al Carnell University Library Ithaca, Nem York C,-E,..College. Cornell University Library QB 301.H41 ; Geodesy.The figure of the earth and isos AVI 3 1924 004 094 565 aw DEPARTMENT OF COMMERCE AND LABOR COAST AND GEODETIC SURVEY O. H. TITTMANN, SUPERINTENDENT GEODESY THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN THE UNITED STATES BY JOHN F. HAYFORD Inspector of Geodetic Work, and Chief, Computing Division Coast and Geodetic Survey WASHINGTON GOVERNMENT PRINTING OFFICE 1909 LETTER OF TRANSMITTAL. DEPARTMENT OF COMMERCE AND LABOR, Coast AND GEODETIC SURVEY, Washington, April 80, 1909. Hon. Cuartes NaGEt, Secretary of Commerce and Labor, Washington. Sir: I have the honor to submit for your approval for publication the accompanying report by the Chief of the Computing Division of this Service, entitled ‘‘The Figure of the Earth and Isostasy from Measurements in the United States.’’ It gives the necessary details as to principles and method of an investigation of which the results have been for some time before the world. All the geodetic work in this country prior to the year 1906 which was sufficiently far advanced to lend itself to the purposes of this discussion has been utilized. The effect of the accumulation of further data on the values obtained by this investigation will be shown in a subsequent publication. Very respectfully, O. H. Tirrmann, Superintendent. Approved: CHARLES NAGEL, Secretary. CONTENTS. Page. General Statement osc siducccn wie Wasrceenne aman pa he's oalauaars oars eammnde eas Cae ooo oe Saenos odes os: 9 Previous preliminary statements........ 2.222.200.0000 cece cece ence c eee cece cece eect eeee sence 9 Determinations of gravity not used........ 2.2.2.0 022 ee nee eee 10 Noteworthy features of the investigation............. 0000020 c eee ec cecee eee nee e cence cence eeeeee 10 Order of presentation..............000000 02002 cece cece eee sd sea clei hate ce aest pda tend Aon ed Pe ented et Teena AY 10 Data ised incth en nvesti Pati OMe. vlessa te wnpes ces wemavers eee eo uti ee ey Feedulau 5 eed Meakasaes ena aay Re eee SS 10 MMANGUla (ON pegess Sele, ceaieie i cia erekid cp nso sac Gqstenssnies 2 Srese oneal op Gd atti eis BA ines wel NMEiceeks encase Boo 9 Shadi 2s 10 Deflections in Mer GaN ci ccc0 oc wdewe dsp numease ka cs weounen suas Hobane pees AoedamuEE Gee SomENe ae aeRO ES 12 Deflections in prime vertical......... 2.2.00. 0 cee ee ee eee eee eee eee e cere ee eeeeeeecaees 17 . Computation of topographic deflections.................0 000000002 oe eee ee 20 Formula... .. pe They oa VSS pa Spe vasus cos 2h hep Sse gait, | acted SOE A Een CSE De EIS A See hatte A 20 Selection’ Of rACMsOE TIMES S «oc oc ecennieagaee Made tees oot onck aden tameMe Wien aOEReLae Peeame te Seems 22 UWsesoP templates: jc cas scecccil tone eeslsarcte to bicialncard eect aiakialeherc laid cleatetlaai alors aenasettoaleld clegutlinlatle Ja haters 23 Example of eomputation—Calais, Maine........ 2.20.00. 2 2020 e eee 26 Water compartments. sem. .s 2 cage sdicneies actecncioete sacmeisGre 25 Garcletnvgulae Sa dadnmnd ss pete Ahad cacemi es 27 Adaptation of method to Mercator charts........... 20000020 ec eee eee ee e eee eee 28 Examples of computation— Uncompahere:: Colorado sess vecenecexs ax Mucekines samen oes eRe SE HES po inland des aemeres 29 North End Knott Island, Virginia.............20 202200. 0 eee eee eee 31 PomibsArvena,, Caliornita 2 oo: lecsclaieoa2 A yaeeemcdaeyn ss ne av seped ae eee ees See Re Ree Seen See 32 Mount Ouray, ColoradO sasckcscawies or cttctenutsisae gitsdniweite's Sale donde gai ee abba pasaies gaa eek eeees 33 Correction: for slope nccescusua vies epee: oo cineeawis eos cinae 82 a SENOME BLES ome as oscar ed oe eee 34 Possible interpolation for outer rings..........-..-22--2--002 20220022 cee ee eee 36 Method of interpolating for outer rings. ........-..2.-.22.-2020 22220020 cece ee eee 39 Hxampléiokinterpolation csccesaigy esteem sic becsgmenlll doteoaweldipe opine ees etebemly oe ee nSkee ase ees = 40 Criteria: of accepted interpolation: «22.022 -s.ea.c2seree keeles ee eieee tase thee eee seeneagee ye ames 43 Saving by interpolation of outer rings. ./...... 2.22.2. 2. 0220222222 e eee eee . 44 Value ot: them eth od) cciccnactmtncecunctee se eeeoas ued secganuen oraeie Ie eae ea a Saree Garaneteadets 45 The economics of this research s...cssc2erssse ness eee eens vee tewee eye ss eee Sy Ge re oweaE ve es MRRe Ess GAKec Se 46 Values of topographic deflections. i-....2...ce.e0cc0u 2 ese wie cin sueekeime die Sab mtuauinie Das aeiegen SEs Louie ge se gimeleled - 17 Tables of deflections observed and computed.................2.-..- 20222022 cece eee eee eee eee ee pereneren ae 48 es Contoura of the géoid. cgi. oe satieenie cs adneudd sicigemeaase vives need ae dame ees eeeert car samyies aidemeels 57 Definition of geoid surfaces. a<2s20<2ccceeece+ ss vena ese eeeceuser: sy eeedee sss seewe see ee cce ee eee eee 58 Construction of geoid contours.............. S Lhep Repay eae cy OT Go eee AE asa INS aa vee 58 Advantages and limitations of method...............0-. 20202002002 eee eee eee eee eee ees 62 Comparison‘of geoid and topography: os secres seve vig gee ges Horkn ne ee iacemiuans oe Seveey ree E eee EEE kawee ed 63 Isdstasy mitist beconsid ered ajc. jece sto ata wie Redeedie cen eeaaneds Poe eeeees ta cemennes ce a8 Rusia ac Munaed 65 Tsostasy efitied oases 2 scenes nodeanee earache eee te webetind sa Mele glvieasmoneacnd sacipsemeieuy a Ses siiassels 66 = Computations of deflections, isostatic compensation considered ...............--- z\eishaleled aye Bech aieiek aa 7 sone ws 68 Derivation Of fortiu leet jo esc co2 con outa hens a eeilage x coe ine ve < Mekein sae BOER eat kee oan cases 69 Table of TECUCHONAACIONS 5... oscauc es oe tviedin sot taleuS we remeueeiee ob daldetoee oe Dede se a eweaes ates 70 Comparison of topographic deflections, with and without isostatic compensation.......---...-......2..2-2.2--- 7 Area method of determining the figure of the earth.......... 22.2... 0 222 eee eee eee 73 Form of observation equation ...........22.- 222-2002 eee eee eee eee ee VEE tie Ree aManehe bose eeee 74 Formule for coefficients........ tet dle tgieicss fd arc chatnania) Ss PERMEATE RSS Sal Poe Mie eau is UE egy ges « 76 Derivation of formule for coefficients............ 22.2.2 - 202 e een eee eee a Examples of computation of coefficients........-..---. +--+ +2222 5222 ee eee eee cee ee eee eee eee eee 88 ‘The observation equations... scese.: vjdckenes ocwcoms npr eeteews subsea tiie Ped eeeseeeka cee eelee ee eeeEs 93 Why two corrections to the initial azimuth...... Cia ae nlp hat beter dace stall hie on w pnlatepaie are eth Sein Oue nays fs 101 Meanings of coefficients in observation equations........-.-.-------- +--+. 2-2-0 e eee eee eee eee eee eee eee ee 101 The normal equations and the values of the unknowns.......-..---------- 2-2-2220 222 eee eee eee eee eee 105 The residuals. ...--..------- +2222 22 eee ee ee ee center ete eee eee 106 6 CONTENTS. Page Reasons jor the adoption of solution G............c0eccceeccceecccceece cece ect eee cence eee esceeeeeeeeenees 114 INGO PLE Wall Wes ceck te ence elena taal ed bd An ae OR tile tak ea at teh Nel te cag pa apeie hain dn Sain rata capers tated aed 115 Effects of errors in data and methods...........0.2.000 000000 e cece cee c eee e een ece ence bebe e eee eee 116 Hrronstd 1G: tOta ll Case Bis avs enaceesesioeecs mesh cteh ua onapeera eee ate enlace ncaa whlch eae Scie Seana eet ere a Laser 116 Effect of errors in astronomic observations. ........-....-..2.00000 222 c eee eee ee eee e eet e eee ee cee ee 117 Hifectotterrorsim:distam cess cats os! ss operas eS Syhcspeens aS 2d elias tle ogee Oaaanadaly poareesee e smmeetaaws 118 Effect of errors in geodetic azimuths............. 2.22.22 e eee eee e cece e eee eee e ence c eee eee eee 120 ° STE ei) UC ES NOR IIUR cacti ection heen ec he Povavee hat ese ley ohecal Re AAS a in dchiosh HSIAVD Seep o aon BE eh a eeyew ores een Sante Cea 121 Accuracy of computations of topographic deflections................0 22.0022 -0 22 cee eee eee cee cence ee 123 Errors in topographic deflections due to maps........ 2.220200 2000-0 o eee eee eee te cette eens 124 Errors in topographic deflections due to omission of inner rings. ............--- 2-0-2220 2 202 cece ee eee eee 125 Table of maximum value of computed deflection, each ring, prime vertical................--22--0---0-- 126 Errors in topographic deflections due to errors in assumed mean densities.................2--+++-2+eee eee 128 Errors in topographic deflections due to method of computation. ......... 2.2.20... 200-22 cece eee eee eee 129 Effect of errors in coefficients. . ... Seen tee Beek Ske ta gtic a Baal ilabios da dnaast babi eee eneeens haps dse 132 General conclusion as to cause of residuals... 2.2.2.0. 020200 nee nents 132 Explanation of illustration No. 10: Residuals of solution G...........0. 00200022 c eee ee ene eee eee 133 Teste of reliability of conclusions. ...2c deoccces x se aauuivadee seee eau ed £2 oc dea ees Ls eeaeE meee: Yee eeeeees 135, Statistics of residuals, ten groups...............2- 2-000 e ee ee eee eee 135 Statistics of residuals, four groups. .--... 2.22... ee ee ce ne eee eee eee eee 138 Conelisions from ReSId WAL 3 s.c44-5 eo Sersceeie es comers earemernnts warp alguedete eis eens ec cieiereaiahany alee So & © a me 8 © Ss @ %@© e NS 2S a S a » 5 e iS) m oo 2 oS 2 8 S s | a = eo | e 2 on xq ‘os | 1g | 1.15 | 1.00 | 165 | 102 | 103 | 211 | 203 | lot | 510 | :05 | 206 | 260 | {77 | —4)51 | —98,71 The sector corresponding to each column of the above form may be identified from illus- tration No. 2. The sector numbered 1 is limited on its northern side by the eastern line from the station; that numbered 5 is limited on its eastern side by the southern line from the station, and so on. Sectors 1 to 8 represent areas farther south than the station; therefore the deflec- tions corresponding to land areas within these sectors have the plus sign. For land areas within sectors 9-16, and therefore north of the station, the deflection has the minus sign. A plus sign means that the zenith is deflected to the north and the astronomic latitude increased. - The italic figures* represent water compartments in which the mean elevations are nega- tive. The signs of these italic quantities are the reverse of those for other quantities in the same sectors, as indicated by the heading. An unusual process is necessary in deriving the italic figures, which will be explained later. For each compartment in rings 28 to 14, comprising all topography at distances from 0.19 to 39.22 kilometers from the station, the deflection is between ’’.000 and ’’.04, and no italics occur—that is, the mean elevation for each compartment is positive and not greater than 400 feet. The italic figures representing negative mean elevations begin to occur in ring 13. The italic number 1.18 in ring 2 and sector 3 expresses the fact that in that compartment, which lies in the western part of the south Atlantic, the mean depth is more than 3 000 fathoms. Each entry in the column headed “Horizontal sum”’ is the algebraic sum of the quanti- ties in that line, and represents, therefore, the deflection due to a whole ring. At this station no ring of topography smaller than ring 13 produces a deflection greater than '’.06. Ring 4 has the largest effect, —6’’.48. This ring includes oceanic compartments having mean depths of nearly 3 000 fathoms, and a land compartment with a mean elevation of 1 300 feet. In each line of the column headed ‘‘Continuous sum” is shown the sum down to that point of the column headed ‘‘ Horizontal sum.’’ Hence each of these values is the topographic deflection due to all rings of topography from the station out to and including the ring indi- cated. This column serves (though that is not its main purpose) to indicate how important *In the computations as made these were red figures, but, for convenience in printing, italics, instead of red, have been used. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 27 it is—at this station, for example—to extend the computation to a great distance if it is desired to secure even a fair approximation to the topographic deflection. The compartment which is in ring 13 and sector 1, southeastward from the station, from 24 to 35 kilometers from it, is the nearest compartment which has a negative mean elevation and therefore is represented by an italic figure in the table. It comprises a part of the Bay of Fundy. A large group of italic figures, which are principally in rings 11 to 1 and sectors 1 to 6, correspond to the Atlantic Ocean compartments. These italic figures for ocean compartments are much larger for this station than are the values for the land compartments. That is the case for many stations. The maximum italic number is 1.18. The maximum value (not in italics) for any land compartment is ’’.37 for ring 1, sector 9, a compartment comprising practically all of the Cordilleras, the western mountain system of North America. The compartment in ring 3, sector 13, for which the computed deflection is ’’.20, com- prises the high mountain mass east of Pitdson Bay and-southeast of Hudson Strait. The compartment in ring 3, sector 10, contains part of Hudson Bay, as indicated by the italic number ’’.02. The compartment in ring 6, sector 15 (in italics ’’.04), contains the Gulf of St. Law- rence, and the compartment in ring 5, sector 16 (in italics, ’’.05), contains Newfoundland and the Great Bank of Newfoundland. There is a group of italic numbers in rings 3 to 1 and sectors 14 to 16. These compart- ments cover the northern part of the Atlantic Ocean. WATER COMPARTMENTS. To obtain the italic figures representing the deflections for compartments which include . oceanic areas, a variation from the procedure for land compartments is necessary for two reasons. First, the depths are, as a rule, expressed in fathoms rather than feet on the charts which were used. No depths were expressed in feet on these charts, as a rule, unless they were less than 18 feet. These small depths are, for the purpose of computing topographic deflections, almost or quite negligible. Second, to treat the depths below mean sea level in the same manner as the land elevations above mean sea level, with only a change in the alge- braic sign, would be equivalent to assuming the space between sea level and the ocean bottom to be void, whereas, in fact, it is filled with sea water having a density of 1.027. If the sea water were to increase in density from 1.027 to 2.67 (the mean surface density of the earth) by simply decreasing in volume and depth, without any horizontal transfer of material, and remaining in contact with the original sea bottom, the new depth of material 1.027 2.67 everywhere be .615 (=1-.385) below the original sea level. Hence, in the computation of topographic deflections, each o¢ean compartment has been considered as if it were a void from sea level down to a depth .615 of the actual mean depth of that compartment. Taking also into account the reduction of fathoms to feet the special factor necessary to put oceanic compartments on the same basis as land compartments is 3.69 (=six times .615), or, in other words, for a sea compartment the deflection produced is D =0’’.0003690 (depth in fathoms). That is, the mean depth of each oceanic compartment expressed in fathoms was multiplied by 3.69 and the result then treated as if it were a negative elevation of a land surface, expressed. in feet. For compartments which are partly oceanic areas and partly land areas especial care must be taken in estimating the mean elevation to keep in mind the negative sign and the factor 3.69, in connection with the water portion of the compartment. In practice these mixed compartments were found to give little trouble. In compartments containing deep fresh water lakes a process somewhat similar to that followed for ocean areas is necessary. Account must be taken of the fact that the density of the lake water is 1.000 and that the lake surface is at a certain known elevation above mean would everywhere become .385 (- of the former depth and the new surface would 28 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. sea level. The Great Lakes were the only important cases of this kind which occurred in con- nection with the present investigation. For each of these lakes a special table was used, giving the values in seconds of arc to be entered in the computation for various depths in that lake. The computation of the topographic deflection was commenced, in each case, with the areas nearest the station which were shown clearly on the contour maps of the largest scale available. As the computation progressed outward to larger rings, when the limits of the maps of this largest scale were passed, other contour maps of the next smaller scale available were used and the templates of corresponding scale applied and the process continued until the limiting distance, 4 126 kilometers, was reached. For example, in the computation of the topographic deflection in the meridian at Calais (No. 164) the maps and charts used were the following, in order proceeding outward: 1. Coast and Geodetic Survey chart No. 300, scale z5455, showing principally the St. Croix River and Passamaquoddy Bay. 2. The post-route map of Maine, scale 772555, approximately, showing drainage, but no contours, and used in connection with a few known elevations. 3. Coast and Geodetic Survey chart No. 301, scale ¢3493, showing Cobscook Bay and part of the Bay of Fundy and contours of land in the immediate vicinity. 4. Hydrographic Office chart No. 1412, Mercator projecticn, scale 1° of longitude =1.55 inches. This was used for depths southeast of Nova Scotia. 5. U.S. Geological Survey contour map of the United States in three parts, scale 2554y70- This was used for all land compartments in the United States not covered by the foregoing. 6. Coast and Geodetic Survey chart No. 101, scale z5457, used for the vicinity of Grand Manan Channel. 7. Coast and Geodetic Survey chart No. 6, scale zy7557, used for some of the depths in the Gulf of Mexico. 8. Map of the United States and Canada, published by the Canadian Geological Survey, scale 1 inch =242 miles; British Admiralty chart No. 2059, Mercator projection, scale 1° of longitude =0.3 inch; and Century Atlas map of the region around the North Pole, scale 1 inch =290 miles. These three were used for Canada and contiguous waters and islands, the last being used especially to obtain the approximate elevations for Greenland and other regions near the Pole. 9. British Admiralty charts Nos. 2936 and 2935, Mercator projection, scale 1° of longi- tude =0.2 inch. These were used for ocean depths in the open sea, far from the coast in the Atlantic. 10. The small U. S. Geological Survey map of the United States, scale 1 inch =111.1 miles, and the Century Atlas were used for Mexico. Difficulties were encountered when it was attempted to apply the templates to charts on a small scale, covering very large areas, constructed on a Mercator projection. On such charts no one scale of distances applies to the whole chart, and the radial lines of the ordinary template, representing arcs of great circles on the éarth’s surface, are not straight lines on the chart. These difficulties were overcome by constructing special templates to fit such charts, the tem- plates being distorted in the same manner as the charts. On these special templates the radial lines are curved, and the lines which on ordinary templates are arcs of circles were curves of varying radii of curvature. These special templates were constructed by plotting by latitudes and longitudes a sufficient number of the points representing the corners of compartments to enable one to draw the two sets of curves connecting them with a free hand with sufficient accuracy. The required latitudes and longitudes of corners were obtained by scaling from a large globe, or map with a polyconic projection, on either of which an ordinary template could be used with sufficient accuracy in locating the corners. Such a special template, once con- structed for a given Mercator chart and for a given station on the chart, could be used with sufficient accuracy for any other station within one or two degrees of the same latitude. For other stations, even on the same chart, differing much in latitude from the one for which the template was constructed, it was necessary to construct a new special template. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 29 This difficulty in regard to small scale Mercator charts was not found to be serious for the reason that, as shown later, the direct estimation of mean depths on such charts was found to be necessary in a relatively small number of cases only. For the many other cases the values were derived directly by interpolation. Three considerations operated to fix upon the ring which has for its outside radius 4 126 kilometers (2 564 miles) as the largest ring to be taken into the computation of the topographic deflections: (a) In the next larger ring considerable areas would be included for which our knowledge of the elevations is very limited, as, for example, the unexplored Arctic regions and the interior of South America. (b) The largest ring included in the computation has a sufficiently great outer radius to insure that for stations in the extreme eastern part of the United States the ring reaches to the Pacific, and that for extreme western stations it reaches to the Atlantic. Hence the whole width of the continent is taken in by the computation. In the next larger ring, for any station, portions of each ocean would be included, and in general would tend to balance each other. Hence the total computed effect for each larger ring omitted will be in general considerably less than for each of the last few rings included in the computation. (c) The larger the ring considered the more nearly the computed topographic deflection corresponding to that ring tends to approach a constant value for the whole United States, and, therefore, the less serious is the effect of omitting said ring. EXAMPLES OF TOPOGRAPHIC DEFLECTION COMPUTATIONS. The following four additional examples of computations of topographic deflections are given for widely separated stations. In contrast to the example given on page 26 of the computation of the meridian component of the deflection at Calais, Maine, a station in a comparatively flat country and near the Atlantic Ocean, the following example represents a computation for the station Uncompahgre, Colorado, in a region of steep, high mountains, and far from the nearest ocean. Computation of topographic deflection, latitude station No. 54, Uncompahgre, Colorado. Number of sector. =a : Hori- | Contin- Ring. Plus unless italics. Minus unless italics. zontal uous sum. sum, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 u wy aA MW Ww wy uw wu wv “ Ww u wv yw uw Za ar wt +.02 | +.02 | +.01 | +.01 +.02 | +.03 | +.03 | +.04 | +.03 | +.03 | +.02 | +.02 26 1.32] 1.33] 1.34] 1.36] 1.38] 1.39] 1.38; 1.38) 1.32] 1.30] 1.30] 1.29] 1.30) 1.30] 1.32] 1.32] + .27!] + .97 +.03 | +.01 | +.01 | +.01 | +.01 | +.01 | +.01 +.03 | +.04 | +.03 } +.03 | +.02 | +.02 | +.02 | +.02 25 1.28] 1.382) 1.34] 1.34] 1.84) 1.384] 1.384] 1.37] 1.27] 1.26] 1.27 | 1.28] 1.29] 1.29] 1.30] 1.30] + .29! + .56 +.03 | +.01 | +.01 | +.01 | +.01 | +.01 | +.01 | +.01 | +.02 | +.05 | +.04 | +.02 | +.02 | +.02 | +.02 | +.02 24 124] 1.30) 1.81) 1307 1.30, 1.29) 1.80) 13h) 126) 1.20) 1.22] 1.25 | 1.28] 1:28] 1.971 1.26| 4+ .22| = .7B +.02} +.01 | +.04] +.02 | +.02 | +.02 | +.02 | +.02 | +.01 | +.04/) +.04 | +.02 | +.01 | +.01 | +.01 | +.01 23 122] 129) 1:25) 124) 122] 1.22) 1.24, 1:28) 1.23] 1.17| 1.17] 1.22] 1.26] 1.26] 1.26| 1.96); +.15| + .93 +.01 -00 | +.01 | +.02 | +.02 | +.02 | +.01 | +.01 | +.01 | +.03 } +.02 | +.01 +.01 22 1.20] 1.25] 1.20] 1.17] 1.15] 1.18] 1.21) 1.25] 1.20] 1.12] 1.15] 1.22] 1.26] 1.26] 1.23] 1.26| — .07] + .86 +.01 00 | +.01 | +.02 | +.02 +.01 | +.02 | +.01 | +.01 +.01 21 1.16] 1.23] 1.12) 1.10) 1.08] 1.25) 1.29) 1.29) 1.14] 1.09] 1.14] 1.17] 1.26) 1.24] 1.21] 1.94] + .03] + .89 +.01 +.01 | +.01 +.01 | +.01 | +.01 | +.01 20 1.14] 1.20; 1.10) 1.10] 1.20] 1.25] 1.22] 1.26] 1.15] 1.06] 1.08) 1.16} 1.26] 1.25} 1.20! 1.24] + .06] + .95 +.01 +.01 | +.01 +.01 +.01 | +.01 19 1.06; 1.11) 1.04) 1.08) 1.12] 1.08] 1.19] 1.25) 1.16) 1.08; 1.04} 1.22] 1.24] 1.19] 1.20] 1.20] — .38] + .57 18 1.09 -98 | 1.02) 1.04] 1.05) 1.20) 1.18] 1.22] 1.10]- 1.14) 1.09) 1.20] 1.15] 1.15] 1.17] 1.92] — .44] + .13 17 1.00] 1.10; 1.20] 1.26] 1.20; 1.15] 1.24) 1.16] 1.13] 1.09] 1.10] 1.10] 1.04] 1.14] 1.18] 1.24] + .29] + .42 16 1.00} 1.16] 1.20; 1.16; 1.20} 1.27] 1.28; 1.10 -95 | 1.04] 1.02] 1.14] 1.10] 1.08] 1.10 95 | + .99 | +1.41 15 1.10; 1.13} 1.10} 1.18) 1.20; 1.15; 1.23) 1.20 TT +95 -96] 1.05] 1.04} 1.04 -95 | 1.04] +1.49 | +2.90 14 1.17| 1.07} 1.09} 1.22) 1.19] 1.10] 1.15) 1.14 79 90 91 85 88 -90 -88 | 1.17 | +1.85 | +4.75 13 1.09 -99| 1.10; 1.20; 1.20) 1.08} 1.10] 1.04 82 66 80 86 88 - 80 . 85 78 | +2.35 | +7.10 12 1.05 | 1.06; 1.02] 1.00 +98 -95 | 1.05 | 1.00 -79 58 +72 -80] 1.00} 1.00 85 92] +1.45 | +8.55 11 -96 | 1.00 80 73 76 77 93 85 +74 65 72 85 94 +95 92 90} + .13 | +8.68 10 -82| 1.03 | 78 70 65 65 65 62 73 62 -76 88 86 | 1.00; 1.10 95 | —1.00 | +7.68 9 87 85 73 71 68 59 56 55 - 50 60 +83 -90 90 941 1.10 84 | ~1.07] +6.61 8 - 69 86 74 70 74 66 +70 67 . 69 58 . 62 - 69 84 94 83 61) — .04] +6,57 7 52 53 63 53 73 65 57 57 72 -92 -70 70 76 77 50 50 | — .84] +65.73 6 36 42 54, 50 62 52 49 49 49 64 83 70 61 50 42 35 | — .60] +5.13 5 19 30 41 40 40 38 14 24 58 56 71 53 44 32 27 22 | —1.17] +3.96 34 14 +05 a7 4 08 14 26 43 03 06 28 10 45 35 65 38 26 23 15 13 | —1.64] +2.32 11 00 40 +04 04 45 03 3 02 33 -50 06 40 61 66 77 05 18 48 20 12 08 12 09 | —2.68-| —0.36 01 05 82 26 2 ar 26 18 70 62 74 84 8&4 76 40 15 11 02 05 10 | —4.07 | —4.43 14 : 03 1 24 33 7 17 92 96} 1.07 | 1.00 98 07 30 04 03 02 12 20 | —5.07 | —9.50 30 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. The elevation of the station is 14 289 feet (4 355 meters) and the slopes in its vicinity are very steep. In each compartment space in rings 26 to 19 in which two numbers are given, the upper one is the slope correction. There are slope corrections in every ring from 26 out to 19 (outer radius 6.653 kilometers). The explanation of such corrections will be found on page 34. There are a few compartments in rings 4 to 1 in which two numbers are given, of which one is an italic number. It was found to be more convenient in many cases for compartments which involve both oceanic and land areas to estimate the mean depth for the oceanic area ‘and the mean elevation for the land area separately, than to combine them in a single estimate. This has accordingly been done in many cases and both of the separate estimates entered on the computation form, as, for example, for the compartment in ring 4, sector 6, including parts of Mexico and of Lower California. For this compartment the estimated mean elevation of the land portion is 2 000 feet, corresponding to ’’.20. This, multiplied by 0.7, the ratio of the land area to the total area of the compartment, gave the value ’’.14 (not in italics), entered in the computation for the compartment. Similarly the mean depth of the oceanic portion of the compartment was estimated to be 550 fathoms, corresponding to ’’.20. This, multiplied by 0.3, the ratio of the oceanic area to the total area of the compartment, gave ’’.06, the italic number entered in the computation for the compartment. The ”’.06 (italic) combined by alge- braic addition with the ’’.14 (not italic) gives ’’.08 (not italic), which is precisely the quantity which would have been obtained by making a single estimate for the whole compartment at once. Mount Uncompahgre is near the northern edge of a large mountain mass, extending above the 10 000 foot contour. The Gunnison River lies at the northern edge of this mountain mass. The valley of the river to the northeastward of the station is represented by the values ’’.88, 90, ’’.88, and ’’.80 in rings 14 and 13 and sectors 13 and 14. The larger values 1’’.00 to 1’”.04 in the same sectors in rings 15 and 12 represent the mountains rising above the 10 000 foot contour on the south and north sides of the valley, respectively. Rings 17 to 11, inclusive, lie with their southern parts on the mountain mass and their northern sides on the northward slope of the mountain mass or in the river valley. Hence for each of these rings the total is positive (a deflection of the zenith to the north) as shown in the column headed ‘Horizontal sum.”’ For the larger rings 10 to 1, which lie nearly or quite outside of the mountain mass of which Mount Uncompahgre forms a part, the deflection is negative in each case (zenith deflected to the southward). The italic figures in sectors 1 to 4 and 16 represent portions of the Atlantic Ocean and the Gulf of Mexico. Those in sectors 5 to 10 represent portions of the Pacific Ocean. As shown in the last column, the algebraic sign as well as the amount of the computed topographic deflec- tion depends upon the distance to which the computation is carried. If the computation be stopped with ring 4, outer radius 1 369 kilometers, or at a shorter distance from the station, the computed deflection of the zenith is to the northward, otherwise it is to the southward. The following examples of the computation of the prime vertical component of the topo- graphic deflection are for three especially interesting stations. For the first, No. 115, North End Knott Island, the depression occupied by the Atlantic Ocean has its greatest influence on the computed deflection. For the second, No. 1, Point Arena, the Pacific Occan depression has its greatest influence. At the third, No. 59, Mount Ouray, the computed deflection is com- paratively small, as the two oceanic depressions and the different parts of the continental mass nearly balance each other. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U.S. 381 Computation of topographic deflection, azimuth station No. 115, North End Knott Island, Virginia. Number of sector. Hori- | Contin- Ring. Plus unless italics. Minus unless italics. zontal uous | sum. sum. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 aw wy wu uw uw u" ” ” uw Yt uw uw Ww u uw uw 7 u 25 000 | .000 000; .000 | .000 000 000 | .000{ .000} .000 000 000 000; .000 000 000 - 00 .00 24 000 | .000 000 | .000| .000 000 000 | .000] .000} .000 000 000 000 | .000 060 000 - 00 - 00 23 000 | .000 000} .000) .000 000 000 | .000} .000) .000 000 000 000 | .000 000 000 -00 - 00 22 000 | .000 000 | .000} .000 000 000 | .000{ .000j; .000 000 000 000 | .000 000 000 -00 .00 21 000 | .000 000 | .000| .000 000 000 | .000]) .000} .000 000 000 000 | .000 000 000 - 00 - 00 20 000 | .001 002 | .002; .001 000 000} .000]) .000; .000 000 000 000 | .000 000 000; + .01| + .01 19 001 | .000 001 -001 001 000 001; .000| .000} .000 000 000 000; .000 000 000 00} + .0L 18 001 - 002 002} .002| .002 002 001 -000{ .000] .000 000 000 001 | .001 000 000 | — .O1 . 00 17 002 | .008 003 | .003} .003. 003 002 | .000} .000} .000 000 000 000 | .001 001 000 | — .02} — .02 16 003 | .004 004 | .004| .004 003 004 | .000} .000} .002 000 001 001 | .001 001 001 | — .03! — .05 15 0038 | .004 004 | .004| .004 004 004 | .002| .002} .008 001 001 001 - 001 001 001 | — .04 | — .09 14 004 | .004 004 | .004| .004 005 005 | .003} .002| .003 003 002 001 | .002 002 002; — .05| — .14 13 004 | .004 004 | .004| .005 005 005 | .003|} .001 | .003 004 002 001 - 001 001 000} — .05| — .19 12 004) .006 006; .006| .007 007 007 ; .003 | .000! .003 003 003 005 | .005 001 000; — .07| — .26 11 004 - 008 010) .O011| .08t 031 009 | .004} .002) .002 003 003 008 | .007 003 000 |} — .14] — .40 10 00 -02 13 22 33 28 20 08 - 00 - 00 00 OL 01 -O1 o1 00 | —1.24 | — 1.64 9 00 il 44 53 65 55 61 42 - 00 - 00 00 00 03 04 03 OL —3.32 | — 4.96" 8 | .00 - 29 59 - 64 ~84 87 85 76 -01 -O1 03 05 06 -07 07 04 | —5.16 | —10.12 7 00 -39 66 81 - 90 -92 95 78 06 -O1 04 09 18 -20 23 12 | —6.23 | —16.35 6 03 - 30 74 | 1.07 96 99 | 1.07 94 12 -O1 06 18 -18 14 12 14 | —6.74 | —23.09 5 04 11 84 -99 | 1.08 | 1.05 | 1.07 | 1.07 14 02 08 17 -10 - 08 09 07 —6.62 | —29.71 4 05 -09 84 | 1.01 - 98 -92 | 1.01 | 1.06 03 O4 03 05 04 -07 07 08 | —6.12 , —35.83 3 04 12 98 | 1.05 ~98 | 1.12 | 1.14 94 29 4? 02 03 -10 ll 13 09 | —5.97 | —41.80 2 - 03 21 -89 | 1.08 -95 | 1.10 | 1.07 88 «2h 46 .13 .20 ~42 . 52 +28 :07 | —7.02 | —48.82 1 85 75 70 76 ~ 84 - 88 -92 fl 55 +46 «17 - 08 = 28 -55 -43 -16 | —5.48 | —54. 30 No. 115, North End Knott Island, is situated on a low-lying island in Currituck Sound, Virginia, surrounded by shoal waters and low lands. There is no large change in the topography, until the submerged edge of the continent, as represented by the 1 000-fathom line, is reached, 130 kilometers from the station. The first effect of this is shown by the values in italics, ’”.22, ’ 33, '’.28, and ’’.20, in ring 10, sectors 4 to 7, inclusive. In ring 7, sectors 13, 14, 15, the values ’’.18, ’’.20, ’’.23 show the effect of the main mass of the Blue Ridge Mountains. In ring 4, sectors 11, 12, 13, the values ’’.03, ’’.05, ’’.04 correspond to the Mississippi Valley. In rings 1 and 2, sector 14, the values ’’.55 and ’’.52 show the effect of the Cordilleras, the western mountain system of North America. . < 32 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. Computation of topographic deflection, longitude station No. 1, Point Arena, California. Number of sector. : ees Hori- | Contin- Ring. Plus unless italics. Minus unless italics. zontal uous sum. sum. 1 2 3 : + 5 6 7 8 9 10 ll 12 13 14 | 15 16 : Ph: at Mt at at dt oF 4, af tt dt Mt ce a at #e ov ge 30 020 021 021 020 019 018 O16 | .014} .018 014 015 015 018 018 019} .919)4+ .02/+ .02 29 020 021 021 020 021 019 015; .010} .009 010 010 011 016 018 019 | .019 j+ .04/+ .06 28 020 021 021 020 018 017 017 | .018| .010 009 009 010 015 018 018 | .020/+ .04/+ .10 27 021 022 022 021 019 018 017 | .021 | .010 009 008 007 015 021 021 | .021 |+ .05/+ .15 26 021 022 021 020 019 018 018 | .026; .020 020 018 015 012 020 025) .020/+ .02/+ .17 25 020 022 022 021 021 018 019 | .027} .023 024 024 019 015 022; .012|+ .02/+ .19 24 02 02 02 02 022 020 022 | .026, .022 023 025 024 010 015 022) .018|/+ .O1/+ .20 23 02 02 02 02 02 022 026 | .025} .000 000 000 000 019 018 022) .012|+ .10/+ .30 22 02 02 02 02 02 02 018 | .022] .003 003 002 002 001 010 019 | .012|+ .13/+ .48 21 02 02 02 02 02 02 023 | .008 006 oor oor oor 003 001 012; .010|+ .16/+ .59 20 OL 02 02 02 02 02 01 . 00 - 009 O11 O1t O11 004 002 001 | .012|+ .16/+ .75 19 o1 02 03 04 04 03 01 - 00 ot o1 02 02 o1 ol 00 -00 |+ .26/+ 101 18 00 02 04 05 05 03 ol - 00 Ot 02 02 02 02 ol Ol -00 |+ .3Lj4+ 1.32 17 00 03 05 06 05 04 ol Ot 03 03 03 02 o1 Ot ol -O1 |+ .38)/+ 1.70 16 05 06 06 05 08 05 02 - OL 06 or o7 O4 ol ot o1 -O1 |+ .64/+ 2.34 15 08 12 12 14 10 09 02 -02 22 22 17 11 03 03 02 -O1 }+ 1.46/4+ 3.80 14 10 15 15 15 15 10 03 - O4 37 37 35 18 09 it or 02 (+ 2.385 /+ 6.15 13 15 19 16 15 15 09 04 - 06 66 66 60 55 18 18 18 -04 |+ 3.92 |+ 10.07 12 17 17 17 18 15 07 07 - 06 62 7 73 73 44 44 29 «11 |+ 4.98 |+ 15.05 ll 20 18 17 12 15 09 03 - 06 73 77 77 8&1 70 62 61 11 |+ 5.90 |+ 20.95 10) .29 | .07 | .02 | .03 | .06 | .05 | .04 | .O | .8f | .77 | 177 | .77 | 173 | 168 | 47 | cat \ + 5.74 4 26,69 - 02 | 9] .17 | .11 | .09 | .04 | .02 | .o1 | .05 | .02 | .77 | .84 | .a4 | 184 | .73 | 77 | .47 at \ + 6.00 |-+ 32.69 8} .36 | .50 | .50 | .50 | .50 | .17 | .19 | .o1 | 87 | .88 | .92 | .92 | .e4 | sy | .78 “At \ + 9.16 |4 41.85 7 47 53 48 45 75 55 - 03 “a 92 92 95 95 92 84 44 a + 9.44 |4+ 51.29 6| 51 54 53 74 59 55 | .27 me 92 92 | .92 | .95 92 | .8f 66 si +10. 10 |+ 61 39 5 38 49 58 57 58 33 25 Ghichgo. i Davenport : i Cleve, e Tv eorlan | ‘a Fo re f tr t ’ ae - i 0, Keokuk ~~ ba 2 t = QuincyT shringield) v7! In@ianapolis err autel | Telferson 72 yee e ty st 7h ip Madiso: 70 isvil rs ours Xe ingto ic K &E s ne. wy NO, 4 areal |, ae: os OL t 5 si 3 c Yi -—| ’ =S a Golumpis \ ‘oA RO ‘Augusta, Geo! Dalla = : < Macon oe | 6 ickabu | 7 ! xX A Ss J Sacks 4 Moore Cogvanoe © oO = P| atchez w , 4 > > wf y va " cong | oF Nl Fermaugina y ustin 4¢ Baton 1 ae { ze jensacols 0 a scnell e a my Hoystono a W, Orlea Lz. x jeans i alveato PAS J Apalachicgla, Pov. suaperyy W N A OP wh OTs sO 242 +19. 38 +72. 94 + 9.04 28 229 +15. 04 +73. 75 + 7.18 10 228 +19. 71 +76. 41 + 9.54 52 227 +20. 62 +80. 27 +11. 32 80 226 +14. 67 +78. 05 9. + 224 +20. 15 +80. 78 +10. 79 225 +15. 90 +86. 09 +14. 36 223 +14. 85 +86. 56 +14. 05 241 +29. 93 +84. 44 +12. 15 222 + 7.64 +73. 39 + 4.48 220 +14. 14 +79. 85 + 8.76 221 + 6.56 +88. 30 +12. 33 219 +. 6. 33 +85. 30 + 9.57 239 + 5.52 +94. 98 240 +14. 04 +87, 24 218 +10. 25 +82. 89 17 + 5.63 +82. 78 13 + 3.70 +79. 08 238 — 4.47 +84. 37 5 + 5.67 +83. 62 237 +18. 18 +90. 66 2 +10. 90 +89. 69 236 +18. 82 +85. 19 4 +12. 32 +79. 91 ee ee S LAEHEEFEEE HEE FEFEFEEE VON RDO BPN ANBDN HO BDNW Orb 00D OO N ON HR OD SO ~I oOo ONE NHEHNOONOHWORNNKHWOKRPTUNA DOOD 8 | LREEEEEHEE EET ET+t +++ +444 x = [PRR EEAAEHE EF EEELEETHEEFEEFEFEEEEEFEFEFHFEFEPEF bn oo a Hh e NWR EAPO OWDWORDDOANOrD SE WORMONOOWWNANOODOALD 14 ee [PEER EEHFE EEE EFEEEF TEE THE Il + 2.92 +75. 11 18 64 42 12 + .97 +75. 86 : : 15 + 7.37 +77. 34 08 2.33 2. 05 14 + 7.32 +75. 66 . 28 1. 65 1.40 26 +10. 14 +77. 05 11. 90 9. 99 9. 64 19 + 4.14 +78. 58 23 6. 89 « 6. 45 . 21 + 9.42 +73. 38 42 5. 74 5. 47 18 + 1.29 +76. 73 40 7. 29 6. 91 27 + 3.86 +64. 35 74 8. 48 8. 24 28 + 8.25 +51. 35 . OL 1.33 1.19 29 + 5.48 +47. 85 94 1.38 1. 29 30 + 3.95 +42. 85 27 1. 89 1. 84 34 + 7.43 +39. 57 81 1.53 1.49 35 + 6.28 +28. 54 76 2. 39 2. 28 32 + 2.83 +23. 61 . 44 5. 95 5. 81 38 — 2.41 +29. 48 — 1.00 — 17 — 1.89 42 +16. 25 +44. 72 +12. 84 +11. 35 +11. 05 40 +12. 98 +45. 96 +14. 05 +12. 48 +12. 17 43 +24. 84 +54. 71 +22. 11 +20: 38 +20. 00 45 +18. 15 +48. 49 +16. 12 +14. 45 +14. 10 39 + 7.98 +37. 85 + 6.30 + 5.16 + 4.93 37 +12. 69 +34. 08 + 3.38 + 2.81 + 2.72 46 +18. 69 +42. 03 +10. 93 + 9.75 + 9.51 47 + 8.74 +29. 30 + 1.84 + 1.48 + 1.42 48 + 5.34 +23. 85 — .82 — .55 — .44 49 +10. 50 +27. 24 + 4.38 + 3.16 + 3.15 51 +14. 68 +31. 12 + 7.42 + 6.30 + 6.08 52 + 4,52 +20. 84 + 1.52 + .98 + .87 54 + 6.79 +19. 90 + .86 + .02 — .1 57 + 7.63 +18. 70 + 4,96 + 4.17 + 3.99 56 +11. 36 +10. 31 + 1.76 + 1.43 + 1.33 58 +13. 92 +20. 00 + 8.89 + 8.06 + 7.84 59 + .95 —11. 25 —11. 07 —10. 21 —10. 01 60 — 9.79 —28.23 | —17.00 —16. 02 —14. 62 62 — 4.44 —22. 20 — 8.72 — 7.01 — 6.67 63 — 1.44 —19. 00 — 4.18 — 3.03 — 2.82 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 57 The meridian components of the topographic deflection are all negative, varying from —0’’.53 at the latitude station Porcupine (No. 216), on the south shore of Lake Superior, to —64’’.97 at the latitude station Santa Barbara, California, (No. 238). All these computed topographic deflections have one sign, because the United States is near the southern point of the continent of North America. At the latitude station New Orleans, Louisiana, (No. 135), near the northern shore of the Gulf of Mexico, the meridian component of the topographic deflection is —27’’.35 and at the latitude station Howard, Maine, (No. 158), it is —35’’.94. An unusually rapid change of the meridian component of the topographic deflection between adjacent points is illustrated by the latitude stations Green River, Utah, (No. 50), and Mount Waas, Utah, (No. 51). Although these stations are only 96 kilometers apart the topographic deflections differ by 14.06, being —22’.76 at the former and —8”.70 at the latter. These stations are in a region of high, steep mountains, and the deep, broad valley of the Grand River lies between them. The maximum negative prime vertical component of the topographic deflection occurs at the azimuth station North End Knott Island, Virginia, (No. 115), namely, —54’’.30. This station is only 130 kilometers from the submerged edge of the continent as fixed by the one thousand fathom line. The maximum positive prime vertical component of the topographic deflection occurs at the longitude station Point Arena, California, (No. 1), namely, +104’.63. This station is only 35 kilometers from the submerged edge of the continent as fixed by the one thousand fathom line. For the azimuth station Mount Ouray, Colorado, (No. 59), in longitude 106° 13’ and for all prime vertical stations to the eastward of it, the prime vertical component of the topographic deflection is negative. For all stations to the westward of Mount Ouray it is positive. An unusually rapid change of the prime vertical component of the topographie deflection between adjacent stations is illustrated by the azimuth stations Gunnison, Colorado, (No. 56), and Mount Ouray, Colorado, (No. 59). Although these stations are only 63 kilometers apart the prime vertical components of the topographic deflections differ by 21’.56, being +10’.31 at the former and —11’’.25 at the latter. These stations are in a region of high, steep mountains. These computed topographic deflections must necessarily exist as actual deflections if the material comprising the surface of the earth and down to the level of the lowest point of the ocean floor has a density of 2.67 and if the densities below that level have no relation to the topography of the surface. The irregularities of the surface constituting the topography certainly exist. The computations of the topographic deflections depend upon the well-estab- lished law of gravitation that the attraction between two masses is proportional to their product and inversely proportional to the square of the distance between them. On the other hand, a comparison of these computed topographic deflections, shown in the third column of the preceding tables, with the observed deflections of the vertical as shown in the second column, shows clearly that the latter are much smaller than the former. After a comparison in detail, one finds it difficult to see easily what relation exists between the two sets of quantities. One may possibly begin to doubt that any close relation does exist between them. As the observed deflections evidently do not correspond to the deflections due to the known topography, it is evident that they must be due, in part at least, to variations in density beneath the surface. CONSTRUCTION OF CONTOURS OF THE GEOID. Before proceeding to a study of the possible relation of the distribution of the subsurface densities to observed deflections of the vertical, it is desirable to show the outcome of a subsidiary investigation which was made to develop the extent to which the observed deflec- tions of the vertical are related to the topography. This investigation was made by con- structing the contour lines of the geoid graphically, starting with the observed deflections of the vertical as a basis. 58 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S8. By contour lines of the geoid are meant lines of equal elevation on the geoid surface, referred to the Clarke spheroid of 1866 as a reference surface, the spheroid being supposed to be in the position fixed by the adopted United States Standard (Geodetic) Datum. The contour lines serve to indicate clearly to the eye, and in a comprehensive manner, the depar- ture of the geoid from the spheroid. The geoid surface is a surface which is everywhere normal to the direction of gravity (an equipotential surface), and it is that particular one of many such surfaces, lying at different elevations, which coincides with the mean sea surface over the oceans. Obviously, the mean sea surface must, with considerable accuracy, be everywhere normal to the direction of gravity. The mean sea surface is an existing physical representation of the geoid surfacé for the three-fourths of the earth covered by the oceans. No similar physical representation exists for the areas covered by continents. One may conceive of such a physical representation by supposing that narrow canals, say one foot wide, were cut down to a depth somewhat below mean sea level along the township boundaries of the land system over the United States. Such canals would form a rectangular system following approximately along meridians and parallels and approximately six miles apart in each direction. If the sea water were allowed free access to all these canals and were protected from all disturbances, the surface of the water in the canals would everywhere become normal to the direction of gravity and would be at sea level and, therefore, would be a part of the surface of the geoid. One may think of the surface of the water in these hypothetical canals as forming a concrete representation of that portion of the geoid which lies under the United States. The restriction-in the preceding statement that the supposed canals must be very narrow (1 foot wide) is introduced because if the canals were supposed to be of considerable width the supposed removal of masses to make the canals would change the direction of gravity at various points and so produce a new geoid. ' The problem at present under consideration is that of constructing the contour lines which will represent the relation of the irregular geoid to the regular ellipsoid of revolution known as the Clarke spheroid of 1866, which is supposed to be in the position fixed by the adopted United States Standard Datum. The deflections of the vertical, as observed and shown in the tables on pages 12-19, are slopes of the geoid, at the points of observation, with reference to the spheroid. Having given these slopes in the direction of the meridian and the prime vertical at these few points, the problem in hand is to construct the contour lines of the geoid. It is a problem similar. to that which would be before the topographic draftsman if the topographer in the field furnished to him observed values of the slopes of the land surface in the direction of the meridian and of the prime vertical at a few points. oo separate steps in the construction of the contour lines of the geoid were as follows: . A series of drawings covering the area in question were made on a large scale, showing fees a few meridians and parallels drawn according to the polyconic projection. 2. The figures showing the observed deflections in the meridian were placed on each drawing in their proper positions. 3. Lines of equal deflection in the meridian were drawn by eye, after first locating a few points on each line by assuming that on the straight lines joining adjacent observed values the rate of change of the deflection in the meridian is constant. 4. The figures showing observed deflections in the prime vertical, resulting from both longitude end. azimuth observations, were placed on the drawing in their proper positions. These deflections as placed on the danwite were of course in seconds of arc of the prime vertical great circle, so as to be directly comparable with the deflections in the meridian. 