01 1 n- ■Qs 4\ O CORNELL UNIVERSITY LIBRARY The date shows when this volume was taken. To renew this book copy the call No. and give to the librarian. Wb« ■■f-¥fim~^ -^]7- ,.oeir55M4 mF-wwn H^J.... HOME USE RULES All books subject to recall All borrowers must regis- ter in the library to borrow books for home use. All books must be re- turned at end of college year for inspection and repairs. Limited books must be returned within the four week limit and not renewed. Students must return all books before leaving town Officers should arrange fot II rMP% r* unicers snouia arrange loi ...nOv.. ./ 1S2T- ^^^ return of books wanted - ■ • during their abs^ce from during town. Volumes of periodicals and of pamphlets are held in the library as much as '" oossible. For special pur- "■poses they are given out for a limited time. Borrowers should not use their library privileges for "• the benefit of other persons. Books., of special value and gift books, when the , , . giver wishes it, are not allowed to circulate. ^ Readers are asked to re- port all cases of books marked or mutilated. Cornell University Library QB 415.H31 V.1&3 Manual of tides ... 3 1924 004 941 773 Do not deface books by marks and writiiUE. \< ^« Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004941773 TREASURY DEPARTMENT U. S. COAST AND GEODETIC SURVEY ^W. y^. DUFFIELD SUPBBINTENDKNT PHYSICAL HYDROGRAPHY MANUAL OF TIDES PART I By ROLtillSr A. HARRIS APPENDIX No. 8— REPORT FOR 1897 WASHINGTON aOVE^NMENT PBINTINa OFFICE 1898 TREASURY DEPARTMENT U. S. COAST AND GEODETIC SURYEY ^W. 'W. DUFFIELD SUPERINTENDBNT PHYSICAL HYDROGRAPHY MANUAL OF TIDES Part I By ROLLIN A.. HA-RRIS APPENDIX No. 8— REPORT FOR 1897 WASHINGTON GOVERNMENT PRINTING OPPIOE 1898 (V UH Y K.57lt)6^ B^->i' f^ tii^u- LrorJ^fu^j ^/>^tn;,utfs APPENDIX NO. 8—1897. MANUAL OF TIDES. Part I. INTRODUCTION AND HISTORICAL TREATMENT OF THE SUBJECT. Bv ROLLI]Sr A. HARRIS. SulDinitted for pulDlication. ZSTovember 15, XS97. 319 [MANUAL or' TIDES.] PREFACE TO PART I. When the plan of this Manual of Tides was proposed it was considered best to prepare Part III in advance of Parts I aud II. The reasons for this are stated in the preface to Part HI, which appeared in the Report for 1894, where a brief outline of the several parts may be found. Before attempting to point out the contributions of individuals to the subject of the tides, it has seemed best to give, as an introduction: (1) the defluitions of terms of common occurrence; (2) a clear idea concerning the movements of fluid particles in simple wave motion; (3) a popular account of the cause of the tides; (4) the general properties of tides, and tidal inequalities together with means of ascertaining them. In the chapters which then follow no attempt is made to include the histories of those sciences (e. g., astronomy and hydrodynamics) with which all study of the tides is closely connected, nor is it even attempted to give anything like a catalogue of tidal workers, or a full account of their works; but these chapters aim to give, in a nearly chronological order, some account of such results, work, or theories as may seem worthy of notice, generally because they mark some advance in the development of the subject, but sometimes either because they illustrate certain errors into which individuals have fallen or simply because they show the notions which have been entertained in the past. As a rule direct quotations are taken in preference to comments whenever they seem to serve the purpose in hand. A vague rule for deciding how far to describe or carry out the work of those individuals who have made extensive or profound investigations in connection with the tides, has been to either omit or to barely state such portions of their work as will jirobably be resumed or at least referred to in subsequent parts of this manual. But those portions of their work which will probably not recur, or not recur unchanged in form, have been given or described in greater detail, especially if they are well known or useful. Concerning the work of Thomson and Darwin, it may be said that a large portion of it will of necessity appear as we proceed. When this is not the case, reference will be made at proper times. For this reason it has seemed unwise to here attempt any comprehensive or minute account of their labors. 6584 21 321 [MANUAL OF TIUES.] CONTENTS OF PART I. Chapter I. DEFINITIONS. Page. Tide, range, interval, establishment 32,5 Spring tides, neap tides, phase or semimeusual inequality, parallax and deoliuational inequalities 326 Dinrnal inequality and quantities connected with it 328 Planes of reference 380 Tidal constants, harmonic components 331 Cotidal lines, tide tables 332 Tidal currents, their representation 334 Tidal theories - - 335 Bore, agger, race, seiche, great sea wave 336 Chapter II. DIGRESSION ON PLANE, OK TWO-DIMENSIONAL, WATER WAVES. Fundamental equations 338 Waves in a canal of uniform depth and indefinite length — Displacement equations, paths of water particles, velocity of fluid particles, velocity of propagation 340 Equation of wave profile 34 1 Distinction between ordinary and tidal waves 345 Water waves compared to polarized light and to sound ; reflection of water waves 346 Wave motion propagated up a canal closed at one end, stationary waves, the seiche period, paths of fluid particles 347 Effect of variable cross-section on range of tide 349 Hydraulic considerations - 350 Chapter III. ON THE ORIGIN OF TIDES. Tide-producing force 353 Tides in a small body of water, graphic process 355 Self-returning equatorial canal 357 The tides of the ocean cannot agree with the equilibrium theory 359 Tides in the case of nature 359 Chapter IV. general PROPERTIES OF TIDES AND MODES OF REDUCTION. General properties, the equilibrium theory, harmonic analysis 363 Nouharmonic reductions, first reduction 368 Periods of tidal inequalities 369 Analysis of tidal inequalities 371 Nonharmonic constants 375 Phase reduction, rough predictions 378 Types of tide - 383 322 CONTENTS OF PART I. 323 Chapter V. TIDAL WORK AND KNOWLBDGB BBFOKE THE TIME 01'' NEWTON. Page. Earliest known references to tides by Greek writers 386 Strabo 389 Pliny 392 Other Latin writers 393 Early tide table, Soaliger, extracts from Haklnyt's Collection of Early Voyages 394 Bacon, Gilbert, Kepler, Galileo ! 397 Eiccioli, Balianus, Horrox, Descartes 401 Varenius 402 Wallis... 404 Moray, Colepresse, Philips, Sturmy 406 Childrey, Flamsteed, Halley 407 Chapter VI. NEWTON TO LAPLACE. Newton 410 Bernoulli 415 Maclaurin, Euler 420 Lalande, St. Pierre, Bennett 421 Chapter VII. LAPLACE. Account of the work of Laplace 422 Outlines of his tidal theory 426 Chapter VIII. WORK SINCE THE TIME OF LAPLACE. Young, Weber 438 Palmer, Kennie 439 Lubbock 439 Whewell 440 Fitz Eoy, Challis, Eussell, Kerigan 442 Airy 443 Stokes, Beechey 451 Bache, Meech, Pourtales, Schott, Avery, Mitchell, Marindin 452 Houghton, Parkes, CroU, Abbott, Garbett, Taylor, Chapman, Schmick, Lentz, Gomoy, Houzeau and Lancaster, Basset, Lamb, Harkness, Baird, d'Auria 454 Ferrel 455 Thomson 462 Darwin 464 Historical sketch of the harmonic analysis 465 APPENDIX NO. 8 — 1897. MANUAL OP TIDES— PART I. INTROD UOTION AND HISTORICAL TREATMENT OP THE SUBJECT. By RoLLiN A. Harris. CHAPTER I. DEFINITIONS. 1. The principal movements of the sea may be divided into three classes : Ordinary or wind waves, tidal movements, and ocean currents. The essential feature of any tidal movement is, as the name implies, its periodicity. The period may not be of constant length, bat if variable it must follow some conceivable law. In conformity with this notion the word tide may be defined as the periodic rising and falling of oceanic and other large bodies of water, due mainly to the attraction of the moon and sun as the earth rotates upon its axis. This rising aud falling is accompanied by and depends upon a lateral or horizontal movement of the waters; such move- ments are called tidal currents. Their periodic character distinguishes them from ocean currents. Remarkable stages of the water level at a given place, whether due to earthquakes, gales, or other causes which probably have no definite law of recurrence, although popularly known as "tidal waves," can not be regarded as belonging to tidal phenomena. Ou the other hand, the stages of a river, if periodic in their nature, may with propriety be included in its tides. 2, The tide rises until it reaches a maximum height called high water, and then falls until it reaches a minimum height called low water. These two phases of the tide may be spoken of as the tides. The difference between a high and a low water is called a range of tide, and so is independent of absolute heights; its average value is called the mean range {M.u). Por a few minutes before and after high or low water, it is difflcult to observe any vertical motion in the tide. While thus apparently stationary, the tide is said to stand. In this connection see §15. For reasons to be given later, based upon the fact that the tides are due chiefly to the differ ence between the moon's attraction upon the enveloping sea and the earth as a whole, one would expect that at most tidal stations two high waters and two low waters would occur each lunar day; in other words, to each transit of the moon (inferior as well as superior) there would correspond one high water and one low water. On an average, the time of high water at a given station follows the time of transit by a certain number of hours and miuutes, called the high-water interval (HWI), or high-water lunitidal interval, or corrected establishment. In like manner the low-water interval (LWI), or low-water lunitidal interval, indicates, unless otherwise specified, the average number of hours and minutes between the time of transit and the time of low water. If intervals at some particular time are meant they should be properly distinguished by name or otherwise; or the average values may, for distinction, have the word mean prefixed to their name. The establishment or vulgar establishment is the (apparent local) time of high water occurring at new or full moon; or, preferably, and what is about the same thing, the high-water interval when the transit (just preceding the tide) occurs at noon or midnight.* *Cf. Lalande, Astronomie (1771-1781), Vol. IV., pp. 43, 314-319; Whewell, Phil. Trans., 1833, pp. 163, 229, 230; Lubbock, ibid., pp. 19-21; Eaper, Navigation (1840), p. 261; Darwin, Admiralty Manual, 5t.h ed., p. 55; Wharton, Hydroi'raphic Sur\eying (1882), pp. 145, 149. 325 326 UNITED STATES COAST AND GEODETIC SURVEY. An inequality in interval, range, or height is a systematic departure of the same from its mean value. The extreme amount of this regular departure is sometimes called the coefficient of the inequality. 3. Not long after new or full moon, the tidal effect of the sun is added to that of the moon. When this effect upon the range is greatest the tides are called spring tides {marSes de vive eau). At any given place the retard, or interval between new or full moon and spring tides, may be regarded as constant. Not long after the moon is in quadrature, the tidal effect of the sun is taken away from that of the moon, and when the range becomes a minimum from this cause neap tides {marees de morte eau) occur.* Their retard at a given place may be regarded as constant, and it does not differ much from the retard of the spring tides, unless the water is shallow, in which case the retard (spring or neap), as derived from the high waters alone, will differ from that derived from the low waters. The inequality, or apparent irregularity, in time or height introduced by the sun, and so dependent u]Don the moon's phase, is variously styled the semimenstrual, semimensual, semimonthly, OT phase inequality; the last seems preferable for most purposes, because there are several kinds of month in common use, especially in tidal work, and the word "phase" suggests a connection with the age of the moon.t When the sun's tidal effect shortens the lunitidal intervals, causing the tides to occur earlier than usual, there is said to be & priming of the tide; when, from the same cause, the interval is larger than usual, there is said to be a lagging. A tidal day is the variable interval (24'' 60'" on an average) between two alternate high or low waters. A more accurate definition is the interval between the mean of four consecutive tides and the mean of the succeeding or preceding group of four consecutive tides. The amount by which the tidal day exceeds 24'» 50"' is sometimes called the "lagging of the tide," and the amount which it falls short the "priming." The amount by which corresponding tides grow later day by day (i. e., the amount by which the tidal day exceeds 24'' 00"") may be called the daily retardation.^ The retard, especially spring and more especially spring high-water, has been called the age of the tide.^ If this term is to be retained, it seems desirable to suppose the age to have one value, and that such as to suit the neap as well as the spring tides, the low waters as well as the high. Moreover, instead of " age of the tide," the expression " age of the phase inequality" will generally be used in what follows. It will subsequently appear that, for deep water at least, the lunitidal interval of such tides as happen to occur as many hours after syzygy as represent the age of the phase inequality, have their mean values. In other words, the spring and neap intervals are about equal to the mean intervals. Because of this fact the times of such tides as give mean Intervals may be used in determining the age. Experience has shown that the ages as determined from heights or ranges do not agree with those determined from times (intervals). || For this reason it seems best, wherever possible, to define it by the value obtained from the harmonic constants of the place and explained in Part III.1] * The spring and neap ranges are conveniently denoted by tlie symbols Sg, Np. Their connection with Mn is shown by the approximate expressions (83), (84), Part III. When /(g has a large shallow-water part (2MS), as at Liverpool, so that 2 M°2 — S°2 — ,«°3 is not near zero, it may be worth while to replace jUi in these expressions by jU2 cos (2M°2 — S°2 — «°2). t There are some objections to the term "phase inequality" inasmuch as particular portions or aspects of the tide, such as high water, low water, or an intermediate time, may be referred to as its phases. Again, according to established usage, the phase of a wave, or oscillation, is the angle upon which the displacements depend; e. g., the phase of th6 harmonic oscillation 2/=C cos {ct + y) is the angle ct-^y. t Cf. Laplace, M6o. Cfl., Bk. IV, H 35 et seq. « Cf. Whewell, Darwin, Wharton, loc. cit. ; Perrel, United States Coast Survey Report, 1875, p. 209. II Airy Tides and Waves, Arts. .541-547; Phil. Trans., 1843, pp. 53, 54. Ferrel, United States Coast Survey Report, 1868, pp. 55, 75, 76; Tidal Researches, pp. 174-199; United States Coast Survey Report, 1875, pp. 209-212. IT Age of phase inequality expressed in hours = 984 (SP — Mj°) . Laplace shows that (for Brest) the age obtained from heights near the syzygies (=1-51349 days) is very nearly equal to the age similarly determined from heights REPORT FOR 1897 PART 11. APPENDIX NO. 8. 327 Since successive transits of the moon occur on an average 12^^ 25" apart, the age can be approximately expressed by stating the number of the transit preceding the tide to which the lunitidal intervals are to be applied.* The effect of selecting an earlier transit is to increase the lunitidal interval by 12^^ 25™. Of course, by adapting the transits to a' suitable terrestrial meridian, any age can be allowed for. Another way of reckoning the age is by the hour of the moon's transit. The time of transit increases on an average 50"' daily, so that if the transit used for spring tides occur at 0'' 50"', such transit follows syzygy by 24^^. But the tide follows the transit by the lunitidal interval; 24 X 60 ["hour of transit for XM) Thour of transit fori , ,g^j j^^j, ,^. 50 L maximum range J + t i J=i ^V -i + i^ vv i;, (i). Whenever the "hour of transit" exceeds 12'', 12'' must be rejected. The same formula is adapted to neap tide, by replacing the word " maximum " by the word " minimum," and always discarding 6" or 18'^ from the "hour of transit." t To infer the age from the time when the interval has its mean value, replace "maximum" range by " mean lunitidal interval," = ^ (HWI + LWI). Some writers prefer to increase the age or retard, as defined above, by the high- water interval, because of the fanciful notion that they thereby obtain the interval between the transit of the moon and the appearance of the resulting high water. 4. OtUer things being equal, the range of tide becomes a maximum soon after the moon is in perigee and a minimum soon after she is in apogee. At these times perigean and apogean tides occur. I The amount by which these effects follow their respective causes may be called the age oj the parallax inequality. Like the age of the phase inequality it may be defined in terms of the harmonic constants. § If this age be approximatelj' allowed for by selecting a proper transit, the lunitidal interval will, so far as this inequality is concerned, remain nearly constant throughout its period, which is an anomalistic month. If, however, the intervals be distributed under two arguments, the moon's phase and her parallax or anomaly, the departures from the average values of the intervals will depend upon both arguments. The phase inequality being known for a mean value of the moon's parallax, the tabular values just described give, when diminished thereby, the parallax inequality in interval arranged under two arguments. Even when thus distributed, the parallax inequality in time is small; but the parallax inequality in height is of considerable importance.|| If a wrong age of the parallax inequality be taken (i. e., if the tides be referred to a wrong transit so far as this inequality is concerned), the inequality in interval will become greater.^ If the tides are not classified with respect to the moon's phase (1. e., if they are classified with respect to parallax or anomaly only), the value of the parallax inequality in time will, as already stated, be small if the transit used corresponds well with the age of the parallax inequality.** 5. In a similar way the effect of the moon's declination or longitude may be considered. Soon near the quadratures (1.51116) : M^c. Cfl., Bk. XIII, J 7. Parrel's constants make tlie age from heights 1 '42 days and from intervals 1-87 : United States Coast Survey Report, 1875, pp. 209-212. The harmonic constants make the age 1-63 days. ' Lubbock, Treatise on Tides, pp. 25-29, or Phil. Trans., 1837, p. 97. t Because of the moon's variation, SO™ should be replaced by 51"" for spring tides and by 49'" for neap tides; but, as both spring and neap tides can generally be used in determining the age, this becomes unnecessary. t The perigean and apogean ranges of tide are conveniently denoted by Pn, An. § Age of parallex inequality ^1-837 (Mj^ — N2") hours. II E. g., Lubbock, Phil. Trans., 1836, pp. 58, 59 ; 1837, pp. 119, 133. Ferrel, United States Coast Survey Report, 1868, p. 69. U E. g., Lubbock, Phil. Trans., 1834, pp. 144, 163; 1835, p. 286. ** E. g., Ferrel, United States Coast Survey Report, 1868, pp. 60, 76, especially the low-water intervals. Here the implied (parallax) age is about 50'' for the high waters and 57'' for the low waters. The harmonic constants give 58'' for the age of the parallax inequality. Ibid., 1875, p. 196. Here the implied (parallax) age for the high waters is 8'' and for the low waters lij"". The harmonic constants give 33". As might be expected, the tabulated inequality (in time or interval) is somewhat greater than in the preceding instance. 328 UNITED STATES COAST AND GEODETIC SURVEY after the inoon is upon the equator the greatest semidaily range of tide will occur, other things being equal, and soon after the moon's extreme declination, the smallest. If a transit be selected which corresponds well with the age of this inequality, the intervals will be, as in the case of the parallax inequality, little affected by its presence. There is one difference, however, and that is, the sun's declination has a direct efi'ect upon the lunitidal intervals, even if the proper transit has been selected. The period of the sun's declinational inequality in the tides is, in the long run, the same as that of the moon's, viz., a half tropic month, and so the two can not be sepa- rated in the treatment of a long series of observations all distributed under one argument — the moon's declination or longitude.* This combined effect may be styled the declinational inequality. Its age is pretty nearly equal to that of the phase inequality. 6. Other irregularities in the motions of the moon and sun give rise to corresponding apparent irregularities or inequalities in the tides. Among these may be mentioned one depending upon the longitude of the moon's node, one upon the moon's evection, and one upon the sun's anomaly.t In ascertaining how the tide of a given day is disturbed by the inequalities, care should be taken to observe whether or notone inequality is involved in another. For instance, if the moon's anomaly is the argument for the parallax inequality, the tabular values (if derived from a sufficiently long series of observations) will be free from the inequality due to the evection, smd this latter may be tabulated and used as an independent correction. If, however, the moon's parallax be taken as an argument, the inequality due to evection, i. e., having the evectional period, must be small in comparison with its former value, because it is, for the most part, tabulated under the argument "parallax." So if we use the moon's longitude as an argument, l;he declina- tional inequality (semidiurnal) will, in the long run, be free from the inequality due to the regression of the lunar node. If, however, the declination of the moon, instead of the longitude, be used as an argument, the nodal inequality will be nearly allowed for. 7. Diurnal inequality in height is the difference in height between two consecutive high waters or low waters.! Diurnal inequality in time or interval is the difference in length of two consecutive high water intervals or low water intervals. At most places the high water inequality (in height or time) differs from the low water inequality. If the greater height inequality be iu the high waters, the greater time inequality will be in the low waters, and conversely. Wherever both of the height inequalities are small in comparison with the (semidaily) range of tide, the inequalities in time (interval) are very small. These inequalities vary in value throughout a half tropical month, and also a half tropical year. The portion due to the moon may be computed and tabulated under the argument of the moon's declination (as it was at a time anterior to the time required, determined by the age of the diurnal inequality^). The portion due to the sun may be tabulated with the two arguments, the day of the year and the hour of the moon's (upper) transit reckoned from to 24.|| The combined effect of moon and sun may be made to follow the moon's declination and the day of the year. If this inequality be comparable in size with the phase inequality, the two should be tabulated together, thus necessitating a table of three arguments, the two just mentioned and the hour of transit. If, however, we disregard the variation in the obliquity of the lunar orbit to the plane of the equator, two arguments sufl&ce for the combined effect, viz., the hour of transit and the day of the year; for these two then infer the moon's right ascension and so her longitude or declination. The number of days from the moon's zero or extreme declination is preferable to the day of the * E. g., Ferret, United States Coast Survey Report, 1868, pp. 60, 78; Ibid., 1875, p. 197. t E. g., Ferret, United States Coast Survey Report, 1868, pp. 79-82. } At places wliere the pliase inequality is large in comparison with the diurnal, it becomes necessary to compare a given iigh water, say, with the mean of the immediately preceding and following high waters in order to put iu evidence the high water diurnal inequality; similarly for the low water inequality. * E. g., Ferret, United States Coast Survey Report, 1868, p. 97. II Ibid., pp. 100, lot. REPORT FOR 1897 PART II. APPENDIX NO. 8. 329 year as an argument. But for the variation in the obliquity of the lunar orbit to the plane of the earth's equator, very good predictions could be made from tables having these two arguments.* Th6 diurnal tcave is that portion of the tide whose period is approximately one day. Its range varies throughout the half tropical mouth and half tropical year. The maximum value of this range may be regarded as occurriug, in the long run, a constant number of hours (viz., the age of the diurnal inequality) after the moon reaches her extreme fortnightly declination; at such times tropic tides f are said to occur, because for most places the moon is then near one of the tropics. | The age of the diurnal inequality is such that if the times of zero declination be increased thereby, the range of the diurnal wave will be a minimum. This age, like the ages of other inequalities, may be expressed in terms of the harmonic constants. § The diurnal inequality is due to the presence of the diurnal wave. A.t the time of the tropic tides, the diurnal inequality (time or height) may be spoken of as tropic. The inequality in high water heights is then denoted by HWQ and in low water LWQ. The larger one then has its maximum value very nearly; so has the quantity VhWQ^+LWQ'^, which is an approximate expression for the tropic range of the diurnal wave, and with greater reason. At places where HWQ, say, is several times smaller than LWQ, the high water inequality when tropic tides occur may not have, even approximately, its maximum value. Of the four ranges of tide upon a day when tropic tides occur, the greatest is called the great tropic (Gc) and the least the small tropic (Sc). The mean range from all four tropic tides is the mean tropic range (Mc). || The great [diurnal] range (Gt) is the difference between the mean of all the higher high waters (HHW) and the mean of all the lower low waters (LLW) of each day during one or more half tropical months. The small \diurnal] range (SI) is the difterence between the mean of all the lower high waters (LHW) and the mean of all the higher low waters (HLW) of each day during one or more half tropical months. It is sometimes convenient to distinguish between the four ranges, which at most stations occur upon any given tidal day, by means of the following terms : The great range, the small range, the high range, and the low range. (See Fig. 1.) At stations where the tide is diurnal there are but two ranges each tidal day, the great and the small. The great tropic range and thelunitidal intervals connected with it can be observed even if the tide becomes wholly diurnal in its character. So with the great diurnal. The sequence of tide is the order in which the four tides of a day occur, particularly the tropic tides. It may be expressed thus, "higher high to lower low," or "lower low to higher high," as the case may be. The {^^gi" expression indicates that (tropic) lower low water ^ °^^ (tropic) higher high water without the lesser tides intervening. The time between (tropic) /^ ^^ ,^^ water and (tropic) },jjv,gj. i,jj^ water must be taken as less than a half lunar day. At places where HWQ and LWQ are very unequal, the sequence, even of the tropic tides, may be different for different seasons of the year. The type of tide is its characteristic form. It is generally indicated by the sequence of tides, the ratios of the tropic diurnal inequalities, and of the spring range, to the mean range. For shal- low waters, however, in rivers especially, the duration of rise or fall may become very important. * E. g., Lubbock, Phil. Trans., 1836, pp. 65-73 ; 1837, pp. 109-118, 126-130. Bache, United States Coast Survey Reports, 1854-1864, "Tide tables for the use of navigators." t Cf. Airy, Phil. Trans., 1845, pp. 44-46, whert approximate values of the tropic semirange of the diurnal wave are given on the coasts of Ireland. The word tropic was officially adopted by the Coast and Geodetic Survey, Dec. 19, 1894. tThe coasts of Europe form an exception, the age being from 2 to 6 or more days. Whewell,^Phil. Trans., 1837, p. 81. § Age of diurnal inequality expressed in hours = 0'911 (Ki° — Oi°). II It is equal to i(6c-|-Sc), and is somewhat less than Mn, the relation being Mc=Mn^2K.2C08 [(Ki° — Oi°)~ (K.^*^ — M2°)]. Expression (89), Part III, includes semidiurnal constituents only, and so is notlexactly equal to |Mc; in strictness A the cosine factor should there also be appended to Kj. / 330 UNITED STATES COAST AND GEODETIC SURVEY. Fig. 1 illustrates the tropic tides and quantities connected with them at San Francisco. In this case the tide is largely diurnal, the sequence is HHW to LLW, and LWQ > HWQ. 8. The tide curve or marigram is a curve whose abscissfe increase uniformly with the time, and whose ordinates represent the heights of the tide or sea at the corresponding times. The average value of the ordinates of the tide curve (reckoned from some fixed mark upon the shore) defines the height known as mean sea level or mean water level. The average value of the heights of high and low waters (reckoned from some fixed mark upon the shore) defines the height known as half-tide level. Mean sea level and half-tide level do not differ much from each other except at places where the duration of fall differs by a considerable amount from the duration of rise, or where the diurnal inequality is large.* Mean sea level (MSL) is the most nearly fixed, and therefore the best, of all planes defined by the tide. Planes used in reducing soundings or in reclconing elevations above the sea should alioays have a known relation to mean sea level, or at least to half-tide level (HTL). The plane of average or mean low water is one-half Mn below half-tide level. The soundings Tra-nsit (upper or loww) Transit (lower or upper) Tropic HHWI Tr an sit (upper or lower) Tropic_ Fis on the Coast Survey charts of the Atlantic coast of the United States are reduced to this datum. The plane of mean loiv-water (ordinary) springs is about one-half Sg below half-tide level or mean sea level. This is the plane generally used along the outer coasts of Europe. The soundings upon the charts of the French coast are reduced to the lowest tides observed. The plane of average lower loic loater or average daily low water is used generally upon the Pacific coast of the United States as well as the Gulf of Mexico. The Indian {harmonic) tide plane or Indian spring low-water marh is Mz + Sa -f Ki + O, below mean sea level. These symbols are defined in the next paragraph. The plane of equinoctial low-water springs is the datum sometimes used by the British Admiralty in Indian waters. In many localities the datum of soundings is an arbitrary, but known, distance below a fixed bench mark. In the establishment of any such plane the hydrographer usually aims at some 'Part III, §24. HTL = MSL-|-M4 cos (^251°— M°^— 0-04 (^' + '^'^'' cos (^M°— Kj'— 0°^ REPORT FOR 1897 PART II. APPENDIX NO. 8. 331 plane whicli is capable of deflnition mth respect to the tide; e. g., mean low water, low- water springs, etc. In a few cases the plane of reference is the height of the lowest observed tides. Of course a datum of this kind cannot be recovered unless through an established bench mark. Since all datum planes must be connected with mean sea level, it might be advisable to reduce soundings to a plane an integral number of feet below this level, the same number to be used over a considerable area, and to be determined by the lowest tides likely to occur. Such areas or regions could be indicated upon the charts. Some hydrographers have used the expression " low- water springs" to denote extremely low low waters due to various inequalities in the tide. Such careless usage should be discouraged. Heights on the land when accompanied by the expression " above tide" or "A. T.," usually refer to high water. This is objectionable, because the height of high water depends upon the range of the tide, and this in turn upon the locality of the tidal observations. Above mean sea or mean water level is a much less objectionable signification. In connection with tidal planes, see § 20, Part II, and §§ 42-45, Part III. 9. Tidal constants are certain intervals (angles) and heights (amplitudes) used in describing the tide ; they are absolutely constant, or nearly so, at a given place. Nonharmonie constants are those tidal constants which refer in some way to high and low water instead of to the constituent periodic elements into which, as will be shown in Partll, the tidal wave may be resolved. Harmonic constants refer to these periodic elements. The portion of the tide following any period strictly, can, by Fourier's theorem, be analyzed into one or more simple cosine terms whose angles or arguments (which are proportional to time) go through 360°, and multiples thereof, in the given periodic time. Either the process or the result is spoken of as an harmonic analysis.* Each such term is called an harmonic component, component tide, partial tide, simple tide, or simply a component. The (uniform) hourly change in the angle of any component is called its speed; the value of its angle reckoned from its high water at any given instant is called its phase; its (constant) semirange is called its amplitude; the (constant) angular retard of the maximum of any component G behind its astronomical cause or fictitious moon (as assigned by the uncorrected equilibrium theory in Part II) is its epoch or lag. The amplitude of will be denoted by G, the epoch by G°, and the speed by c. If G have a period of approximately one day or twenty-four hours, it is said to be diurnal and is written Ci ; if it have a period of approximately twelve hours, it is said to be semidiurnal and is written G-i, and so on. At most places the semidiurnal components are so much larger than the diurnals, quarter diurnals, etc., that the tide curve of any particular day approximates toward a sine (or cosine) curve whose period is about twelve lunar hours. In the analysis of tidal currents G and G° may be replaced by G and G°, G denoting the amplitude of the component . velocity and G°!c being the interval between the transit of the fictitious moon (Table 3) and maximum velocity. Tlhe principal lunar component, denoted by Mj, has an hourly speed of 28''-984 1042, and so its period is a half lunar day. The principal solar component, denoted by Sa, has an hourly speed of 30°-000 0000, and so its period is a half solar day. The luni-solar diurnal component, denoted by Kj, has an hourly speed of 15°-0410 686, and so its period is a sidereal day. The principal lunar diurnal component, denoted Oi, has an hourly speed of 13"-943 0356, and so its period exceeds the lunar day by the same amount as the period of Kj falls short of it. For a more extended list of components see §§ 15-18, Part II, and Tables 1, 36. * For analyzing the quality or timbre of a given note, Helmlioltz made use of a series of spiierioal shells or resonators whose periods of vibration were fixed and known. In this way he could pick out the overtones which were present in the note sounded. The object of the harmonic analysis of a series of heights, tabulated and summed according to a given component time, is quite analogous to that of the analysis of a musical note. The harmonic analyzer to be described in Part II may be likened to Rudolph Koenig's combination of resonators (Jamin, Cours de Physique, 4th ed.. Vol. Ill, p. 175; or Ann. de Pogg., Vol. 122 (1868), pp. 666 et seq.), while the tide predictor (Part III) may be likened to the sirens of Seebeck and Koenig (Jamin, Vol. Ill, p. 172). 33"2 UNITED STATES COAST AND GEODETIC SURVEY. Example slioiciiuj hoic one simple wave is displaced by another. — The height of the surface of the sea from mean level dne to the two components A, B is y = A CO A [at + a) + B aos, [bt + fi) (2) Here the amplitudes are .4., B, the speeds «, b, and the initial phases a, y3. If a = b, the resultant wave is harmonic, having as its amplitude Va'+B^+ -J, AiTcos (a-/^) (3) That is, if we form a parallelogram analogous to the parallelogram of forces regarding a, fi as giving the directions oi A, B, the resultant amplitude is the diagonal of the parallelogram, setting out from the intersection of A and B. Or, it is the third side of a triangle whose opposite angle is 180O-(«~/i). The phase of the resultant wave is the angle whose tangent is A sin a + B sin /3 A cos a -{■ B cos p (4) The resultant wave may, therefore, be written In the parallelogram construction just referred to, this angle is the direction of the resultant diagonal. If a (or fi) = 0, the above angle is the angle between the resultant and A' (or B). In fact, it is then the angle adjacent to A (or B) of the triangle referred to above. 10. A cotidal line is an assemblage of points on the earth's surface where tides occur at the same absolute time. The number of each such line is usually taken as the lunar time (i. e.. the lunar hour after upper or lower transit) at Greenwich when high water occurs at stations along the cotidal line. If solar hours are used — reckoned, of course, from the time of the moon's transit — each period of cotidal lines will consist of 12.42 bour-lines instead of 12. The cotidal lunar hour of a place whose west longitude in time is L is 0-966 HWI + L* (6) while the cotidal solar hour is HWI + 1.035 LA (7) If Greenwich transits be used in making a "first reduction," § 51, the interval so obtained + S, the longitude of the time meridian expressed in time, is the cotidal solar hour. If the meridian over which the moon is assumed to pass have a west longitude in time equal to.-E/, then L must, in all cases, be replaced by i — E. If instead of HWI, we write the vulgar establishment or lunitidal interval at full and change, we have the cotidal lines for full and change f and not for spring tides or for tides of mean lunitidal interval. On the other hand, the retard of the spring tides is not the same the world over, and so the cotidal line 0-966 HWI + L does not represent a series of points along which it is simultaneously and exactly high-water springs; in fact, such lines do not exist. For limited areas, lines of equal lunitidal interval may be drawn instead of cotidal lines. This amounts to making i = in (C) and (7).§ *Cf. Whewell, Phil. Trans. 1836, p. 293 and chart opp. p. 306. Baohe, United States Coast Survey Reports: 1854, p. 149 and sketch 26; 1855, p. 339 and sketch 49; 1862, p. 127 and sketch 46. t Cf. A. S. Christie, The Lady Franklin Bay Expedition, Vol. II, pp. 697 et seq. tE. g., Whewell, Phil. Trans. 1833, pp. 148,149; 1848, p. 7. Airy, Tides and Waves, plate 6 ; reproduced in Enc. Brit., Art. "Tides." Haughton, Manual of Tides and Tidal Currents (1870), Plate IV. Berghaus, Physikalischer Atlas (1892). Probably most astronomies and physical geographies adopt this system. 4 E. g., Schott, United States Coast Survey Report, 1854, p. 173, sketch 16. REPORT FOR 1897— PART II. APPENDIX NO. 8. 333 Intervals referring to the diurnal wave can be used iu a similar way for obtaining cotidal lines referring to the diurnal portion of the tide.* If observations were sufficiently extensive it would be possible to draw a set of cotidal lines for each harmonic component of the tide. Accordingly, a cotidal line for the component A; is an assemblage of points on the earth's surface where the J., tide occurs at the same absolute time. Such lines are naturally numbered in Ai hours after the transit of the A^ fictitious moon across some fixed meridian, as that of Greenwich; but they may be numbered in Bi hours after the transit of the Ai moon. The cotidal A hour for the component A is AjO il5' while the cotidal B hour for the component A is ■ L, (8) ''A?+M. (9) a\i 15 The principal tidal components are M2, S2, K],and Oi. If J.i = M2,t the cotidal lunar hour for M2 is ¥+^ , (10, which is, in deep water, nearly equal to 0-966 HWI + i. * If A; = §2, the cotidal .solar hour for S^ is ^ + L. (11) If Ai = Ki,J the cotidal sidereal hour for Ki is ^\L. (1.) 11. Tide tables are ephemeral publications, usually covering a calendar year, showing in tabular form the predicted or computed times and heights of the high and low waters. For certain principal or typical stations such predictions are given in full, i. e., all tides of the year are predicted ; but for most places tidal differences and ratios are given, which enable the user to obtain his tides from the tides at stations having full jjredictious. The following are the principal tide tables : "Tide Tables by the United States Coast and Geodetic Survey." This publication covers quite thoroughly the coasts of the United States and less thoroughly the world at large. "Tide Tables for the British and Irish PortsJ" containing " also the times and heights of high water at full and change for the principal places on the globe," by the British Admiralty. "Tide tables for the Indian Ports," by authority of the Secretary of State for India in Council. " Annuaire des Marees des C6tes de France," by the French hydrographic service. "Gezeitentafeln," by the German admiralty. These include daily predictions for several stations in addition to those of the German coast ; also intervals and ranges for the world at large. * E. g., Bache, United States Coast Survey Report, 1862, p. 127 and sketch 46. Cf. Part III, § 56. tE. g., Van (ler Stok, Stuilien over Getijden in den Indischen Archipel, Xll (1895), p. 23 and Kaart I. The longitude, L, is here reckoned from Batavia. t E, g., ibid., p. 31 and Kaart II. 334 UNITED STATES COAST AND GEODETIC SURVEY. 12. The velocity (drift) of a current is the rate at which the fluid particles move horizontally. It is usually expressed in knots, i. e., nautical miles per hour, but sometimes in feet per second.* The velocity generally differs for different depths, but its value at the surface may be understood unless otherwise specified. The velocity of propagation of the tidal wave is many times greater than the velocity of the current, and the two must not be confounded. The direction [set] of a current is the direction or point of the compass toward which the fluid particles move. The movement of the fluid in one direction, usually inland, is stjled flood, and in the opposite direction, ebb. The two are not always distinct, and, even if they are, it is not always possible to know which movement should be taken for the flood and which for the ebb. Flow or flood and ebb correspond to the French flot and jusant, while rise and fall correspond to montant or gagnant and perdant. The maximum of the flood or ebb current is sometimes called the strength of flood or ebb. The effect of the tidal wave in giving rise to currents is obvious in two extreme cases : (1) Where there is a small tidal basin connected with the sea by a large opening. (2) Where there is a large tidal basin connected with the sea by a very small opening. In the first case the velocity of the current in the opening will have its maximum value when the tide or height of sea is changing most rapidly, i. e., at a time about midway between high and low water. In other words, the water level in the basin keeps at about the same level as the surface of the water outside. Flood corresponds to the rising and ebb to the falling tide. In the second case the velocity of the current in the opening will have its maximum value when it is high water or low water without; for then there is the greatest head of water for producing motion. Flood begins about three hours after low water and ebb begins about three hours after high water, and so slack water occurs at times about midway between the tides. Many currents in nature lie, in a general way, between these two extreme cases; but see §22. Slach water denotes the state of the current when its velocity becomes a minimum. It follows high- or low-water stand by intervals ranging from zero to three hours, depending upon the locality.t Change or turn of tide are expressions sometimes used instead of " slack water." The velocity and direction of tidal currents are much modified by extremely local causes, while the times and heights of the tides are about the same over considerable areas. 13. Representation of currents, etc. The velocity and direction of a current at any given time are often indicated by an arrow. Usually the arrow indicates direction only, the velocity being written just beyond the point. If the currents corresponding to several phases of the tide, or rather tidal current, are shown upon one sheet, several arrows will usually radiate from the same point upon the map. Their numbers indicate the order of occurrence.! A current station is a point where currents have been observed. Unless the station happens to be in the channel, it is obvious that the rising tide will generally reach and swell the water in the channel before its effect is felt at the station. Similarly the fall- ing tide begins to lower in the channel earlier than in the shallower regions. Hence the order, in time, for the pointing of the radiating arrows is — Shoreward, upstream, offshore, downstream. This is evidently clockwise for stations upon the right-hand bank (looking downstream) and counter clockwise for stations upon the left. For au instantaneous representation of the condition of the currents in a given harbor or region it is customary to make use of lines offlotc. A line of flow is such a line that at all of its points the motions of the fluid particles coincide with it. In other words, if we draw at each point of the fluid a very short arrow, whose direction indicates the direction of the current at the given instant, then a curve, coinciding with a series of them, is a line o/ flow.§ At any given *To change velocities given in feet per second to knots per hour, multiply by -^ = 0'5921= \ ■ to statute miles, multiply by — . t Males dejlot et dejusant vs. Stales de pleine mer et de iasae mer. t For examples see Coast Survey charts, also the Reports; for instance 1879, opp. p. 175 and p. 181. § A. set of charts for the Irish Sea and English Channel, by Beechy (q. v.), is given in the Philosophical Transactions for 1848. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 335 instant the motion of surface of the water will be represented by a system of curves covering it. If the motion be steady, i. e., independent of the time, lines of iiow are usually known as stream lines. V If across the lines of flow we draw a line cutting the system everywhere at right angles, the line is a line of equal velocity potential, or an equipotential line. We may assign to one such line any number we please; but having done so, the numbers belonging to other such lines become fixed. Suppose the numbers upon two adjacent lines of equal velocity potential to differ by a small constant quantity d(l>. The distance apart of the two lines (ds) becomes known when the velocity is known (and conversely) through the equation ds =:—dif>-^ V, or Lines of equipotential form a system of curves cutting the system of lines of flow everywhere at right angles. When the motion is steady and the body of water uniform in depth, the two systems are not only orthogonal, but also isothermal; that is, if we construct stream lines and equipotential lines, as above directed, we can select stream lines as far apart as are the equipotential lines in the same vicinity, and so divide the whole surface of the water into elementary squares. If the real part of a function of a complex variable represents one of these systems, the purely imaginary part will represent the other. Such a function is defined by the boundary conditions; that is, the function must be such that the stream lines coincide with the fixed boundaries of the fluid.* Another pair of systems which might be used to represent the motion of the water are lines of equal velocity and lines of equal bearing. All along a line of equal velocity, the velocity of a current is constant; all along a line of equal bearing the direction or set of the current is constant. These two sets of curves often intersect orthogonally or nearly so. For uniform depth and steady motion the systems are also isothermal. The map of a body of water may have drawn upon it a series of lines, along any particular one of which the current turns at a given lunar hour. Such lines are called cocurrent lines; they are quite analogous to cotidal lines, and, like them, admit of numerous varieties.! If we are concerned with the current at one station only, the velocities may conveniently be taken as the (positive) ordinates of a curve, the times being the abscissae. The directions may be written at the feet of the ordinates. At a station where the flood and ebb are distinct, the velocities of the one maybe taken as positive and the other negative. This representation is quite analogous to the tidal sheet or marigram. 14. For explaining the origin, propagation, and properties of the tide wave, numerous theories are required, according to the circumstances. When the explanation is less complete, or when observation is called in to supplement certain of its defects, the underlying and so-called theory is really only a worldng hypothesis. However, since nobody can hope for theories covering all cases, and since the same theory at one place may serve to explain the tides, while at another place it can serve only as a working hypothesis, we shall follow usage and make the word "theory" cover the expression "working hypothesis." The (uncorrected) equilibrium or statical theory assumes that at each instant the surface of the sea is a prolate elipsoid whose longest diameter points at the tidal body. This hypothetical surface is often defined as being one which is everywhere normal to the direction of gravity as perturbed by the tidal body alone. If the ocean covered the entire earth, the effect upon the direction of gravity of the layer of water constituting the hypothetical tide could be computed. The eccentricity of the equilibrium spheroid, and so the range of tide, would then be somewhat increased. * Baohe, United States Coast Survey Report, 1851, pp. 136, 137, and Sketch A, No. 3 ; the latter represents the currents of Boston Harbor by mating the distance between the lines (of flow) inversely proportional to the velocity at the given point. In case of steady motion and uniform depth such lines wovild be continuous. t Schott, United States Coast Survey Report, 1854, pp. 168-179, discusses the currents of Long Island Sound. His cocurrent lines are really lines of equal lunicurrent interval. His "lines of direction" differ from lines of flow in that they are taken not exactly simultaneously bnt at the time of greatest velocity. 336 UNITED STATES COAST AND GEODETIC SURVEY. The corrected equilibrium theory dififers from the former in assuming the earth not wholly covered by water, and so the surface of even a deep sea cannot actually coincide with the spheroid of the uncorrected theory, but will be parallel to it the distance therefrom at any given point varying with time. This is necessitated by the incompressible property of water. The difference between these two theories can be illustrated by means of a small body of water, a lake, or even a pail of water. The uncorrected theory implies that the whole surface of the small body of water rises and falls by the same amount, twice each lunar day. Moreover, this range of tide would be the same as the range of tide in the same latitude were the surface of the earth covered by an ocean. The corrected theory involves the fact that, on the whole, the surface of the small body of water always has the same height, and so its tides are caused by the water at one side being slightly elevated while in another portion it is slightly depressed. But the surface is normal to disturbed gravity, or parallel to the uncorrected tidal spheroid. The cotidal lines will radiate from a no-tide point instead of being arcs of terrestrial meridians as in the case of the uncorrected theory. The equilibrium theories assume that the water surface arranges itself in each locality normal to the force of disturbed gravity, but they do not explain how this arrangement is made, nor whether it is possible with the known properties of water. They avoid altogether the question of depth of the water, the surface alone being considered. Laplace attempted a theory in which not only the disturbing or tidal forces, but also the motion of the water regarded as a heavy or inert body, are taken into account. Such theories are known as dynamic or Icinetic. They should take account of the viscosity or internal friction of the water as well as the friction at the bounding surfaces. The wave theory considers the tide as a wave and develops the properties of such motions. From the nature of the case it is a kinetic theory. The uncorrected equilibrium theory is useful as a working hypothesis in tidal analysis because it enables one to infer suitable forms of expression for the tidal disturbances, knowing the laws of the forces to which they are due. The principle that the disturbances are coperiodic with the forces, whether the tide approach its equilibrium condition or not, is a deduction of dynamics. The corrected equilibrium theory applies to small, deep bodies of water. The kinetic theory enables one to infer that the amplitude of tidal oscillations having sensibly equal periods are to one another as are their forces, and that their epochs are equal. The wave theory is applicable to canals or tidal rivers. MISCELLANEOUS TIDAL PHENOMENA. 15. In shallow estuaries where the range of tide is considerable, the high water is propagated inward faster than the low water, for at high water the greater depth prevails. The high water thus gaining upon the low water causes the duration of rise of tide to become shorter as the wave proceeds; and so the farther the wave goes without breaking, the more abrupt its frout becomes. Finally, it becomes so steep that the top of the wave falls forward (not in the middle of the stream but near the shelving shores) something like the crest of a breaker. This phenomenon, usually accompanied by much noise, is called a hore. Other names for the bore, boor, or boar's head ate eager (England), mascaret or barre (France), prororoca or pororoca (Brazil). The following rivers and arms of the sea have bores: The Amazon,* Tsien-tang,t Brahmaputra, Ganges, Hooghly, Indus, G-aronne, Dordogne, Seine, Trent, Severn, and Wye rivers, Solway Frith: arms or bays at the head of the Bay of Fundy, and perhaps Magellan Strait and Cook Inlet.| An agger is a double-headed' tide; that is, a tide having two maxima or two minima instead of the usual high or low water. § This gives to the tide a long " stand," and so may be of much practical value. At Southampton there is a double high water, at Portland a double low water, * J. C. Branner, Pop. Sci. Monthly, Vol. 38 (1890), pp. 208-215. t See figure 19. t For tidal diagrams of French rivers, see Comoy, Etude Pratique sur les Marees Fluviales. See Airy, Tides and Weaves, Art. 514 ; also this manual under Alexander the Great, Strabo, Hakluyt, and Sturmy. ^ Cf. Airy, Tides and Waves, Art, 518, and Ferrel, Tidal Eesearches, 5 254. EEPOET FOR 1897 — PART II. APPENDIX NO. 8. 337 and at Havre an almost double high water.* This peculiarity of the tide does not generally persist throughout the lunation. It is usually, but not always, the most pronounced at spring tide. At some places the high and low waters may be very sharp, thus making a staiid of short duration. The high waters at Ipswich and the low waters at Philadelphia may be mentioned as cases of this kind.t Whenever the tidal water has to pass through a rather narrow or shallow channel to fill a tidal reservoir beyond, a strong current is necessitated. This is sometimes called a race; e. g., the Eace at the eastern entrance to Long Island Sound. Of course each of the two bodies of water connected by the strait may be tided ; e. g., the Hell Gate, Messina Strait. But a tidal race is more properly defined as a strong current caused by the meeting of two wave systems from different or opposite directions. The effect will be the greatest where the range of tide is most diminished by this meeting; that is, at a place where trough meets crest, as can be seen by a brief study of the water particles in wave motion (Chapter II). The Portland Eace, the Maelstrom, and Seymour Narrows may be instanced. 16. Seiches (sash) are short-peiiod oscillations (usually from about 10 to 60 minutes) existing at times in many (if not all) lakes and landlocked bays. They represent oscillations in which usually the whole body of the liquid swings to and fro. They are caused by sudden changes of atmospheric pressure, or winds which sweep over its surface. The period of such a seiche is twice length of lake where g denotes the acceleration due to gravity, and h the depth of the lake or bay. Seiches may not always be uninodal, as supposed above, nor does the nodal line always run transversely to the body of water. This phenomenon has been observed on Lakes Geneva, Constance, Ontario, Michigan, Caz- enovia (N. Y.), and others; on bays in India; at Swansea (Wales), Malta (Greece), and Bristol (E.I.).| The effect of an earthquake upon the sea is known as an earthquake wave, an earthquaTce sea (or ocean) wuve, or a great sea ivave. The last term serves to distinguish it from the corresponding oscillation in the solid earth which is known as a great earth wave. The great sea wave may sometimes,' perhaps, be due to a tumbling down of submarine cliffs instead of to an earthquake proper. The effect of these waves is often transmitted to distant shores, where it is recognized (although not always with absolute certainty) by the peculiar oscillations which it adds to the record of an automatic or self-registering tide gauge. Peru, Japan, and Malay Archipelago have furnished notable instances of this phenomenon. * See figure 18. For tidal diagrams, Phil. Trans., 1843, opp. p. 46, and Comoy, op. cit. To ascertain from the harmonic constants the natures of the high and low waters at a given place, draw a curve, as in § 63, Part III, consisting of Mj and its harmonics M4, Mg, Ms, .... In computing the mean range of tide the second portion of formula (65), Part III, should not, perhaps, be used, hut rather a value obtained from the drawing just mentioned. t Phil. Trans., 1843, opp. p. 52. See figure 11. \ The following are a few references to this phenomenon : C. B. Comstock, Annual feeport of the Survey of the Northern and Northwestern Lakes, 1872, pp. 14-16 and PI. VI. Airy, Phil. Trans., 1878, pp. 136-138. Giinther, Geophysik, Vol. II (1885), pp. 373-376. Nature, Vol. 14 (1876), p. 164; Vol. 17 (1878), pp. 234, 281; Vol. 18 (1878), pp. 100, 101; Vol. 19 (1879), p. 446; Vol. 21 (1880), pp. 397, 443; Vol. 33 (1885), p. 184. Science, Vol. 7 (1886), p. 412; Vol. 15 (1890), pp. 99, 117. 6584 22 CHAPTER II. DIGRESSION ON PLANE, OR TWO-DIMENSIONAL, WATER WAVES. 17. Fundamental equations. A more exhaustive account of fluid motion will be given in Part IV. In the present chapter the assumptions made are few and simple. The main object is to give an introduction to the study of wave motion, which shall clearly indicate how the water particles behave according to theory, and which shall also show some applications of the results obtained to the water move- ments in nature. By taking the displacement equations (26), (27), and the equation (28) for granted, several of the paragraphs on wave motion can be understood without reading the present paragraph. In any motion of a fluid, the entire volume taken into consideration must not be altered. That is, if we assume any small mass of the fluid bounded by an imaginary surface to be slightly displaced in the motion, its volume will remain as before; or if we assume an imaginary surface fixed in the fluid and inclosing a small mass or volume of it, the amount contained in this surface will be constant, whatever the motion of the fluid, i)rovided only that the surface remain entirely submerged. We shall assume that all motions take place in or parallel to the vertical plane xy, and, for convenience, that the thickness (z) of tlie body of water treated is unity. Then, considering an imaginary rectangular boundary whose edges are dx, dy, in length, and letting u, v denote veloc- ities along X, y, the difference between the entire quantity flowing into and out of this boundary which is supposed to be stationary is, obviously, u -\-'^ dx j dy — u dy + iv -\- — dy\ dx — v dx. This is equal to zero, because as much flows in as out; . ^ . ^ ^ (13) "duc'^ dy is the equation of continuity. In using this equation it is to be remarked that x, y are the true coDrdinates of the particle, whereas in the work about to be given x, y are the coordinates of the particle when in its undis- turbed condition. The true coordinates are « + x, 2/ + y- To find the equation of continuity in terms of small displacements x, y instead of u, v, assume that the elementary rectangle whose corners bad originally the coordinates 0, dx, dx, dy 0, dy becomes so altered by the motion that the coordinates of the corners are Cx (^ + ^^dx + dx C^+^^-dx+^^dy + dx (^ + ^§dy ly Cy + g^* ly + ^^dx+^^dy + dy (y + ^^dy + dy. Let the small change among neighboring particles be such that the elementary rectangle becomes a parallelogram whose sides are approximately parallel to those of the rectangle; its area is approximately equal to 338 REPOET FOR 1897 — PART II. APPENDIX NO. 8. 339 Since this must be equal to dxdy, it follows that or ^ + ^ = 0, ' (17) provided — • ^ may be neglected, as is the case when each factor is small. Either (16) or (17) is the equation of continuity, the former being, of course, the more accurate; = 2/ 'dy -{- & function of x, (18) ■where h denotes the height of the bottom. If p, denotes the value of the displacement x at the bottom, the corresponding value of y is and since at the bottom where 2/ = & the integral is zero, it follows that this is the required function of x; The last term becomes zero for a horizontal bottom. If the motion be such that all particles once in a vertical line always remain so, x can be replaced by S, which is its surface value, and we may take as elementary area a rectangle whose Iff length is dx +— da? and whose height is /i. + 77, a much larger quantity. The area must remain Ti dx. is the equation of continuity, in this case. The dynamical or pressure equation is w^^'-^Ur'"- I ^^'^^' ('') in which ;; denotes the value of y where y = h, the undisturbed depth, and X denotes the intensity of any impressed force acting in the ^-direction. Since x does not vary with t, the value of — ttj — - is -r^, which is the acceleration (or effective force per unit mass due to the horizontal motion) in the x-direction. This must be the result or the equivalent of the a;-component of all other forces connected with the motion. gr] is the disturbing pressure due to height reckoned from the undisturbed surface, and so the partial ^-derivative of — grj is the corresponding accelerating force in the ^-direction. -rr^dy is, since weight or mass is proportional to dy, an element of the pressure due to the ill' vertical velocity of an elementary mass above the point x, y. The aggregate pressure is the same integrated up to the surface, and the corresponding accelerating force is minus the partial 340 UNITED STATES COAST AND GEODETIC SURVEY. a;-derivative. In this integration it is allowable to take the upper limit as h, instead of the slightly different value h + 7;, because the vertical acceleration ^ is assumed to be a moderately small 1)0 quantity.* If X = 0, the motion of the body is " free," not "forced ;" i. e., the body is left to itself. If the vertical acceleration can be omitted, the water must so move as to keep all particles which lie in a given vertical line, always in a vertical line. If the area be divided into elementary vertical strips of length dx and height h + rj, the elementary volume of water, dx x (h + 77) X 1, varies with the instantaneous height; and so the force equivalent to the effective force in the moving element must likewise vary, lf=3^ r, , 3xV ' ' ^^^^ making use of the corresponding equation of continuity and putting X = 0, we have dx which becomes, if —is small, or if the relative displacement of two neighboring elements of the fluid is small in comparison with the distance between them, WAVES IN A CANAL OF UNIFORM BBPTH AND INDEFINITE LENGTH. 18. It is here proposed to give an interpretation of the wave motion deflued by the following equations, and to point out how the long or tidal wave differs from the short, oscillatory, or surface wave. Let us assume that the horizontal and vertical displacements of the fluid are of the respective forms, t X = A cosh ly sin [at —lx-[-a) (26) y = A sinh ly cos {at — Ix + a), (27) where A, a, a, I are constant throughout the canal and for all time; x, y are independent of the time but vary from point to point, x being measured horizontally from an arbitrary origin, and y vertically from the bottom of the canal. These evidently satisfy the equation of continuity (17) ; they also satisfy the dynamical equation (22) provided tanh Ih = -, (28) or a' = gl tanh Ih. (29) Equations (17), (22) imply that x and y are small in comparison with the wave's length and the depth of the water, respectively. The motion defined by (26), (27) is periodic in time and distance. Any increase of time, accompanied by a proper increase of distance, leaves x, y unaltered, showing that the wave motion represented advances uniformly along x increasing, the velocity being — . V The motion represented is evidently such that similar terms involving 2at, Sat, etc., may be disregarded. where J) denotes tbe intensity of pressure per unit area at a given point; p the density of the fluid, i. e.,its mass per unit volume, and which may be taken as unity. t For definitions and numerical values of hyperbolic functions, see Table 46. REPORT FOR 1897 PART II. APPENDIX NO. 8. 341 19. Deductions from (26), (27). The horizontal and vertical component oscillations (displacements) of any given fluid particle are each simple harmonic functions of the time, and of like periods. Eliminating the angle involving t, we have =1. (30) A^ cosh^ ly A'^ sinV^ ly showing that any particle whose (undisturbed) height above the bottom is y describes an ellipse whose major and minor semi-axes are A cosh ly and A siuh ly. Consequently the foci are distant ^ Vcosb^ ly— sinh^ ly, = A, from the center of the ellipse. As this distance is independent of both X and y, it is the same for the orbit of any particle in the fluid mass; i. e., the two foci of any ellipse are 2A apart and lie in a horizontal line. [It may be noted, although it is not important for the present purpose, that the law of description is precisely the same as that of a body revolving about a central force whose intensity increases directly with the distance of the body from the center.] Let X be constant in equations (26), (27). Since the angle at — Ix + a does not involve y, it is obvious that particles originally in the same vertical line are, at any given instant, in the same phase of either the vertical or the horizontal oscillation (displacement). In this respect the motion of a vertical filament of water somewhat resembles that of a stalk of wheat swaying to and fro In the wind. Fig. 2. For illustrating wave motion. 20. Figure 2 illustrates the wave motion in a vertical section of water, the wave being propagated in the direction Ox. This is not a view of a three dimensional volume of water, but consists of a series of instantaneous views of the same plane ; the times at which the views are supposed to be taken are, as indicated upon an arbitrary time axis t, 0, 1, 2, and 3 seconds, respectively. The orbit of any given particle is fixed; i. e., is the same for all values of t; but the particle itself occupies different positions as t varies. In other words, it describes the orbit and in the direction (clockwise) indicated by the arrow. The orbits which are su^ciently far from the bottom to be shown in the figure, are, very nearly, circles. The wavy line having its axis parallel to the <-axis is a view showing how the height of the surface (at x=0) changes as t varies. In other words, it is a view of what would be traced upon a self-registering apparatus at the locality a;=0. 342 UNITED STATES COAST AND GEODETIC SURVEY. Of coarse, the scale in which time along the ^axis is reckoned is arbitrary. The instantaneous wave profiles are the wavy lines parallel to the .r-axis; they approximate closely to curves known as curtate cycloids. To obtain such a curve, let a series of circles be uniformly distributed along a horizontal line; take a point on each a constant angular distance from the position of the point on the adjacent circle to the left, say; join the points thus obtained. In order that the cycloid be curtate, it is necessary that the common distance between the centers of the circles be greater than the arc subtending the constant angular distance just referred to. The wavy line parallel to the <-axis differs from the wave profiles only that the abscissae may be drawn to a different scale. 21. Let us now return to equations (26), (27). If in these two equations we assume two of the coordinates t, sc, y to be constant while one is variable, equations (26), (27) are the two equations of a displacement curve, the non-constant coordinate being the variable parameter. (Or, if we eliminate this variable parameter, an equation in x, y is obtained.) This locus must be such that if we proceed along the ^, x-, or ^/-axis, as the case may be, the successive dis- placed particles must fall upon it; the (x, y)origin is supposed to coincide with the undisturbed position of any particle along the axis in question. By supposing t and x constant, equations (26), (27) represent an hyperbola; or, eliminating y, we obtain the single equation =1. (31) A^sin^ (at—lx+a) A^ cos^ [at—lx + a) By supposing y and t constant, eqaations (26), (27) represent an ellipse. As x increases the ellipse is described counterclockwise. These equations likewise represent an ellipse when y and x are constant. As t increases the ellipse is described clockwise. In either case the resultant equation is ?" + yl =1. (32) A^ cosh^ ly A' sinh^ ly ]Srow regarding x (or t) as the parameter of a system of curves, equation (31) represents a system of confocal hyperbolas — the foci being 2 A asunder. Similarly regarding y as the parameter, equation (32) represents a system of confocal ellipses. These ellipses and hyperbolas are bi-confocal and constitute a pair of orthogonal and isothermal systems. When the ellipses are circles, the hyperbolas become radiating straight lines. To see the system of displacement ellipses (circles) in the figure, drop all orbits which are in the same vertical line to the bottom of the canal ; they will then have a common center. The nearly vertical line joining any originally vertical series of particles becomes an hyperbola (radial line) cutting the ellipses (circles) at right angles. [If we put x' = at — Ix + a, y' = ly, z' = x' -\- iy', Z = x + iy, then (26), (27), are equivalent to the single equation Z=Asmz' ' (33) But if the x' y' plane or the x ^/-plane be divided into a system of squares by means of lines parallel to the coordinate axes, they become in the x y-plane by the transformation (33), the confocal system of ellipses and hyperbolas already described.] 22. A wave whose length is several or many times the depth of the water, is called a long tcave. Such waves form a limiting case of wave-motion in water defined by the displacements (26) and (27). The other limiting case being that of surface or short waves, whose character is shown in Fig. 2. For a long wave, ly is a small quantity, and so cosh ly = 1, sinh ly = ly, s. — A sin {at — lx+ a), (34) y = A ly cos {at — Ix + a). (35) REPOET FOR 1897 PART II. APPENDIX NO. 8. 343 From (34) we see that, to quantities of the secoml order, the horizontal displacements are the same for all depths of the liquid inasmuch as y is not involved. That is, the water must move to and fro as if divided up into vertical slices. The expression for y shows that the vertical displacements increase as the distance from the bottom increases. The orbits of the particles are the extremely elongated ellipses having as their equation A^^AH^y' ■ (36) They have a constant major axis at all depths, but the minor is proportional to the depth taken. At the surface the amplitude (77) of the rise and fall (tide) is Ih times the amplitude of horizontal displacement (current). The velocity of the fluid particles is S=i^ = Aa cos (at— lx+ a); (37) •■•^/'^=zl'=Vl' (38) as follows from (28), (34), and (35). Example. — When the height of the (rising) tide from mean water level is 2 feet, in a long tidal river 30 feet deep, the velocity is 2 x -J 4 or 2-07 feet per second (flood). The maximum flood velocity occurs at the time of high water, and the maximum ebb velocity at the time of low water. Slack water occurs at the time of mean water level. From Fig. 2 it is readily seen that the particles of water in a wave surface may, at certain portions of the wave period, be actually flowing up hill. This is one of the most obvious ways of detecting wave motion. Experience shows that the motion of the water in tidal rivers which are not abruptly termin- ated, is well represented by the wave motion here considered. For in such cases reflection can alter the wave but slightly. If two waves of like periods and moving in opposite directions be superposed, the result will depend upon the manner of the incidence. If high water falls upon high water the range of the wave will be increased, while the velocity of the current may be reduced to almost zero (see Fig. 2). For, the particles at high water in each wave move in the direction of wave propagation, and so the resultant motion is perhaps zero. If a high water fall upon a low water, the range of the wave may be almost reduced to zero while the current will have its velocity increased. West of the Isle of Man the cotidal hour is about ten, whether the tide comes from the north or from the south. The consequence is that the velocity of the current is small. 23. a denotes the number of degrees by which the phase of the component displacements of any particle is altered in a unit of time. When the orbit of the particle is circular, a denotes its angular velocity. .■.^-^°=r (39) where r is the periodic time of the particle or of the wave. I denotes the number of degrees by which the phases of the component displacements of two particles differ — the centers of their orbits being unit distance apart. (See Fig. 2.) 360° , , 360O .-. ^- = A, or « = ~^ (40) where A is the length of the wave (in feet). 360° should of course be replaced by 27r if we wish to reckon I in radians. .'. J = - = velocity of the wave. (41) This is independent of the amplitude. 3J:'i UNITED STATES COAST AND GEODETIC SURVEY. From (29) we have When t - 1 r^ = ^/tanh^^* (42) I > ' Period (seconds), r,= /2^^ d= i -/a/ (43) Telocity (feet per second), ^, = M =9 ^ x" or ^. (^4) T V2n-, 4 ' 27r 9 4 ^ feet per second = ^ nautical miles per hour = 1.53 statute miles per hour. [The period of a wave whose length is A is (2 7r)i /^, while the (complete) period of a pendulum whose length is A is V a g When ^ is several times smaller than unity- Period, T, = -A = 0-17G ,-. (45) Velocity, ^, = Vp; = 5-67 V/i. (46) 5"67 feet i^er second = 3'36 nautical miles per hour = 3'87 statute miles per hour. The equation t'= -/rt/i is known as Lagrange's formula. From (43) and (44) it follows that the period and velocity of short waves in deep water vary as the square root of the wave length and are indei)endent of depth of the water. From (46) it follows that the velocity of a wave very long compared to the depth of the water (as is the free tidal wave) varies as the square root of the depth, and is independent of the wave length. 24. Uquations of the ivave profile. ■Let r denote the number of wave lengths (not necessarily an integral number) from the origin of coordinates to the undisturbed point; then for the x we have and for the true x or the x of the disturbed point, x = vX + x = vX+A cosh Ih sin {at —2TTv-\-a); (47) also, for the true y, y = h + y = h+A sinh Ih cos {at — 27r y + a). (48) By so taking the origin that at + a = and writing 6 for 27rv, we have 27r ■-Li'- y- ,- J. cosh Z7i sin (9 j, (49) * See Tables I, II, III of Airy's Tides and Waves; or, Tables 47, 48, 49 this manual. t Cf. Newton's Principia, Bk. Ill, Props, 41-46. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 345 Now i = ^; and so for water deep in comparison with A, cosh Ih and siuh Ih are sensibly equal {=^e"'). For this reason we may write a? = g— ( —m sin 9), y- li — ^^—m cos 6. (51) (52) These are the equations of a curtate cycloid (or a trochoid) whose generating wheel, of radius -IT T ij— , rolls below a line distant g— above the surface of repose, or /i + g— above the bottom; m is the fraction of the radius from the center to the tracing point. For curves below the surface, li expressed or implied in the above equations should be replaced by y' where y' denotes the height of any surface of repose above the bottom. In Fig. 2, A = 10 feet, ^i = 10 feet, A cosh Ih = 0-5 foot, -h- = 1*6 feet, which is the radius of the generating circle, and m = 0-314. For waves which are long in comparison with the depth, cosh Ih = 1, and sinh Ih = Ih, so that the equations for the wave profile are X0 '2n ■ A sin 6, y = h-{- Alh cos 6. (53) (54) Now if the amplitude of the a;-displacement (A) be small in comparison with A, these two equations represent a, very flat cosine curve. 