. As there were wide fluctuations of temperature in the room used for comparisons, it could not be assumed that the bar and the thermometers, whose masses differed widely, would have very nearly the same temperatures, except at the time of maximum or minimum temperature of the bar, which time would be indicated by its length becoming a maximum 13 LS 98 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. IV, or minimum. Since no comparisons at minima-temperatures were made, the comparisons used in determining the lengths uf the tubes were those in the immediate vicinity of maxima-temperatures. § &. In Chapter III the method has been explained of determining for each day of measure- ment the quantity, +ar, by which the mean length of a tube for that day exceeds the length of the 15-feet brass bar packed in melting ice. In /-+avr, Lis the excess of the tube’s length, both of its component bars having the same temperature over the length of the 15-feet bar in ice, and varies for each base, while xv is the change in the length of a tube arising from difference of tem- perature of its bars, caused by a change of temperature of 1° F., between 8 a. m. and 12 m., and is a constant for each tube, derived fromm comparisons, and a is the temperature-change from 8 to 12 a. in. on the day of measurement. For tube 1, r—0'".000136-£ 0.000009. For tube 2, 2=0".000097 + 0.000008. (See Chapter ITI, § 15.) The temperature for the value / was not defined for the Keweenaw Base, as the length of a tube at its measurement was practically constant for all temperatures so long as its component bars were of eqnal temperatures. But early in July, 1872, prior to the measurement of the Fond du Lav and Keweenaw Bases, and subsequent to the measurement of the Minnesota Point Base, the compensating lever of tube 1 was changed so as to diminish the expansion of this tube by 0.'°00020 for 1° F. Tube 2 remained unchanged. In Chapter III, § 9, it is stated that the expansions of both tubes (the bars in each being at the sane temperature) are now very nearly zero; hence at the measuremeut of the Minnesota Point Base the expansion of tube 1 was 0.00020 for 1° F, The amount of this change in coefficient was computed from careful measurements of the changes in the knife-edges of the compensating-lever. Hence, in speaking of the length of tube 1 for the Minnesota Point Base, it is necessary to define its temperature. Its length when both bars are at 62° F. will be called its normal length. § 6. In reducing the comparisons at Minnesota Point of the 15-feet brass bar with the tubes, the comparator-readings on each were plotted as ordinates, the times being abscissas. The portions of ordinates intercepted between the curves for a tube and for the 15-feet brass bar near the time of maximum bar-length gave differences of length between bar and tube at known temperatures. Using the expansion 0.00020 for 1° F. for tube 1, 0".00000 for tube 2, and 0'.001786* for the brass bar (§ 11, Chapter L,) since the reductions were from near 62° to 32°, the computed values (tube at 62° F.)—(15-feet bar at 32° F.) were obtained. These values are given in the following table: Maxima-Comparisons at Minnesota Point Base. Tube at 62° minus bar at 32° as de- rived from com- parisons at maxi- imum length of bar. Increase of bar- temperaturein 36 before maxi- tioum length of bar. | Time of maxi. Cotrected temper- 2 » ature at maxi- Date. (Tube. Te mum length of | | bar. | 1870. | hom. oF. oF. in, Aug 4) 1] 52pm | 74.6 +0.5 0. 02076 Aug. 9] 1 5 39 p.m. 67.2 +0.9 0. 02118 Sept. 2 | 1, 5 41p.m. BO se aa enim aewasaes 0. 01895 Aug. 6) 2 | 5 30pm. 71.2 +0.9 0. 03867 Aug. 31) 2 4 30 p.m. 55.5 0.0 0. 03821 Sept. 1] 2 1 45p.m. 55.3 | +1.0 0. 03804 Tt will be noticed that the comparisons of September 2, 1870, give a less length for tube 1 than the preceding ones. The temperature-increments in three hours on August 4 and August 9 are not large, aud to give the normal Jength the observed lengths would not need a large negative correction. But a large negative correction would be needed to give the result of September 2. Unless, therefore, the temperature curve on September 2 was such as to give a length mueb less than the normal length, a supposition that is quite improbable, we must conclude that tube 1 changed its length between the comparisons of August 9 and September 2, 1870, and that its length before measurement must be determined from the comparisons of August 4 and August 9, and its * The provisional value (0'".001787) was used instead of U,0017#6 in the computation of the table. The effect on the length of the base, however, is so slight that the original computation has been used. §§ 5-7.] MINNESOTA POINT BASE. 99 length after measurement from those of September 2, 1870. This tube has frequently changed length during the measurement of bases, the cause undoubtedly being the loosening of the numerous screws which combine its parts. The comparisons of tube 2 indicate no change in length during measurement, and hence the mean of their results is taken to determine the normal length of this tube during the whole measurement. The mean results of the maxima-comparisons are, then, as follows: ‘rom comparisons of August 4 and August 9, 1870, before measurement of base— Tube 1, at 62° F.=15-feet bar at 32°+ 0.02097. From comparisons of September 2, 1870, after measurement of base— Tube 1, at 62° F.—15-feet bar at 329+ 0'".01895. From comparisons of August 6 and 31 and September 1, before and after measurement— Tube 2, at 62° F.—=15-feet bar, at 32°4 0.03831. § 7. These results include the effect of the difference of temperatures of the component bars of a tube, on the tube’s length, while the quantities needed in computation of the base are the normal lengths at 62°; that is, the lengths of the tubes when the component bars in each tube are each at 62°, The data fur computing the corrections to the above results to reduce them to normal lengths are not satisfactory, but an approximate estimate can be made of their values. At the Keweenaw Base the tubes were compared with the 15-feet bar when at its maximum lengths, and the normal lengths of the tubes were also determined. The differences are quantities which, applied to the lengths of, the tubes derived from maxima-comparisons, would give their normal lengths. Now, if the daily temperature-curves during maxima-comparisons at Minnesota Point Base had been identical with those of the Keweenaw Base maxima-comparisons, the difference between maximum and normal lepgths would have been the same for the two bases, and the correction above referred to could be at once applied to the results of the Minnesota Point maxima-compari- sons to give the normal lengths of the tubes for that base. Unfortunately, the temperature-curves for the Minnesota Point comparisons are unknown, except for two or three hours before the time of maximum length of the bar, and hence, in the absence of other information, we have to take these portions of the temperature-curves for comparison with those of the Keweenaw Base, in order to estimate the excess of length of a tube over its normal length. The following table gives the results of comparisons of tubes with the 15-feet bar at maxima- temperatures at Keweenaw Base. The expansion of the brass bar between 32° and 62° is 0.001786 for 1° F., as given in Chapter II,§ 11. It also gives for each day the temperature-increase in three hours before the maximum length of the 15 feet bar, and the excess of length of tube found from each day’s maxima-comparisons over the normal lengths of these tubes, which are for Keweenaw Base (Chapter III, § 16), Tube 1—15-feet bar at 329+0'.01701 +0".00012. Tube 2—15-feet bar at 32°+0".04313 + 0".00014. Maxima-comparisons at Keweenaw Base. Tube at 62° minus Increase of bar : ax: a Time of maxi- Corrected temp- temperature ip bar at 32°, as de- Tube-length Dat Tub length erature at max”) 3h before maxi- | TY ed from com- minus nor- ates ane: aE a ene imum length of mum length of parisons at max- mal leneth Okan: bar. bar. 8 imum length of ene . bar. 1873. he om. oF. oF. in. in. 7 Aug. 1! 1 6 08p.m. 64.8 +3.4 0. 01860 +0. 00159 Aug. 2} 1 4 37p.m. 64.5 +1.3 0. 01791 +0. 00090 Aug. 9/ 1 5 09 p.m. 63. 6 +11 0. 01746 +0. 00045 Oct. 13] 1 3 45 p.m. 54.8 +2. 3 0. 01903 +0. 00202 Oct. 14) 1 1 19p.m. 53.0 +5. 4 0. 01831 +0. 00130 : Means, || cvnexraccoaceecs sulhaceseegenesactdiee 0. 01826 +0. 00125 July 25} 2 6 03 p.m. 72.0 +11 9, 04292 —0. 00022 July 26) 2 4 55 p.m. C98 isicraretcs Sewretsrnisaisieis 0. 04349 -+0. 00035 July 29| 2 6 22p.m. 68.9 +2.3 0. 04348 +0. 00034 July 31) 2 6 00p.m. 61.8 +0. 4 0. 04338 +0. 00024 IMGAME: | xasceddoosicoisteria sel ee eeraneemeteses 0. 04332 +0. 00018 100 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IV, § 8. From this table the increments in temperatures for three hours before maximum at the Keweenaw comparisons are taken for each tube and plotted as abscissas, the corresponding excesses of tube-lengths over normal lengths being plotted as ordinates. A broken line will thus be obtained which gives roughly the relation that these quantities bear to each other. They indicate that gen- erally the excess of length of tube over its normal length increased with the increase of tempera- ture in three hours before comparison, but with one largely discrepant result, that on October 13. The morning of this day was rainy or cloudy, but it cleared in the afternoon. There was the rare phenomenon of a minimum in temperature at 10.a.m. Usually the tube reaches its maximum length from one to two hours before the maximum temperature of the bar, and then sinks to its normal length one or two hours later. On this day it did not reach its maximum length till forty- five minutes before the bar-maximum, and so had not yet nearly approached its normal length. As it is an average temperature-condition that is needed, this day’s comparisons might be omitted. In that case, a right line whose slope is 0'°.00025 per degree, and which cuts the ordinate of zero temperature-change at+ 0.00038, will represent the observations for tube 1. If the result of October 13 be retained, leaving the slope unaltered, the quantity 0.00038 would become 0.00048. Not to exclude any comparisons, this value will be used. These results seem to be the best that can be obtained, and may be expressed as follows: When the temperature of the bar does not change in three hours before the time of its maximum length, the average excess of the length of tube 1 over its normal length is 0.00048. This excess increases by 0'",00025 for each degree’s increase in temperature in the three hours. Plotting from the same table the corresponding quantities for tube 2 at the Keweenaw Base, we obtain a line whose slope is 0.00006, and which cuts the axis of ordinates at+0".00007. In the absence of better data, these lines, which are only rough approximations, must be assumed to represent the relations between the same quantities for the Minnesota Point comparisons. From the table in § 6 we have, from comparisons at Minnesota Point Base on August 4 and 9, 1870, before measurement, mean excess of length of tube 1 over 15-feet bar at 32°==+ 0.02097, while the mean temperature-change in three hours before maximum was 0°.7. Hence, the correc- tion to be applied to+0".02097 is—0".00048—0.7 (0.00025) ——0'.00065; and hence, before meas- urement— Normal length of tube 1 at 629 F.—15-feet brass bar at 32° F.+.0'.02032. For the comparisons of September 2, 1870, there are no data for determining the temperature- increase in three hours before naximum; accordingly, the increase and correction are taken the same as in the comparisons before measurement. Subtracting the correction 0.00065 from 0.01895, there results for value after measurement, : Normal length of tube 1 at 62° = 15-feet bar at 32° + 0°.01830. In tube 2 there is no indication of change of length duriug measurement, hence the mean of all comparisons is taken from the table in § 6. The mean value for (tube 2 at 62°) —(15-feet bar at 32°) is 0.03831, and the corresponding mean temperature-change in three hours is 0°.6. Hence from values previously given the correction to reduce to normal length at 62° is —0".00007— 0.6 (0.00006) =— 0°.00011, giving— Normal length of tube 2 at 62°—15-feet brass bar at 32° + 0",03820. ESTIMATE OF ERRORS IN ADOPTED LENGTHS OF TUBES. § 9. The errors in these values of the tnbe-lengths are those which arise in the operation of comparing; from the errors in the adopted expansions of the standard bar and tubes; in the ob- served temperatures; and in the estimate of the correction to be applied on account of the unequal temperatures of the component bars in each tube to reduce the mean of the results of maxima- comparisons to normal lengths. The observation-errors in comparisons are insignificant in com- parison with the uncertainty in the last-named correction, as the comparators, whose screws were well determined, read to 0°.00001, and hence may be neglected. The mean temperature of the tubes during comparisons differed but little from 62° F., and hence any small error in their expan- sions would have little effect on their mean lengths at 62°. The same is true of the eftect of error in the expansion of the 15-feet bar in reducing its length to that at 62° F. Its length at 32° F. depends on that at 62° F., and the effect of error in its expansion is included in the probable error §§ 8-10.] MINNESOTA POINT BASE. 101 of its length at 32°, given in Chapter I, §11. There remain for consideration the errors in the observed differences of lengths of tubes and 15-feet bar, due to uncertainties in the observed tem- perature of the latter, and the errors in the corrections applied to reduce the observed lengths of the tubes to their normal lengths. The first error arises from the fact that the four thermometers lying on the 15-feet bar did not have precisely the temperature of the bar. But as no comparisons were used save those when the bar had its maximum length, at which time its change of tempera- ture as indicated by its change of length did not exceed 0°.3 F. in a period varying from half an hour to au hour, while the change of the thermometer-indications for the same time did not exceed 0°.4 F., the probable error in the observed temperature at comparison nay be taken as not exceed- ing +0°.1 F., corresponding to a change in the length of the 15-feet bar of £0'.00018, But whatever the actual errors due to this cause, as they arose in the same way at the Minne- sota Point and Keweenaw comparisons, and enter the normal length resulting from the Minnesota Point comparisons, first, directly with one sign; and, second, indirectly through the correctiou applied to give normai length with the opposite sign, the resulting error from this cause in the normal length must be a fraction of 0'°.00018. § 10. Considering now the probable error in the corrections adopted to give normal lengths, it may be remarked that they might have been obtained by other methods. 1. The consideration of the temperature-cur\ es on days of maximna-comparisons at Keweenaw and Minnesota Point Bases might have been omitted, and the assumption might have been made that the correction 1eeded to reduce the mean length of a tube derived trom maxima-comparisons at Keweenaw Base to its normal length could be applied unchanged to the results of maxima-com- parisons at Minnesota Point Base te give the normal lengths of the tubes at that base. This method would have given a correction for tube 1 of —0".00125 in place of —0".00065, and for tube 2 a correction of —0".00018 in place of —0*.00011. In reference to the relative values of the corrections obtained by the two methods, it may be said there can be no question that on a day when the temperature-curve has its ordinary form the needed correction increases with the amount of temperature-increase in the three hours preceding the maximum length of the 15-feet bar, and, therefore, the results obtained by taking into account this increase should be the more accurate. 2. To obtain these corrections still another method might have been followed. Instead of using all.the days of maxima-comparisons at Keweenaw Base, as in the preceding paragraph, only those days (August 2 and 9 and July 25 and 31) might have been used on which the temperature- increase in three hours before maximum length of bar was small, not exceeding 1°.3; since at the Minnesota Point comparisons this change did not exceed 1° on any day, and hence, so far as our information goes, the temperature-conditions were very much alike. This process would give a correction of —0".00067 for tube 1, and —0'.00001 for tube 2, quantities which do not differ widely from the values —0".00065 and —0.00011 already adopted. Indeed this method might well have been used. Notwithstanding what has been said, the assignment of probable errors to these corrections, in the absence of sufficient data, must be a matter of judgment, and so subject to large uucertain- ties. The least-square reduction of six days’ comparisons of tube 1 and five days’ comparisons of tube 2 at the Keweenaw Base gave probable errors of +0".00012 and £0".00014, respectively, in their normal lengths. With fewer days of comparisons and more uncertain methods the probable, errors at the Minnesota Point Base must be much greater. Tube 1 was compared on two days before the measurement of the Minnesota Point Base and on one day after, while tube 2 was compared on one day before and on two days after the measure- ment. In view of the small number of days of comparisons and of the uncertainties in the method of obtaining the corrections adopted, it is thought that the probable errors, arising mainly from these causes in the adopted values of the normal length of tube 1, may reach +0”.00040, and +0'.60050 before and after measurement respectively, and that the probable error in the normal length of tube 2 may reach + 0.00035, the standard 15-feet bar being supposed exact. The values for the normal lengths of the tubes at 62° are, then, as follows: Tube 1 before measurement = 15-feet bar at 32° + 0.02032 + 0.00040. Tube 1 after measurement=—=15-feet bar at 32° + 0.01830 + 0.00050. Tube 2 before and after measurement=15-feet bar at 32°+ 0.03820 + 0.00035. 102 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnap. IV, It is assumed that the change in the length of tube 1 was gradual during the whole measure- ment and proportional to the number of tubes measured. The Jength of the base was 1,325 tubes, but as the last 124 tubes were remeasured the total number measured was 1,449. At the 1325th tube, therefore, +225 of the whole change would have taken place, and its length than was, 15-feet bar at 329-40".01847. As the remeasurement is not included in determining the length of the base, there results, mean length (between Ist and 1325th tubes) of Tube 1 during measurement=15-feet bar at 32°--0".01940 + 0.00038. The quantity --0”.00038 is nearly the average of the probable errors, which are 0'.00040 and 0".00050 at beginning and end of measurement, and at the middle 4 V (0.0004)? (0".0005 P=0".00032. To take this value for all the times the tube entered the base as constant is theoretically to over- estimate the resulting probable error if obtained by multiplying it by the number of times tube 1 entered the base. For, to take the probable error of all the tubes as constant assumes in effect that the actual errors in the computed lengths of all tubes intermediate to the comparisons have the same signs, which is not necessarily true. COMPUTATION OF LENGTH OF BASE. § Li. Having obtained values for the normal lengths of the tubes, the mean length of a tube for any day of measurement can be obtained by the method explained in § 16 of Chapter III. But as tube 1 had at the Minnesota Point Base an expansion of 0.00020 for 1° F. each tube-length for this tube will need a correction (¢— 62) 0".00020, in which t is its observed temperature. These corrections can be summed and applied in mass instead of being applied to each tube- length for tube 1. Tube 1 was used in the measurement 663 times, and tube 2 was used 662 times. Multiplying their normal lengths by these numbers we ie 1,325 (15-feet bar at 32° F.)4+38".151 £0".342, This quantity is to be corrected by the following sums whose origin has been explained in Chapter IIT: 1. Correction for inclinations of tubes == — 21,854 + 07.000 2. Correction for errors in sector-level adjustment =— 0.013 + 0.000 3. Correction for expansion of tube 1 =-+ 0°.652 + 0.033 4. Correction for contact-level =— 0,021 + 0.000 5. Correction for cylindrical surface =-+ 0.039 + 0.000 6. Correction for ar =—=+ 27154 + 07.111 7. Correction for backward pressure =— 0,622 + 0°.106 8. Correction for errors of alignment =— 0.179 + 0.000 - 9. Correction for reference of ends of tubes to t= ps 0".000 + 0.047 10. Correction for change of length of tube 1 = 0.000 + 0,182 Sum = — 19'.844 4+ 0.245 The manner in which the quantities in this list and their probable errors have been obtained is as follows: The first quantity is the sui of the corrections needed to reduce the lengths of the tubes in measurement to their horizontal projections. The errors in this quantity arise first from errors in the observed inclinations, and second in their computed corrections. The inclinations can be read with a probable error of a few seconds, and as the corrections on account of errors in reading would on the whole be as often positive as negative, these minute corrections are not cumulative and their stun, which under the usual law of error should be zero, may be neglected. But errors in the adjust- ment of the sector affect the inclinations of many tubes, with the same sign, and so are cumulative when the inclinations of all tubes have the same sign. The sector of tube 1 was found out of adjust- ment only once during the measurement. It was in adjustment at the 1011th tube, but needed a correction of + 50” at the 1325th or last tube. Tube 2 was in adjustment at 2d tube and needed 36” correction at the 164th; was in adjustment at the 166th and needed 0” correction at the 2 272d; was in adjustment at the 274th and needed — 67” correction at the 508th; was in adjustment at the 510th and needed + 22” correction at the 634th; was in adjustment at the 636th, and needed + 30” correction at the 1010th; was in adjustment at tie 1012th, and needed + 10” acrrestion at § 11.4 MINNESOTA POINT BASE. 103 the 1324th. The observed inclinations have been corrected by aid of the assumption that the changes in these errors between two adjustinents of the sector-level were proportional to the number of tubes measured in the intervals. The sectors were adjusted by bringing the two agates which form the ends of a tube into the same horizontal plane by the aid of a leveling instrument, and then, the sector-are reading zero, by bringing the bubble of the sector-level to the middle of its scale. Repeated adjustments in this way give a range in the values of the corrections of less than twenty seconds, and we may take the probable error of adjustment as five seconds. Assuming five seconds as the probable error in any adopted inclination from this cause (which overestimates the probable error in intermediate tubes), and assuming that the average correction to the horizontal projection of the tube occurs when the tube has an inclination of 1° 30! (which is also an over- estimate, since but 63 tubes had an inclination exceeding 1° 30’), we have for the probable error in the reduction of one tube’s length of 180 inches to the horizontal plane, + 0'".000114. As the adjustments of each tube were examined seven times, there are on an average, a = 110.4 tubes (whether No. 1 or No. 2), measured between adjustments. The probable error in one such set will be + 0°.000114 (110.4), and in the whole base + 0.000114 (110.4) /6x2—= + 0'.044, provided, as previously assumed, that all inclinations and hence all corrections in a set of 110 tubes have the same sign. But, in fact, the two ends of the base differ but 6 feet in height, so that the sum of positive corrections must nearly equal the sum of negative corrections, leaving in any case an alge- braic sum of corrections so small in comparison with 0.044 found under the opposite supposition that it may be neglected. An estimate of the error in the corrected inclinations arising from the assumption that the error in adjustment between the adjustments was proportional to the number of tubes measured, is less easy. But since the maximum correction at any time was 67”, aud since the error was zero after every adjustinent, the effect will doubtless be overestimated if the probable constant error of inclination of all tubes between two adjustments to which the assumption is applied is taken as ten seconds. The assumption is made for tube 1 for one set of 157 tubes, and for tube 2 for five sets averaging 120 tubes each. Taking, as before, 1° 30’ as the angle giving an error at least as large as the average error, the errorin the projection of one tube will be + 0.000228. Suppose, as an extreme case, that the actual error of inclination for tube 1 or 2 in each set was constant in each set, and, moreover, that all the inclinations iu each set had the same sign, then for tube 1 the whole probable error would be + 0.000223 (157) = + 0°.036. For tube 2 the whole probable error would be + 0°.000228 (120) V5 —=+ 0.061. Combining these quantities, there would result for the total probable error arising from this class of errors ininclinations- V (0'".036)?+ (0.061)? —+0".071. But, in fact, as the ends of the base differed only 6 feet in height, the number of posi- tive inclinations and corrections would be approximately equal to those of the negative ones, so that the algebraic sum of the corrections would be very small in comparison with the 0".071 found under an opposite supposition, and may be neglected. It is concluded, then, that the probable error in the base arising from errors in inclinations of tubes may be neglected. The second quantity in the preceding list is the sum of the corrections to the lengths of tubes arising from imperfect adjustment of the sector-level. The quantity is very small, and its probable error is insignificant. The third quantity is the sum of the corrections for expansion of tube 1 needed to give its length at temperatures differing from 62°, the expansion being taken as 0.00020 for 1° F. The probable error in this expansion does not exceed 0.00001, which would give a probable error in the third quantity of +0°.033. . The fourth quantity arises from the bubble of the contact-level not being always at the mid- dle of its scale, and is a part of the measurement. Its probable error is insignificant. The fifth quantity arises from errors in the adjustment of the cylindrical surface. It is very small and its probable error is insignificant. The sixth quantity gives the sum of the quantities az, which measure the effects of the unequal temperature-changes of the two bars in a tube, on the tube’s length, for the whole base. Its error arises mainly from the errors in the quantities x. For tube 1, «= 0.000136 + 0.000009, §5, and For tube 2, «= 0'".000097 + 0'°.000008. 104 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IV, 1 1 es Por tube 1, then, the probabie-error of wv is + ps7 # and for tube 2 + 454 «. If[aa] be divided c in the ratio of 136 to 97, the parts of [ar] which arise from each tube will be given. They are. 1".257 aud 0.897. Dividing these quantities by 15.1 and 12.1, respectively, the resulting prob- able errors due to tubes L and 2 are + 0.083 and + 0°.074, respectively, or for both tubes, + ¥(0".083)?+ (0°.074)? = 40.°111, _ The seventh quantity arises from the fact that a tube already in position is slightly moved by the gentle pressure of contact when the next tube is placed in front of and against it, as explained in Chapter III, § 27. The mean of the two values given there for displacement of a tube, namely, 0.000235, corresponds to an average height of the trestles. The total correction is the whole number of tubes, less ove, multiplied by twice this quantity, or 1324 x 2 x 0.000235 = —0".622. As the small displacement is difficult to measure with precision, and as it will vary with the clean- ness of the apparatus, the probable error in its value is taken so Jarge as 3, or + 0.00004, which gives 41324 x 2 x 0.00004 = + 0".106 for the probable error in the quantity — 0°.622. The eighth quantity is the sum of the corrections arising from the fact that a tube was never precisely in the direction.of the base-line. Stakes were carefully set on the line 1,000 feet apart, their deviations from the line not being sufficient to introduce any error that need be con- sidered. To place the.tubes in line the base-line transit-instrument, having a large telescope, was set up over one stake and pointed at the next, and the front agate of the tube was then brought into the vertical plane of the telescope. The least distance of the telescope from any tube was about 200 feet, the greatest about 1,200 feet. At the first distance the maximum error which could be committed in putting the agate in the vertical plane may be taken as 0.05; at the greatest distance it may be taken as 0.5. As the base was 19,870 feet in length, there were twenty sections of about 1,000 feet each in Jength, and so many times will the distance of the tube from the transit vary from 1,200 to 200 feet. The maximum error in twenty observations at a distance of 200 feet being 0.05, by the ordinary law of error the probable error will be £0".0172. The probable devia- tion of a tube at the distance of 200 feet from the theodolite will then be 0.0172 ¥ 2 epee =D + 130sin 17 = 27". In the same way, the probable error in placing the agate at a distance of 1,200 feet from the theodo- lite in the vertical plane through the base-line will be found to be + 0.172, and the probable devia- tion of a tube from that plane will be 0.172 ¥ 2 * * = 180 sin 1’ As the distance of the tube from the theodolite varies from 200 to 1,200 feet, the probable angular deviation of the tube varies from 27/’.9 to 279’. To get an approximate value for the cor- rection due to this deviation, divide the whole number of tubes in the base according to their prob- able deviations into ten classes of 132.5 tubes each, the probable deviation of any tube of the first class being 27.9, that of the second class twice that deviation, and so on. Now, considering the 132.5 tubes whose probable deviation is 27.9, from the law of errors the number of tubes whose deviations lie between 0” and 0.4 (27.9), between 0.4 (27.9) and 0.8 (27.9), &c., can be found. Assuming that the mean of the extreme deviations in each set may be applied to all the tubes of that set, the correction for one tube multiplied by the number of tubes in this set will give approxi- mately the sum of the. corrections of this set. Doing the same for the other sets and summing the results, we have the total correction, a, for the first class of 132.5 tubes whose probable deviation is 27.9. For successive classes of 132.5 tubes the corrections will increase as the squares of the probable deviations. «—0".000466, so that the total correction is 0".179. This correction is always negative. Its probable errur may be neglected. The ninth error arises from references of end of tube to or from the ground whenever work was closed or begun. When, at the approach of night or bad weather, the work was suspended, the terminal point of the measurement was marked by a cut in the head of a copper tack driven into the head of a stake, this head being below the ground. To fix the position of this mark, a theodo- lite was set up ata short distance from the front agate of the foremost tube in a plane through the pS HLTO", §§ 12-14.] MINNESOTA POINT BASE. 105 x end of the agate, normal to the base-line. After leveling the theodolite and pointing at the end of the agate, a mark was made on the head of the tack in a vertical plane through the end of the agate, the tack having been previously placed in the vertical plane through the base-line. In thus reterring the end of a tube to the ground, experience has shown that a greater error than 0'.03 is impossible. There were twenty-two such references of tube to ground (or ground to tube when starting from tack in the morning). Assuming the ordinary law of error, we have + 0".010 for the probable error of one observation when the maximum error in twenty-two observations is 0'.03. The probable error for the whole base will be + 0.01 /22—=+0".047. The tenth error is that which arises from the assumption that the change in length of tube 1 was proportional to the number of tubes measured. The maximum error that could arise from this assumption would occur if the total change in length, namely, 0.00232, had occurred at the first tube measured. In that case, § 10, the adopted length of tube 1 during measurement would have been 0°.01940— 0.01830 —0".00110 too great. As this tube was used 663 times, the total error would be 07.0011 (663)—0".729, giving a length of base that much too great. But as this change occurred probably from the loosening of screws connecting the different parts of the apparatus, a process which would be gradual, the resulting probable error in the length of the base will not be underestimated if we take it as one-fourth of the maximum possible error, or as + 0.182. § 12. Summing these corrections and taking the square root of the sum of the squares of their probable errors, for the probable error of their sum, we have, when this sum-correction is applied to the length previously obtained, namely, 1,325 (15-feet bar at 32° F.) + 38.151 + 0.342, Minnesota-Point Base = 1,325 (15-feet bar at 32° F.) + 18.307 + 0".421., This length is to be reduced to sea-level. The mean height of the base-line tubes during measurement, above mean tide at New York, was 613 feet, with a probable error of about one foot. The correction to sea-level is then—6".993. Applying it, there results— Minnesota Point Base reduced to sea-level = 1,325 (15-feet bar at 32° F.) + 11°.314 + 0°.421. From § 14, Chapter II, we have the length of the 15-feet brass bar at 32° F,=179".95438 + 0.000120. Hence, MINNESOTA POINT BASE REDUCED TO SEA-LEVEL = 238450".867 + 0°.450 in which, however, the computed probable error may be too small, on account of the uncertainties of the comparisons and in the method of obtaining the normal lengths of the tubes. CHECK ON PARTS OF BASE BY TRIANGULATION. § 1B. A point was selected near the middle of the length of the base-line and the angles of the triangles formed by this point, the North and South Base stations at ends of the base and station Superior, were read. This enabled us to compute the length of either portion of the base from the whole length. The measured length of the north part was........-.. i Venn ese a6 114, 277".822 Length computed from whole line ......-.-.--+-+-+2e+ eee eee eee 114, 278.828 Difference......... piece e eee cee ee eeeeee evG.oies Sue oe ate aiecs —0'".506 The measured length of the south part was. .......-..222 +++ ++ 124, 173".046 Computed length from whole line ......--2+----+se0-- ee eee ree 124, 172.539 DifferenGe co.cc cca e se pe es ween Usieee de ee cee ee wees eee +0°.507 MINNESOTA POINT BASE COMPUTED FROM KEWEENAW BASE. § 14, From the length of the Keweenaw Base, and from the adjusted angles of the triangles between the Keweenaw and Minnesota Point Bases, all the sides of these triangles can be com- puted, including the Minnesota Point Base itself. The length of the Keweenaw Base, when reduced to the level of the sea, is given in § 31, Chapter III, as 347,901".50. Computing from it, with 14LS8S 106 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IV, the adjusted angles in Chapter XIV, C, the length of the Minnesota Point Base, there results: Minnesota Point Base computed from Keweenaw Base......--.--+ 238, 448".32 Minnesota Point Base as measured .......----- eee en cee eee eee 238, 450°.87 Measured minus computed value. ....-.---.-...2 ee ee eee eee 4+2'°.55 The distance between the bases, measured along the axis of the triangulation, is, in round numbers, 240 miles. The difference in the measured and computed values may arise either from errors in the measured lengths of the bases or from errors in the adjusted angles, and it is desirable to know, at least approximately, what probable error exists in the computed length of the Minnesota Point Base in consequence of: the probable errors in the adjusted angles. The thorough solution of this question would require that in the process of adjustment the weights of all adjusted angles should be determined, and from them, since the Minnesota Base can be expressed as a function of the adjusted angles and the Keweenaw Base, that the weight of this function should be found. The solution involves a very large amount of tabor, and it is questionable whether the value of the result is commensurate with the cost. For this reason the strict computation of the error to be expected in the computed Jength of the Minnesota Point Base has not been made. But a quantity which that probable error does not exceed can be found. The probable errors of the observed angles are known. Since the observed angles are inde- pendent, the probable error in a length for the Minnesota Point Base, computed with them from the Keweenaw Base, can easily be found. Those angles have less precision before than after adjust- ment; hence, the probabie error in the computed length of the Minnesota Base will be greater when the observed angles are used than when the adjusted angles are used, and we thus have a quantity which the probable error of the base cannot exceed, when it is computed with the adjusted angles. The following method may be used to obtain the probable error in the resulting length of the Min- nesota Point Base if computed with the observed angles, from the Keweenaw Base, taken as exact: — Ao/ Bi Ai» 7 Bs A3 Bn An In a triangulation in which a base a is given, from which, with the observed angles A;, B,, Ao, B,, &c., the side a, is to be computed. we have sin B, sin B, sin B;.... sin B, — : : : " sin A, sin A, sin A3....sin A, or, taking logarithms, log a,=log a+log sin B,+log sin B,.... —log sin A,—log sin A,— &e. Now, when a, B,, B:, . Ar, A,, &e., are all independently observed quantities, where the angles have a common nrobanlé error p, and a ts taken as exact, we have the probable error squared ne = pS) a 6 (log = nf) 6 (log sin A,) of log a = ne) + Ce rt . (een ay on as &e. The square roots of the coefticients of »”, or the rates of change of log sin for change in angle, may be obtained from tables of logarithmic sines. Denote the change of log sin for the angle B, for one second by /,, for B, by (, &e., for Ay by 4, &c.; then the above equation becomes, probable error squared of log a,=+,"(6?)?+21"(4”)¢?, where a, is the required side in the xth triangle. In applying this method, as the adjusted angles differ by a very small quantity from the observed angles, either can be used. Taking the principal chain of triangles between the Keweenaw and Minnesota Point Bases, Chapter XIV, D, it is seen that the probable errors, p, of an observed angle of average weight in the sections east and west of line Split Rock—Detour are £0.58 and +0/’.33, respectively. Computing [2)” (5°)+ +1" (4)] ¢? for each of these sections (in which the numbers from 1 to n refer to the successive triangles in each section), and taking the square root of their sum there results for the probable error of either base computed from the other taken as exact, +54.1 in units Sag) MINNESOTA POINT BASE. 107 of the 7th place of logarithms. Hence, the probable error in the length of the Minnesota Point Base when computed from the Keweenaw Base taken as exact, with the observed angles, is £2.97. Since the adjusted angles are more precise than the angles resulting directly from observation, the probable error of the length of the Minnesota Point Base, computed from Keweenaw Base with the adjusted angles, would be less than 4-2'.97. Comparing this probable error with the 2.55 already found for the difference between the measured and computed lengths of the Minnesota Base, it will be seen that that difference can be attributed to the small inaccuracies in the values of the angles in the chain connecting the two bases. In Chapter XIV, C, the approximate ratios of the probable errors of an observed and an adjusted angle, for the two sections of the triangulation between the Keweenaw and the Minnesota Point Bases are given as 0.61 and 0.60. Ifthe adjusted angles were independent of each other, by using the mean of these two ratios the probable error in the Minnesota Point Base resulting from the probable errors of the adjusted angles would be 1'*.79 in place of the 2'.97 above. 108 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. Y, CHAPTER V. FOND DU LAC BASE. DESCRIPTION. § 1. This base-line is situated about 10 miles southwest from Fond du Lac, Wis., in a gently rolling country with little timber. Its eastern end is approximately in latitude 43° 43’ N. and longitude 88° 29’ W. from Greenwich. The azimuth of the west end of the base from the east end is 78° 15’ west of south, and the distance approximately 24,355 feet. The stones marking the ends and middle of the base are described in Chapter XV, A, §2. The top of the stone marking the east end of the base is 3 feet below ground, and this top is $27.3 feet above mean tide at New York. This elevation is derived from the mean elevation of Lake Michigan above tide, given in the Lake- Survey Report for 1878 as 582 feet, and from three lines of levels from Fond du Lac by river and by railroads respectively to Depere, Sheboygan, and Milwaukee. There is a range in the resulting heights above Lake Michigan of 4 feet and a probable error in the adopted value, 245.3 feet, of about 2 feet. The ground at the west end of the base is 8 feet above that at the east end. The highest point of ground on the line is 33 feet above and the lowest point 1 foot below the ground at East Base. The inclination of no tube exceeded 2°. , This base was measured in August, September and October, 1872, with the Bache-Wiirdemann apparatus of the Lake Survey, already partially described in Chapter III. The measurement was carried on by Assistant Engineer I. 8. Wheeler, aided by Assistant Engineers Clark Olds and C. F. Burton. The method of measurement was essentially the same as for the Minnesota Point Base, given in Chapter IV. The lengths of the measuring-tubes Nos. 1 and 2 were determined by comparisons with the 15- feet standard brass bar of the Lake Survey at the temperature of the air, and at the suggestion of Mr. Wheeler they were made at times when this bar reached its minimum temperature for the day as well as when it reached its maximum temperature. The comparisons of the measuring-tubes with the bar extended from July 29 to August 12, 1872, before the beginning of the measurement of the base; from September 7 to September 11, near the middle of the measurement; and from October 11 to October 18, after its completion, They were made in a small wooden building. The measurement began on August 15, 1872, and closed on October 4, 1,624 tubes, each nearly 15 feet long, giving approximately the length of the base-line. The first 68 tubes were measured twice at the beginning of the measurement and four times more after the close of the main meas- urement, or six times in all. The measuring-tubes were not covered by tents during the measure- ment, and the trestles supporting the tubes rested directly on the earth, which had been rolled after removing the sod, without the intervention of supporting stakes, except for the rear trestles on the west half of the line and for the last three measurements of 68 tubes. The expansions proper of both base-tubes are taken as zero for this base. (Sec Chapter IV, § 5.) The methods of measurement and of comparisons have already been given in Chapters III and IV. METHOD OF OBTAINING NORMAL LENGTHS OF TUBES. § 2. It has previously been stated that comparisons of the measuring-tubes with the 15-feet bar were made at both maximum and minimum daily temperatures for this base. Experience makes it certain that at neither maxima- nor minima-temperatures do the tubes have their normal lengths, i. e., the lengths they have when the brass and iron bars in a tube have precisely the same tempera- ture. The first problem is to find for each tube its normal length frem the comparisons at maxima- and minima-temperatures. Now, the normal lengths of these tubes were found in terms of the brass §$ 1, 2.] FOND DU LAC BASE. 109 bar by an independent process at the Keweenaw Base, and the lengths of the tubes at maxima- and minima-temperatures were also found in terms of the brass bar. Taking for each tube for the Keweenaw Base the mean of its lengths as determined by comparisons at maxima-temperatures and the mean of its lengths resulting from comparisons at minima-temperatures, it was found that the mean of the two differed little from the value found for the normal length of a tube during the period in which there was no permanent change in length of either tube. For the Keweenaw Base, then, if this difference had been applied for a tube to the mean of the len gths derived from maxima- and minima-comparisons, the result would have been the normal length of a tube for that base. For the Fond du Lae Base the normal length of a tube will be found from the mean of its lengths at maxima- and minima-temperatures by applying to that value the correction which would have given at the Keweenaw Base the normal lengths from the mean of the lengths of a tube at maxima- and minima-temperatures. The following table gives the results of comparisons of tubes 1 and 2 with the 15-fect brass bar at maxima- and minima-temperatures at Keweenaw Base. The expansion of the tubes is taken as zero (see Chapter ITI, § 8), and that of the brass bar as 0'.001786* for 1° I’. between 32° and 62°, (See Chapter II, § 14.) With this last value the length of the 15-feet brass bar has been reduced to its length at 32°, . Comparisons at Keweenaw Base. MAXIMA-COMPARISONS. : Corrected temper- Time of max- Tube at 62° ture at max. or * Date. imum length | Tube. au minus bar at of bar. min. length of | 390'F. _ 1873. hom. oF. in. Aug. 1 6 08p.m. 1 64. 8 0. 01860 Aug. 2 4 37 p.m. 1 64.5 0. 01791 Aug. 9 5 09p.m. 1 63.6 0. 01746 Oct. 13 3 45 p.m. 1 54.8 0. 01903 Oct. 14 1 19p.m. 1 53.0 0, 01831 Mean. 60. 2 0. 01826 July 25 6 03 p.m. 2 72.0 0. 04292 July 26 4 55p.m. 2 69.3 - 0.04349 July 29 6 22p.m. 2 68.9 0. 04348 July 31 6 00 p.m. 2 61.8 | 0. 04338 Mean. 68. 0 0. 04332 MINIMA-COMPARISONS. Aug.11] 5 37am. 1 57. 96 0. 01655 Aug. 12 6 06a. m. 1 57.55 0. 01596 Oct. 14 6 06a. m. 1 37. 55 0.01451 Oct. 15 6 24a. m. 1 45.70 0. 01540 Mean. 49. 69 | 0. 01560 July 26 5 18a. m. 2 58. 89 - 0. 04250 . July 30 6 20a. m. a 57.98 0. 04301 Mean. 58.41 | 0. 04275 Taking from this table for tube 1 the mean of the mean maxima- and mean minima-lengths it is seen to be ; Tube 1=(15-feet brass bar at 32° F.)+ 0.01693 In the same way we have Tube 2-=(15-feet brass bar at 32° F.)+ 0.04303 But the normal lengths of these tubes for the Keweenaw base were (Chapter III, § 15): Tube 1—=(15-feet brass bar at 32° F.)+0'°.01701 Tube 2==(15-feet brass bar at 32° F.)-+0".04314 ‘*The misprinted value 0".001787, taken from the last line of page 1127, Lake-Survey Report for 1877, was used instead of 0i".001786. The effect on the length of the base is so slight, however, that the original computation has not been changed. 110 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. V, Hence, to reduce the mean of maxima- and minima-lengths of tube 1 to its normal length the correction 0°.01701—0".01693 — + 0.00008 must be applied to the mean of the maxims- and minima. lengths. For tube 2 the corresponding correction is 0".04314—0".04303 =+ 0°.00011. Applying these corrections to the mean of axima- and minima-comparisons Lor each tube at the Fond du Lac Base, their normal lengths will result. If the mean temperature-curves during maxima- and during minima-comparisons could be shown to have the same form precisely, it might be expected that at maxima- and minima-temperatures the tube-length would differ from the normal length by equal amounts in opposite directions, so that the mean of the mean maxima- and mean minima-lengths might be taken as the normal length. Possibly this method might give results as nearly correct as that followed. The small amounts found for the corrections, namely, 0.00008 and 0.00011, indicate that the assumption would not be much in error. § 3. The following table gives the results of comparisons of tubes with 15-feet bar at the Fond du Lac Base. Comparisons at Fond du Lac Base of tubes 1 and 2 with 15-feet brass bar in air at maxima- and minima-temperatures. [Length of bar reduced to 32° F., taking expansion of bar for 1° F. as 0'",001786.]* Tube 1, (MAxIMa). Corrected Date. Hour. bar-tem- Tae Means. perature. rn 1872. oF. in. in. | July 29 80.12 0. 01893 July 31 79. 00 0. 02000 Aug. 2 «| 70.16 0.01910 Aug. 3/4 57p.m.......... 72, 44 0. 01953 0. 01954 Aug. 3 | O48 p. Mew easexs 77.16 0.01929 Aug. 6| 5 33p.m.......... 87. 00 0. 02005 Aug. 7| 5 46p.m.......... 84. 83 0.01986 |J Sept. 7 | 5 24p.m.......... 83, 85 0. 01994 | Sept. 8/4 12am..........1 70.96 0. 01831 0. 01889 Sept. 8|5 42p.m... .| 69.90 0. 01818 | Sept. 9| 3 56p.m.......... 75. 23 0. 01912 Oct. 11| 5 06p.m.......... 46. 60 0. 01816 Oct. 12) 4 44p.m.......... 52. 55 0. 01870 0. 01857 Oct. 14|5 42p.m.......... 43. 50 0. 01886 é TusE 1, (Mini). July 30 | 67. 94 0.017471) Aug. 1 65. 25 0. 01703 Aug. 3 ---| 59,22 0. 01748 AWTEETE Aug. 5|634am..........| 66.70 0. 01754 Aug. T | 3 28 a. Messcaseuss 71.05 0. OL715 Aug. 8| 6 33a.m.......... 73, 23 0. 01720 Sept. 8| 8 56am.......... 67. 80 0. 01759 Sept. 10|618a.m..........| 61.46 0.01741 ; 0. 01750 Oct. 11|712am.......... 31.19 0. 01645 Oct. 14/7 54am... 30. 32 0.01648 , Quoted? *See foot-note on page 109, §3.] FOND DU LAC BASE. 111 Comparisons at Fond du Lac Base of tubes 1 and 2 with 15-feet brass bar, &c.—Continued. TuBE 2, (MAXIMA). Corrected Date. Hour. bar-tem- Tee Means. perature. : 1872. him. oF, in. in. Aug. 10 | 5 54p.m........4. 74. 83 0. 03844 Aug. 12 | 5 05 p.m.......--. 79.10 0. 03924 ; 9. 03884 Sept. 10 | 5 86p.m........-.. *78, 20 0. 03926 0. 03926 Oct. 15 | 5 34p.m........-. 50. 09 0. 03924 Oct. 16 | 5 12p.m........-. 52. 40 0. 03953 0. 03954 Oct. 17 | 5 30p.m........-. 53. 60 0. 03986 Mean ssc: csise| sacessczescn 0. 03926 * Uncertain. TuBE 2, (MINIMA). , Aug. 10 | 5 50a.m-.......... 67. 75 0. 03812 0. 03804 Aug. 12 | 6 06a.m.......-.. 61. 05 0. 03797 Sept. 11 | 5 20a.m.......... 68. 42 0. 03841 0. 03841 Oct. 16| 6 12a.m.......... 37. 94 0. 03716 ; eer Oct. 18 | 7 24a.m.......... 37. 96 0. 03783 Mean ......0.)0222-00eee2 0.03790 From these tables it will be seen that comparisons were made at three dates, namely, before the beginning of the measurements, during the measurements between the 800th and 801st tubes, and after all measurements and remeasurements had been completed. It will also be seen that the means of the maxima-comparisons at each period show a steady decrease in the length of tube 1, while its minima-comparisons at the first and last periods show the same thing, although the length at the middle period is slightly greater than at the first. The evidence is strong that tube 1 changed length, and hence the comparisons at each period will be taken as fixing its length at that period, and it will be assumed that its change of length between two such periods of comparison was proportional to the number of tubes measured. For tube 1 before August 8, 1872, the mean value of (tube 1—bar at 32° I.) was— From maxima-comparisons ........-..-.--- J Seeds Wb ee eiettis eA yee Mae ee eels 0. 01954 From ‘minima-comparisons: 222s, «<< seeee cose pee cen dese da ce eaineG ca cae ew eeeens secs 0. 01731 Mean ..... Ste daide aging PSAs PORE a ee icles EE SS Eee PER REA ESS eca ee: 0. 01842 Correction given above to reduce to normal length....-.-.-.. 2.2... - +--+ seer eee ee eee +0. 00008 Normal length of tube 1 prior to August 8, 1872= ............-.. 15-feet bar at 32° F. +0.01850 From comparisons between September 7 and 10, inclusive, the mean value of (tube 1— bar at 32°) was— From maxima-comparisons .... 0-2. 026 e eee ee ee ee eee eect cee cee ee ceees 0. 01889 From minima-comparisons .........- ages ciciamans tetas aug terest aceasta eyatinctcuayn Sepa ntery austastianta’s 0. 01750 MCD societies se Sow ite ia ak el led. Pepsin ie 0 a Seo ees Se Lg ie ER, Flee ele a gatas Mae eee 0. 01820 Coi rection to reduce to normal length ......- Sra dab ewe dad Ree geeyeeeck ag elsey wees ee +0. 00008 Normal length of tube 1 from September 7 to September 10=..... 15-feet bar at 32° F, +0. 01828 Fiom comparisons between October 11 and October 14, 1872, the mean value of (tube 1—bar at 32°) was— 3 From maxima-comparisons .......-- itived SebeUs iw aie ae ee ae we eek oa ae mek SE 0. 01857 From minima-comparisous ...... 2.6.22 eee eee eee nee eee eee eee ee ee eee 0. 01647 Me@an: Ss. Gct tosses SoA Sees Gabe wa ae eh eee i ayvasdiees he oad eben 0. 01752 Correction to reduce to normal Jength .... 22... 0.0 -.2. eee eee cee e eee wasn eee Shaew +0. 00008 Normal length of tube 1 from October 11 to October 14= ..-....... 15-feet bar at 32° F. +0. 01760 112 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Crap. V, If the mean results of maxima- or minima-comparisons of tube 2 at the different epochs are examined in a similar way, it will be seen that the maxima-results indicate an increase in length of tube 2, while the minima-comparisons indicate a decrease, the changes in both cases being smaller than the corresponding ones for tube 1. There is no sufficient evidence that this tube changed length during the measurement of the base, and hence the mean of all the excesses of length of tube 2 over bar at 32° F., or 0.63926, is combined with the corresponding mean for minima- comparisons or 0°.03790, giving 0.03858. Applying the correction +0.00011, previously deduced, to give the normal length, there follows for the whole base— Normal length of tube 2 =(15-feet bar at 32° F.)+0'".03869 § &. To obtain the mean normal length of tube 1 for any part of the measurement between the first and second sets of comparisons, the change in the tube’s length for each tube measured is obtained by dividing the amount of change of length of tube 1 between the comparisons by the number of times tube 1 was used in measurement between these comparisons. Supposing that the first time tube 1 was used its length was the same as at the first compari: son, then when used the mth time its increase of length will be (m—1) times the change for a single tube. Finding in this way the length of tube 1, both the first and the last times it was used in a section of the base on which there were no remeasurements, its mean length for that section will be the half sum of the two. cs “=17 -1.8 ~0.8 +0.1 “20:5 +3.2 +3.0 +0.8 +1.0 + WBroa wl rR ow w +0,2 +0.1 +0.4 —0.5 +2.1 +1.2 +0.6 +0.9 +0.2 +0.3 0.0 +1.7 40.9 -1.7 —0.5 159 160 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. Comparisons of 20°" spaces on R1876 and D1876—Continued. Date. Temp. Space 60™™ to 80™"™ on R 1876 minus space 0™™ to 20™ on D 1876. Means. Residuals. 1881. March 26, 12:18—12:36 p.m .. March 26, 4:10—4:24 p.m .... March 27, 10:02—10:25 a.m .. March 28, 9:19—9:38 a.m ... March 28, 12:02—12:20 p m .. ° March 28, 3:58—4:17 p.m... March 28, 7:18—7:38 p.m .... Mean of the 17 means....... Probable error...........+-. One, 42.71 42.91 42, 21 41, 86 42. 06 42. 36 B +1.0 apt “+0.8 +0.8 +14 +1.2 +0.7 +1.2 +0.6 +0.3 0.0 41.5 +0.8 41.1 +14 +1.2 +1.0 +0.3 as ad bo ey coo;yau Fr WO OR FIT OP wOwWwW oO] FPF OF FO +1.0 FOF +1.0 +0.4 +0.6 0.0 +0.5 +0.4 +0.1 0.0 +0.4 [CHar. IX, $5.7. CONSTANTS OF METRICAL STANDARDS AND BASE APPARATUS. 21LS Comparisons of 20°" spaces on R1876 and D1876—Continued. Temp. Spaco 80™™ to 100™™ on R 1876 minus space 0™™ to 20™ on D 1876. Means. . Residuals. May 20, 12:36—12:44 p.m ...- May 20, 4:29—4:37 p.m .....- May 22, 9:05—9:16 a.m ..-.-- May 22, 12:32—12:40 p.m .... May 22, 4:16—4:25 p.m ..-.-- May 23, 9:28—9:36 a.m ....-- May 23, 12:46—12:50 p.m .--- May 23, 4:34—4:41 p.m....-- June 3, 9:16—9:27 a. m..----- June 3, 1439—12:48 p. m .--. oF, 61.2 61.3 61. 45 60. 95 61.1 60. 0 60. 05 + a o | Si -s owns he Bee DOT] FEF OCONM,M Ol] FA we LD | cloo om ow _ anwnoo f]|T ND woo WD —0.6 —1.8 | onwmwnoo — a — a — en ) a oe oO ano o Lal orn —2.0 —0.2 —0. +1.2 +1.3 +0.3 +0.9 41.5 +0.7 +0.6 411 161 162 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. 1X, Comparisons of 20"™ spaces on R1876 and D1876—Continued. Space 80™™ to 100™™ on Date. Temp. ‘21876 minus spaced™ | Means. | Residuals. to 20™™ on D1876. 1879. oF, be be ie June 3, 4:25—4:34 p.m. .----- 64 3 —3.0 1881. March 29, 9:30—9:50 a.m .--.| 42.06 —0. oon —0.1 —0.8 March 29, 12:06—12:23 p.m -.} 42.16 —0. +0.2 -1.1 March 29, 4:10—4:30 p.m-...| 42.16 +40. +0.1 —1.0 March 29, 7:36—7:56 p.m...-| 42.16 : —0. wnnwmnanst Ilan r OAD waawndirowal raw ww March 30, 9:14—9:34 a.m ..-.) 41.51 +0.5 +0. 4 March 30, 12:22—12:43 p.m..| 41.76 —0.2 March 380, 4:22-4:40 p.m....} 41.61 +0.6 March 31, 9:56—10:12 a.m ...| 41.22 $0.3 ate oie Mean of the 20 means..|...-...-- Probable error.....---- al lputs BalbecineMieeaianiza sie dceitts —0.9 seocitemied hee caeg ti eserte swim ecee se +£0.1 §5.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 163 The residuals for the earlier comparisons of the spaces 60™™ to 80™™ and from 80™™ to 100™™ being large, additional comparisons of these spaces were made in March, 1881, and are included in the preceding tables. The expansions of the two scales per degree Fahrenheit per metre can scarcely differ by 1, hence temperatures of comparisons can be neglected. Probable errors of mean results are derived from their differences from individual mean results. Collecting the results and rounding the probable errors to the nearest tenth of a micron, we have— 0™= to 2077 on metre R1876=0™™ to 20™ on decimeter D1876+41+.44£ 04.1 20™™ to 407” on metre R1876=0™™ to 20™™ on decimeter D1876+-0«.6+4 0#.1 40™™ to 60™™ on metre R1876=0™ to 20™ on decimeter D1876—0+.4+ 04.1 60™ to 80™™ on metre R1876=0™™ to 20™™ on decimeter D1876+ 0+.44 0#.1 80™™ to, 100™™ on metre R1876=0™ to 20™™ on decimeter D1876—0«.9+ 0,1 Adding these values, there results— 0™= to 100™™ on metre R1876=5 (0™™ to 20™™ on decimeter D 1876) +1414 0.2 ‘Whence 0™™ to 20™™ on decimeter D1876—+4 (0™™ to 100™™ on metre #1876) —0+#.22+0+.04. But substituting from § 4 the value of (0™™ to 100™™ on metre #1876), namely: zy (21876) —0+.2-40+.06, there follows 0™™= to 20™™ on decimeter D1876 =}; (R1876) —0.26 and hence, from differences of 20™™ spaces on metre and decimeter given above— 0™ to 20™ on R1876=;; (R1876) +141 0=™ to 40™ on R1876=2; (R1876) +1".5 0=™ to 60™ on R1876=2; (R1876) +0#.8 0=™ to 80™ on R1876=-4; (R1876) +1".0 From the method of comparison the probable errors in the differences between the spaces on the metre and that on the decimeter are independent of each other. But in the final values of the errors of the 20™™, 40™™, 60™™, 80", marks on #1876, a part of the error comes from the compari- sons with the 20™™ on the decimeter, and a part from the adopted value for this 20™™ space depend- ing on the same comparisons, so that the errors are entangled, and as they all depend on 0™.0 to 0.1 of R1876, a part of the error in this space will enter also. Denote the 20™™ space on the decimeter by A, the, successive 20™™ space’ on the metre by J, IT, IIT, IV, V, the observed differences of A and 'L, A and IT, &c., by a, f, 7, 6, «, and the actual errors in these observed quantities, or the corrections needed to aie them exact, by a, b, ¢, d, e. The above equations may be written in the form I=A+at+a IT=A+ 6+ ITI=A +y+e - IV=A+sd+d V=A+tete Calling the first decimeter of R1876, D, by summing and dividing by 5 there results— A=1D-L(atBt+y+dte)—% (atb+et+d+te) Substituting in the values of J, IJ, &c., I=1D++4 (4a—8—-y—5—e)+4 (4a—b—c-d—) i TI=1D++ (48—a—-y—6—e)+} (4b—a—c—d-e) [TT=1D++ (4y—a—f—6—e) +4 (4c—a—b—d-e) IV=1D++ (40—a—B—y—e)+4 (4d—a—b—c—e) V=1D+4 (4e—a—f—y—0) +4 (4e—a—b—c—d) (1) XS 164 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cmar. Ix, Or, summing I+IT =2D+1 (3a+33-27—28—2:)+2 (3a+3)—2c—2d—2e) I+TI+ JIT =8D+21 (2a+23+2~—3d—32) +4 (2a+2b+2e—3d—3e) I+II+III+1V=£D4+}( at + y+ d—42)+4 ( at b+ c+ d—4e) ‘Passing now from the actual errors, a, b, ¢, d, ¢, of the comparisons of the space on the deci- meter with those on the metre to their probable errors already given and which may be represented by a’, b’, c’, @, e’, the probable errors due to these comparisons in the different sums are: ( For I: 4h / 16a? +b? 402+ d+" is } For I+IT: Lb A/9al-+90" +4044" +40" For I+JI+ITT: LEN 4a°+4b°+407 49d? 496" Lie I+TI+TI+IV: tha" +b" +e" +d" +16e" But D, whose value is given in § 4, has a probable error of -+ 0.06; hence one-fifth of that error must be combined by summing squares and extracting square root with the first of the above val- ues, two-fifths with the second, three-fifths with the third, and four-fifths with the last. Perform- ing the various substitutions in (1) and (2) and the value of A, there result— B BH 0™™ to 20 on R1876=,, (R1876)+1.1-+£0.09 o™™ to 40™™ on R1876=2, (R1876)+1.5-40.11 0™™ to 60" on R1876=2, (R1876)+0.8+ 0.12 0™™ to 80™ on R1876=¢5 (R1876)+1.040.10 A on decimeter D1876=25 (R1876)—0.3-£0.05 § 6. The preceding work having given the error in the position of the 80™™ mark on metre R1876, the next step was to find the error of the 81st millimeter mark, this being the one used in comparisons with the yard. The error was obtained in two ways, of which the first was by com- paring the space 80™ to 81™ on R1876 with the space 0,95 to 0'".99 on the Troughton and Simms standard inch described in Chapter II, § 3. The metre and inch were both mounted on the iron beam or metre-carriage previously described. The microscopes at a distance from each other of 0™,12 on the microscope-car remained unmoved. The line of collimation was made vertical within 10’. The metre and inch were made level within 3’. Both were so adjusted that when the metre- car was moved under the microscope’s longitudinal thread the longitudinal graduation-lines of inch and metre remained under it. The correction for error in focusing was obtained by numerous readings on millimeter spaces on inch and on metre. The order of pointings was: Microscope 5 at 0.99 on inch. Microscope 6 at 80™™ on metre. Microscope 5 at 0.95 on inch. Microscope 6 at 81™™ on metre. Microscope 6 at 81™™ on metre. Microscope 5 at 0'".95 on inch. Microscope 6 at 80™™ on metre. Microscope 5 at 0.99 on inch. Two pointings of each microscope were thus obtained at each of the division lines. Taking their means, a single value of the difference between the space 0'".95 to 0°99 on inch and the space 80™™ to S1™™ on #1876 results. This was one set of observations. Five sets were obtained at each visit to the comparing-room, and two visits per day were made, one at 9 a. m. and one at 4p. m. In the following table are given the dates of observations, the corrected temperatures of R1876, the results of each s2t in the form 0.95 to 0.99 minus 80™™ to 81™™, and the residuals obtained by subtracticg the individual results from the mean of all of them. 46.) CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 165 Results of comparisons of space 0'".95 to 0'".99 on standard inch and space 80"™ to 81™™ on metre R1876. Thermom.21476| 0'.95 to 0in,99 Date. on metre, minus Residuals. (corrected). | 80™™ to 81™™, 1879. oF Mw B October 4, a.m .......-....-. 68. 22 +16. 8 —0.6 +15. 0 +1.2 +15. 0 +1.2 +17.4 —-12 68, 31 +17.2 —1.0 October 4, p.m ...-...-.----- 68. 22 +16.9 —0.7 +18. 5 —2.3 +16.5 —0.3 +16. 2 0.0 68. 27 +17.0 —0.8 October 7, 0. M cevasecnecnriss 68. 71 +16. 2 0.0 414.7 41.5 +15. 6 + 0.6 +15. 5 +0.7 +15.5 +0.7 68. 81 +17.3 11 October 7, p.m ...-.--------- 69. 01 +16.7 —0.5 +16. 6 —0.4 +18.7 f 2B +15.7 +0.5 69. 21 +15. 2 +10 October 9, a.m .....--------- 70.19 415.7 +0.5 +17.5 | =—1.8 415.7 +0.5 +17.3 -L1 70. 29 +16.5 —0.3 October 9, p. m....---------- 70. 39 -16. 6 —0.4 414.1 +2.1 +16. 6 —0.4 +13.6 +2. 6 70. 48 +15.6 +0.6 October 11,a.m.........---- 71. 87 415.5 +0.7 +17.6 —14 +16.1 40.1 +16. 0 +0.2 71.47 +15.0 +1.2 October 11, p.m .......----- 71. 57 +16.9 —0.7 +18.5 —2.3 +16.8 —0.6 +15.3 +0.9 ‘71. 67 +16.3 -0.1 October 14,a,m ......--.-.-- 71. 57 +16.5 -0.3 +16.7 —0.5 +16.5 —0.3 +18.1 -1.9 71. 67 +15.0 +12 October 14, p.m..........+-- 71. 67 +15. 2 +1.0 +12.5 +3.7 +16.2 0.0 +16.0 +0.2 71.77 +16.1 +0.1 Mean of 51 results .....|...--------.---- ; +16, 2 Probable error....-..--|-------+++-6- oe + 0.11 In Chapter II, § 3, Colonel Clarke’s value of the space 0.95 to 0.99 is given in terms of the space (9.10), which is again given in terms of the Ordnance standard foot, F., at 62°, which is given in terms of the English yard. Deriving the value of the space 0'".95 to.0'°.99 in terms of ‘the English yard from these data, there results— Space 0,95 to 0.99 on inch =07.00111300+ 0¥.00000019 166 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnap. Ix, under the assumption that the probable errors of the hundredths of an inch given by Colonel Clarke are independent. Converting this into microns, with Clarke’s value of the metre (39'°.370432), there results— Space 0.95 to 0*.99=1017+.7-L0#.17 But the comparisons gave— Space 80™™ to 81™™ on R1876=space 0'°.95 to 0.99 on inch—164.210+.11 Hence, finally, Space 80™™ to 81™™ on R1876=1001+.5-L0+.2 at 62° F § 7. The second method of determining the value of the space 80™™ to 81™™ on #1876 was as follows: The value of the space 80™™ to 100™™ on 1876 has already been given in terms of 81876. Each of the four 5™" spaces forming an aliquot part of this distance was compared with the space o™™ to 5™™ on the decimeter. These comparisons gave the value of the space 80™ to 85™™ on #1876. Then the space 0™ to 1”™ on the decimeter, D1876, was compared with each of the milli- meters between the 80™ and the 85™™ on #1876. This gave the value of the space 80™ to 81™™ on #1876. In these comparisons the methods were the same as in comparing the 0™™ to 20™™ space on the decimeter with the similar spaces between 0°" and 100"" on #1876. Each result for difference of length obtained depended on two microscope-pointings at each end of each scale. Six such results were obtained in each visit to the comparing-room, a visit occupying about 20 minutes. Four such visits were made. The mean of the 24 results is taken as the value of the difference between the lengths of the portions of the scales compared. In the following tables of the com- parisons of the 5™" spaces, the first column gives the date, the second the temperatures, the third the resulting difference of lengths, the fourth the mean result of visit, and the fifth the residual or mean of results minus mean result of visit. Results of comparisons of 5" spaces on metre R1876 and decimeter D1876. 80™™ to 85™™ on 1876 minus o»™ te 5™™ on D 1876. Dates. Temp. Means. | Residuals. 1881. oF, be a March 31, 10:19—11:02 a m..ssee-eeee- 41. 42 +2, 98 41.71 +3. 46 March 31, 12:15—12:42 p.m......--4.-- 41.71 +1. 26 +2. 66 —0. 02 42.01 +1. 83 March 21, 2:18—2:36 p.m ..........-.-- 42. 06 +2. 98 +2.16 +0. 48 42.11 +1. 88 March 31, 4:17—4:34 p.m...... Shee sews 42.16 +3. 46 +2. 80 —0. 16 +2. 93 —0. 29 Mean. of 4 Means: ..icccicicsicecnins | sense cusieil seasinovatenocenen +2. 64 PYG GITOR nc axne seeindninescantiodeus peakanescoawdade +£0.11 § 7.) CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. Comparisons of 5°" spaces on R1876 and D1876—Continued. Date. Temp. 85™™ to 90™™ on R1876 minus o™™ to 5™™ on D 1876. : Means. Residuals. 1881. April 1, 9:24—9:42 a.m ....2--2.0--.--- April 1, 11:29—11:45 a. m.....------++-- April 1, 2:18—2:33 p. ml ...-----+-+-+-+ April 1, 4:10-4:27 p.m..........-.-..-- Mean of 4 means ......-.------- Probable error.....--.-.-------- oF. 41. 37 41. 42 41. 42 41. 51 41. 51 41. 61 41. 61 +0.31 0. 00 0. 00 —0. 35 —0. 01 +0. 09 —0. 32 —0. 01 —0. 01 40.34 Date. 90™™ to 95™™ on FR 1876 minus 0™™ to 5™™ on D 1876. Means. Residuals. 1881. April 2, 8:48--9:11 a.m ..---------220- April 2, 11:48—12:07 p.m -.------------ April 2, 2:26—2:45 p.m .--------+--06-- April 2, 4:25—4:39 p. m ...---+----+----- Mean of 4 means. ..-.---s-0---2- Probable errot..-..------------- oF, 40. 72 40, 82 40. 92 41. 02 41.17 41, 22 41. 22 +0. 06 +0.10 —0. 41 40.13 —0, 03 +0.09 —0. 09 —0.13 +0. 38 —0. 16 167 168 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnmar. Ix, Comparisons of 5"" spaces on R1876 and D1876—Continued. 95™™ to 100™™ on Date. Temp. te fae oa Means. | Residuals. D 1876. 1881. oF. “ 7 vs April 4, 9:28-9:44 a.m ..-..-------0+- 39, 52 —1.41 39. 62 —0. 63 —1.02 —0. 01 April 4, 11:24—11:42 a.m ...-....------ 39. 67 —0. 73 39. 77 —1.20 —_—_——| -101 —0. 02 April 4, 2:22—2:37 p.m .....----eeeeeee 39. 87 —0. 68 39. 87 —2. 25 April 4, 4:09-4:25 p.m ........--eeee ee 39. 92 —1.41 —1.09 +0. 06 —_—_———_| —0.98 —0. 05 Meani of 4 meanssiecacs sccccscees hoemaisccwis vl te caweeswanissiice —1. 03 POO Rite a cincwacsccsandeu di loaag sees hidaesedseeeasian +0. 02 Collecting the results and giving to the nearest tenth of a micron the probable errors which have been derived from the discrepancies between the mean and individual mean results, we have— 80™™ to 85™™ on £1876=0™ to 5™™ on D1876+2+.64 04.1 85™™ to 90™™ on R1876=0™ to 5™™ on D1876—0+.0404.1 90™™ to 95™™ on L1876=0™™ to 5™™ on D1876—0",0404.1 95™™ to 100™™ on £1876=0"™ to 5™™ on D1876—1+.0-40+.0 Adding these values, there results after division by 4, 0” to 5°" on D1876=4 (80™ to 100"" on R1876)—0+.4£0+.05 Substituting the value of 80°" to 100"" on &1876, derived from the correction to the 80°" mark and the 100°" mark previously given, this space being 3; (R1876)—1+.2, there follows: Or" to 5°" on D1876=54,5 (R1876)—0#.7 1.04.05 and hence from differences of 5"™" spaces on R1876 and D1876, given above, 80°" to 85" on H1876=535 (R1876)+14.9404.1 85°" to 90°" on B1876=535 (R1876)—0+.7404,1 90°" to 95"" on R1876=34, (R1876)—0#.7-L04.1 95" to 100"" on R1876=34,5 (R1876)—14.740+.1 The probable errors have been derived in the same way as those of the successive double decimeter marks, save that the space 80™° to 100°" on R1876 has itself a probable error. From (1) of § 5 the actual error in the space 80"” to 100"" on R1876 is 4 (4e—a—b—c—A), provided a, b, ¢, d, are now the actual errors of the comparisons of 0" to 20" on D1876 with the successive 20"™ spaces between 0” and 100°" on #1876, or if the same letters primed denote the probable errors, the probable error in the value of the space 80°" to 100" on R1876 will be gM a? +b? +0" +d +166" §7.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 169 in which a’, b’, &e., have already been given. This result is combined with the probable errors in the values of the 5™™ spaces expressed in terms of the space 80"" to 100" on R1876. The next step in finding the value of the space 80°" to 81™ on R1876 was to compare each of the millimeters in this space with the spate 0”".to 1™™ on D1876. The methods and number of comparisons were the same as in the determination of the space 80™ to 85" on R1876. The tables have the same arrangement, and the probable errors have been derived in a similar way. Results of comparisons of millimeter spaces on metre R1876 and decimeter D1876. 80™™ to 81™™ on RK 1876 minus Qum to jun on D 1876. Date. Temp. Means. | Residuals. 1#81. oF, Bw “& i April 5, 10:26—10:40 a. m-.....-.0----- 39, 32 +1. 47 39. 42 HL. 47 April 5, 12:25—12:40 p. mo... 22... 39, 42 +0. 94 1.10 —0.14 39, 42 +0. 84 —_——_—_-—_——_| +0.97 —0.01 April 5, 2:27—2:39 p. m......----..----| 39.47 +1. 41 39. 52 +1. 26 f —————] +0. 94 +0. 02 April 5, 4:13-4:26 p. m.......02202-2-- 39. 62 1. 05 +10. 83 40. 13 +0. 96 £0. 04 Mean of 4 means .......----...2-..--- eater Probable error.....-.--..-.2+-02+0.--- eee 81™™ to 82™™ on R 1876 minus 0™™ to.1™™ on D 1876. Date. Temp. Means. | Residuals, 1881. oF. Mw Bw B April 6, 9:37—9:53 a. m ....-....------- 38, 92 —0,47 +0. 05 —0.79 —0.21 —0. 05 38. 97 +0. 31 —} —0.19 —0. 26 April 6, 12:20-—-12:33 p. m....-.-..-0--- 39. 02 —0. 21 —0.52 —0.42 ~0.94 +0. 10 39. 12 0.63 —0, 44 —0.01 22 LS 170 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Cuar. IX, Results of comparisons of millimeter spaces on metre R1876 and decimeter D1876—Continued. Date. Temp. 81™™ to 82™™ on M1876 mninus om to 1™™ on D 1876. Means. Residuals. 1881. April 6, 2:10—2:21 p. m........-- saeeie April 6, 4:20—4:30 p. m......---++-+-- Mean of 4 means ....--...--.------- Probable error .-....-----+-------+05+ oF, 39.17 39.17 39, 22 —0. 70 —0. 48 —0.45 +£0.07 +40. 25 +0. 03 Date. Temp. 82™™ to 83™™ on R1876 minus om™™ to 1™™ on D 1876. Means. _Residuals. 1881. April 7, 9:19—9:31 a mM .....-.ee.----- Aprii 7, 12:42—12:53 p. m.--------.06- April 7, 2:15-2:26 p. m.......2.eeee- AL 7) S579 80 Be Be oc sc ves es eeewee Mean of 4 means........-.2....22.66 Probable error....... afoiesSiajateldsajaeidinre oh oF. 38. 92 39. 02 . 89,12 39.17 39, 22 39. 27 39, 42 B 41.41 “£0. 99 40. 99 +1. 62 1.15 +0. 58 2.25 +0. 99 +2. 62 +41. 88 11.57 +41. 20 41.15 40. 52 +1 10 +1. 20 +1. 36 1.15 +1. 56 +0. 63 +1. 78 10. 52 11. 99 +1.12 1.75 +1. 08 41.27 +1. 31 +0.10 +0. 19 —0. 44 +0. 23 +0. 04 Date. 83™™ to 84™ on 1876 minus 0mm to 1™™ on D 1876. Means. Residuals. 1881. April 8, 9:09—9;22 a. m..........22.22 oR; 39. 42 ey +0. 10 —0.26 —0.10 0.31 —0. 68 —0. 84 +0. 09 § 7] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 171 Results of comparisons of millimeter spaces on metre R1876 and decimeter D1876—Continued. 83™™ to 84™™ on R1876 minus 0™™ to 1™™ on D 1876. Date. Temp. Means. | Residuals. 1881. oF, Mw & B e April 8, 12:22—12:33 p. m.......--.---- 39. 62 +0. 10 —0. 63 —0.79 —0. 89 —1.05 39, 67 —0. 05 0.55 -10. 29 April 8, 2:21-2:33 p. m..........2226 39.72 —0. 63 +0. 10 —0.31 +0, 21 +0. 37 39. 82 —0.31 —0.09 —0.17 April 8, 4:14—4:27 p. m......--0---.2-. 39. 92 —0.58 —0.16 —0. 26 +0. 05 0. 00 39.97 +40. 79 —0. 03 —0. 23 Mean of 4 means 0.26 | Probable error.......-.--..---- since +0. 08 84r™ to 85"™ on Date. Temp. Ee Means. |} Residuals. D1876. 1881. oF, a be April 9, 9:02—9:16 a. m .-....-- sesso ee 40.17 +1. 26 +0. 89 +0.73 +0. 47 4-0. 84 40.22 11.05 April 9, 12:26—12:38 p. M...e...-2seee 40,42 +0. 47 10. 68 +0. 94 +0. 79 +0. 89 40.47 +0.10 ‘ | 0. 64 +0. 28 +0. 87 10.05 April 9, 1:24—1:37 p. m..-...22--2e- ++ 40. 62 +1. 31 +0. 94 +0. 94 +0. 94 _ +1.36 40.72 —0. 52 : “+0. 83 +0. 09 April 9, 4:15—4:26 p. m..2.-222- esse eee 40. 87 1.4L +1, 62 : .-+H1. 78 hI. 05 -+0.73 40. 92 1.41 +1. 33 0.41 +0. 92 £0.10 Mean of 4 means Probable error....---------ee-e-+e--+ 1i2 STANDARDS OF LENGTH, BASES, AND BASE APPARATUS. [Cuap. IX, Collecting the results we have— 80" to Sle" on R1876 = 0" to 1"™ on D1876 + 0+.96 + 0+, 04 81™" to 82"" on R1876 = 0™ to 1" on D1876— 04.45 4. 04.07 82"™ to 83™ on R1876 =O" to 1” on D1876 + 14.31 + 04.10 83" to 84" on R1876 =O" to 1”™ on D1876— 04.26 + 0.03 _ S84" to 85°" on H1876==0™ to Im on D1876 + 04.92 + 0+.10 Adding these values, and dividing by 5, we have— o~ to 1" on D1876—=+ (80" to 85°" on R1876)—0+#.5 + 04.04 Substituting the value of the space 80"" to es ae in this section, there results— and hence, from the differences between 0" to 1™ on D187 6 and the millimeter spaces on 21876, there follows: 80" to 81" on R18T6 = pop (KR 1876) + 04.9 + 0.05 81™™ to S2™" on R1876 = 7zeeq (R1876) —O+.6 + 0+.07 ( ( 82" to 83°" on R1876—yoo5 (R1876) + 14.2 + 04.09 83"" to 84"" on R1876 = yap (21876) —04.4 + 04.07 84™ to 85™" on R1876 = ygoo (R1876) + 04.8 + 04.09 The probable errors are obtained ‘in a manner similar to that used in obtaining those of the 5™™ spaces just given. § &. Professor Foerster (letter of June 20, 1879) § 67, gives as a closely approximate value— R1876 —1™ + 2484.89 + 104,31 (¢—15) in which ¢ is temperature in Centigrade degrees. This gives— 80°" to 81™" on #1876 = 1,0014.14 04.05 The value found from the inch, § 6, is— 80"™ to 81" on R1876 = 1001+.5+ 0+.2 If the two values were combined with weights depending on the probable errors, the first value would still result. It may therefore be adopted. We now have the data for determining the corrections to certain graduation-marks needed to change their nominal values into exact values in terms of the interval 0™ to 1,000"" on R1876. Collecting them from §§ 3, 4, 5, 7, we have— | Graduation -..! 0| 20 40 60 80 81 82 84 85 90 95 100 | 200 | 400 | 600 | 800 1, 000 Mw “ M : Correction ....; 0 |41.1 |+1.5 +0.8 -0'2 nus -1'9 vor oy 0 : | H Mw M Me M Me “ “1.0 41.9 |41.3 aes +21 |+2.9 ae 1.5 i The probable errors of these corrections resulting from the comparisons have already been given for the most of them. They are usually about + 0+.1, and for none do they exceed + 0+.2. For the 81°" mark, which is a very important one, it has been specially computed and found to be 40.11. The determinations made in the Lake-Survey office of the errors of certain graduation-marks on #1876 in terins of the interval 0" to 1,000", considered as exact, have now been given. This metre was for some time in the hands of the Kaiserliche Normal-Eichungs-Kommission at Berlin, by whom its graduation-errors were also determined. They are given in § 67. These results were not received from Berlin until after the determinations at the Lake-Survey office were far advanced, and the work at Detroit was completed prior to the reception of the details of the work at Berlin, which indeed, August 1, 1881, have uot yet arrived. Accordingly, there are two entirely independent determinations of the errors of some of the graduation-lines on R1876, made at Detroit and Berlin respectively. Their accordance indivates the degree of accuracy of the work. In the following table the first liné gives the name of the graduation-line on R1876; the second gives its correction in terms of the length from 0" to 1,000", as determined by the Kaiser- liche Normal-Eichungs-Kommission, the correction being positive when the graduation-line is too far from 0""; and the third gives the same corrections as determined in the Lake-Survey office. §§8,9.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 173 Corrections to graduations of R1876, the interval between O™™ and 1,000" being taken as one metre. Graduation .../0| 20 40 60 80 81 82 83 84 85 90 95 100 | 200 | 400 | 600 | 800 | 1,000 Eichungs-Amt M M . B M M oe w M o w Mw correction ..|0|+2.4/+2.4/+0.6)—0.5| 0 Oo J+. j44 [41 |+1.0/+1.0] —0.5)—2.5}—1.1)+0.5}/—0.1] 0 Lake - Survey ‘ correction ..)0|+1.1/+1.5/40.8 +1.0]41.9/+1.3/42.5/42.1/42.9)-+42.2/41.5/—0.2]—1.5] -1.0/+0.1|—0.7] .0 Differences... sea ica laps fa nee | ie Sea | eehae | lasers | wtih set orate —0.3|—1.0|—0.1|+0.4/+0.6]...... To the graduation-lines in the table which are multiples of 10™ the Normal-Hichungs-Kom- mission assigns an accuracy of 0+.2 to 0.3 (verbiirgbar), and to the others an accuracy of 1.0 to 14.5. For the Lake-Survey corrections the probable errors do not exceed 04.2. If the graduation- lines whose values are multiples of 20°" be examined, it will be seen that the greatest discrepancy between the independent values is at 80", where it is 1.5. If the means of these values for each of such graduation-lines be taken, and the probable error of one mean be derived from the dis- crepancies between the means and the individual results, it is found to be +0+.27. This process of combination attributes equal weights to the results obtained for these special graduations by the Normal-Eichungs-Kommission and the Lake Survey, although the probable errors of the latter seem to be the less. Since the comparisons of yard A depend on the 8L"™ mark of #1876, its error is of great importance. It must be obtained by taking the mean-of the corrections found at Berlin and iu the Lake-Survey office for the 80" mark, attributing to it the probable error of one such mean, already given, and then adding to it the value of the interval 80"° to 81°", which has been doubly determined in the Lake-Survey office. We thus have— ; 0™ to 80"™" on R1876— 78> (R1876) + 04.2 + 04.27 80°" to 81™ on R1876= 7 gop (21876) + 0+.9 + 0.05 (§ 7) or, adding— : Or" to 81"" on R1876 = , 845 (R1876) + 14.140+.3 _ The probable error here is considerably larger than that which results from the Lake-Survey work alone, namely, £0.11, but depending on entirely independent determinations, the larger value is probably the more accurate. Hence the interval— 81" to 1,000"" on R1876 = yyy (BR 1876) —1+.1 + 04.3 COMPARISONS OF R1876 WITH CLARKE YARD A. § 9. Having thus obtained the value of the space between the 81st and the 100th millimeter- mark on R1876, which space will be designated by “B’187 6,” the details of the comparison of this space with that between the 4™ 6 marks on the auxiliary cylinders (or quills as they have been called) when their axes are in the axis of yard A and their convex ends abut against its ends, may be given. The Clarke yard A was mounted on the iron beam of the comparing-apparatus described in ’ Chapter VII, § 22. The metre R1876 was mounted parallel to yard A ata distance of 30" or 40" on the two supports carried by the iron beam, which were adjustable vertically. The yard A having been adjusted, £1876 could thus be made parallel to it and its graduations brought to the same height as those on the auxiliary cylinders of the yard, nearly enough for microscope-readings. Yard A lay on the flat surface of the iron bar. Two dove-tail slides with spring contacts could be moved along the iron beam so as to admit the yard between them and be clamped to the beam at any place. Each carried wyes 70" apart, in which accurately turned equal rings on the auxiliary cylinders rested. These wyes were separately adjustable vertically and laterally, so that any elevation, inclination, or azimuth, within certain limits, could be given to the auxiliary cylinders with reference to the axis of the yard, and the wyes were so situated with reference to the slides that when the auxiliary cylinders were laid in them they were nearly in the prolongation of the axis of the end-cylinders of yard A when lying on the flat surface of the iron beam. 17 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnap. 1X, The conditions for the yard A and its auxiliary cylinders to fulfill for precise comparisons were, first, that yard A should be horizontal; second, that the axes of its auxiliary cylinders should lie in the prolongation of the axis of the end-cylinders of the yard. Yard A was easily made horizontal within 1/ by the adjustments of the iron bar on which it rested. An examination of the errors arising from not fulfilling the second condition led to the follow- ing results calculated from the most unfavorable case in which the auxiliary cylinders are used, namely, when their convex ends are in contact as in measuring the interval of 4™".93 between their 46th graduations. Since the ends of yard A have less curvature, the errors in using the auxiliary cylinders with it will be less. a. To avoid errors greater than 0.1 in measuring the distance between the 46th graduations on the two auxiliary cylinders or quills, the axes of the quills must have the same azimuth within 15’. This is effected as follows: A piece of brass plate 112™™ by 30°" has a central hole by which it can be slipped snugly upon the end of the auxiliary cylinders, its plane then being normal to the cylinder-axis. There are two small holes in the plate 44"" distant from the center and in line with it. The two slides with their auxiliary cylinders having been placed so that the cylinders are nearly in the axis of the yard and abut against its ends, the yard is then removed, the brass plate is put on one of the cylinders, its two holes being at the same height, a light is held behind one of its small holes while the eye is placed at the other looking at the plane end of the distant auxiliary cylinder. The two cylinders are then adjusted till the eye sees the reflection of the light in the plane surface of either when viewed from the other. The auxiliary cylinders when a metre apart ean thus be brought into the same azimuth within about 1’. When yard A was removed from between the quills, and the latter were brought in contact so that the interval 4"".9 between their 4.6 graduations might be measured, this method could not so well be used, and the following was adopted: A silk thread was fastened to the bar carrying the quills, and adjusted so that it could be run under the longitudinal microscope-wires for a distance of over 100°" without deviating from them by 0"".05. The microscope was then pointed at the side of the quill at the places where the rings gave it accurately equal diameters. This could be done with an accuracy of 0.01. These edges of the quills were then made to run along the longitudinal thread of the microscope. In this way the azimuth of the quills could be made the same within about 3’. Examination showed that the plane ends of the cylinders were normal to its axis within about 40”. b. To avoid errors greater than 0.1, when the cylinders have the same azimuth but differing inclinations the difference in their vertical inclinations must not exceed 6”. The adjustment is effected with a short, delicate striding level (one division =7”), which can be set on the rings of the cylinders. Examination showed that these rings had so nearly the same diameter that the dif- ference can be neglected. c. The auxiliary cylinders having closely the same azimuths and inclinations to the yard, to avoid errors greater than 0#.1 the contacts of cylinders and yard must be central on their end sur- faces within 0"".07. This can be effected within about 0.12 by looking at the point of contact with a lens. d. The graduated surfaces on the cylinders should be nearly horizontal. This is effected by rolling the auxiliary cylinders till a light and its reflection on the graduated surface are both cov- ered by a plumb-line. In comparisons the adjustment-errors were brought within these limits. The programme for comparisons of #1876 and Clarke yard A was as follows: Two visits, twelve hours apart, to comparing-room on each of three days. Then let metre and yard change sides and readjust everything. At a visit the following readings were made: 1. Thermometer-reading on F# 1876. . Thermometers on yard A. . Microscope-pointings at metre. Double microscope-pointing at yard. . Microscope-pointing at metre. Thermometer-reading on metre. bo me oO a ot §9.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. Wes The means of all thermometer-readings on each bar were taken and the proper corrections applied for thermometer-errors. The time occupied by a visit was usually from ten to fifteen min- utes. There were four A-thermometers inside the iron case of the Clarke yard A, and a Casella thermometer lay on R1876. All have been described in Chapter II, §§ 2 and 7. Both bars were inside the comparing-box. In reduction ninety comparisons on sixty-one days were used. No comparisons were used when indicated temperatures of #1876 and Clarke yard A differed by more than 0°.15 F. This rejected eight comparisons. It was assumed that yard and metre had the same temperature, namely, the mean of the observed temperatures of yard and metre. As both rising and falling temperatures entered the work, any slight error in the assumption of equal temperatures tends to eliminate itself. The mean temperature, ¢), of the comparisons used was 579.918 F., and the cor- responding mean value of yard A+ quill interval — R/1876 = 61.36. This will be designated as (A'—f’ 1876). The observation-equations may be written in the form— (A’— R'1876)o+(fo—t) (Beis Ey) — (A!— BR’ 1876)=0 where ¢ is the mean of the temperatures of yard and metre, Ly1.;, and Hy, the expansions for 1° F. of R/1876 and A’, respectively ; and (A’—R’1876) is the difference of lengths resulting from a single visit to the comparing-room. The following table gives the results of the comparisons. The first column gives the date of comparison; the second, the position of R1876 as north (next observer) or south; the third, the corrected mean thermometer-reading for yard 4; the fourth, the corrected thermometer-reading for the metre; the fifth gives t—t; the sixth gives the value of (A/—#/ 1876) resulting from a visit to the comparing-room; and the seventh gives the residuals in the sense computed minus observed. Results of comparisons of R'1876 and Clarke yard A plus quill-interval. Be | ae | 6 aed. aaa gf. se Is8 Date. OB en = to—t yao Residuals. 289 22? 23 = Bb Baw 22 2a. See. mp Of mn O hes one 4 Bes ee BSS 22.8 & Ao 8 oo 8 on & mae ei 1879. oT. oF oF, Be » Mar. 2, 9:28 a. M~.-..-..--.------ North .... 34.91 34. 91 +22. 998 +62. 3 —0. 81 35. 06 34. 91 +22. 928 +63. 3 —1.81 35. 11 35.11 +22. 798 +62. 8 —1.31 35. 58 35. 61 +22. 308 +62. 6 —1.11 35. 84 35. 8L +22. 088 +61. 0 +0. 49 36. 21 36. 21 +21. 698 +61. 6 —0.11 39. 15 39. 10 +18. 788 +62. 6 —1.13 39. 25 39. 15 +18. 708 +61. 8 —0. 33 39, 37 39.25] +18. 598 +60. 9 +0. 57 39. 50 39.50} +18. 408 +60.7 +0. 67 39. 73 39. 70 +18. 188 +61.9 —0. 43 40. 18 40. 10 +17. 768 +61. 6 —0. 14 42.77 42,79 +15. 128 +61. 4 +0. 05 43, 41 43. 39 +14, 508 +61.7 —0, 22 44, 29 44.14 +18, 688 +60. 6 +0. 84 44, 68 44, 59 +13. 268 +59. 6 +1. 84 44, 78 44. 69 +13. 168 +59. 6 +1. 84 45.10 45. 09 -+12. 808 +61. 8 +0. 37 45. 30 45.19 | +12, 668 461.1 +40. 33 16, all eee: aetatres: 41. 99 42. 10 +15. 868 +62. 8 —1.35 wisn dO, seuss 41. 67 41. 60 +16. 268 +62. 8 —1.35 17, 2200 seco 40. 88 | , 40. 85 +17. 048 +60. 5 +0. 86 cori Ol wesc. 39. 98 40. 00 +17. 918 +63. 4 —1. 94 18, or eee 39. 37 39, 50 +18. 468 +62, 2 —0.73 =20O snosoe 38, 79 38. 80 +19. 108 +60. 0 +41. 47 176 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. 1X, Results of comparisons of R'1876 and Clarke yard A plus quill-interval—Continued. ce ; : <5 ee jee le. ges 5 ‘2 ext —| 3 Cad a | a8 ee ae e Red | $6 | 2u 1e8 Date. ‘gue mae are t—t Sa? Residuals. - | gan £38 Bs SES. ees | gece | Bes eae Dad BER Ee a2ee Ay 3 oO e 1879. oF, oF. a Py Mw Mw Aig 22 9:08 S. tmiae wey at eaeecot South..... 70. 71 70.72 | —12. 812 +61. 9 —0.61 i 71. 28 71.17 | 13.312 +61.9 —0. 62 71. 28 71,42 | —13. 442 +61. 8 —0. 52 69.95 69.83 | —11.982 +62.1 —0.81 69. 44 69.43 | —11. 532 +63. 0 —1.71 69. 24 69.23 | —11. 332 +61. 2 +0. 09 68. 96 69.03 | —11. 092 +64. 4 —3.10 68, 84 68.83 | —-10. 932 +63. 4 —2.10 73. 08 73.02 | —15.142 +61.8 —0, 53 73. 01 73.02 | —15.112 +61.3 —-0. 03 72,75 72.72 | —14. 832 +60. 8 +0. 47 72.33 72,32 | 14,412 +59. 8 +1. 48 71. 65 71.72 | —13.772 +63. 2 —1. 92 71.18 71.12 | 13.242 +60. 5 +0. 78 68. 74 68.68 | —10. 802 +61.8 —0. 50 68. 44 68.38 | —10. 502 +61. 9 —0. 60 67. 92 67. 83 —9. 972 +62. 5 —1. 20 67.34 67. 23 —9. 372 +61.6 —0. 29 66. 62 66. 58 —8. 692 +63. 0 —1. 69 65. 86 65. 94 —7. 992 +65. 0 —3. 69 65. 24 65. 24 —T. 332 +62. 3 —0. 98 11, 63. 98 63.94 | — 6.052 +61. 4 —0. 08 12, 63. 45, 63.49 | — 5.562 +60. 4 +40. 93 63.51 63.44] — 5.572 +61. 4 —0. 07 13, 63. 55 63.44 | — 5.592 462.7 —1. 37 14, 63. 55 63.59 | — 5.662 +60. 3 +1. 03 Sept. 16, 62.63 62.65 | — 4.732 +60. 1 +1, 28 62. 63 62.65 | — 4.732 +58. 9. +2. 43 17; 62. 49 62.55] — 4.612 +60.1 +1. 23 18, 62.31 62.25] — 4.372 +60.7 +0. 63 19, 61.79 61.70 | — 3.832 +61.7 —0. 36 61.11 61.15; — 3.222 +62. 0 —0. 66 60. 51 60.55 | — 2.622 +61.9 —0. 56 60. 31 60.30 | — 2.392 +61.4 —0. 05 60. 19 60.15 | — 2.262 +62. 4 —1.05 60. 23 60.25 | — 2.332 +62. 5 1.15 60. 39 60.45] — 2.512 +63. 8 —2. 45 60. 69 60.65 | — 2.762 462.2 —0. 86 59. 32 59.34] — 1,422 159.7 +1.6 59. 30 59.24] — 1.362 +60. 8 +10. 55 58. 90 58.84} — 0.962 59.7 +1. 65 58. 72 58.69 | — 0.792 +61. 6 —0. 24 58. 58 58.59] — 0.672 +61. 2 -40. 16 58. 68 58.64 | — 0.752 +60. 4 +10. 96 59. 89 59.98 | — 2.032 157.9 +3. 45 60. 14 60.11] — 2.212 +59. 3 +2. 05 61.09 61.14] — 3.212 457.7 +13. 64 63. 98 63.91} — 6.032] +60.2 441.12 64. 93 65.07 | — 7.092 4-61.7 —0. 38 64. 60 64.69) — 6.732 +63. 4 —2. 08 63. 47 63.58 | — 5.612 +62. 2 —0. 87 61.57 61.51] — 3.632 +61.6 —0. 26 §§ 10, 11.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. TTT Results of comparisons of R'1876 and Clarke yard A plus quill-interval—Continued. aE Sa = eas 23 ao 5 a Se Ba = 6 cag mes go ES A ee Sed 2 Fs A tp Je3 Date. 33 Bad oe to—t ae Residuals. a7” Ze z 33 . wdo , Gah pong Po. cosa. SEs SEF @ Hes ose a x5e Bow Be Ba Oo 5 Zao Bae BS @aSe a 3 3 5 1880. oF. oF, oF. w& KB May 15, 12:20 p. m.........-..0-04- South ..-. 59. 54 59. 66 — 1.692 -++59. 0 +2. 35 1G, 9:40! eee niciere car cecige’s a4 ¥O0 xectaes 58. 89 58. 82 — 0. 952 +58. 9 +2. 45 18, 9:56 a. Mi snccic sce cecceceee ite UO sere scoiee 59, 62 59. 71 — 1.752 +59. 7 +1. 65 19, 14250 G0: aseeee sce ee eeee acs O pacer 61.49 61.51 — 3.592 +59. 4 +1. 94 22, 9:24 eID on oes sa. seses iviQO weser's 64. 70 64. 82 — 6.852 +61. 0 +0. 32 24, D:48 Be Wl css wcicrsccisiniseareinie nocd sessile 64. 41- 64. 36 — 6.472 +60. 5 +0. 82 25, 9:58 A. TO ccscwcicnicicaasceee eal cine 65. 18 65. 12 — 7.242 +62. 2 —0. 88 26, 9:25. 8. Ms cceeeessexeensse 2400 veces 67. 50 67.47 — 9.572 +61.7 —0. 40 28; D7 Bay MY sowie ee< siersyewainns sal O wisicrarect 69. 15 69. 13 —11. 232 +60. 3 +10. 99 295 ODT A Mes selsieres amrwiacieu fii AO-nincwisicie 68. 36 68. 45 —10. 492 +59. 9 +1. 40 80, 9:16 Be TD aos mesiceeinie, cece! siz aacarareyn 67. 63 67. 52 — 9.672 +61. 0 +0. 30 31, 8:36 a m.....---. eee eee 330 OO wivene 66. 90 66. 97 — 9. 032 4-61. 9 —0. 59 June 1, 9:18 a. m.....-........--- 2a, SLO weeeerete 66. 60 66. 57 — 8.672 +60.1 +1. 21 The results obtained from the foregoing comparisons are as follows: €=57°.918 (A/—R’ 1876))=+ 614.36 + 0.097 Ee yee— Ly = +t 0.0058 + 0.0082 (A’—R’ 1876) at 59° F.==+ 614.3544 0+.097 § 10. In fifty-one visits the thermometers with yard A had the higher temperature, and in thirty-four visits that on metre had the higher. The mean residual for the first class is +0«.017, and forthe second +0+.035. There is, then, no sufficient evidence that the slight thermometer-differences indicated a corresponding difference in the temperature of 1876 and Clarke yard A. In eighteen visits the temperature of yard had risen more than 0°.5 F. in the preceding twenty- four hours. The mean residual for these visits was +0*.42. In twenty-nine visits the temperature of yard had fallen more than 0°.5 F. in the preceding 24 hours. The mean residual for these visits was —0".34. Since the residual is computed (A/—’ 1876) minus observed (A’—R’ 1876), a positive residual indicates that A’ was relatively too short for rising temperatures and relatively too long for falling temperatures. Since R1876 was freely exposed in the air of the comparing-box, while Clarke yard A was inclosed in its iron case, having walls and cover about 3" thick, it was to be expected that A would change temperature more slowly. Further examination will show that in thirty-one visits the temperature had risen in the pre- ceding twenty-four hours, there being 14 plus residuals and 17 minus residuals, the mean of all being +0+.23. In forty-five visits the temperature had fallen in the preceding twenty-four hours, and there were 17 positive to 28 negative residuals, the mean residual being —0".26. Hence this examination also indicates that the temperature of Clarke yard A lagged behind that of R1876 Had the number of visits with rising and falling temperatures been equal, an elimination of any resulting error would have probably occurred. But even if the whole of the mean residuals, +0+.23 and —0#.26, be attributed to temperature differences of the two standards, the inequality of the namber of visits (thirty-one) at rising and (forty-five) at falling temperatures would give in the result for difference in lengths of A’ and #’1876'an uneliminated error of only 0.05, which, in view of the uncertainty as to its true value, may be neglected. An examination has also been made to see if there was any connection between the values of the residuals and the time, varying from six to fifteen minutes, occupied in a visit to the comparing- room. No connection could be perceived. § Li. The determination of the interval of about 4™".93 between the 46th graduations on the two auxiliary cylinders when their convex ends abutted against each other was made by comparing ‘ 23 L$ 178 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. 1x, this interval at one time with the interval between 0.80 and 0°.99 on the standard inch described in Chapter II, § 3, and at another with the interval between 0” and 5"" on D1876. The auxiliary cylinders were mounted and adjusted as in the comparisons of Clarke yard A with 81876, save that the convex ends of these cylinders abutted against each other, instead of against the ends of yard A. The standard with which the interval was compared was mounted in the prolongation of the auxiliary cylinders, was made level, and aligned so that its longitudinal graduation when its supporting car was moved would run along the longitudinal microscope-thread. The two microscopes were pointed at the left-hand ends of the intervals on the standard and on the auxiliary cylinders. The car carrying both was then moved longitudinally under the microscopes till the right-hand ends of the two intervals could be pointed at. These pointings gave a comparison of the lengths. In the comparisons with the inch, which gave twenty-two values for the interval 4"".93 on two days, there was no readjustment and no known space was read by which to determine the error of run of microscope due to focusing. The adjustments were as in comparisons with D1876. In the comparisons of the interval 4"".93 with the space from 0” to 5™" on D1876, the follow- ing method was followed. The auxiliary cylinders or quills were brought to the same azimuth within 10’; to the same inclination within 6”; the contact of convex ends was made central, both horizontally and vertically, within 0".1. After each adjustment spaces of known values about 1™ were read on with both microscopes to determine the error in run due to focusing. By running the truck carrying the auxiliary cylinders and D1876 alternately backward and forward, 24 point- ings to each end of each of the intervals under comparison were made at each visit. After each visit the quills were disarranged in all their adjustments and readjusted. Six visits to the comparing- room were made in six days. In the following table are given the results of the comparisons of the interval between the 46th graduations at the convex ends of the quills when the convex ends abut, with the space 0°.80 to 0.99 on the standard inch. Each result is the mean derived from two pointings at each graduation. The first column gives the date; the second, the temperature; the third, the interval on quills minus the interval on inch as observed; and the fourth, the residuals for this interval in the sense computed minus observed—each is obtained by subtracting the results of individual comparisons from the mean of the mean results derived from the six visits. Comparison of interval between 46th graduations on quills with space 0.80 to 0.99 on standard inch. Observed (inter- Dato, Tempera: | val OB aul | residuals on inch). 1879. or, Be u June 7, 2:50 p.m ....-------- 64 +91. 33 —0. 35 +92. 23 —1, 25 3:16 pI ss 2eeesesese +90. 23 +0. 75 +92. 13 —1.15 +91. 93 —0. 95 +90. 43 +0. 55 3:32 PAD sess caxedeny +89. 73 +1. 25 +90. 83 +0. 15 +90. 73 +40. 25 3:43°), Mt sesesexuunes +90. 73 +10. 25 +90. 13 +0. 85 8:58: P.M) cccseacecans +90. 73 +0. 25 S59" Ds Th sc accinscees +92. 93 —1. 95 +191. 43 —0. 45 4-92. 33 —1. 35 ATA DS TM cpsc,co acheive 65 +89. 63 +1. 35 FNC, 220, GW ccanecinnnw se 63 +90. 63 +10. 35 +91. 93 —0.75 +90. 93 -+0. 05 +9053 “40.45 +489, 03 +41..95 B82 Pi TW send eu nens'ey +91. 33 —0. 35 RIGOR wamtcneencavecewsslseemnsenenw’ +90. 98 § 11.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 179 The following table gives the results of comparisons of the interval between the 46th gradua- tions on the auxiliary cylinders with the space 0™ to 5™ on D1876. The arrangement of the table is the same as that of the preceding one. Comparison of interval between 46th graduations on quills with space 0™" to 5™™ on D1876. Observed (inter- Residvals from Residuals Tempera- val on quills of means Date. ture. minus interval peuiaieli on h from gen- on D 1876). Pp Set | eral mean. 1881. on BX Me May 28, 3:08—3:37 p.m...... 67.45, 74.5 —0.5 —74.2 —0.2 : —73.6 +04 9 —73.5 +0.56 —73.8 40.2 67. 64 —74.7 0.7 Means ...........2200-- 67.54 TAG Ot | | ocsaleenaeats iad June 1, 2:50—3:15 p.m ....-. 68. 68 —72.7 +0. 5 —73.9 —0.7 —74.2 —1.0 —72.5 +0.7 79.8 +0. 4 68. 73 ~—73.1 +01 MGARB!2c.00nodencceeen 68. 70 BZ Nissan mtedsneiiene' +0. 54 June 2, 10:15—10:30 a.m...-. 68. 44 —71.8 +0. 2 —71.8 +0.2 —71.8 +0.2 —71.3 -+0.7 —73.0 —1.0 68. 44 —72.5 —0.5 MCAS Secisccccecincee ces 68. 44 S120 stew anes Denese +0. G6 June 3, 10:30—11:00 a.m-.... 66. 85 —71.6 +0.3 —70.7 +1.2 —71.8 +0.1 —72.8 —0.9 —72.8 —0.9 * 66. 85 —72. 0 —0.1 MGans'...c2ccesexcissces! 66. 85 tO. Neca seidceSigioeye aici +10. 76 June 4, 11:45—11:58 aM... 65. 66 —73. 2 -1.1 —73. 5 —-14 —71.5 +0.6 —72.4 —0.3 —T0. 6 +1.5 65. 66 —71.3 +0.8 MiGanses. 2c2i sens etescas G5. 66° Fav: || weed enwacwedsne 10. 56 June 6, 10:43—11:48 a.m .... 63, 48 —72.8 —0.2 —71.9 +0.7 —72. 2 +0.4 —72.8 —0. 2 —72.7 —0.1 63. 48 —73.5 —0.9 MCAS faiseneiisicdes Seaicmien 63, 48 =72,6- Neeser iver sieeews +0. 06 Mean of mean values for each Visitiicccc. cocoa. 66. 78 —72. 66 180 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IX, If in comparisons of the interval with 0 to 5"" on D1876 the differences between the indi- vidual results of a visit and the mean result be denoted by A, there will result for the probable error of an individual result due to pointing and reading only, 0.6745 ‘ = —70+.69. In the following table the first column gives the date of comparison; the second gives the number of comparisons on that day; the third shows whether tube 1 was north (next observer) or south; the fourth gives the observed temperature; the fifth gives (t—t) or excess of temperature of comparison above the mean of all observed temperatures of comparison, namely, 42°.747 F., (t—to) being the coefficient of the unknown (Hz, — Hz,); the sixth gives, with a negative sign, the observed difference (Z,—Z,); and the seventh gives the residuals in the sense computed ‘minus observed. , $2.) (Z,—Z)v=—# 89-4. C#,60 CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. Results of comparisons of Z, and Z,. Date, |Nooferm| Position of | Ouserredtem- | ety | —i@inz) [patel minus observed. 1878, oF, oR, a Oi Dec. 30 1 North side.. 34. 03 — 8&7 +78. 5 +11.6 1879. Jan. 1 1 ge:dOvs voeees 34.90 1:8 +82, 1 +14.8 2 2 e300. twan soe 35. 28 — 75 +83. 0 “15. 6 3 1 owed nonce 35. 35 —7.4 +80. 5 -H13. 0 9 1 South side .. 33. 63 — 9.1 +73.2 + 6.5 10 2 ng dOnaemenre 34. 46 — 83 +476. 8 +9.7 11 2 2800) osicehcd 34. 51 — 82 +72.0 +49 12 1 dO... ee 34. 46 — 8.3 +75. 9 + 8.8 13 2 oi dl sxcence 34. 44 — 8.3 +176. 6 + 9.5 Tune 26 2 2.0 .eeeee 71. 20 128, 4 +84.2 AL. 27 2 DEMO ckeaal 72, 42 429.7 +84. 3 +0.7 28 1 Leedonsenoatl 73. 42 130.7 +83. 0 1d 29 2 Northside. . 73.45 +230. 7 +82. 6 — 15 30 2 24 d0e wees 72, 95 430.2 +82, 2 1a Oct. 29 2 20 oe. eee 54. 15 HL 4 +78. 6 +29 30 2 S206 faces: 51. 02 +11. 3 +80. 2 + 4.6 31 2 ddO ee sk%e 53. 14 410.4 $78.1 + 2.9 Nov. 1 2 Lad Gt cath 51.71 + 9.0 +79. 0 +44 3 1 dO. eee ee 48. 88 + 61 71.2 = 2:9 7 2 ce dOiexexase! 42.70 + 0.0 +65. 9 — 48 8 1 puddOenecee 44, 28 +15 +719 + 0.5 11 2 seedO'sce ices 52.72 410.0 +73. 6 —15 12 1 nad O eeacsen 53, 94 +112 175.6 0.0 13 1 BeedO Dsoscies 55. 74 +13. 0 L143 = 1880. Nov. 13 2 a. 48. 62 + 5.9 +76. 8 +3.5 14 2 2.00 ....2. 47, 42 +47 17.5 +51 415 2 bexdO Races 45, 84 +31 75. 4 + 3.3 16 2 eeudO rose es 44,29 +15 475.0 + 3.6 17 2 aedoise ead 42. 52 — 0.2 +70.6 0.0 20 2 REDO. coe 36. 53 — 62 465.2 — 2.9 21 2 20. cesses 34.44 — 83 +61.4 — 5.7 23 1 acd 0. eeeees 31.29 11.5 +60. 4 — 5.3 24 1 eed Ausaees 31. 28 11.4 74.5 4+ 8.8 25 2 StdOsemewet 32. 50 —10.2 465.9 — 0.4 26 2 Sedo seectes 33. 08 — 9.7 +69. 3 + 2.8 28 1 Pico. sececed 34.58 — 8.2 +63. 2 — 3.3 - 29 1 08d cece eee 35. 33 — 7.4 +65. 2 ~~ 2.38 30 1 ned Olaeeteee 35. 93 — 6.8 60.7 — 7.0 Dec. 1 1 600. oes 36. 28 — 6.5 158. 4 — 9.5 2 1 3 gidlO: ceckeee 36.45 — 6.3 +60. 5 — 7.5 3 1 esdOl cme euee 37.70 — 5.0 +58. 0 —10.6 4 1 SeOlaiieete 37.95 — 4.8 +61.5 S71 5 1 weid@ wreaes 39. 61 — 31 68.9 — 0.4 6 1 wes WO aceite 38, 84 — 3.9 +60. 3 — 8&7 10 1 id Oia reels 31. 26 11.5 157.2 — 8.5 11 1 20. seen: 30. 98 11.8 460.3 — 5.3 14 1 22dO oeee ee 35. 18 — 7.6 461.7 — 5.7 15 1 nedonnneeses 36.79 — 6.0 +70. 0 +19 16 1 sede sees 37. 90 — 4.8 +63. 6 — 5.0 17 1 a 38. 38 —44 461.2 — 1.6 18 1 ceed One seste 38 47 — 4.3 62.4 — 6.4 19 1 dO aseres 38, 20 — 4.6 +61. 0 —7.7 20 1 0 ..eeee- 37.88 — 49 +65.9 007 21 1 a2 22222. 38.13 — 4.6 +62. 3 — 6.4 dill orc, ce en |toaacee sanans 9308136; lhestadwctetaetwed +3817. 5 Means. | ace ciawe vaiec see 7 SAsGdT—-Boeanemeehcneass 490. 69 (Ez, —Lz,)= —".4366 4 (#.0513 Probable error of result from single visit = + 4+.42 (Z,—Z,) at 320 F.=—6C#. 04 (4.31 (Z,—Z) at t F.=—66".00—0#.4366 (t— 32) 195 196 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IX,, § 24, An examination of the residuals shows that on thirty-three days, when the temperature- increase in twenty-four hours varied between 0°.03 and 1°.8 I’, the residuals were positive on thirteen days, giving a sum of residuals of 854.4; and negative on twenty days, giving a residual sum of — 98.7. For the thirty-three days of rising temperature the algebraic mean residual was -- 04,40. On twenty-one days the temperature-decrease in twenty-four hours varied between 0°.02 and 29.2 F. On twelve days the residuals were positive, giving in sum + 55.1; on nine days they were negative, giving in sum — 4.4.2. The algebraic mean residual for twenty-one days of falling temperature was + 04.60. The residuals for the zinc bars change in the same way, and by nearly the same amounts in increase and decrease of temperature as the steel bars. But these mean residuals are but about zoatvos part of the length of the bars, and are too small to make any certain connection between their values and the sign of the temperature-change. If the residuals of S;—S, be examined, it will be seen that there are eleven residuals numeri- cally greater than.2", while the probable error of one day’s comparisons is but 14.3. The residuals of Z,—Z, had the same sign as those of S,—S,, except on one of the eleven days. Omitting the residuals of both steel and zine bars for this one day, the average S,— 8, residual for the remaining days, regardless of sign, was 3+.1, while that of Z,— Z, was 6.3, or the residuals had similar signs and a ratio nearly that of the expansion of zinc to the expansion of steel. This points strongly for these days to a difference between the true temperatures of the bars in the tubes 34.1 and those given by the thermometers, an erroneous temperature of one of the tubes of —; x 1° F., or about 0°.12, accounting for the mean of the large residuals. In the zine-bar comparisons from December 30, 1878, to January 3, 1879, there are extremely large residuals, amounting to sgq5sa part of the length of the bars, while in the comparisons of the steel bars at the same time the residuals are very small. Nothing is known that throws any light on the cause of this discrepancy. If these four comparisons were rejected, the sum of the squares of the residuals would be reduced about one-third, and the probable error of one day of comparison, now very large (44.4), wonld be reduced. Since the probable error of pointing of one of the micro- scopes is only 0#.4, and but four pointings are required for a comparison, the probable error from this cause would be for a comparison but 0«.8. The fact that the comparisons of the steel bars give for the probable error of one day’s comparison of the same character and at the same time as those of the zine bars and including temperature and all other errors, but 1+.3, indicates that the probable error of comparison due to errors of measurement alone can little exceed 1+.0, and that the rest of the probable error in the difference 4,—Z, must be due to changes in that difference from tempera- ture or other causes. The range in the residuals of 8,—S, in § 21 is 7.9, If this range were due largely to differences of temperature of the two tubes, since the expansion of Z, is about 2.5 times that of §,, residuals of 2.5X 74.9 =20" would be expected in Z,—4, If the days from December 30, 1878, to January 3, 1879, be omitted in the comparisons of Z, and Z, the range of the residuals is reduced from 264.2 to 21".2. It therefore is probable that the large residuals are mainly due to differences of temper- ature of Z, and Z,. Collecting the values found, they are (S\—S,) at 420.747 F=+432,094 04.18 Es, —Ey,=—0#,0605 £ 0“0151 (S,—4,) at 19-32", 74-4 04.24 — (0.0605 +4 04.0151) (t—32) (Z,—Z,) at 429.747 F.—= — 70+.69 4 0+.60 Ez, —Ez, = — 0+.4366 4 04.0513 (Z;—4,) at 9== — 66+.0040+.81 — (0#,.4366-40".0513) (t—32) If the value of N, in terms of R1876, depending solely on its comparisons with R1876, given in § 18, be substituted in the value just given for S,—WS,, there results, after proper reduction, 8,=4 R1876—37+.014 14.2744 (t—59) : The square of the probable error of this value of §,, excluding the probable error of R1876, is, with 14 as unit, 0.1480 + 0.000480 (t—59)?+ 0.000228 (t—42.75)? , $$ 24,25.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 197 DIFFERENCE OF LENGTH AND RELATIVE EXPANSION OF ZINC AND STEEL BARS IN TUBE 2 OF REPSOLD BASE-APPARATUS. § 25. These bars are designated respectively by Z, and 8, A description of them and of the graduation-lines, about 4 metres apart, which limit the parts considered, may be found in Chapter VIII, § 3. Tube 2 has been repeatedly compared at widely differing temperatures, both with tube 1 and with the 15-feet brass bar of the Lake Survey. Such comparisons were made in the comparing- room of the Lake-Survey office, where the daily temperature-change, as indicated by mercurial thermometers in tube 2, rarely exceeded 2° F. Among other observations, those comparisons included the reading with microscopes of the distance between contiguous graduations on Z, and 8, at both ends of the bars, and also the reading of three of the mercurial thermometers, whose errors are given in Chapter II, §§ 15-19, Geissler thermometers being inserted in the tube at its ends and a Casella thermometer at its middle. These microscope- and thermometer-readings for tube 2 com- plete the observations so far as the present investigation is concerned. In each visit to the com- paring-room the thermometers were read on entering and again on leaving the room, about fifteen minutes later. The room was not entered again till the next set of observations was made. In the majority of the observations used in reduction, but one visit to the comparing-room was made in a day, to avoid irregular temperatures, but in some cases the number of visits rose to three. In reduction, however, the mean result for the day was used and the same weight was attributed to it whether it was derived from three or fewer visits in that day, the errors to be feared from tem- perature-differences being much larger than those of observation. After a preliminary reduction, which gave approximate values for the difference of length of Z, and S, at a mean temperature, and for their relative expansion, the residuals of the observations were computed and plotted with dates as abscisse. Below this curve the mercurial temperatures of the tube were plotted with the same abscisse. -A comparison of the two curves showed at once that, whenever the temperature of the tube rose, the residuals (which were computed values of Z,—S, minus observed values) increased algebraically with the temperatures, and vice versa. This preliminary reduction, in which all comparisons prior to its date were included, showed that the residual errors increased as the temperature-change increased, yet never exceeded about 25+; that a temperature-change of about 10° or 15° F. would produce this amount of change in the residuals, and that a further temperature-change did not increase it; that the rate of residual- change was approximately 2 per degree of temperature-change. The closeness with which each fluctuation of any importance in the temperature-curve was repeated in the curve of residuals made it certain that the large residuals were not due to accidental errors of comparisons, but were actually measured quantities. ; In the earlier comparisons of Z and Sin 1877 it is possible that the two bars were not per- fectly free to expand with reference to each other, although the comparisons themselves do not give any positive indication of it. For this reason these comparisons have not been used in the final reductions. With this uncertainty it may be said that the comparisons of 1877 and those of 1881 show that Z has retained unchanged its length with reference to S, within the limits of uncertainty due to the large regular residual errors. Thus ina series of comparisons on twenty-four days between October 22, 1877, and November 17, 1877, the mean temperature of the comparisons being 619.98, the mean value of Z,— 8, was +177+.4, while the value for this temperature resulting from the tinal reduction given immediately hereafter is 174".5. The difference is too small to indicate change of length, but may be attributed entirely to the cause which gives large regular residuals for Z,— 2. If, then, Z,— S, does not permanently change its value at any temperature, but fluctuates in the results of comparisons, from some cause, about a mean value, the question is as to the method of obtaining a mean value for Z,—S, at some temperature, and a mean relative expansion. If we had a long period of daily comparisons at low temperatures fluctuating about a mean temperature nearly the same for the whole period, and a similar long series for high temperatures, we might hope that from such series mean values of Z,—&, for the high and low temperatures and a mean relative expansion might be obtained which should have a good degree of accuracy. This idea governed in the selection of comparisons to be used in the final reductions. 198 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Cuap. IX, Tt has already been stated that when the Repsold base-apparatus was received from its maker in November, 1876, it was noticed that removable plates at 0", 1", 3", and 4", in the tops of the tubes, might be screwed down so tightly as to clamp the steel and zine bars against their supporting rollers, and thus prevent their free expansion. This clamping was first noticed on October 12, 1878, at which time the plates were screwed down so tightly that a pressure of 8 or 10 kilogrammes on the ends of the steel and zine bars in tube 2 would not move them in the tube. The plates were loosened, and they then moved under a pressure of 1 kilogramme. A stress of 1 kilogramme should clongate S, by 04.5 and Z, by 1#.1. These plates had been repeatedly taken off and replaced before this date. The prior comparisons, which were numerous, give no certain evidence that clamping existed at any time. If it did exist, then in expansion, since the zinc bar has a greater expansion than the steel bar and the cast-iron tube which have nearly equal expansions, the zine bar should be compressed and relatively too short. On passing a maximum temperature, a few tenths of a degree of fall in temperature should entirely remove this compression, and the residual curve of Z,—8, should show an abrupt jump at this point. In none of the many maxima observed does this occur, so that it is not probable that clamping existed for any long period. But as the other comparisons of Z, and §, are very numerous, nove prior to October 12, 1878, have been used in the final reductions. § 26. It has been stated that to get a mean value for Z,—S, and the relative expansion, two long series of comparisons with temperatures fluctuating about steady general mean temperatures are desirable, one of these mean temperatures being high and the other low. The comparisons give an excellent low-temperature series of this character, extending from November 25, 1879, to March 17, 1880, with temperatures between 38° F. and 50° F., and including comparisons on ninety days. No such favorable high-temperature series can be selected from the observations, and therefore all comparisons at temperatures above 55° F. not rejected by the following conditions have been used for the high-temperature series. The condition that no comparisons should be used in which the thermometers indicated a difference of temperature of the ends of tube 2, exceeding 0°.2 F., rejected thirteen days; that when the mean temperature of tube 2 differed by more than 0°.15 F. from the temperature of tube 1 or from that of the brass 15-feet bar, if it was being compared with the latter, rejected four days; and three days were rejected because the observers were in the comparing-room for more than forty minutes. These conditions rejected so many of the group of comparisons between January 6 and 22, 1881, as to leave no good distribution of the remaining comparisons with reference to the temperature-changes; hence all were rejected. There remained after these rejections one hundred and fifty days of comparisons. In some cases thermometers were read on entering and on leaving the com paring-room. The first reading was taken, the second being treated as a check. A visit Xsually occupied about fifteen ininutes, and in the majority of days there was but one visit a day to the comparing-room. If more than one comparison was made on a day, the mean of the results and of the temperatures was taken for that day and used in the observation-equations. The form of the observation-equation is (Z,—S,)o+ (t—to) (Ez,—Es,)—(Zz—S2) = 2, where (Z,—¥#,) is the mean difference of lengths of Z, and S, resulting from one day’s comnarisons; (Z,—8,)o is the mean of all such quantities; ¢ is the mean temperature of tube 2 for a day’s com- parisons; t) is the mean of all such temperatures, and v is the residual error in the sense computed (Z,—8,) minus observed (Z,—S8,). The value of t is 499.504 F., and of (Z,—S,)) is —305+.6. In the following table the first column gives the date of the comparison; the second shows when tube 2 was north (next observer); the third, the mean temperature of tube 2; the fourth, the mean of the observed values of Z,—S, for that day; and the fifth, the residuals. The results of the least-square reduction follow the tables. § 26.] CONSTANTS OF METRICAL STANDARDS AND BASH-APPARATUS. Comparisons of the steel and zine bars of tube 2. Date. | Position of tube 2. ee eee oe Residuals. 1879. oF, it 2 June 26 | North side........ 71. 20 + 514.1 +14. 89 72. 42 + 560.0 +15. 91 73.41 + 598.6 +15. 39 73.44 + 601.2 413, 95 72. 95 + 588.3 + 8.00 Nov, 15 | South side........ 59. 68 + 71.6 +14. 27, AG Neve A0eeca nessa: 58. 76 + 412 + 9.29 DD | son dOssccecicees exc 57. 27 — 13.5 + 6.67 18) cc cdOtcacwardaveces 55. 38 — 83.5 + 3.97 Nov, 25 | South side ..-.... 42,15 — 580.6 — 7.82 226 | eras SOO ste cis arsiciaierors'e 42. 30 — 577.0 — 5.65 27 PscedO> veces sia tae 43. 01 — 552.2 — 3.14 28 ean AdOinranceayceswis 44. 58 — 497.2 + 2.25 291s adO: siveiegsisizinesin 45. 61 — 456.7 + 1.37 80 |. 220 seiwsinsasceces 45. 05 — 476.6 — 0.27 DOG ~ Tiles 0 se ters aecisre es 44. 82 — 487.6 + 1.88 2 Vials lO: ste sisistera ies crow 44,92 — 483.4 + 1.53 3 w22@O 22222-2202 45.70 — 455.8 + 3.93 By lo sei hO. adicsdccsinacieis 46. 43 — 426.2 + 2.41 5: [cc OO ncemseckessee 47. 22 — 398.8 + 5.40 GB |seey CO eescesvdacess 48. 46 — 350.7 + 4.99 TP lier @O votes seimicts,cterre 49. 34 — 318.8 + 6.94 Bl ce MO cca sawcnscic: 49, 22 — 323.3 + 6.83 9}... 48. 86 — 338.0 + 7.68 10}... 49. 35 — 317.2 + 5.73 TL, |x. 48. 74 — 339.6 + 4.66 13 |... 45. 01 — 473.8 — 4.6L 14 |... 43. 60 — 529.3 — 3.35 15, tee 43. 26 — 542.1 — 3.62 16 |... 42. 04 — 585.4 — 7.25 17 |..- 40. 94 — 627.3 — 7.66 18 bess i 40.13 — 656.6 — 9.52 19) Jen 39.79 — 667.0 —12. 20 ODF Il crass, 39. 46 — 685.8 — 6.09 DB hace 39. 76 — 675.5 - — 4 85 24 |... 40.14 — 660.2 — 5.53 95 | an 40, 54 — 645.5 — 4.85 926i). was 40, 42 — 649.8 — 5.16 g7 |... 39. 52 — 684.1 — 5.48 99 |... 38, 82 — 707.6 — 891 30 lene 39. 41 — 689.8 — 4.01 BN lecece 39. 72 — 676.4 — 5.49 1880. Jan. 1)... 40. 22 — 657.8 — 4.86 DN sce 40. 74 — 642.3 — 0.36 B |ecis 41. 29 — 620.2 — 1.30 ers 43,31 — 548.8 + 5.00 TF ase 44, 94 — 489.2 + 8.10 Bisse 45. 29 — 476.0 + 8 36 9]... 45. 91 — 455.7 +11. 91 40 |... 46, 96 — 416.7 +13. 30 DDE care, 47.75 — 383.1 +10. 08 18 loo 47. 45 — 395.3 +10. 74 14 |... 46. 75. — 419.0 + 7.52 16 |... 45. 56 — 465.7 + 8.44 17 |... 45. 58 — 461.8 + 5.31 19 |... 46. 60 — 424.7 + 7.45 20: | sae 46. 95 — 413.6 + 9.81 QL aa 46. 93 — 414.1 + 9.54 92. ececdo., 46. 58 — 426.4 + 8.38 199 200 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuap. IX, Comparisons of the steel and zinc bars of tube 2—Coutinued. Date. | Position of tube 2. mae a ee nig age Residuals. 1880. oF. He a pie 46, 45 — 432.2 + 9.18 45.76 — 456.4 + 6.84 45, 88 — 452.2 + 7.25 46. 46 — 432.1 + 9.46 46, 53 — 424.1 + 4.16 45.71 — 455.0 + 3.51 45,58 — 458.1 + 1.61 43, 61 — 531.4 — 0.86 . 42. 04 — 589.4 — 3.25 40. 38 — 651.5 — 5.00 40.14 — 661.2 — 4,83 40, 02 — 663.5 — 6.85 39. 70 — 677.1 — 5.56 39.97 — 664.8 — 7.47 42. 06 — 591.2 — 0.68 43. 23 — 546.2 — 0.68 43. 66 — 534.9 + 4.56 44.72 — 492.0 + 2.43 45. 63 — 461.0 + 6.44 44.26 — 511.0 + 3.74 43.28 — 551.7 + 4.82 42, 86 — 564.6 + 3.49 43.41 — 542.0 + 2.05 44.96 — 486.5 + 6.17 o7 |... 46.12 — 442.6 + 6.89 98 |... 47.18 — 404.0 + 9.06 Mar. 1]... 45.17 — 478.1 + 5.84 2}.. 43. 70 — 530.4 + 1.60 3 |. 43. 68 — 530.7 + 1.13 4}... 44. 66 — 496, 2 + 4,33 Bilas. 45.95 — 447.3 + 5.05 6|.. 45. 58 — 465.0 + 8.51 Sooo 45.10 ~ — 480.7 + 5.75 9)... 43. 53 — 538.0 + 2.66 10 42,13 — 587.1 — 2.09 1 40.96 — 628.4 — 5.79 12 40.41 — 647.2 — 8.15 13 39. 46 — 685.2 — 6.69 15 38.14 — 733.3 — 9.36 17 37.92 — 739.2 —11.93 75.91 + 704.2 + 5.95 78. 83 + 701.4 + 5.67 76.11 + 713.0 + 4.84 76. 72 + 734.4 + 6.91 77.27 + 754.8 + 7.66 77. 36 + 760.0 + 5.93 68. 97 + 448.1 — 4.89 68. 48 + 431.7 — 7.34 68. 53 + 433.1 — 6.82 68. 94 + 448.9 — 6.85 69. 80 + 477.6 — 2.47 70. 68 + 509.0 — 0.02 70. 91 + 521.3 — 3.47 70. 61 + 508. 4 — 2.11 30 70. 26 + 495.4 — 2.57 31 70. 19 + 492.2 — 2.07 Aug. 1/].... 70, 56 + 505.0 — 0.64 2 71.18 + 524.3 + 1.99 Sept. 24 63. 70 + 251.5 —11.00 85, lect dO = seawater: 63.49 + 244.3 ~-11. 88 1 §§ 27,28.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 201 Comparisons of the steel and zinc bars of tube 2—Continued. Date. | Position of tube 2. | Mean temperature Masa Z a Residuals. 1880. oF, ia ii Sept. 26 | South side........ 63. 98 + 261.0 — 9.73 OT Os ee esses 64, 59 ++ 284.1 — 9.37 OB | dO seas: Seaweed 64, 08 + 266.7 —11. 59 29 |. ...dO Lecce eeeeee 62, 81 + 220.4 14.14 80 |...-do ..... 0.20. 61.01 + 156.6 19,57 Oct. 1] Northside........ 59. 66 ++ 103.8 18.70 2 se lose ecanek sate 59, 39 + 93.2 —18, 49 Rene ee ree 59. 62 + 101.0 17.44 ss 4] 00.dO ceeceeceeeee 59, 82 + 110.0 ~18,75 5 lgaaidoiueecesdd 59. 38 + 91.6 ~17.27 G3) sas dole oman | 58.97 + 73.8 —15. 24 T [e220 ceeeeeeeeees 58, 20 + 48.8 —19. 86 8 eactdoricanee es 57. 73 + 29.0 18.14 9: eee SOO creswardecens 57. 60 + 23.7 —17. 84 Dec. 4] South side........ 37. 95 — 758, 8 + 3.83 Bf AsO! ceesecetecs end 39. 61 — 690.9 + 4.78 er 38, 84 — 724.3 + 8.56 Tl secd@ peccreed csc 36. 73 — 797.1 + 0.20 Bs! odo sae seen eee 34.18 — 889.1 — 5.88 9 |r tlo Reesee lien des 82. 02 — 970.1 7.97 TO dist Sos acescnesee 31. 26 — 999.6 —~ 7.70 Wilh es AG soe sce aes 30. 93 —1010.3 — 9.69 1B |.2..d0 vseeceeeceeee 33, 39 — 920.6 4.77 44 |. 3.0: cease ssestis 35.13 — 860.4 + 1.96 15 |..2.€0 oe. eects 36.79 — 796.7 +201 16}... do ....eeeeeee ee 37.90 — 759.2 + 7.20 IE |ScusdO ied sacudtosie 38. 38 — 743.1 + 9,67 1B [oss AO: cveasecacezec 38. 47 —- 737.7 47.73 19 |. 22.40 .seeeeeeceeee 38. 20 — 752.4 +12, 04 eC 37. 88 — 763.8 411.14 BH |e dO: csecenncen'ss 38, 13 — 754.1 +11. 05 150i arte ta tee alot ay! 7425. 66 —45832. 7 + 2.61 Means ...... 49. 504 — 305, 551 (Z,—8,) = —305.551 + 0.463 at 49°.504 F. (H,,—Es,) = + 38.4648 + 0».0388 for 1° F. (Z,— 8) == —978+.857 + 0.823 at 32° F, Probable error of result from single day’s observations = + 5.67. § 27. The connection between the changes in the residuals and those in the temperatures is best shown graphically. Accordingly, in Plate XXIV the temperatures and residuals have been plotted with times as abscissas for the long series of comparisons from November 25, 1879, to March 17, 1880. An examination of the curves shows at once the closest connection between the temper- ature-changes and the residual-changes, and that there is some regular cause other than accidental errors of observation for these residuals, which are often large. The other residuals, not plotted in Plate XXIV, are for shorter periods of comparisons, but their indications are the same. In gen- eral, all show that when the temperature rises, the observed value of Z,—¥, diminishes with refer- ence to the computed value at the rate of about 2 per degree Fahrenheit, or Z, becomes relatively shorter than was to be expected. In falling temperatures the reverse is the case. § 28. A uumber of theories to account for the phenomenon have been examined. A. The graduation-lines pointed at on the ends of Z, and 8, are in the horizontal plane through the neutral axes of the bars; but to bring their ends closely together for microscopic reading at the same time, the tubes are so made that the portions of these lines pointed at are 8"™ distant from the vertical planes through the neutral axes of the two bars. If, then, the bars bend laterally, the lengths of Z, and S, fixed by these lines will change by 2.3 for a change of one minute in the 26LS8 202 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. 1X, relative inclination it a horizontal plane of the ends of a bar. Now, thezince bar, 2, has a large expansion relatively to the steel bar, S,, and is supported by rollers and pins carried by 8, at 0™.5, 1.5, 2".5, and 3".5. If Z, expands and is in some degree constrained by friction in its free motion with reference to S,, it would tend, if laterally free to bend, to become concave towards 8), which would tend to become convex towards Z, so that the distance between the graduation-marks on 2, which lie between its neutral axis and &,, will be less than if measured on its neutral axis, and Z, will become too short with reference to 8, inrising temperatures. Possibly this effect might be aided by the rollers with vertical axes which confine the two bars between them at each metre of the length of the bars. Should the temperature fall the reverse effect would take place. But this supposes that Z,and S, are free to bend laterally. In fact, at each metre they are confined by rollers which do not allow 0™™.1 play. The question as to the amount of the effect of bending has been examined in two ways. In the first place, in all the work since October 12, 1878 (and only this has been used), it has been made certain that the bars in the tubes moved freely in the tube under an end-pressure of one or two kilogrammes. Since a stress of 1*$ extends Z by only 1.1 and S, by 0«.5, no serious errors are to be feared from direct compression or extension without bending. The end metres of each bar are under a slight constraint from friction of vertical rollers confining them laterally at 0™ and 1™, and at 3" and 4~. Under this slight resistance to expansion (about 4*¢) may the end zine metres bend outward slightly?) Computation gave no appreciable bending effect for such a force of compression. The zinc metres of Z are supported at their middles on thin rollers 28"™ in diameter, carried by horizontal pins projecting from S,. May the friction of the end zinc metres in expansion on these rollers or pins slightly bend the steel pins which will carry with them the guiding-wheels turning upon them? In this case, if the zinc bar were in contact with the edge of the wheel at the height of the center of the wheel, before the bending of the pin, the bending would, by the pressure of the edge of the wheel on the zine bar in expansion, press the middle of the zinc bar away from the steel bar and change the azimuth of the end of the zinc bar. Computation indicates that a push of 1*s on this pin at 0™.5 could not shorten Z by so much as 3“ in this way. One metre of the zinc bar weighs 2"¢.5. An attempt was made to bend the rear-end zine metre at its middle slightly away from the steel bar. No change in the difference of readings on Z and S, could be observed. These theoretical estimates of possible errors, especially of the last, are not very precise, on account of the uncertainty of the data used. § 29. The second method of examining the question of bending of Z and S, when expanding and contracting was as follows: Such slight bendings as those now considered, even if they actually existed, would not change the length of the line joining the ends of the neutral axis of either the steel or zine bars by quantities that need to be considered. If, then, lines be drawn at right angles to the neutral axis at each end of each bar, the interval between them in the vertical plane through the neutral axis may be taken as invariable so far as horizontal bending is concerned, and the comparisons of this interval on a bar with the corresponding interval on the bar used in measuring (called S,, S,, 2, or Z) will give the changes in the values of the latter. Accordingly, such lines were graduated on the ends of the four bars, approximately 4" apart. The graduations were suc- cessfully ruled with a diamond by Assistant Engineer E. 8S. Wheeler on the small steel plates which carry the platinum plates on which are the ordinary graduations. Unfortunately, the portions of the steel plates nearest the neutral axes of the bars are about 1™™ lower than the platinum plates carrying the ordinary graduations. The microscopes had, therefore, to be focused for an interme- diate height, 0™.5 from either surface, and so a little outside the range of distinct vision, which is about 0™™.4 on each side of perfect focus. This rendered the errors of pointing much larger than in ordinary work, giving ranges of 4"in 10 pointings. The 4™ intervals which are used in measuring being denoted by Si, 2, &c., these new 4™ intervals will be denoted by Sy’, Zi’, So!, Zo’. Strictly speaking, these new intervals are in the vertical plane through the neutral axis and about 1™™.0 from it, so that they are slightly above the neutral axis. But it is horizontal bending that is feared, and the lengths of the new intervals will not be affected by that. Indeed, their lengths at 1™™ distance jn a vertical plane from the neutral axis would not be sensibly affected by any probable bending. The following table gives the dates of comparisons of intervals of about 4™ in the neutral axes of steel and zinc bars of tube 2 with the intervals on the same bars, also of 4 metres used in base measuring; the temperatures; the observed differences of length of 4, and NS; Zand Z; S,and 8’; 4, and Z,'; and the residuals for Z,—S, and Z,—8,,. § 29.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 208 Synopsis of results of comparisons of tubes 1 and 2 in the comparing-room Nov. 10 to Dec. 21, 1880. Mercurial Computed | Computed wae temperature. oe pS : ; (Zi—Si) (Z2—S:2) ! re bey S2—S2! Z.—Z2 minus minus observed | observed Tube 1. | Tube 2. (Zi—Si). (Ze—S2). 1880. oF, oF. rv KB B we » Me Nov. 10, 3:26—3:58 p.m ....- 51.79 51. €4 +32. 7 SNOT. [ecice a eereetiet acemiatcarieet —1.6 — 5.2 ' 11, 3:08—3:32 p.m ....- 52. 58 52. 65 +26. 7 —89.9 +778. 2 +653. 2 — 1.6 — 9.0 : 12, 9:16—9:38 a.m ....- 50. 98 50. 90 +35. 0 —70.3 +777. 8 +655. 7 —11.6 —14.2 12, 8:18—8:35 p.m ..-.. 49. 86 49. 89 +35.1 —72.7 +776. 9 +655. 8 —10.5 —l1.7 13, 9:48—9:56 a.m ..... 48. 96 48, 93 +35. 3 —72.7 +777. 4 +656. 1 —11.6 —15.7 13, 8:50—9:10 p.m ....- 48. 37 48.31 +32. 4 —T5. 4 +782. 7 +661. 4 —12.0 | —17.3 14, 9:55-—-10:15 a.m ....' 47.75 47.71 +30. 5 —78.5 +779. 9 +655. 5 —11.0 —16.8 14, 8:22—8:36 p.m -...- 47.17 47.15 4-34. 4 —72. 8 +776. 5 +659. 2 —16.4 —19.9 15, 9:183—9:32 a.m ..... “46.48 46. 38 +33. 8 —73. 2 +781.0 +656. 4 —12.9 —17.6 15, 8:27—8:41 p.m..-.. | 45.42 45. 29 +33. 6 —67.0 +779. 5 +657. 0 —17.3 —19. 2 16, 9:05—9:21 a.m ....- 44. 48 44. 42 +34. 9 —71.5 +781. 9 +657. 9 —16. 4 —21.8 16, 8:38—9:00 p.m ..... 44. 07 44.01 +30.1 —71.1 +774. 6 +658. 6 —20.1 —20.4 Yi, 9:06—9:21 a.m ..... 43. 10 42, 98 +34. 5 —63. 3 +779. 4 +656. 7 —20. 8 —20. 3 17, 8:30—8:43 p.m ....- 42.15 42. 05 +36. 3 —64.9 +781. 2 +663. 1 —16.4 —18.9 Means ...-.--.----|.--.2+++-+ ache sitiee Laciantids | Aaawecten sas +779. 0 +657. 4 —12.9 —16.3 Nov. 18, 9:08—9:20 a.m ..... 40.84 | 40.74 | 436.8 —57.6 +778.8 | +660.4 | —25.0 —21.2 18, 8:31—8:44 p.m ..... 39. 51 39. 37 +36. 2 —51.3 +778. 4 +664.1 —30. 0 —21.4 “19, 9:17—9:36 a.m ..... 38.10 37.99 +39, 1 —54.7 +780. 5 +664. 8 —26.4 —23.5 19, 8:28—8:53 p.m ...-. 37. 52 37. 43 +37.7 —56.1 +778. 2 +665. 2 —23. 6 —20.1 20, 9:02—9:18 a.m ..... 36. 87 36. 80 +38. 7 —58. 9 +779. 3 +664. 2 —25. 2 —25. 0 20, 8:24—8:47 p.m ..... 36. 38 36, 27 +30. 7 —60. 2 +776.1 +661. 3 —23. 0 —17.9 21, 10:04—10:20 a.m ... 35. 15 35. 06 +31. 5 —55. 4 4777.1 +663. 4 —28.5 —19.0 21, 8:30—8:48 p.m +660. 7 —24, 9 —19.6 22, 9:12—9:36 a.m +663. 4 —24, 2 —17.3 22, 8:26—8:41 p.m +666. 5 —24.1 —22. 8 23, 9:18—9:36 a.m +668. 6 —23. 2 —20. 0 24, 9:18—9:47 a.m +659. 2 —16.0 —18.3 25, 9:32—9:57 a.m +670. 4 — 9.4 —13. 3 Means ........-- : +664. 0 —23. 3 —20. 0 Nov. 25, 8:30—8:45 p.m +667. 4 15.0 —10.7 26, 9:11—9:25 a.m +665. 6 —11.6 — 91 26, 8:41—9:06 p.m +667.7 iBT —13.3 27, 9:25—9:43 a.m +664. 9 —11.4 — 61 28, 9:56—10:12 a.m .... 34. 53 34, 58 +31. 4 —66. 4 +780. 3 +664. 9 — 7.9 — 41 29, 9:11-——a. m ..... 35. 30 35. 33 -+33. 0 ~ -67.3 +773. 4 +659. 0 — 3.4 — 2.6 30, 9:07—9:26 a.m ..-.. 35. 95 35. 93 +35. 5 —59. 6 +775. 2 +664. 2 — 7.5 — 3.2 Dec. 1, 9:05—9:23 a.m ....- 36. 30 36. 28 +36.7 —57. 3 +771.4 +656. 3 — 44 +11 2, 9:34-9:55 a.m ....- 36. 40- 36. 45 +35. 4 —63.1 4774.1 +659. 0. — 5.8 —21 3, 2:36—2:54 p.m ..... 37. 56 37. 70 +33. 2 —66. 6 +775. 3 +660. 6 —1.3 + 5.1 4, 9:33—9:50 a.m ....- 37. 92 37. 95 +33. 0 —63, 4 +774. 5 +661. 6 — 3.4 + 2.2 5, 3:05—3:30 p.m ....- 39. 62 39. 60 +31. 3 —68.3 4771.7 +653. 5 +18 +29 6, 9:21—9:38 a.m ..... 38. 83 38. 84 428.7 —60. 9 +778. 8 +663, 3 — 51 + 6.9 Wi CANS rcecnccweseel sae per yess [a ceeous aul anscecumeey bas nde conse +775. 8 +662. 2 — 6.4 — 2.5 Dec. 7, 9:35—9:52 a.m .--.. 36. 82 36. 73 +34. 4 —56.9 +-780. 9 +661. 8 — 7.3 —15 8, 9:27—9:41 a.m .--.. 34, 25 34,18 +82. 1 —58. 4 +773. 9 +664. 6 —14. 2 — 7.8 9, 9:30—9:44 a.m ..... 32. 08 32. 02 +437. 4 —54.0 +776. 6 4665.0 | —15.2 —10.1 10, 9:15—9:34 a.m ..... 31. 23 31. 26 +32. 6 —59.1 +177. 3 +665. 5 —17.9 —10.0 11, 9:17—9:38 a.m .-..-. 30. 91 30. 93 +33. 4 —61.5 +780. 2 +665. 1 —16.1 —12.0 13, 9:06—9:20 a.m ....- 33, 33 33. 39 +32. 3 —69.9 +777.9 +664. 4 — 61 — 6.7 14, 9:10—9:22 a.m .-..- 35. 05 35. 13 +31. 6 —67.3 +777.1 +663. 4 — 40 + 0.1 15, 9:53—10:12 a.m....) 36,80 36. 79 +32. 6 —69. 4 +776.1 +661. 5 + 2.2 + 0.4 16, 9:28-9:48 a.m ..... 37. 85 37. 90 +32. 8 —66.7 4776.8 +659. 5 + 2.3 + 5.6 ‘ 17, 9:44—9:58 a.m ..--- 38, 34 38. 38 +34, 9 —64, 3 +776. 3 +660. 8 + 4.7 + 81 18, 10:06—10:16 a.m ...; 38.47 38. 47 +33. 6 —62.4 +778. 3 +662: 8 +11 +61 19, 11:22—11:52 a.m ..-| 38.20 38. 20 +35. 8 —61.0 +780. 1 +661. 1 + 6.2 +10. 4 20, 9:27—9:38 d.m ..... 37. 88 37. 85 +30. 6 —67. 8 +786. 3 +664. 3 + 7.0 + 83 21, 10:27—10:40 a.m ...| 38,12 38. 13 +31. 7 —62.9 +781. 2 +661. 2 4+ 2.7 + 9.5 CMGANG sasawecneaes eesasesscslonntwsdccslensene er wedel eevee rcenes +778. 5 +662. 9 — 3.9 + 0.0 204 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IX, If there were no bending and uno error of observation, the quantities in the columns headed S,—S,/, and Z,—Z,', during the temperature-changes, which, as is seen, gave residuals for Z,—&, of 25", should remain absolutely constant. But as the focusing was on the very limit or slightly out- side the limit of distinct vision, the errors of pointing become large. Dividing the series into 4 groups, of 13 or 14 observations each, the resulting means for the four periods are for— Ist vroup. 2d group. 3dgroup. 4th group. Ny— Sy’ = + 779".0 7794.0 +7754.8 +778".5 A,—Zi = +657+4 +6644.0 +4+6624.2 +6629 while the corresponding means for the residuals of Z,—S, are for the several groups— — 164.3 — 20+-0 — 25 + 04.0 The values found for S,—S,/ and Z,—Z,’ from many other comparisons besides these are +777+.9 and 660+.0, §33. The extreme range in the mean values of S,—S,/ is but 3.2, and in those of Z,—Z,' but 6+.6, while the range in the mean values of the residuals of Z,—S, is 20". Bending does not then account for any large part of the range in residuals, and, indeed, tbe variation in the mean values of &,—S,/ and Z,—Z,' is too small (about ;5g3555 and socac Of the length of the bars, respectively) to indicate with positiveness anything more than errors of observation, due to the necessarily bad focusing of the microscopes. § 30. B. If the theory of bending or of longitudinal strain in the bars does not account for any large part of the residuals, may not the theory of difference of temperatures of Z, and S, do so? Although the volumes of the two bars and the coatings of their surfaces are alike, and though their specific heats and conductivities do not differ widely, yet it is very improbable that they retain precisely equal temperatures when the temperature is rapidly changing. If, during a steady temperature-rise of 2° F. per day, Z, heated more slowly than 8,, its tem- perature would lag behind that of S, till the difference became so great that Z, would receive heat enough from its surroundings to raise its temperature at the same rate as that at which the tem- perature of S, rose. This would be the maximum difference in temperature, and the continuance of the steady temperature-rise of 2° per day would not increase it. If there were a steady temper- ature-fall the reverse would be the case. This difference of temperature would show itself as a residual for the observed lengths of Z,—S,; when Z, was the hotter its observed would be greater than its computed length, and the reverse would be true when it was the cooler. In the results of comparisons already given in the final reductions, no residual exceeds 20+; if this were due solely to Z, being cooler or hotter than S,, siuce Z, expands about 63 for .° F., a temperature-difference of about 0°.3 F. would be necessary to produce it. While this theory would account in part for the dependence of the residuals on the temperature-changes, and would give the residual-curves fluctuations corresponding to those of the temperature-curves, it does not always account for the values of the residuals. Thus, on September 30, 1880, the residual was—19-, and would be accounted for by supposing Z, 0°.3 F. the hotter. But from September 30 to October 9 the tem- perature fell but 3°.4, or at the rate of 0°.38 per day, and yet the residual was in the vicinity of—18 during the whole period. That is, we must suppose that with a daily temperature-fall of 0°.38, Z, and 8,, side by side, could differ about 0°.3 in temperature for nine days while inclosed in a heavy iron tube. This supposition is very improbable. Again, from January 10, 1880, to January 28, 1880, the residuals varied from+413# to+10 falling once to+5+.3. In this period the total temperature-range was 2°.2 F., the temperature being 469.96 on the first and 46°.46 on the last day. With a temperature-range of but 2°.2 F. in eighteen days, it seems very improbable that Z, could, on account of a different rate of heating, have differed in temperature from 8S, by the amount needed to account for the residuals, namely, by from 0°.2 F. to 0°.1 F., for this period. If these differences of temperature between Z, and 8, actually exist, it may be asked if it is not due to unequal exposure to sources of heat. The side walls of the comparing-room are about 1™.3 distant from the comparing-box. One of these walls separates the comparing-room from the adjoin- ing house, is of brick, and has a passage-way on the opposite side communicating freely with a kitchen and warmed by it; the other side wall is a lathed and plastered partition-wall separating the comparing-room from a hall (which is never heated) of the Lake-Survey office. Fearing that §§ 30,31.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 205 these walls might differ in temperature and affect, through the wooden sides of the comparing- box, the standards within it, many experiments and investigations were made to detect such an effect. It will be remembered that the whole interior of the comparing-room is lined with a layer of sawdust 0™.3 thick. Thermometers sunk 5“ in this sawdust on the side walls sometimes showed a difference of temperature of as much as 4°.5 I’. But thermometers on the comparing-room sur- face of these walls showed no certain difference of temperature... No difference ceuld be detected between thermometers in the interior of the comparing-box at its two sides. In the tubes a com- parison of the readings of thermometers on opposite sides of the bars gave no evidence of temper- ature-difference between the sides. Two or three canvas screens a decimeter apart were set up each side of the comparing-box, so as to cut off the radiation from the walls, and comparisons were carried on at a time of large residuals with these screens up for many days. The presence or absence of the screens had, so far-as could be discovered, no effect on the results of the comparisons. It is seen, then, that all attempts to establish the existence of a difference of temperature be- tween the bars under comparison, due to external sources, failed. Moreover, it will be noticed that if such an effect was sensible, it would extend over long periods, and its changes would be slow, while in fact the residuals follow closely small temperature-changes in the comparing-box from day to day. In comparing the two tubes the order of their bars from north to south was: 1, steel; 2 zinc; 3, steel; 4, zinc. If heat had come from the south wall it would have tendedto heat the two bars nearest it the most, especially the zinc bar. But, in fact, the two tubes varied together quite closely. The heating should also have affected for long periods the south thermometer, but this was not observed. The bars under comparison could not of course have absolutely equal exposure to heat on the two sides of the comparing: box, but from the examinations made it would seem that the effects of such unequal exposure were either so small as not to be detected or were masked by other errors. § BL. There is another indication that the theory of difference of temperatures of the two bars is not sufficient to account for the residuals of Z,—S,. In comparisons of tube 1 with the 15-feet brass bar packed in ice, described in §§ 42-46, made on the Cass farm in August and September, 1880, the times of maximum lengths of Z, and 8, and the corresponding readings of thermometers in the tube, resulted directly from the observations. The first series of these comparisons extended continuously from 8 a. m. August 24, 1880, to 7:40 p. m, August 26; the second extended from 8 a. m. August 31 to 5 p. m. September 3, 1880. The comparisons were made once in about 20™ during the periods, and gave the results in the following table, in which dates near maxima- or minima- temperatures are given in the first column, corrected mercurial temperatures in the second, differ- ences of observed lengths of 8; and the brass bar B in the third, and differences of lengyb of Z, and B in the fourth. . Continuous comparisons of tube 1 and B. I.—FOR MAXIMA. [Maxima- and winima-values are in black type.] | Date. eoreer ae . Si— B32? 11~ Bx | 1880. oF oe | Aug, 23, 2:00 p.m...-..-.--- ST60l) | odeaeeeciecuuetselsccdmenoademaee: 2:20 P.M...-.------ 87, 93 2906 3880 2:26 p.M..-.------- 87.96 2906 3883 2:40 p.m.....------ 87. 86 2908 3891 3:00 p.m......----- 87. 65 2913 3900 3:21 P.M. .eeee eee ee 87.49 2913 3905 | 3:31 p.M....------. 87. 35 2912 3906 i 3:40 p.M.....------ 87. 22 2911 3905 | 4:00 p.m ...-..---- 86. 74 2907 3898 | we t20 Pp. M...-------- 86, 22 2902 3881 + 206 Continuous comparisons of tube 1 and B—Continued. I.—FOR MAXIMA—Counutinued. Date Cones ee 1880. or, Aug. 24, 12:20 p.m..........- 88. 32 12:40 p.m 89, 21 12:59 p. m 89. 90 1:00 p.m 89. 87 ; 1:04: p. Miscessinere vee 89, 25 1:06 p.m....-..---- 88. 85 1:20 p.m 86. 07 1:40 p.m 2:00 p.m Aug. 25, 1:20 p.m 1:40 p.m 2:00 p.m. | Ang. 26, 3:00 p.m. 3:20 p.m....-.- | 3:40 p.m.....---- | Si Buy —Bu I 1 M be 2866 3802 2894 3863 2915 3926 2916 3927 | 2917 3934 2916 3935 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. I.—FOR MINIMA. Aug. 24, 4:00 a.m....... 4:20 a. m......- 5:00 a. m......- 6:20 a. m......- 5:38 a. 5:40 a. 5:54 a. Aug. 25, 11:40 a. m....... 12:00 a. m......- 4:40 a m.......---- 6:00 B.Is teense. stdieia 61.15 saa 60.73 60. 43 were 60. 32 imate 60. 32 60. 37 60. 38 60. 57 ase 60.73 en Ba 61. 51 2283 i 2358 2279 2347 2276 2343 2276 2343 2275 2345 2275 2347 2283 2361 2298 2398 2373 2590 2366 2574 | 2360 2555 ! 2355 2544 2355 ~ 2543 2355 2542 2356 2541 2357 2541 2362 2551 2367 2563 [ Crap. IX, § 31.) CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. Continuous comparisons of tube 1 and B—Continued. -I.—FOR MINIMA—Continued. Corrected mer- Date. curial temp. Si— Bs” Zi — B22 1880, oF. va & AGE 26, 1:20R, Winns cecnces 5840! lke siatetinnctacisewms seacieerexeceeeree 1340'Gy Dh cwewancs ee 58:25 —s Inceseeswaaceaeelemieecas bexeee oe 2:00 a. m..-.-2.---- BO! dtiecteseassl-eseoeezeceaecs 2:20 a, m..... 2. eee 58. 05 2205 2161 2:40 a.M....-.2- eee 57. 98 2201 2152 2:56 a. M.....-----. 57.95 2198 2145 58. 01 2196 2141 58. 05 2196 2140 58. 09 2197 2141 58. 16 2199 2144 58, 27 2198 2143 58. 35 2198 2142 58. 44 2200 2144 58.55 2202 2152 IL—FOR MAXIMA. [Se : | | Aug. 31, 1:00 p.m..... a 82. 04 icummteatacdcaed| ee 2ieceeanaeae's 1:20 pom... eee. poe. Meron ssu et ae eS | 1:40 p.m -.-.---ee- kp a80G8TY Mes wt ees ete [bee peyendnnte | 1:52 p.M ..-...---. | BOAO eeeedtetewdlee ch eun ee eees | 2:00 p.m ...-...--- | 82. 40 2784 3588 | 2:20 p.m... .--eee | 82. 35 2789 3599 2:40 p.m -----2e ee 82. 25 2793 3606 2:56 p.m ---------- 82. 10 2794 3608 3:09 p.m .--..----- 82. 02 2794 3608 3:20 p.m .--------- 81. 90 2794 3607 3:40 p.m ..-------- 81.76 2793 3605 4:00 pint sevcuctuns|eedacuoebeesesa 2792 3602 Sept. 1, 1:20p.m....-...-. 89. 85 2950 3989 1:40 p.m .....----- 89. 99 2960 4017 2:00 p.m .....-----| 90. 15 2967 4038 2:16 p.m ..-------- 90.25 2972 4050 2:30 P.M ---.------ 90.15 2974 4053 2:38 P.M .... ---- 90. 00 2975 4052 3:00 p.m ...------ 89. 35 2971 4044 : 3:20 p.m .--- 22 eee [eee eee eee eee eee 2963 4023 9:40 pain sacse nas oe tacceoeessee cei 2951 3998 Sept. 2, 1:00 p.m-....-..-.. S200 |nsseccnnansnaal sanaermenraieenen : 1:20 p.m .....--.-- 84. 08 2812 3652 1:40 p.m -.---- 22. 84. 86 2832 3705 2:00 P.M -----2--+- 85. 48 2851 3748 2:08 P.M -.-------- 85. 51 2858 3764 2:25 p.m .-.------- 85, 32 2864 3782 2:40 p.m ..---2---- 84.77 2863 3777 3:00 ppm ---------- 84, 26 2859 3766 8:20 Pal wesc akogwesleuek derepeticeeee 2854 3753 Sept. 3, 2:20 p.m... 82. 10 2793 3605 2:40 p.m ---------- 82. 33 2798 3618 3:00 p.m..-------- 82. 54 2804 3631 3:20 P.M .--------- 82.75 2809 3643 3:40 p.m ..---- +e. 82. 55 2812 3648 3:48 p.mt..--.---- 82. 49 2812 3649 4:00 p.m .----2- 22 82. 45 2810 3648 _ 4:20 p.m .-ee eee 82. 51 2807 3644 4:40 p.m ...2.2---- 82, 45 2804 3639 207 208 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnap. IX, Continuous comparisons of tube 1 and B—Continued. II.—FOR MINIMA. Date. cia isa Si— Bsr 21 — Bs 1880. oF, ‘a cr Sept. 1, 4:40am .......... O885% leesatidatciemmcnine eadeneciee ees eer 5:00 @.M ....2. 2. 68. 40 2472 2825 5:20 & ML sccnezsss 68. 29 2470 2816 5:36 8M... 68. 25 2468 2810 5:58 AM weseee eee. 68. 82 2466 2807 6:02 a.m 2.2.2.2... 68. 45 2465 2809 6:20 a.m ....... 69. 25 2471 2823 | 6:40 am ...... 2. 70. 22 2485 2852 , Sept. 2, 5:00a.m........ al 73.11 2589 3116 | 5:20 & mM ..2....- 72.98 | 2585 3107 5:40 ath sepa de 72.87 2582 3099 5:56 a.m ...22...-. _ 72.800 | 2579 3095 | 6:06 a.m ...... 2. 72 90 | 2578 3094 | 6:20 a.m ..2....... 73. 23 2581 3096 | 6:40 aM 22.22... 73. 78 | _ 2587 3107 | 1:00 Bs Mil iaawaccecs |ecestasicwateeniiae 2594 3127 ! It should be remarked that the residuals of Z,—S, follow the same law as those of Z,—S,, so that the conclusions concerning one tube may be applied to the other. This is seen in § 29, where the residuals of S,—S, and Z,—Z, are given. There the residuals of S,—S, are no larger than are to be expected from errors of temperature and of observation, and the residuals of Z,—Z,, which are about 2.5 times larger than those of 8,— 8, will be in the main accounted for by the supposition that the residuals in S,—8, are largely due to small differences of temperature of the two tubes. Moreover, the numerous office-comparisons of tubes 1 and 2 give a curve of residuals for Z,—S,, which follows quite closely the residual-curve for 7,—S,, showing that the two tubes behave essen- tially in the same way. An examination of these tables shows that in eight well-marked maxima of S,—B, the time of the corresponding maximum of Z,— B was later by from —18 to +10 minutes, averaging +1 minute; that the average change of S,—B in the intervals was —0+.13, corresponding to —0°.005 F. in its true temperature; that the average change in 7,—B in the intervals was +0+.25, corresponding to +0°.004 F.; and that in these intervals the thermometer-fall varied between —0°.15 F. and +0°.40 F., averaging +0°.04 F. The intervals between the time of the maximum thermometer-reading and the time of maximum value of Z,— B varied from 7™™ to 74", averaging 40", The thermometer-fall in this interval varied from 0°.09 F. to 1°.05 F., averaging 0°.38 F. There were five well-marked minima. The average time of minimum of 8,—B was 2™ later than that of Z,-B. The average change in §,— B in this interval was —0«.05, corresponding to —0°.002 F.; the average change in Z,—B was +0#.55, corresponding to +0°.009 F.; the average rise of ther- mometer in the interval between the minimum of Z,—B and the minimum of S,—B was 0°.03 F. The intervals between the time of ‘minimum thermometer-reading and minimum value of 7,—B averaged 23™™; the average thermometer-rise in this interval was 0°.09 F, ‘When the temperature rises, if 7; heats more slowly than S, (and this supposition is necessary in order to account for the residuals of Z,— 8; by the supposition that Z, and S, are not at the same temperature), the temperature of the place will first reach a maximum and then begin to fall; a little later, S,, still rising in temperature, will equal the falling temperature of the place, and the temperature of 8; will then be at its maximum, and will begin to fall; still later, the temperature of Z, still rising, will equal the falling temperature of the place, and the temperature of Z, will then be at its maximum, and will begin to fall. At the time of the maximum of Z, it will be colder than S), since Z, is at the temperature cf the place and S, has already lagged behind that tempera- ture. By temperature of the place is meant that. which would be indicated by a perfect thermom- eter of infinitely small mass, lying in the place of the steel and zine bars. _Now since the average intervals of time between the maximum and minimum of the thermom- eter and the maximum and minimum length of 7, are +407" and +23", and the corresponding § 32.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 209 average thermometer-changes are 0°.38 F. and 0°.09 F.; while the average intervals of time between the maxima or minima of Z, and those of §, are 1™™ and 2™2, and the corresponding average thermometer-changes are 0°.04 F. and 0°.03 F., we may assume for this short interval that even if the thermometers do not give the true temperatures of the bars at the beginning and end of the interval, the difference in the temperature-errors at these two instants of time is small in com- parison with the temperature-change, 0°.04 F., in that interval, not exceeding, say, 0°.01 or 09.02 F.; so that within that degree of accuracy the thermometers measure the change of temperature of the place in the interval. Hence, since at the maximum or minimum of S| it had the temperature of the place, and at the maximum or minimum of Z, it had the temperature of the place, and the place in the average changed temperature in this interval by but 0°.04 F., it follows that in the average at the later of the times of their maxima or minima Z, and 8, did not differ in temperature by more than 0°.04 F. Now, the daily temperature-range in these comparisons from minimum to maximum or vice versa, in these cases varied between 2° F. and 30° F., averaging 17° F. If such a change in about eight and a half hours produced less than 0°.04 F. difference of temperature in the two bars at maxima- or minima-temperatures, it seems improbable, that in the office comparisons a change of 10° distributed over five or ten days should give a difference of temperature of 0°.3 F. to the bars. It is concluded, then, that the theory of unequal rates of heating of the steel and zinc bars does not datistastorily account for the large residuals. § B32. C. Can the large residuals be accounted for by the theory that the icnipeiaes of the thermometers is greater than that of the bars in rising temperatures and less in falling tempera- tures? Since Z,—S, increases in value by about 38 per degree F. of temperature-increase, if the thermometers in rising temperatures were hotter than Z, and S,, the value of Z,—S8,, computed with these thermometer-temperatures, would be too great, and the residual, which is computed Z,—8, minus observed Z,—8,, would be positive in rising temperatures and negative in falling tempera- tures. If the thermometers preceded the temperature of the bars by 0°.5 F., the residual would be +19, Experiments have been made to ascertain the rate of cooling of the thermometers. Two thermometers were placed in a metal water-cooler of about 20*¢ capacity. To cut off radiation from the heated thermometer they were separated by a board, which allowed communication of air above and below it between the two sides of the screen. The thermometers, Casella 21476 and Baudin 6131, were inserted through a closely-fitting cover till their bulbs were at the same level. After a few readings on each, Casella 21476 was taken out of the water-cooler, and its temperature raised by from 9° to 18° F., when it was replaced in the cooler, and both were then read at short intervals till their jamperatintes became the same within 0°.1 F. The experiments were repeated on three days. After heating on November 3, 1880, the two differed at 2" 21™ 30° by +1°.65 F., Casella 21476 being the hotter. This dittarence ecarlually diminished to +0°.06 at 3 p. m., thie temperature then being 51°.32 F.. Ou November 4 the difference diminished gradually from +1°.22 F, to —0°.03 F.in one hour. On November 5 the difference diminished gradually from +-1°.20 F. to +0°.17 F. in eighteen minutes. These experiments show that when this thermometer differs from the temperature due to its surroundings by 1° F., it may be expected to approach that temperature within 0°.1 F. in less than an hour. The bulbs of the thermometers other than Casellas, used in determining the temperatures of the tubes, had masses considerably less than these, and hence may be expected to take the tem- perature of the space where they may be more rapidly than the Casellas. Now, in the compari- sons of the tubes in the Lake-Survey office, the temperature very rarely rose at the rate of 0°.1 F. per hour or 2°.4 per day. Since in less than one hour the thermometers reduce their temperature- difference from the space they are in from 1° to 0°.1 F., we may expect that with a difference of 0°.5 at the beginning of an hour in which the temperature-rise is 09.1, they will be within 0°.1 F. of the temperature of the space at the end of the hour, or, in other words, that the thermometers give the temperature of the space occupied by them without greater errors than 0°.1 F. in the comparisons under consideration. But if the thermometers are very nearly of the temperature of the space occupied by them, it does not follow that the zinc and steel bars are of this temperature. In rising temperatures they derive their heat from the tube-walls and from the air inside of the tubes, and, as their masses are considerable, their temperature inevitably lags behind that of the 27 Ls 210 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. Ix, tube-wall and of the interior air, depending mainly, it is probable, on the tube-wall, since the interior air communicates with the exterior only by three small openings, the sum of whose areas does not exceed twenty-six square centimeters. But when in comparisons a maximum tempera. ture of a bar is passed, this bar is cooler than the space it is in immediately before the maximum, and is hotter immediately after. Hence, before such a maximum the spaces occupied by the ther- mometers (or the thermometers themselves within 0°.1 F.) are hotter than the bar, and are cooler afterward. Tf, then, the large residuals are due to the lagging of the temperature of the bars behind that of the thermometers, these residuals should change sign at the maxima- or minima-temperatures (or lengths) of the bars. An examination of the comparisons of tube 1 with the 15-feet brass bar in ice, made at the Cass farm in August and September, 1880, has shown that 4 and S reach their maxima on the average within about one minute of each other, even when the daily temperature- fluctuations are 15° or 20° F., and in this interval neither changes temperature on the average more than 0°.04 F. In office-comparisons with daily temperature-ranges rarely exceeding 2°.5, it may be supposed that the differences of maxima-temperatures of Z, and S, will be still less, so that the time of the maximum value of Z,— 8, may be taken as the time of maximum length of either Z, or S, without important error. Now, an examination of the table of comparisons of Z, and 8,, already given in § 29, shows that in no case did the times of maxima or minima values of Z,—S,, and of the thermometer-readings, differ by twenty-four. hours, while in no case did the residuals change sign within as little as twenty-four hours after a maximum or minimum, and in several cases they did not change sign for many days after a maximum or minimum temperature. If the large residuals were due to lagging of temperatures of the bars behind those given by the thermometers in rapidly changing temperatures, residuals of one-third the size running through many days would be ex- pected in the comparisons of 8, and B. None such were observed. It may be concluded, then, that while the theory that the temperature of the bars when they are heating or cooling lags behind that given by the thermometers is undoubtedly true, yet the amount of the lagging is not sufficient to account for the large residuals. § $3. D. Assistant Engineer E. 8. Wheeler first called attention to the fact that the residuals mwnight be accounted for by supposing that Z when heated or cooled took a set so that its length ata given date and temperature might depend on its previous temperatures. When a thermometer of glass which has long remained at ordinary temperatures is heated to 212° F., and is then allowed to cool in air, it is well known that its freezing-point will be found to have sunk several tenths of a degree, indicating an increase in the volume of its bulb, and that during many weeks there is a slow return towards the original volume. The melting-points of glass and zinc do not differ very widely. May zine in this respect behave when heated and then cooled in some degree like glass? To test this question it was decided to compare the zinc and steel bars in tube 2 with those in tube 1; then leaving tube 2 in the steady temperature of the comparing-room, to place tube 1.in another room and heat it through 20° or 30° F., and afterward to recompare its. bars with those of tube 2, to see if any change could be detected in their relative lengths. In order to eliminate any errors which might arise from lateral bending (if possibly any.slight bending existed) during the considerable expansion and contraction, the graduations at the ends of the 4-metre zine and steel bars lying nearly in the neutral axes of the bars were used in comparisons. The 4-metre intervals on these bars used in measurements of bases have already been designated for tube 1 by S, and %, and for tube 2 by 8, and Z. These intervals are parallel to the neutral axes of the bars at a dis- tance of 8°” from them in the same horizontal plane. The 4-metre intervals between the new graduations are parallel to the neutral axes of the bars at a distance of 1™™ below them in the same vertical plane. These intervals are designated by &{/, Z)’, So’, Z’. The new and old graduations differing about 1™™ in level, the microscopes were focased for an intermediate level. This gave some indistinctness to all pointings, increased the probable error of a microscope-pointing from 0.4 to about 1“, gave for the probable error of one comparison due to pointing errors alone 2, and a probable range in fifty results due to pointing errors alone of about 14". Otherwise the comparisons of S,/ and S,', and of Z,' and Z,/, were made like the comparisons of S; and Z, already described. But in the comparisons on and after June 10, 1881, only 81’, 82’, Le §§ 33,34] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 211 4', Z{ were compared. This permitted accurate focusing in all pointings of the microscopes. Temperatures during comparisons were determined as usual by three thermometers in each tube, the mean of the corrected readings in each tube (the thermometer at the middle of the tube having double weight) being taken as the temperature of the bars in that tube. The values of S,—S, and Z,—Z, and the relative expansions have already been given in § 24 as— S,—Sp= +432. 74-4 0. 24—0, 0605 (t—32) ~ Z,—Zy-= —66. 004.0. 81—0. 4366 (t—32) The values of 8,—S;', &,—8', 7.—Z,', 2,—Z,', were derived from numbers of comparisons vary- ing between 48 and 114. Since each of these differences is between intervals on the same bar, no temperature-errors or expansion-errors are to be feared, and the results should be accurate, pro- vided the conclusion already reached, that no serious lateral bending is to be feared, is true. From 48 comparisons on 35 days, S,— 4&1 =a, O7+ 0.41 From 114 comparisons on 76 days, S,—S_’=+777. 9440. 21 From 48 comparisons on 35 days, Z,—Z,'=+ 400. 14+ 0. 30 From 114 comparisons on 76 days, Z,—Z,'=+ 660. 03-40, 27 Combining these values with those of 8,— 8, and Z,—Z, there result— Si! — Sp! =+41517. 70. 5—0.0605 (t—32) ZL! —Z/=+ 193.9-+40.9—0, 4366 (t—32) With these equations, computed values of S,/—S,' and Z,/—Z,' have been deduced for the ob- served temperatures of the tubes. The residuals are obtained by subtracting observed values from the computed values. ; § BSA. In the first heating-experiments the bars in tube 1 were compared on four days with those in tube 2, then tube 1 was taken from the comparing-room to a distant room, where it remained twenty-four hours, the temperature of this room being kept steadily at between 70° and 80° F. Yube-1 was then taken back to the comparing-room and placed in the comparing-box by the side of tube 2; comparisons were then begun and were continued through six days. In reduction, when mercurial temperatures of the two tubes are different, the lengths of 8, and of Z,' are reduced to the temperature of S,/ and Z,/ by means of their coefficients of expansion, namely, Eg =24».866 and iz, =62".955 for 1° F. It is this corrected observed difference of lengths which is subtracted from the computed difference of length for that common temperature to obtain the residuals in the fol- lowing table. The first column gives the date of comparison; the second and third the temperatures, ¢, and t,, of tubes 1 and 2; the fourth gives the observed S,/—S,' corrected for difference of temperature of 8, and S,,; the temperature ¢, being taken as the standard; the fifth gives the residuals of S,/—S,’, computed as already explained; the sixth gives the observed Z,/—Z,' corrected for difference of temperature of the two bars to the temperature ¢,; and the seventh gives the residuals of 2,'—Z,’: Comparisons of S,', 8,', Z,', and Z,!. BEFORE HEATING OF TUBE 1. Fa ann apne, 1 Sh Si — Sx Hee Zi! —Za eared Date. t ta computed. computed Bue, : corrected. | Compu” | corrected. | CODES observed. observed. e 1881. oF, oF, BB Me ' we vs March 14, 9:40 a,m.....-.--- 39.74 | 39.82 | +1517.4 0.2 +195. 4 -5.0 "15, 9:39 a.m........--, 40.58 40. 68 +1520. 4 —3.2 +198.7 | —8.6 16, 3:36p.m.........- | 41.79 | 41.80 | 1525.2 8.1 +194. 8 5.2 17, 9:27 a. 2.0.2 ee ; 41.90 41.90 | +1526.0 —89 | +1947 —5.2 Moaing oct ccenvecsneens|nddedsaedel ecauneree| een teesnics SBT Wend raeaens —6.0~ 212 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnar. Ix, At 9:30 a.m. on March 17 tube 1 was placed in a room the temperature of which was kept constantly between 70° and 80° F. It remained in this room until March 18, 9:15 a. m., when it was again placed in the comparing-box. 7 Comparisons of 8', S:', Z’, and Z,/—Continued. AFTER HEATING OF TUBE 1. 7 eels ty 1 Sof =a 1 Fat oe Date. a te seats. scmputed eek Sommputed observed. observed. ; 1881. oF, oF. B B mv KM | March 18, 10:07 a.m...--..-- 73, 33 43. 64 +1616. 7 —99.7 +357. 4 —168. 6 1 18, 10:33 a.m..---..-- 67. 42 44. 46 +1594. 7 —77.5 +458. 2 —269. 8 | 18, 11:23 a.m...--..-- 61.10 45. 08 -+1602. 1 —85. 2 +389. 4 —201.3 ‘ 18, 12:31 p.m......... 55. 81 45. 76 +1556. 0 —39.1 +876. 5 —188. 6 18, 3:10 p.m....-...- 49. 76 45, 54 +1532. 4 —15.5 +282. 5 — 94.5 | 18, 4:48 p.m.......-. 47.91 45. 47 +1529. 4 12.5 +287. 3 — 99.3 | 18, 8:15 p.m......... 46, 51 45.12 | +1526. 2 ae) +249. 8 — 61.7 | 19, 44, 25 43. 68 +1524. 3 — 7.3 | +2377 — 49.0 | 19; 44. 02 43. 52 +1515. 3 + 1.7 +230. 4 — 41.6 19, 8:08 p.m.......-. 43, 88 43. 50 +1513. 5 + 3.5 +233. 7 — 44.9 | March 20, 9:39 a.m......-.. 43, 66 43, 38 +1511. 8 + 5.2 +225. 9 — 37.0 20, 43, 70 43, 46 +1522. 8 — 5.8 4231.1 — 42.3 | 21, 43. 48 43. 33 +1512. 5 + 4.5 +226. 4 — 37.5 | 2h, 43. 52 43, 32 +1514. 8 + 2.2 +224. 4 — 35.5 22, 43. 33 43.12 +1513. 6 + 3.4 +220. 6 — 31.6 I 22, 43. 03 42. 90 +1521.1 — 4.0 +218. 4 — 29.3 : 23, 42. 76 42, 59 +-1516. 9 + 0.2 +220.7 — 31.5 | Means for dates March 20, 9:39 a. m. to March 23, 9:18 a.m . fi 008 lwadesieenasies — 35.0 § 3%. From March 23, 1881, to April 14, 1881, both tubes remained in the comparing-room, whose temperature rose gradually to 45° F. The tubes were then compared from April 14 to April 18, 1881, and on April 18 tube 1 was again taken into the heated room, whose temperature was kept for twenty-four hours at between 70° and 80° F. Tube 1 was then replaced in the com- paring-box and comparisons were made daily until May 5, 1881. The results are given in the fol- lowing table, which is arranged like the preceding one: Comparisons of 8,', 8,', Z,', and Z,/—Continued. BEFORE HEATING OF TUBE 1. Beiaes sya 1’ — 82! ’ a 1! —Za! Date. tu te S1’—S?_ | computed | 2-22", | computed corrected. aan corrected. Thee observed. observed. oF. oF, kK MM KM Me April 14, 45. 06 45. 11 +1513. 4 +3.5 +218. 3 —30. 2 15, 44. 90 44, 92 +1524. 9 —8.0 +216. 2 —28.0 15, 45. 00 45. 04 +1511. 9 +5.0 +213. 0 —24.8 16, 45.10 45.13 +1523. 6 —6.7 +212. 6 —24.5 16, 45. 60 45. 76* | +1517.2 —0.3 +221. 0 —33. 2 17, 11:14 a.m.... 46. 00 46. 07 +1524. 5 —7.6 +210. 5 —22. 8 * 18, 11:57 a.m... 46.24 46. 21 +1522. 9 —6.0 +204. 2 —12.8 MEANS c2cds.uetomd niall dp aneeoenal damask suanicdenesmenens SH2E9) | [aicccwsrecinjesie —25. 2 * Thermometers disturbed by the proximity of the hand in adjusting tube. At 12:30 p. m., April 18, tube 1 was placed in a room, where it remained until 9:30 a. m., April 19, when it was replaced in the comparing-box. The temperature of the room remained constantly between 70° and 80° F, § 35.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 213 . Comparisons of 8,', 8,', Z,/, and Z,'—Continued. AFTER HEATING OF TUBE 1. | : Reel Resta j te = Pin a * Date. po ta ead: eoiane est cnet. Sonepaes P observed. observed. i 1881. oF, oF, Mw my be Me | : April 19, 10:12 a.m.......... 78. 32 48. 28 +1609. 1 —92.4 +387, 2 —150.5 | : 19, 11:16 & m.......... 68. 40 50. 12 +1607. 4 —90. 8 +487. 3 —301. 4 19, 12:11 p.m.......... 63. 22 50. 76 +1584. 3 —67.7 +384. 0 —198, 4 | d 19, 56. 07 50. 63 +1543. 7 —27.1 +325. 3 —139. 6 19, 58.75 50. 33 +1527. 8 —11.2 +288. 3 —102. 5 19, 51. 22 49. 57 +1515. 6 + 1.1 +261.5 — 75.3 | 20, 49, 08 48,45 +1514.1 + 2.6 +241.7 — 55.1 20, 49, 12 48, 67 -} 1525. 8 — 91 +252. 4 — 65.9 | 20, 49. 23 48. 80 +1521.1 — 44 +239. 7 — 53.2 April 21, 48,72 48. 40 +1515. 3 + 1.4 +236. 5 — 49.8 21, 48. 89 48, 62 +1522. 7 — 6.0 +232. 3 — 45.7 22, 49. 02 48. 87 +1517. 3 — 0.6 +233. 3 — 46.8 22, 49, 82 49, 67 +1521. 9 — 5.3 | +282. 8 — 46.7 5 23, 49. 88 49, 72 +1522. 8 — 6.2 +228. 1 — 42.0 23, 50. 64 50. 50 +1515. 9 + 0.7 4221.7 — 35.9 24, 51. 30 50. 94 +1516. 5 + 0.1 +221. 8 — 36.2 25, 52. 54 52. 58 +1515. 1 +14 +4227. 2 — 42.4 27, 9:24a.m......-... 55. 96 55. 96 +1511.0 |! + 5.3 +214. 2 — 30.9 29, 10:30 a.m.......... 56. 91 56. 89 +1513.7 + 2.5 +213. 7 — 30.8 May 1, 11:31a.m........-. 54, 35 54, 24 +1523. 4 — 7.0 +204. 2 — 20.1 8, 9:14 a,m-.......-. 52. 96 52. 89 +1512. 3 + 4.2 +2138. 5 — 28.8 5, 11:04 a. m....-..-2. 51. 54 51. 50 +1512. 6 + 3.9 +207. 8 — 22.5 Means for dates April 21, 9:15 a.m. to May 5, 11:04am...) — 0.5 |............) — 368 ; At 11:30 a. m., May 5, tube 1 was placed in a room, where it remained until May 7, 9:30 a. m., when it was replaced in the comparing-box. The temperature-range in the room was indicated by maximum and minimum thermometers, whose readings were 58° and 50° F., respectively. It was hoped that the temperature would fall by a larger amount during the night than it did. Comparisons of S,’, S,', Z,', and Z,/—Continued. AFTER COOLING OF TUBE 1. Resldnals, Besuiaela Si/— Sa! 1 — 82! Zi'—Za Maer Date. a corrected. ae corrected. ent observed. observed. 1881, oF. oF, w “ Mw we May 7, 10:16a,.m.....-..... 55. 52 55.16* | +1515. 4 +0.9 +198. 7 —15.0 7, 2:50p.m........... 55. 72 55.88 | 415216 | —5.3 +212. 9 29.5 8, (t) 55. 85 55.92 | +1520.8 | —4.5 +202. 6 —19.3 May 9, (t) 57, 38 57. 46 +1518. 7 —2.5 +210.1 —27.4 10, 9:25a.m........... 59. 54 59. 66 -+ 1519.5 —3.4 +210. 3 —28.6 11, 9:40 a.m........-.. 61. 73 61. 85 +1515. 3 +0.6 +206. 2 —25.5 14, 10:52 a.m......-..-. 65. 27 65. 26 +1513. 5 +2. 2 +196. 5 —17.2 17, 9:144,m........-.-| 62.88 62. 83 +1512.1 +3.8 +196. 2 —15.9 20, 12:14p.m ......-.-- 59. 40 59. 38 +1520. 6 —4.5 +193. 6 —11.8 23, 12:08p.m.......-..- 59. 53 59. 52 +1519. 9 —3.8 +191. 6 — 9.8 25, 9:39 a.m......--6-- 60. 88 60. 86 +1517.7 -1.7 +191.9 —10.7 Means for dates May 9to 25, inclusive .........-.--------- “Lo |vesevimeewe —18. 4 * This observation not to be used, as the thermometers were disturbed by contact with the hand. t Time not recorded. STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuap. IX, 214 § 86. From May 25, 1881, to June 10, 1881, both tubes remained in the comparing-room, when further comparisons were made before and after heating. Prior to June 10, 1881, readings were made both on the proper graduations of the bars and on those nearer the neutral axes, which are about 1™™ lower. Hence, as the microscopes were focused for an intermediate point 0™™.5 distant from that of most distinct vision and near the limit of distinctness, considerable pointing-errors are to be expected. From June 10, 1881, to July 2, 1881, only the neutral-axis graduations were pointed at and the focusing was good. Comparisons of 8,', S,', Z,', and Z,/—Continued. BEFORE HEATING OF TUBE 1. reg | Reels, ‘ S Pos f Pt | Sy'— So! 1 2 Zilon Z,! L ‘2 Date. t te ares computed 1 computed ate a al corrected. | CO Duy. | corrected. | COMPUT observed. observed. 1881. op, | oF, B be b M | June 10, 10:22 a,m......-... 61. 38 61. 34 +1523. 7 ~- 1.7 +195. 3 —14.2 V1, 9:52 @i Mev cescciccis 61. 01 60. 96 +1526.8 | —10.8 +194. 0 —12.9 13, 10:26 a.m .-. 61. 88 61. 90 +1522. 9 — 7.0 | +191. 6 —10.9 14, 9:40 a.m... 63. 56 63. 60 +1522. 8 — 7.0 +191. 3 —11.3 15, 9:08 aoM o2e x yo—s35 would be introduced into the expansion of B. Collecting results, the equations of condition for the values of Hp, Episys, H,, and Hs may now be written, Weights from Adopted probable errors. weights. From § 57, H,— 39#,945 + 0.038 (at 62° F = 0.7 1 From § 67, Br ya¢— 5".728 + 04.019 (at 62° I.) 2 2.8 4 From Chapter II, § 2, HH, — 5.371 + 04.018 (at 62° FB. (=a, 3.1 4 From § 57, Fn iwz¢— 1".0937 HL, — 04.006 + 0+.009 (at 62° FF. ms 12.3 8 From § 18, 4 Brie — Hs, + 1¥.335 + 0".022 (at 62° F.) =v 2.1 3 From § 56, H,— His —14".833 + 04.012 (at 62° F. ae. 7.0 6 The probable error of 39.945 in the first equation has been derived from the probable errors of the two independent values of the expansion H, for different temperature-ranges, given in Chap- ter II, § 14, by expressing the coefficients of the first and second powers of the temperature in the general pale for the length of the brass bar in terms of these independent expansions. In the fourth equation the probable error of 0#.006 is obtained by multiplying the probable error of the relative expansion resulting from the comparisons of the yard with a part of #1876, given in § 9, by the ratio of the prolonged yard to the metre. The probable errors of the other constants are given in the sections from which the values are derived. § 8. The solution by least squares of the equations of condition just given, the adopted weights being applied, gives E, =39.787-4£0.152, at 62° F. for 1° ; Ene 5.885-40.043, at 62° F. for 19 (1) EB, = 5.37440.055, at 620 F. for 1° B,, =24.927 40.160, at 62° F. for 1° The facts that the absolute-expansion determinations of the 15-feet brass bar indicate that the rate of expansion changes with the temperature, while the comparisons of S, and B, of R1876 and S,, and of R1876 and Clarke yard A indicate no dependence of the relative rates of expansion on the temperature, show, so far as the comparisons serve to settle-the question, that the other bars change their rate of expansion with the temperature by the same amount as does the brass bar. The length of any bar at the temperature t° F. may be written in terms of its length Ls), at 32° F., in the form 4 (2) L=Ly [1+ (t—32) +2 (t—32) Its rate of expansion will be aL (3) ap = Er = Lala t+ 28 (t—32)| §§ 58,59.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 239 The sum of its expansions between 62° and ¢° will be— et J Geet Lol (e-+008) (¢ 62) + 2 (¢—62)% 2 and its length will be— (4) L=Li,+ Leal (a-+603) (t—62) + 3 (t—62)'] * The conclusion that the bars &, R1876, and Clarke yard A change their rate of expansion with the temperature at the same rate a8 does Bi in B= By, [149878 (10) (t—32) +1852 (10)-” (¢—32)"] § 57, carries with it the conclusion that when their lengths are written in the form (2), 7=1852 (10)-” for tiem all.. The values of the expansions in (1) for any temperature may be obtained as follows: For t=62°, (3) and (1) give— E, at 620=39.78740.152= By (a-+603) : Brow at 620= 5,885 4.0.043— i (a,-+6033) (5) at 620= 5.37440.055—Ay (a5+608) li Ey, at 620=24.997 4 0.160— . "o(a-+ 60,2) The lengths at 32° F. in aise of these bars are sufficiently well known (as will be seen by comparing the values substituted below with the final values in § 66 following) to write them in the above equations without introducing sensible errors in the a; the value of 7 is known; and hence from the above equations the values of the 2 may be dedueed. Substituting them in (3) there result the final values— (6) E, =3",998765 [9839 (10) °4+2.x 1852 (10)? (t—32)] (7) Erg yo= 1.000002 [5773 (10)-*-+2 x 1852 (10)-* (t— ea (8) E, =0",914209 [5767 (10)-°4-2 x 1852 (10)? (t—32)] (9) Es, =4™.000264 [6120 (10)"-42 x 1852 (10)-” (¢—32)] The coefficients in metres, or the lengths of the different bars at 32° F. in metres, are derived as follows: : In (6), from S,, § 18; §,— B=1499, § 56; and Foerster’s value of R1876, § 67. In (7), from Foerster’s value of the metre, R1876=1™+ 247+4 104.31 (£2 — 15° C.). In (8), from Clarke’s value of yard A, Chapter IJ, § 2, and his ratio of yard to metre, namely, 1.09362311. In (9), from Foerster’s value of 21876, § 67, and the value of S, in terms of F1876, § 18, namely, S—=4 B1876 — 68,124 1+.3349 (t0—59° F.). The difference in computed lengths of 8, at 92° F., when derived first from its length at 62° F. with a constant expansion, namely, that for 629, and when derived with the varying expansion given above is but 6.7, the latter value being the larger. Now, the probable error in the value of Es at 62°, namely, + 0.160, would give a probable error of 4.8 at 929 F, in the length of 8, when ompated from its length at 62° F. Itis then apparent that there is but little precision in the term which gives the variation of # with temperature, and were it not that Fizeau’s work makes such a term payable a priori, it would be better to reject it entirely for all the expansions given above. § 59. Having obtained the necessary rates of expansion, the lengths of the different bars may now be deduced. From § 9 the value of the mean temperature of the comparisons in the Lake-Survey office of Clarke yard A and R1876 is 579.92 F. Hence, at this temperature the relative length of the two is independent of errors in the relative coefficient of expansion. Colonel Clarke’s value for the length of yard A, derived from Ordnance-Survey standard Y;; is, Chapter IJ, § 2— ‘Clarke yard A at 62° F.—0°.99997695 + 0¥.00000013 Expansion — 07.000005874 + 07.0000000195 for 19° F. The mean temperature of Colonel Clarke’s comparisons of yard A and Y;; was 579.71 F. In the computation of probable errors, to simplify the work it is best to count the length of Clarke 240 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. IK, yard A from this temperature. Its length follows at once. The probable error of its length at 57°.71 is derived from that at 62° by allowing for the effect of the probable error in the expansion for this difference of temperature, and we have— Clarke yard A at 579.71 F.=09.99995175 + 0¥.00000010 At 579.92 this value becomes, using his expansion, Clarke yard A at 579.92 F.=07.99995298-L0¥.00000010 But, § 12, at 579.92 F., R1876—129 Clarke yard A+5296+.6£0+.40 Hence, At 579.92, R1876—19.09388063-0¥.00000045 From this value of R1876, at the temperature 579.92, and the general expression for R1876, namely, R1876— R1876, [1+ 4 (t—32) + 6 (t—32)"] where, § 58, a==5773 (10) and #==1.852 (10)-’, there results, R1876,,=15.09371561 and finally, R1876 = 15.09371561 [1+5773 (10)~ (¢—32)4+1.852 (10)-° (t—32)?] The uncertainty in the length of 2 1876 is least for a temperature near 57°.92, the mean tem- perature of comparisons of Clarke yard A with it, since the mean temperature of Colonel Clarke’s comparisons of Clarke yard A with Ordnance-Survey standard yard Y;; was 57°.71, so that slight errors in the values of the rates of expansien have little influence on the value for R1876 at. this temperature. § 6O. The values of §, in yards and in terms of #1876 may next be found. From § 18 M b K R1876—S8,,+45.54-+ 0.20—0.3485 (t—54.16) R1876—=S8,,4+ 4.4640.12—0.3027 (t—57.85) R1876=S8,5+12.78+ 0.18—0.3366 (t—54.11) R1S76=S,,4+ 9.704 0.15—0.3992 (t—56.86) where the relative expansions have been adjusted to conform to the adjusted values of Hpi, and #,, and the numerical temperatures are the mean temperatures of the comparisons of the different metres of 8, Taking the means of these separate means and then, by aid of the separate expan- sion terms, reducing the equations to this mean temperature, which, as the reduction is small, will involve but slight errors arising from the errors in the expansions and insensible changes in prob- able errors, there result— R1876—S, ,+44+.9940+.20 at 55°.74 BF. R1876—S8,,+ 5#.1040.12 at 559.74 F. RK1876=S,,,+ 12.2340+.18 at 559.74 FB. BA1S876—=S,,,4+10".1540+.15 at 55°.74 F, and adding S,—=4 81876 —72.4740+.33 at 55°.74 F. and since the difference of the adjusted expansions of S, and 421876, § 58, is +1.387 per degree F. we may write ‘: 8, = 4 R1876—72#,4740".3341".387 (t—55.74) as the value of S,, not as obtained from direct comparisons of S,, 8,, and R1876, but from an adjusted value of the relative expansion of §, and R1876. From the value of R1876 in § 59 there results #1876 at 559.74 F, —17.09386664 Substituting this in the value of S, there results 8, at 559.74 TF. —49.37538730 From this value and the value for #, at any temperature in § 58, may be written for any tem- perature S,=49.37474713 [146120 (10)-° (t—32)4 1.852 (10)-9 (¢—32)?] §§ 60-63.} CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 241 Clarke yard A was derived from Ordnance-Survey standard Y;; by comparisons whose mean tem- perature was 579.71 F. #1876 was derived from Clarke yard A, § 9, by comparisons whose mean temperature was 579.92 F. 8, was derived from #1876 by comparisons of each metre of S, with R1876. The mean of the mean temperatures of comparisons of the several metres was 55°.74. Hence it is seen that the length of S, at 559.74, as derived from Y,;, is very nearly free from the effect of errors in the rates of expansion of the different bars. § G1. The mean temperature of the comparisons of S, and B, § 56, was 51°.687 I’., for which temperature S,— B= + 1206+.88+.0+,14 The difference between the adjusted values of H, and H,, § 58, is E,—Es, = 14,860 Hence B=S,—1206+.88 + 0#.144 14.860 (t—51.687) or, substituting for 8, its value in terms of 81876 from § 60, B=4 R1876—1284+.97 at 51°.69 F. and substituting the value of #1876 in terms of the yard, § 59, B= 49.37310732 [1+ 9839 (10)-® (t—32)+41.852 (10)-® (t—32)?] § 62, The values of S,, Z,, Z will be derived directly from comparisons with each other and with 8,. From § 60, S, at 55°.74—49.37538730 From § 58, Hs, =24",9274.0+.0148 (t—62) From § 21, Ey, — Es, = +0+.061 The mean temperature of comparisons of S, and 8, was 42°.75 F. From the value at any tem- perature of 8,, given in § 60, there results : S, at 429.75 —49,37503588. But, § 21, (1) S,= S,4+32#,09-+ 0#,.18 — (0".0605 + 0".015) (t—42.75) Substituting the value of S, at 42°.75 F. 8, at 420.75 F.—49.37507085 Combining the values of H;, and HZ, —H, given above, there results Ei, = 24".866+ 0".0148 (t— 62) The expansion of 8, between 42°.75 and any temperature ¢ will be— 42.75 "Bs dt = 244.581 (t- 42,75) 404,074 (t—42.75)? Hence at any temperature, §8,=4".37507085 + 0¥.00002688 (¢— 42.75) + 07.00000000809 (t—42.75)? or ; 8,=49.37478294 [14 6105 (10)—* (¢—32)+1.852 (10)-* (¢— 32)?] In § 60 the value of S, at any temperature in terms of #1876 is given. Substituting that value in (1) and neglecting the probable errors, there results : §,=4 R 1876 —58#.40+1".326 (t—42.75) § 63. Since the lengths of the bases measured with the Repsold apparatus all depend on the length of S, it is important to kuow the probable error in this length. But the error in its length depends on the errors in the value of the Clarke yard A; on the errors of intercomparisons of Clarke yard A, 1876, S,, and 8,; and since the intercomparisons were not all at the same temperature, the errors of the rates of expansion also enter. These rates of expansion have been 31 Ls 242 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuap. Ix, adjusted in § 58 so that their values are not independent. The errors are therefore entangled, and to obtain the probable error in the value of §, it is necessary to express this value in terms of the independently observed quantities on which it depends, and then derive the probable error of 5S, from the function of these observed quantities so-obtained. The computation is somewhat long and hence is not given; the following is the resulting value of the probable error squared for the value of 8, at ¢° F., the unit being the millionth of a yard: (p. €.)?%,=3.56—0.1109 (t—t,) +0.0306 (t—t,)?-+ 0.00027(t— t,)? 4-0.00016 (t+ t, —124) (t— t,) +-0.00000389 (tt, — 124)? (t—1,)2 in which ¢,=55°.74 F. is the mean temperature of comparisons of S, and R 1876, and t,=42°.75 KF, the mean temperature of comparisons of 8, and 8}. The value of 8, is also given in terms of #1876 in §62. The error in this value depends also on entangled errors which are less complex than in the preceding case, as there are fewer steps connecting the standards. Computing the probable error squared of S, at any temperature expressed in terms of #1876 it is found to be, the unit being 1+: (p. e.)? S;=0.1413+0,00952(t- t,)? +0.00023(t—t,)? This is merely the probable error in the derivation of §; from R 1876 through &; and t and t, are the same as stated above, namely, f=55°.74 F., and 1,=42°.75 F, § 64. From § 26 Zn= S.—305+.55 + 04.46 at 49°.50 F. or substituting for 8, its value from § 60, Z,=49,37488398 at 49°.50 Since, § 26, K,,—E,, = + 38.465 + 0#.039 it follows that Z,—S%=0 at 57°.44 BF, The rate of expansion of Z, is, therefore, E,,= Es, +38.465 or, by § 58, E,, =63+.392+ 0+.0148 (t—62) and hence at any temperature Z,=49 37488398 + 07.00006912 (¢— 49.50) + 0.00000000809 (t—49.50)? or, in a different form, Z,=45.37367686 [1415740 (10)-° (t—32)+4+1.852 (10)-* (¢—32)?] From § 23 . 4,=Z,—70".69+ 04.60 at 429.75 F. Substituting for Z, its value above, 4,=45.37434045 at 42°.75 FB. Since, § 23, E,, —E,,=—0".437 E,, =62",955+ 0".0148 ({—62) Hence, at any temperature 4, =45.37434045-+ 0Y.00006854 (t—42.75) + .0¥.00000000809 (t—42.75)? or, in a different form, 4,=49.37360458 [1415632 (10)—9 (t—32)+41.852 (10)—9 (t—32)?] From the values of 8, and Z,, §§ 62, 64, it follows that Z,—8,=0 at 60°,292 F, §§ 64-66.] CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 243 § 65. Approximate values for the lengths of the steel and zinc bars of the standard metre de- scribed in Chapter VIII, § 26, and designated as T1876 have been derived in the following way: Between May 28 and June 28, 1878, both bars of 71876 were compared at high tempera- tures with each of the metres S,,, 8,2, 8,3, and §,, into which the steel bar & is divided. Comparisons with S,, were made on ten days and with each of the other metres on five days. The number of comparisons made on a day varied from one to four, but was usually two. The values of the several metres making up the length of S, in terms of #1876 are given in §60. Combining the observed differences of length between (M71876), and these metres of S, with the values of these metres, the values of (Jf71876), in terms of R1876 resulted. The mean of the comparisons of each day was used as a single result, no matter how many comparisons were made on that day, and the corresponding mean temperature was taken. The temperatures varied between 679.40 F. and 75°.90 F. Taking general means of differences of length and of temperatures, there resulted, (MT 1876) =F 1876—30«.06 at 70°.90 F. By the same number of comparisons there was derived in the same way, . (MT1876),=R1876+4 6324.78 at 70°.90 F. Both bars of MT1876 were compared directly with R1876 at low temperatures between Jan- uary 8 and February 5, 1881, in connection with some comparisons of #1876 with S,. Comparisons were made on twenty-four days, two comparisons usually being made on each day. The tempera- tures varied between 289.72 F. and 34°.44 F. The mean results are as follows: (MT1876),=R1876— 47.96 at 319.21 F. (MT1876),—R1876+-210+.48 at 319.21 F. Combining the results of the high- and low-temperature comparisons, we have: Expansion of (1/71876),=expansion of R1876+4 0.451 for 1° F. Expansion of (£7 1876),=expansion of R1876+10*.639 for 1° F. From these may be written: (MT1876) = R 1876 —47+.6 + 0".451(t—32) (MT1876),= RF 1876+ 218.9 + 10".639(t—32) From these values the residuals of the daily mean observed differences from R1876 have been computed. For the high-temperature comparisons the residuals vary between +3.1 and —4.4 for (MT1876), and between +3+.7 and—3.6 for (J{T1876),. For the low-temperature comparisons the residuals for (1£71876), vary between +11".6 and —10«.8, while for (MT 1876), they vary be- tween +9#.7 and—5«.3. The comparisons of both bars of T1876 with 81876 give plus residuals amounting to from +6 to 11¢ on February 1, 2,3, 4,5. The fact that these residuals are of the same sign for both bars during these days points to a temperature or other error in one of the standards, as this would affect both bars of M7'1876 in the same way. As the temperatures were -very low at this time, and there were three persons in the comparing-room, it is probable that the large residuals are dae in part to differences of temperature of #1876 and MT 1876. If those days had been rejected, the length of (M7T1876), at 32° would have resulted 2.1 less than that given above, and of (71876), 2.4 less, and the range in the residuals, for either, would have been much reduced. An uncertainty of about 1“ therefore exists in the values of these bars in terms of R.1876 at the temperature of 319.21 F. § 66. The resulting lengths and expansion of the different bars may now be collected. Clarke Yard A at 62° F.= 0¥.99997695 + OY.00000013. (Chapter II, § 2.) Clarke Yard 4, adjusted rate of expansion, E,= 0".914209[5767(10)~+-3.704(10)-* (t—32)] (§ 58.) R1876=19.09371561[1 +5773(10)-°(t—32) +.1.852(10)-*(t—32)?] (§ 59.) 8,=49.37474713[1 + 6120(10)-°(t—32) + 1,852(10)-(t—32)"]. (§ 60.) 244 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cnap. IX, Rate of expansion of 8,= Ey, =4".000264 [6120(10)~° + 3.704 (10)-(t—32)]. (§ 58.) §, =4°.37478294{1+4+6105(10)*(t—32) +1.852(10)-*(t—32)"]. (§ 62.) (Si)j-s, = 437554481 at 60°.292 F.=13".12663443. E,, =24".866-+ 0#.0148 (t—62). (§ 62.) Z, =4°,37367686[1 + 15740(10)—*(t—32) + 1.852(10)-*(t—32)"]. (§ 64.) E,,, =63".392+4-0#.0148 (t—62). (§ 64.) Z, =4.37360458 [1415632 (10)-(t—32) + 1.852 (10)-*(t—32)?]. (§ 64.) E,, =62+.9554 04.0148 (t—62). (§ 64.) B =4°.37310732 [149839 (10)-°(t —32)4 1.852 (10)-*(t—382)"]. (§ 61.) The probable error squared in millionths of a yard in the above value of 8, is (p. €.)? 8;=3.56—0.1109 (t— t) + 0.0306 (t—t,)2+0.00027 (t—t,)? +0.00016 (t+ t,—124) (¢—t,) +0.00000389 (¢+ t,—124)? (t—t,)? in which t,=55°.74 F. and t,=42°.75 F. (§ 63.) In terms of R1876, § 62, §,=4 R1876 —58.40+14.326 (t—42.75) this value depending on the adjusted relative expansion of & 1876 and S,; and the square of the probable error of this value, § 63, is in microns, (p. e.)? S:=0.1413 + 0.00952 (t—t,)? + 0.00023 (t— ts)? in which t,=55°.74 F. and t,=42°.75 F. The value of 8, may also be derived solely from the results of comparisons of R 1876 with 8., § 18, namely, S8,=4 R1876— 684.12 + 14.3349 (¢—59) and of S, with S,, § 21, namely, Si =S,+ 32.09 — 0.0605 (¢—42.75) These give S,=4 B1876—37+.01+ 14.2744 (t—59) a value which is independent of the absolute expansion of & 1876 derived by adjustment. The square of the probable error of this value of S,, derived from the probable errors in the results of comparisons of S, and S,, and of S, and 1876 is, in microns, (p. ¢.)? S; = 0.1480 + 0.00048 (t—59)? + 0.00023 (¢— 42.75)? From § 65 we have for the lengths of the bars in metre 171876, (M7T'1876),—R1876— 474.6+ 0.451 (t—32) (MT 1876),— R 1876 + 2184.9 + 10#.639 (t—32) § 67. The lengths of the bases, measured with the Repsold apparatus, will be expressed in terms of S;. The value of S, in terms of R1876, and the probable error of this value, have been’ given in § 66. Hence, when a precise value of 21876 is obtained, the bases can be accurately expressed in terms of the metre. In the mean time the following gives the present information about R1876. Through the kindness of Professor Foerster of the Kaiserliche Normal-Eichungs-Kommission of Berlin, which it is desired warmly to acknowledge, R 1876 was compared in 187879 with an entirely similar steel metre made for the Kommission by Repsold, and designated as R1878. The errors of the subordinate graduations of R1876 and of the decimeter D1876 were also determined by the Normal-Eichungs-Kommission. It was hoped to obtain in time for insertion in this report the details of these comparisons, but they have not yet been received. R1878 has since been compared by the Comité International des Poids et Mesures with a unit which will probably not differ by more than 1 or 2 microns from the metrical prototype (Comité International des Poids et Mesures, Procés Verbaux, 1880, p. 106), and it is supposed that Professor Foerster’s statements in his letter of September 15, 1880, refer to these comparisons. CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 245 § 67.] It is hoped ultimately to obtain and publish the comparisons of #1876 with R1878 and its value in terms of the prototype metre. The following letters give all the information that has thus far been received from the Normal-Eichungs-Kommission with reference to our standards depend- ing on the metre. RESULTS OF COMPARISONS OF LAKE-SURVEY STANDARD METRE (221876), BY PROFESSOR W. FOERSTER, BERLIN. BERLIN, den 16. April 1879. In Beantwortung des gefiilligen Schreibens vom 17. v. M. erlaube ich mir, Ihnen zunichst ein Verzeichviss der blossen Eintheilungsfehler Ihres stiihlernen Meterstabes von Repsold und des zugehérigen Decimeters zu iibersenden, indem ich beziiglich der noch restirenden, von Ihnen dringend gewiinschten anderweitigen Festsetzungen Folgendes ergebenst bemerke: Der hauptsiichliche Grund der Verzégerung besteht darin, dass wir durch dringliche laufende Aufgaben bisher noch immer verhindett worden sind, absolute Ausdehnungsbestimmungen zu machen. Indessen sind wir wenigstens im letzten Winter dazu gelangt, gute relative Ausdehnungsbestimmungen eines Stahlmeters und eines Messingmeters gegen einen Platinstab zu machen, dessen absolute Ausdehnung ziemlich nahe bekannt ist. Die Ergebnisse dieser Bestimmungen werden in zwei bis drei Wochen so weit abgeschlossen sein kénnen, dass ich hoffe in 4 bis 5 Wochen Ihnen die Lingen Ihrer Maassstiibe gegen den erwibnten Platinstab und die zugehérigen Ausdehnungswerthe zu tibersenden. Einige ungefiihre Bestimmungen werden vielleicht schon friiher Ihnen zugehen kénnen. Wir werden zur Beschleunigung der Sache von Ihrer Ermiichtigung einer Kostenliquidation, die wir bisher in allegemein wissenschaftlichem Interesse zu unterlassen beabsichtigt hatten, nunmehr vollen Gebrauch machen. Kaiserliche Normal-Eichungs-Kommission. FOERSTER. To the OFFICE OF UNITED StTaTES LAKE SURVEY, Detroit, Mich. Beglaubigtes Fehlerverzeichniss der stimmtlichen Striche eines biegungsfreien Strichmaasses von Stahl mit Theilung von 1 Meter Lange auf Platin und einer zugehorigen gleichfalls auf Platin getheilten Decimeterskala von Stahl. Verfertigt von A. Repsold Sohne in Hamburg, zur Priifung eingereicht von dem Office of United States Lake Survey, in Detroit, Mich. — I. DAS STRICHMAASS VON 1 METER LANGE. i ne Bos * milimeter b, Die Minimetersriche a leanne eee al : 5 : : tee Fehler. eat Fehler. Jhell- | Febler., ake Fehler. aoe Fehler. ee Fehler. || spell. Fehler. Hi o | 0 0 25 +2 50 41, 5 0 0 0 0 0 0.0 ' 0 1) = 2% 8 +48 51 +1, 76 0 1| 42.6 1| —0.5 401 —1" 2 -1 27 +2 52 42,0777 0 2) +24 2) 25 02, 41 | 3 0 28 «42 53| 41 78 0 3) 429 |), 5 0.3 +1 4 0 29 +43 54 -1/ 79 0 4) 42.4 4) 1d 0.4, +2 1 5 +1 30 +3 55 41) 380 —1 5) 411 5| —1.6 Oo (Z.—S,)? 262 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. X, cut-off marks. The average distance measured per day was 292 metres. The greatest distance measured in one day was 500 metres. The base was remeasured in the opposite direction. The record was in duplicate, as were the principal observations. (See Chapter VIII, §§12 and 13.) The party reached the base-line May 31, 1877, and completed the second measurement August 25. Further details as to the work may be found in the Report of the Chief of Engineers, U. S. Army, for 1878. The computation of the length of the base has been duplicated in all parts where no other efficient check could be obtained. The details of the method of measuring a base are given in Chapter VIII, §12. § 2. In computing the length of a section of a base-line measured with the Repsold apparatus, it may be considered as composed of the following parts : 1. Of as many times the zero-length of the steel bar S;, as it enters the section, the zero-length of §,, being its length when equal to that of the zinc bar Z, by its side; that is, its length when the metallic temperature, Z, — 8S; = 0. The value of the zero-length of §, is given in Chapter IX, § 66, as 4°.37554481. Multiplying the number of times the whole length of 5), at temperature 2, — S; = 0, entered the section by this value, this part of the length of the section results. § 3. 2. Of the sum of the lengths each less than one tube-length, measured with the tube in closing on a marking-stone at the end of a section. 3. Of the sums of the intervals measured with the cut-off scale, either in closing on a section- stone or in stopping at a cut-off plate, corrected for error and temperature of cut-off scale. §4. 4. Of the sum of the excesses of the actual lengths of 8, due to temperatures during measurements over its zero-length, on the assumption that the metallic temperature of S, at any instant of measurement is correctly given by Z, — 8, the difference of lengths of the zine and ‘steel bars, and on the further assumption that for all temperatures the rate of expansion of the S, divided by the difference of rates of expansion of Z, and S; has the same value, 0.6522 (§ 6), as when the metallic temperature or Z, — NS, is zero. : §%. 5. Of the sum of small corrections to the length of a section arising from the fact that the ratio of the difference of rates of expansion of zinc and steel bars to that of steel bar differs slightly from the constant 0.6522 at metallic temperatures different from Z, — Sy; = 0. § 6. The value of 7,"—S,", or the metallic temperature of any tube, results from the corrected observations made during the base-measurement. At each end of the tube the distance (less than 0™".1) by which the 0" or 4" graduation-mark on the steel bar precedes a graduation-mark on the zinc bar is read with the microscope and the number of the graduation pointed at on the zinc bar is noted. The graduation of the zine bar, though very good, is not perfect, hence the number of the graduation-mark does not give precisely the number of tenths of millimeters. between the 0" or 4" mark on the zine bar and the graduation-mark pointed at. Hence a small correction for error of graduation is needed. One turn of the microscope-screw is not exactly O0>™.1, hence there is a correction for run. The data for both corrections are given in Chapter IX, §§ 69, 72. Apply- ing these corrections, there result the true distances by which the 0" and 4" graduation-marks on the steel bar precede the same marks on the zinc bar. The difference of these distances is Z,—S,, the difference of lengths of the two bars, or the metallic temperature. In reducing the base-measure- ments, these metallic temperatures are obtained for every tube measured. From this metallic temperature and the known length of 8, when §,—Z,—0, the length of 8, for any metallic temper- ature, Z,—S,, is derived in the following way : When §, and 4 are expressed in terms of their lengths, §,” and Z,”, at 32° F., if ’=t—32, where tis the Fahrenheit temperature, we have— (1) S,=8,” (1+at’+3t”) (2) Z,=Z (1+e't'+pt") I . (3) é Pe By 8, (a+ 3) IZ, (4) w= By, = LZ, (a! +2 ft; dt! Cuar. X, $1] CHICAGO BASE. 261 CHAPTER xX. CHICAGO BASE. § 1. This straight base was measured in duplicate with the Repsold base-apparatus in 1877 by a party under the charge of Assistaut Engineer E. 8. Wheeler, who had with him Assistant Engi- neers Charles Pratt and I’. W. Lehnartz. The description of the Repsold base-apparatus and of the method of using it has already been given in Chapter VIII. The length of the base is about 43 miles, its west end is in latitude 41° 47’ and in longitude 87° 49’ west from Greenwich. The azimuth of the base from the west end is 290° 59’. The mean elevation of the tube during meas- urement was 584.3 feet above mean tide in New York Harbor. The line lies in an open prairie; no points ot it differ in level by more than 4 feet; the soil is stiff blue clay covered by six or eight inches of black loam. A part of the line ran through culti- vated fields and a part over grassy prairie. The sod was not removed. The ends of the base were marked by small agate hemispheres set in brass cylinders, which were leaded into the tops of granite posts set in brick-work, the agate hemispheres being three feet below the surface of the ground. A description of these marks may be found in the Lake- Survey Report for 1878, and a sketch is given in Plate XII. The base was divided into eight nearly equal sections by marks on stones, as described in Chapter VIII, §10. Counting from the we st end of the base, these stones were approximately at 228, 458, 692, 924, 1154, 1382, and 1682 tube-lengths from it. During the measurement of the base the tube was kept covered by an awning of sail-cloth stretched over a hut-like frame-work, which was moved with it and protected it from the sun. The microscope-stands were protected in the same way except when being moved forward. In measurement the axes of the microscopes are placed in the vertical plane through the base- line by means of a telescope attached to the measuring-tube and made to move in a vertical plane. The forward microscope having been pointed at the front-end zero of the steel bar in the measur- ing-tube, the tube is then carried forward, and its rear-end zero having been placed under the same microscope it is pointed at. This method assumes that in the interval between the pointings of the same microscope at the front and rear ends of the tube, that is, while the tube is being carried forward, the microscope is perfectly stable. To increase the stability of the microscopes, the feet of their tripod-stands were placed on iron pins 15 inches long and 2 inches in diameter, driven into the ground until their heads were only an inch or two above it. The iron legs of the stands were wrapped with felt and covered with canvas to reduce temperature-changes. Experiment having shown that the observers’ weight resting on the ground in changing positions near the iron pins might disturb the microscope, a plank about & feet long, having low supports at its ends was used in all cases for the observers to stand on. Details as to the stability of the microscope-stands will be given in the discussion of the sources of error in the base-measurement. The apparatus, par- tially protected by awnings, proved to have great stability in winds, so that it was possible to measure on many days on which it would have been impossible with the old apparatus. At the close of work on each day marks were placed under the front ends of the last two or three tubes measured, to which the ends of the tubes were referred by the cut-off apparatus (called by Repsold the Absetz Cylinder). (Chapter VIII, §§9 and 13.) On recommencing work the next day, measurements were begun from all these marks, a weighted-mean restt being used in com- putation. The last three microscope-stands are allowed to retain their position, protected by the awnings, during the suspension of work. Their stability is found to be nearly equal to that of the rt 260 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. IX, §§ 74, 75. VALUES OF DIVISIONS OF LEVELS OF BASE-APPARATUS. § 74. The value of one division on each of the levels was determined with a level-trier. The level-tubes are all numbered. The values are as follows: Hand lével NO: 14). cence cesar cece Sek “me sw Chae ab aecw meee: se ene 1"=51"” Fixed levels on measuring-microscopes, Nos. 15 to 22, inclusive ...-.--------------+-- 1° =35”" Sector: level Of tube: LNG: 23) occur ct mec asses eawals let ara pened dase deme te Sais 1 = 33” Sector level of tube 2, No. 24 ...0.... 002. ee eee eee eee eee iiss, eaeeeeneeeaiiass 1" =38" Detached level of measuring microscopes, No. 25 ..... Side waote ee ee deo dead (Cutronh levels IN 02.26 <3 cai easauscopciad ea min waraeewtaketie cia eae aesrou Ale aua et ciated delanaidiecercinois Baeeod P= 1.6 Striding level for quills, No. 27 ....-..2......-.- Ba cleats eather peace ean [eae 5S Cross-level On tibe 1N6:40) 2 sees ssasececsee “eke w in cece BEd ee Sheer aes . 152" Cross level on tube 2, No. 41 -. 2 2222 eee eee ee ee seine eAtests Jicieiaite Micibstal aie aki oieenels 1 =52” § 73. MISCELLANEOUS CONSTANTS. hengthor cutoih Cy NCE 4.402 coctroe dina So Goa Seage awe Seted bekd bean 2a TAR ORRED = 0". 781 Length, with lengthening bar abtedhed,. ao REM HRSG dee ee hay BEl eee as ER eee ae es =1", 698 Distance between axis of tube-telescope and longitudinal graduation on vabeel bar .... =0". 1062 Value of space between 45 and 50, convex end of quill A......................-2---- = 500+.0 Value of space between 45 and 50, convex end of quill B............2..2222.0 22.2... = 499+, 3 Value of space between figure 1 and middle stroke of letter m on metre R1876 ....... = 5114.4 § 73. CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 209) Values of graduated spaces on zine bar of tube 2. On neutral axis. | Spaces at om Spaces at 1™ Spaces at 4™ Spaces at 0m, mm, B mm. Mw mm. B& mm. “w 0 to 0.1= 102.1 0 to 0.1= 101.0 0 to 0.1= 100.9 0 to 0.1 102.7 0 to 0.2= 200.1 0 to 0.2= 201.4 0 to 0.2= 200.2 0 to 0.2= 201.3 0 to 0.3== 301.3 0 to 0.3= 3011 0 to 0.3= 300.7 0 to 0.3= 298.0 0 to 0.4= 401.4 0 to 0.4= 401.9 0 to 0.4= 401.5 0 to 04= 400.4 ; 0 to 0.5= 501.4 0 to 0.5= 500.7 0 to 0.5= 500.6 0 to 0.5= 503.8 0 to 0.6= 602.3 0 to 0.6= 602.1 0 to 0.6= 601.0 0 to 0.6= 602.3 0 to 0.7= 702.5 0 to 0.7= 702.1 0 to 0.7= 700.4 0 to 0.7= 702.2 0 to 0.8= 9800.9 0 to 0.8= 801.5 0 to 0.8= 800.8 0 to 0.8= 797.2 0 to 0.9= 900.5 0 to 0.9= 901.8 0 to 0.9= 900.4 0 to 0.9= 902.7 0 to 1.0=1001. 6 0 to 1.0=1000.4 0 to 1.0=1000.8 0 to 10= 999.3 | 0 to 1.1—1101.4 0 to 11=1100.7 == 0 to 1.2=1200.7 0 to 1.2=1201.3 ; 0 to 1.83=1301.1 0 to 1.3=1301.7 E 0 to 1.4—1401.0 0 to 1.4=1400.9 | | 0 to 1.5=1501.7 epics 0 to 1.5=1501.2 | 0 to 1.6=1601.3 0 to 1.6=1601.7 | | 0 to 1.7=1701.4 | ——| 0 to 1.7=1701.0 | 0 to 1.8=1801.3 Spaces at 3. 0 to 1.8=1801.7 Spaces at 4™ 0 to 1.9=1901.6 0 to 1.9=1901.1 —.- 0 to 2.0= 2001.6 mm, Be 0 to 2.0 2001.6 mm. Bw 0 to 2.1= 2102.3 0 to 0.1= 100.2 0 to 2.1= 2102.0 0 to 01= 104.5 0 to 2.2—= 2202.2 0 to 0.2= 199.9 0 to 2.2 = 2201.7 0 to 0.2= 207.0 0 to 2,3 = 2302. 2 0 to 0.3= 300.3 0 to 2.3 = 2302.6 0 to 0.3= 310.4 0 to 2.4= 2401.6 0 to 0.4= 399.9 0 to 2.4—= 2402. 6 0 to 0.4= 407.2 0 to 2.5 = 2502.3 0 to 0.5= 499.6 0 to 2.5= 2501.7 0 to 0.5= 506.9 0 to 2.6 = 2602.7 0 to 0.6= 599.8 0 to 2.6= 2603.1 0 to .6= 608.0 0 to 2.7 = 2702. 8 0 tu 0.7= 699.1 0 to 2.7 = 2702.7 0 to 0.7= 706.6 0 to 2.8 = 2802.7 0 to 0.8= 800.6 0 to 2.8 = 2802. 6 0 to 0.8= 806.2 0 to 2.9=2901.4 0 to 0.9= 900.0 0 to 2.9 = 2902.3 0 to 0.9= 904.7 0 to 3.0=23001.6 0 to 1.0=1000.1 0 to 3.0= 3002.5 0 to 1.0= 1008.5 Values of graduated spaces on zine bar of MT1876. Goa ae : Spaces at 0". Spaces at 0™. Spaces at 1. mm wm mm Me mm B 0 to 0.1= 101.1 0 to 1.1=1101.3 0 to 0.1= 100.7 0t0 0.2 199.4 0 to 1.2 = 1200. 8 0 to 0.2== 200.5 0 to 0.3= 299.6 0 to 1.3 = 1301.0 0 to 0.3= 300.8 0to0.4= 399.6 0 to 1.4 1400.9 0to0.4= 400.7 0to0.5= 499.5 0 to 1.5=1501.1 0 to 0.5== 500.7 0 to 0.6= 599.2 0 to 1.6= 1600.9 0 t00.6= 601.2 0to0.7= 699.5 0 to 1.7 = 1701.0 0to 0.7 = 702.3 0 to 0.8= 800.1 0 to 1.8 = 1800.9 0 to0.8= 801.2 0t00.9= 900.1 0 to 1.9= 1901.1 0tv0.9= 901.9 O0tol0= 999.9 0 to 2.0 = 2000. 5 0 to 1.0 = 1001.9 VALUES OF SPACES ON CUT-OFF SCALE. § 73. This scaleis 0.13 long and is divided into millimeters. It is described inChapter VIII, § 9. Its whole length was compared with the space 9".07 to 0".20 on #1876 and found to be cor- rect at a temperature of about 38° F. Therelative positions of the intermediate graduations were determined, and no errors greater than 3 were found, Since no great accuracy is required in this scale, it is assumed to be correct at 38° F, ” 8 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. Values of graduated spaces on steel bars. [Crap. IX, | On tube 1. | On tube 2. On tube 1. | On tube 2. | at | mm, B&B Me mm. BK is | 0 to—0.1 101.9 101. 