5. Lines of equal deflection in the prime vertical were then drawn in the same manner as indicated above for deflections in the meridian. 6. A rectangular system of lines was placed on the drawing in such a manner as to divide the earth’s surface into portions which were as nearly as possible squares, 20.6 kilometers on THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 59 each side and with their sides always placed in the meridian and prime vertical. The particular length, 20.6 kilometers, was chosen because a line having a slope of one second of arc rises or falls one decimeter in 20.6 kilometers (sin 1’ = zyyg5y). In other words, to convert the slope of such a line expressed in seconds of are into the difference of elevation of its two ends in meters, all that is necessary is to move the decimal point one place to the left. This simple device eliminated the necessity for multiplications. 7. On the middle of each of the 20.6-kilometer lines of the rectangular system there were then placed the figures indicating. with the proper sign the difference of elevation of the two ends of this line, supposing it to be a line on the geoid. For any 20.6-kilometer line in the direction of the meridian the required difference was obtained by estimation from the lines of equal deflection in the meridian which had already been placed upon the drawing, the estimated deflection in the meridian being converted into a difference of elevation by moving the decimal point one place to the left. Similarly, the difference to be placed on the middle of each 20.6- kilometer line which lay in the direction of the prime vertical was obtained from the lines of equal prime vertical deflection. . 8. Starting with an assumed elevation for one point on the rectangular system, the elevation of all other points of the rectangular system (corners of the squares) were computed by use of the differences of elevation which had already been placed on the drawing, as indicated in the preceding paragraph. These elevations were placed on the drawing as rapidly as they were obtained. The elevations of one row of points were computed at a time. Let it be supposed that the elevations of the row of points along a meridian have been fixed and those of the next row to the westward are to be computed. For each new point three values can in general be obtained for its elevation. First, from the fixed point due eastward by applying the prime vertical difference of elevation. Second, from a fixed point due southeast by applying first a prime vertical difference and then a meridional difference of elevation. Third, from a fixed point due northeast by applying first a prime vertical and then a meridional difference of elevation. The mean of these three values was taken as the required elevation. Thus the process of computing the elevations of successive rows of points went on until the elevation of every intersection of the rectangular system had been fixed and marked on the drawing. For points on the margin only two values for the elevation could be so determined in general, and in some cases only one. When the successive rows of points established were in an oblique line (northeast and southwest or northwest and southeast), instead of a north and south or east and west line, but two values were available for the elevation of each new point, and the mean was taken. : 9. From the figures expressing elevations now upon the drawing the contour lines, or lines of equal elevation of the geoid above the spheroid, were drawn in exact accordance with the figures, without any generalization or smoothing. Illustrations Nos. 5, 6, and 7 show the details of the construction of the contour lines of the geoid for a small area in Kansas in the vicinity of latitude stations Russell Southeast Base, Ellsworth, and Salina West Base (Nos. 63, 64, and 65). Illustration No. 5 shows steps 1-5, inclusive, of the process of constructing the contour lines. Two meridians (98° and 99°) are shown and one parallel (39°). The astronomic stations, at each of which both the meridian and the prime vertical components of the deflection were observed, are indicated by small circles. The serial numbers of the latitude stations (63, 64, and 65) are shown at the left of the respective circles. The observed meridian component of the deflection is shown at the right of each circle. Similarly, if the illustration is held so that the west is at the bottom, the serial number of each longitude or azimuth station shows at the left, of each circle and the observed prime vertical component of each deflection of the vertical shows at the right.* The lines of equal deflection in the meridian and of equal deflection in * The observed deflections are taken directly from pages 12-19 or 48-56, with the exception thata correction has been applied to the prime vertical component of the déflection at azimuth station No. 65 to take account of an accumulated error in the geodetic azimuth, This correction will be explained later. 60 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. 9/9° 4 9|8° k 2a Soh a / // Pipe + ; \ \ 8 Le EFSF FD % \ \ 1 if ff ft 2 SF a ot \ -2 \ | “4 2 £ #2 ff 2 J NS \ ee 1 “A Pf fF £2 oP oy J % \ as , ff fee PS \ NY \ . \ ‘ / GF Jf ?f 7 a 7 -3 . mY ON Oo Fis FO ; | »% \ \ x k fy J 4 ; , MN Sy \ ‘ \ \. \ \ / if / /) a oe } f PION NN AAW gee NC VTE} a ; ee % We, \ we Y TALE ea Tes 3 rane: SSO % \ ht. \ / a “Sem KL | Ok Iver OF TY ON \\ \ \ / /\/ Z / i ff Ji / he 63 Ot FR % . aN \ Ne / / Ys _ if ey, 9074.15 S / wo SRA KIWI OX S XN ANS We Sy OB -*% SANS SOIT LAS Ff IP f \# SSAS SAAS LL LPS ’ 5a SOM op oe fz oe ee Ps Beet fA PT - * \ \ /: / , + Oo \ ee. a“ x J / / / “he x a ASTRONOMIC STATIONS \ a if a eevee a LINES OF EQUAL \ we fl / ae Pa DEFLECTION INTHE MERIDIAN —\' 2 a EP / eee wa be “ ou ___ ___ ___ LINES OF EQUAL owe ae oe / Sood \\ DEFLECTION IN THE PRIME \ 4 %, cy VERTICAL ‘ af tha \ BS 0 No. 5. N oo : \ \ ol" Oe ais a ales Mey 7 4 ~ a o ¢ gE \ \ \ Dos Fi St fe / o an & x g i NX & on = a | a @ & 8 8 e 2 No. 7. illustration, shown by the figures +18.84, +18.45, +18.06, +17.72, +17.49, at the corners of the squares, were fixed from the eastward by constructions which are beyond the eastern limit of this illustration. The numerical work of fixing the next row of elevations, namely, +18.66, +18.16, +17.65, +17.26, and +16.98, is as follows: 18.84 18.45 18.06 17.72 17.49 Row already fixed. 18.73 18.16 17.62 17.23 17.00 From eastward. 18.58 18.02 17.57 17.26 ..... From southeastward. Beret 18.31 17.76 17.28 16.97 From northeastward. 18.66 18.16 17.65 17.26 16.98 New fixed row. In the computation the second, third, and fourth rows of figures marked “From eastward,” “From southeastward,” and ‘‘From northeastward” were obtained from the first row in turn, as indicated in the description of step 8. A plus sign on the difference of elevation written on the middle of any 20.6-kilometer line of the rectangular system which lies in the east and west direction means that the eastern end of the line is higher than the western end. Similarly, a plus sign on the middle of any 20.6-kilometer line in the direction of the meridian indicates a slope upward to the south. The elevation + 18.73 in the row marked ‘From eastward”’ 62 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. was obtained by subtracting + 0.11 (see illustration No. 7) from + 18.84. The elevation + 18.58 in the row marked ‘From southeastward’’ was obtained by subtracting + 0.29 and — 0.42 (see illustration No. 7) from + 18.45. The elevation + 18.31, shown in the row marked “From northeastward,” was obtained by subtracting + 0.11 from + 18.84 and then adding — 0.42. (See illustration 7.) Each of the elevations of the new fixed row was obtained, as indicated in the description of step 8, by taking the mean of the two or three values just above it; that is, 18.66 is the mean of 18.73 and 18.58, and 18.16 is the mean of 18.16, 18.02, and 18.31, and so on. All elevations being fixed in the manner indicated, row by row, proceeding from east to west, the contour lines of the geoid were then drawn as indicated in the description of step 9. In illustration 7 portions of contours 17, 18, 19, and 20 meters are shown. On the 18-meter contour ten points were fixed directly by inisepelation at the ten points within the limits of the illustration at which this contour crosses the lines of the rectangular system. As the purpose of the general process just described is to detect, if possible, whether any relation exists between that portion of the geoid which underlies the United States, on the one hand, and the topography of the United States, on the other hand, it is important to note that each of the nine steps of the process is independent of any consideration of the topography and may be taken without any information whatever in regard to the topography. In the process there are few and unimportant opportunities for bias or prejudgment of the draftsman to affect the location of the contours. It is almost entirely an automatic process. The drawing was not compared with the topography until it was complete. The process insures that the surface represented shall be continuous, and shall have no sudden changes-of slope, even though the deflections in the direction of the meridian and the prime vertical indicated by the observations are apparently inconsistent with the assumption of a continuous surface. In other words, the method of construction is in itself a method of adjustment of the discrepancies in observations. Because it is an adjustment process the slopes indicated by the constructed contour lines of the geoid do not agree exactly with the observed slopes (that is, observed deflections of the vertical) at the observation stations, the disagreement being greater or less according to the degree of inconsistency of the observations with each other in a given locality and with the requirement that the geoid surface shall be continuous. The degree of inconsistency is indicated mainly by the discrepancies between the two or three derived values of which the mean is taken for each elevation in such a computation as that shown on page 61. It is not claimed that the method of construction of the contour lines of the geoid is even approximately perfect. It has grave defects. For example, for points on the margin of the belt covered by the construction there are but one or two determinations, in general, whereas for interior points there are three determinations. Hence the outer portions of the geoid contours frequently show a tendency to sharp curvature which is fictitious, being due to a defect in method, not to the facts of nature. So, too, it will be found that the results secured will be somewhat different if the construction of the contours proceeds from west to east along a given belt instead of from east to west. On account of these and other defects in the process, together with the fact that the problem of constructing a surface, having given a few observed slopes only at widely scattered points, is essentially indeterminate,* the surface represented by the geoid contours, as constructed, is only a rough approximation to the actual geoid surface. It is, however, the best approximation available. Because it is an approximation constructed by methods independent of the known topography it is believed to be valuable for the purpose for which it is to be used—namely, to study the relation between the geoid and the topography. *The problem is essentially indeterminate, because there are only a finite number of observed values of the slopes, whereas the geoid,. an irregular surface, has an infinite number of unknown slopes at its infinite number of points within the United States, which slopes must conform to only a few known conditions. But the process of constructing the geoid contours which has been used gives one solution which is believed to be reasonable because it insures continuity of the surface, insures continuity in the change of slope from point to point, and insures that the rate of change of slope is not much more rapid in any ‘case than the minimum rate of change consistent with the observed ! slopes. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 638 The contour lines of the geoid, as resulting from this construction based on 496 observed deflections in the United States, are shown on illustration No. 17 at the end of this publication. The contour lines of the geoid are shown in red. Certain selected contour lines of the land sur- face are shown in black on the same illustration, so that a comparison of the two kinds of contours may readily be made in order to detect what relation, if any, exists between the shape of the geoid and the topography. The observed deflections used in constructing the contour lines of the geoid are shown on pages 12-19 and again on pages 48-56. The prime vertical components of the deflections, as determined at stations west of Meades Ranch, Kansas, were, however, first corrected in each case by subtracting from 6’’.16 to 7’’.52, to take account of the known accumulated error in geodetic azimuths, as will be explained later. In constructing the contour lines of the geoid the initial point at the extreme northeastern part of the United States, near Calais, Maine, was arbitrarily taken as +10 meters in order to avoid negative contours. COMPARISON OF GEOID AND TOPOGRAPHY. A comparison shows the following relations between the shape of the geoid and the topography: 1. The 10-meter geoid contour is approximately parallel to the Atlantic shore line from the extreme north in Maine to latitude 37°. 2. The effect of the Adirondack Mountains upon the geoid is clearly indicated by the 204-meter oval among the geoid contours. This oval is somewhat too far west to correspond to the summit of the Adirondacks, but it should be remarked that but few astronomic observa- tions were taken in this locality. 3. The effect of Lake Erie upon the geoid contours is clearly shown and the position of Lake Ontario is also indicated by the geoid contours. 4. The lowest point on the geoid occurs in the eastern portion of Lake Superior. The lowest point of the bottom of Lake Superior is in its eastern portion and is more than 100 meters below sea level. 5. The divide between the Ohio River and the Great Lakes in Illinois is roughly indicated by a closed 16-meter contour line of the geoid. 6. The highest point on the geoid along the transcontinental triangulation east of the Mississippi River (17+meters) occurs in latitude 373° and longitude 814° in the Alleghenies, not far from the position in which it should be found if the contour lines of the geoid were controlled entirely by the general features of the topography. 7. After omitting the exceptional region indicated in the following paragraph, the highest point on the geoid east of the Mississippi River is in latitude 35° and longitude 84°, at the southwestern end of the highest portion of the Alleghenies. It is indicated by the contour 204 meters. 8. South of latitude 33° and west of longitude 86° the contour lines on the geoid show a steep slope upward toward the southeast, to which there is nothing corresponding in the topography. This constitutes a most interesting exception to the general rule that the contours of the geoid show a relation to the general features of the topography. No adequate explanation has yet been found for this exception. 9. West of the Mississippi River, in longitude 91° to 93° along the thirty-ninth parallel, the upward slope of the geoid to the southward and the long curves of contours 19, 20, 21, and 22 meters correspond to the fact that there is a large region, comprising one-quarter of the State of Missouri and a part of Arkansas, lying close to the belt of triangulation on the southward: side and having an elevation of more than 1 000 feet, whereas to the northward the elevations are less. 10. From longitude 98° in central Kansas, nearly to longitude 107° in central Colorado, the slope of the geoid is continuously upward to the westward. This statement is also true of the topography. The upward slope of the geoid to the westward gradually increases from a very 64 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. gentle slope near longitude 98° to a maximum in longitude 105°, near Colorado Springs and Pikes Peak, in Colorado, and then decreases gradually to zero (at a summit) in longitude 1064°, northeast of Gunnison, Colorado. The slope of the land surface in central Kansas is upward to the westward, but very small. It increases.steadily as one proceeds westward through western Kansas and eastern Colorado. Between Colorado Springs and Pikes Peak, in a few miles, there is a comparatively sudden rise from elevations of 5 000 to 6 000 feet on the plains to more than 10 000 feet on the mountain tops. Of this group of mountains reaching above the 10 000-foot contour and forming the greatest mountain mass in the United States the center is approximately in the same location as the summit on the geoid indicated by the 394-meter contour. Thus from longitude 98° to longitude 1064° there is a close relation between the geoid and the topography. 11. The valley on the geoid surface in longitude 108° to 112°, indicated by the 34, 35, 36, and 37 meter contours, coincides in position with the valley of the Colorado and Grand rivers, the most important general feature of the topography in this region. 12. The center in latitude 40° to 41° and longitude 110° to 111°, around which contours 40 to 44 meters show a regular curvature, coincides with the summit of the great Uintah group of mountains reaching above the 10 000-foot contour over a large area. The 44-meter contour on the geoid, which indicates the highest part of the geoid within the area covered by this investigation, falls upon one of the greatest of the mountain masses in the United States. 13. The nearly closed oval formed by the 30-meter contour of the geoid, in latitude 39°, longitude 114°, corresponds to the southern portion of the great depression of which the Great Salt Lake occupies the lowest part. The depression on the geoid is less definite than on the land surface and is somewhat to the southwestward of the depression on the land surface. 14. Along the Pacific coast line, from Point Arena in latitude 39° southward to latitude 334°, the geoid contours vary from 20 meters to 24 meters only. There is a general tendency for the contours at the coast to be parallel to the coast. 15. The valley on the geoid in latitude 34° and longitude 118° to 120° corresponds to the fact that there are high islands (San Miguel, Santa Cruz, Santa Catalina, and others) 50 kilo- meters or more from the coast in this region and that there is, therefore, virtually a valley between these islands and the coast line. 16. From longitude 118° to longitude 122° and between latitudes 38° and 40° the geoid has a decided upward slope to the northward of which no counterpart exists in the topography. This constitutes a second important exception to the general rule that the contours of the geoid show a relation to the general features of the topography. Certain local features of the geoid contours have no apparent counterpart in the topographic contours. This is to be expected for three reasons. First, the geoid contours being based on observations at a few scattered points only may be expected to be true to the facts in their general features only, not in their local features. Second, in some cases a local feature of topog- raphy may have had considerable influence in producing local deflection at a given station and this, in turn, may have affected the geoid contours, as constructed, over a considerable area, and the local features of the topography may thus have been exaggerated into a general feature of the geoid. Third, the other acknowledged defects in the method of construction of the geoid contours have probably produced some of the marked irregularities in the geoid con- tours which appear as local features. Among the local features of the geoid contours which are not believed to be of much significance, for the reasons stated, are the following: (a) The break in the 10-meter contour in latitude 40° and longitude 75° and the sharp bend to the westward of the broken ends; (b) The sharp curvature in the 17, 18, ana 19 meter contours in longitude 96° to 98°; (c) The sharp curvature in the 28-meter contour in the vicinity of longitude 117°; (d) The sharp curvature of the 24-meter contour in the immediate vicinity of San Francisco, California, and the presence of the 22}-meter contour in latitude 373°, longitude 122°, breaking the 23-meter contour. These are both believed to be due to an effect of earthquakes in disturb- ing geodetic azimuths which was not recognized until after this computation was made; THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 65 (e) The closed 23-meter contour in latitude 35°, longitude 120°. The average geoid contour at the Atlantic coast line from Maine to Virginia is about 9 meters. The average geoid contour at the Pacific coast line, from latitude 39° at Point Arena, to latitude 324° at San Diego, is about 23 meters, or about 14 meters higher than at the Atlantic coast line. This difference of 14 meters is believed, however, to be of little significance. It may have been due largely to errors in the prime vertical deflections used in constructing the geoid contours and to the acknowledged defects in the method of constructing the geoid con- tours. An average change of +0’’.7 in the prime vertical components of the deflections would cause the 14-meter difference to disappear. Moreover, it is not necessarily true that the geoid must bear the same relation to the spheroid along the two coast lines. A study in detail of the geoid contours as shown on illustration No. 17 shows conclusively that, though the irregularities in the geoid are much too small to correspond to the computed topographic deflections, yet the geoid is not independent of the topography. The general features of the topography which cover large areas are indicated by the geoid contours. On the geoid the greatest elevations correspond approximately in position to the greatest mountain masses. Depressions and valleys in the geoid correspond to the greater depressions and valleys in the land surfaces. The steepest slopes of the geoid tend to correspond in position to the steepest general slopes of the land surface. A contour of the geoid tends to follow each coast line. The smaller features of the topography are not shown in the geoid contours. It is possible that, if a larger number of observed deflections were available and were used in constructing the geoid contours, smaller features of the topography eM show an effect on the geoid contours. The contradictions between the geoid and the ieseeeisis are few in comparison with the agreements. That is, the directions of slopes on the geoid agree generally with the general slopes of the topography. There are only two important contradictions, to both of which especial attention has been called. ISOSTASY MUST BE CONSIDERED. The logical conclusion from the study of the geoid contours for the United States, taken in connection with the fact already noted that the computed topographic deflections are much larger than the observed deflections of the vertical, is that some influence must be in operation which produces an incomplete counterbalancing of the deflections produced by the topography, leaving much smaller deflections in the same direction. There is abundant evidence in the literature of geodesy indicating that this relation of observed deflections of the vertical to the topography is not peculiar to the United States; that, in fact, it exists elsewhere. Any computation, even though quite rough, of the deflections of the vertical which must be produced by the masses constituting the continents considered as excesses of mass, and of the oceans considered as representing defects of mass, shows that said computed deflections are much greater than those which have been observed. Several such computations treating certain continents as approximations to geometric figures have been made.* These computations indicate that deflections of the vertical greater than 30” should be common. The observers do not find them to be so. On the other hand, whenever the directions and magnitudes of observed deflections are carefully studied it becomes evident that, as a rule, the directions of the deflections and their relative magnitudes evidently bear some relation to the topography surrounding the stations. The deflections, as a rule, are in the directions which correspond roughly to those which would be expected on the supposition that they are produced by the topography. Similarly, as to relative magnitudes, in the areas of low relief and slight slope the observed deflections are, in * For example, see Héhere Geodasie, F. R. Helmert, Part II, commencing on page 313, and Bulletin 48, U. 8. Geological Survey, On the Form and Position of Séa Level, R. S. Woodward, pp. 80-85. 78771—09—_5 66 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. general, small, and the areas of high relief and steep slopes and their immediate vicinity are characterized by larger observed deflections. These are the general facts in regard to observed deflections. Many individual exceptions may be cited, but the general statement as made is true to such an extent as to be a strong indication of the direction in which one must look for some general law connecting the observed facts. Deflections of the vertical must be due to irregularities in the distribution of the masses composing the earth. Such irregularities may occur either as a result of irregularities in the surface of the earth (topography) or as a result of irregularities in the distribution of the densi- ties beneath the surface. The deflections can be produced in no other way. The irregularities in the surface of the earth (the topography) are visible on land and are detected by soundings at sea. In either case they are known. In this investigation the deflec- tions which must be produced by these known irregularities have been computed, namely, the topographic deflections. The distribution of the density below the surface of the earth is invisible and unknown. Both the general approximate studies for the whole world of the necessary effects of the known topography in producing deflections of the vertical, and the detailed exact study already made for the United States alone, by means of computed topo- graphic deflections and geoid contours, indicate that one must look to the distribution of the subsurface densities for an explanation of the discrepancies between observed deflections of the vertical and the deflections which must inevitably be produced by the topography. More~- over, from the general considerations set forth in the preceding paragraphs, it seems that there must be some general law of distribution of subsurface densities which fixes a relation between subsurface densities and the surface elevations such as to bring about an incomplete balancing of deflections produced by topography on the one hand against deflections produced by variation in subsurface densities on the other hand. The theory of isostasy postulates precisely such a relation between subsurface densities and surface elevations. These are, briefly, the considerations which led to the determination to investigate thor- oughly the possible relations between the theory of isostasy and deflections of the vertical in connection with the present investigation. ISOSTASY DEFINED. If the earth were composed of homogeneous material, its figure of equilibrium, under the influence of gravity and its own rotation, would be an ellipsoid of revolution.- The earth is composed of heterogeneous material which varies considerably in density. If this heterogeneous material were so arranged that its density at any point depended simply upon the depth of that point below the surface, or, more accurately, if all the material lying at each equipotential surface (rotation considered) was of one density, a state of equilibrium would exist and there would be no tendency toward a rearrangement of masses. If the heterogeneous material composing the earth were not arranged in this manner at the outset, the stresses produced by gravity would tend to bring about such an arrangement; but as the material is not a perfect fluid, as it possesses considerable viscosity, at least near the surface, the rearrangement will be imperfect. In the partial rearrangement some stresses will still remain, different portions of the same horizontal stratum may have somewhat different densities, and the actual surface of the earth will be a slight departure from the ellipsoid of revolution in the sense that above each region of deficient density there will be a bulge or bump on the ellipsoid, and above each region of excessive density there will be a hollow, relatively speaking. The bumps on this supposed earth will be the mountains, the plateaus, the con- tinents; and the hollows will be the oceans. The excess of material represented by that portion of the continent which is above sea level will be compensated for by a defect of density in the underlying material. The continents will be floated,so to speak, because they are composed of relatively light material; and, similarly, the floor of the ocean will, on this supposed earth, be depressed because it is composed of unusually dense material. This particular condition of approximate equilibrium has been given the name isostasy. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U.S. 67 The adjustment of the material toward this condition, which is produced in nature by the stresses due to gravity, may be called the isostatic adjustment. The compensation of the excess of matter at the surface (continents) by the defect of density below, and of surface defect of matter (oceans) by excess of density below, may be called the isostatic compensation. Let the depth within which the isostatic compensation is complete be called the depth of compensation. At and below this depth the condition as to stress of any element of mass is isostatic; that is, any element of mass is subject to equal pressures from all directions as if it were a portion of a perfect fluid. Above this depth, on the other hand, each element of mass is subject in general to different pressures in different directions—to stresses which tend to distort it and to move it. In terms of masses, densities, and volumes, the conditions above the depth of compensation may be expressed as follows: The mass in any prismatic column which has for its base a unit area of the horizontal surface which lies at the depth of compensation, for its edges vertical lines (lines of gravity) and for its upper limit the actual irregular surface of the earth (or the sea surface if the area in question is beneath the ocean) is the same as the mass in any other similar prismatic column having any other unit area of the same surface for its base.* To make the illustration concrete, if the depth of compensation is 114 kilometers below sea level, ‘ any column extending down to this depth below sea evel and having 1 square kilometer for its base has the same mass as any other such column. One such column, located under a moun- tainous region, may be 3 kilometers longer than another located under the seacoast. On the other hand, the solid portion of such a column under one of the deep parts of the ocean may be 5 kilometers shorter than the column. at the coast. Yet, if isostatic compensation is com- plete at the depth 114 kilometers, all three of these columns have the same mass. The water above the suboceanic column is understood to be included in this mass. The masses being equal and the lengths of the columns different, it follows that the mean density of the column beneath the mountainous region is three parts in 114 less than the mean density of the column under the seacoast. So, also, the mean density of the solid portion of the suboceanic column must be greater than the mean density of the seacoast column, the excess being somewhat less than five parts in 114 on account of the sea water being virtually a part of the column. This relation of the masses in various columns, and consequently of the densities, follows from the requirement of the definition of the expression ‘‘depth of compensation”’ that, at that depth, each element of mass is subject to equal pressures from all directions. In order that this may be true the vertical pressures, due to gravity, on the various units of area at that depth must be the same. If this condition of equal pressures, that is of equal superimposed masses, is fully satisfied at a given depth the compensation is said to be complete at that depth. If there is a variation from equality of superimposed masses the differences may be taken as a measure of the degree of incompleteness of the compensation. In the above definitions it has been tacitly assumed that g, the intensity of gravity, is everywhere the same at a given depth. Equal superincumbent masses would produce equal pressures only in case the intensity of gravity is the same in the two cases. The intensity of gravity varies with change of latitude and is subject also to anomalous variations which are to some extent associated with the relation to continents and oceanic areas. But even the extreme variations in the intensities of gravity are small in comparison with the variations in density postulated. The extreme variation of the intensity of gravity at sea level on each side of its mean value is only one part in 400. Even this small range of variation does not occur except between points which are many thousands of kilometers apart. As will be shown later, the postulated variations in mean densities are about one part in 30 on each side of an average value. Hence, it is not advisable to complicate the conception of isostasy and intro- *Jt would be more accurate to use the words ‘‘inverted truncated pyramid” instead of ‘‘prismatic column.’' The latter expression has been selected because it is sufficiently exact for the purpose and corresponds to the allowable approximations actually made in the mathematical part of the investigation. 68 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. duce long circumlocutions into its definition in order to introduce the refinement of considering the variations in the intensity of gravity. The variation of the intensity of gravity with change of depth below the surface need not be considered, as its effect in the various columns of material considered will be substantially the same. The idea implied in this definition of the phrase “depth of compensation,” that the isostatic compensation is complete within some depth much less than the radius of the earth, is not ordinarily expressed in the literature of the subject, but it is an idea which it is difficult to avoid if the subject is studied carefully from any point of view. It is proposed, therefore, in this investigation to assume that the depth of compensation is much less than the radius and to treat it as an unknown to be determined. ’ ‘s In this investigation the attempt is made to ascertain with as much precision as possible the extent to which the condition called isostasy exists, and the manner in which the isostatic compensation is distributed. This attempt is made primarily because it appears that, if suc- cessful, it would lead to considerable increase in the accuracy with which the size and figure of the earth may be derived from geodetic observations. It has been kept in mind that for entirely different reasons it is also important to establish thoroughly, or to disprove, the theory that the condition called isostasy exists. There are many other theories of geology and geophysics which are so intimately related to this theory that they must stand, or fall, or be greatly modified, according to its fate. In passing, it is interesting to note that, though the beginning of the ideas involving the theory of isostasy are found in primitive form at least as long ago as the discussions by Pratt and Airy in connection with deflections of the vertical apparently produced by the Himalayas, the ideas were first presented in such a clear and forceful manner as to attract general atten- tion in an address in 1889 before the Philosophical Society* of Washington by Maj. C. E. Dutton. Since this address was printed the theory has had its ‘present definite name, and its validity, as well as its relation to various geological problems, has been vigorously discussed. COMPUTATIONS OF DEFLECTIONS, ISOSTATIC COMPENSATION CONSIDERED. I In the principal investigation it has been assumed that the isostatic compensation is com- plete and uniformly distributed with respect to depth from the surface down to an unknown depth of compensation which is to be determined from the observations. Let h, be an assumed depth of compensation. Various values for h, are assumed in the investigation and the most probable value derived from the comparison of the computations made on different assumptions. Let 4, be the compensating defect of the density. Then the assumption stated in the preceding paragraph may be expressed mathematically for any par- ticular compartment used in the computation of topographic deflections by the equation. dh=—06,h,. The symbols 6 and h have the same significance as in connection with the com- putation of topographic deflections. 0 is the mean surface density of the earth; that is, the mean density for the first few miles below the surface, and h is the mean elevation above mean sea level of the surface of the earth within the compartment. The area of the compartment times oh is evidently the total mass in the compartment above sea level. The area of the com- partment times 0,h, is the compensating defect of mass assumed to lie below the compartment. In the equation dh = —0,h,, expressing complete and uniformly distributed compensation extending to depth h,, 0 is a constant for all compartments and is assumed to be 2.67. The depth of compensation h, is assumed to be the same for all compartments. Hence, the equation shown above may be written a ao8 i (a constant) = h % On some of the greater problems of physical geology, C. E. Dutton, Bulletin of the Philosophical Society of Washington, Vol. XI, pp. 51-64. THE FIGURE.OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 69 expressing the fact that, under the adopted assumption, the defect of density below a com- partment is directly proportional to the mean elevation of the surface of the compartment above sea level. For a compartment at the sea coast, for which the mean elevation is zero, the compensating defect of density (0,) is also zero. For an oceanic compartment in which h is negative, 0, is also negative. For such a compartment 0, is a compensating excess of density instead of a compensating defect of density. For an oceanic compartment h is not the negative elevation of the bottom; that is, the depth, but is, instead, .615 of the depth. This modification is necessary to take account of the mass of sea water, as explained on page 27. The com- pensating defect of density is, according to the adopted assumption, a anaximum under the high mountains; is zero under the seashore, and is a maximum compensating excess of density under the deepest parts of the ocean. If greater refinement in statement is required, it is necessary to state that the compensation is assumed to extend from the actual surface of the land and from the bottom of the sea down to the depth of compensation, which is assumed to be at the same distance below sea level in each case, It is assumed that below the depth of compensation no excess or defect of density exists or, in other words, that below that depth the density is simply a function of the depth below sea level and has no relation to surface conditions. The assumption that the isostatic compensation is uniformly distributed through a depth which is everywhere the same was adopted in the main investigation from among various reason- able assumptions for two reasons. As far as the writer can determine, this assumption lends itself most readily to computation, that is, gives rise to simple computations which may be most quickly made. Moreover, it seems to the writer to be the most probable one of the simple assumptions. Certain more complicated and less easily stated assumptions may be slightly more probable. A discussion of certain of the many. possible assumptions will be found later in this paper, among various other discussions of subsidiary considerations. The deflection due to the defect or excess of mass beneath the surface which constitutes the isostatic compensation may be computed by the same formula that was used for computing the topographic deflection. It will, however, be necessary in this case to write the formula in its more exact form, as shown on page 34, in which appears the difference of elevation between the station and the surface of the mass considered. The formula, as adapted to this case, is: f 24 h2 D,=12’'.44 7 h, (sin a’ —sin a,) log, a ee D, is one component of the deflection at the station produced by the compensating defect of mass comprised within a stratum h, statute miles thick, lying within a compartment limited as before. The symbols common to this formula and those on pages 20 and 34, 4, a’, a,, r’, and r,, have identical meanings. h, is the depth of compensation. The fact that this depth is large in comparison with many of the values of r’ and r, makes it necessary as above to introduce the radicals involving h,? in the formula.* The derivation for the above formula is precisely the same as for the formula on page 34. Under the adopted assumption as to compensation it has already been shown that dh= —6,h, Making this substitution in the formula for D, it becomes ee r+yr)* the — 19" a f= Bee alle c= —12" 44 5 h (sin a’ —sin a,) log, - ado This is the same as the approximate formula for D given on page 34 except in the last factor (the logarithmic factor) and in having a minus sign. * See Clarke’s Geodesy, p. 295. 70 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. +(e)? +hy —D,. 8 rtyreth?- Hence THe atv ry D lo a g Yr, It is not necessary in this formula to specify that the logarithms refer to the base e, for the ratio of the logarithms is the same whatever the base. Let F be the factor by which D, the topographic deflection, must be multiplied to secure the resultant deflection D+D, due to both the topography and the compensating defect o1 excess of mass below the surface. jon P tNGT Eh? Then F =P +e Sep a i =, r log r Tf this formula be restricted in application to the particular case in hand in which the / : compartments have been so selected that the ratio = is 1.426 (and its logarithm 0.1541) ,* the formula becomes ‘ r+’ +h? Pi M+ 4 r+h/? se 0.1541 The factors F have been computed from this formula for the rings used in computing the topographic deflections and for various assumed depths of compensation (h,). Reduction factors, F, corresponding to various depths of compensation in kilometers. [Depths in kilometers.] Ring. | 329.8 231.3 162.2 120.9 113.7 79.76 | 55.92 | D9, | sadiaiets eicae's Sieve lee caiceeasl te sla so oe allie enna . 997 DBS | Sseevesesis [bin eee.e ea | Sie ceeesidvel clansnagiees Please See 997 - 996 Of je diciemmlenes ae [5 taea se . 997 997 . 996 - 995 OG. \estaeccte|eicne ase . 997 - 996 . 996 + 995 - 992 7 a eee *, 997 . 996 . 995 . 995 . 992 . 988 24 . 997 . 996 . 995 . 993 . 992 - 988 - 983 23 . 996 . 995 . 992 . 989 . 988 - 983 . 976 22 . 995 . 992 - 988 - 984 . 983 . 976 - 965 ZI . 992 . 988 . 983 . 979 - 976 . 965 951 20 . 988 . 983 . 976 . 967 - 965 - 951 . 930 19 . 983 . 976 . 965 . 954 - 951 - 930 - 900 18 . 976 . 965 . 951 . 935 . 930 - 900 . 859 17 . 965 . 951 . 930 . 906 - 900 . 859 - 801 16 . 951 . 930 . 900 . 867 . 859 . 801 21 15 . 930 - 900 . 859 . 813 - 801 721 . 618 14 .900 | .859 . 801 . 736 721 . 618 . 493 13 . 859 . 801 721 . 638 . 618 . 493 . 358 12 . 801 . 721 . 618 .Oo17 - 493 . 358 234 ll 721 . 618 . 493 . 382 . 358 . 234 . 139 10 . 618 493 . 358 . 253 . 234 .. 139 . 077 . 493 . 358 . 234 . 153 . 139 O77 . 040 . 358 234 . 139 . 086 077 . 040 . 020 . 234 . 139 - 077 . 045 . 040 . 020 . 010 . 139 077 . 040 . 022 . 020 . 010 . 005 , . 040 . 020 .O11 - 010 . 005 - 003 . 040 . 020 . 010 . 006 . 005 . 003 - 001 . 020 . 010 - 005 . 003 . 003 . 001 . 001 . 010 . 005 - 003 . 001 . 001 - 001 . 000 - 005 - 003 . 001 . 001 . 001 . 000 - 000 Pw RO RAWDWSO ° a “I * See page 21. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. Ss. 71 The factor, F, is evidently the same for a whole ring, since in the derivation of the formula for F for a given compartment the factor (sin a’—sin a,) has disappeared. The reason for this may be perceived in another way. The mass above sea level in any compartment and the com- pensating defect of mass are equal. The latter has an effect in producing a deflection at the station which is slightly smaller than the former, simply because much of the compensating defect of mass lies far below the horizon of the station. The angles of depression below the horizon of the station to different parts of the compensating defect of mass being the same for all compartments in the same ring, evidently F, depending implicitly as it does upon the depression angles, should be the same. The following simple case serves to illustrate the meaning of the factors: In illustration No. 8 let the area A, shaded by lines running downward to the right, be the vertical section of a mountain standing on a plain of indefinite extent, B, which is practically at sea level. Sup- pose the depth of compensation to be 113.7 kilometers. Then, according to the adopted assump- tion, within the space 113.7 kilometers deep under the mountain, which is represented in vertical section by the area marked C, shown by lines running downward to the left, the density at every level is less than the density at the same level under the plain by a constant amount 6,, such that u No. 8. the whole defect of mass in C, as compared with a corresponding space under B, is exactly equal to the mass of the mountain, A, above sea level. If the mountain is at a great distance, 3 400 kilometers, for example (in ring 1), from the station for which the deflection of the vertical is being computed, the positive mass A and the negative mass C are in nearly the same direction and at nearly the same distance from the stations and, therefore, their effects in producing deflections will be nearly of the same size and, as they are of opposite signs, will nearly counterbalance each other. The effect of C will necessarily be slightly less than that of A. The factor, F, in the last line of the table, in the column headed 113.7 kilometers, expresses the fact that, for this case, the negative deflection produced by C is 0.999 of the positive deflection produced by A, and that, therefore, the resulting deflection is 0.001 of that which would be produced by A alone. : If the station is at a distance of 200 kilometers from the mountain, as at S, (the mountain being, therefore, in ring 9), the angles of depression in various parts of the negative mass C, especially the lower parts, will be sufficiently great to reduce considerably the effects of these parts in producing deflections at the station, whereas all parts of the mass, A, are still practically in the horizon of the station. Moreover, the lower portions of C are at a considerably greater distance from the station than are any parts of A. For both these reasons the negative deflection produced by C is only 0.861 as great as the positive deflection produced by A, and the 72 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. resultant deflections is 0.139 of that produced by A alone. (See factor for ring 9 in the column headed 113.7 in the table.) If the station is at the foot of the mountain at a horizontal distance of 7 kilometers from its summit, as at S,, some portions of the mountain may be at a sufficient angle of elevation from the station to require a slope correction. (See p. 34.) But the mass A will have nearly full effect in producing the deflection of the vertical at the station. On the other hand, much of the negative mass C is at such a large angle of depression from 8, that it is under 8, rather than in its horizon. These portions of C have little effect in producing a deflection of the ver- tical at the station S,. Their principal effect at S, is to decrease the intensity of gravity rather than to change the direction of gravity. For this case, the table of reduction factors shows that if the attention be limited to the compartment of ring 18 (outer radius 9.5 kilometers, inner radius 6.7 kilometers), which includes the summit, the resultant deflection is 0.930 of that produced by A alone, or, in other words, the negative deflection produced by C is only 0.070 of the positive deflection produced by A. For portions of the mountain still nearer to the station the ratio of negative effect of C to the positive effect of A is still smaller and the reduction factor larger. The.table of reduction factors serves to enable one, having made an assumption as to the depth of compensation corresponding to the heading of any one column in the table, to derive from the computed topographic deflections, due directly to the topography, such as those shown in the examples on pages 26, 29, 31, 32, and 33, the resultant computed deflections due to both topography and the assumed isostatic compensation. Each ‘“‘horizontalsum,” in the examples cited, is the deflection for one ring. Each is multiplied by its appropriate factor, F, from the table, and the new sum taken, as illustrated below, for the assumed depth of compensation 113.7: The following example is for azimuth station No. 115, North End Knott Island: - ar Compnted en ___, | Topographic de- i ‘ uniform isostatic com- ming | eee [ee teeny | Peete eengeea 113.7 kilometers. ” Mu 25 . 00 - 995 . 00 24 . 00 - 992 . 00 23 - 00 . 988 00 22 . 00 . 983 . 00 21 . 00 . 976 . 00 20 + .01 - 965 + .01 19 . 00 . 951 . 00 18 — .01 - 930 — .01 17 — .02 . 900 — .02 16 — .08 . 859 — .03 15 — .04 - 801 — .03 14 — .05 721 — .04 13 — .06 . 618 — .03 12 — .07 . 493 — .04 ll —.14 . 398 — .05 10 —1. 24 . 234 — .29 9 —3. 32 . 139 — .46 8 —5. 16 077 — .40 7 —6. 23 . 040 — .25 6 —6. 74 . 020 — .14 5 —6. 62 . 010 — .07 4 —6. 12 . 005 — .03 3 —d. 97 . 003 — .02 2 —7. 02 - 001 — .0o1 1 —5. 48 . 001 — .01 Sum=—1. 92 It may be noticed that in the table of reduction factors F, with the exception of the column headed 120.9, the figures are the same in the various columns. The columns differ from each other simply in having the figures displaced vertically. This arises from the fact that the suc- THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. Ss. 73 cessive assumed depths of compensation, with the one exception stated, are the same as the outer radii of the successive rings. An inspection of the formula shows that the relation stated is true when such a selection has been made. The arbitrary selection of these particular depths, therefore, saved considerable time in computing the factors F, and was allowable in the begin- ning of the investigation when little was known as to the most probable depth of compensa- tion. The selection of the depth, 120.9, to which this time-saving device does not apply, was made later in the investigation after the most probable depth was approximately known. TOPOGRAPHIC DEFLECTIONS WITHOUT AND WITH ISOSTATIC COMPENSATION. The tables on pages 48-56 show in parallel columns, for convenience in comparison, all of the observed deflections, all of the computed topographic deflections, and all of the available values of the computed defleetions with uniform isostatic compensation considered, for various assumed depths of compensation. The assumed depths of compensation, 162.2, 120.9, and 113.7 kilometers, are common to all four of the geographical groups, designated as the northeastern group, southeastern group, central group, and western group. These are the only assumed depths for the western group. For the other three groups certain other assumed depths also appear, these depths having been used in preliminary stages of the investigation. The following nine cases have been selected from the tables on pages 48-56. They are extreme or unusual cases. Taken together they indicate the general tendencies which are exhibited in the complete tables. o Computed deflection, uniform isostatic compensation considered. T 7 ae Name of station. ae : graphic us Depth of compensation (kilometers). 329.8 231.3 162.2 120.9 113.7 79.8 aA Mt Ms ae Mt tt Mt Mt 238 Mer. Santa Barbara, California] —18.38 | — 64. 97 —14.91/ —12. 78} —12. 35 1P.V. | Point Arena, California +16.98 | +104. 63 +20. 39} +16. 45 | +15. 69 115 P. V. | North End Knott Island, | — 6.62 | — 54.30 | — 7.96 — 3.23) — 2.09] — 1.92] —1.09 Virginia 43 P.V Waddoup, Utah +24.84 | + 54.71 +22.11/ +20. 38 | +20. 00 49 Mer. Patmos Head, Utah —13.52 | — 27.20 — 9.42) — 8.66) — 8.53 178 P. V._ | Cheever, New York —14.77 | — 37.46 | —10. 32 — 9.20| — 8.80) — 8.73 216 Mer. Porcupine, Michigan + 2.44 | — 0.53) + 3.98) 4+ 3.70} + 3.30) + 2.93] + 2.85 169 Mer. Howlett, New York + 192 | — 12.96 /+ 3.86 + 3.62) + 3.40) + 3.33 205 P. V. | Gargantua, Canada + 3.47 | — 11.94 | + 3.02) +2.78 | 4+ 2.46) + 2.17} 4 2.11 209 Mer. Chicago L. H., Illinois + 1.96 | — 10.01 | — 0.05] +0.03 | + 0.07) + 0.11] +4 0.10 The computed deflections, with isostatic compensation considered, are, as a rule, much smaller than, and of the same sign as, the topographic deflections. There are some exceptions, such as those shown in the last four lines of the above table, but they are not numerous. The computed deflections, with isostatic compensation considered, ordinarily decrease numerically as the assumed depth of compensation decreases. There are rare exceptions like the one shown in the last line of the above table. The computed deflection, with isostatic compensation considered, ordinarily agrees much more closely with the observed deflection than does the topographic deflection. \ THE AREA METHOD. An unusual method has been followed in this investigation in forming the observation equations connecting the observed deflections of the vertical, on the one hand, with the constants expressing the figure and size of the earth, on the other hand. As the method is unusual, it has seemed best to show it in considerable detail. The method here used is called the area method to contrast it with the usual method, which may be called’the arc method. 74 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. In the area method no attention whatever is paid to the question whether the various astronomic stations are placed approximately along arcs. The only condition required, other than the necessary degree cf accuracy in the observations, is that all the astronomic stations shall be connected with continuous triangulation all computed on one basis, that is, on one assumption as to the equatorial and polar dimensions of the reference spheroid and as to the starting latitude, longitude, and azimuth at some one point. Astronomic latitudes, longitudes, and azimuths are all used in one set of equations. The are method of deducing the figure of the earth may be illustrated by supposing that a skilled model maker is given several stiff wires, each representing a geodetic arc, either of a parallel or a meridian, each bent to the radius deduced from the astronomic observations on that arc, and is told in what latitude each is located on the geoid, and then requested to construct the ellipsoid of revolution which will conform most closely to the bent wires. Similarly, the area method is illustrated by supposing that the model maker is given a piece of sheet metal cut to the outline of the continuous triangulation which is supplied with the necessary astronomic observations, and accurately molded to fit the curvatures of the geoid as shown by the astro- nomic observations, and that he is then requested to construct the ellipsoid of revolution which will conform most accurately to the bent sheet. Such a bent sheet essentially includes within itself the bent wires referred to in the first illustration, and, moreover, the wires are now held rigidly in their proper relative positions. The sheet is much more, however, than this rigid system of bent lines, for each are usually treated as a line is really a belt of considerable width, which is now utilized fully. It is obvious that the model maker would succeed much better in constructing accurately the required ellipsoid of revolution from the one bent sheet than from the several bent wires. In the area method as used, observation equations of the following form were written: For each observation of the astronomic latitude: kf) +h(2) +m,(«) +n,( 755, )+0:(10 000 e*) +(a— $) =Du For each observation of the astronomic longitude: k,(¢) +1,(2) +m,(@) +n( 55) Fou10 000 e) +cosd’(As—”) =Dp For each observation of the astronomic azimuth: kG) +h(2) +m,la) +n,( ;F5 )-+04(10 000 e*) — eotg’(aa~ a”) =Dy In these equations the meanings of the symbols are as follows: The quantities ¢,, As, and @, are observed astronomic values of the latitude and longitude and azimuth, respectively, at the astronomic stations. The quantities ¢’, 1’, and a’ are the values of the geodetic latitude, longitude, and azimuth at the astronomic stations as computed on the United States Standard Datum and the Clarke spheroid of 1866. The statement that the geodetic latitudes, longitudes, and azimuths are upon the United States Standard Datum means that the computation was carried continuously through all the triangulation upon the assumption that the latitude of the triangulation station at Meades Ranch, Kansas, is 39° 13’ 26’.686; its longitude, 98° 32’ 30’’.506, and the azimuth of the line Meades Ranch to Waldo, 75° 28’ 14’’.52. és—¢’, the absolute term in each latitude observation equation is, therefore, the apparent meridian component of the deflection of the vertical at a latitude station. It is the quantity called A-G, shown in the last column of the tables on pages 12-15, and called the observed deflection in the tables on pages 48-56. 1,—2 is the difference between the astronomic longitude and the geodetic longitude. Cos ¢’ (As—1) is this difference reduced from the parallel of latitude to the prime vertical great circle, and is, therefore, the apparent prime vertical component of the deflection of the vertical at a longitude station. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 75 Similarly — cot ¢’ (@,—4’) is the apparent prime vertical component of the deflection of the vertical at an azimuth station as derived from the azimuth observation. Cos $’(4,—2’) and —cot¢’(@,—@”’) are the quantities shown in the last column of the tables on pages 17-19 and also shown in the tables on pages 48-56 under the heading “Observed deflections.” For the explanation of the factors called cos ¢’ and —cot ¢’ in the preceding three para- graphs, see page 16. The three quantities, (¢), (4), and (@) are the required most probable corrections (to be derived from these observation equations) to the initial latitude, longitude, and azimuth (4, A, @), respectively, at the initial station, Meades Ranch, Kansas. Similarly, Ga is the one- hundredth part of the required most probable correction to the Clarke 1866 value of the equa- torial radius and (10 000 e?) is 10 000 times the required most probable correction to the Clarke 1866 value of the square of the eccentricity, e?. * k, is a numerical coefficient, computed by the formula shown later, such that if the initial latitude (at Meades Ranch) were corrected by the amount (¢) the change produced in ¢,— ¢’ would be k,(¢). Or, in other words k, is a numerical coefficient such that if, instead of starting the computation with the initial latitude ¢, it had been started with the initial latitude 6+(¢), the computed value of the latitude at the station considered (at which an astronomic latitude has been observed) would be ¢’ —k,(¢) instead of ¢’. Similarly, k, is a numerical coefficient such that if the initial latitude were corrected by (¢) the change produced in cos $’( A, — 2”) would be k,(¢). So, too, k, expresses the relation between (¢) and —cot ¢’(a,— 4’). The coefficients 1,, 1,, and 1, express corresponding relations between (A), the correction to the initial longitude, and (¢4—¢’), cos¢’(A,— 1’), and —cot¢’(a,—a’) in the observation equations referring, respectively, to latitudes, longitudes, and azimuths. So, too, the coefficients m,, M,, M,, Ny, N,, Nz, 0,, 02, 0, express similar relations between (a), Goa) and (10 000 e?) and the quantities (¢,—¢’), cos¢’(As—4’), and —cot¢’(A,— 1’), forming the absolute terms of the observation equations. The quantities Dy, representing the residuals of the latitude observation equations, are the final unexplained meridian components cf the deflections of the vertical. The quantities Dp, representing the residuals of the longitude and azimuth equations, are the final unexplained prime vertical components of the deflections of the vertical. The least square solution of the problem consists in finding such values for the required quantities (), (4), (4), ( aD and (10 000 e?) as will make SD#4+ 5D a minimum—that is, the solution makes the sum of the squares of the unexplained deflections of the vertical a mini- mum. The quantities (¢,—¢’), cos¢’(As—2’), and —cot (a,—a’), the observed apparent , components of the deflections of the vertical, as given in the second column of the tables on pages 48-56, arise from four principal sources, namely: (1) From the errors in the initial latitude, longitude, and azimuth (at Meades Ranch) used in computing the geodetic positions and from the errors in the assumed elements (a and e?) of the Clarke spheroid of 1866 on which the geodetic positions were computed. (2) From the errors in the astronomic observations, excluding effects of deflections of the vertical. (3) From the errors in the triangulation—that is, in the lengths and angles fixed by the tri- angulation. (4) From the deflections of the vertical. *In each of these cases a parenthesis is used to indicate the correction to the quantity contained within the parenthesis. 76 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. It is proposed to reduce the effects of errors from source (1) to a minimum by deriving from the computation which is to be made the best possible values for the corrections to the initial data and to the elements of the spheroid. A careful examination for the actual case in hand, involving the astronomic observations and triangulation in the United States used in this investigation, shows that the effects of the deflections of the vertical (4) upon the quantities (d,—¢’), cos ¢’(A,— 4’), and — cot f’(a@,—@”) greatly exceed the effects of the errors in the astronomic observations (2) and of errors in the triangulation (3). Hence, it is proper to proceed with a least square solution on the basis stated above—that is, to make YDj,+.2D2 a minimum. The discussion of the evidence that the effects of errors in the astronomic observations and in the triangulation are small in comparison with the errors due to unexplained deflections of the vertical, will be given later in connection with the general discussions of the accuracy of the various steps of this investigation. In the observation equations as written, and in the above statements, it is assumed that there are but five unknowns to be determined. As, however, it is assumed in this investigation that isostatic compensation exists, extending to a depth to be determined, said depth of com-. pensation is a sixth unknown. It is possible to introduce it as a sixth unknown in the observa- tion equations, but it was believed that another method of procedure was advisable, and it has been followed. A separate solution has been made of the observation equations for each of several assumed depths of compensation. In accordance with the general principle that the most probable values of the unknowns are those from the solution which makes the sum of the squares of the residuals a minimum, that one of the solutions for which said sum is least has beon adopted, in this investigation, as the most probable solution. Each set of observation equations differs from the others only in the absolute terms. The differences will be shown clearly later. FORMULA FOR COEFFICIENTS. For convenience of reference, the formule from which the coefficients k,, k,,k,,1,, ...... 0,, 0,, and o,, were computed are given here, together with the meanings of the symbols which enter these formule. Their derivation will be shown later. =: 8, sino i ed FRO +Q)5 sin?4( ay — ay) sind ad k, = —sin ¢’ sinw |, =—cos ¢’ ieee __ cot f” sin’w 1, =zero 3 sin ay sin 8 . 3 s sin a,(1 + cosw) = 100 Ng m,=p(l +Q)osin'h (am ay aA n, asin i” BR? oS” _cos f’ cos 4 sin w f ____ 100 : Ne ane _ _ cot f’sin a cosw __ 100 . ata sina; q asin 1” pes N sin’ ¢ RR 1 2 hs ; 1 =B 90000sin1” (i —e*ain*g) 9008 — Nag o00sinI(1—e)*? p)[1+2sin’d +3 cos( $+ $’)] = sin’ d = (1—e? sin? d)* cos’? ¢ Bos Wiie ts Ose = 59900 sin I" te sing) 07+ GO 000 a sin 171 et) 8% CON Ar SING Se sin’ d . _(1—e? sin’ ¢)3 cos’ } ai Raa csninw °s= — 55-000 sin 11 ~e? in’)? ** * G0 000 a sin 11 —e)* cee THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 77 The symbols in these formule have the following meanings: ¢, 4, « and ¢’, 2’, a have already been defined as referring, respectively, to the adopted latitude, ‘leamede: ind azimuth at the initial station (Meades Ranch, in this case), and the computed geodetic latitude, longitude, and azimuth at a station at which astronomic observa- tions have been made. a is the equatorial radius of the earth, and e is the eccentricity of the ellipse which is the earth’s cross section through the axis of rotation. The adopted values of a and e, used in this connection, are necessarily the Clarke values of 1866, as the geodetic latitudes, longitudes, and azimuths were computed on that spheroid. s is the distance in linear units measured along the surface of the Clarke spheroid of 1866 between the point (¢, 4) and the point (¢’, 4’), that is, between the initial station Meades Ranch and the astronomic station under consideration. w=X —), that is, the difference of longitude of the astronomic station under consideration and Meades Ranch. ay is the azimuth from the point (¢, 4) Meades Ranch, to the point (¢’, 1’), the astronomic station under consideration, and a, is the back azimuth between these same points. R is the radius of curvature of the meridian in latitude 4(¢+ ¢’). ie Be Soe 5 (1—e? sin’ ¢)? e @ 2 2 6? = 75 Con a eat) DERIVATION OF FORMUL FOR COEFFICIENTS. Let (¢), (A), (@), (a), and (e?), respectively, be corrections to the latitude, longitude, and azimuth, ¢, 4, and a, at Meades Ranch (the initial point) on the United States Standard Datum, to a (the equatorial radius of the earth), and to e? (the square of the eccentricity) as fixed by the Clarke spheroid of 1866. Let ¢’, %’, a be the geodetic latitude, longitude, and azimuth at any point occupied as an astronomic station, as computed on the United States Standard Datum and the Clarke spheroid of 1866. It is required, as a preliminary to the derivation of the formule for the coefficients of the observation equations, to derive expressions for ¢’’, 1’’, a” the latitude, longitude, and azimuth at any point occupied as an astronomic station, after the above corrections (¢), (A), (@), (a), and (e?) have been applied. The lengths and the angles at each triangulation station, as fixed by triangulation, are assumed to remain unchanged. The required expressions may be written in symbolic form as follows: fp’ =f’ +, ¢) +2,(A) +h,(@) +1,(a) +j,(e?) (1) =x +E) +94 +h a) +1,(a) +j,(€7) (2) a’ =a’ +£,( 6) +9,( A) +h, @) +1,(a) +js(e*) (3) in which the f’s, g’s, h’s, i’s, and j’s are coefficients for which expressions are yet to be derived. Let it be assumed that the equations numbered (36), on page 249 in the “Account of Prin- cipal Triangulation,’ Capt. A. R. Clarke, London, 1858,* correctly express the relations between ¢, 4, «(referring to the initial station) and $’; ¥, a’,a,ande*. In these equations, as written by Clarke, the wand a’ stand for the azimuths of the line j joining the two points, always counted * The title page of this volume reads as follows: ‘‘Ordnance Trigonometrical Survey of Great Britain and Ireland. Account of the observations and calculations, of the principal triangulation; and of the figure, dimensions, and mean specific gravity of the earth as derived therefrom. Published by order of the Master-General and Board of Ordnance. Drawn up by Capt. Alexander Ross Clarke . . . under the direction of Lieut.-Col. H. James . . . Superintendent of the Ordnance Survey . . . London, Printed by G. E. Eyre and W. Spottiswoode, 1858.”’ 78 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S8. in the nearest way from the north, sometimes clockwise and sometimes counterclockwise. In rewriting the equations for our use and putting them in the notaticn ordinarily used in the Coast and Geodetic Survey, it will be assumed that all azimuths are to be counted from the south around by west, clockwise, in the usual manner, and the necessary changes in signs of terms involving « and a’ will be made. To avoid conflicts in notation the symbol ay will be substituted for a, and ay for a’. The equations as thus rewritten in our usual notation are: 3 cos 4 (90° — 0) (4) cot }(a,+o04+) = “pa | bout a 6) tans ‘sin 4 (90°— f— 6) sin 4 (90°—¢ +0) ,_ y_ ssind(a@gt ar tO) (4 p-— b= Resin }(@,—a,+0) ato) (1 +Q) (6) In these equations w=’— A, the difference of longitude between the astronomic station and the initial station. , s=the distance in linear units (meters) between ths point (¢, 4) and the point (¢’, 4’), that is, between the initial station and the astronomic station. ay is the azimuth from the point (¢, 4) to the point (¢’, i’) and ay is the back azimuth. cot $(@—w+l) = — tan : (5) * / R is the radius of curvature of the meridian in latitude 949 The general expression for R is = a(1 —e?) : 7 hon (eon! (7) 6, ¢, and Q have the following values with sufficient accuracy for our purpose.* 0-5 | 1+ 6 (te) cos’ ¢ cos? a | (8) in which N is the length cf the normal at latitude ¢ limited by the minor axis, or axis of revolution. ef? a Nee eae (9) f=— aa=e 2) Cos’ p sin2 ay (10) Q= Zoosh (ay + ap) (11) The geometrical meaning cf ¢ and 0 may be made clear by the following figure, which will also be useful in later work in deriving required formule. The figure here shown is a spherical (not spheroidal) triangle upon a sphere of which the radius is the normal C H (=) at tho point (¢, 4) of the spheroid, as limited by the mincr axis of the spheroid at H. The point H is the center of the spherc. The sphere is, thercfore, tangent to the spheroid at (¢, 2), but they have few other common points on their surfaces. The bold faced letters A, B, C, a, b, €, are used to designats the parts cf the spherical triangle in the manner which is customary in text-books on spherical trigonometry. The arc BOC or @ joining C or (¢, 2) with the pole of the sphere is made equal to x=90°-¢. The angle C of the spherical triangle is made equal to 180° — ap. The angle B at the pole is made equal to w = (1’— 2). These statements fix the position and size of the sphere and fix all parts of the spherical triangle, since three parts of the spherical triangle have been fixed. The angle A is evidently nearly equal to a;—180°, or the angle indicated on the figure is nearly equal to a. This latter angle has been therefore placed equal to a3+¢, in which ¢ is a very small angle of which the value has been derived by Clarke as expressed in (10), *See ‘‘ Account of Principal Triangulation,’’ etc., p. 249. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 79 and of which he makes the statement that its value does not ‘(amount to a tenth of a second even in a distance of a hundred miles.”’ # is an are on the sphere. Its radius of curvature is necessarily N, that being the radius of the sphere. If its length (linear) were equal to s, the linear distance between the points (¢, A) and (¢’, 4’) as fixed by triangulation, we would have @= = ing the point (¢’, 4’) on the sphere was fixed as being the intersection of the two arcs ¢ and b (or @), which were started in fixed directions from B and C, whereas the true point (¢’, 4’) lies on the spheroid at a certain distance, s, measured on the surface of the spheroid, from (¢, 4). Hence the length (linear) of # is not equal to s, and @ differs slightly from a as shown in (8), which has been derived by Clarke. But the point A represent- ax =I a The arc € (or p) is approximately (not exactly) cqual to 90°— ¢’. The preceding formule have all been proved by Clarke (See ‘Account of Principal Trian- gulation,” etc., p. 249, and preceding), and will here be accepted without question. The derivation cf the formule which follow is, in the main, that given by Clarke cn pages 616-620 of the ‘“‘Account,” somewhat amplified. The latter part of the work of using tho formule, or rather the coefficients expressed by them, in making a least square adjustment, differs radically from Clarke. The relations between ¢, A, 4, $’, 4’, % being those shown in (4), (5), and (6), and on the figure above, it is required to derive expressions for the coefficients in (1), (2), and (3). Let the correction (4) be supposed applied to the initial longitude, 4, while w., 4, and s remain unchanged. In the figure above, it is evident that the spherical triangle will simply be rotated about B, the pole, as a center without changing its dimensions. c, A, B, and 0 80 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. preserve the same values as before, and evidently preserve the relations to the quantities ¢’, a, w, and @ which they had before. ¢’, a, w and @ remain unchanged. The change in 2’ must be equal to (A). Looking now to the formule (4), (5), and (6) for corroboration we find that Adoes not appearinany of them. The only longitude function which appears is w =(2’— 2). Hence a change (4) in 4 produces an equal change in 1’ and affects nothing else. Hence in (1), (2), and (3), g,= 0.000 (12) 2. = +1.000 (13) g,= 0.000 (14) In obtaining the expressions for the coefficients f,, f., f,, h,, h,, and h, it is important to note that a small change(¢) in ¢, or (a) in ap will cause a much smaller change in 0, see (8), which may be neglected. Hence in the figure on page 79, if d be corrected by (¢) while 4, a, and s remain constant, C and 6 remain constant while a, B,¢ and A vary. Similarly, if a» be corrected by (@,) while ¢, A and s remain constant, @ and 6 remain constant while 4, B, C and e vary. Turning to ‘the fundamental equations numbered (4) and (5) it is evident that if ap varies both ay and w must vary. 0, as already indicated, remains practically constant. Similarly, ¢ remains practically constant and zero, see (10). By differentiation of (4) there is obtained day , dw _sin*}(@p+w)cos}(90°— $8) (15) da, 1 Te. da; a E ¥ cos’ "cos i( 90°—¢+ 8) The general relation for any spherical triangle i sin(.4 + _B)cos’4¢ =cos#(a— b)cos}(a+ b)sinC (16) which can be derived from the equation on line 25 of page 161 of Chauvenet’s Trigonometry, 9th edition (using the general formula sin26 =- 2sin@cos@) , and the general relation for any spherical triangle cos}( 4+ B) =SKat Ping (17) osse become, for the spherical triangle shown on page 79, neglecting the very small angle ¢, —sin( a +w) cos? =cos}(90° — — #)cosh(90° — + #)sinay (18) and see cos3(90°— $+ 4) f) sit (19) cos?’ 2 Using (18) and (19) to simplify (15) there is obtained K da, dw be sin( A+) day tday sina, (20) By differentiation of (5) in accordance with the statements above, there is obtained daz _ dw _ sin?’ $( ay —w)sins(90° — d— 6) o 21 day day cos sink(90°—¢+ 4) (21) The general relation for any spherical triangle sin( A — B)sin® $ =sin 3(a—b)sin a+b) sinC (22) THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 81 which can be obtained in the same manner as (16), and the general relation for any spherical triangle cos $(.4— B) = sin HO+D) sin 4 (23) sin — 2 2 become for the spherical triangle shown on page 79, neglecting the very small angle ¢, —sin (a —w) sin? B= sin 4(90° — 4 — 6) sin $(90°— $+ 6) sin a (24) and sin $( a — w) = 810 190° $ FD cose (25) sin E Using (24) and (25) to simplify (21), there is obtained da, _ dw __ sin(@,—w) da, da, sina, ‘ an) . Adding equations (20) and (26), member by member, there is obtained, after simplification, ae =— a =h, [which is a required coefficient in (3)] (27) F F . Similarly, by subtracting (26) from (20) member by member, and simplifying, there is obtained oe =— oc =h, [which is a required coefficient in (2)] (28) F F Turning now to (4) and (5), if ¢ varies while a, remains constant, «; and w must vary. 6 will remain practically constant, see (8). In the following derivation of the values of das d¢ and oe ¢ will be assumed to be zero. Differentiating (4) under the above stated conditions, there is obtained . a : ap da, du sin? 4(@,+w) sin # tan 2 (29) ‘dé tdé cos *4(90°— b+ 0) After simplifying (29) by the use of (19) and of the general relation sin20 =2sin@cos@, it becomes day dw _ _ sin @sin ay d¢ .d¢ 2 cos* (30) Differentiating (5) under the conditions stated above, there is obtained * . a da, dw _ sin?}( a, — w) sind tan=" (31) dé dé — sin’}(90°— $ +8) After simplifying (31) by the use of (25) and the general relation sin2@ =2sin@cos#, it is da, dw __ sin@sina, (32) dg dg 2sin”t 78771—_09-_6 82 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN JU, S. The general relation for any spherical triangle snB _sinC (33) sind sine becomes for the spherical triangle shown on page 79 sinw _ sina, sind sinp (34) Adding (30) and (32), member by member, and simplifying by means of the general relation sin20 =2sin@cos@ and (34), there is obtained a7 = —=-———, =f; [which is a required coefficient in (3)] (35) Subtracting (32) from (30), member by member, and simplifying by means of (34) and the general relation cos26 =cos?@ —sin?0, there is obtained dw dg =cot p sinw (36) Assuming that with sufficient accuracy for the purpose in hand g¢’ =90° —p, (36) becomes aga tang’ sinw =f, [which is a required coefficient in (2)] (37) In (6) it is evident that if ap be corrected by the amount (as) while ¢ and s are held fixed, a, and ¢’ will be changed. In deriving an expression for se it will be assumed that with F sufficient accuracy (1+Q) may be considered constant and the small angle ¢ neglected. By differentiation of (6) and simplification there is obtained s sings — ane dg’__8( SG) = 2 Oey (38) day R 2sin?4( ay — ay) Substituting in this the value of from (27) there is obtained F dg’ _ E (1 + Qe cose =h, [which is one of the required coefficients in (1)] (39) Similarly, if ¢ be corrected by (¢) while ay and s are held fixed, ag and ¢’ will both be changed. Making the same assumptions as were made when: deriving se, there is obtained by F differentiation of (6) i day e dg’ _y +2(1+0) abet d¢ » (40) d¢ R 2sin*}( ap — ay) Substituting in this the value of ie from (35), there is obtained d¢f’_,_s sin?w a Dixs : ee dé a RU +Q) Ps} (aa ane =f, [which is a required coefficient in (1)] (41) To determine the coefficients i,,1,, i,, j,, jz, and j,, which serve to express the effects upon ¢’, 2’, and a, of changes (a) (e*) in a and e’, the constants which fix the size and shape of the spheroid, it is important to note, first, that in the fundamental equations (4), (5), and (6), neglect- THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 83 ing the very small quantity ¢, @ is the only quantity directly involving a and e? which enters the equations explicitly. From (8) and (9), and neglecting the term in (8) containing 6”, there is obtained 6="(1 —e’sin’d)’ (42) Differentiating (42) with respect to @ and a, e? being supposed constant, dé 0 a: da7 (43) Differentiating (42) with respect to @ and e’, a being supposed constant, dd 6 sintd dfe| = ~ 3 (1—etsin®4) une? In (4) and (5), when @ varies @, and w also vary. Neglecting ¢ and differentiating (4), there is obtained ap dw _ sin?4(a@,+w)tan 3 eee (45) dé cos?4(90° — 6+ A) da, a Combining (19) with this, and simplifying, there is obtained da, , dw _sinay cos b dé = dé 2 cos’? (46) Similarly, neglecting ¢ and differentiating (5), there is obtained : ap dds de sin?4( a, — w) tang cos d (47) do ~dd~ ~~ sin’'4(90°— + 0) Combining (25) with this, and simplifying, there is obtained day _ dw Be _ sina@y cos ¢ dé dé 5 aint? (48) Adding (46) and (48), member by member, there is obtained “dap. cot p dg ~~ Sin cos p sin p (49) In the spherical triangle shown on page 79 the law of proportional sines gives sin @ 5 sin @, sinp cos¢ (50) Substituting the value of sin p from (50) in (49), and assuming that with sufficient accuracy cot p=tan ¢’ (that is, that p= 90°— ¢’), there is obtained is tan @! sina, (51) Subtracting (48) from (46), member by member, and substituting the value of sin p from (50), and assuming p=90° — ¢’, there is obtained dw sin a, dé cos ¢’ (52) 84 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. §S. By combining (51) with (43), there is obtained is __o ae tan ¢’ sin a =1, [which is a required coefficient in (3)] (53) Similarly, from (52) and (43) there is obtained ea at See [which is a required coefficient in (2)] (54) Substituting in (8) the value of N from (9), differentiating with respect to e”, and neglecting terms of higher order, there is obtained dé 0 __sin’¢ __—, s#*(1 —e*sin?¢)*cos’¢ cos’, dfe?] 21 —esin’d + 6a(1 —e?)? (55) which is a value for ae somewhat more accurate than that shown in (44), in deriving which the term in (8) involving 6 was neglected. Combining i and (52) there is obtained sind, 0 sin’d s6*(1 —e’sin’¢)! cos’ cos’a | = qa ~ cosd’ | -5 2 1—e’sin’h = 6a(1 —e?)? ~ da (56) [which is a required coefficient in (2)]. Similarly, combining (55) and (51), there is obtained da, sin’? s67(1—e’sin’?d)*cos’¢ coer | Cx dle?) = ang sinas| 21- e'sin’d + 6a(1 —e?)? “Is en {which is a required coefficient in (3)]. (¢’—¢) R is (with sufficient accuracy) the difference of latitude expressed in meters. [See meaning of R as shown in (7).] In the spherical triangle shown on page 79, let it be assumed that ¢’=90°—p. Then evi- dently (90°—p—®)N is also an expression for the difference of latitude expressed in meters. Hence, ($’— ¢)R =(90°—p—¢)N or N ($’— $) =p (90°—p— 4) (58) N lip v accordingly, there is obtained aries while ¢ remains constant, ¢’ and p will vary in (58). Differentiating (58) dg’ = Rap +(90°-p-aa| F | coy) For the spherical triangle shown on page 79, the following general formule for any spherical triangle ; ; cos¢ =cosa@ cosb+sind@ sin’ cosC sinc cosA =sinb cosa — cosb sina cosC become, respectively, ; ; cos p =sin¢ cosé — cos¢ sin# cos&r (60) —sin p cosa, =sind sing + cosé cos¢ cosa, (61) Differentiating (60), ¢ and ay being constant, there is obtained ~sinpdp = —sin¢ sind dé —cos¢ cosascos6 dé (62) THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 5. 85 By multiplying (61) by dé, member by member, and comparing with (62), there is obtained —sinpdp = sinpcosa@,dé, whence dp = —cosa@,d4 (63) N_[1l—e’sin®4(¢+ ¢’)]}? Ri —e)(1—e’sin’¢)! (64) Differentiating (64) and neglecting terms of higher orders, From (7) and (9) N d ‘ ‘Lay = ill +4 sin’ —} sin’'h(p+¢’)] (65) or in slightly more convenient form for computing N af R 1 «5 : , arti $+3 cos( d+ ¢’)] (66) Substituting the values of dp and Le from (63) and (66) in (59), there is obtained d¢’ rN cosa, dd aa Ee nD PO; +2sin?¢ +3 cos( 4+ 6’) ]d[e?] (67) R a e?)? ; By using (43) and (44) this may be written N Osin? wdfe?] 90°—p— dg’ = Fo cosaada— gy mareieig et aC ape ll +2sintp + Bcos( $+ 9) dle" Whence, the required coefficients i, and j, in (1) are = E ecans (68) N¢@ sin’dcosa, 90°-p—¢ i= R2 1—e’sin’?d + 4(1 —e?)? [1 +2sin’¢ +3cos( b+ ¢’)] (69) (69) may be put in a slightly more convenient form by substituting for 90°—p— p its value from (58). j, then becomes N @sin’¢cosa, _R ¢’-¢ Le“R es ian Nasa oe eee (70) This completes the derivation of the coefficients required in (1), (2), and (3). The refer- ences to the equations showing their values are as follows: f,, (41); g1, (12); hy, (39); i,, (68); j,, (70). f,, (37); 2, (13); hy, (28); i,, (54) ; ja, (56). f,, (35) ; $3) (14); h,, (27); os (53) ; Tes (57). 86 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. The coefficients, collected for convenience of reference, are fei SSG Qs : R - Qs sin?4( a, — &) sind f,=tan a sin w ____sin’w . sing, sind g.= 0.000 g.= +1.000 g,= 0.000 8 sina,;(1 + cosw) 27 fete Bhs COSU): (1+Q)) in ae h, = — 0084 sinw sina, h, — — Sin@s cosw ' sind, ee : NO =F 5 Coste ; a? 6 sina, a cos’ i,= -* tand’ sina, - __N@sin’dcosaz ,R ¢$’—¢ r lite 5 1 ean —esint’s *N Ae) — eel +2sin?6+3cos(¢+ ¢’)] _ sind, | -3 O_sin’¢ i s6*(1—e’sin?d)*cos*¢ cos’ a, ~ cos¢’ 1—e’sin’d 6a( 1 —e?)? _# sin’ s#? (1 — e’sin?)icos’¢ cos’a =t F js— tang’ sine, “2 = e’sin’ dh - 6a (1 —e?)? | According to pages 74, 75, k, is a numerical coefficient such that if the initial latitude were corrected by the amount () the change produced in ¢,— ¢’ would be k, (4). Or, in other words, k, is a numerical coefficient such that if, instead of starting the computation with the initial latitude, ¢, it had been started with the initial latitude $+(4), the computed value of the latitude at the station considered (at which an astronomic latitude has been observed) would be ¢’—k,(¢) instead of ¢’. According to equation (1), page 77, f, is a numerical coefficient such that if, instead of starting the computation with the initial latitude ¢, it had been started with the initial latitude o+(¢), the computed value of the latitude at the station being considered would be ¢’ +f, (¢), instead of ¢’. Hence, f, is the negative of k,, or k~-f 72 Similarly, ; ; “ =e 73 and : : (73) m,= —h, (74) To obtain n, from i,, the sign must be changed for the same reason as for k,, l,, and m,. Also, i, must be multiplied by 100 because n, is a coefficient for (95, (see page 74), whereas i, is a coefficient for (a). Moreover, as the absolute terms of the observation equations (p. 74) are all to be expressed in seconds of arc, as being most convenient for the computer, all other terms must be expressed in the same units. Henee, as @ is in radians * in the expression for i, i, must also be divided by sin 1”’. *According to the way in which formule (71) have been derived, involving differentiation, @ and ¢’—¢, wherever they occur, are necessarily in radians. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 87 Hence n= — so yi (75) Similarly, ; Ji = — 10 000 sin 1” (76) In obtaining k,, |,, m,, n,, and o, from f,, g,, h,, i,, and j, the same considerations apply as those noted in connection with k,, 1,, m,, n,, and 0,, but, in addition, it must be noted that in the observation equations for astronomic longitudes (shown on page 74) —J’cos¢’ enters, instead of simply — 1’. Hence, each of the coefficients f,, g,, h,, i,,j,, must be multiplied by cos ¢’. This factor, cos ¢’, arises from the fact that to derive the deflection of the vertical expressed in terms of the prime vertical great circle, one must multiply the difference between the astro- nomic and geodetic longitude by cos ¢’ (see page 74). For a similar reason, the factor —cot ¢’ enters in obtaining k,,1,, m,, n,, and 0, from f,, g4, h,, i,, and j,. The relations are, therefore: k, = —f, cos ¢’ (77) 1,= —g, cos ¢’ (78) m, = —h, cos ¢’ (79) 100 . Dy = — Sy 7 200s ’ (80) _ ja cos ¥ . %=—79 000 sin 1” (81) k, =f, cot ¢’ (82) 1, =g, cot $’ (83) m,=h, cot ¢’ (84) 100 . Ny = S547 bs cot ¢! (85) ___j,cot . °:=10 000sin 1” (86) Using the relations (72) to (86), inclusive, to make the conversions, the formule of (71) become the formule for the coefficients k,, k,, k,, ....-- 01, 02, 03, printed on page 76, of which the derivation was desired. As a check on the correctness of the derivation of these formulz, the dimensions of the so-called coefficients should be examined. The formule on page 76 show that the k’s, l’s, and m’s are all abstract numbers. This is as it should be, since in the observation equations on page 74 the absolute terms are all in seconds of arc and so, also, are (¢),(4), and (a). The formule show that the n’s are each of the dimensions seconds of arc divided by a length. This is as it should be, since each n is to be multiplied by a length, the required correction to the equatorial radius, and the product must be seconds of arc to correspond to the absolute terms. Similarly, each of the o’s is in seconds of are simply, which is as it should be, since it is to be multiplied by the required correction to e*, an abstract number, in order to give products which are seconds of arc corresponding to the absolute terms. The quantities (35) and (10000 e”) were used arbitrarily in the observation equations rather than (a) and (e’), simply for the purpose of making the average values of the n and o coefficients of about the same magnitude as the average values of the k, 1, and m coefficients. Such approximate equality, by insuring that the relation of the decimal point to the significant figures shall be about the same in the different coefficients, facilitates the least square computa- tion, especially when a summation term is used as a check in the formation and solution of the normal equations. 88 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. EXAMPLES OF COMPUTATIONS OF COEFFICIENTS. For use in computing the numerical values of the coefficients in the observation equations, the formule for which are shown on page 76, it is necessary first to compute values of the length of the arc 6 connecting the initial station with each astronomic station and to compute the forward and back azimuths along this arc. To make computations of § and of the azimuths named with the highest attainable degree of accuracy would be a long and difficult process. For the present purpose only a moderate degree of accuracy is necessary and the simple and short process of computation here described was found to be sufficient. Imagine a sphere of which the radius is = RSTO : and of which the center is the point at which the normal to the spheroid at latitude 4(6+¢’) intersects the axis of revolution of the spheroid. Since the value given above for Nu (compare with formula (9) on page 78) is the length of the normal at latitude 4(¢+ ¢’) limited by the axis of revolution, the small circle on the sphere having the latitude 4(¢+ ¢’) will coincide in space with the small circle of that latitude on the spheroid. Near that latitude the surface of the sphere and the surface of the spheroid are nearly in coincidence. In the approximate compu- tations which were made the two points concerned in latitudes ¢ and ¢’ were transferred from the spheroid to the sphere by making the linear distance from latitude ($+ 9’) to each point approximately the same on the sphere as on the spheroid, while their longitudes remain the same on the sphere as on the spheroid. It was then assumed that @ and the forward and back azimuths between the points as computed from a spherical triangle on the sphere are sufficiently close approximations to the true values of these quantities on the spheroid. The approximations are close because the parts of the sphere and spheroid which are concerned nearly coincide. Let y be an arc of the meridian on the sphere described above which has the same length in meters as the arc of the meridian on the spheroid from latitude ¢ to latitude ¢’. The mean radius of curvature of the arc of the meridian between these latitudes on the spheroid is given by formula (7) for R on page 78. Hence the ratio of this radius of curvature to the radius of curvature of the sphere, N,, is R _ 1—c? N, 1—e? sin’4(¢+¢’) 1—e’ y=(p¢p'e sin’s( $+ $’) Let x be TK go- ¢’), that is, x is the correction to the are 3(¢— ¢’) on the spheroid to Hence ety obtain the are 5 Y of the same length i in meters on the sphere. Then ie —e x=—HP— #(1-4 ao e cos?h(+¢’) =-1o-?)i_@ ani(o+e) Let =90°— ¢ and ¢=90°— gp’ Then, after calling the ieee ates 1-e sin?4( $+¢’) equal to rnity, the. shee expression for x becomes x =4(a—e)e? sin?4(a+e) If, then, in a spherical triangle on the sphere described, B= i’ —A=w is the angle at the pole and @ and ¢’ are the adjacent sides ad =a—x=90°—¢—-x Cf =c+x=90°-¢/+x THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 89 The remaining parts of this spherical triangle, A’, C’, and 6 in the ordinary notation of spherical trigonometry, may be computed by Napier’s analogies, namely, tan 4(A’— C’) =sin 4a’ —c’)ese 41 +’) cot $B tan 4(.A’ + C’) =cos 4(a’ —c’)sec 4(a’ +’) cot 4B and the law of sines, namely, sin’d=sin B sina’ csc A’ =sin B sine’ cscC’ using both forms for a check. The required quantities are 6=b a, =180°— C" @, =180°+ A’ in which .C’ and. A’ are to be considered positive or negative according as B= i’ ~—A=w is positive or negative. The following example illustrates the method as applied to the longest line in this investiga- tion, Meades Ranch, Kansas, to Calais, Maine, (latitude station No. 164, longitude station No. 173): ' ° / at ° 4 st dg 39 13 26.7 ad =a—-x 50 45 54 log sin B 9.71511 a 98 32 30.5 ¢=c+x 44 49 34 log sin a’ 9. 88905 ¢’ 45 1105.7 (a/—c) 25810 log csc A’ . 0. 00075 Vv 67 16 52.8 log sin 4(a’—c’) ; 8. 71436 log sin 0 9. 60491 a=90°—¢ 50 46 33.3 log csc Ha’+¢’) 0.13033 log sin B 9. 71511 c=90°—9¢’ 44 48 54.3 log cot 4 0.55319 — log sin ¢’ 9. 84816 4a+c)=}(+4+C’) 47 47 43.8 — log tan 4( 4’—C’) 9.39788 log csc C’ 0. 