25. Distinction betu-een ordinary and tidal waves. The following characteristics are deductions from the preceding paragraph on wave motion in canals. Short waves. Long or tidal waves. [The depth of the water is supposed to exceed the length of the wave, and the rise to be several times less than the wave length.] Particles move in ellipses which are very nearly circles at the surface. The horizontal and vertical displacements of the par- ticles diminish rapidly below the surface. Particles originally in the same vertical line are, at any given instant, in the same phase of oscillation. The wave profile approaches a curtate cycloid. The marigram, or record of a self-registering gauge, is a curtate cycloid or a projection of one. The period or wave length assumed, the other becomes fixed, regardless of the depth of the water or the rise and fall of the surface. The velocity of propagation depends upon the wave length only. [The depth of the water is supposed to exceed, by a con- siderable amount, the rise and fall of the tide, and the length of the wave to much exceed the depth of the water. ] Particles move in ellipses approaching horizontal straight lines. The horizontal displacements of the particles are about the same at the bottom as at the surface; the vertical displacements are proportional to the heights of the par- ticles above the bottom. Particles once in the same vertical line remain so for a long time. The wave profile approaches a cosine curve. The marigram, or record of a self-registering gauge, is a cosine curve. Two of the quantities period, wave length, and depth of water, assumed, the remaining one becomes fixed, regardless of the rise and fall of the surface. The velocity of propagation depends upon the depth only. 346 UNITED STATES COAST AND GEODETIC SURVEY. Some characteristics not deduced from the preceding paragraph, but rather from observation, are added here: Wind waves. Tidal waves. The period of a short wave at a giveo place depends upon the velocity, continuance, and (in limited bodies of water) direction of the wind. The amount of rise and fall at a given place depends upon the velocity, continuance, and direction of the wind.* Wind waves do notarise unless the velocity of the wind exceed a certain value — 0'45 miles per hour for capillary waves, 2 miles for gravity waves. t The period, as well as the amount of rise and fall, may vary rapidly from place to place, as can he seen in pass- ing around a breakwater. Wind waves are much confused, and their period un- certain. Wind waves are soon destroyed by the viscosity of the water. Storm waves at sea (wind 30 or 40 knots) have a rise and fall of 15 or 20 feet, a period of about 10 seconds, and, hy (43), a length of about 500 feet, t The period of a tidal oscillation does not depend upon the given place, hut upon the astronomical forces to which it is due. The amount of rise and fall of the tide at a given place depends upon, or rather varies with, the direction, and Intensity of the astronomical forces to which the tide is due. Tidal oscillations of like periods are very nearly pro- portional to the disturbing causes, however small these latter may be. The period is fixed the world over; the amount of rise and fall changes slowly from place to place. Tidal waves recur with remarkable regularity. Tidal waves move on as free waves through long dis- tances. The rise and fall of the tide at sea is, by the equilib- rium theory about 1'8 feet x (cosine of latitude)^ § and the length of the tidal wave is hundreds, or even thou- sands, of miles. 26. Ordinary water wave,^ compared with polarized light. Imagine the orbits of the surface particles to be not in the plane of the j)aper as shown in Fig. '2, but perxDeudicular to it, the centers of the orbits still occu]iyiug their former positions, and so lying upon the same horizontal straight line. Suppose these orbits circular, and suppose the particles to move clockwise if we look from toward + a?, the polarization is circular and right-handed; if the particles move in the opposite direction, it is left-handed. In either case the particles will lie upon a helix or screw. The shadow of these points upon the horizontal plane, or upon a plane parallel to the plane of the paper, will represent a beam of plane polarized light. A circularly polarized beam is well illus- trated by sticking large-headed pins into a wooden pencil so that their heads lie upon a helix, and then rotating the pencil uniformly upon its axis. The shadow of this shows the wave motion in plane polarized light. When the orbits of the particles are ellipses, they will represent elliptically polarized light, while the shadows or projections represent plane polarized light. 27. Long water waves compared with sound. The horizontal displacement S now represents the longitudinal displacement of the particles constituting the medium through which sound is propagated. Because tj is proportional to da it is proportional to the variation in pressure at a given time at any given cross-section (and so to the variation in density) due to the motion. Where ^ (or ^) = 0, there is a maximum or a ^x Jx minimum. In water waves the values of x satisfying ^ = are evidently the points of high and pX * According to an item published in Van Nostrand's Eng. Mag., Vol. 24 (1881) p. 36, rise and fall in meters=; f u%, V being the velocity of the wind in meters. The rule is probably due to Coupvent Desbois. See Giinther, Geophysik, Vol. II, p. 378. tLamb, Hydrodynamics, ^ 246, 303. Eussell, Report B. A. A. S., 1837, p. 455. tThis length is probably too great. Eussell, Report B. A. A. S., 1837, pp. 446 et seq. W. Walker, ibid., 1842 (II), pp. 21, 22. Captain Stanley, ibid., 1848, pp. 38, 39. W. Scoresby, ibid., 1850 (II), pp. 26-31. C. W. Merrifleld, ibid., 1869, pp. 3-2, 33. Giinther, Geophysik, Vol. II, p. 378. R. Abercromby, Phil. Mag., Vol. 25, 1888, pp. 263-269. Theodore Cooper, Trans. Am. Soc. of Civil Engineers, Vol. 36 (1896), pp. 139 etseq. For velocity of propagation, see Stokes, Mathematical and Physical Papers, Vol. II, pp. 239, 240. § See ? 47. At Honolulu, Hawaiian Islands, the mean range is 1-2 feet; at Easter Island, 2-2 feet; at St. Helena Island, 2-1 feet; at Ascension Island, 1-4 feet. But it is not to be inferred from these values that the tides in extended oceans are in any way explained by the uncorrected or corrected equilibrium hypothesis. REPORT FOR 1897 PART II. APPENDIX NO. 8. 347 low water at the assumed time. To see how the horizoutal projections of fluid iDarticles are crowded together (like a condensation of a sound wave) at the high waters, and drawn apart at the low waters, we may even make use of Fig. 2, Take the front row of surface particles and project them upon the xaxis, remembering that the displacements (S) must be regarded as small in comparison with the length of the wave. 28. On the reflection of plane water waves. The displacements in an infinite fluid are X = J- cosli ly sin {at — Iod+ a], y = A sinh ly cos {at — Ix + a). Now interpose a vertical barrier where x = L, and the displacements of the reflected wave will be X, = —A, cosh ly sin [at — I {2L — x) -{- a], (55) y, = A, sinh ly cos [at — I {2L — x) + a]. (56) That is, horizontal motions after reflection change their direction, while vertical motions do not. This can be easily seen because the reflection is really due to the horizontal motion. x,j y, being- simple harmonic displacements of the same periods as x, y, they combine with the latter to alter the phase of the wave at a given time and place (cf. §9), Let us now suppose a complete reflection to take place so that the amplitude of the reflected portion should, if conditions permitted, be as great as the amplitude of the original wave. Let x' = x — L and a' = a — IL; then X = A cosh ly sin {at — Ix' + a'), y = A sinh ly cos {at — Ix' + a'), X, = — A cosh ly sin {at + Ix' + a'), y, = A sinh ly cos {at + Ix' + a'); .". X + X, = — 2A cosh ly cos (at + a') sin Ix' = — 2A cosh ly cos {at-{- a — IL) sin \l {x — L)\ (57) y + y, = 2A sinh ly cos {at + a') cos Ix' = 2A sinh ly cos {at + a — IL) cos [I {x — L)]. (58) 29. Wave motion propagated up a canal closed at one end. Now in order that the displacements just ■written apply to the case in hand, two conditions must be fulfilled besides the equation of continuity (17) and the dynamical equation (22). First. Where a? = 0, y = A sinh ly cos {at — Ix + a) ; for otherwise, an abrupt change in height would take place as we pass from open water into the mouth of the canal. Second. Where x = L, x = 0; for, at the head of the canal no horizontal motion can take place. If we write X = ^COShJ/ gjj^ ^J ^jr^ _ ^JJ ^^g ^^^^ ^ ^^^ ^gg^ COS tJu ^ A smh ly ^^^ „ ,^ _ .^ ^^^ , ^ _^ ^. .g^, ' cos IL ■" ^ all of these conditions are fulfilled. 30. Application to long or tidal waves. The expressions (59) and (60) no\^ become, since cosh ly = 1, sinh ly = ly, ■^ sin [I {L - x)] cos {at + a), ' (61) cos IL y = ^^y COS [I [L - x)] COS {at + a). (62) COS v-Lj 348 UNITED STATES COAST AND GEODETIC SURVEY. Prom these we see that for all values of x, S and y or /; are simple harmonic functions of the time; their amplitudes, however, depend upon the value given to x. Throughout the canal the tide rises and falls simultaneously; in like manner, it ebbs and flows. Moreover, it is slack water throughout the canal at the time of high or low water. If IL = 90<^, or any other odd multiple of i ^, the value of j^becomes very great, especially as x approaches L, while for the even multiples 5iecom^_zer2,Srall values of t. A is the length of a free tidal wave in a canal not obstructed by a barrier; when expressed in angular measure, IX is, of course, 2 tt or 360°. The average depth of the Bay of Fundy along its axis is 40 or 50 fathoms. Table 50 gives about 800 statute miles for the corresponding A. .-. J A is about 200 miles. Now it happens that the length of the bay is about 150 miles, so that these particular dimensions may in part account for the large tides near its head.* But the wave progresses at about the rate due to depth according to Lagrange's formula, and so it is reasonable to suppose that the effect of the barrier is scarcely felt because of the gradual shoaling in the upper part of the bay. The Gulf of Maine, whose length inward is about 200 miles and whose depth about 75 fathoms, is, by Table 50, nearly J A in length. Hence the stationary character of the wave and the increase in range, t Portland Canal, forming a part of the boundary between Alaska and British America, furnishes a good illustration of a nearly stationary wave. Its width and depth are quite uniform and its termination is sudden. Simultaneous observations show that the tide at Somerville Bay is simul- taTieous with the tide at Halibut Bay, 30 miles farther up the canal. Also that the tide at Ford's Cave, 60 miles above Somerville Bay, is but five minutes later. 'Sow the depth of the canal is about 125 fathoms on an average along its axis, and so the time required for a wave to be transmitted 60 miles would be about half an hour, instead of five minutes. The range of tide is nearly constant, being on an average 13 or 14 feet. For a depth of 125 fathoms A = about 1 300 miles, Table 50. 31. ForeVs seiche period. Let it now be required to find the period in which a body of water, as a lake, whose length is 2i and whose depth is h, will swing when disturbed from its position of equilibrium by a sudden vertical force acting near either end, or a longitudinal horizontal force acting upon intermediate points. Taking the middle point as origin, either half may be treated as a canal closed at one end, and affected at the mouth with a periodic disturbance whose period is determined by its length and depth. We have just seen that L should be ^ A, in order to bring about the greatest rise and fall at the closed end. But the wave-length and depth being fixed, the periodic time becomes fixed by the equation Eeplacing A by 4L, we have _ 4i_twice length of lake. ■^gh Vgh (64) It is an easy matter to test this formula experimentally. Suppose we have a rectangular tray of water 2L inches in length and h inches in depth. Now, suddenly raise one end or otherwise disturb the equilibrium of the fluid. The free wave immediately traverses the length of the tray, returns, sets out again, and so continues to go back and forth until the equilibrium is gradually restored. Next, suppose that as soon as the wave returns to the end of the tray where it was produced a similar disturbance is repeated. The wave will this time set out increased in size. Let the slight disturbance be repeated periodically, according to the period thus determined, until finally the water simply swings, as it were, there being no progressive character of the motion to be seen. Formula (64) gives the period of the oscillation in seconds, provided we express g in inches (= 386). Of course the period will generally be altered when the depth ceases to be uniform. Ferrel's explanation | of the abnormally large semidiurnal tides of the North Atlantic Ocean is based upon the fact that a tray or canal closed at both ends, extending from Europe to America, having the average depth of the ocean along the parallel of about 52° north, would have about *Cf. Airy, Tides and Waves, Art. 506. t Cf. Mitchell, U. S. Coast and Geodetic Survey Eeport, 1879, pp. 175-190. t Tidal Eesearches, pp.237 et seq. REPORT FOR 1897 PART II. APPENDIX NO. 8. 349 twelve lunar hours for its complete period of oscillation. Possibly the periods of free oscillation of certain zones of the Pacific have considerable influence upon the size of the diurnal oscillations in that ocean. It is possible that component tides, whose periods are some fractions like I, ^, ^, . . . of a half tidal day, may be due in part to stationary oscillations of the kind just referred to; that is, they may owe their size to the length and depth of the body of water in which they occur, rather than to the water being so shallow that the range of tide is a considerable fraction of the depth. 32. Eeturning to equations (61), (62), we have That is, the surface particles, or particles originally occupying the same horizontal line, execute simple harmonic oscillations along fixed rectilinear paths, the tangent of whose inclination to the horizontal is ^y (66) tan [l{L-x)] ^ ' Near the head of the canal this becomes y L — so (67) Let us now suppose the displacements of particles in the same horizontal row to take place about the same point; that is, let the distance x be eliminated, then ^ y^ _ F A cos {at+ a-) -|' ,Qg. This shows that if the middle points of all the rectilinear paths be placed at one point, the extremities of these paths will define an ellipse. For different values of t, the size of this ellipse will vary, but its shape will be unaltered. At the time of mean sea level the ellipse becomes a point. For different values of 2/ the major axis of the ellipse will remain unaltered, but the minor will be proportional to the depth taken. Confining ourselves to the horizontal motion, we may liken it to the horizontal motion of the particles in an elastic body fixed at one end, that is where x = L, and to the other end of which a force is applied. It L — xis small, the body is one of rectangular a?2/-section, and the displace- ment of the particles will be proportional to the distance from the fixed end. When the length L — X is not small, the ir!/-sectiou is bounded by the lines X = L X = x y = 2/ = ± fc sec [l(L - x)] (69) where fc is a constant. 33. How the range of tide may be increased ivhen the cross section of the canal becomes smaller. The energy contained in a long wave can be shown to be directly proportional to its length, breadth, and the square of the amplitude of its vertical oscillation. Now, if the cross section varies so slowly that the wave is not disintegrated by reflection, and if other dissipating causes are ignored, the energy will remain constant. .•. X X b X ?/^ = a constant (70) But X is proportional to Vgh, and so a constant ,„,, ^= Mhi • (^1) The effect of gradual shoaling and converging shore lines, is an increase in the amplitude of the tide wave.* Having once been so increased, it is possible for it to be propagated along the shore as a free wave, virtually governing the tide for considerable distances. * See Lamb, Hydrodynamics, J J 171, 181, 182. 350 UNITED STATES COAST AND GEODETIC SUKVET. SOME HYDRAULIC CONSIDERATIONS. 34. A hay, harbor, or tidal river with hut one opening. Let us suppose that the tide is known at a sufficient number of places to enable one to ascertain approximately the height of tide at any given place in the harbor. We are now not concerned with what takes place outside, but simply with the ever-changing tidal volume within. When the volume is a maximum it is clearly " slack-before-ebb " at the opening or mouth of the harbor; when a minimum, " slack-before-fiood." If, in a short canal of uniform width closed at one end, the depth be such that the range and shape of the tide are constant, then it will be slack water at a given cross section when the crest or trougii of the tide wave is midway between the given cross-section and the head of the canal ;* for, the average depth of the water will then be a maximum or a minimum. Supposing the cross section [F) at the mouth of a harbor to be constant, we have for the velocity ' = pf "^' where dY denotes the change in volume during the short interval of time dt (say ten or twenty minutes). If the area of the harbor is the same at high as at low water, and if the rise and fall of the average surface be denoted by 2/ = A cos (a« + arg„ J. — J.O) -f jB cos (&« + arg„ J? — 5°) + . . . , (73) dy then the velocity at the mouth of the harbor is evidently proportional to -jr., and so to Aa sin {at + arg„ A — A°) + Bb sin [ht + argo B — B°) -f- . . . . (74) In other words, the amplitudes of the various current components compare among themselves, not as the amplitudes of the corresponding tidal components of the harbor, but as these latter multiplied by their respective speeds. Hence, the diurnal inequality in the current velocities is less striking than in the heights of the tide. If, for the sake of form, we write cosines in the place of sines, we must apply i 90° to the above angles. Example. — The area of Sau Francisco Bay and tributaries being about 430 square miles, the width of the Golden Gate at Fort Point 1 mile, and the average (mean sea level) depth at this section 30 fathoms, required, the velocity of the current when the height of the bay is changing at the rate of 1^ feet per hour. Here the hourly change of volume is 430 X 5280 X 5280 x 1^^ cubic feet, and so the change per second is about 5 000 000 cubic feet. The area of the cross-section is about 950 000 square feet. .•. v = W- = 5"3 feet per second = 3-1 knots. In a body of water as large as this, ranges of short duration can not conveniently be used with accuracy for estimating the hourly change in height, unless the tide is known at several points in the bay. 35. On the steady flow of streams. The well-known formula due to Brahms and Ch6zy is v = cVB8, (75) or T ., • • 1 i. 4. / area of cross-section head or fall velocity = empirical constant x ^/ , ,,— „ — , , , . -.— X —7 "* ^ V length of wetted-perimeter length = a coefficient V hydraulic radius x slope. (76) * Cf. L. d'Auria, Jour. Franklin Institute, Vol. 131 (1891), p. 267. EEPORT FOR 1897 PAET 11. APPENDIX NO. 8. 351 Experiments show that c, in a measure, depends upon the roughness of the wetted-perimeter, upon the value of B, and of 8. The value of c is often round about 90, when the foot unit is used, and about 50 when the meter; but the values vary widely.* The best known of the more elaborate formulae is the one generally styled Kutter's.f Because of the inertia of the water, it is obvious that no general formula can be consistent for various sections of a large river like the Lower Mississixjpi. 36. On the flow through a small opening connecting two large bodies of water. Suppose we have two large tanks of water connected by a very short horizontal pipe; also suppose the difference in level (2/„, — 2/») of the surfaces of the fluid to remain constant. All particles in this pipe, at whatever depth it may be situated, and whatever may be its dimensions, provided only it is moderately small, should, by Torricelli's theorem, move with a velocity ^ = ^2(/(2/„, — 2/„). (77) If the dimensions of the pipe have to be taken into account, because of friction between it and the water, we have where C is a coefficient supposed constant for a given material. 37. Short tidal river or strait connecting two large bodies of loater, one or both of lohich are tided. "We shall suppose that the horizontal motions of the two bodies is so small that their influence upon the velocity of the water in the strait may be neglected. Considering only one component of the tide, the respective heights at any given time are Vm = -*!«. cos (at + argo A — A,°), (79) y„ = A, cos [at + argo A — A„); (80) ••■ J/» — Vn = [-4.„. cos A„° — J.„ cos A„o] cos (at + arg,, A) + [A^ sin A„, — J.„ sin A„°] sin (at + arg,, A), = VAJ + A/ - 2A„ J.„ cos (A„o ~ A,o) cos (at + arg„ A + S), (81) , „ _ A„ sin AJ^ - ^ „ si n A„° tan _ - ^_^ gQs ^o _ ^^^ (.Qg j^_^o> showing that the difference in level of the two surfaces is a simple harmonic function of the time. Supposing the motion steady for a limited time, the horizontal velocity in the strait should be, at a given point, proportional to Now it can be shown by § 58, Part II, that or where ± v/ I sin (9 I =1-112 sin 61 + 0-155 sin 3 ^+0-066 sin 5 (9 + . . . , (82) ±V\cos6\ =1-112 cos 61 -0-115 cos3 6+ 0-066 cos5 (9- .... (83) consequently the velocity of ebb and flow is not a simple harmonic function, although the tide in either body of water rises and falls according to such law; that is, there are terms whose periods are i , i , . . . part of the period of the fundamental. Their effects upon the current curve, is to give it a less pointed appearance than a curve of sines, i. e., to render it more like a semicircle. An illustration of such a condition is to be found in the East River, which connects New York Bay with Long Island Sound. Diagrams of the currents in this river, off Twenty-third street, New York, are given upon pp. 423, 425 of the Report of the United States Coast and Geodetic Survey for 1886. * For numerical values under various conditions, see Hermg and Trautwiue's translation of Ganguillet and Kutter's work, A General Formula for the Uniform Flow of Water iu Rivers aud in Other Channels, pp. 39, 160-223, 233-236; also Church, Mechanics of Engineering, pp. 758-761. The symbols li, S, and c are here special or temporary notation. t Ganguillet and Kutter, loc. cit., pp. 24 et setx., p. 129. Church, loc. cit., p. 759. 352 UNITED STATES COAST AND GEODETIC SURVEY. Example— Given, at Governors Island, M, = 2-1 feet, 112°== 231°; at Willets Point, M2 = 3-6, MjO = 330°; required, the times of slack water and of maximum velocities in East River. Reckoning from the time of transit, we liave, for the difiference in level, 2/„ — 2/„ = 24 cos [m^t — 231°) — 3-6 cos (m^^ — 330O). This becomes zero when and a maximum or a minimum m4=88o or 268°; m2<=178o or 358°. But when y„, — y„ is zero, so is v of (77) or (78); likewise fur a maximum or a minimum. When it is high water at Governors Island, m4=231°; and when low water, m2if=olo. 88° — 51o=37o=l'3 hours as the time which slack (before flood) in East River should follow the time of low water. The strength of flood (easterly current) should occur when m2/ = 1780; this means 231° — 178° or 530 or 1-8 hours before high water at Governors Island. Similarly, the slack before ebb in East River occurs 1-3 hours after, and the strength of ebb 1-8 hours before, the time of high water at Governors Island. These statements conform well with observed values, the results of which are given in the Coast Survey Tide Tables. That the velocities of the fluid particles inherent in the wave motions do not account for the currents in East River, appears from the fact that at two hours before high water at Governors Island, the current off Old Ferry Point (a few miles west of Willets Point) is flowing westerly at the rate of l-o knots, which is nearly its maximum value at that place. In the Lower Hudson, off Thirty-ninth street, the maximum velocities are about simultaneous with the tides at Governors Island, as the theory of wave propagation in an indefinite canal would require. Two hours before high water the velocity is small (0-7 knots) and in a northerly direction. Now, had the wave been propagated up East River in a similar manner the velocity would be smdll at this hour. As a matter of fact it has very nearly its maximum value throughout the narrow portions of the river. Moreover, it can not be due to the wave motion from the east, because off Old Ferry Point the velocity is small and in an opposite direction. Thus it is seen that the rapid currents of East River, and particularly around Blackwells Island, are not due to the superposition of two horizontal motions of the water as in simple wave motion, but to the difference in head between New York Bay and the western portion of Long Island Sound.* * Cf. Mitchell, United States Coast and Geodetic Survey Report, 1887, p. 311. IZS- Oxii f. ami GmoiigUjc Suj~vmy Report, /br 1897 Appeneda- 3B. 8 /^ — I) + J a a' (3 sin^ A. - 1) (3 sin^ 6-1) = height of tide according to the uncorrected equilibrium theory. (119) For the lunar tide, „ , 1 mass of moon ,, a'' ^ en x- j. ,ini\^ a a' = i -. — X -. , -,. ^ ^, = 0-59 feet; (120) mass ot earth (moon's distance)-* and for the solar tide, mass of earth (sun's distance)^ ^ ' (i) The height of the semidiurnal portion of the lunar or solar tide at a given station is proportional to the cosine of twice the local hour-angle of the moon or sun multiplied by the square of the cosine of its declination. The factor depending upon the declination is always near unity. (ii) The height of the diurnal portion of the lunar or solar tide at a given station is proportional to the cosine of the local hour-angle of the moon or sun multiplied by the sine of twice its declination. The factor depending upon the declination varies almost directly with the declination. (iii) There is a portion of the lunar or solar tide which depends, at a given station, wholly upon the declination of the moon or sun. The height of this portion is proportional to 3 sin^ d — 1 where d represents the declination of the moon or sun. The period of this expression is a half tropical month or year as the case may be. The height of the entire tide, or of the surface of the sea, at any given time and place is the sum of the six terms just referred to — three belonging to the moon and three to the sun. The corrected equilibrium theory.* — To approximately adapt the foregoing theory to the case of nature, we may write the height of the lunar or solar tide in the form 7^2 cos^ S cos [2 {if} — I) — £2] + El sin 2 d Gos[ip — I — ej] (122) + B„ [3 sin^ 6-1] where B and s must be determined from observations at the given stations. Statements (i), (ii), and (iii) require no modification except that for "hour-angle" we must write "hour angle diminished by a constant appropriate for the station in question" and so for "twice the hour angle". This correction is theoretically necessary (even if the water have neither inertia nor friction) because the earth's surface is not wholly covered with water, and the equation of continuity can not generally be satisfied when the rise and fall is as given by equation (119) unless we continually alter the plane of reference. The ^'s, as did the a"s, involve the factor / mean distance of moon \ ^ _ /^c \^ /actual parallax\ '_ .^ „„, \actnal distance of moony ^ ~\rj ' \ mean parallax J ' ' In practice the inertia and friction of the water produce important modifications in the B's and i's from their equilibrium values. Ifevertheless, the form (122) is capable of aipproximately representing the rise and fall of the tide in nature. This is especially true, if we make the further * See preceding footnote. 366 UNITED STATES COAST AND GEODETIC SURVEY modification of taking 3 and r at times anterior to the time of tide. Such times, as well as the -B's and e's, must be determined from observations made at the given station.* 48. Hxplanation of phenomena noted in § 46 by the equilibrium theory. The tides in (i), § 47, are semidiurnal, while those in (ii) are diurnal. Bach may, for any particular day, be represented by a cosine curve of proper length (period) and amplitude. Fow, it IS obvious that the superposition of a diurnal curve upon a semidiurnal, will, in general, cause the alternate maxima or minima of the semidiurnal curve to become more or less unequal in height and unequally displaced in time. These statements account for (1), (2), and (4) of § 46. As noted in (ii), § 47, the amplitude of the diurnal curve (lunar or solar) is nearly proportional to the declination of the moon or sun. This explains property (6), § 46. The superposition of a semidiurnal curve or wave upon another of nearly equal period, but of greater amplitude, simply increases or decreases the amplitude of the latter when approximately like or opposite phases coincide; but when the phases differ by approximately 90° or 270°, the principal wave is displaced in time by the subordinate one— accelerated or retarded according as Midnight Noon Midnight 2'2 0*" 2 Fig. 7. the maximum, say, is 90° in advance or in retard of the maxima of the principal wave. This accounts for properties (3), (5), (7), and (9), § 46. Property (8) has been explained in § 47 where the values of the perigean, apogean, and mean ranges are compared. This amounts to varying the a' or the jB's inversely as the cube of the moon's distance from the earth's center. At a station where observation shows that Bi is several or many times as great as iJj expression (122), the number of maxima and minima of a curve composed of diurnal and semi- diurnal parts will usually depend upon the number of maxima and minima of the diurnal part when the moon's declination is great; but when the moon is near the equator the number may be governed by the semidiurnal part. This accounts for properties [1] and [2], § 46. The moon crosses the equator and reaches its extreme declination at nearly the same points in the heavens as does the sun. This accounts for property [3], A still more perfect form or expression for the equilibrium theory is obtained by developing the tide-producing potential (the principal part of which is inversely proportional to the cube of ■ Cf. Thomson and Tait, Natural Philosophy, §§ 804-811. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 367 the disturbing body's distance from the earth's center, and directly proportional to 3 cos^ ^^^ — !> § 47), into a series of cosine terms. For considerable periods of time the coefficients of these terms remain sensibly constant and their angles or arguments increase uniformly with the time. Having found from the development of the potential what are the more important terms, one then assumes that by leaving all amplitudes and epochs arbitrary the series is, by the principle of forced oscillations,* capable of representing the tide at any given station. The harmonic analysis, § 49, has for its object the determination of these amplitudes and epochs from tidal records. 49. Harmonic analysis, t Since the tide is periodic in its character, and since the periods of its causes are known from astronomical considerations, it ought to be possible to represent the height at any given time by means of the Fourier series, or rather an aggregation of such series y = Acos (at + a) + B cos [bt + /3) + . . . (124) where y is reckoned from main sea level. For aiding the imagination, we may suppose that any given term in this series represents the oscillation caused by a fictitious star, or moon, moving uniformly in the celestial equator around the earth, and at a constant distance therefrom, having the property of producing a maximum of the oscillation, or component tide, a certain number of hours after its meridian passage. If a denote the hourly speed of the component A, or the apparent angular velocity' of its fictitious moon, and A° its epoch or lag expressed in degrees, A°/a is the lag expressed in hours. Also if argo A denote the hour- angle of the fictitious moon at local mean midnight, at + argo A is its hour-angle at any subsequent hour t. Consequently the time of high water of the component A is Ao_^rg^ (125) a a and the height at any time t is so that A cos (at + argo -A. -A°), (126) a = SirgoA-Ao. (127) By replacing A, A°, a, and a by B, B°, b, and /?, the corresponding quantities for any other component, B, are obtained. The heights due to any components may be shown graphically thus (see Fig. 7) ; Lay off the hours of the day according to any convenient scale. Draw cosine curves of amplitudes A, -B, . . . and of periods —-; -r-, . . . hours in length. The first maxima are located upon the hour lines/ J.° argoJ. -BO argo^ a a ' (128) the succeeding maxima are then fixed by the lengths of the several periods. The symbol B may be used to indicate the time of transit of any fictitious moon. To combine these curves, add the ordinates for each hour, thus obtaining the resultant tidal curve from which the times and heights of high water and low water may be obtained. The object of the harmonic analysis is to resolve the observed tide, i. e., observed heights of the surface of the sea, into simple elements or component tides, consisting of simple harmonic oscillations. The quantities a,b, . . . and argo A, argo-B, ... are known from astro- nomical considerations, so that the analysis of the tide at a given place implies only the determination of the amplitudes A, -B, . . . and the epochs A °, 5°, .... To harmonically analyze a given tide, let its height be given at each hour of the day, for a year, say. Sum these ordinates, as nearly as may be, at the component hours of each component (its harmonics excepted). The sums belonging to each component will be 24 in number and * See Laplace, M6o. C^l., Bk. IV, 5 16; or see under Laplace. Also, Part II, § 14. t See Part II. 368 UNITED STATES COAST AND GEODETIC SURVEY. represent sums corresponding to each of the twenty-four hours into which the component day is supposed to be divided. As the summation in each case is made with reference to the component hours, the effect of the other components upon these 24 sums will, in the long run, approach zero. Having found the 24 heights corresponding to these sums, they may be plotted as hourly ordi- nates; such a plotting would represent the required component tide combined with its harmonics. To analyze these 24.heights, ^o, K K ■ ■ ■ hs, assume each to be of the form h = So + A,cosat + Aisinat + A.2COB2at + A^sm2at+ . , . + As cob 8 at + As sin 8 at, (129) where at = 0°, 150, 300, .... 3450, It is not difficult to show that the most probable values of S„, A, Xare given by the equations 24:Ho=ho+ h + lH+ . . . +7i23. 12 Ai = ho cos 0° + hi cos 15° + A-, cos 30° + ■ . . . + h^s cos 345°, 12 A2 = ho cos 0° + hi cos 30° + /t^ cos 60° + . . . + 7*23 cos 330° 12 A3 = ^0 cos 0° + hi cos 450 + 7i2 cos 90° + . . . + k^s cos 315° 12 Ai = ^0 sin 0° + hi sin 15° + h^ sin 30° + 12 A,. = ho sin 0° + hi sin 30° -f h,, sin 60° + 12 3.3 = ho sin 0° + 7i, sin 45° + 7^2 sin 90° + + ^3 sin 3450 + 7i23 sin 330° + ^23 sin 315° (130) (131) (132) From these values of A, A, we find A and a by the relations A = (A^+1y=, tan « = - ^. A A° then becomes known by the equation A° = argo A — a^ argo A being known from astronomical considerations.* So for components B, C, etc. It may be added that because the hourly heights are tabulated in solar time, most of the amplitudes as brought out in the analysis must be increased by a factor a little greater than unity, known as the augmenting factor ; also that most of these amplitudes must be corrected for the longitude of the moon's node by the application of a suitable factor. For series less than about a year in length, still other corrections must be applied. NONHAEMONIC EBDUCTIONS, ETC. 50. The object of these reductions is to determine nonharmonic quantities from observations made upon high and low waters. Of these quantities the most important are the lunitidal intervals (HWI a^d LWI) and the mean range of tide (Mn). The times and heights of the individual tides depart more or less from the values which would be obtained from applying the mean lunitidal intervals to the times of transits and using the mean range. The departures have a certain kind of periodicity depending upon the inequalities to which they are due, and they may be tabulated with reference to each inequality known to exist in the tide at all places 5 that is, so tabulated as to follow some known astronomical argument. A set of tables thus formed may he regarded as a series of corrections to be applied to the mean tides because of the several inequalities. In case of the largest of the inequalities, it is more convenient for some purposes to have tabulated the actual values of the intervals and heights throughout the period of the inequality instead of the departures from their mean values. In case of the phase inequality the "time of tide" may be tabulated directly instead of the lunitidal intervals.! After such tables have been obtained from the observations, one may, if he choose, analyze the tabular values, thus obtaining a set of coefficients (intervals or epochs and amplitudes) from Which these tables could be reproduced. * The arguments for January 1 of each year from 1850 to 1950 are given in Table 3. t Cf. Lubboct, Phil. Trans., 1831, pp. 898, 399. Here the time of the year (i. e., the month) is used for one of the arguments and so no correction is necessary because of the equation of time. For examples of curves of phase inequality, see Phil. Trans., 1868, last plate. EEPORT FOR 1897 PART II. APPENDIX NO. 8. 369 The disadvantage of the nonharmonic treatment lies in the fact that when several considerable corrections are successively applied to the mean tides, the resulting effect is not the same as it would have been had all been applied simultaneously, as in the case of nature. It therefore becomes important to tabulate each inequality not only with respect to its own argument, but also with respect to the argument of the principal or phase inequality. This necessitates a table of double entry. In some cases it might be advisable to have tables of triple entry. For the present we shall assume that the length of the series of observations treated is suffi- cient for separating the various inequalities sought. Then it is obvious that, grouping the obser- vations with respect to the argument (uniformly varying with the time, or nearly so) of any particular inequality, we shall obtain in the long run certain departures from the mean values of the lunitidal intervals and range, which may be tabulated without maldng any assumption as regards the age of the inequality sought, and so using, say, the immediately preceding transit. Such results are of interest for theoretical purposes. For actual prediction, however, where all important inequalities should be properly applied, it is desirable to form tables of double entry. 51. First reduction. The principal object of this reduction is to determine the lunitidal intervals and the mean range of tide. A specimen of first reduction is given in § 60. This shows that, for Tybee Island Light, Georgia, HWI = 7" 11", LWI = l" 00-°, Mu = 6-8 feet. To roughly predict the tides for Tybee Island Light (ia local time), copy down the local times of the moon's transits across the meridian of Tybee Island Light, and add the intervals, 7^ 11™ for the high waters and l*" 00™ for the low waters. To obtain predictions in eighty-first meridian time, using Greenwich transits, add the uncorrected intervals, 1^ 21™ for high waters and I'' 10™ for low waters. Having thus determined the times, the heights, with reference to half- tide level may be roughly determined by calling the high- water heights J of 6-8, or + 3-4, and the low- water heights, — 3-4; or, referred to the particular staff used, they are on an average 9*4 and 2-6 feet, respectively, as found in the reduction. Another specimen of first reduction, with some additional matter not important for the present purpose, is given in § 30, Part III. 52. Determination of the periods of tidal inequalities. The existence of several tidal inequalities can be inferred from very simple considerations; of these may be mentioned the phase, parallax, and declinational inequalities. Others would hardly be suspected without developing the potential of the tide-producing forces. For the present we may assume such a development made in terms strictly periodic, and also assume that terms of like period are to be found in the tide. Such terms, or component tides, as go through their respective periods twice a day, pretty nearly, having the subscript 2 added to the letters used to designate them, are, as already stated, called semldiurnals; and those once a day, having the subscript 1, are called diurnals.* The number of degrees per hour, i. e. the speed, may be denoted in each instance by the corresponding small letter. Looking now at the column headed " Synodic period," Table 1, it will be seen that there are, opposite the semldiurnals, the synodic periods 14-76529, 27-55456, 13-66079, 31-8119 J, 15-38734, days. These are a half synodic month, an anoma- listic month, a half tropical month, the moon's evectional period, etc. To this list we should add such synodic periods as would be obtained by means of the lunar nodal components differing in period very little from M2 and K2, respectively.! The synodic period for the first is the node- equinox period, or about 18-6 years, and the corresponding variation in the tide is called the lunar nodal inequality. In regions where the water is shallow and subject to annual changes, the year should be used as one Of the periods. In this way may be obtained the periods of all sensible inequalities affecting the semidaily tide, i. e., those affecting both high waters of the day alike, also both low waters. The diurnal inequality in the tide is really made up of several partial diurnal inequalities. * See Part II for further explanations of the symbols used to designate these components. tin Ferrel's notation M'(i,2) and M'(3,2), United States Coast and Geodetic Survey Report 1878, p. 270. The speed of the former should be 28-981898, and not 2 x 14-489846, and that of the latter 30-084344, not 2 x 15-043275. 6584 24 370 UNITED STATES COAST AND GEODETIC SURVEY. They affect ia nearly opposite ways the two high waters of a day aud also the two low waters. The periods ia which they should be tabulated are those synodic periods given in Table 1 which stand opposite the diurnal components. To this list of diurnal components should be added the lunar nodals having speeds nearly equal to the speeds of Ki and d, respectively.* The synodic periods just referred to are such that the same transit (upper or lower) should be adhered to throughout. It has been supposed that the inequalities have periods of fixed lengths; but the arguments which divid^e up the periods may not vary exactly uniformly with the time throughout these periods. Such are styled circular arguments. Most inequalities in the moon's motion have little effect upon the lunitidal intervals, especially if the ages of the corresponding tidal inequalities be properly allowed for in the selection of transits. The reason for this is quite obvious. With the amplitude of the tide it is otherwise. By considering the effects upon amplitude only, and dis- regarding the effects upon interval, it is sometimes possible, when non-circular arguments are used, to throw two or more inequalities into one. For instance, if the moon's parallax be the argument (preferably noting whether it is increasing or decreasing, thereby avoiding the consid- eration of age), the parallax and evectional inequalities referred to above will be embraced in the tabulated height or amplitude corrections. The time corrections will be somewhat uncertain. A similar remark applies to the declinational and nodal inequalities. If we take a month's observations about the equinoxes, the diurnal inequality will be due to the moon, and so the lunar part of K; is involved instead of the entire Kj. If groups of observations are taken about the times of zero declination of the moon, the diurnal inequality will be due to the sun, and so, instead of Kj, Oj, OO, the involved components are the solar part of Ki and Pj. In grouping observations with respect to these arguments or periods, the transits should, as a rule, be kept distinct; for, it is an easy matter to subsequently combine values belonging to the upper and lower transits if desired. For instance, when tabulating for the inequality whose period is a tropical month, we should have, opposite the moon's right ascension or longitude, two classes of high-water lunitidal intervals (approximately equal) derived from upper aud lower transits, respectively, or two such classes of intervals (differing by approximately 12 hours) derived from one transit; so for the low- water lunitidal intervals. Each of these four kinds of intervals should be accompanied by a corresponding height. In this case the diurnal and semidiurnal inequality, whose period is a tropical month or a half tropical month, are capable of being tabulated together. The half-tide level is subject to small inequalities having the periods of the preceding tidal inequalities, or some simple fractions or multiples of such periods. These fluctuations in half- tide level may be due to long-period components (which are astronomical, meteorological, or shallow- water) or to fixed speed relations in the components of short period. t The greatest of these inequalities is, at most places, the annual; it involves the components Sa, Ssa. If high water only be observed, these inequalities in half-tide level cannot be separated from the inequalities in mean high-water heights. *In Ferrel's notation MVi.i) and M'(6,,); 1. c. ante. Their speeds are 15-043275 and 13-940829. t For nnmerical examples, see United States Coast Survej- Report, 1868, pp. 80, 81, column headed J (H/ -f- Hj'). REPORT FOR 1897 — PART II. APPENDIX NO. 8. 53. List of inequalities following a single circular argument. 371 Hourly vari- Name. Argument. ation of argument. Period. Components involved.* Numerical examples. In both high zt aters or bot) o I low maters. Phase. Hour of transit, t 1-0158958 i synodic mo. S^, //,. Lubbock, Phil. Trans., 1831, pp. 400-403, cols. A, B. Bache, United States Coast Sur- vey Reports, 1853- 1864. Ferrel, United States Coast Survey Re- port, 1868, p. 74, Table V. Parallax. Moon's anomaly.J 0-5443746 Anomalistic mo. N„ L=, 2N. Ibid., p. 76, Table VI. Declinational. Moon's longi- tude.! 1-0980330 ^ tropical mo. K,. Ibid., p. 78, Table VII, left side. Evectional. Phase arg. -paral- lax arg. 0-47I52I2 Montbly evec- tional period. v^, 'k,. Ibid., p. 82, Table IX. Lunar nodal. Long, of node. 0-0022064 Node - equinox M'(i,2). [See footnote, Ibid., p. 81, Table period. ante. ] VIII. Annual. Day of year. 0-0410686 Tropical year. Component || having a speed 360° Ibid., p. 80, Table VII bis. ° No. hours in a yr. Solar parallac- 0-9748272 Synodic period tic. M„ T,. /n alternale high waters or alte mate Imv waters. Moon's longi- 0-5490165 Tropical mo. K„ 0., 00.1[ Ferrel, 1. c, p. 78, tude.? Table VII, right side. 0-4668793 Synodic period M„ P.. P.. I-09339I2 Synodic period Qx, Jr. 0-5512229 Synodic period M„M'(3,i)Or M'(3,l,, M'(6,.,. * I. e., besides the mean tide, -wbicli consiats chiefly of Ma- (See Tables 1, 2.) t Strictly speaking, the apparent local time. For a table of single entry, and also a table of double entry where the season of tbe year is not one of the arguments, mean time can be used in reductions instead of apparent, and without corrections when the series is long. This argument is for fixing tbe difference between the right ascension of sun and moon. An equivalent argument 1^ the age or phase of the moon. X True or mean. § True or mean, or her right ascension. For abort aeries the component 3K disturbs this inequality, because 2ra^ is very nearly equal to 2n + kj. II Shallow water meteorological. U If we take a month's observations about tbe equinoxes, the diurnal inequality will be due to the moon, and ao the lunar part of Ki is involved instead of the entire Xj. If groups of observations are taken about the times of zero declinations of the moon, the diurnal inequality will be due to tbe sun, and so instead of K„ O, , 00, tbe involved components are the solar part of K, and P, . 54. Analysis of tidal inequalities following a single circular argument. If the luiiitidal intervals and heights be classified according to an argument x, whose period is that of some tidal inequality, the resulting interval and amplitude may, according to Fourier's theorem, be written Bo+M'i sin x+N'i cos x+M'u sin 2 x+N'u cos 2 x+ i Mn + M- cos x+N'i sin x+M,, cos 2 x+Nu sin 2 x+ (133) (134) wliere Bo denotes the mean lunitidal interval, Mn the mean range of tide, and i the characteristic of the inequality. These expressions may be written in the form Bn+B, sin {x—S!)+B;i sin (2 x—e;,)+ . J Mn [1+iJ.. cos {x—ai)+Bi: cos (2 «—«;,) + (135) (136) 372 UNITKD STATES COAST AND GEODETIC SURVEY. Jf', ^ , , ,., „ „, „ M', B,=± VM',^+N'?=^^, B,= ±VWJ+W? = ^^, . . . , (137) *^'''' = T^'*^'''^'==^' ■ • ■ ; (138) i Mn i?,= ± ^j|f;.2+j^.2 = -A_, J Mn B,,= ± VmJ+Wu'='^^ . . . . , (139) tan «,= J, tan «,=^ .... (140) In reducing tides, the observations are taken in groups. For this reason the coefficients B^, Ei, and £„•, E;„ as determined above, should be multiplied by the factors (a little greater than unity) cy---A~,-'' ^^ and ■ ^'"^^ ^ (141) 2 sm i (x-x;) sm {x,-x;} ^ '' where «,, and a;, are the values of x at the two limits of the group of observations. -B; or i?; may be spoken of as the coefficient of the inequality whose characteristic is i. To tnA Bo, M/, N'i'; Mi,', Ni/, . . . from the tabulated or classified intervals, we suppose a; taken to, say, each 15°. Then we have 24 observation equations for determining jBo, 31/, etc. Let expression (133) be denoted hyyj; then these 24 intervals are tj'„, y',5, y'^^y • ■ ■ y'us- It is not difficult to show that the most probable values of the required quantities are given by the equations 24 Bo = 2/'o + 2/'i5 + 3/'3o + . . . +y'zi„ 12 31' i = i/'o sin 00 + 2/',5 sin 15o + y'^ sin 30° + 12 31' a = y'o sin 0° + y'^o sin 30° + y'eo sin 60° + 12 = il/',,i2/'„ sin 0° + y'is sin 45° + y'g„ gin 90° + 12 N'i = y' cos 4- 2/'i5 cos 15° + y'30 cos 30° + + y'345 sin 3450, + y'-ao sin 330, + y'3i5 sin 315, + 2/'345 cos 3450, (142) For finding J Mn, 31^, JV;, TUf;,, N',!, . . . , we have the above equations with all accents dropped, sines and cosines interchanged, and J Mn in the place of B^. (See §§ 49, 55.) If we suppose that the inequality has been analyzed and certain epochs and amplitudes obtained, it is to be noted that the epochs (with single subscript) of the inequalities should be proportional to the respective ages of the latter. This implies that if an astronomical argument be taken out, on an average, a constant amount earlier or later than the particular phase of the tide (generally midway between high and low water) to which the analysis refers, the epochs must be altered by the respective variations in the arguments during this constant interval; 1. e., the position of the tidal inequality is required at the time at which the astronomical argument is taken; or, what amounts to the same thing, the value of the astronomical argument is required at the phase of the tide with respect to which the analysis is carried out. Suppose high water heights to be analyzed; they belong, on an average, to a time a constant number of hours, HWI (uncorrected), after the moon's transit. Suppose the ranges analyzed; they belong to a time J (HWI + LWI) after the moon's transit. For the convenience of many places, all astronomical arguments can be taken out at the times of the moon's transit across, say, the meridian of Greenwich, and the epochs afterwards altered accordingly. The epochs require no alteration if the astronomical arguments be taken out at the particular phase of the tide to which the analysis refers. This latter method, however, seems particularly undesirable, if not practically impossible, in the case of the diurnal inequalities.* *The fourth sentence from the end of § 4fi, Part III, is meaningless, and should he replaced hy something like the following, viz. : The epochs or,-, an, or «;, su, belonging to a tidal inequality whose characteristic is i, and which depend, in a measure, upon the phase of the tide analyzed, must he so modified as to suit the time at which the astronomical argument is taken; e. g., the time of transit across some given meridian, or the time of high water. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 373 55. Example. Eequired R\ and cti for Sitka, from the month of observations tabulated in § 30, Part III. Copy down the transits and the high and low waters, marking, for distinction, the heights which go with the lower transits. Leave no vacancies, but bring consecutive transits into con- secutive positions, i. e., upon consecutive lines; similarly for the heights. This necessitates at times a splitting up of the date at left-hand margin. Combine the ranges in pairs, for the purpose of eliminating the diurnal inequality, and copy down the time of transit corresponding to each four tides so taken. Thus: Date. Transit. HW I,W Range. July I I 34 I 34 (14-6) 3 '4 7-8 I (13 59) I 59 12-8 (8-4) 7 '4 2 ? 24 2 24 (J4-4) 3 '9 7-6 2 (H 48) 2 48 13-0 (8-3) 7 '4 3 3 II 3 II (14-0) 4-0 7-5 3 (15 34) 3 34 12-8 (7-8) yo 4 3 37 3 37 (13-2) 4-3 7'i 4 (16 19) 4 19 I2'9 (7-6) 6-4 5 4 41 4 41 {12-5) ,^'°. 6-8 5 (17 04) 5 04 13-2 (7-2) 6-0 6 .S 26 5 26 (l2-o) 5'9 6-6 6 and 7 (17 48) 5 48 13-6 (6-6) Distribute these combined ranges according to their hour of transit, using always the nearest whole hour and counting the hour from to 12. Take the sums and means of the values so distributed and determine -Bi and ai by equations (136), (139), and (140); or, for convenience, make use of the form labeled "Harmonic analysis of tides," Part II, § 61, The angle (ai or C) thus determined approximately represents, when converted into time at 10'016 per hour, the time which must elapse between new or full moon and spring tides. This must be corrected for the mean lunitidal interval, and, when the series is short, for the equation of time. For Sitka, J (HWI -f LWI) = 9'' 44">-8; and uncorrected Mn = 7-31 feet, from " first reduc- tion." The analysis gives ai = 26o*7 ; the hourly variation in the angle of the semimenstrualinequality is 1°'016, .•. 1'3-016 X 9'747 = 90'9; and ai becomes 360'6. The correction for not having used appa- 24. rent time is — 0-483 (equation of time). [0-483 =-, ^ — ~ — , — ^— x 1-016.1 For July 1-29 ^ ^ lunar day — solar day the equation of time is + 5™-4; and so the correction to ai is — 2o-6, This leaves 340-0 for the true value of ai. In the harmonic notation. Part III, § 47, ai = ^P — MP; (143) .-. Age of the phase inequality = 0-984 x 34 = 33-5 hours. Upon applying the augmenting factor 1-0211, the Ri J Mn from the analysis is 0-968 feet. The factor Ft = 1-36, Table 33, will very nearly reduce Bi J Mn to its mean value. In the harmonic notation i?i J Mn = S2 + fXi. (144) Now, 1A2 is 0-01 Mn, nearly; .-. uncorrected S2 = 0-968 — 0-073 = 0-895 foot, and corrected S2, 0-895 X 1-36, = 1-217 feet. To correct B^ J Mn, add to 0-968, (1-217 - 0-895). .-. Ej J Mn = 1-290, Bi = 1-278 -j- 3-655 = 0-350. If we use an argument varying uniformly with time, it is not necessary to have recourse to a nautical almanac. The (constant) period of the inequality is divided into 12 or 24 equal parts, and O"" of the first day of the series is taken as origin; periods and twelfths or twenty-fourths of periods are laid off thereafter. The times of transits corresponding to the various ranges are dis- 374 UNITED STATES COAST AND GEODETIC SURVEY. tributed among the parts as accurately as possible. The results are then brought together and analyzed, thus giving an amplitude (E,Mn)aud angle (a,); «■; should then be altered by the speed of the inequality times the mean (uncorrected) interval. The corrected a-; denotes, when divided by the speed of the inequality, the time elapsing between O^ of the first day of the series and the time when the inequality becomes maximum. This time, diminished by the time when the corre- sponding forces become a maximum, is the age of the inequality in question.* 56. List of inequalities folloicing a pair of circular arguments. Arguments. Components involved.* In both high waters or both low waters. Nj, Ls, 2N, v^ Aj Hour of transit t (upper and lower) .Moon's anomaly ("Hour of transit (upper < and lower) [Day of year j [Hour of transit (upper \ and lower) [lx)ng. of node Kj, Tj, Rj, and comp'ts, having speeds ra, j- 360° M' No. hours in yr. M' (3,5) In alternate high waters or alternate low waters. Hour of transit (upper K,, O,, P., GO or lower) Day of year J Hour of transit (upper or lower) Moon's anomaly Hour of transit (upper or lower) I/Dng. of node. Q., J., M., 2Q, p. Numerical examples. Ferrel, United States Coast Survey Report, 1868, pp. 69-71. Lubbock, Phil. Trans., 1831, pp. 400- 403, 412 Ferrel, United States Coast Survey Report, 1868, pp. 61-68. Lubbock, Phil. Trans., 1836, pp. 65-73, 245-254- Bache, United States Coast Survey Report, 1854-1864. Ferrel, United States Coast Survey Report, 1868, pp. 61-68, 100, loi. M' M' (6,1) * I. e. besides the mean tide witli pliase inequality or Mj, Sj, and i^.,. If a component have a synodic period with Mj, S^ (for semi- diurnals, M„ S, for diurnals), or any component involved in the ineg[uality defined by either separate argument, equal to the period of either separate argument or the synodic period of the two arguments or the synodic period with Mj, Sj (or M,, S,) of any other involved com- ponent, such component is involved in the tabular values. t See second footnote, preceding list of inequalities. + Or moon's right ascension, longitude, the number of days from extreme declination of the moon, longitude from intersection of orbit and equator, etc. ^ 2 Q, p, are here used to denote components whose speeds are 12-8542862, 13-4715144, respectively. See Table 1. 57. On the use of noncircular arguments. Tidal inequalities depend upon the difference in right ascension of sun and moon (which is usually given by the hour of transit), their parallaxes, and their declinations. Unless the ages of the inequalities be properly allowed for by selecting suitable transits, it is necessary to make a distinction between increasing and decreasing parallax or declination. These arguments are naturally suggested by the equilibrium theory of tides. A set of tables each having the hour of transit as one argument and parallax or declination of the sun or moon for the other, is given in the Tide Tables for the British and Irish Ports, issued by the commissioners of the admiralty; the daily predictions therein published are obtained by "E. r suppose i = 2 denotes the parallax inequality. Then corrected a-2 + argn M2 — argo N.j=M2° — Nj^, where L2 is disregarded. From the amplitude EMn X 1-0138 /(N.2)-|/(L,) = 2N2. (145) (146) EEPORT FOR 1897 PART II. APPENDIX NO. 8. 375 aid of this set of tables." The tides around the British Isles have very little diurnal inequality, and so are susceptible of this simple mode of treatment. See §§ 4-7. 58. Inference of tidal inequalities from observed nonharmonio constants, etc. Owing to the great amount of labor involved in completely deducing tidal inequalities from observations, it does not seem advantageous to work along that line for obtaining practical results, although it may be desirable for certain theoretical purposes. In fact the results of discussions covering a long period of time (say a node-equinox period) are always instructive. If, on the other hand, we confine our attention to determining a few quantities which fix the size and position of each important inequality, a general knowledge of tides ought to enable us to form tables, based upon these determinations, which are sensibly true at most places. For instance, if the spring, mean, and neap ranges, also the age of the phase inequality are known, the phase inequality in interval and range become approximately known at any given time by means of the following general table. The two columns at the right enable one to approximately introduce the declinational and solar parallax inequalities. The second table shows the lunar parallax inequality.^ Table of phase effects. Time. Increase in luni- tidal intervals. Increase in semi- range of tide. Time. Increase in luni- tidal iiitervals. Increase in semi- range of tide. Date. Factor^.* d. h. m. d. A. m. o oo .Sg-Np °MnX? + -23/(Sg-Np) '0 00 Sg-Np ° MnX? -■29/(Sg-Np) Jan. 1 0-82 o o6 - 5 + ■23 06 + 13 " —•29 11 0-88 o 12 — lo " + •23 12 +25 —•28 21 o"96 o i8 —14 + ■22 18 +35 —•27 31 1-04 i I oo —19 + •21 CO I 00 +44 " —•25 Feb. 10 i"i3 1 I o6 — 23 " + •20 ^ I 06 +52 " —■23 20 I -20 -tJ I 12 — 28 " + •19 •J3 I 12 +58 " — ■21 " Mar. 2 1-25 bo B I i8 —32 " + •18 Oh I 18 -f62 " —•18 12 1-27 'u - 2 oo —37 + ■17 rf ^ ■ 2 00 +65 " —■16 22 1-28 & 2 o6 —41 + •15 2 06 +66 -■13 Apr. 1 1-26 u 2 12 -44 " + ■13 B 2 12 +67 ■' —■10 " 11 1-22 i; 2 i8 -49 " + •11 2 18 +67 " —■08 21 I-I4 < 3 oo —52 + "09 3 00 +66 " —■05 May I I -06 3 o6 -56 " + ■07 3 06 +64 " — •02 " 11 0-96 3 12 —59 + •04 3 12 +62 " •00 " 21 0-87 3 i8 — 6i + •02 3 18 +60 " + ■03 31 077 L4 oo -63 " — 'OI " .4 00 +57 " + ■05 June 10 071 20 0-67 30 0-68 July 10 20 074 0-82 'A oo -57 " + -OS '4 00 +63 " —■01 30 092 3 i8 —60 + •03 3 18 +61 " + ■02 Aug. 9 l-QI 3 12 -62 " ■00 3 12 +59 " + ■04 19 i-io 3 o6 -64 " — -02 " 3 06 +56 " + •07 29 ri8 to 3 oo -66 " — •05 i 3 00 +52 " + ■09 Sept. 8 1-23 -S 2 i8 -67 " -■08 .-S 2 18 +49 + -H i8 1-26 •V 2 12 -67 " — 'lO " -4-' 2 12 +44 +•13 28 1-26 ^ 2 o6 —66 —•13 2 06 +41 " +•15 Oct. 8 1-24 (V ■ 2 oo -65 " —•16 'u - 2 00 +37 " +•17 18 I -20 I i8 —62 —■18 & I 18 +32 " +•18 28 I-I4 1 I 12 -58 " — -21 " V. I 12 +28 " +•19 Nov. 7 i-o6 I o6 —52 — •23 I 06 +23 " +•20 17 0-97 PQ I oo —44 — ■25 pq I 00 + 19 " + •21 27 089 O i8 —35 --•27 18 + 14 " + ■22 Dec. 7 0-83 O 12 —25 — ■28 12 + 10 " + ■23 17 079 o o6 -13 — ■29 06 + 5 " + ■23 27 o-8o o oo " — •29 .0 00 " + ■23 Jan. 6 0-85 ^The factor p applies to the "increaae in semirange of tide," "knd not to the ' declinations of the sun and moon and to the solar parallax. 'increase in lunitidal intervals." It ip due f the »In these the phase inequality is obtained from observation. The inequalities due to parallax and decliuatiou of sun and moon are in accordance with Bernoulli's equilibrium theory. They are as tabulated by Lubbock in the Philosophical Transactions for 1836, pp. 58, 59, 257-262. For Devonport these, too, are based upon observations; of course the table for correction for moon's declination would naturally, in this case, involve that due to the sun. ■JThese tables are based upon Tables 24, 25, and 31. 376 UNITED STATES COAST AND GEODETIC SURVEY. Factor expressing the effect of the moon's parallax upon the mean range of tide. Time. Factor g. Time. Factor y. Time. Factor g. Time. Factor ?. d. d. d. d. s I-I7 a 7 099 a ?! 0-86 a [7 0-98 t I i-i6 (U 6 0-96 I 0-86 S 6 I -02 ¥ ■ 2 i'i5 ^ . 5 o'93 5f ■ 2 0-87 5 I -06 S I-I3 i'S 4 0-90 as 3 0-88 4 I '09 4 5 1-09 I -06 13 3 2 0-88 0-87 tat! ■ D 4 0-90 o"93 HWQ; (152) or H WO 1 rGc Mn~i « Gt - Mn 3-^-HWQ L 5 J ' ^^^^^ HWQ>LWQ. (153) J Mn = depression of mean low water below mean sea level. (154) J Sg = depression of low-water springs below mean sea level. (155) 2Di= VHWQ2 + LWQ2. (156) HWI — LWI = duration of rise, (157) LWI - HWI = duration of fall, , (158) adding 12'' 25"" when necessary to make the result positive. Tropic HHWI + tropic LHWI + tropic LLWI + tropic HLWI = 2 (HWI + LWI) ; (159) and, less accurately. Tropic LHWI = 2 HWI - tropic HHWI, (160) Tropic HLWI = 2 LWI - tropic LLWL (161) Mc = ^ (Gc + Sc). (162) 378 UNITED STATES COAST AND GEODETIC SURVEY. ac = Mc + 4 (HWQ + LWQ). (163) Tropic LLW below MSL = J Mc + J LWQ, nearly. (164) Indian harmonic tide plane below MSL = 0-49 (Sg + 2 D,). (165) 59. Prediction direct from observation. Suppose tliat a person start with a blank book containing as many pages as there are days in a year and labeled accordingly. Suppose the page ruled into 12 (or 24) columns, one for each possible degree of the moon's declination, viz., 18° to 29°. Let the page be divided horizontally into 15 equal strips, one for each day of the moon's age, reckoned from full moon as well as from new. In each of the rectangles thus formed let the observed heights and Innitidal intervals (obtained from the observed times and properly marked) be recorded. If observations be made throughout the node-equinox period, or about nineteen years, ancj thus tabulated, predictions can be made by referring to this record in the following manner: Turn to the day of the year for which predictions are required. Select such tabular values as correspond most nearly to the age and declination of the moon for the given day. If no such tabular value can be found upon the page for the day in question, go a few days forward or backward until a tabular value is found which has very nearly the required arguments. The lunitidal intervals there tabulated give, when applied to the moon's transits for the day in question, the times of the tides. The heights are the tabular values unaltered. If in making the tabulation the effect of parallax upon the height of the tide is allowed for, it may be roughly introduced, when predictions are required, by multiplying the range of the tide by the cube of the value of the parallax for the day divided by the cube of its mean value, or by making use of the second table of § 58. The moon's transits, declination, age, and parallax, when used, are sui)posed to be taken from the Nautical Almanac. A very convenient method for obtaining fairly good predictions at a station having either great or small diurnal inequality, is the following: Take a year's observed high and low waters, or ]ireferably a year's accurate predictions, and copy down alongside each date the time of the moon's superior transit. To predict for any given day of any year, use that part of the year tabulated which is about the same season of the year as is the given date. Find a day having, as nearly as may be, the same hour of transit as has the given day. The times and heights of the tides for that day will be approximately the values required. If only very rough predictions are required, all inequalities may be omitted. In this case it is sufQcient to know the high-water lunitidal interval and the mean range of tide. This has been explained in § 51. In regions where the diurnal inequality is small the page should be divided with reference to the various values of the moon's parallax instead of declination. 60. Phase reductions. Along the coast of Europe and the Atlantic coast of America, the region of the West Indies excepted, the tide is of the semidiurnal type. Consequently, the greatest inequality is the phase or semimenstrual, due to the sun. This is more especially true of times than of heights. If a person has secured a month of observations upon high and low water for any place along these coasts, he can, with very little computation, obtain reasonably good predictions. He has only to tabulate the tides, along with the times of the moon's transits, take the lunitidal intervals (using the transit immediately preceding) and distribute the intervals and heights according to the hour of transit, going from to 12. This distribution constitutes a " second " or " phase reduction," an example of which is given below. Or, he may tabulate the times and heights without using intervals, according to the hour of the moon's transit. In the latter case the time of the tide can be ascertained without even adding the interval to the time of transit, because the times of the tide are supposed to be tabulated with reference to the hour of transit as an argument. If the observations are less than a year in extent, their times, as well as the times of the transits, should, strictly speaking, be changed into apparent time before making a phase reduc- tion; or the average value of the equation of time ^Table 30) for the period of observations may REPORT FOE 1897 — PART II. APPENDIX NO. 8. 379 be substracted from the "hour of transit argument" of the reduction results, thus obtaining the same prediction table as before. The predictions made therefrom are, of course, in apparent time and must be changed into mean time by Table 30. On the Atlantic coast of the United States, the phase inequality being small, the kind of time used is quite immaterial, mean time being almost as satisfactory as apparent. The prediction table got out in the kind of mean time used in the observations can be adapted to another kind of mean time by applying a constant (equal to the difference between the two kinds of time) to the tabular values. An example of the method of making a " first" and a " phase " (or " second ") reduction, is given for Tybee Island Light, Savannah Eiver Entrance, Georgia, for May 1-29, 1891. As a matter of convenience, the Greenwich time of the moon's transit is taken directly from the Nautical Almanac, merely changing astronomical to civil time. The observations were made in eighty-first meridian time, which is 5'' 24" west, while the local meridian is 5^ 23"'37 west. In tabulating the observa- tions two lines are given to each day. All hours less than 12 are in the morning ; all greater are in the afternoon, and when diminished by 12 give the usual reckoning; for instance, IS*" is 3 p. m. First reduction. Tybee Island Light, Georgia. Time of — IvUnitidal interval. Height of— Date. Moon's transits. Remarks. HW I N ^f \ ^' y '\ ID ^ 50 O AO 30 'v ,'-' *N ^^ '-' ""■ -,,^ ^,'' J Teet Jc 10.00 fajo cetas lunw this table gives 3^ as it also does at 30* (= — -.^ — = the assumed length of a syDodical month in days). The present value at 0* is about 2". It is to be noticed, however, that no regard is paid to the phase inequality in these tabular values, and that the age of the moon roughly determines the time of transit; and to this a constant lunitidal interval of 3^ has been applied. It appears from Flamsteed's remarks* that up to his time the rule involved in this old table was generally followed although some had calculated the time of transit more carefully. Julius Gcesar Scaliger (1484-1558). Exercitatio LII,t entitled "De maris motu," is a general exposition of the tides, the views therein suggested, taken in connection with the time at which it was written, give to it considerable historic value. Without professing to know the cause of the tides, Scaliger remarks that because the flow and ebb recur at definite and stated times, the moving cause must be definite in character. His approach to the gravitational theory of tides appears in the following quotation : For since [the tide] is observed to follow the course of the moon, they have judged it created by the moon. [You say] but the moon does not touch the waters. This has been a difficulty with some of the Peripatetics : likewise the magnet ought to give them difficulty. Because there is motion in the iron, although not in contact with the stone, wherefore may not the sea follow the body of the noblest orb ? But it seems manifest. Surely there is a stillness of the sea at the times of the quadratures which is called a calm by the people. At the times of full moon the seas are rougher : so that they seem to restrain themselves with the desire of the moon. . . . The tide then, in the usual sense of the word, is a duplex motion, and truly duplex: for in itself is a return. One part is in conformity with the motion of ihe primum mobile, the other is recurrent and contrary thereto: but both occur on definite times. For there are two motions just as in the contraction and expansion of the heart. He speaks of the tides in the Arctic Ocean, around Great Britain, in the South Sea, the Adriatic Sea, the Red Sea, the Indus River, the Garonne River, the Euripus, and elsewhere; and tries to assign causes for their peculiarities. He infers a declinational inequality in the tides because the moon changes her declination and so rises not always in the same place. He believes the long stretch of the Western Continent to explain the alternation of flow and ebb. As with most of the early writers he finds difficulty in accounting for the ebb : Wherefore does the sea ebb? Not only because of a dislike for the shores and a casting back, but also because it follows its loves, viz., the moon. It differs from the movement of the ocean. Extracts from HaMuyfs Collection of the Early Voyages, Travels, and Discoveries of the English Nation. I 72. A whirlpool on the Coast of Norway, called Malestrande. Giraldus Cambrensis (who florished in the yeere 1210, vuder king lohn) in his booke of the miracles of Ireland, hath certaine words altogether alike with these, videlicet : Not farre from these Islands (namely the Hebrides, Island &c.) towards the North there is a certaine woonderful whirlpoole of the sea, whereinto all the waues of the sea from farre haue their course and recourse, as it were without stoppe: which, there conueying themselves into the secret receptacles of nature, are swallowed vp, as it were, into a bottomlease pit, and if it chance that any shippe doe passe this way, it is pulled, and drawen with such a violence of the waves, that eftsoones without remedy, the force of the whirlepoole deuoureth the same. The Philosophers describe foure indraughts of this Ocean sea, in the foure opposite quarters of the world, from whence many doe coniecture that as well the flowing of the sea, as the blasts of the winde, haue their first originall. ^ Instructions and notes very necessary and needfull to be obserued in the purposed voyage for discouery of Cathay Eastwards, by Arthur Pet, and Charles lackman : given by M. William Burrough. 1580. * Phil. Trans., 1682-3 [p. 12] ; Abr. Vol. II, p. 555. t Exotericarum Exercitatlonem ad Cardanum (Frankfort 1592; first published in 1537). t London, 1809-12. A new edition; the original, London, 1599. j, Vol. I. pp. 134-135. 396 UNITED STATES COAST AND GEODETIC SURVEY. And when you come vpon any coast where you find floods and ebs, doe you diligently note the time of the highest and lowest water in euery place, and the slake or still water of full sea, and lowe water, and also which way the flood doeth runne, how the tides doe set, how much water it hieth, and what force the tide hath to driue a ship in one houre, or in the whole tide, as neere as you can iudge it, and what difference in time you finde hetwene the running of the flood, and the ebb. And if you flnde upon any coast the currant to runiie alwayes one way, doe you also note the same duely, how it setteth in euery place, and ohserue what force it hath to driue a ship in one houre &c.* This shows that the writer was aware of the fact that the duration of flood aud ebb currents are not generally equal. 73. The voyage and trauell of M. Caesar Fredericke, Marchant of Venice, into the East India, and beyond the Indies translated out of Italian by M. Thomas Hickocke. [1563.J From Martauan I departed to goe to the chiefest Citie in the kingdome of Pegu, which is also called after the name of the kingdome, which voyage is made by sea in three or foure daies ; they may goe also by lande, but it is better for him that hath marchandize to goe by sea and lesser charge. And in this voyage you shall haue a Macareo, which is one of the most marueilous things in the world that Nature hath wrought, and I neuer saw any thing so hard to be beleeued as this, to wit, the great increasing & diminishing of the water there at one push or instant, and the horri- ble earthquake and great noyse that the said Macareo maketh where it commeth. We departed from Martauan in barkes, which are like to our Pylot boates, with the increase of the water, and they goe as swift as an arrow out of a bow, so long as the tide runneth with them, and when the water is at the highest, then they drawe themselues out of the Chauell towardes some banke, and there they come to anker, aud when the water is diminished, then they rest on dry land : and when the barkes rest dry, they are as high from the bottome of the Chanell, as any house top is high from the ground. They let their barkes lie so high for this respect, that if there should any shippe rest or ride in the Chanell, with such force commeth in the water, that it would ouerthrowe shippe or barke : yet for all this, that the barkes be so farre out of the Chanell, and though the water hath lost her greatest strength and furie before it come so high, yet they make fast their pro we to the streme, and oftentimes it maketh them very fearefull, and if the anker did not holde her prow vp by strength, shee would be ouerthrowen and lost with men and goods. When the water beginneth to increase, it maketh such a noyse and so great that you would thinke it an earthquake, and pres- ently at the first it maketh three wanes. So that the first washeth ouer the barke, from stemme to sterne, the sec- ond is not so furious as the first, and the thirde rayseth the Anker, and then for the space of sixe houres while the water encreaseth, they rowe with such swiftnesse that you would thinke they did fly : in these tydes there must be lost no iot of time, for if you arriue not at the stagions before the tyde be spent, you must turne backe from whence you came. For there is no staying at any place, but at these stagions, and there is more danger at one of these places then at another, as they be higher and lower one then another. When as you returne from Pegu to Martauan, they goe but halfe the tide at a time, because they will lay their barkes vp aloft on the bankes, for the reason afore- sayd. I could neuer gather any rea.son of the noyse that this water maketh in the increase of the tide, and in deminishing of the water. There is another Macareo in Cambaya, but that is nothing in comparison of this.t 74. Another testimonie of the voyage of Sebastian Cabot to the West and l 8, Bk. XIII, § 1, points out the fallacy of it. II Astronomie, Vol. IV, p. 120. 