2 A 0 to —0.1 100.5 100. Spaces at 0™....... Spaces at 27.2..... ; ; | 0 to+0.1 100. 1 99.4 ; 0 to+ 0.1 97.7 99.1 0 to—0.1 98.9 98.8 0 to—0.1 99. 6 99. 4 Spaces at 1".0..... | Spaces at 2™,3.-.... : | " to +-0.1 98. 6 98.5 lo to + 0.1 98.7 99. 0 Spaces ati, 5...... | ; 0 to--0.1 | 100.8 100.3 Spaeed wk Beda, i to —0.1 99.7 100.1 | 0to+0.1 99. 0 99, 2 0 to + 0.1 98.7 99.0 | 0 to—0.1 99. 5 100.1 0 tu —0.1 101.0 100. 8 Spaces at 1.6 .._.! Spaces at 2™.5..... ; : : to+0.1 99. 5 98. 4 0 to+0.1 100. 6 98.9 0 to—0.1 99. 3 101.0 0 to—0.1 99. 8 100.2 Spaces at 1™.7..... § Spaces at 3”.0..... ; 7 Fi (0 to+0.1 99.2 98.1 0 to+01 99.7 99.8 0 to—0.1 98. 8 100. 2 0 to—0.1 99.9 100. 4 Spaces at 1". 8..... : Spaces at 4™.0..-.. ; ; i to+0.1 99. 0 : 99. 6 0 to+0.1 100. 2 98.9 Spaces at 1™.9.....! a4 to—0.1 99. 2 99. 0 -ON NEUTRAL AXIS. i 98. 6 100. 3 ‘ : 7 : : 9 99.7 Spaces at 0™......- —lto+1 197.1 206. 6 SARS Resa cis eas a Spaces at 4.2.2... —1to+1 200.6 198.7 Spaces at 2m.1...... 90 to--0.1 98.8 101.1 0 to+0.1 98. 8 100. 6 Values of graduated spaces on zinc bar of tube 1. On neutral axis. 0 to 2.2 = 2201. 0 to 2.3 = 2302. 0 to 2.4= 2402. 0 to 2.5= 2502. 0 to 2.6 = 2602, 0 to 2.7 = 2702. 0 to 2,8 = 2801. 0 to 2.9= 2902. 4 0 to 3.0 = 3002.4 TONY NYDN OaAOaAENWOwDW 0 to 0.2 = 199.7 0 to 0.3 = 298.3 0 to 0.4=395.9 0 to 0.5 = 496.1 0 to 0.6= 594.8 0 to 0.7 = 693.1 0 to 0,8= 792.8 0 to 0.9 = 891.1 0 to 1.0=990.5 0 to 2,2= 2200 0 to 2.3 = 2301 0 to 2.4= 2400, 0 to 2.5 = 2501.8 0 to 2,6 = 2600.5 0 to 2.7 = 2701.7 0 to 2,8= 2801.5 0 to 2.9— 2901.8 0 to 3.0= 3001.7 0 to 2.2 = 2194.6 0 to 2.3 = 2294.9 0 to 2.4 = 2395.9 0 to 2.5 = 2493. 2 0 to 2.6 = 2596.4 0 to 2.7 = 2693. 4 0 to 2.8 == 2792.6 0 to 2. 9 = 2893. 8 0 to 3.0 = 2991.7 0 to 2.2= 2182.5 0 to 2.3 = 2287.4 0 to 2.4 = 2385.7 0 to 2.5 = 2485.9 0 to 2. 6 = 2582. 8 0 to 2.7 = 2683. 2 0 to 2.8= 2788.9 0 to 2. 9 = 2887.0 0 to 3.0 = 2985.4 Spaces at 0™. Spaces at 1”, Spaces at 4", Spaces at 0™. Spaces at 4™, mm, Bb mm. Bw mm. Bw mm. a mm. 7 0 to 0.1= 101.8 0 to 0.1 = 100.2 0 to 0.1= 100.6 0to0.1= 98.4 0 to 0.1= 96.2 0 to 0.2= 202.4 0 to 0.2=199.1 0 to 0.2= 200.9 0 to 0.2= 199.8 0 to 0.2—= 198.1 0 to 0.3= 302.3 0 to 0.3 = 299.2 0 to 0.3= 300.4 0 to 0.3= 296.2 0 to 0.3= 295.4 0 to 0.4= 401.6 0 to 0.4 = 398. 6 0 to 0.4= 400.6 0 to 0.4= 396.0 0 to 0.4—= 397.5 0 to 0.5= 5010 0 to 0.5 = 499.1 0 to 0.5= 500.4 0 to 0.5= 497.1 0 to 0.5= 495.2 0 to 0.6= 602.6 0 to 0.6 = 597.1 0 to 0.6= 601.2 0 to 0.6= 594.7 0 to 0.6 593.8 0 to 0.7= 701.9 0 to 0.7 = 697.1 0 to 0.7= 701.1 0 to 0.7= 696.1 0 to 0.7= 697.2 0 to 0.8= 802.4 0 to 0.8 = 796.8 0 to 0.8= 800.6 | 0 to 0.8= 800.5 0 to 0.8= 795.3 0 to 0.9= 902.5 0 to 0.9 = 896.3 0 to 0.9= 901.7 0 to 0.9= 892.2 0 to 0.9= 895.3 0 to 1.0=1001.7 0 to 1.0=994.8 0 to 1.0 = 1000.2 0 to 10= 994.7 0to10= 994.2 0 to L1=1102.0 j—_—-————-———| 0 to 1.1=1101.0 0 to 1.1=1094.6 0 to 1.1= 1092.0 0 to 1.2 = 1203.3 0 to 1.2= 1202.3 0 to 1.2=1192.8 0 to 1.2=1190.9 0 to 1.3 = 1302.8 0 to 1:3 1302. 4 0 to 1.3=1291.0 0 to 1.3= 1292.3 0 to 1.4=1401.7 0 to 1.4 = 1401.6 0 to 1.4=1389.2 | 0 to 1.4=1395.2 0 to 1.5 = 1501. 0 to 1.5 = 1500.8 0 to 1.5=1490.3 0 to 1.5= 1489.1 0 to 1.6 = 1602. 0 to 1.6=1601.8 0 to 1.6=1590.5 | 0 to 1.6= 1588.3 0 to 1.7=1700. 0 to 1.7 =1700.6 0 to 1.7=1692.3 | 0 to 1.7=1685.4 0 to 1.8 = 1801. Spaces at 3m, 0 to 1.8=1801.1 0 to 1.8=1790.9 0 to 1.8=1785.7 0 to 1.9=1901.6 |~ 0 to 1.9= 1900.2 0 to 1.9=1891.8 0 to 1.9=1888.7 0 to 2.02001. mm. bh 0 to 2.0= 2001.0 0 to 2.0== 1995.4 0 to 2.01983. 8 0 to 2.1 = 2102: 0 to 0.1= 99.7 0 to 2.1= 2102.5 0 to 2.1= 2095.5 0 to 2.1= 2086.8 LT 8 9 § 72.4 CONSTANTS OF METRICAL STANDARDS AND BASE-APPARATUS. 257 Table of values of graduated spaces at 4" or non-lettered end of brass bar. Space. Value. Space. Value. Space. Value. i Me | Mo B M M 0to 1 101.0 + 0.19 0 to 12 | 1196.5 + 0, 44 0 to 23 | 2288.4 + 0.46 Oto 2 202.240.22 | 0 to 13 | 1294.7 + 0.45 0 to 24 | 2387.6 + 0.47 0to 3 298.8 40.26 || 0 to 14 | 1394.8 + 0,47 0 to 25 | 2488.4 + 0.43 Oto 4 399. 6 + 0. 32 0 to 15 | 1493.2 + 0.44 0 to 26 | 2582.8 + 0.45 0 to 5 500.0 + 0.31 || 0 to 16 | 1591.5 + 0.48 0 to 27 | 2687.0 + 0, 42 0 to 6 596. 0 + 0.37 0 to 17 | 1693.4 + 0.48 0 to 28 | 2784.9 + 0.40 Oto 7 696.1 + 0,38 0 to 18 | 1789.5 + 0.48 0 to 29 | 2885.6 + 0.39 0to 8 794.3 + 0.39 0 to 19 | 1888.2 + 0.50 0 to 30 | 2986, 2 + 0.36 0to 9 894.5 + 0.43 0 to 20 | 1988.0 + 0.45 0 to 35 | 3485.6 + 0. 63 0 to 10 995.5 + 0.40 0 to 21 | 2086.6 + 0.48 0 to 11 | 1096.0 + 0.45 0 to 22 | 2188.2 + 0.46 The values of the graduated spaces at the 0™ or lettered end of the brass bar were determined in the same way as those at the non-lettered end. After the first determination, a second was made using the space (0.92 to 0.94) on T. and 8. inch. Thus, twice as much work was done on the half-millimeter spaces at the lettered end as on those at the non-lettered end. The values and probable errors of the spaces have been accurately computed in the same manner as for the non-lettered end. The computation is quite long and is omitted, the results only being given in the following table: Table of values of graduated spaces at 0™ or lettered end of brass bar. Space. Value. Space. Value. Space. Value. » Be Me “ M “ 0to 1 101.4 + 0.18 0 to 11 1100.1 + 0.26 Oto 21 | 2100.7 + 0.31 0to 2 200.5 + 0.18 0 to 12 1200. 8 + 0. 26 0 to 22 2200.9 + 0.31 0to 3 301.2 + 0.19 0 to 13 1300. 4 + 0.26 0 to 23 2298.2 + 0.32 Oto 4 402.3 + 0.21 0 to 14 | 1400.6 + 0.28 0 to 24 2398. 9 +. 0. 33 0to 5 500.0 + 0.13 0 to 15 1496. 4 + 0. 22 0 to 25 2497.5 + 0. 29 0to 6 600. 6 + 0.22 0 to 16 1599.5 + 0. 29 0 to 26 | 2597.9 + 0.34 0to 7 700. 8 + 0, 22 0 to 17 1702.4 + 0.27 0 to 27 2698.1 + 0.34 0 to 8 799.9 + 0.23 0 to 18 1800.4 + 0.29 0 to 28 | 2796.3 + 0.34 0to 9 900.4 + 0.22 0 to 19 1901.5 + 0.30 0 to 29 | 2896.5 + 0.35 0 to 10 1000.1 + 0.18 0 to 20 2000. 3 +: 0. 26 0 to 30 2995.9 + 0.31 VALUES OF GRADUATED SPACES ON &, &, 71, 2, AND (MT1876)z. § 72. The spaces on Z, Z%, and (IfT1876), were determined in the same manner as those on the 4™ end of the brass bar. The values are given in the following tables. The probable errors have not been computed, but since the work was of the same character and equal in amount to that done at the 4™ end of brass bar, it is likely that the probable errors do not exceed + 0+.5. The spaces on S, and S, were determined with a microscope only and therefore have an un- certainty of at least 1*. These spaces are unimportant since the zero-mark is always used. In 1880 graduations were made on the neutral axes of the steel and zine bars of both tubes. Their values have also been determined, and are given below. The values of the spaces on the zinc bar enter the determination of temperatures by the me- tallic thermometer formed by the steel and zinc bars. But no very great accuracy is needed for the values of the zinc spaces, since an error of 1“ in them would give an error of but 34 of a degree Fahrenheit in temperature. 33 LS §§ 7-14] CHICAGO BASE. 265 ing, the measurement is made from each of these three marks. Since the intervals between the marks when measured the second time will not agree precisely with those obtained on the preceding day, either from errors of obsérvation or from slight changes in the positions of the marks, it is assumed in computation that the center of gravity of the three marks has not moved in the direc- tion of the base-line, and with this condition all the measurements are referred to the mark last placed. If a, b, ¢ are the distances from the beginning of the base of three cut-off marks made when stopping work, and a+da, b+0b, and c+dc the corresponding distances when beginning work again, the abscissa of c, with reference to the center of gravity of the three marks will be at stopping atbt+e 9 oO and at starting atdat+b+6b+e+6e 3 Subtracting the first quantity from the second there results for the motion of ¢ in the direction of measurement with reference to the center of gravity ae Bea These variations follow directly from the two measurements of the intervals between the cut-off marks, and the resulting motion of ¢ is to be added to the length of the base, since the condition that the center of gravity of the three marks remains unchanged requires that the front mark c shall have moved forward by that amount during the suspension of the work. The sum of these movements is the cut-off correction for the whole base. Habitually the microscopes were left in their positions over the cut-off marks, so that they also could be treated as stable marks in the same way as the cut-off marks were. Such treatment showed that they had about the same sta- bility as the marks, and also that when the cut-off marks showed a good deal of relative motion, the microscope showed the same thing in the same direction. This would indicate that the motion of the cut-off marks when large was not due to slight motions in the ground of the stakes sup- porting the cut-off marks, but to a motion of the ground itself, which carried. with it not only the stakes supporting the cut-off plate but also the heavy iron pins several decimeters distant, which supported the microscope-stands and penetrated the ground about 30 centimeters. In stopping for short periods, as at noon for dinner, but a single cut-off mark was placed. A discussion of these displacements is given later in §§ 24-28. 9, Of the sum of corrections to S, arising from the fact that the observed Z,—S, in base- measurements does not at all times give the true metallic temperature of S,, and hence does not give its true length when taken in connection with the known expansions of the zinc and steel bars. The derivation.of these corrections has been fully explained in Chapter IX, §§ 42-53, and “the method of applying them has been given. § U4. As an example of the method of reduction, copies of the notes of measurement on the Chicago Base from the 1784th to the 1791st tube are given, including the notes of a cut-off. Fol- lowing the record of measurement, a reduction of the measurement with explanatory notes and a reduction of the cut-off with explanatory notes are given. CHICAGO BASE—FIRST MEASUREMENT, 1. Notes of measurement of line. c+dc— ~ | : Grade. Rear end. Front end. i ee ene e |e le, "ee. (Pe ea z gold (e¢1eti8 Ep |e [gel etls z | a8 N agiles | 2 Mu . Na gk ios | s Date. © 4, | a3 a 24 | 34 | fo |Meantime 6 B BA | °A| 3s ; 7 | Sra on | Blo | Be a os qH | Bo! ae a P ee | gS | Be | bo) ss as | go | RH) oa | 38 a 7 : Ae a 4] 4 3 zold |/es i fa) = g= | og jek | Ae] oe | a | 2 | #3 | 2 |eelaels es | 3 |$8| 28] 6 wz | 4 A | & & |A |e |A 4 a 14 |e iA el = = : 1877. | | o 4 | div. | div. rev, rev. = rev. rev. ° Inly 21 | 1784 536/ 66] 3.8/ 10.110! 10.110] 17 | 737] 4 | 11:22a.m..| 10.000) 9.588) 18 ) 73.4) 1 ; 1785 500) 61) 4.1] 9.987 | 9.891} 17 | 74.2 1 |11:30a.m..; 9.987 | 9.692 | 13 | 73.8 2 1786 449} 5.3] 5.0 | 10.034 9.858 | 17 | 74.6 2 |11:38a.m..) 9.990 | 9.777 | 18 | 74.2 3 17#7 529| 4.51 5.8) 9.835; 9.617) 17 | 74.8 3 | 11:45 a.m-.-} 10.000 | 9.810 | 13 | 74.5 4 | 1788 | 5538 | 4.9/ 5.4! 9.960) 9.669) 17 | 75.0 4 [11:48 a,mn..] 10.012 | 9.870) 13 | 74.5 1 34 LS 266 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Car X, 2. Notes of cut-offs at stopping and starting. | | ; Level. Elevation | Date Aiea ’ Mean time _No. of | ee eat a Diagren . Reading of | Veter OF F ae tube. _ . F Se of tube. forward. °! ROME MIGTUSCONY. “Tete Right cut-off | ! | : ; | (vear). (front)., cylinder. | | oe - | cal | = 1877, rev. div. © div, | in, | | B 0 9.925 | 122 | 11.0 2: "Pp 5 July 21 1786 | 12:00 m ae 3 Front.....-- ; A 0 9. 669 79. 15.4 : 1.5 | ag r aa § 8B 0 9.535 23.6 | 07 | a : 1 1 Wee is i vont. -..-- - sd i Be tA 0 9.254 | 19.0 | 5.0 f A a 13.555 13%0 | 11.0 1788 | 12:21 p.m. 1 | PRONE eeees ; cay a | B | 0 | 13.810 14.0 | 10.0 § | 6 2 July 281788 | 9:08.a.m. 1 | Front.....-. § B OR gaye, eps plesD ie ae tai) 0 | 12.2994 18,2 | 10.8 1787 9:13 a.m 4 Front..-..-- j A ; 9 | 11. 565 720) Ud B 0 11.898 | 12.3 6.9 | 1786 9:18 a.m . 3 | Front....--. } Bs 0 | 10.306) 12.7 a | A 0 + 10.000, 120, 60, ! | 3. Measurement of tine—Continued. Grade. Rear end. Front end. ¢ Te (-o- S. e eme I Ee Le af ad = (8, 48 eel es é ig, 8 |e, e8/8 5 : : = e = S as eo 88 Date. goog, «628 | BA | Bs ans gio 8. '588 | o2 | Bs | H eee uae 8g | o8 5 ue sie || BS oe S| | 5 ; a @ a2 Be! So] es | a} & a a. ae ga zs es fs 8 as! ag a | 2s E> | 2s) ee 52) BI Es z eo S 28° 8S); ze z Vea | Be | : ip A wy ago oO oF oH o i ae = 67: 25,8 [44,4 8 a a eB ieee a je eo iF lect ae Pophep aa a ee oy ae a Se Py a ee | 1877. ;o ' div, div. rev rev. ° | rev. rev. | # |) 30) ao July 23/1787 529!) 5.0) 51 9.997 9.137 16 73.3 ‘3798! 553| 4.0| 5.9} 10.350) 9.436) 16 73.5 | 6 9:22 a.m ..| 8.306 | 7.806 1 i | : 9 3.0 11,132 10.100) 16 | 73. 3 | 9:20am ..' 10.788 | 10.268] 13 72, 4 1 | 9:26a.m.. 10.030 9.648 i" 3 1 i | 1789) 6 10/ 6 9) | 1790 | 6 01 | 5.0} 5.0 10.015 10.015 | 17 74.2 9:29a.m..' 9.995! 9.704! 13 | 73.8 vin |446! 5.0; 5 9:33 a.m... 9.996 9.706 | 13 42 | | , 30; 5.0 10.015 9.906, 17 74. | I t REDUCTION OF THE NOTES OF THE PART OF THE BASE-LINE GIVEN IN THE PRECEDING TABLE, Column 2 in the following table gives the measured value of the grade-angle. The value of one division of the level is 33’, and the level correction is positive when the front end of the bubble is the higher. Column 3 gives the grade-angle as found from measured value of grade-angle minus index- correction. The index-correction was 5° 30/ 27", Column 4 gives the correction for grade as found from the formula —2~x length of tube x sin’ $ grade-angle, or—8™ sin? $ grade-angle. Column 5 gives the difference between the reading on the zero of the steel bar, and the reading on the graduation of the zine bar pointed at, expressed in terms of a revolution of the micrometer- screws. Column 6 gives the sum of the two corrections; (a) the correction for the graduation-errors of the divisions of the zinc bar, and (b) the correction to one-tenth the quantity in column 5 to reduce it to millimeters. Column 7 gives the sum of (a) the zine-bar graduation read on, (b) one-tenth the quantity in column 5, (e) the quantity in column 6 for each tube. Columns 8, 9, and 10 give the quantities for the front end of the tube corresponding to those that columns 5, 6, and 7 give for the rear or zero-end. Column 11 gives the difference between the quantities in columns 7 and 10 for each tube which is the Z—S for the whole tube. § 11] CHICAGO BASE. Leduction of the notes of the measurement of the line. 1. BEFORE THE CUT-OFF. 267 | Grade. Rear end, Front end. | [- kin Soeeess 2 7 pAnen a ees | No. of | Difference of ; ; Difference of | ; A Metallic | tube. | Reading Grade ‘ Correction | ™crometer- cespodan | Corrected | micrometer- Sai en Corrected oe : “corrected | ancle | for eraile. readings on wines zine minus | readings on 0 4ING | gine minus me | ' as level. Bory steel and east steel. steel and me steel. i te: steel. ing, steel. | ‘ seed? Mis | : | | 4; 2, 3. 4. 5 ; 6 7. 8. | 9. 10. iy | x il | ae Bee Ue mm. rev. mm. mm. rev | mm. mm. mm. 1784 | 5 35 14 , 004 47 —0.00389 | 0. 000 4-0. 0002 1. 7002 40.412 | +0. 0030 1.3442 +0. 3560 | 1785 459 27 0 31 00 | —0. 1626 +0. 096 | -+0. 0003 1.7099 +0.295 | +0. 0027 1. 3322 --0. 3777 | | 1786 | 4 48 55 ; 0 41 32 ~ —0, 2920 +0.176 | +0. 0004 1. 7180 +0, 213 | +0. 0026 1. 3239 4-0. 8941 | | 1787 5 29 21 | 0 01 06 | —0. 0002 -|-0. 218 +-0. 0004 1. 7222 +0. 190 -| 0. 0026 1.3216 +0. 4006 | | 1788 | 5 53 08, 0 22 41 | —0. 0871 --0, 291 +0. 0005 1. 7296 | +0. 142 | +0. 0026 1. 3168 . +0. 4128 | ; I ' ! - 2, AFTER THE CUT-OFF. jecernes = i age = : - _ 187 5 29 00; 0 01 27 | —0. 0004 +0. 860 +0.0029 | 1.6889 +0. 525 +0. 0030 1. 3555 --0. 3334 : | , 1788 5 53 31 | 0 23 04 | —0. 0900 +0. 814 +0. 0030 | 1. 6944 --0. 500 +-0. 0032 1. 8532 4-0. 3412 ; : ; a 4 e | : 1789 608 56 0 38 29; —0. 2507 +1. 032 +0. 0036 1. 7068 +0. 382 +0. 0027 1,3409 | 4-0. 3659 © 1790 | 6 01 00 | 0 30 33 ~~0. 1580 0. 000 --0. 0002 1. 7002 , 0. 291 +0. 0027 | 1.3318 | +0, 3684 1791 | 0 44 27, —0.3344 40.109 | +-0. 0008 1. 7112 -+-0.290 | |-0. 0027 1, 3317 : 4 46 00 | +0. 3795 i 1 REDUCTION OF NOTES OF CUT-OFF MEASUREMENTS. Column 1 in the following table gives the number of the cut-off set, the numbers running in the direction of the measurement. Column 2 gives the level-correction to the sector-reading, which is positive when the rear end of the bubble is the higher, the axis of the cut-off being then in advance of the cut-off sphere. Column 3 gives the quantities in column 2 reduced to seconds of arc. The value of one division of the level is -1/’.6. Column 4 gives the measured elevation of the slow motion of the cut-off cylinder reduced to millimeters. Column 5 gives the height of the cut-off scale above the center of the sphere of the cut-off plate, found by adding 0.781 (the length of the cut-off cylinder) to the quantity in column 4, Column 6 gives the distance of the zero of the micrometer from the axis of the cut-off cylinder, found by subtracting the mean of the two micrometer-readings on the cut-off scale, from ten revo- lutions, and reduced to millimeters by the relation, 1 revolution of micrometer-screw=0"™.1011. Column 7 gives the product of the quantity in column 5 by the sine of the angle given in column 3. This is the horizontal distance of the intersection of the axis of the cut-off cylinder with the cut-off scale from the vertical through the cut-off sphere. Column 8 gives the sui of the quantities in columns 6 and 7. zoutal distance froin the zero of the micrometer to the point vertically over the cut-off sphere. sign is positive when the zero of the micrometer is in advance of the sphere. Column 9 gives the difference between the rear and front readings on the steel bar in the tube, that is, the difference of the distance between the microscopes and [S; wax t (28), to obtain the whole distance It gives, therefore, the hori- The Columns 10 and 11 contain the distances to be added to 8, 's,=2 between the cut-off spheres. These quantities are formed from columns 8 and 9, with the addition -of the terms (Z—S) e, for the tube measured between the cut-offs obtained from column 11 of the preceding table. The correction to the base given underneath the table is the movement of the forward cut-off sphere plus the movement of the forward microscope with reference to that sphere. It is, therefore, formed from columns 8,10, and 11, the first two terms being taken from column 8, and the last two 268 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. X, being obtained from columns 10 and 11, by the formula given in §10. The value of e is formed from the equation and is numerically equal to + 0.6522. Reduction of the notes of cut-off measurements. ' | 3 | Rear minus | Distance between cut- . Level correction. | fiievation ; 1 athe é 10 rev om ene Scale in Micrometer, front readings L oft spheres. No. of _of the slow , ear ota autos ter. | advance in advance on steel bar in) __ = cut-off. =e “| motion. | ae Bei nh €I- | of cut-off. of cut-off. | tube reduced | | \ | Divisions. Seconds. « | eylin ers rearing’: ‘to millimeters.) 1 and 2. 2 and 3. eae nee aes : | Oe na tte eae : ey flip aoe \ \ , 1 ; 2. 3 Ges 6 7. 8 | 9 | 10. ' 11, | | m. lm mm. mm. ann. | mm. | mm. | mm. 1 —1.6 — 2.6 0. 038 | 0. 819 --0. 0205 —0. 0103 | --0. 0102 epigadeseaciae Se ‘+40. 0102 -+-0.1192 - 2 +9.2 +14.7 0. 030 0. 811 +0. 0613 +0. 0579 . 4-0. 1192 —0. 0167 i —0. 1192 / +0. 3627 3 +1.5 +24 0. 036 0. 817 —0, 3722 +0.0095 | —0. 3627 © —0. 0052 ; —0. 0167 . —0.0052 ; | | "0.1257 | -+0.4767 | +0.40062 | 4-0. 4128e Hl ; i y | | i j t i | I 1 +3.3 + 5.3 0. 038 0. 819 | —0. 0155 +0. 0211 -+ 0. 0056 | Btecrct ete -- 0. 0056 | —0. 1726 2 5 +0. 4 + 0.6 0. 030 0. 811 —0. 1750 +0. 0024 0, 1726” i —0. 0800 | +0. 1726 | +0. 2351 Sal $2.1 + 3.4 0. 036 0. 817 | —~0. 2486 +0. 0135 —0. 2351 i --0. 2067 » —0, 0800 | -+0. 2067 | | 1 Samet ee 1 i I +0. 0982 +0. 2692 | ; 2 oe 0. 3334e +0. 3412¢ | | : \ I mm. mm. mm. mm, Correction to base = +- 0.3627 — 0.2351 — 0.0637 — 0.0701e. § 12. The method of computing the measurement of a base measured with the Repsold appa- ratus having been explained, the results of the measurement and remeasurement of the different sections of the Chicago Base may be given. In the following table the first column gives the number of the section of the base; the second shows by the numbers 1 and 2 whether the first measurement or the remeasurement in the oppo- site direction is given on that horizontal line; the third gives the entire number of times that the steel bar at the temperature 60°.292 F. (S,_,=8,") entered into the measurement of the section; the fourth gives the fractional parts of S, which entered the measurement when closing on the permanent mark at the end of a section, this interval being measured by means of the intermediate graduations on S,, except in a single case, when it was measured with an accurately divided leveling- rod, the value of whose unit was determined by office comparisons with a metre of known length, while other office comparisons have shown that the intermediate graduations on S, give tlie frac- tional parts of the whole tube which they represent without a greater error than 10+, a quantity that can be neglected in comparison with other errors in the length of a section of the base; the fifth gives the sums of the parts of the base measured with the cut-off scale in closing on section- stones; (the graduations of this cut-off scale have been examined and found sufficiently accurate to introduce no error that need be considered); the sixth gives for each section the suins of the 7 corrections to S,° for metallic temperature, on the assumption that - set equals its value when wz) — ts, 4,=8, or 0.6522; the seventh gives the sums of corrections to the length of each section on account of the inaccuracy of this last supposition; the eighth gives the sums of the corrections for the sections on account of inclination of S,; the ninth gives the sums of the small distances measured with the microscopes in each section ; the tenth gives the sums of the movements of the front cut-off plate in each section; the eleventh gives the sums of the corrections for each section on account of the observed Z,—S, not being the correct metallic temperature of S,; the twelfth gives the sum of the different corrections for each section. The duplicate computation gave 1 maximun difference in the lengths of sections of 34.4, a quantity which can be entirely neglected. §§ 12, 13.] CHICAGO BASH. 269 Results of measurement of Chicago Buse-line. so Pe RS ae ed ag j : - re ; ; > | | g¢ |e8% | a lage | 28 | & | 6 | ek ae ae « = b lees 8 Bors | 25 ne | og | ‘8° 23 Fs gw | S38 dg ifn 2) 8s Be 2 ga | me) es . 23 Ba Soc. s Bros | b So . ga | | & S| Fractional tubes. S32 ge B257 82 “gee, se, 88 ae a fee | af | Be |SBaa) Ba [sees | fee | Be | BE e (815 PBS. ll 3Be wee) 2ee $83/ os ge - 4” oS o) 5 = 1 BS PERG E BSaa| sea ES . & | op " _& > ae 5 5 a a 5 a | I [eerie ore : ait . | vee See | — 1. i 3 4 5 6 1 8 Ig, 10 11 12 : a —| i tare) Sele ! f. Me B me Me Me & # Me I 1 228 | —0. 75 + 60020.0 } + 73257.2 | + 362.1} —119484.0 | + 402.0] + 66.4 | +-1266.3 | + 15890. 0 77 2 297 | 10. 25 + 64028.5 | 4- 46112.1 | + 157.5] — 940306) + 499.4) — 25.2) + 477.4 | 4 17219.1 dif “61 230 | +0. 25 + 4005.5 | + 43480.5 | + 181.0 | — 30256.9 | + 161.6) — 205.4) + 896.1} + 18262.4 meee | 230 | +0. 25 | 57022.4 | +108751.6 | +- 661.0 ) — 34525.6 | +2005.4 | + 218.3 | | 717.5 | + 15805.8 . t (1) 234 | +0. 50 — 99089.4 | + 97214.5 | + 536.0 | —-106907.8 | ++ 603.3 | — 140.8 | --1310.8 | —106423. 4 Tiles, 2 | 234 | +-0.50 — 54018.0 | ++ 70797.5 | + 339.0 | --126204.7] — 35.0 + 29.1 | 4. 391.3 | —108700.8 f . mu : 2 | [G1 i 232 —1156. 0251*| + 34013.9 | +-103487.5 | + 586.6 | — 49009.5 | + 576.7 | + 188.6 | +- 955.4 | }- 90749.2 | EU sess 1 Sep i 8 + 92033,3 | 4+ 914414) + 500.9) — 52503.1 | +-1088.3 | + 501.9 | +1293.9 +134356. 6 : mm 1’ 994 —1156.0251 | — 1000.0 ~-317389.7 | +-1665.7 | —305658.2 | +1743.6 | — 91.2 | +4428.6 | + 18478.2 | wesepaet 2) 924 | —0.30 “+ 45021. 4 | j-312102.6 | 1-1658.4 | —307264.0 | +3558.1 | +- 724.1 | +2880.1 | -+ 58680. 7 | mm { 1} 230 | +-0.85 +1156.0251 — 99036.8 | 4+ 70607.2 | + 262.3 | — 74350.8/ +-1001.5 | + 43.4] + 9384 | —100534. 8 V----99 | 931] 40.10 — 5003.2 | +-110232.4 | + 665.8 | — 55758.1 +2219.5 | + 109.8 +1233.6 , + 53699. 8 1 226 | —0. 40 +-140052. 6 | ++ 98906.5 | + 530.1 | — 63979.2 | + 146.5} + 218.8 | +1043.0 | +176918. 3 | VE---}99] 995 | +0.65 — 27011.5 | + "70535.2 | + 298.7 | — 69402.2 | + 379.8] + 70.3 | + 935.9 | — 24193. 8 | 161) 301 | —0. 45 — 90029.8 | +-118829.7 | + 681.6 | —122664.3 | + 823.4 | + 378.6) +1819.6 | — 90161.2 | VII.--- 9} 301 | —0.50 +- 54023.8 ° +-142115.7 | + 926.7 | — 90878.8 | + 556.8} + 295.2 | +1055.6 | +108595. 0 | | ‘(1 | 196} +-0.80 | + 39009.9 | + 53655.8 | - 250.4 | — 445214) 4 932.5 | — 79.5) 411769) + 50424.6 | VIn....'$ 196 | --0.80 | = 11005.5 +102747.9 | + 692.2 | — 43454.5 | + 540.3 — 86.1 4+1188.6 + 50622.9 | un 1 | 953 | 40.80 +1156. 0251 ; — 1004.1 +841999,2 | +1724.4 | —305515.7 | +2903.9 | + 561.3 | +-4977.9 | ++ 36646, 9 Hast part | 953 | +1. 05 "4 11008.6 © +425681.2 | +-2583.4 | —258993.6 | +2696.4 | + 389.2 | |-4413.7 | +-188723.9 | | | ; a |¢1 | 1877 | +0. 80 — 11004.1 + 659388.9 +3390.1 | —611173.9 +-4647.5 | + 470.1 | +-9406.5 | + 55125.1 | Total. 5 1877 | --0.75 | -.56025.0 | +-737788.8 | +4241. 8 | —566257.6 | -7254.5 | +1113.3 | +-7293.8 | -+-247404. 6 * Measured with leveling-rod. MB The mean value of (Zi—S1) for the two measurements of the base was +-570.4, corresponding to 75°.27 F. § 18. In the following table, derived from the preceding one by expressing the two measure- ments of a section with the same fraction of a tube, the principal results are given, and also the differences between the two measurements of each section. To express the sections of the base in terms of entire numbers of ,° and equal fractions of it, the value of the fractions of a tube in metres is needed. Tie value of S| is given in Chapter IX, § 62, and from that its value at 60°.292, for which Z,— N,=0, may be computed. Transforming it into metres by Clarke’s value of the metre (metre— 1".093623) there results— 8, at 609.292 F.—4".0009624 as an approximation more than sufficient for obtaining the values of the fractious of the tube in metres. It has already been stated that these fractions of tubes are the exact fractions which they purport to be within 10+, a quantity that is insignificant in comparison with the length of a section, and which enters a section but once. 270 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuar. X, Principal results of measurement of Chicago Base, | | | . Measure: = Number of Sum of cor- Rie t | Liat tD ! Section. “ent. (Si) =, rections. | Mens. Yiflercuces. by : in 237, 265 (Si) ,=z, | + 15890. 0 1329.1 997,95 a + 17219.1 4 18262. 4 + | 230.25 + 1703410 | 4 2456.6 + o | 930.25 15805. 8 34.5000 — 106423. 4 | 234.50 « — 108700. 8 fs. | 3 232.00 ** 1065275. 9 = Cancecomna | gees | : | -|-2277, 4 232. 00 ae 1065932. 1 co 1s es Dt 1 gpg.o0 —1137546. 9 | yoo 8 —1141608.0 —1139577. 45. +4061.1 ' 23100 | + 455345. 9 < nee ae NAN wu Nr imho —_ he et at ms : ———~ ier a SG, pats, a : ++ 454570. 95 $159.9 2 231,00 ‘ | ++ 453796. 0 | 1 225.50 | + 577014.5 ae ; VI. ee A + 5764 1063.9 | . 2 225, 50 , -- 575950. 6 (ste 7 300.50“ ++ 109887.3 | 6 iggedi. 15 4192.3 | . 2 300.50 | + 108595. 0 , | \ “ | wie ee te 50523,75 | — 1983 | | 2 | 19 196. 40 | + 506229 if "95 953,80 1192672. 3 , | East part 2 | 95.80, 1t8806. 5S 4-1190818. 40 +3707. 8 === 1 | 1877.80 + 55125. 4 st hd + 51240. 95 +7768. 9 Foal 2 |is7e0 | 47856.5 | Substituting in 924 (S,),_., —1139"".58 and in 953.80 (S1)z,-s, + 1190.82, which are respectively the measured lengths of the west and east halves of the base, given in the preceding table, the value of (4),,-s, derived from Chapter IX, § 66, namely: (81) gas, = 487554481 = 13".12663443 there result 12125'.2714 and 12524'.0909 as the measured lengths of the west and cast halves of the base ae and Chicago Base as measured (mean of two measures) = 24649".3623 § £4. The mean level of Lake Michigan from 1860 to 1875, inclusive, is given in Chapter XXII, § 15, as 581.28 feet above mean tide at New York. The mean height of the tubes during the measurement of the Chicago Base above the mean level of the lake was determined with a leveling-instrument, and is 36.7 feet. The mean heights of the east and west halves of the base differ so little that the difference may be neglected. Hence there results mean height of Chicago Base above mean tide, 618.0 feet, with a probable error not greater than +£1.0 foot. The approximate length of the west half of the base is 12,125 feet. Taking 7.321128 as the logarithm of the radius of curvature for the base, there results a correction of —0‘.3577 to the west half of the base to reduce it to mean tide at New York. The approximate length of the east half of the base is 12,524 feet, and there results a correction to the east half of the base to reduce it to mean tide of —0*.3695. Applying these corrections, there result at sea-level, West half of Chicago Base= 924 (8,°)—1™.24861 East half of Chicago Base = 953 (S8,°) + 4.27898 Chicago Base at sea-level ==1877 (8,°) + 3".03037 From Chapter IX, § 66, §P== 1312665445 §§ 14,15.) CTHCAGO BASE, 271 and 8,° when expressed in terms of BR 1876, is S\o== +R 1876 — 584.4 + 14.326 (60.29 —42.75) Making the proper substitutions there result, Chicago Base at sea-level 7508 (R1876 at 60°.29 F.) + 2.96368. West half of Chicago Base at sea-level —12124.9137 English feet. Kast half of Chicago Base at sea-level =12523.7214 English feet. Chicago Base at sea-level —=24648.6351 English feet. The value of the base given above in terms of #1876 depends on adjusted expansions of 1876 and 8,. But in Chapter IX, § 66, a value for S, in terms of R1876 is given, which depends solely on comparisons of #1876 with S, and of 8, with S,. This value is— S,=4 R1876—374.01 4 14.2744 (t—59) Expressing the base by this value it beecomes— 7508 (1876 at 60°.29 F.) + 2".96398 This value will be of use when the length of £1876 is accurately known. The Chicago Base was divided into two nearly equal parts by the middle section-stone. The angles of the triangles formed by these halves of the base and station Willow Springs were read with the same care as primary angles. They are given in Chapter XVI, C, and enable either half of the base to be computed from the whole base. The computed values of the east and west halves are, respectively, 12523.6833 and 12124.9519, and their excesses over the measured lengths are —0*.038 and +0*.038, respectively. § 15. In § 6 a value for depending on the temperature has been obtained. Con- SL H,—Es, sidering this quantity as independent of the temperature, that is, as constant, an approximate value for it might have been obtained from the base-measurements. There were quite large ditfer- ences of temperature between the measurement and remeasurement of several of the twenty sec- tions into which the Chicago, Sandusky, and Olney Bases are divided. If the measurements have all corrections applied to them save those for temperature of S,, the remaining differences between the measurement and remeasurement of a section will be attributable to the changes in length of 8, from temperature and to accidental errors. It will be seen hereafter, in considering the discrep- ancies between measurement and remeasurement, that the accidental errors are very small, and, therefore, that the differences between two measures of the same section uncorrected for differences of temperature, taken in connection with the values of the metallic temperatures Z,—S; for these measures, should give a value for a= - From twenty sections, with ranges in the mean tem- A 1 peratures of the two measurements of a section which sometimes reached 23°.4 F., a pretty good iY mean value of pig Should result. For each measure of each section of the three bases, an A 7S) observation-equation was written in the form a — ae im which «@ is the sum of the ob- —Hs, served Z,—S, for the first measure minus that for the second; 0 is the length of the section uncor- rected for temperature resulting from first measure minus that for the second measure; and v is the residual. Solving by least squares the twenty equations of condition thus obtained, there results Es, Es, aS __ i __ 0.658 40.002. This value corresponds nearly to the mean temperature of all the —Ls, Es, measures of the three bases, which was 74°.47 F', In § 6 the value of TEs comparisons, is given as 0.64119 4-0.000389 (t—32), which at 74°.47 F. becomes 0.65771. derived from office 2i2 STANDARDS OF LENGTIL, BASES, AND BASKE-APPARATUS, [Ciar. X, SOURCES OF ERROR. § 1G. The sources of error in the measurement of a base-line with the Repsold base-apparatus may now he considered. 1. Errors of alignment.—The Chicago base-line was divided into eight nearly equal sections hy seven marking-stones intermediate to the stones which mark the ends of the base-line. A point at about the middle of the base was first obtained approximately on the line, and the angle at it was then read between the ends of the base-line to an accuracy of one or two seconds. This gave the means of computing the distance by which the point should be moved to make the angle at it 180°, and the middle section-stone was then placed with its mark at this point. The other section- stones were then set in a similar way from the stones at the middle and ends of the base. This made it certain that the different sections were parallel to the line joining the ends of the base, within 5’, a precision much greater than is necessary. The average length of a section was about 940 metres. Between two section-stones the measuring-tube was brought into line by setting its 0" mark on the steel bar over a point already fixed (a section-stone at starting), and then moving the front end of the tube in azimuth till the alignment-telescope, firmly connected with the tube, points to a target at the next section-stone, as distant from the line of the base as the vertical plane through the line of collimation of the alignment-telescope is from the vertical plane through the longitudinal line marked on the steel bar. It is evident, if the longitudinal line on the steel bar is not parallel to the vertical plane of the telescope’s motion, that when the telescope points at the distant target the steel bar will not be parallel to the base, its front end each time the tube is set up changing its distance from the base but always remaining on the same side of the base-line, the distance from the base-line of either rear or front end being zero at the beginning and end of the section. The curve drawn through the successive positions of the tube resembles the path of a boat steered toward a fixed point across a river. An investigation by Assistant Engineer J. B. Johnson shows that the curve in polar co-ordinates with the forward section-stone as an origin and with angles counted from the section line may be represented closely enough by ) = —Nep. log (Gj) tan « where A is the length of the section, « the deviation from parallelism of the alignment-telescope, and ry and @ the co-ordinates. The rectification of this curve gives for its length between two sec- tion-stones, a if a, is the actual angular deviation of the steel bar from the telescope plane of if OS a motion for this section. Hence the correction to the measured length of the section is — A/(1 — cos a,) if A’ be the measured length. This may be written / S sin? 1” o2 Assuming that the sections are of equal length, as is nearly the case, the sum of the actual correc- tions for all the sections would be / — (a; a2? nee .+a,?) a sin? 1” in which a, is the actual deviation of the telescope from parallelism for the nth section. Approxi- mately we shall have where ¢ is the mean error for the actual errors u,, a, &c. Hence, [o?] = ne A value of ¢ has been determined from numerous repetitious of the adjustment, each followed by a measurement of the distance on a target attached to the tube, between the line of sight of the telescope in its different adjustments and a fixed point of the target. The resulting value for e, cr the mean error of adjustment found, was 22’, Hence the total correction to the base for this mean error of adjustment would be AL 5 —n sin? 1! (22)? §§ 16-18, ] CHICAGO BASE. 273 As this amounts to less than 0"™.1 for either the Chicago, Sandusky, or Olney Base, it may be neglected. As the adjustments were only made about once in a week, while «a section was meas- ured in two or three days, it has been assumed as an approximation that the adjustment-error was constant for each section. Experience on the Chicago and Sandusky Bases showed that the tele- scope when once adjusted retained its position with great stability, and that the changes produced by a new adjustment were due to the errors in the adjustment itself and not to change in the posi- tion of the telescope. On the Sandusky and Olney Bases, measured in 1878 and 1879, respectively, points were placed accurately on the base-line at intervals of about 300" intermediate to the section- stones and by the method used for placing the section-stones. Then, in measuring the base, whenever one of these points was passed the actual deviation from the base-line was measured. The greatest deviation in any case was 0".015 excluding one case of 0".191, which arose from an error of adjust- ment. See Chapter NII, § 2. On the Olney Base, after adjusting the telescope to parallelism with the steel bar as nearly as possible, by the method given in Chapter VIII, § 11, further changes by computed amouiuts read on the target were made in the adjustment of the telescope until the tubes would pass within 1°" of the intermediate marks. This once accomplished, no further adjustinent was needed during the measurement of the base, although the adjustment was frequently tested. There is of course a slight error in alignment arising from error in pointing telescope at the signal, but as the tele- scope was a good one with an object-glass about 30" in diameter, the error from this cause was entirely insignificant in comparison with that already discussed. ' § 17. 2. Errors of inclination.—In § 8, under (6), it has been stated that seven determinations of index-error of the sector, distributed through the measurement, gave a range of but 26”, and that their mean was taken as the true value of the index-error. We may assume, then, that the probable error in this correction did not exceed 5”, This would introduce an error into the length of the base which would change sign whenever the inclination changed sign, and so, on a nearly level base, would be largely eliminated. Even if there had been no changes of sign of this corree- tion, since the average inclination was much less than 1°, an error of not more than goago00 part of the distance would have been introduced. With the changes of sign the error may be neglected. The sector can be read to 30”, but its vernier was habitually set at exact minutes, and the level, of which one division equals 33”, was read to tenths of a division. These errors of reading are accidental, and so as often positive as negative, and their probable value does not exceed 5”. The resulting corrections change sign with the error, so that the total correction to a section or to the base from this error would be very nearly zero. Taking 1° as the inclination, which would give the average error for a tube from error in reading inclination (and this value is too large), this error would be about 24. The probable error in the length of the base from this cause would be about 2H V 1878 or +86+.6, a quantity which may be neglected. § BS. 3. Stability of microscopes. —It has already been stated that the interval between the position of the 4" mark on the front end of the tube and the position of the 0™ mark on the rear end of the tube, when these marks were brought successively under the same microscope, was measured with the microscope. As the distance to be measured was habitually less than 0™™.1, and as the microscopes can be pointed with a probable error of about + 0+.4 in the comparing-room, which may be increased to 0“.7 or 0«.8 in the field, and since the errors in measuring this quantity are as likely to be positive as negative, their resultant effect on the length of the base does not exceed 0™™,04, and is entirely insignificant. But the method of measurement supposes absolute stability in the microscope in the interval between its pointings at the front and rear ends of the tube. Numerous experiments on this point were made on the Sandusky and Olney Bases. The microscope may from temperature-changes in itself and in the stand which carries it, change its pointing. This it undoubtedly does, but experi- ence in open-air comparisons has shown that such changes are very slow, and that in the period of one or two minutes elapsing between successive pointings with the same microscope, the change is entirely insignificant. On the Olney Base, work was suspended at noon for dinner. On twenty- four days the average length of the stop was 15 49", The relative movement of the microscopes in this interval, from all causes (excluding one day on which it was +174".2), varied between 35 LS 274 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. (Cuar, X, $274.5 and —18-.5, giving in mean +1+.6 as the average motion of the microscopes from each other during this stop. The microscope may also change its pointing from disturbances trans- mitted to it through its stand from the earth, these disturbances arising either from the weight of the men employed in the work, when they change positions in the vicinity of the microscope, or from the disturbance produced by the removal of the front end of the tube from a stand and the placing of the rear end of the tube in the same stand, this disturbance of the tube-stand being transmitted through the ground to the microscope-stand. If by any of these causes the microscope is displaced subsequently to the pointing at the front end of the tube, the displacement will intro- duce no error, provided the microscope returns to its previous position by the time it is pointed at the rear end of the tube next in order of measurement. It will be well to restate the position of the microscope-stand with reference to the tube-stand, and the varying positions of the men employed in the work. The vertical axis of the stand sup- porting the end of the tube is directly under the microscope, so that its feet are not far from the corresponding feet of the microscope-stand. The feet of both rest on the heads of iron pins two inches in diameter and fifteen inches long, shown in Figure 1, Plate VII, driven down to the ground. The axes of the pins supporting corresponding legs of the two stands vary in distance from each other between 7 and 12 inches. When the men approach a microscope to remove the tube from or to place the tube on its stands, they are required to keep 16 inches away from the pins supporting the microscope, or, what is the same thing, 12 inches away from the foot-plate. On the Sandusky and Olney Bases this distance was secured by placiug wooden cross-bars on the platform which supports the observer, at the right places to keep the men away from the micro- scopes. The question arises, and it is a very important one, whether, when a tube-carrier ap- proaches a microscope, keeping at the prescribed distance from its supporting pins, his weight, acting through the ground, displaces the microscope by amounts that need to be considered, and, if so, whether, when he withdraws, the displacement disappears. Two men carry the tube by means of straps passing over their shoulders, one being at the front end and one at the rear. A inicroscope having been pointed at the front end of a tube, the frout carrier then approaches the microscope, lifts the tube from its stand, steps sidewise, and then moves forward. Immediately after the rear carrier approaches the same microscope, places the rear end of the tube on the stand under the microscope, and then withdraws. As at other times the men are two or three feet or - more away from the microscope, the danger of disturbance exists only when the tube is being placed on or lifted from its stand. Numerous experiments were made on the Sandusky and the Olney Base to decide the question whether the microscopes were permanently disturbed by the motions of the tube-carriers. The following method was used: The tube being placed under the inicroscopes as iu measuring, pointings were made at both ends of the tube; then a man approached till his foot, on which he bore his weight, was within a known distance, usually six inches or less, of that foot-plate supporting the front or rear microscope to which he approached most nearly in measuring; then both microscopes were read; the man then withdrew and both microscopes were again read. The readings on the tube, corrected for temperature-change, gave the change of int r- val between the microscopes due to the approach or recession of the man. Assuming that the distant microscope was stable for this short time, about one minute, the change in the interval was the motion of the other microscope in the direction of the base. On the Sandusky Base experi- ments were only made with the front microscope. The following tables give the results. The first column gives the date; the second gives the distance of the man’s foot from the foot-plate of the microscope-stand, and should be increased by three inches to give the distance from the supporting pin; the third column gives the displacement when the man approached the foot-plate; and the fourth, the displacement when he withdrew. A plus sign indicates motion of the microscope in the direction of measurement. s 19] CLUICAGO BASE. 275 SANDUSKY BASE-LINE. -lpparent movement of microscope at front end of tube. . - Apparent move- | Apparent move- |' 1, i e- | Apparent move- Date. a enh hes ee on ap- | ment on reced- | D ane eo ge oa ea ieee i ene ininehes. roachingrear! ing from rear , Gaanches: & roaching front) ing from front eg. | leg. | eg. leg. 1878. | t i Aug. 12° 6 + 84 43, 2 6 ite + 3.8 15 6 — 6.3 +8.5 : 6 — 9.3 + 3.3 16 3 + 9.3 —0.3 | 3 35.7 + 8.7 19 1 — 2.7 —0.1 | 1 —19.7 —13.1 20 2 — 4.9 +4.9 ¢ 2 — 4.2 — 0.5 Be 2 + 1.7 +3.5 | 2 +10.1 --13.3 23 6 — 2.4 +3.3 ! 6 — 7.0 + 0.8 27 3 — 4.4 +6. 3 | 3 + 0.7 — 0.1 28 2 — 3.0 +-0.4 2 — 4.9 + 3.6 29 6 — 4.6 +1.7 | 6 — 89 + 6.8 Sept. 6 6 — 7.4 +2.0 i 6 —25. 8 + 0.9 9 6 —17 +1.4 6 —13. 2 + 4.2 16 3 —12.5 +9.9 3 — 3.0 — 4.6 17 6 ' —11.5 +9.1 6. —16.3 + 93 18 | 6 = 68 46.1 6 —12.1 — 0.3 30, 6 — 47 | 43.7 6 — 8&7 + 4.0 Oct. 1) 6 +10. 2 ‘ +3.1 6 — 5.8 — 0.9 2 6 \ — 0.4 | +7.0 6 — 5.2 —19 31 6 | — 0.4 +17 6 | —18. 8 +16 a 6 | +15 +2.1 6 = 435 ! +14 7 | 6 — 80 +6.1 6 47.5 | — 5.0 8 | 6 08 | 42.7 6 —10.7 | 10.2 12 | 6 +114 45.9 6 410.1 — 4.2 | Means....... —~18 | +4.0 Means ....... — 86 | +19 An examination of these tables for cases in which the man’s foot was within 6 inches or less of the foot-plate, shows that when the rear foot of the microscope-stand at the front end of the tube was approached, the microscope moved to the rear in seventeen cases out of twenty-three, and in twenty-one cases out of twenty-three moved in the opposite direction when he stepped away. The mean movement in the first case was —1“.8, and in the second case +4#.0, so that the reaction would appear from the observations to be greater than the action, giving a displacement of 4+2#.2. When the front leg of the front microscope was approached, the microscope moved backward in nineteen cases out of twenty-three, and when the man withdrew it moved forward in fourteen cases out of twenty-three. The mean movement in the first case was —8#.6, and in the last case +1+.9, indicating a permanent set of —6«.7. Since in measurement the front and rear legs are both approached, the mean effect of the approaches will be —1+,8—8+.6 Sg and of the withdrawals eet au so that the permanent displacement of the microscope would be —2.2. If the two tables be com- pared, it will be noticed that in approaching the rear leg but one displacement greater than 12+.0 was observed out of twenty-three, while in approaching the front leg eight greater displacements were observed, one reaching 35.7. As these legs are but about two feet apart, it is difficult to suppose there was habitually so great a difference in the character of the soil about them as to give such differing results, and there may be some other unknown source of error in this work about the front leg. The work on the rear leg and all the work on the Olney Base show that the micro- scope habitually moved toward the man, as would be the case were his weight to slightly depress the soil about the pin nearest him. The work on the front leg, if it were correct, would indicate that the front leg rose under his pressure. § 19. On the Olney Base, similar experiments were made, and with both the front and rear microscopes. Their results are given in the following tables, in which the first column gives the dates; the second gives the displacement produced by man standing within 6 inches of rear foot- plate; the third, the displacement on withdrawing; and the fourth and fifth the like data for man near front foot-plate. OLNEY BASE-LINIE. STANDARDS OF LENGTH, BASES, AND BASH-APPARATUS. Apparent morement of microscope at front end of tube. [-+ indicates forward, — indicates backward movement. | Date, | vith 6 inches | Oman reeoding | O',i0, G inches | OB man rereding of rear leg. . of front leg. 1879, be be b be | July 25 = 62 + 4.9 ae Bie —- 5.2 28 595 + 5.6 + 4.5 — 4.0 30 OF + 6.7 +11.7 = 92 Annoy Th nemesis visicieiemimaser: -+-14.1 HDI. = sb ebantiandidinrgiorniians eet 6 — 9.9 + 8.0 — 3.0 + 4.0 7 — 9.0 + 8.7 --16.1 —18. 2 8 — 9.5 412.5 -F 11.5 — 83 W —11.6 + 8.6 as To8 — 6.9 14 20.2 +13.6 414.7 —16,2 i 18 — 9.6 + 6.1 + 6.8 = 84 : 19 —10.3 + 9.1 +|-13.3 13.1 20 | TB + 5.7 +13. 9 —11.8 21 — 3.5 + 2.0 49.4 22059) 22 — 8.5 4 4.9 +10. 4 — 83 i 28 — 9.8 4 6.9 + 9.7 = 9.8 29. —12.0 + 9.5 : + 46 — 6.0 Sept. 1 —12.6 L725 : +-11.9 —10.4 2 | —14.1 : 413.5 112.4 S487 | 3 —10.2 + 6.0 + 9.6 — 4.6 4 | 12:1 ; 12.6 + 6.5 a Gp Sj 16.7 | 4.13.0 + 9.5 87 : 8 | --17.0 413.7 119.8 12.2 1 9 11.5 + 8.8 |; 4.0 — £0 | 10 a 16.1 | 415.6 | +223 18.6 i ul —12.2 ! 4-11 f 415.1 — 9.7 2 —10.6 414.5 | (119) — 3.5 Means. -| aa + 9.4 --10. 2 85 i 1 Apparent movement of microscope at rear end of tube. On man standing | On man standing ceili | Date. ee e satan | Ae Rea | WOR 6 ldies Ea | 1879. a Bh B bh Ang. 4 —12.9 + 9.3 ; -| 1.3 —11.9 5 —14.5 + 9.7 +13. 8 — 9.0 | 6 9.9 + 6.7 + 3.9 | ~~ 6.0 : 7 18.4 4.12.0 + 6.9 | = T6 8 —~24 21 ; + 8.9 | st ‘ VW —15.4 VIL 5 213.9 Ti 14 —44.3 +39.4 112.8 12.3 18 — 9.8 ot 229 2355 19 10.4 +65 + 9.2 — 9.5 20 11,3 + 9.0 11.3 | —10.6 21 3.9 441 + 5.6 ~ 5.9 29 ~13.8 49.5 10.3 ee 23 22.8 118.7 +10. 8 1.7 29 | 9 412.5 +13.1 10.7 Sept. 1 —14.8 13.2 +64 = 45 o —19.6 +413, 2 +97 289 3 | —12.5 410.7 +10. 8 —10.2 | 4 — 86 410.8 + 6.8 — 86 5 —19.0 +16. 6 + 4.9 — 4.9 8 —15.0 +13.9 412.7 —10.6 | | 9 =A + 7.2 + 6.7 | — 8.8 | | 10 | —18.4 412.5 +-11.5 — 7.9 | | 11 | 24 | +10.3 +10. 5 —7.6 12, + 2.0 | + 4.3 + 61 | — 4.8 | . Means .. —13.4 | +10.8 + 9.3 = B98 (Cirar. X, § 20.] CITIICAGO BASE. 27 Irom these tables it appears that when a man stood within 6 inches of either front or rear foot-plate of a microscope it moved toward him in 97 cases out of 99; and when the man withdrew the microscope moved in the opposite direction in 97 cases out of 99. Mean displacements of microscopes. « | ais At front end | At rear end Position of man. | of tube. of tube. tenement 1 x 1 , [ : » i ue _ Within 6 inches of rear foot-plate........--.--.- : —l11.1 —13. 4 | WHRGPAWH .cccisciecccasce wesaneecwioncztoernn, =e O04 | 10.8 | Within 6 inches of front foot-plate......-...---- 4-10. 2 | + 9.3 WGI OIPANYD siecists Sere tues cede see os ae ae | = 85 La — 8.2 Sum oo... eee Bieide teeta aeeeecs 20 of see 4 For the front microscope there results then as the permanent displacement resulting from an approach within 6 inches of and a withdrawal from both its front and rear foot-plates (as is the case in measuring, although the approach is not so close) a permanent displacement of the microscope by 0+.0, and for the rear microscope a permanent displacement of —1+.5, or, in mean, —6+.7. The average displacement produced by a man’s weight within 6 inches of either front or rear foot-plate on the sandy Sandusky Base was 5.2, the displacements being irregular in amount and sign. On the Olney Base, much of which was pe ed with sod, displacements were quite regular in amount and in sign, the average value being 11.0. The permanent displacements of the microscope by the approach of men to and their recession from both front and rear foot-plates were —2«.2 for the Sandusky and —0“.7 for the Olney base- line. Considering the result on the Sandusky base-line, it is evident that a few additional obser- vations giving displacements no larger than some observed would seriously modify its value, while the Olney permanent displacement is no larger than its probable error. On the Sandusky Base the recovery exceeded the observed displacement for the rear foot-plate while the mean recovery was slight for the front foot-plate. It is difficult to believe that these two foot-plates should differ widely in their average conduct. The experiments do not then, in average, establish any permanent displacement, though they indicate about —1+.5. But in all or nearly all of these experiments the man’s foot was but 6 inches or less from the foot-plate instead of being a foot or more from it as in actual measurements. We may conclude, then, that in actual measurement the average permanent displacements were so small that the experiments do not enable us to assign a reliable value to them. § 20. Experiments were made on 6 days on the Sandusky Base to see if the pressure on the ground of the tube-carriers between the microscopes and alongside the tube, affected the in- terval between the microscopes. The tube was first read on with the two microscopes; then the tube-carriers stood at their places beside the tube, and the microscopes were again read, then the tube-carriers stepped away and the microscopes were again read. The observed relative displace- ments of the microscopes with men beside the tube yaried from +2+.1 to—9“.6, their mean being —2,5, But two measures of displacement when men stepped away were made; their mean was +140, a plus sign indicating motion of microscopes from each other. On the Olney Base, such measures were made on twenty-four days. The results are given in the following tables: 27S STANDARDS OF LENGTIL, BASES, AND BASE-APPARATUS. [Ciar. X, OLNEY BASE-LINE. Relatire movement of microscopes when tube-carriers come into and go out of position. [— indicates that microscopes move towards, -+- that they move from each other. ] | Date. Relative movement on Kelative movement on coming into position. | yoing out of position. 1879. mn be | July 30 - 55 +5.4 ' Aug. 1 = Ts8 +7.5 4 —11.1 $3.5 5 —10.0 +-5.4 hi - 7.6 6.3 8 “17 4-2. 5 11 — 7.6 +-8.6 ; 14 | ai 45.7 : 18 — 3.9 2.0 19 ad 47.7 20> — 2.8 +24 21 2 +1.0 22 = 16 | $7.8 28 — 2.9 -3.1 29 = 8s 44.5 Sept. 1 + 0.4 4-2.0 2 = 148 99 3 Tah 13.5 5 2.5 +3.7 8 — 3.4 45.0 9 220859 +3.9 10 — 4.3 +-1.8 11 — 4.1 44.7 12 —13 1.7 | Means. - — 4.6 | +\-4.2 On twenty-three days the microscopes moved toward each other when the men stepped beside the tube, and on all days the microscopes moved apart when the men stepped away, as they always do when measuring is going on. The maximum and minimum movements in the first case were —114.7 and —0#.2; and in the latter case +8+.6 and +140, respectively. The mean move- ments were —4".6 and +4#.2, respectively. The mean permanent set was but —0#.4, a quantity not greater than its probable error. It may, then, be said that the experiments of both kinds at both bases indicate in the aver- age no certain permanent displacement of the microscopes by movements of the tube-carriers which need be taken into special account. Such displacements occur, but they seem to be of the nature of accidental errors which eliminate themselves in many observations. § 2k. There is another way in which the stability of the microscopes might be affected. In the measurement of a base, immediately after a microscope has been pointed at the front end of a tube, the tube is carried forward and the rear end is placed under the microscope. The question arises whether the removal of the half weight of the tube (about 90 pounds) from the tube-stand, and its replacement, together with the slight shocks incident to the operation, may not transmit some per- manent disturbance through the ground to the microscope. Two kinds of experiments were made to ascertain this. In the first kind the tube, after hav- ing been read on by the microscopes, was taken a few steps away, brought back and placed on its stands again, and then read on. Experiments were made in this way on the Sandusky Base on 8 days. The observed relative movement of the microscopes varied from +64.8 to —5#.1, having a mean value of 0.0. On four days the microscopes appeared to move towards each other, and on tour to separate. The experiments do not establish any displacement. In the second kind of experiments to determine the effect of weight on tube-stand in displacing microscopes, the tube was first read on with both microscopes, then iron pins were piled among the braces of a tube- §§ 21,22.) CIUCAGO BASE. ot) stand symmetrically about its axis, and the microscopes were again read, aud finally the pius were removed, and the microscopes were again read in some cases. At the Sandusky base 14 experi- ments were made on the front microscope on 13 days. The results are given in the following table: SANDUSKY BASE-LINE. zlpparent movement of microscope at front end of tube when weight was placed on or taken off stand under microscope at front end, (4- indicates forward, — indicates backward movement. | 1 Apparent move- Apparent move- Date. Weight. ‘quent. on placing ment on removing is weighton stand. weight from stand, 1878. i Aug. 6 | 50 pounds....-- | — 3.5 + 3,3 i 9 | Man's weight... -- 4.5 -- 08 9 MOeseicece ce = + ld 1 O4 15 QU) eases eheens — 7.6 i 39 16 WO scenes | 7.6 = 0) 19 (UG) 3 geeks wore —35. 2 \-11.7 22 Ovitacens xis’ 41.6 — 0.5 | oe Ol etuecise secs = BN6 svete Seca eee seus “Sept. 9 |... do 2.22.22... SS ct saieidd LUE LEER SER rf 161... doe... PGBs Aas dde sen he ect es 17 wil O: sreeoe weet pets ge WA zaps dee S xeGnGsee OB as MO ate da Se ants ae ayG — Mawathnchothente vas Oct. 2 50 pounds....-- — 2.6 je doll 12 120 pounds....-.- DBS a Beales breaks ebvatered vee Means ..-.... — 4.5 ie De On two days the weight placed on the stand was 50 pounds; on the others it was 120 or 150 pounds. The extreme displacements observed when tube-stand was loaded were —35.2 and +11+.0, the mean displacement being —4#.5. If the single observation which gave a displacement —35.2 were rejected (and it is the only one exceeding 14.5), the mean displacement would be reduced to —2+.1. The displacement result- ing from the removal of the weights was observed seven times on six days; its extreme values were +11+.7 and —2+.0; its mean value was +2+.3, essentially the same as the displacement pro- duced by putting on the weights if the one observation referred to, be rejected. The experiments do not establish any permanent displacement on the average produced by adding the weights. Nine experiments on the same day were made by loading a rear tube-stand in three different positions with 90 pounds, and observing the displacement of the microscope over it on adding and on removing the weights. The mean displacement when weight was added was +4#.1, and —0#.1 when weight was removed; but as the microscopes occupied but three positions, the results are not sufficient to justify any conclusions from them. Similar experiments were made on the Olney Base, the loads being 90 pounds. They were made on but two days. For the front microscope the loading was repeated (the stands remaining in one position each day) four times on the first day, and six times on the second day. They, therefore, show mainly the results of repeating experiments on the same ground. On loading the stand the displacements varied between +4".5 and —3.3, giving +0#.2in mean; on unloading, the displacements varied between +4”.3 and —4+.2, giving 0.0 in the mean. As the result of the experiments to determine whether a microscope was permanently dis- placed, either by the removal of the front end of the tube weighing 90 pounds, and replacing it by the rear end having the same weight, or by loading the tube-stand by weights ranging from 50 to 150 pounds, and then removing them, it may be said that no permanent displacement of the micro- scope is established in either case, and that if such displacement exist it must be very small in comparison with other errors of measurement. § 22. It has already been stated that, to prevent disturbance of the microscope by the observer, he stood at the middle of a plank 8 feet long, shown in Plate XIII, supported at the ends, which were thus about 24 feet from the microscope foot-plates. At the Olney base-line the experiment 280) STANDARDS OF LENGTH, BASES, AND BASE-\PDPARATUS. [Cirar. X, was made op 7 days of having a man, in addition to the observer, stand first on the rear and then on the front end of the platform. When the man stepped on the rear end of the front platform, the motion of the microscope varied between —11".4 and +1+.9, being —1#.7 in mean; when he stepped from the rear to the front end of the front platform, the motions of the microscope varied between +942 and +14.7, being in mean +644. Experiments on 6 days, in which the man shifted from front to rear end of rear platform, gave motions varying between —6+.2 and +1#.4, or in mean —44.3, These displacements were the temporary displacements; the permanent ones, as in the case of the tube-carriers, were doubtless much smaller; but the work indicates the necessity for a platforin to support the observer. § 23. From the results of these experiments on the stability of the microscopes, it may finally be concluded that while the microscopes nay be displaced by small quantities by the motions of the tube-earriers, the average of such displacements must be much less for the tube-carriers, when at their usual distance of 12 inches from the foot-plates, than in the experiments when the distance was 6 inches, and the average displacement for the base on which it was greatest was 114.0; that these displacements almost entirely disappear when the pressure is removed, the mean permanent displacement for the two bases being but —1“.5 for a man 6 inches from a foot-plate, a quantity too small to be accurately determined by the experiments made, and which would be still smaller for the distance of 12 inches preserved in measurement. This elimination of permanent displacement wight have been anticipated, since the displacement would naturally be toward the man causing it, aud the permanent displacement, if any, should be in the same direction. But im a long series there is no reason for expecting a greater permanent displacement from approach to the front foot- plate than to the rear, and, since the number of approaches to each is the same in measuring, elimination of the effect should be expected. It is also concluded that the movement of the tube trom microscope to microscope causes in the average no permanent displacement in the microscopes, aud that the observer standing on his platform does not permanently affect the microscope by measurable quantities. Slight permanent disturbances of the microscopes und oubtedly occur, but it seems safe to assume that they may be considered as accidental errors, which will show their full effect in remeasurements of the different sectious of each base. As the Chicago Base like the Olney Base was on prairie soil, underlaid within 6 or 12 inches by clay, the conclusions as to the stability of microscopes on the Olney Base can be applied to it. On the Chicago Base, however, the clay was softer, and there were some periods of rain which made the clay quite soft. § 24. 4. Errors in cut-offs, that is in referring ends of tube to marks on ground.—The method of making cut-offs has already been described in Chapter VIII, §§ 9, 13. After the microscope has been pointed at the end of a tube, the tube is removed and a horizontal graduated scale, described in Chapter VIII, § 9, and shown in Plate X, Fig. 4, is put in its place, this scale being fixed on top of an axis which can be made vertical, the lower end of this axis having a conical socket which rests on a hemisphere of steel fixed in the ground-plate. The steel axis is made closely vertical with a level of. which one division = 1.6, and the small deviations from verticality are read. As the greatest distance of the graduated scale from the ground plate does not exceed 1.785, the error in fixing the position of the scale with reference to the ground-mark from this cause cannot exceed 0.1, and as it is as likely to be positive as negative, its cumulated effect in a section or in the base is insignificant. The longitudinal graduation-line of the scale is brought into the direc- tion of the base by the longitudinal microscope-wire. The scale has been compared with a standard and is found to be accurately graduated, so that its errors can be neglected. Its divisions are milli- meters at 38° F. In fact, the zero-mark near the axis of revolution was habitually pointed at with seale pointing to rear and front end of section, so that graduation-errors did not usually enter. The error in determining the horizontal distance in the direction of the base between the 4™ mark on the end of the tube and the cut-off sphere in the ground-plate being insignificant, the question arises whether the cut-off sphere itself remained immovable in the interval of time between clos- ing work in the evening and beginning it again in the morning, or sometimes for longer intervals. * The cut-off sphere is held in the cut-off plate by stout abutting-screws which give it rectan- gular horizontal motions. It may he considered as fixed with reference to the cut-off plate. This isa cirewar disk of iron weighing about 9 kilograins. In using it, three broad-headed wooden §§ 28-25.) CHICAGO BASE, 281 stakes were driven about three decimeters into the ground, leaving their heads projecting slightly, and it was placed on the heads of these stakes and snugly confined by nails driven into them. as three such cut-off plates were placed on stopping work at night (except in a part of the first measurement of the Chicago Base when but two were used) and as their distances apart were again measured on recommencing the work, an idea of their stability can be formed from the re- sults. It has already been stated that in reduction of the base-measurement the assumption is made that the center of gravity of three cut-off spheres does not move and that the measurement is referred to the front cut-off sphere of the three. The motions that we detect, then, are those of, this front cut-off sphere with reference to the center of gravity of three. If, as a part of their motion, all three move by equal amounts in the same direction, this part of their motion is un- detected. § 25. The following table gives for each cut-off on each measurement of the Chicago, San- dusky, and Olney Bases, the date, the number of the tube at which the cut-off was made, the time- interval between stopping and starting, and the change in the distance between the first and sec- ond and the second and third cut-off plates. When the cut-off plates separate from each other the plus sign is given to the change. Relative movements of cut-off plates. TWO CUT-OFF PLATES: CHICAGO BASE, FIRST MEASUREMENT. = | Intervals of | Relative movement of cut-off | time be- plates. Dates. : Nos. oftubes. | tween cut-)| : off read- | ' ings. 1-2 | 2-3 | L t 2 7 1877. ; h. min. mm. : June 13-14..... ... | 78 16 —0.0479 | VS eee 139 15 +0. 1150 15-185 2.0 210 63 +0. 0740 18-20......... 228 43 —0. 2534 | Dy \ 280 20 +0. 1136 i 29-28... | 367 14 —0, 2766 | DSO coke sed ; 458 87 +-0.0078 | 27-28......--. | 513 16 +0. 0879 28-28......--. 513 6 40.0805 | 98-90. aden ves 536 15 +0. 0394 29-30.....---- 636 16 +0. 1414 30-July 2..-. 636 50 +0.0047 | July 2 3......-.. 712 15 +0. 0885 BS25 a asacnnn 772 42 | 40.1242 | BS 6 i Saee ais 843 18 +0.1687 | ee 928 15 0.0473. | TA senccss 966 68 —0. 0117 é WAL 2 ences 1066 16 —0.0455 | ; VW-12. 2.222 = 1154 | 16 +0. 0552 ISAS asmece 1240 | 16 -+0. 0528 23-24..22222-. 1877 18 +0. 1464 TWO CUT-OFF PLATES: CHICAGO BASE, SECOND MEASUREMENT. Aug. 22-24......--. 1761 40 | —0.1271 |; THREE CUT-OFF PLATES: CHICAGO BASE, FIRST MEASUREMENT. July 18-14..-..---.1 1339-40 | 16 +0. 2417 | 40.1816 | TH 6) e css 177-8 45 | +0. 2133 0.1203 16-17-0200. 1495-6 Ww —0. 0701 | 40.1766 | VI-AB. 22. 1525- 6 It | 0.0981 40,1816 | 18-19.... 22. | 1591- 2 a +0. 1018 | 40.0981 19-20..2...2-- 1678- 9 19 —0. 0106 +0,1792 | rs 1751- 2 16 —0. 1248 | —0.0026 | 21-23.......- | 1787- 8 45 | 0.1799 —0.2573. | 36 LS 282 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. X, Relative movements of cut-off plates—Continued. THREE CUT-OFF PLATES: CHICAGO BASE, SECOND MEASUREMENT. | Intervals ot | Relative movement of cut-off time be- plates. 1 Dates. Nos. of tubes. tweenent-- off read- | ings. 1-2 2-3 rage erenge ss Doe ete ey i) einige em =i . h. | mm. mm. | July 42- 3 17 | —0.1108 | -+0. 0233 | 199-200 17 | +0.0700 +0. 1318 233-4 45 | —0.0051 | 40. 0720 330- 1 17 | —0.3266 | +0. 1723 ; 441- 2 40 | +0. 0574 40.1738 Avg. 524-5 16 —0. 1163 0.1234; 647- 8 16 —0. 1018 —0. 0849 - 700- 1 45 | —0. 1963 4-0.0278 | 721- 2 22 +0. 0080 +0. 2410 | * 952- 3 ie | —0. 0788 +0.1331 | 1042- 3 16 | -+-0. 0465 0.1010 | 1160- 1 64 —0. 0760 +0. 6659 | 1216- 7 41 | —0. 2874 +0. 0981 1237- 8 1 —0. 0281 —0.0561 | 1341- 2 Wo —0. 0078 -{-0. 0859 1379- 80 20 | +-0, 0904 -+0. 0296 1423- 4 42 —0. 0108 4-0. 3944 1546- 7 is | --0. 1201 —0.1194 | 1653- 4 4, —0. 1867 +0. 1388, 1720- 1 1 | +0. 0792 —0.0324 | 1806- 7 18 —0. 0319 +0. 0229 | THREE CUT-OFF PLATES: SANDUSKY BASE, FIRST MEASUREMENT. Aug. 6-9 42-3 62 | +0. 0989 +0. 0792 95- 6 63 +0. 0071 —0. 0154 173+ 4 38 +0. 0954 —0. 1875 242- 3 17 | —0. 1190 +40. 0414 323- 4 16 +0. 0407 +0. 1430 406-7 6 | +0. 0720 -+0. 0355 489-90 1 | —0. 0065 +0. 0383 544-5 40 -+-0. 0476 --0. 0681 647- 8 16 —0. 1982 +0. 1995 857- 8 16 +0. 0464 +-0. 0049 934-5 17 —0. 0705 —0. 1107 Sept. 1041- 2 17 —0. 1229 +0. 1557 Bo Wee age 141- 2 16 —0. 1901 +0. 1856 a 1195- 6 45 +0. 1789 —0. 0187 | Wal Qizes ree 1353- 4 46 +0. 0906 —0. 1009 | DIG wesc 1463- 4 161 —0. 4603 +0. 2222 THREE CUT-OFF PLATES: SANDUSKY BASE, SECOND MEASUREMENT. 93-4 16 +0.0610 | —0. 1096 194-5! 112 —0.1736 | +0. 0093 332-3 16 —9.0462 —0. 0141 484-5 15 —0. 2430 | +0. 1071 640-1 16 +0, 0885. | “0. 0159 | 741-2 7 —0.0605 | 40. 3173 841-2 | 16 +0.1729 | —0. 1593 966-7 16 -+0. 0295 | —0. 1290 1157-8 | 15 —0. 1112 +0. 1528 S80 i ecacarue 1263-4 | 40 —0. 0512 | +0.1416 | 0-1 a enrect, 1352-8 15 —0.0367 | +-0.0055 TDA ster 28 : 1424-5 | 20 +0. 1434 , +0. 1112 | § 26.) CHICAGO BASE. 283 . Relative movements of cut-off plates—Continued. THREE CUT-OFF PLATES: OLNEY BASE, FIRST MEASUREMENT. Intervals of | Relative movements of cut-off ° time be- plates. Dates. Nos. of tubes. tween cut- off read- ings. 1-2 2-3 1879. h. mm. mm. July 28-29 é 92- 3 ! 14 —0. 0017 +0. 0591 148-4 | 17 —0. 0846 +0, 0578 248-4 | 14 +0. 1192 —0. 1431 272-3 |! 22 —0. 0619 —0. 0770 Aug. 362-3 |: 62 -++0. 1547 —0. 2592 463-4 ' 15 —0. 0924 +0. 1534 547-8 16 +0. 0333 +0. 0434 593- 4 2k —0. 0334 +0. 0039 648- 9 15 +0. 0478 -++0. 0786 18-9 | 15 —0. 0065 +0. 0100 821- 2 64 +0. 2043 —0. 13854 939-40 15 —0. 0340 +0. 0827 994- 5 23 —0. 1215 -+0. 0306 994- 5 21 —0. 0148 —0. 0085 13-14....-.-.. 1009-10 19 —0. 1984 +0. 1111 14-16.......--. 1079-80 42 +0. 1029 —0. 2572 IG=lWSeeeccaccs 1079-80 46 +0. 1188 —0. 1679 TSA ae escine eee 1175- 6 16 -—0. 0919 +0. 0711 19-20......... 1297- 8 16 +0. 0387 —0. 0806 BOHOL. ciazieiasa otis 1398- 9 16 —0. 0161 +0. 0534 1H OD aati Saye 1519-20 16 +0. 0545 +0. 0382 THREE CUT-OFF PLATES: OLNEY BASE, SECOND MEASUREMENT. 1877. 70-1 16 +0. 0786 +0. 0269 207-8 16 +0. 0273 —0. 0400 312-3 64 +0. 2170 —0. 1407 424-5 16 —0. 0300 —0. 0605 546-7 16 —0. 0612 +0. 0551 641-2 17 +0. 1150 —0. 0490 756-7 16 +0. 0442 —0. 0332 atsaie 872-3 64 —0. 1507 +0. 1628 B= Dcamensie 1040-1 15 —0. 0847 +0. 0325 9-10...-..--- 1204-5 15 +0. 0557 +0. 0098 10-11 eee 1871-2 15 +0. 1019 —0. 0433 11-12.-.-.-... 1523-4 16 —0. 0675 —0. 0037 These tables, giving changes in interval between cut-off plates during the suspension of work at night or during storms, give the data for an estimate of the probable error in the assumption that the center of position of the two or three cut-off plates remained unchanged during the suspension. § 26. If 4; represent the change between the first and second cut-off plates, and 4, that between the second and third; and if 62, dx, 6#3, represent the change in the position of the sepa- rate plates taken as positive when they move in the direction of measurement, we shall have A, =6a,—0a, 4, = 8x,— 0a, and squaring and summing for the base, including both measurement and remeasurement, with three cut-offs, , [4,7] +| 4,7] = [da? 4+ 2d2,? + dag? —25x,0a0, —20a30a,] Now, as we do not know the laws which control the small disturbances, da, of the cut-off plates, it must be assumed that they follow the ordinary law of error. The products of the displacements in the second member of the equation just written will then disappear, leaving [4,2] + [4,9] = [2] + 2[ 0202] + [0ar 284 STANDARDS OF LENGTH, BASES, AND BASH-APPARATUS, [Cuapr. X, But there is no reason for supposing the sum of the squares of the displacements greater for one cut-off plate than for another. Taking them to be equal to that for the first cut off plate, 4{ox,?]=[4°]+[4.7] If there are n cut-offs the square root of the mean square of displacement will be and the probable displacement of any cut-off plate, dx = + 0.6745 flaerl44) 4n For the center of position of three cut-off plates the probable displacement is then Oxy +t 0.6745, [4+ 140 12n or, for two cut-off plates, duty 0.6745 J e [4,"] vi) These probable displacements include the probable errors of the two measurements of the intervals between the cut-off plates, and hence overestimate the disturbance. But as the errors of measurement are small in comparison with the displacements, it is not necessary to specially con- sider them. The probable error of a measurement scarcely exceeds 5“, and this would affect but very slightly the probable value of the displacement. Finding now the values of 6x or the proba- ble movement of a single cut-off plate, during the suspension of work, habitually for 15 hours or more, there results For Chicago Base, 6av=-t- 84" For Sandusky Base, év=-+ 61 For Olney Base, Oa + 42" Deriving the values of the 6x, the probable motion of the center of position of two or three cut-off plates, they are found to be Chicago Base, 1st ineasurement, two cut-off plates, or= £60"; number of stops, 21. Chicago Base, both measurements, three cut-off plates, ox2.=149+; numbers of stops, 8 and 21. Sandusky Base, both measurements, three cut-off plates, 0v;=+35"; numbers of stops, 16 and 12. Olney Base, both measurements, three cut-off plates, day= + 24¢; numbers of stops, 21 and 12. The probable error of the cut-off correction, that is, of the relative movement of the last micro- scope, and the center of position will differ from these only by quantities which may be neglected. Since the observations only gave relative movements of the cut-off plates, they did not detect a common movement of all the plates in the same direction, which without doubt sometimes occurred. Hence the above probable displacements are somewhat too small. Multiplying each of the 6.) by the square root of the number of stops, values varying between £0.27 and +0”™.08 are found for the probable errors introduced into a measurement of a base by the instability of the cut-off plates. They are so small that even when increased somewhat on account of the fact that the method of making the cut-off measurements does not detect a common movement of all the cut off plates in the same direction, it will yet be sufficient to assume that these errors develop their full effect in the differences of length of the sections of the base given by the measurement and remeasurement, and therefore do not need a separate estimate. § 27. The following table gives the numbers of times the signs of the change in interval between the first and second, and second and third cut-off plates and microscopes over them were positive and negative for each base and each measurement, a positive sign indicating that the plates or microscopes moved apart. §§ 27,28.] “~~ CHICAGO BASE. 985 Number of + and — signs in relative movements of cut-off plates and microscopes. TWO CUT-OFF PLATES: CHICAGO BASE. Pair of cut-off Cut-ofts. Microscopes. plates and Pehegcee ected MICTOSCOPES. | No. + | No. — | Whole No.| No. + | No. — | Whole No. 1-2 16 6 22 11 7 18 THREE CUT-OFF PLATES: CHICAGO BASE, FIRST MEASUREMENT. 1-2 4 4 8 2 5 7 2-3 5 3 8 5 2 7 BUTE sescsacce 9 7 16 7 7 14 THREE CUT-OFF PLATES: CHICAGO BASE, SECOND MEASUREMENT. 1-2 7 14 21 9 11 20 2-3 17 4 21 i 9 20 Sums ......... 24 18 42 20 20 | 40 I THREE CUT-OFF PLATES: SANDUSKY BASE, FIRST MEASUREMENT. 1-2 9 7 16 5 7 12 | 2-3 11 5 16 9 3 12 | 1 BUMS 2.2. 20 12 32 14 10 24 | I : THREE CUT-OFF PLATES: SANDUSKY BASE, SECOND MEASUREMENT. 6 ul | 1-2 1 5 7 | 12 5 | 2-3 3 4 | 12 6 | 4 10 | | a. oy n | | Sums ......... | 13 ry 10 21 | = / 9 12 21 9 eo) ee |] 923 13 8 21 9 7 | 1 Sums ........- 22 | 20 | 42 | 18 14 | 32 | THREE CUT-OFF PLATES: OLNEY BASE, SECOND MEASUREMENT. 1-2 7 5 12 6 4 0 | a nn) 7 12 9 1 10 Sums ........- 12 | 12 24 15 5 | 20 I This table shows on the whole a preponderance of cases where there was separation of both cut-off plates and microscopes. § 28. In the measurement of the bases the microscopes usually remained in position over the cut-off plates, and as the measurement of the distances between the cut-off plates gave also the distances between the microscopes, an examination as to their relative motion during suspension of work has also been made. There is a source of error entering this work, however, which does not affect the other. Different parts of the microscope-tripod may have different temperatures at night when the work stops and in the morning when the work begins, due to the changed position of the sun or other causes. This might cause slight motions in the microscopes even when the tripod- feet did not change their position. But this would be partially eliminated from the fact that the sun would change position with reference to both microscopes. 226 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Curap. X, aw The result of the examination is that the cut-off plates have a somewhat greater stability than the microscopes. Since, however, both cut-off plates and microscopes are most unstable during and after rains, and since at that time the greater weight of the microscope-tripod would tend to make its changes of position the greater, it is probable that in good soil and dry weather the micro- scopes and the cut-off plates have about the same stability. As a rule, when the change in the interval between cut-off plates was large, the change in the interval between the microscopes was large and usually in the same direction. Thus on the Chicago Base there were sixteen cut-offs in which thechange of interval of cut-off platesexceeded 0"".15 out of a total of sixty-three. In nine of these the microscope-interval changed by more than 0"".1 in the same direction, and in three cases it changed by more than 0.1 in the opposite direction. On the Sandusky Base the interval between cut-off plates changed in thirteen cases out of a total of forty-seven by more than 0”".15. In ten of these cases the microscope-interval changed by more then 0.1 in the same direction, and in none of the cases did it change in the opposite direction by 0"".1. On the Olney Base the interval between cut-off plates changed by more than 0.15 in four cases out of a total of fifty-one cases. In two of these the microscope-interval changed by more than 0”".1 in the same direction, and in none of these cases did it change in the opposite direction by 0™”.1. Besides the cut-offs, two or three in number, made when the work was to be suspended for many hours, there was a cut-off made on a single plate on stopping for about 1.9 hours on the average at noon. The measurements made on recommencing the work gave the change in relative position of microscope and cut-off plate from all causes. That change was of course due only in part to change of position of the cut-off plate. - The following are the probable changes in position of cut-off plate or microscope for the different bases, derived on the assumption that microscopes and eut-off plates were equally stable from the observed changes in relative position. Chicago Base, +16 Sandusky Base, £134 Olney Base, 410+ The number of cut-offs of this kind for both measurements for all bases varied between twenty-one and twelve, so that the maximum probable error introduced into the measurement of a base from this cause would not exceed 16# /21=0"".07. This error is so smnall that it may be included in the discrepancy of the two measures of each section of a base. § 29. From the discussion of errors of alignment, of inclination, of stability of microscopes and of stability of cut-off plates, §§ 16-28, it is concluded that these errors are either so small that their effect on the whole base can be neglected, or that their effect is sufficiently shown in the discrepancies between the two measurements of each section. The errors which will enter the final value of the Chicago Base will be that in the adopted length of the steel bar 8, at a chosen temper- ature, and those due to the determination of the nunber of times this length of bar is contained in the length of the base, that is, to errois of measurement. If the full effect of the errors of measurement was shown by the discrepancies between the two measures of each section, a pretty good idea of the resulting probable error of measurement in the mean of two measurements of the base could be obtained in the following way. From § 13 the fol- lowing table is derived: I Section. : No. of tubes. DE reucen ot | , mm. Te Sao eet Seeley 297, 25 1.3 TY 2 conus asta erat j 230. 25 +25 Desc he aussie ce aoe 234. 50 +2.8 fei rstacldet se acetate es 232. 00 +0.7 I) Nea sh samces ans 231. 00 +15 | Ih ANilecea Sate Coane aes: 225, 00 +11 | VID cere eeeceeeeeeee reese] * 800.60 41.3 |) WH esececcreiues ude coenewas 196. 80 0.2 §§ 29-31.] CHICAGO BASE. 287 The positive sign prefixed to a difference indicates that the first measurement was the greater. There are but two negative signs to six positive. If the positive signs all corresponded to cases where the temperature of the first measurement differed in the same direction from that of the second, the positive signs might be attributed to an error in the value adopted for the expansions of Z, and 8,; but of the six positive differences three, as will be seen by reference to the temper- ature corrections in § 12, are when the first measurement was hotter and three when it was cooler. ‘Since the first measure of the base was from west to east, the zine bar was on the north side; in the second measurement it was on the south side. If the position of the sun affected the deduced length of S, by slightly heating the nearer bar, the second measurement should have given the greater length. Call d the difference between two measures of the same section and suppose the errors of the different sections independent of each other, then there results for the probable error in the mean of the two measures of the base p. 6.=0. 6745, [(@1— + 1°".46 = mer part of the base. Besides the errors of measurement already mentioned, this probable error includes the other accidental errors of measurement, and a part of the errors due to erroneous corrections for temper- ature. But the discrepancies between the measures of a section do not fully develop the errors arising from temperature. § $0. Neglecting for the present the consideration of the errors arising from the fact that the observed Z,— 8, is not always the true metallic temperature of S,, and leaving this error to be con- sidered later, the errors now to be considered are those in the part of the base expressed by § 6. (1) n(S,),| 540) +0,250) -+ 2000.6 +144588.1| + 761.2 | —154827.9| -+.26337.4|/ —196.3 | +2524.6 | + 21257.7 1/1556 | 40.825 | +21005.3 | +689623.4 | +4928.8 | —467378.5] +64442.4 | +387.5] +7956.1 | +320265.0 Total.-/) 9 | 1556 | +0.925| — 4010.9 | +245516.8] -+1522.6 | —394214.5| --61706.0| +244.8 | +4869.2 . — 84366. 0 iT The mean value of Zi— Si for the two measurements of the base was + 460.5, corresponding to 72°.38 F. § 4. In the following table, derived from the preceding one by expressing the two measure- ments of a section with the same fraction of a tube, the principal results are given, and also the differences between the two measurements of each section. To express the section of the base in terms of entire numbers of §,° and equal fractions of it, the values of the fractions of a tube in metres are needed. The value of S, is given in Chapter IX, § 66, and from that its value at 609.292, for which Z,—S,;=0, may be computed. Transforming it into metres by Clarke’s value of the metre (metre = 1.093623 yard) there results— 8, at 60°.292 I. =4™.0009624 as an approximation more than sufficient for obtaining the values of the fractions of the tube in metres. It has already been stated that these fractions of tubes are the exact fractions which they purport to be within 10“, a quantity that is insignificant in comparison with the length of a section, and which enters a section but once. §§ 4,5. J SANDUSKY BASE. 293 Principal results of measurement of Sandusky Base. A Measure-| Number of Sum of cor- + PP Section. meant. ( Sines, Toctions, Mean. Difference. Me Mw | | 1 243.7 —108300. 2 wie ae dees 2 243.7 —1o8003.5 | ¢ * pee fe ' | A aed 1 259. 3 + 69443. 5 i 4 69379. 0 api | 2 259. 3 + 69814. 5 1 503. 0 — 38856.7 — 38772, — 167. SroseAEy ; 2 503.0 — 38689. 0 j bias sae 1 247, 4 4215394. 3 ere one | Aieols g 247. 4 +215792. 6 CFE — 1 20012 + 19206. 6 t + 18275.8 +1861. 6 anes 2 266. 2 ++ 17345. 0 Middle 1 513.6 “234600. 9 j 42m sas ae part 1 513.6 + 233137. 6 + re - 1 217.7 +127126.1 _ Acat a ee +123329. 8 1125228. 0 Teiies I 1 262. 18. ain | ae PaaS t + 97697. 4 — 587.4 } B wRs + 97976.1 1 540. 2 “}224544, 8 , ; Hinst part } 5 oe Leite bs 2 202905. 4 +3238.9 | Total 1 1556. 8 +-420289. 0 base 2 1556. 8 4415754. 5 ; -+418021. 8 arose § &. From the preceding table there result, for the mean lengths as measured of the three parts of the Sandusky Base, the following values, in which (8,)z,-s==4\° is the length of the steel bar 8; when at the temperature for which Z;=S, or 60°.292 F. mm. West part —503.0(§;),,-s,— 38.77= 6602. 5699 Middle part—513.6 (8;),,-s, + 233.87 =6742.6067 East part —=540.2(S,),,~s, + 222.093 = 7091.7393 the values in English feet being derived from the value (S,),,_s,—= 13".12663443, given in Chap- ter IX, § 66. Computing for Clarke’s spheroid with the azimuth of the base and the mean heights of the tube for each of the three parts of the base given in § 2 the reduction to sea-level, it is found to be for west part 0.1840; for middle part 0.1869; for east part 0.1966; so that at sea level the parts of the base in terms of (S;),,-s, and in feet are mm. Jt. West part =503.0(8;),-s,— 94.86 6602.3859 Middle part=513.6(S,),,-s, + 176.89 =6742.4198 Hast part —540.2(S,),., + 163.01 —7091.5427 As stated in § 2, the adjusted angle between the west and middle parts of the base is 177° 45’ 41.9, and between the middle and east parts of the base is 177° 39/ 52.2, the opening of each angle being toward the north. Projecting the parts of the base after reduction to sea-level on the line joining the western and eastern ends of the base, also at sea-level, the projections, which will be given both in terms of (S;),,~s, and in English feet, are ft. Projection of west part —503.0(8;)z,-s,—1. 75242 — 6596.94.77 Projection of middle part=513.6 (81)z,~s, + 0.17450 — 6742.4120 Projection of east part ==540.2 (8))z,~s,— 1.50075 = 7086.0841 294 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuar. X1, Calling the interval between the west and east ends of the base the Sandusky Base, by sum- ming the projections just given, there results— Sandusky Base at level of mean tide, New York 1556.8 (;),,-s, —3".07867 = 20425" 4438 Substituting for (‘,),~s, its value in terms of #1876, derived from Chapter IX, § 66, namely S,=+4 R1876—58+.404 14.326 (¢—42.75 F.), since Z,=S8, at 609.292 F., Sandusky Base at level of mean tide at New York=6227.2 (41876 at 609.292 F.) —3".13338 This last value of the length of the basis, dependent on the adjusted expansion of #1876 and S, If the length in terms of 81876 of S; dependent alone on intercomparisons of #1876, 8, and S,, be used, viz: S,\=4 R1876—37#.01+1".2744 (t—59) (see Chapter LX, § 66), the resulting length of the base is 6227.2 (R1876 at 60°.292 I.) —3".13372 § 6. The errors in projecting the parts of the base on the line joining the ends of the base and those in determining the difference of level of the base and of the sea are so small that they may be neglected. Indeed, the corrections from these causes are so small that we may take for the proba- ble error in the final value of the base, the probable error before these corrections are made. The probable error in the length of the Sandusky Base will be derived in a manner entirely similar to that used in obtaining the probable error in the Chicago Base which is given in Chapter X, §§ 29-32. From the column of differences, d, in the table given in § 4, there results for the part of the probable error obtained from these differences, a 0.6745, fT 4.1.45 or to9g5yo Bat of the base. Following the process in Chapter X, § 30, to find the probable error in the base arising from uncertainty in the length of §, (1) of that section becomes x (S1)z,-s, =m + 0.000005106 [(Z,— 8,)’]. For the Sandusky Base, § 3, m=+460«.5, corresponding to 72°.38 IF. Substituting this in the expression for the probable error in the adopted length of 8, at any temperature, given in Chapter IX, § 63, there results for its probable error at 72°.38 F., +. 24.96. Since n=1556.8, there fesults for the probable error in the base arising from the first term of (1) above, +4"".61. Taking the prob- able error in the coefficient of the second term at one-fourth of its value, as in Chapter X, § 30, there results as the probable error of the second term +0™".72. Combining it with the +4™".61 already found, there results +4™™,67 as the probable error in the value of the base due to uncertainty in the length of S,. For the probable error in the adopted lengths of §,, due to the observed Z,—S,, not giving the true temperature of S,, 1.5 will be taken as was done, Chapter X, § 31, for the Chicago Base. Multiplying this by the number of tubes in the base, 1556, there results for this probable error, 2"™,33 The probable error in the height of the base, § 2, may be taken as £0.25. This gives a proba- ble error in the reduction to the sea-level of 40,24 Collecting the various probable errors, there results— From discrepancies of measurement.............2..2.ce.cceeeceeee +£1.45 From probable error in value of S,....-....... .022 cece cece ecec eens 44.67 From errors in temperatures Z,—S,.... 0.0.0.0 cee cee cece ee cee cece +2.33 From errors in mean elevation of tubes ...............--.ceececeee +0.24 Combining, there results, : 1 Probabl = at ae robable error in Sandusky Base= + 5™™,42 1148600 part of base. §§ 6-8.] SANDUSKY BASE. 295 Hence from § 5 there results, SANDUSKY BASE AT SEA-LEVEL=20425't,4438 + 0f,0178 § 7. By a process precisely like that followed in Chapter X, § 33, the probable error of San- dusky Base expressed in terms of the metre R1876, § 5, is found to be +3™".03 or sosto05 part of the length of the base. Hence ‘ Sandusky Base at sea-level—6227.2(R1876 at 609.29 F.)~—3133™™,724.3"™,03 5 When the length of R1876 in metres is substituted in this expression, the probable error of the resulting numerical value will be + [(6227.2 «)*4 (3.03)]% in which «=the probable error of the length of R1876 at the mean temperature (72°.38 F.) of Sandusky Base. § 8. The Sandusky Base containing two angles differing little trom 180°, a new station called Check Base was erected, so as to give triangles of good shape when connected with these angular points and with the ends of the base-line. The angles of the lines joining these points and afew to other stations were read. These angles were adjusted by least squares according to the method given in Chapter XIII, with the conditions introduced needed to prevent change in any of the ' primary angles previously adjusted. The angles of this subordinate triangulation were read by Lieutenant P. M. Price, Corps of Engineers, in 1878, with the 14-inch Pistor and Martin’s theodolite No. 2, the number of combined results obtained for each angle being usually 24. The following tables give the names of the stations and angles, the observed angles, their notation, number of combined results, range in the results, their assigned weights, the local and general corrections for each angle, and the adjusted angles. The normal equations for local adjustment are also given, and in tables following the numerical equations of condition, the general corrections in terms of the correlates, the normal equations for the determination of the correlates, the values of the correlates and their logarithms, the values of the general corrections, the residuals resulting from substitu- tion of the general corrections in the equations of condition, and the derivation of the probable error of an observed angle of weight 1. Computing with these angles and the value of the whole base given in § 5, the lengths of the three parts of the base, the excesses of the computed over the measured parts are found to be, for the west part, —0.0485; for the middle part, +0.0151; for the east part, + 0.0333. Triangulation about Sandusky Base. WEST BASE—1. [Observer, First Lieutenant P.M. Price. Instrument, Pistor & Martin’s theodolite, No.2. Date, September and October, 1878. ] Angle as measured between— Notation. | No. meas. Range. Wt. | (v) | [v]} Corrected angle. of aw “ “ “a or a | East Base and East Angle... 1 11 40.376 | li 24 6.2 1 +0. 172 +0. 483 1 11 41.031 | East Base and Sandusky....-....-.---.--- Vpopapa [ecce ce cece e|eee eee cee ee [enon ee ce cecwer ere eeclemeene ceceee 74 52 17.009 East Angle and West Angle 1 07 50.£94 | 12 24 4.3 1 +0. 172 +0. 780 1 07 51.346 | West Angle and Check Base 35 55 27.588 | 1s 24 8.3 1 +0.172 —1. 867 35 55 25. 893 | Check Base and Sandusky.. 36 37 18.254 | Ja 18 6.3 0. 75 +0. 229 +0. 256 36 37 18.739 | Sandusky and Steeple..----- 2 53 34.229 | Is 11 3.8 0.5 --0. 694 0. 000 2 53 33. 535 Sandusky and East Base... 285 07 40.916 | 1,46 7 6.6 0.3 +1. 727 +0. 348 285 07 42. 991 Steeple and East Base...... 282 14 09. 802 | le 14 4.5 0.5 » 70. 694 +0. 348 282 14 09. 456 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1,8(11)+-0.8(12)+0.8(13)--0.8 (14)4-0.5(1s) —0.420=0 0.8(11) +-1.8(12) +-0.8(13)-++0.8 (14) -+0.5(1s) —0.420=0 0.8(11)-+ 0.8(12)-++1.8(13)-+-0.8 (14)+-0.5(1s)—0.420=0 0.8(11)-+0.8(12)+ 0.8(13) +-1.55(14)-+-0.5(15) —0.420=0 z. 0.5(11) + 0.5(12)+-0.5(13)+0.5 (14)+1.0(1s)4-0.322=0 Note.—The value of 1, 424344 is taken as exact, being the sum of the adjusted values of the angles 693 and 694 of Section X of the adjustment of the principal triangulation, Chapter XVII, C, § 4. 296 Triangulation about Sandusky Base—Continued. EAST BASE—2. (Observer, First Lieutenant P. M. Price. STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. Instrument, Pistor & Martin’s theodolite, No. 2. Date, September, 1878.] (Crap. x, Angle as measured between— Notation. | No. meas. '* Range. | Wt. | (v) [v] Corrected angle. oO t wy “a : | a aw of we i t Sandusky and Check Base.. 29 45 32,728 | 21 20 | 6.2 1 -++0. 030 +0. 039 29 45 32. 797 Sandusky and West Base...--.. --.-.---- Ziyots4445 | 69 32 30. 524 Check Base and Steeple.... 2 18 30.504| 22 | 4, j 2 : 2 18 29, 828 Steeple and East Angle .... 35 13 35. 046 | 2 24 | 4.1 1 +0. 030 —0. 707 35 13 34. 369 East Angle and West Angle. 1 08 16.291) 24 4 24 | 4.8 | al +0. 030 +1. 476 1 08 17,797 West Angle and West Base. 1 06 36.904 | 25 24 7.2 1 +0. 030 —1. 201 1 06 35. 733 West Base and Sandusky .. 290 27 28.348 | 26 20 9.1 1 +0. 029 +1. 099 290 27 29.476 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(21)+ (22) (23)-+ (24)+- (25)—0.179=0 (21)-+2(22)-++ (23)-+ (24)+4- (25) —0.179=0 (21) (22)-+-2(23)-++ (24)-++ (25) —0.179=0 (21) (22)+- (23) 4-2(24)-+ (25) —0.179=0 (21) (22)4- (23)4- (2a) 4-2(26) —0.179=0 Note.—The value of 2), 0454445 adjustinent of the principal triangulation, Chap. XVIT, C, § 4. ° CHECK BASE—3. (Observer, First Lieutenant P. M. Price. Instrument, Pistor & Martin’s theodolite, No.2. Date, October, 1878., is taken as exact, being the sum of the adjusted values of the angles 681 and 685 of Section X of tho Angle as measured between— Notation. | No. meas. Range. Wt. (v) {v] Corrected angle. | o ‘ a | a “we “a D8 a West Base and West Angle. 25 47 54.542 31 | 24 4.6 1 +0. 639 —1. 048 25 47 54. 133 West Angleand East Angle. 45 33 32.923 32 | 24 5.8 1 +0. 639 —0. 061 45 33 33.501 EastAngle and East Base .. 30 36 35, 552 33 24 6.8 1 +0. 639 +0. 218 30 36 36, 409 East Base and West Base. . 258 01 54. 425 34 | 24 71 1 +0. 641 +0. 891 258 01 55. 957 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(31)+ (82)+- (33) —2.558=0 (31)+-2(32)4- (33)—2.558=0 (31)+ (82) -++-2(33) —2.558=0 \ WEST ANGLE—4. {Observer, First Lieutenant P. M. Price. Instrument, Pistor & Martin's theodolite, No. 2. Date, August, 1878.] Angle as measured between— | Notation. | No. Meas. | Range. Wt. (v) [v] Corrected angle. t or aw a“ a“ a Oo A “ East Base and East Angle... 1 11 51.120, 41 24 7.8 0.8 +0. 093 —1.191 1 11 50. 022 East Angleand Check Base. 63 57 37.433 42 24 8.6 1 +0. 075 +0. 592 63 57 38. 100 Check Base and West Base. 118 16 40. 646 | 43 24 6.2 0.7 +0. 107 —0. 767 118 16 39. 986 West Base and East Base.. 176 33 50. 402 44 24 8.9 0.6 +0. 124 +1. 366 176 33 51. 892 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 4 (41) +0. 6 (42) +0. 6 (42) —0. 239=0 0. 6 (41)-+1. 6 (42) 4-0. 6 (43) —0. 239=0 0. 6 (41) +-0. 6 (42) +1. 3 (43) —0. 239—0 § 8.) SANDUSKY BASE. 297 Triangulation about Sandusky Base—Continued. EAST ANGLE—5. (Observer, First Lieutenant P. M. Price. Instrument, Pistor & Martin's theodolite No. 2, Date, August, 1878.] Angle as measured between— Notation. | No meas. Range. Wt. (v) (v) Corrected angle. | ! fo} t aw “wt we aw o ¥ “a East Base and Check Base 111 51 19.729 di 24 10.7 1 —0. 474 +0. 151 111 51 19. 406 Check Baseand West Angle 70 28 48. 902 ; 52 24 11.6 1 —0. 474 —0. 016 70 28 48, 412 West Angle and West Base 1 06 28. 158 53 24 7.6 0.8 —0. 593 —0. 825 1 06 26.740 West Base and East Base.. 176 33 25. 344 | 5a 24 13.4 0.8 —0, 592 +0. 690 176 33 25. 442 NORMAL EQUATIONS FOL LOCAL ADJUSTMENT. 1. 8 (51)-+-0. 8 (5) +0. 8 (5s) -+ 1. 706=0 0. 8 (5!) +1. 8 (52) +0. 8 (53) +1. 706=0 0. 8 (51) £0. 8 (52)-+-1. 6 (53)-++1. 706=0 : Numerical equations of condition in the triangulation about the Sandusky Base-line. SIDE-EQUATIONS. IX. (10) +26. 7086 [11] —1.1768 [1,] — 1.1768 [15] + 2.1187 [22] +2. 1187 [23] —25. 2863 [2,] —25. 2863 [25] +8. 4451 [51] + 7.0090 [52] + 7.0090 [53] + 0.410=0 X. (10) +27, 8854 [1,] —1.1751 [13] —10. 2873 [42] —11. 3267 [43] +0. 4553 [52] — 7.0090 [53] —32. 336=0 XI. (10) ++ 1.0982 [2,] +1. 0982 [23] —26. 3068 [2,] + 9.7477 [41] —0.5396 [42] + 8.4451 [5,] + 7. 4643 [52] +51. 151=0 ANGLE-EQUATIONS. . I (hh) 4+ (12) + 015] + (J + 0.348=0 IT. £21] + [22] + [2s] + [2s] + [25] + 1.099=0 UT. Cli] + (Ce) + [1s] + £22] + [25] + [21] + [25] +031] + [82] + [83] + 2. 6383=0 IV. [12] + (1s) + [3:] + [92] + [52] + [5s] + 3.037=0 V. (1s) + [51] + [4s] + 3. 682=0 VI. [22] + [2s] + [24] + £321 + [35] + [41] + [42] + 0.379=0 VIL. [20] + [23] + [85] + [51] + 1.043=0 VU. [32] + [42] + [52] — 0.515=0 Notr.—In the solution of the equations the side-equations were divided by the numbers in parentheses placed opposite them. General corrections in terms of the correlates. {li] = +0, 2956 I +0.5123 IIT —0.32511IV —0.1626V +42.2752IX —0.4342X [1p] = +0. 2956 I +0,.5123 IIT +0.6749IV —0.1626V —0.5138 IX +2,3550 X [1s] = +0. 2956 I +0.5123 TTI +0.67491V +40.8374 Vo =—-—0.5138 IX = =—0,5522 X [14] = +0.3941 1 —0. 6502 III —0.4335 IV —0.2168V —0.5277IX —0.5789 X [2:1] = +0.166711 —0.6667 III —0.5000 VI —0.3333 VII -+-0.7723 TX +0. 4018 XI [2] = 40.16671I +0.3333 TIT 40.5000 VI +0.6667 VII -+-0.9843 TX | +0. 5119 XI [25] = +0.16671I +0.3333 IIL -+0.5000 VI +-0.6667 VII +0.9843 IX -+-0.5119 XI (24) = 40.1667 11 0.3333 IIT = +0.5000 VI. = —0, 3333 VIL —1.7566 IX —2, 2292 XT [25] = +0.1667 II 0.3333 IIL —0.5000 VI = —0, 3333 VIL —1. 7556 IX +0. 4018 XI (3:)] = 40.2500 11T +0.50001IV +0.7500V = —0,5000 VI —0.2500 VII —0, 2500 VIII (32) = 40.2500 TIT +0.5000IV —0.2500V +0.5000 VI —0. 2500 VIT +0. 7500 VIII [3s] = +0.2500 111 —0.50001V —0.2500V = +0,5000 VI +0. 7500 VII —9, 2500 VIII [4:1] = —0.3341V = 0.7238 VI. —0. 2338 VIII +40. 6191 X +0,9463 XT (4,4) = —0.2673 V 40,5791 VI +0. 8129 VIII —0.5337 X —0.2719 XI [4s] = +1 0465 V —0.6013 VI —0, 2673 VIII —0.9110 X —0. 3113 XI [5] = —0.50001V +0.7778 VIL —0.2222 VILL +0. 3067 IX +0.1845X +0,4914 XI [52] = +0.50001V —0,2222 VIL +0.7778 VIII +0. 1627 IX +0.2305 X +0.3924 XI [53] = +0.6250IV 0.2778 VIL —0.2778 VU +0. 2034 IX —0.6456 X —0,4419 XI 38 LS 298 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. XI, Normal equations for determination of the correlates. 1. 0=40.3480 41.28091 +0.88671IT + 0.5912I1V +40.2954V 40.7197 IX 40.7897 X 2, 0=+41.0090 0.833311 +0.6667 II +4 0.5000 VI 0.3333 VIL —0.7723 1X —0, 4018 XI 3. 0=+42.6330 0.88671 +40,6667IT + 3.6Q01 TIT +41.5247IV 40.7622 V 41.5000 VI 40.9167 VIL +0.2500 VIII — 0.2970IX +1.3686X —0. 8036 XI 4. 0=43.0370 +0,59121 9 41.5247 TIT + 3.4748 1V 41.1748 V = —1.0000 VII +1. 0000 VIII —0.6615 1X +1.3877X — 0,0495 XI 5. O=43.6820 +0.29541 +0, 7622TIT + 1.17481V 42.6342V --1.1013 VI —0, 2500 VII —0.5173 VIII —0,5188 1X — 1.4632X —0.3113 XI 6. 0=-+0.3790 40,5001 +1.5000TIT — 1.1013 V 43.8029 VI +1.5000 VIE 41.0791 VIII +0.21201X +0.0854X — 0.5310 XI 7, 0=41.0430 +0.3333 11 +0.9167 IIT — 1,0000IV —0.2500V 41.5000 VI. +2. 8612 VII —0, 4722 VIII 42.2753IX + 0.1845 X 41,5152 XI 8 0=—0.5150 +0,2500 111 41.0000IV — 0.5173 V 41.0791 VI. —0,4722 VII +2.3407 VIII +40. 1627IX —0.3032X + 0.1205 XI 9. O=+40.0410 0.71971 0.772311 — 0.2970 TIT —0,6015IV —0.5138V +0. 2120 VI 42.9753 VIL +0. 1627 VIII +16.01631X —1.5075X +5, 2191 XI 10. 0=—3.2336 +0.78971 +41.368611 + 1.3877IV —1.4632V 40.0854 VI +0. 1845 VI —0. 3032 VII —1.5075 IX + 8.6779X +0.9604 XI ll. 0=+45.1151 —0,401811 —0.80361II — 0.0495IV —0.3113V —0.5310 VI 41.5152 VII +0. 1205 VIII +5.2191 IX + 0.9604X +7.6228 XI VALUES OF THE CORRELATES VALUES OF THE GEN- RESIDUALS RESULTING AND THEIR LOGARITHMS. ERAL CORRECTIONS. FROM SUBSTITUTION OF GENERAL CORREC- ‘i TIONS IN EQUATIONS I = +0. 0006 log 6.77820 {1,) = +0. 483 OF CONDITION. II = —1.0599 log 0. 02527 [12] = -+0.780 No. of Bq. | Residual. || Sp[(v)-+[0]|2=20. 44 TIT = --0, 4958 log 9. 69531 [ls] = —1.867 | ew eens =f IV = —0, 9092 log 9. 95866 [la] = +0. 256 5 tel oe V = —0,5335 log 9. 72713 [21] = --0. 039 | i See VI = —0,5693 log 9. 75534 eis sume | 4 | cto Erobableerrorofan ob- VII = +0. 3929 log 9. 59428 [23] = —0.707 |) ee | Eguan0s served angle of weight VIII = +1. 0224 log 0. 00962 (24) = 41.476 || a 4.0. 0008 unity=0.6745 «/ 20.44 IX = +0. 2943 log 9.46879 is=nen | 5 + _® oo aah a, X = +0.7268 log 9. 86141 [3] = 1018 | an ae XL = —1. 2838 log 0. 09125 [32] = —0. 061 | 5 Sn anis [33] = +0218 | 49 | 0.0070 (41] = 1,191 us| -.0. 0030 [42] = +0.592 [43] = —0.767 (al = +0.151 [52] = —0. 016 [5s] = —0. 825 COMPARISON OF THE MEASURED LENGTH OF SANDUSKY BASE WITH ITS LENGTH COMPUTED FROM CHICAGO BASE. § 9. The adjusted values of the angles of the triangulation connecting Sandusky and Chicago Bases, given in Chapters XVI, C, and XVII, C, furnish for the logarithm of the ratio, Sandusky Base divided by Chicago Base, 9.9183689. The logarithm of Chicago Base expressed in feet is (Chap- ter X, § 33) 4.3917929. Adding this to the preceding logarithm, there results for the logarithm of Sandusky Base computed from Chicago Base, 4.3101618. From § 6 the logarithm of the measured length of Sandusky Base expressed in feet is found to be 4.3101715. The discrepancy between these two logarithms is 97 units of the seventh place, and is equivalent to 0.456 feet=13.9 centi- meters or 77475 part of the Sandusky Base. The probable error of this discrepancy, dependent alone on probable errors of observed angles in the principal chain connecting the two bases (Chap- ter XVII, D), is found to be by the method explained in Chapter IV, § 14, £75.23 units in the seventh place of logarithms, and corresponds to +0.353 feet=+10.8 centimeters. This probable §§ 9,10. SANDUSKY BASE. 299 error, which is too large since the adjusted angles are used in computing the ratio of the bases, is about three-fourths the actual discrepancy. If the above probable error be multiplied by the mean of the ratios of probable errors of observed to probable errors of adjusted angles in the trian- gulation between the two bases, viz., 0.60 (see Chapters XVI, ©, § 11, and XVII, C, § 5), there results as the approximate probable error of the above discrepancy, due to errors in the adjusted angles of the principal chain, +0212 Measured by this probable error the actual discrepancy is still not greater than can be safely attributed to the small errors in the adjusted angles of the principal chain connecting the bases. This chain embraces forty-five triangles, and, measured along its axis, is about 280 miles in length. COMPARISON OF THE MEASURED LENGTH OF SANDUSKY BASE WITH ITS LENGTH COMPUTED FROM BUFFALO BASE. § 10. The logarithm of the ratio, Sandusky Base divided by Buffalo Base, given by the adjusted angles (Chapters X VII, C, and XVIIT, C), of the triangulation connecting these bases is 9.9628985. The logarithm of the length of Buffalo Base, expressed in feet, is (Chapter VII, § 6) 4.3472757. Adding this to the above logarithm of the ratio, there results for the logarithm of Sandusky Base computed from Buffalo Base 4.3101742. The logarithm of the measured length of Sandusky Base expressed in feet, § 6, is 4.3101715. The difference between the two logarithms gives, in units of the seventh place, 27+ 60.88, the probable error being computed from the probable errors of observed angles of the principal chain (Chapters XVII, D, XVIII, D, and XIX, D), according to the method given in Chapter IV, § 14. The actual discrepancy is thus about one-half its limiting probable error. ‘The linear equivalent is Of 127 + 0,287 =3e™,.94 8°™,7 If in place of the probable errors of observed angles, the approximate probable errors of adjusted angles (Chapters XVII, ©, § 5, and XVIII, C, § 5) be used in computing the probable error of the discrepancy, there results, ‘0,127 + Of.171 =3-™.9+ 5,2 The length of the principal chain of triangles joining Sandusky and Buffalo Bases is about 250 miles. The number of triangles in this chain is thirty-six. 300 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS, — [Ciap. XII, CHAPTER XII. OLNEY BASE. § 1. This base-line is situated in the southern part of Jasper County, Illinois, about 8 miles from Olney, on a prairie, about one-half the length of the line being on cultivated ground and the other on unbroken prairie sod. The base-line is a straight ove, and the greatest difference of eleva- tion of its points is 23 feet. Its length is approximately 6,589 metres; the longitude of its west end is 88° 06’ west from Greenwich; its latitude, 38° 52’ north ; and the azimuth of the base-line at the west end is 268° 30’ west of south. The surface soil along the base was underlaid at a depth of about a decimeter by fine clay, which had great stability. Indeed, the base-line as a whole was much firmer than the Chicago Base, which was also on a prairie, with its surface soil underlaid by clay. The base had marking-stones at its ends similar to those of the Chicago Base, described in Chapter X, §1. It was divided into six nearly equal sections by marks on stones two feet square and one foot thick, sunk three feet below the surface of the ground, the mark being a small drill- hole in the top of « copper bolt leaded into the stone. The section-stones were placed on the line by first obtaining an approximate position and then reading to within 2” the angles to the ends of the base. Their deviation was thus known and the stones were then put on the line. Each of these sections was measured in duplicate in opposite directions. The measurement was made, with the Repsold apparatus, by a party under the charge of As- sistant Engineer E. 8S. Wheeler, between July 9 and September 15,1879. The methods of measure- ment and reduction were the same essentially as for the Chicago Base, given in Chapter X. The average number of tubes measured per day, including remeasurements, was 105, and the greatest number measured in one day was 168. Measuring was done on thirty-two days. Guide-points were fixed on the base-line between the section-stones at intervals of 300 metres by first putting them approximately in position and then reading the angles at them subtended by the two ends of the base to an accuracy of about 2’. Their distances from the line were then computed and they were placed upon the line. The adjustment of the tube-telescope and of the sector were tested daily and the other adjustments of the base-apparatus weekly. The adjustments were very stable. Details of the work may be found in the Annual Report of the Chief of Engineers for 1880, p. 2408. § 2. An error of 4’ or 5’ was made in the first adjustment of the tube-!elescope to parallelism with the steel bar in beginning the first measurement of the first section at the west end of the base, and as the daily tests showed that the tube-telescope had not changed with reference to the tube, the error was not at first discovered. The marks on the base-line at which the tube-telescope was directed were about 300 metres apart and when the tube approached within about 30™ of such amark it was moved forward 300". In this first section it was noticed that the tube was off the line when it passed these marks, the miximum deviation being 0".191. Its deviation was known at three points of the first section and from these and from the form of the curves produced by a constant deviation of the tube-telescope from parallelism with the steel bar, Chapter X, § 16, the correction to the first measurement of the first section has been obtained. The first section was about 1096™.5 long, and the resulting correction to its first measurement is—0™™.85. This correc- tion has been applied. §3. As on the Sandusky base-line, the mean elevation of the tube above a bench-mark was §9 1,4.) OLNEY BASE. 301 determined both by a line of ordinary levels run along the line and by means of the observed inclinations of the tube. There were, however, three cases in which the duplicate note-books gave records discordant by 1° for grade-angles, viz., for tubes 7, 17, and 1618 of the first measurement, and one case, tube 1645 of the second measurement, in which they differed by 50’. The probably correct values in these discordant cases were determined by comparing results for differences of height given by the two measurements and a preliminary measurement of forty-six tubes, all of the discrepancies, except that of tube 1618 of the first measurement, falling within forty-six tube- lengths of West Base. An error of 5° in the recorded grade-angle of tube 9 of the first measure- ment, both records being erroneous, was also detected by means of the second measurement and the preliminary measurement. The adopted grade-angles gave relative elevations of the section- stones agreeing well with the corresponding relative elevations determined by spirit-level. The height of the base-tubes during measurement above the sea has been derived by several routes which show considerable discrepancies in their results. The mean heights of the Great Lakes above mean tide at New York are given in Chapter XXII, §13. 1. Starting from Lake Michigan, zenith distances of all stations south to the Olney Base have been observed. This route gives station West Base above mean tide, 485.6 feet. 2. The levels of the Illinois Central Railroad from Chicago to Odin, of the Ohio and Missis- sippi Railroad from Odin to Olney, and the leveling of the base, give 482.6 feet. 3. Railroad levels from Lake Erie at Cleveland to Olney via Vincennes give 490.6 feet. 4, Levels by the Coast Survey and the Mississippi River Commission from gauge at Carroll- ton, La., to Cairo, IL, and then railroad levels from Cairo to Odin, and thence to Olney, give 503.1 feet. 5. The same levels from Carrollton to Cairo, and thence by Vincennes to Olney, give 495.0 feet. In the absence of data for determining the relative weights of these values, their mean has been taken, and as a probable error of not less than + 5 feet should be assigned to it, there results, Mean height of agate marking west end of base above mean tide = 491'.38 + 5t0 The levels over the base-line give mean height of tubes above west end of base during measure- ment of west half of base, — 1.65, and during measurement of east half, — 10.90. There result: West part of base above mean tide, 489.73 + 5.0 East part of base above mean tide, 480.48 + 5'.0 §4. In the following table the first column gives the number of the section of the base; the second shows by the numbers 1 and 2 whether the first measurement or the remeasurement in the opposite direction is given on that horizontal line; the third gives the entire number of times that the steel bar S,° at the temperature of 60°.292 I’. entered into the measurement of the section; the fourth gives the fractional parts of S, which entered the measurement when closing on the permanent mark at the end of a section, this interval being measured by means of the intermediate graduations on S,; the fifth gives the sums of the parts of the base measured with the cut-off scale in closing on section-stones (the graduations of this cut-off scale have been examined and found sufficiently accurate to introduce no error that need be considered); the sixth gives for each sec- tion the sums of the corrections to S,° for metallic temperature on the assumption that Es 9.__= 0.6522 Fiz, a Eis, its value when Z,=S,; the seventh gives the sums of the corrections to the length of each section on account of the inaccuracy of this last supposition; the eighth gives the sums of the corrections for the sections on account of inclination of §,; the ninth gives the sums of the small distances meas- ured with the microscopes in each section; the tenth gives the sums of the movements of the front cut-off plate in each section; the eleventh gives the sums of the corrections for each section on account of the observed Z,—JS, not being the correct metallic temperature of S,; the twelfth gives the sum of the different corrections for each section. 302 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. [Cuap. XI, Results of measurement of Olney Base. 2 Temperature correc- 4 ss ee a 3 ; % tions. a oF oe z, B g a a es Ae a 3 A a2 3 A r 5. 7% eS F oni f|2| & eg “s | 288 |S.2| 83 ss Section. als = ran ak S a oR, ag =o o g a 4 2) 8 iS cae Terms in | Terms in Ba a a 2 Sak 3 $ y* B| 2 2 g (ZAi—S). | (Zi—S1)2. 3 BOR 3a? 3 | 2] 3 3 5 Soe | bH8 | & A a) 4 & 5 5 a 5 8 R 1 2; 3 4 5 6 7 8. 9 10. 11 12, “ Me M # “ # M K — 847. 6* 1] 274) +0.150 | —50026.6 | +131522.1) + 903.6 | —171583.3] — 3491] — 47.1] + 607. 8§ rif — 88912. 6 l2 274} 40.150} + 70021] + 55964.5 | + 210.0 | —153327.6) + 256.8) — 13.0} +41017.4 | — 88889.8 1} 274} +0.400 | +89046.5 | +-201741.2 | +1870.7 | — 61850.1| — 2088.7} — 5.5) +1082.3 | 4229796. 4 II ‘3 ' 974} 40.450} +47011.3 | + 41676.0| + 2084 | — 62609.9] + 1232.4; + 29.0] +1712.0 | + 292589 1| 274) +0.275 | — 8008.1 | +130338.8 | + 894.2 | — 282545) — 2043.6 | + 40.9; +1055.6 | + 94023.3 HIE \ 274 | +0.300| + 5007.2] + 20405.6) + 202.7 | — 32931.6| — 138¢.7| + 47.3] +41197.3 | — 7530.2 1| 822 | 40.825 | 4310118 | +-463602.1 | +3668.5 | —261687.9| — 4481.4 | — 11.7] +2805.7 | +234050.5 pean \ 822 | 40.900 | +59020.6 | +118046.2 + 621.1 | —248869.1] + 100.1| + 63.3] +3856.7 | — 67161.1 [SS SS == \ | | 1| 273 | +0.625| + 998.4 | -+128491.2 | -+ 885.6 | —102438.0| — 5569.7/ —167.8| + 906.9 | + 23106.6 ny: ‘5 273 | +0.650 | —25010.7 | + 6044.2 + 266.7 | —110782.1| — 18296] — 27} +10986 | -- 76215.6 | 1| 275 | -+0.950 | —33010.7 | +127403.8| + 796.4 | — 96114.7| — 3182] — 23.6] +1455.0 |+ 188.0 - ss 275} 40.950} + 8007.9) + 880312 +4 449.9 | — 96022.6| — 342.5] — 54.1] +41850.6 | + 1920.4 1] 274; +0.825 | —19010.2 | +162858.9 +1233.4 | — 97604.0| + 347.6| + 75.2| +1356.9 | 4+ 87278.2 vi 3 274 | -+0.850 | —30014.1 | +116619.9 | + 671.9 | —101114.4| + 9260] + 26.6] +1361.8 | — 11522.3 1} 824] +0.400 | —13002.1 | +418753.9 ; +2915.4 | —296156.7| — 5540.3| —116.2| +43718.8 | +110572.8 Eastpart/)9| goa] 40.450 | —47016.9 | -1-264695. 3 | +1388.5 | —307919.1] — 1246.1} ~— 30.2] +4311.0 | — 85817.5 Total |¢1| 1647) +0.225 | +18009.7 | +882356.0 | +6583.9 | —557844.6 | —10021.7 |] —127.9| +6524.5 | +344632.3 base . 1647 | +40.350 | +.12003.7 | +382741.5 | +2009.6 | —556788.2 | .— 1146.0] + 321! +8167.7 | —152978.6 *The correction 847.6 to the first measurement of the first section is due to errors of alignment, discussed in § 2. In the following table, derived from the preceding one by expressing the two measurements of a section with the same fraction of a tube, the principal results are given, and also the differences between the two measurements of each section. To express the section of the base in terms of entire numbers of S,° and equal fractions of it, the values of the fraction of a tube in metres are needed. The value of §, is given in Chapter LX, § 66, and from that its value at 60°.292, for which Z, — 8, =0, may be computed. Transforming it into metres by Clarke’s value of the metre (metre — 1.093623 yard), there results §, at 609.292 F. = 4™.0009624, as an approximation more than sufficient for obtaining the values of the fractions of the tube in metres. It has already been stated that these fractions of tubes are the exact fractions which they purport to be within 10“, a quantity that is insignificant in comparison with the length of a sec- tion, and which enters a section but once. $6 5,6.] OLNEY BASE. 303 Principal results of measurement of Olney Base. : Measure- : i Section. ont No. of (S1)z,=s, ane Mean. Difference. M io I ; d 274-150 = aro. 7 — 89325. 00 — 870.4 2 274, 150 — 88889. 8 II ; 1 274. 400 +229796. 4 ; -+229551. 70 + 489.4 2 274. 400 4229307. 0 1 ; 7 aie 300 a> CONNIE ; — 6765.45 +1529. 5 2 274 300 — 7530.2 1 822, 850 134035, 5 West part! ; 5 5 x Heir Jaaeiee } ++183461. 25 41148. 5 Iv ; 1 278. 650 == ISOLA ; — 76566. 50 — 701.8 2 273. 650 — 76215. 6 ! Vv j 1 275. 950 ae (188.0 + 1054.20 | —1782.4 2 275, 950 + 1920.4 vI } 1 274, 850 — 12745. 8 ; — 12134. 05 1228. 5 2 274. 850 — 11522. 3 1 824, 450 — 89475. 2 East part, ; ae 5 = 2 824. 450 — 85817. 5 ; ert. 3 re Total 1 1647. 300 ++ 44560. 3 he i t + 45814. 90 —2509. 2 2 1647. 300 + 47069. 5 § &. From the preceding table the mean measured lengths of the parts of the Olney Base are, West part=822.9 (S,)z,-s,-0™.06659=10801'.6890 East part =824.4 (8,),,-s, +0™.11240= 10821".9662 the values in English feet being derived from (S,),,~s,=13*.1266344, given in Chapter IX, § 66. Computing for Clarke’s spheroid for the azimuth of the base and for the mean heights of the tubes during measurement of the west and east parts respectively, given in § 3, the reductions to mean tide at New York, they are found to be, for the west part, 0.2525, and for the east part 0.2482. Applying these corrections, there result, West part at sea-level 822.9 (,),,-s,—0™.14354 = 10801*.4365 East part at sea-level— 824.4 (8,)z—s,+0™.03676=10821".7180 Olney Base at sea-level =1647.3 (8;)z,-s,—0™.10678=21623",1545 If the length of S, in terms of #1876 and its adjusted expansion be used, viz., 8, =4( 1876) — 58+.40-+ 1.326 (t—42.75), Chapter IX, § 62, there results for the length of Olney Base at sea-level, 6589.2 (#1876) —0™.16467 The base may also be expressed in terms of #1876 by substituting for (Si)n=s, its value in terms of R1876 dependent alone on intercomparisons of #1876, 8,, and S,, derived from Chapter IX, § 24, namely, (S;),,-s,==4( 1876) —37+.01+414.2744(t—59). This gives Olney Base at sea-level=6589.2 (1876 at 60°.292 F.)—0™.16504 § 6. The probable error in the length of the Olney Base will be derived in the same way as that of the Chicago Base given in Chapter X. From the differences of measurements of the sections of the Olney Base, § 4, and called d, the probable error in the base due to these differences, considered as the result of accidental errors of measurement, is [@]_ 4 9mm 97 =f 0.6745, | “= £0.97 =e779000 304 STANDARDS OF LENGTH, BASES, AND BASE-APPARATUS. — [Cuap. XII, The probable error depending on the error in the length of S, is obtained from the expression giving the approximate length of the base, (1) n(S,) 2, —s, = m+ 0.000005106[(Z,—8,)"] For the Olney Base the mean metallic temperature of both measures was m=588+.7, correspond- ing to 75°.75 F. Substituting this value in degrees F. in the expression for the probable error in the adopted length of 8, for any temperature, given in Chapter LX, § 63, there results for the probable error in S, at 75°.75 F.,13.42. Since n=1647 for the Olney Base there results for the probable error due to the first term of (1) above, +5™".63. Taking the probable error of the coefficient of the second term as one-fourth of its value, as was done in Chapter X, § 30, there results for the probable error of the second term, +1"".07. Combining the last two probable errors, there results, +5"".73 as the probable error in the value of the base due to uncertainty in the length of &,. For the probable error in the adopted lengths of S, due to the fact that the observed values of Z,—8, are not the true temperatures of S,, +1+.5 will be taken as was done for the Chicago Base, Chapter X, § 31. Multiplving this by 1647, the number of S, in the base, there results a prob- able error in the base of + 2™™.47. The probable error in the mean height of the tubes during measurement above sea-level has been taken as + 5%. This introduces a probable error of + 1™™.57 into the correction for reduction to sea-level. Collecting now the various probable errors in the length of the base, there results, min, From discrepancies of measurements ..... 0-222. 22220 cece ee ce eee + 0.97 From error in value of length of S|............ .....0 22 ee- eee eee + 5.73 From errors in temperature Z,—S,...... 0.20220 cee eee eee Jefe ses + 2.47 From error in mean elevation of tubes above sea ...-....-.-..-..-- + 1.57 Combining these, there results, + 6"™.51 or aor4ea7 part of the base. Hence, from § 5, OLNEY BASE AT SEA-LEVEL=21623".1545 + 0 .0214 § ‘7. The length of Olney Base in terms of the metre R1876 deduced from the length of S; de- pendent alone on intercomparisons of #1876, &, and S, is, § 5, 6589.2 (& 1876 at 60°.29 F.) —0™.16504 By a process precisely like that followed in Chapter X, § 33, the probable error of this length is found to be + 3™".48, or gg¢oo0 Dart of the base. Hence, Olney Base at sea-level = 6589.2 (R 1876 at 60°.29 I.) —165".0443"".48 When the length of #1876 in metres is substituted in this expression, the probable error of the resulting numerical value will be -£ [(6589.2 ¢)?4 (3.48)?]% in which e=the probable error of the length of R1876 at the mean temperature (759.75 F.) of Olney Base. § 8. The middle section-stone divided the Olney Base into two nearly equal parts. Station Mound, with these parts, formed two triangles, whose angles were read with the same care as pri- mary angles, and they were adjusted with the primary angles. Their adjusted values are given in Chapter XX, C,§ 4. With the value for the whole base given above and these angles, the parts of the base have been computed. The excess of the computed over the measured length was For the west part of the base, + 0.0327 For the east part of the base, —0".0327 §§ 7-9.) OLNEY BASE. 305 COMPARISON OF THE MEASURED LENGTH OF OLNEY BASE WITH ITS LENGTH COMPUTED FROM CHICAGO BASE. § 9. The logarithm of the ratio, Olney Base divided by Chicago Base, derived from the adjusted angles in the triangulation (see Chapters XVI, C, and XX, C) connecting them, is 9.9431302. The logarithm of Chicago Base expressed in feet (Chapter X, § 14) is 4.3917929. Hence, by addition, the logarithm of Olney Base computed from Chicago Base is 4.3349231. The logarithm of the measured length of Olney Base expressed in feet is (§ 5) 4.3349191. The discrepancy between the two logarithms is 40, which corresponds to 0.199—6™.1 or to -ggsza part of Olney Base. The probable error of this discrepancy, computed from the probable errors of the observed angles of the principal chain of triangles joining the bases (Chapters X VII, D, and XX, D), accord- ing to the method given in Chapter IV, § 14, is +55.40 units in the seventh place of logarithms or +0*.276=48™.4. The actual discrepancy is then about two-thirds of its limiting probable error. If the approximate probable errors of adjusted angles in the principal chain connecting the bases be used instead of probable errors of observed angles, the probable error of the discrepancy is +0157=+4°™.8, It would appear quite safe, therefore, to attribute the actual discrepancy between the measured and computed lengths entirely to remaining small errors in the adjusted angles in the principal chain of triangles joining the two bases. This chain is about 200 miles long and embraces thirty-five triangles. 39 LS 306 PRIMARY TRIANGULATION, fCuar XW] PART ILL. PRIMARY TRIANGULATION. CHAPTER XIII. ADJUSTMENT OF A TRIANGULATION HAVING BUT ONE MEASURED BASE WHICH IS TAKEN AS EXACT. § I. Ina triangulation-net of p points connected by / lines, if we start from the measured base, two angles observed there will fix a third point. At any two of these three points, two more ob- served, independent angles will fix a fourth point, and so on. Hence to fix p points of the triangu- lation, 2p—4 independent angles will be needed. If m angles are observed, m—2p-+4 of them will be superfluous. Independent angles are selected measured angles on which all other measured angles can be made to depend by means of the conditions that the net must geometrically fulfil. Thus, if at a station two angles were measured, and afterward their sum, this last would be a dependent angle if the others were taken as independent. In triangulations there are more angles measured than the 2»—4 which are necessary to com- pletely determine all the parts of the net. Hence, when one more angle is measured its value will be known in two ways, (1) by direct observation, (2) by computation from the other 2p—4 angles. As no measurement is perfectly exact, these two values will differ by a small amonnt. An equa- tion which expresses a relation, which of geometrical necessity must exist between the different parts of a triangulation-net in order that the net may be a series of points connected by lines, is called an equation of condition. In the case just supposed, the equation of condition is that equa- tion which expresses by syinbols the necessary relations between the 2p—4 angles and the one new angle. When the observed 2p—4 angles and the observed new angle are substituted in it tor their symbols, the resulting equation generally will not be perfectly satisfied, showing that the observed angles do not have precisely the values which could give a possible geometrical net. Tn order that the triangulation may give a geometrical net, the values of the observed angles must be so corrected as to satisfy all the equations of condition that can be formed in the net; and these corrections, in accordance with the principle of least squares, must also have such values that the sun of their weighted squares shall be a minimum. The adjustment of a triangulation consists in determining and applying these corrections. EQUATIONS OF CONDITION. § 2. Equations of condition are usually divided into two classes, namely, angle-equations and side-equations. Into the first, angles enter directly; into the second, tiie sides of triangles or the sines of their opposite angles. Ist. Angle-equations—Let V’, 1’, &c., represent the most probable values of the angles, these being the final values which we have to find; let Al’, M”, &c., represent the values of these angles which result directly from observation, being the means of the observed angles, corrected for instru- mental errors, errors of centering, and errors of signals; and »’, v’ the corrections to be found, which, applied to WW’, M’, Xc., will give V’, 1, &c., so that V’/ = M+’, &e. §§ 1,2.) ADJUSTMENT OF A TRIANGULATION. 307 Angle-equations may arise— (a) At a station from the observation of separate angles, and of their sum or difference. Thus, if V’’=V"+ V’, when we substitute M+ v for V we have M40 SM" 4 0" 4M 4! which may be written in the form Whale palo! tally! =0 where nv’, a’, a’, a’, are known quantities, and the equation expresses a relation between the cor- rections vr, which must be satisfied. It is called a numerical equation of condition to distinguish it from the geometrical one from which it is derived. Again, (bo) If all the angles which make up the circumference at a station be measured, we shall have Vit V+ V+ Viv &6., =360° or substituting the observed values, with their corrections M+», for V, M+! AM Holt MM + Met pie &e., =3602 This may also be written in the form v+alo! tale! +aly"+ &e., =0 where n’ and a’, a”, a’, are known. Again, (ce) In an uncrossed polygon of m interior angles, V’, V7’... Vi, we have WV" oo. FV =(m—2) 180°—e where ¢ is the spherical excess of the polygon. Spherical excesses in the adjustment of triangula- tion result from the formula — ae ABsine — = 2a? (1 — $e cos 2.0) sin 1” in which the constants are those of Bessel’s spheroid. Substituting for V, + v, we again have an equation of condition of the form n/+a/v'+a"v"+ &e., =0. 2d. Side-equations.—These arise from the necessity that in any geometrical net any side com- puted from another side shall be the same by whatever route computed. : : Thus, in the figure, the same value must be found for cd when computed by either of the following routes: bo co cd __,,a0 e€0 do cd cd= are bo co od= a0 a0 €0 do Hence, dividing one equation by the other, je ee ao bo co do eo or, from the proportionality of sines of the angles to the opposite sides, j—Sin A, sin A, sin A, sin A, sin A, sin B, sin B, sin B, sin B, sin B, Substituting for these angles, which are the yet-to-be-found most probable angles, their values in terms of the observed angles M and their needed corrections v, there results— es sin (M’+0’) sin (M'"+0'") sin (We+ vv) sip (Mvi+ vv) sin (Mx+ vi) ~sin( M0”) * sin (A+ 0%) * sin (M4 ov) * sin (APH yet) * sin (M07) which is the equation of condition that the corrections v must rigidly satisfy. For application of the method of least squares this equation must be given a linear form, which can be effected by taking its logarithm, giving, (1) log sin (M’+v’)+log sin (M+!) +log sin (M'+v")+log sin (.Me'+o™)+log sin ("+ 0%) log sin (M’+v’)+log sin (M*+v")+log sin (M“+v")+log sin (Mi +0"")+log sin (M*+e") Since v’, v’’, &c., are only one or two seconds, each term may be developed by Taylor’s theorem, stopping at the first power of v and obtaining thus, log sin (Md’+v’)=log sin m+(" a. mye! log sin (d+) =log sin Ms (Oe ” )o 308 PRIMARY TRIANGULATION. (Cap. XIII, and so on, in which, if 7 is expressed in seconds, cee a may be replaced by the difference in a table of logarithmic sines for a change of one second in the angle M. Calling these tabular differences 0’, 0”, &c., and substituting them in the developments above, and the latter in equation (1) there results, log sin M/+0'w’+log sin M’/+6/o""-+ + + + + =log sin M"+0" vo! +og sin M"+d"us+ + ++ +s which may be written in the form ni alo pale tal v4 &e., =0* Suppose there are in the triangulation-net p points connected by / lines over which observa- tions have been made; I’ of these, however, having been observed over in but one direction. Since the ends of the base are known points, there will be p—2 points to be determined, requiring 2(p—2) observed angles. If m angles are observed,m—2 p44 will be superfluous, and will give (including all kinds) as many equations of condition. In order that the proper number of equations of condition may be obtained, it is desirable to know the number of each kind, and for this purpose they may be divided into local and general equations of condition. The local equations of condition are those which arise at a station when more angles are measured at the station than are necessary to determine all the angles at that station. If there are n distant stations pointed at from a given station, n—1 measured angles will determine all angles at this station. If s angles are measured, there will be, at this station, s—n+1 equations of condition. The general equations of condition are those involving angles at three or more stations, and are either side-equations or polygon-equations, triangle-equations being included under the last head. Starting from the base, three lines observed over fix the first three points, and to fix the remaining p—3 points, two lines are sufficient for each. Hence, 2 p—3 lines will fix the whole net, and if another line is observed over, its value can be obtained by two routes, whence results a side-equation. Jf there are J lines observed over in one or both directions, there will be I—2p+3 side-equations. To obtain the number of polygon-equations, suppose the p points to be connected by p lines, all observed over in both directions. In the resulting polygon, the sum of the angles is known in advance in terms of the angles between its sides. This is a first polygon-equa- tion of condition. Ifan additional line between two of the p points be observed over in both direc- tions, it will cut the preceding polygon into two new ones and will give a new polygon-equation of condition. It will give but one independent equation of condition, for the sum of the angles of the second new polygon can be derived from that of the first new one and of the primitive polygon. Each new dividing-line giving thus a new polygon-equation of condition, and l’ of the whole num- ber 1 of lines having been observed over in but one direction, there results for the whole number of polygon equations of condition, 1+1—lU—p To recapitulate, the whole number of equations of condition in the net is m—2p+4 The number of local equations of condition at each station is s—n4+1 The total number of side-equations in the net is I—2p+3 and of polygon equations = 1—V’—p+1 GENERAL METHOD. * § 3. Having shown how to obtain the numerical equations of condition in the form / Voyt Matt Watt a Va -are" ay" 4. &e., =0, *Side-equations are computed with Vlacq’s 10-place logarithms, §3.] ADJUSTMENT OF A TRIANGULATION, 309 the method of determining the corrections e’, ’’, &c., to the mean observed angles M', M’, &c., having weights p’, p’’, &c., may be given. Suppose » angles are measured in all, and that there are in the whole net equations of con- dition. As already stated, we must have (2) DOP tpl yl? ppl yey (mm) =a minimum, - : the m quantities v’, v’, &c., being subject to the n rigid conditions WM EDO ADM OM 4 & (m)=0 Qv'+ cult clo te, = ely (MW) a ee serene Differentiating (2) and (3), there result— oH tale tao + ene =0 @ (2') Sp de’ +2Qplolde'+ .. 6. (m)=0 and (3/) Bde’ +b de" 40du'+._—, cdeu't edu" +edu' "+ ., PFE Na esac a ioral There are two methods of finding from these equations the values of the unknowns, '. First. From (3’) the values of n of the dv may be found in terms of the others. These values substituted in (2’) eliminate n of the dv in that equation and leave m—n, all independent of each other. The minimum condition of (2) will now be satisfied by placing the coefficients of the (m—n) dv in (2’), which have not been eliminated, separately equal to zero, since these are the partial differ- ential coefficients of (2) with reference to the remaining variables. The resulting m—n equations will give the values of m—n corrections v; the values of the others have already been obtained in terms of these, so that the values of the corrections for both independent and depenflent angles will then be known. Second. The elimination may be effected by undetermined multipliers or correlates, as follows : Multiply each of the x equations (3’) in order by the correlates —I,—II,—III, &c., and add the products to (2’), arranging with reference to dv. Assign such values to the » correlates as to make n of the coefficients of dv zero, leaving m—n coefficients, which, as m—n of the dv are inde- pendent of each other, can be separately placed equal to zero. This process then amounts to multiplying each differential equation of condition (3’) by an undetermined multiplier, adding the products to (2’), and then placing the coefficients of all dr separately equal to zero, giving m equations containing as unknowns m corrections and n correlates. From these equations the values of the m corrections can be found in terms of the » correlates. These values of the cor- rections substituted in the n equations of condition (3) will give n equations containing n correlates as unknowns, from which the values of the correlates can be found. Substituting their resulting values in the expressions for the values of the corrections in terms of the correlates, thé corrections themselves become known. In the adjustment of the Lake-Survey triangulation the two methods have been combined. The labor of adjustment lies largely in the solution of the equations which give the values of the correlates, and increases very rapidly with their number, that is, with the number of equations of condition. Now the local equations of condition, or those which exist between the angles at any one station, usually contain but few variables, so that with little labor, by means of each equation of tondition, one correction can be eliminated by the first method from the differential equations (2’) fora minimum.. In this way as many corrections can be eliminated from (2) as there are local equations of condition, and the second method can then be applied to the new form of (2’), and the remaining differential equations of condition in (3/). If in the n equations of condition there are n’ local equations of condition, the number of corre- lates will thus be reduced from » to n—n’. ade’ +aldo'+taae+ (m) a10 PRIMARY TRIANGULATION. [Cuap. XIII, Going now somewhat more into detail, if the mth angle 1 entered a local equation of con- (lition, these equations being all of the same forin, Vou Vey ye. Vic where C is a constant, there results on substituting for the angles their observed values plus their most probable corrections, i. AO = Mt AM 404 Mitr whence, ni) = al Lalli Lyiv+ OY and vim = dol! +d! +d Substituting the value of dv” in (2’), there results, (27) Qplaldal tape det Fp (0 ho! Heh + 0) (do +d +du)+ 6 6 1 = 0 Each additional local equation of condition will insert a similar term in (2”). Eliminating in this way from (2/) as many of the corrections v as there are independent local equations of condition, (2) may then be written in the following form, in which «, 3,7, &c., are known quantities. pany POT MANE sg ERE TO ROR B OE 6 oy y teil TPA ol by Oly AMI (mam) www. = 0 This is the differential equation for the minimum. The n’ local equations of condition having been already used in eliminating corrections from (2') there remain to be satisfied in (3’) only the n—n/ side- and polygon-equations. Multiplying each of them in order by—I,—II,—III, We., there results, CIT (ade! talde! ta dv+ 6 1 www. =O (“4 . IT (b/d! +0" dol 0 dol" + )=0 / —LT(e'dv'+e"do"! +e dul + )=0 (n—n’) Adding (4) to (2’”), arranging with reference to dv and then placing the coetticients of all dv equal to zero, the following equations result: g aleltally tala + 6 ww FA =a! 40! T+ TII+ (5) Ute elt aot 6 eww. HAY ST +0"TI+0"TTI+ (m—1’) . There are as many of these equations as there are corrections remaining to be found. Solving, they give the values of the corrections in terms of the correlates, and these values substituted in the side- and polygon-equations of condition in (3) give as many equations as correlates, from which the latter can be found as already explained. It will be noticed that thus far all numerical equations of condition have been computed from the geometrical ones by using the means of the observed angles. MODIFICATION OF GENERAL METHOD. § &. The foregoing exhibits the general method followed in the adjustment of the Lake-Survey triangulation, but it has been modified in practice. Instead of determining the total corrections v from equations (5) and (3) these corrections have been determined in two parts such that v=(v)+[2], where the (rv) are the corrections that result from making a first partial adjustment in which the side- and polygon-equations of condition are neglected and only the local equations are included. The [v] are quantities to be added to the corresponding (v) in order to give the total corrections when all equations of condition are considered, and are obtained by a second adjustment. §4.] ADJUSTMENT OF A TRIANGULATION, 311 From the theory of the first method it is seen that the values of the (v7) will at once result by placing the coefficients of the dv in the (2”) or (2’) separately equal to zero, which will give as many equations as there are v, and hence the value of the (v). The resulting values of all (v) which enter no local. equation of condition will be zero. The values of the (v) might just as well be found by treating separately each station and its local equations of condition by the first method, and in practice this has been done. Having found the (v) it remains to find the corrections [v], which added to the (v) shall give the total corrections to the measured angles. It has already been seen that equations (5) and (3) contain the whole solution of the problem of adjustment. Substituting in them for v its value (v)+{v], they become al $(e')4 [oi] galls ot [el t+ 2. HA Sal Ith! TI+e¢ ITI+ &e. (5’) BS Ce fot] bees (Oak) be wee BAS AT +b" T1406" TTT4+ Xe. ee MEF ERENSR RTE e (NY so Se SE BGG ASE RR w +a’ Sry et] ret Fete") be GET cusk SW ease aN cna cee =0 (3’) ne +h Pe aL |e eee ee ee ee ae =(0) Ney aie BOE Regen eee eat ee GU) ae a ek a ee ae ha eS Now the equations by which the (v) were determined are a! (v!)+a" (vt 2.2. +2 30 BW) (vl) +... HA =0 Es eacen ceraruers (ny... hence, substituting the known values of the (v) in (5’) it becomes a’ [a |e” [oe [el oe eel dao Lite’ OT . (5") _ [art e” fo] Hee [el | se Sa DIT HITS ss : Sahih ean ee (NHN) a ce we ee ee The numerical equations of condition (3’’) or (3) were obtained by substituting in the geomet- rical equations of condition for each most probable angle V its observed value M plus its most probable correction, or M+ for V, or, what is the same, M+(v)+[v] for V. But since the local corrections (v) are known, instead of computing the numerical side- and polygon-equations of con- dition (3) with the observed values M of the angles, the locally corrected values of + (v) may be used. The only change it wakes in (3”’) is that it combines all the known terms in each equation (3) into a single term m, such that nol =n! +al(v' +a (ote... ng =n! 4 (W ADO )+ and (3/’) then becomes io t+a'[v'|+a[ol J+... =0 (3!) nl 4D! [vJ4O" [0 J+ 2. 0 ( home ray eng (N— WM). wee cee (5) and (3’’) are of the same form as (5) and (3). If the same process that was prescribed to find the values of the v from (5) and (3) be applied to (5’’) and (3’”), the values of the [v] will result, and (v)+[v] =» gives the total corrections » which, applied to the observed angles, will make them satisfy all the geometrical conditions of the net, and also make the sum of the weighted squares of the corrections to all measured angles a minimum. ; A synopsis of the different steps actually followed in the adjustment can now be given. 1. The means of the observed angles are tabulated with their weights. To each angle is assigned a symbol, which is a number with a subscript number. The main number is that of the D1? PRIMARY TRIANGULATION, (Cuar. XII, § 4, ca station, the stations being numbered consecutively through the portion of the triangulation under consideration. The subscript is the number of the angle at the station, these being numbered from the south round by the west. Thus, 12; is the to-be-found most probable value of the third angle at the twelfth station, 12,’ is the observed angle, (12;) is its local correction, and [12;] the further correction needed to give the total correction (12)+{12s]. ». For each station the observed angles are locally adjusted by the first method. The minimum expression for each station will only include the angles observed there. 3. The numerical side- and polygon-equations of condition are then computed with the locally corrected fundamental angles, and their correlates are written beside thei. 4. Equations (5”) are written out, and from them the values of the general corrections are determined in terms of the correlates. The coefficients a’, a”, . . (7, 3, .. . &e., are derived from the local adjustment. 5, These values of the general corrections, substituted in the general equations of condition, give the equations for determining the correlates. These equations are solved. The correlates thus found are substituted in the values of the general corrections previously determined, whence the values of the general corrections [v] result. Adding these to the local corrections for the same angles, the total corrections are found. The corrections for angles at a station eliminated in the local adjustment, are derived from those of the angles on which they depend. The corrections are checked by substitution in the equations of condition, the residuals rarely exceeding 0.02. In the computation of the side-equations the eighth place of logarithms has been retained. Crap, XIV, A,§§1,2.] MINNESOTA POINT BASE TO KEWEENAW BASR. Le CHAPTER XIV. TRIANGULATION FROM MINNESOTA POINT BASE TO KEWEENAW BASE. A.—DESCRIPTIONS OF STATIONS. NOTE RELATIVE TO ELEVATIONS. § f. The heights of ground at stations in the triangulation of Lake Superior were determined partly by spirit and partly by trigonometrical leveling, but on the whole with no high degree of precision. They are referred to the surface of the lake at the times of determination. As these times differed in some cases by several years, relative heights are subject to uncertainties arising from fluctuations in elevation of the lake surface. Except for the stations of Minnesota and Kewee- naw Bases, a probable error of + 5 feet may not be too great to assign to these heights. DESCRIPTIONS OF STATIONS. § 2. NortH Bass, 1870~’71.*—This station is situated on Minnesota Point, about 2 miles southeast from Duluth, and marks the north end of Minnesota base-line. The height of station used was 50 feet. The geodetic point is marked by the intersection of two lines cut in the surface of a brass frustum, which is leaded into the end of a stone post 1 ft.x1 ft.x3 ft. This post is set so that its upper end is about 1 foot below the surface of the ground. Three reference-stones, similar to the above, are set, each 15 feet from the geodetic point, one being in the prolongation of the base-line and one on either side in the line passing through the geodetic point at right-angles to the base-line. A large pine post, used in 1871 as a latitude post, bears north 63° 53/ east, and is 389.17 feet from the geodetic point. Height of ground at station, 2 feet (estimated). SoutH BASE, 1870-’71.—This station is situated on Minnesota Point, near its southern ex- tremity, and marks the south end of Minnesota Point base-line. The height of station used was 20 feet. The geodetic point is marked by the intersection of two lines cut in the surface of a brass frustum, which is secured by lead in the end of a stone post 1 ft. x1 ft.x3 ft. This post is set so that its upper end is about 2 feet below the surface of the ground. Three reference-stones, similar to the above, are set, each 15 feet from the geodetic point, one being in the prolongation of the base-line and one on either side in the line passing through the geodetic point at right-angles to the base-line. A pine latitude post, used in 1871, was set on line to North Base and 30.8 feet from the geodetic point. The approximate bearing of North Base from South Base is north 36° 06’ west. Height of ground at station, 3.5 feet. MIDDLE BASE, 1870~71.—This station is situated on Minnesota Point, near the middle point of Minnesota Point base-line. The geodetic point is marked by a stone of the usual form, set so that its upper end is about 2 feet below the surface of the ground. The height of station used was 10 feet. : OnzEoTA, 1870-7 1.—This station is situated on the north shore of Saint Louis Bay, about three-quarters of a mile east of the village of Oneota, and on the east bank of a small stream flow- ing into the bay. The height of station used was 15 feet. The geodetic point is marked by a stone post of-the usual form, set so that its surface is about 3 feet below the surface of the ground. A second post, projecting about 6 inches above the ground, is set directly over the first. | The geodetic point was 61 feet south of the south rail of the Lake Superior and Mississippi Railway track, about 60 feet east of the stream above named, and about 30 feet from the shore of the bay. Height of ground at station, 15 feet (estimated). *Dates indicate years when stations were occupied. 40 LS 5 314 PRIMARY TRIANGULATION. [Curar. XIV, A, LESTER RiIvER, 1871.—This station is situated on the north shore of Lake Superior, about 7 miles northeast of Duluth, about 2 miles northeast of the mouth of Lester River, and about 1 mile back from the lake shore. The height of station used was 15 feet. The geodetic point is marked by a brass frustum leaded into the solid surface rock. Three reference pine stumps are blazed on the sides facing the geodetic point, and the initials N., S., W., respectively, formed on them by means of nails driven into the wood. Below each initial is driven another nail, and from the heads of the latter the distances to the geodetic point are: To the one marked N., 16 feet 5 inches; to the one marked S., 15 feet 5 inches; and to the one marked W., 17 feet 4 inches. The bearings of the stumps are approximate. Leight of hill at station, 555 feet. AMINICON RIVER, 1871.—This station is situated on the south shore of Lake Superior, about one-fourth of a mile west of the Aminicon River. The height of station used was 10 feet. The geodetic point is marked by a stone post of the usual form, set so that its upper end is about 3 feet below the surface of the ground. A second stone rising nearly to the ground surface is set directly over the first. Three reference birch stumps are blazed and marked N., 8., W., respectively, by means of nails driven into the wood. The stump marked N. is distant from the geodetic point 14 feet 1 inch; the one marked §., 4 feet 3 inches; and the one marked W., 9 feet 7 inches. A large pine post, used in 1871 as a latitude and azimuth post, bears south 50° 54’ east, and is distant 30.43 feet from the geodetic point. Height of ground at station, 50 feet (estimated). BUCHANAN, 1871.—This station is situated on the north shore of Lake Superior, about 1 mile southwest of Granite Point. It is near the site of the now deserted village of Buchanan, and is not more than 100 feet back from the lake shore. The height of station used was 90 feet. The geodetic point is referred to a brass frustum and an iron bolt leaded into the solid surface rock. The frustum bears south 36° 30/ east and is 2 feet distant from the geodetic point. The iron bolt bears north 36° 30’ west and is 3 feet from the geodetic point. A large bowlder, used as a latitude post in 1871, bears south 78° 44’ west and is 1,194.1 feet distant from the geodetic point. Height of rock at station, 10 feet (estimated). BRULE RIVER, 1871.—This station is situated on the south shore of Lake Superior, about 1 mile east of the mouth of Bois Brulé River. The height of station used was 35 feet. The geodetic point is referred to a latitude post used in 1871, this post bearing south 80° 00’ west and being 69 feet distant from the geodetic point; 69.3 feet directly south of the latitude post is set another post, under which is placed a bowlder, having a cross cut on its upper surface. Height of ground at station, 19 feet. BURLINGTON, 1871.—This station is situated on the north shore of Lake Superior on the point of land bounding the northeast side of Burlington Bay. The height of station used was 50 feet. The geodetic point is marked by a brass frustum leaded into the solid surface rock. On the north, south, and west, respectively, are set three similar frusta, each being 50 feet distant from the geo- detic point. Height of rock at station, 141.5 feet. CLay BANKS, 1871.—This station is situated ou the south shore of Lake Superior, about 6 miles west of Bark Point and about one-half mile back from the lake shore. The height of station used was 60 feet. The geodetic point is marked by a wrought-iron spike driven into the sandstone sur- face rock. Centrally over the spike is set a stone marking-post of the usual form. To the west of the geodetic point 23 feet 9 inches is a pine stump marked W., to the south 37 feet is a poplar stump marked §., and to the north 27 feet is a granite bowlder. Height of ground at station, 250 feet (estimated). SPLit Rock, 1870, ’71.—This station is situated on the north shore of Lake Superior on Split Rock Point. The height of station used was 58 feet. The geodetic point is marked by a cross cut in the solid surface rock. Three reference-marks of the same kind as the above are cut in the rock—one east, one south, and one west, each being 15 feet from the geodetic point. Height of rock at station, 163 feet. DETOUR, 1870, *71.—This station is situated on the south shore of Lake Superior, on the point of land nearly due south of Sand Island. The height of station used was 30 feet. The geodetic point is marked by a stone post of the usual form. Three reference-stones are set—one east, one south, and one west, each 15 feet from-the geodetic point. Height of ground at station, 12 feet. s¢ ~ §2.] MINNESOTA POINT BASE TOU KEWEENAW BASE. 315 WEST SAWTEETH, 1870.—This station is situated on the north shore of Lake Superior, on the highest of the Sawteeth Mountains. It is nearly due north from the mouth of Baptism River and about 2 miles back from the lake shore. The height of station used was 25 feet. The geodetic point is marked by a brass frustum leaded into the solid surface rock. Three pine trees were blazed for references, one being northeast 67.5 feet from geodetic point, one northwest 25 feet, and one south 95 feet. Height of rock at station, 930 feet. East SAWTEETH, 1870.—This station is situated on one of the Sawteeth Mountains, about 135 miles northeast of West Sawteeth station, and about 2 miles back from the lake shore. The height of station used was 30 feet. The geodetic point is marked by a stone post of the usual form. An astronomical wooden post, used in 187071, bears north 23° 25/ west, and is 50.13 feet distant from the geodetic point. Height of ground at station, 903 feet. OUTER ISLAND, 1870.—This station is situated on the northwest side of Outer Island, Lake Superior. The height of station used was 51 feet. The geodetic point is marked by a stone post of the usual form, set so that its upper surface is about 3 feet below the surface of the ground. Three similar stones are set for references—one north, one south, and oue west, each 9 feet distant from the geodetic point. A stone post, rising about even with the ground-surface, is set directly over the stone marking the geodetic point. An astronomical wooden post, used in 1870, bears north 17° 17’ 50” east, and is 37 feet distant from the geodetic point. Height of ground at station, 50 feet. FARQUHAR’S Knop, 1869, ’70.—This station is situated on the north shore of Lake Superior, and is about 4 miles northwest of Sand Bay, whence the trail leading to the station starts. The height of station used was 24 feet. The geodetic point is marked by a cross cut on the solid rock, which is about 3 feet below the surface of the ground. Above this cross is set a stone post of the usual form. An astronomical wooden post, used in 1869, ’70, is south 5.9 feet and east 24.9 feet from the geodetic point. Height of ground at station, 1,113.3 feet. PORCUPINE MOUNTAINS, 1569, ’70.—This station is situated on the south shore of Lake Supe- rior, on one of the highest ridges of the Porcupine Mountains. Itis about 2 miles south of Carp Lake and about 4 miles south of the “Carp Lake Mining Company’s” landing. The height of station used was 25 feet. The geodetic point is marked by a single stone of the usual form, An astro- nomical wooden post, used in 1869, ’70, is 45 feet north and 14.5 feet east from the geodetic point. Height of ground at station, 1,421 feet. . ISLE ROYALE, 1869.—This station is situated near the east end of Isle Royale, about 1} miles north from the old mining company’s landing in Rock Harbor. The height of station used was 34 feet. The geodetic point is marked by a stone post of the usual form. An astronomical wooden post, used in 1866 and 1869, bears north 88° 3/30” eass and is 88 feet distant from the geodetic point. Height of ground at station, 460 feet. WHEAL KATE, 1869,’70,’71.—This station is situated on Keweenaw Point, about7 miles southwest of the town of Houghton, about one-quarter of a mile west of the old Ontonagon road and 14 miles southwest of Wheal Kate mine. The height of station used was 34 feet. The geodetic point is inarked by a stone post of the usual form. An astronomical wooden post, used in 1866, ’69, 70, bears north 70° 51/ east, and is 98.5 feet distant from the geodetic point. Height of ground at station, 906.5 feet. IsLE Saint IGNACE, 1867, ’69,’71,’72.—This station is situated near the east end and on the most elevated part of Isle Saint Ignace. It is approached from Saint Ignace Harbor, which lies on the southeast side of the island, and affurds safe anchorage in any weather. The height of station used was 9 feet. The geodetic point is marked by a cross cut in the solid surface rock. No refer- ence marks were made. The corners of the station pyramids, however, were surrounded by piles of stones, and as the rock in the immediate vicinity of the station is destitute of vegetation the geodetic point can be easily identified. Height of rock at station, 1,263 feet. Tip Top, 1867, ’69,’71.—This station is situated on the northeast shore of Lake Superior, about 8 miles northeast of “ Simmons’ Harbor.” This harbor is abuut 10 miles north of Otter Head. The geodetic point is marked by a brass frustum leaded into the solid surface rock. Over this frustum is placed a tripod about 3 feet high and loaded with stones to keep it in position. Three large bowlders, used to support astronomical instruments, occupy the following positions, respectively: One north 46° 24’ west and 94.6 feet from geodetic point, one north 85° west and 68.5 feet dis- 316 PRIMARY TRIANGULATION. [Cuar. XIV, B, tant, and one south 71° 28’ west and 274.0 feet distant. The latter stone is at the highest point of the hill, and there is some soil in its vicinity. Near the geodetic point the rock is slightly lower and destitute of soil. Height of rock at station, 1,520 feet. MICHIPICOTEN, 1869, ’71.—This station is situated on thenorthwest side of Michipicoten Island. It is on the highest part of the island and is approached from a mining compiny’s landing on the northwest coast. The height of station used was 9 feet. The geodetic point is marked by a cross cutin the solid surface rock. An astronomical stone post, used in 1869, bears south 44° 04’ west, and is 120.25 feet distant from the geodetic point. Height of rock at station, 937 feet. VULCAN, 1869, 71, ’772.—This station is on Keweenaw Point, about 6 miles southeast of the village of Copper Harbor. It is approached by way of an old mining road leading off to the south- east of the east end of Lake Fanny Hooe. The height of statiog used was 75 feet. The geodetic point is marked by a single stone of the usual form. An astronomical wooden post bears north 83° 16’ west, and is 48.3 feet distant from the geodetic point. A second astronomical post is set directly south of and 156.16 feet from the former. Height of ground at station, 726 feet. TRAVERSE POINT, 1871.—This station is situated on the extreme end of Traverse Point. The height of station used was 43 feet. The geodetic point is marked by a stone of the usual form. Height of ground at station, 20.5 feet. CREBASSA, 1871.—This station is situated on the east side of Keweenaw Point, about one-half mile northeast of the Portage Entry light-house and not more than 10 feet back from the edge of a bluff rising nearly vertically 44 feet above the water of the lake. The height of station used was ?4 feet. The geodetic point is marked by a stone post of the usual form. The head of a nail driven into a blazed hemlock tree bears north 43° 53’ west, and is 59.75 feet from the geodetic point. Height of ground at station, 44 feet. NortH BASE, 1871.—This station is situated on Keweenaw Point, about 15 miles south of Portage Entry. It marks the north end of Keweenaw base-line, this line being on the State road connecting Houghton and L’Anse. The height of station used was 30 feet. The geodetic point is marked by the intersection of two lines cut on the end of a copper bolt three-fourths of an inch in diameter and 10 inches long, leaded into the solid sandstone rock, which is about 4 feet below the surface of the overlying soil. The end of the bolt, which projects slightly above the rock, is cov- ered by a telegraph-wire insulator. Directly above is set a marking-stone of the usual form, which rises nearly to the ground surface and is covered by a flat stone about 1 foot square. Another stone post of the usual form, rising about even with the ground surface, is set directly in line to South Base, the distance between the geodetic point and the center mark of the latter stone being 28.37 inches. Height of ground at station, 67.8 feet. SoutH BASE, 1871.—This station is situated on Keweenaw Point, about 64 miles south of Por- tage Entry, and about one-half mile back from the lake shore. It marks the south end of Kewee- naw base-line, and is about 50 feet east of the State road running north along the base-line. The height of the station used was 30 feet. The geodetic point is marked by the intersection of two lines cut on the surface of a brass frustum leaded into the end of a stone post 6 feet long, set so that its upper end is about even with the ground surface. A second stone, 24 feet long and sim- ilarly marked, is set one yard south of the first and exactly in the prolongation of the base-line. The top of this stone is 6 feet below the surface of the ground. Two reference-stones are set, one to the east and one to the west of the line, each about 100 feet distant, and the line joining them passes through the geodetic point at right-angles to the base-line. The east reference is 6 feet Jong and projects about 1 foot above the ground. The west reference was originally the same as the east one, but in 1873 was found to have been broken off at its upper end. Height of ground at station, 70.2 feet. MIDDLE BASE, 1871.—This station is situated on Keweenaw Point, about midway between North Base andSouth Base. The height of station used was 15 feet. The geodetic point is marked by a single stone of the usual form. Height of ground at station, 55.6 feet. QUAQUAMING, 1871.—This station is situated on the northeast extremity of Pe-qua-qua-wa-ming Point, on the east side of Keweenaw Bay. The height of station used was 25 feet. The geodetic point is marked by astone of the usual form. Height of ground at station, 7.2 feet. 43.) MINNESOTA POINT BASE TO KEWEENAW BASE. 317 MIDDLE, 1871.—This station is situated on the east shore of Keweenaw Bay, about midway between Pe-qua-qua-wa-ming Point and Point Abbayé. The height of station used was 25 feet. The geodetic point is marked by a stone of the usual form. Height of ground at station, 10.5 feet. Huron Motunrains, 1871, ’72.—This station is situated on one of the Huron Mountains, about three miles southwest of the mouth of Pine River, and near the north end of Mountain Lake. The height of station used was 15 feet. The geodetic point is marked by a stone post of the usual form. An astronomical stone post, used in 1866, bears south 36° 30’ west, and is 97.5 feet distant from the geodetic point. Height of ground at station, 930 feet. B.—HISTORICAL NOTE, STATIONS, SIGNALS, INSTRUMENTS, AND METHODS OF OBSERVATION. HISTORICAL NOTE. § 3. The general triangulation of Lake Superior was begun in 1861 by Captain (afterward General) G G. Meade, Corps of Topographical Engineers. A preliminary measurement of a base on Minnesota Point at the west end of the lake was made with wooden rods on a stretched rope by Assistant W. H. Hearding. Stations were built and angles read east from the base as far as the line Brulé—Buchanan, shown on Plate I. - In 1865 Colonel W. F. Raynolds, Corps of Engineers, began the triangulation of Keweenaw Bay. Stations were built in the bay, and also station Wheal Kate, by Assistant D. F. Henry. In 1866 the building of stations was continued by Mr. Henry. The work during the vear -was principally astronomical, under Lieutenant M. R. Brown, with Assistants O. B. Wheeler and 8. W. Robinson. Stations Vulcan, Northeast, Saint Ignace, Isle Royale, Wheal Kate, and Huron Mountains were occupied. Colonel F, U. Farquhar and Lieutenant J. F. Gregory read the angles in Keweenaw Bay from South Base station to the line Traverse Island- Huron Island. In 1867 the triangulation of Keweenaw Bay was carried on by Assistant O. N. Chaffee. Keweenaw Base was measured with the Bache-Wiirdemann apparatus by Assistant Henry. The angles of the triangle, Vulean—Tip Top—Saint Ignace, were read by Lieutenant J. Mercur, Corps of Engineers, Assistant O. B. Wheeler, and Assistant G. Y. Wisner, respectively. In 1869 three triangulation and astronomical parties, under Lieutenant E. H. Ruffner, Assistant EK. 8S. Wheeler, and Assistant G. Y. Wisner, worked westwardly from the line Vulcan—Saint Ignace to the line Porcupine Mountains—Farquhar’s Knob. Three other triangulation and astronomical parties, under Assistants O. B. Wheeler, G. A. Marr, and A. R. Flint, worked from the line Vul- can—Saint Ignace eastwardly to station Mamainse. In 1870 the work in Lake Superior was continued with General C..B. Comstock, Corps of Engi- neers, in charge of the survey. As several of the triangles previously measured had too large errors in closure to be of geodetic value, those lying in the chain between Keweenaw aud Minne- sota Point Bases, whose values were thought defective, were remeasured. The Minnesota Point Base was measured by General Comstock, Lieutenant J. H. Weeden, and Assistant E. 8. Wheeler. Triangulation and astronomical work were carried by Lieutenant E. H. Ruffuer and Assistants O. B. Wheeler, G. Y. Wisner, and A. R. Flint from the line Wheal Kate-Farquhar’s Knob to the line Detour—Split Rock. In 1871 the triangulation was completed from the line Detour—Split Rock to the Minnesota Point Base. The triangles from the Keweenaw Baseito the line Wheal Kate- Vulcan were reread, as also the angles of the triangle Vulean-Tip Top-Saint Ignace, thus completing the trian gula- tion between the Keweenaw and Minnesota Point Bases. Angles were read by Assistants Wisner, Marr, Flint, Jones, and General Comstock. The telegraphic longitude of North Base station, Minnesota Point, was determined by General Comstock and Assistants Wisner and O. B. Wheeler. The latitude of the same station, of South Base station, Minnesota Point, and the azimuths of Minnesota Point base-line and of Keweenaw base-line were determined by Assistant Engineer Wisner. The final adjustment of the triangulation was made by Assistant Engineers T. W. Wright and C. H. Kummell. 318 PRIMARY TRIANGULATION. [Cuar. XIV, B, STATIONS AND SIGNALS. § A. In consequence of the mountainous character of much of the shore of Lake Superior, it was sometimes practicable to place the instrument and the signal to be observed on very low structures resting on the rock, thus giving great stability. But in most cases, to avoid cutting lanes for sights through forests, it became necessary to raise the instrument far above the ground. This was usually effected by building a triangular pyramid, the edges of this pyramid being the trunks of stout trees, 12 or 18 inches in diameter at the ground, where they were set far enough apart to give the pyramid stability, and meeting at the vertex of the pyramid, which was cut off to form a stand for the instrament. The legs of this pyramid were strongly braced to each other to give stiffness. Immediately outside of this pyramid, and far enough from it never to come in con- tact with it, was a second pyramid of quadrangular form, carrying a small platform at a height 2 or 3 feet less than that of the top of the inner pyramid. This gave the observer a place to stand on and move about on without in any way disturbing the instrument. Some of these structures were 75 feet in height, and in one case, where the instrument was supported by a central post, it was 90 feet above the ground. They were built by a steamer’s crew. In the early part of the work the instrument was in the open air; but after 1870, tents were used in all cases to shield it. As many of the lines to be observed over were long, a form of heliotrope was ordinarily used as a signal. For lines of moderate length Gauss’ heliotropes gave beams of light of sufficient size, but on the longer lines they were insufficient. The triangle Vulean—Tiptop—Saint Ignace had sides whose lengths in order were in round numbers 100, 92, and 93 miles, the first being the length of the line Vulean-Tiptop. Over long lines a beam of sunlight was flashed to the distant station by means of ordinary mirrors 12 inches high by 9 wide, mounted on wooden stands which permitted their rotation »bout both a horizontal and a vertical axis. A pole with a wooden screen at its end was run out horizontally from the platform. Through the screen there was a circular hole whose diameter, sometimes reaching 8 or 10 inches, varied with the length of the lines over which the flash was (o be sent. The mirror was then placed in such a position that a line through its center and the center of the hole in the screen, 15 or 20 feet distant, pointed at the distant station. The direction of this line was determined with a theodolite when necessary. As the sun’s diameter is about 32’ the cone of rays diverging from the screen will have that angle, and great precision in establishing the direction of the cone of light is not necessary. The mirror was constantly turned by an attendant, called a flasher, so as to keep the reflection of the sun on the screen concentric with the hole init. The sun’s rays then reached the distant station. When the air was steady and the opening in the screen was of such size as to give neither too much nor too little light, these flashes appeared at the distant station as bright and perfectly steady points of light, at which a theodolite could be pointed with great precision. The steadiest air was usually between one and three hours before sunset. When the air was unsteady, these points of light frequently expanded, as seen in the telescope to pale disks of a minute or more in diameter, of varying position and frequently changing outlines. Sometimes they would be small, but vibrating through many seconds of arc, and sometimes two or three would be superposed. As some of the lines could only be seen over when refraction had a value greater than its ordinary value, on these the unsteadiness and fluctuations of the heliotrope lights was great, and a station had to be occupied for many weeks to get a few good hours for work. It was only on rare occasions that station Vulcan could be seen from Saint Ignace. Observers uscd the flashes obtained by cutting off the light sent to a distant station for a shorter or longer interval in place of the dots and dashes of the Morse telegraph code to send messages to that station. This method of sending messages was introduced on the Lake ~ Survey by Assistant Engineers O. B. Wheeler and S. W. Robinson, in 1865, and has since been much used. INSTRUMENTS AND METHODS OF OBSERVATION OF ANGLES. § 3. In the measurement of the angles of the primary triangulation lying between the Kewee- naw Point base-line and the Minnesota Point base-line six different theodolites were used. Three of these theodolites, numbered 2561, 2562, and 2563, were made by Oertling, of Berlin, in 1868, and were used in 1869 and 1870 in the absence of better ones. They are all alike in dimensions and §§ 4,5.) MINNESOTA POINT BASE TO KEWEENAW BASE. 319 form, are non-repeating, and although of large size are very poor instruments. Their horizontal circles are 20 inches in diameter, are graduated to five minutes, and are read by three micrometer- microscopes, one division of the micrometer-head being one second. Their telescopes have a focal length of 30 inches, with object-glasses of 24 inches diameter. They have bad axes, bad micro- scopes, and accidental errors of graduation amounting to 20 seconds. From a comparison of the individual results of the measurements of angles with their means, the following mean errors for one measurement of an angle with these theodolites were found: No. 2561, 4.48; No. 2562, 3.10; No. 2563, 3.91. Of the twenty-eight stations in this section of the triangulation, nine were occu- pied in whole or in part by these instruments. Other and better ones were obtained as soon as was practicable. None of the Oertling theodolites were used after 1871, and only one, No. 2562, after 1870. In the spring of 1871 two new theodolites were received, one from Troughton & Simms, of London, and one from Fistor & Martins, of Berlin. Troughton & Simms’ theodolite No. 1, shown in Plate XVI, is a non-repeating instrument, with a horizontal limb 14 inches in diameter, divided to five minutes, and read by three micrometer-imicroscopes, the divisions of the micrometer-heads giving single seconds. Its telescope has an object-glass 24 inches in diameter, with a focal length of 24 inches, and an eye-piece micrometer. Its 12-inch vertical circle is graduated to ten minutes, and reads by two verniers to ten seconds.. Itis an excellent instrument. Pistor & Martins’ theodo- lite No. 2, shown in Plate XIV, is a non-repeating theodolite, whose horizontal limb is 14 inches in diameter, is graduated to five minutes, and is read by two micrometer-microscopes, the divisions of whose heads equal two seconds. Its object-glass has a diameter of 24 inches and a focal length of 25 inches. It has a watch-telescope which has not been used. Its vertical circle is 10 inches in diameter, is graduated to five minutes, and is read to five seconds by four verniers. It is a good ‘instrument. The sixth theodolite used is a repeating instrument by Gambey, of Paris, with a 10. inch horizontal limb, graduated to five minutes, and reading by two verniers to five seconds. Its telescope has an object-glass 14 inches in diameter and 19 inches focal length. ‘here is no ver- tical circle, but it has a watch-telescope of 16 inches focal length, which, however, has not been used. It has been on the Survey since 1851, and in 1871 was a good instrument. In all primary work this instrument has been used as a repeating instrument, five repetitions being obtained in a set, and their result constituting one measurement. An examinat on of more than twenty-five angles, observed prior to 1872 with each of the last-named theodolites, gives the following values for the mean error of one measure of an angle derived from one pointing at each station for the non-repeating theodolites, and from the result of five repetitions for the repeating theodolite: Troughton & Simms’ No. 1, 1.75; Pistor & Martins’ No. 2, 2/.24; 10-inch Gambey, 1/’.77. These values, compared with those previously given for the 20-inch Oertling instruments, show how decidedly inferior the latter were. In these values, any twisting in azimuth of instrument or station has not been eliminated. From the table of observed angles in C, § 7, it will be seen that in this triangulation the instruments were used as follows in measurement: : a Total number| Total number Theodolite used. of stations. | of angles read. Gambey No.1 ..-..--.----2--+-2-2 2-5) ‘ _ Troughton & Simms’ No.1.-.. --..--. : _ Pistor & Martins’ No.2 .........-..-- Oertling No. 2562 ....-..--------+-+--- | Oertling No. 2563.....--2..-02-22000e2 Ocriling No. 250). .cccccsseesee mentees 50 44 25 24 12 4 DD Wann o For the measurement of the angles given in C, § 7, four stations were occupied in 1869, eight in 1870, twenty in 1871, and one in 1872, Prior to 1872, in the measurement of angles, much had been left to the discretion of the indi- vidual observers. A part of the angles had been read previous to 1870, but not all with the pre- cision deemed necessary. In selecting those to be re-read the tests of closure of triangles and of sum-angles were used. As the shore of Lake Superior is largely a wilderness, the observing parties were moved by a steamer from station to station, the steamer being notified in advance when it 320 PRIMARY TRIANGULATION. (Crap. XIV, B, was supposed the readings at a given station would be completed. If the intervening weather was unfavorable a small number of measures of the angles would be obtained. If very favorable a very large number might be got. Thus, at Outer Island, one hundred and twenty measures of the angle Detour-Sawteeth West were obtained. As the observing parties were also heliotrope parties for other observing parties, and as there were never more than four, and frequently not more than three, observing parties, a given observing party could only read to two or three stations when the heliotropes were necessary. One of the instruments was a repeating instrument; with non-repeat- ing instruments angles were usually separately read, the method of reiteration not being system- atically used. The aim was to get many measurements, especially with the inferior Oertling instruments, on the principal angles of the triangulation. When this was accomplished sum-angles were freely read if there was spare time. When, on short lines, heliotropes were not used, a pole formed the signal to which readings were made. In the few angles read in 1869 there were some cases in which the number of measures of the angle obtained by turning the inicroscopes in the direction of positive graduation differed much from the number obtained by moving the microscopes in the negative direction. Subsequently to 1869 nearly equal numbers of these positive and negative measures were obtained on each angle. The object in taking both positive and negative measures is to eliminate any motion of the horizontal limb of the instrument in azimuth, arising usually from a steady twisting of its support. When the instrument is on a high wooden structure this twisting may exceed 1” per minute of time. An exanination of this question at station Brulé River, in 1871, and at station Vulcan, in 1872, gave the following results. The twist is called + when in the direction of the sun’s daily motion. STATION BRULE RIVER. [Instrument on post 12 inches square and 35 feet high.] 1871. Mean twist in 1”. Time. | Sky. i hm. hem. | July 22 ..... +1.10 5 07 to 716 p.m. » Clear. | July 22.2... 1.34 716 to10 27 p.m. | Clear. | July 23 ..-.. +1. 51 3.17 to 5 51 p.m. Clear. ! j Tuly 28... 40.14 5 51 to 735 p.m. STATION VULCAN. {Instrument on wooden tripod 75 feet high.] | 1872. Mean twist in 1™. Time. | Sky. _— Pan ex Meek Ne ee | H hom, hom. July 24.2... +0. 98 10 40 a.m. to 11 23 a.m. | Clear. July 24.2... 40.57 11 23a. m. to 255 p.m. Clear. | August 27 .. +0. 72 9 00a.m.to 945 a.m. Bright sunshine. , August 27 .. +40. 30 945a.m.toll 42 a.m. Cloudy. August 27 .. +-0. 55 11 42a.m.to 156p.m. Faint sunshine. | This twisting is doubtless mainly due to the sun’s action on the wooden support of the instru- ment. During the day the twist is usually in the direction of the sun’s motion, while during the night it is in the reverse direction. Thirty angles, each having at least sixteen measures in both positive and negative directions, were examined to find the difference in the results of the positive and negative measures. It was found that when the microscopes were moved in the direction of the sun’s motion, the resulting measure of the angle was on the average 0’”.08 smaller than when the microscopes were moved in the opposite direction, indicating that as double the avera ge amount of twist in the interval between pointings at consecutive stations. : § 6.] MINNESOTA POINT BASE TO KEWEENAW BASE. B21 While prior to 1872 the readings of angles were not so arranged as to systematically eliminate periodic errors of graduation, yet angles were read on so many parts of the horizontal limb as to approximate to the same result. It was usually the practice after about four measures of an angle had been obtained to turn the horizontal limb in azimuth. An examination of thirteen angles read by different observers showed that each angle was read on an average on thirteen different parts of the limb. Readings on so many portions of the limb with each of two or three microscopes give for a good instrument a satistactory elimination of accidental, and a partial elimination of periodic, errors of graduation. NOTE ON PERIODIC ERRORS AND ERRORS OF MICROSCOPE- AND TELESCOPE-POINTINGS. § G6. A single measurement of an' angle between two objects is made with a theodolite by pointing the telescope first to one object and noting the readings of the equidistant microscopes on the circle, and then pointing to the other object and again noting the readings of the equidistant inicroscopes. The difference between the means of the microscope-readings in the two positions is the observed value of the angle. Every such observed value is subject to a correction for periodic error. Denoting this correction by ¢, it may be expressed by the equation (1) c=2 uw cos (qz+4 gat) sin $ ga+2 u cos (2 gz+qa+/2) sin ga +2 us cos (3 qz+% ga+ 5) sin $ qa+, &e. In this equation g is the number of equidistant microscopes by which the circle is read, z is the reading of either microscope on the left-hand object (supposing the graduation to increase from left to right), a the angle observed, and 4, %%, &c., 1, 22, &c., are constants. z and a need be taken only to the nearest degree. For a circle read by two opposite microscopes, equation (1) becomes (2) c=2 u, cos (2 2+a+/)) sin 442 uw cos (4242 4+f2) sin 2a+, &e. For a circle read by three equidistant microscopes (3) ‘e=2 uw cos (3 24-3 44) Sin $ a+2 u, cos (6243 44) sin 3a+, &e. For the Pistor & Martius theodolite No. 2, the following weighted mean values of the con- stants %, Us, 4, and % result from computations made in 1876 and 1880. wu, and f, are derived from the observations on ten different angles, and wu, and. £ from the observations on six different angles. ut, =0'.998 + 0.034 fy= 499460 U,=0".4244 0/033 fg=152° + 25° These values, substituted in equation (2), give ce=2'.00 cos (2 2+a+49°) sin a+0”.85 cos (42+ 2a+152°) sin 2a The numerical maximum of ¢ resulting from this expression is + 2/.35. c=+2".35 for z=116° and a= 62° or 240° « g=296° and a= 62° or 240° c= — 2.35 for 2=178° and a=118° or 298° “ g=858° and a=118° or 298° For the Troughton & Simms theodolite No. 1, the following weighted mean values of the constants %, Ws, fi, and %, are the results of a computation. made in 1880. «, and 4, were com- puted from the observations on five different angles and u, and #, from the observations on six _ different angles: u,=0".1724. 0.031 Py =88° 4+ 24° U,=0.436 + 0.053 figa=18° +13° 418 329 PRIMARY TRIANGULATION. [Crrap. XIV,C, These values, substituted in equation (3), give c=0".34 cos (3 2-+ 3 a+88°) sin $a+0.87 cos (624+3 a418°) sin 3a This gives, for a numerical maximum of ¢, +1//.12. ¢= 41.12 for 2=44°, 1649, or 2849; and a=87°, 2079, or 327° e=—1"12 for 2=119, 1319, or 2519; and a=33°, 153°, or 2730 For the details of such a computation reference may be made to the Lake-Survey reports in the Reports of the Chief of Engineers for 1876 and 1879. A determination of the probable error of pointing a microscope to a division-line of Troughton & Simms’ theodolite No. 1 gave For microscope A, +0/.21 B, +£0.23 C0, +018 giving for probable error of a pointing in the mean of the three microscopes +0/.12. Numerous pointings with the telescope at a time of steady atmosphere were made to a distant object, and the probable error of one such pointing, the three microscopes being read, was +0/.39. Freeing this from the probable error due to microscope-pointings, namely, +0/’.12, there re- sults for probable error of telescope-pointing, due to lack of optical power, unsteadiness of instru- ment and object, &c., £0.37. For Pistor & Martins’ theodolite No. 2, with two microscopes, there was found in the same way, Probable error of mean pointing on limb with two microscopes, +0/.14. Probable error of one telescope-pointing, when freed from error of microscope-pointings, + 0.38. C.—MEASURED AND ADJUSTED ANGLES BETWEEN MINNESOTA POINT AND KEWEENAW BASES. § 7. In the adjustment of the triangulation between Minnesota and Keweenaw Bases it hap- pened that it could be divided into two entirely independent sections by the line Split Rock - Detour. Hence those two sections were adjusted independently. A sketch of the triangulation will be found in Plate I. The notation. is explained in Chapter XIII. In the following pages the adjustment of the western section is given first. For each station the name of the observer, of the instrument used, and the date of observation are given. Following this, for each station, a table is given, of which the first column shows the names of the stations between which the angle is observed and the means of observed angles cor- rected for errors of centering of signal pointed at and of instrument, and for instrumental errors, if there are any known; the second gives the notation for the angles; the third gives the number ot measures of the angle, a measure being derived from a single pointing at each of the two sta- tions; the fourth gives the difference in seconds between the least and greatest measure, or the range; the fifth gives the weight, the weights being approximately as the reciprocals of the squares -of the mean errors of a single measure, this mean error being derived from the discrepancies between the individual measures and their means. This method gave in some cases exaggerated. weights to angles, but with instruments which varied very widely in their excellence, some having such accidental errors of graduation as to make angles read with the same instrument, and having the same number of measures differ widely in accuracy, it was difficult to adopt any other method. The sixth column gives the local correction to each angle; the seventh the additional or general correction, and the eighth column gives the most probable angle. Below each table are given the normal equations for the local corrections at each station. In the first section there are eleven stations and eleven of the tables. §7.) MINNESOTA POINT BASE TO KEWEENAW BASE. 323 They are followed by tables giving successively (a) the numerical side- and polygon-equations of condition; (b) the general corrections in terms of the correlates; (c) the normal equations for the correlates; (d) the values of the correlates; (e) the values of the general corrections; (/) the resi- duals of the general equations of condition. This completes the first section of the adjustment. The second section follows, arranged in the same way. Section I—Triangulation from Minnesota Point base-line to line Split Rock—Detour. NORTH BASE, MINNESOTA POINT—1. LObserver, G. Y. Wisner. Instrument, Troughton & Simms’ 14-inch theodolite, No. 1, Date, June and July, 1871.] | Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) (v] | Corrected angles.’ ae en es, ' io 4 “ “we “w ” | So! “uw Oneota and Lester -....-..-. 124 09 40.69 | li 7 ; 30 | 20 ; 1.055 +0. 165 | 124 09 39.800 | Lester and Aminicon ..-.... 85 26 13.14 | lo 21 ee 5.0 —0.353 —1. 207 83 26 11.580 — Aminicon and South Base .. 30 12 52.43 | 13 9 | 385 5.7 | 0.308 | 41.056 30 12 53.178 | Lester and South Base...... 113 39 05.07 | 1243 13 : TG , M2 —0.161 —0.151 | 113 39 04.758 | | South Base and Oneota..... 122 1L 15.61 | 1-1-z-s 30 5.9 13.7 —0. 154 —0. 014 | 122 11 15. 442 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 15. 7(1i) +18. 7(12)-++13. 7 (13) +25. 619=0 13. 7(1,)+20. 9(12) +15. 9(13) +26. 719=0 18. 7(1,) +15. 9(12)+-21. 6(13) +26. 719=0 SOUTH BASE, MINNESOTA POINT—2. (Observer, G. ¥. Wisner. Instrument, Troughton & Simms’ 14-inch theodolite, No.1. Date, July, 1871.} Angle as measured between— Notation. No. meas. | Range. ; Wt. | (v) [v] Corrected angles. 1 oO ¥ au “ “uw “aw oO ‘ a“ : Oneota and North Base ...-.-. 23 08 05. 26 21 29 » 51 23 —0. 182 —0.170 23 08 64. 958 North Base and Lester ...--- 47 31 20.41 22 22 * Fe 6 —0. 505 —0. 298 47 31 19. 607 ' Lester and Aminicon ....---- 88 55 19. 31 23 34 | 7.8 6 —0. 000 +0. 270 88 55 19. 580 Oneota and Lester ...-------- 70 39 24. 60 2142 17 5.7 7 +0. 433 —0. 468 70 39 24. 565 NORMAL EQUATIONS FOR LOCAL AQJUSTMENT. 30 (21)-+ 7 (22) +7.49=0 7 (21)-++-18 (22) +7.49=0 ONEOTA—3. (Observer, G. ¥. Wisner. Instrument, Troughton & Simms’ 14-inch theodolite, No.1. Date, June and July, 1871.) ‘ é fis adic is i Sma | Angle as measured between— Notation. | No, meas. | Range. Wt. (v) {v] Corrected angles C oO a “a “ aw a ° f wt | ' South Base and North Base.. 34 40 39. 66 32 19 2.3 31 —0. 025 -+-0. 019 34 40 39, 654 | South Base and Lester..-.---- 78 27 05.17 3142 29 71 8 +0. 096 —0. 639 78 27 04. 627 Lester and North Base..-.--- 43 46 26. 40 31 6 6.5 1 —0. 769 | —0. 658 43 46 24,978 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 9(31)-++ 8(32)-+7. 12=0 8(31)4-39(32) +-7.12=0 324 PRIMARY TRIANGULATION. [Cuar. XIV, C, Srecrion I.—Triangulation from Minnesota Point base-line, &e.—Continued. | LESTER—4. , (Observer, J. ©. Jones. Instrument, Oertling 20-inch theodolite, No. 2562. Date, July, 1871.) =" tae 3 ; aeee | | Angle as measured hetween— Notation. | No. meas. | Range. Wt. | (v) (v] Corrected angles. -- See Ey ee -|- “ ! @ ¥ " “ ! " | " | on u ; Oneota and South Base ..... 80 63 30.81 | 41+2 22 69 | 8 | +0.507 —0.141 | 30 53 31. 180 South Base and Aminicon .. 37 56 06.78 | 45 35 8.0 | 7 , --0. 579 —0. 505 37 56 06. 854 ' Aminicon and Brulé ......-.- 41 43 08.97 | 44 39 i * 6 +0. 464 +0. 629 41 43 10. 063 | Brulé and Buchanan........ 51 00 39.77 | 43 35 122 | 8; +0.927 —0. 562 51 00 40, 135 \ Aminicon and Buchanan.... 92 43 49.42 | 4344 21 13.1 | 2 | +0.711 ' +0. 067 92 43 50.198 | North Base and Aminicon .. 56 45 45.12 | 4541 5 5.8 | 1 | +0.150 ; 2.585 ! 56 45 42. 685 . Buchanan and Oneota....... 198 26 27.14 | 4—-1-2-3-1—-5 14 13.1 1 +4. 049 -++0. 579 198 26 31. 768 . North Base and Buchanan .. 149 29 35.55 | 414344+5 4 5.2 1 —0, 149 —2. 518 149 29 32. 883 | 1 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 9(4) +2) + (45)+ (44)+ (43)—6. 53=0 2(4¢ta)+ + (44)+ (43)—1.69=0 (4142) +8(4,)-+ (44) (43) —6. 53=0 (ite) + (4541) (45)--10(44)+-4(43) —9. 58=0 (4142) + (4st1)+ (45(+ 4(44)+-7 (43) —9. 58=0 AMINICON—5. (Observer, G. Y. Wisner. Instrument, Troughton & Simms’ 14-inch theodolite, No. 1. Date, July, 1871.] Angle as measured between— Notation. | No. meas. | Range. Wt, (v) | {v)} Corrected angles. ye ds y ! at pattie 7 Oo c: a | | th | a | “wn | °o 4 aw | South Base and North Base. 13 2030.11) 51 [ -s 1.7 2 | —o.940 | —1.389 | 13 20 27.781 | North Base and Lester...... 39 48 06.39 52 ; 10 6.8 2 | —0. 940 | +0. 912 i 39 48 06, 262 | South Base and Lester...... 53 08 34.44 51i+2 ’ 34 7.6 9 +0. 180 —0. 477 | 53 08 34.143 Lester and Buchanan..... 87 47 05. 51 53 | 44 | 8.7 : 8 | —0. 032 +0.209 | 37 47 05. 687 ’ Buchanan and Brulé ........ 60 03 58. 62 54 57 6.1 31 —0. 008 —0. 251 60 03 58. 361 | Brulé and South Base....... 209 00 21.42} 5, / ow | 10.4 2 | 0.130 | +0.519 | 209 00 21.809 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 13 (51) +11(5,) + 2(53)-+ 2(54) +22. 640 11(51) +13 (5_)+ 2(53)-+ 2(54)-+22. 64=0 2(51) + 2(52)-+10(53) + 2(54)+ 4.10=0 2(51) + 2(5,)+ 2(5s)-+ 33(54)-+ 4.10—0 i ' 1 | BUCHANAN—6. (Observer, A. R. Flint. Instrument, Gambey 10-inch repeating theodolite. Date, J uly, 1871.) | Angle as measured between— Notation. | No.meas. | Range. | Wt. (v) (v) Corrected sie Ss mae ered, 5m eat 1 Oo t aw “uw aw uw o ‘ uw | Brulé and Burlington....... 95 03 59.79 | 62 37 5.5 14 +0. 150 +0.204 | 95 04 00,144 Brulé and Aminicon........ 47 57 36.25 63 | 35 3.2 37 +0. 019 —0. 260 47 57 36. 009 1 sue Aminicon and Lester ....... 49 29 05.15 61 1 34 5.7 12 +0. 075 —0. 101 49 29 05. 124 a ‘ 4 | Aminicon and Burlington .. 143 01 36.20 645 | 27 | 5.0 23 +0. 009 —0. 056 143 01 36.153 Brulé and Lester.... ....-. 97 26 41.29 834) 16 1» 6.0 7 +0. 204 —0. 361 97 26 41.133 Burlington and Lester...... 192 20 41,90 Sjyoyg | 15 | 4.9 5 | —0.466 —-0. 157 192 30 41.277 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 24(61)+ 5(62)-+12(63)—2. 78=0 5(61) +-42(62) + 28(63)—7 .23=0 12(61) +28 (62) +72(63)—6. 46=0 $7] MINNESOTA POINT BASE TU KEWEENAW BASE. 220 SECTION 1.—Triangulation from Minnesota Point base-Une, d&e.—Continued. BRULE—7. \ (Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite. Date, July, 1871.] Yt 1 i Angle as measured between— Notation. No. meas. | Range. | Wt. (v) {v] Corrected angles. os | | a a OF " ! } I " | | un " or “ Aminicon and Lester...... - 4025.47.49 Mo) 64 |B | 0.817 | 0.091 . 40 25 47.082 Lester and Buchanan. ...... 31 32 40, 22 | 72 26 | 3.2 21 —0, 045 —0. 256 31 32 39.919 Aminicon and Buchanan ... 71 58 27.31 | Ti42 39 5.3 25 +0. 038 —0.347 71 58 27.001 Buchanan and Burlington .. 28 52 27.23 7 39 5.2 | 21 | 0.336 +0.013 | 28 52 26,897 Burlington and Clay Banks. 79 00 44.40 - Ta ‘ 31 6.7 | 16 —0. 441 +0.152 79 00 44.111 Buchanan and Clay Banks.. 107 53 0849 = 7344, 14 8.2 8 “+2. 853 +0. 165 107 53 11. 008 1 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 28(71) + 25(72) --10. 00-0 23(71) +-46(72) +10. 00==0 24(73)-+ 3(74)+- 9.39=0 3(73) 4-19(74)-++ 9.39=0 BURLINGTON—8. (Observer, J.C. Jones. Instrument, Oertling 20-inch theodolite, No. 2562. Date, July, 1871.) i Angle as measured between— | Notation. | No. meas. Range. | wt. (v) (v] ‘Corrected angles.: | — [|{<———_—-_ | fe} a “a | | “a | | “a a ° t “ | Brulé and Buchanan.......-. 56 03 33.85 | 81 32 | 6.3 9 | 0. 000 | +0.085 | 56 03 33. 935 Brulé and Clay Banks ...-.-. 39 25 53.51 83 54 9.8 7 0. 000 | +0. 419 39 25 53.929 | | Clay Banks and Split Rock .. 85 14 19.15 | 82 | 67 i 101 | 8 0. 000 —0. 593 85 14 18. 557 : ' : | CLAY BANKS—9. (Observer, A. R. Flint. Instrument, Gambey 10-inch repeating theodolite. Date, June and July, 1871.) | Angle as measured between— Notation. No. meas. | Range. Wt. (v) [v] Corrected angles. has | ao °o , “a | au “a - a ° ‘ a Burlington and Brulé, ....-- 61 33 23. 42 1 | 76 5.7 ) 381 +0.170 | +0. 082 61 33 23. 672 Split Rock and Burlingtou.. 38 49 38.70 92 ; 72 5.2 44 +0.074 —0.116 38 49 38. 658 Split Rock and Detour...... 69 38 60.47 93 | 72 6.3 | 39 +0. 077 -+-0. 114 69 39 00. 661 Detour and Burlington ..... 108 26 38.98 9243 ‘ 14 50 |) 6 | 40.341 ' —0. 002 108 28 39.319 Split Rock and Brulé .---.-- 100 23 02. 38 9142 26 © 4.6 | 15 —0. 016 —0. 034 100 23 02. 330 Detour and Brulé.....-..--- 170 02 03.37 914243 23 , 45 | 11 —0.459 | +0. 080 170 02 02. 991 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 57 (91) +-26(92) +-11(93) —12. 48=0 26(91) +-76(92) +-17 (93) —11. 34=0 11(91) -+-17(92) +56(93)— 7. 44=0 SPLIT ROCK—19. (Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite, No. 2. Date, June and July, 1871.] Angle as measured between— | Notation. : No. meas. Range. i i Wt. | (v) I [v] | Corrected angles. | I \ | or ” | " - u \ u OF un | f Detour and Clay Banks.....- 41 36 52.11, 103 54 5.0 28 0. 000 +0. 190 41 36 52. 300 Clay Banks and Burlington.. 55 56 05. 46 101 64 ! 9. 4 12 0.000 | —0.395 55 56 05.065 | PRIMARY TRIANGULATION. [UHap. XIV, C, SECTION L.— Triangulation from Minnesota Point base-line, de.—Continued, DETOUR—11. Instrument, Oertling 20-inch theodolite, No, 2562. {Observer, J. C. Jones. Date, June, 1871.] Notation. | No.meas. Range... Wt. (v) {v) Corrected angle. Angle as measured between— ee eee 4 Reshaaect og veutgien (lead Ai tle a | | “wn | ae ae \ =] 68 44.09.43) 9 lh = 0. 800 +0. 532 68 44 09. 962 i ! { \ Clay Banks and Split Rock .. Numerical equations of condition in the triangulation from Minnesota Point base-line to the line Split Rock—- Detour. SIDE-EQUATIONS. I. $14. 2k8 [1,] —9, 221 [1] — 9.221 [1s] +7. 391 [2,] —11. 888 [22] +17. 674 [31] —4. 302 [32] ++ 5. 666=0 IL. —11.643 [1.] —9. 221 [1,] —19. 279 [2)] +0. 396 [25] —15. 784 [51] + 9. 486 [52] —40. 762=0 III. —30, 062 [55] —2. 903 [5,] +20. 744 [6,] +2. 751 [63] —24.714 [7,] --34. 299 [72] +14, 863=0 ANGLE-EQUATIONS. (11) =+0. 37738 I +0.13879 IT =—0.13204 V- —0. 04687 VI —0. 06170 VII {l.] =—0. 17483 I —0. 10737 II +0.08061 V —0.01027 VI —0. 05579 VII (1,5) =—0. 15337 I —-0.05169 II =-+0.07071 V =—0. 00901 VI +0. 12650 VII (2:] =-++0. 05258 I +0. 03958 IT =--++0.01759 IV =—0.02053 Vs +0. 03812 VI —0. 02053 VII [2.] =—0.11976 I —0.16962 II -+0.06745IV -+0.08798 V —0. 02053 VI +0. 08798 VII [3] = +0. 00660 IT +0, 16667 VII +-0. 16667 VIII [31] =-+0. 25216 I +0.10801 IV —0. 02788 VI [32] =—0. 06276 I +0. 00348 IV -+L0. 03136 VI [45] =—0.01243IV —0.04877 V = —0. 01420 VITI +0.19514 IX —0. 06910 X [44] =—0.00621TV —0.02437 V -—0. 00710 VIII —0. 06910 IX +0. 13212 X (4;] =—0.01198IV —0,11853 V4.0, 12918 VIII —0.01420 1X —0. 00710 X [4145] =+0. 00932 IV -+0.58662 V = +0. 01065 VIII —0. 06297 IX —0.03147 X [4142] =+0.11451 IV +0. 02130°V —0. 01198 VIII —0. 01243 IX —0. 00621 X (5:] =—0.645421L +0. 02543 I +0.27170 VIT_ +0. 04340 VIII —0. 01038 X —0,00213 XI (5:] =+0.61808 TI +10. 02543 IIL —0. 22830 VII +40. 04340 VIII —0.01038 X —0. 00213 XI [53] =+0.00520II —0,31220 III —0. 00825 VII —0.01651 VIII +0.09904 X —0, 00532 XI [54] =+0,00134 IT +0. 00704 III ~-0.00213 VII —0, 00426 VIII -L0, 02556 X +0, 03088 XI IV. + (2%) +02] + [3] +082] + [4:42] + 1.248=0 Ve. + [lo] +015] + [22] + [445] —[45] + 2.530=0 VI. —[hJ]—(le] — [1s] +[2] +0] + 0.164=0 VU. +[1s}+[2:] + [%]+ [5] + 0. 362=0 VIII. + [25] + [45] + [51] + [52] + 0.71250 IX. + [4:)+[6] + [63] + [72] + 1.179=0 X. +(4) +053) + (i) +[% — 0.496=0 XI. + (51) + [63] + (1) +(%] + 0. 858=0 XIL + [6] + (%] + [8] — 0.301=0 XI. + (%] + [8] + [9] — 0.653=0 XIV. + [8] + [9%] +110) + 1.104=0 XV. +93] + [103] + [111] — 0,836=0 General corrections in terms of correlates. <2 et MINNESOTA POINT BASE TO KEWERKENAW BASE. 327 General corrections in terms of eorrelates—Continued. [a] = +0. 09227 IIT =-+0,03807 IX —0.00739 XI —0,00049 XII [&] = —0.00443 IIT =—0.01291 1X —0.01242 XI +0. 03215 XII [6] = —0. 00984 III = +0.012561X +0,01995 XI —0. 01242 XII 71) = 0.30080 ITI —0.03771 1X -+0.06938 X +0, 03167 XI [72] =+0.23805 TTT +0.04223 1X = —0.03771 X +0. 00453 XI [73] = +0, 04251 XII —0. 00671 XIII (w= —0. 00671 XII +40. 05369 XITI [8] =+0.11111 XII {®] = +0. 12500 XIV [a] = +40. 14286 XTIT {2] = +0. 02101 XIII —0. 00672 XIV —0. 00209 XV [%] = —0. 00672 XIII +0. 01627 XIV —0. 00362 XV [93] = —0. 00209 XIII —0. 00362 XIV +0. 01937 XV (10,] = -40. 08333 XIV [10;] = +40. 03571 XV (nj = +40. 10000 XV Normal equations for determining the correlates. io anee ~ 1. 040.5666 41.495731 = +0.57586 11 = + 0.12222 IV. —0.44796 V = —0. 05936 VI —0, 27313 VII 2. 0 — 4.0762 -10.575861 42.10496 IE —0,01602 II] —0.13004 IV —0.32868.V +0. 05985 VI —0, 86013 VIL —0. 02074 VIII +0.00654 X +0.00134 XI 3. 0 =+1.4863 —0.01602 II +2.68507 IIL +0.02543 VII +0. 05086 VIII +0. 32048 IX —0.60596 X —0. 06555 XI —0, 00443 XIT 4. 0 —41.2480 +0.12222I1 —0.13004II +0.31104IV +0.08875 VV +0.02107 VI +0. 06745 VII —0, 01198 VIII —0.01243 IX —0. 00621 X 5. «0 = 42.5300 —0.44796 1 0.32868 11 +0.088751V 4+0.89445 V —0.03981 VI +0. 15869 VII —0. 11853 VIIT —0.04877 IX —0. 02437 X 6. 0 —+40.1640 —0.059361 +-0.05985 IT = +0. 02107 IV —0.03981 V +0. 13563 VI. —0. 02954 VII 7 0 =+40.3620 —0.273131 —O0.86013 II +0.02543 II +0,067451V +0.15869V —-0. 02954 VI +0. 65285 VII +0. 21007 VII —0. 01038 X —0, 00213 XI x 0 40.7120 —0.02074 IT 0.05086 IIT —0.01198 IV —0.11853 V +0, 21007 VII +0. 38265 VIII : —0.01420 IX —0.02787 KX —0. 00426 XI 9. 0 =41.1790 +0.32048 IIT —0.012431V —0.04877 V —0. 01420 VIII +0. 28800 IX —0. 10681 X +0.01709 XI —0. 01291 XII 10, 0 =—0,4960 +0.00654 II —0.60596 IIT —0.006211V —0.02437 VV —0. 01038 VII —0. 02787 VIII - 010681 IX ++0.32610 X +0. 05723 XI ll. 0 =-+40, 8580 +0, 00134 IL =—0, 06555 TIT + —0, 00213 VIL —0, 00426 VIII +0. 01709 IX +0, 05723 X +40. 08703 XI —0. 01242 XI 2. 0 = 0.3010 —0.00443 III —0.01291 IX —0.01242 XI +0.18577 XIJ_ —0. 00671 XIII 13. 0 ——0.6530 —0. 00671 XII -+-0. 21756 XIII —0. 00672 XIV —0. 00209 XV 14. 0 =41.1040 —0. 00672 XIII +0, 22460 XIV —0. 00362 XV 15. 0 =—0. 8360 —0, 00209 XIIT —0. 00362 XIV +0. 15508 XV * Each of the side-equations was divided by 10 for the purpose of avoiding large numbers in the solution. 2B9R PRIMARY TRIANGULATION, [Cnar. XIV. CG, ae Values of the correlates and their logarithms. 1 =— 2.2987 log 0. 3614823-- IX =— 4.6470 log 0. 6671727 _— Il =+ 6.0147 log 0,7792140 ,. X =+4 0.8166 log 9,91200944. TIT =+ 0.0620 log 8. 7923917 ; XI =— 9.6120 log 0. 9828138 — 1V =— 1.7040 log 0, 2314696. - XIT =+ 0.7620 log 9. 88195504 V =-— 5.1014 log 0. 7076804 — XU =+ 2.9300 log 0. 4668676. VI =— 3.7945 log 0.5791546— XIV =— 4.7420 log 0. 6759615_— VII =+10, 5966 log 1. 02516654. XV =+ 5.3200 log 0. 72591164. VIIT =— 9.2137 log 0, 9644341— Nore.—The subscripts + and — attached to the last figure of the logarithms indicate the signs of the corre- sponding numbers. Values of the general corrections. we u “ aw {lJ =+0.165 | [44] =+0. 629 [6] =—(),101 [83] =—0, 593 (1e] =—1. 207 (4,] =—0.505 [6.] =+0. 204 [x] =-+0. 419 [13] =-+1. 056 [4:42] =—0. 141 [6;] =—0. 260 [9] =+0. 082 [21] =—0.170 ° [4145] =—2. 585 {7,] =—0. 091 (9:] =—0. 116 [22] =—U. 298 (J) =—-1.389 | [7%] =--0. 256 (93] =+0.114 [29] =+0.270 | [ho] =+0.912 [7%] =+0.013 — [10,] =—0.395 [3,] =—0.658 | [53] =+0.209 [7%] =+0.152 — [10,] = +0. 190 [%] =+0.019 © [54] =—0.251 [8] =+0.085 — [11,] =+0.532 [45] =—0.562 |; Residuals resulting from substitution of corrections in the numerical equations of condition. | No.of | No.of | eynation. Residual. | equation.’ Residual. : 1 40.0010 9 -+-0. 0000 2 0.0022 =) 10 -+0. 0000 3 —0.0094 , IW | +0,0001 4 ' 0,000 | 2 —0. 0000 5 0.0001 || 8 1-0. 0001 6 —0. 0001 14 —0. 0000 7 -+0. 0001 15 +0. 0001 8 0. 0001 SEcTION I.—Triangulation from line Split Rock - Detour to Keweenaw Base-line. SPLIT ROCK—10. (Observers, O. B. Wheeler and J. C. Jones. Instrument, Oertling 20-inch theodolite, No. 2563. Date, September and October, 1870.] ‘ | | Angle as measured between— : Notation. No. meas. Range. Wt. | (v) | [v] | Corrected angle. poo 7 Soe a arent eee pew oe pe i or w"W au ; “ ! “ | or “ Sawteeth West and Detour. .....-. 95 18 31.11 . 102 111 © 27.4 4 | 0. 000 | | | —0. 598 | 95 18 30.512 | - §7.] MINNESOTA POINT BASE TO KEWEENAW BASE. SEcTION IL—Triangulation from line Split Rock, d&c.—Continued. DETOUR—11. 329 (Observer, Lieut. E. H. Ruffner. Instrument, Oertling 20-inch theodolite, No. 2561. Dates, September and October, 1870. ] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v) (Corrected angles. 1 \ oO iF a we a Ww Oo t a | Split Rock and Sawteeth West... 29 31 46.20 112 60 14.0 4.5") +0.390 | —0. 427 29 31 46.163 | Sawteeth West and Outer Island . 90 02 33.58 113 87 23.5 4.0 | +0. 438 +0. 091 90 02 34.109 Split Rock and Outer Island.-..... 119 34 21. 86 11243 20 12.4 1.4 | —1.252 | —0.3386 | 119 34 20.272 \ NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 5. 9(112)+-1. 4(113)—2. 912=0 1. 4(112)-++5. 4(113) —2. 912=0 SAWTEETH WEST—12. [Observer, A.R. Flint. Instrument, Gambey 10-inch repeating theodolite. Dates, August, September, and October, 1870.] Angle as measured between— | Notation. No. meas. : Range.| Wt. | (vy) | [v] |Corrected angles. a 4 uw u" 4 aw “ o Detour and Split Rock.........--- 55 09 45. 69 | 121 46 | 9.0 | 10 | +0.380 | --0.107 | 55 09 45. ‘963 | Sawteeth East and Outer Island .. 75 10 43.97 12243 12 5.3 | 27 0. 000 - 0. 007 75 10 43. 963 Farqubar's Knoband Outer Island 62 18 23.81 123 72 10.5 17 | —0.148 | +0.590 62 18 24, 252 Outer Island and Detour...-...-. 40 04 34.73 | 124 71 12.2 16 | +0.080 | +0. 037 40 04 34, 847 | Outer Island and Split Rock...... 95 14 21.08 12144 81 10.0 19 | —0.200 | —0. 070 95 14 20. 810 | Farquhar's Knob and Detour..... 102 22 55.95 | 1234-4 6 6.3 1 | +2.522 | +0.627 | 102 22 59.099 = NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 29(121) +19(124)—12. 54=0 4-18(123)+ (124)4 2. 89=0 19(121)4+ (123) +36(124)— 9. 95=0 OUTER ISLAND—13. [Observer, G. Y. Wisner. Instrument, Oertling 20-inch theodolite, No. 2562. Dates, August, September, and October, 1870. ] i ail = | Angle as measured between— | Notation. No. meas. | Range.| Wt. (v) | {v] pare angles. oO # “ “ un | |e x “we Detour and Sawteeth West .-.----- 49 52 56.52 131 120 12,1 17 0. 000 | 24 ‘006 49 52 56. 514 o | Sawteeth West and Farquhar’s Knob 80 00 33. 86 13243 67 14.7 5 0.000 . +1. 868 80 00 35.728 Sawteeth East and Farquhar’s Knob 77 48 12. 97 133 7 12.4 9 0. 000 | —0. 329 77 48 12. 641 SAWTEETH EAST—14. (Observer, A. R. Flint. Instrument, Gambey 10-inch repeating theodolite. Dates, July and August, 1870.] Angle as measured between— Notation. No. meas. | Range.| Wt. (v) [v] (Corrected angles. | | fo] t “ a“ at ch | fo} t “a Farquhar’s Knob and Porcupine | i i 14: 51 13.5 5 | —0. 046 0. 609 62 59 40. 893 Mountains .-.--. namslacbatnaleie aeiehies 62 59 40.33 2 + | Farquhar’s Knob and Ou‘er Island. 64 1) 34.92 14243 83 13.5 7 | —0.356 | —0.367, 64 11 34.197 | Outer Island and Bayfield. -..---. 36 08 55, 86 144 30 95 4 | —0.623 | +0,.383 . 36 08 55. 620 Porcupine Mts. and Bayfield ....-- 37 20 49. 55 14344 35 9.9 7 | —0.033 | —0.593 37 20 48. 924 Farqubar's Knob and Bayfield .... 100 20 29.12 1424344 66 16.0 4 | +0.681 | 40.016 100 20 29, 817 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 12(142)— 7(142+3)— 7(144)— 6.30=0 — 7(14:) 4+-18(1424+3) +11 (144) --12. 94=0 — 7(142)-|-11(142+43) -+-15(144) |- 12. 94=0 PRIMARY TRIANGULATION. [Cuap. XIV, C, 330 Section L.—Triangulation from line Split Rock, &e.—Continued. FARQUHAR’S KNOB—15. Tnstruments, Oertling 20-inch theodolites, Nos. 2561 and 2563. Dates, September, 1869, and July and August, 1870. 1 (Observers, Lieut. E. H. Ruffner and O. B. Wheeler. | Funes, | Angle as measured between— | Notation E No. meas. Wt. : (v) [v] igeoniasacee anales: = ——= = Ese ee OSS os eats, 25 eat aera ! o f a“ | fe “we | | “ “uw °o t “we | Outer Island and Sawteeth West .. 37 4] 15.42 15142 78 16.2 6 . 0.000 +0445 37 41 15. 865 Outer Island and Sawteeth East .. 38 00 28.22 1514243 91 W205; 7 0.000 4-0. 520 38 00 28. 740 f Bayfield and Sawtecth East......-. 29 19 38. 66 15243 37 17.2 ' 2 | 1.762 —0,211 29 19 36. 687 | Isle Royale East and Wheal Kate.. 56 31 44.44) Ida : 70 28.8 2 | 0.000 = +-1. 388 | 56 31 45. 828 | Wheal Kate and Porcupine Mts ... 