04165 3(a—e) 258 49.5 log cos $a’ TS 9.99942 log sin @ 9. 60492 B=V—h=w —31 15 37.7 log sec ia te 0.17277 ar =180°— C’ 245° 18’ 26’ 4B —15 37 48.9 log cot 4 0. 55319 a@3=180°+ A’ 86 37 26 4(a—c) in seconds 10729.’5 ~— log tan 4(A’+ C’) 0. 72538 log 4(a—c) in seconds 4.0306 4(.4’—C’) —14° 02’ 04” 6 23° 44’ 36” log sin? $(a-+c) 9.7393 4(.4’+0") —79 20 30 6 in seconds 85476/” log e? 7. 8305 / —93 22 34 log(@ in seconds) 4, 93184 log x 1. 6004 Cc —65 18 26 log A 8. 50906 x +39./8 log s 6. 42278 The length in meters,of the arc 0 on the sphere in question is s=N,J, or if 0 be expressed in seconds of are, s=N,, (@ in seconds) sin 1’ =(@ in seconds) (i) in which jaz [1—e* sin?4(6+¢’)} - asin 1” The logarithm of A is tabulated for 3(¢+ ¢’) as an argument in Appendix 9 of the Coast and Geodetic Survey Report for 1894. No complete proof is here offered as to the closeness with which the #0 and s so computed approach to the true 0 ands. The considerations given on page 88 indicate that the approxi- mation isclose. Moreover, for the longest line concerned in this investigation the values of log s, a,, and @, were computed by accurate formule. Their values as computed were a, = 245° 18’ 28’'.01 a,= 86° 37’ 32”.87 log s =6.4227742 Hence the errors of the approximate computation of these quantities, shown above, are: For @,,—2’, for a,,—7’’, and for log s, less than one unit in the last decimal place computed, the fifth. | These errors are so small as to have effects on the computed coefficients of the observation equations which are negligible in the present investigation. From formula (11) page 78 log (1+Q) =log| 1+ e cos*4( a, + a) 12 90 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Expanding the right-hand member by the formula 3 xt x5 3 = 4 + x ce ee - ee which is an expansion by Maclaurin’s Theorem, and neglecting higher powers than the first, 2 log (1 +x) =x-5+ 2 log (1+Q) -5 cos?4( a+ a;)M sin? 1” where M is the modulus of the common system of logarithms and sin*1” is introduced because 6 is to be expressed in seconds. Therefore log [log(1 + Q)]=log 6? + log cos?4( a, + ay) +7.92975 — 20 tn21// where 7.92975 —20 is log Msim‘I” 12 Applied to the particular example in hand this becomes Op + Ap 331° 55’ 52” (4p + ap) 165 57 56 log cos?4(@+a@,) 9.9737 log [6? in seconds] 9.8637 log constant 7.9298 log [log (1+Q)] 7.7672 log (1+Q) 0.00585 s(1+Q) Let Dh aut 402, a be denoted by W, then sin k,+1=— *© Wand m,=sin ay (1-+cos w) W sin @ in which [1 —e? sin?4 (6+ ¢’)]}3 B sini” and B=*—> To sind” Sis R- The logarithm of B is tabulated for + ($+¢’) as an argument in Appendix 9 of the Coast and Geodetic Survey Report for 1894. Therefore logs =log B+4.38454 — 10 The coefficients k,, m,, and n,, for the case in hand, may now be computed as follows: Oy — Ap —158° 41’ 00” cos w 0.85482 4 (a, — a) — 79 20 30 log (1-+eosw) 0.26830 log esc? 4 (a, — @,) 0.01512 log W 9.33897 log B 8.51068 log sin &, 9.99925 log s 6.42278 log m, 9.60652 log (1+Q) 0.00585 m, 0.4041 log constant 4.38454 log W 9.33897 log [@ in seconds] 4.93184 log sin? w 9.43022 log cos a, 8.77004 log esc 0 0.39508 log B 8.51068 log (k, +1) 9.16427 log (constant*) 6.68616 ke — 0.8540 log ny 8.89872 n, 0.0792 The first term of 0, is n, times a constant, which constant is a sin’ pe a 1787 Ge sin’ d) 2000000 | a 7 yt * This constant is Wve. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. 91 The second term may be computed as follows: 2nd term= —[1+2 sin?#+3 cos(¢’+ ¢)] + (¢’—¢) z (constant*) = —[1.79975+8 cos (¢’+ ¢)] 4 (¢’— ¢) z (constant*) e+¢ 84° 24° 32” +(¢’-$) =4(a-€ 10729’.5 log cos (¢’+ 4) 8.98869 log 3 cos (¢’+ ¢) 9.46581 3 cos (¢’4+ ¢) 0.29229 log [1+2 sin?f6 +3 cos (¢’+¢)] 0.32057 ; log [4 (¢’ — ¢) in seconds] 4.03058 log z 1.48932 log constant 4.21401 log 2nd term 0.05448 2nd term — 1.1337 1.2787 n,=1st term +0.1013 oO, — 1.0324 The following is the computation of the coefficients k,, 1,, m,, n,, and o, for this same station,, Calais: log sing’ 9.85088 log [6 in seconds] 4.93184 log sin w 9.71511 log sin a, 9.99925 log k, 9.56599 log (constant) 5.19530 k, __+0.3681 log n, 0.12639 _ n, — 1.3378 l,= —cos $’ —0.7048 it log cos? ay 9.2418 log cos¢’ 9.84808 log sin @ 9.9992 log sin w 9.71511 log s 6.4228 log cos &, 8.77004 log [@ in seconds] 9.8637 log csc oy 0.04165 log (constantt) 2.8863 log m, 8.37488 log 2nd term of 0, 8.4138 m, +0.0237_ 2nd term of 0, +0.0259 ——{ 1.2787 n,=1st term of 0, —1.7107 0, — 1.6848 As an example of the computation of the coefficients k,, 1,, m,, n,, and 0,, take the azimuth station Cooper, Maine, for which gp’ =44° 59’ 13’ 48 V =67 28 02.25 w= —31° 04' 28” : : 1 5 : : * This constant is 20000 N (ie)? sin 1” and its logarithm is 4.21401-10. ‘ . 100, { This constant is ~~ : . (l—e? sin? )} cos? ¢ sin 1” {This constant is nO eee 92 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. @p, 4, @ and s, computed in the same way as for Calais, are, in this case, Oy = 245° 46’ 48” az= 86 55 30 6= 23 36 04=84964” logs= 6.42018 _cot¢’ sintw_ _-cos¢’ sinw sin w sin @ sin? sin @sin 0 sin ¢’ In the spherical triangle shown on p. 79, the law of sines gives sin w sin a, sind sin é If ¢ is replaced in this by its approximate value 90 — ¢’ this becomes 1, is zero and k, = sin w _ sin a, cos ¢’ sin w _ 1 5 sin @ cosq’ sin @, sin 0 Therefore, k, = —“!" © with sufficient accuracy. sin ¢ log sin w 9.71278 log [0 in seconds] 4.92924 log sin 9’ 9.84939 log sin @& 9.99937 log k, 9.86339 log constant * 5.19530 k, +0.7301 log n, 0.12391 D, = 1.3302 log cos w 9.93272 log cos? ay 9.2261 log cot d’ 0.00020 log sin ag 9.9994 log csc &p 0.04001 log s 6.4202 log sina, 9.99937 log [8° in seconds] 9.8585 log m, 9.97231 log constant} 2.8863 m, + 0.9382 log 2"4 term 8.3905 SS ———— 2°¢ term +0.0246 1.2787n, ~ 1.7009 05 — 1.6763 FIVE LEAST SQUARE SOLUTIONS. Five complete least square solutions were made of the problem of determining the most probable values of the constants representing the figure and size of the earth. From a compari- son of the five solutions the most probable depth of compensation has also been derived. For convenience, the five solutions have been designated by the letters B, E, H, G, and A. For each solution the 507 observation equations, corresponding to the 507 astronomic observa- tions, were written in a form similar to that shown on page 74, which is the form for solution A. The forms for the equations for the other four solutions differ only in the absolute term, the last term on the left-hand side of each equation. In solution B, the absolute term of each observation equation was the observed apparent deflection of the vertical, as shown in the second column of the tables on pages 48-56, minus the topographic deflection, as shown in the third column of those tables. If the computed topo- graphic deflection were the actual deflection and if there were no errors of observation and no - errors in the assumed values of the latitude, longitude, and azimuth at the initial station, Meades Ranch, and if the a and e? of the Clarke spheroid of 1866 expressed the figure and size of the earth precisely, each one of the absolute terms in these observation equations of the solution B would be zero. Solution B is, therefore, made upon the assumption that no isostatic compensation exists, that the portions of the continent above sea level are excesses of mass, and the oceans represent defects of mass. It may convenientiy be considered to be a solution based upon the * This constant is the same as that in n,. { This constant is the same as that in 0,. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 93 supposition that if isostatic compensation exists it is uniformly distributed through an infinite depth. In solution E, the absolute term of each observation equation is the observed apparent deflection minus the deflection computed on the assumption that the isostatic compensation is complete and uniformly distributed through the depth 162.2 kilometers, as shown in the column so headed in the tables on pages 48-56. In solution H, the assumption is that the compensation is complete and uniformly dis- tributed through the depth 120.9 kilometers, and in solution G that the depth is 113.7 kilometers. These solutions arc similar to solution E, the computed deflection which is subtracted from the observed apparent deflection being taken from a different column of the tables on pages 48-56. If the isostatic compensation were complete and uniformly distributed through the depth assumed in connection with any one of the solutions E, H, or G, the residuals from that solution would be very small, being due simply to the errors of observation. The excess of the residuals above the average magnitude due to errors of observation is a measure of the degree of agreement between the assumption as to isostatic compensation and the fact. In solution A, the absolute terms are simply the observed apparent deflections as shown in the tables on pages 48-56. The observation equations were, therefore, precisely as indicated on page 74, including the absolute terms. This is the solution usually made in connection with the arc method. No relation is postulated between the deflections of the vertical and the topography. This is equivalent to the assumption that there is complete isostatic compensa- tion at depth zero; that there exists immediately below every elevation (either mountain or continent) the full compensating defect of density, and that at the very surface of the ocean floor there lies material of the excessive density necessary to compensate for the depression of this floor. Under no other condition can it be true that the observed deflections of the vertical are independent of the known topography. The particular depths of compensation assumed in solutions E, H, and G depend mainly upon extensive preliminary investigations made largely for the purpose of obtaining an approxi- mate idea of the most probable depth of compensation. : THE OBSERVATION EQUATIONS. - The observation equations for solution G are given below. In each column the symbols ($), (A), (42), (@w), (05) and (10000e*) should be considered as repeated down the column. The observation equations for the other four solutions differ from these only in the absolute terms, as already stated. The statements of the preceding paragraphs will enable one to repro- duce the absolute terms of any equation of any solution as desired by using the tables on pages 12-19 and 48-56. The equations are printed in four groups. The same geographic order is used in each group as in the tables on pages 48-56. The number of each station as printed enables one to identify the station in the tables on pages 12-19 and on illustration No. 13 at the end of this publication. Observation Equations. NORTHEASTERN GROUP—LATITUDES. Station. . u" 241 —0. 961(¢) +0. 216( ag) +0. 147(.2,) —0.569(10 000e?) — 0.80 =Dm 188 —0. 964 +0. 208 +0. 115 —0. 486 — 155 =D 185 —0. 965 +0. 203 +0. 094 —0. 433 — 3.60 =Dm 184 —0. 966 +0. 201 +0. 081 —0. 399 — 0.69 =Dm 183 —0. 962 +0. 213 +0. 065 —0. 386 + 166 =Dm 182 —0. 957 +0. 225 +0. 058 —0. 399 + 2.55 =Dm 186 —0. 957 +0. 225 +0. 058 —0. 400 + 2.56 =D 187 —0. 956 +0. 227 +0. 057 —0. 400 +179 =D 181 —0. 948 +0. 246 +0. 079 —0. 498 + 3.13 =Dm 180 —0. 944 +0. 256 +0. 091 —0. 547 + 3.32 =Dm 179 —0. 941 +0. 262 +0. 108 —0. 597 + 2.73 =Dm 178 —0. 941 +0. 262 +0. 114 —0. 611 + 2.58 =Dm 177 —0. 934 +0. 278 +0. 110 —0. 645 + 1.55 =Dm 176 —0. 927 +0. 292 +0. 113 —0. 692 + 0.35 =Dm 175 —-0. 925 +0. 296 +0. 123 —0. 722 — 2.59 =Dm 94 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Station. 174 173 169 170 168 166 171 172 165 167 164 163 158 162 157 161 159 156 160 155 }- 153 152 154 149 151 150 148 147 144 145 142 248 146 143 141 140 139 138 137 136 107 109 108 Station. 232 196 195 192 191 194 190 188 187 185 184 183 182 181 179 173 172 157 158 233 108 110 Station. 193 186 189 180 175 —0. 923() +0. 300(ap) +0. 136(,2,) | —0.757(10 000e?) —0. 920 0. 305 +0. 170 —0, 836 —0. 925 +0. 295 +0. 086 —0. 646 0. 919 +0. 307 +0. 072 —0. 654 —0. 908 +0. 325 +0. 015 —0. 595 —0. 907 +0. 327 +0. 040 —0. 656 —0. 908 +0. 326 +0. 079 —0. 732 —0. 905 +0. 330 +0. 111 —0. 807 —0. 903 +0. 333 +0. 129 —0. 850 —0. 904 +0. 331 +0. 160 —0. 900 —0. 854 +0. 404 +0. 079 —1. 032 —0. 856 +0. 402 +0. 072 —1.010 —0. 855 +0. 403 +0. 052 —0. 976 —0. 862 +0. 394 +0. 076 —0. 984 —0. 863 +0. 393 +0. 050 —0. 931 —0. 867 +0. 386 +0. 083 —0. 965 —0. 871 +0. 382 +0, 081 —0. 944 —0. 871 +0, 382 +0. 057 —0. 899 —0. 879 +0. 370 +6. 097 —0. 926 —0. 879 +0. 371 +0. 067 —0. 874 —0. 877 +0. 374 +0. 044 —0. 840 —0. 881 +0. 368 +0. 050 —0. 829 —0. 885 +0. 362 +0. 072 —0. 848 —0. 889 +0. 355 +0. 052 —0. 784 —0. 884 +0. 364 +0. 027 —0. 763 —0. 883 +0. 364 +0. 013 —0. 738 —0. 891 +0. 353 +0. 026 —0. 720 —0, 884 +0. 363 —0. 006 —0. 693 —0. 887 +0. 358 —0.013 —0. 658 —0. 887 +0. 358 —0. 012 —0. 658 —0. 883 +0. 365 —0. 044 —0. 612 —0. 880 +0. 369 —0. 044 —0. 635 —0. 893 +0, 349 +0. 004 —0. 657 —0. 899 +0. 340 +0. 001 —0. 615 —0. 901 +0. 336 —0. 038 —0.513 —0. 905 +0. 330 —0. 067 —0. 421 —0. 909 +0. 323 —0. 064 —0. 401 —0, O11 +0. 320 —0. 081 —0. 348 —0.914 +0. 314 —0. 076 —0. 340 —0.919 +0. 306 —0. 089 —0. 275 —0. 920 +0. 304 —0. 132 —0. 155 —0. 916 +0. 312 —0. 152 —0. 129 —0. 917 +0. 310 —0. 158 —0. 104 NORTHEASTERN GROUP—LONGITUDES. +0.189(¢) —O0.731(A) +0. 045(ap) —0.701(,2,) | —0. 894(10 000¢2) +0. 180 —0. 739 +0. 035 —0. 674 —0. 860 +0. 174 —0. 744 +0. 029 —0. 659 —0. 842 +0. 172 —0. 747 40. 025 —0. 653 —0. 834 +0. 192 —0. 749 +0. 018 —0. 732 —0. 935 +0. 192 —0. 749 +0. 018 —0. 732 —0. 936 +0. 213 —0. 741 +0. 024 —0. 802 —1. 023 +0, 222 =0. 737 +0. 028 —0. 833 —1. 062 +0, 229 =O, 783 +0. 033 —0. 853 —1. 088 +0. 229 =O, 731 +0. 035 » —0. 852 —1. 086 +0, 244 —0. 730 +0. 033 —0. 907 —1, 155 +0, 258 —0. 726 +0. 034 0. 954 =1,913 +0. 264 =f), 723 +0. 037 —0..973 = 1.087 +0. 267 —0. 720 +0. 041 —0. 979 —1. 244 +0. 275 =0.711. +0. 052 —0. 996 —1. 264 +0. 368 —0. 705 +0. 024 —1. 338 —1. 685 +0. 350 —0.710 +0. 025 1276 4612 +0. 310 —0. 739 —0. 004 ==], 179 —1.500 +0. 313 —0. 743 —0. 011 —1, 198 —1.525 +0. 318 —0. 742 —0.013 —1,218 —1.550 +0. 247 —0.775 —0. 040 —0. 996 oe +0, 252 —0.778 0.046. —1. 021 —1. 306 NORTHEASTERN GROUP—AZIMUTHS. +0. 413(¢) +1. 125(ag) —0. 690(,2,) —0. 881(10 000e?) +0. 493 +1. 069 —0. 852 —1. 086 +0. 550 Bae —0. 967 —1. 230 +0. 556 +1. 013 —0. 996 —1. 264 +40. 555 +1. 051 —0. 963 —1. 226 Observation Equations—Continued. NORTHEASTERN GROUP—LATITUDES—Continued. Lob PebseR bse Eh ete eases ee ee) el set pe | PYAAR OOW RO RO SOSH SPN ESHOP NE NWORWSSWOSP ESN b) L+t++4+4+4+4+4+ 14440410 1 1 Le l Isk “4 22 32 41 15 26 17 31 16 78 29 26 “09 45 59 01 31 06 81 43, 94 92 58 91 76 98 71 53 07 61 67 51 84 27 44 27 27 60 68 94 72 25 65 z 6. 58 wow ERO awonw aang 95 Po RN IRIS Ou Olin > 6 H Oo 0. 93 49. =Dm =Dm =Dm =D =D =Dm =Dm =D =Dm =Dm =D =D =D =D =Dm =D =Dm =D =Dm =Dm =Dm =Dm =D =Dm =Dm =Dm =D =Dm =D =Dm =Dm =D =D =D =D =D =D =D =D =D =Dm =D =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dr THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 95 Station. 176 174 177 178 171 170 169 168 167 166 164 165 163 162 160 161 159 156 147 154 150 149 1651 152 153 155 148 146 145 144 143 142 109 Station. 80 81 82 83 +0. 579(¢) +0. 622 +0. 610 +0. 610 +0. 730 +0. 736 +0. 718 +40. 722 +0. 698 +0. 704 +40. 684 +40. 694 -+0. 684 +0. 669 +40. 663 +0. 682 +40. 665 +0. 689 +0. 683 +0. 686 +0. 681 +40. 684 +0. 693 +0. 706 +0. 710 +0. 664 +40. 650 +40. 652 +0. 649 +0. 636 -+0. 624 +0. 612 +0. 636 —0. 963(¢) —0. 959 —0. 957 —0. 952 —0. 944 —0. 943 —0. 939 —0. 936 —0. 934 —0. 938 —0. 934 —0. 932 —0. 931 —0. 930 —0. 930 —0. 930 —0. 930 —0. 930 —0. 930 —0. 930 —0. 929 —0. 927 —0. 926 —0. 928 —0. 925 —0. 923 —0. 926 —0. 923 —0. 925 —0. 923 —0. 923 —0. 943 —0. 949 —0. 951 —0. 955 —0. 960 —0. 965 —0. 968 —0. 969 Observation Equations—Continued. NORTHEASTERN GROUP—AZIMUTHS—Continued. +1. 045(ap) +1. 047 +1. 023 +1. 008 +0. 938 +0, 944 +0. 947 +40. 956 +0. 960 +0. 968 +0. 978 +40. 982 +0. 987 +0. 986 +1. 001 +1, 000 +1. 013 +1. 012 +1. 020 +1, 023 +1, 031 +1. 037 +1. 032 +1, 035 +1. 026 +1, 025 +1. 036 +1. 055 +1. 073 +1, 090 +1. 094 +1. 109 +1. 135 SOUTHEASTERN GROUP—LATITUDES. —0. 092(,8,) 0. 109 —0. =th —0. —0. =o —( —0. 0. 6 —0. —0. —0. = ce =O. —0. —0. —0. —D. —0. —0. —0. —0. =H; —0. —0. —0. —0. —0. =f. —0. —0. —0. —0. =O. +0. 211(ap) +0. 221 +40, 226 +0. 238 +0. 256 +40, 259 +0, 266 +40. 273 +0. 277 +0. 268 +0. 276 +0. 281 +0, 284 +0, 286 +0, 285 +0, 285 +0, 285 +0. 285 +0. 286 +0, 286 +0, 287 +0. 290 +0. 292 +0. 290 +0. 295 +0, 298 +0. 293 +0, 298 +40, 295 +0, 298 +0. 299 +0. 259 +0, 244 +0, 239 +0. 231 +0. 216 +0, 204 +0. 194 +0. 192 —1. 004(,%,) —1. 067 —1. —1. —l. —l. —l. —l1. —1. —1. —l. —1. —1. —1. —l. —1. 1. —1. —l. —l. —1. —l. —1. 1 —1. —1. —l. —1. —l1. —1. —1. —1. —1. —0. —0. 069 082 330 333 304 299 261 261 223 232 213 192 169 197 160 196 179 180 166 164 182 198 212 148 116 103 084 050 029 001 014 118 158 148 151 163 153 124 112 099 106 119 121 128 129 129 130 131 131 132 136 132 123 117 103 166 202 221 249 270 198 237 271 295 301 317 328 346 —1. 278(10 000¢?) —1. 359 —1. 359 —1.374 —1. 678 —1. 683 —1. 646 —1. 643 —1.594 —1.596 —1.549 —1.562 =1,598 —1.522 —1. 494 —1.530 —1. 482 —1, 528 —1.507 —1.502 —1. 485 —1. 483 —1.506 —1.526 —1.543 —1. 460 —1, 421 —1. 407 —1. 383 1.341 —1.314 —1. 280 —1. 298 +0. 011(10 000¢?) +0. 033 +0, 043 +0, 124 +0, 046 +0. 047 +0. 058 +0, 008 —0. 084 —0. 092 —0, 152 —0. 145 —0. 122 —0. 125 —0. 099 —0. 098 —0. 096 —0. 094 —0. 094 —0. 094 —0. 097 —0. 097 —0.119 —0. 129 —0. 164 —0. 212 —0.021 +0. 062 +0. 130 +0, 207 +0. 267 +0. 185 +40. 350 +0. 470 +0. 574 +0. 633 +0. 720 +0. 782 +0. 847 FHEHE PL LE LPH EH+tHHE Ft PEEP FEET PRON PACK RAP ARWOWRAWOSMNS POP WM or FHEHHHH4+ 0 PEL EEE DEEL EL bet +++ PtH 444 NENTAWO ANE NNONMNOP NE NWENPOSOSMAE RE RONSON SS 8 44 =Dp 01 =Dp 32 =Dp 04 =Dp 22 =Dp 88 =Dp 41 =Dp 39 =Dp 39 =Dp 01 =Dp 04 =Dp 70 =Dpe 52 =Dp 82 =Dp 06 =Dpr 22 =Dp 73 =Dp 02 =Dp 11 =Dp 55 =Dp 99 =Dp 44 =Dp 08 =Dp 96 =Dre 59 =Dp 30 =Dp 79 =Dp 48 =Dp 57 =Dp 45 =Dp 83 =Dp 23 =Dp 20 =Dpe 63 =Dm 42 =Dmu 64 =D 23 =D 74 =D 03 =D 51 =Dm 40 =Dm 51 =D 62 =D 83 =Dm 36 =Dm 99 =Dm 39 =D 84 =D 09 =Dm 04 =Dmu 91 =D 39 =Dm 67 =D 43 =Dm 18 =D 56 =D 02 =Dm 97 =Dm 33 =D 03 =Dm 58 =D 10 =D 12 =Dm 54 =Dm 70 =DMm 02 =Dm 41 =D 41 =Dm .80 =D .97 =Dm .80 =Dm .27 =D 96 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Station. 126 122 119 118 131 124 127 128 129 130 132 134 133 135 Station. 87 114 90 92 99 100 101 118 113 119 130 123 131 132 137 141 Station. 139 —0. 970() —0. 973 —0.977 —0. 979 —0. 981 —0.977 —0.977 —0. 977 —0. 982 —0. 983 —0. 983 —0. 983 —0. 985 —0. 989 +0. 180(¢) +0. 206 +0, 211 +0. 217 +0. 230 +0. 230 +0. 230 +0. 230 +0. 218 +0. 177 +0. 136 +0. 120 +0. 114 +0. 101 +0. 093 +0. 074 +0. 434(d) +0. 456 +40. 500 +0. 533 +40. 566 +0. 567 +0. 561 +40. 570 10. 584 +40. 582 +0. 587 +40. 595 +40. 591 +0. 601 +0. 641 40. 608 -+0. 626 40. 625 +0. 638 40. 646 +40. 550 +0. 528 +0. 526 +0. 514 10. 485 -L0. 462 +0. 442 +40. 442 +0. 407 +0. 359 40. 340 0. 381 +40. 385 +40. 375 +40. 363 +40. 344 —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. 784(2) 786 788 777 778 778 778 778 796 811 832 807 844 850 860 866 Observation Equations—Continued. SOUTHEASTERN GROUP—LATITU DES—Continued. +0. 191(a,) +0. 179 -L0. 165 +40. 159 +0. 151 +0. 167 +0. 164 -40. 165 +0. 149 +0. 142 +0. 142 +0. 143 +0. 136 +0. 115 SOUTHEASTERN GROUP—LONGITUDES. —0. 036(ag) —0. 046 —0. 050 —0. 034 —0. 039 —0. 040 —0. 040 —0. 040 —0. 066 —0. 082 —0. 108 —0. 063 —0. 128 —0. 136 —0. 154 —0. 165 —0. 354(,45) —0. 316 —0. 205 —0. 205 —0. 286 0, 821 —0. 372 —0. 419 —0. 443 —0. 474 —0. 505 —0. 531 —0. 523 —0. 538 —0. 735(,2,) —0.845 —0. 868 —0. 875 —0. 930 —0. 931 —0. 933 —0. 933 —0.917 —0. 768 —0. 618 —0.513 —0. 535 —0. 481 —0. 460 —0. 372 SOUTHEASTERN GROUP—AZIMUTHS. +1. 194( ag) 1.195 +1. 204 +1. 184 +1. 170 +1. 153 +1. 142 +1. 142 44,149 +1. 147 +1. 147 +1. 145 +1. 140 +1. 123 +1. 146 +1. 156 1. 167 +1. 180 +1. 191 +41. 200 +1. 206 +1. 240 1. 262 +1. 284 +1. 300 +1, 319 +1. 334 +1. 346 +1. 338 +1, 282 +1. 337 1. 348 +1. 382 +1. 430 +1. 514 +1. 511 —0. 683(,85) 0, 717 —0.775 —0. 835 —0. 890 —0. 903 —0. 902 —0. 916 —0. 935 —0. 930 —0. 938 —0. 950 —0. 947 —0. 975 1,013 —0. 959 —0. 976 —0. 966 —0. 976 —0. 980 —0. 845 —0. 794 —0.778 —0. 750 —0. 703 —0. 661 —9. 627 —0. 622 —0. 578 —0. 534 —0. 487 —0. 540 —0. 532 —0. 502 —0. 461 —0. 438 +0. 878(10 0006”) 40. 775 +0. +0. +1. 437 448 . 726 . 815 . 997 171 . 292 . 423 . 545 . 650 627 +1, 722 —0. —1. —1. —i1. —l1. —1. —1. —1. —1. —0. —0. —0. —0. —0. —0. —0. 0. —0. —0. —1. —l1. —1. —1. —1. =I, —1. -1. -1. <1, —l1. -1. —1L —l. —1. -1. —l1. —1. -1. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. 941(10 000e?) 081 112 120 190 191 193 194 173 982 788 656 681 612 583 470 875(10 000e?) 918 992 068 140 156 154 172 196 190 200 215 211 247 297 227 250 236 249 254 081 016 995 959 898 844 801 794 738 682 622 689 678 639 584 —0. 555 FHt+ | ++t44+44 1+ +4] $4441 11 1 )++++ FH+HEHIFFHEIFH IEE IEEE EEE LE EE tHtHI TL | H SPARES Cte iH ON FIs Oi te ih NOR OR Ss NS O00 ovo 3 > S o DR Wr Dr rs won S po 8 SEP HPOP ASP RAP Shwe S SPARWKRODDOSCOHMARMOWW BOON WNENNANL KHON y 2 we o NO o> 43 =Dm =Dm =Dm =D =Dmu =Dmu =Dm =Dmu =Dm =D =D =Dm =Dm =D =Dp =Dp =Dp =Dp = Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp = Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp THE FIGURE OF THE BARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 97 Observation Equations— Continued. CENTRAL GROUP—LATITUDES. Station. u 227 | —0.972(d) +0. 183( ap) +0, 422(,%,) —0.955(10 000e2) — 3.08 =Dm 228 | —0.976 +0.171 +0. 438 —0. 958 — 1.06 =D 231 | —0.976 +0. 169 40. 468 —0. 990 + 2.00 =D 932 | —0. 982 +0. 145 +0, 508 —1. 005 +11.86 =Dm 230 | —0. 985 +0. 135 +40. 475 —0. 961 +11.80 =Dm 229 | —0. 989 +0. 116 +0. 468 —0. 936 — 1.47 =D 223 | —0. 992 +0. 100 40. 446. —0. 897 — 7.34 =D 219) —0. 993 +0. 092 +0, 423 —0. 862 ~ 2.05 =D 213 | —0. 994 +0. 088 +0. 414 —0. 846 + 1.80 =D 212 | —0. 994 +0. 088 +0. 411 —0. 844 +199 =D 211 | —0.993 +0. 091 +0. 410 —0. 843 + 3.14 =D 214 | —0.993 -40. 094 +0. 412 —0. 849 + 0.81 =D 221 | —0.990 +0. 110 +0. 425 —0. 878 — 3.80 =Du 216 | —0.988 +0. 119 +0. 406 —0. 861 — 0.41 =D 226 | —0. 983 +0. 144 +0. 434 —0. 923 — 7.62 =D 225) —0. 982 +0. 145 +0. 433 —0. 922 — 3.49 =D 224] —0. 983 +0. 144 +0. 431 —0. 918 —11.64 =D 222 | —0.985 +0. 134 +0. 417 —0. 889 — 4.03 =D 220 | —0. 984 +0. 137 +0. 410 —0. 884 — 1.44 =D 217 | —0.985 +0. 136 +0. 404 —0. 876 + 2.17 =D 218 | —0. 983 +0. 144 -40. 402 —0. 880 + 7.909 =D 189 | —0.981 +0. 150 -+0. 380 —0. 858 + 4.31 =D 191 | —0.981 +0. 151 -+0. 380 —0. 859 + 4.35 =D 215 | —0.972 +0. 183 +0. 376 —0. 897 + 0.31 =D 190 | —0.972 +0. 183 +0. 363 —0. 879 + 1.36 =D 194 | —0.970 -L0. 187 +0. 358 —0. 878 + 0.37 =D 192 | —0.969 +0. 191 -L0. 357 —0. 882 + 0.92 =D 193 | —0.970 +0. 188 +0. 322 —0. 825 — 1.28 =Du 197 | —0.974 +0. 175 +0. 314 —0. 794 — 3.80 =D 196 | —0.979 +0. 160 +0. 328 —0. 794 — 5.29 =Du 210 | —0.980 +0. 155 +0. 333 —0. 797 — 5.71 =D 195 | —0. 980 40. 154 +0. 330 —0. 791 — 611 =Du 198 | —0.981 +0. 152 +0. 317 —0. 767 — 4,94 =D 200 | —0.979 +0. 157 +0. 314 —0. 769 — 3.25 =D 199 | —0.979 +0. 158 +0. 314 —0. 771 — 3.72 =D 201 | —0.980 +0. 155 +0. 308 —0. 757 —~ 1.21 =Du 202 | 0.982 +0. 148 +0. 299 —0. 734 + 0.44 =D 203 | .—0. 981 +40. 149 +0. 297 —0. 732 + 2.05 =D 204 | —0.983 . +0. 142 +0. 270 —0. 677 — 0.05 =D 205 | —0. 985 +0. 133 +0. 215 —0. 567 + 3.42 =D 206 | —0.977 +0. 164 +0. 134 —0. 449 + 3.51 =D 209 | —0. 982 +0. 148 +0. 120 —~0. 394 +186 =D 207 | —0.983 +0. 144 +0. 113 —0. 373 + 1.50 =D 247'| —0. 984 +0. 139 +0. 024 —0. 158 +142 =D 208 | —0. 983 +0. 144 +0. 017 —0. 150 + 1.24 =D 62} —0.999 —0. 041( ay) —0. 020 +0. 043 — 0.23 =D 63 | —1.000 —0. 003 —0. 021 +40. 054 + 0.19 =D 64] —1.000 -0. 004 —0. 028 +40. 073 — 133 =D 65 | —1.000 +0. 013 —0. 021 +0. 055 — 4.31 =D 66 | —0.999 40. 034 —0. 012 -40. 024 — 0.10 =Du 67 | —0.998 +0. 054 —0.011 +0. 014 + 0.06 =D 68; —0.995 +0. 078 —0. 053 +0. 109 + 2.98 =D 69 | —0.994 40. 086 —0. 047: +0. 086 + 2.23 =D 70 | —0.989 +0. 113 —0. 048 +0. 069 + 4.66 =D 71 | —0.987 +0. 124 —0. 055 +0. 065 + 6.04 =D 72| —0.984 +0. 140 —0. 043 +0. 011 + 2.86 =D 73; —0.983 +0. 141 —0. 047 -+0. 020 + 3.09 =D 74] —0.983 +0. 142 —0. 063 +0. 062 +179 =Du 75} —0. 982 +0. 149 —0. 060 +0. 044 + 1.31 =D 76 | —0.977 -+0. 166 —0. 040 —0. 039 + 0.34 =D 77} —0.973 +0. 180 —0. 053 —0. 030 + 0.89 =D 78| —0.970 +0. 190 —0. 053 —0. 050 — 0.83 =D 79 | —0.967 --0. 196 —0. 081 +0. 010 — 0.20 =D ‘CENTRAL GROUP—LONGITUDES. Station. u 213 | +0.082(¢) —0.685(4) +0. 127(a@z) —0. 283(,%,) —0.359(10 000e2) — 4.30 =Dr 202 | +0. 140 —0. 688 +0. 116 —0. 485 —0. 614 — 7.50 =Dp 198 | +0.178 —0. 688 +0. 109 —0. 619 —0. 784 — 5.85 =Dp 201 | +0. 142 —0. 699 +0. 101 —0. 500 —0. 634 — 4.34 =Dpe 206 | +0. 134 —0. 706 0. 092 —0. 478 —0. 608 — 3.45 =Dp 207 | +0. 128 —0. 713 +0. 083 —0. 458 —0. 583 + 0.95 =Dp 210 | +0. 127 —0. 744 +0. 037 —0.477 —0. 609 — 1.95 =Dp 211} +0. 124 —0. 746 +6. 034 —0. 466 —0. 596 + 0.93 =Dp. 78771—09-—7 98 THE FIGURE Station. Station. 240 239 267 266 265 246 245 236 237 264 238 235 262 234 243 242 261 260 257 258 259 256 255 244 233 254 253 19 21 16 13 12 10 11 “TORO —0. 033(¢) +0. -A0: +0. +0. 40, +0. +0. 318() +0. +0. AG +0. 1G, A: +0. +0. 0. —0, +0. +0. +0. +0. 40, +0. +0. +0. +0. +0. +0. +0. +0. —0. 947(¢) 6, —O, eff, 6, at: =0. ah 2, =O. —6, —0. =O. fl. mee —0. =; =f, cai, Hl; —0. =f, =f, afl, —0. —0. —0. =: =O: 26 Hh =O) —0: =0; =i =); =f) —0. 003 044 090 114 119 154 155 160 239 338 332 277 263 248 279 007 - 026 068 162 178 202 230 256 287 292 322 338 368 404 947 947 940 941 941 941 931 928 935 932 929 927 927 929 926 926 926 923 922 922 928 925 921 920 918 922 920 920 918 914 914 914 914 913 910 910 909 OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Observation Equations—Continued. CENTRAL GROUP—LONGITUDES—Continued. 8 —0. 778(A) —0.006(aw) +0. 133(,2,) -+0.170(10 000e?) — 1.47 —0. 780 —0. 009(ap) —0. 014 —0. 018 — 5.02 —0.776 —0. 003 —0. 173 =0, 221 — 0.89 =), 78 —0. 015 —0. 364 —0. 466 2, OF —0. 782 —0. 019 —0. 459 —0. 586 — 0.60 —0), 781 —0. 018 —0. 481 —0. 614 a2 4,12 —0.776 —0. 016 —0. 615 —0. 786 — 0.19 CENTRAL GROUP—AZIMUTHS. j +1. 020(ap) —0. 592(,#,) —0.748(10 000e") + 1.36 +1. 056 —0. 283 —0. 359 — 5.48 +1. 058 —0. 293 —0. 371 — 8.28 +1, 045 —0. 440 —0. 557 ee +1. 035 —0. 619 —0. 784 — 5,28 +1. 049 —0. 601 —0. 763 — 3.80 +1. 061 —0. 500 —0. 634 == 8, 29 +1. 085 —0. 465 —0. 592 4: 294 +1. 109 —0. 429 —0. 546 +415 +1. 144 —0. 466 —0. 596 + 0.25 +1.235(aw) +0. 011 +0. 014 + 8.36 +1, 235 —0. 041 —0. 052 + 6.13 +1. 229 —0. 108 —0. 138 — 0.64 +1, 240 —0. 252 —0. 322 de 1,62 +1. 235 —0. 279 —0. 356 — 3.60 +1, 282 —0. 317 —0. 405 ae +1. 232 —0. 359 —0. 459 — 9,82 +1, 225 —0. 402 —0.514 ws AL 4-1, 218 —0. 453 —0.579 — 0.78 +1. 221 —0. 459 —0. 586 + 0.06 +1, 209 —0. 509 —0. 650 a +1. 198 —0. 537 —0. 687 — 1.60 +1. 196 —0. 583 —0. 745 +. 2,01 +1. 199 —0. 637 —0. 814 + 4.39 WESTERN GROUP—LATITUDES. 8 —0.249(aw) —0.443(,2,) +1.043(10 000e?) — 3. 65 —0. 250 —0. 445 +1. 047 =— 1,84 —0. 250 —0. 438 +1. 019 + 0.17 —0. 266 —0. 415 +0. 883 + 5.97 —0. 263 —0. 398 +0. 830 + 6.02 —0. 263 —0. 390 +0. 802 + 6.38 —0. 263 —0. 380 +0. 765 = 7.04 —0. 284 —0. 394 +0. 741 ee —0. 290 —0. 400 +0. 742 4. 2.48 —0. 276 —0. 377 +0. 712 — 0.26 —0. 281 —0. 374 +0. 680 — 6.03 —0. 288 —0. 373 +40. 655 — 4.93 —0. 291 —0. 378 +0. 661 — 3.02 —0. 292 —0. 372 +40. 635 + 2.95 —0. 287 —0. 350 +0. 578 — 6.75 —0. 293 —0. 356 +0. 576 + 0.55 —0. 294 —0. 341 +0. 521 + 0.98 —0. 294 —0. 341 +0. 521 + 3.02 —0. 298 —0. 318 +0. 428 — 6.00 —0. 300 —0. 321 +0. 430 — 3.30 —0. 300 —0. 321 +0. 430 — 5.83 —0. 289 —0. 296 +0. 388 * — 0.08 —0. 295 —0. 280 +0. 315 — 8.36 —0. 303 —0. 295 +0. 337 = #6 —0. 305 =), 277 +0. 268 + 5.89 —0. 309 —0. 274 +0. 245 + 3.27 —0. 300 =) 52 +0. 208 — 1.85 —0. 305 ==. 282 +0. 129 — 2.98 —0. 304 —0. 224 +0, 108 — 5.40 —0. 309 —0. 206 +0. 037 — 6.60 —0. 315 —0. 216 +0. 043 — 5.99 —0. 315 =O; 217 +0, 044 == 9.10 —0.315 =O; 217 +0. 043 — 1.55 —0. 315 0; 217 +0. 044 = 1.81 —0. 317 —0. 211 +0. 021 — 3.91 —0. 322 —0, 212 +0. 004 — 6.59 —0. 323 —0. 196 —0. 047 — 4.89 —0. 324 —0. 186 —0. 079 = 6.01 =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =Dp =D =D =Dm =Dm =Dm =D =D =Dm =D =D =Dm =Dm =D =D =D =Dm =D =Du =D =D =D =Dm =D =Dm =Dm =Dm =Dm =Dm =D =Du =Dm =D =D =D =D =D =D =Dm THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 99 : Observation Hquations—Continued. WESTERN GROUP—LATITUDES—Continued. Station. a 1] —0. 905(¢) —0.331(aw) —0.171(,4,) —0. 152(10 000e?) — 0.96 =Dm 2 —0. 904 —0. 332 —0. 151 —0. 210 — 3.46 =Dm 252 —0. 900 —0. 339 —0. 097 —0. 382 — 0.98 =Dmu 251 —0. 901 —0. 337 —0. 078 —0. 421 — 4.27 =D 249 —0. 901 —0. 337 —0. 023 —0. 550 — 3.58 =Dm 5 —0. 908 —0. 325 —0. 152 —0. 178 — 1.08 =Dmu 6 —0. 911 —0. 320 —0. 168 —0. 115 — 3.49 =Dm 8 —0. 912 —0. 318 —0.171 —0. 097 — 4.48 =Dm 14 —0. 916 —0. 312 —0. 166 —0. 089 — 184 =Dm 15 —0. 916 —0. 311 —0. 181 —0. 043 — 4.94 =Dm 17 —0. 918 —0. 308 —0. 162 —0. 087 — 5.66 =Dm 18 —0. 918 —0. 307 —O0. 169 —0. 063 — 6.40 =Dm 20 —0. 920 —0. 305 —0. 134 —0. 152 — 4.52 =D 24 —0. 935 —0. 276 - —0.174 +0. 059 — 0.57 =D 23 —0. 930 —0. 285 —0. 143 —0. 059 — 152 =Dm 25 —0. 931 —0. 284 —0. 126 —0. 102 — 2.76 =D 27 —0. 932 —0. 282 —0. 113 —0.131 — 5.53 =Dm 22 —0. 928 —0. 289 —0. 104 —0.179 — 3.60 =Dm 26 —0. 931 —0. 284 —0. 095 —0. 186 — 4.29 =Dm 28 —0. 941 —0. 262 —0. 074 —0. 173 — 6.05 =Dm 29 —0. 946 —0. 251 —0. 108 —0. 052 — 4.45 =Dm 30 —0. 949 —0. 246 —0. 055 —0. 176 — 5.56 =D 31 —0. 955 —0. 231 —0. 052 —0. 146 —16.96 =Dm 32 —0. 962 —0. 213 —0. 134 +0. 120 —10.24 =Dm 33 —0. 963 —0. 208 —0. 127 +0. 114 ' — 129 =D 37 —0. 970 —0. 190 —0. 101 +0. 082 — 4.33 =Dm 48 —0. 977 —0. 165 —0. 098 +0. 121 — 0.83 =Dm 35 —0. 964 —0. 206 —0, 024 —0. 160 — 4.39 =Dmu 34 —0. 963 —0. 208 +0. 041 —0. 324 — 3.49 =Dmu 39 —0. 971 —0. 187 +0. 069. —0. 344 — 3.34 =Dm 42 —0. 972 —0. 181 +0. 068 —0. 330 — 3.18 =Dm 44 —0. 973 —0. 180 +0. 067 —0. 327 — 167 =Dmu 40 —0. 972 —0. 184 +0. 052 —0. 299 — 4.72 =D 41 —0. 973 —0. 180 +0. 051 —0. 289 — 2.18 =D 43 —0. 973 —0. 180 +0. 043 —0. 271 — 6.01 =Dm 38 —0. 970 —0. 189 +0. 021 —0. 235 — 5.35 =D 46 —0. 973 —0. 178 —0. 010 —0. 139 — 6.08 =Dm 36 —0. 970 —0. 189 —0. 044 —0. 072 — 5.08 =Dmu 45 —0. 973 —0. 179 —0. 046 —0. 045 — 6.12 =D 47 —0. 975 —0. 174 —0. 046 —0. 036 — 0.61 =D 49 —0. 979 —0. 159 —0. 018 —0. 083 — 4.99 =Dm 50 —0. 979 —0. 157 —0. 046 —0. 008 — 3.23 =Dm 51 —0. 983 —0. 144 —0. 066 +0. 067 + 1.72 =Dm 52 —0. 983 —0.141 —0. 009 —0. 080 — 0.66 =Dm 53 —0. 985 —0. 135 —0. 033 —0. 008 — 102 =Dm 54 —0. 988 +0. 121 —0. 084 +0. 150 — 7.93 =Dmu 55 —0. 989 —0. 113 —0. 055 +0. 080 — 173 =Dmu 56 —0. 989 —0. 116 —0. 030 +0. 008 — 810 =Dm 57 —0. 991 —0. 104 —0. 059 +0. 102 — 2.86 =Dm 58 —0. 994 —0, 088 —0. 032 +0. 044 — 2.51 =Dmu 59 —0. 994 —0. 085 —0. 032 +0. 046 — 1.05 =Dm 60 —0.995 ° —0. 080 —0. 024 +0. 028 — 2.39 =Dm 61 —0. 996 —0. 068 —0. 037 +0. 073 — 106 =Dm WESTERN GROUP—LONGITUDES. Station. i 246 | —0.172(6) —0.841() —0.135(a@w) +0.811(,8,) +1.034(10 000e2) + 1.89 =Dp 216 —0. 173 —0. 841 —0. 135 +0. 811 +1. 034 + 0.66 =Dp 243 —0. 189 —0. 828 —0. 115 +0. 857 +1. 094 + 1.57 =Dp 217 —0. 189 —0. 829 —0. 115 +0. 857 +1. 095 + 141 =Dp 9 —0, 238 —0. 795 —0. 070 +0. 999 +1. 279 — 0.21 =Dp 8 —0. 248 —0. 790 —0, 065 +1. 032 +1. 321 — 0.29 =Dp 7 —0. 248 —0. 790 —0. 066 +1. 033 +1, 322 — 0.12 =Dp 6 —0. 248 —0. 790 —0. 066 +1. 034 +1. 323 + 0.78 =Dp 1 —0. 267 —0. 778 —0. 052 +1. 085 +1. 388 + 1.29 =Dp 3 —0. 264 —0. 796 —0. 046 +1. 065 +1. 362 — 0.82 =Dp 16 —0. 243 —0. 782 —0. 050 +0. 993 +1. 270 + 0.18 =Dpr 10 —0. 247 —0. 776 —0, 040 +0. 997 +1. 275 + 0.49 =Dp 20 —0. 229 —0. 778 —0. 038 +0. 927 +1. 186 — 2.72 =Dp 23 —0, 229 —0. 777 —0, 037 +0. 923 +1. 181 — 6.15 =Dp 24 —0, 229 —0. 775 —0. 034 +0. 919 +1. 176 + 1.00 =Dp 25 —0. 228 —0. 774 —0. 031 +0. 914 +1.170 + 1.54 =Dp 22 —0. 233 —0. 771 —0. 029 +0. 929 +1. 188 — 2.27 =Dp 33 —0. 202 —0. 772 —0, 021 +0. 804 +1. 029 + 8.95 =Dp 31 —0. 190 —0. 772 —0. 018 +0. 757 +0. 968 + 0.51 =Dp 100 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Station. 41 44 36 50 53 55 61 Station. 245 244 231 230 234 242 229 228 227 226 224 —0 —0. —0. —0. —0 —0. —0 —0 —0 —0 —0 —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. —0. . 153(p) 151 154 127 “110 091 069 . 593() 592 . O97 . 599 605 628 639 651 665 652 644 660 657 655 633 640 659 658 665 663 642 642 646 660 663 668 664 657 653 642 643 634 633 577 586 580 585° 529 514 494 466 434 414 408 363 393 350 352 354 361 375 357 354 344 321 298 285 276 251 234 236 215 181 164 128 —0, 752(A) —0. 757 =O, 774 —0.777 —0.776 —0. 782 —0.779 Observation Equations—Continued, WESTERN GROUP—LONGITUDES—Continued. +0.021(aw) +0. 586(,4,) 13 +0, 582 +0.0 —0. 014 —0. 014 —0. 010 —0. 017 —0. 010 +0. 614 -L0. 507 +0. 437 -+0. 366 +40. 275 WESTERN GROUP—AZIMUTHS. + +1. 314 +1, 309 +1, 309 +1, 287 1. 279 +1, 288 +41, 285 +1, 271 +41, 258 +1, 266 +1, 255 +1, 245 41, 226 +1, 211 1. 210 +1. 198 +1. 193 +1. 181 +1. 190 +1. 172 -LY, 158 +1. 158 +1. 151 j 41. 132 +1, 114 +1. 128 +1. 132 1. 136 +11, 144 +1, 143 +1. 138 41,177 +1. 154 +1, 147. +1. 132 1. 145 1.170 +1. 150 +1. 161 +1, 213 +1, 166 LY. 137 +1, 139 +1. 143 LL. 144 41.151 +1, 154 HL. 148 41, 158 +1. 178 +1. 197 LI, 226 -L1, 192 +1, 222 LY. 197 +1.359(aw) +0. 814(,3,) 1. 354 +0. 815 +0. 844 +0. 849 +0. 856 +0. 900 +0.'919 -L0. 928 +0. 948 +0. 940 +0. 939 +40. 955 10. 957 +0. 962 +0, 945 +40. 965 +0, 991 +40, 999 +1. 010 41.017 40. 982 +0. 996 +1011 +1. 030 +1. 040 +1, 062 +1. 069 -L1. 050 +1. 041 +1, 022 +1. 018 +1, 006 +1, 008 +40. 901 +0. 930 +40, 927 40. 945 0. 854 +0. 817 40.799 +0. 751 +40. 675 +40. 669 +0. 676 +0. 604 +0. 586 +0. 581 +0. 581 +40, 582 +0, 596 +40. 613 0.576 +0. 563 +40. 535 +40, 513 40. 466 +40, 456 +40, 437 +40. 389 +0. 366 +0. 374 +40. 336 0, 284 259 +0. 749(10 000e?) +0. 744 +0. 785 +0. 649 +0. 560 +0. 468 +0, 352 +1. 039(10 000e?) +1. 039 +1. 078 +1. 084 +1. 094 +1. 151 41.175 +1, 186 LT, 212 +1, 202 +1. 201 +1, 222 +1, 224 +1, 230 +1, 209 +1, 236 +1, 268 -L1. 278 +1, 293 11. 302 +1, 258 41,275 +1294 41,318 +1. 330 +1, 358 +1. 367 +1, 343 +1, 332 +1, 308 +1, 302 +1, 287 +1, 290 +1, 153 +1. 189 +1. 186 +1, 208 +1, 093 +1, 045 +1, 022 -40. 960 +0. 864 +0. 856 +0. 864 +0. 773 +0. 749 +40. 743 +40. 743 +0. 744 +0. 762 +0. 784 +0. 737 +40. 720 +0. 684 +40. 657 -40. 597 +40. 584 +40. 560 +0. 498 +40. 468 +40. 478 +0. 429 +0. 364 +0. 331 +0. 259 LAEEEEEEFEFHE EEE SHEESH | ttt) 144444 [++ 1 +++ ea ore hs co~mkoOoNwpy Lone One + 4. ty tet F1t$ 1414444 tH HITICGCOO A ROMSH OTN ANA TON OTNMSNON WON SHOW SHON MS B® e me PD ROD SW 8 =Dp THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 101 WHY TWO CORRECTIONS TO THE INITIAL AZIMUTH. Thus far in this publication there has been no apparent recognition of the Laplace equa- tion connecting longitude and azimuth observations. The usual form of expression of this equation is (astronomic azimuth—geodetic azimuth) +sin ¢ (astronomic longitude—geodetic longitude) =0. In the present investigation the same principle may be expressed by the state- ment that the two observation equations, one a longitude equation and the other an azimuth equation, for a given station at which both the astronomic longitude and astronomic azimuth were observed, should show the same residual, or unexplained prime vertical component of the deflection of the vertical. Small differences between such pairs of residuals will exist on account of the errors of the astronomic observations and other accidental errors. If, however, there has been a considerable accumulation of error in the geodetic azimuth, as carried through the triangulation by the adjusted angles, it will be put in evidence by differences between the longi- tude and azimuth residuals at coincident stations too large to be accounted for by the accidental errors, and by a systematic tendency for such differences to be of one sign for considerable areas. There are 11 stations concerned in the present investigation at which a longitude station coincides with an azimuth station. A preliminary examination made in the manner just indicated showed that there had been, somewhere between Illinois and Colorado, in the trans- continental triangulation, an accumulation of about 5” of error in the geodetic azimuth, carried through the adjusted § angles. The evidence available was not sufficient to determine the places at which the accumulation occurred with any greater accuracy than indicated in the preceding sentence. The station already adopted as the initial, Meades Ranch, happened to be in the middle portion of this section within which it was apparent that the accumulation took place. Under these conditions it seemed that the simple device of introducing into the equations two unknowns, representing corrections to the initial azimuth at Meades Ranch, instead of one, would, with little additional work, take account automatically of the supposed twist in azimuth, and determine its amount substantially as well as any more complicated method. The required correction (a,) to the initial azimuth was introduced into all equations pertaining to stations east of Meades Ranch, and the required correction (@,,) into all equations pertaining to points west of Meades Ranch. This is equivalent to assuming that at Meades Ranch, in carrying the computation of the azimuth through the triangulation from east to west, an error of (a,) —(@,) was suddenly introduced into the geodetic azimuth at Meades Ranch. The necessary introduction of an extra unknown into the equations in the manner indi- cated, thus virtually inserting a hinge in. the triangulation midway between the Atlantic and the Pacific, has, of course, made the determination of the figure and size of the earth weaker than it otherwise would have been. The weakening has been properly taken into account in deriving the probable errors. These probable errors show this to be a very strong determi- nation in spite of this weakening. MEANINGS OF COEFFICIENTS IN OBSERVATION EQUATIONS. The meanings of the coefficients in the observation equations have already been explained in a general way (pp. 74,75). With the numerical values of the coefficients before one in the printed observation equations it is possible to make this explanation more definite and concrete. It is assumed in fixing the form of the observation equations that a small correction (¢) is to be applied to the initial geodetic latitude at Meades Ranch, Kansas, and it is required to express by the proper coefficients k,, k,, k,, the effect which this change would have on the apparent deflections of the vertical, such as are expressed in the absolute terms of the observa- tion equations. A similar treatment is to be given to the initial longitude, initial azimuth, the assumed equatorial radius, and the assumed flattening. The relations expressed by these coefficients may be seen by examining the formule from which the coefficients are computed (see p. 76). To visualize these relations and to obtain a more concrete conception of their meanings and laws of variation, imagine a model spheroid to be made to scale to represent the earth and imagine the 507 observation stations corre- 102. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. sponding to the observation equations to be plotted on the surface of this spheroid in their proper relative positions. Imagine also that a frame or cage is made to scale of wires so as to fit closely over this model spheroid, each meridian and each parallel of latitude being repre- sented by a wire. By the adoption in the computations of geodetic positions of certain values for the initial latitude, initial longitude, and initial azimuth at Meades Ranch, thus fixing the United States Standard Datum, and the adoption of the Clarke spheroid of 1866, the exact posi- tion of the supposed wire cage and its size and shape have been fixed. Imagine the wire cage to be in position on the model spheroid. Suppose, now, that without any other change being made the cage is so moved that the wire representing the meridian through Meades Ranch is moved southward by the amount (¢), without change of direction of this wire (which would correspond to a change of the initial azimuth) and without any lateral shift (which would correspond to a change of the initial longitude). Such a motion will consist, evidently, of a rotation of the cage about one of its diameters terminating in its equator at two points which are 90° from Meades Ranch in longitude. The geodetic latitude of Meades Ranch, read from the cage, will be increased by (¢). Evidently the geodetic latitude of any other point in the meridian of Meades Ranch will be increased by the same amount. So, also, it is evident that for other points on either side of this meridian, the change will be smaller the greater the distance from this meridian, to longitude differences as great as 90°. Note that, in the latitude observation equations, the coefficient k, is sensibly unity (—1.000) at latitude stations Nos. 63, 64, and 65 (p. 97), which are nearly in the same meridian as Meades Ranch, and that k, decreases numerically in proceeding either to the east- ward or westward. The minimum value of k, (—.854) occurs at latitude station 164, Calais, Maine (p. 94), and it is also small (—.900) at latitude station No. 252, Cape Mendocino, Cali- fornia (p. 99). The minus sign in the coefficients arises from the fact that — ¢’ enters in the observation equations, not ¢’ directly. The wire cage device gives one a good idea of the essential meaning of each coefficient and an approximate conception of its laws of variation. In this device it is not easy to take into account the small departures of the spheroid from the sphere, but said departures are accurately taken into account in the formule for the coefficients. The small displacement of the wire cage, as described, will evidently not change the longi- tude of any point on the spheroid which lies in the same meridian as Meades Ranch, and will not change the longitude of any point on the equator (because every meridian line will move across the equator parallel to that portion of itself). Virtually the meridians will be crowded more closely together everywhere in the United States as the north pole of the cage is made to approach Meades Ranch and all differences of longitude, reckoned from Meades Ranch, will be increased, the changes tending to be numerically greater the greater the longitude difference from Meades Ranch and the greater the latitude of the station. Hence, k,, expressing the relation between (¢) and 1’ cos ¢’ (see form of longitude equation, p. 74), changes sign through zero in the meridian of Meades Ranch, as indicated by the longitude observation equations for longitude stations 64 and 66, in Kansas (p. 98), has a maximum positive value (+.368) at longitude station 173, Calais, Maine (see p. 94), far to the eastward and in a high latitude, and has a maximum negative value (—.267) at longitude station No. 1, Point Arena, California . 