420 UNITED STATES COAST AND GEODETIC SURVEY. [The truth of this is obvious. For, consider the condition of the tide on a sphere covered with water at a given instant. Eeclioning distance from the point of instantaneous half-tide level, the height at a neighboring point on the equator is height 450 E. or W. x sin 2 { .J^^ZLs )- ^'''^ » But an inclosed sea will keep its surface parallel to the surface in the supposed case, and its length will be twice the above " distance;" . ■ . height at extremities = height 45o E. or W. x sin ( earth'sTadTu^s ) = height 450 E. or W. x earth^^iTs ' ^'^^^^ ^^^ length is not great.] (198) 95. Golin Maclaurin (1698-1746). Neither Maclaurin nor Euler, in the prize essays already referred to, developed methods for the reduction or prediction of tides, but each added to the theoretical side of the tidal problem. Maclaurin demonstrates for the first time (what Newton had assumed without demonstration) that a homogeneous sphere when disturbed by the moon or sun becomes, upon the equilibrium hypothesis, a prolate ellipsoid. So far as known to the present writer, Maclaurin is the first to call attention to an effect upon the water which the earth's rotation might i^roduce. He says in Proposition YII of his essay : If water be carried from the south toward the north, either by the general motion of the tide or by any other cause whatever, the course of the water will thereby be deflected little by little toward the east, because the water at a prior time wad carried, by the diurnal motion, toward this sea with a greater velocity than pertains to the more northerly place. Conversely, if the water be carried from the north toward the south, the course of the water, on account of a similar cause, will be deflected toward the west. From this source I suspect various phenomena of the motion of the sea to arise. He suspects the winds are also affected by the same diurnal motion. This effect of the earth's rotation forms an essential part of Laplace's dynamical theory of tides. A somewhat analogous question, viz., the effect of the earth's rotation upon a boJy falling freely from a great height, was discussed by Newton and Dr. Hooke in 1679. 96. Leonard Euler (1707-1783). Euler discusses the tidal problem upon the correct assumption that the tides are caused by the horizontal component of the moon's disturbing force. But if the water have a density com- parable to that of the earth, it is necessary to take into account the horizontal attraction of the two instantaneous high- water regions. Euler neglects this and obtains a smaller elevating effect than that given by the equilibrium hypothesis where the density of the earth is assumed to be that of water, and the mutual attraction of the water is properly allowed for. He expresses (§ 44 of his essay) the height of the tide due to sun and moon in spherical harmonic functions of their zenith distances, carrying the expression to the fourth power of the parallaxes. Later on, he attempts to treat the tides as a problem of fluid motion; that is, he attributes to the fluid particles the property of inertia which the equilibrium theory does not imply. It is now known that the fundamental tidal equations (first obtained by Laplace) have reference to the horizontal motions of the fluid, and to the invariability of its volume. None of these equations were obtained by Euler; he took into account the vertical oscillation only, and neglected the condition of continuity. For a somewhat more detailed review of Euler's essay than is here given, the reader is referred to the introduction of Ferrel's Tidal Eesearches, which includes the most of Laplace's criticisms upon it. 97. Joseph Jerome Lefrangais de Lalande (1732-1807). In the fourth volume of his Astronomy, pages 1 to 348, Lalande gives an exhaustive survey of all available tidal knowledge up to the close of the year 1780; in other words, his treatise covers nearly all that was known on the subject prior to the investigations of Laplace. His great familiarity with sources of information can be inferred from the fact that he had been paying attention to the subject during the seventeen years preceding 1780. He was the editor of the 9/ EEPOET FOR 1897 — PART II. APPENDIX NO. 8. 421 Connaissance des Temps from 1759 to 1774, and again from 1794 to 1807, in which publication a few of his tidal papers appear. He contributed also to the academies at Paris and Dijon. Among the matters treated or included in his astronomy the following may be mentioned : The knowledge of tides possessed by the ancient Greeks and Romans. Their theories and other theories (or hypotheses) before the time of Newton. Newton's theory. Work of Maclaurin, d'Alembert, Euler, and Bernoulli. The equilibrium theory. Tidal phenomena or Inequalities- phase, parallax, diurnal, and declinational. Observed equinoctial spring tides and other remarka- bly high tides. Cassini's discussions in the memoirs of the French Academy. Tides in closed seas, especially in the Mediterranean. River tides. Observations at Brest — times and heights of two or three tides daily, from June 10, 1711, to December 31, 1712, and from January 1, 1714, to August 31, 1716. Some observations at Toulon, 1777-78; Rochefort, 1771-72; St. Malo, 1775-76; Havre, 1701-2; Dunkirk, 1701-2, and Katwyk (Holland), 1766. A collection of information bearing upon tides the world over, including sources of information. General circulation of the sea. Earthquake waves. Tides in lakes, including seiches. Intermittent springs. Table of establishments for places in all countries, with authorities and dates of determination. A^ glance at this tabfe will shoy a sudde;^ activitj' in tidal/Sbservations a)rong the coa»t o^ Europe, beghfriing witl/the year l/oi; one = c sin {it + srzS+ e), (211) V y - -j= a' cos {U+sti3 + £), = y', (212) Cf. equation (106), Part II. t ^''' is proportional to Ft (5) or F; {/j, u) in recent notation. t See § 36, Part II. 4'^ 8 UNITED STATES COAST AND GEODETIC SURVEY. «, h, c, a', being rational functions of /a aud Vl — //, s being an integer; ?/' is the excess of tlie true height of the tide (y) above the equilibrium height ( V'/g or V/g)* These values substituted in (203), (204) give -^(l-;,^)^+2_-^^,«' l^ 3m ^ (21o) (i2_4wV')-/l-yW' ' _ i ^^ ^ ''^ ^ . (214) ^~ [i' - 4 wV') (1 - /^') ' and these values of 6 and c substituted in (202) give '•=»|;^[T'-'-, vary with extreme slowness, in comparison with the rotary motion of the earth; the three preceding terms produce three different spedes of oscillations. The periods of the oscillations of the first Tcind are very long; they are independent of the rotary motion of the earth, and depend wholly upon the motion of the hody L in its orbit. The piriods of the oscillations of the second species depend chiefly on the rotary motion of the earth nt; their duration is nearly one day. Lastly the periods of the oscillations of the third kind depend chiefly on the angle Snt; they are completed in about half a day. The equation (215) is a linear differential equation; hence it follows, he remarks, that these three species of oscillations can exist together, without being confounded with each other; there- fore, we may consider them separately. These three kinds of tides are then discussed by putting s successively equal to 0, 1, and 2. 104. Oscillation of the first species. — The expression for c indicates that for small values of i as in case of oscillation of the first species, the east and- west motion of the fluid i)article may be great in comparison with the other motions, provided there is no fluid friction involved. Laplace believes that the resistances which the waters encounter will, in the case under consideration, reduce the oscillations to their equilibrium values, especially in the case of the sun [Bk. IV, § 6J.* The oscillation of long period thus becomes _ L (sin '^v — ^ cos ^v) (l-|-3 cos W). Oscillations of the second species. — Here i=M, and .s=l ; / is assumed to be equal to I (1 — 2/'^), I and q being constant for all latitudes. He finds for the height of the oscillation of the second species an expression which becomes fi 7" ~3- Iq sin d cos 9 sin v cos v y= cos {nt+TS — tp) (221) • 2 Igq—n^ 3 when we neglect the small fraction ^. The coefQcient is a, or the amplitude of the oscillation. This coefficient diminished by the (astronomical) equilibrium value is the value of a'. These values of i, s, y, a and a' satisfy the fundamental equation (215) ; moreover, y has a form analogous to the corresponding term in the tide-producing potential. This is therefore a solution of Laplace's tidal equation of the required form. When the depth is such that q is small this oscillation, and so the diurnal inequality becomes small. When the depth is constant all over the sphere q=0, and so oscillations of the second species vanish everywhere.t The coefficients b and c become known by (213), (214) as soon a,s a' has been determined ; the horizontal oscillations become 3L . 3 sm V cos V u = cos (nt + ^ — f), (222) 2lgq—n^ 3L cos fl . r^-sin:~W«"^^°°'^^ . V = sm {nt + is — ip). (223) 2lgq— n' These do not vanish when q = 0; i. e., tidal currents of a daily period would exist in the case of an ocean of uniform depth covering the sphere. In the second as well as in the third species, Laplace finds a value of q which will enable one to determine the oscillation even if i be not exactly equal to n or to 2n. * Darwin concludes [Proc. Roy. Soc, Vol. 41 (1886), pp. 339, 342, and Eiio. Brit. art. "Tides"] that this hypothesis is untenable unless the period he Tery long, as in the case of the minute oscillation whose period is nearly 19 years, f See under Newton and Bernoulli. ■130 UNITED STA.TES COAST AND GEODETIC SURVEY. 105. Oscillations of the third species, the depth being constant. The quantities r, ip, and v vary slowly in comparison with 2nt, and so may be treated as con- stants. The small fraction 1/p, which expresses the ratio of the density of the sea to that of the earth, may be regarded as negligible. Putting i = 2ii and s = '2, the oscillation corresponding to the semidiurnal term of T'', or V2', = f -3 sin ''ti cos 2 V cos 2 (nt + cJ - f), (324) is 2/ ^ a cos {2nt + 2 w - 2,'/-. (225) ••• a' = « - f ;57/l - /^n cos = V. (226) The depth being constant, y = 1. Substituting these values of a' and y in Laplace's equation (215), it becomes or putting -^ a (1 - Mr= - f^ra - M'r + (6 + 2 z.^) « - ^-^ (l - m') cos ■' v; (227) 1 — /A^, or sin ^(9, = a^, ^Ml - ^) - ^-g - 2 a (l-x' - ^^'x^~^ + ^.T^cosM' = 0. (228) To satisfy this equation, replace a by the assumed value a = A"'a^ + J.'2'ip^ + A'^'ip" + . . . (229) and compare coefficients of lilie powers of sc. .: J.(i>=-i-:^ cos 2 v; (230) 4c r^ g ^ ' the comparison of the coefficients of x* gives an identity; but all following coefficients can be expressed iu terms of J.''' and A'^'. Laplace's determination of the coefficient of «* has led to discussions by Airy,* Ferrel,t Thomson, i: Darwin, § and G. H. Ling.|| Laplace assigns (Bk. IV, § 10) values to I such that 2 n^/lg = 20, 5, aud 5/2 successively; that is, since n^/g = 1/289, 111 1 = 2890' 722-5' 361-25' the earth's radius being unity. The value of «' becomes a series in powers of a;^, each having a numerical coefficient, and (230) as a general coefficient. For places on the equator the three corresponding spring ranges of the semidiurnal tide are, if A'" = 0-12316 meters, 7-34, 11-05, 1-90 meters. The expressions show that the tides are inverted iu the'flrst instance but direct in the second and third. In high/^l^tudes, however, the tides are always direct. For a great dejjth, the equi- librium spring range is approached which is 0-98528 meter, assuming the tidal effect of the moon thrice that of the sun and disregarding the density of the water. * Tides aud Waves, Arts. 110-113. Phil. Mag., Vol.50 (1875), pp. 277-279, Tidal Researches, p. 154. tPhil. Mag., Vol. 1 (1876), pp. 182-187. The Astronomical Journal, Vol. IX (1889), pp. 41-44. The Astronomical Journal, A^ol. X (1890), pp. 121-123. The Astronomical Journal, Vol. IV (1856), pp. 173-176. t Phil. Mag., Vol. 50 (18751, pp. 227-242. § Encyclopedia Britannica, article Tides. II Annals of Math., Vol. 10 (1896), pp. 95-125. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 431 106. lu the chapter on the equilibrium of the sea he finds that — The equilihrium of the sea is stable, if Us density he less than the mean density of the earth. If the density of the sea exceed the mean density of the earth, its figure ceases to be stable in many cases. In the chapter on particular circumstances of each port he finds that — Tfte oscillations of tlie second Mnd cannot vanish, for the whole earth, except iii the single case where the depth of the sea is constant. ... ' Xo admissible law of the depth of the sea can make oscillations of the third Mnd vanish in all parts of the earth. At the end of § 15, Book IV, he says: The irregularity of the depth of the ocean, the manner in which it is spread over the earth, the position and deolivity of the shores, their connexions with the adjoining coasts, the currents, and the resistances which the waters suffer, cannot possibly be submitted to an accurate calculation, though these causes modify the oscillations of this great fluid mass. All we can do is to analyze the general phenomena ivhich must result from the attractions of the sun and moon, and to deduce from the observa- tions such data as are indispensable, for completing in each port the theory of the ebb and floxo of the tides. These data are the arbitrary quantities, depending on the extent of the surface of the sea, its depth, and the local circumstances of the port. At this point he returns to an empirical or modified equilibrium theory, taken in connection with a dynamical principle which he here lays down, viz. : The state of a system of bodies, in which the primitive conditions of the motion have disappeared by the resistances it suffers, is periodical, Ulce the forces whicJi act on it. Knowing the terms of the potential to be simply harmonic, and having found the terms of the component forces to be of the same character, Laplace concludes, by an argument based upon the introduction of another equal sun, that the oscillation is simply harmonic. Consequently the expression y=^cos{2nt + 2z3-2f — 2X), (231) B and A. being two constants dependent upon the particular port, gives the law by which the solar tide rises and falls, the sun being in the equator. By considering a tide propagated up a canal having two mouths, Laplace notes that the phase and amplitude of the resulting tide are dependent upon the rapidity of the motion of the body in its orbit; i. e., upon the period of the oscillation, and so Therefore the ratio of the coefficients — -- and — j—, given by observation of the tides, is not exactly that of the L L' forces — and — j- ,.3 ,.'3 But since the constant quantities B and A would be the same for the sun and moon, if the motions of these bodies were equal, it is natural to suppose that their differences are proportional to the differences of these motions; therefore we shall adopt this hypothesis, and we shall iind that it satislies the observations with remarkable exactness. Hence we shall put X=0—mT; (232) S=P il—2mQ); (233) O, T, P and Q being the same for the sun and moon.* In § 19, Bk. IV, Laplace investigates inequalities in the motions of the sun and moon moving in the equator by means of fictitious bodies. He says: The most important of these terms is that depending on the angle 2nt — 2mt + 2 as, (234) which produces the ebb and flow of the tide, in the case we have just examined, where the sun is supposed to move in the plane of the equator, and to be always at the same distance from the earth. The other terms jnay be considered as the result of the actions of as many other bodies, moving uniformly in the plane of the equator. Combining together the partial ebb and flow, corresponding to each of these bodies, we shall obtain the total ebb and flow arising from the action of the sun. • • Ferrel early came to the conclusion that the change in the tidal coefficients due to a change of velocity of the disturbing body in right ascension is not generally proportional to the amount of change in this velocity. ^32 UNITED STATES COAST AND GEODETIC SURVEY, If we put I for the mass of the fictitious body, whose action produces the term depending on the angle 2nt — 2qt + 2e,and.a for its distance from the centre of the earth; we shall have 31 12, 2--, = fc, or ^-.,= 3 7,;. (235) We have seen in the preceding article, that the sun being supposed to move uniformly iu the plane of the equator, with an angular motion equal to mt, the part of the expression of the height of the sea, depending on the angle 2 nt — 2 mt + 2 ra, is equal to P {l — 2mQ) -f cos2 (nt—mt + a—O + mT). (236) The constant quantities /', Q, O, T, are the same for all the heavenly bodies, whatever be their proper motions; therefore the sum of the partial tides, arising from the action of all the bodies I, I', I", &c., will l.e, 2P {1 — 2 qQ) ^^cos2(nt — qt + s-0+qT); (237) consequently it will be, • ' I P2fcco8 2 (nt — qt+s — O + qT) + I P§ 1^ Sic 8iii2(ni — qt + e — + qT) ; (238) the (liffermtial heing taken supposing rit to be constant. But by what precedes, we have 2kcos 2 (nt — qt + £ — + qT} = ^^G0s2 (nt + a ~ if, — A) ; (239) ■the time t being decreased by T, in the variable quantities nt, ip, r, of the second member of this equation, and 'l = — « r. Therefore the part of the height of the tide depending upon the action of the sun, and also upon the angle 2 «< + 2 ro — 2 t/,, with the preceding conditions, is represented by P 4 cos 2 (n« + tB — ^— A) + Pe-$ ^-sin2 (nt+a—Tb — X) I. )•■' at ( r^ ) (240) If we transfer to the moon what we have said relative to the snn, we shall find, that the part of the height of the tide depending upon the lunar action, and the rotatory motion of the earth, is P^cos 2{nt + a — tp' — X)+PQ ^-\^ sin 2 (m« + ca — V' — A) I T * at ( )• ■' ) (241) in which expression the time t must also be decreased by T Introducing the declination and the part independent of the rotary motion of the earth, the general expression for the height of the tide is fonnd to be ^, = _ i±3^2i|^ j ^ (1 _ 3 sin^ V) + ^; (1 - 3 sin-^ V) I 5p -\-A < — sin v cos v cos {nt -\-cs — ^ — >')+ , sin v' cos v' cos {nt -{- ai — ip' — y) i -|-ii— j — sin V cos V sin (h< + co — ip — x) + ,j siii v' cos v' sin (mt + cb — ip' — y) i +P \ — cos- V cos 2 (nt-\-a> — ip — A) + ''. cos'^ v' cos 2 {nt -|- ro — ip' — X) > +PQ ''' \ ^"- COS* V sin 2 (»« + o — ^ — A) + "^, cos"^ v' sin 2{nt + a>~-ip' — X)l. at ( ?•'• r' ) (242) In this expression the differentials must he taken supposing nt to he constant; and the time t must he diminished hy a constant quantity T', in the terms 7uultiplied hy A, B; and by the constant quantity T, in the terms multiplied hy P, Q; these constant quantities, as well as A, B, y, P, Q, X, must be determined, in each part, by observation. 107. 'Now observations made at Brest show that the terms in B and Q are small and may be neglected, and so the preceding formula becomes that resulting from the equilibrium theory.* * See below; also under Ferrel. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 433 This formula gives the height of the tide at any instant. [Laplace finds for the tide at Brest (Bk. IV, § 41) a3/= — 0»'-02745 [P (1—3 sin'^ v + 3 t" (1—3 sin^ v')] +0'"'07179 [f sin v cos v cos (;;— 66° 5')*4-3 i" sin v' cos v' cos (u+f/-— ;/,-'— 66° 5')] 4-0"'-78112 [P cos^ V cos 2 (f— 66° 5')+3 i" cos^ v cos 2 (v+ip—ip'—66o 5% (243) Here v denotes the sun's hour angle; i, the ratio of the sun's mean distance to its actual distance.] To find the times of high or low water all but the terms in P can be omitted, and we have upon equating -1. to zero (to ,-5 tos' V sin 2 i'p—ip') ta.ng2{nt+TS-f-X) = j^, j (244) ^3 cos^ V' + -3 cos^ V cos 2{tp- ip') Eange of tide = 2 P-Jfi, cos^ v Y+?-^ cos^ v —^ cos'' v' cos 2 (i/y - f) + (^^^ cos^ v' Y (245) Eange of tide near the ^.^Tures =^'=^^1^ ^^^' ^'± ^ «o«^ ^' } 4 P-, cos' V "'„ cos' V ^i cos' V ± -p, cos' V ^^ {{'P'-i^Y + iq'}' (246) 5 being the variation of the arc ip'—tp in the interval of two consecutive high waters near syzygy or quadrature; ip'—tp is taken at a time of a low water, and so for the mean of two adjacent high waters i[{f-f-W+{^''-i--+ipr+iq'. (247) rr,, , , , . ,f,-/,v . ., decrease near the syzygies. The last term of (246) IS the • , ^i, j ^ ^ ' increment near the quadratures. The time of high water is also given by the equation —3 cos' V sin 2 [tp'—f) ta,n2 (nt+z3—tp~X)= '^ , (248) ^cos'v+±;3Cos' V' cos 2 (^'-^.) which near the syzygies may be written {,p'-tp)I^^ cos' V' nt+TS-ip-X=— ^ ; (249) -r. cos' V' + -^ cos' V similarly near the quadratures (^i_^)__ cos' v' nt+tS—^-X=^^ ^ (230) These expressions give the retardation of the tide near tlie syzygies and near the quad- ratures. * Sexagesimal. 6584 28 434 UNITED STATES COAST AND GEODETIC SUEVEY. Observations show that at Brest ^;=3^. (251) very nearly (§ 31, Bk. IV). With this assumption, the above formulae, all of which follow from the equilibrium theory, enable one to verify the theoretical part of the statements here quoted : We shall now recapitulate in a few words the principal phenomena of the tides, and their relation with the laws of universal gravitatioji. We have generally considered these phenomena near their maxima and minima, and we have divided them into two claases ; the one relative to the heights of the tides, the other relative to the hours of the tides, and their intervals. We shall now examine separately these two classes of phenomena. The heights of the tides in. each port, at their maximum near the syzygies, and at their TOinrnwrn near the quadra- tures, are the data of the ohservations, which best show the ratio of the actions of the sun and moon upon the tides; and hy means of this ratio, the various phenomena of the tides, which result from the theory of universal gravitation. One of these phenomena, which io very proper for the verification of the theory, is the law of the diminution of the tides from the time oi maximum, or the law of their increase from the minimum. We have seen in [2593', 2716],* that the theory of gravity accords perfectly with the observations in this respect. These laws of the decrease and increase of the tides vary with the declinations of the sun and moon : we have seen in [2590, 2592], that their decrease near the syzygies of the equinoxes, is to their corresponding decrease near the syzygies of the solstices, in the ratio of 13 to 8 ; and that this result is conformable to the theory of gravity.t We have seen likewise, [2717', 2718'], that the increment of the tides, counted from the minimum near the quadratures of the equinoxes, is to the corresponding increment near the quadratures of the solstices, as 2 to 1 ; and that the theory of gravity gives nearly the same ratio. + According to this theory, the height of the total tide, at its maximum near the syzygies of the equinoxes, is to the corresponding height near the syzygies of the solstices, nearly as the square of the radius is to the square of the cosines of the declinations of these bodies near the solstices; and we have seen in [2590, 2592], that this differs but little from the result of observations. By the same theory, the excess of the heights of the total tides, in their minimum, near the quad- ratures of the solstices, above their corresponding heights, near the quadratures of the equinoxes, is the same as the excess of the heights of the total tides in their maximum, near the syzygies of the equinoxes, above their corresponding heights near the syzygies of the solstices; and we have seen [2590, 2592, 2717', 2718'] that this is exactly conformable to the theory.} The influence of the moon upon the tides inc7-eases, by the principle of gravity, as the cube of its parallax; and by [2608, 2623, &c], this is so exactly conformable to observation, that we might have deduced from observations the law of this influence. The phenomena of the intervals of the tides accord equally well with the theory, as those of their heights. According to the theory, the daily retardation of the tides, at their maximum, near the syzygies, is only about half what it is at their minimum, near the quadratures. In the firrtt case it is nearly 27' $ [2757], and in the last case 55' [2831]. We have seen in [2745, 2809], that the observations differ but little from this result of the theory. || The retardation of the tides varies with the declinations of the bodies. According to the theory, it is greater in the syzygies of the solstices, than in those of the equinoxes, in the ratio of 8 to 7. In the quadratures of the equinoxes, it is greater than in ih ose of the solstices, in the ratio of 13 to 9. We have seen in [2777, 2839'], that the observations give nearly the same ratios. 1] The distance of the moon from the earth has an influence on the retardation of the tides. According to the theory, an increase of one minute in the semi-diameter of the moon, produces an increase of 258" [2783] in this retardation, in the syzygies, and only 90" [2847] in the quadratures; and we have seen in [2847], that this agrees with the observations, and con- forms in every respect, relatively to the tides, to the law of universal gravitation. If We have treated fully on the ebb and flow of the sea; because it is one of the results of the attraction of the heavenly bodies most obvious to us, and the law which regulates it can be examined at every moment. It is hoped that the theory of the tides here giiven will induce observers to attend to the subject, in ports which, like Brest, are well situated for such observations. Accurate observations, continued during a period of the revolution of the moon's nodes, might fix with precision the elements of the theory of the ebb and flow of the tide, and perhaps make sensible the small oscillations depending on the inverse ratio of the fourth power of the distance of the moon from the earth, which have heretofore been confounded with the errors of the observations.** * Keferences marked with brackets here refer to Bowditch's Laplace. tLast term of equation (246). J Equation (246). 5 27 minutes decimal time = 39" ordinary time; and 55 decimal minutes := 79 ordinary. V' dec. =tV day = 2'' -4 ordinary. 1°=t4^ rt. angle = 0-9 degree, sexagesimal. l" dec. = (-i-iu)'' = Tti'ini- day = 1"''44 ordinary. .'^' '={-chs)°^TshTn! rt. angle = 0-54 minute of arc, sexa- 1= dec. = (-rk)"'=TTniWday = 0»-864 ordinary. ^"=(.ths)' = looiooo rt- angle:=0-324 second of arc, sex- agesimal. II Equations (248)-(250). ITEquations (248)-(250). **The quotations from Bk. IV follow Bowditch's translation and italicizing. REPOET FOR 1897 — PART II. APPENDIX NO. 8. 435 108. lu the thirteenth book, Laplace writes the semidiurnal tides in the form AL -p- cos^ i s COS {2 nt + 2 zs — 2 mt — 2 X) BL + I -^ siu^ E COS. {2 nt-\- 2 xs- 2 y) A'L' + -jn- COS^ ^ e' ca&{2nt + 2xS — 2m't — 2X') BL' + i -773- sin^ e' cos (2 w< + gT — 2 y) (252) when the constants A, A', B, y, A, V are 'known only from observation.* At a time T when the cosine of the first angle is unity, and the sun is distant m T from the equinox, the range of the solar tide is 2A-^cos,^^e + B-^s,VD^Ecos2mT. _ (253) If p denotes the square of the cosine of the declination at syzygies, cos^Y=j. = l-!igi + ""^^"^^; (254) 2 2 and since cos^ i e = i±|^^, (255) the above may be written 2 A -3 2) — (A - B) -5 sin* s cos 2 mT. (256) Similarly, the high water of the lunar tide may be written 2 A' ~ f - {A' - B) ^3 sin* s' cos (2 m'T - 26), (257) S being the right ascension of the "intersection." If P, Q denote the sums of the squares of the cosine of the sun's declinations at equinoctial and solstitial syzygy, respectively, and P', Q' similar quantities for the moon, then the value of i ranges at equinoctial syzygies is 2ia = 2A^P+'2'A' 1-02734 :^' P' - (A - P) ^ (P - <3) - {A' - B) 1-02734 ^; (P' - Q') (258) where 1-02734 is written instead of unity because the inequality of variation increases the tide- producing force of the moon in the syzygies by 2-734 per cent. For solsticial syzygies 2ia' =2A-0+2A' 1-02734 :^' Q' + (^ - -B) p (P - ^) + {A.' - B) 1-02734 ^ (P' - Q') (259) 2a may be supposed to refer to equinoctial syzygy and 2a' to solstitial syzygy. For equinoctial quadratures 2 ia" = 2 A' 0-97266 ~ O/ -2A-, Pi -+■ {A' - B) 0-97266 ^3 (P/ - Q/) + (^ - P) ^ (Pi - ft). (260) "The terms in B are solar and lunar Ka, the orbits being circular. «, e' denote inclinations of the orbits to the plane of the eciuator. 436 UNITED STATES COAST AND GEODETIC SUKVEY. For solstitial quadratures 2 ia'" = 2 A' 0-97266 ^' Fi' -2 A^. Qi - {A' - B) 0-97266 ^^{P,' _ Q,') -(A-B)^^ {P, - ft)- (261) Suppose the values 2 ia, 2 ia', 2 ia", 2 ia'" to be known from observation, and P, Q, P', Q', . . . to be taken from the almanac; the unknown quantities are " a4 A'^., B^„, B^' ylil .13' (262) If the two latter could be found, then the ratio ~/, ^ would become known; and, taking from astronomy the distances of sun and moon, the relative mass of the moon (L') would become known. ■^^* -^^' -^^3 cannot be determined because we cannot eliminate the one without eliminating the other. Accordingly Laplace assumes, as in his fourth book,* A = {l + mx)B, A' = {1 + ni'x) B, (263) wherein m/m' = sun's motion -^ moon's motion = 0-0748. From these relations and the four observation equations, the four unknown quantities and m'x or mx become known, and so the ratio jy jL *»3/ / A»3 . For Brest Laplace finds New observations (1807-1822) Old observations (1711-1716) EQUINOCTlAI< SYZYGIES. met, 48-a =153-711 48-6 = 3-388 met, 150-235 3-163 SOI3T1TIAI, SYZYGIES. 48-0;'= 134-325 48-(5' = 2-078 132-371 1-945 EQUINOCTIAI, QUADRATURES. 48-a"= 56-561 48-«"= 7-744 58-033 7-495 SOI^TITIAl, QUADRATURES. 48-0:"''= 74-769 48-«"'= 3-394 75-517 3-410 m'x — 0-25291 L' IL' r'V r^ = 2-35333 [old observations gave about 3] d. 1-48013 Mass of moon = 1/74-946. Age of phase inequality in these four cases : d. ,».„„. ( 1'5134:9 = the quantity by which maximum tides follow syzygy; 1'51^69 ( I'^^ll^ = *^® quantity by which minimum tides follow quadrature. * Cf. equation (233); also Ferrel, United States Coast Survey Report, 1870, pp. 190-199. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 437 He adds that the interval from syzygy to maximum tides and the interval from quadrature to minimum tides may he regarded as equal. This is in accord with the old observations, which gave 1-50724 and 1-5077 (Bk. lY, §§ 24, 31). (The variation in height (of which d' is a coefficient) near spring or neap tides is as the square of the time from them.] In the fifth chapter, Laplace writes the diurnal portion of the sun's disturbing force in the form |^sin^cos^P*'if^i°('*^ + ^) ^.\, (264) 2r' ( — sm s sm (»i^ + cr — ^ ^ S ^ ' thus bringing to light the two components which have since been styled solar Ki and Pi. Simi- larly the lunar attraction gives rise to two waves since styled lunar Ki and Oi. He determines the amplitude of the diurnal tide by means of observations around the solsticial syzygies. He finds its range to be 0">'=t'2134, while that of the semidaily tide is S'^o'-GO, the equilibrium theory giving 0"<'''674 and 0""'*-350, respectively; also that the high water of the dirunal wave precedes that of the semidirunal by 0''-095.* He says : Thus, by the effect of the rotation of the earth and accessory circumstances, the diurnal tide Is reduced to nearly one- third [of its theoretical value], while the semidiurnal tide becomes multiplied by 16. However, this great differ- ence ought not surprise us, if we consider that, by Book IV, the rotation of the earth destroys the diurnal tide in a sea everywhere of equal depth ; and that if the depth of the sea is rkvi of the terrestrial radius, or about 9,000 metres, the heights of the semidiurnal tide in the syzygies is 11 metres. In the sixth chapter he detects a tide depending upon the fourth power of the moon's parallax and having a period of one-third lunar day. * From recentharmonic analyses, 2 (K^-fOi-f Pi) = 1.00 foot=0.30 meter ;M2°—K,°—Oi°=66<^=2.3hours=0.09 day. CHAPTER VIII. WORK SINCE THE TIME OF LAPLACE. 109. Br. Thomas Young (1773-1829). The tidal theory of Dr. Young is contained in the Encyclopaedia Britannica, 8th edition; article, Tides.* He seems to have beeii the first to distinctly suggest an extensive system of cotidal lines. But in reference to it he adds : If, however, we actually make such an attempt, we shall soon find how utterly Inadequate the observations that have been recorded are, for the purpose of tracing the forms of the lines of contemporary high water with accuracy or with certainty although they are abundantly sufficient to show the impossibility of deducing the time of high water at any given place from the Newtonian hypothesis, or even from that of Laplace, without some direct observation. Somewhat after the manner of Euler he first treats the tidal problem by the equilibrium theory, using the horizontal component of the tidal force. He does not fail to notice the necessity of taking into account the attraction due to the high-water regions when the density of the water is considerable. He states as theorems that the (horizontal) disturbing force of a body varies as the sine of twice its altitude; that an oblong spheroid with its axis passing through the disturbing body is a form of equilibrium; that the tide will be propagated ) nearly. The velocity with which a shallow wave of great length would travel along the surface of water, whose depth = depth here at high water, would, by (172.), J be Vgx depth at high waters Vgxh (1+6). Consequently, the phase of high water travels along the canal with a velocity greater than that of a shallow wave on water of the same depth as the high water. In like manner, the phase of low water travels along the canal with the velocity ■l/[fffe(l — 36)] nearly, which is less than that of a long shallow wave on water of the same depth as the low water. The following theorem will be easily remembered. If Da be the depth at low water, Da thab at high water, and if Di, Di, D3, Dj, are in arithmetical progression; then the phase of low water travels with the velocity due to the depth Di, and the phase of high water with the velocity due to the depth D.i.|| After showing that the ebb-stream should be swifter than the flood-stream, and also giving a solution to the third approximation, he takes up the problem — To investigate the motion of the tide-wave under the same circumstances, when the water of the canal is supposed also to have a current-flow (independent of fluctuations of tide) towards the sea. He finds that the duration of fall exceeds the duration of rise by a quantity greater than in the case of no current. The subsection concludes with an investigation for long waves in a canal whose section is invariable, but of any form, and here the velocity of propagation is found to agree with the rules of Kelland and Green. 118. Subsection 4 supposes the water acted upon by an extraneous force and has applications to a solitary wave, tides, and wind waves. Thus it appears, that a single discontinuous wave of any degree of complexity may travel on water without any force to maintain it, provided, in the first place, that it satisfies the conditions laid down with regard to the differential coefScients at its terminations, and in the next place, that the wave is so long that a succession of simple waves, each of that length, would travel sensibly with the velocity duo to waves of infinite length.il If the single wave is moderately long, a small force will maintain it as a discontinuous wave : but if it be short, the force must be (in proportion to the various pressures acting on the water) considerable. In fact, each of the different terms in the wave-function represents a wave of different length ; and, when the waves are short, each of these would tend to travel on with its own peculiar velocity, which velocities are very different for the different waves. But when the waves are long, the peculiar velocities are very nearly the same for the different waves. * Tides and Waves, Art. 203. The tides at Wilmington, N. C, show signs of this, t Phil. Mag., Vol. 33 (1892), pp. 251, 265. t Tides and Waves, Art. 207. J Numbers thus inclosed, in quotations from Airy, refer to articles or paragraphs in his Tides and Waves. II Ibid, Art. 208. See nnder Russell. IT Tides and Waves, Art. 234. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 447 When a long wave is propagated along a canal of non-uniform depth, Airy's investigations show that the amplitude of the horizontal displacement varies inversely as the fourth root of the cuLe of the depth, while the amplitude of the vertical displacement (i. e., the rise and fall of the tide) varies inversely as the fourth root of the depth.* For a canal of non-uniform breadth, the amplitudes of the horizontal displacement will be inversely as the square root of the breadth of the canal; and the same law holds for the amplitude of the vertical displacement.t In the case of long waves in shallow water, where the depth diminishes, the water is sensibly elevated above its meamheight when the flow ceases; and in like manner it is sensibly depressed below its mean height when the ebb ceases.} I. e., slack-before-ebb or flood occurs earlier because of this shoaling. Where the breadth diminishes, the water is sensibly elevated above its mean height when the flow ceases; and in like manner, the water is sensibly depressed below its mean height when the ebb ceases. I. e., slack-before-ebb or flood occurs earlier because of this contracting. H and G being the amplitudes, or coefficients, of the horizontal and vertical periodic forces acting upon the waters, he finds the following result agreeing with the equilibrium theory as well as with Laplace's theory : If we consider G and H to be quantities not very dissimilar in magnitude, (which we shall find to be true,) the term depending on G in each of these expressions is wholly insignificant in comparison with that depending on H; and thus we arrive at this remarkable conclusion, that the effect of the vertical disturbing force upon, the phwnomena of the tides is insignificant, the whole of the sensible effect being due to the horizontal force.§ Near the end of this subsection he says : The preceding conclusions are very important, as showing that the amount of elevation of the water under the action of forces depends in a most remarkable degree upon other circumstances than the magnitude of the forces. One is, the depth of the sea : another is, the periodic time of the forces. As depending upon the former, it appears that, if there were two parallel canals of different depths acted on by precisely the same forces, there might be high water in one when there was low water in the adjacent part of the other: or there might be elevations and depressions at the same time in both, but their magnitudes might have any proportion whatever. As depending upon the latter, it appears that, if there were two forces acting simultaneously upon the water in the same canal, the periods of those forces being different, (as, for instance, the forces depending upon the action of the Sun and the Moon,) the high water produced by one force might bear the same relation to the phases of that force which the low water produced by the. other bears to the phases of that other force : (thus low water of the solar tide might accompany the transit of the Sun, and high water of the lunar tide might accompany the transit of the Moon, in the same canal.) Or the phases of the two tides might stand in the same relation to the phases of the two forces, but the proportion of their magnitudes to the magnitudes of the fences might differ in any degree whatever. 119. In subsection 5, Airy investigates the tides produced by the moon in a canal, friction being still left out of consideration. He finds, for a canal bounded at both ends, — If the length of the canal is any multiple of half the length of the free tide- wave, this expression || becomes infinite. In reality the wave will become so large that the amount of friction, &c., will be so great as to neutralize the moon's force. U But when such a canal is short he concludes from the expressions for the horizontal and vertical displacements — The first of these expressions shows that the horizontal motion will be greatest in the middle of the canal's length, and will diminish gradually both ways to the ends, where it is 0. The second shows that there is no variation of level at the middle of the canal's length, but that the variation of level in other parts is proportional to the distance from the middle, elevation taking place on one side of the middle at the same time as depression on the other side, so that the surface of the fluid remains sensibly plane, though inclined to the horizon. The law of motion as regards the time is the usual oscillatory law expressed by cos it; but the motion of every particle differs in this respect from the motion of particles in an open sea affected by the tide : that here, the greatest horizontal displacement happens at the same time as the greatest vertical displacement ; whereas, in an open sea, the greatest horizontal displacement happens when the vertical displacement is 0, and vice versa. For a canal of any assumed length, and bounded at both ends, the expressions for the • displacements are generally complicated. * Tides and Waves, Art. 247. § Tides and Waves, Art. 279. t Cf. Lamb, Hydrodynamics, § 181 ; or § 33 ; Part I, this manual. || Vertical displacement. T tides and Waves, Art. 256. ^ Tides and Waves, Art. 299. 448 UNITED STATES COAST AND GEODETIC SURVEY. Airy next supposes the case of a canal closed at oae end whose waters are acted on by the forces of the moon and which communicates with a tidal sea. The result of this supposition is complicated ; but if the moon's force in the canal is insensible, it follows that all the oscillations in different parts of the canal take place at the same time.* When the elevation of the water bears a sensible proportion to the whole depth he finds for a canal opening at one end into a tidal sea — The law of the rise and fall of the water, at every part of the oaual except its month, is now different from that which holds on the supposition that the oscillation is small in proportion to the depth of the canal. But the times of high water and of low water are still the same as before, and the high water and the low water are still simultaneous through the whole length of the canal.t Brief mention is then made of a canal connecting two seas, both tided, or one may be tideless. 120. In subsection 6 friction is taken into account; it is assumed to be proportional to the velocity of the fluid particles, or — /^^ , since the motion is chiefly horizontal. In an indefinite canal, friction shortens the horizontal displacement; it causes the horizontal disturbing force to become zero at a point farther east, and so accelerates the times of the tides. Tides of longer period are more accelerated. Considering the coefficients of the tidal force as variable, it appears that the greatest tide follows the greatest force by the time/ x (constant)^. He says: This appears to us an important result, and one which no other theory has obtained. The equilibrium-theory of tides necessarily makes the tides to be greatest upon the same day on which the force is greatest. Laplace's theory, and the theory of waves in canals without friction, give the same result. But here we find a retardation accounted for by friction; and moreover this retardation is considerable.t For a tide propagated up a river of indefinite length, he finds that, because of friction, the vertical and horizontal motions of the particles diminish continually as the wave travels up the river; also that the flow ceases before the water has dropped to its mean height, and so turns earlier than in the case of no friction. In a tidal river stopped by a barrier, he finds that the slack before ebb is not simultaneous with the time of high water, but somewhat later. This interval may be considerable near the mouth, but it is small near the head.§ Also that when a canal bounded at both ends is acted upon by an external force, the rise and fall of the tide is greater at the ends than at the middle. In regard to a river of indefinite length running on a declivity toward a tidal sea, he concludes that — The circumstance that low water on a tidal river may be higher than high water on the sea, paradoxical as it may appear, is therefore a simple consequence of theory. 121. Subsection 7 contemplates the motion of water in three dimensions. The equation of continuity is symmetrical in X, x and Z, z; there are two equations of equal pressure, the one in X, x; the other is similar in Z, z. He finds solutions for annular and parallel waves, noting the effect of reflection from a straight boundary. Leaving for the present the consideration of the motion of the waves as determined by the differential equations, we shall consider one case in which we seem to derive some assistance from general reasoning. Suppose that a tide-wave is travelling along a canal of large dimensions, and of variable depth in its cross section, the depth diminishiug gradually to both shores. (We may suppose the dimensions to be such as those of the English Channel, or any similar arm of the sea.) It is evident that the investigation of (218.) || does not apply here : for, on account of the shallowness of the water at the sides, the velocity of flow towards both sides to produce the elevation of water there must be comparable with, perhaps equal to, the velocity of flow at mid-channel in the * Cf. § 30, Part I, this manual. t Tides and Waves, Art. 309. X Tides and Waves, Art. 329. § In the Philosophical Magazine, Vol. 12 (1856), pp. 184-188, C. Marret gives a popular explanation of how high- water occurs before the tnrn of the current, and of how the current near the shore turns before it turns in the offing. See Art. 507 of Tides and Waves. II Numbers thus inclosed, in quotations from Airy, refer to articles or paragraphs in his Tides and Waves. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 449 direction of the canal's length. Moreover, as the slope of the bottom is exceedingly email, the waves iu every part of the channel will be travelling in nearly the same manner as if the extent of sea of the same depth were infinitely great, and will therefore travel with the velocity dne to that depth: and, therefore, the ridge of wave cannot possibly stretch transversely to the channel, and travel along with uniform velocity lengthways of the channel. The state of things, then, will be this : the central part of the wave will advance rapidly (171.) along the middle of the channel; the lateral parts will not advance so rapidly; and the whole ridge will assume a curved shape, its con- vex side preceding. When this form is once acquired, it may perhaps proceed with little alteration; for if . . . we suppose two such curves exactly similar, but one a little in advance of the other, the space which separates the wings of the two curves, measured perpendicularly to the curves, (the direction in which that part of the wave must really travel,) is much less than the space which separates the centres of the curves, and by proper inclination may be less in any proportion ; and, therefore, may represent exactly the space travelled over by the wave at that depth while the wave at the greater depth travels over the greater space. That part of the ridge of the wave which is nearest to the coast will, therefore, assume a position nearly parallel to the line of coast. Now the wave whose ridge is nearly parallel to the coast, or which advances almost directly towards the coast, will be a wave of the same character as that treated of ini(307.). For the slope of the beach adds to the surface of the sea a very insignificant quantity, as compared with the breadth of the tide-wave, and the general effect is the same as if a perpendicular cliff terminated the sea on that side. Therefore, for those parts of the sea which are near to the coasts the law of (307.) holds ; namely, the greatest horizontal displacement of the particles occurs at the same time as the greatest vertical displacement; and, therefore, when the sea is rising, the water is, for some distance from the coast, flowing towards the coast, and when it is falling, the water is ilowing from the coast. In mid-channel, the motion of the water will be such as is described in (184.), &c. ; that is, the water will be ilowing most rapidly np the channel at the time of high water, and its motion upwards will cease when the water has dropped to its mean height. From this there follows a curious consequence with regard to the currents at an intermediate distance from the shore, where the effects of these two motions may be conceived to be combined. At high water the water is not flowing to or from the shore, but is flowing up the channel. "When the water has dropped to its mean elevation, the water is ebbing from the shore, but is stationary with regard to motion up or down the channel. At low water, the water is not flowing to or from the shore, but is running down the channel. When the water has risen to its mean height, the water is flowing to the shore, but is stationary with regard to motion up or down the channel. Consequently, in the course of one complete tide, the direction of the current will have changed through 360°, the water never having been stationary. And the direction of the change of current will be of such a kind that, if we suppose ourselves sailing up the mid-channel, the tide-current will turn, in those parts which are on the left hand, iu the same direction as the hands of a watch; and in those parts which are on the right hand, in the direction opposite to that of the hands of a watch.* Beyond this we can add little to the Theory of Waves upon a sea extended in both dimensions. But the follow- ing remarks will be found important with reference to the method of determining from observations some of the phenomena of tides : In tracing the progress of the tide across an extended sea, we cannot observe the different waves as we can those upon a small piece of water. We can do nothing but make observations of the time of the rise and fall of the sea at many different points along the shores of the bounding continents, or at islands in different parts of the sea: and when we have thus ascertained the absolute time of high water at many different points, if they are sufficiently numerous, we may draw lines over the surface of the sea passing through all the points at which high water takes place at the same absolute instant. These lines (adopting the word introduced into general use by the highest authority on the discussion of tide-observations) we shall call cotidal lines. The tracing out the ootid al lines in different seas is the greatest advance that has yet been made in the discussion of the phenomena of the tides in open seas. Now when the series of waves is single, the cotidal lines correspond exactly with the lines marking the position of the ridge of the wave at different times. But when the series of waves is compound, it may happen that the form of the cotidal lines will not present to the eye the smallest analogy with the forms of the ridges of the mingled waves. t The fifth section of the essay is devoted to an account of experiments on waves and to comparisons with theory. He finds a general agreement between the theoretical and observed velocity of propagation, but attributes the want of close agreement to the fact that Mr. Eussell neglected to observe the length of the waves in his experiments. 122. In the sixth section he applies Laplace's equations of motion to tides in narrow canals. In these cases it is unnecessary to consider the forces arising from the earth's rotation. The prob- lem thus simplified admits of solutions which take into account the motion in right ascension of the tidal body. The cases especially considered are a canal along a parallel of latitude, and a great circle in any position. For an equatorial canal, the tide is equal throughout its whole extent, and * Cf. § 13, Part I, this manual. t Tides and Waves, Arts. 358-366. 6584 29 450 UNITED STATES COAST AND GEODETIC SURVEY. the depth will decide whether high water or low water occurs under the moon.* For a canal passing through the poles the tide wave is a stationary wave. In considering the effect of sun and moon he concludes: Ist. If the depth of the sea is less than 14 miles, the mass of the moou inferred from the tides is inevitably too great. 2d. The error will he different (or the moon's mass ■will appear different) in canals of different depths. t Having introduced the effect of friction, he says : Thus it appears that for computing the time of high water it is necessary to use, not the positions of the sun and moon at the true time of the tide, nor the positions at that anterior time which is employed in computing the height of high water, hut a time later than that which is used for computing the height, and therefore a time which is nearer to the true time of high water If we investigated the effect of the passage up the shallow river upon the time of low water, we should find that the positions of the sun and moon corresponding toVin earlier time than that used for the height of the high water must be employed ; but we should still find that the mass of the moon inferred from the variations of the time of low water as referred to the moon's transit is too small. We shall here close our exposition of the Wave-Theory as applied to the tides. As nearly the whole of this theory is published for the first time in the present treatise, we shall not remark upon it at great length.' We think it right, however, to point out to the reader its great and important defect as applied to the explanation of tides upon the earth, namely, that in the case of nature the water is not distributed over the surface of the globe in canals of uniform breadth and depth, or in any form very nearly resembling them. In this regard its fundamental suppositions are probably as much, or nearly as much, in error as those of Laplace's theory. But we also think it right to point out that in regard to the completeness of detail with which the principles can be followed out, there is no comparison between the two theories. This will be seen by the reader who has remarked the facility with which the results of "difference between the angular velocities of the sun and moon," "variable coefficients of force," and " friction," are obtained in finite form. For these, Laplace's theory is quite useless. And though (as we have stated) the fundamental suppositions differ much from the real state of the seas, yet no one can hesitate to admit that the same general conclusions will apply : — for instance, that the moon's mass inferred from the height of the tides is too great, and by different degrees in different places : that the effect of friction will be a retroposition of tides in reference to the places of the sun and moon, &c. The peculiarities of river-tides, which no other theory has touched upon, are almost completely mastered by this. 123. In the seventh section is described Bunt's self-registering tide gauge; the methods of discussion adopted by Laplace, Lubbock, and Whewell; but of special interest is his description of a process of harmonic analysis which he had already applied to the tide curves at Deptford, | and which he is about to apply to the tide curves at Southampton and Ipswich. § In brief, he expresses the depression of the surface of the sea below a fixed mark for any given phase of the tide in the form Ao + Ai COS. phase + A2 cos. 2 phase + &c., + Bi sin. phase + B2 sin. 2 phase + &c.. (269) which, it is well known, is sufficient for the representation of a function which is periodical for 360° of phase. Then follow directions for determining the A's and B's. 124. In Section YIII Airy brings the tidal theories (equilibrium, Laplace's, and wave theories) to bear upon many questions connected with tides the nature of which are indicated by the following topics : Variation in range and shape of the tide as it progresses; the bore; tides in small seas; revolution of tidal currents; races; mean level of the sea little affected by the range of tide; the ratio of the solar to the lunar wave (coefQcient of semimenstrual inequality) varies from place to place, and also depends upon whether it is obtained from heights or from times ; similarly for the age of this inequality; the necessity of using different transits for different inequalities; the diurnal tide; and cotidal lines. The essay concludes with a statement of the present desiderata in the theory and observation of tides. 125. In the Philosophical Transactions for 1845 Airy makes a study of the tide at about twenty stations scattered around the coast of Ireland. He ascertains the times when the high and low water inequalites become zero and when a * Cf. § 41, Part I, this manual. t Phil. Trans., 1842, pp. 1-8. t Tides and Waves, Art. 455. § Ibid., 1843, pp. 45-54. REPORT FOR 1897 — PART II. APPENDIX NO. 8. 451 maximum. He is tlie first writer to make special use of the diurual wave at the time of its maximum amplitude. Its amplitude, and position with respect to the semidiurnal wave are found from the height inequalities.* He shows that the diurnal and semidiurnal waves do not travel alike either in direction or in velocity. The bigh water at Kingstown coincides predsely witli tlie low water at Dunmore East, and rice versa. More- over, between these two stations occurs the station Courtown; and here . . . the semidiurnal tide is nearly insensible. The difference in the times at Dunmore East and Kingstown does not therefore arise from a slow transmission of tide; but arises from a sudden inversion, of the wave, the point which separates elevation from depression being not far from Courtown. And the question now is, whether, on the supposition that the tide- wave enters the Irish Sea by this southern entrance, It is possible to explain the existence of this neutral point and the inversion of the tide beyond it. This he believes may be explained by a result established in "Tides and Waves," where a uniform canal closed at one end communicates with a tidal sea ; and which is, that the oscillation is simultaneous throughout the canal. In case of the Irish Sea representing such a canal, the open end is to the south and the closed end to the north; and if the depth be such that the coefficient (amplitude) of the simultaneous oscillation have opposite signs at Kingstown and Dunmore East, the phenomenon is, in a general way, explained. (See §30, this manual.) He has some further discussion upon the coefficient and age of the semimenstrual inequality as determined from time and height. He determines the coefficients of each individual tide at the various stations, the period covered being two months.! Having discussed the tides at Courtown, he says : Both the semidiurnal tides are very much diminished, the lunar so much that its range is rather less than that of the solar tide. The quarto-diumal tide exists in nearly its greatest magnitude. The geometrical representation is perfect; the mechanical explanation is not complete. In both respects, as regards what is reduced to law and what is yet incomplete, the Courtown tides must be regarded as the most remarkable that have ever been examined. He is inclined to believe that the tertio-diurnal tide is not sensible on the coast of Ireland. At the close of his discussion of the tides at Malta (Phil. Trans., 1878), Airy gives some account of the seiches as observed at that place. 126. Among the Mathematical and Physical Papers (1880) of Prof. G. G. StoTces several relate to the subject of wave motion. He treats the "long wave" in the paper entitled "Eecent researches in hydrodynamics" (B. A. A. S. Report, 1846), and in the one entitled "Notes on hydrodynamics. TV — On Waves" (Gamb. and Dub. Math. Jour., 1849). Airy's work upon tides in canals and certain of Russell's experiments are considered in this connection. Besides these may be mentioned a third paper entitled "On the theory of oscillating waves" (Trans. Gamb. Phil. Soc, 1847). Capt. F. W. Beeehey, Phil. Trans., 1848, discusses the tidal currents in the Irish Sea and English Channel. He draws upon a series of maps lines indicating the direction of the current (lines of flow) at stated hours and lines of equal range, the moon being new or full. Upon a chart of the Irish Sea showing "the set and rate of the flood stream" he has indicated a region of no current, which is caused by the meeting of the waters from north and south. Upon the chart of ranges of tide, the range nearly vanishes at Goartown while on the opposite coast of Wales it is 15 or 16 feet. In the Philosophical Transactions for 1851 he gives charts of the English Channel showing the lines of flow for each hour before and after high water at Dover. These charts have been copied in several publications. Quite recently (1891) M. H6douin of the Service hydrographique de la marine has designed similar charts for this region, the currents being referred to the tides at Cherbourg. * 2Di = l/HWQ^ + LWQ ^ LWQ tan (HW phase) =2wq Of course the amplitudes obtained have not been corrected for the time of year or the longitude of the moon's node, i. e., each value is really Bi — Fi, Table 32. t In the mean, these coefficients nearly coincide with Ms, Mi, Me, and Ms, excepting at Courtown, where the lunar tide does not predominate. The angular constants correspond to 0, 2 M2° — M4°, 3 M..° — iie°, 4 Ma° —Ms". 452 UNITED STATES COAST AND GEODETIC SURVEY. 127. Frof. Alexander D. Bache (1806-1867). As Superintendent of the Coast Survey, Bache caused many tidal observations to be made all along the coasts of the United States. He also gave his personal attention to the discussion of the observations; and among those who assisted in this work were L. P. Pourtales, L. W. Meech, Henry Mitchell, Charles A. Schott, and R. S. Avery. His writings on tides are contained in the Coast Survey Reports (1851-1866) and in the Proceedings of the American Association (1850-1857). As shown by these writings, the chief purpose of his work was the construction of cotidal lines and the obtaining of suitable elements for prediction of tides. This implied, besides many and extended observations, suitable modes of classification according to proper astronomical arguments. The principal numerical results are given in the Reports (1853-1864) under the title "Tide tables for the use of navigators," the most complete of these tables being found in the Report for 1864, pp. 58-90. The constants given for nearly all except Gulf stations are: Mean high-water interval, extreme phase inequality in time, mean range of tide, spring range, neap range, duration of rise, of fall, and of stand. For the Gulf stations the constants are: The average range, the range at greatest declination and at zero declination. For numerous stations the phase inequality is tabulated according to the single argument — the time of the moon's transit. For several Pacific stations tables of double argument are provided — the time of the moon's transit and the number of days from her extreme declination. Tables of single argument — the number of days from greatest declination — are also given. The double-argument table must give predictions superior to those given by means of two tables of single argument successively applied. And so it seems that the ijredictions for the Pacific Coast issued by the Survey for the years 1867-1870, should, on this account, be more accurate than the predictions for the years 1871-1884 where single-argument tables were used, in accordance with Avery's paper published in the Report for 1868 and entitled "Mode of forming a brief tide table for a chart." In fact, if a series of such double-argument tables were prepared for the different years or portions of the node-equinox period, the resulting predictions must accord well with the tides in nature, especially if the heights are corrected for parallax. Bache contemplated correction s for "the solar and lunar parallax and declination," but they were never extensively introduced into the computed predictions of the Coast Survey tide tables. The work of Meech was largely directed along this line which had already been opened up by Bernoulli, Laplace, Lubbock, Whewell, and Airy. Some account of his work is given in the Proceedings of the American Association, 1856, 1, pp. 166-170, and in the Coast Survey Report, 1856, pp. 249-251. An obvious fault of his treatment, where the diurnal inequality is large, is the neglecting of the motion of the moon's node. Bache paid considerable attention to the diurnal inequality along the Pacific coast.* His manner of treatment was essentially that of Lubbock and Whewell, and he shows in the report for 1854 that even where the diurnal inequality is large, the height inequalities are nearly proportional to the sine of twice the moon's declination. . For the Gulf tides, Bache constructed two sets of cotidal lines, one for the semidiurnal tide, and one for the diurnal at the times of extreme declination. In the report for 1856, p. 254 and sketch 35, he shows the semiannual variation of the lunitidal interval of the diurnal wave at extreme declination. This variation he finds to accord in a general way with that given by a formula of Airy'st for the displacement of the lunar diurnal tide by the solar. In fact the varia- tion is zero at the equinoxes and solstices, the interval being longest in February and August. But he infers that extreme variation differs greatly for different places. This conclusion is doubt- less based upon too few determinations. In fact the variation in interval is nearly alike in amount the world over, or at least wherever the age of the diurnal inequality is small. It is simply the perturbation in the Ki Oj wave, at the time when Ki and Oi conspire, due to Pi, or it is very nearly the perturbation in Ki due to Pi (Table 31) multiplied by Ki/(Ki + d). This gives the extreme variation for mean years as about ± 46". In the same report Bache notices the important fact that for several days at the time of »See also discussions of Gulf tides, U. S. Coast Survey Eeports, 1851, pp. 127-136 or 1866, pp. 113-119; 1852, pp. 111-122. t Tides and Waves, Art. 46. REPORT FOR 1897 PART II. APPENDIX NO. 8. 453 maximum diurnal tides, the lunitidal interval of the diurnal wave varies slowly, but that it varies rapidly from day to day as the moon approaches the equator. This is shown by a diagram on "Sketch 35," In a general way, this accords with a remark made in § 13, Part III, or with results obtainable from Table 27. In the Survey Eeport for 1857, also in the Proceedings of the American Association for the same year, Bache shows, by aid of diagrams, the effects which the three great bays of the Atlantic coasts of the United States have upon the range of tide; also how the range of tide increases in passing up the Bay of Fundy, In the reports for 1856 and 1858 he discusses the tidal currents near Sandy Hook and their effect upon the growth of the hook. His explanation of the fact that the velocity of the ebb stream generally exceeds that of the flood i's as follows :* Since the tide wave is propagated most rapidly in deep water, it follows that the fall of the tide takes place earlier in the channel than upon the shore ; hence the water tends to flow laterally from, the shore towards the channel. In this way a convergence of the ebb streams may be expected, especially in shallow bays. With the flood streams the reverse must be true, and the tide wave rising earlier in the channel a flow of water takes place toward the shore. In consequence of these distinctive characteristics the ebb and flood assume an unequal share in the molding of sandy coasts. The ebb current, with its concentration of forces, is a far more powerful agent than the flood ; its scouring capacity along its normal course must be more considerable, and it creates more extensive draft cur- rents .... But the ebb is the primary working agent, and the characteristic features of all channels and basins, on alluvial tidal coasts, miist, as a rule, reflect the effects of the ebb current. 128. An account of Pourtales' method for finding the diurnal wave is given by Charles A. Schott in a discussion of Kane's tidal observations in the Arctic Seas, Smithsonian Contributions to Knowledge, Vol. XIII (1863), p. 78: The process of decomposition in use in the U. S. Coast Survey was at first an analytical one, by computing sine curves; since 1855, however, a graphical process, equivalent thereto, was substituted; this latter method, as introduced by Assistant L. F. Pourtales, may be briefly explained as follows : After the observations are plotted and a tracing is taken, the traced curves are shifted in epoch 12 (lunar) hours forward, when a mean curve is pricked off between the observed and traced curves; this process is repeated after the tracing paper has been shifted 12 hours backward; the average or mean pricked curve thus obtained represents the semi-diurnal wave. On an axis parallel with that on which the time is counted, the differences between the originally observed and the constructed semi-diurnal wave were laid olf ; this constitutes the diurnal curve. In the case in hand I have simplified the process of separation by blackening the under surface of the tracing paper with a lead pencil, and running In with a free hand; the intermediate curves by the pressure of a style, an average of the two traces thus left on the lower paper, gave the semi-diurnal wave in quite an expeditious manner. On the diagram, the diurnal curve with its epoch of high water nearly coinciding with that of the semi-diurnal wave, appears plainly with its variation in size depending on the moon's declination. Besides the Arctic tides just referred to and those observed by Dr. Hayes, Schott has discussed, in the Coast Survey Eeport for 1854, the tidal currents around Nantucket and Marthas Vineyard, also the tides and currents of Long Island Sound. Largely through the exertions of Avery, the Survey commenced the annual publication of tables of predicted tides. These tables, already alluded to, began with the year 1867 and con- tinue up to the present time. In a paper entitled " Methods of registering tidal observations," found in the Survey Eeport for 1876, Avery gives a considerable amount of practical information in regard to observing tides ; also a description of a self- registering gauge of his own design. The principal writings of Mitchell are found in the Coast Survey Eeports from the year 1854 to the year 1887. They deal mostly with the effects of tidal currents ui^on harbors and shore lines. Incidental to such work, he devised a tide gauge for exposed stations,! and an apparatus for observing currents below the surface.^ The localities treated at some length are: Marthas Vine- yard and Nantucket, New York Harbor, Monomoy Peninsula, Portland Harbor, Greytown and Uraba, the Lower Mississippi Eiver, the Delaware Eiver, and the Gulf of Maine. His principal papers of a general character are : " On the reclamation of tide lands and its relation to navigation" (Eeport, 1869); "Location of harbor lines" (Eeport, 1871); "Notes con- cerning alleged changes in the relative elevation of land and sea" (Eeport, 1877). They contain * Cf. Mitchell, United States Coast Survey Report, 1869. t United States Coast Survey Report, 1854, pp. 190, 191; 1857, pp. 403, 404. } Ibid., 1859, pp. 315-317. 454 UNITED STATES COAST AND GEODETIC SURVEY. numerous rules and practical suggestions, which belong to the art of hydrographic engineering rather than to the study of the tides. To these we may add his pamphlet, issued by the Navy Department in 1868, entitled "Tides and tidal phenomena." Hydrographic work of a similar character has been since carried on by Henry L. Marindin. His papers upon the same are to be found in the Survey Eeports since 1880; in particular those for 1888 and 1892. 129. Rev. Samuel Hauglitoii's principal writings upon tides are to be found in the Philosophical Magazine (1856, 1863), the Philosophical Transactions (1863, 1866, 1875, 1877, 1878), and the Transactions of the Royal Irish Academy (1854, 1893, 1895), Besides these may be mentioned brief notices in the Proceedings of the Koyal Society of London (1860-1877) and a small book entitled Manual of Tides and Tidal Currents (1870). The tides discussed by him are those around the coasts of Ireland and in the Arctic Seas. He has, in a general way, followed the methods of Airy, and among the quantities worked for are the mass of the moon, the eccentricity of the lunar orbit, the mean depth of the Atlantic Ocean regarded as a canal running north and south. William ParJces, in the Philosophical Transactions for 1860, treats the high and low waters at Bombay and Karachi, where the diurnal inequality is large. He combines the two waves of variable amplitudes — the diurnal and semidiurnal — and obtains results agreeing fairly well with observation. In the British Association Eeport for 1870 (I, p. 150), Thomson makes some mention of Parkes' work, comparing with observations predictions made by the latter's method, those made by Thomson's method, and those according to the Admiralty method. James Groll has written upon the influence of the tidal wave on the earth's rotation, and upon the causes and climatic effects of ocean currents. These writings are found in the Philosophical Magazine, American Journal of Science (1864-1876), and British Association Eeport (1876). T. K. Abbott has contributed brief papers on tidal theory to the Philosophical Magazine (1870-1872), the Quarterly Journal (1872), and Hermathena (1882). His small book, based upon the foregoing papers, entitled Elementary Theory of Tides (1888), gives a popular treatment of a few fundamental questions in the kinetic, or rather the canal theory. H. Lacy Garbett, somewhat after the manner of Abbott, gives a popular exposition of several difficulties in the kinetic theory of tides. He says (Phil. Mag., 1870) : It appears that, without supposing tlie remark to be in anywise new, I happened in 1853 to make the first English mention that tidal friction must increase the length of the day . . . and to suggest (what Delaunay is now considered to have verified) that this cause might have counteracted and masked the shortening due to con- traction, so as to account for the non-diminution (or, as now admitted, lengthening) of the day since Hipparchus's time. [See under Ferrel, "Questions of priority."] In the Philosophical Magazine for 1874 Alfred Taylor has a paper entitled "On tides and waves, — deflection theory." He advocates the view "that the level of the ocean is nearly repre- sented by high- water mark on coasts and bays where there is free access of the tide and a channel without a sudden taper," instead of being about half-tide level as it would be natural to suppose. He does not believe that tidal action has the smallest effect on the rotation of the earth. His "Deflection theory" takes its name from a supposed deflection (refraction?) which the attractive rays experience in passing through the earth. He deduces from experiments by J. S. Russell and by Darcy a new formula for wave propagation in any depth, j?, viz. : v=Z v'p feet per second. E. J. Chapman, in the Philosophical Magazine -for 1874, proposes the theory that the tides result from the compression of the earth's nucleus, which is surrounded with a layer of incom- pressible water. 130. J. Heinrich SchmicTc is the author of a book entitled Das Flutphanomen und sein Zusam- menhang mit den sakularen Schwankungen des Seespiegels (1874). Besides treating the matter indicated by the title one part is devoted to earthquake, sea, or ocean waves. Hugo Lentz is the author of a book entitled Fluth und Ebbe und die Wirkungen des Windes auf den Meersspiegel (1879), gives among other things an intelligent account of tidal inequalities, and shows that the notion of the " age " of the tide is quite untenable. As indicated by the title, a portion of the work is devoted to wind effects on the height of the sea, the stations considered being along the North and Baltic seas. REPORT FOR 1897 PART II. APPENDIX NO. 8. 455 Gomoy, in his book entitled ^tude pratique sur les Marees riuviales et notamment sur le Mascaret (1881), gives an account of wave motion, particularly waves of translation, as propagated up tidal rivers; a study of the mascaret in the rivers of Prance; and the effects of river improvements. J. G. Houzeau and A. Lancaster in their Bibliographie g6n6rale de 1' Astronomic,* Vol. II, pp. 626-635, give a complete list of papers upon the theory of tides, including the effect of lunar attraction upon gravity, the effect of the tides upon the earth's rotation, also atmospheric or aerial tides. The list begins with the writings of Wallis and continues through the year 1880. 131. A. B. Basset is the author of a treatise on hydrodynamics (1888), the seventeenth chapter of which is upon liquM waves. Chapter XIX is devoted to the theory of tides. He treats in brief the equilibrium theory, gives Darwin's development of Laplace's theory, and also portions of Airy's canal theory. Prof. Horace Lamb in A Treatise of the Mathematical Theory of the Motion of Fluids (1879), in addition to a general exposition of his subject, discusses "waves of small vertical displacement" (i. e., "long waves" or "waves of translation"), and illustrates by examples drawn from Airy's treatment of tides in canals. Fear the close of his book is a "List of memoirs and treatises" pertaining to fluid motion. In his Hydrodynamics (1895) the tidal theory is set forth in a concise and masterly manner. It is the best exposition of the theory known to the writer. The chapters entitled "Viscosity" and "Equilibrium of rotating masses of liquid," involve the principal discoveries along these lines made by Stokes, Eayleigh, Kelvin, Darwin, Love, Lamb, Poiucare, and others. 