39 25 02.18 155 39 18.0 2 | +-0.000 | 1. 383 39 25 03. 563 Porcupine Mts. and Bayfield. ....-- 38 51 19.20 | 151-46 36 9.9 6 | —0.587 | +0.792 | 38 51 19.405 Porcupine Mts. and Sawteeth East. 68 10 54. 63 | 151424346 49 15.8 4 | +0. 881 | +0. 581 | 68 10 56. 092 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 6(15243) 4 4(15146) +12. 92=0 4(15243) +-10(151-46) |-12. 92=0 NotTe.—The angle 154 was read by Lieutenant Ruffner with Oertling, No. 2561. No. 2563. All the other angles were read by O. B. Wheeler with Oertling, PORCUPINE MOUNTAINS—16. (Observer, G. Y. Wisner. Instrument, Oertling 20-inch theodolite, No. 2562. Dates, October, 1869, and August, 1870.] Angle as measured between— Notation. | No. meas. | Range.| Wt. e| (v) | [v] |Corrected angles. Somers ees { 1 oO # we | a “uw a oO € “ao | Bayfield and Sawteeth East ........ 27 56 54. 54 161 60 18.1 3 | 40.173 | +0. 258 27 56 54.971 | Sawteeth Eastand Farquhar’s Knob. 48 49 52. 86 162 72 11.7 9 | +0.058 | +0. 278 48 49 53.196 Farquhat's Knob and Wheal Kate.. 77 08 08. 64 163 66 16.5 4 +-0. 030 -++0. 446 77 08 v9. 116 Bayfield and Farquhar's Knob.....- 76 46 47. 83 16142 57 19. 4 2 | —0.199 | +0. 536 76 46 48. 167 Bayfield and Wheal Kate.......... 153 54 56.42 1614243 14 13.8 1 | —0.119 | +0.982 | 153 54 57, 283 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 6(161)+ 3(162)+ (163)—1.24=0 3(161)-}-12(162)+4+ (163)—1. 24=0 (161)-+ (162) +-5(163) —0. 838=0 WHEAL KATE—17. (Observers, E. 8. Wheeler and A. R. Flint. Instruments, Oertling 20-inch theodolite, No. 2563, and Gambey 10-inch repeating theodolite. Dates, July, August, September, and October, 1869; June, 1870; and Angust, 1871.] Angle as measured between— Notation. _ No. meas. Range.’ Wt. | (v) | {v] icorseexea angles. | Bae nie oa Oy oe eee | oO t wn at | wn | aw oO A aw : | Porcupine Mts.and Farquhar'sK’b. 63 27 12.34) 171 30 / 127 =. 2 | 40.408 | +1.199 63 27 13. 947 Farquhar’s K’band Isle Royale East 51 22 53.64] 172 53 | 18.0 1 3 | +0. 342 | -| 0. 823 51 22 54, 805 | Isle Royale East and Vulcan ....-. 53 40 16.84; 173 ; 41 14.0 , 6 » 40.171 | +0. 788 53 40 17, 799 Huron Island and Porcupine Mts.. 214 24 18.40) 171424344 6 , BS | 0.5! -|-0. 511 +0.200 | 214 2419111 | Farquhar’s Knoband Huron Island. 150 57 08.72| 1724344 6 | 18.5 | 0.1 | 2.557 | —0.999 | 150 57 05.164 | Vulean and Huron Mts ......-..-- 53 16 36.98) 17145 33 ' 63 13 -+-0. 000 -|-0. 487 53 16 37. 467 Vulcan and Quaquaming .......--. 77 45 17.39] 1744546 28 13.0 2 | 40.513 | —1.478 77 45 16. 425 Farquhar’s Knob and Quaquaming. 182 48 28.75} 17243444546 | 4 i 7.0 0.3 -|-0. 146 [ -+0.183 | 182 48 29.029 Porcupine Mts. and Quaqnaming.. 246 15 47.00 | 1714243444546 | 5 | 14.0 0.2 | -s —5. 356 | 246 15 42.976 | | I +-1. 332 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2.7(171) +0. 2(172)+-0. 2(173) +0. 5(172434-4) +0. 2(1744546) —0. 028=0 0, 2(171) +3. 5(172) 4-0. 5(173) +0. 5(17445+46) —1. 622=0 0. 2(171) +0. 5(172) +6. 5(173) 0. 5(1744546) —1. 622=0 0. 5(171) -+-0. 6(1724344) +1. 330=0 0. 2(171) 4-0. 5(172) +0. 5(173) +2. 5(1744+5+46) —1. 622=0 NoteE.—The angles 172, 173, 1744546, 1714243444546, and 17243444546 were read by E. S. Wheeler with the Oertling theodolite, No. 2563 §7.] MINNESOTA POINT BASE TO KEWEENAW BASE. 331 SEcrion I1.—Triangulation from line Split Rock, &e.—Coutinued. ISLE ROYALE EAST—18. [Observer, G. ¥. Wisner. Instrument, Oertling 20-inch theodolite, No. 2562. Dates, Juno, July, August, and September, 1£69. | Angle as measured between— Notation. No. meas. | Range.| Wt. (v) (v] Connected angles. | oO 8 “ “uw “ u“ | oF uw" Wheal Kate and Farquhar's Knob. 72 05 50.11 181 39 12. 2 5 0. 000 +0.555 | 72 05 50. €65 Isle St. Ignace and Vulcan........ 107 19 06.78 184 41 14.0 4 —0. 018 +1.689 | 107 19 08. 451 Vulcan and Wheal Kate .....-.... 41 11 08.52 185 14 10.4 2 —0.035 | —0. 287 41 11 08.198 Isle St. Ignace and Wheal Kate .. 148 30 15. 23 18445 33 12,2 4 | 40.017 | +1.402 | 148 30 16. 649 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 8(181) +4 (185) +0. 28=0 4(184) +6(185) +0. 28=0 |Observers, G. Y. Wisner and A. R. Flint. VULCAN—19. theodolite. Dates, August and September, 1871.] Instruments, Troughton & Simms’ 14-inch theodolite, No. 1, and Gambey 10-inch repeating Angle as measured between— Notation. No. meas. | Range.| Wt. (v) {v] | Corrected angles. ° ‘ aw a ze “ “a ° * a Huron Mts. and Crebassa ........ 33 26 31.73 | 191 19 5.0 19 0.000 | —0. 848 33 26 30. 882 Huron Mts. and Traverse Point .. 36 10 31.56 | 191+2 9 4.3 4 0.000 | —1. 208 36 10 30. 357 Huron Mts, and Wheal Kate...... 49 14 53.72 | 191+2+3 41 6.7 19 —0.181 | —0. 235 49 14 53. 304 Huron Mts. and Mt. Houghton..-. 59 59 15.77 | 191424344 16 7.8 3 +1. 448 | +0. 129 59 59 17.347 Huron Mts. and Isle Royale East.. 134 23 46.08 | 19142434445 18 4.2 11 +0.160 | —0.237 | 134 23 46.003 Isle St. Ignace and Huron Mts.... 169 35 59.61 | 1914243444546 5 4.0 6 —0. 328 | +0. 662 | 169 35 59.944 Wheal Kate and Mt Houghton ... 10 44 22.46 | 194 15 8.5 4 +1. 219 | +0. 364 10 44 24.043 | Mt. Houghton and Isle Royale East 74 24 27.08 | 19s 20 TT 5 +1. 942 | —0. 366 74 24 28.656 «4 . Isle Royale East andIsle St.Ignace 35 12 14.14 | 196 5 4.5 2 —1. 098 | +0. 899 35 12 18. 941 | Wheal Kate and Isle Royale East. 85 08 53.63 | 194145 18 6.7 7 —0. 929 | —0. 002 85 08 52. 699 : Mt. Houghton and Isle St. Ignace. 109 36 41.54 | 19546 16 5.3 7 +0. 524 | 40.533 ; 109 36 42. 597 | Wheal Kate and Isle St. Ignace... 120 21 06.65 | 1914546 6 5.2 2 —0.907 | +0.897 120 21 06. 640 Isle St. Ignace and Tip Top.....-- 56 30 26.98 | 197 6 4.4 2 —1.161 | —0. 537 56 30 25. 282 Isle Royale East and Tip Top .... 91 42 36.47 | 19647 a2 6.3 3 +2. 391 | +0. 362 91 42 39. 2238 | Mt. Houghton and Tip Top.....-- 166 07 08.26 | 19546+7 23 5.0 11 --0.377 | —0.004 166 07 07.879 | Huron Mts. and Tip Top....-..--. 226 06 27.40 | 1914243444546+47 4 8.8 0.3| —2. 299 | +0. 125 | 226 06 25. 226 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 39. 3(191-4243)-420. 3(194) +17. 3(195)-+ 6. 3(196)-+ 0. 3(197) 43. 956=-0 20. 3(191-42+2)+33. 3(194) +26. 3(195)-+ 8. 3(196)-+ 0.3(197)—78 526=0 17. 3(19142-43)-+26. 3(194)-+-49. 3(195)-+26. 3(195)-+-11. 3(197) 82. 656—0 6. 3(1914243)-+ 8. 3(194)-+26. 3(195)-+31. 3[19s)-+14. 3(197)— 9, 056=0 0. 8(1914243)+ 0. 3(194)-111. 3(195)+-14. 3)196)-+16. 3(197)-4-12. 384 =0 Nore.—Angles 1914243, 191424344, 194, 194+5, 195-46, 191+5-+6, 197, 19546+7, and 19142+344+5+6+7 were read by Wisner with the Troughton & Simms’ instrument. Angles 191 and 19142 were read by G. Y. Wisner with the Gambey instrument. Angles 1914+2+43; 445, 1914243444546, 195, 196, dnd 19647 were read by A. R. Flint with the Gambey instrument. ISLE ST. IGNACE—20, (Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite. Date, July, 1872.] Angle as measured between— | o 4 “ ' Vulean and Isle Royal East i j | 37 28 57.72 Notation. 20 No. meas. Range. Corrected angle. “ 40. 651 ov u 37 28 58. 368 ol | | | 332 PRIMARY TRIANGULATION. [Cuar. XIV, C, Srecrion Il.—Triangulation from line Split Rock, &ce.—Continued. HURON MOUNTAINS—21. (Observer, A. R. Flint. Instrument, Troughton & Simms’ 14-inch theodolite, No, 1. Date, September, 1871.] i | | Angle as measured between — | Notation. No. meas. | Range.) Wt. (v) [v] Corrected angles. \ ° ‘ “ a“ We “ws or “we Wheal Kate and Traverse Island .. 17 40 43.39 2h 20 81 6 —0.120 | +0. 473 17 40 43.743 ‘Traverse Islandand Traverse Point. 12 35 27.79 21s 32 6.8 16 —0.045 | —0, 284 12 35 27. 461 Traverse Point and Vulean...-.... 47 12 28.20 2la 41 6.3 15 +0. 094 | —0. 729 47 12 27. 565 Crebassa and Traverse Point ....-- 33 41 03.16 2114243 46 7.3 15 +0. 142 | +0. 257 33 41 03. 559 Crebassa and Vulcan... .. paiesmyyce 80 53 31. 81 211424344 30 7.9 10 —0.214 | —0.472 80 53 31.124 Wheal Kate and Vulean. -. 77 28 39. 26 21243+4 35 8.2 15 +0. 049 | —0. 540 77 28 38.769 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 25(2114+2+43) +10(2la)—4. 50=0 4-21 (212) +15 (213) -+15(214)-+1. 80=0 4-15 (21y) +31 (213) +15(214) 41. 80=0 10(21142-+43) +15 (212) +15 (213) +-40(214) —2. 70=0 TRAVERSE POINT—22. [Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite, No. 2. Date, October, 1871.] Angle as measured between — | Notation. | No. meas. ; Range. Wt. | (v) | {v] | Corrected angles. : \ ; oO t a | | oO | | aw wt | ° t “a | Vulcan and Huron Mts..... ---+ 96 87 07.44 221 44 ) 48 |! 28 i —0.093 | —0.760 96 37 06.587 Huron Mts. and Traverse Island.. 56 49 44.50 22243 8 | 4.4 2 — —0.216 | +0.079 56 49 44. 363 Huron Mts. and Crebassa ......-. 74 49 51.96 2294344 29 4.9 16 = —0.109 | —0. 092 74 49 51.759 Middle and Crebassa ....... ..-.- 34 53 31.37 | 22344 33 11.8 4 | +0.000 | +0. 026 34 53 31.396 Crebassa and Vulean ........-.-.- 171 26 68.96 | 221424544 | 44 7.3 “9 +10. 238 | — 0.852 | 171 26 58,316 Traverse Island and Crebassa .... 18 00 07.61 224 i 14 3.5 10 | i i | —0.043 | —0.171 18 00 07. 396 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 32(221)+ 9(22243)-+ 9(224)-+5. 31=0 MIDDLE—23. 9(221)-+-27 (2224-3)+25(224)+-7. 71=0 9(221)+4-25(222+43)-+ 35(224)-++7. 71=0 {Observer, G. Y. Wisner. Instrument, ‘Troughton & Simms’ 14-inch theodolite, No. 1. Date, October, 1871.] (v) [v] | Corrected angles. Angle as measured between— Notation. No. meas. Range. Wt. ' Oe uw | um" | “ “ or u Quaquaming and South Base ...... 13 42 48.28 231 24 STs Og +40. 030 | +0. 694 13 42 49. 004 South Base and North Base ...... 25 20 34.44 232 15 16:2 4 +0. 486 | +1.383 25 20 36.309 North Base and Crebassa ....-..... 18 36 46. 34 233 17 6.9 4 | +0. 486 | —1. 679 18 36 45.147 Crebassa and Quaquaming -. .. 57 40 10.08 2314243 24 | 5.9 1 12 | —0.018 | +0. 398 57 40 10. 460 South Base and Crebassa.......... 43 57 21.91 23243 25 ' 63 ' 11) —0.158 —0.296 ° 43 57 21.456 Crebassa and Traverse Island ..... 57 04 51.37» 234 48 8.6 1. 0. 000 | —0, 282 57 04 51. 088 23145 43 a) , Crebassa and Traverse Point ...... 66 38 51.13 I 0. 000 +0. 112 66 38 51. 242 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 19(23,)-++12(232)--12(233) — 12. 24=0 12(231)-+-27 (232)-+23 (233) —24. 67=0 12(231)--28) 222) -L27 (233) — 24, . 67=0 $7) MINNESOTA POINT BASE TO KEWEENAW BASE. ooo SECTION IL.—Triangulation from line Split Rock, &c.—Continued. QUAQUAMING—25. (Observer, A. R. Flint. Instrument, Gambey 10-inch repeating theodolite. Date, October, 1871.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] |Corrected angles. ° a “a “a uw we ° t a South Base and Middle Base ...... 29 13 58. 02 251 20 3.5 19 +0. 118 | —0. 330 29 13 57. 808 Middle Base and North Base. .-..-. 25 51 07. 32 252 . 20 4.6 : 10 +0. 224 | —0. 627 25 51 06. 917 South Base and North Base. ..... 55 05 05. 93 25142 | 24 7.0 | 9 —0. 248 | —0. 957 55 05 04. 725 South Base and Crebassa ...-..... 79 12 09.33 251424344 23 6.3 10 0.000 | +1. 191 79 12 10.521 Wheal Kate and Traverse Island. 64 21 56.78 25445 8 4.0 3 —0.112 | -L0. 675 64 21 57. 343 Wheal Kate and Middle 101 24 50.72 2544546 8 4.5 4 +0. 083 | +-0.067 } 101 24 50. 870 Crebassa and Middle -.. 68 00 15.85 255+6 25 4.2 19 —0.018 | —0. 231 68 00 15 603 Crebassa and Traverse Island .... 30 57 21.58 255 4 2.8 2 +0. 167 | +0. 377 30 57 22. 074 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 7 (254-45) 4(25546) —4(25s)-H1. 52=0 4(254+5)+-23 (25546) —4(25s)4-1. 52=0 —4(25445)— 4(255-46)-+6(255) 1. 52=0 CREBASSA—26. (Observers, G. A. Marr, A. R. Flint, and G. Y. Wisner. Instruments, Pistor & Martins’ 14-inch theodolite No. 2. and Gambey 10-inch repeating theodolite. Dates, September and October, 1871.] Angle as measured between— : | Notation. No. meas. | Range.| Wt. (v) [v] peo angles. Oo + “ ” u u oo: " Traverse Point and Vulcan ...... 5 49 02. 08 261 ' 22 6.5 9 +1. 201 | —0. 708 5 49 02.573 Vulcan and Traverse Island ...... 5 22 56.43 262 | 29 7.3 8 +0. 343 | +0. 310 5 22 57. 083 Traverse Island and Huron Mts-.. 60 17 07.29 263 4 54 8.3 12 +0. 193 | —0. 396 60 17 07. 087 Traverse Island and Middle .---.- 67 15 39.15 26344 64 9.5 12 —0.575 | —0.107 67 15 38. 468 Middle and Quaquaming.......-.. 54 19 34. 59 265 | 49 1.9 15 —0.570 | +0. 278 54 19 34. 298 Quaquaming and South Base ..-.. 38 13 52. 80 266 7 10.6 | 10 0.000 | +0.798 | 38 13 53.598 Traverse Point and Traverse I’d.. 11 11 59.81 26142 : 23 6.8 5 +0. 244 | —0. 398 11 11 59. 656 Traverse Island and Quaquaming 121 35 12. C6 2634445 | 42 5.7 16 +0.535 | +0.171 | 121 35 12.766 Quaquaming and North Base ..-. 51 05 40.71 26647 | 63 11.6 7 0.000 | +0. 371 51 05 41.081 Traverse Point and Huron Mts... 71 29 09.02 2614243 66 11.7 7 —1. 483 | —0. 794 71 29 06.743 Vulcan and Huron Mts .....--.-- 65 40 03. 36 26243 36 6.9 9 +0. 896 | —0. 086 65 40 04.170 Traverse Point and Middle ...-...-. 78 27 39.04 261424+3-+4+4 44 13.5 4 —0.411 | —0. 505 78 27 38.124 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 25 (261) +16(262) + 7(263)-++ 4(263+4) —34, 56=0 16(261) +33(262) +16(263)-+ 4(263+44) —31.32=0 7(261) +16(26,)-+-28 (263) —19. 30=0 4(261)+ 4(26,) +32(263+4) + 16(265) +21. 36=0 +16(26344) +31 (265)-+26. 28=0 Nore.—Angles 261, 262+3, 263, 26344, 26142+3+4, 265, 263+4+5, and 266+7 were read by G. A. Marr with the Pistor & Martins theodolite. Angles 26,, 26142, and 26, were read by G. A. Marr with the Pistor & Martins and Gambey theodolites. Angle 26142+3 was read partly by G. A. Marr with the Pistor & Martins theodolite, and partly by A. R. Flint and G. Y. Wisner with Gambey theodolite. SOUTH BASE, KEWEENAW POINT—27. (Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No.1. Date, October, 1871.] ; i Angle as measured between— | Notation. | No. meas. | Range. Wt. (v) | {v] Corrected angles. ! on ‘ nw | uw “ " eo iG | North Base and Crebassa.... -- 7 42 42,94 on 36 83 10 0.000 | +0. 964 7 42 43,904 | Crebassa and Middle.......------ 43 29 10.65 . 272 | ay 5.6 “16 +0.277 | +0. 210 43 29 11.137 1 | Middle and Quaquaming....---.- 19 04 45.35 | 273 39 Sel 15 +0.277 | —0. 607 19 04 te a | Crebassa and Quaquaming ..-.-- 62 33 59. 33 | 27243 4 3.6 1.5 | —2.776 | —0.397 62 83 56. 157 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 16.5(279)-+ 1 5(273) 4: 995=0 1. 5(272)-+16. 5(273) —4. 995=0 ae PRIMARY TRIANGULATION. [Cuar. XIV, 6, SECTION I].—Triangulation from line Split Rock, &c.—Continued. NORTH BASE, KEWEENAW POINT—28. (Observer, A. R. Flint. Instrument, Gambey 10-inch repeating theodolite. Date, October, 1871.) Angle as measured between— | Notation. | No. meas. | Range. | Wt. | (v) | (v] Corrected angles. | F oS ee if ° f uw 1 ‘ wa uw a“ fo} a “a ' Crebassa and Traverse Island.. 5 18 34.38 281 i 4 24 | 3 0. 000 —0. 298 5 18 34. 087 | Crebassa and Middle...... ... 55 57 58. 68 28142 : 4 2.2 3 +0, 285 +0. 689 55 57 59.654 | Middle and Quaquaming...... 48 49 13. 64 283 d 12 ; 25 | 19 +0.005 —0.044° 48 49 13.601 | Crebassa and Quaquaming.. ... 104 47 12. 66 2814-243 | 8 1.8 17 —0. 050 +0.645 104 47 13.255 | Quaquaming and South Base... 54 38 15. 53 284 24 4.2 13 | —0.059 —0. 072 54 38 15.399 - Middle and South Base .... .103 27 28.99 28344 2 286) (61 6 | +0. 126 | —0.116 103 27 29.000 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 20(281-+2)-+-17 (283) —5.78=0 17(28142)+42(283) +6 (28s) —4. 70=0 +6(283) + 19(28,) +1. 08=0 Numerical equations of condition in the triangulation from the line Split Rock -Detour to Keweenaw Base-line. SIDE-EQUATIONS. I. (10) + 5.5578 [2842] + 5.5573 [28] + 14.9423[2%] — 7.5481 [271] 4+ 3.3819 [2%] + 3.3819 [27%] + 16.9927 [26547] — 26.7263 [26;] + 21. 135=0 Il. (30) — 18.4192 [2%] + 14.9423[28,) — 7.5481[27,] — 7.5481 [272] + 53.3266 [275] — 60.3353[23,] + 25. 9487 [230] + 47,475=0 IID. (10) + 5.5578 [28,42] ++ 23.9770 [28] — 15.1151 [26s] + 16.9927 [26547] — 12.6225 [23,] — 12.6225 [23] + 13. 3262 [235] + 43. 726=0 LV. (20) — 21.7711 [26.44] — 12.9466 [265] + 27.8922 [25545] — 62.9949 [255] + 9.7068 [23] + 9.7068[2%] + 9.7068[23,;] + 23.3380[23%] + 34.179=0 V. (100) —243.8175 [28,] + 17.2600 [28,42] —172. 7654 [26544] —163. 9409 [265] —163/9409 [26547] — 5.3693 [235] + 8.2619 [23,] — 2.147=0 VI. (100) —106, 3363 [26;] —106. 3363 [26.]° + 8, 8245 [26344] —138, 5598 [23] +124, 9286 [23145] — 69.3446 [22:44] -+-134. 1877 [22] — 69.737=0 VII. (100) —106. 3363 [26,] —106. 3363 [26,] + 12. 0168 [263] — 13.7629 [22245] + 64.7931 [224] — 54,5851 [214045] +148. 8498 [215] + 30.913=0 VIII. (10) — 7.0509 [26,] + 2.4701 [26] + 2.4701 [265] ++ 2.4431 [22,] + 5.7083 [22.43] + 5.7083 [224] + 28.7942 [1942] — 31.8808 [19] + 5,198=0 IX. (100) — 24.3099 [26] —137. 8826 [265] +103. 5727 [26344] +103. 5727 [265] — 71.6531 [25445] — 71. 6531 [255] — 8.5050 [25545] + 18.3262 [23,] -+ 13, 3262 [23,2] + 13.3262 [233] — 9.0907 [23145] — 5.7083 [22.45] — 5.7083 [22,] + 30.1908 [2254,] ++ 31.5897 [21.4045] +114. 8858 [19,] — 65, 3818 (19 4045] +223. 2634 [17145] —166. 6608 [1744546] —328, 971=0 X. (30) — 34.7388 [16)] + 4.9462 [16. | + 37,4786 [152453] — 26.1358 [15146] + 31.4340 [lto4g] — 27.5918 [142] + 31.4340 [14] + 52. 485—0 XI. (10) + 27.2544 [15,42] — 26,9418 (1di4e43] + 10.1817 [14043] — 4.7123 [18045] . + 4.7123 [135] — 11.0511 [125] + 0, 8590 [12243] + 22, 502=0 ANGLE-EQUATIONS. Mil. (ed) = feng “Ue HPs) +[25)] [2527 +0. 462=0 XU. [272] +[275] +266] +128 1424344] —1.592=0, XIV. [2842] -+[ 2881 +2647] FE r4g24 44] [25] [250] —3. 1640 XV. [22 ye] 4265] +126647] +[235] +0. 341=0 XVI. [28] = + [251424344] —[21] [250] +[25546] [2%] +232] i —3. 950=0 XVIL [28] 9 4-[24] Ray =i +[ 222 ] —2, 441=0 NotE.—In the solution for determining the general corrections, each of the side-eyuations was divided by the number inclosed in parenthesis and placed opposite it. » XVIII. XIX. XX. XAT. XXII. XAIIL. XXIV. XXV. XXXVI. XXVIT. XXVIII. XXIX, XXX. XXXI. XXXIL. [28] [28142] [283] [284] [271] [272] [273] [26,] [262] [263] [26544] [265] [266] [26647] [25,] [252] [25424344] [25445] [255] [25546] [23] [235] MINNESOTA POINT BASE TO KEWEENAW BASRE. [261] [261] [262] [221] [2h] +[262] +[ 262 ] +[263] +[21i] +[213] [1744546] —D17145] +[263] +[195] +[194} +[172] +[165] [19%] [20] [181] {17,] [162] [15142] (131] [121] +[124] +[1l2] =+0. 04066 I +0. 07816 XV =—0.01514 I —0, 03313 XV =-+0. 08342 I -++0. 01046 XV =—0, 07548 I =+0, 01879 I =+0. 01879 I =—0. 00394 III +0. 00495 IX =—0. 00374 III +0. 01313 IX =+0. 00313 III —0. 05798 1X =+0. 03415 LI +0. 01806 IX =—0, 06639 III +0, 02409 IX =—0, 26726 I =+0, 24275 I =+0, 02217 XII =+0. 04213 XII =-+0. 10000 XIIT =—0, 48366 IV =—0, 84938 1V =—0, 00297 IV =—0, 16758 II +0. 05740 XVI =+0. 15699 II +0. 12122 XVI +[15i46] [1514243] +[ 14043] +[13243] Numerical equations of condition, ke. ANGLE-EQUATIONS—Continned, +[ 26344] +[19% 42] + [214] +L25i45] —[265] +185] + [18] +[154] +([155] +[15243] +[133] +[123] +[113] +[102] +1245] +[22243] +[2h4e43] +[214] +L 4243] —[2h5 ] +[17s] +[ 142] +[ 22344] +[221] —[19] +[21igo4s] +[17445] +£21,4243] —[2l2] Continued. —[213 ] —[26s44] General corrections in terms of the correlates. +40. 02555 II —0. 03313 XVI —0. 03006 II +40. 03898 XVI +0. 03571 II —0. 01231 XVI —0. 02516 II —0, 02525 IT +40. 11003 II +0. 00381 1V +40, 00261 XV +40. 00362 IV +0. 00248 XV —0, 00302 1V —. 00207 XV —0. 03303 IV —0. 02260 XV —0. 00383 IV +40. 04392 XV +0, 24275 LIT —0. 02217 XIV —0. 04213 XIV -++9, 1000 XIV —0, 06232 1X -+0, 09348 IX -+0, 02340 IX —0, 09661 TIT —0, 01813 XVII —0. 30047 III +40. 13935 XVII —0.03600 III —0. 02267 XVII +0. 07504 III +0, 02267 XVII —0. 02370 III +0, 04421 XVII +40. 10000 XII +40. 05556 XII 0, 05556 XII +0. 00446 V +0. 02495 XVIII +40. 00423 V +0. 02368 XVIII —0. 00353 V —v.01977 XVIII —0. 03859 V +0. 03393 X VIII —0. 02296 V —0. 01751 XVI —0, 23425 V —0. 02217 XVI —0. 04213 XVI +0. 10000 XVI —0. 01544 XVI +0. 02317 XVI +0. 05019 XVI +0. 01906 IV +0. 00513 IV —0. $1273 V +40. 01349 V —0. 00572 V +40, 00181 V +40. 05556 XIII +40. 05556 XIII —0. 03236 V1, -+0. 03172 XIX —0, 03070 VI +0. 00299 XIX -0. 02563 VI +0, 02607 XIX +0. 01434 VI —0. 00585 XIX —0. 00740 VI +0, 00302 XIX +0. 10000 XIII -+40. 08880 XXIII —0. 13321 XXIII —0. 03861 XXIII -40. 00097 V -++0. 00594 V -+0. 01046 XII --0. 01231 XII +40. 05652 XII +0. 10000 XVII +0. 06111 XVIT —0. 00556 XVIT . = 0.08171 VIL -—0. 02695 XX —0. 03334 VII +0. 03166 XX +0. 03127 VII +0. 02436 XX +0. 01096 VII —0. 00079 XX —U. 00566 VII +0, 00041 XX -++0. 14285 XIV +0. 00523 IX +0, 00140 IX +0, +0. —0. —0. +0, +0. —0, 00481 VIIT +0. +0, .G1718 XXII —0. . 02339 XXIII —0 +0. —0. 33D +0. 367=0 +0. 629=0 +1.407=0 +2. 692=0 +0, 288=0 +2. 166=0 ~0, 499=0 —3. 238=0 —2. 766=-0 —3. 028=0 —1. 468=0 +0. 177=0 —2. 903=0 —0. 121=0 +1.131=0 04503 XIV 00585 X1V 00185 XIV 04803 VIII 00416 XXIIT 02803 VIII 02316 XXIII 04791 XXIII 00337 VIII 00174 VIII 14286 XV . 01813 XV 11065 XV 336 [233] p23] [23445] [221] 22243] [22,4] [22544] [2his24s] [2h] [215] [21] [20] [191] [19142] [194243] [194] 195] L196] (197] [184] [185] (171) (172] {173] [1724544] [17s45+6] [17445] {161] [162] [163] [15:42] [15i42+45] (15243] [12541] [154] [155] [142] [14243] L144] [131] [13:43] [133] (121] [123] [124] (12242) {11.]J [11s] [102] PRIMARY TRIANGULATION. General corrections in terms of the correlates—Continued. 05925 IT 12878 AVI 10608 1V =—), —0. =+0. 00261 VI 10406 VI 11333 VI 17336 VI 02026 VII =—(). a=, =f), =—0. =—0. =—0. —O. =-+0. —0. =), +0. =+0. =—0. =+0. 05019 VII 07421 VII 00394 VII 16779 VIII 71986 VITI I . 04924 TX . 09040 IX . 04520 TX . 04103 IX . 69770 IX . 17174 IX . 23310 X =-+0. 06928 X =+0. 03276 X =-+0. 45424 XT =—0. 38488 XI =+0. 36313 X =— 0. 23237 X =+0. =+0. =—0. =-+0. =-+0. =+0. =—0. =+0. 04512 X 01965 X 03439 X 09425 XI 05236 XI =—0. 00171 XI =—0. 06154 XI =+0. 00261 XI =+0. 00318 XI . 03385 XXIII 04622 XXIIT 01733 XXIII 015387 XXIII 05263 XXV 50000 XX VI 50000 XX VII 05882 XXXI +0 +0. ie +0. —0. +0. +0. +0. —0. —0 —0 =20: +40. 40. 40. —0. 00990 XX VIE +40. 20792 XX VII —0. —0. +40. +0, +0. +0. +0. +9, +0. —0. +0. +0. +0. -+0. —0 3da25 IL HU. 11065 XVII 00751 V —0. +40 00008 VIE = +0. 06570 VIL. +40. 06542 VIL +40. O1471IX +4. ooz922IX +40 OOLI0IX =—-4.0. 00518IX —0. o60a7IX 4.0, .02437IX 4.0. 01393IX 0, 00082IX —0. .00116IX —0, o0139IX 4.0. . 02954 XXIII —0. 05424 XXII —0, 02712 XXIII +40. 02462 XXIII +10. 41864 XXTII —0. 07692 XXII —0. 02970 XXVII —0. 16667 XXX 14286 XXIX 13636 XXVIII 04545 XXVIHI 02283XI +40. 10679X1 +0. 06766 XI = -L0. 20000 XXX 11111 XXIX 00155 XXX —0 05569 XXX —0. 00236 XXX +0. —0. +0. 00513 1V 12596 VI . 13881 VI 00179 VIII 01734 VIIL 00347 VIII 04656 XIX . 00924 XIX 00347 XIX 01641 XTX 05263 XX 03727 XXII 02131 XXII 00126 XXII 00177 XXII 00213 XXII 00985 XXIV 01808 XXIV 15763 XXIV 00821 XXIV 02712 XXIV 07692 XXIII 04620 XXVIII 09571 XXVIII 00990 XXVIII 11930 XX VIIT 02242 XXVIII 03923 XXVIII . 02788 XXXI 00236 XXXI 04256 XXXI 04682 XXXI 19732 XXXI —0. 00748 V —0.01010 IX Lu. 00067 IX —0, 00197 IX —0, 00039 1X +40. 07548 1X 40. 03016 XX —0. 01386 XX —0. 00520 XX +0. 02461 XX £0. 25000 XXI —0, 02256 XXIV +0, 03098 XXIV +0, 02489 XXIV —0, 02776 XXIV +0, 00694 XXIV —0. 12500 XXIV +40. 25000 XXIV —0, 01970 XXVI 0, 29717 XXVI —0. 01808 XXVI +0, 01641 XXVI —0, 05424 XXVI +0. 02242 XXIX -LO. 10488 XXIX —0. 06645 XXIX +0. 05275 XXXII +0. 00155 XXXIT —0. 02788 XXXII +0. 18060 XXXIT —0. 04682 XXXII +0. 25000 XXXIT +0, 00140 IX +0. 11111 XVII —0. 01165 XIX +0. 03453 XIX +40. 00691 XIX +40. 25000 XVIII —0. 01641 XXI —0. 02311 XXI —0. 00867 XXI +0. 04102 XXI —0,00177XXV +0. 01227 XXV —0. 04003 XXV 0. 08361 KXV —0, 04579XXV +0. 18750XXV —0, 12500XXV +0. 44313 XXVII —0, 01970 XXVII --0, 00985 XXVILI —0, 36928 XX VII —0. 02954 XXVII [Cuap. XIV, C, +0. 13935 XV +40. 03453 XXI —0. 00971 XXI —0. 00194 XXI —0. 00370 XXII +0. 03235 XXII +0, 01213 XXIT +0. 00924 XXII all §7.] No. of equation. 1. 6. 10. 11. 12. 13. 14, 16. 17. 18. Ss co Oo coo =-++2, 11350 =-+1. 58250 =+4, 37260 =+1. 70985 =—0. 02147 =—0. 69737 =+0. 30913 =-40. 51980 =—3. 28971 =+1.74950 =+2. 25020 =+0. 46200 =—1.59200 ——3. 16400 =+0. 34100 =—3. 95000 =—2. 44100 =+0. 36700 43 LS MINNESOTA POINT BASE TO KEWEENAW BASE. Normal equations for determining the correlates. +1. 33531 I —0. 22968 XIII +0, 09851 I —0, 00931 IX —0. 04065 XVI £0. 39879 I +0, 01118 VI +0. 28179 XIV —0. 00455 XIX —0. 03390 II —0, 00826 VII +0, 00513 XVII —0. 39095 I —0, 02305 VI —0. 22643 XIV +0, 00516 XIX +0. 01118 IIT- +0, 02622 VIII —0. 00507 KX +0, 00855 ILI +40. 02171 VIII —0, 02627 XX +0. 00263 III +2, 66200 VIII —0. 13495 XX —0, 00931 1 —0. 03257 VIL. +0.00140 XVII +0. 14621 XXII +0, 04924 XXVII +1. 03556 X +40. 02001 X -40, 00261 XXXI +40, 04552 I +0. 11111 XIII —0, 22968 I +0. 10000 XVI +40. 26827 I +40, 10000 XIII +40, 28341 I —0. 00740 VI +0. 18789 XIV +0. 00302 XIX —0.01514 I +0. 03003 IX +0. 43209 XVI +0. 01159 I +0. 00140 IX +0. 14789 XVI 0, 02647 III —0. 01663 VIII +0, 00392 XX +0. 09851 IL +40. 26827 XIV +0, 71733 I +40, 09533 XII +40, 11223 XVII —0. 12347 II +40. 00855 VII +0. 48861 XV —0. 00061 XX —0. 01790 III —0. 00254 VIII —6. 02560 XVIII +0. 00760 IT —0, 00966 VIL —0, 26115 XV +0, 00070 XX —0. 15750 IV —0. 08311 IX —0. 00261 XXI —0. 00826 IV —0. 03257 IX —0, 00386 XXI —0. 00254 IV —0, 20853 IX +0. 72165 XXI —0. 04292 IIL —0, 20853 VIII +0, 10152 XVIII —1. 11468 XXIII +0. 02001 XI +2. 52104 XI —0. 00171 XXXII +0. 09533 II —0. 06515 XIV +40. 08478 II +40. 05556 XVII —0. 00451 II +-0. 35804 XIV —0. 03370 I —0. 00566 VII +0. 40429 XV +0, 00041 XX —0. 04065 II —0, 07661 XII +0. 14789 XVLL +0, 11223 IL +40. 19976 XII +0. 37134 XVII —0. 02560 IV +40. 10152 IX —0. 03618 XXIII +40, 39879 ILI +0. 28341 XV —0. 12347 III +40. 08478 XIII +1. 63806 III +0, 00263 VII —0. 32204 XVI +0. 03537 XXIII +2. 84763 IV —0. 29315 IX +0. 00441 XIX —0. 37305 IIL —0, 00297 VIII +0. 00119 XVI +0. 06802 XXIII —0. 02305 V —0. 00740 XV -++0. 01869 XXIII —0. 00966 V —0. 00566 XV +0. 02008 XXII —0. 00297 V —0. 00174 XV +0. 00318 XXIII —0. 29315 IV +1. 88993 IX —0. 02755 XIX +0. 05995 XXIV +0. 03276 XX VIT +0. 02283 XXVIII —0, 02370 III +0. 01046 XV +40. 11111 XII +40, 28179 III -+0. 18789 XV +40. 48831 III —0, 00174 VIL —0, 16191 XVI —0. 02339 XXIII —0. 32204 III +0, 1000 XIII —0, 03861 XXIIL —0. 24913 IIL +40. 05556 XIII —0. 02990 V —0, 01751 XV —0, 39095 V —0, 01514 XVI —0. 03390 LV —0. 00451 XIV —0. 01790 IV —0. 04292 1X —0, 24913 XVII +0, 07183 V +0. 00130 XV +0, 00060 XX +40. 07183 IV —0.07077 IX +40. 00203 XVII 40, 68850 VI —0. 08327 XVIII +0 15789 VI —0. 05409 XVIII —0. 01831 XXIII -+0. 02622 VI —0. 01663 XVIII —0. 07077 V +0, 02549 XV +0. 02515 XX +0. 00116 XXV 04552 XII 0, 01159 XVII +0. 00760 V —0, 03370 XV —0. 37305 V . 02370 XIT +0. 02647 XVIII . 15780 VI ). 02122 XVI . 39956 XXIII . 48959 V , 00181 XII +0, 02990 XVIII 15789 VII 02816 XIX a, = -L0. 24588 VII —0, 05432 XIX +0. 02171 VIL +0. 00562 XIX —0. 08311 VI 0. 03003 XVI —(. 00451 XXI +0. 09040 XXVI +0. 15492 XXVIII -L0. 01965 XXIX —0, 22573 XXIX +0. 00181 V —0. 07651 XVI +0. 31111 XIII —0. 22643 V +40. 17015 XVI +0. 00130 IV +40. 02549 IX —0. 13332 XVII +0, 02122 1V +40. 17015 XIV +40. 00513 IV +0. 00400 XIV —0, 08327 VI +0. 44367 XVIIT +0. 29345 XXX +40. 33194 XII +40. 19976 XVII +0, 10000 XIV —0. 06615 XII +40. 00400 XVII —0. 26115 V +0. 01046 XII —0. 01751 XVIII +0. 00119 V —0. 16191 XV +0. 00203 V —0. 13332 XV —0, 05409 VII +0. 02886 XIX 338 No. of equation. 19. 20. 21. 22. 24. o. 26. 27. 20. 0 =-b0. 62900 0 =-+1, 40700 0 =+2. 69200 0 =-+40. 28800 0 =+2. 16600 0 =—0, 49900 U0 =— 3, 23800 () =—2, 76600 0 =—3. 02800 U =—1. 46400 0 =+40, 17700 0 =—2. 90300 0 =—0. 12100 0 =-++1. 13100 PRIMARY TRIANGULATION. Normal cauations for determining the correlates—Continued. — 0.00455 III +0, 00562 VIIT +40, 05923 XX —0, 00061 ILL —0, 13495 VIII +0, 16342 XX —0, 00261 VI +40, 02461 XX +0, 02008 VII +0, 16791 XXII +0, 03537 III +40. 00318 VIII +0, 06276 XIX —0, 02712 XXIV +0. 05995 IX —0.01808 XXVI +0, 00116 IX +0. 09040 IX 40, 04924 1X 41. 15105 XXVII +0, 15492 X +0. 01965 X 10. 29845 XI +0. 00261 XI —0. 00171 XI +0. 00441 1V —0, 02755 IX —0, 02806 XXI +40. 00060 IV +0. 02515 IX +0, 02461 XXI —0. 00386 VII +40, 32555 XXI +0. 14621 IX —0, 12510 XXIII +0. 39956 IV —1. 11468 LX +0. 07397 XX . 05424 XXVI 02256 XXII . 00177 XXII . 05424 XXIIT . 03276 X . 02283 XI . 22573 XI . 42236 XXX . 00236 XXX +0. 00155 XXX . 00985 XX VIT . 00990 XXVIII +0. 00516 V -—0, 023816 VI +0, 00302 XV +0. 02886 XVIII —0. 00370 XXII +0. 06276 XXIII +0. 00070 V —N. 00507 VI +0. 00011 XV +0. 00392 XVIII +0. 00554 XXII +0. 07397 XXIII +0. 72165 VIIT —0. 00451 IX +0. 00924 XXII -+-0. 01537 XXTIT —0. 00370 XIX +0. 00554 XX —0. 02256 XXIV —0. 00177 XXV +0. 06802 V +0. 01869 VI —0. 02339 XV —0. 03861 XVI +0. 01537 XXI —0. 12510 XXII —0. 02954 XX VII —0. 02712 XXIITIT_ +0. 46350 XXIV —0. 15276 XXIV —0. 01808 XXIV —0. 02954 XXTIT -0. 32374 XXV +0. 99717 XXVI -—0, 00985 XXIV —0. 00990 XX VII +0. 39682 XXVIII +0. 02242 XXVIII +40. 35885 XIX —0. 00236 XXXI +0. 00155 XXXII +0. 29870 XXXI —0, 07470 XXXII —0. 07470 XXXI_-+-0. 48335 XXXIT Values of the correlates and their logarithms. tS II Hl IV y 4.002 log 0, 6022771 3.023 log 0. 4804381— =— 4.562 log 0. 6591553 =-- 0.483 log =-+ 0.360 log 9. 5563025 9. 9459607 VI =+ 1.515 log 0. 18041264 VIL =— 4.963 log 0. 6957443_ VILL =+ 7.716 log 0,88742044. IX =+ 5.201 log 0.7160869. X =— 2.440 log 0.38738938_ XI =— 2.086 log 0. 3193143 XII =— 0.259 log 9. 4132998 XII] =— 2.714 log 0. 4336098 XIV =-+17, 084 log 1. 2325896. XV =+ 0.652 log 9.8142476. XVI =— 2.459 log 0.3907585_ XVII =+ 6.117 log 0.7865385 4 XVIII =— 0.414 log 9.6170003_ XIX =—12. 759 log 1. 1385869_ XX =+4 2.512 log 0. 40001964 XXI =—27.030 log 1. 4318460_ XXII =+ 0.742 log 9. 87040394. XXIII =+ 6.019 log 0.77952434. XXIV =+ 5.034 log 0.7019132., XXV =+12. 363 log 1. 0921239, XXVI =+ 2.776 log 0. 44341954. XXVIL =+ 2.765 log 0. 44169514. XXVIII =+ 4.953 log 0. 6948683, XXIX =— 1.921 log 0,2964845_ XXX =+ 8.355 log 0,9219465,. XXXI =— 0.108 log 9.0334938_ XXXII =— 2.391 log 0.3785796— —0. 05432 VIT +0. 14878 XIX —0. 02627 VII +0. 05923 XIX —0. 02806 XIX +0. 00924 XXI —0. 01831 VII —0. 03618 XVIII +0. 90345 XXIII —0. 15276 XXV —0. 01970 XXVII —0. 01970 XXVI +0. 02242 XXIX 6x] MINNESOTA POINT BASE TO KEWEENAW BASE. 339 Values of the general corrections. [231] =—0.293 ' [23.45] =+4+0.112 [174s] =+0. 437 [2842] =+0.689 [221] =—0. 760 [16,] =-10, 258 [28:] =—0.044 | [2245] = 40.079 | [162] =-+0, 278 [284] =—0,072 | [224] =—0.171 [165] =-+0. 446 [271] =+0,954 | [224.] =+0. 026 [142] =+0. 445 [272] =+0. 210 [2lit2¢3] =-+0. 257 [1514043] =+0.520 [273] =—0.607 [212] =+0.473 | [141s] =—0.211 [26,] =—0.708 | [215] =—0. 284 [154] =+1. 388 [262] =-+0. 310 [214] =—0.729 | [155] =+1.383 [265] =—0, 396 [20] =-+0. 651 [15s] =+9.792 [26:44] =—0.107 [191] =—0, 848 [140] =-40. 609 [265] =-++0, 278 [1942] =—1.203 [14.43] =—0.367 [265] =+0.798 | [194043] =—0.235 | [144] =-+0, 383 [26647] =+0.371 [194] =+0.364 | [13)] =—0. 006 [251] =—0, 330 [195] "=—0.366 | [13.43] =+1.868 [252] =—0.627 | [195] =-+0, 899 [135] =—0. 329 [251424044] =+1.191 [19;] =—0, 537 [121] =—0. 107 [25a45] =+0.675 | [18] =-+1. 689 [125] =+0.590 [255] =-++0. 377 | [185] =—0, 287 [124] =-10. 037 [25546] ——0.931 | [171] =+1. 199 [12.43] =—0.007 [23] =+0.694 | [17%] =+0. 823 [112] =—0, 427 [232] =+1.383 | [175] =+0.788 | [11s] =-+0, 091 [233] =—1.679 [174944] =—0. 999 [10.] =—0, 598 [234] =—0.282 | [l%ts¢6] =—1.478 | Residuals resulting from substitution of general corrections in numerical equations of condition. No. equation. | Residual. | No. equation. | Residual. “1 - =0.0069 17 : 0.0000 =| 2 x0. 0068 18 / =0.0001— | 3 0.0085 19 —0. 0002 4 —0.0108 = || 20 40.0002} 5 | -0.0180 || 21 ! 0.0000 | 6 | 70.0680 22 0.0000 | 7 +0.0160 | 23 / £0. 0004 8 40.0015 | 24 | +0.0001 | 9 +0.0220 25 | 40.0001 | 10 ! 0. 0000 26 | +0. 0004 1 , 0.0081 27 | —0. 0003 12 0.0002 28 0.0000 | 13 40.0001! 29 | 40.0001 | 14 | 40.0001; 30 | 0.0002 | 15 —0.0002 31 | +0. 0001 ! 16 0. 0000 f 32 ~0,0001 I § &. It is of much interest to know, at least approximately, the probable errors of the observed angles in this triangulation, and the corrections to them which have been deduced afford the means of finding those errors. In a series of m observations giving values / for a linear function of » unknowns, if v is the most probable correction to | we have the equations vw =—l ,au+b yroey ... weight p’ (1) ) aa ies Pysees ... weight p’’ (m) a 4g If the variables are independent it is well known that, p being the probable error of an observation whose weight is unity, (2) p=0.6745 / [per] per] B40) PRIMARY TRIANGULATION. [Cuar. XIV, C, D, On the other hand, if the variables are connected by + equations of condition, these equations can be used to eliminate » unknowns from (1), leaving m equations with n—r independent unknowns, to which (2) will apply, becoming then, (3) p=o.0745 | [eer =r aoe m—(n—r r This last equation follows from the fact that in a triangulation each observed angle may be used to give an equation of the form (1), containing only the most probable angle as an unknown, thus giving m=n. If there are r equations of condition, (3) gives at once the probable error of an observed angle whose weight equals unity. It has already been stated that in adjustment the chain of triangles connecting the Keweenaw and Minnesota Point Bases was broken into two sections at the line Split Rock - Detour, the sec- tions being adjusted separately. The first. section includes the stations North Base (Minnesota Point), Oneota, South Base (Minnesota Point), Lester, Aminicon, Buchanan, Brulé, Burlington, Clay Banks, Sawteeth West, Detour, and Split Rock. For this section 7 in equation (3)—= 32. Theaverage weight of an angle in this section is 12.01, the probable error of an angle of weight unity is 1/.36, and hence the probable error of an angle of average weight is £0.39. Selecting the chain joining the tio bases, whose triangles have the best forin and their angles the greatest weight, the average weight of an angle in those triangles of this section which enter this principal chain is 17.35, and hence the probable error of an observed angle of average weight in this part of the principal chain is +0/.33, The second section includes the following stations: Split Rock, Detour, Outer Island, Sawteeth East, Porcupine Mountains, Farquhar’s Knob, Isle Royale, Wheal Kate, Vulean, Huron Mount- ains, Traverse Point, Middle, Crebassa, Quaquaming, South Base (Keweenaw Point) and North Base (Keweenaw Point). For this section r in equation (3) is 69, and there results p=+1/.72 as the probable error for this section of the angle whose weight is unity. It will be remembered that the weight unity is that of an angle for which the mean error for one measure of it as derived from observations at the station alone is 1’. The average weight of the angles in this sectionis 7.96, and hence the probable error of an angle of average weight is £0.61. The average weight of an angle in those triangles of this section which enter the principal chain is 8.84, and hence the probable error of an observed angle of average weight in this part of the principal chain is -L 0.58. The probable error of an observed angle of the whole principal chain may be derived from the excess of the suin of the observed angles of a triangle over 180° plus the spherical excess by using to obtain the probable error of an observed angle, the expression 0.6745, | ve in which the v are the excesses just referred to, and 7 is the number of triangles. In this way the value £0.58 is found as the probable error of any angle in the principal chain connecting Keweenaw and Minne- sota Point Bases. Colonel Walker, in Survey of India, Vol. II, page 195, shows, for a triangulation whose angles have nearly equal weight, that the probable error of any adjusted angle may be approximately m—r . : : , in which m is the Wi ; number of observed angles and r the number of rigid equations of condition which they must sat- isfy. In this chain, for the section extending from Keweenaw Base to the line Split Rock - Detour, derived froin the probable error of an observed angle by multiplying it by,/ m=109 and r=69, hence J a =0.61. For the section between Split Rock-Detour and Minne- sota Base m—=50, r=32, and i" =0.60, Multiplying the probable error of an observed angle of the main chain in each of the two sections by 0.61 and 0.60, respectively, there result for the probable error of an adjusted angle in the selected chain between Keweenaw Base and the line Detour — Split Rock, + 0/58 x 0.61 = 4 0.35, and between this line and the Minnesota Base £0.33 x 0.60 = £0.20, Since the weights of angles between the Keweenaw and Minnesota Point Bases Spi at 0 RE ; _ m—Fr ‘ CR differ widely, the values of the ratio a can only be considered as rough approximations. In the remainder of the triangulation the angles in each section have nearly equal weights. §§ 9,10.] MINNESOTA POINT BASE TO KEWEENAW BASE. 341 PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. §9. The values just deduced are collected in tabular form, as follows: Let m=whole number of observed angles in a section (one adjustment). r=whole number of rigid conditions in a section. n=number of triangles in principal chain. [per]=sum of weighted squares of corrections to observed angles. p,=probable error of an observed angle of weight unity. p,=probable error of an observed angle of average weight in whole section. p,=probable error of an adjusted angle of average weight in whole section. p,=average weight of an observed angle in whole section. p,=average weight of an observed angle in principal chain. p-=probable error of an observed angle of average weight in principal chain. p. =probable error of an adjusted angle of average weight in principal chain. {vrv]=sum of squares of closing errors of triangles in principal chain. p.=probable error of au observed angle in principal chain as derived from the closing errors of triangles. FOR THE ENTIRE SECTIONS IN THIS CHAPTER. igs | | fiat | Section. Extent of section. mir | [pv] p, | Ds Ps a vf | a |e | | | | i | i ” feo | I | Minnesota Point Base to Split Rock -Detour “| 50 | 32 | 130.56, 1.36 | 12.01 | 0.39 0.60} 0.24 | | II | Split Rock-Detour to Keweenaw Base .....---- » 109 | 69 | 447.68 | 1.72) 7.96] 0.61 0.61 | 0.37 | ! ; } FOR THE PRINCIPAL CHAIN CONNECTING THE MINNESOTA POINT AND KEWEENAW BASES.* | | | From closing errors of triangles. ‘ Section. Divisions of principal chain. Pe | Pe | py 7 ies i - 1 [vo] | 2 Average | Greatest ' : ! | Pe error. error. ' | j ! | * a“ | uw a“ a“ | I | Minnesota Point Base to Split Rock-Detour ...; 17.35 | 0.383 | 0.20! 3.99] 9 | 0.26 0. 55 1. 03 | II | Split Rock-Detour to Keweenaw Base ......... 8.84 | 0.58 | 0.35 ; 49.13 | 15 | 0.70 | 1.38 3. 47 | 1 ‘ Entire principal chain ..........2-2.2-.0-2-)ee.222 : ra | sewielse | 53.12 | 24 | 0.58 | 1.07 3.47 | : i I *Given in D, § 10 following. D.—PRINCIPAL CHAIN OF TRIANGLES BETWEEN KEWEENAW BASE AND MIN- NESOTA POINT BASE. § LO. In a triangulation lying between two bases and adjusted without reference to them, the values of any side will differ when computed first from one base and then from the other. If proper weights can be assigned to the two values, and their weighted mean be taken, that mean will have a greater accuracy than either of the separate values. But as these separate values differ but slightly, great accuracy in determining their weights is not necessary. As an approxi- mation we may compute from each base, with the probable errors of the observed angles, the weights of a triangle side. The ratio of these two weights will approximate to the ratio of the true weights of this side and may be used in combining the two values of the side computed with the adjusted angles from the two bases: The adoption of such mean values for the sides of the triangles will carry with it new values for the angles, no longer perfectly satisfying their equations of condition, but these changes in the angles will usually be within the probable errors of the angles and so may be neglected. To obtain approximate weights for the two computed values of a triangle side, the method given in Chapter LV, § 14, may be adopted. Ina triangle A B C, if the side B C be computed from BAD PRIMARY TRIANGULATION. [Cuav. XIV, D, L1G, the square of the probable error in BC, expressed in units of the seventh place of logarithms arising from the errors of the angles, will be p’(#+,3°), where pis the probable error of any observed angle in this part of the triangulation and @ and ,7 are changes for 1’ in the seventh place of the logarithmic sines of A and B. For a succession of transverse sides with the same value for p this probable error squared will be p*[a?+,7] for the last side. For the side A B the probable error of the side B CO may be taken as a sufficient approximation. If » differs for different parts of the triangulation, these sums will be formed for each value of p and their sum will be taken. If to this sum there be added the square of the probable error in the logarithm of the base after the error of the standard of length is excluded, the reciprocal of the sum will be the weight to assign to the logarithm of the side computed from the first base. Finding in the same way the weight of the logarithm of the saine side when computed from the second base, the weighted mean of the two values will be the value to be adopted. The reciprocals of the weights are derived from the v4.7 for cach triangle, given in the following table, by adding to p*[4?+ 7], 19.01 for the computa- tions depending on the Keweenaw Base and 58.68 for the Minnesota Point Base, those being the probable errors squared in the seventh place of logarithms of the bases after the error of the standard is exc uded, corresponding to the probable errors 0'".349 and 0.421 respectively. The bases will be left unchanged, their probable errors being used only in computing the weights of intermediate sides. Calling the weight of the lesser logarithmic value of the side p, where vy p is the reciprocal of the probable error, and of the greater, p’, the correction to the lesser value to give / the weighted mean value will be +5 co dif d be the difference between the two values. The weight of any such mean value will be ptp’. When p and p/ are equal the corresponding mean side will be that with the greatest probable error, and the probable error will decrease toward each base. By this method unequal corrections are applied to the logarithms of the different sides of a triangle, but the deviation of the correction to any side from the mean of all the corrections to the sides of that triangle rarely equals 2 in the seventh place of logarithms. This quantity is small compared with the probable error in the logarithms of the sides and if its consideration be neg- lected we may then suppose that the mean value is applied to the logarithms of all the sides of this triangle. Within the degree of approximation stated, this method of making the two bases agree amounts then in effect to the following: Starting from one base the logarithms of all the sides of each triangle receive the same correction. This correction varies frow triangle to triangle, increas- ing from one base toward the other. It leaves all angles unchanged, but it effects the agreement of the bases by using values differing by small quantities, sometimes by 6 units in the seventh place of logarithms for the same transverse side of the triangulation, as it forms a part of the pre- ceding or following triangle. It gives a very nearly correct value for the distance between the bases and affects the azimuths of long lines only by insignificant quantities. Instead of distributing the error gradually through the triangulation, as a strict adjustment would do, it does it by small abrupt steps in passing from one triangle to the next and leaves the triangulation a series of slightly disconnected triangles. The method actually followed of giving unequal corrections to the different sides of the samme triangle, and of leaving the previously adjusted angles unchanged, gives the same results as those just stated within the degree of approximation previously mentioned (2 units in the seventh place of the logarithms of the sides). In the following table, giving the principal chain of triangles between Keweenaw and Minne- sota Point Bases, the first column gives the names of the stations; the second gives the adjusted angles taken from C, § 7; the third gives the triangle-error or the excess of the sum of the observed angles over 180° plus the spherical excess; the fourth gives the logarithm of the side, expressed in feet, opposite the station in the same line, the value being computed from the Keweenaw Base with the adjusted augles and the length of the base given in Chapter ILI, § 31; the fifth gives for each triangle o’ and 7; the sixth gives the sum of «?-+,? from Keweenaw Base up to the opposite triangle, inclusive; the seventh gives the quantities l ; and the eighth the weighted mean loga- D rithms of the sides as depending on both bases and derived by the method just explained. Both bases depend on the mean of Clarke yards A and B. § 10.) MINNESOTA POINT BASE TO KEWEENAW BASE. From Keweenaw Base to Split Rock - Detour p=+40.58, and from Split Rock - Detour to Minnesota Point Base p=+0.33. These values are taken from C,§ 9. The logarithm of the measured value of the length of Minnesota Point Base, expressed in feet, is 4.2982176, and the logarithm of the same, as computed trom Keweenaw Base, is 4.2982130; hence, ¢=-+0.0000046 The constant for the system ( : + a) = 3034. Chain of principal triangles between Minnesota Point and Keweenaw Bases. 1 | Weighted mean 1 Logarithms Stations. Angles. | Errors of of sides in a? and B2 -& (a2+B2) logarithms of , closure. feet. sides in feet. of “ uw t Quaquaming.......-...... | 55 05 04.725 | { 4. 4622750 4. 4622750 North Base -.....-..-.---- | 54.38. 15.399 | $ —0.215 44598901. | ssscnieseres| ages tie wees nic ly coe eeds 4. 4598903 South Base ......-..-...- 70 16 40. 061 If | 4. 5222084 57. 76 273. 85 111 4. 5222086 \ z : | Crebassa..........--..--. | 51 05 41. 081 | | 4. 5222084 289. 00 | sees feces (nema \ 4. 5222086 Quaquaming.........-.... | 2407 05.796 |) +2.772. 4,2424465 [lle eee eee eee 4, 2424468 North Base ........--..--- 104 47 13. 255 J | 4. 6164985 30. 25 * 593. 10 219 4. 6164988 \ i Middle.........2...22-.-2- . 57 40 10.460 |) (' 4.6164985 1FGV8D: inka Seeder areas 4. 6164988 Crebassassnisecc cienesiecxicaw 54 19 34. 298 i —0. in | 4. 5993964 jreaas Wikedesecees 5 Hales: 4. 5993969 Quaquaming........-..-.- 68 00 15. 601 | 4, 6568323 72,25 | 842.24 | 302 4. 6568328 l i Traverse Point ..-.....--- | 84 53 31.396 l f 4. 6568323 DIZ 04. Ye'sccererveinets: |x oa aiecred 4, 6558328 Crebassaiec.cccceese woes a5 78 27 38.124 —0. 778 4.8905444 [oases cesses} sccsewac cascleses anes 4, 8905454 Middle iszesiciin veeenceie z--- 66 38 51. 242 J | 4, 8622949 82. 81 1837. 09 637 4. 8622959 Huron Mountains........- 33 41 03.559 |) | 4. 8622949 4. 8622959 Crebassa.......22-22...00- 71 29 06.743 | —2.079 | 5.0952826 5, 0952241 Traverse Point ........--. 74 49 51.759 |J ' 5, 1029028 5. 1029038 i Willan ue daedseedsccs a0 26 an 682 |) | 5. 1029028 5. 1029038 Huron Mountains. .....--. 80 53 31.124 | > —0. 724 | 5. 3561741 5. 3561762 5 GrOPASEl. .wicstceneeman 65 40 04.170 J 5, 8212837 5. 3212858 | Wheal Kate ..- * 53 16 37.467 |) ( 5. 3212837 5. 8212858 Vidleanstsctetaasane cess 49 14 53.304 |b —0, i 5. 2967674 5, 2967796 Huron Mountains.....---- 77 28 38. 769 | J 5. 4069075 5. 4069097 ; I Tele Royale evc-2 ses e2-5 * 41 11 08.198 i ( 5. 4069075 5. 4069097 Wheal Kate ..-....-- .--- 53 40 17.799 —0. 294 ) 5. 4944950 5. 4944975 Waltaii <22ee3 seecccanercan 85 08 52. 699 | | 5. 5868062 5. 5868087 . \ sie eee siltoe “Ameitacersics Farquhat’s Knob ..-.. 56 31 45. 828 l f 5. 5868062 | MOS 20. he vepenscerares|| oa sien 5. 5868087 “Wheal Kate ...--.--. ---- 51 22 54. 805 —3. 108 BxGOS8T00. (1) coccnsamatalt aecusanedxal| pen mua 5, 5583825 Isle Royale ...-...-------- 72 05 50. 665 J | 5. 6440052 46, 24 5067. 18 1724 5. 6440078 * | eee valli raat 5 Se Pa ae eee EPP ~ }: oy. 3 - 1 | ; Porcupine Mountains. ... | 77 08 09.116 if 5. 6440052 23,04 |evecee’neomeelzeeeeas x | 5. 6440078 Farquhat’s Knob .... ..-- ; 59 25 03, 563 +3. 466 - DvASTTABS: | swale sjeieceg |emeese ceecce|aeee eines 5. 4577812 Wheal Kate ..-...-------- | 63 27 13. 947 | 5. 6065564 112. 36 5202. 58 1769 5. 6066591 Sawteeth East .-.-.. 62 59 40. 893 | f 5. 6066564 AGS | twmeomewns. Las cenens 5. 6066591 Farquhar's Knob _ 68 10 56.092 |; | 2. 361, 5. 6245197 2-2-2222 22. T gene GiceaeaAlaedeeia 2 5. 6245226 Porcupine Mountains... . : 48 49 53.196 J 5, 5334542 342.25 5659. 32 1923 5, 5884571 | i Outer Island....-...- ---- 77 48 12.641 |) (; 5. 5834542 5. 5834571 Farquhar’s Knob .......-- 38 00 28.740 |! —0, sa 5, 8327768 5, 3327798 Sawteeth East ---.-- wee 64 11 34.197 J 5. 4977361 102. 01 5782. 49 1964 5. 4977391 \ S : 1 | Sawteeth West...-..-.--. 62 18 24, 252 { 5. 4977361 5. 4977391 Outer Island...........-.-| 80 00 35,728 +2, 755 5. 5489415 5. 5439449 | Farquhar’s Knob.......-. 37 41 15. 865 5. 3368598 745, 29 6648. 78 2256 5. 3368632 : Farquhar’s Knob 344 PRIMARY TRIANGULATION. (Citar. XIV, D, 6 10, Chain of principal triangles between Minnesota Point and Keweenaw Bases—Continued. i : | | 1 Logarithms | | Weighted mean | ‘ Errors of Doe i er 2 21 R2 1 ¢ “ a Stations. 4 Angles. closure. os in | a? and B = (a?-+ 82) a | leo De | i i i oO t a uw Detour seechscsnenentaraad | 90 02 34. 109 | { 5. 8368598 0.00 | - : 5. 3368632 | Sawteeth West ........... ' 40 04 34.847 p +0.6404 | 5.1456116 ..-.-2..0 2. feeeeee seen 5.1456152 Onter Island.........-..-- , 49 52 56.514 J {| 5.2203609 313. 29 5. 2203645 Petr, eh a ee ee ie eo | | Split Rocke. .ccccs) wexsee 95 18 30. 512 | {, 5.2203609 4.00 [eeese ee Tesciasic tans 5. 2203645 De tout sccsssasenes sa whe 29 31 46. 163 —0. 362); 5, 9149578 ...--------- U esac mmuaeaaiecs Sela 4. 9149615 Sawteeth West..... ...- | 55 09 45. 963 J (\ 5,1364519 213. 16 7179.23 2434 5. 1364556 Sees sre poe alas Segre cnt eae ] Clay Banks .........-2-.-. 69 39 00. 661 | i 5. 1364519 | BONS ole ekeidebssomacdedas 5. 1364556 Split Rock..........--.+-. 41 36 52.300 | +0.9134 4, 9866826 J..... -.- sleet eee eee eee 4, 9866863 Detour aces acc aendis eee as 68 44 09.962 |J {. 5. 1838189 | 67. 24 7307.31 | 2448 5. 1338226 mae a . Burlington ....-....-..-- | 85 14 18, 557 i f 5. 1338189 4.00 nearer eter 5. 1338226 Clay Banks........-. .-- | 88 49 38.658 17 —1. 030 4. 9325697 4. 9325734 Split Rock........---2---- | 56 56 05. 065 J i 5. 0535595 5. 0535632 Brul6.... ...-2220-22eee 8 79 00 44.111 l 5. 0535595 TEBE lisse cceiteremenee fectdisssions 5. 0535632 Burlington...........-...- 39 25 53, 929 4-0. 382 4 BG4A749: | ewiesrecszew as) soma perience [ee cadens 4. 8644787 Clay Banks ...........-.-. 61 33 23. 672 J Lj} 5.0057256 129. 96 7662.57 | 2487 5. 0057294 | © 95 04 00.144 | 5. 0057256 Bi00) -|esseamipacee sense 5. 0017294 28 52 26. 897 001164)| 496912702) “veesednescccdaneencainxans | aaxesad 5. 6912740 56 03 33.935 (| 4. 9268031 4. 9263069 { 60 03 58.361 l if 4. 9263031 4, 9263069 71 58 27.001 |; —0. 809 4. 9666259 4, 9666398 47 57 36.009 |J | 4, 8592830 4, 8592869 41 43 10. 063 4. 8592830 B5696. lasasipiceccer ll weuenes 4. 8592869 97 51 04, 048 --1, 008 50820064. |x ccrovieectene Sastdearnesieln) soeu eas 5. 0320605 40 25 47.082 |J 4, 8480655 610. 09 9538.88 | 2691 4. 8480696 e 1 South Base .............. 88 55 19.580 i (| 4,8480775 0:00 | eeseesweeess|xovenes 4, 8480696 Lester - +0. 4 AZO SOSBT: © | ciecis arotesic ctaterlstoterateitaseasl| waste cing 4, 6368589 Aminicon ......2..22..--- 53 08 34.143 | J 4. 7513044 249, 64 9788.52 | 2718 4. 7513085 Qneota .aeciecae cae swaces 78 27 04. 627 | (| 4.7513044 AS49: 1} cases weve sn eeeenen 4. 7518085 South Base ........ ....-- 70 39 24.565 | > 0. 212 J} 4.97349526 |.-0..2..-0ee|seee ee ceeeeefeeee eee 4. 7349569 ViBSCEF seca Sacnieinie amines 30 53 31.180 | J 4.4706605 | 1239. 04 11046.05 | 2855 4. 4706648 North Base .............. 122 11 15. 442 i ( 4. 4706605 4. 4706648 South Base ............... 23 08 04. 958 jf 0-476). 4.1374075 4, 1374120 OneOtaseieseveeesescaseees 34 40 39. 654 i [ 4, 2982130 924.16 | 12147.10 | 2975 | 4, 2982176 For the side Vulcan - Wheal Kate p becomes nearly equal to p’, giving for the probable error squared of the weighted mean logarithm of this side 756.97. Adding to this the probable error squared of the logarithm of the Jength of the standard, namely, 8.39 (Chapter II, §14), there results for the probable error of the logarithm of this side + 27.66, corresponding to + ;s@sso part of its length. From this side the probable errors decrease toward the Minnesota Point and Ke- weenaw Bases, where they are respectively + s¢2so9 and 4 gy¢eop- The probable error + 27.66, found for the logarithm of the side Vulcan — Wheal Kate, since it has been derived from the probable error of au observed angle, is too great and should be diminished in a ratio somewhat less than that of the probable error of an observed to that of an adjusted angle. Cuar.XV,A,§$1,2.) VULCAN-IURON MOUNTAINS TO FOND DU LAC BASE. 345 CHAPTER AY. TRIANGULATION FROM LINE VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. A.—DESCRIPTIONS OF STATIONS. NOTE RELATIVE TO ELEVATIONS. § 1. The heights of ground at stations described in this chapter were determined chiefly by trigonometrical leveling. The heights at stations about Lake Superior and between it and Green Bay are reterred to the mean surface of Lake Superior from 1871 to 1875. (See Chapter XXII, § 13.) Heights along Green Bay are referred to its surface at the times of determination and are there- fore subject relatively to some uncertainties arising from fluctuations in elevation of the lake-sur- face. For any of the group of heights referred to Lake Superior the probable error may be esti- mated as not exceeding + 5 feet. For the group referred to Green Bay + 2 feet would appear to be a sufficiently large probable error, since these heights do not depend on trigonometrical levels over long lines. Heights in the vicinity of and south of Fond du Lac Base are referred to the mean surface of Lake Michigan for 1860 to 1875. (See Chapter XXII, §13.) They were computed from zenith-distances observed in the triangulation and connecting with a point of known height on Milwaukee court-house. The height of East Base given herein is 8 feet greater than that de- rived by spirit-leveliug, as stated in Chapter V,§6. As no special precision was aimed at in the trigonometrical leveling of this part of the triangulation, a probable error of not less than + 5 feet may be assigned to the heights. DESCRIPTIONS OF STATIONS. § 2. Ives HILL, 1872.--This station is situated on one of the Huron Mountains, about + miles southeast of the mouth of Pine River, and about one-half mile east of Ives Lake. The height of station used was 38 feet. The geodetic point is marked by a nail leaded into the solid surface- rock. The geodetic point is 11 feet 2 inches distant from each of three small triangles cut in the rock—one south, ene west, and one east—the bearings being approximate. Height of rock at station, 1,030 feet (estimated). GRANITE ISLAND, 1872,’73.—This station is situated on the north side of Granite Island, Lake Superior. The height of station used was 36 feet. The geodetic point is marked by a cross cut in the solid surface rock. The center of Granite Island light-house tower bears south 73° 33/ 46” west and is 171.32 feet distant froin the geodetic point. Height of rock at station, 38.4 feet. TRILOBA, 1873.—This station is situated on a bald granite knob, about 3 miles southwest of Granite Point. The height of station used was 17 feet. The geodetic point is marked by a brass frustum leaded into the solid rock. No reference-marks aremade. The hill, however, can be easily identified as it is the highest one in the immediate vicinity. Height of rock at station, 635 feet. MESNARD, 1873.—This station is situated about 2 miles south of Marquette, on a granite knob called Mount Mesnard. ‘The height of station used was about 4 feet. The geodetic point is marked by a cross cut in the solid surface rock, which is about 3 feet below the surface soil. Around and above this mark a pile of small stones was nade. Three reference-croxses are cut in the solid sur- face-rock in the following positions: one north 45 feet distant, one east 27 feet distant, and one west 17.3 feet distant from the geodetic point. Height of ground at station, 522.1 feet. SHELTER Bay, 187:.—This station is situated on the south shore of Lake Superior, about 2 miles southwest of Shelter Bay aud about one-fourth of a mile south of the west end of Deer Lake. 44L 8 346 PRIMARY TRIANGULATION. [Cnap. XV, A, The height of station used was 30 feet. The geodetic point is marked by a single stone of the usual form. Teight of ground at station, 451.7 feet. GRAND ISLAND, 1872, 773.—This station is situated on Grand Island, about 2 miles south of the light-house, and on the highest part of the island. The height of station used was 50 feet. The geodetic point is marked by a stone of the usual form, set so that its upper end is about 2 feet below the ground surface. Three reference-stones are set, one north, one east, and one south, each 10 feet distant from the geodetic point, the bearings being approximate. Height of ground at sta- tion, 388.2 feet. DivipF, 1873, ’74.—This station is situated on the dividing ridge of land between Lake Su- perior and Green Bay, about 64 miles south of Munising, and about 1,000 feet east of the old Bay de Noquette and Lake Superior road. The height of station used was 110 feet. The gecdetic point is marked by a stone of the usual form, set so that its upper end is 34 feet below the ground surface. A second stone of the same form, rising flush with the surface of the ground, is set directly above the first. The geodetic point is south 67° west and 760 feet distant from the northeast cor- ner of section 27, township 46 north, range 19 west. Height of ground at station, 426.7 feet. Mup LAkg, 1874.—This station is situated in the Northern Peninsula of Michigan, in the southeast corner of section 4, township 45 north, range 20 west. The height of station used was $5 feet. The geodetic point is referred to a stone which bears south 29° 19/ east and is 31.02 feet distant. This stone is set so that its upper end is about 3 feet below the ground surface. A. sw- face stone marked with a cross is set directly over the first stone. Two refvrence-stones are set, one approximately north 5.75 feet distant, and one approximately south 2.83 feet distant. The geodetic point is also marked by a spike driven into the root of the tree used as the station center- post. The southeast corner of section 4 bears south 44° 06’ east and is 1,061.8 feet distant from the geodetic point. Height of ground at station, 403.5 feet. STURGEON RIVER, 1874.—This station is situated in the Northern Peninsula of Michigan, in section 16, township 42 north, range 19 west, being south 40° east and 705 feet distant from the quarter-stake on the north side of the section. The height of station used wis 30 feet. The geo- detic point is marked by a stone set so that its upper end is about 24 feet below the surface of the ground. Above the latter is set a reference-stone rising 4 to 6 inches above ground. Height of ground at station, 345.4 feet. MONISTIQUE, 1874.—This station is situated in the Northern Peninsula of Michigan, 975 feet southwest of the northeast corner of section 7, township 43 north, range 18 west. The height of station used was 30 feet, the theodolite having been supported on a tree cut off at this height. The geodetic point is the middle point of the line joining two stones set 5 feet apart in an east-and- west line. Height of ground at station, 394.7 feet. FisupAm River, 1874.—This station is situated in the Northern Peninsula of Michigan, about 5 miles northeast of the mouth of Fishdam River, or the head of Big Bay de Noquette. The height of the station used was 48 feet. The geodetic point is marked by a single stone, set so that its upper end is about 2 feet below the ground surface. References were made to the three tres supporting the platform. The tree to the north is distant 8 feet 1 inch; the one cast, 6 feet 6 inches; and the one northwest, 12 feet 6 inches. Height of ground at station, 253.6 feet. Pint& HILL, 1874.—This station is situated on Peninsula Point. It bears north 65° east and is 54 miles distant from Squaw Point. Bt 4 1 +0. 203 | —0.715 | 13 16 32. 088 | Shelter Bay and Mount Mesnard .. ..... 35 02 50.513 | 55 ‘ 15 | 7.5 | 1 | +0.157 | +0. 084 35 02 50. 754 Ives Hal and Granite Island.........--- .- 72 00 57.018 ; ii4e ‘ 17 » 40 | 1 | +0. 237 | 40.179 | 72 00 57.434 | Granite Tsland and Shelter Bay .......... 86 05 50.280 | 5344 8 ' 298 ' 05° 0.092) —0.864 | 86 05 49. 324 Granite Island and Mount Mesnard...... 121 08 41.380 | 534445 5 3.0 0.3 | —0.522 | —0.780 | 121 08 40. 078 ‘ I NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(51)+1 (59) +0.7090=0 1(51)-+2(52) +0. 7090=0 - 1. 8(53) +0. 8(54)+0. 3(55) —0. 57450 0. 8(54)-{-1. 8(54) +0. 3(5s) 0. 57450 0.3(5s) + 0. 8(54)-4-1. 3 (5s) —0. 3255=-0 MOUNT MESNARD—6. {Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite. Date, August, 1873.] Angle as measured between— | Notation. | No, meas. Range. Wt. (v) (v] Corrected angles. setts + i | Seats \ i Pe eS ; ° , u | “ | uw u “ oO t “ | Triloba and Granite Island ...... 31 46 32.995 | 6 27 2:8 1) “+ 0.148 | —0. 022 31 46 33.119 | Granite Island and Shelter Bay .. 100 25 34. 308 62 26 5.7 | 1 +0.148 | —0.341] 100 25 34.115 Shelter Bay and Triloba .......-. 227 47 52. 255 63 25 , 54 f 1 +-0.148 | +10. 364 227 47 52.767 t | f NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(61)+ (62)—0. 444=—0 (61) -+- 2(62) -- 0. 444=0 SHELTER BAY—7. [Observers, G. Y. Wisner and G. A. Marr. Instruments, Troughton & Simms’ 14-inch theodolite No. 1; Pistor & Martins’ 14-inch theodo- lite No. 2. Dates, July, 1873, and June and July, 1874. ] : Angle as measured between— | Notation. | No. meas. | Range. wt. | (v) | {v) Corrected angles. fe} ‘ “uw | wn | ° t aw Mount Mesnard and Triloba..... 12 48 03. 781 hh 16 4.1 gS 12 45 02.748 Triloba and Granite Island...... . 19 23 53. 400 | 72 15 2.1 1 19 23 52, 287 - Granite Island and Grand Island 123 34 58. 812 | 73 17 2.0 1 123 34 58. 404 | Grand Island and Mud Lake..... 66 00 07.190 74 21 2.8 1 | +0.155 | 0.072 66 00 07. 273 | | Mud Lake and Grand Island..... 293 59 52. 500 To 4 21 : 3.1 1 | {0.155 | +0.072 | 293 59 52.727 t t ie endl 535 . . Norte.—The angles 71, 72, 73, were read by G. Y. Wisner with Troughton & Simms’ instrument. The angles 74, 7-4, were read by G. A. Marr with Pistor & Martins’ instrument. 358 PRIMARY TRIANGULATION. [Cuar. XV.C, Suction HWL—Triangulation from the line Vulean- Huron Mountains to the line Burnt Bluff- Pine Hill—Continued. GRAND ISLAND—s. (Observers, G. Y. Wisner and G. A. Marr. Instruments, Troughton & Simms’ 14 inch theodolite No. 1; Pistor & Martins’ 14-inch theodolite No. 2. Dates, August and September, 1872, and September and October, 1873.] Angle as measured between— Notation. | No. meas. ‘Range.’ Wt. | (v) | {v] Corrected angles. IL err eee ee age lee i OF Of I “ | “ “ Or uu ' Divide and Mud Lake .......---26.---+ 28 25 45.526 81 | 22 BBO | 41.027 | —0.017') 28 25 46, 536 | Mud Lake and Shelter Bay .........--- 53 44 02.303 8243 15 3.7 1 | + 1.027 | —0. 029 53 44 03.301 ' Wood Island and Shelter Bay.-..------- 2 31 50.686 8 a2 4.5 1 | —0,590 | 4-0. 231 ; 2 31 50. 327 | Shelter Bay and Triloba ............--. 23 44 38.766 | 84 | 17 3.9 9 1 | 0.085 | 0.019 | 23 44 38.700 | Triloba and Granite Island 13 17 41. 452 85 23 "3.1 1 | —0. 085 | —0. 060 183 17 41,307 Wood Island and Huron Mountains.... 41 00 18.690 83444546 | 17 | We) pSndeiteete: —0. 614 41 00 18.079 Divide and Shelter Bay ....-..-...-.--- 82 09 50. 753 814243 12 4.9 0. 6 | —0. 870 | —0. 046 82 09 49. 837 Granite Island and Divide ...........-- 240 47 49. 565 | 86-47 : 23 6 42 1. 40.504 | --0.087 240 47 50. 156 Wood Island and Granite Island ....... 39 34 09. 849 834445 ' 27 7.3 2 +0. 295 | +-0. 190 39 34 10. 334 4 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 6(81) --1. 6(82+43) -+1(84) +-1(85) —4, 1424=0 L. 6(81)-++-2. 6(82+3) -{-1(84) +-1(85) —4. 1424=0 -+3(83) + 2(84) + 2(85)-+2. 1100=0 1. 0(81) +-1. 0(8243) 4+-2(83) +-4(84) +3 (85)—0. 2780=0 1. 0(81)-++1. 0(82483) + 2(83) -+-3(84)-+4(85) —0. 2780=0 Nork.—Angles 83, 83444546, and 834445 were read by G. Y. Wisner with the Troughton & Simms instrument. All the others were read vy G. A. Marr with the Pistor & Martins instrument. MUD LAKE—9, (Observer, G. A. Marr. Instrument, Pistor & Martins’ 14-inch theodolite No. 2. Date. June, 1874.] Hae Sig Degen a wa” igor ge=se es, . Angles as measured between— Notation. | No. meas. Range.| Wt. (v) | [v] |Corrected angles 1 = Se —— ov “" “" i “ , u" Ou “ ! Shelter Bay and Grand Island .......-- 60 15 50.759 9 25 34/41 0.000 —0. 145 60 15 50.614 | Grand Island and Divide.............-- 47 20 58.062 | 92 | a 2.8 | 1 40.119 | +0. 105 47 20 58, 286 Divide and Monistique...........--.--- 72 56 50.257 | 9s { Oy 6.0 1 +0119 | +0, 109 72 56 50.485 | Monistique and Sturgeon ......- . 22 93 22.588 | 94 21 | 3.8 1» +0.119 | —0, 453 22 23 22. 204 : Sturgeon and Grand Island ..........-- 217 18 48.667 9541 21 5.0 1 | +0.119 | +0. 239 217 18 49. 025 . ee a ee ee ee fe Clie a Nee ae $ NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(92)-+1(93) +1(94) —0. 476=0 1(92) + 2(93) 4-1(94) —0. 476=0 1(92) +1 (93) +2(94)—0. 476=0 DIVIDE—10. ' (Observers, G. Y. Wisner and G. A. Marr. Instruments, Troughton & Simms’ 14-inch theodolite No. 1; Pistor & Martins’ 14-inch theodolite No. 2. Dates, September, 1873, and May and June, 1874.] Angle as measured between— Notation. No. meas. | same: | wt. | (v) {v] Corrected angles. fe) ede Y yg oan mae) — eee a or u" | “ | " | u or “ Monistique and Mud Lake .......-.... 79 11 00.064 101 22 29 i 1 +0. 413 | —0. 064 79 11 00.418 Mud Lake and Grand Island........... 104 13 15.870 102 20 | 24 TS seagate —0. 132 104 13 15.738 Mud Lake and Monistique .........-.. 280 48 59.110 | 10243 | 21 | 3.7 1 | +0.413 | +0. 064 280 48 59. 587 ; i i Novre.—The angle 10, was read by G. Y. Wisner with the Troughton & Simms instrument. The others were read by G. A. Marr with the Pistor & Martins instrument. §6.] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 359 SEcTION HT.— Triangulation from the line Vulean— Huron Mountains to the line Burnt Bluff— Pine Hiti—Continued. MONISTIQUE—1i1. {Observer, G. Y. Wisner. Instrument, Troughton & Simms’ 12-inch theodolite No. 2. Date, May, 1874.] Angle as measured between— Notation. _ No. meas. Range. Wt. (v) , Corrected angles. i @. €sec2 Ss. pox. gesy eyjaecnas eraser Reet a Ze I, ° a “wn at a wt oO t aw Fishdam and St.rgeon ......--.. 53 09 20.379 ili 16 1.0 1 —0. 249 ; +0. 066 53 09 20.156 ' Sturgeon and Mud Lake........-. 110 01 52.571 lle 17 76 1 —0.245 —0. 376 ; 110 01 51. 950 Mud Lake and Divide ........... 27 52 09,912 lls WW 5.9 al: —0. 245 | +0. 186 | 27 52 09. 853 Divide and Fishdam 168 56 38. 205 lla 17 5.9 1 —0. 288 4-0, 124 | 168 56 38. 041 6.0 1 +-0. 044 : ' 0.190 222 05 58.197 Divide and Stirgeon..........--- 222 05 57, 963 Ua+i 16 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2111) 4-1 (112) -| 1118) 4-1.067=0 1(111)-4 3(112)-4 2(11s) + 1.513=0 1(11) +2(11z) | 3(113)--1.513=0 STURGEON—12. [Observer, G. Y. Wisner. Instrument, Troughton & Simms’ 12-inch theodolite No. 2. Date, June, 1874.] Angle as measured between— Notation. No. meas. | Range. Wt. (v) (v] Corrected angles. | oF ” “ow uu ‘i uy | ou “ Mud Lake and Monistique.....-. 47 34 47. 267 121 21 6.4 1, —0.335 —0. 286 47 34 46. 646 | Monistique and Fishdaw ...--... 89 40 37. 148 12) 21 5.6 1 —0.335 4-0. 346 89 40 37.159 Fishdam and Burnt Bluff......-. 62 07 43.173 12, 22 4.1 1 —0.335 —0. 464 62 07 42.374 Burnt Blutt and Pine Hill ....... 36 41 19. 933 124 24 » 5.1 To jeeeeeeeeee , +0. 738 36 41 20.671 | Burnt Bluff and Mud Lake ...... 160 36 53.752 | 12445 20 4.8 1 | —0.335 = -;-0. 404 160 36 53. 821 : NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2 (121) +1 (122)-|-1 (123) +1. 340=0 1 (121)-+2 (122) 1 (12a) 1. 340=0 1 (121) 4-1 (122) 4 2 (122) +1. 340=0 FISHDAM—13. (Observer, G. Y. Wisner. Instrument, Troughton & Simms’ 12-inch theodolite No. 2. Date, June, 1874.} } } ! 3 i Angle as measured between— Notation. . No. meas. penne ‘wt. (v) {v] | Corrected angles. ; mee ge fa *. snes fe Simei as Fe shell ec yhl ' oO # ” | “" u" ” ‘ oO 2 “uw i _ Burnt Bluff and Sturgeon -..---- 91 38 39.171 131 20 | 7.0 1 —0. 282 —0, 559 91 38 38.330 | Sturgeon and Monistique -.--.-.. 37 10 03. 299 13g 20 1 25 1 —0.282 +0. 250 37 10 03. 267 Monistique and Burnt Bluff....-- 231 10 18. 376 133 17 5. 6 1 —0. 282 4-0. 309 , 231 10 18. 403 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2 (181) +1 (182) +0. 846=0 1 (181) +2 (132)-+0, 846=0 360 PRIMARY TRIANGULATION. [Cnap. XV, C, SECTION TL Triangulation from the line Vilcean— Huron Mountains to the line Burnt Bluff— Pine Hill—Coutimued. BURNT BLUFI—14. (Observer, RLS. Woodward. Tnstrument, Troughton & Simms’ 14-inch theodolite No.1. Date, June, 1874.] Angle as measured hetween— | Notation. _ No. meas. “Range. Wt. | (v) | {v] | Corrected angles. ° € “we 1 i ¥? : uw * aw O° ‘ wt Boyer's Bhif'and Ford River... 59 24 3%, 666 i 16 =6| 66 | 1 | 40.4901 | 10.038) 59 24 39.190 Ford River and Pine Hill.....--- 62 46 81.875 142 16 "4.4 1 40.491 | +0. 867 52 46 33. 233 Pine Hill and Sturgeon .....----- 39 43 08.725 | 14; 16 88 1 +0.491 | 40.591 39 4:35 09. 807 Stureseon aud Fishdam 2. ..---- 26 13 41.043 , 14, ; 20 ; 6.1 / t,he , tt 694 | 26 13 40. 838 Fishdam and Boyer’s Bluff ..-.-- 181 51 57, 238 145 | 16 | 74 4 +0. 491 | es ee it perme cae a theta Se a ; | } \ NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 214i) | 1142) -4-1(143) | 1(14y) 2. 458 =0 1 (141) -|-2(142) | 1(143) +-1(144) —2. 458=0 1(141) | 1(14)) 4 2(144) 4 1144) 2. 458=0 1(141) | 1(142)-|-1(143)-} 2(14q) —2. 458=0 Norg.—The adjustment of the triangulation was divided at the line Burnt Bluff-Pine Hill. ILence the general corrections to 14) and 142 ave derived from the snececding section of the adjustment. PINE HILL—15. {Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, June, 1874.] Angle as measured between— A Notation. _ No. meas. , Range.| Wt. (v) (v)] Corrected angles. 2 | eee Oo. “ " i “ “uw eo # “ Sturgeon and Burnt Bluff...-..-- 103 35 30, 334 Vy ‘ 20 17 | 1 | 0.000 | -|-0.7388 | 103 35 31.072 | ’ Burnt Bluffand Ford River .... 76 34 15. 669 Lz i 19 81 ; 1 | 0. 000 +1. 083 76 34 16.752 | Numerical equations of condition in the triangulation from the line Vulean— Huron Mountains to the line Burnt Bluff— Pine Hill. SIDE-EQUATIONS. I. (10) — 5.7798 [2421+ 14.0967 (24245) 17. 2425 [35] = + 6.3490 [440] wn -1316(3,] 9 + 11.3037 [4)] — 17,961=0 Il. (30) — 21.4090 [3,] — 38. 6515 [35] — 32,6515 [35] — 11.3012[4,]) + 11.3012 (4)49] + 17.6502 [4,] + 31.8992 [5,] — 20,3858 [52] + 57.162=0 LD. (500) 4+607.7145 [22] + 63.0998 [2142] 9 — 63.0998 [24045] — 49.1339 [4142] — 49. 1433 [45] + G.NBdN [5] + 6.8348 [52] — 6,5090-[55] +929. 1941 [8] +840. 0884 [83] +840. 084 [5y] —840. 0884 [8:4s4546] —1149, 813=0 IV. (20) + 13.5068 [3,]) — 5.8369 [33] + 1.4364 [5s] + 1.4364[5,] — 30.0170([5,] — 19.0931 [6] — 15.2188 [62] + 5.717=0 V. (40) 4 61.2899 [3,] + 1.4296 [32] + 1.4296[3,] + 6.5090[5;] — 89,2341 [54] — 27.9221 [72] — 13.9421 [75] — 108. 276=0 Notr.—In the solution for determining the general corrections, each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. ANGLE-EQUATIONS. VIL [hi] + (le) + [ls] + [242] + [35] — 1.427=0 VI. (11) + (lJ + (35) + [85] + [4:42] — [4.] — 1.888=0 VII. [1s] + [2i4243] + [4] 0. 516=0 § 6.) VULCAN-HURON MOUNTAINS TO FOND DU LAG BASE. Numerical equations of condition, &e.—Continued. ANGLE-EQUATIONS—Continued. IX. (1) + [8,)] + [35] + [86] + [52] X. {h] + Lite] — [4] + [43] + [ir] XI. (3) + [33] + [33] + [3] + [85] XU. [2%] + (&] — [8] — [35] ~— [8.] —[35] 183] — (8) — £85] + [8pits46] XU. [33] + [5s] + [54] + [5s] + [6] XIV. [3:1 + [33] + [53] + [54] + [72] XV. [3:2] + [62] + (7%) + [7%] XVI [3] + (%] + [8] + [85] XVIL. (74) + [8e43] + [9] XVI. [&] + L%] + [10] XIX. [95] + [10] + [11s] XX. [9%J + [11:] + [12] Xk. Dh + fia reg XXII [12] + [13%] + [144] XXII. [12] + [14s] + [151] General corrections in terms of the correlates. {hj =-+0, 30471 VI +0.38750 VIL —0. 08279 VITI+0. 51132 IX —0. 12382 X [12] =-40. 00428 VI +0.08016 VII —0. 07588 VITI—0.128821X +0. 20398 X [13] =-+0. 00399 VI —0.15867 VII +0. 16266 VIII—0. 08279 TX —0. 07588 X {14] =—0.00714 VI +0. 03382 VII —0. 04096 VIII+0. 042451X —0. 00863 X L21+42] =—0. 57798 I +0. 12620 1II +1.00000 VI [22] = +1.21543 TTI +1. 00000 XIT [2i4e+3] =-+1. 40967 I —0.12620TII +1. 00000 VITI [31] =—0. 05250 I +0. 03062 IT —0.04200IV +0.64991 VV +0.01149 VI —0. 01687 VII +0. 21024 XI —0.16943 XII —0. 10951 XIII—0, 21902 X1V—0. 10951 XV -++0, 42926 XVI [32] =-+0. 00144 I +-0. 00217 IT +0.512471V —0.16242V +0.00252 VI +0. 00383 VII +0. 04105 XI —0.03240 XII —0. 42472 XTII+0. 15056 XIV+0. 57528 XV —0. 10951 XVI [3s] =-+0. 001441 +0. 00217 II —0.454731V —0.16242 VV +0.00253 VI +0. 00383 VII -+0. 04105 XI —0.03240 XII +0,.57528 XIII+0. 15056 XIV—O. 42472 XV —0. 10951 XVI [34] =—0. 60021 I +0. 19394 IT —0. 00382 1V —0.01979V —0.01666 VI —0. 36802 VII —0. 03235 XI —0.00915 XII —0. 00995 XITI—0. 01990 XIV—0. 00995 XV —0. 01245 XVI [35] =-+1. 11381 1 —0. 40423 IT +0.000501V —0.04336V —0.11654 VI +0, 50826 VII —0. 02576 XI —0.247e8 XII +0. 00130 XITII+0. 00260 XIV+0. 00130 XV —0. 02836 XVI [36] =—0. 39765 I —0. 64709 II +0.000971IV +0.01779V +0.62813 VI +0.51159 VII +0. 01655 XI +0.11685 XII +0. 00253 XITI+0. 00506 XIV+0. 00253 XV +0. 01149 XVI [41] =-+0. $9910 I —0.192661I —0. 01787 II] —0. 71428 VII +0. 76623 VIII —0. 58441 X [4142] =+0. 545201 —0. 07476 II —0. 01787 III +0.71428 VII +0. 05195 VIII +-0. 12987 X [43] =—0, 22424 I +0. 34218 II —0. 04469 III —0. 71428 VII +0. 12987 VIII +-0. 32467 X [51] =+1.002051II -+0.0045611I —0.33333 1X +0. 66667 X [52] =—0.94079 II +0.0045611I +0. 66667 1X —0, 33333 X [5s] =—0. 00916 III -+0.169881V +0.77673V +0.70313 XI +0.31250 XIII +0. 40625 XIV [54] =+0. 00387 III +0.16988IV —1.61695V —0. 29688 XI +0. 31250 XIII +0. 40625 XIV [55] =+0. 00122 1II —1.23291IV +0.19390V —0.09275 XI +0. 62500 XIII —0. 18750 XIV [6] =—0. 38279IV +0. 66667 XIII —0. 33333 XV [62] .=—0.18907IV —0.33333 XIII +0. 66667 XV [71] = +1.00000 XV [72] =—0.69805V -+1.00000 XIV +1.00000XV 46 LS o61 2. 013=0 0. 648=0 0. 383=0 0. 140—0 0. 636=0 2. 300=0 2. 976=0 0. 300=0 0. 246=0 0. 044=0 0. 231=0 1.115=0 0. 662=0 1.717=0 2.06 =0 +++ +44 44 + —0. 02932 IX — 0. 00612 IX —0. 90612 IX +40, 02464 IX +0. 15710 IX +0. 49473 IX 362 [75] [74] [e] [82+] [83] [4] [83] PRIMARY TRIANGULATION, LCirap. XV, C, General corrections in terms of the correlates—Continued. =—0, 34855 V =—0.09143 IIT —0, 07042 XI =—0.09143 IIT —0, 07042 XI =+0. 30405 IIT —0, 19718 XI =-+0, 10290 III —0, 35212 XI =-10. 28111 IIT (S3-4445+6 ]=— 1. 68018 ITT (9]J [92] [9s] [94] (10,] [102] [11] (112) (11s] (121] [122] [12s] [124] (13)] [132] [145] [144] [15.) No. of equa- tion. 1. =} +0, 64789 XI +1. 00000 XVI +0.50000 XVIT +0. 04604 XIL —0, 14084 X VI-0. 34742 XVII+0. 65258 XVIII +0. 04694 XIL —O. 14084 XVI+0. 65258 X VIT—0. 34742 XVIII —0, 20188 XII —0. 39436 XVI+0. 09390 XVII+0. 09390 XVITI —0. 09859 XIL +0. 29577 XVI—0. 07042 XVII—0. 07042 XVIII —0. 09859 XII +0. 29577 XVI—0. 07042 X VII-—0. 07042 XVIII +1. 00000 XIT ll +1. 60000 XVIT +0. 75000 X VIII—0. 25000 XIX —0. 25000 XX —0. 25000 XVIII+ 0. 75000 XIX —0. 25000 XX —0. 25000 X VIIT— 0. 25000 XIX + 0.75000 XX +0. 50000 XIX +1. 00000 XVIII —0. 12500 XIX —0. 12500 XX +0. 62500 XXI —0. 37500 XIX +0. 62500 XX7—0. 12500 XXI +0. 62500 XIX —0. 37500 XX —0. 12500 XXI +0. 75000 XX —0. 25000 XX —0. 25000 XX —0. 25000 XXI —0. 25000 XXIT +0. 75000 XXI —0, 25000 XXII —0. 25000 XXI +0. 75000 XXII +1. 00000 XXIII —0, 32333 XXI -+-0. 66667 XXIT +0. 66667 XXI —0. 33333 XXII +40, 80000 XXIII +0. 80000 XXIT Normal equations for determining the correlates. =—1. 79610 +45. 72871 I —0, 75959 IL +1. 00000 XXII —0. 28239 III +40.00055 IV = —0, 08033 V —0. 97563 VI +0, 36226 VIE +2. 30877 VIII +0.11595 IX —0.57814 Xx —0.04960 XI —0. 46400 XII +0,00145 XIII +0. 00289 XIV +0.00145 XV —0. 05250 XVI 2, 0=+1.90542 —0. 75959 I -13.35478 IL —0.02545 111 +0,000841V -++0.04706 V —0. 64709 VI —0. 93342 VII —0. 19266 VII] —1.798171IX +41. 46213 X +0,03496 XI +0, 17534 XII +0. 00217 XIII +0. 00433 XIV +0.00217 XV +0. 03062 XVI 3. 0=—2, 29963 —0. 28239 I —0.02545 11 +5.54467 IIT —0.00221IV —0.0J011 V +0. 12620 VI —0,14407 VIII -+0.00456IX --9.04013X +40.27195 XI -—1.15281 XII. = —0, 00407 XIII —0, 00529 XIV -+0. 38401 XVI —0. 09143 XVII —0. 09143 XVIII 4. 0=-10. 28585 +0, 00055 I +40, 00084 II —0, 00221111 +42.86291IV —0,41364 V +0, 00097 VI +0,00147 VIL = —0.002351X +0.18562 XI —0,01242 XII —1.73067 XIII +0.39750 XIV +0, 32340 XV —0, 04200 XVI A. 0=—2. 70690 —0. 08033 I +0.04706 11 —0.01011 IIT —0.41364IV +5. 32671 V +0. 01779 VI —0, 02557 VIE —0.04536 IX +1.10180 XI —0.26192 XII —0.80874 XIII —1.86311 XIV —0,86047 XV +0. 30136 XVI 6. 0=—1. 42700 —0. 97563 I —0.64709 II +0.12620 TIT +0.00097IV. = 4.0, 01779 V 4+1.94111 VI +0. 82058 VII 0.00399 VIII +0.79944IX +40. 00428 X +0.01655 XI +40, 11685 XII +0. 00253 XIII -++0.00506 XIV +0.00253 XV 0.01149 XVI 0=—1. 88800 +0. 36226 I —0.93342 IT +0.00147 IV —0. 02557 V +0. 82058 VI +2. 91607 VII —0. 87295 VIIL +1.03933IX -+0.79444X —0.00921XI —0.13103 XII +0. 00383 XIII 0.00766 XIV +10, 00383 XV —0.01687 XVI §6.] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 363 Normal equations for determining the correlates—Continued. No. of equa- tion. 8 0=—0.51600 +2. 30877 I —0,19266 IT —0.14407 IIIT +0.00399 VI —0,87295 VII +1. 92889 VIII —0. 08279 IX —0. 66029 X 9. 0=—2. 01300 +10. 11595 I —1.79817 II +0. 00456 III —0. 00235 IV —0. 04536 V +0. 79944 VI +1. 03933 VII —0.08279 VIII +1.85446IX —0, 45715 X —0.04156 XI —0.14018 XII —0. 00612 XIII —0. 01224 XIV —0. 00612 XV —0, 02932 XVI 10. 0=-+0. 64800 —0. 57814 I +1.46213 IT = —0.04013 IIE +0.00428 VI +0.79444 VII —0, 66029 VIII —0. 45715 IX +1. 90960 X 11. 0=-+40. 38300 —0. 04960 I +0.03496 IT -+40.27195 IIT =+-0.18562 IVs +1. 10180 V +0. 01655 VI —0, 00921 VII —0.04156IX +1.64336 XI —0.33282 XII +0. 35355 XIII +0. 4-835 XIV +0.04105 XV +.0, 50601 XVI —0. 07042 XVII —0. 07042 XVIII 12. 0=-+0. 14000 —0. 46400 I -+0.17534 II —1,15281 III —0.012421V —0. 26192 V +0. 11685 VI —0.13103 VII —0,14018IX —0,33282XI +2.89032 XII —0.03240 XIII —0. 06480 XIV —0. 03240 XV —0, 36661 XVI -++0. 04694 XVII +40. 04694 XVIII 13. 0=-+0. 63600 +-0. 00145 I +0.00217 IT —0.00407 ITI —1.730671IV —0.80874 V +0. 00253 VI +0. 00383 VII —0.0061I2IX -+0.35355 XI —0.03240 XII +2.49195 XIII +0. 77556 XIV —0.75805 XV —0. 10951 XVI 14. 0=-+2. 30000 +40. 00289 I +0. 00433 IT —0.00529 TTI +0,397501IV —1.86311 V +40. 00506 VI +0.00766 VIL —0.01224IX +0.48835 XI —0.06480 XII -+0.77556 XIII +2. 11362 XIV +1.15056 KV —0.21902 XVI 15. 0=-+2. 97600 +-0. 00145 I +0.00217 II +0,323401V —0.86047 V +0. 00253 VI -++.0. 00383 VII —0, 00612 IX +0. 04105 XI —0.03240 XIT_ —0.75805 XIII +1.15056 XIV +3. 24195 XV —0. 10951 XVI 16. 0=-+0. 30000 —0. 05250 I +0.03062 IT +0.384011JT —0.04200IV -+0.30136 V +0. 01149 VI —0,01687 VII —0,02932IX -+0.50601 XI —0.36661 XII —0.10951 XIII —0.21902 XIV —0.10951 XV 4-2, 02080 XVI —0. 14084 XVII —0. 14084 XVIII 17. 0=-+0. 24660 —0.09143 II] —0.07042 XI -+0,04694 XII —0.14084 XVI +2. 15258 XVII —0.34742 XVIII 18. 0=+0. 04400 —0.09143 IIT —0.07042 XI +0.04694 XIT —0.14084 XVI —0,.34742 XVII +2. 40258 XVIII —0. 25000 XIX —0, 25000 XX 19. 0=—0. 23100 —0, 25000 XVIII +-1.87500 XIX —0. 62500 XX —0. 12500 XXI 20. O=+1. 11500 —0. 25000 XVIII —0. 62500 XIX +2.12500 XX —0.37500 XXI —0. 25000 XXII 21. 0=—0. 66200 —0. 12500 XIX —0.37500 XX +2. 04167 XXI —0.58333 XXII 22. 0=+1.71700 —0. 25000 XX —0.58333 XXI +2. 21667 XXII 23. 0=—2. 06700 +2, 80000 XXIII Values of the correlates and their logarithms. I =+0. 6352 log 9. 80291054 II =+0. 4550 log 9. 65801144 IIL =+0. 4831 log 9. 68403704 IV =—0. 4314 log 9. 6348801_ V =+0. 3081 log 9. 48869174 VI =+0. 6976 log 9. 84360654 VIL =-+40. 1938 log 9. 28735384 XIII =—0. 7969 log 9. 9014038_ XIV =+0. 1349 log 9. 13001194 XV =—1. 0330 log 0, 0141003_ XVI =—0. 3003 log 9. 4775553_ XVII =—0. 1446 log 9. 16016383_ XVIII =-— 0.1324 log 9. 1218880_ XIX =—0, 1285 log 9, 1089031_ VIII =—0. 4556 log 9. 6585837_ IX =+0. 9415 log 9. 97382034 X =—0. 4996 log 9. 6986224_ XI =—0.1651 log 9. 2177471_ XII =-+0. 2008 log 9. 30276374 XX =—0, 6905 log 9. 8391637_ XXI =—0. 0584 log 8. 7664128_ XXII =—0. 8678 log 9. 9384196_ XXIII =-10. 7382 log 1), 86817404 364 PRIMARY TRIANGULATION. [CHap. XV, C, Values of the general corrections. “uw “ “ [lh] =-+0.869 [5.] =+0. 368 [%] =+0. 105 [lk] =-0.165 [53] =—0.149 [93] =—0. 109 [15] =—0.142 [54] =--0.715 [94] =—0. 453 (lu) = +40. 065 [5s] =+0. 084 [10,] =—0. 064 [2142] =+0.391 [61] =—0, 022 [10,] =—0. 132 [2] =+0.788 [6] ——0. 342 [11,1] =-L0. 066 (2i4248] =-+0. 379 [71] =—1. 033 [112] =—0. 376 [3] =+0.149 [72] =—1.113 [11s] =+0. 186 [3] =—0.488 [7s] =—0, 408 [12] =—0. 286 [33] =-+0. 165 [74] =—0.072 [12:] =-10. 346 [34] =—0.335 [81] =—0. 017 [123] =—0. 464 [33] =-10. 636 [843] =—0.029 [12,] =-+0. 738 [33] =-+0.475 [83] =+0, 231 [13:] =—0. 559 [4] =+0.279 [84] =+0. 019 [132] =-+0. 250 [dae] = =+0. 353 [8s] =—0. 060 [143] =-+0. 591 [44] =—0.368 [Ss444546] =—0. 611 [144] =—0. 694 [5:1] =—0.189 [9] =+0, 145 [151] =+0. 738 Residuals resulting from substitution of general corrections in numerical equations of condition. sew Residual. ! é ee. Residual. | 1 +40. 0010 | 13 —0. 0001 2 —0.0060 |! 4 —0. 0001 3 +0.0500 |) 15 —0. 0001 4 +0.0020 = |, 16 0. 0000 5 —0. 0040 WwW —0. 0002 6 40. 0004 18 —0. 0001 7 -+0. 0003 19 —0. 0001 8 +0. 0001 20 | ~0, 0001 9 0. 0000 2 ©, 40.0001 10 —0.0001 | 22 | +0. 0001 1 —0. 0001 23 +40. 0000 12 +0. 0000 SECTION 1V.—Triangulation from the line Burnt Bluff— Pine Hill to the line Eldorado - Taycheedah. FORD RIVER—16. [Observers, A. R. Flint and G. ¥. Wisner. Instruments, Repsold 10-inch theodolite, Troughton & Simms’ 14-inch theodolite. Dates, June and July, 1874.) Angle as measured between— Notation. | No. meas. Range. ; Wt. (v) (v] Corrected aagiea| ——~ —— a oi: “a “a mi a “ oe “a | Pine Hill and Burnt Bluff........ 50 39 10. 853 161 16 5.1 1 —0. 225 | +-0. 880 50 39 11.508 Burnt Bluff and Boyer's Bluff.... 66 17 33.377 162 15 Be: 1 +0.320 | +0, 074 66 17 33.771 Boyer’s Bluff and Cedar River... 55 17 15.508 163 16 4.7 1 -+0. 320 —0. 546 55 17 15. 282 Cedar River and Azimuth-Mark. 187 04 00.321 164 16 48 1. —0. 225 | —0. 204 187 03 59, 892 | Cedar River and Burnt Bluff .... 238 25 09.385 16—2—3 10 6.6 0.5 +1. 090 +0. 472 238 25 10. 947 Azimutb-Mark and Pine Hili.... 0 41 59.975 165 16 5.6 | 1 —0. 224 | —0. 204 0 41 59. 547 2 ff NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(161)-+-1. 0(162) +1. 0(163) --1(164) +0. 084=0 1(161)+-2. 5(162) -+1. 5(163) +-1(164) —0. 881 =0 1(161) +1. 0(162) +2. 5(163)+-1(164) —0. 831 =0 1(161) +1. 0(162)-++1. 0(163) +2(164) +0. 0834—=0 NotE.—Angles 162, 163, and 16-23 were read by A. R. Flint with Repsold instrument. § 6.] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 365 SECTION 1V.—Triangulation from the line Burnt Bluff- Pine Hill to the line Eldorado - Taycheedah— Continued. BOYER’S BLUFF—17. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Dates, June and July, 1874.) Angle as measured between— | Notation. No.meas. | Range. | Wt. | (v) fo} Corrected angles. iB , l ay a i oF ” | “ | : ” u : Oo. 2 “uw | i : ; Door Bluff and Eagle Bluff -....-. 3 58 07. 984 171 12 3.6 | 08 , 0.000 +-1.429 3 58 09. 413 Door Bluff and Cedar River . 55 56 57. 640 17i+2 20 [S620 uke a » 0.126 +0.166 55 56 57. 680 | Cedar River and Ford River ..... 64 11 43.751 | 173 20 5 1 —0.126 | —0. 393 64 11 43. 232 | : Burnt Bluff and Door Bluff ....-- 185 33 29. 802 175 ‘ 20 | 7.0 1 0.000 | —0. 042 185 33 29.760 8.0 1 Ford River and Door Bluff ....-. 239 5118.987 Itats | 20 i —0.126 4-0. 227 239 51 19.088 | 2 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(171-42)-+1(172) +0. 378=0 1(171-42)-++2(172) +0. 378=0 CEDAR RIVER—18. {Observer, R.S. Woodward. Instrument, Troughton & Simms’ 12-inch theodolite No.2. Date, July, 1874.] > ; Angle as measured between— | Notation. No.meas. Range. Wt. | (v) (v] Corrected angles. i | °o t ua , ra | aw “a ° # we Ford River and Boyer’s Bluff.... 60 31 03.300 | 181 ' 20 12.4 1 +1.061 | —0. 623 60 31 03. 738 ee I Boyer’s Bluff and Door Bluff... -. 33 57 51. 840 182 | 20 8.8 1 +1.061 | +0. 081 33 57 52. 982 Door Bluff and Eagle Bluff ..... - 40 31 28.188 183 . 20 6.3 1 +1.061 | +0. 283 40 31 29. 532 Rocbereau and Ford River .....- 181 29 15.115» 18 | 20 97 | 1 +0.000 | —0. 303 181 29 14. 812 © Eagle Bluff and Ford River...... 224 59 32. 427 | 184+5 20 11.6 1 +1. 062 | +0. 259 -224 59 33.748 i 1 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(181) +-1(182) -+-1(183) —4. 245=0 1(181)-F 2(182) + 1(182) —4. 245—0 1(181) + 1(182) -+ 2(183) —4. 245=0 DOOR BLUFF —19. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1874.) Angle as measured between— Notation. No. meas. | Range. wt. (») {v] ‘Corrected angles. ah | | fe oO t “a | aw “a Ww oO aw Eagle Bluff and Rochereau ..... 46 11 07.120 19 19 i 68 . 1 —0 357 | —0. 466 46 11 06. 297 Rochereau and Uedar River...-.-. 36 18 22.123 192 20 | 45 1 1 | —-0. 357 | +9. 565 36 18 22.331 Cedar River and Boyer’s Bluff... 90 05 10. 883 - 193 20 ree 1 | —0. 357 | —0. 094 90 05 10. 432 Boyer’s Bluff and Eagle Bluff ... 187 25 20. 888 : 194 19 | 5.7 1 -++0. 057 | —0. 005 187 25 20. 940 Eagle Bluff and Boyer’s Bluff. ... 172 34 38.230 | 1914243 10 | 4.9 0.5 +0. 825 | +0. 005 172 34 39. 060 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(191)---1. 5(192)-1. 5(193) +1. 962=0 1. 5(191)-+ 2. 5(192) +1. 5(193) 4-1. 962=0 1. 5(191) 4-1. 5(192) +-2. 5(192) -[-1. 962=0 366 PRIMARY TRIANGULATION. [Cuar. XV, C, SECTION [V.—Triangulation from the line Burnt Bluff- Pine Hill to the line Eldorado — Taycheedah— Continued. ROCHEREAU—20. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1874.) Angle as measured between— | Notation. | No. meas. | Range. | Wt. (v) | (v] Corrected angles, oO t uw | uw a“ uw ° t “ Cedar River and Door Blutf...... 59 39 49.775 201 | 10 5.9 0.5 | —0.206 | +0. 740 59 39 50.309 Door Bluff and Eagle Bluff ...... 42 14 14.191 202 | 1 6.6 0.5 | —0.206 | —0. 442 42 14 13. 543 Eagle Bluff and South Egg .....- 37 03 14.150 | 203 12 7.7 0.5 | —0.693 | +0, 922 37 03 14.379 South Egg and Menomonee...... 39 12 56.175 203 12 7.1 0.5 | —0.693 | —1. 617 89 12 53. 865 Cedar River and Eagle Bluff .... 101 54 03. 631 20142 10 5.7 0.5 | —0.077 | +-0. 298 101 54 03, 852 Eagle Bluff and Menomonee. ..-. 76 16 08.529 20344 ' 12 TA 0.5) -+-0.410 | —0. 695 76 16 08. 244 Menomonee and Cedar River .... 181 49 47. 647 205 i 20 6.5 1.0 | —0.140 | +0. 397 181 49 47.904 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 0(20,) +1. 5(202) -+1. 0(203)-+1. 0(204) +2. 10550 1. 5(201)-+2. 0(202) +1. 0(203)-+1. 0(204) +2. 1055= 1. 0(203)-+1. 0(20,) -+2. 0(20s) +1. 5(204) +2. 8360=0 1. 0(20,)-+1. 0(20,) +1. 5(203) +-2. 0(20,) +2. 8360=0 EAGLE BLUFF—21, (Observer, R. S. Woodward. Instrument, Troughton & Simms’ 12-inch theodolite. Date, July, 1874.] Angle as measured between— Notation. No. meas. | Range.| Wt. (v) | (v] Corrected angles. oO a aw “un “uw a“ ° 1 aw South Egg and Menomonee ...-.. 47 48 5,515 21 19 6.3 1 +0. 212 | —0. 197 47 48 5,530 Menomonee and Rochereau...-... 59 43 41.413 212 20 11.8 1 +0, 211 | —0. 297 59 43 41, 327 | Rochereau and Cedar River...... 34 35 38.198 213 20 6.3 1 +0. 212 | —0. 292 34 35 38.118 | Cedar River and Door Bluff...... 56 59 02. 546 21445 20 7.9 1 +0. 211 | +0. 330 56 59 03. 087 Boyer’s Bluff and Door Bluff. .-.. 3 27 14. 093 215 9 3.3 0.5 0.000 | —2. 456 | 3 27 1). 637 | Door Bluff and South Egg ....... 160 53 31,270 | 216 25 15.0 1 40.212 | +0. 456 160 53 31. 938 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(211)-+1(212) +1 (213) +1(21445)—1. 058=0 1 (211) + 2(212) 4-1(213) +-1(21445) —1. 058=0 1(211) +-1(212) +-2(213) +-1(21445)—1. 058=0 1(211)+-1 (212) +-1(213) +-2(214-4+5) —1. 058=0 MENOMONEE—22. [Observer, R. S. Woodward. Instrument, Troughton & Simms’ 14-inch theodolite No. 1. Date, June, 1874.] Angle as measured between— | Notation. | No. meas. | Range.| Wt. | (v) | {v] Corrected angles. ‘ | Oo. & a I “ “uw Oo ay Rochereau and Eagle Bluff....... 44 00 11. 966 221 | 16 3.8 1 +0. 039 | —0. 220 44 00 11.785 Eagle Bluff and South Egg ...-.. 47 44 26, 542 | 222 | 16 | 5.1 1 | +0. 039 | —0. 966 47 44 25.615 South Egg and Débroux ..... - 62 48 53.615 \ 223 : 16 4.8 1 +0. 039 | —0. 013 62 48 53, 641 Débroux and Peshtigo .... - 11 31 52.860» 224 16 8.1 1 +0. 039 | +0. 424 11 31 53. 323 Peshtigo and Rochereau.......... 193 54 34, 822 | 225 | 20 6.8 1 +0.039 | 40.775 193 54 35. 636 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(221) +-1(222) +1(223) +1 (224)--0. 195=0 1(221) +2(222) +1(22s) -+1(224)—0. 195=0. 1(221) +-1(222) +2(223) +.1(224) —0. 195=0 1(221)+-1(222) +1(223)-+-2(224)- 0, 195=0 VULCAN- HURON MOUNTAINS TO FOND DU LAC BASE. Continued. SOUTH EGG—23. 367 SEcrion IV.—TZriangulation from the line Burnt Bluff- Pine Hill to the line Eldorado — Taycheedah— (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1874.] Angle as measured between — Notation. No. meas. | Range. | Wt. (v) {v] Corrected angles. ° # wt | aw aw “a oO ‘ uw Débroux and Peshtigo ........... 38 24 18,808 | 231 20 6.5 i. -+0. 033 | +0, 029 38 24 18. 870 Peshtigo and Menomonee...... — 380 51 58,738 232 20 ; 5.6 1 4-0. 033 | —0. 035 30 51 58. 736 -| Menomonee and Rochereau...... 49 02 30. 448 | 233 20 7.8 1 -+0.018 | —0. 280 49 02 30.186 aochereau and Eagle Bluff....... 35 25 00.239 234 12 4.6 0.5 | -+0.037 | —0. 398 35 24 59, 878 Menomonee and Eagle Bluff ..... 84 27 30.713 233-+4 8 5.5 0.5; -+-0.029 | —0. 678 84 27 30. 064 Eagle Bluff and Débroux ........ 206 16 11. 614 236 18 4 54 +0. 032 | -+0. 684 206 16 12.330 ! h NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(231) +-1(232)-+1 (283)+1 (234) —0. 153—0 1(23)+2(232)-++1 (283)+1 (234) —0. 153=0 1(281) +1(232)-+2. 5(233) +1. 5(234) 0. 166=0 1(281) +1(232) +1. 5(233)-+2 (234) —0. 166—=0 PESHTIGO—24. (Observer, A.R. Flint. Instrument, Repsold 10-inch theodolite. Date, May, 1874.) Angle as measured between— Notation. No. meas. | Range. | Wt. (v) {v] | Corrected angles. Oo Ww “a a aw Oo t a“ Menomonee and South Egg ...... 74 47 15.143 241 20 6.4 1 —0. 004 | - -0. 216 74 47 14.923 South Egg and Débroux ......... 85 30 00. 708 240 20 8.3 1 —0. 004 | —0. 048 85 30 00. 656 Débroux and Red River ......... 15 56 16. 302 243 20 4.4 1 —0.311 | +0.778 15 56 16. 769 Débroux and Gales .......--..--. 56 54 36. 578 24344 20 4.8 1 +-0. 307 | —0. 849 56 54 36.036 | Red River and Menomonee ...... 183 46 28. 7&9 2412-3 14 5.2 0.5 | —0.623 | —0. 514 183 46 27. 652 Gales and Menomonee ....---..-. 142 48 06. 658 245 11 4.7 0.5 +0. 614 | 41.113 142 48 08. 385 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. (Observer, R. S. Woodward. 2 (241)4+1 (242)-+0. 5(243)+-0. 5(24344)-+-0. 0145=0 1 (241)+2 (242)+-0. 5(243)+-0. 5(243+4)-++0. 0145—0 0. 5(241) +0. 5(242)-+1. 5(243) +0. 4710=0 0. 5(241) +0. 5(242) + +1. 5(243+4)—0. 4565=0 DEBROUX—25. ~ Instrument, Troughton & Simms’ 14-inch theodolite No.1. Date, May, 1874.] : Angle as measured between— Notation. No. meas. | Range.| Wt. | (v) {[v] | Corrected angles. °o ww a | “wn “ oO od “ Gales and Peshtigo -..----------- 79 16 11. 664 25, 16 65 / 1 —0. 291 | —0. 051 TY 16 11. 322 Peshtigo and Menomonee ....... 8 10 51. 244 25 15 3.4 1 0.000 | +0. 017 8 10 51. 261 Peshtigo and South Egg.-.....--- 56 05 41.937 Bete 16 5.1) 1 —0. 291 | —0. 268 56 05 41.378 ‘| South Egg and Gales ....-..-.--- 224 38 07. 273 254 16 3.0 1 | —0. 292 | -+0. 319 224 38 07.300 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2 (251) +1 (25,43) +0. 874=0 1 (25) +-2 (252-+42)-+0. 874=0 368 PRIMARY TRIANGULATION. [Cuar, XV, C, SEcTION IV.— Triangulation from the line Burnt Bluff- Pine Hill to the line Eldorado - Taycheedah— Continued. GALES—26. [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite, Date, September, 1873.] Angle as measured between— Notation. | No. meas. | Range. | Wt. | (v) fp] Corrected angles. ; “ is eh) te 7 aoe ov uu i “we | “" | “uw Oe “ Peshtigo and Débroux......----. 43 49 13. 840 | 26; | 20 5.1 1 40.018 | —0. 409 43 49 13. 449 | Débroux and Red River ..-.-..-.-. 32 15 59. 106 265 | 20 6.1 | 1 +0.017 | —0. 039 | 32 15 59. 084 | Red River and Red Banks.. 54 10 34.377 263 ‘ 20 67 | 1 0. 000 | --0. 864 | 54 10 35. 241 : Red River and Little Tail ....... 82 26 20. 882 263+4 17 62/141 -+-0.018 | —0. 547 82 26 20. 353 | Little Tai\ and Peshtigo ........- 201 28 26. 102 265 20 | 6.7 1 -|-0 017 | 4-0. 995 201 28 27.114 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2 (26,)4-1 (262)4-1 (263+4)—0.070=0 1 (261)+2 (262)-+-1 (263+4)—0.070=0 1 (261)+-1 (269)-+-2 (2634+-4)—0.070=0 RED RIVER—27. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, October, 1873.] : | Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {vl Corrected angles.' ! 5 ee | pee ee ee | o ” | ” " ” : o 7 ” Red Banks and Little Tail ...... 30 45 05. 680 ; 21 | 10 6.8 0.5 4-0.463 | +0. 825 30 45 06. 968 Little Tail and Gales, .-...--..--- 55 51 59. 675 272 10 4.2 0.5 | +0.463 , —0. 839 55 51 59. 299 Gales and Peshtigo ..-..-..-.---- 62 56 29. 510 273 20 7.0 1 | +0.177 | —0. 510 62 56 29.177 Red Banks and Gales ..-..------- 86 37 06.390 27142 10 7.2 0.5; —0.109 | —0. 013 86 37 06. 268 Peshtigo and Red Banks .....-.. 210 26 23. 856 274 20 6.8 1 | +0.176 | +0. 523 210 26 24.555 { | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 0(271) +1. 5(272) +1(273)—1. 7965=0 1. 5(271) 4-2. 0(272) 4+-1(273) —1. 7965=0 1. 0(271) +1. 0(272) 4-2(273)—1, 2790=0 LITTLE TAIL—2s. (Observer, R.S. Woodward. Instrument, Gambey 10-inch repeating theodolite No.1. Date, September, 1873.] Angle as measured between— Notation. No. meas. | Range. | Wt. (v) (v] Corrected angles. | or aw | “ ww . “uw : or uw ; Gales and Red River......-...--. 41 41 41.155 28) 16 | 3. 6 1 |, +0. 290 | —0. 183 41 41 41. 262 | ' Red River and Red Banks ....-.. 59 23 47.595 282 16 vo 249 a | +0. 290 : —0. 140 59 23 47.745 | , Red Banks and Bruce.....--..--- 27 27 27.537 282 16 | 5.0 1 | +0.290 , —0. 463 | 97 27 27. 364 Bruce and Long Tail P't L't-House 19 00 08.437 | 284 4 | 27 | @8* 0.980 | 0.501 | 19 00 08. 875 Long Tail Point Light-House and ! | | Fort Howard ............--.--- 14 58 22.308 285 3 4 33° 062 | +0.949 | +0.516 | 14 58 23.773 | | Bruce and Fort Howard ......... 33 58 32.542 | Wits | 16 2.7 1 | +0.101| +0.005| 33 58 32. 648 | Fort Howard and Gales........-- 197 28 29.911 | 286 | 16 39 | 1 | +0.289° 40.781 197 28 30.981 | | i I | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 7, 2(281) +1(282)-1(283)+4-1 (284)4-1 (285) —3. 057=0 1(281) +-2(282)-+1(283)+1 (281)+1 (285)—3. 057=0 1(281) 4-1(282)-+-2(283)+1 (284)-+-1 (285)—3. 057=0 1(281) +-1(282)+-1(283) +2. 2(284)-+2 (285)—4. 854—0 1(281) +1(28,)+1(283)+2 (284)-+2. 2(285)-—-4. 854-0 §6.] VULCAN-HURON MOUNTAINS TO FOND DU LAG BASE. 369 SECTION 1V.—Triangulation from the line Burnt Bluff— Pine Hill to the line Eldorado — Taycheedah— Continued. RED BANKS—29. {Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, October, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v) Corrected angles. neeee eee enti, tne ected fet eae a 1 6 a “a | “a awn wt fo} t we | Fort Howard and Little Tail. .... 79 31 42. 980 29142 | 10 3.2 0.5 | +4-0.251 | —0. 441 79 81 42. 790 | Long Tail Point Light-House and | Little Vail o.cores sassacg oahes 50 09 30. 819 292 | 8 4.4 0.5 -+-0. 307 | —0. 599 50 09 30. 527 Little Tail and Gales ............ 50 38 46.750 293 i 10 a7 0.5 | —0.043 | --0, 137 50 38 46.570 ° | Gales and Red River ............ 39 12 19. 686 294 11 5.8 0.5 —0. 043 | —0. 202 39 12 19. 441 | Red River and Court-House...... 187 05 46. 092 295 20 5.5 1 +0. 275 | +0. 411 187 05 46.778 | Little Tait and Red River. .....-. 89 51 05. 756 293+4 11 6.8 0.5 +0.594 | —0. 339 89 51 06.011 Court-House and Long Tail Point i | ‘ Light-House ..............22... 32 53 35. 850 296+1 | 8 4.9 | 0.5 +0. 307 | +0. 527 32 53 36. 684 | Court-House and Little Tail.... 83 03 07.290 296+1+4+2 | 12 7.3 | 0.5 | —0.007 | —0. 072 83 03 07. 211 Court-House and Fort Howard -. 3 31 23.800 296 | 10 4.6 \ 0.5 | +0. 252 | +0. 369 3 81 24.421 1 i ‘ NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1 (29)-+) +40. 5(293) +0. 5(294) +0. 5(295) —0. 3460=0 4-1 (292) +0. 5(292)-+-0. 5(294)+0. 5(295)—0. 4015=0 0. 5(29;-+2)-+0. 5(292) +2, 5(293) +2. 0(294)-+1. 5 (295) —0. 4985=0 0. 5(29)-42) +-0. 5(292) +2. 0(292)-+2, 5(294) +1. 5(295)—0. 4985=0 0. 5 (29-42) +0. 5(292) +1. 5(293) +1. 5(294)-+2. 5(29s) —0. 8385=0 BRUCE—31. [Observers, A. R. Flint and R.S. Woodward. Instruments, Repsold 10-inch theodolite, and Gambey 10-inch repeating theodolite. Dates, July, 1872, and October, 1873. ] Angle as measured between— Notation. No. meas. | Range. | Wt. (v) {v] I taneenbaa angles. ov “ | “ I u “u ow “" _ East Depere and West Depere... 32 44 08.500 3li 4 2.7 | 0.2 | —0.646 | —0.155 32 44 07. 699 West Depere and Fort Howard .. 14 18 44.480 312 5 4.5 | 0. 2 —0. 646 | +0. 001 14 18 43. 835 Fort Howard and Oneida ........ 10 24 40. 160 31z i 12 7.3 9 0.5! —0. 963 | --0. 037 10 24 39. 234 Oneida and Long Tail Point Light- \ | - MOWsCsensicencoy es seescieaccen 49 24 26.915 3la 13 7.4 , 0.5} —0,206 | —0. 091 49 24 26. 618 | Long Tail Point Light-House and ' : : Little Wail ccccossvccscnscacanes 18 47 35. 738 315 | 8 3.4 | 0.5} —1.071 | -++0. 087 18 47 34.754 . Little Tail and Red Banks ....--- 38 20 04.017 316 6 3.1 | 0.2 | +0. 520 | +0. 893 38 20 05. 430 East Depere and Fort Howard... 47 02 51. 693 3lit2 17 93 5 1 —0. 005 | —0. 154 47 02 51. 534 Fort Howard and East Depere... 312 57 08.335 31—1-z 8 4.3 1 0.5 | —0. 023 | +0. 154 312 57 08. 466 East Depere and Oneida .. .-..-. 57 27 31.167 31lit2+3 6 BT 0.2 | —0. 282 | —0.117 57 27 30. 768 ' Oneida and East Depere ..-..---- 302 32 30. 038 311-23 8 | 6.5 0.5 | —0,923 | +0.117 302 32 29, 232 Long Tail Point Light-House and | | / Haat DOpere occccs seco ceacw ance 253 08 01.814 31—1—2—3—-4 7 , 46 0.5 | +6. 592 | +0. 208 253 08 02. 614 Red Banks and East Depere ...-- 196 00 23. 225 317 8 | 28 0.5 | --0 023 | —0.772 196 00 22. 430 Fort Howard and Long Tail Point | Light-House ...---------.------ 59 49 06.140 313+4 15 ; 81 1 —0. 234 | —0. 054 59 49 05. 852 Long Tail Point Light-House and Port Howard ...... snes sess 300 10 54. 000 31-4 10 6.9 0.5} +€. 094 | +0. 054 300 10 54.148 Fort Howard and Little Tail,* ... 78 36 39.791 313+ats 16 4.5 x +0. 782 | +0, 033 78 36 40. 606 Little Tail and Fort Howard. -.. 281 23 18.000 31—s—a—s 2 14 0.1 | +1. 427 | —0. 033 281 23 19. 394 Long Tail Point Light-House and | CIN, caxeseunence cednwnasasan 310 35 33.567 | 31—4 3 8.6 0.1) —C 276 | +0. 091 310 35 33. 382 Long Tail Point Light-House and Red Banks.....- -------------- 57 07 40. 367 315+6 ! 3 2.7 0.1 | —1.163 | —0. 980 57 07 40. 184 ‘ | * All angles except this one were read by A. R. Flint with the Repsold theo lolite. NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 4(81)) +3. 2(812) +1. 7(81s)-- 1. 0(314) 4-0. 5(31s) +0. 5(315) +6. 3801=0 3. 2(31,) +3. 4(312) +1. 7(31s)-+1. 0(814) +0. 5(31s) +-0. 5(316) +6. 3801=0 1. 7(311)-+1. 7(312) +4. 8(81s) +3. 6(314)--L. 6(31s)-+ 0. 5(316)-+9. 0114 =0 1. 0(81;)-+1. 0(212) +3. 6(312) +4. 2(314)-+1. 6(315) +0. 5(316)-+7. 0760=0 0. 5(31,)-+ 0. 5(312) +1. 6(313)-F-1. 6(814) +2. 2(315)-+ 0. 6(316)-+-4. 5596=0 0. 5(31,)-+ 0. 5(81z) +0. 5(81a) +-0. 5(314)-+0. 6(31s) +-0. 8(316)-+ 1. 4563=0 47 LS 370 PRIMARY TRIANGULATION, [CHar. XV, ¢, SECTION LV.—Triangulation from the line Burnt Bluff- Pine Hill to the line Eldorado - Taycheedah— Continued. FORT HOWARD—32. (Observers, A. R. Flint and R. 8. Woodward. Instruments, Repsold 10-inch theodolite, Gambey 10-inch repeating theodolite. Dates, July, 1872, October, 1873, and June, 1874.] : 7 | Angle as measured between— Notation. | No. meas. Range. 1 Wt (r) {v] Corrected angles, i | oO / a“ a“ “ “un oO ® aw | Little Tail and Red Banks ....... 39 02 19. 064 321 23 8.2 J 0.000 —1.275 » 39 02 17.789 ' Red Banks and Court-House .... 23 50 44. 025 322 ' 4 a2 0.2 40.370 +0. 209 23 50 44. 604 ; Bruce and Court-House.-......-. 4 31 44.325 323 : 4 6.4 0.2 40.370 "4-0. 209 4 31 44. 904 Bruce and East Depere ..-..----- 86 53 16.500 | 324 | 19 7.2 1 +0, 307 | —0. 273 | 86 53 16. 534 Red Banks and Bruce..... ....-. 28 22 28.784 32243 | is 3.6 1 +0. 306 40.418 | 28 22 29. 508 East Depere and Bruce ..-......- 273 06 43.338 | 32-4 | 13 86 | 0.5) —0.145 | +0. 273 | 273 06 43. 466 | East Depere and Red Banks ..... 244 44 13.343 | 32—2—s—a 7 4.5 | 0.5 40.760 0.145! 244 44 13, 958 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 7(322)-+1. 5(323) +0. 5(324) —1. 3375=0 1. 5(322) 4-1. 7(323) +-0. 5(324)—1. 3375=0 0. 5(322)-|-0. 5(323) +2. 0(324) —0. 9845=0 Notr.—Angles 32) and 32243 were read partly by A. R. Flint with the Repsold instrument and partly by R. 5. Woodward with the Gambey instrument. All the others were read by A. R, Flint with the Repsold instrument. EAST DEPERE—33. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1872.] Angle as measured between— Notation. No. meas. | Range. wt. (v) {v] ‘Corrected angles. oO rw awn : | “ “a ? oe ¢ aw Calumet and Freedom ........-.. 35 05 43. 582 331 11 5.1) 0.5 +0.076 | —0.772 i 35 05 42. 886 | Freedom and West Depere ..-..-- 14 38 27.975 332 4 2.5 0.2) +0.300 | +0. 662 14 38 28, 987 | West Depere and Oneida ........ 46 52 38. 681 333 16 ; eee ed +0. 040 | —0. 370 46 52 38.351 | Oneida and Fort Howard .......-. 29 51 48, 264 334 11 5.1) 0.5 | +0. 539 | +0. 347 29 51 49.150 | Fort Howard and Bruce .......-. 46 03 52.047 | 335-+6 17 5.0 1 | +0.270 | —0. 039 | 46 03 52. 278 | Long Tail Point Light-House and Bre seica zene dacaadiastescse 27 06 16. 000 336 2 2.0 0.1! 0.000 | +0. 156 | 27 06 16. 156 | Freedom and Calumet ......-..-- 324 54 16.760 33-1 10 5.8 0.5 —0.418 | +0. 772 324 54 17.116 | Calumet and West Depere....-.. 49 44 11. 854 33142 10 5.2 | 0.5) 40.079 | —0.110 49 44 11. 823 | West Depere and Calumet ....- 310 15 47. 083 33—1—2 f 6 4.5 0.3 | +0. 984 | +0. 110 i 310 15 48.177 | | Bruce and Calumet ............-- 187 27 28. 256 337 7 4.6 0.3) —0.030 | +0.172 | 187 27 28. 398 Weet Depere and Freedom ...... 345 21 31.300 | 33-2 2 0.2 0.1} +0.425 | —0. 662 345 21 31. 063 | Oneida and Freedom ............ 298 28 55. 300 33—2—3 2 0.4 0.1 | —2.296 | —0. 290 298 28 52.712 Oneida and West Depere .....-... 313 07 20.722 33-3 9 5.0 0.5 +0.557 | +0, 370 | 313 07 21. 649 | Oneida and Bruce..........-...-- 75 55 42.667 _ 334+5+6 3 3.7 0.1) —1.547 | +0.308 | 75 55 41.428 | Bruce and Oneida.........---.--- 284 04 18.467 33_a—p—» 6 4.5 | 0.3 | 40.413 | —0, 308 | 284 04 18.572 Bruce and Fort Howard .-.-....--. 313 56 07. 683 | 33—5—6 6 4.7 0.3 0.000 | +0. 039 313 56 07.722 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 1(881) +1. 1(332) +0. 3(333) + 0. 3(334) + 0. 3(335-+6) —0. 7440=0 1. 1(531) +1. 5(332) 4-0. 4(333) +0. 3(334)+-0. 3(335+6)—0. 7919=0 0. 3(331) +0. 4(332)+ 1. 9(333) 4-0. 3(334) +0. 3(335+6) —0. 4614=0 0. 3(331) +0. 3(332) + 0. 3(33 3) 4-1. 2(334) + 0. 7(335-+6)—0. 9607=0 0, 3(331) +0. 3 (332) +0. 3(333) + 0. 7(334) +2. 0(3354+6) —1. 0417=0 § 6.}. VULCAN - HURON MOUNTAINS TO FOND DU LAU BASE. 371 SECTION IV.—Triangulation from the line Burnt Bluff— Pine Hill to the line Eldorado — Taycheedah— Continued. ONEIDA—34. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1872.) 1 | Angle as measured between— | Notation. No. meas. | Range.| Wt. (9) {v] Corrected angles. 1 | — | fo} t aw | aw aw “a oO e “ | Red Banks and Bruce............ 20 58 35.155 | 341 9 3.6 0.5 | +0.123 | +9, 819 20 58 36. 097 | Bruce and East Depere ........-. 46 36 47.953» 3do 15 5.9 | 1 +-0. 061 | +0. 527 46 36 48,541 | East Depere and West Depere... 74 20 27.256 | 343 16 6.5 | 1 +0. 061 | —0 423 74 20 26, 894 | West Depere and Long Tail Point : | | Light-House .......-........-- 211 31 45: 063 | 344 | 16 97 / 1 +0. 061 | +0.135 | 211 31 45,259 | Long Tail Point Light-House and | | Red Banks .....-...-......-.-. 6 82 24.144 | 345 9 6.2 | 0.5] +0.123 | —1. 058 6 32 23.209 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 0(341) +0. 5(342) 4-0. 5(343)+-0. 5(344)—0, 2145=0 0. 5(341) 4-1. 5(842)-++0. 5(343)-++ 0. 5(344) 0. 2145=0 0. 5(341)-+ 0. 5(342) + 1. 5(343) +0. 5(344)—0. 2145=0 0. 5(341)-++0. 5(342) 4-0. 5(343) +1. 5(344)—0. 2145=0 WEST DEPERE—35. [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, August, 1872.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | [v] | Corrected angles. 1 | fan \ on u } “ “" on " Oneida and East Depere ......... 58 46 55. 840 351 | 18 fo. —0.123 | —0. 237 58 46 55. 480 East Depere and Calumet ....... 92 07 28. 216 352 ! 14 5.6 | 0.7 | —0.312 | —0. 624 92 07 27. 280 Calumet and Freedom .........-- 43 43 20.387 353 | 12 7.0 | 0.6) —0.548 | +0. 558 43 43 20.397 East Depere and Oneida..... -.. 301 13 04, 213 35-1 | 3 | 3.8 ae 0.1 | +0. 070 | +0. 237 | 301 13 04. 520 | Freedom and Oneida..-..-..-..-. 165 22 16. 669 354 18 7.0 | 1.0) 0.129 | +0. 303 165 22 16. 843 Calumet and East Depere.....-..-. 267 52 31.744 35—2 | 9 6.2 0.5 | +0,352 | +0, 624 267 52 32.720 East Depere and Freedom ......- 135 50 47.727 35243 | 11 oe 7 0.5 | +0.016 | —0. 066 135 50 47. 677 Freedom and East Depere --.-.-- 224 09 14. 650 | 352-3 | 2 i 0.1) —2.393 | +0. 066 224 09 12. 323 Freedom and Calumet ......- --. 316 16 39. 920 35—s | 5 | 6. 0.2) +0.241 | —0. 558 | 316 16 39. 603 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 1(351) +1. 0(352)-+-1. 0(353) +1. 1173=0 1. 0(351) +2. 8(352) +1. 6(352) +1. 8733 =0 1. 0(351) +1. 6(352)-+2. 4(358) +1. 9367=0 CALUMET—36. [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, August, 1872.] | Angle as measured between— Notation. No. meas. | Range.| Wt. (v) {v] |Corrected angles. o ov “uw Oo “ Oshkosh and Clayton ....-...--- 47 38 42.917 361 12 5.4 0.6 —0. 284 | —1.005 47 38 41. 628 ' Clayton and Freedom ......-.--- 59 51 21. 038 362 13 6.2 0.6 —0. 286 | +0. 741 59, 51 21. 4938 Freedom and West Depere....-- 22 17 56. 814 363 7 2.8 0.3 —0. 027 | —0. 334 22 17 56. 453 West Depere and East Depere .. 38 C8 22.120 364 10 3.8 0.5 ~-0. 042 | +-0. 200 38 08 22. 278 | Clayton and Oshkosh ....-.-..--- 312 21 16. 050 36-1 6 6.1 0.3 | +1.317 | 41.005 312 21 18.372 | Oshkosh and Freedom........... 107 30 03. 225 361+2 12 7.8 0.6 | -+-0.160 | —0. 264 107 30 03.121 Freedom and Oshkosh....-. .-.. 252 29 56. 542 36—1—2 12 6.7 0. 6 -+0. 073 | --0. 264 252 29 56. 879 Freedom and Clayton .--.....--- 300 08 37. 933 | 36—2 6 4.0 0.3) +1.315 | —0. 741 300 08 38. 507 West Depere and Freedom ....-- 337 42 03. 200 | 36—3 2 3.6 0.1 --0. 0138 | +0. 334 337 42 03. 547 Freedom and East Depere ....-- 60 26 19.710 | 36344 10 4.9 0.5 | —0.845 | —0. 134 60 26 18.731 | East Depere and West Depere .. 321 51 37.980 36-4 5 14 0.2 | —0. 058 | —0. 200 821 51 37. 722 | East Depere and Oshkosh ..-...- 192 03 40. 320 | 365 5 4.9 0,2 —2. 570 | +0. 398 192 03 38. 148 | East Depere and Fréedom....... 299 33 40.971 | 36—s—a 9. 3.5 | 0.5 | +0,164| +0.134] 299 33 41.269 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2, 3(361) +1. 4 (36,) +0. 2(36a) +0. 2(364) +1. 0681=0 _ 1.4(861) +2. 3(862) +0. 2(363) +0. 2(364)-+ 1. 0693=0 0. 2(361) +0. 2(362) +1. 6(863) +1. 2(364) +0. 2077=0 0. 2(861) +-0, 2(362) +1. 2(363) +1. 9(36s) + 0. 2263=0 372 PRIMARY TRIANGULATION. (Crap. XV, C, SECTION LV.— Triangulation from the line Burnt Bluff-Pine Hill to the line Eldorado—Taycheedah— Continued. FREEDOM—37. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, August, 1872.] Angle as measured between— Notation. No. meas. Range. | Wt. (v) (v]) [Corrected angles. Z bee eget ye fh (SG ee |e? Hcy poe peers | eee I oO 4 au" a “ uw or “uw | West Depere and East Depere.. 29 30 44.764 , 371 11 7.4 | 0.5 +0.180 | —1.158 29 30 43, 786 | East Depere and Calumet. .....-. 84 27 59. 015 | 372 13 7.5 | 0.6 —0. 066 | +0. 905 84 27 59. 854 Calumet and Appleton ...-.-.--- 38 29 01. 382 373 13 6.3 0.6 +0. 054 | +0. 113 88 29 01.549 . Appleton and Clayton ...---.--- 26 0€ 56. 021 | 374 9 5.0 | 0.5 —0. 255 | +0. 108 26 06 55. 874 | Clayton and West Depere ..-...- 181 25 19.150 375 16 , 8.4 0.8 | —0. 245 | +0. 032 181 25 18. 937 | West Depere and Calumet....--. 113 58 43. 990 | 37142 10 4.3 0.5 —0. 097 | —0. 253 113 58 43. 640 Calumet and West Depere .....- 246 01 15. 233 } 87-1-2 3 | 4.3 0.1 +0. 774 | +0, 253 246 01 16. 360 | Appleton and West Depere...... 207 32 13.077 | 371-23 3 | 1.9 0.1 | +1.594 | +0. 140 207 32 14. 811 | Calumet and Clayton .......----- 64 35 57.150 | 37344 14 | 6.4 0.7 | +0.052 | -+-0. 221 64 35 57. 423 ! Clayton and Calumet ....-------- 295 24 02. 537 37—s—4 8 5.9 0.4 -|-0. 261 | —0. 221 295 24 02.577 ' Calumet and East Depere.....--- 275 32 02. 350 37-2 2 0.9 | 0.1 | —1.299 | —0. 905 275 32 00. 146 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 0(371) +1. 5(372)-+0. 9(373) +0, 8(374) —0. 1049=0 1. 5(371) +2. 2(372) +0. 9(373) +-0. 8(374) +0. 0316=0 0. 9(37)) +0. 9(372) +2. 6(373) +1. 9(374)-+ 0. 2425=0 0, 8(871) +0. 8(372) 4-1. 9(373) + 2. 4(374) + 0. 4187=0 CLAYTON—38. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, August, 1872.] nay Angle as measured between— Notation. No. meas. Range., Wt. (v) {v] | Corrected angles. : ote. ag eel es cael oO ‘ a“ | Ww ri } w wt Oo t aw Freedom and Appleton ......---. 16 34 38.700 381 2 | 1.8 | 0.1 0. 000 0. 000 16 34 38.700 Freedom and Calumet .....-..--- 55 32 41.113 381-42 15 6.2 . 0.7) +0. 265 | +0. 632 55 32 42.010 ' Calumet and Stockbridge ......-- 22 05 59.092 |. 383 | 12 3.8 ' 0.6 | —0.426 —0. 938 22 05 57.728 | Stockbridge and Oshkosh......-. 57 57 40. 058 384 , 2 | 564 , 0.6] —0.432 —0. 085 57 57 39,541 _ Calumet and Freedom ........-.- 304 27 18. 050 381-2 2 | 0.5. 0.1} 40.572 | 0.632) 304 27 17.990 ' Oshkosh and Freedom ........-- 224 23 40.115 385 13 6.1 | 0.6| 40.215 | +0.391 | 224 23 40.721 | Stockbridge and Calumet........ 337 54 01. 300 38s 2 0.8 ! 0.1 | +0. 034 | +0, 938 337 54 02. 272 , Calumet and Oshkosh...... ..--- 80 03 37. 443 38344 7 i B.4 0.3 | +0.849 | —1. 023 80 03 37. 269 | Oshkosh and Calumet.......-.... 279 56 23, 033 38—s—a : 3 2.8 0.1} --1.325 | +1. 023 279 56 22. 731 | Oshkosh and Stockbridge.... .. 302 02 20, 375 38—4 | 4 | 3.8 0.2 | —0.001 | +0. 085 | 302 02 20. 459 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 4(38142) 1-0. 6(383) +0. 6(38,) +0. 1431=0 0. 6(38142) +-1. 7 (383) +1. 0(38,) +0. 9964 =0 0. 6(3814-2)+-1. 0(383) 4-1. 8(384) +1. 0438=0 OSHKOSH—3). [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Dates, August and September, 1872. ] 1 7 Angle as measured between— | Notation. No. meas. | Range.| Wt. | (v) [v7] | Corrected angles. | | ° a uw ae | a n aw fe} A “ Clayton and Calumet ......-.---- 52 17 42. 953 \ 391 15 4.4 0.8 | —0.005 —0.769 52 17 42,181 . Calumet and Stockbridge... ..-. 10 14 20. 625 39, 16 5.5 0.8 | —0.102 | +0. 678 | 10 14 21. 201 Stockbridge and Taycheedah .... 84 48 34.536 | 393 | 11 AT 0.5! +0. 399 | +0. 056 | 84 48 34. 991 Calumet and Clayton .......--... 307 42 17. 100 39-1 2 1.2 0.1 —0.048 +0.767 , 307 42 17.819 Stockbridge and Clayton......... 297 27 57.325 | 39—1—2 | 4 5.0 0.2. —0.796 | +0. 089 297 27 56. 618 ‘Taycheedah and Clayton ... ...- 212 39 20. 000 | 38,4 | 2 | 1.0 0.1 | +1.594 | +0. 033 ‘212 39 21. 627 Stockbridge and Calumet......-. 349 45 40.300 39-2 | 2 3.4 0.1, —0,823 | —0. 678 849 45 38, 799 Taycheedab and Stockbridge .. 275 11 24. 933 | 39-5 | 6 | 5.3 0.3! +0. 132 —0. 056 275 11 25. 009 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1, 2(39,) +0. 3(392) +0. 1(393) —0. 0027=0 0. 3(391) +1. 2(392) +0. 1(393)-+0. 0845=0 0, 1(391) +0. 1(392) +0. 9(893)—0. 3479=0 § 6:] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 373 SEctTion IV.—Triangulation from the line Burnt Bluff— Pine Hill to the line Eldorado — Taycheedah— Continued. STOCKBRIDGE—40. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, September, 1872.] Angle as measured between— Notation. No. meas. | Range.| Wt. : (v) | {v] Corrected angles. i e oo ° ‘ “ “ i ww aw oF " Taycheedah and Oshkosh -- 54 30 03.274 40i+2 19. 5.8 1 | 0. 000 | +0, 215 54 30 03. 489 : Eldorado and Oshkosh..-........ 23 00 55.006 40. 16 6.5 1 : 0. 000 | —0. 174 23 00 54. 832 | Oshkosh and Clayton .... ...... 59 30 17.380 403 15 3.7 1 | 0.000 | +0. 589 59 30 17. 969 | . ‘i | TAYCHEEDAH—41. {Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, September, 1872. ] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | [v) Corrected angles. | Oo f a a a uw oO ‘ a Catholic Church and Oakfield... 17 35 03.586 4lite | 14 5.5 | 0.7 0.000 ; —0. 022 17 35 03. 564 East Base and Oakfield ......... 5 19 00.825 412 > | 4 2.9 0.2 0.000 | —1. 413 5 18 59. 412 Oakfield and Eldorado .......--. 66 40 21.939 413 ‘18 80 | 1.0 0.000 | —0. 073 66 40 21. 866 Eldorado and Oshkosh.......--- 42 46 57.786 414 | 14 6.8 | 0.7 | -+0.074 | +0. 196 42 46 58. 056 Oshkosh and Stockbridge. .-.... 40 41 22.521 41s ; 19 5.1 | 1.0 | +0. 052 | +0.178 40 41 22.751 Eldorado and Stockbridge ...-.. 83 28 20.950 4lats | 2 3.9 | 0.1 | —O0.517 | +0. 374 83 28 20. 807 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 0. 8(414) +0. 1(415) —0. 0643=0 0. 1(414)-++1. 1(415)—0. 0648—=0 Nore.—The corrections [vJ fur 411+2, 412, and 413 will be found in the adjustment of the triangulation from the line Eldorado - Taycheedah to the line Minnesota Junction - Horicon. ELDORADO—42. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, September, 1872.] 1 ‘ Angle as measured between— | Notation. No. meas. | Range.| Wt. (v) (v] Corrected angles. % J aw | a “ “ oO ‘ u Stockbridge and Taycheedah... 65 02 31.683 421 | 18 4.7 1.0 0.000 | +0. 174 65 02 31. 857 Taycheedah and CatholicChurch 50 33 26.000 422 8 7.6 0.5 | +0. 688 | —0. 227 50 33 26.461 Catholic Church and Oakfield .. 24 49 03. 680 423 10 5.4 0.5 | +0. 688 | +-0. 287 24 49 04. 655 Oakfield and Springvale ...-...- 47 04 33, 855 424 20 5.9 1.0 0.000 | —0. 003 47 04 33. 852 Taycheedah and Oakfield. .-.... 75 22 31.744 42243 9 4.6 0.5 | —0.688 | +0. 060 75 22 31.116 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 0(42,)-+0. 5(423)—1. 032—=0 0, 5(422)-+1. 0(423)--1. 0320 Notr.—The corrections [v] for 422, 423, 424, 422+3 will be found in the adjustment of the triangulation from the line Eldorado - Taycheedah to the line Minnesota Junction - Horicon. at4 Numerical equations of condition in the triangulation from the line Pine Hill- Burnt Bluff, to the line PRIMARY TRIANGULATION, Eldorado — Taycheedah. SIDE-EQUATIONS. (Cuar, XV, ¢,. XXX. (7) —16.4617[17,]4+ 2. 2527[17)42]— 2. 7753[19,] — 2.7753[19,] — 0.0317[193] — 1. 881202146) 415, S690 215] XXNII. (20) + 2..7753[19,]-25. 8815[19.] XXXVIT. (20) —27. 8667[203] +25, R020[20,] XLT. (9) + 4. 9140[22.J— 5. 9000[22,] + XLVIL. (20) + 1.2441[27,]—13. 0292[272] L. (25) —29. O55R[28,]-+11. 46380285] 4-37. BB05[ 316] —25. 965232] +38. LI. (30) — 9. 7614[24:]+41. 1313[2%,] +27. LIT. (20) — 2. 7470[28,]+31. 2441[28] +20. 427, 1486 315]— 2%. 7587[32,] — & +61, 4492[ 335] LIV. (40) +22. 9867(283]+22. 9867[29,] —21. +16. — 25. an +27. +11. 7589[ 201] + 4. 4374[20.] +30. 5281[215] — 13. 6817[21,45]+18. 880—0 7413[21,] — 6. 6507[212] + 0. 6411[22,] +19.7725[22,] +79. 594—0 .7255[24,] — 1. 6570[242] +19. 0155[ 252] — 4. 8643[25045]-+ 2. 106=0 7637[28)] + 4. 1276[28)] +17. 2664[293] — 25. 6112[ 294] 9818[ 322] +38. 9818[ 325] 8153[ 292] —48. 2672[315] +23. 8539[3145] 3116[31;] +20. 3116[ 312] +20. 3116[315] +20. 3116[31,] 4638[24;] +10. 7031[315] +10. 7031[31,] +10. 7031[315] 7587[322] — 8. 7587[325] + 1. 1447[32,] —20. 2870[ 33:46] 35, 5278[34]—35. 5278[34,] —35. 5278[ 344] LVI. (30) + 6. 3834[31,)]-+ 6.3834[31,] + 6. +50, 5547[335]-+40. 4182[34.] +34. LVIII. (25) —32. 7522031) ]+-45. 7288312] +45. +12. 628434, ]+ 1. 5305[ 345] LXII. (12.5) +12. 1306(33,]—17 Norr.—lIn the solution for determining the general corrections, each of the side-equations was divided by the .8330[332] — 0. 4323[345] +34. 43230344] number inclosed in parenthesis and placed opposite it. XXIV. XXYV. XXVI. XXVILI. XXVIII. XXIX. XXXI. XXXII. XXXIV. XXXV. XXXVI. XXXVIII. XXXIX. XL. XLII. XLIII. XLIV. XLV. XLVI. XLVIII. XLIX. LII. LY. LVI. LIX. ANGLE-EQUATIONS. (147] +[15.]) +[16,] (14) +162] —(1%142] —[173] —[1%s] (163] +173] +[18:] (17:42] +18] +[193] [183] +191] +[192] (17142) —[171] +[182] —({18] —[18] —[18] [19.] +[202] +[213] [203] + [204] +[21o] [21,1] +222] +[233] [20,] +[221] +[222] [223] +[224] +232] + [2a +[18:] +[2li4s] —[21s] +[192] +[20,] +[21its] +[221] +[23,] +[ 233] +[241] [231] an [242] + [25245] [223] +231] +[232] [24544] +(25,] +[261] [2444] —[248] +1261) [26344] +(272] +[28)] [263] +[2%1] +[272] [26344] —(263] +[231] (28) +[28] +285] (284) +[26] +[31s] [311] +312] +[321] (i 4B 42) [333] +343] +[351] [33:] +[332] +[852] + [25243] —[252] +[262] +[273] +[ 295] +[282] +[293] +([29140] +[321] 0532[31,] —28, 9025[315] — 5.9158[315] —71. 0809[34,] 3034[31y] +24. 4252[31,] —18. 4171[33,] —18. 4171[33;46] 7288| 315] —33. 2907[33,] —13. 5720[33,] —13. 5720[ 33546] 7810[35.] —22. 0159[35,] — 9. 3651[37;] —11. 4050[37.] +32. 444=0 —2, 830=0 —0.376=0 +1.561=0 —0.153=0 —0. 712=0 —1.887=0 —2. 149=0 +0. 869=0 +1.212=0 +1.841=0 +3. 082=0 —0. 160=9 +0. 287=0 +0. 304=0 +1.309=0 +2. 585=0 +1.569=0 — 0. 648=0 +1. 871=0 +2.175=0 +(31.] +(315] +[32)] +[322] +[32s]-+40. 8180 +[33546] +(331] +[33346] +[342] +(365] +0. 466=0 —0.718=0 +1. 030=0 +0. 535=0 §6.] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 375 Numerical equations of condition—Continued. LX, (36) +(36,] +(84) 4(8i9 +0. 002=0 LXL (8%) +36) +(3nj +4372] +0, 029=0 LETT. (365) (8% +0971] +[3840] —1.593=0 LXIV. [36] +[38] +([38,] +[39] +2. 795=0 - LXV. [38] +[39] +[39] +[405] —0.415=0 LXVE. [30] 2l40.ae) Et] —0. 449=0 LXVII. [40:42] —[402] +[41.] +[41,] +[42:] —0.938=0 General corrections in terms of the correlates. }14] = [14.] =+0.80000 XXIV [15.] =+1. 00000 XXIV +9. 80000 XXV [16,] =+0.75000 XXIV —0, 12500 XXV —0. 12500 XXVI [162] =—0. 12500 XXIV +0. 68750 XXV —0. 31250 XXVI [16,5] =—0. 12500 XXIV —0. 31250 XXV +0, 68750 XXVI [16,] =—0. 25000 XXIV —0. 12500 XXV —0. 12500 XX VI (14) = —1. 66667 XXIX —3.91945 XXX [17,42]=—0. 33333 XXV —0. 33333 XXVI +0. 66667 XX VII +40. 66667 XXIX +0. 21264 XXX (17,] =—0.33333XXV +0, 66667 XXVI —0.33333 KXVII —0.33333 XXIX —0. 10632 XXX [17;] =—1. 00000 XxV. (18] =+-0.75000 XXVI —0.25000 XXVII —0.25000 XXVIII —0. 50000 XXIX —0.50000 XXXI . (1r2] =—0.25000XXVI 40. 75000 XXVII_ —0.25000 XXVIII 0.50000 XXIX —0, 50000 XXXI [18] =—0.25000 XX VI —0, 25000 XXVIL -++0.75000 XXVIII +0.50000 XXIX +0,50000 XXXI [18;] =—1. 00000 XXXI [19,] =—0.27273XXVII +0. 45455 XXVIII —0.17898 XXX —0.27273 XXXI +0. 72727 XXXII +0. 45385 XXXIII [192] =—0.27273XXVII +0, 45455 XXVIII —0.17808 XXX 0.72727 XXXI_ —0, 27273 XXXII —0. 97900 XXXII [195] =+0.72727 XXVII --0.54545 XXVIII +0.21-96 XXX —0.27273 XXXI_ —0. 27273 XXXII +0. 31508 XXXIII [20,] =+0.21212XXXI —0,78788 XXXII +0. #4059 XXXIII —0, 24242 XXXIV —0. 12121 XXXVI +0. 01264 XXXVII [20.] =—0. 78788 XXXI +1, 21212 XXXII —0.39127 XXXIII —0, 24242 XXXIV +0, 01264 XXXVII \ —0. 12121 XXXVI —0. 78788 XXXVI [203] =—0. 12121 XXXI —0. 12121 XXXII —0, 12846 XXXITI +-0. 42424 XXXIV —2. 70655 XX XVII [204] =—0. 12121 XXXI —0. 12121 XXXII —0. 12846 XXXIII +0. 42424 XXXIV +1. 21212 XXXVI +2. 66233 XXXVII [21,] =—0.20000 XXVIII —0.2u000 XXIX +0. 05375 XXX —0, 40000 XXXII —0. 16846 XXXTII —0. 20000 XXXIV +0.80000 XXXV —0.96315 XXXVII (21,] =-—0. 20000 XXVIII —0.20000 XXIX +0. 05375 XXX —-0. 40000 XXXII —0. 16846 XXXITI +0. 80000 XXXIV —0. 20000 XXXV —O. 00862 XXXVIT [215] =—0.20000 XXVIII —0. 20000 XXIX +0.05375 XXX +0. 60000 XXXII +1. 35795 XXXIII —0. 20000 XXXV +0. 80000 XXIX —0. 20000 XXXV —2. 00000 XXIX +0. 32392 XXXVII —0, 21499 XXX +40. 32392 XXXVII +4. 44654 XXX —0. 20000 XXXIV [2l145]=+0. 80000 XXVII —0. 20000 XXXIV [215] = +0. 60000 XXXII —0, 85255 X XXIII 3 ( 6 [221] [222] L225] [224] [231] [232] [233] [231] [241] [242] [243] [24544] [251] [252] PRIMARY TRIANGULATION, [Cnar. XV,C, General corrections in terms of the correlutes—Continued. =+0. 80000 XXXIV —0, 20000 XL =—0. 20000 XXXIV —0. 20000 XL =—0, 20000 XXXIV +0. 80000 XL =—0, 20000 XXXIV —0. 20000 XL =—0, 28571 XXXV =—0, 28571 XXXV =+-0, 28571 XXXV =+0,57143 XXXV =+0. 71429 XXXVII1 =--0, 28571 XXXVUI =—0. 14286 XXXVIII =—0. 14286 XXXVIII =— 0. 33333 XXXIX [2504s]=+0. 66667 XXXIX [261] [262] [265] [26345] (271] [272] [27s] [28] [ 282] [28] [283] [285] [29142] [292] [293] [294] [295] (311] =+40.75000 XLII ——0, 25000 XLI =—0, 25000 XLII =—0. 20000 XLIII ——0.20000 XLII =-+40. 70000 XLII =+10. 79630 XLIV +0, 15183 L =—0. 20370 XLIV +0, 15183 L =—0. 20370 XLIV —1,01040 L =—0. 09259 XLIV +10. 27745 L ——0. 09259 XLIV “0.27745 L =—0, 10000 XLV +0, 10057 LIV =—0. 10000 XLV +0. 67524 LIV =—0. 80000 XLV —0, 05747 LIV =+1, 20000 XLV —0. 05747 LIV =—0. 20000 XLV —0. 08620 LIV =—0, 03246 XLIX —0. 02409 LIV —0. 20000 XXXV_ +0. 60000 XXXVI +0. 02191 XLI +0, 80000 XXXV +0. 02191 XLI —0. 20000 XXXV +0. 56791 XLI —0. 20000 XXX V —0, 63365 XLI —0. 09524 XXXVI —0. 09524 XXXVI +0. 76190 XXXVI —U. 47619 XXXVI —0, 28571 XXXIX +0. 71429 XXXIX —0. 14286 XXXIX —0. 14286 XXXIX —0. 33333 XL —1. 00000 XL +0. 66667 XL +0. 50000 XLIII +0.50000 XLITI —0, 50000 XLIII —0. 80000 XLIV +1. 20000 XLIV —0. 20000 XLIV +0, 59259 XLVI —0. 06067 LI +0. 59259 XLVI —0. 06067 LI —0. 40741 XLVI —0. 38605 LI —0. 18519 XLVI +3. 71162 LI —0. 18519 XLVI —3. 14358 LI —0. 10000 XLVI —0. 10000 XLVI +1. 20000 XLVI —0. 80000 XLVI —0. 20000 XLVI —0. 17832 L +0. 10422 LV +0. 60000 XXXVI —U. 40000 XXXVI —U, 40000 XXXVI —U. 23810 XXXVIII +0. 76190 XXXVITI . 09524 XXXVITI . 19048 XXXVITI . 50701 XLI . 31327 XLI . 06458 XLI . 06458 XLI - 18016 XLI . 11283 XLI . 36032 XLI . 25000 XLIV . 25000 XLIV . 75000 XLIV 40000 XLV 40000 XLV . 40000 XLV . 06337 XLVIL . 18193 LIIL . 11844 XLVII . 13193 LIL . 32482 XLVII . 13193 LIL 14765 XLVIL . 98501 LIIL —0. 14765 XLVIIL +4, 51274 LUI +0. 04272 XLVII +0. 04272 XLVIL +2. 06343 XLVIL —2, 23933 XLVIL +0. 08545 XLVII —0. 13160 LI +0. 05404 LVI —0. 17208 XXXVIT —0. 40000 XXXVIIT +0, 7849 NXXVIL —0. 40000 XXX VIII —0. 20414 XXXVIL +0, 60000 XXXVIII —0. 20414 XXXVI +-0. 60000 XXXVIII +0.76190 XXXIX = +0. 52381 XL —0,23810 XXXIX +0.52381 XL —0, 09524 XXXIX —0,19048 XL —0.19048 XXXIX —0. 38095 XL —0. 14286 XLII —0, 14286 XL +40. 09524 XLIL = —0. 66667 XLII +0,.76190 XLIE +0. 66667 XLII +0, 66667 XLII —0. 33333 XLII —0. 25000 XLVI —0. 25000 XLVI +1.00000 XLV —1. 00000 XLVI +0, 75000 XLVI +40, 59582 XLVILI —0, 83152 XLVII +0. 11785 XLVIL —0. 38889 XLVIII —0, 18519 XLIX —0. 11706 LIV —0, 38869 XLVILI —0. 18519 XLIX —0. 11706 LIV +0.61111 XLVI —0. 18519 XLIX 40. 45761 LIV +0.27778 XLVI +0. 37037 XLIX —0. 05321 LIV 0.27778 XLVI +40. 37037 XLIX —0. 05321 LIV +1.17500 XLVIII +0. 16226 LI 0.17500 XLVIIL +1. 08944 LI —0.10000 XLVIII —0, 09272 LI —0.10000 XLVIII —0, 09272 LI —0. 15000 XLVIIL —0. 13908 LI +0. 20318 LII —7. 97637 LVIIL +0, 18273 LIIT v6] [31.] [31s] [31,4] [315] [315] [32.] [322] [825] [3243 [331] [332] [333] [334] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. General corrections in terms of the correlates—Continued. =—0, 03246 XLIX —0, 02409 LIV =-+40. 06695 XLIX +40, 33586 LIV =+0. 05817 XLIX —0, 26171 LIV =+0. 48848 XLIX —0, 39797 LIV =—0, 40398 XLIX —0. 09736 LIV =+1. 00000 XLVINI =—0, 01755 LII +40. 02782 LVIII =—0. 03509 LIL +0. 24319 LVIII =—0, 03509 LII —0. 69458 LVIII =—0, 34858 LII —0. 28301 LVI [33545]=+0. 63516 LIT [336] [341] [342] [343] [344] (351) [352] [353] [361] [362] [365] [364] [371] [372] [373] [374] —0, 10885 LVI =—1.77729 LIV —+0. 00018 LIV —-+0, 00018 LIV —-+10. 00018 LIV —-10. 61758 LVI =—0, 11876 LVI ——0. 17815 LVI ——0. 01372 LIX ——0, 01372 LIX ——0.74760 LIX —=-++1, 00137 LIX ——0, 63687 LX —-+10, 95224 LX ——0, 06944 LX =—0. 04015 LX [3842]=-+0. 87884 LXIII [383] [384] [39;] [392] [393] =—0. 20478 LXIII =—0, 17918 LXIII =-+40, 89390 LXIV —=—0, 21721 LXIV =—0.07519 LXIV (40,43)= [402] [403] [414] [415] [421] =+1. 00000 LXV 48 LS —0. 174832 L +0. 10422 LY +0. 17125 L +0, 49330 LV —0. 07360 L —0. 43689 LV —0. 43818 L —0. 05437 LV +2. 35706 L —0, 12475 LV +1. 00000 XLIX +0, 33898 XLIX +0, 33898 XLIX —0. 16949 XLIX +0. 01780 LIII +40. 21856 LIX +40. 03559 LIII 40, 57796 LIX “0, 03559 LIII —0, 12627 LIX +0. 35358 LITT —0. 13686 LIX —0. 64427 LIII —0. 05264 LIX +30. 72460 LIII —0, 28571 LV +0. 85714 LV —0. 14286 LV —0. 14286 LV —0, 11876 LIX +40. 59976 LIX —0. 35036 LIX —0, 03773 LX —0. 03773 LX 0, 44410 LX 0, 25377 LX +0. 39951 LXI +0, 28537 LXI —0. 16666 LXI —0. 09635 LXI —0, 38396 LXIV +0, 47781 LXIV +0. 41809 LXIV +0. 67669 LXV +0. 67669 LXV —0. 15038 LXV +1. 00000 LXVI —0, 11494 LXVI +0. 91954 LXVI —0. 13160 LI +0. 05404 LVI +0, 25148 LI —0. 31445 LVI +0. 19590-LI +0. 49949 LVI —1,55451 LI --0, 13725 LVI +2. 04467 LI —0. 08025 LVI —1. 03861 L +40. 52857 L -40, 52857 L —0, 26428 L —0. 06317 LV +0.77708 LX —0. 12633 LV —0. 55852 LX —0. 12633 LV +0. 00486 LX 40. 74512 LV —0, 04562 LX 0. 28658 LV —0. 01755 LX —1. 04079 LVI +40. 82688 LVI 0. 62735 LVI +40. 62735 LVI —0. 17815 LXI —0. 35036 LXI -L0, 72447 LXI —0. 02401 LXI —0, 02401 LXI +1. 19170 LXI —0. 74760 LXI —0. 19049 LXII —0, 36919 LXII +0. 13620 LXII +40. 07874 LXII —0,17918 LXV —0. 44369 LXV +0. 86178 LXV —0. 07519 LXVI —0. 07519 LXVI +1. 12782 LXVI +1. 00000 LX VII —1. 00000 LX VII +1. 14943 LXVIT +0, 30460 LX VII +1, 00000 LXVILI « + 0.20318 LI + 7.71983 LVIIL — 0.19791 LII + 1.21297 LVI + 0.07826 LII — 0.92198 LVIII + 0.05474 LIT ‘— 0.18538 LVIIL — 0.22024 LII + 0.11751 LVIII — 0.08475 LILI — 0.08475 LII + 0.54237 LIL + 0.03878 LVI + 1.55093 LXII + 0.07755 LVI — 2.16336 LXII + 0.07755 LVI + 0.19178 LXIL — 0.45743 LVI + 0.08590 LXII — 0.17593 LVI + 0.03303 LXII +19. 85160 LVI — 0.28571 LVII — 0.14286 LVII + 0.85714 LVII — 0.14286 LVII + 0.32119 LXIL + 0.57961 LXII — 1.25409 LXII — 0, 41838 LXIII — 0.69273 LXIII — 0.02401 LXIII — 0, 01372 LXIII — 0.15342 LXIII — 0.10959 LXIII + 0.25411 LXIII + 0, 30316 LXII 0, 18273 LHI —0, 17031 LUI +40, 08579 LUI +40, 80874 LIII —0. 78214 LIL —0, 43794 LIII —0. 15331 LILI —0, 15331 LI +0. 10527 LIII +0. 00486 LVII —0. 13113 LVII +40. 57310 LVII —0. 09124 LVII —0. 03509 LVII —0. 35610 LVI +0. 32709 LVI —0, 56317 LVIII —0. 17805 LVIII +0. 69273 LXIV —0, 41838 LXIV —0. 02401 LXIV —0. 01372°LXIV B77 378 No. of equation. 24. 25. ~ 26. 27, 28. 29. 30, 31. 33. 34, 35, 36. 37. 38. 39, 40. 41. 0=—2. 83000 +2. . 12500 XXIV . 38333 XXIX . 12500 XXIV . 25000 XXVIII . 33333 XXV . 16667 XXIX . 31508 XXXII . 25000 XXVI 57295 XXX 20000 XXXIV . 33333 XXV . 13333 XXIX . 20000 XXXIV . 10632 XXV . 52044 XXIX . 43589 XXXII 50000 XXVI . 93939 XXXI . 12121 XXXVI , 27273 XXVII . 06061 XXXI . 40000 XXXV . 31508 XXVII 13611 XXXI . 16846 XXXV 20000 XX VIII . 64242 XXXII . 02424 XXXVI . 02191 XLI —0, . 16846 XX XIII . 17866 XXXVII . 02191 XLI . 12121 XXXI . 88571 XXXV . 09524 XXKIX +0. . 66045 XXXII 3.27474 XXXVI . 02236 XLI —0. . 67619 XXXVIIJ — 0.52381 XXXIX . 14286 XLII = +1, —0. 0=—-0. 37600 0=+1. 56100 0=—0, 15300 0=—0. 71200 0=—1. 88700 0=—38. 89743 0=—2. 14900 0=-+0. 86900 0=-+0, 94400 0=+1. 21200 0—-11. 84100 0=-+3. 08200 0=+.3. 97970 0=—0. 16000 0=-+0, 28700 0=4-0. 30400 0-10. 23400 PRIMARY TRIANGULATION, Normal equations for determining the correlates. 55000 XXIV 20000 XX VIIT 32392 XXVIII 40000 XXXIV 28571 XXXV 19048 XL 20000 XXXIV . 12500 XXV . 15417 XXV . 10632 XXX . 64583 XXV . 83333 XXIX 58333 XXVI 42560 XXX aootr ow S . 79545 XXVII . 95455 XXXI 20000 XXXV 83333 XXVI 52044 XXX 20000 XXXV 10632 XXVI . 36967 XXX 0, 05375 XXXIV 0.77273 XXVII 1. 06061 XXXII eseescees 0. 01264 XXXVIT . 05455 XXVIII a 3. 13939 XXXII 0.12121 XXXVI 1.37770 XXVIII 0. 56798 XXXII 0. 12846 XXXVI 0. 20000 XXIX 0. 42538 XX XIII 0, 22492 XXX VII — 0.20000 XXIX + + +. + 0. 40000 XXXIV 0. 68571 XXX VIII 0. 12121 XXXII 3. 17402 XXXVI 0.59048 XL 0. 32392 XXIX 0. 28624 XX XIII 9, 22108X XX VII 0. 68571 XXXV 0, 09524 XXXVI 0. 67359 XLI 0.77143 XXXV +1. 12381 XXXVIIT + 1.19048 XXXIX == (i) ih 33333 XLIT 02191 XXXIV + 0.02191 XXXV +10. 44127 XXX VIII — 0. 67359 XXXIX 10, 11558 XLII Oo . 12500 XX VI . 64583 XXVI 2 +2. 10417 XXVI . 10632 XXX +2. 14394 XXVIT —0. 77273 XXXI > —2. 45909 XXVIIT +1. 05455 XXXIT . 32392 XXXVIT +1. 16667 XX VII +0. 60000 XXXII , 82392 XXXVI +40. 42560 XX VII —0. 17898 XXXT +0. 05375 XXXV +0. 95455 XXVIII —0, 13811 XXXII 60000 XXIX 56798 XXXIII 66048 XXX VII 85255 XXITX 60371 XX XIII 28624 XXXVII 05375 XXX 44848 XXXIV 40000 XXX VIII -L0. +0. +40. —0, 44, +0. +40, +2, =); +0. 05375 XXX 42. 45714 XXXV —0, 28571 XXKIX —0. +3. 10. —0. —0. —0. 12846 XXXIII 27474 XXXVII 04382 XLI 08705 XXX 22492 XXXIV 40828 XXX VIII 0: 4) 89524 XXXVI 12381 XL —0. 52381 XXXVIII —0. 47619 XLII —0, 59048 XXXVI +3, 51429 XL +0. 04382 XXXVI . 90524 XL e [Cuap. XV, C, —0, 33333 XXVII —0. 58333 XX VIT —0, 50000 XXXI —0. 79545 XXVIII —0, 27273 XXXII +1.30000 XXIX —1. 37770 XXXII +1. 30000 XXVIII —0, 85255 XX XIII —0, 57295 XXVIII —0. 34022 XXXII —0. 08705 XX XVII —0, 17898 XXX —0, 24242 XXXIV —0. 34022 XXX —0, 64242 XXXIV +0. 43589 XXX —0. 42538 XXXIV —0. 24242 XXXT — 0.40000 XXXV — 0.20000 XL —0, 40000 XXXII +0. 88571 XXXVI —0.77143 XL +1. 02424 XXXIV —0. 89524 XXXVIIT +40. 01264 XXXI —0, 17866 XXXV —0. 20414 XL —0. 40828 XXX VII +0. 44127 XLI +2. 14286 XX XIX —0. 20414 XXXVII —1. 90524 XLI +0, 02236 XXXVII +5. 76449 XLI § 6.] No. of equation. 42. 43. 44. 45. 46. 47. 48. 49. 51. 52. 54, 56. VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 379 Normal equations for determining the correlates—Continued. 0=-+1. 30900 — + 0=-+42. 58500 + | OW SOP SSS SS OMS SP re Sr YES 0=-+41. 56900 0=—0. 64800 0=-+1. 87100 + 0=—0. 45625 + + + +2. 17500 0=-+0. 81800 0=—3. 86724 )+++4+H Se ee Sei he 2 Oe SS 0=-40, 53547 0=-+0. 46600 0=—1. 66020 0=+2. 42628 0=—0. 71800 0=—0. 40307 . 17857 XLIT . 16667 XLII 50000 XLVI 25000 XLII 34259 XLVI 15183 L 40000 XLII 47503 XLVIT 25000 XLII 13518 XLVI 30366 L 11785 XLII 70619 XLVIT 05713 LI 388e9 XLIV 34167 XLVIII 04214 LIIT 18519 XLIV 03230 XLIX 50739 LIII 10561 LVIII 15183 XLIV 23289 XLIX 48577 LIIL 22068 LVIIT 06067 XLIV 34426 XLVIIT 26319 LIT 15701 LVI 23441 XLIX 17353 LIII 03509 LVII 03303 LXII 13193 XLIV 0.50739 XLIX . 36597 LITT 0. 03559 LVII 0. 03350 LXIT 0. 11706 XLIV 0.45175 XLVIII 0. 04817 LIT 0 0 0 0 x . 15121 LVI . 00204 XLIX . 09553 LITT . 26919 LVII 0.11693 LXIT 0. 04779 XLIX 61. 28307 LIIT 0. 70490 LVII 0, 07301 LXII sect; . 14286 XXX VIII —0, 47619 XX XIX +1. 16667 XLII +3. 03333 XLIII +0. 11785 XLVII —0.70000 XLII +0. 23185 XLVILI —0. 06067 LI + 0.40000 XLIV —0, 10000 XLVIII —0.50000 XLIII +43, 01336 XLVIL —0. 21406 LI +0, 23185 XLIV —0.57740 XLVI . 21037 LIL . 10000 XLV . 65556 XLIX .45175 LIV . 37038 XLVI . 23289 L . 43025 LIV . 30366 XLVI . 52967 L . 57390 LIV . 09272 XLV —0. 53909 XLIX —7. 34378 LIII +0. 39169 LVIII —0. 62093 L 04817 LIV —0. 36539 LVIIT —0. —0. —0. —0. 26387 XLVI 48577 L 58965 LIV 10625 LVITI —0, —0. —0. 05747 XLV 43025 XLIX 58965 LIII +0. 00018 LVII —0. 18540 L +0. 28786 LIV +0. 89165 LVIII —0. 09937 L —0. 15121 LIV +0. 57613 LVIII -—U. 33333 XL —0. 25000 XLIV —0. 70000 XLIV +2. 74630 XLIV —0, 38889 XLVIII —0. 13193 LUI +3. 00000 XLV —0. 09272 LI +1, 34259 XLIV —0. 87778 XLVIII —0. 26387 LILI —2, 47503 XLV —0, 29530 XLIX —0, 16211 LIV —0. 87778 XLVI —1, 49411 L —0. 29530 XLVII —0.53909 LI +0. 00204 LV +0. 24210 XLVII +3. 28832 LI —0. 18540 LV —0. 21406 XLVI +3, 28832 L +1. 12193 LIV —0. 26319 LI -0. 49502 LV —0, 05264 LIX —0, 21037 XLVII —7. 34378 LI —0. 09553 LV +0. 05340 LIX —0. 29159 XLVI —0. 57390 L +4. 21972 LIV +0. 60205 LVIII —0, 01172 LI +2. 59058 LV —0. 18950 LIX +0. 15701 LI —0. 01286 LV +0. 11633 LIX + 0.11555 XLI 0. 25000 XLVI 0. 40000 XLV 0. 40000 XLV 0. 18519 XLIX 0.11706 LIV 1. 80000 XLVI 0.05747 LIV 1.80000 XLV — 0.37038 XLIX 0. 29159 LIV 3. 01336 XLVI 0. 24210 L + + + 0.57740 XLVITI 0. 34426 LI 1.55556 XLIX 0. 23441 LIT + 0.04779 LVI + + 1.49411 XLVIII 0. 62093 LII 0. 09937 LVI 0. 05713 XLVIT 10, 35134 LI 0.01172 LV + 1.58389 LIT 0. 06786 LVI 0. 01755 LX + 0. 04214 XLVIII 0.17353 LIT +461. 28307 LVI + 0.01780 LX — 0.16211 XLVII + 1.12193 LI + 0, 28786 LV + 0.49502 LII — 0.01286 LVI — 0.06317 LX — 0.06786 LIL +42. 71428 LVI + 0.03878 LX 380 PRIMARY TRIANGULATION. [Cuar. XV, C, Normal equations for determining the correlates—Continued. No. of equation. 57, O=+1. 03000 —0.03509 LIT + 0.03559 LITT +0.00019 LIV —0, 26919 LV +0.70490 LVI + 2.04782 LVIT = —0.13141 LVIII —0. 24503 LIX +0. 00486 LX — 0.17815 LXI +0. 51297 LXII da. O=—O0. 55224 +0. 10561 XLIX + 0.22068 L +0. 39169 LI —0. 36539 LI —0, 10625 LIII + 0.60205 LIV -+0. 89165 Lv +0. 57613 LVI —0.13141 LVII +28.50951 LVIII -+0.27102 LIX +0. 02782 LX —0. 31995 LXII 59. 0=-+40.53500 —0. 05264 LIT + 0.05340 LITI —0.18950 LV -40. 11633 LVI —0. 24503 LVIL + 0.27102 LVUI +42.39765 LIX +0. 47233 LX —1.09796 LXI — 0.03282 LXII —0. 01372 LXIIJ —0. 01372 LXIV 60. 0=-+0. 00200 —0. 01755 LII 0.01780 LITT = —0. 06317 LV +0. 03878 LVI +0. 00486 LVIT 0.02782 LVIII -++0.47233 LIX +2. 42719 LX +0. 72947 LXI 1.18174 LXIIT = —0, 14732 LXIII —0, 03773 LXIV 61. 0=-+0. 02900 —0.17815 LVIIT — 1.09796 LIX +0. 72947 LX +2. 60105 LXI —1.81377 LXII — 0.28702 LXIII —0. 02401 LXIV 62, 0=+2.59552 -+-0. 03303 LIT — 0.03350 LITT -+0.11893 LV —0.07301 LVI +0.51297 LVII — 0.31995 LVIII —0. 03282 LIX -+1.18174 LX —1.81377 LXI + 7.24358 LXII = +-0. 21493 LXIII +++ 63. 0=—1.59300 —0. 01372 LIX — 0.14732 LX —0. 28702 LXI +0. 21493 LXII +2. 12884 LXII — 0.80234 LXIV —0.17918 LXV 64. 0=+2.79500 —0.01372 LIX — 0.03773 LX —0. 02401 LXI —0. 80234 LXITI +2. 48253 LXIV + 1.09478 LXV —0.07519 LXVI 65. 0=—0.41500 —0.17918 LXIII + 1.09478 LXIV +3.21516 LXV —0. 15038 LXVI 66. 0=—0. 44900 —0. 07519 LXIV — 0.15038 LXV = +4. 04736 LXVI +1. 80460 LXVII 67. 0=—0. 93800 +-1.80460 LXVI + 4.95403 LXVII Values of the correlates and of their logarithms. XXIV =+1. 0835 log 0. 03482894 XLVI =—1. 0081 log 0. 0035036— XXV =+0.0418 log 8. 62117634 _ XLVII =+0. 4159 log 9. 6189889. XXVI =—0.5777 log 9.7618527_ XLVIIL =—0. 4045 log 9. 6069185 XXVII =—0. 0886 log 8. 9474337_ XLIX =—0, 3550 log 9, 5502284_ XXVIII =—0. 1894 log 9.2773800— L =+0.5407 log 9. 73295644 XXIX =+0.2145 log 9. 33142734 LI =—0.2793 log 9. 4460709_ XXX =—0. 4559 log 9. 6588696 LIL =—0. 3305 log 9.5191715— XXXI =+0, 3029 log 9, 4412993. LIT =—0.1045 log 9. 0191163 XXXII =—0. 4201 log 9, 6233527 LIV =—0.5284 log 9, 7229628— XXXII =—0, 2145 log 9. 3314273... LV =+0, 3537 log 9. 54863514. XXXIV =—0.9001 log 9,9542908_ LVI =+0.1696 log 9, 2294258 XXXV =—1. 2225 log 0, 0872480_ LVIT =—0.5645 log 9.7516639_ XXXVI =—0. 0807 log 8. 9068735— LVIIL =+0. 0099 log 7. 9956352 XXXVII =—0. 4427 log 9. 6461095 _ LIX ==—1. 1454 log 0. 0589572— XXXVIII =—0. 4342 log 9. 6376898_ LX =+1.0712 log 0. 02987064 XXXIX =—0,3705 log 9, 5687882_ LXI =—1. 4184 log 0. 1517987— XL =-~0, 2850 log 9, 4548449_ LXII =—0. 8641 log 9.9365640— XLI =—0, 1270 log 9. 1038037_ LXIII =+0.2704 log 9. 43200674. XLII =—0. 3702 log 9, 5684364_ LXIV =—1, 3007 log 0.1141771_ XLII =—1. 0346 log 0, 0147725 LXV =+0.5890 log 9.77011534. XLIV =—0,5348 log 9.7281914_ LXVI =+0. 0411 log 8, 61384184 XLV =—0. 1446 log 9. 1601683_ LXVII =+0.1744 log 9. 24156654 §6.] VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. a (14.] =-+0. 033 (142] =+-0. 867 [152] =+1.083 [16.] =-+0. 880 [16] =-+0.074 [163] =—0, 546 (16,] =—0.204 {1%1] =+1. 429 [t7%140] =-+0. 166 (173] = 0.393 {17;] =—0. 042 [18] = -0. 623 [18] =+0. 081 [18] =-+0,283 [18;] =—0.303 [12] =—0. 466 [192] =-++0.565 [19,] =—0. 094 [20] =+0.740 (20:] =—0. 442 [203] =-+0. 922 Residuals resulting from substitution of general ee a pe Sr ey spss ate (20. (21) [212] [21s] [21445] [21s] [221] [222] [225] [224] [231] [232] [283] [234] [241] [242] [243] [24544] = [251] [252] [25245] Values of the general corrections. “a =—1.617 =—0. 197 =—0. 297 =—0, 292 =+40. 330 =—2, 456 =—0, 220 =+0.017 ——0. 268 vee [26.] =—0. 409 [262] =—0. 039 [263] =-++0. 864 [26544] =—0. 547 [271] =+0. 825 [27] =—0.839 [273] =—0.510 [2&] =—0.183 [28] =—0.140 (28] =—0. 463 [28,] =—0.511 [285] =+-0.516 [2949] =—0. 441 [29] = —0.599 [293] =—0. 137 [294] =—0.202 [295] =+0. 411 [311] =—0. 155 [312] =-+0.001 [315] +0. 037 [31,4] =—0. 091 corrections in the [315] =-+0. 087 [3l6] =+0.893 [32,] =—1.275 [32] =+0.209 [32.] =4-0. 209 [32] =—0.273 [33;] =—0.772 [332] =-10. 662 [335] =—0.370 [33,] =+0.347 [33.46] =—0. 039 | [385] =-+0. 156 | [8h] =+0. 619 [34.] =++0.527 [345] =—0. 423 [34,] =+0.135 [35:] =—0.237 [352] =—0. 624 [355] =-10.558 [36,] =—1. 005 [362] =+0.741 covatim,| Residual. |} No.of, | Residual. || N09, | Residual. 24 —0. 00006 39 +0. 00004 54 —0. 00080 25 +0. 00014 40 —0. 00012 55 +0. 00012 26 —0. 00014 41 +0. 00126 56 —0. 01290 27 +0. 00003 42 —0. 00003 57 +0. 00009 28 -+0. 00001 43 +0. 00009 58 —0. 01825 “29 —0. 00003 44 | ~—0. 00011 59 -++0. 00010 30 +0. 00490 45 +0. 00004 60 +0. 00016 31 -+0. 00003 46 —0. 00016 61 +0. 00012 32 —0. 00009 | 47 —0. 00080 62 +0. 00262 33 —0. 00040 48 —0. 00003 63 —0. 00002 34 +0. 00011 49 | +0.00005 64 +0. 00011 35 +0. 00012 50 —0. 00775 | 65 +0. 00012 36 —0. 00009 51 +0. 00930 66 +0. 00019 37 —0. 00980 52 —0. 00012 67 +0. 00015 38 | —0.00005 | 53 | +0. 05080 “a [36;] —=—0. 334 [36,] —=--0. 199 7 tise [372] =++0.905 [373] =+0.113 [374] =+0. 108 [38142] =++0. 632 [38%] =—0.938 [38] —=—0.085 [39] =—0.767 [39%] -—=-10. 678 [395] =+0. 056 [40:42] =+0. 215 [40.] =—0.174 [403] =+0.589 [41,4] =-++0.196 [415] =-+10.178 [42] =-+0.174 381 numerical equations of condition. 382 PRIMARY TRIANGULATION. [Cnap. XV, SECTION V.—TZriangulation from the line Eldorado—Taycheedah to the line Minnesota Junction - Horicon. OAKFIELD—43. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Dates, October and November, 1872.] J 1 Angle as measured between— | Notation. | No. meas. Range.| Wt. | (v) (v] Corrected angles. ° 8 aw | w " wn Oo “ Horicon and Minnesota Junction... 32 54 41.589 | 43) 9 ‘ 3.6 0.5 | —0.099 | —0, 259 32 54 41. 231 Minnesota Junction and Waupun .. 50 55 03.433 | 432 15 5.2 0.7 | —0.175 | +0. 593 50 55 03, 851 Waupun and Springvale 58 57 55.920 | 483 150A T | 7, —0.450} —0. 719 53 87 54.751 Springvale and Eldorado 40 02 22.733 | 434 18 ;| 63 1.0 0.000 | +0. 256 40 02 22. 989 Springvale and West Base.....-.--- 46 49 51.936 | 43146 14 | 7.0 0.7 | —0.345 | +40.132 46 49 51.723 Eldorado and Taycheedah .....-.-.. 37 57 07.987 | 435+6 15 | 6.3 0.7 0.000 | +0.138 87 57 08.125 West Base and East Base........... 38 49 29.631 | 43647 16 | 5.8 0.8 | —0.109 | +0. 437 38 49 29. 959 Horicon and Waupun....--.--.-.--- 83 49 44.843 | 43:42 7 | 2.8 0.4 | —0.095) +0.334 83 49 45. 082 East Base and Horicon ........---.. 186 32 58,844 | 43g 9 ' 6.6 0.5 | --0.175 | —0,184 136 32 58. 485 Minnesota Junction and Springvale. 104 52 58.000 | 43243 3 | 6.3 | 0.1) +0.728 | —0.126 104 52 58. 602 Waupun and West Base ....-..----- 100 47 46.675 | 4334445 8 5.3 | 0.4 | +0.386 | —0. 587 100 47 46.474 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 4(431) +0. 9(432) + 0. 5(433) +0. 5(434-45) +0. 5(43647) +0. 7481=0 0. 9(431) +1. 7(432) +0. 6(433) +-0. 5(43a+5) +-0. 5(436-+7) +0. 8834 =0 0. 5(431) +0. 6(432) +1. 7(433)-+0. 9(43445) +0. 5(43647) +1. 2842=0 0. 5(431) +0. 5(432) +0. 9(433) +1. 6(43445) +0. 5(43647) +1. 1489=0 0. 5(481) +0. 5 (432) +0. 5(433) +0. 5(434-45) 4-1. 3(43647) +0. 6765=0 SPRINGVALE—44. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Dates, September and October, 1872.] Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) | [v] Corrected angles. fo} # “ “a ‘ a“ “ Oo t aw Eldorado and East Base...---.------ 61 07 44.980 | 441 10 5.7 0.5 | —0.199 | +0.133 61 07 44. 914 East Base and Oakfield -- 8145 19.178 | 442 13 5.2 0.6 | —0.166 | -+0.039 31 45 19. 051 Oakfield and Waupun......... -- 40 42 14.262 | 443 16 6.3 O58" ewerariccvsess —0. 825 40 42 13.437 Eldorado and Oakfield ........--.--- 92 58 02.800 | 441+2 3 3.9 0.1 | +0.993 | +0.172 92 53 03. 965 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 0. 6(44,) +0. 1(449) +0. 1358=0 0. 1(44))-+0. 7(449)+0. 1358=0 EAST BASE—45. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Dates, October and November, 1872: ] I Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | [2] Corrected angles. ° a uw “we aw | “a oO ‘ 4 Oakfield and West Base .........-.. 33 28 54.437 | 451 ! 24 6.4 1,0) +40.042 | +0.309 33 28 54. 788 West Base and Middle Base ........ 0 00 03.797 | 452 ! 13 2.8 0.6 —0.251 | —0. 081 0 00 03. 465 Middle Base and Check Base ....... 24 31 07.875 | 453 | 17 7.8 1.0) —0.150 | —0. 046 24 31 07.679 Check Base and Springvale-........ 4 35 14.693 | 454 6 4.5 0.3 | —0.851 | +0. 036 4 35 13. 878 Check Base and ‘l'aycheedah........ 109 01 04.340 | 45445 IL 9.0 0.5 | -+0.167 | —0.315 109 01 04.192 Taycheedah and Catholic Church... 119 15 57.067 | 456 9 6.0 0.5 | +0.168 | +0.130 119 15 57.365 Oakfield and Springvale ...........- 62 25 19.396 | 45)+24+3+44 16 a7 0.8 | +0.196 | +0.218 62 35 19.810 Catholic Church and Oakfield....... 73 42 50.520 | 457 2 6.1 6.1; +1.988 | +0. 003 73 42 52. 511 West Base and Check Base ...-...-.. 24 31:10.800 | 45243 { 5 8.0 0.2 | +0.471 | 0.127 24 31 11.144 West Base and Springvale... ..-... 29 06 24.620 | 4524344 4 3.1 0.2 | -+-0.493 | —0. 091 29 06 25. 022 Check Base and Catholic Church ... 228 17 00.590 | 4544+5+6 2 2.9 0.1 | +1.152 | —0.185 228 17 01. 557 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 9(45,) +0. 9(452) +0. 9(45) +0. 8(454) +0. 1(454-+5) +0. 1(45,) +0. 9284=0 0. 9(45,) +1. 9(459) +1. 3(45g) +1. 0(454) +0. 1(454-+5) +0. 1(45g) +1. 4518=0 0. 9(45;)+-1. 3 (459) +2. 3(45g) +1. 0(454) +0. 1(4544+5) +0. 1(45g)+-1. 4518=0 0. 8(45,) +1. 0(452) +1. 0(453) 4+-1. 3(45,) +1. 4738=0 0. 1(45,)+-0. 1(45,) +0. 1(45,) +0. 7(454-+5) +0. 2(454)—0. 1147=0 0, 1(45;) +-0. 1(452) +0. 1(453) +0. 2(4544+5) +0. 7(45¢)—0. 1147=0 § 6.) VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 383 Section V.—Triangulation from the line Eldorado-— Taycheedah to the line Minnesota Junction - Horicon—Continued. WEST BASE—46. [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, October, 1872.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | {v] Corrected angles. x ey | Oo F aw | “ a“ | ae D@ #£ “ | Check Base and Middle Base....... 25 05 25. 357 | 461 7 5.4 O98 Vases ees Y | +0. 162 25 05 25.519 Check Base and East Base........-. 25 05 29.775 46142 8 4.7 0.4 —1.005 | +0. 188 25 05 28. 958 East Base and Oakfield ............. 107 41 35. 342 463 21 8.0 1.0 —0.402 | +0. 430 107 41 35.370 Check Base and Oakfield ........... 132 46 59. 690 401+243 2 1.0 i 0.1 | +4.020 | +0. 618 132 47 04, 328 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 0. 5(46142)-+0. 1(463)-+0. 5427 =0 0. 1(461-+42) +1. 1(463)-0. 5427 =0 MIDDLE BASE—43. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, November, 1872.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. i i | °° # aw ! ! aw | | uw °o é “ | West Base and Check Base.......... 87 55 04. 350 481 12 5.5 0.6 j.-.------- -+0. 146 87 55 04. 496 | Check Base and East Base........... 92 05 02. 436 i 482 14 _ 61 OeF | ecemcescns | - 0. 028 92 05 02. 408 , | ! I CHECK BASE—49. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, November, 1872.] : i i Angle as measured between— -Notation. | No. meas. | Range. é Wt. ! (v) [v] Corrected angles. pa Deere os \ ' °o a aw “ { aw fe} ‘ a East Base and Middle Base .......-.. 63 23 49. 835 491 17 BA TB Aseccseencse +0. 094 63 23 49. 929 Middle Base and West Base ......--. 66 59 29. 800 492 17 656° | A. Beshcwenae +0. 201 66 59 30. 001 WAUPUN—50. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, October, 1872. ] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. uF wn “ a“ Go F “ Springvale and Oakfield .....--.-..--- 85 19 52. 958 501 19 6.8 1 lecceaseees —0. 660 85 19 52, 298 Oakfield and Horicon .........------- 60 50 13. 287 502 19 12.1 1 ---- | +0.307 60 50 13. 594 Horicon and Minnesota Junction .... 34 00 48.328 503 16 8.3 Ll eeceadsen +0. 278 34 00 48, 601 MINNESOTA JUNCTION—51. {Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, May, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. or “ “ “ Ck “ Waupun and Oakfield .......-------- 34 13 54.910 5h 20 5.0 Te la tmeeoecs —0. 416 34 13 54. 494 Oakfield and Horicon.....----.------ 58 27 05. 085 512 20 8.0 To eteaeeceae +0. 089 58 27 05.174 Horicon and Lebanon........-..----- 60 40 56. 071 51s 20 6.5 MT eve sees ecaeea +0. 215 | 60 40 56. 286 ol i Note.—The correction [v] for 51s is derived from the succeeding section of the adjustment. 513 above is designated 11 in the succeeding section. 384 PRIMARY TRIANGULATION, - [Cuap. XV, C, SECTION V.—Triangulation from the line Eldorado-Taycheedah to the line Minnesota Junction- Horicon—Continued. HORICON—82. (Observer, R. 8. Woodward. Instrument, 10-inch Gambey repeating theodolite. Date, May, 1873.] | Angle as measured between— | Notation. | No. meas. | Range.| Wt | (v) | (v] Corrected angles. — | | | | Oo ‘ aw “a Oo FF a | Minnesota Junction and Oakfield.... 88 38 14.394 52142 16 2.3 W, hdseiorarerctoiec! —0, 228 88 38 14.166 Waupun and Oakfield ........-....-. 35 20 00. 900 522 16 2.9 dh Seaton iste +1. 014 35 20 01. 914 | Woodland and Lebanon ...... ......- 34 19 08. 384 523 16 3.0 lL) socmceree —0. 295 34 19 08.089 | Lebanon and Minnesota Junction ... 90 13 47. 021 524 | 16 5.0 A. iseececae: +0. 215 90 13 47. 236 Notr.—Angles 523 and 52s are the same as 2: and 22 of the succeeding section of the adjustment, from which their corrections [v] are derived. Numerical equations of condition in the triangulation from the line Eldorado—Taycheedah to the line LXVIIL (100) — LXIX. LXXV. LXXXI. (30) + 5, 4942 [425] 2, 2692 [4li4o] — 18.6499 [410] —156. 4680 [43,] +1156, 4680 [43547] + 91. 3420 [451] SIDE-EQUATIONS. + 91.3420 [45415] -+ 6. 8508 [455] (70) +226. 2427 [419] — 19.5821 [424] +102. 2612 [45] — 46.1591 [459] + 0.7655 [48)] (20) + 30, 2555 [43,] — 0.9868 [511] 9, 0796 [415] Minnesota Junction— Horicon. — 20.9191 [415] +156. 4680 [43445] —183. 4638 [43546] ++ 91.3420 [452] + 5.4942 [425] — 32. 7460 [422] + 91. 3420 [455] + 5, 4942 [425] — 1.0608 [44,] +102. 2612 [455] + 0.0021 [455] + 0.7662 [48] —. 2.2765 [43)] — 13.9140 [512] — 35.0786 [44] + 10.9192 [454] — 44.9676 [46,] ++ 11.7495 [502] —102, 2612 [45:] + 91.3420 [45445] +329. 841=0 + 44, 9657 [46,42] — 31.1998 [505] — 49,271=0 — »5.016—0 + 14. 934=0 Notre.—In the solution for determining the general corrections, each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. ANGLE-EQUATIONS. LXX. [4c] + [41s] -+ [420] ++ [425] LXXI. [41s] + [42] + (4%) + [43546] LXXIL [4%] + (43) + (44) + [440] LXXIIL. [43445] + (48c¢7] + £440] + (451) LXXIV. [4347] + [451] + [465] LXXVL [453] + [45s] + (46149) + [49] LXXVII. [453] + [4%] + [4%] LXXVIII. [46.] + [48%] + [49] TXXIX. [48)] + [4G] 4 150] LXXX. [43] ++ (50:) ++ [503] + (811) LXXXIL [43] + (43:) + [50] + [522] LXXXUI. [43] + [5le] + [52142] + [44] + [44] — 1454] + [45445] +1. 647=0 + [452] + [453] + [454] + [492] N —0, 125=0 --0. 425=0 —0, 826=0 —1.176=0 —0. 356=0 —0. 020=0 —0, 509 =0 2, 204=0 —0. 758=0 —1. 655=0 +40. 398=0 § 6.] VULCAN-HURON MOUNTAINS TO FOND DU LAO BASE. General corrections in terms of the correlates. (4ligo] =—0. 03241 LXVIIT [41-] =—0. 93250 LXVUI [415] =—0.20919 LXVIII [42] =—0. 47324 LXVIII [423] =+0. 29156 LXNVIIL [424] [43,] =—0. 48773 LXVILI —0,51298 LXXX [43.] =—0.20059 LXVIII +0. 94687 LXXX [435] =—0. 83383 LXVIII —0, 13983 LXXX [43.] =—1.56468 LXV [43.45] =+1. 25365 LXVILL —0. 02454 LXXX [43546] =—2. 62091 LXVILI [43c47] =+1. 30686 LX VIII —i), 10366 LXXX [44,] =+0. 09635 LXIX [44] =—0.72964 LXIX [445] = [45:] =+0. 43246 LXV +0. 76479 LXXIV [45.] =-10. 42869 LXVIII —0. 14195 LX XIV [455] =+0.29320 LXVIII —0, 08517 LXXIV [454] =—0.86759 LXVIII —0, 29593 LX XIV [45445] =-++1. 25549 LXVIII —0, 05974 LXXIV [454] =—0. 43433 LXVIII —0, 05974 LXXIV [461] = [46142] =—0. 18519 LXXIV [463] =+0. 92593 LXXIV [48,] =+0. 04253 LAXV [48] =+0. 03649 LXXV [49.] =+1. 00000 LXXVI [492] =+1.00000 LXXVI (50,] =+1.00000 LXXIX [50.) = [503] = (51) = [512] = [52142] = [522] =+1. 00000 LXXXII 49LS +16. ++ S + | | | ++ 144 eroerse cf ¢ Ss 2 So fF ©& + + + 16020 LXIX 12971 LXIX 05232 LXTX 05232 LXIX . 27974 LXIX . 31171 LXXIII . 82936 LXXXI . ved LXXIIL . 88380 LXXXI . 53291 LXXITI pe < . 04887 LXAXXI - 00000 LXXII . 80122 LXXITL . 18212 LXXXI - 42857 LAX 283523 LXILI . 27484 LXAXXI . 70732 LXX . 24390 LAX 0, 66136 LXIX 0. 213840 LXXV 0. 72320 LXIX 1. 68246 LAXV 0. 43392 LXIX 0. 52924 LXXV . 17708 LXIX 0.75270 LXXV 1.82779 LXIX 0. 10387 LXAXV 0.78199 LXIX 0, 10387 LXXV 3.74730 LXXV 3. 05323 LXXV 0. 27757 LXXV e 1. 66667 LX XVIII 1. 42857 LXXVITI 1, 00000 LX XVII 1. 00000 LXXX 1.00000 LXXX 1.00000 LXXX +5, +1. +40, +0. 4, —0. +40. —0. ab, at. +0. 42. =i, cit +0 —i —U, —0 00000 LXX (0000 LXX 66667 LXX 66667 LXX 00000 LXX 18048 LXXIV 65770 LXXXII 10366 LXXIV . 43390 LXXXIL . 11810 LXXIV . 18266 LXANIT . 14973 LANIV . 14677 LXXXIT . 98501 LXXLV . 29314 LXAXXIT 46341 LXXIT 21951 LXXII 23619 LXX 20712 LXXVI . 42170 LXX . 74953 LXXVI 25302 LXX . 44972 LXXVI1 . 43359 LXX ~ 78274 LXXVI .45434 LXX . 10802 LXXVI 54566 LXX . 10802 LXXVI 03704 LXXVI 18519 LXXVI . 00000 LXXVITT . 58748 LXXXI . 55999 LXXXIT . 04934 LXXXT . 69570 LXXXI S 00000 LXXI +0. +0, +1. 04283 LNXIX eT —0. .51298 LXXXUL +0. 220) 66667 LXXI 66667 LXXI 00000 LXXII 17067 LXXXIII 13983 LXXIX 90452 LXXIX 04283 LXXXIIT . 41481 LXXIX . e223 LXXXUL ). 11810 LXXIX 18948 LNN NIL 24390 LXXIII . 46341 LXXIIL . 25000 LXXIX 24174 LXXIIL . 08517 LXXVIL . 11838 LXXII . 34393 LXXVIL .07102 LXXIIL 79365 LXXVI . 47477 LXXIIL . 29353 LXX VIL . 04791 LNXIIT . 04051 LXX VIL —U. oa +2. 04791 LXXIIT 04051 LX XVII 50000 LXXVIIT +1. 00000 LXXXII +1. 00000 LXXXIII AA. 00000 LXXXIII OR PRIMARY TRIANGULATION, [Cmar. XV, C, Normal equations for determining the correlates. agp Gx, OO, 49271 13. 87867 LXVIIDT + 0.27597 LXIX + 0.79970 LXX —3. 01178 LXXI "be 1, 56462 LXAXIT + 2.90727 LXXIIE + 1.730232 LAXIV = —0.75190 LXXV + 0.78189 LXXVI + 0.29820 LANVIT) — 0.838383 LXXIX — —0. 20059 LXXX — 0.71500 LXXXT — 0. 68832 LXXXIT) — 0.48773 LXXXIIT 6, O=+4.71201 + 0.27597 LXVIIT 9 --57.55592 LXIX +18. 95661 LXX —0. 02507 LXXI — 0.91303 LXXIT — 0.08824 LXXII + 0.66136 LXXTV 9 —1.11271 LXXV + 4.15712 LXXVI + 0.43392 LXXVIT 70, O=+1.64700 + 0.79970 LXAVITET = +18. 95661 LXIX +12. O2kbe LXX +2. 33333 LXXI + 2.46342 LXXII — 0.76658 LXXIII + 0.238619 LXXIV = —0. 64883 LXXV + 0.67472 LXXVI + 0.25302 LXAXVIT 71. 0=—0.12500 -- 3.01172 LXVIII = — 0.02507 LXIX + 2.33333 LAX +3. 76190 LXXT 72. 0=—0. 42500 — 1.56468 LXVIII = — 0.91303 LXTX + 2. 46342 LXX +4. 68292 LXXII + 1.21951 LXNITI 73. 0O=—0, 82600 + 2.90727 LXVIII — 0. 08824 LXIX — 0, 76658 LXX +1. 21951 LXXII + 4.00577 LXXIIIT + 1.07697 LXXIV — 0, 18213 LXXV +0. 12940 LXXVI + 0.07102 LXXVII — 0.53291 LXXIX = — 0, 12820 LXXX —0. 45696 LXXXI — 0.43991 LXXXII — 0.31171 LAXXIIT 74. O=—1.17600 + 1.73932 LXVIIL + 0.66136 LXIX + 0.236519 LXX +1. 07697 LXXUI + 2.67573 LXXIV — 0.05917 LXXV — 0.41231 LXXVI —0. 08517 LXXVII — 0.11810 LXXIX — 0.10366 LXXX — 0.27484 LXXXI = =-—0. 29314 LXXXIT — 0.18948 LXXXITI 75, O=—0. 16720 — 0.75190 LXVIIT = — 1.11271 LXIX — 0, 64883 LXX —0. 18213 LX XIII — 0.05917 LXXTV = +-12.78402 LXXV + 1.90001 LXXVI +0,56573 LXXVII — 3.70477 LXXVIII 76, 0=—0,35600 + 0.78189 LXVII +4 1.15712 LXIX + 0.67472 LXX +0. 18940 LXXIIT — 0.41231 LXXIV + 1.90001 LXXV + 5.23629 LXXVI +1.44972 LXXVII + 1.00000 LXXVIII 77. 0=—0. 02000 + 0.29320 LXVITI + 0.43392 LXIX + 0, 25302 LXX +0. 07102 LXXIIF — 0.08517 LXXIV + 0.56573 LXXV + 1.44972 LXXVI +3, 22222 LXXVII 78, O0=—0. 50900 + 3.70477 LXXV + 1.00000 LXXVI + 5.16667 LXX VIII 79, O=+2. 20400 — 0.83383 LXVITI — 0.53291 LXXIII = — 0.11810 LXXIV 9 +3. 15452 LXXIX — 0.13983 LXXX — 0.04887 LXXXI — 0.18266 LXXXIIT —0, 04283 LXXXIII 80, 0=—0.75800 — 0.20059 LXVIII — 0.12820 LXXITI — 0.10366 LXXIV —0. 13983 LXXIX + 3.94687 LXXX — 1.90565 LXXXI + 1.43390 LXXXII —0.51298 LXXXII 81. 0=-+0. 74670 — 0.71500 LXVIIT — 0.45696 LXXIIL — 0.27484 LXXIV —0, 04887 LXXIX — 1.90565 LXXX + 6.13315 LXXXI + 1.53304 LAXXIT +1. 13366 LXXXIII sv. J0=—1, 65500 — 0.68832 LAVITT — 0.4399, LXXTIT — 0.29314 LXXIV = —0. 18266 LXAXIX + 1.43390 LXXX + 1.53304 LAXXT = + 3,09160 LXAXXIT +0. 65769 LXXXITI 3. 0=-++0. 39800 — 0.48773 LXVITI = — 0.31171 LXXIIT = — 0.18948 LXAXIV = —9. 04283 LXXIX — 0.51298 LXXX + 1.13366 LXXX1] + 0.65769 LXXXIT +3. 17067 LXXXIIT “alues of the correlates and their logarithms. LXNVIIL =-+0. 6715 log 9. 82702014 LXXVI =-+0.1142 log 9. 05759004 LXIX =-+0, 3282 log 9. 5160989, LXXVIT =—0. 0202 log 8.3049212_ LXX =—1. 2180 log 0. 0356330_ LXXVITI =-+0. 0872 log 8. 94076544 LXXT =-+1. 3284 log 0. 12333874 LXXIX =—0, 6593 log 9, 8193860_ LXXIT =+1. 3067 log 0. 11617264 LXXX =—0. 4382 log 9. 6416922_ LXXIIT =—1. 1017 log 0. 0420673_ LXXXI =—0. 4561 log 9. 6590886_ LXXIV =+0.4921 low 9. 69204454. LNXXI =+41.0137 log 0. 00590944 LXXV =+0. 0151 log 3. 17897694 UXXXITI =-- 0.2281 log 9. 3581778_ $7]. VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 387 Values of the general corrections. ‘ | “Wt a [Alize] =—0.022 [43547] =+0.437 | [46,] =-++0. 430 [41] =—1.413 | [44] =+0.133 | [48] =-+0.146 [41s] =—0.073 | [440] =+0.039 | [48] =—0. 028 [422] =—0.227 | [44] =—0.825 | [49] =--0. 094 [423] =-+0.287 | (4)) =+0.209 | [49] =+0. 201 [425] =—0.003 | [45.] =—0.081 | [50,] =—0. 660 [4] =—0.259 | [453] =—0.046 | [50.] =+0.308 [43.] =+40.593 | [45,] =+0.036 | [50;] =+0.273 [433] =—0.719 + [4544;]=—0.315 | [51,] =—0. 416 [434] =+40.256 | [45;] =-++0.130 | [512] =+0.089 [43.45] =+0.132 , [46] =+0.162 | [5242] =—0, 228 [43546] =+0.138 [46.42] =+40.188 | [52] =+1.014 Residuals resulting from substitution of general corrections in the numerical equations of condition. 1 ae Residual. | sae Residual. | = 1 68 —0. 00600 76 —0. 00004 | | 69 0.00089 | 7 —0. 00001 | / 70 —0. 00005 | 78 -+.0. 00004 | i oa +0.00001) 79 +0. 00006 | 72 —0. 00002 | 80 —0. 00002 | 73 —0. 00005 81 +0. 00100 74 —0. 00002 82 +0. 00001 | | 75 —0. 00360 | 83 +0. 00003 : | PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § 7. Let— m = whole number of observed angles in a section (oue adjustment). r = whole number of rigid conditions in a section. n = number of triangles in principal chain. [pov] = sum of weighted squares of corrections to observed angles. , =probable error of an observed angle of weight unity. Pa = probable error of an observed angle of average weight in whole section. p, == probable error of an adjusted angle of average weight in whole section. p, =average weight of an observed angle in whole section. p, =average weight of an observed angle in principal chain. Pe = probable error of an observed angle of average weight in principal chain. p/ = probable error of an adjusted angle of average weight in principal chain. [vv] = sum of squares of closing errors of triangles in principal chain. fe = probable error of an observed angle in principal chain as derived from the closing ‘errors of triangles. Proceeding as in Chapter XIV, C, § 8, there are found the following values: FOR THE ENTIRE SECTIONS. | | | = ‘Section, Extent of section. | milion | [pvv] | pi | Dy | By me u Vn _————o |___--- : | | “we | “ ! | w III | Vulcan-Huron Mountains to Pine Hill-Burnt Bluff..' 80 52 21.12 | 0.43 | 1.09 | 0.41 | 0.89 0. 24 IV | Pine Hill-Burnt Bluff to Eldorado-Taycheedah ...... 188 | 129 | 55.80 | 0.44] 0.64; 0.55 | 0.56 | 0.31. V_| Bldorado-Taycheedah to MinnesotaJunction-Horicon| 48 28 | 8,88 | 0.28 | 0.63 0.48) 0.65 | a1 | eer | ! 388 PRIMARY TRIANGULATION. [Cuap. XV, D, FOR THE PRINCIPAL CHAIN CONNECTING THE LINE VULCAN-HURON MOUNTAINS WITH FOND DU LAC BASE, GIVEN IN D, § &, FOLLOWING. : ; From closing errors of triangles. | . t | Sn en ee Section. Extent of principal chain in each section. Pe oo Ps ! Po! 1 j ; : ; [ve] py tt ‘Average Greatest ae ee: error. error, | | | i : SS +e aes & se, |i | ae 1 1 uo | “uw ; “ | Hn | “uw | TIL = Vulean- Huron Mountains to Pine Hill-Burnt Bluff... 1.03 0.42) 0.25 ' 2451 13 | 0.54) 1.21 2. 56 IV Pine Hill- Burnt Bluff to Eldorado-Taycheedah...... 0.79 0.50 | 0,28 | 40.02 26 1 0.48 "1.04 3. 10 V Eldorado - Taycheedah to Fond du Lac Base... ..... 0.76 0.44 | 0.28 0.98, 4 0.19 0.43 0.71 | | | | | Entire principal chain ........-...... .--. ---- Neietee tenes ieeaeee 65, 51 | 43 | 0.48 1.03 | 3.10 D—CHAIN OF PRINCIPAL TRIANGLES BECWEEN THE LINE VULCAN-HURON MOUNTAINS AND THE FOND DU LAC BASE. § 8. In Chapter XIV, D, a method of computing the weighted mean logarithms of the sides in a chain of triangles depending on two bases has been given, and this method is there applied to’ the system of triangles between Keweenaw and Minnesota Point Bases. The two systems con- necting the Keweenaw with the Minnesota and the Fond du Lac Bases have a common part, embracing six triangles lying between Keweenaw Base and the line Vulcan- Huron Mountains. Computing the common line Vulcan —- Huron Mountains from each of the three bases and taking the weighted mean, it is found that the effect of the Fond du Lac Base on the logarithin of this line is less than unity in the seventh place. The probable error of the logarithm of this weighted mean line is (probable error of standard being excluded) 26.24 in the seventh place, while the correspond- ing value obtained from Chapter XIV, D, where Keweenaw and Minnesota Point Bases are alone considered, is 27.58. The weighted mean logarithms of the sides of the common triangles given in Chapter XIV, D, therefore need no modification on account of the Fond du Lac base, and for com- puting the weighted mean logarithms of the triangle-sides between the Fond du Lac Base and the line Vulcan - Huron Mountains, the latter line may be taken as a base, using the weighted mean logarithm and probable error of its length derived from Chapter XIV, D. The adjustment of the triangulation between the line Vulean—Huron Mountains and the Fond du Lac Base gives the following values for the probable error, », of au observed angle of average weight in the respective sections of the principal chain. See Chapter XV, ©, § 7. From the line Vulean- Huron Mountains to the line Pine Hill—-Burnt Bluff........-. p=+0".42 From the line Pine Hill- Burnt Blutf to the line Eldorado —Taycheedah -........... p=+0".50 From the line Eldorado -Taycheedah to the line Minnesota Junction -Horicon. ..... p=t0" 43 Using the notation of Chapter XIV, D, For the first section, 2 (@+4,’)v?= 4021 For the second section, 2 (2+)p?= 3874 For the third section, 2 (a?+,3)e?—= 586 Sum.... 2... S481 The square of the probable error in the logarithm of the weighted mean value of the side Vulean - Huron Mountains, obtained from Chapter XTV, D, is 750, the error of standard bar being excluded, and the corresponding quantity for the Fond du Lac Base is 36, both being expressed, as are the above sums, in units of the seventh decimal place. Hence, the constant 1 Dp The logarithms of the Fond du Lac Base resulting by computation from the weighted mean length of the line Vulcan — Huron Mountains, given in Chapter XIV, D, and from the direct measure- ment, are— $y =SAS14 150-4 36-0267 From computation (feet) 2.22... Menysaalay eee eee ae aoe 4. 3865922 HOM MCHSUTEMIONL TEE) gesvek ne avns ee densny ecvauced saaees 4. 3865918 $2] VULCAN-HURON MOUNTAINS TO FOND DU LAG BASE. 389 The difference between — aoe gives d=—4, in units e the seventh decimal place. From these values of e ptp y jana d, and from the values of given in the table, the weighted mean logarithms result. The line in ae system having the maximum probable error is Burnt Bluff—Sturgeon. For this 1 line — =4622, yand 5, 4645, giving for the square of the probable error of the weighted mean loga- rithm of this side, 2317, Adding to this the square of the probable error of the logarithm of the length of the standard 15-feet bar, viz, 8.39 (Chapter IJ, § 14), there results for the probable error of the logarithm of the above line, + 48.22, corresponding to the 55t¢¢ part of its length. In the table which follows, the first column gives the names of the stations; the second gives the adjusted values of the angles, taken from Chapter XV, C, § 6; the third gives the error of closure; the fourth gives the logarithm of the side in feet opposite the station on the same line, this value being computed from the weighted mean value of the logarithm of the line Vulcan - Huron Mountains, and the angles in the second column; the fifth gives for each triangle 0? and /; the sixth gives the sum of («?+,3°) from the first triangle in a group having the same probable error for an observed angle, up to the opposite triangle, inclusive, in the direction of the Fond du Lac Base; the seventh gives the quantities; and the eighth gives the weighted mean logarithms of the sides. Chain of principal triangles between Vulcan - Huron Mountains and Fond du Lac Base. ' i : Logarithms Weighted mean ! Stations. i Angles. | Barone of of sides in | a? and B?| © (a?+p?) © iL logarithms of ' | Closure. feet. sides in feet. , { | “ _ Granite Island.............- 45 56 31. "286 i 5. 3212858 412.09 |. ececcccecce) wissen 5. 3212858 ON IGIY aa eavicn de hens aati 28 42 35. 249 +1. 217 STSGSG14 | lpocccmmaeea lt eac0d necen dees ee 5. 1463514 Huron Mountains........... 105 21 00. 131 | = 5. 4490065 33. 64 445. 73 830 5. 4490065 Fves Hill: jiszie visitecinitainctiis 106 46 48. 229 1 if 5. 4490065 BOD: |laside cies oe cease! areal arta 5. 4490065 Granite Island.........----. 50 41 08. 002 +1. 504 HB56460L- [asecescaial| aamex.cecaei|s Penmaes 5, 3564610 Vul0an.c.oceseas hasta soesede 22 32 09. 536 | if 5. 0513906 2580. 64 3066. 06 1301 5. 0513905 "TRUODS vase ves tee wee wene ox 72 00 57.434 l (| 5.0513906 BT, wecntanany cal veweeecun 5. 0513905 Granite Island ..-..--.-..-. 84 47 31. 359 —0. 606 } Di OTS4SG & Priciicnc: sire |ascinimuins sacliieee iste 5. 0718485 ves Hills. uiwecise caececmeae 23 11 32. 437 | J | 4. 6684400 2410. 81 5523. 11 1743 4. 6684399 : Mount Mesuard ..........-. 31 46 33.119 |} f 4. 6684400 UWSOO0 feinieee cee ee atesee oer, 4. 6684399 | Granite Island......... ..-. 27 04 47.180 +0. 794 SGONOEE ewcveneun i[eeweee sexes |k cones es 4, 6051923 TRIOUS vostxnsx seta aosesesent 121 08 40. 078 | 4, 8793674 161. 29 6840. 40 1980 4. 8793673 ss Shelter Bay... cseecnctne ns 32 08 55. 035 | l ( 4, 8793674 TUBE 2G timacde Chemise doninnence 4, 8793673 Granite Island.-..-..----..- | 47 25 32. 690 —2.107 | Da OZOGT4 | ccc sescuscrciailleaicieices: gkieieta: 4 Syeicle nee 5. 0204753 Mount Mesnard ..........-. 100 25 34.115 J | 5. 1461318 15. 21 7977. 86 2184 5. 1461317 Grand Island ......-----.---| 37 02 20.007 |) ( 5. 1461318 TIS AL co satee eciniees a alssisls 5. 1461317 { | Shelter Bay .....-. -2- ee: 123 34 58.404)! +0.691/| 5.2869708 |...... pete icone ee 5, 2869707 Granite Island.........-...- 19 22 43. 707 J 4. 8871679 3588. 0 OL | 12344.28 | 2969 4, 8871678 1 Mud Lake.......-..-------- 60 15 50. 614 | { 4. 8871679 VAG AT, cease crtgteceass| ie ast adie 4. 8871678 Grand Island ......--------- 53 44 03. 301 +0. 936 48549747. legis etetgcies) cewsts sog08]|t veg eses 4, 8549746 Shelter Bay......2022< a4 66 00 07. 273 j | 4. 9092250 86.49 | 12577. 18 3011 4, 9092249 DTS: sewasae eyeaees cee eas 104 13 15. 738 f | 4. 9092250 4, 9092249 Mud Lake....-.------ ie 47 20 58. 286 --1. 102 4, 7893245 4, 7893244 Grand Island 28 25 46. 536 \ 4, 6004197 1513. 21 14118, 48 | 3288 4, 6004196 390 PRIMARY TRIANGULATION. [Cuap. XV, D, Chain of principal triangles between Vulcan- Huron Mountains and Fond du Lae Base—Continued. \ | | \ ' ‘ : Logarithms : Weighted mean Stations. Angles. oo : of sides in | a2 and B? & (a?+f?) i logarithms of | . | feet. p sides in feet. to Pai Soo Sh a ay | oO t we ua | Monistique ...............-- 27 52 09. 853 il (46004197 | 1584.04 )...... 22. | oe 4. 6004196 Mud: Dake 2 s2- cele teens’ | 7956.50.45 | 40.5182, 49111593 |... fee aa Saks 4. 9111521 Divides :secessepeaseeeneses 79 11 00.413 J t 4, 9228928 16.00 | 1571852 | 3576 4. 9228926 An NSS Eesha da Ve leh oe Cumeeeee 2 | Sturgeon .......-0...2200--- | 47 34 46. 646 |, ( 4.9228928 | 368.64 |............ pases 4. 9228926 Monistique ....... .....---- 110 01 51.950 | P —1.5764| 5.0276103 |... joes 5. 0276101 Mnd Lake....... 2.22... | 22 23 22.204 J | 4, 6355208 | 2611.21 | 18698. 37 | 4111 4, 6355206 Seed aoemonliie Sit a - anit VRS aE: _ 2 SF sae a oe Fishdam.............020- »..| 87 10 03,267 |) f 4.6355208 | 767.29 |.....2.0.2..).008 2 4. 6355206 SUANSEOM z 2c0ceiteniencccsies | 89 40.37. 159 | Lo, 244, 4, 8543709 |..------ 2 fe ee eee eee neon ee eee 4. 8543707 Monistique ..... 2... 2... 53 09 20. 156 | J | 4.7576124 | 249,64) 1971530 | 4294 4. 7576122 De NN egbtner ep ae Burnt Bluff........... 2-2-2. | 26 13 40.838)” (| 4.7576124 | 1823.29 |.......22--.[.-2.-22- 4, 7576122 Sturgeon 2.22. ..ceee ees) 62 07 42.374 | A Agassi 25: 05R6978) ids casen.[ecuece ue: sos lgencees 5. 0586973 Fishdam.........2...2-. + 91 38 38.330 , | d 5. 1120682 1.00 | 21539.59 | 4622 5. 1120680 a | net Pie eee eased | 103 35 31.072 |) (| 5.120682 BEiOL | eeatns eeek asoedae _ 5.1120680 Burnt Bluff..........------- 39 43 09, 807 j 42.5584] 4,9299230 |........ se eeeeee cecee[eeeee ees 4. 9299228 Sturgeon ......2.-0... 2220 | 36 41 20. 671 | | 4.9007206 | 800.89; 22366.49-) 4771 4, 9007204 Ford River ..... 22.2. -.2--- 50 39 11.508) « f 4.9007206 | 295,84 |.....2.2222.]----.--- 4. 9007204 Burnt Bluff .. ......2.2---- 52 46 33.233 |' +3.096/| 4. 9134084 |...2.. 22. feces 4. 9134232 Pine Hille: o- siusasaaceccsac 76 34 16,752 J L 5. 0003215 25.00} 320.84 | 4851 5. 0003213 Boyer's Bluff ......... ...- 5417 49.328 |) ( 50008215 | 281.04 .......2 2. Sete 5. 0008213 Ford River ..... 02.2. 2-22 6617.38.71 |4 41.0612) 5. 0524486 |.......... Bisiicie na be |e 5. 0524484 Burnt Bluff.... ...2.2-.2--- 59 24 39.190 J 3 5.0256589 | 153.76 | 705.64 | 4947 5. 0256587 __ — poo 2 Qedar River ........--...2+- | 60 31 03. 738 | (| 50256589 | 141.61 1... seep eee oe 5. 0256587 Boyer’s Bluff . 64 11 43.232 |) 0.3074] 5. 0402656 | ......... insets eRe! 5. 0402654 Ford River ........-.------- | 55 17 15. 282 | | | 5.0007688 | 213.16 | 1060.41 | 5036 5. 0007686 ! Door Bluff ......2222- 20000 | 90 05 10.432 °) (| 5.0007688 OOO: ee eae 5, 0007686 Cedar River ......2.------- | 88:57 62, 982 poate) 4.747980 | oo... ee. esse eeeeee| ce eee 4. 7479328 Boyer’s Bluff .....--..-..--- | 55 56 57. 680 J | 4.9190838 | 204.49} 1264.90 | 5087 4. 9190836 | = ee Me poetctae tet, Sasa! Eagle Bluff .......1...0.---- 56 59 03, 087 t ( 4.9190888 | 187.69 |...2.. J.J. 4. 9190836 Door Blutf.........---.----- 82 29 28, 628 | aay 270 | . 49918805 |... ef. 4, 9918303 Cedar River 40 31 29.532 J L, 4.8083348 | 605.16 | 2057.75 | 5285 4. 8083346 ae i. seal Rochereau ......-..-2--2-+-- 42 14 13,543 |) fi 4,8088348 | 538.24 | ......22-0-[.0ee eee 4, 8083346 Eagle Bluff 91 34 41.205 |' —1.010! Go 4. 9806721 : Door Bluff.......-. -------- 46 11 06.297 |} ( 4.9391208 | 408.04 | 3004.03 | 5522 4. 8391206 Menomonee 44 00 11.785 }) ( 4.8891208 | 475.24 |... eee 4, 8391206 Eagle Bluff —— 59 43 41. 927 { _osse) 4.9336587 |........2.)...02.220-- Lee oh 4, 9336585 Rochereatt...-.--..2-- -2e+- | 76 16 08. 244 | | | 4. 9847319 26.01 | 3505.28 | 5648 4, 9847317 Sean etn aes | 1 5s oe I South Egg .......-..2-.2.05- | 84 27 30.064) f) 4.9847319 4.00 |... 22. Naeem 4. 9847317 Menomonee........ 2-22.22. 40 44.25.6105 $ 1,561!) 4.8560507 | ....2...-[eseeesceceee[sceceeee 4, 8560595 Eagle Blutf .............---- 47 48 05.530 | [ 4, 8564800 4, 8564798 pe eee a ie 5; pone Peshtigo ... 2.0... -2-..--- | 74 47 14. 928 7 ( 4,8564800 | 4, 8564798 ‘South Egg...............-..| 30 51 58.736! ' 40.267/, 4. 5821189 | 4. 5821187 Menomonee..........--.---- | 74 20 46.964 |} [1 4. 8555570 34.81 | 3941.39 | 5756 4, 8555568 ' Débroux . 56 05 41.378 | (| 4.8555570 | 201.64 | 2.2.0.2... belied 4. 8555568 Peshtigo 85 30 00.656 '' —0.549/| 4.9351584 [0002.22 | cece 4, 9351581 South Egg ...-...-.--...2--- 38 24 18. 870 | | 4,7297435 | 707.56 | 4850.59 | 5984 4, 7297432 § 8.) VULCAN-HURON MOUNTAINS TO FOND DU LAC BASE. 391 Chain of principal triangles between Vulean- Huron Mountains and Fond du Lac Base—Coutinued. aay tT a oe NG | - a Fas Edie of 3 30) i Logarithms | ! | Weighted mean Stations. Angles. eae as sides in | a? and 62] & (a?+) | ds | lo ue of : feet. ‘ sides in feet. ae | | NOG es ee ansiews enwas ceeeees _ 43 49 13.449 l { 4. 7297435 ATO. GV ccscins tage Geese us 4, 7297432 Peshtigo: sic) se cts ns wetsws | 56 54 36. 036 S127S 5 458920843 foeeces ec ceen ses ewes see we te 4, 8125340 Dé6broux...........220-20002 | 79 16 11.322 ] | 4, 8817260 16.00 | 5346.20 | 6107 ' 4. 8817257 are re ht emt, | I = ! Red River ...... 2... .2--.- 62 56 29.177 \ (. 4.8817260 NAA! | ccsechu seis Seetee orteeee 4, 8817257 Gales 25¢ Soeaes. aul Veda e584 76 05 12.533 | p 1-755 4 ' 4,.9191394 | aaNet ls tad | Paces uy Memeeetee 4.9191391 ' Peshtigo: sss: ctecackeceueene 40 58 19. 267 a) | 4.7487699 | 590. 49 6051. 18 6284 4, 7487696 ‘ soeee ats : b Aad Reel snc oe ahd te AE SR in oe Tee : ly kas ee al | 1 Little. Daiby es cece eid | 41 41 41. 262 | f 4. 7487699 | 566.96 |...22.. 02... | ealeses | 4.7487696 (ARE BUIOE icin conte abioeneere 55 51 59. 300 i —0. 797 1 4, 8437322 |. ....e. eee eee cee eicibaee atest ' 4, 8437319 CABS cs cckcundss te ccdonanmes | 82 26 20. 353 | U4, 9220502 | 7.84 | 6615.98 6425 4, 9220499 | LE a a | . { : 1 Red Banks...-.-........---- 89 51 06. 011 | ( 4, 9220502 Os 00: |isceisex resco cic | a siete ae 4, 9220499 ae jAittle Tall... sccasencsanece 59 23 47.745 | > --1. 693 | | 48569001: |ecvecs vnalseeere tees. |ameresed 4. 8569088 (ROG RIVER jacsiie aot pacaecnee 30 45 06. 968 J ( 4. 6307454 1253. 16 7869. 14 6738 4. 6807451 a_i Iti Fe eS peas, | eben, Grasse no ene oa meintede aes se csi*h wots ee Fort Howard ...-. ......-.. 39 02 17. 789 \ { 4. 63807454 G7 0B, [jae ste ecicee Ne. ate yagi | 4. 6307451 Little Tail .... ....-..----- 61 26 00. 012 ( —l. 5324 4. 7751396 is fees. 2 Benes alae sete 4. 7751393 Red Banks..........-.....-. 79 31 42.790 | J l 4, 8242222 15. 21 8555. 16 | 6910 | 4, 8242219 Es, tinged Ys bs, fess hi pitta ana ec okt | Brueé: ccercssccccseuusus cere 78 36 40. 606 . } 4, 8242222 | 068: | eaerrewe yx eersll s eeiye: 2 4, 8242219 Fort Howard .....--.....--. 67 24 47.297 | > -;-0.370 4. 7982007 ,.----- -- |------- 2-22 -Jeeee eee 4. 7982004 EX 6 Va ices cetrseeda nated 33 58 32. 648 J () 4,5801471 | 973.44) 9546, 24 7158 4. 5801468 eater ea SASS er tei eas apie cement East Depere ....++....-+-++- 46 05 52. 278 4,5801471 | 412,09 | wining Hage GS aes se | 4. 5801468 BTUG622:ccs2.cctnsesatetees 47 02 51.534 +0. 106 4. 5872052 [eaaeesatalee tats oe ys 4, 5872049 Fort Howard ...-.------ -- 86 53 16.534 | J l 4, 7221005 1.00, 9959.3 7261 | 4. 7221002 * ' j ae sie | (ee Fe z ONCE B30. csinte: aid hee emeecm cis 46 36 48. 541 ) 4. 7221005 | POG, 01 biseave ooawel sound nox 4. 7221002 East Depere ..---- ---..---- 75 55 41. 428 +40. 6204 A, BITAD20 | xawwewes ns Goeth ce tsk toma 4, 8474917 Brie enciccecccsoetast nee 57 27 30. 768 f 4. 7865526 182. 25 ! 10537, 59 7405 4. 7865528 West Depere ....... ....--- 58 46 55. 480 | f 4. 7865526 N68) S41 oc ceaigannda|seeihese 4. 7865523 East Depere.......-------+- 46 52 38.351 | > —1. 052 4STUTIA2L lens becsies | brea vinta cna grad ee 4. 7177418 ON Be es oie a vc oem eciscnwm ee 74 20 26. 894 | J | 4. 8380581 34,81 10736, 24 7455 4. 8380578 Galuimétinnccsene eeickseess 38 08 22. 278 | 4. 8380581 TUB OAS emsaren waee~ |oemee 4, 8380578 West Depere .......--- -- 2 92 07 27. 280 —0. 809 5. 0470686 |.......--./..2..2.-26- dliciehieegused de 5. 0470683 East Depere ......--.-----+- 49 44.11.8238 |) —- 4, 9299372 316.84 | 11771.32 | 7713 4, 9299369 Breed Oi ache cotceweeecae os 113 58 43. 640 l (| 4,9299372 BRED lassi amceweadiae an wane 4, 9299369 Calumet .......------------- 22:17 56. 453 —0, 701 J A DABQTTD: sceceatexe lhe sdecsReavenllecies 4 =k 4, 5482776 West Depere .....--------+ 43 43 20.397 J 2 | 4, 8087162 | 484.00 | 12841.81 | ~ 7857 4. 8087159 et on ——$—— . i Aare en teal peices ipl s aceoberande ene 55 82 42. 010 (| 48087162 20736: soveeceaaey| iran sar 4. 8087159 Galnioleeexoepeecet tects 59 51 21.493 +1. 625 4, 8293868 | ...------|--22 22 eee ee] ee ee 4, 8293565 Freedom....------2-200--200+ G64 35 57.423 4. 8483348 100.00 | 12649.17 | 7933 4, 8483345 = to | ! | Oshkosh .....22.-22+2--0200 52 17 42.181 4, 8483348 POGIRO (i cceun cardadle dea Soke 4, 8483345 Clayton ...-2.2-2--22--2--+- 80 03 37. 269 —2, 285 4.9434969 |.-.---2---)---e ee eee Jenne eee 4, 9434966 Calumet ........2-.0-2-0--5 47 38 41. 628 4, 8186992 368. 64 | 13283.50 | 8092 4, 8186989 es Stockbridge .-.....--------- | 59 30 17. 969 4, 8186992 LBB. FGt lciceisicre catereiell cleapateis 4. 8186989 Oshkosh | 62 32 03,382 | > —0. 124 4. 8314207 | .--------|-e- eee eee e fee eres 4, 8314203 | Clayton. ......- arp _ 57 57 39. 541 4, 8115921 174, 24 13611. 50 8174 4, 8115917 Seay ed aS at = ef We be os a es | Paycheedah ......------2++- 40 41 22.751 4, 8115921 600. 25 |.......----- preteeeee 4, 8115917 | Stockbridge -- .-.| 54 30 03.489 | > +-0. 900 4.9080618 /...-------|eeeeee eee ee epeee eee 4. 9080614 Oshkosh ....------++-+------ 84 48 34.991 4, 9955867 4,00 | 14215.75 8325 4, 9955863 ») 39 Chain of principal triangles between Vulcan— Huron Mountains and Fond du Lac Base—Continued. PRIMARY TRIANGULATION, [Cuar. XV, D, § x, Stations. | Angles. ee = Moetaeots a? and ms (24-82) 1 Teeaton of | feet. sides in feet. cr Fe all ee ees “Stockbridge | at ao osear | L) atennna iiss an 6405.18 8oas Tome ae a ! = oaths & gesee * a 6 a ee eee oe helps ee |) ee i Taydheedith ae 66 40 21, 866 ; , | 4, 9301566 "sL.90 | 805.51 | 8703. | 4. 9301562 f ee re a, 8 ae ees ea. Bh a 2 being the reading of the finding-microscope on the left-hand object, the graduation increasing from left to right. The probable errors of the coefficient, 1/.87, and of the constant angle, 3279, resulting in their determination are +0”.02 and £1°, respectively. For No. 4 the corresponding correction is— ¢=+1/.59 cos (243 at 67°) sin 5a +1/.16 cos (6243443299) sin 3a +0/.74 cos (92+ ; 5 4-4-3939) sin 2a «being here the reading of the finder, which for either instrument is 180° from microscope A. §6. 7,8.) FOND DU LAC BASE TO CHICAGO BASE. 401 The probable errors of the first, second, and third coefficients in this series are +0/’.05, £0.08, and +0/.16, respectively; and the probable errors of the corresponding constant angles are +3°, +149, and +16°. The numerical maximum of ¢ in the last expression is— €=42".45 c=+42".45 for z=either 65°, 185°, or 305° and a=either 71°, 191°, or 311° e=—2".45 for z=either 16°, 136°, or 256° and a=either 49°, 169°, or 289° The independent probable errors of a single micrometer-reading on the horizontal circle and a single bisection of an object with the cross-wires of the telescope have been found to be for No. 3 + 0.22 and +0’.34, respectively. The corresponding probable errors for No. 4 have been found to be +0.27 and £0.36, re- spectively. The theodolites when in use were mounted on trivets or stands, which were securely attached to the station tripod by means of screws or bolts. For those instruments having no provision for shifting-their horizontal circles, stands with movable parts were provided, so that the zeroes of the cireles could be fixed in any desired azimuth. The position of the center of the instrument, with reference to the vertical line through the geodetic point, was determined with the same care and in the same manner as the position of a signal. MEASUREMENT OF HORIZONTAL ANGLES. § 8. The main features of the methods followed in the measurement of horizontal angles were in accordance with the requirements stated in Chapter XV, B, § 5, and need be only brietly recapit- ulated here. In all cases combined results, 7. e., means of a positive and an immediately succeed- ing negative measure of an angle, were obtained. With non-repeating instruments the stations situated about the horizon in the order A, B, C,... 2, of their azimuths were usually pointed to in the order A, B, C,...H#; H,...C, B, A, when the horizon was not closed. When the horizon was closed, they were pointed to in the order A, B, C,...H, A; A, H,...C, B, A, in case they were all visible, and not so numerous as to make too long an interval between consecutive point- ings to any station. Otherwise, as many consecutive stations were pointed to as could be con- veniently at one time, and the remaining angles making up the horizon were measured as oppor- tunity offered. In the later work, in order to avoid an excess of pointings to any one station, the initial station was changed from A to B, B to C, &c., whenever the circle was shifted in azimuth. With the only repeating instrument used the angles between consecutive stations were measured separately in sets of five repetitions. Tle mean of a set obtained by shifting the verniers in one- direction and an immediately succeeding set obtained by shifting them in the opposite direction was always taken in order to eliminate as well as possible any effect of twist of station or instru- ment. Except in that part of the triangulation lying between the lines Minnesota Junction - Horicon and Warren—Fremont the horizons at all statious were closed. Prior to 1877 few angles other than those between consecutive stations about the horizon were observed. During 1877, 1878, and 1879, however, the work was strengthened by the measurement of many sum-angles. In instrumental] manipulation errors due to Jack of collimation and inequality in heights of wyes were eliminated by frequently transiting the telescope and obtaining the same number of measures in the two positions of the telescope. To eliminate the effect of accidental and periodic errors of graduation the zero of the circle was frequently changed in azimuth, the number of such changes varying from six to twenty. Instructions given in 1872 required the elimination of periodic error by systematic changes of zero, but they were not fully complied with by all the observers before 1876. On this account the changes in position of the azimuth circle and the number of measures in those different positions were not for all observers such as to secure the best elimination of periodic errors before that year. Au idea of the extent to which angles meas- ured previous to 1876 may involve errors of the periodic class can be formed from the following statements relative to the manipulation of each instrument, and froin the periodic errors of these instruments already referred to. 51L 8 402 PRIMARY TRIANGULATION. [Cuap. XVI, B, Repsold theodolite No. 1, 1873 and 1874.—Elimination of periodic error not systematic. Angles were observed on eight to sixteen different parts of circle. Repsold theodolite No. 1, 1875.—Elimination of periodic ertor nearly complete except at Stony Point, N. Y., Duck Island, and Scottsville stations. Angles observed on twelve or more different parts of the circle. Troughton & Simms theodolite No. 1, 1874 and 1875.—Elimination of periodic error partial. Angles observed on eight to ten different parts of circle. Troughton «& Simms theodolite No. 2, 1874.—Elimination of periodic error not generally com- plete. Where elimination was not complete, angles were corrected before using them in adjust- ment. With same instrument in 1875, elimination was systematic. Pistor & Martins theodolite No. 2, 1875.—Elimination generally systematic; when not so, cor- rections were applied to angles before using thein in adjustment. Gambey repeating theodolite No. 1, 1873.—Elimination systematic, except at stations Horicon and Lebanon. Angles observed on eight to sixteen different parts of circle. Gambey repeating theodolite No. 1, 1874 and 1875.—Elimination not generally systematic. Angles observed on ten or more different parts of the circle. It is thus seen that the angles likely to be vitiated by periodic errors are those measured with the Repsold, Troughton & Simms No. 1, and the Gambey theodolites prior to 1876. With the Rep- sold and Troughton & Simms theodolites the maximum errors possible from this cause in an angle are +2/.25 and +1/.12, respectively. (See Chapter XV, B, § 4, and Chapter XIV, B, § 6.) If we suppose only such angles as might produce these maximum errors had been observed with these instruments, the most unfavorable of all suppositions, and that all errors between zero and the limiting values, + 2/.25 and +1/.12, are equally probable (also an unfavorable supposition, since an inspection of the functions expressing the periodic errors shows that small errors are more likely to occur than large errors), the probable error of any such angle would be, if measured with the : DOD a) sl! ; 3 “12 Repsold theodolite, + 5 . or if measured with the Troughton & Simms theodolite, ie . These probable errors are applicable, however, only to the measurement or mean of the measurements of the angle on one arc of the circle. In case the angle were measured the same number of times on n different but random ares, the probable error of the mean angle would be, under the same unfavorable Dil & “ supposition as to the law of error, approximately Lee in the first case and + 9 oe in the second. For n=9, which is about the minimum number of changes of the circle with the Repsold and about the average number with the Troughton & Simms theodolite, the above expressions give £0/.37 and +0/’.19, respectively. The angles which could produce the first of these results, with the Repsold theodolite, are 84°, 96°, 264°, and 276° (Chapter XV, B, § 4); those which could produce the second result, with the Troughton & Simms theodolite, are 33°, 87°, 153°, 207°, 273°, and 327° (Chapter XIV, B, § 6). The angles actually measured with these instruments were of all magni- tudes from 0° up to 360°, but the greater portion of them were near 60°. It may be concluded, therefore, that the probable errors, arising from periodic errors, of angles measured with the Rep- sold and Troughton & Simms No. 1 instruments in this triangulation are on the average consider- ably less than +0/.37 and +0.19, respectively. The periodic errors of the Gambey repeating theodolite have not been determined, but judging from the discrepancies between the mean and individual results for angles so observed as to give means free from such errors, they cannot be large, and the probable error from this cause of an angle measured with this instrument may be safely assumed as not exceeding the larger of the two probable errors derived above, viz, +0/.37. The angles measured prior to 1876 and likely to be affected by periodic errors are confined chiefly to the systems of triangles between Fond du Lac and Chicago Bases and between Buffalo and Sandy Creek Bases. About one-third of the stations between the last two bases were reoccupied, however, during 1877 and 1878. Nearly all angles that had been measured previously at these stations were remeasured, and many new angles were also observed, care being taken to make the elimination of periodic errors in the mean angles certain. The method of eliminating periodic errors which has been carefully adhered to since 1876, is to obtain the same number of measures with the zero of the azimuth circle in each of n successive § 8.) FOND DU LAC BASE TO CHICAGO BASE. 403 positions such that 2 times the angular distance between any two consecutive positions equals the angular distance between consecutive microscopes. The number x is thus determined by the total pumber of measures or results to be obtained. If, for example, 16 combined measures are to be made, x must be 16, 8, 4, or 2. Usually »=8 was used for 16 results. Errors of run were determined by frequent comparisons of the micrometer-screws with spaces of known values on the circles, and corrections to the observed angles were applied whenever the effect of such errors was appreciable. The number of combined results required in the measurement of angles varied with the dif- ferent instruments, the aim being to have these numbers such that the mean values of all angles would have the same weight, and that the probable error of a mean angle as derived from the dis- crepancies between it and the individual measures should be about £0.25. The following table shows the number of combined results required with each instrument in each of the different years during which it was used. These numbers were derived iu the manner explained in Chapter XV, B, § 5, from the probable errors of single combined results. With Troughton & Simms theodolites Nos. 3 and 4, in 1876 and the early part of 1877, while they were under examination, it was deemed best to obtain twenty-four combined results on a fully measured angle. After being thoroughly tested their performance was found equal to that of Troughton & Simms theodolite No. 1, and the number of results required was reduced to 16. Number of results required for weight 1. Instruments. 2) SS Se mapas Se 1873. | 1874. | 1875. | 1876. | 1877. | 1878. | 1879. Troughton & Simms No. 1........----+-+--+-2+-/------ 16 16 16 16| 16 16 Troughton & Simms No. 2........---------+-++-|--+--- 20 Ge win chor awaosilt peal ees. -| Troughton & Simms No. 3 .. ...---.--.-------- faebias| nese [reniiee 24 24,16 | 16 16 Troughton & Simms No. 4......------.-----220-|---22+ [seen epee ee ee 24 pe 16 Pistor &, Martinis No.2 .223.22seceseesececnaegexexeyee|=sese% 16 WB [es eceees| essen: 16 Repsold Need. eccsneccvaceeyscseees ae weeeee sn 20 20 24 D4 lssasicealicceces 16 Gambey No.1. ....-..---------- eee eee eee eee eee 16 20 WG. onc arcis acdccinss 2) ewes] esses During the measurement of angles the theodolite was always shielded from the sun by a tent specially designed for the purpose, or by an equivalent covering of canvas. Observations were made only when the atmosphere was steady, so as to give sharply defiued images of signals. On high stations not sheltered by timber, angles were measured only when the wind was very light or calm, a condition which in some cases had to be waited for many days. The rapidity attained in observing has varied somewhat with the conditions, distinctness of signals, &c., and with the ob- servers. Using non-repeating instruments, however, under ordinarily favorable circumstances skilled observers have made an average of one pointing per minute, such pointing including the fixing of the cross wires of the telescope on the image of a signal and readings of the two or three micrometers. With the repeating instrument used, the average time required to make a set of five repetitions was four minutes. As a rule, the days when any observations could be made did not furnish suitable conditions for a longer interval than two hours, and this occurred usually between 2p. m. and sundown. Occasionally, cloudy days furnished an interval of steady atmosphere long enough to enable the observer to make all the requisite measures at a station, but advantage was seldom taken of such opportunities, as it has been deemed best not to have important angles depend wholly or mainly on measures made during one day. Below are given two tables, the first showing the number of stations occupied and the number of angles measured by the different observers in different years, and the second the number of stations occupied and the number of angles measured with the different instruments in different years. 404 PRIMARY TRIANGULATION. [Cuap. XVI, B, C, Table showing number of stations occupicd and number of angles measured by different observers in different years, 1873 1874 1875. | 1876. 1877. 1878; 1879. 28 | | te a oa oe oa oa os om oe et oa . OG. SO oOo. 190 eo. | 6 on: 8 e,/e,]/e.,/6. Observers. ealegi ed agi ed | eg) séle¢|e2le¢) salsa] 58 | a3 i 22|\)26 28,2512 45 '242/)4n/22 26 )242) 24a) 22) 25 : Be l|@ae 2s a|)eé€n|2a/| 28 | da] 88 aq Bal aa Bal dq Belge. se a1 ES | 22 | 28] Be) 28 | 2 24) Bt) 5S ps 1 A’ |a [A 4 4 4 A A A A A A” \% Gi My Wisnt®s 2a. neloavccectecc|sccissss| Fences 7 29 9 45 4 17 15 76 11 53 16 87 Any Ris, PMN Givccexre cic aciates eas 10 34 8 BS. peseees| oat cee 3 17 13 60.) seecsteesics 6 47 GAs MaErh cs mceceoass vase el acacecle cena eeeetl aseseal ee Saee (aeeeee 1 TZ Wiseeece wjarmat, | aaa all melee! Users alerts E. S. Wheeler ...--.-...------|------|---- 2+ [eee ee [eee ee e]ee eee [eee ep eee eee pene eee eee ee [eee ee] eee eee fees 1 3 R. S. Woodward .-...--------- 9 26 5 20 10 33 3 12 VL. 68 4 35 16 | 100 J. H. Darling..--. --..-- .+--|------|------|+-- 2+ + [eee 2 6 26 2 9 8 RN eames | eres 14 94 Th RUSSO sexarccossteccietndag [be reer Samet 5 15 8 DDS sare faa oudeteae a icvbvcvdag lisers: avesse evoiehelased ateie'zheny | andi eeSsa NV; Ay Motealb. scecsccescse sen |eeds ce] ecccucecusess [ateces 6 BAP | ecetees is) recta Makes elicit aca lee en cle cise (Pesce Table showing number of stations occupied and nunber of angles ments in different years. 1873. 1874. 1875. 1876. 1877. 1878. 1879. t eo. \5 2, (Sele .| Sele, | eele.| oes. |S. 18 Instruments. oe a2 we |e Bl ug ae ug a2 na Pa aD fe aa 2H) 22) 25/28/55) 22) 52) 22) 24) 42 | 24 | 22 | 3H 2a|\22)|2e|22|2a|22)88! 22/22) 28) ee) 28 | 22! 5a) Se | Se) Fo | 2s | Sn | 22 | Fal es) Fa) 2s | ba | Be | A A|A a |A A |A 4 A a A a a Trou hton & Simms No. 7 29 9 45 4 17 15 76 11 53 8 50 Troughton & Simms No. 2..-.|.-.---|------ 5 20 10 OO leeisecu (ioc. sa] eeeteeoxeeed Troughton & Simms No. 3 Troughton & Simms No. 4 gore sen tecthi| sarsia-cie'l sa oieiac | sieie srasai| Simeise [areata ss 2 9 8 AG Jsacccslescons| 14 94 Pistor & Martins No. 2 .....--|.-----|------ ar 6 34 1 MR Nrasicie di | araraicyarsy ee scieel| eke 14 84 Repsold No. 1...-------------- 10 34 8 38 6 26 3 17 Gambey No. 1 .-.---.--.------ 9 26 5 15 8 29 MEASUREMENT OF ZENITH DISTANCES. § 9. Previous to the year 1876 but little attention was paid to the measurement of zenith dis- tances, although enough were observed to give the elevations of nearly all stations occupied. On nearly al! Jines of the triangulation measured during 1876 aud subsequent years, however, reciprocal zenith distances were observed. These were not generally simultaneous, no attempt having been made to render them such. They were observed usually during the afternoon from 1 to 4 o’clock, or before the atmosphere had become sufficiently steady for the measurement of horizontal angles. Ten separate measures of tbe zenith distance of each station visible from the one occupied were required, and they were generally distributed over several days, so as to eliminate as much as possible any unusual effects of refraction. C.—MEASURED AND ADJUSTED ANGLES BETWEEN THE LINES MINNESOTA JUNC TION-HORICON AND MICHIGAN CITY-BALD TOM. § 10. The following tables give an abstract of the adjustment of the triangulation comprised within the above-stated limits. A sketch of this triangulation is given in Plate III. The adjustment is made in two parts, viz: Section VI, extending from the line Minnesota Junction —Horicon to the line Warren—Fremont, and Section VII, extending from War-en- Fremont to Michigan City-Bald Tom. The scale of weights assigned to observed angles is as fallows: Weight 1 to mean of— 16 combined results by Gambey theodolite No. 1 20 combined results by Repsold theodolite No. 1 20 combined results by Troughton & Simms theodolite No. 2 measured with the different instru §§ 9, 10.] FOND DU LAC BASE TO CHICAGO BASE. 405 Weight 1 was also assigned when the number of combined results differed by no more than one-fourth of the above numbers, but when the number of combined results differed from the staudard number by more than one-fourth that number, a weight equal to the ratio of the given number to the staudard number was assigned. For a detailed explanation of the tables see Chapter XIV, C,§ 7. Here, however, as in Chapter XV, ©, the column headed “No. meas.” gives the number of combined results, a combined result being the mean of a positive and a negative measure. The quantity designated “Range” is the difference between the greatest and least combined results. One condition, viz., that of a sum-angle at station Michigan City was violated by the above division of the triangulation. This sum-angle formed the link connecting Section VII and Section VIII of Chapter XVII, C. The locally adjusted angles, with their resulting weights, were used in computing the general adjustments of Sections VII and VIII. SECTION VI.—Triangulation from the line Minnesota Junction— Horicon to the line Warren— Fremont. Note.—For angles at stations 1 and 2 see stations 51 and 52 of the preceding section of the adjustment. LEBANON—3. {Observer, R.S. Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, May, 1873.] Angle as measured between— Notation. | No. meas. | Range.) Wt. (v) {v] Corrected angles. { @ 4: “ “" ” ov “ | Minnesota Sunction and Woodland. 63 10 05.353 Bite 16 3.3 1 |eseeeeeee +0. 215 63 10 05. 568 Horicon and Woodland ....-...----- 34 04 48. H6 32 16 3.5 De | sieccieraiz: paisa . —0.510 34 04 48. 436 Woodland and Erin.....-..----..--. 62 54 02. 743 33 16 4.0 OL fee eee eee —0.131 62 54 02. 612 Woodland and Delafield ..-..-.-.--- 112 28 47. 435 3344 | 16 3.1 | ban epee re +0. 255 112 28 47. 690 WOODLAND—4. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, May, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. or “" ” or ” | Erin and Lebanon ........---------- 80 57 44. 418 4 20 3.3 i 80 57 44.543 Lebanon and Horicon........-.-. -- 111 36 04. 168 42 20 5.9 1 111 36 03. 873 ERIN—5. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, May, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | Corrected angles. o “ “w or a" | Milwaukee Court-House and Lisbon.. 15 26 29. 895 51 11 4.5 0.5 15 26 30. 224 Lisbon and Delafield...-....--------- 54 16 34.192 5g 20 4.8 54 16 34. 501 Delafield and Lebanon........------- 86 50 06. 367 5s 20 4.8 86 50 06. 622 Lebanon and Woodland......-------- 36 08 13. 362 54 20 6.0 1 36 08 13. 487 DELAFIELD—6. (Observer, R.S, Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, June, 1873.) Angle as measured between— - Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. @ # “ “we mw oF “" a Lebanon and Lisbon...--..--.-- ee 84 12 49.215 6142 16 3.6 K ipwwanieegas +0. 255 84 12 49.470 Evin.and Lishew ..<«-~0-.cawiwes sxe 40 37 39. 671 62 17 6.3 D |hetecaiesouserene +0. 167 40 37 39. 838 Lisbon and New Berlin.-..-..------- 58 18 36.172 6 16 3.4 1. lesen smaccan +0. 871 58 18 37. 043 New Berlin and Waterford .......--. 41 54 11.647 64 16 4.9 TL |eamcsmcnan +0. 002 41 54 11. 649 406 PRIMARY TRIANGULATION. [Cuar. XVI, C, SECTION VI.— Triangulation from the line Minnesota Junction— Horicon to the line Warren-Fremont— Continued. LISBON—7. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, May and June, 1873.) i Angle as measured between— Notation. | No meas. | Range.| Wt. (v) {v) Corrected angles. oO t “a a “ a oO # “we Milwaukee Court-House and New BGP co cieiad decent” wesin sarees Cskeihes: 62 14 26. 525 vil 10 2.7 0.5 —0. 062 —0. 371 62 14 26. 092 New Berlin and Delafield........... 57 18 45. 457 72 20 5.6 1.0}; —0.031 | +0. 624 57 18 46. 050 Delafield and Erin .-.....-..-....--. 85 05 46. 407 73 20 6.0 1.0); —0.031 ] +0. 068 85 05 46.444 Erin and Milwaukee Court-House .. 155 21 0], 888 74 4 6.6 0.2 | —0.153 | —0. 321 155 21 01.414 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 0.7(71) +0. 2(72) +0. 2(73) +0. 0554=0 0. 2(71) +1. 2(72) +0. 2(73) +0. 0554=0 0. 2(71) +0. 2(72) +1. 2(7a) +0. 0554=0 NEW BERLIN—8. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, June, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) °o a “a “w Calcdonia and Waterford .... ..-.- 50 25 16.426 8&1 20 7.3 VO vs csevins ¢ Waterford and Delafield) .-.....--.- 100 47 03. 312 82 15 6.4 1.0) panzsae Delafield and Erin .......-.-.....-. 47 09 47. 945 83 9 4.8 OB ive eeeesavece Delafield and Lisbon.............-.. 64 22 37. 004 83-4 18 5.2 | 10 |......--- Lisbon and Milwaukee Court-House. 74 44 34. 565 85 li 7.5 Corrected angles. ° t wo 50 25 16. 364 100 47 03. 314 47 09 48. 456 64 22 37. 685 74 44 34,520 WATERFORD~—9. (Observer, R. S. Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, June, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. {v) [v] Corrected angles. ° ¥ uw aw aw fo} a “ Delafield and New Berlin ....--..-.. 37 18 45. 959 91 16 3.6 lo |aeeeeeseicd +0. 002 37 18 45. 961 New Berlin and Caledonia .....-..-. 88 10 53. 829 92 16 2.8 1 +0. 251 88 10 54. 080 New Berlin and Dover.......-...-.. 126 53 43. 625 9243 16 3.2 Lo |ksesee ----| 0.313 126 53 43, 312 CALEDONIA—10. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. oO ‘ a a uw fe} t aw Somers and Dover ..........-------.- 70 28 18. 293 101 20 6.1 De etwatet sees +0. 855 70 28 14.148 Dover and Waterford -- 49 21 10.503 102 20 5.5 Te faScetesese —0. 313 49 21 10. 190 Waterford and New Berlin ....-..... 41 23 50. 823 103 20 WA De) iparoeesge —0. 062 41 23 50. 761 DOVER—11. {Observer, R. S. Woodward. Instrument, Gambey 10-inch theodolite. Date, July, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. is Se ° a wn a uw fo} t . ww Waterford and Somers........-...- 151 26 04. 950 1li4e2 16 4.6 Uo VS ecereeteoy —0, 313 151 26 04. 637 Caledonia and Somers ...........-.. 59 30 02. 222 ll2 16 4.1 Me bees eed +1. 168 59 30 03. 390 Somers and Bristol ................. 54 44 08,372 11s 16 3.1 D beues execs =1, 124 54 44 07. 248 § 10.] FOND DU LAC BASE TO CHICAGO BASE. 407 SECTION VI.—Triangulation from the line Minnesota Junction - Horicon to the line Warren—Fremont— Continued. ® SOMERS—12. {Obsorver, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1873.] | Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v) Corrected angles. ° ¥ “a a a“ ° t : a Benton and Bristol .....--.......... 57 50 16. 650 121 20 1.7 1 +0. 052 57 50 16. 702 Bristol and Dover... - -. 50 44 36. 660 122 20 6.2 1 —1L.124 50 44 35. 536 Dover and Caledonia..............-. 50 01 42. 189 123 18 8.5 1 +0. 855 50 01 43, 044 BRISTOL—13. (Observer, R. S. Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, July, 1873.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. °o # a “uw aw ° ‘ a“ Dover and Somers .....-....--..---- 74 31 18. 887 131 16 3.1 1 asegeecccef 1124 74 31 17. 763 Somers and Antioch .-......-.-----. 127 57 24.116 13e48 16 5.2 1 127 57 24. 168 Benton and Antioch .............--- 67 57 04. 784 133 16 2.9 1 67 57 04. 394 BENTON—14. (Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, July, 1873.} Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v) Corrected angles. Oo e aw “ oO yd “ Warren and Bristol .... -.-.----.--- 115 07 28. 300 14i42 20 5.0 1 115 07 28, 182 Antioch and Somers .-.---.--------- 111 54 54. 150 14243 20 5.8 1 111 54 53. 930 Bristol and Somers ..-..-......-..-. 62 09 23.748 143 20 5.0 1 62 09 24. 020 ANTIOCH—15. [Observer, R. S. Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, J uly, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v7] Corrected angles. : °o t uw aw a“ o t “ Bristol and Warren.......-.-------- 115 44 30. 809 15i+e 16 4.3 Bo leesxideekal —0. 337 115 44 30. 472 Benton and Warren ...---.-.-.----- 53 27 04.119 152 16 2.9 1 53 27 04. 339 Warren and Fremont..-..-..-..---. 55 31 49. 442 153 14 4.4 1 55 31 50. 086 WARREN—16. [Observer, A. R. Flint. Instrument, Repsold 10-inch theodolite. Date, August, 1873.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. oO ‘ a“ a“ Oo °o ‘ “a Fremont and Antioch......--.-.---- 67 11 46, 258 162 20 5.1 D laceaeareea a +0. 644 67 11 46. 902 Antioch and Benton ......-..-.----- 61 10 58. 008 163 20 6.5 M, |laeriesereseisians —0. 118 61 10 57. 890 FREMONT—17. [Observer, R. S. Woodward. Instrument, Gambey 10-inch repeating theodolite. Date, August, 1873.) Angle as measured bgtween— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angle. “ or un" +0. 644 57 16 23. 597 eo uw ‘ Antioch and Warren ..-.----------- 57 16 22. 953 17, 16 3.1 A.) oiave eeiseutad 408 PRIMARY TRIANGULATION. (Cuap. XVI, C, Numerical equations of condition in the triangulation from the line Minnesota Junction — Horicon to the line Warren - Fremont. € SIDE-EQUATIONS. VIT. (40) — 27.8529 [6.] — 3.3114 [63] +27. 3494 [7] +29. 1559 [75] — 87. 4835 [23] + 67.9611 [8344] —13. 061-0 VIII. (100) +206. 1077 [5)] —15. 1430 [52] —129. 8837 [7a] +10. 0984 [544] —28. 3088 [85] +24. 5415 [6:] —12. 9988 [65] —152. 4494 [7] —129. 8837 [72] —-30. 831=0 NotE.—In the solution for determining the general corrections each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. I. II. Ill. Iv. — Vv. VI. IX. X. XI. — XII. XIII. XIV. XV. XVI. XVII. [11] +[22] [21] +32] [35] +41] ANGLE-EQUATIONS, +[3i42] —[32] +[42] +154] [33] +0344] +[53] +[6i42] —[62] [52] +[62] . [65] +[72] {64] +[8:] [8] +[92] +[73] +8344] +£91] +[103] —1.154=0 +1. 100=0 —0. 119=0 —0.730=0 —0. 543=0 —2.175=0 —0. 005=0 —0. 128=0 [92] 49245] +10] +[11i40] —[1l2] +2. 359=-0 [10,] +112] [125] (1e] +1122] +L131] [321] +(13:43] —[133] +[145] —2,879=0 +3, 373=0 —0.766=0 C13.) [145] + 14e48] +£15)4] —[ 150] +1. 438=0 (143] —[14243] +0 14142] +[152] {153} +[16.] +[171] +[16;] —0. 476=0 —1.931=—0 Values of the general corrections in terms of the correlates. (tJ =+1. [a] = 2] =+1 [3142] =+1. (3:] =—1 [33] =+1 [3344] = [4] =41. [42] =+1. [5.] =+4. [i] =—0. [53] =+1. (54) =+41. (6.42) =+1. [6] =—1. (6) = [6,] =+1. {%1] =—0. [%2) =—0. [75] =+0. (8) =+41. [82] =+1. 00000 I - 00000 T 00000 I . 00000 T - 00000 TII 00000 III 00000 II 12216 VIII 15143 VIII 00000 IV 00000 IIT 00000 IV 00000 IV 00000 IX 22222 V 11111 V 88888 V 00000 X 00000 TX +1. 00000 II +1. 00000 I —1. 00000 IV +1. 00000 IV +1. 00000 V +1. 00000 V +1. 00000 VI —0. 22222 VI +0. 88888 VI —0.11111 VI —0. 69632 VII —0. 08279 VII —0. 31392 VII --0. 52678 VII +0.57194 VII -40. 24542 VIII —0. 12999 VIII —1.79417 VIII —0. 67142 VIII —0. 67142 VIII 3% § 10. | No. of equation. 1. meee PES ae So NS PON aAnEen 52 eg i FOND DU LAC BASE TO CHICAGO BASE, Values of the general corrections in terms of the correlates—Continued. =+42, =~2, =+3, =—0. [&] =—4. [8344] =-+1. eT 4, (2) =+1. [9%] =+1. [9245] =-+1. (og =, (iG) eed, [L0s;] =+1. (lhi4:)=+1. iia: =a (ig) sk: [12] =+1. (13) =, (125] =+1. fia Set [13243] =+1. (ie 1, (14i42] =+1. ( 14243] =-+1. [143] =+1. (15.42] =+1. (iter =a, (i) =e, [ie =e. [163] =+1. (1%) =+1. 37418 VII 00000 VI +1. 69903 VII -+0. 10098 VIII 56618 VIII 00000 IX 00000 X —1. 00000 XI 00000 XI 00000 XII 00000 XI 00000 X 000U0 XT 00000 XT +1. 00000 XII 00000 XII 00000 XIV 00000 XIII 00000 XII 00000 XIIT 00000 XIV 00000 XIV +1. 00000 XV 00000 XVI 00000 XV —1. 00000 XVI 00000 XIV. —1.00000 XV _+1.00000 XVI 00000 XV 00000 XV +1. 00000 XVI 00000 XVII 00000 X VII 00000 XVI 00000 XVII Normal equations for determining the correlates. . 15400 +4, 00000 I . 10000 —1. 00000 I . 11900 +3, 00000 TIT . 73000 —1. 00000 IIT . 54300 —1. 00000 IV .17500 —0,.11111 V . 32653 -L0. 69632 IV . 30831 —0, 24542 IV . 00500 -++3. 00000 IX . 12800 -+3. 00000 X 35900 —1. 00000 X 87100 —1. 00000 XT —1. 00000 II +43. 00000 II —1, 00000 IV +5. 00000 IV +2. 88889 V +2. 88889 VI —0. 12438 V —0. 57743 V —1, 00000 XI +45. 00000 XT +3. 00000 XII 37300 +3. 00000 XTIT 76600 +4. 00000 XIV =-+1. 43800 —2. 00000 XIV =—0. 47600 +1. 00000 XIV =—1. 93100 +3, 00000 XVII Ss —1.00000 Vs + =+0,.69632 VIL — 0.24542 VIII —0.11111 VI — 0.12438 VII — 0.57743 VIII +2. 14302 VII — 0.70043 VIIL +2. 14502 VI +13. 72220 VIL — 0.93704 VIII —0.70043 VI — 0.93704 VII +13. 24598 VIII —1. 00000 XII —2. 00000 XV +1. 00000 XVI +5. 00000 XV —3. 00000 XVI —3. 00000 XV +5. 00000 XVI 409 PRIMARY TRIANGULATION, Values of the correlates. I =+0. 2147 Il =—0. 2951 IIL =-+40. 124% 1V =+0, 2553 X =—0. 0617 XI =—0. 3131 XII =-+0. 8553 XIII =—1. 1243 | Cuap. XVI, C, Vi =+0. 3208 AIV =-10, 0523 VI =-+0. 8713 AV =—0. 3372 VIL =—0. 1169 XVI =—0, 1176 VII =-+0. 0798 XVIT =-+0. 6437 IX =-+0. 0017 Values of the general corrections. [1] =+0.215 [6142] =-10. 255 [2] =+0. 002 (13,) =—1. 124 [2] =—0.295 [6.] =+0.166 [9%] =+0.251 [13243] =++0, 052 ° [2] =+0.215 [63] =-++0.871 [945] =—0. 313 [13;] =—0. 389 [3:42] =++0. 215 [6,] =+0. 002 [10,] ==-+0. 855 [14142] =—0. 118 [3] =~0.510 [41] =—0.371 [10.] =—0.313 [14243] =—0. 220 [3] =—0,131 [72] =-+40. 624 [105] =—0. 062 [145] =-+0.272 [3544] =-+0. 255 [73] =+0.068 [1142] =—0. 313 [15:42] =—0. 337 [41] =+0,125 [8:] =—0. 062 [lg] =+1. 168 [15.] =+0.220 [42] =—0. 295 [%&} =+0. 002 [11s] =—1.124 [153] =+0. 644 [5,] =+0,329 [ts] =+0.511 [121] =+0. 052 [16,] =--0. 644 [52] =++0. 309 [844] =-+0. 681 [lv] =—1. 124 [165] =—0.118 [53] =+10. 255 [8] =—0.045 [123] =+0. 855 [171] =+0.664 [54] =++0. 125 SECTION VII.—Triangulation from the line Warren- Fremont to the line Michigan City-Bald Tom WARREN—16. (Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Date, August, 1873.) Angle as measured between-- Notation. | No. meas. | Range. | Wt. | (v) (v] Corrected angle. “ uw oT uw | oF u 65 12 13. 589 “ Deerfield and Fremont .............. 161 | 20 5 1 —0. 352 65 12 13, 237 FREMONT—17. (Observer, It. S. Woodward. Instrument, Gambey theodolite No. 1. Date, August, 1573.] Angle as measured between— Notation. | No. meas. | Range. | Wt. | (v) {v] Corrected angles. a heat or uw “uw | | uw" “u o f a Warren and Deerfield ........ .2--- 70 00 09. 894 liz 16 4.9 1 0. 000 —0. 972 70 00 08, 922 Warren and Palatine ..........-.-.- 123 34 05.391 17243 16 2.9 1 0. 000 +0. 620 123 34 06.011 $ 10. ] FOND DU LAC BASE TO CHICAGO BASE. 411 Secrion VII.—Triangulation from the line Warren—Fremont to the line Michigan City- Bala Tom— Continued. DEERFIELD—12. {Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Dates, August, 1873, and July, I+74.] i | Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) [r] Corrected angles. 1 esas Z oa fo} ‘ " “a a a Co t a“ Park Ridge (new) and Palatine ----. 74 16 19. 722 18i+2 20 5.8 | 1 —0.217 | —0. 868 74 16 18. 637 | Palatine and Park Ridge (new) ...-. 285 43 40. 711 18—1—2 20 6.0 | 1 —0.216 | +0.868 285 43 41.363 ' Park Ridge (old) and Palatine ...... 7A 15 03. 666 182 15 7.9 | 0.75 | —0.025 |} --0.120 74 15 03.521 Park Ridge (old) and Fremont. . 130 12 54.805 18243 5 2.7 | 0.25 | +0.750) +0.462 130 12 56.017 | Palatine and Fremont ..........---- 55 57 51.939 183 15 7.6 | 0.75 —0. 025 -+-0. 582 55 57 52. 496 : Fremont and Warren.......-....--. 44 47 38. 845 184 20 6.6 | 1 +0.169 | —0.420 44 47 38. 594 | Warren and Evanston University -. 143 29 19.733 185 20 6.3 | 1 +0.169 | —0.068 143 29 19. 834 _ Evanston University and Park Ridge t (Old) satccedewecawsasseueatech ones 41 30 05.360 186+1 20 5.4 ]1 +0.169 | +0.026 41 30 05, 555 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(181+42) --0. 433=0 2. 00(182) 4-1. 25(183)-+4+ (184)-+- (185)—0. 257=0 1, 25(182) +2. 00(183)-- (181)-+ (185) —0. 257=0 (182) + (183) + 2(181)+ (185) -0.457=0 (182) + (183)+- (184)-+2(185) —0. 457=0 NOTE.—18i+2 and 18—1--2 were read in 1874, the remainder in 1873. PALATINE—19. {Observer, R.S. Woodward. Instruments, Gambey repeating theodolite No. 1, and Troughton & Simms theodolite No. 2. Dates, September and October, 1873, and July and August, 1874.] | Angle as measured between— Notation. | No. meas. | Range.} Wt. | (v) {vl Corrected angles. oF “" | | “ “ " ou u" Fremont and Deerfield..-.....-..--.- 70 28 10. 622 191 39 7.6 2 —0.047 | +0. 569 70 28 11.144 Deerfield and Park Ridge (new) ..-. 53 19 05.178 199 21 8.8 1 —0.094 | - 1.067 53 19 04.017 Park Ridge (new) and Lombard .... 47 34 27.447 19344 20 8.3 1 —0.094 | —0.019 47 34 27.334 Lombard and Fremont ........----- 188 38 17. 081 19, 20 4.1 1 ~-0.093 | 40.517 188 38 17. 505 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(191) + (192)+ (19344) +0. 328=0 (191)+2(192)-+ (19344) +0. 328=0 (191) + (192)-+2(193+44) + 0. 328=0 NoTeE.—19i was partly measured with the Gambey instrument in 1873; the remainder of the angles were read with the Troughton & Simms instrament in 1874. PARK RIDGE (new)—20. (Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Dates, July and August, 1874.] Angle as measured bet ween— Notation. | No. meas. | Range. | Wt. () (v] Corrected angles. | oO t uw a“ a“ aw oO ‘ “ Shot Tower and Lombard........--- 93 50 42. 697 201° 20 6.7 1 +0. 082 -+-0. 921 93 50 43.700 Lombard and Palatine ....-..---.--- 75 50 49. 080 202 10 5.8 0.5 | —0.695 | +0.530 75 50 48. 915 Lombard and Deerfield .-........--- 128 15 27.356 20243 10 5.5 0.5; +0.859 | —1.292 128 15 26.993 Palatine and Deerfield ...-..... ---. 52 24 40.525 203 10 11.8 0.5 | —0.695 | —1.752 52 24 38, 078 Deerfield and Evanston University . -77 24 56.159 204 17 6.7 1 +0.082 | +0.170 77 24 56.411 Evanston University andShot Tower 60. 28 52. 682 205 18 6.9 1 40.083 | +0.131 GO 28 82. 896 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(201) + — (202)+ (203) + (204) +1. 143=0 (201) 2. 0(202) +1. 5(203)-+ (204) +-2. 267=0 (201) +-1. 5(20,) +-2. 0(20,)-+- (20,)-+2. 267=0 (201) (203) (20) + 2(204) | L. 143=0 412 PRIMARY TRIANGULATION. [Ciap. XVI, C, SECTION VIL—TZriangulation from the line Warren-Fremont to the line Michigan City- Bald Tom— Continued. . LOMBARD—21. (Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No. 2. Date, August, 1874.] Angle as measured het ween— | Notation. | No. meas. | Range. | Wet. (v) | (v]} Corrected angles. . a t “ ' uw aw | “ oO Z aw Palatine and Park Ridge (new) ..--- 56 34 44.857 . 2h 20 8.1 1 +0492 | —0. 648 56 34 44.701 Park Ridge (uew) and Shot. Tower 46 37 58.092 2Qle 25 V9 [42h 0. 