99). o zie relations hold with respect to the coefficient k,, expressing the relation between (¢), the correction to the initial latitude, and @’cot¢’ ,except that on account of the factor cot¢’ there is a tendency for k, to increase with decrease of latitude. Between azimuth stations 65 and 67, in Kansas (p. 98), k, changes sign at the meridian of Meades Ranch; it has a maximum positive value (+.736) at azimuth station No. 170, Howard, Maine (p. 95), the azimuth station farthest to the eastward, although this station is in a high latitude, and has a maximum nega- tive value ( —.668) at azimuth station No. 237, Ross Mountain, California (p. 100). Let the wire cage be imagined to be again in its position as fixed by the adopted United States Standard Datum. Suppose, now, that, without any other change being made, the cage THE FIGURE OF THE RARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 103 is so moved that the wire representing the parallel of latitude through Meades Ranch is moved eastward, along itself, by the amount (4), without change of direction of the meridian through Meades Ranch (corresponding to a change of the initial azimuth), and without any lateral shift in the parallel of latitude (corresponding to a change in the initial latitude). Such a motion will consist, evidently, of a rotation of the wire cage about its polar axis through an angle (A). Such a motion will produce no change in latitude nor in azimuth at any point. Hence, the coefficients 1, and |, are zero for all stations. The longitudes of all points will be changed by the same amount (4). Hence, |,, expressing the relation between (4), the correction to the initial longitude, and — d’cos¢’, is simply —cos¢’. This coefficient, 1,, is therefore negative in all parts of the area covered by this investigation, is a maximum (—.866) at the longitude station which is farthest south, No. 141, New Orleans, Louisiana (p. 96), and a minimum (—.685) at the longitude station which is farthest north, No. 213, Minnesota Point North Base, Minnesota (p. 97). To shift the wire cage from its position as fixed by the United States Standard Datum in such a way as to correspond to a change (@) in the initial azimuth without any other change, one must evidently rotate the cage countet-clockwise about its diameter passing through Meades Ranch. This will decrease all latitudes to the eastward, and increase all latitudes to the west- ward of Meades Ranch, the change being greater the greater the distance and the difference of longitude from Meades Ranch. The coefficient m,, expressing the relation between (a) and — ¢’, is found, therefore, to pass through zero between latitude stations 63 and 64, in Kansas (p. 97), to have a positive maximum (+.404) at latitude’station No. 164, Calais, Maine (p. 94), this being the station farthest east and most distant from Meades Ranch, and to have a negative maximum ( —.339) at latitude station 252, Cape Mendocino, California (p. 99). So, too, by considering the motion of the wire cage described, it is evident that one should expect the coefficient m,, expressing the relation between (a) and — i’cos¢’, to have a positive maximum ( +.127) at the longitude station farthest north, No. 213, Minnesota Point North Base, Minnesota (p. 97), and to have a negative maximum (—.165) at the longitude station farthest south, No. 141, New Orleans, Louisiana (p. 96). The stations at which the values of m, should be zero are evidently those at which the arc of a great circle from Meades Ranch to the station is perpendicular to the meridian at the station, or, in other words, at which the azimuth of Meades Ranch from the station is either 90° or 270°. Such stations are all farther north than Meades Ranch, the excess of latitude being greater the greater the difference of longitude between the station and Meades Ranch. For these changes of sign, see page 94 for the north- eastern group, pages 97 and 98 for the central group, and pages 99 and 100 for the western group. Consult also illustration No. 13 at the end of the volume. Similarly, if one considers the motion of the wire cage corresponding to a change (a) in the initial azimuth, it is evident that the azimuth at all stations tends to change by amounts nearly equal to (a), but decreasing somewhat as the distance from Meades Ranch increases, since for points at 90° on a great circle from Meades Ranch the change becomes zero. There all wires of the cage move to new positions parallel to their old ones. On account of the factor cotd’ the coefficient m,, expressing the relation between (a) and @’cot¢’, tends to be smaller the greater the latitude. One should expect, therefore, that m, would vary nearly as cot¢’ but tend to be smaller for the more distant stations. The minimum value ( +.938) is found at azimuth station No. 171, Cooper, Maine (p. 95), though there are other azimuth stations, such as No. 205, Gar- gantua, Canada (p. 98), (for which m, = + 1.020), in much higher latitudes but nearer to Meades Ranch. The maximum value (+1.514) occurs at azimuth station No. 136, Fort Morgan, Ala- bama (p. 96). This is the azimuth station which is farthest south. An increase in a, the equatorial radius of the spheroid, without change of flattening and without any change in the initial latitude, longitude, or azimuth, is evidently expressed by an increase in size of the wire cage without change of shape or position. Such an increase in size would evidently tend to make all geodetic latitudes approach that of Meades Ranch. Hence, one should expect that n,, expressing the relation between (a3) and —9/, to have a iacimum positive value (+.508) for the latitude station which is farthest north, No. 232, St. Ignace, 104 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. Canada (p. 97), and a maximum negative value ( —.538) for the latitude station which is farthest south, No. 135, New Orleans, Louisiana (p. 96). The zero values of n, should be expected at stations at which the parallel of latitude points directly toward Meades Ranch; that is, at which the azimuth of Meades Ranch from the station is 90° or 270°. For the points at which n, changes sign, see page 94 for the northeastern group, page 97 for the central group, and page 99 for the western group. Similarly, from a consideration of the wire cage it is evident that an increase in the equa- torial radius would tend to make the longitude of all stations approach that of Meades Ranch. One should expect, therefore, n,, expressing the relation of (3) to — d’cos¢’, to pass through zero at the meridian of Meades Ranch (see longitude stations 64 and 66 in Kansas, p. 98), to be a negative maximum ( — 1.338) for the longitude station farthest to the northeast, No. 173, Calais, Maine (p. 94), and to be a positive maximum ( + 1.085) for the longitude station farthest to the west, No. 1, Point Arena, California (p. 99). The effect of a change in the equatorial radius upon the azimuth at any station is sing’ times the change in longitude produced at that station, the factor sing’ serving always to convert the difference of longitude of two meridians into their convergence. The change from the factor cos¢’, in the absolute term of the longitude observation equation, to the factor cot¢’, in the abso- lute term of the azimuth equation, neutralizes the factor sing’ and, therefore, the law of varia- tion of n,, expressing the relation between (aos a0 ) and a’cot¢’, is precisely the same as ee for n, Hence, it is found that Dy changes sign between azimuth stations Nos. 65 and 67, in Kansas (p. 98), has a negative maximum ( — 1.333) at the azimuth station farthest to the north- east, No. 170, Howard, Maine (p. 95), and has a positive maximum (+1.069) at the azimuth station farthest west, No. 2, Paxton, California (p. 100). An increase in the flattening of the spheroid without change of equatorial radius and with- out any change in the initial latitude, longitude, or azimuth, is evidently expressed by a corre- sponding flattening of the wire cage. But in this case, the conception of the wire cage will help one but little in forming a concrete idea of the meaning of the coefficients 0,, 0,,and0,. It may, however, aid one in recognizing the complexity of the relations which they express, for if one attempts to study the nature of the change of shape of the cage corresponding to an increase in the flattening it is soon evident that the radii of curvature of its different wires are all changed and the parallels of latitude are all shifted in position. Some radii of curvature are increased, as, for example, of given parallels of latitude near the equator and of arcs of the meridian near the pole, and some radii of curvature are decreased, as, for example, of arcs of the meridian near the equator. The laws of variation of 0,, 02, 0,, a8 determined by an inspection of their numerical values in the observation equations and checked by an examination of the formule from which these coefficients were computed, are here stated for comparison with the preceding paragraphs. The coefficient 0, depends mainly upon the latitude, but also tends to‘ have greater negative or smaller positive values the greater the departure in longitude from Meades Ranch in either direction. For the latitude station No. 232, St. Ignace, Canada (p. 97), which has the highest latitude of any station, 0, is —1.005, but this is slightly exceeded by 0, (—1.032) at latitude station No. 164, Calais, Maine (p. 94), the fact that this station is in a considerably lower latitude than No. 232, being offset by the fact that it is much farther east. The maximum positive value of 0, (+1.722) occurs at the latitude station which is farthest south, No. 135, New Orleans, Louisiana (p. 96). In the longitude of Meades Ranch the sign of 01 shenwes at the latitude of Meades Ranch (39° 13’), but for stations to the eastward or to the westward it changes sign at a lower latitude, the extreme case being at the Pacific coast, where it changes sign in latitude 38°01’. The coefficients 0, and o, follow nearly the same law of variation as the coefficients n, and n,, respectively. They have their positive maximum and negative maximum values at the same stations and change sign between the same stations as n, and n,;. The extreme values for o, are —1.685 at longitude station No. 173, Calais, Maine (p. 94), and +1.388 at longitude station THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. &8. 105 No. 1, Point Arena, California (p. 99) ; the extreme values for 0, are —1.683 at azimuth station No. 170, Howard, Maine (p. 95), and +1.367 at azimuth station No. 2, Paxton, California(p. 100). THE NORMAL EQUATIONS AND VALUES OF UNKNOWNS. The normal equations formed in the usual way from the observation equations for solution G are as follows: Normal equations.—Solution G. a +286.69(¢)— 3.32(2)-+ 17.63(a,)— 18.06(ay,)— 77.58( 49.) — 71.75(10 000e2)— 22.95=0 18.06()-+ 0.97(A) 17.63(¢)+ 0.28(2)-+137.29(a,) 3.32($)+46.75(2)-+ 0.28(a,)+ 0.97(ay)+ 12.03( 555 ) + 15.25(10 000e?)— 0.78=0 = 88.48( 595 ) —127.73(10 000e2)— 29.28=0 +101.61(a,,)+ 64.18( 396 ,) + 73.25(10 000¢?) +543.63=0 77.58(¢)+12.03(2)— 88.43(a,)-+ 64.13( ty) +185.15( 735 ) +187.82(10 000¢?)+-319.99=0 71.75( $)-+15.25(A)—127.73(a@,)-+ 73.25(cty)-+187.82( 85 ) +371.18(10 000e?)+-492.79=0 For convenience of comparison, the absolute terms of the normal equations for all five solutions are given here together. Absolute terms. | Solution B. Solution E. Solution H. Solution G. Solution A. —1738. 89 — 43.14. — 24.85 — 22.95 — 8.59 + 142.81 + 23.55 + 3.26 — 0.78 — 45. 87 +3962. 61 + 23.95 — 22.63 — 29. 28 —168. 10 —4292. 95 +385. 74 +519. 94 +543. 63 +964. 19 —8596. 18 + 93.85 +286. 97 +319. 99 +879. 34 —9861. 02 +274. 01 +461. 86 +492. 79 +998. 15 The values of the unknowns derived from the five solutions are given below. The solutions gave directly the values of (ga) and (10 000e”), but for convenience of reference there are given in these tables (a) and (e’), the corrections to the Clarke equatorial radius and square of the eccentricity. | ($) (A) (ap) (aw) (a) (e*) aw aw aw aw Meters. Solution B +21.04 | —16. 54 +6.10} +10.53]} +4 890 | +0. 000659 Solution E + .25) — .76 + .63| — 4.68} + 222; — .000064 Solution H — .16) — .O1 +.21} — 5.29} + 98]. — .000066 Solution G — .22); 4+ .183 +.13} — 5.40} + 76!) — .000065 Solution A — 1.20; + 1.85 — .638 | — 7.82} — 261) — .000035 Applying the corrections (a) and (e’) to the Clarke 1866 values, and converting the results into the more common forms of expression, there are obtained the following values expressing the figure and size of the earth. Equatorial radius. Reciprocal of flattening. Polar semi- diameter. Solution B (extreme rigidity) depth of compensation 162.2 kilometers) Solution H (depth of compensation 120.9 kilometers) depth of compensation 113.7 kilometers) depth of compensation zero) Solution E Solution G f Solution A 6 383 096 6 378 428 6 378 304 6 378 283 | i Meters. 6 3877 945 268. 7 297.7 297.9 297.8: 296. 5 Meters. 6 359 344 6 357 006 6 356 890 6 356 868 6 356 435 106 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. THE RESIDUALS. For the five solutions, the residuals corresponding to the quantities Dp and Dy in the second member of the observation equations, as indicated on page 74, are given in the following tables. These residuals are the portions of the deflections of the vertical which remain unexplained after each solution. The comparison of the residuals from different solutions furnishes the basis for a selection of the most probable solution. The residuals of the selected most probable solution contain the information as to the manner and extent to which the assumptions of that solution differ from the facts of nature. A plus sign on the residual at a latitude station for a given solution indicates that the zenith at that station is deflected to the northward of the position in which it would be if the assump- tions on which that solution is based were correct. Similarly a plus sign on the residual at a longitude station or an azimuth station for a given solution indicates that the zenith at that station is deflected to the westward of the position in which it would be if the assumptic ns on which that solution is based were a correct representation of all the facts. Residuals at latitude stations of northeastern group. Station. Solution B. Solution E. Solution H. Solution G. Solution A. uv aw uw ws wv 241 — 4.76 — .22 — .08 — .08 + .25 188 — 6.32 —1.02 — .92 — .91 —1.04 185 — 9.19 —3. 21 —3. 03 —3. 01 —2. 88 184 — 6.69 — .37 — .16 — .13 22 183 — 4.81 +1. 81 +2. 16 +2. 20 +3. 43 182 — 3.97 +2. 60 +3. 03 +3. 09 +5. 70 186 — 3.96 +2. 61 +3. 04 +3. 10 +5. 78 187 — 4.69 +1. 83 +2. 27 +2. 33 +5. 14 181 — 1738 +3. 25 +3. 68 +3. 76 +6. 75 180 — 107 +3. 41 +3. 89 +3. 99 +7. 52 179 — .63 +2. 95 +3. 36 +3. 44 +4. 54 178 — .66 +2. 80 +3. 23 +3. 31 +4, 97 177 — 1.02 +1. 70 +2. 20 +2. 30 +5. 26 176 — .1l7 + .81 +1. 09 +1.18 +3. 13 175 — 2.05 —2.00 —1. 82 —1.78 —1.30 174 — 1.41 —1. 69 —1. 42 —1.38 +1. 39 173 + 2.50 +1. 10 +1. 22 +1. 24 +1. 50 169 — 2.83 —1.14 — .76 — .68 +2. 85 170 + 2.98 —1.59 +1. 82 +1. 87 +4. 56 168 + 8.15 +1. 36 + .94 + .90 78 166 + 6.23 + .86 + .86 + .87 +2. 10 171 + 2.36 —2. 35 —2. 49 —2. 53 —4. 41 172 + 4.32 + .91 + .98 +1. 01 23 165 + 2.96 — .16 . 09 + .12 +1. 25 167 + 7.44 +3. 95 +4, 22 +4. 24 +4. 72 164 +17. 53 +5. 61 +5. 31 +5. 23 +4. 64 163 +13. 53 +1. 26 . 94 87 00 158 +18. 55 +4. 79 +4, 45 +4. 37 +3. 02 162 +10. 24 —1. 24 —1.59 —1.65 ~—2. 82 157 +14. 40 +2. 38 +1. 98 +1. 90 + .39 161 + 9.34 — .98 —1.31 —1. 38 —2. 33 159 +12. 45 +2. 36 +2. 04 +1. 98 + .94 156 +11. 62 + .45 + .l1 + .06 —1.08 160 + 9.56 —.01 — .41 —.61 —2. 53 155 +12. 41 +2. 21 +1. 87 +1. 80 + .64 153 +14. 42 +8. 14 +2. 81 +2. 74 +1. 27 152 +14. 42 +3. 82 +3. 46 +3. 40 +1.77 154 +12. 80 +3. 24 +2. 85 +2. 76 + .54 149 +11. 18 +1. 94 +1. 64 +1. 55 +1. 73 151 +13. 93 +3. 08 +2. 81 42.74 +1. 32 150 +11. 66 + .33 + .07 + .02 — .95 148 +11. 23 +1. 60 +1. 34 +1. 26 + .51 147 + 9.52 —2.14 —2. 36 —2. 38 —2. 85 144 + 7.70 —3.71 —3. 91 —3. 95 —4, 23 145 + 7.65 —3.77 —3. 97 —4, 01 —4, 35 142 +10. 71 —2. 67 —2. 86 —2. 90 —2.13 248 + 9.50 —4. 03 —4.19 —4, 22 —4. 06 146 + 7. 67 —2, 28 —2. 55 —2. 60 —2.75 143 +10. 04 + .63 + .29 + .20 —1.19 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 107 Residuals at latitude stations of northeastern growup—Continued. Solution H. Station. Solution B. Solution E. Solution G. Solution A. uw Ww “us uw Ww 141 +12. 12 +1. 32 + .90 + .82 —1.47 140 + 7.12 —3. 41 —3. 74 —3. 80 —4, 23 139 + 7.53 — .72 —1.00 —1. 03 —1.78 138 + 9.79 +2. 26 +2. 04 +2. 01 +2. 60 137 +11. 95 +4. 39 +4. 13 +4. 09 +3. 11 136 +14.18 +7. 58 +7. 32 +7. 29 +5. 83 107 — 3.07 —6. 50 —6. 49 —6, 48 —5. 97 109 + .63 —3.05 —3. 06 —3. 04 —2. 72 108 + 1.77 —1.51 —1.49 —1. 46 — .70 Residuals at longitude stations of northeastern group. 232 196 195 192 191 194 190 188 187 185 184 183 182 181 179 173 172 157 158 233 108 110 GOD OH BDO NYO ¢ oO featy Plier i dd | wo a> bd Ltd Bee Nrors so a —21. 07 —18. Ts — 7. —li. a 72 40 63 84 63 27 +L EEEEAALETHHHTHHH+ | , ©, Sr See ne le. oO = Residuals at azimuth stations of northeastern group. 193 186 189 180 175 176 174 177 178 171 170. 169 168 167 166 * 164 165 163 162 160 161 159 156 147 154 150 149 151 WW + 1.49 —13. 23 —20. 64 —14. 53 Vv +5. 06 —ll —13 —10 —19 —20 —21 —20 —22 —17 —20 . 84 . 16 . 83 . 46 . 29 . 69 . 59 . 64 45 . 46 —18.17 —15. 03 —16. — 8. —14. —16. FP, WOWN WAH 56 —4, =i. -11 —l. —3, —4, =. . 30 61 99 57 03 15 66 + .52 + .14 +5. 28 —4.15 —7.12 + .54 — .96 —2. 49 —3. 95 —7. 33 —6. 02 +1.14 + .79 —1.49 +3. 31 —1. 46 41103 45.54 4 +5. 31 —4. 06 —6. 95 + . 64 . 85 —2, —3. —/, 36 » N H FOU pe: 100% OURO Rs Pee ee OS Rn, NINN ot 20 a AHEEEEEHH | F+tH+4+4444+ bE T+ 108 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U.S. Residuals at azimuth stations of northeastern group—Continued. Station. Solution B. Solution ls. Solution H. Solution G. Solution A. ” u” ” ” ” 152 — .98 +6. 64 +6. 98 +7. 02 + 8.40 153 —10. 27 —1, 93 —1.59 —1.53 — .51 155 —11. 85 —3. 41 —3. 26 —3. 24 — 5.38 148 —11.39 —i. 94 —3. 76 —3. 72 — 3.00 146 —10. 74 —5. 59 —5. 45 —5, 41 — 7.58 145 — 4.08 —1, 64 —1.52 —1.50 — 1.10 144 + 2.35 +2. 48 +2. 53 4-2. 53 + 3.35 143 + 6.94 +6. 84 +6. 91 +6. 91 + 7.11 142 + 4.52 +2. 33 +4. 32 +2. 31 + 2.07 109 + 8.80 +1. 50 +1. 3i +1. 28 + 1.28 Residuals at latitude stations of southeastern group. 119 118 131 124 127 128 129 130 132 134 133 135 Fen i Sa hs Je Ri ON Oo es bo aS 6 a Pred tbeb bbb dee tttti bd bb tb batt tlt bl a RO ORNL ¢ 8 S a i ee eee » N 1] rH Sr ° © PRD oes me e RY Sr rs 69 Go's O's a> Qo AAD “ut acl +2, +5. aa —1. +5. 44, 48 64 96 16 19 53 43 40 . 81 . 90 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Residuals at longitude stations of southeastern group. Station. | Solution B. |. Solution E. Solution H. | Solution G. Solution A. aw oF ut Vf i 87 — 1.70 — .39 + ..17 27 +2. 76 114 + 8.35 +2. 85 +2. 36 +2. 27 —1, 33 90 +10. 59 +4. 44 +3. 96 +3. 85 + .57 92 + 5.46 +2. 06 +1. 65 +1. 55 —1.57 99 + 1.55 —1.77 —1.90 -1.91 —3. 10 100 + 2.37 = 112 —1. 24 —1. 25 —1.80 101 — .41 —3. 93 —4, 04 —4, 06 —3. 91 118 — .62 —4. 03 —4.15 —4.16 —4, 21 113 + 5.52 —1. 82 —1.97 —2. 00 —2. 46 119 + 8.99 +3. 24 +2. 96 +2. 92 +1. 83 130 + 3.71 + .83 + .90 + .92 +1. 22 123 + 1.09 + .15 + .48 + .37 + .77 131 + .47 —1.56 —1.34 —1. 46 —101 132 + .56 —1.93 —1.80 —1. 92 —1.78 137 + 7.43 +3. 64 +3. 65 +38. 53 +3, 21 141 + 7.99 + .30 + .10 — .05 — .64 Residuals at azimuth stations of southeastern group. 4 a M/ ‘/ a 85 — 6.10 —5.14 — 4.91 — 4.85 — 3.82 86 — 2.40 —1.52 — 112 — 1.05 + .31 88 — 144 —3.47 — 3.30 — 3.22 — 3.34 89 + 5.34 — .78 — 1.24 — 1.33 — 3.13 “91 + 9.79 +2. 67 + 2.22 + 2.13 + .93 94 + 7.49 +8. 12 + 2.93 + 2.90 + 2.59 95 + 8.58 +5. 22 + 5.01 + 4.98 + 4.09 96 + 9.05 +5. 65 + 5.52 + 5.50 + 6.54 102 — 1.70 —d.71 — 5.90 — 5.90 — 6.61 97 + 1.24 —3. 58 — 3.73 — 3.77 — 4.61 103 — 3.59 —8. 72 —- 8: 87 — 8.88 — 8.55 105 — 4.66 —9.91 —10. 02 —10. 08 —10. 23 104 — 3.10 —7. 53 — 7.66 — 7.68 — 8.07 107 — 4,30 —7. 25 — 7.27 — 7.28 — 7.02 235 +11. 27 2. 05 + 174 + 1.70 + 1.80. 106 — 2.16 —0d. 68 — 9.81 — 9.80 —10. 02 117 + 6.65 —3. 89 — 4,22 — 4.25 — 4.04 116 +15. 99 +3. 95 + 3.58 + 3.53 + 3.59 Ill +17. 45 +2. 95 + 2.38 + 2.31 + 1.88 115 +12, 41 —3. 84 — 4.62 — 4.61 — 6.15 93 +15. 47 +7. 62 + 7.17 + 7.09 + 6.15 120 +10. 87 +2. 60 + 2.15 + 2:04 + .62 121 7.05 —1.03 — 1.48 — 1.52 — 2.19 122 +10. 94 +3. 59 + 3.23 + 3.17 + 2.57 125 + 8.26 +2. 18 + 1.73 + 1.67 — .02 126 + 3.57 —1.48 — 1.74 -— 1.79 — 4.00 128 + 6.34 +2, 44 + 2,43 + 2.42 + 2.72 129 + 7.96 +3. 85 + 3.81 + 3.81 + 4.17 127 + 3.27 +1.91 + 1.98 + 2.01 + 1.30 124 — 1.38 —1. 36 == 1.17 — 1.12 — 12 138 + 6.57 +5. 52 + 5.59 + 5.60 + 6.23 133 + 2.19 ea 7 + .81 + .86 + 1.31 134 + 3.59 +1. 30 + 1.33 + 1.34 + 2.21 135 + 5.04 +2. 00 + 1.94 + 1.91 + 2.27 136 +11. 90 6. 85 + 6.56 + 6.50 + 6.28 139 + 7.18 +1. 47 + 1.16 +111 Se sc HO Residuals at latitude stations of central group. ys ee ur VE “SP 227 + .01 — 147 — 1.81 — 1.90 — 3.78 228 + 2,20 + .60 + .23 + .14 — 108 231 + 6.45 + 3.86 + 3.35 + 3.24 + 1.07 232 +16. 77 +13. 86 +13. 26 +13. 14 + 9. 50 230 +14, 77 +13. 56 +13. 11 +13. 02 +11. 48 229 + 1.16 + .43 - .B — .27 — 5.06 223 — 6.77 — 5.61 — 6.07 — 6.18 —10. 11 109 110 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. §. Residuals at latitude stations of. central group—Continued. Station. Solution B. Solution LC. | Solution H. Solution G. Solution A. tf a dt of Mt 219 — 2.81 — .53 — .86 — .98 — 3.66 213 + .63 + 3.24 + 2.95 + 2.90 + .42 212) + .42 + 3.34 + 3.12 + 3.08 + 1.98 211 + 1.27 + 4.40 + 4.27 + 4,23 + 4.18 214 — .97 + 2.03 + 1.94 + 1.91 + 2,23 221 — 4.95 — 2.60 — 2.65 — 2.67 — 2.60 216 — 2.91 + .42 + .65 + .69 + 2.79 226 — 7.26 — 6.41 — 6.43 — 6.45 — 5.65 225 — 3.06 — 2.26 — 2.30 — 2.32 — 2.09 224 —11. 43 —10. 43 —10. 46 —10. 48 —10. 18 222 — 6.29 — 3.05 — 2.91 — 2.90 — 2.12 220 = 2.97 — .49 — .33 — .32 + .20 217 + .40 + 3.05 + 3.24 + 3.28 + 4.65 218 + 6.46 + 8.78 + 8.98 + 9.02 +10. 85 189 + 3.15 + 5.26 + 5.36 + 5.40 + 6.84 191 + 3.25 + 5.30 + 5.40 + 5.44 + 6.77 215 + 1.60 - 1. 69 + 1.46 + 1.42 + .76 190 + 2.19 2. 65 + 2.49 + 2.465 + 2.06 194 + 1.33 + 1.69 + 1.49 + 1.45 + 1.62 192 + 2.11 + 2.29 + 2.05 + 2.01 + .89 193 — 1.06 — .05 = 22 — .26 — 1.09 197 — 4.35 — 2.68 = 2.77 — 2.80 — 3.58 196 — 5.68 — 4.04 — 4,23 — 4,29 — 5.60 210 — 6.27 — 4.46 — 4.65 — 4.70 — 5.92 195 — 6.75 — 4,87 — 5.05 — 6.11 — 6.51 198 — 5.99 — 3.72 — 3.92 — 3.96 — 5.14 200 — 4.16 — 2.06 — 2.23 — 2.27 — 3.17 199 — 4.55 — 2.53 — 2.70 — 2.74 — 3.39 201 — 2,24 — .05 — .20 — .25 — .91 202 — .92 + 1.57 + 1.43 + 1.38 + .41 203 + .68 + 3.18 + 3.04 + 2.99 + 2.09 204 — 2.48 + .94 + .85 + .83 + .47 205 — .83 + 4.17 + 4.19 + 4.19 + 3.98 206 — 2.39 + 3.91 + 4.14 + 4.14 + 4.96 209 — 4.52 + 2.26 + 2.41 + 2.44 + 2.87 207 — 5.18 + 1.87 + 2.06 + 2.06 + 2.62 247 — 7.03 + 1.47 + 1.73 + 1.78 + 2.34 208 — 7.30 + 1.30 + 1.55 + 1.59 + 2.05 62 —10.51 — .32 + .10 + .18 + 1.46 63 — 9.31 — .14 + .31 + .37 + 1.69 64 —10. 84 — 1.69 — 1.24 — 1.18 — .08 65 —13. 56 — 4.62 — 4.20 — 4.14 — 2.92 66 — 9.10 — .37 + .04 : + .10 + 1.97 67 — 8.76 — .15 + .21 SPs + 1.91 68 — 7.56 + 2.48 + 3.00 + 3.10 + 4.03 69 — 7.95 + 1.84 + 2.31 + 2.37 + 4.02 70 — 6.19 + 4.37 + 4.75 + 4.81 + 5.63 71 — 3.60 + 5.83 + 6.13 + 6.19 + 6.86 72 — 6.77 + 2.72 + 3.00 + 3.05 + 3.71 73 — 6.59 + 2.94 + 3.22 + 3.27 + 3.92 74 — 8.26 + 1.54 + 1.87 + 1.92 + 2.76 75 — 8.67 + 1.08 + 1.41 + 1.47 + 2.33 76 — 9.16 + .25 + .52 + .57 + 1.22 77 — 8.62 + .82 + 1.06 + 1.10 + 1.12 78 —10. 54 — .91 — .62 — .60 — .06 79 —10. 42 — .46 — .09 — .03 + .97 Residuals at longitude stations of central group. MT a“ wt a af 213 + 7.03 —3. 76 —4. 27 —4. 37 —7. 55 202 — 3.73 —7.13. —7.50 —7.57 —9. 67 198 — 9.30 —6. 22 —5. 96 —5. 92 —A4. 86 201 —114 —4,12 —4. 38 —4, 42 —5. 41 206 + .97 —3.16 —3. 48 —3. 53 —4. 73 207 + 6.09 +1. 21 + .93 + .87 + .01 210 + 2.18 —1. 88 —2. 02 —2. 04 —2.49 211 + 5.43 +1. 00 + .86 + .84 + .22 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 111 Residuals at longitude stations of central group—Continued. Station. Solution B. Solution E. Solution H. Solution G. Solution A, uw wt uw wt w 64 +36. 98 + .55 —1. 23 —1.54 6.07 , 66 +24. 18 —3. 74 —4, 92 —5. 12 —8. 12 69 +18. 88 — .27 — .88 — .99 —1.81 74 +11. 64 +2. 35 +2. 01 +1. 97 + .66 77 + 4.03 — .62 — .68 — .70 — .72 79 — .60 —4, 21 —4. 20 —4, 22 —3. 94 83 — 1.38 — .52 — .3l1 — .28 — .04 Residuals at azimuth stations of central group. at at a tt Mt 205 — 5.55 + .90 +1. 37 +1. 46 +4, 25 214 + 1.72 —4, 85 —5. 28 —5. 36 —8. 14 208 — 174 —7.74 —8. 11 —8. 16 —9.03 * 204 — 7.96 —9. 01 —9, 21 —9. 21 —9. 53 200 —11. 04 —5. 50 —5. 18 —5. 13 —3. 75 199 — 7.95 —3. 90 —3. 72 —3. 70 —2. 92 203 — 2.96 —2.97 —3. 16 —3. 18 —3. 83 209 + 4.74 +3. 13 +2. 90 +2. 86 +2. 45 215 + 7.16 +4. 45 +4, 31 +4. 27 +4. 50 212 ‘4+ 2.44 + .49 + .38 + .37 + .03 65 +39. 16 +3. 34 +1. 96 +1. 69 —3. 21 67 +28. 49 +7. 43 +6. 45 +6. 29 +4, 21 68 +18. 31 + .33 =, 37 — .49 —2. 33 70 +13. 41 +2. 22 +1. 81 +1. 76 + .96 71 + 7.13 —3. 01 —3. 41 —3. 46 —4, 25 72 + 4.50 —4.16 —4. 55 —4. 59 —5. 45 73 + 4.58 —2. 34 —2. 65 —2. 68 —3. 42 75 + 1.05 —4. 07 —4, 25 —4, 28 —4.13 76 + 2.59 a4, — .59 — .60 — .60 78 + 3.06 + .27 + .20 + .19 + .39 80 — 1.06 —1.73 —1. 60 —1. 60 — .60 81 — 2.08 —1. 67 —1. 48 —1. 48 — .50 82 + 1.07 +2. 02 +2. 12 +2.13 +2. 42 84 + 2.97 +4, 21 +4. 48 +4. 50 +45. 44 Residuals at latitude stations of western group. uy My “ a yt 240 + 8.82 — 2.66 — 3.02 — 3.12 — 5.29 239 +10. 51 — .85 — 121 — 1.31 — 2.40 267 +12. 78 + 1.22 + .83 + .73 +114 266 +20. 85 + 7.51 + 6.86 + 6.72 + 9.30 265 +22. 05 + 7.99 + 7.08 + 6.80 + 1.69 246 +22. 68 + 8.47 + 7.42 + 7.19 + 3.47 245 +10. 24 — 4.62 — 5.92 — 6.20 —14. 08 236 +19. 66 + 6.72 + 5.84 + 5. 66 + 8.00 237 +18. 05 + 4.95 + 3.96 + 3.76 + 2.12 264 +16. 97 + 2.18 + .95 + .68 — 6.92 238 +10. 21 — 3.66 — 4.78 — 5.04 —14. 34 235 + 9.94 — 2.73 — 3.68 — 3.88 — 8.47 262 +12. 25 — .70 — 174 — 1.96 — 9.08 234 +17. 38 + 5.10 + 4,21 + 4.03 — 1.06 243 + 6.46 — 4,95 — 5.53 — 5.64 — 7.10 242 +14. 90 + 2. 62 + 1.85 + 1.69 — 1.46 261 +14. 40 + 2.98 + 2.30 + 2.17 — 3.35 260 +16. 59 + 5.03 + 4,33 + 421 = 2.19 257 + 6.44 — 4.08 — 4.60 — 471 — 7.96 258 +10. 02 — 1.16 — 1.83 — 2.00 — 7.78 259 + 7.62 — 3.59 — 4.36 — 4.53 —10. 24 256 + 9.25 + 109 + 1.18 + 1.21 + 2.24 255 + 1.34 — 6.92 — 6.98 — 6.98 — 7.40 244 + 8.77 — 2.06 — 2.64 — 2.76 — 3.41 233 +17. 99 + 7.87 + 7.45 + 7.35 + 9. 28 254 +16. 65 + 5. 63 + 4.94 + 4.77 + 4,44 253 + 8 81 — .21 —_ .34 — .35 + .10 112 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Residuals at latitude stations of western group—Continued. Station. Solution B. Solution E. | Solution TI. Solution G. Solution A. Vt af af Ff “st 19 + 7.89 — 1.09 — 1.35 — 1.39 — 1.91 1. + 3.85 — 3.87 — 3.84 — 3.80 + 1.04 16 + 2.97 — 4.80 — 4.91 — 4.91 — 1.82 13 + 5.38 — 3.69 — 4.16 — 4,21 — 3.70 12 + 9. 24 + .15 — .34 — .40 + .50 10 + 9.84 + .72 de + .16 + 1.08 11 + 9.60 + .45 — .04 . 10 + .92 9 + 8.73 — 1.56 — 2.10 = 27 — 4.66 4 + 5. 44 — 3.96 — 4.69 — 4.82 6. 52 3 ‘+ 7. 86 2.12 2.91 — 3.06 — 5.58. 7 + 7. 88 — 2.16 — 2.98 — 3.15 — 6.48 HE +13. 37 + 2.42 + 1.25 + .99 4. 60 2 + 8.78 — .55 — 1.29 — 1.44 3. 8] 252 + 4,29 + 1.06 + 1.18 ae 22 — .72 251 + .26 — 2.21 — 2.05 — 2.04 + .05 249 — 1.17 — 1.35 — 1.26 — 1.22 — 1.32 5 + 9.90 + 1.62 + 1.00 + .88 — .20 6 + 8. 24 — .77 — 145 — 1.61 5. 79 8 + 6.43 — 1.99 — 2.52 — 2.63 — 6.02 14 + 7.82 + .37 + .04 — .02 .70 15 + 4.65 — 2.84 — 3.13 — 3.17 — 6.15 7 + 3.27 — 3.56 — 3.81 — 3.86 — 3.72 18 + 2.31 — 4,43 — 4.62 — 4.63 — 411 20 + 4.51 — 2.16 — 2.60 — 2.67 . — 3.47 24 + 6.18 + 1.19 + .99 + .95 + 45 23 + 3.84 + .15 + #.17 + .15 — .29 25 — .08 — 1.55 — 1.13 — 1.05 + 6.37 27 — 4.61 — 4.41 — 3.92 — 3.80 — 134 22 — .57 2. 06 — 1.84 — 1.80 + .80 26 — 4,43 — 3.27 — 2.66 — 2.50 + 1.83 28 — 9.86 — 6.24 — 4.49 — 4.37 + 40 29 — 4.00 3. 26 — 2.97 — 2.93 — 2.19 30 —12. 42 5. 07 4.14 — 3.95 — .23 31 —23. 74 —16. 47 —15. 61 —15. 45 —10. 40 32 — 6.37 — 8.72 — 9.01 — 9.06 — 9.60 33 + 1.18 + .04 — .10 — .13 — 1.31 37 — 9.03 — 4.06 — 3.35 — 3.22 + .17 48 — 6.60 — .55 .O1 + .12 +10. 20 35 —12. 33 — 3.80 — 3.13 — 2.98 — .82 34 —13.05 — 2,41 — 1.99 — 191 — 6.57 39 —13. 10 — 2.14 — 1.86 — 1.84 — 3.33 42 —12. 99 — 2.15 — 1.76 — 172 — 1.69 44 —11. 51 — .66 — .26 AP, OD: + .96 40 —14. 28 — 3.80 — 3.35 — 3.28 — 131 41 —12.03 — 137 — .86 — .77 + .27 43 —15. 46 — 5.17 — 471 — 4.61 — 6.46 38 —15. 00 4.69 — 4.06 — 3.95 — 1.07 46 —15. 03 — 6.51 — 4.94 — 4,82 — 3.57 36 —13. 52 4.78 — 3.99 — 3.83 + .07 45 —14. 87 — 5.88 — 6.13 — 4,94 — 2.88 47 — 7.69 . 05 + .45 + .53 — .94 49 —10. 02 — 3.59 — 3.83 — 3.88 —11. 03 50 — 8.44 — 2.11 — 2.15 — 2.20 — 4.12 51 — 7.88 + 1.55 + 2.47 + 2.62 +10. 87 52 — 8.83 + .33 + .39 + .36 — 6.12. 53 —11. 43 — .88 — .20 — .09 + 6.15 54 —18.07 — 8.32 — 7.42 — 7,22 — .09 55 —11. 90 — 1.56 — 1.08 — 1.00 — .51 56 —20.79 — 8.16 — 7.42 — 7.28 — 1.69 57 —12. 42 — 2.48 — 2.21 — 2.19 — 2.74 58 —13. 39 2:97. — 1.91 1. 87 + 2.91 59 —11. 74 — .71 45 — .48 — 1.33 60 —13. 88 — 2.19 — 181 — 177 — 4.01 61 —10. 84 — .74 — .56 — .55 — 1.62 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. Residuals at longitude stations of western group. Station. | Solution B. Solution E. Solution II. Solution G. Solution A. za a “ “i uf 246 — 7.15 +1. 42 +2. 29 +2. 49 + 9.40 216 — 8.37 + .19 +1. 06 +1. 26 + 7.96 243 — 3.64 +2. 00 +2. 06 +2. 07 + 2.66 217 — 3.87 +1. 83 +1. 89 +1. 90 + 2.74 9 —13. 16 — .59 — .04 + .04 + 6.36 8 —13. 92 — .90 — .18 — .06 + 23 <é —13. 96 —.79 — .0ol + .11 + .69 6 —13. 03 + .12 + .88 +1.01 + 1.70 1 —18. 75 —1.12 +1. 00 +1. 45 +12. 95 3 —138. 66 —1.53 — .80 — .69 + .15 16 —10. 85 . 00 + .30 + .33 + .37 10 —11. 38 + .05 + .51 + .59 + 1.90 20 —10. 08 —2.71 —2.61 —2. 63 + .08 23 —11.10 —5. 60 —5. 96 —6. 07 —22. 35 24 — 1.93 +1. 91 +1. 24 +1. 07 — 7.14 25 + .37 +2. 65 +1. 78 +1. 59 — 9.82 22 — 5.41 —1. 67 —2.13 —2, 23 — 7.81 33 +15. 77 +9. 96 +9. 13 +8. 95 + 8.12 31 +10. 42 +1. 66 + .69 + .49 + 1.95 41 +11. 56 +2. 67 +1. 88 +1. 98 +10. 09 44 + 9.72 + .54 + .96 +1. 09 +12. 27 36 +17. 57 +6. 68 +6. 00 +5. 86 + 5.52 50 +14. 46 — .41 —1.36 —1.54 — 4.75 53 +16. 86 +1. 70 +1. 46 +1. 41 — 2.08 55 +24. 60 +2. 50 +1. 82 +1. 74 .77 61 +37. 14 —1. 38 —4,31 —4, 89 —20. 86 Residuals at azimuth stations of western group. at SE tt Mt Mt 245 + 6.28 +14. 89 +15. 74° +15. 94 +20. 03 244 + .88 + 8.54 + 9.24 + 9.41 +15. 69 231 + 3.55 + 8.58 + 8.71 + 8.73 + 9.41 230 + 1.46 + 6.78 + 6.91 + 6.94 + 7.45 234 — 7.89 — 194 — 1.85 — 1.82 — 2.37 242 — 1.63 + 5.42 + 5.52 + 5.55 + 7.32 229 — 6.00 + 3.05 + 3.41 + 3.45 + 2.99 228 — 3.64 + 5. 28 + 5.61 + 5.63 + 7.58 227 — 5.77 + 4.45 + 5.26 + 5.43 + 8.47 226 — 9.83 + .77 + 1.38 + 1.50 + 2. 64 224 — 7.10 + 4.63 + 5.29 + 5.40 + 8.21 225 —15. 99 — 3.22 — 2.09 — 1.85 + 3.87 223 —17. 45 — 3.90 — 2.85 — 2.64 + 2.90 241 — .03 +13. 14 +14. 13 +14. 34 +18. 04 222 —11. 98 — 1.36 — 1.40 — 1.45 — 4.08 220 —11. 09 + .90 + 1.15 + 1.21 + 2.49 221 —26. 05 —10. 21 — 8.96 — 871 — 5.14 219 —22.93 — 7.61 — 6.59 — 6.38 — 5.30 239 —32. 98 —14. 86 —13. 31 —12. 98 — 6.10 240 —16. 40 + .29 + 1.50 + 1.72 + 2.49 218 —17.31 — 3.68 — 3.30 — 3.25 — 1.29 17 —21, 21 — 7.61 — 7.16 — 7.12 — 5.81 13 —18. 81 — 6.11 — 4,88 — 4,87 — 7.67 238 —31. 48 —15. 89 —15. 12 —14. 99 —15. 88 5 —20. 16 — 4,32 — 3.47 — 3.34 — 5.71 237 —13. 73 + 2.59 + 3.76 + 3.98 + 6.88 2 —19. 75 — 3.46 — 2.42 — 2.25 — .28 236 — 8.12 + 6.24 + 6.76 + 6.82 + 7.58 4 — 9.73 + 3.61 + 3.77 + 3.78 + 1.07 11 —15. 14 — 3.31 — 3.48 — 3.57 — 8.32 12 —18.01 — 5.37 — 5.39 — 5.48 —10. 32 15 —13. 60 — .8l — .76 — .78 — 3.88 14 —11. 88 — .03 — .il — .16 — 3.90 78771—09——8 1138 114 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Residuals at azimuth stations of western group—Continued. Station. Solution B. Solution E. Solution H. Solution G. Solution A. | | wy wy ” uy u , 26 —15. 00 — 6.15 — 5.85 — 5.80 — 1.138 » 19 —21.30 — 9.34 — 8.62 — 8.48 — 7.02 21 —10. 94 — 2,22 — 2.16 — 2.18 — 1.69 18 —21.66 —12. 23 —11.76 —11. 67 — 9.75 i —10. 60 —10.18 —10. 47 —10. 51 — 7.07 28 + 5.24 + 1.78 + .93 + .80 — 2.78 29 + 5.15 — .8 — 1.79 — 1.97 — 5.36 30 + 6.57 — 2.82 — 3.90 — 4.11 — 6.87 34 +10. 20 + .78 — .35 — .56 — 3.60 35 +19. 66 + 4.41 + 2.66 + 2.31 — 4.39 32 +21. 36 + 4.79 + 2.93 + 2.54 — 7.64 38 + 7.10 — 5.99 — 6.54 — 6.64 —12. 73 42 + 9.73 — 1.21 — 1.01 — .94 + 5.94 40 + 5.01 — 5.70 — 5.41 — 5.33 + 2.68 43 + 8.15 — 1.93 — 1.49 — 1.34 +14. 48 45 + 7.73 — 2.65 — 2.26 — 2.15 + 7.77 39 + 8.79 — 2.95 — 3.11 — 3.11 — 2.39 37 +18. 06 + 4. 66 + 3.90 + 3.75 + 2.21 46 +14. 58 + 2.96 + 2.85 + 2.86 + 8.14 47 +16. 87 + 2.00 + 1.06 + .89 — 1.92 48 +17. 83 + 1.09 — .46 — .80 — 5.47 49 +18. 47 + 1.18 + 1.16 + = .95 — .01 dl +16. 88 + 2.12 + 2.03 + 2.03 + 4.06 52 +16. 44 — 2.04 — 2.68 — 2.79 — 5.89 54 +18. 88 + .81 + .47 + .41 — 3.68 57 +19. 02 — 2.66 — 3.01 — 3.03 — 2.96 56 +30. 06 + 4.30 + 3.51 + 3.41 + .90 58 +23. 21 — .20 — .49 — .A7 + 3.54 59 +39. 94 + 6.66 + 4.71 + 4.31 — 9.48 60 +43. 84 + 1.82 — 119 — 1.78 —20. 04 62 +42. 06 —114 — 3.85 — 4,37 —14. 62 63 +39. 36 — 2.75 — 4.85 — 56.23 —11. 49 REASONS FOR ADOPTION OF SOLUTION G. The sums of the squares of the residuals of the different solutions were as follows: Solution: B (extreme TIGIMItY) sc. css ceeeieeele docs etude dees se Dhuedammud Bre eke emeeeeie es da. ca eaedlecand 65 434 Solution E (depth of compensation 162.2 kilometers) ............--..- 22-22-2222 200 eee eee ee eee 8 220 Solution H (depth of compensation 120.9 kilometers) ............------.------+--- seb dle) eRe leis te Wel ea aba 8 020 Solution G (depth of compensation 113.7 kilometers) ..................----- Semank syed eemenes ys ia, Gaasaees 8 013 Solution. A (depth, of compensation Zera) .c-s.02scsea ssa secvase es ssts semzeensad és esse cepeedens yess pee ues 13 922 Solution G, having the smallest sum of the squares of residuals, is probably the closest approximation to the truth. Solution B is evidently far from the truth. The sum of the squares of the residuals is more than eight times as large as for solution G. This comparison constitutes a very strong proof that the assumption of an isostatic compensation which is complete and uniformly dis- tributed within the depth 113.7 kilometers is a very much closer approximation to the truth than the assumption of extreme rigidity. The very wide departure of the equatorial radius, flattening, and polar semidiameter, as derived from solution B (see page 105), from all previous derived values is in itself a strong indication that the assumption of extreme rigidity is far from the truth. In fact, observations 1 > 368.77 “An of gravity show positively that the value of the flattening given by solution B not exist. Solution A is also evidently a considerably wider departure from the truth than solution G. The introduction of the assumption of complete isostatic compensation uniformly distributed within the depth 113.7 kilometers in the place of the assumption that no relation exists between deflections of the vertical and topography, has reduced the sum of the squares of the residuals from 13 922 to 8013. In other words, the introduction of the assumption of isostasy in a THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 115 definite and reasonable form has eliminated 42 per cent from the sum of the squares of the residuals. Solution G is apparently a closer approximation to the truth than solution E. Solution G is apparently slightly nearer the truth than solution H, but there is little basis here shown for a choice between these two. The following table shows other means of comparison among the five solutions: Solution B. | Solution E. | Solution I. Solution G. | Solution A. | Maximum residual +43.84 | —16.47 | +15.74 |} +415.94 | —22.35 Percentage of residuals greater than 5”.00 66 18 18 18 29 Percentage of residuals Jess than 2”.00 15 41 43 43 35 Mean residual, without regard to sign 8. 86 3. 06 3. 04 3. 04 3. 92 These comparisons confirm the statements already made, based on the sums of the squares of the residuals. ‘ The results of the five solutions make it clear that if the assumed depth of compensation is made to vary from infinity to zero the sum of the squares of the residuals decreases from 65 434 for the assumed depth infinity to a minimum value of about 8 010 for some assumed depth not differing greatly from 114 kilometers, and then increases again to 13 922 as the assumed depth is decreased to zero. The depth for which the sum of the squares of the residuals would be a minimum is the ideal most probable depth of compensation. To ascertain this ideal most probable depth. with great accuracy is not important, for two reasons: (a) It is evident from a comparison of the sums of the squares of the residuals for solutions E, H, and G that the ideal minimum sum, when ascertained, would be found to be very little less than 8 013, the sum corresponding to solution G, and that therefore the corre- - sponding solution would be a very slight improvement on solution G. (b) It-is evident from a comparison of the values of the equatorial radius, flattening, and polar semidiameter, as derived from solutions G and H, that a change in the depth of compensation adopted as most probable introduces but little change into the corresponding most probable values for the equatorial radius, flattening, and polar semidiameter. From the residuals of solutions E, H, and G, the conclusion was reached that the most probable depth of compensation is 112.9 kilometers, which agrees so closely with the depth used in solution G, 113.7 kilometers, that it is not certain that solution G can be improved upon. The approximate process by which the value 112.9 was derived is easy of application. But the explanation of it is so long that it is deemed best not to insert it here because of the break in the continuity which would result. It is given later in this publication with the dis- cussion of various subsidiary questions. ADOPTED VALUES. For these reasons solution G is adopted as the most probable solution. This fixes upon the following values as the most probable which can be derived at present from observations in the United States: ° / “4 a Latitude-of Meadés Ranch). o.c.c00.5 suis cages itacunadtciwnrmcatetemienned oerwoice ene nae 39 13 26.47 +0.17 Longitude of Meades Ranch..............---22+-222++022+00+ tptantea asec 22g Wee es exes 98 32 30.64 +0. 40 Azimuth of line Meades Ranch to Waldo, to be used in computations extending eastward from Meat es (Ranch. 252s 2's sete whiue oa astnc bes Coenen Deeenedase faa hier ee ecaasoneeasae 75 28 14.65 40.32 Azimuth of line Meades Ranch to Waldo, to be used in 1 computations extending westward from Meéades Ranch «..o2. scene ssceseves vee dewesd idee amees ves svaloeded: ceceuegsseetee de ee. cee 75 28 09.12 40.33 Equatorial radius of the earth, meters, 6 378 28334. Reciprocal of flattening, 297.8.0.9. Polar semidiameter, meters, 6 356 868. 116 THE FIGURE-OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. The probable errors were computed in the usual way. They are somewhat too small, as they are based upon the assumption that the residuals are all accidental in character and independent of each other. It is evident from an examination of the residuals of solution G, as shown on pages 106-114, that there is a sufficient tendency to geographic grouping of residuals of like sign to indicate that the residuals are not entirely independent of each other. It is extremely difficult to estimate the amount by which the computed probable errors should be increased to represent the real uncertainties in the values. It is believed, however, that the real probable errors are not very much larger than those shown above. The United States Standard Datum is evidently a very near approach to the ideal. The derived correction to the initial latitude () is but little greater than its own probable error, and the derived corrections to the initial longitude (A) and to the initial azimuth for compu- tations extending eastward (a) are much smaller than their respective probable errors. The sign of each of these three required corrections is therefore still in doubt. For the correction (aw) to the initial azimuth for computations cxtending westward, the relatively large value —5’’.40 is in accordance with the preliminary investigation made, which indicated an accumulated error in geodetic azimuth in the transcontinental triangulation somewhere between Illinois and Colorado. As stated on page 101, the preliminary investiga- tion indicated this accumulated error to be about 5”. Solution G makes it 5’’.53, (@w)—-(@z). ERRORS DUE TO ALL CAUSES. Thus far this publication has consisted, in the main, of a statement of the data used and of the methods of investigation and computation, accompanied by barely sufficient illustration and explanation to enable the reader to understand what was done and, in a general way only, to understand why it was done. In the pages which follow will be found a systematic discus- sion of the accuracy of the data used and of the accuracy of the various parts of the computation. In connection with this discussion the reasons for various features of the methods of computa- tion will appear. This separation of the bare statement of what was done, on the one hand, from the discussion of accuracy and of reasons for methods, on the other hand, has two advan- tages. It enables the writer to secure greater continuity in the statement of what was done, with consequent gain in clearness; it also enables him to make the discussions of accuracy and of reasons for methods, more concrete than would otherwise be possible, as by virtue of the separation they are, in form, discussions of matters already presented, of something concrete already done, rather than of something abstract to be done. Because the methods of this investigation are unusual, it is especially important to consider carefully the accuracy of each step of the process and to indicate the reasons for decisions as to methods. Moreover, as a definite and sustained effort has been made to keep the economics of the problem in view and to simplify and shorten each step of the process as much as possible without appreciable loss of ultimate accuracy, it is especially important, on the one hand, to ascertain whether this has been carried too far and, on the other hand, to ascertain whether it is advisable in any future investigation to simplify and shorten still more. The combined errors due to all causes are represented by the residuals in the observation equations. The residuals of the adopted solution—G—will be taken as the basis of the following discussion. As the residuals in this solution are smaller on an average than in any of the other four solutions, it sets a higher standard and furnishes a more severe test of the methods than any of the other solutions. As already stated (p. 115), in solution G the mean residual without regard to sign is 3’.04, 43 per cent of the residuals are less than 2’.00, 18 per cent are greater than 5’’.00, and the maximum residual is +15’’.94. For the residuals of solution G, considered in three separate classes, the principal statistics llows: are as fo Mean without re- gard to sign. Maximum. Wore wsidielsvswcsucte ies eteeece. OG —15/7.45 For A residuals ___....._...-------------- 2 .34 + 8 .95 Fora residuals _____...-...-------------- 3.85 +15 .94 f THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. 114 EFFECTS OF ERRORS IN ASTRONOMIC OBSERVATIONS. A summary of the latitude observations for each of 109 of the latitude stations involved in this investigation has been published.* These may fairly be taken as representative of the 265 latitude stations involved in this investigation, at 261 of which the observations were made with zenith telescopes. At the 4 remaining stations the observations were made with Airy’s zenith sector. Of these 109 latitudes there are only 29 for which the probable error of the derived latitude is as great as +0’’.10, the largest probable error is +0”.66,} and the average probable error is +0’.09. The average error of observation of an astronomic latitude without regard to sign is probably 0’’.11,{ or one twenty-fifth of the average residual without regard to sign for latitude stations in solution G, namely, 2’’.76. It is reasonably certain that the error of no astronomic latitude among these 109 is greater than 2’’.31,§ which is but little more than one-seventh of the maximum latitude residual in solution G. These statements include the errors due to inaccuracies in the declinations of the stars observed. There are many astronomic latitudes used in this investigation to which no correction has been applied for variation of the pole. Out of 109 such corrections in the list above referred to, 60 are less than 0’’.10, and the maximum is +0’.27. Hence the fact that such corrections have been omitted at many stations is of little significance in the present investigation. The omitted corrections constitute errors of the accidental class, which have evidently contributed but a small percentage of the mean residual, 2.76. The astronomic longitude of Fort Gratiot, Michigan, No. 232, was determined by trans- portation of chronometers between that point and Detroit, Michigan. All of the other astro- nomic longitudes, including Detroit, were determined by the telegraphic method. Many of the longitude stations are in the longitude net of the United States, and the remainder are each connected with one or more stations of the net. The probable error of the average longi- tude determination in the telegraphic net is +0°%.024= +0’.36.|| This probable error is derived from the adjustment, and includes the systematic error peculiar to each station and not exhibited in the probable errors computed from the discrepancies between determinations of the same longitude difference on various nights. For many of the longitude determinations ° concerned in this investigation but not included in the telegraphic net, summaries of the results, showing their accuracy, have been published.{/ Any error of the astronomic longitudes which is common to all the stations does not affect the accuracy of the results obtained from the present investigation, since this investigation is concerned simply with differences of longitude between points connected by continuous triangulation. The mean error made in transferring the longitude from Greenwich across the Atlantic, therefore, has no effect on this investigation. On account of the complicated manner in which the various longitude determinations are interrelated in the longitude net and with other determinations connected with it, to determine the probable error of each derived longitude accurately would require too great an expenditure of time to be warranted by the value of the result. Accordingly, an approximate computation of the probable errors of the differences of longitude between Kansas City, a point in the central *See The Transcontinental Triangulation, Special Publication No. 4, Coast and Geodetic Survey, pp. 626-737. Others of the astronomic latitudes used in this investigation are also published in a similar manner in The Eastern Oblique Arc, Special Publication No. 7, Coast and Geodetic Survey, pp. 253-316. {This large probable error occurred at latitude station No. 13, Washington Square, San Francisco. There is no other probable error among the 109 greater than +0’’.23. t According to the theory of errors of observation the average error without regard to sign is 1.2 times the probable error. Thus, if the probable error is +0’’.09, the average error without regard to sign is 0’’.11. §In this and in various other parts of this investigation, it isassumed to be reasonably certain that the error does not in any case exceed 3.5 times the maximum probable error of any observation. According to the theory of errors, an error as great as 3.5 times the probable error should occur on an average once in 55 times. In the above case it isstated as reasonably certain that the maximum error is not greater than 2/”.31, which is 3.5 times the maximum probable error, +0’’.66. || See p. 255 of The Telegraphic Longitude Net of the United States and its Connection with that of Europe, by C. A. Schott, Appendix 2, Coast and Geodetic Survey Report for 1897. {See the Transcontinental Triangulation, pp. 807-826, and The Eastern Oblique Arc, pp. 317-326. 118 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. portion of the longitude net, and each of ten stations selected at random out of the 79 concerned in this investigation, has been made. For these ten stations the probable error varied from +0°.010 to +0°.124, with a mean of +0%.036. Hence the average error, without regard to sign of the longitudes as referred to Kansas City, is probably about 0°.043. Reducing this to are and multiplying by cos¢’ for latitude 39°, this being the mean latitude of the longitude sta- tions, there is obtained for the average error in the apparent prime vertical deflection at longitude stations 0’’.50, which is slightly more than one-fifth of the average residual without regard to sign for longitude stations in solution G. It is reasonably certain that the maximum error of any observed astronomic longitude is not greater than 0°.43, which corresponds to a maximum error in the apparent prime vertical deflection of 5’’.0, which is somewhat more than one-half of the maximum residual for longitude stations in solution G, namely, +8’’.95. For 73 of the 163 azimuth stations involved in this investigation, a summary for each station has been published.