132. Prof. Wm. RarTcness, Washington Observations for 1885, App. Ill, gives a collection of determinations of the mass of the moon since the time of Newton, adding thereto determinations made by himself from the harmonic constants of over thirty stations. His concluded value from the tides is 0-012714±0-000222=l/78-653± 1.374 (=gQ+ Sja). At the close he gives a "List of works consulted in the preparation of the foregoing paper," and here are given numerous refer- ences to recent papers on tides. Maj. A. W. Baird is the author of a book entitled Manual for Tidal Observations (1886). In the lirst part are practical directions for locating stations, setting up and caring for tide gauges, and auxiliary (meteorological) instruments. In the second part are directions for carrying out the harmonic analysis in accordance with the system of Thomson and Darwin. The appendix consists of auxiliary tables used in connection with the analysis. L. d'Auria has contributed several articles to the Journal of the PrankUn Institute, among which are the following: "On the measurement of tidal heights" (1879); " On the force of impact of waves," etc. (1890); "A new theory of the propagation of waves in liquids" (1890); "Analytical discussion of the tidal volume admitted into bays and rivers," etc. (1891) ; " The law of variation of the theoretical amplitude of tidal oscillation," etc. (1891). In these the meaning of the author is not always clearly set forth; consequently it seems impossible to ascertain just what he has in mind, and why he believes that certain relations obtain. The subjects of these papers are important and his treatment is suggestive; for these reasons they may be worth consulting. Prof. William Ferrel (1817-1891). 133. Perrel's Tidal Eesearches, published by the Coast Survey in 1874, include the greater part of his theoretical work. One of the principal objects of these investigations is the deter- mination of the effects resulting from fluid friction when assumed to vary according to a power of the velocity greater than the first (friction =— / F"). Laplace had generally altogether ignored friction, and Airy had assumed it to be proportional to the first power of the velocity.* In either of these cases the fundamental differential equations of motion are linear, but upon Perrel's assumption they no longer remain so. Dr. Young had assumed friction to be as the square of the velocity, but his treatment is imperfect, inasmuch as it does not involve the equation of con- tinuity. The important and then new subject of shallow-water components is treated at some length in the Tidal Eesearches, but much more fuUy in his " Discussion of tides in Penobscot Bay."t * Brussels, 1882. + United States Coast and Geodetic Survey Report, 1878. 456 UNITED STATES COAST AND GEODETIC SURVEY. Ferrel was the first to give, in 1868, any considerable development of the tide-producing potential. This development he reproduces, with some modifications, in his Tidal Eesearches. Confining our attention to this later treatment, we may describe it as follows: Laplace's expression for this potential, when developed in multiples of the body's hour angle, gives rise to several distinct parts or classes of terms, which may be written in the general form N,cos s {nt + zS — ^;), or N/ cos s {nt + ^ — f), according as the moon or sua is considered, s taking the values 0, 1, 2, . . . . The first part does not contain the hour angle of the disturb- ing body, the moon, say; the second class has a period a lunar day in length; the third class a half lunar <'iay; the fourth class one- third of a lunar day; on account of the smallness of the last it may be disregarded, for the present at least. The coefficients of these periodic functions of the moon's hour angle have, at a given place, two elements of variability; the one being the factor l/>\ the other some sine or cosine function of S, the moon's declination, l/r^ is equivalent to a constant quantity (which is slightly greater than 1/p', p being the meail value of r) plus comparatively small periodic terms whose arguments or peripds readily follow from the expression for the moon's parallax. Here and elsewhere Ferrel employs circular arguments which vary uniformly with the time, or nearly so. The arguments of the principal periodic terms in 1/r^ are the moon's mean anomaly, the argument of evection, of variation, twice the moon's mean anomaly, and the mean anomaly of the sun. The circular argu- ments belonging to the functions of S, already referred to, are the longitude of the moon and of the lunar node. When Ifr^ is multiplied by these functions of d, terms naturally arise whose arguments are simple combinations of those in the two factors. In this manner the coefficients of the three principal parts of the moon's tide-producing potential are each developed into a number of terms constant or periodic. The periodic terms in that part (No) of the potential which does not involve the moon's hour angle, and which give rise to oscillations in the sea level of long period, really constitute a harmonic development. The coefficient (Ni) of the function whose I)eriod is one lunar day has no constant term, but its principal term has as argument the longi- tude of the moon reckoned from the solstice. This coefficient or amplitude is therefore negative during half of each month. The coefficient (N2) of the function whose period is a half lunar day consists of a constant term together with numerous periodic terms. The most important of these have as arguments the moon's mean anomaly, twice the longitude, the arguments of evection and of variation. Of course the tide-producing potential of the sun admits of a similar development. The tide producing potential due to the attraction of both sun and moon may likewise be developed. The terms which do not involve the hour angle of either body are simply added together algebraically. The parts having a half-day period, K2 cos 2 [nt + zS — ip) and ^2' cos 2 (nt + u — 1//), give, when combined, a resultant amplitude of the form obtained when two cosine curves are combined into one. The angle or argument, which is twice the moon's hour angle, becomes in the resultant somewhat altered; but this, too, is in accordance with the combination of two simple cosine curves. The expansion of the resultant amplitude [Ni) gives rise to a constant term and to numerous periodic terms, the chief of which has as argument twice the angle between the sun and moon. The arguments of several others have already been mentioned. If in the diurnal part, the sidereal (or, more properly, tropical) day had been used instead of the lunar, then the coefficient would have had a constant term, and numerous periodic terms; the arguments of the two principal periodic terms being twice the ' longitude of the moon, and twice the longitude of the sun,"botli reckoned from the solstice, say. This nonharmonic development of the potential is in a form for application to observations made upon high and low \^aters. It shows the theoretical proportions between the various inequalities iu the tide. The non-harmonic or inequality methods of Ferrel form an extension to the works of Laplace and Lubbock. He makes use of all observations, and not of certain groups selected for particular purposes as did Laplace; he distributes the observations according to the inequality sought, usually dividing its period into 12 or 24 nearly equal parts; he analyzes the corresponding 12 or 24 values of the ranges or intervals, thereby determining the most probable value of the amplitude and position of the inequality; he compares the ratio of the coefficient to the range of tide with the corresponding ratio in the tide producing potential; the failure of these to agree implies the existence of what Laplace calls "accessory circumstances," or an REPORT FOR 1897 — PART II. APPENDIX NO. 8. 457 Incorrect assumed mass of the moon, or both. The greater the number of inequalities treated, the more of th6se constants can be determined. Ferrel usually determines two besides the correction to the mass of the moon, using therefor the three largest inequalities in the (semidaily) tide.* As some account of his method of determining the coefiicients and epochs of the inequalities appears in § 64, also in § 46, Part III, no further notice will be taken of it here than to refer to Chapter VI of the Tidal Eesearches, where the tides at Brest are discussed; to his " Discussion of tides in Boston Harbor;" t and particularly to his " Discussion of tides in New York Harbor ."| In regard to Ferrel's harmonic development of the potential of the tide-producing forces, we will only remark that it is the first ever made — at least with any tolerable completeness; that a number of lunar nodal terms are given which arise from the varying inclination of the lunar orbit to the plane of the equator; and that his numerical values of the coefficients of the sun's tide-producing potential are each affected by a term in S/x.^ For, the coefficients of the tide- producing potential of the sun, when expressed as fractions of certain parts of the tide-producing potential of the moon, must involve some assumption regarding the mass of the moon relative to the mass of the earth or sun. Ferrel assumes the mas^ of the moon 1/80 that of the earth plus another very small fraction dpi of the earth's mass. 134. The fundamental tidal equations are satisfied by assuming the vertical and horizontal displacements of the fluid particle which result from a harmonic term of the potential to be simple harmonic functions with constant coefiicients and coperiodic with the term of the potential, friction being ignored or taken to be proportional to the first power of the velocity. Bnt if friction be as a higher power of the velocity, then, although the water be deep, the simple harmonic functions just referred to no longer satisfy the tidal equations, and simple harmonic functions of one-third the period of the others must be included in the expressions for the displacements. Hence we have obtained as a first result of the effect' of friction, which must be regarded as new and important, that when friction is as a higher power than the first power of the velocity, it produces, in either diurnal or semidiurnal tides, small oscillations with a period which is one-third of that of the principal tide. In case of very shallow water, where the amplitude of the vertical oscillation bears a sensible proportion to the depth, quarter-day oscillations must be included in the expressions for the dis- placements of the particle, the resistance due to friction being either included or igQored.|| 135. FerreVs method of determining the moon^s mass from harmonic components. By the equilibrium theory the amplitudes of all components of the same class (long-period, diurnal, or semidiurnal) should have fixed ratios to one another and so to any one of them. The epochs of all components of a class should be equal to one another. If the speeds of the compo- nents were very nearly equal this would, undoubtedly, be very nearly the case ; and constant use is made of this fact in inferring one component from another. In passing from one component to another of sensibly different speed, Laplace assumed that the amplitude is altered by a small quantity, proportional to the difference in their speeds. As will presently be seen, this agrees with Ferrel's work only to the first approximation. Ferrel assumes that the change in the tidal coefficient due to a change of velocity of the disturbing body in right ascension, is not generally proportional to the amount of change in this velocity, as Laplace had assumed. fl Let io denote the speed per day, expressed in radians, of a component Ao; let i, or i, the speed of another component A„ to be compared with Ao ; and so i = io+u, (270) where % denotes the daily difference in speeds expressed in radians. Let the coefficients of the corresponding terms of the tide-producing potential be H, and Hg (=!)• Let the ratio A,/Ao be denoted by B,; the question arises, how does B, differ from ff„ because i, is not exactly equal to to ? * Tidal Researches, U 19, 25, 56, 73, 182-196. t United States Coast Survey Report, 1868. tibid., 1875. § Ibid., 1878, p. 270; Tidal Researches, U 28, 29. II Cf. Airy, Tides and Waves, Art. 198; or see under Airy. IT United States Coast Survey Report, 1868. # = 1. (271) 458 UNITED STATES COAST AND GEODETIC SURVEY. By the equilibrium theory But -^ being a function o", are constants which Ferrel denotes by IS, E'; .•.B,= H,{l + u,Il+u\E'). (273) Similarly for the epoch, s, = s„ + u,G + u\Q'. (274) In case of the semidiurnal components, Ferrel's equations for determining -B, J7', dpi. are, adding equivalents in the harmonic notation,* S2/M2 = A = (0-4582 - 36-2 dfi) (1 + 0-4255 E + 0-181 E'), (275) //j/Mz = J22 = 0-0240 (1 - 0-4255 E + 0-181 E'\ (276) Ka/M- = i23 = (0-1256 - 3-2 S^i) (1 + 0-4599 E + 0-212 E'), (277) L2/M2 = E4 = - 0-0286 (1 + 0-288 E + 0-052 ^')> (278) ]Sr2/M2=rE5=0-1922 (1-0-228 -E7+0-052 E'\ (279) lunar-|E6= -0-0359 (1-0-001 E), (280) (-0359 (1+0-461 _E7+0-212 E'). (281) t lunar "1^6=—' nodaljE,=O-0 " Where the amplitudes of all the principal components are determined from observation, we get B,, by dividing A, by Aq, and hence An is thus eliminated . . . from the preceding equa- tions. The first members being thus determined from observation, these equations, or a sufficient number of them for the purpose, can be used in determining the unknown constants in the case of nature, and the correction of the moon's mass. It is readily seen that in these equations, . . . the determination of Sjx depends almost entirely upon the first and third, and that . . . the neglect of the terms depending upon E\ unless they are large, can have no sensible effect upon the value of S^x, and that the effect of neglecting them is thrown almost entirely upon the value of JE7. When, therefore, the principal object is to obtain the correction of the moon's mass, and a very accurate value of E is not desired, the terms in the equations depending upon E' may be neglected, and then the first and third equations are sufQcient for the purpose. All, however, can be used and the most probable values obtained by the method of least squares." The equations between the amplitudes of the diurnals for the determination of the constants Ao, E, E', and djx are Ki=Ai=(0-5306-13-l dfj) (1+0-230 ^+0-053 E') Ao, (282) Oi=A2=0-3813 (1-0-230 ^+0-053 ^')Ao, (283) Pi=A3=(0-1730-13-6 dfA) (1+0-196 ^+0-040 E') Ao, (284) [lunar ']A4=0-084 (1+0-231 ^+0-053 E') Ao, (285) nodal J A5=0-070 (1-0-231 ^+0-053 E') Ao. (286) In §§ 197-228 of his Tidal Researches, Ferrel applies these formulae to the tides at Liverpool, Portland, Fort Point (Cal.), and Kurrachee. * Tidal Researches, pp. 91, 92. REPORT FOR 1897 PART II. APPENDIX NO. 8. 459 In his "Discussion of the tides in Penobscot Bay," United States Coast Survey Eeport for 1878, he replaces the ,two lunar nodal components of the diurnal group by the lunar elliptic com- ponents Qi and Ji, and omits E'. These formulae become K, = (0-5306 - 13-1 S/x) (1 + 0-230 E) A„, (287) Oi = 0-3813 (1 - 0-230 E) Ao, (288) Pi = (0-1730 - 13-6 dp.) (1 + 0-196 E) Ao, (289) Ji = 0-011 (1 + 0-458 E) Ao, (290) Qi = 0-052 (1 - 0-458 E) A,. (291) In a paper by Prof. "William Harkness (q. v.), entitled "The solar parallax and its related con- stants,"* the author has put Perrel's equations for the moon's mass, etc., into forms better adapted to computation. 136. On inferring small components. Of course the solution of the equations in Sp, E and G {E', O' being neglected) render it theo- retically possible to infer the amplitudes and epochs of other small components which may be required in the representation of the tide. To illustrate, suppose we wish the amplitude and epoch of Qi. From the equations in K,, d, and Pj the quantities S/j, E, and Ao are obtained. These values for E and Ao being substituted in the equation for Qi give its theoretical amplitude. On page 448 of the United States Coast and Geodetic Survey Keport for 1882, Perrel thus finds Qi for Port Townsend and Astoria. On account of the large positive value of E, the formula Qi = 0-052 (1 - 0-458 E) Ao gives in each case a value for Qi much smaller than that obtained from harmonic analysis. In fact, Qi could have been inferred from Oi or Kj by means of its equilibrium ratio much closer than by Perrel's process. Consequently his remark that his small inferred value of Qi is due to a certain shallow water component combining with Qi can hardly seem probable. The epoch of a small diurnal component like Qi is inferred by putting e„=i=J (K,o+0iO) (292) G=-^^^~=0-911 (K,o-OiO) hours (293) = 0-038 (KiO-OiO) days. (294) Then QiO=i-26-25 G (295) where G is expressed in days. On the next page of the Eeport (1. c.) Ferrel thus determines Qi° for Port Townsend, Astoria, and San Diego. The agreement with the analysis is very satisfactory. Similarly he makes use of the equations in Sz, K2, and N2, determining Sju and E from the semidiurnal group, but with the modification noted below.t It is readily seen from an inspection of these eqnations that they can he satisfied only very imperfectly for Pulpit Core, within any determined Talues of Sfi and E, and that they can he much hetter satisfied hy multiplying the first members of the equations hy an unknown constant. This constant is introduced upon the hypothesis that the tidal components are diminished by the effect of friction which is as a higher power than the first power of the velocity, as I have at various times explained. Upon this hypothesis large tides are diminished by friction more than small ones in proportion to their amplitudes, and hence where there is one large component, as the mean lunar, and a number of much smaller ones, since the amplitudes of the latter are obtained by analysis from the differences between the larger and smaller resultant tides, the smaller oomponentB are diminished more than the larger ones in.proportion to their magnitudes, unless friction is as the first power of the velocity. If we take the first, third, and fifth of the preceding equations for Pulpit Cove, and introduce a constant factor c, we have — 0-1574c = (0-4582 — 36-2 5/() (1 + 0-4255 -B) . (296) 0-0469c= (0-1256 — 3-2 Sjn) (1 + 0-4599 E) (297) 0-2082C = 0-1922(1- 0-228 -E) (298) 'Washington Observations for 1885, App. III. t United States Coast and Geodetic Survey Eeport, 1878, p. 297. 460 UNITED STATES COAST AND GEODETIC SURVEY, The solution of these equations gives — 5/j = 0-00263 -E = — 1-164 o = l-166 ' (299) The solution of all the equations by the method of least squares would give values for these constants differing but little from those above on account of the smallness of the amplitudes in the neglected equations, which gives them little weight. The value of Sfi above gives for the moon's mass fi^-^, which is much too large, as is usually the case where the relations differ much from those of the equilibrium theory. The equations for Liverpool give /d = -jV) s-ntl for Kurraohee, where the relations approximate more nearly to those of the equilibrium theory, /u = j-^.^, which is perbaps not very much in error. In regard to the epoch, we have L = MjO, (300) and G is determined from the values of 82°, '^2°, and K^o. In regard to the effects of shallow-water components upon such quantities as these and S//, he says :* From the preceding investigation of the shallow-water tides, I think that we can now see clearly why it is that satisfactory and consistent values of the moon's mass have not in general been obtained from the relations of the semidiurnal tides; for these relations are disturbed by the various shallow-water components, which do not enter into the theory of deep-water tides, which has been used in determining the moon's mass. The perfection of the tidal theory, so as to represent accurately the results of observation at all tide stations, and give a correct mass of the moon, depends now mainly upon the study of the shallow-water terms. With regard to the determination of the moon's mass, from the results so far as obtained the relations of the diurnal tides promise better success in the future than those of the semidiurnal tides. The diurnal tides are not affected by so many of the shallow-water components, and it is probable that these can be determined from the analysis of the observations, since there are two comparatively quite large components with periods differing from those of any others, and hence can be determined by analysis of the observations ; and then from the theoretical relations given in Schedule III the others can be, at least approximately, determined, and the components of deep- water tides which they affect can be corrected for their effect. The relations of these corrected results, obtained from the analysis of the observations, should then agree with the theoretical relatives, and give a correct mass of the moon. 137. Chapter IV of the Tidal Researches treats of tides in canals. He naturally goes over much of the ground previously gone over by Airy. As already stated, Ferrel assumed a more general law of iluid friction so that many of Airy's results follow as special cases of Ferrel's. The subjects here considered are canals extending east and west along the equator or parallels of latitude; canals coinciding with a meridian; and shallow canals extending from the sea inland. Under east and west canals Ferrel notices that : In the case of friction, . . . the osoillaUons of each separate component cannot in general vanish, and give rise to a complete nodal point. There cannot , . . ie in general any place in the canal where the vertical oscillations completely vanish. We might . . . have two canals near each other, extending east and west, of the same length and depth, such as to satisfy (206)t approximately for either the moon or sun, or both, if the canals were not very long and shallow, and if we should suppose the lunar forces to act upon the one and the solar forces upon the other, the lunar and solar tides in the two canals would not only not be at all in proportion to the forces, which is the effect of a large value of JE, but also thet epochs might be very different in the two, upon which the value of G depends. If we therefore suppose the lunar and solar forces to act upon the same canal, we have the two tides coexisting without interference, at least when friction is as the first power of the velocity, but the epochs of the two differing, that Is, the times of high water occurring at different intervals from the times of transit of the moon and sun over some assumed meridian, the high waters of the two do not coincide generally at the times of the syzygies of the moon and sun, and cause the greatest tides, but some time before or after. . . . It is evident from a mere inspec- tion of the expressions, that it depends entirely upon circumstances whether IS and G are positive or negative, that is, whether the lunar or the solar tide is the greater in proportion to the forces, and whether the maximum of the result- ant tide happens before or after the syzygies. This will be also shown in a subsequent part of the chapter by means of actual computations in various assumed cases. In § 145 Ferrel gives a table for various assumed conditions, or rather constants, relating to tides in canals, such as the length, depth, and friction constants. From these he computes the constants Aq, Lq, U, G, F, F'. From the computed values he notices that friction may increase the amplitude of the tide; the amplitudes for different assumed conditions may vary widely; it may be high water at one end of a canal while it is almost low water at the other end; the values * United States Coast and Geodetic Survey Report, 1878, p. 299. tl. e., such length and depth as will give very large vertical oscillations. REPOET FOR 1897 PART II. APPENDIX NO. 8. 461 Of £", Q, and F' may be either positive or negative, and their values may vary greatly for tbe (liflwent assumed conditions. The equations belonging to a canal coinciding with the equator apply to shallow canals extending from the sea inward, "by neglecting the forces in these equations, and regarding m as expressing distance in terms of the earth's radius, instead of longitude." This renders the fundamental equations of motion very simple, especially if friction be also neglected. Ferrel finds, in § 143— That the equatiou of continuity in a shallow canal cannot be satisfied without a change of mean level, and that the periodic vertical oscillations are about this disturbed mean level, instead of the undisturbed in the case of no oscillations. This is a now and important result, and shows that where the water is shallow the true undis- turbed level cannot be obtained from any number of tidal observations taken at equal intervals through all parts of the phase of the tide, but that to the level thus obtained a correction must be applied ... to reduce it to the true level, which . . . is in some places positive and in others negative.* In §§ 248-253, Ferrel notes instances taken from nature to which these statements seem to apply. 138. Chapter V is devoted to the theory of tides upon an ellipsoid of revolution, and is largely devoted to Laplace's solutions of tidal equations! In case of the diurnal tide everywhere vanishing if the depth of the water were uniform, Ferrel contends that although Laplace's result is correct, his manner of solution is incomplete in that it fails to show that the problem remains indeterminate until the proper assumption is made regarding the form of a\ the excess of the height of tide over the equilibrium height. Ferrel first published his views on this subject in Gould's Astronomical Journal, Vol. IV (1856). In regard to the indeterminate coefficient of of, i. e. J.4, he contends that, since ever so little friction must destroy the initial conditions, thus making the oscillations depend entirely upon the disturbing force and vanish with it, the value of A4, to be used must be zero. Finally, for various depths assumed by Laplace, Ferrel computes anew the corresponding ranges of tide, and finds them, he believes, much more conformable to nature. For references to the late papers of Ferrel and others on this question, see footnote under Laplace. 139. Chapter VI is devoted to the discussion of high and low waters by the inequality method already referred to. The next chapter gives various comparisons between theory and observation, the most of which have likewise been referred to. Chapter VIII gives some account of the tides of the North Atlantic Ocean (whose size he attributes to the fact that a canal extending from Europe to America, thus closed at both ends, and having the depth of the Atlantic, has for its free period approximately a half lunar day) ; t the tides of the Gulf of Mexico, of the Island of Tahiti, and of Lake Michigan; §§ 248-253, already referred to, are devoted to observed variations in sea level from place to place. The chapter closes with an account of different forms of tidal curves. 140. Chapter IX is upon the tidal retardation of the earth's rotation. This brings the author back to his first scientific paper, and which was published in Goujd's Astronomical Journal, Volume III (1853), pages 138-141. In a note entitled "Questions of priority," published in the Journal, Volume IX (1890), page 189, and quoted below, Ferrel again makes reference to this subject. In this 1853 paper he makes the first numerical estimate of tidal retardation in the earth's axial rotation based upon mathematical principles, although Kant had in 1754 made a rough estimate of it, and it seems that J. B. Mayer | and others had published something upon the sub- ject. At the time of writing this paper, Ferrel supposed that the then observed secular acceleration of the moon was fully accounted for by Laplace's theoretical expression for the same. He was, therefore, led to believe that the tidal retardation was counteracted by a gradual cooling and shrinking of the earth. He points out that if earth and moon are similarly constituted, the retardation in the moon's axial rotation due to the earth must be to the moon's effect upon the earth's rotation as the square of the mass of the earth is to the square of the mass of the moon. Bertrand was the first to show, about 1866, that the real motion of the moon in her orbit * Cf. Tidal Researches, 4 183. t See § 31. X See Phil. Mag., Vol. 25 (1863), p. 403. 462 UNITED STATES COAST AND GEODETIC SURVEY. (not merely the motion as estimated by the period of the earth's rotation) was affected because of the tides upon the earth. This gives a retardation more than one-third as great as the aT)parent acceleration. In his Tidal Eesearches Ferrel puts these two effects together and finds, from the value of the moon's secular acceleration known by other means, that the tidal displacement due to friction ought to be abaut 2°. As noted below, he had previously (in 1864) shown tidal fric- tion to be the probable cause of the small outstanding acceleration of the moon. His note entitled " Questions of priority" is as follows: It is well known that there is a discrepance between the times of the computed and observed phenomena of certain ancient eclipses, which indicates that there has been a retardation of the earth's rotation. A plausible expla- nation of such a retardation is, that it is due to the effect of friction upon the tidal wave. The first suggestion of this explanation is usually attributed to Delaunay. In Thomson and Tait's Natural Philosophy it is stated that " About the beginning of 1866 Delaunay suggested that the true explanation of the discrepance might be a retarda- tion of the earth's rotation by tidal friction." Delaunay's note on this subject was communicated to the Academy at Paris on December 11, 1865 {Comptes Sendus, Vol. CI, p. 1023). My "Note on the Influence of the Tides in Causing an Apparent Acceleration of the Moon's Mean Motion" was read before the American Academy of Arts and Sciences on December 13, 1864 {Proc. Vol. VI, pp. 379-383). In this note it was shown that upon the hypothesis of only a very moderate displacement of the vertex of the tidal wave by friction, the resulting amount of retardation of the earth's rotation would furnish an explanation of the discrepance between the computation and observation of the ancient eclipses. The writer also claims that the first suggestion that the cause of the exact equality between the time of the moon's rotation in its orbit and on its axis is due to the effect of the attracting forces upon the lunar tides, was given in his paper "On the Effect of the Sun and Moon upon the Rotary Motion of the Earth" (Astr. Jour., 1853, III, pp. 138-142).* G. F. Becker, Am. Jour. Scl., Vol. 5 (1898), p. 108, states that Laplace, in the 1824 edition of the System du Monde, refers the equality of the moon's periods of rotation and revolution to tidal action caused by the earth's attraction in the still fluid moon ; and that Kant considered the tidal retardation in the moon's axial rotation as well as that in the earth's. a;^^-*-*.^!^ In a paper published in the Astronomical Journal, Vol. Y (1858), pp. 97-100, ie examines the deflected course taken by a body moving upon or near the earth's surface because of the earth's rotation. -It has an important bearing upon the general circulation of the ocean and atmosphere. Among other things he establishes that a moving body in the northern hemisphere is always deflected to the right, and in the southern to the left. The radius of curvature of the path is always inversely as the sine of the latitude. For a small range of motion the path is circular, but for a large range it is not; the path is, however, self returning. On pages 113, 114 of the same volume is a note supplementary to the preceding paper. He remarks that the deductions from theory have been verified by some delicate experiments of Foucault.t Besides devising numerous methods for the prediction of tides,t Ferrel in 1880 invented a tide-predicting machine. His published account of the machine and its use is found upon pp. 253-272, of the Survey Eeport for 1883. It was designed with special reference to the prediction of high and low waters, thereby differing from Thomson's machine which simply gives the contin- uous curve. The theory of the machine is also given in §§ 58 and 60 of Part III. His paper entitled " Eeport of meteorological effects on tides," found in the Survey Eeport for 1871, refers to observations at Boston. 141. Sir William Thomson {Lord Kelvin). Thomson seems to have been led to the study of tides through his work upon certain physical problems which involve their consideration. Among these problems is that of the rigidity of the earth, which he considers in the Philosophical Transactions of the Eoyal Society for the year * Cf. D. Vaughan, " Secular variation in lunar and terrestrial motion from the influence of tidal action," B. A. A. S. Report, 1857. Thomson, Phil. Mag., Vol. 31 (1886), p. 533, says that Ferrel was the first to evaluate tidal retardation. Abbott, Elementary Theory of Tides, pp. 22 et seq. Kelvin, Popular Lectures and Addresses, Vol. II (1894), pp. 10-44, 64-72. Ball, Time and Tide (1895), pp. 58-68. See under Thomson and under Garbett. t See Am. Jour, of Science and Arts, Vol. XV, p. 263, and Vol. XIX, p. 141. {United States Coast Survey Report, 1868, pp. 87-95; 1875, pp. 215-221; United States Coast and Geodetic Survey Report, 1878, pp. 299-304. REPORT FOR 1897 PART II. APPENDIX NO. 8. 463 1863.* He finds that the earth's mass must have an effective rigidity at least as great as that of steel, otherwise the effect of its yielding would have beea noticeable upon the amount of the pre- cession or the nutation. Moreover, the earth must be for the most part solid and not fluid as had generally been maintained; for, a thin crust would have to be of fabulous rigidity to prevent tides in the molten matter within. The effect of any elastic yielding is, of necessity, to diminish the range of tide. Calling this range unity for an ocean covering a rigid sphere, the elastic yielding of the nucleus would cause the range to become | or | according as the rigidity of the nucleus is assumed to be that of steel or of glass. Supposing the long-period tides to nearly conform to the equilibrium theory, Thomson and subsequently Darwin were led to the careful study of such oscillations. A discussion by the latter of tides observed in European and Indian ports is given in Thomson and Tait's Natural Philosophy.! Poincar6 notices that the results of this discussion would be in error by a f^ part, for a sea covering the entire earth, because the attraction of the disturbed water is there disre- garded.! Darwin concludes that because of the water's inertia these tides (the small nineteen- yearly one excepted) do not conform to the equilibrium theory sufiiciently close for making valid the earth's rigidity derived from them.§ Thomson's paper upon the tidal retardation of the earth's rotation appears in Volume 31 (186(i) of the Philosophical Magazine; also in Thomson and Tait's Katural Philosophy.|| In Vol. II (1894) of Thomson's Popular Lectures and Addresses entitled Geology and General Physics will be found a popular treatment of the tidal retardation of the earth's rotation, given at the close of the address entitled "On geological time." In this volume are papers which treat of the internal constitution and the rigidity of the earth, viz., " Review of evidence regarding the physical condition of the earth" and "The internal condition of the earth; as to temperature, fluidity, and rigidity." Another paper is entitled "Polar ice-caps and their influence in changing sea levels." In about 1867 Thomson devised the harmonic analysis for tidal observations. In perfecting it he has been aided by J. C. Adams, E. Roberts, and more particularly by G. H. Darwin. A his- torical sketch of this subject is given beyond. U In about 1872 he invented the tide predicting machine, although the first machine for actual work was not constructed until about 1876. For a brief account of tide-predicting machines and references to writings connected therewith, see § 57, Part III, of this manual. It is hardly neces- sary to say that this invention has proved to be thoroughly practical. The Thomson harmonic analyzer was invented in about 1878. Some account of this machine, along with references pertaining thereto, is given in § 66, Part II. Among the statical problems given in Thomson and Tait's Natural Philosophy are the equilibrium theory of tides, and the effect of lunar and solar attraction on apparent terrestrial gravity. If a sphere be but partially covered with water, its surface of equilibrium, even if the sphere turn upon its axis very slowly, cannot generally coincide with that of a sphere entirely covered with water to the same depth. The surface (or portions of surface) will, however, be parallel at any given instant to the instantaneous surface of the covered sphere. Upon this fact rests Thomson's "corrected equilibrium theory." Bernoulli treated the case of a small inclosed body of water upon this assumption, but Thomson was the first to suggest its application to the ocean. The effect of the land is to modify the amplitudes and epochs in the expressions for the lunar and solar tides. Since the equilibrium theory is not concerned with depths, these modifications depend upon surface integrals or, rather, quadratures.** The work of making these quadratures * "On the rigidity of the earth," pp. 573-582. "Dynamical problems regarding elastic spheroidal shells and spheroids of incompressible liquid," pp. 583-616. Cf. Proc. Roy. Soc, Vol. 12 (1862-63), pp. 103, 104 ; Phil. Mag., Vol. 25 (1863), pp. 149-151. + Ed. 1883, §§ 847,848. t Journal de Mathgmatiques pures et appliqu^es, Vol. 2 (1896), p. 80. § Proc. Roy. Soc, Vol. 41 (1886), pp. 339, 342. See B. A. A. S. Report 1886, pp. 56-58 ; also under Laplacet II Section 830. ^ See B. A. A. S. Reports, 1868, 1, pp. 489-510 ; 1870, 1, pp. 120-151 ; 1871, 1, pp. 201-207 ; 1872, 1, pp. 355-395 ; 1876, 1, pp. 275-307. ** Natural Philosophy, Ed. 1867, or 1883, § 808. 464 UNITED STATES COAST AND GEODETIC SURVEY. has been performed by Darwiu for a long-period oscillation,* and by H. H. Turner for a diurnal and a semidiurnal oscillation, t It was the intention of Thomson to give the dynamical treatment of tides in a subsequent volume of the Natural Philosophy. The continuation of this work beyond the iirst volume has, however, been abandoned. It seems that Darwin's restatement of Laplace's theory is in accordance with suggestious by Thomson.f Thomson has worked out additional solutions for Laplace's tidal equation § and defended Laplace's solution in the case of a semidiurnal tide when the ocean is of uniform depth. || 143. Prof. George E. Darwin.' In the Philosophical Transactions for 1863 Thomson gives, with important physical deduc- tions, an independent solution of a problem previously solved by Lame, viz., the state of strain of an elastic sphere under given stresses. In the Transactions for 1879^ Darwin treats the case of a viscous sphere or spheroid instead of an elastic sphere. He finds that the equations of flow in an incompressible viscous fluid are analogous to those of strain for an incompressible solid. He therefore finds it possible, in a measure, to adapt Thomson's work upon bodily tides in the elastic sphere to his case of a viscous sphere. His results regarding the effective rigidity of the earth are in the main confirmatory of Thomson's. He finds a remarkably simple rule for making a com- parison between tides in a fluid sphere and in a viscous sphere, also the effect of the internal yielding on oceanic tides. ** Eesults from Darwin's paper on the precession of a viscoug spheroid are subsequently adapted by him to the making of a numerical estimate -of the retardation of the earth's axial rotation, ft In the Proceedings of the Eoyal Society for 1879 he gives a paper entitled "The determina- tion of the secular effects of tidal friction by a graphical method," where first appear his well- known diagrams illustrating the evolution of the earth-moon system. ||: Outline of Darwin's theory of tidal evolution. — Suppose the earth to be in liquid or semiliquid condition and to be rotating rapidly upon its axis, the period of rotation being from two to four hours. The centrifugal force may be sufficient of itself to cause the matter now constituting the moon to become detached from the earth, whether as one body or as a chain of meteorites consti- tuting a ring, is immaterial, provided the latter soon come together and make up the moon. If the centrifugal force be not sufficient for the accomplishment of this, it may happen that the length of a half day approximately coincide with the period of free bodily oscillation of the earth, which is probably a little less than two hours. The periodic tidal forces fj^om the sun will cause the successive tides to rise higher and higher, until finally a portion of matter will be detached. At first earth and moon revolve nearly as a single rigid body about their common center of gravity; the earth-day and the moon-day are each equal to their "month." At the time here considered, the energy of the earth-moon system is a maximum ; for as yet no energy has been dissipated by tidal friction due to tidal currents, which each body is to set up in the other just as soon as their periods of axial rotation differ from their "month" or period of revolution about their common center of gravity. It is a principle of dynamics that the sum of the moments of momentum of all rotations and revolutions of a system not influenced by extraneous forces is constant, however the distances, velocities, and the amount of energy may vary. For simplicity of conception, the rotation of the moon upon her axis can at first be ignored or lost sight of. The moon produces tidal currents in the earth, thereby slowing down the earth's axial rotation and lengthening the earth-day. By the *Il3id., Ed. 1883, U 810, 848. t Proo. Roy. Soc, Vol. 40, 1886, pp. 303-315. t PMl. Mag., Vol. 50 (1879), pp. 388-402. i Phil. Mag., Vol. 50 (1875), pp. 279-284, 388^02. li See under Laplace. 1[ " On tlie bodily tides of viscous and semi-elastic spheroids, and on the ocean tides upon a yielding nucleus," pp. 1-35. "On the precession of a viscous spheroid, and on the remote history of the earth," pp. 447-538. "Problems connected with the tides of a viscous spheroid," pp. 539-593. See Proo. Roy. Soc, Vols. 27 (1878), pp. 419-424 ; 28 (1878-79), pp. 194-199. *» Phil. Trans., 1879, pp. 15, 28. B. A. A. S. Report, 1882, pp. 472-475. ttThomson and Tait, Nat. Phil., App. [G. a]. It Also found in Thomson and Tait, Nat. Phil., App. [G. b], and in Eno. Brit., Art. "Tides." EEPOET FOR 1897 — PART II. APPENDIX NO. 8. 465 principle just referred to this necessitates a retreating of the moon and so, by Kepler's third law, an increase in the length of the "month." The length of the day will go on increasing until day and month shall become equal, and computation shows this day or month to be about two of our present months in length. The energy of the system will then be a minimum, for no tides or tidal friction can then exist. The earth-moon system will then be in stable equilibrium, and not in unstable equilibrium as it was when it possessed a maximum amount of energy. The energy curve of the diagrams already referred to, has orbital momenta as abscissae and axial momenta as ordinates. It has one real maximum and one real minimum corresponding to the two critical periods already described. There is one stage in the evolution when the month has a maximum number of days. For the earth-moon system, as here considered, this number is 27, or about the present number. In the paper on the precession of the viscous spheroid (Phil. Trans. 1879), where account is taken of solar tidal friction and the obliquity of the ecliptic, this number is found to be 29. Consequently we have passed through the stage of the greatest number of days in the month, although the month now is really longer than ever before, owing to the increase in the length of the day. The tides in the mopn, due to the earth's attraction, have already caused her day to be one month in length, and tides in the earth due to the sun must finally cause the earth to revolve upon its axis once in a year, whatever length the year may then have. A popular treatment of tidal evolution is-given by Ball in a book entitled Time and Tide. Darwin's work upon the harmonic analysis has been largely in the nature of perfecting methods for its application. He drew up the reports of the Tidal Committee, which appear in the British Association Eeports for the years 1883, 1884, 1885, and 1886. The report for 1883 is intended "to systematize the exposition of the theory of harmonic analysis, to complete the methods of reduction, and to explain the whole process." The report for 1886 contains, among other things, a method devised by Darwin for analyzing a short series of hourly ordinates, and a method of prediction; these were designed for the Admiralty Manual, where they may also be found. His more extended method of tidal prediction appears in the Philosophical Transactions for 1891. He devised a method for the harmonic analysis of high and low waters, which is published in the Proceedings of the Eoyal Society, Volume 48 (1890). A concise treatment of the subject of tides by Darwin is contained in the Encyclopaedia Britannica, ninth edition. The brothers George and Horace Darwin have made a series of interesting experiments for a committee appointed by the British Association on the measurements of the lunar disturbance of gravity, for the purpose of throwing some light on the elastic yielding of the earth. These are described in the Association Eeports for 1881 and 1882; they are also briefly mentioned in Thomson and Tait's Natural Philosophy, § 818'. Historical sTcetch of the harmonic analysis. 143. In the harmonic treatment it is supposed, as indicated by the name, that the tide at any given place consists of simple harmonic oscillations, whose periods and amplitudes remain constant — at least for a considerable time. It now seems almost as natural to adopt a series of periodic terms for the expression of the tide as for the " equations " of the motions of the sun and moon. The reasons why tidal workers before Thomson (in about 1867) did not have recourse to such a series seem to be, 1st, the fact that upon the coasts of Europe, where the tides important to navigation were first carefully studied, the tide wave is almost wholly semidiurnal in its character ; that is, the two high waters or the two low waters of a day are almost equal in every respect, and so the phenomenon of rise and fall is comparatively simple; 2d, the custom of observing only the high and low waters (in many cases only the former) instead of the entire tidal curve; and, 3d, the idea that tidal work meant rough work, and so did not necessitate an elaborate scheme which, upon its face, seemed to involve more labor than did the less systematic methods. 6584 30 466 UNITED STATES COAST AND GEODETIC SURVEY. Accordingly the tide was assumed to be composed of two simple waves, one due to the moon and one to the sun. Each had a variable amplitude and period due to the body's varying parallax and declination. The resultant tide was given (as now in the British Tide Tables) by means of a series of tables, based in part upon observations at the port, which generally had the hour of the moon's transit as one of their arguments. Predictions obtained by means of such tables usually made no distinction between the two tides of a day although we find the diurnal inequality described by Colepresse and Sturmy in the Philosophical Transactions for 1668, and Laplace had pointed out that it was due to oscillations of approximately daily periods. But at places where the diurnal inequality is large, the want of a systematic procedure became strongly felt. For, the greater the number of important inequalities in the tide, the greater the difficulty in disentangling them; and, moreover, a long series of observations becomes necessary. The foundations of the harmonic analysis were laid by Laplace. For, he enunciated the principle of forced oscillations; he introduced tidal bodies having uniform motions; he showed how to develop the tide-producing potential into a series of periodic terms, and pointed out the more important harmonic constituents of the astronomical tide; he developed the method of least squares suflflciently far for making it applicable to the determination of the coefScients of a sine- and-cosine function of an angle and its harmonics. But he did not attempt an analysis of equidistant ordinates based upon this knowledge, nor did he completely develop the tide-producing potential. Dr. Thomas Young suggested the importance of observing and analyzing the entire tidal curve, rather than the high and low waters merely. Airy showed that in shallow water the difference between the duration of fall and of rise is due to the presence of an oscillation having half the period of the tide wave. Moreover, he applied an harmonic analysis to the tide wave, thus determining, from day to day, the fundamental oscillation and its numerous harmonics. His method made use of the entire curve, and not the points of maxima and minima merely. 144. In the year 1867 the British Association, upon the motion of Sir William Thomson, appointed a committee for the purpose of promoting the extension, improvement, and harmonic analysis of tidal observations. Thomson's statement to the other members of the committee, with some corrections and additions, is given in the British Association Eeport for 1868, and from this the following, including footnotes, is taken : The chief, it may be almost said the only, practical oouclusion deducible from, or at least hitherto deduced from, the dynamical theory is, that the height of the water at any place may be expressed as the sum of a certain number of simple harmonic functions * of the time, of which the periods are known, being the periods of certain components of the sun's and moon's motions, t Any such harmonic term will be called a tidal constituent, or sometimes, for brevity, a tide. The expression for it in ordinary analytical notation is A cos nt -j-B sin nt ; or R cos (nt — s), if A = R cos «, and B ^ R sin e ; where t denotes time measured in any unit from any era, n the corre- 2 7t spending angular velocity (a quantity such that — is the period of the function), R and s the amplitude and the epoch, and A and B coefficients immediately determined from observation by the proper harmpnio analysis (which consists virtually in the method of least squares applied to deduce the most probable values of these coefficients from the observations). The chief tidal constituents in most localities, indeed in all localities where the tides are comparatively well known, are those whose periods are twelve mean lunar hours, and twelve mean solar hours respectively. Those which probably stand next in importance are the tides whose periods are approximately twenty-four hours. The former are called the lunar semidiurnal tide, and solar semidiurnal tide ; the latter, the lunar diurnal tide and the solar diurnal tide.} There are, besides, the lunar fortnightly tide and the solar semiannual tide.J The diurnal and the semidiurnal tides have inequalities depending on the eccentricity of the moon's orbit round the earth, and of the earth's round the sun, and the semidiurnal have inequalities depending on the varying declinations of the two bodies. Each such inequality of any one of the chief tides may be regarded as a smaller superimposed tide of period approximately equal; producing, with the chief tide, a compound effect which corresponds precisely to the discord of two simple harmonic notes in music approximately in unison with one another. These constituents may be * See Thomson and Tait's 'Natural Philosophy', §§ 53, 54. t See Laplace, ' M^canique Celeste ', liv. iv. § 16. Airy's ' Tides and Waves ', ^ 585. } See Airy's ' Tides and Waves ', §§ 46, 49 ; or Thomson and Tait's ' Natural Philosophy ', $ 808. § See Aii-y's 'Tides and Waves', ^ 45, or Thomson and ' Tait's Natural Philosophy', ^ 808. IvUnar. Solar. 2 fovember 15, 1897. 471 [MANUAL Of TIDES.] PREFACE TO PART II. The object of Part II is to give a sufficient amount of instruction for enabling a person to make reliable observations upon the tides and to reduce them by the harmonic analysis. The system of analysis is that given by Darwin in his report to the British Association for the Advancement of Science at its Southport meeting (1883). The mathematical developments are chiefly those embraced in his report, and its notation has been generally followed. The tables appended to this part have been so numbered as to form a continuation of those appended to Part III, Appendix No. 7, Report for 1894. I have to acknowledge the assistance received from members of the Tidal Division, in the way of suggestions, computations, and the preparation of tables. 473 [MANUAL OP TIDES.] CONTENTS OF PART II. Chapter I. OBSERVATION OF TIDES. Page. Selection of sites for tidal stations 477 Staff gauges or tide staves 477 Box gauges 478 Other non-self-registering gauges , 479 Automatic or self-registering tide gauges 480 Attachments or additions to tide gauges, including indicators 482 Establishment and care of a self-registering tide gauge 484 True scale of the gauge or marigraph 486 Additional directions to the observer 488 Preparation of the marigram for reduction 489 Chapter II. astronomy; tidal COMPONENTS SUGGESTED; ETC. Mean motions of the moon, sun, equinox, etc 490 Principle of forced oscillations 491 What components, or periodic oscillations, should exist in the tide 491 Rules for inferring certain nonharmonic quantities 496 Mean longitude of moon, sun, lunar perigee, etc 497 Formulae for computing I, v, and i 498 Kepler's problem 499 Reduction to the equator, and conversely 500 Approximate expression for the right ascension of the sun and moon 500 Astres fictifs, fictitious moons, etc 502 Sum of a cosine series 503 Note on the determination of empirical constants 503 Chapter III. THE TIDE-PRODUCING POTENTIAL. Its derivation and value 505 An approximate determination obtained by introducing moon and anti-moon 507 An approximate determination from Proctor's construction 508 Spherical harmonic expression of this potential * 509 Spherical harmonic deformations consistent with the condition of continuity 510 Effect of mutual attraction between the fluid particles 514 Chapter IV. DEVELOPMENT OF THE TIDE-PRODUCING POTENTIAL. The problem stated .- 517 Spherical harmonic expressions of the potential 518 Development of these expressions into series of periodic terms '. . 519 Approximations adopted 521 Special treatment for La and Mi 522 The eveotion and variation 523 The equilibrium height of tide 525 475 476 CONTENTS OF PART II. The solar tides _• g|g Tides depending on the fourth power of the moon's parallax 527 On the mean values of the coefficients gyg The factor/ 529 Special treatment for the luni-solar tides Ki, Kj 53Q Meteorological and shallow- water components 532 The corrected equilibrium theory gg^ Cjiapter V. THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. Comparison between component and solar hours 537 Stencils, their construction and use in making summations 539 Adding machines 54Q The Thomson harmonic analyzer 5^j^ Augmenting factors g^3 The process of making the analysis 544 Ferrel's method of eliminating the effects of components 545 The effect of a short-period component upon daily mean sea level 548 Example showing the application of the harmonic analysis 549 FormulfB for inferring amplitudes and epochs 554 On the abbreviation of summations 557 On the harmonic analysis of tides of long period 558 To reproduce the quantities harmonically analyzed 565 Harmonic analysis of a series two weeks in extent 566 Harmonic analysis of high and low waters 567 Interpolation of hourly heights 572 Remarks upon published results and tables 573 Harmonic analysis of tidal currents 573 Prediction of currents 574 Remark on the notation , _ 575 TABLES. 36. Shallow-water components 579 37. The theoretical amplitudes of some of the more important components for every five degrees of latitude.. . 582 38. Augmenting factors 583 39. Values of & — a and of 24 (6 — a) 584 40. Synodic periods in days and hours 586 41. For clearing one component of the effect of others 587 42. Component hours derived from solar hours 589 43. For the summation of long-period tides 604 44. Acceleration in HW and L W of a semidiurnal wave due to a diurnal wave 610 45. Height of HW and LW for a tide composed of a diurnal and a semidiurnal wave 612 46. Hyperbolic functions 6I4 47. Period of a wave 616 48. AVave velocity 617 49. Ratio of vertical to horizontal axes of elliptic orbits of water particles 617 50. Propagation of a free tide wave along a uniform channel 618 APPENDIX NO. 9—1897. MANUAL OF TIDES— PART II. TIDAL OBSEEVATION, EQUILIBRIUM THEORY, AND HARMONIC ANALYSIS. By EOLLiN A. Harris. CHAPTER I. OBSERVATION OF TIDES. 1. Selection of sites for tidal stations. The selection of a site for the observation of tides depends upon the object in view. If a knowledge of the tides at a given point is required, then there is little or no choice in the matter; if, on the other hand, and this is usually the case, a station is to be selected which shall tolerably well represent a considerable area, the following desiderata may govern the selection : Ready communication with the sea, deep water at low tides, shelter from storms, freedom from freshets, and non-proximity to the head of a bay or tidal river. At stations located along straits or upon islands, the tide is liable to be peculiar and so not representative for any considerable region. Freshets may cause great irregularities in the tide, particularly in mean water level,* Near the head of a bay various shallow- water phenomena may occur ; f also seiches in the lesser bays or coves.t Far up a tidal river the duration of rise may be several times less than the duration of fall; and the range of tide may be nearly obliterated.§ A wave cannot be propagated through a very' narrow strait into a large body of water without losing its original form and altering its amplitude. For, there is no cause at work which will impart to the particles of water sufficient velocity for supplying or taking away the volume of water necessary to maintain the wave form unchanged.|| Waves coming around an island from different directions generally produce some kind of inter- ference in certain localities.^ Where the water is deep, it is not likely that the t3rpe of tide change rapidly from point to point, although the shore may be cut up by bays, canals, and straits.** The selection of stations for the prediction of tides may be governed by considerations like the above. 2. Staff gauges or tide staves. A tide staff {tide pole) is a graduated rod, usually made of wood, but sometimes of metal. It is essential to any series of tidal observations, whether the tides are observed directly upon it or not. It should be carefully divided into feet and tenths, by aid of a steel tape or otherwise,' and fixed in a truly vertical position to some object affording a steady and permanent support; for example, to a solid wall or pile. Its zero should be set below the lowest low water likely to occur, and its length should be more than sufficient for measuring the height of the highest tides known at the station.!! Where the beach slopes very gently, or where there are great changes in * E. g., the lower Mississippi; the Delaware. tE. g., at Providence, E. I., there are double-headed tides. At the heads of the arms Petit-Coudiac and Avon, Bay of Fundy, bores sometimes occur. t E. g., Bristol, E. I. ; Karwar and Beypore, India. § E. g., Eivers Adour, Dordogne, Garonne, Charente, Loire, and Seine, of France; also Cape Fear Eiver, North Carolina. . II E. g., Strait of Gibraltar; the East Eiver, New York. U E. g., east of Ireland, and Nantucket Island, Massachusetts. **E. g., southern portion of Alaska. +t Sometimes the graduations increase downward, the zero being a fixed point above the surface of the water; e. g., Airy, Phil. Trans., 1842, p. 1. •477 478 UNITED STATES COAST AND GEODETIC SURVEY. level, as in rivers, several staves may be required for obtaining all readings. They should, of course, be so set, by means of levels, as to virtually constitute a single staff. If made of wood, the staff should be an inch or more in thickness and four or more in width— say not less than gig- part of its length. To prevent the staff from becoming coated and hard to read, it is sometimes well to provide a ready means of removing the same from its support when not in actual use. This is easily done by leaving such small projections and definite marks upon the support as will enable one to readily return the staff to the same position whenever it is to be read. Metal staves, coated with porcelain, are more durable than those made of wood, and do not require removing from their supports when not in use. Bench marhs. — In order to detect any settling or rising in the support of a tide staff, and to enable a person to recover the plane of reference at any future time, several tidal bench marks of a permanent character and situated at various distances from the staff should be established. These marks usually consist of the bottoms of holes drilled into rocks; projections or markings upon rocks or walls; a certain portion of a step, door, or window sill: in the absence of such objects, a buried stone, or a deeply driven stake, pile, or iron pipe may be used. All bench marks should be carefully described (usually by aid of diagrams) for identification. The zero of the staff should be referred to the bench marks by sets of levels run from time to time while tidal observa- tions are in progress. Ordinarily the levels should be reliable to about too of a foot. Tidal bench marks should, whenever feasible, be connected with transcontinental and other long lines of levels. It is particularly desirable to have all tidal stations which are located upon the same body of water accurately connected with one another in order that all heights may eventually be referred to a common datum. Directions for observing. — The staff and bench marks established, the observer should read the height of the tide at even intervals of time. Eeadings at the exact hours throughout the twenty- four hours of each day are preferable for most purposes. The kind of time used is immaterial, provided that it be the same throughout the series of observations. It should always be specified - in the record. In making such observations it is of importance to know the time to within about one minute. In high and low water observations readings should be made every ten minutes, say, for about forty minutes before to forty minutes after each of the four tides of the day. For tides of large range, less than forty minutes will suffice, while for tides of small range more time will be required. In reading a height upon the staff, unless the surface of the water be perfectly smooth, note a point midway between the crest and trough of the waves. A glass tube, partially closed at the ends by notched corks, and held alongside the staff, will facilitate making these readings; or glass tubing may be fastened alongside the staff. In either case the opening in the lower end should not be too large. A small floating body will be found serviceable in marking the surface of the water, or a few enclosed drops of colored oil may be dropped into the tube. 3. Box gauges. A box gauge consists of a long vertical box inclosing a float which rises and falls with the tide. By this arrangement observations may be made when the sea is comparatively rough. The bottom of this box may be pointed or funnel-shaped or, for ease of construction, simply slanted, with a small opening at the lowest part, in order to prevent the accumulation of mud or sand. Besides this, other openings should be made near the lower end of the box. These should be provided with slides for closing such a number of them as will give steady motion to the float without causing the level of the confined column of water to differ sensibly from the mean level of the water surface on the outside. The area of the holes left open should usually be between 2-^- and 1^0 of the cross-section of the float box, aad the lower end of the box should be several feet below the lowest low water. Of course the farther the box extends below the surface of the water the larger may be the openings, as the amplitudes of wind waves decrease rapidly in going downward. (See Fig. 2, Part I.) In some cases the float carries a vertical rod which may itself be graduated,* or it may simply point to graduations upon a fixed scale; t in other cases the float is attached to a wire or varnished cord which passes over ont! or more pulleys, moves an index * United States Coast Survey Reports, 1854, pp. 190, 191; and 1876, p. 131. tBaird's Manual for Tidal Observations, p. 5. Phil. Trans., 1893, pp. 55, 56'. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 479 along a graduated scale,* or rotates a drum carrying a pointer,! and terminates in a counterpoise; in still other cases the cord is replaced by a flexible tape upon which graduations are made.f Various combinations and modifications of these styles readily suggest themselves.§ A simple staff gauge should always be located near a box gauge, and the readings of the two should be frequently compared : for, it is obvious that the line of flotation may in time become altered; or the access of water may be clogged by sand or marine growths, so that the box gauge does not give the true range of tide. In such gauges as have a graduated vertical rod or tape attached to the float, the reading point — i. e., the point of the float box opposite which the movable graduated rod or tape is read — should be referred to a bench mark on the shore. The vertical dis- tance of the line of flotation from the zero of the rod or tape should be ascertained and given in the description of the gauge. When the graduations are upon a fixed vertical rod or the float box itself, one of these graduations should be referred to bench mark. The distance of the movable pointer above the line of flotation should be given. The obvious rule covering all cases, even when the graduations are upon a horizontal or oblique scale, is: Measure the vertical distance from the bench marh to the water, at the same instant noting the reading of the gauge. The difference between the two values should remain constant. Such reference will detect variations in the working of the gauge, and enable one to recover the plane of reference determined from the series of observations, if such plane should be required in the future. 4. Other non-self -registering gauges. A siphon gauge consists of a box gauge upon the shore communicating with the off-shore water by means of a pipe laid along the bottom forming a siphon.|| While observations are in progress, care must be taken that all air in the highest part of the siphon be frequently expelled; otherwise the flow in the pipe will be decreased. This may be accomplished by there inserting a stopcock to be opened at high water or whenever the surface of the water is above it. A closed standpipe or other vessel, preferably of glass, attached to the highest point and filled with water, will serve as a reservoir for the accumulated air. The advantage of this arrangement is that it needs filling with water only occasionally, because the accumulated air does not then immediately decrease the cross-section of the pipe.^ A pressure gauge ** is an instrument, somewhat analogous to a barometer, for measuring the pressure of the water at the bottom of a harbor, in order to ascertain the depths of water and so the height of the tide. The pressure, when the depths are not too great, may be exerted upon a bag filled with air which communicates by means of a hose with a manometer located on board a vessel or on the shore. Such gauges have heretofore generally proved unsatisfactory for long- continued records, owing to the difflculty of preventing sand or shellfish from increasing the normal pressure of the water.® Thomson's depth recorder, which indicates depths by the compression of air, can be used for measuring the tides in very deep water, ff A spar gauge j:|: consists of a long spar bolted at the foot with a universal joint to a block or stone, having attached to the portion above the surface of the water an arc of a circle over which passes a plummet line which indicates the inclination of the spar to the vertical. The gradua- tions upon the spar and this angle of inclination give, by aid of a table of sines, the depth of the water at any time. This gauge may be used for off-shore observations and where the current is strong. Owing to changes which are likely to occur in the sea bottom, the readings of this gauge should be frequently referred to a bench mark upon the shore. 'United States Coast and Geodetic Survey Bulletin No. 12 (1889), p. 143. tZeitschrift fiir Instrumentenkunde, Vol. IV (1884), p. 439. t United States Coast Survey Report, 1857, pp. 402, 403; 1876, p. 131. 5 Phil. Trans., 1831, pp. 174, 175. United States Coast Survey Report, 1876, p. 131. II United States Coast and Geodetic Survey Bulletin No. 12 (1889), pp. 143-146. Baird's Manual for Tidal Observations, pp. 3-6. ITCf. Cliurch, Mechanics of Engineering, p. 736; Trautwine, The Civil Engineer's Pocket-Book, 16th*. ed.. Art. "Syphon"; Knight, American Mechanical Dictionary, Art., "Siphon." ''United States Coast Survey Report, 1858, pp. 247, 248. This gauge as used at Boston was supplied with glycerine instead of air; it was connected with a self-registering apparatus. See an article entitled "Description of a tide gauge for cold climates," by .John M. Batchelder, Am. Jour. Sci. and Arts, Vol. 2 (1871), pp. 67, 68. ft Thomson, Popular Lectures and Addresses, Vol. Ill, p. 54. tl United States Coast Survey Report, 1857, pp. 403, 404. 480 UNITED STATES COAST AND GEODETIC SURVEY. A tripod or pulley gauge, sometimes used where observations are made upon ice rising and falling witli the tide, has a flexible cord made fast to an anchor on the bottom, and which, passing over a pulley directly above, terminates in a counterpoise. The heights may be indicated by the movement of the counterpoise over a graduated scale; or by the number of revolutions of the pulley*, or by a graduated scale securely fastened to the vertical portion of the rope. AUTOMATIC OK SELP-KEGISTBEING TIDE OAUGES. 5. The object of these gauges is to trace a curve, or leave some other record, which will enable one to readily find the height of the sea corresponding to any given instant of time covered by the period of observation. Many forms have been proposed or constructed from time to time; most of them, however, more or less resemble the one described by Henry R. Palmer in 183 l,t which is probably the first self-registering tide gauge ever constructed. The essential parts of any form may be said to be, (1) a float and box similar to those employed in a box gauge, (2) a time piece, and (3) some means of recording the height, either in a continuous manner or at short discrete intervals of time. Usually the motion of the float as it rises and falls with the tide is communicated to the recording portion of the gauge by means of a flexible cord which passes over a grooved wheel called afloat wlieel.X Thence the motion is transferred, but usually on a reduced scale, through some mechanism depending upon the particular kind of gauge, to a pencil which traces a curve upon a moving sheet of paper. The paper is driven or carried along by means of a cylinder connected with a well-regulated clock. The pencil is free to move in a direction perpendicular to the line of motion of the paper. In some gauges the paper used is in the form of a long band, and usually of sufiicient length for containing a month's record; it is paid out from one cylinder, passes over a second upon which the tracing pencil rests, and is received upon a third. In others there is but one cylinder and this usually revolves once in twenty-four hours. For gauges of this kind, the sheet of paper is first dampened with a wet sponge or cloth, then wrapped about the cylinder and its edge pasted down; when dry it will so hug the roller as not to slip. In some cases it is made fast with rubber bands. One or several days' record are made upon each sheet; but care must be taken to change the sheet before the record becomes confused by many tracings. In order to keep the float cord and the bands of paper taut, it is necessary to have some arrangement for counterpoising; either weights or springs may be used for this purpose.§ * Smithsonian Contributions to Knowledge, Vol. 13 (1863), pp. 1, 2. t Pha. Trans., 1831, pp. 209-213. tThe word "cord" may be used as a general term Including wire, tape, chain, etc. In many forms of gauge a rack-and-pinion takes the place of cord and float wheel. § For description of several self-registering gauges, the following references may be consulted : "Description of the self-registering tide-gauge arranged for' the Coast Survey," by Joseph Saxton; United States Coast Survey Report, 1853, pp. 94-96. "Methods of registering tidal observations," by R. S. Avery; ibid., 1876, pp. 130-142. "The self-registering tide-gauge;" Baird's Manual for Tidal Observations, pp. 10-14. (Thomson's gauge); Minutes of Proceedings of the Institution of Civil Engineers (London), Vol. 63 (1881), pp. 2-10. "Notes relating to self-registering tide-gauges as used by the United States Coast and Geodet.'.c Survey," by J. F. Pratt; Report, 1897, App. No. 7. "Ueber einen elektrisch registrirenden Flnthmesser der Telegraphen-Bauanstalt von Siemens & Halske"; Zeitschrift fur Instrumentenkunde, Vol. IV (1884), pp. 95-99. " Registrirender FluthmesSer" (F. R. Reitz's) ; ibid.. Vol. V (1885), pp. 165-168. "Ueber Fluthmesser," by Prof. Eugen Gelcich; ibid.. Vol. VI (1886), pp. 86-89. "Der selbstregistrirende Pegel zn Travenmunde," by Prof. W. Seibt; ibid.. Vol. VII (1887), pp. 7-14. "Der selbstregistrirende Fluthmesser von R. Fuess;" ibid.. Vol. VII (1887), pp. 243-246; Engineering News, July 28, 1888. " Der selbthatige Universalpegel zu Swinemiinde, System Seibt-Fuess ; " ibid.. Vol. XI (1891), pp. 351-365; also ibid.. Vol. XIV (1894), pp. 41-45; ibid.. Vol. XV (1895), pp. 193-203. " Neuer Mareograph," by L. Fau6 ; ibid.. Vol. XII (1892), pp.-171, 172. This is a self registering pressure gauge described in the Journal de Physique, II, Vol. 10, p. 404. Other references may be found in the index of Zeitschrift fur Instrumentenkunde under the heads " Wasser- standsanzeiger," " Fluthmesser," and " Pegel." It may be added that several designs for new tide gauges, also for attachments and improvements to older forms, are on file at the office of this Survey. 480 UNITED STATES COAST AND GEODETIC SURVEY. A tripod or pulley gauge, sometimes used where observations are made upon ice rising and falling witli the tide, has a flexible cord made fast to an anchor on the bottom, and which, passing over a pulley directly above, terminates in a counterpoise. The heights may be indicated by the movement of the counterpoise over a graduated scale; or by the number of revolutions of the pulley*, or by a graduated scale securely fastened to the vertical portion of the rope. AUTOMATIC OR SELF-REGISTERING- TIDE GAUGES. 5. The object of these gauges is to trace a curve, or leave some other record, which will enable one to readily find the height of the sea corresponding to any given instant of time covered by the period of observation. Many forms have been proposed or constructed from time to time ; most of them, however, more or less resemble the one described by Henry E. Palmer in 183 l,t which is probably the first self-registering tide gauge ever constructed. The essential parts of any form may be said to be, (1) a float and box similar to those employed in a box gauge, (2) a time piece, and (3) some means of recording the height, either in a continuous manner or at short discrete intervals of time. Usually the motion of the float as it rises and falls with the tide is communicated to the recording portion of the gauge by means of a flexible cord which passes over a grooved wheel called a float wheel.X Thence the motion is transferred, but usually on a reduced scale, through some mechanism depending upon the particular kind of gauge, to a pencil which traces a curve upon a moving sheet of paper. The paper is driven or carried along by means of a cylinder connected with a well-regulated clock. The pencil is free to move in a direction perpendicular to the line of motion of the paper. In some gauges the paper used is in the form of a long band, and usually of sufiBcient length for containing a month's record; it is paid out from one cylinder, passes over a second upon which the tracing pencil rests, and is received upon a third. In others there is but one cylinder and this usually revolves once in twenty- four hours. For gauges of this kind, the sheet of paper is first dampened with a wet sponge or cloth, then wrapped about the cylinder and its edge pasted down; when dry it will so hug the roller as not to slip. In some cases it is made fast with rubber bands. One or several days' record are made upon each sheet; but care must be taken to change the sheet before the record becomes confused by many tracings. In order to keep the float cord and the bands of paper taut, it is necessary to have some arrangement for counterpoising; either weights or springs may be used for this purpose.