000 —0. 140 46 37 57. 952 Park Ridge(uew) and Willow Springs 102 39 56.945 | 2lo+3 20 72 121 +0.492 | -£ 0. 849 102 39 58. 286 » Willow Springs and Palatine ....--- 200 45 16,722 SIy 20 8.0 | 1 +0.492 | —0. 201 200 45 17.013 | | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(211)+ (212+3) —1. 476=0 (211) + 2(21243)—1. 476=0 SHOT TOWER—22. [Observer, A. R. Flint. Instrument, Repsold theodolite, No. 1. Dates, August and September, 1874, and September, 1877.] Angle as measured between— , Notation. oe Range.| Wt. (v) [v] ce °o A aw aw “a we °o € a Michigan City and Millers .......-. 27 20 26.443 | 221 16 5.5 1 +0. 337 | —0.547 | 27 20 26, 233 Miilers and Willow Spvings .....--- 92 50 52.692 | 222434446546 9 6.7 0.5 | +1.688 | —0.047 | 92 50 54. 333 Morgan Park and East Base..--..-- 19 46 29.144 | 22344 20 8.0 1 —0. 036 | —O 216 | 19 46 28. 892 West Base anl Morgan Park.....-. 315 48 37.587 | 22—s—a—»—s—7 20 5.7 1 —0, 034 | —0, 149 315 48 37.404 Military Acad. and Willow Springs 87 14 29.470 | 22u46+6 10- 4.0 0.5 | —0.030 | —0.758 | 37 14 28. 682 East Base and Middle Base.-....... 13 16 16.074 | 225 mal 6.9 1 —0, 036 | —0.592 | 13 16 15. 446 Middle Base and Willow Springs... 4 36 20.391 | 226 19 6.0 1 —0. 036 | +0.181 | 4 36 20.536 “Willow Springs and West Base..... 6 32 17.031 | 227 20 4.8 1 —0.085 | +0.776 | 6 32 17.722 Willow Springs and Lombard .....- 48 55 01.519 | 22748 28 7.7 1.5 | —0.072 | +0.451 | 48 55 01. 898 Willow Springsand Park Ridge(new) 88 26 19 537 | 2274+8+9 10 4.9 0.5 +1. 054 +0. 644 | §8 26 21,235 West Base and Lombard....-....... 42 22 44.601 | 228 8 10.7 0.5 | —0.100 | —0.325 | 42 22 44.176 Lombard and Park Ridge (new)..... 39 31 19.301 | 229 20 6.0 1 —0.157 | 4-0.193 | 39 31 19. 337 Park Ridge (new) and Water Works. 97 12 21.930 | 2210411 20 9.2 i +0.371 | —0.268 | 97 12 22.033 Evanston University and Water Width Soaps Kote cdsaue noes 59 06 46.639 | 22n 19 9.0 1 0. 000 +0. 079 | 59 06 46.718 » Water Works and Michigan City... 54 09 55,612 | 2212 20 7.5 1 40.336) +0.218 , 54 09 56.166 Water Works and Millers 22.2.2... 81 30 22.322 | 221241 25 6.5 1.25) 40.406 | —0.329 | 81 30 22.399 Water Works and Military Academy 137 06 47. 682 | 22:2+14+2+3 20 1T. 1 —0. 014 +0. 382 |137 06 48. 050 | Water Works and Willow Springs.. 174 21 17.566 | 22124142943444+5+46 24 5.9 1 —0.458 | —0.376 |174 21 16.732 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(221) + (22243444646) + (227) (228) + (229)-++ (2210-++11) —2. 390=0 (221) +2. 75(2224+344 +546) +2. 25(227)4-2. 25(228)--2. 25(229)4-2. 25(2210-411) —5. 044=0 -+-2(223 44)-++ + (225)4+ (226)+ (227) +0. 227=0 +1. 5(2244+6+6) + (227)-+ (228) (229) + (22104+11)-+0. 015=0 b (22844) 2(225)-+ (226) (222) +0, 227=0 + (22244) s- (225) -4-2(226) + 227) +0. 227=0 22) 42. 25(22242 4445 46)4- (22344)+ (224+6+6)-+ (225) (226)4+8. 25(227) +6. 25(22,)-44. 75 (220) --4. 25 (2210411) —3. 5050 (22))+2. 25(222 +344 4546) + (2244546) +6. 25(227) 4-6. 75(224) 4-4. 75(229)+-4. 25 (2210411) —3. 732=0 (221) 42. 25(2224344 4546) + (2244546) 4. 75(227)-+-4. 75(22s)+-5, 75(229)44, 25(2210-411) —3. 902=0 (221) £2, 25(22243 fats +6) + (2244+5-+6) 4. 25(227)4-4. 25 (228)-4-4. 25(229)+-5, 25(2210-4+11) —4. 600=0 Nore.—22344, 225, 226, 227, 22-34 5-6-7, 228, and part of 22743 were measured in 1877; the remainder in 1874. § 10.] FOND DU LAC BASE TO CHICAGO BASE. 413 Section VII—TZriangulation from the line Warren—Fremont to the line Michigan City- Bala Tom— Continued. WILLOW SPRINGS—23. [Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Dates, August, September, and October, 1874, and August, 1877.] Angle as measured between— Notation. be meas.) Range. | Wt. | (v) | [v]} pocam oO t a“ H tw “a oO a “a Lombard and Park Ridge (new) .... 34 04 25.366 | 23, 8 6.9 | 0.5 | 4-0.415 | 40.439 | 34 04 26. 220 Lombard and West Base. ......--.-- 58 26 51. 887 | 23142 24 6.3 | 1 —0.019 | 40.394 | 58 26 52. 262 Lombard and Military Academy.... 136 22 54. 882 | 2314243444546+7 12 4.0 | 0.5 | —0,149 | +0. 228 | 136 22 54. 961 Park Ridge (new) and Military Acad. 102 18 28.538 | 2324344454647" 8 4.7 | 0.5 | +0.414 | —0.211 | 102 18 28.741 West Base and Shot Tower... ..... 16 36 06. 564 | 233 20 4.3 /1 —0.124 | 40.445 |. 16 36 06. 885 West Base and Middle Base .-....... 25 36 29.405 | 23344 12 5.1 | 0.5 | 40.211 | —0.010 | 25 36 29. 606 Shot Tower and Middle Base ....-... 9 00 28.250 | 234 20 5.8 ]1 —0. 074 | —0. 455 9 00 22.721 Shot Tower and Military Academy . 61 19 57.750 2344-5+46+7 6 6.2 | 0.25 | —1.325 | —0.611 | 1 19 55.814 Shot Tower and Azimuth Mark...-. 70 32 47.323 | 23445+6+7+8 20 5 | 1 +0. 352 | —0.362 | 70 32 47.313 Azimuth Mark and Shot ‘Tower...-. 289 27 12.253 | 23-4-s—6—7—-8 20 6.4 | 1 +0. 072 | +0. 362 | 289 27 12. 687 Middle Base and East Base .. -- 16 58 34.415 | 235 24 6.4 | 1 +0. 031 +0. 518 | 16 58 34. 964 East Base and Morgan Park........ 34 24 49.432 | 286 24 10.2 | 1 +0. 032 | —0.668 | 34 24 48.796 Morgan Park and Military Academy — 0-56 09.308 | 237 24 4.7 | 1 +0. 031 | —0. 006 0 56 09. 333 Military Acad. and Azimuth Mark.. 9 12 51.416 | 238 22 4.7 | 1 —0.166 | +0. 249 9 12 51.497 Azimuth Mark and Lombard .--.-.-. 214 24 13.902 | 289 20 6.0 1 +0.115 | —0.477 | 214 24 138. 540 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. (231) —0. 5(231-42) —0. 5(233) —0. 50(284)—0. 50(23) —0. 50(236) —0. 50(237) —0.476=0 . 5(231) +3. (231-42) +2. 0(233) +2. 00(231) +2. 00(235) +2. 00(2386) +2. 00(237) + (23s) 4-0. 637=0 . (231) +2. 0(231-42) +3. 5(233) +2. 50(284) +-2. 00(235) +-2. 00(286) +2. 00(237) + (23s) +0. 241=0 . 5 (281) +-2. 0(23142) +2. 5(233) +5. 75 (284) +4. 25(285) 4-4. 25(236) +4. 25(237)-+3(23a) +1. 0770 . 5(231) +2. 0(231-42) +2, 0(232) +4. 25(231) +5. 25(23,) +4. 25(285) + 4. 25 (237) +73 (28a) 40. 873-=0 . (281) +2. 0(231-42) +2. 0(23s) +-4. 25(28a) + 4. 25 (235) 4 5. 25(235) 4-4. 25 (237)41-3 (23x) 4-0. 873-0 —0. 5(231)-+2. 0(23142)-+2. 0(233) +4. 25(234) +4. 25(235)-44, 25(236) 5. 25(237)-+3(23,) 10. 873=0 ’ + (23142) + — (232)-+3. 00(23,) +3. 00(235) +3. 00(2386)+3, 00(237)-+-4(238)-+0. 746=0 the AE SS oS SS Nore.—231+2, 233, 234, 23344, 285, 236, 237, and 231454647 were measured in 1877; the remainder in 1874. MILITARY ACADEMY—24, (Observer, A. R. Flint. Instrument, Repsold theodolite No.1. Date, October, 1874.] Angle as measured bet ween— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. oO ‘ “a u Pid “a Oo Z Ww Willow Springs and Shot Tower.... 81 25 36. 653 21 20 7.2 1 +0. 420 | —0. 725 81 25 36. 348 | Willow Springs and Millers --.-.--. 176 37 34.549 24142 12 4.5 0.5} ~—0.310 | —0O 096 176 37 34.143 Shot Tower and Millers ...-.------- 95 11 56. 327 242 11 5.3 0.5 | +0.839 | +0. 629 95 11 57.795 ' Millers and Willow Springs -.--.--- 183 22 25. 497 243 21 5.0 1 +0, 264 | --0. 096 183 22 25. 857 | dh eue, git NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(241) +1. 5(242) —2. 307=0 1. 5(241)-++-2. 0(242) —2. 307=0 414 PRIMARY TRIANGULATION. [Cuap. XVI, C, Secrion VIL—Triangulation from the line Warren- Fremont to the line Michigan City- Bald Tom— Continued. WEST BASE—25. (Observer, A. R. Flint. Instrument, Repsold theodolite No.1. Dates, June and July, 1877.) , = 1 Angle as measured between— Notation. | No. meas. | Range.| Wt.) (v) {v] Corrected angles. | oO ‘ w a 1 aw “a fe} ‘ “wn | Shot Tower and East Base.....----- 59 55 44. 873 QWi+e 24 &1 | I —0.160 , +0.219 59 55 44. 932 Middle Base and East Base .-.....-- 00 00 00. 815 252 24 3.0 | 1 +0.161 ' +0. 290 00 00 01. 266 Middle Base and Morgan Park... --. 21 01 39. 159 25243 24 7.2) 1 | 0.160 | 10. 636 21 01 39. 635 Morgan Park and Military Academy — 1 02 00. 014 254 24 | 5.5: L . -0.161) —1.102 1 01 58.751 Military Academy and Willow | | Spritige dccesaccteceteos gees . 74 52 13,298 256 24 6.6 1 -0.160 | +40. 333 74 62 13.471 Willow Springs and Shot Tower.... 203 08 24. 434 256 24 6.7 1 | —0.161 | —0. 204 203 08 24. 069 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. +-2(25142)— (252)+ (25242) (254) (255) +0. 963=0 — (25142)-++2(252)— (25243)— (254)— (255) —0. 963=0 + (25i4-2)— (252)42(25242)4+ (251) (255) +0. 963=0 + (25142)— (252) (25243)-+4-2(251)4- (255)+-0. 963-0 + (25142)— (252)4+ (25243)4- (254)-+-2(255)--0. 963=0 EAST BASE—26. [Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Date, July, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. oO #£ a “a aw aw ° t wf Willow Springs and West Base...-. 40 29 05,139 261 24 Tok a +0.064 | —0. 227 40 29 04. 976 West Base and Middle Base ..-..--. 00 00 01.185 262 24 2.4 1 +0. 065 | —C. 023 00 00 01. 227 Middle Base and Shot Tower ..-.-.-- 95 39 20. 289 263 24 5.6 1 +0.064 | +0. 082 95 29 20.435 Shot Tower and Morgan Park ..-.-- 123 09 05. 901 264 24 5.4 1 +0.065 | +0.540 | 123 09 06.506 Morgan Park and Military Academy 1 41 46.055 265 24 5.1 1 +0. 064 | +0.158 1 41 46.277 Military Academy and Willow SUNOS wesewaseseenisonuweackwase 99 00 41. 045 266 24 6.2 1 40.064 | —0. 530 99 00 40. 579 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(261)4+ (262)+4+ (263)+ (261)+ (265)—0. 386=0 (261) -+-2(262)-+ (263)+ (261)+ (265)—0.386=0 (261) + (262) +2(263)+. (261)+ (265)—0. 386=0 (261)+ (262) + (263) +-2(264) + (265) —0. 386=0 (261) + (262)+ (263)+4 (261) +2(266)—0. 386=0 MIDDLE BASE—27. [Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Date, July, 1877.| Angle as measured between— Notation. | No.meas. | Range.| Wt. (v) [v] Corrected angles. o ‘ uo aan , aw wt oO t aw Willow Springs and West Base...-. 57 27 38. 801 271 24 6.4 | 1 —0.091 | —0.106 57 27 38. 604 West Base and Shot Tower. -...... 108 55 38. 692 272 24 65 | 1 —0.091 | —0.378 | 108 55 38,223 Shot Tower and East Base.......... 71 04 24.575 273 25 7.2 | 1.25 | —0.073 | —0. 230 71 04 24, 272 East Base and Morgan Park....--.. 27 20 56. 669 274 22, 88 | 1 —0.091 | +0. 283 27 20 56. 861 Morgan Park and Willow Springs.. 95 11 21.700 2765 21 83 | 1 —0.091 | +0. 431 95 11 22. 040 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2Q%)+ Cid + (273)+ (274) +0. 437=0 (271) -+2(272)+ (273) (274)-++0. 487=0 (271)4 (272) +2. 25(273) 4+ (274) +0. 487=0 (271) 4 (272)+ (272) 4+2(27a) +0. 437=0 § 10.] FOND DU LAC BASE TO CHICAGO BASE. 415 SEction VII.— Triangulation from the line Warren— Fremont to the line Michigan City- Bald Tom— Continued. MORGAN PARK—28, {Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Date, August, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. o t uw “we a aw Oo t a“ Willow Springs and West Base..... 27 05 54. 324 281 24 6.2 1 +0. 040 | . +0. 321 27 05 54. 685 West Base and East Baso......-..-. 17 46 51.014 282 24 7.5 1 +0.041 | —1.150 17 46 49. 905 East Base and Shot Tower...-. ... 37 04 24.114 283 24 9.4 1 +0. 040 +0. 742 37 04 24. 896 Shot Tower and Willow Springs. ... 278 02 50. 386 284 24 7.4 id -+0.041 | +0.087 | 278 02 50.514 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(281) 4+ (282)-+4+ (283)—0. 162=0 (281) +2(282)+4+ (283) —0. 162=0 (281) + (282) +2(283)—0. 162=0 MILLERS—29, (Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No.2. Date, August and September, 1874.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. °o , “ aw “a “a o -- wo Military Academy and Shot Tower 29 11 38.326 291 21 10.0 1 —0.006 | +40.317 29 11 38. 637 Shot Tower and Michigan City ..... 114 11 26. 524 292 | 18 12.3 1 —0.006 | —0. 618 114 11 25.900 Michigan City and Otis....... --- 2917 11.870 293 20 10.4 1 | —0.006 | 40.154 29 17 12.018 Otis and Military Academy. .. .... 187 19 43.303 294 | 20 6.8 1 —0.005 | 40.147 187 19 43. 445 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(291)+ (292)-+ (293)+0.023=0 (291) +2(292)-++ (293)-++-0.023=0 (291) + (292) +2(293)-++0.023=0 MICHIGAN CITY—30. (Observers, R.S. Woodward and A. R. Flint, ‘Instruments, Troughton and Simms theodolite No. 2, and Repsold theodolite No. 1. Dates, October and November, 1874, and May, 1877.] Angle as measured between— s Notation. | No. meas. | Range.| Wt. (v) (v7) Corrected angles. oO ‘ “ we aw aw or “a Bald Tom and Galena. ..-...--.-.--- 54 47 01.324 | 301 16 5.2 1 4:0:496 |osexexnccs)s vevins eereeyeeeees Bald Tom and Springville ..-....---- 75 43 51.528 | 30:42 20 5.7 1 +0.003 | +0,117 75 43 51. 648 Galena and Millers .......--...----- 141 85 29.122 | 302+3+4 16 4.6 a, AOPEOG | cccss am ecisialesmems aeesiseimenins Springville and Otis ........-.-.---. 55 23 13.333 | 303 20 8.9 1 +0. 285 | —0.181 55 23 13. 437 Otis and Millers 65 15 26.004 | 304 20 10.9 1 40.285 | +0.172 65 15 26. 461 Millers and Shot Tower... .-...--. 38 28 11.265 | 30s 28 6.1 1.5 +0. 333 | —0. 260 38 28 11.338 Millers and Bald Tom (W.) ..------ 161 01 12.248 | 305+6 20 8.7 1 +0. 283 +0. 004 161 01 12. 5385 Shot Tower and Bald Tom...-....... 125 09 16.465 | 3064-7 18 5.9 1 +0. 499 +0. 152 125 09 17.116 ‘ Bald Tom (W.) and Springville ..... 78 20 07.278 | 307+1+42 20 12.9 1 ; +0, 284 +0. 005 78 20 07. 567 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(301)— (3014+2)— (303)— (304) —0.419=0 — (301) -+3(3014+2)-+-2(3032)+2(304)-++ — (305) —0.986=0 — (301)-+-2(30142) +4 (303)-+3(304)4+ — (305) (806-4+6)—2.123—0 — (301)-+-2(301+2)-+3(303)-+4(304)+ (805) + (306+6)—2.123=0 (301+2)+ (303)+ (304)-+-2.5(305) —1.405=0 (303)-+ (304) +2(305+6)—1.137—0 Nore.—30i+2 was measured in 1874; 301, 302+344, 305, and 306+7, were measured in 1877, by A. R. Flint, with the Repsold instru- ment. The remainder were read in 1874, by R. S. Woodward, with the Troughton & Simms instrument. x 416 PRIMARY TRIANGULATION, [Cuap. XVI, C, Section VIL—Tricangulation from the line Warren-Fremont to the line Michigan City- Bald Tom— Continued. OTIS—31. (Observer, A. Rt. Flint. Instrument, Repsold theodolite No. 1, Dates, October and November, 1874.] Angle as measured bet ween— | Notation. | No. meas. | Range.| Wt. (r) [eo] Corrected angles. oi \ ecxtgpaiud Samia eins Be ae ms rains Ale ‘cpeeminaais) as oO it uw aw “wn “we oO f a“ Millers and Michigan City. ..... ... 85) 27° 22.787 3h. 20 6.7 1 4-0.042 | +0. 004 85 27 22.773 Michigan City and Springville... ... 51 45 07.262, 812 21 5.4 1 0.000 | —0, 524 51 45 06, 738 Michigan City and Millers ..... .. 274 32 47.189 | Slots 20 6.1 1 +0. 042 | —0. 004 274 32 37. 227 SPRINGVILLE—32. {Observer, R. S. Woodward. Instrument, Troughton and Simms theodolite No. 2. Date, November, 1874. | Angle as measured between — Notation. | No. meas. |Range.| Wt. (v) | {v] Corrected angles. °o + aw a“ Ww a“ ° é “a Otis and Michigan City ...........-- 72 51 40.470 321 20 12.7 1 -+0.053 | —0, 242 72 51 40, 281 Michigan City and Bald Tom ....... 77 50 41. 254 822 oe i 141 1 0. 000 — 0. 067 77:50 41. 187 Michigan City and Bald Tum(W.).. 77 59 62. 240 32243 10 86 | 0.5 +-0. 150 +0. 080 77 59 02.470 Michigan City and Otis...... ..... 287 08 19. 500 3224344 16 13.2 | 1 —0.023 | +0, 242 287 08 19.719 Bald Tom (W.) and Otis ........-..- 209 09 16. 787 324 4 1.8 | 0.25 +-0, 300 +0. 162 209 09 17. 249 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2.25 (821) -4-0.25(32243) —0.156=0 0.25(321) 4+-0.75 (32243) —0,126=-0 BALD TOM—33. [Observer, A. R. Flint. -Instrument, Repsold theodolite No. 1. Date, November, 1874.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angle. Oo , uy “a “we “we ° é aw Springville and Michigan City....... 26 25 28. 206 334 20 5.0 1 0. 000 —0. 067 26 25 28.139 Numerical equations of condition in the triangulation from the line Warren — Fremont to the line Michigan City - Bald Tom. SIDE-EQUATIONS, — XXII. (20) —41. 3655 (18.4:] —35. 4361 [1%,] —35. 4361 [18,] —35. 4361 [18;] +15. 6837 [199] 19.2434 [19344] 3.1711 [20] + 3.1714 [20.] + 3.1714 [20,] 14.7958 [204] +13, 8943 [21] 19. M881 [Zl] +25, 5220 [229] —30. 0339 [220411] +30. 0339 [22,1] —26. 859=0 NXV. (20) 19.5881 [2] 9 — 4.7318 (2loys] — 0.5740 [227] -— 0.5740 [225] 24.9479 [224] c 55.3710 [23] Fed. 2420 [23.49] 24. 2420 [239] + 0.677=0 XNIX. (7) + 5.6306 [2244] + 5.6306 [22,] + 5.6306 [225] 21.6594 [22,] 4. 8617 [235] — 7.1010 [23,] — 7.1010 [235] — 7.1010 (23) + 3.3512 [25.40) — 3.3512 [259] + 3.3512 (25.49) — 5.2873 [25,] — 5.2873 [25,) + 7. 922=0 XXXIID. (20) —3L. 2400 [23] 31. 2409 [25,] 11. 9527 [23g] 21.9108 [26,] —21. 9108 [262] 21.9108 26,] FL 7527 [26,] — 2.9768 (24, ] 2.0768 P28] E24. x899 [xy] —2e. 093=0 XXXIV. (8) = M807 [2542] — BLY (2%) +f 3.3512 [25.45] — 2.0853 (26) — 2, 0853 [265] 113.7528 [26] — 14. X230 [2K] +13. 0438 [285] —-33, 2660 XXXV, (25) — 4.5007 [2%] = 4.5007 [23,] — 4.5007 [235] +25. 1821 [2%] +25. 1821 [237] +51.9554 [252] —51. 9554 [25245] —51. 9554 [25,] + 5.6928 [255] — 3. 3391 [26] 421.3101 [26.] 21.3101 [263] 21.3101 [26,] 4:21. 3101 [265] —3s. 707=0 8 10.) FOND DU LAC BASE TO CHICAGO BASE. 417 Numerical equations of condition, &e—Continued. SIDE-EQUATIONS—Continued. XXXIX. (20) —29. 3470 [25.45] 7-69 £25,] 7.69 [255] 11.5877 [27,) Fd. 9048 [27] $29, 0048 [272] +29. 9943 [27,] — 9.2415 [22,] 32. 697=0 XL. (10) — &.83900 [242] - S.s809 T2AZ] 8.8512 [253] 418.6073 P26.) EIS. 6073 (26,] — 7.2200 [272] + B1178 F273] 15.4897 [27] TA S230 (28)] 0 3.7813 [28,] —231. 8160 ALI. (20) -E 31.9066 [25.4] 20. 6841 (26) ] +20, 6841 [25,] — 3.0818 £263] — 3.9813 [264] 115211 [27)] AS. 4277 [272] 48. 4277 [e735] 29.9043 [e7y]- —10. 7626 (28, J +21. 1440 [2"2] — 4,059=0 XLII. (20) —2. 2095 [2244546] $59. 9069 [225] 449.9069 [22,] 11.5119 [23,] —11. 5119 [22, 18.1708 [235] 18.1708 (28;] — 3.3391 [26,] — 3.3391 [26,]) — 3.3301 [26,] --17,. 008" (26, —17. 995% [265] +25. 163=0 Note.—In the solution for determining the general corrections, each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. ANGLE-EQUATIONS, XVOL +016) +17] + [12] 41. 745=0 XIX. — [172] + [17243] +13] + [19%] —2. 7430 XX. + (lRige] + [192] + [23] +3. G&7=0 XXT + (19344) + [202] + [21] +0.137=0 XNIT. + (20,J + [rl] + [220] —U. 973=0 XXIV. — [212] + [2l43] + £227] + [22s] + [2542] + [23] —2. 279=0 XNVI. 4 [22544] + [225] +22) +23) + C28) + (28.7 + Pa] + P28] + [22s] +1. 318=0 XXVI. + [22:45] + [22s +22) + E227] + 25142] — [252] + [25248] + [22] + [2s] —0. 807=0 XXVIII. + [233] + [234] + (2335) + 02%] + [254] + [255] + [22] -40, GOR=0 XXX. + [225] + [225] +227] + [2542] + [262] + [265] —0. 643=0 XXXII. + [225] + [22. + (231) +1285] + [26] + [262] + [263] +0. 516=0 XXXIL + [23] — [26] —{26.] — [26] — (26) + P]+ 22) +1. 868=0 XXXVI. + (22. + [22 + [2842] — [252] + [272] —0. 508=0 XXXVI -+ (22: + [262] +(e] +0. 741=0 XXXVI. + (23) + [231] + [B42] +12] +0253) + (27) +0. 249=0 XL. + [2214540] + [234] + (235) +123.) +[23]) +[24) +2. 095=0 XLIV. — [2244546] + [22e¢ststs¢6] + [242] + [29] —1.657=0 XLV. + [22)] + [292] + [305] ; +1. 426=0 XLVI + [2%] + [304] + [311] < —0. 3300 XLVI. +.[303] + [312] + [32] +0. 946=0 XLVI. + [342+ + [322] +[3%] @ +0. 018=0 General corrections in terms of the correlates. {16,] =+1.00000 XVIII [172] =+1. 00000 XVIII —1. 00000 XIX [17243] =+1. 00000 XIX [i342] =+0. 50000 XX —1.03414 XXII [1} =—O.17301 XVI 0.40580 XIX 1.33526 XXIII [1X] =—0.17391 XVII +1, 02714 XXIII +0, 92753 XIX 5 [1%s4445] =$0.39131 XVIIT = -40.57971 XIX 2, 41382 XXIII {1-4] =+0.72261 XVITI = —0,. 17391 NIX —0. 60234 XXII [183] =—?.21739 XVIII —0. 17301 XIX —0. 69334 XNTIT [19,] =) 42857 XIX —0. Leb XX —0, 14225 XXT +0. 02542 AXTIL [192] =—0. 1426 NIX 0. 71429 XX —0, 22571 XXT +0, 28501 XXIII 53 LS 418 PRIMARY TRIANGULATION, [Crar. XVI, C, General corrections in terms of the correlates—Continued. [19541) =—0.14285 XIX — —0, 28571 XX +0. 71429 XX1 —, 91132 XXTIT [20] =—0, 15385 XX —0.1838) XXI 4.0. 76923 XXII —0. 09745 XXUIT [2404] =-+0, 30769 XX 40.3070 XXI 0.46153 XXII —0, 22730 XXIII (20. =—0, T6023 XX 41,23077 XXI -- —0,15385 XXII. —0. 06497 XXIII [20] =+1, 22077 XX —0. 76023 XXI -- —0, 15385 XXIL- —0, 06497 XXII] [20,) =—(), 14345 XX 0.15385 XNT 0, 93077 XXII +40, 48360 XIII [2h] =+0. 66667 XXI +40, 46315 XXIII 0.33333 XXIV +0. 07480 XXV [21.] —-0. 0000 XXIT 0.79552 XXTILT = —0.80000 XXIV —0, 79552 XXV [21 esa) =—0.33333 XXI 0.23157 XXIII +0. 66667 XXIV —0. 15773 XXV [220] ——0,0M88XXTL 0.02055 NXIIT = —0. 00801 XXIV. —0. 01833 XXV 40, 00171 XXVI —0,00058 XXVIE 0,00847 XNIX = —0.00115 XXX 0.00114 XNXI -0. 00172 XXXVI 10, 00057 XNXVIL +0,082x2 XLIT —-+0.04944 XLUIL 0, 21676 XLIV— -L0. 61658 XLV [2easpapngn] =O. MIR NNIL © 0.14885 XNIIT =—0.05610 XXIV —0,19R34 XXV +0, 01203 XXVI —0,00400 XNVIE -+40.05928 XNIX —0.00801 XXX 0, 00802 XXXII —0, 01202 XXXVI +0, 00401 XXXVI +0,22973 XLIT 4.34596 XLII. +40. 48273 XLIV. —0. 18394 XLV iad =-+0. 01588 XXII --0, 21215 XX VIL —0, 21215 XXXVIT +0.00594 XXIII —0.02991 XXIV — +0. 02067 XXV + +0. 76088 XXIX —0. 57570 XXX —( +0. 00343 X LIT +1. 27644 XLII + . 36355 XXVI 242430 XXXI- —0.36355 XXXVI - 00058 XLIV = +0. 00057 XLV — [22eesge] == O.047G4 NXT = £0. 02082 XXIII. —0,08973 XXIV. -£0.06201 XXV_——- 1, 09065 XXVI 40. 63645 XXVIL $2. 28204 XNIX — -+£0.972900 XXX -.0.72710 XXXI-- —0. 09065 XXXVI +40, 36355 XXNVIT +0.01029 XLIL = 2.16138 XLII = -+40,00174 XLIV. +0. 00171 XLV [22.4540] =—0.08030XXIT +40, 12329 XXIII = —0. 04809 XXIV. —0.11001 XXV_—- 0.01029 XXVI —t.00845 XNVIL +40.05079 XXIX = —-0,00688 XXX ——-L0,.00685 XXXNI —0, 01031 XXXVI +0. 00343 XXXVI +0. 86358 XL 4-1.37024 XLIIE = —0. 63385 NLIV +0. 03282 XLV [22] =+0,01588 XXII 0.00694 XXIIE =—0.02991 XXIV. +.0,02067 XXV_——-+£0, 36355 XXVI 0.21215 XXVIL_ -+10.76088 NNIX = +.0,42430 XNX -.0,57570 XXXI-- —0. 36355 XXXVI +40. 78785 XXXVI +10. 00843 XLIE = 1.71891 XLIIJ 0.00058 XLIV. +0. 00057 XLV P22s40] =+0,03176 XXIL 4.0. 01383 XNIIL = —0.05982 XXIV. 4.0.04184 XXV_ 0.72710 XXVI +-0.42430 XXVIL 1.59176 NXIX = 0.84860 XNNX -+41.15140 XXXII 0, 27200 XNXVI +0.57570 XXXVII 0.00686 XLIL = —3. 43782 XLII +40,.00116 XLIV. +0. 00114 XLV (22..] =+0,01588 XXIZ -40.000804 XXIII = —0.02991 XXIV 0.02067 XXV_—-{0, 36355 XX-VI +0. 21215 XXVIT —0, 21215 XXXVII (225 =—f), 06354 XXIT --0. 15141 XXVIT —0. 15140 XXXVIT +0. 76088 XXIX +0, 00343 XLIL -+0,.42430 XXX +0. 57570 XXXI —1.71891 XLITI = +0. 00058 XLIV —0. 02778 XXIII - 11963 XXIV —0. 08269 XXV —2. 23922 XXTIX = -L0.30281 XXX —0, 30280 XXXI —0. 01374 XLIT +0. 88487 XLII —0. 00229 XLIV +0. 63645 XXXVI +0. 00057 XLV —0, 45420 XXVI +0. 45421 XXXVI —0. 00229 XLV [22745] =— 0, 22238 XXII —0.09725 XXIII +10. 41871 XXIV —0, 28941 XXV —0. 08973 XXVI +0,02990 XXVIT —0. 44234 XNTX +0. 05901 XXX —0, 05982 XXXI -++0. 08972 XXXVI —t), 02991 NANXVIT —0. 04809 XLIT +0.10174 XLII = —0. 00801 XLIV —0, 00801 XLV [22] =—0, 15884 XXT —0. 00947 XXNTIT +0. 29908 XXIV —0. 20672 XXV +0. 30447 XXVI —0, P2E5L NNVIT | 41.7968" XXNTX —0, 24300 XXX 0, 24202 XXXI —0, 36449 XXXVI . 12149 NXXVIT —0. 03435 X LIT —0.78313 XLII = =—0.00572 XLIV =—0. 00572 XLV | .O8TOL NXIL +1. 09549 XXIII =—0, 22238 NNIV +0. 73862 XXV +0. 04764 XXVI1 - 01590 XNVIT = +0. 23492 XNXIX —0, 03178 XXX +0. 03176 XNXT —0. 04766 XXXVI . 01588 XXXVIT —0. 02930 XLIT —0, 23896 NLILE = —0. 01488 XLIV —0. 01488 XLV [22a] 0, 23068 XXIT —2.084d8= XXNTT = —0.12421 XXIV —0. 28418 XXV +0. 02661 XX VI 0.00888 XXVIT +0.131z1 XXIX = —0.01775 XXX +0. 01774 XXXI —0. 02662 XXXVI .00RT XXX VIT —0. 15799 X LIL —0. 30758 XLITL = —0. 02633 XLIV —0. 02633 XLV [2210411] D. 23068 XXIT —lJ.IsBI8 XXII = =—0. 12421 XXIV —O0. 28418 XXV +0, 02661 XXVI JOSS XNVIT --0. 13121 XXIX — 0.01775 XXX +0. 01774 XXXI —0, 02662 XXXVI 00857 XNXVIT —, 15799 XLIT 230758 XLT —0, 02633 XLIV —0 . 02633 XLV § 10.] (22,1) [231] [23142] FOND DU LAC BASE TO CHICAGO BASE. General corrections in terms of the correlates—Continued. =+1,50170 XXII =+0. 15741 XXIV +0. 06482 XXXI +40. 16663 XLII =-+0. 36111 XXIV —0. 02777 XXXI —0. 16665 XLII 22.4243] =+0. 64815 XXIV [283] [234] [23145] —0. 20370 XXXI —0, 33332 XLII =+0, 28704 XXIV —0, 17583 XXXI —0. 16667 XLII =—0. 13839 XXIV -L0, 47223 XXXI +0, 16669 XLII =—0. 20370 XXIV +1. 09259 XXXI 40. 33333 XLII [2314546] =O. 26451 XXIV [235] [2365] [23647] [23] [23s] [241] [242] [25:42] +0. 71296 XXXI +0. 50000 XLII =—0. 06481 XXIV +0. 62037 XXXI -L0, 16667 XLII =—0. 06481 XXIV —0. 37963 XXXI +0. 16667 XLII =—0. 12962 XXIV —0, 75926 XXXI +0. 33334 XLII ——0. 06481 XXIV —0, 37963 XXXI +0. 16667 XLII —+0. 08796 XXIV —0, 19908 XXXI —0. 41663 XLII +0, 72727 XLII =—0, 54545 XLII =+0, 50000 XXVII +1, 00116 XXXV —0, 26589 XLI [2514243] =+1. 50000 XXVII [252] [25242] —1, 15294 XXXV +0, 77665 XLI =—0.50000 XX VII 41, 07705 XXXV +0. 26589 XLI —-+0, 50000 XXVII —1. 67705 XXXV ++1, 32944 XLI —3, 02637 XXV +40, 05093 XXXII —0. 05522 XLIII +0. 16854 XXV —0. 06944 XXXII -L0. 11020 XLII +0. 34982 XXV —v. 06481 XXXII +0. 00052 XLIII +0. 18128 XXV +0. 00463 XXXII —0. 10968 XLII —0, 20681 XXV —0. 15277 XXXII +0, 54942 XLII —0. 42637 XXV —0. 37963 XXXII +1. 31872 XLII —0, 64593 XXV +0. 39352 XXXII +0. 60387 XLIII —-0. 21956 XXV —0. 22685 XXXII +0. 76929 XLIII —0. 21956 XXV 4+0.77315 XXXII —0. 71485 XLII —0, 43912 XXV +0. 54630 XXXII —1, 42970 XLII —0. 21956 XXV —0, 22685 XXXII —0. 71485 XLIII +0. 56162 XXV —0. 10880 XXXII +0, 08310 XLIII —0. 54545 XLIV +0. 90909 XLIV —0, 33333 XXVILL +0. 66667 XXXVI —1. 00000 XXVIIT +1, 00000 XXXVI +0. 33333 XXVIII —0. 66667 XXXVI —0. 33333 XX VIL —0. 33333 XXXVI -+0. 11575 XXVI —0. 07079 XXXIII —0. 09721 XX VI +0. 00185 XXXII —-0. 26851 XXVI +0. 27942 XXXII —0. 17130 XXVI +0. 27757 XX XIII +0. 31944 XXVI —0. 62900 XXNIII +-0.71296 XXVI —1.93370 XXXUI +1. 10648 XXVI —0. 87827 XXXIIT +0. 39352 XXVI —1. 1471 XXXII +0. 39352 XXVI +1. 05543 XXXIIT —0, 21296 XXVI +1.51272 XXXII —0. 60648 XXVI +0. 45729 XXXII —0. 30788 XXVI +0, 24589 XXXTII +40, 49115 XXIX —0. 50000 XXX VIII +1. 47345 XXIX —(). 50000 XXX VIII —0, 49115 XXIX +0, 50000 XXX VIII +0, 49115 XXIX +0. 50000 XXXVIII +0. 17594 XXVIII 4-0. 08009 XXXV —0, 36110 XXVIII —0. 08737 XXXV +0. 01853 XXVIII --0. 14557 XXXV +0. 37963 XXVIII —0, 05818 XXXV +0. 13890 XXVIII —0. 36027 XXXV +40, 53703 XXVIII —0. 92979 XXXV +0. 93518 XXVIII —0. 31200 XXAXV +0, 39815 XXVIII —0. 56952 XXXV +0. 39215 XXVIII +0. 61779 XXXV —0, 20370 XXVIII +1. 23558 XXXV —0. 60185 XXVIII —0. 61779 XXXV —0, 25464 XXVIII —0. 19293 XXXV --0, 83333 XXX 0, 11378 XXXIX +0. 50000 XXX —1, 12601 XXXIX +0. 16667 XXX —0. 11378 XXXINX — 0. 16667 XXX —1, 35357 XX XTX 419 —0. 07502 XXIX +0. 07408 XXXVIII —0. 08467 XNXNIX —0, 22222 XXXVUI “0.47174 XXIX +0. 14815 XXXVIUI +0. 55641 XNIX +0. 37037 XXX VIII —-0, 44946 XXIX +0. 44444 XXXVIII —0. 84541 XXIX 40, 29629 XXXVIIL —1, 24138 XXIX +0. 14814 XXXVHI —0. 39597 XXIX —0. 14815 XXXVIIT —0. 39597 XXIX —0. 14815 XXXVIII +0, 22250 XXIX —0. 29630 XNAVIT1 +0. 61847 XXIX -—0, 14815 XXXVIII +0. 34930 XX1X —0. 03704 XXXVIII —1. 06044 XXXIV —0. 64518 XL —0. 13360 XXXIV —0. 71643 XL —0. 46342 XXXIV +40. 64518 XL +40. 46342 XXXIV +40, 5733 XL 420 PRIMARY TRIANGULATION, General corrections in terms of the correlates—Continued. Yn paqs0] = - 0.50000 XXVIL —0. +0 [25,] =.—K) —1 —0 [25145] =—1 +0 —0 [en =—0) +1 —0 (26) ] =—0 —0 [2614248] =+1. —0. [262] =-+0. 40. [Migs] = 41. +0. [2624541] =-+1. +40, [265] =+0, +0, [spi] = +0. +0 [264] =—0, +-0 [20;] =—0. +0 [27,] ig +0. [272] Sih), —0. [27248] =+0. —1 [275] =—0 —0 [274] =—0 —0 [28] =+0 +40 [2A 42] =+0 —0 [282] =+0 —1 [2s] =+0 as | [29] =+0 [293] == [295] =f 92523 XXXV . 79706 XLL ), 50000 NX VIT . 07705 XXXV . 26580 XLI . 00000 XX VII . 15182 XXAV . 53178 XLT . 50000 XXVIT .2ve87 XXAV . 26589 ALI . 33333 XXX . 67959 XXAXV 00000 XXX 06683 XXXV 66667 XXX 30638 XXXV 33333 XXX 61276 XXXV 00000 XXX 91914 XXXV 66677 XXX 30638 XXXV 33333 XXX . 66667 XXAVIT 33333 XXX . 30638 XXAV . 33333 AXA . 30638 XXXV . 20833 XXXVI 67066 XLI 79167 XXXVI 92467 XLI 62500 XXXVI . 66440 XLI . L667 XXXVI . 73973 XLI . 20833 XXXVI . 253800 XLI . 25000 XXVI . 05560 XXXIV - 90000 XXVI . 23104 XXXIX . 25000 XXVI . 79728 XXXIV - 25000 XX VI . 68608 XXXIV . 75000 XLIV . 25000 XLIV » 25000 XLIV Ak: =A, —0. +0, —0. +2. —0. —0, —0. —0. =0: —0, +0. +0. —0. —0. —0. +0. —0. +0. . 11552 XNNLX » 50000 XXVIT 211552 XXXIX . 25000 XLV - 75000 XLV 00000 XXVIII 00000 XXXVI . 66667 XXVIOTL 38353 XXXVI . 33333 NAVI . 66667 XXXVI . 66667 XXVIII . 33333 XXXVI . 60000 XXXI . 16667 XX XVII . 50000 XXXT . 50000 XXAVIT . 50000 XXXAT . 16667 XXAVIL . 00000 XXXT . 66667 XXXVII . 50000 XAXT . 50000 XXXVII . 50000 NAAT 58533 SAA VIT 66667 XXXIT 48098 XL 50000 XX XT 16667 XXAVII 50000 XXXI 16667 XXXVII - 16667 XXXVIT 16667 XXXVII 50000 XXX VII 66667 XXXVII 16667 XXXVII 50000 XX VIT 34656 XXXIX 50000 XXVIIL 93036 XL 50000 XX VII 25000 XLV —0, 99469 XXTX —0. 50000 XXX +1. 50000 NXXXVIII—0. 34133 XXXIX 2] _ 74292 XXIN +40. _ . 48584 XXIX me + . 74292 XNIX 4.0. —0, 33333 XXXII —0. 62024 XL . 00000 XXNII +40, 00001 XL —0, 33333 NXXIL —-0, 62024 XL —0, 66667 XXXII 40. 62025 XL —1. 00000 XXXII +1. 86074 XL —0. 33333 XNNIL +41. 24049 XL . 45842 XXXII . 95488 XLI . 33333 XXXII . 24049 XL . 6667 XXXII . 62024 XL ee So —0. —0. 75000 XXVUOI +0. 27597 XL +1. 00000 XX AIT +0, 25953 XLI ), 25000 XX VIIT —1. 20633 XL —0. 25000 XXVIII 10. 65440 XL —0, 25000 XLVI —0. 25000 XLVI +0. 75000 XLVI +0. = —0. 16667 XXX 50000 XXX VIJI-L0. 50612 NXATX —0. 33333 XXX . 00000 XXXVITI-+1. 01224 XXXIX —0. 16667 XXX 50000 XXX VIJI+0. 50612 XXXIX —0, 66238 XXXII -L0. 75581 XLI —1.98714 XXXII +1. 03418 XLI —0, 66238 XNXIII +0, 75581 XLI —1, 32476 XXXII +0, 27837 XLI —0, 20396 XXXII —0, 19907 XLI —0, 66238 XXXII —0, 47744 XLI +1. 01916 XXXIV +0, 29996 XLII +1. 12080 XX XIII —0. 47744 XLI +0, 43316 XXNUI —0, 27838 XLI .79167 XXX VILI—0. 41876 XXXIX . 20833 XXX VIII-+0. 50488 XXXIX . 37500 XXX VITI-+0. 90878 XXXIX 16667 XXX VIII-+-0. 40390 XX XTX 20833 XXX VIII--0. 50488 XX XTX +0. 50000 XXXII —0. 65790 XLI —0. 77110 XXXII 40. 50000 XXXII +0. 92743 XLI —0, 50000 XXXII —0, 12977 XLI [Cuap. XVI, C, 40, 5245 XXXIV +41. 05155 XL +0, 04402 XXNIV 40, 23881 XL +40. 08904 XXXIV +40. 47762 XL +40. 04452 XXXIV -10, 23841 XL —0, 19964 XXXIV —0. 2165 XLII —1. 12024 XXXIV —0. 64953 XLULI —0, 46030 XXXIV —0, 21651 XLUL —0. 92060 XXXIV —0. 43302 XLUI -L0, 59886 XXXIV -4.0, 08345 XLII —0. 46030 XXXIV —0, 21651 XLII +0, 61276 XXXV 1.51946 XXXIV +0. 51647 XLID —0, 19964 XXXIV -40. 51647 XLUI 40, 42115 XL —0. 30085 XL +0, 28548 XL -L0. 58633 XL —1. 12782 XL —0. 34555 XXXII —1.74168 XXXIV —0, 38555 XXXITI +1, 00720 XXNNITI § 10.] General corrections in terms of the correlates—Continued. [304:] =—0. 13418 XLV [305] =—0. 06789 XLV [30,] =—0. 06789 XLV [30;] =+0.30777 XLV [30346] =40. 06789 XLV [3l.] =+0. 50000 XLVI —0. 08911 XLVI —0. 21109 XLVI +0. 45296 NLVI —0, 06789 XLVI —0. 12094 XLVI —0, 08911 XLVII +0, 45206 XLVII —0. 21109 XLVII —0. 06789 XLVII —0. 12094 XLVII FOND DU LAC BASE TO CHICAGO BASE. +0. 48013 XLVIIT —0, 08911 XLVIII —0. 08911 XLVIII —0, 13418 XLVIII +0. 08911 XLVIIT [31,] “=++1. 00000 XLVII [321] [322] =+0. 46154 XLVII =+1. 00000 XLVIII [32043] =—0, 15385 XLVII Normal equations for determining the correlates. —1. 17391 XIX +3. 35610 XIX +2, 44506 XX —1. 05494 XX +0. 07886 XXV —0. 15385 X XT +0.04764 XAVI —0, 04766 XXXVI —0. 01488 XLV —1. 00172 XIX +0. 46670 XXIV —0. 01390 XXX +0. 15695 XLIIT —1. 02238 XXII +0. 02990 XXVIT —0. 06481 XXXII —0, 05690 XXIT —0. 02068 XXVII —0, 21956 XXXII +0, 02082 XXIII —0. 69334 XXTIT —0. 14286 XX —1. 05494 XXI +2.61173 XXI +2. 15624 XXIT —0. 01590 XXVIT +0. 015€8 XXXVIT —0, 26407 XX +2, 21516 XAV +0, 01388 XXXI +0. 02056 XLIV +0. 46670 XXTIT +0. 01853 XXVIII +0, 27942 XXXII . 02991 XXXVII +0, 14815 XXX VIII —0. 38141 XLIT +2, 21516 XXIII —0. 46465 XX VITI +0, 53467 XXXIIT . 02067 XXX VII—0. 02553 XXX VITI—0. 97550 XLIT —0, 35824 XXIV +1,.18518 XXVIII +1. 04126 XXIX ~-0, 64157 XX XIII - 0. 05560 XXXIV -+0 36355 XXXVII +0. 14814 XXX VIII—O, 11552 XXNNIX —1.55752 XLIIL —0. 00696 XX TTT —1,50000 XXVIII + .62225 XXXIII —0. 24480 XXXIV —0, 50000 XXX VITI—0. 89497 XXXIX [33,] =+1. 00000 XLVIII No. of equation. 18. 0=+1.74500 + 2.78261 XVII 19. 0=—2. 74300 — 1.17391 XVIII 20. 0=+3. 68700 — 0.14286 XIX 21. 0=+0. 13700 — 0. 14286 XIX — 0.33333 XXIV 22. 0=—0.97300 — 0.15385 XX — 0.05690 XXV + 0.03176 XXXI — 0.01488 XLIV 23. 0=—1. 34290 — 0.69334 XVIII +14. 811638 XXIII + 0.10274 XXIX’ + 0.12329 XLII 24, 0O=—2, 27900 — 0.33333 XXI — 0.35824 XXVI — 0, 26352 XXXI — 0 — 0, 00801 XLV 25. 0=+0. 03395 + 0, 07886 XXT — 0.58392 XXVI — 0, 38503 XXXI + 0 — 0.01833 XLV 26. 0=+1.31800 +0. 04764 XXII +1. 13645 XXVII +0, 89352 XXXIT +0. 51029 XLII 27. 0=—0. 30700 — 0.01590 XXII -+3. 28786 XXVIT —1. 27651 XLII 28. 0=+0. 60800 +0. 01853 XXIV —2, 17081 XX1X +0. 14464 XXXIV 4-0. 75359 XL —0. 00055 XLIV —0. 46465 XXV —0, 33333 XXX —0. 21836 XXXV —1. 19968 XLI +0.00174 XLIV +0. 02090 XXTV +1.51687 XXTX —1. 15294 NXXV —1. 26836 XL —0, 00058 XLV +1, 18518 XXVI +0. 53705 XXXAI —0. 66667 XXXVI +0. 93333 XLII —0. 14286 XXI —0. 15385 XXII —0. 15385 XXII +0, 20252 XXIII +0, 23492 XXIX —0. 08930 XLIL —0, 51314 XXI +0. 02082 XXVI —0, 02084 XXXVI +0. 02056 XLV +2. 53353 XXIV +0. 02940 XXTX —0, 14555 XXXV +0. 10226 XLIII 4-0. 69820 XXIV +1. 08690 XXIX —0, 39814 XXXV —0. 14749 XLIII 0.58392 XXV +0. 27290 XXX —0. 31200 XXXV —0, 27596 XL +0. 00171 XLV —0. 02068 XXV +1. 07571 XXX +1. 36856 XXXVI +1.59532 XLI —1.50000 XX VII +0. 89815 XNNIL 421 — 1.00172 XXIII “—~ 0.26407 XXIIT — 9.51314 XXIII — 1, 02238 XXIV — 0.031783 XXX — 0, 23896 XLII + 0, 20252 XXII — 0.00696 XXVII + 0.00694 XXXVIT 0. 69820 XXV 0.05981 XXX 0.03972 XXXVI — 0.00801 XLIV +++ +10. 56072 XXV — 0.04135 XXX — 0, 06202 XXXVI — 0.01833 XLIV +2, 94713 XNVI +1.44006 XXXI —0. 09065 XXXVI 4-0, 12976 XLI +1. 13655 XXVI +0, 42430 XXNI +0, 21215 XXXVII —0. 00345 XLII $3. 39814 XXVIII —0. D625 XXXL +1,51851 XXX VIII+0. 66568 XX XIX +0. 49418 XLIIT No. of cquation. 20, O=-41. 30. BL. 0O=-L0. ay, 33. 36. 37. 39, 40. V=-+1, 0=—4, 1 0=+0. =+1. — . Us), PRIMARY TRIANGULATION. Normal equations for determining the correlates—Continued. 13171 +0, 23492 XXII . H4300 51600 RGS00 - L065 . 50800 74100 24900 aie 3. 1S 150 a | oa + — 41. +40. 51647 XNVIL . 39597 XXNII _ 76088 XXXVIL 57212 XLII 03178 XXIT 07571 XXVII . 66667 XXXL . 09097 XXXVI . 00688 XLIL . 03176 XXII . 42430 XXVIL . 37963 XXXII . 07570 XXXVII 00116 XLIV . 06481 XXIV . 66667 XXX . 37824 XXXV . 20721 XLI . 27942 KXIV . 03373 XXIX . 62968 XXXIV _ 80548 XL . 05560 XXVI .12024 XXXI . 59702 XXXVI . 08402 XLI . 14555 XXIV . 39053 XIX . 64162 XXXIV . 63996 XXXIX =) . 36356 XXVIL . 59702 XXXIV 73244 XXXIX 00171 XLIV +0. . 21215 XXVII . 66238 XXXII . 16667 XXX VIII-L0. . 93542 XLII . 14815 XXIV = 0: +0. =: =O, —0. +40. —0. 04766 XXII 01588 XXIT 88775 XXIX 55246 NXXIV 76009 XXXIX 11552 XXVI 23104 XXXII 40390 XXXVIT 27596 XXVI 17086 XXXIT 82032 AXXVIL do8ld XLITI +0 ale L88773 +40, . 10274 XXIIT . 01390 XXIII . 33333 XXVIOL . 82476 XXXII - 50000 XXX VIII-+0. . 98597 XLIIT . 01388 XXIII 58703 XXVIII —3. 92084 XXNIIT +0, 29629 XXX VIII-+1. +40 —0 =f. —0. +0. 40. . 32476 NXX 2.10574 XXXV 40, -L0. 0, +40, 41 . 00114 XLV . 21956 XXV 37963 XXXI 33333 XXXVIT 16667 XLII 53467 XX'V . 66143 XLT | . 24480 XXVII 14090 XXNII . 46030 XXX VII . 00073 XLITI . 39814 XXV . 61392 XXX . 95267 XXXV . 84641 XL . 02084 XXIII . 66667 XXVIII 07589 XXXV . 59121 XL . 00172 XLV 00694 XXIII 76088 X XTX 46030 XXXIV 40390 XXXIX 00058 XLIV . 02553 XXV 50000 XXX - 34368 XXXV - 47270 XL /n9497 XXVIL +0. 02940 XXIV 40. +0. 0. i; + + + + + + +0, 0. 0. 0. 1 0. 2-3) 0) mae 0 0. 2.17031 XXVITT +12.23814 XNIX 08373 XXNIIT —0. 13121 XXXIV XXAVITI—1. . 10528 XLUOT 30365 XXNIX 00849 XLIV 05981 XXIV 22031 XXIX 98104 XXXIV 11373 XXXIX 00113 XLIV 26352 AXTV 67634 XXIX . 12024 XXXIV 03418 XLI 89352 XA VI 10648 XXXII 14315 XXXVIII—0. . 98179 XLII —0. 3. 0. 64157 XX VI 92084 XXXT 66238 XXXVIT - 42098 XLIL 0. 4. 14464 XXVIII 62968 XXXII +1. 08690 XXV [Cuav. XV1,C, +1. 04126 XXVI 0.2631 XXX £0. 67634 XXXI —0. 39053 XXNNV —0, 49604 XXXVI —0, 70374 XL 40. 78354 XLI 40, 0UNd7 XLV —0.04135 XXV +0, 27290 XXVI +43.31807 XNX 4.1, 84860 XXXI +41. 61392 XXXV_— 1.39378 XXXVI —0, 02493 NL +40. 01248 XLI —0. 00115 XLV —0.38503XXV 4-1, 44006 NNVI +1.84860 XXX +43. 74400 XXXI —0.99662 XXXV_ +0, 27290 XXXVI +0.34019 XLII —2. 76864 XLIIT +0. 89215 NXVIIL —0, 39597 XNIX +1. 15067 XXXIII —2. 14090 XXXIV 23104 XXXIX —2. 17086 XL +10, 62225 XXVII_ —0.93625 XXVIII +1,15067 XXXII 7.96842 XXXII —0, 56142 XXXVIII-L0. 17816 XXXIX —1. 42067 XLIII —0. 13121 XXIX +10, 49132 XXXIV 56246 XXX VITI—0. 67076 XX XIX . 31200 XXVI . 99662 XXATI . 07589 XXXVI - . 22618 XLI . 08972 XXIV 49604 XXIX’ . 21567 XXXVI . 45645 XLI . 02991 XXIV . 09097 XXX . 30638 XXXV . 82682 XL 00057 XLV . 14814 XXVI . 20629 XNNI . 20833 XXXVI . 46832 XLI 0. .17816 XXXII —0, . 76009 XXXVI-44. . 26836 XXVII 66568 XXVIIT 67076 XXXIV 42220 XXNNTX 75399 XXVIII .80588 XXXTIT -+-5.956°8 XXXIV 17270 NXNVIUI—1L. 80204 XXNXNIX . 15294 XX VII +0. 37824 XANIT +0. 30638 XXXVII +0. 30579 XLIT —0. 06202 XXV +1. 39378 XXX —0, 53022 XXXVII —0. 01031 XLIT re = 02067 XXV 07570 XXXI 53022 XXXVI 21717 XLI +0. 41. —0. =f —0, 50000 XX VIT —0. 14815 XXXII . 16667 XXXVII +0. 43974 XLII . 80365 XXIX . 63996 XXXV . 80294 XL —0. 70374 XXIX +0. 34611 XXXV +10. 13219 XL _ + Se 98104 XXX 04162 XXAXV “45. 95608 XL a —0, 21836 XXVIUI +2. 10574 XXXII . 34368 XXXVI . 11747 XLIIT . 09065 XX VI +0. 27290 XXXI —1, 20833 XXXVI —0. 83404 XLII — +0, 36355 XXVI —0. 33333 XXXII +12. 28785 XXXVII 0. 00343 XLII +1. 51851 XXVIIT —0, 55142 XXXII +3. 10648 XXXVI +0, 11378 XXX +0, 73244 XXXVI —4, 33998 XLI —(), 02493 XXX —1.59121 XXXVI —1. 45613 XLI § 10.] No. of equation. 41. 0=—0. 20260 42, N=+42. 09500 43. 0O=—1. 5810 44. 0=—1, 65700 45. 0=+1. 42600 46. 0=—0. 33000 47. 0=-+0. 94600 48, 0=-+0. 01800 FOND DU LAC BASE TO CHICAGO BASE. Normal equations for determining the correlates—Continued. +40. 12976 XXVI +1. 03418 XXXI 1.45645 XNXVI +8. 82134 XLI —0, 0930 XXII —0. 00345 XXVII +40. 16667 XXXII +2. 25752 XLII —0. 23896 XXII —1. 27651 XXVII —0. 58179 XXXII —1, 93542 XXXVII +15. 38335 XLIII —9. 01433 XXII —0, 00055 XXVII +6. 00058 XXXVII —0, 25000 XLVI —0. 01488 XXII —0. 00058 XXVII +40. 00057 XXXVII —0. 31720 XLVI —0, 25000 XLIV —0, 06789 XLV =. 42 ALY 59532 XXVII —0. 20721 XXXII —1. 21717 XXXVII —, 80734 XLII +0, 12329 XXIII +0. 33333 XXVIII —0. 42098 XXXIII 41. 25926 XLII 4-0, 15695 XXII +40, 49418 XXVIII —1, 42067 XXXII 41. —1,19968 XXVIII +0. 78354 XXIX —1. 06143 XXX +1. 46832 XXX VITT —4. 33998 XXXIX —0.38141 XXIV —0, 57212 XXIX 4-0, 30579 XXXV —1. 17930 XLIV 40, 10226 XXIV —5. 16528 XXLX +1. 00073 XXXIV +0. 43974 XXXVIII-+0. 55814 XL —1, 02428 XLIV +0. 02056 XNIIL +0. 00849 XXIX —1.17930 XLII +0. 02056 XXIII 0. 00847 XXIX -L0. 03282 XLII —0. 06789 XLVIL —0. 31739 XLV —0,21109 XLVI —0, 08911 XLVI +40. 04944 XLV —0. 00801 XXIV —0, 00113 XXX —1, 02428 XLIII —0. 00801 XXIV —0. 00115 XXX +0. 04944 XLITT —0,. 13418 XLVIII +1. 70295 XLVI +1.91450 XLVIT —0. 08911 XLVIT +40 —2, 08402 XXXIV —2 24 —0,97550XXV +40 —0.00688 XXX +40 —0.01031 XXXVI -L0 -+0. 03282 XLV —0.14749XXV 1 2.93597 XXX 2 —1.11747 XXXV_—0 —0,50754 XLI ety —0.01333XXV +40. -++0,00116 XXXI —0 ++2.77567 XLIV —0. —0,01833 XXV 4.0. +0,00114 XXXI —0. —0.46676 XLIV. 441 —0.21109 XLVIL —0 —0. 08911 XLVII +2 48013 XLVIHI Values of the correlates and their logarithms. XVUI XIX XX =—0. 3524 log 9,5469989_ =-1+0. 6200 log 9. 79238474 =— 1.6261 log 0. 2111552_ XXI =—0. 4851 log 9. 6858134_ XXII XXIV XAV XXVI XXVIII XXIX XXX XXXI XXXII XXXII =-+0.7814 log XXIII =+0. 0529 log =-+1. 0210 log =—0.1176 log 9, 0704812_ =—3. 6391 log XXVII =+3. 5744 log =+1. 1362 log =—0.0018 log =—0.3317 log =+3. 4073 log =+1. 8721 log =—-), 0641 log 9. 89289014. 8, 72346004, 0. 00903004 0. 5610000_ 0, 55320324 0. 0554433. 7. 2480000_ 9, 5207324_ 0. 53241554 0. 27372334 X, 8067200_. XLV XXXIV =-+0. 2969 log XXXV =-+0. 6223 log XXXIX =— 0.4391 log XL =-+0, 8354 log XLI =—1. 2026 log XLII =—0. 8693 log XLII =-+40. 1732 log XLIV =-+40.1700 log =—0.7653 log XLVI =+0. 0075 log XLVII =—0. XLVITL =—0, 0672 log 9, 47258094. 9, 79398584. XXXVI =—2. 1601 log 0. 3344759_ XXXVIT =—2. 9333 log 0. 4673580_ XXXVIII =—0. 9778 log 9. 9902456_ 9, 6425931_ 9, 92188414 0. 0801284_ 9..9394444_ 9, 23844764 9, 23039784 9. 8838260_ 7. 87330004. 9, TISOTIS_. 8. R2739000_ 5236 low 425 . 01245 XXX 2 22618 AXNAV - 45613 XL . 51029 XXVI . 34019 XXXIT - 00343 XXXVII nh 299792 XXVI . 10864 XXXI . 83404 XXXVI . 25926 XLII 00174 XXVI . 00171 XXXVI 46676 XLV 00371 XXVI 00172 XXXVI . 76435 XLV . 08911 XLVIII 404 PRIMARY TRIANGULATION. [Cuap. XVI, C, Values of the general corrections. uw bf ee at at (16) -=- 0.382 | C20.) S=+H0.170 © [22] =+0. 079 | [26,J=—1. 102 (29.)) =+0.317 [i] = OMe {21,] =—0, 618 | [28.) =-+0. 439 [25,] =—0. 333 | (2) =-0. 618 (17-45) =-+0. 620 [212] =—0. 140 | 23142) =O. IM (26, J =-0.227 | [29] =+0. 154 Limp} = O88) P2h4¢5] =-40. 819 [283] = 0. 15 (26.] =—0. 023 (30) 42] =-+-0. 117 (in-]) =—0.120 [221] SLAG | [234] =—0, 45 {26,] =--0. 082 [30;] =—0. 181 [1x;]) =--0, bx2 (22 pug epage | =O, O47 [28,] =O. 518 (26) =+0. 540 [30,] =-++0. 172 (iy) =e deO 22a] =—0. 216 [23,] =—0. 668 (26; ]=-+0. 158 (30;] =—0. 260 [1ix.] =—0. 068 ae | Sh The, [237] =—0, 006 {27,]=—0. 106 [30546] =-+0. 004 [19 } =O. 560 | Pees =—0. 192 | [28s] = FO. 240 (27:] =—0. 370 [31] =--0. 004 (1) =HLo7 Dey, =+0.181 | [240] =—0.725 [275] =—0. 230 [3lo] =—0.524 Cip/]=—0.010 | [225] =40.776 | [242] = +0. 620 [27,] =O. 283 [32,] .=—0. 242 Feo] bOI Ie S=—0.325 | [25,42] =-+0. 219 [2s] =--0. 321 [32] =—0, 067 p20.) =+40. 530 [224] a4, 108 | [25.] =-+0. 290 [2%] =—1. 150 [32245] =+0. 0x0 (203) =—-1.782 1) [Rao] =—0. 268 (28243 ] =--0. 636 {225 ] =-++0. 742 [334] =—0. 067 Residuals resulting Jrom substitution of general corrections in numerical equations of condition. ee Residual, | atte Residual. See | 2 | lh eee | 18 —0. 0001 | aM | 0, 0112 19 £00002) BB 4.0,.0585 20 40.0001 | 36 —0. 0001 21 —0.0002 | 37 +0. 0003 oe) +40. 0001 | 38 — 0. 0001 | 23 +0. 0060 | 39 +0, 0240 | 24 0.0008 | 40 =, —0.0080 1 9h +0.0100 © 41. | 0.0160 | 26 -+0. 0004 42 | -40. 0002 27 0. 0007 43 0.0260 | 28 —0. 0001 44 0.0001, 29 +0. 0021 45° | 40.0003 | . 80 -0. 0006 46 | 0. 0000 : ; BL | +0. 0005 47 | ~—:0, 0000 | 82 | ~ 0, 0005 48 | +0. 0009 33 | —0. 0040 ; PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § 08. Let m=whole number of observed angles in a section. r=whole number of rigid conditions in a section. n=nuinber of triangles in principal chain. [per]=sum of weighted squares of corrections to observed angles. py=probahle error of an observed angle of weight unity. p.=probable error of an observed angle of average weight in whole section. pf=pvrobable error of an adjusted angle of average weight in whole section. p=average weight of an observed angle in whole section. p,=average weight of an observed angle in principal chain. p.=probable error of an observed angle of average weight in principal chain. p/=probable error of an adjusted angle of average weight in principal chain. ~ p,=probable error of an observed angle in the principal chain as derived from the errors in closing of triangles. [erJ=sum of squares of closing errors ip triangles. $§ 11, 12.j , FOND DU LAC BASE TO CHICAGO BASE. 425 Proceeding as in § 8, Chapter XIV, C, there are found the following values: FOR THE ENTIRE SECTIONS OF THIS CHAPTER. Section. Extent of section. m r [pvr] p, Dy Ps / — p,! | | at “a “ “ VI | Minnesota Junction -Horicon to Warren-Fremont...| 50 | 18 11.53 | 0.54| 0.94] 0.56 0. 80 0. 45 VII | Warren-Fremont to Michigan City-Bald Tom.....-. 104 | 64 29.06 | 0.45; 0.93] 0.49 0. 62 0. 31 “FOR THE PRINCIPAL CHAIN CONNECTING THE FOND DU LAC AND CHICAGO BASES, GIVEN IN D, § 12, FOLLOWING. From closing errors of triangles. Section. Extent of principal chain in each section. Pe Po p! ° [ve] | 2 Average | Greatest Pe error. error. “ au uw “a V |. Fond du Lac Base to Minnesota Junction-Horicon...| 0.77 | 0.43 | 0.26 | 11. 81 5 | 0.60 1.32 2. 65 VI Minnesota Junction -Horicon to Warren-Fremont...; 0.90 | 0.57 | 0.46 | 39.80 | 15 | 0.63 1. 28 3.37 VII | Warren-Fremont to Chicago Base ........---.------- 0. 84 | 0.48 | 0.30 | 43,61 9 | 0.86 1.80 4. 69 | Entire principal chain ........02.00--00eee00e-0-[eeeeeee|eeee-[ee eee 95.22} 29 |o71] 1.45 4. 69 D.—PRINCIPAL CHAIN OF TRIANGLES BETWEEN FOND DU LAC AND CHICAGO BASES. § 12. The principal chain of triangles between Fond du Lac and Chicago Bases has two tri- angles which enter the principal chain between Fond du Lac and Keweenaw Bases, and three tri- angles which enter the principal chain between Chicago and Sandusky Bases. The complication involved in making the sides of these common triangles depend on three or more bases has been avoided without material loss of accuracy in the following manner: The chains running north and south from Fond du Lac Base diverge from the line Oakfield—Springvale. The chains running north and east from Chicago Base diverge from the line Willow Springs—Shot Tower. The ad- justed value of the length of the first line has been taken from the principal chain between Kewee- naw and Fond du Lac Bases, Chapter XV, D, § 8; the adjusted value of the second has been taken from the principal chain between Chicago and Sandusky Bases, Chapter XVII, D, § 6; and these two lines have been used as bases in adjusting the intervening triangle-sides. The loga- rithms of these adjusted lines and their probable errors expressed in units of the seventh place of decimals, as derived from Chapter XV, D, and Chapter XVII, D, are, the unit of length being one foot— Oakfield—Springvale...... 0... 6.2.2... cece ee eee ee eee 4.7953707 +19.65 Willow Springs—Shot Tower........-.----.-----+-++5: -.. 49117967 +19.65 These sides will not have their values changed in obtaining weighted mean sides for the inter- vening triangulation. The logarithm of the second of these lines computed from the first through the intervening triangles is 4.9118042, giving a discrepancy of 75 to be distributed among the logarithms of the intermediate sides. The probable errors of observed angles of average weight in this principal chain of triangles are, between the lines Fond du Lac Base and Minnesota Junc- tion-Horicon, + 0.43; between the lines Minnesota Junction-Horicon and Warren- Fremont, +0”.57; and between the lines Warren—Fremont and Willow Springs—Shot Tower, £0.48. See Chapters XV, C, and XVI, C. Using the notation of Chapter XIV, D, we have— pty (22+) p?-386-+ 386 == With the above values of » and the corresponding sums of («+ 4°) for the parts of the chain, b4L 8 4 426 PRIMARY TRIANGULATION. (Car. XVI, D, 2 (a’?+ 7) p? is found to be 4101 in units of the seventh decimal place of logarithms. Therefore the constant Lid —+4+ — = 4873. pp The additional data required in computing the changes in the logarithms of the lines are given in the following tables, which are arranged in the same manner as those in Chapter XIV, D, and show the principal triangles, &c., between the lines Oakfield—Springvale and Willow Springs—Shot Tower. The logarithms of the sides in the fourth column are computed from the logarithm of the line Oakfield—Springvale and the angles of the triangles given in the second . column. The line in the chain having the maximum probable error is Delafield—New Berlin, for which 3 =2390 and pr 2883, giving for the probable error of the logarithm of this line, neglect- ing error of standard, which is here insignificant, + 34.9, corresponding to y7ys0 part of its length. Principal chain of triangles between Fond du Lac and Chicago Bases. Logarithms Weighted mean Stations. Angles. AErOES OE : sides in | a2 and f2| 3 (a2+ 82) . lo arithme ” . ‘ect. sides in feet. fo} ‘ “ a“ Waupun 85 19 52. 298 4. 7953707 BOON seciseiccteietle sis S52 4, 7953707 Oakfield 53 57 54.751 —2, 654 4.°7045798) |... 2 ts.5e-|sevece setecaewedee: 4. 7045790 Springvale 40 42 13. 437 4, 6111598 600. 25 604. 25 499 4. 6111590 FROVICOM: i siscccns sateiiesece< 35 20 01.914 4. 6111598 F=f 17148 || ene ar] re 4. 6111590 Waupun..........-.---.---- 60 50 13. 594 +1. 560 4, 7901096) |axcnsees va) sseesseeeces|sexenees 4. 7901086 Oakfield ..22s20cescecesecces 83 49 45. 082 4. 8464535 4, 84 1491.18 666 4. 8464525 Minnesota Junction .....-.. 92 40 59. 668 4. 8464535 1; 00: | 5 2-ccenekess eee eeees 4. 8464525 Mori¢on: 5555-2 -a-as sesene 53 18 12. 252 —1. 296 4, T51OOLT | sexsseseseleseesesceces|easseee 4. 7510004 Wanpunvecsesssescceseseses 34 00 48. 601 4, 5946427 973. 44 2465. 62 848 4, 5946414 Lebanon .........----+.----- 29 05 17.132 f 4. 5946427 1428; 84 |scecciswiesiee: fSeeesees 4, 5946414 Horicon . ss..seesesseessesse 90 13 47. 236 +1. 155 49078664. |ezxnctesoenaceeecaedeas laeievisens 4, 9078643 Minnesota Junction .-.....-- | 60 40 56. 286 | 4. 8483453 139. 24 1568. 08 1356 4. 8483432 | Woodland .......-.-....---- 111 36 03. 873 | 4. 8483453 68:89) | cscacecssgecfemsesicss 4. 8483432 Lebanon osoee cc eiseics cose 34 04 48. 436 —1. 100 AMG2B4801 cist iccinied sears spancsaiall vormennia 4, 6284275 Horicon 2-225: ssesse02ss00% 34 19 08. 089 4. 6310935 954. 81 2591. 78 1687 4, 6310909 EYIn oe eices secaseokecpesngeens 36 08 13. 487 { 4. 6310935 B29 4d, | cceccie safes ciallnaree seis 4. 6310909 DebanOn «ape scsencaccciec cs 62 54 02. 612 +0.119 4, 8099453 4, 8099423 Woodland .........--.....-- 80 57 44. 543 | 4. 8550235 11. 56 3432. 78 1959 4. 8550205 Delafield ....-..--..---.00085 43 35 09. 632 { 4. 8550235, 4880816 lsc anacewe ac] ences eee 4, 8550205 BVI os sis esas cosawane ewes 86 50 06. 622 +0. 729 5: OVS8684, | ecccsesccellsececepscag el ta aweace 5. 0158600 Lebanon ......-...-.-20.--- : 49 34 45. 078 | 4. 8980829 324. 00 4245.19 2222 4. 8980795 New Lisbon ......-......--. 85 05 46. 444 4, 8980829 8500) | nocisaies cena] teailesxaes 4, 8980795 2 Delafield. .....-..-....--- aaie 40 37 39. 838 +0, 513 A: TVB35048:- | | occoced cis acecmee cecal aumiacee 4. 7133469 OPH, 22g aise Steet eines 3. 54 16 34.501 4. 8091465 228. 01 4477. 20 2298 4, 8091430 New Berlin .....-......----- 64 22 37. 685 ( 4. 8091465 4, 8091430 Delafield... c:22 se scctsjaenies wie 58 18 37. 048 +2. 145 ) 4. 7839849 4, 7839812 New Lisbon. .-.-........---- 57 18 46. 050 | 4.7792255 4, 7792218 Waterford ........-.----..-- 37 18 45. 961 4, 7792255 167.29 | acs seee vee |secnnees 4. 7792218 “New Berlin .........2......- 100 47 03.314 +0. 006 A OBBB9G4s | noes ied calico sacardasulicones.cd 4. 9888921 Delafield 22. scsasecessecces 41 54 11. 649 4. 8213292 547. 56 6079. 02 2816 4, 8213249 . | Caledonia .......---..--..--. 41 23 50. 761 | 4, 8213292 BTL OI sacs ered | cents 4, 8218249 Waterford .......-.......--. 88 10 54. 080 +0. 127 BLQ00T 272: | ease nccewerers| greiecare are weet loneriee sre 5. 0007224 New Berlin ........-.. saseneh: 50 25 16. 364 4, 8878583 302. 76 6952. 99 3099 4, 8878535 § 12.] FOND DU LAC BASE TO CHICAGO BASE. Principal. chain of triangles between Fond du Lac and Chicago Bases—Continued. Logarithms Weighted mean Stations. Angles. ne of | “of sides in | a? and B2| & (02+?) = logarithms of sure. | feet. : P sides in feet. oro4w “ 91 56 01. 247 4. 8878583 4. 8878535, 49 21 10.190 —2. 358 4. 7681955 4, 7681904 38 42 49, 232 4. 6842830 4. 6842779 50 01 43. 044 l 4. 6842830 4, 6842779 59 30 03. 390 +2. 878 4. 7351718 4. 7351665 70 28 14.148 J 4. 7741149 4. 7741096 BHO] cswascawne cone acennns 74 31 17.763 4. 7741149 4. 7741096 50 44 35. 536 —3. 372 4. 6790779 4. 6790725 54 44 07, 248 4. 7021118 4. 7021064 BOMON dc cccinerssn went 62 09 24. 020 4. 7021118 T2821. ios) cte assmasane octsianeciaie 4. 7021064 csincn nica iarnsas ped 60 00 19.774 +0. 766 4. 6931021 4. 6930965 SOMCTS ss ececcseees Henecoees 57 50 16. 702 4, 6831981 174, 24 8555. 08 3618 4, 6831925 BRYON. c0sccecicas sevens ces 62 17 26.133 4, 6831981 TOS SQ |licsinrs $$2-50:5:c0ll aeicizteioe 4, 6831925 Benton .-2cceny-ciessews seen 49 45 29. 910 —1. 439 4, 6188002 ~ scans seias tecmewa laced coieecimed 4. 6188035 BYIGtOL < jefe cis dessa crdaise oa 67 57 04. 394 4. 7031155 72. 25 8750. 54 3681 4. 7031098 Warten: sic. ssi ies cesnseriseiciet 61 10 57. 890 l 4. 7031155 194056: |evaies axes batten seeeee! 4. 7031098 ANTOGD. o:22 sisiersia aveisid e aciee-cier 53 27 04. 339 +0. 476 4. 6654360 |..-.------|--------.---[----- eee 4, 6654302 Benton «0-00 6.2605 eodsecaree 65 21 58. 272 J L 4. 7190905 94. 09 8979. 19 3755 4. 7190847 Fremont s< +0. 898 4. 8405974 4, 8405906 Park Ridge (new)----------- 93 50 43. 700 4. 9781042 4. 9780974 Willow Springs. .-...------- 75 02 59, 147 4. 9781042 31.36 |....--------|----.--- 4, 9780974 Shot Tower.....-...----.---| 48 55 01.898 +2. 558 4, 8702924 |..2-------[oee--- eee eee [eee ee eee 4, 8702855 Lombard ..... saGadweeeeetee 56 02 00. 334 4, 9118042 201.64) 2587.69 | 4487 4, 9117967 427 428 PRIMARY TRIANGULATION, [Cuar. XVII, A, CHAPTER XVII TRIANGULATION FROM CHICAGO BASE TO SANDUSKY BASE. A.—DESCRIPTIONS OF STATIONS. 2. NOTE RELATIVE TO ELEVATIONS. § 1. The elevations of the surface of the ground at the stations between Lakes Michigan and Erie depend upon the heights of the ground at stations Cedar Point and Stony Point, near the shore of Lake Erie, and at stations Bald Tom and Michigan City, near Lake Michigan. The heights of stations Cedar Point and Stony Point above Lake Erie were accurately determined by spirit leveling. The heights of stations Bald Tom and Michigan City above Lake Michigan were determined by single zenith distances over short lines. The relative heights of the intermediate stations were derived by trigonometrical leveling, reciprocal, non-simultaneous zenith distances having been observed over fifty-one lines, and single zenith distances over thirteen lines of the triangulation. The coefficient of refraction used in computations depending on single zenith dis- tances was 0.06, a mean of the values deduced from the trigonometrical leveling in Illinois. The total discrepancy shown by comparing the relative elevation of Lakes Michigan and Erie, computed by way of the triangulation, with their relative elevation determined by precise leveling (Chapter XXII, § 13), was 2.75 feet. This discrepancy was distributed amongst the heights of stations between the lakes proportiovally to their distances from either lake, the heights of sta- tions Cedar Point, Stony Point, Bald Tom, and Michigan City, derived in the manner stated above, being taken as exact. Excepting stations Cedar Point and Stony Point, the heights are all re- ferred to the mean level of Lake Michigan, which is 8.4 feet higher than the mean level of Lake Erie. (See Chapter XXII, § 13.) An estimate of the probable errors of these heights may be formed thus: First. Considering errors of closure, found by summing relative heights around tri- angles, and disregarding distance as in this case unimportant, since the lines do not vary greatly in length, the probable error of the relative height of two stations is found to be about + 1 foot. The height of any station in the chain can in general be determined by at least four independent _ routes, and as the number of lines along either flank may be taken as 14 (the actual numbers are 15 and 13) the probable error of any height, i. ¢., of the mean of the independent values, cannot exceed +1 foot x Je= . +£1.3 feet. The heights having the greatest probable errors are those of stations near the middle of the chain between the lakes, and the probable errors of heights in either direction from the middle part of the chain decrease from station to station, becom- ing zero for the heights of stations on the lake shore. Second. Using the above discrepancy in the relative heights of Lakes Michigan and Erie, viz, 2.75 teet, as a basis, the maximum prob- able error of a height would be about 4 x 2.75 feet or + 1 foot, and the probable errors at nearly all stations would be less than +1 foot. It may be concluded, therefore, that the probable errors of the heights of these stations do not on the average exceed + 1 foot. The elevations of stations east of the line Cedar Point-Stony Point described in this chapter are referred to the mean surface of Lake Erie, and were determined either by direct connection with the surface of the lake or by trigonometric leveling, as stated in greater detail in Chapter XVIII, A,§1. The precision of these elevations may be taken as equal to that of the heights of stations = Southern Michigan. $§ 1,2.] ; CHICAGO BASE TO SANDUSKY BASE, 429 DESCRIPTIONS OF STATIONS. § 2. GALENA, 1877.*—This station is situated in the southwest quarter of section 32, township 38 north, range 2 west, Galena Township, La Porte County, Indiana, about 4 miles northwest of the village of Rolling Prairie, a station on the Lake Shore and Michigan Southern Railway, 4 mile northeast of a school-house and cemetery at the corner of sections 31 and 32, and } mile east and 4 mile north of the western and southern section-lines. The height of station used was 65 feet. The geodetic point is marked by a stone post of the usual form set 24 feet below the surface of the ground. Three stone reference-posts are set as follows: One bearing south 32° 24’ west, distant 60.0 metres; one bearing south 84° 19’ west, distant 62.1 metres; and one bearing south 83° 48’ east, distant 36.6 metres from the geodetic point; a land-survey stone on the south line of section 32, 4 mile east of the southwest corner of that section, bears south 8° 06’ west, and is dis- tant 214.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 375.9 feet. BERTRAND, 1877.—This station is situated in section 15, Bertrand Township, Berrien County, Michigan, about 4 miles south of Buchanan, a station on the Michigan Central Railroad. The height of station used was 72 feet. The geodetic point is marked by a stone post of the usual form set 3 feet below the surface of the ground, with a stone post set directly over it as a surface- mark. Three stone reference-posts are set as follows: One bearing north 2° 13’ west, distant 13.11 metres; one-on the section-line between sections 14 and 15, bearing south 84° 13/ east, dis- tant 9.69 metres; and one at the corner of sections 14, 15, 22, and 23 of township 8 south, range 18 west, bearing south 2° 46’ 45” east, distant 309.7 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 356.9 feet. ¥ CARLISLE, 1877.—This station is situated in the southeast quarter of section 11, township 37 north, range 1 west, Olive Township, Saint Joseph County, Indiana, about 2 miles southwest of Carlisle village, on the Lake Shore and Michigan Southern Railroad, on what is known as Jar- rett’s Hill. The height of station used was 75 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Three stone reference-posts are set as follows: One bearing north 32° 16/ west, distant 16.1 metres; one bearing south 86° 16’ east, distant 14.4 metres; and one bearing south 18° 24’ west, distant 8.7 metres from the geodetic point. The corner of sections 2, 3, 10, and 11 bears north 39° 09’ 30” west, and is distant 1640.9 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 332.2 feet. MILTON, 1877.—This station is situated in the southeast quarter of section 4, township 8 south, range 16 west, Milton Township, Cass County, Michigan, about 4 miles east of the town of Niles, and 230 metres southeasterly from the center of the section, on the highest ground in the vicinity. The height of station used was 66 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: Two on the east side of the road just west of the station, one bearing south 62° 30’ west, distant 131.8 metres, and one bearing north 62° 03’ west, distant 130.9 metres; and one at the center of section 4, bearing north 32° 06/ west, distant 230.4 metres from the geodetic point. The corner of sections 4, 5, 8, and 9 bears south 57° 12’ 45” west, and is distant 1113.7 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 314.1 feet. ; PENN, 1877.— This station is situated in the northwest quarter of section 20, township 37 north, range 3 east, Penn Township, Saint Joseph County, Indiana, about 3 miles southeast of the city of South Bend. The height of station used was 40 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing north 86° 30’ east, distant 99.7 metres; one bearing south 7° 37’ east, distant 41.9 metres; and one bearing south 51° 56’ west, distant 69.6 metres from the geodetic point. The corner of sections 17, 18, 19, and 20 bears north 63° 51’ west, and is distant 788.4 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 309.3 feet. CALVIN, 1878.— This station is situated in the northeast quarter of section 27, township 7 south, range 14 west, Calvin Township, Cass County, Michigan, about 7 miles south of Vandalia, a station * Sze note relative to topographical sketches of stations, under Burnt Bluff, Chap. XV, A, § 2. 430 PRIMARY TRIANGULATION. [CHap. XVII, A, on the Michigan Central Railroad, and about 7 miles southeast of Cassopolis, a station on the same road. The height of station used was 110 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing north 59° 28’ east, distant 44.6 metres; one bearing south 21° 36’ east, distant 87.5 metres; and one bearing south 9° 47’ west, distant 82.1 metres from the geodetic point. The quarter-section corner on the west side of section 27 bears south 67° 07’ 30” west, and is distant 1004.5 metres from the geodetic point. The height of ground at the station above the mean level of Lake Michigan is 429.3 feet. JEFFERSON, 1878.— This station is situated in the southwest quarter of section 2, township 37 north, range 6 east, Jefferson Township, Elkhart County, Indiana, about 3 miles south of Bristol, a station on the Lake Shore and Michigan Southern Railway. The height of station used was 64 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference- posts are set as follows: One bearing south 27° 28’ west, distant 13.5 metres; one bearing north 15° 01’ west, distant 23.6 metres; and one bearing north 25° 44’ east, distant 95.0 metres from the geodetic point. The corner of sections 2, 3, 10, and 11 bears south 62° 43’ west, and is distant 350 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 388.3 feet. PorTER, 1878.— This station is situated in section 36, township 7 south, range 13 west, Porter Township, Cass County, Michigan, about 8 miles west of White Pigeon village. The height of station used was 73 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts were set as follows: One bearing south 48° 02/ east, distant 156.5 metres; one bearing north 73° 29’ east, distant 120.6 metres; and one bearing north 73° 24’ west, distant 108.3 metres from the geodetic point. The quarter-section corner on the west side of section 36 bears south 56° 55’ 03 west, and is distant 1304.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 368.4 feet. VAN BUREN, 1878.—This station is situated in the northwest quarter of section 33, township 38 north, range 8 east, Van Buren Township, LaGrange County, Indiana. The height of station used was 64 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing south 3° 20’ west, distant 130.0 metres; one bearing north 22° 59’ west, distant 27.6 metres; and one bearing south 89° 11’ east, distant 175.9 metres from the geodetic point. The quarter-section corner on the west side of section 33 bears south 50° 44’ west, and is distant 752.4 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 369.7 feet. SHERMAN, 1878.—This station is situated in the northeast quarter of section 22, township 7 south, range 10 west, Sherman Township, Saint Joseph County, Michigan, about 4 miles north- westerly from Sturgis, Michigan. The height of station used was 64 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing south 28° 58’ east, distant 45.9 metres; one bearing south 21° 16’ west, distant 41.7 metres; one bearing north 70° 17’ west, distant 26.8 metres from the geodetic point. The quarter- section corner on the south side of section 22 bears south 24° 12’ 30” west, and is distant 921.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 455.7 feet. Moneo, 1878.— This station is situated in section 4, township 37 north, range 11 east, Springfield Township, LaGrange County, Indiana, about 3 mile northeast of Mongo Mills. The height of station used was 66 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing south 59° 13/ east, distant 28.6 metres; one bearing south 14° 19’ east, distaut 16.7 metres; and one bearing south 57° 03! west, distant 30.1 metres. The corner of sections 4, 5, 8, and 9 bears south 22° 44’ 00” west, distant 873.5 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 445.5 feet. BRONSON, 1878.—This station is situated in the southeast quarter of the southwest quarter of section 16, township 7 south, range 8 west, Bronson Township, Branch County, Michigan, about three miles in a westerly direction from Bronson railway-station, on the Lake Shore and Michigan Southern Railway. The height of station used was 74 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing e $2.4 CHICAGO BASE TO SANDUSKY BASE. 431 north 40° 13’ east, distant 106.5 metres ; one bearing north 85° 08/ east, distant 68.6 metres; and one bearing south 36° 12’ east, distant 116.4 metres from the geodetic point. The quarter-section corner on the south side of section 16 bears south 19° 07’ east, and is distant 208.9 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 416.4 feet. FREMONT, 1878.— This station is situated in section 31, township 38 north, range 14 east, “ Fremont Township, Steuben County, Indiana, about 3 miles southwest of the village of Fremont, a station on the Fort Wayne, Jackson and Saginaw Railroad. The height of station used was 62 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference- posts are set as follows: One bearing north 22° 00/ west, distant 46.5 metres; one bearing north 42° 29’ east, distant 22.9 metres; and one bearing south 5° 15’ west, distant 55.7 metres from the geodetic point. The northwest corner of section 31 bears north 28° 10’ west, and is distant 834.6 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 560.2 feet. Quincy, 1878.— This station is situated in the southwest quarter of section 8, township 6 south, range 5 west, Quincy Township, Branch County, Michigan, about 2 miles northwesterly from Quincey railroad-station, on the Lake Shore and Michigan Southern Railway. The height of station used was 104 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: Two approximately on the section line between sections 7 and 8, one bearing south 18° 02/ west, distant 41.5 metres, and one bearing north 16° 53’ west, distant 46.25 metres; and one bearing north 35° 13’ east, distant 181.5 metres from the geodetic point. The quarter-section corner at the middle of the south side of section 7 bears south 56° 55/ 10” west, and is distant 980.5 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 478.9 feet. READING, 1878, 1879.— This station is situated in section 26, township 7 south, range 4 west, Reading Township, Hillsdale County, Michigen, on the grounds of the factory of the Colby Wringer Company, in the village of Reading, on the fort Wayne, Jackson and Saginaw Railroad. The height of station used was 115 feet. The geodetic point is marked by two stone posts in the usual manner. Three stone reference-posts are set as follows: One bearing north 48° 28’ east, distant 135.8 metres; one bearing south 83° 12’ east, distant 102.9 metres; and one bearing north 86° 20’ west, distant 147.8 metres from the geodetic point. The southeast corner of the factory of the Colby Wringer Company bears north 51° 05’ west, and is distant 50.25 metres. The northwest corner of section 26, being the corner of sections 22, 23, 26, and 27, bears north 44° 11’ west, and is distant 715.32 metres. The northeast corner of section 26, being the corner of sections 23, 24, 25, and 26, bears north 65° 56’ 50” east, and is distant 1235.9 metres frdm the geodetic point. The height of ground at the station above mean level of Lake Michigan is 627.2 feet. BuNDAY, 1879.— This station is situated in section 7, township 5 south, range 1 west, Somerset Township, Hillsdale County, Michigan, about 2 miles west of Somerset Center, on a hill known as Bunday’s Hill. The height of station used was 65 feet. The geodetic point is marked by a stone post set 24 feet below the ground-surface. Three stone reference-posts are set as follows : Two approxi- mately on the section-line on the east side of section 7, one bearing north 85° 56’ east, distant 132.0 metres, and one bearing south 54° 50’ east, distant 161.7 metres; and one on the north side of the road south of the station, bearing south 14° 47’ east, distant 210.9 metres from the geodetic point, The southeast corner of section 8 bears south 56° 15’ 45” east, and is distant 2166.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 702.2 feet. HILLSDALE, 1878, 1879.— This station is situated in the northwest quarter of section 27, town- ship 6 south, range 3 west, Hillsdale Township, Hillsdale County, Michigan, about one mile west of the town of Hillsdale. The height of station used was 65 feet. The geodetic point is marked by a stone post buried in the ground, with another stone post set directly over it as a surface- mark. Three stone reference-posts are set, as follows: Two on the section-line on the west side of section 27; one bearing north 35° 03’ west, distant 163.3 metres, and one bearing north 74° 02/ west, distant 98.3 metres; and one on the north side of the road south of the station, bearing south 1° 33/ west, distant 64.4 metres from the geodetic point. The clock-tower of Hillsdale College 432 PRIMARY TRIANGULATION. {Cuar. XVII, A, bears north 45° 52/ 28” east, and is distant 1898.5 metres from the geodetic point. The northwest corner of section 27 bears north 6° 59’ 18” west, and is distant 742.5 metres. The quarter-section corner on the west side of section 27 bears south 51° 51’ west, and is distant 118.8 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 657.1 feet. WHEATLAND, 1879.—This station is situated in section 29, township 6 south, range 1 west, Wheatland Township, Hillsdale County, Michigan, about 4 miles in a north-easterly direction from Pittsford, a station on the Lake Shore and Michigan Southern Railway. The height of station used was 110 feet. The geodetic point is marked by a stone post buried in the ground, with another stone post set directly over it as a surface-mark. Three stone reference-posts are set as follows: Two in the road on the west line of section 29, one bearing north 79° 13’ west, distant 351.6 metres, and one bearing south 54° 04’ west, distant 411.2 metres; and one in the road on the south line of section 29, bearing south 24° 35/ east, distant 662.2 metres from the geodetic point. The corner of sections 19, 20, 29, and 30 bears north 18° 50/ 20” west, and is distant 1053.9 metres; the quarter-section corner on the west side of section 29 bears north 59° 46’ 20” west, and is dis- tant 392.3 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 665.8 feet. PITTSFORD, 1879.— This station is situated about 25 miles, in a southerly direction, from Pitts- ford, a station on the Lake Shore and Michigan Southern Railway, in section 25, township 7 south, range 2 west, Jefferson Township, Hillsdale County, Michigan, about one-third mile west of the east line and a few metres north of the south line of the section. The height of station used was 74 feet. The geodetic point is marked by a stone post set three feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Three stone reference-posts are set as follows: One bearing south 81° 35’ east, distant 39.97 metres; one bearing south 5° 45’ west, distant 16.09 metres; and one bearing north 77° 47’ west, distant 18.70 metres from the geodetic point. The southeast corner of section 36, Jefferson Township, being the common corner of Jetterson, Pittsford, Wright, and Ransom Townships, bears south 20° 12’ 39” east, and is distant 1719.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 567.1 feet. Woopstock, 1879.—This station is situated near the south line of section 36, township 5 south, range 1 east, Woodstock Township, Lenawee County, Michigan. The height of station used was 60 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set on the south line of section 36 as follows: One bearing south 54° 41’ east, distant 59.7 metres; one bearing south 0° 16’ west, distant 35.2 metres; and one bearing south 35° 00’ west, distant 43.5 metres from the geodetic point. The northeast corner of section 1, town- ship 6 south, range 1 east, Rollin Township, bears south 88° 57’ 35” east, and is distant 1132.9 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 608.6 feet. FAIRFIELD, 1879.— This station is situated about one-fourth mile north of the village of Fair- field, about 14 miles north of the railway station Fairfield, on the Chicago and Canada Southern Railroad, and about 6 miles south of Adrian, on a slight rise of ground near the northeast corner of section 3, township $ south, range 3 east, Fairfield Township, Lenawee County, Michigan. The- height of station used was 100 feet. The geodetic point is marked by a small hole drilled in the top of a stone post set 24 feet below the surface of the ground, with another stone post set directly over it as a surface-mark. Two stone reference-posts, with the letters U. S. cut on their tops, are set as follows: One bearing north 81° 01’ east, distant 21.5 metres, and one bearing north 86° 25/ west, distant 36.0 metres from the geodetic point. A stone post of the same description, set 3 inches underground at the corner of sections 34 and 35, township 7 south, range 3 east, and sec- tions 2 and 3, township 8 south, range 3 east, bears north 51° 43’ 58” east, and is distant 203.0 metres from the geodetic poiut. The height of ground at the station above mean level of Lake Michigan is 216.9 feet. RaIsIn, 1879.—This station is situated about 6 miles northeasterly from Adrian, and 4 miles south of Tecumseh, in section 14, township 6 south, range 4 east, Raisin Township, Lenawee County, Michigan. The height of station used was 41 feet. The geodetic point is marked by a small hole § 2.) CHICAGO BASE TO SANDUSKY BASE. 433 drilled in the top of a stone post set 24 feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Two stone reference-posts, marked U.S. on their tops, are set along the road fence west of the station as follows: One bearing north 35° 41’ west, distant 172.9 metres; and one bearing south 49° 54’ west, distant 127.7 metres from the geodetic point. A stone post of the same description, set 3 inches below the surface, at the corner of sections 14, 15, 22, and 23, of township 6 south, range £ east, bears south 12° 04/ 18” west, and is distant 454.8 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 269.3 feet. . : BLISSFIELD, 1879.—This station is situated in section 22, township 7 south, range 5 east, Bliss- field Township, Lenawee County, Michigan, about 3 miles southwest of the village of Deerfield, and about 3 miles in a northeasterly direction from the village of Blissfield. The height of station used was 117 feet. The geodetic point is marked by a stone post set 3 feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Three stone reference-posts are set along the road east of the station, as follows: One on the east side of the road, bearing north 45° 11/ east, distant 39.0 metres; and two on the west side of the road, one bearing north 83° 25 east, distant 10.86 metres, and one bearing south 21° 01/ east, distant 37.0 metres from the geodetic point. The corner of sections 22, 23, 26, and 27 bears south 3° 35/ 15” east, and is distant 351.4 metres from tbe geodetic point. The height of ground at the station above mean level of Lake Michigan is 114.7 feet. DUNDEE, 1879.—This station is situated within a few metres of the line between sections 25 and 36, in township 6 south, range 6 east, this being also the line between Dundee and Summer- field Townships, Monroe Gounite Michigan, on the northwest side of the road between Petersburg and Dundee, and about 24 miles from each place. The height of station used was 106 feet. The geodetic point is marked by stone posts in the usual manner. Three stone reference-posts are set as follows: Two on the northwest side of the road above mentioned, one bearing north 76° 51’ east, distant 22.4 metres, and one bearing south £° 49/ east, distant 17.55 metres; and one on the south- east side of the road, bearing south 18° 53’ east, distant 34.05 metres from the geodetic point. The corner of sections 25, 26, 35, and 36, bears north 87° 32’ west, and is distant 110.25 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 99.5 feet. BEDFORD, 1879.— This station is situated in section 5, township 8 south, range 7 east, Bedford Township, Monroe County, Michigan, about 250 metres in a southerly direction from the quarter- section corner on the north side of the section. The height of station used was 94 feet. The geo- detic point is marked by a small hole drilled in the top of a stone post set 24 feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Three stone reference. posts are set as follows: One bearing south 30° 17’ east, distant 32.8 metres; one. bearing south 49° 44/ west, distant 33.0 metres; and one bearing north 39° 12’ west, distant 41.0 metres from the geodetic point. The corner of sections 4 and 5, township 8 south, range 7 east, and sections 32 and 33, township 7 south, range 7 east, bears nih 70° 59’ 40” east, and is distant 823.2 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 96.6 feet. Mownrok, 1879.—This station is situated in the cemetery of the Monroe County Poor-farm, in Monroe Township, Monroe County, Michigan, about 4 miles west of the city of Monroe. The height of station used was 115 feet. The geodetic point is marked by a small hole drilled in the top of a stone post set 3 feet below the surface of the ground, with a stone post set directly over it as a surface-mark. Three stone reference-posts are set in the corners of the cemetery lot, as fol- lows: One bearing north 13° 32’ west, distant 42.0 metres; one bearing south 5° 22’ east, distant 20.0 metres; and one bearing south 73° 36’ west, distant 31.4 metres from the geodetic point. The station is 34.6 metres from the center of the road which runs on the line between Raisinville and Monroe Townships. The southwest corner of private claim No. 432 bears south 25° 07’ west, and is distant 745.3 metres from the geodetic point. The height of ground at the station above mean level of Lake Michigan is 43.9 feet. Stony Pont, 1877, 1879.— This station is situated on the end of Stony Point, a point of land projecting into Lake Erie at its western end, in Monroe County, Michigan. The height of station 55 LS 434 PRIMARY TRIANGULATION. (Car. XVI, A, used was 55 feet. The geodetic point is marked by a hole drilled in the top of a stone buried beneath the surface of the ground. Another stone post is set directly over the geodetic point for a surface-mark, Three stone reference-posts are set as follows: One bearing south 15° 03’ west, distant 22.5 metres; one bearing south 84° 14’ west, distant 5.7 metres; and one bearing north 25° 03’ west, distant 16.7 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 5.2 feet. CEDAR Pornt, 1877, 1879.—This station is situated near the end of Cedar Point, Lucas County, Ohio, on the east side of Maumee Bay. The height of station used was 70 feet. The geo- detic point is marked by a stone post of the usual form, set about 3 feet below the surface of the ground. A stone post, projecting about 6 inches above the surface, is set directly over the geodetic point as a surface-mark. Three stone reference-posts are set as follows: One bearing south 61° 05/ east, distant 28.8 metres; one bearing north 61° 00’ east, distant 30.2 metres; and one bearing north 13° 28 east, distant 35.5 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 4.5 feet. MIDDLE SISTER, 1877.— This station is situated on the west side of Middle Sister Island, near the western end of Lake Erie, about 150 metres north of the south end of the island and 15 metres from the lake shore. The height of station used was 55 feet. The geodetic point is marked by a nail leaded into the solid rock, which is one or two feet below the surface of the ground. Three stone reference-posts are set as follows: One bearing south 39° 57’ west, distant 21.9 metres; one bearing north 33° 49’ east, distant 32.8 metres; and one bearing south 43° 23/ east, distant 21.9 metres from the geodetic point. The height of ground at the station above Lake Erie is about 10 feet. Locust Potnr, 1877.—This station is situated on the south shore of Lake Erie, on Locust Point, in Ottawa County, Obio, about 10 metres back from the lake. The height of station used was 41 feet. The geodetic point is marked by a stone post of the usual form, set 3 feet below the surface of the ground. Three stone reference-posts are set as follows: One bearing south 649 41’ east, distant 45.0 metres; one bearing south 18° 26’ west, distant 24.3 metres; and one bearing north 86° 30/ west, distant 24.3 metres from the geodetic point. The height of ground at the sta- tion above Lake Erie is about 2 feet. PorntTE PELEE, 1877.—This station is situated on the northwest point of Pointe Pelée Island, an island in the west end of Lake Erie, in the Province of Ontario. It stands in a grove about 210 metres in a southerly direction from the lake shore. The height of station used was 59 feet. The geodetic point is marked by a hole drilled in a brass bolt leaded into the solid limestone rock between the letters U.S. cut in the rock, 19 inches below the surface of the ground. A stone post is set directly over the geodetic point for a surface-mark. Three stone reference-posts are set as follows: One bearing north 22° 44’ east, distant 61.7 metres; one bearing south 80° 12’ east, dis- tant 35.9 metres; and one bearing south 44° 47’ west, distant 31.5 metres from the geodetic point. The east corner of a stone house bears south 53° 36’ west, and is distant 194.8 metres. The above bearings are magnetic. The height of ground at the station above mean level of Lake Hrie is 28.4 feet. MIDDLE BAss, 1877.— This station is situated on Middle Bass Island, in the western end of Lake Erie. It is on the north side of the island, about 650 metres from the eastern end. The height of station used was 75 feet. The geodetic point is marked by a nail leaded into the solid rock about 24 feet below the surface of the ground. Two stone reference-posts are set as follows: One bearing north 79° 26’ east, distant 17.5 metres; and one bearing south 81° 06’ west, distant 23.05 metres from the geodetic point. The height of ground at the station above the mean level of Lake Erie is 7.8 feet. KELLEY’S ISLAND, 1877.—This station is situated on Kelley’s Island, Erie County, Ohio, somewhat to the north and east of the center of the island, in a piece of open woods. It is about 670 metres south of the bay on the north shore of the island, 243 metres south of a road running east and west, and about 1080 metres east of the road running north from the landing on the south side of the island. The height of station used was 76 feet. The geodetic point is marked by a brass wire leaded into the solid rock between the letters U. 8. cut in the rock, 16 inches below the surface. A stone post is set for a surface reference-mark directly over the geodetic point. Three stone reference-posts, marked U.S. on top, are set as follows: One bearing north 20° 07’ § 2. CHICAGO BASE TO SANDUSKY BASE. 435 east, distant 32.7 metres; one bearing south 31° 32’ east, distant 30.3 metres; and one bearing north 87° 11’ west, distant 45.9 metres from the geodetic point. The Island House, a hotel at the landing on the south shore of the island, bears south 45° 55’ west, and is distant 1720 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 41.1 feet. DANBURY, 1877.—This station is situated in Danbury Township, Ottawa County, Ohio, about two miles northeast of Danbury railway station, on the Lake Shore and Michigan Southern Rail- way, and about two-thirds of a mile southwest of the corner of lots 3, 4, 9, and 10 in said township. The height of station used was 116 feet. The geodetic point is marked by a stone post of the usual form. set so that its top is 3 feet below the surface of the ground. Another stone post is set directly above the geodetic point as a surface-mark. Three stone reference-posts are set as fol- lows: One in a fence corner, bearing south 82° 20/ east, distant 61.57 metres; one by a fence on the east of the station, bearing north 57° 24’ east, distant 67.57 metres; and one by a fence on the south of the station, bearing south 3° 36’ west, distant 9.84 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 21.4 feet. WEST BASE, 1877, 1878.— This station, marking the northwest end of the Sandusky base-line, is situated on the northeast side of Cedar Point, about 40 metres trom the lake shore and 1016 metres from the light-house at the end of the Point. The height of station used was 70 feet. The geodetic point is marked by a stone post of the usual form, set in a bed of brickwork 3 feet square by + feet deep. Three stone reference-posts are set as follows: One Fearing ‘south 8° 02’ east, dis- tant 26.3 metres; one bearing south 55° 11’ west, distant 19.5 metres; and one bearing north 64° 21’ west, distant 38.8 metres from the geodetic point. An azimuth post on the line to East Base bears south 40° 23’ east, and is distant 46.91 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 6.6 feet. East BASE, 1877, 1878.—This station, marking the southeast end of the Sandusky base-line, is situated on the northeast side of Cedar Point, northeasterly from Sandusky, Ohio, about 43 miles from the Cedar Point light and 32 metres back from the lake shore. The height of station used was 65 feet. The geodetic point is marked by a stone post of the usual form, set in a bed of brickwork 3 feet square by about 3 feet deep. Three stone reference-posts are set as follows: One bearing south 13° 52’ east, distant 29.87 metres; one’ bearing south 69° 36’ west, distant 24.78 metres; and one bearing north 50° 57’ west, distant 47.7 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 5.8 feet. SANDUSKY, 1877.—This station is situated about 34 miles southwest of Sandusky, near the eastern border of Margaretta Township, Erie County, Ohio, about 14 miles south of the west corner of Perkins and Portland Townships. The height of station used was 115 feet. The geodetic point is marked by a brass wire leaded into the solid rock, between the letters U. 8. cut in the rock, about 3 feet below the surface of the ground. A stone post is set directly over the geodetic point for a surface mark. Three stone reference-posts are set as follows: One bearing north 3° 34 west, distant 69.0 metres; one bearing south 69° 12’ east, distant 50.4 metres; and one bearing south 71° 40’ west, distant 43.3 metres from the geodetic point. The Sandusky court-house bears north 33° 37’ east, and is distant 5907.0 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 59.3 feet. TOWNSEND, 1877.—This station is situated in lot 56, Townsend Township, Huron County, Ohio, about half a mile northeast of the village of Collins, and one-eighth of a mile south of the corner of lots 41, 42, 55, and 56. The height of station used was 105 feet. The geodetic point is marked by a brass bolt leadéd into the solid rock between the letters U.S. cut therein, about 23 feet below the surface of the ground. Three stone referencé-posts are set as follows: One bearing south 14° 10’ west, distant 42.1 metres; one bearing north 9° 10/ west, distant 64.0 metres; and one bearing north 82° 52/ east, distant 44.1 metres. The height of ground at the station above mean level of Lake Erie is 349.6 feet. BROWNHELM, 1877.—This station is situated near the center of Brownhelm Township, Lorain County, Ohio, about 14 miles in a southerly direction from Brownhelm railway depot, and one-half mile northwesterly from the corner of lots 44, 45, 52, and 53 in Brownhelm Township. The height of station used was 124 feet. The geodetic point is marked in the usual manner. Three stone 436 PRIMARY TRIANGULATION. [Cuap. XVI, B, C, refereuce-posts are set as follows: One bearing south 87° 31’ east, distant 57.5 metres; one bearing south 10° 06’ west, distant 14.65 metres; and one bearing south 53° 41’ west, distant 42.35 metres. The height of ground at the station above mean level of Lake Erie is 180.8 feet. CAMDEN, 1877.—This station is situated near the north side of Camden Township, Lorain County, Ohio, about one-half mile northeast of the village of Kipton and one-third mile southeast- erly from the corner of lots 5 and 6 in Camden Township and 15 and 16 in Henrietta Township. The height of station used was 106 feet. The geodetic point is marked by a brass wire leaded into the solid sandstone rock, between the letters U.S. cut therein, 24 feet below the surface of the ground. A stone post is set directly over the geodetic point as a surface-mark. Three stone ref: erence-posts are set as follows: One bearing north 83° 18’ east, distant 76.8 metres; one bearing south 8° 19 west, distant 99.8 metres; and one bearing north 37° 35/ west, distant 44.4 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 318.5 feet. ELyRiA, 1877.—This station is situated in the northwest corner of Elyria Township, Lorain County, Ohio, about 44 miles west of north of the village of Elyria, about 1 mile west of the Cleve- land, Tuscarawas Valley and Wheeling Railroad, and within the fork of two public roads leading to Charleston and North Amherst. The height of station used was 116 feet. The geodetic point is marked in the usual manner. Three stone reference-posts are set along the fence on the east of the station as follows: One bearing north 19° 44’ east, distant 27.1 metres; one bearing north 86° 16’ east, distant 9.1 metres; and one bearing south 12° 35’ east, distant 32.3 metres from the geo- detic point. The height of ground at the station above mean level of Lake Erie is 182.4 feet. GRAFTON, 1877.— This station is situated in the northwest part of section 89, in the southeast part of Grafton Township, Lorain County, Ohio, about 1 mile northwest of the village of Erhart, and 184.5 metres southwest of the track of the Cleveland, Tuscarawas Valley and Wheeling Rail- road. The height of station used was 110 feet. The geodetic point is marked by a hole drilled in the top of a stone post set 27 inches below the surface, a stone post being set directly over it for a surface-mark. Three stone reference-posts are set as follows: One bearing north 46° 18’ west, dis- tant 52.0 metres; one bearing north 9° 04’ east, distant 71.8 metres; and one bearing south 72° 03/ east, distant 43.6 metres from the geodetic point. The first and last reference-stones mentioned are set along a fence running northwest and southeast, 10 metres northeast of the station. The second is set by a fence running in a northeasterly direction from the station. The height of ground at the station above mean level of Lake Erie is 380.7 feet. OLMSTED, 1877.— This station is situated in section 13, Olmsted Township, Cuyahoga County, Ohio, about 2 miles west of the village of Olmsted Falls, and one-third of a mile south of the corner of sections 8, 14, and 13. The height of station used was 119 feet. The geodetic point is marked by a stone post of the usual form set 2.3 feet below the surface, with a stone post set directly above for a surface-mark. Three stone reference-posts are set as follows: One bearing north 8° 10! west, distant 90.9 metres; one bearing south 80° 26/ east, distant 67.7 metres; and one bearing south 13° 07’ west, distant 71.7 metres from the geodetic point. An east-and-west rail fence is south 17.1 metres; a north-and-south line fence is 15 metres west; an east-and-west highway is north 321 metres; and the Lake Shore and Michigan Southern Railway is south 653 metres. The first and last mentioned reference-stones are set by the north-and-south line fence, and the second is set by the east-and-west rail fence named above. The height of ground at the station above mean level of Lake Erie is 221.5 feet. RoyaLron, 1877.— This station is situated in section 12, Royalton Township, Cuyahoga County, Ohio, about half a mile north of the village of North Royalton, on the east side of # north-and-south highway. The height of station used was 35 feet. The geodetic point is marked by a hole drilled in the top of a stone post set 28 inches below the surface of the ground, a stone post being placed directly over it as a surface-mark. Three stone reference-posts are set as follows: One on the east side of the north-and-south highway, bearing north 4° 14/ west, distant 50.6 metres; one an the north side of a northeast-and-southwest highway, bearing north 80° 57’ east, distant 73.2 metres; and one on the west side of the nurth-and-south highway, bearing south 35° 28’ west, distant 41.4 metres from the geodetic point. The northwest corner of a church with a stone foundation, at the Junction of the two roads, bears south 11° 57’ east, and is distant 17.7 metres from the geodetic §6 3,4.) CHICAGO BASE TO SANDUSKY BASE. 437 point. The road fence on the west of the station is distant 5.5 metres. The height of ground at the station above mean level of Lake Erie is 699.1 feet. Rocxrort, 1877.—This statiou is situated in section 19, in Rockport Township, Cuyahoga County, Ohio, about 6 miles westerly from Cleveland, 4 mile northwest of the village of Rockport, and 4 mile northwest of the Lake Shore and Michigan Southern Railway. The intersection of the Lake Shore and Michigan Southern Railway and the Rocky River Railroad is north 2004.11 metres and east 3124.98 metres from the station. The height of station used was 95 feet. The geodetic point is marked by a stone post of the usual form set 27 inches below the surface. Three stone reference-posts are set as follows: One bearing south 52° 32’ west, distant 364.1 metres; one bear- ing north 19° 19’ west, distant 22.7 metres; and one bearing north 59° 47’ east, distant 194.3 metres from the geodetic point. The first and last named stones are set on the southeast side of the road just northwest of the station, the first being set in the corner of the road fence and a line fence south of the station, and the last in the corner of the road fence and a line fence northeast of the station. The second stone is set by the fence on the northwest side of the road. The height of ground at the station above mean level of Lake Erie is 208.5 feet. WARRENSVILLE, 1877.— This station is situated in Warrensville Township, Cuyahoga County, Ohio, about 3 miles northeast of Randall railway station on the Mahoning Division of the New York, Pennsylvania and Ohio Railway. The height of station used was 92 feet. The geodetic point is marked by a stone post of the usual form set 20 inches below the surface. Three stone reference-posts are set as follows: One bearing south 73° 46’ east, distant 56.7 metres; one bearing south 89° 13’ west, distant 75.5 metres; and one bearing north 55° 54’ east, distant 66.4 metres from the geodetic point. The first and last mentioned reference-stones are set by a line fence east of the station, and the second by a line fence west of the station. The height of ground at the station above mean level of Lake Erie is 641.5 feet. . WILLOUGHBY, 1877.—This station is situated on the shore of Lake Erie near the mouth of Chagrin River on the west side, and about 3 miles west of north of Willoughby railway station, in Willoughby Township, Lake County, Ohio. The height of station used was 107 feet. The geo- détic point is marked in the usual manner. Three stone reference-posts are set as follows: One bearing south 86° 40’ east, distant 10.1 metres; one bearing north 74° 44’ east, distant 45.6 metres ; and one bearing north.13° 25’ west, distant 27.0 metres from the geodetic point. The lake shore on the northwest is distant about 80 metres. The height of ground at the station above mean level of Lake Erie is 39.7 feet. CHESTER, 1877.— This station is situated in Chester Township, Geauga County, Ohio, about 2 mile west of Chester Cross Roads, and about 9 metres north of the east-and-west road from that place. The height of station used was 89 feet. The geodetic point is marked in the usual man- ner. Three stone reference-posts are set along the road fence south of the station as follows: One bearing south 63° 03’ west, distant 19.25 metres; one bearing south 6° 23’ west, distant 9.0 metres; and one bearing south 59° 30/ east, distant 17.75 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 719.2 feet. B.—STATIONS, SIGNALS, INSTRUMENTS, AND METHODS OF OBSERVATION. § B. See Chapter XVI, B. C.—MEASURED AND ADJUSTED ANGLES BETWEEN THE LINES MICHIGAN CITY- BALD TOM AND WILLOUGHBY-CHESTER. § 4. The following tables contain an abstract of the adjustment of the triangulation com- prised within the above-stated limits. A sketch of this triangulation is given in PlateTV. The adjustment is made in three parts or sections, namely: Section VIII, extending from line Michigan City-Bald Tom to line Fremont-Quincy. Section IX, extending from line Fremont—Quincy to line Cedar Point-—Stony Point. Section X, extending from line Cedar Point-Stony Point to line Willoughby -Chester. The scale of weights assigned to observed angles is as follows: On the same line with the name of each instrument here given is the number of combined results to the mean of which was assigned weight unity. 438 PRIMARY TRIANGULATION. [Cuar. XVII, C, Troughton & Simms theodolite No. 1, 16. Troughton & Simms theodolite No. 3, 16. Troughton & Simms theodolite No. 4, 24 at Olmsted, Royalton, Grafton, and Camden. Troughton & Simms theodolite No. 4, 20 at Townsend. Troughton & Simms theodolite No. 4, 16 at remaining stations where it was used. Repsold theodolite No. 1, 20. Pistor & Martins theodolite No. 2, 16. When the number of results obtained for an angle differed from the standard number by more than one-fourth of the latter, the weight assigned was the ratio of the former to the latter, rounded to the nearest one-fourth. For a detailed explanation of the tables see Chapter XIV, C, § 7, and the remark in Chapter XV, C, § 6, relating to “No. meas.” By the above division of the triangulation, a sum-angle condi- tion was disregarded in the general adjustment at each of stations Michigan City, Fremont, Quincy, Cedar Point, Willoughby, and Chester. The locally adjusted angles at these stations, with result- ing weights, were used in computing the general adjustment. t Section VIII.— Triangulation from the line Michigan City- Bald Tom to the line Fremont— Quincy. MICHIGAN CITY—30. (Observer, A. Rt. Flint. Instrument, Repsold theodolite No.1. Date, May, 1877.] Notation. | No. meas. Angle as measured between— Range. Wt. | (v) [v] Corrected angle. " “ Or’ lh uu | +0.496 | —0, 262 54 47 01. 558 Notr.—The weight and local correction of 30: are taken from the previous section of the adjustment. se °o ¢ a“ | ae lM Bald Tom and Galona ..+e ects 3 ie a 1 47 01. 324 a 301 iL BALD TOM—33. [Obseryer, A. R. Flint. Instrument, Repsold theodolite No.1. Date, June, 1877.} | Angle as measured between— Notation. | No. meas. | Range. Wt. wel @ {v] | Corrected angles. | ©). oF “u | | “W i Tee | “wt | o # “aw | Bertrand and Carlisle. ......-.- ieee 34 20 43. 604 331 20 7.9 | 1 +0. “008 +0. 549 | 34 20 44.201 Carlisle and Galena ........-- -- 33 39 32.466 332 20 | 88 1 | 40.048 | —1.052) 33 39 31. 462 Galena and Michigan City...-...--- 35 06 57. 523 333 20 | 7.5 1 +0.048 |) +0. 052 35 06 57. 623 Michigan City and Bertrand........ 256 52 46. 214 335 20 | 13.3 1 | +0. 049 | +0. 451 | 256 52 46,714 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(331)-+ (382) (333) — 0.193 =0 (331) 4+-2(332) 4+- (833) —0.193=0 (331)-+ (382) -|-2(333) —0.193=0 GALENA—34, (Observer, A. R. Flint. Instrument, Pistor & Martins theodolite No.2. Date, September, 1877.] j | | ' Angle as measured between— Notation. No, meas. Range. Wt. | (v) | {v] Corrected angles. ete e ey a aoe | oe, | fe La J ow oem | Michigan City and Bald Tom....... 90 06 01. 579 Bh: 16 Hal 1 | 40.087, +0.188 90 06 01. 854 | Bald Tom and Bertrand.....-....... o4 20 26. 885 342 l 16 57 1 | + 0. 087 —0.337) 54 20 26.635 Bertrand and Carlisle ..-.. 32 05 80. 872 B43 | 16 | 66 | 1 -{-0. 087 —0. 439 | 32 05 30. 520 | Carlisle and Michigan City 3 28 00. 316 344 i 16 66 1 +0. 087 +0. 588 183 28 00. 991 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(841)-+ (342) + (343) —0. HHS =0 (341) +2(842)-+ (843) —0. 348=0 (341)+ (842) +-2(343)—0.348=0 i 94.) CHICAGO BASE TO SANDUSKY BASE. 439 SEcTION VIII.—Triangulation from the line Michigan City-Bald Tom to the line Fremont— Quincy— Continued. BERTRAND —35. : {Observer, A. R. Flint. Instrument, Repsold theodolite No.1. Date, June, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. : (v) | {v] Corrected angles. \ °o f a ye $E | a oO f uw Milton and Penn --.....--.......--.- 54 31 31. 269 351 16 8.6 1 —0. 029 | +0. 258 54 31 31. 498 | Penn and Carlisle.......-..--. ...65 82 27 30. 229 35, 16 11.3 1 —0. 030 —0. 904 82 27 29,295 | Carlisle and Galena .......-.......- 31 31 31. 206 353 16 5.3 I —0.029 40. 794 l 81 31 31.971 | Galena and Bald Tom ........ ...-. 57 39 19. 343 B54 16 8.0 | 1 —0.030 , —0.299 57 39 19. 014 | Bald Tom and Milton.-.-........-.. 133 50 08.101 355 16 | 6.3 \ 1 —0. 030 | +0. 151 133 50 08. 222 | . 1 / NORMAL EQUATIONS FOR LOCAL ADJUSIMENT. 2(351)+ (352)-+ (353)+4+ (354) +0. 148=0 (351) +-2(352)-+- (353)+- (354) +0. 148=0 (351)+ (352) 4-2(353)+ (354) 4-0. 148=0 (351) + (352)-+ (353) + 2(354) +0. 148=0 CARLISLE—36._ [Observer, A.R. Flint. Instrument, Repsold theodolite No.1. Date, September, 1877.] 1 Angle as measured between— | Notation. | No. meas. “Range. ws Wt. (v) | {v] Corrected angles, I oj: u" | ” | no u" A oO nr Galena and Bald Tom..........---.- 59 54 32.221 361 16 : 8 | 1 +0.046 | +0. 057 59 54 32. 324 Bald Tom and Bertrand.........-.-. 56 28 25.129 362 16 bs eb +0.047 | +0.559 | 56 28 25.735 Bertrand and Milton.........-...-.. 22 28 11.313 363 16 oe 1 +0.046 | —0.918 | 22 28 10. 441 Milton and Penn...-.......-. 34 41 13. 601 364 16 6.9 1 +0.047 | —0, 212 84 41 13.436. Penn and Galena 186 27 37. 503 365 16 3.4 i +0.047 ) 40.514 | 186 27 38. 064 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(361)-+ (362)-+ (363)-+ (36a) —0. 233=0 (361) +2(362)-- (362)-+ (364) —0. 233=0 (361)-+ (362)-+2(36,) + (364) 0. 233—0 (361)-+ (362)-++ (36g) +2(361) —0. 223-0 MILTON—37. (Observer, A. R. Flint. - Instrument, Repsold theodolite No. 1. Date, October, 1877.] Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) | {»] : Corrected angles. | ° f # wt “a “a oO A “ Calvin and Jefferson...-..-..-.-.--- 33 36 05. 626 371 16 6.2 1 —0.096 | —0.375 33 36 05. 155 Jefferson and Penn .....------...-- 75 36 29.914 372 16 6.4 1 —0. 096 —0.315 | 75 36 29.508 Penn and Carlisle....--..--------.-- 51 46 54. 881 372 16 9.0 1. —0.096 40.179 , 51 46 54. 964 Carlisle and Bertrand........-...--- 20 82 48.765 371 16 7.2 1 —0,096 , +10. 560 | 20 32 49, 229 Bertrand and Calvin - 178 27 41. 293 375 16 7.5 1 —0.095 | —0. 049 | 178 27 41.149 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(371) + (872) + (3873)+ (3874) +0. 479=0 (371) -+2(37,) + (373)+ (3874) +0. 479=0 (871)+ (872) 4+2(873)+ (374)+0. 479=0 (871) + (37y)-+ (373) -| 2(874) +0. 479=0 440 PRIMARY TRIANGULATION. [Cuar. XVI, C, SEecTION VILL—TZriangulation from the line Michigan City—Balad Tom to the line Fremont- Quincy— Continued. PENN—3x. [Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Date, October, 1877.] | Angle as measured between— | Notation. | No. meas. eda | Wt. oo | oo ‘Corrected angles. ah ee cea ane Geen teal eee (ate | |b ' or “a | : ca | oF a“ Carlisle and Beitrand.... 2... ....- 40 23 08. 566 381 f 16 ‘ af A sa. ‘058 4, 's79 40 23 07. 629 | Bertrand and Milton... 0. .......- 53 08 44.389 | B82 18 x | 1 | —0. 058 -++0. 695 53 08 45. 026 Miltomand Caltin...ccosccusncossces 39 23 36. 605 | 383 oi i | 0.058 | 4-0. 029 39 23 36.576 Calvin and Jefferson ......--0. 0.2.2. 84 48 45, 446 384 17 , 5&7 1 —0. 058 —0. 083 34 48 45.305 | Jetferson and Carlisle... .......--.-- 192 15 45, 283 , 385 1 17 85 8 1 | —0.057 ar | TE | 4-0. 238 192 15 45. 464 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 2(381)-+ (382)-- (383)-++ (384)-+-0. 289=0 (381)-2(382)-- (883) (384)+-0. 289=0 (381)-+ (382)-++2(383)-++ (384)-++0. 2890 (38:)-+ (382)-++ (383)-+-2(381)-+-0. 289=0 CALVIN—39. [Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, May, 1878.] Angle as measured between— | Notation. | No. meas. | Range,; Wt. | (v) | [v7] Corrected angles. oO t a aw “a aw o. £ “we Sherman and Porter 8 16 26.722 391 16 5.7 1 +0. 016 —1.112 8 16 25. 626 Porter and Van Buren 22 16 39. 841 392 18 3.9 1 +0.017 | +-0.933 22 16 40.791 Van Buren and Jefferson ...-.-.---- 31 43 07. 476 393 18 3.1 1 4-0.016 | +0. 049 31 43 07. 541 Jefferson and Penn ......-.-..-----+ 78 08 16. 569 394 16 4.4 1 +0.017 | —0. 064 78 08 16.522 Penn and Milton...-.....--..--..--. 31 28 49.758 395 16 3.7 1 +0.016 | ~—0.127 31 23 49. 647 | Milton and Sherman 188 11 39. 536 396 16 4.4 1 +0.016 | +0. 321 188 11 39. 873 | ‘i NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(391) + (392)+ (393)-+ (394)4- (395) —0. 098=0 (391) +2(39,)+ (393)+ (394)+ (395) —0. 098=0 (391) + (892)-+2(393)+ (394)-++ (395)—0. 098=0 (391) + (392)+4- (393) 4+-2(394)-+ (395)-- 0. 098=0 (391) + (392)-+ (393)+ (391)-+-2(395)--0. 098-0 JEFFERSON—40. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Dates, April and May, 1878.] : | Angle as measured between— Notation. ho No. meas. “Range. Wt. (v) {v] Corrected angles. | = a aw =i | a uw a o t wn Penn and Milton................---- 30 11 09.818 401 | 16 ; Bl 1 +0. 065 +0. 152 30 11 10. 035 Miltonand (Calvin so cces netue deters se 36 51 49. 694 402 16 | 1 1 +0.066 | —0. 096 36 51 49. 664 Calvin and Porter....... -- -. 42 07 22,333 | 403 | 17 62) 1 +0. 065 +40. 308 42 07 22.706 Porter and Sherman 45 41 34. 091 404 16 5.6 1 +0.066 | +0. 083 45 41 34. 240 Sherman and Van Buren......-.---. 21 59 39.920 405 16 4.6 1 --0. 066 —0. 797 21 59 39.189 Van Buren and Penn ......-...----. 183 08 23.750 406 ; 16 | 7 a a | +0.066 | +-0.350 183 08 24, 166 ——e- ! . NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 7 2(401)+ (402)++ (402)+ (404)+ (405)—0.394=0 (401) +-2(402) + (403)-+ (40a)+4- (405)--0.894=0 (401)-+ (402)-+2(403) + (404)+- (405)—0.394=0 (401)+ (402)4- (403)+-2(404)-+ (405) —0.394=0 (401)+- (402)-+ (403)4- (404) +-2(405) —0.394=0 §4.] CHICAGO BASE TO SANDUSKY BASE. 441 SEcTION VIIL—Triangulation from the line Michigan City- Bald Tom to the line Fremont- Quincy— Continued. PORTER—4I1. [Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, May, 1878.] Angle as measured between— | Notation. | No. meas. | Range. | Wt. (v) (v] Corrected angles. oO « aw “ “a “ °o t aw Sherman and Van Buren..... Sede 50 46 16. 654 41, 18 4.3 1 —0. 242 | —0. 077 50 46 16. 335 ‘Van Buren and Jefferson ..........- 57 46 38. 243 41, 16 4.4 1 —0.242 | —0. 023 57 46 37.978 Jefferson and Calvin.-...-.. .....-. 83 52 49. 259 41, 16 3.3 1 —0. 242 | +0.473 83 52 49. 490 Calvin and Sherman ................ 167 34 16. 812 41, 18 4.3 1 —0.242 | —0.373 | 167 34 16.197 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(41,)+ (412)-+ (412)+0.968=0 (411) +2(412)+ (413)+0.968=0 (411)+ (41,)-+2(418)+0.968=0 VAN BUREN—42. (Observer, G. Y. Wisner. Instrnment, Troughton & Simms theodolite No.1. Date, May, 1878.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. °o a Ww ow aw aw Oo . ig Jefferson and Calvin .-....-.--------- 38 28 16. 908 42, 16 Bo 1 | —0.058 | +-0. 243 38 28 17. 093 Calvin and Porter .....-- ..-..----- 16 03 52. 416 42, 16 2.6 1 | 0.058} —0. 243 16 03 52,115 Porter and Sherman ..-.-..---..---- 85 45 49.792 42, 16 6.1 1 | —0.058 | —0. 216 85 45 49. 518 Sherman and Mongo .......--..--..- 52 35 17. 153 42, | 16 6.0 1 | —0.058 ; —0.116 52 35 16. 979 Mongo and Jefferson. ........-...--- 167 06 44.021 42, . 16 68 | 1 ! —0.058 | +0. 332 167 06 44. 295 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(42,)+ (422)+ (423)+ (424) +0. 290=0 (421) +2(422)-4+ (42s) (42a) +0. 290== (421)+ (429)4+-2(423)+ (424)-+0. 290—0 (42;)+ (422)+ (423) -+2(424) +0. 290=0 SHERMAN—43. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite, No.1. Date, September, 1878.] Angle as measured between— Notation. | No. meas. | Range.) Wt, (v) | {v] Corrected angles. oO # aw a“ “mw “ ° t aw Bronson and Mongo .....----------- 51 33 31. 608 48, 16 2.2 1 —0.041 | +0. 068 51 33 31. 635 Mongo and Van Buren....-..------- 82 45 41.030 439 16 6.4 1 —0.041 | —0.301 82 45 40, 688 Van Buren and Jefferson ......----- 17 42 22. 268 433 16 4.2 1 —0.041 | +0. 401 17 42 22. 628 Jefferson and Porter .....--.-------- 25 45 33. 693 43,4 16 3.6 1 —0.041 | —1. 206 25 45 32. 446 Porter and Calvin.....---.---------- 4 09 17.514 435 16 3.9 1 —0.041 | +0. 892 4 09 18.365 Calvin and Bronson......----------- 178 03 34.133 43, 16 4.5 1 —0.041 | 40.146] 178 03 34. 238 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(431) + (43)+ (438)-+ (434) (435)-+ 0.246=0 (431)-+2(482)-+ (483)+ (484)-+ (436) +0.246=0 (431)-+ (439) +2(43s)-+ (434)+ (488)-+0.246=0 : (431)-+ (439)+ (482) +2(43a)-+ (43s) +0.246=0 (481)+ (432)-+ (483)-+ (434) +2(435)-++0.246=0 56 LS 4492 PRIMARY TRIANGULATION. [CHar. XVII, ©, SECTION VIII.— Triangulation from the line Michigan City- Bald Tom to the line Fremont- Quincy— Continued. MONGO—44. [Observer, G. Y. Wisner. Tustrumenut, Troughton & Simms theodolite No. 1. Date, September, 1878. ] \ ov “ uw u or “ _ Van Buren and Sherman.......--- - 44 39 03.939 441 18 3.9 1 --0.155 | —0. 277 44 39 03, 507 Sherman and Bronson .... 48 20 14.027 440 18 5.1 1 —0.156 | 0. 092 48 20 13. 963 Bronson and Fremont .....--..----. 79 21 42. 987 443 18 8.1 | 1 —0.155 | +-0.016 79 21 42. 848 3.0 | 1; --0.156}| +0.169| 187 38 59. 682 Angle as measured between— Notation. | No. meas. Range. | Wt. (v) {v] Corrected angles. | | | | | Fremont and Van Buren..........-. 187 38 59. 669 444 i8 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(441)-+ (442)4- (443)-+0. 622=0 (441) +2(442)+ (443)+0. 622=0 (441)-+ (444) -4+-2(443) + 0. 622=0 BRONSON—45, (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, September, 1878.] Angle as measured between— i Notation. | No. meas. | Range. Wet. | (v) : [v] (Correcten angles. rae | 1 or “ “" | | “uw a oa. 2 “ ! Quincy and Fremont........--.----- 58 53 49.490 45, 16 6.0 1 —0. 076 -|-0. 042 58 53 49.456 | . Fremont and Mongo ........-------- 61 17 52. 291 45, | 16 4.1 1 | —0. 077 —-0. 091 61 17 52.123 1 Mongo and Sherman............ -.. 80 06 15. 286 45; | 16 3.9 1 —0. 076 0. 014 80 06 15, 196 Sherman and Quincy. .....-...------ 159 42 03. 239 45s : 16 5.6 1 —0. 077 --0 063 159 42 03.225 | t NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(451)-+- (452)-++ (453) 4-0. 306=0 (451) +2(452)+- (453) +0. 306=0 (451)-+ (452)+2(453) +0. 306=0 FREMONT—46. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, October, 1878.] Augle as measured between— Notation. | No. meas. Range. | Wt. (v) [v] Corrected angles. Se sisieccea ose) | teense aie ectcaese ! Kecan Beee oee J! oO t uw | | aw | aw “uw oO t “ ' Mongo and Bronson ....-..-.------- 39 20 26. 133 46 16 | 46) 1 +0.161 | —0.115 39 20 26.179 | Bronson and Quincy -----.---------- 62 01 52. 306 162 16 ; 3.6 | 1 -+-0.161 | —0.016 62 01 52. 451 | Quincy and Reading......--..------ 45 34 24, 583 463 16 | 4.6 | 1 SU GE dca cawmclaneeneeeesol encwxe | Reading and Mongo .....-.-.---.--- 213 03 16.334 — 464 16 | 6.0 | 1 HOIGL |scsesccaee| beursemeserececese NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(461)+ (462)+ (463) —0. 644-0 (461)-+2(462)+ (463)—0. 644=0 (461)+ (462) +-2(463) —0. 644=0 QUINCY—47. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, October, 1878.) Angle as measured between— Notation. | No. meas. | Range. wt. (v) [2] Corrected angles. | oes & tf wt wn WwW a“ c t “ | Hillsdale and Reading .....-......-. 31 53 15. 287 471 17 3.5 1 = 0108) iecesiees| pccccceecmeeree rs Reading and Fremont ...... be ttenn 54 23 55. 762 4, 17 | 4.4 Te | AOS) oc cka | asie utokenal Fremont and Bronson ..........---- 59 04 20. 005 473 17 5.9 1 —0.103 | —0. 016 59 04 19. 886 Bronson and Hillsdale .-....-.--.--.. 214 38 29. 359 474 17 | 8.5 1 0.104 | nici ciecezc acs ewaiew an naees cies NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(471)+ (472)+ (473) +0. 418=0 (471) +2(47,)+ (47s)+0. 413 =0 (471) + (472) -+2(473) +0. 413 =0 §4.] CHICAGO BASE TO SANDUSKY BASE. 443 Numerical equations of condition in the triangulation from the line Michigan City- Bald Tom to the LIIT. (10) —13. 7944 [34o]+ 1.3126 [343] — 0.3011[35,] +13. 0824[35,] (10) —15. 0045 [35,]+ 2.7875 [35] —13.5915[36,] +16. 8307[36,] LVIII. (143) —31. 688 [37)]+ 5. 4028 [872] — 5. 9556[38)] +24. 3246[38] LXIII. (10) +11. 0680 [39,J— 4.2315 [39] — 4. 2315(39;] + 7.0648[ 411] LIV. line Fremont — Quincy. + 7.0386 [43,]—36. 5951 [435] LXV. SIDE-EQUATIONS. (124) —15. 2995 [39,]+18. 7668 [39] —13.2709[41)] + 2.2575[412] —12. 2009[36,] +13. 9501[36,] — 9,8717[373] + 6.7079(374] —11. 8931[ 39,] — 7. 4706[395] + 7.0648[ 412] + 2.2575 415] —11. 4986[ 42, ] +14. 9986[ 422] LXVIT. (124) — 8.3876 [403;]— 8.3876 [40,] — 7.5845[40,] —26, 4972[42,] — 4. 4095[42,] — 4. 4095[423] —19. 2123 [433]+-17. 3828 [43,] +17. 3828[43,] — 7,043=0 — 4,501=0 — 9,694=0 +57, 247=0 +18. 427=0 +14. 805=0 Norte.—In the solution for determining the general corrections, each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. XLIX. L. LI. LII. LV. LVI. LVII. LIX, LX. LXxI. LXII. LXIV. LXVI. LXVIII. LXIX. LXX. LXXI. LXXII. LXXIII. LXXIV. [30] + [333] + [841] ANGLE-EQUATIONS. [33,] + [3%] + [342] + [354] (33,] + [353] + [35,] + [362] (33,] + [842] + [343] + [86] (35,] + [373] + [87,] + [88] [352] + [363] + [36,] + [38] [364] + [373] + [88] + [382] [371] + [894] + [395] + [402] [37.] + [383] + [38,] + [40,] [38,] + [894] + [40:] + [402] [39] + [397 + [895] + [403] + [40.] + [43,] + [43,] + 0.053=0 (39,] + [393] + [403] + [41s] [393] + [402] + [40,] + [405] + [421] [40,] + [405] + [412] + [42] + [42] [40,] + [411] + [412] + [43,4] (41,] + [42s] + [433] + [484] [42,] + [482] + [44] [43:] + [442] + [453] [443] + [452] + [461] [45,] + [46.] + [47s] + 0.021=0 + 1.139=0 — 1.603=0 + 1.771=0 — 1.692=0 + 2.913=0 + 0.217=0 + 0.662=0 + 0.217=0 + 0.091=0 — 1.763=0 + 0.114=0 + 0.737=0 +1. 223=0 + 1.098=0 + 0.694=0 — 0.146=—0 + 0.190=0 — 0 009=0 General corrections in terms of the correlates. [30,] =+0. 65625 XLIX [33,] =—0. 25000 XLIX +0.50000 L [33] =—0.25000 XLIX +0.50000 L [333] —=+0.75000 XLIX —0. 50000 L [344] =+0.75000 XLIX —0. 25000 L [342] =—0.25000 XLIX +0.75000 L [343] =—0.25000 XLIX —0. 25000 L [351] =—0.20000 L [352] =—0. 20000 L [353] =—0.20000 L +0. 75000 LI —0. 25000 LI —0. 50000 LIT +40. 50000 LIT +0. 50000 LIT +0. 60000 LI —0. 28473 LIII —0. 25000 LIT —0.25000 LI ~-+0.75000 LII —0, 25000 LI +0, 31204 LUT —1, 06740 LIII +0, 44331 LIIL 0, 40000 LI —0, 26463 LIL —1.25611 LIV +0.80000 LV —0. 20000 LVI 0, 40000 LI —0, 25463 LILL -+0.52309 LIV —0.20000 LV +0. 80000 LVI +0, 24434 LIV —0,20000 LV —0. 20000 LVI 444 General corrections in terms of the correlates—Continued. [354] =+0. 80000 L —0, 20000 LVI [36,] =—0. 20000 LI —0, 20000 LVIT [36] =-+0. 80000 LI —0. 20000 LVII [36,] ——0. 20000 LI —0. 20000 LVII [36,] =—0. 20000 LI + 0, 80000 LVII [371] =+0. 06397 LIV —0, 20000 LX [3%] —-+0. 06397 LIV +0. 80000 LX [373] =-- 0.99390 LIV —0, 20000 LX [374] =+0.73406 LIV —0, 20000 LX [38] ——0. 20000 LV — 0, 20000 LXI [38] =+0. 80000 LV —0, 20000 LXT [32] =—0. 20000 LV —0. 20000 LXI [38,] =—0. 20000 LV +0. 80000 LXI [39,] ~-+0. 22591 LVIIT —0, 33333 LXIV [39] =+0. 22591 LVIII +0, 66667 LXIV [39,] =+0, 22591 LVII +0, 66667 LXIV [39,] =—0. 60661 LVIII —0. 33333 LXIV [39;] ——0. 29703 LVITI —0, 33333 LXIV [40,] =—0. 16667 LIX —0,50000 LXVI (40,] —-++0, 83333 LIX —0.50000 LXVI [403] =—-0. 16667 LIX +0, 50000 LXVI [40,] =—0. 16667 LIX +0. 50000 LXVI [40;] =—0. 16667 LIX +0, 50000 LXVI (41,] =+0. 29680 LXIII +0. 75000 LXX [41,.] = +0. 29680 LXTII —0, 25000 LXX [41;] =—0. 18393 LXIII —0. 25000 LXX PRIMARY TRIANGULATION. +0, 60000 LI +0, 80000 LIE —0. 20000 LIT —0. 20000 LIL —0. 20000 LI —0, 40000 LV \ —0. 40000 LY +0. 60000 LV +0. 60000 LV +0, 80000 LVI —0. 20000 LVI —0, 20000 LVI 2 —0. 20000 LVI —0. 33333 LIX —0, 04623 LXV —0, 33333 LIX —1. 27019 LXV —0, 33333 LIX +1. 45511 LXV +0. 66667 LIX —0, 04623 LXV +0. 66667 LIX —0. 04623 LXV +0. 83333 LX +0. 32480 LXVII . 16667 LX = => +0, 32480 LXVIT —0. 16667 LX —0. 34621 LXVIT —0. 16667 LX —0, 34621 LXVIT —0. 16667 LX —0, 28196 LXVIT —0, 25000 LXTIV —0. 25000 LXIV +0. 75000 LXIV +1. 04861 LIT —1. 26508 LIT +1. 36003 LIII - 0.08498 LIII ~ 0.03498 LIII —0, 20000 LVII f —0. 20000 LVII +0. 80000 LVII —0, 20000 LVII +0, 60000 LVI +0. 60000 LVIL —0, 40000 LVII -—0. 40000 LVIT —0. 16607 LXI —0, 16667 LXVI —0, 16667 LXI —0. 16667 LXVI —0, 16667 LXI +0, 83333 LX VI +0, 83333 LXI —0. 16667 LXVI —0. 16667 LXI —0. 16667 LX VI +0. 66667 LXI —0. 33333 LX VIII +0. 66667 LXI —0. 33333 LX VIT —0, 33333 LXI —0. 33333 LXVITI —0. 33333 LXI +0. 66667 LXVIII —0. 33333 LXI 4-0. 66667 LX VIII +0. 22027 LXV —0. 84140 LXV +0, 40087 LXV +0, 24434 LIV —0. 06478 LIV —U. 06478 LIV —1. 42393 LIV +1. 61829 LIV —1. 85022 LVIITI +0. 74620 LVITI +0. 36800 LVIII +0, 36800 LVIIT —0, 25716 LVIIT —0. 25716 LVITI —0, 67405 LVIII +1. 44556 LVIII +0.50000 LXIT +0.50000 LXII +0.50000 LXII —0. 50000 LXII —0. 50000 LXIT —0. 33333 LXII —0. 16667 LXTX —0. 33333 LXIT —0. 16667 LXIX +0. 66667 LXII —0. 16667 LXIX +0. 66667 LXII +0, 83333 LXTX —0, 33333 LXIT —-0. 16667 LXIX —0. 25000 LX VIII +0. 75000 LXVIIT —0. 25000 LX VIII (Cap. XVII, C, —0. 20000 LV —0. 40000 LVI —0, 40000 LVI +0. 60000 LVI +0. 60000 LVI +0. 80000 LIX —0. 20000 LIX —0. 20000 LIX —0. 20000 LIX —0. 40000 LX —0. 40000 LX +0. 60000 LX +0. 60000 LX +1. 06339 LXIII —0. 46657 LXUI —0. 46657 LXIII —0, 04341 LXIII —0, 04341 LXIII —0. 16667 LXIV —0. 16667 LXIV 4-0, 83333 LXIV —0, 16667 LXIV —0, 16667 LXIV +0, 50000 LXIX +0, 50000 LXIX e —0. 50000 LXIX § 4.] (42.] =—0.97589 LXV —- +0, 80000 LX VI —0. 20000 LXXI [422] =+1.14389 LXV —0. 20000 LXVI — 5 0.20000 LXXT [423] =—0.05600 LXV = —0, 20000 LX VI —0. 20000 LXXI [424] =—0.05600 LXV — —0. 20000 LX VI +0. 80000 LXXI [43] =—0.33333 LXII +0. 49261 LXIII —0.16667 LXXI +10, 83333 LXXII [432] =—0.33333 LXIL 0. 49261 LXIII +0. 83333 LXXI = —0. 16667 LXXII [433] =—0.33333 LXII +40. 49261 LXIII —0. 16667 LXXI —0. 16667 LXXII [43,] =+0.66667 LXIE = +-1. 19647 LXITI —0. 16667 LXXI —0. 16667 LX XII [43;] =-+10. 66667 LXIZ —3. 16690 LXIII —0.16667 LXXI = —0. 1667 LXXII (44,] =+0.75000 LXXI —0. 25000 LXXII [442] =—0.25000 LXXI +0. 75000 LXXII [443] =—0.25000 LXXI —0. 25006 LXXII [45,] =—0.25000 LXXIT + —0. 25000 LXXIII [45.] =—0. 25000 LXXII +0. 75000 LX XIII . [453] =+40.75000 LXXII —0, 25000 LX XIII (46,] =+0. 75000 LXXTII [46,] =+0. 75000 LXXIV [473] =+0. 75000 LXXIV No. of equation. 49, O=-+0. 02100 +2. 15625 KLIX —0.75000 L 50. O=-++1. 13900 —0.75000 XLIX +2.55000 L +0. 24434 LIV —0. 20000 LV 51. 0——1. 60300 —0.25000 XLIX +1. 10000 L +0. 42390 LIV —0. 40000 LV 52. 0=-+£1.77100 —0.75000 XLIX +1. 00000 L : —0. 06478 LIV —0, 40000 LVI 53. 0——0. 70430 +0.31204 XLIX —0,01879 L +0. 29974 LIV —0. 25463 LV 54, 0=—0. 45010 +0. 24434 L +0. 42390 LI —1. 44595 LV +0. 71745 LVI +0. 06327 LX 55. 0——1. 69200 —v. 20000 L —0. 40000 LI —0. 40000 LVI +1. 20000 LVII —0. 20000 LXI 56. 0=-++2. 91300 —0. 20000 L —0. 80000 LI —0. 40000 LV +2. 80000 LVI —0. 20000 LXI “57. 0=-+0.21700 —0, 20000 LI —0. 20000 LII CHICAGO BASE TO SANDUSKY BASE. General corrections in terms of the correlates—Continued. —1. +0. +0, 21230 LXVIT +0. a 44. <1. . 20738 LXVIL » 20738 LXVIT . 74436 LXVIT 55472 LXVII 21230 LXAVIT 56506 LXVIT 18324 LAVIT 18324 LXVII 40. 60000 LXVUI +0. 60000 LX VIII —0. 40000 LX VIII -—0. 40000 LX VIII —0. 16667 LXIX —0. 16667 LXATX —0. 16667 LXIX +40. 83333 LXIX —0. 16667 LXIX- —0. 25000 LNXTI —0. 25000 LXXUI +40. 75000 LXXUI -40.75000 LXXIV —0, 25000 LXXIV —0. 25600 LXXIV Normal equations for determining the correlates. +1. 20000 LVI —0. 40000 LXI +2. 80000 LVII —0. 25000 LI +1. 10000 LI —0. 20000 LVI +2. 75000 LI —0. 80000 LVI —0, 45000 LT —0.20000 LVII -2. 12391 LI —0, 32459 LVI —0. 06478 LII 0. 69439 LVII —0, 25463 LIII +0. 478354 LVITI —0. 40000 LIT +1. 20000 LVII —0. 03498 LITT —0. 14632 LVIII. —0. 75000 LII +1. 00000 LIT —0. 45000 LIT —0. 20000 LVII +2. 55000 LIT —-1. 87917 LIT —0. 03498 LVIT +0. 29974 LIT —0, 11643 LVITT —1, 44595 LIV —0. 40000 LIX —0. 32459 LITI —0. 25716 LVIII +0. 69439 LIV —0. 20000 LIX 445 —0, 20000 LXX —0. 20000 LXX +0, 80000 LXX —0. 20000 LXX —0. 33333 LXX —0. 33333 LXX +0. 66667 LXX +0. 66667 LXX —0. 33333 LXX +40. 31204 LIII —0, 01879 LUI +2, 12391 LI —1.87917 LIII +6, 33433 LIII +8, 09402 LIV +40. 08327 LIX +42. 80000 LV —0. 80000 LX +0.71745 LIV —0. 40000 LX +1, 20000 LV —1. 00000 LX 446 No. of equation. 58. 0=—0. 67858 —0. 11643 LIV 59. 60. 61. 62. 63. 64. 65, 66. 67. 6R. 69. 70. We 72. 73. 74. ‘ ~ +0. 0=-+0, 66200 -L0. —0. —0. 0=+0. 21700 0=-+40, 09100 V=-+0, 05300 0=-+5. 72470 - 0=—-1. 76300 0=-+1. 47416 a 0=+0. 11400 + —0. +1. +0. —0. 0, —0, 33333 LIX Q=-+1, e440 U=-40. 73700 PRIMARY TRIANGULATION. Normal equations for determining the correlates—Continued. 2.75386 LIX 45182 LXTV 06327 LIV 36667 LX 09246 LXY . 06327 LIV 36667 LIX . 50000 LXVI —0. 41. 04623 LXV 20000 LV 26667 LX . 67773 LVIIT . 84018 LXTIT . 33333 LXVIT . 05884 LVIIT . 11707 LXIV . 79007 LXIX . 45182 LVITI . 11707 LXIIT . 58333 LAVIT . 06267 LVIII . 58579 LXTV ~ 62113 LAIX . 22591 LVI 46657 LXTIT 60000 LXAVITT 32480 LIX 34621 LXIV 83703 LXTX —0. 58333 LXTIV 41 0 —0 42 0-=-++1. 22300 . 16667 LXTX . 16667 LIX . 66667 LXIV . 66667 LXIX 0=-++1. 098N0 --0. 33333 LXII —0. 34882 LXVIT —0. 33333 LXXII 0=-L0. 69400 —0. 33333 LXII —0. 40000 LX VIII —0 0=—0. 14600 —0 . 25000 LXXTIT . 33333 LXIT —0. 41667 LXXT 0=-+0. 19000 0=—0. 00900 —0, 25000 LXXI —0, 25000 LXXIT -L.0. 47884 LV 441. 51771 LX -L0, 06267 LXV —0. 40000 LV +1, 33333 LXI -—-0, 83333 LXVI 0, 80600 LV 4-2, 83333 LX +0, 32480 LXVII —0. 20000 LVI +2. 96667 LXI —1. 16667 LXVI —1. 33333 LIX -1. 66667 LXIV -L1, 33333 LXIX —0. 08682 LIX —0. 47773 LXV -41. 98588 LXX —0, 83338 LIX 2, 91667 LXIV —0, 66667 LXIX —0, 09246 LIX -46, 97522 LXV +0. 16427 LXX —0, 83333 LIX -41. 16667 LXIV +40, 50000 LXIX +40, 32480 LX +1, 68491 LXV —0, 34882 LXX —0, 33333 LX —0. 67340 LXV —0, 65000 LXX —0, 16667 LX —0, 62113 LXV +41. 16667 LXX 41. 98588 LXITI —0, 65000 LX VIII +0. 49261 LXTIT —0. 16667 LXIX +0. 49261 LXTIT +2. 33333 LXXII —-0. 50000 LX XII —0. 26000 LXXTIT —0. 25716 LVI +40, 83895 LNI +0. 22591 LXVI —0, 20000 LVIT —1. 33333 LXII +40. 32480 LX VIT —0. 40000 LVI +1. 26667 LXI —0. —0. 40000 LVII —1. 16667 LXII +0. 64960 LX VII —0. 33333 LX +40. 13869 LXV -40. 33333 LXX —0, 04341 LXI —0, 46657 LXVI -40, 49261 LXXI —0, 16667 LX -40, 58579 LXV —0. 25000 LXX —0, 04623 LXI +40, 47922 LXVI —0. 05600 LXXf —0, 50000 LX +40. 47922 LXV —0, 20000 LX.X +0. 64960 LXI —2.52910 LXV1 +0. 35768 LXXI . 66667 LXI +1. 60000 LX VI —0. 40000 LX XI —0, 33333 LXI +40. 50000 LX VI —0. 16667 LXXI —0. 25000 LXIV +1. 16667 LXIX cS —0. 05600 LXV —0. 53333 LXX —0. 20738 LXVII —0,50000 LX XIII +2. 25000 LXXIII +42. 25000 LXXIV 33333 LX VIII —0. 14632 LVIL +0. 67773 LXII —2. 75386 LVIT —0. 08682 LXIIT —0), 33333 LX VITIT —1. 00000 LVI —0. 33333 LXI —0. 16667 LXIX 0. 83895 LVIII —0, 04341 LXII —0. 66667 LXVIII —1. 16667 LXI +1. 50000 LXVI —0, 33333 LXXI —1. 84018 LXII —3, 49724 LXVII +0. 49261 LXXII —0. 66667 LXI +1. 16667 LXVI +0, 18869 LXII +41. 68491 LXVIT —1. 16667 LXI -43, 18333 LX VI —0. 20000 LNXI +1. 67406 LXII 9, 75351 LX VIL -—0, 20738 LXI 40. 33333 LNII —1.97059 LX VII 1.38333 LXII -L0, 83703 LX VII —0. 16667 LX XII +40, 16427 LXV +12, 88333 LXX —0. 20000 LXVI +2. 38333 LXXT —0. 16667 LXTX —0. 25000 LXXTV —0, 25000 LXXIV [Cuap. XVII, C, +7. 78913 LVIII +0. 05884 LXIIT + 296667 LIX — 0.83333 LXIV — 0.16667 LXIX + 1.51771 LVIII — 0.16667 LXIV 4. 1, 23233 LIX — 0.66667 LXIV — 0.33333 LXIX + 4.16667 LXII + 1.67406 LXVIL — 0.33333 LXXII +14, 38111 LXIII + 0.29680 LXVIII + 1.66667 LXII — 0.34621 LXVII — 0.47773 LXIIT + 1.50000 LNII 2.52910 LXVII —3. 49724 LXITI —1. 97059 LXVIII +0. 29680 LXITI +3. 28333 LX VIII +1, 79007 LXIII ++1. 16667 LXVIII —0. 20000 LX VI —0. 53333 LXXI -L.0, 35768 LXVILI —0. 41667 LXXII —0: 33333 LXX 0. 67340 LXVIIT _ 64.) CHICAGO BASE TO SANDUSKY BASR. 447 Values of the correlates and their logarithms. XLIX =—0. 3987 log 9. 6006898 LXII =—0. 7087 log 9, 8504379 L =—0. 5326 log 9. 7264175_— LXITT =—0. 6548 log 9.8161418_ LI =+0. 6304 log 9. 79964384 LXIV =+0. 9998 log 9. 99991314. LIT =—0, 9707 log 9, 9870806. LXV =—0. 0358 log 8. 5543680 LIL =—0. 4201 log 9. 6233630_ LXVI =—0. 7868 log Y. 8958644 — LIV =+0. 2312 log 9. 36393154 LXVII =—0. 6771 log 9. 8306721— LV =+0, 4542 log 9. 65723764 LXVIII =—0.7703 log 9. 8866768_— LVI =—1. 1189 log 0. 0487991_ LXIX =+1.5449 log 0. 18290884. LVI =+0. 0026 log 7. 42160394. LXX =—0.7860 log 9. 8954502— LVIIT = —0, 0724 log 8. 8596186_— LXXI =—0. 4470 log 9. 6503561-- LIX =—0), 4863 log 0. 6868953_ LXXII =—0. 0776 log 8. 8900855 LX =—0, 2385 log 9. 3775248_ LXXIIT =—0. 1538 log 9. 1869281_ LXI =+0. 0409 log 8. 61182954. LAXIV =—0. 0217 log &. 3367176._ Values of the general corrections. Wy “t ‘ “a [30,] =—0.262 . [36,]=+0.057 | [38,]=—0.083 | (41,]J=— 0.077 | [43;] =+0. 892 [33:] =+0. 549 (362]=+0.559 = (39. J=—-1.112 | [4]=—0.023 | [44,]=—0.277 [33.J=—1.052 | [36J=—O.918 | [39%]=+0.933 | [415]=4+0.473 | [44,]=+0.092 | [36,] =—0. 212 | [393] =+0.049 | [42,]=+0.243 | [445]=+0.016 | [335] =+0. 052 (34,J=+0.188 | [37,J=—0.375 | [39,J=—0.064 | [42,]=—0.243 | [45,]=+0. 042 [34] =—0.337 (3%2J=—0.315 [39] =—0.127 | [42,]=—0.216 | [45,] =—0. 001 [345] =—0. 439 | [3%5J=+0.179 [40,J=+0.15e (42, =—0.116 [453] =—0. 014 [35.]=+0.258 | [37%]=+0.560 | [40.]=—0.096 . [43,]=+0.068 | [46,]——0.115 [35] =—0.904 | [38J=—O.479 — [40,]=+0.30R [43,7 =—0.301 — [45,]=—0, 016 [355]=+0.794 | [38,]=+0. 65 | [40] =+0.083 [43] =+0.401 | [47%] =—0. 016 [35,5] =—0.299 | [34]J=+0.029 | [40] =—0.797 | [43,J=—1.206 | Residuals resulting from substitution of general corrections in numerical equations of condition. d ce oa Residual. Sarat Residual. J 49 0. 0000 62 0. 0000 50 0. 0000 63 +0. 0040 51 —0. 0001 64 —0. 0001 52 0.0000 65 +0. 0002 . 53 —0. 0002 66 0. 0000 54 —0. 0001 67 —0. 0012 55 0. 0000 68 +0. 0001 56 0. 0000 69 +0. 0001 57 0. 0000 70 +0. 0001 58 —0. 0003 1 0.0000 59 0. 0000 72 0. 0000 60 0. 0000 73 0. 0000 61 0. 0000 74 0.0000 SEcTION IX.—Triangulation from the line Fremont — Quincy to the line Cedar Point — Stony Point. FREMONT—46. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, October, 1878. ] o£ “a Quincy and Reading.......-....---.. 45 34 24. 583 “ a“ Ww or “ Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) | Corrected angle. | 465 16 4.6 1 +0.161 | —0. 588 45 34 24.156 Note.—The weight and local correction of 46; are taken from the previous section of the adjustment. 448 PRIMARY TRIANGULATION. [CHar. XVII, ¢, Srcrion IX.—Triangulation from the line Fremont — Quincy to the line Cedar Point - Stony Point— Continued. QUINCY—47. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, October, 1878.] Augle as measured between— | Notation. | No.'meas. Range.| Wt. (v) {v] Corrected angles. Oi “" 4 i “ uu “u ou: “ Hillsdale and Reading .....-....--.-- 31 53 15. 287 47 / 17 3.5 1 —0.103 | —0. 216 31 53 14. 968 Reading and Fremont ......- sete 54 23 55, 762 | 472 , 17 4.4 1 —0.103 | —0. 588 54 23 55. 071 Novre.—The weights and local corrections of 47; and 472 are taken frow the previous section of the adjustment. READING—48. (Observers, G.Y. Wisner and R. 8. Woodward. Instruments, Troughton & Simms theodolites Nos.1and 3. Dates, October, 1878, and July, 1879.] Ae | | Angle as measured. between— Notation. | No. meas. | Range. Wt. (v) [v] - |Corrected angles. oe ——. Sal . t aw aw Ww “a oO a au Fremont and Quincy. -..-..-------+-- 01 42.379 | 481 16 4.4 | 1 +-0.018 | —0. 430 80 01 41. 967 | Pittstord and Quincy .--.-..--.----- 35 01.027 | 481+5 ' 4 1.3 | 0.25 ; —0.199 | —0.078 220 35 00. 750 | Quincy and Hillsdale . 29 02.121 | 482 21 5.7 | 1.25 | —0.016 | +0. 021 85 29 02. 126 | Quincy and Pittsford ........2..---. 39 24.59.448 | d8epaq4 | 3.9 0,25 | —0.276 | 40.078 | 139 24 59.250 | Tlillsdale and Wheatland .....-...-- 29 57 24.894 | Aes I 16 I> 5.0 | 1 +0.106 | 40.004 | 29 57 25.004 | Hillsdale and Pittsford ........-2..- 53 50 57. 1O8 4e344 21 ¢ ESS 625 0.101 | 0. 057 53 55 57. 124 | | Wheatland and Pittsford ...-------. 23 58 31. 961 | ARs 16 4.0 1) 40106 | 0.058 | 23 58 32.120 ' Pittsford and Fremont....-..---.--- 140 83.18.413 | 485 16 "4.0 | 1 | --0. 018 --0. 352 140 33 18.783 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 2(481)+4+ (482) + (483) -+ (484) —0. 282=0 (481) +2. 75 (482) +1. 50 (483) +1. 50(484) —0. 291=0 (481) 4-1. 50(482) 1-3. 75(483) +2. 75(484) —0. 682=0 (481) +1. 50(482) 4-2. 75 (483) + 3. 75 (484) — 0. 682=0 Nove.—483 and 48 were measured by Mr. Woodward with the Troughton & Simms No, 3, in July, 1879. The remainder were measured by Mr. Wisner with the Troughton & Simms No. 1, in October, 1878. BUNDAY—49. (Observer, G. Y. Wisner. Instruments, Pistor & Martins theodolite No. 1, ‘and Troughton & Simms theodolite No 1. Dates, May and June, 1879.) Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. oF “uw | u" | “ “ oF un" Woodstock and Wheatland ......... 63 53 06. 544 41 2 7.2 | 1 —0.018 | —0.535 63 53 05. 991 Wheatland and Hillsdale ........... 48 28 35.971 499 20 49 | 1 —0.018 | +0. 046 48 28 35.999 Hillsdale and Woodstock ........... 247 38 17. 534 493 a2 8.7 1.5} —0.013 | +0.489 | 247 38 18.010 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(491) +1. 5(492) +0. 073=0 1. 5(491) +2. 5(492) +0. 073=0 Noru.—493 was partly measured with the Troughton & Simms instrument, the remainder of the angles with the Pistor & Martins instru- ment. §4.] CHICAGO BASE TO SANDUSKY BASE. 449 SECTION IX.—Triangulation from the line Fremont — Quincy to the line Cedar Point— Stony Point— Continued. HILLSDALE—50. (Observer, G..Y. Wisner. Instruments, Pistor & Martins tieodolite No. 2,and Troughton & Simms theodolite No.1. Dates, November 1878, and June, 1879.] Angle as measured between— Notation. | No. meas. | Rang, | W t. | (v) {v] Corrected angles. fo} ft uw |» | wy a oO a aw Bunday and Wheatland............. 44 25 40.613 | 501 16 ! 5. 5 1 | —0.053 | —0. 607 44 25 39.953 Bunday and Pittsford .............. 80 47 32.819 | 50142 18 | 6.8 1 | —0.315 | —0.140 80 47 32. 364 Wheatland and Pittsford .... ...... 36 21 51.996 | 502 16 {| 5.8 1 | —0.052 | +0. 467 “36 21 52. 411 Pittsford and Reading .............. 91 34 24.758 | 50s 18 5.8 1 | +0.175 | —-0. 064 91 34 24. 869 Pittsford and Bunday ..-... ........ 279 12 27.619 | 5034445 16 | 6.3 1 | —9.123 | +0.140 279 12 27. 636 Reading and Quincy ...--........--. 62 37 43.405 | 504 18 4.3 1 | 40.175 | —0. 043 62 37 43. 537 Quincy and Bunday ........-..-..-. 125 00 18.808 | 505 17 | 4.2 1 | 4+0.175 | 40.247 125 00 19. 230 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(501)+3(502)-++ (50s) (504)-+-0. 018=0 3(501)-+4(502)-+ (503)-+ (504)-+0. 018=0 (501)-+ (502)-+2(50s)-++ (504)—0. 420=0 (501)-++ (£02)+ (503)-+2(504) —0. 420=0 NorTe.—501, 502, and 5034445 were measured with the Pistor & Martins instrument, in 1879; the remainder with the Troughton & Simms instrument, in 1878. WHEATLAND—51. (Observer, R.S. Woodward.’ Instrument, Troughton & Simms theodolite No. 3. Date, June, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. ° t “ a“ a “a o A “we Woodstock and Fairfield ....-.---- 43 30 48.485 | 5li 16 3.8 1 —0.031 | —0. 069 43 30 48, 385 Woodstock and Pittsford ........... 127 57 45.129 | 5li4e 8 4.5 0.5 | +0.148; +4-0.010 127 57 45, 282 Fairfield and Pittsford......... .... 84 26 56.849 | 5le 16 6.9 1 —0.031 ) +0.079 84 26 56. 897 Pittsford and Reading 56 02 33.220, 513 16 8.3 1 —0.007 | +0.319 56 02 33. 532 Pittsford and Hillsdale . 78 08 51.025 | 51344 8 3.9 0.5 | +0.317! +0.301 78 08 51. 643 Pittsford and Bunday...- 165 14 36.313 | 5134445 8 4.8 0.5 | —0,224 | +0, 221 165 14 36, 310 Reading and Hillsdale .. 22 06 18.1386 | 514 16 4.7 1 —0.007 | —0, 018 22 06 18.111 Hillsdale and Bunday...-.-.----..-- 87 05 44.569 | 51s 16 4.9 1 +0.178 | —0. 080 87 05 44. 667 Hillsdale and Woodstock ....--..-.- 153 58 23.272 | 515+6 8 4.0 0.5 | +0114} —0.311 153 53 23.075 Bunday and Wooéstock ........---- 66 47 38.573 | 516 16 5.7 1 +0. 066 | —0.231 66 47 38. 408 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 0(511)+2. 0(512)+1. 5(51,)-+1. 5(514)-+ (51s) —0. 000=0 “2, 0(511)-+3. 0(512)-L1. 5(513)H1.5(514)+ (51s) —0. 000=0 1. 5(51a)-+1. 5(51g)-+8. 5(513)-+2. 5(514)-+1. 5(515)—0. 131=0 1. 5(51,)-+1. 5(51g) +2. 5(51s)-8. 5(514)-+1. 5(515) —0. 131=0 (511)-+ — (51g)-+1. 5(51g)-+1. 5(514)-+2. 5(516) —0. 362=0 3X =a me CR 450 PRIMARY TRIANGULATION. [Cuap. XVII, C, SEcTION JX.—Triangulation from the line Fremont — Quincy to the line Cedar Point — Stony Point— Continued. PITTSFORD—52. [Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Dates, May, and June, 1879. J Angle as measnred between— Notation. | No. meas. | Range.| Wt. | (v) | (v] Corectea angles. | ! | oj “ | “ : " | | ov u" Reading and Hillsdale ...........--- 34 29 38, 427 521 22 4.4 | 1.25 | .+0. 059 | 40, ‘050 34 29 38. 545 | Reading and Wheatland .........-.. 99 58 54. 846 52142 16 4.4 /1 | +0. 037 | +0. 031 99 58 54.914 Hillsdale and Wheatland ........... 65 29 16.120] 52, | 16 | 54 {1 | —0.977} +0. 028 | 65 29 16,369 Hillsdale and Woodstock ......-..-.. 98 42 36.930 j 52243 16 | 4.0 | 1 : —0.203 | —0. 200 98 42 36. 527 | Wheatland and Woodstock ......-.. 33 13 20. 268 | 523 i 16 | 42 /1 +0. 062 | —0.172 | 33 13 20. 158 Wheatland and Fairfield........-... 78 52 58. 851 | 523-44 | 16 | 4.3 | 1 | +0.253 | —0,.179 78 52 58.925 Woodstock and Fairfield ..:........ 45 39 38.915 | 524 | 16 4.5 | 1 --0. 141 | —0. 007 45 39 38. 767 Fairfield and Reading...-..-....---- 181 08 05.901 | 52s 16 | 3.6 | 1 +0. 112 | +0.148 | 181 08 06.161 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 25(521) +2(52,)-+ (523)-++ (524) —0. 668=0 2, 00(521) +4 (522) 4+2(52g) + (524) 1. 210=0 (521) 4+2(522) +4 (523) +2(524) 0. 579=0 (521) + (529) +2(523) +3 (52s) -0. 037=0 WOODSTOCK—53. (Observer, G. Y. Wisner. Instrument, Pistor & Martins theodolite No. 2. Date, June, 1879.] Angle as measured between-— Notation. | No. meas. | Range. | Wt. | (») | |v] [Corrected angles. ° t “a - aw “a Oo i “ Raisin and Blissfield ..........-...-. 14 00 39, 893 | 531 16 5. 5 1 —0.414 | +0. 009 14 00 39. 488 Raisin and Fairfield ..........-.... 89 01 55.187 | 53142 8 4.7 0.5} —0.662] —0.148 39 01 54. 376 Bunday and Pittsford .............. 291 51 49.097 | 531424346 16 v1 1 —0.511 | +0. 246 291 51 48. 832 Blissfield and Fairfield.............. 25 01 15. 459 | 53, 16 6.5 1 —0.414 | —0.157 25 01 14. 888 Fairfield and Pittsford.........-.... 89 33 18.150 | 53, 16 4.9 1 —0.745 | —0.025 89 33 17.380 Pittsford and Wheatland ... -- 18 48 54.536 | 534 16 6.0 1 +0. 354 | 4-0. 046 18 48 54. 936 Wheatland and Bunday. .. 49 19 16.170 | 53, 16 5.0 1 +0.354 | --0. 292 | 49 19 16. 232 Bunday and Raisin ....-...-..--.--. 163 16 37.401 | 536 16 9.0 ud —0.744 | 40.419 363 16 37. 076 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(531) +1. 5(532)+ (532)-+ (58a)-+ (585)-+1. 691=0 1, 5(531)+2. 5(532)-+ (583)-+ (584)-+ (585) -+1. 691=0 (531)+ — (582)-+2(532)-+ (534)+ (58s)+-1. 609=0 (531) + (582)-+ (53g)-+-3(584)-+-2(535) —0. 197=0 2 (531)+ — (582)-+ (583)-+2(58a) +-3(535) —0. 197-0 FAIRFIELD—54. (Observer. J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Dates, May, June, and July, 1879.] Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) [v] Corrected angles. @ fF uo uo ue a“ oF “ Pittsford and Wheatland ........... 16 40 05.321 | 541 20 5.9 | 1 +0. 367 | —0. 552 16 40 05. 136 Pittsford and Woodstock.........-- 44 47 05.503 | 54142 16 5.7 | 1 +0.192 | —0.133 44 47 05. 562 Wheatland and Woodstock ......... 28 06 59.798 | 542 28 7.38 | 1.75 | +0.209]) +0.419 28 07 00. 426 Woodstock and Raisin.......-. ..-. 76 00 48.800 | 543 16 5.5 | 1 —0. 257 | —0. 336 76 00 48. 207 Woodstock and Pittsford ....... -- 315 12 53.685 | 5434445 20 7.2 |1 +0. 620 | +0.133 315 12 54, 438 Raisin and Blissfield.........- 45 24 59.650 | 544 16 41 {1 —0. 258 | —0, 018 45 24 59, 374 Blissfield and Pittsford ............. 193 47 06.627 | 545 16 3.3 | 1 —0.257 | +0. 487 193 47 06. 857 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(541) 4-3. 00(542)4+ (543)+ (544)—1. 580=0 3(541)+-4. 75(542)4+ (543)+ (544)—1. 580=0 (541) + (542) + 2(543)+ (844)+-0. 196=0 (541) + (542)-+ (543) -+2(544) +0. 196=0 § 4.) CHICAGO BASE TO SANDUSKY BASE. 451 SEcTION IX.—Triangulation from the line Fremont - Quincy to the line Cedar Point — Stony Point— | Continued. RAISIN—55. LObserver, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Date, June, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | (v] | Corrected angles. o 4 “ : “" “ “ ov 4" Dundee and Bedford...-..........-. ~ 21 35 46, 551 55, : 19 65 | 1 —0. 005 —0. 454 21 35 46, 092 Dundee and Blissfield .-..... -..... 40 27 53. 449 55142 17 5.2 | 1 —0. 158 +0. 330 40 27 53. 621 Bedford and Blisstield -............. 18 52 06. 748 552 23 6.8 | 1.25} —0.004 | +0. 784 18 52 07. 528 Blissfield and Fairfield.............. 73 50 35. 936 553 20 8.2 {1 +0.123 | +0. 006 73 50 36. 065 Blissfield and Woodstock ........... 138 47 55, 099 553+4 16, 5.5 | 1 —0. 285 —0. 161 138 47 54. 653 Fairfield and Woodstock............ 64 57 18.632 55, 20 V9 V1 +0.123 | —0. 167 64 57 18. 588 Woodstock and Dundee............. 180 44 12. 059 556 20 “| 125 | 1 —0.163 | —0. 169 180 44 31,727 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(55,)-+2. 00(55_)-+ (55,)-+ (554)—0. 224=0 2(551)-+3. 25(55,)-+ (55g)-+ (554) —0. 224=0 (551) (55q)-+8(554)-+-2(554)—0. 605=0 (551)+ — (55,)-+2(55,)-+3(55a) —0. 605=0 BLISSFIELD—56. {Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, June, 1879.] Angle as measured between— Notation. | No. meas. Range.| Wt. (v) [v] Corrected angles. oO a wm a“ “ aw ° ‘ “wn Fairfield and Woodstock .........-. 33 32 59. 957 561 3 8 5.7 0.5 —0.646 | —0. 647 33 32 58, 664 Fairfield and Raisin ..-.... .....--. 60 44 25. 267 56142 18 9.5 1 +0.325 | —0.374 60 44 25. 218 Fairfield and Dundee .........--.--- 156 30 34. 879 56142+2 8 3.8 0.5 | +0.046 | —0.109 156 30 34. 816 Woodstock and Raisin ............. 27 11 26.972 562 12 4.6 1 —0. 691 +0. 273 27:11 26. 554 Woodstock and Dundee ............ 122 57 34. 879 56243 8 4.0 0.5 | +0.735 | +0.538 | | 122 57 36,152 Raisin and Dundee ....-.........-.. 95 46 09. 169 563 18 9.2 1 +0.164 | +0. 265 95 46 09. 598 Raisin and Bedford .....-..-... .--. 143 49 56. 766 56344 8 3.4 0.5 —1. 059 +0. 439 143 49 56.146 Dundee and Bedford ....-...-...--.. 48 03 45. 820 564 18 4.7 1 +0.554 | +40.174 48 03 46. 548 Bedford and Fairfield..-......--.--- 155 25 38. 676 565 18 7.6 1 "40.025 | —0. 065 155 25 38. 636 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 0(561) +2; 5(562)-+1. 5(563)-+ (56a)-+2. 865=0 2. 5(561)-+4. 0(56,)-+2. 0(56,)-+ (564)-+3. 4960 1. 5(561)-+-2. 0(56)-+3. 5(563)-+1. 5(56,)-+0. 946=0 (561)-+ — (562)-++1. 5(563)-+2. 5(564)—0. 294=0 DUNDEE—57. (Observer, G. Y. Wisner. Instrument, Pistor & Martins theedolite No. 2. Date, J ‘une, 1879.] Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) (v) Corrected angles. o - " | " | “ “ or uu Morroe and Bedford ........... -- 67 06 25, 495 571 20 87) 1 —0.134 | —0.749 67 06 24. 612 Bedford and Blissfield 75 21 26. 527 57, 20 4.6 1 —0.134 | +0. 007 75 21 26.400 Blissfield and Raisin 43 45 56. 591 573 20 1.3 1 —0.135 | +0. 847 43.45 57. 303 Raisin and Monroe....-..--..-.....- 178 46 11. 925 574 20 8.1 1 —0.155 | —0.105 173 46 11. 685 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(571)-+ (572)-+ (573)-+-0. 588=0 (571) +2(57)+ (57g)-+0. 588=0 (571)-+ {572)-+2(573) +0. 538=0 452 PRIMARY TRIANGULATION. [Cuap. XVII, C, SEcTION IX.—Triangulation from the line Fremont — Quincy to the line Cedar Point - Stony Point— (Observer, J. 1. Darling. Continued. BEDFORD—5z. Instrument, Troughton & Simms theodolite No. 4. Date, June, 1879.] 1 Angle as measured between— Notation. | No. meas. | Range. | Wt. | (v) (v] | Corrected angles. i | \ ° Ss aw | i “a “a oO e tt | Blissfield and Raisin ......-....--.- 17 17 56.801 | 58 28 10.5 | 1.75 | +0.165 | —0.281 | 17 17 56. 685 | Blissfield and Dundee.......... ..-- 56 34 47.822 | 58i+2 16 4.1 {1 —0.397 | +0. 051 56 34 47.476 _ Cedar Point and Dundee............ 222 25 18.581 | 58i+2+5 16 4.3 {1 +0. 237 +0. 246 222 25 19. 064 Raisin and Dundee..--.-......--..---- 39 16 50.294 | 582 28 7.9 | 1.75; +0.165 | +0.332 39 16 50.791 | Dundee and Monroo...........-.--- 68 37 31.705 | 583 16 4.8 | 1 —0.016 | +0. 044 68 37 31,733 | Dundee and Cedar Point............ 137 34 41.229 | 58344 16 4.9 | 1 0.047 | —0. 246 187 34 40. 936 Dundee and Blissfield........-...-.. 303 25 12.384 | 5834445 20 7.3 = )1 +0.191 | —0. 051 803 25 12. 524 Monroe and Cedar Point.........--. 68 57 09.509 | 584 16 tek | 2 —0.016 | —0, 290 68 57 09. 208 Cedar Point and Blissfield . --- 165 50 31.180 | 585 16 10.9 | 1 +0.213 | 40.195 165 50 31. 588 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4. 75(581) +3. 00(58,)+ (583)+4+ (584) —1. 248=0 3. 00(581)+-4. 75(582)4- (583)4- (58,)—1. 248=0 (581) + (582) +-4(58,) +-3(58,) —0. 220=0 (58,)+ (582) +3(583) +4 (584) —0. 220=0 MONROE—59. (Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Dates, May and June, 1879.] i : 5 | Angle as measured between— | Notation. | No. meas. Range. | Wt. | (v) {v] Corrected angles. ! i OF aw | 1 uw . " a" or uw : Stony Point and Cedar Point ....... 68 58 06.371 | = 591 20 10. 6 1 +0.114 | —0.556 68 58 05. 929 Stony Point and Bedford............ 143 27 17. 610 | 59142 | 20 5. 2 1 —0.416 | —0.485 143 27 16.709 Cedar Point and Bedford ......-.--. 74 29 10. 595 | 592 20 | 4.2. 1 -+0. 114 +0. 071 74 29 10.780 Bedford and Dundec......-.....-.-- 44 16 04.515 593 20 7.9 1 —0.301 | —0.079 44 16 04.135 | | Dundee and Stony Point.....-...... 172 16 38. 893 594 | 20 7.0 1 —0.301 | +0. 564 172 16 39. 156 \ ! : NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. (Observer, R. S. Woodward. 3(591)4+-2(59,)-+ (593) —0. 270=0 2(591)+3(59)-+ (593)—0. 270=0- (59))-+ (592) + 2(592) +0. 374=0 CEDAR POINT—60. Instrument, Troughton & Simms theodolite No. 3. Dates, May and June, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. ee fe} t “we uw “aw “a oO # “w Bedford and Monioe...........----- 36 33 41. 076 601 16 5.6 1 +0. 084 | —0.117 36 33 41. 043 Monroe and Stony Point.-....-..... 39 32 51.317 602 16 5.4 1 +0. 084 | --0. 744 39 32 50. 657 Stony Point and Bedford......-..... 283 53 27.356 603 16 5.7 1 +0. 083 | -++0. 861 283 53 28. 300 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(601)-+ (60.)—0. 251=0 (601)-++2 (602) —0. 251=0 STONY POINT—61]. (Observer, G. Y. Wisner. Instrument, Troughton & Simms cheodolite No. 1. Dates, May and June, 1879.) Angle as measured between— Notation. | No.meas. | Range.| Wt. (v) (v) Corrected angles. ! £67736 LAXXVHUI .05154 XCI1 . 05154 XCT +0. 68931 XCV 2 . 46014 XC +0. 51480 XCIV . 25053 XCI . 52816 XCV . 07389 XCI . 58527 XCV . 35092 XCIV . 12200 XCVIIT . 03980 XCV . 79167 XCVIIT Values of the correlates and their logarithms. —0. 09091 LXXX +0, 12203 LXXXYV —0, 236086 LANXIX +40. 25874 LXXXV +0, 77180 LXXXIX —0. 067 XCIIT +40, 12987 LXXXY +2. 22720 LXXXIX -L0, 13033 XC1V — a) So —2. 11010 LAXXIX - 46014 XCIII = = » 72343 LUXXXIX . 29475 XCITT + _ . 94299 XCIT 51990 XCVI + —v. -—0. 29475 XCI —0, 46182 XCY . 54920 XCII . 35092 XCVI = +0. 63931 XCII . 28980 XCVI —0, 28980 XCV —0. 54789 XCVI LXXV =—0. 7887 log 9. 8941338_ LXXVI =—0, 2884 log 9, 4600103_ LXXVIL =—0. 2403 log 9, 3807538_ LXXVIII =-40. 0489 log 8. 68904224 LXXIX =-10, 2760 log 9. 44089334. LXXX =—1, 1560 log 0. 0629391_ LXXXI =—0. 8947 log 9. 9516677_ LXXXIT = —0, 3015 log 9, 4792873_ LXXXIII =—0. 0874 log 8. 9413623_ LXNXIV =—0, 8066 log 9. 9066690_ LXNXV =+0,5915 log 9.719621 4 LXNXVI =-41. 2845 log 0, 10872734 LXXXVIT =—1. 3050 log 0. 1156172_ LXXXVIII =—0. 4845 log 9. 6852669_ LXXXIX =—0, 1653 log 9, 2182729_ XC =-+0. 2404 log 9, 38102484 XCI =—0. 2165 log 9. 3353977_ XCII =—1. 4338 log 0. 1564916_ XCIII =-+1. 4807 log 0. 17046714. XCIV =-+0. 8059 log 9. 90625964 XCV =-+1. 2315 log 0. 09043794 XCVI =—0. 6431 log 9. 8082515_ XCVIT =—0. 9769 log 9. 9898457_ XCVIIT =—1. 6043 log 0. 2052821— * §4.] [46s] [471] [47] [48] [482] [483] [481] [491] [492] [501] [502] [503] [50.] CHICAGO BASE TO SANDUSKY BASE. =—0. 588 =—0. 216 =—0. 588 =—0. 430 =-0. 021 =-+0. 004 =-+0. 053 =—0. 535 =-+0. 046 =—0. 607 =-+0. 467 =—0. 064 =—0. 043 Values of’ the general corrections. Lol] [512] [51s] [514] [515] [521] [529] [523] [524] [531] [532] [53s ] [534] " =—0. 069 =+0. 079 =-+0. 319 =—0. 018 =—0. 080 =-40. 009 =—0, 157 =—0, 025 =+0, 046 [535] [541] [542] [545] [544] [551] [552] [553] [554] (56, ] [562] [563] [564] “ =—0. 292 =—0. 552 =-40. 419 =—0. 336 =—0.018 =—0, 454 =+0. 784 =+0. 006 =—0. 167 =—0. 647 =-+0, 273 =-40, 265 =+0.174 [571] [572] [575] [581] [582] [583] [584] [59)] [592] [593] [60,] [602] [61,] 0.749 =-+40. 007 =+0. 847 ——0, 281 —=+0. 332 —=+0. 044 ——0, 290 =—0, 556 —=-+40. 071 =—0. 079 =—0,117 =+0.744 =—0, 802 457 Py * . . . . . . ange Residuals resulting from substitution of general corrections in numerical equations of condition. é Ee Residual. coe. Residual. 75 0. 0000 87 0. 0000 76 0. 0000 88 0. 0000 W7 0. 0000 89 —0. 0001 78 0. 0000 90 —0. 0017 79 —0. 0025 91 0. 0000 80 0. 0000 92 0. 0000 81 0. 0000 93 0. 0000 82 0. 0000 94 —0. 0009 83 +0. 0006 95 0. 0000 84 0. 0000 96 0. 0000 85 —0. 0002 97 0. 0000 86 +0. 0001 98 0. 0000 SECTION X.—Triangulation from the line Willoughby - Chester to the line Stony Point- Cedar Point. CHESTER—56. [Observer, G. ¥Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, June, 1877.] Angle as measured between— Notation. No. meas. Range. | Wt. (v) [2] Corrected angle. Warrensville and Willoughby or " 92 12 11. 653 561 18 “ 3.5 1 “" +0. 052 u +0. 104 of uw 92 12 11. 809 Norte.~—The weight and local correction of 56 are taken from Section XI of the adjustment. WILLOUGHBY—57. {Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, June, 1877.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] (Correctea angles. or Ww “ aw “ ofr “ Chester and Warrenaville.....-...... 38 10 03. 094 572 20 2.7 | 1.25 | —0.155 | +0. 085 38 10 03. 024 Warrensville and Rockport. -........- 37 51 09. 507 573 * 21 4.3 | 1.25]. —0.156 | +0. 262 37 51 09. 613 Note.—The weights and local corrections of 572 and 573 are taken from Section XI of the adjustment, 58L 8 458 PRIMARY TRIANGULATION. [Cuap. XVII, C, SECTION X.—Triangulation from the line Willoughby — Chester to the line Stony Point - Cedar Point— Continued. WARRENSVILLE—58, (Observer, G. ¥. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, July, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v) Corrected angles. oO t a | “a a uw ° # wn Royalton and Olmsted .. 25.19 42.976 | 58: : 16 5.6 1 +0.500 | —0. 088 25 19 43. 388 Royalton and Rockport ..--....-..-- 37 10 08.566 | 58)+42 16 4.1 1 —0. 262 | +0.139 37 10 08, 443 . Olmsted and Rockport.....--.--..... 11 50 24.328 | 582 16 5.5 1 +0.500 ; +0. 227 11 50 25, 055 Rockport and Willoughby -.......-- 107 17 58.683 | 583 | 16 3.8 1 +0. 239 | +-0. 193 107 17 59.115 Willoughby and Chester ....-...--- 49 37 45.632 | 58 | 16 4.6 1 +-0.239 | —0, 097 49 37 45,774 | Chester and Royalton............--. 165 54 06.663 | 585 | 16 5.6 1 +0. 240 | —0, 235 165 54 06. 668 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(581) 4+-2(582)-+- (583)4- (584) 2. 980=0 2(581)-+3(582)+ (583)+4- (581)—2. 980=0 (581)4- (582)-4+-2(583)4+ (584)—1. 718=0 (581)-+ (582)-++ (583) 4+-2(584)—1. 718=0 ROCKPORT—59. {Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, July 1877.] | Angle as measured between— | Notation. | No. meas. | Range. Wt. | (v) (v] Corrected angles. | | | or “" ‘ | | a“ “ “ ov a" Willoughby and Warrensville ...... 34.50 52.431) 50 —-16 | a4 | +0.030 | +0. 130 34 50 52. 591 | Warrensville and Royalton ......-.. 76 57 22.951 | 592 | 16 5.6 1 +0. 030 ; +0. 088 76 57 23. 069 Royalton aud Olmsted ..... oh, Sigreeiors 73 38 29. 469 | 593 | 16 4.7 1 +0. 030 } +0. 081 73 38 29. 580 Olmsted and Willoughby. .- --- 174 33 15. 028 594 16 3.3 i 1 +0. 031 | —0, 299 174 33 14. 760 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 2(591) + (592)-+ (593)—0. 121=0 (591)+2(592) + (593)—0. 121=0 ‘(591)-+ (592)-}2(583)—0. 121=0 ROYALTON—60. (Observer, J. H. Darling. Instrument, Troughton and Simms theodolite No.4. Date, July, 1877.] Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) (v] Corrected angles. fe) ‘ a“ uw | | “a a“ ° “wn Grafton and Camden........-..---.- 23 18 50. 804 601 24 4.9 1 —0.011 | —0.079 23 18 50.714 Camden and Elyria ........-.--....- 23 11 22. 884 602 | 24 60 | 1 —0.011 | —0 316 23 11 22, 557 | Elyria and Olmsted .........-....--. 2 18 39.079 603 ; 24 3.1 1 —0.011 | —0. 046 2 18 39. 022 Olmsted and Rockport...---......-. 54 12 25.161 604 24 5.3 1 —0.011 | --0. 226 54 12 25. 376 Rockport and Warrensville......... 65 52 29.150 605 24 6.6 | tr —0.011 | +0. 206 65 52 29, 445 Warrensville and Grafton...... ---. 191 06 12.989 606 26 9.9 | 1 —0.012 | —0. 091 191 06 12. 886 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(601)+ (602)+- (603)-++- (604)+ (60s)-+0. 067=0 (601) +2(602)-+ (603)+ (604)+ (60s)-++0. 067=0 (601)+- (602) +2(603)-+ .(604)-+ (605) -++0. 067=0 (601) + (602)+ (603)+2(604)-+ (605) 4-0. 067=0 (601)+ (602) + (603)+ (604)-+2(605) +0. 067=0 §4.] CHICAGO BASE TO SANDUSKY BASE. 459 SECTION X.—Triangulation from the line Willoughby — Chester to the line Stony Point — Cedar Point— Continued. OLMSTED—61. 7 (Observer, J. H. Darling. Instrument, Troughton and Simms theodolite No.4. Date, July, 1877.) Angle as measured between— Notation. | No. meas. | Range.| Wet. (v) (v] Corrected angles. ~ ¢ am “uw aw “ ° , a“ Rockport and Warrenesville......... 17 33 42.514 61 22 6.4 1 +0.141 | +0.136 17 33 42.791 Rockport and Royalton 52 09 05. 508 611+2 4 2.5 0.2 —0. 278 +0. 450 52 09 05. 685 Warrensville and Royalton.. - 84 35 22.439 612 22 3.3 1 +0. 141 +0. 314 34 35 22. 894 Royalton and Grafton...... -. 80 52 00. 469 613 24 5.5 1 4-0. 087 | -+ 0. 084 80 52 00. 640 Grafton and Camden...... 57 53 14. 881 Gls 24 5.9 1 +0. 087 —0. 465 57 53 14. 503 Camden and Elyria.......... -- 386 10 22. 838 61s 24 5.6 1 +0.087 | —0. 064 36 10 22. 861 Elyria and Rockport .........,-...-. 182 55 16. 230 6lé 25 5.0 1 +0. 086 | —0, 005 132 55 16,311 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2, 2(61,) +1. 2(612)-+ (612)-+ (61a)-+ (615) —0. 740=0 1. 2(61,) +2. 2(612)-+ (613)+4- (614)+- (615)—0. 740=0 (61,) | — (612)+4+-2(613)+- (614a)+ (615) —0. 629=0 (611)4- (612)4- (612) 4+-2(614)+ (615)—0. 629=0 (611)-{- (612)-+ (613)+4+ (614a)-+-2(61s)—0. 629=0 GRAFTON—62. (Observer, J. H. Darling. Instrument, Troughton and Simms theodolite No.4. Date, July, 1877.] Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. . : | oO é a | | “ ar a oO f M“ Camden and Elyria .........-...--.- 45 27 23.543 621 24 6.3 1 +0.099 | -++0.179 45 27 23. 821 Elyria and Olmsted ..............--- 38 49 36. 563 622 | 24 7.5 L +0.099 | —0, 261 38 49 36. 401 Olmsted and Royalton .......--...-- 50 19 07. 951 623 ! 24 5.4 1 +0.099 | —0. 056 50 19 07, 994 Royalton and Camden .-- --.--.---- 225 23 51. 548 624 | 24 %.2 1 +0.098 | +0. 138 225 23 51. 784 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(621) + (622)-+ (623) —0. 395=0 (62,)-+2(622)-+ (62g) —0, 395=0 (62;)-++ (62) -+2(62s) —0. 395=0 , ELYRIA—63. (Observer, G. Y. Wisner. Instrument, Troughton and Simms theodolite No.1. Dates, July and August, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (2) Corrected angles. | or “ | “" “" “ of “ | Olmsted and Royalton .........----- 2 45 43,415 631 16 |; 48 1 —0.044 | —0.326 2 45 43.045 Royalton and Grafton .......------- 44 21 04.409 632 16 | 5.8 1 —0, 044 | —0.392 44 21 03,973 Grafton and Camden...-...--------- 68 15 03.279 633 18 3.8 1 —0.044 | +0. 490 68 15 03.725 Camden and Brownhelm...... ...-.- 38 49 27. 559 634 18 3.6 1 —0.044 | +0.443 38 49 27. 958 Brownhelm and East Base....-.---- 16 32 52. 678 635 16 5.3 1 —0.044 | +0.102 16 32 52.736 East Base and West Base....-..----- 5 37 21. 814 fi36 16 7.1 L —0.044 | —0. 429 5 37 21,341 West Base and Olmsted.....-.------ 183 38 27. 153 637 16 5.0 1 —0.043 | -+0,112 188 38 27. 222 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(631)+ (632)4- (633) + (634)-+ (635)-+ (63g)+-0. 307=0 (631) +2(632)-+ (63,)+ (634)4- (635) + (635) +0. 307=0 (631) ++ (632) -+2(634)-+ (634)-+ (635) + (636) 1-0. 307=0 (631) ++ (632)-+ (63,)-+2(634)-+ (63s)+4 (636)+-0. 307=0 (631) + (632)+ (633)+ (63a) + 2(635)- (636) 4-0. 307=0 (631) + (632)+ (f33)-+ (634)-+ (635)-+-2(63¢) 4-0. 307=0 460 PRIMARY TRIANGULATION. [Cuar. XVIT, C, Continued. CAMDEN—64. \ (Observer, J. H. Darling. Instrument, Troughton and Simms theodolite No. 4. Date, August, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v) (Corrected angles. % . I ~ ° ‘ a“ | “a aw | a Oo * a“ | Townsend and Brownhelm.......... 90 19 61.029 | 641 \ 24 4.9 1 —0.022 | 40.425 90 20 01. 432 Brownhelm and Elyria .......-..--. 43 06 02. 093 642 24 6.9 1 —0.022 : —0.407 43 06 01. 664 Elyria and Olmsted .-...........-.-. 28 27 47.300 643 | 25 7.5 1 —0. 022 | —0.130 28 27 47.148 Olmsted and Royalton 15 44 43. 903 644 24 7.0 1 —0.022 | +0.401 15 44 44, 282 Royalton and Grafton 22 05 02. 342 645 24 75 1 —0.022 | —0. 066 22 05 02. 254 Grafton and Townsend....-.--- .--. 160 16 23. 466 646 22 4.7 1 —0.023 | —0. 223 160 16 23. 220 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(64,)-+ (642)-+ (643)-|- (644)-+ (645) +0. 138=0 (64) +2(642)-+ (643)-++ (644)+- (645) +0. 183-0 (64,)+ (642) -+2(643) + (644)-+ (645) +0. 133=0 (641)+- (642)-+ (643) +-2(64a)+ (64s) 4-0. 188=0 (641)-+ (642)+ (643)-+ (644)-+2(64s) +0. 133=0 [Observer, G. Y. Wisner. BROWNHELM—65. Instrument, Troughton and Simms theodolite No. 1. Date, August, 1877.] Angle as measured between— Notation. ;| No.meas. | Range.| Wt. (v) (2) | Corrected angles. of aw a“ “ aw ° ‘ mw Elyria and Camden ....-+--.--5--+<+ 98 04 30. 919 651 16 i 5.2 1 +0.158 | —0. 254 98 04 80. 823 Camden and Townsend ........----. 50 46 04. 864 652 16 | 4.3 1 +0.158 | +0. 341 50 46 05. 363 Townsend and Sandusky ......---.- 47 23 45. 986 653 16 3.9 1 +0.158 | +0. 290 47 23 46. 434 Sandusky and East Base........-..- 8 27 39. 686 654 16 6.3 1 +0.158 | —0. 356 8 27 39. 488 East Base and West Base ....-. 6 35 53. 970 655 16 ' 10.5 1 +0.158 | —1. 206 6 35 52. 922 West Base and Kelley’s -.-- 17 53 03. 552 656 18 ( 7S9r td +0. 158 | +0. 925 17 53 04. 635 Kelley’s and Elyria ..- seee--- 130 48 59. 916 657 16 | 9.9 | 1 --0. 159 -++0. 260 130 49 00. 335 2(651)+ (652) (653)4+ (654)+ (655)-++ (65¢)—1. 107=0 (651) +2(652)+ (653)+ (654)+ (655)+ (656)—1.107=0 (651)+ (652) -+2(653)+ (654)-+ (655)+ (654)—1. 107=0 (651)-+ (652)-+ (653)-+-2(654)+- (65s)-++ (65c)—1. 107=0 (651)-+ (652)-+ (653)+4 (654)-+-2(655)-+ (656) —1. 107=0 (651)-+ (652)+ (653)-+ (654)-++ (65s) 4-2(65c)—1. 107—0 TOWNSEND—66. (Observer, J. H. Darling. Instrument, Tronghtou & Simms theodolite No. 4. NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. Date, August, 1877.] = | Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. oO t aw | “ a “ oO 2 “a Sandusky and West Base 19 40 28.184 661 20 5.0 1 —0. 333 | +0. 296 19 40 28. 147 Sandusky and East Base 21 18 58.049 66142 10 5.2 0.5 | +0.371 | —0.164 21 18 58. 256 West Base and East Base....---..-. 1 38 30, 902 662 20 6.6 1 —0. 333 | —0. 460 1 38 30.109 © | East Base and Kelley’s 7 27 32, 553 663 : 20- 4.9 4° —0.148 | +0.178 7 27 32, 583 Kelley's and Brownhelm 71 10 56, 529 664 ‘ 20 6.7 1 —0.148 | +0. 206 71 10 56. 587 Brownhelm and Camden......-..--. 38 53 53. 787 665 20 5.7 1 —0.148 | +0.077 38 53 53,716 Camden and Sandusky ....---.....- 221 08 39, 304 666 | 20 5.6 1 —0.149 | —0. 297 221 08 38, 858 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(661)+1. 5(662)+ (663)-+ (664)-+ (665)+-1. 777=0 1. 5(661) +2. 5(66,)+ (663)+ (66,)+ (665)+-1.777=0 (661) + (661) + (661)+ (662) -+2(663)+ (664)+ (66s) +1. 259=0 (66,)+ (663)4+2(664)+4+ (665)-+1. 259=0 (662)+ (663)-+ (664)-+2(666)+1. 259—0 SECTION X.— Triangulation from the line Willoughby — Chester to the line Stony Point - Cedar Point— 64.] CHICAGO BASE TO SANDUSKY BASE. 461 SECTION X.—Triangulation from the line Willoughby — Chester to the line Stony Point - Cedar Point— Continued. SANDUSKY—67. (Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No.4. Date, September, 1877. ] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v] Corrected angles. oo “ uw | “a a ofr uw Danbury and Kelley's .-............ 42 52 18, 222 671 20 61 ) 1 -+0.079 | —0.118 © 42 52 18.183 Kelley’s and West Base ............ 22 32 18. 600 67, 20 4.6 | 1 | +0.079} +0.260; 22 32 18.939 West Base and East Base .-........ 35 35 11.917 673 20 4.9; 1 +0.079 | 40.624 35 35 12. 620 ' East Base and Brownhelm.......... 23 27 01.315 | 674 20 4 2 7 | +0.079 | —0.381 | 23 27 01.013 ; Brownhelm and Townsend........-- 32 38 47.519 675 20 | 4.3 | 1 | +0.079 | —0. 050 | 32 38 47.548 Townsend and Danbury ........-.-- 202 54 21. 952 676 20 | 6.0 1 | --0.080 | —0.335 202 54 21. 697 t NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(671)+ (672)+ (673)+ (674)+ (67) —0. 475=0 (671) + 2(672)-+ (673)+ (674)-+ (675)—0. 475=0 (671) + (67»)-+ 2(673)-+ (674)-+ (676) —0. 475—0° (671) + (672)-+ (673)-+2(67a) + (67) —0. 475=0 (671) + (679)-+ (67s) + (674)-+2(67%6) —0. 4750 EAST BASE—62. [Observer, G. Y. Wisner. rs Instrument, Troughton & Simns theodolite No. Date, August, 1877.] Angle as measured between— Notation. DB No. meas. | Range.| Wt. (v) (v] Corrected angles. oO d ow |. aw “ oO t “ Elyria and Brownhelm .... ...-.--- 8 09 09. 213 681 i: 16 a4 | 1 +0.106 | +0. 465 8 U9 09, 784 Brownhelm and Townsend .....--- 45 30 06. 026 682 16 3.0 1 | 40.106] —0.033 45 30 06. 099 2 ‘Townsend and Sandusky .......--.- 102 35 14, 384 683 | 18 | 1 +0.107 | —0. 655 102 35 13.786 Sandusky and Danbury .........--. 47 05 16. 167 684 18 i ae +0.106 | —0.108 | 47 05 16. 165 Danbury and West Base.... ...--. 22 27 13. 678 68 | wo i 0 | 1 | 40.107! 40.574 22 27 14.359 Weat Base and Kelley’s -......--.-. 24 39 32. 309 686 18 5,2 A: || +0.106 | +0. 156 24 39 32.571 Kelley’s and Elyria .....-....-..---- 109 33 27. 528 687 16 5.5 |. 1 | +0.107 | —0.399 109 33 27, 236 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(681) + (682)-+ (683)+ (684)-+ (685)-++ (686)—0. 745=0 (683) +-2(68,)+ (683)+ (684)-+ (685)+ (686) —0.745=0 (681)-+ (682)+2(683)+ (684)+ (685)+ (686)—0. 745=0 (681) + (68,)+ (683)-+2(684)+ (685)+ (686)—0. 745=0 (68;)+ (682)+ (683)+ (684)-+2(685)+ (686) —0. 745=0 (681)-+ (68,)-++ (633)-+ (684)+ (68s) +2(686)—0. 745=0 WEST BASE—69. [Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1, Dates, August and September, 1877.] x Angle as measured between— Notation. No. meas. | Range.| Wt. | (vy) | [a Corrected angles. | 7 oF " “ | “ ” oF “ Elyria and Brownhelm ...-.-------- 9 07 40. 373 691 7 4.9 | 1 +0.075 | 41.113 9 07 41. 561 Brownhelm and East Base 31 01 58. 603 692 " 16 4.1 |1 +0.076 | —0. 924 31 01 57. 755 East Base and Townsend ....------- 6 13 46,424 693 16 3.6 | 1 +0.075 | —0. 865 6 13 45. 634 Townsend and Sandusky .-..------- 68 38 30. 724 694 16 6.3 ; 1 +0.076 | +0. 575 68 38 31.375 Sandusky and Danbury ..-..------- 72 37 42. 849 | 695 16 | 5.7 | 1 +0.075 | -+0. 848 © 72 37 43.772 Danbury and Middle Bass. -.......- 52 38 51.957 696 | 16 | 62 ]1 +0.076| 40.362! 52 38 52.395 Middle Bags and Kelley’s.-.---.---- 15 37 21.499 697 16 6.3 ,1 +0.075 | —1. 108 15 37 20. 466 Kelley's and Elyria ........-..--.-- 104 04 06. 983 695 | 21 , 93 | 1.25] +0.060 | —0.001 | 104 04 07. 042 462 SECTION X.—Triangulation from the line Willoughby — Chester to the line Stony Point - Cedar Point— PRIMARY TRIANGULATION. Continued. WEST BASE—69—Continued. NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 25(691) |-1. 25(692) 4-1. 25(693) -+1. 25(694) +1. 25 (695) 4-1. 25(69e) -+-1. 25(697) —0. 735=0 1. 25(691) 4-2. 25(692) +1. 25 (693) 4-1. 25 (694) 4-1. 25(695) +1. 25 (696) 4-1. 25(697) —0. 735=0 . 1, 25(691) 4-1. 25(692) +2. 25(693) +1. 25(694) +1. 25(695] 4-1. 25(696) -+ 1. 25(697) —0. 735=0 it be ee 25(691) -++-1. 25(692) +-1. 25 (693) +2. 25 (694) +-1. 25(695) +1. 25(696) +1. 25(697) —0. 735=0 25(691) +-1. 25 (692) -|- 1. 25(693) +1. 25(694) +2. 25(69,) +1. 25(696) +1. 25(697) —0. 735=0 25(69,) +1, 25(692) +-1. 25 (693) +1. 25 (694) +-1. 25 (695) 4-2. 25 (696) +-1. 25(697) —0. 735=0 25 (691) 4-1. 25(692) +-1. 25(693) +1. 25(694) +1. 25 (695) 4-1. 25(696) +2. 25(697) —0. 7350 [Cuar. XVII, C, DANBURY—70. [Observer, R.S. Woodward. Instrument, Tronghton & Simms theodolite No. 3. Date, September, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | {vl | Corrected angles. Oo t a “ a “a Oo t aw Locust Point and Middle Bass ...... 76 48 48.084 | 701 16 8.2 1 —0.016 | —0. 421 76 48 47. 647 Middle Bass and Kelley’s ..-.-....-.. 39 18 37.729 | 702 16 5.5 1 +0. 321 40.171 39 18 38. 221 Middle Bass and Sandusky ......--. 136 39 28.718 | 7024+3+4+5 6 7.4 0.25 | —1.356 | +0. 527 136 39 27. 889 Kelley's and West Base ............ 55 23 09.948 | 703 16 5.8 1 +0.321 | —0. 038 55 23 10. 231 West Base and East Base ....--..-. 10 02 44.696 | 704 16 3.7 1 -+0.171 | -+0.110 10 02 44. 977 West Base and Locust Point.... ... 188 29 23. 052 ! 7044546 6 5.4 0.25 | 4-0.561 | +0. 288 188 29 23.901 East Base and Sandusky..-...-..--. 31 54 54. 055 i 105 16 4.8 1 +0. 121 +0. 284 31 54 54. 460 East Base and Locust Point ........ 178 26 38.498 . 70546 6 5.6 0.25 | +0.248/ +0. 178 178 26 38. 924 Sandusky and Locust Point .-...... 146 31 44.775 | 706 16 | 6.3 1 —0. 205 | —0. 106 | 146 31 44. 464 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 50(701) +1. 50(702) +1. 50(703)+-1. 25(704) + (705) —0. 914=0 1. 50(701) 4-2. 75(70.) +1. 75(703) +1. 50(704) +-1. 25(706) —1. 487=0 1. 50(701) -+ 1. 75(702' 4-2. 75(703) +1. 50(704) + 1. 25(705) —1. 487=0 1. 25(701) +1. 50(702) +1. 50(703) +2. 50(704) +1. 25(705) —1. 368=0 (701) +1. 25(70.) +1. 25(703) +1. 25(704) +2. 25(705) —1. 285=0 KELLEY’S—71. [Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Date, September, 1877.] Angle as measured between— Notation. | No. meas. | Range.) Wt. (v) [v] Corrected angles. oO ‘ uw ht a“ WwW or : 4 Brownhelm and Townsend. .-..----- 28 28 40. 951 Tl 18 4.4 1 +0.190 | +0. 843 28 28 41. 984 Townsend and East Base .......--. 9 19 44. 669 712 20 - 3.3 1 +0.190 | —0. 236 | 919 44. 623 East Base and West Base...... ----- 1] 06 40. 660 713 20 | 4.2° 1 +0.189 | +0. 347 | 11 06 41.196 West Base and Sandusky.......-.-. 16 33 44. 363 Ta 16 | 40 1 +0.190 | +0.091 16 33 44. 644 Sandusky and Danbury..........--. 39 46 53. 135 Ts 18 6.8 1 +0.189 | —0. 627 39 46 52.697 | Danbury and Middle Bass .......... 91 27 49, 334 Tle 16 4.8 1 +0.190 | —0. 695 91 27 48.829, | Middle Bass and Brownhelm......-. 163 16 25. 561 Tr 18 5.9 1 +0.189 | +0.277 | 163 16 26.027 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(7la)+ (Tl2)+ (71s)+ (71a)+ (71s)+ (716)—1. 327=0 7 (711) +2(71_)+ (T1a)+ (Tla)+ (715)-+ (716)—1. 327=0 (71,)+ (T1_)+2(71g)+ (71a)-+ (71s)+ (71s) —1. 327=0 (71,)-+ (T1g)+ (T1a)+2(71a)-+ (71s)+ (T16)—1. 327=0 (Tn) + (712)+ (T13)-+ (714) +2(715)+ (71s) —2. 327=0 (Thi) + (7s)+ (71a) (Tq) + (716) +2(716)—1. 327=0 $4.) CHICAGO BASE TO SANDUSKY BASE. 463 SECTION X.—Triangulation from the line Willoughby - Chester to the line Stony Point — Cedar Point— Continued. MIDDLE BASS—72. . [Observer, R.S, Woodward. Instrument, Troughton & Simms theodolite No. 3. Dates, September and October, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. Co % “" | “ “" “ o 4 “" Kelley's and West Base....-........ 16 34 14.051 | 721 16 | 3.4 | 1 —0.018 | —0. 437 16 34 13. 596 Kelley's and Danbury ...... ... .... 49 13 32. 816 | 72142 4 | 4.7 | 0.25); +0.641; —0.016 49 13 33. 441 Kel'ey's and Locust Point .......... 108 09 49.039 | 721+2+3 10 3.2 | 0.5 —0, 048 | —0. 182 108 09 48. 809 Pelée and Danbury ..... -.----..-- 158 57 48. 3%5 | 721+2+6 8 3.8 | 0.5 +0. 868 | —0 033 158 57 49. 220 West Base and Danbury.....-.-..-. 32 39 19.442 | 722 15 | 6.9 11 —0.018 | -+0.421 32 39 19. 845 Danbury and Locust Point ......... 58 56 14.751 | 723 9 6.0 10.5 +0. 783 , —0. 166 58 56 15, 368 Danbury and Middle Sister ......... 121 01 44.126 | 72344 4 1.4 | 0.25} +0.736 | —0, 223 121 01 44. 639 Locust Point and Middle Sister..... 62 05 30.426 724 9 81 | 0.5 —J.098 | —-0. 057 62 05 29. 271 Locust Point and Pelée ...........-. 142 05 58.819 | 72446 8 4.1 | 0.5 +1.364 | +0.199 142 05 55, 412 Pelée and Locust Point ............. 217 54 05.668 | 724-5 ; 6 3.5 | 0.25 | —0.881 | —0. 199 217 54 04. 588 Middle Sister and Pelée ............ 80 00 26.058 735 a 16 46 1% —0.173 | +0. 256 80 00 26.141 Middle Sister and Kelley’s......... 189 44 41.877 | 725+6 8 6.0 | 0.5 —0.196 | +0, 239 189 44 41, 920 Kelley's and Middle Sister.......... 170 15 18.129 | 72-s—6 Boy 3.7 | 0.5 +0.190 | —0. 239 170 15 18. 080 Pelée and Kelley’s...............--. 109 44 15. 488 | 726 ; 16 29 |}1 +0. 308 | —0.017 109 44 15.779 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 75(72)) +2. 75 (722) + 2. 50(723) +2. 00(724) + (725) +0. 5382=0 2. 75(72)) + 3. 75(722) 4-2. 50(723) +2. 00(724) + (725) +0. 532=0 2. 50(721) + 2. 50(722) +3. 75(723) 4-2. 75(724) + 1. 50(7 25) +0. 435=0 2. 00(72,) +2. 00(722)+-2. 75(723) +4. 00(724) +2. 25(725) +2. 703=0 (72) 4- (722) +1. 50 (723) +2. 25(724) +3. 25(725) +1. 896=0 PELEE—73. (Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No.4. Date, October, 1877. ] T I | Angle as measured between— Notation. No. meas. | Range. | Wt. (vy) | [ev] Corrected angles. | | oj. “ : | | u" | “" | “ Oj uh " Middle Bass and Middle Sister...... 64 46 53. 046 | 731 24 7.7 1.25 | +0.051 +0. 252 64 46 53. 348 Middle Sister and Kingsville........ 70 48 10. 143 732 ! 16 4.5 | 1 +0. 064 | —0.126 70 48 10. 08L Kingsville and Middle Bass.... .... 224 24 56. 632 | 733 | 16 3.5 | 1 +0.065 | —0.126 224 24 56.571 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 25(731)-+ (732)—0. 180=0 (731) + 2(73a) —0. 180=0 LOCUST POINT—74. (Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, October, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. | ° i “a aw “ue aw ° ‘ ao Cedar Point and Middle Sister....-. 79 21 34.373 Th 16 7.4 1 +0.207 | +1. 001 79 21 35. 581 Middle Sister and Middle Bass....-- 52 38 36,770 742 16 5,9 1 +0.207 | —0. 282 52 38 36. 695 Middle Bass and Danbury .-...--.-- 44 14 58.577 743 16 5.1 1 +0.207 | —0.576 44 14 58. 208 Danbury and Cedar Point ....-..--. 183 44 49. 452 744 16 5.7 1 +0.207 | —0. 143 183 44 49.516 { NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(74;)+ (742)-+ (743) —0. 828=0 (741) +2(742)+ (743)—0. 828=0 (74,)+ (742) +2(742)—0. 828=0 464 PRIMARY TRIANGULATION. [Cnap. XVII, 6, SECTION X.— Triangulation from the line Willoughby — Chester to the line Stony Point - Cedar Point— Continued. MIDDLE SISTER—7%. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, October, 1877.] Angle as neasured between— | Notation. | No. meas. | Range. | Wt. (v) | (v] Corrected angles. ee —— _ +0. 11840 VI +40. 81579 XLV +40, 25000 X +40. 25000 X +0. 25000 X —0, 28571 XI —0. 10643 XX +40, 42857 XII —0, 21286 XX +0. 71429 XIII —0, 10643 XX -L0. 71429 XILT —0. 10643 XX — 0.20571 XIII —0. 54251 XX .28571 XLLL . 07463 XX . 28571 XLII , 10643 XX . 01947 XIL . 63333 XVIT . 01947 XII . 16667 XVII . 01947 XII . 16667 XVLE . 33630 XII . 16667 XVI 07788 XII . 33333 XVIL . 25842 XII . 16867 XVII 42357 XVUL 42857 XXVI .71429 XVIII .28571 XXVI . 42857 XVIII . 42857 XXVI 42357 XVILL .57143 XXVI . 42857 X VIL 57143 XXVI +0. 57143 XVIII .57143 XXVI +0.57143 X VEIL . 42857 XXVI —0, 02939 XX —0. 05317 XXVIII t | o —0, 05878 XX —0, 20000 XXXI —0. —0, +40, +0, -0. +0, =<: +1. —0. +0. —0, +10. 18421 VII 11921 XV 50000 XI 50000 XI 50000 XI 57143 X1V 24571 XXI 14286 XIV 57143 XXI 57113 XIV 286571 XXL 57143 XIV —0, 28571 XXI —0. 42857 XIV 28571 XXI —0. 42857 XIV . 71429 XXI 42857 XLV 71429 XXI . 33333 XIII . 06080 XX . 33333 NUL . 31215 XX . 66667 XLII . 06284 XX . 66667 XIII . 06284 XX +0, 33333 XLII . 12563 XX . 33333 XIII . 06234 XX 23571 XIX 29571 XXVIII 85714 XIX . 14283 XXVI —0, 28571 XIX —0, 28571 XX VII +0. 71429 XIX —0. 28571 XX VIL +0, 71429 XIX +0, 71429 XXVIL —0, 28571 XIX 40. 71429 XX VII —0, 28571 XIX —0. 28571 XXVIL +0. 20000 XXIII —0, 10000 XXXI 2 = +0. 40000 XXIII -++0. 60000 XXXIT 467 -L0, 03205 1X +1. 21352 XV +2, 42704 XV LL. 21352 XV —1.02299 XV —0, 35101 XV —0, 35101 XV —0. 35101 XV — 0. 16667 XIV —0. 16657 XIV -40, 83333 XIV —0. 16657 XIV —0. 33333 XLV —0. 16667 XIV 0.23571 XXI —0.57143 XXXI —1. 14286 XXI —0. 28571 XXXI —0. 28571 XXI —0, 57143 XXXI —0. 23571 XXI +40, 42057 XXXI —0. 25571 XXI +40, 422857 XXXI —0.28571 XXI -40. 42857 XXXI +0. 71429 XXI +0. 42857 XXXI +0, 40000 XXIV 40, 30000 XXXII +0, 80000 XXIV. -+0, 00262 XXXIII 468 [6614243] =—0. +0. +10. =—0. —0. 40. = [662] ° [66:44] =—0 +0. +40. [6624544] =—0. Lt, —0. sa), [665] +0. +0. =—0. +0. +0. =—0. [66345] (65, [535] [671] Boa) [672] [67245] [67.4040] = [675] [67244] [675] [675] [681] General corrections in terms of the correlates—Continued. -40 85714 NVIIL = —0. 14286 XIX —2. 42054 XXV —0,28571 XXIX = +0. 14286 AXAVIL -L0, 43753 XXX VIII PRIMARY TRIANGULATION. 40000 XVIE 0.20000 XIX —0. 1672 XXV +0. 50000 XNVI 00524 NNXULE —0. 32709 XXX1V 10000 XVII. —0.20000 XIX —0 00355 XXV -.0,50000 XXVI +0 00131 XXXIM —0.17642 XXXIV 30000 XVII. 4.0.40000 NIX = —0 13033XXV 4.1, 00000 XXVI-+E0 00393 XXXII —0, 40306 XXXIV 50000 XVIL_ 1.00000 XIX. —0 50000 XX VI +0. 45524 XXVITL +0 12745 XXXIV 20000 XVIE -+.0.60000 XIX = —0 13388 XXV —-4.0.50000 XXVI_ +0 00262 XXXII —0, 22664 XXXIV 40000 XVIT_ 1.20000 XIX = —0 26776 XXV_-4.1.00000 XXVI +0 43574 XXXIII_ +0. 04897 XXXIV 20000 X VIT +0. 60000 XIX — 1, 13388 XXV +0. 50000 XXVI +0 43312 XXXIIT +0. 27561 XXXIV . 80000 XVII —, 40000 XIX +0. 13085 ARV —, 50000 XX VI —0, 21917 XXXIIL +0. 02574 XXXIV (aga? NNT OI XXIV | +40. 16667 XXIX -40,11637 XXX —0 50000 XXXVIT +1, 62853 XNXVII—0. 165167 NXITL = 0.33833 NNIV +40. 16867 XXNIX 40.1637 XXX LO. 50000 XXXVIL —0. 87561 XXXVIII-L0. 33333 XXL = 0.66667 XNIV. +40. 66667 XXIX 0.61172 XXX -L0. . 00000 XXXVIL —1. 75122 XXXVIIL-L0, 50000 NXIIT = 1.35671 XXV 1 .34909 NNN -41.00000 XNXIE —1 . 71033 XXXVIIL-+0. 50000 XNXIX —-1 .16667 XXII = —0.33833 XXIV +0. , 83333 XXIX -+0.49535 XXX +0. . 90000 XXX VII —0. 87561 XXXVITI—0 . 33333 XXITILT = +0. 33333 XXIV -—1. . 66667 XXIX =—0, 46546 XXX 4 23472 XXX VITI—0. 33333 XXXIX —0. ). 16667 XXIILT -++-0.66657 XXIV —1. . 16667 XXIX = —0, 96081 XXX 4-0. . 10000 XXXNVIL +0, 04089 XXX VIII—0. . 83333 XXIII 16667 XXIX LO 50000 NXXVIT -L0 +0. +0. —0, . 66667 AXTV . 11637 XXX 41. +0. . 04089 NNNVIN—0, 18S07 XX -L0, 23800 XX VIIL 020389 XX . 12710 XXVIII . 15868 XX 0, 29117 XXVIUI . 28797 XX - 50000 XXXT 12929 XX 0.16407 XXVIII . 20858 XX 82814 XXVOL . 12029 XX . 16107 XXVUOTI 37614 XX 20103 XXVIII 08708 XXV . 66667 XXXII 16667 XXXIX 08708 XXV 33333 XXXIL 83333 XXXIX 17416 XXV 66667 XXXII 66667 XXXTX . 00000 XXVII 26657 XXXII . 22359 XLII 08708 XXV 33333 XXXIT . 16607 XXXIX 44379 XXV . 66667 XXXII 66667 XL 53087 XXV 33333 XXXIT 16667 XXXIX 18255 XXV 33333 XXXIT 16667 XXXIX 14889 XX 75800 XXX +40. £0000 XNHIT 0, 40000 XXXI +0. 20000 XXIII —0. 10000 XXXI +40. 60000 XXIII —0, 30000 XXXI +1, 00000 XXTIE +40, 50000 XXXII +0. 40000 XXIII —0. 20000 XXXI +0. 80000 XXIII +0. 60000 XXXI +0, 40000 XXTIT +0. 80000 XNXI . 60000 XXIIT 20000 XXNXI 33333 XXVIL . 23509 XXXII . 66667 XL . 33333 XAXVIT . 42219 XXAIIT . 66667 XL . 33333 XXVIL . 34438 XXXII . 33333 XL . 45324 XX VIL - 00000 XXXVI = +0, 66667 XXVIL —0. 42219 XXXII —0. 33333 XL +1. 33333 XXVII —0, 84438 XXNUI +0, 31172 XLU +0, 66667 XXVII —0, 42219 XNXIII —0. 33333 XL —0, 33333 XXVII 41. 73676 XXXII —0. 33333 XL. +2. 10249 XXII -+0. 46240 XXXV [Cnap. XVII, C, +0, 60000 XXIV +1. 20000 XXXIT +0. 40000 XXTV +0, 30000 XXXIT +0. 20000 XXIV +40. 90000 XXXII +0. 26421 XXV +0, 43705 XXXIIT —0, 20000 XXIV +0. 60000 XXXII —0. 40000 XXIV +0, 20000 XXXIT —0, 20000 XXIV —-0, 40000 XXXII 9, 20000 XXIV —, 40000 XXXII —0, 06332 XXVIIL —0. 33333 XXXVI +40, 91184 XLII —0, 06332 XXVIII +0. 66667 XXXVI —1.53531 XLII —0, 75828 XXVIII +41. 33833 XXXVI .37945 XLII 50000 XXIX +40. 50000 XXXVII —0. 69496 XXVIII +0. 66667 XXXVI +0. 15586 XLIT —1. 38992 XXVIII +40, 33333 XXXVI -- 0, 69496 XX VUI —0, 33333 XXXVI +0, 15586 XLII +1. 57989 XXVUI —0. 33333 XXXVI +0, 15586 XLII —0. 14266 XXIV —0. 42857 XXXVI §4.] [6e2] [68243] [682434445] [633] [625] [68.45] [68] [546] [625] (591] [693] [69245] [693] [69544] CHICAGO BASE TO SANDUSKY BASE. General corrections in terms of the correlates—Continued. =—0. 14286 XVII —1.39619 XXV —0. 14286 XXXVI =—0, 28571 XVIII —2. 10272 XXV —0, 28571 XXXVI =—0.57143 XVII —1.26164 XXV +0, 49857 XXXVII =—0. 14286 XVIII —0. 70653 XXV —0, 14286 NNNVII =—0. 14286 XVIII +40, 42054 XXV +0, 85714 XXXVI =+0. 28571 XVIII -++0. 84108 XXV +0. 71429 XXXVII =—0. Mes6 XVII +0. 42054 XXV —0. 14226 XXXVIL =—0, 28671 XVII +40, B4108 XXV —0. 28571 XXXVI =—0, 14285 XVII +40. 42054 XXV —0. 14286 XXXVI =+40.87179 XXI —0, 25641 XXIX —0, 12821 XL =—0, 12821 XXI —0. 25641 XXIX —0. 12 XL =--0. 25642 XXI 40, 48718 XXIX —0, 25642 XL =—0. x21 XXI +40, 74359 XNIX —0, 12821 XL =—0,25642 XXI +1. 48718 XXIX —0, 25642 XL [6944454047] =—0. 64105 XXI [694] [69s454647] +0, 71795 XXIX +0), 35895 XM =—0.12821 XXI +0. 74359 XXIX —0, 12821 XL =—1, 5123-4 XXI —0, 02564 XXIX +0, 48716 XL +40. 85714 XIX —(), 28571 XXIX +0. 64551 XX —1. 07568 XXX +0. 43753 XXXVUT +0. 71429 XIX —0. 57143 XXIX 87506 XX XVIII 40, +0, +0. +0. —0. 42857 XIX l4eo6 XTX —0. 28571 XXIX 43753 XXXVI +0. —0, 14286 XIX +0. 71429 XXIX 32927 XXXVIIT +0. =0i 41. —0. _Ltz86 NIX +0. 28571 XIX —\) 85714 XXLX 16462 XXXVIIL 42857 XN XIX 70H XNXVITT 71429 NXIX +0. 94619 XX —2, 15136 XXX +1. 54755 XX —2. 27308 XXN -40. 30068 XX —1. 07568 XXX +0. 30068 XX —0. 06131 XXX +0. 60136 XX --0, 12262 XXX -b0. COOGR XX --0. 06131 XXX —1. 08971 NXAXVII —0, 28671 XIX 40, we . 14286 XIX 91381 XXII —0. 203383 NXX .25641 XLI 0.70706 XXIE . 02604 XXX -—0. 25641 XLI 91512 XXII . 00220 XXX _ 51282 XLI +0. 20806 XXII 02324 XXX 25041 XLI .41312 XXII 04763 XXX 51232 XLI . 04030 XXII 0, 65917 XXX +9. 71795 XLI +40, 20803 XXII —0, 02384 XXX —0, 25641 XLI +0. 83224 XXII —0, 63533 XXX +0. 97436 XLI 42267 XXIX 07912 NXXVIET 2287) XXX . 03971 XXXVIIL +0, 60136 XX 0, 635) XXX --0, 300:8 XX 0, 75300 XXN --0. 25641 XXVI —0. 01091 XXXIV +0. 40387 XLIL +0. 74359 XXVI +1. 20484 XXXIV -40, 40387 XLI 4-1. 48718 XXVI +2. 58968 XXXIV +0. £0774 XLU +0. 74359 XXVI +1. 20484 XXXIV +40, 40387 XLIL +0, 48718 XXVI LU, 65233 XXXIV +0, 80774 XLIL —0. 28205 XXVI —1, 27520 XXXIV —1, 13083 XLIL —0. 25641 XXVI —0. 64251 XXXIV 0, 40387 XLII —1. 02564 XXVI —2.57004 XXXIV —1.53470 XLII —0. —0, 538699 XXIL 51203 XXXV as . 07393 XXIL - 02536 XXXV —1 ~-2, 14796 XXII —2, 05072 XXXV . 53699 XXIT . 51268 XXXV — 0.53699 XXIT 2 91268 AXAV . 07398 X NIT . 02936 XXAV . 93699 XXIT .o1268 XXXV .Ob4I ANIT » C1326 XXXV . 12725 XXII --1. 12504 XXAV 8462 XXVIL 14724 XXXV . 12821 XLT +0. 61538 XXVII 70564 XXXV . 12821 XLIIL 23076 XXVII .81151 XXXV —0, 25642 XLIIL . 61538 XXVIL 39413 XXXV . R21 X LIT 23076 XX VIL . 78826 XXXV 25642 XLT . 07690 XXVII 97065 XXXV .35805 XLII 61533 XXVIL .39413 XXXV . 12821 XLII .53348 XXVIL —1.57652 XXXV +0, 48716 XLII | Ss t 469 —0, 14286 XXIV —0. 42857 XXXVI 71429 XXIV 85714 XXXVI +0. —0. +0. 42857 XXIV +0, 28571 AXXVI . 85714 XXTV . 42857 XXXVI . 14286 XXIV .57143 XXXVI . 28571 XXIV . 14286 XXAVI - 1426 XXIV - 97143 XXXVI 2 2Ro7L A XIV - 142386 XXXVI . 14286 XXIV O71} XXXVI 47020 XXNVIIT _ 38462 XXXIX . 18983 XLV 21360 XXVIII 38462 XNXIX . 18983 XLV . 42720 XXVIII . 76924 XXXIX . 37966 XLV .21360 XXVIII 38462 XNXIX . 18983 XLV 04338 XXVIII —0. 76924 XXXIX . 37963 XLV . 35722 XXVIII . 07690 XXXIX 53151 XLV . 17022 NXVIIL 38462 NNNIX —. 13933 XLV +1. 58032 XXVIU +1. 46152 XX XIX -40. 72134 XLV = = General corrections in terms of the correlates —Continued. 470 [695] =—-0, 12821 XXI —0. 25641 XXIX +0. 87179 XL (69,,] =—0. 12421 XXI —0, 25641 XNIX —0, 12821 XL [69.47] =—0. 25612 XXT —0. 51282 XXIX —0, 25642 XL [695] =—0. 12821 XXI —0. 25641 XXIX —0, 12821 XL (70.] =—0. 04651 XX XVII —0, 36544 XLIII [703] =—0, 11628 XXXVII +0, 51496 X LIL [705] =—0. 11628 XXXVII +0, 51195 XLIIL [70544] =—0, 30233 XXNVIL +40, 19602 XLII [705] =—0. 18605 XXXVII —0. 31894 XLII [70.45] =+0.51162 XXXVII —0, 55150 X LUI [705] ==, 69767 XXXVII —0, 23256 NLIIL (71) 9 =+0.385714 XXXI —0. 28571 XXXVI [71,42] =+0.71429 XXXI —0. 57143 XXXVI (71.] =—0, 14286 XXXI —0, 23571 XXXVI (7lo43] =—0. 28571 XXXI +0, 42857 XXXVI L7lo4s44] =—0, 42857 XXXI +1. 14246 XXXVI [715] = —0. 14286 XXXI +0, 71429 XXXVI [714] =—0, 14286 XXXI -+0.71429 XXXVI (7ligs] =—0. 28571 XXXI +0. 42857 XXXVI [715] =—0, 14286 XXXI —0, 23571 XXXVI [714] =—0. 14283 XXXI —0. 23571 XXXVI [721] =—0, 34022 XLII [722] =-+0, 65978 XLII [725] =—0. 19296 XLIII [724] =—0. 03588 XLIII [72s] =-+0. 01557 XLIII [73;] =-++0.57143 XLVII PRIMARY TRIANGULATION. -40, 20806 XXII — 0, 20383 XXX —0, 25641 X LI +40. 20306 XXII —0, 25641 XXVI --0, 64251 XXXIV . 45971 XLIL » 25641 XXVI —0, 20383 XXX —0. 64251 XXXIV +0. 74359 XLI —0. 73943 XLII +0. 41612 XXII —0. 51282 XXVI —0. 40766 XXX —1, 28502 XXXIV +1. 42718 XLI —1. 47886 XLIT +0, 20306 XXII —0, 25641 XXVI —0, 20383 XXX —0. 64251 XXXIV +0. 74359 XLT —0, 73943 X LIL —0, 25469 XNXNVIU—0O. 14839 XL 0. 18272 XLIV +0. 13627 XLV —0. 36935 XXX VII—0. 27575 XL +0. 75748 XLIV =—0.50923 XLV +0, 49448 XXXVITI-—-0. 27575 XL » 24252 NULLV +0, —0, 40199 XLIV +40, 46209 XXX VIIL-L0. —0, 15947 XLIV +0, 33251 XXXVITI-+1. —0,. 27675 XALIV —0, 12858 XXXVI -pu. —0, 11628 XLIV —0, 42357 XXXII —0. 14286 XXXTX +0. 14286 XXXII ‘0, 2857L XXXIX +40, 57143 XXXII —0, 14286 XXXIX +1. 14286 XN XII —0. 28571 XXXIX +1.71429 XXXII +0.57143 XXXIX +10. 57143 XXXII —0, 14286 XXXIX +L0.57143 XXXII +40. 85714 XXXIX +0. 14286 XXXII +0.71429 XXXIX —0, 42857 XXXII —0, 14286 XXXIX —0, 42857 XXXII —0. 14286 XXXIX 0.31956 XLIV +0.31956 XLIV —0, 38592 XLIV —0.07176 XLIV +0. 03114 XLIV 95657 XXXVITI--0. “40. 13561 XLV 27353 XL 24996 XLV 5AV28 AL 11632 XLV ai090 XL 20113 XLV 51162 XL +0, 08481 XLV . 74920 XXXII 28571 XLI .16955 XXXII 57143 XLI 57905 XXXILL 28571 XLI . 15930 XXXIII (57143 XLI . 73895 NXXIIL / 14286 XLI 57965 XXNNUI .28571 XLI 57965 NXXUI . 71429 XLI . 58307 XXXII . 42857 XLI . 00342 XXXII .71429 XLI +0, 00342 XNXUI —0. 28571 XLI —0, 19296 XLVI —0. 19296 XLVI +0. 77116 XLVI —0. 33378 XLVI —0, 00609 XLVI +40. +0, 40. — . 38462 XX VII 39413 XXXV . 12821 XLII . 38462 XX VII 39413 XXXV 87179 XLII . 76924 XXVIL —0. 78826 XXXV 0.74358 XLULL —0. 38462 XX VIL 39413 XXXV —-0, 12821 XLII —0. 18272 XLI -0. 68831 XLVI -—0, 24252 XLI —0. 18272 XLVI +0. 75748 XLI —0. 18272 XLVI 0.59801 XLI —0. 28160 XLVI —0, 15947 XLI —0, 10188 XLVI —0.27575 XLI —0. 14839 XLVI —0. 11628 XLI —0, 04651 XLVI —1,59463 XXXIV . 14286 XLIV —0, 47227 XXXIV —0. 28571 XLIV +1. 12236 XXXIV —0. 14286 XLIV +42. 24472 XXXIV —0, 28571 XLIV +2. 08219 XXXIV —0. 42857 XLIV +1, 12236 XXXIV —0. 14286 XLIV —0. 16253 XXXIV —0. 14286 XLIV —0. 32506 XXXIV —0. 28571 XLIV —0, 16253 XXXIV —0, 14286 XLIV —0, 16253 XXXIV -40. 85714 XLIV +40. 01557 XLVII +0. 01557 XLVII —0. 00609 XLVII —0, 29587 XLVII +0, 50575 XLVIL = > (Cuap. XVIL C, +0. 47020 XXVIII +40. 61538 XXXIX —-0. 18983 XLV +40. 47020 XXVUI -40, 61538 XX XIX —0. 59158 XLV -++0, 94040 XXVIII +1. 23076 XXXIX +1. 10100 XLV +0, 47020 XXVIII -40. 61538 XXXIX 4-1. 69258 XLV —0. 11818 XLIL --0. 16438 XLU -L0. 41053 XLII +40. 36355 XLIL —0, 04198 XLII —0. 06254 XLII —0. 020.6 XLII — (53368 XXXV —0. 35642 XLV —1. (6736 XXXV —0.71284 XLV —0. 53368 XXXV —0, 35642 XLV 0.56087 XXXV —0.71284 XLV +0, 55408 XXXVI —0. 23314 XLV +1. 09455 XXXV —0, 35642 XLV —0, 00679 XXXV +40, 47970 XLV —0. 01358 XXXV +40, 95940 XLV —0. 00679 XXXV +0. 47970 XLV —0, 00679 XXXV -40, 46625 XLV —0. 03588 XLVHI —0, 03588 XLVIIL —0. 33378 XLVIII -40. 68179 XLVI —0, 29587 XLVIII $4.1 No. of equation. 1. 2. ~ 10. I, CHICAGO BASE TO SANDUSKY BASE. General corrections in terms of the correlates—Continued. [722] =—0. 28571 XLVII [741] =--0. 25000 XLVI [742] =—0. 25000 XLVI [743] =+0. 75000 XLVI [7%] =-+40. 78571 XLVI [753] =—0. 21429 XLVII [75,4] =—0. 21429 XLVIT [75,] =--0. 21499 XLVII [76,] =—0. 33333 XLIX [762] =-+0. 66667 XLIX (77,] =-+0. 44444 L --0. 25000 XLVIII +0, 75000 XLVIII —0. 25000 XLVIII —0. 21429 XLVIII 0.78571 XLVI —0. 21429 XLVUI —0, 21429 XLVIII +0. 66667 L —0, 33333 L +0. 75000 XLIX —0, 25000 XLIX —0. 25000 XLIX —0. 21429 XLIX —0. 21429 XLIX +0. 78571 XLIX --0. 21429 XLIX Normal equations for determining the correlates. 0=—0. 09100 +2. 08903 I 0——0. 58500 —0. 27273 I —0. 25000 V 0=—0. 53300 —0, 18182 I —0. 41667 V +40, 25521 IX 0——0. 75700 —0. 09091 I +1, 19298 V 9=—, 07785 0=+0. 41300 0=-+1. 10400 * 0=-+40. 58760 0=+0. 28300 0=-+0, 21200 -0, 42655 IX —0, 33333 XIII —0. 25000 TI —0. 74619 VI —0. 16667 X —0. 26316 XIV +40, 49754 IT 4-4, 42082 VI — 0.14093 X +40, 11840 XIV 0, 50000 II +3. 06579 VII —0. 68421 XI —0, 23879 XV —0. 33333 IIT +1. 50000 VII -+40.70741 XII —0. 14286 XVI +0, 25521 III —0. 54456 VII 40. 03205 XI +2. 17438 XV —0. 50216 XXI —0. 16667 IIL +0.75000 VII +1. 16667 XI —1. 07632 XV —0. 13158 IV -+0. 03205 1X +0. 33333 XIII —0, 33333 XVII —0, 27273 II +2. 08903 II 40, 49754 VI —0. 43182 IL —0, 15878 VI —0! 16667 X —0. 09091 IT +1,54016 VI —0, 33333 X —0, 13158 XIV —0, 41667 III —0. 76316 VIL —0, 26316 XI —0.34114 XV —0, 15878 III —0, 30439 VII +40. 11840 XI +0. 15349 XV —1.13158 IV -+1. 50000 VIII +0. 09948 XII —0. 66667 IV +3. 19048 VIII +1. 38095 XIII —0, 14286 XVIII +40. 42655 IV +0. 37227 VIII +1. 72483 XII —0. 25108 XVI —0. 33333 IV +1, 16667 VIII -L0. 31670 XII —0. 16667 XVI —0. 26316 V +1. 16667 X —0, 51754 XIV —0, 12568 XX —0. 18182 TIT —0. 43182 III +2. 12879 IIT —0. 50000 VII +40. 05344 XII +40, 93939 III —1, 13158 VII —0. 13158 XI --0.17057 XV +1. 19298 1V —0. 33333 VIII +0. 18278 XII -+1. 54016 IV —0, 28186 VIII —0. 01301 XII —0. 76316 V —0. 54456 IX +0. 50000 XIII —0. 33333 V +0, 37227 IX 0.57143 XIV —0, 10643 XX +0, 08747 V +6. 55858 1X —0. 28953 XIII —0, 25108 XVII —0. 16667 V +0. 41072 IX —0, 50000 XIIT —0. 16667 XVII +0. 11840 VI +3, 14912 XI —1. 21471 XV —0, 21429 L —0. 21429 L 0. 21429 L +0. 78571 L —0. 09091 IV —0. 09091 IV +0, 93939 IV —0. 33333 VIII —0, 16667 XIII +2, 73286 IV —0. 66667 VIIT +40. 17155 XII +2. 63597 V +0. 08747 IX ~-0. 16667 XIII —0.74619 V +0, 29127 IX —0. 14093 XT —0, 30439 VI +0. 75000 X —0. 18421 XIV —0, 28186 VI +1, 16667 X +1. 21352 XV —0. 28571 XXI +0, 29127 VI +0, 41072 X +1. 03638 XIV —0. 18706 XX —0. 14093 VI +2. 41667 X —0. 16667 XIV —0. 06284 XX —0. 68421 VII —0. 49231 XII —0. 33333 XVI 471 472 No. of equation. 12% 13. 14. 16. 17. 19. 20. PRIMARY TRIANGULATION, [Cuar. XVI, C, Normal equations for determining the correlates—Continued. 0=—1. 16045 +40. 0=—0. 05300 0=+0. 42200 0=42. 58216 0=+0, 21700 (0=—0, 84200 0=—0. 54600 0=—0. 28600 0=-+1. 36900 sy +0 +0 +0 05344 TIT . 09948 VIT A9231 XT . 7855 XV . 16667 TT . 50000 VIT . 33333 XT . 20712 XV 28884 XX 13158 TV 257143 VIL . 07106 XII . 59824 XVI . 85714 XXT 17057 IV 21352 VITT . 78586 XIT . 00224 XVI . 70202 XXI . 14286 VIII . 01947 XII 2.54762 XVI 2 23036 XX 1. 42857 XXVI . 16667 X . 16667 XIV . 42857 XVITT . 19447 XXIT » 92857 XXVI - 40000 XXXII » 14286 VIL . 89101 XV - 00000 XTX .42807 XXTIT . 14286 XXVIT - 46240 XXXV . 28571 XVI » BR693 XOX - 54286 XXTV . 82214 NXVL - 20000 XXXIT . 42857 XXNVT . 10643 VOI . 00734 XIT - 23036 XVI . 29098 XX . 24190 XXIV . 60136 XXIX . 12604 XXXII . 80068 XXXVIL oS 40. . 17155 IV T0T41 VIAL . 72696 XIT OL AVI . 33333 TV 238095 VITT - 44912 XI . 61906 XVI 257143 XXT . 26316 V . 03638 TX . 80952 XTIT . 16667 XVII . 34114 V . 17438 1X . 20712 XT 36325 XVIT . 25108 1X . 61906 XTIT - 30953 XVIT .97143 XXT 22RO7L AXVIT - 33333 XT . 35825 XV ORBTL XIX . 74286 XXIII 2 2k571 XXVIT . 21917 XXXII . 25108 IX - 57143 XVI - 07426 XX - 14286 XXIV . 285671 XXIX . 42867 XXXVI . 68571 XVIT ~ 57143 XXT . 12843 XXV . 28571 XXTX . 43574 XXXL . 14286 XXXVIT . 18706 TX 33854 XTIT . 31534 XVI . 96820 XXT 298715 XXV » 22770 XXX - 00775 XXXIV . G20R9 XXXVI 40. +1. +0. L L 42 +0. —0 —0 —!1 12278 V 72483 IX 44912 XIII 01947 XVI . 16667 V 28953 IX 59525 XIII 33333 XVII . 11840 VI . 16667 X . 36341 XPV . 42857 XVIIT . 15349 VI - 07632 X 2.37775 XTV . 85101 XVITIT . 16667 X . 69524 XTV .97143 XVOT . 19447 XXII - 57143 XXXT . 01947 XIT . 80953 XVI . 315384 XX . 20000 XXIV . 20103 XXVIII . 02574 XXXIV 228571 NIIT 42857 XVIT 280714 XAT 42054 XXV . 75800 XXX . 14286 XXXVIT - 00000 XVIIT - 92593 XXII . 14286 XXVI . 07568 XXX . 04897 XXXIV 43753 XXXVI . 06284 X . 88213 XTV . 07426 XVII . 07024 XXIT . 28797 XXVI . 12929 XXXI - 30947 XXXV —0, 01301 VI +0, 31670 X +40. 07106 XIV —0, 00734 XX —0. 14093 VI —0. 50000 X +41. 80952 XIV —0, 28571 XVIII 18421 VII 51754 XI 37775 XV 38213 XX —U. —0. +42, =i), — 0. 23879 VIT —1. 21471 XI . 45976 XV . 12832 XX . 33333 X1 ). 00224 XV . 26571 XIX . 14286 XXII . 33333 X TIT . 4904 XVIT . 2807L AXT . 13083 XXV 77143 XXXI , 42357 XIV -13. 42857 XVI +3, 18037 XXII —0, 28571 XXVI +0, 28571 XXXI +0. 43753 XXXVI +13. 48571 XIX +1.51429 XXUI +0, 42257 XXVIT +1. 45714 XXXI —0. 5128 XXXV —0. 12568 XI —0, 12832 XV +0, 38593 XIX —0), 31736 XXIII —0. 11056 NXVIIL —0. 18207 XXXII +46, 90204 XXXVI §4.] CHICAGO BASE TO SANDUSKY BASE. Normal cauations for determining the correlates—Continued, No. of equation. 21. 0=—1. 97000 22.0 O=42. 39356 23. =—0. 46000 24. 0=+1. 25000 25, 0=—1, 44003 26. 0=-+3. 13600 27, 0=+2. 53100 28, 0=—!. 62990 GOL s —0, 28571 VUI —0. 70202 XV . 97143 XIX 228071 XXII . 2641 NAXITX / 1724 XAXAV . £0887 XN LIT . 19447 XVI . 07024 XX . 538699 XXIV - 60329 XXVIII . 66928 XXXIV - 03939 XXXVITI . 65543 XLIU - 14286 AVI 31736 XX . 06667 XXIV - 92196 XXVIII . 13333 XXXIT . 50000 XXAVIL . 15586 XLIT . 20000 XVIL - 53699 XXIT . 33333 AXVIL . 20000 XXXI 251268 XXXV . 33333 XXX EX . 13033 AVIT .5€0c0 XXIT . 26421 XXVI 04861 XXX 00056 XXXIV . 309387 XXAVIU . £2857 XVI ), 28797 XX 1.26421 XXV .48718 X AIX . 43705 XXXII . 25642 AL . 37966 XLV . 28571 XVI . 95605 XXT 44379 XXV 40744 X XTX 0, 84438 XXXII . 83472 XXAVIT . 52333 XLII . 20103 XVIT . 60329 XXII . 97196 XXVI » 19513 XXX . 20025 XXXIV . 59443 XXXVI —1, 42200 XLII | me La = = .o0216 IX -57143 XVI . 96220 XX . 11355 XXVI . 20383 XXX . 88462 AAXIX 1221 XLUI . 19447 XVIT . 85698 XXI . 58080 XXV . 65786 X AIX . 54870 XXXV - 62418 XAXIX . 20806 XLII 742286 XVIL . 28571 XXT 44321 XXV . 16667 XXIX . 17512 XXXL C1089 XXXVI 1426 XVII 06667 XXIII _ 95886 XXVILL . 26667 XXXII 09524 XXXVI 66667 XL 42014 XVII 44321 XXII 44379 XXVIL- . 13388 XXXI 50995 XXXV 08708 XXXIX 92857 XVII 11355 XXI 70147 XXVI 00220 XXX 46223 XXXIV 51282 XLI . 28571 XVII . 64976 XXII . 37363 XXVI 51290 XXX 94717 XXXIV —1, 48719 XNXIX —0. 38463 XLUI -0, 32814 XIX +1. 98196 XXIII —3. 64690 XXVII +0. 16407 XXXI —2. 09009 XXXV +1. 34728 XXXIX +40. 47020 XLII .STI3 XIII 0.28571 XVII 3.72803 XXI 95605 XXVII 14286 XXX 12821 XL 18083 XLV 18037 XVII 04427 XXIL 24723 XXVI 79377 XXX 20193 XXXVI 202806 XL 30807 XLV 42057 XVI 19447 XXII + 1.57143 XXVI + 0, 11637 XXX 0.05148 XXXIV 0, 16667 XXXIX | = ecsere rk xesese +444 — 0,54246 XIX 2.99048 XXIV — 0.61905 XXIX 1.31719 XXXII 1. 14286 XXXVI 0.31172 XLIL 1, 12843 XIX - 1.06195 XXIV 2.95094 XXVIII 0, 04738 XXXII 1.43578 XXXVI 0.17416 XL — 0.28571 XVII 24723 XXII 37363 XXVIU . 78571 XXXI .31151 XXXV ). 80774 XLII 14286 XVIII 61905 XXIII + 4.60805 XXVII + 0.85714 XXXI + 0.91738 XXXV 1.05130 XL 0.56949 XLV 0. 11056 XX 0, 95886 XXIV 7. 66626 XXVIII 0.36165 XXXII 0.75828 XXXVI + 0.34356 XL + 0.69622 XLV ah Gy =o + + + =i, Ao, so71t ATV a5714 XVIII —1.85695 XNIT +0. 47020 XXVIIL —0), 01091 XXXIV 2 25G4EL XLT eS => = —0, ~-0. +1. +0, —0, +40. 92693 XTX 19447 XXIII 64976 XXVII 3€054 XXXI 53699 XXXVII 41612 XLI 251429 XIX . 39048 XXII - 61905 XXVIT . O2857 XXXI . 33333 AXAVI . 33333 XL = 1, 24190 XX . 06195 XXV . 92012 XXX . 10045 XXXIV . 51931 XXXVIII 58715 XX .17500 XXV . 92816 XXIX 2.70110 XXXII . 68178 XXXVII . 08143 XLII +2. 14286 XIX +1.57143 XXII —1,97196 XXVUI +0. 50000 XXXII —0. 76924 XXXIX —0, 25642 XLII 40, 42857 XIX +0. 33333 XXIV —3. 64690 XXVIII +40. 66667 XXXII +0, 33333 XNXVI —0. 76923 XLI 40, 47020 XXI +2. 95094 XXV —1, 73834 XXIX +3.54701 XXXIII —0. 82160 XXXVII +0. 94040 XLI 474 No. of equation. 29, 31. 32. Bae PRIMARY TRIANGULATION. [Cuar. XVII, C, Normal equations for determining the correlates—Continued. 0=—0. 80000 — 0.28571 XVIII 0=—0, 33843 0=—0, 70300 0=— 0. 67000 O=-+1. 41528 O43. 10447 0=+40, 95484 0=—1. 94400 b+ + escorts resosres + +++4H+ ervenroocreorss | ++4 + | Sf oS SS eS oe SS SS + ob + + S 5786 XXIL J8718 XXVI 32505 XXX 81362 NNXV 93591 XXXIX 25642 XLUI 75800 XVII 79377 XXII . 00220 XXVI . 52138 XXX 67611 XNXV 49512 XXXINX 20383 XLII 57143 XVI 0, 12029 XX 0.20000 XXIV 0.16407 XXVIII 1.31902 XYXIV 28571 XLI 40000 XVIL 26667 XXIV 36469 XXVIII 24762 XXNIL 80952 NNXVI 14286 XLI 21917 XVI . 31719 XXIV 54701 XXVIII . 21438 XXXII 31492 XXXVI 0. 65728 XL 0. 49033 XLV 0.02574 XVII 0. 66928 XNIL 2, 46223 XXVI 1.70989 XXX 34319 XXXIV . 64251 XL 16253 XLIV 46240 XVIII 54870 XNII 91738 XXVIU 53368 XXXI 23950 XXXV 18918 XXXIX 30413 XLUI 42857 XVII 33333 XXII 75828 XXVIII . 80952 XXXII . 47619 XXXVI . 33333 XL 0. 12328 XLV —0, 28571 XIX —0. 16667 XXIII +1. 89744 XXVII 0, 33333 XXXII +1. 80953 XXXVI —0. 58975 XL 37966 XLV . 07568 XIX . 11687 XXII +0. 51290 XXVII —0, 23272 XXXII +1. 24710 XXXVI -L0. 02891 XL —0. 30181 XLV —0.77143 XVII —0. 14286 XXI +0. 13388 XXV +43.37143 NXXI —0, 53368 XNXV —0. 11285 XLIV 20000 XTX . 04738 XXV 33233 XXIX 21438 XXXII . 66944 XXXVI 06773 XLIL -L0, 43574 XIX 70110 XXV 42219 XXIX 01301 XX XIII . 07947 XXXVII 58307 XLI _ . 04897 XIX . 05148 XXII . 94717 XXVIT . 81902 XXXI . 19317 XXXV . 61008 XLI . 35673 XLV . 51268 XIX » 51268 XXIV . 09009 XXVIII . 55408 XXXIT . 18834 XXXVI . 39413 XL 00679 X LIV - 42857 XIX . 09524 XXIV . 80953 XXTX . 31492 XXXII . 57143 XXXVII . 42857 XLI +0. 60136 XX —0.61905 XXIV —1. 73834 XXVIII —0, 42219 XN NIIT +41, 21429 XXXVII —0. 51282 XLI —2. 22770 XX 1. 92012 XXTV . 19513 XXVIIT . 29477 XXXIIT . 66678 XXXVIT . 40766 XLI . 28571 XVIII . 36054 XXII .78571 XXVI . 82857 XXXII .QR57L NXXVI . 35642 XLV . 18807 XX 50000 XXVI . 23272 XXX 41.75510 XXXIV +0, 90476 XXXIX —0, 42857 XLIV —0, 12604 XX +0. 43705 XXVI 40. 29477 XXX +4, 10183 XXXIV +0, 40063 XXXVIII +0, 53626 XLII 00775 XX 10045 XXIV 20025 XXVIII 75510 XXXIT 95983 XXXVI 02403 XLIT +0. —U. aad, A, +0. +2. —1. 30947 XX +1. 50923 XXV —1. 81362 XXIX +1, 25462 XX XIII —0.51268 XXXVII —0. 80184 XLI —0. 60052 XLV +0. 90204 XX +1. 43578 XXV +1. 24710 XXX +0. 95983 XXXIV —3.50137 XXX VIII —1.37945 XLII —0, 25641 XXI +0, 92816 XXV +3. 74908 XXIX 0, 65233 XXXIV —1.58505 XXXVI +40, 96360 XLII —0, 20383 XXI +4. 84861 XXV +0. 32505 XXIX +1. 70989 XXXIV —1. 39840 XXXVIIT +0. 53328 XLII 45714 XIX +0. 82857 XXIII +0. 85714 XXVII —1. 31608 XXXII . 14286 XX XIX tA, . 13333 XXIII . 66667 XX VIT 282847 XXXI . 55408 XXXV . 33333 XL —0. 23314 XLV +2. 17512 XXIII —0(. 84438 XXVIT —1. 31608 XXXI +1. 25462 XXXV +0. 15746 XXXIX +0. 00342 XLIV = —0.01091 XXI +0, 00056 XXV +0, 65233 XXIX +4, 10183 XXXII —2. 09006 XXXIX —0. 64251 XLIII +0, 14724 XXI 41.31151 XXVI +4, 67611 XXX +4, 19317 XXXIV —0, 85041 XXXVIII +1. 24159 XLII —1. 20128 XXII -L0, 33333 XXVII —0, 28571 XXXI +1. 18834 XXXV +1. 38095 KXXIX —0. 28571 XLIV §4.] No. of equation. 37, 38. 40. 41. 42. 43. 44, CHICAGO BASE TO SANDUSKY BASE. 475 Normal equations for determining the correlates—Continued. 0=—0. 94300 O=+1. 94747 0=—0. 45300 0=—1. 38400 0=+1. 32000 0=-10. 39463 0=—0. 91600 _ 0=-+0. 54000 —0. 14286 XVIII —0. 50000 XXIII. +1. 21429 XXIX +1. 57143 XXXVI 1.51162 XL —0. 11626 XLIV +0. 43753 XVII +0. 04089 XXIII +40, 59443 XXVIII +0, 40063 XXXIII a8; 43 zl, =, +0. —, +40. +0, tt. ze == At i), —0. act . 14568 XLII —0. 14839 XLVI —0, 25641 XXI +40. 94040 XXVIII 0, 14286 XXXII +10, 42857 XXXVI —0. 53216 XL —0, 52823 XLIV +0, 40387 XXI —0. 08143 XXV +0, 96360 XXIX +2, 02403 XXXIV ae +5. =f, =f +0. = 02646 XLIT 25469 XLVI 38462 XXIT 08708 XXV 93591 XXIX 15746 XXXTII 50000 XXXVII 94506 XLI 39087 XLV 12821 XXT 17416 XXV 58975 XXIX 64251 XXXIV _ 60401 XLII 11848 XLVI 12821 XXI 47020 XXVIII —0, 39413 XXXV —0, 67971 XL “10, 83452 XLIV —~0, 03588 XLVIII —0, 14286 XXXI —0. 00679 XXXV —0. 14286 XXXIX +40, 83452 XLIII -40. 03114 XLVIL 02854 XX XVIII 08643 XXX VIII 02646 XXXVIII —0. 14286 XIX —1. 14286 XXIV +0, 66678 XXX +3. 05481 XXXVII —0. 11628 XLI -0. 08481 XLV -40, 43753 XIX +0, 51931 XXIV —1. 58605 XNXIX —0. 85041 XXXV —0. 87561 XXXIX -++0.12513 XLII -++0. 62418 XXIT —0. 76924 XXVI —0, 49512 XXX —2, 09006 XXXIV —0, 87561 XXXVUI —3. 47388 XLIT +0, 20806 XXIT —0. 25642 XXVI +0. 02891 XXX —0. 39413 XXXV +1. 28204 XXXIX . 67971 XLIT . 41612 XXIT . 51282 XXIX . 58307 XXNXIT . 11628 XXXVIL . 67323 XLI . 19404 XLV . 65543 XXIT . 80774 XXVI . 53328 XXX . 24159 XXXV . 47388 XXXIX . 49328 XLII . 20806 XXIT — 0. 25642 XXIX . 23256 XXXVIT . 20854 XLI . 96717 XLV —0, 42857 XXXIL —0. 28571 XXXVI —0, 27575 XL +2. 25374 XLIV —0, 07176 XLVIII +40, 30068 XX -L0. 68178 XXV —1, 07947 XX XIII +0. 07800 XXXVIIT —0, 48813 XLII —0. 04651 XLVI —0, 92089 XX —1. 30937 XXV —1.39840 XXX it S018 RAK VI +1. 08643 XL —0. 36935 XLIV —0, 16667 XXII —1, 48719 XXVII —0. 14286 XXXI —1, 18914 XXXV +3, 53661 XXXIX +0.61537 XLII —0. 33333 XXIIL —1. 05130 XXVII —0. 33333 XXXII -40, 33333 XXXVI +43, 26603 XL —0, 27575 XLIV —0. 51282 XXVI —0. 40766 XXX —1. 61008 XXXTV +0, 49448 XXXVIII —1. 06833 XLIT —0. 18272 XLVI +0. 15586 XXTIT +1. 52333 XX VIT —1. 06773 XXXII —1. 37945 XXXVI —1. 14568 XL —0. 16438 XLIV —0. 25642 XXVI —0, 20383 XXX +0, 12513 XXXVIIT —0. 49328 XLIT —0, 55840 XLVI +0. 00342 XXXTIT —0, 11628 XXXVIT —0. 52823 XLI —0, 04298 XLV —0. 53699 XXIT —0, 82160 XXVIII —0. 51268 XXXV +0. 50000 XX XTX —0, 23256 XLIIT +1. 03939 XXII —06. 83472 XXVIII —1, 66944 XXXII +0, 07800 XXXVII +0, 49448 XLI 0, 23204 XLV —0, 33333 XXIV +1. 34728 XXVIII +0, 90476 XXXII +1. 38095 XXXVI +1. 28204 XL —0. 14286 XLIV —0. 66667 XXIV +0, 34356 XXVIII —0, 65728 XXXIII +1,51162 XXXVII —0, 53216 XLI +0. 01130 XLV —0. 76923 XXVII —0. 28571 XXXI —0. 80184 XXXV +1. 94506 XX XTX +1. 25854 XLITT +40. 31172 XXIV —1. 42200 XXVIII +40. 53626 XXXIII —0. 48813 XXXVITI —1, 06833 XLI —0. 99981 XLV- —0, 38463 XXVII —0. 64251 XXXIV +0. 61537 XX XIX +2. 56149 XLII +40. 01557 XLVII —0, 16253 XXXIV —0, 36935 XXXVIIT —0. 16438 XLIT —0, 56864 XLVI 476 No. of equation. qin, dG, 4, 49, 5. PRIMARY TRIANGULATION. (Cuap. XVII, C, Normal equations for determining the correlates—Continued. O=-+3. 36715 0=+-1, 16300 0=—0, £4800 O=-+0. S78U0 - J=—2, 36760 C==—0, 10900 . 18983 XXT . 69622 XXVIOL . 23314 XXXIT . 12825 XXXVI . 01130 XL . 04298 XLIV . 04651 XXXVIT .11Rde X LIT 2. 20997 XLVI . 01557 XLT . 51016 XLYTIT . 03588 XLII +2. 21750 XLVHT —0, 25000 XLVI —0, 54762 L — 0.21429 XLVIT +40, 30807 XXII _ 37966 XNIX +0, 49038 XXXII +40, 08481 XXXVIT +42. 19404 XLI 44.95315 XLV —0. 25469 XXXVIT 0, 55840 XLII — 0. 00609 XLVIT -40, 03114 XLIV —0. 21429 XLIX —U. 07176 XLIV —0. 46429 XLIX —0, 21429 XLVIT = = —0, 21420 XLVI . 37966 XXVI . 80181 XXX . 35683 XX ATV 223204 XXXVIIT . 99981 X LIT . 13327 XLVI T —0, 14889 XL . 56864 XNLIV 58378 XLVIIT 00609 XLVI . 21429 L . 58378 XLVI —0. 21429 L —0, 46429 XLVITI —0, 64762 XLIX Values of the correlates and of their logarithms. T =-40. 1381 log 9, 1403195 4 XXVI =—0, #850 log TL =-+0, 4285 log 9. 63197114 | XXVII =—1,2970 log IIT =+40. 6017 log 9. 77938724 | XXVIIL =—0. 4713 log IV =—0,3155 log 9. 4989994_ | XXIX =-11. 6223 log V =-10.5697 log 9, 7550767 4 XXX =-10. 4516 log VI =+0, 2031 log 9, 30777414. | XXXI =-+40. 6668 log VIT =-10, 0825 log 8. 91629604 XXXII = 40,4572 log VIII =—0, 4406 log 9. 6440051_ XXXII =-+40. 3762 log IX =—0. 0492 log 8, 6918768_— . XXXIV =—0, 6489 log X =-+40. 1644 log 9. 21584904 | XXXV =+0, 7044 log XI =—0. 1287 log 9, 0887030— XXXVI =—0. 5639 log XII =+0, 1398 log 9. 1454450. XXXVII =—0. 0780 log XIIL =+40, 2689 log 9, 4296715, XXNVIIL =—0, 4408 log XIV =—0. 0267 log 8. 4260230_ | XXXIX =-10.5897 log XV =—0.2016 log 9.3044474_ | XL =+0, 6394 log XVI =-10. 0545 log 8, 73623714 | XLI =—0,6796 log XVII =+0. 6490 log 9, 81221794. XLII =-10. 4797 log XVIE =+1.2818 log 0, 1078271). | XLIIL =-+0, 8581 log XIX =+1. 417% log 0, 1516272. | XLIV =—0.7518 log XX =—0. 6352 log 9, £020173— | XLV =—0, 2682 log XXI =—0.5411 log 9.7332775— XLVI =—0, 4331 log XXII =—0,7801 log 9, ¥921670_ XLVII =+0. 4402 log XXII =—0.6511 log 9. 8136544_ XLVIIL =—0. 1388 log XXIV =+0.7139 log 9. 8536495 XLIX =-+41.1440 log XXV =—0.1996 log 9. 3002041— “L =-+40, 4218 log 0. —0, 56949 XXVIT 35642 XXXI —0. 60052 XXXV +1. 39087 XXXIX . 96717 XLII So . 18272 XLI . 13327 XLY . 25000 X LIX . 86229 XLVIT . 51016 XLVII +2, 20238 X LIX a LL. 0682 L 9, 9216813 0. 1129433 9, 6732974 0, 2101365. 9, 6547155 4. 9, 8240151. 9, 67685824 9, 5754188, 9, 8122045 9, SA7B132 4 9, 7512329_ 8, R920946 — 9, 6442810_ 9.706017 4 9, 80579984 9, 8329795 9, 6746040 4. 9, 9335328 9. 8761139— 9, 4283940— 9, 6366082 9, 6436994 4. 9, 1424833 0, 0584374 4 9, 62510924 £ §4.] [56,] [572] [573] (58) [582] [583] [58,] [59] [595] [593] [60,] [60,] [605] [604] [605] ' (61,] [61;] [613] [61,] [615] [62,] [622] [625] CHICAGO BASE TO SANDUSKY BASE. ae =+0. 104 =-+40, 085 =-L0. 262 ——0, 088 =-10. 227 =+0. 193 =—0. 097 =-+40. 130 =-+40, 088 =+0. 081 =—0,079 - =—0. 316 ——0. 046 =+0. 226 =-++10, 306 —=-L0. 136 =-+0.314 =-10. 084 ——0. 465 =—0. 064 =+0.179 =—0, 261 ——0, 056 Values of the general corrections. Mt [63,] =—0.326 [63,] =—0.392 [63;] =-+0. 490 [63,] =+0. 443 [63;] =+9. 102 [63.5] =—0, 429 (64,] =-+40. 425 [64,] =—0, 407 [643] =—0. 130 [64,] =+0. 401 [64;] ——0. 066 [65:] =—0.254 [65.] =-+10.341 [653] =-+.0.290 [65,] =—0.356 [65;] =—1.206 [656] =+0.925 [66,] =++0.296 [66,] =——0. 460 (66,) =+0.178 [66,] =-+0.206 [665] =+0.077 (67,] =—0.118 [672] [675] [67,] [675 ] (68) ] | 682] [683] [68] [635] [685] [69,] [692] [693] [694] [695] [695] [69;] [70,] [702] [70s] [70,) 705] [711] u =-+0. 260 =-+0. 624 =—0. 381 =—0, 050 =-+40. 465 =—0. 033 =—0. 655 =—0. 108 =+0. 574 =+0. 156 =+1.113 =—0. 924 =—0. 865 =-40, 575 =-+0. 848 =-+0. 362 =—1. 108 =—0. 421 =-+0.171 =— 0, 038 =-+0. 110 =-+0, 284 = +0. 843 [712] [7153] [714] [71s] [716] [721] [722] [723] [724] [725] [731] [732] [741] [742] [745] [75142] | [752] | [753] [754] [755] [76,] [762] [77] Residuals ‘resulting from substitution of general corrections in numerical equations of condition. ] - ee Residual. | rank Residual. catia Residual. i 1 0. 0000 18 | +0.0001 35 0. 0000 2 0. 0000 19 —0.0001 | 36 —0.00C1 3 0. 0000 20 +0.0024 | 37 —0. 0001 4 0. 0000 21 +0. 0001 | 38 —0. 0004 5 0. 0000 22 +-0.0055 |; 39 0. 0000 6 0. 0000 23 —0. 0001 _ 40 0. 0000 7 0. 0000 24 0.0000 || = 41 —0.0001 | 8 —0. 0001 25 0.0009 | 42 0. 0000 9 —0. 0005 26 0.0001 | 48 0. 0000 10 0.0000 | 27 0. 0000 44 0. 0000 i 0. 0000 | 28 0. 0000 45 —-0. 0016 12 0. 0006 | . 29 0. 0000 46 0. 0000 13 0. 0000 30 _ +0. 0007 47 0. 0000 ut 0.0000, 81 —0. 0001 48 0. 0000 15 —0.0008 | 32 —0. 0001 49 0. 0000 16 0.0000 ; = 38 —0. 0020 50 0. 0000 17 +0. 0001 ; 34 -0. 0004 478 PRIMARY TRIANGULATION. {Cnapr. XVII,C, D, PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § &. Let— m r n = whole number of observed angles in a section (one adjustment). = whole number of rigid conditions in a section. = number of triangles in principal chain. [pov] = sum of weighted squares of corrections to observed angles. Py Ps Pa Ds Pe Pe p. if [vv Pt = probable error of an observed angle of weight unity. = probable error of an observed angle of average weight in whole section. = probable error of an adjusted angle of average weight in whole section. = average weight of an observed angle in whole section. = average weight of an observed angle in principal chain. = probable error of an observed angle of average weight in principal chain. = probable error of an adjusted angle of average weight in principal chain. | =sum of squares of closing errors of triangles in principal chain. = probable error of an observed angle in principal chain as derived from the closing errors of triangles. Proceeding as in Chapter XIV, C, § 8, there are found the following values: FOR THE ENTIRE SECTIONS OF THIS CHAPTER. a Extent of section. m a [pov] | pr | py | pg WEE) pe m | aw “uw a VII | Michigan City - Bald Tom to Fremont - Quincy ..---. 67 39 14.42 | 0.41 11.02 | 0.41! 0.65 | 0.26 IX | Fremont - Quincy to Cedar Point -Stony Point ....-.. 93 65 17.78 | 0.35 | 0.99 | 0.36 | 0.15 | 0.19 X | Cedar Point - Stony Point to Chester - Willoughby. -.| 126 84 25.00 | 0.37 | 0.95 | 0.38 | 0.58 | 0.22 FOR THE PRINCIPAL CHAIN CONNECTING THE CHICAGO AND SANDUSKY BASES, GIVEN IN D, § 6, FOLLOWING. \ i - From closing errors of triangles. | Section. Extent of principal chain in each section. De p. |e! | or A ge | Greatest | [vv] 7 ’ verag a | t error. error. | “wo | “wu a“ he a VII | Chicago Base to Michigan City - Bald Tom ........... 0.82 | 0.50 | 0.31 | 11.52 9 | 0.44 0. 96 2.07 VIII | Michigan City - Bald Tom to Fremont -Quincy....-.. 0.88 | 0.44 | 0.28 | 22.70 | 14 | 0.50 1.00 2.91 IX | Fremont - Quincy to Cedar Point - Stony Point ....... 1.00 | 0.35 | 0.19] 13.73 | 14 | 0.39 0.78 1. 86 X | Cedar Point -Stony Point to Sandusky Base. ....--.-- 0. 82 | 0.41 | 0.23 | 15.64 8 | 0.54 1.07 2.71 Entire principal chain + 210 scssccessasacatinessas barnes secciee eaeees ~ 63. 59 | 45 | 0.46 0. 94 2.91 | D.—PRINCIPAL CHAIN § 6. The and form a ch OF TRIANGLES BETWEEN CHICAGO AND SANDUSKY BASES. principal triangles connecting Chicago and Sandusky Bases are forty-five in number, ain about 280 miles in length. In the adjustment of this chain for the discrepancy between the measured value of either base and its value computed from the other base through the triangulation, the bases are considered exact. Were they not considered exact, their probable errors would not in this chain require corrections to the logarithms of their lengths as great as a unit in the seventh place of decimals. The triangles fall in four different sections of the adjust- ment, the dividing lines of which, lying between the bases, are Michigan City -Bald Tom, Fremont- Quincy, and Stony Point-Cedar Point. Starting from Chicago Base, the probable errors of observed angles of average weight in the parts of this chain separated by the above lines are (see Chapters XV. I, C, § 11, and XVII, ©, § 5) £0.49, £07.44, £0’,.35, £0.41. With these values §9 5, 6. CHICAGO BASE TO SANDUSKY BASE. 479 we find, using the notation of Chapter XIV, D, § 10, and the values of «? and , given in the following tables, ot yee (A+B) pt 0659 The logarithm of the measured value of Sandusky Base expressed in feet is 4.3101715. The logarithm of Sandusky Base, computed from Chicago Base through the triangulation, is 4.3101618. Hence, d=+497. The additional data required in deriving the corrections to the logarithms of the lines, as com- puted from the Chicago Base, are given in the tables which follow, these being arranged in the same manner as those in Chapter XIV, D, § 10. The line in the system having the greatest prob- able error is Jefferson-Milton. Yor this line — =2923 and +, =2736, giving for the probable error of its logarithm + 37.6 in units of the seventh decimal place, which is equivalent to ;;53, part of the line’s length. This probable error is, however, independent of the probable errors of the bases, and is, therefore, somewhat too small. But since the probable error of either base is only about toovon Of its length, the above result would not be materially changed by taking the probable errors of the bases into account. Principal chain of triangles between Chicago and Sandusky Bases. - 7 eens : Logarithms Weighted mean — Stations. Angles. Eee of sides in | a? and B2|} ¥ (a2-+£?) 3 loveritome of | ; feet. sides in feet. oO é uw aw | Willow Springs ....-..----- 42 35 04. 570 | (| 4.3917929 4. 3917929 : East Base. ...- Seeoes setion aatiee 40 29 64.976 | > +0. 484 J 4. 3738195 4, 3738197 | ' West Base.....-..........-. 96 55 50. 591 J | 4. 5582257 4. 5582259 | | é Morgan Park ..-.-...-..---. 44 52 44.590 | ( 4. 5582257 445, 21 sini Gidehai orava cota |lotaeverersnata, 4. 5582259 Willow Springs...-.-.------ 34 24 48.796 —1. 6274 44618324 |e enn vote soles veee tee rel istiosss 4. 4618328 PASG BAGO ee cence. cciedieinies ee 100 42 26. 856 i | 4. 7020314 16. 00 | 991. 87 237 4. 7020318 I Shot: Towersecee. cece cuss 37 39 04. 874 { 4. 7020314 146: 29: Vccwicictcieicte oa Msilce. aes ties 4. 70203818 Willow Springs...-.-.--..-. 60 23 46. 481 —1. 251 4.855845 | sceccreacslseces cesses [pee rseee 4, 8553452 Morgan Park ...-....--..--- 81 57 09. 486 | 4. 9117960 9. 00 1746. 16 417 4. 9117967 i Military Academy...-....-. 81 25 36. 348 4. 9117960 10. 24 Latent sinsigh Awe Aehk 4, 9117967 Shot Tower....--.--.------. 37 14 28. 682 —1. 187 46986950: |wcimns craves lees cies cioma [beierds zie 4. 6985558 1 - , Willow Springs.......------ 61 19 55. 814 l 4. 8598813 182. 25 1888. 65 452 4. 8598821 MUl@tS o ccccc.scaveeee! ease 29 11 38. 637 |) { 4. 8598813 1421/29 fescewexissepitieee ede 4, 8598821 1 Shot Tower........--.-.---. 55 36 25. 651 i +2. 071 50882191, | souvey sie .s| ioteviervensec|| Meas ond 5. 0882205 Military Academy ...------. 95 11 57.795 | 5. 1698790 3. 61 3313. 55 793 5. 1698804 Michigan City .-........-.-- 38 28 11.238 | | f 5. 1698790 OQBI Fl) caccer - sjeiectes eid *resais sions 5. 1698804 Millers ccnecceensereveecisa-| 114 11 26.900! ( —0. 762 B, SBGTOGG. lisiwiaraweisjeiea| 2s aciande simeis |niwin ee ecag's 5. 3361079 Shot Tower. .-...----.----- 27 20 26.233 ) | 5. 0380925 1656. 49 5672. 29 1358 5. 0380948 fe t pein Glee ceacece Obsaesicwseseeets saver seer | 85 27 92.773 l f 5. 0380925 DO. |isiciaiseersistercinia a haisierarg escin 5. 0380948 Michigan City o< <. ..<0-0<00- 65 15 26.461 +0. 650; 4. SO7GBOS | icsecscawclesasaavecnen fews canes 4, 9976421 Millers: .sccc2ccccesas. seeen' 29 17 12.018 J | 4, 7289264 1413. 76 7088. 94 1696 4. 7289293 Springville ........-....----| 72 51 40. 281 { 4, 7289264 B22) |wassicwxwnwes mans oes 4, 7289293 Michigan City ......- ---| 55 23 13.437 —0. 608 BS. O0000T2 |exsevencenlerevensvenns|exand oad 4, 6640602 QWi8xosee cide scuiteesaaseee 51 45 06. 738 | 4, 6437091 275. 56 7406. 75 1772 4. 6437121 Bald Tom...-.....- ie 26 25 28.139 4. 6437091 TOT TG | acccnasasmwe ts seeses 4, 6437121 Michigan City -| 75 48 51. 648 —0.015 RAIGERE Wsawveceie Wewsadetenachar tiie: 4, 9817274 BpTNEVIN: cv cancerwugn ncn 77 50 41.187 | J 4, 9854855 20. 25 9224, 76 2208 4. 9854893 480 PRIMARY TRIANGULATION. [Cuar. XVII, D, Principal chain of triangles beticeen Chicago and Sandusky Bases—Continued. Hl ari | 5 ! Stations. Angles. La at Ta sides in a? and B?| & (a?-} 8?) b | eeactige vt : *e feet. sides in feet. | E i : Oo ¢ “ | " | Galena.........eee eee 90 06 01. 854 | ( 4. 9854855 4 9854893 : Bald Tom... 35 06 57.623 § 40.1134] 4.7453296 4. 7453335 Michigan (ity ......-....--- 54 47 01. 558 | | 4. 8976979 4. 8977018--~ ee ee re no ree oe ree Carli sles. x dhacceoke meooede 59 54 32. 304 il ii 4. 8976979 WAGHBS corey Se cemiee 3 by geeaess 4. 8977018 ‘Bald Tom...... 2222-22-20 33 39 31.462 |S —1.5032! 4.7042677 |.....--2. je.--- eee eee berccudete 4. 7042716 Galena 86 25 57.155 J tl 4. 9597243 108 | 372. 54 | 2279 4.9597282 Bertrand......-....----+---- 89 10 50. 985 | (| 4.997248 0.09 Jediuiaie ee ween 4, 9597282 Carlisle ....-2.0.2222.-- 206+ 56 28 25.735 |> 41.6392 | 4. 8807434 asd duseroiawiay: I vtdctog us 4. 8807476 Bald Tom.........-.-.2-2.+- | 34 20 44. 201 | 4. 7111882 2461 4. 7111924 Penn sckaceaetsckceheasae 40 28 07. 629 l (| 4. 7111882 GLO 09 sacadevs visas cece ‘ 4. 7111924 Bertrand ......------------- se 27 29.295 | —2.908/] 4. 8958806 |......... [alee vata teciclata hte 4. 8958941 @arlisles oc osc seas cesce seeks 57 09 23.877 , J 4, 8240228 184.96 2116. 32 2612 4, 8240273 | s Ws, oe ss pmpeeo hy tenes 72 19 44. 193 (| 4. 8240228 ASDA.) ccs. etead ancnbacs 4, 8240273 Penit cercd 2 paaseesaecoe sonst 5308 45.026 | 41.413/ | 4. 7481985 [....... fee aan ee - 4. 7481981 Bertrand .......... --.---- 54 31 31, 498 | 4. 7558374 225.00 | 2387.56 : 2664 47558420 Ie zeal Bees 73 1b We ian totes ni Jefherson. <222 202d eevee ! 30 11 10. 035 al 4. 7558374 4, 7558420 | Milton...........-- eneeenGs 75 36 29.503 | —0. 364. | 50405874... 5.0405924 | ODM Lc cc; scas sewer ee es 74:12 21.881 J A 5.037210 | 5.037260 | CalWini sce 2 cane nieaene eae 109 32 06. 169° ( 5..0377210 5. 0377260 | Jefferson ...2....2200220002- 36 51 49.664 ' —0.650/! 4. 8415570 ; 4.8415624 | NiLOn ese ate sateen ce 33 36 05.155 | ( 4. 8065162 | 1004. a9 4793.95 | 2124 4,8065216 | i een aa} Joiners panera, pais Poiberees aggicda ae sdasge aces 83 52 49. 490 | ( 4, 8065162 BBM |p racers carte ae steams 4. 8065216 Jefferson ......-2..2220-2065 42 07 22.706 |» +1. 619 BGB55416 Vesccateascnsyci: (veicis sv Sarcans [eaters 4. 6355470 53 59 48.332 J | 4. 7169376 234.09 | 5032.88 | 3170 4. 7169430 La : ak. | Van Buren .. .....----.---- 54 32 09. 208 1 (4.769376 W500: ee scwmaneean| daneaas Fa | 4. 7169430 Porter ...- 57 46 37.978 | + —0. 963 , I A738 TSE - |g cacccseiad laseceveciaceil wise ce | 4. 7334239 Jefferson 67 41 13, 429 : J | 4..7722577 73.96 | 5331.84 | 3227 | 4. 7722632 etree iy Sey i Sherman.........-- : | 43 27 55.074 ] { 4. 7722577 4. 7722682 Van Buren ...2. 22-22-2222: 85 45 49.518 |) —1.480, | 4, 9335054 4. 9335412 Porter asco camcenmseeemars 50 46 16.335 J Ul 4, 8238155 295.84 | 6120.52 | 3377 4, 8238213 Mong oss ceesceveesserces vs 44 39 03. 507 (| 4. 8238155 452. 69 eer Sei 4, 8238213 Sherman.....--..---.