* These may fairly be taken as representative of the whole 163 stations. Of the 73 there are only 12 for which the probable error of the result is greater than +0’’.30, the largest probable error is +0’.79, and the average probable error is +0’.23. The observations were all made upon stars with a variety of theodolites, some repeating theod- olites and some direction theodolites. The average error of the observed astronomic azimuth, without regard to sign, is probably about 6’’.28. This, reduced to a prime vertical compo- nent of the deflection of the vertical, by multiplying by 1.23, the factor cot ¢’ for the assumed mean latitude, 39°, is 0’’.34, or less than one-eleventh of the average residual without regard to sign for azimuth stations in solution G, namely, 3.85. It is reasonably certain that the error of no observed astronomic azimuth among these 73 is greater than 2.77, corresponding to a prime vertical deflection of 3’.48 at that station; which is less than one-fourth of the maximum azimuth residual in solution G, namely, +15/’.94. From the preceding three paragraphs it appears that but a small portion, certainly less than one-tenth part on an average, of the residuals of the latitude and azimuth observation equations in solution G is due to errors in the astronomic determinations. The errors in the observed astronomic longitudes are relatively larger—probably furnishing about one-fifth part of the residuals of the longitude observation equations in solution G. Even in connection with the residuals of the longitude equations it is evident that the errors of the astronomic observa- tions have but a minor, though not an insignificant, part. The portion of the residuals produced by errors of the observed astronomic latitudes should have no relation to the geographic distribution of the stations. On the other hand, there should be a slight tendency for residuals produced in part by errors in the observed astronomic longitudes to be greater for extreme eastern or extreme western stations than for stations in the middle part of the country. Also, there should be a slight tendency for the effects of errors in the astronomic longitudes to be greater in the south- ern part of the country than in the northern part, on account of the variation of the factor cos ¢’ from .866 for the extreme southern station (No. 141) to .685 for the extreme northern station (No. 213). The errors in the observed astronomic azimuths have no relation to geographic distribu- tion, but their effects in producing residuals should tend to be considerably greater for southern than for northern stations on account of the variation of the factor cot ¢’ from 1.72 at the extreme southern station (No. 136) to .91 at the extreme northern station (No. 205). No such laws of distribution are observable in the residuals. EFFECTS OF ERRORS IN DISTANCES. The errors in the computed lengths of the lines joining triangulation stations depend upon the errors in the angle measurements and the errors in the measurements of base lines. The errors in length due to the second cause are much smaller than those due to the first. * See The Transcontinental Triangulation, pp. 743-801. Similar summaries for 56 astronomic stations also involved in this investigation are published in The Eastern Oblique Arc, pp. 328-366, 375-376. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 119 To make a careful estimate in detail of the effect of the errors in length upon the present investigation would be a complicated, difficult, and tedious matter. The present purpose will be served by a brief consideration of the more important principles and facts concerned, from which it will appear that errors in length, that is, in the computed distances between triangula- tion stations, contribute a minor part only to the errors of the final results of this investigation. The laws of accumulation of error in triangulation are such that the error of distance, expressed in meters, between any two triangulation stations tends to be proportional to the square root of the distance between the stations. The error in distance, when expressed as a proportional part of the distance, therefore, tends to vary inversely as the square root of the distance. The transcontinental triangulation may be taken as typical of all the triangulation involved in this investigation. The probable errors of length in the various parts of this triangulation have been computed and published.* The largest probable error in length, when expressed as a proportional part, occurs between the line Hortley-Keut: in Maryland, and the line joining Cape May and Cape Henlopen light-houses, at the eastern end of the triangulation. The probable error of length of this section of the triangulation is +1.1 meters on 56 kilometers, or one part in 50 000. This probable error corresponds to a probable error of the difference of longitude between the two ends of the section of only +’.05. In such a section of triangula- tion, lying along a meridian, the probable error of the difference of latitude of the two ends produced by such an error in length would be only +’’.04. Of the 11 sections into which the transcontinental triangulation is divided by bases, there are only two in which the probable error of length exceeds one part in 100 000. Combining the probable errors of the various sections of the transcontinental triangula- tion as published, by taking the square root of the sum of their squares, it becomes evident that the probable error of the distance from the initial station Meades Ranch to either the eastern or the western end of the triangulation is only +6 meters, corresponding to a probable error in the difference of longitude of +’’.25, or toa probable error of a difference of latitude of +’’.20 in a similar triangulation lying along a meridian. It must be kept in mind, also, that the effect on differences of longitude is somewhat reduced in its effect on apparent deflections of the vertical by multiplication by the factor cos ¢’. A comparison of these small possible effects of errors of length upon the apparent deflec- tions of the vertical at latitude and longitude stations with the average residual of more than 2” (see p. 116) which has been found in this investigation, shows that the errors of length have but a small part in producing those residuals. The average residual would be reduced by less than one-tenth part if absolutely exact lengths were substituted for the best lengths now available. The effects upon the apparent deflections of the vertical of errors in the computed geodetic azimuths as produced by errors of length are of the same magnitude as the similar effects produced through the computed geodetic longitudes. The dimensions of the earth derived from this investigation, the equatorial radius and the polar semidiameter, are necessarily in error by the same proportional part as the average distance between the astronomic stations involved in this investigation. By the methods indicated above, it is found that the probable error of the total length of the transcontinental triangulation is only +8 meters, or one part in 500 000. The writer estimates from this and other considerations that the probable error of the average distance involved in this investiga- tion is about one part in 500000. From this cause, then, the computed equatorial radius may have a probable error of this proportional part, or +12 meters. The computed probable error of the equatorial radius, derived from this investigation, is +34 meters (see p. 115), indicat- ing that errors due to other causes are much larger than those arising from errors in the com- puted lengths. There may be constant errors as great as one part in 500 000 in the measurement of each of several of the bases concerned in this triangulation. But as the various bases were meas- ured under a variety of conditions with sets of base apparatus varying widely in design, the *See The Transcontinental Triangulation, pp. 368, 395, 417, 434, 451, 480, 514, BBL, 567, 592, and 611. - 120 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. errors which were constant for each base were probably different in sign, as well as magnitude, for different bases. These errors are therefore in the accidental class in so far as this investi- gation, as a whole, is concerned. EFFECTS OF ERRORS IN GEODETIC AZIMUTHS. The errors in the computed geodetic azimuths used in this investigation are due mainly ‘to the errors in the angle measurements, as crrors in the angles used in the computation enter with full value into the computed azimuth. As already indicated, the effects of errors of length upon the computed geodetic azimuths are very small. The errors in the geodetic azimuths affect directly the absolute terms in the azimuth observation equations and the corresponding residuals. Through these equations the final results of the investigation are affected. The errors in the geodetic azimuths also necessarily affect.the computed geodetic latitudes and geodetic longitudes; but the effects upon the latitude and longitude observation equations are very small in comparison with the effects upon the azimuth observation equations. The accuracy of the angle measurements has been determined from the adjustment of the triangulation. The probable errors of the computed geodetic azimuths may be computed. Let d be the probable error of an observed direction in any section of triangulation. This has been computed from the results of the adjustment of various sections of triangulation* involved in this triangulation by the formula d= 6754/27" in which each v is the correction to an observed direction as derived from the adjustment, and ¢ is the number of rigid conditions satisfied by the adjustment. The probable error of an adjusted direction which will be called d, is necessarily less than the probable error of an observed direction. The relation between the two is given by the formula n—c wo) in which n is the number of observed directions in the section of triangulation considered. Or, expressed in general terms in words, the average value of the ratio of the weight of the observed value of a quantity to that of its adjusted value equals the ratio of the number of independent unknowns to the number of observed quantities. | The geodetic azimuth may be computed through a chain of adjusted triangulation by using any series of lines between terminal points of the section, each selected line being one over which observations were made in both directions. The same computed azimuth will be secured, : whatever series of lines is selected, the triangulation having been completely adjusted. The accuracy with which the azimuth is carried through the triangulation will depend, therefore, upon the least number of lines that can be arbitrarily selected which will connect the terminal points of the section and over which observations have been made in both directions. Let 1 be this minimum number of lines. Then the number of adjusted directions involved in carry- ing the computed geodetic azimuth through the section is 21, and the error introduced into the computed geodetic azimuth is d.J2l n—-c 2 a a yp *See The Transcontinental Triangulation, p. 613; The Eastern Oblique Arc, p. 235; and Triangulation in California, Part I, Appendix 9, of the Report of the Coast and Geodetic Survey for 1904, p. 520. + See the Adjustment of Observations, by T. W. Wright, second edition, p. 143. or its square is THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 121 This formula has been applied to successive adjusted sections of nearly all of the triangulation involved in this investigation.* If the geodetic azimuth be considered as absolutely fixed at Meades Ranch, the initial point used in this investigation, the computed probable error of the geodetic azimuth is +2’’.9 at San Diego, California, +3’’.4 at the point in Virginia at which the eastern oblique arc triangulation to the southward leaves the transcontinental triangulation, +3’.9 at Cape May, +4’’.1 at Calais, Maine, and +5’’.0 at New Orleans, Louisiana. These are so large as to indicate that the major portion of the residuals from the azimuth observation equations may be due to the errors in the geodetic azimuths resulting from the errors of measurement of the horizontal angles in the triangulation, the average value of such residuals without regard to sign being only 3’’.85. (See p. 116.) But dependence need not be placed on these computed Piolebts errors alone to determine the accuracy of the geodetic azimuths. Other strong evidence is available. Wherever both the astronomic longitude and the astronomic azimuth are determined at the same triangulation station in such a triangulation as that under consideration, of the primary grade of accuracy and of large extent, it is possible to determine with considerable accuracy the true geodetic azimuth and, therefore, the accumulated error in the geodetic azimuth. For each such point it is evident that the prime vertical component of the deflection of the vertical should be the same, except for errors of observation, whether derived from the observed astronomic longi- tude or from the observed astronomic azimuth. Expressed algebraically, this is (see pp. 74, 75) cos f(A, — 1’) = —cot f(a, — a’) This relation is evidently the same as that expressed by the well-known Laplace equation (astronomic azimuth — geodetic azimuth) +sin ¢’ (astronomic longitude — geodetic longitude) =0. A point at which coincident astronomic longitude and astronomic azimuth observations have been made, has already been given by others the appropriate name of Laplace point. There are 11 Laplace points in the present investigation. It has already been stated (see p. 101) that a preliminary .investigation making use of these points indicated that there had been, somewhere between Illinois and Colorado, in the transcontinental triangulation, an accu- mulation of about 5” of error in the geodetic azimuth, carried through the adjusted angles. The evidence available was not sufficient to determine the places at which the accumulation occurred with any greater accuracy than indicated in the preceding sentence. The investiga- tion did not show, with certainty, that any error in geodetic azimuth had accumulated in any other part of the triangulation. The device was, therefore, adopted of virtually introducing a hinge into the triangulation at Meades Ranch by putting into the observation equations and, therefore, also into the normal equations, two separate required corrections to the initial azimuth at Meades Ranch, one (@,) pertaining to triangulation to the eastward of Meades Ranch, and the other (@,) pertaining to triangulation to the westward. This is equivalent to assuming that the error in geodetic azimuth all occurred suddenly at Meades Ranch, instead of gradually at unknown points between Illinois and Colorado. The values of (@,) and (a) at Meades Ranch, as derived from the adopted best solution, G, are (see p. 105) +’’.13 and —5”’.40, differing by 5’’.53, thus confirming the conclusion reached from the preliminary examination that the accumulated error in geodetic azimuth between Illinois and Colorado was about 5”. As an indication of. the errors which still remain in the results due to errors in the cor- rected geodetic azimuths, the values of the residuals at the 11 Laplace points, as given sep- arately by the longitude and the azimuth observation equations in solution G, are given in the table which follows. These residuals are the unexplained portions of the prime vertical com- ponents of the deflection of the vertical, and in each case their difference, shown in the sixth * The formula has not been applied to the Lake Survey triangulation as the values of d are not available for it. 122 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. column, should be zero to satisfy the Laplace equation and thereby indicate that there were no errors of observation and no errors in the geodetic azimuths. Place. Longitude Azimuth Longitude Azimuth Difference station. station. residual. residual. long—az. 7 Mt wt Cambridge, Massachusetts 157 147 +8, 21 +0. 18 +38, 03 Ogdensburg, New York 179 180 +2. 14 + .64 +1.50 Tonawanda, New York : 185 186 — .84 —4. 06 +3. 22 Minnesota Point N. B., Minnesota 213 214 —4, 37 —5. 36 + .99 Sault Ste. Marie, Michigan 198 200 —5. 92 —5. 13 —.79 Ford River 2, Michigan 201 203 —4, 42 —3, 18 —1. 24 Willow Springs, Illinois 211 212 + .84 + .37 + .47 Parkersburg, Illinois 17 78 —.70° | + .19 — .89 Gunnison, Colorado 55 56 +41. 74 +3.°41 —1. 67 Salt Lake City, Utah 44 45 +1. 09 —2.15 +3. 24 Ogden, Utah 41 42. +1. 98 — .94 +2. 92 The differences show too little tendency to grouping of signs, and nearly all of them are too small, to warrant the conclusion that they are due to anything else than accidental errors in the astronomic observations and triangulation. It is possible that the four values greater than 2” are indications of twist in azimuth, but it is not certain that even these are not due to accidental errors. The differences are so small as to shcw that only a minor part of the residuals of the azimuth observation equations are due to uncorrected errors of geodetic azimuth. They are of such a magnitude as to indicate, however, that the small excess of the average residual of the azimuth equations (3’’.85) over the average residuals of either the latitude equations (2.76) or the longitude equations (2’’.34) (see p. 116) is probably due to the errors of the geodetic azimuths. An approximate examination shows that the possible accumulated errors of the geodetic azimuth corresponding to the differences in the table are so small that their effects on the computed geodetic latitudes and longitudes are probably everywhere less than 1” and, as a rule, are less than 0.2. Hence, the errors in the geodetic latitudes and longitudes, due to this cause, contribute a minor, but not negligible, part of the residuals of the latitude and longitude equations. It is evidently desirable that more Laplace points be introduced into this triangulation before it is again used for a study of the figure of the earth. The fact that the average residual in the azimuth equation is considerably larger than in either the latitude or longitude equation, and the evidence indicated above that this is probably due to accumulated errors in the geodetic azimuths used, make it certain that to secure the theoretically best results the azimuth equations should be assigned less weight than the latitude and longitude equations. The equations have all been given equal weight because it was not certain until the investigation was nearly complete that the azimuth equations should be given less weight than the others, and it is even now very difficult to decide how much the weights of the azimuth equations should be reduced. Considering it granted that they should be reduced somewhat in weight, has the failure to make the reduction seriously affected the adopted results? To test this question the G solution has been repeated after assigning to each azimuth equation east of Meades Ranch the weight 0.7 and to each azimuth equation west of Meades Ranch the weight 0.4, the weights of the latitude and longitude equations remaining unity, as before. The assigned weights 0.7 and 0.4 are based upon ratios of mean squares of residuals in the different groups. The amount of change in the derived results produced by this reduction of the weights of the azimuth equations is shown below. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 1238 Adopted solu- | Soluti ith re- « nptad wl | Soule | imeense st 4, 4 Correction to initial latitude — 0.22 — 0.20 0. 02 Correction to initial longitude + .13 + .17 . 04 Correction to initial azimuth for eastward computations | + .13 — .06 07 Correction to initial azimuth for westward computations | — 5.40 — 5.81 41 Correction to Clarke 1866 equatorial radius +76 meters | +72 meters | 4 meters Correction to Clarke 1866 square of eccentricity — .000065 | — .000072 . 000007 The changes in the final results produced by the reduction of weights of the azimuth equations are so small as to be of little consequence. For only one of the six unknowns (ay) is the change as great as the probable error of the unknown as derived from solution G. The changes produced in the other five unknowns are from ,, to 4 of the probable errors of those quantities derived from solution G. Hence, it is immaterial, in so far as the final results are concerned, whether the weights of the azimuth equations are reduced or not. As it is evident, however, that the azimuth equations are affected by greater errors than the latitude and longitude equations, in drawing each conclusion which is based in part on the residuals of the azimuth equations (as to the depth of compensation, for example) it is important to note the degree to which the evidence given by the azimuth residuals is cor- roborated by the evidence from the latitude and longitude residuals. Later in this publication it will appear that this precaution has been taken. THE ACCURACY OF THE COMPUTATIONS OF TOPOGRAPHIC DEFLECTIONS. The accuracy of the results from solution B (see p. 92), in which solution the absolute term of the observation equation is, in each case, the observed apparent deflection minus the topographic deflection, depends directly upon the accuracy of the computation of topographic deflections. All errors in the computed topographic deflections enter with full value into the observation equations. But solution B is relatively unimportant as compared with solution G, the adopted best solution. In solution G, which has been ascertained to represent the truth more accurately than any of the other solutions, the absolute term of the observation equation is in each case (see p. 93) the observed apparent deflection minus the deflection computed upon the supposition that the isostatic compensation is complete and uniformly distributed throughout the depth 113.7 kilometers. These computed deflections are obtained from the computations of topo- graphic deflections by multiplying the topographic deflection derived separately for each ring of topography by a factor pertaining to that ring. These reduction factors (see p. 70), are nearly unity for inner rings, are smaller the larger the ring, and become nearly zero for the largest ring uscd, No. 1, being only .001 for that ring. The errors of the computed topographic deflection, therefore, enter into solution G with sensibly full value for inner rings, with a decreasing percentage of effect for successively larger rings (as fixed by the reduction factors), and for the outer ring (No. 1) have practically no effect. Errors in the computed topographic deflections arise from a variety of causes and follow laws having a similar degree of variety. In discussing the separate classes of errors affecting the computed topographic deflections it is important to note for each, not only its full effect on the comparatively unimportant solution B, but also its reduced effect on solution G, from which the more important conclusions of this investigation are drawn. It is important to note, also, with reference to each class of errors, whether they are of the accidental type or of the systematic or constant types. The effects of accidental errors are, to a large extent, eliminated from the final results and conclusions in such an investigation as this, which depends upon a large number of observations. The uneliminated effects of the 124 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. accidental errors are properly measured by the probable errors attached to the final results. On the other hand, the effects of constant or systematic errors may or may not be eliminated, and the computed probable errors do not furnish a reliable measure of their effects. Therefore it is especially important to scrutinize carefully possible constant or systematic errors. ERRORS IN TOPOGRAPHIC DEFLECTIONS DUE TO MAPS. 1 The accuracy with which contour lines are located on a given map depends fundamentally upon (a) the number of points within the area of which both the elevation and horizontal position are fixed by instrumental determinations, (b) the accuracy with which the instrumental observations are made, and (c) the accuracy with which the contours as drawn conform to the instrumental determinations. It is probable that the errors arising in the computed deflections from errors in the instrumental observations referred to (b) are negligible; that they affect the hundredths of seconds only. For some areas touched by this investigation the number of points instrumentally deter- mined is so small that there is considerable uncertainty in the mean elevations of those’ areas. Greenland is an extreme example, and Guatemala is a typical example. As a rule, the contours as drawn on the original field topographic sheets probably conform _ to the instrumental determinations so closely as to introduce no appreciable errors into the computed deflections, except for topography very close to the station. But many of the maps actually used are on a much smaller scale than the field sheets. As a necessary accom-_ paniment of the reduction in scale the contours are generalized. In the process of generaliza- tion the apparent mean elevation of a given area, as read from the contours, may possibly be changed, and probably is changed in some cases, enough to affect the computed deflections appreciably. The errors in the computed topographic deflections so produced are of the accidental class. As their principal effect is upon the inner rings they affect solution G with substantially full value, as well as solution B. Similarly, errors in the mean elevations, as shown by the maps, for areas near the station, due to an'insufficient number of points fixed directly by instrumental determinations, are of the accidental class and affect solution G as well as solution B with full value. On the other hand, an error in the mean elevation, as read from the map, for an area at a considerable distance from all of the stations at which astronomic observations were made, and outside the area covered by these stations, produces errors in the computed deflections which tend to be small. But such errors are of the systematic kind, the effects of which are not easily eliminated from the results of an investigation, even though it is based on observa- tions at many stations. For example, from the best information available, it was decided to treat Guatemala as having a mean elevation of 3 000 feet (914 meters). It is possible that this mean elevation is actually as small as 2 000 feet (610 meters). If so, the computed prime vertical components of the topographic deflections are in error by about 0’’.01 near the Atlantic and Gulf coasts and by about 0’’.01 with the reverse sign at San Diego, California, these being the extreme values of the error. Similarly, at New Orleans, in the supposed case, the meridian component of the topographic deflection is in error by 0’’.04, this being the maximum error from this cause in the United States. The minimum is 0’’.01, with the same sign, at Calais, Maine. The reduction factors (see p. 70) are, however, so small for all compartments which include portions of Guatemala, that errors from this cause do not affect the hundredths of seconds in the observation equations of solution G. In general, it is believed that the errors in mean elevation of distant topography produce small but systematic errors in solution B and inappreciable errors in solution G. The nearness of approach of the computations of topographic deflection to the station is limited by the scale or contour interval of the best map or field sheet available. The scale of the map fixes the size of the smallest ring of topography which can be definitely located on the map by means of the template and for each compartment of which an estimate of the THE FIGURE OF THE EARTH AND ISOSTASY FROM MEAUREMENTS IN U. 8. 125 mean elevation can be made. If the contour interval is large, or the topography is nearly flat, or both, it may happen that, though a small ring and the compartments comprising it may be definitely located on the map, the whole ring may fall between one pair of contours and, therefore, in so far as that map is concerned, may all seem to lie at one elevation. In this case the computation is virtually limited to larger rings by the contour interval of the map, rather than its scale. In such a case, if a map of smaller contour interval, and possibly also larger scale, later becomes available the computation may be extended inward for several more rings, and for these rings the computed topographic deflection may be found to differ considerably from zero. This limitation of nearness of approach to the station, as fixed by available maps and field sheets, introduces much larger errors than arise in any other way from the incompleteness or inaccuracy of the mapping. At the station Cape Mendocino (No. 252) the.computation of the meridian component of the topographic deflection was carried to ring 37, of which the inner radius is 0.0079 kilo- meter. At Rouse Point (No. 167) the computation of the meridian component was stopped at ring 13, of which the inner radius is 39.22 kilometers. These are the extremes in this respect between which all other cases lie. In 322 cases out of 496 the computation was carried inward toward the station at least to ring 26, of which the inner radius is 0.389 kilometer. The question arises: How large are the errors in the computed deflections due to the omitted topog- raphy nearer to the station than the nearest ring used in the computation? There are 16 cases out of 496 in which the computation was carried inward at least to ring 35. For these cases a small absolute limit can be set to the possible error due to omitted topography in smaller rings. Let it be assumed that, in ring 35, the mean elevation of each of the eight compartments south of the station is 15 000 feet (4572 meters) above the station and for each of the eight compartments north of the station, the mean elevation is 15 000 feet below the station (or the equivalent of that after the density of sea water is considered). As the outer radius of ring 35 is only 22.8 meters, the supposed condition represents a more abrupt slope than it is possible to find on the earth. But the meridian component of the topographic deflection at the station computed by the more exact formula on page 34 corresponding to this extreme condition is only 0’’.0065 for each compartment, or 0’’.10 for the whole ring. Obviously the deflection produced by the topography in ring 35 at any station may approach but not exceed 0’’.10. If the same computation be made for successively smaller rings it will be found that the successive limits computed are in decreasing geometric progression in which the fixed ratio is 7;3. It follows from this law that the absolute limit to the sum of the topo- graphic deflections for ring 35 and all smaller rings is 0’’.25. The actual neglected topography inside ring 35 probably corresponds to topographic deflections which are on an average much smaller than this absolute limit. All of the computations of topographic deflections were examined to ascertain the maxi- mum computed value for each ring in the 496 computations. The following table shows these maxima for rings 37 to 30 in comparison with the absolute limits computed as indicated in the preceding paragraph: Maximum value “Maximum value cy a ee eS deflection. deflection. imit. 37 ” 01 tee ”.05 36 . OL ied . 07 35 . 03 "01 .10 34 . 04 . 02 215 33 . 07 . 03 . 20 82 .11 . 05 .29 31 .13 . 04 -42 30 .18 . 40 . 59 With one exception the observed maxima are less than one-half the absolute limit. In the exceptional case it is only two-thirds the absolute limit. 126 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. For the hypothetical case of a compartment in ring 35 (outer radius 22.8 meters) having a mean elevation of 15000 feet above the station, the computed deflection (see p. 125) is only 0’’.0065, corresponding to an effective mean elevation of only 65 feet above the station. The reason for the very large discrepancy between the effective elevation and the actual elevation is that in such a compartment much of the attracting material is above the station at such a large angle of elevation from the station that it is ineffective in producing a deflection of the vertical.* Computations for various rings made as indicated on page 125 have, as there stated, shown that the absolute limit varies from ring to ring in geometric progression with a constant ratio of 1.43. This is the ratio between the outer radii of successive rings. (See p. 21.) Hence the maximum effective elevation of a compartment is proportional to the outer radius of the ring in which it is located.t The absolute limit 0’.59 for ring 30, increasing in geometric ratio 1.43, becomes 43” for ring 18, of which the outer radius is only 2 295 meters. The maximum value of the topographic deflection for ring 18 in this investigation was 1’’.80 in the meridian compu- tations, and in the prime vertical computations, 2’’.88. In general it is found that deflections for large rings as actually computed from the maps fall much farther below the absolute limit than do those for small rings. Hence, in estimating the effect of neglected topography for larger rings than ring 35, it is desirable to find some closer limit, some limit approaching actual conditions more closely, than does the absolute limit. The following table is the basis of such an attempt to find a closer limit: Maximum value in Sum of maxima Ring. mies vertical ica Station at which said maximum was observed. dence wad inane putations. ing this ring. 35 — ’.01 No. 8, San Francisco, Washington Square, California “01 34 — .02 . 03 33 — .03;No. 241, Avila, California . 06 32 — .05 11 31 — .04No. 225, Arguello, California 15 30 + .40 . 55 29 + .81\No. 228, Santa Cruz West, California . 86 28 + .29 1.2 27 + .26 14 26 + .82}iNo. 26, Mount Conness, California 1.7 25 + .46 ; 2,2 a 7 1. = No. 25, Virginia City, Nevada n 22, — 1.39 5.3 Bh a De iNe: 23, Genoa, Nevada Hee Ae 20 — 2.50 9.6 19 — 2.32 12 : : ze aa,No. 43, Waddoup, Utah = 16 + 2.86 21 15 + 2.73/No. 44, Salt Lake City, Utah 23 14 + 2.39 26 13 + 3.92 30 12 + 498INo. 1, Point Arena, California me 11 + 5.90 : 41 10 + 5.74 46 *It would be effective in producing a change of the intensity of gravity at the station. + This conclusion is reached here by the use of the regular formula used in this investigation. One who cares to do so may prove it more elegantly by deriving the formula which is directly applicable to the limiting case. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 127 Maximum value in Sum of maxima Ring. pre, pe ae Station at which said maximum was observed. aa putations. ing this ring. 9 + 7'7.24 No. 222, Castle Mount, California 53” 8 + 9 .72 No. 239, Monterey Bay, California 63 7 + 9.47 No. 4, Mount Helena, California 73 6 +10 .15 No. 3, Ukiah, California 83 tp gd|No- 19, Round Top, California e 3 +10 .04 No. 1, Point Arena, California 113 2 — 8 .57 No. 85, Gould, Ohio 122 1 — 7 .40 No. 233, Provincetown, Massachusetts 129 For the present purpose three facts are especially significant in this table: First, the values in the second column increase downward much more slowly than the absolute limits already discussed, which are in geometric progression with a constant ratio 1.43. Second, among the maxima both algebraic signs arefound. Third, the different maxima occur at different stations. The sums in the last column may be taken as furnishing a limit of the deflections due to neglected topography which will probably not be reached in any actual case. Thus, for example, the deflection corresponding to the neglected topography for all rings smaller than ring 17 will probably in no case exceed 15’’, for in order to be 15” each ring smaller than ring 17 must pro- duce a deflection as great as the maximum observed for that ring at any station, and moreover all of these extraordinarily large values must have one algebraic sign. Such a coincidence is not likely to occur in the United States.* To obtain a reliable estimate of the errors probably introduced by the neglected topography in the inner rings it is evidently desirable to go one step further, for it is probable that the limits fixed in the preceding paragraph are approached but seldom. It is desirable to secure some estimate of the average effect, as distinguished from the maximum effect, of the neglected topography. Ten prime vertical stations were selected at random by taking the twenty-third, forty- sixth, sixty-ninth, etc., stations in the prime vertical list as arranged 1 in geographic order in the observation equations. The numbers of the selected stations in the prime vertical list are 193, 147, 113, 106, 198, 65, 3, 229, 11, and 47. At No. 193 the smallest ring computed was ring 30, and the algebraic sum of the coranutied topographic deflections for rings 30 to 18, inclusive, was —0’.20. The sum without regard to sign of the values in the second column of the table above for these rings is 14’’.66, of which the observed —0’’.20 is but 0.01 part. In other words, the actual computed deflection for these rings at station No. 193 is only 0.01 of the probable maximum for these rings. Similar comparisons were made for the other nine stations selected at random named above. In no case was the ratio of the actual value to the probable maximum greater than 0.11, and the average ratio was0.03. Itisfair to estimate, then, that though the probable maximum topographic deflection due to all topography in rings smaller than ring 17 is 15”, the probable average effect of such topography is only 0’’.45 (0.03 of 15’’). There were 117 of the 496 computations of topographic deflections which included no ring smaller than ring 17. There were 322 of the 496 computations which were carried inward toward the station at least to ring 26. By the same reasoning as in the preceding paragraphs it may be estimated that for computations carried to ring 26 the probable maximum topographic deflection corre- sponding to the neglected smaller rings is 1’’.4, and the probable average topographic deflec- tion only 0’.04. *It should be noted that in such a table as that above, constructed from the meridian components of the deflections of the vertical, instead of the prime vertical components, the values are in general much smaller than in the table shown. For example, the 15’ limit for all rings smaller than ring 17 is, in the table, based on meridian deflections only 9’’. 128 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U.~ 8. Taking into account all of the considerations which have been given, and guided by judgment based upon much study of the details of the problem, it is estimated by the writer that the topographic deflection corresponding to the topography of neglected inner rings is about 0’.3 on an average; that it is reasonably certain that in about two-thirds of the 496 separate cases it does not exceed 1’’.5; that it probably exceeds 5” in a few cases only; and that it is improbable that it is as great as 15” in any case. The effect of errors from this source is nearly as great on solution G as on solution B, but in either case the errors are in the accidental class and their effect upon the final results is very small. If it were possible to take all of the topography into the computation, to neglect no inner rings, the average of the residuals would probably be reduced by about 10 per cent. Errors due to the shrinkage and distortion of the maps used are negligible. The errors of each of the classes of errors discussed in the preceding paragraphs, and due to the errors of maps, have been made as small as possible by using the best maps for the purpose which are available. The errors of these classes still remaining could not be made smaller by any method of computation. They are inherent in the data, not due to the method of computation. ERRORS IN TOPOGRAPHIC DEFLECTIONS DUE TO ERRORS IN ASSUMED MEAN DENSITIES. So, too, the errors in the computed deflections due to the error in the adopted ratio of the surface density of the earth to the mean density—namely, 1 to 2.09*—are inherent in the data and can not be reduced by adopting any more refined methods of computation. It is unlikely that this ratio is in error by as much as one-fifteenth of itself. Any error, expressed as a proportional part in the adopted value of this ratio, produces an error of the same pro- portional part in the computed topographic deflections and in the reduced deflections when compensation is considered (see the formule on pp. 20 and 69). Hence it is unlikely that the computed deflections involved in solutions B and G are in error by as much as one-fifteenth part. The computed deflections involved in solution G, corresponding to the assumption that the depth of compensation is 113.7 kilometers, are, at more than two-thirds of the stations, less than 3.00 and the maximum is 20’’.00 (at prime vertical station No. 43, Waddoup, Utah). Hence an assumed change of one-fifteenth part in the adopted ratio 1 to 2.09 would produce £ ‘t 2 changes in the residuals of solution G which would be less than 0’’.20( = os) for more than ‘ ee two-thirds of the stations, and the maximum change would be about 1’’.3 700 An error in this ratio introduces errors of the systematic class into the observation equations which will not be effectively eliminated from the final results. For example, if the adopted ratio is one-fifteenth part too large the computed prime vertical deflections west of Colorado, which are all positive, are each too large by one-fifteenth, and those for stations east of Colorado, which are all negative, are also too large by one-fifteenth. These will both tend to make the derived equatorial radius and polar semidiameter from solution G too large. If the adopted ratio as used in solution Gis changed from 1 to 2.09 to zero to 2.09, that is, is reduced by 100 per cent of itself, the computed deflection would become zero as in solution A. In other words, such a reduction of 100 per cent in the assumed ratio in solution G reduces that solution to solution A. Hence, a reduction of one-fifteenth in the adopted ratio would change the final results from solution G by approximately one-fifteenth of the difference between those results and the corresponding ones from solution A, making them approach the results from solution A. Solution A gave an equatorial radius 337 meters less than solution G, and a polar semidiameter 433 meters less. Hence, an error of one-fifteenth in the adopted ratio, ® This ratio is based upon the supposition that the surface density of the solid portion of the earth is 2.67 and the mean density of the earth is 5.576. For the data and considerations upon which these values are based, see The Solar Parallax and its Related Constants, by William Harkness, Washington, Government Printing Office, 1891, pp. 89-92, 139. The data there given is also a part of the basis for the above estimate of the uncertainty in the adopted ratio 1 to 2.09. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. 129 which is improbable, would produce changes of only about 22 and 29 meters, respectively, in the equatorial radius and polar semidiameter, as derived from solution G. It appears, therefore, that the error arising from this source is probably insignificant. ERRORS IN TOPOGRAPHIC DEFLECTIONS DUE TO METHOD OF COMPUTATION. The remaining errors affecting the computed topographic deflections, which are discussed herein, are capable of being further diminished by using a more precise method of computation or by introducing greater refinements into the method which has been used. In assuming, as has been assumed in the computation, that the deflection produced by the masses within a given compartment is proportional to the mean elevation of the surface in that compartment, the fact is neglected that equal masses in different portions of the com- partment produce different deflections at the station. Of two equal masses within a given compartment and lying on the same radial line from the station, the one farthest from the station produces least effect. Of two equal masses within a given compartment and at the same distance from the station, the one which lies more nearly at right angles to the reference line of the template (more nearly in the prime vertical if the meridian component of the deflec- tion is being computed) produces least effect. The errors arising from this source are evidently smaller the smaller the compartments used. In this investigation it is believed that the com- partments are sufficiently small to insure that this error will, as a rule, be less than 0’’.02 for each compartment, though in rare cases it may be greater than 0’’.05. Moreover, in occasional cases when it was noticed, during the progress of the computation, that large and abrupt varia- tions of elevations occurred within a given compartment, especially within one of the larger compartments shown on the template, errors from this cause were guarded against by sub- dividing this particular compartment into smaller compartments. The smaller compartments, or subdivisions, were bounded by radial lines and circles which were located in accordance with the principles used in fixing the boundaries of the large com- partments.* ; The errors under discussion arising from variations of elevations within each compartment belong mainly in the accidental class as there is great variety in the direction and steepness of slopes within the area covered by the computation for each station. Though there are a large number of compartments involved in the computation for a station, nearly 600 in the extreme case, it is believed that the errors are so small for each compartment and so nearly accidental in character that their effect upon the computed deflection for a station is ordinarily less than 0.10, though it may occasionally exceed 0’.20. The basis for this estimate is an: examination in detail of the differences of computed deflections for adjacent compartments as shown in computations such as are reproduced on pages 29-33. The errors from this source tend to be less for inner rings with small compartments than for outer rings. Hence they are much smaller in their effect on solution G than on solution B, the reduction factor F (see p. 70), being smaller for outer rings, and therefore effective in reducing the errors for those rings in solution G. Errors arise from the computer’s inability to estimate the mean elevation within a com- partment with absolute accuracy. The difficulty of making the estimate increases with the increase in the size of the compartment, with increase in the total range of variation of eleva- tion within the compartment, and with increased irregularity of the contours. In making the computation for any given station there is no difficulty in estimating the mean elevation within less than 100 feet for the greater number of compartments, and thus securing each of the deflec- tions corresponding to these compartments within less than 0/’.01. There were, as a rule, however, a few compartments concerned in each computation in which there was difficulty in making the estimate. To secure greater accuracy in connection with these difficult com- partments, to obtain a check on the results, and to obtain a measure of the accuracy attained, a second computer was required to make independent estimates for each station of the mean *See footnote on p. 33. 