§ * Smithsonian Contributions to Knowledge, Vol. 13 (1863), pp. 1, 2. t Phil. Trans., 1831, pp. 209-213. t The word " cord" may be used as a general term including wire, tape, chain, etc. In many forms of gauge a rack-and-pinion takes the place of cord and float wheel. § For description of several self-registering gauges, the following references may be consulted : "Description of the self-registering tide-gauge arranged for' the Coast Survey," by Joseph Saxton; United States Coast Survey Report, 1853, pp. 94-96. "Methods of registering tidal observations," by R. S. Avery; ibid., 1876, pp. 130-142. "The self-registering tide-gauge;" Baird's Manual for Tidal Observations, pp. 10-14. (Thomson's gauge); Minutes of Proceedings of the Institution of Civil Engineers (London), Vol. 65 (1881), pp. 2-10. "Notes relating to self-registering tide-gauges as used by the United States Coast and Geodet.'.c Survey," by J. F. Pratt; Report, 1897, App. No. 7. "Ueber einen elektrisoh registrirenden Fluthmesser der Telegraphen-Bauanstalt von Siemens & Halske"; Zeitschrift fiir Instrumentenkunde, Vol. IV (1884), pp. 95-99. " Registrirender FluthmesBer" (F. R. Reitz's); ibid.. Vol. V (1885), pp. 165-168. "Ueber Fluthmesser," by Prof. Eugen Gelcich; ibid.. Vol. VI (1886), pp. 86-89. " Der selbstregistrirende Pegel zu Travenmiinde," by Prof. W. Seibt; ibid.. Vol. VII (1887), pp. 7-14. "Der selbstregistrirende Fluthmesser von R. Fuess;" ibid.. Vol. VII (1887), pp. 243-246; Engineering News, July 28, 1888. " Der selbthatige Universalpegel zu Swinemiinde, System Seibt-Fuess ; " ibid., Vol. XI (1891), pp. 351-365; also ibid.. Vol. XIV (1894), pp. 41-45; ibid.. Vol. XV (1895), pp. 193-203. " Nener Mareograph," by L. Fau6 ; ibid., Vol. XII (1892), pp.-171, 172. This is a self registering pressure gauge described in the Journal de Physique, II, Vol. 10, p. 404. Other references may be found in the index of Zeitschrift fiir Instrumentenkunde under the heads "Wasser- standsanzeiger," "Fluthmesser," and "Pegel." It may be added that several designs for new tide gauges, also for attachments and improvements to older forms, are on file at the office of this Survey. REPORT FOE 1897 PART II. APPENDIX NO. 9. 481 6. Pig. 1, taken from the Coast Survey Report for 1853, shows a three-roller gauge designed by Joseph Saxton. F denotes the float, W the float wheel, P the tracing pencil, and the clock. The paper is paid out from the roller E,, is pulled forward by means of pin points in the cylinder E^ which is driven by the clock, passes under E^ which pressing down upon it enables the points to puncture the sheet, and is finally received upon R''. E, E', B", E'" denote various counterpoises. Fig. 2.— Avery's one-roller tide gange. This gauge was used by the Survey for many years, the most important change made being the substitution by R. S. Avery of a balance clock for a pendulum clock. This change was neces- sitated because of the shocks of the waves against the supports of the tide house when in au exposed location. Several other improvements were made by Avery. One was the putting of band wheels at 6584 31 482 UNITED STATES COAST AND GEODETIC SURVEY. the ends of E, R^, E*, so that E and E* could be moved by E^ by means of an endless band. This necessitated some means for keeping the paper taut and at the proper tension, which was accom- plished by allowing the cores of E and R* to revolve with some friction in their cylinders. The friction was produced by friction plates which could be adjusted at will. Another improvement was in the mode of clamping the clock to the cylinder. Fig. 2 shows a one-roller gauge devised by Avery and described in the Coast Survey Report for 1876. 7. Pigs. 3 and 4 show a form of gauge now constructed in the Instrument Division of the Survey. This is a three-roller gauge capable of receiving one month's record without change of paper. The float wheel has a spiral groove in its periphery so that the float wire, having been made fast at a given point, may be wound around it a number of times without causing any crossing or piling up. A shoulder of the float wheel has a similar spiral groove for the wire or cord of the float counterpoise. The float wheel communicates motion to the recording pencil by means of a coarse screw working in a nut. The float- wheel end of the screw rests upon ball bearings; this secures accuracy as well as ease of working. The recording pencil can be accurately set at any distance from the base line by loosening the float wheel from the coarse screw and rotating the latter, thus driving the nut and pencil. An attachment is provided for marking the exact hours upon the sheet. This result is accomplished by making use of an additional clock, and adapting its striking api)aratus to the sudden movement of the recording pencil each hour. The paper is kept taut by means of a spring pressing against the roller from which the paper is paid out, and a weight attached to the receiving roller. The distance between the flanges of the rollers, that is, the width of the sheet, is thirteen inches. 8. For special purposes many attachments or additions have been devised. Of these may be mentioned integrators, indicators, time-marking and printing attachments. In the more recent gauges, electricity often plays an important part. An integrator is an arrangement for continuously summing the heights' of the sea for the purpose of finding mean sea level. One form of integrator may be described thus:* The cylinder upon which the curve is traced has attached to one end of its axle a smooth disk. A small friction wheel with sharp edge has its plane perpendicular to the plane of the disk, and its axis in the same plane as the axis of the cylinder. This small wheel moves across the face of the disk as the recording pencil moves across the recording sheet. The fiiction between this disk and the small wheel causes the latter to rotate. When the phase of the tide is at about mean sea level, the edge of the wheel should be set at the center of the disk; then for phases higher than this, it will rotate in one direction; and for lower phases, in the opposite direction. The resulting number of rotations is ascertained by additional wheelwork. This number divided by a quantity proportional to the length of the period covered by the observations, will show how much the observed mean sea level differs from the one assumed. Another form of integrator t consists essentially of a pendulum whose variable length is dependent upon the height of the sea for the particular instant. A cam connected with the float wheel alters this length in a suitable manner, while a clockwork registers the entire number of beats. An indicator is either an independent instrument or an attachment to a tide gauge, which shows ujjon a large dial, or otherwise, the height of the sea at any given instant. Tlie indicator may stand close to the float bos and have a mechanical connection therewith; or, it may be located many miles distant and have an electric connection. A broad and clearly-painted tide staff, situated in a conspicuous place, constitutes an indicator of the simplest form.:j: '"RegistrirenderFluthmesser" (F. R. Reitz's) Zeitschrift fur In8triimenteiik;unde,Vol. V (1885), pp. 165-168. t"Der selbthatige Universalpegel zu Swinemiinde, System Sei bt-Fuess ; " Zeitschrift f'ir Instrumentenkunde, Vol. XI (1891), pp. 351-365. t For descriptions and examples of indicators the following references may be consulted : "The tide indicator at Rouen;" Scientific American Supplement, September 23 (1893), p. 14785. "Tidal indicator, New York Harbor;" Scientific American, March 3 (1894), p. 133. Coast and Geodetic Survey, REPOKT FOR 1897 PART II. APPENDIX NO. 9. 483 The Coast and Geodetic Survey has recently erected two tidal indicators, one at Fort Hamilton, N. Y., and one at Reedy Island, Delaware Eiver. The latter, shown in Fig. 5, has a face thirty feet in diameter. It is proposed to erect a similar indicator at Presidio, Oal. In these indicators the rise and fall of the tide is communicated to a large float wheel carrying a pointer, by means of a flexible wire cord. The float wheel is so counterpoised as to keep this cord constantly taut. The end of the pointer moves along a semicircle whose divisions represent feet and half feet of rise and fall, the zero denoting the position of the pointer when the tide is at the plane of mean low water. The arrow head has its two barbs so hinged as to enable it to point either upward or downward. The upward pointing indicates a rising tide, the downward, a falling. Fig. 5.— Tidal indicator, Delaware Eirer, Debiware. At the time shown iu the figure, the tide is IJ feet above mean low water and is still falling, as indicated by pointing of the arrow. Each triangular barb of the arrow head rotates about a point near its obtuse angle when a reversal takes place. The power accomplishing this comes from the float and is communicated to a lever by causing the float cord to pass between a group of three pulleys, thus giving rise to some "Notice to mariners" No. 177. Harpers Weekly, Vol. 38 (1894), p. 96. United States Coast and Geodetic Survey- Report (1893;, pp. 27, 28. ^ A similar indicator is located at Reedy Island, near Philadelphia. " Notice to mariners" No. 202. '•'Elektrisolier Tiefwasserstandsmessermit Zifferblatt" (A. Grabi^'s); Zeitschrift fiir Instrumentenkunde, Vol. IV( 1884), p. 439. " Elektrischer Wasserstandsanzeiger" (A. Hempel's); ibid.. Vol. VIII (1888), p. 224. " Elektrischer Wasserstandsanzeiger mit Registrirvorrichtung," by W. E. Fein ; ibid.. Vol. IX (1889), pp. 338-343. "Der selbthiitige Universalpegel zu Swinemiinde, System Seibt-Fness;" ibid.. Vol. XI (1891), pp. 351-365. 484 UNITED STATES COAST AND GEODETIC SURVEY. small amount of friction. The lever actuates a vertical rod in the shaft of the arrow, so to speak, and this in turn actuates an arm or projection attached to each barb. From the figure it is evident that the barbs are of sufficient size to require counterpoising in order that as little work as possible may be required of the float. The object of time-marking attachments is to indicate upon the record sheet the exact hour as given by a clock, either near or distant, in order to avoid errors which might result from using hour marks made by points fixed upon the cylinder. The gauge of Mr. Palmer, already referred to, marked the positions of the hours by means of a punch actuated by a toothed wheel. The figure was in the form of an arrow whose direction was always that of the wind-vane above the tide house. A cross-mark showed the exact position of the arrow to be taken as the hour mark. Since the invention of the electric telegraph, the principle of having a clock make (or break) a circuit has beeji at the foundation of nearly all schemes for the transference of a particular instant of time, whether for astronomical or other purposes. In this way it is easy to have one standard clock mark the exact hours upon several gauges simultaneously.* The time-marking attachment now used upon the gauges belonging to this survey, and shown in Figs. 3, 4, causes the pencil which traces the tidal curve to suddenly move back and forth, thus dawing a short horizontal mark from the curve at each exact hour. The advantage of horizontal marks over vertical ones is that the hourly heights can be more easily read from the sheet. Some gauges cause hour marks to be made at both edges of the sheet, thus precluding such error as may arise from slanting the height scale when the sheet is being read. Printing attachments are designed for the purpose of saving time in the reading of a tidal record. One form leaves a record consisting of two rows of printed figures, the one being the hours of the day uniformly distribuied, and so easily made; the other, the heights of the sea at the exact hours, or at certain fractious of hours.t For ascertaining the meteorological effects upon the tides and the condition or density of the water, the following auxiliary instruments may be provided : A self-registering aneroid barometer, a mercurial barometer for checking the aneroid, a self-registering anemometer, a thermometer, and a densimeter or salimeter. Establishment and care of a self-registering tide gauge; also the reading of the record. 9. A small house or closed shed must be provided for sheltering the registering portion of the gauge. With the most common forms of automatic gauges this house must be located upon a wharf or other staging directly over the float box; but some forms have been devised in which the recording portion of the gauge may be placed wherever convenient. When a special house has to be constructed, it should be large enough to afford room to get at the registering part of the gauge, and be provided with a window for light and ventilation. The site of the gauge should be so selected as to afford as much protection from violent storm waves as possible, while at the same time not so far removed from the port for which the tidal observations are wanted as to introduce any material alteration in the time or range of the tide. The location should be reasonably accessible, and the structure upon which a gauge is placed should be as firm as is practicable. The siphon arrangement already described will be found to be very serviceable where deep water lies far out from the shore line.f In setting up the gauge, care must be taken to have it properly leveled in order that all parts may work freely. The float box should be vertical and the float suspended in its center. The statements made in § 3 concerning the size of the openings apply here fairly well. The scale to be used can be decided upon as soon as the range of tide at the place is approximately known. * Some references to time-marking attachments : "Elelitrischer Wasserstandsanzeiger mit Eegistrirvorrichtung," by W. E. Fein; Zeitschrift fur Instrumenten- tunde, Vol. IX, (1889), pp. 338-343. "Der selbthatige Universalpegel zn Swinemiinde, System Seibt-Fuess ;" ibid., Vol. XI (1891), pp. 351-365. "Der kurvenzeichnende Kontrolpegel, System Seibt-Fuess," by Wm. Seibt; ibid.. Vol. XIV (1894), pp. 41-45. "Notes relating to self-registering tide gauges as used by the United States Coast and Geodetic Survey," by J. F. Pratt; report (1897), App. No. 7. f'Ueber einen elektrisch registrirenden Fluthmesser der Telegraphen-Bauanstalt von Siemens & Halske;" loc. cit. ante. t Cf. Baird, Manual for Tidal Observations, pp. 2-4. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 485 The zero of the gauge (i. e., the position of the tracing pencil) can be approximately fixed by noting a reading of the water upon the tide staff. A closer approximation is afterwards made in the manner described beyond. An automatic tide gauge of any sort is sometimes called a marif/raph, and the record pro- duced by it a marigram; these names can be readily distinguished as to their application by the more common words "telegraph," the instrument, and "telegram," the message sent. In some forms of marigraphs there is a row of steel pins at each end of a cylinder which make small perforations in the paper, the distance between them indicating hours or half hours of time. After putting the paper on such gauges, and before connecting with the clock, draw a straight line between the pins at opposite ends of the cylinder, place the recording pencil upon this line, and then when it is an exact hour (or half hour, if the pins are close enough to measure that inter- val of time) connect the driving clock with the apparatus, and note the time along the ruled line. In other forms of gauges the hour lines and their numbers are i^reviously marked upon the edge of the cylinder, or upon profile paper stretched around it^ but care must be taken in starting to make the record and actual time agree. In the most improved forms of gauge the hours are marked upon the paper by a special clock, or by the driving clock itself, using either electrical or mechanical means. In such gauges there is no need of starting the record at any whole hour or half hour. On the blank paper, at the beginning and end of each marigram, a note, similar to that below, should be written and filled out: Station Latitude Loiisitiule The time used is Tidal record from to Marigram No Marigraplx No Scale Observer In connection with every marigraph there should be a fixed tide staff, in order to have a check upon the working of the apparatus; and the readings of this fixed ^t&S, cikW&A. staff readings, together with the times of making them, must be recorded on the marigram. The best time for taking a staff reading is generally at or near the time of high or low water, because the height of the water surface then changes slowly; but it is also desirable that several staff readings be made, at least once a month, when the water is about at its mean level, on both arising and a falling tide, in order to show that the gauge is working freely and accurately. A general form for making such entries upon the marigram is as follows, the whole being connected, by means of an arrow or other device, with the exact position of the recording pencil of the gauge at the time : Monday, Bee. 21, 1896, 9^ 35'^ A. M. ) Gauge eloclc correct. Staff 6.34 ft.] Such a note should be made at the beginning and end of each sheet or roll of paper, without regard to the phase of the tide. When there is a time-marking attachment to the gauge, time comparisons should be made at the instant when the hour mark is made. Then a note like the following may be used : P. M. Wed., Sept. 25, 1895^ Correct time, 1:59 Gloclc set right At 2:01, staff reads 6.34. By clock is here meant the clock which makes the hour marks, and the above note implies that this clock made the 2 o'clock hour mark 1 minute too soon. If the hour marks are made by means of electric connections with a standard timepiece, no such time comparison is necessary. * Whenever anything unusual happens to the record, such as disturbances caused by great storms, or a stoppage of the gauge, an explanatory statement should be written upon the mari- gram. In fact, a marigram shonld be a complete record in Itself, as notes made elsewhere are liable to become permanently separated from ii. 486 UNITED STATES COAST AND GEODETIC SUEVEY. A graduated piece of paper, wood, metal, glass, or other material, called a height scale, is used for reading the marigram. This scale shows the relation between the marigram and nature; for instance, a scale of one-tenth mean^ that a variation of 10 inches in the height of the water surface is indicated by an inch of change on the record; and with such a scale 1*2 inches on the marigran corresponds to a foot on the fixed staff. Gauges can be made to work upon any desired scale, according to the range of tide at the place; but while it is desirable that the scale used should be as near as possible to nature, the average curve produced ought not to occupy much more than one half of the paper, for unexpected variations of surface level are sure to occur, and these are oftentimes matteis of much interest. Upon starting the gauge care should be taken to so adjust the pencil that half-tide level will fall as nearly as possible in the middle of the record paper. In order to make scale readings it is necessary to have some datum line upon the marigram ; the zero of the scale is often placed upon the line of punctures made in the paper by the row of steel pins for marking time, or preferably upon a line traced by a stationary pencil. When the gauge is first started this datum line may be placed anywhere, and arbitrarily called such a division of the scale as will insure positive read- ings for the curve, but after sufQcient record has been obtained to properly determine the relation between such assumed datum line and the fixed staff, it is desirable to so change the datum as to make the scale and staff readings agree as closely as possible. In the manufacture of marigraphs it frequently happens that mechanical difflculties prevent the obtaining of the exact scale desired, and hence with a new machine one must always find out trom the record itself what its working scale really is. 10. True scale of a marigraph. The true or working scale of the tide gauge is ascertained by comparing the staff readings with the readings of the curve corresponding to these times. The sum of the staff readings near high water, less the sum of a like number of staff readings near low water, gives, when divided by this number, a certain range of observed tide. Treating the corresponding scale readings in the same manner, a certain range of recorded tide is obtained. The ratio of these two ranges shows how much the true or working scale of the tide gauge differs from the assumed scale used in reading the curve. For instance, the sum of the three observed ranges in the fourth column iu the example below is 9-7 feet, while the sum of the corresponding scale ranges in the seventh column is 9'9. .•.9'7-^9-9=0'980; and, since the assumed scale is 10, the working scale is 0-980 x 10=9-80. Having thus found that scale heights (by assumed scale of 10) must be multiplied by 0-980 in order to give true heights from base line or scale datum, we multiply 26-5 by it, thus obtaining 2«-0 feet. Subtracting 26-0 feet from the corresponding staff heights, we obtain — 3-3 feet. Since 6 heights are taken, this must be divided by 6, giving —0-55 feet, which the staff will read when the tracing pencil crosses the base line. Or, if we construct a scale of 9-8, i. e., a scale such that one unit of the scale =9^g foot, then it should have its division —0-55 marked as the one to be applied to the base line of the marigram iu making all height readings. The following method amounts to giving larger individual ranges greater weights in the determination than is given to the smaller ranges, and so is applicable where the diurnal inequality is large or where a more elaborate determination may be desired. The notation employed is of a temporary nature. Let ^' staff" be a staff reading taken at or near high or low water. Let "scale" be a reading with the assumed scale at the time of a staff reading. The expression for the scale, true or assumed, as here used, is supposed to be greater thau unity; that is, if the record has been reduced ten times, it is called a scale of ten, instead of one-tenth. The assumed scale is supposed to be so taken that the ratio B differs little from unity, .Let A, B, C, I), etc., be successive.ranges (HW-LW) on staff. Let. A', B', C, D', etc., be successive ranges (HW-LW) on sheet by assumed scale. If the staff and scale readings involved no error, we should have 22-^^--^-^-^-^ -etc KEPOKT FOR 1897 — PART II. APPENDIX NO. 9. 487 But as an error is likely to occur in each comparison between staff and scale, the precision with which B is determined from the above fractions is in each case proportional to the range. That is, the precisions are as A: J5: 0: . . . , or a.s A' : B' : G' : .... If we weight the determinations according to the precisions, we have, as before, B= ^,A'+^,B'+gG'+ . . . B C In other words, A' +B' +0' + . true scale A+B+G + -A>+B'+G'+ (1) assumed scale' .•. True scale = -E x assumed scale. But it is reasonable to suppose that the weights given to different determinations ought to be proportional to some power of the precision greater than unity; that is, to J.'", 5'", (7'" .... , where «>!. From the law of accidental errors, it is seen that n = 2;* A B therefore, giving to -jj, -_^, etc., the weights A'^, B'^, etc., we have B = AA' + BB' + GG' + A'^ + B'^ + G'-' + . . . A'''+ B'^+ C"^+ . . . ■ The reading on staff when the pencil crosses the base line of the sheet is y = ~^ [staff] - B [scale] I where v is the number of staff or scale readings and the square brackets denote their sums. Form for computing the true or worlcing scale of a marigraph. Date. staff. Scale. Staff range X scale range. H W I,W Range. H W I,W Range. (Range)2. 1896. Dec. I 2 3 Sum Feet. 5-0 5-8 5-4 Feet. I '9 2-8 1-8 Feet. 3'i 3-0 3-6 Feet. 5-6 6-5 6i Feet. 2-4 3-5 2-4 Feet. 3'2 3-0 37 Feet. 10-24 9'00 I3'69 9-92 9'oo i3'32 i6-2 6-5 6-5 97 i8-2 8-3 8-3 9-9 32-93 32-24 227 26-5 7? _ 32-24 0.070 True scale = B x assumed scale = is! x 10 = 9'79 y=W 22-7 - 0-979 X 26-5 \ = - 0-53 feet. (2) (3) •Merriman, A Text-Book on the Method of Least Squares, p. 41. 488 UNITED STATES COAST AND GEODETIC SURVEY. That is, if we construct a true scale we must then raise the base line of the sheet by an amount representing 0-53 feet according to true scale, in order that all readings made thereafter may be reduced to staff. It is to be noted that B should remain the same as long as the same gauge is employed; but the value of y must change and so necessitate a new determination whenever the relation between scale and staff zeros is altered. The number of observations necessary to make a good determination of the true or working scale and position of the datum line depends upon the value of this scale or ratio of reduction, as well as upon the smoothness of the water, the cross-section of the float used, and the care taken in indicating the exact place on the tide curve which corresponds to the time of reading the staff; as a general rule, however, about thirty high-water and as many low-water comparative readings will suffice. The time and height of these staff readings should be recorded on the marigram, and afterwards copied into a separate register similar to the above form for computa- tion. The scale readings are made by placing the assumed scale so that it corresponds to the arbitrary datum line upon the marigram, and noting where the curve crosses the edge of the scale, care being taken that the place of crossing is exactly at the point marked as being the position of the recording pencil at the instant of making the staff reading, and that the scale is perpen- dicular to the datum line. Having thus found the true scale of the record, it is well to have a paper scale constructed for making subsequent readings of the curve. The arbitrary datum line, or the pencil upon the gauge which traces it, should be changed by the value of y, so that the scale readings may agree with the staff readings as nearly as possible. During the progress of the series it frequently happens that the relation between this datum line and the staff is altered by some adjustment of the marigraph, and hence the value of a new y must be computed whenever there is the least suspicion that any alteration has taken place in the relation between scale and staff readings. If the observer makes any change in the relation he should record the time and amount of such change upon the marigram. It should be made only after he has carefully determined the relation between the datum line and the zero of the tide staff. 11. Additional directions to the ohserver. In most localities the time may be obtained from railroad or telegraph stations and carried to the gauge by means of a well-regulated watch which has been compared with this standard not many, hours before. The kind of time used is unimportant so long as it is defined upon the marigram and maintained for a considerable period, but it is troublesome to have one kind of time at the beginning and another kind at the end of a series. If for any reason it is found desirable to cliange the kind of time used, it should be done at some convenient epoch, such as the begin- ning of a calendar year or the beginning of some month, and ample notes should be made upon the record calling attention to such change. When distant from telegraph stations, a carefully constructed sundial will be of use in setting the watch, for a good sundial will give the time of apparent noon within one minute, which is Bufllciently close for any marigraph. Sun time is converted into standard time by adding (Part HI, §27) L— 8+ equation of time, Table 30. (4) The gauge should be visited once or twice every day, especially at about the times of those high and low waters which occur during daylight, so as to make staff readings, time comparisons, and to attend to the various details necessary to keep the gauge running. In cold weather there is often much annoyance and loss of record caused by the water freezing. At permanent stations a system of pipes passing through the float box and carrying hot water is sometimes provided, thus imparting warmth enough to the confined water to prevent the formation of ice around the float. In some cases a few gallons of kerosene oil poured into the float box will prevent freezing, but in this case there is a change in the line of flotation which must be allowed for in tabulating the record. Moreover, unless the float box has been constructed with great care to secure tight joints, the oil is sure to soon leak out. If nothing better can be done when the gauge is frozen up, let the observer secure as many readings of the staff as may be practicable, particularly at the REPORT FOR 1897 — PART II. APPENDIX NO. 9. 489 exact hours, for even one or two readings a day will be better than nothing for interpolating the break in the record preparatory to analysis. 12. Preparation of the marigram for reduction. In those forms of gauge without a time-marking attachment it is diiBcult to secure correct time throughout the diflferent hours of each day, because there is generally some eccentricity in the connection of the driving clock with the axis of the gauge cylinder, thus causing the paper to move irregularly, while the clock may keep correct time. If such a gauge has its axle marked so that whenever connection is made with the driving clock a given hour will always correspond to a fixed portion of the surface of the roller, a scale may be made which will represent quite closely the true length of each hour, no matter how unequal the spaces occupied by them may be. Such scale should be copied upon the marigram.* If the cylinder is driven at a sensibly uniform rate it is often convenient to subdivide the space between the time notes made by the observer into equal hour spaces. This may be done by iirst making a paper scale with uniform hour spaces a little longer than the average hour of the marigram, the scale being long enough to reach between successive time notes; and then placing it upon the tide curve, so moving and slanting the scale as to make it exactly agree with two ver- tical lines drawn through that part of the curve referred to by each of two consecutive time notes. The hours of the time scale, while held in tbis position, are transferred to the marigram by succes- sive dots, through which vertical lines may be drawn, intersecting the curve at each hour, and subsequently numbered to agree with the notes. With gauges having steel pins for marking the hours, the perforations in the paper may be used as hour marks, provided the gauge clock is kept always correct and care is taken to start the record exactly on a line joining two steel pins on opposite ends of the roller. As it is practically impossible to maintain correct time without disarranging the relation of the punctures to the clock, and since there is likely to be some eccentricity in the connection between the driving clock and the roller, it will generally be more exact to use one of the methods just described for ascer- taining the hours. When the gauge has a time marking attachment, it is only necessary to number the marks made by the mechanism. Sigh and low waters. — It is customary to tabulate the time and height of the high and low waters, but there is often considerable difBculty in fixing ujion the proper part of the curve in making these readings. The aim should usually be to select the highest and lowest points of what appears to be the true tidal curve. Some persons select the highest and lowest portions of the curve, regardless of accidental disturbances, as well as of peculiarly shaped high or low waters, but even this latter is likely to introduce considerable irregularity in the times, because the tide curve is generally not symmetrical about its extreme points, and these points are liable to swing back and forth during a lunation. In such cases the usage has sometimes been to imagine a small por- tion of the curv^near high or low waters cut off by a horizontal chord, and to take the i^oint where the perpendicular from the middle of the chord cuts the curve as the point of high or low water. It is convenient to have a small scale, equal to one hour, subdivided into six parts, so that the number of minutes beyond any hour mark may be easily estimated. The height is read by so placing the scale as to make it agree with the datum line, holding it perpendicular to this line, and noting where the curve at the point selected crosses the edge of the scale. Hourly heights or ordinates. — It is very important that the height of the sea at each exact hour should be ascertained and recorded. As the marigram has been already subdivided into hours, it is only necessary to see that the scale agrees with the datum line, is perpendicular to that line, and intersects the curve exactly at the hour mark. A form for tabulating hourly heights is given in § 61. A correct datum line upon the marigram, and the relation between staff and scale can be found from the staff readings by aid of § 10. Instead of actually ruling in a new datum line, it is well to mark upon the scale the value by which the assumed datum has been found to differ from the one corresponding to staff; by placing this mark of the scale upon the datum line, the scale readings will then approximately agree with the staff readings. *Cf. Phil. Trans. 1838, p. 250. CHAPTEIE II. ASTRONOMY; TIDAL COMPONENTS SUGGESTED; ETC. 13. Mean motions. In the harmonic treatment of tides it is important to know the mean sidereal motions of the moon, sun, equinox, lunar perigee, solar perigee, and the moon's node. Upon these depend the periods of the tidal components and tidal inequalities. The values given below are taken from various authorities, as indicated in the right-hand margin. Mean sidereal motion. Per Julian year (epoch, Jan. 0, 1900). Moon 4812° -6649577 + o°-oooo462 f £922\ /^ i — 1900 >, \ 100 y i9oo\ Sun 359'99373ii — O'OOOCXX)! Equinox — o'oi3958i — 0-0000062 (50" -2493) (o'''-0222) Lunar perigee 40'6763487 ■ — 0-0002070 ('■ 100 J — i900\ 100 J Solar perigee 0-0032336 + 0-0000029 ( — ^^ ) (ii'''-64io) (o'''-oio4) Moon's node — 19'3553827+ 0-0000393 f - - igoox 100 J Hansen, Tables de la Lune, pp. 15, 16, -with. Newcomb's correction, Researches on the Motion of the Moon, p. 268; or Harkness, Solar Parallax, p. 14, equation (33), omitting term of second power. Newcomb, Tables of the Sun, p. 9. There denoted by n. Newcomb, Tables of the Sun, p. 9, ('re — ^Y Hansen, Tables de la Lune, pp. 15, 16. Newcomb, Tables of the Sun, p. 9, ('n — '^\ Hansen, Tables de la Lune, pp. 15, 16, increased by o"-io, Newcomb's correction. Researches on the Motion of the Moon, p. 274. [Cf. Harkness, Solar Parallax, pp. 16, 17, 140.] Mean sidereal motion. Per mean solar day. Temporary symbol. Numerical value (1900). Moon D* 13-176358543 Sun ©* 0-985609120 Equinox Tx — 0-000038215 Lunar perigee <"» 0- 1 1 1365773 Solar perigee ** 0-000008853 Moon'! 3 node S3* — 0-052992149 Earth'. s meridian* ®* 360-985609120 Mean motion relative to the equinox, i. e., mean motion in longitude. Per mean solar day. Per mean solar hour. Formula. Numerical value {1900). Numerical value. Symbol. Moon ])*-T*=3)T 13-176396758 0-5490165316 d Sun ©*-T» = ©T 0-985647335 0-0410686390 V Equinox T*-T* = o'o 0-0 Lunar perigee 0*— T»=raT 0-111403988 0-0046418328 m Solar perigee 5r* — T*=?f T 0-000047068 0-0000019612 Moon's node «*-T* = fiT - 0-052953934 — 0-0022064139 Earth's meridian ©*-T* = ©T 360-985647335 15-0410686390 y ' I. e., its motion upon the celestial sphere, as seen from the earth's center, 360°=® , — ©,. 490 REPORT FOR 1897 — PART II. APPENDIX NO. 9. 491 Astronomical periods obtained from mean motions. — If we divide 360° by one of the above sidereal motions, or by a combination of them, the length of some mean astronomical period will be obtained. The following list inclades the more important of such periods: Mean astronomical periods. Formula. 360°- ©* [D»-©*] or Do ^[D*-Tx] or Jcp ^[©*-T*] or ©cip [D*— raj or Jct [O*-**] or©;r ^[3)*-«J*] or 5n [0*-fi*] or ©Q — [©*— «*] or ©w -[])*-©*-(0*-ra*)] or [50- ffi* [e«-T*] or ©cp [©»-]) J or®;,' [©*-©*] or ©0 ■ -^[S3*— T»] or S3t Name. Sidereal month Sidereal year Synodical month Tropical month Tropical year Anomalistic month Anomalistic year Nodical month Eclipse year Evectional period in moon's parallax •©(«>] Moon's evectional period Sidereal day Tropical day* Lunar day Solar day Revolution of equinox Revolution of lunar perigee or line of apsides Revolution of solar perigee or line of apsides Revolution of moon's node Node-equinox period Numerical value (1900), d 273216609 365'25636o5 29'530588i , 27-3215816 365-2421989 27'5545503 365-2596413 27-2122191 346-6200271 411-7846609 31-8119389 0-9972696723 0-9972695663 i'03505oioi2 10 9420384-666 3232-591040 40664181-63 6793 '45916 6798-36171 14. Principle of forced oscillations. This principle, due to Laplace,t is of fundamental importance in the analysis and prediction of tides; it may be stated thus: The state of any system of bodies in which the primitive conditions of the motion have disappeared through the resistances which the motion encounters, is coperiodic with the forces acting on the system. If there were but a single strictly periodic force acting upon the given system, the effects of successive periodic actions must eventually become identical, and their periods become that of the force. Now the magnitude of any tide-producing force being very small in comparison with the force of terrestrial gravity, the accelerations imparted to the fluid particles must be very small, and so must be the resulting displacements. Therefore if several such forces act simultaneously, they act as if totally independent of one another and so their effects permit of superposition. This being so, the disturbance may be regarded as made up of terms whose periods are the periods of the several forces. Here is the clue to what periodic terms ought to be found in the tidal wave; for, there ought to be an oscillation corresponding to each term of the causes producing the tide. Such terms follow from the development of the tide-producing ijotentials of the moon and sun. Before l)roceeding to this development, it may be well to consider what periodic terms are suggested Croin a superficial view of the nature of the tide-producing causes. WHAT COMPONENTS, OR PERIODIC OSCILLATIONS, SHOULD EXIST IN THE TIDE. 15. Since the lunar tide is due to the difference between the moon's attraction upon the earth as a whole and the enveloping sea, there ought to be set up an oscillation whose period is a half lunar day; and likewise, because of the sun's attraction, an oscillation whose period is a half solar day. Confining our attention to the case of the moon, it may be observed that the actual lunar day is not of constant length, because the moon's orbit is an ellipse, not a circle; because it is * Generally, but improperly, called "sidereal day." tM^c. C^L, Book IV, J§ 16, 17. 492 UNITED STATES COAST AND GEODETIC SURVEY. inclined to the plane of the earth's equator, and this inclination is not constant; also because the sun disturbs the moon, producing evection and variation. These irregularities iu the moon's apparent diurnal motion will, sooner or later, be in certain ways reflected in the lunar tide, which but for them might be assumed to be a wave of uniform period for all places, and of constant amplitude at any given place. Denoting the strictly periodic portion of the tide by M^, let us inquire, what other strictly periodic oscillations or components of about the same period ought to exist in the lunar tide? Let ma denote the hourly speed of M2, and n the period, in hours, of some inequality or irregu- larity in the moon's motion. If a component have the speed -.±f (5) it will gain or lose on M2 one period during the time IT, as can be seen by multiplying this speed by 77.* In other words, JT is a synodic period for Ma and a component with either of the above speeds. The nature of the Inequality must be taken into account in order to ascertain which sign to take, and also whether, for the present purpose, 7T should include the whole or the half period as usually given. The development of the tide-producing potential shows that the two components, one for each sign, are sometimes required. If /I=an anomalistic month, then "^ + f =28.9841042+ggj||0_ ^ 29-5284789, m.-§^0^28.9841042-^g3^||^ = 28.4397295.^ This suggests that there should exist because of the irregularity in the moon's motion due to its varying parallax, one or both of these components, which may be denoted by L2 and Nj. Since the moon's apparent diurnal motion is slowest when she is in perigree, the perigean tides should have a longer period than the mean tide; and because of the nearness of the moon to the earth, the range should be increased. The most of the parallax inequality in the tide must be due to that component which, when coinciding with M2, increases the length of the period; in other words, N2 is the larger lunar elliptic component, and hi the smaller lunar elliptic component. If 77= a half tropical montb, then m2 + f = 28-9841042 +^^^^-^^ = 30-0821373, m2 - ^-^= 28-9841042 -^.^^^^^ = 27-8860711. Since the moon's apparent diurnal motion is, cceteris paribus, greatest when she is in the equator, and since her tendency to produce tides is then greatest (Principia, Bk. Ill, Prop. XXIV), the speed of the component causing the declinational inequality in the semidiurnal tide is greater than the speed of M2. This component is the lunar part of K2. The other component is not required in connection with this inequality. If 77 = a half synodic month, i. e., the moon's variational period, then m, + ^ = 28-9841042 + gg^lg^Qg^ = 30-0000000, m2 - 3f = 28-9841042 - gg^M_ = 27-9682084. » Cf. Laplace, M^c. 061., Bk. XIII, § 3. EEPOET FOR 1897 — PART II. APPEWDIX NO. 9. 493 Cceteris paribus, the apparent diurnal motion of the moon is least when she is in the syzygies, and lier distance to the earth is then least. (Principia, Bk. I, Prop. LXVI, or § 87, Part I.) This shows that the lunar tide is then greater than usual and of longer period; consequently the com- ponent causing the variational inequality in the semidiurnal tide is the one whose speed is less than that of M2. It is denoted by p2. If the other component, whose speed is 30° per hour, exist at all it will unite with S2. Let n = half of the evectional period in the moon's parallax. Every time the line of apsides passes the sun, the eccentricity of the lunar orbit becomes a maximum (Principia, Bk. I, Prop. LXVI). That is, the amplitude of the oscillation (F2) which mainly accounts for the parallax effect upon the tide, would, if no additional component were introduced, have an inequality of a period of about 206 days. A component which would probably represent such an inequality must have for its speed one of the two values. n2 + ?g = 28-4397295 + ,^^^^^^^^^3^ = 28-5125831, n2 - ^ = 28-4397295 - ^^^^^ = 28-3668759. (6) At such times of greatest eccentricity the progression of the line of apsides becomes a maximum. This increases the anomalistic month, and so, by the above formula for the speed of N^, must increase the speed of the oscillation representing the most of the parallax efl'ect at the time when its amplitude becomes a maximum. Consequently, the principal component due to the moon's evection should have the speed 28-5125831. This component is designated as v^. 16, Whenever the moon is not upon the equator, the two tides of a day will generally differ because the moon's north polar distance at the time of a superior transit is not equal to her south polar distance at the time of an inferior transit. In other words, if we suppose the moon and anti- moon to successively cross the meridian of a place, the one will be a body of north declination and the other a body of south declination. It would be natural to try to represent this inequality by a wave of variable amplitude attaining a maximum when the moon is far from the equator and vanishing at about the time when the moon crosses the equator. The speed of such a wave should be ffli , or the apparent diurnal motion of the moon about the earth. When the amplitude becomes zero the phase of the wave should suddenly change by 180°, To avoid this great variability in amplitude and this sudden change of phase, a component, lunar Ki , is suggested which shall gain 360° on the moon in a tropical month, and another component, Oi, of about equal amplitude, which shall lose the same amount ; for, the speed of the resultant wave will evidently be that of the moon or mi , its amplitude will be a maximum when the moon is far from the equator, either north or south, and zero when she is upon or near the equator. If 71 = a tropical month, then m. + f = 14-4920521 + ggg^^j^ = 15-0410686 = k„ m, - ?g = 14-4920521 - gg^^ = 13-9430356 = o. (7) The same line of reasoning would lead one to infer that all lunar components might be accompanied by small components* of almost the same speeds as themselves for taking into account the effects of the regression of the moon's node. For most purposes it is preferable to suppose this inequality accounted for by means of slight variations in the amplitude and epochs of the components which do not owe their origin to the movement of the node. * Styled lunar nodal components. See Ferrel's Tidal Researches, p. 43; also United States Coast apd Geodetic Survey Report, 1878, pp. 270 et seq. For example, let 77= the node-equinox period ; then m^ — r°^ = 28-9818978 ^ the speed of Ferrel's M' (i, 2). Again, ki -f ^ = 15'0432750 = the speed of Ferrel's M' (3, ,), and oi — ^ = 13'9408292 = the speed of Ferrel's M' (6, ij. 494 UNITED STATES COAST AND GEODETIC SURVEY. Considering now the portion of the tide due to the sun, and placing ZZ" = an anomalistic year, then s. + ^-§ = 30-0000000 + g^^ = m-ommi, (8) s. - f = 30-0000000 - g^|L_ = 29-9589333, and we obtain two speeds which are denoted by V2 and ts, respectively. By placing 17= a half tropical year, the first speed becomes equivalent to ka. By placing il = a tropical year, then s, + ?g = 15-0000000 + g^s^^ = 15-0410686, (9) s, - ^0 = 15-0000000 - s^,^ = 14-9589314, and we obtain two speeds of which the first is ki and the second is denoted by p,. There are other oscillations having a truly astronomical origin, but their speeds, as a rule, are less readily obtained from the mean motions of the moon and sun than from the speeds of certain components like Oi, Ki, etc. (See Table 2, already referred to.) 17. Overtides. So far we have been maiuly concerned with the periods of the oscillations. That they aie simply harmonic is not self evident. In fact, where the water is shallow the crest of the tidal wave must generally move faster than the trough. This, as the wave proceeds, causes the duration of fall to exceed the duration of rise. If upon a simple harmonic oscillation we super- pose in a suitable manner a small one of double the speed, we obtain the effect desired. We are therefore led to infer that there probably exist along with M2, S2, N2, eti;., the components M4, S4, N4,etc. Again, it seems natural to suppose that, for instance, the principal lunar part of the tide may have other departures than the kind jast noted from a simple harmonic form. But whatever its shape may be, it can be represented by a Poarii-r series of terms whose speeds are simple multiples, like 2, 3, 4, etc., of the speed of the principal component. That is, M2 may naturally enough in shallow water be accompanied by M4, Mg, Mg, etc. So for S2, and other components. These are sometimes called overtides because of their analogy to overtones in musical sounds. 18. Compound tides. In shallow water there maybe sensible compound tides; that is, components whose speeds are the sums or differences of the speeds of the principal components. These were suggested by Helmholtz's theory of compound sounds. In fact, we have only to multiply together, in pairs, the principal simple harmonic (cosine) terms which constitute the tide in order to ascertain the theoretical relations between their amplitudes, arguments, and epochs. Ferrel gives quite extended lists of such components in the Report of the United States Coast and Geodetic Survey for 1878, pp. 274-276 (about equivalent to Table 36), and Darwin, a list of the more important cases in the British Association Eeport for 1883, pp. 74-78. Tiie velocity of propagation for a high water of, say, the component M2 is Vg(h + 3 M2) = {9h)i + f ^ . (10) and of the trough Vg(h-3M2) = {gh)i-i^^, (11) g being the acceleration of gravity and h the depth at half-tide level. The difierence between these two velocities is proportional to a: REPORT FOR 1897 PART II. APPENDIX NO. 9. 495 At a given place (since g and/i are constant) the time distortion is evidently proportional to Mz; for, to a first approximation, all components, whatever their amplitudes, require about the same time in their propagation from point to point — the velocity being {gh)^. For convenience, suppose M2 to be a somewhat varying amplitude, that is, let M2 not stand for the true amplitude of M2, but the resultant amplitude when combined with another component of about equal speed. Let its value on two different occasions be denoted by (M2),, (M2),,. The question now arises, how do the corresponding amplitudes of M4 compare with each other? In the first case the duration of fall exceeds one-fourth lunar day by an amount proportional to (M2),, and in the second, to (M2),,, according to the equations just obtained But it can be readily seen by combining waves [eqs. (60), (61), Part III] that this excess in the one case is proportional to ,^^v- and in the other to (M4),, (M2),, ■ (M4), . (M,),, • • (M2), • (M2)„-'-^^)' • (^^)" or (M,), :(M4)„=(M2);:(M,V; (13) that is, if the amplitude of a component be different on two occasions, the amplitude of the overtide having double the speed of the fundamental, will vary as the square of the amplitude of the fundamental. (Applying this to the diurnal and semidiurnal waves, one might perhaps surmise that the amplitude of the diurnal wave, and so the diurnal inequalities, vary, from place to place, as the square root of the semidiurnal range.) Suppose that the cause of the variation in M2 is the addition of S2. producing spring tides. At the times of the spring tides, what is the amplitude x of the overtide M4? M4 : x=M/ : (M2+S2)^ At the neap tides we have '=^^ (l-^+l?)- (15) Thus we see that if the quarter diurnal component is to have an approximately fixed position upon the semidiurnal wave it must have a variable amplitude. But, just as in § 15, we infer the speeds of one or two additional quarter diurnals by putting 11 = a, half synodic month expressed in hours. m4+^= 57-9682084+ 3g^.3^^Qg^ = 58-9841044, (16) m4— ^ =57-9682084- 3g^.gg^Qg^ = 66-9523124. The component defined by the first speed (=m24-S2) is MS or (MS)4. The values of a; just written show that the amplitude is given by the relation MS=^xM4. (17) The speed of MS being the arithmetical mean between the speeds of M4 and S4, the period of MS is, of course, the harmonic mean between the periods of M4 and 84. By putting JT = one-fourth of a synodic month, the speed defining the component S4 is qj)tained. The values of x show that S4 - 54 X M4, (18) which might have been inferred from (13) by writing 82 for one of the Mz's. 496 UNITED STATES COAST AND GEODETIC SURVEY. Similarly, tnere should be compound tides dependent upon Mj, N2; M2, Kg; etc. So for S^, N2; etc. The speed of MS being the speed of M2 plus the speed of S2, its phase must of course vary as the sum of the phases of M2 and S2 varies, and so it is customary to take its initial argument equal to the sum of the initial equilibrium arguments of M2 and S2. At spring tides MS should conspire with M4, and at neap tides it should interfere. The speed of M4 is equal to twice the speed of 512, and it is customary to take for its initial argument twice the initial equilibrium argument of M2. .-. ms = m2 + S2; (19) argo MS = argo M2 + argo S2 ; (20) phase MS ( = phase M4, at springs) = phase M4 + phase S2 — phase M2 ; (21) phase MS (= phase M4 ± 180°, at neaps) = phase M4 + phase S^ — phase M2; (22) ms* + argu MS - M.S° = m,t + arg„ M4 - M4O + s^t + arg, S2 - 82° - (m2< + argo M2 - M^o) • (23) and so MS° = M4O + S2O - M2°. (24) The more general treatment of this subject is given in § 48. 19. Meteorological tides. The land and sea breezes, and the daily variation in atmospheric pressure, may give rise to a tide whose period is a solar day; and, as the cause is not directly astronomical, it is natural to suppose that overtides would have to accompany the simple harmonic form in order to represent a tide whose origin is so remote. The change of seasons gives rise to an annual tide, Sa. Such a tide must represent the stages of rivers at river stations. As very high stages areusually of comparatively short duration, it is not reasonable to suppose that a single component can represent the annual changes in river level. In other words, the overtides become comparatively large. It is hardly necessary to add that while the determination of meteorological tides from long series of observations is valuable for some purposes, their recurrence is not generally certain enough to make them of much value in tidal predictions. The foregoing has led to, or at least suggested, the most of the components which are to be sought in the process of analyzing the tidal wave. Some of those already brought to notice are, from the nature of their origin, too small to be included in a working schedule. It may be worth while to here call attention to a practical application of § 18, or rather to empiral rules there hinted at. 20. Rules for inferring certain nonharmonic quantities from values at a neighboring station. Suppose that for a secondary station we know the value of a semidiurnal range of tide, say Mn, and wish to estimate the value of another semidiurnal range, say Sg, or Np from a neighbor- ing principal station where the tide is supposed to be fully known. We naturally put (Sg)„ = (Sg), (^"^^'S .-. (Mn)„ = (Mn), i|g'; (25) (Np)„ = (Np),W^'. (26) Similarly for perigean and apogean ranges. But § 18 suggests the hypothesis which observations have in a measure corroborated, that the amplitude of the diurnal wave and all quantities pro- portional thereto vary between neighboring stations not as the ratio of the meau ranges of tide varies, but rather as the square root of this ratio. Consequently, putting temporarily k for .fiz^Rhj V (Mn), we have* (D,)„ = (D,),/^, (27) (HWQ)„ = (HWQ),A;, (28) (LWQ)„ = {LWQ),1c, (29) (depression of mean LLW below MSL)„ = J (Mn)„ + [(ditto), - i (Mn),]fc, (30) * For notation see 5$ 3-7, Part I. REPOET FOR 1897 — PART II. APPENDIX NO. 9. 497 (depression of Indian or harmonic tide, plane below MSL)„ = 0-49 {8g)„ + [(ditto), - 0-49 (Sg),] 7c, (31) = 0-49 [(Sg)„ + (Di)„J, approximately. (32) Here "ditto" refers to the left-hand member. The Indian tide plane is M2 + S2 + K, + Oj below mean sea level (MSL). (33) (depression of tropic LLW below MSL)„ = [(ditto),-J(Mn),]l^^^ (34) (Gc)„ = (Mn)„ + [(GO, - (Mn),l (E^^L^j. = (Mn),, + [(Gc), - (Mn),] 1c, (35) (tropic HHWI),, = (HWI)„ + [(tropic HHWI), - (HWI),] 1ME)/_ . (P')// ^ (36) (Mn)„ (Di), (tropic LLWI),, = (L WI), + [(tropic LL WI), - (LWI),] ^^°)/ . (^0// . (37) (Mn)„ (Di), When (Di)„ is known from observation, the value of Ic from (27) should be used. 21. Mean longitude. The mean longitude of a body may be defined as the distance along any given great circle moved over by a uniformly moving or "mean" body from the intersection of its orbit aiid the circle. It is simply the time since the body passed such intersection multiplied by the aver- age angular velocity of the body. For a fixed body or point the mean, as well as the true, longi- tude is measured from the foot of the perpendic- ular let fall from the body or point in question upon the circle in which longitude is reckoned. If two or more great circles meet in a point, each circle will have a mean body of its own; but all these mean bodies will be at the same distance, at any given instant, from the origin; that is, the real body will have the same mean longitude reckoned in whichever circle. If origins in the several circles be taken a constant distance from the common intersection, the mean longitude in each circle will be altered by the same amount. The mean longitude of the sun {h) from T is the same in the equator as in the ecliptic. If v denote the E. A. or mean longitude of I in the equator (see Fig. 6), then h — vis, the sun's mean longitude from the point I. It we take a point T ' upon the moon's orbit such that Q T ' = Q T , and denote the distance T ' J by 5', then the moon's mean longitude from I, whether in the orbit or the equator, i& s — S, s being her mean longitude from T ' or T in the orbit or the equator. Pig. 6 supposes the observer to be within the celestial sphere looking outward. The ecliptic, moon's orbit, and the celestial equator are supposed to be fixed, while the projection of the terres- trial meridian upon the celestial sphere moves eastward, as indicated by the arrows, and at the rate of 15° per sidereal hour. 8' is a slowly moving point upon the equator, distant 180° from the mean sun (S), and so T S'= h ± 180°. When the meridian passes over the point 8' it is mean local midnight, where we shall assume t, the mean solar time, to be zero. W^hen the meridian is at X, then t, or 15 t, = 8'x. The hour angle of T may be temporarily denoted by g. M denotes the 6684 32 Fig. 6. TJ= r, r'i=s, rs = h. T'M=s, rS' = hA: 180°, r'P=p, rx = g = t + h±180o, T'J2 =N, rsi^w, S'x = t, 498 UNITED STATES COAST AND GEODETIC SURVEY. position of the mean moon, and P that of the lunar perigee. The following equations are given for convenience of reference; t is the time expressed in degrees instead of hours: Ix = x = t+h± 180O - V, IM=3, = s — S, IP = rSi=p — S. (38) Most tables of the moon and sun give mean longitude of the lunar and solar elements reckoned in the orbit or the ecliptic and from the origin T . To refer any of them to the equator, using I as origin, it is necessary to know the mean longitude of the moon's node (N); then v and 5 can be obtained by solving the spherical triangle T Ja. To avoid this labor the values of v and S for any value of N may be taken from Table 7. In all of the work pertaining to the reduction and prediction of tides, we shall adopt the fol- lowing (Hansen's) values for s, p, h, pi, and N, which are those used by Darwin in his report for 1883: s = 150O-0419 + [13 X 360° + 132o-67900]T+ 130-1764Z) + 0o-5490165fl, 2) == 240O-6322 + 40O-69035T -f 0o-1114i) + 0o-0046418ff, h = 280O-5287 + 360O-00769T + Oo-9856i) + 00-04106865, (39) Pi = 280O-8748 + 0O-01711T+ 0o-000047i), W = 2850-9569 - 190-34146T - 0°-052954D. where T is the number of Julian years of 365| mean solar days each ; I) the number of mean solar days; H the number of mean solar hours after Greenwich mean noon January 1, 1880. On account of the slowness of the secular changes in the coefScients of T, J), or H the epoch of this table may, for tidal purposes, be regarded as 1900. See Hansen's Tables de la Lune, p. 15, from which these formulae may be obtained by putting t—80. Kewcomb's corjections are not of sufiScient magnitude tt seriously alter these values. Affecting the symbols employed by Hansen with a zero subscript, to indicate that Newcomb's corrections have been applied, we have* e =360°— ©„ + cos JV^ ' ^^''^ tan S = i cot co sin ^ — J *^ sin 2 JV — -^2 , (47) tan V = i cosec go sin JT — i i^ sin 2 N ^y-i — » (4:8) ^ sm^ co' ^ ' cos 7 = (1 — J i^) cos CO — i sin w cos ^. (49) Table 7, taken from Baird's Manual, was computed upon the assumption that oo=2^° 2T'3 and 1=5° 8'-8. 23. Kepler^ s problem. By analytical geometry the vectorial angle (»),* reckoned from the perigee, is connected with the eccentric angle (E) by the equation cos,= cosJ;-6 5 1— e cos -E' ^ ' e being the eccentricity of the orbit. By Kepler's second law equal areas are described in equal times, and so the mean {M) and eccentric anomalies are connected by the equation M=E—e sin E. (51) The elimination of E from these two equations constitutes the solution of Kepler's problem. For a complete solution see books on mathematical astronomy, such as Dziobek's Mathematical Theories of Planetary Motions, E^sal's M^canique Celeste, etc. For most tidal purposes it is sufficient to carry the solution to the second powers of e, and this may be easily accomplished in the following manner: If we develop by Taylor's theorem the value of v from the first equation written in the form 'i)=cos-i (cos EJrh) (52) where h= _ e^in^ (53) 1— ecos-B ^ ' we obtain ■»=^+esin.B+4e2 sin2^+ .... (54) The second equation gives M=E—e sin {M+e sin M), or .27=M+esin (M+esin Jf); (55) .-. E=M+e sin M+i e^ sin 2 M, to the second power of e, (56) e sin E=e sin M+^ ^ sin 2 M, to the second power of e, (57) * The present use of the letters v, E, M, and h will probably be confined to this paragraph. 500 UNITED STATES COAST AND GEODETIC SURVEY. :| e^ sin 2 11=1 ^ sin 2 M, to the second power of e; (58) .w = Jf+2 e sin ilf+f e^ sin 2 ilf, to the second power of e. (59) 24. Reduction to the equator, and conversely. Let A denote the inclination of an orbit to the equator, and B a right angle formed by the equator and the hour circle passing through the body. By spherical trigonometry we have tan b = -^^5^ = tan e (1 - sin^ J.)""*, (60) = tan c (1 + J sin^ A + f sin" A+ . . . ); (61) . • . & = tan-^ (tan c + Tc), say. (62) By Taylor's theorem & = c + Jsin 2c sin^ A + (^sin2c+ gVsinie) sin" J. -t^ .... (63) The usual formula for this is, unless the exact solution by trigonometry be preferred, b = c + tatf ^A sin 2c — J tan" ^A sin 4c + ^ tan" ^A sinGc - . . . ; (64) but for certain purposes it is convenient to expand in powers of sin A. Also, tan c = tan h cos A = tan Z> (1 — sin^ J.)*> (65) = tan6(l — Jsin^ J- — Jsin"A); (66) . ■ . c = tan-' (tan b + W), say, (67) = b — lsm2b sin^ A — [^ sin 2b — -^ sin 46) sin" A. (68) 25. Approximate expressions for the right ascension of the sun or moon. The object of this paragraph is to show that if we displace in time the strictly periodic portion of the equilibrium lunar (solar) tide by the lunar (solar) components, the resulting wave will have its crest beneath the tidal body or 180° therefrom. The true longitude of the sun is, by Kepler's problem, § 23, l=h-\-2ei sin {h — pi) + |e/ sin 2 (h —pi), (69) where h denotes the mean longitude of the sun, pi that of the solar perigee, and Ci the eccentricity of the earth's orbit = 0-01679. The coefficients 2e], f Ci^, are converted into degrees by means of the factor 57-3, thus giving l = h + 10-924 sin (h -pi) + Oo-020 sin 2 {h -p{). (70) The right ascension corresponding to I is, § 24, a = 1 — \ sin'^ w sin 2 I — sin" oo (J- sin 2 1 — -^ sin 4 I), (71) = h+ 10-924 sin {h —pi) + 0o-020 sin 2 (h -p^) - 20-450 sin 2h + 0O-045 sin 4 h, (72) GO, the obliquity of the ecliptic being taken as 23° 27''3. l^fow since all solar tides are due to the sun, the crest of the solar wave composed of these partial tides ought, in the case of semidiurnals, to have its phase equal to twice the hour angle of the sun. Sun's right ascension x 2 = 2 h — acceleration in S2 due to Tj - " " S2 " " Ka - " " S2 " " solar K 2, Table 15. (73) EEPORT FOR. 1897 — PAET II. APPENDIX NO. 9. 501 This is approximately equal to 2 A - ^ ^ sin (arg T^ - arg S2, =Pi - h) 02 S2 _ Jj b sin (arg R^ - arg S„=h-p, + 180°) 02 S2 _'^h. sin (arg K2 - arg S^, = 2 h). (74) 02 S2 Here "arg" stands for the equilibrium argument, V + u, of Table 1; it is equivalent to the phase of the oscillations, if its epoch (or lag) be zero; Kj here denotes the solar part of K2. Putting in the theoretical values of these component ratios, we have for twice the sun's right ascension * 2 1i- 30-32 sin {pi -h) + 00-47 sin {h -pi) - 4o-95 sin 2 h; or 2h + 30-79 sin {h-pi) - 4o-95 sin 2 h, (75) the half of which is h + 10-90 sin {h -pi) - 2o-48 sin 2 h. (76) This value of the sun's right ascension agrees well with value (72). Thus it is seen that T2, E2, and solar K2 account for irregularities in the solar wave corresponding well with the irregularities in the motion of the sun. The quantity + lo-90 sin {h —Pi) — 2o-48 sin 2 h is, when converted into time, very nearly the equation of time (to change apparent to mean time), Table 30. For, Sun's right ascension = right ascension of mean sun + equation of time. In a precisely similar manner the right ascension of the moon reckoned from I is ff, + 60-292 sin (cr, — cJ,) + 0O-216 sin 2 (o-, — cTj — 140-324 sin^ I sin 2 c, — sin* I (7o-162 sin 2 ff,~ lo-790 sin 4 ff,), (77) (T, denoting the mean longitude of the moon from I, tzI, that of the lunar perigee, the eccentricity of the orbit being 0-05491. A niore accurate value from Hansen's tables is, taking into account the higher powers of e, the evection, variation, and the annual inequality, ff, + 60-290 sin {a, — a,) + 0o-216 sin 2 (ff,-zs,) — 140-324 sin^ I sin 2 c, — sin* I (7o-162 sin 2 o", — lo-790 sin 4 ff,) + 10-241 sin {s — 2h+p) + 00-596 sin 2{s — h) — 00-183 sin (^—i>i). (78) The two terms in sin 2 ff, may be added together by giving to sin I a mean value. This expres- sion increased by v is the moon's right ascension reckoned from' T . By § 21 the above expression may be written S — S+ 60-290 sin (s —p) + O0-2I6 sin 2 (s—p) — 150-458 sin^ I sin 2{s — S) + sin" I [lo-790 sin 4 (s — 5)] + 10-241 sin {s—2h+p) + 0o-596sin(2s — 2 A) — 00-183 sin (h—Pi). (79) * I. e., making use of Table 1 and multiplying by 57-3. 502 UNITED STATES COAST AND GEODETIC SURVEY. The harmonic components give for twice the right ascension of the moon, likewise reckoned from f 2 s — 2 r — 120-50 sin (s—^) — 10-42 sin 2 (i) — s) ^2, La; 2N — 320-47 sin^ J sin 2 (s — £ ) lunar Kj — 2o-54sin(2 7i — s-j?) v^^\^ — lo-33sin2(7i-s). ^, (80) Beckoned from T , the double right ascension of the moon is this expression with the term 2 v omitted. Since v is nearly equal to 5, this agrees well with twice the preceding expression, which shows that the mean lunar tide is so perturbed by additional components as to follow the true moon. Either enables a person to compute the approximate time of the moon's transit by means of Table 3. In a similar manner we might compare ordinary periodic expressions for the moon's parallax or radius vector with the expression for the amplitude of the semidiurnal wave regarded as the Mj wave perturbed by 1^2 > Lz , etc. 26. Astres flctifs, fictitious moons, etc. For aiding the imagination, Laplace* introduced a set of fictitious bodies having certain analogies to the mean sun and mean moon. Such bodies move uniformly in the celestial equator and at a constant distance from the earth. The successive transits of any astre fictif across a given terrestrial meridian defloes a corresponding day, differing in length but little from twenty- four mean solar hours. The Tidal Committee of the British Association made use of such fictitious bodies as the harmonic analysis of tides seemed to demand.t By the introduction of suitable bodies, which were not used by Laplace, the parallaxes of the mean sun and mean moon become constant. The assigned position of a fictitious body has, of course, a direct influence upon the epoch of any component tide determined with reference to the body. The following quotation from a paper by Thomson is found upon page 481 of the British Association Eeport for 1878: s (technically called the epoch) is the angle, reckoned in degrees, which an arm revolving uniformly in the period of the particular tide has to run through till high -water of this constituent, from a certain instant or era of reckoning defined for each constituent as follows : — Definition of E.t — To explain the meaning of the values of e given in the following table of results, it is conven- ient to use Laplace's "astres fictifs," or ideal stars. Let them be as follows: — Mthe "mean moon." /S the "mean sun." K for diurnal tides, a star whose right ascension is 90°. KfoT semi-diurnal tide, the "first point of Aries," or f • a point moving with angular velocity 2 d, and having 270° of right ascension when 3f is in f. Q a point moving with angular velocity 2 6 — a, and 270° before M in right ascension when the longitude of Jlf is half the longitude of the perigee. P a point moving with angular velocity 2 77, having 270° of right ascension when S is in f. N a point moving with angular velocity, | (J — ^ o, and passing alternately through the perigee and apogee of the moon's orbit when M is in perigee. X a point moving with angular velocity, ^6 -\-as, and passing alternately through 90° on either side of the perigee of the moon's orbit when M is in perigee. The value of s in each case above means the number of 360ths of its period which the corresponding tidal constituent has still to execute till its high-water from the instant when the ideal star crosses the meridian of the place. Thus if n denote the periodic speed of the particular tide in degrees per mean solar hour, its time of high- water is -, reckoned in mean solar hours after the transit of the ideal star. n * M^c. C^l., Bk. IV, § § 17, 19. Cf. expression for the tide in Bk. XIII. See under Laplace. t B. A. A. S. Reports, 1868, p. 496; 1876, p. 293; 1878, p. 481. Thomson and Tait, Natural Philosophy (Ed. 1883), § 848. t "This definition for the several cases of K diurnal, and 0, F, Q, and L differs by 90° or 180° or 270° from the definition given in the British Association Eeport (1876) for a reason obvious on inspection of Tables I. and II., pp. 304 and 305 of that report, which (except in respect to the longitude of perigee and perihelion) show e as previously reckoned for the several constituents." REPORT FOR 1897 PART II. APPENDIX NO. 9. 503 It is to be remarked tliat there are two K bodies always 90° apart in the celestial equator. Darwin discards the astres fictifs, and says in his report for 1883 : In tlie present Report tlie method of mathematical treatment diifers considerably from that of Sir William Thomson. In particular, he has followed, and extended to the diurnal tides, Laplace's method of referring each tide to the motion of an asti-e fiutif in the heavens, and he considers that these fictitious satellites are helpful in forming a clear conception of the equilibrium theory of tides. As, however, I have found the fiction rather a hindrance than otherwise, I have ventured to depart from this method, and have connected each tide with an ' argument,' * or an angle increasing uniformly with the time and giving by its hourly increase the 'speed' of the tide. In the method of the astres Jibtifs, the speed is the difference between the earth's angular velocity of rotation and the motion of the fictitious satellite amongst the stars. It is a consequence of the difference in the mode of treatment, and of the fact that the elliptic tides are here developed to a higher degree of approximation, that none of the present Report is quoted from the previous ones. In case of a diurnal component, the argument is evidently the hour angle of the corresponding astrefictif; in case of a semidiurnal, twice the hour angle; etc. Since these bodies do not really exist in nature, but are created for aiding the imagination, it seems justifiable to carry the flctioa one step farther if simplicity can be thereby attained. Imagine a system of bodies to have an apparent and uniform motion around the earth from east to west. Let their periods be equivalent to the periods of the various short-period tidal compo- nents. That is, an M2 moon or body will cross the meridian of a place twice each mean lunar day; an M4 moon, four times; etc. They are to be so placed in the celestial equator that the equilib- rium arguments of the components ( V+ u) are the hour angles of the fictitious bodies. Bach body has the property of producing a ™^p/^,?,^ of ^^^ corresponding component tide a certain number of hours (= epoch expressed in time) after its VQwor ti'^'^^it. (See Pig. 7, Part I.) 27. Sum of the series cos 6^ + cos 2 (9 + cos 3 i^ + . . . + cos n6. (81) If n denote any positive integer and d any angle, then 2 cos e cos nd = cos {n+1) + cos (n — 1) 6 (82) because the second member of this equation is equal to cos nd cos 6 — sin nd sin 6 + cos nd cos 6 + sin nff sin d. Now giving to n the values 1, 2, 3, . . . n, we have n = n w = M n = n 2 cos 6* 2 COS nd= 2 cos(m + 1)^H- 2 cos{n — l)d. (83) n=l n=l n=l n = n Calling 2 cos nff, S, we obviously have w=l 2 /S cos ^ = 2 aS -f 1 - cos (9 -f cos (w + 1) ^ - cos nd; (84) o_ -1 , cos w^ — cos(« + 1) ^ „_, •■*-"*+ 2 (1 - cos d) ' ^^^^ sin(2w+l)| sin(n-fl)| = -i + i ^ = -1 + ^cos«^. (86) sin I sin I 2 Special case nff = 360°. Either value of /S just obtained reduces to zero. 28. iVbte on the determination of empirical constants. Let there be three empirical constants {x, y, z) t to be determined from n observations upon * This we have generally called ' ' equilibrium argument," denoting it by aig. We then have arg — epoch = phase . t The notation used in this paragraph is temporary. 604 UNITED STATES COAST AND GEODETIC SURVEY. the value of a liuear function involving them (a,a; + 6,1/ + CiZ). This function changes its value for different assigned values of the coefQcients [a, h, c) which are absolutely known from theory, and which are made to vary in accordance with the circumstances of the observations. Let — )!:, denote an observed value under the circumstances whose characteristic is i. If there is no error in this measurement, then a,x + b,y + c,z + fc; = 0. (87) "; If there were three accurate measurements (t,= 1, 2, 3), then x, y, z would be accurately deter- mined; if the three measurements were not accurate, ,r, 2/, z would still be definitely, though hot accurately, determined. ISfow when there are more than three (i = 1, 2, 3, . . . n) slightly inaccurate measurements, the values of a;, 2/, z determined from any three observation equations will not exactly satisfy the others. As it is not known which observation is in error, all may be assumed to be slightly in error, and the values of a?, ?/, z might be found by taking the mean values of all determinations. But such a process would not give the most probable values of the unknown quantities. Instead of actually solving all equations for x, ly, «, suppose these latter to have such values as are the most probable under the given set of measurements. Clearly none of the observation equations will now have zero as its right-hand member, but the one whose characteristic is i will have a small residual t)„ say, instead of zero. If «, y, z have their most probable values, it can be shown from the theory of accidental errors that they render the sum of the squares of the residuals (fi^ + «2^ + . . . + »;^ + • • . + ^/) of the n observation equations a minimum.* But the minimum of the function :S (a.flj + l},y + e,z + kf (88) i = l is found by equating each of the partial derivatives to zero; for, one can show that Lagrange's conditions are satisfied for this function. t The three resulting equations (which are the three normal equations) are i = n 2 a^a^x + 6;2/ + c-^ + A;,) = 0, ^=l i = n 2 biittiX + h,y + dz + fc,) = 0, (89) ^ = l i = 11 2 Ci{aiX + b;y + o^z + Tc,) = 0. i = l For any number of unknown quantities the normal equations will have the above form, and their number will be equal to the number of the unknowns. Special case. If a,, b„ c,, . . . be of the form ( ri— \ where r is an integer ranging from to w — 1, all coefficients in the normal equations will be zero except those of the form Sa^', 2hf, 'Sc^. . . . which are respectively equal to J except for the case r = 0, which gives 1 instead of J. The products «;&„ &,ei . . . must each be zero because they can be written in the form cos J. cos £ = J { cos {A — B) + cos {A + B) |, cos A sin B = i{ sin (A + B)—sm {A — B)], sinAcosB = i\sin{A-B) + sm{A + B)\, ^ (^0) sin A sin B = i\ cos {A — B)—cos {A + B)\, and because the algebraic sum of the sines or cosines of angles which divide a circle into a whole number (n) of equal parts is zero (§ 27). * See any treatise on least squares. t See any standard treatise on differential calculus. CHAPTER III. THE TIDE-PRODUCING POTENTIAL. 29. The attracting force of the moon upon any particle of unit mass whose distance is D from the moon's center is ^ (91) where M is the moon's mass and fx the attraction between unit massed unit distance apart.* E"ow if W be such a function that 1Z_ f^ 3D~~ D (92) it is, by definition, the gravitational potential of the moon at the point where the particle is situated; for, in the direction of D increasing, the force is negative. From this equation it is seen that the moon's potential decreases as the distance of the particle from the moon increases. If W=-^+ constant, (93) equation (92) is satisfled.t Let r = the distance of the moon's center from the center of the earth, p = the distance of the disturbed particle from the center of the earth, 6 = the angle at the earth's center between the disturbed particle and the moon's center. In the plane triangle defined by the earth's center, the moon's center, and the disturbed particle, the lengths of two sides are r and p, while the included angle is 6. Consequently the remaining side, whose length is B, has the value Vr^ - 2 rp cos 6 + p^. (94) Replacing D by this value and making the constant zero, (93) becomes W = ^^ (95) Vr^ — 2rp cos 6 + p^ Suppose the earth and the particle P to constitute a system not subject to deformation by the moon. The whole system is urged toward the moon just as if each unit particle of the system had applied to it the force /if ■ (96) acting in a direction parallel to the line joining the centers of the earth and moon. The components of this force are -^^, -^^, -3^, (97) where Wi, TOz, w^s are direction-cosines of the line joining the centers of the earth and moon referring to axes fixed in the earth, the origin being the earth's center. If U denote the potential *y = ^^/ii; .