------- g2 45 40.688 |$ —0.948)| 4.973517 |..2.2222. Jenene Seema 4. 9735237 Van Buren...........----+-- 52 35 16.979 | 4. 8769704 259.21 | 6883.42 | 3514 4. 8769764 igi oases ante at Bronson west see cseauee 80 06 15.196 1 J 4. 8769704 1BA69. | is Sacer Sesewtee 48769764 Mons. 22 nseceaaesaacecntan 48 20 13.963 '' —0. 127 GBGGHE vce ee ohadices sagcek ee 4.7568475 Sherman sa2:2cbeeccneseatan 51 33 31.635 J | 4. 7773785 278.89 |° 7126. 00 3570 4.773846 sean Be Se Jj 1 ) Fremontccccccwicseeseewceess 39 20 26.179 | (| 4..7773785 660149 see geten aes eineteen 4. 7773846 | Bronson .....---+-200222220- 6117 52.123 | —o.261/| 4.9184012 j........- eee wea eee 4.9184075 | | Mongo «....-.-0------2---- 79 21 42. 848 | J | 4. 9678094 15.21! 7801.70 | 3699 4. 9678157 | | QUIDCY -.----.---- eee eee ee | 59 04 19. 886 | { 4. 9678094 TOSATO Nace ce ac alate ates h 4.9678157 | Fremont.....--..--2--222+++ 62 01 52.451 | —0.008 4. 9804764 |..-.---2--[eeceeeccence [ee eee eee 4.9804828 | | Bronson yxeeuesedecess een 58 53 49. 456 J i 4. 9670114 161.29 | 8121.75 | 3760 4. 9670178 | | Reading ....--...-.-----.--. 80 01 41.967 } { 4. 9670114 79600 | sateen tooo Sl oeceeee 4. 9670178 Quincy .......2....22220.00- 54 28 55.071 | \ —1.530 A jSESTOSE: iocanccden(otnacda ose Illveeene 4, 8837651 Fremont...-....-..--------- 45 34 24.156 | 4. 8274094 428. 49 442.18 | 3815 4. 8274159 CHICAGO BASE TO SANDUSKY BASE. 481 Principal chain of triangles between ee and Sandusky Bases—Continued. Stations. | ! | Logarithms Weighted mean | Angles ae oe of sides in | a? and 6?) & (a2+ 2) 1 logarithms of | | . ; feet. P| sides in feet. | eaten he Serene 5p, Se PASE tS. os, ee SP PTS oe. boa ie ni Fillsdales cone scexace 2 --{ 62.37 43.537 _ | | 4, 8274094 118. 81 |.-202...----]-- 222-8 4. 8274159 Reading .................... | g529 02126 § —otge) 4.9770285 ooo cece eeeeeche cece 4. 8776303 QUINCY’ siccs se oer ntn emed 31 53 14, one ; s 4 4. 6018152 | 1149, 21 1710. 20 3973 | 4, 6018220 et me ces af a book Ah Poe Aes Bir BF PPM 2h creme aoe nated } i 7 . : POG, con pasicvs ane wane | 84 29 38, 545 eo te 4, 6018152 936. 36 prrteteteeceefe eer ee es 4, 6018220 * Billsdalee. cscsescewnaccecces | 91 34 24. 869 4, 8485897 |...-....-. Naas cantina cae 2a ojeud 4. 8485968 | | POMBE was was eeickee mene 53 55 57.124 J 4, 7563389 287. 16 1 2883. 72 4119 | 4. 7563460 | ! Wheatland ......-...-2...... 78 08 51. 643 {| 4. 7563389 A ee ee 4. 7563460 Hillsdale 36 21 52.411 +1. 282 ; SNORE boncinnsowndleneuen raknae le nmesests 4. 5387016 Pittsford 65 29 16. 369 | | 4. 7246790 92.16 2995. 24 4133 4, 7246861 = = Ig Bundays. cece. sesceveexayes | 48 28 35. 999 ] I 4, 7246790 349. 69 | fratess giniioa eats |e eye 4. 7246861 Wheatland . 87 05 44. 667 —0. 534 4. 8498216 |.......... | raisins sivia ase) 202 cwleeign 4. 8498289 Hillsdale. .............2.2.2. 44 25 39.953 J | 4, 6954833 462. 25 3807. 18 4204 4. 6954906 | : [= ceed Woodstock 49 19 16, 232 ' | f 4, 6954833 Bate OL | cuquse cases loa cincwe 4, 6954906 Wheatland .. 66 47 38. 408 | —0. 656 \ BOTTBQODD jarcias wena ol exwenenwemwels annie ale 4. 7789667 Bhnay sos eeeeeesavanerasy 63 53 05. 991 | ( 4. 7688332 106. 09 4240. 88 4288 4. 7688406 : | 5 POP ElG cos cmon se oseneincs 28 07 00. 426 | { 4. 7688332 1552. 36 |...-.2-2.0e- [eee e ee 4. 7688406 Woodstock .. 108 22 12. 316 +0. 158 4 DO OT28907 bececen vewsl seawendiwemd ences sce 5, 0728575 Wheatland 43 30 48. 385 J l 4. 9334833 492. 84 6286. 08 4543 4, 9334911 ROTI cco ucines wen yews mer 64 57 18. 588 4. 9334833 QBVOL, | recone eecinns| ceeneasa 4. 9334911 Fairfield 76 00 48. 206 —1. | FOGB 2058 leone tac-'| wicicitssioasre stad oeacaenarce 4, 9633038 Woodstock 39 01 54. 376 4. 7755346 670. 81 7054. 90 4639 4. 7755426 Blissfield .... 60 44 25, 218 4. 7755346 139: 24 | ons sia tcroisi aei[slciei slates 4. 7755426 Raisin ..... 73 50 36. 065 6} AESIIBIIBS:. | 2252-0 oscfocned cede sien) tasemces! 4, 8173199 Fairfield 45 24 59. 374 4, 6874312 428, 49 7622. 63 4710 4, 6874393 Dundee..-...---.2-.---..--- 43 45 57. 303 4. 6874312 POLO lacs ceeccssss awveneey 4. 6874393 Blissfield 95 46 09. 598 +1. as) 4 BASQ9OT acnsrcernc|awe cna sacede|ecounang 4. 8453080 RABI 5.5 2c -Scicasinscaisin 40 27 58. 621 4, 6597874 610.09 | 8716.72 | 4846 4. 6597457 Bedford. . 20:22. 62scccwsorees 56 34 47. 176 | 4. 6597374 WOSE2T facins wee Pca has cg dieisie 4. 6597457 Dundee.... 75 21 26. re +0. xsl TRIO: | rage ewan es ewanaevaawes leiiedin ss 4, 7238998 Blissfield ..-.....-..-2...--- 48 03 46. 548 | 4, 6097332 357.21 | 9267.14 4915 4, 6097416 Monroe 44 16 04. 135 4. 6097332 466. 56 |..----.---- lee ee ee. 4. 6097416 Bedford 68 37 31.733 —1. 235 Me TBAOZT2: I icisinisiciciarwies|ciemains dwoaies lseseresveves 4. 7349297 Dundee. .-.-.- 2.2202 - + reeeee 67 06 24, 612 4. 7302387 79. 21 9812. 91 4983 4. 7302472 Cedar Point ....--.----..--. 36 33 41. 044 | { 4, 7302387 806.56: |on nes veecse|s cesses 4, 7302472 Monr0@s. es2-a2222006 se cecemce 74 29 10.780 —0. 154 49891054 lise cc watesins freee sees eoell peice ane 4. 9391141 Bed fOr ison weeny esinocesen 68 57 09. 203 J | 4, 9252371 65.61 | 10685, 08 5091 4, 9252458 Stony Point....-.....-.----- 71 29 OL. 461 4, 9252371 PRY | nentnecadeusternenane 4. 9252458 Cedar Point.....--.-.-.-.--. 39 32 50. 657 —1. 862 4, 7522650 |. --.2 2-2 e- [eee cee ceweee [eee ee eee 4, 7522738 Monroe ...----.0----+-2---+- 68 58 05. 929 4. 9183791 65.61 | 10799. 69 5106 4, 9183879 Middle Sister ...--.--.------ 51 58 36. 201 4. 9183791 272620 I sistecioracaareianael sists’ neces 4, 9183879 Cedar Point.........-.---.-- 47 31 39.178 +0. 359 A, BBQBUGR le cewcs death enee an nemeael ned nouns 4, 8898156 Stony Point.....--.--..----- 80 29 46.116 5. 0159833 12. 96 285. 21 5153 5. 0159921 Locust Point ...--.--.------ 79 21 35. 581 l 5. 0159833 5, 0159921 Middle Sister ....-....-.-.-- 42 43 14, 154 +2. 706 4. 8550155 4, 8550244 Cedar Point........--------- 57 55 11.749 J 4. 9515555 4, 9515644 61 LS 482 PRIMARY TRIANGULATION. [Cuar. XVI, D, § 6. Prine ‘pel chain of triangles between Chicago and Sandusky Bases—Continued. eras — cients Be | Logarithms | Weighted mean Stations. | Angles. Mipors ef at sides in | a? and " % (a?4-B2) . logarithme of - eet. 2 sides in feet. | | F Middle Bass .....--......--- | ® a 28, ‘on il { 4. 9515555 AQ3G 21, asecaduenas separate | 4. 9515644 Locust Point.-...--...-..-..) 52 38 36. 695 nr —1. 647 ACQOSSSIT nie assecieide shes netcceanisiellS saree eee 4. 9055607 Middle Sister ............- 65 15 55. 575 | 4. 9634610 94. 09 | 692. 75 5220 4. 9634700 ean J i Danbury. w22. cesses cesses 76 48 47. 647 | ( 4. 9634610 24. 01 |...... Welasids Iemsinience 4. 9634700 Middle Bass ...........-.--- 58 56 15.368 | —-0,189 | ATO0TEATT: cermeancalooreomedceas [eoeneiees 4, 907562 Locust Point .........-...-. 44.14 58. 208 J | 4. 8187868 466. 56 1183. 32 5301 4. 8187959 | ‘ ou Kelley's cisccicc secswae snscisins OL 27 48, 829 | fF 4. 8187868 4. 8187959 Danbury vsicccc sev ccccneniccss 39 18 38, 221 —0. 065 4, 6206913 4. 6207005 Middie Bass ........-..----- 49 13 33. 441 i | * 4, 6981910 331. 24 1514, 92 5356 4, 6982002 | \ West Base ..........0. 20-28 68 16 12. 861 } a 4. 6981919 TO SDB lsinic ciaterote ait icit| dio aretersiere 4, 6982002 Dan dary sesn0ax sacsnecassian 55 23 10.231 | > —0. 469 J 4.°6956025 fence cccncleeveee emesis |scasises 4. 6456118 Kelley's ....--.--20-00---8- 56 20 37.341 i | 4. 6505232 198. 81 1784. 29 5401 4. 6505325, Sandusky ..-.---.2..2. 0-2-2. 65 24 37.122 4. 6505232 92. 16 | Sodemegeetea ds scum t 4, 6505325 West Base...........-.2---. 72 37 43.772 +1. 909 4, 6715369 *.2.2.-..-- paises F sceseaisinl|s Siomersie 4. 6715463 Dan DUT ys os: cscc5scineesecicicis | 41 57 39. 437: 4. 5169926 | 547. 56 i 2424. 01 5506 4. 5170020 ; f | _ -f ' | | East Base.....-...--.-..--- 69 32 30.524 | ( 4. 5169926 : 254 cncisiaciesciont ayy obmreter : 4. 5170020 West Base...........------. | 74 52 17.009 \ +1. 243 4.5299680 ....-..... snes enmanaals oaeee 4. 5299777 ‘ 1 Sandusky ...---.....-..----- ' 85 35 12. 620 | ( 4, 3101618 | 864. 36 | 3350. 78 5659 4. 3101715 : { Cuar. XVIII, A, §§ 1,2.) SANDUSKY BASE TO BUFFALO BASE. 43 CHAPTER XVIII. TRIANGULATION FROM SANDUSKY BASE TO BUFFALO BASE. A.— DESCRIPTIONS OF STATIONS. NOTE RELATIVE TO ELEVATIONS. § 1. The elevations of the surface of the ground at stations west of Font Hill, described in this chapter and in Chapter XVII, A, § 2, were determined for the most part by trigonometrical leveling. The heights of those stations near the shore of Lake Erie were, in nearly every case, determined by means of a single zenith distance to some point of known height above the surface of the lake. For stations remote from the lake shore, heights were computed from non-simulta- neous, reciprocal zenith distances. The latter data were sufficiently numerous, also, to give a good mean value for the coefficient of refraction. Fifty-one independent values were obtained from observations over as many different lines, which varied in length from 6 to 44 miles, averaging 18 niles. The greatest and least values for the coefficient of refraction were 0.145 and 0.042. Only six values exceeded 0.09, and they were obtained from observations made either early in the spring or late in the fall on lines extending over water. The weighted mean of the remaining coefficients, which were obtained from observations made during the summer season, on 37 lines extending over land and 8 lines extending over water, was 0.061. This value was used in computing heights dependent on single zenith distances. No rigorous adjustment of these heights has been made. The adopted values are quite accordant, however, as shown by the numerous checks resulting both from the connections with the lake surface and from the independent rotites along which the computation may be carried. The probable error of any height may be estimated as not exceeding on the average +1 foot except for stations Grand River, Silver Creek, and Sturgeon Point, to whose heights may be assigned a probable error of + 4 feet. The heights of stations referred to Lake Erie are above the mean level of that lake given in Chapter XXII, § 13. The heights of stations east of Font Hill referred to the mean level of Lake Ontario are above the mean level of that lake given in Chapter XXII, § 13, and were determined as stated in Chapter XIX, § 1. DESCRIPTIONS OF STATIONS. §2, LirrLe Mountain, 1877.*—This station is situated on a high hill, called Little Mount- ain, in Mentor Township, Lake County, Ohio, about 4 miles south of Mentor railway station on the Lake Shore and Michigan Southern Railway. The height of station used was 104 feet. The geodetic point is marked by an iron bolt leaded into the solid rock. Three stone reference-posts are set, as follows: one bearing north 28° 11’ east, distant 95 metres; one bearing south 81° 41’ east, distant 97.3 metres; and one bearing south 5° 53/ east, distant 124.2 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 674.7 feet. CLARIDON, 1877.—This station is situated in lot 1, section 14, Claridon Township, Geauga County, Ohio, about 14 miles in a southerly direction from the village of East Claridon, and 2 miles from the railway station. The height of station used was 100 fect. The geodetic point is marked in the usual manner. Three stone reference-posts were set as follows: one bearing south 83° 38’ east, distant 87.8 metres; one bearing south 23° 11/ east, distant 65.5 metres; and one bearing south 3° 41’ west, distant 61.2 metres from the geodetic puis. The first-mentioned refer- ence- “stone is set by the norte and-south line-fence east of the station, and the last two by the east- * See note relative to topographical sketches of stations under Burnt Bluff, Chapter XV, A, § 2. 484 PRIMARY TRIANGULATION. [Cuar. XVIII, A, and-west line-fence south of the station. The height of ground at the station above mean level of Lake Erie is 793.0 feet. MEsSoporaMtA, 1877.—This station is situated in section 57, Mesopotamia Township, Trumbull County, Ohio, about one-third of a mile east of the west line, and one-half mile south of the center line of the township, and about 4 miles east of the village of Middlefield. The height of statior used was 70 feet. The geodetic point was marked in the usual manner. Three stone reference- posts were set as follows: two along a line-fence to the north of the station, one bearing north 23° 23/ west, distant 102.6 metres, one bearing north 9° 51/ east, distant 96.0 metres; and one by a north- sent south line-fence, bearing south 20° 40/ east, distant ‘29. 4 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 599.2 feet. THOMPSON, 1876, 1877.—This station is situated in Thompson Township, Geauga County, Ohio, about one-half mile southeast of the village of Thompson, on Thompson Ledge, 313 metres south along the ledge from the road leading east from Thompson. The height of station used was 88.3 feet. The geodetic point is marked by a brass rod leaded into the solid rock, about 6 inches below the surface of the coarse, compact, pebbly soil. Five stone reference-posts are set as follows: one bearing north 34° 02/ west, distant 85.4 metres; one bearing north 81° 42’ east, dis- tant 6.2 metres; one bearing south 25° 06! east, distant 36.0 metres; one bearing south 19° 34’ west, distant 34.2 mene: and oue bearing south 770 02’ west, distant 35.0 metres from the geodetic point. The bluff ig distant 12 metres on the east. The height of ground at the station above mean level of Lake Erie is 734.5 feet. ANDOVER, 1876, 1877.—This station is situated on the highest point of Owen’s Mound, in Andover Township, Ashtabula County, Ohio, about 1 mile north of the village of West Andover, and about 24 miles west of Andover railway station on the Franklin division of the Lake Shore and Michigan Southern Railway. The height of station used was 113 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet below the surface, with a stone post set directly over it as a surface-mark. Three stone reference-posts are set along the fence south of the station, as follows: One bearing south 73° 59’ east, distant 32.38 metres; one bearing south 4° 04/ east, distant 10,56 metres; and one bearing south 75° 07 west, distant 39.0 metres from the geo- detic point. The height of ground at the station above mean level of Lake Erie is 617.8 feet. CONNEAUT, 1876, 1877.—This station is situated on the South Ridge, a height of land running in an easterly and westerly direction, in the southwest corner of Conneaut Township, Ashtabula County, Ohio, about 5 miles southwest of the village of Conneaut, 2 miles west of the- village’ of South Ridge, and 3 miles east of the village of Kingsville, on the north side of the Ridge Road. The height of station used was 115 feet. The geodetic point was marked by a stone post of the usual form, set 2.6 feet below the surface of the ground. Three stone reference-posts are set along the road-fence, as follows: One bearing south 75° 04’ west, distant 29.56 metres; one bearing south 14° 05/ east, distant 8.1 metres; and one bearing south 750 43’ east, distant 33. '88 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 308.9 feet. EDINBORO, 1876.— This station is situated in the western part of Washington Township, Erie County, Pennsylvania, about 3 miles west of Conneauttee Lake and the village of Edinboro. The height of station used was 115 fect. The geodetic point is marked by a stone post of the usual form, set 2 feet below the surface of the ground, with a stone post set directly over it for a surface- mark. Three stone reference-posts are set as follows: One bearing west, distant 8.5 metres; one east, distant 5.81 metres; and one south, distant 5.1 metres from the geodetic point. These bearings are approximate. The height of ground at the station above mean level of Lake Erie is 929.6 feet. HovuGuHton, 1876, 1877.—This station is situated on a large sand hill about 7 miles east of Port Burwell, Proruce of Ontario, and 1 mile east of Houghton. The height of station used was 53 feet. The geodetic point is marked by a stone post, set 3 feet below the surface of the ground, with the letters U.S. cut thereon, the tops of the letters being north. Three stone reference-posts are set as follows: One in a clump of willows, bearing north 51° 04’ west, distant 117.75 metres; one north, distant 4.4 metres, with letters U. s, cut in top, the tops of the letters being west; and one south, distant 4.5 metres from the geodetic point. The directions of the last two stones men- tioned are approximate. The height of ground at the station above mean level of Lake Erie is 187.1 feet. §2] SANDUSKY BASE TO BUFFALO BASE. 485 ERIE, 1876.—This station is situated in the western end of Mill Creek Township, Erie County, Pennsylvania, about 8 miles by the roads southwesterly from the city of Erie, and 2 miles easterly from Swan railway station. The height of station used was 114 feet. The geodetic point is marked by a stone of the usual form, set 24 feet below the surface of the ground. Three stone reference- posts are set along a fence west of the station, as follows: One bearing north 54° 54’ west, distant 56.39 metres; one bearing south 56° 59’ west, distant 31.29 metres; and one bearing south 2° 38/ west, distant 56.34 metres from the geodetic point. The road southeast of the station is distant about 244 metres. The height of ground at the station above mean level of Lake Erie is 285.9 feet. Lone PoInT, 1876.—This station is situated on the north side of Long Point, in the county of Norfolk, Province of Ontario. The height of station used was 123 feet. The geodetic point is marked by a hole drilled in the top of a stone post, set 2 feet below the sandy surface of the ground. Three stone reference-posts were set as follows: One bearing south 52° 27’ east, distant 49.4 metres; one bearing north 82° 51’ west, distant 72.3 metres; and one bearing north 33° 47’ west, distant 63.7 metres from the geodetic point. An oak tree, 1.7 feet in diameter, bears south 56° 45/ east, distant 45.7 metrés; a pine tree, 1.4 feet in diameter, bears south 52° 08! east, distant 51.7 metres; one, 1.5 feet in diameter, bears north 23° 25’ east, distant 27.4 metres; one, 1 foot in diam- eter, bears north 81° 25’ east, and is distant 12.5 metres from the geodetic point. The water’s edge, northeast, is distant about 44 metres. The light-house at the eastern extremity of the point bears south 76° 31’ east, and is distant 5828 metres. The height of ground at the station above mean level of Lake Erie is 5.1 feet. WESTFIELD, 1876.— This station is situated in Westfield Township, Chautauqua County, New York, on a hill, at the junction of three wagon roads, about 5 miles southwest of the village of Westfield. The height of station used was 90 feet. The geodetic point is marked by a stone post of the usual forin, set 20 inches below the surface of the ground. Three stone reference-posts are set as follows: One by the road-fence on the south of the station, bearing south 1° 40’ west, distant 12.06 metres; and two along the road-fence on the northwest of the station, one bearing north 81° 08’ west, distant 9.02 metres, and one bearing north 15° 29’ west, distant 21.18 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 886.4 feet. GRAND RIVER, 1876.—This station is situated in the west part of lot 17 of the 5th concession, Township of Dunn, County of Haldimand, Province of Ontario, 24 miles west of Port Maitland, and about half a mile back from the lake shore. The height of station used was 100 feet. The geo- detic point is marked by a stove post of the usual form buried in the ground, with a stone post set directly over it for a surface-mark. Three stone reference-posts are set as follows: One bear- ing south 84° 00’ west, distant 98.05 metres; one bearing south 0° 57’ west, distant 702.4 metres; and one bearing south 15° 44’ east, distant 679.4 metres from the geodetic point. Mohawk light- house bears south 79° 52’ east. The height of ground at the station above mean level of Lake Frie is 50.3 feet. SILVER CREEK, 1876.— This station is situated in Sheridan Township, Chautauqua County, New aaa York, about 3 miles west of the village of Silver Creek, three-fourths of a mile east of Sheridan railway pce station, and just south of the track of the Lake Shore and Michigan Southern Railway at a deep cut. in slate rock. The height of station used was 75 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet below the surface. Three stone reference-posts are set as follows: ie One bearing north 7° 12’ east, distant 30.66 metres; one bearing north 34° 30’ east, distant 62.84 metres; and one bearing north 72° 23/ east, distant 89.89 metres from the geodetic point. The first two reference-stones mentioned are set along the fence south of the railroad, the third by a fence east of the station. The height of ground at the station above mean level of Lake Erie is 153.3 feet. StuRGEON PoINnt, 1876.—This station is situated in Evans Township, Erie County, New York, at the extreme end of Sturgeon Point, about 3 miles northwest of Angola, a village on the Lake Shore and Michigan Southern Railway, and about 3 miles west of Derby, at the end of the road leading from Derby to the lake shore. It is about 10 metres back from the edge of the bluff. The height of station used was 36 feet. The geodetic point is marked by a stone post of the usual form, set 14 feet below the surface, with a stone post set directly over it for a surface-mark. Two stone reference-posts are set by the fences on the northeast and northwest of the station, as fol- lows: One bearing north 45° 52’ east, distant 32.75 metres; and one bearing north 75° 03’ west, 486 PRIMARY TRIANGULATION, [Crap. XVII, A, distant 5.5 metres from the geodetie point. The height of ground at the station above mean level of Lake Erie is 37.9 fect. SuGAR LOAF, 1876.— This station is situated on Sugar Loaf Hill, about 13 miles southwest of Port Colborne, Province of Ontario, about 600 metres northwesterly from Sugar Loaf Point, and 160 metres north from the lake shore. The height of station used was 50 feet. The geodetic point is marked by a stone post of the usual form, set 2 feet below the surface of the ground. Three stone reference-posts are set as follows: One, marked U.S. on top, on the east side of the town- line road between the towns of Wamfleet and Humberstone, projecting 8 inches above the surface, bearing north 0° 19’ east, distant 252.31 metres; one, marked with « cross on top and L. 8. on south face near top, projecting 9 inches above the surface, at the foot of the hill, near the corner of w field, bearing north 64° 04’ cast, distant 82.21 metres; and one, marked with a cross on top, projecting 6 inches above the surface on top of the hill, bearing south 43° 11/ east, distant 16.9 metres from the geodetic point. The Port Colborne light-house bears north 84° £9’ east. The height of ground at the station above mean level of Lake Erie is 141.2 feet. Font HILt, 1875.—This station is situated in lot 5 of the 7th concession, township of Pel- ham, county of Welland, Province of Ontario, on a hill about 1 mile northwest of the village of Font Hill. The height of station used was 474 feet. The geodetic point is marked by a stone post of the usual form, set 3 feet below the surface of the ground, with a stone post set directly over it for a surface-mark. Three stone reference-posts are set as follows: One bearing north 27° 18’ east, distant 303.91 metres; one bearing north 32° 32’ west, distant 330.75 metres; and one bearing north 89° 54’ west, distant 51.75 metres from the geodetic point. The height of ground at the sta- tion above the mean level of Lake Ontario from 1860 to 1875 is 618.6 feet. Hampura, 1875.—This station is situated in Hamburg Township, Erie County, New York, on the shore of Lake Erie, about 7 miles south of Buffalo, 2 miles northeast of Hamburg railway sta- tion on the Lake Shore and Michigan Southern Railway, and three-fourths of a mile northwest of Bay View railway station on the same railway. It is 295 metres west of the Bay View Hotel, a summer resort, and 15 metres back from the edge of the bluff. The height of station used was 5 feet. The geodetic point is marked by a stone post.of the usual form, set 28 inches below the surface of the ground. Three stone reference posts are set as follows: One bearing south 12° 14’ east, distant 18.13 metres; one bearing south 81° OS’ east, distaut 14.12 metres; and one bearing north 64° 32’ east, distant 42.5 metres from the geodetic point. The height of ground at the station above Lake Erie is about 20 feet; above mean level of Lake Ontario, 347.6 feet. MIDDLE BASE, 1875.— This station, near the middle of the Buffalo base-line, is situated in Tonawanda Township, Erie County, New York, on the east side of the track of the Erie Railway, about 1 metre from the fence. The height of station used was 3 feet. The geodetic pointis marked by a stone post 5 feet long, marked in the usual manner, set 345 feet below the surface of the ground, A stone reference-post, projecting 6 inches above the ground, is set on a line through the geodetic point, parallel to the railway, about 1.2 metres in a northerly direction from the point. The height of ground at the station above mean level of Lake Erie is 29.3 feet. West Base, 1875.— This station, marking the southwestern end of the Buffalo base line, is situated in Buffalo Township, Erie County, New York, about 5 miles nearly north of the Buffalo City Hall, on the north side of the Erie Railway, aud the west side of Delaware street. The height of station used was 87 feet. The geodetic point is marked by a small hole drilled in a piece of brass leaded into the top of a stone post, set 34 feet below the surface of the ground. Two stone posts are set on a line at right angles to the base-line through the geodetic point, at a depth below the surface of the ground about the same as that of the geodetic point, as follows: one bear- ing south 32° 49/ east, distant 3.68 metres ; and one bearing north 32° 49’ west, distant 3.71 metres from the geodetic point. Three stone reference-posts are set as follows: one at the intersection of the base-line and the town-line between Tonawanda and Buffalo on the west of Delaware street, bearing north 57° 11’ east, distant 605.02 metres ; one 1.8 metres east of the east side of Delaware street, and 8.02 metres south of the Erie Railway track, bearing south 84° 46’ east, distant 775.54 metres; and one in the edge of a piece of woods, bearing south 25° 36/ east, distant 88.23 metres from the geodetic point. The height of ground at the station above mean level of Lake Erie is 32.3 feet; above mean level of Lake Ontario, 358.4 feet. §2.] SANDUSKY BASE TO BUFFALO BASE. 487 East BASE, 1815.— This station, marking the northeastern end of the Buffalo base-line, is situated in the southwestern part of Amherst Township, Erie County, New York, about 8 miles northeast of the Buffalo City Hall, 45 miles southeast of Tonawanda, and about 12 miles in a northerly direction from the post-office of Eggertsville. The height of station used was 75 feet. The geodetic point is marked by a cross on a piece of brass leaded into a stone post set 3 feet below the surface. Two stone posts are set on a line at right angles to the base-line through the geodetic point, at a depth below the surface of the ground about the same as that of the geodetic point, as follows: one bearing north 32° 46’ west, distant 3.66 metres; and one bearing south 32° 46’ east, distant 3.86 metres from the geodetic point. Three stone reference-posts are set as follows: one oa the north side of the road south of the station, at a bend towards the east, bearing south 49° 10’ east, distant 105.0 metres; one on the north side of the same road, 65.0 metres northeast of the first, bearing south 77° 08’ east, distant 135.1 metres from the geodetic point; and one on the west side of the same road 60.96 metres south of the first, bearing south 29° 37’ east, and distant 148.8 metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 316.2 feet. BUFFALO PLAINS, 1875.—This station is situated in the northeast corver of Buffalo Township, Erie County, New York, on the grounds of the Erie County Poor House. The height of station used was 50 feet. The geodetic point is marked by a stone post of the usual form, set 2 feet below the surface of the ground. Three stone reference-posts are set as follows: one bearing south 75° 12’ west, distant 22.10 metres; one by the reservoir of the County House, bearing north 34° 17’ east, distant 127.56 metres; and one bearing south 82° 38 east, distant 28.19 metres from the geodetic point. The buildings of the County House are distant about 470 metres in a north- westerly direction. The height of ground at the station above mean level of Lake Ontario is 435 feet. BUFFALO, 1875.— The new City Hall tower was used as astation at this point. The City Hall is a large granite building on the square inclosed by Eagle, Franklin, Church, and Delaware streets in the city of Buffalo, New York. The geodetic point is marked by a cross cut in a piece of brass leaded into the granite shelf at the base of the pedestals of the statues on the top of the tower on the north side. It is 0.175 metres from the inner edge of the stone shelf, 0.744 metres from the outer edge, and 1.212 metres from the base of the pedestal of the statue on the northeast corner of the tower. Its height above the ground is 160.5 feet. The height of ground at the station above the mean level of Lake Ontario is 358.6 feet. RIDGEWAY, 1875.—This station is situated in an open cultivated field, just southwest of the intersection of two roads, on a slight rise of round about half a mile southwest of the village of Ridgeway, in the township of Bertie, county of Welland, Province of Ontario. The height of station used was 100 feet. The geodetic point is marked by a hole drilled in the top of a stoue post set 20 inches below the surface of the ground. A stone post is set directly over the geodetic point as a surface-mark. Three stone reference-posts are set along the highway fence east of the station, as follows: one bearing north 18° 14’ east, distant 24.1 metres; one bearing south 14° 14’ east, distant 33.4 metres; and one bearing south 6° 23’ east, distant 86.2 metres from the geodetic point. The fence north of the station is distant 27 metres, that east of the station 9 metres. The Methodist church, Ridgeway, bears north 47° 44’ east, and is distant about half a mile. The height of ground at the station above mean level of Lake Erie is 89.6 feet; above mean level of Lake Ontario, 416 feet. DRUMMONDVILLE, 1875.—This station stands on a sand-bank in the village of Drummond- ville, Province of Ontario, on the north side of Lundy’s Lane, about 20 metres east of a frame tower called the General Scott monument. The height of station used was 44 feet. The ground on which the station was built is not permanent, but the reference-stones are so set as not to be disturbed. The geodetic point is marked by a stone post of the usual form set 4 feet below the surface. A stone post is set directly over it as a surface-mark, and three reference-stones are set as follows; one in the southwest corner of the lot, distant 8.07 metres; one by the fence south of the station, distant 10.10 metres trom the geodetic point, and 10.3 metres from the first reference- stone; and one by the fence west of the station, distant 6.76 metres from the geodetic point, and 12.44 metres from the first reference-stone. The height of ground at the station above Lake Erie is 136.6 feet; above Lake Ontario, 462.8 feet. 488 PRIMARY TRIANGULATION. [Cuap. XVIII, B, C, TONAWANDA, 1875.—This station is situated in Tonawanda Township, Erie County, New York, about 2 miles south of the village of Tonawanda. The height of station used was 85 feet. The geodetic point is marked by a hole drilled in the top of a stone post set 2 feet below the surface of the ground. Three stone reference-posts are set as follows: One bearing north 69° 07/ east, distant, 25.2 metres; one bearing north 48° 43’ west, distant 10.65 metres; and one bearing south 74° 93/ west, distant 18.7 metres from the geodetic point. An azimuth post, used in 1875, bears north 55° 18’ east, and is distant 9.24 metres. The height of ground at the station above Lake Erie is 52 feet; above mean level of Lake Ontario, 378.1 feet. PEKIN, 1875.— This station is situated on the western border of Cambria Township, Niagara County, New York, in lot 25, about half a mile east of the village of Pekin, on the north side of the street, just east of a cemetery. The height of station used was 101 feet. The geodetic point is marked by a piece of iron leaded into a stone post of the usual form set 2.2 feet below the surface, with a stone post set directly over it for a surfave-mark. Three stone reference-posts are set as follows: one on the north side of the street and on the west boundary of the cemetery, bearing south 59° 22 west, distant 286.36 metres; one ou the same side of the street, and on a farm boundary, bearing south 2° 54’ west, distant 148.25 metres from the geodetic point, and 8.47 inetres distant from a stone in the middle of the street marking the corner of the farms; and one on the same farm boundary, bearing north 12° 28/ west, distant 22.58 metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 410.6 feet. FALKIRK, 1875.— This station is situated in Newstead Township, Krie County, New York, about one-fourth of a mile northeast of the village of Falkirk, 1 mile east of the village of Akron, and 80 metres south of the edge of a limestone blutt 60 feet or more in height. The height of station used was 75 feet. The geodetic point is marked by a nail leaded into the solid rock about 2 feet below the surface of the ground. Three stone reference-posts are set as follows: One bearing south 229 12’ west, distant 25.9 metres; one bearing south 36° 52/ east, distant 29.2 metres; and one bearing south 59° 04/ east, distant 44.75 metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 593.2 feet. B.—STATIONS, SIGNALS, INSTRUMENTS, AND METHODS OF OBSERVATION, § B. See Chapter XVI, B. C.—MEASURED AND ADJUSTED ANGLES BETWEEN THE LINES WILLOUGHBY- CHESTER AND FALKIRK-PEKIN. § 4. The following tables give an abstract of the adjustment of the triangulation comprised within the above-stated limits. A sketch to accompany it is given in Plate V. The adjustment is nade in two sections, namely: Section XI, extending from Willoughby —-—Chester to Grand River— Westfield. Section XI, extending from Grand River— Westfield to Falkirk —Pekin. Weight unity was assigned to the following number of combined results for the respective instruments used in measuring the angles: Troughton & Simms No. 1, 16. Troughton & Simms No, 2, 24. Troughton & Simms No. 3, 24. Troughton & Simms No. 4, 24. Pistor & Martins No. 2, 24, at Drummondville and Buffalo. Pistor & Martins No. 2, 20, at Sugar Loaf. Gambey No. 1, 16. Repsold No. 1, 20. For a detailed explanation of the tables, see Chapter XIV, C, § 7, and see the remark in Chapter XV, ©, § 6, relating to the column headed “No. meas.” At each of the above-named stations marking the lines of division of the adjustment, a sum- angle condition was disregarded in deriving the general corrections to the angles. The locally- adjusted angles, with their resulting weights, at these stations, were used in computing the general corrections. §§ 3, 4.] SEcTION XI.—Triangulation from the line Grand River — Westfield to the line Willoughby — Chester. GRAND RIVER—44. (Observer, R.S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, August, 1876,] SANDUSKY BASE TO BUFFALO BASE. 489 Angle as measured between— Notation. No. meas. Range. wt. (v) (v] Corrected angle. ou Westfield and Long Point. ......--..- “" 52 26 09. 982 44a 14 uu 7.2 “ —0. 201 u --0. 011 o “ 52 26 09.770 Note.—The weight and local correction of 444 are taken from Section XIT of the adjustment. WESTFIELD—45. [Observers, G. Y. Wisner and R. S. Woodward. Instruments, Troughton & Simms theodolites Nos. 1 and 3, Dates, June and October, 1876.] Angle as measured between— Notation. | No. meas. |Range. | Wt. (v) {v] Corrected angles. Oo # a “ a a“ ao Ff uw Erie and Long Point..........---..-. 64 51 01.072 451 14 57 1 —0.170 | 40.011 64 51 00. 913 Long Point and Grand River ......- 52 02 06. 509 452 16 4.8 1 —0. 171 --0. 009 52 02 06. 329 * NotTes.—The weights and local corrections of 45; and 452 are taken from Section XII of the adjustment. 451 and 452 were partly measured by Mr. Wisner with the Troughton & Simms No. 1, in June, 1876, and partly by Mr. Woodward with the Troughton & Simms No. 3, in October, 1876. (Observer, J. H. Darling. LONG POINT—46. Instrument, Troughton & Simms 14-inch theodolite No. 4. Date, August, 1876.] | Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) (v) Corrected angles. | of “ a“ “a au ° f a ; Grand River and Westfield ......--. 75 31 50. 868 461 16 7.6 | 0.67} —0.117 | —0. 204 75 31 50. 547 Westfield and Frie..........-..-..-- 57 28 15. 676 462 28 7.4 1 —0,078 | —0.108 57 28 15. 490 Erie and Houghton .........---..--- 88 43 44. 666 463 24 7.9 1 —0.079 | +0, 435 88 43 45, 022 138 16 09. 142 464 26 7.3 1 —0.078 | —0.123 | 138 16 08.941 | Houghton and Grand River. .....-.- NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 67(461)+ (462)-+ (463)-} 0. 352=0 (461) +2(462)-+ (463)-+0. 352=0 (461) + (462)-+2(46s) +0. 352=0 ERIE—47, (Observer,"G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Dates, September and October, 1876.] Corrected angles. Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] or “ “ 4 u °o ‘ a Edinboro and Conneaut...---------- 64 35 04.513 411 16 2.2 1 —0.190 | —0.110 64 35 04. 213 Conneaut and Houghton..--.------- 84 19 49, 098 472 17 2.1 1 | —0.190 | +0.260] 84 19 49.168 Houghton and Long Point..-..--.--- 37 51 30. 347 473 18 3.7 1 —0.190 | +0.316 37 51 30.473 Long Point and Westfield.-....---- - 57 40 50, 245 474 18 4.5 1 —0.190 | —0. 226 57 40 49. 829 Westfield and Edinboro. . 115 32 46.747 475 17 2.9 1 —0.190 | —0.240 | 115 32 46.317 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(47,)+ (472) + (473)+ (474) +0. 950=0 (47,)+2(472) + (473)+ (47,) +0. 950=0 (47,)-+ (472)-+2(47,)-+ (474) +0. 950220 (47,)-+ (47a). (472) +2(744) +0. 950=0 62 LS 490 PRIMARY TRIANGULATION. [Cuap. XVIII, C, SECTION XI.—Triangulation from the line Grand River — Westfield to the line Willoughby — Chester— Continued. HOUGHTON—48, {Observers, A. R. Flint and G. Y. Wisner. Instruments, Repsold theodolite No. 1, and Troughton & Simms theodolite No. 1. Dates, August and September, 1876, and May, 1877.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. o£ a“ wm aw “ o £ mw Long Point and Erie............-.-- 53 24 50. 557 481 26 5.2 | 1.25 | —0.025 |} +0. 023 53 24 50, 555 Long Point and Edinboro - 60 24 52. 304 481+2 16 4.7 |1 +0.474 | +40. 250 60 24 53. 028 Edinboro and Long Point. .. -- 299 35 07. 699 481-2 16 6.3 | 1 —0.477 | —0. 250 299 35 06. 972 Erie and Edinboro............2..... 7 00 02. 617 482 18 4.5 [1 —0.371 | +0.277 7 00 02. 473 Edinboro and Erie............. ----. 852 59 57.175 48_2 18 4.2 [1 +0.579 | —0. 227 352 59 57. 527 Erie and Conneaut. ................. 31 08 13. 648 48243 16 4.0 /1 +0.045 | +0.300 "81 08 13, 993 Conneaut and Erie.................- 328 51 46. 263 482s 15 3.4 71 +0.044 | —0,300 328 51 46. 007 Erie and Long Point..........--.-.. 306 35 08.732 4824344 26 7.5 | 1.25 | +0.736 | —0. 023 306 35 09. 445 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4. 5(481)-+-2(482) +0. 854=0 2(481)-+-4(482) +11. 535=0 2(48243)—0. 089-=0 Norer.—48e+3 and 48—2—3 were measured by G. Y. Wisner with the Troughton & Simms instrument in 1877. The remainder were read by Mr. Flint with the Repsold instrument in 1876. EDINBORO—49. (Observer, A. R. Flint. Instrument, Repsold theodolite No.1. Dates, September and October, 1876.] * Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. ao ¢ “ uw a : “ Qo # “ Andover and Conneaut ............. 42 42 44. 255 49) 28 7.3 1 —0.136 | —0.110 42 42 44, 009 Conneaut and Houghton............ 60 04 52. 940 49, 22 V1 1 +0.128 | +0. 417 60 04 53. 485 Conneaut and Erie...........--..-.. 84 09 59. 681 49043 28 5.5 1 —0. 263 | +0.104 84 09 59, 522 Houghton and Erie .............-..- 24 05 06. 222 493 25 3.7 1 +0.128 | —0.313 24 05 06. 037 Erie and Andover .......--.--.--.-- 233 07 16, 599 494 28 8.0 1 —0.136 | +0. 006 233 07 16. 469 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(49,)+- (492)-+ (492)-++0. 016=0 (49,)-+3(492)-+-2(493) —0. 503=0 (49,)-+2(492)-+3(49,)—0. 503=0 CONNEAUT—50. {Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Dates, August and September, 1876, and May, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. Oo 7 “we | aw “a aw ° ‘ “ Houghton and Erie.-....-.---.----- 64 32 03. 469 50) 16 3.8 1 —0.107 | +0.473 64 32 03. 836 Erie and Houghton .........--...--- 295 27 56.744 50-1 16 2.2 1 —0.106 | —0.473 295 27 56. 165 Thompson and Erie. --........------ 185 49 12. 366 50146 16 6.2 1 -+0.070 | —0. 187 185 49 12, 249 Erie and Edinboro. ....-...-...----- 31 14 58. 049 502 16 2.3 1 +0. 069 | +0. 043 31 14 58,161 Edinboro and Andover .....-....--- 85 07 58. 672 503 17 7.0 1 +0.070 | —0. 303 85 07 58.439 Andover and Thompson ............ 57 47 50. 635 50a 17 5.4 i +0. 069 | +0. 447 57 47 51.151 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(501) +0. 213=0 2(502)+ (503)+ (504)—0. 278=0 (502)-+2(503)+ (504)—0. 278=0 (502)+ (503) -+2(504)—0. 278=0 NoTE.—50; and 50—1 were measured in 1877, the remainder in 1876. § 4.) SANDUSKY BASE TO BUFFALO BASE. 491 SEcTION XI.—Triangulation from the line Grand River — Westfield to the line Willoughby — Chester— Continued. ANDOVER—51. (Observer, A. R. Flint. Instrument, Repsold theodolite No. 1. Dates, October and November, 1876.] Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) [v] Corrected angles. oe a“ aw au aw ° ‘ “ Mesopotamia and Thompson. .....-. 41 49 59, 265 5lit+e2 24 6.1 1 +0.144 | —0. 034 41 49 59. 375 Thompson and Mesopotamia. . . 318 10 00. 448 51—1—2 24 7.3 1 +0.143 | +0. 034 318 10 00. 625 Edinboro and Claridon -.. - 203 15 31.780 Slits 21 71 1 +0.404 | —0.151 203 15 32. 033 Claridon and Thompson .. -- 27 33 58.518 52 24 4.3 1 +0.403 | —0. 065 27 33 58. 856 Thompson and Conneaut -.- ---- 77 01 07.833 513 24 6.5 1 +0. 403 | +0. 483 77 01 08.719 Conneaut and Edinboro .....-...... 52 09 20. 256 5la 26 4.7 1 +0.403 | —0. 267 52 09 20. 392 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(51142) —0. 287=0 2(51g)+ (513)+ (514)—1. 613=0 (512)-+2(51a)-+ (51a)—1. 613=0 (Slg)+ (513)-+2(514)—1. 613=0 THOMPSON—82. (Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Date, November, 1876.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. °o ¥ “a wn uw aw oO - uw Conneaut and Andover ........----- 45 11 02.049 521 27 6.3 1 +0.141 | +0. 787 45 11 02. 977 Andover and Mesopotamia .-....... 70 00 41. 285 522 28 6.6 1 +0,141 | —0.104 70 00 41.322 Mesopotamia and Claridon.......-.. 25 50 49. 951 523 28 8.1 1 +0. 141 —0. 228 25 50 49.864 Claridon and Little Mountain-....-. 57 19 50. 582 524 28 6.2 1 +0.141 | —0. 607 57 19 50.116 Little Mountain and Conneaut...... 167 37 35. 427 526 26 6.8 1 +0.142 | +0. 152 167 37 35. 721 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(52;)-+ (52_)-+ (523)-+ (524)—0. 706=0 (52,) +-2(522)-+ (523)-+ (524) —0. 706=0 (52) (529)-+2(52)-+ (524) —0. 706=0 (52,)+ (522)-+ (523) +2(524)—0. 706=0 MESOPOTAMIA—53. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, May, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. °o Ly a“ ao “ a“ Oo x g. AF Claridon and Thompson .-.-....----- 47 45 15.101 53) 16 3.2 1 +0. 060 —0. 231 47 45 14.930 Thompson and Andover -- 68 09 21. 256 532 16 3.0 1 +0.060 | +0.341 68 09 21. 657 ‘Andover and Claridon .....--..----- 244 05 23, 462 533 16 2.7 1 +0. 061 | —0.110 244 05 23, 413 & NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(53,)+ (532)—0. 181=0 (53;) 4+-2(532) —0. 181=0 492 PRIMARY TRIANGULATION. [Cuar. XVII, 6, SECTION XI.—Triangulation from the line Grand River — Westfield to the line Willoughby — Chester— Continued. CLARIDON—54, (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1. Date, June, 1877.] Angle as measured bet ween— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. ° £ “uw uw “we a“ oO t uw Chester and Little Mountain. ....... 40 25 02. 053 54 17 4.8 1 +0.155 | +0.368 40 25 02. 596 Little Mountain and Thompson. .... 59 41 45, 047 542 17 3.5 1 +0.155 | —0,512 AQ 41 44, 690 Thompson and Andover ...--.----.- 56 34 32,185 543 16 4.3 1 +0.155 | —0. 460 56 34 31. 880 Andover and Mesopotamia ......... 49 49 23. 439 54g 16 6.4 1 +0.155 | +0. 336 49 49 23. 930 Mesopotamia and Chester .......-.- 153 29 16. 500 5465 17 6.5 1 +0.156 | +0, 248 153 29 16. 904 NORMAL EQUATIONS FOR LOCAL ADJ USLMENT. 2(541)+ (542)+- (543) (544) —0. 776=0 (541) 4-2(542)-+4+ (543)4+ (544)—0. 776=0 ° (541)+4+ (542)4-2(543)+ (544)—0. 776=0 (541)-+ (542)4- (543) -+2(544)—0. 776=0 LITTLE MOUNTAIN—55. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, June, 1877.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | ‘[v) Corrected angles.! ' : are 1 | eat ‘ fo} t a Ww aw “a ! oO ‘ at . Thompson and Claridon ...... -.--- 62 58 26. 422 55, 16 2.2 1 4-0.059 | —0. 414 62 58 26. 067 Claridon and Chester..-..-..------- 71 41 06. 338 559 16 4.2 1 +0.059 | +0. 487 71 41 06. 884 | Chester and Willoughby....-.....-. 75 50 29.709 553 16 4.6 1 +0.059 | —0.418 | 75 50 29. 350 Willoughby and Thompson ......... 149 29 57.294 554 16 2.9 1 | 40.060] +0.345 149 29 57. 699 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(551)+ (552) (55s) 0. 2370 (551) +2(552)+ (553) —0. 237=0 (551)+ (552) +2(553)—0. 2370 CHESTER—56. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. L Date, June, 1877.] | Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) [v} Corrected angles. | ° t uo | “we “uw “wn ° I “we Warrensville and Willoughby ..-.-. 92 12 11. 653 561 18 3.5 1 405052: |cceninecuicelaesceeeeetes eeeeels Willoughby and Little Mountain... 54 01 00.333 562 18 5.8 1 +0.052 | —0.573 54 00 59. 812 Little Mountain and Claridon 67 53 50. 998 563 18 4.4 1 +0.052 | +0. 106 67 53 51.156 Claridon and Warrensville.......... 145 52 56. 808 564 18 3.2 1 A02052 jo anciccieecie|isccceeciiececseece NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(561)+ (562)4+ (563) —0. 208=0 (561) +2(562)4+ (562)—0. 208=0 (561)-+ (562) +2(563) —0. 208=0 § 4] SANDUSKY BASE TO BUFFALO BASE. 493 SECTION XI.—Triangulation from the line Grand River — Westfield to the line Willoughby — Chester— Continued. WILLOUGHBY—57. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No.1. Date, June, 1877.] | Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) [v] Corrected angle. or wo ws “a uo oO f Ww | Little Mountain and Chester -...... 50 08 31.940 571 20 | 5.6 1.25 | —0.156 | —0. 467 50 08 31.317 | Chester and Warrensville ........- 38 10 03.094! 572 20 eT 0 |) eb: || 01 ab seek racial ae we weehene eon | Warrensville and Rockport......... 37 51 09. 507 573 21 4.3 T2650] OBE: | yececcece'sa [cian ae nediacrigeiers | Rockport and Little Mountain...... 233 50 16.120. 574 16 3.7 [1 090 eek Sa eaten } | soe NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2.25(571)-+ (B72), (573)-+0. 661=0 (571) +2. 25(572)+ (574) -+ 0. 661=0 (571)+ — (572)-++2. 25(573)-+ 0. 661=0 Numerical equations of condition in the triangulation from the line Grand River — Westfield to the line Willoughby — Chester. SIDE-EQUATIONS. VIL. (30) +34, 9240 [47,] +37. 0144 [47,] —12. 1161 [49,] +47. 1023 [49,] —12. 1601 [50,] — 2. 1327 [50,.] +19. 819=0 XIII. (15) — 9.8192 [52.] — 2. 1605 [52,] +10. 22x5 [53,] 4-18. 6688 [53,] —13. 8962 [543] +17. 7784 [54,] —17. 878=C Notr.—In the solution for determining the general corrections, each of the side-equations was divided by the nuinber inclused in parenthesis and placed opposite it. ANGLE-EQUATIONS. I. [444] + [452] + [46,] +0, 224=0 I. (45) + [46] + [47] -40. 322=0 TI. = (463) - + [47s] + [481] —0.774=0 IV. [472] + [48243] + [501] -1. 033=0 Ve. — [48] + [4848] + [49] + [50.] + [502] —1. 005=0 1 1472) +12) e+ —0.037=0 VU. [4%] + [503] + [51y] -40. 679=0 IX. [504] + [51s] + [52] —1.717=0 X. [5li4e] + [52] + [532] —0. 2020 XI [525] + (53] + [545] + [544] -40, 582=0 XI = [5le] ++ [52] + [52s] + [543] -40. 857=0 XIV. [52] + [54] + [55] +1. 532=0 XV. [54] +552] + [56s] —0, 981=0 XVL [555] + [562] +57] +1. 458=0 General corrections in terms of the correlates. (44,] =+0. 83333 I : (45,] =+0.73333 II [452] =-40. 73333 I [46.] =+1.000001 —0.33333 II —0, 33333 IIL (46.] =—0.33333I -+10.77778 11 —0, 22222 III [465] =—0.33333I —0,22222II -+0.77778 III [47,] =—0.200001IT —0.20000 IIL --0.20000IV -+0.80000 VI +40. 68454 VII [472] =—0.200001I —0.20000 III +0.80000IV —0.20000 VI 0.75422 VII [475] =—0.200001I -+0.80000 III —0.200001IV —0.20000 VI —0. 47959 VII [474] =+0.8000011 —0.20000 IIL —0.20000IV —0,20000 VI —0.47959 VII [48:] =+0.28571 III -+0. 14286 V : [482] =—0.14286 III —0.32143 V [48:43] =+0.50000IV -+0.50000 V ; [49,] =—0.12500V 0.25000 VI —0.14578 VII -+0. 62500 VIII [494] =-+0.62500V -+0.25000 VI —0.84121 VII —0. 12500 VIII 494 PRIMARY TRIANGULATION. General corrections in terms of the correlates—Continued. (Cuap. XVIII, C, [49] =—0.37500 VV -++0.25000 VI. +1. 13276 VIL —0. 12500 VIII [50,] =+0.500001V -+-0,50000 V —0, 20267 VII (50;]) =+0.75000 V—- -+£0..75000 VI. —0, 05332 VII —0. 25000 VIII —0. 25000 IX [503] =—0.25000V = —0. 25000 VI -+-0.01777 VIL +0.75000 VIII —0. 25000 IX ¥ [50,] =—0.25000 VV = —0, 25000 VI. +0.01777 VII —0, 25000 VIII -+-0. 75000 IX [51142] =-++0. 50000 X° [512] =—0. 25000 VILL —0. 25000 IX -+-0. 75000 XII [51,] =— 0.25000 VII --0, 75000 TX —0, 25000 XII [514] =-+0. 75000 VILE —0, 25000 LX —0, 25000 XII (52.} =+0.80000 1X —0,20000 X —0, 20000 XI —0. 40000 XII +0. 15973 XIII —0. 20000 XIV [52:] =—0.200001X +0.80000X —- 0, 20000 XI. +0. 60000 XII —0. 49488 XIII —90. 20000 XIV [523] =—0.200001X —0,20000X +0.80000 XI -+0,60000 XII +-0.01570 XIII —0. 20000 XIV [52,4] =—0.200001IX —0,20000X 0.20000 XI —0,46000 XII -++0. 15973 XIII +-0. 80000 XIV [53,] =—0.33333 X +40. 66667 XI_ +0. 03974 XIII [53] =-++0. 66667 X —0.33333 XI -10. 60243 XIII : [54,] =—0.40000 XI —0. 20000 XII —0, 05176 XIII —0. 20000 XIV -0. 80000 XV [542] =—0.40000 XI —0. 20000 XII —0. 05176 XIII +0. 80000 XIV —0, 20000 XV [543] =-++0.60000 XI +0. 80000 XII —0. 97817 XIII —0. 20000 XIV —0. 20000 XV [544] =-++0.60000 XI —0. 20000 XII +1. 13347 XIII —0. 20000 XIV —0. 20000 XV (55,] =-+0.75000 XIV —0, 25000 XV —0,25000 XVI [Af2] = -0.25000 XIV -+40.75000 XV —0. 25000 XVI [553] =—0. 25000 XIV —0. 25000 XV +0, 75000 XVI [5t,] =++0.75000 XVI - [563] =--0. 75000 XV (57,] =+0.61176 XVI Normal equations for determining the correlates. sanction 1. 0==+0, 22400 +2.566671 —0,33333 1 —0. 33333 IIT 2. 0=-++0, 32200 —0. 333331 +2. 3111112 —0, 42222 TT =—0.20000 IV. —0.20000 VI —0. 47959 VII 3. O=—0,77400 —0, 333331 —0.42222 11 +41.86349 IIT —0.20000 IV +0.14286 V = —0. 20000 VI —0. 47959 VII 4, 0=—1.03300 —0.20000 11 —0.20000 IIT -+1.80000IV +1.00000V —0.20000 VI +0.55155 VII 5. O=—1. 00500 +0.14286 III +1.00000IV +2.69643 VV +1.00000 VI —1. 09720 VII —0.37500 VIII —0. 25000 IX 6. 0=—0. 03700 —0.20000 IT —0.20000 IIE —0.20000TV +41.00000V +42.05000 VI +0. 92277 VII —0. 50000 VIII —0. 25000 LX 7. 0=+0. 66063 —0.47959 IIT —0.47959 TIT +0.55155IV —1.09720 V4.0. 92277 VI. -++3. 93167 VII —0. 12800 VIIL -+-0. 01777 IX 8. 0=+0. 67900 —0.37500 V =—0,50000 VI —0. 12800 VIL +2. 12500 VIIL —0.50000 IX 0.25000 XII 9. 0=—1.71700 —0.25000 V = —0.25000 VI +0. 01777 VIL —0.50000 VIII +2.30000 IX —0. 20000 X —0. 20000 XI —0.65000 XII -+-0, 15973 XIII —0. 20000 XIV 10. 0=—0, 20200 —0.20000 IX -++1.96667 X —0.53333 XI -++0.60000 XIZ_ +0. 10755 XIII —0. 20000 XIV 11. 0=-10, 58200 —0, 20000 1X —0.53333X 4-2, 66667 XI +1. 20000 XII_ +0. 21074 XIII —0, 60000 XIV —0. 40000 XV 12. 0=-+-0. 85700 —0. 25000 VIII —0.65000 1X -++0.60000X +1.20000XI +2.75000 XII —1. 45735 XIII —0. 60000 XIV —0. 20000 XV 13. 0=—1.19187 +-0.15973 IX +-0.10755 X= +0. 21074 XI. —1. 45735 XII +3. 34818 XIII +0. 10796 XIV —0. 05176 XV 14, 0=+1.53200 —0.20000 IX —0.20000X —0.60000 XI —0,60000 XII -+0. 10796 XIII +2. 35000 XIV —0, 45000 XV —0. 25000 XVI 15. 0=-0. 98100 —0. 40000 XI —0.20000 XII —0. 05176 XIII —0. 45000 XIV +2.30000 XV —0.25000 XVI 16. 0=+1. 45800 —0.25000 XIV —0.25000 XV +2.11176 XVI § 4.] SANDUSKY BASE TO BUFFALO BASE. Values of the correlates and their logarithms. I =—0. 0128 log 8, 1082267— II =+0.0155 log 8, 19061184 III =-40. 5577 log 9. 7464006 4 IV =+1.5544 log 0, 19156004 V =—0.9548 log 9. 9798942_ VI =+1. 1246 log 0, 05101354. VIL =—0. 8532 log 9.9310458_ VIII =—0.1154 log 9. 0621305_ IX =+0. 6345 log 9. 80245904. X =—0. 0689 log 8. 8384083_ XI =—0, 4056 log 9. 6080765— XII =-10. 0858 log 8. 93374034 XIII =+0. 4175 log 9. 62062534 XIV =—0.7587 log 9. 8800873 XV =+0,. 1414 log 9, 1504801 + XVI =—0. 7635 log 9. 8828106— 495 Values of the general corrections. “ a“ a“ “ [444] =—0.011 | [48,] =+0.023 | [51i42]=—0.034 | [54] =+0.388 (45:] . =+40.011 | [4%] =+0.227 | [512] =—0.065 | [54] =—0.512 [452] =—0.009 | [48%43]=+0.300 | [513] =+0.483 | [54] =—0. 460 (46.] =—0.204 | [49,]) =—0.110 | [514] =—0.267 | [54,] =+0.336 [46] =—0.108 | [49] .=+0.417 | [52] =+0.787 | [55,] =—0.414 [46s] =+0.435 | [49,] =—0.313 | [52] =—0.104 | [55.] =-+0. 487 [47] =-0.110 | [50] =+0.473 | [52,5] =—9.298 | [55,] =—0.418 [47] =+0.260 | [50,) =+0.043 | [52,] =-—0.607 | [56] =—0.573 [47%] =+0.316 | [503] =—0.302 9 [oh] =—O.231 5 [56,] =-+10. 106 [474]. =—0.226 | [50,) =-+0,447 | [532] =+0.341 | [57] =—0.467 Residuals resulting from substitution of general corrections in numerical equations of condition. See. Residual. Sle Residual. 1 0. 00000 9 +0. 00001 2 0. 00000 10 0. 00000 3 0. 00000 11 0. 00000 4 0. 00000 12 0. 00000 5 0. 00000 13 +0. 00003 6 0. 00000 14 0. 00000 7 0. 00000 15 0. 00000 8 0. 00000 16 0. 00000 SEcTION XII.—Triangulation from the line Falkirk - Pekin to the line Grand River -— Westfield. FALKIRK—29, (Observer, T. Russell. Instrument, Gambey theodolite No.1. Dates, July and August, 1875.] Angle as measured between— Notation. | No. meas. | Range.} Wt. (v) [v] Corrected angles. a + “ “uo a “a Oo , uo Tonawanda and Pekin..........-..-- 81 23 10. 691 291 17 3.1 1 +0.082 | --0.457 31 23 11. 230 Notz.—The weight and local correction of 29: are taken from Section XIII of the adjustment. 496 PRIMARY TRIANGULATION. [Cuap, XVIII, C, SECTION XII.—Triangulation from the line Fulkirk— Pekin to the line Grand River — Westfield— Continued. PEKIN—30. (Observer, G. ¥. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Dates, July and August, 1875.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | (v] Corrected angles. G. ¥ “ ww | “ “a o # aw Falkirk and Tonawanda.........---- 66 25 49. 294 302 18 5.9 1 +0.065 | +0. 471 66 25 49, 830 Tonawanda and Drummondville . .--.. 59 58 19. 872 303 18 5.6 1 -+0.065 | -- 0,121 59 58 20. 058 Norr.—The weights and local corrections of 302 and 303 are taken from Section XIII of the adjustment. |Observers, T. Russell and G. Y. Wisner. TONAWANDA—31. August, September, and October, 1875.] Instruments, Gambey theodolite No. 1, and Troughton & Simms 14-inch theodolite No.1. Dates, Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) {v] Corrected angles. oO / a “ aw at o 4 “wn Buffalo and Ridgeway ......-....--- 49 07 09.914 | 31) 16 2 43 1 +0. 202 | +0. 963 49 07 11.079 Ridgeway and Drummondville...... 75 12 24.354 | 3le 16 2.6 ne --0.019 | ---0. 833 75 12 23.540 Ridgeway and Falkirk -.....-...--. 220 32 32.861 | 31lo43+44 16 5.4 1 —0.507 | +0.019 220 32 32. 873 Falkirk and Ridgeway.......--.---. 139 27 28.337 | 31—z—s—a 16 3.0 1 —0.691 | —0.019 189 27 27. 627 Drummondville and Pekin.......... 63 09 08.104 | 313 | 16 2.9 1 -L0. 018 +0. 198 | 63 09 08. 315 Pekin and Falkirk.............-..-- 82 10 59,840 | Bla 17 2.5 1 +0.019 | -+ 0. 659 82 11 00.518 Falkirk and East Base -...--.-..... 16 06 31.787 | 316 16 5.2 1 +0. 202 | -+0. 067 16 06 32. 056 East Base and Buffalo Plains ....... 82 36 50.065 | 31¢ 28 4.8 2 +0.101 | +0. 348 32 36 50. 514 Buttalo Plains and West Base ...... 36 04 44.909 | 317 16 4.6 1 +0. 202 | —1. 027 36 04 44, 084 West Base and Buffalo ...........-. 5 82 10. 063 31s 16 4.8 1 +0. 201 —0.370 5 32 09. 894 NORMAL EQUATIONS FOL LOCAL ADJUSTMENT. 2(311)+ (312)+ (313)+ (314)4+ (815)+ (316)+ (817)—0. 964=0 (311) +4 (312) 4-3(31g) 4+-3(814)+ (318)+ (Ble)+ (817) —0. 892=0 (311) +3(312) +4 (31g) +-3(314)-+ (315) + (816)+ (317) —0. 892=0 (311) +3(312)+-3(313)+4(31a)+ (816)+ (316)+ (317)—0. 892=0 (311)-+ (812)4- (813)+ (314)-+-2(315)+ (316) (317)—0, 964=0 (3li)+ (Bl2)+ (81z)+ (814)4 (815)+3(316)+ (317)—0. 964=0 (811)-+ (812)+ (313)-+ (814)-++ (816)4- (316) +4+2(317)—0. 964=0 NOTE.—3hy, 3124344, 31s, part of 316, 317, and 31s, were read by Mr. Wisner with the Troughton & Simms instrument, in September, 1875 The remainder were read by Mr. Russell with the Gambey instrument, in August and October, 1875. DRUMMONDVILLE—32. (Observer, W. A. Metcalf. Instrument, Pistor & Martins_14-inch theodolite No. 2. Dates, August and September, 1875.] Angle as measured between — Notation. | No. meas. | Range.) Wt. (v) [v] Corrected angles. oO # “ “ “ a“ ° % w Pekin and Tonawanda. ........-.--. 56 52 32. 523 321 24 3.9 1 +0.041 | —0.103 56 52 32. 461 ‘Tonawanda and Bultalo. ......--..-. 21 48 17.006 322 16 6.0 1 —0.059 | +0. 541 21 48 17. 488 Tonawanda and Ridgeway..-....--. 52 38 14. 004 32243 10 3.3 0.5 | +0.200 | +0. 597 52 38 14. 801 Buttalo and Ridgeway ...-........-. 30 49 57. 316 323 22 5.2 1 —0.059 | +0. 056 30 49 57.318 Ridgeway and Sugar Loaf .......... 39 16 00. 424 324 24 6.4 1 +0.042 | —0.531 39 15 59, 935 Sugar Loaf and Font Gill...-..-..-. 43 43 09.498 32, 24 4.4 1 +0.041 | +0. 302 43 43 09. 841 Font Hill and Pekin ................ 167 30 03.185 326 17 3.0 1 +0. 042 | —0.265 | 167 30 02.962 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(821) + (32) + — (323)-+ (324)-+ (825) —0. 048=0 (321) +2. 5(82p) +1. 5(32s)-+ (324)-+ (32,)-+0. 1110 (32;) 4-1. 5(325) +2. 5(324)-+ (324) + (325) +0. 111—0 (321)+ — (322)+ — (32g)+2(324)+ (325)—0. 048=0 (32;)+ —(82g)+ (323) (824) +2(325) —0. 048=0 § 4.) SANDUSKY BASE TO BUFFALO BASE. AQT Section XIII.—Triangulation from the line Falkirk— Pekin to the line Grand River — Westfield— Continued. RIDGEWAY—33. [Observer, J. H. Darliug.. Instrument, Repsold theodolite No.1. Dates, August, September, and October, 1875. | (v) [v] Coireeted angles. Angle as measured bet ween— | Notation. _ No. meas. | acazes| Wt. | | 4 ese oO , uw ' uw “we wn . ° a wt Drummondville and Tonawanda... 52 (9 22. 166 | 33, 22 : Se * Fe | —0.089 | 0.523 52:09 22. 600 Tonawanda and West Base ]4 00 57. 972 | 33, 24 61 121 | » —0.089 | +40. 26) 14 00 58. 144 West Base and Buffalo ........---.. 27 27 02. 939 | 333 a4 57 et | —0.088 | —1.096 | 27 27 01.755 | Buffalo and Hamburg....-..-....--- 36 17 07. 977 | 334 40 | 7.6 1.5 | —0.059 | +0. 058 _ 86-17 07.976 Hamburg and Sturgeon Point....... 5438 21.840 | 335 24 \ G4 1» —0.089 | 40.422 54 38 22.173 Sturgeon Point and Silver Creek.... 21 31 10. 234 | 336 : 26 fA OD, | —0. 088 +-0. 205 21 31 10. 351 Silver Creek and Sugar Loaf........ 70 07 44, 262 337 26 4.7 | 1 | —0.089 | —0. 169 70 07 44.004 | Sugar Loaf and Font Hill........... 4439-44.577 33g u 57 | 1 | —0.u89] —0.845 | 44.39 43.613 | Font Hill and Drummondville .-.... 39 08 28. 801 ; | 5.6 1 —0.088 | 40. 641 | 39 08 29. 354 | 339 ' 24 | 3 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(831) + (382)+ (333) + — (33a) + (335)4+ (336) (337)-+ (338) 4-0 768=0 (331) +2(332) + (333)+ — (334)+- (335) 4- (336)-+ (337)-+ (33a)-+.0. 768=0 (381) -- (882) 4+2(333)-+ — (334) 4+ (335) + (336)-+ (337)-+ (33a)-+-0. 768=0 (331) + (332)+ (333) +2. 5(334)+ (335) + (33s) (33) 4- (33a) +-0. 768=0 (331)+ (382)+ (333)+ — (384) 4+-2(33s)-+ (336)4- (337) (33a) -+-0. 7683=0 (33;)+ (832)-+ (333)4+ — (334) + (335)-+-2(836)4- (337)-|- (338) +0. 768=0 (331) + (332)+ (333)4 — (334)4+ (335)-+ (336) +2(337)-|- (338) +0. 768 =0 (331) + (332) + (833)+ — (334)+ (335) + (386) + (337) +2(33a) +0. 768=0 BUFFALO—34. [Observers, Ww. A. Metcalf and J. H. Darling. Instruments, Pistor & Martius 14-inch theodolite No. 2, and Repsold theodolite No.1. Dates, September, October, and November, 1875.] 1 1 Angle as measured between— | Notation. | No. meas. | Range.) Wt. | (v) [v} lGiomentutl angles. vee ! i : ; or e “ | | | ” | " “u | orn | Hamburg and Sturgeon Point....-.. 40 46 37.878 | 341 24 | 5.8 1 +0. 067 | —0. 871 40 46 37.069 | Hamburg and Ridgeway .... -..--- 94 27 25, 537 | res 24 , &2 9 1 | 0.178 | +0, 229 94 27 25, 588 | Ridgeway and Hamburg .........-. 265 32 34.751 | 341-2 | 4 | 24) 41 | _o.110 | —0. 229 | 265 32 34.412 Sturgeon Point and Ridgeway ..-.-- 53 40 47. 352 | 342 24 | 6.8 | 1 | +0. 067 | +1. 100 | 53 40 48.519 Ridgeway and Drummondville -..-. 55 32 41.027 | 34, | 24 | 3.8) 1 40.188 | —0.128 | 55 82 41. 087 Drummondville and Ridgeway ..... 304 27 18.598 | od—s | 24 | 3.6 1 +0. 187 | +0. 128 304 27 18.913 | Drummondville and Tonawanda.... 33 52 09.146 | 344 24 3.1 | 1 +0. 062 | —0. 782 \ 33 52 08. 426 Buffalo Plains and Drummondville. . 287 51 43. 622 | 34—4 -»-s—7 24 6.4 | J +0. 062 | +1.160 ' 287 51 44. 844 Tonawanda and West Base..-.....- 3 33 27.755 | 34, 25 5.2 1 +0. 062 | -+ 0.136 | 3 33 27.953 West Base and East Base...... ---- 26 19 44. 087 | 346 24 4.5 1 +0. 062 | +0. 511 26 19 44. 660 East Base and Buffalo Plains.......-. 8 22 55. 080 | 347 25 84) 1 +0. 062 | —1. 025 8 22 54.117 + NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(341) +2(342) —0. 336=0 2(841) +3(34,) —0. 3386=0 ‘ 2(343) —0, 375=0 2(344)-++ (346)-+ (346)-- (347) —0. 310=—0 (344) +2(345)-+ (346)-+ (347) —0. 310=0 (344)-+ (345) -+2(346)-- (347) —v. 310=0 (344)+ (345)-+ (246) -+2(347)—0. 310=0 Nore —341, 34142, 84—1- 2, and 342 were read by Mr. Darling with the Repsold instrument, in November, 1875. The remainder were read by Mr. Metcalf with the Pistor & Martius instrument, in September and October, 1875. 63 LS 498 PRIMARY TRIANGULATION, [Cuar. XVII, C, SECTION XUI1.—Triangulation from the line Falkirk-Pehin to the line Grand River - Westfield— Continued. BUFFALO PLAINS—35. (Observer, (i. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Date, October, 1875.] Angle as measured between— Notation. | No. meas. | Range.) Wt. | (v) [v] Corrected angles, oO ¢ te wm wt aw °o A uw Buffalo and West Base......-..----- 63 33 31. 558 351 16 37 1 +0.030 | —0. 871 63 33 30.717 West Base and Tonawanda ...-..--. 36 33 28. 541 352 16 34] 21 +0.031 | +0.180 36 33 28.752 Tonawanda and Middle Base ......- 3 39 43. 466 353 16 | 47 1 +0.0380 | +0.451 3 39 43.947 Middle Base and East Base ..-..-.... 51 00 04. 288 354 16 3.0 1 +0.031 | —0.489 51 00 03. 830 East Base and Buffalo ........-----. 205 13 11.995 355 16 | 3.8 1 +0.080 | +0.729 | 205 13 12,754 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(351)-+ (B52)-+ (353)-+ (354) —0. 1520 (351) -+2(35y)+ (853)+- (354) —0. 152=0 (351)+ (352) -+2(353)-+ (354) —0. 152=0 (35))-+ (852)4 (353) + 2(354) —0. 1520 EAST BASE—30, (Observer, G. ¥. Wisner. Instrument, Tioughton & Simms 14-inch theodvlite No.1. Date, September, 1875. J | Angle as measured betweeu— | Notation. | No. meas. Range. Wt. | (v) | (v7) leonte cted angles. | ov u 1 “ { | “ | u" oO 4 ” Buffalo Plains and Buffalo ......-.-. 16 50 19. 156 36, 16 » 4.0 1 | --0 089 - 0.398 16 50 18. 678 Buffalo and Midlle Base ....--...--. 31 18 60. 494 | 362 16 | 5.0 | 1 - 0 080} 49.222 | 31 18 60. 636 Middle Base and West Base .......- 0 00 00.172 36, 16 | 2.8 | 1 | —0.080} -+0,192 | 0 00 00. 284 West Base and Tonawania . 44 35 02. 372 364 16 | 3.3 1 ~—@.080 | --0.105 44 35 02. 187 Tonawanda and Buffalo Plains...... 267 16 38. 206 365 16 : 4.0 1 —0.080 | +0. 089 267 16 38. 215 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(36,)+ (36,)-+ (363)4- (364)-+0. 400=0 (361) -+2(362)-+ (363)+ (364)+0. 400-=0 (361)+ (36) +-2(363)+ (364) +0. 400=0 (361) +- (362)+ (363)+2(364) + 0. 400—0 WEST BASE—37. (Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No.1. Date, September, 1875.] a Angle as measured between — Notation. | No. meas. | Range.| Wt. | (v) | {v] leiesvacbet scutes as rr gee oa 5 Sees a i o 7 “uw | | ” | “ “ } or uo East Base and Buffalo Plains ....... 40 38 23.731 87142 | 16 | gett | +0.065 | +0. 134 | 40 38 23. 930 Middle Base anid Buffalo Plains. ..-. 40 38 23. 336 t 379 i 16 ' Wey" 0.000 | +0, 295 40 38 23. 631 Buffalo Plains and Buffalo .......... BL 43 50. 576 373 16 1 43 1 | +0.065 | - 0. 034 81 43 50. 607 Buffalo and Ridgeway .... 59 34 41. C99 374 16 - 4.0 1 +0.065 | —0. 686 59 34 41.078 Ridgeway and Tonawanda.. 111 19 40. 533 3765 16 . 1.7 1 +0.065 | +0.493 111 19 41.091 z +0.065 | +0. 093 66 43 23. 294 Tonawanda and East Base.......--. 66 43 23.136: 37, 16 ee) NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(87142)+ (373)4- (874) + (375) --0. 325=0 (37142) 4-2(373)+4+ (374) 4- (375) —0. 325=0 (37142) + (3873)-+2(374) + (375)--0. 325=0 (8714+2)-++ (373)-+ (374) 4 2(375) —0. 325=0 § 4.] Scv1on XIII.—Triangulation from the line Falkirk — Pekin to the line Grand River - Westfield— {Observer, G. Y. Wisner. Continued. MIDDLE BASE—38. SANDUSKY BASE TO BUFFALO BASE. Instrument, Troughton & Simms 14-inch theodolite No, 1. Date, October, 1875.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Coriected angles. fe} ‘ “a aw ow “a o / “a West Base and East Base .......... 179 59 59, 643 381 16 1.6 1 --0.124 | --0.102 , 179 59 59.417 East Base and Buffalo Plains -. - 80 51 37.429 38, 16 3.3 1 —0.125 | —0.419 | 80 51 36. 885 Baffalo Plains and West Base. .....- 99 08 23. 302 38, 16 3.9 1 —-0, 125 4-0. 521 99 08 23. 698 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. [Ovserver, T. Russ oll. 2(382)+ (383) +0. 3740 (385) + 2(383) +0. 374=0 HAMBURG— 39. Instrument, Gambey theodolite Nu. 1. Date, October, 1875.] Angle as measured between-— | Notation. | No, meas. | Range. i oO ; ‘ tt “Ww Sturgeon Point and Rilgeway .-.... 65 45 29. 9&9 3% j 17 1.4 Ridgeway and Baffalo .... 49 15 26.218 392 be 3.6 Buffalo and Sturgeon Point ......-.. 244 59 02. 536 | 393 ! 17 2.7 Wt. (v) | (v] * | Corrected angles. a | wt Oo f “at | 1 +0. 419 --0. 340 65 45 30. 068 | 1 +0.419 | --0. 235 49 15 26. 872 1 +0. 419 +0. 105 : 244 59 03. 060 NORMAL EQUATIONS FUR LOCAL ADJUSTMENT. 2(391) 4+. (39,) —1. 257=0 (391) +.2(392) —1. 257==0 FONT HILL—40. | Observer, R.S. Woodward. Instrument, Troughton & Simms 12-inch theodolite No.2. Dates, August and September, 1875.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | [v] Corrected aneleas Oo * “wn | | | a | “ur | “uw fo} - rd Drummondville and Ridgeway 57 52 21.275 40) | 24 10.4 1 +0. 099 +0. 569 | 57 52 21. 943 Ridgeway and Sugar Loaf ........-. 39 54 28, 595 402 i 24 9.1 1 +0. 099 —0. 103 39 54 28.591 ugar Loaf and Grand River ....-.. 55 25 23. 697 403 24 11.7 1 +0.099 | —0.372 55 25 23. 424 Grand River and Drummondville... 206 47 46. 037 404 24 9.8 1 +0.099 | ---0. 094 206 47 46. 042 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2 (401)+ (402) (403) —0. 396=0 (401) +2(402) + (403)—0. 396=0 (401) + (402) +2(403)—0. 396=0 , 499 d00 PRIMARY TRIANGULATION. (Cuar, XVIII, ©, Section XUT.—Triangulation from the line Fulkirk- Pekin to the line Grand River - Westfield— Continued. SUGAR LOAF—41. [Observer, G. A. Marr. Instrument, Pistor and Martins 14-inch thoodolite No. 2. Dates, May and June, 1876.} 1 Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) | (v] Corrected angles. a a pee — Sees ce a | cea sti ig art data hla eet ae liieegacicmacliic ictal Et oO t “a “wr . “we “ oo t “ | Font Hill and Drummondville. ....-- 38 30 00. 840 4 ! 23 8.8 ' 1 | 40.290 | —0.119 38 30 00.511 | Drummondville aud Ridgeway ...--- 56 55 47. 188 412 | 21 5.0 1: 40.290] 4-0.679 56 55 48,157 | Ridgeway and Sturgeon Point ....... 47 55 45. 200 413 | 21 i 4.4 | 1 : +0. 290 -—0. 132 47 55 45. 358 Sturgeon Point and Silver Creek .... 36 26 39. 381 Ala 20 » 8B 1) 40.291 -+0. 369 36 26 40. 041 | Silver Creck and Westfield .......... 29 52 10. 264 41s | 2t 5.7 | 1 | +0290} —1.351| — 29 52 09. 203 ! Westficld and Grand River ....-...-. 61 06 29. 172 4le6 18 6.1 1. +0,290 | -£0, 189 | 61 06 29 651°. Grand River and Font Hill.......--- 89 13 06. 423 417 | 22 6.7 1 +0. 291 +0. 365 89 13 07. 079 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(41,) + (412)4+ (413)-+ (4la)4+ (416) 4- (416) —2. 032=0 (41,)+2(412)-+4+ (413) + (414)+- (415)-+ (416)-—2. 033=0 (41,)4- (412) 4+2(413)+ (414) + (416)-- (416) —2. 032=0 (4li)+ (412) + (413) 4+2(414) + (415) + (416) 2. 032=0 (41,)+ (Al2)+ (413)+ (41a) +2(415)+ (416) —2. 032=0 : (41,)+ (412)+ (413)-+ (41a)+ (415)-+2(416) --2 032=0 STURGEON POINT—42. [Observer, R. S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, Tune, 1876.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | [v] ‘Gorreutea angles. \ —_ menenned peace ee i | | oo t wm | a“ a se Oo V “wn Silver Creek and Sugar Loaf........ 99 26 38. 009 42) { 22. 13.4 1 +0. 406 -+-0. 300 99 26 38.715 | Sugar Loaf and Ridgeway..----.--- 40 25 20.551 422 ; 29 _ 59 1 +0.405 | --0.277 40 25 21. 233 , Ridgeway and Buffalo ......-.-. -.- 35 23 42. 820 423 | 22 i 7.4 1 | 40.406 | —1.115 | 85 23 42.111 _ Ridgeway anil Hamburg...--..----- 59 36 08. 457 42344 | 338 | 13. 2 2 | 0.000 | 40,138 ° 59 36 08. 595 Butfalo and Silver Creek ~ 184 44 16.998 42445 | 22. | 14.2 1 {-0. 405 | 1-0. 488 184 44 17. 941 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(421)-+ (42,)+ (423)—1. 622—0 (421) +-2(422)+ (423) —1. 622=0 (42))4 (422) 4-2(423) —1. 622=-0 SILVER CREEK—43. (Observer, G. Y. Wisuer. Instrament, Troughton & Simms theodolite No. 1. Dates, My and June, 1876. ] i " ; : ; ' | Angle as measured between— . Notation. | No. meas. | Range., Wt. (v) | (v) Corrected nieen! Z oes igh Heke hee DEP oe d eek | o 4 “u ’ wo 1 | “" SH “ | | Westfield and Grand River.......-. 87 42 19. 668 431 19 ' yo1 | 1 +0. 130 | —0, 356 87 42 19.442 Grand River and Sugar Loaf....--.- 35 10 45. 699 432 19 6.9 I. +4-0.129 | —0.172 35 10 45. 656 | Sugar Loaf and Ridgeway ......-.-. 25 29 51. 060 433 ; 19 5.3 1 +0.130 | +4-1.205 25 29 52. 395 | Ridgeway and Sturgeon Point .....- 18 36 51.514 434 i 19 G7 1 --0.129 | —1,.141 18 36 50. 502 | | Sturgeon Point and Westflelil ...... 193 0) 11. 411- 435 17 6.3 uf -+0.130 | +0. 464 193 00 12. 005 | ! | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT, 2(43,)-+ (43,)-+ (483) 4- (434) —0. 648-=0 (481) +2(43,) + (433)-+ (434)—0, 648=0 (481) + (432) +2(433) + (434)—0. 648=0 (431)+ (482)-++ (482) +2 (484) —0. 6480 §4) SANDUSKY BASE TO BUFFALO BASE. 501 Section XTII.—Triangulation from the line Falkirk - Pekin to the line Grand River — Westfield— Continued. GRAND RIVER—44, (Observer, R. S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, August, 1876.] | | | | i ! Angle as measured between— | Notation. | No. meas. | Range. Wt. | (v) {vJ ee angles. | ec art | Q t a | “a | “we a oO if a | Font Hill and Sugar Loaf.... ....-. 35 21 30,513 44, | 17 5.0 1 —0.201 | +0, 596 | 35 21 30. 908 | Sugar Loaf and Silver Creek........ 53 50 39. 025 | 442 16 6.7 1 | ~ 0,201 | —0,518 53 50 38. 306 | Silver Creek and Westfield .....-... 42 17 37. 987 443 | 16 | 4.5 1 | ~—0.201 |} +0.491 | 42.17 38.277 | . Westfield and Long Point --.....-.. 52 26 09. 982 444 14 ny hace, 1 00200 igse once nos ea teens aoe i ' Long Point and Font Hill .......... 176 04 03, 699 445 8 | 7.6 0.5 —0,409 |... 222... | Siotels So eisibewie sia | . i NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 1. 5(441)+-0. 5(442) +0. 5(443)-L0. 5(444)+-0. 603-0 0. 5(441)-++1. 5(442)-+0. 5(443)-++0. 5(444)-+-0. 6083=0 0. 5(441)-++0. 5(442)-+-1. 5(443)-+-0. 5(444)+-0. 603=0 0. 5(441) 4-0. 5(442)-+-0. 5(443)-+1. 5(444)-+-0. 603=0 WESTFIELD—45, [Observers, G. ¥. Wisner and R.S.Woodward. Instruments, Troughton & Simms theodolites Nos.1 and 3. Dates, June and October, 1876. | \ ' i Augle as measured between— Notation. | No. meas. | Range. : Wt. (v) [v) Corrected angles. 1 o # “ | uw “ | “ ov “ Eric and Long Point .........-....- 64 51 01.072 451 14 5.7 1 S05 17004)! at done rellaae cs eathaetclelcae ' Long Point and Grand River ....-.. 52 02 06. 509 4b, 16 4.8 1 POs Mals Ne car ehisis dclllstaGeore memes erase Grand River and Sugar Loaf .....- . 22 45 18.919 453 9 3.8 0.5 | —0.808 | 40.178 22 45 18. 289 Grand Riyer and Silver Creek ...... 50 00 07. 618 453+4 18 5.2 1 10.234 | —0. 257 50 00 07. 595 Sugar Loaf and Silver Creck....-.-- 27 14 50.145 454 16 2.1 1 —0.404 | —0.435 27 14 49. 306 1 LOM i ts seiissieccta ly seen dete beemaeaee | Silver Creek and Eri sc.c002 a2s00-s- 193 06 45. 079 456 17 6.8 | ‘ { NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(45i)-1- (452)+ — (453)-+ (454)-+1. 7240 (451) +2(45)4+ — (453)-++ (454)-+1. 724—-0 (451)-+ (452)+-2. 5(453) +2(454)+3. 170-0 (451)-+ (452)-+ — (453)-+3(454)-+3, 170=0 Norr.—45344 and parts of 451, 452, and 45s, were read by Mr. Woodward with the Troughton & Simms No.3, in October, 1876. The remainder were read by Mr. Wisner with the Troughton and Simms No. 1, in June, 1876. ’ Numerical equations of condition in the triangulation from the line Fatkirk— Pekin to the line Grand River- Westfield. SIDE-EQUATIONS. IV. (10) -+ 0.0001 [36,42] — 18.8660 [363] 24.5309 [87,] ++ 0. 0001 [372] — 3.3874 [38] — 3.3875 [385] + 7.918=0 VII. (40) — 55.1748 [351] — 44.7040 [3524544] —69. 5686 [36,] +34. 6295 [3045] +4 13,3469 [37142] + 16. 4070 [375] — 84.971=0 TX. (25) — 17.7558 [Bligeyspats] — 23.7022 [3]e] —40. 9471 [35142] —44. 7040 [3ho44] — 69.5686 [36;] + 5.2949 [6.4544] — 32. 466—0 "XT. (150)4193, 5262 [Sligessqatete] +217. 2284 [31;] 14.2877 [35)] + 3.7569 [354] 3. 0601 [37s] 131.5433 [37145] — 60. 883—0 XIV. (10) + 18.2261 [311] ++ 14,9324 [31o] +14. 9324 [315] +14. 9324 [314] + 14,9324 [315] + 14,9324 [316] +14. 9324 [31,7] — 1.3088 (34,] — 1,3083 [34,] — 1.0930 [34;] —12.3639[37,] — 8.2209 [37,] -- 14. 154—0 XX. (10) —- 18.2261 [311] + 5.5604 [312] —16. 0762 [32] +19. 1986 [325] — 14, 2313 [345] + 0,2153 [344] “++ 28, 137—0 502 PRIMARY TRIANGULATION. [Cuar. XVIII, ¢, Numerical equations of condition, &e.—Continued. SIDE-EQUATIONS—Continued. XXIV. (20) 1.6412 [34,] + 17.1192 (34.) —- — 9.4209 (39,] +18. 1376 (392) — 29, 6326 [425] 4+ 12.3519 [42544] — 59. 630=0 XXVI. (20) — 35.2748 [32;] + 25.7548 [32,] —15. 4780 [349] +14. 4466 [345] — 13.7101 [41] + 19.0053 [41,] 24. 7201 [42,] -+ 29. 6326 [423] -- 86, 242=0 XNIX. (40) — 16.9310 [415] + 2.0743 [41,) 4:24.9742 [42,] 4.49. 6948 [42,] — 44,1480 [435] + 62.5125 (43,] +100. 239=0 XXXIT. (15) — 23, 1644 [32,] + 2.5904 [32,) 13. 2219 f40,] +25. 1746 [40,] + 2.0014 [41,] + 15.7115 [412] — 13.403=0 XXXV. (25) — 7.6098 [33] + 21.3048 [33] —25. 1746 [40.] +14, 5124 [405] — 29.8704 [43:] + 44.1480 [439] 29. 6731 [44] +115. 3852 [44,] — 13. 151-0 XXXIX. (30) + 13.6133 [43:] + 43. 4837 (43,] — —17. 6495 [44g] — 2. 2643 [445] — 50.1936 [455] + 40. 8862 [45,] + 31. 036=0 Note.—In the solution for determining the general corrections, each of the side-eqnations was divided by the number inclosed in parenthesis and placed opposite it. ANGLE-EQUATIONS, 1. [354] + [36)] + [36.] + [82] +1. 084=0 Il. [352] + [353] + [354] + [36,] + [36.] + 236.) + [3714] —0, 292—0 III. (352) + [8%] + [37%] +. [383] —1.447=0 Vo (316) 4+ [383] + (354) + [361] + [862] + [365] + [36] —0, 220=0 VI. [846] + [345] + [(85.] - [375] +1. 419=0 VOL — [3h] — [8h] — [31s] — [314] — [815] + 0345) + 34.) + £86.) + (363) + [36] +0, 092=0 xX. — (8h) — Bh) — [81s] — [314] — [31s] — B3ts) + (845) + [34a] + Bh] + [851] + [352] +2. 46450 XII. — [312] — [31s] -- (814) — [815] — [816] — [31] + [332] + [375] —1.347=0 XII. [385] + [843] + [844] + [84,5] + [387] +2.556=0 XV. (Qt) + [802] + [314] —1,586=0 XVI. [303] + [31s] + [321] —0, 211=0 XVII. [3le] + [322] + [323]. + [331] —0, 287=0 XVII. (823) + [33:] + [88] + [332] + [342] +0, 384=0 XIX. (31,) + [322] -+ [33s] + [343] + [344] +0. 733=0 XXE. [334] + [34.] + [842] + [39%] —0. 522=0 XXIL—[(33,] + [336] + [842] + [42s] —0, 465=0 XXIII. [326] + [39:] + [42544] —0. 220=0 XXV. [336] + [83;] + [41s] + [420] —0. 181=0 XXXVI. [837] + [413] + [414] + [43s] —1.273=0 XXVIII. [414] + [42:1] + [483] + [484] —0. 734=0 XXX. [32,] + [325] — [33] — [83.] — [33s] — [33,] — [33;] — [83,] — [33;] — [33.] + [40,] —0. 980=0 XXXT. [32] — [83:] — [33] — [335] — [33,] — [335] — [33] — [33,) + [41] +0, 057=0 XXXII. [823] + [401] + [40.] + [41,] —0. 649=0 XXXIV. [403] — [41,] — [41.] — [41s] — [414] — [44] — [415] + [440] —0, 590=0 XXXVI. (415) + [43:1] + [432] + [454] +2, 315=0 XXXVIT. (416) + [442] + [443] + [453] —0. 340=0 XXXVIIE— (43:] + [445] 4+ [453] ++ [454] +0. 123=0 General corrections in terms of the correlates. ’ [29] =+40.72727 XV [302] =--0. 75000 XV [30;] =+0.75000 XVI [31] =—0.10145 V —0, 50723 VIII - 0.26406 IX —0, 40578 X +0, 22968 XI +0, 59422 XII +0.56548 XIV —0,02899 XV —0.02899XVI —0, 02899 XVII 40,79710 XIX —1. 46892 XX $4] [3litetatite] [3ligz¢s+itste] [3),] ([3lopa4445-+6+7 ] [313] B31] [31s] [Sle] [31] (32)] [320] [32] [324] [325] [331] [33,] [335] [33,] [33,5]. SANDUSKY BASE TO BUFFALO BASE. General corrections in terms of the correlates—Continued. ==—0, 24637 V +0, 01459 XII +0. 50723 XIX =+0. 20291 V —0. 18832 XII +0. 40578 XIX =—0.01449 V —0, 05795 XII —0. 02899 XIX =+0. 20291 V —1, 18832 XII —0, 50422 XIX =—0, 01449 V —0. 05795 XII —0, 02899 XIX =—0. 01449 V —0. 05795 XII —0. 02899 XIX =—0.10145 V —0, 40578 XII —0, 20290 XIX =+0. 44928 V —0, 20291 XII —0.10145 XIX =—0), 10145 V —0. 40578 XII —0, 20290 XIX =+0. 80000 XVI —0, 40000 XXX =—0, 10000 XVI —0, 20000 XXX =—0, 10000 XVI —0. 20000 XXX =—0. 20000 XVI +0. 60000 XXX =—0. 20000 XVI +0, 60000 XXX ==:—0, 11538 XII —0. 0769! XXI —0, 11538 XXX = +0. 83462 XII —0. 07692 XXI —0. 11533 XXX =—0, 11533 XII —0. 07692 XXI —0, 11538 XXX = -0, 07692 XII +0. 61538 XXI —0. 07692 XXX =—0, 11538 XII —0. 07692 XXI —0, 11538 XXX —1, 23178 VIII -40, 90272 XIV — 0, 88418 XX —0,98541 VIII +1. 02080 XIV —0. 70734 XX —0, 07244 VILL +40. 03371 XIV +40, 44771 XX +0. 01459 VIII +0. 69143 XIV +1. 11527 XX —0, 07244 VIII +0, 03371 XIV —0. 10833 XX —0. 07244 VIII +40, 08371 XIV —0. 10833 XX —0,50723 VIII £0, 23611 XIV 0, 35369 XX +0, 24637 VILI +40. 11808 XIV 0, 17634 XX +0. 49277 VIII 0, 2361L XIV +0, 35369 XX —0. 20000 XVII —0, 20000 XXXI +0, 40000 XVII —0, 10000 XXXI +0. 40000 XVII —0. 10000 XXXI —0, 20000 XVII +0. 80000 XXXI —0, 20000 XVII —0. 20000 XXXI —0, 11538 XIII —1,1923L XXII —0. 23077 XXXI —0, 11538 XIII —0, 19231 XXII —0, 23077 XXXI +0, 83462 XIII —0, 19231 XXII —0, 23077 XXXI —0, 07692 XLII +0,53-46 XXII —0, 15385 XXXI —0. 11538 XIII +0, 80769 XXII —0, 23077 XXXI —0, 64122 1X +0. 07244 XV —0, 89221 1X +40. 05795 XV —0. 03770 IX —0, 28986 XV —0. 18198 IX +0, 05795 XV —0. 03770 IX —0, 28986 XV —0. 03770 IX +0. 71014 XV —0, 26406 IX —0, 02899 XV —0, 25099 IX —0. 01149 XV +0, 44617 IX —0, 02899 XV —0. 10000 XVII —0, 20000 XXXII —0. 30000 XVIII —0, 10000 XXXII +0. 70000 XVIII —0. 10000 XXXII —0. 10000 X VIII —0. 20000 XXXII —0. 10900 XVIII +0, 30000 XXXII +0. 83462 XVII —0. 11533 XXIII —0. 06321 XXXV —0, 11538 XVII —0, 11538 XXIII —0. 06321 XXXV —0, 11538 X VIL —0. 11538 XXII —0. 06321 XXXV —0. 07692 XVII —0. 07692 XXIII —0. 04214 XXXY —0. 11538 XVII +0. 88462 XXIII —0, 06321 XXXV —0, 98541 X +0, 07244 XVI —1. 18832 X +0, 05795 XVI —0, 05795 X —0, 28986 XVI —0, 18832 X +0, 05795 XVI —0. 05795 X +0.71014 XVI —0, 05795 X —0, 28986 XVI —0, 40578 X —0, 02899 XVI —0, 20291 X —0, 01449 XVI +0), 1 9422 X -—0, 02899 XVI —0, 03123 XX +0, 27432 XXXII —~1, 70129 XX +0. 13716 XXXIII +1. 82619 XX +0. 13716 XXXII —0. 03123 XX —1. 26997 XXXII —0, 03123 XX +0. 44701 XXXILL +40. 65355 X VILL —0.23077 XXV +0. 65385 X VIII —0, 23077 XXV -40, 65345 XVIII —0, 23077 XXV —-0, 23077 XVIII —0. 15335 XXV —0. 34615 XVIIT —0, 23077 XXV 503 +0. 55773 XI +0, 07244 XVII +0, 67261 XI +40, 05795 XVI +0, 03270 XI +0, 71014 XVII -40. 83063 XI +40. 05795 XVII +0, 03279 XI —0. 28986 XVII +0, 03279 XI —0, 28986 XVII +0, 22968 XI —0, 02899 XVII +40, 11488 XI —0. 01449 XVII +0, 38770 XI —0, 02899 XVII —0. 03118 XX VI +0, 40035 XXVI —1. 36339 XXVI +1, 20656 XXVI —0, 031138 XXVI —0. 23077 XIX —0. 11538 XX VIT +0. 76923 XIX —0, 11538 XXVIII +0. 76923 XIX —0, 11533 XXVIT —0, 15385 XIX —0, 07692 XXVIT —0, 23077 XIX —0, 11538 XX VII 504 PRIMARY TRIANGULATION. [Cuar. XVIII, ¢, General corrections in terms of the correlates—Continued. (33,.] 9 =—0. 1136 XII —O.11588 XE 0.11538 XVID —0. 34615 NXVL — 0, 25077 XIX —0, 07692 XXI —0,19231 XXIT 0, 11538 NNUE 0.76923 XXV —0, 11538 XX VII —0.11533NXX — —0,28077 NNNI—- —0. 06321 NXNXV [38;] =—0. 11538 XII 0.11538 XL 0, LIS88XVIL = 0.34615 XVITT = —0. 23077 XIX —0, 07692 XXI —0,1923L XXIL 0.11588 XXIIL 0.76923 XXV_—-L.0. 88462 NXNVII 0.11586 XXX 0.23077 XXNXI 0. 36759 XXXV [33] =—0. 11538 NII -0.11538XHI 0.11588 XVIL —0, 34615 XVII —0, 23077 XIX —0, 07692 XXI —0.19231 XXIL 0. HS38 XNUL = 0, 28077 XXNV —0. 11538 XXVII —0.11533XNX 4.0, 76923 XNNNI +10, 78808 XNKV [34] =+0.20000 XXI —0.40000 XXIL = —0.29315 XXIV +0. 30956 XXVI [34] =+0.20000 XXI +0.60000 XXIL +0. 48075 XXIV —0, 46434 XXVI [345] =+0.50000XIIT = --0. 06542 XIV 0.50000 XVIIL +0. 50000 XIX —0. 71157 XX +0. 36117 XXVI [34} =—0. 40000 VI 0.40000 VIE = —0. 60000 X 0.60000 XII = —0, 08280 XIV +40, 30000 XIX +0, 01722 XX [34] =—0.40000 VI +0.60000 VIET = +.0. 40000 X -40.60000 XIIL = —0. 06127 XIV —0. 20000 XIX —0. 00431 XX [34,] -=-+0. 60000 VI ++0.60000 VILL +0, 40000 X —0,40000 XIII 4.0, 04803 XIV —0. 20000 XIX —0. 00431 XX . [34,] =-++0. 60000 VI —0. 40000 VILE = 0. 40000 X —0,40000 XII +0, 04803 XIV —0. 20000 XIX —0. W431 XX [35)] =—0. 20000 I —0. 60000 IL —0. 40000 LI —0. 40000 V -0. 30000 VI —0. 43294 VII —0. 26746 IX +40. 60000 X +0, 07087 XI [35142] =—0. 40000 I —0. 20000 II +40, 20000 III —0, 30000 V 40. 63000 VI —0. G041L VIL —0. 53492 1X +1. 20000 X +0. 07194 XI [3%] =—0. 200001 +0. 40000 II 40. 60000 LIL —0, 40000 V —0, 20000 VI —0, 17117 VIL —0. 26746 1X +40. 60000 X +0. 00L07 XI [3524345] =+0. 40000 T +1, 20900 I +0. 80000 III 0, 30000 V —0, 60000 VI —0.51351 VIL —1. 10294 IX —0, 20000 X —0, 04689 XI [35,] =—0. 20000 I +0. 40000 II +0. 60000 III 4-0. 60000 V —, 20000 VI —0.17117 VU —0. 41774 1X —0. 40000 X —0, 02398 XI [35344] =+0. 60000 I 40. 80000 II +0. 20000 III -++1. 20000 V —0, 40000 VI —0, 34234 VIL —0, 83548 1X —0. 80000 X —0, 04796 XI [354] =-+40. 30000 I +0. 40000 I —0, 40000 IIL -L0. 60000 V —0. 20000 VI —0. 17117 VI —0.41774 1X —0, 40000 X —0, 02398 XI [36,] =-+0. 60000 I -40. 40000 IL +40. 37733 LV +0. 20000 V —1. 73767 VIL —0.60000 VIE = —2. 35327 IX [3%] =-+40.60000 I +0. 40000 II +40. 37733 1V 40, 20000 V 0, 86729 VII +0.40000 VIET = 4-0, 64127 IX [36243] =+0. 20000 I -+0, 8000 II —1. 13195 IV +0. 40000 V +1. 73458 VIL +0,80000 VIII = 1.28254 1X [3604544] =—0. 20000 I +40, 20000 II —0. 75463 1V +0. 60000 V +1. 73613 VIL +1.20000 VIEL =—-.:1. 92381 IX [363] =—0. 40000 1 +40. 40000 IT —1. 50928 IV +40, 20000 V -L0. 86729 VII +0.40000 VIIL £0, 64127 IX . [364] =—0. 40000 I —0. 60000 II +0. 37732 IV +40. 20000 V +0. 00155 VII -+0.40000 VIII 4.0. 64197 IX [37142] =-+0. 20000 11 +1.96247 IV —0. 20000 VI +40. 18490 VII +9, 35486 XI —0, 20000 XII —0.20000 XIII +0. 41170 XIV [3%] +1. 00000 IIT —2. 45308 IV [375] =—0.20000 II —0. 49062 IV +0, £0000 VI +0. 26141 VII 0, 33446 XI —0. 20000 XII —0. 20000 XIII +0: 41170 XIV §4.] [37] [875] [382] [383] [89] [392] [401] [402] [403] [41] =—0, 20000 II Al —0, 20000 XII “40. [37445] =—0. 40000 IT —0 +40, 60000 XII ti, ——0, 20000 II af, -+0, 80000 XII ail, =-+40. 66667 I i =——). 59983 I i, =o ei [412] [413] [414] [41s] [416] (421] [422] [42] [421] [482] =+0.75000XXX 40 =—0.25000 XXX +40 =—0.25000XXK 0 =—0.14286XKV —0 —0, 14286 XXXI-L0 —0. 14286 XXXVII =—0, 14286 XXV - —0, +0, 85714 XXXI —0. —0. 14286 XXXVII =+0.85714 XXV +40. —0, 14286 XXXI —0. —0. 14286 XXXVII =—0,14286XKV 0. —0, 14286 XXXI —0. —0, 14286 XXXVII ——0.14286XXV_ —0, —0.14286 XXXI —0 —0. 14286 XXXVII ——0.142865XXV —0 —0, 14286 XXXI —0. -40, 85714 XXXVII =—0,25000 XXII +0. +0, 15768 XXIX ——0.25000 XXII 0. +40. 77568 XXIX =+0.75000 XXII —1 —0. 46668 XXIX [42544] =+0.50000 XXIII +0. =~ 0.20000 XXVII_ —0. +40, 20000 XXX VIII +0. =—0.20000 XXVII_ —0. —0, 20000 XXX VIJI-+1. [43s] [434] [441] [442] [445] [453] (454) =-+0, 80000 XX VII =—0. 20000 XX VII SANDUSKY BASE TO BUFFALO BASE. General corrections in terms of the correlates—Continued. =+0, 66667 XXI 40, 20000 XXX VII —0. 40. 20000 XKXVIII—0. —0. —90. =+0, 23333 XXXIV —0. =+0. 64 L 49062 IV 80000 XTIT . 98124 TV 60000 XIII 49062 IV 20000 XIIT 33333 III 66667 III 66667 XXIII . 33333 XATILT - 50000 XXNIT . 50000 XXXIT . 50000 XXXII . 03782 XXVI . 85714 XXXNIT 72333 XXVI 14286 XXXII 91245 XXVI 14286 XXXII 03782 XXVI 14236 XXXII 03782 XXVI . 14236 XXXIT . 03782 XXVI 14286 XXXII 37041 XXIV 37041 XXIV . 11122 XXIV 30880 XXIV 40000 XXVIII 07313 XXXIX 40000 XXVIII 05381 XXXAIX 60000 XXVIII 38045 XX XIX 60000 XXVIII 35065 XXXIX 98910 XXXV —0. 20000 VI —0 —0. —l. —0. —0. —0. +0. —0. +40. —0. oS —0. =0; —0. —0. ai, . 82469 XIV 40000 VI 23508 XIV 20000 VI 41039 XIV 11291 1V . 11292 IV 61833 XXTV . 76260 XXTV . 08067 XXXIIL .47910 XXXTIT . 19921 XXXII . 28571 XAXVIT . 03525 XXXII . 28571 XX VII . 87874 XXXII 71429 XXVIL 16869 XXXII 71429 XXVIT . 16869 XXXIIT 2e571 XXVIT . 16869 XXXII 28571 XXVII 16869 XXXII 25000 XXV . 75000 XXV 25000 XXV . 09182 XXTX - O91R2 XXTX . 19552 XXIX 47099 XXIX 51284 XXXV +0. 83333 XXX VII —0. 49027 XXXIX =+0, 83333 XXXVII_ -10. 83333 XXX VITI—0. 06290 XXNIX =+0, 93333 XXXVII +0, 93333 XXXVITI—1. 56174 XXXIX =+0. 73333 XXXVI +0. 73333 XXX VIII -+0, 99944 XXXIX s —0, 14877 VII —0. 29754 VII —0. 14877 VII —0, 25000 XXXIV —0. 25000 XXXIV +0. 75000 XXXIV —0. 14286 XXVIII —0. 14286 XXXIV —0. 14286 XXVIII —0. 14285 XXXIV —0. 14236 XXVIII —0. 14286 XXXIV +0. 85714 XX VITI —0. 14286 XXX1V —0. 14286 XXVUI -—0, 14286 XXXIV —0. 14286 XXVIII —0. 14286 XXXIV —0. 06141 XX VI —1. 29742 XXVI +1. 42023 XXVI —0. 11422 XXXV --1. 30904 XXAV +1. 65170 XXXV ---0, 11422 XXXV 505 . 62210 XI . 0420 XI 2 . 52210 XI . 10662 XXXV . 90036 XXXV +0, 68712 XXXV 05306 XXIX . 14236 XXXVI . 05306 XXIX . 14286 XXXVI . 37022 XXIX . 14286 XXXVI . 10492 XXIX . 14286 XXXVI . 05306 XXIX . 95714 XXXVI . 05306 XXIX . 14286 XXXVI +0. 75000 XXVIII —0, 25000 XXVIII —0, 25000 XX VITI +0. 60000 XXXVI +0. 60000 XXXVI —0, 40000 XXXVI —0, 10000 XXXVI DOG PRIMARY TRIANGULATION. [Cuar. XVII, C, Normal equations for determining the correlates. guts le 0=-+1. 08400 + 2, 66667 I + 1.20000 IT — 0. 73333 III + 0.64175 IV + 1.00000 V 0. 20000 VI = 1,04155 VII — 0.20000 VIII — 2, 12974 IX — 0, 40000 X — 0.02398 XI sD + 1.2¢000 I + 3.20000 II + 0.80000 III + 1.20785 IV + 1.40000 V — 0.80000 VI — 0.33170 VII + 0.20000 VIII — 2.17367 IX — 0.20000 X + 0.30797 XI — 0.20000 XII — 0.20000 XIII + 0.41170 XIV 3 — 0, 733383.1 + 0.80000 II + 2.86667 III — 2.56600 IV + 0.20000 V — 0.40000 VI — 0.34234 VII —- 0. 68520 IX. + 0.20000 X — 0, 02291 XI 4. + 0.64175 I + 1.20785 II — 2.56600 III +13, 75563 IV — 0.37732 V — 0.49062 VI — 1.18266 VII — 0.75464 VIII — 1,20984 Ix + 0.87050 XI — 0.49062 XIT — 0.49062 XIII + 1.00994 XIV 5. + 1.00000 I + 1.40000 II + 0.20000 III — 0, 37732 IV + 2.44928 V — 0.40000 VI — 0.34388 VII + 0.84637 VIII — 1.51593 1X — 1.00291 X + 0.06692 XI — 0, 20291 XII + 0.11808 XIV — 0.01449 XV — 0.01449 XVI — 0.01449 XVII — 0, 10145 KIX + 0.17684 XX 6. — 0.20000 I — 0.80000 II — 0.40000 III — 0. 49062 IV — 0.40000 V + 2.80000 VI — 0.17153 VII + 0.20000 VIII — 0.26746 IX + 1.40000 X + 0.40533 XI — 0.20000 XII — 1.00000 XIII + 0.50776 XIV — 0.40000 XIX — 0.00862 XX % — 1.041551 — 0.33170 II — 0.34234 III — 1.18266 IV — 0, 34388 V — 0, 17158 VI -+ 5.86389 VII + 1.73612 VIII + 6.80477 IX — 0.60410 X + 0, 21024 XI — 0.14877 XII == 0, 14877 XIII + 0.30624 XIV 8. — 0.20000 I + 0.20000 II — 0.75464 IV + 0.84637 V + 0.20000 VI + 1.73612 VII + 3.63178 VII 4- 2.56503 IX + 1.78541 X — 0.55773 XI — 0.01459 XII + 0.20000 XIII — 0.91596 XIV — 0.07244 XV — 0.07244 XVI — 0.07244 XVII — 0, 90723 XIX + 0.87556 XX 9 — 2, 129741 — 2.17367 II — 0. 68520 III — 1.20984 1V — 1.51593 V — 0.26746 VI + 6.80477 VII + 2.56503 VIII +10. 01948 IX + 0.35730 X — 0.53710 XI + 0.18202 XII — 0.75309 XIV —- 0.03771 XV — 0.03771 XVI — 0.03771 XVII — 0.26407 XIX + 0.46031 XX 10. — 0.40000 I — 0.20000 II + 0.20000 III — 1.00291 V + 1.40000 VI — 0.60410 VII + 1.78541 VIII + 0.35730 IX + 3.58832 X — 0.60067 XI + 0.18832 XII — 0.20000 XIII — 0.98601 XIV — 0.05795 XV — 0.05795 XVI — 0.05795 XVII — 1.00578 XIX + 0.69441 XX 11. — 0. 02398 I + 0.30797 II — 0.02291 III + 0.87050 IV + 0.06692 V + 0.40533 VI + 0.21024 VII — 0.55773 VIII —« 0.53710 1X — 0.60067 X + 2.34489 XI — 1.35271 XII — 0.52210 XIII + 2.73365 XIV + 0.03278 XV + 0.03278 XVI + 0, 03278 XVII + 0, 22969 XIX — 0.40038 XX 12. — 0.20000 II — 0, 49062 IV — 0, 20291 V — 0.20000 VI — 0.14877 VII — 0.01459 VIII + 0.18202 Ix + 0, 18832 X° — 1.35271 XI + 2.87294 XII — 0.31538 XIII — 1.10182 XIV -— 0.05795 XV — 0.05795 XVI — 0.17333 XVII + 0.65385 XVIII + 1.36345 XIX — 4.11527 XX — 0.07692 XXI — 0.19231 XXII — 0.11538 XXIII — 0.23077 XXV — 0.11538 XXVIT — 0.11538 XXX — 0, 23077 XXXI — 0.0632] XXXV 4] No. of equation. 13. 14, 15. 16. 17. 18. 19. 20. 21. 22. SANDUSKY BASE TO BUFFALO BASE. 507 Normal equations for determining the correlates—Continued. 0=+2. 55600 0——-1. 41540 0——1. 58600 0——0. 21100 0=—0. 28700 0—-L0. 38400 0=-10. 78300 0=+2. 81370 0——0. 52200 0=—0. 46500 — 0.20000 II + 0.20000 VIII + 3.38462 XIII -+ 1.86923 XIX 0. 11538 XXIII 0. 11538 XXX 0. 41170 II 0, 30624 VIT 2.73365 XT 0. 03370 XV 0. 41727 XIX 0.01449 V 0. 03278 XI 0. 28986 XVI 0.01449 V 0. 03278 XI 2.26014 XVI 0. 13956 XX 0.20000 XXXII 0.01449 V 0. 03278 XI 0. 28986 XV- 0. 25976 XIX 0. 11538 XXITIT 0. 51538 XXX 0. 06321 XXXV 0. 65385 XIT 1. 05385 XVII 0. 23076 XXI 1. 00222 XXVI 0. 10000 XXXII 0. 1. 0. 1. 0. 0. 0. 0. 0. 1. 0. 0. + + + + + + ll t+ + 10145 V 00578 X 41727 XIV 80769 XVIII 38462 XXII 23076 XXVII 17684 V 69441 X 92059 XIV 11462 XVIII 06245 XXX 07692 XII — 0. 15384 XIX + 0.95020 XXIV — 0.07692 XXX — 0.19231 XII — 0.38462 XIX 0.63047 XXIV ~ 0. 25000 XXVIII 0. 10535 XXXV — 0, 49062 IV — 0.20000 X 1.03418 XIV 0. 69866 XX — 0.23077 XXV 0.23077 XXXI 1.00994 IV 0.91596 VIII 1, 10182 XII 0.03370 XVI 0.92059 XX 0.07244 VIII — 0.05795 XII 0. 28986 XVII — 0.07244 VIL 0.05795 XII — 0.48986 XVII 0.08118 XXVI + 0.27432 XXXII — 0.07244 VIII — 0.17333 XII 0. 48986 XVI + 0.57261 XX — 0.23077 XXV — 0.43077 XXXI + + + 1.15385 XIII ++ 3.16155 XVIII — 0.57693 XXII — 0.34614 XXVII + 0.13716 XXXIII — 0.40000 VI + 0.22969 XI — 0.02899 XV -L 3.63556 XIX — 0.23076 XXIII — 0.23076 XXX — 0.00862 VI 0. 40038 XI — 0.10833 XV — 2.16327 XIX 0, 03122 XXXI — 0.07692 XIII + 1.68205 XXI — 0.15385 XXV — 0.15385 XXXI — 0.19231 XUI + 0.73846 XXI — 0.63462 XXV 0. 46668 XXIX — 1.00000 VI 0.52210 XI 0. 11538 XVII 0.07692 XXI + 0.36117 XXVI — 0.06321 KXXXV + 0.11808 V — 0, 75309 IX — 1.03418 XIII + 0. 03370 XVII 0.04725 XXVI 03771 IX . 03370 XIV . 02899 XIX . 03771 IX . 03370 XIV . 10000 XVIII — 0.40000 XXX | 03771 IX 11538 XIIL . 39476 XVII 07692 XXI 96304 XXVI 20000 XXXII | S lie St eos 06542 XIV 80769 XIX 34614 XXIII 54614 XXX 18963 XXXV 90723 VIII 36345 XII 02899 XVI 16327 XX 46154 XXV 46154 XXXI 87556 VIIL 11527 XII — 0.13956 XVI +10, 18030 XX — 0.03122 XXXII — 0.07692 XVII + 0.73816 XXII — 0.15478 XXVI — 0.04214 XXXV — 0.19231 XVII + 2.69615 XXII + 0.95589 XXVI — 0.19230 XXX SPSS 2 MS SS OS PS — 0.14877 VII — 0.31538 XII + 1.15385 XVIII — 0.19231 XXII — 0.11538 XXVII + 0.50776 VI — 0.98601 X + 3, 44623 XIV — 0.06542 XVIII 0.05795 X 2. 18741 XV 0. 10823 XX 0, 05795 X — 0.28986 XV — 0.02899 XIX — 0.20000 XXXI + — 0.05795 X + 0.03370 XIV + 1.05385 XVIII — 0.19231 XXII — 0.11538 XXVII + 0.27432 XXXII + 0.10000 XVI 1. 11462 XX 0. 69231 XXV 0. 79231 XXXI 0. 26407 IX 1, 86923 XIII 0. 25976 XVII 0. 15384 XXI 0.36117 XX VI 0. 12642 XXXV 0. 46031 IX 0. 69866 XIII 0. 57261 XVII 3.77512 XXVI 0. 04283 XX XIII 0. 23076 XVIII 0. 41025 XXTII 0, 07692 XX VII 0, 57693 XVIII + 0.80770 XXII — 0.19230 XXVII — 0.35462 XXXI PRIMARY TRIANGULATION. [Cuap. XVII, C, Normal equations for determining the correlates—Continued. en ON i= —0, 22000 —- 7, L638 AI. ae (), T1558 XT — 0.11538 XVII — 0.34614 XVIII — 0, 23076 XIX — 0.41025 XXI + 0.80770 XXII + 2.05129 XXJIT — 0, 30953 XXIV — 0.3077 XXV — 0.11538 XXVII — 0.11538 XXX — 0, 23077 XXXI — 0§,06321 XXAV Q4, 0=—2, 98150 + 0,95020 XXI1 — 0.63047 XXII — 0.30953 XXIII + 3.20929 XXIV + 0.37041 XXV = 2 d763L XK VIL + 0.37041 XXVIII + 0.69145 XXIX 25 b==—U, 18100 — 0, 23077 NIT — 0. 23077 XIII — 0, 93077 S VIL — 0, 69231 XVIII — 0.46154 XIX — 0.15385 XXI — 0. 63462 XXII — 0. 23077 XXIIT + 0.37041 XXIV + 3.14560 XXV — 0.38497 XXVI + 1.48353 XXVII — 0.39286 XXVIII + 0.40546 XXIX — 0.23076 XXX — 0.60440 XXXI — 0.14286 XXXII — 0.16869 XXXIII — 0.14286 XXXIV — 0.43080 XXXV — §, JAVeO XXXVI — 0.14886 SXXVIT 26, 0=+4. 31210 + 0.36117 XIII — 0, 04785 XIV — 0.08118 XVI — 0.96304 XVII — 1.00222 XVIII + 0.36117 XIX — £772 5x = 0,15478 XXT + 0.95589 XXII — 2.47631 XXIV — 0.38497 XKV + 9.64944 XXVI + 0.87463 XXVII — 0.09923 XXVIII — 2. 03888 XXIX + 1.12539 XXX + 0. 48323 XXXI — 0.11900 XXXII — 2.638999 XXXIII — 0.03782 XXXIV — 0.087682 XXAVI. — ©, 03782 AAKVIT OT, 0=—1. 27300 — 0. 11538 NIL — 0.11538 XIII — 0.115388 XVIE — 0.34614 XVIIL — 0.23076 XIX — 0.07692 XXI — 0.19230 XXII — 0.115388 XXIII + 1.48353 XXV + 0.87463 XXVI + 3.113820 XXVII + 1.31423 XXVIII — 1.46082 XXIX — 0.11538 XXX — 0.51649 XXXIT — 0.28572 XXXIT — 0.33738 XXXIII — 0.28572 XXXIV + 1.28411 XXXV — 0,68572 XXXVI — 0, 28572 XXXVII — 0.20000 XXXVII— 0.38065 XXXIX Qe, 0=—0. 73400 — 0. 25000 XXII + 0.37041 XXIV — 0.39286 XXV — 0, 09923 XXVI + 1.31428 XXVIL + 2.80714 XXVIII + 0.53807 XXIX — 0.14286 XX XI — 0.14266 XXXII — 0.16869 XXXIII — 0.14286 XXXIV + 1.53748 XXXV — 0.94286 XXXVI —- 0.14286 XXXVII — 0.40000 XXXVIII— 0.76130 XXXIX 29, 0=42.50598 — - 46668 XXIT + 0.69145 XXIV + 0.40546 XXV — 2, 03838 XXVI — 1.46082 XXVII + 0.53807 XXVIII + 4.84264 XXIX + 0.05306 XXXI + 0.05306 XXXII + 0.06266 XXXIII + 0.05306 XXXIV — 2.00149 XXXV — 0.13058 XXXVI + 0.05306 XXXVII — 0.09182 XXXVIII— 0.17476 XXXIX 30. 0=—0. 98000 — 0.11538 XII — 0.11538 XTII — 0.40000 XVI — 0.51533 XVII — 0.54614 XVIII — 0.23076 XIX — U. 06245 XX — 0.07692 XXI — 0.19230 XXII — 0.11538 XXIII = — 0.23076 XXV + 1.12539 XXVI ~ 0.115338 XXVIT + 2.83462 XXX + 1.36923 XXXI -+ 1.1000u XXXII — 1.90363 XXXTII_ — 0.25000 XXXIV + 0.04342 XXXV 31. 0=--0. 05700 — 0, 23077 XII — 0. 23077 XIII — 0.20000 XVI — 0.43077 XVII — 0.79231 XVIII — 0.46154 XIX — 0.03122 XX — 0.15385 XXT 0. 38462 XXIT — 0.23077 XXIT1 — 0.60440 XXV + 0. 48323 XXVI 0.51649 XXVIII — 0.14286 XXVIII + 0.05306 XXIX + 1.36923 XXX + 3.19560 XXXI -— 0.34286 XXXII -- 0.39123 XXXIII — 0. 14286 XXXIV + 0.72578 XXXV — 0.14286 XXXVI — 0.14286 XXXVII 22 O=—0. 64900 — 0, 20000 XVI — 0.20000 XVII — 0.10000 XVIII — 0.03122 XX — (). 14286 XXV — 0.11900 XXVI — 0.28572 XXVIII — 0.14286 XXVIIT + 0.05306 XXTX + 1.10000 XXX — 0.34286 XXXI +: - 2.65714 XXXII + 0.81018 XXXII — 0.64286 XXXIV — 0.79374 XXXV — 0.14286 XXXVI | . 14256 XAXAVIT 33. 0=—O. 80853 + 0.27432 XVI S + 0, 27432 XVII + 0.13716 XVIII + 0.04283 XX 16869 X XV — 2.63999 XXVI — 0.33738 XXVIII — 0.16869 XXVIII . 06266 XXIX — 1.90363 XXX — 0.39123 XXXI + 0.81018 XXXII 6, 32907 XXXIIL — 0.36791 XXXIV — 1.60506 XXXV_ — 0.16369 XXXVI — 0.16869 XXXVIL ++ oe 2 §4.] SANDUSKY BASE TO BUFFALO BASE. 509 Normal equations for determining the correlates—Continued. auation 34. 0=—0. 59000 — 0. 14286 XXV — 0.03782 XXVI — 0.28572 XXVII — 0.14286 XXVIII + 0.05306 XXIX — 0.25000 XXX — 0.14286 XXXI — 0.64286 XXXII — 0.36791 XXXIII + 2.44048 XXXIV -- 0.30198 XXXV — 0, 14286 XXXVI — 0.14286 XXXVII 35. O0=—0.52604 — 0.06321 XII — 0.06321 XIII — 0.06321 XVIT — 0.18963 XVIII — 0.12642 XIX — 0.04214 XXI — 0.10535 XXII — 0.06321 XXIII — 0, 43080 XXV + 1.28411 XXVIII + 1.53748 XXVIII — 2.00149 XXIX + 0.04342 XXX = ++. 0.72578 XXXII — 0.79374 XXXII — 1.60506 XXXIII — 0.30198 XXXIV + 8.06020 XXXV — 1.42326 XXXVI + 0.51264 XXXVII — 0.11422 XXXVIII— 2.25095 XX XIX 36. O=+2.31500 — 0.14286 XXV — 0.03782 XXVI — 0.68572 XXVII — 0.94286 XXVIII — 0.13058 XXIX — 0.14286 XXXI — 0.14286 XXXII — 0.16869 XXXIIT — 0.14286 XXXIV — 1.42326 XXXV + 2.79048 XXXVI — 0.14286 XXXVII + 1.33333 XXXVIII+ 2.14138 XXXIX 37. 0=—0. 34000 — 0.142286 XXV — 0.03782 XX VI — 0.28572 XXVII — 0.14286 XXVIII + 0.05306 XXIX — 0. 14286 XXXI — 0.14286 XXXII — 0,16869 XXXIII — 0.14286 XXXIV + 0.51284 XXXV — 0.14286 XXXVI +4. 3.45714 XXXVI + 1.76667 XXXVIII— 2.11491 XXXIX 38. 0=-10. 12300 — 0.20000 XXVII — 0.40000 XXVIII — 0.09182 XXIX — 0.11422 XXXV + 1.33333 XXXVI + 1.76667 XXXVII + 3.30000 XXXVIII— 0.55207 XXXIX 39. 0=+1.03453 — 0.38065 XXVIL — 0.761380 XXVIII — 0.17476 XXIX — 2,25095 XXXV + 2.141388 XXXVI — 2.11491 XXXVII — 0.55207 XXXVIII-+ 5, 85092 XXXIX Values of the correlates and their logarithms. I =—0.2635 log 9. 4208136_ XXI =—0. 4676 log 9. 6698373_ Il =—0. 9252 log 9. 9662356- XXII ==-+0.7274 log 9. 66177934 III =-++0. 6763 log 9. 8301458; XXIII =—0. 1320 log 9. 1206068_ IV =+0. 1555 log 9. 19167454 XXIV =+0. 6596 log 9. 8193044 V =—1.3431 log 0. 1281115_ XXV =+0.3783 log 9.5778363, VI =—0. 8451 log 9. 9269081_ XXVI =—0.9468 log 9.9762766_ VIL =+1.7511 log 0.243233} XXVII =—0.5451 log 9.7364443_ VIII =+1.5360 log 0. 18640544 XXVIII =+0. 6721 log 9. 82744044 IX =—1. 8292 log 0, 2622564_ XXIX =—1:4566 log 0. 1633492_ X =—1.3397 log 0.1269978_ XXX =+1. 0065 log 0. 00281814 XI =-+0. 0507 log 8. 70483664 XXXI =—0. 1930 log 9. 2855798_. XII =-L0. 0902 log 8. 95501394 XXXII =—0. 4240 log 9. 6273863_ XIII =—1. 2666 log 0. 1026360_ XXXIII =—0. 0904 log 8. 9560723_ XIV =—0. 4297 log 9. 6331451_ XXXIV =+0. 0481 log 8. 68232564 XV =-10.6277 log 9. 79774524 XXXV =—0.5624 log 9.7500839_ XVI =-+0.1617 log 9. 20879064 XXXVI =—1. 6689 log 0.2224225_ XVII. ——0. 3586 log 9. 5546224_ XXXVI =—0. 1284 log 9. 1085312_ XVIII =-+1. 0550 log 0. 02325254 XXXVIII =-40.7359 log 9, 86679524 XIX =—0.7116 log 9, 8522604_ XXXIX =+0.2491 log 9. 39637494. XX =—0, 9098 log 9.9589555_ 510 “u PRIMARY TRIANGULATION. Values of the general corrections. a uw a“ [Cuar. XVIII, C, D, [29.J=+0.457 | [32] =+0.302 | [846] =+0.511 | [37;]=+0.493 | [42,] =+0. 300 [30.J]=+0.471 | [33,]=+0.523 | [84,7] =—1.025 | [38,j;=--0.419 | [42.] =+0.277 [305] =-+0.121 | [33,]=-+0.261 | [35:] =—0.871 | [88] =+0.521 | [42] =—1.115 [31] =+0.963 | [33]=—1.096 | [352] =+0.180 | [39] =—0.340 | [4254.] =+0. 138 [3ly] =—0.833 | [33,]=-+0.058 | [353] =+0.451 | [39,) =+0.235 | [43,] =—0.356 [31s] =+0.193 | [33;]=-++0.422 | [35,] =—0.489 | [40,] =+0.569 | [43] =—0.172 [314] =+0.659 | [335] =+0.205 | [36] =—0.398 | [40,]=—0.103 | [435] =+1.205 [31s] =+0.067 | [33,]=—0.169 | [36,] =+0.222 | [40,5] =—0.372 | [43,) =—1.141 [316] =+0.348 | -[335]=—0.845 | [36,] =+40.192 | [41,]=—0.119 | [44] =-+10.596 [317] =—1.027 | [34,5] =—0.871 | [36,) =—0.105 | [41o] =-+0.679 | [44,] =—0.518 [32,] =—0.103 | [34.]=41.100 | [3742] =-+0.134 | [415] =—0.132 | [445] =-+10. 491 [32] —=+0.541 | (345) =—0.128 | [37%] =+40.295 | [414] =+0.369 | [45,3] =+0.178 [325] =+0.056 | [34,4] =—0.782 | [37] =—0.034 | [41,]=—1.351 | [45,] =—0. 435 [325] =—0.531 | [34] =-++0.136 | [37] =—0.686 | [415] =-++0. 189 Residuals resulting from substitution of general corrections in numerical equations of condition. ene a Residual. ee Residual. 1 0. 0000 21 0. 0000 2 0. 0000 22 0. 0000 3 +0. 0001 23 -L0. 0001 4 +0. 0006 24 —0. 0004 5 0. 0000 25 0. 0000 6 0. 0000 26 +0. 0002 7 +40. 0024 27 +0. 0001 8 +0. 0001 28 0. 0000 9 40. 0032 29 —0. 0060 10 0. 0000 30 +0. 0001 11 —0. 0120 31 +0. 0001 12 0. 0000 32 0. 0000 13 0. 0000 33 +0. 0006 14 —0. 0012 34 +0. 0001 15 0. 0000 35 +0. 0017 16 0. 0000 36 0. 0000 17 —0. 0001 37 0. 0000 18 —0. 0001 38 0. 0000 19 0. 0000 39 +0. 0003 20 —0. 0008 PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § d. Let m= whole number of observed angles in a section (one adjustment). r=whole number of rigid conditions in a section. n=number of triangles in principal chain. [ pvv]=sum of weighted squares of corrections to observed angles. ¢i=probable error of an observed angle of weight unity. p.=probable error of an observed angle of average weight in whole section. p.=probable error of an adjusted angle of average weight in whole section. p,=average weight of an observed angle in whole section. p,=average weight of an observed angle in principal chain. p.=probable error of an observed angle of average weight in principal chain. p.=probable error of an adjusted angle of average weight in principal chain. [vv]=sum of squares of closing errors of triangles in principal chain. #,=probable error of an observed angle in principal chain as derived from the closing errors of triangles. §§ 5,6.] SANDUSKY BASE TO BUFFALO BASE. 511 Proceeding as in Chapter XIV, ©, § 8, there are found the following values: FOR THE ENTIRE SECTIONS OF THIS CHAPTER. m | 7 | [pov] | p, D, Ps Af H— | py Section. Extent of section. os ™m “ “ uw 57 33 8.72 0.385 | 1.01 | 0.34 0. 65 0. 22 89 58 | 29. 66 0.48} 1.02 | 0,48 0, 59 0, 28 XI | Chester - Willoughby to Grand River - Westfield ..... XII | Grand River- Westfield to Falkirk -Pekin..-.-.. acai iaraes FOR THE PRINCIPAL CHAIN CONNECTING THE SANDUSKY AND BUFFALO BASES, GIVEN IN D, § 6, FOLLOWING. From closing errors of triangles. Section. Extent of principal chain in each section. Pe Pe Po! [vv] ; Average | Greatest ks Pe error. error. Ww Ww a “we “a X | Sandusky Base to Chester - Willoughby .-..-......--- 0.80 | 0.41 | 0.24 | 12.43] 14 | 0.37 0. 82 191 XI | Chester- Willoughby to Grand River- Westfield ..-.. 0.99 | 0.35 | 0.23 | 12.49 | 12 | 0.40 0, 81 2. 33 XII | Grand River- Westfield to Buffalo Base .......-....-- 0. 84 | 0.53 | 0.31 | 10.50] 10 | 0.40 0. 78 2.18 Entire principal chain .......-.-.. 2.2.2. 020. eee lee eee] eens] ee eee 35.42 | 36 | 0.39 0. 80 2.33 D.—PRINCIPAL CHAIN OF TRIANGLES BETWEEN SANDUSKY AND BUFFALO : BASES. § 6. The principal chain of triangles joining Sandusky and Buffalo Bases has two triangles in common with the chain joining Sandusky and Chicago Bases, and five triangles in common with the chain joining Buffalo and Sandy Creek Bases. As the latter two chains were adjusted inde- pendently of the first, the complication to which the adjustment of the three chains would lead has been avoided as unimportant. The adjustment of the present chain, therefore, is made between the lines Danbury -Sandusky and Drummondville- Ridgeway, these being the lines separating this chain from the other two, the adjusted length of the first being taken from the chain joining San- dusky and Chicago Bases, and the adjusted length of the second from the chain joining Buffalo and Sandy Creek Bases. From Chapter XVII, D, the weighted mean value of the logarithm of the line Danbury -Sandusky is 4.6715463+ 12.21, the unit of length being the English foot, and the probable error being in units of the seventh decimal place. From Chapter XIX, D, the weighted mean value of the logarithm of the line Drummondville-Ridgeway is 4.8944527422.45. These sides will not have their values changed in obtaining weighted mean sides for the intervening triangulation. The logarithm of the latter line as computed from the line Danbury—-Sandusky and the connecting triangles is 4.8944500, giving a discrepancy of 27 units in the seventh decimal place. The triangles of the chain fall in three different sections of the general adjustment, the dividing lines of which between the two bases are Chester— Willoughby and Grand River—West- field. The probable errors of observed angles of average weight in these parts of the principal chain are in order trom Sandusky Base east, +0’.41, 10.35, and +0/.53, respectively. (See Chap- ter XVIII, C,§ 5.) With these data and those furnished in the following tables we have, using the notation of Chapter XIV, D, d=+27 a+5=2 (02 +f) p?+1494.504—=3579 ae es : : ‘ From these quantities and the values of D- given in the tables, the corrections to the logarithms of the sides computed from the line Danbury-Sandusky are readily derived. The arrangement of the tables is the same as that of the tables in Chapter XIV, D, to which reference may be made for a detailed explanation. The line in the system having the minimum weight is Claridon—Little 512 PRIMARY TRIANGULATION. [Cuar. XVIII, D, 1 1 Mountain, for which oo and moe 95. These values give for the probable error of the log- arithm of this line in units of the seventh decimal place, +29.9, which corresponds to zzs!-yo part of the Jength of the line. Principal chain of triangles between Sandusky and Buffalo Bases. Logarithms Weighted mean Stations. Angles. rons Be of sides in | a? and B2| X (a?4+?) 2X logarithms of feet. P sides in feet. ° t Wt “a ANONOY Bic con S nemeroicd masa 39 46 52. 697 ( 4, 6715463 B40.109): | isco sacorediencic [aisizisie nie 4. 6715463 SaAnMUSKY csc ssesces wees 42 52 18.183 +0. = A6982004. | care wes | bow eee ciseee Heenesed 4. 6982006 Danbury 97 20 49. 668 4, 8618821 7.29 647. 38 258 4, 8618823 Brownhelm ........--------- 82 56 387. 045 j 4, 8618821 WOSG125° | css eo crsceccoleede ote 4. 8618823 Sandusky...-2.-----.---.--- 81 34 32.572 +1. 054 5. 1217227 5. 1217230 Rely s:. ss sccnseceveecexs 65 28 52. 447 | 5. 0853917 92. 16 1795. 79 452 5. 0853920 ¢ Townsend .......---...--4+ 99 57 27.426 5. 0853917 + DB S09 loss .2 cic cidseitreSal] sed cities 5. 0853920 BrO wi heliaywciceeicins vsccesies 47 23 46. 434 +0. 772 49588915 I) <2cecneer jpaeereceeen|Seed aces 4, 9588920 Sandusky isesccec se: cee seeer 82 38 47.548 | J 4. 8239372 1075. 84 2885, 32 636 4. 8239377 COICO oictee!sg.cresciersiscieicbess oe78 90 20 01.432 |} { 4, 8239372 OOM seer uirael | Steet | 4. 8239377 Brownbhelm .3240s:-2vsscasece 50 46 05. 363 +0. 831 4, TIBOLT9 |esvsstexes| wees caceeaxel> | 170L 4. 6491472 Claridonl:.,cc.2cas.eseceeeates 40 25 02. 596 | { 4. 6491459 610. 09 | nocuccarsecimials [Peed aed 4. 6491472 Little Mountain ....-.-.--.- 71 41 06.884 | +1.247 4,8147600 |.........- ie aw aniaraie aaa belek a 4. 8147613 CHEStEP a sceereteegeceearess 67 53 51.156 | 4. 8041874 72. 25 1020. 19 1784 4. 8041887 § 6.] SANDUSKY BASE TO BUFFALO BASE. Principal chain of triangles between Sandusky and Buffulo Bases—Coutinued. | Logarithms | , Weighted mean Stations. Angles. Errors of of sides iu | a? and 62] X (a24-62) 1 logarithins of | closure. | feet P sides in feet | | i fe} - a aw Thompson .................. 57 19 50. 116 4. 8041874 4. 8041887 : | [. ClATIGON sec iccsts eet saccmce 59 41 44. 690 i —1.178,, 4, 8151699 4, 8151713 Little Mountain ............ 62 58 26. 067 | J | | 4. 8287590 4 8287604 ' aes Mesopotamia -.......-...... 47 45 14.930 { 4. 8287590 BOB 5642 lion teaser Rett se 4. 8287604 Thompson ..........-..--.-- 25 50 49. 864 —0. 0724 PODS BOO. aac ad wats mein tesa see ec accra 4. 5988306 OD eck eascctoutonrace 23 55.810 « 9413347 38. 44 726. 16 1870 4. 9413361 Clartd 106 23 ( 4.9 1 4 ANOVED: 222.02 icendstcaeede x 41 49 59. 375 | f 4. 9413347 GOS 2S: Nectar stein sccelnls esc 4.9413361 Thompson.......--..-..... 70 00 41.322 +0. 548, SO902SLL. | acicusstecrctull Ov aes Aoces ee eaves 5. 0902526 Mesopotamia -.............. 68,09 21. 657 i l 5. 0848754 70. 56 2318. 97 1946 5. 0848769 Conneaut 57 47 51.151 f 5. 0848754 TES Vovsisisierl celetcteel| See esceys 5. 0848769 Andover... 77 OL 08.719 +2. 330 DB AGGLIOR Woveendoessleserenameceeil|a':scmcee 5. 1461772 Thompson 45 11 02.977 | 5. 0082933 436. 81 2960. 02 2021 5. V082948 Edinboro ...----..........-- 42 42 44. 009 { 5. 0082933 519. 84 ; adle presen few cake ss 5. 0082948 Conneaut .................-- 85 07 58. 439 —0. 343 5. 1752941 : 5. 1752957 Andover ......-.----.....24. 52 09 20.392 | 5. 07431380 265 69° 3745.55 2116 5. 0743146 L . BIO: sss dacdodosnsdadaessess 64 35 04.213 { 5. 07431 30 100. 00 5. 0743146 Conneaut -..--.-..--..- -.-. 31 14 58.161 —0. 347 He 8334804" bec ccwaccas|oecsosecese |seeves <4 4. 8334910 Edinboro ........-.-.---.... 84 09 59. 522 t 5. 1162654 4. 84 3850. 39 2129 5. 1162670 Houghton ......-....-..2--- 31 08 13.993 { 5. 1162654 VO OSs bo caciclscote.nellaceabete 5. 1162670 BMi6: conse sensectrcsecacec ae 84 19 49. 168 +0. 781 64005716: levenorcwss|escaeeeass nal secetecs 5. 4005793 Conneaut ......-...----..--- 64 32 03. &35 | J | 5. 3583181 100. 00 5161. 43 2287 5. 3583198 | Long Point ............ .-.. 88 43 45. 022 { 5. 36 83181 5. 3583198 Erie ..--.-- 37 51 30.473 + 0. 480 5. 1463849 5. 1463866 Houghton 53 24 50. 555 | 5. 2631177 5. 2631194 Westfield pc ssnieccscezcace: 64 51 00.913 | { 5. 2631177 8201. | cm asctcome el aes 5. 2631194 Long Point .-....--. pore, dist S 57 28 15. 490 —0. 591 5. 2822613 |...------- bs eed capslincdetets 5. 2322631 BVi@. saree eeone cence eracnces 57 40 49. 829 J | 5. 2332702 176. 89 5679. 94 2352 5, 2332720 Grand River’, 20: cesaccesacxx 52 26 09.770 l { 5. 2332702 262 EE | rotarsiersiciccinizraftachina do 5. 2332720 Westfield .........2-..------ 52 02 06. 329 —0. 341 D209 ST a accneneesl senor scmsecnl meee ewer 5. 2309175 Long Point .......---.--2+-+ 75 31 50. 547 J | 5. 3201800 30. 25 5972. 63 2387 5. 3201818 a Silver Creek ......---------- 87 42 19. 442 l ( 5. 3201800 0.81) |exseexuawess beaed sek 5. 3201818 Grand River......---------- 42 17 38.277 +0. 5.1484973 |... vortecs [cess eceeeee|e teense 5. 1484992 Westfield ...- 50 00 07. 595 J 5, 2047928 309, 76 310. 57 2473 5. 2047947 Brigae LOA! ccc: eseacessuns 90 58 38. 854 | { 5. 2047928 OLMGr We cscee Bawete been Saad 5. 2047947 Silver Creek .....--..--.---- 35 10 45. 656 4 --1. 344) HOCH STIS! i seen Sec ute arcis lowie oa! eassecid cre 4. 9653814 Grand River ....-..-----.--. 53 50 38.306 |) l 5. 1119505 234. 09 544, 82 2538 5. 1119524 Sturgeon Point .-..-----.--. 99 26 38.715 { 5. 1119505 FB 2D lossy scutes tine nes 5. 1119524 Sugar Loaf ......-.--.----- 36 26 40. 041 -+ 1. 689 4. S91GOBE. Pisscirce cess) ssssms tect scenes 4. 8916954 Silver Creek ......----...--.| 44 06 42. 897 | 4. 9605237 470. 89 1027. 96 2671 4. 9605257 Ridgeway ...--.------------ 91 88 54. 355 | f 4. 9605237 OE SGN. Faces ecco celacec 4, 9605257 Sugar Loal ..--<2 20000. 4069 47 55 45. 358 ( +0. =| 4, 8312020 gaps eee eens : ecussiacnigelte mrciesgiage 4. eee Sturgeon Point ..---...-.--. 40 25 21. 233 | J 4, 7725589 610. 09 1638. 41 2840 4. 7725610 Drummondville 39 15 59. 935 { 4. 7725589 4, 7725610 Ridgeway .........--------- 83 48 12.997 +0. 099 4. 9686592 - 4. 9686615 56 55 48. 157 4. 8944500 187, 69 2486. 59 3075 4, 8944527 Sugar Loaf .....-.---..----- 65 LS 513 514 PRIVARY TRIANGULATION. [Cnap. XIX, A, \ CHAPTER XIX. TRIANGULATION FROM BUFFALO BASE TO SANDY CREEK BASE. A—DESURIPTIONS OF STATIONS. NOTE RELATIVE TO ELEVATIONS. § L. The heights of ground at stations deseribed in this chapter and in Chapter X VIII, from station Font Hill to station Oswego, both inclusive, depend upon connections with the surface of Lake Ontario made at Oswego by spirit-levef and at Sodus by a single zenith distance over a line about 3 miles long, and upou connections with the surface of Lake Erie at West Base (Buffalo Base) by spirit-level and at Ridgeway by single zenith distances over two short lines. East of Oswego the heights depend on connections with the water surface at North and South Base (Sandy Creek Base) and Stony Point. Heights of stations not directly referred to the lake surface were computed mostly from non-simultaneous, reciprocal zenith distances. No exact adjustment of these heights has been made, but judging from discrepancies in heights computed by inde- pendent routes, the probable error of any height does not exceed +4 feet. Heights of stations on the islands of Lake Ontario and on the north shore were obtained chiefly from the detail topo- graphical charts, and may be erroneous to an extent indicated by the above probable error. DESCRIPTIONS OF STATIONS. § 2. GAspont, 1875, ’78.*—This station is situated in Royalton Township, Niagara County, New York, about 14 miles east and one-fourth mile south of the village of Gasport, and about 15 metres south of the limestone ledge. The height of station used was 102 feet. The geodetic point is marked by a nail leaded into the solid rock, about 1 foot below the surface of the ground. A stone post is set directly over the geodetic point as a surface-inark, rising 8 inches above the ground. Three stone reference-posts are set as follows: One bearing north 8° OL west, distaiit 9 metres; one bearing north 49° 31/ east, distant 21.75 metres; and one bearing north 80° 11/ east, distant 22.1L metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 397.8 feet. . BATAVIA, 1875.—This station is situated in the township of Batavia, Genesee County, New York, about + miles northwest of the city of Batavia, on a slight rise of ground about one-half mile north of an east-and-west highway. The height of station used was 89 feet. The geodetic point is marked by a hole drilled in the top of a stone post set 2 feet below the surface ef the ground. A stone post is set directly over the geodetic point for a surface-mark. Three stone reference-posts are set as follows: One bearing south 80° 05/ east, distant 31.0 metres; one bearing south 36° 12/ east, distant 30.6 metres; and one bearing north 34° 55’ west, distant 21.9 metres from the geo- deti¢ point. The Blind Asylum at Batavia bears scuth 74° 38/ east, and is distant 3 or 4 miles; a beech tree 18 inches in diameter bears south 10° 20/ east, distant 8.3 metres; a hemlock tree 30 inches in diameter bears north 79° 05/ west, distant 12.7 metres; and a rail fence is distant 93 metres on the east. The height of ground at the station above mean. level of Lake Ontario is 711.5 feet. ALBION, 1875.—This station is the soldiers’ monument in the cemetery 2 miles east of Albion, Orleans County, New York. The monument stands on the highest point in the cemetery, is built of sandstone, and its tower is 60.5 feet high. The geodetic point is on the west side of the center *See note concerning topographical sketches of stations under Burnt Bluff, Chapter XV, A, § 2. $§ 1,2] : BUFFALO BASE TO SANDY CREEK BASE, 515 or “well” of the monument, and is marked by a one-fourth inch hole drilled in the coping-stone. A stone reference-post 2 feet long and 6 inches square is set with its top 4 feet below the surface of the ground vertically under a point 3.524 metres from the geodetic point on the line to Batavia station, bearing south 19° 54/ 46.07 west. This reference-stone is in turn referred to three holes drilled in the southwest side of the monument, at a height of 3 feet above the ground, distant respectively 1.65 metres, 1.03 metres, and 1.47 metres, the first mentioned being farthest south. Distances from the reference-stone to marble monuments with square granite bases were taken as follows: Dyer, 3.9 metres; Hill, 34.78 metres; Smith, 25.0 metres; French, 29.3 metres. The height of ground at the station above mean level of Lake Ontario is 461.1 feet. MORGANVILLE, 1875.—This station is situated in Stafford Township, Genesee County, New York, about 7 miles cast of the city of Batavia and 14 miles east of the village of Morganville. The height of station used was 75 feet. The geodetic point is marked by a hole drilled in the top of a stone post and filled with lead, set 2 feet below the surface. Three stone reference-posts are set as follows: One bearing north 62° 33’ east, distant 33.35 metres; one bearing north 2° 23’ east, distant 15.75 metres; and one bearing north 60° 59 west, distant 30.1 metres from the geodetic point. The refererce-stones are set along the road-fence on the north of the station. The height of the ground at the station above mean level of Lake Ontario is 635.1 feet. Brockport, 1875.—This station is situated in lot 13, section 7, district No. 3, Sweden Town- ship, Monroe County, New York, about 14 miles south and half a mile east of the village of Brock- port. The height of station used was 38 feet. The geodetic pointis marked by a cut marble post of the usual form, set 3 feet below the surface, with a stone of the same form set directly over it and projecting 6 inches above the surface. Two stone reference posts are set in the road to the south of the station as follows: One bearing south 22° 24’ west, distant 471.64 metres; and one bearing south 0° 31’ east, distant 451.51 metres from the geodetic point. The intersection of an east-and-west road about half a mile north of the station, and a north-and-south road, which is a prolongation of Main street, of Brockport, is at the corner of lots 5, 11, 8, and 14, and bears north 45° 55’ west, and is distant 1118.8 metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 481.1 feet. ScorrsviLLE, 1875, ’77.—This station is situated in Rush Township, Monroe County, New York, on a steep, wooded hill rising 150 feet above the general level, about 25 miles southeast of the village of Scottsville and about Lalf a mile east of Scottsville station, on the Rochester branch of the Erie Railway. The height of station used was 35 feet. The geodetic point is marked by a hole drilled in the top of a marble post set 14 feet below the surface. A marble jost is set directly over the geodetic point as a surface-mark. Three stone reference-posts are set as follows: One bearing south 24° 48’ west, distant 16.6 metres; one bearing south 72° 58’ west, distant 21.3 metres; and one bearing north 31° 08 east, distant 16.2 metres from the geodetic point. The church at North Rush bears north 33° 24’ east, distant about one-third of a mile; an oak tree 18 inches in diameter bears south 76° 28’ east, distant 5.6 metres; and an.oak tree 12 inches in diam- eter bears north 19.32’ east, distant 6.1 metres from the geodetic point; The height of ground at the station above mean level of Lake Ontario is 545.7 feet. PINNACLE HILL, 1875, ’77.—This station is situated in the township of Brighton, Monroe County, New York, in a Catholic cemetery, on the summit of a hill known as Pinnacle Hill, about 2 miles southeast of the center of the city of Rochester. The height of station used was 33 feet. The geodetic point is marked by a cross and the letters U. S. cut in a bowlder 3 feet below the surface, with an ordinary marking-stone set directly over it, rising to the surface of the ground. JT'wo stone reference-posts are set as follows: One bearing north 15° 23/ west, distant 22.56 metres; and one bearing south 13° 12/ east, distant 31.73 metres from the geodetic point. A large marble monument, marked “Mahon” on the base, bears south 80° 55’ west, distant 87.97 metres; a large granite monument, marked “Cummings” on the base, bears south 70° 26/ west, distant 117.75 metres from the geodetic point. A black-oak tree bears north 82° 02’ west, distant 25.54 metres. The Rochester court-house bears north 45° 38’ west, distant about 2 miles. The height of ground at the station above inean level of Lake Ontario is 502.1 feet. TuRK’s Hint, 1875, 77.—This station is situated near the southern boundary of Perrinton Township, Monroe County, New York, about 2 miles south of the village of Egypt, and about 3 516 PRIMARY TRIANGULATION, ECrap. XIX, A, miles northeast of Fisher's, a station on the Auburn branch of the New York Central Railroad. It is on a hill about 1 mile southeast of the hill generally known as Turk’s Hill. The height of sta- tion used was 28 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet below the surface of the ground. Three stone reference-posts are set by the road-fence on the east of the station as follows: One bearing north 49° 23/ east, distant 21.2 metres; oné bearing north 85° 10/ east, distant 17.9 metres; and one bearing south 56° east, approximately, distant 20.9 metres. The northeast corner of a house with permanent stone foundation is 16.4 metres distant on the south. The height of ground at the station above mean level of Lake Ontario is 680.9 feet. WALWORTH, 1875, ’77.—This station is situated in lot 121, Walworth Township, Wayne County, New York, about three-fourths of a mile northeast of the village of West Walworth. A tripod about 4 feet high was used to support. the instruinent while observing. The geodetic point is marked by a single stone post of the usual form, set 24 feet below the surface of the ground. Three stone reference-posts are set as follows: One by the fence on the east of the station, bearing south 26° 51’ east, distant 82.39 metres; and two in the north-and-south road about one-quarter ofa mile west of the station, one bearing south 62° 14’ west, distant 530.76 metres; and one bearing north 86° 1 west, distant 480.9 metres. The height of ground at the station above the mean level of Lake Ontario is 406.2 feet. PALMYRA, 1875, °77.—This station is situated in lot 29, Palmyra Township, Wayne County, New York, about 3 miles east of the village of Palmyra and 14 miles southwest of East Palmyra, on the highest point of a long, narrow hill extending north and south. The height of station used was 49 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet. below the surface of the ground. Three stone reference-posts are set along the line-fence west of the station as follows: One bearing north 60° 29’ west, 49.8 metres distant; one bearing south 77° 41’ west, 40 metres distant; and one bearing south 40° 174’ west, 51.7 metres distant from the geodetic point. The distance from the northwest stone to the west stone is 32.8 metres, and from the west stone to the southwest stone is 31.6 metres. A hard-maple tree 9 inches in diameter stands 29.1 metres approximately north of the geodetic point. The height of ground at the station above mean level of Lake Ontario is 418.7 feet. CLYDE, 1875.—This station is sitnated in lot 73, Galen Township, Wayne County, New York, about 2 miles south of the village of Clyde, on a high, narrow ridge running approximately north and south. The height of station used was 4 feet. The geodetic point is marked by a stone post of the usual form, set 24 feet below the surface of the ground. Two stone reference-posts are set by the line-fence running north and south about 2 metres west of the station, as follows: One bearing north 23° 52/ west, distant 8.0 inetres, and one bearing south 1° 304’ west, distant 5.6 metres from the geodetic point, the distance between the two reference-stones being 14.88 metres. A third reference-stone is set by the road-fence west of the station, bearing north 89° 30’ west, distant 158 metres. The ground at the station is 168 feet above the road. A basswood tree, 14 inches in diameter, the only tree on the hill, stands 5.43 metres north of the geodetic point. The height of ground at the station above mean level of Lake Ontario is 391.9 feet. Vicrory, 1875.—This station is situated near the eastern boundary of Victory Township, Cayuga County, New York, about half a mile southwest of Ira railway station, on the Southern Central Railroad. The height of station used was 85 feet. The geodetic point is marked by a cross cut in lead in the top of a stone post of the usual form, set 2 feet below the surface of the ground. Three stone reference-posts are set as follows, the bearings being approximate: One bearing south 22° 10/ east, distant 16.7 metres; one bearing north 31° east, distant 16.9 metres; and one bearing north 44° 57’ west, distant 15.3 metres from the geodetic point. The station stands on a hill about 75 feet above the general level of the ground. The height of ground at the station above mean level of Lake Ontario is 326 feet. Sopus, 1875, ’77.—This station is situated in lot 6, Sodus Township, Wayne County, New York, on Greene’s Hill, about half a mile west of the village of Sodus and one-fourth mile south of the Lake Ontario Shore Railroad. The height of station used was 75 feet. The geodetic point is marked by a stone post of the usual form, set 3 feet below the surface of the ground, with a stone post set directly over it for a surface-mark. Two stone reference-posts are set as follows: One §2.] BUFFALO BASE TO SANDY CREEK BASE. 517 bearing north 47° 15/ west, distant 11.25 metres; and one bearing south 34° 50’ west, distant 12 metres from the geodetic point. Sodus Academy bears south 78° 28/ east, and is distant 815.3 metres. The height of ground at the station above mean level of Lake Ontario is 347.8 feet. VANDERLIP, 1875, ’77.—This station is situated about 450 metres frum the shore of South Bay, Lake Ontario.s in Prince-Edwards County, Province of Ontario, about 8 miles southeast of the town of Picton, half a mile south of the post-office of Cardwell, and half a mile southwest of the mouth of Black Creek. The height of station used was 75 feet. The geodetic point is marked by a dentin lead run into the solid rock about 1 foot below the surface of the ground. Three stone refer- ence-posts are set as follows: One bearing north 18° 32’ east, distant 141.73 metres; one bearing north 33° 30/ west, distant 92.95 metres; and one bearing south 51° 26/ west, distant 29.5 metres. The height of ground at the station above Lake Ontario is about 150 feet. SoutH BAsE, 1874, 75, ’78.—This station, marking the south end of the Sandy Creek base- line, is situated close to the lake shore, near the southern end of a sand point lying between North Pond and Lake Ontario, in Sandy Creek Township, Oswego County, New York. The height of station used was 72 feet. The geodetic point is marked by a brass frustum leaded into a stone 8 inches by 8 inches by 3 feet, between the letters U. 8. cut in the stone, the top of the stone being 3 feet below the surface of the ground. Two stone posts are set on a line to the east approximately at right angles to the base-line, one 3 feet long and set 33 feet below the surface, distant 3.39 metres from the geodetic point, and one 5 feet long set 4 feet below the surface, and distaut 7.98: metres from the geodetic point. Three stone reference-posts are set as follows: One bearing north 71° 38’ east, distant 34 metres; one near the hotel on the south side of Little Sandy Creek, bearing south 32° 01’ east, distant 543.6 metres; and one on the southeast side of North Pond, bearing south 70° 58 east, distant 842.9 metres from the geodetic point. The height of ground at the station above the lake is 5.7 feet. MIpDLE BASE, 1874.—This station is at the end of the 523d tube of the Sandy Creek base- line from South Base. A small tripod about 4 feet high was used to support the instrument during observations. The geodetic point is marked by a single stone of the usual form, 5 feet long, and set 5 feet below the surface. The héight of ground at the station is about 11.5 feet above Lake Ontario. | .NortH Bass, 1874.—This station, marking the north end of the Sandy Creek base line, is situated on the lake shore, near the boundary line between Oswego and Jefferson Counties. The height of station used was 60 feet. The geodetic point is marked by a brass frustum leaded into a stone post 8 inches by 8 inches by 5 feet, set 5 fect below the surface of the ground. Two stone posts are set on a line through the. geodetic point perpendicular to the direction of the base-line; one 5 feet long and set 5 feet below the surface, bearing north 87° 14’ cast, distant 3.79 metres; and one 5 feet long and set 5 feet below the surface, bearing south 87° 14’ west, distant 3.70 metres from the geodetic point. Two stone reference-posts are set as follows: One bearing north 3° 09/ east, distant 60.17 metres; and one bearing south 82° 14’ west, distant 20.87 metres from the geodetic point. A tree blazed on the side toward the station bears north 2° 55’ east, distant 60.5 metres. The height of ground at the station above mean level of Lake Ontario is 11.9 feet. Sanpy CREEK, 1874.—This station is situated in Sandy Creek Township, Oswego County, New York, about 1 mile northwesterly from the village of Sandy Creek. The height of station used was 40 feet. The geodetic point is marked by a stone post of the usual form, set 3 feet below the surface of the ground, with a stone post set directly over it as a surface- iat: Two stone reference posts are set as follows: One bearing south 29° 16’ west, distant 69.5 metres; and one bearing south 28° 59’ east, distant 65.5 metres. The sefarance-atones are set in a line of maple trees on the south side of the road running by the station on the south. The height of ground at the station above mean level of Lake Ontario is 235.7 feet. MANNSVILLE, 1874, ’75.—This station is situated about half a mile northeast of the village of Mannsville, in lot 177, Ellisburg Township, Jefferson County, New York, on the highest land in the immediate vicinity. A tripod about 4 feet high supported the instrument during observations. The geodetic point is marked by a single post of cut marble, set 18 inches below the surface of the ground. Three marble reference-posts are set as follows: One bearing north 67° 48’ east, distant 91.79 metres; one bearing south 54° 30’ east, distant 107.84 metres; and one bearing south 34° 06/ west, distant 393.1 metres from the geodetic point. The first two ‘reference: stones mentioned are 518 PRIMARY TRIANGULATION, [CHap. XIX, B, ©, set close to a stotie wall east of the station, and the third in the road (Lorain street) on the south of the station. The northeast corner of lot 177 bears north 23° 44’ east, and is distant 190.7 metres. A stone astronomical post, occupied in 1875, standing on the north side of Railroad street, near its janection with Lorain street, bears south 68° 28’ west, and is distant 630.98 metres froin the geodetic point. The height of ground at the station above mean level of Lake Ontario is 474.8 feet. OSWEGO, 1875, °78.—This station is situated in the southwest part of the city of Oswego, New York, one the grounds of the Orphan Asylum, corner of Ellen and West Seventh streets. The height of station used was 100 feet. The geodetic point is marked by a stone post of the usual form, set 2 feet below the surface of the ground, a stone post being set directly over it as a surface- mark. Three stone reference-posts are set as follows: one in the northeast corner of the lot on which the asylum stands, bearing north 50° east, distant 52.50 metres; one in the northwest cor- ner of the lot, bearing north 54° west, distant 49.65 metres; and one bearing south 6° west, distant 49.38 metres from the geodetic point. The bearings given above are magnetic. The Oswego court- house bears north 22° 24 east (true), and is distant 1627 metres. An astronomical post, ocenpie.l in 1868, bears north 16° 58 east, distant 2242 metres from the geodetic point. The height of ground at the station above mean level of Lake Ontario is 168 feet. Stony Pornt, 1874, 75, ’+8.—This station is situated on Stony Point, Lake Ontario, in Hen- derson Township, Jefferson County, New York. The height of station used was 74 feet. The geodetic point is marked by a hole in the solid rock, filled with lead, 20 inches below the surface of the ground. A stone post is set over the geodetic point asa surface-mark. Three stone refer- ence-posts are set as follows: One bearing south 69° 30’ east, distant 28.9 metres; one bearing north 63° 30/ east, distant 26.0 metres; and cone bearing north 73° 48’ west, distant 30.6 metres from the geodetic point. An oak tree bears south 5° 30/ west, anc is distant 29 metres; a bass- wood tree bears north 45° 30/ west, and is distant 19 metres. The lake shore on the west is distant 391 metres. Stony Point light-house bears south 38° 38/ west, and is distant 1025.8 metres. The height of ground at the station above the mean level of Lake Ontario from 1860 to 1875 is 26.3 feet. : GRENADIER, 1875.—This station is situated on the southwest side, near the west end of Grenadier Island, in the lower end of Lake Ontario. The station is about 170 metres back from the head of a small bay. The height of station used was 33.5 feet. The geodetic point is marked by a stone post of the usual form set 2 feet below the surface. Three stone reference-posts are set as follows: One approximately north, distant 40.05 metres; one northwest, distant 50.25 metres ; and one northeast, distant 54.25 metres from the geodetic point. The northwest stone is distant 31.1 metres from the north stone, and the northeast stone is distant 46 metres from the north stone. The height of ground at the station above the lake is 18.4 feet. . Duck ISLAND, 1874, ’75.—This station is situated on the north side of Main Ducks Island. The height of station used was 45 feet. The geodetic point is marked by a hole drilled in the solid rock and filled with lead, about 1 foot below the surface of the ground. A stone post is set over the geodetic point as a surface-mark. Three stone reference-posts are set as follows: One bearing south 66° 41’ west, distant 28.3 metres; one bearing north 24° 34’ west, distant 35.4 metres; and and one bearing north 66° 07’ east, distant 33.2 metres from the geodetic point. False Ducks light- house bears north 83° 56’ west, distant about 9 miles. The height of ground at the station above the lake is 21.8 feet. AMHERST, 1874.—This station is situated on the south side of Amherst Island, about midway between the east and west ends, and 116 metres from the lake shore. A roadway runs along the shore in front of the station at a height of 10 feet above the level of the water. The height of station used was 23.5 feet. The geodetic point is marked by a nail leaded into the solid rock about 18 inches below the surface of the ground. Three stone reference-posts are set along the fence southwest of the station, as follows: One bearing north 72° 13’ west, distant 14.2 metres; one bearing south 39° 48’ west, distant 9.1 metres ; and one bearing south 16° 44’ east, distant 28.45 metres from the geodetic point. The distance from the intersection of the above fence with the road to a house owned by Mr. McCormick is 138.5 metres in a southwesterly direction along’ the road. The height of ground at the station above Lake Ontario is about 10 feet. §§ 3,4.) BUFFALO BASE TO SANDY CREEK BASE. 519 WOLFE, 1874.—This station is situated on the southeast shore of Wolfe or Long Island, about 135 metres back from the water’s edge, in a sandy field on a bluff about 60 feet high. The height of station used was 33.5 feet. The geodetic point is marked by a stone set 1.7 feet below the sur- face of the ground. Three stone reference-posts are set by the road-fence northeast of the station, as follows: One bearing south 88° 28’ east, distant 100 metres; one bearing north 34° 17’ east, distant $5.17 metres; and one bearing north 1° 10/ west, distant 139.38 metres from the geodetic point. The height of ground at the station above the lake is 57.9 feet. KINGSTON, 1874.—This station is the square tower of the Wesleyan Methodist Church, facing east, on the corner of Williams street and Sydenham Place, in the city of Kingston, Province of Ontario. The geodetic point is on the second stone pier from the south front pinnacle, and is marked by a hole drilled into the stone. On the stone is inscribed U.S. L. 8., 1874. The height of the instrument above the ground was 108.5 feet. The northwest corner of the residence, on the southeast corner of Johnson street and Sydenham Place, bears north 54° 15’ east, and is distant 76.16 metres; the southeast corner of Regiopolis College, on the north side of Johnson street at the north end of Sydenham Place, bears north 37° 54/ east, and is distant 114.6 metres; the south- west corner of Regiopolis College bears north 32° 07’ east, and is distant 109.8 metres from the geodetic point. The transit-pier of the Kingston Observatory is situated south 527.2 metres and west 101.9 metres from the geodetic point. CARLTON, 1874.—This station is situated on Carlton Island, in the upper end of the Saint Law- rence River, about one-half mile from the north shore of the island, in a piece of open.woods on a ridge about 80 feet above the water. The height of station used was 33 feet. A triangular hole about 1 inch deep on the south side of a large irregularly-shaped hole cut in the solid rock marks the geodetic point. A stone post 5$ inches square, with a 14-inch triangle drilled in the center, is placed over the geodetic point for a surface-mark. The station is referred to a blaze on a pine tree bearing south 70° 28’ west, distant 7.65 metres, and to a blaze on a stump bearing south 42° 32/ east, distant 4.65 metres from the geodetic point. An east-and-west rail-fence is south of the sta- tion 93 metres. The height of ground at the station above the Saint Lawrence River is about 80 feet. Sir Jonn, 1874.—This station is situated on the southwest side of Howe’s Island, Saint Law- rence River, in a pasture field about 60 metres northeast of a road running northwest and south- east, and about 535 metres from the intersection of the road with the water’s edge. The height of station used was 23 feet. The geodetic point is marked by a square stone post rising to the sur- face of the ground. Three stone reference-posts are set as follows: One bearing south 86° 33! west, distant 69.15 metres; one bearing south 60° 35’ west, distant 63.78 metres; and one bearing south 35° 28’ west, distant 71.91 metres from the geodetic point. The stones are set along the road-fence. The bearings given are approximate. B.—STATIONS, SIGNALS, INSTRUMENTS, AND METHODS OF OBSERVATION. § B. See Chapter XVI, B. C.—MEASURED AND ADJUSTED ANGLES BETWEEN THE LINES FALKIRK - PEKIN AND CARLTON -SIR JOHN. § 4. The angles within the above limits were adjusted in one group. It is designated Section XII. The sketch to accompany it is in Plate V. , An abstract of the adjustment is given in the fol- lowing tables. Weight unity was assigned to the following number of combined results for the respective instruments: Troughton & Simms No. 1, 16. Troughton & Simms No. 2, 24. Troughton & Simms No. 3, 24 in 1877. Troughton & Simms No. 3, 16 in 1878. Pistor & Martins No. 2, 16. Gambey No. 1, 16. Repsold No. 1, 24. 520 For a detailed explanation of the tables see Chapter XIV, ©, § 7, and see the remark in Chap- ter XV, ©, § 6, relating to the column headed “ No meas.” The locally adjusted angles, with their resulting weights at stations Falkirk and Pekin, were used in computing the general adjustment PRIMARY TRIANGULATION. which disregarded a sum-angle condition at each of those stations. Cuap. XIX, C, SECTION NIII.—Triangulation from the line Sir John - Carlton to the line Falkirk — Pekin. (Observer, T. Russell. SIR JOHN—1. Instrument, Gambey theodolite No. 1. Dates, June and July, 1874.) i" Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) (v] Corrected angles. oO , wm “ aw uw Oo , “a Carlton and Kingston..........----- 90 17 44.914 lite 20 5.4 1 0. 000 +0. 951 90 17 45. 865 Wolfe and Kingston .....-.......--- 56 24 09.774 lez | 20 5.0 1 0. 000 +0. 850 56 24 10. 624 CARLTON—2. (Okserver, T. Russell. Instrument, Gambey theodolite No. 1. Date, June, 1874.] Angle as measured bet ween— | Notation. No. meas. | Range.} Wt. (v) (v] Corrected angles. i | a o y “ 7 | uw “ aw oF “ | Wolfe and Sir Jolin 2.22.0 -2..----- 120 48 06. 544 2ife | 20 4.0 1 0. 000 +1. 342 120 48 07. 886 { Kingston and Sir Jolin .......-..--. 62 03 27. 564 22 | 2 4.1 1 0. 000 +0. 058 62 03 27, 622 | (Observer, T. Russell. Instrument, Gambey theodolite No. 1. KINGSTON—3. Datcs, July and August, 1874.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) p) Corrected angles. x | | oF aw “uw aw aw ° ‘ “ Sir John and Wolfe. ......f...-....- 64 40 50. 914 dite 20 4.6 1 0. 000 +1. 228 64 40 52. 142 Carlton and Wolfe..........-------- 37 02 04. 434 32 20 2.5 1 0. 000 +0. 915 37 02 05. 349 |. Wolfe and Amherst .......-...---.- 88 19 14. 697 33 23 5.2 1 0. 000 +1316 88 19 16.013 (Observer, T. Russell. WOLYE—4. Instrument, Gambey theodolite No. 1. Dates, June, August, and September, 1874.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. ° , u aw uw “uw oO % Galloe and Carlito... ..+.2.2 scene 225 32 01.901 | 4142434445 16 3.9 1 0. 000 0. 000 225 32 01.901 Duck Island and Carlton ...-..--... 188 07 18.535 | 42434446 21 5.4 1 0. 000 —0.011 188 07 18. 524 Amherst and Carlton ......-......-- 140 12 34.439 | 4ata+s 20 2.6 1 0. 000 +1. 327 140 12 35. 766 Kingston and Carltun........-.----- 84 13 14.344 dats 26 4.9 1 0.000 | +0. 397 84 13 14.741 Sir John and Carlton ......-.....-.. 25 18 16.804 | 45 21 3.6 1 0.000. | +0. 234 25 18 17. 038 coe BUFFALO BASE TO SANDY CREEK BASE. 521 SECTION XIIL.—Triangulation from the line Sir John— Carlton to the line Fatkirk—Pekin—Coutinued. [Observer, T. Russell. Instrument, Gambey theodolite No. 1. AMHERST—5. Dates, October and November, 1874, and May, 1875. ] Angle as measured between— Notation. | No. meas. | Range. | Wt. _ k [v] | Corrected angles. | ° ‘ a “ uw o a “a { Kingston and Wolfe ...............- 35 41 23.025) bi | 16 Tl 1 | i, "239 4-0. 984 35 41 23. 67 | Kingston and Duck Island .......-. 1l1 45 28. 463 Sipe+3 20 ; 85 1) +0.219 | +0. 641 | 111 45 29. 323 Wolfe and Duck Island. ..........-- 76 04 06. 321 52+3 20 | 3.4 1 | —0.332 | 0.343 | 76 04 05. 646 | Grenadier and Duck Island. -. 54 38 00. 343 53 18 | 26 1 0.000 | —0. 636 | 54 37 59. 707 Duck Island and Vanderlip =. TL 15 25. 434 54 21 3.4 1 —0.112 | —0.951 TL 15 24. 3871 | Vanderlip and Kingston .........- 176 59 06. 108 55 20 3.6 1 | —0.112 | +0.310 176 59 06. 306 3(5,)+2(524+8)+ (54) +1.771=0 2(5))+3(52+3)+ (54)+1.771=0 (5) + (52+) 4+2(54)+0. 888=0 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. Novre.—di+3+2, 52+3, and parts of 5, and 54 were read in 1874; the remainder in 1875 DUCK ISLAND—6. [Observers, G. Y. Wisner and J. H. Darling. Instruments, Troughton & Simms 14-inch theodolite No. 1 and Repsold theodolite No.1. Dates, August, 1874, and June and July, 1875.] Angle as measured between— Notation. No.meas. | Range.| Wt. (v) | (v] Corrected angles. : oO ‘ aw “ “a a °o t “a Oswego and Vanderlip...-...--..--- 104 08 58. 927 | 6 i 40 | 10.4 2 | +0.102| +0.625 | 104 08 59. 654 Vanderlip and Amherst.....-..----. 70 26 31.989 62 | 40 8.4 2 +0.10L | —0. 854 70 26 31. 236 Awherst and Wolfe.....-.--.------- 56 01 12.475 63 : 16 3.8 1 +0. 2038 © +40. 203 | 56 OL 12. 881 Wolfe and Grenadier ..-..------.--. 18 45 43. 359 64 | 16 4.1 1 +0. 2038 | +0. 213 | 18 45 43.775 | Grenadier and Stony Point --.--..-. 49 53 12.770 65 16 4.5 1 +0. 203 | +-0. 856 | 49 53 13. 829 Stony Point and Oswego.........-.. 60 44 19. 465 66 * 16 4.5 1 +0. 203 | -1 044 | 60 44 18. 624 | { NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(6,)+ (6:)+ (63)+ (64)-+ (65)—1. 015=0 (61) +3(62)-+ (63)+ (64)+ (65)—1. 015=0 (61)-+ (62) +2(62)+ (64)+ (65) —1. 015=0 (61)+ (62)+ (63) -+2(64) + (65)—1. 015=0 (61) + (62)+ (63)+ (64)-+-2(65)—1. 015=0 Nore.—61 and 62 were read by Mr. Darling with the Repsold instrument in 1875, the remainder by Mr. Wisner, with the Troughton & Simms instrument in 1874. 66 LS Dae PRIMARY TRIANGULATION. [Cuar. XIX, C, Secrion XU1L—Triangulation from the line Siv John— Carlton to the line Fatkirk—Pekin—Continued. GRENADIER—7, (Observers, T. Russell and W. A. Metcalf. Instruments, Gambey theodolite No. 1 and Pistor & Martins 14-inch theodolite No.2. Dates, May and September, 1875. ] Angle as measured bet ween— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. Oo Z wn | | at “a ° ‘ a ‘Stony Point and Galloo ...........-- 29 42 14. 537 1 i 47°) 0.5) —0,042 0. 000 29 42 14. 495 Stony Point and Duck Island....... 78 13 33. 638 Tite 27 7.6 1.5: 40.170 0. 000 78 13 33.808 | Gallow and Duck Island........-.--- 48 31 19. 355 Te 11 5.8 0.5 —0.042 0. 000 48 31 19.313 | Duck Island and Amherst .-.---..--- 50 35 04. 280 | 73 | 18 GL 1 | —0.065 | +0. 629 50 35 04.844 | Duck Island and Stony Point......- 281 46 25. 893 | T344 16 2:7 1 +0. 299 0. 000 281 46 26.192 Amherst and Stony Point ...-...--- 231 11 22. 043 | 74 15 5.8 1 —0. 066 | —0. 629 281 11 21.348 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4. 0(71) +3. 5(72)-+ (73)+0. 381=0 3. 5(71) +4. 0(72)-+ (73) +0. 381=0 (T1)+ (72) +2(73) +0. 215=0 Nore.—7344 and part of 7142 were read by Mr. Russell with the Gambey instrument in September, 1875; the remainder were read by Mr. Metcalf with the Pistur & Martins instrument in May, 1875. ‘ STONY POINT—8. [Observers, G. Y. Wisner, T. Russell, J. H. Darling, and R.S. Woodward. Instruments, Troughton & Simms theodolites Nos. 1 and 3, Gambey theodolite No.1, and Repsold theodolite No.1. Dates, August and September, 1874, May and September, 1875, and September, 1878. ] Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) | {v] Corrected angles. ° A £7 “a wu fo} ‘ a Maunsville and Sandy Creek .....-- 17 28 38.635 | 81 24 4.7 | 15 —0.526 | +0.272 17 28 38. 381 Mannsville and North Base..... .-. 81 15 26. 612 8142 16 4.2 1 —0. 169 +0, 585 31 15 26.978 Mannsville and South Base.......-.. 34 52 40.320 | 8i+e+3 16 5.0 | 1 +0.025 | +0. 433 34 52 40.778 Mannsville and Oswego...-...------- 74 387 34.739 | Biteteta 2 0.9 | 0.1 +1.518 | +0.549 | 74 37 36. 806 Sandy Creek and North Base 46 48.588 | 8 48 GF | cBeb —0.254 | +0. 263 13 46 48. 597 South Base and Sandy Creek ....-. 35 57.660 | 8-2-3 30 7.2 11.5 +0.104 | —0. 161 342 35 57. 603 North Base and South Base ........ 37 13.905 | 83 66 91 13.5 —0.003 | —0. 102 3 37 13. 800 North Base and Oswego -...-..-.--- 22 10.606 | 8344 16 3.6 | 1 —0.792 | +0. 014 | 43 22 09. 828 South Base and Oswego.......-..--. 39 44 55. 496 8a 21 5.9 | 1.25 | +0.416 | +0. 116 39 44 56. 028 South Base and Duck Island........ 128 06 57.374 | 84+6 14 5.4 | 1 —0.658 | —0.303 128 06 56. 413 Oswego and Duck Island . 88 22 00.858 | 85 a7 5.7 | 2.25) —0.054 —0.419 88 22 00. 385 Duck Island and Grenadier (W.)..-. 51 53 11.681 | 86 16 3.0 | 1 0.059 | —0. 065 | 51 53 11. 557 Duck Island and Grenadier. ......-. 51 53 12.597 | 8647 17 2.9 [1 +-0.096 | +0. 927 | 51 53 13. 620 Grenadier and Duck Island.-........ 308 06 47. 211 86-7 17 3.6 | 1 +0.096 | -—0. 927 | 308 06 46. 380 Duck Island and Mannsville......-. 197 00 23.659 | 86+7+8 16 7.4 (1 —0.720 | —0. 130 197 00 22. 809 Grenadier (W.) and Mannsville. .. 145 07 11.375 87+8 16 3.6 | 1 —0.058 | —0. 065 145 07 11. 252 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 5. 6(81)-+-4. 1(82)-++3. 1(83)+-2. 10(84)-++2. 00(86)-+ (86) +3, 286=0 4. 1(81)-++8. 1(82)-+4. 6(&3)+-2. 10(84)+-2. 00(85)-+ (86) +3. 516=0 3. 1(81)+4. 6(82)4-9. 1(83)++3. 10(84)-+-2. 00(85)-++ (86) +1. 700=0 2. 1(81)+2. 1(82)+-3. 1(83)-+5. 35(841)-3. 00(85)-+ (86) —0.357=0' 2. 0(81)-+-2. 0(82)+2. 0(83)-+-3. 00(84)-++5. 25(85)-+- (86) +0. 659=0 (8i)+ (82) (83)-+ (84)-+ — (86)++2(86) +0. 538=0 2(86+7) —0. 192=0 NOTE.+-8i42, 8344, 86, 8748, and part of 8 were read by Mr. Wisner with the Troughton & Simms instrument No.1 in 1874. 8—2—s, and parts of 82 and 8 were read by Mr. Darling, with the Repsold instrument in May, 1875. 8647 and 8—s—7 were read by Mr. Russell with the Gambey instrument in September, 1875, The remainder were read by Mr. Woodward in 1878. § 4] BUFFALO BASE TO SANDY CREEK BASR. 523 SECTION XILI.— Triangulation from the line Nir John- Carlton to the line Fulkirk—Pekin—Continued. . OSWEGO—29. [Observers, G. Y. Wisner and R.S. Woodward. Instruments, Troughton & Simms theodolites Nos. 1 and 3. Dates, June and July, 1875, and July, August and September, 1878.] 1 Angle as measured between— | Notation. No, meas. | Rauge.}| Wt. | (v) {v] Corrected angles. 2 1 (‘Es ke ov uo | “we u “ Gee! u Victory and Sodus ...-............ 46 52 09.575 91 40 6.0 2.5 +0. 038 | —0. 058 46 52 09. 555 Victory and Vanderlip............ 127 21 57.017 | 9142 6 2.9 | 0.5 | —0.169 | —0. 893 127 21 55. 955 Victory and Duck Island . .....-. 154 11 14.118 | 914243 8 6.2 | 0.5 | +0,494 | —1. 000 154 11 13. 612 Sodus and Vanderlip...... .. .... 80 29 46.104 | 92 20 4.2 [1 +1.131 | —0. 835 80 29 46.400 Sodus and Duck Island ....... .-. 107 19 03. 283 | 9243 3 3.7 10.2 | 41.716 —0, 912 107 19 04, 057 Sodus and Stony Point..... ie tees 138 12 49.435 | 924344 16 6.6 {1 —0. 683 | —0. 232 138 12 48. 520 Vanderlip and Duck Island ....... 26 49 16.607 | 93 21 5.9 | 1 -+1.157 | —0. 107 26 49 17. 657 Vanderlip and Stony Point . ..... 57 43 01. 960 ; 9344 4 4.9 | 0,25 | —0,443 | +0, 603 57 43 02. 120 Duck Island and Stony Point ..... 30 53 42.880 , 94 33 ES °2 +0. 873 | +0.710 30 53 44. 463 Stony Point and North Base ...... 22 11 56.638 | %5 16 2.8 | 1 —0. 359 | +0. 166 22 11 56,445 * Stony Point and South Base ..... 28 22 51.215 | 95460 6 3.6 | 0.5 0,000 | +0. 113 28 22 51, 328 Stony Point and Maunsville ..-.-. 28 57 58.515 | 95460+60 24 5.5 | 1.5 | --0.264 | +0. 047 28 57 58. 298 Stony Point and Victory.......... 174 55 00.285 | 95460+6047+8 12 10.1 | 0.75 | 41.350 | +0,290 | 174 55 01.925 Stony Point and Sodius............ 221 47 10.554 | 9546a+604+74+8+41 18 91/1 , +0, 694 | +0. 232 | 221 47 11.480 North Base and Mannsville ..... 6 46 02.177 | 96a+6n 28 4.7 (1.75 —0. 205 | --0.119 6 46 01. 853 Mannsville and Sandy Creek ..... 416 06, 292 | 97 30 5.0 | 2 | 0, 612 +0. 526 | 4 16 06. 206 ' | Mannsville and VIClOLY scieccaress 145 57 02.449 | 97+8 6 1.8 | 0.5 | +0. 935 | +0. 243 | 145 57 03. 627 Sandy Creek and Victory ......... 141 40 58.931 | 9a 16 | 5.0 | 1 | 1.227 | —0.283 ) 141 40 57.421 NORMAL EQUATIONS FOR LOUAL ADJUSTMENT. 5. 75(91)-+3. 25 (92)-+2. 75 (93) +2. 25(94) +1. 50(9546a-+65 )-+ (97)— 4. 028=0 3. 25(91)-+6. 45(92)+-4. 95(93)-4+4. 25(94) HAL. 50(95-46-+60 )-+ (97) —15. 841=0 2.75(91) +4. 95(92) +6. 20(93)-+4. 50(94) +1. 50(95-46u-460 )-+ (97)--15. 790—0 2. 25(91)-44. 25(92)-+-4. 50(9s) +6. 50(9a) +1. 50(95460-+6) + (97) —L4. 760-+0 2. 75(95) —1. 75(9o-46a+6b ) + 0,525=) 1. 50(91) +1. 50(92)-+1. 50(92)-+-1. 50(94) 1. 75(95) +4. 75 (95460460 )+ (97)— 3.556=0 (Q+ = (2) (93) (94) + — (95+6u+6s )+3(97)— 1.096=0 NOTE.—9,, 93, 95, 9s, and parts of 9), 94, 96a+6s, and 97 were read by Mr. Wisner withthe Troughtun & Simms instrument No.1 in 1875. The remainder were read by Mr. Woodward with the Troughton & Simms instrument No. 3 in 1878. MANNSVILLE—10. (Observers, G. Y. Wisner and R.S. Woodward. Instruments, Troughton & Simms theodolites, 14-inch No.1, and 12-inch No. 2. Dates, Octo- ber, 1874, and May, 1875.] Angle as measured betweep— Notation. No. meas. Range. | Wt. (v) {y) Corrected angles. oa 8 “aw | | “wn aw “a | of “a Sandy Creck and Oswego ..--.--.--- 22 04 07. 750 101 | 16 5.3 1 +0.186 | ---0. 229 22 04 07, 707 Sandy Creek and South Base ....-.-. 23 23 57. 446 10i+2 | 25 7.9 1 —0.435 | +0. 467 23 23 57.478 Oswego and North Base ....-..--.-- 20 05 15. 706 10243 16 6.3 1 +0.186 | +0. 757 20 05 16. 649 South Base and North Base. . 18 45 27. 252 10; 25 6.6 1 —0.435 | +0. 061 18 45 26, 878 North Base and Stony Point ..--..-.. 56 19 10. 600 104 16 2.7 1 —0.249 | +0. 731 56 19 11. 082 Stony Point and Sandy Creek...-...- 261 31 26. 069 106 16 ~ 3.5 1 —0. 248 | —1. 259 261 31 24. 562 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. +2(101) — (101-42)— (103) —1.242=0 — (101) +3(10142)-+2(103)-+ (104)4+2. 609=0 — (101) +2(10142)-+-3(103)-+ (104) +2. 609=0 + (10i+2)+ (103)-+2(104) +1. 367=0 NoTE.—101+2 and 103 were read by Mr. Woodward with the Troughton & Simms instrument No.2 in 1875. The remainder were read by Mr. Wisner with the Troughton & Simms instrument No. 1 in 1874. 524 PRIMARY TRIANGULATION, [Cuar. XIX, C, SECTION NILL— Triangulation from the line Sir John— Carlton to the line Falkirk—Pekin—Continued. SANDY CREEK—11. [Observer, G. ¥. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Date, October, 1874.] | Angle as measured between— | Notation. ' No. meas, | Range.| Wt. @) —— [v) Corrected angles. | - Learners at ee Hg _ oF mt an ™ | " u" as pi ~ Oswego and South Base ... .---.--. 20 47 17,225 li * 16 5.3 1 —0. 051 +0. 312 | 20 47 17. 486 South Base and North Base.......-. 34 33 10. 406 lle 16 2.3 1 | —0.051 | +0. 404 34 33 10.759 | North Base and Stony Point......-- 34 16 31.912 ll 16 4.2 1 —0. 051 —0. 318 | 34 16 31.543 Stony Po'nt and Mannsville .... -.. 64 02 45. 975 lla 16 4.1 1 | —0.051 | 40.741 64 02 46, 665 | Mannsville and Oswego. ...-..------ 206 20 14. 737 11; 16 4.3 1 | —0.051 | —1.139 206 20 13.547 | | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 211i) + (11g) (11g)+ (114) +0. 255=0 (111) +2(112) + (11g)+ (114) 4-0. 255 (11+ (11,)4-2(1 1g) (11,)-+-0. 255 (11) (11y)-F (113) +-2(114) 4-0. 255 yi oO ll 0 0 0 NORTH BASE—12. (Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Date, September, 1874. ] ' Angle as measured between— Notation. | No meas. | Range.) Wt. (v) | [v] Corrected angles. °o a “uw aw a“ a Oo t a“ South Base and Oswego.....-..----- 46 58 56. 275 121 | 16 3.9 1 | +0.053 | +0. 247 46 58 56.575 | Oswego and Stony Point....-....--- 114 25 55. 644 122 16 4.1 1 +0.052 | —0.287' 114 25 55. 409 Stony Point and Manneaville .-....-. 92 25 22. 631 123 | 16 5.5 | 1 | +0.053 | —0.111 92 25 22. 573 Mannsville and Sandy Creek. .-....- 39 31 17.700 124 16 3.6 a -+-0.052 | —0. 146 39 31 17. 606 Sandy Creek and South Base ......- 66 38 27. 487 125+6 16 3.0 1 | +0. 053 | +0. 297 66 38 27. 837 Middle Base and South Base. .:...-. 0 51 04. 800 126 lu 1.6 1 | 0.000 | +0.172 0 51 04. 972 ! i NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(121)-+ (122) + (123)-+ (124) —0. 263=0 (121) +2(122)+ (128)-++ (12) —0. 263= 0 (121) + (122) +2(123) + (124)—0. 263 =0 (121)-+ (122)+ (122) -+2(124)—0. 263 =0 MIDDLE BASE—13. [Observer, G. Y. Wisner. Instrument, Troughton & Simtas 14-inch theodolite No. 1. Date, October, 1874.] Angle as measured between-- Notation. | No. meas. | Range. | Wt. (v) {v] Correctedangle. | 6 4 u | | u" a “ oo: " South Base and North Base .....-. 178 15 30. 533 13, \ 16 15 1 0. 000 +0. 172 178 15 30.705 | § 4] BUFFALO BASE TO SANDY CREEK BASE. B25 SEcTION XUI— Triangulation from the line Sir John - Carlton to the line Falkirk —Pekin—Continued. SOUTH BASE—14. {Observers, G. Y. Wisner and R. S. Woodward. Instruments, Troughton and Simms theodolites Nos. 1, 2, and 3. Dates, October, 1874, May 1875, and October, 1878.] | Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) [v) Corrected angles. : I : oj “u | “ ” u ou “" | North Base and Middle Base........ 0 53 24.150 | 141 16 1.8 | 1 0.000 | +0.172 0 53 24. 322 North Base and Mannsville......... 55 04 47, 942 | 14142 20 6.3 | 1.25 | —0.018 | —0. 096 55 04 47.828 | North Base and Sandy Creek ....... 78 48 21.455 | 14i4v43 26 7.6 | 1.5" —0. 044) +0.090 78 48 21.501 Mannsville and Sandy Creek ....... 23 43 33.285 | 143 45 7.6 | 2.25) +0.202 | +0.186 23 43 33. 673 ‘| Sandy Creek and Oswego ........... 154 21 28.845) 144a 16 4.2 | 1 +0. 367 | +0,417 154 21 29. 629 Sandy Creek and Stony Point......- 266 13 44.369 | 144a+4p 16 5.9 | 1 +0.022 | —0. 186 266 13 44. 205 Oswego and Stony Point............ 111 52 14.812 | 14a, 16 5.2 | 1 +0. 367 | —-0. 603 111 52 14. 576 Stony Point and North Base......-. 14 57 54.258 | 145 22 3.9 | 1.5 —0. 060 | +0. 096 14 57 54, 294 Stony Point and Mannsville ........ 70 02 41.646 | 1454142 16 90 | 1 +0, 476 0. 000 70 02 42.122 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4, 25(141-+42) +3. 00(143) +1. 5(144a)-+1. 5(1445)—1. 629=0 3. 00(141-4+2) +6. 25(143) +2. 5(144a) +2. 5(144%)-—3. 041=0 1. 50(141+2)+-2. 50(143) +-4. 5(144a) +3. 5(1440)—3. 411=0 1. 50(14142)+-2. 50(143) +3. 5(1440) +4. 5(1440) —3. 411=0 Nore.—141, 1440+ 4s, and parts of 1414243 and 145 were read by Mr. Wisner with the Troughton and Simms instrument No. 1 in 1874. Part of 143 was read by Mr. Woodward with the Troughton & Simms No. 2 in 1875. The remainder were read by Mr. Woodward in 1878 with the Troughton & Simms No. 3. VANDERLIP—15. (Observers, T. Russell and R.S. Woodward. Instruments, Gambey theodolite No. 1 and Troughton and Simms theodolite No. 3. Dates, June and July, 1875, and May and June, 1877.] Angle as measured between— Notation. | No meas. | Range.| Wt. (v) {v] Corrected angles. . ° i a“ “ “ a ow Ww Awherst and Duck Island......-..-.-. 38 18 07.120 | 151 47 87 12 +0.094 | —1.116 38 18 06. 098 | Amherst and Oswego...........--.- 87 19 53.467 | 151+2 6 5.2 10.25 | —0. 546 +0. 355 87 19 53.276 Duck Island and Oswego ..-..--.---. 49 01 45. 538 |} 15, 44 11.9 | 2 +0. 169 +1. 471 49 01 47.178 Duck Island and Sodus ........-.--- 87 59 12.554 | 152+ 16 4.9 | 0.75 | —0.200 | +0.565 87 59 12.919 Oswego and Sodus.............-.-.- 38 57 26.546 | 153 40 13.3 | 2 +0.101 | —0. 906 38 57 25.741 Sodus and Amherst ..-........-...- 233 42 40.406 | 15, 39 14.8 | 2 +0. 026 +0.551 | 283 42 40.983 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4, 25(151) +2. 25(152) +2. 00(154) —0. 982—=0 2, 25(151) +5. 00(152) + 2. 75(154) —1. 335=0 2. 00(151) +2. 75(15,) +4. 75(153) —1. 133=0 Nore.—151, 15», 15242, 152, and 15, were partly read by Mr. Russell with the Gambey instrument in 1875. The remainder of the angles were read by Mr. Woodward with the Troughton and Simms instrament in 1877. 526 PRIMARY TRIANGULATION, (Cirar. XIX, C, SECTION NILL—Triangulation from the line Sir John—Cariten to the line Falkirk—Pekin—Continued. SODUS—16. (Observer, R. 8. Woodward. Instruments, Troughton & Simms 12-inch theodolite No. 2, and 14-inch theodolite No.3. Dates, June an July, 1875, and July, 1877.] Y Angle as measured between— Notation. No. meas, } Range. | Wt. (v) {v] | Corrected angles. Oo ? a“ | Ww Ww au | fo} t “a Vanderlip and Oswego .. .......--- 60 32 57.548 | 16, 28 10.7 | 1.5 | 40.095 | —0.684 60 32 56. 959 Oswego and Victory .. 29 42 35.931 | 162 24 86 | 1 —0.103 | 40.466 | 29 42 36, 294 Victory and Clyde ..... .--.....-+-: 46 43 03.786 | 163 25 11.3 / 1 —0. 103 | —0. 384 | 46 43 03. 299 Clyde and Palmyra ....-...--- ...-- 59 50 58.700 | 16,4 26 11.10) 1.5 —0. 068 | —0. 394 59 50 58. 238 Palmyra and Turk’s Hill ........-. 32 65 38.709 | 1654 24 6.0 | 1 —0.171 | —0. 069 32 05 38, 469 Palmyra and Walworth .. .....---- 47 07 14.585 | 165 ; 5b 26 7.4 1.5 +0. 291 | +0. 020 47 07 14. 896 Walworth and Palmyra........ ---- 312 52 44.821 | 16—5a—sp 24 5.0 | 1 +0. 303 | —0.020 | 312 52 45.104 Palmyra and Pinnacle Hill ..-...... 55 19 10.358 | 165u+5b+6a 8 4.0 | 0.25 | —0.073 | +0. 658 55 19 10. 943 Pinnacle Hill and Palmyra........-. 304 40 49.523 | 16—5a—sr—6a 8 4.0 | 0.25 | +0.192 | —0.658 | 304 40 49. 057 Turk's Hill and Walworth. ......... 15 01 36.509 | 165y 24 3.4 | 1 —0.171 | +0. 089 15 01 36. 427 Walworth and Pinnacle Hill -. ..... 8 11 55.277 | 166a 12 3.6 | 0.5 | 40.132 | +0. 638 8 11 56. 047 Walworth and Vanderlip. 116 03 09. 242 | 166a+6d 27 85 | 1.5 | +0.096 | +0.976 | 116 03 10.314 Walworth and Oswego. .....-...---- 176 36 07.471 | 166a+6d+4 10 3.6 | 0.5 | —0.490 | +0.292 | 176 86 07. 273 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 0(16,) +1. 5(162) +1. 5(16,) +1. 5(164) +1. 5(165u) +1. 5(1658) +0. 637=0 1. 5(16,) +3. 0(162)-+2. 0(162) + 2. 0(164) +2. 0(165«) +2. 0(1650) +1.190=0 1. 5(16,) +2. 0(162) +3. 0(164) +2. 0(164) +2. 0(165a) +2. 0(1653) +1.190=0 1, 5(16,) +2. 0(162)-+2. 0(163) + 3. 5(164)-+2. 0(165u) +2. 0(1658) +1. 190=0 1. 5(16,) +2. 0(16,) +2 0(163)+2. 0(164) +6. 0(165a) +5. 0(1645) +0. 5(166a) +2. 218=0 1. 5(16,) +2. 0(162) +-2. 0(163) +2. 0(164) +5. 0(165u) +6. 0(1650) +0. 5(166a) +2. 218=0 0. 5(165a)+0. 5(1652)+ (166a)+4+0. 039=0 NOTE.—16,, 16), 163, 161, 165a+5), 166a+6v, and 166a46)+1 were read in 1875 with the Troughton & Simms No. 2; the remainder in 1877 with the Troughton & Simms No. 3. VICTORY—17. [Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Date, July, 1875.) -— A Angle as measured between— Notation. | No. meas. | Range.) Wt. (v) | {v] Corrected angles. 1 oO t a“ uy “uw | “wn o i we Clyde.and Sodus :....-0+ +0401 -e0+0+ 44 41 46. 463 17, 16 6.8 1 +0.121 —0, 949 44 41 45. 635 Sodus and Oswego .....------------- 103 25 16. 606 172 16 6.3 1 -+0.121 | —0. 099 103 25 16. 628 | Oswego and Clyde .........--.------ 211 52 56. 569 173 16 3.8 1 +0. 120 | +1. 048 211 52 57. 737 | ‘ si NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(17))+ (172)—0. 362=0 (171) +2(172) —0. 362=0 CLYDE—18. (Observer, W. A. Metcalf. Instrument, Pistor & Martins 14-inch theodolite No. 2. Dates, June and July, 1875.] ' ! Angle as measured between— Notation. | No. meas. |Range.| Wt. (v) (v) Corrected angles. ! 2 oO t a | aw aw a oO f a Palmyra and Sodus ................- 45 23 43. 875 18, 16 2.0 1 0.489 | —0. 804 45 23 43. 560 | Sodus and Victory.... ..........--- 88 35 13. 095 182 16 Ted 1 +0.489 | —0. 597 88 35 12. 987 Victory and Palmyra 226 01 01. 563 183 16 5.1 1 -+0.489 | +1.401 226 01 03, 453 I | > NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(18,)+ (18.)—1. 467=0 (18,) +2(182) — 1. 467=0 §4.] BUFFALO BASE TO SANDY CREEK BASE. 527 Section XIII.— Triangulation from the line Sir John— Carlton to the line Falkirk - Pekin—Continued. PALMYRA—19, (Observers, W. A. Metcalf and R. S. Woodward. Instruments, Pistor & Martins 14-inch theodolite No. 2, and Troughton & Simms 14-inch theodolite No. 3. Dates, June, 1875, and July, 1877.] Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) (v] Corrected angles. ov “ “ uw “ o # “we Turk’s Hill and Walworth.......... 4412 02.712) 19% 40 5.9 2 +0. 036 | +0. 672 44 12 03. 420 Walworth and Sodus ...... .-...--. 74 25 05. 020 | 19, 40 71 2 +0. 036 | —0.397 74 25 04. 659 Sodus aud Clyde: oc sewers seeecevens 74 45 20.509 193 16 5.9 1 +0.118 | —1,240 74 45 19. 387 Sodus and Turk’s Hill.....--. -.-.. 241 22 52. 242 | 19344 24 3.0 i —0.046 | —0.275 241 22 51.921 Clyde and Turk’s Hill ..........--.. 166 387 31. 452 194 16 5.7 1 +0117 | +0. 965 166 37 32. 534 [ NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(191)-+2(19,) + (19,)—0. 3330 2(191) +4(192) + (19,)—0. 333=0 (191)+ (192) -+2(19,)—0. 307=0 NOTE.—193, 194, and parts of 19: and 192 were read by Mr. Metcalf with the Pistor & Martins instrument in 1875. The remainder wer read by Mr. Woodward with the Troughton & Simins instrument in 1877. WALWORTH—20, {Observer, R.S. Woodward. Instruments, Troughton & Simms 12-inch theodolite No. 2, and 14-inch theodolite No. 3. Dates, July, 1875, and July, 1877.] Angle as measured between— | Notation. | No. meas. | Range. | Wt. | (v) [v] Corrected angles. Oo f “a | aw | “ aw o / aw Sodus and Palmyra .......... ---.. 58 27 41.923 201 | 49 9.7 2 | —0.011 | —0. 629 58 27 41. 283 Palmyra and Turk’s Hill.....-....-. 83 52 54. 744 202 : 49 10.0 2 —0.011 | +0.343 83 52 55. 076 Turk’s Hill and Pinnacle Hill. ...--. 54 48 16. 562 203 48 8.1 2 —0.011 | +0. 016 54 48 16. 567 Pinnacle Hill and Sodus ....-..----. 162 51 06. 814 20, - 48 8.2 2 —0.010 | +0. 270 162 51 07. 074 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(201) +2(202) + 2(203) +0. 086=0 2(201) +4 (202) +-2(203) + 0. 086—=0 2(201) + 2( 202) +4 (203) +0. 086 =0 Nore.—Parts of all the angles at this station were read in 1875 with the Troughton & Simms instrument No. 2, and the remainder in 1877 with the Troughton & Simms No. 3. TURK’S HILL—21. [Observers, W. A. Metcalf and R. S. Woodward. Instruments, Pistor & Martins theodolite No. 2, and Troughton & Simms theodolites Nos. 2and 3. Dates, July and September, 1875, and July, 1877.] Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) (v] Corrected angles. oo é a aw Ww “ fo} t “ Scottsville and Pinnacle Hill ........ 49 25 11.353 211 64 7.5 3 —0. 030 +0. 418 49 25 11.741 Pinnacle Hill and Walworth... ..-.- 79 00 51. 562 212 64 8.0 3 —0. 030 | —0.135 79 00 51.397 Walworth and Sodus ...-.-.--.------+ 22 37 46.702 2130 24 4.6 1 +0. 637 +0. 407 22 37 47.746 Walworth and Palmyra..-. .-..----- 51 55 02. 534 213a+3b 40 9.9 2 —0. 363 | —0. 009 5L 55 02. 162 Sodus and Palmyra ..-....--.--------- 29 17 14.195 223d 24 4.4 i +0. 637 | —0. 416 29 17 14.416 Palmyra and Scottsville .....-------- 179 38 55. 004 214 64 8.8 2 —0. 030 |- —0. 274 179 38 54.700 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 6(211) +3 (212) +3(213«) 4+3(213p)—3. 552=0 3(211) + 6(212) +3(2132) +3(213)—3. 552=0 3(21,) +3(212) +6 (2134) +5(2135)—6. 826=0 3 (21) +3(212) +5(213a) + 6(2135)—6. 826=0 NoreE.—21), 212, 2130430, and 21, were partly measured by Mr. Metcalf with the Pistor & Martins instrument in July, 1875. 211, 212, and 214 were partly measured and the remainder of 213a+3) was measured by Mr. Woodward with the Troughton & Simms instrument No. 2 in September, 1875, The remainder of the angles were read by Mr. Woodward with the Troughton & Simms instrument No. 3 in July, 1877. 528 PRIMARY TRIANGULATION. [Cuar. XIX, C, SECTION NUL.—Triangulation from the line Sir John—Cartton to the line Falkirk— Pekin—Continued. PINNACLE HILL—22. (Observers, G. Y. Wisner and R.S. Woodward. Instruments, Troughton & Simms theodolites Nos. 1,2,and 3. Dates, July and October, 1875, and August, 1877. ] Angle as measured between— Notation. | No. meas. | Range. Wt. | (v) [v7] | eormsntaa ‘cua | ans [dete e| ay i { haan | oO / “we “ } | aw “we | of aw | Walworth and Turk’s Hill .......-- 46 10 53.196 221 40 6.5 2 —0.316 | —0.192 : 46 10 52. 688 | Turk’s Hill and Scottsville .-....... 84 22 48,171 | 222 64 12.7 3 +0. 037 | +0. 510 | 84 22 48.718 | Scottsville and Brockport .....-.--- 68 35 26.907 | 223 . 16 4.9 | 1 | -+0.264 ; —0. 658 | 68 35 26.513 — ' Scottsville and Sodus .........----- 220 29 21. 760 | 223440 10 3.2 | 0.5 | +0. 536 | 0.930 220 29 21.366 | Scottsville and Walworth ......-..- 229 26 19.581 | 223+4u+40 18 4.1 | 0.75 | —0.669 |; —0.318 | 229 26 18. 594 | Brockport and Walworth. ..-..----- 160 50 51.476 | 224044 16 4.7 | 1 +0.265 | +0. 340 160 50 52.0%) | Sodus and Walworth..-......--...-. 8 56 57.405 | 2245 14 3.5 | 0.5 —0.789 | +0. 612 8 56 57. 228 Sodus and Turk's Hill......-..-.... 55 07 48.053 | 224041 6 5. 2 0. 25 +1. 443 +0. 420 55 07 49. 916 Turk’s Hill and Sodus .......------- 304 52 12.038 | 22—1-1 6 4.2 | 0.25; —1.534]} —0.420 304 52 10. 084 Sodus and Scottsville .......----..-- 139 30 38,112 | 22454142 4 3.0 | 0.2 —0.408 | +40. 930 139 30 38. 634 j NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4, 95(221) +2. 45(222)-4+ (225) +1. 2(224,) +2. 156=0 2, 45(221) 4-5. 45(222)-+4+ (223)-+-0. 7(224) +0. 859=0 (22))+ (222) -4+2(223) —0. 250=0 1. 20(221) +0. 70(223) +1. 7(2242)-+1. 695=0 Norn,—22z, 2240442, and parts of 221 and 222 were read by Mr. Wisner with the Troughton & Simms instrument No. 1 in July, 1875, Partof 222 was read by Mr. Woodward with the Troughton & Simms instrument No. 2 in October, 1875. ‘The remainder of the angles were read by Mr. Woodward with the Troughton & Simms instrument No. 3 in 1877. SCOTTSVILLE—23. (Observers, J. H. Darling and R. 8. Woodward. Instruments, Repsold theodolite No. 1, and Troughton & Simms theodolitcs Nos. 2and3. Dates, July and October, 1875, and August, 1877.} Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) (v] Corrected angles. oj. i “ “uw oO # “uw Morganville and Brockport..,. ..-. 46 45 36.095 231 34 71 (15 —0.531 | +0. 627 46 45 36.191 Brockport and Pinnacle Hill. --. 74 23 46.295 232 34 922 | 1.5 —0,531 | —0. 894 74 23 44. 870 Pinnacle Hill and Turk’s Hill. .-..-- 46 11 59. 830 233 66 8.9 | 2.75 -+0.004 | --0.495 46 12 00. 329 Turk’s Hill'and Pinnacle Hill...... 313 47 59. 091 23-3 16 4.4 | 0.75) +1.075 | —0, 495 313 47 59. 671 Turk’s Hill and Morganville........ 192 38 39. 368 234 36 10.9 | 1.5 —0.530 | —0, 228 192 38 38. 616 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 0(231) +1. 5(282) +1. 5(233) +2. 382=0 1. 5(231) +3. 0(232) +1. 5(283) +2. 382=0 1. 5 (23) +1. 5(232) +5. 0(233)-+1. 572=0 Norr.—231, 23,, 23,, and part of 233 were read by Mr. Darling with the Repsold theodolite, in July, 1875. Part of 233 was read by Mr. Woodward with the Troughton & Simms instrument No.2, in October, 1875. 23—3, and the remainder of 23; were read by Mr. Woodward with the Troughton & Simms instrument No. 3, in 1877. BROCKPORT—24. [Observer, R.S. Woodward. Instrument, Troughton & Simms 12-inch theodolite No.2. Dates, July and August, 1875. ] Angle as measured between— Notation. | No. meas. | Range. | Wt. | (v) | (v] Corrected angles. ov “uw “ | a“ “uw | oO ¢ " Pinnacle Hill and Scottsville ......- 37 00 50. 270 24) 27 9.3 1.5) 40.118} —0.532 37 00 49. 856 Scottsville and Morganville......... 65 53 17. 304 24, 27 13.6 1.5 | +0.118 $0. 988 - 65 53 18, 410 | Morganville and Albion .... - 81 28 42.767 243 i 28 | 17 1.5} 40.118 | —0.590 ° 81 28 42.295 | 1.5 +0. 117 +0. 134 175 37 09. 439 Albion and Pinnacle Hill ........--. 175 37 09. 188 244 28 85 ! NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 0(24)) +1. 5 (24g) +1. 5 (244) —0. 706=0 1. 5(24,) +3. 0(242)-+ 1. 5(243) —0. 706=0 1. 5(241) +1. 5(242) +8. 0(244) — 0. 706 =0, §4.j BUFFALO BASE TO SANDY CREEK BASE. 529 SECTION XIII.— Triangulation from the line Sir John- Carlton to the line Falkirk— Pekin—Continued. {Observer, T. Russell. Instrument, Gambey theodolite No. 1. MORGANVILLE—235, Date, July, 1875.] Angle as measured between— | Notation. | No. meas. | Range.} Wt. (v) Lv) \eosreetedl angles. i °o t wy a“ Ww uw o A “wt Batavia and Albion .....-.... -..--- 67 24 57. 250 251 18 6.1 1 —0. 053 +0.171 67 24 57. 368 | Batavia and Brockport’ aaeteed tesene 111 12 48, 354 25i+2 6 2.6 0.5 | —0.459 | —0.778 111 12 47.117 Albion and Brockport ......--...... 43 47 50. 751 252 18 3.0 1 —0.053 | —0.949 43 47 49.749 Brockport and Scottsville .......... 67 21 06. 087 253 16 4.4 1 —0, 283 +41. 030 67 21 06. 834 Scottsville and Batavia ............. 181 26 06.565 254 16 6.0 1 — 0.284 | —0. 252 181 26 06. 049 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(251) +1. 5(25y)-+ (254) +0. 496=0 1. 5(25,) +2. 5(25y) + (255) +0. 496=0 (Observer, W. A. Metcalf. Instiument, Pistor & Martins 14-inch theodolite No. 2. (251) — (25,) + 2(254) +0. 673=0 ALBION—26. Dates, July and August, 1875.] Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) | [v] Corrected angles oO ft wa | “a uw | “we oO t a Brockport and Morganville..-....-. 54 43 28.981 261 | 16 5.5 1 | +0. 459 | —0. 444 54 43 28. 996 Morgauville and Batavia. - -- 38 38 19. 080 26, | 16 4.3 1 0.459 | +0, 676 38 38 20. 215 Batavia and Gasport..........------ 62 03 57. 772 263 16 6.0 1 | 4+ 0.459 | —0. 872 62 03 57. 359 Gasport and Brockport .-.-..-.--.-. 204 34 12,332 264 16 5.0 1 | +0.458 | +406. 640 204 34 18.430 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT.” 2(261) + (262)-+ (263) —1.835—=0 (261) +2(26)+ (263)—1.835=0 (261) + (262) +2(263)—1.835=0 BATAVIA—27. {Observer, J. H. Darling. Instrument, Repsold theodvlite No. 1. Dates, July and August, 1875.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) | (v} Corrected angles. | ° t “ | un aw we oO A “a Falkirk and Gasport 40 29 30. 684 271 30 6.6 1.5} +0.108 | +0,072 40 29. 30.720 |. Gasport and Albion.....----.-------- 68 48 39. 518 272 28 8.0 1.51 0.108! +40.747 68 48 28. &79 Albion and Morganville....--.----..- 73 56 42.919 273 26 9.2 1 +0.163 | +0.427 73 56 43. 5u9 Morganville and Falkirk..--.-...---- 176 45 06.337 274 24 | 4.4 | 1 | +0.163 | +0. 392 | 176 45 06. 892 2.5271) + (272) + (27,)—0. 542=0 (271) +2. 5(272) + (273) —0. 542=0 (27:)+ — (272)4+-2(27,)—0. 542=-0 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 530 PRIMARY TRIANGULATION, [Cuap. XIX, ©, Section XIII.—Triangulation from the line Sir John-Carlion to the line Falkirk— Pekin--Continued. GASPORT—28. (Observer, R. S. Woodward. Instruments, Troughton & Simms theodolites Nos. 2 and 3. Dates, August, 1875, and October and Novem ber, 1878. ] Angle as measured between— | Notation. _ No. meas. | Range. Wt. (v) |v] le eiestel cae ae pee er eee > om i : * . oO t aw ¢ a“ wa “ Oo f wt Albion and Batavia....-....-.-.---- 49 07 26.051 | 281 48 _ Bt 25 —0.071 | —. 387 49 07 25. 593 | | Batavia and Falkirk.........-2..--. 35 16 09. 094 | 282 : 48 | 8 2.5 —0.057 | +0. 007 35 16 09. 044 | Batavia and Pekin...--....-...-..... 132 38 32. 257 28243 4 2.8 0.25; —0.132) 40.113 132 38 32. 238 Falkirk and Pekin ....--.--....-.-- 97 22 23.145 | 283 : 48 12.6 i 2.5 —0.057 | +0. 106 97 22 23.194 284 ' 43 63 2 —0.089 | +0. 274 178 14 02. 169 Pekin and Albion...........-....--- 178 14 O01. 984 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4. 5(281) +2. 00(282) +2. 00(283) 4-0. 548=0 2. 0(281)-++4. 75 (282) -+ 2. 25(283) + 0. 543=0 2. 0(281) +2. 25(282)+ 4. 75(28,) 4-0. 543=0 Note. —Parts of 281, 282, 283, and 284 were read with the Troughton & Simms No. 2, in 1875; the remainder of the angles in 1878, with Troughton & Simms No. 3. FALKIRK—29. [Observer, T. Russell. Instrument gGambey theodolite No.1. Dates, July and August, 1875.] Angle as measured between— | Notation. | No. meas. Range.| Wt. (v) {v] Corrected angles. | | —— | Of u” ! uw “ow “ oF “ | Tonawanda and Pekin ....- .--.--- 31 23 10. 691 291 17 j 3.1 I =E0. 082 | sees cescnc| ces cccence ees ees i Pekin and Gasport...--...-2..------ 48 49 46. 890 | 29, 19 3.9 1 +0. 082 —0.185 48 49 46. 787 | Gasport and Batavia ......-.-..- w-- 104 14 21.342 292 24 6.5 1.5 | +0. 054 —0. 273 104 14 21.123 j | Batavia and Tonawanda ....--...-. 175 32 40.778 | 294 18 39 | 1 AiO) OBI oak a's view) deets angatetuelsieta cite | | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(291)-+ (292)-+ (293) 0. 299 =0 (291) +2(292)+4+ — (293)—0.299=0" (291) (292)+2. 5(29,) —0. 299=0 PEKIN—30. {Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1. Dates, July and August, 1875.) | Angle as measured between— | Notation. | No. meas. | Range.| Wt. (v) | | Corrected angle. | oO a aw | | a “ | a oO t “a Gasport and Falkirk..-.....-..----- 33 47 51.526 301 18 4.0 1 +0. 065 —0. 190 33 47 51.401 Falkirk and Tonawanda 66 25 49.291 | 302 18 | 5.9 1 “PO, 068 |i eco friaciesienienin tee’ ’ Tonawanda and Drummondville .... 59 58 19.872, 303 18 | 5.6 Di fS0L 065" bates sen tle tnee tate enue Drummondville and Gasport........ 199 47 59. 017 304 18 | 46 1 | 40.066 |.......... sacicle estate ea | I NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(301)-+ (302)+ (303) —0. 2610 (301) +2(302)+ (30,) 0. 261=9 (301) + (302) +2(30,)—0. 261=0 §4.] BUFFALO BASE TO SANDY CREEK BASE. 531 Numerical equations of condition in the triangulation from the line Sir John— Carlton to the line Fatkirk - Pekin. I VII XV AVITI XXIII. . (10) » (20) . (20) . (20) (50) . (20) XXVII. (10) XXIX. XXXVII. (20) (25) XXXIX. (30) + 13.9876 [12] — 10.5623 [4s45] — 14,9447 [53] + 17.3045 [73] + 35. 1870 [94] — 5.7690 [8&1] 57. 0246 [10,] — 42. 6729 [11] + 13.5954 [8] 0.1409 [112] +109. 5078 [10] + 57.0890 [112] + 32.3738 [125] SIDE-EQUATIONS. + 0.1087 [ida] +12. 6933 [45] + 7.1444 [54] — 0.6003 [5] —26. 6586 [15)] + 7.8064 [8] + 5.0910 [10,42] 42, 6729 [119] +13. 5954 [85] — 0.1409 [115] —57, 5742 [1040] +42. 5320 [115] — 9.2090 [124] + 3.1378 [10,42] + 3.1378 [105] + 10.2483 [114] —18. 0304 [125] + 3.1378 [10,42] + 8.7492 [105] = 8.1548 [115] ++ 9.0825 [1445] + 19,5482 [11] + 6.1022 [122] + 47.9060 [143] + 29.3673 [165,] + 14, 8503 [203] + 11. 4886 [19,] +10. 2483 [114] +16. 4685 [11g] + 6, 1022 [123] —49, 0652 [1€5.] — 5.5327 [22] +17. 3602 [192] — 50.5077 [21a] +87. 5392 [215,] 12.7795 [2142] — 4.3887 [7)] +16. 5172 [847] -+F18. 2841 [159] + 7.8064 [8s] + 5, 0910 [105] —42. 6729 [115] —11.7217 [8] +43, 8633 [1440] —57..5742 [105] +42. 5320 [114] +17. 1695 [104] —18. 9213 [124] + 8.7492 [104] + 7, 6447 [145] +16. 4685 [11,] -=25, 5224 [124] 127. 2850 [20,] +14. 6717 [22s] —40. 2070 [201] +23. 9476 [22] + 4.987=0 — 4.3887 [72] —41, 6432 [95] —115, 270=0 + 7.8064 [8,] + 5.0910 [104] — 42,5320 [114] + 28. 443=0 455. 3220 [11,] — 8.4517 [1445] — 41. 468—0 +57, 0890 [11] +32, 3738 [122] + 8.1650 —30, 6941 [115] — 36.379=0 — 8.1548 [1lo] + 9.0325 [1440] — 15.027=0 + 6, 1022 [12)] —14. 6993 [14142] — 27:977=0 +27, 2860 [20.] + 26, 223=0 —27, 2850 [202] + 19.378=0 Nore.—In the solution for determining the general corrections each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. ANGLE-EQUATIONS. il. [12] +0542] +[4s45] —[45] IIL. [lige] +122] +Bite] —[32] IV. [2142] —[%] +132] +[4s45] Vv. [33] + [434545] —[4s45] +151] VI. [4o4s¢4+56] —[4s44+5] +[52+3] + [6s] VIL [Fs] +fFs} +664) +175] IX. [54] +62] +[15)] X. {6.] +[9s] +[152] XI —{6] —[6.] —[6s] —[64] —[65] +[%s] XII. [65] +i] AL%e) = +1 86+7] XIU. = [RJ +(e] +[8] -+[103] +[104] —[143] XIV. [81] +[82] +(83] +81] +([954+a4 06] —[101] XVL > [8] +f] — +£10.] +0120) XVII. [#2] +(8%] +[& J +09s+ea+60] +[97] 41h] XIX. = [8s] +8): +95] +[122] XX. [8] +[9546a] + [1440] XXL = [9%] +[16.] +[17] XXII. [97] +(1%] +[1L) +[1le] +[11s] +114] XXIV. [92] +[15s] +[16:] " XXV. [10,40] = + [103] = + [1s] +114) +[124] XXVIU = [10,40] + [le] «= F015) +0114] +[14s] —2, 240=0 —1. 322=0 —2.595=0 —3, 231=0 +1. 478=0 —0. 409=0 +2. 920=0 —1.989=0 +0. 753=0 —1.782=—0 —1.226=0 —2. 084=0 —1. 155=0 —1. 248=9 +0. 107=0 +0. 374=0 . —0.309=0 —1.436=0 +2. 425—0 —0. 806=—0 —1. 480—0 +[94] —[144a) [1440] +1042] + [1s] + [104] +(1ls} [14s] 532 PRIMARY TRIANGULATION. Numerical equations of condition, Ge.—Continued. XXX, XXXI. XXXIL XXXIIL XXXIV. XXXV. XXXVI. XXXVI. XL. XLI. XLII. XLII. XLIV. XLY. XLVI. XLVII. XLVIII. [lite] =-+0. 01087 I [le] =+1, 398761 [2142] =—1, 27795 I [22] =+2, 39476 I [3142] =+1. 00000 II [32] =—1. 00000 III [33] =-+1. 00000 V [4o4a44¢5] =41. 00000 VI [434445] =+1. 00000 V [4445] =—1. 05623 I [45] =+1. 26933 I [51] =-+0, 62500 V [5243] =—0. 37500 V [5s] =+1.00000 VII [4] =—0, 12500 V [1] =—0. 10090 VI —0, 10000 XII [62] =—0. 10000 VI —0. 10000 XII [65] =-++40. 80000 VI —0, 20000 XII [64] =—0, 20000 VI —0. 20000 XII (6;] =—0. 20000 VI ++, 80000 XII (71] =—0. 07692 VIL [7142] =— 0.15384 VII [72] =—0, 07692 VII [73] =+0.57692 VIL [Pi] =+0, 00109 VII +0. 18058 XVI ANGLE-EQUATIONS—Continued. {12} —[l21] —[12] —[123] —[124] +£14i42] +[143] (126) +£18.] 4014.) (163) +[171.] +[182] (16,) +[18,} +[193] (165«¢} +[ 16s] +€192] [2%] Le) EEO KCI] Let [1655] + C160] +[2lo] +0 2bse] +0221] +0220) +213.) +[2h2] (19,]° [203] {21,J [225] [231] [245] [25.] [265] [271] [285] +202] +[212] +[222] +[ 28: ] +[242] +[252] +[ 262] +1272] +[282] +129] +[221] +[235] +[241] +125] +[ 261] +£275] +£231] +293] +301] General corrections in terms of the correlates. +1. 00000 111° +1. 00000 11 1. 0U00u IV 1, 00000 III 1. 00000 III 1.00000 IV + + + + 1.0000u VI 1.00000 II 1.00000 II . 37500 VI . 62500 VI 0.74724 VIIT . 12500 VE . 20000 VII . 20000 VII . 60000 VIE . 60000 VII - 40000 VIL . 10032 VILL - 20064 VIL -—0, 1082 VII +0, 53293 VIL —0, 03028 XI —0.17771 XV —1. 00000 IV +1,00000 IV —0. 04465 VIIT —0, 04465 VIIT 40, 22326 VIIT -—0, 05000 TX +0, 45000 IX —0. 10000 IX —0, 10000 IX —0, 10000 IX +40, 15385 XII +0. 30770 XII +40, 15385 XII —0. 15385 XII +0. 17124 XIII —0, 07802 XVIII —1. 00000 V —0, 12500 IX —0, 12500 IX +40. 62500 TX +0, 45000 X —0. 05000 X —0. 10000 X —. 10000 X —0. 10000 X +9, 13744 XIV —0. 05314 XIX —0.790=0 —0, +1. +2. +1. +0. =, = +0. 1. +2. —2, ALT: .274=0 +2 +0. +0. —l 517=0 930=0 438=0 007=0 210=0 418=0 006=0 311=0 423=0 083=0 645=0 983=0 006=0 337=0 269=0 (Crap. XIX, C, —0. 10000 XI —0. 10000 XT —0. 20000 XT —0. 20000 XI —0. 20000 XI —U. 16058 XV —, 03380 XX §4.] BUFFALO BASE TO SANDY CREEK BASE. General corrections in terms of the correlates—Continued. [8] =-+40, 00052 VIII +40, 10621 XVI [&+3] =+0, 00022 VIII +40, 01353 XVI [Sssee] =+0. 00447 VIII —0, 00705 XVI [es] ——0. 00030 VIII —0, 09268 XVI [84] =-+-0. 00425 VIII —0, 02058 XVI [83] ——0, 00897 VIIE —0, 05362 XVI [35] —+0,00170 VII —0, 06496 XVI [Sca7] -=-40. 41298 VIII [91] =-10. 06289 VIII —0. 03194 XIX [93] =-40.39702 VIL —0. 00347 XIX [95] =—1, 34975 VIII —0, 01291 XIX [24] =-+0, 92812 VIIE —0. 02017 XIX [2s] ——0, 00860 VIII 40.0164 XIX [9s4er] =. 00000 XX [Yereur6> ] =—0. 01349 VIII +0.21635 XIX [97] =—0. 00826 VIII —0. 04945 XIX (11) =—0.50000 XIV +0. 33333 XXV [19:2] =—0.£0000 XIII +0. 33333 XXV (10) 213] =—0. 83556 XV —0. 18158 XXVI (10. 2 s4¢4J=+0. 50000 XIII —,01879 XXTIT [105 ] =+0.20000 XIII +40. 33333 XXV [13 4] =+41.00000 XIII +0. 71303 XXVII [10] =+0. 50000 XIII -40. 01380 XXIII [111] =—0, 42814 XV —0. 40000 XXV —0. 20000 XXX [Iigops] =—1. 28442 XV —0. 20000 XXV +0. 40600 XXX =—0, 42814 XV —0. 40000 XXV +0. 80000 XXX [112] —0, 01734 XI +0, 17066 XVII — 0, 00739 XI +0, 20453 XVII —0, 14911 XI +0, 47176 XVII +0, 00995 XI +0. 03887 XVIL —0, 14172 XI +0. 26723 XVIL +0, 29891 XI —0.14911 XVII —0. 05676 XI —0, 07246 XVII +0,50000 XII —0. 02208 X -40, 25706 XXI —0. 26327 X —0, 10250 XXI +0, 50835 X —0, 02208 XXI —0, 16556 X +0,-00962 XXI —0, 01291 X —0, 03194 XXI —0. 02029 X —0, 05019 XXI —0. 01238 X —0. 03064 XXI —1,85¢41 XV —0. 09078 XXVI —0. 43278 XV —0, 09079 XXVI —0, 33333 XVI +0, 10460 XXVIL +40. 50000 XIV -L0, 33333 XXV —0, 43278 XV —0, 09079 XXVI +0. 50000 X1V —0.50000 XXVIII +0. 50000 XLV —0, 33333 XXV +40, 40000 XVII +0, 20646 XXVI +1. 20000 XVII —0, 92533 XXVI --0. 40000 XVIT +0. 20645 XXVI +0, 03287 XIII +0, 09928 XVIII +40. 11642 XILL +0. 20969 XVIII -40, 02682 XIIL —0, 01756 XVIII +0, 08355 XIII 40, 11041 XVIII —0. 08960 XTIT —0, 22725 XVII —0, 04367 XIII +40, 07803 XVIII —0, 07719 XIII +0. 00879 XVIII +40. 00962 XI —0, 03064 XXII —0, 08591 XI —0. 00333 XXII —0, 16556 XI —0, 01233 XXII +0. 33160 31 —0. 01935 XXII —0. 02017 XI —0. 04945 XXIL —0. 03169 XI —0.07771 XXII —0.01935 XI +0. 38114 XXII —0. 16667 XVI +40, 05230 XXVII —0. 16667 XVI —0, 22827 XXVIL -L0, 33333 XXII +0. 33333 XXVIII —0, 30550 XV +40, 33844 XXVI —0. 16667 XVI +0, 33287 XXVII -40. 12728 KV +40. 56006 XV +0. 52002 XXVI +42. 21570 XVII +0, 12123 XXVII +1. 10082 XVUI —1. 26727 XXVII —0. 55744 XVII —0, 69425 XXVIII +0. 04609 XIV —0, 06012 XIX +0. 06062 XIV +0. 04709 XIX +0, 29405 XIV +0, 30110 XIX +40, 01453 XIV +40, 10721 XIX +40. 23343 XIV 40, 25401 XIX —0, 18539 XIV —0, 13177 XIX —0, 12304 XIV —0, 05808 XIX 05019 XIV . 10250 XXIV . 00545 XIV . 46712 XXIV . 02029 XLV —0, 26327 XXIV —0, 03169 XIV —0, 08591 XXIV +40, 21685 XIV —0, 00317 XXIV 40. 34077 XIV —0, 00545 XXIV —0.07771 XIV —0, 00333 XXIV +0. 66667 XXII +0. +0. +0. = 16667 XXII 03758 XXIII +0,33333 XVI +40, 48976 XXVIL +0. 16667 XXII =" —0. 33353 XXVIII +40. 50000 XVI +40, (6667 XVI +0. 38516 XXVIL +0. 20000 XXII —0. C0000 XXVIII +0. 60000 XXII +0, 20000 XXVHI +0, 20000 XXIT 16667 XXVIII 66667 XXVIII +0. 10267 XV -L0, 01322 XX +0. 12149 XV —0. 05580 XX +0. 23558 XV —0, 26723 XX 0, 01882 XV —0. 06902 XX +0,11409 XV +0, 32303 XX —0.04771 XV —0, 14172 XX —0. 01364 XV —0, 04585 XX —0, 08083 XVII —0. 00878 XVII —0. 03267 XVI —0. 05104 XVIL- +0. 16740 XVII -+0, 26306 XVII -+40, 30343 XVII +1. 07629 XXIII —0. 01879 XXII -40. 66667 XXV +0. 16667 XXII 533 40, 16667 XXVIII —0. 01879 XXIII +0. 42923 XXVI —-0. 16667 XXII —0. 16667 XXVHUT +0. 34481 XXIIL —0. 02405 XNXIX +40. 74329 XXII +0, 22629 XXIX +0, 34481 X NTI +0, 40000 XXVIII -10, 45256 XXIX 9 ov 4 (11s) (114) [121] [1214245] [122] [12243] [123] [124] [26] (13,] (14: | [14:42] [lds] (l4ie] [14445] (144, J (15,] [152] [153] (16,] [162] [163] [164] [1652] [len] [16.4] ({171] [172] [18] [18] PRIMARY TRIANGULATION. General corrections in terms of the correlates—Continued. =—0, RL XV +40, 60000 XXV 20000 XXX 42109 XV 60000 XXV . 20000 XXX . 20000 XVI . 37727 XXIX =+0, 40000 XVI 13181 XXIX =——0, 20000 XVI +0, 37727 XXIX +0, 60000 XVI 74d XXINX . 80000 XVI 37727 XX1X . 20000 XVI 20296 XXX 00000 XXXI . 00000 XXXI 00000 XXXI . 19424 XII . 64350 XXIX 16545 XUL 79412 XXIX . 10970 XIIL _ t75u XXIX 21940 NUL 25504 NXIX . 10970 XIII . 19758 XXIX 53170 VUE 0, 43866 VILL 03008 VIII . 10598 XXI 01631 XXXV © 73506 XXI 04076 XXXV . 26404 XXI 04076 XXXV =—0, 17663 XXI —0, 02717 XXXV ——0, 04075 XXI -L.0,57065 XXXV =—0, 04076 XXI —0, 42935 XXXV =-+0, 04076 XXI —0, 07065 XXXV ——-0, 33333 XXI =-+40, 66667 XXI ——0, 33333 XXXII =-40, 66667 XXXII lI L _ +0. 40000 XVII —1, 33825 XXVI =f, 40. 60000 XVIT 71888 XXVI —0. 20000 XIX . 20000 XXX . 40000 XIX . 60000 XXX . 80000 XIX . 20000 XXX . 60000 XTX . 40000 XXX . 20000 XTX —0. 20000 XXX —0. 20000 XIX —F. 20000 XXX —0, 03183 XVII +0. 20144 XXX —0. 10403 X VILL +0. 12469 XXX +1, 45700 XVII —0, 7674 XXX 40, 29828 XVIII —0, 15348 XXX —1, 15872 XVIII —0, 07674 XXX +0, 32702 IX —0. 10480 IX —0, 07702 1X +0, 49098 XXIV —0, 05115 XXXVII —0, 10598 XXIV —0. 12787 XXXVII --0. 10598 XXIV —0, 12787 XXXVI —0, 07064 XXIV —0. 08525 XXXVII —0, 01631 XXIV 0.50000 XXXVI —0. 01631 XXIV +0, £0000 XXXVI +40. 01631 XXIV +1. 00000 XXXVI +0, 66667 XXXII —0, 33333 XXXII +40, 66667 XXXHI —0. 33333 XXXIIL —0. 55744 XVITT —0, 69425 XXVIL —0, 55039 XVII +1. 14606 XXVII . 22216 XXII S +0. 624d XXII So +0. 42532 X NII 85064 XNTIT . 42532 XALIT . 10634 XXL E —0. 01798 XX 2 —0. 05876 XX . 41577 XX . 16846 XX . 58423 XX . 10480 X . 82702 X . 14520 X . 10598 XXXIT » 26494 XXNIT +0. 73506 XXXII . 17663 XXXIT 2 —0, 04076 NXXIT . 36570 XX XVII —0, 04076 XXXII 80900 XXXVIT +0. 04076 XXXII 18425 XXXVIL | ia 20000 XXIL 40000 XXVIII nas 40), 20000 XXIT 40000 XXVIII 10. 4:0 —0. 20000 XXV —0. 60000 XXV —0. 20000 XXV —0. 40000 XXV —0, 20000 AXV +0. 0000 XNV —0. 15348 XXVII +0. 11018 XXVIL +0. 10724 XXVII --0, 21448 XXVIII +0, 10724 XXVII —0. 07702 XXIV —0, 14520 XXIV -40. 32702 XXIV —0. 07064 XXXII —0, 17663 XXXII —0. 17663 XXXII +40, 54890 XXXII —0. 02717 XXXII —0, 02717 XXXII +0, 02717 XXXIIT [Cuapr. XIX, C, +0. 05367 XXII1 +0, 29e58 XXIX +0, 05367 XNHI +0, 29858 XXIX +0. 36952 XXVI +40. 20704 XXVI +40, 36952 XXVI —0. 16248 XXVI —0. 53200 XXVI —0. 57655 XXVI —0, 16828 XXVIII +0. 28297 XXVIII —0, 05876 XXVIII —0, 11752 XXVIII —(, 05876 XXVIII —0. 03262 XXXIV —0. 08152 XXXIV —0. 08152 XXXIV —0. 05434 XXXIV +0. 14130 XXXIV +0. 14130 XXXIV —(. 14130 XXXIV 94.7 BUFFALO BASE TO SANDY CREEK BASE. as General corrections in terms of the correlates—Continued. [19] =—9. 10000 XXXIIL —0. 15000 XXXIV +0.20000 XXXV_ 4-0. 35000 XXXVIIE +-0. 04723 XXXIX (19.] =—0.10000 XXXITI ++0.35000 XXXIV _ -++-0. 20000 XXXV —0, 15000 XXXVITI +0. 14509 XXXIX [195] =+0. 60000 XXXIIT —0. 10000 XXXIV —0.20000XXXV_ — —0. 10000 XXXVIIL —0. 09617 XXXIX (20,.) =+0.37500 XXXIV +40. 19861 XXXVI —0. 12500 XXXVIIL —0. 38890 XXXIX —0. 12500 XL [20142] =+0.25000 XXXIV -10, 39722 XXXVIT +0. 25000 XXXVIIL —0. 56243 XXXIX — —0, 25000 XL [20.] =—0.12500 XXXIV +0, 19861 XXXVII +0. 37500 XXXVUI —0. 17353 XXXIX — —0, 12500 XL [20,] =—0.12500 XXXIV —0, 05009 XXXVII —0, 12500 XXXVIIL -+0, 28122 XXNIX 0.37500 XL (21) =—0.u4762 XXXV —U. 12699 XXXVI —0, 09524 XXXVIIL +0.02058 XXXIX = —0.07937 XL +0. 25397 XLI [212] =—0,04762 XXXV_— +0,20635 XXXVI —0.09524 XXXVIII +0. 02058 XXXIX +0. 25397 XL —0. 07937 XLI [2lsa] —=—0.42857 XXXV_ = +£.0,52381 XXXVI +0. 14286 XXXVIII_ —1. 49833 XXXIX —0. 04762 XL —0. 04762 XLI [Ql] =+0.57143 XXXV_ —0.47619 XXXVI +0. 14286 XXXVIII +1. 43657 XXXIX —0. 04762 XL —0. 04762 XLI (22.] =+0. 13427 XXXVI —0. 17790 XXXVII +0. 31763 XL —0. 09920 XLI —0. 10922 XLII [222] =—0.13026 XXXVI +40. 00372 XXXVII —0. 09920 XL +0. 24550 XLI —0. 07315 XLII [225] =—0.00201 XXXVI +0. 08709 XXXVII —0, 10922 XL —0. 07315 XLI +0. 59118 XLIL [22] =+0.51710 XXXVI +10. 46926 XXX VII —0, 15336 XL —0. 3106 XLI +0. 10721 XLII (23,] =—0. 08333 XLI —0. 19444 XLIT +0. 47222 XLUOI (22] =—0. 08333 XLI +0. 47222 XLII = —0. 19444 XLIII [233] =++0.25000 XLI —0. 08333 XLIT = —0, 08333 XLIII [24] =+0.50000 XLII —0.16667 XLIII —0. 16667 XLIV [24] =—0.16667 XLII = -++0.50000 XLIILT =—0. 16667 XLIV (24s] =—0.16667 XLIT —0.16667 XLIII +0.50000 XLIV -(25:] =—0.16667 XLIILT = —0.33333 XLIV- +0. 66667 XLV [252] =—0.16867 XLIIL = +0.66667 XLIV. _ —0, 33333 XLV [253] =+0.66567 XLIIE —0,16667 XLIV —0. 16667 XLV [26,] =+0.75000 XLIV —0,25000 XLV = —0. 25000 XLVI (26.] =—0.25000 XLIV +0.75000 XLV —0. 250°0 XLVI [263] 0.25000 XLIV —0.25000XLV — 40. 75000 XLVI (27.] =—0.20000XLV = —0.13333 XLVI 0.53333 XLVII. [272] =—0. 20000 XLV -++0. 53333 XLVI =—0. 13333 XLVIT [273] =+0.70000 XLV —0,20000 XLVI —0.20000 XLVII (23:] =+0. 29788 XLVI —0.08511 XLVII —0. 08511 XLVIII [28] =—0. 08511 XLVI +0. 29574 XLVII —0. 10426 XLVIII [28] =—0.08511 XLVI —0.10426 XLVII +0.29574 XLVI (29.] =+0.27727 XLVIII [29%] =+0.54545 XLVII (30,] =+0.75000 XLVIIT Normal equations for determining the correlates. No. of equation. 1, 0=:-+0. 49870 +12. 05150 I —0. 92680 IT +2. 40563 ITI —4. 72894 1V , + 1.05623 V 2, 0=—2. 24000 — 0.92680 I +4. 00000 IT +1. 00000 III +1. 00000 TV — 1.00000 V 3. 0=—1. 32200 + 2.40563 I +1. 00000 IT +4. 00000 ITI —2. 00000 IV 4. 0=—2. 59500 — 4.72894 I +1. 00000 II —2. 00000 III +4. 00000 IV — 1.00000 V 5. 0=—3, 23100 + 1.05623 I —1. 00000 IT —1. 00000 IV +3. 62500 V — 1.37500 VI —0, 04465 VIIT — 0, 12500 IX ~ ro) 36 No. of equation. G. ‘ane 10. 11. 13. 14. 16. 17, 18. PRIMARY TRIANGULATION. [Cuap. XIX, C, Normal equations for determining the correlates—Continued. rt Q=+1. 47200 —1. 37500 V +3. 42500 VE +0. 60000 VIT — 0.04465 VHT —0, 22500 TX —. 10000 X —0. 20000 XI — 0.20000 XII 0=—0. 40900 +0, GO000 VI +2. 77692 VII —0. 21431 VIII ~- 0,20000 1X —0. 20000 X —0. 40000 XI —0. 55385 XIT 0=—5 76345 —0. 04465 V —0. 04465 VI —0. 21431 VII + 7.03756 VITT —0. 30844 TX —0. 91109 X +0. 91915 XI + 0.21220 XIT +0. 00131 NTI —0. 00793 XIV +0. 00143 XV + 0.00161 XVI —-0, 01728 XVI —), 00234 XVIOIT —0, 00465 XTX + 0.004125 XX 0=+2. 92000 0=—1. 98900 0=+.0, 75300 O=--1.78200 0=—1. 22600 0——2. 08400 O=41. 42215 0=—1. 15500 0-=—1, 24800 0=—2. 07340 +0, 06229 XXT —v, 12500 V . 40202 1X —0. 07702 XXIV . 10000 VI +11. 28537 X —0. 03267 XVIT —0. 40847 XXIV —0. 20000 VI —0. 26556 X —0. 21708 XIV 7303 XVIII —0. 01935 XXII —0. 20000 VI —-0, 10000 X +0, 00131 VIII -+40. 04819 XV —0. 00605 XIX —0. 66545 XXVIII —0. 00793 VIII 77226 XIV —0. 09558 XVIII .57771 XXII 43746 XXVII +0. 00143 VIII +8. 99543 XV +0. 13291 XIX .71479 XXV —1.11778 XXIX +0. 00161 VIII +40, 50215 XV —0. 31326 XIX —0. 53333 XXV +0. 37727 XXIX —0. 01728 VIII +0.55711 XIV . 08326 XVIII +0, 90343 XXII —0. 92533 XXVI +0. 40000 XXX —0. 00234 VIII . 16257 XV —0. 11684 XIX —1. 10783 XXV —1, 68286 XXIX De --0. 00826 XXII —0. 22500 VI —0. 15480 X —0. 20000 VIT —0. 26556 XI —0. 01291 XTX —0. 40000 VII . 43051 XI —0. 04771 XV —0.15194 XIX —0, 08591 XXIV —0. 56385 VII —0, 20000 XI —0, 04367 XI +0. 70411 XVI —0. 19930 XX —0.53908 XNIX —0, 02029 X 62791 XV +0, 46481 XIX . 09508 XXIII —0. 04771 XI 50215 XVI +40. 11409 XX +0. 79055 XXVI —0, 42814 XXX —0, 05362 XI . 76346 XVI —0, 02058 XX —0, 01198 XXVI —0. 20000 XXX —0, 03267 X . 04884 XV -+40. 46850 XIX 0.74329 XNIIL —1. 26727 XXVII £ +0. 07803 XI +0. 02126 XVI —1. 38597 XX. +0. 57907 XXVI —0, 69330 XXX +0. 36694 XXTV --0. 20000 VIT —0. 10000 XT —0.91109 VIII . 10000 XII —0. 02208 XXI +0. 91915 VIII —0. 20000 XII . 05362 XVI —0, 14172 XX => +0. 21229 VIII +1. 60770 XII +1. 67251 XI 40. 02682 XVII +0, 42923 XXVI +0, 02879 XXX —0, 21708 XI +40. 68353 XVI +0, 23343 XX —0. 00545 XXIV +0. 08819 XIII —1. 04884 XVII —3, 56392 XXII +0. 24225 XXVIT +0. 70411 XTIT —0. 00705 XVIT —0. 16667 XXII +0. 38516 XN VIT —0. 20015 XI —0, 00705 XVI +40. 26723 XX —0, 00878 XXIV +0, 20000 XXVUI +0. 06258 XIII +1. 08326 XVIT +0. 55043 XXIT +0. 53499 XX VIL — 0.30844 VIIT — 0.10000 XIT — 0.15480 IX — 0, 02029 XIV — 0.01238 XXII — 0.10000 IX — 0. 04367 XIII — 0.20015 XVII + 0, 00962 XXI — 0.10000 1X + 0.69306 XIV — 0.06278 XVHT 0, 39337 XX VII + + 0, 69806 XIII + 0.f5711 XVI — 0.05019 XXI 0. 42922 XXVI + + 1.62791 XIV . 16257 XVIII . 77359 XXIII — 1.71015 XXVHI ++ 0.68353 XIV + 0.02126 XVIII + 0.41412 XXIII — 0.16667 XXVIII + 0.02682 XIII + 2.23825 XVI — 0.08083 XXI — 0.20000 XXV + 0.22629 XXIX + 0. 09558 XIV +10, 09750 XVIII + 0.95097 XXIIT — 1.76930 XXVIII §4.] No. of equation. 19, 20. 21. 22. 23, 24, 26. 28. 29. 30. 31... 32. BUFFALO BASE TO SANDY CREEK BASE. O37 Normal equations for determining the correlates—Continued. 0=-+40. 10700 —0. 00465 VIII +0. 46481 XIV —0, 11684 XVIII --0, 04945 XXII +0. 36952 XXVI 0=-+0. 37400 +0, 00425 VIII +0, 11409 XV +0, 25401 XIX —9, 12752 XXIX 0=—0. 30900 +0. 06289 VIII —0. 08083 XVII —0. 20848 XXIV —0. 04076 XXXV —0, 00826 VIII —3. 56392 XV —0, 01945 XIX —0. 00333 XXIV +0.76667 XXVIII —1. 09508 XIV +0. 95097 XVIII —0. 33658 XXV 41. 13573 XXIX O=-12. 42500 +0. 36694 VIII —0. 00545 XIV —0, 00333 XXII —0, 03262 XXXIV 0=—1. 43600 0=-40. 16330 0=—0. 80600 —1.71479 XV —0. 20000 XIX —1. 37750 XXVI —0. 60000 XXX 0=—1. 81890 +0. 42923 XIIT —0, 92533 XVII —0, 04414 XXIII —0. 50370 XXVIII 0——1. 50270 +0. 39337 XIII —1, 26727 XVII — 0. 27582 XXIII —0. 36053 XXVIII —0, 66545 XITI —1. 76930 XVilI +1. 13333 XXV +1. 84384 XX1X —0. 53908 XIII —1, 68286 XVIII +1. 13573 XXIII +1, 84384 XXVIII +0. 02879 XIII —0. 69330 XVIII +10. 12267 XXIII +0, 52469 XXVIII +3. 00000 XXXI —0. 59827 XXI —0, 08152 XXXIV 0=— 1. 48000 0=—1. 39885 0==— 0. 79000 0=—0, 51700 0=-+11. 93000 68 L 8 —0. 01291 X +40, 13291 XV +1. 66286 XIX +0. 42532 XXIII +0. 37727 XXIX —0. 14172 XI —0. 02058 XVI +42. 90726 XX —0. 07674 XXX — 0 02208 X —0, 03194 XIX —0.59827 XXXII —0. 12787 XXXVII —0. 01238 X —0. 16667 XVI —0. 03064 XXI +0. 73333 XXV +0, 52487 XXIX —4, 77359 XV +0, 42532 XIX —0. 04414 XXVI +0. 12267 XXX —0. 07702 IX -—0, 00878 XVII +1, 28512 XXIV —0, 01631 XXXV —0, 53333 XVI _+-0, 73333 XXII +0. 55641 XX VII +0. 42922 XIV +40. 57907 XVIII —1.37750 XXV +0. 49069 XXIX +0. 43746 XIV -L0. 53499 XVIII +0. 55641 XXV +0. 07017 XXIX —1.71015 XV —0. 05876 XX —0. 50370 XXVI +0. 52469 XXX =) ey +0. 37727 XIX —0, 60680 XXV +5. 19089 XXIX —06, 42814 XV —0. 26000 XIX —0, 60000 XXV +0, 67533 XXIX —0, 10598 XXIV —0. 04076 XXXV —0. 15194 XI —0. 31326 XVI +0. 25401 XX —0. 00347 XXIV —0. 20000 XXX —0. 19930 XIII +0. 26723 XVIT +0. 10724 XXVIT +0, 00962 XI +1. 65879 XXI —0. 17663 XXXII —0. 01935 XI +0. 90343 XVIL +1, 84781 XXII -—0, 29723 XXVI +0. 20000 XXX +0. 44412 XVI +1. 87325 XXII —0. 27582 XXVII —0. 40847 X —0. 00347 XIX —0, 10598 XXXIT —0. 05115 XXXVII —0. 20000 XVII —0. 33658 XXIIT +1, 13333 XXVIII +0, 79055 XV +10. 36952 XIX +3. 87859 XXVI +0, 57597 XXX +0, 24225 XV +0. 10724 XX +2. 00673 XXVI —0. 73755 XXX —0. 16667 XVI 4-0. 76667 XXII —-0, 36053 XX VII +0. 37727 XVI —0. 12752 XX +0, 49069 XXVI +0. 67533 XXX —0. 20000 XVI —0. 07674 XX +0.57597 XXVI +1. 92613 XXX +2. 06840 XXXIT —0. 12787 XXXVIT —0. 00605 XIIL +0, 46850 XVII —0, 03194 XXI —0, 20000 XXV +40. 23343 XIV —1.38597 XVIII —0, 05876 XXVIII 0.05019 X1V —0, 03064 XXII —U. 08152 XXXIV —0.57771 XIV +0.55043 XVIII +1. 87325 XXIII —0. 06891 XXVII 0.74329 XVII +3, 90483 XXIII +0, 43336 XXVIII —0. 08591 XT —0, 20848 XXT —0. 07064 X XXIII —1, 10783 XVIII +2. 66667 XXV —0. 60680 XXIX —0. 01198 XVI —0, 29723 XXII +2. 00673 XX VII +40. 38516 XVI --0. 06891 XXII +3. 14136 XXVII +0. 20000 XVII +0, 43336 XXIII +2. 14964 XXVIII 4.0. 22629 XVII +40, 52487 XXII +0. 07017 XXVIL +0. 40000 XVII +0. 20000 XXIT —0. 73755 XXVIT —0, 50996 XX XIII 538 No. of equation. 33. 34. 30. 37. 38. 40. 41. 42. 43. 44. 46. 47, 48. 0=42. PRIMARY TRIANGULATION. [CwHap. XIX, C, Normal equations for determining the correlates—Continued. 43800 0=+41, 00700 0=-+0, 21000 0=—1 0=+41 0=—1 . 41200 . 01892 . 00600 . 64600 . 31100 - - 42300 . 08300 0=—2. 64500 0=+1 0=—1. 0=+42 0=+0 0=-+40 . 98300 7400 . 00600 . 338700 . 26900 —0, 17663 XXI —0. 15434 XXXIV —0. 09617 XXXIX —0. 08152 XXT +1. 00760 XXXIV —0. 24381 XXXIX —0, 04076 XXTI . 34130 XXXIV . 34286 XXXVI . 97619 XXXV 47775 XXXIX . 12787 XXI . 64191 XXXIV . 19861 XXXVIIT . 08709 XLIL . 10000 XXXIIT . 19861 XXXVIT . 09524 XLI . 09617 XXXII . 44679 XX XVII - 02058 XLI . 12500 XXXIV . 22024 XXXVIII . 10922 XLIT . 04762 XXXV . 02058 XX XIX . 08333 XLII . 00201 XXXVI . 56340 XLII - 03333 XLI . 16667 XLV . 16667 XLII . 25000 XLVI —0. 16667 XLIII —0. 20000 XLVII —0. 25000 XLIV —0. 08511 XLVIII —0. 20000 XLV —0. 08511 XLVI —0. 07064 XXIV —0. 22717 XXXV —0. 03262 XXIV +0. 34130 XXXV —0, 12500 XL —0. 01631 XXIV +1.54208 XXXV +1. 62889 XXXIX +2, 91153 XXXVI 0.34062 XL —0. 05115 XXIV . 36570 XXXV —0, 44679 XXXIX —0. 27500 XXX1V +1. 01072 XXXVIIT —0, 24381 XXXIV —0. 18806 XX XVIII —0. 04762 XXXV +0. 30180 XXXIX —0. 25725 XXXVI. —0. 17557 XL +0. 03709 XXX VII —0. 36111 XLII —0. 36111 XLII —0. 33333 XLII —0, 58333 XLIV —0. 45000 XLV —0, 213844 XLVI —0. 10426 XLVII —0. 50996 XXXII —0. 08525 XXXVIT —(. 08152 XXXII +0. 64191 XXXNVIT —0. 04076 XXXIT —0. 97619 XXXVI —0. 04762 XL —1. 08389 XXXVII —0, 25725 XLI —0. 12787 XXXII —1. 08389 XXXVI —0, 22799 XL +0. 34286 XXXV —0. 18806 XX XIX +1. 62889 XXXV +5. 10126 XXXIX +0, 34062 XXXVI 40. 94660 XL +0. 00372 XXXVII +0. 74947 XLI —0. 10922 XL —0. 16667 XLIV +1. 63889 XLITT +1. 91667 XLIV +2. 11667 XLV +1. 58121 XLVI +1. 37452 XLVII +1. 77301 XLVI +1. 81557 XX XIII —0, 10000 XXXVIIT —0, 15434 XXXII —0. 27500 XXX VIII —0. 22717 XXXII —0. 36570 XXXVIL —0, 04762 XLI +0. 04762 XXX VIII --0, 00201 XLII —0, 08525 XX XIII +5, 95568 XXXVII +0, 00372 XLI +0. 04762 XXXVI —0. 22024 XL —1.47775 XXXVI +0. 30180 XL —0, 22799 XXX VII —0. 17857 XLI —0, 09524 XXXVIIL —0. 15648 XLII —0. 15648 XLI —0. 33333 XLIV — 0, 58333 XLV —0. 45000 XLVI —0. 21844 XLVII —0. 10426 XLVIIT §4.] BUFFALO BASE TO SANDY CREEK BASE. 539 Values of the correlates and their logarithms. I =+0. 4062 log 9. 6087185.. XXV =-+1. 4399 log 0° 15834444. II =+0. 2814 log 9. 44938584 XXVI =+1. 5074 log 0. 17823144 III =+0. 9463 log 9. 97601974 XXVII =—0. 8934 log 9, 9510411_ IV =-+1. 8608 log 0. 26970904. XXVIII _ =—0. 0834 log 8. 9212181_ V =+1. 3164 log 0. 1193945. XXIX =-10. 4205 log 9. 62376604 VI =—0. 0107 log 8, 0301948_ XXX =+U. 1777 log 9. 24961414 VII =+0. 2413 log 9. 38252134 XXXI =-10. 1723 log 9. 23636094. VIII =+1. 1743 log 0. 0697939 XXXII =—1. 9971 log 0. 3003976_— IX =—1. 6794 log (0), 2251515 XXXITI =—2. 2044 log 0. 3432806 X_=+1. 2787 log 0. 10677554 XXXIV =—2. 4332 log 0. 3861850— XI =—1. 0161 log 0. 0069536 XXXV =+1. 1281 log 0. 05233224 XII +10, 8838 log 9. 94635404, XXXVI =-+1. 0468 log 0. 0198844 4 XII =-+0. 0123 log 8. 0899051+ XXXVIT =-+0. 2034 log 9. 30841504 XIV =—0. 3766 log 9. 5752340— XXXVIII =—0. 3553 log 9. 5506441_ XV =—0. 2722 log 9. 4348403_— XXXIX =—0. 3079 log 9. 4883815 -- XVI =+1. 9104 log 0. 28111754. XL =—0. 6287 log 9. 7984573 XVII =-+0. 3982 log 9. 6001340+ XLI =-+2. 0764 log 0. 31730684 XVIII =-+0. 4117 log 9. 6146020. XLIT =—0. 9979 log 9. 9990827 XIX =-+0. 3747 log 9. 57371854 XLITI =-+1. 2825 log 0. 10805064. XX =-0. 0567 log 8. 75350654 XLIV =—1. 0850 log 0. 0354457_ XXI =—1. 1471 log 0. 0595899_ XLV =-+0. 0350 log 8. 5438198. XXII =-+0. 9359 log 9. 9712434 + XLVI =—1.5129 log 0. 1798217 XXIII ——1. 4037 log 0. 1472898 XLVII =—0. 4998 log 9. 6987702. . XXIV =—2. 4911 log 0. 3963929_ XLVIII =—0, 2537 log 9. 4043798... Values of the general corrections. aw ‘ &f ut a uw [lise] =-+0. 951 [7s] =-+0. 629 {1l;] =—0.318 [166] =-+0. 638 [24.] =—0.532 [12] =+0.850 | [8] =+0.272 | [Ils] =+40.741 | [1%] =—0.949 | [242] =4+0.988 [Qiso] = 41.342 | [&] =+0.263 | [12] =+0.247 | [1%] =—0.099 | [24,] ——0.590 [22] =+0.058 | [8%] =- 0.102 | [1%] =—0.287 | [18] =—0.804 | [2]. =+40.171 [340] = 41.228 | [&] =+0.116 | [12] =—0.111 | [18%] =—0.597 | [25.] =—0.949 [32] =+0.915 | [85] =—0.419 | [12] =—0.146 | [1%] =+0.672 | [255] =-+1.030 [33] =+1. 316 [es] =—0. 065 [125] =-+0.172 [19,] =—0. 397 [26;] =—0, 444 [4o494445] =—0.011 | [8ctr] =+40.927 | [131] =-+0.172 | [19%] =—1.240 | [26] =+0,676 [4o4s¢5] =+1.327 | [9] =—6.058 | [14] =+0.172 | [20] =—0.629 | [26,] =—0.872 [4446] =-+0. 397 [92] =—0. 835 [14142] =—0. 096 (2C2] =+0.343 [27,] =—0.072 [45] =-40.234 | [95] =—0.107 | [14s] =+0.186 | [20;] =+0.016 | [2%] =—0.747 [51] =-40. 984 [94] =-+0.710 [1442] =+0. 417 [21,] =-+0. 418 [275] =-+0. 427 [Sets] =—0.343 | [95] =+0.166 | [14,] =—0.603 | [21] =—0.135 | [28] =—0.387 [5s] =—0.636 | [9s4ea] ==+0.113 | [151] =--1.116 | [2ls0] =+0.407 | [28] =-+10.007 [54] =—0.951 | [9steape.J=+0.047 | [152] =+41.471 | [21a] =—0.416 | [283] =+0. 106 [6] =+0.625 | [97] =+0.526 | [155] =—0.906 | [22] =—0.192 | [2%] =—0.185 [62] =—0. 854 [10,] =— 0, 229 [16,] =—0. 684 [22.] =-+0.510 [£93] =—0.273 [63] =+0.203 | [1042] =-+0.467 | [16] =+40.466 | [22] =—0.658 | [70] =—0.190 [64] =-+0, 213 [105] =-+0. 061 [16;] =—0. 384 [224,] =-+0. 612 [65] =-+0. 856 [104] =+0. 731 (16,] =—0.394 L23,] =+0. 627 [11] = 0.000 | [11] =0.312 | [164] =—0.069 | [23,] --—0.894 [72] = 0.000 |. [1] =+40.404 | [16;,) =+0.089 | [23:1] =+0. 495 : 540 PRIMARY TRIANGULATION. [Cuar. XIX, C, D, Residuals resulting from substitution of general corrections in numerical equations of condition. saa: Residual, enean Residual. equation. Residual. 1 -0.0003 =| 17 0. 0001 33 | 0. 0000 a) 0.0000 = «18 —0. 0048 34 0. 0000 3 0.0000 =; 19 0. 0000 33 | 0.0001 4 0.0000 | 20 0.0000 36 0.0000 5 0.0000 || at 0.0000 37 —0. 0010 6 0. 0000 2 0.0000 38 +0 0001 7 0.6000 | 23 —0. 0020 39 —0. 0003 8 0.0000 | (24 4-0, 0001 40 0.0000 9 0.0000 | 0. 0000 41 0.0000 10 0.0000 =| = 26 0. 0010 42 +0. 0001 u +0. 0001 | 27 -40. 0009 43 0. 0000 12 0.0000 | 28 0. 0000 44 0. 0000 1B 0.0000 || = 29 +0. 0008 45 0.0000 14 o.o000 =|) 30 —0. 0001 46 0. 0000 15 +6. 0004 31 0. 0000 47 —0. 0001 16 0. 0000 | 32 0. 0000 48 0.0000 PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § 3. Let m=whole number of observed angles in a section (one adjustment). y=whole number of rigid conditions in a section. n=number of triangles in principal chain. [ prv]=sum of weighted squares of corrections to observed angles. =probable error of an observed angle of weight unity. p,=probable error of an observed angle of average weight in whole section. p=probable error of an adjusted angle of average weight in whole section. p,=average weight of an observed angle in whole section. p.=average weight of an observed angle in principal chain. p.-=probable error of an observed angle of average weight in principal chain. p.=probable error of an adjusted angle of average weight in principal chain. [rr]=sum of squares of closing errors of triangles in principal chain. p,=probable error of an observed angle in principal chain as derived from closing errors of triangles. . Proceeding as in Ciapter XIV, C, § 8, there are found the following values: FOR THE ENTIRE SECTION OF THIS CHAPTER. Section. Extent of section. m r [pvv] Py D, Ps fmt P, “uw “ “wn XIII | Falkirk-Pekin to Carlton-Sir John.-..-..--.......--| 170 | 112 | 90.61 | 0.61} 1.22] 0.55 0. 58 0, 32 FOR THE PRINCIPAL CHAIN CONNECTING THE BUFFALO AND SANDY CREEK BASES, GIVEN IN D, § 6, « FOLLOWING. From closing errors of triangles. Section. Extent of principal chain in each section. Po Po p! . [vv] | 2 Average | Greatest Pe error. error. | | “a a“ Ww aw a“ .« XII} Buffalo Base to Falkirk-Pekin ......-....22.02-000005 0.87 | 0.52 | 0.30} 8.00 7 | 0.42 0. 83 2.13 XIII | Falkirk-Pekin to Sandy Creek Base ......-......--.. 1.46 | 9.50 | 0.29 | 45.36 | 22 | 0.56 1.21 3. 42 Entire principal chain .........22.220..22220.22-|-c0e0ee ltieeeleeenes | 53.36 | 29 | 0.53] 111 3.42 §§.5,6.] BUFFALO BASE TO SANDY CREEK BASE, 541 BASES. § G. In adjusting the sides of the principal triangles joining Ruffalo and Sandy Creek Bases, the bases have been considered exact. The probable error of an observed angle of average weight in that part of this chain between Buffalo Base and the line Pekin—Falkirk is +0/.52 (Chapter XIX, C, §5), and for the remainder of the chain the corresponding probable error is 0.50 (Chap- ter XIX, C, §5). The logarithm of the measured length of Sandy Creek Base with the English foot as unit is 4.2057301. As computed from Buffalo Base through the intervening triangulation, the logarithm of Sandy Creek base is 4.2057324. The difference of these two logarithms gives, in the notation of Chapter XIV, D, § 10, d=—23. With the above probable errors, and the values of (c?+?) for the several triangles given in the following tables, the constant 1,1 + p’ = F(w+ fi?) p? = 4249 These quantities and the values of 5 supplied by the tables readily give the corrections to the log- arithms of the sides computed from Buffalo Base. The line of least weight in the chain is Pinnacle Hill-Turk’s Hill, for whieh | = 2123 and z = 2126, giving for the probable error of the logarithm of its length +32.6 in units of the seventh decimal place. This probable error, which is independent of the probable errors of the bases, corresponds to zyss55 of the line’s length, a fraction about seven times as great as the ratio of the probable error of either base to its length. Principal chain of triangles between Buffalo and Sandy Creek Bases. Logarithms Weighted mean Stations. Angles. cee of sides in | a2 and B? | & (242) a logarithms of : fect. Pp sides in feet. oF % “ “ Tonawanda ...........------| 68 41 34.598 |) ( 4. 3472757 Ok lasucesiendians lesecees : 4. 3472757 East Base. ..-....2..2.------ 44 35 02.187 \ —0. 4034 4, 2243328. loci scaciz ox lec cnemeuse|omeawess 4, 2243328 West Base..........-...---- 66 43 23. 294 J | 4. 3411538 82. 81 150. 05 40 4. 3411538 Buffalo Plains .......--..--. 54 39 47.777 |) fc 4, 3411538 4. 3411538 32 36 50. 514 +0. a 4. 1611550 4. 1611579 92 43 21.785 J 4, 4290972 4. 4290971 Buffalo) c.s0senaccaaieaciencnsee 38 16 06.730 l f 4. 4290972 T1289) | eens oasacins |samaiten 4. 4290971 Tonawanda .......----.-.--. 41 36 53.978 —2. 126 4.4594103 |....-..... Ecanteerte ale ge heeiaie ere 4. 4594101 Buffalo Plains .-..-..--..--. 100 06 59. 469 J | 4, 6303575 14. 44 1100, 39 294 4. 6303573 Ridgeway ...------.-..----- 41 27 59. 899 | { 4. 6303575 50644) | ossiec mecca ste seesaw: 4. 6303573 Tonawanda .......-...-..+-- 49 07 11.079 —0. 507 A GBT94GC2 asain winisieis x besadiontie'anecysil| vrata clarion 4. 6879160 Buta lo wis wie cscs cineca inane) 89 24 49. 513 f | 4. 8093565 0. 04 1666. 87 445 4, 8093563 52 38 14. 801 4, 8093565 256.00) |/:.0sicce's sere orel| Sace oder 4. 8093563 75 12 23.540 | - +0.417 4, 8944530 |...-2..2--|.eee ee eeeeee[ee eee eee 4, 8944527 52 09 22. 600 J f 4, 8065475 265. 69 2188. 56 584 4. 8065472 59 58 20. 058 l ( 4. 8065475 4. 8065472 63 09 08.315 +0. a 4.8196056 |.......... eiatatea gare widi| aisteisiarsis 4. 8196052 56 52 32.461 J 4. 7921164 190. 44 2527. 84 675 4. 7921160 Palkeitke nc cis scuinceenaienien vein 31 23 11. 220 | 4. 7921164 1190. 25 |... 2. ceaneelecnsenee 4. 7921160 Pekin ics. se6aqaiceaisioae ws Silaroe 66 25 49.830 | > +1. 606 5. 0376086 |.--.--.---|-------eeeee[e eee eee 5. 0376081 Tonawanda.......-.---.---- 82 11 00.518 J 5. 0713865 8.41 3726. 50 995 5. 0713860 9) 42 PRIMARY TRIANGULATION. (Cuap. . Principal chain of triangles between Buffalo and Sandy Creek Bases—Continued. XIX, D, Logarithms 1 Weighted mean Stations. Augles. Hee of sidesin | a? and B?| & (a?+ 8?) = logarithms of : feet. P sides in feet. Oo # " “a Gasponticcssacces siese toads 97 22 23,194 l ( 5. 0713865 E20 Wiis Casecseos lacs t aaa! 5 0713860 Paitin: ocstocwdgan decades 48 49 46. 787 —0. 179 4.9516456 |.......---|- eerste |e dee cies 4 9516449 POKiliemsicas exevessseece 33 47 51.401 | | 4. 8202694 985. 96 993. 25 1245 4 8202687 Batavia. scscceccceensce ones 40 29 30.720 |} i 4. 8202694 G1OS09 || cweiae view ne nd [esieseca: een 4. 8202687 Gasporbs: hn cuar Podacissectndic 35 16 09. 044 j —0. 233 A(GQ2B7G) lewsciccerssehiiscizis goes sicwita Has 4. 7692870 Falkirk. ......---..---.----- 104 14 21.123 | 4. 9942461 28. 09 1631. 43 1406 4. 9942453 AlDIOMiecce sae saved eases 62 03 57. 359 | } (| 4.9942461 128021 |ac0ceeeesec] ss ccncce 4. 9942453 Batani@s: .cecex raceces sateen 68 48 38. 879 ( -~1.510 J 1B, OLIC4AG. | swesee sca feces seciecinenil: neeeces 5. 0176437 Gaspottscoie2oncekegerce ieee 49 07 25, 593 J i 4. 9266391 331. 24 2085. 88 1521 4. 9266383 Morganville ......---------- 67 24 57. 368 { 4, 9266391 TIAA saci aeons | Scares 4, 9266383 Albion. -......-.....-.------| 38 38 20.215 +1. 8435 4GSOTHID. asses cee as bires caseccleteesees 4. 7567571 BatavlBiiecec ec snondetionseee 73 56 43. 509 tl 4. 9440111 36. 00 2199. 32 1549 4, 9140103 Brockport .........---...+- 81 28 42, 295 |} 4, 9440111 QaGD eimai asec lsiey ares 4. 94401038 Morganville ..-.-..-......-.| 43 47 49.749 { —1, 459 ACT890051L. lecctcoeedc|eseted tena sa llepaeomce 4. 7890042 ANbiODs. eccee sus cesceenetace 54 43 28. 996 | J 4, 8607279 222.01} 2430.94 | 1608 4, 8607270 Scottsville ............2.... 46 45 36.191 t 4. 8607279 39204. [acces ves sais bem ee aie 4. 8607270 Brockport ..............2--- 65 53 18.410 | > +1. 949 49586568 |...---.--.|-------- ema) se ecusad 4. 9586559 Morganville .......--...---- 67 21 06. 834 J 4. 9634528 77. 44 2900. 42 1726 4. 9634519 Pinnacle Hill .........------ 68 35 26.513 | ) { 4, 9634528 88:89: | eccmnmictsare|saanexer 4. 9634519 Scottsville .........-......-. 74 23 44. 870 L —2, 233 5 ASQTBIOGD |asissinecce: [esac cde ewan Seesczen 4. 9781945 Brockport: sss sincieiee sie 255% 37 00 49. 856 J 4. 7741062 778. (Lb 3747. 72 1939 4. 7741052 DLork'’s Hill ..ccsescacvecsces 49 25 11.741 |) (| 4.7741062 324.00: [essence versne|eeceseee 4. 7741052 Pinnacle Hill ........--....- 8422 48.718 | 41.4344] 4. gorde7g |.....2..2 [eee ee eee efeeee eee 4, 8914867 Scottsville .......--.....---. 46 12 00. 329 J 4. 7519733 404. 01 4475. 73 2123 4. 7519722 Walwortthsontcsceseeca te 54 48 16.567 | (| 4..7519733 29901. | ecoos setewatiandesa 4. 7519722 Turk’s Hill.......2.22.22-05 79 00 51.397 | —o.6083| 4.8316175 |.....2.0..|----2-ceeeee]eeeeeeee 4. 8316163 Pinnacle. Hill exe -sseeveees 46 10 52. 688 J | 4. 6979065 408.04 | 5105.78 | 2282 4. 6979053 WWE PR sence ce sccee sede te 44 12 08. 420 (| 4.6979065. ROBE ener smes bina | eo ceecde 4. 6979053 Walworth, 20; 823) | paceacanall en tetemamatensvess Kansas and Westfield......-.....--- 35 56 01.443 233 | 16 3.4 1 +0.115 | +0. 087 35 56 01. 645 Westtield and Lynn Grove........-. 168 29 51,880 | 284. | 16 5.3 1 0} 453 |eicecex woee| seeceemerees sees oy Westfield and Palermo........-..--. 218 38 50. 669 23441 | 8 4.2 O55) 20! 136 cece melee cece aaiaenrnesntned Is NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(231) + (232)4- (223) +0. 677=0 (231) +3 (282) + 2(233)—0. 120=0 (231) 4+ 2(282) 4+-3(233)—0. 120=0 Note —The general correction of 23; is taken from Section XV of the adjustment. Numerical equations of condition in the triangulation from the line Willow Springs —- Morgan Park to the line Oakland -— Kansas. SIDE-EQUATIONS. IX. (12) —11. 