78771—09-—_9 130 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. elevation of at least 10 per cent of the compartments estimated by the original computer. The compartments for which it was presumably most difficult to make the estimates were selected, by inspection of the maps, to be subjected to this second estimate. Slight differences, of course, developed between the two computers for many of the compartments. If the estimate of the second computer agreed with that of the first computer within less than 0’’.20 on each com- partment, and also within 0’’.20 on the total topographic deflection for the station, the work of the first computer was allowed to stand unchanged. Otherwise, the computers revised their work together with extra care, subdividing into smaller compartments, if necessary, until they had agreed within the specified limits. In 58 per cent of all the cases it was not found necessary to reexamine together any compartments. It is believed that these precautions insure that the errors due to inaccuracies in estimates ‘are so small that the computed topographic deflection for any station is ordinarily in error from this cause by less than 0’.20, though occasionally the error may be as great at 0’’.30. These errors are, apparently, of the accidental class. They tend to be larger for outer rings than for inner rings, and therefore have much less effect on solution G than on solution B. How large are the errors introduced into the computed topographic deflections by the interpolation of values corresponding to outer rings? The complete computation was made for 68 stations. Each new station to be computed was so chosen, if possible, as to lie within the triangle defined by the nearest three stations for which the computation had already been made, and near the center of the said triangle. From these three surrounding stations the interpolation, if any, was made. The computation was commenced with the inner smaller rings and proceeded outward. The three rules used by the computers in deciding at what ring it was allowable to begin to accept the interpolated values and to accept them for all larger rings were, as stated on page 43, as follows: Rule 1. Commence to accept the interpolated values as final with the first ring for which such interpolation is allowable under either rule 2 or rule 3, and which is beyond the one containing the nearest of the three stations from which the interpolation is made. Rule 2.—Let 1'’.00 divided by the number of a ring be called the interpolation limit for that ring. Subject to rule 1, acceptance of the interpolation may begin with a given ring if the three rings next within it each shows an agreement between the interpolated and computed values which is within the interpolation limit. Rule 8.—Subject to rule 1, acceptance of the interpolation may begin with a given ring, if the next ring inside of it shows an agreement within the interpolation limit and if at the near- est of the three stations from which the interpolation is made, the agreement was also within the interpolation limit for the corresponding ring and for all rings farther out for which the comparison was made. Under these rules the total error made by accepting interpolated values would always be less than 1’’.00, if the error of interpolation was of the same sign and magnitude for all larger rings as was I-C (interpolated minus computed) on the last ring for which the comparison was made. It was believed, however, that the agreement between the interpolated and computed values (commencing with rings not smaller than those contemplated under rule 1) would tend strongly to be closer and closer for successive rings proceeding outward. It was also believed that there would be a strong tendency for the various differences between interpolated and computed values for several rings such as are interpolated under the rules to include values having both the plus and minus signs, and, therefore, for the errors in the accepted interpola- tions to tend to be eliminated from the final result for the station. Both. of these beliefs were based, at first, on theory only. If they are correct the total error introduced at any station by accepting interpolated values will be, in general, much less than 1’’.00. The correctness of these beliefs is established by the results secured during the progress of the computations. During the progress of 479 computations a comparison between the com- puted and interpolated values was secured on from 2 to 15 rings. In 73 per cent of the cases the average value, without regard to sign, of I-C (interpolated minus computed) was less for the outer one-half of the rings on which both interpolation and computation was made at that THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 1381 station than for the inner half of such rings. So, too, in 84 per cent of the cases there were found to be both plus and minus signs of the values of I-C at the station. These tests confirm the theory to such an extent that it is believed that the total error introduced into the computed topographic deflection at a station by the acceptance of inter- polated values is seldom greater than 0’’.50 and is, as a rule, less than 0.25. More than 90 per cent of the accepted interpolations were for rings larger than ring 13. For these the reduction factors (see p. 70), used in connection with solution G, are all less | than 0’’.50. Hence the effect upon the absolute terms in the observation equations of solution G of errors of interpolation is probably seldom greater than 0.20, and is, as a rule, less than 0’’.10. The errors due to the accepted interpolation must be almost entirely in the accidental class. The possible magnitude of the following errors has been considered and in each case the conclusion reached that these errors are negligible; in other words, that they affect the hun- dredths of seconds only, not the tenths: (1) Errors due to inaccuracy in the construction of templates. (2) Errors due to inaccuracy in placing the templates on the maps. (3) Errors due to the omission of slope corrections known to be each less than 0’’.01. The slope corrections which were larger than 0’’.01 were computed. The slope corrections referred to are those necessary to take account of the fact that in certain cases the mean surface within the compartment is so far above or below the station that the masses concerned in the computation can not be considered to be.sensibly in the horizon of the station. (4) Errors due to omitted decimal places. The comprehensive conclusion is that the total error in the computed value of one com- ponent of the topographic deflection at a station, due to all errors of such a character that they are capable of being further diminished by using a more precise method of computation or by introducing additional refinements into the method which has been used, is seldom more than 0’’.60 and is probably less than 0’’.30 in about one-half of the cases. This total error affects each absolute term of the observation equation in solution B. Similarly, it is believed that the total error of this character which affects solution G is seldom greater than 0’’.30 and is less than 0’’.15 in about one-half of the observation equations. The average residual from solution G being more than 3”, it is believed that the accuracy with which the topographic deflections have been computed is amply sufficient and that any changes or refinements in this part of the computations which would increase the time required to make the computations would not be warranted by the slight additional accuracy secured. Would a gain in accuracy result from extending the computation of topographic deflections to a greater distance from each station? The considerations which led to stopping the com- putations at the ring (No. 1) having an outer radius of 4 126 kilometers (2 564 miles) instead of extending it to the antipodes have been given on page 29. Since the computations have been completed it has been estimated that not more than four more such rings as have been used would be necessary to extend the computations to the antipodes. In this connection it must be recalled that the rings rapidly increase in width at great distances from the station, partly on account of curvature (see p. 23). It was also estimated that the sum of the computed topo- graphic deflections for these three or four rings would not in any case exceed 20’’, and would probably be less than 10’, as a rule, for each station. The basis for this estimate is indicated on page 29, and again in the table on page 127, in which it will be noted that the observed maxi- mum for ring 1 is much less than for either ring 4 or ring 3. The omitted portion, probably less than 10”, of each computed topographic deflection introduces errors into solution B which are of the systematic class since, for example, the omitted portion of the prime vertical deflec- tion will have a decided tendency to be of one sign for stations near the Atlantic coast and to be of the opposite algebraic sign for stations near the Pacific coast. The computed results from solution B might, therefore, be appreciably altered by extending the computation to the antipodes. It is reasonably certain, however, that such an extension of the computation would not affect any residual in solution G by more than 0’’.01, and that it would have no effect upon the last, significant figure retained in each final result from this solution. This certainty arises 132 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. from the fact that the reduction factor F (see p. 70) for ring 1 is only 0’.001, and would be still smaller for larger rings. The reduced topographic deflection with isostatic compensation extending to a depth of 113.7 kilometers is only 0.01 at any station for ring 1, the largest ring for which the computation was made, and would be still less for each of the larger rings which have been omitted. EFFECTS OF ERRORS IN COEFFICIENTS. The coefficients are used’in forming the normal equations and in computing the residuals of the observation equations. The accuracy necessary in connection with these uses depends mainly upon the magnitude of the derived values of the unknown quantities and of the abso- lute terms in the observation equations. In solution G the derived values of the unknowns are so small as to contain two significant figures only, except (aw), which is 5’’.40 (see p. 105), and the absolute terms of the observation equations are as a rule less than 10’. Under these con- ditions an examination of the manner in which the coefficients are used shows that if each were correct to the nearest unit in the second decimal place the derived values of the unknowns would probably be correct to the last decimal place now retained and the computed residuals would ordinarily differ from their true values by hundredths of seconds only, not tenths. As each unknown as now computed has two uncertain figures (two significant figures in its prob- able error), and as the average residual is more than 2’’, this grade of accuracy in the coefficients would be sufficient. As a matter of fact, however, each coefficient was computed to three or more decimal places and was so used. Therefore the effect of omitted decimal places in the coefficients, in the computation as made, is so small as to be of no consequence. The coefficients have been computed by formule which are known to be approximate. An exhaustive examination has not been made of the magnitude of the quantities neglected in this approximation, but the extensive investigation which has been made enables the writer to state with considerable confidence that the errors in the coefficients due to approximations in deriving the formule from which they are computed probably do not affect the second decimal place in any case and, as a rule, do not affect even the third decimal place. Therefore the errors due to this cause are of no consequence. As an illustration it may be mentioned that the errors of—2” in a, —7” in 4s, and of one in the fifth decimal place in log s, as computed for latitude station No. 164, from the adopted approximate formule for that purpose (see p. 89), produce errors in the fifth decimal place only in any of the coefficients for that station. GENERAL CONCLUSION AS TO CAUSE OF RESIDUALS. Various sources of error, each of which contributes something to the residuals of the adopted solution G, have been discussed somewhat in detail. Among these sources have been the astronomic observations of latitude, longitude, and azimuth, the observations of angles, the measurement of base lines in the triangulation, the errors in the computations of topo- graphic deflections due to the errors and incompleteness of available maps and also to the method of computation, the errors in the assumed mean densities, and, finally, errors in the computed coefficients used in the equations. The detailed examination of these separate sources of error has shown that no one class of errors nor the combination of all classes enumerated is sufficient to account for such large residuals as those in solution G. These residuals must be due mainly to some other cause. Neither does an examination in detail of these various classes of errors show any reason why the residuals of solution G should be smaller than those from solution A. From a long and careful study of the details of the evidence the writer is firmly convinced on two main points. : The first is that the residuals of solution G are smaller than those of any other solution made, simply because the assumption upon which solution G is based is a closer approxima- tion to the truth, as a statement of a general law controlling the variation of subsurface densities, than any of the other assumptions made. The assumption for solution G is that the isostatic compensation is complete and uniformly distributed throughout the depth 113.7 kilometers. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 183 The second main point is that the residuals of solution G are mainly due to irregular varia- tions of subsurface densities from the law indicated in the preceding paragraph, the densities being excessive for various areas of greater or less extent and deficient for various other areas. No general law controlling these variations has, as yet, been discovered by the writer. As these two conclusions are of fundamental importance, some of the evidence on which they are based will now be set forth in considerable detail. Illustration No. 10, at the end of the volume, shows the residuals of solution G, of which the numerical values are tabulated on pages 106-114. The arrows are drawn to:the scale indicated on the sketch, to represent the residuals of solution G. Arrows representing residuals which are almost zero have been lengthened suffi- ciently to make them show. The butt of each arrow is placed as nearly as possible in the position of the station of observation. In some cases it has béen necessary to displace the arrows from their true position in order to avoid confusion with other arrows. The number on each arrow is the same as the number given to the corresponding observa- tion in the list printed on pages 12-19. An arrow pointing to the southward indicates that the observed astronomic latitude is greater than the computed latitude, and an arrow pointing to the northward indicates a differ- ence of the reverse sign.~--In other words, the arrow indicates the direction of the nadir point from its normal position as fixed by solutionG. The arrow must tend, therefore, to point toward regions in which the density is in excess of that postulated in solution G, and away from regions in which it is less than that postulated. Similarly, an arrow pointing to the eastward or westward indicates the prime vertical component of the deflection of the nadir point, as fixed by a determination of astronomic longi- tude or astronomic azimuth, said deflection being expressed with reference to the normal posi- tion of the nadir as fixed by solution G. At stations where both longitude and azimuth were observed, only the residual corresponding to the longitude determination was plotted. The residuals represented by these arrows are due, in part, to errors of observation and computation. Subject to this reservation, they represent the effects of the departures of the actual distribution of densities from that postulated in solution G. To facilitate a comparison of the residuals with the topography with a view to detecting any possible relation, certain contour lines have been drawn, namely, the shore line, the 100-foot ‘contour, the 1 000-foot contour, and the 4 000-foot contour, in the eastern part of the United States; and in the western part, the shore line, the 1 000-foot, 5 000-foot, and 10 000-foot contours. In each ocean the 1 000-fathom curve has also been drawn. This serves to indicate the position of the submerged edges of the continent. This illustration No. 10 contains a great mass of material whichis availablefor extensive study. Even a hasty examination indicates that for certain areas there is a decided tendency for all of the residuals to be of one sign. For example, in a large continuous area comprising nearly all of Utah, all of Nevada, and nearly all of that part of California which lies between latitudes 37° and 40°, all of the meridian residuals are negative, indicating either an excess of density ‘to the northward or a defect of density to the southward, or both. The grouping together of residuals of the same sign is more clearly shown in illustrations Nos. 11 and 12, at the end of the volume. Within each area on illustration No. 11 which is inclosed by a solid black line and marked ¢+, the meridian residual of solution G is positive for every latitude station, corresponding to an excess of density to the southward of the station. Similarly, in areas marked ¢— all meridian residuals in solution G are negative, corresponding to an excess of density to the northward. Note the large ¢— area in Utah, Nevada, and California to which attention has already been called,* and the very large $+ area covering Missouri, Illinois, Indiana, and a part of Wisconsin *A single latitude station, No. 10, New Presidio, San Francisco, Cal., within this area has a small positive meridian residual +/”.16. The illustration, No. 11, isso small that this could not be conveniently shown as distinct from other San Francisco latitude stations. 134 THE FIGURE OF THE EARTH AND ISOSTASY FROM MHASUREMENTS IN U.S. Note also that three ¢+ areas together make a nearly continuous ¢+ area extending from northern Maryland to Louisiana, along the Allegheny Mountains and the Gulf coast. These three areas are separated by two ¢— areas, each of which contains but a single latitude station. Aside from these exceptionally large groups each group shown in illustration No. 11 is too large, as a rule, to be probably due merely to accidental errors. For example, it is improb- able that the large $+ group containing 10 stations on the shores of Lake Ontario and Lake Erie is due to accidental errors. Neither will any of the classes of errors previously discussed to which the observations and computations are subject, produce such a geographic grouping of residuals. The grouping must be due to irregular variations of densities from the general law postulated as the basis of solution G. Within each area on illustration No. 12 which is inclosed by a solid black line and marked PV +, the prime vertical residual of solution G is positive for every longitude and every azimuth station, corresponding to an excess of density to the eastward of the station. Similarly, in areas marked PV — all prime vertical residuals are negative, corresponding to an excess of density to the westward.* Two PV+ areas together extend almost without interruption from western Maryland to southern Mississippi; another PV + area contains nearly all of the stations on the shores of Lake Erie,Lake Ontario, and the St. Lawrence River; and 8 stations are grouped in one PV — area in northern Wisconsin and the northern peninsula of Michigan. There are various other groups too large to be due to accidental errors. They must be due to variations in density from the law postulated as the basis of solution G, the departures being of one sign for consid- erable areas. Illustrations Nos. 11 and 12 are a graphic proof that the residuals of like sign are grouped together geographically to such an extent as to show that they are due largely not to accidental errors but to systematic disturbances having a regional distribution. An analytical proof of the same point follows. The 507 astronomic stations concerned in this investigation have been divided into ten groups, each geographically as compact as possible and each containing about 50 stations. The limits of these groups are indicated on illustration No. 13 at the end of the volume. In each group the mean of the latitude residuals with regard to sign was taken. The probable error of that mean was also computed upon the supposition that the residuals represent accidental errors only, the basis of the computation being the prob- able error of one latitude observation equation as computed from the 265 latitude residuals. The longitude residuals and the azimuth residuals were treated in the same manner. The results are given below in tabular form. The probable error of a single latitude observation equation is +2’’.38, of a single longitude equation +2’’.00, and of a single azimuth observation equation +3’’.26. Latitude residuals. Longitude residuals. Azimuth residuals. Group. Mean Mean Mean P| Number | ‘with | Probable | Number with, | Probable Namber with, | Probable ‘ regard error ‘ regar' error. ' Tegar error. residuals. | > cies, residuals. | {5 Sign. residuals. | {5 Sign, Mf “ut ea uy TAZA a 1 24 +0.30 | +. 49 5 +2.58 | +. 89 23 +1.40 | +. 68 2 28 +0. 87 . 45 15 +0. 63 . 52 il —2. 94 . 98 3 26 —0. 44 .47 6 —2. 48 . 82 17 —2.13 .79 4 24 +2. 63 . 49 9 +0. 68 . 67 17 +2. 19 .79 5 39 +0. 44 . 38 6 —4. 16 . 82 8 —3. 80 1.15 6 25 +2. 36 - 48 9 —0. 04 . 67 17 —0. 37 79 7 22 —1.10 - ol 7 —1. 56 75 17 —0. 09 79 8 21 —8. 45 . 52 5 +3. 67 . 89 17 —1. 72 .79 9 29 —1.97 | -44 13 —0. 42 55 15 —3. 35 . 84 10 27 +0. 31 . 46 4 +1. 93 1. 00 2) +2. 20 atl *There are five prime vertical stations, viz, Nos. 8, 42, 44, 78, and 194, each of which has a residual of opposite sign from that-for other stations in the group surrounding it. In each of these cases the exceptional station is either coin- cident with another station or so nearly so that no distinction can be made on the small scale of illustration No. 12. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 1385 If the residuals were entirely due to accidental errors, only about one-half of the means with regard to sign would exceed the corresponding probable errors. In the above table in 23 cases out of 30 the mean exceeds its probable error, and in 3 cases it is more than 5 times its probable error, namely, groups 4 and 8 of the latitude residuals and group 5 of the longitude residuals. This test agrees, therefore, with the graphic test in showing that the residuals are due to regional disturbances. TESTS OF RELIABILITY OF CONCLUSIONS. Are the conclusions that the E, H, and G solutions are much nearer the truth than the B and A solutions, and that solution G is nearest the truth, dependent on the effects of a few large residuals? Are these conclusions dependent upon and vitiated by the geographic grouping of residuals of like sign to which attention has been directed? Do astronomic observations in all parts of the area treated and all three classes of astronomic observations cooperate in confirming the conclusions reached? In answer to these questions the following tables are presented with comments: Mean values of the squares of the residuals in various groups.* Solution B. | Solution E. | Solution H. | Solution G. Solution A. United States group.—All the observa- tions, 507 residuals 129.06 | 16.21 15. 82 15.81 27. 46. All latitude observations, 265 residuals 79.37 | 13.34 |} 12.91 | 12.86 18. 96 All longitude observations, 79 residuals 132. 91 8.47 8.81 9.01 35. 51 All azimuth observations, 163 residuals 207.98 | 24.63 | 23.94 | 23.89 37. 38 Group’ 1 (Maine, New Hampshire, Massa- chusetts, Rhode Island), 52 residuals 168. 18 8.98 9.14 9.18 9. 46 Group 2 (Connecticut, New York, Pennsyl- ’ vania, Ohio, Michigan), 54 residuals 72. 26 9.45 9. 65 9.71 21. 32 Group 3 (New Jersey, Pennsylvania, Dela- ware, Maryland, Virginia), 49 residuals 36.62 | 17.38 | 17.49 | 17.48 17.71 Group 4 (Virginia, North Carolina, Tennes- see, Georgia, Alabama, Mississippi, Lou- isiana), 50 residuals 50.23 | 10.83 | 10.83 | 10.86 13. 80 Group 5 (Michigan, Minnesota, Wisconsin), 53 residuals 31.52 | 23.94 | 23.96 | 24.00 28. 44 Group 6 (Virginia, West Virginia, Ken- tucky, Ohio, Indiana, Illinois, Missouri, Wisconsin), 51 residuals 35. 45 8.16 8. 52 8. 58 10. 29 Group 7 (Missouri, Kansas, Colorado, Utah), 46 residuals 453. 35 8.92 8.53 8. 70 45. 50 Group 8 (Utah, Nevada, California), 43 re- siduals 157.25 | 25.25 21.83 21. 28 84. 09 Group 9 (California, northern part), 57 re- siduals 138.85 | 16.63 15. 97 15. 94 37. 64 Group 10 (California, southern part), 52 re- siduals 182.16 | 33.10 82. 29 32, 28 57.39 * Certain numerical values printed in this table and on the following pages differ slightly from those given in ‘‘ Geodetic Operations in the United States, 1903-1906. A Report to the Fifteenth General Conference of the International Geodetic Association,” because of the correction of an error in computation which was discovered after that report went to the printer. 136 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. Mean value of residuals without regard to sign. Solution A. Solution B. | Solution E. | Solution H. | Solution G. United States group.—All the observa- - ut * a e tions, 507 residuals 8. 86 3. 06 3. 04 3. 04 3.92 All latitude observations, 265 residuals 7.29 2.78 2.76 2.76 3. 34 All longitude observations, 79 residuals 8. 85 2.22 2.30 2.34 4.18 All azimuth observations, 163 residuals 11. 42 3.91 3. 85 3. 85 4.72 Group 1, 52 residuals 11.95 2.50 2.50 2.50 2.46 Group 2, 54 residuals 6. 83 2. 48 2.51 2. 53 3.69 Group 3, 49 residuals 4.15 3. 35 3. 34 3.33 3. 33 Group 4, 50 residuals 6. 00 2. 69 2.71 | + 2.72 2.94 Group 5, 53 residuals 4, 33 3. 83 3. 85 3. 86 4. 32 Group 6, 51 residuals 5.11 2.31 2.36 2.37 2. 66 Group 7, 46 residuals 18.47 2.19 2.21 2. 26 4, 83 Group 8, 43 residuals 11. 59 3.95 3. 54 3. 48 4.41 Group 9, 57 residuals 9. 83 2.86 2. 82 2. 83 4,42 Group 10, 52 residuals 11. 56 4.55 4.56 4. 57. 6.17 Percentage of residuals less than 2’’.00. Solution B. | Solution E. | Solution H. | Solution G. | Solution A. United States group.—All the observa- tions, 507 residuals 15 41 43 43 34 All latitude observations, 265 residuals 18 43 45 45 40 All longitude observations, 79 residuals 18 57 56 56 39 All azimuth observations, 163 residuals 10 29 34 34 23 Group 1, 52 residuals 4 40 46 48 46 Group 2, 54 residuals 15 52 48 48 33 Group 3, 49 residuals 45 35 41 41 39 Group 4, 50 residuals 16 46 48 44 38 Group 5, 53 residuals 30 26 28 28 25 Group 6, 51 residuals 20 53 . 49 51 37 Group 7, 46 residuals 0 54 57 57 33 Group 8, 43 residuals 2 28 40 40 37 Group 9, 57 residuals 11 46 46 46 44 Group 10, 52 residuals 10 27 29 29 12 Percentage of residuals greater than 5’’.00. Solution B. | Solution E. | Solution H. | Solution G. | Solution A. United States group.—All the observa- tions, 507 residuals 66 18 18 18 29 All latitude observations, 265 residuals 61 14 13 13 23 All longitude observations, 79 residuals 66 8 9 10 30 All azimuth observations, 163 residuals 75 29 30 30 38 Group 1, 52 residuals 94 10 12 12 10 Group 2, 54 residuals 52 11 13 13 28 Group 3, 49 residuals 27 22 22 22 24 Group 4, 50 residuals 58 14 16 16 i8 Group 5, 53 residuals 36 25 28 28 34 Group 6, 51 residuals 53 10 8 6 14 Group 7, 46 residuals 100 9 7 11 30 Group 8, 43 residuals 95 30 21 19 40 Group 9, 57 residuals 72 14 12 12 40 Group 10, 52 residuals 83 37 38 40 54 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. §. 137 Maximum residual in each group. ‘Solution B, Solution E. Solution H. Solution G. Solution A. United States group.—All the obser- i e ne . - vations, 507 residuals +43.84 | —16.47 | +15.74 | +15.94 | —22.35 All latitude observations, 265 resid- uals — 23.74 | —16.47 | —15.61 | —15.45 | —14. 34 All longitude observations, 79 resid- uals +37.14 | + 9.96 | + 9.13 | + 8.95 | —22.35 All azimuth observations, 163 resid- uals : +43.84 | —15.89 | +15.74 | +15.94 | —20. 04 Group 1, 52 residuals —22.64 | + 6.64] + 6.98 | + 7.02] + 8.40 Group 2, 54 residuals —20.64 | — 7.99 | — 7.33 | — 7.25 | —12.83 Group 3, 49 residuals +17.45 | — 9.91 | —10.02 | —10.03 | —10. 23 Group 4, 50 residuals +15.47 | + 8.07 | + 8.12} + 8.14} + 8.89 Group 5, 53 residuals +16.77 | +13.86 | +13.26 | +13.14 | +11. 48 Group 6, 51 residuals +11.64 | + 6.01 | + 6.13 | + 6.19] + 6.86 Group 7, 46 residuals +43.84 | — 8.32 | — 7.42 | — 7.28 | —20. 86 Group 8, 43 residuals —23.74 | —16.47 | —15.61 | —15.45 | +14. 48 Group 9, 57 residuals —31.48 | —15.89 | —15.12 | —14.99 | —22.35 Group 10, 52 residuals —32.98 | +14.89 | +15.74 | +15.94 | +20. 03 s The above tables are placed in the order of the reliability of the tests furnished by them, the most reliable tests being placed first. The geographic limits of groups 1 to 10 are shown on illustration No. 13 at the end of the volume. In the first table, giving mean values of the squares of the residuals, every geographic group and every class of astronomic observations all cooperate in indicating that the E, H, and G solutions, representing isostatic compensation, are much nearer the truth than either solution B, on the basis of extreme rigidity, or solution A, based upon the assumption that the deflections of the vertical are independent.of the topography. The first table indicates but slight differences between the E, H, and Gsolutions. Though the mean square is less for solution G than for E or,H in the group comprising all observations (507 residuals) and also in the group of latitude observations (265 residuals) and the group of azimuth observations (163 residuals), this is not true for the smaller group of longitude observations (79 residuals) and the geographic groups 1 to 7, inclusive (46 to 54 residuals each). It is noticeable, however, that the differences in favor of solution G, as against solution E, in groups 8, 9, and 10 are much larger (from .69 to 3.97) than the differences against solution G in groups 1 to 7 (from .03 to .42).. While the preponderance of evidence is in favor of solution G, the evidence contains contradictions. The evidence shown in the second table, of mean values of residuals without regard to sign, corroborates closely that shown in the first table. The only exceptional features are that in group 1 the mean residual for solution A is slightly smaller than for the G, H, and E solutions, and in group 8 it is the same as for the G solution. The evidence in the third table, the percentage of residuals less than 2’’.00, corroborates, in a general way, that in the first table. It is, however, stronger in favor of solution G than is the first table. Of the 14 groups there are but three in which the percentage of residuals less than 2’.00 is not at least as large in the G solution as in the H and E solutions. An excep- tional feature of the table is that for groups 3 and 5 the percentage of residuals less than 2’’.00 is greater for solution B than for any other solution. This is, however, offset by the fact that in group 7 not a single residual is less than 2’’.00 in solution B, all being in fact greater than 5’’.00. The fourth table, showing the percentage of residuals greater than 5’’.00, furnishes a general corroboration of the first table and contains no exceptional features. 138 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. The last table, showing the maximum residual in each group, corroborates the first table. It is slightly stronger than the first table in favor of solution G as against solutions E and H. Its exceptional features are that in groups 5 and 8 and in the group of latitude residuals the maximum residual is, in each case, smaller for solution A than for any other solution. The residuals have also been studied in four geographic groups, known as the northeastern, southeastern, central, and western groups, of which the limits are shown on illustration No. 13. The tables for these groups are given below in the same form as the tables already shown. For convenience of comparison the line for “‘all the observations” is repeated from above in each table. In the first table, moreover, the subdivision of each geographic group into lati- tude, longitude, and azimuth observations is also shown. Mean value of the squares of the residuals in various groups. Solution B. | Solution E. | Solution H. | Solution G. | Solution A. United. States group, all the observations, 507 residuals. 129.06 |. 16.21 15. 82 15. 81 27. 46 Northeastern group, 118 residuals 112.46 | 10.03 | 10.14 | 10.17 15. 37 Latitude observations, 58 residuals 81.16 7.71 7.77 7.78 10. 99 Longitude observations, 22 residuals 93.14 7.26 8.52 8. 82 20. 97 _ Azimuth observations, 38 residuals 171.41 | 15.19 | 14.70 | 14.61 | 18.80 Southeastern group, 105 residuals 41.57 | 13.02 | 13.09 | 13.11 15. 09 Latitude observations, 53 residuals 29. 35 8. 84 9. 36 9.47 13.11 Longitude observations, 16 residuals 29. 30 6. 39 5.91 5. 83 5. 28 Azimuth observations, 36 residuals 65.03 | 22.12 | 21.78 | 21.69 22.37 Central group, 102 residuals 86.22 | 16.01 } 16.17 | 16.23 20. 07 Latitude observations, 63 residuals 43.64 ) 16.84 | 16.81 | 16.84 19. 45 Longitude observations, 15 residuals 179.07 | 11.66 | 13.05 | 13.39 23. 64 Azimuth observations, 24 residuals 139.94 | 16.55 | 16.42 | 16.38 19.48 Western group, 182 residuals 214.31 | 22.18 | 20.88 | 20.78 46.57 Latitude observations, 91 residuals 132.09 | 17.14 | 15.56 | 15.32 27.10 Longitude observations, 26 residuals 203. 70 8. 93 8.41 8. 59 73. 26 Azimuth observations, 65 residuals 333.66 | 34.53 | 33.31 | 33.30 63. 16 Mean value of residuals without regard to sign. Solution B. Solution E. | Solution H. | Solution G. | Solution A. United States group, all the observations, a ns if ee Me 507 residuals, 8.86 | 3.06 3.04 3. 04 3. 92 Northeastern group, 118 residuals 8.97 | 2.57 2. 60 2. 60 3. 09 Southeastern group, 105 residuals 5.07 | 2.88 2.89 2.89 3.10 Central group, 102 residuals 6.44 | 3.02 3. 06 3.07 3. 54 Western group, 182 residuals 12.34 3. 50 3. 40 3. 40 5.14 Percentage of residuals less than 2’’.00. Solution B. | Solution E. | Solution H. | Solution G. | Solution A. United States group, all the observations, 507 residuals. 15 41 43 43 34 Northeastern group, 118 residuals il 44 45 46 38 Southeastern group, 105 residuals 30. 44 48 46 39 Central group, 102 residuals 22 40 40 41 32 Western group, 182 residuals 7 37 41 41 30 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8S. 139 Percentage of residuals greater than 5’’.00. Solution B. Solution E. | Solution H. | Solution G. Solution A. | United States group, all the observations, 507 residuals. 66 18 18 18 29 Northeastern group, 118 residuals 69 12 14 14 19 Southeastern group, 105 residuals 47 18 18 17 20 Central group, 102 residuals 49 15 17 18 24 Western group, 182 residuals 85 24 21 24 44 Maximum residual vn each group. Solution B. | Solution E. | Solution H. | Solution G. Solution A. at uy uw uy vr United States group, all the observations, 507 residuals. +43.84 |—16.47 |+15.74 |+15.94 |—22.35 Northeastern group, 118 residuals —22.64 |— 7.99 |— 7.33 |+ 7.29 |—12. 83 Southeastern group, 105 residuals +17.45 |— 9.91 |—10.02 }—10.03 |—10. 23 Central group, 102 residuals +39.16 |+13.86 |+13.26 |+13.14 )+11.48 Western group, 182 residuals +43.84 |—16.47 |+15.74 |4+15.94 |—22. 35 These tables corroborate the conclusions already drawn. The exceptional features of the tables are that, in the longitude observations of the south- eastern group, the mean value of the square of the residuals is less in solution A than in any of the other solutions, and that in the central group the maximum residual is less for solution A than for any other solution. From these statistical studies of the residuals taken in various groups the following three conclusions may be drawn. (1) That solution B (extreme rigidity) is much farther from the truth than any of the other solutions and that this conclusion is not dependent upon nor vitiated by a few large residuals or by the geographic grouping of the residuals and is a necessary conclusion regardless of what class of observations is utilized (latitudes, longitudes, or azimuths). The evidence is prac- tically unanimous on this point. (2) That solution A (depth of compensation zero) is farther from the truth than any of the three solutions G, H, and E (depth of compensation 113.7, 120.9, and 162.2 kilometers, respec- tively), but much nearer the truth than solution B (extreme rigidity), and that this conclusion is nearly free from doubt due to the influence of exceptionally large residuals, to geographic grouping of residuals, or to contradiction between different classes of astronomic observations. (3) That the preponderance of evidence isin favor of solution G (depth of compensation 113.7), being nearer the truth than either solutions H or E (depths of compensation 120.9 and 162.2 kilometers), but that this conclusion is drawn from conflicting evidence indicating that the nearness of approach to the truth is so nearly the same in these three solutions that the choice between them is made uncertain by the influence of a few unusually large residuals, and by the influence of the geographic grouping of residuals. Moreover, the three classes of astronomic observations do not agree in the choice among these three. In other words, though it is certain that an approach to perfect isostatic compensation exists extending to a moderate depth (cer- tainly not greater than 200 kilometers) if it is uniformly distributed, the precise depth is diffi- cult to determine from the data in hand. 140 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. SOLUTIONS OF EQUATIONS IN FOUR SEPARATE GEOGRAPHIC GROUPS. During the progress of this investigation as the data gradually became available the normal equations corresponding in form to those shown on page 105 were formed separately for each of the four geographic groups known as the northeastern, southeastern, central, and western groups (see illustration No. 13). For some of these, other depths of compensation were assumed than those involved in solutions G, H, and E. These various solutions served as a recon- naissance of the problem. They are now superseded by the much stronger solutions which involve all of the data at one time.’ They are much weaker than the final solutions not simply or mainly because each involved only about one-quarter of all the data. The fact that each was restricted, roughly speaking, to one-quarter of the total area is much more important in reducing the strength of the conclusions drawn from each solution. The results of the various solutions are, however, of value as an indication of the degree of instability in derived results and conclusions which may be expected whenever studies are made of the figure of the earth and isostasy from data confined to a small area, and as an indi- cation of possible characteristic differences between different parts of the United States as to the depth and completeness of isostatic compensation. Comparison of various solutions made separately for four groups. (Corrections to the assumed United States Standard Datum and Clarke 1866 values.] (€2)in units of Mean A a a the sixth | squareof (¢) ( ) ( ) ( ) decimal | residuals. place. NORTHEASTERN GROUP. Seconds. Seconds. Seconds. Meters. Solution B (extreme rigidity) +19. 13+ 1.60 1. 8744.13 9. 344-3. 34 +22764358 | +8554 212 34. 53 Solution D (depth ofcompensation 329.8 km.) | + 1.3240. 77 Solution E (depth ofcompensation 162.2km.) | — 0.1340. 73 Solution A (depth of compensation zero) — 0.6340. 88 6. 5941.99 + 5.0441.61 + 3794172 | +2984102| 10.31 4.62+1.89 | + 4.6341. 52 + 1264164 | +2894 97 9.31 0. 3842.28 | + 3.4341.84 — 1164197 | +2094117| 13.52 Prit SOUTHEASTERN GROUP. Solution B (extreme rigidity) +19.1440.C38 | —11.2442.22 | + 2.2541.31 | +44944165 | +9224 82] 23.40 Solution D (depth ofcompensation 329.8km.) | + 1.4740.46 | + 2.2641.61 | — 1.6140.95 | + 3434120 | —21U4+ 60) 12.33 Solution E (depth of compensation 162.2 km.) | + 0.29+0.43 | + 5.92+1.52 | — 3.57+0.89 — 1424113 | —317+ 56 10. 89 Solution F (depth of compensation 79.8km.) | — 0.3040.42 | + 7.8241.46 | — 4.6540. 86 — 4014108 | —3654 54 10. 10 Solution A (depth of compensation zero) ~— 1.2040.42 ! + 9.0841.47 | — 5.4440.86 | — 5944109 | —4384+ 54! 10.16 CENTRAL GROUP. Solution B (extreme rigidity ) +11.59+0.59 | +18.5041.51 | —14. 4240.93 + 8564163 | —4734 87] 16.80 Solution D (depth of compensation 329.8km.) | + 1.183+0.56 | + 2.7041.44 | — 2.9140.89 | — 3764155 | —3814 83] 15.28 Solution E (depth ofcompensation 162.2 km.) | + 1.05+0. 55 0. 0041.43 | — 1.3240.88 | — 18834154 | —267+ 82] 15.05 Solution A (depth of compensation zero) + 1.0740.59 | — 3.6741.52 | + 0.6010.94 + 724164 | —1264 88 17.04 WESTERN GROUP. Solution B (extreme rigidity) 4+21.1540.62 | +53.7442.58 | —31. 8741.42 | +115404189 | +6084111| 49.87 Solution E (depth of compensation 162.2km.) | — 1.8640.38 | + 5.424+1.58 | — 8.02+0.87 | + 8664116 | —2264 68] 18.72 Solution G baepth ofcompensation 113.7 km.) | — 2.3940.38 | + 2.8041.57 | — 6.6640.87 | + 3744115 | —2084 67| 18.47 Solution A (depth of compensation zero) — 3.2540.58 | — 2.2642.40 | — 5.0641.83 | — 7404176 | — 434103 | 43.29 It is to be noted that, in each group, some solution for a moderate depth of-compensation gave a smaller mean square of the residuals than did solution A. This shows that it is possible to derive the depth of compensation separately from each group. From the results here given, supplemented by certain approximate computations, the most probable depth of compensation was derived from each of the first three groups in June, 1905, before the solution for the western group or for all groups combined had been made. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U.S. 141 Most probable depth of compensation. From northeastern group, 146 kilometers, estimated weight 1 From southeastern group, 48 kilometers, estimated weight 4 From central group, 127 kilometers, estimated weight $ Weighted mean, 117 kilometers The weight estimated for the most probable depth of compensation from the northeastern group was made double that for each of the other two groups because the solutions for different depths showed a tendency to a more rapid variation in the mean square of the residuals with respect to assumed depth of compensation. These three values of the most probable depth of compensation differ considerably. But the weighted mean, which was a prediction, is very close to the most probable value derived a year later from the complete investigation, namely, 112.9 kilometers. . No estimate of the depth of compensation was made from the western group. If made it would apparently have been not far from 117 kilometers. Though the derived depth of compensation is much smaller for the southeastern group than for the others, it is not certain that this difference is real. It may be merely a result of the unavoidable errors in the derivation. The writer considers it merely an indication, not a proof, that the depth of compensation is smaller for this region than for the remainder of the United States. , The degree of instability in the determination of the equatorial radius and the square of the eccentricity is shown by the four corrections to the Clarke 1866 values as derived from the best solution for each group. The best solution is considered to be in each case that for which the mean square of the residuals is least. (a) (e*) From the northeastern group +1264+164 meters +.000280+.000097 From the southeastern group —401+108 meters —.000365+.000054 From the central group —183+154 meters —.000267 + .000082 From the western group +374+115 meters —.000208 + .000067 The adopted final results from the complete investigation involving all the observations are (a) = +76+34 meters and (e”) = —.000065 +.000031. If weights be fixed, as is usual, in inverse proportion to the squares of the probable errors, no one of the values of the hen radius derived from separate groups is to be assigned a weight as great as 75 (an 03 of that assigned to the final adopted value. Similarly, no one of the values of the correction to the square of the eccentricity derived from a separate we ‘ * 2 ' group is to be assigned a weight as great as } er of that of the final adopted value. In other words, though the combined group on which the final results depend contains only about four times as many observations as each of the four Beparate groups, because of the effect of increased area, the weights of its results are from 6 to 23 Goes 2 =3) times as great as the weights of those from the separate groups. The total range of the four values of the equatorial radius from the four groups is 775 meters. The difference between the Bessel and the Clarke 1866 values of the equatorial radius is 809 meters. The total range in the four values of e’ is .000645, corresponding to a range of 28 in the reciprocal of the flattening. The difference between the Bessel and the Clarke 1866 values of the reciprocal of flattening is only 4.2. 