-./( = 3 -g = i ^Tg-, since E = earth's mass = volume X density = ^ Tta^Sg. tit is customary to give the potential a sign such that its value will decrease when we go in the direction indicated hy an arrow representing the force; i. e., the partial derivatives are not the forces in the corresponding directions as contemplated in this chapter, but minus such forces. 505 506 UNITED STATES COAST AND GEODETIC SURVEY. at P of this force, it must be such a function ot x, y, z, the coordinates of P, that its partial derivations shall be the above component forces. Such a function is U = ^^ (mi 00 + niiy + m3z)+ constant. (98) If|),,jj2,i'3 denote the direction-cosines of P referring to the axes mentioned in connection with nii, mi, m^, we have x=p^p, y=P2p, z=p\p; (99) .•. cos 6 =pi mi 4-^2 WJ2 +P3 »»3, (100) = sin X sin S + cos A. cos 6 cos (moon's hour angle), (101) where \ — the latitude of P, 6 = the declination of the moon. .-. Cr=^i' cos (^ + constant. (102) Now let the system be subject to deformation; in other words, let there be an opportunity for P to move relatively to the earth's center or to a rigid nucleus which may surround the center. The force causing this movement has for its potential Tr-U'=y,say; (103) or , , „-^ — ^ ,^ cos e — constant = V. (104) Vr-^ — 2rpcos6' + p2 »" ^ ' At the earth's center the potential of the tide-producing force must be zero because W and TJ are there equal. Making p = and T^ = 0, the constant becomes equal to + ^; (105) Y_ m M ixM_fxMp^^^^^ Vr^ — 2rp cos 6 + ff r r^ The expression may be written Vr^ — 2rp cos + p' (106) (107) J-(l-2^cos^ + j;) \ (108) This expanded in powers of — becomes ^(P„ + P:J + P2$ + P3^3+ • ■ •) (109) * Cf. Laplace M^c. C^l., Bfe. Ill, § 23, and Ferrel, Tidal Researches, p. 25. This expression, without the term — f^MIr, is the disturbing function il in the astronomical problem of three bodies. REPOKT FOE 1897 — PART II. APPENDIX NO. 9. 507 where Pi = cos (9, „ 3 cos^ e-1 -C2 = n ) T> 5 cos^ 6 — 3 cos ff r,:= 2 ' (110) The P's are functions of 6 alone and are called zonal harmonics or Legendre's coefficients. Equation (106) now becomes or 30. Approximate determination of the tide-producing potential.* (Ill) (112) SiM Fig. 7. Imagine the moon to be divided into two equal parts, the one occupying the moon's position, the other a position diametrically opposite but at an equal distance from the earth's center. The potential of the entire moon at any point P now becomes ^=iMM(^, + ^)=^-^^ /xM p cos ,+ jmM e + p^ Vr' — 2rpcosd+ p^' = i^ilfi(p„-Pi| + P,^_ . . . +P„ + P,| + P,£; + -^r^o+p.^:+p.^v . . . ); > the P's have been defined in § 29. .■.w^tM + .Ml^Q-^^^^y . . . ]. (113) (114) (115) (116) Since one-half of the moon is upon one side of the earth's center and the other half upon the other side, and at an equal distance from it, the earth's center, and so the solid portion of the earth, has no tendency to move. That is, the force tending to move the earth's center is zero ; ^ = i?=^ = o- dx dy 3e ' .•. JJ = constant. (117) (118) ' See Thomson aud Tait, Natural Philosophy, ^804. OUO UNITED STATES COAST AND GEODETIC SURVEY. The tide-producing potential is V=W-[I: /uM + mM fi fZ cos^ B ■ y_r^\ -)- ] constant. At the center of the earth y = and p = ; constant = + ^LM:. T (119) (120) The required tide-producing potential is, then, F=;.ilf[g( 3cos^.-l -^-| (,,,^ to the third power of-, or of the moon's parallax. 31. Proctor's construction. The following is Proctor's construction for showing graphically the disturbing influence of the sun upon the moon; it is here used to show the tide-producing force of the moon M upon a unit particle P. It is assumed that the line PM M whose length is D, represents a force acting on P toward ilf, the magnitude of this force really be- ing fjiMlIP; that is, the attraction of the moon on a unit mass situ- ated at P. Upon the same scale BM, where TB = 2UT, represents the attraction, in magnitude and direction, of the moon upon a unit mass at H, the earth's center; or, the attraction of the moon upon the entire earth divided by the earth's mass. The line PB represents both in magnitude and direction the disturbing force of M upon P. For, the attraction at ^ is PMx P3P (122) or, since PM and TM are very nearly equal and EM = ET + TM, it is or TM — 2ET. (123) The line P2?, completing the triangle of forces, represents the force tending to move P relatively to^. The potential of this force, since its action is relative to E, can be found by integrating the force PB or its components along the directions of their action. We shall integrate the component PQ along the line EQ. Upon the same scale as before, the line TB represents, because of the remoteness of M, the force By construction, jjM fiM tan £ = i tan 9. (124) (125) REPORT FOR 1897 — PART II. APPENDIX NO. 9. 609 The angle BPQ is equal to 6 + s. The force represented by PQ must be (^-^)«ec.cos(.+ ^), (126) _fnM_ p.M \ 3 cos' d-1 - ( 7)2 ^2 ) 2 cos (9 ' (127) = f,M(\+'^4co^9+ . . . -\)^^^JL=^, (128) \r^ r^ T^J 2 cos B = t^ (3 cos' e - 1)* (129) neglecting higher powers of the moon's parallax. ... r=f^i3 cos' e-i)3p=^ scos^^e-i _^ ^^^.t^^,. (130) The tangential component at P of the force PB has for its value QJR, or fM^ _ ^^ sec s sin (fi + ^) = 3 >^ sin 6 cos ^. * (131) But this is pdd (132) Hence the constant in F contains neither p nor (9; in other words, it is the same for all points of the earth. At the earth's center it is zero, and must remain so for all positions of P. -ry- yUilfp' 3 cos' ^ — 1 ,-,QO> •■■ y = -p 2 ^ ' 32. Spherical harmonic expression for cos' e-i. (134) By § 29, cos =pimi+p-i TO2 + pi m-3. (135) From analytical geometry we have K+i'2'+i)3' = l, m]' + TO2'+.m3' = l; (136) and so of course {Pi + V% + -pi) (»»i' + W2' + mi) = 1. (137) Prom these relations it follows that p/' — jUj' wii' — mj' cos' — i = 2pip2 rnxw,! + 2 -- — ^ — 2 — "^ " -^'^^ ***'***' ,0 , -i i'l^ +J? 2'-2j?3 ' »ii' + «i2' - 2 ms' + 2 j?i^3 miwis 4- f 3 3 (138) * Cf. values of — P, — T, p. 25, Godfray's Lunar Theory; Thomson and Tait, Natural Philosophy, § 812. il^ UNITED STATES COAST AND GEODETIC SURVEY. The reason why CDs'* d — i should be expressible in this form is given below, but the verification can be made in the manner just indicated. It may be noted that because of the symmetry in (135), (138) must be symmetric mpi,p2,p3 and mj, mj, m.3. The reason why , /3 cos ^ -1\ P" [^ 2 ) (139) should be expressible in the terms of 00^ -y^, xy, xz, yz, x' + y^-2z\ (140) f^{Pi^—P2% p'piPi, p'piPs, ffpiPi, p\p^+Pz^-'ip3% (141) Since by (99), (100), p cos d = niix + m2y + m^z, (142) and since p' = af + y^+ z\ (143) p^ (3 cos^ 9 — 1) must be a (rational, integral) homogeneous function in x, y, z of the second order. The property possessed by the functions (140) is that they (and so any function of them of the form A^ {a? — y") -Y B^ x y -\- C^xz ^ D^yz + E^ (x^ +y^-2 z% the coefiacients being any constants) satisfy Laplace's equation ;fF ^F fF ^ where F stands for any one of the quantities x^ — y', etc., or any combination of them of the kind just given. Other homogeneous functions analogous to (140), such as a? - z^, y^ — z^, 3?-\-z^ -2 2/2, y^^+Gyz- z\ which are made up from (140) by using suitable coefflcients Ao, Bq, etc., are not independent solu- tions of (144) if (140) be regarded as such. It can be shown that if ^=p2 ^ 3cos^g-l \ ^j, ^,p^ . ^j,^ ^^^^ generally, p^'P, ; (145) then But this equation is a particular form of so that any solution of (146) is also a solution of (144). Since ^ or p'( "'"'^^ j is a solution of (146) and, as has just been shown, is a homogeneous function in x, y, z of the second order, must be some form of F. In like manner it can be shown that because p' (| cos^ ^— f cos 6) satisfies (146) and is a homo- geneous function in x, y, z of the third order, it must satisfy (144). 33. To show that, by the equilibrium theory, the attraction of the moon produces spherical har- monic deformations of the ocean and also that such deformations are consistent with the equation of continuity. * Easily proved by direct substitution in (146) for any particular case required in tidal work. For the general proof, see Ferrers, Spherical Harmonics, pp. 4-10. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 611 Since the tide- producing potential represents work, viz., tlie force of terrestrial gravity multi- plied by the height of the tide, the height at various places must be the tide-producing potential divided by g. But g may be regarded as constant, and so the height of the tide will be propor- tional to V, and so a spherical harmonic deformation. Suppose everything stationary, and let p denote the radius rector of the free surface of the sea; the earth will assume the form of a solid of revolution whose equation is , rp' /3 cos^ e-l\ , p Y5 cos= ^ - 3 cos 5i\ , n P = «+«Lr3(;^^ 2 J + F^ 2 j+ ■ ■ -J, (148) or ,3 cos^ 6 — 1 . „ „3 5 cos^ ^ — 3 cos 6 ,-, aq\ where Ci, C2, Cs are small quantities. But this value of p satisfies (146), showing, as will be noted in § 35, that the equation of continuity is satisfied, or that no volume is lost or gained by the deformation. That is, the volume generated by revolving the curve o 3 cos^ — 1 ,-i Kftx p = a + C2p^ g , (150) or , 5 cos' (9 — 3 cos 6 /i ki \ p = a + C3p^ 2 » (1^^) or ^ 1 „ ^2 3 cos^ ^ — 1 . ,5 cos' 6 — 3 COS ,-, k^. p = a + (hp' 2 + 03 p' 2 ' ^ ' about the a;-axi8* is equivalent to the volume of the undisturbed sphere of radius a. This may be shown independently and as follows : In the first instance, since p is nearly equal to a, p= y/ x'^ + y'' = a + C2{%oif^ — ^a^); .\ x^ + y^ = a''+ ac2 (3 x^ - a'), (153) or «■' y^ (154) a^ {1+ 2 aca) "^ a^ (1 — ac^) The volume of the ellipsoid generated by revolving this ellipse is 3 ^*' Vr+Waci Vl — ac2 Vl — aci, = f 7ra', neglecting terms in Cz^. (155) In the second instance we have sc'' + y^ = a^+ ac3 (5 a;3 - 3 a^x). (156) The required volume is taken between the limits This is, neglecting terms in Cs^, nfyHx (157) (158) x = a-\- a?Cj, X = —a -)- a?C2. ^Tt a?. * In this and the next paragraph, the coordinate axes are taken with reference to the instantaneous spheroid, the i:-axis passing through the disturbing body, and are not fixed in the earth as is usually the case. 512 UNITED STATES COAST AND GEODETIC SURVEY. 34. Further illustration. A surface of equilibrium or a level surface is one which has the same potential at all its points. Supposing the earth to be a sphere without rotation, and the moon divided as in § 30; any surface of equilibrium has for its equation JJ.F P + ipiM r^^+^1 = constant,' (159) E denoting the earth's mass. Because the water of the sea is incompressible and because the action of the moon is symmetric about a line joining the centers of earth and moon, the surface whose equation is (159) must cut a perfect or undisturbed sphere in two small circles whose planes are perpendicular to the line joining the centers. Let a = the radius vector of the sur- face (159) at these small cir- cles, = the mean' radius of the undisturbed sphere. If now we write p = a{l + u), (160) au is a very small quantity in comparison with a and represents the variation of p from a constant value a. Since (159) is true for all possible values of p, it is true when p = « or «. = 0. Developing (159) and putting p = a we have, as in (116), Fig. 9. ft. fv \ fy'- -K* r \__ ] : constant. (161) JSTow 2 P2 = 3 cos^ (9 — 1, . • . if we make cos 9 = —=, P^ — 0. Hence, if we omit all terms beyond P2 -J- in the brackets as being comparatively small, (161) becomes -^ -f -^ = constant. (162) Writing this value for the constant and putting p = a (1 + m), equation (159) gives upon development - (1 _ M + M^ _ )+?[P.+ P/-^fii±^"+ . . . ] = E M a "^ r' (163) Omitting the second and higher powers of « as being very small, we have from (163) Ma:' = ^ p P2, = 2 «' P2 = a' (3 cos^ e - 1), say. (164) When = O,u = " = 2 a' ' U r^ *See Thomson and Tait, Natural Philosophy, U 800, 804. Taking into account the earth's axial rotation, the potential of the centriftigal force is i oo^d' where 00 is put temporarily for angular velocity and d for the distance of any point from the axis. This shotild be added to the left side of (159). REPORT FOR 1897 — PART II. APPENDIX NO. 9. 513 when (165) e = 90°, M = - i r^ ^' = - «'. ISTow au represents the inequality in the radius Aector of the surface of equilibrium. If the surface be an ellipsoid of revolution, the section made by a plane passing through the centers of the earth and moon must be an ellipse having semiaxes «(! + 2 a') and a(l — a'), respectively. For the lunar tide, aa' = 0.59 feet; for the solar, 0.27 feet. The equation of such an ellipse is + ^^-TX^-=^^ (166) Now write a2(l + 2a'f ^a^(l-a')' X = p cos 6, 2/ = p sin e, (167) p = a {1 + u), and (166) becomes (1 + uf cos' , (1 + uf sia^ _ ., ,„„. (1 + 2 a')' {1-a'f - ' ^ '' .-. M = a' (3 cos^ 6 - 1), (169) which agrees with (164), and shows that the variation in p is such that the section made by the plane passing through the centers of the earth and moon is an ellipse. To show that the condition of continuity is fulfilled, that is, that no volume has been lost or gained by the deforming action of the moon, it is only necessary to remember that the volume of the ellipsoid is ■Itt «5 (1 + 2 a') (1 — a') (1 - «'), = f7ra3 (1 + a'-3 «''+ . . . ); (170) when a' is so small that all powers beyond the first may be neglected. But this is the volume of a sphere whose radius is a. In obtaining (163), cos 6 was arbitrarily put equal to ;= in order that P-i might be equal to zero. The reason for this is that for some value of we know that the tide-producing potential must be zero; and to the degree of approximation here assumed, this potential is ^ P. (171) 35. Given the displacement of sea level, to show that the displacement potential is of the same form as the tide-producing potential and satisfies the equation of continuity. We have seen that the displacement of the sea level in a vertical direction, i. e., along a radius of the earth, is aa' (3 cos^ 6 — 1), (172) or pa' (3 cos^ ^ - 1), . (173) since a and p are sensibly equal. The volume of water moved in any small displacement of a level surface is its thickness multiplied by its horizontal area; that is, in the case under consideration, pa' (3 cos^ e-1) dS, fl74) 6584 33 514 UNITED STATES COAST AND GEODETIC SURVEY. where dS represents the area of an elementary portion of the surface. The entire amount of outward and inward displacement must be equal to zero, since the volume inclosed by the level surface remains unaltered. .-. ff pa' (3 cos" e - 1) dS = 0. (175) The displacement potential * is such a function of the coordinates that the differential coefficient with respect to a coordinate represents the displacement along that coordinate. Denoting this function by (f), it must be such a function of the coordinates that In this case we have -^ = displacement along radius. (l'^6) -^ = pa' (3 cos^ d - 1), (177) jp •. = ^a'{3 cos^ ^ - 1), = p^a'Pi, (178) to which any constant may be added. Thus it is seen that ^ satisfies (147). In general, letting n denote the normal to the surface S, fr>='' (i^«) when the integration extends over the entire closed surface S; that is, there is as much outward as inward displacement when the volume inclosed by S remains unaltered. To show that (170) is satisfied, first determine an expression for dS. The element of surface generated by revolving the short line of length p d 6 about the aj-axis is equal to The question is. Is p2 sin 6 de, = d8. (180) when the integral Is taken over SI This Integral becomes 2 p'>a' I sin d (3 cos^ 6 -V) dd, = 2/3V 2 / smdde-3 I sm^ 6 dd , (182) = 2pW _ 2 cos (9 -2pW J cos 3 (9-1 cos (9 =0, (183) as might have been inferred from the fact that (j) satisfied (147). (147) is another form of the equation of continuity, as can be shown by considering a small displacement of an elementary volume. The higher spherical harmonic deformations admit of similar treatment. 36. Alteration in the tide-producing potential, and so in the height of the tide, caused hy the mutual attraction between the fluid particles constituting the tide wave. The figures of the hearenly bodies depend on the law of gravity at their surfaces, and as this gravity is the resultant of the attraction of all their particles, it must also depend on their figures; therefore the law of gravity, at the surfaces of the heavenly bodies, and their figures, have a mutual connexion, which renders the knowledge of the one necessary for the determination of the other. In consequence of this, the investigation becomes very difficult, and it seems to require an analysis specially adapted to the subject.— (Laplace, Book III.) * See Ibbetson, Mathematical Theory of Elasticity, p. 57. REPORT FOR 1897 PART II. APPENDIX NO. 9. 515 So far in this chapter, the mutual attraction of the water particles has been disregarded in the tide-producing potential. Practically nothing is lost by so proceeding, because the continents and the inertia of the water do not permit the tide to assume a spherical harmonic deformation. A complete statement of the equilibrium hypothesis requires this source of disturbance to be noticed, and to that end an application of the special analysis referred to by Laplace will be given. It may be here noted that Newton took account of the mutual attraction of the particles in his theory of the figure of the earth, and in his tidal theory derived therefrom.* A spherical harmonic distribution of density of attracting matter or, tchat amounts to. the same thing, a thin attracting layer of corresponding thickness, on a spherical surface, produces a similar and similarly placed spherical harmonic distribution of potential over any concentric spherical surface throughout space, external and internal. Let us ascertain the potential of such a spherical surface at points along the axis of symmetry for the distribution (or thickness), taking the center of the sphere as origin; then by replacing the distance on the axis from the origin by p, the distance of any point in space from the origin, and multiplying each term of the developed potential by a surface harmonic of the proper order, the resulting expression of the potential is true for points whether upon the axis or not. For an internal point we know that the equation (147), often written v^ 1^=0, must be satisiied because (see § 32) any term of the form p" P„ is a solution; consequently an expression involving the sum of such terms must be a solution; and it is the required solution because when (9 = it reduces to the expression known to be true along the axis. The expression applying to external points is a solution of v^ ^ = whose terms are of the form K^.PJ P Denoting the equilibrium height of the tide by %azP„ (KS4) the density of the water by ff, then the mass of tidal water in a zone whose width is a dd is evidently ^a'^zn Gsmd P^de. (185) The distance from this zone to a point in the axis distant » from the origin is yz—a cos 6 +a sin 6 J j or (z^-2 az cos 6+0")^. (186) Writing temporarily fx' for cos 6, the quotient of (185) by (186), i. e., the gravitational potential due to the elementary zone, becomes _i a^Z7taP2d)A.' ,jg^> ^ (2^-2 az /t+a^)* ^' z for an external point on the axis, or ^a^ z n a Pi f T^ , „rt, „«' 3 " P.^P^^^P^'l^ . . . ^djx' (188) z z J ^2 -^a^Zn aPA P„+Pi-+P2-,+ ... \d ^i' (18!)) \ (X (X J ^ * Prineipia, Bk. I, Prop. 91 ; Bk. Ill, Props. 19, 36, and 37. t Ferrers, Spherical Harmonics, pp. 1, 2. X The quantity /i of 5 29 does not, of course, enter into the expression for », and so is in this paragraph taken as unity. It may be explicitly introduced as a factor into (192) and (194). 516 UNITED STATES COAST AND GEODETIC SURVEY. for an internal point. Integrating (188) between the limits ju'^ + l and M'=-h ^e liave, since I 1 P„P,„d /.i'=0 (190) +1 provided n and m are unequal positive integers, and since J •— 1 P/d^'=-^:^^,* (191) +1 Ha'^- (192) Now, taking any external point, replace z by p, its distance from the origin, and multiply by Pj. Thus we obtain for the potential of the spherical shell ^"^P^- (193) Similarly for the higher harmonic deformations of water. This, added to the tide-producing poten- tial of the moon gives for the entire tide-producing potential, to terms of the third order in 1/r, ^^+-.^$.^^F. (194) This must be equal to jr f a je Pz or the work accomplished in elevating the water; .'. putting p = a - " ' • (195) <^-*-*~) Bat a since JS = ^ tt a^d^, S, being the density of the earth, = g^^a ' 1 2 Er^l-^ff/S,' .-. The equilibrium height of tide, or 5' = 5 = I- ^ « '^e (196) (197) When the density of the fluid (a) is zero, the equilibrium height of tide or the tide-producing potentials is that found in the preceding paragraphs, or Euler's result. When S, is taken equal to ff, the equilibrium height is f times as great, or Kewton's result. As a matter of fact cr/S, is only about 2/11, and so the equilibrium tide would not be greatly increased because of this mutual attraction.! For ff= 0, range of semidiurnal tide at the equator = a z = H= 3 aa', §§ 41, 47, Part I. * Ferrers, Spherical Harmonics, p. 17. t Cf. Thomson and Tait, Natural Philosophy, j^ 815-817. Harkness, Solar Parallax, p. 139, makes 6%/tf =5-576 ^^ 0-016. CHAPTER IV. DEVELOPMENT OF THE TIDE -PRODUCING POTENTIAL. 37. The tide-producing potential of the moon at any given point depends upon the geographic position of the point and upon the particular time chosen. Of course the direction and distance of the moon enter into the value of this potential, but they are both functions of the time. Con- sequently the expression of this potential should be in terms of the coordinates of the influenced point and functions of time. Moreover, since the tide-producing causes are periodic in their character, the time functions should be simple harmonic functions; that is, functions consisting of terms of the form A cos (at + a) or A sin [at + a) where A, a, and a are constant, while t, the time, varies uniformly. If c denote the moon's mean distance (i. e., the semiaxis major of the orbit), and e the eccen- tricity of the orbit, the latus rectum has for its value c (1 — e^). If xs, denote the longitude of the perigee from the intersection of the orbit with the plane of the equator and I the longitude of the moon from the same origin, both reckoned in the plane of the orbit, I — 73, is the moon's true anomaly. The polar equation of the ellipse representing the orbit, taking the origin at the focus, is ^^""•"^'^ = 1 -f e cos (i! - cr,). (199) Since the most important part of the tide-producing potential 7, § 29, depends upon the third power of - , we cube both members of this equation and obtain re_(l-^-j _ -L ^ 3 g COS (J _ ^^) _^ 3 g2 gog2 (j _ ^^) ^ gs ^ogs (j _ ^r^) _|. . _ _ ^ = 1 4- 3 e cos (i - GJ,) + 3 e^ ^ — ^—^ + . . . , (200) where terms having the factor e^ are oniitted. If ff, denote the moon's mean longitude measured in her orbit from the intersection, and zs, the mean, as well as the true, longitude of the perigee measured in the same way, the mean anomaly will be a, — tS,. The solution of Kepler's problem, § 23, gives the equation l=ff, + 2esin{ff, — a,)+ie^sm2{ff, — 'CS,)+ . . (201) This value of I substituted in (200) gives an expression for [}^^y (202) in terms whose angles or arguments vary uniformly with the time. The next step is to express cos'* d — J in functions whose angles vary uniformly with the time. Its value as given by (138) consists of functions of p,, p-i, pj and m,, mz, wtj; but as pi, p^, p^ depend only upon the position of the disturbed particle with respect to axes fixed in the earth, they do not involve the time. We have, then, to express the m- functions, or quantities proportional to them, by means of functions whose angles vary uniformly with the time. 517 518 UNITED STATES COAST AND GEODETIC SURVEY. Suppose the moon's orbit to be a fixed circle of the celestial sphere concentric with the earth; then its intersection with the celestial equator is a fixed point to which the rotation of the earth maybe referred.* Let the ^-axis be the one about which the earth rotates, carrying with it the x- axis and the y-axis which lie in the plane of the equator. Let J denote the right ascension of the extremity of the aj-axis reckoned from the intersection. Fig. 10 shows the appearance of the celestial sphere viewed from without. The meridional arcs xZj yz revolve in the direction indicated by the arrows. If M denote the position of the moon, her true longitude from I being I, then wii = cos Mx, nil = cos My, nh — cos Mz ; .•. m,i = cos I cos X + sin I sin x cos I, rrii = —cos I sin j + sin I cos x cos I, (203) W3 = sin I sin I. Fig. 10. then We may observe that nii is derivable from m^ by putting a; + ^ ;r in place of x, Kow, for brevity, let p = cos J J, g = sin ^ I ; (204) (205) mi =i)2 cos {x — l) + (f cos {x + I), rrh = — # sin (z — ') — [«-iJ^']*».i!=["!.(i=3]V.; equations (138) and (211) then give r^ i , ,^^ ,,, p^=2 p, p. XY+2Pl=Sl' ^^+2 p. p. TZ (211) (212) c' (l-e^) +2p,p3XZ+i 2 2 j,i2+jp/_2 pi Z^+ T^-2 Z' (213) • The intersection is here supposed to move only as the equinox moves. REPOKT FOR 1897 — PART II. APPENDIX NO. 9. 519 38. Before completely expressing the functions, X^— Y^ XY, etc., as simple harmonic functions of the time, it is of interest to examine the special case where the moon's orbit is assumed to be circular instead of elliptical. Since e=0, equation (201) gives l=o-,; .'.X^-Y^=p* cos 2 {x-ff,)+2p''q^ cos 2 x+g* cos 2 (x+ff,), (214) -2 XY=p' sin 2 (x-(T,)+2pY sin 2 x+g^ sin 2 (x+ff,), (215) YZ=—p^ q cos {x—2 ff,)+pq (#—9^) cos j+i>g^ cos (^+2 a,), (216) XZ^=-ij3g sin {x-2ff,)+pq {f-q^) sin j+m' sin (X+2 ff,), (217) i(X^+r2-2^2)=i (jp^-4j)Y+^)+2jp2 g2cos2 cr,. (218) From this it appears that there are three classes of tidal causes, and so (§ 14) three classes of tides:* Semidiurnal tides, period about one-half day. Diurnal tides, period about one day. A fortnightly tide, period one-half tropical month. The constant term in (218) indicates a permanent change in sea level because of the existence of the moon. From § 13, the hourly variation in x is x=15-0410686, and in a, it is ff=0'5490165; consequently the component tides have for speeds the following values : 2{y — ff) = 28-9841042 = m2, 2 ;/ = 30-0821372 = k2, 2 (;/ +ff)= 31-1801702 = ;/ - 2 0- = 13-9430356 = o„ (219) X = 15-0410686 = ki, ;/ + 2 0- = 16-1391016 = 00, 2 0= 1-0980330 = mf. The relative size of the components occurring in any particular X- Y-Z function can be roughly determined by putting j> = cos J (23° 27'-3)= 0-979 and g = sin J (23° 27'-3) = 0-203. Upon inspect- ing YZ or XZ it is seen that the amplitude of Oi is a trifle greater than that of lunar Ki. 39. A natural way for developing the fnnction X^— Y^, say, is indicated here, the work being carried to terms in e'^ : x^-r2 = ^'^•^3 ^'^ f# cos 2{x — l)-\- 2 j>Y cos 2 X + q^ cos 2 (j + I)], (220) = [l + f e^-f 3ecos(J-Gy) +f e2cos2(i-cJ,)+ . . .' ] X [p^ cos 2 {x-l) + 2 p^ q^ COB 2 x+ q' cos 2 (^ + I)]. (221) It is obvious that if I in this product be replaced by its value i=o-, + 2 esin (** into — p^q, 2 p\^ into pq (p'^ — q^), and q* intoj) q^. Because of the choice of axes, the term of T^ containing Z^ will also disappear. The expression for XZ is obtained from that of YZ by writing sines in the place of cosines. .-. XZ=-{l-i^^) [p^q sin (X - 2 ff,) -pq^ sin (x + 2 Y + <^) [(1 - f e^) + 3 e cos {a, - cr,) + f e^ cos 2 (ff, - t3,)] + 2 j?Y [(1 — ¥ 0^) cos 3 (T, + f e cos (3 ff, — cJ,) — J e cos (c, + CJ,) + V- e^ cos (4 ff, — 2 zs,)]. (240) These, then, are the required developments as far as terms in e^. 40. The obliquity of the ecliptic is 23° 27' -3, and I oscillates between 5° 8' -8 greater and 5<= 8' -8 less than that value. The value of q or sin ^ /, when I is 23° 27'-3, is •203, and its square is •041, and its cube •0084. The eccen- tricity of the lunar orbit e^ '0549; hence g^ is a little smaller than e. The preceding developments have been carried as far as e", principally on account of the terms involving V b'', which, as e is about iV, have nearly the same magnitude as if the coefficient had been ^ e. It is proposed, then, to regard g' and q' as of the same order as e, and to drop all terms of the order e', except in the case where the numerical factor is large. This rule will be neglected with regard to one term for a special reason, which appears below ; and for another, because the numerical coefficient is just sufficiently large to make it worth retaining. 522 UNITED STATES COAST AND GEODETIC SURVEY. Adopting this approximation, we may write (237), (239), (240), thus, — , J2_ Y"-= {l — \^ e^) p-i COS 2 (^ — (J,) + (l — i e^) 2p'' q^oos2 x + 1 ep^ COS (2x — 3 (>, + ro,) — \ef- [p^cos (2;i; — (5, — ra,)-6 3^cos (2 ;t — <5, + «,)] + V e^i'''cos (2;t— 4(S, + 2ra ), (241) XZ=-(1- V-e^) [p''g8in(;t:-2(J,)-25238in(;f4.2 sin 2 ts , ic,An\ icot^ JJ— cos2Gr, 2 X - ff, + ct. = (2 Z - ff; - «,) + 2 tiJ„ i?2 cos (2 ;); - ^^•^ ^ - jp-^-6g^cos2cT, ~ i cotH J" - cos 2 «/ ^ ""-^^ /' 2 = j)2 - 12 g^ cos 2 cy, + 36 ?^, (252) /' = -/ p^ — 12 g'^ cos 2 cr„ approximately. (253) * B. A. A. S. Report 1883, pp. 57, 58. Small type in the text generally implies direct quotation, as above. EEPOKT FOR 1897-=PAET II. APPENDIX NO. 9. 523 The two terms thus combined may be written - i ep* ■/1-12 tan^ J J cos 2 cr, cos (2 j - o", - cr, - 22) ; (254) that is, the term having 2 j — ff, + nJ, for argument simply produces a slow variation in the amplitude and period of the predominating one whose argument is 2 ;i; — a, — tU,. Since J J is always less than 15°, the denominator of tan B is always positive and so B must always lie in the first or fourth quadrant. In X Z occur the terms i epq [i>^ sin {x-ff,- rSj + 3 {p^-q^) sin {^-ff^ + rs,)]. , (255) These might be combined into one having x — a, + tS, as argument, and whose amplitude and period would be subject to slow variations. But as either argument is nearly equal to ;t; — ff,, it is convenient, as will afterwards appear, to suppose a component of argument x — '^i having a slowly varying amplitude and period. We are to transform the expression 4 cos tS, sin (x — ff,) + 2 sin tS, cos (x — 0,) (25C) into the form fornix — ff, + Q). (257) Comparing this with (266), we have tan Q = ^tan txr,, (258) /"2 = 16 — 12 sin^ 7S„ (259) /" = 2Vt + |cos2cx,. (260) Consequently the two terms become ep'g. /f + f cos 2 rs, sin (z — c, + Q) (261 ) where tan Q =i tan rs,. Since tan Q passes through zero or infinity with J tan tSj, Q must always lie in the same quadrant as cf,. The object of the transformations (254), (261), which may seem theoretically undesirable, is as follows: — The numerical harmonic analysis of the tides is made to extend over one year, and this period is not long enough to distinguish completely a tide whose argument is 2x — „ called the smaller elliptic semidiurnal tide ; such irregularity has in fact been noted, but no explanation has previously been given of it. Again, the pair of terms with arguments x — o', J^ c3, will appear in the harmonic analysis with the single argu- ment X — <*/) 3,nd the resulting numbers will necessarily appear very irregular, unless compared with the theoretical expression (261). 41. The evection and variation. To the first power of e, the inequality in the moon's longitude due to evection is repre- sented by d = s + ^^me sin (s — 2 7i -f ^*),t (262) and in radius vector '^i^-^) = 1 + Jg6- me cos (s - 2 7t +p*), (263) where 6 is put for the moon's longitude in the ecliptic, aud m for the ratio of the sun'& to the moon.'s mean angular motion. 'p, denoting mean longitude, will be marked for the present with an asterisk. tSee Godfray, An Elementary Treatise on the Lunar Tlieory, §§ 71, 92. For more accurate values of the coefficients of sin (« — 2 h +p*) and cos (s — 2 h+p*), see Hansen, Tables de la Lune, J 1. 524 UNITED STATES COAST AND GEODETIC SURVEY. If we neglect tlie distinction between longitudes in the orbit and in the ecliptic [which is in effect neglecting a term with coefficient sin ^(i x 5'^ 9')], we have from (262), l = d, + ifmesm (s — 21i+p*); (264) whence cos (2 ! + a) = cos (2 6, + a) + ^i me [cos (26, + s — 2h+p* + a) — eoB(2d, — s + 2h—p* + a)]. (265) And from (263) and the definitions of iJ, W, $ in (226), JJ=r^iL_?!iT=i4.iji^ecos(s — 2A+JB*), (266) 'P{a)=coaa + flmelco8(s — 21i+p* + a) + cos(s — 2h+p* — a), (267) $(a) = Goa(2d, + a) + \<^meco&{26,j-s — 2h+p* + a)—^mecoa(2d,^-s + 2h—p*-j-a). (268) Then substituting from (266), (267), (268) in (2271-(231), and dropping the terms which are merely a reproduction of those already obtained, and neglecting terms in q'^ and q^, we have ^- — ^'' = \'^ mep* cos {2 X — 2 d. — s + 2 h—p*) —^ mep* cos {2 x — 2 d, + s — 2h+p-), (269) XZ= — \<'g^ mep^ 2 sin (x — 2 (J, — s + 2 7^— j)») + \^ mep" g sin (;r — 2(J, + s — 2 ft +p*) + tf mepg (p'^ — ^) [sin (x + s — 2h+p*) + ain {x — ^+^^—P*)], (270) HX^+T^ — 2 Z-i) =^ (p4— 4i>2 g2 _(. g4) if. „,e cos (s — 2 ft +j3*). (271) It must be noticed that ^^ me arises by the addition of the coefficient of the Eveotion in longitude to three halves of that in the reciprocal of the radius vector; that \% me is the difference of the same two quantities; and that ^/ me is three times the coefficient in the reciprocal of radius vector. When the development of the lunar theory is carried to higher orders these coefficients differ considerably from the amounts computed from the first term, which alone occurs in the above analysis. Hence, when these coefficients are computed, the full values of the coefficients in longitude and reciprocal of radius vector must be introduced. According to Professor Adams, the full values of the coefficients are, In longitude -022233, and in cjr -010022. The ratio of the mean motions m is about i\, and is therefore a little greater than «, heace me is somewhat greater than e^. Thus we may abridge (269)-(271), and -write the expression thus : — X- — Y-^=^ mep* aoa (2 j — 2 (?, — s + 2 ft — jj*) — |f me^J^ cos (2 ;( — 2 (?, + 3— 2 ft +i)'), ' (272) A:Z=— Yff »»ep' 2 sin (;i; — 2 tf, — s + 2 ft —p*), (273) \ (X«+ y^ — 2Z'') = \{p* — lLp'^2 = 0, p3~ sin i . (283) •■■ Pi^ —Pi = cos^ A, pipt = 0, 2>2i?3 = 0, 2 ^,^3 = sin 2 A, (284) h{P^^+ Pi -'ipi)=\- sin^ A. (285) Now supposing the place to be at the earth's surface, then p = a, the earth's radius. c\l—e^f ■■■ ^= t 3n ^^3 [* <'*'®' ^ (^' - ^') + ^^'^ 2 ^^^ + t (i - sin' ^) i (^"^^ + T'-2 Z')]. (286) The XT- Z functions being simple time harmonics, the principle of forced vibrations (§14) allows us to conclude that the forces corresponding to Fwill generate oscillations in the ocean of the same periods as the terms in V, but of unknown amplitudes and epochs. Now the work represented by V must clearly be equal to hg, where h is the height of the tide from the undis- turbed sea level and g the force of gravity. ...T = %,or;. = I=^, <287) where JH is the mass of the earth. •■■'' = * wOy (T^^ t* ''°^' ^ (^' -Y') + sin 2 \XZ+ f (i - sin^ A) ^ (X^ + r^ _ 2 Z')]. (288) It is convenient to have the quantities X'- r^ xz ijxM-r^ — 2z^ (l_e2)3» (i_e2)3' expressed as a series of cosine terms, each sign being positive. Since (289) — cos x= + cos [x + Tt), — sin a? =+ cos ( a? + ^ ), (290) -f sin x= + cos and (*-l); (1 _ e2)3 - 1 + 3e^ approximately, we have (T^)' ^ (^ - *^') P' ^»s 2 (J - ff,) -f (1 + f e^) 2 p2q^ cos 2 X M2, K; + 1 ep* cos (2 X - 3 (T, + zs,) N 2 + J ep* V |1 — 12 tan^ J J cos 2 CJ,| cos (2 x — ff, — tS, — R + tt) L^ -f ^ ey coa{2x-4:ff, + 2a,) ' 2 N -f \^^ mep* c,os{2 x — 2 0, — s + 2h —p n*\ V2 + \%mep*QO&{2x-20, + 8-2h+p* + 7i) x. + ^ my cos (2 J - 2 0-, - 2 s + 2 7i), Mi 526 UNITED STATES COAST AND GEODETIC SURVEY. p-^^-^=(L-f e2)[jj3gcos(z-2G-, + J;r)+2)33cos(;i: + 2(r,-^7r)] 0„ 00 + (1 + f e^)M{i'^-2^) cos (x — J tt) Ki + J ef^2 cos (;(; — 2 0-, — s + 2 7i — _p*+ J tt), /3i | (292) ^^ (l_e2)3 ^ = * (# - ^i^Y + g^) [1 + 1 e^ + 3 e cos (ff, - cJ,) Mm + ^ me cos (s - 2 7i +i)*) + 3 m^ cos (2 s - 2 A)] MSf + 2 ^2g,2 [(1 _ I e2) cos 2 0-, + ^ e cos (3 ff, - ts)\. Mf (293) In these expressions *^" ^ = icotH'j-?os2^/ tan Q = J tan ts,. (294) i'; 2) X) ^/; ^/ should now be replaced by their values in §§ 21, 37, thus giving the general expres- sions for the equilibrium tidal coefi&cients and arguments of Table 1, Part III. All tides have the universal coefficient f ^ ( t ) «; which is about If feet in value.§ By (288), the semidiurnals, diur- nals, and tides of long period have cos' A, sin 2 A, and J — f sin' A. as general coefficients. 43. The solar tides. Expressions for the sol£)(r components follow, because of symmetry, from those of the lunar. To pass from the latter to the former we have to put s = h, p* =pi*, S = r = 0, ff = rj, I = GO, e = dt, a = a^. (:295) In order that the relative values of the theoretical amplitudes of solar and lunar components may be readily seen, the universal coefficient f ri ( ~ ) * ^iH be retained. This involves the introduction of a factor Ti _ mass of sun /mean dist. of mooh\^ _ f).4f;noK _ 1 r 2Qf5\ r mass of moon ^ \ mean dist. of sun J ~ ~~ 2-17226 ^ ' where the mass of the moon is assumed to be 1/81-5 that of the earth. A tide of greater importance than some of those given in (291), (292), and (293) is one whose argument in (237) is 2 j + it, — zs,. The mean value of its coefficient is 0-00323. There is also the larger variational diurnal tide, which has been omitted: it would have a coefficient 0'00450; also an evectional termensaal tide, ^ff- me i sin' I cob {3 s — 2 7j+j)*), with coeiificient of magnitude 0-00292. All other tides in a complete development as far as the second order of small quantities, without any approximation as to the obliquity of the lunar orbit, would have smaller coefficients than those comprised in the above list. Such a develop- ment has been made by Professor J. C. Adams, and the values of all the coefficients computed therefrom, in com- parison with the above. t The symbol 2 Q is here adopted because Qi and 2 Q are analogous to Ns and 2 N. t A Greeli letter is here adopted because A2 and y-i denote other evectional components. § If we assume (of. Harkness, Solar Parallax, pp. 138, 140) k ==81^' c ~ 60-34 "■ = ^^ ^^^ ^^ ^^®*' ^'^'^ coefficient M 1 becomes 1-760 feet; if w = jjttc, it becomes 1-751 feet. This is approximately the theoretical range of the lunar tide at the equator. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 627 44. Tides depending on the fourth power of the moon^s parallax; M„ M,, etc. By equation (111) this portion of the tide-producing potential in 1/r* is F= ^p" (I cos^ (9 - f cos 6). (297) Neglecting the eccentricity of the lunar orbit, as well as its inclination to the plane of the earth's equator, we obtain V-~ tL^p" = I {pi^ - 3 p,pi) (m,3 - 3 mymi) c + i (P2^—Spi^Pi) {mi'— 3 m,2 Mi) + I (Pi^ +P\ pi— ^V\Pi) {ini^ + mi rn^) + I (i'a^ + P\ iJz — 4 2>2 pi) (mi' ma + mi>). (298) We have seen in § 33 that a harmonic deformation of the form of Fabove, represents a possible shape of a sphere covered by water; that is, the equation of continuity is satisfied. By (206) mi = p'^ cos (x — I), m2= —p^ sin {x — I)', (299) and so m,' — 3 mi mz^ = p^ cos 3 (j — Z), (300) Wi' + mi mi' =p^ cos (x — I). (301) Pi = cos A, Pi = 0,p3 = sin A, Now put, as before, and we have h = i~ (^* YJ^ f-s-g cos^ A jp« cos 3 (x-l) + iV cos A (1 — 5 sin^ X).p^ cos (x — «)"] (302) Now, cos A (5 sin^ A — 1) has its maximum value „ 7== when cos X = -(^-\/15: that is to say, when A. ^58° 54'; thus we may write (302) h = i^ f^Ya Tcos-' A -fi f^^ cos'^i Icosl3t+3(h — v)—3{s — i)} + i%-v/l5co8A (1 — 5sin2A)Tjjl/i5 (-\ COS" J / cos [t + (A — t) — (s— ^)] 1. (303) In this expression observe that there is the same ' general * coefficient' outside [ ] as in the previous develop- ment; that the spherical harmonics cos' A, i^u l/lo cos A (5 sin^ A — 1) have the nlaximum values unity, the first at the equator and the second in latitude 58° 54'. The 'speeds' of these two tides are respectively 3 (x — 6) or 43°'4761563 per mean solar hour, and y — d, or 14°-4920521 per mean solar hour. The coefficient of the tide 3 {x — 6), which is comparable with those in (288), is A [jj cos'^ i I, (304) and the mean value of this function multiplied by cos 3 (v — ?) is -00599; also the coefficient of the tide (y — e), likewise comparable with previous coefficient, is -iin \/15 (-\ COS" i /, (305) and the mean value of this function multiplied by cos (v — |) is -OOIBS. * I. e. universal. 528 UNITED STATES COAST AND GEODETIC SURA'EY. The expression for the tides is written in the form applicable to the equatorial belt bounded by latitudes 26° 34' N. and S. (viz. where sin l = i -/g). Outsideof this belt, what may be called high tide, will correspond with low water. The distribution of land on the earth will probably, however, seriously disturb the latitude of evanescent tide. It must be noticed that the y— 6 tide is comparatively small in the equatorial belt, having at the equator only I of its value in latitude 58^ 54'. Referring to the schedule of theoretical importance, " we see that the ter-diurnal tide M.i would come in last but four on the list, and the diurnal tide M, (with rigorous speed y — (?) would only be about a half of the synodic fortnightly variational tide. It thus appears that the ter-diurnal tide is smaller than some of the tides not included in our approximation, and that the diurnal tide should certainly be negligeable. The value of the Mi tide, however, is found with scarcely any trouble, from the numerical analysis of the tidal observations, and therefore it is proposed that it should still be evaluated. 45. On the mean values of the coefficients. Any of the preceding lunar tides may be written in the form Jcos(r+M) (306) where t7is a function of I, and u a function and v and ^; this may be seen upon referring to Table 1. Now since I is by equations (44) or (49) a function of w, i, and N, so also is J. The expression (306) when developed will give a term independent of N, which may be written in the form Ji cos I (307) wherein Ji is the mean value of the semirange in question. It may be proved (see Table 1 and § 23) that in no case does J involve a term with a sine of an odd multiple of N, and it may also be shown that in every term of sin u there will occur a sine of an odd multiple of W; whence it follows that J sin u has mean value zero, and Ji is the term independent of N' in J cos ii. It may also be proved that in no case does cos u involve a term in cos N, and that the terms in cos 2 N are all of order i'; also it appears that J always involves a term in cos JSf, and also terms in cos 2 N of order i^. Hence to the degree of approximation adopted, J, is equal to Ji, cos Kn, where Jn is the mean value of ./, aud cos «,i the mean value of cos u. In evaluating cos Mo from the formulae (47)-(49), we may observe that wherever sin^ iVoccurs it may be replaced by i ; for sin'- N='} — i cos 2 JS', and the cos 2 jN'has mean value zero. The following are the values of cos ito thus determined from (43), (46) : — (a) cos2 {r ~i)„=l — i'(^ (/?) cos 2 y„ = 1 — i- ^- sin 00 1 (X) cos (2| — j-)o = l — i»M i ) \ sm oo J (5)co3(2? + ^)o = l-Ji^(i±J^^y - (308) (£) e0SJ'n=l — 1^'—.—„ — ■ ' sin ^ Gli (£) fOs2|n=l — «- C0t»(B. The suffix indicating the mean value. Similarly the following are the J,i's or mean values of J:— , ,, ,11- J 1 fi I 1 ,•■ sin-^ \ oa — CO (a') cos^J-io=cos^ica 1 + -J- 1- • — ^ ., 1 — J sin'^ (bI (^') &(?') sin^/„ = sin2 &J fl + ii „ , -r ■ J 1 r -I I T -i! /cos 2 03 2 cos (ax"! /or.n\ (r') sin 7oCosH-fo = sin (a cos^ J CD l + J»''(~^-^s — — p-^ ) (309) ^' I " ' |_ \^ siu^ (B cos-=-J(a /J , ^ , . , . , , T ■ -91 r -r 1 T •! A COS 2 m 2 cos (ax"! (5') sin /„ sin^ \ /„= sin m sin^ i c» \-\ri%- i 5 — + . , , ) (£') sin /„cos io = sin ra cos ca [1 + Ji^ (cot= oa — 5)]. ' This embraces the astronomical tides given in Table 1; also a termensual and an evictional monthly. REPOET FOR 1897 — PART II. APPENDIX NO. 9. 529 On referring to schedules [B],* it appears that (a) multiplied by (a') is the mean value of the cos^^J cos 2 {v — I) which occurs in the semidiurnal terms; and so on with the other letters, two and two. Performing these multiplications, and putting 1 — J i^ in the results as equal to cos* ^ i, and 1 — | i^ as equal to 1 — | sin'-' i, we 15nd that the mean values are all unity for the following functions, viz. : cos'i-i- J cos 2 {r — I) sin° I cos 2 y sin I c os' 1 7 cos (2 g — v-j cos-t + 00 C08''i i ' sin'^ co {1 — f sin" i)' sin oa cos--J^ ao cos' + i ' (310) sin Z sin- i / cos (2 1 + '') sin / cos 7 cos y sin^/cos2^ sin w sin'^ i co cos* i i ' sin la cos co (1 — | sin-" i' sin'" oa cos* ^ i ' Lastly, it is easy to show rigorously that the mean value of 1 — 1 sin^ I (311) . (1 — 'i sin^ 03) (1—1 sin^! i) is also unity. If we write CO = COS i ca cos ^i — sin ^ od sin i ie^^ ■ (312) H = sin I 03 COS J i + cos ^ a; sin i je^Vi (313) where z stands for i/ — 1 ; and let ra,, x, denote the same functions with the sign of Jff^ changed, then it may be proved rigorously that C08-'-i/C08 2(3'— ?)=i(cB-' + CBi-<) (314) sin" /cos 2r = 2(ro2 3<;i2 + rai2«=) (315) sin I eoa'i Icos {2i — v)=^a>^ K + Wi^ Ki (316) sin / sin' ^ / cos (2 | + r ) = csk^ + a>i x,' (317) sin 1 cos I cos v = (mxi + roi«) (cbcbi — /cwi) (318) sin2 7cos2 4 = 2(cB=' 3cs> + rai2 KiS) * (319) 1 — i sin' i'=ro=' roi» — 4 raoi 3<;k, + ;<:* jci'. (320) The proof of these formulije, and the subsequent development of the functions of the co's and k's, constitute the rigorous proof of the formulae, of which the approximate proof has been indicated above. The analogy between the cb's and 3c's, and the p, q of the earlier developments of this Report, is that if i vanishes a = a}i^p, K=rHt^q. [See a paper in the Phil. Trans. R. S. Part II. 1880, p. 713.] Mean sea level varies slightly on account of the regression of the lunar node. The mean value of the coefficient of change in mean level due to the existence of the moon (cf. § .38) is ^ (1+1 e") (1-f sinH) (1— | sin^ f»)=0-25224, and the variable part is, approximately, — (1+f e^) sin i cos i sin oo cos ca cos N, = — 0'0328 cos N, N being the longitude of the ascending node, which decreases at the rate of 19o-34 per annum or 0O'0529539 per day (§ 13). 46. The factor f. Since v, S are always small, the mean values of the expressions cos* J I cos* J I cos* J w cos* I i 0-91538 sin^ I SID? I siv? w (1 sin I cos^ J J sin J cos^ J I sin GO cos'' J w cos* J i 0-38005 sin I sin^ ^ I _ sin I sin^ J I sin CD sitf J 00 cos* ^i~ 0-01638 (321) (322) (323) (324) * B. A. A. S. Report, 1883, p. 66, or Table 1, this manual. 6584 34 530 UNITED STATES COAST AND GEODETIC SURVEY. sin I cos I sin 2 I sin 03 cos 09 (1 — f sin^ i) ^ 0-72147 sin^ I sin^ I are always near unity, wbile the mean value of sin^ CO cos* i i ~ 0-15779 (325) (326) 1 — t sin^ I 1 — -I sin^ I .^2^\ (1 - f sin^ go) (1 - I sin'* i) ~ 0-75316 is exactly unity. But these expressions are functions of J proportional to those functions of J which are labeled "coeiHcients" in Table 1. Therefore they may be taken as factors / by which the mean valnes of the coefficients are to be multiplied in order to produce a value for a partic- ular time. The luni-solar tides. — In combining two waves, A and B, of equal speeds, the resultant amplitude is, § 4, Part III, ^A^ +2AB cos (phase A ~ phase B) + B\ (328) and the displacement or alteration in the phase of A due to B is an angle whose tangent is sin (phase B — phase A) cos (phase B — phase A)-\- -=. B and this is so regardless of the relative sizes of A and B. Denoting, for the moment, lunar Ki by [Ki] and solar Ki by JKi|, and letting the accent signify that the longitude of the lunar node is involved, these two waves may be written |Ki| cos (« + ;i. + Jtt-KiO), (330) [Ki'] cQs{t+h + ^n-v- KiO). (331) The iirst is displaced by the second by an angle — r', where tan v' = ,Tt/, i^xr n (332) cosi'+ )Kij/[Ki'] ^ and the resultant amplitude is K,' = ^/[Ki'f + 2 [Ki'] \K^\ cos r + {Kip. (333) The phase of the resultant wave is t + h+^Tt-'KP-y'. (334) Now t varies 15° per mean solar hour and h, 0-0410686, and so the hourly variation in < + fe is ki; . • . the resultant oscillation is, reckoning from « = on a day when h = ho, K/ cos (ki* - C) (335) where Q=x-i7t-ho+r'. (336) But Z = H-{To+u) (337) and To + u = i7i + ho — v'. (338) REPORT FOR 1897 — PART II. APPENDIX NO, 9. The context shows whether t is expressed in degrees or hours. From (325), L^lI = .. ^mli^osl =/of lunar K, =/([K,]). [Ki] sin 05 cos ftj (1 - I sm^ t) •' ■ y \l ij; [K{\ ^ t(1 + I e') sin I cos I . {Ki| ri(l + I Ci^) sin oo cos oo' .•. /of lunisolar K] or/(K]) Ki' [K,']' where SK,! + [K,] |KiJ _r^(l + fV) 1 + -^-^. = 0-46407, Similarly for Kj [K,] T(l + |e2) l-|sinM jKij j Ki I sin CB cos 0!3(1 — fsin^t) [K7] ~ [KiJ sin J cos J " tan 2 r" = sin 2 V cos2y+\K^\/[K2'] factor /for lunisolar K2 Vl + 2-g^,cos2.+ 5^^ '[K2' 1 + 5KJ. where [K, jKaj rill + fei") 1 [K2] ~ r (1 + fe^) 1 - f sin^ i 0-46407, jKzj _ jKjij 8in^a3(l-|sin'i) [KTJ" [K2I sin^J 531 (339) (340) (341) (342) (343) (344) (345) (346) (347) The tides L2 and Mj. — By § 42 the L2 tide is proportional to cos — 5 at the middle of the series tan Q = i tan P. (352) (353) 532 UNITED STATES COAST AND GEODETIC SURVEY. Since tan J P and tan Q pass through zero or infinity simultaneously, it follows that they always lie in the same quadrant. The /of M] may betaken as sinJcosHJ , sin-<.7^^os^I^cos^ V| + I cos 2 P. (354) The average' value of this expression is not very near to unity as is the average value of most of the/'s of the other components. It is, in fact, about l-550o, as is shown at the end of Table 10. For cos* i I, f = 1-0003 - 0-0373 cos W + 0-0002 cos 2 JST; sin / cos' il, f= 1-0088 + 0-1886 cos 2V - 0-0146 cos 2 iV^; K2, / = 1-0243 + 0-2847 cos N + 0-0080 cos 2 N; Ki, / =1-0060 + 0-1166 cos iV- 0-0088 cos 2 iV; (355) sin^ I, f= 1-0429 + 0-4135 cos JST - 0-0040 cos 2 N; 1-1 sin^ /, / = 1-0000 - 0-1299 cos ¥ + 0-0013 cos 2 JSf; L2, / = 0-9780 + terms in N and P; M„ / = 1-5505 + terms in N and P. 47. Table 37 shows the equilibrium amplitudes of several components (disregarding, as usual, the mutual attraction of the fluid) for various latitudes. In order to see what type of tide may belong to a particular latitude, draw the M2, K,, and Oi waves, with the amplitudes given in the table, upon separate pieces of paper. Ki and Oi generally conspire for extreme declinations of the moon and interfere when she is near the equator. The resultant Kid wave combined with the M2 wave will show the diurnal inequality peculiar to the latitude selected. Meteorological tides. — As already stated, there must generally be a tidal component Si whose period is a mean solar day ; for, the daily variation of the barometer is a well-established fact. At some places the land and sea breezes may also give a component of this speed which, of course, combines with the one answering to the variation of the barometer. In regard to the annual component Sa, it may be said that even if it repeat itself reasonably well at a given place, there is no reason for supposing its curve to be nearly harmonic. Conse- quently we should expect terms higher than the first to appear in the Fourier series I'epresenting it. The semiannual Ssa (partly astronomical and partly meteorological) is the only harmonic usually worked for. The (equilibrium) argument of Sa is h or the mean longitude of the sun and of Ssa, it is 2 h. These arguments become zero at the time of the vernal equinox; arguments of Sa, Ssa might be so taken as to become zero at the beginning, say, of the calendar year. 48. Overtides or shalloio-ivater components. Let the height of the tide, exclusive of shallow water components, be denoted by y'; let the total height be, as usual, denoted by y. Then y should be some function of y' such that y=K,y'+K,y''+E3y"+ . . . (356) where ^1, JCj, lu are the numerical coefftcients of the powers of y'. isTow we know that where y' is small, 2/=2/'; thatis, jBTi^l. Therefore we shall write y=A cos (arg A-Ao) + '& cos (arg B-Bo)+ . . . + E^ y''+K^ yi'\ (357) wherein -A, P, . . . are not shallow-water components. For the shallow-water tides con- stituting K2 y''\ we are not concerned with the absolute magnitude of the coefficient JQ, bnt REPORT FOR 1897 — PART II. APPENDIX NO. 9. 533 « rather with the relative values of the coefiBcients of the coustituent terms. Squaring y', taken equal to y, we have, besides the constant terms A^, B^, . . . , J A^ cos (arg 2 A -2 Ao) + ^ B' cos (arg 2 B-2 B°). +AB cos (arg ^ + arg B-A°—Bo)-\-AB cos (arg A-arg B—Ao-{-B°) + similar terms whose coefficients are J C^, A G, BO, .... (358) Now the shallow-water component whose argument is arg A ± arg B, must have as its speed aAih. In this manner the following tables of speeds, arguments, and what may be called primitive amplitudes and epochs have been obtained. Having obtained the principal terms of 2/'^, one cah then proceed to the terms of 2/" or y'xy'^. (See Table 36* for a list of the principal shallow- water components. Considering a group of shallow-water tides whose speeds are approximately equal, let us assume that each theoretical epoch differs from each primitive epoch by a quantity ^o which is constant for the group. Then A^'+BO-K={AB)^ (359) where {AB)° is the theoretical epoch of a component whose speed is a-\-b. In like manner A°+AO-Il,=(AA)° (360) Bo+BO-Eo={BB)°- (361) Suppose A to be larger than B, and suppose that {AA)° and of course A°, B°, have been deter- mined from observation. Then it is possible to infer {AB)o and {BB)°. In fact -Eo,=2 A°—(AA)°, substituted in the expressions for {AB)°, (BB)° gives {AB)°={AA)°+BO-Ao, (362) {BB)o=(AA)o+2 B°-2 A°. (363) Let it likewise be assumed that in each group the primitive range must be multiplied by the same constant Co; .: iG^-A-A={AA), (364) Go-A-B^{AB), (365) iG„-B-B=(BB), (366) Go-A-G=(AC), (367) where (AA), (AB) . . . denote the theoretical amplitudes of components whose speeds are «+a or 2 a, and a±6, . . . If we happen to know (A J.) (and of course .4., -B, . . . ) from observation, then {AB), {BB) . . . can be inferred. In fact (AB)=2 (AA)-~ (368) {BB)={AAy^„ (369) Applying these rules to (MS)4 and 84 the values of their coefficients given in Table 1 may be obtained. This agrees with the inferences made in § 18. For applications to nature, see Ferrel, in the Survey Eeport for 1882, pp. 442, 443, 445, 447. * Adapted from Terrel, United States Coast and Geodetic Survey Eeport, 1878, pp. 273-276. 534 UNITED STATES COAST AND GEODETIC SURVEY. The ^^ corrected'" equilibrium theory. 49. It has been already noticed (§ 40, Part I) that small bodies of water may obey the "corrected" equilibrium theory. That is, their surfaces may be everywhere perpendicular to the force of gravity as perturbed by the moon. It remains to develop the disturbing force into a series of harmonic terms. The value of the potential of the tidal forces is where v=,M\e,Q-J^2l^-y^ ''-"e- f- -p —3 cos^ A cos^ 8 cos 2 ((/) — I) + 2 sin X cos /\ sin 2S cos ('/' — I) + H3 sin^ A - 1) (3 sin^ d-l)~\, (370) Here I is used to denote the west longitude of the point, its former significations (§§ 25, 37) being no longer necessary. The slopes of the disturbed spherical layer to the surface of the undisturbed sphere are ^ ^^' fS71^ a^A. ' acosXdl' ^ ' the former being the slope (elevation) in the south-to-north direction, the latter in the east-to-west. These slopes are the deviations of the plumb line from the vertical, or they are the forces caus- ing its deviation where g is the vertical force.* — ^— — ^ -. ^ =^ eastward .component = 2 ;w ^ — cos A cos^ S sin 2 (f — I) — sin A sin 2 ^ sin [^ — I) . (372) — -yr = southward component = | ^^ -J sin 2 A cos^ c^ cos 2 (^ — i) — 2 cos 2 A sin 2 (J cos {'p — I) -f sin 2 A (1 - 3 sin^ 6)~\ . (373) Let e denote the easting of a given point from the no-tide point and s the southing, expressed in feet; then at any instant the height of the tide (H) is e x eastward component + s x southward component. Let the height due to semidiurnal eastward component be denoted by JI2,, and simi- larly for southward component by H^,, we have ^^_2e tan 2 (,/.-/) hi a cos A ' ^ ' 5'= + — tan A. (375) hi a ^ ' But from §42 we have for the height of the (uncorrected) equilibrium semidiurnal lunar tide J (1 - 1 e^) cos* iIcos[2t + 2(h-v)-2(s- 5)] + i {l + %e^)i sin^ J cos [2 f -f 2 {h - v)] + J.fecos* J Jcos [2t + 2{h-v) -2 (s - f,) - {s - p)] + . . . . ■ (376) * The diurnal term of Eq. 23", § 812, Thomson and Tait, should be multiplied by 2; and in Eq. 23", the diurnal term shguld have its sign changed. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 535 lu this equation e denotes the eccentricity of the lunar orbit. The time of the maximum of a single periodic disturbance of level can be found as follows : It is obvious that H^, has its maximum simultaneous with the maximum or minimum of h-z; Hi, has its maximum three hours, or 90°, later or earlier. The resultant disturbance has its max- imum between these two times. The hour angle {Cit) reckoned from the transit of the fictitious body or the maximum of hi, is The corresponding height is For M2, tan-'^'. (377) -ff. Hi, cos Cit + Hi, sin dt. (378) ^' = ^ tan A : .-. H^, = 0-800 i sin 2 A, (379) M2 a a ' Hi, = - 0-800 — cos ^ ; (380) a, the earth's mean radius, is 20 902 000 feet, and 0-800 cos^ A the equilibrium value of Mj. Uxample. — From a map of Lake Superior we see that the no-tide point (center of gravity of the surface) is 6 miles north of Keweenaw Point (Lat. 47° 32', Lon. 87 ^. The line .ioining this 4^^ point to Duluth is 210 miles in length and bears S. 76Jo W. The no- tide point is 4° 19' (or 17") E. of Duluth. Required the amplitude and epoch of Mi at Duluth. Here s= 49 X 5280 feet, ^ = - 204 X 5280 " Hi, = 00099, fl2. = 0-0557; .-. C2t = 80°, Hi, cos 80° + Hi, sin 80° = 0-0566 = M2, 80° ___ = 2'^ 46" = time of HW at Duluth in no-tide point time. . • . 2i> 46" - 17" = 2" 29" = HWI for Duluth M2O = 80° - 8° = 72°. Observation gives* M2 = 0-063 feet, M2° = 81°. 50. The "corrected" equilibrium tide can be obtained from the uncorrected in the following manner whether the sea be small or large: In the first place make two stereographic projections, one of the northern and one of the southern hemispheres, the pole in each case being at the center. Upon these mark the outlines of the sea in question. Upon a partially transparent sheet, using the same kind and size of projec- tion, let the equilibrium heights of a given component, say of Mj, be written in their proper places. Let the center of this sheet be placed upon the center of either hemisphere, and place the radiat- ing line of greatest height upon a given terrestrial meridian. The surface of the sea is divided by the meridians and parallels into rectangles whose areas are proportional to the cosines of their lati- tudes. The volume of the uncorrected equilibrium tide is found by multiplying the elementary areas into their respective thicknesses at the given time. The transparent sheet shows the thick- ness for the assumed time. The volume divided by the area shows how much the (uncorrected) equilibrium spheroid lies above the "corrected" equilibrium spheroid at the assumed component hour. At another hour it will have another value. These values tabulated with opposite signs will define a curve drawn once for all for the sea in question, which when added to the (uncor- rected) equilibrium curve at any place will give the "corrected" equilibrium tide at that place. * Obtained by analyzing the heavy curve shown in Plate III, App. BB, Report of the Survey of the Northern and Northwestern Lakes (1873). 536 UNITED STATES COAST AND GEODETIC SURVEY. So proceed with each of the important components. We may not completely separate the height into components if we, for a given declination of the moon, construct a stereographic pro- jection with the moon distant S from the bounding circle and the (uncorrected) equilibrium heights written upon it. But this process is less convenient than the former. The same theory is expressed analytically after Thomson and Tait in the following manner : If h or au denote the (uncorrected) equilibrium height of the tide, then au - a (381) will denote the "corrected" height, wherein tidal volume over sea ■ area of sea (382) ^ a f/vdrr (383) where gi = area of sea and the elementary area dff — cos X d X d I, and 3Ma^ 2 E"r' « = ^^« (cos^^-i). (384) This gives for the "corrected" height where aa - o! = i~~ [(cos^ A. cos 2 Z - |i) cos 2 ^ + (cos^ A sin 2 J - ^) sin 2 f] cos^ S + 3 — ^ [(sin A cos A cos Z — @) cos f + (sin A cos A sin i — g) sin (/'] sin S cos d + 4 f ^ (3 sin^ A - 1 - 1) (3 sin-^ S ^ 1), (385) Jij T • ^ = ^ r r cos^ A cos 2 IdG, ^ = (^ \\ cos' ^ sin 2 Ids, ® = ^ [ r sin A cos A cos Ida, S = 77 j sin A cos A sin Ida, % =A~JI ^^ ^*°' A - 1) dff. In the integrations or quadratures for a, f and d are regarded as constants, and so are taken from beneath the integration signs. This height may be written in the form Ko [3 sin^ d - 1] + JBi sin 2 tf cos [tj}-l- «,] + -B2 cos^ 6 cos [2 (5/; — I)- £2]. (386) CHAPTER V. THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. ON THE SUMMATION OP HOURLY OEDINATBS. 51. In the harmonic analysis it is convenient to consider component days, whose lengths are the periods of the various diurnal components o;^ twice the periods of the semidiurnals.* Such days are divided into twenty-four equal parts called component hours. If the tidal curve be read at the component hours and sums made by combining for each hour all readings belonging to it, twenty-four sums will be obtained. These sums are then analyzed in the manner described in § 58. To avoid tabulating the curve for each kind of component time, the tabulation in mean solar time is made to serve for all. This is done by distributing the (solar) hourly heights among the component hours as nearly as possible. The speeds or periods of the components determine where the various component hours fall upon the solar hours. The manner being always the same for a given component, tables of such correspondences between component and solar hours may be prepared as follows: If we put Si = 15° for the hourly speed of the mean sun or diurnal solar component, and Ci for the speed of any other diurnal component, then ii=i5 (387) Si 15 will represent the portion of any component hour corresponding to a, solar hour. While this com- parison of component and solar hours is only required to the nearest whole component hour, in order to secure even this degree of approximation throughout the hours of a whole year, it is desir- able to carry the value of ~ out to about eight decimal places. For all components, zero hour is always taken to coincide with zero hour of the first day of the series. By successive additions of — the component hour corresponding to any solar hour may be found ; the first solar hour of the first 15 day of series corresponds to the :-^ component hour, the second solar hour to the -— component nci 15 hour will be lost or gained according as C: is less or greater than s, when hour, and the wth solar hour from the beginning to the ~ component hour. A half component * X.O solar hours shall have elapsed from the beginning; that is, the solar and component hours agree from the beginning of the series until this number of solar hours has b6en reckoned, when the component hour taken to the nearest whole hour will differ by one from the solar hour. At subse- quent regular intervals of Si 15 Si ~ Ci 15 ~ Ci (389) solar hours, a whole component hour will be lost or gained, that is, the difference between compo- nent and solar hours will increase one at each such time. If Ci < Si, as is usually the c&.se, two adjacent solar hours at one of these times fall upon the same component hour, i. e., within a half com- " The length of a component day in solar hours may be found by dividing 360° by the speed per hour (c,), Table 1. 537 538 UNITED STATES COAST AND GEODETIC SURVEY. ponent hour of the time aimed at ; but if Ci > Si, a component hour will be skipped, because no solar hour occurs within a half component hour of it. In either case all the solar hours are represented by component hours, the maximum divergence being a half component hour. If the maximum divergence allowed be assumed to be a half solar hour, then all solar hours are not represented by component hours when Ci < Sij and when Ci > Si, a solar hour may occasionally be taken to represent each of two consecutive component hours. The times when the difierences between solar and component hours change, are given in Table 42, designated " Component hours derived from solar hours." In this table, the values on the left hand of each column denote mean solar hours, and those on the right hand show how much the numbering of these hours must be altered in order to obtain the corresponding number- ing of the component hours. The component and solar hours, as we have seen, start together with zero hour at the beginning of the series, and, reckoned to the nearest whole component hour, they continue to agree until the first tabular val^e is reached, when the component hour is the sum or the difference (according to whether a plus or minus sign is used) of the two given values. After this the component hours continue to differ from the solar hours by the first right-hand value until the next tabular value is reached, when the difference becomes that right-hand value, which is to be used until the next given value, and so on. Whenever the value to be added to the solar hour is such that the sum is equal to or exceeds 24, the sum should be diminished by 24; and whenever the solar hour is less than the value to be subtracted from it, increase the solar hour by 24 before making the subtraction; it is unnecessary to keep track of the component days. For the long period components, where the change during a solar hour is very small, it is proposed to use the daily sums of the hourly heights as the quantities to be operated upon, and Table 43 shows what component hour each one of the daily sums corresponds to, reckoned to the nearest whole component hour. 52. As an example of the use of Table 42, find the M hours on the fourth day of series corresponding to the first five hours of solar time. Entering the table for the fourth day of series and in the column M, one sees that for the second solar hour the difference between solar and M hours is —3, which difference will continue from that time until the next tabular value; but for that portion of the day preceding the second solar hour on the fourth day of series we must look to the tabular value next above, which is —2 at the twenty-first hour of the second day of series, and as this difference continues until the next tabular value, it is the difference for the first and second M hours required. The resulting M hours are therefore as shown above. It will be noticed that the first and second solar hours both correspond to the twenty-third M hour; this is due to using whole M hours, as may be seen by multiplying 0-9661368, which is the portion of an M hour corresponding to a solar hour, by the number of solar hours from the beginning of the series up to these hours, showing that on the fourth day the first solar hour corresponds to 22'53 M hours, while the second solar hour corresponds to 23-49 M hours. Whenever the difference between the solar and component hour is such that it changes in a period less than a solar day, the table gives two or more columns to the component; but the above example will suffice to explain the use of the table even in such cases. Having tabulated the component hours which most closely correspond to the solar hours, the hourly heights of any series to be analyzed may be distributed in accordance with this relation. This, however, is a laborious process, for it not only requires as many copies of the hourly heights as there are components sought, but each copy must have its heights arranged differently, according to the relation existing between the hours of the component being worked for and the solar hours, as shown by the table. Instead of using a table, blank forms may be made out once for all indicating where the hourly heights are to be written. Darwin in his report for 1883 gives a sample of such form. He ha? since prepared and published a set of blank forms for eighteen kinds of summation. Before making a particular summation the hourly heights of the series to be analyzed are to be copied into the form in the way indicated by certain marks thereon. This method requires as many copies of the hourly heights as there are components to be worked for, but does away with the inconvenience of following a table to find the order of arrangement. Fourth day of series. Solar hours. M hours. o 22 I 23 2- 23 . 3 o 4 I us. Coast arod, GeodjeticSui'veylteport foT'ldd?. AppemcLtx Nb.9 m>ji M Date Series THE VORRlS TETefiS CO.,. PHOTO- LITHO,. WASHINGTON, D. C. Fio-.IL. Stendl for theM summation, sheet for tiie even hours. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 539 Darwin* has recently devised a set of movable scales for saving labor in distributing the hourly heights for each component; but in matters of simplicity, convenience, cheapness, and rapidity of use, the apparatus does not compare favorably with the stencils described below. 53. StencilsA This term has been applied to a series of sheets so perforated as to mechanically indicate what observed hourly heights belong to the various component hours. Their use does away with the necessity of copying or rearranging the tabulated hourly heights. Directions for constructing stencils for any given component. — Select blank forms similar to those upon which the hourly height to be summed have been tabulated. These forms should be so contrived as to cause the heights to stand sufficiently far apart from one another that no two of them can ever be seen through the same opening when a stencil is applied; the mere leaving of a large space in which to write each height will not answer, for the heights must be always found in a definite place, which may be designated by light ruled lines. In additiow to the usual way of denoting the date by the day of the month, it is desirable to use a series of consecutive numerals, known as " days of series," which always begin with 1 on the first day of the record used and end so that its last value indicates the number of days taken. The stencil sheets, being intended for use upon any series, have merely the " days of series" upon them as dates. For the sake of clearness in using the stencils it has been found desirable to separate the component hours into even and odd, thus making two stencil sheets for each page of tabulation, and the sheets which thus constitute a pair must have the same days of series. Having thus prepared blank forms with a duplicate set of days of series, and having written the symbol of the component for which it is designed at the head of each page, turn to the table designated " Component hours derived from solar hours " and in the manner already explained proceed to make the blank forms into a table showing what component hour corresponds to each solar hour throughout the series, entering the even component hours on one blank and the odd hours on the other. Join those spaces containing the same component hours by a broken line, and write upon this line, at suitable distances, the number of the component hour it represents. The spaces where the component hours fall are then stamped with a steel punch which makes openings of sufficient size for showing the tabulated heights. The accompanying figure shows the first seven days of the even hour of the M stencil. The portions inclosed with lines represent openings which are cut through the sheet so as to show the tabulated hourly heights to be summed, when the stencil is placed over them. The size of these sheets is governed by the size used in tabulating the hourly heights. The marginal arguments are the solar days of the series and the solar hours, reckoned from midnight. The broken lines joining openings show that the heights appearing through all openings so connected are to be added together, and each such sum belongs to the M hour written upon the line. It is generally desirable, particularly in summing for smaller components, to so omit or repeat certain hourly heights that each component hour of the period covered receive one, and but one, hourly height; in other words, to make the maximum divergence between the two kinds of time a half solar, instead of a half component, hour. To construct stencils suitable for this purpose use the same table as before, omitting the unmarked hours when cSi. By marked values are meant those pointed out by the arrow, Table 42. Directions for using the stencils. — The stencils are to be applied one sheet at a time to the tabulated hourly heights. Care must be taken to see that the proper sheet is applied in each instance, which is done by making the days of series upon the stencils agree with the corresponding days upon the sheets of hourly heights. The stencil must be placed carefully so as to accurately coincide with the tabulation beneath it, using paper weights to hold it in position while making the summations. If a broken line for any component hour runs out at the top or bottom of the stencil, there will in general be another portion of the same hour on the opposite edge, which should be included in taking the sum. As the hourly heights are summed through the stencil openings for each component hour, the sums are set down in a suitable form having the 24 component hours and pages of the tabulated heights as arguments. After all the stencils have been applied, * B. A. A. S. Report, 1892, pp. 345-389. t See United States Coast and Geodetic Survey Report, 1893, I, p. 108. Also this manual. Part I, {i 145. 540 UNITKD STATES COAST AND GEODETIC SURVEY. the sums for each component hour on the summation form, are combined into a single sum for each of the 24 component hours throughout the period of observation used. The divisors for these final sums are obtained by counting the number of openings in the stencils for each component hour; and as a convenience these divisors may be written, once for all, ou the left margin of the stencils, or giveTi in a table. The twenty-four means thus obtained may then be converted into residuals by subtracting from each the mean of all, and these residuals are analyzed in the way explained under harmonic analysis; or, the twenty-four means may be analyzed directly. Sum checks. — Bach page of the hourly heights of the sea should be summed horizontally and vertically before any of the stencils are applied. Any stencil sum tor the whole page (adding together the sums belonging to the odd and even hours) should be the same as the sum of the vertical columns or horizontal lines, provided all hourly heights are used once and but once. But when stencils are constructed with reference to the marked values of Table 42 an additional or third stencil sheet should accompany each pair which will point out the hourly heights omitted or used twice according as Cj < s. The sum obtained by aid of this sheet must be added to or subtracted from the total sum obtained from the even and odd hour sheets in order to check the work. For a component like K, P, E, or T, whose speed differs little from that of S, lines joining the openings will frequently become horizontal. When this happens openings should be made in the right-hand margin of the stencil sheets, so that the horizontal sums already made may be simply copied upon the proper component hours. In this connection see § 66. 54. Adding machines. Several varieties of adding machines are used by the Survey in making these and other sum- mations, viz., the "Comptometer" and the " Comptograph," manufactured by the Felt & Tarrant Manufacturing Company, Chicago, 111. ; the "Burroughs Eegistering Accountant," by the American Arithmometer Company, St. Louis, Mo., and a computing machine made by A. Burkhardt, Glass- hiitte, Germany. The machine last mentioned is designed more especially for multiplication and division. 55. A jjroposed machine for obtaining component sums. Having seen that the stencils mechanically point out where the hourly values must go in making up the partial sums, the idea naturally suggests itself of having the equivalent of stencils so control a registering apparatus as to simultaneously give all the required summations. Let there be as many cylinders— each, say, 26 inches long and 10 inches in diameter— as there are independent summations to be made. Each cylinder will represent the stencil of a single component for, say, 370 days. The circumference of each cylinder should be divided into 370 equal parts, each division fixing a line or element which represents a day. All cylinders are supposed to have a common movement in the direction of their axes an inch or so in extent for bringing the holes about to be mentioned, into their proper positions for the various hours of the day. Each day line contains 24 holes in the surface of the cylinder, determined by the cor- respondence between solar and component hours for the day in question, the small movement along the axis having been taken into account. The recording or adding apparatus for each kind of summation consists of two series of toothed wheels, all wheels of a series being upon a common shaft. The number of teeth upon each wheel of the first series may be taken as 300, and of the second series 299. The number of wheels in each series is 25, one for each component hour and one for those few hourly heights which are used only in checking the sums of the 24 partial sums. The number of revolutions made by the 300tooth wheels can be found by subtracting the readings of the 300 tooth wheels from the readings of the 299-tooth wheels. The parts of revolutions are, of course, the direct readings. The cylinders are placed side by side in a horizontal frame, all axes being parallel. This frame is supported by a table or framework and is capable of the small amount of motion already referred to. All cylinders are made to rotate together by means of a rack and spur wheels. Above these cylinders, or above the intervening spaces, the two sets of toothed wheels serving as counters are mounted. The shafts bearing the 300-tooth wheels can all be made to rotate the same amount by means of parallel rods and cranks. The operator imparts motion to the mechanism by means of a crank at one end of the framework. This carries a pointer which. REPORT FOE 1897 — PART II. APPENDIX NO. 9. 541 moving over a graduated dial, indicates the amount of its rotation and of the 300-tooth wheels, which are not held fast by the levers about to be mentioned; that is, it indicates what number or hourly height is being entered. The crank can be released and returned to its initial position without causing any of the shafts to rotate. The cylinders control the 300-tooth wheels by means of levers, one for each wheel. The 25 levers for each component are upon a common axis parallel to the axis of the cylinder. At one end of each lever is a needle-like projection for entering the perforations in the surface of the cylinder, while at the other end is a sharj) edge, extending upward, for engaging the wheel above, thereby preventing its rotation. Since the preventing of a wheel from revolving with its shaft must give rise to friction and wear, it seems best to stop but one out of each 25 rather than to stop 24 of them. This involves no extra work on the part of the operator except the subtracting of the final machine readings from a constant number— the grand total of all hourly heights. For each succeeding day, the cylinders are all turned forward one notch; and for each succeeding hour of the day, they are all carried forward automatically a small but constant amount along the line of their axes. As already intimated, the hourly heights when entered once are to be simultaneously summed in all the kinds of summations required in analysis, thus enabling a person to sum for all components almost as quickly as he now sums for one upon an ordinary adding machine. This machine is designed to take the place of an harmonic analyzer in tidal work. Its merits are its positive workings, the great number of components which can be included, and its simplicity of construction, in that hundreds of its parts are exactly alike. 56. The Thomson harmonic analyzer. _ _ The immediate object of the harmonic analyzer is to determine the coefficients Ho, A, A, B, B, G, G, . . . from the observed tidal carve, whose equation may be written y = H„ + A cos at + B cos bt + G cos ct + . . . + A sin at +B sin bt+V sin ct+ . . . (390) The average value of y is ■ ■— (391) 1 /•«=< Eeplacing y by its value given above the result is readily integrated, giving ID A Ti C Hat A — sin at + -^ sin ht + - sin et + a b c — - cos at — 'T cos bt — — cos ct— . . . • (392) — v^v/o wc/ — -7— \jyji^ f f — — a c When t is large, the average value of y approaches -Ho- -^ny planimeter whicb enables one to find the area of the curve, and so the average value of y, can be used for finding the value of .Ho. For instance, if a disk rotate uniformly with t, and has upon its face a small friction wheel whose axis intersects the axis of the disk perpendicularly, and if this friction wheel be moved inward and outward according to the value of y at each instant, the number of rotations of the friction wheel will be proportional to the area of the curve. Suppose that the rotation or angular velocity of the disk be a more complicated function of the time than t multiplied by a constant, say I ^ {t) dt. Let the equation of the ordinate of the curve he y = f (t). Now, if the rotation of the disk be proportional to the ordinate of a curve whose equation is y' = r (j) (t) dt, (393) the number of revolutions of the friction wheel will be proportional to ./: (t) (^ {t) dt.* (394) /■ ' ThomsoTi and Tait's Natural Philosophy, Part I, pp. 493-495, and Proc. Key. Roc, Vol. 24 (187fi), pp. 266-268. 542 UNITED STATES COAST AND GEODETIC SUEVEY. For, in the place othtwe now have / (f> (<) dt,Jc being a constant, and so in the place ot k d t, {t) d t. In the harmonic analysis of the tide curve y = rp{t), the function ^, as will be presently explained, is of the form m = Z{nt), (395) and soy' = I