28363 [7,] — 2.98067 [75] —-11. 35925 [8] +18. 95853 [8,] —22. 67882 [91] + 3.41374 [9,] +112. 546=0 XIV. (12) —14.51686 [12,] + 9.30196 [12,] — 7.32972 [12,] +17. 66514 [13,] —13. 89060 [145] + 5. 75918 [144] —21.739=0 XXL. (10) —15. 55703 [15s] ++ 4. 44373 [15,] —10. 59439 [16;] +11. 53102 [16;] — 9. 04167 [171] 411, 23329 [172] +11. 730=0 XXIL. (11) — 9.34115 [16,] — 9.34115 [16,] — 9.34115 [16] +12. 27872 [16,] — 7. 66533 [175] +13. 86060 [17,] +13. 86060 [17,] —16. 22979 [19,] + 7.71952 [18,] —24, 832=0 XXIV. (8) —25. 31763 [16,] + 1.51673 [16,] — 8.38928 [18,] — 6.92922 §18,] — 6.92923 [185] — 6.92923 [18,] — 0.79815 [20,] +. 7.91878 [204] + 7.2510 XXIX. (7) — 7.91763 [17;] +21.52593 [175] + 1.58811 [19] + 1.58811 [19.] +10. 39144 [195] — 4.50710 [21] + 5.41820 [21,] + 5. 41820 [215] — 9. 781=0 XXXIII. (18) + 6.92923 [18] +14. 64875 [183] +14. 64875 [18,] —28. 51493 [192] — 6.75645 [19.] —20. 41688 [21,] —20, 41688 [21,] +33. 64864 [215] +33. 018=0 Notr.—In the solution for determining the general corrections, each of the side-equations was divided by the number inclosed in parenthesis and placed opposite it. eANGLE-EQUATIONS, Io f4i) + (2) + Pi) —0. 415=0 TI. [2:) + [82] + [44] —0. 483=0 Wf. [33] + (43) + [5] +40. 198=0 IV. [41] +[63] + [7] —1.114=0 V. [42] +[5:] + [7%] —0.570=0 VI. [6] + [62] + [7%] + [8:] —0.485=0 VIL. [6] +83] + [8s] + [9%] —1.207=0 VII. [62] + [7%] +074] + [91] —1.915=0 X. [8] + [103] + [112] —0.383=0 XI. [8] +9) + [1h] +1. 009=0 XII. [10.] + [125] + [132] —2, 547=0 §4.] (1J [21] [22] [31] [32] [3s] [41] [42] [43] [45] [51] [52] [6] [62] [63] (71} [72] [7s] [74] [81] [2] [es] ("J [9] [92] [93] [10;] [102] [103] [11,] (11. =+0. =—t) =4U). =-+0. =+0. ==; =+0. 71 LS CHICAGO BASE TO OLNEY BASI. Numerical equations of condition, &e,—Continued. NIL. XV. XVI. XVII, XVIII. XIX. XX. XXIII. XXV. XXVI-. XXVII. XXVIIL. XXX. XXNXI. XXXII. XXXIV. XXXYV. 50000 I . 33333 I 66667 I 58333 1 . 25000 T . 08333 I . 12500 IL . 12500 II . 25000 IL 75000 II . 66667 IIT 33333 ILL . 16667 IV . 16667 IV . 66667 IV . 38095 1V . 61905 IV 04762 IV . 04762 IV 09091 VI 227273 VI 227273 VI 72727 VI 20167 VII 54167 VII 20833 VIL ). 25000 X . 25000 X 75000 X 12500 X 62500 X ANGLE-EQUATIONS—Continued, (10.] + [Us] + [131] (121) + CMs] + C4) + [182] (122) + (135) + (13d + (15) (13,] + CMa] + (15) + [152] [14] + (14-] + [155] + (165) [1] + C165] + (166) + 017) (142) + [1%] + (15s) + (17) (16) + [L682] + Cts.) + CRs] + fk) + flee] 1. x01—o [162] + [20,] + [214] + [210] [16] + [l6.) + [173] + (17.) + [2l)J [165] + [173] + [174] + [175] + [1%] (174) + (17) 4+ [18s] + [19] + [192] (174) + (18) + [bs] + [2h] + [2h] [1s] + [203] + (20) + [21 (18) + (195) + 2b) + (ete) + L2ts] [201] + 224) + [232] [20,] + (26) + [231] 2. 0090 4-0, F230 +40. U86—0 40, 303-50 0, 3030 -bb, 7090 —0, 2670) —0,710=0 0. 803=0 +1. 1160 +40. 5020 -b2. 1232s0 —1. 42850 +1. 0070 +0. 102=0 +0, YRL=) General corrections in terms of the correlates. +0. 66067 IL —0, 33333 IL —0. 25000 IT 40. 5S333 TI —0, 25000 IT —-0, 12500 TIT —0. 12500 III —0. 75000 ILI —0. 25000 III —0, 33333 V +0. 66667 V +0. 33333 VI -40. 33333 VI —0, 33333 VI +0. 61905 V —0. 38095 V —0, 04762 V —0. 04762 V —0, 18182 VII 0.54545 VIT +0.45455 VIL +0, 45455 VIL 0, 54167 VILL 029.67 VIO —0, 04167 VHT +0. 75000 XII —0, 25000 XII —0, 25000 XII +40. 62500 XI —0, 12500 XI -—0, 08333 TIT —0. . 25000 LL +40. +0. 58333 I 68750 LV . 31250 1V . 12500 1V 12500 TV 06567 VIL 33333 VIT . 16667 VII . 04742 VI 1, 0A762 VI . 61905 VI . 88095 VI 0575S TX 17272 IX 11031 IX OTTO IX . LOGG7 IX / 70531 LX 01048 IX . 25000 NTI . 75000 XIII . 25000 XTIT . 12500 XU . 37500 XII —0, 33333 VILL 40. 66667 VIEL” —0. 16667 VIII —0.09524 VIE +40 0.09524 VIL 40. +0,23810 VUE 0. 0.23810 VIE -L0. 40, 63636 X =): —0. OU09L X 40, ==, QQOIL X —0, —0, O90DL X —0. —0, 04167 XI —(), YOR3B3 XI Lal Vv / O70 V . 12500 V » 12500 V ). 54167 XT 05047 TX 0564 L IX $R4so TX 20286 TX OgOUT XI 562 (11a) (121) (122) [12s] (131) {182] [13s] (134) [14] (142) [16.] [163] [16,] [165] (16, [17] (17. (175) [174] [175] [ie] [1°2] [125] PRIMARY TRIANGULATION. [Cuar. XX, C, General corrections in terms of the correlates—Continued. . 37500 X . TIRGE XIV -0. 25000 XIT - 50000 XII 227273 XIL . 72727 XIT . 09091 XII - O9U9T XIL . 06160 XIV 27273 XX . 06160 XTV » 72727 XX - 111d XTV . 09091 XX . 72634 X1V . 09091 XX . 27273 XV . 10103 XXI » 72727 XV . 10103 XXI = =—0. 09091 XV —1, 15158 XXI —0, 09091 XV +40. 84849 XXI =—0, 01266 XVIII —-2. 03492 XXIV =—0. 01266 XVIII +1. 31938 XXIV =--0, 03797 XVIII -L0. 41425 XXIV =— 0.10127 XVIII +0, 11298 XXIV =—0. 36709 XVIII +0. 03766 XXIV =-+0. 63291 XVIIL +9. 03766 XXIV =—0, 26027 XIX —0. 12327 XXVII =-0. 68286 XIX —0, 20547 XX VII =—0. 06849 XTX +0, 17809 XXVII =—0. 06849 XIX +0. 17809 XXVIT =—(). 06849 XIX +0. 17809 XXVIT =— 0. 16586 XXII +0, 57971 XXXI +0, 36143 XXII —0, 23188 XXXI -—0, 88871 XXII +0, 4348 XNXT —0, 12500 XI +0, 62500 XII +0.50000 XV —0, 25000 XVI +0. 69002 XIV. —0. 25000 XV +40, 50000 XVI —0.19379 XIV —0. 25000 XVI . +0, 72727 XL = 0.07830 XIV 0. 1NIR2 XVI —0. 09091 XVIET —0.27273 XT. 0, 07830 XIV. 0. IKIRR XVI. —0, 09091 XVII —0. 09091 XI 0, 92401 XIV L0.87273 XVI 0. 36364 XVI —0, 09091 XIE LL 1SeRY XIV | -LO.VTZIB AVE £0. 63635 XVII —0.181-2 XV 0. 00091 XVIE 10. 45455 XVHIE +0, 72727 XIX <1), INLR2 XV —0. 09091 XVI 4-0,45455 XVI —0, 27273 XIX 0.27273 XV 4.0.63686 XVIT 0. 18182 XVIII —0. 00091 XIX 0.27273 XV 0.36864 NVIT 0. 18182 NVITL «0, 09091 XIX 40.72727 XVI 0.45455 XVIT = —0, 09091 XVII —0, Isis XX 0.27273 XVI $0.45 155 XVI 0.09091 XVII —0. 181*2 XX —0, 09091 XVI —0, 1812 XVI +40. 63636 XVII 0, 27273 XX —0, 09091 XVI 0.1812 XVIT 0, 36364 XVIIT 027273 XX —0, 02532 XIX = —0,00119 XXI-- —0, 22830 NNYL +0. 10127 NXIII —0.37975 XXV - —0.17721 XXVI —0, 63797 XXVI —0. 02532 XIX —0,00119XXI = —0, 12839 XXIJ_— +0. 10127 XXIII +0, 62025 XXV 0.17721 XXVI- —0. 03797 XXVII —0,07505 XIX - —0,00356 XXI-—- —0. 38516 XXIZ— +0. 30380 XXII —0,13924 XXV +40. 46885 XXVI —0, 11392 XXVII —0,20253 XIX = —0, 00948 XXI-—- 4.0. 9383R NNIE = —0. 18987 XXIII —0,03797 XXV 0.58928 XXVI +0. 69620 XXVII +0, 26582 XIX —1.09382XXI = —0. 05929 XXIZ = —0, 06329 XXIII —0, 01266 XXV —0,13924 XXVI —0. 10127 XXVII +-0.26582 NIX = 4.1.11872 XXI.-—- —0, 05929 XXII. —0. 06329 XXIII —0.01266 XXV_ 0. 13924 XXVI__ —0, 10127 XXII 0.64381 XX —0, 8744 NXE = —0,07494 XXIT_ —0, 08218 XXVI —0, 08218 XXVILI —0. 07990 XXIX —0, 04109 XXX —0.26027 XX 40,9424 XXI-- —0,12487 XXII —0. 13698 XXVI —0. 13698 XXVIII —0. 13315 XXIX —0, 06849 XXX —0.04109XX 0, 03978 XXI-- 1.19637 XXII +0. 45206 XXVI —0. 54794 XXVIII —0. 53261 XXIX —0. 27397 XXX —0,04109XX —- —0.03978 XXI-— 4.0. 76053 XXII +0. 45206 XXVI +40. 45206 XXVUI —1. 66370 XNIX 0.72603 XXX —0,04109 XX 0.03978 XNI--- 4.0. 76053 NNIL = —0;54794 XXVI 40. 45206 XXVHT 42.54252 XXIX —0, 27307 XXX —0.10145 XXUI —0.31921 XXIV +40. 04348 XXVIII —0. Ins40 XXX —0, 23188 XXXII —0. 17183 XXXII : —0. 15942 XXIII —0.04556 XXIV —0. 21739 XXVIII +10. 27537 XXX +0. 49276 XXXII +0. 21739 XXIII —0, 21739 XXXII +0, 05995 XXXIII —0, 04560 XXIV +10. 47826 XXVIII +0, 26087 XXX +0, 09324 XXXII CHICAGO BASE TO OLNEY BASE. 63 General corrections in terms of the correlates—Continued, +0. 27536 XXIII +40. 05797 XXXII =—0, 25000 XXIII +0. 75000 XXVIT —0. 13674 XXIV +0. 24641 XX XIII +0. 50000 XXVIII — 0, 25769 XXIX 0. —0. 26087 XXVIII —0. 20290 XXX 25000 XXXII —0. 25000 XXVII —0, 25000 XX VII +0. 50000 XXVIII —0. 25769 XXIX —0. 04762 XXV —0. 14286 XXV +0. 61905 XXV —0. 38095 XXV —0, 09524 XXXI —0, 28571 XXXI +0. 23810 XXXI +40, 23810 XXXI +0. 61905 XXXIV —0. —0. 14286 XXXIV +0.57143 XXXV —0. 04762 XXXIV —0. 04762 XXXIV --0. 16667 XXVI- —0. 79457 XXIX —1. 06774 XXXIII —0. 16667 XXXV +0. 83333 XXVI _+-0. 62333 XXIX —1. 06774 XX XIII — 0. 16667 XXXV —0. 16667 XXVI +0. 62333 XXTX +1. 93591 XX XIII —0. 16667 XXXV —0. 16667 XXVI —0. 15070 XXIX +0. 06654 X XXIII —0. 16667 XXXV —0. 16667 XXVI —0. 15070 XXIX +0. 06654 XXXIIT +0. 83333 XXXV —0. 33333 XXX +0. 66667 XXX +0. 66667 XXX —0. 33333 XXX -—0. 33333 XXX § 4.) [14] =+0.76121 XXII —0. 14493 XXXI [19] +0. 48988 XXXIII [19] =+0.75000 XXIII . —1. 09428 XX XIII [193] =—0. 25000 XXIII +0, 11452 XXXIII [20] =—0, 04239 XXIV [20.] =—0, 12716 XXIV [203] =—0, 43385 XXIV (20,] =+0.65077 XXIV (21) =—0.33333 XXV +0. 50000 XXXII (2lo] =-—0. 33333 XXV +0. 50000 XXXII [21s] =-++0. 66667 XXV +0. 50000 XXXII [214] =+0. 66667 XXV —0. 50000 XXXII [21s] =—0. 33333 XXV —0.50000 XXXII [22,4] =+1. 00000 XXXIV [23:]) =+1. 00000 XXXV [23] =+1.00000 XXXIV : —0, 25000 XXXII —0. 50000 XXVITI +-0. 99993 XXIX -++0. 75000 XXXII 14286 XXXV —0. —0. —0. 14286 XXXV 14286 XXXV 16667 XXXI . 16667 XXXI —0. 16667 XXXI +0. 83333 KXXI —0. 16667 XXXI Normal equations for determining the correlates. No. of equation 1. , 0=—0. 41500 +1. 75000 I —0. 58333 IL —0. 08333 LIT 2, 0=—0.48300 —0. 58333 I +2. 00000 II —0. 50000 III —0. 12500 V 3. 0=-+0.19800 —0. 08333 I —0,50000 II +2, 00000 III —0. 45833 V 4. 0=—1. 11400 —0. 12500 II —0.12500 IIT = +1. 97322 IV —0.38095 VI —0.16667 VII —0.26191 VIII 5 0=—0. 57000 —0. 12500 II —0. 45833 III —0. 69345 1V —0.04762 VI —0.09524 VIII +0. 05641 IX 6. UV—— 0.48500 —0.380951IV —0. 04762 V +2. 01299 VI +0.57143 VIII +0.92227IX —0,09091 X 7 6=—1.20700 —0.16667IV +0.78788 VI +2.11743 VII +0.99316 IX —0, 18182 X —0. 75378 XI x O=—1. 91500 —0.261911V —0.09524 V +0, 57143 VI +1. 68453 VIII —1.38870IX —0.04167 XI 9, 0—+1.54550 +0.056411IV +0. 05641 V +0. 92227 VI —1, 38870 VIII -++5.97836 IX —0.05758X 10. 0=—0, 38300 —0.09091 VI —0.18182 VII —0. 05758 IX —0.21591 XI —0.25000 XII —0. 62500 XIII 11. 0=+1.00900 —0.27273 VI —0.75378 VIL —0. 04167 VIII —0, 21591 X +1. 89394 XI —0, 12500 XIII 12, 0=—2. 54700 —0. 25000 X 41.97727 XII = —0. 52273 XIII / —0.43182 XVI —0.09091 XVII 13, 0=+42.00900 —0. 62500 X —0.12500 XI —0, 52273 XII —0. 07830 XIV. —0.18182 XVI —0.09091 XVII —0. 12500 IV —0. 12500 IV —0. 69345 V +0, 05641 1X +1. 97322 V +40. 78788 VII —0, 27273 XI —0, 62500 VIII —0. 62500 VII +0. 99316 VII —0, 15324 XI +2. 01136 X —0, 15324 IX —0.27209 XIV +2. 10227 XTIT Ot No. ol eqnation. 14.. VW, 20. 2h Use = i 0=-L0. O=-++. 0=-+0. 0=-+0. 0=-L0. O=-H0. O= - iS — O=— - 17300 25800 /SoL0d 0==-b0, . 7 Le00 0=--0. 0=-+1. Normal A1160 —0, + H2500 — 0, —0. —(), sp. +40. OSG00 SUB00 Bonde TO900 | =] 28700 | | 2° 906 10 bo + —. So + 1 CS St iS ee So300 rary + | = + +0 +e 11600 - +40, =); +0. 42. PRIMARY TRIANGULATION. , [Citap. XX, C equations for determining the correlates—Conutinued. eee ATI S190 XVI 06160 XX O83 16 XTV AAD XV OT 48182 XII 77273 XVI 10103 XX1 . 09091 XTI . 72727 XVI . 45455 XX . 12220 NT . 17887 XVIII UR NNIT Lae RANT . 06160 XTYV 80177 XIX _ 12658 NNIII 40801 XXVII +0, 06160 XTV . 72727 XVIII 740d XXIL . 07990 XXIX 10103 XV 97114 XTX . 00594 XXIII . 12882 XXVIT 2 U5020 SVL . 18835 XXII - 11738 XXVI 28325 XN - 06829 XVITL . 74909 XXIII 2ABORG ANVIL 0.1014 XNXI +0. 03706 NVIIL 248363 XXILT . Live AA VII 01556 XXXI1 201266 XVITI . 10126 XXII 03797 XXVIT leds SARUIT 13024 XVII Ligss SALT gees XKVI JLRTS ANN 2 WF AKAY AN? XV OI 226307 XXIL 386 XXVI . 17809 XXX --0. 07830 NIL OMT NVOET HL. 77278 NV ) Pelee ATS -O, TRIS2 XTIT SIE NN LL 22 0, 09001 XTIL 2, TR XVIT 20206 XXI AAS RY . P2030 ATX , Uh2290 XT Tes AAV . 18182 XV —(, 53300 XX UT582 XXIV . 13698 XX VITI . 86364 XV 63300 XIX . 08218 XXVI —-0, 04109 XXX +0.10103 XVI 17757 XX +0). 00353 XXIV —(. 07956 XXVIII 0, 244 XTX —0.76944 XXIII BOF RAVIE . 16586 XXXI1 . 12658 XIX . 48863 XXIV . 71739 XXVUI . 40942 XXXIT 07522 SIX . 91013 XXIV . 04560 XXVIII . 16593 XXX . 02532 XIX Js8053 XXIV DAP SAI — 0, 04762 XXXIV 241546 XTX 11898 SOIT 938846 XN VII 16667 XXXI ih 2 t . / A001 XTX - 43086 XXIEL . 98047 XXVIL —0, 25000 XXXII +45.17471 XIV +40. 12320 XVI 0. F27B XVI —0. 36364 XX 0, 92.190 XIV —0, 09091 X VITT +40. 24775 XV —0, 36364 XVII —0. 09091 XVI +0. 72727 XX +40. 03766 XNIV — . 09091 XVII - 97114 XXTI . 02532 XXV . 13315 XNIX . 18182 XVI . 91654 XX £12327 XXVIL . 20206 XVIT 5.47100 XX1I 001 XXV . 07733 XNIX 07494 XX _ 35328 XXIV . 63235 XNVIOI .36143 XXXII 00594 XXT +0, 10126 XXV . 25769 XXX . 75463 XXXII . 00353 XXI +0. 88053 XXV —0, 09116 XXX —0, 04239 XXXIV —0, 00119 XXJ 42. 57264 XXV +0. 33383 XXX 47619 XXXV —0, 08212 XX 0. 52793 XXIV —0, 09583 XXVILL -L0,50000 XXX w = —(). 12327 XX +40, 11208 XXIV +40, 85618 XXVIUL +40, Jn9x8 XXXIIT 0 +40 +0. +0, —0. d, u 29346 XV . 06160 XIX 72727 XVI 10103 XXT 52273 XV 181Rk2 XX Lt 2727 NV .Q9091 XIX - 86304 XVIT . O82R6 XXT . 01266 XXV . 72036 XVIII 224345 XXIT - 41546 XXVI . 06849 XXX 245455 XVIT .17757 XXI 08218 XXVITI . 03286 XVII . 07807 XXIT 1, 09260 XXVI . 08978 XXX . OTR07 XXT . 12889 XXV /ATR50 XXINX +0. 03585 XX XTIT —0 . 76944 XXII . 118938 AXVI +0. 05797 XXX +0. 35328 XXII +0. —0 52723 XXVI1 . 10729 XXXI - 0.12716 XXXV —0 —0 +0 —U . 12839 XXTI . 51054 XXVI . 90476 XXXI . U9260 XXT ~-0.51054 XXV —1. —1 —0 224 +40 57298 XXIX . 06774 XXXII 2 WR AXT . 03797 XXV . U8852 XXTX 64. Normal equations for determining the No. of equation. 2, =-+10. 50200 —0. 18698 XIX +10. 71739 XXIII 422. BHR NXVITL 0, 71739 XXXII —0. 13815 XIX —0, 25769 XXIII 0. 36344 XXVIII +41. 45202 XXXII —0, 06849 XIX +40, 05797 XXIII +40. 17809 XXVIII —0, 52174 XXX1 —0. 16586 XXII —0. 16667 XXVI +1, 88924 XXXI —0. 45239 XXXV +40, 36143 XXII —0. 25000 XXVII _73188 XXXI 03535 XXII 06774 XXVI +11. 02136 XXX +40, 08654 XXXV —0, 04239 XXIV —0, 14286 XXXV —(), 12716 XXIV —0, 33333 XXX —0. 14256 XXXIV Oa=—1. 89730 = 30, 0=+42. 1200 31. O==—1. 42800 m 0=-++1, 00700 = Ly = 0=-++1. 83430 = + —1. 34. 0=++0. 19200 0=-+0, 28200) — 0. R218 XX — 0.04560 XXIV -b 0.36344 XXIX — 0.51116 XXXII — 0.07990 XX + 0.47263 XXV +12. 54440 XNIX + 1.39236 XXXII — 0,04109 XX — 0.09116 XX1V + 0.71293 XXXID + 1.27537 XXXII — 0.10145 XXII + 0.04348 XXVHUI — 0.73188 XXXII — 0.40942 XXIL — 0.71739 XXVIII + 2.74276 XXXII — 0.75463 XXII 0, ARk0R8 XXVII — 0.10529 XXXI — 0, 04762 XXV — 0.47619 XXV — 045239 XXX1 + 2.40476 XXXV CIUCAGO BASE TO OLNEY BASE. —U. 07056 XX1 —. 09588 XXVI 7129083 XNX 207733 XXI . 57298 XXVI .41704 XXX . 15070 XXXV . OBUTS AXT . 33333 XAXV -A1704 XXIX . 02136 XXXII . 10729 XXIV . 15070 XXTX 0, 10529 XXXII . 04556 XXTV . 45202 XXIX . 02510 XXXITI . 16593 XXIV .§1116 XXVIII (1, 02510 XXXIL . 09524 XXXI = . 16667 XXVI . 60000 XXXII Values of the correlates and their logarithms. 1 =-+40, 4553 IL =-+0. 6023 TI =+0. 3645 IV =+1. 2477 V =+-0. 9425 VI =—0.9194 VIL =41. 7227 VIL =+2. 4643 IX =+0. 1541 X =-+0. 3071 XI = +40, 0RGH XID =+1. 2501 XIV =—0. 5362 XIV =-+0. 4120 XV =-+0. 0151 XVI =-+0. 0924 XVII =—-0. 2601 XVID =—0. 0126 log 9. 65826004. XIX =—0.309) log § log 9. 7798201 + XX =—0.1205 log log 9. 56173334 XXI =—0. 1608 log log 0, 09612414 XXII =-40. 7226 log log 9, 97429524 XXII =41. 3188 log log 9,9635139— : ~ XXIV =—0.0949 log log 0, 23622234 XXV_ =-10. 2054 log log 0. 39169714 XXVI =-40. 2069 log log 9. 18774634. XXVII =—U. 6694 log log 9. 48722824 XXVIIL =—0. 3736 log log 8. 93851974 XXIX =-+10. 0644 log log 0. 09695174 XXX =—U.7626 log log 9. 7293592 XXXII =-+40.5902 log log 9. 61493944 » XXXII =+40. 0061 log log 8, 17926454 XXXII] =—0. 0214 log log 8. 9658599. XXXIV =—0. 0529 log Jog 9. 4151737 XXXV =—0. 0600 log log 8. 1003705— x 565 correlates—Continued. +0. 63235 XXTI +0. 84618 XXVIT +0. 04348 XXXI +1. 47850 XXII +40, 03852 XXVII —0, 15070 XXXF +0, 23295 XXII +1, 11873 XXVI +2, 59561 XXX —0. 33333 XXXV +0. 90476 XXV —0,52174 XXX -—0, 09524 XXXIV +0. 50000 XXVI - 27537 XXX . 50000 XX XV .00245 XXV . 39236 XXIX RB. 03114 XXXII fs . 61905 XXXIV =i, +0. 15070 XXIX 06654 XXXIII . 1912216... 9, 1128368. . 2064211... 85802204. 1201822 4 9773120 — 3125581 4 3157185 4 8256662 . 5723720_- S09U881 4 8857286 — 9. 77099024 7860412 5 . 3312248 . 7237019 — 7784269 966 {11] =+0. [2] =+0 [22] =+0 3] =+0 [32] =+0. [33] =+0 [41] =+0 [4.] =+0 (45] =—0 [44] =+0 [5] =—0 [52] =-++0 {6,] =—0 [6.) =+0. [63] =-+0 [71] =—0 [72] =+0 {%] =—0 (74) =+0. (&] =—0 227 . 250 . 103 . ORD 146 . 024 - 442 . 137 . 151 . 087 O71 . 507 . 187 . 004 . 440 . O74 . 231 . 161 364 . OO1 PRIMARY TRIANGULATION. Values of the general corrections. vw [2] =—0. 6x0 [&] =-+0.810 [8] = +0. 280 [9] =-++0. 658 [99] =-+0. 305 [93] =—0. 412 (10,] =+0. 995 (10.] =—0.791 [10;] =-+0. 052 (11,J =-++0. 033 [1].] =-+0. 382 {11;] =-— 0. 461 [2] =—0. 345 [122] =-+0. 014 [123] =-+0. 522 (1%] =—0. 756 (13,] =+1. 030 [13,3] =—0. 326 [13] =-++0.272 | wu [14;] ——0. 150 [14)] =—0. 031 [14,] =—0, 495 [14s] =+-0. 440 [15] =—0. 047 [12] =—0. 124 [15s] =+0.179 [154] ——0. 130 [16,] =-+40. 153 [16,] =+40. 040 [16,] —-+-0, 252 {16,] =+0. 129 [165] =-+-0. 005 [16.] ——0.364 [171] =-+40. 206 [172] ——0. 200 [173] =—0. 476 (1%4] ——0.277 [175] =+0.555 de [1x,] =+0. 250 [1k] =—0, 210 (183;] =—0. 708 (18,] =+41. 089 (19,] ——1. 047 [19.] =-+0.975 [193] —=-40. 091 (20,] —-—-0. 086 [202] = --0, 213 [203] =-10. 320 [204] =+0. 012 (21,] =+0. 040 [212] =—0. 431 [213] =—0. 497 (21,] =+0. 846 [21,] =—0. 009 [224] =—0. 053 (23,] =—v. (60 [232] =—0. 053 (Citar. XX, C, Residuals resulting from substitution of general corrections in numerical equations of condition. | | | | ae Residual. r ee Residual. 1 \ 0. 0000 19 —0. 0001 2 0. 0000 20 0. 0000 3 C. 0000 21 0. 0000 4 0. 0000 22 —0. 0016 5 0. 0000 23 0. 0000 6 0. 0000 24 —0. 0006 7 0. 0000 25 0. 0000 8 —0. 0001 26 0. 0000 9 +0. 0007 27 —0. 0001 10 0. 0000 28 0. 0000 11 0. 0000 29 —0. 0012 12 0. 0000 30 0. 0000 13 0.0000 ~ 31 0. 0000 14 +0. 0001 32 —0. 0001 15 0. 0000 33 —0, 0007 16 0. 0000 34 0. 0000 17 0. 0000 35 0. 0000 18 0. 0000 64. CHICAGO BASE TO OLNEY BASE. 567 SECTION XV.—Triangulation from the line Oakland — Kansas to the line Denver - Parkersburg. KANSAS—22. (Observer, G. Y. Wisner. Instrument, Pistor & Martins theodolite No. 2. Dates, September and October, 1879.] Angle as measured between— Notation. | No. meas. | Range.} Wt. (v) {v] {Corrected angles. fo] é aw aw aw wn fo} a wa Martinsville and Casey ...-.-.....-. 24 45 32.416 | 22) 24 7.8 1 +0.220 | —0.770 24 45 31. 866 Martinsville and Westfield. ....... 68 59 53.026 | 22142 16 | 4.3 1 —0. 522 | — 0.020 68 59 52. 484 Martinsville and Oakland. ........-- 146 31 08.215 | 2214243 8 2.4 0.5 | +0.367 | 4-0. 045 146 31 08. 627 Casey and Westfield. ...-........... 44 14 19.648 | 22, 24 9.7 1 +0.220! +0. 750 44 14 20.618 Westtield and Oakland .....-. 22... 77 31 15.986 | 22, 16 5.1 1 +0. 092 | --+-0. 065 77 31 16.143 Westfield and Palermo ............- 112 31 13.142 | 22344 8 6.0 0.5 | —0.792 | -}0.012 112 31 12. 362 Oakland and Paleimo..........-.--. 34 59 55.996 | 224 16 4.1 1 +0.276 | —0. 053 34 59 56. 219 Palermo and Martinsville .........- 178 28 55.266 | 22, 16 6.3 1 05120 acute see va |eanecicceeicteseais NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3. 5(221) 4-2. 5(222)-+1. 5(223)+ — (224)—1. 732=0 2. 5(221) +3. 5(222) +1, 5(223)4+ (224) —1. 732=0 1. 5(221) + 1. 5(222) +3. 0(223) +1. 5(224) —1. 350=0 (23) + (22g) +1. 5(223) + 2. 5(224) ~—1. 268=0 NotTe.—The general correction (v] to 224 is taken from Section XIV of the adjustment. OAKLAND—23. (Observer, A. R. Flint. Instrument, Troughton & Simms theodolite No.1. Date, September, 1879.] Angle as measured between— Notation. | No. meas. Range. | Wt. (v) [v] | Corrected angle or wu” uw uw " | i Kansas and Westfield ........-..-.-. 35 56 O01. 443 233 16 | 3.4 1 +0.115 | +0, 087 or mu 35 56 01. 645 Note.—The local correction of 233 is taken from Section X1V of tho adjustment. MARTINSVILLE—24. {Observer, G. Y. Wisner. Instrument, Pistor & Martins theodolite No.2, Date, October, 1879.) Angle as measured between— Notation. | No, meas. | Range.| Wt. (v) [v] Corrected angles. oO ‘ “a a“ | uw “a fe} y a | Belle Air and Casey ....---.-----+-- 50 09 38, 937 241 16 5.0 1 —.008 | +40. 096 50 09 39. 025 | Casey and Westfield ...-....-------- 78 15 51.875 242 16 5.9 1 0.009 | --0.371 78 15 51.495 | Westfield and Kansas ....--.------- 44 45 46.184 243 16 4.3 1 —0. 008 | —0. 036 44 45 46.140 Kansas and Belle Air....-.-.--.---- 186 48 43. 038 244 16 6.7 1 —0.009 | +0.311 186 48 43, 340 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(241)-+ (242)+4+ (243) 4 0. 034=0 ‘ (241) 4+-2(242)+4+ (243) 4 0. 034=0 (241)+ (242) +2(243) +0. 084=0 568 SECTION NV.—Zriangulation from the line Oakland — Nansas to the Continued. WESTFIELI—25. [Observer R. 8. Woodward Tustrument, Troughton & Simms theodolite No. 3. PRIMARY TRIANGULATION. Date, October, (Cuar. XX, C, line Denver — Parkersburg— 1879. | Angle as measured between - : tintin » | No. meas. Range.} Wt. (v) | {v] (Corectea angles. a alas cera — amit = ' on aS: o- “ ! un “ | " | oF u | Oakjand and Kansas.............--. 66 82 43. 702 251 | 16 i 5.2 1,1 —0. 240, —0.133 66 32 43. 329 Kansas and Martinsville...........- 66 14 22.099 Wye , 16 A 2 +0. 004 | -- 0.161 66 14 22. 264 | Kansas and Casey ..----.-. .--.--- 108 02 29. 619 ' Lha+3 | 8 | 3.4 0.5 »} —0.488 +0. 354 108 02 29.485 | Martinsville and Casey .......-.-.- 41 48 07. 024 254 , 16 ) £6 71 +0.004 | --0. 193 41 48 07. 221 | Casey and Oakland.....0 02.2.0 0--- 185 24 47. O17 | 254 i 39] 1 —0.240 | —0. 221 | 185 24 47. 186 NORMAL EQUATIONS FOR LOCAL ADJUSTMEN'P. Y251)+ (252) + (251) +1. 5(252) 4+ 253) 4-0. (251) +2. 5(252) +1. 5 (25s) + 0. 2. 5 (253) +0. 472=0 224=0 224=0 BELLE AIR—26. (Observer, J. HW. Darling. Instrument, Troughton & Simms theodolite No. 4. Date, October, 1879.) Notation. | No. meas. Range. Wt. | Angle as measured between— ; (v) | {v] Corrected anyles. | Bal oO a aw ws “a we i oO t a a Obleng and Hunt City .... 22.2... 44 32 07. 589 261 16 5.0} 1 | +0. 017 +0. 024 d4 32 07.630 | Oblong and Casey .......-2..2-2.-+5 111 30 22.168 | 26.42 12 1.3 1) 40.135} 0.067! 111 30 22.370 | Hunt City and Casey.......-2....--- 66 58 14. 509 262 16 5.2 1 40.188 ; +0. 043 66 58 14.740 | Hunt City and Martinsville. .... 2... 138 45 56. 669 26243 16 3.4 A —0.171 | —0. 065 188 45 56. 433 Casey and Martinsville ,..... 02.2... 71 47 41.477 263 | 16 5.4 1 +0. 324 | —0, 108 71 47 41. 693 Martinsville and Oblong .........--- 176 41 55, 744 204 | 16 49 1 | -+0, 152 -+-0. 041 176 41 55. 987 NORMAL KQUATIONS FOR LOCAL ADJUSTMENT. 3(261) + 2(262)-+4 (263) - 0.751 =0 2(261) +-4 (262) +2 (26s) —1.434 =0 (261) +2 (262) --3 (263) —1.364=0 CASEY— (Observer, ALR. hut. Instrument, Troughton & Simms theodvlite No.1, Date, October, 1879. ] an = 7 a sr. | { SS Ane +3 ‘ Angle as measured between— Notation. | No. meas. | Range. | Wt. (v) | [v] Corrected angles. of spate hee, pare aa oo eackein da Beet aa = oO ‘ uw “we wm aw | oO ¥ wn Westfield and Kansas...-.-......... 27 43 10. 611 2m 17 3.9 1 40.052 | 40,278 27 43 10. 941 e » Westtield and Martinsville ......... 59 56 02. 405 27142 | 16 24 1 —0. 072 —0. 079 59 56 02. 254 » Kansas and Martinsville............ 32.12 .51.618] 272 | 17 44 1 40.052 | --0. 357 ge 12 51.313 | Martinsville and Belle Air......-. -- 58 02 39. 812 27s | 16 3.1 1 +0. 017 —-0. 024 58 02 39. 805 Martirsville and Oblong 98 56 00. 024 27341 | a | 58 1 ~. 039 | —0. 182 98 55 59. 8083 Bele Air and Oblong ...2-..--. -.. 40 53 20.134} 274 : 8 3506] 605] poone | —0.158 40 53 19, 998 Belle Air and Hunt City....2. 02.22. 66 20 11. 094 27445 | Is 4,3 1 —0.319 | 40.195 66 20 10. 970 | Belle Air and Westfield .2.0........ v4e 01 17. 514 Sheet, | 16 4.1 1 +0. 324 +0. 103 242 01 17. 941 ; Oblong and Hunt City 22.2022... -. 26 50, 646 275 ji 17 4,2 1 --0.027 | 40,353 v5 26 50. 972 ' Hunt City and Westticld 5 41 07.409 | 276 | 18 39 | 1 —0.346 | —0.092 | 175 41 06.971 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(27)) 4-8272)42(27a)-+ (27a) (275) — 0. 391 =-0 3(271) +4 (272) 4+-2(273)-+ (27a) 4. (27s) — 0. 3910 2(27)) 4-2(272) 4-4 (273) +2. (27) 4 (275) —-0. 23 = 0 (271) + (272) 4-227) +3. 54274) +2(275)—0. 162=0 (271). (272) (27s) -+2. 0(2T4) +3(275)— 0. 084=0 §4.] CHICAGO BASE TO OLNEY BASE. 569 SECTION XV.—Triangulation from the line Oakland — Kansas to the line Denver — Parkersburg— Continued. OBLONG—28. [Observer, G. Y. Wisner. Instrument, Pistor & Martins theodolite No.2. Dates, October and November, 1879.] Angle as measured between— | Notation. | No. meas. | Range.; Wt. (v) [v] Corrected angles. ° # a“ wn “a “a ° A aw Claremont and Mound............-. 34 36 31.101 281 20 4.8 1 +0.099 | —0.157 34 36 31. 043 Mound and Hunt City --........... 65 50 49.477 28, 16 3.5 1 +0.100 | +0. 404 65 50 49. 981 | Hunt City and Casey ..--.. ...-..-. 32 06 47.113 283 16 5.4 | -1 +0.143 | +0. 389 32 06 47. 645 Hunt City and Belle Air............ 59 43 05. 962 28344 8 3.7 0.5 —0.088 | +0. 059 59 43 05. 933 Casey and Belle Air ................ 27 36 18.474 284 | 16 3.6 1 +0.144 | —0.330 27 36 18. 288 Belle Air and Claremont............ 199 49 33. 249 285 | 16 5.9 1 +0.100 | —0. 306 199 49 33. 043 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. Q 2(281)-+ (282)4+ — (283)-+ (284) —0. 586=0 (281) +-2(282)+ (283)4+ (284)—0. 586=0 (281) + (282) +2. 5(283) +1. 5(284) —0. 773=0 (281) + (282)-+1. 5(28,) +2. 5(28,)—0. 773=0 HUNT CITY—29. [Observer, R.S. Woodward. Instrument, Troughton & Simms thevdolite No. 3. Date, October, 1879. } 1 : Angle as measured between— Notation. | No. meas. pte (v) {v] loaractaa angles. | ° ‘ a“ : a ! aw c " a ee ase eT Casey and Belle Air ..............-- 46 41 34.723 | 291 16 ee L —0.054 - +0.198 | 46 41 34. 867 | ; Belle Air and Oblong..--..- ev aes fou eaieiare: 75 44 47, 237 295 16 4.5 1 —0, 208 | +0. 026 75 44 47. 060 Belle Air an Mound .............. 145 05 08 755 292+3 12 1.7 1 +0.150 | +0.179 145 05 09. 084 | Oblong and Mound ....-..-....--.-- 69 20 21. 828 293 16 6.1 1 +0.043 | +0.153 69 20 22. 024 Oblong and Casey .. .....--......-. 237 33 38.790 29344 8 2.2 0.5 | —0. 493 —0. 224 237 33 38.073 | Mound and Casey .........-...----- 168 13 16. 233 294 16 7.3 ; 1 +0. 193 —0. 377 | 168 13 16. 049 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2, 5(291)-+1. 5(292)-+ (293)-+0. 396=0 1. 5 (291) -+3. 5(292) +2(293) +0. 706=0 (291)-+2 (292)+3(293) +0. 331=0 CLAREMONT—30. (Observer, G. Y. Wisner. Instrument, Pistor & Martins theodolite No. 2. Date, November, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] | Goerectell angles. ‘or “" “" “u a“ or " Parkersburg and Denver ..........- 85 42 19.356 | 30) 16 7.7 1 —0. 215 | +0.198 85 42 19, 339 Parkersburg and Mound ........... 152 31 16.747 | 30142434445 8 4.8 0.5 | +0.548 | —0, 272 152 31 17. 023 Denver and Onion Hill ...........-. 17 49 15.536 | 302 16 8.2 1 —0. 144 | —0. 437 17 49 14, 955 Denver and Mound ..........-....-- 66 48 58.298 | 302434445 8 4.8 0.5 | —0.144 | —0. 470 66 48 57. 684 | Onion Hill and West Base ......-... 28 12 13. 803 | 303 16 5.6 1 —0. 144 | —0, 286 28 12 13.373 West Base and Check Base.......-.. 7 24 42.167 | 304 24 ‘ 6.2 1 —0. 144 | +0. 220 7 24 42, 243 | Check Base and Mound .........---- 18 22 47.112 | 306 16 8.8 1 —0. 032 | +0, 033 13 22 47.118 Check Base and East Base.......-.. 18 30 33. 645 | 306+6 8 4.8 0.5 | —0,222 | +0 028 18 30 33. 451 | Mound and East Base .......-...-. . 5 07 46.173 | 306 16 7.6 1 +0.170 | —0. 005” 5 07 46. 338 | East Base and Oblong....-...--.--.. 34 36 06.997 | 307 16 8.8 1 +0.059 | +0. 212 34 36 07. 268 Oblong and Parkersburg......-..... 167 44 49. 247 | 308 16 7.2 1 +0. 059 | +0. 065 167 44 49. 371 | NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(301) +1. 5(302) +1. 5(303) +1. 5(304)+-1. 5(305)4+ — (306) B, (307) +1. 004=0 1. 5(301)+-3. 0(302) +2. 0(303)-+-2. 0(304)+2. 0(305)-+ — (306)+ (307)+1.164=0 1. 5(301) +2. 0(302) +3. 0(303) +2. 0(304) +2. 0(305)+ — (80g) + (307) +1. 164—0 1. 5(30;) +2. 0(30,) +2. 0(803) +3. 0(304)-+2. 0(305)-+ (366) (307) +1. 164—0 1. 5(301) +2. 0(302) +2. 0(303)-+2. 0(304) +3. 5(30s)+-1. 5(306)-+ (307) +0. 984=0 (301)+ (302)+ (303)+ (304)+1. 5(305)+2. 5(306)-+ (307) +0. 211=0 (301)+ (302)-+ (303)4+ (304)-+ (305) — (806) +2(307) +0. 391=0 i2L8 570 PRIMARY TRIANGULATION. (Cuap. XX, C, SECTION XV.—Triangulation from the line Oakland - Kansas to the line Denver - Parkersburg— Continued. MOUND—31. |Observer, A.R. Flint. Instrument, Troughton & Simms theodolite No. 1. Dates, October and November, 1879. ] Angle as measured between— Notation. No. meas. | Range.| Wt. (v) [v) Corrected angles. ° # we “uw wu “ oe ‘ “uw Hunt City and Oblong......-..-. 44 48 48.472 | 3li 16 4.1 1 —0.148 | +0. 302 44 48 48. 626 Oblong and East Base............ 93 44 38.110 | 312 16 4.4 1 - 0.016 | —0.199 93 44 37. 895 Oblong and Claremont........... 105 39 36.322 | 312+3 8 4.3 0.5 | —0.064 | —0.121 105 39 36, 137 East Base and Claremont........ 11 54 58.147 | 31s 16 2.6 1 +0.017 | +40. 078 11 54 58, 242 East Base and Check Base....... 19 31 25.758 | 31344 18 4.0 1 +0.171 | —0. 154 19 31 25.775 East Base and West Base........ 71:13 08.118 | 313444+5+6 16 5.0 1 —0,400 | —0,185 71 13 07. 533 Claremont and Check Base...... 7 36 27. 861 | 31a 16 3.7 1 0.096 | —0. 232 7°36 27. 533 Claremont and Denver ....... --- 61 34 30.826 | 3144+5+6+7 16 3.6 1 +0.300 | +0. 099 61 34 31. 225 Claremont and Onion Hill . 65 25 55.665 | 314+5+6+7+8 8 3.7 0.5 | —0.439 | +0.073 65 25 55, 299 Check Base and Middle Base 19 31 55.608 | 31s 16 3.8 1 +0.077 | —0,455 19 31 55. 230 Middle Base and West Base ..... 32 09 46.027 | 316 18 3.1 1 +0.077 | +0. 424 82 09 46 528 West Base and Denver..... -... 2.16 21.701 | 317 16 3.4 1 —0.129 | +0.362 2 16 21. 934 West Base and East Base - 288 46 52.475 | 317+8+9+142 16 2.0 1 —0.193 | +0.185 288 46 52. 467 Denver and Onion Hill........... 3 51 24.315 | 318 18 3.4 1 —0.215 | —0. 026 3 51 24. 074 Denver and Hunt City........-.. 147 57 03. 807 | 3le+9 16 4.2 1 +0.485 | —0. 280 147 57 04.012 Onion Hill and Hunt City........ 144 05 40.961 | 319 8 3.7 0.5 | - 0.769 | —0, 254 144 05 39. 938 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 5(31,)- 1. 5(31a)-41. 5(313)-1. 5(314)-+1. 5(815)+-1. 5(816)+1. 5(317)+0. 5(31s)-+0. 334=0 1. 5(31,)-+3. 0(31g) 4-2. 0(31s)-H1. 5 (314)-+-1. 5(31s)-+1. 5(316)-+1. 5(817)-+0. 5(31s) +0. 301=0 1, 5(3L1) +2. 0(312)-16. 0(313)-+-4. 5(314)-+-3. 5(31s) +3. 5(316)+1. 5(317)+0. 5(31s)+0. 194=0 1. 5(811)-H1. 5(812)-44. 5(813)-+7. 0(814)-+5. 0(316) +5. 0(316)+3. 0(317)-+ (31s)-++0. 521=0 1. 5(311)-H1. 5(31a)-+3.5(31s)-+5. 0(314) +6. 0(31s)+5. 0(316)-+3. 0(317)-+ — (31s)-+0. 271=0 1. 5(311)-H1. 5(31a)-+3. 5(31s)-+5. 0(314)-+5. 0(315)-+6. 0(316)+3. 0(317)+ — (31s)--0. 271=0 1. 5(311)-+1. 5(31a)-+1. 5(31s)-+3. 0(314) +3. 0(315)+3. 0(316)+4. 0(317)-+ — (318)+0. 628=0 0. 5(311) +0, 5(812)4-0. 5(31s)+ (814)-+ —(315)-+ (316) (317) +2. 0(318)-+0. 524=0 ONION HILL—22. (Observer, R. S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, November, 1879.) Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v] Corrected angles. °o iF uo “ aw a“ ° ‘ “a Mound and West Base. -......-...--. 4 40 28. 048 321 16 3.5 1 —0. 087 | —0.365 4 40 27.596 West Base and East Base........... 16 39 28. 348 32, 16 5.2 1 +0. 083 | —0.320 16 39 28.111 West Base and-Check Base....-.... 43 46 06. 687 32243 16 5.3 1 —0.170 | +0.424 43 46 06. 941 East Base and Check Base......... . 27 06 38.004 | * 323 16 8.0 1 +0. 082 | +0. 744 27 06 38. 830 Check Base and Claremont ......... 17 07 48.677 324 16 3.8 1 —0. 087 | —0. 609 17 07 47. 981 Claremont and Denver ............. 101 24 49.225 325 16 3.7 1 —0.087 | —0.441 101 24 48. 697 Denver and Mound................. 193 00 47. 882 326 16 4.4 1 —0.088 | +0. 991 193 00 48 785 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(321)4+ (322)-+ (32g)+ (324)4+ (325)+4+0. 184=0 (321) 4+3(322)-4+2(32,)+ (324)4+ (32,)—0. 151=0 (321) + 2(322)-+3(323)+ (324)-+ (32s)—0. 151=0 (321)-+ (322)4+ (323)+2(32,)4+ (325) +0. 184=0 (321) (32g)+ (323)4+ (324)-+2(32s) +0. 184=0 § 4.) CHICAGO BASE TO OLNEY BASE. 571 SECTION XV.—Triangulation from the line Oakland - Kansas to the line Denver - Parkersburg— Continued. WEST BASE—33. |Observers, R.S. Woodward and J.H. Darling. Instruments, Troughton & Simms theodolites Nos. 3 and 4. Dates, October and Novem- ber, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) {v) Corrected angles. ° t a“ a“ “wn uw nf ow Mound and East Base...........---. 47 46 00. 480 33, 16 4.5 1 +0.049 | +0. 065 47 46 00. 594 Mound and Middle Base ............ 47 46 02. 866 33142 16 4.9 1 +0.307 | +0. 436 47 46 03. 609 East Base and Middle Base ......... 0 00 02. 638 33 8 1.4 0.5] +40.006 | +40.371 0 00 03, 015 East Base and Check Base .......-.. 46 45 34.513 332+8 16 6.0 1 —0.328 | +0. 068 46 45 34. 253 East Base and Claremont .........-. 52 08 20. 994 3324844 16 5.3 1 +0, 221; —0. 230 52 08 20. 985 Middle Base and Check Base ....... 46 45 31. 232 333 16 5.7 1 +0.309 | —0.303 46 45 31. 238 Check Base and Denver .......-.... 88 45 13.338 334+5 16 5.2 1 —0. 049 | —0.074 88 45 13.215 Check Base and Onion Hill ......... 96 16 39. 302 3344546 16 4.1 1 +0.029 | —0.595 96 16 38. 736 Claremont and Denver.............. 83 22 26. 038 335 , 16 6.4 1 +0.221 | +0. 224 83 22 26. 483 Denver and Onion Hill.............. 7 31 26, 225 33 16 ot aT 1 —0.183 | —0, 521 7 31 25. 521 Denver and Mound ....-............ 176 43 11. 642 33647 16 6.2 1 +0. 355 | —0. 059 176 43 11.938 Onion Hill and East Base ........... 216 57 46. 637 33741 16 4.6 1 —0.153 | +0. 527 216 57 47.011 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 3(331) +2. 0(33q)-+ (333)-+ (38445) —0.418=0 2(331) +5. 5(339)-+4(33s)+3(33¢+5)— (33s) + (336) —0. 815=0 (331) +4. 0(33,)-+5(383)-+3(38a+5)— (33s)-+ (336) —1. 067=0 (331) +3. 0(33q)-+3(333)-+5(334+5)— (335) +2(336) —0. 163=0 — — (33a)— (33s)— (38446) -+2(33s) —0.176=0 + (83q)-+ (383) +2(384-45) +-3(336) +0. 381=0 NOTE.—332+3+44, 334+6+6, 335, 336, and 33741 were measured by Mr. Darling with the Troughton & Simms No.4. The remainder wero read by Mr. Woodward with the Troughton & Simms No. 3. EAST BASE—34. [Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Date, November, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (v) [v] Corrected angles. oO ‘ ww “ a fo} t a Claremont and Check Base....--.--- 18 20 42.331 | 341 16 2.5 1 —0. 220 | +0.708 18 20 42, 819 Claremont and West Base .......--- 101 56 22.781 | 34:42+044 16 5.0 1 +0.220} +0.516 101 56 23. 517 Check Base and Onion Hill ......... 63 17 21. 868 | 342 28 1.3 1 —0.439 | +0. 287 63 17 21.716 Check Base and Middle Base .....- 83 35 37.101 | 342+8 16 6.1 1 +0. 830 | —0. 242 83 35 37. 689 Check Base and West Base.....-..- 83 35 40.512 | 342+s+4 34 7.8 2 +0.378 | —0.192 88 35 40. 698 Onion Hill and West Base .-. 20 18 19.900 | 343s+a 28 6.2 1 —0.439 | —0.479 20 18 18. 982 Middle Base and West Base . 0 00 02.497 | 344 36 3.9 2 +0. 462 | +0. 050 0 00 03. 009 Middle Base and Mound ... - 61 00 54.948 | 344+5 20 5.3 1 —0.094 | +0. 108 61 00 54. 957 West Base and Mound... 61 00 51. 467 | 34, 16 4.3 1 +0.423 | +0. 058 61 00 51. 948 West Base and Check Base. - . 276 24 18.069 | 345+6+41 28 9.2 1 +1.041} +40. 192 276 24 19. 302 Mound and Check Base..........-.- 215 23 26. 890 | 34641 20 5.7 A: +0. 330 | +40. 134 215 23 27.354 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(841) + (84248)-+ (344) — 0.852=0 +-2(842)— (34a+8)— (844) + 2.170=0 (341) — (342) +7 (842-48) -+6(84a)+ (845) — 9.228=0 (341)— (342) +-6(342+a) +9(844) +2 (34,) —10.207=0 (3424-8) -+2(344)-+3(345)— 3.0240 572 PRIMARY TRIANGULATION. (Cuap. XX, C, SECTION XV.—Triangulation from the line Oakland - Kansas to the line Denver - Parkersburg— Continued. MIDDLE BASE—35. [Observer, E. S. Wheeler. Instrument, Repsold theodolite No, 1. Date, October, 1879.] Angle as measured between— | Notation. | No. meas. | Range.) Wt. (v) [v] Corrected angles. ov " un" “uw a" or “u" West Base and Mound...........--- 100 04 09. 081 351 16 7.8 1 +0.152 | +0.669 |} 100 04 09. 902 Mound and East Base .....-.-.- -- 79 55 44. 134 35g 16 4.2 1 +0. 152 —0. 210 79 55 44. 076 East Base and West Base........... 180 00 06. 328 353 16 . 92 1 +0. 153 —0.459 | 180 00 06. 022 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(351)+ (352)—0, 457=0 (351) +2(35)—0. 457=0 CHECK BASE—36. (Observer, J. H. Darling. Instrument, Troughton & Simms theodolite No. 4. Dates, November and December, 1879.] ! 1 J ; Angle as measured between— Notation. | No. meas. | Range.| Wt. | (v) [v] Corrected angles. * & 8 a“ i aw | “ 4 a“ a t uw Claremont and Onion Hill -. - 127 15 17.079 361 16 4.2 1 | 40.090 | —0.611 127 15 16. 558 / Onion Hill and West Base .. -. 39 57 14.179 36, 16 4.4 1 +0. 380 | —0. 062 39 57 14. 497 ' Onion Hill and East Base - 89 36 00.229 | 3624344 8 4.5 0.5) —0.581 } +0. 005 89 35 59. 653 | West Base and Mound............-. 33 46 43.582 | 363 21 6.7 1 +0. 132 | —0.185 33 46 43. 529 | West Base and East Base........... 49 38 44.842 | 36344 12 6.5 1; 40.247) +0.067 49 38 45. 156 Mound and East Base ............-. 15 52 01.243 | 364 21 4.7 1 +0.132 | +0. 252 15 52 01. 627 East Base and Claremont .......-.-. 143 08 43. 092 365 16 | 6.1 1 +0.091 | +0. 606 143 08 43. 789 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2(361)+ — (362)4+ = (363)+ — (364) —0. 825=0 (361) +2. 5(36,) +1. 5(363) +1. 5(364)—1. 437=0 (36) +1. 5(362) +3. 5(363) +2. 5(364)—1. 454=0 (36,) +1. 5(362)+ 2. 5(363)+3. 5(364)—1. 454=0 DENVER—37. [Observer, R. S. Woodward. Instrument, Troughton & Simms theodolite No. 3. Date, November, 1879.] Angle as measured between— Notation. | No. meas. | Range.| Wt. (2) (v] Corrected angles. ° t aw 7 “wo “a uw ° t a“ Onion Hill and Mound.............. 9 09 24.748 371 16 4.8 1 +0.014 | —0. 003 9 09 24.759 Mound and West Base........-..--. 1 00 26.277 37, 16 2.1 1 +0.014 | —0.153 1 00 26.138 West Base and Claremont .-.....-.. 50 36 06.370 373 16 5.8 1 +0.014 | —0, 684 50 36 05. 700 Claremont and Parkersburg ........ 48 34 58. 482 374 16 3.5 1 —0.028 | +0. 278 48 36 58. 732 Claremont and Onion Hill .......... 299 14 02. 480 37446 8 3.6 0.5 | +0.083 | +0. 840 299 14 03. 403 Parkersburg and Onion Hill ........ 250 37 04. 167 3765 8 2.9 0.5 | —0.058 | +0. 562 250 37 04. 671 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 2. 0(371)-+ — (872)4+ — (373) +0. 5(374)—0. 040=0 (371) +2. 0(372)4+ — (373)+-0. 5(374) —0. 040=0 (371)+ — (872) +-2. 0(373) +0. 5(374) —0. 040=0 0. 5(371) +-0. 5(372) +0. 5(373)4-1. 5(374) +0. 022=0 § 4.) CHICAGO BASE TO OLNEY BASE. 573 SECTION XV.—Triangulation from the line Oakland - Kansas to the line Denver - Parkersburg— Continued. PARKERSBURG—38. (Observer, A. R. Flint. Instrument, Troughton & Simms theodolite No. 1. Date, November, 1879.] 1 | | Angle as measured between— | Notation. | No. meas. | Range. |. Wt. | (v) (v] | Corrected angles. | Gi ek u" | u“ | “ ou u" West Azimuth Mark and Denver... 31 43 41.471 | 38, 16 4.2 1 +0. 268 | +0. 001 31 43 41.740 West Azimuth Mark and East Azi- : ; ; J muth Mark ........--....--..----- 178 33 56.512 f 3814243 | 30 6.9 1.5 : —0. 019 0. 000 178 33 56.493 Denver and Claremont............-. 45 40 42.617 38, 16 6.0 |-1 | +0,268 | —0. 002 45 40 42. 833 Claremont and East Azimuth Mark. 101 09 31.601 | 38, \ 16 4.4 1 | 40.268} +0.061 101 09 31.870 East Azimuth Mark and West Azi- | ! | Tenth Mark .212- 00: sesxsaewavernas 181 26 03, 345 38, | 30 5.7 1.5 | +0. 162 0. 000 181 26 03. 507 NORMAL EQUATIONS FOR LOCAL ADJUSTMENT. 4(381)-+3(38,)-+3(383) —2. 683-0 3(381)+4(38,)-+-3(38,)—2. 683=0 3(38,)-+3(38,)-4+4(38,)—2. 683=0 Numerical equations of condition in the triangulation from the line Oakland—Kansas to the line : Denyer—Parkersburg. XXXIX. XLV. LI. LIV. LVI. LIX. LXII. LXIV. LXVI. LXIxX. LXXI. (25) (25) (5) (6) (10) (10) (10) (10) (10) (6) (6) —45, 6532 [221] +430. 4054 [255]* + 8.2965 (26,] —13. 3824 [292] + 2.8748 [314] + 9.5630 [32)] + 3.6770 [332] — 3.0785 [303] + 4.8992 [31,4] — 3.0963 [34,] —11. 7198 [322] +21. 6222 [222] +17, 2466 [262] + 2, 8748 [315] — 2.1568 [322] + 3.6770 [333] — 3.0785 [304] + 4, 8992 [315] — 6.3107 [342] + 9.9009 [323] —16, 3678 [38.45] +16. 0378 [335] —~ 4, 4523 [34245] — 4. 4523 [344] — 0,0004 [33,] + 3.7395 [352] +11. 0867 [303] — 3.6210 [316] + 6.1438 [363] + 3.3848 [315] — 3.3149 [322] +419. 1134 [332] -4-11. 0867 [30,] — 3.6210 [317] + 3, 3848 [316] — 3.3149 [32s] — 2.3162 [33445] — 2.3162 [335] +14, 1652 [303] — 7.4983 [349] + 1.7706 [314] + 9.5630 [321] +-11. 7838 [37,] —11, 7198 [322] — 0.3300 [33¢] 414. 1652 [30,] +16. 0135 [36,] + 1.7706 [31s] -- 9, 5130 [322] — 4,8989 [372] —11.7198 [32s] —11. 7838 [37;] SIDE-EQUATIONS. —18. 0616 [242] —33.5478 [285] ++ 2.8748 [315] — 2. 1568 [325] —13. 6876 [243] +40. 2660 [28,4] — 9, 6256 [317] — 2.1568 [324] + 3.6770 [33.45] — 4.0070 [33,] — 3.0785 [30,] 4 4, 3992 [31] 15. 2276 [305] + 4.8992 [317] ++ 3.2144 [34.45] + 3.2144 [344] + 9.9009 [324] — 0.3300 [336] —11. 6638 [344] —18. 3061 [305] — 3.6210 [31s] —13, 2466 [317] + 1.6668 [33,] —15, 2276 [305] + 0.1470 [362] ++ 1.7706 [3le] + 9.5630 [32s] . — 4.8989 [375] —11.7198 [324] —11. 7838 [37] —16. 3678 [332] — 7.5486 [34] + 0.0004 [345] + 9.6256 [314] -++16, 0135 [361] —13. 2466 [315] + 1.6668 [332] —15, 2276 [305] ++ 0.1470 [365] + 1.7706 [311] + 9.5630 [32,] — 4.2506 [325] + 5.5100 [375] ++ 6.8580 [252] —33. 2287 [291] — 9, 6256 [31s] + 3.6770 [33)] — 0.3300 [335] — 4.7264 [315] ++ 4,8992 [316] + 3.2144 [345] —16. 3678 [335] — 3.0963 [34] + 3.7388 [351] — 3.6210 [315] + 6.1438 [36,] +18. 6655 [321] + 1. 6668 [335] + 3.0963 [34] + 0.1470 [364] — 9, 6256 [31s] + 4, 2506 [325] — 2.7759 [33s] —65. 546=0 +32. 349=0 + 7.594 =0 + 4.414=0 — 3,586=0 — 8,210=0 +15. 960=0 +11.171=0 +11. 086=0 + 2.637=0 — 1.661=0 Nors.—In the solution for determining the general corrections, each of the side-equations was divided by the num- ber in parenthesis placed opposite it. 574 XXXVI. XXXVII. XXXVIII. XL. XLI. XLII. XLII. XLIV. XLVI. XLVII. XLVIII. XLIX. L. LII. LIII. LY. LVII. LVIII. LX. LXI. LXIII. LXY. LXVII. LXVIII. Lxx. LXXII. [221] [22s] [22s] [233] (24:] [249] [243] (251) [252] [25s] [261] [262] [263] (27) [272] [27s] [274] [275] PRIMARY TRIANGULATION. Numerical equations of condition, &e.—Continued. [223] + [28s] + [251] [221] + [222] + [243] [222] + [252] + [253] [242] + [253] + [271] [241] + [263] + [27s] [262] + [274] + [275] [26] + [262] + [274] [26] + [283] + [28.] [28] + [293] + [3h] [28:] ++ [306] -+ [307] [303] ++ [304] -+ [305] + [824] [304] + [305] + [314] [303] + [322] + [323] [303] + [304] + [306] [31s] + [314] + [31s] + [345] [31s] + [814] + [31s] [316] + [331] + [832] (31s] + [314] + [815] [303] + [80s] + [324] [31s] + [316] + [817] [82s] + [342] + [362] [303] + [30s] + [304] + [30s] + [314] + [Ble] + [316] + [3te] + [372] [302] + [325] + [371] [303] + [305] + [335] [30.] + [374] + [38] ANGLE-EQUATIONS. + [252] + Pi] + [272] + [29] + [284] + [2%] + [Ble] + [81s] + [314] + [31s] + [316] + [817] + [81s] + [821] [8g] 4+ [316] + £831] + (38:] + (88)) + [88e067 — 133) + [824] + [385] + [336] + [306] + [32s] + [32s] -+ [34] + [342] + [Ble] + [3] + [316] + [32] + [322] — [342] + [8le] + [8h] + [346] + [851] + [844] + [345] + [852] + [361] + [Ble] + £32:] + [322] + [32s] + [362] + [365] [32] ++ [32s] + (38446] + [885] + [362] + [36s] + [364] + [872] + [375] + [87s] General corrections in terms of the correlates. ——0. 09091 XXXVI —=—0, 09091 XXXVI =40. 72727 XXXVI =41. 00000 XXXVI =—0. 25000 XXX VII =—0. 25000 XXXVILI =+0. 75000 XXXVILI =+0, 66667 XXXVI =—0. 16667 XXXVI =—0. 16667 XXXVI ——0, 25000 XLII =—0, 25000 XLI =+0. 50000 XLI +0. 27273 XXXVII —0. 36364 XXXVIII—1. 47659 XXXIX +0. 27273 XXXVII +0. 63636 XXX VIII -+1. 21443 XXXIX —0. 18182 XXXVII —0. 09091 XXX VIII-+0. 08739 XXXIX +0. 31749 XX XIX —0. 40497 XXXIX —0. 23001 XX XIX —0. 16667 XXX VII +0. 66667 XXXVIT —0. 33333 XXXVIT +40. 25000 XLII +40. 50000 XLII —0. 25000 XLII =-+0. 60571 XXX VITI-+-0. 21143 XL =—0, 39429 XXXVITI +0. 21143 XL =—0. 10857 XXX VITI —0. 21714 XL =+0. 03429 XXX VITI-+0. 06857 XL =—0, 05714 XXX VITI—0. 11429 XL —0. 25000 XL +0. 75000 XL —0. 25000 XL —0, 33333 XXX VIII —0. 24842 XXXIX +0. 33333 XXX VITI —0. 22253 XXXIX +0. 33333 XXX VITI--0. 71937 XXXIX 0.50000 XLIV +0, 25000 XLIII —0, 25000 XLII —0. 10857 XLI —0, 10857 XLI +0. 46286 XLI —0. 25143 XLI +40. 08571 XLI +0, 75000 XLI —0. 25000 XLI —0. 25000 XLI —0. 00654 XLV —0. 25000 XLIV —0. 17246 XLV —0, 02286 XLII —0. 02286 XLII —0. 16571 XLII +40. 26286 XLII -40, 22857 XLII + [875] [Cuap. XX, C, —0.019=0 —0. 105=0 —1. 382=0 10,2570 +0, 036=0 —0. 436=0 +0, 422=0 —0. 108=0 —0, 859=0 +0. 071=0 + [3%] + [82s] +40. 509=0 +0. 176=0 +0. 768=0 —1.092=0 + [34243] + [344] +40. 954=0 +0. 062==0 —1.528=0 +0. 711=0 +1. 285=0 —0.117=0 +0. 234=0 —1.035=0 +1. 207=0 +1. 718=0 +1. 183=0 —0. 475=0 —0. 16667 XL —0. 33333 XL +0. 66667 XL +0. 26196 XLV -++0. 03429 XLUT +0, 03429 XLITT —0. 25143 XLII +0. 60571 XLIII —0. 34286 XLII § 4.] (281) [282] [283] [284] [29] [292] [29s] [30,] [803] [303] [30344] [3054445] [304] [805] [30s+6] [306] [307] (31. J [312] [31s] [314] [3l4t5+6] General corrections in terms of the correlates—Continued. =—0. 12500 XLII =—0. 12500 XLUI =—0. 31250 XLII =-+0. 68750 XLIII =-++0. 54167 XLII =—0, 20833 XLII =—0. 04167 XLII =—0. 20054 XLVII —0. 04255 LIV —0. 10840 LX VIII =—0. 04607 XLVII -+.0. 32228 LIV +40. 78591 LX VIII =—0, 04607 XLVII —0. 19080 LIV —0. 21409 LX VIII =—0. 09214 XLVII —0. 38160 LIV —0. 42818 LX VIII =—0, 26287 XLVIT —1. 05201 LIV —0. 57452 LXVIII =—0. 04607 XLVII —0. 19080 LIV —0. 21409 LX VIII =—0. 17073 XLVII —0. 67041 LIV —0, 14634 LXVIIT =-+0, 24933 XLVIT 40. 95923 LIV —0. 13550 LX VIIT =-+0. 42006 XLVII +1. 62964 LIV +40, 01084 LX VIII =-+0, 54472 XLVII —0. 42868 LIV —0, 05691 LXVIIT =+0, 64994 XLVI —0. 22032 LILI —0. 18506 LXI ——0, 21402 XLVI —0, 24631 LIII —0, 15942 LXI ——0. 05802 XLVI +0. 29832 LIII +0. 10820 LXI =+0, 02276 XLVI —0. 14845 LIIL —0, 45683 LXI =—0, 00236 XLVI —0. 14557 LIII —0. 06769 LXI CHICAGO BASE TO OLNEY BASE. —0, 25000 XLIV —0. 25000 XLIV +0. 37500 XLIV +40. 37500 XLIV —0. 20833 XLIV —0, 54167 XLIV —0, 29167 XLIV —0. 26558 XLVIII —0. 21680 LX —0. 21680 LXX —0. 57452 XLVIII —0. 42818 LX +40. 57182 LXX 40, 42548 XLVIII +0, 57182 LX +0, 57182 LXX +0. 85096 XLVIII +1. 14364 LX +0. 14364 LXX +1. 19243 XLVIII 40. 85096 LX —0. 14904 LXX —0. 03359 XLV —0. 03359 XLV —1. 42588 XLV +1. 52666 XLV —0. 60844 XLV —0. 01305 XLV +0. 21151 XLV —0, 15718 XLIX —0.15106 LXII +40. 70461 LXXII —0, 36043 XLIX —0, 20682 LXII —0, 10840 LXXII —0, 36043 XLIX +40. 90784 LXII —0, 10840 LX XII +0. 27914 XLIX +1. 80368 LXII —0. 21680 LX XII +40. 76695 XLIX +40. 31832 LXII —0, 26558 LX XII +0. 42548 XLVIII +0. 63957 XLIX +0. 57182 LX —0. 42818 LXX -L0. 34147 XLVIII —0. 29268 LX —0, 29268 LXX +0. 16803 XLVIII —0. 27100 LX —0. 27100 LXX —0. 17344 XLVI +40, 02168 LX +0, 02168 LXX —0, 08943 XLVIII —0. 11382 LX —0. 11382 LXX —0. 27204 XLVII —0. 08684 LIV -++0. 08892 LXII +0. 40402 XLVI +40. 08754 LIV +0. 14730 LXII +40. 24382 XLVI —0. 43624 LIV —0, 26407 LXII —0. 14059 XLVII -+L-0. 25360 LIV -0. 68715 LXII —0. 21533 XLVII +40. 39554 LIV +40. 25825 LXII +0. 90184 LXII —0. 10840 LXXII +0. 48781 XLIX —1. 48536 LXII —0. 04878 LX XII +40. 30353 XLIX —1. 10413 LXII —0. 11924 LXXII —0. 18428 XLIX +0, 38123 LXII » —0. 07046 LXXIT —0. 03252 XLIX —0. 17084 LX —0. 13008 LX XII —0. 16230 XLVIII —0. 06038 LV -+40. 20336 LXIV —0. 06637 XLVIII —0.07461 LV -0, 23160 LXIV —0, 12544 XLVI +40. 10310 LV —0, 28806 LXIV +0. 08519 XLVIII -+0. 00914 LV +40. 10746 LXIV +0. 17509 XLVIII +40. 25192 LV -++0. 63816 LXIV —0, 25000 XLVI -++0. 75000 XLVI —0. 12500 XLVI —0. 12500 XLVI —0. 04167 XLVI —0, 29167 XLVI -++0. 54167 XLVI —0. 10840 L —0, 12553 LXVI —0, 21409 L —0. 40019 LX VI +40. 78591 L +1. 01633 LXVI +40, 57182 L +2. 03266 LXVI +40, 42548 L +40, 94954 LX VI —0. 21409 L +1. 01633 LXVI —0. 14634 L —1. 08312 LXVI —0. 13550 L —1. 67968 LXVI +0. 01084 L —0. 59656 LXVI —0, 05691 L +0. 08637 LXVI —0, 00236 XLIX —0. 01256 LVII —0, 13564 LXVII +40. 10533 XLIX +0, 00614 LVIL —0. 03775 LXVII —0, 32066 XLIX —0. 04351 LVII —0, 15798 LXVILI +0, 24278 XLIX —0, 14962 LVI +40, 11146 LXVII +0, 57258 XLIX +0. 16490 LVII +40, 24134 LXVIL +0. 75000 XLVIT —0. 25000 XLVII —0. 12500 XLVII —0, 12500 XLVIT —0, 33604 LIT —0. 37398 LXVII —0. 56368 LIT +0. 21139 LXVII +0. 43632 LII +40. 21139 LXVII +087264 LII +0. 42278 LXVII +1. 01899 LIL +0. 61791 LX VI +0, 43632 LII +0. 21139 LX VII +40. 14635 LIL +0. 19513 LXVII -40. 57996 LII +0. 03253 LX VII +0. 43361 LII —0. 16260 LX VII —0. 27642 LIT —0. 14634 LXVII +0, 30655 LI —0, 04782 LVIII +0. 00274 LXIX +40. 39110 LI —0, 03075 LVUI +40. 03477 LXIX —0. 56019 LI +0. 14661 LVIII —0, 09882 LXIX +40. 44297 LI +0. 15876 LVIUL +40. 07504 LXIX +1. 09443 LI +0. 08702 LVIII +40. 17752 LXIX 575 576 =—0 +40. +0. [3latste+74x] =—O. +0. [8la45+6+7] =—0. +0. +0. =—0. +0. +0. =—U. +0. [315] [3ls+6] [8ls4o+7+8] [315] [317] [3hi+s] [31s] 40, =+0. —0. —0. [321 | [32424544] +40. +0. [322] —1. —0. bi, —9. “40. =40, +0. 4%, 10, +0, +0, =i), ae. i, [32945] [324344] [323] [32344] [324] +0. —0. =+1. =+0. =+0. PRIMARY TRIANGULATION. General corrections in terms of the correltates—Continued. . 13564 XLVI 16138 LIII 20790 LXI 16230 XLVI 62278 LIIL . 66303 LXT 01256 XLVI 00144 LIII 19457 LXT 02512 XLVI 00288 LIIT 38914 LXI 18506 XLVI 77123 LIT . 11986 LXI . 01256 XLVI . 00144 LITT . 19457 LXI . 18328 XLVI . 30695 LITT . 27559 LXI . 15994 XLVI . 76835 LITT . 73072 LXT . 02666 XLVI . 46140 LOT 45513 LXI 42857 XLVIII 19917 LVI 07143 LXV 14286 XLVIIT 39834 LVI 14286 LXV 14286 XLVITI 17774 LVI 35714 LXV 28572 XLVIII 19341 LVI 28572 LXV 71429 XLVITI 59751 LVI 21429 LXV 14286 XLVIII 98433 LVI 64286 LXV 57143 XLVIII 77525 LVI 57143 LXV 42857 XLVITI 79092 LVI 07143 LXV —0. 19573 XLVII -40, 38519 LIV +40. 03205 LXII —0,19181 XLVII +40, 70975 LIV —0. 15803 LXII —0, 03737 XLVII +0.07097 LIV —0, 21446 LXII —0,07474 XLVII +0. 14194 LIV —0, 42892 LXII —0, 05122 XLVII +40. 45615 LIV —0, 84518 LXII —0, 03737 XLVII +0. 07097 LIV —0, 21446 LX +40. 01960 XLVII —0, 01035 LIV —0, 22618 LXII +0, 02352 XLVII +0. 31421 LIV —0, 41626 LXII +0. 00392 XLVII -40. 32456 LIV —0. 19008 LXII —0, 35714 L —0, 21429 LX —0, 21429 LXVIUI +0.71430 L -0, 42856 LX —0.57144 LX VIII +0. 21429 L —0. 07143 LX —0, 07143 LXVIII --0, 42858 L —9, 14235 LX —0, 14286 LX VUI +1.07144 L --0. 64285 LX —0, 35715 LX VIII -40. 21429 L —0. 07143 LX —0. 07143 LXVII +0. 85715 L +0. 71428 LX —0, 28572 LX VIII -L0. 64286 L +0. 78571 LX —0. 21429 LX VII +0, 31936 XLVIII +0. 08336 LV —0, 05939 LXIV +40, 74822 XLVOL 0, 04965 LV —0. 72877 LXIV +0, 04495 XLVIII +0. 12139 LV +0, 26535 LXIV +40. 08990 XLVIII -40, 24278 LV +0. 53070 LXIV -40. 66303 XLVI +40. 04051 LV —0, 83623 LXIV +0. 04495 XLVIII 0. 12139 LV 0. 26535 LXIV 40, 14427 XLVIII —0, 16856 LV —0, 69755 LXIV 0.57313 XLVI —0, 20227 LY —1. 36693 LXIV +0. 42886 XLVIII —0, 03371 LV —0, 66938 LXIV +1. 65681 LI -+0, 64286 LXI +0, 53126 LXIX +0, 51156 LI +0. 71430 LXI 41. 41670 LXIX —0, 22905 LI +0, 21429 LXI -+-0. 17709 LXIX . —9. 45810 LI +0. 42358 LXI +0. 35418 LXIX . 14525 LI +0. 07144 LXI +40, 88544 LXIX —0, 22905 LI +0. 21429 LXI +0. 17709 LXIX —0. 91620 LI —0. 14285 LXI +0. 70835 LXIX —0, 68715 LI —0, 35714 LXI +0,53126 LXIX —_ 10, 24134 XLIX +0. 06494 LVI +0, 47302 LXVII +0, 17509 XLIX +0, 04495 LVII +40. 31936 LXVII +40. 16490 XLIX —0. 34274 LVII +0, 06494 LXVII -0. 82980 XLIX -40. 31452 LVII 40. 12988 LXVII —0. 06769 XLIX +0. 19457 LVIL +0, 20790 LXVII +0. 16490 XLIX +0, 65726 LVII +0. 06494 LX VII . 33124 XLIX --0, 09996 LVII +0, 23168 LXVII —0. 39749 XLIX —0. 11995 LVII 0. 073802 LXVII —0, 06625 XLIX —0, 01999 LVII —0, 15366 LX VII —0. 28571 LII —0, 14286 LXIII 40. 84941 LXXI +40. 57144 LII +0. 28570 LXUI —0, 99040 LXXI —0, 42857 LIT +0, 28571 LXIII —0. 36796 LXXI +0, 14286 LII +40. 57142 LXIIL —0. 73592 LXXI 40, 85715 LII +40. 42856 LXIII —1. 83981 LXXI +0, 57143 LII +0. 28571 LXIII —0. 36796 LXXI -+1. 28572 LII +0, 14285 LXIII —1, 47185 LXXI -+-0. 71429 LII —0. 14286 LXIII —1. 10389 LXXI > oS [Cnar. XX, C, —0, 01143 LI +0, 01842 LVIII -L0, 38612 LXIX —1. 00266 LI +40. 00470 LVIII —0. 59374 LXIX +40. 32573 LI +0. 46413 LVIII +0, 05124 LXIX +0. 65146 LI —0.07174 LVIII -0, 10248 LXIX —1. 44563 LI —0. 15406 LVII —0, 66878 LXIX -0. 32573 LI —0, 53587 LVILI +40, 05124 LXIX —1. 10586 LI —0. 06860 LY LI +40, 20860 LXIX —2, 09709 LI —0, 08232 LVIII —0. 77126 LXIX —0. 99123 LI —0. 01372 LVI —0, 97986 LXIX +40. 71429 LIT +1.51393 LXIV 40, 57144 LIL +0, 70523 LXIV +0. 57143 LIIT —), 22804 LXIV +0. 14286 LIT —0. 45608 LXIV —0. 14285 LIT —0. 80870 LXIV —0, 42857 LIII —0, 22804 LXIV —0. 71428 LIIT —0, 58066 LXIV —0, 28571 LIT —0, 35262 LXIV CHICAGO BASE TO OLNEY BASE. General corrections in terms of the correlates—Continued. § 4.) [325] =—0.57f43 XLVI — 0.19917 LVI —0, 07143 LXV [33,] =-+10. 38241 XLIX -10, 23519 LVII +40, 04516 LXXI [Shters] = +0. 94419 XLIX +0. 31169 LVII —0. 00563 LX XT (Stiasiei jae) =O, toads KIX +0, 19314 LVII —0. 05197 LXXI [332] =—0. 16824 KLIX -40. 33654 LVIL —0. 03838 LXXI (Pse) Sos KLIS —0. 04205 LVILI —0, 09713 LXXI [333] =-+0. 13002 XLIX —0, 26004 LVII —0, 01241 LXXI peu =-+0, 05927 XLIX —0. 11855 LVILI —0, 04634 LXXI [33i¢s¢6] = PU. Ux249 XLIX —0. 06501 LVII : —0. 01686 LXXI [335] =—0, 48948 XLIX —0, 02103 LVIL —0), 27989 LXXI [336] =—0, 02678 XLIX +40. 05354 LVIL +40, 02948 LXXI [341] =-0, 50000 LII -0. 00877 LVIIL [349] =-+0,50000 LII —0, 00377 LVIII (Rapes =-+0, 09350 LIIL +40, 26302 LIX (34o4344] =-+0. 00877 LIII —0, 08185 LIX [Boorse] = +0. 36842 LILI -40, 06141 LIX [344] =—0, 08773 LUI —0, 34787 LIX [345] =, 35965 LIII 40, 14326 LIX [351] —+0, 66667 LVII [352] =——0, 33333 LVII [361] =-40. 65625 LX +1. 04629 LXVI [362] =—0. 18750 LX —0. 29474 LXVI i3Ls —0. 35714 L —0, 21429 LX -+40. 78571 LX VIII —0. 01338 L —0. 48970 LIX +6. 08796 L —0, 05845 LIX —0. 01912 L —0. 15346 LIX -40, 04589 L +1, 13204 LIX —0. 00574 L +0. 33624 LIX +0. 05545 L —0.70169 LIX —0. 10708 L —0. 09501 LIX +0, 26386 L —0. 17541 LIX +0. 49714 L +0. 16810 LIX -L0. 37094 L —0, 08040 LIX —0. 00438 LIII +0. 04093 LIX —0. 49562 LILI —0. 04093 LIX +40. 05171 LIV +0. 06140 LXV +40. 00473 LIV +40. 09649 LXV +0, 19741 LIV -40. 05263 LXV —0. 04698 LIV +40. 03509 LXV -L0. 19268 LIV —0, 04386 LKV —0. 33333 LVIII 4.0. 66667 LVIII —0, 25000 LXI +40. 50000 LXI —0. 25579 LI —0. 35714 LXI —0. 35414 LXIX 0. 28212 LI +0. 02676 LXIII +0, 24732 LI —0.17591 LXIII +40, 29797 LI 0. 03823 LXIII —0, 12675 LI —0. 09177 LXITII +40. 01585 LI +0. 01147 LX III +40. 09195 LI —0. 11090 LXIII +40. 05065 LI +49. 21414 LXIII 40. 00648 LI +40. 47226 LXIII —0. 39276 LI -40. 00574 LXIII —0. 04417 LI -L0, 25812 LXIII —0, 26038 LIV —0, 04825 LXV —0.52353 LIV +40. 54825 LXV 40, 03509 LV —0, 06505 LXVI —0. 08772 LV —0, 10228 LXVI +0, 31579 LV —0, 05575 LXVI —0, 12281 LV —0. 03717 LXVI +40, 40351 LV +0, 04647 LXVI +0, 12460 LIX “0, 12467 LIX +0, 89729 LXIL +0, 00694 LXII —0. 28571 LI —0. 14286 LXIII +40. 14098 LXXI +40, 49139 LV 40. 05085 LXIV +40. 34227 LV 40, 14209 LXIV +0. 27343 LV -L0. 04883 LXIV —0. 25620 LV -++0. 01616 LXIV —0. 21796 LV —0. 00202 LXIV +40. 10708 LV -++0. 07508 LXIV —0. 06884 LV —0. 09326 LXIV +0. 02676 LV —0, 13871 LXIV —0. 10898 LV —0. 00100 LXIV +0. 09560 LV —0. 04545 LXIV +40. 04386 LV +0, 20593 LXVI —0. 04386 LV —0. 42603 LXVI —0, 02734 LVI —0. 04296 LVI —0. 02343 LVI —0. 01562 LVI +0. 01953 LVI —0. 18750 LXIII +0. 62500 LXIIT 577 —0. 28571 LIT —0. 35262 LXIV --0. 17882 LVI —0. 10898 LXX —0. 00605 LVI +0, 00192 LXX —0. 21221 LVI +0. 13003 LXX —0. 14549 LVI +0. 08795 LXX —0. 39103 LVI +0, 23901 LXX —0. 03938 LVI -+0. 02295 LXX —0. 20616 LVI +0. 12811 LXX --0, 01811 LVI -0. 00574 LXX +0. 60640 LVI -+0. 61951 LXX -L0. 18805 LVI —0. 12237 LXX —0, 35599 LVI —0. 17630 LVI —0. 19298 LVIII —0. 01755 LVIII +0. 26315 LVIII +0. 17543 LVIII +0. 28070 LVIII —0, 31250 LXV +0. 37500 LXV [35243] =—0. —0 (36.4941) =—0. —0. [363] =—0. —0. [364] =—0. —0 (87,] =—0 —0 [37i+2] =-+0 —U [372] =+0 ==() [37243] =+1 —0 [373] =+0, —, [374] =—0 +0. (33) ] =—0 [332] [383] No. of equation. PRIMARY TRIANGULATION. General corrections in terms of the correlates—Continued. 25000 LX . 80208 LXVI 31250 LX 49122 LXVI 06250 LX 09824 LXVI 06250 LX 09824 LAVI .47619 LXVII . 09524 LXXIL . 04762 LXVIT . 19048 LXXIT . 52381 LAVIT . 09524 LXXIT . 04762 LXAVIT +1. 00000 LXI +40. 50000 LXI +0. 50000 LXI —0, 50000 LXI +0, 28571 LX VIII +0, 57142 LX VIII +0. 28571 LXVIII +0, 57142 LXVIUI +0. 21405 LXII —0. 19322 LXII +0. 20711 LXII —0. 40727 LXII +1. 88516 LXIX 0, 98987 LXIX —0. 89529 LXIX —1. 79058 LXIX -++0. 50000 LNIII -+-0. 37500 LXIII —0, 12500 LXTIT —0. 12500 LXTII —0, 23810 LXX —0. 47620 LXX —0. 23810 LXX +0. 52380 LXX [Cuap. XX, C, +40. 50000 LXV +40. 62500 LXV -++0. 12500 LXV +0, 12500 LXV —1, 24740 LXXI —2. 49480 LXXI —1, 24740 LXXI +0. 38750 LXXI 19048 LXAXIT . 52381 LXVIT - 09524 LXXNIT . 19048 LXVII . 76190 LXXIT . 30000 LXXIT =-+0. 70000 LXXIL =—0. 30000 LXAXNIT +0. 22571 LXVIIL —0, 23571 LX VIII —0. 89529 LXIX —0. 03152 LXIX +0. 76190 LXX —0. 09524 LXX Normal equations for determining the correlates. 36. O0=—0. 01900 +2 37. 0=—0. 10500 ta 43. 44. 46. 1. 38200 0=—2. 62184 0=-L0. 25700 0=-40. 03600 —0 as + =o) 0——0. 10800 —0. —0. 0=+1. 29396 —0. +5. 0=—0. 85900 —0, Als 10. —0 0, —0. —0. —0. +40. =), +0. —0. ei, ==0: el. 0=—0, 43600 —0. +0. 0=+0. 42200 +40. 39394 XXXVI 16687 XL 20336 LATV —0. 314819 XXXVIT . 13564 LXVIT —0, 42424 XXXVIII —0. 16103 XXXIX 34849 XXXVI -+41.96213 XXXVIIT +0. . 58333 XL —0. 25000 XLI 42424 XXXVI +0.60605 XXXVIJ_ +41. 54477 XL —0. 10857 XLI =0 16103 XXXVI —0.71470 XXXVII +1 31440 XL +0. 31749 XLI 16667 XXXVI —0.58333 XXXVII_ +10. 83953 XL —0. 46714 XLI —0 25000 XXXVII —0. 16857 XXXVIII +40. 71286 XLI —0. 41571 XLII —0 02226 XNXVIIL -- 0.94572 XL —0 51285 XLIII —0.45833 XLIV —0 03429 XXXVIII -L0. 06858 XL —0 .79321 XLII +0. 62500 XLIV +1. 12500 XLVII 45833 XLII +0.62500 XLII 4-1. 54167 XLVI —0.25000 XLVII 17246 XLI —0. 34648 XLII +1. 36655 XLV +0.17792 XLVI —0. 04167 XLII —0.12500 XLIII —0 94161 XLVI —0.52204XLVII —0. 30655 LI —0, 22032 LIIL —0. 01256 LVII —0.04782 LVI = —0. +0. 60605 XXX VIII —0,.71470 XXXIX 90873 XXXVIII +1.71127 XXXIX . 02286 XLII +0. 03429 XLIIT . 71127 XXXVIII +4. 97917 XXXIX 54477 XXXVIII +0. 31440 XXXIX . 04572 XLIT 31749 XXXIX . 50143 XLII - 41571 XLI . 34648 XLV . 50143 XLI 78208 XLV 79167 XLIV 78208 XLII 03359 XLVII . 54167 XLIV 16230 XLVIIT 08684 LIV 12506 LXI 00274 LXIX +0. 06858 XLIII —0. 46714 XL —0. 17246 XLV +1. 53310 XLII —0,. 04167 XLVI +40. 51285 XLII —0. 12500 XLVI +0. 08119 XLV +40. 08119 XLIV -0. 17792 XLV —0, 00236 XLIX —0. 06038 LV +0, 08892 LXII +41. 63490 LXXI +0, 28663 LXXI §4.] No. of equation. 47. 48. 49. 50. 52. 53. 54, CHICAGO BASE TO OLNEY BASE. Normal equations for determining the correlates—Continued. 0=-+0. 07100 0=-+0. 50900 =-+0. 17600 0=-+40. 76800 0=+1. 51880 0=—1. 09200 0=-+0. 95400 0=+0. 73567 —0. 12500 XLIII +2. 36262 XLVII —0. 16909 LI -L0. 02849 LV —0. 05122 LXI —0,50467 LXVII —0. 20054 LXXII —0. 16230 XLVI 44, 13078 —-0. 34226 LIV +0. 00470 LVIII -4.0. 28570 LXIII +40. 93727 LXVII —0. 99040 LXXI —0. 00236 XLVI —0. 87669 L —0. 46567 LIV +0, 08702 LVIII —0, 32529 LXII +0. 64786 LXVII +0. 22792 LXXI —0. 04607 XLVILI —1.58218 LI —0. 01338 LV +1, 21467 LX —0, 85515 LXIV —0,57124 LXVIU —0. 10840 LXXII +40. 30655 XLVI —1.58218 L —0, 37747 LIV -.0. 20851 LVIII +40, 94982 LXII —0. 01143 LXVII -42. 60233 LXXI -L0. 15719 XLVILI —0, 91620 LI +1, 24300 LVI +40. 14285 LXIII +0. 45531 LX VII —1, 47185 LXXI —0, 22032 XLVI —0, 14285 L +40, 99445 LIV -+L0, 42323 LVIII —0. 42210 LXII +0. 37028 LXVI 40. 48145 LXXI —0, 08684 XLVI —0. 19080 L +6, 38962 LIV +40, 03403 LVIII -L.0, 88320 LXII —0. 34454 LXVII —0, 04255 LXXII —0. 25000 XLIV —0, 45468 XLVIII +40. 15719 LIL —0, 03737 LVII +40. 09362 LXII —0. 04607 LX VIII —0. 45468 XLVII —0. 49110 LI +40, 04965 LV +1. 27952 LX —0, 02354 LXIV —1. 14596 LX VIII —0, 26558 LXXII —0, 43213 XLVII +1. 78516 LI +40. 63433 LV —0, 32156 LIX +0. 03249 LXIII —0, 36043 LXVIII - —0, 15718 LXXII +1. 13978 XLVIII +41. 29347 LIL +1. 39196 LVI +0, 07144 LXI +0, 21429 LXV +0, 88544 LXIX —0. 06909 XLVII +8. 86605 LI +40. 81636 LV —0. 24226 LIX —0. 45162 LXIII —0. 25579 LX VIII 4+1.59043 XLVIII -++3. 73832 LII +1, 58692 LX —0, 58066 LXIV —0, 84940 LXVIII —0.33604 LX XII +0, 05201 XLVII —0, 13509 LI +40. 51240 LV +0. 10234 LIX +0, 14285 LXIII +0, 16138 LXVII 0. 85226 XLVILI —0. 37747 LI +0, 15198 LV 40, 05483 LIX —0. 36815 LXIV 40, 32228 LXVIII —0, 03359 XLV —90, 43213 XLIX +40. 05202 LITT +0. 06586 LVIIT —0, 05646 LXIV —0, 06405 LXIX +3. 08351 XLVIII +41. 59043 LII +0. 39834 LVI 441,39983 LET +0. 14286 LXV 0, 82296 LXIX 40. 94204 XLVIII -40, 58267 LII —0. 81861 LVI 40. 27914 LX +40. 68799 LXIV 0. 17752 LXIX —0. 87669 XLIX —0. 14285 LIE -40. 03251 LVII +0, 90184 LXII +1. 01633 LXVI +1. 06896 LXX —0. 49110 XLVI —0. 91620 LII —1.29311 LVI —0. 68715 LX +16, 28251 LXIV +2. 22099 LXIX 40. 58267 XLIX —1, 21428 LII —0, 14285 LXI +1. 07143 LXV +0. 70835 LXIX +1, 19422 XLVITI —1, 21428 LIT —1. 22404 LVI —0, 28572 LX 10, 26906 LXIV —0, 28572 LX VIII —0, 34226 XLVIIL —0. 20628 LII +0. 35657 LVI —0. 38160 LX —0, 52353 LXV —0, 40696 LXIX —0, 52204 XLVI —0, 04607 L +40. 85226 LIV —0. 09214 LX —0.51019 LXVI —0. 09214 LXX +0. 94204 XLIX +1, 19422 LUI +0. 04495 LVII +0. 16029 LXII +0, 94954 LXVI —0. 14904 LXX +2. 59290 XLIX —0. 14557 LIII +40. 37907 LVII —0. 06769 LXI —0, 06679 LXVI —1. 21034 LXX +2. 72543 L —0. 19080 LIV +0. 08770 LIX +0. 69242 LXIII +0. 21139 LX VII —2. 09022 LXXI +1. 78516 XLIX —0, 13509 LIII +0. 48110 LVII —0, 24692 LXI —0, 22905 LXV —0, 39276 LXX 41. 29347 L —0. 20628 LIV -40, 69955 LXII +0. 13288 LXVI —0, 12736 LXX —0. 14557 XLIX +3. 07086 LILI +0. 00144 LVII +1. 62838 LXI —0. 92419 LXV +0, 01579 LXIX —0. 46567 XLIX +40, 99445 LIIL +40. 07097 LVII +40. 45615 LXI —1. 68927 LXVI -+-0, 13148 LXX O79 580 No. of equation. 55. 56. 57. 58. 59. 60, 61. 63. PRIMARY TRIANGULATION. [Cuap. XX, C, Normal equations for determining the correlates—Continued. 0-40. 06200 0=—0, 35°60 (=—1. 52800 (=-+40. 71100 0=—0, 82100 0=-+1. 28500 0=—0. 11700 0=-+1. 59600 0=-+0. 23400 —0. 06038 XLVI —0, 01338 L +1. 24102 LV —0. 444 LIX +40. 40095 LXIV +0, 07870 LXIX +40. 39834 XLVIII +1, 24300 LII +5, 08671 LVI +0, 79092 LX +40, 80803 LXV 0, 60640 LXX —0, 01256 XLVI -L0, 03251 L -.0, 35658 LV +0, 76784 LIX +, 33236 LXIV 0. 00678 LX XI —0. 01782 XLVI -0, 20851 LI 0, 00391 LVI —0, 15406 LXI +40, 00930 LXVI —0, 32156 XLIX -L0, 05433 LIV —0, 07994 LVIII —0. 04093 LXV —0. 09214 XLVII —0. 68715 LI -L0. 79092 LVI —0, 33036 LXIII +0, 42278 LXVII 1.10389 LXXI —0. 18506 XLVI +0, 07144 L +0, 45615 LIV —0. 15406 LVIIL -L0, 92856 LXIII +40. 20790 LXVII +0, 08892 XLVI +0, 90184 L 0. 88320 LIV +2. 70097 LX -+0. 40624 LXIV —0, 20682 LX VIII +0, 28570 XLVIII +0. 14285 LI —0, 06501 LVII +0. 00694 LXII —0. 29474 LXVI —0, 75278 LXXI > +40, 02849 XLVII 40. 81636 LI +40, 19435 LVI +0, 04051 LXI —0, 04386 LXV . 10898 LXX —0, 81861 XLIX —1, 22404 LIII +0, 03333 LVII —0. 39258 LXI +9. 02197 LXVI . 31689 LXXI —0. 03737 XLVILI +0. 48110 LI +0. 03333 LVI 40. 19457 LXI +0. 06494 LX VII _ +0, 06586 XLVIL +0. 42323 LIII —0. 86920 LVII +0, 20862 LXII 0, 01842 LXVII 0. 08770 L —0. 34644 LY -+2. 66442 LIX +0, 04335 LXVI -1. 27952 XLVI +11. 58692 LIL +2, 58560 LX —0, 35262 LXIV —0. 64247 LX VIII —0, 21680 LX XII —0, 05122 XLVII —0, 24692 LI +0, 04051 LV —0. 60714 LX -+0, 22162 LXIV —0, 35715 LXVIII -40. 09362 XLVII +0, 94982 LI —0. 00584 LV —0. 63113 LXI —0, 19322 LXV +0. 31440 LXIX -+0, 03249 XLIX +0, 14285 LIII —0. 17541 LIX +1. 66868 LXIII —0, 14286 LX VIII -40, 01965 XLVIII +40, 51240 LIII -0, 35658 LVII —0. 005e4 LXIL -40. 04647 LXVI +0, 04516 LXXI +1. 39196 L +0. 35657 LIV +0, 00391 LVIII —0, 21152 LXIII -—0, 19917 LX VIII -L0, 04495 XLVIII +0, 00144 LUI +41. 89566 LVII —0. 21446 LXII +0, 05124 LXIX +0, 00470 XLVIII +40, 03403 LIV +1. 89230 LVIII +0. 08475 LXIV +0, 02746 LXIX —0. 24226 LI —0. 25989 LVI —0. 17541 LXIII -L0, 16810 LXX -10, 27914 XLIX —0, 28572 LIII —0, 60714 LXI —0, 38393 LXV 0, 53126 LXIX +1, 37733 XLVIII —0. 14285 LIT —0, 39258 LVI +3. 19130 LXI +40.71429 LXV +0, 21666 LXIX -40. 16029 XLVIII 0. 69955 LII —0, 21446 LVII +7. 25466 LXII +5. 67036 LKVI +0, 69502 LXX 0. 69242 L £0. 02676 LV —0. 33036 LX - —0, 59479 LXIV -L0, 35418 LXIX +40. 63433 XLIX -L0. 15198 LIV +0. 51433 LVI -L0. 02676 LXIII 0. 08336 LX VIL —1, 29311 LI +0. 19835 LV —0, 25989 LIX —0, 30444 LXIV +0. 49378 LXIX +0, 37907 XLIX 0.07097 LIV —0. 86920 LVIII —0, 06501 LXIII — 0.02103 LXX +0, 08702 XLIX -+0, 51433 LV —0. 07994 LIX —0. 00877 LXV +0. 10234 LILI 0. 76784 LVII +0. 03089 LXIV —0, 07336 LXXI 4.1. 21467 L —0. 38160 LIV +2. 70097 LXII +3. 07895 LXVI +0. 14364 LXX —0, 06769 XLIX +1. 62838 LUI +0. 19457 LVII —0. 63113 LXII —0, 39298 LXVI +0, 11349 LXXI —0. 32529 XLIX —0, 42210 LIII +0. 20862 LVIII +0, 00694 LXIII +0, 14355 LXVIL —0, 15106 LXXII —0, 45162 LI —0, 21152 LVI +0. 92856 LXI +0. 66072 LXV +0, 00574 LXX §4.] No. of equation. 64. 65. 66. 67. 68. 69. 70. 71. 72. CHICAGO BASE TO OLNEY BASE. Normal equations for determining the correlates—Continued. 0=+1. 11710 0=—1. 03500 0=-+1. 10860 0=+1. 20700 0=+1. 71800 0=+0. 43950 0=-+1. 18300 0=—0. 27683 0=—0. 47500 —0. 20386 XLVI —0. #5515 L —0. 36815 LIV +40. 08475 LVI +0, 40624 LXII —0. 05939 LXVII +1, 83238 LXXI +0, 14286 XLVI —0. 92419 LIII —0. 00877 LVIII —0. 19322 LXII —0, 91725 LXVI —0.51019 XLVII +40. 13288 LIT -40. 02197 LVI —0. 39298 LXI +7. 48852 LKVI —0, 12553 LXXII —0. 13564 XLVI +10. 21139 L —0. 34454 LIV -L0. 42278 LX +10. 54935 LXVI -0, 94658 LXX —0, 04607 XLVILI —0. 25579 LI —0. 19917 LVI —0. 14236 LXIII. +40, 78281 LXVII —0.71892 LXXI +40, 00274 XLVI 40, 88544 L —0. 40696 LIV -L.0. 02746 LVIII -L0. 35418 LXIII —0. 25956 LX VIII —0, 03152 LXXII —0. 09214 XLVII —0. 39276 LI +0. 60640 LVI -L0, 69502 LXII +0, 94658 LXVII +41. 35501 LXXI —0. 99040 XLVIII —1. 47185 LII . 00678 LVIL, —0. 75278 LXIII . 71892 LX VII . 28663 LXXII —0. 20054 XLVII . 33604 LIT . 12553 LXVI , 31204 LXX —0. 05646 XLVILI +6, 28251 LI -40. 40095 LV +0, 03089 LIX —0.59479 LXIII —0, 35262 LX VIII +0, 21429 L —0. 52353 LIV —0. 04093 LIX 0, 66072 LXIII —0. 07143 LX VII 40. 94954 XLVIII 0. 37028 LIII +40. 00930 LVIII 4-5. 67036 LXII +0, 54935 LXVII —0. 50467 XLVII —0; 01143 LI -++0. 08336 LV +40. 20790 LXI +2. 34994 LXVILI -.0, 38750 LXXI —1, 14596 XLVIII —0. 84940 LII —0, 64247 LX —0. 35262 LXIV —2. 42875 LXVIII —0, 39412 LXXII 0. 06405 XLVII +2, 22099 LI +0. 07870 LV 4-0. 53126 LX +11, 93056 LXIV +18. 85737 LXIX —0. 14904 XLVIII —0. 12736 LIL —0, 02103 LVII 40. 00574 LXIII -L0. 85752 LX VIII —0. 31204 LXXII +40, 22792 XLIX +40, 48145 LIIL —0, 07336 LIX 1, 83238 LXIV —4, 24491 LXIX —0, 26558 XLVIII —0. 04255 LIV —0.56446 LXVII -L0, 28663 LX XI —0, 02354 XLVIII —0, 58066 LIL —0. 30444 LVI —0, 35262 LX +15, 02318 LXIV +1, 93056 LXIX —0, 22905 LI —0. 04386 LV —0. 38393 LX —0, 22804 LXIV +0. 17709 LXTX —0. 06679 XLIX . 68927 LIV . 04335 LIX . 29474 LXITIT . 40019 LX VIII 93727 XLVIII 45531 LIT 06494 LVII +0. 14355 LXII +0. 78281 LXVIII . 56446 LXXIT —0. 36043 XLIX —0, 28572 LITI —0. 35715 LXI —0. 07143 LXV —0. 25956 LXIX +0, 0, +0. +0, 82296 XLVIII +0, 70835 LII 40, 49378 LVI -L0, 21666 LXI +40. 17709 LXV —0, 89529 LXX —1, 21034 XLIX +0, 13148 LIV -10, 16810 LIX . 00100 LXIV . 89529 LXIX . 09022 L . 04516 LV . 10389 LX . 36796 LXV . 35501 LXX —0,. 15718 XLIX —0. 21680 LX —0, 39412 LXVII +2. 16651 LXXII + 0.68799 XLIX + 0.26906 LIII ++ 0.33236 LVIL + 0.22162 LXI — 0.22804 LXV — 0.00100 LXX + 1.07143 LII + 0.80303 LVI + 0.71429 LXI + 1,81611 LXV — 0.36796 LXXI + 1.01633 L + 0.04647 LV + 3.07895 LX’ — 0.91725 LXV + 0.61614 LXX + 0.64786 XLIX ++ 0.16138 LIII 4+ 0.01842 LVIII — 0.05939 LXIV — 1,40446 LXIX — 0.57124 L + 0.32228 LIV — 0, 20682 LXII — 0.40019 LXVI + 0.85752 LXX + 0.17752 XLIX + 0.01579 LIII + 0.05124 LVII + 0.31440 LXII — 1.40446 LXVII — 4,24491 LXXI + 1.06896 L — 0.10898 LV + 0.14364 LX + 0.61614 LXVI + 2.52505 LXX ++ 2.60233 LI — 1.31689 LVI + 0.11349 LXI + 0.38750 LXVILI +10, 02279 LXXI — 0.10840 L — 0.15106 LXII — 0, 03152 LXIX o8L PRIMARY TRIANGULATION. Values of the correlates and of their logarithms. XXXVI =-+0. 0873 log 8.9410640+. | XXXVII =—0, 1668 log 9. 2223002_ XXXVIII =-+0. 6348 log 9. 40263014 XXXIX =+0,3291 log 9. 51731474 XL =—0. 4448 log 9. 6481550- XLI =—0. 2153 log 9.3330440— XLII =+0. 6408 log 9. 80669544. XLII =—0. 6142 log 9. 7882886— XLIV =-+40. 6751 log 9, 82038744 XLV =—0. 0355 log 8.5504730_ XLVI =+0. 7093 log 9. 85083614 XLVII =+0. 1479 log 9. 16990944 XLVII =+0,5721 log 9. 75746444. XLIX =—0. 4693 log 9. 6714320 — L =—1.5482 log 0, 1698383_ LV =-40.7732 log 9. 88831994. LVI =—0. 2766 log 9, 4418993_ LVII =+1.1760 log 0. 07042584 LVII =-+0. 2972 log 9. 47299034 LIX =—0,1318 log 9, 1199813_ LX =—2. 8810 log 0. 4595373_ LXI =—0. 4526 log 9. 6557433_ LXII =+0. 0247 log 8. 39199314 LXIIL =-+0. 0559 log 8, 74741184 LXIV =+1.2300 log 0..08990514. LXV =—0. 2997 log 9. 4766867_ LXVI =+1. 0147 log 0. 00632054. LXVII =-+0. 3963 log 9. 59802414 LXVIII =—1. 8415 log 0. 2651811 LXIX =-+0. 1964 log 9, 29307514 LXX =+0, 5724 log 9. 75773764 (Cuap. XX, C, LI =—1.5331 log 0. 1855733_ LII =+0. 8096 log 9. 90829204 LIII =—1.1513 log 0.0611988— LIV =+0.2096 log 9. 3214949 LXXI =—0, 3828 log 9, 5829833_— LXXII =—0. 0023 log 7. 3677369_ Values of the general corrections. [22] =—0.770 | [29.] =+0.026 | [33] =+0.371 [22] =+40.750 | [29] =+0.153 | [33] ——0.303 [225] =+0.065 | [30] =+0.198 | [33,45] =—0.074 [233] =-+0.087 | [302] =—0.437 | [33,] +0. 224 [24] =-10.096 | [30:] =—0.286 | [33] ——0.591 [24.] =—0.371 | [30,] =+0.220 | [34] —+0.708 [245] =—0.036 | [305] =+0.033 | [34] 40.287 [25,] =—0.133 | [305] =—0,005 | [34:45] =—0,242 [252] =+0.161 | [30;] =+0.212 | [344] =-+40.050 [253] =+0.193 | [31] =+0.302 | [34,] —=-L0. 058 [26] =+0.024 | [31] =—0.199'! [35,] +0. 669 [26.] =+0.043 | [315] =--0.078 | [35.] = —0.210 [26,] =—0.108 | [31,] =—0.232 | [36] =—0.611 [271] =+40.278 | [31,5] =—0.455 | [36] =—0. 062 [272] =—0.357 | [316] =+0.424 | [36,] =—0. 185 [275] =—0.024 | [31,7] =4-0.362 | [36,] =+0.252 [274] =—0.158 | [3ls] =—0.026 [37,] =—0.003 [275] =+0.353 | [3%] =—0.365 | [37] =—0.153 [28%] =—0.157 | [32,] =—0.320 | [37,] =—0.684 [28] =+0.404 | [32] =+0.744 | [374] =+0.278 [28%] =+0.389 | [32] =—0.609 | [38,] =+0.001 [28] =—0.330 | [32,] =—0.441 | [38,] —~—0. 002 (29.] =+0.198 | [33] =+0.065 | [3%] =+0.001 § 5.) CHICAGO BASE TO OLNEY BASE. 583 Residuals resulting from substitution of general corrections in numerical equations of condition. see ik Residual. a ane Residual. | Ces : 36 0. 0000 55 0. 0000 37 | 0. 0000 56 —0. 0013 38 0. 0000 57 +0. 0001 i 39 +40. 0007 58 0. 0000 | 40 0. 0000 59 —0. 0001 41 ; 0. 0000 60 —0. 0001 a 0.0000 61 —0. 0001 43 0.0000 |. 62 —0. 0007 44 0. 0000 63 —0. 0001 45 0. 0000 64 +0. 0013 46 0. 0000 65 —0. 0001 47 l 0. 0000 66 +0. 0003 i 48 0. 0000 67 0. 0000 49 +0. 0001 68 —0. 0001 50 | 0. 0000 69 +40. 0004 51 +0. 0011 70 0. 0000 52 —0.0001 71 —0. 0005 53 0.0000 72 0. 0000 54 | 0.0007 | | ‘ PROBABLE ERRORS OF OBSERVED AND ADJUSTED ANGLES. § a. Let— m = whole number of observed angles in a section (one adjustment). r = whole number of rigid conditions in a section. n = number of triangles in principal chain. [pvv] = sum of weighted squares of corrections to observed angles. p; = probable error of an observed angle of weight unity. p, = probable error of an observed angle of average weight in whole section. p,/ = probable error of an adjusted angle of average weight in whole section. p, = average weight of an observed angle in whole section. p, = average weight of an observed angle in principal chain. p. = probable error of an observed angle of average weight in principal chain. p.! = probable error of an adjusted augle of average weight in principal chain. [vv] = sum of squares of closing errors of triangles in principal chain. p, = probable error of an observed angle in principal chain as derived from the closing errors of triangles. Proceeding as in Chapter XIV, C, § 8, there are found the following values: FOR THE ENTIRE SECTIONS OF THIS CHAPTER. Section. Extent of section. m r [pvv] | py Ps | Ps ER. so “ u" a“ XIV | Willow Springs-Morgan Park to Oakland-Kansas.-.| 125 | 83 30.18 | 0.41 | 0.96 | 0.42 | 0.58 | 0.24 XV | Oakland-Kansas to Denver-Parkersburg.....-.---.- 121 | 89 19. 69 | 0.32 | 0.95 | 0.33 | 0.51 | 0.17 584 PRIMARY TRIANGULATION. [Cuar. XX, D, FOR THE PRINCIPAL CHAIN CONNECTING THE CHICAGO AND OLNEY BASES, GIVEN IN D, § 6, FOLLOWING. From closing errors of triangles. Section. Extent of principal chain in each section. Pe Po Po! | [v7] aie |p Average | Greatest t error. error. | etc ! uw “a a“ uw a VII | Chicago Base to Willow Springs - Morgan Park ...... 0.63 | 0.57 | 0.35 2. 89 2 | 0.47 1.06 ° 1. 63 XIV | Willow Springs - Morgan Park to Oakland-Kansas...| 0.98 | 0.41 | 0.24 | 40,70 | 21 | 0.54 LL 2. 85 | XV | Oakland-Kansas to Olney Base........-...----------- 0.89 | 0.34 | 0.17 2.96 | 12 | 0.19 0. 34 1, 29 | | Entire principal chain ..........c000e0ceeecseee[oeceeelieeeeelecee ee 46.55/35 |0.45| 0.84 | 2.85 D.—PRINCIPAL CHAIN OF TRIANGLES BETWEEN CHICAGO AND OLNEY BASES. § 6.— The two principal chains of triangles connecting Chicago Base with Sandusky and Olney Bases diverge from the line Morgan Park— Willow Springs. In adjusting the sides of the chain joining Chicago and Olney Bases, it has been deemed sufficiently accurate to take this common line in place of Chicago Base as the terminal line, and use its adjusted length, derived from the chain joining Chicago and Sandusky Bases, Chapter X VII, D, §6. The last-named chain gives for the weighted mean logarithm of the line Morgan Park-~Willow Springs, expressed in feet 4.70203184 15.07, the probable error being in units of the seventh decimal place. _ With this loga- rithm and the angles of the triangles given in the following table, the logarithm of the Olney Base is found to be 4.3349234. The logarithm of the measured length of Olney Base, expressed in feet, is from Chapter XII, §5, 4.3349191. The discrepancy between this and the value derived by the triangulation is —43 units in the seventh place of decimals. The probable errors of observed angles of average weight in the chain are for the parts lying north and south of the line Kansas- Oakland, -—0’.41 and +0.34, respectively. (See Chapter XX, C, §5.) With these values and the values of o? and /, given ii the tables which follow, there results for the entire chain, using the notation of Chapter XIV, D, a (a+) p?= 2745. Considering Olney Base as exact, the constant for the system— 1, 4 97454 (15.07)?=2972. P PB The logarithmic discrepancy derived above is d= —43. From these data and the values of 5 given in the tables, the corrections to the logarithms of the several sides as computed from the line Morgan Park—Willow Springs are readily derived. The logarithm of the line Morgan Park—Willow Springs is, however, left unchanged. The arrangement of the tables is the same as that of the tables in Chapter XIV, D, to which reference may be made for a detailed explanation. The line in the system having the naximum probable error is Ash Grove-Spring Creek, for whieh 7 =1466 and r= 1006, giving for the probable error of the logarithm of the line, + 27.26 in units of the seventh decimal place. This probable error corresponds to the ysgsao part of the line’s length. 9 6.] CHICAGO BASE TO OLNEY BASE. Principal chain of triangles between Chicago and Olney Bases. Logarithms | } S (a2-+B2) | Weighted mean | . Errors of zarithm 1 Stations. Angles. Anes. eg, aides in | a? and B2 ip een es - | | °o - ua ut Fi | Orland ...2.2..2.22.02 weeeee 47 50 00. 040 ( 4.7020818 | 364.81 |.......202. we wees 4, 7020318 Morgan Park 2. ........2-- 56 04 39.416 | 40.6144! 4.7510696 |...2.. 2.2 [iecee cece ee ce eveees 4. 7510692 Willow Springs ............. 76 05 21.193 | 4. 8191715 27.04} 391.85 . 293 4. 8191711 ae, So ee Cretenescge cee tenance 40 37 23. 366 4.8191715 600.25 |.....e verses pagrus | 48191711 85 55 33.910 |S 40.0804) 5. 0044387 |.......... -) 5,0044881 53 27 03. 986 49104411 , 243.36 1235.46 | 436 | 4. 9104405 : oa ee Garden .........22-02000e000- 109 37 38. 643 | (/ 4.o1oqt | 56.25 Before dle? 4.910445 Or eee ee 34 22 30.693 | —o.684/) 4.6881854 |.......... Peele ; 4. 6681946 Orland .....2.00.0.cee0eeeeee 35 59 51.214 |} | 4.7056903 | 841. 00 ; 2132.71 | 587 4. 7036295 38 49 41. 651 1 4. 7056303 | 681.21 | a eS ete aoa 4, 7056295 59 34 41.400} 4 40.2384) 4. 8440404 |...0....-. Locee oun ato 4, 8440394 81 35 37.776 | 4. 9036808 9.61 2823.53 | 704 4. 9036798 | | | <= * i ae 90 01 34. 805 | (, 4.9036808 i | 4. 9036798 49 41 31.863 |S +0. 9115 4, 7859658 | .... .... i acrernee | renee 4. 7859646 40 16 54.077 J [| 4.7142796 | 615.04 3438.58 | 808 | 4. 7142784 i Pe eee Lata: 98 mare eesnleite gs Kankakee ........02.2002-0- 42 52 26.551 f #77 | 518.20 |... ee etL 4. 7142784 Grant....0.02.0ceceeeeeee eee 39 04 06.286 $ 42.6072, 4.6810387 |...02..22.| cee. eee eee | oslo’ 4. 6810324 Manteno.......2.22-2-22ee0- 98 03 27, 743 | 4, 8772140 9.00! 3962.87 | 896 | 4.877227 F pete oe St te oe et hee | ' St. Anne .. ce. eeeee.! GL 39 12.566 (4.872140 4. 8772127 Kankakee ...-.. 02222220005. 80 47 26.510 | 41. ay 49270517 | 4, 9270502 Grant...2...0200ceeeee cee 37 33 21. 840 | 4.717687 4. 7176842 if 1 Glifignteeate econ cecde ace 44 53 01. 124 4.7176857 | 445.21) ..2- ceeeeedeeeeeees 4, 7176842 St Apne 20.00 .ee eee weeeee 69 29 53.588} $ --1.429/| 4.2406689 | 2... ....|.22--2 eee fee eee 4, 8406673 Kankakee .. 65 37 04.065 4, 8285135 g2.16| 5378.69 | 1136 4, 8285119 Watseka ose eee 42 13 18.664 |) c| 4.e2estas | s4ag9 eee] eee 4. 8285119 Clifton 85 41 52.338 | 0. = 4.999917 |.... .... : 4. 9999150 St. Anne 52 04 50.251 | J 4,8981514 | 268.96, 6190.54 | 1272 4. 8981496 ! __! sug} gecaencee seuss Spring Ureek ....2...2.0000+ 52 25 00.970 |), {48081514 | 262.44 | acenenezea call 4. 8981496 Watseka.........22-2-020-- 87 19 51.025 |$ —2.260/' 49986078 |... ..-.. Ep sucdagnake lereunesl 4. 9986957 Clifton....2.2.2.2. -2.2eeee 4¢ 15 09.207 |) F 4.095070 | 615.04) 7068.02 | 1421 4. 8095049 ali Grove. scesy aeveceees 64 29 39.946 |} ( 4.8093070 | 102. v1 | Se teneecelll tae - 4, 8095049 Spring Creek 56 48 17.986 | +2.853/ 4. 7760067 |.......... Des sceadocie tt aA 4, 7766646 Watseka....2.2. ceceeeeeeee 58 42 02. 845 | J l 4.7857336 | 163.84 | 7333.87 | 1466 4. 7857315 eo Is sie rye cree Ser) tenet renee : Pasion ecw eee es 43 01 46.737 |) 4. 7857336 | 510.76 | .... 22.22. |eeeeeee: 4.7857315 Ash Grove......... -...--. 66 09 52,840 |$ +0.2342| 4.91v9987 [0.000 lle. sah es abe aa 4. 9129914 Spring Creek | 70 48 21.535 4. 9268709 53.29) 7897.92 | 1561 4. 9268686 t 2 £ = ae 7 ae, “a. t - 1 LIBRA eeeaeteima ne 74 42 07.877 (| 4.926709 | 32. 49 , eee yee 4, 9268686 Pixon. ccceansseaoawesact: 49 52 59.518 |$ —o. =) 4,8260474 |... ceceee ceececceeeesleeeeees 4 8260451 WeliGrove. Asees soceneseseis 55 24 53. 703 4, 8580877 | 210,25! 8140.66 | 1602 | 4,8580854 ig See hy one Han Loui ooe esteceaee eee 66 45 36.274 |) (| 4..8580877 81.00 LS OA Dr cake 4, 8580854 Butler ......--- 35 09 27.322 j —0 = 4,6351298 |........ peers be 4. 6561275 PUNE nieces “arated dees 78 04 57.135 4, 8853751 20.25 | g241.91 | 1619 | 4, 8853728 Se eA eran eae oo Sete eice et Wak ee ge oa Pilot Grovo...see0---2-+----| 63 17 22.618 (| * 8853751 | DONG cco tndas'd) dares 4, 8853728 Rantoul cocccs toueseeeeets 61 55 09.429 |' —1.582| 4.8799917 |... ..... [erste tees feseeee: 4. 8799893 Dttlens eielAn aes aces 54 47 29.073 | 5, 8466356 | 219.04} 8573.31 | 1675 4, 8466332 74LS 535 D86 PRIMARY TRIANGULATION. [Cnar. XX, D, Principal chain of triangles between Chicago and Olney Bases—Continued. 1 ' i RB r Logarithms | | : 1 Weighted mean : Stations. Angles. ; Mai. | of sides in a? and B2| & (a?+-f?) logarithms of ; : | erste feet. sides in fect. : | nt a ———— oO ¥' aw | wo May VICW cc an dews seta tae : 50 15 15.217 | l 48466356, 306:25 |scweenergens wecenes 4, 8466332 i 5 02 ‘L _o.goge: 4.8972061 |......2.2.| -2-. -.- ee ee 4, 8972036 | Pilot: Grove soscese cece cee 59 45 02. 354 f 0. 898 |: 4. 8972061 : : Rantoul, gccgosaid Saxecigecny 69 59 43. 657 | 4. 9337457 59. 29 8938. 85 1737 4. 9337432 I ete, Seteae te aor ee 4 et eee eee Fairmount. oi cssecscs0: sues | 69 51 55. 884 | f 4, 9337457 4. 9337432 [ Maywiew? cccases soasieedecices i 44 03 33. 039 +2. 459- 4, 8033669 4. 8033644 Pilot. Grow e .c2400e0c2cocds 00 | 66 04 32. 254 J | 4. 9221171 4. 9221146 Lynn Grove . 5 77 55 02. 230 | } | 4.9221171_ ; 20/25. sa. ecreeesiees lee se rate 4, 9221146 Fairmount w2 220550525. 022 202 38 21 02. 955 —0.6012' 4.7245699 jtteer oes ed 4. 7245673 EUNDAMMIOW oda mesents dapeediee | 63 43 55.751 | 4, 8845103 i 108. 16 9214.91 |° 1784. 4. 8845077 i I | : - Sa ile com ge ate See Sree rom a eee Coleg! SFE » Palerm@:.ccsecs oases assess 69 23 20. 403 | {| 4. 8845103 62: EL | ezeceeweeek poeweces 4. 8845077 ‘Lynn Grove ....2...206e+++ 42 20 08.198 § 41.965). 4.7415573 |... es eel eee eee ee fee eee (4, 7415547 yp Raa ON N Gis s.c2 e.cies be de sient 68 16 32. 326 J | 4, 8812423 70. 56 9347. 88 1806 4. 8812397 Oakland s<2e.cseseccnces as | 50 08 59. 318 l f 4. 8812423 | 309. 76 aaesed Seeaeeweest 4 4. 8812397 Palermo ........020-0+ 202-2 77 34.05.556 \ —0.770'| 4.9857345 [........-- eee eset or sees 4. 9857317 ’ Lynn Grove .....--2-...2+-5 52 16 56. 499 J | 4, 8942340 | 265. 69 9923.33 | 1903 | 4, 8942312 t k Wansalsneutaasd kenkd Sareea 2 ‘34 59 56.219 { 4. 8942340 906. 01 | ace yal. “ayeaisya [vernal aly 4, 8942312 Oaklands... nsec seess see © 105 25 07. 585 | +0. 4514 §.11973867 |....------ dioarisiisinjeed al aceaihe Re 5. 1197336 ' i Palermo ' 39 34 57.748 J l 4. 9399242 650.25 11479. 59 2166 4, 9399211 : I Westfield .............-2.225 66 32 43. 329 (| 4.9399242 | 82.81 .....2-.22-e fee eee | 4. 9399211 Kansas ...: 77 81 16. 143 —0. 014 ) 4.9669942 feces cece. cece temacceslseens ene 4. 9669909 Oa and os swiss creer swine na sais 35 56 01. 645 | 4. 7459034 846. 81 929. 62 2271 | “4, 7459001 Martinsville . 44 45 46.140 i 4. 7459034 AAD 44 | cesziscemeroiccaie a) prarae aria 4. 7459001 Westfield nc csiciecesesis ax aes 66 14 22. 264 —0. 421 4: 85075804 Jian ted tlaseorasete shaeasaacs 4. 8597546 FRANSES aceacedstee pesca ciel 68 59 52. 484 i 4. 8683697 64. 00 1443. 06 2329 4, 8683663 CRB OY ae ninicites die ciarcroretes cigzecte 59 56 02. 254 | { 4. 8683697 148. 84 Wcctebescaa ert tae 4, 8683663 Martinsville ..........-...-. 78 15 51. 495 —6. 334 4 OO1GGEL. | | oins cares es heeiccemaccthacn| camisetas 4, 9219506 Wreestfieldl : cccc6 ses ececnsicices 41 48 07. 221 | 4. 7549664 556. 96 2148. 86 2409 4. 7549629 i Belle Air saecewie sevececcns 71 47 41. 693 | 4. 7549664 APB cra Sok ts Sante ‘ on ae: 7 4. 7549629 CASON? oie dicen clears dicands 58 02 39. 805 +0, 297 4 MOSSE” rian viele tees! Bowie ee 4. 7058952 1 Martinsville ...--...---..-- 50 09 .89. 025 { 4. 6625491 309.76 | 2506. 23 © 2449. 4. 6625386 : | Bi Citys cc2cts2 hoyeareeeee 46 41 34. 867 | i 4. 6625421 9204 «|. 23 dc08 Scene ache tees 4. 6625386 | Belle Aire: set cevcees seceasn 66 58 14.740 |$ 40.2512 4.7645284 |..2..22. 2.) eee eee Ream 4. 7645248 | Casey... 66 20 10.970 J if 4. 7624528 84. 64 2982.91 . 2503 | 4. 7624492 | ODOR E 2-4 voce eee eqoasecvond 59 43 05.933 ( 4..7624528 TH1O9 se cee wens lige | 4,7624492 Hunt City 75 44 47. 060 —0.165/' 4, 8125824 le 4, 8125787 Belle Air ........0.00000.00- 44 32 07.630 : 4, 6720968 | 4. 6720931 | 44 48 48. 626 (| 4.6720968 4, 6720931 i 65 50 49. 981 l +0. a 4. 7842430 4, 7842392 69 20 22.024 |J 4.7951610 62.41 | 4104.01 | 2629 4. 7951572 Claremont .....22..-...2-2+ 39 43 53. 606 | (| 4.7951610 645.16 |....2.--2..- lees, 4,.7951572 | Mound. 222c2eeeezcesseeccees 105 39 36.137 |! +0.193)| 4.9731031 |......2.. |eeeeee eee _-. | 4,.9730990 p IOblon eta laawadecersaoncecees 34 36 81, 043 j | 4. 7438535 930.25 | 5679.42 | 2807 = 4. 7438494 7 gee OOF ap maeeietiee eh Pe ap ee gy Onion Hill ..........0--222-- 65 34 22. 518 | ( 4.438535 90.25 L222 eee eee piste 3 | 4. 7438494 Mound......-2---22002 -220 65 25 55.299 $ —1,293/. 4.7438668 |.2.. 022 eee eee ee eee 4. 7433627 Claremont .......2-. =. --+- 48 59 42.729 J | 4, 6623272 388.56 6108.23 2856 | 4, 6623231 § 6.) GHICAGO BASE TO OLNEY BASE. . 587 Prineipal chain of triangles between Chicago and Olney Bases—Continued. i oy i jo" tt ' Stations. Angles. es : eds ia a? and Bp? | Z (2+?) MS natinne oF fevt. | p =| sides in feet. _ <= ! Check Base ...--.. . .. -.../ 3 43 58. 026 ( 4. 6623272 TROY | easeateene (aiadse 4. 6623231 Onion Hill ...... Oe ches Gen 48 26 34. 537 a +0. | 4. 5541443 |.......... Rcsmetunanie eacenies 4. 5541401 | Mound.............25.-2-24. 57 49 27 766 4.6076572 | 176,89} 6322.33 | 2880 4. 6076539 1 ( ie = West Base ..... ...-. 22... ' 96 16 38.736 8 { 4. 6076572 5:00" |aslebecleneiasics| aagaeictis 4. 6076530 Check Base.............---. ' 39 57 14.497 | +0. 006 4.4179207 |........-.[.-222222225- Ladkeness 4. 4179165 Onion Hill...... ........--- 43 46 06, 941 j | 4. 4502164 484.00 | 6811.62 | 2935 44502122 | Fast Base. .-....-.. 83 35 40. 698 4 { 4, 4502164 4,4502122 West Base......- ele ts 46 45 34.253 |} +40. 240 4. 3153564 ue canoes 4, 3153521 Check Base ..... ...-.-.---- 49 38 45.156 | J | 4. 3349234 320.41 | 7137.79 | 2972 | 4. 3349191 . ‘ | 5838 PRIMARY TRIANGULATION. [Cuar. XXI, CHAPTER XX. TRIANGULATION NOT FORMING A PART OF THE MAIN SYSTEM. § I. Besides the main primary triangulation for which the adjusted angles and mean sides have already been given in Chapters XIV to XX, there are two lateral chains depending on these mean sides, and one independent triangulation depending on a base measured with the primary base-apparatus of Wiirdemann. The lateral chains are: One in the north end of Lake Michigan, extending from the line Door Bluff—Cedar River of the main triangulation in Green Bay to Spec- tacle Reef in Lake Huron; and the other in Lake Superior, extending from the line Vulcan-St. Ignace of the main triangulation east to station Mamainse. In both chains there are many angles not well measured. The independent triangulation is in Saginaw Bay. In these triangulations each triangle has been adjusted by making the sum of its angles equal to 180° plus the spherical excess, and for some stations, where sum-angles were read, a local adjust- ment was made. § 2. The triangulation extending from the line Door Bluff—Cedar River, in Green Bay, to Spectacle Reef, in Lake Huron, was measured in three sections. The section between Spectacle Reef and High Island had its angles read by Captain J. N. Macomb, between 1849 and 1855, with the 10-inch repeating theodolite Gambey No. 1, and the angles were well measured. The section between the line Hat Island— Pointe aux Chénes and Pointe aux Bees Sciés was measured in 1860 by Lieutenant J. L. K. Smith and Assistant James Carr, the theodolites used being Gambey No. 1 ane Wiirdemann No. 65. The section between the lines South Fox— North Manitou and Cedar River- Door Bluff was measured in 1864, the observers being O. B. Wheeler, O. N. Chaffee, G. E. Swin- scoe, and W. T. Casgrain. The theodolites were Gambey Nos. 1 and 2, Wiirdemann-Gambey No. 1, and Wiirdemann No. 65. ¢ A base-line, about 4 miles long, in the first section, was measured by Captain T. J. Lee, topo- graphical engineers, in 1854, on the south side of the Straits of Mackinac. (See Report of Chief of Topographical Engineers, U. S. A., for 1854.) Its length depended on that of the 15-feet brass bar of the Lake Survey, whose length at the time was not known with great precision. Wiirdemann had assigned a length and a coefficient of expansion to this bar, but nothing is known of the methods by which he determined them, and it is now known that the expansion assigned by hin, namely, 0.0017 per degree Fahrenheit, is largely in error. As the length of this bar has been changed since the measurement of the base by the insertion of agate plates in its ends, it is impos- sible now to determine its original length. Accordingly, the sides of the triangulation have been made to depend on the side Door Bluff-Cedar River of the main triangulation, both for length and azimuth, this being the nearest side for which identity between the old and new stations was certain. The azimuth Door Bluff—Cedar River is given in Chapter X XVII, § 3, as 126° 05! 24.45, and the logarithm of the length in feet (Chapter XV, D, § 8) as 4.9190836. § 3. In the following table, the first column gives the names of stations in groups of threes for each triangle; the second gives the date; the third the observer; the fourth gives the seconds of the mean observed angle at the station for that triangle; the fifth gives the adjusted angle; the sixth gives the logarithm of the side in feet which is opposite to the station on the same horizontal line. §p 1-3]. TRIANGULATION NOT INCLUDED IN MAIN SYSTEM. 589 Triangulation from Green Buy to Lake Huron. | 1 ! Seconds of Adjusted 4 | Name of station. Date. Observer. measured spherical ee hme c { angle. angle. See nee: . | * I “w oO wu | Boyer's Bluff....-. .. ...--- 1864 00. 60 55 57 01,14 4. 9190835 Cedar River 1864 08. 24 33 58 06. 56 4. 7479703 | Door Bluff ..-..-...-...---- 1864 55. 72 90 04 53. 40 5. 0007638 | 1 - | Bark River ...-....-..-.... 1864 05. 28 76 48 06. 08 5. 0007638 Boyer’s Bluff.......... aden 1864 58. 73 33 50 00. 63 4. 7580733 Cedar River -............... 1864 54. 99 | 69 21 54.55 4. 9835935 1 Burnt Bluff ....-.........--. 1864 O.N. Chatfee.......--- 23. 65 42 42 33. 60 4, 9835935 1 Bark River .......-..-...--. 1864 49. 33 52 37 49.15 5. 0524311 2 | Boyer’s Bluff. .....-.-...-.-- 1864 50. 89 84 39 49. 80 5. 1503232 1 Rock Island .......-......-- 1864 O. N. Chaffee. .-.....- 59. 47 76 48 58,43 5. 1503232 | Burnt Blof cscocceccce es eee: 1864 38. 05 58 14 37.01 | 5. 0914914 . Bark River ....-.-------.+-- 1864 26. 50 44 56 27.47 | — 5,0109587 i 1 s South FOC cecces soeeseavvens ' 1864 | O.B. Wheeler..... .. 58. 00 24 22 59.18 5. 0109514 t » Rock Island ...........-.-.- | 1864 | O. N. Chaffee.......... 00. 28 74 16 00. 44 5. 3786337 Burnt Bluff ....-....----.--. l 1864 | W. T. Casgrain ....... 04. 90 81 20 06. 08 5. 3901967 Northwest Manitou.......-. 1864 | G.E. Swinscoe......-. 57.75 91 02 58.39 5. 8901967 South Fox ......--.-22---..- 1864 | O. B. Wheeler......... 31.19 63 21 32. 42 5. 3415242 Rock Island ..........-----. 1864 | O.N. Chaffee........-- 34. 00 25 35 34. 68 5. 0257201 GRE OAS id diciarsscdvo par droinrs ears icasidacesaee | oi 5sizigigre ais eles Gis badtete, pinvsrela alligisanerelé wiesigsSle 58 41 40.44 5. 0257201 South Fox ......-.---------- 1864 | O.B. Wheeler ........ 32, 32 63 10 33.40 5. 0446113 Northwest Manitou ......-.. 1864 | G.E.Swinscoe........ 48. 50 58 07 48. 50 5. 0230893 North Manitou ...--..----.- 1860,’64; James Carr and O.B. 19. 27 50 48 19. 24 5. 0230893 Wheeler. | Cat Head ........-.- ----| 1860 | Lieut. J. L. K. Smith -.; 38. 50 77 38 33.59 5. 1236067 South Fox 22: -s. - 1st pair, Ap=—lgt+y 4,=—l,+y > 2d pair, &e., &e., &e., where lis the latitude resulting from one observation ona pair, and a, y, &¢., are the means of the latitudes from each pair. If there are m observations on n pairs, we shall have for the prob- able observation-error for that instrument (3) E,=0.6745. [ [44] m—n Chauvenet in his astronomy finds £,, in an example taken from the Coast Survey, equal to 0.30. The Coast-Survey Report for 1866 states that its usual value is below 0.50. Professor Gould, however, in the Coast-Survey Report for 1865, gives a long series of observations with an average of seventeen nights’ work on nineteen pairs, and finds Z)=0”.80. The following results have been obtained from the following Lake-Survey instruments: 32-inch zenith-telescope No. 12, about 15 years old, and 24-inch Pistor & Martins transit No. 1, twenty-three years old: 1.—ZENITH-TELESCOPE No. 12. Seacseitl apace Probable error of fs . a 0. 0 0. 0 servation in e Station. Date. Observer. results.| pairs. half difference of zenith distances. a“ North Base, Minnesota Point ..-.---..-. 1871 | G. Y. Wisner...... 55 23 0. 89 South Base, Minnesota Point .---..-..-. ABT Ws ccod Otsesevnsdseede 60 30 0. 78 Aminicon River .gseconcacesnsenseatsreu BST) ascescO: wissesseetiesserees 54 27 0. 62 Isle St. Ignace, south latitude post ..... 1871 | G. A. Marr......-. 73 83 0. 62 Cre@bassa sc: sacsicaniecncsceewiescmecs sn BEL. | Leni acme aicaics oyu 128 47 0. 54 South Base, Keweenaw Point........... 1871 | A.R.Flint........ 29 14 0. 95 DOOD sesccaduduotananecacns iweeneade 1872 | C.B.Conrstock....| 63 23 0. 69 SUNIS sens ste misannemcwscnessnusxoul scaveeue| thee etiekeceapepes 462 197 Probable error of observation in the whole series, 0.69. §3.J : LATITUDES. 621 . 2.—PISTOR & MARTINS TRANSIT No. 1. des cela ae Probable error of ob- . : o. of | No. 0 servation in the Station. Date. Observer. results.| pairs. half difference of zenith distances. “a Brulé RivePes:-:ascuxeseaveneewen cecees 1871 | G. A. Marr......-. 90 34 0. 99 Isle St. Ignace, north latitude post .....- AST) [s2ecdO weeseseccsccc| 119 52 1.19 DGt0ib caiscscwccwisistsveensanvaeyescaasis se 1872, | C.B.Comstock....| 79 30 0. 89 SUMS sccecetsweeciesecmeccecece isoleaicend hei eneawes Spacek 288 116 Probable error of observation in the whole series, 1/.05. Subsequently to the above determinations of E, for zenith-telescope No. 12, it was found that its micrometer-screw was not of uniform value throughout its length. Determining and using its correct variable value, the following results were subsequently obtained for it and for zenith-tele- scope No. 19: ae Station. Observer and date. me Ed aieie Eo | Mean Eo * a“ u 12 South Base, Keweenaw Point ...-...--. G. Y. Wisner, 1873 .... 51 4 0. 48 12 North Base, Sandy Creek............... G. Y. Wisner, 1874 .... 32 4 | 0.43 } ae 19 Minnesota Junction ........-..--.--.--- A. R. Flint, 1873....... 36 5 0. 41 l 19 ord! Rivers stecizis sceleca’e Mais donot sWejetececine! G. Y. Wisner, 1874 .... 34 4 0. 49 0. 43 19 Willow Springs soc ccnucccswosscacawuens A. R. Flint, 1874....... 33 4 0. 39 J ERRORS OF DECLINATION. § B. From (1) we have ; Ey (4) E}=E,—, If the errors of declination in a catalogue are accidental, we may find the probable error EH; for that catalogue by observing stars contained in the catalogue and finding, first, #, as follows: Observe many different pairs of stars an equal number of nights n at one or several stations, taking care that no star enters more than one pair, and call the mean of the results from a single pair an individual result; these individual results depending on equal numbers of observations will have equal weights. Take the differences 4’ between each individual result at a station, and the mean of them for that station. Suppose there are m/ individual results or pairs, and n/ stations, and for the same ° reason as in (3) we shall have Al (5) E,=0.6745 el as the probable error of the latitude resulting from x observations on one pair of stars. Substi- tuting the value of H, thus found and of n in (4), we have the value of E; for the catalogue used, and substituting it and the value of /, already found in (2), we have the combining weight to be given to a latitude result from one pair of stars of the given catalogue, when observed » times, Combining at a station the results from each pair with the weights thus determined, we have the final result for the latitude of the station. It will be noticed that all the 4’ in (5) are independent of each other, not being connected either by observation or declination error. This would not be the case if any star had been used at two stations or in two or more pairs at the same station; and then the ordinary method of least squares could not have been applied. If pairs had been observed an unequal number of times, the indi- vidual results would have had different weights which could not then be assigned, as Hs was as 622 ASTRONOMICAL DETERMINATIONS. (Cuap, XXIII yet not known. On applying the above method to 132 stars whose declinations are given in “Mean Declinations of'981 stars, 1875,” there is found £;= +0/.53. This value of Ls; gives 0.75 as the probable error of a declination in that catalogue. Another determination, in 1875, of Hs from latitude-work at stations Ford River, Minnesota Junction, and Willow Springs gave H;—0’.60. Taking for No. 12 the latest determination of EH), namely, +0/.45, and for Es, +0/’.60, there results for the probable error H, in the latitude derived from observing a single pair of stars once, E,= V (0".45)?-+ (0”.60)?= + 0.75 The determination of the values of one revolution at different parts of the micrometer-screw, and of its periodic inequality, is so easily made by observing transits near elongation of a close polar star across the horizontal, movable, micrometer-wire set in advance at every quarter-turn of the micrometer-head, that it should never be omitted when the highest precision is desired. The comparison of the mean value of the first quarter-turn derived from many turns of the screw, with similar values for the second, third, and fourth quarters, will indicate any large periodic inequality, while the comparison of the mean values of one turn derived from the first ten turns, with a similar value for the second, third, &c., ten turns, will show the irregularity of the screw if itis serious. As any errors in change of inclination of telescope are measured with the level, the values of one division of the level should be determined for different parts of the level-tube, with cither a level-trier or the nicrometer. If these differ much the separate values, and not their means, should be used. To get the best declinations practicable, Prof. T. H. Safford was authorized in 1872 to prepare alist, already referred to, whose title is “‘Mean Declinations of 981 Stars for January 1, 1875,” which has furnished the star places used in the computation of latitudes given in this chapter, unless otherwise specially stated. In giving the results of the latitude work connected with the triangulation of Lake Superior, to avoid occupying too much space, the separate results will not be given, except for South Base, Keweenaw Point. In Chapter XXVIII all the latitudes observed near primary triangulation sta- tions are collected to show the effect of local attractions. LATITUDE OF ST. IGNACE. § 4. Station St. Ignace, on the highest point of the island of that name, is 1263 feet above the level of Lake Superior. Its rocks are igneons, large masses of basalt being found in the immediate vicinity. The island is 15 miles long from east to west, seven miles wide, and the station is near the longitudinal axis of the island, at about 5 miles from its eastern end. To the south of the island Lake Superior deepens to 600 feet at a distance of fifty miles; on the north a shallow strait, from 6 to 10 miles wide, separates it from the main land, which rises as it recedes from the strait, in 3 or + miles, to hills 1500 feet in height. Tor further details Plate VI may be referred to, which gives outlines and contours that are, however, only roughly sketched, and shows the positions of the six points at which latitude was observed. From this description it will be seen that in the surface of the earth in the vicinity of the station there are considerable irregularities, and that the geological formation would indicate unequal densities. For both reasons the station is one at which deviation of the plumb-line might be anticipated. In order to form an idea of the amount of local deviation of the plumb-line, lati- tude determinations have been made at six points in the vicinity of the trigonometrical station St. Ignace. The resulting latitudes of the station St. Ignace are given in the following table, which also gives the rectangular coérdinates of the points at which latitude determinations were made, station St. Ignace being taken as the origin: $$ 4,5.] LATITUDES. 623 fable giving observed latitudes in vicinity of station St. Ignace. - Zenith : Coérdinates of post. Resulting Post. Observer. Date.| telescope Seg oe ee of | pop. |————__ spline - No. Latitude. | Departure. station. e Metres. Metres. SteTenaGe soo |ecesciecseesecesiccis'| ssckicg lveamecenst lecceawesile Sediase| sareteaaesvees | eeessean 000. 00 000. 00 No. 1...-- O. B. Wheeler ...| 1866 12 17 41 48 47 28.55 | +0.14 1S. 16.89 W. 28.71] 48 47 29.10 No. 2...-- G. Y. Wisner. ...} 1867 12 21 56 48 47 25.03 | +0.25| 8S 93.76 | W. 78.97] 48 47 28.07 No.3...-- acd saasiiwxcis'sl 1867 12 16 33 48 47 27.67} +0.18/S 16.98 |W. 28.56] 48 47 28. 22 No. 4...-- O. B. Wheeler -..| 1866 12 16 29 48 46 15.60 | +£0.11 | S. 2217.09 | EB. 3374.28] 48 47 27.35 No.4..--- G. A. Marr...... 1871 12 5 5 48 46 16.65 +0. 31 | S. 2217.09 | E. 3374.28} 48 47 28.40 No. 5....- ses O waciesiceeces 1871 12 25 40 48 44 52.00 | +0.15 | S. 4687.66 | W.1010.10] 48 47 23.76 12 No. 6..--- ---.do spetteeeees 1871 § P&M1 } 30 73 48 51 19.00 | +0.17 | N.7104.03 | E. 11.86] 48 47 29.00 One revolution of micrometer-screw of zenith-telescope No. 12 was 63.07 at 5th turn, and 63”.87 at 45th turn, the middle notch of comb- scale being taken as the 25th turn, and the value of a revolution was assumed to increase uniformly from the 5th turn to the 45th. One division of level equals 1”.00. One revolution of micrometer of Pistor & Martins No.1 was taken,as 85.226 and constant, and one division of level equals 2/.29. It will be seen that latitude posts Nos. 1, 2, and 3 were in the immediate vicinity of station St. Ignace, the most distant one being within 130 feet. Giving weights derived from their probable errors to the three results we have for their weighted mean 48° 17 28.65 which is adopted as the observed latitude of station St. Ignace. Post No. 5, nearly a mile south-southwest from St. Ignace, and about 1200 feet below it, being near the level of the lake, gives for the latitude of station St. Ignace a value 4’’.89 less than that adopted above. Latitude post No. 6, 44 miles north of station St. Ignace, gives for the latitude of station St. Ignace a value 0.35 greater than that adopted above. Post No. 5 is one at which a deflection of the plumb-line to the north would be expected, since immediately to its north the island St. Ignace rises 1200 feet above it, and at a distance of 15 miles the high mainland rises ; while to its south the depth of the lake becomes 500 feet within 10 miles, and for 60 miles averages 600 or 700 feet. LATITUDE OF SOUTH BASE, KEWEENAW POINT. § §. This station, near the head of Keweenaw Bay, has within a radius of 3 or 4 miles only moderate elevations, not exceeding two or three hundred feet, and as small depressions below the water surface in the lake. But a radius of 25 miles would include much of the central ridge of Keweenaw Point, which rises to a height of about 1000 feet and contains eruptive rocks; it would include a part of the Huron Mountains to the east, rising to about the same height, the high ground to the south of Keweenaw Bay, and a portion of the lake, out to the depth of 400 feet. Aside, then, from the eruptive character of a part of this region, which would give varying densities, the surface is itself quite irregular, so that a sensible deviation of the vertical from the normal to the mean ellipsoid is not improbable. The following table gives the individual results for the latitude of South Base, Keweenaw Point. The star numbers are those of the British Association Catalogue unless otherwise indicated, In determining weights the values H; =+0/.56, and H, =+ 0.48, have been used. The instrument used was Wiirdemann zenith-telescope No. 12. The value of one revolution of micrometer was taken to vary uniformly from 63.07 at the fifth turn to 63.87 at the 45th turn, the middle notch of comb-scale being the 25th turn. One division of level equals 1/.00. The instrument was mounted on a post, which was 25.3 feet distant, bearing north 89° 28’ east from station South Base. 624 ASTRONOMICAL DETERMINATIONS. Latitude of South Base, Keweenaw Point. S [Instrument, Wiirdemann zenith-telescope No. 12. Observer, G. Y. Wisner.] Resulting latitude. Pairs of stars observed. |~ Means. | Weights. Tul? 23, 1873. | July 27, 1873.| July 29, 1873. Sagas 3, fe} t 46 52 “a “we Ww wy “ 5568 HEBB! ||, sisscienamesais 20. 82 20. 79 20. 81 1.55 5568 BOG | 2 aceccecieawcs 21. 00 22. 01 21.51 1.55 5873 TUUD ledesiateeriayste sis 24.17 21.77 22. 74 2. 56 B Draco. BOGS il eccrcesateiass cieveransiy 23. 66 22. 59 22. 68 2. 56 6036 $9900 [ie ccisiceeicceiaeeslewe nes seece cee 21. 89 22. 27 2.33 6056: “Gr2908 Meas jesse de Sameer emenee 21.98 22.09 2. 33 6129 GROW? Nepoascienr ietetee 22. 89 23.19 23, 14 2. 28 R. C. 3820 GLO9. eremcoresetne Q8G9T — Vacsacseswis exisiston auatcateleeisers wiz 23. 47 41228 6246 6203 24.75 22. 03 22. 34 22. 07 22.79 2. 69 6258: «Gre 2568. Vecacescenessce 23. 74 23. 46 23. 20 23. 47 2. 56 6368 a Lyrae j....---------- 22, 07 22. 55 22. 68 22, 43 2. 56 6419 6404 23. 02 2. 56 6452 Gr. 2693 22.16 2. 56 6522. Gr. 2770 21. 40 2, 56 6626 Gr. 2844 21. 93 2. 56 XIX 193 6711 21.11 2. 56 6741 6731 23. 54 2. 56 6852 6851 21. 64 2. 56 6867 6915 23, 36 2. 33 6959 6963 22. 57 2. 33 7035 7022 22. 23 1. 82 70385 6986 21. 52 1. 36 7035 7048 21. 42 0. 92 7182 7241 21. 76 2. 33 7243 7253 23. 38 2.33 7278 XX 401 23. 25 2. 33 7294 7290 21. 54 2. 33 7294 7241 22. 98 1.84 7377 7399 22. 63 2. 33 7411 7503 22. 52 2. 56 7431 7501 22. 86 2. 33 7560 7544 22.17 2. 33 7589 7593 21.72 1. 84 Gr. 3601 7614 20. 36 1. 84 3680 7695 21, 22 1. 23 7727 7695 21. 64 1, 23 T7154 TTT 22. 34 2. 56 7824 - 7803 23. 33 1. 84 7855 7913 22. 51 1. 84 7882 7894 2178 nswdoweueseinss| a2 inecnmme tes |eemmages ened 21.78 1. 84 F855) Grss8ds | scse cee ce cedars dcr ecletiese 20. 92 20. 92 1. 84 Gr. 3901 Gr. 3867 22. 80 22, 80 1. 84 Gr. 3913 7948 22. 72 22. 72 1. 84 8036 7983 21.07 21. 83 2. 33 7995 Gr. 3947 22. 36 22. 36 1. 23 8082 8056 2279 Nia ceciaince siesta 22.79 1. 23 8082 8058 22048 leowsececcewes 22. 43 1. 23 8107 8118 2.9L” issdinscemeniss 21.91 1. 23 8107 $128 A182 Nl esicceud onsen 21, 82 1, 23 8231 8223: | |lpcmesamece sacs secacdneckenes 2284 | edits wisne cermndicie: 22. 84 1. 84 Gr. 4125 8245) || cece deccecsn| ssaiesccesnece 232716. lesaaaeerysess 23. 76 1. 84 Weighted mean of results Reduction to station Latitude of station of au _. 46 52 22.350 % 0.072 = 0. 002 -- 46 52 22.348 + 0.072 [Cnar. XXIII, §§ 6,7.) LATITUDES. 625 Hence we have for the latitude of South Base, Keweenaw Point, 46° 52 22".3540".07 NotE.—In a few cases a single star on one side of the zenith has been used in combination with two or three on the other side. The weights of the results, derived by (2) § 2, have in such cases been multiplied by the factor 3 ip n being the number of Stars on the one side of the zenith combined with one on the other. There are also a few cases in which a latitude result has been obtained by observing a single star, first as a N. star and immediately afterwards as aS. star, or in the reverse order. The weight assigned to such a result is the weight of an ordinary result 2 2 (derived by (2) § 2) multiplied by ae LATITUDE OF NORTH BASE, MINNESOTA POINT. § 6. This station is at the west end of Lake Superior. To the west and the north the ground soon rises to heights of 500 feet. To the south the country is more level, while to the east the lake deepens to 400 feet within 30 miles. Latitude was observed here with zenith-telescope No. 12, by Assistant G. Y. Wisner, in 1871, on three nights. ‘Twenty-three pairs of stars were observed and 46 results for latitude obtained. The value of one revolution of micrometer-screw was taken as increasing uniformly from 63.19 at 5th turn to 63.81 at 45th, the middle notch of comb-scale being the 25th turn. One division of level was equal to 1.00. The instrument was mounted on a post 389.2 feet distant and bearing north 63° 53/ 24” east from station North Base. Star places were taken from “ Declinations of 981 Stars, 1875,” and weights were determined as previously explained, the value of Ey being +0/.48, and of H;,+0.53. The resulting latitude of the post was 46° 45’ 30’.0140’.17, and applying the correction —1’.69, to reduce to the trigonometrical station, there results for latitude of North Base, Minnesota Point, 46° 45 28".3240'.17 LATITUDE OF VULCAN. § 7. This station is on the high promontory named Keweenaw Point, and is at an elevation of 726 feet above Lake Superior. To the north the lake deepens to 600 feet within 10 miles, and to the south it deepens to 300 feet within 18 miles. Observations for latitude were made at this station by Lieutenant James Mercur, Corps of Engineers, in August, 1867, the instrument used being Wiirdemann zenith-telescope No. 1, a small instrument, whose object-glass is 24 inches in diameter and focal length 24inches. One revolution of micrometer-screw was taken as 69.20, and one division of level as 1.08. Two observation- posts, differing 1/’.541 in latitude, were occupied. Post No. 1, 48.5 feet distant, and bearing north 83° 16’ west from station Vulcan, was occupied on August 9, 14, and 15,1867. Twenty-seven pairs of stars were observed, giving 41 results for latitude. Star places were taken from Safford’s Cat- alogue of 2018 stars, 1875.0. In weighting results, H) was taken as +1.35 and H; as +0”.53. The weighted mean latitude of Post No. 1 was.... ..--.----.-----.--- 47° 26! 44.871 40.27 Reduction to trigonometrical station Vulcan .-.....-... 62 eee ee 6,056 Latitude of Vulcan from Post No.1 ......----- +0250 foe eee eee 479 26' 44/.8154.0.27 At Post No. 2, 156.2 feet due south from Post No. 1, 27 pairs of stars were observed on July 24, 25, 26, August 2 and 4, 1867, giving 47 results for latitude. 47° 26) 42/771 40.32 1.485 47° 26° 44/,256-4.0'.32 The weighted mean latitude for Post No. 2 was ... ......-.....------- Reduction to trigonometrical station Vulcan ........-.-..-------+. ++. + Latitude of Vulcan from Post No. 2......----.--------- ecules See ete aie Weighting the results according to their probable errors there results finally for latitude of tation Vulcan, ; are 47° 26’ 44”,.58240".20 79 LS 626 ASTRONOMICAL DETERMINATIONS. (Cuar. XXIII, LATITUDE OF HURON MOUNTAINS. § 8. This station is on the south shore of Lake Superior, about 35 miles northwest from Mar- quette. The shore rises rapidly to hills of a thousand feet in height, while to the north the lake reaches a depth of 300 feet within 15 miles. Observations for latitude were made at a point 97.5 feet distant and bearing south 36° 30/ west from the station, which is 932 feet above the lake, by Mr. S. W. Robinson, in September, 1866. The instrument used was the Pistor & Martins broken transit previously described, one revolution of micrometer being 85/’.22 and one division of level being 2/.29. Observations were made on 32 pairs of stars on September 20, 21, 24, and 25, 1866, but out of the 54 results for lati- tude all but 7 were on the last two nights. The star places were taken from Safford’s Catalogue of 2018 Stars for 1875.0. The results from each pair were weighted by using the values y= +1/.05 and H;=+0".57, although the latter is probably too large, giving too small weights. The resulting latitude of the observing post is 46° 52/ 52’.304+0”.13. Applying correction to trigonometrical point, + 0.77, there results for latitude of trigonometrical station Huron Mountains, 46° 32’ 33”".0710".13 LATITUDE OF FORD RIVER. § 9. Ford River station is situated on the west shore of Green Bay, about +4 miles south of Escanaba. In its immediate vicinity there is a thin surface-soil underlaid by limestone rock, whose strata are nearly horizontal. To the southwest, west, and north, within fifty miles, there are no marked surface irregularities. On the east and south, however, lies Green Bay, 15 to 20 miles wide and 40 to 120 feet deep, and within fifty miles the broken peninsula lying between Green Bay and Lake Michigan rises at intervals to a height of 200 feet and Lake Michigan attains a depth of 400 to 500 feet. Latitude was observed by Assistant Engineer G. Y. Wisner, with Wiirdemann zenith-telescope No. 19, the instrument being mounted on a heavy wooden post 29.5 feet distant, bearing south 76° 00/ west from the station. One revolution of micrometer was taken as 62” .241, and one division of level as 0.86. Observations were made on four different nights, in July, 1874, as shown by the table following. Thirty-four pairs of stars were observed, giving in all 115 individual results. Latitude of Ford River. [Instrument, Wiirdemann zenith-telescope No. 19. Observer, G. Y. Wisner. ] Resulting latitude. ait OF reyes ob- - | Means. | Weights. July 3, 1874. | July 4, 1874. | July 7, 1874. | July 8, 1874. | o- 45 41 uw a“ “aw “aw “ XV 39 5076 06. 92 06. 64 OG. 96) jaccedmscssres 06.84 | 3.08 5113 BIDE. ress seee esr 03. 21 02, 82 amis eacakes 03. 02 2.74 5168 5181 05. 58 05. 49 07. 65 05. 74 06.11 3. 28 5210 5295 06. 46 06. 86 05. 48 05. 28 06. 02 3. 28 5338 5388 02. 49 04. 50 04. 29 05. 32 04.15 3. 28 5463 5497 04. 99 05, 27 04, 55 04. 80 04. 90 3. 28 5523 5535 07.19 07. 20 07. 38 07, 35 07. 28 3. 28 5546 5574 04. 13 Hts eno arera vaio oon 01.70 04. 71 03. 51 3. 08 5596 5644 04. 51 05. 20 04.79 05. 58 05. 02 3. 28 5790 5795 05. 02 05. 49 06. 06 05. 93 05. 62 3. 28 5929 5937 05. 72 04. 69 04. 27 03. 98 04. 67 3. 28 5997 6036 0474: lesswsiemasursiy 04. 56 05. 21 04. 84 3. 08 | 6068 6091 05. 88 05. 23 05. 45 05, 18 OR 44 3. 28 ' 2494 6227 03. 31 02. 76 05. 61 04. 90 04, 15 3. 28 6109 6264 07. 06 QS 5S Mase esses beh aara 06. 86 06. 50 3. 08 §§ 8-10. | LATITUDES. 627 Latitude of Ford River—Continued. I \ : Resulting latitude. | | oh ol: : Means. | Weights.. | July 3, 1874. | July 4, 1874. | July 7, 1874. | July 8, 1874. | a wa Wt “ a | 6203 6252 03. 53 03. 98 03. 63 05. 03 04, 04 3. 28 6268 6335 05. 09 06. 22 05. 06 05. 25 05. 41 3. 28 ais 6350 6355 07.16 | 06. 72 05. 66 : 05. 14 06. 14 3. 28 6372 i fa Led eemrenmat ea Mme tenreats i 05.50 05.50 | 2.07 | 6522 6556 04. 65 05. 47 04. 68 04. 97 04. 94 3. 28 | 6599 6623 05, 21 04. 65 05. 13 05. 24 05. 06 3. 28 6651 XIX 193 |....-......-.. 04. 96 05.60 05. 84 05. 47 3. 08 : 6721 6728 OB.67 lace emceceeesas 06. 20 06. 59 06. 49 3. 08 6741 6745 05. 99 05. 68 06. 38 | 05. 34 05. 85 3, 28 | 6768 6769 03.12 O08 kenteevecoeics ; 06.02 04.92 | 3,08 6806 6824 06. 43 05. 05 05. 31 04. 58 05. 34 3. 28 | 6851 XTX 370 05. 80 05. 92 04. 66 05. 98 05. 59 3. 28 6915 6924 lncetencrees es 03. 40 03. 56 04. 36 03. 77 3. 08 6965 BTLO: ascwslensrste sreidcieea 06. 40 05. 88 06. 12 06. 13 3. 08 6997 M085 PM seeweryemcamsicie 04. 42 03. 72 04. 76 04. 30 3. 08 7055 MOBS Miia apraophiccmeer 04. 50 05. 51 05. 15 05. 05 3. 08 7120 W108 4 senidemegisemed 05. 16 05. 94 Bo nacseniyanaes 05. 55 2.74 7171 NY98:> ‘letwgesseeean 06. 08 06. 14 05. 84 06. 02 3. 08 7254 C268: is eeametccwuems 05. 65 O4TL | exeveceseeesery 05. 18 2.74 ° ‘ “a “a Weighted mean of results.... -....0 - 20.22 eee ee eee eee cee eee 45 41 05. 265+0. 110 Reduction to station cocscass cavceseasewrdsacencane nesenecens aoe we + 0. O71 Latitude of station. ..- 12.02.02. 22 cece ee cece eee eee seen 45 41 05.386+0. 110 These results combined in the manner explained in §§ 2, 3, the values Ey, =+0".487 and H;— +0”.496 being derived from the observations, give 45° 41’ 05/.265407.110 for the latitude of the observing-post. Adding to this value +0/.071, to reduce to the triangulation station, there results for the latitude of Ford River station, 45° 4 05".3410".11 LATITUDE AT FORT HOWARD. § 10. Station Fort Howard is situated near the southern extremity of Green Bay, in Wis- consin. There is a limestone ledge about 4 miles southeast of the observing-post, which has a general direction of northeast by north, and an elevation of 50 to 200 feet above the bay. Between the astronomical post and this ledge, and for 25 miles or more to the west and north, the country is flat, rising nowhere higher than perhaps 25 feet above the bay. The ground at the post is about 20 feet above the bay. : Observations for latitude were made at a point nearly two miles northeast of the station, by James Carr, assistant engineer, in October, 1862, on seven nights, with zenith-telescope Wiirde- mann No. 1, having a focal length of 24 inches and an aperture of 23 inches. One revolution of . the micrometer-screw was equal to 69.14, and one division of the level to 1.08. The instrument was mounted on a wooden post 21 inches in diameter, set 4 feet deep in sandy soil, which was made firm by wetting the sand as it was filled in round the post. The number of pairs of stars observed was 30, and 77 individual results for latitude were obtained. In reducing the observations, declinations from Safford’s Catalogue of 981 Stars (Washington, Government Printing Office, 1873) were used. Weights for the results from each pair of stars were assigned according to the method described in §§ 2 and 3, and note added to § 5, with y= +1”.92 and 2; =+0”.53. The value of 2 was derived from the observations at this station. The following table gives the pairs of stars observed, the individual results, the mean result for each pair, and the weights assigned to each pair. The resulting latitude of the observing- post is 44° 31’ 187.624 07.10. 628 ASTRONOMICAL DETERMINATIONS. [Cuar. XXIII, The position of the observing-post, referred to station and other marks, is as follows: 8465.5 feet distant, bearing north 54° 44’ 43” east from station Fort: Howard; north, 2798.7 feet. west, 775.4 feet from court-house, Green Bay (center of cupola); 272.8 feet due north of a stone meridian- post; and 1612.6 feet due south of another stone meridian-post. Applying a correction of —48/.24 to the above result to reduce to the trigonometric station, there results for the latitude of station Fort Howard, 44° 30’ 30".28+0".10 Latitude at Fort Howard. (Observer, James Carr. Instrument, Wiirdemann zenith-telescope No. 1.] Resulting latitude. Pairs of stars with British Association | _| Means. | Weights. Catalogue numbers. | October 11, , October 14, October 16, | October 17, | October 19, | October 21, | October 22, 1862. 1862. 1862. 1862. 1862. 1862. 1862. ov “ 6865 6813 18. 23 0. 25 6928 6937 17. 06 0. 25 7076 W114 16. 34 0.44 7085 7114 : 16. 44 0. 44 7166 7204 * 17.99 0.47 . 7233 7241 18, 20 0. 83 7254 7273 17. 45 0. 25 7278 7313 18. 07 0. 44 7294 7313 18. 29 0. 55 7365 7373 18, 87 0. 66 477 7501 16, 65 0. 25 7560 7524 17. 37 0. 55 7560 7602 19. 89 0. 44 7746 7765 19. 92 0. 83 7787 T7717 17. 60 0. 66 7824 7901 15. 21 .0.81 7888 7901 17, 87 0. 65 7847 XXII 113 15.75 0.31 7848 =X XIT 113 16. 63 0. 44 7999 7917 21, 85 0.12 7999 7932. 22. 49 I 22. 49 0.12 7999 7962 22. 55 18. 65 0. 33 Ll 44750 1982 Vieseuentias sel beeen oxicuus [ee denieeeeses 18:48 |e cacee ony [acsearle eae dats 18. 42 18. 43 0.31 8118 BIDS: liccsieicissieiesseisjei| Scsorwtaarsewmee g lascaleveerterd ares, 16. 08 ABB: jes scctreccas 19.11 17.79 0. 66 8224 8237 TES” \esenansenzcus| Conamciemmane 24.76 MOTD: | yeeeecesiciesc 16. 21 18.74 0. 83 92 DOr aiekiackeresie ica ee cians oe cl Gece sreueeeiese (Meee meas 22. 37 5 2s 17.11 19. 74 0.47 169 155 18. 01 WBE8T +l sgreciesse es 15. 21 17. 20 0. 66 314 334 20.61 |exewesearese (Asie sreisiciond are 25. 81 23. 21 0.47 441 425 20560 |acs 0 aoe etck [nce Secret 24, 72 22.66 | 0.47 558 BUG) sidsatisec teal Madensaaccerl maces ocean BOGE) loscasauhksaven teneeercmaenis 22, 33 22. 51 0. 47 Oo -# a“ aw Weel shied means. ache iinavitndetcdhilers de cme aask Sulen minth een ats eel mon at ahha theresa tts 44 31 18.5234 0. 103 Reduction to trigonometrical station ..0..0c5ssasese gegesxt uccexsasenee/eees seeeceesaseexeeese _ 48, 24 Tiatiuder ok Rove Howard 22sec aa esaeeccnecndatcun ted sadducusuvataucutceascaccens ~~ 44 80 30, 283 0. 108 LATITUDE OF MINNESOTA JUNCTION. § EA. Minnesota Junction station is situated near the center of Dodge County, Wisconsin, on a mound rising about 40 feet above the general surface of the land, which, within a radius of 5 miles, is slightly rolling prairie. On the south, west, and north this prairie extends upwards of 20 miles. On the east the surface is more irregular. At a distance of 10 miles the limestone ledge extending approximately northeast by north one-fourth north, across the State of Wisconsin, rises to a height 120 feet greater than that at the station, while between the latter and the ledge the basin of Horicon Lake forms a depression of 250 feet. Between the ledge and Lake Michigan the surface is broken by numerous hills of drift origin. The mound at the station is 416 feet above §11.J LATITUDES. 629 the surface of Lake Michigan, which attains a depth of 350 feet within 60 miles of the station, the shore of the lake being about 45 miles from the station. The latitude was observed by Assistant Engineer A. R. Flint, with Wiirdemann zenith-tele- scope No. 19, on five different nights, during August and September, 1873, as shown in the table following. The instrument was mounted on «a heavy oak post. There were obtained in all 156 individual results, 36 different pairs of stars being observed. One revolution of the micrometer was taken as 62/251, and one division of level as 0.86. Star places were taken from “ Mean Declinations of 981 Stars for January 1, 1875.” . Latitude of Minnesota Junction. {Instrument, Wiirdemann zenith-telescope No. 19. Observer, A. R. Flint.] Resulting latitude. CC ob ; Means. | Weights. August 26, | August 27, | August 28, | August 29, | September 1, 1873. 1873. 1873. 1873. 1873. Oo Ff 43 28 a “a “a aw a“ aw 6095 6162. |jecsen cen geass 33. 43 33. 03 33. 38 33.77 33. 40 2.74 Gr. 2597 6364. | eeceseeaves veliescxseeeceees 33. 89 82. 99 32. 95 33. 28 2. 64 6392 B42. foes ssleeenries 32. 54 31. 72 32. 81 3L. 71 32. 20 2.74 Gr. 2701 6475 33. 92 32. 86 32.19 B12 920 |sceiciasecle se, 32. 72 2.74 6495 6516 31. 08 30. 60 30. 20 31. 37 30. 37 30. 72 2. 81 6599 6626 29. 60 29. 97 28. 57 29.75 29. 37 29. 45 2. 81 6687 6698 30. 48 31. 59 BANAT: Wee ice cena dec, 31, 24 31. 48 2.74 6720 6728 31. 28 31. 39 30. 81 30. 70 30. 84 31. 00 2. 81 6754 6769 32.19 82, 49 29. 15 32.17 31. 96 31. 59 2. 81 6817 Gr. 2957 |...-...-....-. 30. 35 30. 10 30. 19 31. 03 30. 42 2.74 6851 6856 33. 52 32. 43 32. 50 32.16 32. 57 32. 64 2. 81 6881 6915 31. 37 BOWGT (etic ctetipey Stee 30. 65 31. 98 31.17 2.74 6962 6996 31. 62 32. 79 32. 53 82:30. |eswececeeeeczd 32. 31 2.74 » 7061 WOT6. ierossaintetortoe: 33. 30 33:11 “82. 08 32. 20 32. 67 2.74 7112 T158 ts vasexceessesic 32.14 33. 15 81. 72 32, 40 32. 35 2.74 7260 7268 30. 67 30. 40 30. 64 30. 51 30. 72 30. 59 2. 81 7274 7820 | eccinsimacacie 33. 05 32, 89 32. 94 32. 44 32, 83 2.74 * 7333 TAO! |. onion Aemrerciee 32.11 31. 76 31, 20 31. 85 31. 73 2.74 7548 TOOG) |122 Sre,2 Soccesiee 31. 04 31. 89 32. 30 31. 57 31.70 2.74 7598 FOOD, | eraser cis tect 31,15 31. 60 32. 02 31. 08 31. 46 2.74 7679 TIOD |, eres vows ean 33.31 33. 66 33. 56 33. 00 33, 38 2.74 Gr. 3717 7770 29. 74 30. 44 81,14 31. 33 30, 45 30. 62 2. 81 7850 7894 31. 28 31. 40 31.40 31. 53 31. 80 31.48 2, 81 7932 7950: | csnvtvcsatnce® ! 33. 13 32. 63 | 32. 94 33. 08 32. 94 2.74 7972 OBS} |aeacie Sacmacins 32. 89 32. 27 82. 42 30. 87 32.11 2.74 8023 S056) sceeh5 oseesenicein 30. 80 30. 30 29. 98 30. 32 30. 35 2. 74 8076 8110! | seen sewees cet 32. 10 31. 28 31, 48 32.75 31. 90 2.74 Gr. 4052 Gr. 4074 32.70 33. 28 8207 nc eenenae st 33. 59 32, 91 2.74 8223 8237 33. 29 32. 33 30. 68 33. 29 31. 05 32.11 2. 81 8345 16 31. 41 30. 78 31. 40 30. 49 30. 74 30. 96 2. 81 60 78 B2I000 | lsedsaree oat csi mane aaSslnadneved BB OOF Nese ain iniaaincie sie 82. 91 2. 44 189 227 33. 13 32. 46 33. 49 33. 45 33. 32 33.17 2. 81 283 330 30. 48 30. 29 31, 42 31,13 30. 45 30. 75 2. 81 352 377 32. 70 31.91 31. 23 32, 49 32. 38 32. 14 2. 81 441 480 31. 91 32. 31 31, 89 30, 95 32. 01 31. 83 2. 81 560 579 31.17 30. 24 29. 79 31. 37 30. 67 30. 65 2. 81 Weighted mean of results...-...---------- +--+ 220 see eee cee eect teeter eee 43° 28/ 31.82 4+ 0.11 From the observations, the value +0/.425 was found for Mp (see §§ 2, 3) and Hs = -+£0/.565 was derived from the observations at both Ford River and Minnesota Junction stations. The combining weights of the individual results being deduced from these values of /, and H,, there results for the latitude of Minnesota Junction station, the instrument being centered vertically over the triangulation station, 43° 28’ 31".8240".11 630 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIII, LATITUDE OF WILLOW SPRINGS. § 12. This station is situated in Cook County, Illinois, about 1 mile southeast of Mount Forest, a railway station on the Chicago and Alton Railway, and is about 16 miles southwest of Chicago. The ground at the station is about 150 feet above the surface of Lake Michigan, and is considerably undulating in the vicinity. To the northward from station Willow Springs the ground declines until it reaches a river and canal about 2 miles distant, where it is about 150 feet below the station. Continuing northward, the ground rises 30 feet or so above the river, and remains at about this elevation for many miles. To the southward from the station the land for a mile or so has about the same elevation as at the station, then there is a depression of from 50 to 100 feet, alter which the ground gradually rises until at a distance of 10 miles it reaches the level of the station, and continues at about this level farther south. Latitude was observed here by Assistant Engineer A. R. Flint in September and October, 1874, on five nights, with the Wiirdemann zenith-telescope No. 19, having a focal length of 32 inches and a 3-inch object-glass. One revolution of the micrometer-screw was equal to 62.251, and one division of the level to 0.861. The instrument rested on a solid oak stump, about 3 feet in diam- eter, around which a platform was built. It was situated 38.4 feet distant, bearing north 22° 50’ west from the station. The number of pairs of stars observed was 33, and 132 individual results for latitude were obtained. For reducing the work, declinations were taken from Safford’s Catalogue of 981 Stars. Weights were assigned to the results for each pair of stars according to the adopted method (§§ 2, 3) with Ey=+0".426 and Bs = +0/.57. The following table gives in the successive columns the British Association Catalogue numbers of the stars observed, those forming a pair being placed on the same horizontal line; the individ- ual results for each pair on the separate nights; the mean result for each pair; and the weights assigned to those means. The resulting latitude of the observing-post is 41° 43/ 38/.97940/".117, and applying a correc- tion of —0.350 to reduce to the trigonometrical station, there results for the latitude of station Willow Springs, 48° 42’ 38".629140".117 Latitude of Willow Springs. (Observer, A. R. Flint. Instrument, Wiirdemann zenith-telescope No. 19.] | alan ass Ge: | Segoe | Repeat | September | etaber ty | Octeherh premus, arate | 7 a | oO dk 41 43 | aw aw uw" uw uw u ' 6470 40. 48 SOB fo seatecemstoe 40.10 | 1.30 6530 39. 63 39.97 39.80 1.22 | 1656: 66H |aseecmenacezee) BOMT | eviedezeeerces 40.15 go.71 | 1.22 6717 40. 33 40. 20 40.28 | 1.30 6764 39. 08 38.17 - 38.36) 1.30 6865 38. 90 BBN062 “WaSedit aah aac i 88.22 | 1,30 | 6962 40. 30 40. 58 40.14 | 40,05 (1.37 | 6983 36. 95 36. 94 36.31 | 37.19 | 1.37 7083 38. 54 39.75 39. 76 39. 35 1.30 7091 37.81 37.41 36. 70 37. 31 1.30 7120 39. 08 39. 46 40.25 39. 37 1.39 7182 38. 16 37. 59 39. 38 38. 34 1.37 Tal P60 SOO.) BR BO ede seceelie 37. 24 37.95 37.79 1.37 7290 7207 37.16 36.91 36. $2 37.28 37. 34 37. 00 1.39 A 7345 7378 | 39. 64 39.10 39. 62 40. 20 40. 10 39.73 1.39 7411 7399 | 38. 83 89.53 | 89.58 39. 81 39. 62 39.47 1.39 §§ 12,13.) LATITUDES. 631 Latitude of Willow Springs—Continued. Pansorved | RIB | Sahiera | share | OC tg | Otten & | Means. | Weights, “ " “ ” “" uw : 7488 7465 37. 73 38. 37 38. 65 38.56 |lsnceuwcacweses 38. 33 37. | 7501 7505 ST. BS) cjessinsietsines aie.s 39. 75 38. 87 37.75 38. 56 1.37) | T544 7559 38. 92 38. 30 39. 30 38. 49 38. 70 38. 74 1.39 7593 7565 38. 65 38, 28 39, 69 O828i- eeeoameakanad 38.71 1.37 7681 7614 40. 31 40. 64 40. 04 40.12 40, 62 40. 35 1.39 7800 7770 39. 13 39, 63 40. 11 39. 92 40. 69 39. 90 1.39 7815 7843 37, 80 39. 76 38, 12 38. 60 39. 00 38, 66 1.39 7894 7858 38. 33 39. 87 38, 05 38. 47 38. 88 38. 72 1.39 7948 7915 39. 92 39. 78 39. 67 39. 28 39.13 39. 56 1.39 7972 7962 37. 33 38. 51 87. 85 BT OS Weawie wee aes 37, 92 1. 37 8028 7994 40.79 39.10 39. 15 39.17 38. 90 39. 42 1.39 8107 8141 37,12 37. 87 37, 58 37, 83 37. 20 37, 52 1.39 8171 Gr. 4052 40, 28 40. 50 40507 laseswseatercos|seees sccemeeee 40, 28 1.30 8223 8212 39. 69 40. 47 39. 95 40. 61 40. 00 40.14 1.39 Gr. 4172 8345 38. 97 38. 58 37. 31 88.382: | eecsccenecwces 38. 30 1,37 16 52 38.15 39. 47 89. 45 39-69) Weioess Seeesmede 39.19 1.37 165 TDS) | icccisrssre 40. 88 BOSTO? — Wrcezercaelsscincea dicuelfieisuers Sassereiarsigete 40. 29 1, 22 oO t “a ve Weighted Meanis<.ccccccc sens soe yese sccm eeg ys yee esedeeed peeeds eoseeneagasatoaeet 41 43 38. 979+0. 117 RedUCtiONn: tO SATION: <2; nj 2s) aces sets se dicisiaicinien ddeeuk ue scetieecedeeetemd cee _ 0. 350 Latitude of Willow Springs || Snes sooxeels 37.81 38. 19 1.30 394 410 34. 47 35.50 36. 33 35. 43 1.30 422 429 35. 86 37. 55 36, 52 36. 64 1.30 431 443 37. 20 37. 92 36. 90 | 87. 34 1.30 464 475 |..eee eee eee 37. 60 37.65, 37. 62 1.21 471 482 37. 65 38, 23 37.79 37. 89 1.30 494 503 37. 84 38.29 37.00 37.71 1.30 518 521 36. 06 37.01 36.15 36. 41 1.30 527 537 38. 84 39. 31 37.14 38. 26 1.30 549: 1662 i gzssneacd oes 36. 93 37. 08 37. 00 1.21 570 585 36. 21 BTBo: seen ae tts 36. 69 36. 74 1.30 604 620 36. 09 87.08 | sens execs 35. 98 26. 47 1.300. 681 682.0 | sssees eecesae Bda7H) || shoe cee: Geos 35.47 35. 63 128R 3 635 636 |... eee ass BERS (emegeence een 37. 07 36. 66 ier) | — evens oe | oO t aw “a Woigbted mean.....-.--...----- +. 2-22 cee eee eee ee reece eee 40 01 36. 737+0. 108 = Correction to stationwcsccc. 22. seven oceceus coccwies cauleceescas -- 0. 033 Latitude of Fairmount.......-.--..----------++------++- 40 01 36. 704 £0. 108 LATITUDE OF WEST BASE, OLNEY BASE-LINE. § 14. This station, forming the western extremity of the Olney base-line, is situated in Jasper County, Ilinois, about 9 miles north of the town of Olney. The land in the vicinity of the station, and for several miles to both the north and the south, is very nearly level, and for 20 miles in. every direction it probably does not vary 100 feet in elevation from that of this station. Latitude was observed here in May, 1880, by Assistant Engineer G. Y. Wisner on four nights with the Wiirdemann zenith-telescope No. 19, having a foval length of 32 inches and a 3-inch objective. A revolution of the micrometer-screw was equal to 62.224, and one division of the level to 1/.292. The observations were made on a post of white oak, 17 inches in diameter, and set 54 feet in the ground, with the earth well tamped around it. This post was situated 53.5 feet distant, bear- ing north 88° 36’ east from the station. The number of pairs of stars observed was 30, and 115 individual results for latitude were obtained. Declinatious were taken from Safford’s Catalogue of 2018 Stars, and weights were assigned to the results from the different pairs according to the adopted method (§§ 2, 3) with Ay=+0.42 and Hs; =+ 0.57. §§ 14, 15.] LATITUDES. 633 The following table gives in the successive columns the British Association Catalogue numbers of the stars observed, with those forming a pair placed on the same horizontal line, the individual results for each pair on the separate nights, the mean results for each pair, and the weights assigned to those means. The resulting latitude for the observing post is 38° 51/ 41/’.216+ 0.064, and applying a correc- tion of +0/.013 to reduce to the trigonometrical station, there results for the latitude of station West Base 38° SL’ 417.2294 0".064 Latitude of West Base, Olney. [Observer, G. Y. Wisner. Instrument, Wiirdemann zenith-telescopo No. 19.] ae stars | May 13, 1880. | May 14, 1880.| May 15, 1880. | May 16,1880.) Means. | Weights. So # 38 51 “ a“ “ a“ “ 10 19 40. 42 39.77 41. 88 40. 44 40. 628 1.35 33 29 41. 58 40. 56 41, 89 40.13 41. 040 1.35 40 50 40. 54 39. 95 41.18 42.00 40. 918 1.35 3 67 66 41. 64 40. 39 41.44 41.53 41. 250 1.35 92 97 42.14 40.26 41. 84 42. 02 41. 565 1.35 125 114 41.10 41. 68 40. 83 40. 05 40. 902 1.35 135 132 41.39 40. 89 41.98 41. 69 41. 489 1.35 145 153 40. 96 40. 86 40. 51 41. 46 40. 948 1.35 183 170 41. 38 40. 06 40.50 |eeeeeeeeeeeees 40. 647 1.30 208 217 42, 82 41. 32 41. 66 41.76 41. 890 1.35 OOF OO Ne 2as curenrth e clecerech een 42, 24 41.70 41.970 L21 252 257 41.73 41. 85 41. 86 41. 68 41.780 1.35 267 275 40. 38 39, 82 40.07 40. 55 40, 205 1.35 286 297 40. 60 40. 85 41, 68 41. 02 41, 038 1.35 312 302 40.73 40.97 40, 41 39. 80 40. 478 1.35 345 334 41. 04 41.42 41, 50 42. 37 41. 582 1.35 356 366 41.27 40. 90 40. 50 41,07 40. 935 1.35 384 382 41. 94 41.16 41, 46 41. 98 41. 635 1.35 405 410 4.11 41.24 41, 20 40. 62 41. 042 1.35 429 433 |. 40.92 40. 93 40. 38 40. 66 40. 722 1.35 458 447 41.98 41, 61 41,76 40. 56 41. 478 1.35 459 466 |...-.--- eee 42. 67 42. 41 41, 42 42.170 1.30 474. 478 42. 44 39, 80 42, 07 41, 04 41, 338 1.35 516 490 41.74 41. 64 41, 53 40. 80 41, 428 1.35 B19 52k, egsesivaccecte 41.07 41,27 41.30 41, 213 1.30 526 524 42,11 42. 58 43, 07 41.75 42, 378 1.35 529 541 41.04 40.73 41.16 40. 38 40, 828 1.35 564 565 41.98 41.76 40.78 41. 62 41. 510 1.35 585 591 40. 60 39. 87 40.99 40. 43 40, 472 1.35 597 604 41.14 41. 56 40.78 40. 87 41. 088 1.35 Oo # “ W Weighted mean ....--.----------- +--+ +2 tere ee eereeeeeeeee 38 51 41. 216 + 0. 064 « Reduction to station ....--.---.----- 20sec ee cence eee e ee + 0. 018 Latitude of West Base, Olney....------.--+--+ee5+- 38 51 41. 229+ 0. 064 é LATITUDE OF PARKERSBURG. § 15. This is the terminal station at the south end of the triangulation extending south from Lake Michigan. It is situated about 11 miles south of Olney, Ill., on a rise of ground about 80 feet above the general level. The surface of. the country in this region is level or rolling, there being no hills with an elevation exceeding one or two hundred feet within a radius of perhaps 50 miles. Latitude was observed here by Lieutenant P. M. Price, in August, 1879, on five nights, with zenith-telescope No. 19, made by Wiirdemann, having a focal length of 32 inches and an aperture of 3 inches. The value of one revolution of its micrometer-screw was 62/’.224, and of one division of its level, 1.292. The instrument was mounted on a heavy oak post 22 inches in diameter, sunk 3 feet in the ground, situated 113.8 feet distant, bearing north 87° 43’ west from the station. The number of pairs of stars observed was 38, and 126 individual results for latitude were obtained. 80 LS 634 ASTRONOMICAL DETERMINATIONS. [Cuar. XXIII, In reducing the observations, declinations from Safford’s Catalogue of 2018 Stars (Washington, Government Printing Office, 1879) were used, except for two stars not in that catalogue, which were taken from the United States Coast-Survey Catalogue, in Appendix 7 of the Report for 1876. The constants for reducing from mean to apparent declinations were taken from the American Ephemeris for 1879. Weights for the results from each pair of stars were assigned according to the adopted method, previously described (§§ 2, 3), with Zyp=+40.42 and Hy; = +£0.57. Following herewith is the usual table giving the pairs of stars observed, with the British Association Catalogue number, the individual results, &e. The resulting latitude of the observing-post is 38° 34/ 53/.249-+ 0.09, and applying a correc- tion of —0’’.045 to reduce to the trigonometrical station, there results for the latitude of Parkers- burg, 38° 34’ 53”.204+10".09 Latitude of Parkersburg. (Observer, Lieutenant P.M. Price. Instrument, Wiirdemann zenith-telescope No. 19.] Ponte OF ates ob- Busse 9, a 10, sary 11, ae rd 12, ery 13, Means. | Weights. oO # 38 34 “a a“ aw an “uw a 5552 OED! I iis ascre iGeizicieid love oiersicnstsicihisial sea Reshearels, eis 5869 lice eacmiesvelee 53. 69 1. 00 5652 * 5706 S047 Newsweceowees 53. 66 BAAG2> «eee cecees 53. 25 1. 30 5775 5842 DD LDL > |Ihstaintawrersisvinime) tru isccteetercre cise 53. 27 53. 07 538.16 1.30 5874 5886 53. 89 he set aieione 53. 98 52.71 53. 06 53. 41 1. 36 5962 5990 58365. ese sedeices: GB 24 i eeticwse-nen 52. 52 53. 14 1.30 6068 6082 D808 leczseasssuce 53. 29 52.77 52. 95 53. 21 1. 36 6129 6150 52. 92 51.91 53,12 54.18 54.11 53. 25 1. 39 6218 6235 49. 71 53. 32 54.13 D2LAG! hedieajoiewcwiea’s 52. 40 1. 36 6355 6365 53. 49 55. 45 54. 04 55. 25 54, 52 54. 55 1.39 6429 6475 52. 72 49. 63 51. 29 54. 90 53. 87 52. 48 1.39 6581 G59: iste sierwiatescceiase 54. 43 . 52, 85 52. 95 52. 94 53. 29 1. 36 6656 6698 52. 90 52.77 52. 60 53. 51 52. 72 52. 90 1.39 6714 6721 52. 50 56. 94 53. 03 55. 46 54. 78 54. 54 1.39 6745 6777 52.79 53. 85 53. 47 53. 37 52. 79 53. 25 1.39 6817 6875 53. 64 53. 89 53. 38 53. 94 54. 46> 53. 86 1.39 6996 7006 53. 57 53. 47 48. 92 D5; 25' lsnemereeeine.’ 52. 80 1. 36 7073 7101 53. 29 53.19 Da OL- |leeiterceSometata| tind sie cereale 58. 50 1.30 7112 UBL, | ccceew corse OED lil xe Saiiusd es cuteden Doe2O) Veena seems 53. 79 1.21 7161 7164 52. 65 98.37 |sewsescccens 53. 68 55. 95 53. 91 1. 36 7204 7241 57.75 S245) lossesseeee ce 52. 80 52. 47 53. 87 1. 36 7313 7336 : 52. 58 1.30 7383 =C.S. 1924 51. 43 1. 00 7465 7501 51. 97 1. 30 7602 7614 53. 02 1.21 7705 7721 54. 07 1.30 7705 7731 50. 28 1.00 7731 = Gr. 3717 52. 28 1.30 7879 7901 54. 08 1.00 7880 7901 52. 62 1.30 8058 8141 54, 24 1.30 8211 8245 54. 07 1,30 C.S. 2176 58 53. 64 1.36 166 197 52.17 1.36 259 297 53. 91 1.30 330 345 53.04 | 1.36 465 480 53. 89 1.30 501 516 54. 02 1. 00 544 575 53.19 1.30 Oo f “ a Weighted mean ttre tte eee eee ee crete cee nee e ene nee weeeae 38 34 53. 24940. 09 es — 0. 045 ded elem awe siae ace cape ccpespetaeses 38 34 53. 204+0.09 § 16.] LATITUDES. 635 LATITUDE OF TOLEDO. § 16. Observations for latitude of Toledo, Ohio, were made in 1868 by Assistant Engineer O. B. Wheeler at a point 185.2 feet west and 188.4 feet north of the stone which marks the intersec. tion of Monroe and Ontario streets in that city, and 54.5 feet east and 44.6 feet south of the longi- tude stone post occupied in 1881. The land is quite flat in this vicinity, and there is no marked elevation of ground within a ~adius of 50 miles. In the vicinity of Toledo the soil is deep and underlaid by limestone rock. The instrument used was Pistor & Martins trausit No. 1, of 24 inches focal length and 24-inch object-glass. It was mounted on a large wooden post. One revolution of the micrometer was equal to 85’.220, and one division of the level equals 2.293. Observations were made on five nights. Twenty-four pairs of stars were observed and 44 individual results were obtained. Declinations were taken from Safford’s Catalogue of Mean Declinations of 2018 Stars (Washington, Government Printing Office, 1879). Weights were assigned to the result for each pair of stars by formula (2) § 2, with Ay>=+1.05 and #,=-+0".53. Following is the usual table of results, giving the British Association Catalogue numbers of the stars, the individual and mean results for each pair, and the weight for each mean result. The resulting latitude of the observing-post is 41° 39/ 03/.2140/.15, and applying a correction of +00/.44 to reduce to the longitude-post of 1881, there results for latitude of ‘Toledo (longitude- post of 1881) 41° 39 03".6540".15 : Latitude of Toledo. (Observer, 0. B. Wheeler. Instrument, Pistor & Martins transit No. 1.] ce ee ob- ee 28, Oseee 29, aoe 2, Hovenee 4, Noveniiee 5,! Means. | Wei hts. o fF 41 39 “a “a aw an “7 “we 6962 6997 || ceeoccrecisy|Soesekeedsoeee 02,46 acteneeecaceealens Satanic 02. 44 0.72 6965 FOOG: | crore, siciavesa'oe2i2ie | sisaw sawieswnaies OL 26: | swteccccineeccs| sceccceticcen ce 01. 26 0. 72 7027 7100 |.... 04. 50 0. 72 7114. Gr. 3243 02, 43 0.72 7194 7218 |.... 00. 81 0. 72 XX 401 7297 04, 11 0.72 R. C. 5132 7373 03. 74 1. 20 7399 7411 |.... 04. 26 1.54 7501 7566 |. 04. 37 1. 54 7607 7637 |.- 02. 92 1. 20 7607 7642 03. 30 0. 72 7155 7706 02. 54 0.72 7117 7800 |. 01. 53 1, 54 7815 XXII 113 }.. 01. 54 1. 20 7850 7917 |. 04. 02 1, 54 7962 8028 03. 20 1.54 8083 8091 03. 78 1, 20 8118 8171 02. 93 1. 54 16 52 03. 36 1. 20 109 121 02. 69 1, 20 169 229 03. 48 1. 54 377 BBE |e ac sroe rer 05. 80 0. 72 425 575 04, 18 0. 72 501 566 03,17 0.72 oO z “a “a Weighted mean....-....--.2 ecee cece eee e te eet e eet tenet tester ene eeeaee 41 39 03. 21 4: 0.15 Correction to reduce to longitude-post of 1881........----++++++eeee seer eres eee + 00. 44 Latitude of Toledo (longitude-post of 1881)..---.--.--2.s2e-+eeeeeeeeeeee 41 39 03.65 40,15 636 ASTRONOMICAL DETERMINATIONS. Cuap. XXIII, LATITUDE OF WEST BASE, SANDUSKY. . § 17. The observations from which the latitude of West Base was determined were made in the city of Sandusky, about 24 miles nearly south of station West Base. The post was situated 1022.7 feet south and 2599.5 feet west of the northeast corner of the custom-house. In this vicinity the soil is underlaid by limestone rock, which reaches nearly to the surface of the ground. There is no marked elevation of ground for 40 or 50 miles in any direction. To the northward lie in succession, first, Sandusky Bay, then Marblehead and Cedar Point peninsulas, and then Lake Erie, with comparatively shallow water and numerous islands. Latitude was observed by Assistant Engineer G. Y. Wisner on four nights in October, 1868. The instrument used was Wiirdemann zenith-telescope No. 12, having a focal length of 32 inches and a 24-inch object-glass. One revolution of the micrometer-head at turn No. 5 = 63.07, and at turn No. 45 = 63.87, and was assumed to change uniformly between these values. The number- ing of the turns was such as to make the middle notch of the comb-scale No. 25. One division of the level = 1.03. The instrument was mounted on a heavy wooden post. Thirty-one pairs of stars were observed, giving 69 individual results for latitude. In reducing the observations Saf- ford’s Catalogue of Mean Declinations of 981 Stars (Washington, Government Printing Office, 1873), was used for declinations. Weights were assigned to the results for each pair by the formula (2) § 2, with Hp =+0.80 and #;=+0".53. In one case a single star on one side of the zenith was combined with 4 stars on the other side, thus forming 4 pairs. In another instance 1 was com- bined with 3; and there were 6 cases where 1 was combined with 2. These were weighted accord- ing to the formula given in the note to § 5. Following is the usual table, giving in its successive columns the pairs of stars with their British Association Catalogue numbers, the results for each pair on each night, the mean result for each pair, and the weights assigned to the mean results. The resulting latitude of the observing-post is 41° 27’ 13.49+ 0.15, and applying a correction of +1/ 51.10 to reduce to West Base (determined by a seeondary triangulation), there results for latitude of West Base, Sandusky, 41° 29 04".59140".15 Latitude of Sandusky. (Observer, G. Y. Wisner. Instrument, Wiirdemann zenith-telescope No. 12.] Paleudeias. | OP ner Mi Octet | Gataher 1, October 18, areans. | Weights. O° 7. 41 27 a uw a “a wy 7112 6967 LGE28t Whecsieisteatacrdsictd| etensictee ene eeeeem erence 17. 23 0. 44 7112 6997 DD cdGr: Wl aoe atte oll Umacetcicsci cl ledciaunacteleeta 15.70 0. 44 7112 6969 TG3T4- Wasser sascecislace Ce Laateratecerane 16.14 0. 44 7112 1084 cratsretsrocemete 14s 39. aimee eco, 14. 23 14, 28 0. 67 7171 T7167 13. 28 13. 34 14. 38 13. 42 13. 61 2.27 7254 7336 11. 96 13. 04 12. 46 12. 04 12. 38 2.27 7317 7337 14, 24 14,18 11. 69 12. 65 13.19 2.27 7273 O20! |S eeaccneaeie ss 13. 60 13,12 12. 55 13. 09 2. 03 7345 7373 12. 43 13.41 V3e380 | eeciecce tes aie 12. 38 1.35 R. 5132 7373 11. 94 10. 85 13262: seeesargescas 12.14 1.35 7411 GOGO oss cdscamis WASOG) jrevsrasewisgercracctecsl| ichanne bedrs is avs 14. 06 0.72 7431 1309 Cee sige W430", |coceed cchsmnlvetencaaiccee 14. 30 0.72 7480 TAGD liaise qaporerateclees 13. 03 neEi ose ce eetanouus ip 13. 03 0.72 7469 PAG 2i| emrccamasinrdlersrd cies sa meaels TBSG8.) Naess teva 13. 73 0.72 7501 OUD | isecoraiad vaeee WBE5O? | evict Aakaiciiel pedaeavta len © 13. 50 1.08 §§ 17,18. | LATITUDES. 637 Latitude of Sandusky—Continued. Pairs of stars. Deveber 10, Gotaber il, oe 12, one 13! Means. | Wei ghts. a“ “ uw “we Ww 7503 F566: ec. seseceans TB, 79). lsseecn seevlen| emacs weiss 15. 79 0. 72 7503 R. 9430 |.----..-.--- UNO |kessed secasellacceesnexace’ 13. 01 0.72 7681 7602 16. 08 14, 99 16, 15 eine ais stnctora 15. 74 2. 03 7705 7614 11. 64 13. 54 4s AT | eect esion crn 13.22 2. 03 7746 7721 16. 27 22. 43 O9;TL. |eeece aeneies 15. 94 2. 03 7800 VW1T 13. 71 13. 38 1398) Vecemes sevice 13. 69 2. 08 7815 XXIT113 13. 06 13. 19 ADBS lecwces catecias 12. 71 2. 03 Gr. 3779 7843, 13. 58 T4386) |esescssecas s Baie were 13. 97 1. 66 7894 7879 12. 35 11.75 TW SS haces: pened! 11. 82 1. 35 7894 7880 TQS8E «| est eicie ae ci DD AB. ciciciam cciecnis. 12. 64 1.11 7913 7901 13. 58 13. 34 12.405 ll seins csceie ne 13, 11 2. 03 8028 7932 13. 03 12. 68 12.50 [ecsces cesece 12. 74 1.01 8028 7962 12. 68 12. 03 VOT Wass were omissesen 12. 33 1.01 8028 7994 13. 63 12. 87 T2c3L. | lesicermcawccaie! 12. 94 1.01 8261 52 TH99)” lececdketdecsheesesmesesa|seexmewesens 11. 99 0. 72 16 52 LQRSD< || siateraimensenid oGialltiooia gamye dal pomcasaaeeee 12. 85 0. 72 fo} + Ww uw Wreeis hited, i Oinn aise oes ciaiecctcistaisisiepctei hereateseiaie atcicta cle Suieleicleescter sists 41 27 13.491 4+ 0.15 Correction to reduce to West Base........2....2.020-00200 eee + 151.10 — Latitude of West Base. ............2. cece cece eee cece eens 41 29 04.59 + 0.15 LATITUDE OF TONAWANDA. § 18. This station is situated in Erie County, New York, about 2 miles south of the village of Tonawanda. To the east, south, and west of the station, within a radius of 10 miles, there are no marked changes of elevation. Fifty miles to the south, however, the hills rise to a height 500 to 1000 feet greater than that at the station. Northward, the surface declines very gradually for twelve miles to the limestone ledge crossing Niagara River near Lewiston, N. Y., and visible in a line nearly parallel to the south shore of Lake Ontario, between Rochester, N. Y., and Hamilton, Ontario. From this ledge, which is about 250 feet high in the vicinity of Lewiston, the ground slopes off gradually to the basin of Lake Ontario. Latitude was observed here by Assistant Engineer A. R. Flint in October, 1875, on six nights with the Wiirdemann zenith-telescope No. 19, having a focal length of 32 inches and a 3-inch object-glass. One revolution of the micrometer-screw was equal to 62/.224, and one division of the Jevel to 1.292. The first night’s observations were made with the instrument supported by a wooden post used previously as an azimuth post (see Chapter XXIV, §11). On the remaining nights a stone pier was used. This stone pier was situated 31.9 feet distant, bearing north 86° 30! east from the station. The number of pairs of stars observed was 40, and 156 individual results for latitude were obtained. Declinations were taken from Safford’s Catalogue of 981 Stars, and weights were assigned té& results from each pair according to the adopted method (§§ 2, 3) with Hy=+0".42 and H; =+0”.57. : Herewith is given the usual table, showing the British Association Catalogue numbers of stars, the individual results for each pair on separate nights, the mean result for each pair, and the weights assigned to the latter. In this table the individual results for October 2 are corrected to reduce them to the position of the instrument on the remaining nights. The resulting latitude for the stone pier is 43° 00/ 07.8454 0.085, and applying a correction of —0’’.019 to reduce to the trigonometric station, there results for the latitude of Tonawanda. 43° 00 07'.82610".085 638 ASTRONOMICAL DETERMIN ATIONS. [Cuap. XXIII, Latitude of Tonawanda. (Observer, A. R. Flint. Instrument, Wiirdemann zenith-telescope No. 19.] 5 | ; ; bias 7 | ee sir ob- peter 2, rer 14, Osea 19, Uetater 20, aN 21, Cereer 22; wy eans. Weights. “ 6556 6566 06. 260 1.00 6603 6651 06. 450 1.00 6711 6721 08, 410 1.30 6728 6745 07. 760 1.21 6769 6779 07. 776 1.39 799 €806 08, 322 1.39 6799 6813 08. 150 1.00 6830 6849 07. 908 1.39 6876 Gr. 3013 08. 401 1.41 6967 6985 08. 108 1.39 7029 7035 07. 853 1.41 7083 7114 08. 070 1.41 7171 7174 07, 468 1.39 7182 7213 07, 515 1.41 7233 7260 06. 940 1. 36 7277 7306 08. 427 1.41 7345 7398 07. 528 1.39 7411 7462 07. 983 1. 36 7496 7505 08, 488 1. 36 7566 7598 07. 408 1.39 1753 7787 08. 274 1.30 7845 7858 - a5 07.394 . 1.30 7894 M932. |stsisare wits Sere OF OL, leeeieer screenees 06. 85 08. 69 07. 92 07. 618 1.36 7983 8023) |ancss sewscaga’ OT 21 pacieicemcerccss 07.13 09. 33 07. 57 07. 810 1.36 8076 BONG! || eds ierg i eaveie 09:00" | |occumiscwees 08. 26 09. 03 06. 91 08. 300 0, 72 8110 BIIB? ic cea cra Besa MOSB4 4h cece cies acess 08. 52 06. 27 08. 84 08. 493 1.36 8229 8237 | asaseis cevesn, OB. 62: |xcwsescecsxs 07. 76 08. 07 07. 31 07. 940 1.36 16 28 07. 51 07.45 07. 733 1.36 60 60 Ot 1S eet aeencas 07. 813 0. 70 120 123 setenbee te leu taCeigeses 07. 000 1.00 120 146 07. 03 07. 65 07. 180 1.30 166 169 06. 57 05. 88 06. 783 1. 36 245 259 « 08.70 06. 00 07. 318 1.36 310 314 08. 35 06. 82 07, 232 1.36 337 352 06. 87 06. 97 07. 075 1,36 294 294 10. 01 09. 87 09. 400 0. 70 299 299 09. 11 08.75 ° 08. 933 0.70 425 Gr. 317 07. 80 07. 40 07. 540 1. 36 492 501 08.60 | 07. 52 08. 205 1.36 516 555 11, 27 10. 51 10. 870 1.36 ° t “ a WIGIENEOG MEAN css.42)2 oe winwinedciciis cic sevacinee cin’ Se seiie kee een Meets ciemen eee 43 00 07. 845 +0. 850 Reduction to station ..... 2.22.22. cence cece cece e cee eee cece cece ceseeeeeees -- 0. 019 Latitude of Tonawand asi. sccc20c ceeeics eee cccesceccces cinseoeasceesee 43 00 07. 826 +0. 085 LATITUDE OF NORTH BASE, SANDY CREEK. § 19. This station forms the northern extremity of the Sandy Creek base-line, and is situated on a sandy beach at the eastern end of Lake Ontario. The line of the beach at this point, and for 10 miles in either direction, runs nearly due north and south. To the west the lake bottom declines at the rate of about 40 feet to 1 mile. To the east there is a sand dune about 50 feet in height, 100 metres distant, then a pond and marsh extending about 1 mile, followed by rising ground. The ground at the station is scarcely above the surface of the lake. Latitude was observed here by Assistant Engineer G. Y. Wisner, in September, 1874, on four nights, with Wiirdemann zenith-telescope No. 12, having a focal length of 32 inches and a 23-inch object-glass. One revolution of the micrometer at turn No. 0—63’'.07, and at turn No. 40=63"".87, and was assumed to change uniformly between those values, the middle notch of the comb-scale §19.] LATITUDES. | 639 being taken as the 20th turn. One division of the level was equal to 1.00. The instrument rested on a large pine post set firmly in the ground, situated 81.1 feet distant, bearing south 13° 24’ 26” west from the station. The number of pairs of stars observed was 32, and 120 individual results tor latitude were obtained. For reducing the work declinations from Safford’s Catalogue of 981 Stars were used. Weights were assigned to the results for each pair according to the adopted method (§§ 2, 3) with BHt0".45 and Ly= + 0".57, The accompanying table gives in the successive columns the British Association Catalogue numbers of the stars observed, those forming a pair being placed on the same horizontal line; the individual results for each pair on the separate nights; the mean result for each pair; and the weights assigned to those means. The resulting latitude for the observing-post is 43° 40’ 40.7414. 0.07, and applying a correc- tion of +0.779 to reduce to the trigonometrical station, there results for the latitude otf North Base, Sandy Creek, 43° 40’ 41”.52010".07 Latitude of North Base, Sandy Creek. (Observer, G. Y. Wisner. Instrument, Wiirdemann zenith-telescope No. 12.] Pairs stars observed.| September | September | September | Septemer | Means. | Weights. | oF 43 40 “a a aw aw “uw 6520 6493 41. 02 41. 07 40. 31 40. 05 40. 61 1.33 6579 6599 39. 98 40.15 38. 49 39. 79 39. 60 1,33 6697 6651 40.70 40.12 40, 94 39, 57 40. 33 1.33 6717 6711 40. 80 40. 24 40. 48 39. 93 40. 36 1.33 6754 6745 39.15 41. 50 41.09 40. 69 40. 61 1.33 6764 GRiL.. |eocomemanccase 40. 62 41,20, | scsaseacicacis 40. 91 1.18 Gr. 2957 6817 40. 45 40. 88 40.79 42.19 41. 08 1.33 6865 6875 41. 25 41. 54 41.41 40.77 41, 24 1.33 6959 G98T |} ceeciavceeexs 42. 08 42.73 41,21 42. 00 1. 27 6985 6990 39. 51 40. 71 41.01 40. 27 40. 37 1. 33 7062 7001 40. 38 41.45 40. 53 40. 47 40.71 1.33 7083 Gr. 3215 40. 99 41.31 41.13 40. 64 41 02 1.33 7112 7114 ALTE Wasrercre erase riersdel| cetera aiecmass 40.17 40. 97 1.18 7161 Gr. 3243 40. 61 40. 43 40. 55 40. 97 40. 64 1, 33 7215 7194 41. 97 42.51 41, 38 41. 33 | 41.80 1.33 7253 7241 40. 68 41.18 40. 91 41. 30 41. 02 1.33 7268 v Cygni 41.04 42.41 42. 42 40.65 | 41.63 1.33 “BIT 7333 40. 54 41.27 41.49 40. 2 41. 03 1.33 7345 7383 40. 45 40. 40 40. 08 40. 06 40, 25 1. 33 7448 7453 42. 05 40. 61 41. 69 38. 98 40. 83 1. 33 Gr. 3524 7505 41. 93 40. 86 40. 80 41, 72 41. 33 1.33 7598 7602 40. 90 41. 81 41.15 39. 49 40. 84 1.33 7695 R.C. 5408 40.63 - 39. 85 40. 66 40.19 40, 33 1.33 Gr. 3717 7737 40. 53 39. 72 40. 05 40,45 40.19 1, 33 7800 Gr. 3750 40. 36 40. &9 41, 53 40. 61 40. 85 1.33 7813 7843 40. 47 39. 78 40.59 | 40. 48 40. 33 1.33 7906 Gr. 3843 38. 78 40. 69 40. 61 | 39. 26 39. 83 1,33 Li. 44750 7978 |.------ 22 ee 40. 45 40. 64 41.07 40. 72 1.27 7999 7984 39. 88 41.11 39. 45 | 39, 87 40. 08 1.33 8058 8023 39, 43 39. 66 39, 84 | 39. 47 39. 60 1.33 8110 8076 40. 43 41. 30 41. 21 ; 41. 45 41.10 1.33 8223 82387 | lawiswiciercenceee 41. 50 41,94 | erence apaguees 41. 72 1.18 ‘i Oot “ “ Weighted mean.......-.-.------seeee rene eee ec ee ener encom anes 43 40 40.7410. 07 Reduction to station ........2.---0- 22sec e eee e ee ee eee ener teens + 0.779 43 40 41.520+0.07 Latitude of North Base 640 ASTRONOMICAL DETERMINATIONS, [Cuap. XXIV, CHAPTER XXIV. AZIMUTHS. § 1. In the triangulation between Keweenaw and Minnesota Point Bases, azimuth determina- tions were made at North Base, Minnesota Point (§ 2), at Aminicon (§ 3), and at South Base, Keweenaw Point (§ 4). In the triangulation between Keweenaw and Fond du Lac Bases, azimuth determinations were made at Ford River (§ 5), and at Bruce (§ 6); in that between Fond du Lac and Chicago Bases, at Minnesota Junction (§ 7); in that south of Chicago Base, at Willow Springs (§ 8), and at Parkersburg (§ 9). Azimuth determinations were also made at West Base, Sandusky (§ 10), and in the triangulation between Buffalo and Sandy Creek Bases, at Tonawanda (§ 11), and at North Base, Sandy Creek (§ 12). AZIMUTH AT NORTH BASE, MINNESOTA POINT. § 2 Assistant G. Y. Wisner observed at this station for azimuth on one night, the instrument used being Troughton & Simms 14-inch theodolite No.1. The azimuth of the line North Base— South Base was to be found. The instrument was on a stone post, firmly set in the ground, which had been previously used in longitude determinations. The azimuth mark observed-was a light shining through a narrow slit in a board set up on the post, 3145 meters distant, which had previously served as support for a meridian mark. Time was given by Bond & Son’s sidereal clock No. 256. The observations consisted in measuring the angle between a close circumpolar star and the azimuth mark, and noting the time. The programme was to point alternately to the star and mark till from two to six pointings at the star had been obtained, then to reverse the telescope and get as many additional pointings. The level was read for each pointing at star. In some cases two pointings at the star were made for a single pointing at the azimuth mark. The instrument was very stable, the means of pointings at mark before and after reversal differing in no case by one second of arc. In computation, the pointing at each star was used separately to obtain a result. The mean of pointings at the mark before reversal was used with pointings at the star before reversal, and the mean after reversal with pointings at star after reversal. Azimuths were com- puted with the formula sin t tan,.4 =. ee cos ¢ tan 6—sin ¢ cos t in which ¢g, 5, and ¢ are the latitude, star’s declination, and star’s hour-angle. Corrections for level were computed separately. Star places for Polaris and 2 Urse Minoris were taken from the American Nautical Almanac, and for 6 Ursie Minoris and 39 Cephei, which are not given in the American Ephemeris, from Saftord’s Catalogue of Time Stars. In a set, equal weights were given to means of results before and after reversal, although for 39 Cephei, on July 13, there were four pointings at star before reversal and but two after. In combining the mean results derived from different sets of observations, a weight propor- tional to the total nuniber of pointings at the star is attributed to each result. The probable error is derived from the discrepancies between the results from the different sets and their weighted mean. §§ 1,2] AZIMUTHS. 641 The origin of the horizontal circle remained in essentially the same position throughout the work, but the microscopes were temporarily changed 180° in reversal. Periodic error is then only so far eliminated as is effected with three microscopes when turned 180°. The following table gives the results of the separate observations: Azimuth at North Base, Minnesota Point. AZIMUTH OF LINE AZIMUTH POST— MERIDIAN MARK. (Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1.] Star, date, &c. Telescope— | Azimuth of mark. Means. Result for star. fo} a a °o a am oO tf at Polaris, near Lower Culmi- | Direct ...... 00 00 01.6 nation, July 13, 1871. 359 59 57.3 359 59 59, 45 eal 11° a Reversed ... 00 00 01.0 = oO it hi Se BRO 87 ONES 359 59 59.1 00 00 00.05 | 359 59 59.75 | 39 Cephei, near East Elonga- | Direct ...... 359 59 59.1 tion, July 13, 1871. 60.9 a=23h 28™ 01°,4 56.7 5=86° 35/ 30.84 62.3 359 59 59.75 Reversed .-. 58. 0 57.2 359 59 57. 60 359 59 58. 68 6 Ursa Minoris, near West | Direct ...... 359 59 59.0 Elongation, July 13, 1871. 58.7 359 59 58. 85 a 1ah1aY Oats Reversed -.. 56. 8 a OH ey 5=88° 25/ 09”.77 59. 4 359 59 58.10 359 59 58.48 Polaris, near East Elorga- | Direct .-.... 359 59 60.0 tion, July 13, 1871. 61.6 a=1 11” 445.9 56. 0 5=88° 37/ 01.2 59. 8 59.1 60. 9 359 59 59, 57 Reversed ... 56. 8 59. 8 56.2 57.1 | 57.8 | 57.7 359 59 57. 57 359 5Y 58. 57 A Urse Minoris, near Upper | Direct ...-.. 359 59 57.2 Culmination, July 13, 1871. / 58.3 359 59 58.25 a=19" 54™ 17*.2 Reversed ... 59. 2 3 B= 889 Ob) 127 58.9 359 59 59.05 359 59 58.65 I The second set of observations on Polaris required the maximum time, sixty minutes, while the set on 2. Urse Minoris required but sixteen minutes. The following table gives the mean result for each set of observations, the number of pointings to mark and to star in each set, and the weights attributed to the mean results. The results derived from Polaris and 4 Urs Minoris are corrected to conform to Auwers’ declinations (see § 4), and the result from 39 Cephei (which is not given in Auwers’ Catalogue), is corrected to agree with the declinations as given in the General Bericht der Europiische Gradmessung for 1874. 6 Urs Minoris is given in neither of these catalogues, and receives no correction in the table. The results given in the table have also been corrected for diurnal aberration and for the effect of a correction, + 2’.34, to the latitude, 46° 45! 27/0, with which the azimuths in the preceding table were computed, 8LLs 642 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, Azimuth at North Base, Minnesota Point. SUMMARY OF RESULTS. AZIMUTH OF LINE AZIMUTH POST— MERIDIAN MARK. 7 moaon oe | Pungo at— gst) elation Date Star, &e. tpleseaing: aren © ene — Corrected Weight. Mark.) Star. | Ephemeris. ai Aé6 AA | ° ‘ oO t 359 59 359 59 1871. un " u u" July 13 | Polaris, near Lower Culmi- | Direct ..-... 2 2 nation. Reversed -.- 2 2 60. 05 —0.04 | +0. 04 0. 00 60. 05 1 13 | 39 Cephei, near East Elonga- | Direct ...-.. 4 4 tion. Reversed . .- 2 2 59, 21 —1.44 | +0. 30*) —0. 43* 58.78 . LS 13 | 6 Urse Minoris, near West | Direct ...... 1 2 Elongation. Reversed ... a 2 BBG eacanielaesmay del enmeses 58. 69 1 13 | Polaris, near East Elonga- | Direct .....- 3 6 tion. Reversed ... 3 6 58. 97 —1,44 | +0. 04 | --0. 06 58. 91 3 13 | A Urs Minoris, near Upper | Direct ...... 2 2 Culmination. Reversed .-.. 2 2 58, 97 +0. 01 | +0.55 | +0. 01 58. 98 1 W GiphtediOan vijaje sie szeccsiciei ane die teats teeraieizvalsi starch) aicloperid clawgiajaiateanats ie el ciaipiaie oie sth /eid cles sie Sretsueiginve 359° 59’ 59.0240", 14. «Corrections to reduce to declination as given in General Bericht der Europiische Gradmessung for 1874. In the previous longitude work at this station the instrument was set, each of six nights, at the beginning of the work, on the meridian mark. If the air had been steady, and the instrument had not changed its deviation during the night, a value of the azimuth of the meridian mark would have resulted from the time-reductions. The value which did result was 0/.22 west of south, differing 1’’.09 from the value resulting from the azimuth work. It is a rough check on the latter. The azimuth of the line Azimuth Post— Meridian Mark has now been given. From it is to be derived— 1. That of the line North Base—Meridian Mark, and 2. From this, that of North Base—South Base. The distance of the-azimuth post from North Base was measured five times with rods, depend- ing on a standard yard, along a stretched wire, giving a mean distance of 186.651, the measures having a range of 0.15. The angle Meridian Mark—Azimuth Post—North Base was measured with Troughton & Simms theodolite No. 1. This gave the means of computing with accuracy the correction to the azimuth of line Azimuth Post—Meridian Mark, to get that of line North Base— Meridian Mark. Its value is —51/ 04.13. In addition the angle North Base—Meridian Mark— Azimuth Post was measured on a cloudy day, eight single or four combined measures being obtained with Troughton & Simms theodolite No. 1, giving in the mean 51/ 02/.23 +0’.4. Adding 1.58 for convergence of meridians at Meridian Mark and North Base, this becomes 51/ 03.81 +0/".4, differing but 0.32 from the preceding value. Attributing double weight to the first value, there results for the mean correction —51’ 04.02 +0.23. This gives for azimuth of line North Base to Meridian Mark, 359° 08! 55/00 +0/.27 The angle at North Base between the meridian mark and South Base was measured with Troughton & Simms theodolite No. 1, sixteen single or eight combined measures being obtained. The mean value of the angle is, 35° 16’ 30.68 £07.27 the probable error being derived from the discrepancies between the separate combined results and their mean. Subtracting this from the azimuth of North Base—Meridian Mark, there results for azimuth of line North Base—South Base, B2B° 32 24” .3210".38 west of south. §3.] AZIMUTHS. 643 AZIMUTH AT STATION AMINICON. § 3. At station Aminicon the same observer as at North Base, Minnesota Point, determined the azimuth of the triangle side Aminicon—Lester River, with the same instrument and essentially the same methods and programme. There were usually, however, for each pointing at the mark, two consecutive pointings at the star, and for these two pointings at the star the level was read but once. The instrument was mounted on a wooden post sunk 5 feet in the ground. Time was given by a chronometer. The azimuth mark was a light on station Lester River, 13 miles distant. In reduction, the same method was followed as for North Base, Minnesota Point. The microscopes were turned 180° in azimuth by reversal, on July 22, but not on July 23. The shortest time occu- pied in a set of observations was 24 minutes, on 6 Urs Minoris, July 22, and the longest time was 57 minutes, on Polaris, on July 22. The following table, arranged like that for North Base, gives the result for each observation on a star, and the mean result for each star. Azimuth at Aminicon River. AZIMUTH OF LINE, AZIMUTH POST— LESTER RIVER, (Observer, G. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1.] | Azimuth of Lester River station. Means. Result for star. Star, date, &c. Telescope— | t ov “ | ‘or, + uw ° 1 “ | 39 Cephei, near East Elonga- Direct .....- ! 153 36 31.6 | tion, July 22, 1871. | 28.0 : a=23) 28" 039.6 f 29.4 | 5=86° 35/ 33.17 : 28.7 i 29.4 27.6 153 36 29.12 Reversed . -- 27.3 | 28. 8 28. 05 153 36 28, 58 6 Urse Minoris, near West Direct .-..-. : 27.0 Elongation, July 22,1871. | | 30.2 ' 28. 60 ! a=12h 18m 58.63 ' Reversed ... 30.8 5=88° 25/ 07,98 : 27.3 = 31.6 29.9 29. 90 29. 25 Polaris, near East Elonga- | Direct ...-.-. 28.8 } tion, July 22, 1871. 26.1 a=1 11™ 528.26 30.1 5=88° 37/ 02/.4° 27.1 ; 28,3 ' 30.0 28. 40 1 Reversed . .. 30.5 30.7 30. 3 28.9 26.6 26.7 29. 4 29. 61 28. 70 39 Cephei, near East Elonga- | Direct ...-. ' 28.1 tion, July 23, 1871. 27.4 a=23" 28™ 03,9 30.8 6=86° 35! 33/44 ; 33. 4 29. 92 Reversed .-. 33.1 31.7 : 29.0 ' 29, 2 30. 75 30. 34 644 ASTRONOMICAL DETERMINATIONS. [Curar. XXIV, Azimuth at Aminicon River—Continued. Azimuth of Lester, River station. Oo # t fe) t “ fo} t wn 6 Ursx Minoris, near West | Direct ...... 153 36 33.3 Elongation, July 23,187]. 37 2 a=12" 13™ 58%.01 32.6 8==88° 25! 07.77 29.9 153 36 33.25 Reversed ... 28.7 28. 5 26.6 28.4 28. 05 153 36 30. 65 Star, date, &c. Telescope— Means. Result for star. Azimuth of Bu- chanan ‘station. Means. Result for star. | Star, date, &c. Telescope— : ° t a“ oO + “uw oo “uw Polaris, near East Elonga- | Direct ..-... | 191 23 12.3 tion, July 23, 1871. 10.2 a=1» 11™ 539.0 9.0 S=88° 37! 02.5 10.8 10.4 12.7 191 23 10.90 Reversed ... 8.8 111 9.5 09.27 ; 119 23 10.09 | The results in the preceding table have each to be corrected for diurnal aberration and for the effect of a correction, + 4/’.93, to the latitude, 46° 41’ 31.2, with which they were originally computed. Applying these corrections, and giving weights to the results from each set proportional to the number of pointings at a star in the set, the following table results. In it, instead of giving the weight 1.2 to Polaris on July 23, but half that weight has been assigned. The reason for this is that in this set the angle from the star was not read as in other sets to Lester River station since the light had become dim there, but toa light on Buchanan station. This azimuth was reduced to Lester River by using the finally adjusted angle given in Chapter XIV, C, between Lester River and Buchanan. The results in the following table are also corrected for errors in declination, in the same man- ner as the results at Minnesota Point (see §§ 2 and 4), Azimuth at Aminicon River. SUMMARY OF RESULTS. AZIMUTH OF LINE AZIMUTIL POST—LESTER RIVER. | Se eee ee | |, Date. | Star, &e. oe " ate tions from : a eee Comied | Weight | Light. Star. (Ephemeris. “@g As AA ! | ° , | ° ’ i 1871. | 153 36 153 36 i as ; : | aw i a aw “a | | July 22 | 39 Cephei, near East Elonga- | Direct ....-- 4 6 ; | | i tion. Reversed .. a | 2 29.384 © —1.43 ) 40.30, —0. 43", 28.91, 8 22 | 6 Urse Minoris, near West | Direct ...... 1 | 2 , i Elongation. Reversed...) 2 4 29.35 0 $1.45 |... eee | 29. 35 | .6 22.) Polaris, near East Elonga- | Direct ..--.- 2 6 i I \ tion. Reversed ... 3 | 7 29. 03 —1.43) -|-0.04 | —0.06 28. 97 | 1.3 23 | 39 Cephei, near East Elonga- | Direct ...-. 3 | 4 f i tion. Reversed ... 2 4 31.10 —1.43 40.30", —0, 43* 30.67 | 8 23 | 6 Urs Minoris, near West | Direct ..... 2 4 i | Elongation. Reversed ... 3 | 4 80075 ANAS cee free mtnies | 30. 75 8 23 Polaris, near East Elonga- | Direct ...... 3 | 6 \ | tion. Reversed“... 3 6 27.94 ° 1.43 40.04 | —0.06 27. 88 6 WGI EAC a ceed cranes Sidemndseddsaamereeme cine saseweniaed aterate venidesblaeid atenieramdemmeirreie ciara 153° 36/ 29.44” + 0,29 * Correction to reduce to declinations given in General Bericht der Europiiische Gradmessung for 1874. §4] AZIMUTHS. 645 The azimuth just given is that of Lester River from the azimuth post on which the instrument stood. It is to be corrected so that the azimuth of Lester River from station Aminicon may result. To obtain this correction the distance from instrument on azimuth post to Aminicon station was measured as at North Base, three times, the range in measument being 0.10, giving a mean dis- tance of 30.432. The angle at azimuth post between Aminicon and Lester River was measured with the Troughton & Simms theodolite No. 1, giving 24° 32/ 42.1. The resulting correction is +36".71. Applying it to the azimuth of line Azimuth Post to Lester River, there results for azimuth of the triangle side, Aminicon—Lester River, 153° 37 06".1540".29 west of south. AZIMUTH AT SOUTH BASE, KEWEENAW POINT. § 4. The observer for azimuth and the instrument were the same at this station as at Amini- con, and the method was generally the same. But instead of pointing at the azimuth mark after each one or two pointings at the star, no pointings at the mark (with a single exception) were made during the pointings at the star in a given set. Instead, five or more pointings at the mark were made immediately before the first pointings at the star, and as many more after the last pointing at the star. The programme was then as follows: 1. Five or more pointings at mark. 2. Four or five pointings at the star, and levels read for each two pointings. 3. Reversal. 4, Four or five pointings at star, and levels read for each two pointings. 5. Five or more pointings at mark. For one set, that on Urs Minoris on July 25, there were in addition two pointings at mark immediately before and immediately after reversal. On the other nights the method of observation requires the assumption that any change of position of the instrument in the period, varying from 18 to 35 minutes, occupied by the star pointings was proportional to the time. Such an assumption is doubtless inadmissible for. so long a period, and is probably one of the causes of the discrep- ancies in the results obtained in each set from observations made before and after reversal. The instrument was mounted on a wooden post, 2 feet in diameter, sunk 5 feet in the ground, which had been used by Assistant Flint as a latitude postin 1871. The distance of the instrument on its azimuth post from station South Base was measured with a standard scale on the edge of a level board. The mean of three measures having a range of 0.004, was 26.570. The angle at South Base between North Base and Azimuth Post was measured three times with Troughton & Simms theodolite No. 1, and found to be 10° 22/46”. These codrdinates give a correction of 33.92 to reduce the observed azimuth of line Azimuth Post — North Base to that of South Base — North Base. The difference between the mean pointings at the mark before and after star observations varied between 0.72 and 7.6, except on July 26 in the set of observations on Polaris, when it amounted to 13.86. If the change had been due to a twisting of the instrument in azimuth during the period of observation, it should have shown itself in comparing the five azimuths deduced from the five observations on the star before reversal with each other, or in a similar comparison of the azimuths resulting from observations after reversal. Giving seconds alone, the results from the five observations on Polaris before reversal and the mean of the five pointings at the mark before reversals were 29.24, 32.19, 32.39, 30’.49, 28.38, while those from observations after reversal were 34.34, 34/’.44, 34.64, 34.94, 35.54, Neither set shows any marked change in azimuth with the time, as would be the case if the reading of the mark on the limb had been changing instead of fixed, as the computations assume. The difference of 13/’.86 in the mean pointings at the mark before and after reversal must then be attributed to instrumental errors and to disturbances arising from reversal or other cause. The reversal is effected by hand, and the telescope and vertical circle are very heavy, so that there is danger of disturbance. There was no certain disturbance in level. There was no systematic elimination of periodic error, the horizontal limb remaining in essentially the same position throughout the observations. The microscopes were turned 180° in azimuth in reversal on July 23, but not on July 25 and 26. 646 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, The following tables give the results for each observation on each star: ¢ Azimuth at South Base, Keweenaw Point. AZIMUTIL OF LINE AZIMUTIL POST —NORTIT BASE, (Observer, G@. Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1.] Star, date, &c. Tolescope— Acunutl or North | 1 | Base. Means. Result for star. | I Polaris, near East Elonga- . Direct ....-.. i 199 10 338.47, tion, July 23, 1873. 31.42 : a=1"12"215,5 35. 37 | & - 882.287! 447.2 34.07 199 10 33.58 | o£ is TD eh wu ° o.% u | | Reversed ... 31, 60 31.17 | | 3.45 | 32.460 31.67 | 199 10 32.63 | | . Polaris, at East Elongation, Reversed -..! 34. 00 { | duly 25, 1873. 35.38 | l a=" 12m 238.0 29. 78 5-880 37/ 44".5 31. 28 33. 00 32. 69 | | Direct. ..... 35. 28 34.91 34. 80 35. 86 35. 66 35. 80 34. 00 ¢ Ursw Minoris, at West Direct ...... 34, 37 Elongation, July 25, 1873. 35. 70 a=15" 48" 415.6 : 31.11 == 78° 11 12.7 | 35. 51 31.71 33. 68 31.37 | 35. 07 ' 38. 47 36.17 34. 87 35. 19 34. 44 1 Reversed. .. \ | | | | _€ Urs Minoris, at West Reversed Elongation, July 25, 1873. a=16" 59" 095.1 5==82° 14’ 40.9 Direct ..-... 32. 85 34. 40 \ _ 6 Ursw Minoris, at West | Direct ...... 36. 33 | Elongation, July 25, 1873. 42.78 a =18 13" 315.2 40.51 886° 36/ 297.3 | 40. 87 | | | | | | | 35.47 39.18 / Reversed... 32, 30 30. 07 31. 84 ; 30. 50 | 29. 57 30. 86 35. 02 64.) AZIMUTHS. 647 Azimuth at South Base, Keweenaw Point—Continued. ‘ | Azimuth of North | M Star, date, &c. Telescope— Baas. eans. Result for star. Oo 7 “ i Oo f Gr! 7 a” Polaris at East Elongation, Reversed aoe 199 10 29.24 | Tuly 26, 1873. 32.19 a= 1 12" 938,7 32.39 8=880 37’ 44.65 30. 49 28.39 | 199 10 30. 54 Direct ....-. 34. 34 | | 34.44 | 34. 64 | 34.94 | 35.54 | 34.78 | 19910 32.66 ; =e : es Sond neadee eh '¢ Urse Minoris at West | Direct ..... 199 10 32. 66 i Elongation, July 26.1873. | : 35. 56 ; a=15> 48" 415.5 34.06 3=780 11 12.8 34.76 34.26 , Reversed . es, 28. 26 26. 41 29. 81 33. 06 29. 39 31. 82 | An examination of this table shows considerable differences in the results on the same night, before and after reversal on a given star. Part of the discrepancies arise without doubt from the long interval between the pointings at the mark, during which it has to be assumed that the change in reading on the mark is proportional to the time. It is possible that a part may come from some other cause. The difference in the results from 6 Urse Minoris, July 25, before and after reversal is so much greater than any possible error of observation that the results are not used. To the results from the other stars weights are assigned proportional to the number of pointings at the star. The following table gives in the first six columns a summary of the results in the preceding table, the results in the sixth column being further corrected for diurnal aberration. The results in the sixth column depend upon the declinations given in the American Ephemeris,, which has heretofore been the preferred authority for star-places in Lake-Survey reductions. There is, however, a later and thoroughly revised catalogue—the Fund. Cat. fiir die Zonen-Beobachtungen, Leipzig, 1879—compiled by Dr. Auwers under the direction of the Zonen-Commission der Astrono- mischen Gesellschaft, which may now be taken as the best authority available. Columns giving the quantities used in deriving corrections to reduce the results in the sixth column to conformity with Auwers’ declinations, and a column of corrected results, are, therefore, added to the following table, the seventh column giving the differential coefficient of each azimuth result with respect to the declinations, denoted by s the eighth column giving the corrections to the declinations given in the American Ephemeris to reduce them to Auwers’ declinations, denoted by J6; the ninth column giving the corresponding corrections to apply to the azimuth results in the sixth column, denoted by 44 (=%4)s and the tenth column giving the azimuth results corrected to conform to Auwers’ declinations. The quantities me can be used hereafter, if so desired, to further correct these azimuth results, when the star-places become known with greater precision. The probable error is derived from the differences between the separate results and their weighted mean. 648 ASTRONOMICAL DETERMINATIONS, [Cuap. XXIV, alcimuth at South Base, Keweenaw Point. SUMMARY OF RESULTS. 2 AZIMUTH OF LINE AZIMUTIT POST — NORTH BASE. | Number of uf Reduction to Auwers’ | ee r pointings at— ae declinations. is bes _ Date. Star, &e. eon ean - —— abe oe eae Weight. | Light. | Star. |Ephemeris. as AS AA he bake I : Co or or 199 10 199 10 1873. uw “uw “a “uw July 23 | Polaris, near East Elonga- | Direct ...... 8 4 tion. Reversed ..-. 5 4 32. 63 ~—1.46 | +0.04 | —0.06 32. 57 0.8 5 | vase d Orccsécrgessdacsae ead Reversed ... 9 5 | Direct ...... 5 5 33. 98 —1.46 | +0, 04 | —0. 06 33. 92 1 26: |eeyness GO apseeucerscee seuss 3 Reversed ... 5 5 : Direct .-..-- 5 5 32. 66 —1.46 | +0.04 | —0. 06 32. 60 1 25 | ¢ Ursa Minoris, near West | Direct ...... 7 5 ; Elongation. Reversed ..- t 5 34. 44 +1.50 | +9. 02 | +0. 03 34, 47 1 | 26: seece 0G? is a qrotibisini weciasisiertts ie Direct ...... 5 4 : Reversed ... 5 4 31. 82 +1.50 | +0.02 +0. 03 31. 85 0.8 25 | e Urs Minoris, near West | Reversed -.. 5 5 Elongation. Direct ...-.- 5 5 34. 40 +1.48 | —0.05 | —0.07 34. 33 1 WieiGbtedsm Gan cies ences veces tyes cee ue pee adabinetoe tenet chen avs Se meweseseeawies 199° 10’ 33.37 + 0.30. To reduce this azimuth to the azimuth of triangle-side South Base — North Base, a correction, +33.92, is to be applied, derived from the coérdinates of the azimuth post referred to South Base, already given. There results for azimuth of North Base, Keweenaw Point, from South Base, Keweenaw Point, 199° 1 07".2940".30 west of south. AZIMUTH AT FORD RIVER. § &. Observations for azimuth were made at Ford River station, July 10, 11, 12, and 13, 1874, by Assistant Engineer G. Y. Wisner. The instrument used was Troughton & Simms 14-inch theodolite No. 1. It was mounted on a heavy wooden post, set firmly in the ground. The stars observed were: 32 Camelopardalis, near Western Elongation. Polaris, near Eastern Elongation. e Urs Minoris, near Western Elongation. A Urse Minoris, near Upper Culmination. Their places were taken from the American Ephemeris. Time was given by a chronometer. The observations were made according to the following programme: : Five or more readings on azimuth mark. . The same number of readings on star, with level rallies, 3, Reversal of telescope. 4. Five or more readings on star, with level readings. 5. The same number of readings on azimuth mark. The zero-line of the horizontal circle was changed but once during the series of observations. Twice, however, the microscopes were turned 180° when the telescope was reversed, so that micro- scope A had, in all, the followirg positions for readings on mark: On July 10 and 11, 28°; on July 12, 2080; and on July 13, 358° and 178°. This disposition was not such as to secure a good elimination of periodic or accidental errors of graduation, though both are known to be small with the instrument used. A determination of the value of one division of the striding level used was made on July 11 by comparing it with the well-determined level of Wiirdemann zenith-telescope No. 19, compar- isons being made at temperatures varying from 50° to 82° Fahrenheit. These determinations, which indicate a nearly uniform increase in the value of one division of the level with the tem- perature-increase, were used in the reduction of the observations. : §5.] AZIMUTHS. 649 The azimuth mark was a light limited by a slit about one-fourth inch wide in the box contain ing the lantern, and 1.75,miles distant from the azimuth post. The azimuth of the star for each observation was computed by the usual formula sin t cos g tan 6 —sin @ cos 0’ wherein A, t, and 6 are the azimuth, hour angle, and declination of star, respectively, and @ is the latitude of the place. In the reduction, the differences between the individual readings on the star and the mean of the readings on the mark for one position of the telescope were taken to obtain individual results for azimuth of the mark. The following tables give the individual and mean results for azimuth of mark from each star for the separate nights. They are corrected for diurnal aberration, and give the azimuth of the mark from Ford River trigonometrical station, the instrument being centered vertically over the geodetic point of the latter. The individual results are affected by collimation, periodic and acci- dental errors of graduation, and by errors due to instability of the theodolite during the interval etween pointings to the star and mark. tan A = Azimuth at Ford River. AZIMUTH OF LINE FORD RIVER— AZIMUTH MARK. (Observer, G. ¥Y. Wisner. Instrument, Troughton & Simms 14-inch theodolite No. 1.] Star, date, &c. | Telescope— | Azimuth of mark. Means. Result for star. a | Od “ o oF “ Oo ff a“ 32 Camelopardalis, near West | Direct ...... 218 38 63. 68 Elongation, July 10, 1874. 59. 27 a=12> 48™ 175.0 62. 07 5=84° 06 02/.38 63. 89 61. 63 218 38 62.108 Reversed - -- 57. 26 55.88, - Ba. Bd 55. 3€ me . ge 53.78 55. 364 218 38 58.736 ‘ Polaris, near East Elonga- ; Reversed ...' 60. 14 | tion, July 10, 1874. | 58. 00 a=1 12™ 268.5 58. 48 6=88° 38/ 03/.78 | 55. 96 56. 96 60. 97 62, 21 58, 960 Direct ....-. 62. 12 64. 56 63. 77 63, 67 62. 86 61. 85 62. 83 63, 094 61, 027 s ee teak ee 32 Camelopardalis,near West | Direct ...... 59. 41 Elongation, July‘11, 1874. 64, 99 a=12h 48™ 169.8 63. 12 5=84° 06/ 02/.25 64. 49 62, 23 = 62. 94 62. 863 Reversed ... 64. 31 63. 04 61, 91 59. 52 61. 43 63. 48 62, 282 62. 572 82 Ls 650 ASTRONOMICAL DETERMINATIONS. Azimuth at Ford River—Continued. Star, date, &e. zimuth of mark. Means. Result for star. Polaris, near East Elonga- tion, July 11, 1874. a=1"12™ 279.5 5=88° 38/ 30/’.88 | Telescope— | A | Reversed -..| Direct .-.... ° + aw 218 38 63.79 63. 74 64. 46 66. 43 65. 98 67. 31 65.71 64. 64 62. 25 63. 55 56. 43 56.19 56. 83 58. 57 57. 84 55. 94 57. 42 59. 37 59, 27 59. 62 218 38 64,786 57. 748 218 38 61. 267 A Urs Minoris, near Upper Culmination, July 11, 1874. a=19" 50™ 555.3 6=88° 55! 35.3 Direct .....- : Reversed .- 59. 53 61. 68 61. 01 62.18 62. 33 - 61. 63 60. 92 61. 69 61. 57 60. 18 61. 346 61. 198 61. 272 | e Urs Minoris, near West Elongation, July 11, 1874. a=16" 59™ 045.2 5=82° 14’ 30'.43 Reversed ... Direct ...... 59. 28 61. 38 60. 75 58. 75 61.18 61, 42 63. 38 63. 28 60. 75 61. 69 60. 268 62. 104 | 61. 186 Polaris, near East Elonga- tion, July 12, 1874. a=1h 12m 289.4 5=88° 38' 03.97 Direct ...... Reversed ... 55. 98 56. 63 55. 59 54.10 54. 60 54. 56 54. 96 56. 80 54. 61 55.10 58. 62 58, 20 59, 37 59. 24 58. 57 59. 33 59. 08 57. 26 57. 43 56. 30 58. 340 56. 791 ([Cuar. XXIV, §5.] AZIMUTHS. 651 Azimuth at Ford River—Continued. Star, date, &c. Telescope— | Azimuth of mark. Means. Result for star. | | ° Z uy o # “ fo} y “a A Urs Minoris, near Upper) Reversed ... 218 38 53. 41 Culmination, July 12, 1874. 58, 55 a=19> 50™ 558.1 i 57. 43 8= 88° 55! 35.7 | 56. 98 i q 56. 02 218 38 55.478 Direct ....-. 61. 68 : 58. 83 | 62. 98 4 62. 50 | 60. 18 61. 234 218 38 58. 356 ' 1 e Urse Minoris, near West | Direct ..---. 53. 00 | Elongation, July 12, 1874. ; 57. 67 a=16" 59” 048.1 55. 13 | 8=820 14’ 30/.66 ‘ 52. 59 52. 68 54. 214 Reversed ...' 58. 49 57. 834 | , 56. 024 32Camelopardalis, near West) Direct ...... 62.88 | Elongation, July 13, 1874. 64. 59 a=12 48m 168.5 66. 75 ; 5=84° 06/ 02/.0 65. 84 \ 64. 26 64. 855 Reversed . .- 56.16 52. 05 53. 382 |. 59. 118 Polaris, near East Elonga- | Reversed .-. tion, July 13, 1874. a=1h12™ 298.5 6=88° 38/ 04.1 58. 611 a n 2 oS | | “59.00 | | 58. 493 | 58. 552 A summary of the preceding results is given in the table following. The individual results in column 6 are corrected for periodic error by the formula given in Chapter XIV, B, § 6, and are further corrected in the table for the errors in the declinations taken from the American Ephemeris in the manner explained in § 4. The mean result for azimuth of mark from a star on any date is weighted according to the number of pointings to the star. Hence, there results for weighted mean azimuth of mark, 218° 39! 00/.17+0".50 west of south, the probable error being derived from the discrepancies between the individual results and the weighted mean. 652 ASTRONOMICAL DETERMINATIONS. [Ciap. XX1V, Azimuth at Ford River. SUMMARY OF RESULTS. AZIMUTH OF LINE FORD RIVER— AZIMUTH MARK. Number of | Animuth. | Reduction to Auwers’ pointings— | 42Imuth. declinations. Position of ' Declina- | Corrected : Date. Star, &c. telescope. | | . tions from one agimuth of | Weight. | a ee | mark, | star. | a As 44 | aogier Santen i Nate coe AT I oo; oF 1874. 218 38 218 38 a aw a “ July 10 | 32Camelopardalis, near West | Direct ...... 5 5 Elongation. Reversed -.. 5 5 59. 06 +1.48 | 41.55 | 42.22 61. 28 0.5 V1, |eestece AO swaaatatiaeeatecs sas Direct ....-. 6 6 Reversed ... 6 6 62. 89 41.43 | £1.55 | 42,22 65.11 0.6 WD) encase 16” .ctccetcagy seeeemees Direct ...... 6 6 Reversed er 6 6 69.05 | 41.43 | 41.55 | +2. 22 61. 27 0.6 10 | Polaris,near East Elongation | Reversed ... 7 7 | Direct .....- 7 7 61.13 —1.42 | +0.04 | —0. 06 61. 07 0.7 Dl ccsex 06 cecccteuecesdae s2eEe3 Reversed ...| 10 10 Direct ...... 10 10 61. 37 —1,42 | +0.04 | —0. 06 | 61.31 1.0 12: | seeees Oi cis Sclsicth ewiecieniaae Direct ...--. 10 10 Reversed ... 10 10 57.13 —1.42 | +0.04 | —0. 06 57.07 1.0 13) |csees DO) added sieieniaicrsece Reversed ... 8 8 Direct ...-.. 8 8 58. 32 —1.42 | +0.04 | —0. 06 58, 26 0.8 11 | A Urse Minoris, near Upper | Direct .....-. 5 5 Culmination. Reversed ..- 5 5 61.45 +0.01 | -+0.59 | +0. 01 61. 46 0.5 1D lakes DO) sascdiesatesied see sige Reversed ... 5 5 Direct .---.. 5 5 58. 79 +0.01 | +0.59 | +0. 01 58. 80 0.5 11 | « Urse Minoris, near West} Reversed ..-. 5 5 Elongation. Direct ...... 5 5 61. 52 +1.45 | —0.05 | —0. 07 61. 45 0.5 12 Jeosmes OO. ar celasnndciniscemeccy Direct ...... 5 5 Reversed .. 5 5 56. 43 +1.45 | —0.05 | —0.07 56. 36 0.5 WieightedimeaMcccciic cen siasinse ac dnsarciscaeewsan soe db slemtcesesenwaes Hot obnnsecdsatwoemts 218° 39’ 00’.17+0".50. The angles between Cedar River station and Azimuth Mark, and between Azimuth Mark and Pine Hill station, were observed with the angles of the triangles at Ford River station, 16 com- bined measures of each being obtained. The adjusted value of the angle Cedar River — Ford River — Azimuth Mark, is 187° 03’ 59.89 (Chapter XV, C). The probable error of the measured value of this angle resulting from the adjustment is + 0.44 (Chapter XV, C,§7). Using this value for the probable error of the above adjusted angle, we have for the azimuth of the primary side Ford River — Cedar River, 81° 33' 00".28+ 0.67 west of south. AZIMUTH AT BRUCE. § 6. Azimuth was observed at Bruce station on three nights in July, 1872, by Assistant Engineer A. R. Flint. The instrument used was theodolite No. 1, made by Repsold & Sons. It was mounted on a heavy oak post vertically over the geodetic point of Bruce triangulation station. The method of observation was the same as that followed at Ford River, and already explained, except that the numbers of pointings on the mark and on the star for one position of the telescope were not generally equal. Time was given by a chronometer. The stars observed were a, 0, and e Urse Minoris. During the series of observations the horizontal circle of the theodolite had three different positions, as indicated by the following readings of Microscope A on the azimuth mark, viz: on July 12, 61°; on July 14, 175°; and on July 16, 270°. The light of Long Tail Point light- house, about 6 miles distant, was used as an azimuth mark. The position of the ball on the dome vertically over the lamp of this light-house was accurately determined by pointings from primary stations Bruce, East Depere, Oneida, Little Tail Point, and Red Banks (see Chapter XV, C). § 6.) AZIMUTHS. 653 The first of the following tables gives the individual and mean results for azimuth of the mark from each star for each night. The second table gives a summary of the results in the first table, corrected for periodic error according to the formula given in Chapter XV, B, § 4. These results are further corrected in the table for errors in the declinations taken from the American Ephemeris, as explained in § 4. Weights proportional to the number of pointings to the star are assigned to the means given in the last column of the first table, except for the result from 5 Urse Minoris. The pointings to this star for telescope reversed were referred to station Fort Howard, and the above-named result is found by applying the adjusted value of the angle at Bruce between Fort Howard and the light-house, viz: 59° 49’ 05.85 (Chapter XV, C), to the azimuth of Fort Howard computed from these pointings. For this reason the result from 5 Urse Minoris is given a weight equal to three-fourths that indicated by the number of pointings. Azimuth at Bruce. AZIMUTIL OF LINE BRUCE—LONG TAIL POINT LIGIIT. (Observer, A. R. Flint. Instrument, Repsold theodolite No. 1.] 1 { | Star, date, &c. Telescope— | Azimuth of mark. | Means. | Result for star. | ° ' u oS # “ Polaris, near East Elonga- | Direct ...... 139 16 05. 34 tion, July 12, 1872. 05. 47 a=1" 11” 575,05 05. 80 5=88° 37! 21.7 06. 13 ; 03. 69 | 05. 43 05. 58 08. 02 04. 21 06. 35 07. 04 05. 42 139 16 05.71 . | | | | \ ‘ Reversed -.- 06. 50 05. 69 07. 33 1 08. 28 08. 40 | 10. 18 06. 04 | 06. 82 06. 91 06.49 | | 07. 13 ' 07. 02 07. 23 139 16 06.47 Direct ...... 00. 09 Wa! oe cA | tion, July 14,1872. 00. 61 a=1" 11™ 588,76 02. 62 8=88° 37! 21.8 03, 30 OL. 64 03.62 | 04. 53 04. 60 02. 63 06. 24 06.12 | 06. 54 05.79 | | 04. 66 04.570 j | Polaris, near East Elonga- | . | | Reversed -.. 02. 81 03. 84 05. 07 03. 85 ASTRONOMICAL DETERMINATIONS. Azimuth at Bruce—Continued. Star, date, &c. | 8 Ursa Minoris, near West | Elongation, July 14, 1872. a= 18" 13™ 549.53 8= 86° 36! 27".7 | e Urse Minoris, near West Elongation, July 14, 1872. a=16> 59™ 16*.97 6=820 14 47.0 Polaris, near East Elonga- tion, July 16, 1872. a=1" 12” 008,72 5=88° 37/ 22.1 Telescope— | Azimuth of mark, | Ol” uw 139 16 06.26 04.79 | 03. 65 | 04. 24 02. 67 04. 63 03.16 02, 83 05. 59 07. 54 06.15 07. 29 06. 49 07.76 | 09.21 | 09. 64 | | | Reversed ...: 06. 90 06. 20 04. 40 04, 00 05.10 03. 60 05. 00 06. 50 Reversed . .. 04. 89 04. 63 05. 06 05.17 04. 51 03. 25 04. 10 03. OL Direct ...... 05. 04 05. 23 05. 98 05. 33 04. 22 02. 82 05. 20 04, 44 Reversed ..- 139 16 04. 03 07. 46 05. 38 05. 05 04, 33 04. 78 | Result for star. ° ‘ uo | | | 139 16 054. 74 05. 21 04. 55 [Cuar, XXIV, $7. AZIMUTHS. 655 o aAsimuth at Bruce. SUMMARY OF RESULTS. AZIMUTH OF LINE BRUCE—LONG TAIL POINT LIGJIT. Number of in Reduction to Auwers’ pointings— a declinations. es ; Position of Hi a _| Corrected | Date. Star, &e. telescope. Hons from azimuth. “Weight. To To |Ephemeris.| 24 | i mark. | star. | dé ae 44 | oe | | Aine | + 1872. * 139 16 | 139 16 | \ ” | ‘July 12 | Polaris, near East Elonga- | Direct ...... 7 12 ‘ tion. Reversed... 7 12 05.87 | —1.87 | +0.04 | 0.05.) 05, 32 3 | TA. [ones se OG eepetdeeegd Mecca! Direct ....— 8 | Reversed . .. 4 8 03. 72 —1.37 | +0.04 | —0.05 | 03. 67 2 : 16 |..---- OO. secieaiseincmmuinwases ses Direct ...... 6 8 ; i j i Reversed ... 8 8 03. 87 —1.37 | +0.04 | —0.05 ! 03. 82 2 4 | 6 Urse Minoris, near West | Direct ..---. 3 8 a ‘Elongation. Reversed ... x 7 8 05. 96 +1.39 | +0. 33 | +0. 46 06. 42 1.5 | 14 ¢ Urse Minoris, near West Direct ...... 3 4 i Elongation. Reversed...) 4 4 05.68 | +142; —0.04| -0.06' 05.62 1 | { : : . Oo ft a “ate Wieighted mean ssa. ja25 skescgecadcanesan sass eeethamedeeeaecae- des oben atame ee 139 16 04. 59+ 0. 33. + 0.31 139 16 04. 90+0. 33 west of south. Correction for diurnal aberration Azimuth of line Bruce—Long Tail Point Light......-...-..--. .-----+----- The weighted mean of the results in the above table is 139° 16’ 04.594 0/’.33, the probable error being derived from the discrepancies between the separate results and the weighted mean. Applying to this mean the correction +0”.31 for diurnal aberration, there results for the azimuth of the line Bruce—Long Tail Point Light-house, 139° 16’ 04”.90+ 0.33 west of south. AZIMUTH AT MINNESOTA JUNCTION. § 7. Observations for azimuth were made at Minnesota Junction station on September 3, 4, 6, 7, and 8, 1873, by Assistant Engineer A. R. Flint. The instrument used was the theodolite No. 1, by Repsold & Sons. It was mounted vertically over the geodetic point of the triangulation station on the post previously used in latitude determinations. The stars observed were a, 6, and 2 Urse Minoris and 51 Cephei. The method of observation was substantially the same as that followed at Ford River and Bruce stations, §§ 5, 6, except that the telescope was usually reversed midway between each series of pointings to the azimuth mark. Time was given by a chronometer. The azimuth mark was a light limited by a slit about one-half inch wide in the box containing the lamp, and about 2 miles distant from the azimuth post. In the reduction, the differences between the azimuths of the star at the times of observation and at elongation were computed, and these differences were applied to the circle-readings on the star to give the corresponding circle-readings for azimuth at elongation. The mean of these read- ings for one position of the telescope, subtracted from the mean of the corresponding readings on the mark, gives the horizontal angle between the star at elongation and the mark. In the following tables the first column gives the date of observation, the star observed, its right ascension, declination, and azimuth at elongation, the latter being denoted by A,. The second column gives the readings on the mark for each position of the telescope and their means. The third gives the readings on the star reduced to elongation and their means. The fourth gives the mean angles between the star at elongation and the mark for each position of the telescope and their means, and the fifth gives the result from each star for azimuth of mark. The individual results.in the fourth column.are affected by collimation, accidental and periodic errors of gradua- tion, and by such errors as arise from lack of stability of the instrument or the post supporting it dacing the interval between the pointings to the star and to the mark. The position of the tele- scope is indicated in the tables by the letters D and R. ‘ 656 ASTRONOMICAL DETERMINATIONS. (Cuar. XXIV, Ls Azimuth at Minnesota Junction. AZIMUTH] OF LINE MINNESOTA JUNCTION—AZIMUTH MARK. (Observer, A. ht. Flint. Instrument, Repsold theodolite No. 1.] 180 24 21.85 110 58 59. 75 | Readings on star | Angle between i Star, date, &e. Readings on mark. reduced to elon- mark and star ‘Result for star. | gation. at elongation. - | oO d “uw ° a we °o y “we | Oo f a“ Polaris, near East Elonga- D. 291 23 23.00 D. 180 24 21.43 : | tion, September 3, 1873. 21, 25 22. 68 i a = Lh 12 628, 12 21. 50 21, 23 | 5 = 88° 87! 54.7 21320. 21.46 | : Ap = 1819 53° 07". 84 21. 00 22, 50 | | 21. 60 21. 80 ; | Means ..291 23 21. 60 R. 111 23 19.95 | i} | | | RB. 00 24 13.34 19. 00 ‘i 14.19 | 20.50 12. 32 | | 19.70 12, 21 | | | 20. 00 12. 37 19.25 | 11.79 | Means ..111 23 19.73 | 00 24 12.70 110 59 07.03 Mean os ccsnencsisarecs | edehenekcee nena 110 59 03.39 | 292 5211.23 | | Polaris, near East Elonga- D. 111 54 42.40 R. 180 55 44. 27 | tion, September 4, 1873. 39. 00 43.11 | ; a = 1" 12™ 52°, 55 40.20 43.07 | 8 = 88° 37 55.1 40. 50 41.98 | ; _ Ap = 1819 53! 07", 29 R. 291 54 45.00 41.58 | | 45.75 43.11 | ‘ 43.50 43.16 ‘ 45.25 41.98; | eee TE ent 42, 31 | Means ..291 54 42.70 180 55 42.73 110 58 59.97 | | D. 111 54 43.75 | D. 00 55 44.19 | | ; 44, 50 44. 87 | 45, 25 45, 81 45.00, 46, 52 | | | R201 54 48.50 46.61 | | 49.70 46. 32 | 50.75 46.96 | ! | | 50.00 | 47.06 | hesiovs sting wvdcanste=espwae 47.78 | | ' | | Means ..111 54 47.18 00 55 46.24 | 110 59 00.94 | ie ih RG asia acer “110-59 00.46 | 292 52 07.75 | $7] AZIMUTHS. Azimuth at Minnesota Junction—Continued. Readings on star Angle between Star, date, &c. Readings on mark. reduced to elon- mark and star | Result for star. gation. ; at elongation. | °o , “a ° d AF °o f de fe} * $Urse Minoris, near West R. 232 28 05.50 | D. 294 55 29.85 Elongation, September 4, 07. 00 29, 83 1873. 06. 25 28. 34 a=18) 13™ 175,20 05. 50 28. 18 6=86° 36’ 36.9 D. 52 28 02.25 29.10 Ag=175° 19 34.80 02. 00 29, 23 03. 10 03. 60 Means .. 52 28 04.40 294 55 29. 09 117 32 35.31 R. 232 28 05. 25 R. 114 55 27.16 07. 50 26. 86 07. 85 28. 73 07. 25 28, 98 D. 52 28 05. 25 29. 28 06. 50 26. 76 05. 75 05. 75 Means . .232 28 06.39 114 55 27.96 117 32 38.43 NLGRN one tenen ds Sacnen ial hecaasicaeantnaes 117 32 36. 87 292 52 11.67 51Cephei, near East Elonga- D. 351 57 56.25 R. 62 54 34.26 tion, September 4, 1873. 56. 00 33. 36 a=6) 40™ 16".42 57. 50 33. 85 5=87° 14’ 00”.4 R. 171 58 00.75 33. 49 A e=183° 48/ 49/.52 00. 55 33. 39 01. 45 32. 80 33. 98 34. 36 Means .-171 57 58.75 62 54 33. 69 109 03 25. 06 D. 351 57 58.25 D. 242 54 33.96 59. 50 33. 97 59. 50 35.15 R. 171 58 02. 25 36.77 03. 50 34. 57 03. 50 34, 63 36.71 36, 55 Means -.351 58 01.08 242 54 35. 29 109 03 25.79 MMR co cpinnsteccceese b coiiabnee uasodedeats 109 03 25. 43 292 52 14.95 83LSs 657 658 ASTRONOMICAL DETERMINATIONS. Azimuth at Minnesota Junction—Continued. (Cuap. XXIV, Readings on star Angle between Star, date, &c. Readings on mark. reduced to elon- mark and star | Result for star. gation. at elongation. ° ¥ “ue o f a“ ° ‘ ue o t a“ A Urs Minoris, near West D. 351 57 58.25 | D. ° 237 37 13.68 Elongation, September 4, 59. 50 13, 99 1873. 59. 50 15. 39 a = 195 51™ 29%, 90 R. 171 58 02. 25 14. 39 5 = 88° 55/ 43.8 03. 50 14. 48 Ag = 178° 31! 25”. 73 03. 50 14, 21 15. 96 16. 31 Means... 351 58 01.08 237 37 14. 80 114 20 46. 28 | R. 171 5804.75 | R. 57 37 16.42 | 05. 50 . 14. 59 : 05. 50 15.13 D. 351 58 01.50 15. 81 02. 00 15. 64 01. 25 15. 87 16. 26 15. 54 Means.. 171 58. 03. 42 . 57 37 15. 66 114 20 47.76 Mie and vaiie.tccis uae sien | scene ewan sats 114 20 47. 02 292 52 12.75 Polaris, near East Elonga- D. 8811 02.75 | R. 157 12 05. 66 tion, September 6, 1873. 03. 65 05.77 a = 1) 12™ 538, 40 03. 00 05. 74 & = 88° 387 55.7 03. 75 05. 77 Ag = 181° 53’ 06”. 56 R. 268 11 12.75 06. 66 12. 45 05. 69 13. 00 07. 35 13. 00 06. 88 06. 05 05. 59 06. 52 Means.. 268 11 08. 04 157 12 06.15 110 59 01.89 D. 88 11 06.00 D. 337 12 03.38 06. 55 03. 82 06. 35 04. 89 07. 25 05. 42 R. 268 11 17.00 05. 77 17. 00 05. 26 16.75 04. 68 16. 25 04. 52 04. 34 05.18 06. 27 Means.. 88 11 11.64 337 12 04. 87 110 59 06 77 . Méanl soos 20 Yee oc s|eseeenindion Fsceaine 110 59 04.33 | 292 52 10.89 AZIMUTHS. Azimuth*at Minnesota Tunction—Continued. 659 Star, date, &c. Readings on mark. Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result_for star. fe} t a 5° A “a o , “a fe} 7 we A Urse Minoris, near R. 208 35 17.75 D. 274 14 27.80 ; West Elongation, Septem- 18. 95 28. 61 ber 6, 1873. 19. 00 27. 03 ' a= 19h 51™ 278, 81 D. 28 35 12.50 27. 05 5 = 88° 55! 44,2 ~ 12.75 27. 30 Ag=178° 31 26,28 13. 50 26. 91 Means .. 28 35 15.74 274 14 27.45 114 20 48, 29 * R. 208 35 21.00 R. 94 14 29.76 ; 22. 00 31. 66 21. 50 31.10 D. 28 35 14.00 30. 73 13. 75 32,17 13.70 30. 94 Means . .208 35 17. 66 94 14 31.06 114 20 46.60 MCAM sacsusccste vec end| ois scteredossecsees 114 20 47.45 292 52 13.73 Polaris, near East Elonga- R.” 176 29 39.75 | D. 245 30 35.89 tion, September 7, 1873. 38.70 34. 73 a=1 12™ 53s. 86 39. 20 36. 59 6 = 88° 37’ 56.0 38. 95 37. 22 Ag=181° 53’ 06”, 14 D. 356 29 40.75 37. 02 40. 05 35. 45 40. 25 36. 28 40. 75 36. 09 40. 50 37.18 40. 50 35. 85 Means ..356 29 39.94 245 30 36. 23 110 59 03.71 R. 176 29 40.75 R. 65 30 30.50 41.10 30. 91 41. 25 30. 60 40. 75 30. 53 41. 40 30. 35 41. 60 30. 13 D, 356 29 43.00 30, 38 42, 25 31. 32 42.15 31. 37 41. 50 31. 51 Means ..176 29 41. 56 65 30 30.76 110 59 10. 80 Mea ...-00ccccceeccns|ccocesccsncncesentace 110 59 07. 26 292 52 13. 40 660 ASTRONOMICAL DETERMINATIONS. Azimuth at Minnesota Junction—Continued. (Cuap. XXIV, Star, date, &c. Readings on mark. Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result for star. 6 Urs Minoris, near West Elongation, September 7, 1873. a=18> 13™ 16*.04 §=86° 36/ 37.1 Ae=175° 19! 35.08 ior it “ R. 262 38 43.75 oO t “ D. 325 06 10. 22 43. 50 09. 03 44,75 09. 53 44, 50 10. 06 D. 82 38 45. 60 08. 93 45. 20 09. 06 45.10 09. 07 45. 80 09. 07 Means .. 82 38 44.78 325 06 09. 37 117 32 35. 41° R. 262 38 46.50 R. 145 06 07. 42 46. 90 08. 10 48, 20 07. 81 47. 00 09. 02 D. 82 38 47.00 08, 22 46. 50 07.74 46.75 06. 55 47.00 08. 22 Means. .262 38 46.98 145 06 07. 89 117 32 39.09 Mean s22cceseas wiseldsjoceneecameaeereesess 117 32 37.25 292 52 12, 33 51Cephei, rear East Elonga- R. 262 38 46.50 | D. 333 35 27.20 tion, September 7, 1873. 46. 90 27. 85 a=65 40™ 175.81 48. 20 27. 94 5=87° 14’ 00’.0 47. 00 28.17 A ¢=183° 48’ 50.08 D. 82 38 47.00 28. 38 46. 50 29, 92 46.75 28. 96 47. 00 28. 29 47.75 28. 58 47.75 28. 16 48. 20 28. 50 Means.. 82 38 47.23 333 35 28. 36 109 03 18. 87 R. 262 38 47.75 R. 153 35 27.97 49. 50 28. 59 49. 00 27. 98 48.75 27. 84 49. 05 28.19 49. 00 28. 42 49. 00 27. 65 D. 82 88 48.75 27, 25 49. 25 27,27 48. 95 26. 54 48.75 27. 33 Means ..262 38 48. 89 153 35 27.73 109 03 21.16 MGA cocescacscecreens lheeetentateteeseig se 109 03 20. 02 292 52 10.10 §7.] AZIMUTHS. Azimuth at Minnesota Junction—Continued. Star, date, &c. Readings on mark. Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result for star. : 110 59 07. 96 ° ‘ a“ fo} t uw o ‘ uw oO x “ A Urs Minoris, near West D. 30 54 56.05 R. 96 34 05.17 Elongation, September 7, 56. 25 05. 94 1873. 57. 50 07. 03 a=194 51™ 26%, 83 57. 00 07. 04 5=88° 55/ 44.4 R. 210 54 54.50 05. 46 Ae=178° 31! 26.55 54. 50 05, 27 54.75 08. 28 55. 00 08. 13 54. 50 08. 19 53, 75 07.79 07. 44 Means ..210 54 55 38 96 34 06, 89 114 20 48. 49 D. 30 54 58.75 D. 276 34 13,42 58.75 13. 42 59. 00 13. 91 59. 50 13. 81 59. 25 13. 30 60. 00 13. 39 ' R. 210 54 56.95 13. 64 ! 57, 25 13. 72 | 57. 50 12. 75 | 57. 30 12. 94 14. 26 | Means ...30 54 58.48 276 34 13.51 114 20 44. 92 1 | MICA Y: jcscd wes odeetel| sec eedijeayeaeee’ Sate 114 20 46.70 292 52 13. 25 Polaris, near East Elonga- D. 29 50 12.00 D. 278 51 08.19 tion, September 8, 1873. 12, 50 07. 70 a=15 12™ 545, 36 11. 50 08. 83 5=88° 37’ 56.3 11. 50 08. 60 Ag=1819 53’ 05”. 73 12. 50 09. 13 12, 50 09. 50 12. 00 09. 75 ‘ 09. 52 Means ...29 50 12.07 278 51 08.90 110 59 03.17 R. 209 50 04. 50 R. 98 50 52, 92 04.75 53. 90 04, 35 53. 02 05. 50 53.11 06. 75 52. 43 06. 75 54. 03 06. 00 51. 45 06. 50 52. 26 Means ..209 50 05. 63 98 50 52. 89 110 59 12.74 292 52 13.69 662 ASTRONOMICAL DETERMINATIONS. zimuth at Minnesota Junction—Continued. (Crap. XXIV, Star, date, &c. Readings on mark. 6 Ursae Minoris, near West of u D. 252 49 50.00 Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result for star. oj “uo D. 135 17 19.76 Elongation, September 8, 49. 50 21.03 1873. 49. 75 19. 74 a=18" 13™ 15%, 66 49. 60 20. 62 & = 86° 36’ 37.2 21. 33 Aeg=1T5° 19’ 35”, 22 19. 67 19. 59 Means ..252 49 49.71 135 17 20.25 117 32 29.46 R. 72 49 44.65 | R. 315 17 05.63 45.75 06. 53 45. 75 04. 34 45. 25 05. 80 04.18 06. 25 05. 41 Means .. 72 49 45,35 315 17 05. 45 117 32 39.90 MOAN crstsisis caricisiowis cletsall dace e isan tesecaen 117 32 34. 68 292 52 09. 90 51Cephei, near East Elonga- R. 72 49 46.25 R. 323 46 19. 98 tion, September 8, 1873. 46. 20 18. 23 a=6b 40™ 18%, 26 45.75 17. 98 & = 87° 13’ 59/9 46. 00 20. 25 A = 1839 48’ 50”. 21 19. 93 Means .. 72 49 46.05 323 46 19.27 109 03 26.78 D. 252 49 54.00 | D. 143 46 34.50 54. 25 33. 27 53. 75 33. 64, 53. 60 33, 51 33. 65 Means . .252 49 53.90 143 46 33.71 109 03 20.19 Meanie ieesic escaain seis 109 03 23.49 292 52 13.70 §7.] AZIMUTHS. 663 The next table gives a summary of the preceding table, the results in the sixth column being corrected for periodic error by means of the formula given in Chapter XV, B, § 4. A correction is also applied in this table on account of errors in the declinations taken from the American Ephem- eris, aS explained in § 4. Weights proportional to the number of pointings on the star are assigned to the separate results. Azimuth at Minnesota Junction. SUMMARY OF RESULTS. AZIMUTH OF LINE MINNESOTA JUNCTION — AZIMUTH MARK. i Number of : Reduction to Auwers' t pointings— | Azimuth. declinations. | Position of Declina- Corrected : Date. Star, &c. telescope. one from azimuth | Wei, ght. i To To |, mena dA 6 ae of mark. mark. | star, |~“Pemeris. a, | ‘ Ore Go. F | 1873. : 292 52 292 52 | | Sept. 3 Polaris, near East Elonga- | Direct ....../ 6 6 | . ‘ - 7 ee | tion. Reversed...| 6 6 12.93 | —1.38 | +0.04 | —@.06 12. 87 06 | 4 Neecteste OO escseos seerseceseicse Direct .----.- 8 9 | i | Reversed ... 8 Oo 4 09. 45 | —1.38 | +0.04 | —0. 06 09. 39 0.9 6 |e... do sccetuigeereaneed Direct ...... 8 un Reversed ... 8 11 | 10. 90 ; —1.38 | +0.04 | —0. 06 10. 84 11 | © axed OO uscd tandetaaeinons Direct ...... 10 1 3 : | | Reversed . . + 10 10 13. 42 —1.38 | +0.04 | —0. 06 13. 36 1.0 \ 8 lsassee OO eps.s'clee sentewes seees5 Direct .....- i 7 8 ; | Reversed .. | 8 8 12. 57 —1.38 , +0.04 | —0. 06 12. 51 0.8 4.6 Urae Minoris, near West | Direct .... | 8 6 | Elongation. Reversed . - zl 8 6 10. 45 -+1.88 | +0.32 | +0. 44 10.89 0.6 W Neseiniess GO pceemes Setetees sunset Direct .....-. | 8 8 | Reversed... 8 | 8 11.98 | 41.38 | 40.32 | +0. 44 12, 42 0.8 Slabs lid doosiniasnae Direct ...... be fh ge | Reversed ... 4 7 | 09. 02 +1.38 | +0.32 | +0. 44 09. 46 0.7 4 | A Urse Minoris, near West | Direct ..... 6 8 a F Elongation. Reversed .. + 6 8 | 12. 46 +1.38 | +0.59 | +0. 81 13. 27 0.8 Cihisekee OG Accaisiinesteniece seer Direct ..--.. 6 6 | Reversed ... 6 6 | 12. 99 +1.38 | +0.59 | +0. 81 13. 80 0.6 Th beens GO wee mast secex cards ses Direct ...... 10 11 Reversed ...| 10 11 | (12.36 +1. 38 | +0.59 | +0.81 13.17 Li 4 | 51 Cephei, near East Elonga- | Direct ...... 6 8 | tion. Reversed -.. 8 | 14.99 —1.38 | +0.22 | —0.30 14. 69 0.8 © anon Woche ta eoaniniane nie Direct ...... [a 11 | Reversed .-.| 11 11 | 09. 87 —1.38 | +0.22 | —0.30 | 09. 57 11 B liesu ce A .pceagoacsdeees acne Direct 20140. 4 5 Reversed ... 4 5 | 13,81 —1.38 | +0. 22 | —0.30 13. 01 0.5 : | | fe} / uw a“ Weighted moan........- --.000 secnecceeeee conte e cee cere ee cece ee cecece cee ee cneeer eee eeees 292 52 11.99+0. 32 Diurnal aberration 0.31 Aim wth Of MAT es. icsice cicasincwaildsineaniewwieecvidas pene pws Vewee Rees Caegetuneseseoeee 292 52 12.3010. 32 From these results and weights the weighted mean 292° 52’ 11’.99+. 0.32 is derived, the prob- able error being computed from the differences between the separate results and weighted mean. Applying to this mean the correction +0’.31 for diurnal aberration, there results for azimuth of mark 2929 52/ 12/.30+0/.32 west of south. 664 ASTRONOMICAL DETERMINATIONS. [CHar. XXIV, The angle between the primary line Minnesota Junction-Horicon and the line Minnesota Junction— Azimuth Mark was carefully measured with the instrument used in the azimuth work, twenty-two combined measures of it being made, giving the mean value 20° 21! 27/7270. In the adjustment of the triangulation between Fond du Lac Base and the line Minnesota Junction-Horicon, a weight unity was assigned to the mean value of an angle resulting from twenty combined measures; and the probable error of the measured value of an angle of weight unity, shown by this adjustment, is 10.38. (Chapter XV,C,§7.) Assigning this probable error to the above angle between Horicon station and the azimuth mark, there results for the azimuth of the line Minnesota Junction—Horicon, 272° 30’ 45” .03+40".350 west of south. AZIMUTH AT WILLOW SPRINGS. § §. Azimuth determinations were made at Willow Springs station on the nights of October 2, 3, 4, and 5, 1874, by Assistant Engineer A. R. Flint. The instrument used was the Repsold theodolite No.1. It was mounted vertically over the geodetic point of the triangulation station on a heavy oak post. The stars observed were u, 6, and 2 Urse Minoris and 51 Cephei. The method of observation was precisely the same as that followed at Minnesota Junction (§ 7). Time was given by a chronometer. The azimuth mark was a light limited by a vertical slit about one- fourth inch wide in the box containing the lamp, the mark being about 1 mile distant from the azimuth post. The observations were reduced in the same manner as those made at Minnesota Junction, and the following tables give the results arranged in the same form as those shown in § 7, to which reference may be made for a more detailed explanation. The mean result for a star given in the fifth column, however, is here corrected for diurnal aberration. v3] AZIMUTHS. 665 : Azimuth at Willow Springs. AZIMUTH OF LINE WILLOW SPRINGS— AZIMUTH MARK. (Observer, A.R. Flint. Instrument, Repsold theodolite No. 1.] Readings on star An gle between | Star, date, &c. Readings on mark. reduced to elon- mark and star Result for star. | gation. at elongation. | ° i “ oO t aw oO - uw ° i uw ' Polaris,near East Elongation, | D. 46 10 03.90 R. 112 56 54.51 | October 2, 1874. , 04. 25 53. 45 a=1" 13™ 215, 08 ! 04. 10 55. 16 5=88° 38/ 26. 4 ; 04. 