142 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. CONSTANCY OF.THE DEPTH OF COMPENSATION. It has been assumed in this investigation that the depth of compensation is the same in all parts of the area concerned in this investigation. This area includes all of the United States and large adjacent areas. The area concerned is that over which the computation of topographic deflections was extended, the areas covered by rings of moderate radius (neither very small or very large) being concerned to the greater extent, while the outer larger rings are concerned to a progressively smaller extent. It matters little in this investigation whether the actual depth of compensation is great or small in the distant areas covered only by outer rings in the computation of topographic deflections. This investigation can furnish little or no information as to the depth of compensation in such distant areas. The investigation does, however, furnish some indications of the possible variation of depth of compensation within the limits of the United States. In connection with the preceding paragraph, note that in the table on page 70 of reduction factors a change in the assumed depth of compensation produces a very small change in the factor for the outer ring (No. 1). Note also that for such a ring as No. 11 (outer radius 113.7 kilometers) a change in the assumed depth produces a comparatively large change in the factor F. Thus, for the two cases cited a change of the assumed.depth from 120.9 to 113.7 kilometers makes changes in F of less than .001 and of .024, respectively. Similarly, it should be noted that, for the small ring No. 24 (outer radius 1.13 kilometers), this change of assumed depth makes a change of only .001 in F, and for smaller rings the corresponding change is still smaller. The more sensitive are the factors to change of assumed depth of compensation the more sensitive are the computed deflections to such changes of assumption, and therefore the stronger the determination of the depth of compensation in the area covered by the ring in question. The most reliable evidence obtained in this investigation in regard to the possible varia- tion in the depth of compensation is contained in the residuals of the A, G, H, and E solutions for all of the observations treated as one group. These residuals are tabulated on pages 106-114 and statistics in regard to them are given in the tables on pages 135-139. For any given region the most probable depth of compensation is that for which the mean square of the residuals of the corresponding solution is least. In the table on page 138 showing the statistics of the residuals indicated above, as sepa- rated into four geographic groups, solution E has the smallest mean square of the residuals in the first three groups, and for the western group solution G has the smallest. As solution E was made for an assumed depth 162.2 kilometers, and solution G for 113.7 kilometers, this indicates that the depth of compensation is greater for the other three groups than for the western group. The evidence is very weak, however, since the mean square of the residuals ‘varies less than 2 per cent among solutions E, H, and G, in the northeastern group, and still less in the southeastern and central groups. The determination of the depth of compensation in these three groups is, therefore, weak. The depth of compensation is much more strongly determined in the western group, in which the mean square of the residuals shows a range of variation among solutions E, H, and G of more than 6 per cent. If in the same table on page 138 the mean squares of the residuals for the latitude, longi- tude, and azimuth equations of each group separately are studied, the weakness of the evidence becomes still more apparent. Of the nine tests possible in this way for the three groups, northeastern, southeastern, and central, there are five which show the H or G solution having smaller values than the E solution and thus contradicting the result for the group as a whole. Similarly, if the evidence given by the tables on pages 138, 139 of mean residuals without regard to sign, of percentage of residuals less than 2”.00, of percentage of residuals greater than 5”.00, and of maximum residuals, is studied, it will be found to be weak and to contain many contradictions within itself though its general tendency is to indicate that the depth of compensation is somewhat greater for the other three groups than for the western group. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. S. 143 If the evidence from the ten geographic groups given in the tables on pages 135-137 is examined, it will be found to lead to the same conclusion as above. In the first table, giving the mean squares of the residuals, groups 8, 9, and 10,in the far west, show a minimum for solu- tion G and each of the remaining groups show a minimum for either solution E or solution H. The evidence is weak, as it consists of small differences. Moreover, the evidence from the four tables following this first one is both weak and contradictory, though in general trend it cor- roborates that from the first table. Finally, it may be noted that, as shown on page 141, from the preliminary solutions made separately for the northeastern, southeastern, and central groups, it was predicted that the depth of compensation was 117 kilometers and that this afterwards proved to agree almost exactly with the final result. These solutions indicated, therefore, that the mean depth of compensation is substantially the same for these three groups as far the western group. The evidence from these preliminary solutions for separate groups is weaker than that already discussed, depending upon the final solutions, because in the preliminary solutions the values of the bre unknowns, including (a) and (e”), were determined with much less accuracy than in the final solution. The general conclusion is that while there are indications that the depth of compensation is greater in the eastern and central portions of the United States than in the western portion, the evidence is not strong enough to prove that there is a real difference in depth of compen- sation in the different regions. Possibly such a difference may exist, but it is not safe now to assert that it exists. METHOD OF COMPUTING THE MOST PROBABLE DEPTH OF COMPENSATION. On page 115, after calling attention to the relative values of the sums of the squares of the residuals for various solutions and to the fact that this sum is less for solution G than for any of the others, it was stated that the most probable depth of compensation, as derived from these solutions, is 112.9 kilometers. The method by which this particular value, 112.9 kilo- meters, was derived was not there stated. It will now be given. Having made three complete solutions for three assumed depths of compensation, and having obtained the sums of the squares of the residuals for each, the problem is to compute the most probable depth of compensation, or that depth for which the sums of the squares of the residuals would be a minimum. The concrete case in hand is, having made the three solutions EF, H, and G, for which the sums of the squares of the residuals are, respectively, 8 220, 8 020, and 8 013, and the corresponding assumed depths are 162.2, 120.9, and 113.7, to com- pute the most probable depth. The fundamental assumption made for this purpose is that the curve which has for abscissze the logarithms of the depths corresponding to the three solutions and for ordinates the sums of the squares of the residuals in those solutions, is a parabola, with its axis vertical. This assumption is based on three considerations. First, for small changes of assumed depth of compensation the computed deflection varies nearly proportionally to the variation in the logarithm of the depth; second, the consideration just stated being approximately true, it follows that when the assumed depth is changed the residual for any one station will vary nearly in proportion to the change in the computed deflection, and the curve having logarithms of depth for abscisse and that residual squarea for ordinates will therefore be a parabola with its axis vertical; third, any short portion of the curve referred to at the beginning of this paragraph, which is a composite of many such parabolas, will be nearly a parabola with its axis vertical. This assumption is a generalization of which no proof is offered. Its safety depends upon the fact that it is applied to short portions only of the curve. The proposition made above, that for small changes of assumed depth of compensation the computed deflection varies nearly proportionally to the variation in the logarithm of the depth, is one which it is not easy to prove from the formule. It may be made evident in another way. In the table of reduction factors, see page 70, three columns are headed 162.2, 113.7, and 79.76 kilometers. The logarithms of these Hinde depths are 2.210, 2.056, 144 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. and 1.902, with the successive differences .154. The two successive differences of the factors for corresponding rings, in these columns are Differences of Factors F. Bing. |antiny ouat tors apt tik. etn Coe ee apt 73.76. 26 . 001 . 001 25 . 001 . 003 24 . 003 . 004 23 . 004 . 005 : 22 .005 . . 007 21 . 007 .O11 20 .O11 . 014 19 .014 . 021 18 . 021 . 030 17 . 030 . 041 16 . 041 . 058 15 . 058 . 080 14 . 080 . 103 13 . 103 125 12 .125 135 11 135 124 10 . 124 . 095 9 . 095 . 062 8 . 062 . 037 7 . 037 . 020 6 . 020 . 010 5 . 010 . 005 4 . 005 . 002 3 . 002 . 002 2 . 002 . 000 1 . 000 . 001 These two columns of differences are identical except that in the second column each differ- ence is raised one line higher than in the first.* Hence the mean difference in the second column is identical with the mean difference in the first column. If the values of the computed topo- graphic deflections for separate rings, such as are shown on page 72, into which these factors are multiplied to obtain the computed deflection when isostatic compensation: is considered, were all equal, it is evident that the two successive differences between the computed deflections corresponding to these three depths would be equal, just as the differences of the logarithms of the depths are equal. In other words, the computed deflections would be proportional to the logarithms of the depths. But the computed topographic deflections for the separate rings at a station are not equal. They ordinarily vary somewhat irregularly from ring to ring, with a tendency to increase, as a rule, with increase in the size of the rings, except for the last few outer rings. Hence in the table above, in addition to comparing mean differences, the actual , corresponding differences must be compared. The greatest disparity between corresponding _ differences occurs on ring 9, for which the smaller difference (.062) is nearly two-thirds as great as the larger difference (.095). For other rings the disparity is less. Hence, even though the computed topographic deflection is different for different rings, it is still approximately true that for small changes of assumed depth of compensation the computed deflection varies nearly proportionally to the logarithm of the depth. Of course the smaller the assumed change of depth the closer is the approach to proportionality. Finally, it should be stated that for many cases the law stated has been tested by comparing the deflections computed for various assumed depths of compensation and has been found upon an average to be approximately true (see tables on pp. 48-56). * This arises from the peculiar relation of the factors, in such columns as the three selected, which has already been pointed out on p. 72. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 145 The proposition which has been made, that when the assumed depth of compensation is changed the residual at a station varies nearly in proportion to the change in the computed deflection, may be tested by comparing the computed deflections on pages 48-56 with the residuals on pages 106-114. The abscissa of the lowest point on the actual curve having logarithms of depths for abscisse and sums of squares of residuals for ordinates is obviously the logarithm of the most probable depth of compensation, as for that depth the sum of the squares of the residuals is a minimum. To find the abscissa of the lowest point of a parabola with its axis vertical, having given three points on the curve, two properties of such a parabola were used. The slope of the chord joining two points on the curve is the same as the slope of the curve at the point of which the 8240 ” Z / 8180 o = a oO 8120 . o SUMS OF SQARES OF RESIDUALS Z) 8060 7 ea 8030 al [en H —=O-—7 » MG 2.040 2.060 2.080 2.100 2.120 2.140 2.160 2.180 2.200 2.220 LOGARITHMS OF ASSUMED DEPTHS OF COMPENSATION No. 14. abscissa is the mean of the abscisse of the two points. The slope of the curve at any point is proportional to the distance measured along the horizontal from the lowest point of the curve to that point. The method may be illustrated numerically by the case at hand, namely, the sums of the squares of the residuals in the three solutions E, H, and G being 8 220, 8 020, and 8 013, respec- tively, what is the most probable depth of compensation? On the curve shown in illustration No. 14 having for abscisse the logarithms of the assumed depths of compensation and for ordinates the sum of the squares of the corresponding residuals, the three abscisse corresponding, respectively, to the ordinates 8 220, 8 020, and 8 013 are 2.210, 2.082, and 2.056, these being the logarithms of the depths 162.2, 120.9, and 113.7 kilometers, assumed in solutions E, H, and G. 78771—09—10 146 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Hence the slope of the chord from the point H to the point E on the curve is 8 220—8020 + 200 3.210 — 2.082" +.128 1 * °0° and this is assumed to be the slope of the curve itself at the point of which the abscissa is 2.146 (the mean of 2.210 and 2.082). Similarly the slope of the chord from the point G to the point H is 8020—8013 + 7 3.082 —2.056~ +006 17° and this is assumed to be the slope of the curve at the point of which the abscisse is 2.069. Hence the rate of change of slope of the curve is 1 560 — 269 +1291 2.146—2.069 +.077 The distance from the point having the abscisse 2.069 back to the point at which the slope = +16 800 ; : 269 : é ae “ of the curve is zero is, therefore, 7-399 = .016, and the abscisse of this minimum point (M) on the curve is 2.069 — 0.016 =2.053. The most probable depth of compensation has this logarithm and is 113.0 kilometers.* A consideration of the approximations involved in such a computation shows that the accuracy of the determination of the most probable depth will be greater the nearer are the three points (such as E, H, and G) for which the solutions are made, to each other and to the point sought, and that it is desirable that the point sought shall be included between two of the three points. For these reasons it is important to make a good prediction as early as possible in the investigation of the most probable depth of compensation. The method of determining the most probable depth here described was evolved gradually during the progress of this investigation. The best proof of the general soundness of the method is the fact that successive computations of the most probable depth showed a rapid convergence upon one value, that which has been adopted as final. To obtain an idea of the accuracy with which the most probable depth of compensation is determined by this investigation, seven other computations of it were made in addition to that shown above, using in each case a part of the residuals only, as indicated in the following table: Probable depth of compensation . (kilometers). From all latitude residuals 98 From all longitude residuals 156 From all azimuth residuals 105 From all residuals of the central group 174 From all residuals of the northeastern group 187 From all residuals of the southeastern group (t) From all residuals of the western group 107 This indicates that the probable error of the derived depth of compensation is a few kilometers, not tens of kilometers nor tenths of kilometers. Other tests were made with still smaller groups of residuals which confirmed this conclusion. The general conclusion reached on this point, from all the evidence available, is that for the United States and adjacent areas, if the isostatic compensation is uniformly distributed with respect to depth, the most probable value of the limiting depth is 70 miles (113 kilometers), and it is practically certain that the limiting depth is not less than 50 miles (80 kilometers) nor more than 100 miles (160 kilometers). * The difference, 0.1 kilometer, between this value and the value 112.9 kilometers (see p. 115) obtained from the original computation is too small to be of any importance and is due to a slight difference in the arrangement of the two computations and to the effect of omitted decimal places. + The observations indicated this part of the curve, near the points corresponding to E, H, and G of illustration No. 14, to be nearly straight, but convex upward instead of downward. Hence, no determination of the probable depth of compensation could be made from these three points alone. The A and B solutions (depth zero and depth infinite) for the southeastern group show the general shape of the curve to be concave upward. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 147 UNIFORM DISTRIBUTION OF COMPENSATION, WHY ASSUMED. In order to make it feasible to compute the deflections of the vertical, taking into account both the topography and the isostatic compensation, it has been assumed in the principal portion of the investigation that the compensation is uniformly distributed with respect to depth from the surface to the limiting depth of compensation; that is, it is assumed that the defect of density below any givén portion of an elevated continental area, as compared with the density at the same level below a coast area lying at sea level, is a constant for all levels between the surface and the limiting depth of compensation. It is assumed that below that depth no defect or excess of density exists. This assumption was adopted as a working hypothesis, because it happens to be that one of the reasonable assumptions which lends itself most readily to computation, and because it seemed to be the most probable simple assumption. It was necessary to make some assumption as to the distribution of the compensation with respect to depth. Various considerations should influence one in deciding which of various possible assumptions is probably nearest the truth. Near the surface of the earth to the depth to which the observations of temperature have been extended, the temperature has been found to increase about 1° C., upon an average, for each 30 meters increase in depth. The temperature probably continues to increase down to the limiting depth ‘of compensation, though possibly the average rate of increase in the lower half of the interval is less than that stated. At 100 kilometers below the surface very high temperatures must exist, so high as to tend to decrease the strength of the material, assuming for the moment that the material is all solid. With increasing temperatures and, consequently, decreasing strength, as the depth below the surface increases, one should expect stress differences of a given intensity to produce motion, or stresses to produce a change of volume, at the greater depths rather than near the surface. If this consideration alone be taken into account, the readjustment of material to eliminate stress differences and equalize stresses, and, consequently, the isostatic compensation, should be expected to increase with increase of depth. If, however, a consideration of the probable increase of temperature with increase of depth leads one to believe that, in spite of the tendency of increase of pressure to raise the melting points of the various materials composing the earth, the material below a critical depth is liquid while that above is solid, his belief as to the probable distribution of compensation with respect to depth must be correspondingly modified. In this case one may conceive that a light solid crust is floating on a heavier liquid substratum. On this basis the thinner portions of the crust (oceanic areas) should have their upper surfaces lower and their lower surfaces higher than the corresponding parts of the thicker portions of the crust (continents), this being the necessary condition of stability, as for an indefinitely extended floe of ice. It is here assumed that the crust itself has the same density at all parts at a given level, this being the assumption ordinarily made by those who deal with a crust floating in a liquid substratum. The isostatic compensa- tion in this case all occurs near the bottom of the crust. It extends, below a continental mass, from the level of the lower surface of the crust under the seacoast down to the bottom of the actual crust under the continental mass. It is represented by the protuberance on the under side of the crust, which corresponds for equilibrium to the continental protuberance above sea level. The isostatic compensation on this basis is therefore zero down to a deep level, and from that point is uniform down to a still deeper level, at which it suddenly changes to zero again. G. H. Darwin, in his memoir entitled ‘‘On the Stresses Caused in the Interior of the Earth by the Weight of Continents and Mountains,’* assumes that the earth is a competent, elastic structure, solid throughout. Upon that assumption the stresses which must exist in it, due to the weight of the continents and mountains, are computed. The computation indicates that continents of such dimensions and form as those now in existence would produce stress-differ- ences which would increase from the surface downward to a maximum at a depth of from 600 to 1 000 miles, said maximum being as great as 4 tons per square inch. On this basis the failure of the material on account of the stress-differences, due to the continents, would be more apt to * Philosophical Transactions of the Royal Society of London, 1882, vol. 173, pp. 187-230. 148 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. occur at greater depths than near the surface, and therefore the readjustment of material and the isostatic compensation should increase with increase of depth for the first 600 or 1 000 miles below the surface. Darwin’s computation also indicates that the stress differences produced by parallel mountain chains, separated by valleys, increases rapidly with increase of depth, down to a maximum at a depth which is ;, part of the distance between adjacent mountain chains. Thus, if two parallel mountain chains are 500 kilometers apart, the depth at which the maximum occurs will be 80 kilometers. Two such mountain chains, rising 4 000 meters above the intervening valley bottom, would produce maximum stress differences of 2.6 tons per square inch on the assumption stated. According to this point of view, the stress differences due to the existing inequalities of the earth’s surface must increase with increasing depth to several tens of kilometers below the surface. Beyond this depth at which the maximum stress differ- ences must occur, according to Darwin’s computation, and which is directly proportional to the distance between successive ridges, the stress differences must decrease slowly with increase of depth. On this basis the failures under stress, readjustment, and consequent isostatic com- pensation, due to the inequalities in the surface of the continents, should increase with increase of depth to a level far below the surface and then slowly decrease. The distribution of stress differences was computed by Darwin, with the results stated above, upon the assumption that every part of the earth is a competent, elastic structure, that is, that no failure under stress occurs at any point. If failure does occur, at any point, per- manent deformation of the material takes place, there is readjustment in form and position, and the stress differences below that point are reduced in amount, the stress-differences pro- duced by the inequalities of load at the surface not being transmitted below that level with their full value corresponding to purely elastic deformation. Hence, in so far as failure and readjustment occur at a given level, they tend to prevent similar failure and readjustment at lower levels. From this point of view it appears that isostatic compensation should increase with increase of depth to a short distance only below the surface and beyond that should decrease with increase of depth. In the preceding paragraphs it is tacitly assumed that failure (yielding) under the stress differences at certain levels is accompanied by isostatic readjustment produced either by bodily transfer of material or by changes of density, or by both, at that level. Prof. T. C. Chamberlin has reached the conclusion that, according to the planetesimal theory of the formation of the earth, the isostatic compensation would be greatest at a level slightly below the surface, and from that point would decline at a varying rate, which rate increases rapidly at first and at greater depths decreases slowly, approaching zero at great depths. This peculiar distribution is indicated as being due to differential weathering and vulcanism during the process of growth of the earth.* These different considerations as to the probable distribution of isostatic compensation with respect to depth are not mutually exclusive.. It may be that the actual distribution is a resultant of several or all of the actions and modifying influences which have been indicated briefly. Some of these tend to produce a uniform distribution of isostatic compensation with respect to depth, some to produce a maximum of isostatic compensation near the surface, some to produce it at moderate depths neither very near the surface nor very near the limiting depth of compensation, and some tend to produce a maximum near the limiting depth. There- fore it has seemed that the most probable simple assumption is that the compensation is uni- formly distributed from the surface to the limiting depth. It is not supposed that at the limiting depth of compensation there exists a perfectly abrupt change of conditions with respect to compensation. But it is believed to be possible that the decrease of isostatic compensation from its mean value to zero may all take place within so small a range of depth that the difference between the actual mode of distribution and the abrupt change postulated in the stated assumption may not be capable of detection by the geodetic observations. Hence, it is deemed justifiable to make the assumption in the form stated, which is such as to lend itself most readily to computation. * Journal of Geology, 1907, p. 76, and Geology, by T. C. Chamberlin and R. D. Salisbury, Vol. II, pp. 107-111. THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. 149 It should be noted that each of the considerations brought forward in the preceding para- graphs indicates that the isostatic compensation must be sensibly limited to some finite depth much less than the radius of the earth. The writer knows of no plausible conception of the conditions within the earth that would lead one to believe that isostatic compensation extends to the center. Hence, throughout this investigation it is assumed that, whatever the mode of distribution of the isostatic compensation with respect to depth, it extends to a limiting depth which is but a small fraction of the radius. While it seems desirable, in connection with the present investigation, that the first assump- tion made as to the distribution of compensation should be a reasonable one and, preferably, that it should be the most probable simple assumption, it was evident, at the outset, that a failure to make the best selection would not be fatal to ultimate success and would probably not even hamper the investigation. It appeared on the preliminary reconnaissance of the problem that the most efficient method of attack is probably to make some one assumption as to the distribution of the isostatic compensation with respect to depth, to make full compu- tations on this assumption, and then to test other assumptions by comparison with this one; not by complete new computations, but by a computation of the small differences in computed deflections produced by the change from one assumption to the other. As various assumptions were to be tested, it was not of paramount importance which should be tested first. Relative ease of computation was properly one of the controlling elements in making the choice. POSSIBLE COMPENSATION IN A 10-MILE STRATUM. It seemed desirable to test the assumption that possibly the compensation is complete and uniformly distributed through a comparatively thin stratum, lying at a considerable depth below the surface. It was decided to assume the thickness of said stratum to, be 10 miles (16 kilometers) and to derive the most probable depth for it. Such an assumption corresponds roughly (see p. 147) to the hypothesis that the outer por- tion of the earth is a light solid crust floating on a heavier liquid substratum. The correspond- ence is not exact, for under said hypothesis the thickness of the stratum within which the compensation occurs is variable, and the mean depth of compensation is variable, as indicated in the discussion later in this publication under the heading, “The floating crust hypothesis,” page 163, whereas, under the assumption to be tested, the thickness is constant (10 miles), and the depth (to be derived) is assumed to be constant. As a first approximation, it was assumed that the compensation occurs in a stratum 10 miles thick of which the bottom lies at the depth 35 miles. Let A be a uniform distribution of defect (or excess) of density through the depth 35 miles from the surface and with the defect (or excess) 3.5 times as great as that necessary for complete compensation within said depth. Let B be a uniform distribution of defect (or excess) of density through the depth 25 miles from the surface and with the defect (or excess) 2.5 times as great as that necessary for com- plete compensation within said depth. Then the uniform distribution of density which will produce complete compensation within the specified stratum lying between depths 25 and 35 miles, and which will for convenience be called X, is the exact equivalent of the difference, A-B, of the two distributions, A and B, or X = A-B. This may be made clear by a reference to page 68. It is there shown that for complete compensation between the surface and depth h,, dh=—0,h,, or 6,= aid in which 0, is the 1 compensating defect of density, h is the mean elevation above mean sea level of the surface in the area under consideration, and 0 is the mean surface density of the earth (2.67). In the assumed distribution A, h,=35 miles and the corresponding value of 0, is— 0: The assumed defect of density in distribution A is stated to be 3.5 times this and is, therefore, h h 0,= 8.5350 = — 758 Similarly, the assumed defect of density in distribution B is h h =—2.5—6= 0 Os 2.5550 ~ 10 150 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. Therefore, when the distribution B is subtracted from distribution A, it leaves no compen- sation between the surface and the depth 25 miles, but does leave the compensating defect of density, 0,= 19% uniformly distributed between depth 25 and depth 35 miles. This is exactly the value of 0, necessary for complete compensation in a stratum 10 miles thick, as may be seen by comparing with the formula 6,= — as with h, =10 miles. hy If the area in question is a plateau at an elevation of one mile above mean sea level —2. : : i. ‘ 0, =03= — a = a =: —.267. This compensating defect of density in the stratum ten miles thick between depth 25 and depth 35 miles, under one square mile of the surface, is exactly equivalent to a defect of mass of one cubic mile of material of density 2.67, which i is equal to the excess of mass above mean sea level. The compensation is, therefore, complete. Let Fx be a reduction factor, similar to the reduction factors F shown on page 70, such that if the topographic deflection D is multiplied by F, the product DF, is the resultant deflec- tion D+Dx, due to both the topography and the compensating defects or excesses of mass supposed to lie between depths 25 and 35 miles, according to the assumed distribution of density which has been called X. As indicated on pages 69, 70, the deflection due to the compensation is necessarily opposite in sign to the topographic deflection, As there the deflection due to compensation alone is —D(1—F), so, in the case in hand, it is D, = —D(1—F,). Let F,, and F,, be such factors F as are shown on page 70 for complete compensation extending from the surface down to the depths 35 and 25 miles, respectively. Then the deflec- tion due to complete uniform compensation extending to depths 35 miles would be — D(1—F,,) and that due to the assumed compensation called A would be 3.5 times this, or —3.5D(1—F;,,). Similarly the deflection due to the assumed compensation called B would be —2.5D(1—F,,). Hence the deflection due to the assumed compensation called X, which is equivalent to A—B, is D,= —{3.5D(1 —F,,) —2.5D(1 —F,,)} = — Dj1 —(3.5F,; —2.5F,;)} By placing this equal to the value of Dx, already derived, namely, —D(1—F;), it appears that F, =3.5F,, — 2.5F 95 The factor F, was computed by this formula for each ring of topography. To do this it was necessary first to obtain F,, and F,, for each ring of topography, as they had not yet been computed, though F had been computed for various other depths, as shown on page 70. It was possible to compute F,, and F,, directly from the formula for F, shown on page 70. The final computations of their values were made, however, by a much shorter process, given later in this publication under the heading, “Various reduction factors obtained graphically,’’ see page 154. The reduction factors F, so computed are shown below in comparison with the factors F, taken from page 70, for uniform compensation extending to the depth 113.7 kilometers (70.67 miles). tk Factor F for Factor F for Ring. Factor Fx. | depth 113.7 | Difference. Ring. Factor Fx. | depth 113.7 | Difference. kilometers. kilometers. 27 1. 000 - 997 +. 003 13 . 656 . 618 ++. 038 26 1. 000 - 996 +. 004 12. . 464 . 493 —. 029 25. 1. 000 - 995 +. 005 11 . 292 . 358 —. 066 24 . 999 . 992 +. 007 10 . 168 . 234 —. 066 23 - 998 - 988 +. 010 9 . 092 . 139 —. 047 22 . 996 . 983 +. 013 8 . 045 077 —. 032 21 . 995 . 976 +. 019 7 . 023 . 040 —.017 20 . 994 - 965 +. 029 6 . 010 . 020 —. 010 19 - 993 . 951 +. 042 5 . 008 . 010 —. 002 18 - 988 . 930 +. 058 4 . 004 . 005 —.001- 17 . 982 . 900 +. 082 3 . 002 . 003 —. 001 16 . 966 . 859 -+. 107 2 . 000 . 001 —. 001 15 - 912 . 801 +. 111 1 . 000 - 001 —. 001 14 . 816 -721 +. 095 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. Ss. 151 It appears that the factor F, for each ring of topography is not very different from the factor F for depth 113.7 kilometers for that same ring. If the agreement were cxact for each ring, the computed total deflection for a station with the compensation assumed to be complete and uniformly distributed to depth 113.7 kilometers—namely, the sum of the quantities DF (one for each ring), see page 72—would be identical with the computed total deflection for a station with the assumed compensation X—namely, the sum of the quantities DF. It would then be absolutely impossible by observations of deflections of the vertical to determine which of these two assumed compensations is nearer the truth. As the agreement of the two factors for each ring is close, it is evident that it will be difficult to discriminate by observation between the two kinds of compensation. To obtain more concrete evidence on this point, it was decided to compute the resultant deflections due to topography and to the assumed compensation X for ten meridian and ten prime vertical stations, selected as being fair representatives of the 496 such stations. The following table shows the selection of stations made, the computed deflections with the X compensation considered, the computed deflections corresponding to solution G—that is, with complete compensation uniformly distributed from the surface to depth 113.7 kilometers—and the small differences between the two sets of computed deflections. Computed deflection. MERIDIAN STATIONS. e Com; pensation No. Name. compensation. depth 113.7 Difference. kilometers. 152 | Mount Independence, Maine — 1.69 — 1.80 +.11 169 | Howlett, New York + 3.48 + 3.33 +. 15 100 | Hill, Maryland — .45 — .59 +.14 123 | Sawnee, Georgia — 2.66 — 2.72 +. 06 189 | Thones Hill, Michigan + 2.27 + 2.14 +.13 76 | Weed Patch, Indiana — .29 — .30 +.01 60 | El Paso E. B., Colorado — 3.80 — 3.49 —.31 35 | Ibepah, Utah + .56 + .68 =, 12 17 | Yolo NW. B., California — 1.51 — 2.03 +. 52 238 | Santa Barbara, California —12.13 —12. 35 +. 22 PRIME VERTICAL STATIONS. 163 | Mount Independence, Maine —3.14 —3. 33 + .19 175 | Howlett, New York + .70 + .66 + .04 103 | Hill, i — .75 —1, 02 ae 27 128 | Sawnee, Georgia — .14 — .15 + .01 202 | Thones Hill, Michigan —2, 23 —2.14 — .09 81 | Weed Patch, Indiana + .61 + .62 —.01 62 | El Paso E. B., Colorado E —6. 03 —6. 67 + .64 35 | Ibepah, Utah : —2. 37 —2. 28 — .09 14 | Yolo NW. B., California + .27 +1. 40 —1.13 229 | Santa Barbara, California +3. 95 +4. 76 — .8l Sum without regard to sign, 49. 03 52. 46 Of the twenty differences for these representative stations, only three are greater than 0”.50, and only one is greater than 1”.00, whereas the average residual without regard to sign from the G solution, in which the isostatic compensation is assumed to be complete and uni- formly distributed from the surface to the depth 113.7 kilometers, is 3”.04 (see p. 136), and but 43 per cent of these residuals are less than 2”.00 (see p. 136). It is evident that the agreement of the deflections computed with the X compensation with those computed with the G com- pensation is well within the limits of the accidental errors, and that it is therefore very difficult to ascertain which of the assumed compensations is nearer the truth. The assumption of the particular depth 35 miles for the bottom of the stratum 10 miles thick within which compensation X is assumed to lie, was made after a preliminary reconnais- 152 THE FIGURE OF THE EARTH AND ISOSTASY FROM MEASUREMENTS IN U. 8. sance of the problem, in which an attempt was made to predict the assumed depth for which the agreement with the G compensation would be closest. An examination of the preceding table shows that the deflections computed with the X compensation are on an average slightly smaller than those computed with the G compensation. Since, with any assumed distribution of compensation, the greater the assumed depth the larger are the computed deflections (see pp. 70-72), it appears that a closer agreement would be obtained if the assumed depth were made slightly greater. It was decided as a second approximation to increase the assumed depth by 2 miles—that is, to try a compensation, which will be called X,, complete and uniformly dis- tributed through a stratum 10 miles thick with its bottom at the depth 37 miles. The computations were then made for this compensation, X,, precisely as outlined above for compensation X. The factors F,, and the corresponding deflections were computed and the deflections compared with those for compensation G. The factors F,, were found to be as follows: Factor F for Factor F for Ring. | Factor Fx. | depth 113.7 | Difference. Ring. | Factor Fx. | depth 113.7 | Difference. kilometers. kilometers. 27 1. 000 . 997 +. 003 13 . 693 . 618 +. 075 26 1. 000 - 996 +. 004 12 . 504 . 493 +. 011 25 1. 000 - 995 +. 005 11 3815 . 358 —. 043 24 1. 000 . 992 +. 008 10 . 187 . 234 —. 047, 23 ~ 1. 000 - 988 +.012 9 .101 . 189 —. 038 22 1. 000 - 983 +. 017 8 . 049 077 —. 028 21 1. 000 . 976 +. 024 7 . 025 . 040 —.015 20 1. 000 . 965 +. 035 6 . 011 . 020 —. 009 19 - 998 . 951 +. 047 5 . 006 . 010 —. 004 18 . 993 . 930 +. 063 4 . 003 . 005 —. 002 17 - 987 - 900 +. 087 3 . 001 . 003 —. 002 16 . 976 . 859 +.117 2 . 000 - 001 —.001 15 - 925 . 801 +. 124 1 . 000 . 001 —. 001 14 . 840 .721 +. 119 These factors are also shown graphically on illustration No. 15 at the end of the volume. The differences, taken in the same manner as in the table on page 151, between the deflec- tions computed with compensation X, and the corresponding deflections computed with com- pensation G, were found to be as follows: Xe deflection X2 defiection minus minus G deflection. G deflection. Meridian stations: Prime vertical stations: 177 178 INDEX. Page. Page. Creoid contours, CONSITUCTION Oleic... ces ia ceanecemcnees vhndines dond Be | PeOsals, SUID PE Olin isccmss scannne darned teunvs Rolwane eUUaaa? 132 illustration showing...........-.-....--.-20--22000+ sum of squares from various solutions.............- Oe ieccksh gov 114 Iimitedaln accubaeyas.s cee osci sec casement een seReee table of statistics............-2---..-..20-- 135 relation to topography Rigidity versus isostasy...........-..----+-- 166 Geoid surface, definition of 58 | Rings, inner, effect of omission of. 125 Graphic method of obtaining various factors...............-.---- 144 outer, effect of errors in interpolation of....................---- 131 Gravity determinations not used..............-... 02-200 e eee eee 10 outer, interpolation for..............-2.-2eee eee eee eee eee 39 Grouping of signs of residuals 133 PALO Ol Pals osceen 4 sxeessen: oh vino een see eel ddlidneietis seeautealeaTeins 21 illustrations showing................-...-- Nos. 11 and 12, in pocket | Rules for acceptance of interpolation...........-.2--..22ssseee0++ 43 Initial azimuth, why two corrections to.............-...--+.---7. 101 | Slope,.correction for ............02-sececeececeeeeeeccececeeeecees 34 Inner rings, average value of computed topographic deflections... 127 correction for, not often required...... . 36 effect of omission of..................- 124 PABLO Secisuneoa ec cane ha Wench ees o 35 Interpolation, effect of errors due to 130 | Solutions, assumptions for each..........2-2-2+2e2eeeeeeeree seer 92 for outer rings, criteria for acceptance..............--.-..2+-- +++ 43 | Solution C, land only uncompensated.................2-202+-202+ 168 XAMD|C Of. 20... 02. - sees sere ee eee eee eter cet ete ee etree cena 40 ) Solution G, adopted............2....00eececeeeeeeeeeeeeeeeeenees 114 illustrations of 41 | Solutions in four geographic groups............220202000e00s0eeeee 140 ‘ method of 39 | Station errors. See Deflections. possible 36 | Stations, astronomic.............22--22202eeceeeeeeeeeeeeeeteeeee 12,17 saving of labor 44 | Statistics of residuals...............2-02e00e0220eeeeceeeseeeeeeees 135 Isostasy, defined ......-...+...2-2.-+2+ 01s 0ee eee e eters eee eecee eee 66 | Stress-differences, defined............. 164 increased accuracy due to recognition of....................+--- 171 reduced by compensation 166 must be considered............-+-+-++2+2e0eeeeeeeetees esse esee 85 | Summary of conclusions..........2..20-2--02ceceeeeeeeeeeeeeeees 174 versus extreme rigidity Isostatic compensation, defined Tables, astronomic observations ...........-.-..2-.-2+2+2+-e2e00+ 12,17 deflections computed with. ...........0..... 02.222 eee eee eee eee coincident longitude and azimuth stations ..........-.........- 122 degree of completeness of.......-... Fialacstacshaisiafesaesansnavers esavatbrnids saci 164 completeness of compensation..........- -- 165 readjustment in progress..........-.--2.220 eee eee eee eee eee 166 deflections, observed and computed . eis 48 Land only uncompensated, solution C............2-.......0ee00+ 168 various depths of compensation....... -- 48 TADIACS POLS: ccc jac ansecais acdc didecoseaiaidecw iaeaccaceade 121 dimensions of the earth, various values............--------+--+- 172 Latitudes, astronomic and geodetic... 12 geodetic latitudes. ..............-2--+-2e-sseeeee Sass roceieia 12 Limit of topography considered ....... 29 geodetic azimuths and longitudes. 17 Longitudes, astronomic and geodetic 17 Laplace points ; 122 limit of topographic deflections from outer rings...............- 125 Maps and charts, effect of errors in....-...-.2.---22+---22e0-e eee 124 maximum value of topographic deflection from each ring...... 126 Maximum residuals in various groups........-...--.---------+- 137,139 mean residual, various groups Maximum value of deflection from each ring, table of........-...- 126 normal equations............--.--- Mean square of residuals in various groups.....-.....-...---.-+ 135,138 observation equations. ............2.2--2202eeeeeeeeeeeeeeeceees Meaning of coefficients in observation equations............--...- 101 observed deflections.........-----2-ceecece cee ceccecececcecenees Mercator charts, ROW US8dsiec 0. 0ccuawiex eeeeaayeres - 28 PACA OfTINBS cisweserc xeraanasnaanearaes aaecesatecmaaar bens Method of constructing geoid contours.........-.----------+++--+- 57 reduction factors for Chamberlin compensation advantages and limitations of. ..........-..--2-2-22---e- eee eee 62 for compensation in 10-mile stratum...................- Normal Gq uiationss 2ai.cauyyh dele deeiidan cheese eoowtace eee 105 uniformly decreasing compensation wwueeuwen dew iy veimenda eee for various depths of compensation...........-.......-----+-- Observed deflections at 507 stations. .........2....-...2.2--0220-- 48 residuals from five solutions Observation equations, from 507 observations.............-.-.--- 93 residuals, statistics............ FOTIMAORS oi. cisisreasie cee nese sidees sow iSin ees eee ae eine arcsec 74 results from solutions in four geographic groups... Outer rings, criteria for interpolation for.. 43 results of solution C.........-.2-2- 0c cee ce cece e cece eeeececerenee 168 effect of errors due to omission of. . --- 131] saving of labor by interpolation of outer rings................ 44 interpolation for’: ..:...)..0i.eeraxteves neeeeeees Seas saeeseeemens ¥ 36 slope corrections 35 Planetesimal hypothesis, compensation according to........--- 159, 167 statistics of residuals -- 185 Probable errors of adopted values of unknowns..............----- 115 topographic deflections : abe Gan ; Poem plapess.pxc2vecseciweves wasaeiesass peeeawee siecle cece gates ence eels 23 22 | Ten-mile stratum, compensation in-...........-.-2.-22.0.2 eee e ee 149 - 22] Pests of reliability of conclusions _ 185 Ratio of radii of rings, arbitrarily adopted.............--..------- 21 | Topographic deflections, accuracy of computation................ 123 Reasons for adoption of solution G...........---.---------++++++- 114 At 507 Statlonsy ccc deccas teeaiocetascdenaesaeecece deer endeameaun 48 Reasons for assuming uniform depth of compensation. . --- 147 effect of errors due to method of computation .. 129 Reduction factor curves, illustration show ing........ No. 15, in pocket examples of computation........ 1, 32, 33 Reduction factors, for Chamberlin compensation 161 formule for computation . . . om 20 for compensation in a 10-mile stratum.........-..--+---+----- 150, 152 method of computing......-.-.--- 222-22 0..2seeeee eee e cece 20 for uniform compensation..........--.------2.eee eee eee eee ee 70 use of templates in compuling............22222.22-2.eeeeeeeeeee 23 for uniformly decreasing compensation 158 valuable features of method of computing 45 for various depths of compensation...........---- 70 | Topography, distant, effect of. 131 obtained graphically .........--.------+-++++++- 154 included, limit of distance... .. ‘ 29 Reliability of conclusions, test of...........-----+-----+++---+2e- 135 | Triangulation involved in this investigation.....................- 10 Reports on triangulation involved in this discussion.........-...- 10 seals : Residuals, conclusions from 132,139 | Uniform distribution of compensation, why assumed............. 147 from adopted solution, illustration showing........ No. 10, in pocket | Uniformly decreasing compensation...............- --- 156 from five solutions of 507 observation equations..........---.-- 106 | Unknowns, adopted values OL ssecdansaspivniet at steteci ootentadne cee 115 BLOUIPINE Of og So-cis eins coil cave PRM ReEeeR sepa sens HeoCRES 133 values of, from various solutions. ............--..-+-.-+4- 105,140, 168 maximum, in various groupS......-.-----------+--22 2-2 eee 137,139 ; ; ; ; percentage less than 2”.00. . . ‘Water compartments, in computation of topographic deflections.. 27 percentage greater than 5”.00 Weight of azimuth equations, reduced, effect of................-- 122 srs eer mainly eee mene pan anes