] ; (422) ... _ 6 A = ^_^i(b-a)T ^ ^^^ ^^^j,_a)r + ^] + ^'^f}' 'J'}^ Gcos[Uc - a)r + y]+ (423) J(6 — «)r i(c — «)T in like manner the average value of B sin (6 — at + fd] is B (b-a)T [cos (6 — « T + /?) — cos /?] , (424) * This holds for any value of r heoause the height of .the tide wave is of necessity a single-valued function of t. REPORT FOR 1897 — PART 11. APPENDIX NO, 9. 547 which is equal to '%^-~a^l' ^ «i^ \iih-a)r + /3]. (425) + ''^ 1^-1)^ ^ '^° ^^^' ~ "^^ + ^J + • • • • ^^^^^ lu findiug the effects of B, G, . . . upon the amplitude aud phase of J. we are concerned only with the length of the series; let us therefore suppose the initial phase of A„ that is a„ to be zero; then Z, = 0, A, = il,. _ _ .•.tanrfa=- ^^ _ ; or, tan c^C = — — = (427) A, + 6A' B,(A)+ SA where Co {A) +6Z = Q (A). (428) A = 4.+^; or, B (A) = ^-(^)+/-^ (429) cos da ' ^ ' cos dZ ^ ' The required values of 6 A, 6 A are easily determined. At a time when a, = 0, /S becomes p — a^ and y, y — a,. . • . — 6A and dA may be written + ^'l \c-~a^r' ^ ^"" TM^ - «)^ + C (A) - C (0)] + . . . , (430) ^ (c — a)r Special case. — Suppose that we are concerned with two waves, A and -B, whose speeds are exactly equal. Then formulae (430) aud (431) give - SA = B cos [dA) - Z {B)], (432) Si. = B sin [C„ {A) - C (B)]. ' (433) These values substituted in (427) and (429) clear the A of the effects of B. Table 41 is given for the purpose of showing how the disturbing effects may be tabulated once for all for an observation period of iixed length. Somewhat smaller effects could have been obtained by selecting lengths suitable for the several components, but covering nearly the same period. The length best adapted to the determination of a particular component would generally be a synodic period of that component with one or more of the largest components of its class, i. e., with diurnals or semidiurnals according as the component is diurnal or semidiurnal. Each tabular value consists of two parts, the first is the amplitude, or the numerical value of 180 sin J(6 -a)r ,^ ,„,^, ^*^ (^ ' 4 ^ r-^, or 57-29578 ; (434) ^ ^^^-"^^ ' ih-a)l ^ the second is the angle J(6 - a)T (435) but with multiples of 180° rejected or added so as to leave the angle between and +360° and 548 UNITED STATES COAST AND GEODETIC SURVEY. when substituted in the numerator, to render the above amplitude positive; when n is written underneath it indicates that an odd multiple of 180° has been rejected or added. It will be noticed that the amplitude (A,) and phase (aj of the component sought are taken as they come out from the analysis; while for the components (£, (7, . . . ) whose effects are to be eliminated, the best amplitudes and phases obtainable are to be used. In the above work A^B^G, . . . denote the iJ's of A, B, G, . . . instead of the E's, and this fact may be signified by accenting the A, B, G, . . . . 60. The effect of a short-period component upon daily mean sea level. By '• daily mean sea level" we shall generally imply that the 24 hourly heights corresponding to 0'-, 1^, 2^, . . . 23'^ are simply added together and the mean taken. The value will obvi- ously pertain to 1 1^^ 30" a. m. instead of noon. The sum or mean could be made to pertain to noon by including the 0^ value of the next day, in which case half weight should be given to this valus and half to the 0"^ value of the day in question. Let the equation of the short-period wave be y=^ A cos (at + a) (436) in which the time is supposed to be reckoned from O'' a. m. of the first day of the series as usual. The average height of the surface of the sea for any day (rth day of series) is, so far, as dependent upon A, 2/, = 25«^ cos [^(24 r-i «) + llj « + a j sin 12ft (437) since t is taken between 24 (/• — 1) — J- and 24 (r — 1) + 23J hours. This is rendered a maximum or minimum according as sin 12a is positive or negative by putting « = — 24a (r — 1) — ll^a; (438) this gives for the maximum elevation or depression 2^'= 2ift^"''^^^«- Assuming that 24« is not far from some multiple of 360°, equation (437) may be represented by a curve whose abscissae are proportional to 24 (r — 1) + 11^. The amplitude of this long- period wave, which we may for the present call L, is 2^ 1 2 sin 12a I; (439) the speed is s ~ ft; the phase is X = a [and so C (L) = Z (A) ], (440) when sin 12 « is positive, and 1 = « ± 180° [and so C (i) = C {A) -± 180°] (441) when sin 12 a is negative. Again, suppose that a' represents the phase of J. at any given midnight. The mean of the 24 hourly heights for the following day is, § 27, ^'^^^ A cos (at + a') = 4-si5^ ^°' (11 i« + «') ; (442) that is, discrete intervals increase the value of L.* The following mechanical means of determining the average height of the sea for any given day has been suggested by Prof. J. 0. Adams: t • See Laska, Sammlung von Formeln, pp. 409,417. t Report B. A. A. S., 1883, p. 104. Or H 68,69, below. KEPORT FOE 1897 — PAKT II. APPENDIX NO. 9. 549 The value of y, the average daily height for midnight preceding the rth day, dependent upon several short period components A, B, ... is, y = ^ Sin 12a ^^^ [24 r - 1 a + «] + 24 sm ^« ■■ or if i be written for r — 1 A sin 12a B sin 12& 24 sin ^b cos [24 r - 1 6 + yS] + 2/ = cos [24ia + «] + f/^"/f,^ cos [24^& + y3] + 24 sin Ja ^ ' "•• ' 24 sin J& while the expression for the height of the tide at any time is y = A cos (at + a) -\- B cos [M + /S) + (443) (444) (445) If a tide predictor which mechanically sums (445) when the amplitudes introduced into it . . , be set with amplitudes are A, B, and with phases A sin 12a B sin 12b 24 sin ia ' 24 sin ^b ' ' ' a = argo A — A°, J3 = argo B — B°, (446) (447) (to any of which 180° should be applied when the amplitude (446) is negative) it will give at ll* 30"" of each day the average value of the 24 hourly heights of that day so far as these heights depend upon A, B, . , , , 61. Example showing the application of the harmonic analysis. Hourly tidal ordinates. Station, Sitka, Alaska. Observer, Fremont Morse. •Tabulator, A. F. Z. Kind of time used, mean local civil. Lat. 57° 4' N. ; long. 135° 20' W. Date, 1893. Tide gauge No. 34; scale, Vff- Readings are reduced to staff. Day of month. July I 2 3 4 5 6 7 Horizontal sums. Day of series. I 2 3 4 5 6 7 h. m. Feet. Feel. Feet. Feet. Feet. Feet. Feet. 00 I3'9 I3'i 11-6 lO-Q 8-4 7-3 6-8 71-1 I 00 i4"5 14-1 13-0 II-4 9-6 8-0 67 77-3 2 00 14-2 i4'4 13-8 12-5 10-9 9-2 7 '3 82-3 3 00 13-0 13-8 13-8 13-2 120 10-4 8-5 84-7 4 00 io'9 12-2 12-9 13-0 12-4 "■5 97 82-6 5 00 8-5 lO'I 11-2 II-9 12-3 120 10-8 76-8 6 00 60 7-5 8-9 IQ-I II-I II-7 11-3 . 66-6 7 00 4-3 5 '5 6-8 8-0 9-5 10-8 11-2 56-1 8 00 3 '5 4'i 49 6-1 7-6 9'3 IO-5 46-0 9 00 3 '9 3-9 4-1 4-8 6-0 7-8 9-3 39-8 10 00 5-3 f7 4-2 4-3 5-1 6-5 8-1 38-2 11 00 7-5 6-5 5-3 4-9 5"i 5-9 7-1 42-3 Noon. 9-5 8-4 7'i 6-2 57 6-0 6-7 49-6 13 00 II-4 lo-s 9-2 8-2 72 6-9 6-9 60-3 14 00 12-5 12-1 H-2 10-2 9-2 8-4 7'9 71-5 15 00 12-8 12-9 12-4 II-7 10-9 IO-2 9-3 80-2 16 00 12-3 12-8 12-8 12-7 12-4 11-9 ii-o 85-9 17 00 II'I I2-0 12-4 '12 -8 i3"i i3"i 12-6 87-1 18 00 9-8 107 "•3 I2'2 13-0 13-6 137 84-3 19 00 8-8 9"4 IQ-Q 10-9 I2-I 13-2 14-0 y8-4 20 00 8-4 8-5 8-8 9"5 10-7 12-1 i3'5 71-5 21 00 8-8 ^3 7-9 . 8-3 9-1 10-5 I2-I 65-0 22 00 10 -Q 8-9 8-0 7-6 8-0 8-8 10-4 61-7 23 00 Sums. "•5 IO-2 8-7 77 7-3 7-4 8-5 61-3 232-4 234-6 230-3 228-2 228-7 232-5 233-9 I 620-6 550 UNITED STATES COAST AND GEODETIC SURVEY. Hourly tidal ordinates. Station, Sitka, Alaska. Observer, Fremont Morse. Tabulator, A. F. Z. Kind of time used, mean local civil. Lat. 57° 4' N. ; long. 135° ac/ W. Date, 1893. Tide gauge No. 34 ; scale, iV. Readings are reduced to staff. Day of month. Julys 9 10 II 12 13 14 Horizontal sums. Day of series. 8 9 ID n 12 13 14 h. m. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 00 7-0 8-3 lO'O 12-5 14-4 16-0 16-2 84-4 I 00 61 6-4 ■7-5 lO'O 12-3 14-8 i6-i 73 '3 2 00 5-8 5 '3 5-6 7-3 9-6 12-3 14-6 60-5 3 00 6-5 5'o 4-2 5'o 6-6 9'3 I2-I 48-7 4 00 7'5 5'5 3'9 3'5 4'i 6-1 8-9 39'.5 5 00 8-9 6-8 4'5 3-0 2-5 3-6 57 35 -o 6 00 IO-2 8-4 60 3-8 2-2 2-0 32 35-8 7 00 II-Q 9-9 7-9 5-5 3-3 2-0 1-9 41-5 8 00 II-2 lo-g 97 7-6 5-2 3'2 21 49'9 9 00 107 "•5 ii-i 97 7-8 5-6 3-8 60-2 10 00 9'9 ii'3 irS "■5 IO-2 8-4 6-4 69-5 II GO 8-9 IO-6 11-8 12-4 I2-0 ii-o 93 76-0 Noon. 8-0 9-6 H'2 12-4 12-9 12-6 11-7 78-4 13 00 7-6 8-9 • 10-3 11-6 12-8 13-5 13-3 78-0 14 00 7-9 8-4 9-2 10-5 II-9 13-2 13-9 75-0 15 00 8-8 8-5 8-6 9-2 IO-5 12-1 i3'4 71-1 16 00 10-2 9'3 8-6 8-5 9-1 10-5 11-9 68-1 17 00 11-8 107 9-5 8.4 8-2 8-9 lo-i 67-6 18 00 13-4 12-4 IO-8 9-4 8-2 Z'9 8-5 70-6 19 00 14-4 13-9 12-6 ii"o 9-2 8-1 77 76-9 20 00 14-5 14-8 14-3 12-8 10-9 9-1 7-9 84-3 21 00 13-8 14-9 i5'3 14-5 13-0 ii-i 9-1 917 22 00 12-3 i4'o i5'4 15-6 14-8 I3'3 II-2 96-6 23 00 IO-2 12-2 14-4 15-5 i6-o 15-2 13-5 97-0 Sums. 236-6 237'5 234-3 231-2 227-7 229-8 232-5 I 629-6 Hourly tidal ordinates. Station, Sitka, Alaska. Observer, Fremont Morse. Tabulator, A. F. Z. Kind of time used, mean local civil. Lat. 57° 4' N., long. 135° 20' W. Date, 1893. Tide gauge No. 34; scale, -i";;. Readings are reduced to staff. Day of montli. July 15 16 17 18 19 20 21 Horizontal sums. Day of series. 15 16 17 18 19 20 21 h. m. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 00 15-2 13-9 II-8 9-3 7-6 6-8 7-2 71-8 I 00 i6-2 15-6 13-8 II -0 8-8 7-3 7-0 79-7 2 00 i5'9 16-3 15-2 12-7 10-3 8-3 7-3 86-0 3 00 14-3 15-9 15-6 13-7 11-7 9-6 8-1 88-9 4 00 11-6 14-0 14-8 13-9 12-5 10-7 9-2 86-7 5 00 8-5 11-4 13-2 13-2 12-7 11-5 IO-2 80-7 6 00 5 '4 8-3 10-5 II-4 11-9 11-6 II-O 70-1 7 00 3'i 5-6 7-7 9-1 10-4 II-Q II-2 58-1 8 00 2-4 4-1 5-6 7-1 87 9-9 10-9 48-7 9 00 3'i 3-7 4-4 5-6 7-1 8-8 10-2 42-9 10 00 4-9 47 4-5 5-0 6-1 7-7 9-4 42-3 11 00 7-8 6-8 5-7 5-5 5-9 7-1 8-7 47-5 Noon. 10-5 9-4 7-9 6-8 6-5 7'J 8-3 56-5 13 00 12-8 12-0 10-2 8-8 7-8 7-8 8-5 67-9 14 00 i4"3 13-9 12-6 ii-o 97 9-1 9-1 79-7 15 00 14-5 14-9 14-1 12-8 11-5 10-6 IO-2 88-6 16 00 137 14-9 14-7 •14-0 13-0 12-2 II-4 93-9 17 00 12-0 13-6 14-1 14-2 13-7 13-2 12-7 93-5 18 00 IO-2 11-8 12-7 13-4 13-6 13-6 13-5 88-8 19 00 8-6 9-9 10-7 II-8 12-6 13-4 13-8 80-8 20 00 7'9 8-4 9-0 IQ-I II-I 12-4 13-4 72-3 21 00 8-2 7-9 7-7 8-4 9-4 10-9 12-4 64-9 22 00 9-6 8-5 7-3 7-3 8-1 9-4 10-9 61-1 23 00 Sums. 11-8 lO-Q 7-9 7-1 7-1 8-0 9-4 6i-3 242-5 255-5 251-7 243-2 237-8 238-0 244-0 I 712.7 KEPOKT FOR 1897 PART II. APPENDIX NO. 9. 551 Hourly tidal ordinates. Station, Sitka, Alaska. Observer, Fremont Morse. Tabulator, A. F. Z. Kind of time used, mean local civil. Lat. 57° 4' N., long. 135° 20' W. Date, 1893. Tide gauge No. 34; scale iV. Readings are reduced to staff. Day of month. July 22 23 24 25 26 27 28 Horizontal sums. Day of series. 22 23 24 25 26 27 28 h. m. 00 1 00 2 00 3 00 4 00 5 00 6 00 7 00 8 00 9 00 10 00 11 00 Noon. 13 00 14 00 15 00 16 00 17 00 18 00 19 00 20 00 21 00 22 00 23 00 Sums. Feet. 80 72 6-8 7-0 77 8-8 9-8 io'6 II'O io'9 10-5 99 9'3 9-0 91 9-6 IO-5 "■5 12-6 i3"3 13-3 12-9 11-9 10-4 Feet. 8-8 7-6 6-5 6-0 6-2 7-0 8-1 9"3 IO'2 10-8 io'9 107 IO'2 97 9-3 9-3 97 IO-5 11-6 12-6 13-2 i3"3 12-8 II-6 Feet. lO'I 8-4 6-9 5-8 5-4 57 6-5 7-9 92 io'4 ii-i ir3 10-9 10-4 9-8 9-3 9'3 57 IO-6 117 12-8 13-4 i3'4 127 Feet. 11-4 97 7-8 6-2 ^^8 %\ 8-2 97 II-O 117 117 II-4 IO-6 9-8 9-3 9-3 lo-o ii-i 12-4 13-4 14-0 13-8 Feet. 12-9 ii"3 9-2 7-2 5-5 4-8 4-7 5-8 7-4 90 IO-8 12-2 12-6 12-5 ii-S io'9 lOI 9-6 9-9 10-8 12-2 13-7 14-9 15-3 Feet. 14-9 13-5 11-6 9-2 7-1 5-4 4-7 • 5-1 6-4 , 8-2 lo-i I2'0 13-0 13-2 12-6 11-6 IO-3 9-4 9-1 9-5 107 12-3 13-8 14-9 Feet. 14-8 14-3 12-5 10-3 7-8 5-7 4-3 4-0 4-9 6-6 8-8 10-9 12-5 13-2 13-0 I2-I IO-7 9-4 8-6 8-6 9-4 10-9 12-7 14-2 80-9 72-0 61-3 51-7 44-8 42-2 43-4 49-3 57-3 65-6 73-2 78-7 8o-2 79'4 76-2 72-6 69-9 69-4 72-4 77-6 84-0 89-9 93-5 92-9 241-6 235-9 2327 234-3 245-1 248-6 240-2 I 678-4 Hourly tidal ordinates. Station, Sitka, Alaska. Observer, Fremont Morse. Tabulator, A. F. Z. Kind of time used, mean local civil. Lat. 57° 4' N., long. 135° 20' W. Date, 1893. Tide gauge No. 34; scale ■^. Readings are reduced to stail. Day of month. July 29 Day of mouth. July 29 Day of month. July 29 Day of month. July 29 Day of series. 29 Day of series. 29 Day of series. 29 Day of series. 29 h. fit. 00 1 00 2 00 3 00 4 00 5 00 6 00 Feet. 15-0 14-9 13-8 II-8 9-1 6-6 47 h. m. 7 00 8 00 9 00 10 00 11 00 Noon. 13 00 Feet. 3-8 4-0 5-4 7-5 9-8 II-9 I3-I h. m. 14 00 15 00 16 00 17 00 18 00 19 00 Feet. 12-8 II-5 10-0 8-8 8-2 k. m. 20 00 21 00 22 00 23 00 Sums. Feet. 8-6 9-8 117 13-4 239-6 552 UNITED STATES COAST AND GEODETIC > SURVEY. Hourly sums. Station, Sitka, Alaska. Component, M. Computer, D. S. B. Kind of time used, mean local- civil. Lat. 57° 4' N., long. 135° 20' W. Year. Month. Day. Hour. Observations begin 1893 July i o Observations end 1893 July 29 23 Page. 0" ik ak 3' 4' 5' 6' 7" » 9" lo' II' 1 89-3 91-4 88-0 78-9 73-6 50-8 39-9 34-5 42-0 43-6 ,55-8 69-1 2 62-9 7,V6 83-9 98-9 82-7 75-2 66-4 67-8 57-2 62-2 71-8 84-9 3 8o-i 105-9 99-0 98-1 90-8 79-2 75-8 48-1 44-5 46-0 60-6 62-6 4 I02'6 82-5 84-1 82-1 86-5 63-4 51-2 48-7 39-8 41-2 40-0 48-2 5 Sums. i5'o 14-9 13-8 11-8 9-1 6-6 4-7 3-8 4-0 5-4 7-5 272-3 334-9 370-4 369-9 371-8 345-4 277-7 239-9 203-8 187-3 197-0 233-6 Divisors. 29 29 28 29 30 28 29 29 29 29 29 28 Means. "•55 12-77 13-21 12-82 11-51 9-92 8-27 7-03 6-46 6-79 8-06 9-72 Residuals. +1-66 +2-88 +3-32 +2-93 + 1-62 +0-03 -1-62 -2-86 —3-43 -3-10 -1-83 -0-17 Page. 12' ijh 14" 15" 16" 171. IS' 19' aot 21' 22h 23' I 8i-o 103-0 103-5 86-0 76-9 66-4 50-7 54-8 47-4 52-6 60-4 81-0 2 II2'9 92-3 92-4 86-0 73-7 65-3 46-7 32-9 24-9 25-2 38-9 47-9 3 72-0 93-9 97-3 104-8 95-0 66-4 520 41-4 37-4 44-9 51-3 65-6 4 59'o 81 -8 79-9 83-6 82-9 79-0 82-4 75-7 64-3 66-3 72-3 80-9 5 Sums. 9-8 11-9 I3-I 13-4 . 12-8 11-5 lo-o 8-8 8-2 8-6 21-5 13-4 3347 382-9 386-2 373-8 341-3 289-6 241-8 213-6 182-2 197-6 244-4 288-8 Divisors. 29 30 29 29 29 29 29 30 28 29 30 29 Means. 1 1 "54 12-76 13-32 12-89 11-77 9-99 8-34 7-12 6-51 6-81 8-15 9-96 Residuals. +1-65 +2-87 +3-43 +3-00 +1-88 +0-10 —1-55 —2-77 -3-38 -3-08 -1-74 +0-07 Sum of means = Mean = = 237-27 = 9 REPORT FOR 1897 — PART II. APPENDIX NO. 9. 653 Component M. Harmonic analysis of tides. Station, Sitka, Alaska. Beginning of obs'ns, July i, 1893, o''. Lat., 57° 4' N.; long., 135° 20' W. Middle, July 15, 1893, 12^. + 1-66 +2-88 +3'32 +2-93 + 1-62 +0-03 -1-62 -2-86 -3'43 -3-10 -1-83 — o'I7 +1-65 +2-87 +3 '43 +3'oo + 1-88 +0'I0 -1'55 -277 -3-38 -3-08 -174 +0'07 -i-o"oi — 0*11 — o"o7 — 0'26 — o"o7 —0*07 —0*09 -005 — 0'02 —0*09 — o'24 2dJ^3 TeverBed -0'24 -o"09 -0'02 -0-05 -0-09 •966 •865 707 ■5 ■259 707 o -707 707 •259 -•866 -•707 ■5 •966 H-o^oi +o^25 —0^02 —0^05 — 0^21 -f0^02 —0^07 5x8 12 Ci -t-o^olo -t-0^242 —o'oi-j -o^o35 -o^ios. +o^oo5 -{-©•loo 4-o^oo83 6X8 12 £3 +o^olo +0-177 +0-035 +0^210 — 0^014 +0^4l8 7X8 12 C5 +0^010 +o^o65 +o'oi7 +0^035 -o^ios +o^oig +0^041 +0^0034 •259 •5 •707 •866 •966 259 3+4 +o^ol —0^23 — 0^20 —0^09 — 0^31 — o'l6 — 0'07 12x15 12 ^i -o^o6o -o^ioo -o'o64 -o^268 -0T55 -o^070 -o^7i7 -0^0598 13X15 12 i3 -o^i63 — 0^200 —0^064 +0T13 +0^070 -0-244 -0'0203 14x15 12 J5 — 0"222 — OTOO +o"o64 +0^268 — 0^041 —0^070 -O^IOI -o^oo84 19X11 12 C7 +0^010 —©•065 +0^017 -0-035 —0-105 —0-019 -0-197 -0-0164 21x18 12 ^7 —0-222 +0-100 +0-064 -0-268 —0-041 +0-070 -0-297 -0-0248 1+2 2d half ot 23 +3'3i +5-75 +6-75 +5-93 +3-50 +0-J3 -3'i7 -5-63 -6-81 -6-l8 -3'57 — o-io 25- 26. 27. 28. I + -866 -5 + ■5 -866 — I — -■5 -866 + --866 •5 + 23-24 + 6-48 +11-38 +13-56 + I2-II + 7 '07 + 0-23 25x29 12 £-2 + 6-4S0 + 9-855 + 6-780 - 3-535 0-199 + 19-381 + 1-6151 26X29 12 ^2 + 5-690 + 11-743 + I2-II0 + 6-123 + 0-II5 +35-781 + 2-981I 27x29 12 C6 + 6-480 13-560 + 7-070 — O'OIO — 0-0008 28x29 12 S6 + 11-380 + 0-230 0-500 ■ 0-0417 23+24 24 6A +0-14 +0-12 — o-o6 -0-25 —0-07 +0-03 35- 2d half of 34 -0-25 —0-07 +0-03 36- ■866 -866 38. +0-39 +0-19 —0-09 39- 36x38 12 C4 +0-390 +0-095 +0-045 +0-530 +0-0442 37x38 12 J4 +0-165 -0-078 +0087 +0-0072 -866 -866 34+35 — o-ii +0-05 —0-03 41x43 12 Cj — o-iio —0-025 ,+0-015 -0-120 — o-oioo 42X43 12^8 +0-043 + 0-026 +0-0698 +0-005 Log.i Log.c ]>g. tan i Vo + u....'.'.'.'.'.. k' K I,og. [cos] sin ^ ■IvOg. aug. fac .. I = 0. 554 UNITED STATES COAST AND GEODETIC SURVEY. 62. Formulae for inferring amplitudes and epochs. Amplitudes. J. foo79 O ' lo-o, '56 Ki 2 Q = 0'026 Oi /Ji = 0-038 Oi] 00= 0-043 Oi P,= 0-331 K, Qi = 0-194 Oi K2 = 0-272 S2 _fo-i45 N2 lo-o ^ — lo-028 Mj 2N^ 0-133 N2 R„ = o-ooS S3 T2 = 0-059 S2 X2 = 0-007 M2 Ml = 0-024 M2 J'2= 0-194 Ni 2Q° = Pi° = 00° = P.° = k;2° = Iv2° = 2N° = R2° = Tj° = V = J'2° = Epochs, K,° + 0-496 (Ki°-0,°) K,° -1-992 (K,°-0,°) K,° - 1-429 (K,°-0,°)^ 2Ki° -Oi° K.° Ki° - 1-496 (K,°-Oi°) 2M2° — N2° 82° -0-464(82° -M.2°) 2 N2° — M2° 82° Sj° 8-2° -0-536 (82° -M2°) 2 M2° — 82° M2° — 0-866 (M2°—N2°) 63. Clearance from the effects of other components. Computation of equilibrium arguments Va-\-u. Sitka, Alaska, July 1-29, 1893. Beginning of series, Tables 3 and 4. ELEMENTS. Middle of series, Tables 6, 7, 8, and 9. A«i>p, N I P V i v' iv" B Q 99°-27 3030-09 69°-79 281°-H 24°-152 280-225 67°-405 4o-540 4°-090 3°-238 6°-856 ll°-97 50°-23 Com- Va+u . LokK po- nent. From Table 1 (Greenwich) t^o, midnight. Numerical values. Correc- tion from Table 5. Local Vo+u. Tables 10, II, 12, 13- 00 00000 Ji h+s—p+go°—v 99-27-t-303-09- 69-794- 9ooo-4-54 58-03 + 4-27 62-30 2Q h-is+2p-go°+2l-v 99-27— l32-36-(-l39-58- 90oo-|-8-l8- 4-54 20-13 -19-31 0-82 pi 3 A-3 s-p-i)o°+2 l-v 297-81- 189-27— 69-79- 90-00-H&-I8— 4-54 312-39 -13-76 298-63 Ki k+go°-v' 99-27-f 90-00- 3-24 186-03 + 0-37 186-40 9-95652 Ot h—2S—ga°+2 l—v 99-27— 246-18- 90-00-H 8-l8— 4-54 126-73 - 9'52 II7-2I 9-93166 00 h + 2S+qo°-2$~V 99"27-F246-l8-f 90-00— 8'l8— 4'54 62-73 4-10-26 72-99 Pj -A -90° - 99-27- 90-00 170-33 - 0-37 169-96 0-00000 Qi A-3 s+p-g^+2 l-v 99-27- l89-27-f 69-79— 90-00-1-8-I8- 4-54 253-43 -14-42 239-01 9-93166 K2 2A-2 v" 198-54- 6-86 191-68 4- 0-74 192-42 9-88937 ^ 2h-s-p+i8o°+2 {-2 iz-ZE I98-S4-303-09— 69-794- i8o-oo-l-8-i8— 9-08-11-97 352-79 - 4-25 348-73 9-92319 :mi A-i-f90°-l-J-^-|-j3 99-27—303-09-1- 90-00-1- 4-09— 4-54-1-50-23 295-87 - 4-57 291-30 9-92037 M, 2 A— 2 J-t-2^— 2 I* 198-54-246-18-I- 8-i8- 908 311-46 - 9-15 302-31 0-01483 M4 4A-4i-l-4f-4>' 37-08-I32-36-F 16-36- l8-l6 262-92 -18-28 244-64 0-02966 M6 6 A-6 j-l-6 1—6 v 235-62- 18-544- 24-54- 27-24 214-38 -27-50 186-88 0-04449 N= 2/l-iS+p + 2(-2 1> 198-54-189-274- 69-794- 8-l8-9-o8 78-16 -14-04 64-12 0-01483 2N 2 ft— 4 S+2p + 2 {— 2 V 198-54-132-364-139-584- 8-18-9-08 204-86 -18-94 185-92 S;, 1 0-00 0-00 0-00 0-00000 T2 -h+pi - 99-274-281-11 181-84 - 0-37 181-47 lil 4A— 4 J-H2 J-2 I- 37-08-132-364- 8-i8- 9-08 263-82 -18-28 •245-54 0-01483 »'2 4A— 3i— /-f2 1—2 V 37-08- 189-27— 69-794- 8-18— 9-0B 137-12 -13-39 123-73 0-01483 REPORT FOR 1897 — PART II. APPENDIX NO. 9. 556 When the series analyzed is practically a calendar year, the value of Vo + u can be taken, without computation, from Table 3, and adapted to the time meridian by Table 5; the F and /can be taken, -svithout modification, from Table 10. Com- po- nent. From auxiliary tables. From analysis and inference. Re R ^c i Fo-t-M F f H K From analysis fxH From analysis K-(yo+u) o, CO Pi K2 Lz Mz Nz 2N Sz Tz Mz 62-30 00-82 298-63 186-40 117-21 169-96 23901 192-42 348-73 302-31 64-12 185-92 0-00 181-47 245-54 12373 0-86569 0-85439 0-85439 0-90474 0-85439 0-58221 I -ooooo 085439 0-77512 0-83790 103475 1-03475 1-03475 I -ooooo r-00000 1-03475 103475 1-15515 1-17043 1-17043 I -10528 I- 17043 1-71759 I -ooooo I-I7043 I 29014 I -19346 0-96641 0-96641 096641 r-00000 I -ooooo 0-96641 0- 9664 1 ft. 0-079 Oi =0-0755 0026 Oi =0-0249 0-038 Oi =0-0363 1-8173X0-8230 =1-4956 0-9561 0-043 Oi =0-0411 0-331 Ki =0-4948 0-194 Oi =0-1855 0-272 Sz =0-2933 0-145 Nz =0-0897 3'5493 0-6:83 0-133 Nz =0-0822 0-8046x1-3408 =1-0788 0-059 Sz =0-0636 0-024 Mz =0-0852 0-194 Nz =0-1200 KiO-l-0-496 (Kio-Oio) =I37°-7 Kio- 1-992 (KiO-OiO) = 83°-2 K,o-i-43o(KiO-0,o) = 95°-5 I36°-3-9°'5 =1260-8 2 KiO-OiO =1480-7 PiO=KiO =1260-8 K,o-i-495{KiO-OiO) = 94°-I KzO=S20 = 390-0 2 M2O-N2O = 33O-J 3°-9 334 5 2NzO-MzO =3050-1 530-6-140-6 = 390-0 TzO=S20 = 3gO-o 2 MzO-SzO =3280-8 M20-o-868(M20-NzO) =3380-4 2-0087 1-1190 1-8772 0-2113 1-2792 0-1762 3 '4301 0-5976 0-8046 0-1523 1-0559 0-0872 0-0291 0-0425 I '653 1 I'llgo 0-0706 0-4948 0-2171 0-3784 0-1071 3-4301 0-5976 0-0794 1-0788 0-0616 0-0823 0-1160 3099 347-7 278-7 238-7 85;4 270-4 53-6 I33'6 277-4 75-4 82-4 156-9 300-4 3477 75'7 316-8 215-1 206-6 44-6 61-6 270-4 119-2 39'0 217-5 83-3 214-7 From Table 31, mean of four values, June 30-July SO. Acceleration of Ki due to Pi = — 10° -45 X -^ ( K, ) = — 9° -455 Resultant amplitude K and P = [(1-2378 — i") X -^ (Ki)] + i- = 1-2151; recip. =0-8230 Acceleration of S2 due to Kz = — n°"825 X/(K2) = — i5'256 " SzduetoTz = + o°-675 " Sz due to Kz and Tz = - i5°-256 -f o°-675 = — i4°-58i Resultant amplitude Sz and Kz = [(0-8475 — i') X/(K2)] + i = 0-8033 " " SzandTz =0-9425 " " Sz and Kz and Tz = 0-8033 + o'9425 — i = 0-7458; recip. = i'34o8 64. Application of elimination tables. -tfKi=+0-050 J,' cos[ 904-C. (Ki)-C ( Ji ] + 0-0565 Oi' cos [3380+C. (K,)-C (O,)] + 0-03(iOO' cos[220-)-f^ (Ki)-C (OO)]-h0-959 Pi cos [331°+^ {K,)-C (Pr)] + 0-052 Qi' cos[328°+C,(K,)-C ( Qi)] + 0-049(2Q')cos [SlQo+C. (Ki)-C(2Q)] +0-011 pi' cos[354o+C(Ki)-C ( a)]. rfKi=+0-050 Ji' sin [9° +C„ (Ki)-C ( Ji)l+0-0565 O/ sin [3380+C. (Ki)-C ( O,)] + 0-056OO'sm [22° +Ce (Ki)-C (OO)]+0-9o9 P, sin [331o+r. (Ki)-C (Pi)] + 0-052 Qi'sin[328o+C(Ki)-C ( Q,)J+0-049(2Q')sin [319o + C. (Ki)-C(2Q)] +0-011 pi' sin [3540+C. (Ki)-C ( Pi)]. Here J/ signifies the R of J,, or B {Si), and not Ji (or the S" of J,) nor B, (J,), which means the R direct from analysis and so before the effects of the other scheduled components upon it have been eliminated. Similarly for Oi', Pj, Qi', etc. -tyK, =+ 0-050x0-0872 cos ( 9o+310o_ 75o)+0-0565xl-1190 cos (3380+3100-348°) +0-056x0-0706 cos ( 22o+310o_ 76O)+0-959 x 0-4948 cos (331o+310o_317o) + 0-052x0-2171 cos (328o+310o_215O) +0-049 x0-02!)l cos (319o+310o_ 82°) +0-011x0-0425 cos (3540+3100-1570). = +0-0043 cos 244O+0-0632 cos 300O+0-0040 cos 256O+0-4745 cos 324O+0-0114 cos 63o+ 0-0014 cos 187O+0-0005 cos l47o, = -0-0043 X 0-438+0-0632 x 0-500-0-0040 x 0-242+0-4745 x 0-809+0-0114 x 0-454-0-0014 x 0-993 - 0-0005 X 0-839 = - 0-0019 + 0-0316 - 0-0009 + 0-3839 + 0-0051 - 0-0014 - 0-0004 = +o-4i(;o. 556 UNITED STATES COAST AND GEODETIC SURVEY. (yKi = + 0-0043 sin 244O+0-0632 sin 300O+0-0040 sin 256O+0-4745 sin 324O+0-0114 sin 63O+0-0014 sin 187O+0-0005 sin 147o. = -0.0043 X 0-899-0-0632 x 0-866-0-0040 x 0-970-0-4745 X 0-588 + 0-0114 x 0-891-0-0014 X 0-122+0-0005 X 0-545 = -0-0040 - 0-0547 - 0-0039 - 0-2790 + 0-0100 - 0-0002+0-0003= -0-3315. ^ -^ -0-3315 -0-3315 ^^P '^^= 2-0087-0-4160 = 1-5927 =-0-2081; (m—l) sin (//re — 182 Z)] A + bp(n) ^(m-\-l) cos (/Jn + 182 l) — iP(n) (m — l) cos (z3« — 182J)] B. 6584 36 (471) (472) 562 UNITED STATES COAST AND GEODETIC SURVEY. TUese coe£Boient8 may be written in a form more convenient for computation. For (OTJ:0 = o°^^/'"i^!! =^ 008 182 (m± )- J sin 182 (m ± I) cot ^ (m ± I). 2 sin i (m J^ J) v j_ / i ^ \ -i- / ^ v _i_ / Then let Also let K (B, = (m + J) + (m — l) ' Z (n, = (ni + O — (TO — ' il> (n) COS An = -^ ip (n) sin ^m sin 12 n sin I- 71 cos /^n- The ftmctions K (m, 0, Z (m, 0, C {n), S («) may be easily computed from (473), (474), (475). Then if we denote the additive clearance for 2Sh cos U by and that for 2dh sin U by We have (473) (474) (475) (476) (477) (478) [A, n, I, cos] A 4- [B, n, I, cos] B, [A, n, I, sin] A+[B, n, I, sin] B. [A, n, I, cos] = — (?i) K (», Z) cos 182 J + S («) Z (», sin 182 I [B, n, I, cos] = — S (m) K (n, I) cos 182 ? — C (m) Z (», sin 182 I [A,m, ?,sin]= — S (») Z (m, cos 182 ! — C (m) K (», sin 182 ? [B, n, Z, sin] = C (m) Z (», cos 182 i — S (») K (Ji, sin 182 i We must remark that if i (m + = 360°, (m+0 is equal to 182-5. This case arises when I is the tide MSf of speed 2(^ — 77), and m the tide Ms of speed 2 (f — 6), for m + Hs then 24 X 2 (r — J?) = 720°. The clearance of the long-period tide I from the effects of the short-period tide n requires the computation of these four coefficients. For the clearance of the five long-period tides from the effects of the three tides Ms, N, O, it will be necessary to compute 60 coefficients. If it shall be found convenient to make the initial instant or epoch for the tides of long period different from that chosen in the reductions of those of short period, it will, of course, be necessary to compute the values which A and B would have had if the two epochs had been identical. A and B are, of course, the component semi-ranges of the tide of short period at the epoch chosen for the tides of long period ; to determine them it is necessary to multiply E by the cosine and si^e of V-{-u — k at the epoch. [Q.] Schedule of Coefficients for Clearance of Daily Means in the Final Equations. / = (J — as 2d 2(6-7)) 27) (M2) w=2(;k — (J). "A, «, I, cos" B, n. /, COS A, n /, sin B, n /, sin -0-05S57 -0-I7036 -0-17075 ho-04410 -fo -00302 —0-03773 +0-04170 -j-o-01052 +5-7393 — 2-9228 —2-8400 — 57271 — 0-10410 -0-07525 — 0-00176 -j-0-00476 -0-01465 -0-07546 -0-00353 ho-00958 (N) « = 2;k— 3(*-|-ca. A «, I, cos" B, ra, I, COS A n /, sin B, n. /, sin —0-05884 —0-07738' —0-02059 -1-0-I138I -1-0-03680 —0-22337 —0-15245 -0-08544 -I-0-02938 —0-19384 — 0-I22IO —0-08081 — 0-01760 +0-00254 +0-00020 -)-o-oooo7 — 0-01760 +0-00254 +0-00041 -t-0-00015 (O) n^=y — 26. A, n. I, cos" B, n. I, cos" A, n, /, sin B, n, /, sin -0-06485 -0-34765 -0-34523 -0-04052 +0-01673 -0-07788 +0-08418 +0-03379 +0-01582 -0-08158 +0-08748 +0-03295 — 0-19240 —0-18260 — 0-00460 +0-00897 -0-19340 — 0-183 II —0-00926 +o-oi8o2 REPOET FOE 1897 — PART II. APPENDIX NO. 9. 563 It may happen from time to time tliat the tide-gauge breaks down for a few days, from the stoppage of the clock, the choking of the tube, or some other such accident. In this case there will be a hiatus in the values of Sh. Now, the whole process employed depends on the existence of 365 continuous values of Sh. Unless, therefore, the year's observations are to be sacrificed, this hiatus must be filled. If not more than three or four days' observations are wanting, it will be best to plot out the values of Sh graphically on each side of the hiatus, and filling in the gap with a curve drawn by hand, use the values of Sh giv^en by the conjectural curve. If the gap is somewhat longer, several plans may be suggested, and judgment must be used as to which of them is to be adopted. If there is another station of observation in the neighborhood, the values of Sh for that station may be inserted. The values of Sh fftr another part of the year, in which the moon's and sun's declinations are as nearly as may be the same as they were during the gap, may be used. It may be, however, that the hiatus is of considerable length, so that the preceding methods are inapplicable : as when in 1882 the tidal record for Vizagapatam is wanting for 67 days The following method of treatment will then be applicable : — We find approximate values of the tidal constituents of long period, and fill in the hiatus, so as to complete the 365 values, with the computed height of the tide during the hiatus. To find these approximate values we form 2Sh cos It and 2Sh sin It for the days of observation; next, in the ten final equations of Schedule P we neglect all the terms with small coefficients, and in the terms whose coefficients are approximately 182*5, we substitute a coefficient equal to 182"5 diminished by half the number of days of hiatus. For example, for Vizagapatam in 1882 we have 182'5 — | X 67 = 149, and, e. g., 2Sh cos (d — a) <^149A approximately. After the approximate values of A, B, C, D, &o., have been found, it is easy to find the approximate height of tide for the days of the hiatus. This plan will also apply where the hiatus is of short duration. It may be pursued whether or not we are working with cleared daily means; for if the daily means are uncleared, as will henceforth be the case, we import with the numbers by which the hiatus is filled exactly those fictitious tides of long period which are cleared away by the use of the "clearance coefficients," in preparing the ten final equations for solution. Other methods of treating a stoppage of the record may be devised. If the stoppage be near the beginning of the year, or near the end, we may neglect the observations before or after the gap, and compute afresh the 100 coefficients of Schedule P, and the clearance coefficients of Schedule Q for the number of days remaining. If the gap is in the middle we might compute the values of the coefficients of Schedules P and Q ae though the days of hiatus were days of observation, bearing in mind that the formulae are to be altered by the consideration that time is to be measured from the initial 11" 30"" of the year, instead of from the initial lli" 30™ of the days of hiatus. The so computed coefficients are then to be subtracted from the values given in Schedules P and Q, and the amended final equations and amended clearance coefficients to be used. It must remain a matter of judgment as to which of these various methods is to be adopted in each case. 70. Method of Equivalent Multipliers for the Harmonic Analysis for the Tides of Long Period. Up to the present time the harmonic analysis for these tides has been conducted on a plan which seems to involve a great deal of unnecessary labour. If I be the speed of any one of the iive tides for which the analysis has been carried out, in degrees per m. s. day, the values of cos It and sin It have been computed for < = 0, 1, 2 . . . 364, so that there are 730 values for each of the five tides. These 730 values have then been multiplied by the 365 Sh'B corresponding to each value of t, and the summations gave 2Sh cos It and 2Sh sin It, the numerical results being the left-hand sides .of one pair of the ten final equations explained in § 68. Now, it appears that this labour may be largely abridged, without any substantial loss of accuracy. The plan proposed by Professor Adams is that of equivalent multipliers. The values of cos It may be divided into eleven groups, according as they fall nearest to 1-0, -9, -8, -7 ... . -2, -1, 0. Then, as all the values of Sh are to be multiplied by some value of cos It, and that value of cos It must fall into one of these groups, we collect together all the values of Sh which belong to one of these groups, sum them, and multiply the sum by the corre- sponding multiplier, 1-0, -9, -8, &c., as the case may be. Since there are as many values of cos It which are negative as positive, we must change the sign of half of the Sh'e. This changing of sign may be effected mechanically as follows:— In the spaces for entry of the Sh'a, those Sh'a whose sign is to be unchanged are to be entered on the left side of the space if positive, and to the right if negative; when the sign is to be altered this order of entry is to be reversed. Thus in the column corresponding to each multiplier we shall have two sub-columns, on the left all the Sh'B which, when the signs are appropriately altered, are +, and on the right those which are — . The sub-columns are to be separately summed, and their difference gives the total of the column, which is to be multiplied by the multiplier appropriate to the column. The treatment for the formation of 2Sh sin It is precisely similar. The annexed form [Schedule R] is designed for entry for determination of SSh cos {d—?f) t. The entries of Sh are to be made continuously in the marked squares from left to right, and back again from right to left. The numbers in the squares, which in the computation forms are to be printed small and put in the corner, indicate the days of observation. The rows are arranged in sets of four corresponding to each complete period of 2((3 — 7). In the middle pair for each period the -f values of Sh are to be written on the right, and in the rest on the left. The word 'change' opposite half the rows is to show the computer that he is to change the mode of entry. Each column, excepting that for zero, is to be summed at the foot of the page, and multiplied by the multiplier corresponding to its column. A pair of forms is required for each tide of long period; they are very easily prepared from the existing forms, in which the values of the multipliers are already computed. 564 UNITED STATES COAST AND GEODETIC SURVEY. [E] Total + Total — Total Multiply Results Sum laterally Form for Reduction of the Tide MSf. + - I'O + - •9 + - •8 + - •7 + - •6 + - ■5 + - •4 + - •3 + - •2 + - "I No entries. o I 2 3 1 1 6 5 4 8 9 10 II H 13 12 15 1 16 17 18 21 20 19 22 23 24 25 29 .28 27 26 30 31 32 33 • 36 35 34 37 38 39 40 44 43 42 41 45 46 47 51 50 49 48 52 53 54 55 58 57 56 59 60 61 62 66 65 64 63 * 67 68 69 70 74 73 72 71 &c. &c. &c. No entries. Xro X-9 X-8 X7 X-6 X-5 X-4 X-3 X-2 x-i X-o 1 change. change. change, change. change, change. change, change. change, change. Sum of + = Sum of — = 2Skcos 2 {d — r/) t= . . . REPORT FOR 1897 — PART II. APPENDIX NO. 9. 565 71. To reproduce the quantities harmonically analyzed. Having analyzed a set of partial or hour sums and determined the e's and s's, or iJ's and C's, it is sometimes desirable to recombine the partial tides of the analysis sheet into a resultant curve in order to see how this curve compares with the curve obtained by plotting the hour sums; or it may be done for other purposes. The nature of the case will suggest how many of the harmonics should be retained. The same process is applicable to inequality curves, certain constants having been found from the analysis of the heights or intervals involving the particular inequality in question. (See § 54, Part I.) The accompanying form will show how this combination may be made. Form for reproducing observed quantities from the results of an analysis. y=y^ + C, COS^ + Si sin X -\- Ci COS IX + fj sin 2x -{- c-^ cos 'ix -\- jjsin 3^ + A X — x=i^ = 15° JIT = 2h = 30° x = -^ = ^i° x = n^ = 60° ^ = 5" = 75= x=ea = 90° ;r = 7I1 = 105° (0) ^X(o) (I) ^X(l) (2) ^X(2) (3) ^X(3) (4) ^X{4) (5) ^X(5) (6) ^XC6) (7) ^X(7) Ci = I 966 ■866 •707 ■5 259 - -259 Ji = ' 259 •5 •707 •866 966 1 •966 C2 = I 866 •5 - '5 - 866 —I - -866 Sl = o 5 •866 I ■866 5 . — ^5 C3- I 707 - •707 —I - 707 •707 i3 = 707 I •707 - 707 —I - ^707 H = I 5 - '5 —I - '5 5 I J4 = o •866 •866 - •866 - ^866 •866 C6 = I — I I —I S6 = I — I I — I C8 = I -■5 - '5 I - '5 - '5 I - '5 i8 = Sum •866 - ^866 •866 - ^866 •866 — ;)'=J'o + su)n A X = 9<-= 120° ;ir = 9li=i35° x = v:^— 150° X= III" = 165° jr = I2h = 180° jr = I3h = 1950 jr = i4h = 210° ^=15'' = 225° (8) ^X(8) (9) ^X(9) (10) ^X(IO) (") ^X(ll) (12) ^X(I2) (13) ^X(13) (14) -4x(i4) '15) ^X(i5) Ci = - '5 - -707 - ^866 - 966 —I - 966 - ^866 - 707 ft = •866 •707 ■5 259 - 259 - '5 - '707 Cu = - "5 ■5 866 I 866 ■5 Ss = - -866 —I - ^866 - 5 5 ■866 I C3 = I ■707 - 707 —I - 707 ■707 ^3 = o •707 I 707 - 707 —I - -707 C4 = - '5 —I - '5 5 I 5 - 5 — I H = ■866 - .866 - •866 ■866 ■866 C6 = I —I I — I S6 = I —I I -1 CS = - '5 I -5 - '5 I - '5 - "S 1 SS = Sum - -866 •866 - ^866 ■866 - ■866 L-«..-i y =^o + sum A jr = i61> = 240° jr=i7h = 255° x = \%^ = 270° j: = igh = 285° X = 20*^ = 300° jr = 2ll> = 315O X = 22h = 330° ^ = 231 = 345O (16) ^X(l6) (17) -4X(I7) (18) ^X(l8) (19) ^X(I9) (20) ^X(20) (21) .^X(2l) (22) ^X(22) (23) A X (23) c% = - '5 - '259 259 •5 •707 ■866 966 Si = - -866 966 —I - 966 - •866 - ^707 - '5 - 259 Cs = - '5 — 866 —I - 866 - '5 •5 866 S2 = •866 5 - 5 - •866 — I - -866 - 5 ^3 = I 707 - 707 -I - -707 707 ^3 = 707 I 707 - -707 — I - 707 Q = - '5 5 I 5 - '5 —I - '5 5 •!4 = - •866 - •866 •866 ■866 - ^866 - ^866 C6 = I — I I —I S6 = I —I I -I f CS = - -5 - '5 1 - '5 - '5 I - '5 - '5 Suin •866 - -866 •866 - ^866 ■866 - •see ji=yo + sum y=ya-\-R-i COs(:f — C, )+7?, C0S(2^— Cs)+-/?3C0S(3^— C3) - • R = ly^ -\- s^ , tan !i = s/c, c=:J?cosZ, s = jRsmZ. 566 UNITED STATES COAST AND GEODETIC SURVEY. 72. To combine the various tidal components for any given future time. This may be illustrated by au example: To find the height of the tide at San Francisco (Fort Point), Cal., at 1 o'clock p. m. (standard time) March 1, 1910, the principal tidal components being Mj=l-69ft., M20:=332, 8^= 0-39 ft., 82° = 330% N2 = 0-37 ft., IST^o = -305°, Ki = 1-22 ft., KiO = 107°, Oi = 0-78 ft., OjO = 0-87°, Pi = 0-37 ft., Pi^ = 105°. We are to find the value of y = M2' cos (m2^ + argo M2 — M^P) + S2 cos (Sjit + argo S2 - 82°) + N2' cos (n2« + argo S^ - ^2°) + K/ cos {\it + arg„ Kj - KP) (479) + Oi' cos (Oii+ argo Oi- OjO) + Pj cos (pi« + argo Pi - Pi°) where the accent denotes that the factor/, Table 10, has been applied to the amplitude. Now Table 3 gives Ya + u^ or arg, for Greenwich midnight, while the above equation supposes it to belong to meridian 01 San Francisco and the time to be local. According to section 62, Part III, we may use the Greenwich values directly, provided we modify the epochs so as to take into account the longitude of the place and of the time meridian. The epochs modified once for all are 3440, 340°, 322°, IO80, 990, 106° respectively. By Tables 3 and 4, we have for March 1, 1910, argo M2 = 2430 + 1° = 2440, argo Kj = 3° + 58° = 61°, argo 82 = 4- = 0, argo O, = 243 + 303 = 186, argo ^2 = 107 + 311 = 58, arg„ Pi = 350 + 302 = 292. By Table 1, we have for 13 hours (midnight to 1 p. m.) m^t = n°, s^t = 30°, n^t = 10°, kit = 196°, Oit = 181°, pi^ = 194°; . • . 2/ = M2' cos 277° + 82 cos 50° + W cos 106° + Ki' cos 149° + Oi' cos 268° + Pi cos 210°. By Table 10, the values off are 0-98, 1-00, 0-98, 1-07, 1-12, 1-00 respectively. Mz' = 1-67, 82 = 0-38, N2' = 0-35, K,' = 1-31, d' = 0-86, Pi = 0-37 feet. y= + 0-20 + 0-24 - 0-10 - 1-10 - 0-03 + 0.35 = - 0-44 feet as the height of the sea, reckoned from mean sea level, at the given time. At 2, 3, 4, 5, and 6 o'clock, p.m., the heights are iu like manner found to be —0-64, —1-33, —2-32, -2-55, -1-14 feet. 73. Harmonic analysis of a series two weeTcs in extent. In the Manual of Scientific Enquiry* Darwin shows how to analyze a short series of hourly readings extending over a fortnight or a month. Summations are made for M, 8, and O. The M and O sums are analyzed in the usual way, giving the amplitudes and epochs of M2 and O,. Tbe 8 sums give an affected 82 and Ki; the summation extends over slightly different periods in the two cases. The lengths of series used are as follows : d. h. d. h. For M 14 12 or 29 00, " 8 (diurnal) 14 00 '< 28 GO, " 8 (semidiurnal) 15 00 " 30 00, " 14 00 " 20 21. ' Or B. A. A. S. Report, 1896. REPORT FOR 1897 — PART II. APPENDIX NO. 9. 567 If, for the moment, we put t„n,/,_ / (Ka) Kg sin ( arg* K^ - arg Sg) /^^q. '' ~ Sg (withTj) +/ (Kj) Ka cos (arg K^ - arg Sg)' ^ in whicli the arguments are to be taken at the middle of the series, we obtain the amount that the Sz corresponding to a given solar parallax is accelerated by Kg (see § 2, Part III). From Table 1 ^ = 3-67, arg Kg - arg S2 = 2 (^ - v"). ' <481) K2 The formula for tp may now be written tan i: = /(K.)sin2(fe-.") (482) ' 3-67 p, + f (Ka) cos 2 (fe - y") ^ ' where _ /" sun's parallax A^ _^. Sa (with %) • /^g^\ ' ~ V sun's mean parallaxy S2 = tabular value last column, Table 31. .-.820=^(82) + ^. (484) The fifth column of Table 31 gives the correction of 82° due to the direct effect of T2; Darwin has disregarded this correction. The amplitude of 82 is the observed amplitude B (82) multiplied by 3-67 cosj/^ ^ (485) 3-67 _p, +/ (K2) cos 2 (7i - v") The epoch and amplitude of K2 are obtained from the equations K20 = 82°, <486) K. = 34,82. (487) In like manner we may put tan 6 - Pi sin ( arg Pi - arg Ki) ^.gg. *^^ ^ - /(K,)Ki + Picos(argPi-argK,) ^^^^^ where (j) is the amount by which Ki is accelerated because of Pi. From Table 1 ^ = 3, arg Pi - arg Ki = - 2 A + 7-' ± 180° : (489) .-.tan ^=3 g^ ^ (^ ^ - '''} (490) ^ 3/(Ki)-cos(2 A- j^') . ^ ' KiO = C (Ki) + argo Ki + ^, (491) For Pi we have 3/ (Ki) — COS (2 /t— T-') ^ ' Pi° = KiO, ■ (493) Pi=iK,. ' (494) 74. Harmonic analysis of high and low waters.] The components Ki and d can be quite accurately obtained by the following process : Copy the heights of the high and low waters into the form for "hourly ordinates," always putting these values upon the nearest solar hour. Apply the K and O stencils in the usual man- ner, but also keeping track of the number of high waters and the number of low waters that enter into each partial hourly sum. Then bring the partial hourly sums together, and note the differ- ence between the number of low and high waters. Correct the hourly sums by this difference multiplied by one-half of the mean range of tide for the period of the observations. Analyze the * I. e., the eqtuilibrium argument, Table 1. t See Chapter III, Part III. 568 UNITED STATES COAST AND GEODETIC SURVEY. twenty-four hourly means or residuals in the usual manner. K, thus obtained should be corrected for Pi by Table 31. O, does not require correction. The example given below shows the degree of accuracy attained at Sitka from a 29-day series of high and low waters. The corresponding results for Sandy Hook, N. J., are also given. Station, Sitka, Alaska. Component, K. Computer, D. S. B. Kind of time used, mean local civil. Lat. 57° 4' N.; long. 135° 20' W. yr. mo. da. hr. Observations begin 1893, July i o. Observations end 1893, July 29 23. Page. oh ih 2h i^ 4h 5" 6h 7^ 81> gh loh ll" stencil sums 1 CO 21-2 28-4 13-2 12-5 12-0 II-4 0-0 3 '4 7-9 9'3 5-9 No. highs and lows iH, ll- 2H I H I H I H I H I t 2I, 2 I, I 1/ 2 i6-i l6-3 ,5-8 0-0 5-0 3'9 2-9 4-0 13-0 00 ii'S 11-9 I H I H I L I I- I I< I t 2 I< IH.H I H I H 3 70 6-8 23'l l6-3 15-7 14-0 12-7 II -6 II'2 2-4 3-6 9-3 II- I h IH.II, I H I H I H I H I H I H I L I X, 2 I, 4 27-6 15-3 15-1 6-8 6-0 0-0 5-3 9-4 4-7 15-0 0-0 II-O 2H I H I H I I, I I, I t. 2 I, I I< iH, IL I H 5 CO 0-0 15-1 O-Q Q-O CO 0-0 0-0 0-0 3-8 o*o 0-0 Sums I H I L 507 59'6 87-5 36-3 39'2 29-9 32-3 25-0 32-3 29-1 24-4 38-1 Page. I2I1 13" 14" 15" i6h I7h I81> igh 20I1 2ih 22* 23h Stencil suras I 67 0-0 O-Q 25-8 12-8 26-1 13-6 14-0 8-4 16-I 7-6 7-2 No. highs and lows I I, 2 H I H 2 H I H I H I I, 2 I, I I, 1 1< 2 O'O 33'0 I3'5 22-2 8-6 0-0 16-4 7-8 7-6 14-6 15-0 31-2 2H,lI, 1 H iH, iL I L 2 I< I I< I Iv I H I H 2 H 3 13-0 8-3 Q-O 0-0 29-7 14-7 28-0 13-7 13-8 7-8 7-9 7-3 2 I, I I, 2 H I H 2 H I H I H I t I L 1 1, 4 CO "■3 33-5 26-4 9-2 9'3 0-0 18-8 17-6 I3'4 13-4 0-0 I H 2H,lL 2 H I Iv I I< 2 I, 2 I< I H I H 5 CO 0-0 Q-Q O-Q 13-5 0-0 O'O 0-0 0-0 8.2 0-0 0-0 Sums I H I I, 197 52-6 47-0 74-4 73-8 50-1 58-0 54-3 47'4 60-1 43-9 45-7 Page. Qh ll> 2l> 3" 4" 5" 6h 7h 8h gh loh Ilh No. of highs 3 3 5 2 2 2 2 I 2 I 1 2 No. of lows I 2 2 I 2 I 2 4 3 5 3 3 Sums 507 59-6 87-5 36-3 39-2 29'9 32'3 25-0 32-3 29-1 24-4 38-1 J Mn X diff. — n -3-6 — ii-o -3-6 0-0 -3-6 0-0 -t-ii-o + 3-6 -t-14-6 -1- 7-3 +i-6 43'4 56-0 76-5 32-7 39'2 26-3 32-3 36-0 35-9 437 317 41-7 Divisors 4 5 7 3 4 3 4 5 5 6 4 Means 10-85 11-20 10-93 iQ-go 9-80 8-77 8-08 7-20 7-18 7-29 7-92 8-34 Residuals -1- I-oo + 1-35 -1- 1-08 + 1-05 — 0-05 — 1-08 — 1-77 -2-65 -2-67 -2-56 — 1-93 — I-5I Page. I2h 13" I4h 15" 16I1 I7h 18I1 igh 20h 21I1 22l> 23" No. of highs 3 3 5 4 3 3 2 I 2 2 2 No. of lows 3 2 I I 2 I 2 3 4 4 2 2 Sums 197 52-6 47-0 74'4 73-8 5='i 58-0 54'3 47-4 60-1 43'9 457 J Mn X difif. -f II'O -3-6 — 7-3 — 14-6 -J'3 -^i. -3-6 -1- 3-6 ■f ii-o -f 7-3 0-0 O'O 307 49-0 397 59'8 66-5 42-8 54'4 50-7 58-4 67-4 43-9 457 Divisors 3 5 4 6 6 4 5 5 5 6 4 4 Means 10-23 9-80 9-92 9 '97 11-08 10-70 10-88 10-14 11-68 11-23 10-98 11*42 Residuals -1- 0-38 — 0-05 -1- 0-07 -\- 0-12 + 1-23 + 0-85 ^- 1-03 -1- 0-29 + 1-83 -1- 1-38 + I-I3 +1-57 The divisor for each hour = No. of highs + No. of lows. The sum of the hourly means is 2364-9 and the mean 9-85^. The correction J Mn x diff. is one-half the mean range from the "first reduction" multiplied by the difference between the number of highs and lows entering into each hourly sum; sub- tracted when the highs are in excess, and added when the lows are in excess. Mn for this month is 7-31 feet. SITKA, ALASKA. From 29 days' Wgli and lo-vv wateri, ^'analyzed as above. From harmonic analysis of homly ordinates I year, July 1-29, 1893 1893-94 Ki I -602 K, 1-508 Ki° 127-3 Ki° ,125 Oi o'9i7 Oi 0-906 Oi° 957 0'° 123 REPORT FOR 1897 — PART II. APPENDIX NO. 9. 569 SANDY HOOK, N. J. From 29 days' liigh and low waters, analyzed as above, From harmonic analysis of hourly ordinates 2 years, July 1-29, 1893 1887, i888 Ki 0-336 Ki 0-334 K,° 97-4 ■ Ki° loi Oi 0-150 Oi 0-173 0,° 99-6 Oi° 98 If, however, before beginning the summation we subtract J Mn from all high- water heights and add J Mn to all low-water heights, the necessity for keeping count of the number of highs and lows will be avoided and the same result obtained as before. If the high and low water heights be tabulated upon hourly ordinate forms, and summations made with the K and O stencils even for a very long period, there is no guaranty that the effect of M2 will be totally eliminated, although it will be nearly so. To understand this, suppose that at the given station the tropic diurnal inequality be great in low- water heights but small in the high- water heights; also, suppose that observations be confined to high waters. Kow if these heights be tabulated in their proper places upon the hourly ordinate sheets and the K stencil applied to them, it is clear that an amplitude much too small will be obtained for Ki. In other words, M2 will have a pronounced effect upon the result. If, now, lows as well as highs be included in the observations, the effect of M2 upon a diurnal tide will be very much diminished; but there is no reason to suppose that it will ever disappear completely, however long the series. While it is possible to distribute high and low water heights or times according to known arguments and obtain consistent results for a given station, serious difficulties arise when an attempt is made to interpret the results in terms of the harmonic constants or to accurately obtain these constants by any prescribed distributions. For instance, the phase inequality in the range of tide is not a simple harmonic increase and decrease of the range even if fXi be ignored. Its unsymmetrical character can readily be seen by referring to Table 16. The parallax and declinational inequalities are even more complicated than that of phase, as is pointed out in §§ 46, 47 of Part III. Having called attention to some of the difflculties encountered in the harmonic analysis of high and low water observations, we pass to a more thorough and systematic procedure. For convenience, the whole work of making the analysis may be divided into six steps or operations : (1) Making a "first reduction." (2) Finding the mean amplitudes and the time occurrence of the mean ranges using four consecutive tides. (3) Obtaining hourly ordinates from these amplitudes by aid of a system of sine curves drawn upon a transparent sheet. (4) Summing these ordinates with semidiurnal stencils and analyzing the partial sums. (5) Diminishing the heights of the tides by the ordinates belonging to the times of the tides, and tabulating the heights thus altered upon hourly ordinate forms. (6) Summing these values with diurnal stencils, and analyzing the partial sums. (!') The process of making a "first reduction " is fuUy described in § 51, Part I, and § 27, Part III. The epoch of M2 is found from the lunitidal intervals by aid of this last-named paragraph ; the amplitude, from the observed mean range of tide by aid of Tables 23 and 14. Where there is much difference between the duration of rise and of fall, the amplitude of M2 as determined from the range must be affected with M4. To correct for this, divide the Mj as obtained above by the quantity cos ■» + sin 2 ti X ^ (duration fall ~ 6'^-21) (495) where V = (duration fall ~ 6'>-21) x 140-492. (2') In the accompanying tabulation the last column shows the values of the successive mean amplitudes; the fifth, the times of occurrence of the ranges; and the sixth, their duration which is 6-21 hours on an average. The values inclosed in parentheses involve observations prior to July 1; generally they would have been simply inferred from succeeding values. 570 UNITED STATES COAST AND GEODETIC SURVEY. SITKA, ALASKA. Date. 'T'-^,^ «f Sum of tMe.°^ consecutive Mean. 6 hours, etc., applied. Differ- ence. Height of tide. Range. Sum of alternate ranges. Ampli- tude. 1893- h. h. h. h. (227) h. Feet. Feet. Feet. Feet. (3-8) July I. 1-1 (4-8) 6-1 14-6 II-2 (3-9) 8-2 6-2 3-4 44-0 II -o II -o 9-4 15-6 3"9 14-8 6-2 12-8 44-9 II-2 17-2 4-4 15 '4 3-8 19-9 6-2 8-4 45-4 11-4 23-4 6-0 14-9 37 2. 2'0 6-1 14-4 46 'O "■5 5-5 lo-S 15-1 3-8 87 6-3 3'9 46-9 117 II-8 9-1 15-2 3-8 15-4 6-0 13-0 4y4 II-8 17-8 47 14-8 37 20-8 6-2 8-3 48-1 I2'0 o-o 57 147 37 3- 2-5 6-2 i4'o 487 I2'2 6-2 10 -o 14-5 3-6 - 9'4 6-1 4-0 49-1 12-3 12-3 8-8 15-0 3-8 i6-Q 6-1 12-8 49-8 12-4 18-4 5"o 14-2 3-6 21'2 6-2 7-8 50-3 12-6 0-6 5-4 i3'9 3'5 4- 3"2 6-1 13-2 50-9 127 67 8-9 14-0 3-5 9'9 6-3 4-3 51-9 13-0 13-0 8-6 14-2 3-6 i6-6 6-2 I2"9 52-8 13-2 I9'2 5-3 i3'5 3-4 22'2 6-1 7-6 53'3 I3'3 1-3 4-9 12-8 3-2 5- 4-1 6-3 12-5 54'i 13-5 7-6 7-5 I3"i 3 "3 10-4 6-2 5-0 55 '3 13-8 13-8 8-2 13"5 3-4 i7'4 6-3 13-2 56-4 14-1 20-I 6-0 13-0 3'2 23-4 6-2 7-2 57-2 14-3 2-3 4-8 I2-I 3'o 6. 5 '2 6-1 I2-0 57-8 14-4 8-4 6-1 12-5 3'i II'2 6-3 5-9 34'9 87 147 77 13-1 3-3 i8-o 6-3 13-6 36-1 9-0 21'0 7-0 12-5 3-1 7- o'S 6-6 6-6 (3'). To prepare for interpolating hourly ordinates from the successive ranges, first sel cross-section paper whose smallest divisions are about iV inch square. Let an hour of time an inch or more, and let a foot of tide, when convenient, correspond to an inch on the sheet. 1 the upper margin of the sheet number the hours from to 24, or 0, and thence on to 12, maki hours represented. On a sheet of tracing cloth lay out a rectangle, say 10 inches high and enough to represent 6 lunar, or 6-21 solar, hours. The center of this rectangle is the nod( system of, say, 10 sine curves whose amplitudes vary from to 5 inches, and are numbei convenient, from to 5. All curves extend over a half period or 6-21 hours. To intei] hourly heights, place the node of the sine curves at the time of occurrence of a semidiurnal : given in the fifth column of the above tabulation. Select the curve having a number equi\ to the mean amplitude at the time given in the last column. Bead the heights at the points -^ this curve crosses the hour lines. These are the required hourly heights reckoned from mean level. So proceed with each range. If the tide have a very large phase inequality, causir quarter tidal day to depart much from a quarter lunar day, or 6-21 hours, this fact may be REPOET FOR 1897 — PART II. APPENDIX NO. 9. 571 into consideration by reading one or two of the hourly heights on either side of the node as before, and the hourly heights near the times of maxima and minima, when the edges of the rectangle have been made to fall exactly half way between the times given in the fifth column ; or more than one permanent set of curves may be used. Probably this refinement is unnecessary. SITKA, ALASKA, 1893. Interpolated ordinates of semidiurnal wave. Day of month. July I 2 3 4 5 6 7 Day of series. I 2 3 4 5 6 7 h. VI. Feet. Feet. Feet. Feet. Feet. Feet. Feet. O OO 8-3 7-1 6-0 5-0 4-1 3 '3 2-8 I CXD 9"5 87 77 f'7 5-5 4'2 3-2 2 OO 9 "9 9'5 9"i 8-3 7-1 5-6 3 '9 3 OO 9-1 97 9-6 9-3 8'4 7-1 5'2 4 OO 7-6 8-6 92 9-4 9" I 8-3 6-5 5 oo' 57 7-0 8-1 8-6 9'2 g-o 7-9 6 OO 3-8 S'l 6-4 7-2 8-3 8-9 87 7 OO 2-5 3-4 4-6 5'5 7-0 8-0 87 8 OO 21 2-3 3'2 3"9 5-3 6-6 8-0 9 OO 27 2-2 2-5 2-8 3'9 5'o 67 lO OO 4-1 3'0 2'5 2-5 3-0 3-8 5-3 II OO 60 4'5 37 3'o 2-6 3 '2 4-0 Noon. 79 6-4 5-5 4-2 3 '3 2-8 3'3 13 00 9-3 8-2 7-4 6-0 4-6 3-5 3'o 14 00 99 9-4 9-0 7-8 6-3 4-9 ^'% 15 00 9;4 97 9-8 9-0 7-9 6-5 4-8 i5 00 8-9 9'4 9'5 9-0 r^ 6-3 17 00 6-3 7-4 8-3 9-1 9-2 8-9 77 18 00 4-5 5-6 67 7-9 8-8 9"2 87 19 00 3"o 3'9 4-9 6-3 77 87 9"2 20 00 2-2 27 3-4 47 6-2 7"5 87 21 00 2-6 2-4 2-S 3"3 4-6 6-0 7-8 22 00 3-6 3'o 2-6 27 3 '5 4'5 6-3 23 00 5'3 • 4-2 3'5 3-1 2-8 3 '3 4-8 (4') Before summing with the semidiurnal stencils, the hourly heights should have a constant added to them in order to avoid negative quantities ; in the accompanying tabulation 6 feet has been added. The summation and analysis are then to be carried out as in the case of true hourly ordinates. (5') The values of the semidiurnal ordinates referred to mean water level are now known for each hour; they are therefore known for the times of the true high and low waters. Subtracting the appropriate semidiurnal ordinates, we have four heights per lunar day which lie upon the diurnal curve, very nearly. (6') The diurnal heights are then summed with the stencils, and analyzed for Ki and Oi in the usual way excepting that the augmenting factors 1-00287 and 1-00249 are to be used twice instead of once. When the series is very short, and so the divisors generally quite unequal, it may be advisable to write each height of the tide twice, i. e., to regard it as the houTly ordinate immediately preceding and immediately following the time of tide. Below is appended the results obtained from analyzing a series of tidal observations one month in extent. They show how the results obtained from high and low waters agree with the results obtained from regular hourly readings. Using the notation of §§ 59, G4, the uncorrected amplitudes and angles are really the E/s and C/s, since no corrections have been applied to them on account of the disturbing components. 572 UNITED STATES COAST AND GEODETIC SURVEY. SITKA, ALASKA, JULY, 1893. ponent From ordinates obtained from observed high and low waters. Uncorrected Uncorrected amplitude, X. angle, i. From observed hourly ordinates. • Uncorrected Uncorrected Period amplitude, Ji. angle,_^. analyzed. Feet. Feet. d. h. s. 0-897 49 0-873 53 29 13 /"= 0-179 127 0-191 108 29 13 N, o-8i8 263 0-785 264 27 13 L. 0-055 284 0-041 338 27 13 K. 1-907 308 2-018 309 27 8 O, I -017 355 1-097 352 27 8 75. Interpolation ofJwurly heights from tabulated high and low waters. The set of curves drawn upon a transparent slieet and described in the preceding paragraph, may be used for this purpose when the tide is wholly semidiurnal in its character. The hourly heights of a tide of this kind may be computed by the formula, t_ Depression below high water ) ^ r ^^^^^^ ^^^ f I8O0 j^) = ^, approximately, (496) or elevation above low water ) 2 V ^ o-7-4-0'6-- where d is the duration of rise or fall, expressed in minutes, r the corresponding range or amount of the same, and t the number of minutes from high or low water; t < d. If the diurnal inequality is considerable, any process of interpolation is rather laborious and not always accurate, even where the shallow water components are small, because the period of the diurnal wave is not generally exactly twice that of the semidiurnal. The range of the diurnal wave is, approximately, 2 Ji = V(HW ineq.f + (LW ineq.)^ (497) The high-water inequality is found by subtracting a given high-water height from the mean of the two adjacent high-water heights. Similarly for the inequality in the low waters. The range of the semidiurnal wave is, very nearly, .p 4.1, H mW ineq.)' + (LW ineq.)2 ..qsn 2 A = mean range for the day — =-- ^ ' ^ ^ „ — ,, y — (498) '' " "^ 16 X mean range for the day The position of the maximum of the semidiurnal wave can be found from values like those given in the fifth and sixth columns of the second tabulation in the preceding paragraph. These data, with Table 19, enable one to find the position of the diurnal wave with respect to the semidiurnal, also the value of Ji. Eules for determining the quadrant of the HW phase of the diurnal wave, i. e., the angle or phase of the diurnal at the time of HW of the semidiurnal, are as follows : Sequence HHW to LLW Sequence LLW to HHW fFor HHW, HW phase falls in 1st quadrant. [For LHW, HW phase falls in 3d quadrant. 'For HHW, HW phase falls in 4th quadrant. 'For LHW, HW phase falls in 2d quadrant. (499) The HW phase, when converted into time at the rate of 15° per hour, gives the time by which the HW of the diurnal wave precedes that of the semidiurnal. This angle should be taken between - 180° and -f 180°. The sum of the hourly ordinates of the diurnal and semidiurnal waves give the hourly ordinates of the tide; the height of tide at any time t is Ji cos 290 {t - ti) + Ai cos 150 {t - t,) (500) where h, ti denote the times of high water of the two waves. REPORT FOR 1897 — PART 11. APPENDIX NO. 9. 573 76. BemarTcs upon published results and tables. The effects of the motion of the moon's node upon the amplitudes and epochs of the compo- nents, as brought out by the harmonic analysis, were not allowed for in the earlier published results. Consequently, determinations from successive years were not inter se comparable. In this connection see the British Association Eeports, 1878, p. 481, note, and 1883, p. 91. Ferrel corrected the components Mz, K2, K,, and O, for such effects, but omitted the corrections for the other components similarly affected. His tables for this purpose are given upon page 304 of the Survey Report for 1878. It is to be noted that the numerical values of his ^e's for Kj, M2, and K2 have the wrong signs prefixed; he seems, however, to have always corrected this in making analyses. Initial equilibrium arguments.— In the Survey Eeport for 1878, Ferrel uses c, and in the Eeport for 1883 he uses Jc, to denote Fo, or the uniformly varying portion of ¥„ + u. The A;'s (or c's) of Kj, K2, and X2, and perhaps others, are sometimes wrong by 90° or 180°. To ascertain this, compare the ¥s (or c's). Report for 1878, pages 270, 303, and for 1883, page 267, with the Vo+u of Table 3. A leap year is not so convenient as a common year for this purpose, because the ifc's then refer to the 2d instead of the 1st of January. The smaller discrepancies shown by such comparison are due to the u of Table 1. That is Vo + u — Jc= u = — de* 01 + Ae,\ very nearly. (501) In this manual the origin of the day is taken as midnight. Consequently, unless otherwise stated, Fo refers to midnight instead of noon as contemplated by the British Tidal Committee. In Tables 1 and 3 the initial equilibrium argument of E2 is in error by 180°. E. Roberts has noted that the lower half of Table 8 has been incorrectly formed from the upper half. The tabular values will still hold good if for (P = ) 95°, 100°, 105°, etc., there be substituted 175°, 170°, 165°, etc. 2 6 In using Table 31, enter columns g' ^ as many days before the given date as there are degrees in^^l; Tables 6, 7. Enter Table 32 as many days before the given date as there are degrees in \v'; Tables 6, 7. Enter Table 33 as many days before the given date as there are degrees in v' ; Tables 6, 7. 77. Harmonic analysis of tidal currents. The height of the tide or the vertical displacemeat of the surface of the sea at a given point is usually assumed to have for its expression y OT h = M2' cos (m.2t + argo M2 — M2O) + Sj cos (s^t + argo gz — 82°) + Nj' cos {Uit + argo Nj - ^2°) + . . . + Ki' cos (kit + argo Ki — Ki°) + O/ cos {oit + argo Oi — OiO) + Pi' cos (pi^ + argo Pi - ri°) + . . . (502) In a canal of indefinite length it would be reasonable (according to § 22, Part I) to assume the horizontal displacement (f ) to be of the above form, save that sines take the place of cosines, and that the amplitudes are the above amplit^des^ a l ^ malttpliiod by tho- oamo . ponotan t. The velocity of the current is dS/dt, an expression involving cosines instead of sines, and having the horizontal displacement amplitudes multiplied by the respective speeds. ^'Wwr nl inirn W.iat a i^~^^-^^l diurnal component of the velocity is only about one- half as great as a semidiurnal ^hen the twbv ^ corresponding tidal components, or partial tides, are equal. On the other hand, the quarter ^ ^^ e*«ia— .aZ^,^^ diurnals and the sixth diurnals have a tendency to become more pronounced in the velocity. The (^^«* s JA, *&^ ^J theoretical ratio between two velocity amplitudes is not the same as the theoretical ratios between the two corresponding tidal coefficients, even though both partial tides are diurnal or both semidiurnal; but this ratio must be multiplied by the speed ratio. *de as used by Ferrel is the alteration in the epoch of a component which will adapt it to a particular year. It is tabulated for M2, Kj, Kj, and Oi upon page 268 of the Eeport for 1883. t ^e is the correction in the observed epoch of a component, due to the motion of the moon's node. It is tabulated upon page 304 of the Report for 1878, as already stated. 574 UNITED STATES COAST AND GEODETIC SURVEY. It is here supposed that we have hourly observations upon the velocity and direction of the current extending over one or more months. These velocities can be resolved with reference to two fixed directions — say, north and east. The velocities may be denoted by the letters used to denote the partial tides, each with a dot written above each : e. g., M2, S2, N2. The resolved portions may be distinguished by the subscripts n, e, s, or lo, according to the direction — north, east, south, or west — to which they refer. The resolution of the velocities into two portions is readily effected by aid of a circle drawn upon cross-section paper and divided into degrees, together with a scale whose zero is at the center of the circle and whose straight edge always falls upon a radius. The summations and analyses are made in the same manner as for the partial tides, because the velocity in a given direction (north, say) is assumed to be written v„ = M2/ cos (m2< + argo M2 - M2„o) + 82, cos (Sj* + argo S2 - 82, °) + N2„' cos (n2« + argo ^2 - Na.o) +- . . . + Ki„' cos (k,< + argo K, - K,,o) + 6 J cos (Oit + argo O, - 0,„°) + P,„ cos (p,< + argo Pi - Pi, o) + . . . . (503) For the easterly portion of the velocity e simply replaces n. Where the water follows a fixed channel M2=VM,/+M,/. (504) If a direction indicated by the subscript approximately coincides with the direction of flood, then for long tidal rivers we should expect M2° to be approximately equal to M.i°; but for a small bay or harbor it should be nearer M2°— 90°. If the direction indicated by the subscript approxi- mately coincides with the direction of the ebb, M.2° should be approximately equal to M2° i 180° for a long tidal river and M2° + 90° for a small bay or harbor. The same statements hold true when M3 is replaced by S2, ^2; Ki, etc. In tidal rivers or in Straits the currents have a nearly fixed line of motion, and so It is hardly necessary to decompose the velocities into two parts, as indicated above. 78. To combine the various current components for any given future time. Combine, as in § 72, all partial north-and-south currents; likewise combine all partial east- and-west currents. With the two velocities thus obtained a rectangle of velocities can be constructed whose diagonal represents the direction and velocity of the total current. Prediction of currents. — Suppose all partial north-and-south currents to have been properly combined, and a curve drawn representing the result from hour to hour. A tide predictor which traces a continuous curve can be used for this purpose and so the labor of computation be avoided. Suppose the partial east-and-west currents to have been similarly combined and a REPORT FOR 1897 — PART II. APPENDIX NO. 9. 575 curve drawn. Let the two curves be placed side by side. At any given instant the direction and velocity of the total current become known upon constructing a rectangle of velocities by aid of the two curves. The time of minimum velocity can be found either by trial or directly by the following process : Upon glancing at the two curves the approximate time of minimum velocity can be found; it will generally lie near the point where the curve of greater amplitude crosses its axis. A straight line can be drawn closely coinciding with this curve for some distance either way from the assumed time of minimum velocity. Another straight line can be drawn nearly coinciding with the other curve for the time in question. The question then reduces to the simple geometrical problem (which has an application in § 36, Part III) : In a plane are given two straight lines referred to rectangular coordinates; it is required to find geometrically an abscissa such that the sum of the squares of the two corresponding ordinates shall be a minimum. Suppose the equation y = mx (505) to represent the line having the greater inclination to the a!-axis and the equation 1+1 = 1, (606) the other line. When (507) the sum of the squares of the two ^'s becomes a minimum. The problem may now bo stated: Given a, 6, and m, to find x geometrically. Let 00 and AB denote the given lines; draw AO parallel to the y-axis and lay off AI) = BO = b. Bisect AO, thus determining U; then with U as center and OD as radius, describe an arc FOR. Draw BG parallel to the ic-axis; take GM=FO and draw the line EHI. Take EI=EO = i a, and project I upon the as-axis in J: then is OJ the required abscissa. 79. Rules governing the choice between Roman anddtalio letters in tidal work. Considerable confusion having already arisen among writers upon tides in regard to the nota- tion employed, it has been thought best to here state certain rules which have been generally followed in this manual. Eoman letters are used to denote — 1st. Quantities which are in themselves particular or definite tidal quantities at a given station. E. g., Mz, MO2, S2, 83°, Mn, HWI. 2d. Definite quantities intimately connected or associated with those of the kind just referred to. E. g., m2, S2, meaning speeds; M, S, denoting particular series of lunar and solar tides. Italic letters are used to denote — 1st. Quantities not tidal. B. g., /S = the longitude in time of the time meridian ; a — the earth's radius; V = the moon's distance; I, i, = inclinations of lunar orbit; V= potential. 2d. Indefinite tidal quantities; i. e., such as must be connected with definite tidal quantities before they have a meaning. E. g., S or R used for amplitude, V^ + u, F, f. 3d. Temporary or general symbols whether tidal or not. E. g., X, Y, Z, x, y, z, A, B, 0, a, b, c, AO, C2, Ci. AUXILIARY TABLES REDUCTION AND PREDICTIOISr TIDES, [Tables 1 to 35 are appended to Part III, Appendix No. 7, Report for 1894.] 6584 37 577 REPORT FOE 1897 — PART II. APPENDIX NO. 9. 679 Table 36. — Shallow-tvater components. [Terms from y'^.] SEMIDITJENAL COMPONENTS. Designation of component. Primitive amplitude. Speed. Argument. Primitive epoch. (K.K.) ■ K,0,) (KxP.) M, K,0, K.P, k. + k, = k, kj + Oi = m, k, + p, = Sj 30-0821372 28-9841042 30-0000000 2 arg K, arg Ki + arg O, arg Ki + arg P, > 2K,° Ki° + 0,° Ki° + P.° (OiO,) (O.Pi) 0, KOi= O.P, 0, + 0,= 0j Oi+pi 27-8860712 28-9019670 2 arg Oi arg Oi + arg Pi 20i° Oi° + Pi° (P.P.) P= ^p.= Pi + Pi = P= 29-9178628 ' 2 arg P, 2Pi= COMPONENTS OF LONG PERIOD. (Ki~Ki) (Ki~Oi) (K.-~Pi) Mf Ssa ^Ki= K.Oi KiP, ki — ki = k, — Oi = mf ki — pi = ssa 1-0980330 0-0821372 arg Ki — arg Oi arg Ki — arg Pi Ki° - Oi° Ki° - Pi° (0,~0i) (o.~p.) MSf }40^ OiP. Oi — Oi = pi — o, = msf 1-0158958 arg Pi — arg Oi Pi° - Oi° (Pi~Pi) ^Pi' Pi-p, = o TEEDIUENAL COMPONENTS. (M=Ki) (M.O') (M,Pi) MK 2MK M,Ki M=Oi M,Pi nia + ki = mk ffls+Oi m, + pi 44-0251728 42-9271398 43^9430356 arg M= + arg K, arg Mj + arg d arg Ma -|- ai^g Pi Ma° + Ki° Ma° + Oi° M,° + Pi" (S,Ki) S.O1) (S.Pi) S,Ki S.O1 S.P. Sa+ki S, + Oi s^ + p. 45-0410686 43 '9430356 44-9589314 arg S2 + arg K, arg S» + arg Oi arg Sa + arg Pi Sa° + Ki° Sa° + Oi° Sa° + P.° (N,Ki) (N.Oi) (N.Pi) N^Ki N,Oi N=Pi n^ + ki n^ + Oi Hs + p. 43-4807982 42-3827652 43-3986610 arg Na + arg K. arg Na + arg d arg Na + arg Pi Na° + Ki° Na° + Oi° Na° + Pi° (K,Ki) (K.O.) (K,P.) MK K^Ki K,Oi K,Pi 3k. ks + Oi = mk ks + pi 45-1232058 44-0251728 45-0410686 3 arg Ki arg Ka + arg Oi arg Ka + arg P. 3Ki° Ka° + Oi° Ka° + Pi° (L.Ki) it?:! L,Ki Iv.Oi LaPi l= + k. l= + Oi 44'5695474 43-4715144 44-4874102 arg L2 + arg Ki arg La + arg d arg Iva + arg P, U° + K.° La° + Oi° . I/a° + Pi° DITJENAL COMPONENTS. (Ma~Ki) (Ma~Oi) (Ma~Pi) Oi Ki MaKi MaOi MaP. ma — ki^ Oi ma — Oi = k, ma — Pi 13 '9430356 15-0410686 14-0251728 arg Ma — arg Ki arg Ma — arg Oi arg Ma — arg Pi Ma° — Ki" Ma" ~ Oi° Ma" ~ Pi" (Sa~Ki) Sa~OiJ (Sa~Pi) P. Ki SaKi SaOi SaPi , Sa — ki = Pi Sa — Oi Sa — Pi = ki I4'95893i4 16-0569644 15-0410686 arg Sa — arg Ki arg Sa — arg d arg Sa — arg Pi Sa^-Ki" Sa- - Oi° Sa° - Pi" (Na~Ki) Na-^Oi) (Na-Pl) [Mi] NaKi NaO, NaPi ria — ki = q, na— o.= [m,] Ha — Pi 13-3986610 14-4966940 13-4807982 arg Na — arg Ki arg Na — arg O, arg Na — arg P, Na° - Ki= Na" - Oi° Na" - Pi° (Ka~Ki) Ka~0,) (Ka-Pi) Ki KaKi KaO, KaP. ka-ki = ki kj— Oi ka-pi 15-0410686 16-1391016 15-1232058 arg Ka — arg Ki arg Ka — arg Oi arg Ka — arg Pi Ka° — Ki" Ka" - Oi= Ka" — P." (La~Ki) La~Oi) La~Pi) J. LaK, LaOi LaPi la-ki la — Oi = j, la -Pi 14-4874102 15 '5854432 14-5695474 arg La — arg K. arg La — arg O. arg La - arg Pi La° - Ki- La" - Oi- ^ La° - Pi° For a description of this table, see ? 48, Part II. For sake of clearness we have supposed {AB) to denote a component -whose speed is a + b, and {A- component whose .speed is a~5. ,B) 580 UNITED STATES COAST AND GEODETIC SURVEY. Table 36. — Shallow-water components — Continued. [Terms from y'=.] QUAETEE-DIUENAL COMPONENTS. Designation of component. Primitive amplitude. Speed. Argument. primitive epoch. {MM,) (MsSs) (MsNs) (MsK,) (MsLs) M, MS MN 'AM,' MS, MsNs MsKs MsLs ma -)- ma = tn^ ms + Ss ma -\- n,^ mn ins + ks ms + ls 57-9682084 58-9841042 57-4238338 59-0662414 58-5125830 2 arg Ma arg Ms + arg Ss arg Ms + arg Ns arg Ms + arg Ks arg Ms + arg Ls 2 Ms° Ms° + Ss° Ms° + Ns° Ms° + Ks° Ms° + Ls° (SsSO (SsNs) (SsKs) (SsLs) S4 R4 ^Ss= SsNs SsKs SsLs Ss + Ss = S4 Ss + ks = r4 Ss + ls 60-0000000 58-4397296 60-0821372 59-5284788 2 arg Ss arg Ss + arg N, arg Ss + arg Ks arg Ss + arg Ls 2Ss° Ss° + Ns° S,° + Ks" Ss° + Ls° (NsNs) (NsKs) (NsLs) N4 ^Ns= NsK, NsLs Us + n, = n4 ns + ks Ds + ls 56-8794592 58-5218668 57-9682084 2 arg Ns arg Ns + arg Ks arg Ns + arg Ls 2 Ns° Ns° + Ks° Ns° + Ls° (KsKs) (K.I^=) K4 >^Ks= ks+ks = k4 ks+ls 60-1642744 59-6106160 2 arg Ks arg Ks + arg Ls 2 Ks° Ks° + Ls° (Lsivs) L4 KI^/ ls + ls=l4 59-0569576 2 arg Ls 2sLs° COMPONENTS OF LONG PERIOD. (Ms -Ms) Ms~Ss) (Ms ~ Ns) (Ms ~ Ks) (Ms~Ls) MSf Mm Mf Mm AM,' MsSs MsNs MsKs MsLs ms — ms ^ Ss — ffls = msf ms — Hs = mm ms — ks = mf Is — ms=mm 1-0158958 0-5443746 1-0980330 0-5443746 arg Ss — arg Ms arg Ms — arg Ns arg Ms — arg Ks arg Ls — arg Ms Ss° - Ms° Ms" - Ns° Ms° - Ks° Ls° - Ms° (Ss~Ss) (Ss-Ns) (Ss~Ks) (Ss~Ls) Ssa KSs= SsNs SsKs SsLs Ss — Ss = Ss — ns ks — Ss = ssa Ss-ls 1-5602704 0-0821372 0-4715212 arg Ss — arg N, arg Ks — arg Ss arg Ss — arg Ls Ss° - Ns° Ks°-Ss° Ss° - Ls° (Ns~Ns) (Ns ~ Ks) (Ns~Ls) >^Ns= NsKs KsLs Hs — Hs = ks — Hs Is — Hs I -6424076 I -0887492 arg Ks — arg Ns arg Ls — arg Ns Ks° — Ns° Ls° - Ns° (Ks~Ks) (Ks~Ls) ^Ks= K2L/2 ks - ks = ks-ls 0-5536584 arg Ks — arg Ls Ks° - Ls° (Ls-Ls) >^Ls' ls-ls = REPORT FOR 1897 — PART 11. APPENDIX NO. 9. Table 36. — Shallow-ioater components — Continued. [Terms from j/'3 or j)/' Xj''^.] ONE-SIXTH-DIUENAL COMPONENTS. 581 Designation of component. (M,M,M,) (M^M^S.) MMA2) [MSA) (S=M,K,) (S=M^1.4 ( S2S2S2 ) 'S2S2N2) b2!52-*^2 } Oabz J-/2 } Ms Primitive amplitude. KM,3 M/K, M/L, K M.S/ YzSM^ KS=3 Speed. 3 m^ = mg 2 m^ + Sj 2 m^ -f n^ 2 m^-j-kj 2 m^ 4- 12 2 Sj + m^ m^ + S2 + n^ His -j- Sj + ka m^ -(- Sj + 1, 2 m^ + Sj m^ -j- 2 S2 m= + Ss + n^ IDs + Sj + k, m^ + S2 + U 3 Ss = S6 2 Sa + "2 2 S2 -|- ka 2 S2 + I2 86-9523126 87-9682084 86-4079380 88-0503456 87-4966872 88-9841042 87-4238338 89-0662414 88-5125830 87-9682084 88-9841042 87-4238338 89-0662414 88-5125830 90-0000000 88-4397296 90-0821372 89-5284788 Argument. „N2 argKs argL2 3 arg M2 2 arg M2 + arg S2 2 arg M2 -|- arg N2 2 arg M2 + arg K, 2 arg M2 4- arg L2 2 arg S2 + arg M^ argM2 + argS2 + arg arg M2 + arg S2 + arg arg M2 + arg S2 2 arg M2 + arg S2 arg M2 + 2 arg S2 arg M2 + arg S2 + arg N2 argM2 + argS2 + argK2 arg M2 + arg S2 + arg L2 3 arg S2 2 arg S2 + arg N, 2 arg S2 + arg K^ 2 arg S2 + arg L,^ Primitive epoch. 3M2° 2 M,° + 82° 2 M2° + N2° 2 M2° + K2° 2 M2° + L2° M2' + 2 82° M2° + 82° + N2° M2° + 82° + K2° M2° + 82° + L2° 2 M2° + 82° M2° + 2 82° M2° + 82° + N2° M2° 4 82° + K2° M2° + 82° + h,° 382° 2 82° + N2° 2 82° + K2° 2 82° + Iv2° SEMIDIURNAL COMPONENTS. fM3~M2M2) M2 KM23 m2 28-9841042 argMj M2° (M,~M2S2) S2 M2»82 S2 30-0000000 arg 82 82° (M2-M2N2) N2 M2=N2 n. 28-4397296 arg N2 N2° (M2-M2K2) K2 M2=K2 k2 30-0821372 argKj K2° (M2-M2L2) L2 M2%2 12 29-5284788 argLe L2° (M2-8282) 28M KM282- 2 S2 — m2 31-0158958 2 arg 82 — arg M, 2 82° - M2° (M2~82N2) A2 M282N2 S2+n2— m2=A2 29-4556254 arg 82 + arg Nj — arg M2 82° + N2° - M2° (Ma-SaK,) M2S2K2 S2+k2— m2 31-0980330 arg 82 + arg K, — arg Mj 82° + K2° — M," (M2~S2L2) M2S2L2 S2+I2 — ms 30-5443746 arg 82 + arg L2 — arg M, 82° + 1.2° - M2° (S.~M2S2) 2M8 ^S2M2= 2 m2-S2=/i2 27-9682084 2 arg M2 - arg 82 2 M2° - 82° M2S2 m. 28-9841042 arg Mj M2° (82~M2N2) M2S2N2 mj+Hj— S2 27-4238338 arg M2 + arg N, — arg 82 M2° + N2° - 82° (Sa-MsKj) M282K2 m2+k2— S2 29-0662414 arg M2 + arg K2 — arg 82 M2° + K2° - 82° (82 -Mai,,) T^ M282l,2 ni2 + l2 — S2=J'2 28-5125830 arg M2 + arg K^ — arg 82 M2° + L2° - 82° (82-8282) S2 >^S23 , ■ Sj 30-0000000 arg 82 82° (82~82N2) S2~82K2) N2 82=N2 Hj 28-4397296 argN2 N2° K2 S2=K2 k2 30-0821372 argK2 K2° (82-821,2) L2 8.