25 54, 65 ' Ap=181° 49 17". 46 52, 98 | R. 226 10 08.00 54.31 08. 00 53, 82 08. 70 54, 32 07. 50 Means... 10 06. 09 56 54.17 113 13 11.92 D. 46 10 04.50 D. 292 56 53.99 05. 15 54. 07 04. 33 54,19 04. 50 53. 67 54, 51 R. 226 10 08.50 54. 66 09. 75 52, 92 10. 25 53. 88 09. 50 Means.. 10 07. 06 56 53, 99 13. 07 Meat. civecesaseewscaclensans eewewens eneaee 113 13 12.50 295 02 30.27 & Ursz Minoris, near West R. 226 10 04.00 D. 286 34 58.73 Elongation, October 2,-1874. 04. 00 59. 76 : a==18h 12™ 438, 97 03. 50 58. 10 : 5=86° 36’ 37”. 5 05. 00 35 01.92 Ag=175° 27! 22.17 34 59. 59 t D. 4610 00.25 35 00.37 2 02. 50 34 59.93 00. 75 35 00.45 03. 00 Means. . 10 02, 88 34 59, 86 119 35 03. 02 R. 226 10 07.50 R. 106 34 55.03 03. 30 55, 39 04. 00 55. 04 05. 25 56. 25 56. 03 D. 46 10 00.30 55. 36 02.70 55. 90 02. 90 55. 92 02. 50 Means. . 10 03. 56 34 55. 62 07. 94 MGOH os cixacmevaxeenwel senna cnenenenas see 119 85 05,48 295 02 27.96 84 Ls 666 ASTRONOMICAL DETERMINATIONS. Asimuth at Willow Springs—Continued. [Cuar. XXI1V, | Readings on star; Angle between Star, date, &c. Readings on mark. reduced to elon- mark and_ star | Result for star. | gation. at elongation. | Poecea natal natn ee ‘ ofr “we Oo ‘ “ oO # “a | Oo t “a | 51 Cephei, near East Elonga- D. 0 03 20.00 | R. 68 43 30.72 | | tion, October 2, 1874. 20. 50 29, 03 a = 6 41™ 049.33 21.00 | 30. 88 i § = 87° 13! 55.90 20. 50 28. 86 Ay = 183° 42/ 35.11 28. 70 R. 180 03 23.50 29. 41 21.75 29. 45 23. 00 29, 32 23, 50 Means. . 03 21. 72 43 29.55 | 111 19 52.17 D. 0 03 19.00 D. 248 43 30,12 21. 00 30. 30 22. 00 31. 04 : 19. 50 31. 54 ; 33. 72 R. 180 03 25.30 29, 94 25. 50 32. 99 24. 75 30. 89 24. 50 Means 03 22. 69 43 31.32 51. 37 Mieiinia <2 jo ccokce anette eceetecat aedacdncee 111 19 51.77 | 295 02 27.19 | ra i | A Urse Minoris, near West R. 18u 03 23.20 D. 243 35 08.33 | Elongation, October 2, 1874. 24.70 08. 45 a = 19> 49™ 475,74 24. 50 07. 73 | 8 = 88° 55! 57.70 23. 00 08. 12 A y= 178° 34! 11.42 09.17 | D. 0 08 21.50 13. 71 \ 20. 50 10. 66 23. 50 19. 01 23. 00 Means.. 03 22. 99 09. 40 116 28 13.59 1 R. 180 03 24.25 R. 68 35 08.53 23. 30 08. 54 22. 50 05. 84 26. 00 03. 68 03. 12 D. 0 03 22.00 04. 39 23.70 | 04.17 23. 00 | 05. 56 22.00 | Means 23. 34 05. 48 17. 86 MGAR 32 escijeecme net a seemenaee cunerens 116 28 15.73 295 02 27.46 §2] AZIMUTHS. Azimuth at Willow Springs—Continued. Star, date, &c. Readings on mark. o | | Readings on star reduced to elon- gation. | Angle between mark and';star at elongation. Polaris, near East Elonga- ‘ i 38. 00 R. 136 ‘i 21. 30 tion, October 3, 1874. 37. 50 25. 35 a= 1) 13 215,97 ; 38.25 ° 23.76 | 3 = 88° 38/ 26”, 80 | 38. 00 22. 76 Ag = 1819 49' 167.92 | ' OR, 240 04 37.00 | 24.40 | ‘ 36.50 | 22. 85 i 35.50 | 23. 07 34,80 | 25.07 | Means.. 04 36.94 23.57 | 113 13 13,37 D. 60 04 40.00 | D. 306 51 27.48" 38. 50 29.58 | 39. 70 29.26 | 35. 00 28 61 | 27.33 R. 240 04 35.80 27. 24 35.00 26. 92 39. 50 26.99 35. 50 Means.. 04 37.38 51 27.89 09. 49 Meant. cjocsscecesviesese:|nseeyceusese des abet 113 13 11.43 nn ol 6 Urse Minoris, near West | R. 240 04 33.50 | D. 300 29 33.53 Elongation, October 3, 1874. 35. 30 30.13 a = 18> 12” 435, 51 34.90 30. 31 3 = 86° 36’ 37". 40 34. 80 31.49 | Ag = 1750 27 22.09 28, 89 | D. 60 04 36.75 32.63 35. 50 31.55 | 37. 80 31.66 | 37. 30 /Means.. 04 35.98 31.27 | 119 35 04.71 1 R. 240 04 35.00 | R. 120 29 31.24 35. 00 30.24 | 36.70 30.71 35. 00 29.03 | 30.32 | D. 60 04 35.50 27.00 | 36. 00 29, 87 36. 00 30. 43 35. 90 Means .. 04 35. 64 29. 86 05. 78 MGA ii nin ined ohne eee 119 35 05.24 Fo a rll Result for star. | 295 02 28. 66 295 02 27. 64 667 668 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, alzimuth at Willow Springs—Continued. Readings on star | Angle between Star, date, &c. Readings on mark. reduced to elon- mark and star | Result for star. | gation. at elongation. | Oi 3% “ oF au Of “ oF “ 51 Cephei, near East Elonga-— D. 90 01 15.50 | R. 158 41 21.19 tion, October 3, 1874. : 16. 30 16. 06 a = 6h 41 049,94 15.90 | 14. 01 § = 87° 13! 55/90 15. 30 14. 01 A= 183° 42! 35/07 21.31 R. 270 01 16.50 20. 34 \ 16.60 23, 44 14.50 | 22.12 | | 15.40 | “Means.. —01:15..75 41 19.07 111 19 56.68 | a : | D. 90 0114.50 D. 338 41 22.68 15. 00 21.06 | 12. 00 19. 69 13. 50 21.32 19.75 ; R. 270 01/15. 50 20. 66 14.50 21.21 | 13, 30 21.23 | 14.00 | ‘Means... 01 14.04 41 20.96 53. 08 | Meenas esc cna Leseonccaes eaters 111 19 54.88 | 295 02 30.26 | A Urs Minoris, near West | R. 270 01 16. 50 | TD. 333 32 57.84 Elongation, October 3, 1874. | 14.00 | 57. 88 a = 19» 49™ 46,30 | 14. 50 | 58. 27 5 = 88° 55’ 57’’.80 | 14.75 | 59. 82 , Ag =178° 34/ 11.55 58. 00 D. 90 01 11.50 | 57.44 13.50 | 58.56 | 14.00 | 59.16 14. 00 Means.. 01:14. 09 32 58.57 | 116 28 15.72 i \ | | R. 270 01 14.50 R. 153 32 58,79 14. 60 56, 28 11€ 28 16.08 295 02 27.94 i 14. 50 57. 66 | 14. 30 57. 69 | 57.18 |! | D. 90 01 13.50 57.42 | | 13. 00 57.55 | | 13, 50 58.34 | 14. 50 | Means.. 01 14.05 32 57.61 | 16. 44 | | §8.] AZIMUTHS. Azimuth at Willow Springs—Coutinued. Readings on star | Angle between Star, date, &c. Readings on mark. reduced to elon- mark and starat | Result for star. gation. elongation. oO a a oO t a | Oo A a | oO t aw Polaris, near East Elonga- D. 140 01 32.50 | R. 206 48 14.08 | tion, October 4, 1874. 30. 00 14. 54 | a=1 13™ 219,41 29. 50 13, 27 8=88° 38! 27.20 29. 30 13. 92 Ag=181° 49! 16.40 15.11 . R. 320 01 27.00 14.55 28. 00 15. 52 ; 28. 00 15. 94 27. 80 I | Means... 01 29. 03 | 48 14. 62 113 13 14.41 | ! | D. 140 01 31.50 D. 26 48 18.83 | : 29. 00 17. 80 | 29. 50 17. 50 31.50 18. 01 | 18.32 R. 320 01 28.50 18. 34 | . 27. 00 17. 98 27. 20 17.95 27.00 | | Means. . 01 28.90 | 48 18.09 10.81 | | Mean. 1... ceecseeeee lee oleate sOacd 113 13 12. 61 | 295 02 29.32 \ 5Ursx Minoris, near West D. 140 01 32.50 R. 200 26 23.17 | | Elongation October4,1874. 33. 00 22. 32 , a.=18 12” 439.05 32. 00 24.98 5=86° 36/ 37.4 33. 30 22, 22 | A e=175° 27! 22.00 22.37 | R. 320 01 29.00 24. 57 | 27. 20 22.45 | 30. 00 20. 03 | 31.15 | Means.. 01 31.02 26 22.76 | 119 35 08, 26 | or = D. 140 01 32.50] ° D. 20 26 26.87 | 32. 00 26. 96 | 33. 00 26. 34 | | 34.00 26. 61 | 26. 55 R. 320 01 27.50 27. 87 27. 20 28. 29 : 28. 00 28, 67 29.50 Means.. 01 30. 46 26 27.27 03. 19 Mean .........0..220+- Son PRA 119 35 05.72] 295 02 28.03 670 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, Azimuth at Willow Springs—Continued. Readings on star | Angle between Star, date, &c. Readings on mark. reduced to elon- | mark and. star | Result for star. gation. | at elongation. oO t “a ° f a“ °o f a oO t uw 51 Cephei, near East Elon- R. 339 59 45.20 D. 48 89 55. 64 gation, October 4, 1874. » 43. 50 56. 08 ! a=6> 41™ 05.52 44. 50 57. 30 5=87° 13' 56’.00 44.70 57, 22 Ap=183° 42/ 35/.02 56. 38 3 D. 159 59 47.20 55. 70 | 46. 00 55. 60 ! | 48. 50 55.93 | | 47. 20 | Means... 59 45.85 | 39 56. 25 111 19 49. 60 ! R. 339 59 41.50 R. 228 39 43.74 : | 40. 70 45, 87 41. 90 47. 32 | | 41. 00 45. 80 48, 91 | D. 159 59 45.70 45.74 | 44. 50 47.35 | 44. 80 47. 50 45.50 | | Means.. 59 43. 20 | 39 46.53 | 56. 67 j Maite age sle vention peecteeeeenaams 111 19 53.13 | 295 02 28.46 | 8 Ursa Minoris, near West D. 159 59 45. 00 R. 220 24 27.70 ‘ Elongation, October 5, 1874. | 46. 00 28.09 * a=18" 12” 42,61 47. 00 26. 62 | d=-86° 36 37.3 47.50 27.10 | A e=175° 27! 21,91 26. 04 , | R. 339 59 87.00 26. 82 | | 37. 50 26.86 | 36. 00 26.46 , | 35. 00 | Means.. 59 41.38 | 24 26.96 | 119 35 14.42» oo : =| . D. 150504600/ D, 40 249810 | 47. 00 38. 45 | 46 00. 37. 87 | | 46.00 35. 28 37. 27 R. 339 59 35. 00 37. 59 | 34. 50 38, 62 | 34.00 | 38.71 | 35. 50 | Means.. 59 40.50 24 37.74 35.02.76 i | Mean ...---.eeeee A Nery 119 35 08.59 | 295 02 30.81 , $8.) AZIMUTHS. Azimuth at Willow Springs—Coutinued. Readings on star Angle between Star, date, Kc. Readings on mark. reduced to elon- mark and star | Result for star. gation. at elongation. { O° gk Oo ¢ af O oth Op Th 51 Cephei, near East Elon- R. 339 59 38.50 D. 48 39 56, 42 gation, October 5, 1874. 38. 00 57. 03 a= 6" 41™ 06*.08 38. 00 57. 20 3= 87° 13’ 56.0 39. 00 40 00. 27 i Ag= 183° 42) 34” .93 39 58.79 D. 159 59 49.00 57. 32 ° 49. 80 57. 86 50. 00 58. 22 . 49. 20 i Means .. 59 43.94 | 39 57.90 111 19 46. 04 | | OR. 839 59.38.00 R. 228 39 40.19 | 38. 80 42. 98 | 39. 20 | 43. 00 39. 20 43, 21 | 43, 21 D. 159 59 49. 00 42.91 | 50. 30 | 43.19 49. 00 43,17 49. 20 Means.. 59 44.09 | 39 42.73 111 20 01.36 Mean ¢eccciveneceastvn hee eee 111 19 53.70 | 295 02 28.99 A Urs Minoris, near West | D. 170 02 09.00 | R. 233 33 39.02 Elongation, October 5, | 09. 80 | 40. 08 1874. | (9.00 j 39. 35 a= 19» 49m 435.45 09. 00 39. 41 8= 88° 55’ 58.00 | 39. 53 A ==1780 34 11” .82 | R. 350 01 58.50 40. 16 | 58, 50 40. 32 | 58. 50 40.79 58. 00 | Means.. 02 03.79 33 39, 83 116 28 23.96 | | D. 170 02 09.20] D. 58 33 56.95 08. 50 56. 53 09. 00 55.70 09. 00 54. 49 55. 46 R. 350 01 58.30 53. 85 56. 90 56.14 ! 58. 70 56. 64 58. 50 _ Means .. 02 03. 51 33 55. 72 07.79 EM Gail oeicaatenare atte Aas seed clase stae 116 28 15.88 | 295 02 28.01 671 672 ASTRONOMICAL DETERMINATIONS. [Ciap. XXIV, The next table gives a summary of the results in the preceding. The results for azimuth of mark in this table have each been corrected for periodic error by the formula given in Chapter XV, B,§ 4. They are also corrected for the errors in the declinations taken from the American Ephemeris, as explained in § 4. They still involve such residual errors as are not eliminated by reversal of telescope, as may be due to inaccurate indications of the striding level, and such as arise from instability of the instrument or its support between the pointings to the star and to the mark- As may be seen from the readings on the azimuth mark, the collimation of the instrument remained very steady or changed very slowly during any night, and its effect must therefore have been well eliminated by the reversals. The corrections for inclination of telescope axis were usually small; the maximum correction was 6.6, and the average, 2’’.1. Azimuth at Willow Springs. SUMMARY OF RESULTS. AZIMUTH OF LINE WILLOW SPRINGS — AZIMUTH MARK. Number of : Reduction to Auwers’ ! pointings— eae declinations. C ted Date. Star, &c. Position of tel- StS aon fron) tS aetinith escope. s To To | American dA of mark. mark. || Star, Ephemeris. a Aé AA c ‘ ° t 1874. 295 02 295 02 uu . “" “ u Oct. 2 Polaris, near East Elongation ..... Diveet reveeeszxe 8 8 Reversed ..-.-.- 8 8 29. OL —1.32 | +0.04 | —0.05 28. 96 | 3 leseces do. te 5 le .| Direct ......---- 8 8 | Reversed ....-.- 8 8 27. 68 —1.32 | +0.04 | —0.05 27. 63 | AEN weeccs. DO ssoelcre gohan ee ennaced tive osd Direct 8 8 ; Reversed ..-.-.- 8 8 29. 93 —1,32 | +0.04 | —0.05 29. 88 2 | § Urse Minoris, near West Elonga- | Direct ..-..-.--. 8 8 : tion. Reversed .....-. 8 8 26. 95 +1.35 | +0.32 | +0.43 27. 38 © lewancs IUD Sclmope seco uaaiuas gecaes car Direct .......--. 8 8 F ! Reversed ......- 8 & 26, 31 +1.35 | +0.32 | +0.43 26. 74 ‘ 4: laseee GO oe cess eee ees sess eevee sey Direct: -osjsciars.ciae 8 8 ' Reversed ...-.-- 8 8 28. 80 +-1.35 | +0.32 | +0. 43 29. 23 DS lac .oie Oise Blscsmaecees Rdoeatees Direct .... ..-- 8 8 | Reversed ....... gs | 8 30.53 | +1.35 | +0.32 | +0.43} 30.98 | 2 | 51 Cephei, near East Elongation...| Direct ..-..-...- 8. t 8 | Reversed .....-- 8 8 27, 28 —1.35 | +0.21 | —0. 28 27. 00 | 8: | cman i do snsaasnae he ane scan taee Direct .......... 8 8 Reversed ....... 8 8 30, 31 —1.35 | +0.21 | —0.28 30. 03 | 4 |spensn WO se sunederecueaeea wereee ae Direct .......--- 8 8 | Reversed ....... 8 8 27.96 | —1.35 | -+0.21 | —0. 28 27. 68 i 5) | ences WOimcacomenpase pees eamenased Direct ..-....... 8 8 Reversed 8 8 28.50 | —1.35 | +0.21 | —0, 28 28, 22 . | 2 | A Urse Minoris, near West Elonga- | Direct .. 8 8 7 | tion. Reversed 8 8 27.31 +1.35 | +0.59 | +0. 80 28.11 iB. | eee mae OO) ese scree ssmmeeeseenaaeeeee Direct .........- 8 8 Reversed ....--- 8 8 28. 09 +1.35 | +0.59 | +0.80 28, 89 iD? | neal OO) seccwcaneseaisecenicives tees Direct ....--.-.. 8 8 Reversed .....-- 8 8 27. 63 +1.35 | +0.59 | +0.80 28. 43 1 "Wrelghted means ain ose cccicene jccmestiow cade supibccies Sadecicinaes wesesdemeecemeents 295° 02! 28. 51140". 223 Assigning equal weights to the individual results in the last table, their mean is 295° 02! 28/.5140".22, the probable error being derived from the discrepancies between the individual results and their mean. This is the azimuth of the mark from the azimuth post or Willow Springs station. In the $9.) AZIMUTHS. 673 measurement of the angles of the primary triangulation at this station, the direction of the azimuth mark was determined by the following angles: Military Academy — Willow Springs — Azimuth Mark. Azimuth Mark —W illow Springs — Lombard. Shot Tower— Willow Springs — Azimuth Mark. Azimuth Mark — Willow Springs — Shot Tower. The adjusted value of the last of these angles is 289° 27 12/.73 ° (see Chapter XVI, C). The weight assigned to the observed value of this angle in the adjustment is 1, and the probable error of an observed angle of weight 1 shown by this adjustment is +0/.44 (Chapter XVI, C, § 11). The probable error of this angle is somewhat less after adjustment than before, but is not as small as the probable error of an adjusted angle of average weight, inasmuch as it does not enter directly into so many equations of condition. It will be sufficient if we assign +044 as its probable error. There results for the azimuth of the triangle-side Willow Springs — Shot Tower, 2VAO WD’ 41".2440".49 west of south. AZIMUTH AT PARKERSBURG. § 9. Two series of azimuth determinations were made at Parkersburg station in the year 1879. The first series was made during the nights of August 9, 10, 11, 12, 13, 16, and 17, and during the days of August 11,12, and 17. The second series was made during the nights of November 20, 23, 24, 25, and 29, and during the days of November 20, 21, 24, and 25. The observations were made by Assistant Engineer A. R. Flint, with the Troughton & Simms theodolite No.1. Time was given by a chronometer. The stars observed were a, 6, and 2 Urse Minoris and 51 Cephei- The theodolite was mounted on a cut limestone post 5 feet long, 22 inches square, and set so as to pro- ject about 2 feet above the surface of the ground. For the night observations the azimuth mark was a light set vertically over a reference-point on the upper surface of a cut-stone post 3 feet long and 1 foot square. The latter stone was set so that its upper surface was about 15 inches below the ground surface, and it was about 2 miles distant from the theodolite, in a westerly direction. The mark to which the day observations were referred was a target nailed to a tree, about 11 miles distant from the theodolite, in an easterly direction. At the time of the first series of determina- tions the stations in that part of the triangulation had not been erected, and for this reason the azimuth could not then be referred to a triangulation-line. In order to check any possible move- ments of the observing-post and of the post over which the west mark was placed, stakes with - small-headed tacks set in their tops were driven about them, and careful measurements of the rela- tive positions of the points of reference on the posts and the tacks were made. For the observing- post seven such stakes were driven, four so that the heads of the tacks lay in the direction of the west azimuth mark, and three so that the heads of the tacks lay in a line at right angles to that direction. For the reference-post four stakes were set in such a manner that two lines defined by the tacks intersected vertically over the point fixed on the stone. The target constituting the east azimuth mark was referred to a spike driven into the base of the tree to which the target was nailed. Early in November, in erecting the triangulation-station at Parkersburg, the observing- post was disturbed so as to change the azimuth of the line from it to the west reference-stone by about 1”. As this disturbance made the connection of the azimuth observed in August, with a tri- angulation-line, depend wholly on the stability of the reference stakes, a second series of azimuth determinations was ordered and made. When the tacks were examined, however, their relative positions were-found unchanged, and Assistant Engineer Flint, who made the examination, was of the opinion that the position of the instrument in August could be redetermined within 0.05 inch. In the November series the theodolite was set as nearly as possible in that position, which was taken as the center also of the trigonometrical station. As no change in the positions of the azimuth marks from August to November was indicated by their references above named, the direc- tions of the lines from the theodolite to the marks are assumed to have been the same, respectively, 85 LS 674 ASTRONOMICAL DETERMINATIONS. [Cuar. XXIV, during both series of determinations. In making the observations the following programme was adhered to as nearly as circumstances would permit: . Reading on mark. . Reading on star, with corresponding chronometer time. . Level readings, direct and reversed. Reading on star as above. . Reading on mark. . Telescope transited and alidade revolved 180°. ' Reading on mark. Reading on star. . Level readings. 10. Reading on star. 11. Reading on mark. This programme was then repeated in the reverse order. The readings on the mark for each POD DNATP WHE le} star were the same during one night. On each night the circle reading was changed ue where n represents the number of nights intended for a series, being five for the August series and four tor the November series. Observations were made each night to determine chronometer error. The reduction of observations was performed in a manner similar to that followed at West Base, Sandusky, § 10. The cdordinates of the stars were taken from the American Ephemeris for 1879. In the following table the first column gives the star, date, and céordinates of star; the second, the corrected sidereal times of observation; the third, the readings on the mark; the fourth, the readings on the star; the fifth, the resulting angles between the star and the mark; the sixth, the computed azimuths of the star at the times of observation; the seventh, the correction for inclina- tion ot, telescope axis; the eighth, the resulting azimuth of the mark for each observation and the means for each star; the ninth, the resulting azimuth of the mark for each star on correcting the mean for aberration, run of micrometers, and periodic error when it is not eliminated in the mean result for the several nights. The first section of the table gives the observations on the west azimuth mark, made at night; and the secund. section of the table gives the observations on the east azimuth mark, made in the day time. §9.] AZIMUTHS. 675 Azimuth at Parkersburg. J.— AZIMUTH OF LINE PARKERSBURG— WEST AZIMUTH MARK. (Observer, A. R. Flint. Instrument, Troughton & Simms theodolite No. 1.] Time of ob-| Reading on | Reading on Angle | Azimuth of | Level | asimuth of | Result f Star. date. &e. s 2 £ between mark Zin 0. correc: azimuth 0 SU. or ‘ _ . servation. mark. star, and-atar: star. tion: mark. star. . he m. 3. oO fF Mw oO tf aw ° : : Ww °o t a a oO # wt oO f aw Polaris, near East | 18 59 01.58 | D. 1 28 04.10 | D. 71 37 35.33 70 09 31.23 | 181 42 07.59 | +-0.18 | 111 32 36. 54 Elongation, Au- | 19 02 59.57 02,13 43, 00 40. 87 16.03 | +0.18 35. 34 gust 9, 1879. 15 35.54 | R. 181 27 59.13 | R. 251 37 57.17 58. 04 30.70 | —0. 34 32.32: ~| a=1h 14m 595,04 18 39. 54 28 00. 80 | 58. 00 57. 20 31.44 | —0. 34 33. 90 Ran ‘> a 5=88° 89 51”.53 28 57.52 R. 181 28 00.30 | R. 251 37 52.97 52. 67 25.96 | —0. 89 82. 40 32 13.51 27 59.33 47. 93 48. 60 21.62 | —0.89 32, 13 s 42 42.49| D. 128 00.17 D. 71 37 26.23 26. 06 41 59.21 | —0.16 32. 99 45 30.48 27 59.13 18. 30 19.17 51.67 | --0.16 31.74 AM aims cat Sat] ACoA estes cet a a inn 6 Th Ey, cea see ON cata alto d 111 32 33,42 | 111 32 32.96 - 3 Polaris, near East | 18 55 00.51 D. 25 55 06.67 | D. 96 04 23.80 70 09 17.13. 181 41 56.77 | —0.92 | 111 32 38.72 Elongation, Au- | 57 32.59 04. 03 31.07 27. 04 42 03.54 | —0. 92 35. 58 : gust 10, 1879. 19 06 20.49 | R. 205 55 03.00 | R. 276 04 44. 80 41. 80 21.34 | --5.75 33.79 a=1> 15" 005.17 08 58.48 04. 23 49, 30 45.07 24,92 | —5.75 34. 10 5=88° 39" 51.82 | 15 16.47 | R. 205 55 06.50 | R. 276 04 54.18 47. 63 30.17 | —4. 60 37. 94 18 56.47 03, 40 55. 03 51. 63 31,10 | —4. 60 34. 87 38 26.44 | D. 25 55 08.37] D. 96 04 52.40 ° 44. 03 19.33 | 0.00 35. 30 35 47.44 10. 54 49. 23 38, 69 15.11 | 0.00 36. 42 Sua cimeansisea wig Meosieinesncsbretmrinlaelweies Meee Dee seacerccseccee[secceees| 111 32 85, 84 | 111 32 86,38 Polaris, near East | 19 09 44.02 | D. 229 59 12.60 | D. 300 09 05.13 | 70 09 52.53 | 181 42 25.50 | 41.04 | 111 32 34.01 Elongation, Au-| 11 54.01 15. 57 09. 60 54. 03 27.62 | 41.04 34. 63 gust 11, 1879. 18 49.99 | R. 49 59 16,63 R. 120 09 05,43 48. 80 30.81 | —0.41 41. 60 a=1" 15 015,09 21 28, 99 13. 90 02. 53 48. 63 30.53 | —0.41 41.49 BBE so) bar 05 26 36.98 | R. 49 59 12.83) R. 120 09 02.30 49, 47. 27.67 | —0.69 37. 51 29 04.97 14.57 08 58.23 43. 66 25.20 | —0. 69 40. 85 35 49.95 , D. 229 59 13.60 | D. 300 08 58,73 | 45.18 14.77 | -+2.26 31.90 39 12,95 14.77 50.07 | 35. 30 07.53 | -+2.26 34, 49 iMicam eee sedition seen lls extras arate salt Ns ie oh te eeeaehacaett dl easaaeasa dh eae 111 32 37.06 | 111 32 37.35 Polaris, near East | 18 58 42.73 | R. 253 27 18.40 | R. 323 36 51.30 | 70 09 32.90 | 181 42 05.72 | —0.58 | 111 32 32 24 Elongation, Au- | 19 01 08.73 16. 90 58, 27 | 41.37| , 11.28 | —0.58 29.27 gust 12,1879 | 08 56.71 , D. 73 27 17.67 | D. 143 87 01.80 | 44,18 24,20 | —0. 58 39. 49 a=Ih15" 014.97 12-19,70 12.73 02. 83 | 50. 10 27.62 | —0. 38 36. 94 5=88° 39/ 52.82 | 48 30.69 | D. 73 27 14.23 | D. 143 37 08.10 | 58. 87 30.44 | --0.46 37. 03 | 28 03,68 14. 60 08.27 53. 67 30. 02 | --0. 46 36. 81 | : 99 44.67 | R, 258 27 21.77 | R. 323 37 11.60. 49. 83 24.06 | —1. 84 32, 39 : 3217.66 20. 90 06.63 45.73 20.57 | —1. 84 33. 00 Meats osc tates eid heed tai ee A tees ht EPRI ER RTE ere eee aH ee eae Ie 111 32 34.65 | 111 32 34.96 | | : Peet hellg. Poet I { | Polaris, neat East | 19 03 45.90 RB. 97 26 20.43 | R. 167 36 04.00 | 70 09 43.57 181 42 16.05 —0.92 | 111 32 31.56 | Elongation, Au-; 05 57.90 19.97 01. 90 41. 93 19.76 | —0. 92 | 36. 91 Bust 1951879, 16 05.87 ; D, 277 26 16.55 | D. 347 36 14.73 58.18 29,58 | 0.35 | 31. 05 a=1" 15 025,79 1757.87, 16.97 17.20 | 10 0.23 30.06 | — 0.35, 29,48 | 5=88° 39 521.55 24 57.86 | D. 277 26 18.93 | D. 347 36 12.60 09 53. 67 28.31 | —0.46 34.18 | 26 49.85 | 18. 83 12, 57 53, 74 26.87 | —0.46 | 32, 67 | | 33 27.84 | R. 97 26 22.77 | D. 167 36 01.07 | 38. 30 18.45 | —2.76 | 37. 39 35 47.84 / 21. 80 * 85 58.03 | 36. 23 14.95 | —2.76 | 33. 26 1 nee ee Dee Arad | eed eared boc ae Net Aes A kedal iced 111 32 83.56 111 32 33.87 676 ASTRONOMICAL DETERMINATIONS. [Cuar. XXIV, L— eae Liege ae ce adl [eS site | Saeueaeseaaen| cots 111 32 32.45 | 111 32 32.87 § Urs Minoris, | 23 41 17.38 | D. 229 59 15.90 | D. 294 07 33.80 | 64 08 18.01* 175 40 46.09 | +2.65 | 111 32 30.73 near West Elon-| 45 11.37 14.43 | 13. 50 07 59.19 26.12 | +2. 65 29, 58 gation, August | 59 31.36 | R. 49 59 15.63 | R.114 96 39.37 23,90 . 00.66 | +2.76 39.52" 1, 2819s 55 08. 35 15. 87 | 38. 63 22.92! 39 55.64 | 42.76 35. 48 ante oe 0 04 23.33 | R. 49 59 15.57 , R. 114 06 36. 43 21. 02 | 53.45 | 13.15 35. 58 hae lgo anieio | S86 BOAR 87 | oy 1882 16.20. 39.47 23, 43 58,05 | +3.15 37.77 15 36.30 | D. 229 59 16.13 | D. 294 07 13.10 57.10 40 25.09 | $5.18 Th 07 | 13 18.30 12. 83 26.90 08 14.18 37.74 | 45.18 28. 74 Mean ......-. osiuclechsluntee 3 ranee age age eeee aaa hereaaeta sates “on Lee soon 111 32 33.81 | 111 32 34.12 | (8 Ursm Minoris, | 23 47 58.18 | D. 73 27 13.50 D. 137 84 61.70 | 64 07 38.20 | 175 40 15.03 | 1.56 | 111 32 35.27 2 | | near West Elon-| 50 36.13 16.20, 44. 43 28. 28 06.22 | —1. 56 36. 43 I é } ee August | 99 24.11 | R.253 27 20.93 R. 317 84 39.03 18. 10 39 51.69 | —1.50 32.09 | , Ay 1879 02 55. 10 20.77 ; 40, 90 20. 13 52.65 | —1.50 31.02 | =1gb 5, & | pase oe 10 06.09 | R. 258 27 21.43 R. 317 34 54. 80 33.87 40 05.41 | 2.49 29, 85 ; b= 860 36 43.15 12 20.08 21.10 , 35 00. 03 38, 93 12, 53 | —2.19 31.41 2114.07 | D. 73 27 14.93 D.137 35 30.60 08 15.67 | 55.26 | —0. 46 39.18 24 04. 06 15.43 52. 23 | 36. 80 41 13.76 | --0.46 36. 50 Means chi lee.cuaeeeeselt soeucecuacsereae [etereseetanesens feresteetseees esac aimaaes [oan tae 111 32 33.96 | 111 32 34.27 | ‘Ineludes correction for run of micrometers. 679 ‘ { 99] AZIMUTHS. I.— Azimuth at Parkersburg—Continued. Star, date, &c. Time of ob- Reading on Reading on b ‘pet 8 oral Azimuth of oO. Azimuth of — Result for u : servation. mark. star. and stat: | star. tions. mark. star. Suede aes hom. 8. oF u” or a“ ‘OY “ Gv “ a“ oO # au | Or 6 uo 8 Urse Minoris, | 23 45 37.34 | R. 97 26 23.57 | R.161 34 14.00 | 64 07 50.43 | 175 40 24.83 | +0.46 | 111 32 34.86. near West Elon- | 48 27.33 26. 23 01.97 35.74 13.41 | -£0. 46 38. 13 gation, August | 9 9) 34.31 | D.277 26 22.43 | D. 341 93 41.03 18. 60 39 52.11 | —1. 84 31. 67 13,1819. 05 54. 30 23, 33 45.77 92. 44 56.33 | —1.84 32. 05 —18b S17 ' ea le Te ele 11 32.29 | D.277 26 23.73 | D. 341 33 59.73 | 36. 00 40 10.03 | —2.88 31.15 i 2 AQ Qn 1 5=86° 36! 43".35 13 57.28 24, 33 84 09.77 | 45.44 18.81 | —2.88 30.49 | 34 46.24 | R. 97 26 28.38 | R. 161 36 32.30 10 03.97 42 44.85 | 40.12 41. 00 36 50.28 26. 67 53. 00 26. 33 43 06.20 | +0.12 39.99 Mia sees oie Seed teed | | Ss teeselll aie aetsa lca o| aad Josey peace adee | etna ae 111 82 34.92. 111 92 35.23. Hees to : bx 1 os \ I ; : \ 8 Urse Minoris, | 23 42 56.22 | R. 134 14.97|R. 65 42 15.97 | 64 08 01.70, 175 4037.82) 0.00 | 111 92 36.12 | near West Elon 45 40. 22 | 14. 60 06.33. 07 SLB 5,05! 0,00 33. 82 | ‘3 \ ue : ; gation, August 53 28.21 D. 181 34 05.50 | D. 245 41 24.00 | 18. 50 39 59.74 —3.97 | 37. 27 | 16, 1879. | 56 58.20. 05. 50 21.73 | 16, 23 54.41 —3.97 | 34, 21 | | a=18h 11" 175.28 9 93 14.18 | D. 181 34 04.80 | D245 41 19.77 | 14.97 53,71 —3.91 34, 83 | 5=86° 36° 43.85 06 02.17 | 06. 00 23.13 17.13 87.15 | —3.91 36.11 i 150815) BR. 134 12.33 | R. 65 42 02.43 50.10; 40 24.43 | 5 33.18 | | | 17 52.15 13.07 16. 43 08 03.36 | 37.82 1.15 | 32, 81 I Seabee ie | Mean........ sinicaci vee ate Gee al aeeee Ahi | aceasta Pease ti pp dees HT 38 aba | 1H 8 111 32.35.13 i | | 1 | <= ——s ce ee nee Se 8 Urse Minoris, 23 48 01.11 D. 25 27 07.67 D. 89 34 47.80 | 64 07 40.13 175 40 15.76 —2.76 | 111 32 32.87 i near West Elon- 50 18. 11 | 08. 60 | 42. 23 38. 63 08. 10 | —2. 76 31.71 | : | z gation, August 56 57.10 R. 205 27 11.37 | R, 269 34 39. 93 28. 56 39 54.57) 40.12 26.13 17, 1879. |, eae 14. 87 | 39.17 24.30 53. 18 | 40.12 | 29.00 | ( a=18h 11" 16.74 |g g9 43.08 | R. 205 27 14.90 | R. 269 34 87-77 22.87) 53.55 | +0.46 aL | | 8=86° 36 447.02 | 4 25. 08 15.43 | 40 20 ‘04.77 55. 09 | $0.46 | 30.78 10 35.07 D. 25.27 10.77 | D. 89 34 41.43 | 30. 66 40 07.96 | --1.50 35. 80 13 31.06 05.57 52.40 46. 83 18.05 , —1.50 29.72 Mean .....-. dmiectin’ ent aa E Santas paunisesiee [ees Sacer | Geeta tial Seta sot ihiete i, | 111 32 30.89 | 111 32 31.09 8 Urs# Minoris, | 23 47 47.72 D.117 22 48.80 D.181 30 28.20. 64 07 39.40 | 175 40 10.75 | 48.80 111 32 33.15 pear West Elon-| 54 04.73 51.87 27.77 | 35.90 39 52,93 | +3. 80 20. 83 gation, Novem- | 9 95 00.75 , R. 297 21 29.87. R. 1 28 40.67 ; 10. 80 53. 54 | +3.34 46. 08 1 1 s ber 20, 1879. 07 59.75 24. 63 | 45.97 | 21.34 59.08 | 43.34 | 41.03 a=18! 10” 395.21 14 51.76 R.297 21 26.13 R. 1 29 07.27 | 41.14 40 22.75 | 44.14 45.75 | 8=86° 36/ 41.22 18 13.76 ' 27.40 | 25,37 | 57. 97 39.29 | +4. 14 45.47 / 27 23.77 D.117 22 52.03 | D.181 32 12.90 | 9920.87; 41 41.26 +2.99 | 24, 38 | 29 07.78 | 49. 80 | 25, 57 | 35.77 | 55.72 | +2.99 | 22, 94 | Neat cena al aes eee chaeheuee ol feel ee Jstteeeseescees ee Ae La | 111 32 34.96 | 111 32 35.27 al Bi ediotee At act ge . 8 Urse Minoris, | 118 21.40 D.326 55 27.47! D. 3117 43.90 64 22 16.43 | 175 54 40.59 | +4. 48 | J11 32 28, 64 near West Elon-| 2117.41 25,27 | 18 47.33; -23-22,06 | 55 47.28) +4,48 | 29. 65 gation, Novem- | 39 37.44 | R.146 55 24.00 R.211 22.17.67/ 26 53.67 59 34.55 —1.57 | 39. 31 ber 23, 1879. 32 38.95 ' 23. 50 23 15. 40 | 27 51.90 | 176 00 26.79 | —1.57 33. 32 a=18" 10” 38°.43 38 19.47 | R. 146 55 23.70 | R.211 25 48, 83 | 30 25.13 02 59.04 | +011 | 34. 02 8=86° 36! 407.34 40 45.98 23.93 | 26 56.80) 3132.87 04 07.17 | 40.11 ; 34, 41 48 42,49 | 1D. 326 55 23.23 | D. 31 30 57.30 | 35 34. 07 07 58.88 | +5. 04 29, 85 51 44. 50 24. 60 | 32 28, 58 37 03.93 09 31.56 | +5. 04 _ 32. 67 Mean | spualie eee Labeahadecacenetess oSee tea 111 32 82.78 | 111 32 32.94 : 680 ASTRONOMICAL DETERMINATIONS. (Cuap. XXIV, I—Avimuth at Parkersburg—Continued. fees ; f | " Angle . | Level ‘ | s . | Timeofob- Readingon | Reading on = Azimuth of ‘eq.| Zimuth of | Result for Star, date, &e. servation. mark. . star. Dei gen marke star. See mark, star. eed | ieesee 8 43 I | hom. 8. Oo We Oo " ov u" Oo “ “ oj: “ oor “ 8 Urs Minoris, | 23 55 26.95 D.177 02 04.40 D. 241 69 21.20) 64 07 16.80 | 175 39 50.66 | 40.46 | 111 32 34.32 near West Elon: | 58 27.95 00. 50 15. 20 14. 70 48.10 | +0.46 33. 86. gation, Novem- 9 q4 53.95 | R. 357 02 01.43 | R. 61 09 24.90 23.47 51.69 | +228 30.50 berean eyo 06 54. 95 | 01. 93 30. 10 28,17 55,35 | +2.28 29.46 ale oe 12 32.95 | R. 357 02 01.73 | R. 61 09 46.00 44.07 | 40.11.86 | 41.71 29, 30 BBP BO A0N Te. | agua On 01. 27 51. 60 50. 33 17.55 | $1.71 28.93 | 23 48.95 D.177 02 61.17 | D. 241 10 39.83 08 38. 66 41 12.83 | +1. 60 35.77 25 40. 95 02, 23 54. 10 51. 87 26.55 | +1.60. 36. 28 Means: acca |ecgtieaiies cond psaiwbid agece dics) seve beeses satoney| es aberece see) secesaene never sears ete 111 32 82.31 111 32 32.59 1 = a ‘ : ; ae oe { 8 Urew Minoris, 23 49 59.88 D. 27 15 42.57 | D. 91 23 19.47 | 64 07 36.90 175 40 01.74 41.87 | 111 82 26.21 | near WestElon- 52.53.38 42, 23 13.33 8110 39 54.62 | 41.37 24. 89 gation, Novem-' 9 99 92,88 R. 207 15 42.20 | R. 271 22 54.73 12.58 47.48 —0.91 34, 04 ‘ber 25, 1879. 02 37.88 42, 83 55.73 12.90 | 48.67 | —0.91 34. 86 = 18" L088. 00 08 28.87 ' R. 207 15 41.67.| R. 271 23 04.17 22.50 | 58.70 | —2. 44 33.76 §=86° 36' 39.84 | 49 41.87 44. 20 11.27 27.07' 40 05.07 2.44 35. 56 i i | 1753.37) D. 27 15 42.90 | D. 91 28 55.50 08 12. 60 | 35.77 | 41.14 24, 31 111 32 29,58 | 111 32 29,97 20 22. 87 41. 50 24 09. 63 28.13 50. 00 \ +1.14 23. 01 | i Mean .......- | 8 Urse Minoris, 23 58 46.40 | D. 60 32 20.60 D.124 39 41.43 | 64 07 20.83 | 175 39 46.35 | 42.72 | 111 32 28, 24 i near West Elon- 0 01 22, 90 20. 33 | 43.43 28.10 46.59 | 12.72 26, 21 gation, Novem- 9g 4g.99 | R, 240 32 26.00 | R. 304 39 48, 63 22. 63 58. 26 | -+1. 36 36. 99 ber 29, 1879. 10 26.90 27. 63 53, 80 26.17 40 02.97 | +1.36 38. 16 a=18" 10" 36°.97 15 19.90 | R. 240 32 26.67 | R. 304 40 12, 23 | 45. 56 | 21.89 | +0.28 36. 56 nae 0. / tt SBOE BOE BEBE a7 19.00 27. 67 21.20 | 53. 53 | BL.14 | 40.28 37, 84 23 20.40 | D. 60 32 21.90 D.124 41 04.13 08 42.23 4108.05 | 44.77 30. 59 24 38.90 22, 70 12. 80 50. 10 17.41 | 44.77 32, 08 Meum se ae AA dete ale ue ahaa li a Nae he Ue Aes eerste ile 111 32 33.33 | 111 32 33.16 | = ' I os, Be Mergent eae teed Wh | | . | A Urse Minoris, | 125 19.94 | D. 73 27 15.83 | D.140 83 49.33 67 06 33.50 178 39 10.62 —0.11 | 111 32 37.01 near West Elon-| 27 45.93 12.17 47.13 34.96 | 07.18 —0.11 | 32.11 | gation, August | 36 99 99 | R. 253 27 21.23 | R. 320 33 50.17 92.94! 38 59.33 41.50 31. 89 12; 1879, 38 46.91 23. 63 | 48. 43 24. 80 | 58.43 41.50 35.18 | a=19" 45™ 138.47 : \ 8=880 56’ 39”'.40 ; Sih Seat hel Aon lapel oll, aie OS hc Sn Tt eRe © ard te ah 111 32 34.04 111 32 34.51 a. Sc I A Ursa Minoris, | 128 01.00, R. 1 34 12.03 | R. 68 40 48.03 | 67 06 36.00 | 178 39 08.32 , —2.84 111 32 29.48 pear West Elon-| 29 43.99 11. 63 43.73 |” 32 10 06.26 | —2.84 31. 33 . ' - 1 gation, August | 49 52,97 | D. 181 34 03.13 | D. 248 40 26. 40 28. 27 38 59.73 | —4.00 32. 46 TG; AE79: 44 13.96 05. 30 25.77 20.47 59. 83 | —4. 00 35. 36 = h 3 | . alo" fom 10.12 | 5517.94 | D. 181 34 04.10 | D. 248 40 35.33 31.23} 39 07.94 | —3.20 | 33.51 5=88° 56’ 40.58 rine | =a Pts 58 24. 93 04. 07 41. 07 37. 00 12.27 | —3.20 32.07 209 41.91! R. 134 11.73 | R. 68 41 15757 07 03. 84 35.42 | —2.15 29. 43 | | 13 15.90 ' 13. 63 24. 67 11. 04 45.16 | 2.15 31.97 | Me Fa aes clesh ahegs oateell tae dee resene 2a eee he ae sane |e SA 111 32 31.95 | 111 32 31.76 § 9.] AZIMUTHS. I.—Azimuth at Parkersburg—Continued. 681 — aan | Angl Level | , Silk Hons Sava Time of ob- =‘ Reading on Reading on ue © Azimuth of eve’ | Azimuth of | Result for Star, date, &c. ‘ between mark correc- servation. — mark. star. eeand star. star. Hone: mark, star. hom. 8. of “ | o a | oF uy o F u a“ o “ A Urse Minoris, | 1 28 36.90 D. 25 27 09.07 | D. 92 33 40.73 | 67 06 31.66 | 178 39 07.92 | —1.84 | 111 32 34,42 near West Elon- | 33 09. 89 07, 57 37.53 | 29. 96 08.33 | —-1. 84 31. 53 gation, August 41 04.87 R.205 27 12.07 | R. 272. 38 43. 30 | 31.23 38 59.91 | --1.72 26. 96 17, 1879. 43 18.87 10.97 43.47 | 32.50 59.99 | —1.72 25.77 a=19" 45° 09.31 48 28.85 R.205 27 12.90 | R. 272 38 45.23 | 32, 33 39 01.97 | —2.30 27. 34 il yr 1 3=88° 56’ 40.88 50 42.85 | 13. 87 48,47 | 34. 60 03.57 | —2. 30 26. 67 bc : 2 01 44.82 D. 25 27 09.00 D. 92 33 50.50 ! 41, 50 18.28 —2.88 33. 90 04 45.81) 07.18 55.67 | 48. 54 24.26 | —2. 88 32. 84 OU etait Re insce SOsnce ateN taae neaee | LesdulretiR aS aA ea aI | 111 32 29.93 | 111 82 30.26 A Urse Minoris, | 125 07.90 | D.117 22 08.67 D.184 28 57.63 | 67 06 48.96 | 178 39 23.81 | —0.11 | 111 32 34.74 near West Elon. 28 13.91 10. 03 51.80 41.77 | 20.64 | —0.11 38.16 gation, Novem- 3g 19, 92 | R. 297 22 09.93 | R. 4 28 56, 00 46. 07 14.40 | +2. 62 30. 95 ber 20, 1879. 40 29, 93 | 11. 37 55.50 44.13 | 13, 82 | +2, 62 32. 31 jas lowed 3.38 45 57.98 | BR. 297 22 12.00} BR. 4 28 54.70 42.70 15.52 | +1. 60 34. 42 hae 56/ a ae a 48 27.94. 12. 33 56. 43 44.10, 17.11 | +1.60 34.61 ' =“ 56 8295 | D. 117 22 10.80 D. 184 29 01. 97 51.17 26.62 | +1.60 37. 05 58 29.95 08. 83 08.77 9 59. 94 29.79 | +1. 60 31.45 Mean eticics.todenna nen eee Al sdavonee once aie eaeeeaeey hee psSeaestan eget (aeacseed 111 32 34.21 | 111 32 33.89 ents A Urse Minoris, | 1 21 09.85) D. 27 15 43.97| D. 94 22 49.83 | 67 07 05.86 | 178 30 98.77 41.82 | 111 32 24.73 near West Elon-| 25 19. 35 47.17 45. 00 06 37.83 | 29,43 | 41.82 26. 42 # 4 gation, Novem- 31 43.35 | R.207 15 44.03 | R. 274 22 22. 90 | 38. 87 | 15.79 | —2.37 34, 55 ber 25, 1879. 35 01.85 45. 33 19. 40 34. 07 13, 84 | —2.37 37.40 a=19" 43m 07.71 42 05.35 | R.207 15 43.77 | R. 274 22 22. 30 38. 58 | 13.05 | —3. 82 30.70 3=88° 56’ 50.97 45 30.35 43.58 21.17 37. 64 | 14.31 | —3. 82 32. 85 54 56.85 | D. 27 15 46.33| D. 94 22 42.53 56. 20 | 23.42 | 42.68 29. 90 59 24. 85 46. 67 50. 67 07 04. 00 30.57 | +2. 68 29, 25 i 1 ee | Meanie veils aus We atari Ratesheet [ae cose Dae Nateaccanl Seeded 111 32 30.73 | 111 32 31.51 | | d Urse# Minoris, ' 127 46.90 D. 60 32 20.03 | D. 127 39 16.00 | 67 06 55.97 | 178 39 18.68 | 43.20 | 111 32 25.91 near WestElon-; 29 28.90 21.17 10. 67 | 49.50 16.94 | +3. 20 30. 64 gation, Novem- 35 26.90 R. 240 32 26.17 | R. 307 39 04.73 38. 56 12.96 | +1. 32 35. 72 ber 29, 1879. 37 13.90 : 28.07 04. 33 36. 26 12.41 | 41.32 37.47 a==19% 43" 03.28 41 37.40 R. 240 32 27.13 | R. 307 39 02.93 | 35. 80 12,26 | 40.91 37.37 SHEP OH AEE | a BOO 25. 83 | 03. 97 | 38. 14 12.97 | +0. 91 35. 74 50 03.90 D. 60 32 19.50 | D. 127 39 14.43 | 54.98 17.05 | £5.47 27. 59 ! 51 45,90 21.73 15.17 | 53. 44 18.84 | 45.47 30. 87 | . 1 Mean ........ Te Leche ana det ey side Aaalthe techn oheaes paeerriealnee | netraceneet Mobeaadeneses apr aced 111 32 32.65 | 111 32 32. 34 i ! as 682 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, Azimuth at Parkersburg. ° IL.—aziMUuTH OF THE LINE PARKERSBURG—EAST AZIMUTH MARK. 7 | Angi ih of Level | Time of ob-; Reading on Reading on | ne,° Azimuth of | Azimuth of | Result for Star, date, &c. : servation. | mark. star. petraea mar star. Sone i mark. star. h. mM. 8. | oO t Ww | oO f at oO # wt ° , uw | Ww | oO t wu oO Z “a Polaris, nearLower’ 15 15 20. 54 | R. 24 29 00.50 | R.275 18 06.47 | 109 15 54.03 | 180 50 34. 36 | 8.28 290 06 25.16 Culmination, Au- 17 17.53 00.27 | 51.20 | 09. 07 51 19.04 | —3. 28 | 24, 88 used) 1879; 27 04.51 D. 204 28 58.37, D. 9517 23,23, 11 35.14 55 00.02 —4.15 | 31. 01 a=1 15™ 008.92 29 46.51 | 58.10 | * 18 22. 50 10 35. 60 56 00.00 | —4.15 | 31. 45 PEBEP 901 HBN-00 84 54. 50 | D. 204 28 58.37 D. 95 20 13. 80 08 44.57} "57 52.82 —5,24 32, 15 37 35, 50 | 57. 60 | 21 12. 50 07 45.10 58 51.13 | —5. 24! 30. 99 47 02.48 R, 24 28 59.60 R. 275 24 42.78 04 16.87 | 181 02 12.76 | —4.35 25, 28 50 05.48 + 58. 73 25 46.17 03 12. 56 03 16. 56 | —4.35 24.77 AVE Cates Sorel vf zal Sacdalacia hk hc cad cla Dect ET al des A A er ee | 290 06 28.21 | 290 06 28. 67 ea ae : ! Polaris, near Lower| 15 28 16.22 | D. 48 38 10.03 | D.299 22 08.57 109 11 01.46 180 55 26.07 | —0.88 | 290 06 26.65 | Culmination, Au-| 30 23. 22 08. 90 | 56. 18 1012.77, 5612.98 | —0.88 24. 87 gust 12, 1879. 42 49.19 BR. 228 33 10.58 R.119 27 15.90 05 54.63 | 181 00 42.88 | —0.77 36 74 , | a=1> 15m 019.85 44 39,18 | 09.27 56.17 13. 10 01 21.78 | —0.77 34,21 » 8889 3952.30 | 51 25.17 | R. 228 88.10.17 R.119 3018.67, 02 51.5003. 43.57 | —1.98 33. 09 55 41.15 | 07. 20 | 3146.59 01.20.70, 05 11.34 | 1.98 | 30. 06 16 02 35.14 | D. 48 33 08.73 | D.299 3417.23 | 108 58.51.50 07 30.50 40.33 22, 33 05 37.14 | 09.31 35.16.97 57 52.34 08 30.56 +0.33 | 23, 23 | | ' ceca gl Po oiBarssss acs eee cater Reece Sata LEO iat ena este | 290 06 28.90 | 290 06 29.04 \ ! | teks’ | m , Polaris, near Lower! 14 46 06.29 | D. 180 08 03.40} D. 70 40 36.80 109 27 26.60 180 38 58,27 | +237 | 290 06 27.24 Culmination, Au.) 51 44.28 04.27 42 53.50 25 10.77 41 15.01 | $3.16 ' 28,94 gust 17, 1879. 56 32.27 | R. 0 08 07.87 | R.250 44 47.50' 28 20.37 43 10.41) —1.13 29. 65 * q=1" 15" 055.54 | 15 01 34. 25 07.97 46 46.77 21 21.20 45 10.07 | £0.90 | 32.17 | 5=88° 39 53".52 og 15.25 | R. 0 08 08.67 | B.250 47 51.37' 2017.30 46 13.64 | —0.11 | 30. 83 "09 51. 24 08. 60 50 01.20 18 07. 40 48 24.75 | —0.23 | 31.92 | 18: 11.28 | D. 180 08 04.93 | D. 70 51 22.03 16 42, 90 49 42.01 | +2. 94 27, 85 | 19 89.22 04, 20 58 50. 50 14 13.70 52 10.17 | +2. 82 | 26. 69 DGB Rp reece Sess aoe | REESE Lee ll meonat sere eg aaa gues ask DERE ee wnat [seeesece [290 06 29.41 | 290 06 29. 29 | | Polaris,nearLower| 15 41 39.17 | D. 72 00 54.23 | D, 322 54 44. 63 | 109 06 09.60 | 181 00 15.79 | -+1.70 | 290 06 27.09 | Culmination, Au-: 42 59.17 53. 53 55 11.43 | 05 42.10 44,23 | 41.70 28. 03 j Sust17, 1879. | 47 50,16 | R. 252 00 59.43 | R142 56 52.10) 0407.38 02 26.71 | —0.46 33. 58 | a=1 150 059,54 49 45,15 59. 08 57 88.47 | 03 25.56 03 06.75 | —0.46 31. 85 1 —_ ; | bP HBRrSO OSU A | 58 92.15 | TR 252 00 59.93 | 142 98 51.0002 08. 98 04 25.09 | —0.11 34.01 | | 55 36.14 01 01. 98 5933.77 1 28.16 05 07.41 | —0.11 35. 46 | 16 00 37.13 | D. 72 00 57.13 | R. 323 01 15.60 108 59 41.53 06 48.99 | +0. 46 30. 98 ‘02 23,12 54. 80 58. 58 01.27 07 24.22 | +0.46 25.95 Mean ........ PR APULIEA ©) red Oke one ita nm MeeeRn Meena ereee [is Ae tha ddl sheen eee 290 06 30.87 | 290 06 31.00 il | Polaris, near East | 19 46 05.35 | R.115 56 02.30 BR. 7 30 98.23 108 25 24.07 | 181 41 05.57 2.65 | 290 06 26.99 | | Blongation, No- | 51 82.36, 01. 53 19. 53 42.00 40.47.82 | —2.07 27.75 vember 20, 1879. 55 07.36 | D, 295 55 56.90 D. 187 29 54.17 26 02.73 | 33.06 | —4. 95 30. 84 | a=1 15" 25.94 | 20 00 42.37 57.77 31.17 | 26. 60 08.19 | —6.33 28, 46 ; — 889 40/ 207" ; | B= ROAD! 204-86 03 47.37 | D. 295 55 54.20 D. 187 29 15.73 38. 47 39 52,97 | —0.46 30. 98 11 11.38 | 58. 83 28 32.87 | 27 20.96 | 11.92 | -+0. 46 33. 44 17 59.40 R.115 56 02.23 R. 7 28 05.20 ' 37.0338 28.63 | 44.14 29, 80 24 49.41 00.97 27.17. 50 2843.47! 37 39,82 | $5.29 28. 58 Meith sicseeeedl dnvawe vewioctajaucnses vectsuueatesectauesednanens Pinner WL ode lundeuaene taseeuee 290 06 29.61 | 290 06 29. 92 i ‘ §9.] AZIMUTHS. Il.— Azimuth at Parkersburg—Continued. 683 ? | x . Angle . Level . Time of ob- Reading on Reading on 8 3 | Azimuth of Azimuth of | Result for Star, date, &c. servation. | mark. star. aaa star. pong mark. star. hom. s. of “ oF u OP “ oi: a“ “ or a o 7 “ Polaris, near East | 18 59 45.15 | R.145 48 53.20 | R. 37 23 50.27 | 108 25 02. 93 | 181 41 22.98 | —0. 46 | 290 06 25. 45 Elongation, No. | 19 09 44.17 56. 63 2405.00 | 24 51.68 40.25 | —2. 28 29. 60 vember 21,1879. | 44 43.18 | D. 325 48 52.77] D. 217 24 02.37 50. 40 44.53 | —4.56 30.37 a=1 15m 255,40 21 15.19 ! 53.00 01.97 51.08 45.80 | —5.36 31.47 SS eeraonetnle 25 11.20 D. 825 48 52.10 | D. 217 28 59.97 52.13 | 44.17 | 5.36 | 30. 94 29 16.21 | 50, 43 58. 23 52. 20 40.55 | —4.90 | 27. 85 33 22. 22 | R. 145 48 59.87 | R. 37 24 03.30 56. 57 35.00 | —3.76 27. 81 39 23,23 | 58. 87 28 52.33 25 06. 54 23,30 | —2.28 27. 56 Mean ........|.....0-.2004- | Ais ORES Areas AA rental tO 1 te | state, Sealer 290 06 28. 88 | 290 06 29.19 i Polaris, near East 19 19 45.91 , D. 325 29 21.57 D.217 04 87.67 | 108 24 43.90 | 181 41 45.38 | —0.68 | 290 06 28.60 | Elongation, No- 24.99.91, 21.70 | 35. 90 45. 80 43.96 | —1.14 28, 62 | vember 23,1879. 9g 33.91 | R.145 29 23.90 R. 37 04 35.47 48. 43 40.53 | -+0.91 29. 87 | a=1 15 245.30 34 27.91 | 23. 57 | 29. 03 54. 54 32.37 | --0. 34 | 26.57 = j i 3=88° 40°27".73 87 33.92 | R. 145 29 25.10 | R. 37 04 19.83 25 05.27 26.47 | +0.46 | 52.20 44 35, 92 22. 67 | 05. 50 17.37 08.91 | 0.23 25. 85 : 1 47 29.92 | D. 325 29 22.03 D217 08 49.57 32. 46 00.02 | —1.48 31. 00 53 09. 92 22,17 | 27. 80 54. 37 40 39.79 | —3. 08 81.08 Mean... .... je abet ciate! iecadperiaeeabe sate, errata atk fea ae ee en eee cca oats sae 290 06 29.22 | 290 06 29. 85 | Polaris, near East | 18 58 42.26 D.175 36 05.80 | D. 67 10 54.33 | 108 25 11.47, 181 41 19.50 | —0.57 | 290 06 30. 40 Elongation, No- | 19 03 32.27 , 03. 93 11 07.50 24 56.43 29.85 | 10.11 26. 39 1 1 vember 24, 1879. | 97 17.77 | R. 355 36 06.60 | R. 247 11 05.67 25. 00. 98 36.06 | —2, 62 | 34, 37 a=1" 15" 235.80 | 12 46.78 | 06. 67 11. 43 24 55,24 42.16 | —2.96 | 34. 44 8=88° 40/27".96 4g 04.78 R. 355 36 07.30 | R. 247 11 17.37 49. 93 | 44.28 | £0.57 | 34. 78 21 32.79 06. 90 20. 10 46.80 | 44.65 | 0.91 | 32. 36 | | 25 27.29. D.175 36 06.90 | D. 67 11 24.33 42. 57 | 42.89 | 43.53 | 28.99 20 44. 80 05. 63 18.47 47.16 | 37.68 | +274 27.58 | .——, EMT ates ah Feks F son e8 Naha ll Slates aad atonal ea | gasehas tes Ne sd | 200 06 81.16 | 200 06 81.47 Px cee ee i santa oe aa j 1 t Polaris, near East | 19 03 43.03 D. 205 49 26.83 | D. 97 24 30.27 108 24 56.56 181 41 29.84 | +0.68 ! 290 06 27.08 Elongation, No- 10 29. 02 | 99.37 | 39.37 | 53. 00 | 39.68 | 11.60 | 34. 28 : \ ; 4 vember 25,1879. 43 59,02 | R. 25 49 20.90 | R. 277 24 29.70 | 51. 20 | 42.67 | —0. 80 | 33.07 | axlb 15™ 239,33 19 21.02 | 21. 53 30. 83 50.70 | 44,51 | 40.11 | 35. 32 S= 889 40" 2808 21 07.02 , R. 25 49 21.20 R.277 24 32.57 , 48. 63 | 44,39 | —1.14 31. 88 ‘96 17.01 | 22. 53 | 29.90 . 52. 63 41.92 | —0. 75 33. 80 | 29 10.01 | D.205 49 25.37 D. 97 24 39.87. 45. 50 | 39.22 | 40.91 25. 63 38. 37.00 22, 33 ! 31.93 | 50.40 | 33.13 | 41.71 25, 24 ! | ren | 290 06 30.79 | 290 06 31.10 A summary of the data in the preceding tables is given in the following table. As already mentioned, the individual results for azimuth of mark have been corrected for periodic error when it is not eliminated by the shifting of the circle, for ran of micrometers, and for diurnal aberration. They are further corrected in the table following for the errors in the declinations taken from the American Ephemeris (see § 4). They still involve such errors as are not eliminated by reversal of telescope, such as may be due to inaccurate indications of the striding level, and such as arise from instability of the theodolite or its support between pointings to the star and to the mark. The level corrections were usually quite small, the maximum being 6.84 and the average 2.06. The first section of the table gives the results for azimuth of the west mark; the second, for azimuth of the east mark. 684 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, Azimuth at Parkersburg. SUMMARY OF RESULTS. I,—Werst AZIMUTH MARK. | os at | Reduction to Auwers’ / eee No. of pointings— pate | Declivations, Gorreoted Date. Star, &c. tions from 7 | azimuth of | Weight. : telescope. American | dA AS AA mark, Tomark.| To star. |Ephemeris.. “gj. ea yaaa he Ee eee aes | Se aoe oO f °o ‘ ' 111 32 | 111 32 1879. mL i " " " Aug. 9 | Polaris, uearEast Elongation) Direct ....-- 8 8 ' Reversed .-.. 8 8 32. 96 —1.27 | 40.03 | —0. 04 32, 92 1 i 10 licesea OY 2 svweakwonsernen Direct .. 8 8 Reversed -.- 8 8 36. 38 —1.27 | +0.03 | —0. 04 36. 34 1 T) jasscce DO: stsics vies esise esate Direct ....-- 8 8 Reversed .-. 8 8 37. 35 —1.27 | +0.03 | —0. 04 37.31 1 12 | icin sie. OO sciie cece cst zticans], DITGEt soccer 8 8 !* Reversed .-. 8 8 34. 96 —J.27 | +0.03 | --0. 04 34. 92 1 19 Ne netsts Oss sheiciere stele a aichapelataisteteie.e Dire ca. o0c 8 8 Reversed ...! 8 8 33.87 1.27} +0.03 | —0.04 | 33.83 1 16 |..-.-- OO! secsicden teed PSiiee ss Direct .....- 8 8 ! Reversed ... 8 8 | 34.382 ; —1.27} 40.03 | 0.04, 34.28 1 Noy. 23 | Polaris, wear West Elonga- | Direct ...... 8 8 | tion. Reversed -..|: 8 8 t 33. 01 | +1.27 | +0.03 | +0. 04 33. 05 1 Aug. 9 | 51Cephei, near East Elonga- | Direct ...... 4 4 ' ! ; tion. Reversed ..- 4 4 34,11 —1.28 | +0.17 | —0. 22 © 33. 89 0.5 J) dd decease 0: eg sa eeerceewne~s Direct .....- 8 8 ! Reversed ... 8 8 i 35.49 | —1.28 | +0.17 | —0. 22 35, 27 1 12} aiina OO) senate neckocome shakes Direct .....- 8 8 | ' : Reversed .. 8 8 35. 39 | ~—1.28 | +0.17 | —0. 22 35.17 1 13: eee OGh si Sra eee dress Direct ...... 8 g . Reversed ... 8 8 35.08 | —1.28 | +0.17 | —0. 22 | 34. 86 1 1G) esac OG Semis esusasesskesee Direct ...... 8 8 | Reversed ... 8 8 34. 43 1.28 40.17) —0.22 | = 34.21 1 U7 |xsexes 00) s:isertceoteesses uxeses Direct ...... 8 8 ' Reversed ... 8 8 31.37 | —1.98| 40.17/ 0.22 31.15 1 Nov. 20 '....-- praise Nous Baagutuass Direct ...... Be 8 Reversed ... 8 8 33.63 | —1.28 | +0.17 | —0. 22 33.41 1 24 |...--- GO! ea anes) neers eeeeee Direct ...... 8 8 : \ Reversed --. Bet 8 32.67 ° —1.28| 40.17) —0.22! 32.45 1 ae NessO GE arcs "Direct ...... f ) 2 | Reversed ...: 8 8 31. 82 —1.28 ) 40.17 | —0. 22 | 31. 60 i 20 jeasees WO. diinciedig dace eees | Direct ...... | 8 8 | | Reversed . ..! 8 8 34. 63 —1.28) +0.17 —0. 22 34.41 1 Aug.10 | 8 Urs# Minoris, near West | Direct ...... ; 8 | 8 Elongation. , Reversed .. z, 8 8 32.87 | +1.27 | 40.29 | +0.37 33, 24 1 | gee dove ee ied ide aad Direct ...... 8° 8 ' Reversed a 8 8 34,12 +1. 27 | +0.29 | -+0.37 34. 49 1 12 | ee Abi ane eae eee ete Direct ...... » 8 8 | Reversed ...| 8 8 34.27 | 41.27 | 40.29 | +0.37 34. 64 1 13 | ele diaw CO! 2:2 tai garckle orate meats Direct ...... ! 8 8 | Reversed... 8 8 35.23 | $1.27] 40.29) 40.37/ 35.60 1 i 16 |...-.. GO 22:2 sodaawasescemeuees Direct ....-. ( 8 8 i Reversed .- i 8 8 35. 13 +1. 27 | 40.29 | 40.37 35. 50 1 19 [bees Oishi oe eet eos ences Direct ...... 8 8 : | Reversed ... 8 8 31. 09 +-1.27 | +0.29 | +0. 37 31. 46 1 Nov. 20 |.....- 0} coc ccyhartes eee eases Direct .....-. 8 8 f | Reversed ..., 8 8 35. 27 +1.27 | +0.29 | 40.37 | 35. 64 1 aaa doomed! Direct ......) 8 8 | Reversed .. 8 8 32.94 -+1.17| +0.29 | 4.0.34) 33.28 1 24 | eae WO oil eas OS oe Direct ...... 1g 8 : Reversed... = 8, 8 32.59 | 41.27) 40.29) 40.37) 32.96 my $9] AZIMUTHS., 685 alzimuth at Parkersburg—Summary of results—Continued. IL—west AZIMUTH MAkK—Continued. No. of pointings— Azimuth. | Reduction to Auwers’ : | epi 5 ina- Declinations. me : Date. Star, &e. pia eka a = Sa ee dons hen esi teeee oer azimuth of Weight. Tomark. To star. toneneae ee 48 Ady mark. 1 = . 1879. } ” | uw uw ” Nov.25 | 5 Ursw Minoris, near West | Direct -..--. 8 8 i 4 | Elongation. Reversed... 8 8 29.97 $41.27 40.29 40.37! 30.34 1 OF cebu EEE no scckcala brioumeaeenen ged | Direct ...... 8 8 : : Reversed ... 8 8 33. 16 +1.27° 40.29 40.37 33.53 | 1 i Aug. 12 | A Urse Minoris, near West Direct ....-. 4 4 r : Elongation. | Reversed .../ 4 4 34. 51 +1.29 40.57 +0. 74 35.25 | 0.5 TO tases MGs teemencestne aati cater Direct ..---- 8 8 ; Reversed 8 8 31.76 | 41.29 | +0. 57 ! +0. 74 32, 50 1 AW sceaiced AG iislchiclcleleystersted eedes Direct ...... 8 8 Reversed - 5 8 30.26 +41.29 +0. 57 | +0. 74 31.00 *« 2 Nov. 20 ...... douse cased denteivecieces Direct ....... 8 8 Reversed ... 8 8 33.89 | 41.29 40.57 +0. 74 34. 63 1 2 eee. dota 8 | Direct ...... gg 8 | | Reversed... 8 5 8 31.51, 41.29 40.57 | +0.74 32.25 | 1 OO sandler WUE Sen wertiadla ian Pm es Viger Direct vesee- 8 8 , Y ‘ _ Reversed ... 8 8 32. 34 | +1.29 +0.57 +0.74 33. 08 1 Weighted mans... snmsceuesate Shee awake eee uae Seed Bee emeeeee 111° 32! 838.754 + 0.189 IJ.—EAST AZIMUTH MARK. _ No.of pointings— , Azimuth. Reduction to Auwers’ “ a ' ' ina- Declinations. ; Date. Star, &¢. | Bosttion of |_________ ona from amimnathor, Weight. . ' | American dA mark. | | To mark.’ To star. ‘Ephemeris.| gg Ad AA a |} oo ' OF | o ' | 290 06 | 290 06 1879. ‘i | nm . Aug.11 Polaris, near Lower Culmi- ; Direct ...-.- 8 8 | / : nation. Reversed ...| 8 8 28.68 —0.70' 40.03 0.02; 2866 | 1 | 2. db siete ed | Direct ...... 1 8 8 | | Reversed... 8 8 29.04 | —0.77| 40.03 —0.02/ 29.02 1 AD sucess dosed eergee (Direct tees y 8 8 | | | Reversed ... 8 8 29.29, —0.56 | +0.03 —0.02 29.27 lie hi} AT se fio coke kena san etee Direct ...... 8 8 | Reversed... 8 8 + 31.00 | —0.79: 40.08 —0.02 30. 98 1 | ' Nov. 20 Polaris, near East Elongation Direct ...--- 8 8 | Reversed ... 8 8 29.92 | 1.27, 40.03 —0.04| 29.88 st i OY os as Orso k soe Seeee ee toey _ Direct ...... 8 8 | | . | | Reversed... 8 8 29.19 | 1.27 40.03 0.04 29.15 1 | 28 lewsn OOsansene scans eintaleietais | Direct ...-.- i 8 8- . Reversed... 8 8 , 29.85 1.27 +0.03 , —0.04 | 29. 81 1 } DA as cicteae Oi vis cisicl erect navacee eee 8 8 1 - | \ 8 8 | 81.47 1.27 | 40.03 | —0. 04 | 31.43 1 i 25: cee OO sccccsderecason Aeee 8 8 ' ' | ! | Reversed .. 8 8 81.10 —1.27 0.08 | —0. 04 | 31.06 1 t 1 I Weeialite an Gah peae sevs'vaic entalacee ad yasa heed Adeeb eat Aekets te 290° 06! 29'.918 + 0.297 Assigning weights to the individual results for azimuth of mark proportional to the number of pointings made to the star, there results for azimuth of the west mark 111° 32/ 33.75 west of south, and for azimuth of the east mark 290° 06/ 29.92 west of south. 686 ASTRONOMIOAL DETERMINATIONS. (Crap. XXIV, The probable errors of these two results, as derived from the discrepancies between the individual results and their weighted means, are + 0.10 and +-0/’.23, respectively. The former of these may be considered the best value attainable for the probable error of the result for the west mark, but the latter (+0//.23) is probably too small for the result for the east mark, since this result depends on observations to but one star, Polaris, which was observed at but one elongation, the eastern, and one culmination, the lower. It will, therefore, be better to assign relative weights to the mean results for the west and east marks proportional to the number of individual results to them respectively. This gives to the latter a weight =4;,times that of the former, so that the results, with their probable errors, are as follows: (1) Azimuth of west mark, 111° 32’ 33/.75 4.0.19. (2) Azimuth of east mark, 290° 06’ 29/7.92-+ 0/.36. These two azimuths were connected with the triangulation by the following angles, read by Assistant Engineer Flint, with the same theodolite used in the azimuth work. (Chapter XX, C.) The weights given are those of the observed angles. West Azimuth Mark—Parkersburg—Denver, weight 1. Denver—Parkersburg—Claremont, weight 1. Claremont— Parkersburg— East Azimuth Mark, weight 1. East Azimuth Mark—Parkersburg— West Azimuth Mark, weight 1.5. West Azimuth Mark—Parkersburg—East Azimuth Mark, weight 1.5. The first and fourth of these angles will be used in deriving the azimuth of the line Parkers- burg—Denver. Their adjusted values are: (3) East Azimuth Mark—Parkersburg— West Azimuth Mark, 181° 26’ 03/.51. (4) West Azimuth Mark —Parkersburg—Denver, 31° 43/ 41/.74, The probable errors of these angles are less after adjustment than before, but are not as small as those of an average adjusted angle, since one of them (3) enters into but one equation of condi- tion directly, and the other (4) into but two. In view of this, it will probably be best to assign to them the probable errors of their observed values. The adjustment of that portion of the triangulation to which these angles belong shows a probable error of +0/.32 for an observed angle of weight 1. Hence, there are obtained + 0.26 and +0.32 for the probable errors of (3) and (4) respectively. Adding (3) to (2) gives (5) 111° 32/ 33.43 40.44 for the azimuth of the west mark as derived trom the east mark; and combining (5) and (1) with weights derived from their probable errors, the following weighted mean azimuth of the west mark is obtained: 111° 32/ 33.70 £0.17. Adding to this the angle (4) whose probable error is +0”.32, there results for the azimuth of the line Parkersburg— Denver, 143° 16 15”.44+4 0”.36 west of south. AZIMUTH AT WEST BASE, SANDUSKY BASE-LINE. § 10. Azimuth determinations were made at West Base station, on the Sandusky base-line, on the nights of September 13, 14, 17, 18, and 21, 1877, by Assistant Engineer G. Y. Wisner. The instrument used was the Troughton & Simms 14-inch theodoilite No.1. It was mounted ona heavy wooden post set about 5 feet in the ground, and nearly on the line between East and West Base stations, being 153.9 feet from West Base, and 0.447 inch northeast of the line joining the two Base stations. The stars observed were a, 6, and 4 Urse Minoris and 51 Cephei. Time was given by Bond & Sons chronometer No. 255. The azimuth mark was a light set directly over the geodetic point at East Base, about 3.8 miles distant. The observations were made according to the follow- ing programme: 1. Two readings on the azimuth mark. 2. Four readings on the star with level readings. 3. Two readings on the mark. § 10.] AZIMUTHS. 687 4, Reversal of telescope, the pivots remaining on the same wyes respectively. 5. Two readings on the mark. 6. Four readings on the star with level readings. 7. Two readings on the mark. A second set of readings was then taken in the reverse order of this programme. There were thus obtained 16 results for azimuth of mark from each star observed on any date. The circle remained in the same position throughout each night’s work, but was set forward 24° on each succeeding night. ‘ In the reduction the differences between the individual readings on the star and on the mark in the order of observation were taken to obtain individual results for the angle between the mark and the star. The azimuth of the star was then computed for each observation thereon by the formula sin t tan A= is o tan d—sin @ os t in which A, t, g, 6, are respectively the azimuth of the star at the time of observation, the hour- angle, the latitude of the station, and the declination of the star. From these quantities and the corresponding level corrections are obtained the results for the azimuth of the mark. In the following table the first column gives the star, date cf observation, and the codrdinates of the star as taken from the American Ephemeris; the second, the corrected time of observation ; the third, the readings on the azimuth-light; the fourth, the readings on the star; the fifth, the observed angles between the mark and the star; the sixth, the computed azimuths of the star at the times of observation; the seventh, the correction for inclination of telescope axis; the eighth, the resulting azimuth of the mark for each observation; the ninth, the mean result for the star. The relative positions of the telescope are indicated by the letters D and R. 688 | Polaris, near eo Sw 1 Star, date, &e. | i ' Elongation, Sep- tember 13, 1877. a=l1h 14m 255.) 5=88° 39 221.5 Polaris, near East Elongation, Sep- ' tember 14, 1877. a=1) 14™ 251.7 5=88° 89! 22.9 Polaris, near East Elongation, Sep- tember 17, 1877. a=) 14™ 275,0 5=88° 39/ 24.1 ASTRONOMICAL DETERMINATIONS. [Cuap, XXIV, alzimuth at West Base, Sandusky, AZIMUTH OF LINE AZIMUTH POST — EAST BASE, SANDUSKY. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1.] ig ; Pe Angle. 4.3 Level |... p Geto RG) CU tree meds OAS eon, OU ean h. an. & oO , “ o a a“ 2 oO f “a Oo t aw aw ° ¥ “a O «# “ 19 05 4.8 BR. 392 28 21.17 R. 184 98 67.5. 137 49 23.64 181 47 26.26 — 7.33 | 319 36 42.57 08 27.8 20.27 | 39 04. 00 | 16. 27 30. 85 i= 7.33 39.79 09 15.3 21. 0 | 04.17 16.93 31.85 | — 7.12 41. 66 10 30.8 20. 30 04.77 | 15.53 33.29 — 7.12 41.70 18 30.8 D.142 28 20.43 D. 4 39 18.50 01.98 37.88 + 2.83 | 42. 64 20 03.8 20. 63 21.90 48 58.73 37.84 ++ 2.83 39. 40 2L 13.8 21.50 | 21.50 4900.00 | 87.63 | + 2.17 | 39. 80 22 27.3 22.70 20.93 | 01.77 37.22 | + 2.17! 41.16 | 28 37.8 D. 142 28 28.30, D. 4 39 12.53 | 10.77 | 82.40 + 2.25" 45.42 | 30 03.3 22.50 13.17 09. 33 30.62 | + 2.25 42.20 31 04.3 | 21.03. 12, 00 09. 03 | 29. 18 + 2.48 40. 69 32 19.3 20.00 10.70 09.30 | 27.25 +f 2.48 39. 08 4118.8 R.322 28 20.10 | R, 184 38 35. 83 | 44.297 07.74 | — 8.62 43, 39 | 42 54.8 19. 20 31.73 | 47.47 | 03.20) — 8.62 42.07 | 44 00.3 18. 87 27,33 BL. 54 46 59.95) — 8.45 43. 04 45 22.8 18. 60 23, 03 55.57 | 55.62 — 8.45 42.74 | 319 36 41.70 ; | 3 o | 18 58 25.6 R,346 34 26.20 R. 208 44 54.67 | 187 49 31.53 | 381 47 10.96 | — 2.18 | 319 36 40. 46 19 00 01.1 | 25.50 56. 33 29.17 14.85 — 2.13 | 41. 89 56.6 23.17 45 00. 50 22. 67 16.96 — 1.63 38. 00 01 57.1 ' 23.60 02. 10 21.50 19.16 | — 1.63 | 39. 03 08 51.1 | D. 166 34 27.47 D. 28 45 25.83 OL. 64 30.82 | + 6.00 38. 46 10 07.6 27.57 28. 10 48 59.47 32.34 | + 6.00 37. 81 57.6 28. 43 28.77 59. 66 33.22 | + 6.20 39. 08 11 54.1 28.47 , 29. 10 59. 37 34.11 | + 6.20. 39. 68 17 17.6 D. 166 34 28.50 D. 28 45 29.58 58. 97 37.15 | + 6.13 42. 25 18 47.1 | 28, 83 | 29. 67 59.16 37.36 | + 6.13 42. 65 19 38.1 | 27.98 | 31. 60 56. 33 37. 34 | + 6.07 | 39.74 20 32.6 | 28. 80 | 33. 10 55.70 87.26 ' + 6.07. 39. 03 29 27.1 R.346 34 23.83 R.208 45 10.93. 49:12. 90 30.88 | — 3,10 40. 68 30 55.6 , 21,93 06.10 ° 15, 83 28.87) — 3.101 41. 60 31 55.6 | 21.43 05.37 | 16. 06 27.37 | — 2.70 | 40.73 32 59.6 | 20.77 00.77 20. 00 25.63 | — 2.70 | 42.93 | 319 36 40. 25 —<—<$$—aee- i ee —- tse se ay AS eee eee Ss te ee ee Dey yey Bess, pete piel 19 02 32.5 D.190 01 35.17 | D. 5212 15.93 187 49 19.24 | 181 47 18.73 | + 4.38 319 36 42.35 03 53.0 35, 30 21,23 | 14.07 21.36 + 4.37, 39, 80 04 53.0 34, 63 21.93 12.70 23.16 + 4.13 | 39. 99 07 04.0 34,73 28,00 06.73 26.72 + 4.12 | 37.57 1420.0 KB. 10 01 41.83 | R. 232 12 32.43 | 09. 40 34.30 — 8.25 35,45 15 41.5 41.27 34, 50 06.77 34.99 © -- 8.25 ' 33.51, 16 29.5 43.17 34, 60 08, 57 35.30 — 7.45 | 36, 42 | 17 28.0 | 41, 60 35.40 | 06. 20 35.56 — 7.45 34. 31 21 53.0 | R. 10 01 42.10 | R. 232 12 33. 50 08. 60 35. 31 | — 7.38 36, 53 23 14.0 42, 27 29. 45 | 12. 82 34,75 ° — 7.37 | 40, 20 24 11.0 41.17 | 29. 60 | 11.57 34, 22 | — 7.20, 38, 59 25 02.0 42.07 29. 60 | 12. 47 33.66 | — 7.20 | 38. 93 32 33.0 D.190 01 40.43 | D. 52 12 26.20 | 14,23 24, 80 4270" 41.73 , 33 55.5 | 40. 43 24. 90 | 15. 53 22. 42 | + 2.70 | 40. 65 34 51.0 | 40. 53 19. 67 | 20. 86 20.69) + 1.75 | 43. 30 36 06.5 39.77 18, 23 | 21. 54 18.17 | + 1.75 | 41.46 | 319 86 38, 80 | | i I § 10.) AZIMUTHS. 689 Azimuth at West Base, Sandusky—Continued. ‘. a BA ae Angle ‘ “| Level , ae Star, date, Se. Tiation, mare | eaten” betweenmark AMHEED OS | correc: | Attar | Star |: hom. 8. Ot OR! die Oh IE Be A IR. Tf bh | a & op te | Polaris, near East 19 02020 R214 11 35.43 R. 76 2211.6 137 4928.90 181 47 17.26 + 3.93, 310 86 44.49 Elongation, Sep- 03: 16.0 36.07 13. 50 22. 57 19.77 | + 3.32 45. 66 tember 18,1877. | 04. 08.0 30.77 | 15.77 15. 00 21.41 | + 3.88 46.29 a=1> 14m 275.3 51.0 30. 90 | 16.53 | 14.37 | 22.68 | + 3.87 40. 92 [| 8=889 3924.4 | 4910.5 DD. 34 11 26.40 | Dr256 22 10.03 16. 37 | 32.30 | — 6.88 | 41.79 | 13 24.0 26, 33- 13.10 | 13. 23 ° 33.28 ! — 6.87 39. 64 | | 14.180 27. 88 14.17 13. 66 | 33.87 | — 7.25 40. 28 | 15 20.0 24. 88 14. 60 10. 23 | 34.41 | — 7.25 37.39 | 20 32.0 'D. 34 11 26.60 | D. 256 22 13. 70 12.90 | 35,24 | — 6.88: 41.96 | 21 40.0 26.28 12. 83 13.40, 34.96 | — 6.87, 41.49 , 22 30.5 26. 70 11. 98 14.77 34. 67 | — 7.88 | 41. 56 | 28 21.0 26. 90 09.73 17.17! 34.29 | — 7.87) 43. 59 29 57.5 R.214 11 30.83 | R. 76 22 16.37 14.46 28.24 | + 1.00 | 44. 20 31 06.5 31.43 13.73 17.70 26.63 | + 1.50 45, 83 55.0 30. 28 13.43 16. 80 25.40 | + 1.50 43.70 32 45.0 29, 50 11.67 17. 83 24.06 + 1.50 43.39 | 319 36 42, 22 Qe twa ake if sao. cad naan [ese aac la : | ! Polaris, near East’ 19 03 06.2 | R. 238 16 55.67 | R. 100 27 38.87 | 137 49 16.80 181 4717.95 | + 6.25 | 319 36 41.00 Elongation, Sep- 04 19.7 55. 73 | 43.07 12.66 20.27 | + 6.25 | 39. 18 tember 21, 1877. 05 08.7 59. 35 45.17 14.16 | 21.73 | + 6,75 | 42. 64 a=1» 14m 285.2 06 08.2 55.17 | 47.98 07. 24 23.87 | + 6.75 37. 36 BS ERESO! 2505 13 55.2 | D. 58 17 06.17 | D. 280 28 01. 60 04. 57 32.15 | — 4.87 | 31. 85 15 12.2 06.50 , 03. 37 03.13. 32,89 | — 4.87 | 31.15 ' 16 04.2 04.70 03. 33 01. 87 33.26 | — 4.80 | 29. 83 (17 08.7 1.07 03. 88 03. 24 33.59 | — 4.80 | 32. 03 21 16.2 D. 58.17 05.00 | D. 280 28 00. 80 04. 20 83.65 | — 4.95 | 32. 90 2227.7 05.83 | 00. 70 , 05.13 83.23) — 4.95 | 33.41 23 26.7 04.23 | 01.27 02.96, 32.77 | — 5.37 | 30. 36 24 28.2 06.70 00.33 06.37 32.18 | — 5.37 33. 18 , »31 00.2 R. 238 17 02.00 | R100 27 47.50 | 14.50. 25.87 | + 8.05 44. 92 32 14.2 01. 00 45.60 | 15.40 | 23.46 | + 5.05 43.91 33 08.7 16 56.60 | 43. 60 13.00 | 21.95 | + 5.75 40.70 34 04.7 58.33 | 41.67 | 16. 66 | 20.29) + 5.75 42.70 | 819 36 36.70 eh ae oe OS eo Dh Rectan et Ss eee aed Jt a lil eres | 8 Urse Minoris, 28 41 51.5 R, 322 28 15.17 | R.178 20 42.73 | 144 07 32.44 | 175 29 18.42 | —10.75 | 319 36 40.11 near West Elon-, 43 26.5 14.50 | 33.10 , 41.40 09.93 —10.75 40. 52 - gation, Septem-; 44 43.5 14.93 | 28.73 46. 20 | 02. 52 ; ~10.38 | 38. 34 | ber 13,1877. | 46 01.5 1.80 | 22. 33 52.47 98 57.94 10.38 | 40.03 | a=1811e 47.5 | 56 37.5 D. 142 28 19.93 | D.u5x 20 15.40 08 04.58 | 30.06. 0.00 | 34. 59 | RE BO OHIO 5810.5 20. 30 | 11.40 08. 90 28.95 0.00 37. 85 | / 59 19.5 19.90 | 10.77 09.18 28.45) + 0.50 | 38. 08 "9 00. 25.5 20. 20 | 1.77 08. 43 28.61) + 0.50 37. 54 1 i + | 06 46.0 D.142 28 20.73 | D. 358 20 20. 83 07 59. 90: 36.07 | -- 1.68 37. 65 , 08 04.55 19. 93 24.37 55. 56 | 39.17 | + 1.68 26.41 09 05.5 22, 20 , 26. 88 55.37 | 41.88 | + 0.98 38. 23 | 55.0 20.98 27. 80 53.13 | 44.38 + 0.98 38.49 90 08.0 RB. 322 28 17.80 ' R. 17% 20 56.63 21.17 29 82.26’ —12. 00 | 41.43 ) 91 38.0 15. 50 21 07.28 08.27 41.94 | 12, 00 38.21 | i 22 50.5 16.10 , 19. 90 06 56.20: 50.29 | 11.18 35, 36 28 55.5 16.33: 24.17 52.16. 58.10 11.13 39.13 | 319 36 38.27 | De a I es oe, pl eee Sekeisa! pene as 2 87 LS 690 ASTRONOMICAL DETERMINATIONS. [Cuar. XXIV, Azimuth at West Base, Sandusky—Continued. r : at es Angle =! “i »| Level . Star, date, ao, | Timegfob-] Reading on | Reading on Iuetaenamark| Atizath of | apace. | Aaimuth of | Regt fr h. m. 8. 6. oF “ or 7 o fF “ Oo. oF “ a“ O. 4 uw of “ 8 Urs Minoris, | 23 42 39.5 | R. 346 34 16.77 | R. 202 26 49.70 | 144 07 27.07 | 175 29 14.13 | — 4.25 | 319 36 36.95 near West Elon- 44 14.0 17. 00 36. 97 40. 03 06.23 | — 4.25 42. 01 gation, Septem- 45 36.0 15. 20 33. 50 41.70 28 59.89 | — 3,85 37.74 ber 14, 1877. 46 51.5 17, 20 25.77 51. 43 54,55 | — 3.85 42. 13 a=18" 11™ 473.0 56 49.0 | D. 160 34 23.33 | D. 22 26 23. 23 08 00.10 30.02 | + 6.50 36. 62 6=86° 36! 41”.1 58 16.5 24. 10 16. 83 07. 27 29.00 | + 6.50 42.77 59 15.5 25. 37 15. 40 09. 97 28.65 | + 6.07 44. 69 0 00 38.5 23. 57 18. 40 05. 17 28.74 | + 6.07 39. 98 08 10.5 | D.166 34 24.13 | D. 22 26 30.53 07 53. 60 39.56 | + 7.25 40.41 09 30.0 24. 57 35. 27 49. 30 43.23 | + 7.25 39. 78 10 32.5 25. 20 37 33 | 47. 87 46.51 | + 7.00 41. 38 11 30.0 24, 27 41.70 | 42. 57 49.87 | + 7.00 39. 44 20 03.5 | R. 346 34 17.57 | R. 202 27 06. 20 11. 37 29 31.97 | — 4.38 38. 96 21 23.5 15. 60 14. 88 00.77 40.60 | — 4.38 36, 99 22 13.0 17. 70 21.10 | 06 56. 60 46.08 | — 4,37 38, 31 23 37.5 15. 23 27. 33 | 47. 90 56.12 | — 4.37 39.65 | 319 36 39. 87 6 Urs Minoris, | 23 43 51.7 | D.190 01 38.57 | D. 45 54 05.40 | 144 07 33.17 | 175 29 08.07 | + 2.00 | 319 36 43. 24 near West Elon- 45 22.7 39. 10 00. 57 38. 53 00.85 | + 2.00 41. 38 gation, Septem- 46 31.2 38. 83 53 56. 47 42. 36 28 55.98 | + 2.00 40. 34 ber 17, 1877. 47 59.2 37.77 51.77 46. 00 50.26 | + 2. 00 38. 26 a=18! 11™ 458.7 56 50.2 | R. 10 01 45.43 | R. 225 53 32. 03 08 13. 40 30.08 | — 9.70 33. 98 6=86° 36’ 41.2 58 30.7 45. 60 30. 57 15. 03 28.97 | — 9.50 34. 50 0 00 00.2 46. 43 31. 00 15. 43 28,73 | — 8.13 36. 03 02 46.7 46. 27 32. 60 13. 67 30.12 | — 8.12 35. 67 | 08 48.2; R. 10 01 46.30 | R. 225 53 42. 87 03. 43 41.34 | — 8.98 35. 79 10 23.7 46. 43 47. 00 07 59. 43 46.30 | — 8.97 36. 76 11 24.2 45. 93 51. 40 54, 53 49.75 | — 9.45 34. 83 12 41.7 | 46. 60 55. 23 51. 37 54.60 | — 9.45 36. 52 | 21 37.2 | D.190 01 37.33 | D. 45 54 38,83. 06 58. 50 29 42.34 | + 2.02 42. 86 | 23 03.2 38. 10 50. 00 48. 10 52.26 | + 2,03 42. 39 | | 24 04.7 36. 63 58. 77 | 37, 86 59.80 | + 1.30 38. 96 | | 25 32.7 | 38. 83 55 11.27 | 27. 56 30 11.02 | + 1.30 39. 88 | 319 36 38. 22 | 1 | 6 Urse Minoris, 23 46 09.9 | R. 214 11 30.60 | R. 70 03 44.63 | 144 07 45.97 | 175 28 57.56 | + 1.40 | 319 36 44, 93 near West Elon- 47 36.9 30. 00 87. 37 52. 68 51.68 | + 1.40 45. 71 gation, Septem- 48 31.9 26. 57 32. 43 54.14 48.43 | + 1.52 44. 09 ber 18, 1877. 49 49.4 28. 67 27.93 08 00. 74 44,17 | + 1.52 46. 43 a=18) 11" 45+.3 58 12.9 | D. 34 11 19.27 | D. 250 02 57. 83 21. 44 29.27 ; —11.93 38. 78 5=86° 36/ 41.3 59 30.4 19. 20 58. 07 21.13 28. 88 | —11. 92 38. 09 0 00 33.9 19. 17 57. 23 21. 94 29.03 | —12.75 38. 22 01 45.9 18. 60 56. 50 22. 10 29.47 | --12.75 38. 82 07 43.4 | D. 34 11 17.67 | D. 250 08 13.57 04. 10 38.65 | — 6.62 36. 13 09 09.9 18. 03 15. 10 02. 98 42.56 | — 6.63 38. 86 10 16.9 18. 63 17. 40 01. 23 46.00 | — 6.65 40. 58 11 25.9 18. 10 23.17 07 54. 93 50.00 | — 6.65 38. 28 19 02.9 | R. 214 11 27.07 | R. 70 04 15.57 11. 50 29 26.30 | + 3.50 41. 30 20 16.9 27.73 22.97 04. 96 33.76 | + 3.50 42. 22 21 05.9 29. 67 29. 43 00. 24 39.07 | + 4.50 43, 81 22 09.9 27. 60 33. 67 06 53. 93 46.26 | + 4.50 44,69 | 319 36 41.31 | =~ I § 10.) AZIMUTHS. 691 Azimuth at West Base, Sandusky—Continued. Time of ob-| Reading on Reading on Angle | Azimuth of Level | 4,imuth of Result for Star, date, &o. servation. mark. star. [Pane star. | star. Sonar mark, star. h.m. 5s. o u o 7 u o 4 " o -: " u o on “ oF " 6 Urse Minoris, 23 42 14.0 | R. 288 16 59.27 | R. 94 09 38.60 | 144 07 20.67 | 175 29 16.33 | + 3.88 | 319 36 40. 88 near West Elon- 43 36.5 59. 30 27. 33 31. 97 09.12 | + 3.87 44, 96 gation, Septem- 44 35.0 56. 07 20. 20 35. 87 04.37 | + 4.13 44, 37 ber 21, 1877. 45 34.5 57. 43 16. 60 40. 83 28 59.93 | + 4.12 44, 88 a=18" 11™ 449.0 52 24.0 | D. 58 17 05.33 | D. 274 08 58. 67 08 06. 66 37.11 | — 7.93 35. 84 5=86° 36/ 41.2 53 50.5 05. 63 57. 80 07. 83 34.08 | — 7.92 33. 99 54 46.0 05. 27 56. 90 08. 37 32.57 | — 8.13 32. 81 55 43.0 04. 90 56. 07 | 08. 83 31.19 | — 8,12 31. 90 0 00 52.5 | D. 58 17 04.50 | D. 274 08 50.57 13. 93 28.93 | -- 7.08 35. 78 02 16.5 04, 27 52. 07 12. 20 \ 29.80 | — 7.07 34. 93 03 04.0 04.77 56. 83 07. 94 30.54 | — 7.08 31. 40 04 37.0 05. 27 57. 73 07. 54 32,41 | — 7.07 32, 88 12 15.0 | R. 238 16 55.73 | R. 94 09 09. 03 07 46.70 52.97 | + 2.75 42. 42 13 44.0 56. 40 15. 67 40. 73 59.05 | + 2.75 42. 53 14 39.5 57. 90 22. 07 35. 83 29 03.08 | + 2.13 41. 04 15 55.0 57. 93 29. 90 28. 03 09.19 | + 2.13 39. 35 | 319 36 36.12 51 Cephei, near 0 83 53.2 | R. 322 28 16.07 | R. 186 32 38.53 | 185 55 37. 54 | 183 41 16.48 | —11.38 | 319 36 42. 64 East Elongation, 35 23.7 15. 67 47.47 28. 20 23.17 | --11.38 39. 99 September 13, 36 26.7 16. 23 51. 90 24, 33 27.58 | —10. 50 41.41 1877. 37 48.2 15. 53 55. 67 19. 86 32. 80 | —10. 50 42.16 a=6b 42m 345.5 48 49.7 | D.142 28 22.48 | D. 6 33 44.70 54 37.73 58.07 | + 1.25 37. 05 5=87° 13/ 44.8 50 11.2 21. 80 47.43 34, 37 59.05 | + 1.25 34. 67 51 10.2 19. 50 47. 40 32. 10 59.44 | + 1.00 32. 54 52 09.7 21. 67 47. 50 34.17 59.65 | + 1.00 34. 82 56 48.2 | D.142 28 21.27) D. 6 38 41.70 39. 57 57.18 | + 1.25 38. 00 59 13,2 21. 57 42. 07 “39. 50 53.76 | +- 1.25 34. 51 1 00 13.2 22. 07 38. 00 44. 07 51.83 | + 1.25 37.15 01 18.2 22.77 34. 60 48.17 49,59 | + 1.25 39. 01 10 20.7 | R. 322 28 15.43 | R.186 32 40.27 55 35.16 18.71 | —10.75 43.12 11 29.7 14. 60 34. 67 39. 93 13.29 | —10. 75 42.47 12 22.2 14. 27 30. 90 43, 37 08.94 | —10. 95 41. 36 13 32.7 13. 33 24. 83 48, 50 02.72 | —10. 95 40. 27 | 319 36 38. 82 51 Cephei, near} 0 30 39.1 | BR. 346 34 15.90 | R. 210 38 30.67 | 135 55 45.23 | 183 41 00.11 | — 3.88 | 319 36 41.46 East Elongation, 32 09.6 16. 87 39. 33 37. 54 08.11 | -- 3.87 41.78 x“ September 14, 33 22.6 17.37 48. 83 28. 54 14.11 | — 3.00 39. 65 1877. 34 34.1 17.17 53. 73 23, 44 19.60 | — 3.00 40. 04 a=6> 42 358.1 43 49.1 | D. 166 34 20.10 | D.. 30 39 38.70 54 41. 40 50.46 | + 5.88 387. 74 §=87° 13/ 44".7 44 59.6 24.73 41, 63 43,10 52.84 | + 5.87 41. 81 46 02.6 21. 27 42.00 39. 27 54.61) + 5.48 39. 31 47 11.1 21. 40 42, 33 39. 07 56.31 | + 5,42 40. 80 52 32.1 | D.166 34 22.40 | D. 30 39 47.87 34, 53 59.69 | + 6.25 40. 47 53 57.1 21. 33 50. 33 31. 00 59.40 | + 6.25 36. 65 54 49.1 20.17 47. 23 32, 94 58.96 | + 5.50 37. 40 55 48.6 21. 80 45. 97 35. 83 58.26 | + 5.50 39. 59 1 03 51.6 | R. 346 34 15.33 | R. 210 39 18.10 57. 23 43.05 | — 3.37 36. 91 04 58.1 15.77 10. 53 55 05, 24 39.68 | — 3,38 41. 54 05 51.6 16. 93 06. 50 10. 43 36.68 | — 3,25 43, 86 06 59.6 15, 23 OL. 83 13. 40 32.67 | — 3.25 42. 82 | 319 36 40.11 . Gd2 ASTRONOMICAL DETERMINATIONS. [Cuap. XXIV, Azimuth at West Base, Sandusky—Continued. on : " [Soa ane Angle Pr Level yi 7 Sian, date; be. Tinagtol:) eadingom ) Reewoeen efwced mck AMGEN OT | apres. | Aunt, Bae h. m. 8. oO ‘ “a Oo t uw °o ca ao ° * a“ ua oO t wt o f “ 51 Cephei, near 0 82 52.5 | D.190 01 38.07 | D. 54 06 08.43 135 55 29.64 183 41 11.99 + 2.55 | 319 36 44,18 East Elongation, 34 11.5 37. 87 15.07 22. 80 18.25, + 2.55 43. 60 September 17, 35 13.5 37.17 29,17 15. 00 | 29.79 | + 2.55 40. 34 1877. 36 27.0 36.70 27. 10 09. 60 | 27.96 | + 2.55 40.11 a=6" 42" 36.8 43 59.5 | R. 10 01 46.37 | R. 234 06 51.10 54 55.27 “51.19 | — 9.15 37.31 | | 6=879 13/444 45 16.5 45. 63 58. 67 51. 96 | 53.70 | — 9.15 36.51 , 46 30.0 47.27 5D. 57 51.70 | 55.70 | — 8.63 38.77 . 47 41.0 47.50 07 00.17 47. 33 | 57.36 | — 8.62 36.07 i 5312.5 | R. 10 01 47.70 | R. 234 06 59. 53 48.17 42 00.09 | — 9.50 38.76 54 23.5 46. 27 07 02.43 43. 84 | 41 59.65 | — 9.50 33.99 | 55 38.5 45.73 01.77 43.96 58.82 | — 8.45 34. 33 56 45.0 45.77 06 59.17 46. 60 | 57.77 | — 8.45 35. 92 1 04 37.0 | D.190 01 87.27 | D. 54 06 36.57 55 00.70 | 41.30 | + 1.93 43. 98 | 06 50. 0 36.17 32. 70 03.47 | 33,83 | + 1.92 ° 39.22 | 07 41.0 35.97 | 26. 10 09. 87 30.59 | + 0.75 | 41.21 ; | 08 55.0 35. 50 22. 40 13.10 25.66} + 0.75 ' 39.51 319 36 38.98 Pe ga a ee weet Stas 2 : i | | 51 Cephei, near | 0 32 44.2 | R. 214 11 27.07 | R. 78 15 57.73 | 135 55 29.34 | 183 41 11.20 | + 3.38 | 319 36 43,92 East Elongation, 34 04.2 27. 20 | 16 05.13 22. 07 17.63 | + 3.37 43,07 | September 18, 35 00.2 | 25. 83 08. 27 17.56 21.84 | + 4.20 43. 60 1877. 36 05.7 25.07 12. 87 12. 20 26.44 | + 4.20 | 42. 84 | a=6? 42™ 375.3 43 24.7 | D. 34 11 18.57 D. 258 16 23.07 54 55, 50 49.89 | — 5.25, 40.14 8=879 13/44/.4 44 39.2 19. 40. 29, 23 50.17 52.53 | — 5.25 | 37.45 45 50.7 | 20.17 : 30.27 49.90 . 54.65 | — 5.50 | 39.05. 47 11.7) 20.50 31.43 49. 07 56.68 | — 5.50 40, 25 52 32.7 | D. 34 11 20.00 | D. 258 16 31. 87 48.13 42 00.19 | — 6.13 ! 42.19 58 48.7 19. 90 34. 60 45.30 41 59.90 | — 6.12 ' 39.08 | ; 54 42.2 | 17.57 34. 83 42. 74 59.51 | — 7.05 35.20 i 55 53.2 | 18.57 33. 67 44,90 | 58. 64 | — 7.05 36.49 1 03 03.7 R. 214 11 23.93 | R. 78 16 31.57 52. 26 45.83 | + 4.58) 42.77 04 22.2 22.90 28. 08 54. 87 42.11 | 4 4.57— 41.55 05 16.2 | 22. 07 24. 10 57.97 39,21 | 4+ 2.83 40.01 | 06 26.2 | 23, 83 19. 50 55 03. 88 35.26 | + 2.82 | 41.91 , 319 36 40. 60 : 51 Cephei, near) 0 32 16.0) R.238 16 58.13 | R.102 21 23.73 185 55 34.40 | 183 41 09.00 | + 3.00 319 36 46.40 | East Elongation, 33 24.5 58. 03 29. 33 28.70 14.64 | + 3.00 46. 34 | September 21, 34 23.0 56. 90 37. 67 19.28 19.17 | 4+ 4.00 | 42,40 | | 3877. 35 26.0 57. 80 41.47 16. 33 23,88 | -- 4.00 44.21 | a=6" 42m 388.8 42 49.0} D. 58 17 06.33 | D. 282 22 12.40 54 53. 98 48.68 | — 6.00 | 36. 61 | 8=87° 13' 44.9 44 12.0 06. 50 15. 83 50. 67 51.81, — 6.00 | 36.48 | 45 07.5 06. 70 18, 27 48.43 53.65 — 6.00 36. 08 46 09.5 06. 90 21.93 44.97 55.45 — 6.00: 34. 42 | 51 00.5 | D. 58 17 06,27 | D. 282 22 24.20 42.07 42 00.20 | — 5.75 | 36, 52 | 52 18.0 07. 10 23.57 43. 58 00.48 — 5.75 | 38, 21 | | 53 12.5 05. 93 22. 50 43. 43 00.34 | — 7.00 36.77 54 25.5 05. 87 20. 00 45, 87 41 59.91 — 7.00" 38. 78 | 1 01 43.5 | R. 238 16 58.00 | R. 102 22 04. 93 53. 07 49.55 -- 1.75 44.37 | 02 49.0 55. 93 00.17 55. 76 46.79 + 1.75 44.30 | 03 45.5 56. 93 21 58. 60 58. 33 | 44.19 + 3.00 | 45. 52 / ; | | 04 46.5 56.17 53. 30 ! 55 02. 87 | 41. 21 | + 3.00 l 47.08 | 319 36 40.91 § 10.) AZIMUTHS. 693 Azimuth at West Base, Sandusky—Conutinued. Anglo » Level | sya 1 Time of ob- Reading on Reading on ee aioe of : Azimuth of | Result for | : Star, date, &c. servation. mark. star be star. ae ; mark. | star. Q | a fakin aia aig |e pe . oe Saad ae +, — 7 him. 8s. Oo 7 “ : o fF uw Oo F uw | o 8 au “ ob “ lo A Urse Minoris,! 1 24 01.5) R. 322 28 13.83 | R.181 26 48.77. 141 01 25.06 | 178 35 24.70 —11.65 | 319 36 38,11 | 5 son! oe . | near West Elon- | 25 21.0. 13. 93 44, 97 28. 96 22.34) —11. 65 39. 65 | gation, Septem- 26 26.5 | 14.40 | 42.77 31. 63 | 20.51 | —11.25 | 40. 89 | ber 18, 1877. 27 44.0 13. 43 | 42. 27 31.16 | 18. 50 | ~11. 25 | 38.41 \ | a=198 46" 48.3 37 12.0 | D.142 28.19.93, D. 1 26 53.43 26. 50 | 08.78 | + 0.18 | 35.41 | 5=880 56 24”.9 38 34.5 21.27 | 53. 07 28.20 | 08.09 | + 0.13 | 36. 42 i 39 42.0 | 23.50 53.17 30. 33 07.67 | — 0.45 | 37. 55 | 40 44.5 22. 27 | 54. 67 27. 60 | 07.38 | — 0.455 34. 54: 4 i | | 46 54.0 | D. 142 28 23.00 | D. 1 26 53.03 29. 97 07.84, 0.00 37.81 | ‘ | 48 10.0 23, 83 54.27 29. 56 08.39 0.00 | 37. 95 | 49 10.0 22.77 56. 83 25.94 08.92 | + 0.15 35. 01 | 50 25.0 21. 63 58.27 | 23. 36 | 09.73 + 0.15 | 33, 24 | ‘ i 59 14.5 | R. 322 28 12. 67 | R. 181 26 41.70 | 30. 97 | 19.79 | —11. 25 : 39.51 | 2 00 45.0 13. 33 44. 43 28. 90 | 29. 27 | —11. 25 | 39. 92 | | 01 53.0 12. 50 | 46. 97 25.53 | 24.26 | —10.70 | 39.09 | ‘ 02 45.0 12. 60 49. 50 23.10 | 25.89 , —10.70 38.29 . 319 36 37. 60 | cones pee aon a at es poe ene : | fide oss | ! | 1 | A Urse Minoris,| 1 29 49.6 | R. 346 34 15.03 | R, 205 82 46.03 | 141 01 29.00 | 178 35 15.84 | — 4.80 | 319 36 40. 04 | ‘near West Elon- 31 20.1 |, 14. 43 43. 78 30.70, 14.02 — 4.80 | 39. 92 | gation, Septem. 32. 27. 6 15. 10 43.57 31. 53 12.80 | — 3.83 | 40. 50 ber 14, 1877. 33 36.6 15.13 43. 00 32. 13 | 11.68 | — 3.82 39. 98 | | a=19" 46" 475.1 [| 42. 86.1 | D.166 34 19.87 | D. 25 32 52.67 27. 20 | 07.39 | + 4.50 | 39.09 | | 5=88° 56 25".1 44 12.6 19. 03 54.27 24. 76 | 07.46 | + 4.50 | 36.72 i 45 22.1 21.18 54. 63 26. 50 07.65 | + 3.98 38.13 | | ‘ 46 42.6 19. 50 54.17 25. 33 | 08.04 | + 3.97 37. 34 | | 51 59.6 D.166 34 19.67 D. 25 32 55.53 24.14 11.27 | + 5.82 41.23 ‘i 53 15.1 19.77 56. 60 23.17 12.44 | + 5.82 41.43 | | 54 15.6 19. 50 58. 87 20. 63 13.48 | + 5.38 39. 49 | ! | 55 26.6 19. 60 33 00. 40 19. 20 | 14.83 | + 5.37 |" 39. 40 | | 2 05 44.1 | R. 346 34 06.37 | R. 205 32 58.53 07. 84 | 32.29 | — 4.33 | 35. 80 | | | | 08 04.1 06 07 33 03. 03 | 03. 04 | 37.67 | — 4.32 36. 39 09 11.1 | 06. 93 05. 83 | 01.10 | 40.44 | — 3.68 37. 86 | 10 30.1 06. 07 08. 88 00 57. 24 43.83 | — 3.67 | 37. 40 319-36 38.79 1 : | Cassia Ha eee I a ee = ine 1 is = | —— snk “ ra, are _ A Ursa Minoris, 115 44.0 D.190 01 36.67 D. 49 00 44.20 141 00 52.47 | 178 35 44.17 + 1.50 319 36 38.14 - : 5 36. 43 38. 83 57. 60 | 41.14 1.50 | 40. 24 sland West Elon 16 54.5 BAS ; F een gation, Septem- 18 03.0 35.27 | 38. 08 57. 24 38.32 + 1.50 37.06 | ber 17, 1877. 19 20.5! 35. 83° 33.47 ° 01 02. 36 35.27 + 1,50 | 39. 13 1 ! a=19" 46m 43.2 26 47.5 R. 10 01 45.00 R. 229 00 22. 60» 22. 40 20.89 — 9.93 ; 43, 36 | $=880 56! 25.7 27 54.0 44.43 21.10 23. 33 19.21 — 9.92 | 32. 62 | ' 28 46.0 46.20. 19.40 26. 80 17.98 | — 9.75 35. 03 7 29 48.5 : 45.93 | 16.43 | 29. 50 16.60 -- 9.75 36. 35 | 34 09.5 I. 10 01 46.47 | R. 229 00 11.40 35. 07 11.95 | —10, 25 36.77 | 35 19.0 46. 37 11. 20 | 35.17 11,05 | —-10. 25 | 35. 97 36 05.0 45, 23 09.17 | 36. 06 10.50 | —10.13 | 36. 43 37 04.0 45. 37 © 08. 50 | 36. 87 09.87 , —10.12 | 36. 62 44 85.0 | D.190 01 28.00 | D, 49 00 05.48 | 32. 57 08.31 + 2.30 43,18 46 03.5 ° 35. 53 | 06. 10 29. 43 08. 65 | + 2.30 40. 38 46 59.0 | 36. 27 06. 20 30. 07 08.97 + 1.60 40. 64 ' | 48 18.0 | 35.77 | 06.77 29. 00 09. 57 | + 1.60 40.17 | 319 36 37. 63 | | : I 694 ASTRONOMICAL DETERMINATIONS. (Cwap. XXIV, Azimuth at West Base, Sandusky—Continued. Level i se : dine \ a | Angele i | 8 ; j Stinaatese ition, aun °* Stn eae “te ee Mae ee e enews tee 4 pies Zl poss =: i geea: eee eS | ; hone. 8 ° i uw o 7 uw Oo fF " | uo oO 4% uw Of uw [A Ursie Minoris, 124 03.4 R. 214 11 21.73 | R. 73.10 10.93 | 141 01 10.80 | 178 35 25. 63 + 4.50 319 36 40, 93 | near West Elon-! 25 26.9 22. 83 07. 90 14. 93 | 23.17 + 4.50, 42.60 gation, Septem- | 26 16.9 23.77 05.17 18. 60 | 21.79 | + 5.58 45.97 | 18, 187. , er oro 24. 60 02.70 21. 90 19.95 | + 5.57 , 47.42 a=19' 4641.9 ---35 44.4 dD. 84:11 19.63 | D. 253 09 47.50 32.13 10.83 | — 6.70. 36. 26 | | 4=88956'25".8 87 14.4 19. 27 46. 27 33. 00 09.90 | — 6.70 36. 20 | 3810.9 20.17 45. 10 35. 07 09.41 | — 7.58 | 36. 90 39 26.9 18.77 44.13 | 34, 64 08,89 | — 7.57 35. 96 | | 4455.4] D. 34 11 18.73 | D. 253 09 43.07 | 35. 66 | 08.50 | — 7.25 | 36.91), , $6 20.4 18.43 45.17 33. 26 08.87 | — 7.25 34, 88 | AT BBS 18.77 46.27 | 32.50 , 09.27 | — 7.63 | 34.14 | "4847.4 19. 43 46, 53 ! 32.90 , 09.96 | — 7.62 | 35. 24 | 57 09.4 | R. 214 11 22.60 | R. 73 10 06.60 — 16. 00 18.08 | + 5.30 | 39. 38 | | 59 49.9 24.17 09. 37 14. 80 | 22.09 | + 5.30 | 42.19 | ! | 2 0057.9 23. 60 10.77 | 12. 83 24.00 | + 4.88 | 41.71 | | 02 86.4 23.77 13.83 09. 94 27.00 | + 4.88 41, 82 | 819 36 39.28 | bedi s Se Pcie ph eens | A Urse Minoris, | 1 18 28.8 | R. 238 16 56.30 R. 97 15 52.20 | 141 01 04, 10 | 178 35 37.88 | + 0.75 | 319 36 42.73 | near West Elon- 19 54.8 56. 27 50. 43 05. 84 34.59 | + 0.75 | 41. 18 gation, Septem- | 20 47.8 | 57. 87 47.17 | 10.70 32.67 | + 1.50 | 44. 87 | ber 21, 1877. | 2157.8 | 57. 57 44.83 12.74 30.23 | + 1.50 | 44.47 | | a= 19 46” 38.3 | 29 03.8 | D. 58 17 04.50 | D.277 15 39.00 25. 50 18.26 | — 7.00 | 36.76 | 868° 56°267.3 | 30 20.8 04. 43 38.77 25. 67 16.63 | — 7.00 | 35. 30 : 31 20.3 | 04. 27 38. 67 25. 60 15.46 | — 7.55 33. 51 | {32 81.8 | 03. 60 38, 23 25. 37 14.20 | — 7.55 32.02 38 14.8 D. 58 17 03.60 | D. 277 15 33. 23 30. 37 10.03 | — 7.05 | 33. 35 | 39 39.8 | 02. 50 32. 00 30. 50 09.49 | — 7.05 | 32. 94 | 40 41.3 | 05.10 29. 60 35.50 | 09.22 | — 7.25 | 37.47 42 00.8 | 04. 27 32. 83 | 31. 94 | 09.02 | — 7.25 | 33, 71 ! ' 50 18.3 | R. 238 16 56.83 | R. 97 15 26.43 30. 40 11.65 | + 0.83 | 42. 88 51 40.3 | 56. 47 28. 23 28, 24 12.73 | + 0.83 41, 80 | 52 39.8 | 54. 50 29. 57 24. 98 13.61 | + 1.03 | 39. 57 | ' 53 59.8 | 54. 90 31.40 23. 50 14.96 + 1.08 | 39,49 | 319 36 38, 25 The next table gives a summary of the results in the preceding table. Periodic error is eliminated from the mean result by the shifting of the circle previously mentioned. The results are corrected in this table for errors in the declinations taken from the American Ephemeris, as explained in § 4. The individual results still involve such residual errors as are not eliminated by reversal of telescope, such as may be due to inaccurate indications of the striding-level, and such as arise from instability of the instrument or its support between pointings to star and to mark. The corrections for inclination of telescope axis are, in some cases, greater than is desirable in such work, but are not beyond the range of accurate determination with a good level. The maximum correction, as will be seen by the preceding table, was 12.75, and the average 5/’.27. § 10.] AZIMUTHS, 695 Azimuth at West Base, Sandusky. SUMMARY OF RESULTS. AZIMUTIL OF LINE AZIMUTH POST— EAST BASE, B Number of point- ‘ Reduction to Auwers’ | I ings— Azimuth. Declinations. Position of Deciina- ' Corrected _ Date. Star, &c. telescone: —— a Hons from |———— | azimuth \ merican aA : of mark. | To mark. To star. | Ephemeris. ae a5 | AA | | > ! ee Se ee ee : | 319 36 319 36 1877. | ‘ wa ! Ww “a a“ : Sept.13 | Polaris, near East Elongation ..... Direct .....- & ; 8 | ; Reversed . -. 8 | 8 41.70 —1.31 | +0. 04 | —0.05 41. 65 | 1¢ eaises GO eo eee Seeee Sem ats Direct ..-..- | 8 | 8 i Reversed ...! a 8 40, 25 | —1.31 } +0. 04 | —0, 05 40,20 | TP shee dee ea Direct ...... ~ MB gn RE | Reversed...) 8 ' 8 | 3880 |—L3L| 40.04 —0.05, 38.75 | |) We asillb enee nanos vations: | Direct ...... a ae ae | | (Reversed...) 8 | 8 | 42.92 © —21.81/ 40.04) -0.05; 42.17 | BEV scisates do ...... S doacdau asses oeast ' Direct ....-- | 8 | 8 : | | | : Reversed | 8B 36.70 '—1.31| 40.04} 0.05; 3665 | | 13 | Ursa Minoris, near West Elonga- Direct ......! 8 | 8 j | . | | tion. ' Reversed I 8 | 8 38. 27 ' +1. 32 | 40.31 +0. 41 38. 68 fy ad ae aloe sees hen erent ke Direct ...... | 8 i 8B | | | ‘Reversed-..; 8 | 8 | 3987 | 4132) +40.31| +041; 40,28 © TG oes J schiouasadeee eat ee A domks Direct ...... 8 1 Bs | | | Reversed _ g | 8 |} 3822 | 41.32] +031) +4041! 38.63 [) iatee eiot Aditi nae ue eae i! Direct...... 8 + 8 | Reversed 5) BU) 41.31 41.32/40.31] 40.41 4179 | | 2h. een neinn COL rs ceneeetesneties eee. aemee tld Direct ..--.-) 8 ' 8 Reversed... 8 | 8 38. 12 | 41.32 | 40.31] 40.41 38,53 | ; 13 , 51 Cephei, near East Elongation ..; Direct ...... ; 8 ; 8 Reversed ..- 8 | 8 38, 82 —1.32 | +0.19 | —0. 25 | 38. 57 | | Te coces WE Lie aeed Sua vosceonwnn aon DitGGt aes xy 8 8 Reversed... 8 8 40.11 | —1.32! +0.19] —0.25| 39.86 | he Ota. c pieeoa nasa een.e weecats Direct ...... 8 8 | | \ ss | Reversed: i+ 8 8 38.98 | —1.32 ; 40.19 | —0.25 38. 73 | | 18. | eenee OO vasepansa awd ciwien eer temmis ee ' Direct ...--- 8 8 | Reversed . . | 8 8 40. 60 —1.32 | +0.19 | —0. 25 40.385 | Pe soieee Occ teat eset ese eeeeeseee Direct ......: 8 8 | | ; ‘ Reversed ...i 8 8 40, 91 —1.32 | 40.19 | —0.25 40.66 | 13 | AUrse Minoris, near West Elonga- | Direct ...-.. | 8 8 | | : tion. | Reversed ... 8 i 8 37. 60 i +1. 32 | +0.59 | +0. 78 38, 38 \ Be occa 16 causes 5G Saeaxenense ' Direct ..... 8 | 8 |Reversed..., 8 | 8 | 38.79 | 41.32 40.59! 40.78 39.57 | 17! ose lot. Pare eRe aoat A arete Direct ...... re +o |g | Reversed Bike 8 | 8 37. 63 +1 32 | +0.59 | +0, 78 38.41 18) 2285 G0 asnecdices ei de ce ouesee Direct ...... be Bee) Be | | Reversed .. “ 8 | 8 i 39. 28 --1. 32 | +0. 59 | +0. 78 40. 06 | ' ZL, | pscrsion Gia esis, G26 ceRchee Bee ees | Direct ...--- 8 | 8 | ' Reversed oe 8 8 | 86.25 | 41.92 | 40.59] +078 39.03 1 1 1 syateitedl’ sauna otto itccpannadapapssiiee sauauee ssuee’ dos heals at yaactenlenence vos wer suey Se: 319° 36 89”, 544.0”, 209, Assigning equal weights to the individual results in the last table, their mean is 319° 36/.39’”.54+40.21, the probable error being derived from the discrepancies between the indi- vidual results and the mean. This is the azimuth of East Base from the azimuth post. The correction to reduce this azimuth to the azimuth of East Base from West Base is —01/.25; the correction for diurnal, aberration is +00”.31. There thus results as the azimuth of the line West Base — East Base, 319° 36’ 38”.6040."21 west of south. 696 ASTRONOMICAL DETERMINATIONS. [Cuap, XXIV, AZIMUTH AT TONAWANDA. § 2. Azimuth determinations were made at Tonawanda station on the nights of August 24, 25, 26, 27, and 28, 1875, by Assistant Engineer G. Y. Wisner, The instrument used was Troughton & Simms 14-inch theodolite No.1. It was mounted on a solid wooden post set 54 feet in the ground, and nearly on the line between Tonawanda station and the azimuth-mark, being (.05 foot north of that line and 30.34 feet from the geodetic point. The stars observed were «, 5, and 4 Urs Minoris, and 51 Cephei. On the first three nights observations were made in the following order for each star: 1. S readings to azimuth light. ‘ », 8 readings to star with 3 level readings. . Reversal of telescope. . Sreadings to star with 3 level readings. . S readings to azimuth light. On the last two nights the programme was changed to the following, which gave two reversals during the observations on each star and diminished the interval between pointings to the star and mark: pointings to mark, to star, level readings, pointings to star, to mark, reversal, pointings to mark, to star, level readings, pointings to star, to mark—two consecutive pointings being made to each object—this comprising one-half the programme. The circle remained in the same position throughout each night’s observations, but was changed in position on each succeeding night, the readings of microscope A on the azimuth-mark being in order 1879, 172°, 2179, 257°, 297°, Time was given by Bond & Sons’ chronometer No. 206. The azimuth-mark was a light limited by a vertical slit 4 inch wide, ina box fastened firmly on a post set in the ground 1.3 miles distant from the azimuth-post. In the reductions for a part of the observations, the differences between the azimuth of the star at the times of observation and at elongation were computed, and these differences were applied to the circle readings on the star, to give the corresponding readings for azimuth of the star at elongation. The mean of these readings for one position of the telescope, subtracted from the mean of the corresponding readings on the mark, gives the horizontal angle between the star at elonga- tion and the mark. In the reduction of the rest of the observations the circle readings on the star were reduced to the pole, the mean of these reduced readings for one position of the telescope, sub- tracted from the mean of the corresponding readings on the mark, giving the horizontal angle between the mark and the pole. In the first part of the table following, the first column gives the date of observation, the star observed, and its right ascension, declination, and azimuth at elongation, the latter being denoted by A, and including the correction for diurnal aberration; the second gives the readings on the mark for each position of the telescope and their means; the third gives the readings on the star, reduced to elongation, for each position of the telescope and their means; the fourth gives the mean angles between the mark and the star at elongation for each position of the telescope and their means; the fifth gives the result from each star for azimuth of the mark. In the latter part of the table, the first column gives the date, star, and its codrdinates; the second, the readings on the mark; the third, the readings on the star reduced to the pole; the fourth, the angles between the mark and the. pole for each position of the telescope and their means; and the fifth, the resulting azimuth of the star, corrected for diurnal aberration. The position of the telescope is indicated by the letters D and R. wm wile ou AZIMUTHS. 697 § 11] Azimuth at Tonawanda. AZIMUTH OF LINE AZIMUTH POST— AZIMUTH MARK. (Observer, G. Y. Wisner. Instrument, Troughton & Simms theodolite No. 1.] Star, date, &c. Readings on mark. Readings on etar | reduced to elon- ; gation. stall Angle between | mark and star | Result for star. | at elongation. i Polaris, near East Elonga- tion, August 24, 1875. asih 13 239.57 5=88° 38’ 34’’.03 A g=181° 51! 21.82 | Oo t rie | fo} + a“ D. 187 24 09.60) D. 138 57 41.30 ° 09.17 | 38. 69 | 08. 60 | 37. 66 08.50 | 37. 42 | 08. 87 ! 39. 46 | 08. 97 38.61 08. 27 | 38. 08 | 08.13 | 38. 30 | Means... 38. 69 | R. 7 23 57. 67 | R. 313 57 31.88 58.17 28.47 | 57. 20 26. 87 | 56.77 | 24. 35 | 56.70 22.47 | 57. 17 23. 54 | 56. 87 23. 96 57. 53 22, 61 Means.. * 57.26 25. 52 OM CAI oot ad rcsincy corsecoe spices eee eee 53 26 30. 07 31.74 Polaris, uear East Elonga- tion, August 25, 1875. a=1 13™ 249,29 5=88° 38’ 34.32 Ag=181° 51 21.42 D. 172 24 03 50 D. 118 57 34.53 04. 30 34. 68 02. 90 33. 75 03. 70 32. 66 | 04. 18 32. 71 03. 60 32. 84 04. 67 32. 73 03. 63 32. 31 Means .. 03. 80 33. 28 R. 352 23 59. 50 R. 298 57 30.09 53 26 80.91 | 235 17 52.73 53 26 30. 52 59. 97 29.70 58. 50 29.71 59. 97 | 27. 84 60. 43 27. 31 59. 50 | 26. 64 60. 03 | 27.09 59.97 | 27. 42 | Means.. 59.73 | 28, 22 Mean ...... ..-225.6-- | acini neat, Ge aded aoe 31.51 | 58 26 a1. 01 235 17 52. 43 88 LS 698 ASTRONOMICAL DETERMINATIONS. Azimuth at Tonawanda—Continued. ([Cnap. XXIV, Star, date, &c. Readings on mark. Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result for star. Polaris, near East Elonga- ° t uw D. 217 24 04.53 OW “ D. 163 57 32.06 tion, August 26, 1875. 04. 20 33. 70 a=1" 13™ 25%.00 03. 63 33. 64 5=88° 38! 34.60 03. 93 34. 05 Ag=181° 51’ 21.04 04. 20 34, 88 04. 20 34.13 04. 33 ° 35. 29 04, 90 35. 37 Means .- 04, 24 34. 14 53 26 30.10 R. 87 24 03.57 R. 343 57 31.34 02. 60 30. 64 02. 77 30. 28 03. 20 29. 04 03. 37 28, 70 01. 67 29. 01 03. 33 28. 07 02. 67 28. 99 Means .. 02. 90 29. 51 33. 39 MGaN..c.. ccs -cessetseas| seas sseweren ss eed 53 26 31.74 235 17 52.78 Polaris, near Eastern Elon- D. 257 24 27.50 D. 203 57 59.57 gation, August 27, 1875. 28. 00 59.17 a=1) 13™ 25*.62 28, 83 58. 81 6=88° 38' 35/'.02 27. 33 61. 04 Ag=181° 51! 20.46 Means .. 27.92 59, 65 58 26 28,27 R. 77 24 24.93 R. 23 57 57.46 24.70 55. 41 26. 60 56. 57 25, 27 57. 37 Means -. 25. 38 56. 70 28, 68 R. 77 24 26.50 R. 23 57 57. 46. 27.18 56. 25 28.17 54, 29 27. 60 54, 22 Means .. 27. 35 55. 55 31. 80 D. 257 24 28.23 D. 203 57 58,57 28. 43 57.75 26. 67 58. 40 27.47 57, 24 Means .. 27. 70 57. 99 29.71 Meant ose seins -secrionca |e veaceisiaceies Sees 53 26 29.61 235 17 50.07 $11] AZIM UTHS. Azimuth at Tonawanda—Continued. ; Readings on star | Angle between Star, date, &c. Readings on mark. reduced to elon- mark and star | Result for star. gation. at elongation. ° a u ° J “a oO ‘ Wn ° t “a Polaris, near East Elonga- D. 297 24 56.70 | D. 248 58 25,22 tion, August 28, 1875. 57.27 25. 64 a=1> 13™ 26,20 56. 37 26. 04 5=88° 38! 35.33 55. 97 26. 59 Age teers Means ... 56. 58 25. 87 53 26 30.71 R. 117 24 55. 83 R. 63 58 24.70 55. 97 24, 59 56. 67 25. 72 ‘| 56. 40 24. 50 Means .. 56. 22 24, 88 31, 34 R. 117 24 57.10 R. 63 58 28. 82 56, 37 24. 28 55. 87 25. 46 56. 37 24.49 Means . 56. 43 24. 51 31. 92 D. 297 24 57.40 D. 243 58 24.78 e 57. 33 24, 59 _ 58.08 24, 47 56. 97 21. 99 Means .. 57. 43 23. 96 33. 47 : Moaiias us saevcaden vue laadupemaceher enero 53 26 31.86 | 235 17 51.90 ~ 6Ursx Minoris, near West R. 352 23 58. 93 R. 292 27 52.24 Elongation, . August 25, 58. 97 49. 20 . 1875. 59. 60 50. 75 a=18> 12™ 389,52 58. 57 "49, 98 5=86° 36! 35.43 58, 43 51.14 Ag=175° 21! 43.56 58. 23 50. 72 58. 20 50. 45 58. 40 50, 53 e Means .. 58. 67 50. 62 59 56 08.05 D. 172 24 07.40 D. (112 27 59.97 06. 70 57. 27 07. 97 57. 20 06. 37 56. 51 05. 90 56. 50 : 06. 57 56.79 05. 73 59. 72 05. 77 61. 26 Means ..~ 06. 55 58.15 08, 40 Moai ssiver igs ci] Ateneo 59 56 08.22 | 235 17 51.78 700 ASTRONOMICAL DETERMINATIONS. Azimuth at Tonawanda—Continued. [Cnap. XXIV, | Readings on star | Angle between Star, date, &c. Readings on mark. ‘reduced to elon- mark and _ star | Result for star. | gation. at elongation. - °o f “ fo} t “ fo} ¥ “ | ° A wn 5Ursx Minoris, near West R. 37 24 04.60 , R. 337 27 57. 64 Elongation, August 26, 03.43 55.77 1875. 04.67 | 54, 90 a=18" 12™ 385.09 03. 13 55. 98 ‘ 6=86° 36! 35/’.52 03. 90 55. 87 Ag=175° 21! 43.69 03. 07 56. 41 04. 40 56. 35 04, 67 56. 21 Means .. 03.98 | 56.14 59 56 07.84 | : D. 217 24 06. 67 D. 157 27 57.05 | 07. 88 57.76 08.60 | 58.65 | 08.43. | 59.17 | i 08. 83 58. 16 08. 40 57.44 , 07. 43 60. 01 | 07. 80 | 64. 70 | Means 08.00 | 59.12 | 08.88 | Mean .....2.22-26- 2-55 : epee aeuies ake | 59 56 08.26 | 235 17 52.05 ® 1 51Cephei, near East Elonga- R. 187 24 02.938 | R. 185 53 18.23 | tion, August 24, 1875. 02. 67 16.71 a=6) 41™ 173.73 03. 33 16. 50 §=87° 13/ 57.15 03. 73 15. 24 A g=183° 47! 07.91 02. 73 18. 00 01. 80 18. 51 02. 33 18.00 ‘ 03.17 18. 86 Means .. 02. 84 17.51 51 30 45. 33 D. 7 24 05.17 D. 315 53 27.27 06. 07 23.19 ‘ i 06. 03 24. 58 | 06.70 21. 04 | 06. 03 22. 46 | 05.03 25. 08 | 05. 27 26. 90 | 05.17 27.07 | Means 05. 68 24. 70 40. 98 | Meany .notesee tick dadlsadecs veeawedaeoaed 51 30 43.16 | 235 17 51.07 §11.J AZIMUTHS. Azimuth at Tonawanda—Continued. Readings on star Angle between Star, date, &c. Readings on mark. reduced to elon- mark and star | Result for star. ‘ _ gation. at elongation. ° t “ o a a ° t “a °o cs aw A Urs Minoris, near West D. 7 24 05. 33 D. 310 38 46.74 Elongation, August 24, 05. 90 42.76 1875. 05. 58 41, 55 a=19> 49™ 235.66 05. 73 40. 37 5=88° 55! 58.67 06. 30 40, 49 A e=178° 32! 27.50 06. 83 41. 36 x 07. 33 41, 48 06. 63 41.94 Means -- 06. 20 42.09 56 45 24.11 R. 187 24 05.98 R. 180 38 41.99 05. 30 41. 67 06. 00 40. 86 06. 07 41. 04 06. 33 40. 90 06. 03 40.79 06. 23 42. 89 06. 70 43. 73 Means 06. 07 41.73 24, 34 Mean i253 /5-Gas sated lasso ence 56 45 24.22 | 235 17 51.72 A Urse Minoris, near West R. 357 23 59.33 | R. 295 88 36.77 Elongation, August 25, 59. 23 39. 08 1875. 59. 80 38, 82 a=19) 49™ 295,66 : 60. 23 36. 98 6=88° 55! 58.98 58. 90 37. 45 Ag=178° 32’ 27".92 59. 90 37. 35 60. 10 37. 51 59. 63 38. 21 Means .. 59. 64 37.77 56 45 21. 87 “ D. 172 24 02.37 D. 115 38 40.10 03. 90 42, 96 04, 23 45. 65 03. 83 45. 60 04, 43 44. 03 02. 97 44, 31 03. 33 44. 66 03. 63 44.18 “Means 03. 59 43. 94 19. 65 Mean sisinesenceamantilensis ocean cow erecmte 235 17 48. 68 56 45 20.76 701 T02 ASTRONOMICAL DETERMINATIONS. Azimuth at Tonawanda—Continued. ([Cuap. XXIV, Star, date, &e. Readings on mark. Readings on star reduced to elon- gation. Angle between | mark and star , Result forstar. at elongation. | oO A a ° t “uw ° t aw ° ¢ a A Urs Minoris, near West R. 87 24 05. 67 R. 340 38 42.19 Elongation, August 26, 04. 93 43. 76 1875. 05. 27 41, 28 a=195 49™ 215,61 04. 70 41.34 6=88° 55’ 59".27 04. 50 40. 76 A e=178° 32' 28.32 05. 00 40. 43 05. 43 40. 67 05.17 40, 68 Means .. 05. 08 41.39 56 45 23. 69 D. 217 24 02.50 D. 160 38 40.99 05. 00 45. 34 05. 20 44. 07 05. 18 45. 88 05. 03 47. 83 05. 67 47.17 05. 97 47.79 05. 43 47.05 Means .. 04. 99 45.77 19, 22 Mans .cecsegscexaxess|ceteeensacueeeerinese 56 45 21.45 235 17 49.77 A Urse Minoris, near West D. 257 24 34.10 D. 200 39 11.08 Elongation, August 27, 33. 90 09. 84 1875, “34. 13 10. 02 a=19 49" 209,55 ee 11. 25 6=88° 55! 59/55 Means .. 34. 06 10. 55 56 45 23, 51 A p=1780 32/ 28.70 = R. 77 24 32.03 R. 20 39 07.88 32. 33 08. 64 32. 23 09. 31 31. 93 07. 70 Means .. 32. 18 08. 38 23.75 R. 77 24 31.90 R. 20 39 08. 44 31.77 07. 85 30. 87 08. 55 31.17 08. 34 Means .. 31, 43 08, 29 23.14 D. 257 24 35.07 D. 200 39 10. 94 33.17 13. 60 33. 63 14. 57 33.17 14.16 Means 33. 76 13.32 © 20. 44 235 17 51. 41 Mean sce svat os saccclesmescincemeuee tenes 56 45 22.71 § 11] AZIMUTHS. Azimuth at Tonawanda—Continued. Star, date, &c. Readings on mark. Readings on star reduced to elon- gation. Angle between mark and star at elongation. Result for star. A Urse Minoris, near West Elongation, August 27, 1875. a=195 49 195.49 5=88° 55/ 59.77 A g=178° 32 29.00 Oo + u" D. 297 25 08.37 o fr “ D. 240 39 46.63 08. 80 47. 30 09. 10 45. 65 08, 23 45.74 Means.. 08. 62 46, 33 R. 117 25 06.60 R. 60 39 45. 45 07. 43 46. 09 05. 63 45,14 06. 90 45. 01 Means. . 06. 64 45. 42 R. 117 25 06.90 R. 60 39 46,18 06. 87 45. 53 08. 37 43, 93 08. 23 44, 33 Means.. 07. 59 44, 99 D. 297 25 09.83 D. 240 39 47.10 08. 90 ’ 45, 25 08. 23 44, 69 08. 50 45,41 Means 08. 87 45. 61 MCAT 3 cicii2 hud nemntaed| Pobeaecodwaantea ae 56 45 22. 29 21, 22 22. 60 23, 26 56 45 22,34 | 235 17 51.34 Star, date, &c. Readings on mark. Readings on star reduced to pole. Angle between mark and pole. Result for star. 6 Urs# Minoris, near West Elongation, August 24, 1875. a=18) 12™ 389.95 6=86° 36/35/,25 or " Oo 4 a“ D. 7 24 04. 67 D. 312 06 15. 61 04, 23 12. 03 04. 93 14. 05 03. 67 13. 75 03. 97 15. 00 03, 83 13. 12 03. 43 13. 01 03. 67 13. 43 Means... 04. 05 13. 75 55 17 50.30 R. 187 24 02.93 R. 132 06 11.61 02. 67 09. 59 03. 33 09. 39 03.73 08. 50 : 02. 73 08, 51 01. 80 09. 98 02. 33 10. 39 03.17 10. 54 Means.. 02. 84 09. 81 55 17 51. 66 235 17 51.97 703 704 * ASTRONOMICAL DETERMINATIONS. Azimuth at Tonawanda—Continued. [Cuap. XXIV, . ; Readi ti Angle bet Star, date, &c. Readings on mark. padtieed Garis as Wane oa anole, Result for star. fo} t aw ° , “uw ° é “w fo} , a“ 6 Urs Minoris, near West D. 257 24 32.00 . D. 202 06 42.31 Elongation, August 27, 32. 40 37. 03 1875. 32. 93 40. 33 a=18) 12™ 37°.67 33. 10 39. 78 C= ERe eb gor eS Means .. 32. 61 39. 86 55 17 52.75 R. 77 24 27.43 R. 22 06 38. 67 28, 97 38, 73 29. 83 37. 93 31.17 38. 03 Means .. 29. 35 38. 34 51. 01 R. 77 24 31.00 R. 22 06 36.99 29. 43 39. 51 29. 73 38. 78 30. 10 38. 85 Means .. 30. 06 38. 53 51. 53 D. 257 24 34.00 D. 202 06 42. 22 34. 27 39. 88 34. 23 40.19 33. 47 37. 67 Means... 33. 99 39. 99 54. 00 Meal. .soniateesuiteds3| uy Reese, 55 17 52.32 235 17 52. 63 6 Urse Minoris, near West D. 297 25 00.50 | D. 242 07 08.10 Elongation, August 28, 00. 13 07. 68 1875. 01. 50 08. 76 a=18> 12™ 378,25 00. 37 09. 53 B= 86° 367°35"".72 Means .. 00. 62 08. 52 55 17 52.10 R. 117 24 56.50 R. 62 07 05. 25 57. 93 08.72 59. 10 09. 99 58. 93 09. 82 Means .. 58.11 | 08. 44 49. 67 R. 117 24 58. 33 R. 62 07 04.91 59. 17 07. 94 58. 60 04. 38 58. 63 04. 84 Means .. 58.68 | 05. 52 53. 16 D. 297 25 01.33 D. 242 07 07.70 01. 30 11. 42 00.17 09. 67 01. 27 10. 70 Means 01. 02 09. 87 51.15 POD cteiciacre naan mae tiviad we oem wake ean ean waleee 65 17 51.52 35 17 51. 83 § 11] AZIMUTHS. Azimuth at Tonawanda—Coutinued. Star, date, &c. Readings on mark. ' Readings on star reduced to pole. | | ! a t a oO ‘ aw 51 Cephei, near East Elonga- D. 172 2406.03 D. 117 06 14.92 | tion, August 25, 1875. 07. 33 12, 84 a=6" 41™ 189,24 07. 60 12.76 8=87° 18! 56.95 07.70 13.41 | : 08.17 15.41 ° 06. 83 15.97 | 07. 30 15. 57 | ‘ 06.17 15.54 | Means .. 07.14 14.55 | 5 R. 352 24 01.63 02. 07 01.47 00. 63 00.77 01. 97 s 00. 47 00.73 | Means.. 01. 22 R. 297 06 09.81 12.12 10 02 09. 79 09. 23 09. 06 10. 42 09. 77 10. 0 Angle. between mark and pole. 55 17 52.59 51.19 "85 17 61.89 ‘Result for star. 51 Cephei, near East Elon- D. 217 24 07.17 D. 162 06 12.03 gation, August 26, 1875. 07. 43 17. 62 a=6) 41™ 185.77 - | 08.70 20.17 §=87° 13! 56.75 08. 37 20. 10 08. 40 18. 00 08. 73 17. 90 07.10 16. 98 07. 47 15. 16 , Means .. 07. 92 R. 37 24 03.50 17. 24 QR. 342 06 11.51 03, 33 12.17 04. 23 11. 33 04. 57 11. 55 04. 93 12. 26 | 04. 23 12. 72 } 04. 97 12. 62 04. 90 11. 02 Means .. 04. 33 11. 90 > Meat occagcasnaeneese 55 17 50. 68 52. 43. 5517 51.55 23517 51.86 | 235 17 52. 20 | | | | 4 | ' I | i 1 1 TW5 TO6 ASTRONOMICAL DETERMINATIONS. (Cuar. XXIV, Azimuth at Tonawanda—Continued. ' Readings on star | Angle between Star, date, &e. Readings on mark. Result for star. reduced to pole. | mark and pole. i orn | orn ow Oak ie 4 51 Cephei, near East Elon- D. 257 24 33.73 | D. 202 06 45. 33 | i | gation, August 27, 1875. 34.17 | 42.17 a=6" 41™ 19*,30 35.53 43. 04 8=87° 13! 56.63 33.17 | 45, 12 1 Means .. 34. 15 43, 91 55 17 50. 24 ae R. 7724.20.17 | R. 22 06 37.52 | a | | 29. 33 37.68 | | 30. 07 36. 78 ; 30.53 35. 93 Means .. 29.39. | 36.98 | 52.41 ie 4 t R. 77 2430.10 ) RR. 22 06 38.32 | . 30.50 | 37. 27 29.33 © 37. 43 29.17 37, 82 ' ; Means .. 29.77 37.35, 52. 42 S a \ } 1 "oD, 287 2434.93 -D. 202 06 44.27 | 34.73 40. 38 | ; 32.77 40. 65 34.40 | 39.74 7 | “Means .. 34.21 41.26 | 52, 95 Meat A jcf-. 9 sea.e3 Aeeed ‘cheats wonder emaneeh 55 17 52.01 | 235 17 52.32 \ 51 Cephei, near East Elonga- D, 297 25 00.77 D. 242 07 11.98 | tion, August 28, 1875. 01. 88 11. 04 I a=6" 41” 19°80 01. 83 08. 74 | 2 5=87° 13’ 56.45 02. 47 09.40 js | Means .. 01.72 10.29 | 5517 51.43 | R. 117 245817 | R. 6207 08.74 | 59.03 07. 88 | 58.67 | 07.61 | 59.30. | 03, 95 | Means 58, 79 07.54 51.25 a R. 117 24 59.95 oR. 62 07 04.65 | 59. 30 04.76 \ 60. 40 05. 66 | 59. 03 06. 02 ' | Means .. 59. 66 05. 27 54.39 | —— D. 29795 08.17 D. 242 07 12.68 05.10 14. 08 08.17 12. 90 , 08.17 12.51 Means .. 07.40 | 13. 03 54. 37 Mean 205.20. acdc data ee 5517.52.86 | 285 17 53.17 The next table gives a summary of the data in the preceding table, each result for azimuth of mark being corrected for periodic error, by the formula given in Chapter XIV, B, § 6. These results are further corrected in the table for errors in the declinations taken from the Amer- ican Ephemeris, as explained in § 4. They still involve such residual errors as are not eliminated by reversal of telescope, such aS may be due to inaccurate indications of the striding-level, and such as arise from instability of the instrument or its support between pointings to star and to is v1. AZIMUTHS. 707 mark. The greatest and least intervals of time between the first pointing to the mark and the last pointing to the star und vice versa, on the first three nights, were respectively 22 minutes and 12 minutes, the average interval being about 17 ininutes, so that the stability of the instra- ment and its support was assumed for about 8 minutes. There being two reversals and fewer readings to mark and star ina set on the last two nights, the stability of the instrument was assumed for a much less time. The corrections. for inclination of telescope axis were generally not large, the maximum being 9.87 and the average 3.73. The inicrometers were thoroughly adjusted for error of run before any observations were made, and readings we:e made each night to test their stability. Azimuth at Tonawanda. SUMMARY OF RESULTS. AZIMUTH OF LINE AZIMUTH POST— AZIMUTH MARK. Number of point- Avi Reduction to Auwers’ ‘ ings— | Azimuth. Declinations, Position of | | Declina- Corrected | Date. Star, &c. telescope. fe tis if, ES, ye from eS azimuth of et | American dA | mark, ' iTo mark. To star. ; Ephemeris. w " As 4A — - a - | : OF, ou ' ou i ‘ 235 17 1 235 17 | 1875. | j ” “" u" uw Aug. 24 ,; Polaris, near East Elongation. .. | Direct...... 8 8 Reversed ...! 8 8 52. 69 —1.37 | +0.04 4 —0.05 52. 64 25 |... doa hesegeaeoateeeu tte cerned Direct ...... Le 8 Reversed...) 8 8 52.14 | 1.37 | 40.04) ~0.05] 52.09 2B) \a2eecs OG) snseceen vaeseiseeesecemcee: Direct ..... 8 8 | Reversed ... 8 8 52. 82 1.37 | +0.04 | —0..05 52. 77 27 |.----- CO Acieicde aise: ie aaeiniemieenieniies Direct ...... 8 8 Reversed ... 8 8 50. 30 —1,37 | +0.04 | —0. 05 50. 25 28. | saacaie Ors: aaaeit seesee. dats: sar ecsy Direct ..... 8 8 | Reversed ...; 8 8 51. 63 —1.37 | +0.04 | —0. 05 51. 58 | 24 | 6 Urs Minoris, near West Elonga-| Direct ...-.. : 8 8 tion. Reversed ... 8 8 51. 97 +1.38 | --0.31 | +0. 48 52. 40 | 25 I ase HOEY. Lt SIO a ite Kin dance Direct ...-- 8 8 : | Reversed .... 8 8 61.78 | 41.38 +0.31| +043) 52.21 BE laaxaais 00 Secasar Paemeleeccen 8 8, Raynolds — Young .. = + 0.258 + 0.022, Young — Wheeler... = — 0.017 + 0.024, Wheeler — Robinson. = — 0.005 + 0.015, Raynolds — Robinson = + 0.093 + 0.016, the probable error of each being derived from the discrepancies between the individual and mean values. The { ba \ sign before the numerical vaiue of any equation indicates that the first of the later earlier termined at Detroit. The first two were determined partly by comparison of clock-errors obtained when the observers used separate instruments and the same stars, and partly by comparison of the mean times of transit over five or more wires reduced to the middle wire, obtained when the ob- servers alternated in using the same instrument and in observing the same stars. The last two equations were determined entirely by the second method just mentioned. For the first two the Troughton & Simms and Wiirdemann No. 15 transits were used, for the third Wiirdemann No. 15, aud for the fourth the Troughton & Simms transit. Professor Young observed only with Wiirde- mann transit No. 15. two observers observed a star’s transit { } than the second. These equations were all de- ~ Table of results for personal equation. . No. of stars | Mean for | Total No. of | Mean ofall | Probable Equation. Date. observed. date. stars observed.| results. error. 1864. Ss. s. 8. Raynolds — Young.......... Duly 18! |. scsieccs ciwseelsaasicecenest ; DULY? DO, fs ctieseceise pena Wewcinagee cde 11 +0. 258 +0, 022 Young — Wheeler .......-.. July 18 i Wy! WO. eis iscstcinsore ieisis [isis a bie creitrese 24 —0.017 | +0. 024 Wheeler — Robinson ....... Mar. 3/12 +0.017 Mar. 24 | 24 +0. 036 Apr. 18) 30 —0. 047 66 —0. 005 +0. 015 Raynolds -- Robinson ......- July 27 12 +0. 101 July 28 16 +0. 140 Aug. 2 | 7 —0. 004 Aug. 7 | 3 +0. 043 38 +40. 093 +0. 016 The application of these equations requires the assumption that they were the same when ob- served as when the longitude determinations were made. As will be seen from the above and the preceding table, the first three equations were determined on dates somewhat remote from the dates when the difference of longitude was observed, and on this account they should have less weight as affecting that difference than the fourth equation. However, in view of the objections to assigning such weights by any arbitrary process and of the fact that the four equations are sub- ject to a rigorous condition, it seems best to give each equation the weight indicated by its prob- able error derived as stated above. The observed differences of longitude shown in the first table, will also be given weights indicated by their probable errors. In order, now, from the preceding data, to derive the most probable value of the difference of longitude, let it be denoted by y+(y), and let the most probable values of the aFove personal equa- tions be denoted in order by a+(a), at+(a), v+(x3), and a,+(24). Then substituting for y, §3.] LONGITUDES. 721 19™ 51*.0 and for 2, x, #3, and #,, their observed values, there result the following equations: (y) —(#))=+0°.839-+ 4,, with weight o-(sam ) (y) —(a#,)=+0°.956+4,, with weight »-(oor) (Wa) +(ay)= +0082 dy, with weight r-( gia30 ) (a1)= 0 +4,, with weight p-( oases.) (x)= 0 +4;, with weight o-( sami) (v)= 0 +4, with weight o-( ons) . is ; (x)= 0 +4, with weight o-( war) In addition we must have— Ly + (a) +21 (a2) -+03+ (a3) =94+(¥4) (&1)+(@2)+(a3)—(a4)=—0°.143 or Substituting the value of (a,) from this last equation in the seventh of the above equations and solving so as to make [p4?] a minimum, there results (y)=+08.840+08.059, y+(y)=+19™ 518,840-40°.059. and hence The wave and armature time was computed from the differences between the two clock times of transit of those stars whose transits at the two stations were recorded on both chronographs. The difference between such differences for any star gives double the wave and armature time. The results obtained are given in the table below, the separate values being weighted inversely as the squares of their probable errors, which latter are derived from the discrepancies between the individual values and their mean for any date. Table of results for wave and armature time. Date Wave and arma-| Probable | No. of stars ‘ ture time. error. observed. | 1864. 2. os August 8 ......--...--- 0. 0462 +0. 0017 29 10 i csnzee saaetee 0. 0434 +0. 0017 19 1D csnceecare ones 0. 0461 +0. 0020 15 Weighted mean .... 0. 045 +0. 001 Subtracting the mean value of the wave and armature time from the above value of y+(y), there results for the difference of longitude between the transit-post at Detroit and the transit- post at Fort Howard, 19™ 51%.80- 0.06. The Detroit transit-post of 1864 was 0°.127 east of the east pier of the present Lake-Survey Observatory, and the post at Fort Howard was 6910.8 feet or 6°.360 east, and north 4884.2 feet from Fort Howard triangulation station. Applying these corrections, we have Fort Howard station west of east pier of Lake-Survey Observatory at Detroit, oO! 197 58°.0340°.06. 91 L 422 ASTRONOMICAL DETERMINATIONS. Cuap. XXV, DIFFERENCE OF LONGITUDE BETWEEN FORT HOWARD AND ESCANABA, AND LONGITUDE OF STATION FORD RIVER. § 4. The differences of longitude between Fort Howard and Menomonee and Menomonee and Escanaba were determined telegraphically in 1865, and the results were published in the Report of the Chief of Engineers for 1866. Neither of these differences was corrected for personal equation, and no precise data exist for making such correction. Their sum, however, or the difference of longitude between Fort Howard and Escanaba, can be derived so as to give a good elimination of personal equation as well as of wave and armature time. In these determinations the two extreme stations, Fort, Howard and Escanaba, were each pro- vided with a chronograph and sidereal clock made by Bond & Sons. Wiirdemann transit No. 1 was used at Fort Howard, and No. 15, by the same maker, at Escanaba. Hach of these transits has a focal length of 31 inches and an. objective 2.5 inches in diameter. At the intermediate station, Menomonee, Pistor & Martins transit No. 1 of 24 inches focal length and 2.5-inch objective was used. The transit-key of each observer was connected with the main circuit, so that the star tran- sits at each station were recorded on both chronographs. The same list of stars was observed at the three stations. Observations were made on the nights of August 11, 12, 14,16, 17, 18, and 22. On the first three nights the observers at Escanaba and Fort Howard were Assistant O. B. Wheeler and Professor C. A. Young, respectively. Wheeler and Young then exchanged stations and observed at Fort, Howard and Escanaba, respectively, on the last four nights. The observer at Menomonee on each date was Assistant 8. W. Robinson. The results of the determinations are given in the following tables condensed from the account of the work given by Assistant O. B. Wheeler in the Report of 1866 previously referred to. The difference for each night is the mean of the results computed from the two chronographic records, and is given a weight equal to the number of stars observed: Fort Howard — Menomonee. ‘ Computed differ-| No. of stars Date “ence. observed. 1865. mM. ». AUQUSEAL scoscveicce cones ceeriecomnses 1 34, 421 i 39 V2 esses steeessiceanaeeses 34. 348 47 | Mas 2.5 sis ieyayeeiaieie Statik coeeseig cele 34, 437 39 Weighted mean............-.- 1 34, 399 125 ANPUBUIG sree ere ee teed ss se, 1 34.577 33 j AG, sieorsnncuasctetavesee pate 34. 389 27 | HG ce core tah eam toes 34. 465 50 : 02) socseeratemamiangiamsoce 34, 516 25 | Weighted mean..........-.... 1 34. 487 135 ! Menomonee — Escanaba. Computed differ- | No. of stars Date. ence. observed. 1865. m. 8. PSUBUSE Mo 2/2.o cect cece a asrenama hae dels 2 17. 361 44 DD iectieciers Yaoi, Sioa eet ses 17. 372 49 De, dscteleeciie stems worden weceas 17. 354 45 Weighted mean .............. 2 17. 363 138 AUGUBO AG drciidv seeceeaaid vicnseoses 2 17.305 33 AM natatentusreseane Saercsseas 17. 406 22 DS 2icisiecciSs J delserage reqats aoe Seed 17. 395 49 dn ia kee eta ciate a 2 17. 294 27 Weighted mean. sssccd vac cscs 2 17.376 131 §§ 4,5.) LONGITUDES. 723 It will be seen from these tables that the number of stars observed on any date was about the same for each of the two differences Fort Howard— Menomonee and Menomonee—Escanaba. As those observed were also mostly the same stars at the three stations, the half sum of the four means in the table will give a result for the difference Fort Howard—Escanaba nearly free from the errors of observation at Menomonee, and free from the relative personal equations between the observers. The result thus derived makes the transit-post at Fort Howard west of the transit- post at Escanaba 3™ 51°.81. The data from which this result is derived do not afford a measure of its probable error. Judging from similar work, however, such probable error may be safely estimated as not exceeding+0°.1. The transit-post at Fort Howard was 6°%.36 east of the triangulation station. Applying this correction and the longitude of Fort Howard triangulation station, given in §3, to the above differ- ence, there results fur the longitude of the transit-post at Escanaba, 15™ 59°.86-+ 08.12 west from Detroit. The transit-post at Escanaba was 11°.99 east of Ford River triangulation station. Hence the longitude of Ford River is west of Detroit, 16" 1L1°.8540°.12. DIFFERENCE OF LONGITUDE BETWEEN MARQUETTE AND ESCANABA.—LONGITUDE OF STATION TRILOBA. §%. The difference in longitude between Marquette and Escanaba was determined by the tel- egraphic method in July, 1865. Each station was provided with a chronograph and sidereal clock. Wiirdemann transit No. 1 was used at Marquette and No. 15 at Escanaba, these instruments being those described in §4. The same list of stars was observed at both stations, the star transits being recorded on both chronographs. Observations were made on the nights of July 20, 21, 22, 27, 29, and 31. The observers were Assistants O. B. Wheeler and 8. W. Robinson, Mr. Wheeler observing dur- ing the first three nights at Marquette and during the last three at Escanaba, Mr. Robinson observ- ing on the same dates at Escanaba and Marquette, respectively. The results of these determina- tions were published in the Report of the Chief of Engineers for 1866, from which the following table of results is abridged. The result for each night is the mean computed from both chrono- graphic records, the wave and armature time being thus eliminated: Difference of longitude, Marquette— Escanaba. Computed dif- | Number stars Date. ference. observed. 1865. m »&. July 20 1 35, 245 46 | 21 35, 216 42 | 22 35. 191 33 | 27 35, 216 42 29 35. 274 55 31 35, 210 51 Putting now for the most probable value of the difference in longitude, Marquette — Escanaba, 1™ 35°+y, and for the relative personal equation, &c., eliminated by the interchange of observers, x, the following equations result: 8. 0. 245= 4,, with weight 46 —0.216=4,, with weight 42 0.191=4;, with weight 33 y+ae—0. 216= 4,, with weight 42 y+x—0, 274=4;, with weight 55 yta—0. 210= 4¢, with weight 51 From these the normal equations are: eee | 269y+ 27#=+61. 497 2Ty4+-2692=+ 8. 207 whence y=-+08.228 + 08.009 x=-+05.008 + 02.009 [24 ASTRONOMICAL DETERMINATIONS. [Cuap. XXV, Hence the difference in longitude between the transit-posts at the two stations is 1™ 35%.23+ 05.01. Adding to this the longitude of the Escanaba transit-post, § 4, the longitude of the Marquette tran- sit-post is west of Detroit, 0° 17™ 35*.09+ 08.12, The Marquette transit-post was situated on Thone’s Hill, 28422.8 feet distant, bearing south 19° 46’ 28.65 cast of triangulation station Triloba. This requires a correction of +9*.17 to reduce the above longitude to Triloba. The longitude of Triloba is, therefore, west of Detroit, Oo 17" 44°.264.0°.12, LONGITUDE OF WILLOW SPRINGS. § G. The longitude of primary station Willow Springs was determined in 1876 by telegraphic exchange of signals with the Lake-Survey Observatory at Detroit. The observer at Detroit was Captain H.M. Adams. The instruments used were Buff & Berger astronomical transit No. 2, with focal length of 39 inches, aperture of 3 inches, and eye-piece giving a magnifying power of 87 diameters; Bond & Sons sidereal clock No. 256; and Bond & Sons chrono- graph No. 216. The observer at Willow Springs (Mount Forest) was Lieutenant D. W. Lockwood. The instru- ments used were Wiirdemann astronomical transit No. 1, with focal length of 31 mches, object-glass 24 inches in diameter, and eyepiece giving a magnifying power of 100 diameters; Negus sidereal break-circuit chronometer No. 1524; and Bond & Sons chronograph No. 245. At both stations the transits were mounted on stone piers, that at Willow Spring's being 5716.1 feet distant, bearing north 38° 09/ 01’ west from the trigonometrical station, near the railway station Mount Forest, Illinois, on the Chicago and Alton Railway; and that at Detroit being the east pier of Lake-Survey Observatory. The land within a radius of fifteen miles from Mount Forest does not vary from the general level more than about 100 feet at any point. About 16 miles to the northeast is Lake Michigan, with a low shore and a bottom gradually declining at a rate of about 75 feet in 10 miles. Wire-intervals, inequalities of pivots, and level values for the Buff & Berger transit were de- termined in the summer of 1876. Wire-intervals of the Wiirdemann transit were determined in the winter of 1875-6, and inequalities of pivots and level values in the summer of 1875. Observations for difference of time between Detroit and Mount Forest, Illinois, were made on four nights in August, 1876, automatic clock and chronometer signals being sent with complete time-determinations preceding and following according to a programme similar to that followed in determining the difference of longitude between Detroit and Tonawanda. (See § 9.) The length of the telegraphic line was about 300 miles. The time-determinations were reduced by the method of least squares, the weights of the obser- vation-equations being derived by the process explained in § 9, the value of ¢, being taken as + 0°.049 and « from observations of equatorial stars as + 0°.066, and weight unity being given to the mean of eleven wires for equatorial stars. The codrdinates of the stars used were taken from the American Ephemeris, from the catalogue of ‘539 Sterne,” Berlin, 1876, and from “General Bericht Europadische Gradmessung,” 1874, preference being given to the catalogues in the order named. The following results for difference of time between Mount Forest and Detroit were obtained : Detroit and Mount Forest. Difference of observed local sidereal times. Date. OE oduct oS ee eee Means. Bab and arma: | est to Detroit. Mount Forest. re-Umed | 1876. hm, 8. hm. ». hm 3». &. Aug. 17 0:19 14. 944 0 19 15.128 0 19 15. 036 0. 092 23 15. 023 15. 216 15. 120 0. 096 25 15. 036 15. 234 15. 135 0. 099 | 26 14. 963 15.141 15. 052 0. 089 - LONGITUDES. 725 §§ 6,7.] Assigiing equal weights to the results for the separate nights, their mean is 0% 19™ 15*,086 +0%.017, the probable error being derived from the discrepancies between this mean and the four individual results. The relative personal equation of Captain Adams and Lieutenant Lockwood was determined by observations on two nights before the departure of the field-party from Detroit and on one night after their return. The observations were made in the same manner as those in the longitude work, each observer using the same transit that he used in the longitude work, the transits being set on stone piers in meridians 5 feet apart, Lieutenant Lockwood’s being the westerly one, and the ob- servers working independently. The observations were reduced in the same manner as the field observations, and the following results were obtained : Relative personal equation, Captain Adams and Theutenant Lockwood. Difference of observed local time. Date. | By signals sent | By signals sont from Lockwood from Adams to | Means. to Adams. Lockwood. 1876. & | o July 15 +0. 243 + ; "186 +0, 215 18 +0. 195 +0. 160 +0. 178 Oct. 23 +0. 315 +0. 245 +0. 280 1 A result for difference of time in this table is Lieutenant Lockwood’s observed local time of a given signal minus Captain Adams’ observed local time of the same signal. Assigning equal weights to the results for the separate dates, their mean is +0°.224+0%.020, the probable error being derived from the discrepancies between the mean and the results for the separate dates. Applying the known difference of longitude of the two instruments, +0°.005, to this quantity, there results for the. relative personal equation between Captain Adams and Lieutenant Lockwvod +0°.229+0*.020, which is to be added as a correction to the observed difference of longitude between Detroit and Mount Forest. From a system of secondary triangulation measured by Lieutenant Lockwood, the position of the transit at Mount Forest was found to be 3.105 west of the triangulation station Willow Springs. The difference of longitude between Detroit and Willow Springs is then obtained by adding the following terms: he m. 8. Observed difference of time bétween Mount Forest and Detroit.... .. .... 0 19 15. 08640. ‘O17 Relative personal equation of observers .........- 020.0. 002 020 cece ee eens + 0. 229-+0. 020 Willow Springs triangulation station east of Mount Forest ...... 0 -....... — 3. 105 There results, Williow Springs west of Detroit, O° 19™ 12°.210+05.026 Further details may be found in the reports of Captain Adams and Lieutenant Lockwood, pub- lished in the Report of the Chief of Engineers for 1877. LONGITUDE OF PARKERSBURG. § 7. The longitude of primary station Parkersburg was determined in 1879 by telegraphic exchange of signals with the Lake-Survey Observatory at Detroit. The observing-pier for Parkersburg was set about 10 miles north of the trigonometrical station in the city of Olney, on the Ohio and Mississippi and the Grayville and Mattoon Railroads, this being the nearest available telegraph station. The transit was 124.2 feet distant, bearing south 56° 00’ east from the southeast corner of the High School building, Olney, and 55861.2 feet dis- tant, bearing north 16° 38’ 35.4 west from station Parkersburg. The land in this vicinity is slightly undulating, but there is no variation from the general level to exceed one or two hundred feet within a radius of perhaps 50 miles. The observer at Detroit was Captain D. W. Lockwood. The instruments used were Buff & Berger astronomical transit No. 2 (mounted on the east pier of the Lake-Survey Observatory) with 726 ASTRONOMICAL DETERMINATIONS. [Cuar. XXV, focal length of 39 inches, aperture of 3 inches, and eye-piece giving a magnifying power of 87 diameters; Bond and Sons sidereal clock No, 256; and Bond and Sons chronograph No. 216. The observer at Olney was Lieutenant P. M. Price. The instruments used were Wiirdemann astronomical transit No.1, with focal length of 31 inches, aperture of 25 inches, and eye-piece giving a magnifying power of about 100 diameters ; Negus break-circuit sidereal chronometer No. 1524; and Bond and Sons chronograph No. 245. At both stations the instruments were mounted on stone piers. Wire-intervals, inequalities of pivots, and level values are those used in the reduction of the observations for longitude of Willow Springs, excepting the wire-intervals of the Buff & Berger transit. The spider-thread diaphragm of that instrument was replaced by one of glass, in March, 1877, and the values of the intervals graduated thereon were determined at that time. Observations for difference of time between Detroit and Olney were made on four nights in July, 1879, automatic clock and chronometer signals being sent, with complete time-determina- tions preceding and following, according to a programme in all respects similar to that followed in determining the longitude of Tonawanda. (See §9.) Stars were selected from the American Ephemeris, from “ Mittlere und Scheinbare Oerter von 539 Sternen” for 1879, and from the “Gene- ral Bericht Europiische Gradmessung” for 1874, preference being given to the catalogues in the order named. The time-determinations were reduced by the method of least squares, weights being assigned to the observation-equations as in the reduction of observations for longitude of Willow Springs. (See § 6.) The following results were obtained: Detroit and Olney. / i Difference of observed local sidereal times. a ' ¢ | a From signals sent from | From signals sent from | | Wave and arma- | ‘Olney to Detroit. Detroit to Olney. | Means. | ture time. | 7 ry , 28795 hm». hm 8. | hom. 8. % | July 26. 0 20 08, 582 0 20 08. 404 | 0 20 08. 493 | 0. 089 ! 28 | 08. 602 08. 409 | 08. 505 0. 097 | 29 08. 645 08. 410 | 08. 527 0.117 | 30 08. 498 08. 308 | 08. 403 | 0. 095 Assigning equal weights to the results for the separate nights their mean is 0% 20™ 08.482 + 05.018, the probable error being derived from the discrepancies between the mean and individual results. The relative personal equation of Captain Lockwood and Lieutenant Price was determined by observations on two nights before the departure of the field-party from Detroit, and on two nights after their return. The observations were made in the same manner as those in the field, Lieuten- ant Price’s transit being set on a stone post outside of the Lake-Survey Observatory, a few feet to the south and west, and the observers working independently. The observations were reduced in the same manner as the field observations, and the following results were obtained : Relative personal equation, Captain Lockwood and Lieutenant Price. Difference of observed local times. | Date. | ; . omievand weetiee Dy signe se from | Means. = = = 1879. , . s. i July 5 —0. 009 —0. 037 | 0. 028 | 8 —0. 104 —0.176 —0. 140 | Ang. 26 +0. 143 : +0. 078 | 40,110 27 +40. 054 +0. 032 / 40.043 A result in this table for difference of time is Lieutenant Price’s observed local time of a given signal minus Captain Lockwood's observed local time of the same signal. Assigning equal weights $8.] LONGITUDES. a ee to the results for the separate dates their mean is— 0°.002+ 0°.036, the probable error being derived from the discrepancies between this mean and the results for the separate nights. Applying the known difference of longitude between the two instruments, + 0°.005, there results for the personal equation between Captain Lockwood and Lieutenant Price, + 0%.003+ 0°.036, which is to be added as a correction to the observed difference of time between Detroit and Olney. The position of the transit at Olney with reference to the spire of the “Immanuels Kirche der Ev. Gemeinschaft,” was determined by Lieutenant Price (1442.8 feet distant, bearing south 81° 49/04” west from the spire), and the spire was located with reference to the main triangulation by obser- vations from stations Parkersburg, Claremont, Check Base, and Denver. The reduction of these observations places the observing-post at Olney 138.461 west of triangulation station Parkersburg. The difference of longitude between Detroit and Parkersburg is then obtained by adding the following terms : he om. 8. Observed difference of time between Detroit and Olney...... ......-...- 0 20 08. 48240. ‘O18 Relative personal equation of observers ....... © --..0 2-2. ee eee ee eee + 0. 003+ 0. 036 Parkersburg east of Ulney observing-post....... ..- ashechitat9 ciate legates La, — 13. 461 There results, Parkersburg west of Detroit, 0" 19” 55°.0244 0°.040 Further details of this work may be found in the reports of Captain Lockwood and Lieutenant Price, published in the Report of the Chief of Engineers for 1880. LONGITUDE OF TOLEDO. § 8. The longitude of Toledo was determined in June and July, 1881, by telegraphic exchange of signals with the Lake Survey Observatory at Detroit. The stone observing-post used at Toledo was set near the north corner of Monroe and Michigan streets, and was 233.0 feet north and 239.7 feet west from a stone which marks the intersection of Monroe and Ontario streets. The land in the vicinity of Toledo is low, flat, and marshy, a large part of the ground on which the city is built being filled in. The observing-post, however, stands in the natural bed-clay. The instru- ment at Detroit was set on the west stone pier in the Lake-Survey Observatory. The observations were made by Assistant Engineers O. B. Wheeler and Thomas Russell. Bond & Sons sidereal clock No. 256 and spring-governor chronograph No. 216 were used at Detroit, and Negus sidereal break-circuit chronometer No. 1524 and Bond & Sons electro-motor chronograph No. 316 were used at Toledo. The transit used by Assistant Engineer O. B. Wheeler was Wiirdemann No. 15, of 31-inch focal length and 23-inch objective; that used by Assistant Engineer T. Russell was Troughton & Simms transit No. 2, of 29-inch focal length and 24-inch objective. Wire-intervals were determined for a new set of wires inserted in Wiirdemann transit No. 15 by Buff & Berger, in April, 1881, and were redetermined for Troughton & Simms transit No. 2. Pivot corrections and level valores were redetermined for Wiirdemann transit No. 15, and for Troughton & Simms transit No. 2 the values found in 1875 were used, after testing their accuracy by a few additional observations. Observations for difference of time were made on four nights in June, 1881, by Mr. Wheeler at Detroit, and Mr. Russell at Toledo. The observers then changed places, taking with them their transits and break-circuit keys, and observations were made on four nights in July, 1881, personal equation and kindred instrumental errors being thus eliminated from the mean result for the eight nights. Automatic clock and chronometer signals were exchanged between the observers on each night of observation, preceded and followed by complete time-determinations, according to a pro- gramme quite similar to that followed in determining the longitude of Tonawanda (§ 9). Stars were selected and star-places were taken from “ Mittlere und Scheinbare Oerter von 539 Sternen” for 1881, with the exception of two stars taken from the American Ephemeris by Mr. Russell. The length of the telegraph line was about 60 miles. The time-determinations were reduced by the method of least squares, the weights of the obser- vation-equations being derived by the process explained in § 9, the value of <; being taken as +.05.056 and « as +0°.08, and weight unity being given to the mean of eleven wires for equatorial stars. 728 ASTRONOMICAL DETERMINATIONS. (Cuar. XXV, The following results for difference of time between the observing-piers at Toledo and Detroit were obtained : Detroit and Toledo. 0. B. WHEELER, AT DETROIT. T. RUSSELL, AT TOLEDO. | Difference of observed local sidercal times. ' Date. oatench || daartieda Means. | - | 1881. hm 8 hom». hom». | June 23 0 1 57.968 0 1 57.966 0 1 57.967 / 24 57. 792 57. 848 57. 820 { 28 57. 760 57. 761 57. 760 i 29 57, 789 57. 810 57. 800 T. RUSSELL, AT DETROIT. O. B. WHEELER, AT TOLEDO. July 1 0 1 57.748 0 1 57.717 0 1 57.732 2 57. 724 57. 722 57. 723 4 57. 773 57. 763 57. 768 5 57. 824 57. 172 57. 798 Putting now 1™ 57s+y for the value of the difference in longitude (Toledo minus Detroit), and x for the quantity eliminated by the exchange of position of observers, the following observation- equations result from the above column of means: 8 y—x—0. 967= 4, y—ax—0. 820= 4, y—a—0. 760=4, y—a«—0. 800= 4, yte—0. 732= 45 ytu—0. 723= 46 y+a—0. T68= 4; y+ae—0. T98= 4, Assigning equal weights to these equations, the normal equations are 8y + 6. 368 8x2 ——0. 326 whence 8 y=+0. 79640. 016 a——0. 04140. 016 Hence, Toledo observing-post is west of the west post in Detroit Observatory 1™ 57*.796 + 0°.016. The reduction to the east post in the Detroit Observatory is +0°.004. There results, Toledo west of Detroit, O OL 57.80010°.016. The longitude of Toledo was also determined in 1868, Assistant Engineer O. B. Wheeler making the observations at Toledo on a wooden post 16.6 east and 13.6 south of the stone post occupied in 1881. Lieutenant E. H. Ruffner made the observations at Detroit. Chronographs were used and automatic signals were exchanged. No correction for personal equation was applied to the result of 1868, nor was there any attempt at its elimination. The results of the two determinations agree quite well, however. The result obtained in 1868, reduced to the post occupied in 1881, is, Toledo west of Detroit, 0° 01” 57°.693. The result obtained in 1881 is adopted. Further details may be found in the reports of Assistant Engineer A. R. Flint on longitude work in the Report of the Chief of Engineers for 1881. The longitude of Toledo was referred to station Cedar Point of the primary triangulation in 69, LONGITUDES. 729 the following manner: By a secondary triangulation the longitude-post of 1881 was found to be 879.0 metres south and 650.5 metres west of the spire of Saint Mary’s Church, situated on the cor- ner of Michigan and Cherry streets, Toledo. From angles observed at stations Bedford and Cedar Point, the spire of Saint Mary’s Church was found to be 17535.95 metres distant in azimuth 71° 05/ 56.27 trom Cedar Point station. These codrdinates give longitude-post of 181 28.11 west of church spire, and church spire 11! 57’.05 west of station Cedar Point. Therefore, Cedar Point is west of Detroit, O° OL 08.123. LONGITUDES OF TONAWANDA AND MANNSVILLE. § 9. The longitudes of primary stations Tonawanda and Mannsville, New York, were deter- mined, in 1875, by telegraphic exchange of signals with the Lake-Survey Observatory at Detroit. The observer at Detroit was Lieutenant D. W. Lockwood. The instruments used were Wiirde- mann astronomical transit No. 1, with focal length of 31 inches, object-glass 24 inches in diameter, and eye-piece giving a magnifying power of 100 diameters; Bond & Sons sidereal clock No. 256, with break-circuit attachment; and Bond & Sons chronograph No. 216. The transit was mounted on the east pier of the Lake-Survey Observatory. The field observer was Assistant Engineer A. R. Flint. The instruments used by him were Troughton & Simms astronomical transit No. 2, with focal length of 29 inches, object-glass 24 inches in diameter, and eye-piece giving a magnifying power of about 80 diameters ; Negus sidereal break-circuit chrounmeter No. 1524; and Bond & Sons chronograph No. 245. Wire-intervals, pivot inequalities, and level values for both instruments were redetermined. Observations were commenced on August 30, but owing to unfavorable weather were ot com- pleted until December 27, 1875. The programme included two nights’ observations to determine the relative personal equation between Lieutenant Lockwood and Assistant Engineer Flint; ob- servations to determine the difference of time between Detroit and Tonawanda; observations to determine the difference of time between Detroit and Mannsville; and two nights’ observations to determine relative personal equation on the return of Assistant Engineer Flint from the field. The following programme was followed as closely as the weather would permit on each night of obser- vations for difference of time, the sathe stars being observed at both stations. when possible : Circumpolar star, reversed on, with level readings. Equatorial stars, with level readings. Reversal of telescope. Equatorial stars, with level readings. Circumpolar star, reversed on, with level readings. Clock and chronometer comparisons in duplicate. Circumpolar star, reversed on, with level readings. Equatorial stars, with level readings. Reversal of telescope. Equatorial stars, with level readings. Circumpolar star, reversed on, with level readings. Clock and chronometer signals were seut by placing the Detroit clock and the field chronome- ter alternately in circuit, each for 2™ 208, two sets of signals being sent from each station. The time-determinations were reduced by the method of least squares. For observations on stars south of the zenith, weight unity was assigned to each equation resulting from a complete transit over seven wires. The weight resulting from a transit over any other number of wires was computed from the formula : ei +7 p= : ee pag | where ¢, is the probable error of culmination; «, the probable error of an observed transit over a single wire; and », the number of wires over which the star’s transit was observed. ¢, was taken as +08,056 trom Appendix 12, United States Coast-Survey Report for 1872, and ¢ from observa- 72 LS 7 730 ASTRONOMICAL DETERMINATIONS. (CHap. XXV tions of equatorial stars was found to be +0%.07. For equations resulting from transits of polar stars, weights were computed by the formula 2 a ie PHY e! « in which e’ is the probable error of an observed transit of a polar star over a single wire, and ¢ and p ave the same as defined above, p being computed for x, the number of wires over which the polar star was observed. The codrdinates of the stars were taken from the Americau Ephemeris for 1875, and from the Catalogue of ‘529 Sterne,” Berlin, 1875. 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Wet canes oe ges ss : | 1 | | | Wie [Ae pe TP 204 ween | av | ¢ TUBUOPAN AL TOOT VO “OLY shee eee sini Ts ggg FT| BOO TT ‘PaLTSL 90 | : | | | gust i (D2 ASTRONOMICAL DETERMINATIONS. [Cuar. XXVI, § 2. Longitudes other than those given in Chapter XX V were determined telegraphically for the Lake Survey at many points along the lakes between 1863 and 1870. In this work personal equation was only occasionally eliminated. The methods were essentially the same as those described in Chapter XXV, save that in some cases telegraphic signals were only sent when the same star crossed each wire of the instrument at the first and second station. In the following table the first column gives the name of the station whose longitude was determined; the second gives the date of observation; the third, the number of stars observed in common at both stations; the fourth, the observed difference of time between the two stations occupied; the fifth, the reduction to fixed point at each station; the sixth, the difference of time between the station and Detroit; the seventh, the longitude of the station west of Greenwich, in are, derived by applying to the result in the sixth column, the longitude of Detroit west of Green- wich (5" 32™ 128,24); the eighth column gives the seconds of geodetic longitude computed through the triangulation from the longitude observed at Fort Howard; the ninth, the discrepancy between the geodetic and observed longitudes; the tenth, the names of the observers; the eleventh, the in- struments used; the twelfth, data as to personal equation, catalogues used, points of reference, &c. : NON-FUNDAMENTAL. 153 §2 “SINQsSUIPSO TV VP ‘neg WIG yV 0 ‘gaToMoUa A 4¥ 9 ‘yonHed +¥ Dv 06 ‘ST 08 GL 86 OT O&— | PLSB T+ | Ther O€— Jone "7" KN ‘Smmgqsuepso ‘ydeizouoigo pue ‘yo0[9 (-A0Taq 998 TBOLOPIs ‘T “ON PISTRIA [VOT ‘OB9MSO DLZL‘O— | ZS ‘OT 93— ‘q10130({ ULI OF aKNSQ) “OIST Ut paysrqug | WOUousE sUIIETL 7 10VSTE | 98 ‘s9TPT.AA “A “O 61°20 b — Gai yan? wean “poyeUIUNye st WoIeNbe [euosIeg ‘qdeisouo1g9 pue ‘yo0[9 “Sanq, SI 30 F £@ &T ‘osnoy [eorepis ‘cl ‘ON yIsueIR -SUsPIO ye ‘Iow Ie °20 F c It (oSoa JUST St Sinqsuepso 3e yutod poxty | [eormoucryse uUuvOTepin A, | -PUy “WA “WerT |*--"- PEARS AN AREA ET ARE Shek ec nen pie | ape iY ee ee - OL ATug | -sO mor Sinqsuepso) "898T “qderZouorg9 pue ‘yD0]9 G0 FE GO 6 £0 OL OF+ | O8TRO+ | 19°60 OFT [--TT ttt fr ttt ttf UU ‘Ned WUIes Yeosepls ‘GT “ON gSueT. | “YQN[NC 4 ‘yxO0}s C28 ‘2ugt ur peysrqug | oFoucTse uUEmOpIT AL 09 “g “OQ ‘Way ‘AXX sajdeqg os * perdde uomenbe peuoss0g ‘zoqourowosqo 20000 | GB"GO GET TTT |e ‘yosyaq, Woy qINTAC) ‘osnog-m04 =| om UOTE §=‘o}9TIOMOITO GL 80. Fe ose e eres seeessee *|""Weeur pazqste MA -8119 §9981G PoeyIUG JO IoI09 Y8eO Teolopis ‘T ‘ON 4yisuesy ‘Neg PUIG 99 80 F Semen GT Cagny “qynos Ss} [NYT yuyeg ye yurod poxty | [eomousyse UULMOpPIN AA | Fe UN WM Wren eetrrresseesecces RRS RAS) GEG Be rere PI oun | -uq Wor [neg UTES) “TL80 $3 ‘9+ 09 ‘FE GL‘LZ SE L8 19°60 8I+ | 9689 °LZ— | 82°LT 8I+ |--777 Te srrteseessl-oe* TOT ‘GaMOMOUaPL (8 § ‘AX 107 -deqg 03s ‘yroeq EET Or | OS TS GLb [Freee ees see ee« Woy prlVKoH 3107) “OLET PUL OBI UT peysTTaNE “(mopenbe “sonUaAsR IOATY puvly puy u0SUt Teuosirad = 10g -YS@M JO Soxe OY JO TOI}008107UT eo PE T — ne Pelee racy pe}0e1109) TLOPL oy} Jo [INOS Joos GE PUL 4sau yooq ‘qdviSouo1g9 pue ‘yqo0[9 =| ‘:pavmo 10,7 9B eaten ee a $0 St Yor ‘zegT-Tust Jo Luoqea | [OMPPIS ‘T “ON F18UMIZ | TAPPOTAN “AO oF Fe T og | st -z0sqQ Leaang-oxer] Jo 101d yee0 st yeotmouo1se uuvMmopina | pur Sunoy “Wy ‘Oo . Gone RZ ee O1q2} STU} UT JOIJ9q 4e yuTod poxtT “eqeuBos” PU PABAOH eo Fs T 88 9T ‘porjdde uorjyenbea teuosi0g | 4104 4% sydvasou01g9 pue PFE T 68 *L “(P181) Toes UoYe[Dsueray Aseur SH0TO ‘T ‘ON g{smeay [wor “‘SoTOTIOTO TT gee T Ly SL (preaoy VOT +d st oouomoueyZ ye yuIod poxig | -mouoIjse sUIZIE 7 IOPSIG | 4 ‘MosTIQoYy A'S [TTT fs SR ER EES Se ee a 68 Il ‘Suny | worz oououo0usy_) - é eb Abe 9 ue "9 2 UL “Cost Q QR et Pee | Fe | 82 | 248 | 28 | de | Ede “Sy IVMley *s}U9UIDIAgUy *S194.108qQ 23 ze qe E 5 8 BS go = = re ‘oye “woneg ue | 82 i Bn& | Fs ge | ee¢ sé & 43 She |e g3 Ses BE 2 Se aoe 3 gh | Fs ne o vt x aAe a Bo BS yoyuaunpunf rou ‘sapnzibuoT pnonuouc.usy 95 LS [Cuap. 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Bue R es | 282 ee er Pe Fag & BS | "8 “ponuywog—vjusunpunf you ‘sapnpbuoT ynonuwou0s.usp cists) NON-FUNDAMENTAL. § 2.) “yaryung 3¥ Y ‘oregng yy 6 ‘WOARA purly ¢7 *‘IoySOOOY YW a ‘OSOMSO UVP - “Youepoy VW 9 "UM0FIOIV AM FV 9 “‘qorjeq FV V 66°6 — | 18°ST 08 °SG 16 6L GS 9h PI— /YTOL'L + | 60 "FS FI— 1 7 AON Sapoaqg (‘9A0qv 998 “ydeisouorygo pure ‘y90]9 = DLT ‘0 — | 66°SP QT— [ttf ‘yromeq Wosy oTeBN_) [eolopis ‘Gt ‘ON yisuesy ‘ojeyng }e ‘10a 06 6R Tb crt certs fire ce ecee fee eee eee eee ueeyt [eormouoIjsv UUTMepInA | -puy'H “| “‘gneryT |_—__ —_—_ “OLST UT poysTTqng ‘ely puv 66 “6F T 08 &T “peyeuranyya you uolyenbe jeuosiseg | ojeyng 4v sydvasouoryo e 18°67 T 8& L ‘(9L8T) esnoy PUB SHOOT ‘TON qIsueIy “pay 66 6F T at % (copes “-WUST] St yryUnq ye yurod poxrg | Teormouonse WULOLepIn A | 9 ‘ToUSTM “A “HD [ot Fae, Seal ae niger pate Hope eines ead ela reece ar 89 6h T + SL & ‘sdeg | Jug woz yiryunq) - *898T CL'€ — | 83 'FT 008 oS 8L 708 9I— [4180 T + | 66 °8h 9T— jon” sereeeesestesesecccs NT ‘opegng | (‘KOTO 098 ‘41013 DLET'0 — | 93 FBS — “eq WoIZ puLpeas[D) &L6T II— coeeseeeess: TROL ‘ydeisouoiqo pure ‘x9070 “oLeT UT poystiqng | MO7OPIS TON F8aeT} Teor Reesesisineacs 8661 IT og |e@ 390 “peyeurmye st woyenbe yeuoszag | WOME SUIZIVTT p 10ySTq | ye ‘IOPIOT AM “EL “O 09 6 IL ot 62 *sqoor48 ‘ydvisouorgs pur ‘x9019 €L°6T IT b & UBS pue esuryoxg josexejo | twosopis ‘ct ‘on yisuedy ‘oreyug ye ‘loa ZP6L IL IZ IZ (‘put WorjoosIE} UT ST OTeBN 4 yutod poxty | [VoyouNse wUMOEpPINA, | BUY A “Gyorgy j++ 7 ate ts 18 ‘61 TI— v8 BI ‘ydog | -es0TQ wmoay opegng) ‘8981 ‘qdeisouo1q9 pur ‘yo0]9 “T9BT UT poysrqng | OPIS T “ON HSTVIy eee pueiy ST'TS FI 98 SI'Lb BI+ [Sesh T + | Ggcch SEF Jono enc (‘198t Jeormoucrse UUvMepInA | 4v ‘VQ sours pigt ‘0 — | $8 -SP at ot ae ‘uep ‘vg ‘f) ,,“worRenbe jeuosidd oy,, ‘ydvisouorqo pue ‘yo019 *TOST. “esnoy Teosepis ‘cT “ON rsueIy “HorgoqT 4B $8 SP GT 08 &T “74ST St UosvY puery ye qurod poxty | [eorwouoysy uuemopimm | ‘oog ‘W ‘O “ynorT Pre iene ee ore cemerer feels seesetyereseess =") 98 'eh 8E+ Te 8T “AON |" WOTAL ‘Weaey puviy 7 “098T Té'0 — | 69°0S 00 ‘TS 98 LL ¥8 ‘FP TS— | 9198S — | ShLP TZ Jot fee ne "7" HIOX AON Toysoqooy (‘A.0Toq 908 . 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' i ‘TL8T a ( 66 ASTRONOMICAL DETERMINATIONS. (Crap, XXVI, § & For convenience of reference, the results for astronomical latitudes and longitudes given in this chapter and in Chapters XXTIT and XNYV will be collected in a single table arranged in the order of latitudes. Observatory of 1871 at Detroit, given in Chapter NXY, § 1. Table of Astronomical Latitudes and Longitudes. Station. St. Ignace, Ontario Dip Lop; OntanOncssccaseeceees. sees: Isle Royale, Mich ' Farquhar's Knob, Minu Michipicoten, Ontario Gargantua, Ontario ...-.......--.---- | Copper Harbor (Fort Wilkins), Mich... Vulean, Mich... ...-. ebds sisie ake shel Mount Houghton, Mich Sawteeth East, Minn Wheal Kate, Mich | Onter Island, Wis-.........-..--.- ---- Portage Entry, Mich Crebassa, Mich Buehapan, Minn Huron Mountains, Mich South Base, Keweenaw Point, Mich... Porcupine Mountains, Mich Whitefish Point, Mich North Base, Minnesota Point, Minn -. Brulé River, Wis ..... s ae Sonth Base, Minnesota Point, Minn -. | Amainicon, Wise soc ses ce ccies ecinesee ene Grand Island, Mich ' Marquette, Mich....-..-.------..----- ; | Sault Ste. Marie, Mich | Troquois Point, Mich | Escanaba, Mich | Ford River, Mich | Burnt Bluff, Mich | Beaver Island, Mich . Cedar River, Mich ' Rock Island, Wis Boyer’s Bluff, Wis Door Bluff, Wis ' Menomonee, Mich. (near Marinette, ¢ NVAS: a crosts crclecceich teeters cesta lataeiersiane Green Island, Wis ‘Thunder Bay Island, Mich ..-. « Saint{Paul Mintecsceecnts oes ceisct iss Ogdensburg, N. ¥ Clay Banks, Door County, Wis Frankfort, Mich Red Wing, Minn. - » Fort Howard, Wis Kewaunee, Wis Deadwood, Dak Manistee, Mich Valley Junction, Wis Valley Junction, Wis ......-----.-.--- = Thone’s Hill, Mich s..02 sc s20. -peea cae te Y | Details fl 1 t | | | | ' Latitude. Longitude. | given on | page— oO y wt 48 47 28.65 |... 622 48 16 25.78 ° 733 48 07 55.22 |....2. eee ee ee 733 AT BIBS ON dsacnzasscsen 738 47 45 20.58 0-22... wee | 733 47 34 38,80 734 47 28 03.15 | 734 47 26 44.58 '...0.. eee 625 47 24. 15,88... ceeceeene 734 47 23 09. 734 | 47 04 18. 734 47 04 14, 735 46 58 47. 735 46 58 41. 735 46 56 24. 735 46 52 53. 626 46 52 22. 623 46 47 03. 735 46 46 17. 736 46 45 28% (25, 717 46 45 20, 736 46 42 51, 736 46 41 36. 736 46 33 736 ae ioe nage 723 46 32 737 46 30 737 46 29 137 45 44 737 45 41 722 45 41 03. 737 45 34 28,78 | PL eee 738 45 25 4: 738 45 738 45 : 738 4517 AS GU veces iin 738 45 05 12.78 | 87 35 27.75 | 739, 753 : 739 : 739 . 739, 753 F 739, 758 44 4119.45 |......02..00. 740 44 38 53.06 |........2.... 740 44 33 44.64 | 92 31 49.65 | 740, 760 44 30 30.28 | 88 02 34.05 | 627, 718 4425 41.12 .......0..004, 740 gee a iy 103 43 18.6 766 44 16 45.53 ,......00..006 741 44 14 57.24 |.. dais 741- eechaele te ay 90 25 56.70 760 Poiut of reference. Triangulation station. Do. : Do. Do. Do. t Do. : Stone post marked §, ; Triangulation station. Do. Do. Do. Stone post marked S. Light-house. Triangulation station. Do. Do. Do. Light-house. Triangulation station. Do. Do. Do. Light-house. Stone post. Light-house. Do. Do. Do. Triangulation station. : Do. i Light-house. Triangulation station. Light-house. Triangulation station. Do. Do. Liglt-house. Do. Custom-bouse. Light-house. Azimuth station. Do. Spire of Catholic church. ‘friangulation station. Azimuth station. Post-office. Azimuth station. Southwest corner of sec- tion 31,township 21 north, range 3 cast. » Northeast corner of sec- tion 1, township 18 north, range 1 west. -\ll longitudes depend on the longitude of the east pier of the Lake-Survey Rawley’s Point, Wis , Mannsville, N.Y... ' Minnesota Junction, Wis NON-FUNDAMENTAL. Table of Astronomical Latitudes and Longitudes—Continued. Station. Big Pointe au Sable, Mich Writertowai, Ni VP sceccuied otesauneene | Point Creek, Wis -.---.. North Sheboygan, Wis | Goderich, Ontario..-............-.---- South Sheboygan, Wis ! North Base, Sandy Creek, N.Y .....-. ; Clay Banks, Mich........-..-..--.---- ' Aa TWAS soe ose cruel Shee weree : Oswego, N.Y .@.......2----06- coeas : Newaygo, MW acs xan soccaiaae tenn SACTINAW ANE. 355 Sch ea eSoradsenden t SHE LAs, MC cc cows ves te cen camces ' Duele Dake, MCW sc00 cevics ves peace vee Stanton, Mich, s<..s20% 4055 swsescw. se0 ; Fax Poink, Wastscec.ccceselseinescesatd ROCHESTER, Niu Wee ce ncisietiectine caceteee Lapeer, Mich Bln E MNCL c arate cinia ss cieidciewrereersieciniees tae Fort Gratiot, Mich..........--.-- ---- Tonawanda, N.Y Saint Johns, Mich Grand Haven, Mich.............------ t Tonia, Michticcicn-usncsseeumcemettcaad Corunna, Mich..........-.------------ South Twin Sisters, Mich......--..--. Grand Rapids, Mich Buffalo, N. Y.....--.-------- 202 ee ee eee Fort Fetterman, Wyo Lansing, Mich .........----0----.-0++- Camp Robinson, Nebr.....---- .-..--- Hastings, Mich .........-.....--..---- Howell Micht ssesovesense sens csiacec'sres'e Charlotte, Mich. .-...-...-...-.2.------ Allegan; Mich: .2 ccc sce sceccswassecces Galena, TMs sc ciccncates accawiaataesnncane 43 20 03.07 43 10 13. 26 43 09 22. 44 43 03 07. 26 43 01 11.71 43 00 21.0 | 43 00 07. 83 43 00 02.39 42 58 37.92 42 58 34, 88 42 57 47.04 42 52 44. 66 42 38 54.13 42 38 11.62 | 42 38 06.95 42 36 11.18 42 33 58.96 42 31 44.74 42 30 26.10 81 42 44. 85 76 02 26.70 85 48 04.35 83.57 50.4 84 36 24. o 85 04 52.50 77 36 51.00 83 18 53.40 83 41 06.45 | 78 58 21.15 | Bt 33 36.60 85 03 53.25 ' 84 07 02. 85 85 40 06.15 78 52 18.00 105 29 12.6 84 33 16.95 103 27 44.1 85 17 17.55 83 17 15. 60 83 55 50, 85 84 50 01.20 85 51 08, 55 90 25 33. 60 | 86:14 51.15. | Details | given on page— Point of reference. 741 | Azimuth station. 741 Light-house. 742, 754 | Court-house. 742 Azimuth station. 742 Light-house. | 742, 754 | Court-house. | 729) Triangulation station. { 743 | Azimuth station. 638 | Triangulation station. 743 | Azimuth station. 743 | Do. 628 Triangulation station. i : Tad Do. ' | 743,760 Intersection of Wood and | State streets. | 743,760 | Court:house. 744,760 Intersection of Washine- ton and Mill streets. 744 Azimuth station. 744,761 Iutersection of Main and Camburn streets. } 744 | Azimuth station. 744,755 | City Hall. | 745, 761 | Intersection of Calhoun and Franklin streets. | 745, 761 | Intersection of Keasley : | and East streets. : i 745 | Light-house i 637, 729 | Triangulation station. | 745,761 . Intersection of Clinton { and State streets. 755 745, 761 Light-house. Intersection of Main and First streets. 745, 761 | Intersection of Shiawassee avenue and Flint road. 746 | Azimuth station. 746, 761 | 3-section corner on west side of section 30, town. | ship 7 north, range 11 | west. : 746,755 Intersection of Exchange ‘and Michigan streets. 766 ; At military post. 746,762 | Intersection of Michigan and Capitol avenues. | 766 | Flag-staff at military post. 747, 762 | Intersection of State and | Church streets. 747 | Azimuth station. 747, 762 | Southeast corner section 29, township 3 north, : range 10 east. i 747,762 | Intersection of Higgins and Monroe streets. | 747, 762 | Intersection of Oliver and Stoddard streets. 748,763 | Northwest corner of county building. 748,763 | Intersection of 4th prin- cipal meridian with State line. T07 768 ASTRONOMICAL DETERMINATIONS. (Citar. XXVI, Tudble of Astronomical Latitudes and Longitudes—Coutinued. _ Details | Station. Latitude. Longitude. © given on Point of reference. | page— o a ut oO é wo = Dankitl NiV + ccscc vate emeswass ces 42 29 44.30; 79 21 25. 80 748,755 | Light-house (1876). Detroit MiCiiccacs: caeis aw. xeeeeeemes 42 19 58.65 83 03 03.60 | 748,715 | East pier of Lake-Survey i = Observatory. Wve; Mich: «0 pac.e.cennies cakes 42 17 24.46 | 85 35.11.70 | 749,763 | Intersection of South and | Park streets. | NEAPSH AM, MUCH saree eacicce seamen, 42 16 24.67 | 84 57 49.05°) 749,763 | Intersectionof Kalamazoo | . avenue and Mansion street. ' South Haven, Mich ..-..-.----.------ 42 16 13.48 |...--.-.------ 749 | Azimuth station. | | Jackson, Mich occssscseecacss eseceeqes 42 14 49.40 | 84 24 29.85 | 749,763 | Intersection of Main and ' | Jackson streets. | | Pan? Pia, Michie: = camescsssecdenen 42 13 01.66 | 85 53 03.30] 749, 763 Intersection of Main and : Van Buren streets. Port araimie, Wf02sc.s2ses tecazees|yeneeereeerscs 104 33 21.6 766 | Flag-staff at military post. rie, Paes sssexecen< sues yReecesiteses _.| 42 09 17.49 | 80 04 46.95 | 750, 756 Light-house. > Fort McDermit, Nev. .-...-..--.--+2++|------ 0-25-25 117 387 32.7 766 | At military post. Monroe; -Michis 2% tiamsscneieaeccceceee ee 41 54 48.65 | 83 23 46.35 | 750,756 | Court-house. Chicago; DN .scc.cce.c-es 20d ein set 41 53 22.49 | 87 36 48.90 | 750,756 | Light-house. Ashtabula, Ohio ...-.-.--------------- 41 52 36.13 | 80 47 38.55 | 750,757 | Southwest corner of pas- senger depot. Willow Springs, Ill ........----------- 41 43 38.63 | 87 51 06.75 | 630,724 | Triangulation station. i Toledo, Ohio « 20..2cseeces seme be neeees 41 39 03.65 | 83 32 30,60 | 635,727 | Stone longitude-post of ! 1881. | Rock Island, W w.2: 20 s2.0see22~, selauarecie 41 31 01.51 | 90 33 45.90 | 751,764 | Soutbeast corner of guard- house. Cleveland, Ohio ....------- --+--++---- 41 30 06.18 , 81 42 13.20 | 751,757 | Light-house. West Base, Sandusky, Ohio......---. | 41 29 04.59 - : 636 | Triangulation station. Sandisk; OliG.. ccxes sedeeaserce deer tense sis oeaeccns "82 42 47.85 757 | Northeast corner of cus- tom-house. Ogden, Uta vcnccscxsnes tecwacenmantes cemngeccensaary LET 159) 06,10 766 Stone post in observatory. Hudson, Ohio ..-...--.----.eeeee eee ee "gtd ehessta meng 81 26 16.80 737 Do. Fairmount, IL ....5..acexevore cosner ses 40 UL S670) (ocncncenccasne 631 | Triangulation station. Louisiana, Mo .-..-. ---. 2-225 +-+22- 39 27 12.65 91 03 08.25 | 751,764 | Northeast corner of high . school building. | Fort Leavenworth,Kans .....-..---- |.-+---------+- 94 54 50. 25 767 , Observatory. West Base, Olney, Il..--..------------ 8 SL 4128 tacee onwsameen 632 | Lriangulation station. Parkersburg, Ill ..-.---..------- ‘Li. | 88 84.53.20 88 01 48.96 | 633,725 | Do. i CATO) TU. aetenie ncimiaeaixcenmaeetiogs 36 59 47.99 89 09 31.20 | 751, 757 Triangulation station De- , : | | fiance. Memphis, Veni. ...22..ccexseeseszes: 35 08 39.85 | 90 03 20.25! 751,758 | Triangulation station. Fort Richardson, Tex.......----------| see---eeeee ee 98 09 41.1 767 | At military post. ort Guiffin Pex s.ceceos saeco ctewneeca(Ledewse ainee'es - 99 18 48.6 767 Do. Fort Concho, Tex ..--..-.--------+--+-[eeee eee eee | 100 25 42.4 767 Do. Fort Stockton, Tex .-- 1 767 Do. 1 Pott MeKavett, Tex oo. 2. ancneccsesss [eceeeeeeeese-+, 100 06 23.1 | 167 Do. y er , 1 §G. The following are descriptions of such of the stations in the foregoing tables as are not sufficiently described therein, The years indicated are those iu which the stations were oceupied: Copper Harpor (Fort Wilkins), 1864.—Observations for latitude were made on a pine post situated just north of Fort Wilkins, about one mile east of Copper Harbor, Keweenaw County, Michigan. A stone post marked 8, just outside the fort fence, is 233.9 feet south of the observa- tory-post, and a cross on the rock near the water’s edge is 515.9 feet north. Mount Hoventon, 1865.—This station was situated on Mount Houghton, a mountain on the south side of Keweenaw Point, about 14 miles north of Béte Grise Bay, Keweenaw County, Mich- igan. Observations for latitude were made in 1865 from a wooden post set about 100 feet northeast from the cliff on the south side of the summit. Observations for longitude were made from a wooden post distant 38.4 feet in a southeasterly direction from the latitude-post. A secondary tri- angulation-station, with tripod 20 feet high for support of instrument, and marked by a stone § 6.] NON-FUNDAMENTAL. 769 buried in the ground, was distant 76 feet, bearing south 85°30’ west. The height of the mountain above Lake Superior is 877 feet. PortaGE EntTRY, 1864.—Observations for latitude were made on a pine observatory-post set near the west end of the island at the mouth of Portage River, Houghton County, Michigan, about 1100 feet west of the canal. A stone marked 8 stands 17.0 feet north of the observatory- post. CLAY BANKS (on west shore of Lake Michigan), 1866, ’72.—This station is situated near the shore of the lake, 85 miles northward from Ahnepee village, Door County, Wisconsin. As de- scribed in 1873, it is about 15 feet back from the edge of the bank, about 120 feet from the edge of the lake, and about 330 feet from the wagon-road running from Ahnepee to Clay Banks. The height of station used was 48 feet. This station was occupied in making azimuth observations and in reading a horizontal angle for carrying the azimuth southward. The geodetic point is marked by a stone post of the usual form, set 34 feet below the ground surface. Three stone reference- posts, with square tops rising 10 inches above ground, with holes in their centers and the letters U.S. cut on them, were set as follows: One bearing south 39° 38’ west, distant 154™.66; one bear- ing north 27° 15’ east, distant 145".42; and one bearing north 53° 16’ west, distant 46™.12. A lat- itude-post occupied by Mr. Robinson in 1866 bears south 39° 38’ west, distant 155™.50, and one oc- cupied by Lieutenant Powell in 1872 bears north 27°15’ east, distant 530™.60. The height of ground at the station above Lake Michigan is 60.7 feet. FRANKFORT, 1871, ’73.—This station, on the east shore of Lake Michigan, is in Benzie County, Michigan, about 14+ miles north of the village of Frankfort, 3 miles south of the Pointe aux Bees Scies light-house, and about 170 metres back from the shore. It is on a high and steep hill, the highest land in the vicinity. The height of station used was about 10 feet. There is no record of stone markings, but it is probable that the geodetic point was marked by stone posts in the usual manner. Horizontal angles were measured at this station in 1871 to carry azimuth southward. Latitude observations were made in 1871, and azimuth observations in 1873 from a post bearing north 74° 55’ east, distant 27™.75 from the station. The old triangulation-station, Pointe aux Bees Scies, bears north 1° 16/32” east, distant 5570™.8. The height of ground at the station above Lake Michigan is 323 feet. KEWAUNEE, 1873.—This station is situated on the west shore of Lake Michigan, in Kewaunee County, Wisconsin, near the mouth of the Kewaunee River. It was occupied for measuring a horizontal angle to carry azimuth southward. Latitude was observed at a post near by, and referred to the station. The height of station used was 8.1 feet. The geodetic point was marked in the usual manner. Two stone reference-posts, 2 feet long and 6 inches square, projecting 10 inches above the surface of the ground, with a small hole and the letters U.S. cut in the top of each, are set as follows: One bearing south 26° 13’ west, distant 36.76; the other bearing north 32° 29/ west, distant 34™.78 from the geodetic point. The latitude-post bears south 89° 31’ west, and is distant 13".08. The station was about 6 metres from the edge of the bank, and about 40 metres from the water’s edge. The height of ground at the station above Lake Michigan was 64.9 feet. : MANISTEE, 1871.—This station, on the east shore of Lake Michigan, is in Manistee County, Michigan, 24 miles north along the shore from the mouth of Manistee River, and about 260 metres north of a cross-road leading down to the beach from the main road. A large pine post was occu- pied as the station in 1871 for measuring horizontal angles for carrying azimuth. No stone mark- ings nor references are recorded. Latitude observations were made in 1871 at a post which bears north 23° 14/ east, distant 30".3 from the station. Hydrographical station, Rush Lake, 30 feet high, built in 1866, and used for locating off-shore soundings, on ground 74 feet above the lake, and probably having a stone marking, bears north 15° 59’ 50” east, distant 475™.5 from the station. The height of ground at the station above Lake Michigan is about 65 feet. RAWLEY’s Point (or Two RIvERS), 1866, ’72,’73.—This station is situated on the west shore of Lake Michigan, in Manitowoc County, Wisconsin, about 5 miles north of the village of Two Rivers, and about 1000 metres south of the Twin River Point light-house. A horizontal angle was measured in 1873 for carrying azimuth southward, and latitude observations were made near 97 LS 770 ASTRONOMICAL DETERMINATIONS. (Cuar. XXVI, by in 1866 and in 1872. As described in 1873, the station was about 36 metres distant from the edge of the lake, and the height of its center-post was 65.4 feet. The geodetic point was marked by «a stone post set 2 feet below the ground surface. Three stone reference-posts, rising 10 inches above ground, and having the letters U.S. cut on them, were set as follows: One bearing north 15° 49 east, distant 48".34; one bearing north 82° 23’ west, distant 56.02; and one bearing south 17° 49’ west, distant 44".93. The latitude-post of 1866 bears south 14° 12/30” west, distant 941.80, and the latitude-post of 1872 bears north 89° 15’ west, distant 12™.04 from the geodetic point. The height of ground at the station above Lake Michigan is 14.7 feet. Bie PoINnTE AU SABLE, 1866, 71,’72.—This station, on the east shore of Lake Michigan, is situated in Mason County, Michigan, on the end of the point nearly southwest from the Big Pointe au Sable light-house. The height of station used was 20 feet. The geodetic point was marked by a stone post set 3 feet below the ground surface, with another stone post set directly over it for a surface-mark. The station was occupied in 1871 and 1872 for measuring horizontal angles, and latitude observations were made on a post near by in 1866 and 1872. The light-house bears north 49° 28/ east, distant 109.5, and the latitude-post bears north 4° 31’ east, distant 96.3 from the geodetic point. PoInt CREEK, 1873.—This station, on the west shore of Lake Michigan, is in Manitowoc County, Wisconsin, about 15 miles north of Sheboygan, 9 miles south of Manitowoc village, and about 151 metres north of the mouth of Point Creek. It was occupied in 1873 for reading horizon- tal angles, and latitude and azimuth observations were made on posts near by. The height of station used was 26.8 feet. The geodetic point is marked by two stone posts set one above the other, the upper rising 1 foot above the ground surface. Three stone reference-posts, rising about 10 inches above the surface, and having the letters U. S. cut on them, are set as follows: One bearing north 5° 12’ east, distant 48".65; one bearing north 78° 28’ west, distant 35".87; and one bearing south 37° 05’ west, distant 37".12. The latitude-post bears south 22° 32/ west, distant 18.62, and the azimuth-post bears north 0° 19 west, distant 4™.91 from the geodetic point. The station is about 36 metres distant from the edge of the lake, and 64 feet above it. Nortu SHEBOYGAN, 1871, ’73.—This station, on the west shore of Lake Michigan, is situated to the northeast of Sheboygan, Wis., near the light-house. 101 38 16.09 | Paxton ..........-....| 281 28 23.63 4. 8580853 13. 6600 See ecenu| treet lane 148 36 54.41 | Spring Creek .........] 328 28 43.35 | 50484638 | 21.1753 flpecaiaa eli te iS ae Siete cticcead 176 20 23.97 | Ash Grove..........-.) 356 19 47. 97 4. 8260451 12, 6885 9 25 59.98 | Lynn Grove........... 189 23 04. U4 5. 1127321 24. 5526 18 13 53.09 | Mayview ..| 198 10 27.89 | 4,8972036 | 14.9475 179 33 (23.72 | Paxton: . ..-20-00ce00<% 359 33 20.79 4. 6551275 8. 5604 2 32 14.37 | Fairmount............ 182 31 51.06 | 4. 2033644 | 12. 0429 10 00.09 | Palermo .......2... .. 184 08 48. 94 5. 0743857 22.4777 17 08.49 | Lynn Grove .......... 222 06 34.89 | 5.0563949 | 21. 5656 36 46.62 | Mayview ...........-. 248 25 43.10 4. 9337432 16. 2596 21 48.97 | Rantoul........22..... 308 14 09.43 4. 8466332 18, 3045 04 23.14 | Paxton ...-. 2,-2.2. 327 56 40. 08 5. 0196348 19. 8153 Mayview .......-..--. 4.0; 06/5295) | 8807 QUA | ccc sa cceiaceen| sorceambcen deeaseeemas ale auattatecdeds loose seseeecsme eee eecee 40 01 35.84 | 87 50 48.75 6 02 19.89 | Palermo ......-..-..-. 186 01 32. 10- 4. 7415547 10. 4453 Lynn Grove .....-.--. 254 08 43. 02 4. 8845077 14, 5169 Mayview ....---...... 278 18 09. 67 4. 9221146 15. 8300 Rantoul .....2- .22--- 333 51 52.98 | 5.0762526 | 22.5745 Lynn Grove.......... 39 58 09.97 | 88 06 36.56 | 176 13 40.79 | Mayview ............. 356 13 11.90} 4. 7245673 | 10. 0446 Palermo ....-....-..- 39 52 33. 82 |. 87 52 03.20 | 39 04 06.14 | Oakland ............. 218 57 21. 60 4. 8942312 14. 8456 Dye: secre Gsia stess eiticvar| lesanale taynsttemiece | ereiseeees' se ast 116 38 11.71 | Lynn Grove ........-. 296 28 51.22 4. 8812397 14. 4081 Oakland ....2...-..-.. 39 42 31.90 | 88 02 35.30) 0 18 30.88 | Westfield............. 180 18 26.77] 4. 9669909 | 17. 5532 DG. sened cay eer, eeeeeewees 4% iesede ser, ance 168 48 22 28) Lynn Grove... .. 348 45 47.72 4..9857317 18. 3273 Kansas . 39 30 51.85 | 87 51 48 11! 22 43 45.05 | Casey.......22-2.22.-. 202 387 50. 42 5. 0564159 21. 5666 TAS, epee heatl lean erates orave 66 58 05.67 | Westheld. ....222... 246 51 10.09 4. 7459001 10. 5503 udabeiseet ahead hmasecn estates 144 29 21.81 | Oakland ....2. 022.222, 324 22 29.18] 4. 9399211 | 16. 4925 : nebiapceseenl ye taccuenceeee 179 29 18.03 | Palermo .............] 359 29 08.39 | 5.1197336 | 24.9517 39) 27° 15.94 |) $8802: 41266: |ecswecaxvonsan|eese + onc cows sanse veyags leseneeosien goee|eee ned yeeeem eels ves sede vee 39 18 56.71 | 8751 15.48 | 4 47 17.23 | Belle Air ............. 184 46 43.16 | 4..7058952 9. 6219 Md eta ade | aac eee 54 56 56,26 | Casey..............--.| 284 50 41.73] 4. 7549629 | 10.7728 Scag iets delsas\eit | neeee seine amis 133 12 47.75 | Westfield............. 313 05 32. 35 4, 8683663 13. 9873 eee eaecr cel cada ea 177 58 33.90 | Kansas ...............| 357 58 13.19 | 4.8597546 | 13.7126 39 13 33.42 | 88 01 07.18 | 174 54 39.48 | Westfield ............. 354 53 39.58 | 4.9219506 | 15, 8240 39 10 36.33 | 87 52 09.33] 128 39.10] Oblong ...........-... 181 28 25.73 | 4.8125787 | 12. 3011 assole abed| Si Deni mkte 46 00 46.73 | Hunt City ............] 225 55 13.66 | 4. 7624492 | 10.9601 mite Ansel ace Man hth 112 59 01.48 | Casey ................] 292 53 21.54] 4, 6625386 8.7077 Hunt City ..........- 39 03 58.76 | 88 00 57.21) 11 00 22.74) Mound ............... 190 58 50.30 | 4.7842302 | 11. 5241 Dio sneeesagce oes Skedoet ek pn peel nos dee ate 179 13 88.79 | Casoy....--...22--.24- 359 18 32,50 | 4.7645248 | 11.0126 21 17 58.78 | Clavemont.........-.. 201 13 28.25 | 4.9730990 | 17. 8019 55 54 29.82 | Mound .........-....- 235 47 38.93 | 4.7951572 | 11.8174 121 45 19.80 | Hunt City ..........-. 301 40 00.72 | 4. 6720931 8. 9014 153 52 07.44 | Casey... 333 46 41.53 | 4,9652666 | ~ 17.4837 40 45 24.36 | West Base 220 43 40.86 | 4. 3005614 3. 7838 - 43 01 46.29 | Denver ...........---- 222 55 54,97| 4.8130817 | 12.3154 46 53 10.37 | Onion Hill............ 226 48-44.46 | 4. 6623231 8. 7034 4 56 52.21 | Check Base ........... 184 56 38,08} 4,3153521 3, 9149 68 14 13.93 | Onion Hill ....... -.. 248 08 40.17 | 4. 6566510 8. 5905 88 32 32.91 | West Base............ 268 29 41.45 | 4. 3349191 4, 0953 149 33 24.86 | Mound ............-.. 329 32 16.82 | 4. 2281571 3, 2028 44 00 28.92 | Denver ........-22-24- 223 56 21.11 | 4. 6538821 8. 5359 51 31 54.44 | Onion Hill............ 231 29 12.06 | 4. 4179165 4, 9577 730 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuap. XXVII, Geographical Coordinates of Primary Triangulation Stations—-Continued. : : ; ; ee Reverse Logarithms | Distances Stations. Latitudes. | Longitudes. | Azimuths. Stations. azimuths; of distances | in English in feet. niles. ee eee o * “w ¢ “we oO t “w oO a “ 38 48 20.73 | 88 01 58.29) 95 20 38.42 | Onion Hill............ 275 15 19.00 4. 6076530 7. 6740 piSutefae bie atehars| Wee grees edie 135 17 52.92 | West Base........-.-.| 315 15 15. 69 4, 4502122 5. 3405 Baad dpeeecassstenk ar Giahsvaicionte® 169 04 36.45 | Mound settee cee ee eee 349 03 42. 60 4. 5541401 6. 7813 38 48 57.72 | 88 10 27.97 | 33 47 55.67 | Denver .. -+se--| 213 46 30. 20 4. 2883121 3. 6786 SO AG TRU | BA US AA ER tiie cennctcu ce vesueu inaed sicamssnaiadlonemas cackan xa anoaeimtoeemes|seimmem seven 38 45 28.75 | 87 59 41.54 8 58 17.62 | Parkersburg........- 188 56 57.57 4, 8145247 12, 3564 Hinvsse ete Vie elan| eemSee ee mere 94 40 36.96 | Denver .....-...-...-.| 274 32 26.80 4. 7938582 11. 7821 Saute aiptdid areresepl| bdercretcla Sd. seats G 112 29 51.91 | Onion Hill ............) 292 23 06.97 4. 7433627 10. 4889 Seca ueeese seas uae seeenceks: 140 42 05.28 | West Base. 320 38 02. 44 4, 6848081 9. 1659 Spackussadascs|eenensera yest 148 06 17.53 | Check Base 328 05 21. 86 4. 3116199 3. 8814 pechsin'siahas sevens) nse hema anbiann cont 161 29 34.65 | Mound ...............| 341 27 15.07 4. 7438494 10. 5006 ai Cuneo Aes PBR eaaee ne aad 166 37 20.99 | East Base.............| 346 36 09. 39 4. 5916612 7. 3965 38 34 51.73 | 88 01 49.66 | 143 16 14.68 | Denver .....-...-.... 323 09 25. 53 4. 9380695 16, 4224 V.— WILLOW SPRINGS — SHOT TOWER TO SANDUSKY BASE. 6985558 9. 4607 Military Auadeniy sl 41 41 21.85 | 87 40 382.96 | 105 56 36.49 Willow Springs...... 285 49 35.07 4. MMOS 2 cncnctesnces neieraree 4136 55.42) 87 14 18.63 | 102 51 36.95 Military Academy ....° 282 34 10. 64 5. 0882205 23, 2053 DY nd da es he RSs HS SGSHS het ot gia ay be dass 132 03 15.59 Shot Power ....------- BLL 47 09, 04 5. 1698804 28, 0057 OUiScecstca cee ieee 41 35 18.65 | 86 52 35.94 | 95 46 18.44 Millers .. 275 31 53.51 4, 9976421 18. 8369 Michigan City .....-.. 41 44 07.79 | 86 52 20.79 115 51.27 | Otis. ....---2-- +e eee ee 181 13 41. 20 4. 7289293 10. 1460 | Diss sccnavs vatnre seaseasee ands al rawsme Besressytess 66 29 17.74 | Millers ........--.---- 246 14 41.49 5. 0380948 20, 6708 Darga &uomiereseltategte saan: thembaoa ates 104 57 29. 06 | Shot ‘Tower. .-.. ...- 284 26 42. 81 5. 3361079 41. 0652 Springville ....-...--- 41 39 52.82 | 86 44 30.61 | 53 04 10.33 Otis ...............-- 232 58 47994 4. 6610602 8. 7383 DO sez seqcseuseses (vere vce degen seeie ceeding de 125 55 50.61 | Michigan City ........ 305 50 37. 83 4. 6437121 8. 3383 GAO eccpecaas esumes 41 41 45.88 | 86 40 32.13 | 105 01 39.31 | Michigan City ....... 284 53 47.74 4. 7453335 10. 5366 Bald Tom......--..--- 41 54 19.34 | 86 35 59.45 | 15 10 42.92 Galena ....-........- 195 07 41.16 4. 8977018 14. 9647 23 52 12.41 | Springville...-....-.-. 203 46 31. 80 4, 9817274 18. 1591 50 17 40.54 | Michigan City .....-.. 230 06 46.18 4, 9854893 18.3170 | 101 40 52.77 | Galena._....---....--- 281 33 38. 31 4. 7042716 9. 5860 161 35 25.10) Bald ‘Yom............ | 341 31 11.46 4. 9597282 17. 2621 38 08 29. 34 | Carlisle........--..... 218 03 50.83 4. 7111924 9.7400 69 40 01.31 | Galena.. ............. 249 28 07.79 4, 9381048 16, 4237 DOcascsxve sed sien setdieee Seratae all iedacseern pants 127 19 20.33 | Bald Tom..........-.. 307 10 27. 26 4. 8807476 14. 3918 POND. s2s-scsdeedecees 41 38 53.13 | 86 12 26.72) 95 24 40.72 | Carlisle.............-. 275 13 14.71 4, 8958941 14, 9025 135 47 48.35 | Bertrand............-. 315 41 00. 04 4. 8240273 12. 6297 8 57 51.23 | Penn .... 188 56 33. 38 4. 7558420 10. 7946 60 44 46.19 | Carlisle.............-. 240 32 01.27 4. 9998338 18. 9322 81 17 35.41 | Bertrand...... ceeee---| 261 09 28, 54 4, 7481981 10. 6063 48 31 28,74 228 20 09. 96 5. 0141538 19. 5668 79 55 18.39 -| 259 45 16.58 4. 8415624 13. 1501 88 24 50. 31 263 08 55. 26 5. 0405924 20. 7950 113 36 00, 34 293 21 21.73 5, 0377260 20. 6582 150 27 50. 00 330 23 12, 21 4. 8065216 12. 1308 12 36 52.59 | Jefferson...-....-..--. 192 35 12.71 4. 7169430 9. 8698 96 29 42.08 | Calvin ........2..2..6- 276 23 23. 88 4, 6355470 8. 1830 80 24 13.99 | Jefferson -........-..- 260 16 26.14 4, 7334239 10, 2516 118 52 31.08 | Calvin ...........---.. 298 40 04. 67 4, 9861527 18. 3450 134 56 23.20 | Porter ..-...-.......-- 314 50 14.61 4, 77226323 11. 2106 40 48 35.34 | Van Buren ........... 220 42 12,72 4, 8238213 12. 6237 58 30 57.97 | Jefferson .-......-..-. 238 16 46.95 5. 0557017 21, 5312 84 16 30.41 | Porter ......-..-.-.-.. 264 03 58. 28 4. 9335412 16, 2520 * 88 25 48.77 | Calvin ........-.--.--. 268 06 58. 25 5. 1083675 24, 3071 93 31 13.29 | Van Buren ........--- 273 17 29.70 4, 9735237 17. 8193 138 10 16.80 | Sherman..-...--...... 318 02 54. 65 4. 8769764 14. 2673 6 31 30.52 | Mongo.......-.---..-. 186 30 30. 76 4. 7773846 11. 3436 86 37 45.73 | Sherman.............. 266 29 23. 02 4. 7568475 10. 8196 86 04 18. 54 265 52 13.61 4. 9184075 15, 6954 DOs ssee tas ctetess| seceeeccaneeas|sasehenceseway 125 24 44.71 305 13 38.40 4. 9678157 17. 5866 v §3.] GEOGRAPHICAL COORDINATES. 781 Geographical Codrdinates of Primary Triangulation Stations—Continued. Stations. Latitudes. | Longitudes. | Azimuths. Stations, Reverse aes in taeten azimuihe: in feet. miles. o ¥ “ oO t aw °o f a“ oO t aw Quincy .............. + 4) 57 40.01 | 84 55 20.05 7 28 23.17 | Fremont...--....---- 187 26 37.17 4, 9670178 17. 5543 66 32 43.05 | Bronson . seceee-| 246 19 48, 94 4. 9804828 18, 1071 53 09 59.21 | Fremont...........-- 233 01 01.33 4, 8837651 14. 4921 133 11 41.17 | Quincy -...2..2.2.... 313 04 28.10 4. 8274159 12. 7286 38 44 25.90 | Reading .-.......2.... 218 40 43. 30 4, 6018220 7. 5716 101 22 07.43 | Quiney ..... 22.2222. 1! 13.18 4, 8776303 14. 2888 Pittsford ........-.... 41 49 33.25 | 84 29 00.51 | 92 47 01.36 | Reading .. 2 36 40.42 4. 8485968 13. 3648 DOs. cence aeeees eRiemmipehaases 16 39.90 | Hillsdale...... 2.2.2... 09 59. 03 4. 7563460 10. 8072 Bunday .....--.-----++ 42 03 16. 30 00.89 | Hillsdale.......2.2....) 226 22 26. 67 +. 8498289 13. 4028 Wheatland ........-.- 41 55 06. 47 03.73 | Pittsford ........222.. 192 45 56. 27 4, 5387016 6. 5474 DO eisiaicisiersseissnrecdisicr| crarcie cides eke | 68 49 37.26 | Reading ............. 248 38 08. 30 4. 9231806 15, 8689 Dib c200200 vese weal eoaneseciae ce : 55 55.36 | Hillsdale.......2..... 270 48 06. 62 4. 7246861 10. 0474 ! Do......-.--- ese | bok les enages 5 01 40.03 | Bunday........2..... 358 01 24.90 | 4, 6954906 9. 3941 Woodstock - ae 41 59 12 08 13.95 | Pittsford.............. 225 59 16. 43 4, 9269027 16, 0055 57 08.89 | Wheatland .......... 244 49 18. 44 4, 7688406 11. 1226 4 16 25.12 | Bunday..... 22.22... 294 08 18. 91 4. 7789667 11. 3850 56 31.09 | Pittsford. 2.2.2.0 0.2... 271 38 55. 20 5. 0790430 22.7200 macinetic saree te (Ruand aa fitevdcuc.3k | 108 36 36.23 | Wheatlaud........ 2. | 288 20 06. 82 5, 0728570 22. 3987 Meme DIE TAG | Antes seve hia mee 136 43 36.66 | Woodstock - .-. | 816 34 56.57 4, 9334911 16. 2501 | 41 57 11.50 | 83.45 30.14 32 49 09.86 | Fairfield 2.222022. 22. 212 44 24. 87 4.7755426 V1 2956 97 46 28.44 | Woudstock | 277 33 02.19 4. 9633038 17. 4048 78 18 50.59 | Fairfield. 2.22. 2. 22. 258 69 24.24 4. 8173199 12 4361 WGeemencuecP,. 1.2 + 42.1 —180° Sage ee In these formule J is the projection of the line 1:2 on the meridian; s is 1ts length; a is the semi-axis-major of Clarke’s spheroid=20926062 English feet; ¢ is its eccentricity=0.082272; B, and B, are the latitudes of 1 and 2; and a.2 a2, are the azimuths of the line 1:2. Gl; represents terms of the 6th order. Using this formula and the radii-vectores and azimuths given in § 1, the lines Fort Howard — Ford River, Ford River— Vulcan, Vulean—Saint Ignace, and Vulean— Huron Mountains were projected on the meridian of Fort Howard, giving the following results: , - English feet. Projection of line Fort Howard — Ford River. ...... eveasea tease = 429622, 52 Projection of line Ford River — Vulean .... .--.- eee eeee ees 641731. 58 Projection of line Vulcan — Saint Ignace. ......-.-.--.----- 02 ++: = 489677, 82 Projection of line Vulcan — Huron Mountains...... ......-------- =207130. 21 These lines were also projected on the Fort Howard meridian by Andre’s methods. (See Zacharie Die Geoditischen Hauptpunkte, p. 197 .) The projections thus found were substantially the same as those given above, the greatest difference between the two results for the same pro- jection being less than spsd000 Of the projection. § B. The second method used was to project by Helmert’s formula each trian gle-side separately on the meridian of Fort Howard, using the latitudes and azimuths obtained by the geodetic com- putation. Between Ford River and Vulcan the sides of the triangulation were separately pro- jected on the meridian, using Helmert’s formula for M. The sum of these projections was 0".06 larger than the value of the projection of the radius-vector from Ford River to Vulcan given above, a difference that is insignificant in comparison with the errors arising from the triangles. The com- putation of the spherical excesses of the polar triangles, and of the triangles themselves, involves more work than does the projection of the separate triangle-sides. Accordingly, from Fort Howard 822 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuap. XXIX, southward to the end of the meridian chain at Parkersburg the separate triangle-sides were pro- jected by Helmert’s formula (1) given above. The lengths of the triangle-sides were taken from Chapters XV, D, § 8; XVI, D, § 12; NVII, D, § 6; and XX, D, § 6; and the latitudes and azi- muths at their ends from Chapter XXVIT, § 3. Summing the values found for the projections between the parallels of Fort Howard, Minne- sota Junction, Willow Springs, Fairmount, West Base (Olney), Parkersburg, and including in the table the values previously given for Ford River, Huron Mountains, Vulcan, and Saint Ignace, the following table results, in which the first column gives the name of the station, the second its ob- served latitude taken from Chapter XXIII, the third the interval on the meridian of Fort Howard between the parallels of the stations, derived as just explained, and the fourth the sums of these intervals, counting from Parkersburg. Distances between paratlels along the meridian passing through Fort Howard. seeaeg, | Bitaliels passing | THECTrale from Far Stations. Observed latitudes. chee nes econ several % arallels, \ f high feot. ANE: in English feet. js eae ee | Q@ % “ | Parkersburg ..--...---.----- 38 34 53. 20 00. 00 | 101878. 83 | West Base (Olney) ..-.-.---- 38 51 41.23 101878. 83 | 424660, 27 | Fairmount....-..-....-..--- 40 01 36.70 526539. 10 ! 619488, 24 Willow Springs......-..-... 41 43 38. 63 1146027. 34 636930. 58 Minnesota Junction ......-. 43 28 31.82 1782957. 92 376877. 73 _Fort Howard ....-..--.----. 44 30 30. 28 2159835. 65 429622, 52 FOL RAVE ej2: sesso es cies 45 41 05. 34 2589458. 17 ‘ 434601. 37 Huron Mountains.......--.. 46 52 53.07 3024059. 54 207130. 21 Vil@ am sin cscis: Saittasebdiwne 47 26 44. 58 3231189. 75 489677. 82 St. Ignace. ....-- eae. ceeee- 48 47 28.65 3720867. 57 | é , The preceding table gives the intervals on the meridian of Fort Howard in English feet. As itis expected ultimately to obtain the length of #1876 in terms of the International metre with great precision, it will be well to give these intervals in terms of #1876 at some defined tem- perature. The bases measured with the Bache-Wiirdemann base-apparatus depend for their length on that of the 15-feet brass bar, which depends on brass yard No. 6, which again depends on the mean of the values assigned by Colonel Clarke to the lengths of the Clarke yards A and B. (See Chap- ter II, § 9.) A-value of the length of the 15-feet brass bar has been used for the Buffalo and Sandy Creek Bases, differing slightly from that used for the Minnesota Point, Keweenaw, and Fond du Lae Bases; but as the change in length at 32° F. or 62° F. did not exceed z;st555 part, a small quantity compared with the probable errors in these bases, this change may be neglected. (See Chapter II, § 19.) The lengths in feet of the Chicago, Sandusky, and Olney Bases depend on Clarke yard A alone, instead of on the mean of A and B, but it is shown in Chapter II, § 2, that the discrepancy between the two values is insignificant. It may then be said that the intervals on the meridian given in § 3, which depend on the bases measured with the Bache-Wiirdemann. base- apparatus, depend on Colonel Clarke’s value of Clarke yard A, and on its expansion given in Chapter II, § 2. $4. ARCS OF MERIDIAN AND PARALLEL. 823 For the portions of these intervals which depend on bases measured with the Repsold base- apparatus it may be said that their lengths depend on that of Clarke yard A through metre R1876. The mean temperature of the comparisons of £1876 and A is given in Chapter IX, § 12, as 57°.92 F., and there is also given z ee 1000 R876 at 579.92=—-; “O19 substituting for A its value at 579.92, as given by Colonel Clarke, R1876 at 57°.92—19.09388063 + 0y.00000045, this last expression gives : 1¥,000000 =0.91417653 1876 at 579.92, and enables us to transform the feet in the table of § 3 into multiples of R1876 at 57°92 F. In the following table, then, the intervals depending on bases measured with the Repsold apparatus have, for errors depending on the standard, only errors of comparison of & 1876 with ¥, and of 8, with S,, while those depending on bases measured with the Bache-Wiirdemann apparatus have in addition, for errors depending on standards, the errors of comparison of Clarke yard A with 1876. “A +5™™,2966-£ 04.40; or, Distances between parallels along the meridian of Fort Howard. t | ' | eat cor aeeen Intervals fi rom Parkers- Stations. | Observed latitudes. | adjacent stations, in| burg to the several! : multiples of #1876 at| Patallels, in nvultiples 570.92 Fy of R1876 at 579.92 F, oO .# “uw Parkersburg... eeejewe Siz dieteicsaictars 38 34 53. 20 0. 00 31045. 08 1876 West Base (Olney) ..-..---- 88 51 41. 23 31045. 08 R 1876 129404. 82 R1876 Fairmount.........--------- 40 01 36.70 160449. 90 R1876 188773. 87 11876 Willow Springs...........-- r 41 43 38. 63 349223. 77 R1876 194089. 00 21876 Minnesota Junction ......-. 43 28 31.82 543312.77 R1876 114844. 26 1876 Fort Howard ......--------- 44 30 30. 28 658157. 03 #1876 130916. 94 R1876 Ford River ..-.-...-----.--- 45 41 05. 34 789073. 97 Fe 1876 182434. 12 121876 Huron Mountains..-....-.--- 46 52 53.07 921508. 09 R1876 iS 63117. 86 R1876 VOOR cccaaew sean se eescaae 47 26 44, 58 984625. 95 121876 149217. 32 1876 St. Tonace:s sa saciecceseedce 48 47 28.65 1133843. 27 R1876 § 4. The chain of triangles running from Chicago east to the eastern end of Lake Ontario does not deviate widely from the parallel of 42°, and hence may be used to obtain the length of the are of the parallel of 42° which lies between the meridians of Willow Springs and Mannsville, these meridians differing in longitude by 11° 48’. The method followed has been to project, by meridians through its ends, each triangle-side on the parallel whose latitude is the mean of the latitudes of the ends of the side, and then to project chosen ones of these projections on the parallel of 42°. In selecting these projections to be projected on the parallel of 42°, those were taken corresponding to a continuous broken line of triangle-sides connecting the end stations, this broken line being made up of sides so chosen as to run as nearly east and west, and to be as near the parallel of 42° as was practicable. The three sides of each triangle were ‘projeated on the parallel of 42° also, in order to ‘check the computation, since the projection of one side should be nearly equal to the sum of the projections of the other two. Helmert, Héhere Geodiisie, erster Theil, 311, gives the following formula for the difference of longitude in seconds of are, of the ends of a weodotit line on a spheroid, the length of the line 824 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuap. XXIX, being s the latitudes of its ends B, and B,, and the azimuths at its ends a,. and 4.4. The equatorial radius and the eccentricity of the nee are d@ and ¢. The formula is— QQ) Dye =p! - W sec B sina [1-2 es W? [1— sec’ B sin’ a] — & [10 sin’? B — 1] cos’ a). 0 OL ay st + i950 & , (L—sec’ B sin? a) (1—9 sec? B sin? a) + crt] Fis By, oe In this equation W°—1—¢ sin By; B= y= 5 ; pis the number of seconds “a in radius; and Gl, represents terms of the 6th order, = being of the first. 0 Representing the are of the parallel of Bincluded between two meridiaus by P;.2 its value will be Ee af cos B Py. 2=A- pl We and (1) may be written— (2) Pro=s sin a [1 sya ®* (W:[1—sec? B sin’? a]—e [10 sin? B—1] cos’ a) +950 , (1—sec? B sin? a) (1—9 sec’ B sin’ «)+ Gi, | In the chain between Willow Springs and Mannsville there is but a single side having an approximately eastern and western direction whose length exceeds 63 kilometers, and that one does not exceed 70 kilometers. Hence * may be taken as always less than aa or of the second Mo order, i0 1 being the maximum value of small quantities of the first order, and P;.. will be aceurate to quantities of the 10th or 11th order. The value of P;.2 would be little changed, B and a being observed eee by small changes in @ and e, which determine the spheroid. A change of + dy i dg, ANG i ; ein e, both of which 5000 0 are greater than the differences between Clarke’s and Bessel’s spheroids, would only change the logarithm of P,.. in the 8th place. 4 Since in (2) terms multiplied by a are at least of the 8th order, they may be omitted in the A computation of projections, Having referred the triangle-sides to the parallels of 4 (B, + B,) for each, it remains to refer these projections to the parallel of 42°. To do this, it is only necessary to multiply the first pro- jection by.the ratio of the radius of the parallel of 42° to the radius of the parallel of B, that is by cos 42° ei 1l—¢ sin’? B cos BV 1—€ sin? 42° Effecting the division under the radical it becomes [1+¢ sin? 42° — sin? B+e' sin? 42° (sin? 42°—sin’ B)+ ete., |! or developing by the binomial formula Es 2 ue a ’ es (3) — oe [1+ 4 & (sin? 42° — sin? B) + § e! sin? 42° (sin? 42° — sin’ B) PP” cosB — 4 é (sim? 42° — sin’ BY + Glg| To find the change in this ratio caused by a change ine for the reference-spheroid, differentiate with reference to e, and there results P® cos 42° . ae ‘ : pe = Cos B lede (sin? 42° —sin’ B) + 2¢"de ee °—sin? B)— ede (sin? 42°—sin’ B)| To get the maximum change for this chain take de = (4) d Os =i! ; B=44°; and there results = cos 42° cae oh : me cos 449 wy (34) +255 oo Sin’ 42 (.034) — 3 549 (.034) | v4 ARCS OF MERIDIAN AND PARALLEL. 825 Since ¢ =.0068, the second and third terms in the brackets do not exceed 10-° and may be neglected. But the first term becomes .000068 (.034) = 0.0000023, or the ratio in the extreme case is changed about zgoeso0 part. As this wonld be an extreme value, and would apply to but few sides, no serious error would be produced in the length of the parallel by neglecting it. But it can be easily taken into account by computing to three significant places the coefficient of de in the tirst term of (4) for each of the selected sides forming the continuous broken line between two longitude stations. The projection of one of these triangle-sides on the parallel of the mean lati- tude of its ends will then be projected on the parallel of 42° by multiplying it by eta 5e, where ¢ is the value of the ratio in (3) for Clarke’s spheroid, a is the coefficient of se computed as just stated, and de is a symbol for the correction to e= 0.082272 required when a more accurate value of ¢ is found and used. ; The following is the list of stations determining the broken line of triangle-sides between Willow Springs and Mannsville, which have been projected on the parallel of 42°: - List of the stations determining the lines projected on the parallel of 42°. Willow Springs to * Cedar Point to Tonawanda to Cedar Point. Tonawanda. Mannsville. Willow Springs. Cedar Point. Tonawanda. Shot Tower. Middle Sister. Falkirk. Michigan City. Middle Bass. Batavia. Bald Tom. Kelley's. Mrganville. Bertrand. Brownhelm. Scottsville. . Milton. Elyria. Turk’s Hill. Calvin. Olmsted. Palmyra. Porter. Rockport. Clyde. Sherman. Willoughby. Victory. Bronson. Little Mountain. Oswego. Quincy. Thompson. | Sandy Creek. Hillsdale. Conneaut. , Mannsville. @ Dunday. Erie. Woodstock. Westfield. Raisin. Silver Creek. Dundee. Sturgeon Point. Monroe. Ridgeway. ® Cedar Point. Tonawanda. The lengths of the sides are taken from Chapters XVII, D, to XIX, D, and the latitudes and azimuths at their ends from Chapter X XVII. The following table contains the summation of these projections on the parallel of 42° for the intervals between the principal longitude stations. The first column gives the name of the longi- tude station, the second its observed longitude from Detroit derived from Chapter XXV, the third the included are of the parallel of 42°, and the fourth the sum of these ares, counted from Willow Springs as an origin. Distances between meridians along parallel of 42°. Observed longitudes reforred to Detroit. | tatervals between meridians | Intervals from Willow Stations. of adjacent stations, in Springs to the several + west. English feet. meridians, in English feet. — east. 2 eo »? “ Willow Springs...-.----.--. +4 48 03.15 0. 00 1227742. 62+- 224.2 de < Cedar Poithe.-..ss casseccnee + 17 01.84 1227742. 624224. 2 S¢ 1208492. 334 204.5 Se Tonawanda.....------------ —4 09 42.44 2436234. 954428. 7 Se 770735. 68—1248, 5 de Mannsville ....---.--.------ —6 59 36. 86 3206970. 63819. 8 Se 826 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuap. XXIX, By means of the relation given in § 3, namely, 1 yard = 0.91417653 R1876 at 579.92 F., the distances expressed in feet in the above table may be converted into multiples of the metre R1876 at tlre temperature 57°.92 F. The following table gives those distances expressed as multiples of R876: Distances between meridians along parallel of 42°. ' Observed longitudes referred to Detroit. Intervals between meridians of , Interval from Willow Springs Stations. a adjacent stations, expressed in | are oadouset wiet | went, multiples of 21876 at 579.92 F. | at 579.92 F. —east. | | =< or aw | Willow Springs..... +4 48 03.15 00.0 374124.50 218764 68.3 R1876 de Cedar Point......... + 17 01.84 | 374124.50 2218764 68.3 R1876 6e 368258.44 R1876+ 62.3 R1876 de Tonawanda ......-.. —4 09 42.44 742382.94 218764 130.6 121876 de ( 234862.82 R 1876—380.4 21876 6¢ Mannsville ........ | —6 59 36. 86 977245.76 R1876—249.8 P1876 Se § 4. It is of interest to form an approximate idea of the precision of the triangulation and of the arcs of the meridian and of the parallel of 42°. In computing the weighted mean sides of the triangulation in Chapters XIV, D, to XX, D, the quantities | are given for each triangle side. These quantities are the squares of the probable errors of the logarithms of the sides when com- puted from the first base with the observed angles. South and east of the Chicago Base the - in- clude only errors arising from the observed angles, and-do not include errors in stapdards nor in measurement of bases. North of Chicago Base they exclude errors of standard, but include errors in measurement of base. South and east of Chicago the errors in meagurement of the bases did not sensibly affect the approximate weights for triangle-sides when computed from two such bases, and hence those errors were neglected in computing the weighted mean logarithms of the sides. In the chapters referred to, if 3 and a represent the squares of such probable errors in the loga- rithin of a side when computed from the the two adjacent bases, ty a constant whose value is given in each of the chapters referred to, and hence from the tabulated values of 3 and the value of this constant, the values of 7 follow at once. The weight of the logarithm of a weighted mean side in Chapters XIV, D, to XX, D, is then p+p’, and the square of the probable error in its loga- But this square of the probable error in the logarithm of a side has been obtained rithm is 1. ptp' on the assumption that the side has been computed from the bases with observed angles; in fact, the sides have been computed with adjusted angles, and the question at once arises as to the reduction in the probable error of the logarithm of a side in consequence, In Chapters XIV, ©, to XX, C, it may be seen that the probable errors of adjusted angles are approximately 0.6 of the probable errors of the observed angles. If, then, the adjusted angles and their errors were inde- pendent of each other, in order to obtain the probable error squared in the logarithm pf a side computed from a base, it would only be necessary to replace the p? [2+/°] of Chapters XIV, D, to XX, D, by (0.6) p* [+]. But, in fact, the errors of the adjusted angles depend on each other in consequence of the rigid equations of condition of the triangulation, and hence this value is not exact, and is probably too small. On the other hand, it is evident that the computation of a side with the adjusted augles must give their values much more accurately than when the observed angles are used, and hence that p* [a?+,%] is much too large. The factor by which p? [@+#] should 65.) ARCS OF MERIDIAN AND PARALLEL. 827 be multiplied in order to give the proper result for adjusted angles, lies then between (0.6) and the square of a quantity considerably less than unity. It seems probable that its value is in the vicinity of (0.7)*, but to avoid overestimating the accuracy of the work it will be taken as (0.8)’, so that (0.8)’p?[a?-++ 3°] will be taken as the square of the probable error arising from angle errors in the logarithm of a side when ‘computed from one base with the adjusted angles. To get the reduced values of the probable errors of the weighted mean logarithms of sides given in Chapters XIV, D— XX, D, we must multiply the values of 5 and a by (0.8)? after sub- tracting the square of the probable error of measurement of the base where such error is included in 3 or > and add to the resulting values the squares of the probable errors of the bases due to measurement. The reciprocal of. the sum of p and p’, thus modified, plus the square of the prob- able error of the logarithm of the standard, will give the square of the probable error required. Where consecutive bases depend on different standards, a mean value of the probable errors in the logarithms of the lengths of the standards has been used for the intermediate sides, since the errors of the two standards are not independent. In this way the total probable error in the sev- enth place of the weighted mean logarithms of all the sides of the principal chains of triangulation have been computed, and the means of these have been taken for the sides included between each of the pairs of parallels of latitude given in the table of § 3. From the probable errors in the mean logarithms, the probable errors in the mean sides may be obtained, and from these an estimate of the probable error in the interval between two parallels, along the meridional arc, or between two meridians, along the parallel of 420, may be obtained. The following tables give the results. The first column gives the names of the stations limiting the sections of the chain; the second the number of the included triangle-sides of the principal chain; the third and fourth give the average probable error of a triangle-side in the section; Average probable errors of triangle-sides projected on the meridian to obtain the intervals between parallels. ' Average probable error of a triangle-side, N aia of sane , oath sides projectedon ‘ Sections between stations. meridian in there-| Expressed in units | Expressed as a frac- ‘sl spective sections.| of seventh place | tionalpartoflength of logarithms. of side. 1 Parkersburg to West Base (Olney) ..-...--- 4 10.0 #34000 1 West Base (Olney) to Fairmount .......-... 8 15.2 586000 : . 1 Fairmount to Willow Springs...-.-..-.----- 9 20.0 ai 000 1 Willow Springs to Minnesota Junction ..... 10 23.6 184000 1 Minnesota Junction to Fort Howard.....--. 8 24.4 178000 : 1 Fort Howard to Ford River. ...--..--------- 8 _ 37.0 17000 1 Ford River to Huron Mountains.......-.--- 8 33.9 738000 ; 1 Huron Mountains to Vulcan .............--- 1 22.2 796000 1 Vulcan to St. Ignace......-----e-eeeeee eee 1 25. 2 73000 828 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuap. XXIX, average probable errors of triangle-sides projected on parallel of 429. Average probable error of a triangle-side, | A % Number of uate g 5. Sections between stations. sides projectéd ou | py 4 pegsed in units | Ex d : os pressed as a frac- parallel in section. | "OF seventh place | tional part oflength of logarithms. of side. - * ‘ 1 Willow Springs to Cedar Point --. 17 26.1 166000 1 Cedar Point to Tonawanda ...-..- a, 20.8 309000 Tonawanda to Mannsville .....--. 11 24.1 {80000 §6. In the chain of triangles connecting two bases the probable error of a triangle-side is a minimum at the bases and increases to a maximum for a side about half way between the bases. The tables which have just been given show the average probable errors of triangle-sides for cer- tain sections. It is also of interest to know the maximum and minimum probable errors of sides in the principal chain; accordingly, the following table gives in order the names of the bases, the names of ‘the mean sides of maximum probable error lying between each pair of consecutive bases, and the probable errors of such bases and lines, expressed in units of the seventh place of logarithms and in fractions of their Jengths. The probable errors of the bases are taken from Chapters III—VIT and X—NXIII, and the probable errors of the sides of maximum probable error are derived from the | of Chapters XIV, D—XX, D, by applying the changes for probable error of standard and of measurement of base, when necessary, as in §5. Table of probable errors of base-lines and of lines of maximum probable error in each section of the chain of principal triangles. I.—OLNEY BASE TO MINNESOTA POINT BASE. 3 ' Probable error of loga- | Probable error as frac- Base. Line of maximum probable error. | rithm of line in units tion of length of ° ' of seventh place. line. 1 Ole y-Base sn22 sc vessel ewaeaesteercweea Seoec ieee? ewes aes ‘ 4.2 7013000 | 4 Ash Grove—Spring Creek.........-.- 22.2 796000 ‘ ' i CHICA EO: BES cossicisicipaiccieistels| eaeeetia ay Seep wine ete oe mies ee ' 4.1 7952000 rie New Berlin — Delafield ..-...........- 28.2 754000 i Pond OR LAG DAG «versssessl canasa eumey saSsaeme Rane s RRR AM RMES Aen 6.7 649000 1 Burnt Bluff— Sturgeon .............-- 38.7 12000 1 Keweenaw Bas sesacccccdse|sepeces eeebexuwre eccent weeded ceegasened 5.2 330000 1 Vulcan — Wheal Kate ..-........2.2.. 22.3 195000 : : 1 Minnesota: Point: Base. sesce0) :exmesg es stews scoceeceeus sceweee eee ces 8.2 530000 §§ C7.) ARCS OF MERIDIAN AND PARALLEL. Table of probable errors of base-lines, &e.—Continued. Il.—CHICAGO BASE TO SANDY CREEK BASE. | Base. Lino of maximum probable error. “rif it ela oa ; Chicdieo Bases: 24cccni6.c50hsleneeodiecnceedcuucawe ace cx deeecnsaeccessels 41 — Jefferson —Milton .......... 222.22... 30.3 1 oh 00 PAT BBY TING, s.o0n ae ean wnion:ts a nied Unk Selegya cus odds Maewenenad \guekie 3.8 Tia005 Claridon — Little Mountain ........... 24.2 Tas Birffulo: Base: v.csciicacegs dey seewses venioeneneniee sea uadeuesecdoacssds 4.9 sant Turk’s Hill— Pinnacle Hill........... 26.4 a Sandy Creek Base ........2.).2222..ccccccenceee ceccccccececcccecece 4.8 aiaaea §7. As the methods used in estimating probable errors of triangle-sides can only give rough approximations to the true values, any check on the results is valuable. Such a check is afforded by the comparison of the difference between the measured length of a base and its length computed from the next base by means of the adjusted angles, with the estimated probable error in that com- puted length. In the following table the data for such comparisons are given. The first column gives the name of, a base and the second that from which its length has been computed with the adjusted angles given in Chapters XIV, C—XX, ©; the third gives the distance between the bases meas- ured along the axis of the chain of triangles connecting them; the fourth gives the excess of the length of the base computed with adjusted angles over its measured length expressed in parts_of the length of the base; the fifth gives the same excess in feet; the sixth gives the probable error in the computed length of the base derived from the probable errors of the observed angles between the two bases; the seventh gives this probable error when derived from the probable errors of the adjusted angles between the two bases by the method given in §5; the eighth gives references. Comparison of computed lengths of bases with measured lengths. Computed length minus Probable error of computed measured length. length expressed in parts of base. Distance be- Name of the base Reference : tween bases ‘ % Base. from which com- along axis of | a 4 Using prob- Uae 0.8 of Bole to my ter puted. chain xpressed in | y. pressed | able errors le errors of 0 and §. arts of the in feet Sfobserved served angles as ase. i dneles probable error of gies. adjusted angles. Miles. 2 1 1 1 Minnesota Point..; Keweenaw -....-- 240 — 954000 0. 21 50006 snub IV, §14 1 1 1 gee ‘ aoe wees Vv, §12 Fond du Lac.....- Keweenaw ....... 320 355000 10 Hw #5000 § é _1 a aA X, § 35 Chicago .......-.. Fond du Lac....--. 150 + 54000 46 62000 77000 $ 1 1 1 i i) ‘ a enaee XI,§ 9 Sandusky..-...... Chicago ..csie0 sas 280 5000 46 55000 woo § t ds ab XI, §.10 Sandusky...-..... Buffalo ..--..----- 250 + Tei000 +18 71000 90000 § 7 7 obs VIL, § 8 Buffalo ........... Sandy Creek...... 210 — a85000 oe 87000 84000 r§ ‘ 2 9 1 1 XIL§ 9 PUROV xn svevdeve xe Chicago .........- 200 + 709000 +20 78000 97000 1§ | 830 PRINCIPAL RESULTS OF THE GEODETIC WORK. [Cuar. XXIX, An examination of this table shows that on comparing the measured length of a base with its length computed from the nest base with the adjusted angles of the intermediate -triangulation, the differences are in three cases greater than the probable error in this computed length, estimated from the probable errors of the adjusted angles, and in four cases less, and hcuce that the discrepan- cies between the observed and computed lengths of the bases are such as are to be expected if the probable errors assigned to the adjusted angles and sides are nearly correct. In other words, these discrepancies indicate that those probable errors are nearly the true ones, and, so far as the nuinber of bases is sufficient to justify a conclusion, they indicate that the estimated probable errors are rather too large than too small. § 8. It is desirable to know approximately the accuracy of the values which have been arsigned in §§ 3 and 4 to the intervals on the meridian of Fort Howard, between the parallels through cer- ‘tain astronomical] stations; and to similar intervals on the parallel of 42°. In §§ 5 and 6 data have been given as to the maxima, minima, and average probable errors of triangle sides in these in- tervals. From § 5 it will be seen that for intervals containing eight or more triangles, either along the meridian of Fort Howard or on the parallel of 42°, the average probable error in the length of a triangle-side varied from zz)557 to sgaoua part. These errors enter by equal amounts in the projec. tions of a side on the meridian or parallel. If in the projections the errors were all of the same sign for each interval, the sum of the projections would be in error by an equal part of its value. On the other hand if the projections were min number, of equal lengths, and their errors were independent of each other, the resulting probable error in the sum of the projections would be the probable error of a side, expressed in parts of its length, divfded by Vx, Neglecting the fact that the sides of the triangles in each of the intervals are of unequal length, and treating them as if of equal length, an approximation to the probable error in the interval arising from the triangula- 1 tion errors would be obtained by multiplying the average probable errors in § 5 by 4+ ava (except when there is but a single triangle-side, where it will remain as given). But an error in the projection of the triangle-sides on the meridian and parallel also arises from the errors in the azimuths of the sides. The amount of error from this cause will be essentially the same as if the lines joining the astronomical stations in the tables of §§ 2 and 4 were projected directly, and the errors would mainly arise from the errors in the values of the azimuths of these lines. An examination of the discrepancies given in Chapter XXVIII, § 2, between computed azi. muths and those observed at Bruce, Minnesota Junction, Willow Springs, and Parkersburg, where there are no inequalities in the earth’s surface, which would indicate local deviations in azimuth, seems to indicate that the azimuth observed at Bruce and adopted as the fundamental one for the whole triangwation was not subject to local error of any great amount. The question then arises as to the amount of error which accumulates in carrying the azimuth through the triangles from Fort Howard to distant points, such as North Base, Minuesota Point, Parkersburg, or Tonawanda. Some information on this point is given by the second members of the numerical equations in Chapter XX VIII, §3. Those second members are mainly the accumulated errors in azimuth, pro- vided the observed longitudes are exact; if inexact, the values of the second members will be in part due to the longitude errors. From the table it will be seen that in the part of the triangula- tion belonging either to the arc of the meridian or of the parallel the greatest discrepancy is —3/.1. So far as these results can furnish the basis for a conclusion, it would seem safe to assume that the probable error in the azimuth of any line connecting adjacent astronomical stations in the tables’ of §§ 3 and + does not exceed +2”, It may be noticed that at North Base, Sandy Oreek, but 8 miles from Mannsville, the discrepancy between the observed and computed azimuths is but +1/79. The effect of au error of 2” in azimuth on the projection of the line on the meridian or parallel will depend essentially on the cosine of the angle of projection. In the following table are given the principal astronomical statious for the meridian of Fort Howard and the parallel of 42°, the ap}%roximate probable error in their intervals obtained as ex- plained in the first part of this section, expressed as a fraction of the interval; the approximate angles made by the lines joining the astronomical stations with the meridians or parallel; the s §§8,9.) ARCS OF MERIDIAN AND PARALLEL. 831 probable errors in the projection due to a probable error of +2” in this angle expressed in parts of the length of the line; and the combined probable error arising from both linear and azimuth errors. y Azimuthal | Combined Intervals between ae png Sea eae: probable probable 6 Oo Parkersburg to West Base ......-.-...-..--++- e300 11 24 aot seem West Base to Fairmount ...-..++--++20--++++++. Baan 9 39 wants sera Fairmount to Willow Springs .......-----...--- a 0 07 0 A000 Willow Springs to Minnesota Junction ....-...- sagan 20 22 aaa a Sci Minnesota Junction to Fort Howard ........--- on 25 38 a aman Fort Howard to Ford River ..-.........--...--. aa 29 28 ! ian aa Ford River to Huron Mountains..............-. a 25 29 am anos Huron Mountains to Vulcan ...........-------.- imam 8 43 man vt Vuléan to St. Ignace... 02.02.02 02eeeceeeeec eens a 140 | ee a Willow Springs to Cedar Point ..- ...-...-.... sain 017 0 sa7050 Cedar Point to Tonawanda ....--.------.--.---- ssap0 21 22 | aan Sao Tonawanda to Mannsville. ......-.-22----+--+-: sarah 19 02 | 257000 aaa | § 9. The principal intervals on the are of the meridian through Fort Howard and on the parallel of 42°, with their approximate probable errors, may now be given. It will be remembered that the probable errors are only rough approximations. Are of meridian through Fort Howard. Intervals be- | Approximate probable error— 1 eee Leas ‘ Observed lati- els throug! Stations. tude. adjacent sta- | pressed in | Expressed as tions,in Eng- P 56 fractional part lish feet. : of interval. oO t uw - Parkersburg....-----.---- 38 34 53. 20 i 101878. 83 ; £0. 26 385000 s West Base (Olney) -.---4- 38 51 41.23 ‘ 424660. 27 +1, 23 346000 Fairmount...--..--..----- 40 01 36.70 ‘ 619488, 24 +1.90 326000 Willow Springs...-..----- 41 43 38. 63 ‘i 636930. 58 +8, 25 196000 Minnesota Junction ...... 43 28 31. 82 : 376877. 73 +2, 26 167000 Fort Howard 2210.04 s50200 44 30 30. 28 i 429622. 52 +3. 41 736000 Ford River .......-----.--| 45 41 05,34 ‘ 434601. 37 +3, 06 {42000 Huron Mountains...-...--. 46 52 53.07 : 207130. 21 +1.10 788000 WOH socks ceascannennnes 47 26 44. 58 i 489677. 82 +2, 85 | 773000 St. Ignace......-------+0-+ 48 47 28.65 832 PRINCIPAL RESULTS OF THE GEODETIC WORK. — [Cuap. XXIX, §¥. Are of Parallel of 42°. Observed Jon Bi Approximate probable sie (tude referre 3 | | . Intervals between meri- _- ——~-———- Stations. : to Detroit. dians of adiagent Bie - 4 Expressed as : tions, in English feet. xpresse - | +west } of Detroit. : : | in feet. eeaoue park | —east of interval. t ie spent aeaiag: | ov ” 4 Willow Springs..........--- +4 48 03.15 ‘| 1227742, 62+ 224. 2 6¢ +4, 60 567000 Cedar Point ..-... Kjovicinie teeick +0 17 01.84 1 ’ 1208492. 334 204.58¢ | 45.81 08000 | Tonawanda..........-.---.. --4 09 42.44 i | 770735. 68—1248. 5 de | +3. 80 203000 Mannsville .........2-2-.-5- —6 59 36.86 \ | s « - APPENDICES. 833 105 Ls App. I, §§1,2.] ADDITIONAL DATA RELATIVE TO R1876. 835 APPENDIX I. ADDITIONAL DATA RELATIVE TO METRE B1876. §1. The data given in Chapter IX, § 57, showing a considerable difference between the expan- sion of £1876, given by the Kaiserliche Normal-Eichungs-Kommission, Chapter IX, § 67, and those obtained by the Lake Survey, the following letter was addressed to the Kommission: In your esteemed letter of June 20, 1879, you gave me as a preliminary value for the Lake-Survey metre, 21876, ; #1876=1 metre+248".89-4.0".25-+-(10".310".034)(t—15) and in your letter of September 15, 1880, you advised me that the error in this value, with reference to the true metric measure, would not exceed 1” or 24. By true metric measure, I understand the metre which will be fixed by the International Standard Metre, which is to be, or has been, derived from the Métre des Archives. Since the receipt of your letter a long series of comparisons at high and low temperatures, in this office, of R1876 with steel Clarke yard 4, whose coéfficient of expansion was carefully determined in 1874 by Colonel Clarke at the Ord- nance-Survey office, Southampton, England, have given a value for the expansion of 21876 as depending on that found by Colonel Clarke for the Clarke yard. That value is Expansion of 21876=10".59 for 1° C. derived fro u Clarke yard A. £1876 has been also carefully compared witb each metre of a steel bar 4 metres long, belonging to a base-measuring apparatus by Repsold, at both high and low temperatures. This 4-metre steel bar has been well compared at high and low temperatures with four metres of a brass bar 15 feet long, whose absolute expansion has been carefully determined directly in this office. Combining the two sets of comparisons there results : Expansion of &1876=10".64 for 1° C. derived from absolute expansion of brass bar. These two values derived by entirely independent methods agree well, and it is difficult to suppose them more than 0".1 in error, while their probable error is much less, Their mean differs by 0*.3 from the value, 10.31, assigned in your letter. This leads me to ask whether any error of copying may have slipped into the value 104.31, and, if not, whether you now think it possible that the true value may be 0*.1 or 0¥.2 greater than 10".31. I would also ask whether in the value assigned in your letter, namely, R1876=1 metre+248",894.0".25-+(10".310".0.34)(¢—15) the value of 21876 is free at 15° C. from any small error that may exist in this value of its expansion. FEBRUARY 26, 1881. As this letter was not immediately answered, and the preparation of this report could not be further delayed, the value of the expansion of #1876, derived from the adjustment of different values, Chapter IX, § 58, namely, Ep ie=1™.000092[5773(10)°+ 2 x 1852(10)-"(¢—32)] for 19 F.=5«.8854 0.043 at 62° F. was adopted. §2. On September 12, 1881, the following letter was received, but too late to incorporate its results in the main report. It is, therefore, given in this appendix: BERLIN, den 10. August 1881. Die Kommission ist nunmehr in der Lage Ihnen niihere Details iiber die hier angestellten Bestimmungen der Linge - und der Ausdehnung sowie der Theilungsfehler Ihres von Repsold in Hamburg verfertigten biegungsfreien Stahlmeters mit der Theilung auf Platin, welches hier mit 21876 bezeichnet wurde, mitzutheilen und dadurch die in Ihrem Schrei- ben vom 20. Juni 1879 angegebenen Resultate zu erliiutern und theilweise zu modificiren. Es diirften damit gleich- zeitig die in Ihren gefiilligen Schreiben vom 16. Octcber 1880, 16. Februar und 27. April 1881 angeregten Fragen, soweit sich dieselben auf dieses Meter beziehen, ihre Beantwortung finden. Die Hauptursache der bisherigen Verzégerung besteht darin, dass wir die genaueren Resultate der in Breteuil angestellten Vergleichungen abwarten wollten, um Ihnen mdglichst definitive Resultate mittheilen zu kénnen, und dass uns diese Vergleichungen erst seit kurzer Zeit bekannt sind. Die ersten Bestimmungen der Gesammtlinge des Stabes wurden im Jahre 1876 angestellt, indem 15 Vergleichun- gen mit dem Messingmeter 1605 bei Temperaturen zwischen 11° und 20° ausgefiihrt wurden. Diese Vergleichungen verdienen aber nur ein sehr geringes Vertrauen, weil sie zu den ersten mit dem betreffenden Komparator angestellten Beobachtungen gehéren, und die Methode der Beobachtung noch nicht geniigend durchgebildet war. Namentlich die 336 APPENDIX I. [ App. I, Bestimmung der Temperatur der Stiibe liess in dieser Versuchsreihe noch viel zu wiinschen iibrig. Es herrscht daher auch einige Unsicherheit dariiber, wie die Vergleichungen am besten zu berechnen sind, und die Resultate der Rech- nungen sind innerhalb weiter Grenzen unsicher. In der Beilage A sind die einzelnen Resultate dieser Vergleichungen aufgefiihrt: niémlich die Temperaturen der beiden Stiibe tig05 und tr nach den Angaben der auf ihnen liegenden Thermometer und die nnmittelbar gefundene Liin- gendifierenz. Die Berechnung ist nur beiliiufig mit den spiiter gefnndenen Ausdehnungskvefticienten ausgefiihrt und zwar einmal unter der Annabme, dass die Temperaturen der Stiibe der Angabe der aufliegenden Thermometer ent- sprechen (I), bei welcher Annahme im Mittel R—1605=+21".9 bei +15°.00 folgt, zweitens unter ‘der Aunahme, dass die Temperaturen beider Stiibe durch t,w5 allein gegeben sind, weil die Beriih. rungsfliiche des anderen Thermometers mit £1876 eine verhaltnissmiissig zu kleine war, wm sichere Resultate erwarten zu lassen (II). Aus diescr zweiten Annahme folgt im Mittel R1876—1605=-+-25".7 bei +15°.00. Endlich wiirde noch die Annahme in Frage kommen und den vorerwiknten vielleicht vorzuzichen sein, dass als Temperatur der beiden Stiibe das Mittel aus den beiden Thermometerangaben anzusehen ist. Aus dieser Annahme wiirde folgen R1876—1605=+ 27.2 bei 15°.00. Das Messingmeter 1605 ist mehrfach mit dem deutschen Urnormal (Pt) verglichen worden. Aus den neuesten besten Messungen folgt 1605— Pt=—63".14.0".79 + t(9".653-L.0".044), also bei Anwendung der gesetzmissigen Gleichung des Urnormals 1605=1™—60".13-+-18".253%, d.h, 1605=1™"-+213".7 bei+15°.00. Die Verbindung dieses Werthes mit den obigen Differenzen von RF 1876—1605 ergiebt viel kleinere Werthe fiir die Lange von £1876 als die spiiteren Vergleichungen, doch liegen, wie schon erwihnt, gréssere Bedenken gegen die Genauigkeit dieser ersten Messungsreihe vor. ‘ .Eine Bestimmung der Liinge von 21876, welche sehr viel gréssere Garantieen fiir ihre Richtigkeit bietet, beruht auf der Vergleichung von 1876 mit einem diesem Stabe ganz gleichartigen, mit #1878 bezeichneten Meterstabe von Repsold. Es sind im Miirz 1878 drei Reihen bei den Temperaturen von 8°.60, 15°.75, und 21°.49 angestellt worden, deren Details sich in der Beilage befinden, und welche, wenn man die sehr kleinen Differenzen zwischen den Tempera- turen der beiden Stiibe beriicksichtigt, auf das Resultat fiihren: R1876—R 1878=+21".86 bei &°.602 _ =+22 .88 bei 15 .751 =+22 .81,bei 21 .489. Das Mittel simmtlicher Reihen giebt R1876—R 1878=-+-22".5240".21 / Vorzuziehen ist aber dasjenige Resultat, welches man erhilt, wenn die Ausdehnungsdifferenz der beiden Stiibe beriick- sichtigt wird: R1876 — R 18738 = 22.52 4 0".17 + (t— 15°.281) (0.076 +1 04.029) oder in einer anderen Form * R1876 — R1878 = 21".36 + 04.484 ¢ (0".076 4 04.029). , Der Stab £1878 ist in zwei Reihen in den Jahren 1878 und 1879 mit dem deutschen Urmeter verglichen worden- Die erste dieser Reihen liess nur die Berechnung der relativen Ausdehnung zu. Als Resultat ergab sich: 1. R1878—Pt=const +1".68-+ ¢ (1".676 + 0¥.074) 2. R1878— Pt=+ 684.79 4 0".49 + ¢ (14.643 + 0*.032). Anzuwenden ist die aus der Verbindung der beiden Beobachtungsreihen sich ergebende Gleichung R1878— Pt=+ 68".71 + 0".49-+ ¢ (1".648 + 0.029) aus deren Verbindung mit der obigen sich ergiebt R1876— Pt=+90".07 + 0.68 + t (1".724 + 0¥.041). Setzt man nun fiir Pt seinen legalen Werth, wie er aus.der Vergleichung mit dem Métre des Archives unter der Annahme folgt, dass die Ausdehnung der beiden Meter 8.6 betriigt, so wird R1876 = 1™ + 93".08-+ 10,324 t. Der—1".16-+ 0".014 t betragende Unterschied gegen die Ihnen unter dem 20. Juni 1879 gemachte Angabe £1876 = 1™ + 248".89 4 OF .25-+4 (104.31 + 0".034) (t— 15°) riihrt von uachtriiglichen kleinen Verbesserungen her, die sich bei einer Superrevision der thermometrischen Rech- nungen ergaben. Bei den damals angegebenen wahrscheinlichen Feblern ist ausserdem nur die Unsicherheit der Ver- gleichung der beiden Repsoldmeter unter einander beriicksichtigt worden. Die Ihnen unterm 15, September 1880 gemachte Angabe, dass die gegebene Gleichung von £1876 nur um 1* bis 2" von dem wabren metrischen Maasse abweichen diirfte, hat sich leider, soweit es sich um niedere Temperaturen handelt, nicht bestiitigt. Dieselbe stiitzte sich auf eine ia Breteuil ausgefiihrte Vergleichung von 21878 mit dem Platin- Iridiummeter type I bei 30°, welche ein mit den bisherigen Annahmen sehr gut iibereinstimmendes Resultat ergeben ~ §2.] ADDITIONAL DATA RELATIVE TO R 1876. 837 hatte. Vergleichungen bei 0° und 15° haben dies Resultat nich bestiitigt. Als Resultat der drei Vergleichungsreihen ergab sich vielmehr #1878 = 1™ + 644.82 + 104.371 t-+ 0".0969 #2. Dabei ist angenommen, dass die Gleichung von type I gegeben ist durch IT=1™ + 674.6 + 84.525 t+ 0".0039 ¢. _ Der Fehler von type I bei 0° ist durch indirekte Vergleichung mit dem Métre des Archives, die Ausdehnung des Stabes als Mittelwerth aus verschiedenen absoluten Bestimmungen abgeleitet worden. In der Gleichung fiir #1878 ist der Koefficient von f nach den Fizeau’schen Bestimmungen angenommen, und die Grésse der anderen Glieder aus den Bestimmungen bei 0° und 30° berechnet. Da die Vergleichung bei 15° ein nur um 0*.05 von der Forme! abwei- chendes Resultat ergab, so wird man bei derselben stehen bleiben kénnen. Unter Fortlassung des in ¢ multiplicirten Gliedes wird in der Niihe von 15° nach den Bestimmungen in Breteuil zu setzen sein R1878=1™ + 64".82 + 10".578 ¢. Die Verbesserung welche hiernach an die aus der Vergleichung mit dem deutschen Meter gefolgerte Gleichung anzu- bringen wiire, ist — 9 + 04.3304. Mit dieser Verbesserung geht die Gleichung fiir #1876 tiber in . R1876 = 1” + 86".18 + 10".654 t. Dies diirfte augenblicklich der wahrscheinlichste Werth von 1876 in wahrem metrischen Maasse sein. Unter wabrem metrischen Maasse miissen wir das durch das Métre des Archives als Einheit dargestellte Maasssystem verstehen, da eine von diesem unabbingige Definition bisher nicht existirt, und auch bei einer spaiteren Festsetzung einer solchen jedenfalls ein méglichst enger Anschluss an das bisherige System erstrebt werden wird. Die grosse Differenz des in Breteuil gefundenen und anch durch die von Ihnen mitgetheilten Versuche bestiatigten Ausdehnungskoefficienten von R1878, gegen den von uns mitgetheilten Werth muss wohl grésstentheils dem Um- stande zur Last gelegt werden, dass die nach Borda fiir Platin angenommene Ausdehnung zu klein ist. In der That ergiebt sich nach Fizeau die Ausdehnung des Platinmeters zu 3 84.68 ¢ +0".0030 2, also in der Niihe von 15° zu 8.80 ¢. Eine kleine Differenz der Ausdehnungen ist auch zwischen dem Métre des Archives und dem deutschen Urnreter angedeutet, da die Vergleichungen dieser Meter sich auch durch die Gleichung darstellen lassen Pt—A=1".14-++0",094 ¢, oder, wenn man nur die Vergleichungen durch Brix beriicksichtigt, Pi—A=0".39-+0",136 t. Doch sind diese Vergleichungen nur bei wenig von einander verschiedenen Temperaturen angestellt. Wir hoffen, auch unsererseits in wenigen Monaten durch hier angestellte absolute Ausdehnungsbestimmungen zur Entscheidung der noch bestehenden Unsicherheiten beitragen zu kénuen. Ein Anschiuss an das aus der Toise abgeleitete System, insbesondere an die fiir geodatische Operationen wichtig gewordene Bessel’sche Kopie der Toise ist fiir Ihren Stab durch zwei in den Jahren 1878 und 1879 ausgefiihrte Ver- gleichungen von & 1878 mit der Toise Lenoir gegeben. Es ergab sich 1. Z=1.948 570 76 R 1878 bei 18°.255 2, [=1.948 575 53 R 1878 bei 159.74, Verbindet man die beiden Gleichungen, indem man dieselben durch Interpolation nach der Temperatur auf die Normaltemperatur reducirt, so erhilt man D=1.948,574 57 R1878 bei 169.25. Die Toise Lenoir, im Jahre 1852 von Baumann umgearbeitet, ist in den Jahren 1852 und 1872 und auf unserem Repsold’schen Komparator in den Jahren 1877 und 1878 mit der Bessel’schen Toise verglichen worden. Die Resultate dieser Vergleichungen sind 7 1. L—B=—0.0043 2, L—B=—0.0066 3. L—B=—0.003724_0.00013 4, L—B=—0.00336-L0.90022. Als bestes Resultat diirfte das Mittel aus den beiden letzten Vergleichungen gelten kénnen ; also L—B=—0'".00354. Die Bessel’sche Toise wird allgemein nach Bessel um 0/’,00080 kiirzer angenommen als ‘‘die Toise.” Nach ihrem eigenen Certifikate wiirde sie um 0.00078 zu kurz sein; aus einer von Bessel ausgefiihrten Vergleichung mit zwei anderen mit Certifikaten versehenen Toisen ergiebt sie sich um 0'.00333 zu kurz und um 0.00411 zulang. Im Mittel aus den drei Certifikaten wiirde sich daber ihre villige Identiti’ mit der Toise du Pérou ergeben. Bleibt man bei der stets benutzten Gleichung stehen ; B=1T — 0'’’.00080 bei 16°.25, - eee L=1' —0'.00434 bei 16°.25. 838 APPENDIX I. [Apr. I, und durch Verbindung mit der friiher abgeleiteten Gleichung L=1,948 574 57 R1878 bei 16°.25, aus welcher R 1878 = 48-7 1.000 236 96 L bei 16°.25 folgt, R 1878 = 443/296 + 231".94 bei 16°.25 ‘ oder mit dem friiher gefundenen Ausdehnungskoefficienten R1878 = 443’ 296 + 65".41 + 10".248t R 1876 = 443" 296 + 86.77 + 104.324 t. um — 4".2-+4 0,.014t von der vorliufigen Angabe vom 20, Juni 1879 R1876 = 443/296 + 245".6 + 10.31 (t—15°) verschieden. Mit dem Pariser Ausdehnungskoefficienten hitte man R 1876 = 443''.296 + 240".1 4+ 10".55 (t— 15°). Die wichtigeren Messungen zur Bestimmung der Theilungsfebler von #1876 griinden sich ebenfalls vorzugsweise auf Verglcichungen mit 21878. Es sind daher in der Beilage B auch die Messungen und Rechnungen zur Bestimmung der Theilungsfeller dieses letzteren Stabes ausftihrlicher mitgetheilt. Die Theilungsfehler sind siimmtlich auf einer Liingentheilmaschine von Repsold untersucht worden, bei welcher zwei mit cinander fest verbundene Mikrometermikroskope in der Lingsrichtung eines Stabes verschoben werden kénnev, wiihrend entweder jedes der beiden Mikroskope auf denselben Stab oder je ein Mikroskop anf einen von zwei nebeneinander liegenden Stiiben pointirt. Die erstere Einrichtung wurde bei der Untersuchung ucx Fehler der Deci- meterstriche von #1878 angewandt, indem man den beiden Mikroskopen Abstiinde von resp. 5, 2 und 1 Decimeter gab und iit diesen festen Abstiinden die Intervalle des Stabes verglich. Die Beobachtur gsresultate und der Gang der Rechnung sind in den Beilagen vollstindig mitgetheilt—abgeselen davon, dass die Ablesungen der Mikrometer schon wegen der periodischen Ungleichheit der Schranben verbessert sind. Zum vollen Verstindniss ist nur noch zu be- merken, dass bei beiden Mikroskopen einer grésseren Ablesung ein kleinerer Werth des zwischen den Mikroskopen ein- gestellten Intervalles entsprach, und «dass die Einheit, =19, in welcher siimmtliche Ablesungen gegeben sind, und dice Rechnung durchgefiihrt wurde, dem tausendsten Theile eines Umlaufs der Mikrometerschrauben entspricht und gleich 0".05 gesetzt werden kann. Gegen die Art und Weise, in welcher aus den drei Beobachtungsreihen die definitiven Fehler der Decimeter- striche abgeleitet sind, wiirden sich Einwendungen erheben lassen. Eine strenge Ausgleichung nach der Methode der kleinsten Quadrate, welche nachtriiglich ausgefiihrt wurde, fiihrte jedoch zu den folgenden ganz unerheblich abwei chenden Resultaten : und demnach | Negativer Febler— , Decimeterstrich. x44 angenaherter; Nach strenger Rechnung. Rechnung. p. Dp. 0 0 0 1 +12 +12 2 +10 +10 3 +5 +4 4 +27 +25 5 +9 +10 6 —25 / —23 9 —26 : —25 8 —23 —23 9 —4 i —3 « 10 0 | 0 Von einer Beriicksichtigung dieser Abweichungen, welche nur in wenigen Fillen 0".1 betragen, wurde Abstand genommen, . Es ist aber noch ein anderer Umstand in Betracht zu ziehen. Anch bei der zweiten strengeren Rechnung ist an- genommen worden, dass die Fehler der Messun¥ nur ans den zufilligen Poiutirungsfehlern bestehen und unabhingig sind von der Liinge des gemessenen Intervalls und von der Zeit der Pointirung. Dies ist uicht strenge richtig, da durch die Temperaturvariationen wiihrend der Messung Aenderungen des Unterschiedes zwischen der Entfernung der Mikroskopachsen und den entsprechenden Intervallen des Stabes eintreten kénnen. Diesem Umstande ist leicht Rechnung zu tragen, wenn die Aenderungen gleichmiissig zwischen den einzelnen Messungen erfolgen. Ist nimlich die relative Verliingerung des zu bestimmenden Stabes zwischen je zwei Messungen gleich y, die Zahl der Theilinter- valle, welche direkt bestimmt werden, n, so muss dem auf die gewédhnliche Weise berechneten Fehler des Striches, welcher das ite Theilintervall abschliest, noch die Korrektion +! . Dy hinzugefiigt werdeu. Diese regelmassigen Aenderungen sind nun bei den vorliegenden Messungen stets dadurch voll- stiindiyg c:imivirt worden, dass an jede Mcssnngsreihe sich eine Zweite unwittelbar anschloss, by welcher die Einstel- §2.] ADDITIONAL DATA RELATIVE TO R1876. 839 lungen aber in der entgegengesetzten Richtung erfolgten. Die hierbei noch zuriickbleibenden Fehler, welche davon herriihren, dass die Voraussetzung einer gleichmissigen Aenderung der Entfernung der Mikroskope und der Liinge des Stabes nicht strenge erfiillt ist, konnen als zufiillige mit den Pointirungsfehlern vereinigt werden, und bleibt daher die gewihlte Berechnung gerechtfertigt. Die Centimeterstriche von 1878 sind in ganz iihnlicher Weise wie die Decimeterstriche bestimmt, nur treten hier Hiilfsintervalle von resp. 5,2 und 1 cm. Linge an Stelle des konstanten Abstandes der Mikroskope. Fiir die Berechnung gelten daher auch hier dieselben Bemerkungen wie fiir die Decimeter. Die in Beilage C gegebenen Resultate fiir die Bestimmung der Theilungsfehler von R1876 durch Vergleichung der Decimeter- und Centimeterstriche mit #1878, der Millimeterstriche mit einer ibrerseits absolut bestimmten Silber- theilung auf dem Messingstabe Nr. 6, und der Zebntelmillimeter durch direkte mikrometrische Vergleichung werden ohne weiteres verstindlich sein. Als Einheit gilt hier 1#—0.001 mm. Die zum Theil statttindenden Abweichungen der Resultate vop den Ihnen unterm 16. April 1879 gemachten Mittheilungen riihren von der genaueren Bestimmung der Vergleichsstiibe’ her. * » : * * Kaiserliche Normal-Aichung-Kommission. FOERSTER. BEILAGE A. Bestimmung der Gesammtlinge wnd der Ausdehnung des Stahlmeters R1876. a, RESULTATE DER VERGLEICHUNG DES STAHLMETERS R1876 MIT DEM MESSINGMETER 1605. ‘ R—1605 bei 15°. Datum der Vergleichung. t 1605. tr. R—1605. 18.25 (ticos—tr). | 7.921 (tieos—15). L ti. ° ° & “ w w 12. 28 12.51 +57.7 —4. 20 —21.55 +32.0 | +36.2 12.26 , 12.51 +66. 6 —4, 56 —21.70 +40.3 | 444.9 11. 36 11.39 +56. 0 —0. 55 —28. 83 +26.6 | +27.2 18.72 18. 96 — 4.2 —4. 38 +29. 47 +20.9 | +25.3 18. 82 19.06 —7.3 —4. 38 +30. 26 +18.6 | +23.0 18. 80 19.10 ; —1.8 —5.47 +30. 10 422.8 | +283 19250 paaeena seu 18. 90 19.15 = 452 —4. 56 +30. 89 +19.1 | 423.7 DSi an caulach ee depths 19.59 19.71 —16.3 —2.19 +36. 36 +17.9 | +4201 DBM aie sencidicieiae kesten neko ae 19. 61 19.77 —16 ° —2, 92 +36. 52 +22. 0 +24. 9 OSE ieihaemeneurcieanee 19. 69 19. 87 —14.5 —3. 28 +37.15 +19.4 | 422.7 OB onl onsen 19. 69 19. 91 —15.3 —4.01 +37.15 +17.8 | +219 Septcd san. oss hlesevents 17. 63 17. 85 — 0.8 4.01 +20. 83 +16.0 | +420.0 We 5.26pee oisdeesenees 17.73 17. 92 +27 —3. 65 +21. 63 +207 | 424.3 hs cre eee 17.71 17. 96 — 0.7 —4, 56 +21. 47 +162 | +20.5 A ah SAR tee ta panel 17.77 17.96 — 0.5 —3. 46 +21. 94 418.0 | +214 Die Stiibe liegen auf je*zwei Rollen, welche 0.23 der Stablingen von den Enden entfernt sind. Auf jedem der Stiibe liegt ein Thermometer mit/seiner Metallskale auf. 6. VERGLEICHUNG DER REPSOLD’SCHEN STAHLMETER F£1876 UND #1878. I Reihe. I Reihe. IT Reihe. tis76. tiazs. R 1876—R 1878. tis76. tis7s. R1876—F 1878. tis76. tis7s. R 1876—R 1878. ° ° “ ° @ bh Me ° ° ps a 8. 592 8. 608 422, 7 15, 740 15. 758 +23. 2 21. 489 21. 487 22.1 8. 596 8. 607 20.9 15. 749 15. 760 22.0 21. 489 21. 486 22.9 8. 605 8. 619 21.9 15. 758 15. 766 23.3 21. 490 21. 486 22.7 8. 614 8. 628 21.4 15. 757 15.775 22.4 21. 487 21. 487 23.6 Mittel ..8. 602 8. 615 21.7340.2 || 15.751 15. 765 22,73 +0. 2 21. 489 21. 487 22. 8340.2 7 | Auf jedem der Stiibe liegen zwei Thermometer Fuess’scher Konstruktion, deren Gefisse in mit Quecksilber gefiillten Kupferklétzen stecken und mittelst der letzteren auf den Stiben aufliegen. Jede Beobachtung besteht aus einer doppelten Einstellung auf die Striche der Stiibe und einer doppelten Ablesung der Thermometer. Der Direktor der Kaiserlichen Normal-Aichungs-Kommission. 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JOP SNY WEP TWayOstaz zZuaayi eYAvjs orp JUV YPIsyoNY JIT Tour orp ‘g— wn SunaassoqiaA oo OOM Toqep WyYle TOUS Lod “WoHvyte Xopuy Wopeses 41 aOIIsSIOJoTMILIVIq_ VIP AN SunaassoqssA TOIUP A OLY Jap suv IVq[oypIWUN Vp Woq[as1op TOYO UWIAYVSoU o1q—'rapuy wapv.labun yuu ayorgsiajamwad AT “E — os ton @ —= —___—_-+_—__. — Jo|[]Ja 9Al}Bvooll 19 . ected P09 pe OSE __ 19]]9,J OAT}BVSOU 19 or cetiexe eee P OF Ee -*-YOUTS IMJ parr aS Sens 2 : &@ er ee0e IO[AJ SATPVSOT AVP 0 gi+os = tS naga, oares @ oI+ exer-boeXog [TAT oatqeson sop 0% FXO ‘og PUN OF EMILY orp ny os X OT ITAEDH Sep OB PUN OB OPUIYG ofp ANZ L9[YaT woaTpeBon wog]oqqTUII FjaswOa ap wasesep uyeyie sajyemoepreddog sap UoTlaIssunsseyy Uayyorp usp suy O='xor ‘og pun OF ETOYS Ap op =F Xol JYIIMOY) SVP OB pUu OZ 9S a Lap Ja[GaJ Waatjvesot Vopael[Usar suesaiqy stp way[eyqro GW) 49M Woselp Ny ospLysi1opamM19aq_ Jep SunszsssaqiaA TOUN— ‘wapuy wapyiabh pw ayo.gscazamwad aT “% 7 TATILUBaP SOYIIPSPOHUPL SOP JOLT] OAVSOU Lop OSTR PATA 8a ‘az ea qYPTME) SVP IojouTLooq, Woupezale Joep WITLIAYoa[H19 A OT wep IPXG+P'EXOT Or Sue [aT Wop PULUTOY Tep ‘OT FPTMIH sep IapuvUle JoyUN JoJoy Moq el, Jop WOYLorTo1e[S19 A OT Lop EIA Wotay[Vyso Syypos10A Wap ULM IQeTH—"YoLasNUT 9 “T “ONNHOIGTIOSAY | ; | | | | 0 O PO 1 oO 0 o ;0 i |o 0 0 0 | 0 oe | TL 920 +6 Gol $6 99 92 } 0g 00% oot r- e- fet) et o-|st| r+ | mtye- oF eho tae OFL FOL Vel 88 6oT #8 £9 9b 8é est 06 oo 0z— ga SS 1 8 =. Re PB | p+ | wo | ate | atm | etn | z+ lt | 68 cot 19 16 eg 96 OF 1B zor 08 m= ge b= 1 a> ore pm | ee | | a> tee | at i+ | gor | 88 sh | 89 ch | 88 ce BE ze ver | Ob eg | wee | t= | t= peti e- ler [et] e= sI- | G- | F— | ost | F6 | ob eh “9 wo | as Te “eo | 98 ort | 09 ee Pot oe VY oteb oo Pee Pte fe Fer | Trt | oer | cet | seb | Lr Tet | FOr 16 ¥6 66 68 &¢ e, | eg PLL 0g wet | wet | ect | cet ; ect | set , fOr © bet . os oF fest | cet | Pe gt | TOL 98 cor lor | 6 99 we 08 PLU oF | @¢— ort ug eet, let » Sth | 9st | set "tet | ett att | eet | 6+ 9L 1g es #9 o9 09 €f oo! bE GIL og | oI+ 08 cer &% cet ott | gé+ eet | oakt wt ocrt | fet | eet eg 6¢ IF uc tig ! 19 oF 09 | FP eb oz | ae atk ie eek | ete get att | eet | att | eet | st i set cot | le | OF Te ap 6a 68 st te 6a oF or a 16 | 0 f 0 0 1 9 0 0 0 0 | 0 0 0 PR Te 0 0 0 ; 0 0 0 0 | 0 0 | t I | i 1 — | —=—— er ir. => are eee 2 od a te | nit | Wt | nN | 1d 6 8 | “a nat | *a ta | ta | *a ‘a | 9 | ve) 8D a 99 "9 9 "9 | z) | wy | ug *ponutyuo0 )— AMO Ise a LAN IOA NUNIGZNIA WIC ‘)D *ponuljuOQ—SLST Y 8.azaupyv7y sap daqyafsbhunpayy dap bunwmurssg—q TVA 845 ADDITIONAL DATA RELATIVE TO R£ 1876. ~§2.] 0 0 0 0 0 Qi [Braet lee geet esisnalas fom tines Re cemeis ze | 99 79 zor |sor |e t— |99 0 | €@9 T+ | ZoT0 | 96 ett] oF ro PLS (et at |p | a 88 6F 19 OL FOL 76 9- | Fo o— | 99 F- | Tere + | VIL Ss — Lee Nabeat (Sei Gest: PETS eka © [GPRS BRA ees PAROS RA ISICARES EES | 8 ¥6 | IF LL 90T ToT 96 2— | 0¢ 6— | 69 Bt | SII sI—| TOL 0 sIr_ | 09 96 Zot #81 | eal 6— | 99 9- | 96 I— | BOLL —| FL 8 — o— |e= |The | s= Jee fée— | 9 66 8g 16 60T FOE =|, QOL. ZA— | 6S I— | 86 s— | SOLT + / weTat GIF | H+ | et |e + feet stl | & eI FIT SLT Tol €— | 9L #— | BOTS | CLIT | TT + Si [ee Ee ag ere pg ees eee ee eee SSisestens ses aia 1 + LIT | 18 OT git B06 SILI— | 08 I+ | 401 €— | LeL 6 — | 6st eT+ e— joet |@- Jo-— Jet FIT | TIT Zor ZIT 11 | LOT 2+ | SIL 4— | cot e— | TTT + | Loeb + ol ST 08 Te BS fee ree tes a fpre eee rate te eec|somewnes lasemunne 3 a— |e let Jere Jer PIL | TIT FIL oat s6q || TIL s+ | TIT 0 SILI— | UL6 + | “ET + 0 0 0 0 0 Qo, |fPeeeees bpeccsten po sseaneleasarca|es neces 66 [46 | 96 = j or jest ors joore- | 26 T- | ss 2+ oste +| 0 \ 1 . Ww °q | eq tq Iq 89 wr £9 wo id 9g ry eg sg | 1g i °— ry ey ty IT m9 f “XHQNI WHCVYTD LIN AHOIMLSUTLANILNAQ TIG—‘¢ ‘TOpPIOa UossemMes FY Siq VY Ieq OSeT] OIYT USS odoysoryIPL Toepusssowl oIp JuY Snzog UI oqryg rap SunyosnejzI9A J9yUN puis “y sIq Sy “Ng ‘L@F seTyo,7 War9]44101 wep q1UI G'z — oy[VYyIE [9991TT STB ueNT eI9RY OS ‘TOTUTUOTAS 1M 9YIIE AA TeMesso[TOsassne Uepreq oIp Uv oyVH ‘PL F J0[YA,q WoteT}41MI Diep 4IUI G — : QT UOSTqn Uap sue [oIIPL s[e UU 4[VY10 Os ‘sn¥ SID pun "fY oyIIE AA USPUE][VJsnvi9y] YIV}s Uepleq olp Uru 4ssaT[qog | : + G9— | (G2e—) | ¢— | T+ aap ae (oe — ae E90: == | ete) | s— |e- | Ort ‘¢e¢+ @2-— | gI- ler so- | ‘a : | | | ot 9s— | 9¢-- 9+ | 98- | Ig—- | g2— | S8— | BSe- | GE- | Be- | OF oe— | s€— | 8e—~ | BF- | Ls— | Os— | LI— | Be— | Ge g ee— | 69— 6}— | 9F— | GR- | T9— | BG— | GE- | GO- | FO- | TE 66— | 6 | we 82—- | 9F- | ge— | Bb— | Te- | os— 0 | ie 1 I oy 6I9 sly ay ry | sty wy | ely | ay | ny og, 69 | ay | 49 | 99 99 wt &) zy wy “muaID | | 866 | CUE | 0S | T6T | 80% | LET | GEE | LST | FEE | Lee || Coe L— | Toes +} sosT+ | esr st | ser 9+ | 9026— | coae— | 916 —| Ieee + ozect | oT oe | Lhe GOG | CST | BET | GOT | GOT | GST | Gs | LG | Leas + | Les ot—| tosst | cotet | tert | TArs— | wre | wre—|oeee+ | geet 2 6h | FSG | BBL | ZT | OST | OST | OOT | FOZ | Lez | PLS || Les st | GEES +) O6T 8— | Ost St | FOL F— | OFEI+ | GeL I+ | ees att} seztI—{ weet (3° LOT | S8— | 6€L | 38 | LOT : 68 | cor | 612 | stz | oz 8816 + | O8L F—- | cel b+ | FL St | OTLE— | 46 S— | F6 St | AIZa+ > 2ta6— | Oz0 0 | ) ee | lg | te | Ag | sg | tg | Hg | sig | ag | ug oy oly sly ay oy sty ty sty ay ly ‘wg yee | OE | Let | LIT | Ge | s2t | sot | oor | er | LIT || tee 0 6ig I+ | 9ZL I+ | LITO cet 9+ | szto oo ¢ + | INL I— | set e— | siItI— | or ¥62 | 062 |.09T | SOL | L8T | SB | GT | 2eT | ST | OFT |] F620 063 0 Fel OF | POT T+ | eet e+ | Let cI— | eet ctt | ost e— | tor z— | Grr e— [2 oP ns £96 | 968 | 806 | BOT | S6T | 92t | 84 | LOT | SBE | G98 | S9SS— | soGet | OIBT— | eATG— | oGTI— cate + PLE + | GOLS— | strat (Feet |S! ez | L6L | 292 | oe | gee BLT | SIT | 01% | 90% | GEE |! Iza I+ | 96r I+ | 6S s— | OVO ceo Tt | Sol t + cILa—| tos6t+ | costt+ | oceet+ | 0 ogg ogy 8gT 40 9 og rg eg og Tey oy 6 8y Lv 9—v sv ty Sy Vv I? ‘mdD ‘WO G HOIULG waAgq—v ‘MALAWIINGAO GId—'II 844 APPENDIX I. [App. 1, BEILAGE B—Bestimmung der Theilungsfehler des Stahlmeters R 1878—Continued. ce. DI® EINZELNEN CENTIMETERSTRICIIS. Cm. Ai Ag As Bi | Be Bs |, Ci C2 cs | Di De | D, Mo Mi M | 0 —3 210 | —3 209 | + 1172 || 207, 206 VB) | ssiosoe| seicnewioseueis . 0 0 0 0 0 0 0 if + 3246] —5 271] — 2 226 |) 249| 266 | 224 i be ze : j ‘s Ly — 6250] —3 266 | — 9 236 | 244] 263] 2e7s Pp | Teter —40 | —51 | —50 || 2f + 4280} 41313) — 2279 |) 284] 314) 277) i si call sett aah Sa ae = 2 go] siz] aos |p i - 6 286 | +3 311] + 2 29 2 cr ae pare af + 3315 | +5370) 414 343 || 318 | 375 ae vege | Gall oat Ae Ay 0 320) —2 370) — 7 351 |) 320] 368] 3d4$fo py pp penne ae 8 355 | —2 428 | —11 405 || 347] 426 agalt eaig lt Soel el cas t) +1 378 0 430 | —11 415 | 379) 430) 4045), 56| +7 397 | +2 462) — 1445 || 404] 464) 444 3 | alae ‘ ‘ — 1400] -—3 474 | + 7 448 || 399) 471 455 | elt ae OA de ees ' eh lst 3 439 | +9520] — 7519 || 442] 529 ae Ee ace Spa : fl ne a t + 5449} —6 519} 4+ 4529] 454) 513 | 5339) | —25 | — = ea 1480 | —5 583 0 597 || 479] 578 oF 448 gil ae ; Ble Ne ‘ — 4475 | —7 296 | — 5 598 || 471] 289 593 op a ican ale ers ia giles Coed | «tte ater +13 484 | —2 311] + 3610 || 497] 309] 613 Bu ee eal) Sco eceet [tf ee anaes) owrseal|l mace || — sain | latacee asieMele Moeeae 24 | +25 5 18 14 7 10 ' — 5490] —9 326 | + 2619 |) 485] 317 at sit as ha F we ao a = 7 a = _ =36| — 5 541 | —3 391 | — 4 661 |! 536] 388] 657 Qe E ee sheen or Metal etre peed ee All| 2 encill | Memmilhaa ee head ee 9 7} +19 12 10! | s.ceted 9 ' — 1540} —9 385} + 6 669 || 539] 376 ac el ele + + a © t % 10 | + 4580] —1 438) + 6738 || 584| 437] 744 |I...--.]..---.] ---.- 0 0 0 0 0 0 0 . Fiir periodische Fehler der Schrauben korrigirte Ablesungen beider Mikroskope. Summe der beiden Ablesungen. . . Relative negative Fehler der einzelnen Intervalle auf das Hiilfsintervall. . Negative innere Fehler der Theilstriche innerhalb des Decimeters. tonb AUSGLEICHUNG. u. Der 5 cm-Strich.—Giebt mandem Mittel aus den 18 wirklich benutzten Vergleichungen der beiden halben Deci- meter unter einander das Gewicht 18, dann kommt dem Mitte] My aus den auf der vorhergehenden Seite berechneten 3 Vergleichreihen der einzelnen Centimeter das Gewicht # zu, es wird also der negative Fehler des Mittelstriches des ersten Decimeters SOKO ENE b. Die iibrigen Centimeterstriche.—Nach Verbesserung der Mittel Mo fiir diesen Werth des 5 cm-Strichs erhalten die Fehler M, fiir die Centimeterstriche mit geradem Index Werthe, die von denen aus der direkten Vergleichreihe der Doppelcentimeter nur um wenige Einheiten abweichen. Nimmt man deshalb ohne Riicksicht auf die nicht sehr verschieden ausfallenden Gewichte die Mittel aus beiden und verbessert dafiir die Centimeterstriche mit ungeradem Index, so wird ein wesentlicher Fehler nicht begangen. Der Direktor der Kaiserlichen, Normal-Aicoungs-Kommission. Berlin, den 10, August 1881. [L. s.J FOERSTER. BEILAGE ©. 7 Festimmung der Theilungsfehler von Repsold’s Meter aus Stahl fiir Amerika, R 1876. Fiir die Bestimmung der Thkeilungsfehler der Decimeter sind diese 20 mal mit Doppeleinstellungen beider Mikro- meter mit denen auf dem der Kommission gehérigen Stabe von Repsold & 1878 verglichen worden. Die Ergebnisse der einzelnen Messungsreihen, von periodischen Fehler der Schrauben und den Konstanten der Anfangs- und Schluss- cinstellung befreit, sind, ausgedriickt in Tanusendtelmillimetern, im Sinne relativer Fehler gegen den Vergleichstab in den Kolumnen 1 bis 20 zusammengestellt. Kolumne 21 enthiilt das Mittel der Kolumnen 1 bis 20, Kolumne 22 die inneren Fehler des Vergleichstabes, Kolumne 23=21-+22 die absoluten inneren Fehler des amerikanischen Stabes. Zur allgemeinen Vergleichung ist unter 24 noch das Resultat einer iilteren, aber unvolikommeneren Vergleichungsreihe mit einem anderen Stabe mitgetheilt, deren relatives Gewicht aber so gering ist, dass sie keinen Einfluss mehr anf die Werthe der Kolumne 23 zu iiussern vermag. § 2.] BEILAGE C—Bestimmung der Theilungsfehler von Repsolid’s Meter, R1876—Continued. ADDITIONAL DATA RELATIVE TO RF 1876. Demeter) 4 2 3 4 5 6 7 8 9 | 10 u 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 —0.1) 40.7] 406) 40.7) —02) 403) -0.5) 40.7) +05] 41.1) -14] 40.1 3 —21) -04] -07; ~0.7] ~06] -0.4) -1.3]) -1.3/ -08| -0.7) —41] —-16 3 -1.0) ~0.1] 40.1) +06 0.0}--0.9| -1.5) ~0.7 0! 40.5} -24] —19 4 -1.8) 40.4] 404] +01 0.0} 40.3) 402) 409 0} 40.9} -1.3] —0.7 5 -13] -0.4] -04!] -0.3] -05] -04!] -1.8] -14) -07] -05] -18] -13 6 —-24/ —L1] -19] ~1.7] ~-1.5] -—04] -1.6] -16] -13] -1.3] —03 +05) 7 —3.2/ =-22! -27/ -31]/ -24] -—27/ -2%9/ -20] -21] -27] -18] —13 8 -1.9} -19] -18) -14] +10] -1.4] -24] -13] -o8] -11] —138 0.5 | 9 06} —O1/ 408) -04} —06| -07) -1.9) -11] -08) —07)> -10] —0.4 | 10 0 0 0 0 0 0 0 0 0 | 0 e Decimeter-) 43 14 15 16 17 18 19 20 21 22 23 24 strich. 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -0.5} —05] -—06] —0.3] +04] -05] -0.4|] -06 0} -0.6} —0.6} —1.2 2 —31] —3.3] -~35|] -24/] -—20] -29] -33] -32]} -19] -05] -24] -2.0 3 —17/ -24] -26] -15] -14, -14] -23] -23/) —11] -02] ~1.3] —0.5 4 —13/ +01] —12 0} +05; —06] ~1.0] -08 |) -0.2] -1.3] -15] -1.0 5 —1.7] -0.7) -11] -1.0] -1.1} -19] -11 | -1.0) -0.4} 14 0.8 6 —0.5) 40.1} -03] 407] 401] +04) +01! +402]] -0.7) +412) +05] —0.8 7 —1.6) -16] -1.7] -08] —09} -0.8] —-1.4 aia —2.0| 41.3) —0.7] —2.0 8- -10; -13] -—09{ -04] -1.2) -06] —0.9 0.8 | -1.2! 41.1] -01] -o1 9 40.2) -1.1] —0.9! -0.2/ 404} ~03] -—03] -0.6 |} -0.5| +02] -0.3] —-1.3 10 0 0 0 0 0 0 0 0 0 0 0 0 845 Fiir die Bestimmung der Theilungsfehler der Centimeter des ersten Decimeters sind diese ebenfalls 20 mal unter Doppeleinstellung beider Mikroskope mit denen auf dem der Kommission gehérigen Stabe von Repsold verglichen Kolumne 23 enthiilt hier die inneren worden. Die Ergebnisse sind analog denen fiir die Decimeter zusammengestellt. Theilungsfehler der Centimeter innerhalb des Decimeters, Kolumne 24 dieseiben innerhalb des ganzen Meters. Meneberiee) <3 2 3 4 5 6 7 8 9 w | u |] wp 0 0 0 0 0 0 0 0 0 0 0 0 0 1 415] 425} 423] 41.5] 42.4] 423] 429] 423! 422] 432] 416] 43.0 2 41n0| 421) +21] 404] 421.7] 416] +09] +1.0!1 41.0) 40.9] 42.0] 42.6 3 42.4] 423} 420) 424] 422] 423) 421] 423] +21] 423) 42.8] 411 4 414] 416] 42.8] 415| 42.7] 420] 41.4) 418) 41.9] 420] 412.3] 40.9 5 414] 416] 41.6) 428] 420] 41.3} 412] 414] 42.3! 415] 41.3] +06 6 404] +07] 409] -02} +05] +06] 40.8 0} 40.5) 41.0) 411] 404 7 40.4] +10] 414] +04] 416] +408] +06] 40.3) -04] 422] 41.3] +10 8 —0.2| 403/ —03) +08] —02] 408] 404] —05} —04) -02] +407] +07 9 4.7/ 422! 41.5] 421] 424] +21] 42.0) 41.8] 424] 422] 414] 41.2 10 0 0 0 0 0 0 0 0 0 0 0 0 Centimeter-) 13 4 15 16 7 18 19 20 21 22 23 24 strich. | | 0 0 0 0 0 0 0 0 0; 0 0 0 0 1 424] 420] 425] +24! 426] 414] 420) 423) 42.27 +04] +26! 425 2 425] +16] +22] 425] 424] 41.5] 424! 423/1 41.7] 407] +24] +23 3 41.8] +12] 426] 421] 427]) +25] +24] 425) 421] 409] 42.0] +28 4 414] +10] +12] +12] 429] 408) +20) +2.9]/ 41.5] 408) 423] 423 5 41.3] 413] 407) +10! 405) +05) 408) 41.8] 412] ~o02] 414) 422 6 41.4] +19] +07] +05) +09! —04) 402) +18] +07] +03] +01] +06 7 407| 422] 409] 413] 413] +09] 41.3) 42411 41.0] 408] +18] 414 8 40.8! 41.5] +10] 409] 41.2] +07] 40.5] 41.6) +05] -05] 00) —05 9 424] 424] 417{ +25] 415) 42.7] 423] +29]] +20] -05] 41.5] 41.0 10 0 0 0 0 0 0 0 0 0 0 0| -0.6 846 BEILAGE C—Bestimmung der Theilungsfehler von Repsold’s Meter aus Stahl fiir Amerika, R18S76~— Continued. Die Millimeter sind zweimal mit denen auf Hauptnormal Nr. 6 verglichen worden und zwar in Gruppen von je Die folgende Zusammenstellung enthilt in den Kolumnen 1 und 2 die vom periodischen Fehler der Schraube, den Konstanten und den Theilungsfehlern des Vergleichstabes befreiten Resultate, Kolumne 3, das Mittel aus 1 und 2, stellt dic inneren Theilungsfebler der Millimeter innerhalb der einzelnen Centimeter, Kolumne 4 innerhalb des ganzen zelin, Meters dar. APPENDIX I. é as 13. 2. £5 : 134 ta Be] 1. | 2 3. 4 | BB) OL 95 3. 4, || BE] 4 2, 3. 4. a Bs = A | | A a | I 0 0 0 | 0 0), 20 0 0 0} 42.3 |] 40 0 0 oO; 42.1 1 | stil -ne, <2) -oso |) or | soe} =oe! ov] 426] ar | =o.5' | —18;) =0.9 | saa a} -1ol -1.0/ —10 ~0.5 | 22 | -01| -08] —0.4] 420]/ 42 | 408) -05] +02] +a1 3 | -14 -o5/) -10/ -02} 23 | 401] 401) +01]! 425]] 48 | -o6) -17} -1.1] 40.7 4 | -07, —L8] -12/ -o27 24 | -o09} —1.5] 12] +23] 44 | +01] -16/ -08!] +09 5 | +10) -L7] -03/ +10) 25 | 08] —15| —11) +14 45 | -0.2! —08] -05] 411 6} She” 18, =1:6 0.0) 26 | 408] 05) +02] 428/] 46 | 407] -11, -02] 41.3 7) -1L2) “18 | -15) 103]! 27 | -03| —10} -07] 419]) 47 | 401] -13] -06] +08 8 | -07) -10) -09]} 411} 28 | +02! -1.5] -07] 420]! 48 | 404] -25] -10] +403 9 | -1.6 ~2.0 | -1.8| 40.4 || 299 | +05) —01] +02) +29)] 49 | -1.2] -20] -16] —04 10 0 0! o0| +25 '} 30 0 0 0} +2.8 |} 50 0 0 0} +141 10 0 0 o0| +25 || 30 0 0 0) +2.8 |) 50 0 0 O| +11 lL) -0.6 00{ —03] 422) 31 | 421.8 0; +09] 43.6! 51 | -02] —05] —or3] +08 12 | -11] -02] -06] 412] 32 | 401] -03) —o01] +25 || 52 | 41.2] 42.2] +2.2] +22 13 | -05] —05] —05} +20]/ 33 | —o1] 401 0.0] 4+26/| 53 |} -01] ~02] ~02] +08 4) 40.1} -13) -06] 41.8] 34 | -03] -06] —04] +4211) 54 | -1.6} -1.9] -18] -o9 15 | +0.6{ 41.1] +08) 43.2 || 35 0.0} 407} +03] 427] 55 | 40.6] —04] +01) 4210 16 | 40.3} —08] —02] 42.2]! 36 | -1.5] -o1] --08] 41.6] 56 | -o03] —1.2] -07] -40.1 17 | +06) +14) 410} +3.3]/ 37 | -02] 403) 401) 424] 57 | -o4] -1.1] -o7] +01 is |:+0.4] +09) 40.7) +3.0]] 38 | -o.6! —o6] —o6}] 41.6] 58 | -o1] -—12] -07 0.0 i | 41.1] +09] +10] 43.3 ]] 39 | -11] -11]) -1.1] 421]) 59 | -o04}] 40.6] +01] +08 20 0 0 0| +23 |] 40 0 0 0} 42.1] 60 0 0 0} +406 £3 $4 Be 1. 2. 3. 4. Bs 1. 2. 3. 4, Rin ma A A 60 0 0 0| +0.6/| 80 0 0 0| —0.5 61 | 40.6} 41.7) 411] 418]/ 81] 412] 401) 406] +03 62 o| +04] +02] +410) 82] 402} +04] +403] +01 63 | -0.1| +05] 402] 42.0]] 83 | 41.6] 41.1] 413) +12 64 | 40.2] +07] 404] 41.3] 984] 403] +06] +04] +05 65 | +25) 41.5] 420] +30] 85] +04] 42.7] 4121] 413 66 | +04} +11] 408] 41.9] 96] -o7] 412] 403] -40.7 67 } 41.3) 41.7] 41.5] 426] 87] -11]} 41.9] 404) +10 68 | 41.1) +09] 410] 422] 98 | 413] 428] 421] 42.8 co | 414] 41.3] +414 274 89 | —31] -01] -16] -07 70 0 0 o0| +14]! 90 0 0 0} +1.0 70 0 0 0) +214 {| 90 0 0 o| 41.0 71 «| 426{ 41.4) 415 27]) 91] —14] -02] —08 0.0 72 | 410! 408] +09] +19] 92} -1.6] -1.0] -13] -06 73 | 41.0] +0.9] 41.0] 41.8] 93 0.0! +08] 40.4] +09 ™4 | -07]| -02] -05] +01]) 94 | -07] 402] —02] +0.2 7 = Oy8! | 056 | one 0.0|! 95 | 403} 41.5} 409) +11 7% | —0.2) 40.4] +01] 40.3]/ 96] —02/) 41.0] 40.4] 40.4 7 | 0.1] 402) 401] 401 ]}) 97} -01] 408! 40.4] +03 7 | +04] 404) 404] 40.2] 98] 404] 412.5] 41.0] +07 7 | —02] +04] 401] -03]} 99 | 404] 41.2] +08] +04 80 0 0 0} —0.5 |) 100 0 0 0| -06 §2.] ADDITIONAL DATA RELATIVE TO R 1876. 847 Die Zehntelmillimeter des ersten Millimeters nebst den beiden Zusatzzehnteln sind zweimal mit einem Hiilfsin- tervall im Gesichtsfeld des einen Mikroskopes verglichen, dic Resultate sind in den Reihen I nud IT enthalten ; III, das Mittel aus I und H, enthiilt die inneren Theilungsfehler innerhalb des ganzen Millimeters, LV innerhalb des Meter toretice. | it, | Bis | baie, —0.1 0.7 —0.2 +0.2 +0.4 0.0 0 0 0 0 +001 —0.8 —0.1 —0.4 =e 0.2 +0.4 +0.9 +0.6 40.5 0.3 +0.9 +1.6 ++1.2 40.9 0.4 +24 +2.3 +23 +2.0 Oe —03 | +403 0 | -05 0.6 4-20 +0.6 +1.3 40.7 0.7 +1.0 +0.5 +0.7 40.1 Oe +0.4 | +403 +03 | —0.3 0.9 —0.9 —0.6 -0.7 ~1.6 1.0 0 0 0 0.9 11 +15 +1.8 +17 +0.7 Der Parecior der Kaiserlichen Normal-Aichungs-Kommission. Berlin, den 10. August 1881. [L. s.] ForRSsTER. [TRANSLATION. ] BERLIN, August 10, 1881. The Commission is now enabled to communicate to you more complete details concerning the determinations made here of the length and the expansion, and also the errors of graduation of the steel metre, graduated on the neutral axis on platinum, made for you by Repsold, Hamburg, which is known here as #1876, and to explain the results given in letter of June 20, 1879, and partially modify them. This will also answer the questions raised in your letters of October 16, 1880, February 16, and April 27, 1881, so far as they relate to this metre. The chief cause of delay has been that we waited for the more exact results of the comparisons made at Breteuil, in order to be able to send you as definitive results as possible, and that these comparisons have only lately been made known to us. ‘The first determinations of the total length of the bar were made in 1876 by 15 comparisons with brass metre 1605 at temperatures between 11° and 20°. These comparisons, however, are worthy of but little confidence, because they were among the first made on the comparator, and the method of observation was not at that time sufficiently elaborated. Especially did the deter- mination of the temperature of the bar in this series of comparisons leave much to be desired. There is also on this account some uncertainty as to the best method of reducing the comparisons, and the results of the computation are unreliable within wide limits. The individual results of these comparisons are given in Appendix A, viz: the temperature of the two bars tos and t, as given by the readings of the thermometers lying upon them, and the difference of length found directly. The computation was made incidentally with the coéfficient of expansion found later, and, in fact, under the assumption (I) that the temperatures of the bars corresponded to the readings of the thermometers lying upon them, from which assumption results in the mean— R—1605= +4 214.9 at +15°.00 In the second place it is assumed that the temperatures of both bars are given by tieos alone, because the surface of the other thermometer in contact with #1876 was proportionally too small to expect reliable results. (II) From this second assumption there results in the mean— #1876 —1605 = 4+ 25.7 at +15°.00 Finally, the assumption would come under consideration, and would perhaps be preferable to the foregoing ones, that the mean of the temperature-indications of both thermometers be taken as the temperature of both bars. From this assumption would follow— R1876—1605= 4 27.3 at 15°.00 S48 APPENDIX I. (App. J, srass metre 1605 has been often compared with the German original standard (Pt). From the latest best measurement results— 1605— Pt = — 18", 144 04.79-+¢ (94,653 + 0#.044) and therefore, by applying the legal equation of the original standard,— 1605=1" — 604.134 184.253 ¢ that. is— 1605=1"+213+4.7 at +15°.00 The combination of this value with the above differences of R1876—1605 gives much smaller values for the length of 1876 than the later comparisons, but there is, as already remarked, greater doubt as to the accuracy of this first series of measurements. A determination of the length of #1876 which affords very. much better guarantees for its correctness, rests on the comparison of #1876 with a Repsold metre exactly similar to it, desig- nated R1878. In March, 1878, three series were made at temperatures 8°.60, 159.75 and 21°.49, the details of which are given in the appendix, and which, if we take into account the very small dif- ferences between the temperatures of the two bars, lead to the result— R1876 — R1878=+421.86 at 8.602 +:22.28 at 15.751 +22.81 at 21.489 The mean of all the series gives R876 — R878 =+224,.524 04,21. ~ That result, however, is to be preferred which is obtained by taking into account the difference of expansion of both bars, . W1S76 — RASTSH22.524- 0#.17-+(t—15. 281) (0.076 4: 04.029), or, in another form, R1876 — RUST8=214.36 4 0#.48+¢ (04.076 + 0.029) The bar R1878 was compared with the German original metre in two series, in the years 1878 and 1879. The first of these series permits only of the computation of the relative expansion. The result was— 1. R1S78— Pt=Const. +1.68+¢ (14.676 + 04.074) 2. R1AST8S—Pt=+68".79 + 04.49+¢ (14.643 + 04,032) The following equation, resulting from the combination of both series of observations, is to be used: BR 1878 — Pt=+68+.71 + 0#.49-+¢ (14.648 + 0.029) Combining this with the above we have 1876 — Pt=+90+.07 + 04.68-+¢ (14.7244 04.041) If we place for P¢ its legal value, resulting, from the comparison with the Métre des Archives, on the assumption that the expansion of both metres is 8.6t, then R187T6E=1"+934,.08+104.324¢ The considerable difference, —1+.16+0+.014¢, from the result given to you by letter of June 20, 1879, FASTO=1"+248+,89 + 04.25+(10".31 + 0".034) (¢ — 15) is caused by small supplementary corrections which resulted from a further revision of the ther- mometer computations. Moreover, in the probable errors there given, only the uncertainty of the comparison of the two Repsold metres with each other was taken into account. The statement made to you in letter of September 15, 1880, that the given equation of k1876 ought to differ only from 14 to 2+ from the true metric measure, has, unfortunately, not been con- firmed, so far as low temperatures are concerned. It was founded on a comparison made at Breteuil §2.] ADDITIONAL DATA RELATIVE TO R1876. 849 of R1878 with the platinum-iridiam metre, type J, at 30° C., which had given a result very con- sistent with assumptions previously made. Comparisons at 0° and 15° C. have not confirmed this result. As the result of the three series of comparisons we now have RAST8=1"+G4«.824104.371 t4-0#.00608 In this it is assumed that the equation of type J is T=1"4+674.6+8+.525t+0+.00390? The error of type J at 0° has been derived by indireet comparison with the Métre des Archives, and the expansion of the bar is the mean value of various absolute determinations. 1n the equa- tion of R1878, the coétticient of ? is assumed trom Fizeau’s determitations, and the amounts of the other terms are computed from the determinations at 0° and 30°. As the comparison at 15° gave a result differing only about 0+#.05 from the formula the latter is retained. Neglecting the term involving f we can place, when the temperature is near 15°, according to the determinations at Breteuil, RAST2=1"4 G44. $2+104,578¢ The correction which according to this should be applied to the equation derived from the com- parison with the German metre is —64,9-£ 04.330 ¢ With this correction, the equation for R 1876 becomes FAST6=1"4 864,184 164.65 1 t This may, for the present, be considered the most probable value of R1876 in true metric measure. By true metric measure must be understood that system of measure represented by the Metre des Archives as unit, as a definition independent of this does not as yet exist, and in the future establishment of such a definition, as close a junction as possible with this system will certainly be adopted. - 8 The great difference in the coéfficient of expansion of A 1878, found at Breteuil, and also con- firmed by the investigations communicated by you, from the value sent by us, must be attributed mainly to the circumstance that the assumed expansion as given by Borda for platinum is too small. In fact, according to Fizeau, the expansion of the platinum metre is 84.68 (+. 0+.0039 and, therefore, 8.80¢ in the vicinity of 15°. A small difference of expansion is also indicated between the Métre des Archives and the German original metre, as the comparison of these metres may also be represented by the equation Pt—A=14.144 04.004¢ or, if we consider only the comparisons by Brix, Pt—A=04.39+04.136¢ These comparisons, however, were made at only slightly different temperatures. From absolute expansion determinations which are being carried on here, we are in hopes of being able in a few months to contribute to a decision of the uncertainties yet existing. A union with the system derived from the toise, especially with Bessel’s copy of the toise, which has become so important in geodetic work, has been effected for your bar through two comparisons in 1878 and 1879 of 21878 and the Lenoir toise. The results were: ‘ 1. L=1.948 570 76 #1878 at 18°.255 2. D=1.948 575 53 R1878 at 15°.74 Combining the two equations, that is, reducing them by interpolation according to tempera. ture to the standard temperature, we have L=1.948 574 57 R1S7S at 169.25 107 LS 850 APPENDIX I. LArr. 1, The Lenoir toise, as repaired (amgearbeited) in 1852, by Baumann, was compared with the Bessel toise in the years 1852 and 1872, and on our Repsold comparator in the years 1877 and 1873. The results of these comparisons are: Mt Ws 1 L—-B= —0.0048 2. L—B= —0.0066 3. L—B= —0.00372 + 0.00013 4. L—B= —0.00336+ 0.00022 As the best result, the mean of the last two comparisons may be taken; hence L-B= —0' 00354 The Bessel toise is generally taken, according to Bessel, as 0/’.00080 shorter than “the toise.” According to its own certificate it would be 0/.00078 too short. From a comparison made by Bessel with two other certified toises it was found to be 0//7.00333 too short and-0/".00411 too Jong. In the mean froin the three certificates, its complete identity with the toise of Peru would result. If we adhere to the equation always used B=1"—0/".00080 at 169.25 then there follows L=1"*— 0.00484 at 169.25 and, in combination with the equation previously derived, L=1.948 574 57 R1S78 at 169.25 fiom which 443.296 RKA878= - aGq 1-000 236 96 ZL at 169.25 whence follows FR 1878 =443".296+2314,94 at 16°25 or, with the earlier-found coéfficient of expansion RAST3= 443. 296-+65 41+104.248 ¢ and, consequently BR AS8T6=443/",2964864.774+104.324 ¢ differing by —44.2+0+.014¢ from the preliminary result of June 20, 1879, which is RAST6=443,296+2454.6410.31 (t—15) With the Paris coéfficient of expansion we would have had RASTCO= 443". 2964 2404.14 104.55 (t—15) The more important measurements for the determination of the graduation errors of #1876 are also based preferably on the comparisons with R1878. We have therefore also given in Appendix B the measurements and computations for the determination of the graduation errors of the last-named bar in more detail. The errors of graduation have all been examined on a Repsold graduating machine, which was so arranged that two micrometer-microscopes rigidly connected with each other could be moved in the direction of the length of a bar, while each of the two micro- scopes points at the same bar, or a single microscope points at one of two bars lying side by side. The first arrangement was adopted in the investigation of the errors of the decimeter marks of #1878 by making the distances between the microscopes respectively 5, 2, and 1 decimeters, and then comparing the intervals on the bar with these fixed distances. The results of the observa- tions and the process of computation are given in full in the appendices, except that the micrometer readings are first corrected for periodic inequality of screws. For the complete understanding of the subject, it only remains to be stated that for both microscopes a greater reading corresponded to a smaller value of the interval at which the microscopes were placed, and that the unit, =1’, in which all the readings are given, and the computation carried out, corresponds to the thousandth part of a revolution of the micrometer-screws, and may be placed equal to 0+.05. §2.] ADDITIONAL DATA RELATIVE TO £1876. 831 - Objections may possibly be raised against the mauner in which the definitive errors of the deci- meter marks have been derived from the three series of observations. A rigid adjustment accord- ing to the method of least squares, which was made in addition, led to the following only slightly differing results: Negative error— Decimeter mark. | = By approximate | By rigid computa- computation. — | tiou. Dp. Ps 0 0 ; 0 1 +12 412 2 +410 4-10 3 + 5 | -- 4 4 +27 | +25 5 +9 +10 6 —25 —23 aU —26 —25 = 8 —23 : —23 9 aA 33 ' 10 0 | 0 These differences, which in only a few cases amounted to 0#.1, were neglected. Another circumstance must be taken into account. In the second rigorous computation it has been assumed that the errors of measurement consist only of accidental errors of pointing, and are independent of the length of the measured interval and of the tine of pointing. This is uot strictly correct, since from temperature variations during measurement, changes may occur in the difference between the distance of the microscope-axes and the corresponding interval of the bar. It is easy to take account of these changes if they occur uniformly between the single readings, for if the relative lengthening of the bar between any two measurements equals y, and the number of gradua- tion intervals, which are directly observed, is n, then there must still be applied to the error of graduation which closes the ith interval, derived in the ordinary way, the further correction a (n—t) y Zn + These regular changes were completely eliminated from the measurements in question by com- bining directly with each series of measurements a second immediately following, in which the pointings were made in the reverse order. The errors yet remaining, which result from the fact that the supposition of a regular change of distance of microscopes aud of the length of the bar is not strictly fulfilled, may be combined with the errors of pointing as accidental errors, and so the method of computation chosen is justified. The centimeter marks on #1878 were determined in a way quite similar to that of the decime- ter marks, except that auxiliary intervals of 5, 2, and 1 in length were used instead of the con- stant distance between the microscopes. ‘The same remarks therefore apply to the computation as to that of the decimeter. The results given in Appendix C for the determination of the graduation errors of #1876 by comparison of the decimeter and centimeter marks with #1878, of the millimeter marks with an absolutely determined silver scale on brass bar No. 6, and of the tenths of a millimeter by di- rect micrometric comparison, will be understood without further explanation, The unit here is 1*=0™",001. The differences of these results, in some instances, from those communicated to you on April 16, 1879, arise from the more accurate determination of the comparison bars. * * * * * * * * * Kaiserliche Normal-Aichung-Kommission. FOERSTER. APPENDIX I. (App. I, oe) CH te APPENDIX A. Determination of the total length and expansion of steel metre R1876. a. RESULTS OF THE COMPARISON OF STEEL METRE R1876 WITIL BRASS METRE 1605. | Datiecot | | | R — 1605 at 15° | compari- t 1605 te R — 1605 | 18.25 (¢ 16u5—t x) | 7.921 (¢ 1605—15) |—- ---- a | sun. | lL. II. e | 1876. | ° ° B BK Bb Bw we ; May & 12:28 12; 51 +57.7 - 4.2 —21.55 +32.0 +36. 2 8 | 12526 12. 51 + 66. 6 —4.56 —21.70 +40. 3 +44.9 | 9) 11.36 11.39 +56. 0 —0. 55 —28. 83 4-26. 6 +27. 2 | July 18 18.72 18, 06 — 4.2 —4. 38 +29. 47 +20.9 +25. 3 18 18, 82 19. 06 — 7.3 —4. 38 +30. 26 +18.6 +23. 0 19 | = 18.80 19.10 — 1.8 —5. 47 +30. 10 +22. 8 428.3 19 18. 90 19.15 Td —4, 56 +30. 89 +19.1 +23.7 28 19. 59 19.71 --16.3 aae19 +36. 36 417.9 : +20.1 28 19. 61 19.77 116 , —2. 92 + 36, 52 +22.0 +24.9 5 28 |" 19.69 19. 87 —14.5 | —3. 28 +87. 15 +19.4 + 22.7 zs) 19.69 19. 91 —15.3 : —4.01 +37. 15 -+17.8 +21.9 ; Sept. 4 | 17. 63 17. 85 —0.8 | —4. 01 +20. 83 +16. 0 4-20. 0 4 17.73 17. 92 + 2.7 | —3. 65 : +21. 63 +20.7 +24. 3 4 1:72 17. 96 — 0.7 —4.56 +21. 47 +16. 2 +20. 8 4 | 17.77 17. 96 — 0.5 | —3. 46 | +21, 94 4-18. 0 +21.4 The bars lie each on two rollers, which are distant 0.23 of the bar-lengths from the ends. A. thermometer lies on each of the bars with its metal scale in contact. b. COMPARISON OF THE REPSOLD STEEL METRES R1876 AND 2.1878. tr t7s8 Ri76— R78 oO °o fee Tete’ sccs ese s sauce 8. 592 8. 608 +22.7 8. 596 8.607 20.9 1 8. 605 8.619 21,9 G14 8, 628 21.4 Means 2.2... wh 8. 602 8. 615 21.7340. 2 TT SCVICR verses esi 15. 740 15. 758 +23. 2 15. 749 15. 760 22.0 ° 15. 758 15. 766 23.3 15. 757 15.775 22.4 Means... ....-- 15.751 15. 765 22.73 40.2 TU] S6riesi. exec 2sexse 21.489 21.487 22.1 ! 21, 489 21. 486 22.9 } | 21.490 21. 486 227 | 21.487 21, 487 23.6 | Means .......-.- | 21.489 21. 487 22, $340.2 On each of the bars lie two Fuess thermometers, whose cups are inserted in copper blocks filled with mereury, and, by means of the latter, lie on the bars. Tach observation consists of a double pointing to the marks on the bars and a double reading of the thermometers. Berlin, August 10, 1881. Director of the Kaiserliche Normal Aichungs-Kommission : [L. 8.] FOERSTER. 853 ADDITIONAL DATA RELATIVE TO R 1876. § 2.) 0 | 0 0 0 0 0 0 & 6F ae | 09 +9 | 00r Beh at ae 8cF sep LGR SG «6 eR I+ «| OG F- | FORD — | Geb L — : ee 8e— | 4I-— | ze— | 92—- | 9r—- |oI—- T+ Juz 0 1g ze i g 08 | ep 868 TOF 91% slp =| GIR Bt | GEE T— | HELt | HL —| 1h 6 — ce 8— | HE— | O3— | os— | os— | sI— | oz— |e T— |#t- | 02 4a | - sai ; 09 | oce roe 68E ale eos =| ere Lk 798 0 cee F e98 OT+ | gue 9 — oo+ Tet | + | cot | zet+ | OI— | est |! og oP 09 1 Tg | | di ¢ oF 0GF 63h Ser €lP 9eF Gb G— =| Tee 8 sth P— | GIF 9 — | BEF Ee — eb ee) ee | Ge 9It | II- | 3+ flor 91 & T ST | | 03 | | | oF oF Gar 10F Slt || OF 0 elp e+ | LebS— | 96 ¢ + | cop T+ 0 jo |o 0 0 0 0 0 0 0 0 | 0 | | 0 (—— : | UvolT [R1IMe4) | wn vad 6q 8q “qd 9d Fe) | 69 te) | 49 | 89 og 6g 8g 4g | °g | oy | oy | sy | LV | 9— “Up ‘OOT JO SUIPROI Jo SIOII9 U1e1N0D sol1as oT) JO STOTFVAIASGO O44 Se] OT} fpaqoafor st g SoTIEg | | 0 0 88 FL 6eI | y, | 00t hp cor 9IF coh I) | eh b+ SIb T+ | oze g— one oI- ott o 6F elt | 22 | os | || itp 68 Tep ‘ tlpe + | 06g #+ eeh o— seg L— ga t+ ¢+ 0g cg ag | | | BE | 09 | | | 698 Le 968 | e3— | ge I— 668 €— OF B+ coe ot tet | 19 $8 98 | i | “eS OF ‘| FBF StF Tob 98h Z — | GSB AL 19h 9— | FOB S— ee ret I- 1 Le 68 & i | : 5B | 0 Ler 6e% car Teh b— | Lp g— StF 0 | O8F — ZB 0 0 | 0 0 0 | i 0 7 1 | | | | | sq | tq | tq | tq | 99 | tg | &9 | ay | sq | re tg | tg 8V | yw | ty | ow Ww mie) “SMUVN-UALGNIONG NAA AH], “¢ #'G+ :uvoeul [vionex) a ot+ Sot got e+ Set 6+ or+ gt SSE HI RSS AS Sete BEES YAVUL-O]PPLU Jo JOIIe eaTyefor eayVSON “q x 00T 868 006 688 688 288 988 268 £18 318 898 468 I+ | cog st | sss I+ | ess 0 Iss 9+ | 928 01+ | 2680 | g98g9 +) B18 L-— | FB 9— 0 _ £06 906 668 F68 006 68 106 168 68 £8 806 S~ | L066 I~ | £06 9— | 806 6—- | CoG S— | 8689 — | ces 9+ | 106 OI— | ces e— | 06g 9— E 1 0 [eae 7 | a a a | a a a a a a a Vv Vv Vv Vv Vv ¥ ay Vv y¥ osfoy»+ wD ‘WAVIN-ATqdy AW, ‘Dv ‘SUALANIONUd AHL—I "SLST Y atzau 2978 fo ssouda-volmnpud ayy fo vor vurnuazaqy “ad XIGNGddV [Apr. I, APPENDIX I. B54 ! ete | Le soe 908 00% | 1a £08 008 a1 LZ 96 b+ | gce6 — | 90ET—~- | 6osE— | 9619 + | GOTT + | BIZSI— | GOVE + | IEE + oh cee le oie Tze 81% | 6cz LBZ Lise 11Z Tes le2 €I— | FEG— | oleg + (ooas— Es— | LTS OI+ | BBG — | LIZ9-- | T1ZOI+ | og 80g Tos Lee 66T 91 L1Z FIZ 11% ger 11z set + | Loeg— | ser — | sere + jsiest jose + | TIG0 wOZL— | SIZL— |) Gog ale soe £08 808 98T To 10 908 11a ros b+. coe, + | ote TI— . g02g— | Este + | ToZ0 102 0 GOT + | FIZE— | og £18 | BB 92% TAT 89T 891 891 ¥9T eLT 991 €11¢ — 1836 — | GLEE FTL + | Sts + | UWL P— | esT6 + | GOTH + | GIT E— | o¢ soe e0e 11€ cor 86 cer 181. T6T T61 00% : SOLE + 66S E+ | CGS TIT | BT 9 + | 29 — | I F— | 00% 6— | CBT 9+ | LET E+ | oF ore | RE | fee 88% ge eB £83 Liz ee | goa eo 8 — Lee — | oeee— GFE TT— | GEES — | FEET— | TIZ9 + | LIZO + | WSZI—- | Oo ce | 862 aie 0% sie 66T S61 0 Ser OFZ gIeT— eoes~ |Ise6+ ‘se8— | Tes— | HEE— | B\0BE~ | BL OT+ | GEFE+ | og te | «GTE ore ge Gee aes Ge LIZ gT@ 8% — | eet — , peg — | Loe e + | eee 8 — | Gz 0 1€2.9 — | 28 0I— | ties + | 8280 a Tee ore Tee abs Wee | 68 SI ers 6cG ote Wes— HEP | TeeO Fees + | OT — | ozs — | gee L + | 1868 — | 9ZT— | 4 | | we eee need | | Ee “g sg 4g sq | aeL | rE | ®q *q ta sy | ov yo 4 ! oy | Ue | ty “Vv | tw mp ING, “FT OF TT Sates JO URIUT OT} 0} F IYSIBA 9} ‘OT 0} g S9TI9S Jo UBATH BT} 0} NBATS ST G \ J ONTVA [VUTTIII} ATVIJIQIV YALA SYAVM zojouroap- “SMAVIN-YALANIONG e AIDNIG GH] “9 ‘sonTea UIT “JT ‘TONVNPBIS JO SIOIIO DATIVTAI VANVSIN “GC G VY} JO SIOIIO VONYRNPBIS OANVSON “CO ‘ITO TOR WoOIF TT pure JT SoTOdSOIOIU JO SOX¥ VY} JO VOULISIP JULISUOD JY} YA UOSTIVATIOD UL S[VATA]TI PoINsvoul IT} JO saotareyIp ‘st gvTy ‘owes oq} Jo UNG ‘g ‘SdCOOSOIOIM OA OY} JO SOUIPBAI PI}DIIION “VW ‘I[QVIPIITN BIB ‘IDAIISQO 94} JO 9jJOU ¥V 0} SUIPIOIOV ‘s}UATMAINSVATT IT] osnvdEq ‘Ee 07 E SOTIBS JO TVAM 9} 07 T IBSIoM oy ATTO JUSIOA IY] UBS [RIOTS VT} JO UONLMIOJ 94} UT 0 0 0 98 | | | 00T cer gee gee) cep T+ | rept | cep 6 — | Tor E— so— w— | te | as : | og Fob 968 9b | Sab Gt | Leb E— | OI— | Soh T+ | Lo—- ¥— =| 0c— || ! ei 09 | &LE gor | £98 69g ¢+ | ore 3— ie 698 0 ett ue Pe. eer 8h OF ‘ | ag Ob FER) EF Te | 8h | Ltt | Ber P #20 oI-— | 9I+ |i se 02 | LOF 68 REF | BERL | SOP 1 | OOF TI— | Gab B+ 0 0 0 0 | | | 0 | | Ww gq | ng | rd elgg ag | ug | ny v | ev ¥ | ay ny | artt@) “ponuljyuoyj—suuVN-YaALANIOAd NaAGQ any “panuryjuog—ersel ay adgaut jaais fo s.10...0a-voyonpHnas ay? fO WOloULUsIIag &55 ADDITIONAL DATA RELATIVE TO R 1876. § 2] ‘ga1]OM1 J[eY oq} JO TostAVdMos oN MOI SUNTMSaI YVY} PUL Jd}aLMLDep oq} JO SMOSTIVATOD JO Sa]1I8 9} ULOIF POTTBIQO on[vA oY} UaaANjoq GoMAIATIp payVur oy} Suttepisuos opqeyysnl Larue sures Forza ‘G— qnoqe Jo WoHoat1oo oqIANF [LYS SIY} WorF Seryooor WOH} -VNPRIS BIPPLU ST, ‘SUONVNPVss-1aJaMILDop MAAS oq} JO UOTDeIIOD Aq, SALas oY} WoIF ATJOOITP PUNOF OLIV OSG} JO SIOLLO aATBVSeU OY, ‘sUuoUMNPDAD sajaUdep Ppo AUT “ qe+er eL+oe ‘ a ‘ Ca Fs ea 109 CO Fr ‘todd O.A1}VSOU 9G} ‘OS aan ei oe oe sae =0e ++ WONNpRIS VY} OJ $}]NSEI 910q} VOUT ct=— 08 Tt : = fL70¢ __ eorz9 as yeSou oyy ‘ Let=Oexoettexc: ‘OF OI+= exer toaXoe H Th 06 ‘EL=51 x OT ‘09 pUv OF SUOTJENPLIS OT} roy {ZL=EXOT IYS10a oT ‘Og pue OZ SMONLNpLAS oT Joy ‘ASed snoracrd v UO psoupep ‘s1OLIE ANVSOU OY} OF ‘puvy JeqJO oy} UO ‘pouTezqo QIv S19] ITMLVAp-e[qnop oY} JO S]MOMIAINSvOM Jo SeT1as JOoMIp oy} MOI “F=EXOT ‘09 PUL OF SUONVUpLIS oy) Jo esoqy $SE=51 X QT TYSIOM 9} GALT YS PUB OG SUOYNPLAS oY} Jo s1OIsE DAVSOU Suyuser oy} (JL) ONIVA SIYI LOF SUOTZVUPLIS TajJoMMIOIp 94} JO WOLoeI100 IIFV “SUOLMNPDAD AaZaULIAP UIA AUT, °G ay a ee Oe ign, i ° Tt xXEtPeXor oq [TMA TON -BUpPRIS a[PPUL OT} JO JOIII BANIVTEU OT} BOUDA, ‘SIIJOMLp 9[SUIS 94} JO SUOSTIVATIOD U9} JO SoTIOS OT} JO ULOT OT} OF WAALS 9q YsnU G=; qqSiaa oy) uy} ‘eased Sutpaosad v uo punoy ‘1eT}0 Pos YIM sorjOUT J[VY OT} JO SMOSIAVAMOD UA} JO Sal1ES OT} OF UWOALS ST OT JUSTIA 94} JT ‘WON PVA ayppiwUt AUT *T “INANLSALay 0 0 0 0 0 0 0 0 0 0 0 0 9 6S : ToL 9BE 6 601 $6 99 9L os 008 00 ro Pe et | Ga | Sab | ee | an | ese eae oP esh be e ee ip Ope, EOL Ter 88 60T 8 49 OL ge es 06 ee— | oz— a st—- | s— | r++ | 8-— | e+ ww | su | SIS | 8 | ee IIL 68 SOT 19 16 eg 98 OF 1B cor 03 92—- | se= b= s— | e+ | ot- | e+ | T— | Ge |} F- | SI~ | ET E+ got 88 8L 89 eo sé c 8 ae ISL OL se— co t— | et }]e-]}] 6+] e+ | ¢— | ot | 6~- | &— | 08+ #6 OL &L ¢9 19 ag 13 1g 9% OFT 09 6+ Trt | Tet Tet fet pet Let Ppt Tet 06+ eet gor ght Tol FOL 16 $6 66 68 g¢ SL &¢ BLT 0s Let Te+ got ag egt+ e+ so+ ig+ 9g+ ort (At set ot 9IT TOL 98 Zot TOL $6 99 8 9 PEL OF gt+ art | og+ got Tot 7 og+ ee+ Te+ cir abt fet 6gt+ 9L ug €¢ i) 99 09 es ¢9 68 6IT 0g ort ost |i cet sot set 9tt+ ge+ 63+ apt Lot opt pet eet gg 6¢ IF Lg 1G 19 OF 09 TF eh 03 ait || att | ect | ott | ect | stt | eet | att | ozt | 8+ | set | tet | cat 1g OF Tg oF 6% 6E St &F 62 op 91 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vA Ww | nw 1g ‘q «L ‘a oq 1g ’a *¢ “a ‘a ny 89 89 9 99 89 D 89 eo) 9 wo APPENDIX TI. (App. I, 856 Fo7 SE ae | o | | a6 99 #9 Lor | gor | e6 I- | 99 0 e9 I+ | Loto 96 ett i Bt Ms 9+ 3 § 88 6F T9 fol FOL #6 9 Fo ¢— co F— Ter @ + SIT 8 ~t | L6G Tf LL 90T TOT 96 3— 0¢ 6— 69 8+ SIL Gi— TOT 0 g— 09 c6 OL FEL @ol 6— 99 9— 96 I— 60T L — GFL 8 mk 66 ge 16 60L #OL 900 L- 6g I— €6 67 gol t + esl o + teem ; gett | err | pir | ger | tte- | on r— | gors+ | ect | wettt ! | 2 LIT 18 FOL SIT G06 SIL I— 08 I+ LOT €— Lot 6 — 68T weet ; lo- oe TIT GOL GIT Tle Lot 2+ SIT £- cot €— TIT Tt + L0G F + i | | FIT TIL FIL O03T 861 TIT e+ TIT 0 crt I- Tt6 + L6T T at 0 | | 66 16 96 gor | est || zors— | oore— | 26 1- | 66 2+ | ostrg + h uo 1 i | eq | lose | re | eg | cg | og vo oF y ty ly or oO © ~~ “SNOILVOAGVYUY UALAWNIINYD NEAW TAL “@ , “YW 02 "V 28 SsuOltsod Iraq] UOIZ SaMOOSOIOLWML SULIMSVATT BY} 0} ADUITIJOI TILA SIVq OT} JO WSUVYOIIJUL Lay paInsvaul 919M “W 0 SY— Ng ‘TOF Jo119 Uvaut oy} TILA ‘o-y— ‘TvaNT oy? SV pauTE}qoO sey phon 9a Uaqy ‘pepupur usaq pey sonjea paroofer om JL “FI IOLA ULI OT} YQIAL ‘c— ‘eT SULTIVML OY JO WRaUT aT] Sv pauIEIqo st o10q} WAY ‘poyoofor or¥ "OQ puv sM soNTVA JUBPIODSIP AIDA ON) JT i e+ lee —!| (ete) ¢— Tsk | e-or— seI— 38 est— be | co | (g'Fe—-)| 8 | 6 OI+ | og + IG. — ek= | Srl | co —-l'a : | j | or 9g— 9g— or) ee | IS 8o—: 8&— | 86—| GF- | se NG os 0&=| “8 | 8h= | sk— Lo— |} 0@—- | LI- Ge— | 6e— : os 69— ‘6e— | 99— | GF | 19— | 8S— | ¢ch—| 69- | Fe Tr 66—| 6F—- | GS— |} 86— 9¢—- | S8—} Sf 1s 0E— é dae | | | | | | 0D Bi) gly | alg oly stg | rly alg t ay | uy) | op 69 8) | 49 99 99 ry Bg | wy | 9 my l eed 1h Spat Wee : pe ag 866 £0€ £06 I6L £06 | L6T 66T LST PEE 188 sog L — Toe 2 + Z0o I+ €8T 8+ 261 9+ 906 6— G0G €— 961 6 — Teg ¢ + gee ¢+ or ; OF LEG 60G ect est | 69T 99T 6¢T GHG 64g | 286g + | Leo 01— 10s 8+ Got €+ Tet t+ TLT 6— TLL G¢— col € — 062 3 + 918 e+ ; 2 | 66 Foc S21 Bel Ost | oct O9T $96 L386 PLE 286 I+ | 6F2G oe 960 8— OSE O+ ree 6FT T+ 6cT [+ ees I+ 866 TI— TL3 6+ L6T est 6EL 8 TOL | 68 col 616 8Ie 066 sere + | 681 # — eet F+ $L 8+ Ort 6— 46 8— $6 Bt eet L6G 6 — 062 0 0 (Aes | ogy ogg | aig | ag | sg sg | ng | ig | ag ug oy | sy sly Ly oy ay boy sw ay uy ‘uIg FEE 0GE Lot LIT 681 | gel cot O9T cel LIT | FEE 0 61g I+ ger I+ LIT 6 eet 9+ 8cI 0 o9t¢ + PC t= cel §€— sit I— or FOS 066 O9T cot L8T Bt cet LLT el OFT |, FbG 0 U6G 0 FST 9+ FOL T+ G8T G+ LOT GI— SLT GI+ Ost €— T9T L— 6FI €— } S £96 966 806 89T S6T 961 gL LOT cel 698 89g ¢— 666 6+ 01s G— 8LT ¢— 96T I— cal & + fl + 69T G— grr 2+ Foe ot G66 L6T Loo 066 66 GLT SIT Os 906 66g |) Tos T+ 961 T+ 696 G— 0c6 0 ees T+ sort + SsIlc — Tos 6+ GOs Ft 968 ¢+ 0 | panes : oes i i | cacti a is KS | al) ig a | 6v 8Vv Ly ov ay ty ry aw a ‘ug ‘SNOLLVAGVUY) UALANILNAO-AALY AU “D ‘pInUyUOO—RIsl gy agau 19978 9y2 fO $10.49 U ‘SYALAWILINGAD AHL— IL ODUPDAD YZ LO WONDMUUAIIAT 857 ADDITIONAL DATA RELATIVE TO R1876. §2] ‘TAOWLaP IY} UITIIA SUOTUPLAS OT} JO SIOLIA AIPV[IL VAY}VSIN ‘TVAIOUT LIVIPIXNL aT]} 0} STVAIO}UL O[SAIS VY} JO SIOIIA VAT}VSIT VATPLOY. ‘SSUIpvor [OQ Jo ung "SMOIOS OY} JO SOLID OTpOLIad IOF pa}oat109 Sadooso1dIM Y}Oq JO SSuIpVay 6+ ort 9I—- LI— 8I— FI— 8I— 6r— 0 0 0 0 Qh) Peeaeei es ga rete | FS isk 89 SL 9 + | 8th I— oss # + | OT 69— 19— co | 4 it §| sz9 gue oes 6999 + | cse6— | OFT - 4 ort ait ort L ict [eer aes as tee (| aso 8ge ges 199 — | Tee e- | Fog — ge— | um | te- | é Teo Lie cer mee | Bee | Ee att t+ g get fete |p Safeses cs |e eee | et9 608 LOF ol9e + | Tlez— | Fer ett oz— o2— 93— £68 682 Tp ses ¢ — | 96GL— | SLE —Q eis Le so— 6- ise oes cree vrier4 $| 4 26s gis oLF Ls 0 egg s— | 08h T — 19— so— se— | | ees sig #6 eect + | GIg9— | oro + 6— ao ti oor Brey a ere ee ae 7 9 i| ots 62s ap sig L— | oot | 6ere + Lg— 8g— er | 93h TLp G6e SIL + | FLEE | OOF T — g— a+ r- L+ 6+ , | PeF POF FOF steT— | sopot | ween t 0b— e— so—- | | FOF ofr 6Le Sp II— | ogh 6 gue tT + LI- or— #I— tI- 25 ae * #68 oF L¥E Soh TI— | Sche—- | soeg — og— 8g— Te | PFE 898 oze Teg — | 01863- | o2e0 91— Ti $I— aS iets ip eeeketer ete ees ae e 2% gue Bie ere FI+ | olest+ | siege + e9— 8g— 8e— | 6e Lie 082 sez + | fles+ | 9829 — 6— g— t- P- Or— freer cece et fer eee Siaie palsdiaeeiere | z Ls rie 588 6468 — | stlett+ | os + os— Is— oF— Kota £90 Fre 9826 — | 993e—- | ossa — 9 — pe ae 9- ey) Parse are a see sek ener T Fes 998 GPa 925 T— | Tkag— | Pee + 1g— 09— ah 0 0 0 0 Oi leaeeee ae eee “ 8At 908 102 srt + | 6ose— | ole— jo A oT eg +d veg 80 ty ty | tg ay 1g Sy ow iv “ap “‘SNOILVOAGVUY UALAWILNGO ATONIS GH, ‘9 a mao < 108 LS 898 APPENDIX I. (App. I, ADJUSTMENT. a. The 5™ graduation. If the weight 18 is given to the mean of the 18 comparisons actually used, of the two half de- cimeters with each other, then to the mean M, of the three series of comparisons of the single cen- timeters computed on the preceding page the weight 2 must be assigned. The negative error of the middle graduation of the first decimeter will then be —18x5+43x4_ Se he +f b. The remaining centimeter graduations. By correction of the means M, for this value of the 5° graduation, the errors M, of the even centimeter graduations obtain values which differ by only a few units from those of the direct series of comparisons of the double centimeters. If, therefore, the mean of both is taken without ref- erence to the weights, which do not come out very different, and the odd-centimeter graduations are corrected for this, no essential error will be committed. Director of the Kaiserliche Normal-Aichungs-Kommission. Berlin, August 10, 1881. [L. 8.] FOERSTER. APPENDIX C. DETERMINATION OF TILE GRADUATION ERRORS OF REPSOLD STEEL METRE, #1876. For the determination of the graduation errors of the decimeters they have been compared 20 times by double pointings of both micrometers with those on the bar belonging to the Commission, made by Repsold, &1S78. The results of the single series of measurements, freed from periodic errors of the screws and the constants of the initial and final pointings, are collected together in coluinns 1 to 20, and are expressed in thousandths of millimeters in the sense of errors in relation to the comparison-bar. Column 21 contains the mean of the columns 1 to 20, column 22 the relative errors of the comparison-bar, column 23 = 21 + 22 the absolute relative (ineren) errors of the Ameri- ean bar. For general comparison there is besides, given under 24, the result of an older but more imperfect series of comparisons with another bar, of which, however, the relative weight is so small 02 that it exerts no influence on the values of column 23. Deci- | | meter 1 2 3 4 5 6 7 8 9 10 11 12 mark. | | 0 0 0 0 | 0. 0 Ol v 0 0 0 0 0 0 1 | ~o1! 407) 406] 407] —02! 403] -o5! +07/ 405] 411) —1.1] 401 2 2.1 | =048 | =0-7 | <0 ore!) Soa) 13) S39 Sore) Sa7) Shi} Sie Bo | AO —0.1 | 40.1) 406° 0.0) —09} —1.5]) —0.7 0} +05) —24] —1.9 4) -1.8! 404; 404] 401, 00! 403] +02] +0.9 0} +09] —1.3| —0.7 5 “3 ~0.4) —04] ~o03] -o5/ ~o4/ —18] -1.4! -o7/ -o05] 1.8] —13 6) 8.40 ial Sh) he” be) 04) =a6) 18) 1a) 18) soe) 408 7) +32; —22 -27) -31, ~24] ~27] —29] —20/ -21] —27] —18] —1.3 go eg lee) 8) ang? eho) al) 8.4.) ns te) ee |) ee | 08 9. -06 -01- 408, -04 06; -07} —19!} —1.1] —08!} —07/ —1.0] —0.4 Ww! 0; 0 0 0 0 0 0 0 0 0 0 0 Deci-| | meter, 13 14 15 1 | #417 18 19 20 21 22 23 24 hark, | 0 0 0 | 0 0 0 0 0 0 0 0 0 0 1} —0.5) —0.5) —0.6) —0.3) 404] —05] 04, —0.6 0; —0.6| —0.6} —1.2 2) Bal a 3.5] —24' —20} —29] —33) —3.2]/] —19] —05] —24/) —20 S|) SL) Hah) 26) a6) te | aaa) —e.3:| 9184) aa | 08) 2e8:) 0:5 4 | —1.3]/ +401] —12 0 +0.5} —0.6} —1.0} —0.8 || —0.2} —13} 1.5} —1.0 5 | —L7) —07) —L1) —10) —11] —19;/ 11) —15]) -1.0]) —04] —14) —08 6 | —0.5) 401] —0.3) +407) 401] +04] +01] +02] —o07] +02] +05] —08 7 | —16} —1.6]) —17}] —0.8) —09}) —0.8} —1.4] -1.1]] —20] 41.3] —0.7] —20 8 | —10} —1.3/ —09] —04! —12] —06} —09] —03]/ —1.2] 411] —01] —o1 9 | 40.2} —L1); —09}) —0.2) 404) —0.3}) —0.3} —0.6]) —05] +02] —o03! —13 10 0 0 0 0 0 0 0 0 0 0 0 | ADDITIONAL DATA RELATIVE TO FR 1876. 859 For the determination of the graduation errors of the centimeters of the first decimeter they have also been compared 20 times by double pointings of both microscopes with those on the Repsold bar belonging to the Commission. The results are collected analogously to those for the deci- meter. Column 23 contains here the relative graduation errors of the centimeters with respect to the decimeter, column 24 the same with respect to the whole metre. bob . | 2 og I fg] 1 2 3 4 5 6 7 8 9 w , U Ii, aa | ; y 1 0 =, he aS, 3 0 0 0 0 0 0 0 0 0 oj 0 0 0 1 | 425) 425] 423] 41.5] 41.4] 423) 419! 423} 422) 432. 416] +30 a | 420) 4249] 401) +04) a7] 4n0] 409) 420] a0! sae! 490] 420 3 | +24) 429] +20] 424] 422) 423] 422] 423] 422] +23/ +18] 421 4) 444] 426) 41.8! 415) 41.7) 410] 414) 418). +19) 420) 413) 40.9 5 | 414/ 416) 426) 428] +20] 42.3] 42.2] +14] 413) 415] 413) +06 6 | 40.4) 40.7) +09} —02] 40.5] +06] +0.8 o} 40.5! 410) 412] 40.4 7 | 404) $10) 414] 404] 41.6] 408] +06] 403) 04] 412) 413] +10 8 | 02) +03] —03/ 40.8) 02) +08] 404) 05 > 04) 02) 40.7] 40.7 9 | 41.7; 422] 415] 422] +24) 422] 420] 41.8) 424] 422] 414] 412 10 0 0 0 0 0 0 C 0 0 0 0 0 - Wg a 3 au #3/ 13 14 15 16 7 18 19 20 21 22 23 24 aa : D oO 0 0 0 0 0 0 0 0 0 0 0 0 0 1] 424] 420] 425) 424] 426) 414] 420/ 4231] 422) 40.4) +26] 42.5 2h 425] 41.6] 422) +25) 424] 415} 424) 423] 41.7) 40.7] 424] 42.3 3 | 418) 41.2] 426) 422] 427] 425] 424, 423] 421] 409) 430] 42.8 4 | 414) 410] 412) 412) 419!) 408] 420) 419) 41.5] 408) 423) 42.1 5 | 413] 41.3!) 407] 410] 405] 405] 408) 418) 412) —02 424] 411 6 | +14! 41.9) 407) 405] 409] —04] 40.2) 41.8 || 40.7! 463] +10] +0.6 7 | 407) 4+2.2{ 409] 42.3] 41.3! 409] 413] 424] 410] +08] +18] +14 8 | 408) 41.5) 410) 400) 41.2] 40.7] 40.5) +416] 40.5) —0.5) 00} —05 9 | 424) 424] 427/ 425] 41.5) 41.7] 423) 429] 420) 0.5} 415] +10 10 0 0 0 0 0 0 0 0 0 0 o| 0.6 860 APPENDIX I. [App. 1, The millimeters have heen compared twice in groups of ten each with those on the principal standard No. 6. The following collection contains in columns 1 and 2 the results freed from the periodic error of the screw, the constants and graduation errors of the comparison-bar; column 3, the mean of 1 and 2, presents the relative graduation errors of the millimeters with respect to the single centimeter, column 4 with respect to the whole metre. | | i 4 i | | & ie 5 Bu Bu Daj Bs 1 2 3 4 BS 1 2 3 4 BS 1 2 3 4 =e ma Ba 2 } | eB 5 0 0 0 0 0 || 20 0 0 0) 42.3/] 40 0 0 o| 42.1 1] -1.1} —12° —1L2} —o9j/ 21] -o8] —o06} —o07} +216]/ 41] ~—05) -1.3] —o9] 411 2} 10] -1.0) -10) —05|| 22} —o1! -08| -04) 420] 42} 408] —o5] 402] 42.1 3} -14) —05| 10} -02] 23] 401} +01} 401) 425]} 43; ~o6] -17] —11] 40.7 4] —07}; -18. —12] —o2|) 24} —o9} —1.5] 12] 423]] 44] +012); —1.6] —08] +0.9 5|} +1.0 ht —0.3] 421.0]) 251] —og! —1.5] —11! 42.4] 45} -02] -08} —o05] 421.1 6; —1.5] -1.8/ —1.6! oof 26] +408! 05] 402! 428]] 46) 407/ -11) —02] 413 7| —1.2] —18/ 1.5) 403] 27] -03] —-1.0] ~07] 419]| 47| +01] -13] —06} +08 s| —o07] —1.0) —o09] 42.1]] 28] +02] -15] —07] 420]/ 48] +04] —25] —1.0] +03 o| —1.6| —2.0] —18| 404] 29] +05] —o1] 402] +29] 49| -1.2] -20! -16] —o4 10 0 0 o| +25) 30 0 0 o| 4+28]} 50 0 0 o| 414 10 0 0 o| 42.5 | 30 0 0 o| +281] 50 0 0 o| +11 i} —o06} 00] —03/ 422] 31) 418 o| +09] +36!) 51] —02] -05} —03] +08 wz] —1.1] —02}] —o6} 412 ') 32] +01] -o03] —o1] 425 |) 52] 42.2] 422] +12] 422 13/ -05| 05] —05 +2.0| 33/ —01| 401} 00] +26] 53) —01] —02] —o2] +08 iu} 401/ -13] —06] 4181 34] —o3/ -o6/ —o4) 42.1]) 54] —16] —1.9] -1.8] —o9 | 406) 411] 408/ 432] 35) 0.0] +07] 403) 427] 55] +06} 04) +01] 41.0 ie} +03! —08| —o2/ +22] 36! -15] —o01! —o8! 41.61) 56! 03! -12} —07] +01 17| 40.6] 414] +10/ 433) 37] 02) 406] 402] 4241) 57) -04] -11! —07! 401 is} +04] 40.9) +07] +80) 38) —0.6] -0.6] —06/ 416) 58) —0.1/ —1.2| —07] 0.0 19) 41.1] +09] 410) 43.3) 39] —11] -12/ —11] 421]) 59] -o4] +06! 40.1] 40.8 20 0 0 0, 423) 40 0 0 0 $2.1) 60 0 0 0| 40.6 | ge | 4 2 3 a | Ba] a 2 3 4 =a =e b= a 60 0 0 0| 40.6) 80 0 0 0| —05 et} 40.6! 41.7] 411] 42.8!) 81] 41.1! 401) +406) +03 e| o | 404] 402] 41.0]/ 82] 402! 404] +03] 401 63} —01]/ 40.5] +02] 41.0]) 83) +16; 41.1] 413] 412 64] 402) 40.7] +04] 41.3) 84) 403! 406] +04! 405 65) +25] +15] 420] 430] 85]/ +04 41.7] 411] 413 66] 404! 411] 408! 41.9; 86{ -07) 41.2] 403! 40.7 e7{/ 41.3] 41.7| 415) 426]| 87] —1a] 419} +04] 410 68} 411) 40.9] 410] 422]) 98) 413] 428] 421) 42.8 69| 41.4! 41.3] 414] 427!) 89] ~31] 01] —16] —07 70 0 0 o| 41.4 ]) 90 0 0 o| 410 70 0 0| 0] +14) 90) 0 0 o| +1.0 71) +16) +14 +413] 4273) of 14] -02] -08' 0 72) +10] 408) 40.9} 419) 92; —L6] —10] -13; —06 73) 410 40.9 410} +18] 93) oo! 408! 404) 40.9 74; 07] —02! —o05| +01! 94° 071 +02] —02| 402 io, 02! -06 04! 060) 95: 403] 415] 409) 411 76) 0.2) 404! 401} +03; 96 —0.2| 410) 404) 40.4 77-01} 402. 401] 401] 97; —01] 408] 404) 40.3 78 404 404 404) 402) 98 404) 41.5) 41.0) 40.7 79, 0.2) 404. 401) --03') 99 40.4) 41.2] 408] 40.4 80 | 0 | 0 0 —0.5 100” 0 0 o| -0.6 §3.] ADDITIONAL DATA RELATIVE TO £1876. 861 The tenths of millimeters of the first millimeter, besides the two additional tenths, were com- pared twice with an auxiliary interval in the field of view of a microscope. The results are con- tained in columns [and IT; I], the mean of I and LU, contains the relative graduation errors with respect to the whole millimeter, LV with respect to the metre. Millimeter a I IL UI IV —0.1 +0.7 —0.2 +0.2 +0. 4 0.0 ' 0 0 0 0 +0.1 —0.8 01 —0.4 —0.6 0.2 +0. 4 +0. 9 4-0. 6 +0.5 0.3 +0.9 +1. 6 +1. 2 +0. 9 0.4 +2.4 +2. 3 +2. 3 +2.0 0.5 —0.3 +0. 3 0 —0.5 0.6 +2.0 +40. 6 41.3 +0.7 0.7 +1.0 +0.5 +0. 7 +0.1 0.8 +0. 4 +0. 3 +0. 3 —0.3 0.9 —0.9 —0.6 —0.7 —1.6 1.0 0° 0 0 —0.9 Li 41.5 +1.8 7 +0.7 Director of the Kaiserliche Normal-Aichungs-Kommission: Berlin, August 10, 1881. [L. 8.] FOERSTER. § &. From Professor Foerster’s letter of August 10, 1881, it will be seen that he gives as at present the most probable value of #1876 HAST6=1"+864.18+104.654 ¢ where ¢ is the temperature in centigrade degrees. A more general value for #1876 may be obtained by substituting in the value given by Professor Foerster #1876 — R1878=214.364 04.48-+1(0+.076 + 0.29) the value of R1878, given by him and derived from the platinum-iridium metre, type I, of the In- ternational Metre Committee, namely, RASTS=1"+644,824+10+.371¢+0+,00697 There results B187T6=1"+864.18+ 10444714 0#.0069F t being in degrees centigrade. This expression gives to 21876, at 15° C., a length 1.6 less than that derived from Professor Foerster’s approximate formula above. The later expression involving #? should be more accurate, and, until the equation of the platinum-iridium metre, type I, is more accurately known in terms of the prototype metre yet to be adopted by the International Metre Committee, is the best that can be obtained. The Procés Verbaux of the International Committee of Weights and Measures of 1880, p. 103, states that the metre type I will probably differ by but one or two microns from the prototype yet to be adopted. The value for the expansion of #1876, derived from the adjustment in Chapter LX, § 58, is for centigrade degrees, at 15° C., 10#.572, while the new value given by the Eichungs Amt is 10+.654. Between 0° C. and 30° C. the greatest dif- ference of length of R1876, computed from 15° C., with one or the other expansion, would be 1.2. The letter of Professor Foerster makes some slight changes in the values for the errors of the subordinate graduations of R1876, previously furnished by him, and given in Chapter EX, § 67. For the 80"™ graduation, which is a very important one in determining the length of #1876 from Clarke Yard A, the correction is changed by 0.5, bringing it into closer agreement with the cor- rection found for this graduation in the Lake-Survey oftice. 862 APPENDIX I. (App. I, § 3, A vew adjustment of /,, £,, By, and Fy, Similar to that given in Chapter IX, § 57, can now be made by substituting the new value assigned by Professor Foerster in his letter August 10, 1881, for the expansion of #1876. Differentiating the value RLA1ST6=1"+864,18+ 10", 447¢4-04.0069P there results 10+.447+0+,0138¢ as the rate of expansion, and at 62° IF, (=169.67 C.), which is the temperature adopted in Chapter EX, § 57, it becomes 10#.677 for 1° C., or for 1° Fi, By iy =5*.932. The weight of this value is unknown. It is undoubtedly great, and on the scale of weights used in Chapter IX, § 57, a weight of 16 will be assigned to it. The equations of condition of Chapter IX, § 57, for a temperature of 62° F'., may now be written with a column added for their residuals. w M Weights. E, — 39.945 =r, = — 0.032 1 Boreas — 5.932=7,=—0.000 16 E, — 5.371 =0,= +0.034 A Epsae— 1.0937E,— 0.006=0,= + 0.015 8 Ah 0, + 1.335=v3;=—0.012 3 By sy —14.833=v5—= +0.005 6 The following are the values resulting from solving these equations by least squares. The residuals result from substituting these values in the equations of condition: Mw Mw E,, =39.913 4 0.033 E, = 5.932-40.008 E, = 5.40540.013 E,=25.075 40.033 These values are the best that can be obtained at present. The residuals in the preceding are very small, except in the case of #,, where it is 347 part of the expansion, a quantity that is not large. The length of Clarke Yard .4 in terms of the Ordnance-Survey standard, Y;;, is best known at the mean temperature of their comparisons, which was 57°.71 F. A change of 747 in the ex- pansion of Clarke Yard .1 would only change its length, as assigned by Colonel Clarke in Chapter TI, § 2, by scotooo at a distance of 25° F. from the mean temperature. The change in the relative expansion of #1876 and Clarke Yard A, in passing from the adjustment of Chapter LX, § 57, to this new adjustment, is but 0+.016, a quantity so small that it practically leaves the relation between these standards, given in Chapter IX, § 12, unchanged. In April, 1882, the metre £1876 was sent to the International Bureau of Weights and Meas. ures, at Sevres, France, for comparison with the standards of that Bureau. App. IT, § 1.] EFFECT OF HEATING AND COOLING ZINC BAR Z,. 863 APPENDIX II. SLOW RETURN OF ZINC BAR Z, TO ITS ORIGINAL LENGTH, AFTER BEING HEATED. By E. S. WuEELer, Assistant Engineer. § 1. A series of comparisons of §,, Z,, S:,, and Z, was made from Angust 26 to December 15, 1881, for the especial purpose of finding the time required for Z, to return to its normal length after it had been largely disturbed by a rapid change in temperature. These comparisons were made in the comparing-room, which was visited but once a day. . The thermometers used. were Geissler 1, 2, 3, and 4, and Casella 21474 and 21476. The micro- scopes were Repsold Nos. 5 end 6. The observations were made and reduced in the same manner as previous tube-comparisons, described in § 21, Chapter IX. The old graduations on the bars were used throughout, but from time to time observations were made on the neutral-axis graduations to ascertain if there was any bending of the bars. None was observed. The difference in the length of a zinc bar at different times though at the same temperature, as described in Chapter IX, § 33, will be called the set. Theset of Z, and of Z at the time of each observation has been computed in the following manner: The differences at the observed temper- ature between Z, and 8), Z, and &, have been computed with the constants given in Chapter IX, §§ 24, 26. These computed differences were then subtracted from the observed differences, and the remainders taken as values of the set. The set is, therefore, called plus when the zinc bar is too long, and vice versa. The outline of the work is as follows: Tubes 1 and 2 were first compared for ten days, then tube 1 was taken to a warm room and its temperature kept between 115° F. and 120° F. for twenty-four hours. It was then returned to the éom paring-room and allowed to cool for twenty- four hours, when observations were resumed and continued for one hundred days. This heating and cooling left Z, with a set of +734; thatis, Z, was 73+ longer than its temperature and expansion indicated. The subsequent observations have been studied to ascertain at what rate this excess of 73" would disappear provided no new disturbances occurred. The results of the observations are given in the following table: The first column contains the date, the second the temperature, the third and fourth the sets of Z and Z,, respectively, the fifth ordinates of the logarithmie curve defined in § 2 following, with date for abscissa, and the sixth residuals obtained by subtracting the fifth column from the fourth. Set. Ordinates of log- tile | He eee! Za, A. abscissa. 1881. oF. w B Aug. 26 68.5 +1 WB: Vac oue ene ads canes aceumaemneat 27 68.9 + 3 SENT Wsawinneresedaccces ts |Amadicetauisis 29 70. 4 —1 SENG. coors trsrecincisinjsceisiarnies dee] digeaiecaie’ See 31 73.2 — 6 +13 Shara erdpaz is mot ensis ciniaie lhe aise REIT Sept. 2 75.3 —10 GH Get etvene athe galeeeonoaa ts 5 75.0 —7 410 | fcc cemeaceeeeesiee| oveuzeeade se Tube 1 now heated to 115° F, for 24 hours. Sept. 7 77.2 — 8 +73 66 +7 76. 6 —9 +72 66 +6 9 76.3 —12 + 67 65 +2 864 APPENDIX II. (Arr. I, | 4 Set. , Ordinates of log- Bite: a uae ' Z2, Zi. abscissa. 181. OF; Bw B&B Sept. 10 75.2 co's +68 65 a 8 12 92.5 0 + 65 64 +1 14 69. 6 +6 +69 63 + 6 16 68.5 +6 +63 62 4s 4, 19 68.1 + 6 4-59 61 —2 21 67.6 7 +58 60 = 24 69.0 +3 +54 59 —5 27 GL5 —3 +50 58 —8 30 71.6 —2 +50 57 —7 Oct. 3 71.0 —1 +52 55 3 4 70.8 +1 +50 55 = 8 63. 6 +12 +56 54 +2 12 61.4 417 +53 52 +1 17 61.3 +11 +48 50 —2 20 60.3 +12 452 49 +3 24 58.2 +14 +47 47 0 28 57.5 +13 +44 46 —2 Nov. 1 59.3 +9 443 45 a4 5 55.7 +12 +50 44 +6 9 55,2 +13 444 43 41 10 556) ewwinatns a Aon +42 42 0 11 Bad. ileseccnste sian +47 42 +5 15 51.3 +17 +52 41 411 19 50.7 +7 441 40 +1 23 47.0 +17 +46 39 +7 28 44.0 +14. +42 37 +5 Dec. 1 47.2 +11 +28 37 —9 2} 48.3 +6 +30 36 —6 3 48.5 +4 +30 36 —6 10 46.0 +8 437 34 +3 15 46.6 +7 +32 33 —1 ¥ § 2. The heating of tube 1 oceurred between September 5 and 7. The table shows that tube 1, after being heated to 120° F., fell, in twenty-four hours, to 77° F., and that, Z, had a set of +104 before heating and +73 after cooling. Therefore, a fall of 43° F. in twenty-four hours produced a set of +634, The table also shows that the set of Z, at the end of the observations was about half as large as at the beginning, and that the rate of decrease was most rapid at first, thus suggesting that the rate of decrease of the set at any time was proportional to the amount of the set at the time. This law, when expressed graphically (tine being the abscissa and set the ordinate), is the characteristic of the logarithmic curve. The following discussion of the observations was there- fore made. The set of each observation was plotted as ordinate, with the date or time as the abscissa. The most probable logarithmic curve was then passed among the plotted points (the most proba- ble when the errors are those of observation and are wholly in the ordinates). The method of computation was that given in Jordan’s Vermessungskunde, volume 1, page 44. An approximate logarithmic curve was first chosen, from which the values of .Y and Yin the equation h B=NX10" were computed, in which B, the observed quantity, is the ordinate or set expressed in microns, and h is the abscissa expressed in days. The most probable corrections to the approximate values and Y, called JV and 1Y, respectively, were computed as follows: From the above equation is derived ab dB ,+- AB= ay 4X+ 7743 §2.] EFFECT OF HEATING AND COOLING ZINC BAR Z,. 865 Sel a | - or, substituting «@ for ae b for oe and putting / for (410 % —B), there results adX+d5SV¥+l=v in which 1 _ oh 0g d= —y log b=log a+log eo iW being the modulus of common logarithms. Each observation gives an equation of the above form. There were 34 observation-equations of this kind, from which the values 4X and JY were derived and added to the first values of X and Y. A second approximation was not thought neces- sary for this work. The corrected values are A= 66.4 Y=328.6 With these values the ordinates corresponding to the dates of each observation were computed, and are given in the fifth column of the preceding table. These computed ordinates, subtracted from the observed ordinates given in the column headed “Set”, leave the residuals, which are given in the column headed v. These residuals are much larger than the errors of observation. This is probably due in part to the changing temperature of the comparing-room; that is, a set at any time is not alone a function of the first large set and the time which has elapsed, but is also a function of all the small sets which have occurred since the large one. The table shows that in 100 days the set had diminished from +65 to +33+, or one-half. There- fore, at the end of 200 days the set would be +16, and after 300 days it would be +8+ after 400 days it would be +4+ after 500 days it would be +2 2 This law of decrease may be expressed as follows: With a constant temperature Z, will lose one-half its set in one hundred days. This time, 100 days, is probably too great, for during the 100 days of observation the tempera- ture of the comparing-room fell from 77° to £6° F., and this fall of 31° would of itself make a plus set of some amount, as is shown by Z%, the set of thie latter having increased 15" during the time. It is difficult to say how much effect the fall in temperature had upon Z,, but whatever its amount the tendency would be to indicate too great a time for losing one-half of the set. 109 LS 866 APPENDIX III. Apr. ILL, APPENDIX III. DIFFERENCE OF LONGITUDE BETWEEN DETROIT, MICH., AND CAMBRIDGE, MASS. Revort or A. R. FLINT AND O. B. WHEELER, Assistant Engincers. OFFICE UNITED STATES LAKE SURVEY, Detroit, Mich., June 30, 1882. Sir: In compliance with your instructions, we have the honor to make the following report on our recent determination of the difference of longitude between Detroit, Mich., and Cambridge, Mass. The field-work was done in obedience to the following order: * OFFICE UNITED STATES LAKE SURVEY, Detroit, Mich., April 22, 1881. Assistant Engineer A. R. FiLint, Detroit, Mich.: Sir: On May 2 the determination of the difference of longitude of Detroit and Cambridge Observatory will be undertaken. Time-determinations will be made on five nights with yourself at Detroit and Assistant Engineer O. B. Wheeler at Cambridge, using chrouographs aud instruments of the same construction and size; the observers will then change * * x # * * places, and five more nights be obtained. Star-places will be taken from 539 Sterne des Fundameutal-Catalogs, Berlin, 1831. Eleven wires will be observed on for time-stars, aud on slow stars at least seven wires before and after reversal. Programme for time-determination. Level readings. Circumpolar star reversed on. Level readings. Five or more well-determined time-stars. Level readings. Reversal. Level readings. Five or more well-determined time-stars, Level readings. Circumpolar star reversed on. Level readings. : Then will follow the exchange of clock-signals, which will be sent alternately for 1” 20* from each station till two sets have been sent from each station. The observers will carefully.adjust the tensions of their relays and request the repeating stations to do the same before serding clock-signals. After exchange of clock-signals another time- es according to the above programme will be made. Every care will be taken to secure most accurate work. Very respectfully, C. B. COMSTOCK, Major of Engineers and Brevet Brigadier- General. The instructions were followed as strictly as the weather and other cireumstances would permit. z The two sets of transits, clocks, chronographs, observing-keys, and other electrical apparatus were as nearly alike as possible. The observer used the same transit and observing-key at both stations, in order to eliminate any errors peculiar to their construction. §1.] ‘LONGITUDE OF DETROIT FROM CAMBRIDGE. 867 The transits, made by Wiirdemann, and numbered 1 and 15, respectively, were of the same size and similarly mounted. They are of about 31 inches focal length, 24 inches clear aperture of object-glass, are provided with diagonal eye pieces and reticules of nineteen threads, and magnify about sixty-five diameters. Mr. Flint used transit No. 1 and Mr. Wheeler used No.15. The pivot- corrections and level-values were redetermined for each transit. The values of one division of the levels were tested by means of the level trier, with the tubes in their holders, and were found to be the same as formerly used. The wire-intervals for transit No. 1 were redetermined, but not differing materially from the old values, the latter were used. The wire-intervals for a new set of wires in transit No. 15 were determined by at least twenty observations on slow stars for each wire. Bond & Sons sidereal clock No. 256 and Bond & Sons spring-governor chronograph No. 216 were used at Detroit. Frodsham sidereal clock No. 1327, belonging to Harvard College Obserya- tory, and Bond & Sons spring-governor chronograph No. 245 were used at Cambridge. This clock is the same as was used, by the United States Coast Survey in the transatlantic longitude work ot 1872, and was then designated the South Clock. The connection with the clock was through a local circuit with a battery of the same number of cups as was in use on the local circuit of the Observatory. The zine poles of the two circuits were placed toward each other in order to prevent interference of the” circuits with each other. The use of this clock was kindly granted to the Lake Survey by Professor E. C. Pickering, Director of the Observatory, and we may here state that every facility was rendered by the Director and his assistants in the execution of our work at that station. At Cambridge the observations were made on the south pier of the United States Coast-Sur- vey Observatory used in the transatlantic work of 1872. This station is on the grounds of the Harvard College Observatory, 108 feet, or 0°.096 of time, west of the center of the great equatorial dome of the main building (see United States Coast-Survey Report for 1874, p. 164). The two granite piers there described were spanned by a marble slab 3 feet in length and 3 inches in thick- ness to accothmodate our transits. At Detroit the observations were made on the west stone pier of the Lake-Survey Observatory (1871-’82). This pier is 5.1 feet, or in time 0°.004, west of the east pier which was used in the Washington longitude work of 1871. Usually a greater number of stars was observed than was required by the programme, and in selecting for reduction preference was given to those stars observed at both stations on the same night. The signals were made automatically, the clock breaking the circuit every second, except the sixtieth or zero second. The clock in a local circuit repeated to the main line, and the signals were ' again repeated to a local clock-and-chrunograph circuit at the other station. There were two repeaters on the main line, either at Albany and Buffalo or at New York and Buffalo. The signals were transmitted entirely through closed circuits. Care was always taken that the repeaters were kept in close adjustment during the exchange of signals. The observations were made on May 13, 23, 24, 26, June 4 and 11, 1881, with Mr. Flint at Detroit and Mr. Wheeler at Cambridge; and on June 21, 22, 23, 24, and 29, 1881, with Mr. Flint at Cambridge and Mr. Wheeler at Detroit. The nights of May 13 and 23, being partially inter- rupted by clouds, were each considered as of half weight. On May 13, at Detroit, there was no reversal on a slow star, and the stars observed after signals were all with the clamp one way. On May 23, at Cambridge, the stars observed after signals were all with the clamp one way. The chronographic readings were, made in duplicate, and the reductions have been entirely in duplicate, a second person duplicating the work of the first. Collimation was determined from reversals on polar stars, except in two cases, where it was deduced by least squares from the obser- vation-equations. The clock correction at a given epoch, the azimuth, and hourly rate, were deter- mined by least squares from the observed times corrected for errors of level, reduction to middle wire, diurnal aberration, and usually for collimation. The observation-equations from which the most probable values of 40, a, and p were deduced are of the form AatRpt d0+[t+Bb+ Oct di +ab’n)|—(a+ assumed 4t)=v, in which A, B, and C are the azimuth, level, and collimation factors respectively; FR, the interval from a star observation to the given epoch, expressed in tenths of-an hour; a, b, and ¢, the azimuth, level, and collimation corrections, respectively ; p, the rate per hour, + when losing, — when gaining; 4t, the clock correction at the given epoch = assumed dt+d0 j 40, the correction to the assumed 4t; it, the clock-time of transit, usually the mean of 11 wires; 4¢ -~ 868 APPENDIX IIL [App. III, \ the reduction to the middle wire; al’n, the correction for diurnal aberration; «, = A. &., the right ascension, taken exclusively from the catalogue “ Mittlere und scheinbare Oerter fiir das Jahr 1881 von 539 Sternen”; 2, the residual from an observation-equation; ¢, is an abbreviation for the term [t+ Bb+ O(e+ ditab’n)]. In the reticules were three sets of 5 wires each, the wires being numbered from 1to15. Usually the mean of observations on wires 3 to 13 was taken. In some cases the observations were incom- plete on these wires, and then observations on other wires were substituted, or a fewer number was used. In such cases the clock-time of transit is from the mean of observations on wires reduced individually to the middle wire, and 4i=0 in the formula. The size of the correction C(e+J4ital’n) together with the number of wires observed will sufticiently indicate these excep- tions in the reductions. The pivots of transit No. 1 being irregular in form required a pivot-correction varying with the altitude, so that the factor b generally varies for different stars. This\pivot-correction was applied to the observed inclination of the axis as indicated by level-readings, and was taken from a curve plotted from the following data, which were obtained from special observations. Altitude of star. BB | | 90° — 5.63 divisions of level. ! 40° North. — 4.60 divisions of level. 40° South. —10. 17 divisions of level. | 65° North. — 8.67 divisions of level. 65° South. | — 7.04 divisions of level. | | i B! In this table | B his four times the inclination of the axis for clamp { east west special observations, and the value of one division of the level was 0.918. Four times the value of pivot-correction, or B/—B, is tabulated for convenience in computation. For transit No. 15 a constant pivot-correction was applied to the level-readings in obtaining the factor b, and in the reduction the value was carried to three decimal places instead of two, as given in the tables following. All the observations with transit No. 15 have been reduced directly in accordance. with the above formula. The collimation from the mean of 19 reversals on slow stars for the six nights at Cambridge was +0°.130 for clamp east, and from 23 reversals at Detroit, for the five nights at Detroit, was +0%.085 for clamp east. (See Table 12.) -For the observations with transit No. 1, on May 13 and June 4,¢ was deduced from the observation-equations. On May 13 the rate was adopted equal to zero, since it was known to be very small, and the data were not good for determining both collimation and rate, since all the stars after signals were with the clamp one way. On June 29 a’ was introduced in the observation-equations, since the observations extended over a long space of time and a change of azimuth was suspected. (Sce the tables following for the individual values for collimation.) Tables 1 to 11 give the data from which the observation-equations were formed, together with the resulting normal equations and the values of the unknowns, for the observations with transit No.1. Tables 13 to 23 give similar data for the observations with transit No. 15. The weight, p, of an observation-equation is derived from the following formule. Albrecht gives (see Formeln und Hiilfstafeln, &c., Leipzig, 1874, page 7) the probable error of an observed transit over a single wire of a star at any declination ~ 2 # = Ey CS) sec’d in which ¢, = probable error of transit over one wire of a star at the equator, and v = the mag- nifying power of the telescope. We have, from observation, for transits Nos. 1 and 15, ¢., = +0%.06 aud v = 65; therefore ; , referring to the c= j (0.06)?+ (0.049)?sec?a ; t §1.] LONGITUDE OF DETROIT FROM CAMBRIDGE. 869 The United States Coast-Survey Report for 1872, page 224, gives n in which ¢ is as above. ¢ is the probable error of culmination, reduced to the equator, N is the number of wires to whose mean weight unity.is assigned, and » is the number of wires observed. We have used V=11 and »=+0°.056, as given for smaller instruments in the United States Coast-Survey Report referred to above. All stars from 0° to 45° declination have been given weight unity when observed over 11 wires. Tables 24 and 25 give the means of clock-times of comparison, the clock-corrections, and the resulting differences of time for the several nights. The final results are given in Table 26. The mean of the results for difference of time between the two observing-stations for the first six nights, giving weight 4 to results for May 13 and 23, is 47™ 418,046 and for the last five nights 47™ 418.114 One-half the difference of these results, namely, 0.034, which represents the personal and in- strumental equation, when added to each of the first six nights’ results, and subtracted from each of the last five nights’ results, gives the results for difference of longitude for the several nights. Taking a mean of these results, giving weight 4 to results for May 13 und 23, and applying a reduc- tion of — 0°.004 to reduce to the east stone-pier of the'Lake-Survey Observatory, and a reduction of +0°.096 to reduce to the dome of Harvard College Observatory, there results the final weighted mean difference of longitude. The United States Lake-Survey Observatory, 1871~82, east: pier, west from the dome of Harvard College Observatory, 1872~’82 =47™ 419.1724 08.031 The probable error of this mean is derived from the discrepancies between it and the separate results. The difference of longitude between Detroit and Washington, given in, Chapter XXV, is 24” 00°.15. The United States Coast Survey gives for the difference of longitude between Wash- ington and Cambridge, 23” 415.042 (see Report on Telegraphic Longitudes, Appendix No. 6, Coast- Survey Report for 1880). The sum of these gives for a difference of longitude between Detroit and Cambridge 47™ 418.192, a value differing less from the result of the present direct determination given above than the probable error of the latter. The value 47™ 41°.172+0°.031 by the direct connection will then be adopted. From the same authority as above quoted we have also the longitude of the dome of Harvard College Observatory west from Greenwich +" 44™ 30°.994, Adding the result above given, namely, 47™ 41°,172, there results for the longitude of the east pier of the Lake Survey Observatory west from Greenwich 5" 32” 129.166. The value adopted in March, 1874, and upon which all work since reported has been made to depend, was 5" 32" 12°24, All longitudes reported since 1874 in the annual reports and those given in the present volume must, therefore, be diminished by 0°.074. Of this correction 0*.020 is due to the difference between the direct and indirect determinations of the longitude of Detroit from Cambridge, already mentioned, and 0°.054 to a change in the longitude of Cambridge formerly adopted by the United States Coast Survey, this change resulting from an adjustment given in Appendix 6 of Coast-Survey Report for 1880. In the reductions we have been assisted by Assistant Engineers E, 8. Wheeler, T. Russell, and A. Ziwet. r fully submitted. Respectfully su AR ae O. B. WHEELER, Assistant Engineers. General C. B. Comstock, Lieutenant-Colonel of Engineers, and But. Brig.-Gen. U. 8. A. 870 APPENDIX III. | App. III, Difference of Longitude, Detroit, Mich., and Cambridge, Mass. TABLE 1.—TIME DETERMINATION. {Detroit, Mich , May 13, 1881. A. R. Flint, Observer.) Star. CL oe ea ie ! ip OEE OTB) ABBE AREA) | and 8. Bi 8. 8. 8. hom. 8. hom. 8. m, 8 @ Leonis ..........-.--- E. 1 | 40.20 | 40.19 —0. 05 +0.20/ 0.00] 11 09 27.55 | 11 08 02.30] —1 25.44 € Urse Maj. med......: E. 1L | $0.19 | 40.22 - 0.06 +0.09/ 0.00] 11 131802] 1111 52.66] —1 25,58 @ DSO ibeecer oscnaes as E. 11 | 40.23 | 40.19 0.05 +0.26} 0,00] 11 16 27.79; 1115 02.63) —1 25.35 € LebulS vzacea weedeeiowe E. | 11 | $0.22) 40.19 —0. 05 40.23} 0.00] 111911.08! 1117 45.84] —1 25.43 | 58 Urse Maj ...........- E. | 1 | +019} 4.0.26 —0.07 —0.01| 0.00 | 11 25 32.76) 11 24 07.44] —1 25.58 |v Leonis ............... E. | | 40.23 | -£0.17 —0. 05 +0.30} 0.00} 113219.20 11 30 54.05] —1 25.32 | y Cephei, L. C .......... E. | i | —0.08 | +0.17 +0. 34 +1.69 | 0.00 | 11 35 51.54] 23 34 97.8 |, --1 23.9 x Urse Maj.........2.-. E. | 40.19 | 40.28 —0.08 —0.07;, 0.00} 11 41 14.23 | 11 39 48.78] —1 25.73 B LeOnisincssukitesgeees E. ll | 40.20 | 40.18 —-0. 05 40.21} 0.00| 11 44 27.34] 11 43 02.10 | —1 25.42 B Virginis .............. E. | 40.23 | 40.17 —0. 05 40.28] 0.00) 11 45 57.72] 11 44 32.56] —1 25.33 o Virginis .............. w. | at | 40.20) 40.17 +0.02 | 40.24] 0.00) 12 00 36.91) 11 59 11.67! —1 25.41 33 Bootis .... Ww. 11 | 40.08 | 40.11 +0. 03 —0.03| 0.00} 14 35 53.26] 14 34 27.73 | —1 25, 64 pm Virginis .....8........ | Ww. 11 0.02 | 40.01 +0, 02 +0.32/} 0.00] 14 3815.91} 14 36 50.80 | —1 25.12 109 Virginis .... ........- lw. 11 | 40.03 | 40.02 +0. 02 +0.28| 0.00] 14 4142.52] 14 4017.31} —1 25.23 | 47 Hev.Cephei, L.C ..... w. 8 | 0.10} 40.26 —0. 24 41.95} 0.00] 14 5142.10] 250188 | —1 23.6 | B Bootis.........20...+- W. |} ll | +0.07 | 40.09 +0. 03 +0.01| 0.00} 14 58 56.22 | 14 57 31.01 | —1 25.30 3 Serpentis ... w| W. | 1 | 40.04 | +0. 08 +0.05 +0.26| 0.00 15 10 45.02} 15 09 19.84 | —1 25.21 8 Bootis .........-...-.. Ww. 11 | +0.08 | +0.1u. +0. 02 +0.08| 0.00] 15 1210.78 | 15 10 45.52 | —1 25.36 pe Bootis dneadreeeena sas W. |} 11 | 40.08} 4.0.10 +0. 03 +004} 0,00| 15 21 2825] 15 20 02.83 | —1 25.52 B Corone Borealis ...... w. | a1 | +0.08 | +0.09 +0. 02 +011} 0.00] 15 24 23.83} 15 22 5857] —1 25.35 v! Bootis .......-2---..-. Ww. | a | 40.08) +0.11 +0. 03 40.01} 0.00} 15 28 07.96 | 15 26 42.49) —1 25.58 | 4 Bootis Ww. | i | 40.08 | 40.11 +0. 03 +0.01} 0.00) 15 29 00,08] 15 27 34.56 | —1 25.63 | Adopted rate p=0*. 00. OBSERVATION-EQUATIONS. Epoch 13". 0 clock time. At=—1" 25%. 004A. Weight. +0.46a+1. 04¢4A0-+0.45=0= 40.03 1 $0.21 41.18 + +40.59=0=+40.05 1 +0.59 $1.01 + +40.36=0=+40.00 1 +0.63 41.02 + +0.44=0=+40.05 1 —0.08 41.39 4’ 40.59=v=—0.06 1 $0.68 41.00 + +0.33=v=+40.01 1 43,87 4.43 + —1.1 =v=+40.3 0.11 —0.16 $1.51 + +40.75=v=+40.03_ 1 40.47 41.04 + +40.48=0=+40.02° 1 +0.64 $1.00 + +40.34=0=40.00 1 +0.55 —1.01 + +40.87=v=40.11 1 0. 06a—1. 41¢-4+ A0-+0. 58=v=40.08 1 40.74 —1.00 + +40.08=v=—-0.10 1 +0.64 —1.00 + +0.19=0=-0.03 1 +4.47 45.21 + —1.4 =v=-0.3 0.08 +0.03 —1.32 + 40.25=v=—0.22 1 40.60 —1.01 + +40.19=0=—0,05 1 +018 —1.20 + +40.81=v=—0.10 1 +0.10 —1.26 + +40.47=v=+40.03 1 : +0.25 —1.15 + +0.30=0=—0.08 1 +0.03 1.33 + +0.52=0=40.05 1 : 40.02 —f.33 + +0,58=v=+40.11 1 NORMAL EQUATIONS. REsuLts. +6. 97a+ 0.20¢+ 7.2640-+1.04=0 a=+05, 437 +0,20 431.81 — 2.90 +0.24=0 c=—0*. 061 cl. E. 47.26 — 2.90 +20.19 +7.92=0 A@=—0*. 560 §L] LONGITUDE OF DETROIT FROM CAMBRIDGE. 871 Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 2.—TIME DETERMINATION. (Detroit, Mich., May 23, 188]. A. R. Flint, Observer.] Star cu |Maet] o | Bo |oretactavn) ao | Ro |lecktipept | Right asco | 4 ¢ esses 8. 8. 8. 8. 8. hom. s. hom. 2%. mM. &. 4 Hev. Draconis ........ E. 8 +0.14 | 40.56 +0. 00 —0.22 | +0.08 | 12 08 04.78 | 12 06 40.8 —1 24.5 4 Hev. Draconis ....--.- Ww. 8 +0. 02 | +0.08 —0. 13 —0.22 | +0.08 | 12 08 05.20 | 12 06 40.8 —1 24.3 » Virginis ..............| W. 11 +0.13 | +0.10 —0. 05 +0.05 | +0.07} 12 15 16.51 | 12 13 51.89 | —1 24.67 6 Canum Ven .........-| W. 11 +0.19 | +0. 25 —0.07 +0.00 | +0.07 | 12 21 26.47 | 12 20 02.03 | —1 24.62 24 Come seq ...---.----- Ww. 11 +0.16 | +0.16 —0. 06 +0.03 | +0.07 | 12 30 37.10 | 12 29 12.49 | —1 2471 y Virginis med.......... w. 11 +0.13 | +0. 09 -—0. 05 +0.05 | +0.06 | 12 37 05.31 | 12 35 40.78 | —1 24.57 6 Virginis .......-...... w. i +0.14 | +0.11 —0. 05 +0.05 | +0.05 | 12 51 04.11 | 12 49 39.52 | —1 24.65 e Virginis ..-........... Ww. il +0.15 | +0.13 —0. 05 +0.04 | +0.05 | 12 57 42.65) 12 5618.17 | —1 24.56 ie a Urse Min., L.C ...... Ww. 8 +0.14 | —3 97 +1.17 +2.45 | 40.04 | 13 16 10. 60 1 14 48.3 119.5 a Urse Min., L.C....... E. 7 +0.15 | —4. 25 +0. 00 +2.45 | 40.04 | 13 16 13.60 1 14 48.3 —1 21.0 7 Bootis -...-...--.- E. 11 +0. 24 | -- 0.23 +0. 02 +0.03 | -+0.03 | 13 43 04.00 | 13 41 39.53 | —1 24.72 n Urs Maj E. 11 +0. 23 | +0. 35 +0. 03 -—0.02 | +0.03 | 183 44 18.54 | 13 42 54.23 | —1 24.69 y Bootis ...-...----.---- E. 11 +0. 23 | +0. 22 +0, 02 +0.03 | +0.02 | 13 50 2868 | 13 49 04.23 | —1 24.69 + Virginis .............- E. il +0.27 | +0. 21 +0. 02 40.05 | +0.02 | 13 57 03.21 | 13 55 38.68 | —1 24.76 d Bootis ...-.......-.-.. E. 11 +0. 24 | +0. 25 +0. 02 +0.02 | 40.01 | 14 06 25.86; 14 05 01.44 | —1 24.69 x Virginis ..-.....-.--.- E. il 40.29 | +0.18 +0, 02 +0.06 | +0.01 | 14 08 00.88) 14 06 36.87 | —1 24.71 B Corone Bor..-....--.- E. 11 +0.17 | +0.19 +0. 03 +0.02 | —0.03 | 15 24 23.17 | 15 22 58.61 | —1 24.78 v! Bootis ...--..-.---.--- E. 11 +0.16 | +0. 21 +0. 03 +0.00 | —0.03 | 15 28 07.04 | 15 26 42.53 | —1 24.75 v? Bootis -....----.-- : E. 11 +0.16 | +0. 21 +0. 03 +£0.00 | —0.03 | 15 28 59.08 15 27 34.60 | —1 24.72 « Coron Bor E. il +0.17 | 4+0.18 +0. 02 +0.02 | —0.03 | 15 31 06.84] 15 29 42.24 | —1 24.80 @ Bootis ..--..--.---.--- | E. 11 +0.16 | 40.21 +0. 03 +0.00 | —0.04 | 15 35 01.10 | 15 33 36.44 | --1 24.90 ¢ Corone Bor. seq. ..---- j E. li +0.17 | 40.21 +0. 03 +0. 01 | —0.04 | 15 36 21.65 | 15 34 56.99 | —1 24.90 B Serpentis .....-....... l W. 11 +0.08 | +0.07 —0. 05 +0.03 | —0. 04 . 15 42 09.73 | 15 40 45.05 | —1 24.70 wu Serpentis --.----.----- Ww. 11 +0. 05 | +0. 04 —0. 05 +0.05 | —0.04 | 15 44 52.87 | 15 43 2817] —lL 24.69 e Serpentis ...-.-.------ Ww. 11 +0. 07 | +0. 06 —0. 05 +0.05 | —0.04 | 15 46 21.21} 15 44 56.50 | —1 24.72 e« Corone Bor....-.- ewes} We i +0.10 | +0.11 —9. 06 +0.02 | —0.05 | 15 54 07.55 | 15 52 42,90 | —1 24.70 ‘| 750 Groombridge, L.C ....; Ww. 8 —0.03 | +0. 22 +0. 33 +0.72 | —0.05 | 16 00 59. 36 3 59 36.6 —1 23.3 750 Groombridge, L.C ..-.| E. 8 +0.00 | +0.00 +0. 00 +0.72 ) —0.05 | 16 01 00.56 8 59 36.6 —1 23.9 WwW. 11 +0.08 | +0. 06 —0. 05 -+0.05 | —0.06 | 16 09 34.87} 16 0810.17) —1 24.71 Ww. 1 +0.08 | +0. 06 —0. 05 +0.05 | —0.06 | 16 13 29.81 | 16 12 05.12 | .—1 24.70 Ww. 8 +0.05 | +0.17 —0. 11 —0.17 | —0.06 | 16 22 28.86 | 16 21 03.9 --1 25.0 E. 8 +0.10 | +0. 34 +0. 00 —0.17 | —0.06 | 16 22 28.36 | 16 21 03.9 —1 24.8 COLLIMATION (Clamp E.) 8. 1812 Groombridge....-.-------------+-+seereeee 4 Hev. Draconis a Urew Min., LC .---- 22 eee eee cee eee eect tnenes wetter enenec teen eeeee 750 Groombridge, L. C.-...---------eee cece ee cere et cece cnecerert eect error cree steers yn Urse Maj ..---------- 2-222 ee cere rere eee eee ete ee eee eee ee tects teeter eee eer ee nees Mean 222.0. cecc en nce ener ceee enc n ec reser cece nn cereancetsnmercensrccrcrsnerconrcenecescas é Adopted collimation c=+0°.012 cl. E. (Mean of May 23 and 24.) OBSERVATION-EQUATIONS. Epoch 14*.5 clock time. At=—1™ 24,5010. Weight. Weight. Weight. — 2.89a—2. 4p A@—0.1 =v=—0.5 0. 104 +0. 42a—0. 7p+A0-+0. 19=v= +0.00 1 +40. 12a+1. 1p +4040. 40=v=40.138 1 + 0.67 —224+ 40.17=0=40.05 1 40.65 —0.5 + +0.26=v=+40.09 1 40.46 412+ +0.20=v=--0.05 1 + 0.06 —21+ +0.12= 1 +0.32 —0.4 + +40.18=v=—0.03 1 40.71 $1.2 + +40.19=v=—-0.04 1 + 0.42 —2.04+ 40.21=v=+40.07 1 40.80 —0.4 + 40.21=v=40.04 1 40.61 41.3 + +0.22=v=-0.01 1 + 0.68 —1.9 + 1 +029 41.4 + -++0.20=v=—0.07 1 + 0.62 —1.6 + 1 -40. 26a-+0. 9p +4040. 28=v=+0.03 1 49.55 41.54 —0.9 =v=—0.5 0.018 + 0.52 —1.5 + 1 40.03 +1.0 + +0.25=v=—0.02 1 +0.72 41.7 + +0.21=0v=—0.04 1 432.62 —1.2 + 0. 001 40.038 +10 + +40.22=0=—0.05 1 ao 73 417+ +40,20=v=—0.05 1 4+ 0.43 —0.8-+ +0,22=v=40.04 1 +0.29 41.04 +40.30=v=+40.05 1 2.29 +1.9-+4+ 40.4 =v=—0.1 0.143 — 0.20 —0.8 + +0.19=v=—0.04 0.640 40.03 +1.14+ 40.40=0=+0.12 1 NORMAL EQUATIONS. RESULTS. +9. 99a— 0. 67p-+ 9.3240 +1. 47=0 . a=+0". 075 —0.67 +43.48 + 0.10 +1.56=0 p=—0*. 034 Z Ad= —0%, 240 49.32 + 0.10 $23.91 +5. 05=0 872 APPENDIX UI. [App. III, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued., TABLE 3.—TIME DETERMINATION. (Detroit, Mich., May 24, 1881. A. R. Flint, Observer.] | : 1 : - Star. | 1 NEM) | BL Ce+Sitad'n)| Aa pe (Cee tmect| Riehe rae) a i | ae fas pe | Sie : | e: $8. 8. R 8 hom. 8. hom. 8. m. 8. | 4 Hev. Draconis .-....-- Ww 7 | —0.30 ; —1.19 —0.13 + 1.48} 0.00 | 12 08 04.84 | 12 06 40.8 | —1 22.8 | 4 Hev Draconis......-.- E. 7 | —0.18 | —0.71 +0. 00 + 1.48} 0.00} 12 08 04 49| 12 06 40.8 | —1 23.0 : » Virginis ..............| E. ll | --0.09 | —0.07 +0. 02 — 0.34) 0.00} 1215 17.16 | 12 13 51.88 | —1 25.23 | 6 Canum Ven........... E. 11 | —0.14 | —0. 18 +0. 03 — 0.03 | 0.00} 12 21 26.96 | 12 20 02.02 | —1 24.79 20 Come ...... 2.2.22... E. 11 | —0.12 | —0.12 +0. 02 — 0.19 | 0.00} 12 25 12.59 | 12 23 47.40 | —1 25.09 y Virginis med ......-.. E. 11 | —0.09 | —0. 07 +0. 02 — 0.35} 0.00! 12 37 06.09 | 12-35 40.77 | —1 25.27 8 Virginis . E. 11 | —0.10 | —0. 08 +0. 02 -- 0.32 | 0.00 | 12 51 04.80 | 12 49 39.51 | —1 25.23 « Virginis ........- wigel AE 11 | —0.10 | —0.09 40.02 | — 0.27) 0.00) -12 57 43.38 | 12 56 18.16 | —1 25.15 a Ursw Min., L.0.....-. E. 8 | —0.17 | -+4.83 +0. 00 —16.67 | 0.00) 13 16 24.49] 114 49.4 ) —1 39.9 w Urewe Min., L.C....--. w. 7 | 0.25 | 47.09 $1.17 —16.67| 0.00 | 13 16 22.00] 114 49.4 | —1 40.9 + Bootis ..........-...-- Ww. 11 | —0.27 | —0. 26 —0. 05 ~ 0.22} 0.00} 13°43 04.82} 13 41 39,53 ) —1 24.98 1 Bootis . w. 11} —0.26 | —0.25 ~0. 06 — 0.21} 0.00} 18 50 29.54) 13 49 04.23 | —1 25.00 + Virginis w. 11 | —0,29 | —0. 22 —0. 05 — 0.33) 0.00 | 13 57 04.11 | 13 55 38.69} —1 25,15 a Draconis ......-...... w, J1 | —0.29 | —0.65 —0. 12 + 0.47; 0.00 | 14 02 38.61] 14 01.13.58 | —1 24.26 « Virginis .............- w. i | —0.31 | —0.2 —0. 05 ~ 0.41) 0.00 | 14 08 01.83; 14 06 36.36 | —1 25.22 4 Urse Min ............ Ww. 7 | ~0,27 | —1.06 —0. 13 +145) 0.00} 14 10 49.20} 14 09 24.6 | —1 23.4 4 Uisa Min ............ | izE 7 | -O.11 | —0.43 +0.00 +145} 0.00} 14 1048.77 14 09 24.6 | —1 23.7 v! Bootis ...--...--2----- E. 11 | —0.22 | —0.29 +0. 03 — 0.02) 0.00) 15 2807.52 15 26 42.53 | —1 24.73 i Bootisuwasseees Hereulis ........2....2. | E | n- | +009] +012 +0.31 0.00 | —0.01 | 16 05 04.64} 16 05 04.21] —0. 86 6 Ophiuehi. ...c00 2: esceses | E. | 11 +0.09 | +0. 06 +0. 22 +0.01 | —0.01 08 10.29 08 10.05 —0. 52 e Ophbiuchi .............. E} it +0.09 | +0. 06 +0. 22 +0.01 | —0. 01 12 05. 20 12 05.01 | —0.47 y Merculigicc acess xeuss E. | iW +0.09 | +0. 8 +0. 23 0.00 | —0. 02 16 43. 62 16 43.41 | —0.52 @ Hereulis<..2 22240 425% E. 1 +0.09 | +0. 08 +0. 23 0.00 | —0. 02 19 58. 89 19 58 64 —0. 56 » Ophiuchi E. 11 | +0.09 | +0.06 +0. 22 +0.01 | —0. 02 24 58, 40 24 58.08) —0.60 o Herculis.............. E. | 1 +0.07 | +0. 09 +0. 30 0.00 | —0. 02 30 19, 21 30 19.07 | —0.53 ¢ Herculis....... | E | | 40.07) 40.08 +0. 26 0.00 | —0. 02 36 51. 23 36 51.11] —0.46 » Herculis . i 11 +0.07 | +0. 08 +0. 29 0.00 | —0. 02 38 52. 24 38 52.05 | —0.56 « Ophiuchi | E. ll +0. 07 | +0.05 0. 22 0.00 | —0. 02 52 05. 62 52 05. 35 0. 5¢ e Ursa Min.............. E. | i +0.07 | 40.37 +1. 64 —0. 04 | —0. 02 58 16.00 58 17.3 —0.7 Adopted collimation = +-0%.130 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 14".6 clock time. At=+0".00+A8. Weight. Weight. + 0. 68a—3. Ip +040. 54=0=40.09 1 + 0.12a+1. 0p+ A@+0, 42=v=—0.08 1 — 0.16 294+ +40, 60=0=40.14 0.65 + 0.69 41.0 + —0.12 1 +047 —29+4 +0, 1 + 0.47 41.14 +0.48= i 1, — 2.67 —2.6+4 +0, 0.11 +071 41.1 4+ 40.52=v=+40.03 1. + 3.70 —274 + +0. 0.13 — 24 +12 4 0.1 + 0.38 —22+4+ +0. 1 — 0.07 415+ 40.86=v=40.36 0.7 + 0.42 —214 +0. 1 + 0.72 +15 4+ +0.52=v=+0.03 1 + 0.68 —2.0-+ +0. 1 + 0.73 41.6 4+ 1 + 0.62 —1.8+ +0. 1 + 0.42 41.7-+ 40.52=v=40.01 1 +£0.52 —1.7-+4+ +0. 1 + 0.48 41.74 40.56=0=40.05 1 40.74 —-154+ +40.46=0=—-0.01 1 + 0.64 41.8-+ +40.60=v=+40.10 1 432.60 —14 4+ 42.1 =r=41.9 0.901 — 0.01 41.9 + +0.58=7=40.02 1 + 0.43 —09-+ +0: 1 + 0.21 42.04 40.46=v=—0.05 1 + 0.42 —0.8 + +0.29= 1 + 0.07 +204 40.56=v=40.05 1 : 4-055 +234 40.54=0=+40.03 1 + 0. 04a-+1. 0p A0-+0.40=v=—0.10 1 4.74 42.4 + 40.7 =v=40.2 0.04 NORMAL EQUATIONS. RESULTS. +11. 66a— 2. 62p+10. 690+ 5.20=0 Ad=—0*. 493 — 2.62 486.164 1.49 + 1.54=0 p=—0". 009 +10.69 + 1.49 425.73 412. 62=0 a=+-0". 009 §1.] LONGITUDE OF DETROIT FROM CAMBRIDGE. 883 - Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TaBLe 14.—TIME DETERMINATION. (Cambridge, Mass., May 23, 1881. 0. B. Wheeler, Observer. | ! i i ; Star. eh No.of} > | Bo | ceraitatn)| aa | RB honee | guia | eee RC eae Oe Gia’ Owesaens | 8. 8. Bs i 8. h. m. 8. him. 8. 8. x Urse Maj: cc: --0-22..5.| Es ll —0.14 | —0. 22 +0. 34 +0.01 | 40.21 11 39 56.49 | 11 39 48.59 —8. 02 .# Leonie ss. sservenscaaey E. 11 —0.14 | —0.13 +0. 23 —0.02 ) 40.21. 43 10.13 43 02.00 —8. 23 4 Hev. Dracon .......... E. 12 —0.14 | —0. 57 +0. 56 +0.10 | --0.18 12 06 48.96 | 12 06 40.8 —8.1 : BG, Li O cena) guccane E. 10 —0.14 | +0. 29 —0. 48 —0.14 | +0.17 09 39.11 0 09 30.1 8, 8 , 6 Canum Ven E. 11 —0.14 | —0. 18 +0. 29 0.00 | 40.17 20 10.02 | 12 20 02.03 —8. 10 20 Come ........ E. 11 —0.14 | —0. 14 +0. 24 —0.01 ) +0.16 23 55.46 23 47.41 —8.15 24 Comm seq.--.......--- E. ll —0.14 | —0.14 +0.12 —0.02 | 40.15 29 20. 63 29 12.49 —8. 12 y Virginis med .... .... E. 11 —0.14 | —0.10 +0. 22 —0.03 | +0.14 35 48,77 35 40. 78 —811 4 a Urse Min., L.C........ W. 5 —0.09 , +2. 60 +6. 31 —1.21 | +0.10) 13 14 44.96 1 14 48.4 —5.5 ' a Urse Min., L.C ....... E. i —0.20 | +5. 58 —4, 92 —1.21 | +0.10 14 54.99 1 14 48.4 —7.3 4p. (BOOtis xc: soovsinesecmt.ss9 E. 11 —0.14 | —0.13 +0. 23 —0. 02 | +0. 06 41 47. 67 13 41 39.53 —8. 24 , 7 Uren Ma) cscs xe2e caus E. 10 —0.14 |} —0.2L +0. 18 +0.01 | +0. 06 43 02.58 42 54. 23 —8. 32 1 wy Bootis..... -4:.c0ceescuees E. 11 —0. 14 | —0 14 +0. 24 —U.02 | +0. 06 | 49 12.43 49 04. 23 —8. 30 © WAPBINIS! 225.058, ait booe ‘glia 11 —0.14 | —0.11 +0. 22 —0.02 | 40.05 | 55 46 95 55 38. 68 —8. 38 , d Bootis ..........--..-.. Ww. 11 —0. 07 | —0. 08 —0. 28 —0.01 | +0.04 | 14 05 10.00] 14 05 01.44 —8. 20 x Virginis Ww. 1L —0. 07 | —0. 04 —0.26 ° —0.03 | +0. 04 06 44. 98 06 36 37 —8. 31 _ 4 Urse Min . -| W. M1 —0.07 | —0. 26 —1. 23 +0.10 | +0. 03 09 34. 57 09 24.7 —8. 4 AK. Bootis-.i2 ecvaxcs vevicenc W. 11 —0.07 | —0. 09 —0, 2L 0.00 | +0. 03 12 03. 29 11 54. 74 —8. 25 pp Bootis cccceas seesseance W. 11 —0. 07 | —0. 07 —0. 30 —0.01 | +0.01 26 53. 82 26 45.25 —8. 20 ; aw Bootis pr .... --....--.| W. 11 —0. 07 | —0. 06 —0, 27 —0.02 | +0.00 35 19. 90 35 11. 26 —8. 31 @ Hercnlis.-...- --..-.--. W. 11 +0. 02 | +0. 03 —0. 36 0.00 | —0.11 16 05 13. 22 16 05 04. 29 —8. 60 1 8 Ophinehd seco os accce% Ww. 11 +0. 02 | +0.01 —0. 25 —9.03 | —9.11 08 18. 84 08 10.17 —8. 43 e Ophiuchi ......--...... Ww. 10 +0. 02 | +0. 01 —0.15 —0. 03 | —0.12 12 13. 64 12 05.12 —8. 38 y Herculis..............- W. 11 +0, 02 | +0. 02 —0. 27 —0. 02 | —0.12 16 52. 23 16 43.52 —8. 46 4 Ursa: Min.. ayer WW 10 +0. 02 | +0, 07 —0. 6) | $0.08 | —0. 13 21 12.95 21 03.9 —8.5 B HPC 8 y.eicieiaises.255% ose WwW. 1l +0. 02 | +0. 02 —0. 27 ! —) 01 | —9.13 25 18.18 25 09. 52 —8.41 | & Hevreulis .cc2406 5 Shee, WwW. 1L +0. 02 | +0. 02 —0. 35 0.00 | —0O 14 30 27.94 30 19.19 —8. 42 6 Hercolisssccccesceas- 22% i We ge cE -+0.02 | +0. 02 —0. 30 —0.01 | —0.14 36 59. 91 36 51, 23 —8. 10 « Ophiuchi ......-....... Ww. | 1l +0.02 | +0. 02 —0. 26 —0.02 | —0.17 52 14.52 52 05. 50 —8. 58 Adopted collimation= 4-0%.130 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 14,6 clock time. At=—- 8.004480. Weight. : We ght. — 0. 16a—2, 9p +A0+0.02=v=—0.07 0. 65 — 2. 83a—0. 4p+A0+40.4 =v=-+0.2 0.1 + 0.47 —2.9 + +0.283=v=+0.11 1 — 0.10 —0.4 4+ +0.25=v=—0.03 0.7 — 2.89 254+ 40.1 =v=}01 0.1 + 0.23 —0.2 + 40.20=v=—0.11 1 + 3.70 —2.4 4+ 40.8 =v=+0.5 0.13 + 0.45 —0.0 + +40.31=v=—0.02 1 + 0.06 —2.3+4 +40.10=v=—0.04 1 + 0.38 —22+4 40.15=v=—0.01 1 — 0.07441. 5p 4464-0. 60=v=+0.18 0.7 + 0.42 —21+4 40.12=v=—0.06 1 + 0.72 41.5 + +0,48=v=—0.02 1 + 0.68 —2.0 +4 +40.11=v=—0.09 1 + 0.73 41.6 + 40.38=9=—0.08 1 432,59 —144+ —1.6 =v=—3.0 0.001 + 0.42 41.7 + +0.46=0=40.01 1 + 0.43 —0.9 + +40.24=v=—0.03 1 — 2.29 41.8 + 40.5 =v=40.2 0.13 — 0.21 —0.9 + +40.32=v=+40.08 0.63 + 0.38 41.84 +0.41=v=—0.04 1 + 0.42 —0.8 + +40.30=v=+0.03 1 — 0.01 41.9 + +40.42=0=--0.03 1 + 0.65 —0.7 + £0.38=v=+40.10 1 + 0,22 +2.0+ +40.40=v=—0.06 1 + 0.32 —0.5 + +40.20=v=—0.08 1 + 0.55 42.3 + +40,58=v=+40.08 1 + 0.80 —0.5 + +40.31=v=+0.01 1 NORMAL EQUATIONS. RESULTS. +49.70a— 1. 59p+ 7. 6240-42, 58=0 A0=—0°. 307 —1.59 +62.76 — 4.36 +4-3.16=0 p=—0°. 072 47.62 — 4.36 422.14 +6.76=0 a=—0°*. 037 884 APPENDIX IIL [ App. 11, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 15.—TIME DETERMINATION. [Cambridge, Mass., May 24,1881. O.B. Wheeler, Observer. ] \ ‘ Ws Star. ee | b By Ciepatedtk: de | |Ppek iment, Bigns aaeoms, ag, [ice seen ates, Gh ere ace te eA coe eee alceteesy Gcatetcn en | | 8. 8 8 | 8 8 the me 8 hom. 8. 8 © Virginis secs sececasses rE. 11 | —0.13 | —0.11 +0. 22 —0.02 | +0.14 | 11 59 20.82 | 11 59 11.58 — 9.35 4 Hev. Dracon ....--...-- E. 11 —0.13 | —0. 51 +1. 09 +0.08 | +0.13 | 12 06 49,61 12 06 40.76 — 9.4 Bri G6) Tal ies srcerae st aests E. 7 —0.13 | +0. 26: —0. 48 | —0.10 | +0.13 09 40. 30 0 09 30. 23 — 9.8 20 DOE cecncaxneansenses 3 iL —0.13 | —0.18 | +0. 24 | —0. 01 +0. 12 23 56.55 | 12 23 47.40 — 9.26 8 Canum Ven.....-...--- W. 1l —0. 08 | —0. 11 —0. 34 | 0.00 | +0, 12 28 18.07 28 08. 27 — 9.35 y Virginis med .......--. W. ll —0. 08 | —0. 06 | —0. 25 —0.02 | +0. 11 35 50. 42 35 40.77 — 9.34 8 Virgins 2scc onayes neces W 11 —0.08 | —0. 06 : —0. 25 —0. 02 | --0.10 49 49.29 49 39.51 — 9.47 @ Vinginis: ces cece s caves WwW 1l —0. 08 | —0.05 © —0. 25 «=O, 02 +0.09 , 13 04 00. 21 13 03 50. 43 — 9.48 a Urse Min., L C.......- W. e —0.12 | +3. 35 +6. 31 | —0.91 | +0. 09 14 51.70 114 49.4 —12.0 a Ursw Min., L.C........ E 7 --0.20 | +5. 58 : —4, 92 —0.91 | +0. 09 15 04,10 1.14 49.4 —15.4 tT BOOS neces vexesaaeeres E. 11 —0.12 | —0.12 | +0. 23 —0.01 | +0. 07 41 48.82 | 13 41 39.53 — 9.40 a BOOS icc sexe cis cress cee E TL —0.12 ; —0.12 -b0. 24 —0.01 | -++0. 06 49 13.45 49 04. 23 — 9.34 4 Urse Min...-.......-.. E 11 —0.12 | —0. 45 +1. 08 +0. 08 | 40.05 | 14 09 33.74 | 14 09 24.62 — 97 | Virginis ...............| E. 11 —0. 12 —0. 08 +40. 22 —0. 02 | +0. 04 22 17.00 22 07.66 — 9.48 p Bootis ..... E | 11 —0.12 | —0.14 +0. 26 —0. 01 | +0. 04 26 54. 62 26 45, 25 — 9.49 aw Bootis, pr E.; 2t —0.12 | —0,11 +0. 23 —0. 01 | --0. 03 35 20. 63 35 11. 26 — 9.49 47 Hev. Ceph., L.C........ E. 11 —0.12 | +0. 31 —1.16 —0.12 | +0. 02 50 29.71 2 50 19.4 — 95 \ Ge 150; GEG vesesanee wes E 5 —0.10 | +0.77 —1. 37 —0.27 | —0.02 | 15 59 47.86 3 59 36.7 —10.6 e Ophiuchi... W. 11 —0. 01 | —0.01 —0. 25 —0. 02 | —0. 03 16 12 14.95 | 16 12 05.13 — 9.56 Hertulis .sc228 seaesnen W. 11 —0.01 | —0. 01 —0. 27 —0.01 | —0. 04 16 53. 39 16 43. 54 — 9.57 , w Mevialis . 2c... 22ceranse W. 1l —0.01 | —0.01 —0. 26 —0.01 | —0, 04 20 08. 64 19 58.77 — 9.60 B: Herenli8.. ca:aesecegons W. 11 —0. 01 | —0. 01 —0. 27 —0.01 | —0. 04 25 19.44 25 09. 54 — 9.62 | G: Gremlis 2: sneciz seni are 1 OW, 11 —0.01 | —0.01 —0. 30 —0.01 | —0. 05 37 01. 08 36 51, 24 — 9.53 9 Merculis ..2: 22. cescees W. 11 | —0.01 | —0. 01 —0. 33 0.00 | —0. 05 39 02. 02 38 52.18 — 9,50 | 49 Hereulis........-..... ‘OW, 11 —0.01 | —0.01 —0. 26 —0.02 | —0. 06 46 53. 03 46 43.14 — 9.62 e Ursw Min...-....-...-. | Ww 9 0. 00 0. 00 —1. 08 +0.13 | —0. 06 58 28. 40 58 17.6 — 9.7 e Urse Min.............. E. 6 | +0.01 | +0. 07 ; +0. 84 +0.13 | —0. 06 58 26. 62 58 17.6 — 9.9 | « Herculis ............... i #E | 11 | —0. 02 | —0. 03 +0. 34 0.00 | —0. 08 17 28 47.49 17 23 38, 03 — 9.77 | BaDriconisi caret | El —0.02 | —0.03 40.18 +0.01 | —0. 08 27 57.25 27 47.75 | — 9.65 @ Opin his ac eciees wisacrey ' iE | 11 —0.02 ; —0.02 +0. 23 —0.01 | —0.08 | , 29 37.29 29 27. 96 — 9.54 | 6 Opbincbi........... gz. | at | —0.02 | —0.02 40. 22 —0. 02 | —0.09 37 48.39 37 39.03 | — 9.56 | w Herenlis 2: 22s:cccscgeces E./ i —0. 02 | —0. 02 +0. 25 —0.0L | —0. 09 42 00. 56 41 51,23 — 9.56 | @ Herculis -......--.-.--.. E. | 11 | —0.02 | —0.02 | +0. 28 0.00 | —0,10 52 22. 54 52 13.33 —947 67 Ophiuchi......-.-...22. E | 11 | —v. 02 | —0. 02 : +40. 22 —).02 | —0.10 54 53. 86 54. 44, 56 — 9.50 & Urs Min..-...-...---. E. | 8 \ —0. 03 | —0. 32 | -+1. 92 +0. 33 | —0. 11 18 10 56.97 18 10 49.16 — 9.4 6 Ursw Min........-..- ap Wes | 6 —0.07 | —0.79 | —2. 47 +0. 33 | --0.11 11 01.80 10 49.16 — 9.4 Adopted collimation =-+-0*. 130 Clamp East. (See Table 12. ) OBSERVATION-EQUATIONS. Epoch 15". 4 clock time. At=—9*,54A0. Weight. Weight. + 0.55a—3. 4p-+-A0—0. 15=v=—0.02 1 + 9, 55a-£0. 6p-+A9+1.1 =v=40.8 0.01 — 2.89 3.34 0.1 =v=40.1 0.1 + 0.73 +0.8 + +0.06=v=+40.02 1 + 3.70 —3.2 + +0.3 =v=+0.4 0.13 + 0.42 40.9 + +0.07=v=+0.03 1 + 0.38 —3,0 + —0.24=v=—v.12 1 + 0.49 +0.9-4+ +0.10=v=+40.06 1 + 0.01 —2.9 + —0.15=0=—0.02 1 + 0.38 41.0 + 40,12=v=40.08 1 + 0.68 —2.8-4+ —0.16=v=—0.06 1 + 0.22 41.2 + 1 + 0.62 264+ —0.0838=v=+40.06 1 + 0.07 41.3 + 1 + 0.74 —2.3 4 : 1 + 0.47 41.4 + 1 432.59 —22 4 0.001 — 4.74 41.6 4+ 0. 04 +£0.43 —1.7 4 1 — 0.16 42.0 + 0. 65 + 0.42 —1.6 4 1 — 0.28 42.14 0.6 i288 1,245 0.1 + 0.51 $2.1 4+ 1 + 0.70 —1.0 + 1. + 0.62 42,2 + 1 + 0.23 —1.0 + ZL + 0.28 42.3 + 1 + 0.45 —0.8 + 1 + 0.11 42.5 4+ 1 + 4.46 —0.6 + 0. 09 ' + 0.64 42.5 4+ i . 1 11.80 +2.8+ —0.1 =v=40.1 0.01 NORMAL EQUATIONS. RESULTS. +14. 98a— 4, 89p+10. 0240-+0. 12=0 Ad=-+0°. 010 — 4.89 4100.36 — 2.26 +3.92=0 p=—0*. 040 +10.02 — 2.26 +24.73 —0.06=0 a=—0*. 028 I $1] LONGITUDE OF DETROIT FROM CAMBRIDGE. 885 Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 16.—TIME DETERMINATION. (Cambridge, Mass., May 26,1881. O.B. Wheeler, Observer.] Star. Gi. | eet | oe Bb | Ole+Aitab'n)| Aw | Rp Clock time of Hignteaeeey | seis 8. Ss. 8 s & hom. 8. hom. 8. 8 o Virgini®. .........2..5. w.| ou |—ow) ou —0. 26 —0.04 | 40.19 | 11.59 23.56) 1159 11.55] —11. 64 4 Hey. Dracon w.| 9 | 0.13 | —0.52 —0.72 +0.20 | +018} 12 06 53.03 | 1206 40.6 | —11.2 4 Hev. Dracon . E. 4 —0.14 | —0. 57 +0. 56 —0.20 | +0.18 06 52. 19 06 40.6 —11.6 | Virginis....... E. | 11 | —0.13 | —0.10 +0. 22 —0.05 | +0.17 14 03. 34 13 51.86 | —11.60 6 Canum Ven.. E. | m= | —0.13 | —0.17 +0. 29 0.00 | +0.17 20 13.44 20 01.99 | —11.57 | 20 Comets cacsexncescaenn BE. | 12 |-0.13 | —0.13 +0.12 —0.03 | +0.16 23 59. 04 23.47.38 | —11. 65 24 Comm seq.....---.----- E. | i | —0.13 ; —0.13 +0. 24 —0.03 | +0.16 29 24. 05 29 12.46 | —11.70 | y Virginis med .........- E. | 11 | 0.13 —0.09 +0, 22 —0.05 | +0.15 35 52. 21 35 40.76 | —11.58 '¢ Virginis...........----| E | a1 | —0.13 | —0.11 +0.23 —0.04 | +0. 13 56 29. 59 56 18.15 | 11.56 a Urse Min., L.C........ E. 7 | 0.16] +4.46 —4.93 —2.25| 40.11} 13150801) 114509 | —167 a Urse Min.,L.C.......- Ww. 7 | 0.18 | +5.20 $6.31 —2.25 | 40.11 14 57.13 144509 | —17.8 + Bootis ..... w. {| a | —0.13 | —0.12 —0. 27 —0.03 | +0. 08 41 51.55 | 13 4139.52; —11. 64 n Bootis .........22+0+2- W. | 11 | —0.07 | —0.06 —0.27 —0.03 | +0.07 49 16.29 49 04.22 | —11.74 d Bootis .......2-2--++++- Ww. | 1 | —0.05 | —0.05 —0. 28 —0.02 | +0.05 | 1405 13.51) 14 05 01.44) —11.74 4 Ursm Min.......-.2..-- WwW. | 11 | —0.05 | —0.20 —1.24 +0.19 | +0. 05 09 37. 66 09 24.5 | —411.7 Virginis ............-..| W. | 11 | —0.05 | —0.03 —0. 25 —0.05 | +0. 04 22 19.78 22 07.66 | —11. 84 x Bootis, pr........------ w.| 1 | —0.05 | —0.05 —0.27 —0.03 | $0.03 35 23.40 35 11.26 | —11.82 47 Hey. Ceph., L.C ....--- w. | it | —0.03 40.07 +1.33 —0.31 | +0.01 5029.71} 250195 | —11.6 Gr. 750, L. C...2-2-22--+ Ww. 8 | —0.04 | 40.29 41.76 —0.66 | —0.06 | 1559 47.50' 359368 | —128. Gr. 750, L.C.....2-22--+ E. 5 | 0.04 | 40.29 —1.37 —0.66 | —0.06 | 15 59 50.38 5936.8 | —12.5 8 Ophiuchi ......-....... E. | 11 | —0.08 | —0.06 +0. 22 —0.05 | —0.07 | 16 08 21.85} 1608 10.20) —11.81 ¢ Opbiuchi ...........--- E. | 11 | —0.08 | —0.06 +0. 22 —0.05 | —0.07 12 16. 84 12 05.15 | —11. 85 y Herculis....--.---.+-++ E. | 11 | —0.08 | —0.08 +0. 24 —0.03 | —0.08 16 55.26 16 43.54 | —11.88 w Hereulis...........---, E. | 11 | —0.08 | —0.07 +0.12 —0.03 | —0. 08 20 10. 63 19 58.79 | —11.89 B Herculis.........------ E. | 11 | —0.08 | —0.08 +0.24 —0.03 | —0.09 25 21.27 25 09.55 | —11. 88 o Herculis...........-.--| E. | a1 | —0.08|—0.u1 +0.15 0.00 | —0. 09 30 30. 95 30 19.21} —11.78 ¢ Herculis........---.+-+ w. | 1. | —0.03 | —0.03 —0.30 —0.02 | —0.10 37 03.37 36 51.26 | —11.78 Hereulis .......-.-.--++ w. | it | —0.03 | —0. 04 —0. 33 0.00 | —0.11 39 04.52 38 52.19} —11.96 « Ophiuchi. .........+-+- Ww. | 1 | —0.03 | —0. 02 —0. 26 —0.04 | —0.12 52 17.70 52 05.54 | —11. 88 e Urse Min..........--.| W. | 11 | —0.03 | —0%14 —1, 08 "0.33 | —0. 12 58 30. 50 58 17.6 Shes 19 Hev.Camelop.,L.C....| W. | 11 | —0.03 | +0.08 41.34 —0.31 | —0.13 | 17 03°09.57} 502584 | —12.6 a Herculis....-.-..-. --. w. | 11 | —0.03 | —0.02 —0.26 —0.03 | —0.14 09 28.89 17 0916.63! —11.98 mw Herculis ......-..--.+-- W. | 12 | —0.03 | —0. 04 —0.18 —o0.01 | —0.14 11 09. 38 10 57.30 | —11.86 Adopted collimation=+0*. 130 Clamp East. (See Table 12.) : OBSERVATION-EQUATIONS. Epoch 15.0 clock time. At=—11*.7+A0. Weight. Weight. + 0, 55a—3, 0p+A0—0.06=0=+40.04 1 + 4,46a—0.2p+A0—0.1 =v=—0.5 0.09 2.89 2.9 + —03 == 00 O01 | 4 9 55041. 0p4A041.0 =v=40.2 0.01 + 0.67 —2.8 + —0.10=v=—0.03 1 $0.72 $114 4011=v=—0.06 1 + 0.06 —2.7 + —0.13=0=—0.01 1 40.73 $1.2 + +0.15=»=—0.02 1 + 0.38 —2.6 + —0.05=v=40.03 1 40.41 41.3 4 40,18=v=+40.02 1 + 0.42 —25+4+ 0.00=v=+0.08 1 + 0.49 $1.3 + 40.19=v=40.03 1 + 0.68 —24 4+ —0,12=v=—0.07 1 1030 4144 40180-9001 1 + 0,52 —21+4+ —0.14=v=—0.10 1 — 0.01 41.5 + +40,08=»=—0.06 1 432,58 1.7 + 45.5 =0=4+3.3 0.001) 1 g 09 44.64 40.08=v=—0.09 1 + 0.43 —1.3 + —0.06=v=—0.06 1 + 0.07 $1.7 + +40.26=v=40.10 1 40.42 —1.2 4+ +40,04=0=40.03 1 40.55 41.94 40.18=0=—-0.03 1 + 0.32 —0.9 + +40.04=v=+0. 02 #1 Red 4.74 $2.0 + 40.0 =v=-+0.2 0. 04 — 2.83 08 + 00 =v=402 O01 | 4 451 4214 40.9 =v=404 0.09 s + 0.70 —0.6 + 40.14=r=+0.08 1 40.48 4224+ 40.28=v=40.06 1 : +045 04 + 40.12=v=+0.07 1 40,12 +224 40.16=v=—0.04 1 NORMAL EQUATIONS. RESULTS. 413. 28a— 2. 70p+ 9.95A0-+1.27=0 A0=—0". 052 — 2,70 482,22 — 5.21 +4.68=0 p=—0°. 062 : a=—0*, 069 4+ 9,95 — 5,21 423.43 +41.59=0 S86 APPENDIX Hi. (App. IIL, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 17.—TIME DETERMINATION. (Cambridge, Mass , June 4, 1881. O. B. Wheeler, Observer. ] i Star ee | b BL | Olefaitabn) Ad Rp ee TASRENS OIA “gee I ge ae 8. s | a, hom. 8. h.m. 8. 8. a Urse Min.,L C....--. E. 8 —0.09 | +2. 60 | —4. 92 —2.70 | +0. 08 13 15 20. 88 1 14 58, ° —20. 4 a Urse Min, L.C......) W. 7 | 0.03 40.74, 46.30 2.70 | +£0.08 15 08.77 14 58 2 ATG Pp: ADOOUIS: sy cuscmgervnns dat W. 11 0.00 0.00 —0. 30 —0, 02 | +0. 04 14 27 01, 50 14 26 45,19 —16. 01 mw Bootis, pr..--. -.-----| W. 11 0.00 0. 00 —0. 27 —0.04 | +0. 04 35 27. 68 35 11. 24 —16. 17 fe VANEIDIS. weenie ciccees s| We 11 0. 00 0. 00 —0. 25 —0. 06 | -++0. 04 37 07. 14 36 50. 84 —16. 05 109 Virginis ....... -| W. ll 0. 00 0. 00 —0. 25 —0.05 | +0. 04 40 33. 62 40 17. 36 —16. 01 P. XIV, 221 Ww. 1L 0. 00 0. 00 —0, 26 —0. 04 | +10. 03 50 55. 83 50 39. 55 —16. 02 B Bootis ...... --.----..- EK. 11 +0. 03 | +0. 03 +0. 29 0.00 | +0. 03 57 46. 63 57 30.96 —15. 99 P Boous: seescil teens ces E. 11 +0.03 +0. 03 +0. 25 —0. 02 | +0. 03 59 39.71 59 24. 00 —15 99 6 Boots sce ses essecesese E. 11 +0.03 | +0. 03 +0, 27 —0. 01 | +0 02 15 11 01.10 15 10 45. 53 —15. 87 » Bootis E. at | +0. 03 | +0. 03 +0. 28 —0. 01 | +0, 02 20 18.45 20 02. 85 —15. 91 8B Coron. Bor....-.-..-... E. 11 | +0. 03 | +9. 03 +0. 26 —0. 02 | +0. 02 23 14.26 22 58. 62 —15. 93 u Coron. Bor.....-....-. E. 11 +0. 03 | --0. 03 +0. 25 —0. 02 | +0. 01 29 37.91 29 42. 26 —15. 93 ¢ Coron. Bor. seq ..--.-- E. 11 +0. 03 | +0. 03 +0, 28 —0.01 | +0. 01 35 12.58 34 56.99 —15. 90 e y Coron. bor .........-.. E. 11 +0. 03 | +0. 03 +0. 25 —0. 02 | +0. 01 38 03. 63 37 48.02 —15. 89 Gers 100} eC ccsstcct | JE: 9 +0. 03 | —0.19 —1.37 —0.79 | —0. 00 15 59 56. 60 3 59 37.9 —17.1 y Herculis.....-. -| E. 1] +0. 03 | +0. 03 +0. 25 —0. 03 | —0. 01 16 16 59. 44 16 16 43, 61 —16.11 » Ursxe Min E. 11 +0. 03 | +0. 09 +0. 47 +0.19 | —0. 01 21 19.12 21 03.8 —15.9 ¢ Herculis....-..-..-... _E. 11 +0. 03 | +0. 03 +0, 26 —0. 02 | —0. 02 37 06.95 36 51.31 —15, 93 y Herculis..............| E. 11 +0. 03 | -+-0. 04 +0. 29 —0.01 |} —0. 02 39 07. 80 88 52. 26 —15. 87 49 Herculis.........-.-.. .E. 11 +0.03 | +0. 03 +0. 23 —0. 04 | —0. 02 46 59.09 46 43.25 —16. 10 « Ophiuchi -| E. 11 +0. 03 | +0. 03 +0. 22 —0.05 | —0. 03 52 21.46 52 05. 62 —16. 09 e Urse# Min ............ E. 9 +0.08 +0. 45 +0. 84 +0. 39 | —0. 03 58 32.46 5817.5 —16.2 ¢ Urse Min ..... 2.2... w.| 6 | 40.08 | +0.45 —1.08 +40.39 | —0. 03 58 34.75 58 17.5 16.6 «a Herculis...---......-. 7 11 -+0.05 +0. 05 —0. 26 —0. 04 | —0.03 17 09 33, 08 17 09 16.74 —16. 13 6 Herculis.............. i 11 +0. 05 | +0. 05 —0. 16 —0.03 | —0. 03 10 28.10 10 11.96 —16. 03 a Ophiuchi 11 4-0. 05 | +0. 04 —0 26. —0. 04 | —0. 04 29 44,51 29 28.10 —16,19 B Ophiuchi 11 +0.05 | +0. 04 —0. 25 —0.05 | —0. 05 37 55.49 37 39.19 —16. 09 « Herculis . 11 +0.05 | +0. 06 —0. 29 —0. 02 | —0. 05 42 07. 65 41 51.38 —16, 04 @ Herculis.............. Hi 11 -++0.05 | +0. 06 —0. 32 —. 01 | —0. 06 52 29. 75 : 52 13.49 —16. 00 67 Ophiuchi @ 11 j 40.05 | +0. 04 —0. 25 —0.05 | —0. 06 55 01. 07 54 44.73 —16.18 | 6 Urse Min ............ Ww. 7 | +0.04 +0. 48 —2.47 +0. 98 | —0. 06 18 11 07. 56 18 10 50.1 —15.5 | 8 Ursee Min 2.2.2.2... E. | 8 | to01 | +016 41.92 +40.98 | —0. 06 11 03.98 10 50.1 —16.0 Adopted collimation=+0%. 130 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 16". 0 clock time. At=—16*.0+A90. Weight. Weight. +32. 57a—2. Tp |-A6+3.0 —v=+0.4 0.001 | + 0.42a 40. 3p-+46+0. 11=v= +0. 08 1 + 0.23 —-1.5 +4 +40.01l=v=+40.04 1 — 2.294044 —0.1 =v=401 ~ 0.13 +045 —14 + 40.17=v=40.18 * 1 + 0.22 40.6 + —0,07=v=—0.10 1 + 0.74 —14 + 40.05=v=+0. 04 1 + 0.08 +0.7 4+ —0.13=0=—0. 15 1 + 0.64 1.3 + +40.01=v=+0.01 1 + 0.47 40.8 + +40.10=v0=+0.05 1 + 0.48124 +40.02=0=40.02 , 1 + 0.55 +0.9 + +0.09=v=+0. 02 1 + 0.04 —1.0 + —0.01=v= +0. 03 1 —4.74 41.0 + 40.4 =v=40.8 0. 04 + 0.29 10 4 —0,01=»=+0.01 a + 0.48 $1.2 + 40.13=0=+0.07 1 + 0.18 —0.8 + —0.13=»=—0.11 1 + 0.33 41.2 + 40.03=»=—0. 02 1 + 0.10 —0.7 + —0,09=v=—0. 07 1 + 0.51 41.5 + +0.19=y=+40.12 1 + 0.26 —0.6 + —0.07=v——0.06 1 +062 41.64 +0.09=r= 0.00 1 + 0.30 —0.5 + —0.07=v=—0.07 1 + 0.98 41.7 + 40.04=v=—0. 02 1 + 0.12 0.4 + —0.10=v=—0.09 1 + 0.114194 0.00=v=—0. 06 1 . + 0.30044 —0.11=v=—0.11 1 + 0.64 41.9 + 40.13=v=+0. 03 ] + 9.55040. 0p+A041.1 =v=+40.3 0k) te ase St ee ee me NORMAL EQUATIONS. RESULTS. +9.00a+4 0.74p+ 8.3646+0. 65=0 AG=+05. 014 +0.74 +33.67 + 2.21 +1.00=0 p=—0°, 029 +8.36 + 2.21 425.19 +0,40=0 =—0°. 083 st] LONGITUDE OF DETROIT FROM CAMBRIDGE. 887 Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 18.—TIME DETERMINATION. (Cambridge, Mass., June 11, 1881. O. B. Wheeler, Observer. | | N f | i Star. cl. | med | b Bo | Cw+ai+av'ny| Aa | Rp |Clocktimeo!) Rightascon:) — ,_1, | s. o. oo oe roe hom. 3. him. 8. oe a Urse Min., L.C....... E.| 8 0.00} 0.00 —4, 92 —4.62 | —0.02 | 13 15 31.79] 115 04.8 —22.1 a Urse Min., L.C....... W.: 7 | —0.02 | +0.56 +6. 30 —4.62 | --0. 02 15 19.44 15 05.8 —21.5 Tt Bootis deideicktesktents w. | 11 | —0.05 | —0. 04 —0. 27 —0.06 | —0. 02 41 55.52 | 18 41 39.41 | —15.80 n Bootis .-.....--.- wW. JL | —0.05 | —0. 04 -—0, 27 —0.06 | —0. 02 49 20.18 49 04.12 | —15.75 d Bootis sas Baal eee w. | 11 | —0.05 | —0.05 —0.28 —0.05 | —0,02 | 14 05 17.46} 14 05 01.33 | —15. 80 » Virginis .............. W.' 31 | 0.05 | —0.04 —0.25 —0.10 | —0.01 22 23.70 22 07.63 | —15.78 = Bootis pr ...........| W. | 11 | —0.05 | —0.04 ~-0.27 —0.06 | —0. 01 35 27.35 35.11.21} —15. 88 » Virginis Ww. 11; —0.05 | —0.03 —0. 25 —0.11 | —0, 01 37 06. 95 36 50.83 | —15. 84 109 Virginis w. 11 | —0.05 | —0. 04 —0. 25 —0.09 ;} —0. 01 40 33.46 40.17.34 | —15. 83 | 47 Hev.Ceph.,L.C ...... w. 6 | —0.05 | 40.14 +0.73 —0.63 | —0. 01 50 36.17 | 2 50 20.8 —16.3 47 Hev.Ceph., L.C ...... E. 6 | —0.03 | +0.07 —0. 59 —0.63 | —0. 01 50 37.43 50 20.8 —16.1 w Bootis ...-.....---.... E. |. 11 | —0.03 | —0. 03 +0. 25 —0.04 | —0.01 59 39.49 | 14 59 23.96] —15.75 48 Hev.Ceph., L.C ...... E. | 11 | —0.03 | +0.07 —1.00 —0.56 | —0.01 | 15 05 34.55] 3 05 17.3 —16.3 3 Serpentis ........-.--- E. oo —0.03 | —0. 02 +0. 22 —v.09 | —0. 01 09 35.43 | 15 0919.93 | —15.70 u Bootis .-.......--..--- E. 11 | —0.03 | —0. 04 +0. 28 —0.01 | —0.01 20 18.31 20 02.82 | —15. 73 8 Coron. Bor E. 8 | —0.03 | —0.03 +0.13 —0. 04 | —0. 00 23 14.11 22 58.60 | —15.61 v? Bootis ...--..-- E. 11 | —0.03 | —0. 04 -+0. 30 0.00 | —0. 00 27 49.89 27 34.55 | —15. 60 ¢ Coron. Bor E. 11 | —0.03 | —0. 04 +0. 28 —0.02 | —0. 00 35 12.42 34 56.97 | —15.69 y Coron. Bor... sates E. ll —0. 03 | —0, 03 +0. 25 —0. 04 | —0. 00 38 03. 43 37 48. 02 —15. 63 0; Serpentis .........--. E i —0.03 | —0. 03 +0. 23 —0. 07 | —0. 00 41 00. 69 40 45.11 —15. 78 « Serpentis .....-...---- E. WW —0. 03 | —0. 03 +0. 12 —0.06 | —0 00 43 42. 03 43 26. 38 —15. 74 ¢ Urse Min........-... E. 11 | —0.04 | —0. 16 +1. 08 +0.40 | —0: 00 48 38.33 48 23.9 —15.3 '¢ Urs Min .........-.. E. 9 —0.01 | —0. 07 +0. 84 +0.67 | +0.01 | 16 58 31.73 | 16 58 17.3 —15.2 & Ure Mithgoccsasscccs Ww. 6 —0.03 | —0.15 —1.08 +0. 67 | +0. 01 58 33. 87 58 17.3 —15.3 « Herculis........2..--. Ww. 11 | —0.01 | —0.01 —0. 26 —o0.07 | +0.01 | 17 09 32.85] 1709 16.81 | —15.77 aw Herculis ...--.....--.- w. ll —0. 01 | — 0.02 —0. 32 —0. 02 | +0. 01 11 13.53 10 57.45 —15. 74 B Draconis..........---. w. 11 | —0.01 | —0. 02 —0. 42 40.04 | +0. 01 28 04. 06 27 47.94 | —15. 68 a Ophiuchi | W. 11 | —0.01 | —0. 01 —0. 26 —0.07 | +0.01 29 44. 36 29 28.19 | —15.90 « Herculis ..| EB. 11 | —0.03 | —0. 04 +0. 32 +0. 01 | +0. 01 36 25. 01 36 09.66 | —15.63 B Opbiuchi | #. 11 | —0.03 | —0. 02 +0. 22 —0.09 | +0. 01 37 54. 85 37 39.29 | —15.76 » Herculis E. 11 | —0.03 | —0. 03 +0. 25 —0.04 | +0. 01 42 06. 92 41 51.48 | —15.66 6 Herculis . E. 11 | —0.03 | —0. of +0. 28 —0.02 | +0. 02 52 29.10 52 13.59 | —15.75 67 Opbiucbi ao 11 | —0.03 | —0. 02 +0. 22 —0.09 | +0. 02 55 00. 29 54 44.85 | —15.64 6 Urse Min ......- | E. 6 | —0.03 | —0.32 +1. 92 41.68 | +0.02 | 18 11 02.43} 18 10 50.3 —13.7 , 8 Ursew Min........-.--- Ww. 2 | —0.04 | —0.48 —2. 47 41.68 | +0. 02 11 06. 35 10 50.3 —13.1 mites a sity ij : Adopted collimation = + 05.130 Clamp East. (Sce Table 12.) OBSERVATION-EQUATIONS. Epoch 16.0 clock time. At=—15*.7+Aé. Weight. Weight, 4.32. 57a—2 Tp+A046.1 =v=415 0.001 | + 0.30a—0. 4p-+A8—0. 07=v=—0.10 1 + 0.43 —2.3 + +40.10=0=+40.038 1 + 0.46 —0.3 + +0.08=0=40.02 1 + 0.42 —22 + 40.05=0=—0.02 1 + 0.43 —0.3 + 40.04=v=—0.01 1 + 0.38 —19 + +40:10=v=+0.04 1 — 2.84 —0.2 4+ —04 =v= 0.0 O11 +. ig yliest= — 4, Mat. dp+A0—0.5 =v=+0.2 0.04 $0.45 14 4 40.18=0=+0.07 1 40.48 412+ +40.07=0=40.02 1 $074 44 40:M=v=+003 1 40.12 +124 40.04=0=+40.04 1 + 0.64 —1.3 4- 4+0.13=v=+40.04 1 — 0.29 $1.5 + —0,02=v=40.04 0.6 + 4.46 —1.2-+ 40.5 =v=—0.1 0.09 40.51 415 + $0.20=0=4015 1 + 0.29 —10 4+ 40.05=v=40.01 1 Lon oa 408 ces tO Ooll — 0.09 41.6 + —0.07=v=—0.04 0.7 40.60 —0.8 + 0.00=v=—0.09 1 - nek ee ws ee ae ‘ + 0.10 0.7 + +0.08==+40.02 1 40.11 1.9 + 40.05=0=40.06 1 0 a0 eae + 0.64 41.9 -4+ —0.06=v=—0.12 1 + 0.02 —0.5 + —0.10=v=—0.09 1 11.80 $2.2 + —23 =v=—0.6 0.005 + 0.12 —0.4 -+ —0.01=v=—0.02 1 NORMAL EQUATIONS. RESULTS, +11. 52a— 4,75p+ 9. 11A0+1, 55=0 Ad= +0. 013 — 4.75 +46.31 — 4.14 —1.01=0 p=-+08, 008 @ 49.11 — 4.14 424.45 +1.00=0 a=—O*. 142 888 APPENDIX III. LApp. III, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 19.—TIME DETERMINATION. (Detroit, Mich., June 21, 1881. O. B. Wheeler, Observer.] | Star. Cl. Npot » | 2b | Cerartaim| do | za |Clpckiimeot) Bight atcen:) 4, he Lo | i a Bs 8 s & hem. 8. hom. 8 & | a Urse Min., L.C WwW. 8 | 40.05 —1.49 44. 36 42.5L | —0.0l | 13:15 0895 | 115 13.8 +2.0 | a@ Urse Min., L.C.. E. 7 | 40.03 | —0.74 —2, 98 42.51 | —0. 01 15.17.55] 115 13.8 0.0 | 1 Virginis . . cseel E | 11 | 40 04) 40.08 +0.18 +0.05 | —0.01 56 36.55} 13 5538.54] 41.78 d Bootis .........2+. wee) ER} om | 40.05 | 40.05 +0. 20 +0.02 | —0.01 | 14 04 59.14 | 14 05 01.22} +41.83 4 Uree Min .....2...... KE, 9 | 40.06 | +0. 26 +0. 33 —0.22 | —0. 01 09 20.58 09 22.6 414 | ® Virginis E. lt | 40.07 | 40.05 -+0.18 +0.05 | —0. 01 22 05. 53 2207 57 | 41.81 p Bootis ......... K. 11 | 40.07, +0.08 +0.21 0.02, —0.01 26 42, 88 26 45.04 | 41.87 uw Virginis E. 1 | 40.07 | +0. 04 +0. 18 40.06 | —0. 01 36 48,73 36 50.79, 41.84 | 109 Virginis ..............| E. lL | 40.07 | $0.05 +0.18 +0.05 | —0. 01 40 15.19 40 17.30) 41.88 | 47 Hev. Ceph., L.C ....-. E. 7 | 40.08 | —0. 20 —0. 36 40.34 | —0. 01 50 20.36} 2 50 21.9 21 | 47 Hev.Ceph.,L.C ...... Ww. 7 | 40.04 | —0.10 +0. 53 +0. 34 | —0. 01 5019.14) 2 50 21.9 42.3 48 Hev. Ceph., L.C .. w. |} 1 | 40.03 | —0.07 +0..95 40.30 —0.0L | 1505 15.61] 3 05 18.3 41.8 3 Serpentis ...........-. w. | 1 | 40.03 | $0.02 —0.21 +0.05 ; —0 OL 0912.20 1509 19.89] 41.88 6 Bootie ......-....-.... W.) i | 40.01 | 40.01 —0. 25 +001 —0.01 10 43. 61 10 45.42} 42.05 B Coron, Bor......2.--- Ww. 11 | 40.06 | 40.08 —0, 23 40.02 —0.00 22 56, 89 22 58.53 | 41.81 ¢ Coron. Bor. sey.....--- W. out | 40.06 | +0.08 —0.26 +0. 01 | —0,00 BH SRAM) 34 86.88) 41.62 y Coron. Bor ........-.-- w.. 1! 40.06 | +0.07 —0. 23 40.02 —0.00 37 46.23 | 37 47.98) 41.91 8 Serpentis WwW. | i | 40.06! 40.06 —0.22 +004 —0. 00 4043.33! 40.45.09} 41.92 ¢ Urs Miu W.| uo 40.32 | 40.47 —1.02 —0.22 —0. 00 48.23.03, 48 23.4 40.9 Gr 2373 oe] We | oan | 40.05 | 40.20 —0. 98 —0.21 40.00] 1635 49.72 16 35 50.4 415 49 Herculis...........--. Wo) 11 | 40.05) 40.05; —0.22 +0. 04 | +0.00 4641.55! 46 43.37 | 4.1.99 « Urse Min w. 8 | 40.05 | 40.28 0.75 —0.37 | +0.00 58 15.54 58 16.8 417 « Urse Min E. 7 | 40.06 | +0.37 +0. 51 —0. 37 | +0.00 58 14. 66 58 16.8 a8 a Herculis...........--. E. | 1 | 40.06 | +0.05 +0. 18 +0.04 | -+0.01 | 17 09 14.87) 17 0916.88] +1.78 a Herculis..........---- E. | 1 | 40.06 | +0.07 +0. 22 +0.01 | +0. 01 10 55.39 10 57.51} 41.83 x Herculis... .......--. E. | 1 | 40.06] 40.09 40.27 —0.01 | 40.01 23 36.21 23.38.25) 41.68 iP Draconis: <.c0222 220525 E. 11 -+0.06 | -+0.10 +0. 29 —0.02 | +0. 01 27 45.75 27 47.97 +1. 83 a Ophiuchi ............. E. | 1 | 40.05 | 40.04 +0.18 40.04 | +0. 01 29 26.18 29 28,29] +1.89 B Opbiuchi ............. w. | oa | 40.01) +0.01 —0.21 +0.05 | 40.01 37 37. 67 37 39.41 | 41.94 » Herculis... --{ We. | 41 | 40.02 | -F0. 02 —0. 24 +0.02 | +0. 01 41 49.91 41 51.56} 41.87 @ Hercillis.............. w. | iu | 40.02 | +0. 02 —0. 26 +0.0L | 40.01 52 12.05 52.13.68} +187 67 Opbiuchi ............. WwW. | 11 | 40.02 | +0. 02 —0, 21 +0.05 40.01 54 43,31 54 44.98} +1.86 72 Ophiuchi ..........-.- w. | 1 | 40.02] +0. 02 —0.21 +0.04 | 46.01] 18 01 44.63] 1801 46.16] 41.72 8 Ursa Min Ww. 7 | 40.03 | 40.32 —1.71 —0.91 | +0. 01 10 49.74 10 50.1 41.8 8 Urse Min E. 7 {|- 0.00} 0.00 +1.17 —0.91 | +0. 01 10 47.10 10 50.1 +1.8 Adopted collimation =-~+ 0*.085 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 16.2 clock time. At=+1*.8+A@, Weight. Weight. +32. 57a—2. 9p +A0+0.8 =v=+43.3 0.001 | — 2.85a—0.4ptA0+0.9 =v= +07 0.1 gies Coenen chee ee ete OU — 2. Ta+0.4p+A0-40.3 =v=40.1 0.1 y + 0.32 —214+ —0.03=v=—0.01 1 — 2.84 20 + 404 s0=402 01 FOr 3 06 ee 0. patel od Eh ee ie ae — 4.75 +08 + 40.3 =v=—0.1 0.04 ty (ed ieee hae + 0.48 410+ +40.02=r=+40.08 1 40.74 —L6 4+ —0.0d=v=+40.02 1 eb Ole op EO abr 0 OB So 0.00 DA pee eee aaa: 016 412+ 40=v=4013 0.65 — 0.28 413 4+ +0.083=0=—0.03 0.6 BG SLE Ok OL OOD oe BD aed a als, 0! 09= S008 ee ee ee Ne eel pid A lh ee ee A Se roe 40.28 415-4 —0.07=v=—003 1 ee ee + 0.11 41.7 + —0.07=v=—0.04 1 + 0.26 0.8 + —0.01=v=+0.02 1 ahd Aa AMMA A x coe - Se a 40.55 $1.84 40.08=0=4014 1 Fh oy As eee —11,81 420+ 0.0 =0=-0.9 0.01 NORMAL EQUATIONS. RESULTS. +13. 94a— 3. 66p+ 8. 41Ae—1. 18=0 Ag= +0", 014 — 3.66 444.66 + 2.44 +40,09=0 p=-+05. 005 —0.96=0 a= +05. 077 e 4+ 841 4+ 244 422.80 § 1. LONGITUDE OF DETROIT FROM CAMBRIDGE. Difference of Longitude, Detroit, Mich., and Cambridge, Mass,—Continued. TABLE 20.—TIME DETERMINATION. (Detroit, Mich., June 22, 1881. O. B. Wheeler, Observer.) - Star. a Ursv Min., L.C a Urse Min., L.C...... 7 Virginis d Bootis ...-.------ « Virginis ........-.-,-- 4 Urse Min ¢ Virginis p Bootis » Virginis 109 Virginis .--s-02225 «+ 47 Hev. Ceph., L.C 47 Hev. Ceph., L.C wy Bootis 48 Hev.Ceph., L.€ 3 Serpentis 8 Coron. Bor. .-- y Coron. Bor.-....----- a Serpentis ........----- 8 Serpentis ¢ Urse Min e Ursa Min e Urse Min @ Herculliisicssea2 ose. 67 Ophiuchi 72 Ophiuchi o Herculis ...--..--..-- 6 Urse Min 6 Urse Min -- j | O: LBRSE) | me /cvtaiterm | ae | me (Ouakumeet Righteen | ot gassed | Lo a: 8. é: Be | i hom. 8. hm. 8. 8. 1) oe lool —2.98 0.13 40.05 | 13152166) 115148 | —0.5 iw. |} 7 | 40.08 | —2.23 +4. 36 —0.13 | 40.05 1512.43) 115148 | 40.3 .{ ij 40.13 | 40.10 —0.21 0.00 | +0. 04 55 34.85] 13 55 38.54 | 43.80 W. 1 | 40.13 | +0.14 —0.23 0.00} +0.04| 14 0457.40] 14.05 01.22) 43.91 OW.) 11 | $0.13) -+0.08 —0.21 0,00 4.0.04 06 32. 53 06 36.25 | 43.85 | We) ML | 40.13} 40.51 —1.01 +0.01 | +0.03 09 19. 33 0922.5 | +43.6 . We; 2 | 40.13 | 40.09 —0.21 0.00 | 40.03 22 08. 84 22 07.57) -+3.85 JW, | nn | 40.13 | 40.15 —0.%4 0.00 | 40.03 26 41.22 26 45.04 | +3.91 - wi! a | 40.13 | 40.09 0.21 0.00 +£0.03 36 47. 05 36 50.79 | +3. 86 w. | i1 | 0,42) 4010 0.21 0.00 | 0,03 40 13.58 40.17.29} 43.82 We | 7 | 40.12 | —0.32 40.53 —0.02 | 40,03 5017.33 | 250220 | 44.4 | EB} 7 | 40.071 —0.18 —0. 36 —0.02 | 40.03 50 18 92 5022.0 | 43.6 BE. | 11 | +0.07| +0.07 +0. 20 0.00 | 40.02 59 19.74 | 14 59 23.88 © -+3.87 | EB | nm | 4007) 016; —0.81 —0.02 | +002] 150515.35| 305183 | +44.0 | EB a | 40.07) 40.06; +40.18 0.00 | 40.02 09 15.74} 150919.89 +43.91 E. | 1 | $0.07) 40.08) — +0.20 0.00 | +0.02 22 54. 32 22 58.54 | +3.94 -| Ej 41 | -+0.07) +0.08 +0. 20 0.00 | +0. 01 37 43.75 37 47.97 | +3.94 Eu | 40,07, 40.06 = $0.07 0.00 | +0.01 38 23. 88 3827.88 | 43.87 |e. ! a | 40.07 | 40.07 40.18 0.00 | +0.01 40 40. 90 40 45.09 3.94 E. | 1 | 40.08) 40.31 +0. 86 40.01} $0.01 48 18.33 4823.3 | 43.8 |g. | 8 | 40.13; 40.75 40.51 |.40.02 | 0.01] 16581200} 1658167 | +434 JW. 7 | +40.12| +0.67 | a7 =| 4002 | 0.01 58 12. 90 58167 | +43.9 |owoi a | 40.11) 40.13 0.26 0.00 | 0.02] 1752 09.99] 1752 13.68 +48.82 ow. a1 | 40.08} +0. 06 —0.21 0.00 | —0. 02 54 41.23 54 44.98) +3. 90 ow. a | 40.08! 40.07° —0.22 0.00 | —0.02 | 18 01 42.69] 1801 46.17 43.72 Ww. 1 | 40.08!) 40.09, —0,24 6.00 | —0. 03 02 53.78 02 57.52 +3. 89 |W.) 6 | 40.05! 40.63. —1.71 +£0.05 | —0.03 10 46.18 10 50.1 } +5.0 | EO; 7 | 40.05) 40.68) 9 +117 $0.05 | —0,03 10 44.19 10 50.1 | +4.1 BE. ou | 40.05) $0.05; 40.19 0.00 | —0. 03 18 37.10 18 41.18| $3.84 BE.) It | 40.06) 40.08. — -+0.23 0.00} —0.03' 32.53.93 32 57.98) 43.74 E. 11 | 40.06 | +0.06 +0.19 0.00 | —0. 03 40 32. 03 40 35.97 | +8 69 E. | 11 | +0,06| +0.07 $0.21 0.00 | —0. 03 45 40.46 45 44.63 | 43.89 BE. | 1 | +0.06! +0.05 40.18 0.00 | —0.03 50 17.93 50 21.99 | +43. 83 E. | 11 | +0.06| 40.06 40.18 0.00 | —0. 04 54 12.8 54.16.92) 43.80 Adopted collimation= + 0°.085 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 16.5 clock time. 432. 57a—3. 2p +A043.9 =v=+3.9 + 0.65 2.6 + 0.00=v=+40.09 + 0.32 24 + —0.1=v=—0. 02 + 0.80 —2.4 + —0.05=v=+40. 04 — 2.84 2.3 + 40.2 =v=40.3 40.70 21 + —0.05=v=+0.03 + 0.23 —21+4+ —0.11=v=—0.03 + 0.74 —1.9 + —0.06=v=+0. 02 + 0.64 1.8 + —0.(2=0=+40. 06 + 4.46 174+ 0.2 =v=—0.1 40.29 15 + —0.07=v= 0,00 +394 144+ 0.2 =v=—0.2 + 0.60 —1.3 + —0,1l=v=—0. 04 + 0.26 —11 + —0,14=v=—0.07 + 0.30 0.9 + —0.14=v=—0. 08 NORMAL EQUATIONS. Weight. o.oo + 1 + i 1 0.1 of 1 + 1 + 1 + 1 0.09 + 1 wa 0.11 4 1 + a - 1 + At=435. 8446. 0. 59a—0. 9p+A8@ -0.07=v=—0. 0 +14. 04a— 5. 38p-+10. 28A0—0. 56=0 — 6.38 +79.30 -- 3.56 +1.36=0 £10.28 — 3,56 +23. 45 112 LS —1.22=0 0.46 —0.8 + —0.14=v=—0.08 1 2.85 —0.7 + 0.0 =v=+0.1 0.1 -- 4.75a40.5p+A0+0.1 =v=+40.2 0.04 0.11 41.4 + —0.02=0=+40.01 1 0.63 41.4 + —0.10=v=—0.07 1 0.55 +1.5 + +40,08=0=40.11 1 0.28 41.5 + —0.09=v=—0.07 1 —11.81 41.7 + -—0.7 =v=—0.6 0.01 0.39 41.8 + —0.04=v=—0.02 1 0.08 420+ 40.06=0=-}0.08 1 0.40 +2.2 + 4011=v=40.138 1 0.19 +2.3 + —0.09=v=-0.07 1 0.62 42.3 + —0,08=v=--0.01 1 0.47 42.4 -+ 0.00=v=+40.01 1 RESULTS. Ad=+0%. 051 p=—0*. 015 a=—0°, 0U4 Weight. 1 890 APPENDIX III. (App. III, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 21.—TIME DETERMINATION. [Detroit, Mich., June 23, 1881. O. B. Wheeler, Observer. ] Star. a SOY) os Bo | Cictsitad'ny| Aa | Rp | Oe ee arate Sacks baa Sas Dee Pec | | Ss. Se Si 3} Bs him. 8. hom. 8. ss a Urs Min., L.C....... ' E, 9 +0.13 | —3.71 —2. 98 —1.79 | 40.07 | 13 15 22.40 | 13 15 15.9 +0. 2 _ » Urse Min. L.C...-.-. W. 6 +0.07 | —1. 86 +4. 36 —1.79 | +0.07 15 14.93 15 15.9 -1.5 |G Booti8sccccescecaws cnc) We 11 +0.13 ) 40.14 —0. 23 —0.02 | +0.05 | 14 04 57.56 | 14 05 01.20 +3. 73 ; y Virginia .......--...-. WwW. 11 +0.18 | +0. 08 +0. 49 —0. 04 | 4-0.05 06 31. 96 06 36, 24 +3.71 ® Virginis.............. W. 11 +0.13 | +0. 09 —0. 21 —0.04 | +0. 05 22 03. 97 22 07. 56 43.71 Pi BOOS: ox 220 weeeyinxens W. 11 +0.13 | +0.15 —0. 24 —0.01 | +0. 05 26 41.33 26 45.01 43.77 mw Bootis pr .....---.---- W. 11 +0.13 ) 40.12 —0. 22 —-0.02 | +0. 04 35 07. 51 35 11.12 +3. 71 109 Virginis ........-..... Ww. 11 +0.13 | +0.10 —0.21 —0.04 | +0. 04 40 13. 62 40 17. 28 +3.77 47 Hev. Ceph., L.C ...... W. a +0.16 | —0. 43 -+0. 53 —0.25 | +0. 04 50 18. 61 2 50 22.2 43.5 47 Hey. Ceph., L.C -...-. E. 6 +0.13 | —0.35 —0. 36 —0.25 | +0, 04 50 18.45 2 50 22.2 +4.5 | W Bootis .:2.025 520002 20 E. 11 +0.12 | +-0.13 +0. 20 —0. 02 | -- 0.03 59 19.72 | 14 59 23. 87 +3. 82 48 Hev. Ceph., L. C....-- E. 11 +0.12 | —0.27 —0. 31 —0. 22 | 40.03 15 05 16. 06 3 05 18.45 +3.0 3 Serpentis .....-.....-- E. 11 +0.12 | +0.09 +0.18 —0.03 | +0. 03 09 15. 90 15 09 19. 88 +38. 71 SO BOOUA os2's caiciseainose te EK. 11 +0.12 | 40,14 -b0. 24 —0.01 | +0. 03 10 41.23 10 45. 39 +3. 81 @ Bootis . 222.6. s26ec22; E. 11 +0.12 | 40.15 +0, 22 — 0.01 | +0, 02 19 58. 66 20 02.71 +3, 68 @ Coron. Bor... . -00<2s E. 11 +0.12 | 40.14 +0. 21 —0.01 | +0. 02 28 06, 87 28 10.95 +3. 73 | a Coron. BOV.s.ees concn a 11 +0.12 | 40.13 +0. 20 —0.02 | +0. 02 29 38.00 29 42.17 +3. 8t ¢ Coron. Bor. seq --..--. E. 11 +0.12 | 40.15 +0. 22 —0.01 | +0 02 34 52.79 34 56, 87 43.71 | ¢ Ursw Min... 2.2.2... E. 11 -+0.11 | +0.41 +0. 86 +0.16 | +0.01 48 18.32 48 23.2 4-3. 6 | & Urs. Min, sc.2esee2s Tr. 7 +0. 07 | +0. 37 +0. 51 +0. 26 | —0. 02 16 58 11. 47 16 58 16.6 +4.2 * Urs Min ..........-. W. 6 +0.11 | +0. 60 —0.75 +0.26 | —0.02 58 13.05 58 16.6 +3.7 a; Berens acess secccass| Wi 11 -+0.11 | +0.10 —0. 22 —0.03 | —0. 03 17 09 13. 23 17 09 16. 88 43.77 i RCV GUYS sage wrexcsivine.s so Ww. 11 --0.11 | +0.14 — 0. 26 —0.01 | —0. 03 10 53.77 10 57. 50 4-3. 85 ! B Draconis .......-..-. | W. 11 +0. 11 +0. 18 —0. 34 +0. 02 | —0. 03 27 44 46 27 47.94 +3. 64 t Herenlis.c:.:-- esicees W. 11 +0.11 | +0. 16 —0. 30 0.00 | —0. 04 36 06. 06 36 09. 68 +3. 76 B Ophiuchi ....-....-... W. 11 +0.11 | +0. 09 —0.21 —0.03 | —0. 04 37 35. 82 37 39.40 +3.70 i Berens: y2ch. veces ny E. il +0.12 | +0.13 +0. 20 —0.02 | — 0.04 41 47.61 41 51.55 +3. 61 @ Mereliss esc science E. 11 +0.12 | +0.15 +0, 22 —0.01 | —0. 04 52 09. 78 52 13. 67 +3. 52 € Herculis .-....-...... E. 11 +0.12 | +0.13 +0. 20 —0. 01 | —0. 04 53 07. 94 53 11.90 +3. 63 67 Opbiuchi ....-.. ..-... E. 11 +0.12 | +0. 09 +0.18 —0.03 | —0. 94 54 40. 91 54 44. 97 43.79 72 Ophiuchi . ore .| E. 11 +0.12 | +0.10 +0. 18 —0.03 | —0.05 | 18 01 42.28) 18 01 46.17 +3 61 @ Herculis.. c2.525205-85+ E. dL +0.12 | 40.13 +0. 20 —0.01 | —0.05 | 02 53. 55 02 57. 51 +3. 63 6 Urs Min .....-...-- E. 6 +0.12 | 41.43 41.17 +0.65 | —0. 05 10 43. 78 10 50.0 $3.6 6 Urse Min ........---- W. 6 +0.09 | +1.10 —1.71 +0. 65 | —0. 05 10 46. 90 10 50.0 43.7 Adopted collimation = + 05.085 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 16.2 clock time. At=-+3*,5-+-A8@, Weight. Weight. 4$22.58a—2. 9p} A0+4.1 —r=4+26 0.001 | 4 0,12a—0.6p-4+A0—0.29=v=+40.03 1 40.32 —214 —0.93=v=40.04 1 — 2.85 044+ 0.1 =v=+0.3 0.066 +080 --21 + —0.18=v=+0.06 1 — 4.75040. 8p4+-A0—0.5 == 6.0 0.029 recy cameeee | aac | ee ee a 40.12 +104 —035=0=-0.15 1 i ee — 0.28 413+ —0.14=v=+40.09 0.5 PD ; Relate 0; — 0.09 41.44 —0.26=v-—0.06 1 4 . a ; bs ae, . O58 |) 4 o.6l 414 4+ —0.20=v=-0.03 1 : 5 ; + 0.28 41.5 + —0.1=0=40.07 1 +o A POGOe ONE T oa oes Neenah aT 4 + 0.60 -10 4 —0.21=r=+40.03 1 4, ob aa ae. mee GS I 1 +018 —1L0 4 —0.31=v=—0.05 1 iG tia ao oaebae- 98 a ‘ ee ee : aes . ; + 0.55 41.8 4+ —0.11=r=40.05 1 ee oa 40.87 418 4 —0.13=0=40.05 1 Ped: =v. | 11.81 $2.0 + 02 =v=+40.6 0.006 NORMS L EQUATIONS. RESULTS. +9,48a— 3. 62p+ 8,1540—1. 48=0 A@= +09, 235 ~3.62 450.13 — 1.51 +41.40=0 p=—0". 025 +8.15 — 1.51 +23.75 —5.16=0 a=—0'. 055 LONGITUDE OF DETROIT FROM CAMBRIDGE. Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. [Detroit, Mich., June 24, 1881. TABLE 22.—TIME DETERMINATION. O. B. Wheeler, Observer. | 891 Star. Ch | Foe |B | Bb [Oe+aizav'ny| da | Rp |Clocktimeof) Rightawen- | . « Uree Min., L.C....... Ww. 8 | +014 | —4'08 44.36 —2 18 | 4.0.10 me inieag) ae Wo | 47 a Ursa Min., L.C....... E. 7 | 40.11 } —2.97 —2. 98 —2.18 | 40.10 15 25.73 15 17.0 —2.8 d Bootis ...........2.... E. 10 | 40.16 | +0.18 —8. 65 —0.02 | +0.07 | 14 05 06.33 | 1405.01.20] +43.34 x Virginis E. 11s | +0.16 | +0.10 £0.18 —0.05 | +0. 07 06 32. 65 06 36.25 | +3.32 ¢ Virginis ... E. 1L | +0.16 | +0. 12 +0.18 —0.05 | +0. 07 22 04. 05 22.07.56 | 43.21 p Bootis ... E. | +0.16 | +0.19 +0. 21 —0.02 | 40.07 26 41.18 26 45.02} +43.44 w Virginis . E. lL | 40.16 | +0.11 40.18 —0.05 | +0.06 36 47.20 36 50.78 | +3.29 ¥ Bootis .......-.......- E. 11 | +0,16 | +0.18 +0. 20 —0.02 | 49.05 59 20. 30 59 23.88 | +3.20 48 Hev.Ceph., L.C.....-. Ww. 11 | +0.18 | —0. 42 +0. 95 —0.26 | 40.04} 1505 14.99] 3051865] 43.1 3 Serpentis ............. w. 11 | +0.18 | +0.15 —0. 21 —0.04 | +0. 04 09 16.54 | 15 0919.89] +3.41 § Bootis ........---...-- w. 11 | 40.18 | 40.22 —0. 25 ~0.01 | +0. 04 10 41.97 10 45.40 | 43.46 6 Coron. Bor.....--..--- w. 1 | 40.18 | 40.21 —0.25 —0.01 | +0.03 28 07. 69 2810.97] 43.32 a Coron. Bor...--.------ Ww. 11 | +0.18 | 4-0.20 —0. 23 —0.02 | +0. 03 29 38. 90 29 42.19} +3.32 ¢ Coron. Bor. seq.....--. Ww. 11 +018 | +0. 23 —0. 26 —0.01 | +0. 03 34 53.54 34 56.89 | +3.38 « Coron. Bor ....--.----- w. 11 | +0.18 | 40.20 | —0. 23 —0.02 | 4-0. 02 52 39. 59 52 42.92 | +3.36 Gr. 750, L.C . w. 6 | 40.18 | —1. 34 | +1. 22 —0.64 | $0.01 59 38.38] 359 41.1 +2.8 Gis 150,Ds Coase: a sceccnse E. 7 | 40.11 | —0.96 —0. 83 —0. 64 | +0. 01 59 40. 87 59 41.1 +2.0 B Draconis ........----- E. 11 | +0.15 | +0. 24 +0. 29 +0.02 | —0.04 | 17 27 44.296} 17 2747.97 | 43.18 a Opbiuchi ...--........ E. 11 | 40.15 | 40.14 +0. 18 *—0.03 | —0. 04 29 24.70 29 28.29 | +43.27 « Herculis..-........--. E. 1 | 40.15 | 40.22 0. 25 +0.01 | —0. 04 36 06.14 36.09.71 | +3.10 B Ophiuchi ...-........- E. 11 | 40.15 | +0.12 +0. 18 —0.04 | —0. 04 37 35. 84 37 39.42 | +43. 28 wu Herenlis.......-....-. E. 11 +0.15 | +0.17 +0. 20 —0. 02 | —0. 04 41 47. 88 41 51.57 +3. 32 E. Ws | 40.15 | 40.12 +0. 18 —0. 04 | —0.05 54 41.44 54.44.99 | 43,25 E. 6 | +0.12 | 41.43 +117 +0.79 | —0.06 | 18 10 43.90] 18 10 49.9 43.4 Ww. 7 | 40.12 | +1.43 ~1.71 +0.79 | —0. 06 10 46. 40 10 49.9 +3.8 w. 1. | 40.16 | +0.16 —0. 23 —0.03 | —0. 06 18 38.01 18 41.20] 43.26 Ww. IL | +0.16 | +0.20 —0.27 0.00 | —0. 07 40 24.18 40 27.21] +43.10 51 Cephei, L.C....-......| W. 5 | 40.16 | —2.07 +4, 32 —1.07 | —0.07 44 08.70 | 6 44 12.7 +1.8 6 Serpentis ...........-. w. 1 | +0.16 | +0.13 —). 21 —0. 04 | —0. 08 50 18.85 | 18 50 22.02) 43.25 e Aquile ............--. Ww. 11 | 40.16 | —0.14 —0. 22 —0.03 | —0. 08 54 13.70 5416.95 | +493. 33 @ AQUIHBsc. seek eobees w. 9 | +0.16 | +0.14 +8. 27 —0.03 | —0. 08 59 48.62 | 19 00 60.07 | +3. 04 Adopted collimation=+0*% 085 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 16, 4 clock time. At=+3%1+A0. Weight. Weight. +32. 58a—3. Ip-+ Ad 45.4 =v=43.5 0.001 | — 0.28a41.1p+ 46 —0.082v=40.11 0.5 + 0.32 —23-4+ —0.25=v=+40.01 1 + 0.51 +L1+ 1 + 0.80 —-23-+4+ —0,22=v=+0.01 1 — 0.09 412+ 40.01l=v=4019 1 + 0.69 —20+4+ —0.11=v=+40.12 1 +061 44.2+ ~—0.18=v=—0.05 1 +£0.23 —20+ —0.34=v=—0.08 1 +0.283 4134+ —0.24=v=—0.09 1 +0.74 18+ —0.18=v=+40.04 1 +068 415+ —0.15=v=—0.03 1 +°0.29 144+ —0.10=v=40.14 1 —11.81 41.84+ —0.5 =v=+0.4 0. 006 +394 134 +40.09=v=+0.1 0.076 | + 0.38 +1.94+ —0.15=v=—0.038 1 + 0.60 124+ —0.32=v=—-011 1 + 0.06 423+ —0.0l=v=40.13 1 + 0.18 —124+ -0.37=v=—0.13 1 415.93 42.34 41.3 =v=+40.4 0. 003 + 0.22 09+ —0,22=v=40.01 1 + 0.62 424+ —0.16=v=—0.07 1 + 0.29 —-09-+4+ —0.23=v=—0.01 1 +0.47 4254+ —0.23=v=-0.13 1 +012 084+ —0.29=v=—0.06 1 +049 +26+ 40.06=v=+0.16 1 +029 054+ —0.27=v=—0.06 1 +955 044+ 40.7 =v=+0.3 0.011 NORMAL EQUATIONS. RESULTs. +9. 49a— 1.22p+ 9, 00A0—1. 26=0 Ad=-+0*. 208 —1.22 465.66 + 1.17 +1.81=0 p=—0°. 032 +9.00 + 1,17 422.60 —4.04=0 a=—0*. 066 892 APPENDIX III. (Apr. II, Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 23—TIME DETERMINATION. : [Dotroit, Mich,, June 29, 1881. O. B. Wheeler, Observer. ] | Star a. | Nest] a | as ‘oepatpaby| da | ay Cloektiment) Bight ascen) ga, 8. be 8 8. hom 8 hom. 8. 8. | a Urs Min., L. C....-- E. 7) 40.12 3.34 —2. 98 +4.79 | 40.06} 1315 22.86} 1 15 21.6 45.1 | a Ure Min., L. C....-- Ww. 8| +007 / 1.86) — +4.36 +4.79 | -++0.06 15 14.94 15 21.6 +41 a Virginis........-....- Wo IL 40,09} 40.06 = —0.24 +0.11 | +0.04 | 14 36 50,21} 14 36 50.74] +0.68 | 48 Hev. Ceph., L.C ...... W.) 11] 40.09) 0.20! 5.53 +0.58 | 40.03 | 15 05 24.00} 305 19.1 +0.8 | 3 Serpentis ............ W. 11 | 40.09 | +0.07 | —0. 21 +-0.09 | +0. 03 09 19.37 | 1509 19.86} +0.63 § Bootie occcuseesecescme Ww. IL} +0.09 | +0. 10 —0. 25 -+0.03 | +0. 03 10 44.99 10 45.34} -+0.50 1 Serpentis ......-.-++++ w. 11 | +0.09 | +0. 08 | —0, 22 +0.07 | 40.03 20 19.10 2019.50 | +0. 54 6 Coron. Bor ....-------- W. It) 40.09 | +0.10 | —0. 25 +0. 03 | +0. 03 28 10. 66 28 10.91 +0. 40 a Coron. Bor....- Ww. 11 | +0.09 | +0. 09 | —0. 23 +0. 04 | +0. 03 29 41.71 29 42.14 +0. 57 ¢ Coron. Bor. seq E. 11} 40.08) 40.10, 0.22 +0.02 | 40.02 34 55, 87 34.56.83 | +0. 64 y Coron. Bor......-..--- E. 11 | -40.08 | +0. 08 | +0. 20 +0.04 | +0. 02 37 47.03 37 47.93 | -+0. 62 B Serpentis .........-.-. BE. 1 | 40.08) 40.07; 40.18 +0.07 | +0. 02 40 44.17 40 45.06} +0.64 u Serpentis............. E. 1t | +0.08 | +0. 05 +0. 18 +0.10 | 40.02 43 27.41 43 28.26) +40.62 y Serpentis ...... E. 11 | 40.08 | +0. 07 +0. 18 +0 07 | +0.02 50 59.97 51 00.78 | 40.56 « Coron. Bor E. 11 | 40.08 | 4-0. 09 +0. 20 +0.04 | +0.02 52 42.09 5242.89) 40.51 Gre 750, Tk Onc enas scmee E, 7 | 40.08 | —0.58 —0. 88 +1.46 | +0. 02 59 41.90 59 42. 2 41.7 Gr. 750, LC... 0-22.02. W. 7| +0.10 | —0.77 41.22 +1.46 | +0. 02 59 38. 54 59 42.2 +3.2 8 Ophiuchi ...........-. Ww. IL} 40.01 | +0. 01 —0.21 40.09 | —0.01 | 17 37 39.01) 17 37 39.45] +40. 64 » Herevlis W. iL | 40.01) 40.01 0,24 +0.04 | —0..01 41 51.26 41 51.59) 40.56 @ Herculis. W. IL | +0.01 | 40.02 —0. 26 +0.01 | —0. 02 52 13.51 5213.71. 40.44 72 Opbiuebi W. IL, 40.01) 40.01 —0. 21 40.08 ' —0.02 | 18 01 45.87 18 01 46.23] +0.56 o Herculis ..........-.. w./ it} 40.01) 40.01 —0. 24 +0.04 | —0. 02 02 57.23 02 57.56 | 4.0.56 8 Ursw Min ...... 2... W. 7) 40.04, 40.47 171 —1.74 | —0. 02 1052.20) 1049.43) 1.5 8 Urse Min ...... ‘kB. 7 40.07 | +0. 80 +117 1.74 | —0. 02 10 50.09| 1049.43; —2.6 109 Herculis / EL) 40.06 | 40.06 +0.19 +0.06 | —0.02 18 40. 58 18 41.24) 40.41 @ Dy cntscnne. sees: EL 11) 40.06 40.08 +0. 23 +0.01 | —0. 03 32. 57.20 32 58.04 | 40.53 110 Herculis ...........-.. |B.) lL) 40.06 40.06 +0.19 +0.06 | —0. 03 40 35. 49 40 36.06} +0.32 51 Sap heh TiC. sac coc esac E. 6 +0.06 —0 79 —1.4t 42.34 | —0.03 4413.12) 44.18.0 42.1 TTB oe yale de aioe le. | 1) $0.06 +0.07 +0.22 40.02 | —0.04 19 03 06.02 | 19 03 06.85! 40.54 @ Lyre sae EB) ou | 40.06 10.08 4-0. 22 40.01) —0.04, 12 16 £8 12 17.72 | +0.5t 6 Aquile ............... E. | 11) 40.06 40.05 +0.18 +0.09 | —0.04 | 19 32.95 19 33.76 | +0.58 Ce iy. | 7 | 40.08 +0. 34 +0. 38 —0. 48 | —0. 04 | 28 55. 89 2856.4 | —0.2 Gr. 2900 ...... _wW. | 7 +008 40.17 —0.55 —0.48 —0.04 28 57. 23 28 56.4 —0.5 He Og prices swosee sancce: w.} ou | 40.06 40.09 —0.31 —0. 04 | —0. 05 | 52 36.48 52.36.67| 40.44 | ‘ f 1 Adopted collimation= +0*. 085 Clamp East. (See Table 12.) OBSERVATION-EQUATIONS. Epoch 174.0 clock time. At=+0*.5+A0@. Weight. Weight. +32. 58a—3. 7p + AP—4.1 =v=+40.7 0.001 + 0.62440. 69+ S@—0.14=v=—0.06 1 + 0.74 —2.4 4 : 1 + 0.28 40.7 + —0.06=r=—0.03 1 + 3.94 —1.9 4+ 0.3 =v=+40.3 0.076 +011 40.94 +40.05=v=40.04 1 + 0.60 —-1.8 4 —0.13=v=—0.01 1 + 0.55 +1.04+ —0.06—v= 0.00 1 + 0.18 —1.8 4+ 0,00=v=+0.06 1 + 0.27 41.0 4+ —0.06=v=—0,04 1 + 0.46 —1.7 + 1 11.81 41.2 4+ 42.6 =r=+0.8 0.006 “0.22 —1.5 4 1 + 0.38 $134 40,09=v=40.13 1 + 0.30 —1.5 + 1 + 0.08 41.5 + —0.03=r=—0.05 1 + 0.12 —1.4 + p +e0.40 +1.7 +. 40.18=v=40.21 1 + 0.30 —1.4 4 1 415.93 41.74 —16 =v=+0.7 0.004 + 0.46 —1.3 + 1 $6148 424 4- —O0¢er=—0.08 1 0. 71 sels 1 + 0.10 +224 —0.04=v=—0.07 1 Se ger de a: 1 “4 0.64 42.3 4 —0.08=v=--0.03 1 1; 0.29 —L1 + 1 — 3.25 +25 + 40.8 =v=40.3 0.006 4 9.55 —1.0 4 0.011 — 0.28 42.94 +40.06=v=—0.03 1 NORMAL EQUATIONS. TREsuLtTs. +9. 78a— 5. 06p+ 8.35A0—1. 49-0 Ad= +0". 005 --5.06 +63.71 — 0.18 +1.84=0 p=—0". 017 48.35 — 0.18 4+24.16 —1,12=0 a=+0%.147 LONGITUDE OF DETROIT FROM CAMBRIDGE. 893 § 1] 1 SIO TF Le | LLT TP LP LLI ‘TP LP voseurmoerere | 866 EL Lb ot GOL ‘ES T— 000 "LE 8F ST GLE FS FE OT 189 “ST 0-- 9ST ‘OT SE 9T LLI ‘TP LP coseseesseses"!) 866 10 OF ST OL ES I— 000 ‘S@ eh ct GLE 'SP 6G OT | G89"ST O— |, LST'8S 6c 9T IT “7 grorjz9q 8F8 0b LE Poy “""l ECB OF LF FOF PS ES ST €0L ES T— LOT ‘8h FS ST 6Ie "G0 OF 9 | 189 "ST O— 000 ‘1% OF 9T | OF8 ‘OF Lb 81h 8h FP ST SOLES I— | OST ‘ZL OP ST | 8Ie'6s ZE9T c9'GEO- | 000°Sh 2E9T | IT — a mo “ves pLiqmeg ¥80 Th LP 0co "1h LE | (OGG TP LP a = L8€ 96 GO ST £19 ‘Fo T~ 000 “1g €0 €T 109 °L0 0¢ SL | 186 ST 0— | 889 “Ec 0G ST 616 IF LF a ly ee as 68E SO Lo FT 119 ‘$] T— 000 ‘8% RS FI 809 ‘th PP ST | GLEGT O— | 189 °00 Sb ST P PEE —: cee POL SE OF LB) 886 “Ob LF S90 IT F0 ST $19 ‘Fo T— 619 SE CO ST 810 GS TS CT | G86 “ST O— 00080 2¢ ST . h6 0h Lb | «LLO OP 6S FI G19 ‘Fo I> 689 ‘FO 10 ST 0601S LP GT . 086 °ST 0— 000 °LE Le ST % ounpe “ ciate aa ee torts OSpLiqued | OO Th 4b PST IP LP | oct Te ob seesresseeess=) 6901S 88 FT ' Té6 ‘Fo T— 000 9F 6§ FT 166 “G0 96 ST 6LL ‘IT 0— 000 ‘$1 96 ST | OST TP LP sorossssssesss) T10'8h GE FT | Geb FG T— 000 ‘ST FE FT LS 6S 0G ST SLL IT 0— 000 ‘If 02 ST | 9% ; “> 9101397 | 106 ‘OF Lt cossesersccss'! #68 OF Lb Toe OT FR FL | C86 FE I €90 TP CF FT STG ‘LP 8 ST S8L TL 0— 000 60 G& ST ‘ ! 036 ‘OF LP 0€ 6S CE FT 086 ‘Fo I— £86 FG LE FT 06 ‘OF €6 CT LLL IT 0— 000°GS €6 ST | 96 ees srtttss eS priqmeg 8E0 TP LP PPL IP LE ToT Lb) 99 Le 9¢ FT £8 FS I 000 ‘GS LS FT LOE 80 FF ST F0S "60 0O— TIS ‘LT $F ST | LPL TP LP " ae 99T “6h €& FT FE8 Fo T— 000 FL é FT SIE 0E 16 SL | 88h "60 O— 108 6E 14 ST | FS Ou ERT Rhee ote) Coes sos grorqaqy C&6 ‘OF LP ee gets ot 8&6 ‘OF LP GLG "EP OE FL FE8 FS I— 90F 80 8€ FI OIG. F% FS ET 06 ‘60 0— 000 “FE #6 ST 966 ‘OF LP 68S "FG 86 FI EB 'bS I Sch 6h 64 FT | SIS ‘C0 OT ST €8P 60 0— OOOST OT ST | FE Tren So yr eer esprqmey 996 ‘OF LE 660 TF LP LOL ‘TP LE sorceeserseeecl ThA SE 10 ST 6EL ‘to I— 000 °LT €0 ST 8hE “SE GF ST L6E ‘80 0— ChL IP 6F ST T60 TF Le srescrscrersscl OCS "Le SE FT FPL be [-- 000 ‘ce 9€ FT LFE ‘80 SS ST S9E “80 O— GIL ‘OL &3 ST | £6 Roisison ere neceion?S sOTe TES OR LBS [Perera oes ChB OF Lb OLL ‘GO #5 FT €¢L ‘Fo I ECG °L] 8S FT C19 “8h 6E CT S88 80 0O— 000 GS 6€ ¢1 LI8 ‘OF LP 918 “CE LE FI GPL #3 I— 19S 00 6€ FT £89 ‘9T SG CT L9€ ‘80 0— 000 "SZ ¢3 ST | 8 Ketan Kani esplaqmep COT Ih Lb+ | C96 TH LP LOG TP LP Pepe ees Coos OF ‘GP £0 FL 09g "Sa T— 000 ‘89 SO #1 169 "EZ TS FT 96 ‘00 0— S61 Fe IS FT B96 Th Lb) OFF 60 SS ST 09g “GZ I— 000 °¢€ 9¢ ET 80L°0S oF FL £6F ‘00 O— 106 ‘TS ch FL | ET meres * FO1}OCT 90 Th LE eae 4 “"*") €90 TR LP OFF ‘LS LO FL 09g “Se I— 000 “8% 60 FI G06 SE Co FT L6F ‘00 0-- 000 6 GS FT | C90 IF LF OFF 96 00 FL 09g "¢s T— 000 &¢ 10 FL GOS *LO 8h FT C6F 00 0O— 000°80 8b FT | ST ABR port t ttre os prqmrg 8 “Ub 8 “a "8 "ub "UL "8 Ub Sw ‘SS "Ub" Ss tu ‘U “Ae “S$ "UY “TS8T ‘ * avduro9 Jo | , ssuostavdimag “-gtogpredraoa jo “suostiud woo s fem “q1o.110, ‘oSpruquieg | SUOsue sm01}99 1109 a } 3°) -saotjoer109 1 "guROWy : 2 SOWL} [VILEPIY JO SOUL} YOOT[D | SOUT} [VOBOpPIs . jo xouy yop Yyoq UvayT WO} STVUSIG | WoT] sfeasig HID ae H90lD ‘ ! jo suvoyy | jo suvoyy jo survey Jo suvayy | aE —wogy speusig ‘9Ull} JO BVDUDIIOBIC "“IOALOSQGO ‘quit UV ‘qlorqaq *LOATIBQO ‘LoyaoTL AA. ‘ro ‘oSplqueg | ‘SNOSIAVIWOO MOOT AO SLINSAU—"bS LAVAL ‘ponunu0g—'ssvyy ‘abprquoy pup “yoy roca ‘apnabuoT fo aouatag ig : APPENDIX IIL. Are. III, 894 S66 OF LP PEL Ib LE SET Tt LP 096 °CS GO LT &6F ‘O+ L9F CG SO LT 860 °L0 €¢ LI PPE '86-- GhP CE eS LT OSL It LF 8GF ‘9S 8o OT S6F 0+ €£6 €¢ 8¢ OT 89S "LE OF LT OFE “86— 86890 LF LT | 66 : PP aT ode “* qrorgeqr LSS ‘OF LP pee ESET BEB OF: LP 992 91 60 LT C6P 0+ PLL CT 60 LT O@T “LS 9¢ LT LEE “86— LOF ‘SS LG LT 698 ‘OF LF 661 61 G0 LT FGF OF 08 “6T c0 LT 899 00 O¢ LT GPE 8S — 00062 0S LE | 62 yar ; te BS eSpriqmeg OLL Ih LF OFC Th LP 646 ITP LE pa Swans TOE "63 9E OT Toe “e+ 000 ‘9G 98 9T 0¢¢ ‘OT to LT SET '€o— £89 EE FZ LT Go Th Lp 608 9F 12 9T 606 "E+ 000 ‘EF 12 9T TSS "Le 60 LT 9eI “Eo— LL9 ‘0S 60 LT | Fe ; "7" gforzoqy EL6'0F LE “| PL6 OF LP 688 "SE 6E 9T 00g ‘e+ BRC ‘0 6E OT 898 "PI Lo LT LET “86 — 000 "88 46 LE £16 OF LE 668 LT &3 9T 108 "e+ C6¢ FT Co OT GL8 8S GI LT SCL “€o— 00066 ET LT | Fa ae espuquiey SEL IP Lb LOG TF LE LEC‘Th LB” EOS O€L ‘PL €% OT oes ‘e+ 000 IT €& 9T 286 °C OT LT TLE "@3— 8¢¢ ‘8T IT LT LLO TP LE ee ceeeees) ge. Pp LI OT ee. e+ 000 TF LT 9T 01096 SO LT 693 ‘66— 6L¢ 8h CO LT | & we a . . "=" prorqoq O10 ‘Th LP PEE EES LLOTE LP 6IL ‘FO Te 9T Ten “e+ 886 00 16 9T 0€L "SE 80 LL OLG "Go— 000 °80 60 LT 800 Th LF 6IL 6E 9% OT wel es T66 “CE 96 OT LoL 0G FT LT €L6 “GO 000 "EF FIT 2 &% ar pe ere . espuqueg TLI'TP LP 6G ‘1h LE TOS Th Lh ft} egg ‘sg Te OL ecg e+ 000 ‘6G 1% 9T FST “LE 60 LT £0¢ ‘Te— LEQ "8S 60 LT €86 Th LP ee FCS ‘IT 9T 91 peg e+ 000 ‘80 9T 9T LEL “eo $0 LT 66F “Te— 969 FT FO LE | wee he soeeeeses gLorzoqy OSO'Th Lb Tre “| LOTR LP SFP SE FG OT egg e+ 966 “FE Fc OL C6P 6I CT LT Soe ‘16— 000 IF Gt LT SSO TP LP 9FP LE BL OT FO8 E+ G6S "ES ST 9T 66F "80 90 LT 10S ‘T6— 000 ‘0€ 90 LT | peat ay “* eSpraqure dy FOL TP LPT 986 ‘Th LP 096 ‘lh LP FE SS SLB OL. LRAT CTg T+ 000 ET 16 9T GLO ‘8S 80 LT BEL 0G-— €&8 “81 60 LT Sle lh LP rie Ae PIS FE ET OT PIBI+ 000 8% €T OL L60°9@ 10 LT 9¢L ‘06— €88 ‘9F TO LT 1S 5 ala “"* gored: G60 1h Lb pS Eh SGOT vee C1G ‘LE &Z OT GI8 ‘T+ OOF CF €6 OT Go 86 IT LT 8SL 06— 000 “6F IT LT s10'Th LE | SG"TZ LT OT #18 T+ Ith GT AL 9L | ete'ZO CO ZT | zsL0c— | 000°8% GO LT | 1g eng |---- “so s-oSprquiey ‘ew "sw ‘su ‘su su YY ‘¢ ‘Sw Y 8 wy s ‘sum Y ‘T88T } : : ; ‘suostivdmoo jo | ‘suosjivdmoo |-saostirduios yo! , “stostaedwoo a “euLOTT bahay Be | opens saul} [eaepis ote JO Salty YHOTD | SoTMTZ [vaLapts Sos jo saul} yoo[o qyoq uve 31 Ss) if BTSULSTS | josuveyy | » Jo saRe pt jo suvayy - jo suvoyy area —qwory speusig | i 1 “OUIT} JO S9DTIAIEHICG. | “JIAIIEQO “LajIeY MA“ “OE ‘Jlomqyeq “JaAJasqgo ‘VUIL “WY ‘sspuquereg “SNOSIUVdNO)D MOOTO dO SLTISHY— ss ATavV ‘ponuyuog—'ssvyy ‘abpiiquog pun “yay Gvowag ‘apnzbuoy fo suasagig §1.] LONGITUDE OF DETROIT FROM CAMBRIDGE. 895 Difference of Longitude, Detroit, Mich., and Cambridge, Mass.—Continued. TABLE 26.—FINAL RESULTS. 0. B. WHEELER AT CAMBRIDGE. A, R. KLINT AT DETROIT. Difference of time. Double wave- Date. 3 Wt. | Correction. ae ed time, east Signals from | Signals from M reste? minus west. Cambridge. Detroit. can 1881 ae 2 h. m. % hem. 8. 8. h. m. 8. 8. May 13 rcs secceccssenevsctencxt 0 47 41.064} 0 47 41.262] 0 47 41.163] 0.5 +0.034 | 0 47 41.197 +0. 198 40. 831 41. 099 40.966 | 0.5 +0. 034 41, 010 +0. 268 40. 932 41.144 41.038 | 1 + 0. 034 41. 072 +0, 212 26h: scscatemne deanery 40. 907 41. 154 41. 030 1 +0, 034 41. 064 +0. 247 POS Bn teaatincinecwaca omen ded 40. 948 41. 220 41.084 | 1 +0. 034 41.118 +0. 272 Dl pecesrereriageinicie mousse wets adits 40. 848 41.177 41.012 | 1 +0. 034 41. 046 +0. 329 Weighted mean. ..........0. 2000. cence cece ee cee ee oe 0 47 41.046 , A. R. FLINT AT CAMBRIDGE. 0. B. WHEELER AT DETROIT. | PUNO RL nes cettenie sewiaeseeen 0 47 41.022} 0 47 41.286] 0 47 41.154; 1 —0. 0384 | 0 47 41.120 +0. 2€4 ] 41. 050 41. 292 4L.171 | 1 —0. 034 41. 137 +0. 242 | 41.010 41. 267 41.138 1 —0. 034 41. 104 +0. 257 40. 973 41. 246 41.110 | 1 —0. 034 41. 076 +0. 273 40. 857 | 41. 134 40.995 ; 1 —0. 034 40. 961 +0. 277 Gacinat oiiniacempemciassacedl Weighted mean .........0.c02002ececeeeeeeeceeeeeeeees 0 47 41.114 | Difference of weighted means = personal and instrumental | equation ........-.....- isiapacattasd intataistaial ize ainlatet Riaisinian giaiewey sia jors sia ofS 0. 068 Correction = half of difference ..-......-......-02-+.-0. e000 eee eee = 0. 034 ho hom. 8. Weighted mean of corrected results. ........-2-- 2-222 eee eee ee eee cece ce eee e eee e eee eeeeeees 0 47 41.080 Reduction to east pier of Detroit Observatory ....------ 202. cee cece eee ens cee ee eee eee eee eee eee _ 0. 064 . East pier of Lake-Survey Obzervatory, Detroit, west of Coast-Survey post, Cambridge ........-...-- 0 47 41.076 Longitude of Coast-Survey post, Cambridge, Appendix 6, Coast-Survey Report of 1880....-....-..--. 4 44 31.090 Longitude of east pier of Lake-Siurvey Observatory at Detroit west of Greenwich .....----..-----.-+- 5 32 12.166 ot 896 APPENDIN IV. [App. 1V APPENDIX IVY. MAGNETIC WORK OF THE LAKE SURVEY. By T. Russert, Assistant Engineer. § f. The principal observations to determine the amount and direction of the earth’s mag- netic force at various places in the region of the Great Lakes were made in the years 1858, 1859, and 1860, while the survey was under the charge of Captain George G. Meade, and in the years 1870 to 1880. For declination and intensity a magnetometer and declinometer, with suspended collimator-magnets, were used. For the determination of dip, one dip-circle by Barrow and two by Troughton & Simms were in use. At most of the places where observations were made in 1858 and 1859 by Lieutenant William Proctor Smith, the same were repeated by Captain A. N. Lee in 1872 and 1873. The instruments were, when it was practicable, set up in the same places as when the observations were first made. The setting of the instruments on the original sites was prevented in some instances by buildings and other improvements. A comparison of the observations of the two periods is interesting, as showing the changes of the magnetic elements from year to year. A great many determinations of magnetic declination have been made in con- nection with the work of mapping the shore-line of the lakes. These were made with theodolite- compasses by comparing the magnetic bearing of a line with the azimuth determined astronom- ically. The needles are from four to five inches long and supported on a point. The readings of the ends are made by estimation on a graduated circle a little greater in diameter than the length of the needle. DESCRIPTIONS OF INSTRUMENTS. § 2. Wiirdemann magnetometer No. 3, for the measurement of declination and horizontal intensity, has a circle 6 inches in diameter, graduated to 10 minutes, and read by opposite verniers to 30 seconds. The glass tube for the suspending thread of the magnet is 18 inches long. The telescope for viewing the magnet scales is about 8 inches in length. It is mounted on pivots outside the horizontal circle, and swings vertically. A weight on the opposite side of the instru- ment counterbalances the telescope. Collimator-magnet ©; is 3 inches long and 0.75 inch in diameter. The scale is 0.05 inch long, and is divided into thirty parts. Magnet C, is 3.9 inches long and 0.75 inch in diameter. There are two smaller magnets, both 0.3 inch in diameter, the one 3.6 inches long and the other about 3 inches. These latter magnets were first used in 1876. The instrument is provided with two inertia-rings. The old ring has a weight of 397.44 grains, the external radius is 0.10225 foot, and the internal 0.07975 foot. The new ring, No. 2, for the light magnet, has a weight of 487.07 grains; the external and internal diameters are 2.002 and 1.600 inches. The Jones declinometer, also used in measuring declination and horizontal intensity, has a circle of 9.5 inches in diameter, graduated to 20 minutes, and read by two verniers to single minutes. The glass tube that surrounds the suspending thread of the magnet is 9 inches long. The telescope is 8 inches long, and is not movable in altitude. Near the eye-piece end of the tele- scope, and above it, is attached*a graduated ivory are 6 inches long. This scale is used to measure the angle of deflection of the suspended magnet in observations for horizontal intensity by viewing the reflection of the scale in a mirror attached below the suspended magnet. A weight attached to the instrument counterbalances the telescope opposite. Magnet marked “b” isa §§ 1-3.] MAGNETIC WORK OF THE LAKE SURVEY. 897 hollow steel cylinder without a lens. It is 3 inches long and 0.25 inch in diameter, and has attached a mirror 0.5 inch by 0.75 inch. Magnet D; is 3.6 inches long, and 0.25 of an inch in diameter. A brass cylinder, with a weak magnet inside for adjusting the line of detorsion, is 4.1 inches long and 0.6 inch in diameter. The inertia-ring, No. 3, weighs 830.24.grains; the external and internal diameters are 2.933 and 2.354 inches; the thickness of the ring is 0.16 inch. The Barrow dip-circle, No. 26, has a vertical circle 6 inches in diameter, graduated to half degrees and read by two verniers to minutes. To the arm carrying the verniers, two microscopes are attached that have threads in the foci extended in the direction of a diameter of the vertical circle. To observe the position of the needle the image of the point is bisected with the thread. The prolongations of the threads pass through the zeroes of the verniers. Two needles have been used with this instrument, each about 3.5 inches long and in form tapering from the center toward the ends. With needles of this shape the moment of inertia is much less for the same weight of needle, without any considerable diminution of its magnetic moment. The pivots of the needles rest on straight agate edges so that in rotating they roll. A framework with two sides of glass incloses the needle, when mounted, to protect it from currents of air. The instrument has a tele- scope and horizontal circle for bringing the plane of the needle into the magnetic meridian. The Fox dip-circle, Troughton & Simms No. 1, has two vertical circles on which the needle- ends are read. The plane of one circle is 0.25 inch inside that of the other. The outside circle is 6.5 inches in diameter, and is graduated to 15’; the other is graduated to 30’. The reading of the needle-end is made through the glass side of the box that protects the needle from currents of air The needle-axles rest in jeweled hollows. An attached telescope that moves in altitude, and a horizontal circle serve to bring the plane of the needle into the magnetic meridian. The instru- ment has three tapering needles, each about 6.5 inches long. Dip-circle, Troughton & Simms No. 2, has the vertical circle graduated to 15’. The reading of the needle-end is estimated as in Troughton & Simms No. 1. There are three needles, each about 8 inches long. Nos.3 and 4are tapering. Needle ‘“‘A” has the same widtt throughout, with semi- circular ends. The readings-with this needle are made from fine lines on the ends in the direction of its length. The needle-axles rest on agate edges and roll as in the Barrow dip-circie. The instrament has a horizontal circle, but no telescope. There is a compass with a needle about 4 inches long; the box containing the needle fits on two short pins on the horizontal limb. METHOD OF OBSERVING FOR DIP. § 8. After leveling the instrument and bringing the plane in which the needle rotates into the magnetic meridian, the observations for dip proceed as follows: The ends of the needle, after it comes to rest, are read on the vertical circle. The needle is then reversed on its supports, and the ends read again. The mean of the readings of the ends is free from error arising from the axis of the needle not passing through the center of the vertical circle. Reading with the face of the needle both ways eliminates error due to the magnetic axis of the needle not passing through the needle-points. A similar set of observations is made with the plane of the vertical circle turned 180° in azimuth. The mean of the two sets will be free from error arising from the line joining the zeroes of graduation of the vertical circle not being parallel to the plane of the horizon. After several series of observations have been made as above, the polarity of the needle is reversed and the observations repeated. In reversing the polarity of the needle, care is taken to use the same number of strokes of the bar-magnets from the center of the needle outward on both ends. The polarity is reversed to eliminate error due to the center of gravity of the needle not being in the axis around which it revolves. In determining dip, the mean of the results given by two needles was commonly taken. In the dip-observations of 1858, 1859, and 1860, made by Lieutenant William Proctor Smith, the polarity of needle No. 2 was not reversed, as it was used in determining mag- netic intensity by deflections, and for that purpose its magnetic moment had to be kept, so far as possible, constant. — The dip is affected by errors arising from the needle-axles not being truly cylindrical, and also from magnetism of the circle. The method of allowing for this is to determine a constant for the game instrument and needle and apply it to the observed dip. This constant is found by making 113L8 898 APPENDIX IV. [Arr. IV, two complete sets of dip-observations in planes at right-angles to each other, the one 45° east of the magnetic meridian, and the other the same angle west. The correction, c, will then be c=0— 0, in which 6= the true dip, 0,= the observed dip in magnetic ineridian. cot? 0= cot’ 6’+ cot? 0”, in which 0 and 0” are the values of the dip found in the planes at right-angles to each other. This correction was always small. For the Barrow dip-circle the correction found in 1858 for needle No. 1 was + 45”, and for No. 2— 4/37”. For the same needles, in 1873, Capt. A. N. Lee found the corrections Jess than the errors of observation. MAGNETIC DECLINATION. §4. The magnetic declination is the angle made by the vertical plane containing the magnetic axis of a freely suspended needle or magnet with the plane of the true meridian. It varies in the course of a day about 10’ in summer time and half as much in winter. The end of the needle toward the north has its most easterly position, called the morning eastern elongation, about seven o’clock in the morning. The afternoon western elongation is reached about one o’clock. The values of the declination given in Table II following, beginning with 1876, are the means of these two elongations reduced to the hourly mean for the day. The reduction is made by applying a certain part of the difference of the declinations at the clongations, depending on the time of the year when the observations were made. The part of the difference applied varies from +0.01 for the month of June to —0.15 for Devember. The table used to give the reduction to the mean of hourly observations is in the Coast-Survey Report for 1869, and is derived from a series of mag- netic observations made at Girard College Observatory. The declinations given in the table of results for 1858, 1859, and 1860, by Lieutenant William Proctor Smith, are from a single reading of the declinometer taken at any time of the day without regard to the elongation. The declinations of the same years that are marked with a star at Cambridge, Erie, Buffalo, Fort Niagara, Char- lotte, and Sacket’s Harbor, were derived by finding the magnetic meridian with a dip-circle. In this method the magnetic meridian was found by reading the horizontal limb with the face of the needle toward the north and turned so as to give a dip of exactly 90°, and again with the same face of the needle south and the dip 90°. The mean of the two readings gives the reading corre- sponding to a line at right angles to the magnetic meridian. The declinations obtained in this way may be in error by a degree or more. In Captain A. N. Lee’s work the declinations given are the means of the morning eastern elongation and the elongation at one o’clock in the afternoon, and the same are given in General C. B. Comstock’s work in 1870 and 1871. The declination is determined either with the declinometer or with the magnetometer, the tele- scope pointing south. The magnet used is a hollow steel cylinder from 3 to 4 inches long, with an external diameter of 0.3 to 0.7 inch. In the end that points to the north is a lens, and in that toward the south a plane glass on which a scale is graduated. The light from the graduations is parallel after leaving the lens so that when viewed with a telescope at a short distance, adjusted to stellar focus, they are plainly visible. The magnet is suspended by one or more fibers of unspun silk. In the older work, with the magnetometer, heavy magnets were used that required six fibers of silk to support them securely. In 1876, beginning with Lieutenant T. N. Bailey’s work with the lighter magnets, only one thread was necessary. The thread hangs in a glass cylinder about 1 inch in diameter, and there is a small rectangular wooden box that surrounds the magnet, when suspended, to keep off currents of air. The ends of the box have small windows of glass ; through one the magnet-scale is viewed with the telescope, and through the other light from the outside is thrown on the scale by a concave mirror. Before observations are begun the line of detorsion of the threads suspending the magnet, that is, the direction in which the force of torsion of the threads acts, is brought into the magnetic meridian. This is done approximately by suspending a brass cylinder in place of the magnet, and §4.] MAGNETIC WORK OF THE LAKE SURVEY. 899 of about the same weight. After coming to rest the threads are revolved through an angle equal to that which the axis of the cylinder makes with the magnetic meridian. For revolving the threads there is a graduated circle, called the torsion circle, at the top of the glass tube in which the threads hang, and of about the same diameter as the tube. Before making this adjustment the telescope is pointed nearly in the direction of the magnetic meridian, as turning the telescope in azimuth revolves the torsion-circle and changes the plane of detorsion an equal amount. As in declination-observations, the method is to note the scale-reading on the vertical wire and not move the telescope; the torsion of the threads restrains the free motion of the magnet. To allow for this, the declination-magnet is suspended and a reading of the scale is made on the vertical wire; then the torsion-circle is turned 90° and another reading of the scale is made; the torsion-circle is then turned 90° the other side of its first position, and the scale is again read. The mean difference of scale-readings shows the effect on the reading of the scale produced by a twist of the suspending thread of 90°. For the smaller amount of twist of a few minutes or so, such as takes place in changes of declination, the effect is assumed to be in proportion to the angle turned through. For convenience in reduction, the effect of torsion is applied as a correction to the angular value of the seale-division, always increasing that value. The ratio by which the angular value of the scale- interval is increased is usually very small, never being more than 54,. The angle which the magnetic axis of the magnet makes with the meridian is found by noting the reading of the horizontal circle when the vertical wire of the telescope is on a meridian mark or other mark of known azimuth, and again when in coincidence with the division of the scale that marks the magnetic axis of the cylinder. The azimuth of the mark is fixed by observations of the sun or of Polaris. The reading of the scale that corresponds to the magnetic axis is found by noting the reading of the scale when the vertical wire of the telescope is on some division near its middle, then revolving the magnet 180° around its axis and noting the reading of the vertical wire again as soon as the magnet comes to rest. The mean of the two readings gives the reading of the magnetic axis of the cylinder, and is called the scale-zero. The value of a scale-division is between 2/ and 3’. The difference of extreme values in a series of determinations of scale-zero is about 0.2 of a scale-division. As the magnet is nearly always vibrating a little, the mean of the extreme readings of the vertical wire on the scale is taken as the true reading. The viewing telescope when once fixed with the vertical wire near the zero of the scale in a series of observations is no longer moved, but the changes of declination are shown by the differ- ent scale-readings. The angular value of a scale-division must then be known. To find it, the magnet is fixed. in the position it has when suspended for observing declination. The vertical wire is brought into coincidence with a division near one end of the scale and a reading of the horizontal circle made; the wire is then brought to the other end of the scale and the circle is read again. The difference of circle-readings, divided by the number of scale-divisions passed over, gives the angular value of a division. The range in a series of values is usually about 0.06. Beginning with the work done in 1876, and at only a few places previous to that time, before selecting a place for magnetic observations, tests for local attraction were made. The method of doing this was to lay off two distances of 300 to 600 feet from the proposed position of the instru- ment and at right angles to each other. The magnetic bearings of these lines were then observed from both ends, and if the differences were found to be considerable, the existence of some local disturbing cause was taken for granted and the site was not occupied. The greatest difference of bearings at the ends of @ line for any station where the magnetic elements were finally determined was at Galena, Ill., where it was 3’. _— The following were the formule used in the reduction for the determination of declination : H__™ __ _ cosfficient of torsion. F ~90°—u H= force of torsion of the thread. F= horizontal component of the earth’s magnetic force. u= mean difference of scale-readings reduced to angle, caused by turning the suspending thread 90°, first in one direction and then in the other. 900 APPENDIX IV. [Arp. IV, d0=0' +a (14% (e—s+fr) d= value of declination for the day. 6/= angle between the true meridian and the vertical wire of the telescope. It is + when the end of the needle toward the north is west of the meridian. a= angular value of a scale-division. e= mean of scale-readings at the elongations. It is + when greater than s. s= reading of scale corresponding to magnetic axis, or zero of scale, as it is called. It is + when less than e. r= difference of scale-readings at clongations. f= factor for reduction to the mean of hourly observations. The values of f, taken from the United States Coast-Survey Report for 1869, are: VANUATY cee. d anes eda —0. 089 JOY ca nieaie a Maew oad ssa +0. 005 February. gceccevense 220s —0. 040 August ... 0.22... 50) cee ee —0. 023 March séducn joe aweats iis Se —0. 019 September ............ --. —0. 044 April, 0 giesews ewe cedex —0. 068 OCtObOR sce decide eedans —0. 096 MOY scence Gide eens we ee —0. 0138 November............-..-- —0. 096 UMC oy ad saat bicee alae +0. 010 December. ...............- —0. 154 MAGNETIC INTENSITY WITH DIP-CIRCLE. § &. At the stations where observations were made in 1859 and 1860, except Cambridge and Detroit, the magnetic intensity was determined by deflections with the Barrow dip-circle. Detroit was used as the base-station. The intensity there was determined by deflections and vibrations with the Jones declinometer. The values of total intensity for Detroit that were used in the rela- tive determinations of 1859 and 1860 were 13.710 and 13.785, respectively. With the dip-circle the intensity was determined as follows: Needle No. 2 has four small holes, two that are each 0.5 inch from the needle-end, and the others 0.75 inch. In these holes the weights used in deflecting the needle were inserted. ‘Che weights used were small pins. Four angles of deflection as given by different weights were observed at the base-station, and also the angles of deflection produced by the sane weights at the other stations where the relative intensity was required. From these the total intensity was derived, and then the horizontal intensity by dividing by the secant of the dip. The range in the four values for each place is on the average about 0.12 in the observations of 1859, and about 0.21 in those of 1860. The largest range was at Minnesota Point, in 1859, the ex- tremes being 13.95 and 14.25. The formula used in reduction for relative intensity with dip-circle was as follows: i J __ayl R=R SO” cos (4’—v’) sinv’ cos (4—?v) R = total intensity at base-station. v = angle of deflection at base-station. 4 = dip at base-station. FR’ = total intensity at some other station. wv’ = angle of deflection at that: station. 4’ = dip at that station. MAGNETIC INTENSITY FROM VIBRATIONS AND DEFLECTIONS. § 6. The method of determining intensity from vibrations and deflections is far more accurate than that which precedes. This was the method followed in 1858 and in the years 1870 to 1880. It consists in determining the angle by which the magnetic axis of a suspended magnet is deflected by another magnet supported at a short distance, and in observing vibrations of the deflecting magnet when suspended. The axes of both magnets in observing deflections must be in the same horizontal plane, and the prolongation of the axis of the deflecting magnet must pass through the center of the suspended magnet. The vibration- observations consist in determining the time in §§ 5, 6.] . MAGNETIC WORK OF THE LAKE SURVEY. 901 mean solar seconds that it takes the deflecting magnet when suspended to make one vibration. The angle of deflection depends on the ratio of the magnetic moment of the deflecting maguet to the horizontal component of the earth’s intensity. The time of vibration depends on the product of these same quantities. Hence the horizontal intensity can be derived, and, by multiplying by the secant of the dip, the total intensity. f The time of vibration of a magnet depends not only on its magnetic moment and the horizontal intensity of the earth, but also on the moment of inertia of the magnet. To obtain this latter, the times of vibration of the magnet are observed when suspended alone and when weighted with a brass ring of which the moment of inertia is known. The ring used for this purpose is made of brass, the diameter about 3 inches, the width of the metal about 0.3 of. an inch, and the thickness about half the width. The ring is put on the magnet on its flat side, so that its plane is parallel to the plane in which the magnet vibrates, and with its center in the line of the thread suspending the magnet. As the moment of inertia of a revolving body is the sum of all the products formed by multiplying the mass of each particle by the square of its distance from the center around which it rotates, so in this position of the ring the moment of inertia will be its weight multiplied by one- half the sum of the squares of the internal and external radii. These dimensions of the ring are measured, and the weight is found with a balance. The moment used is in terms of the weight in grains and the radii in parts of a foot. The moment of inertia of the ring will vary slightly with temperature, but the change is so small that it has not been taken into account; neither has the change in the moment of inertia of the magnet been considered, as a difference of 50° I. causes a variation of only about z>455, while the difference of extreme values in a series of ten determinations of this moment is usually 745. The time of vibration is corrected for the change of the magnetic moment when the magnet has a different temperature while vibrating from what it had while being used as a deflector. The rate of the chronometer and the effect of the torsion of the suspending- thread are also allowed for. The arc of vibration was always so small, being about 1°, that no reduction to the time for an infinitely small are was necessary. In making vibration experiments the magnet is started to oscillate horizontally by attracting or repelling one of its poles. The oscillation is made at first to surpass the limits of the scale so that it may acquire steadiness of motion before observation is begun. All up-and-down motion is checked. When the motion of the magnet is within the limits of the scale, the extreme readings of the scale on the vertical wire of the telescope are noted. The time that the division of scale half way between these extremes crosses the vertical wire of the telescope is then observed to a fraction of a second. The time of one vibration is derived by dividing the difference of the observed times of two remote crossings of the wire by the number of vibrations the magnet has made in the meantime. An interval of at least 300 vibrations is used in deriving this time. The time of vibra- tion is from five to eight seconds, depending on the magnet. The mean of the times of vibration as given by several sets of observations is used in the reduction. The extreme difference in the values of a vibration in several determinations is usually not more than 0.02 second. The deflection observations made have been of two kinds, depending on the instrument used. With the magnetometer, the deflecting magnet has its axis perpendicular to the axis of the sus- pended magnet when deflected. In this case the ratio of the magnetic moment of the magnet to the earth’s horizontal intensity is computed with the sine of the angle of deflection. In the case of the declinometer the deflecting magnet remains perpendicular to the magnetic meridian, and the tangent of the angle of deflection is used in the reduction. With the magnetometer, the angle of deflection is measured on the horizontal circle, aud with the declinometer, on an ivory scale attached to the telescope. The reflection of the scale is viewed in a mirror fastened to the suspended mag- net below its center. The method of observing with the magnetometer is to bring the vertical wire of the telescope on the scale-zero of the suspended magnet while influenced only by the earth, and then read the horizontal circle. The deflecting magnet is then placed on the deflection-bar at a distance of one foot from the suspended magnet. This distance is measured between the centers of the magnets. The telescope is turned in azimuth until the vertical wire is on the scale-zero, and. the circle read again. The difference of the two circle-readings is the angle of deflection. The motion of the tele- scope also carries the deflecting magnet around. The angle of deflection varies from 10° to 28°, 902 APPENDIX IV. [Arp. IV, depending on the distances of the magnets apart. The usual distances at which deflections were observed were 1.0 and 1.3 feet. The deflections are observed with the north end of the deflecting magnet toward the suspended magnet, and then with its south end in that direction, and also at the same distances on the opposite side with the north end alternately toward and away from the suspended magnet. Observing under these different circumstances eliminates slight errors arising from inaccuracies in the fixed centers of the magnets and errors due to other instrumental peculi- arities. In Lieutenant William Proctor Smitl’s observations with the Jones declinometer, in 1858, the distances between centers of magnets were sometimes as great as two feet. With this instrament, the deflecting magnet remaining perpendicular to the magnetic meridian, the observed angles of deflection vary from 1° to 10°. In placing the magnet-centers at a certain distance apart, the errors in the length of the rod used are known and allowed for at the time. The correction for distribution of magnetism in the pair of magnets used, where any was to be made, was derived from observations made specially for that purpose at the two distances 1.0 and 1.32 feet. The lengths of the magnets used were about in the proportion 1 to 1.22, the longer one being used as the deflector. With magnets in this proportion, the correction for distribution of magnetism is a mininum. The change of magnetic moment of the deflecting magnet, caused by temperature, is determined from observed deflections under widely differing temperatures of the deflecting mag- net, with the distance from the center of the suspended magnet unchanged. In the early work in 1858 and 1859 the temperature-constant used for the deflecting magnet of the Jones declinometer was determined by Professor Kingston, of the Toronto Magnetic Observatory, aud found to be q=0.00015 tor 19 F. In Captain A. N. Lee’s work, in 1872 and 1873, the value used for the deflecting magnet of the Wiirdemann magnetometer was q=0.00022. This value was determined, in 1871, by General C. B. Comstock, from deflections observed when the magnet was at the temperatures 52° and 92° F, in water, both within a quarter of an hour. The coéfficient for the new light magnet first used in 1876 was found to be q=0.00044 for 1° F., from observations made when the magnet was packed in melting ice and when in water at temperatures of 90° and 106° F. The observa. tions at the extreme temperatures followed within five minutes of each other. The effect on the result due to induction is very small, and is neglected. Observations made with the magnetometer require no correction for torsion, as the line of detorsion turns in moving the telescope, and the resistance to the magnet is, therefore, the same when deflected as when hanging freely. In the case of the declinometer, where deflections are measured on the ivory scale by reflection, torsion is taken into account. The following are the formule for magnetic intensity used in reductions: Vibrations. T= TA +) [1—(t’—1)q] T =corrected time of vibration. T) = observed time of vibration. a= coéfficient of torsion, the same as in observations for declination. t’ = temperature of magnet when vibrating. ¢ = temperature of magnet when deflecting. q = temperature-constant or change in magnetic moment of deflecting magnet for a tempera- ture-change of 1° F. The time of vibration is also corrected for rate of chronometer, if considerable. K'= g(r? +r?) w A’ = moment of inertia of ring.” y =outer radius of ring in feet. v’ = inner radius of ring in feet. w = weight of ring in grains. §6.] MAGNETIC WORK OF THE LAKE SURVEY. 903 7? 4 = moment of inertia of magnet and stirrup. T’ = corrected time of single vibration of magnet loaded with ring. T= corrected time of single vibration of unloaded magnet. wait mX = Pr K= KE i m = magnetic moment of the magnet. X = horizontal component of the earth’s magnetic intensity. mz = 3.14159. The temperature-constant, q. _ ancotu a q = temperature-constant. t = higher temperature at which deflection is observed. t) = lower temperature at which deflection is observed. n = difference of scale-readings corresponding to (t—t,). a = angular value of a scale-interval. u = angle of deflection corresponding to ft). Deflections with magnetometer. ar sin ua — =) m = magnetic moment of deflecting magnet. X = horizontal component of the earth’s magnetic intensity. + = distance apart of magnet centers in feet. u = corrected angle of deflection. P= constant depending en distribution of magnetism in the pair of magnets. The value of Me is computed for each distance separately and a mean taken. —A en 5 sin wu! ad ll sin a P= Ue eee =‘ AL ry sin w—r sin wu ye re A = the value of z for the distance in which * is 1 foot. A, = the value of x for the distance in which 7, is 1.32 feet. The angles of deflection from sets of observations made at different temperatures are reduced to the same temperatures by the following formula: uy = observed angle at temperature ft. uw = angle reduced to standard temperature t. q = temperature-constant determined as explained above. Deflections with declinometer. aan tan w(1—%,) The notation is the same as in the formula for the magnetometer. The angle wu must be cor- rected for torsion. P is neglected when the least distance of the centers of magnets is not less than four times the magnet-length, and the magnet-lengths are as 1 to 1.22. When the magnets are not of these lengths, P is computed from the formula pa r?P tan uw —r, r* tan u ie r,) tan u/—?r tan u in which u and wu’ are angles of deflection observed at two distances, r and 7. wo 904 APPENDIX IV. [App. 1V, To find the horizontal intensity of the earth’s magnetic force and the magnetic moment of the deflecting magnet, mA =a from vibrations. me : -= from deflections. m= Vai r a tale The total intensity, ~, is derived from the horizontal intensity and the dip 6. = JY sec 0 To reduce m to a standard temperature, . My=m [1+ (t—t)g] myo=value of m at the standard temperature, t). t=temperature at time of deflections. q=temperature-constant. CHART OF EQUAL DECLINATIONS FOR THE YEAR 1880.—PLATE XXX. § 7. In compiling this chart all the observations of declination made in the lake region with magnetometer or compass were used. It includes a few values of declination given in the list of Canadian light-houses on the lakes for the year 1880; also recent observations at points on the lakes and in the east made by the United States Coast and Geodetic Survey, given in Appendix No. 9 of the report for 1879. In reducing the declinations observed at different times to the com- mon year 1880, the yearly changes as given in Charles A. Schott’s paper in the above-mentioned report were applied. The value of the change for the place nearest the point of observation was the one used. Ina table on the chart are given the yearly changes in 1880 for a few places taken from the same report. The method of placing the lines of equal declination on the chart was as follows: The points where observations were made were marked, and the declinations for the year 1880 written after them. Points were then selected by interpolation which were considered to have a declination of a whole number of degrees. The points of equal declination were united by straight lines. Curves were then drawn, with the zig-zag lines as guides, rounding off the angles and making the area between the zig-zag line and the curve the same on both sides. When the observations were numerous in a particular locality a mean was taken for the center of gravity of the group. § 8. TABLE I.—Relutive total intensity for 1858 and 1859 with weighted dip-circle needles. {Observer, Lieutenant William Proctor Smith.] Station. arrow Uine ipag meen & : Simms No. 1. ~~ COUR, MOSS coecanexncueiwaneevannnduactnne note Binmnianne 1. 7820 1. 7820 Detroit; Mich’s. oc oosewce owcewesevaente cues teases tees veer ace 1. 8157 1. 8275 Borestwille: Mite li. cia sjecscsmecicas weecns clemnistocinisiclonanticapateeneee TO248: —- Marrsateregeractete ges ence Thunder Bay Island, Mich.........-....0.00 20. -2ccececeeneeeeee 1. 8470 a aiciana alanis Bini: Stareeon Point, MiG vc cicccuwnicnvacceceuescacsecauerswerancue ERP Didawsadinesidiees 7 Fort:Gratlot; Wich szsscasscscsnnreecae pecer ean face exee ess eeeieece T8072? xoeneetcesceeos Cambridge was used as a base-station in obtaining the relative intensities in Table I. § 9%. Table II contains the results of the magnetic work done on the Lake Survey with the magnetometer, declinometer, and dip-circles. The total intensities at most of the stations in 1859 and 1850 are derived from deflections observed with the dip-circle. When- not otherwise stated, the intensities are from vibrations and deflections with the declinometer or magnetometer. The dip as given by a single needle is usually the mean of two results with the polarity reversed. In the early work in 1858, 1859, and 1860, the polarity of needle No. 2 was not reversed in observing dip, as its magnetism had to be kept constant, being used in observations for relative total intensity. §$§ 7-9.) MAGNETIC WORK OF THE LAKE SURVEY. 905 TABLE LI. | Lati- | Longi- Declina- . Horizon- Total in- | Instruments; place of observation; Place. tude. | tude. Date. tion. Needle.) Dip. rita tensity. observer. Go Ff oO ‘ ° to o ¥ Detroit ..-..--..- 42 20] 8303] Apr., 1859| 0 42 OE 1 73 47 3.8379 | 13.7102 | Jones declinometer, Barrow dip- 2 73 41 circle, Troughton & Simms dip- Cc 73 35 circle No. 1. West side of Wash- 73 41 ington avenue, 200 feet south of Grand Riveravenue. Lieut. Will- iam Proctor Smith. Detroit ..-....-..- 42 20; 8303 |May, 1860|.........-.-)......-. 73 43 3. 865 13.785 | Jones declinometer, Barrow dip-cir- ; cle, Troughton & Simms dip-circle No. 1. Same location as above. Lieut. William Proctor Smith. Detroit joss cciccicsics 42 20] 8303 |May, 1872| 02513E'........|.......... 3. 881 13.720 | Wirdemann magnetometer, Barrow Si ABT2 | secre HSeiie 2 73 31 dip-circle. In park near Lake V1 UBT 22s weccosens : 1 73 37 Survey office, about 400 feet due 29, IST besscseveceee | 2 73 36 west of the above location. Capt. ' 73 35 A. N. Lee. Detroit ...-....... 42 20 | 83 03 | May 5,1873}............ | vaseverr| maser ace 3.886 |...-.----- Wirdemann magnetometer, Barrow 12, 1873 |.-.--..----- 1 73 31 dip-circle. In park near Lake Sur- 12,4878) scccscmwssers 2 73 33 vey office. Capt. A. N. Lee. 14,1873} 017 40H |........|..---..--- 3. 881 UAABTH ccm meesce sun leek oosalndaniee case 3. 873 16, 1873 |..-..-.-.--- 1 73 37 17,1873 | 016 54E O1717E 73 34 18. 715 DPAIGE a .css vesses 42 20) 83 03 | May 24,1876|....-.-----.;land 2] 73 29.9 |---------- ----------| Wirdemann magnetometer, Barrow 25, 1876 dip-circle. In park near Lake Sur- June 3, 1876 3. 897 vey office ; coérdinates: from lamp- 5 SI6 |urcaraqnsccullasetecas|aceadeosed 3. 899 post at southwest corner of Cliff- 61876! 0 04 S60). 2.2. enn h crm asnee 3. 899 ord street and West Park Place, 8. 004 42 EB 73 34.1 13. 781 18° 06.3 W.; from northwest cor- nerof Lake-Survey building, S. 26° 01.0 W. Lieut. T. N. Bailey. Sacket’s Harbor --| 43 57 | 76 07 | June 22,1859; 818 W) 1 75 47 3.403 | 13.809 | Declination only approximate; ob- 2 75 41 served with dip-circle. Total in- 5 aa tensity from deflections with Bar- row dip-circle. Place of observa- tion near barracks, 313 feet south of flag-staff. Lieut. William Proc- tor Smith. Sacket’s Harbor ..| 43 57 | 76 07|June, 1872) 8 06 02 WI....---.|---------- 3.482 | 13.847 ) Wirdemann magnetometer, Barrow 8, 1872 |eawswseces se 1 75 29 dip-circle. Barracks, 313 feetsouth G, 1872 |on-s eos enue 2 75 24 of flag-staff. Capt. A. N. Lee. a5 ST Sacket’s Harbor ..| 43 57 | 76 07 | May 24,1873} 8 17 22 W].....---|..-------- 3.472 |..-------- Wiirdemann magnetometer, Barrow 26, 1878: |. cesses see 1 75 25 3. 470 dip-circle. Barracks, 313 feetsouth 26, 1878: jenacee stoned 2 75 25 of flag-staff. Capt. A. N. Lee. 27,1873 | 8 12 27W 2 75 23 8 15 00 W 75 24 13.770 Charlotte -..-.---- 43 15| 77 37 | June 18,1859) 431 Wy) 1 75 12 3.561 | 13.942 | Declination only approximate; ob- 2 75 11 served with dip-circle. Total in- 75 12 tensity from deflections with Bar- row dip-circle. Ninety-one feet northeast of light-house. Lieut. William Proctor Smith. Charlotte ........- 43.15 | 7737] June, 1872) 3 44 26W.......-. wesece---.| 3-619 | 13.920 | Wiirdemann magnetometer, Barrow 7, 1872 |.----- cee ee fone eeeee 74 57 dip-circle. Latty street, 377 feet 8.1872: nnnomsse ea ieeeaes or 74 59 southwest from the corner of 10, 1872 |. c.sannes eos] pecuwe~ 74 57 Broadway. Capt. A. N. Lee. | 74 58 114 Ls 906 APPENDIX IV. TABLE II—Continued. (Arr. IV, Charlotte .....---- Buftalo Buffalo Buffalo Cleveland. Cleveland Cleveland . Grand Haven Grand Haven | Michigan City .... Lati- tude. oF 43 15 42. 53 42 55 41 30 41 30 41 30 43 05 41 43 Longi- tude. oF 77 37 78 53 81 40 81 40 81 40 86 13 86 183 86 54 Horizon- | Declina- : i Total in- | Instruments; place of observation; Date. tion. |Needle.| Dip a tensity. observer. , ovww of May 29, 1873 DO84: ineemeres ae Wirdemann magnetometer, Barrow 30, 1873 3. 632 dip-circle. In rear of Methodist 30, 1873 3. 634 Church on Broadway, and about 31,1873 | 3 49 36 W).-...... 74 51 600 to 800 feet southeast otf old 3 46 10 W 74 50 13. 886 station on Latty street. Capt. A. N. Lee. June 11,1859) 256 W 1 74 49 3. 608 13.743 | Declination only approximate; ob- 14,1859 | cia sieeaice oe 2 74 44 served with Barrow dip-circle. 74 47 Total intensity from deflections with Barrow dip-circle. 315 fret south of cast end of south pier, Lieut. William Proctor Smith. June, 1872| 3 52 26W!........). ates 3. 644 13. 823 | Wiirdemann magnetometer, Barrow 14, 1872 |. ......---- 1 74 43 dip-circle. At Fort Porter. Capt. 14 B12 io ae oiaisints Ws 2 74 43 A.N. Lee. 7443 une: 8, ABTS: weno ossc [soot eAleoeeween le 3.667 |...-......| Wiirdemann magnetometer, Barrow 4,1873| 3 55 40 W 1 74 26 dip-circle. At Fort Porter. Capt. 5,1873| 4 01 01 W)... -...|.--------- 3. 667 A.N. Lee. 6, 1873 |....- 1 74 30 6, 1873 2 74 31 74 29 13. 707 aly 41889 | on scasc cans 1 73:19 3. 956 13.794 | Totalintensity from deflections with Ae V859) | mooie 2 73 21 Barrow dip-circle. Jones decli- 5,1859 0 46 00 W nometer, Barrow dip-circle. 166 73 20 feet south-west of northeast cor- ; ner of marine hospital. Lieut. William Proctor Smith. June, 1872! 0 4452W .....-.|. saeee oer 4.017 13.833 | Wiirdemann magnetometer, Barrow 17, 1872 1 73 07 dip-circle. Same location as above. 17, 1872 2 73 09 Capt. A. N. Lee. 18, 1872 |....------- 1 73 07 73 08 June 16, 1873 ' 05117W 1 73 08 2908 | cxceeees Wiirdemann magnetometer, Barrow 17,1873} 0 50 31 W 2 73 08 3. 993 dip-circle. Same location as above. TTBS cece cca 1 73 09 Capt. A. N. Lee. 0 50 54 W 73 08 13. 773 Aug. 18,1859| 4 24 15E 1 74 08 3. 815 13.987 | Totalintensity from deflections with 2 74 12 Barrow dip-circle. Jones decli- 74 10 nometer, Barrow dip-circle. 100 yards west of Detroit and Mil- waukee freight depot. Lieut. William Proctor Smith. Aug. 27, 1873 (PAB seRR es [eeieee ts Sean ee Rs 8.854 |eaceeesece Wiirdemann magnetometer, Barrow 28, 1873 Seliaeusited aaa VI Hi 73 55 3. 848 dip-circle. Station in lot in rear 28, 1873 328 43E 2 74 O1 3. 848 of jail in center of lot. Capt. A. 29, 1873 i 327 40E 1 73 58 N. Lee. 3 2812 E 73 58 13. 940 Aug. 28,1859| 5 22 39 E 1 73 02 4, 018 13.789 | Totalintensity from deflections with i 2 73 02 Barrow dip-circle. Jones decli- 73 02 nometer, Barrow dip-cirele. Northeast corner of light-house inclosure. Lieut. William Proc- tor Smith. § 9.) MAGNETIC WORK OF THE LAKE SURVEY. 907 TABLE JI—Continued. Lati- | Longi- Declini- ‘ Horizon- | potal in- : ‘ Gens Place. tade..| tude. Date. tion. |Needle.| Dip. jal iatet i i, Instruments i pisce.at observation ; °o a oO Ff mo fe o ie Michigan City ....| 4143 | 86 54 | Aug. 25,1873 | 3 58 23 E 1 72 42 4097 Weesciamarate Wiirdemann magnetometer, Barrow 25, ABTS) essences 2 72 45 4. 087 dip-circle. Station on beach 300 26,1873 | 359 41 EB 1 72 43 4, 089 feet north of light-house. Capt. 35902 E 72, 43 13.770 A.N. Lee. Milwaukee ...--.- 43 03 | 87 55 | Aug. 20,1859 | 6 20 06 E aL 73 56 3. 858 13.956 | Totalintensity from deflections with 20; 1859 |-0000s<