=IV2 I2 29-5284788 arg 1,2 L2° [Terms fromjj'''' ox y' Xj''^-] ONE-EIGHTHDI0ENAL COMPONENTS. (M2M6) (M286) (82M,) . (8285) Ma 88 ^M24 Yz M2823 Yl S2M23 ^82-' 4 nia = ma 3s, + m2 3 ms + S2 4S2 = S8 115-9364168 118-9841042 116-9523126 120-0000000 4 arg M2 3 arg 82 + arg M^ 3 arg Ms + arg 82 4 arg 82 4M2° 3 82° + M2° 3M2°+82° 482° QUAETEE-DIUENAL COMPONENTS. (Ms-Me) (M2~86) (82~M6) (82-86) M4 84 J^M2'' Yz M2823 Yz 82M23 >^824 2 fflj = ni4 3 S2 - m, 3 m, - S2 282 = 84 57-9682084 61-0158958 56-9523126 60-0000000 2 arg M2 3 arg 82 - arg M2 3 arg M, — arg 82 2 arg 82 2M,° 3 82° - M2° 3 M,° - 82° 282° 582 UNITED STATES COAST AND GEODETIC SURVEY. Table 37. — Tlie theoretical amplitudes of some of ilie more important components for every 5 degrees of latitude. K cos'X sin 2 K i — i siii^A M2 Nj S2 K, 0, P. Mf o Feet. Feet. Feet. Feet. Feel. Feet. Feet. +90 00000 0-0000 — I'OOOO 0-000 0-000 0-000 O'OOO 0-000 0000 -0-138 +85 +80 0-0076 o'030i 0-1736 0-3420 —0-9886 —0-9548 0-006 0-024 o-ooi 0-005 0-003 o-oii +0-081 -I-O-160 +0-058 +0-II4 +0-027 +0-053 —0-136 -0-132 +75 +70 +65 o"o67o 0-1170 0-1786 0-5000 0-6428 0-7660 -0-8995 —0-8245 -0-7321 0-054 0-094 0-I43 o-oio 0-018 0-028 0-025 0-044 0-066 +0-233 +0-300 +0-358 +0-166 +0-213 +0-254 +0-077 +0-099 +0-118 —0-124 — 0-I14 — o-ioi +60 +55 +50 0-2500 0-3290 0-4132 0-8660 0-9397 0-9848 —0-6250 —0-5065 —0-3802 O-20O 0-263 0-330 0-039 0-051 0-064 0-093 0-122 0-154 +0-404 +0-439 +0-460 +0-287 +0-312 +0-327 +0-134 +0-145 +0-152 —0-086 —0-070 —0-052 +45 +40 +35 0-5000 0-5868 0-6711 i-oooo 0-9848 0-9397 —0-2500 — 0-II98 +0-0066 0-400 0-469 0-537 0-077 0-091 0-104 0-186 0-218 0-250 +0-467 +0-460 +0-439 +0-332 +0-327 +0-312 +0-154 --0-152 -0-145 -0-034 —0-017 +0-001 +30 +25 +20 0-7500 0-8214 0-8830 0-8660 0-7660 0-6428 +0-1250 +0-2321 +0-3245 0-600 0-657 0-706 O-I16 0-127 0-137 0-279 0-306 0-329 +0-404 +0-358 40-300 +0-287 +0-254 +0-213 +0-134 +0-118 +0-099 +0-017 +0-032 +0-045 +15 +10 + 5 0-9330 0-9698 0-9924 0-5000 0-3420 0-1736 +o'3995 +0-4547 +0-4886 0-746 0-776 0-794 0-145 0-150 o'i54 0-347 0-361 0-369 +0-233 +0-160 +0-081 +0-I66 +0-114 +0-058 +0-077 +0-053 +0-027 +0-055 +0-063 +0-067 - 5 — 10 I -0000 0-9924 0-9698 0-0000 -0-1736 —0-3420 +0-5000 +0-4886 +0-4547 0-800 0-794 0-776 0-155 0-154 0-150 0-372 0-369 0-361 +0-000 — o-o8i —0-160 +0-000 -0-058 — 0-114 +0-000 —0-027 -0-053 +0-069 -1-0-067 +0-063 -15 —20 -25 0-9330 0-8830 0-8214 —0-5000 -0-6428 —0-7660 +0-3995 +0-3245 +0-2321 0-746 0-706 0-657 0-145 0-137 0-127 0-347 0-329 0-306 -0-233 —0-300 -0-358 —0-166 —0-213 —0-254 -0-077 -0-099 — 0-118 +0-055 -0-045 +0-032 -30 -35 -40 0-7500 0-6711 0-5868 -0-8660 -0-9397 —0-9848 +0-1250 +0-0066 — 0-1198 0-600 0-537 0-469 0-116 0-104 0-091 0-279 0-250 0-218 —0-404 -0-439 —0-460 —0-287 -0-312 -0-327 —0-134 —0-145 —0-152 +0-0I7 +0-001 —0-017 -45 -50 -55 0-5000 0.4132 0-3290 — I-oooo -0-9848 -0-9397 —0-2500 —0-3802 —0-5065 0-400 0-330 0-263 0-077 0-064 0-051 0-186 0-154 0-I22 -0-467 — 0-460 -0-439 -0-332 -0-327 -0-312 -0-154 —0-152 -0-145 -0-034 —0-052 — 0-070 -60 -65 -70 0-2500 0-1786 0-1170 —0-8660 —0-7660 —0-6428 —0-6250 -0-7321 —0-8245 0.200 0-143 0-094 0-039 0-028 0-018 0-093 0-066 0-044 —0-404 -0-358 -0-300 -0-287 -0-254 —0-213 —0-134 -0-118 -0-099 —0-086 — o-ioi — 0-II4 -75 -80 -85 -90 0-0670 0-0301 0-0076 0-0000 —0-5000 -0-3420 -0-1736 0-0000 -0-8995 -0-9548 —0-9886 — I-oooo 0-054 0-024 0-006 Q-OOO 0010 0-005 o-ooi 0-000 0-025 O-OII 0-003 o'ooo -0-233 —0-160 —0-081 OOOO' — o-i66 — 0-114 —0-058 o-ooo -0-077 -0-053 — 0-027 0-000 —0-124 —0-132 —0-136 -0-138 Tabular value = \l ^(-)'a = 1-760I X latitude factor X coefficient. The latitude factor is given in column 2, 3, or 4; the coefficient in Table i. For this table it is assumed that ^^gpj^. 7 = 6^' '^ = ^° ^'^ °°° ^""^^^ ^''''°'''^- ing to Harkness, Solar Parallax, pages 138, 140, using a mean radius of the earth instead of the equatorial radius. The negative amplitude signifies that the phase of the tide is altered by 180°. The north latitude is +, the south -. REPORT FOR 1897 — PART II. APPENDIX NO. 9. Table 38. — Augmenting factors. £83 Subscript. I 2 3 4 5 6 7 8 s i-oooo O'OOOO I -0000 00000 I'OOOO o-oooo 1-0000 o-oooo I-OOOO 0-0000 I-OOOO O'OOOO I-OOOO O'OOOO I'OOOO o-oooo J 2SM 1-00307 0-OOI33I Group cc 1-01231 0-005313 )vers one so ar hour. K P,R,T 1-00287 0-001246 1-01158 0-004998 I -02632 0-011281 I '04746 0-020138 L A, MS 1-00273 o-oo]'i96 I-OIII6 0-004819 1-02534 0-010868 1-04568 0-019400 M 1-00266 0-OOII53 1-01075 0-004644 I -02440 0-010470 1-04396 0-018683 1-06989 0-029339 I -10283 0-042507 I' 14363 0-058286 1' 19343 o'o76797 N V 1-00256 o-ooiiii I -01033 0-004464 O 2N,/t I -00249 o-ooio8i 1-00994 • 0-004295 I -02256 0-009691 1-04300 0-018285 oo 1-00333 0-001442 Q p 1-00227 0-000983 2Q I -00209 0-000906 MN 2MK 1-00261 0-001132 I -01055 0-004557 1-02394 0-010274 1-04311 0-018331 MK 1-00274 0-001189 I -01102 0-004760 1-02503 0-OI0739 All 1-00286 0-001240 1-01152 0-004974 Grou I -0261 7 0-011219 3 covers one 1-04720 0-020030 component I -07513 0-031461 hour. 1-11072 0-045605 I -15496 0-062571 1-20920 0-082498 The tabular value for any component other than S is chord c r ^jjgj.e r = the length of the group. It is a solar hour when each component hour receives one, and only one, hourly height; it may be regarded as a component hour when all hourly heights are used in the summation. (See \ 57, Part II.) Tides of long period. ■When all daily means are used, the factors given under the heading "Group covers one component hour" are to be applied to the long-period tides, the subscripts referring to the year or month, instead of the day as m the case of tides of short period. When attention is paid to the arrows. Table 43, in making the summations, the augment- ing factors due to using' solar instead of component time, are given by the above formula by putting r = one solar day, and ^=mf, msf, mm. The results are: 1.00887 (log. =0.003835), 1.00759 (log. =0.003282), 1.00217 (log.= 0:000941). In case of any long-period tide there is, besides the_augmenting factor V^^^^^J^J'^^^^^-^Z^^^^^jf^^^^ group factor, due to using the mean of 24 heights each day. ' ' ' " '" '"" ^ factors for Mf, MSf, and Mm. The numerical values just given are also the group 684. UNITED STATES COAST AND GEODETIC SURVEY. Tabi,e 39. — Values qfb—a and o/z4X (d—a). DIURNAI^S. A £ h K, M, 0, 00 Pi Qi 2Q Si pi h - 0-5443747 - I -093391 2 - 1-6424077 + 0-5536583 - 0-6265119 - 2-1867824 - 2-731157I - 0-5854433 2-1139289 -13-064993 -26-241389 -39417785 +13-287799 -15-036286 -52-482778 -65-547770 -14-050639 - 50-734294 K. + 0-5443747 - 0-5490165 - 1-0980330 + 1-0980330 - 0-0821372 - 1-6424077 - 2-1867824 — 0*0410686 - 1-5695542 +13-064993 - 13-176396 -26-352792 +26-352792 - I-97I293 -39-417785 -52-482778 - 0-985646 - 37-669301 M, + I -09339" + 0-5490165 - 0-5490165 + 1-6470495 + 0-4668793 - 1-0933912 - 1-6377659 + 0-5079479 - 1-0205377 +26-241389 +13-176396 -13-176396 +39-529188 + 11-205103 -26-241389 -39-306382 + 12-190750 -24-492905 Oi + 1-6424077 + 1-0980330 + 0-5490165 + 2-1960660 + 1-0158958 - 0-5443747 - 1 0887494 + 1-0569644 — 0-4715212 +39-417785 +26-352792 +13-176396 +52-705584 +24-381499 -13-064993 -26-129986 +25-367146 -11-316509 00 - 0-5536583 - 1-0980330 - 1-6470495 — 2-1960660 — I-1801702 - 2-7404407 - 3-2848154 — 1-1391016 - 2-6675872 -13-287799 -26-352792 -39-529188 -52-705584 -28-324085 -65-770577 -78-835570 -27-338438 -64-022093 Pi + 0-6265119 + 0-0821372 - 0-4668793 - 1-0158958 + 1-1801702 - 1-5602705 - 2-1046452 + 0-0410686 - I-4874I70 +15-036286 + 1-971293 -11-205103 -24-381499 +28-324085 -37-446492 -50-511485 + 0-985646 -35-698008 Q. + 2-1867824 + 1-6424077 + 1-0933912 + 0-5443747 + 2-7404407 + 1-5602705 - 0-5443747 + I-601339I + 0-0728535 +52-482778 +39-417785 +26-241389 +13-064993 +65-770577 +37-446492 • -13-064993 +38-432138 + 1-748484 2Q + 2-7311571 + 2-1867824 + 1-6377659 + 1-0887494 + 3-2848154 + 2-1046452 + 0-5443747 + 2-1457138 + 0-6172282 +65-547770 +52-482778 +39-306382 +26-129986 +78-835570 +50-511485 +13-064993 +51-49713I +14-813477 Si + 0-5854433 + 0-0410686 - 0-5079479 - 1-0569644 + I-139IOI6 — 0-0410686 — 1-6013391 - 2-1457138 - 1-5284856 + 14-050639 + 0-985646 -12-190750 -25-367146 +27-338438 - 0-985646 -38-432138 -51-497131 -36-683654 Pi + 2-1139289 + 1-5695542 + 1-0205377 + 0-4715212 + 2-6675872 + 1-4874170 - 0-0728535 — 0-6172282 + I-5284S56 +50-734294 +37-669301 +24-492905 +11-316509 +64-022093 +35-698008 - 1-748484 -14-813477 +36-683654 REPORT FOR 1897— PART II. APPENDIX NO. 9. 585 I a o o X M- fM U) "^ i4 1 e D « a 1 ^ w ^ m <0 SI ;S s s s « s ^ 00 00 1^ lO HI 3 % 1'* ft 2 S- (O in a? o o a ;f « CO + 1 + $ •to g. + + ° c? + + + + + + l-> t-, to + + to to + + + + s « If S 9 5^ ?> 1? 1 f; VO OD 1 s ID ^ 1^ 00 If It la « 00 3;vS' ll o o CO in M K 1 1 1 1 O H 1 T b H + + ° s + + it CO t 1 Vf 1 1 b Vo + + 1 1 i t g> it 00 o »o in lo 00 tT o l~> w 1 If R S! to N IE Is r- CO o o if II r? 1 r 1 1 O H 1 T b M + + 1 1 " '=3. 1 1 1 r M in 1 7 o "to I 7 ■to to 1 1 !§ m 00 e ? ll t g; to t^ •^ to VO t i Roo ^ 00 1 1 ^.2 ■S ,3- lO lO O vS^ S'g S 3 to CTs a ft ^ N to VD O p ? g "1 % 5 ■i?, ff O t^ jn p o o DO CT> ^■s 'ft 1^ b in b w b M H ^f w t^ b ■■* O fO b w M in b « l_T 1 1 + + + + fO + + 1 T 1 1 1 T to + + + + to 1 1 % ov ti N r-. ro ^ ^ VO 00 a to S VD i% M M 00 « 5 VO 8 S <3 ^ g <;^ lO 00 s s- 'S s^ R K ^"^ ii- JO «g^ -T in as o fO lO W H to Ov o r^ VO w VO t^ I-* 0^ M VD O m 00 t^ ■>1- 00 o r^ g! S 3: !i uS VO fH ■>i- to 0\ to m -t O CTv O C7N o o in o O CO b N ° 2 b Vo ■" •ft ■" ■? b "h o b b W ■" '5 " S " !? 1 1 + + + + + + + + 1 I 1 1 + + + + + + 1 1 s. « m »n If t o lO lO S VO S VO U3 ■S 00 g.00 If ff s- r^ Ov •1" *8 v^" *8 vS^ ■ r- Qv g s ^ o W W M \D O VO "^ O (O ■ « m in vS 00 ^ ^ -ftvO 00 r^ 2 * 00 r^ s a M VO ■* CO r- to t^ Ov m 00 PI i^ Tl- oS M 0^ *:; n ^ '^ in 00 00 rl- ^p 00 m ov iz; ^ %■ if o o S9 s ^ o in If W CO s- s O N 2 S- W « w n O* 1 1 b to W M " b « bv HH t-^ O M o ■* to V^t- 1 1 1 1 1 T in 1 1 tn t 1 1 r 1 r 1 1 1 T 1 1 « VO 00 ON VO ^ 00 00 (VI ■* w ON W ^ o Tj- »0 R §, s g; s ? CO M VO Ov -^ 00 t=-. CTi CO rO O "* m 00 VO 00 (? R to o> ^ N Ei (S 00 TJ- lo m 00 ^ 00 a> Si if O VO Ov o m M H VO -i ro M to to rO « M VO 1 1 1 1 1 1 + + 1 1 1 1 1 1 1 1 + + 1 1 Is, ll to OS ll 1^ 5- ? t« s> § •2 00 R s; Si •* ■* m M S" iQ m *-* M VO g 1 S o o S'8 ■§ 2 pf> s % r- On Ov CO f* 1- s % 5: 3 S"R '" ■§ b ry b to '" ■« " N " jr ■° j? b w '" s- O M V, og. 1 1 1 1 + + + + 1 1 1 1 1 1 1 i + + + + 1 1 4f It o o Si If ' O M 1^ IP 1 B m to ll ■ 18-I-6 t 14+5 *- 9+4 <- 4+3 f 23+2 1" 19+1 <- 14-I-0 <- -2 f 19-4 <- 9-1 4- ■3 <- 14-5 <- 9-6 t 4-7 t 0—8 «- 19—9 *- 14— lot 9-llt 5-l2<- 0-1-iK- ig-t-iot 14+9 t 9+8 5+7 <- 0-1-6 <- 19+5 f 14+4 \ 10+3 <- 5+2 <- o-l-l t ig-Ho t 15-1 «- 10—2 *r- 5-3 <- ••■■ 0-4 t 19- 15-6 *-.... 10—7 19+10' 2+9 ' ■ 9+8 ' ' 16+7 ' 6+5 'I !■ 13+4 ' \- 20+3 t 22— n't* 23+6 t lo+i <- 17+0 <- . 7-2 •«- 14-3 *- II-6 •«- 18—7 ■«- , 8— 9 ^ 15— io<- 12+1K- 19+10^ 9+8 ■<- 16+7 <- 23+6 ■<- 13+4 •«- 20+3 ■«- lo+l ■<- 17+0 <- 7—2 <- 14—3 <- 21—4 <- 11-6 •«- 18-7 *- 3+2 ■«- O— I 12+4 1 1- ,19+3 1 • 9+1 ' > 16+0 ' ' 13-3 ' ' 20—4 ' • 10—6 ' • 17-7 ' [ 7-9 i - 14—10'! 23-1 t 21— lit 4-I2'| [• li+iil 1- I8+I0't' 1+9 ' ' 8+8 ' • 15+7 ' S+'i ' - 12+4 ' ' 19+3 ' 2+2 ' ' 9+1 ' ■ 16+0 ' 6-2 ' t 13-3 1 !> 20-4 t 22+6 'j- 23—1 t 3-5 t '°~* t 0-8 ■^ 7-9 1 4—12*- 1+9 <- 5+5 <- 6 ^ 17-7 t 7-9 t I4-IO*- 11+1K- i8+io<- 8+8 •«- 15+7 ■«- 12+4 ■«- 19+3 *- 22+6 *- 13+4 <- 0+5 '^ 13-4 '^ 0-5 <- jit or 2 MS I3-I2<- 3+iot 10+9 f 17+8 t 0+7 '^ 8+6 <- 15+5 <- 22+4 *- 5+3 <- 12+2 ^ 19+1 '^ 2+0 't' 9-1 '^ 16—2 -f 7-4 <- 14-5 <- 21—6 «- 4-7. 11-8 1 'is-g't I— lo*!* 8-II'^ 15 — 12't' 23+iK- 6+10*- 13+9 <- 20+8 <- 3+7 '^ 10+6 1 17+5 '^ 0+4 '^ 7+3 "f 15+2 «- 7— n't- 22 — 12't- 13 + "*- ■■■ 4+I0<- 18+9 ^ 9+8 t ■••■• 0+7 <- 15+6 <- 5+5 I 20+4 '^ 11+3 ! 2 + 2 8+II' 5+8 t 12+7 -I 2+5 <- 6+1 <- 3-2 <- 0-5 «- 4-9 <- 9+4 <- 13+0 10-3 *- 7-6<- II— lo<- 18-11 15+10 19+6 16+3 20—1 17-4 14-7 18-11 15+10 19+6 16+3 <- 23+2 <- 21—8 t 22+9 ^ <- 20—1 «- 17-4 <- 14-7 <- 18— II*- I— 12<- 8+II*- I5 + I0<- 22+9 <- 5+8 •<- 12+7 <- 19+6 «- 2+5 <- 9+4 *- 16+3 ■«- 23+2 <- 6+1 <- 13+0 ■<- 20— I «- 3— 2 «- 10-3 «- 17— 4 <- 0-5 <- 4-9 *- I— 124- 5+8 <- 2+5 <- 7—6 <- II — 104- 8+iM- 12+7 4- 9+4 <- 14—7 <- 21—8 «- 18— li«- i5+io<- 22+9 «- 19+6 *- 16+3 <- 23+2 <- 6+1 <- 13+0 *- 19— I t 2—2 ' ■ 9-3 ' ' 16—4 ' 6-6 ■ ' 13-7 ' • 20-8 ' 3-9 ' * 10—10' > I7-II' 0— I2i [ 7+lI'l [• 14+lot 23-5 t 21+9 t f II+7 1 I- 18+6 -f • 8+4 ' ' 15+3 ' ■ 12+0 ' • I9-I ' • 9-3 ' ' 16—4 ' ^ 13-7 i [ 20—8 t 22+2 t 23-5 v 4+8 i+S 5+1 2—2 6-6 3-9 0—12 4+8 1+5 5+1 2-2 f 9-3 t 16-4 t 23-5 t 6—6 f 13—7 4- 20—8 «- 3—94- 10—104- 17— 114- 0—124- 7+II4- 14+104- 21+9 4- 4+8 4- 11+74- 18+64- f 10—10^ t 17-iit 7+II' ■ 14+10' ■ 11+7 ' ■ 18+6 ' ' 8+4 ' • 15+3 ' (• 12+0 ' ' 19— I X 21+9 t 22+2 t 1+5 *- s+i *- 2—2 4- 6-6 4- 3-9 <- 8+4 4- 15+3 4- 12+0 4- 19 — I 4- 9-3 4- 16-4 4- 13—7 4- 20—8 4- 10— 104- 17— 114- 23-5 <- O— 124- 7 + II4- 14 + 104- 21+9 *- 4+8 4- 11+74- 18+64- 1+5 *- 8+4 4- 15+3 4- 22+2 4- 5 + 14- 12+0 4- 19—14- 2—2 4- 9—3 4- 16—4 4- 23—5 4- 6—64- 13—74- 20—8 4- 3—9 4- 10—104- 17— 114- 23—12 + 6+Il't' 13 + IO-t' 20+9 -f 3+8 j- 10+7 f 17+6 f 0+5 + 7+4 t 14+3 t 21+2 '^ 4+1 i !■ ii+o ^ f 18-1 ^ 1—2 ' ■ 8-3 ' ' 15-4 ' 5-6 ' • 12—7 ' ' 19-8 ' 2—9 ' ^ 9— lo-" ' 16-11' 6+lli [ 13+10' h 20+9 ^ 22-5 '^ 23-I2t 3+8 ■' 0+5 4+1 1—2 5-6 2-9 t 6+lif 3+8 j- 0+5 4- 4+1 <- 1 — 2 4- 5-6 4- 2—9 4- 6+ii<- 3+8 4- 10+7 7+4 :i+o 8-3 12—7 17+6 .14+3 18- 1 15-4 19-8 21+2 f 22-5 '^ 9— lO-f 16 — lit 23— I2t 13 + iof 20+9 X 10+7 1 17+6 *- 7+4 4- 14+3 4- 21+2 4- II+O 4- 18-I 4- 8-3 4- 15-4 4- 22-5 4- 12—7 4- 19—8 4- 9 — 104- 16 -114- 23—124- 13 + 104- 20+94- 10 + 7 <- 17+6 *" 18+9 t 4+104- 18—9 4- 4-lot 15+11 1 15-1K- 7—104- 14—114- 4+114- 11+104- 18+9 '^ 1+8 ^ 8+7 1 15+6 '^ 22+5 1 6+44- 13+3 <- 20+2 4- 3+1 «- 10+0 't' 17-1 1 0—2 f 7-3 1 14-4 1" 22—5 4- 5-6 4- 12—7 4- 19-8 4- 2-9 t 9— lof 16— lit 23-I2t 6+ilt 14+104- 21+9 <- fi or 2 MS 4+8 • 11+7 <- 18+6 t "i+s't 6-7 t 21—8 t 12—9 4- 3—104- 17— lit 8— I2t 23+114- 14+104- •4+9 t 19+8 t 10+7 <- 1+6 4- 16+5 4- 6+4 t 21+3 t 12 + 2 4- 3+1 4- 17+0 t 8—1 t 23—2 4- 14-3*- •■■ 4-4 t 19-5 t 10—6 4- 1-7 4- 6-9 t 12 — 114- 3-124- 8+lot 14+8 4- 4+7 t 10+5 <- 1+4 <- 6+2 t 12+0 4- 15-8 17+lIt 23+9 4- 19+6 t 15+3 t 21 + 1 4- ■I t 17—2 t 8-3 + 23-4 4- 14-5 <- 4-6 t 19-7 t 10—8 4- 1-9 4- 15 — lot 6— lit 21 — 124- I2 + II4- 2 + Iot 17+9 t 8+8 4- 13+6 t 4+5 t 10+3 4- 1+2 4- 6+0 t 12 — 2 4- 2-3 t 8-5 4- 13-7 t 4-8 t 10—104- I-II4- 6+Ilt 12+9 4- 2+8 t 8+6 4- 13+4 t •4+3 t lO+I 4- 0+0 t 6-2 4- 12—4 4- 2-5 t 8-74- 13-9 t 4- lot 10 — 124- + Ilt 23+7*- 19+4 t I5+I t 21 — I 4- 17-4 t 23-6 4- 19-9. 15— I2t 21 — 104- 17+7 t 23+5 <- 19+2 4- I5-I t 21-3 4- 17-6 t 23-8 4- 19—114- 15 + Iot 6+9 4- 21+8 4- II + 7 t 2+6 t 8+4 4- 13 + 2 t "it 17+5 <- 23+3 *- 4+1 19+0 ■ 10— 1 4- 0-2 t 15-3 t 6-4 4- 21—5 4- 11 — 6 t 2—7 t 17" 8—9 4- 22— lot 13-Ilt 4— 12t 19 + 114- 10+104- 0+9 t 15+8 t 598 UNITED STATES COAST AND GEODETIC SUKVEY. Table 42. — Component hours derived from solar hours — Continued. Day of series 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 371 6-5 t 1-7 <- 5-Iot 0—12^ 4+9 <- 8+6 t 3+4 «- 7+1 <- 2—1 <- 6—4 <- o— 5 f 5-9 <- 9—12^ 3+lot 7+7 f 2+5 6+2 I+O 5-3 10-8 14— II 9 + 11 13+8 20—9 <- I9+I0«- 23+7 «- 17+5 t •■ 12+3 T 21+2 t 16+0 T II — 2 <- 20—3 t 15-5 t 19-8 f 23-11 1 10 — 7 <- 14—10^ l8 + llt 13+9 10-7 j r 17-8 1 0-9 ' ■ 7-10' ' 14— II' 4+11^ • II + IO' ' 18+9 ' 1+8 ^ ' 8+7 ' ' 15+6 ' 5+4 ' [ 12+3 1 \- 19+2 't' 21-12'!' 22+5 1' 2+1 '^ 9+0 1 16— 1 1 23—2 ^ I — 12't 8+4 f 15+3 t 22 + 2 t 6 + 1 13+0 *- 3-2 <- 10-3 «- 17-4 '^ 0-5 1 7-6 1 14-7 1 s-g'*- 12— IQ<- 19—11^ 9+Ilt l6+lof 23+9 't' 6+8 t 14+7 <- 21+6 <- ja or 2 MS 6+7 <- 21+6 <- II+5 t 2+4 't' 17+3 <- 8+2 <- 22+1 'f 13+0 t 4—1 «- 19-2 <- 9-3 t •■• 0-4 t 15-5 t 6—6 <- 21—7 <- II— 8 t 2—9 t 17— io<- 8—11^ 22— I2t 13 + iit 4+io«- 19+9 <- 9+8 t 0+7 '^ 15+6 <- 6+5 <- 20+4 1 II +3 1 •• 2+2 1 17+1 ■<- 8+0 <- 22—1 1 13-2 1 4-3 <- 19-4 <- 9-5 f 0-6 1 15—7 <- 6—8 «- 20— 9 t II — lO'I' 2— ll<- 17— 12<- 8+II<- 22+lof 13+9 t • 4+8 «- 19+7 •«- 9+6 t 0+5 i 15+4 <- 6+3 ^ 20+2 f ll + l t 2+0 23-9 *- 21 — ii<- 8— 12«- 19+11*- 6+10*- 17+9 <- 4+8 <- 15+7 t 2+6 t 13+5 f 0+4 f II +3 t 22+2 f 9+1 t 20+0 f 7-1 t 19—2 *- 6-3 <- 17-4 <- 4-5 <- 15-6 <- 2—7 <- 13-8 *- 15 + 1 *- 21+2 f 2+3 <- 8+4 t 13+5 <- 19+6 t 0+7 <- 6+8 t 12+9 t 17 + 10*- 23+11 1 4+i2<- 10 — Ilf 15-10*- 21—9 t 2-8 *- 8-7 t 13-6 <- 19-5 t 0—4 *- 6-3 t II— 2 *- 17-1 t 22—0 *- 4+1 t 10+2 f 15+3 <- 21+4 t 2+5 *- 8+6 1- 13+7 <- 19+8 t 0+9 *- 6 + lof 11 + 11*- I7+I2t 22—11*- 4-10I- 9-9 <- 15-8 t 20—7 *- 2—6 *- 8-5 t- 13-4 *- 19-3 1" 0—2 *- 6-1 -f ij— o *- 17+1 t 22+2 *- 4+3 I' 9+4 «- 15+5 f 20+6 <- 2+7 t 7+8*- 13+9 1" 19+10 't' O + II*- 6+I2t II — II*- 17— lot 22—9 *- 4-8 t 9-7 <- 600 UNITED STATES COAST AND GEODETIC SUKVEY. Table 42. — Component hours derived from solar hours — Continued. Day of series. 87 83 89 90 91 92 93 94 95 96 97 98 99 100 lOI 102 103 104 105 106 107 108 109 110 III 112 "3 114 "5 116 117 118 119 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 7+0 <- . 3-1 <- 19-3 t • 15-4 t ■ 12—5 <- . 8-6 <- . 4-7 <- ■ 0-8 t 16— ict- . I3-IK- . 9 — 12<- 5+ll<- i + iof 17+8 t 14+7 «- 10+6 «- 6+5 «- 2+4 t 18+2 t 15+1 <- 23-2 <- 20—9 "t" 21+9 t 22+3 ^ -3 t II+O <- 7-1*- .... 3—2 t 23- 19-4 t .... 16-5 <- 12—6 <- 8-7 <- 4-8 I •■ 0—9 X 20— lox 17— II*- 21+8 t 13-12*- , 9 + 11*- . 5+iot 1+9 1" 18+7 *- 14+6 <- 10+5 *- 6+4 j 2+3 t 19+1 <- 15+0 *- II— I «- 7-2 t V 3-3 T 23-4 t 20-5 <- 22+2 f 6 *- 7 <- 8 " 4-9 -9 i -107 21— II*- 17—12*- l3+"<- 9+lot 5+9 t 1+8 t 22+7 *- 18+6 *- 14+5 *- 23+1 <- 10+4 t 6+3 + 3+2 *- 19+0 *- 15-1 t II— 2 f 7-3 "r 4-4 «- -5 <- 16-7 12-8 -9 5-10*- I — II*- 21—12*- 17 + Ilt i3+io|' 9+9 T 6+8 *- -9 T 2+7 *- 18+5 t 14+4 t 10+3 t 7+2 <- 22+6 <- 9-5 <- 5-7 <- 0-9 t 6— 12<- l + iof 7+7 *- 2+5 '^ 8+2 *- 3+0 t 9-3 <- 4-5 t 0-7 t 5-iot I — I2f 7+9 <- 2+7 t 8+4 <- 3+2 t 4-3 t 5-8 I— 10 6+11" 2+9 8+6 *- 3+4 t 9+1 *- 4-1 t 0—3 <- 5-6 t 1-8 *- 6-iit 2+11*- 7+8 t 3+6 t 8+3 t 4+1 t o— I *- 5-4 t 1-6 <- 6-9 t 2— II*- 7+lot 3+8 *- 8+5 t 4+3 t o+I *- 5-2 t 1-4 <- 6-7 t 2—9 *- 7-l2t 3+lo<- 8+7 1- 4+5 <- 0+3 *- 5+0 *- 1-2 *- 6-5 t 2—7 *- 7— lo't' 3-12*- 8+9 t 4+7 <- 9+4 t 5+2 <- l+o *- 6-3 *- 2-5 *- 7-8 t 3-io<- 8+iit 4+9 <- 9+6 t 5+4 *- 0+2 ^ 6-1 *- 1-3 t 7-6 <- 9—6 *- 4-8 t o— 10*- 5+iit 1+9 t 6+6 t 2+4 f 8+1 *- t 9-4 *- 4-6 t 0-8 *- 5-ilt l + ll*- 6+8 t 2+6 <- 7+3 t 3+1 t 6 *- 5-9 1" ■ li<- 6+iot ■ 2+8 *- 7+5 t 3+3 *- 8+0 t 4-2 t 0—4 *- 5-7 t 1-9 *- 6— I2X 2+10*- 7+7 t 3+5 *- 8+2 t 4+0 «- 0—2 *- 5-5 <- 7 *- 6— icj^ 2—12*- 7+9 t 3+7 ■<- 8+4 t 4+2 *- 9+0 t 5-3 •^ 1-5 <- 6— b *- io<- 7+iit 3+9 *- 8+6 t .+4 <- 9+2 t 0-3 T 6-6 *- 1-8 t 7-11*- 3+ll<- 8+8 *- 4+6 *- 9+3 t 6-4 *- 6 t 2— lit 8+l0<- 4+8 *- 9+5 <- 5+3 <- o+i t 6-2 *- 4 t 7-7 <- 20—11*- 21+8 *- 22+3 *- 23-2 <- 20—7 *- 21 — 12*- 22 + 7 ■ 23 + 2 *- 19-5 t 18+5 t 17+4 *- 15+3 <- 13+2 *- 21 — 10*- 22+9 ^ 23+4 ■ 19-3 t 20—8 t 2I + Ilt 23+6 <- I9-I t 20—6 f 21 — Ilf 22+8 t 19+1 t 20-4 t 21-9 t 22 + Iof 23+5 t 20—2 f 21-7 t 22 — I2f 23+7 t 20+0 t 21-5 t II + I t 20—9 *- 9+0 t 2I+I0*- 7-1 t 22+5 <- 6-2*- 23+0 *- 4-3 <- 2-4*- 0-5 t 22—6 f 20—7 'T- IS -8 t 17-9 *- 15- -10*- 13- u*- II- -12^ 9+llt 7+lot 6+9 *- '4+8 <- 2+7 <- 0+6 t 22+5 t 20+4 t 19+3 *- 17+2 <- ■ I5+I <- 13+0 t II — I t 9-2 t 8-3 *- 6-4 <- 4-5 <- 2-6 t 0-7 t 22-8 t 21—9 *- 19—10*- 17— II*- 23+6 t 2+7 «- 21+5 t ■ 19+4 <- . 17+3 *- ■ 14+2 t . 12+r t lo+o *- 7-1 t 5-2 t 3-3 *- 1—4 <- 22—5 t 20—6 *- 18-7 *- 15-8 t 13-9 t II — 10*- 9 — II*- — 12^ 4+II*- 2 + 10*- 23+9 t 21+8 t 19 + 7 *- 16+6 f 14+5 t 12+4 <- 10+3 *- 7+2 t 5 + 1 t 3+0 *- O— 1 f 22—2 t 20-3 *- 18-4 *- 15-5 t 13-6 <- II— 7 *- 8-8 t 6-9 t 4—10*- 2 — II*- 23— I2f 21 + 11*- 19 + 10*- 16+9 t 14+8 t 12+7 <- 10+6 *- 7+5 t 5+4 ^ 3+3 t" V 0+2 t 22+1 t 20+0 *- 17- 1 t 15-2 t 13-3 <- II— 4 *- 8-5 t 6-6 t 4-7 *- 1—8 t 23—9 t 21 — lo<- 19— II*- 16 — I2t I4+II*- 12+10*- 9+9 t 7+8 5+7 *- 3+6 *- 0+5 t 22+4 *- 20+3 <- 17+2 t 15+1 t 13+0 *- 10— I t 8-2 t 6-3*- 4-4 t" 1-5 t 21 — 7 *- 18-8 t 16-9 t 23-6 t 2— II*- I — I2f I + II<- o+lo*- 23+9 t 22+8 t 22 + 7 *- 21+6 *- 20+5 t 20+4 *- 19+3 «- l8+2 t ■ I7 + I -* ■ 17+0 <- . 16-I t . 15-2 t • 15-3 *- ■ 14-4 *- . 13-5 t ■ 12—6 f . 12—7 *- . II-8 f . 10—9 x . 10—10*- . 9— II*- . 8-l2t . 8+Il«- . 7+10*- . 6+9 t ■ 5+8 t . 5+7 *- ■ 4+6 *- 3+5 t ■ 3+4 *- 2+3 *- 1+2 t o+I t 0+0 *- 22—2 t 22-3 *- . 21— 4 *- . 20—5 t ■ 19—6 t . 19-7 *- . 18-8 ■»■ . 17-9 t • 17—10*- . 16— ll<- . I5-I2f . 14 + 11T ■ 14 + 10*- . 13+9 t ■ 12+8 t . 12+7 *- . 11+6 *- . 10+5 t . 10+4 *- . 9+3 t" ■ 8+2 t , 7+1 t ■ 7+0 <- . 6-1 *- , 5-2 t 5-3 <- 4-4 *- 3-5 t 2-6 t 2-7 <- 1-8 *- 0-9 t o — 10*- 22 — I2f 2I + Ilf 21 + 10*- 20+9 t 19+8 f 23-1 *- MS 23-11*- 19+7 *- 18+6 <- 17+5 t 16+4 t 16+3 <- 0—9 *- II— lot 22— llf 9 — I2f 20 + Ilt 7+iot 18+9 t 5+8 t 16+7 t 4+6 *- 15+5 •«- 2+4 *- 13+3 *- 22+0 *- 9-1 <- 20 -2 t 7-3 t 18-4 t 5-5 t 16-6 t 3-7 t 14-8 t 1-9 t 13-10*- 15-6 t 20-5 ■(- 11 — 12*- 22+11*- 9+lo<- 20+9 *- 7+8 <- 18+7 *- 2-4 t 7-3 •<- 13-2 t 18-1 <- 0—0 t 5-1 <- II + 2 f 17+3 T 22+4 *- 4+5 t 9+6 *- 15+7 t 20+8 *- 2+9 t 7+lo<- 13+iit I8 + I2«- O— II t 5—10*- II-9 t 16-8 *- 22—7 t 3-6 <- 9-5 *- 15-4 t 20-3 *- 2—2 t 7-1 *- 13-0 t 18+1 *- 0+2 t 5+3 «- II +4 t 16+5 *- 22+6 f 3+7 *- 9+8 t 14+9 *- 20+iot 2+11 1 7+12*- 13-llt 18—10*- 0-9 t 11-7 t 16-6 *- 22-5 t 9-3 t 14—2 *- I— o *- 7+1 t 12+2 *- 18+3 t 0+4 t 5+5 <- 11+6 t 16+7 *- 22+8 t 3+9 <- 9+iot REPORT FOR 1897 — PART II. APPENDIX NO. 9. 601 Table 42. — Component hours derived from solar hours — Continued. series ibi 162 163 164 165 166 167 l6g 169 170 171 172 173 174 175 176 177 178 179 180 181 182 i8j 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 237 238 239 240 3+1 «- 23+0 <- I9-I -r 15-2 ■(• 11-3 t S-4 «- 4-5 <- 0—6 2+4 <- 7+1 8-4 ■■ 4-6 0-8 *- 5-iit i+ii*- 6+8 t 2+6 *- 7+3 t 3+1 <- 8-2 t 4-4 <- 0—6 *- 5-9 t I — Il<- 6+iof 2+8 *- 7+5 t 3+3 <- 8+0 t 4-2 *- 0—4 *- 5-7 <- 1-9*- 2-9 f 8-12*- 3+iot 9+7 «- 4+5 t °+3 t 6+0 *- 1—2 ^ 7-5 <- 7 t 10*- 3-l2t 9+9 «- 4+7 t 0+5 t 5+2 t 1+0 t 7-3 <- ■5 t 3-lot 9+11*- 4+9 t 0+7 <- 5+4 t 1+2 *- 6—1 ■' 3 ■• 7-6 ■■ 3-8 t 9— II*- 4+iit 0+9 «- 5+6 t 1+4 «- :6+i t 2 — 1 *- 7-4 t 3-6 t 9 T ■lit o+ii<- 5+8 t 1+6 *- 6+3 t 2+1 *- 7-2 t 3-4 *- 8-7 t 9 «- II<- 5+iot 1+8 *- 6+5 f 2+3 «- 7+0 t 3-2 *- 8-5 t 4-7 <- 9 *- 5-i2<- I+IO*- 6+7 1- 2+5 *- 7+2 f 3+0 <- 8-3 t 9-7 t 10*- :o— I2f ;6+9 *- 2+7 *- 7+4 <- 3+2 *- 8-1 t 4-3 «- 9-5 t 5-8 *- 10 -^ 22— lof 23+9 t 20+2 *- 21-3 *- 22-8 t 23 + Ilt 20+4 • 21- -I «- 22- -6 *- 23- -11 t 19+ 1 «- 20-1-6 *- 17+0 <- 2I + I <- 15-1 t 22—4 *- 13-2 t 23-9 <- II-3 t 20+8 *- 21+3 *- 22—2 *- 23-7 *- 20+10*- 21+5 <- 22+0 *- 23-5 *- 19— I2t 20+7 t 22 + 2 ■<- 23-3 <- 19— lot 20+9 t 21+4 t 23-1 *- 19-8 t 20+11 1 21+6 t 22 + 1 t 19—6 t 20—11 f MK I5-I2t 13+lit ii+iot 10+9 *- "s+s'*^ 6+7 <- 4+6" t 2+5 t 0+4 t 22+3 f 21+2 *- 10—4 «- ■'8-5'*^ 6-6 <- 4-7 "t 2-8 t 0-9 t 23—10*- 19—12*- 17+llt 15+lot 13+9 t '12+8*^ 10+7 ■ "k+6 ■ 6+5 t 4+4 t 2+3 t 1+2 *- 23+1 *- 21+0 «- 19-1 t 17-2 t 15-3 t 14-4 *- 12-5 *- 22+7 <- 14-10*- 12— II*- 9 — I2t 7+n<- 5+io<- 2+9 f 0+8 f 20-f 6 *- 17+5 t 15+4 *- 13+3 <- 10+2 f , 8+1 t 6+0 *- 4-1 *- , 1—2 f 21—4 *- 18-5 1 16-6 { 14-7 *- ■ II-8 ^ . 9-9 t . 7—10*- , 5-II*- . 2 — I2t . 0+Ilt 22 + 10*- 19+9 17+8 15 + 7 <- 13+6 <- 23-3 ■ Ilf 9 t 8 + 10+5 t 8+4 *- 6+3 *- 3+2 t i+i f 23+0 *- 21—1 *- , j8— 2 t , 16-3 *- , 14-4 *- . II-5 t 9-6 t ■ 7-7 «- 4-8 t . 2-9 t -10*- 22—11*- 19-1 17+1 15 + 10*- -I2t Hl + 12+9 t 10+8 t 8+7 *- 6+6 *- 3+5 t 1+4 *- 23+3 *- 20+2 t 18+I t 16+0 <- 14— I *- . II— 2 t , 9-3 <- ■ 7-4*- 4-5 ^ 2-6 t 0—7 *- 22—8 *- 19-9 t 17—10*- 15-11*- 12— I2t lo+iix 8+10*- 5+9 t 3+8 t 1+7 ■<- 20+5 f 18+4 t 16+3 *- 13+2 t ii + i T 9+0 *- 7-1 *- 23+6 *- IS + 2 t . 14+1 t 14+0 *- I3-I <- 12—2 t ■ 12-3 <- II-4 <- . 10-5 t . 9-6 t . 9-7 <- . 8-8 *- . 7-9 t ■ 7— lo«- . 6— ii«- . 5-i2t ■ 4+iit . 4+10*- . 3+9 <- ■ 2+8 t ■ 2+7 <- • 1+6 *- 0+5 t 23+4 t 23+3 <- +2 f +1 t 21+0 *- . 20—1 *- , 19—2 f 18-3 { , 18—4 *- , 17-5 f ■ 16-6 t , 16—7 *- , 15-8 *- . '4-9 t ■ 14—10*- , 13-11*- . 12— 12't' . Il + Ilf . 11+10*- . 10+9 «- • 9+8 t . 9+7 *- ■ 8+6 *- . 7+5 t ■ 6+4 t ■ 6+3 *- . 5+2 t . 4+1 + • 4+0 *- . 3-1 ♦- 2—2 t 1-3 + 1-4 «- 0-5 T 23-6 t 23-7 «- 22-8 *- 21—9 ^ , 20— lox . 20—11*- . 19 — I2t . i8+ilf , 18+10*- . 17+9 <- . 16+8 t ■ 16+7 «- . 15+6 *- . 14+5 t . 13+4 T ' 13+3 «- . 12+2 *- . ii+i 1" ii+o *- . 10— I *- , 9-2 t ■ 8-3 t ■ 8-4 *- , 7-5 t 6-6 f . 6-7*- , 5+6 t 16+S t 3+4 t 14+3 t 1+2 t 12 + 1 t 23+0 t 10— I f 21—2 t 9-3 <- 20—4 *- 7-5 «- 18-6 *- 5-7 «- 16-8 «- 3-9 ■<- 14—10*- I — iif 12— I2f 23 + 11 1 lo+iof 21+9 t 8+8 t 19+7 t 6+6 t '18+5'*^ 5+4 <- 16+3 *- »+2 <- 14+1 <- l+o *- 12— I *- 23-2 *- 14+11*- 20 + I2t I-II*- 7— lOf 12—9 *- 18-8 f 23-7 *- 5-6 1- 10-5 *- 16—4 <- 22—3 t 3-2 *- 9-1 f 14—0 *- 20+1 f 1+2 «- 7+3 1- 12+4 *- 18+5 t 23+6 *- 5+7 t 10+8 *- 16+9 t 21+10*- 3+iit 9— 12'f' 3+1 9- 14— II*- 20— lot 1-9*- 7-8 t 12—7 *- 18-6 t 23-5 •<- 5-4 t 10-3 *- 16—2 t 21— I *- 3-0 t 8+1 *- 14+2 t 19+3 «- 1+4 t 7+5 t 12+6 *- 18+7 t 23+8 <- 5+9 t 10+10*- l6+llt 21 + 12*- 3-llt 8-10*- 14-9 t 19-8 *- 1-7 t 6-6 *- 12-5 t 17-4 <- 23-3 *- 5-2 t 10— I *- 16—0 t 2I + I *- 3 + 2 t 8+3 «- 602 UNITED STATES COAST AND GEODETIC SURVEY. Table 42. — Component hours derived from solar hours — Coutinuecl. Day of series. 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 282 2S3 286 287 290 291 292 293 294 295 296 297 29S 299 300 301 302 303 304 305 306 307 308 309 310 3" 3'2 313 314 315 316 317 318 319 320 19+1 <- 15+0 t II— I T- 7-2 t 4-3 <- 0—4 <- 20—5 <- 16-6 t 12—7 f 8-8 t 5-9 -^ I — io<- 21 — iif 17 — I2f 13 + IlT lO + ICX- 6+9 -tr- 2+8 18+6 ^ 14+5 t II+4 <- 7+3 <- 3+2 <- 23+1 t 19+0 f I5-I t 12—2 15. Ci < 15. If the decimal of solar hour in the first equation above fall between 0.0 and 0.5, the arrow should be horizontal; if between 0.5 and 1.0, vertical. ' Ci ]> 15. Reverse this rule. The speeds used are those derived from the mean motions given in § 13. Instead of the above rules, the following may be used: Suppose all solar hours of the series to have been converted into component hours ; in each doubtful case mark that solar hour which lies nearest the component hour thus considered. 604 UNITED STATES COAST AND GEODETIC SURVEY. Tablm 43. — For the summation of long-period tides. Day of series. M£ MSf Mm Sa Day of series. Mf MSf Mm Sa Day of series. Mf MSf Mm Sa I 76 18 I3<- 18 151 12 2 11 2 I 1 I 77 19 14 I9<- 152 13 3 12 3 2 2 2 78 20 15 19 153 14 4 13 4 3 3 3 79 21 • 16 20 154 15 5 14 5 4 4<- 4 80 22 I7<- 21 155 16 * 15 6 5 4 5 81 23 17 22 156 17*- 6<- 15*- 7 6 5 6 82 0«- 18 23 157 17 7 16 8 7<- 6 7 83 19 5 158 18 8 17 9 7 7 7«- I 84 I 20 I 6 159 19 9 18 10 10 8 8 8 85 2 21*- 2<- ' 160 20 I0<- 19 II II 9 9 9 86 3 21 2 161 21 10 20 12 10 9*r 10 87 4 22 3 162 22 11 21 13 II 10 II 83 23 4 163 23 12 22 14 12 II 12 89 6 164 o«- 13 22«- 15 13 12 I3«- 90 7<- I 6 165 14<- 23 16 I4«- 13 13 91 7 2 7 166 I 14 17 14 13*- 14 92 8 2<- 8 167 2 15 1 l8 15 14 15 93 9 3 9 168 3 16 2 19 16 15 16 94 10 4 9*- 169 4 17 3 20 17 16 17 95 II 5 10 170 5 18 4<- 21 18 17<- 18 96 12 6«- 11 171 6 19 4 22 19 17 19 97 13 6 12 172 7 I9«- 5 23 20 18 20<- I 98 14 7 13 6 173 8 20 6 24 21 19 20 2 99 15 8 14 7 174 S*- 21 7 II 25 22 20 21 100 I5«- 9 15 175 9 22 8 12 26 22<- 21 22 lol 16 io<- 16 176 10 23*- 9 27 23 22 23 102 17 10 16*- 177 11 23 10 28 22*- 103 18 II 17 178 12 ii<- 29 I 23 I 104 19 12 18 179 13 I II 30 2 2 105 20 13 19 180 14 2 12 31 3 1 3 106 21 14 20 181 15 3*- 13 32 4 2 3<- 107 22 15 21 182 I5<- 3 14 33 5 2«- 4 108 22<- I5<- 22«- 183 16 4 15 34 5<- 3 5 109 23 l5 22 184 17 5 16 35 6 4 6 110 17 23 185 18 ^ 17 36 7 5 7 III 1 18 186 19 7 iS 37 8 6«- 8 112 2 I9<- I 187 20 8 18«- 38 9 6 9 2 113 3 19 2 7 188 21 8«- 19 39 10 7 10 3 114 4 20 3 8 189 22<- 9 20 12 40 II 8 IO<- 115 5<- 21 4 190 22 10 21 13 41 12 9 II 116 5 22 5«- 191 23 II 22 42 I2«- 10 12 117 6 23<- 5 192 I2«- 23 43 13 II 13 118 7 23 6 193 I 12 44 14 ll«- 14 119 8 7 194 2 13 I 45 15 12 15 120 9 I 8 195 3 14 I<- 46 16 13 l6<- 121 10 2 9 196 4 15 2 47 17 14 16 122 II 3 10 197 5<- 16<- 3 48 . 18 15 17 123 I2«- 4 II 198 5 16 4 49 I9<- I5«- 18 124 12 4<- 12 199 6 'Z 5 50 19 16 19 125 13 5 12«- 200 7 18 6 SI 20 17 20 126 14 6 13 201 8 19 7<- 52 21 18 21 3 127 15 7 14 202 9 20 7 53 22 19*- 22 4 128 16 8<- 15 8 203 10 21 8 54 23 19 23«- 129 'Z 8 16 9 204 II 2I«- 9 13 55 20 23 130 18 9 17 205 I2«- 22 10 H 56 I 21 131 I9<- 10 18 206 12 23 11 57 2*- 22 I 132 19 II 19 207 13 12 58 2 23 2 133 20 12«- I9<- 208 14 I«- 13 3 3 134 21 12 20 209 15 I 14*- 60 4 o«- 4 135 22 13 21 210 16 2 14 61 5 1 5 136 23 14 22 211 17 3 15 62 6 2 6 137 IS 23 212 18 4 16 63 7 3 6«- 138 I 16 213 19 5*- 17 64 8 4 7 139 2 17 1«- 214 20 5 18 65 9 4<- 8 140 3 I7«- I 215 20*- * 19 66 10 5 9 141 3<- 18 2 216 21 7 20 67 IO<- 5 10 142 4 19 3 217 22 8 21 68 II 7 II 4 143 5 20 4 218 23 9 21«- 69 70 12 8«- 12 5 144 6 2I«- 5 9 219 10 22 13 8 13 145 7 21 ^ 10 220 I 10<- 23 14 71 14 9 I3«- 146 8 22 7 221 2 11 15 72 15 10 14 147 9 23 8<- 222 3 12 I 73 16 II 15 148 10 8 223 3<- 13 2 74 75 17 12 16 149 I0<- I«- 9 224 4 14*- 3 I7<- 13 17 150 11 I 10 225 5 H 4 This table gives the nearest component "hour" (i.e., 24th of monthly or yearly period) for each day (11:30 a. m. ) of the series. In Mf MSf, and Mm two days sometimes fall upon the same "hour." The arrow is used to indicate the one making the closer coincidence. Consequently the one so marked, or rather the corresponding daily height, is the one to be taken in preference to the other. See note given below Table 38. REPORT FOR 1897 — PART II. APPENDIX NO. 9. Table 43. — For the summation of long-period tides — Continued. 605 Day of Mf MSf Mm Sa Day of Mf MSf Mm Sa Day of Mf MSf Mm Sa series. series. series. 226 6 15 4<- 3" 9 I2<- 6<- 20 396 ii<- 9 8 227 7 16 312 I0<- 13 7 21 397 12 10 9 228 8 17 6 313 10 14 8 398 13 II to 229 9 18 7 314 II 15 9 399 14 12 II 230 IO<- 19 8 315 12 16 10 400 15 I3<- 12 231 10 19*- 9 316 13 l6«- II 401 16 13 13 232 II 20 I0«- 317 14 17 12 402 17 14 14 2 233 12 21 10 318 15 18 13 403 18 15 15 3 234 13 22 , II 319 16 19 ' I3<- 404 l8<- 16 I5<- 235 14 23 12 15 320 17 20<- 14 405 19 17 16 236 15 23<- 13 16 321 18 20 15 406 20 18 17 237 16 14 322 .8-«- 21 16 407 21 l8<- 18 238 I7<- I 15 323 19 22 17 403 22 19 19 239 17 2 16 324 20 23 18 409 23 20 20 240 18 3^^ I7<- 325 21 I9<- 410 21 21 241 19 3 17 326 22 I 19 21 411 1<- 22<- 22 242 20 4 18 327 23 I<- 20 22 412 I 22 22<- 243 21 19 328 2 21 413 2 23 23 244 22 6 20 329 I 3 22 414 3 245 23 7 21 330 I<- 4 23 415 4 I I 246 8 22 331 2 5 h 416 5 25 16 17 l8«- 18 19 23 9 9<- 13 14 15 16 17 18 23«- 23 I3«- 13 14 15 16 17 16 17 l8«- 18 19 20 23 23<- 3*- 3 4 5 6 7<- 7 12 12<- 13 14 15 I6«- 16 17 18 19 3 4 5<- 5 6 7 9<- 9 12 13 14 I4«- 15 16 17 19 16 I7<- 17 18 19 20 23 IK- 3 4 5 6 7 7<- 14 14*- 15 16 17 18 19 2CX- 3<- 3 4 5 6 13 14 15 16 >7 I7«- 18 19 23<- 23 3 3 4 I 5 2 6«- 3 6 3*- 7 4 8 S 9 6 10 Dayoi series 131 1 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1320 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 I3« 1367 1368 1369 1370 1371 1372 1373 1374 Mf 23 4 4<- 5 6 7 13 14 15 16 17 l3 19 I9<- 2<- 3 4 5 6 7 8 9 9«- 13 14 15 l6<- :6 17 18 19 23<- MSf 1375 7<- 1376 8 1377 9 1378 10 1379 II 1380 12 I18I 13 I3S2 14 1383 I4<- 1384 15 1385 16 1386 17 1387 18 -1388 19 1389 20 1390 21 12 13 14 15 16 l6<- 17 18 19 20<- 20 23 0«- 5<- 6 7 8 9^ I3<- 13 14 15 16 17 18 l8<- 19 20 3 4 5 6 7 7«- 10 ll<- 13 14 15 16 l6<- 17 18 19 23 0*- 13<- 14 15 16 17 18 19 9*- 9 13 14 15 i5 i6<- 17 23 23<- 3 4 5<- 5 6 7 13 14 15 16 17 18 19 I9<- 20 21 2<- 3 7 8<- Day of Mf series 1391 2I<- 1392 22 1393 23 1394 1395 I 1 1396 2 1397 3 1398 4<- 1399 4 1400 5 1401 6 1402 7 . 1403 8 1404 9 1405 10 1406 II*- 1407 II 1408 12 1409 13 1410 14 1411 15 1412 16 1413 17 1414 18 1415 19 1416 I9<- 1417 20 1418 21 1419 22 1420 23 1421 1422 I 1423 2 1424 2<- 1425 3 1426 4 1427 5 1428 6 1429 7 1430 8 143- 9<- 1432 9 1433 10 1434 II 1435 12 1436 13 1437 14 1438 15 1439 l6<- 1440 16 1441 17 1442 18 1443 19 1444 20 1445 21 1446 22 1447 23 1448 1449 o<- 1450 I 145 1 2 1452 3 1453 4 1454 5 1455 6 1456 7 1457 7*- 145S 8 1459 9 1460 10 1461 II MSf 3 4 5 5«- 6 7 8 9 9<- I3<- 13 14 15 16 17 18 l8<- 19 22<- 23 7<- 8 9 13 14 15*- 15 16 17 18 19 3 4<- 4 5 6 9 9<- 13 14 I5<- 15 16 17 18 19 22^ 23 3 4 5 5<- 6 7 9 10 II*- 13 14 15 16 17 l8«- 18 19 8<- 9 13 14* 14 15 16 17 18 19 6584- -39 610 UNITED STATES COAST AND GEODETIC SURVEY. Tablb 44. — Acceleration in HW and LW of [The amplitude of the semidiurnal wave is taken as unity.] ^i HWpl- 1,-Wph ase.* ase.* 0° 180 10° 190 20° 200 30° 210 40° 220 50° 230 60° 240 70° 250 80° 260 90° 270 / ; ; / / 1 / / / / o-o CX3 0, CX3 00 00 00 00 00 00 00 00 O'l 00 29 57 I 24 I 48 2 10 2 27 2 40 2 49 2 52 0'2 00 57 I 52 2 45 3 33 4 15 4 50 5 19 5 36 5 44 0-3 00 I 23 2 45 4 02 5 14 6 17 7 II 7 53 8 22 8 36 0-4 00 I 49 3 35 5 16 651 8 15 9 28 10 26 II 07 II 29 o"5 00 2 13 4 23 6 28 8 24 10 10 II 41 12 56 13 50 14 22 0-6 00 2 36 5 09 7 37 9 55 12 03 13 52 15 24 16 33 17 15 07 00 2 58 5 53 8 43 11 23 13 50 16 00 17 50 19 15 20 09 0-8 00 3 19 636 9 46 12 48 15 35 18 06 20 14 21 56 23 04 0-9 00 3 40 7 17 10 48 14 09 17 18 20 09 22 37 24 36 26 00 I'O 00 4 00 7 56 II 47 15 29 18 58 22 09 24 57 27 ]6 28 57 I'l 00 4 18 8 34 12 44 16 46 20 35 24 06 27 15 29 55 31 55 1-2 00 4 36 9 JO 13 39 18 00 22 09 26 01 29 32 32 33 34 55 I '3 00 4 54 9 46 14 33 19 12 23 41 27 54 31 46 35 II 37 56 I '4 00 5 11 10 20 15 24 20 22 25 10 29 44 34 00 37 48 40 58 I '5 00 5 27 10 52 16 14 21 30 26 37 31 33 36 II 40 24 44 03 1-5 00 5 43 II 24 17 02 22 36 28 02 33 18 38 20 43 00 47 09 17 00 5 58 II 53 17 49 23 40 29 25 35 02 40 28 45 36 50 18 1 1-8 00 6 12 12 24 i8 34 24 41 30 45 36 43 42 34 48 II 53 29 1-9 00 6 26 12 52 19 18 25 42 32 04 38 23 44 38 50 46 56 43 M-4 2-0 CXD 6 40 13 20 20 00 26 40 33 20 40 00 46 40 53 20 60 00 O 2-1 00 653 13 47 20 41 27 37 34 34 41 35 48 40 55 54 63 20 13 2-2 00 7 06 14 13 21 21 28 32 35 47 43 08 50 39 58 27 66 44 ' 5 2-3 00 7 18 14 38. 22 00 29 25 36 57 44 39 52 36 60 59 70 12 ft a 2-4 00 7 30 15 02 22 37 30 17 38 06 46 08 54 32 63 31 73 44 2-5 00 7 42 15 26 23 13 31 08 39 13 47 36 S6 25 66 03 77 22 2-6 00 7 53 15 48 23 49 31 57 40 18 49 01 58 16 6833 81 05 27 00 8 04 16 II 24 23 32 45 41 22 50 24 60 08 71 03 84 54 2-8 00 8 15 16 32 24 56 33 32 42 22 51 45 61 54 73 32 88 51 2-9 00 8 25 16 S3 25 29 34 17 •43 25 53 05 6339 76 00 92 56 3'o 00 f 35 17 14 26 00 35 01 44 22 54 22 65 23 78 26 97 II 3"i CX5 8 44 17 33 26 31 35 44 45 21 55 38 67 05 80 52 lOI 36 3-2 00 8 54 17 52 27 01 36 26 46 17 56 53 68 45 83 16 106 15 3 '3 00 9 03 18 II 27 30 37 06 47 12 5805 70 23 85 39 III 11 3-4 00 9 12 18 30 27 56 37 46 48 05 59 16 71 58 87 59 116 25 3-5 00 9 21 18 47 28 26 38 25 48 57 60 25 73 32 90 19 122 05 3-6 00 9 29 19 05 28 53 39 01 49 47 61 30 75 04 92 35 128 19 37 00 9 38 19 22 29 19 39 39 50 37 62 38 76 34 94 49 135 21 3-8 CX) 9 46 19 38 29 44 40 14 51 25 63 42 78 02 97 01 143 37 3 9 00 9 53 19 54 30 09 40 49 52 12 64 44 79 27 99 10 154 19 4-0 00 10 01 20 09 30 33 41 23 52 58 65 45 80 50 loi 16 180 00 HW Phase. 360° 350° 340° 330° 320° 310° 300° 290° 280° 270° LW Phase. 180 170 160 ■5° 1^0 130 120 no 100 90 *I. e., the phase of the diurnal wa%'e (B) at the time of HWor I,W of the semidiurnal wave (A). HW phase = (time of HW ol A - time of HW of £} b, LW phase = (time of LW ot A - time of HW of B) b. If one of the speeds be somewhat variable, the resultant times and heights will be given more accurately by keeping the phases between - 90° and + 90°. If they do not fall within these limits, this and the following table may be entered with the phases: HW phase = (time of LW oi A - time of LW of B) b, LW phase = (time of HW of ^ - time of LW oi B) b : the resultant heights must, however, have their signs changed. For tropic tides HW phase = n^ + %A°-B°,h^ phase = nir ±^+ y^ A° - B°, n being an integer. (See §§ 5, 20, Part III.) REPORT FOR 1897 — PART II. APPENDIX NO. 9. a semidiurnal wave due to a diurnal wave. [The amplitude of the semidiurnal wave is taken as unity.] 611 HW Phase. tW Phase. 100°, 280 110° 290 120° 300 130° 310 140° 320 150° 330 160° 340 170° 350 180° 360 / / / / / ; / / / o-o 00 00 00 00 00 00 00 00 00 0"1 2 50 2 43 2 31 2 14 I 53 I 28 I 00 31 00 0-2 5 42 5 29 5 05 4 32 3 50 3 00 2 03 I 03 00 o'3 8 35 8 18 7 44 655 5 52 4 36 3 09 I 37 00 0-4 II 3P II 10 10 28 9 24 7 59 6 17 4 20 2 12 00 o'5 14 28 14 06 13 16 II 58 10 12 8 02 5 33 2 50 00 o'6 17 27 17 06 16 10 14 38 12 31 9 54 6 51 3 30 00 07 0-8 20 29 20 10 19 09 17 24 14 57 11 51 8 13 4 13 00 23 34 23 19 22 14 20 18 17 30 .13 55 9 40 4 58 00 0-9 26 42 26 33 25 27 23 20 20 12 16 06 II 13 5 45 00 I'O 29 53 2§ 52 28 47 26 31 23 03 18 26 12 52 6 36 00 n 33 07 33 18 32 16 29 53 26 04 20 54 14 37 7 31 00 I'2 36 25 36 51 35 56 33 27 29 18 23 34 16 30 8 29 00 I "3 39 49 40 32 39 47 37 15 32 46 26 26 18 32 9 32 00 I '4 43 17 44 23 43 52 41 21 36 33 29 32 20 43 10 40 00 > ^'l 46 51 48 24 48 15 45 48 40 40 32 56 23 07 II 53 00 5 i'6 ^ 1-7 50 31 52 40 52 59 50 43 45 17 36 43 25 44 13 13 00 54 19 57 II 58 12 56 17 50 32 40 59 28 38 14 40 00 "3 1-8 58 15 62 03 64 05 62 53 5648 45 55 31 53 16 16 00 £ 1-9 62 21 67 22 71 01 71 19 64 57 51 56 35 39 18 02 00 2 ^-o 66 40 73 20 80 00 86 40 80 00 60 00 40 00 20 00 00 2'I 71 14 80 18 45 24 22 13 00 ^ 2-2 76 07 89 14 52 48 24 45 00 pi 2"k 81 25 27 44 00 U ^-4 87 18 31 17 00 1 2-5 94 08 35 47 00 2-6 102 48 42 08 00 27 00 2-8 00 2-9 \ 00 3'o 00 3"i 00 3-2 CO 3 "3 00 3-4 00 3-5 00 3-6 00 37 00 3-8 00 39 00 4-0 CXJ HW Phase. 260° 250° ■ 240° 230° 220° 210° 200° 190° iSo° I,W Phase. 80 70 60 50 40 3° 20 10 When the top argument is used the tabular values are positive; when the bottom argument, they are negative. To express the acceleration in time divide by a, the speed of the semidiurnal component. To find the acceleration when b is not exactly equal to % a, multiply the tabular values by 2b This acceleration is directly expressed in time by multiplying the tabular values by 2b Table 17 is a graphic form of this table. Rollet de I'lsle has given in the Annales Hydrographique for 1896 (p. 248) a graphic table serving ^e pur- pose of Tables 17, 18, or Tables 44, 45, and which he calls an abacus. It is really the inverse of Tables 17, 18. 612 UNITED STATES COAST' AND GEODETIC SURVEY. Table i?,.— Height of HW and UW for a tide [The amplitude of the semidiurnal wave is taken as unity.] HW Phase.* I,W Phase.* 0° 180 10° 190 20°' 200 30° 210 40° 220 50° 230 60° 240 70° 250 80° 260 90° 270 o-o I "OOOO 1 -oooo 1 -oooo 1 -oooo I -oooo I -oooo I "oooo 1 -oooo 1 -oooo 1 'OOOO O'l 1 -looo I -0985 I -0941 I -0869 I -0771 1 -0650 1 -0509 I -0353 I -0186 1 -0012 O '2 0-3 1 "2000 1-3000 I-I97I I -2958 1 -1885 I -2831 I -1744 I -2624 I -1552 I -2342 I-I3I4 1 -1991 I -1039 I -1581 I -0727 1 -H23 I -0395 1 -0629 1 -0050 1 -OII2 0-4 I "4000 I '3945 1-3780 I -3510 I -3I4I 1 -2681 1 -2143 I -1539 I -0885 I -0200 o"5 0-6 I -5000 1 -4932 1 -4731 I -4401 1 -3948 I -3384 I -2721 1 -1975 1 -1165 I -0312 ' I '6000 1 -5921 1 -5684 I -5296 I -4763 1 -4098 I -3314 1 -2430 1 -1467 1 -0450 07 0-8 I "7000 1 -6908 I -6639 I -6195 I -5585 I -4822 1 -3922 1 -2904 1 -1792 1 -0612 1 "8000 1 -7899 1 7596 1 -7099 1 -6415 I -5558 I '4545 I -3397 1 -2139 1 -0800 0-9 I -9000 I -8888 I -8555 1-8006 I -7253 I -6304 1 -5182 1-3908 1 -2508 1 -I0I2 I -o 2 -0000 I -9878 I '9515 1 -8917 1 -8094 I -7059 I -5832 i -4436 1 -289S 1 -1250 I ■! 2 -lOOO 2 -0869 2 -0477 I -9832 I -8942 I -7824 1 -6496 I -4982 I -3309 I -1512 I '2 2 '2000 2 -i860 2 -1440 2 -0749 I -9797 1 -8598 I 7172 I -5544 I -3741 1-1800 I '3 2 -3000 2 -2851 2 -2405 2 -1670 2 -0656 1 -9380 1-7860 1 -6122 I -4194 I -2112 I '4 2 '4000 2 -3842 2 -3371 2 -2594 2 -1522 2 -0170 1-8560 1 -6716 1 -4668 I -2450 i 1-5 1-6 2 '5000 2 -4834 2 -4339 2 -3520 2 -2392 2 -0970 I -9271 I -7325 1 -5161 1 -2812 & 2 '6000 2 -5826 2 -5307 2-4450 2 -3266 2 -1774 I -9993 I -7949 I -5673 1 -3200 'ri 17 2 7000 2 -6818 2 -6276 2 -5382 2 -4145 2 -2586 2 -0725 1 -8588 1 -6206 I -3612 1 '8 2 -8000 2 -781 1 2 -7247 2 -6316 2 -5029 2 -3406 2 -1468 1 -9241 I -6757 I -4050 I -9 2 -9000 2 -8804 2 -8219 2 7252 2 -5917 2 -4232 2 -2220 I -9908 1 -7327 1 -4512 -s 2 'O 3-0000 2 -9797 2 '9191 2 -8191 2 -6809 2 -5065 2 -2981 2 -0587 I -7915 I -5000 OJ 2 •! 3-1000 3 "0790 3 '0165 2 -9132 2 -7704 2 -5903 2 -3752 2 -1280 I -8521 I -5512 ■ 3 2 '2 3-2000 3 -1784 3-1139 3-0074 2 -8604 2 -6747 2 -4531 2 -198s 1-9144 I -6050 ."t^ 2 '3 3-3000 3 -2778 3-2114 3 -1019 2 -9506 2 -7597 2 -5318 2 -2702 I -9785 1 -6612 a 2-4 3-4000 3 -3772 3 'sogo 3 -1965 3 -0412 2 -8452 2 -6114 2 -3431 2 -0443 1 -7200 < 2-5 3-5000 3 "4766 3 -4067 3 -2913 3 -1321 2 -9312 2 -6917 2-417I 2 -III7 1 -7812 2-6 3 -6000 3 '5760 3-5044 3 -3863 3 -2233 3 -0177 2 -7728 2 -4922 2 -1808 1 -8450 , 27 3 -7000 3 '6755 3 -6023 3 -4814 3 -3148 3 -1047 2 -8545 2 -5683 2-2514 1 -9II2 2-8 3-8000 3 7749 3 -7002 3 -5767 3 -4065 3 -1921 2 -9370 2 -6455 2 -3234 I -9800 29 3 -9000 3 '8744 3 -7981 3 -6721 3 -4985 3-2800 3 -020I 2 -7236 2 -3970 2-0512 ■ 3-0 4 -0000 3 '9739 3 -8961 3 -7677 3-5908 3 -3682 3 -1038 2 -8027 2 -4721 2 -1250 3 "I 4-1000 4 '0734 3 -9941 3 -8634 3 -6833 3 -4569 3 -1882 2 -8827 2 -5485 2 -2012 3-2 4 -2000 4 -1730 4 -0922 3 -9592 3 -7760 3 -5459 3 -2731 2 -9636 2 -6262 2 -2800 3 '3 4-3000 4 -2725 4 -1904 4 -0552 3-8690 3 -6353 3 -3586 3 -0452 2 -7053 2 -3612 ! 3-4 4 -4000 4 -3720 4 -2886 4-1512 3 -9622 3 -7250 3-4446 3 -1277 2 -7856 2-4450 3-5 4-5000 4 "4716 4 -3869 4 -2474 4 -0556 3 -8151 3 -53" 3-2110 2 -8671 2 -5312 3-6 4-6000 4-5712 4 -4852 4 -3437 4 -1492 3-9055 3 -6181 3 -2950 2-9500 2 -6200 3 7 4 -7000 4 -6708 4 -5836 4-4401 4 -2429 3 -9962 3 -7056 3 -3797 3 -0334 2 -7112 3-8 4-8000 4 7704 4 -6820 4 -5366 4 -3369 4 -0872 3 -7936 3 -4652 3-I182 2 -8050 3-9 4-9000 4 -8699 4 -7804 4 -6331 4 -4310 4 -1785 3 -8820 3 -5512 3 -2039 2 -9012 4-0 5 -oooo 4 -9695 4 -8789 4 -7298 4 -5253 4 --2701 3 -9708 3 -6379 3 -2906 3-0000 HW Phase. 3600 35°° 340° 330° 320° 310° 300° 290° 280° 270° I,W Phase. 180° 170° 160° 150° 140° 130° 120° 110° 100° ^° * See footnote, preceding table. For high waters use the tabular values as given; but for low waters, alter their signs. REPORT FOR 1897 — PART 11. APPENDIX NO. 9. composed of a diurnal and semidiurnal wave. [The amplitude of the semidiurnal wave is taken as unity.] 613 HW Phase. LW Phase. 100° 280 110° 290 120° 300 130° 310 140° 320 150° 330 160° 340 170° 350 180° 360 Mean value.f O'O O'l 0'2 0-3 0-4 I 'OOOO -9839 -9702 -9590 -9503 I '6000 '9669 -9361 '9076 -8815 I '0000 0-9510 -9038 -8587 -8158 I -0000 "9365 -8745 -8141 7554 I 'OOOO -9238 -8489 07751 -7025 1 "OOOO -9137 -8281 -7432 -6591 I -OOOO -9062 0-8127 7195 -6267 I -OOOO -9016 -8032 -7049 -6067 I -OOOO -9000 0-8000 -7000 -6000 I -OOOO I -0006 I -0025 1-0056 I -0100 0-5 0-6 07 0-8 0-9 -9442 '9406 -9397 "9415 o'946o -8578 -8367 -8181 '8023 7891 7750 7365 07004 -6667 -6357 -6985 -6436 -5906 -5397 0-491 1 -6313 -5614 -4932 -4262 -3612 -5758 -4933 0-4118 0-33I4 0-2521 -5342 0-4423 -3508 -2598 -1693 -5087 -4107 0-3129 0-2152 0-II74 0-5000 0-4000 0-3000 0-2000 -1000 I -0157 I -0226 I -0308 I -0404 I '0513 i i-o I 'I I -2 1-3 1-4 -9532 -9632 '9760 -9919 1 -0105 7789 7715 7672 7661 7682 -6074 -5819 -5595 -5403 '5245 0-4448 -4012 0-3602 -3223 -2874 -2980 -2368 -1778 -I2I2 -0672 -1739 -0971 "0213 —0 -0520 — -1239 -0794 —0 -0098 — -0982 —0 -1859 —0 -2727 -0202 —0 -0770 —0 -1740 -0 -2709 -0 -3674 o-oooo — -1000 —0-2000 — -3000 —0-4000 I -0635 1 -0772 I -0921 1 -1088 I -1268 1 1-5 1-6 17 1-8 1-9 1 -0322 I -0569 I -0848 I -1159 I -1504 7738 7830 7959 '8129 -8343 -5123 -5041 0-5002 -5012 -5077 -2561 -2287 -2058 0-1879 9 '1765 -0161 —0 -0326 —0 -0756 — 0-1151 —0 -1487 -0 -1939 —0 -2616 —0 -3266 -0 -3886 —0 -4468 -0 -3584 -0 -4429 —0 -5264 —0 -6082 -0 -6882 —0 -4637 -0 -5597 -0 -6554 —0 -7506 -0 -8454 —0-5000 —0 -6000 -0-7000 — o-8ooo —0-9000 I -1465 I -1677 1-1909 1 -2160 1 -2433 1 •2 -o 2 •! 2 -2 2-3 2-4 2-5 2-6 27 2-8 ,2 '9 3"o 3-1 3 "2 3-3 3 '4 3 '5 3-6 3 7 3-8 3-9 4-0 I -1882 I '2296 I -2748 I -3238 I -3770 I -4348 I -4979 '8604 •8919 -9298 -5210 -1744 -0 -1736 —0-5000 —0 -7660 —0 -8412 — -9126 -0 -9397 -I '0333 — I -1262 —I -2182 —I -3090 -I -3984 -I -4855 — I -OOOO —I -1000 — I •2CX50 -1-3000 — I -4000 —I -5000 —I -6000 —I -7CKIO —I -8000 — I -9000 — 2 -OOOO — 2 -lOOO — 2 -2000 — 2-3000 — 2 -4000 — 2-5000 —2 -6000 — 2 -7000 —2 -8000 — 2 -9000 —3-0000 I -2732 HW Phase. LW Phase. 260° 80° 250° 70° 240° 60° 230° 50° 220° 40° 210° 30° 200° 20° 190° 10° 180° 0° ! 4^2 t when b is not exactly equal to J^ a, mean value = i + (tabular value — i ) -^ • Table iS is a graphic form of this table. The above column of mean values may be compared with expression (29), Part III, and with the last column of Table 21. 614 ■UNITED STATES COAST AND GEODETIC SURVEY. Table 46. — Hyperholic fitnciiona. u V sinh u cosh u tanh M In degrees. In degrees. 9 = tan V = sec V = sin V e" £~~" O'OO O'OOOOOO O'OOOO 0-0000 0-000 0-0000 I -0000 O'OOOO I'OOOO I'OOOO 0'02 1-1459156 0-0200 1-1458 I -145 0-0200 1-0002 0-0200 I '0202 0-9802 o'04 2-291831 0-0400 2-2912 2-288 0-0400 1-0008 0-0400 I 'O408 0.9608 o-o6 3-437747 0-0600 3-4357 3-428 0-0600 1-0018 0-0599 i'o6i8 0-9418 o-o8 4-58366 0-0799 4-5788 4-561 0-0801 1-0032 0-0798 1-0833 0-9231 O'lO 5-72958 0-0998 5-720 5-693 0.1002 1-0050 0-0997 1-1052 0-9048 0'12 6-87549 0-II97 6-859 6-811 0-1203 1-0072 0-1194 1-1275 0-8869 0-I4 8-02I4I 0-1395 7-995 7-917 0-1405 1-0098 0-1391 1-1503 0-8694 o-i6 9-16732 0-1593 9-128 9-0H 0-1607 1-0128 0-1586 1-1735 0-8521 o-i8 10-3132 0-1790 10-258 10-100 O-1810 1-0162 0-1781 1-1972 0-8353 0'20 11-4592 0-1987 11-384 11-167 0-2013 1-0201 0-1974 1-2214 0-8187 0'22 12-6051 0-2183 12-505 12-216 0-2218 1-0243 0-2165 1-2461 0-8025 0'24 13-7510 0-2377 13-621 13-254 0-2423 1-0289 0-2355 1-2712 0-7866 0'26 14-8969 0-2571 14-732 14-271 0-2629 1-0340 0-2543 1-2969 0-7711 0-28 16-0428 0-2764 15-837 15-265 0-2837 1-0395 0-2729 1-3231 0-7558 0-30 17-1887 0-2956 16-937 16-245 0-3045 1-0453 0-2913 1-3499 0-7408 0-32 18-3346 0-3147 18-030 17-197 0-3255 1-0516 0-3095 1-3771 0-7261 o'34 19-4806 0-3336 19-116 18-134 0-3466 1-0584 0-3275 1-4049 0-7II8 0-36 20-6265 0-3525 20-195 19-045 0-3678 1-0655 0-3452 1-4333 0-6977 0-38 21-7724 0-3712 21-267 19-935 0-3892 1-0731 0-3627 1-4623 0-6839 0-40 22-9183 0-3894 22-331 20-801 0-4108 I -081 1 0-3799 I '4918 0-6703 0-42 24-0642 0-4082 23-386 21-648 0-4325 1 -0895 0-3969 I '5220 0-6570 0-44 25-2101 0-4264 24-434 22-470 0-4543 1 -0984 0-4136 1-5527 0-6440 0-46 26-3561 o-/\i\/\(> 25-473 23-275 0-4764 1-1077 0-4301 1-5841 0-6313 . 0-48 27-5020 0-4626 26-503 24-045 0-4986 1-1174 0-4462 i-6i6i 0-6188 0-50 28-6479 0-4804 27-524 24-803 0-5211 1-1276 0-4621 1-6487 0-6065 0-52 29-7938 0-4980 28-535 25-533 0-5438 1-1383 o'4777 1-6820 0-5945 0-54 30-9397 0-5155 29-537 26-245 0-5666 1-1494 0-4930 1-7160 0-5827 0-56 32-0856 0-5328 30-529 26-930 0-5897 1-1609 0-5080 1-7507 0-5712 0-58 33-2316 0-5500 31-511 27-595 0-6131 1-1730 0-5227 1-7860 0-5599 o'6o 34-3775 0-5669 32-483 28-237 0-6367 1-1855 0-5370 1-8221 0-5488 0-62 35-5234 0-5837 33-444 28-861 0-6605 1-1984 0-5511 1-8589 0-5379 0-64 36-6693 0-6003 34-395 29-462 0-6846 1-2119 0-5649 1-8965 0-5273 0-66 37-8152 0-6167 35-336 30-045 0-7090 1-2258 0-5784 1-9348 0-5169 0-68 38-9611 0-6329 36-265 30-604 0-7336 1-2402 0-5915 1-9739 0-5066 070 40-1070 0-6489 37-183 31-149 0-7586 1-2552 0-6044 2-0138 0-4966 072 41-2530 0-6648 38-091 31-670 0-7838 1-2706 0-6169 2-0544 0-4868 074 42-3989 0-6804 38-587 32-174 0-8094 1-2865 0-6291 2-0959 0-4771 076 43-5448 0-6958 39-872 32-663 0-8353 1-3030 0-6411 2-1383 0-4677 078 44-6907 0-7III 40-746 33-132 0-8615 1-3199 0-6527 2-1815 0-4584 o-8o 45-8366 0-7261 41-608 33-587 0-8881 1-3374 0-6640 2-2255 0-4493 0-82 46-9825 0-7412 42-460 34-025 0-9150 1-3555 0-6751 2-2705 0-4404 0-84 48-1285 0-7557 43-299 34-446 0-9423 1-3740 0-6858 2-3164 0-4317 0-85 49-2744 0.7702 44-128 34-848 0-9700 1-3932 0-6963 2-3632 0-4232 0-88 50-4203 0-7844 44-944 35-238 0-9981 1-4128 0-7064 2-4109 0-4148 o-go 51-5662 0-7985 45-750 35-613 1-0265 1-4331 0-7163 2-4596 0-4066 o"92 52-7121 0-8123 46-544 35-976 1-0554 1-4539 0-7259 2-5093 0-3985 0-94 53-8580 0-8260 47-326 36-323 1-0847 1-4753 0-7352 2-5600 0-3906 o"96 55-0039 0-8394 48-097 36-660 . 1-II44 1-4973 0-7443 2-6117 0-3829 0-98 56-1499 0-8528 48-857 36-983 1-1446 1-5199 0-7531 2-6645 0-3753 I'OO 57-2958 0-8658 49-605 37-293 1-1752 1-5431 0-7616 2-7183 0-3679 I -02 58-4417 0-8787 50-343 37-593 1-2063 1-5669 0-7699 2-7732 0-36059 I -04 59-5876 0-8913 51-069 37-880 1-2379 1-5913 0-7779 2-8292 0-35345 I -06 60-7335 0-9038 51-783 38-158 1-2700 1-6164 0-7857 2-8864 0-34646 I -08 61-8794 0-9160 52-485 38-423 1-3025 1-6421 0-7932 2 '9447 0-33960 i-io 63-0254 0-9281 53-178 38-677 1-3356 1-6685 0-8005 3'oo42 0-33287 I-I2 64-1713 0-9400 53-860 38-924 1-3693 1-6956 0-8076 3-0649 . 0-32628 I-I4 65-3172 0-9518 54-531 39-160 1-4035 1-7233 0-8144 3'i268 0-31982 ri5 66-4631 0-9632 55-189 39-387 1-4382 1-7517 0-8210 3-1899 0-31349 i-i8 67-6090 0-9745 55-837 39-607 1-4735 1-7808 0-8275 3-2544 0-30728 1-20 68-7549 0-9857 56-476 59-817 1-5095 1-8107 0-8337 3-3201 0-30II9 B = the angle at the center of the hyperbola made by any secant line and the transverse axis of the hyperbola. u = twice the area of the hyperbolic rector thus determined, the length of the semiaxis being unity. tan0 = tanh m. w = an auxiliary angle called the gudermanian* such that the equations of the hyperbola are x = sec v,y = tan v. * For representations of this angle and for further particulars concerning hyperbolic functions see Chapter IV, by James McMahon, in Merriman and Woodward's Higher Mathematics; and Hoiiel, Recueil de Formules et de Tables numfirique. Newman and Glaisher have tabulated «-'' and c* in the Transactions of the Cambridge Phil. Soc, Vol. 13 C1883), III. REPORT FOR 1897 — PART II. APPENDIX NO. 9. Table 46. — Hype^'hoUc fanctiona — Continued. 615 u V e sinh M cosh u tanh u In degrees. [n degrees. =:tan V =:sec V = sin V ^ £~" 1-22 69-9009 0-9967 57-103 40-023 1-5460 1-8412 0-8397 3-3872 0-29523 1-24 71-0468 1-0074 57-721 40-215 1-5831 1-8725 0-8455 3-4556 0-28938 1-26 72-1927 I -0180 58-328 40-401 1 -6209 1-9045 0-8511 3-5254 0-28365 1-28 73 '3386 1-0284 58-925 40-582 1-6593 1-9373 0-8565 3-5966 0-27804 i-30 74'4845 1-0387 59-511 40-753 1-6984 1-9709 0-8617 3-6693 0-27253 1-32 75-6304 1-0490 60-087 40-920 1-7381 2-0053 0-8668 3-7434 0-26714 1-34 76-7763 1-0586 60-654 41-080 1-7786 2-0404 0-8717 3-8190 0-26185 1-36 77-9223 1-0684 61-212 41-232 1-8198 2-0764 0-8764 3-8962 025666 1-3'^ 79-0682 1-0779 61-758 41-380 1-8617 2-II32 0-8810 3-9749 0-25158 1-40 80-2141 1-0873 62-295 41-523 1-9043 2-1509 0-8854 4-0552 0-24660 1-42 81-3600 1-0965 62-823 41-657 1-9477 2-1894 0-8896 4-1371 0-24171 1-44 82-5059 I -1055 63-343 41-788 I-9919 2-2288 0-8937 4-2207 0-23693 1-46 83-6518 I-II45 63-851 41-915 2-0369 2-2691 0-8977 4-3060 0-23224 1-48 84-7978 1-1231 64-351 42-034 2-0827 2-3103 0-9015 43929 0-22764 1-50 85-9437 1-1317 64-843 42-148 2-1293 2-3524 0-9051 4-4817 0-22313 1-52 87-0896 1-1402 65-327 42-261 2-1768 2-3955 0-9087 4-5722 0.21871 1-54 88-2355 1-1484 65-800 42-370 2-2251 2-4395 0-9121 4-6646 0-21438 1-56 89-3814 1-1566 66-265 42-473 2-2743 2-4845 0-9154 4-7588 0-21014 , 1-58 90-5273 1-1646 66-728 42-571 2-3245 2-5305 0-9186 4-8550 0-20598 i-6o 91-6732 1-1724 67-171 42-668 2-3756 2-5775 0-9217 4-9530 0-20190 1-62 92-8192 1-1800 67-612 42-756 2-4276 2-6255 0-9246 5-0531 0-19790 1-64 93-9651 1-1876 68-045 42-846 2-4806 2-6746 0-9275 5-1552 0-19398 1-66 95-1110 I -1953 68-469 42-930 2-5346 2-7247 0-9302 5-2593 0-19014 1-68 96-2569 1-2023 68-885 43-013 2-5896 2-7760 0-9329 5-3656 0-18637 1-70 97-4028 1-2094 69-294 43-090 2-6456 2-8283 0-9354 5-4739 0-18268 1-72 98-5487 1-2164 69-696 43-166 . 2-7027 2-8818 0-9379 5-5845 0-17907 1-74 99-6947 1-2233 70-091 43-233 2-7609 2-9364 0-9402 5-6973 0-17552 1-76 100-8406 1-2300 70-476 43-303 2-8202 2-9922 0-9425 5-8124 0-17204 1-78 101-9865 1-2366 70-856 43-373 2-8806 3-0492 0-9447 5-9299 0-16864 1-80 103-1324 1-2432 71-228 43-433 2-9422 3-1075 0-9468 6-0496 0-16530 1-82 104-2783 1-2495 71-593 43-497 3-0049 3-1669 0-9488 6-1719 0-16203 1-84 105-4242 1-2559 71-952 43-556 3-0689 3-2277 0-9508 6-2965 0-15882 1-86 106-5702 1-2619 72-303 43-615 3-1340 3-2897 0-9527 6,-4237 0-15567 1-88 107-7161 1-2680 72-649 43-666 3-2005 3-3530 0-9545 6-5535 0-15259 1-90 108-8620 1-2739 72-987 43-720 3-2682 3-4177 0.9562 6-6859 0-14957 1-92 110-0079 1-2797 73-319 43-770 3-3372 3-4838 0-9579 6-8210 0-14661 I '94 111-1538 1-2854 73-645 43-816 3-4075 3-5512 0.9595 6-9588 0-14370 1-96 112-2997 1-2910 73-966 43-864 3-4792 3-6201 0-9611 7-0993 0-14086 ' 1-98 113-4456 1-2964 74-274 43-910 3-5523 3-6904 0-9626 7-2427 0-13807 2-00 1 14-5916 1-3017 74-584 43-950 3-6269 3-7622 0-9640 7-3891 .0-13534 2-02 115-7375 ■ 1-3070 74-886 43-993 3-7028 3-8355 0-9654 7-5383 0-13266 2-04' 116-8834 ' 1-3122 75-183 44-032 3-7803 3-9103 0-9667 7-6906 0-13003 2-06 118-0293 1-3173 75-472 44-070 3-8593 3-9867 O-968Q 7-8460 0-12745 2-08 119-1752 1-3222 75-758 44-108 3-9398 4-0647 0-9693 8-0045 0-12493 2-10 1 20-321 1 1-3271 76-037 44-145 4-0219 4-1443 0-9705 8-1662 0-12246 2-12 121 -4671 1-3319 • 76-311 44-177 4-1055 4-2256 0-9716 8-3311 0-12003 2-14 122-6130 1-3365 76-578 44-208 4-1909 4-3085 0-9727 8-4994 0-11765 2-16 123-7589 I -341 2 76-843 44-239 4-2779 4-3932 0-9737 8-6711 0-11533 2-18 124-9048 1-3457 77-102 44-270 4-3666 4-4797 0-9748 8-8463 0-11304 2-20 126-0507 I -3501 77-354 44-297 4-4571 4-5679 0-9757 9-0250 0-11080 2-22 127-1966 1-3544 77-603 44-327 4-5494 4-6580 0-9767- 9-2073 0-10861 2-24 128-3425 1-3587 77-848 44-352 4-6434 4-7499 0-9776 9-3933 0-10646 2-26 129-4885 1-3628 78-084 44-378 4-7394 . 4-8437 , 0-9785 9-5831 0-10435 2-28 130-6344 1-3669 78-320 44-402 4-8372 4-9395 0-9793 9-7767 0-10228 2-30 131-7803 1-3710 78-549 44-425 4-9370 5-0372 0-9801 9-9742 0-10026 2-32 132-9262 1-3748 78-773 44-449 5-0387 5-1370 0-9809 10-1757 0-09827 2-34 134-0721 1-3787 78-996 44-469 5-1424 5-2388 0-9816 10-3812 0-09633 2-36 135-2180 1-3825 79-212 44-490 5-2483 5-3427 0-9823 10-5909 0-09442 2-38 136-3640 1-3862 79-425 44-511 5-3562 5-4487 0-9830 10-8049 0-09255 2-40 137-5099 1-3899 79-633 44-532 5-4662 5-5569 0-9837 11-0232 0-09072 2-42 138-6558 1-3934 79-836 44-549 5-5785 5-6674 0-9843 11-2459 0-08892 2-44 139-8017 1-3969 80-037 44-565 5-6929 5-7801 0-9849 11-4730 0-08716 2-46 140-9476 1-4003 80-233 44-582 5-8097 5-^951 0-9855 11-7048 0-08543 2-48 142-0935 1-4037 80-426 44-598 5-9288 6-0125 0-9861 11-9413 0-08374 2-50 143-2394 1 -4070 80-615 44-616 6-0502 6-1323 0-9866 12-1825 0-08208 2-60 148-9690 1-4227 81-504 44-683 6-6947 6-7690 0-9890 13-4637 0-07427 2-70 154-6986 1-4366 82-310 44-741 7-4063 7-4735 0-9910 14-8797 0-06721 2-8o 160-4282 I '4493 83-040 44-787 8-1919 8-2527 0-9926 16-4446 0-06081 2 -go 166-1578 1-4609 83-701 44-828 9-0596 9-1146 0-9940 18-1741 0-05502 616 UNITED STATES COAST AND GEODETIC SURVEY. Table 46. — Hyperbolio functions — Continued. « V sinh K cosh w tanh M In degrees. In degrees. = ta.nv = sec V = sin V 3-00 171-8873 1-4713 84-301 44-861 10-0179 10-0677 0-9951 20-0855 0-04979 3-IO 177-6169 1^808 84-841 44-883 11-0765 II-1215 0-9959 22-1980 0-04505 3-20 183-3465 1-4894 85-331 44-906 12-2459 12-2866 0-9967 24-5325 0-04076 3-30 189-0761 1-4971 85-775 44-925 13-5379 13-5748 0-9973 27-1126 0-03688 3 "40 194-8057 1-5041 86-177 44-936 14-9654 14-9987 0-9978 29-9641 0-03337 3-50 200-5352 1-5104 86-541 44-948 16-5426 16-5728 0-9982 33-1155 0-03020 3-60 206-2648 1-5162 86-870 44-961 18-2854 18-3128 0-9985 36-5982 0-02732 370 211-9944 1-5214 87-168 44-966 20-2113 20-2360 0-9988 40-4473 0-02472 3-80 217-7240 1-5261 87-445 44-971 22-3394 22-3618 0-9990 44-7012 0-02237 3'9o 223-4535 1-5303 87-681 44-975 24-6911 24-7113 0-9992 49-4024 0-02024 4-00 229-1831 1-5342 87-901 44-980 27-2899 27-3082 0-9993 54-5981 0-01832 5-00 286-4789 1-5573 89-227 44-989 74-202 74-208 0-9999 148-41 0-006738 6-00 3437747 1-5658 89-716 44-993 201-71 201-72 0-9999 403-43 0-002479 7-CX3 401-0705 1-5690 89-895 45-000 548-35 548-35 I -0000 1096-6 0-000912 8-00 458-3662 1-5701 89-960 45-000 1490-5 1490-5 i-oooo 2981 -0 O-O00335 9-00 5i5'662o 1-5705 89-986 45-000 4051-6 4051-6 I -0000 8103 -I 0-000123 10 -CX) 572-9578 1-5706 89-995 45-000 IIOI3-2 11013-2 I-oooo 22026-5 0-000045 00 00 1-5708 90-000 45-000 00 CO I'OOOO CO slnh M = - , cosh 7 ■ cosh u e" + e-' Table 47.^ — Period of a wave. lycngth of wave in feet W- Depth of water {h). I 10 100 I 000 10 000 100 000 I 000 000 10 000 000 100 000 000 Feet. Seconds. Seconds. Seconds. Seconds. Seconds. Seconds. Seconds. Seconds. Seconds. I 0-442 1-873 17-641 176-29 1762-9 17629 176295 1762947 17629473 lO 0-442 1-398 5-922 55-789 557-51 5575 -I 5575 1 557508 5575085 lOO 0-442 1-398 4-419 18-726 176-41 1762-9 17629 176295 1762947 1 ooo 0-442 1-398 4-419 13-975 59-218 557-89 5575-1 55751 557508 ' lo ooo 0-442 1-398 4-419 13-975 44-192 187-26 1764-1 17629 176295 loo ooo 0-442 1-398 4-419 13-975 44-192 139-75 592-18 5579 55751 The period ( r ) of a wave is determined by the equation ,_27CX I . ■21th _ 0-1953 ^ r-=ii^/tanhi4^,= tanh 6-283185-=?- where^ is taken equal to 32-1722 feet per second, as in this table; or r2_ 0-195373^^ j^ tanh 6-283185-' if ^ IS taken equal to 32-16. EEPORT FOR 1897 — PART II. APPENDIX NO. 9. Table iS.— Wave velocity. 617 Uepth of water! i (A)- Length of wave in feet (A). I 10 100 I 000 10 000 100 000 I 000 000 10 000 000 100 000 000 Infinite. i Feet. I i IO lOO I ooo IO ooo loo ooo Ft./ sec. 2-262 2-262 2-262 2-262 2-262 2-262 Ft./ sec. 5-340 7-156 7-156 7-156 7-156 7-156 ; Ft./ sec. 5-668 16-89 22-63 22-63 22-63 22-63 Ft./ sec. 5-672 17-92 53-40 71-56 71-56 71-56 Ft./ sec 5-672 17-94 56-68 168-9 226-3 226-3 Ft./ sec. 5-672 17-94 56-72 179-2 534-0 715-6 Ft./ sec. 5-672 17-94 56-72 179-4 566-8 1689 Ft./ sec. 5-672 17-94 56-72 179.4 567-2 1793 Ft./ sec. 5-672 17-94 56-72 179-4 567-2 1794 A./ sec. 5-672 17-94 56-72 179-4 567-2 1794 The wave velocity, i. e., velocity of propagation is A/r. Tables 47 and 48 are adapted from Airy's Tides and Waves. Table 49. — Jiatio of vertical to horizontal axes of elliptic orbits of water particles. y 2.1 Ratio of y 2nl. Ratio of y 2^^ Ratio of ! K A axes. A A axes. A A axes. 0-00 0-0000 0-0000 o-io 0-6283 0-5568 0-40 2-5133 0-9869 o-oi 0-0628 0-0627 0-12 0-7540 0-6375 0-50 3-1416 0-9962 0-02 0-1257 0-1250 0-14 0-8796 0-7062 0-60 3-7699 0-9989 0-03 0-1885 0-1863 0-16 1-0053 0-7638 0-70 4-3982 0-9995 0-04 0-2513 0-2461 0-18 1-1310 0-8113 080 5-0265 0-9999 0-05 0-3142 0-3042 0-20 1-2566 0-8501 90 5-6549 0-9999 0-06 0-3770 0-3601 I -00 6-2832 0-9999 0-07 0-4398 0-4134 0-25 1-5708 0-9171 0-08 0-5027 0-4642 0-30 1-8850 0-9549 10-00 62-8319 I -0000 0-09 0-5655 0-5120 0-35 2-1991 0-9757 00 00 I -0000 The maxitium horizontal displacement at the bottom (where jy^o) being A, and 2A, the distance between the i bci of the elliptic orbits, the maximum displacements for other depths are: y ~ X = ^ cosh ly^A cosh 2 tt ^, y y = A sinh ly =A sinh 2 ?r ^ ; .-. -^ = tanh2!r4^, X A' i tvhjch is the ratio tabulated above. 618 UNITED STATES COAST AND GEODETIC SURVEY. Table 50. —Propagation of a free tide wave along a uniform channel. Depths. Velocity of propagation. Wave length, Time required to travel Difference in phase of tide wave Knots, or Fath- oms. Feet. Feet per second. nautical miles, per hour. statute miles per hour. statute miles. I foot. I naut. mile I Stat. mile. per statute mile. per foot. J. A. A. Radians. O O'OOO 0-000 0-000 0-00 [0-00 [o-oooo I 5-672 3-358 3-867 48-03 0-1763 0-2978 0-2586 7-4953 14196 2478 2 8 -022 4-750 5-469 67-93 0-1247 0-2105 0-1828 5-2996 10037 1752 3 9-824 5-817 6-698 83-20 o-joiS 0-1719 0-1493 4-3269 08195 •1430 4 1 1 '344 6-717 - 7-735 96-07 0-08818 0-1489 0-1293 3-7472 07097 1239 5 12-683 7-510 8-648 107-41 0-07886 0-1332 0-1156 3-3517 06348 1 108 I 6 13-894 8-226 9-473 117-66 0-07199 0-1216 0-1056 3-0597 05795 lOIl 7 15-007 8-886 10-232 127-09 0-06662 0-1125 0-09775 2-8327 05365 0936 8 16-043 9-499 10-938 135-86 006234 0-1053 0-09141 2-6498 05019 0876 9 17-016 10-075 11 -602 144-10 0-05877 0-09921 0-08621 2-4983 04732 0826 10 17-937 10620 12-230 151-90 0-05574 0-09416 0-08177 2-3700 04489 0783 II 18-812 11-139 12-826 159-31 0-05316 0-08985 0-07794 2-2597 04280 0747 2 12 19-649 11-634 13-397 166-40 0-05089 0-08598 0-07463 2-163S 04098 0715 13 20-451 12-109 13-944 173-19 0-04890 0-08258 0-07174 2-0785 03937 0687 14 21-223 12-566 14-470 179-73 0-04713 0-07955 0-06911 2-0030 03794 0662 15 21-968 13-007 14-978 186-04 0-04552 0-07686 0-06676 1-9351 03665 0640 16 22-688 13-434 15-469 192-14 0-04407 0-07446 0-06464 1-8737 03549 0619 17 23-387 13-847 15-945 198-05 0-04275 0-07220 0-06270 1-8177 03443 0601 3 18 24-065 14-249 16-408 203-79 0-04156 0-07018 0-06094 1-7664 03345 0584 4 24 27-787 16-453 18-946 235-32 0-03598 0-06079 0-05277 1-5298 02897 0506 5 30 31-067 18-395 21-182 263-09 0-03219 0-05435 0-04721 1-3683 02591 0452 6 36 34-032 20-151 23-204 288-21 0-02939 0-04963 0-04310 1-2491 02366 0413 7 42 36-759 21-765 25-063 311-30 0-02720 0-04593 0-03990 1-1564 02190 0382 . 8 48 39-297 23-268 26-794 332-79 0-02545 0-04297 0-03733 1-0818 02049 0358 9 54 41-681 24-680 28-419 352-98 0-02399 0-04052 . 0-03519 1-0199 01932 0337 lO 60 43-936 26-014 29-956 372-07 0-02276 0-03845 0-03338 0-9675 01832 0320 "5 90 53-810 31-861 36-688 455-69 0-01858 0-03139 002726 0-7900 01496 0261 20 120 62-134 36-790 42-364 526-19 o-oi6io 0-02718 0-02361 0-6842 01296 0226 30 180 76-099 45-058 51-885 644-45 0-01314 0-02219 0-01927 0-5586 01058 0185 40 240 87-871 52-029 59-912 744-14 0-01138 0-01922 0-01669 0-4838 00916 0160 50 300 98-243 58-170 66-984 831-98 0-01018 0-01719 0-01493 0-4327 00820 OI43 60 360 107-620 63-722 73-377 911-39 0-009294 0-01569 01363 0-3950 00748 0131 70 420 116-243 68-828 79-256 984-41 0-008606 0-01453 0-01262 0-3657 00693 0121 80 480 124-268 73-580 84-728 1052-38 0-008045 0.01359 01 180 0-3421 00648 0113 90 540 131-807 78-043 89-868 1116-22 0-007587 0-01281 0-01 1 13 0-3225 00611 0107 100 600 138-936 82-265 94-729 1176-60 0-007199 0-01216 0-01056 0-3060 00579 OIOI 150 900 170-162 100-754 116-019 1441-03 0-005877 0-00992 0-00862 0-2498 00473 0083 200 1200 196-486 116-340 133-968 1663-96 0-005089 0-008598 0-007463 0-2163 00410 0072 300 1800 240-645 142-487 164-076 2037-92 0-004156 0-007018 0-006094 0-1766 00334 0058 400 2400 277-873 164-530 189-459 2353-19 0-003598 0-006079 0-005277 0-1530 00290 0051 500 3000 310-671 183-950 211-821 2630-95 0-003219 0-005435 0-004721 C-1368 00259 0045 600 3600 340-323 201-507 232-039 2882-06 0.002939 0-004963 0-004310 0-1249 00237 0041 700 42CX3 367-591 217-653 250-630 3112-98 .0-002720 0-004593 0-003990 0-1156 002I9 0038 800 4800 392-971 232-680 267-935 3327-91 0-002545 0-004297 0-003733 0-1082 00205 0036 900 5400 416-809 246-795 284-188 3529-79 0-002399 0-004052 0-003519 0-1020 00193 0034 1000 6cxx) 439-356 260-145 299-561 3720-72 0-002276 0-003845 0.003338 0-0967 00183 0032 1500 9000 538-098 318-611 366-885 4556-94 0-001858 0-003139 0-002726 0-0790 00150 0026 2000 12000 621-342 367-900 423-643 5261-90 0-001610 0-002718 0-002361 0-0684 00130 0023 3000 18000 760-986 450-584 518-854 6444-48 0-001314 0-002219 0-001927 0-0559 00106 0019 4000 24000 878-711 520-289 599-121 7441-44 0-001138 0-001922 0-001669 0-0484 00092 0016 5000 30000 982-428 581-701 669-838 8319-78 o-ooioiS 0-001719 0-001493 0-0433 00082 0014 6000 36000 1076-20 637-222 733-770 9113-87 0-000929 0-001569 0-001363 0-0395 00075 0013 7000 42000 1162-43 688-278 792-563 9844-10 0-000861 0-001453 0001262 0-0366 00069 0012 8000 48000 1242-68 735-800 847-283 10523-79 0-000804 0-001359 0-001180 0-0342 00065 0011 90CX3 54000 1318-07 780-434 898-682 11162-17 0-000759 0-001281 O-OOIII3 0-0323 00061 0011 lOOOO 60000 1389-36 822-650 947-294 11765-96 0-000720 0-001216 0-001056 0-0306 00058 0010 "Velocity = ^'gfi where A = the undisturbed depth in feet and,^ = the acceleration of gravity, assumed to be 32-1722 feet per second in this computation. If r=the periodic time (= 12-4206012 solar hours or^%, lunar day for the tide -wave), then X, or wave-length, = r V^ feet = — -_ Vgh miles. Difference in phase = ^-^r— . A. The nautical mile is taken as 6080 feet. TEEASURY DBPARTMBlSrT UNITED STATES COAST AND GEODETIC SURVEY W. W. DUFFIELD StrPEEmTENDEKT PHYSICAL HYDROGRAPHY MANUAL OF TIDES FAJEIT III By ROLLIN" -A.. UA-RlilS APPENDIX No. 7— REPORT FOR 1894 WASHINGTON aOVEBNMENT PRINTINa OFFICE 1896 TEEASURT DEPAETMENT UNITED STATES COAST AND GEODETIC SURVEY W. W. DUFFIELD SUPEEINTENDENT PHYSICAL HYDROGRAPHY MANUAL OF TIDES P^ET III By ROLLIN A.. HA-RRIS APPENDIX No. 7— REPORT FOR 1894 WASHINGTON aOVERNMENT PRINTING OFFICE 1895 APPENDIX No. 7—1894. M^J^NJJJ^Ju OF TIDES. PART III. SOME CONNECTIONS BETWEEN HARMONIC AND NONHARMONIC QUANTITIES, INCLUDING APPLICATIONS TO THE REDUCTION AND PREDICTION OF TIDES. Submitted for publication December 22, 1894. 125 PREFACE TO PART III. The following pages have been prepared for the purpose of supplying some of the immediate wants of the Tidal Division. They constitute Part III of a proposed manual of tides, the plan of which may be outlined thus : Part I. History of tides, including old methods of treatment. Part II. Tidal observation, equilibrium theory, and harmonic analysis. Part III. Connections between harmonic and nonharmonic quantities. Part IV. Tidal theory and astronomical quantities deduced from tidal observations. Part V. Tidal currents. Tables. Auxiliary tables and a collection of tidal constants, including cotidal charts. As no considerable portion of the material constituting Part III is elsewhere available, it has seemed desirable to publish it in advance of Parts I and II, whose subject-matter has been largely treated in various publications. The object of this paper is, in a general way, indicated by its title; but it may not be out of place to comment here upon certain of its aims and features. Owing to a lack of reasonably precise definitions and adequate modes of reduction, the nonharmonic constants (i. e. those referring to high and low water) have too often been of little service where an accurate and satisfactory knowledge of the tide was desired. It is true that quite accurate predictions have been made, based upon the tables resulting from nonharmonic discussions; but it is to be observed (1) that the tables so used were designed chiefly for particular stations and so were of little or no use generally; (2) that long series of observations were necessary for the construction of such tables; (3) that most nonharmonic discussions have been confined to stations where the diurnal ii' equality is so small as to give no serious trouble. The present attempt to overcome these objections provides an approximate method of analyzing a single month's observations upon high and low water for obtaining both the harmonic and the nonharmonic constants, regardless of the amount of diurnal inequality, and without dififlcult com- putations. It also provides tables for the analysis and prediction of tides, designed for general use, regardless of the location of the station. Little attention has heretofore been given to the classification of tides for the purpose of selecting the best possible port of reference for a given subordinate station, proximity in geo- graphic position having been the sole criterion with few exceptions. Now, it is believed, are given for the first time such harmonic and nonharmonic criteria as will readily show how the types of tide at different stations compare. In this connection are given rules for referring one station to another. The chapter entitled "Prediction of Tides" describes a proposed tide-predicting machine combining the merits of the Thomson and the Ferrel machines; a general graphic process enaljjling one (by means of a set of curves constructed once for all from the harmonic constants of the given station) to predict tides without a machine and without computation, the amount of work involved depending upon the accuracy desired; also an approximate arithmetical method for predicting high and low waters from certain tidal constants. 127 128 UNITED STATES COAST AND GEODETIC SUEVEY. By correlating harmonic and nonharmouic quantities a better insight into tides is obtained than from the consideration of either alone. In this way observable phenomena may be inferred from harmonic constants, inconsistencies may be detected, and missing quantities supplied; also particular values of nonharmonic quantities may be reduced to their mean values, and conversely. §§ 46, 47 are given mainly for the purpose of furnishing a key to the study of Perrel's non- harmonic analyses. These discussions, covering years of observations, throw much light upon the nature of tidal inequalities. The appended tables, already alluded to, are based principally upon those of Baird and Darwin. The considerable computation involved in their preparation has been performed by members of the Tidal Division. Their scope differs somewhat from that of the paper accompany- ing them. This remark applies especially to Tables 1 to 13, which are designed for the harmonic analysis of tides from hourly ordinates, a subject to be treated in Part II. I have to acknowledge much assistance from Mr. L. P. Shidy, whose long experience in tidal work has rendered his advice particularly valuable, and whose cooperation has enabled the work to go on. [MANUAL OF TIDES.] CONTENTS OF PART III. Chapter I. PROPERTIES OF COMPOUND WAVE HAVING A PREDOMINATING COMPONENT. Pagfc Displacement in time and height of the maxima and minima of the predominating component 131 Approximate average values of resultant heights , 132 Combination of two components of equal periods I33 Combination of two components whose periods are as 1 : 2 I34 Accurate and approximate values of resultant heights for special cases compared 133, 135 Restrictions in the application of approximate rules I35 Effect of components Vvith variable amplitudes I35 Speed of a wave the two components of which have speeds or periods nearly equal 135 Effect of small components having speeds commensurable with that of the predomiuating one 136 Conditions for a predominating component I37 Chapter II. COMPUTATION OF NONHARMONIO QUANTITIES FROM HARMONIC TIDAL CONSTANTS. Definitions ". X39 Computation of mean lunitidal intervals 140 Computation of mean range of tide 140 Computation of hourly heights of the M tide 142 Computation of range and speed of diurnal wave and range of tide near times of moon's extreme declination. 142 Computation of ages of tidal inequalities I43 Computation of spring, neap, perigean, and apogean ranges 144 Computation of quantities connected with extreme declinational tides 145 Computation of average range of diurnal wave and of great and small diurnal ranges of tide 147 Computation of average height inequalities; application to planes of reference 147 Computation of half-tide level 148 Tides chiefly diurnal 148 Chapter III. REDUCTIONS OP OBSERVATIONS MADE UPON HIGH AND LOW WATERS. Object '. 149 First reduction I49 Mean sea level and long-period tides 150 Eeduction of spring and neap tides I53 Eeduction of parallax tides 157 Eeduction of declinational tides.... 160 "Datum planes 167 Ferrel's expressions for tidal inequalities 168 Chapter IV. TO RRDUCB results TO THEIR MEAN VALUES. Effect of the longitude of the moon's node 173 Approximate relations used in reducing quantities to their mean values 173 S, Ex. 8, pt. 2 9 129 ' 130 TTNITED STATES COAST AND GEODETIC SUEVEY. Chapter V. ON THE CLASSIFICATION OP TIDES. Page. Harmonic and nonhannonio criteria 175 Predictions made for a principal station adapted to a subordinate station 176 •Constants inferred by comparison of tidal characteristics 177 Cotidal lines I79 Chaptei: VI. PREDICTION 01' TIDES. British tide-predicting machines 180 Ferrel's tide-predicting machine..... 181 A proposed tide-predicting machine 181 A graphic method of predicting tides 183 An approximate method of predicting tides by computation 186 TABLES. 1. The principal harmonic components, with their speeds, coefficients, etc 190 2. Dependence of component speeds upon certain astronomical quantities 194 3. EquUibrmm arguments (Vg + u) at the midnight preceding January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with the elements used in computing them 195 4. For adapting the uniformly varying portion ( Fq) of the equilibrium arguments of Table 3 to Greenwich midnight, beginning any day throughout the year 205 5. For adapting the uniformly varying portion ( Vo) of the equilibrium arguments of Table 3 to local midnight for any degree of west longitude 207 6. Values of N, I, and P for Greenwich midnight, beginning each month, from 1850 to 1949 209 7. Values of /, v, |, v' , and 2 v ', corresponding to each half degree ot N 214 8 Values of iJ for completing n for Lj 217 9. Values of Q for completing u for Mi 218 10. Factors F and / for reduction and prediction of tides, computed for the middle of each year, or for July 2 at Greenwich mean noon for common years, and at preceding midnight for leap years 219 11. Values of log R' for obtaining the factor Fof L2 from that of Mj 236 12. Values of log Q' for obtaining the factor F of Mi from that of O, 237 13. Factors F and/, corresponding to every tenth of a degree of I, for reduction and prediction of tides 238 14. Factors {F) for reducing Mn, Ki -I- 0., and X to mean values 247 15. Acceleration in H W and L W of a wave, due to another wave of equal speed 248 16. Height of H W and L W for a tide composed of two waves of equal speed 248 17. Acceleration in H W and L W of a semidiurnal wave, due to a diurnal wave 250 18. Height of H W and L W for a tide composed of a diurnal and semidiurnal wave 250 19. High and low water inequalities in height 250 20. Great tropic range and its duration 251 21. Eifects of various tidal components upon the mean semirange of tide 252 22. Value of i Mn when Mi = 1 253 23. Value of M.2 when i Mn = 1 253 24. Variation in lunitidal interval and mean semirange of tide due to the phase wave composed of Si and /ij. . 254 25. Variation in mean semirange of tide due to the parallax wave composed of N2, L-, and 2 N 255 26. Effect of Qi upon the amplitude of Oi i... 256 27. Perturbations in Ki due to Oi 1 258 28. The speed which corresponds to a period of given length 257 29. The sun's mean longitude at Greenwich mean noon 257 30. Approximate equation of time at Greenwich mean noon 258 31. Perturbations in K,, S2 due to the components Pi, K2, and T,j 259 32. Factor Fl for clearing Di of the effects of the longitude of the moon's node and of P, 260 33. Factor i^j for clearing 82 of Kj and T, 261 34. Effect of v-i upon the amplitude of N; 261 35. Group factors 262 OHAPTBE I. PROPERTIES OF A COMPOUND WAVE HAVING A PREDOMINATING COMPONENT. 1. The equation of a compound wave may be written y = A cos {at + a) + B cos {bt + (3) + G cos {ct+ y)+ . . . (1) where A, a, a refer to the predominating component (i. e. the one which determines the number and approximate times of the maximum and minimum values of y) and where B, b, /?; G, c, y; . . . refer to the subordinate components. The values of t rendering y a maximum or a minimum are roots of the derived equation Aa sin {at + a) + Bb sin {bt + /?) + (7c sin {ct + y) + . . . =0. (2) Any root of (2) which causes y to become a maximum may be written t = ^±JL _ ^ _ « , (3) a a a and a minimum (2 ?t + 1) TT ic a ., * = a ¥ ~ a' ^^> where n is an integer or zero and v, to, quantities usually small in comparison with tt. v/a (or w/») is the amount by which the time of any maximum (or minimum) of the compound wave precedes the corresponding time of the predominating component. 2. Approximate expressions for v, w, and y. When V is small and b not many times greater than a, we may assume cos ~ V = cos V (5) a ^ sin - » = — sin v, (6) a a ^ ' and so for cos — v and sin - v. When b differs little from a these are approximations even if v be not so small. Of course the same approximations are true when w replaces v. From equations (2), (3), and (4) we have Aa sin 2 n TT + Bb sin -(2n7r+j^— a J+Gcsiji~(2n7r+jr-aJ+ . . . tan v = a j~-- - ^- ^ ^ ^r , (7) Aa^ cos 2 n 7t+ Bb'^cos -(2n7i:+^ /3—:x] + Gc^ cos ^{^2n7r+—y— a j+ . . . Aasin2'n+iyr+Bbsin-(2n+l7r+j^-aJ + Gcsin^(^2n+l7i+^y-aJ+ . . . tan w=a FT""^"" a \ c / a \ ' (^) Aa'cos2n+l7r+Bb^cos-{^2n+l7t+j/3-aJ + Gc'cos—[2n+l7r+ — /3—aJ+ • ■ ■ 131 132 UNITED STATES COAST AND GEODETIC SURVEY, and the corresponding maximum and minimum values of y are y^\A expresses the average amplitude of a wave composed of the simple waves A, B, and G, provided c differs from b by an infinitesimal and B- + C of (31) is not greater than B^ of (30). The amplitude of B and G combined is VW +C^+2BGcos (h ~ c) t. (32) Replacing B'' of (30) by the square of this expression, the average value of B'' becomes W + GK In like manner, if formula (31) apply to three waves, the formula ^ + iTa> [^' ^' + OU"^ + D' d?] (33) will apply to four waves, provided c and d differ from 6, and so from each other, by an infinitesimal and B^+ G'^ + D^ of (33) is not greater than B^ + (7^ of (31) or W of (30). So one can proceed for any number of simple waves. Tables 16, 18, and 21 show that (30) is nearly true when a differs from h or from 2 6 by an infinitesimal, even if B is not small in comparison with A. In the former case B may be equal to A and in the latter it may be equal to 2 A without sensibly affecting the truth of (30). 7. The effect of a subordinate wave having a variable amplitude of the form B' cos ib't+ /3')+B", (34). where B"^B', in increasing the average atnplitude of a given wave. In deriving formula (18) all amplitudes were assumed to be essentially positive; therefore in applying it to the present case the variable amplitude must not become negative. Since B"^B', all values of cos (6 't + /?') from + 1 to — 1 are to be used, and the average value of B'^, or [B' cos [b't + /J') + B"f, (35). becomes ^B'^ + B"\ (36). When B" = 0, this becomes hB"; (37> i. e. the effect is one-half as much as it would have been had the amplitude of the subordinate wave been constantly equal to B'. (It may be noted here that the average numerical value of B' cos {Vt + /?') is 2 B'JTt, the square of which is about f B' ^ instead of J B' I) 8. To find the speed of a wave composed of two simple waves having approximately equal hut not commensurable speeds. 136 UNITED STATES COAST AND GEODETIC SURVEY. If n' be such a value of n that -{2n'7r + ^^-a)=o, then there is no displacement of the maximum of A due to B; \. e., A and B conspire at this common maximum. For the following minimum, (2n'+l.+ l^-a) = '-.. At this maximum of A the phase of B is zero, at the minimum it is - ;r; the phase of 5 at a time midway between this maximum and minimum of A is equal to 9— n. At the rth subsequent maximum, minimum, and midtime the phases of B are ^2r;r,-2r+l^,2-4r+l^, respectively. If then 6 „ bTt - 2r;r = ^-^2 and ft „ , in -2r+ln=z + ^^- By means of § 2 we obtain for the length of a half period expressed in hours, say, corre- sponding to a phase of B equal to z, 56 sinfa + -90°^ Bh sm( z --QQ°\ \P.= +5^-3 -y T ^+57-3 y -. ^- 39 AO' - B¥ cos (z + -900^ Aa^ + B¥ cos (z - - \^ a y \ U 90° If t denote the number of hours between the time of conspiring of A and B, and the given time for which the length of a half period is required, we have z = {b — a) t. (40) Finally, the speed for this time is equal to 360° -^- (41) At the time of conspiring the speed is approximately equal to « + « ,J , pJ ' (42) Bh {b - a ) ' Aa? + B¥ and, at the time of interfering, Bb lb — a) ,.„, 9. Small components having speeds commensurable with the speed of the predominating one. Suppose now that inequation (1) there are terms of the form A( cos {i at + «() (44) where i is an integer greater than unity. EEPORT FOR 1894— PART II. 137 So long as all speeds are independent of one another the average value of v or w will be zero. When the harmonics of -4. are included, the average values of (7) and (8) become • A, ai sin i f ^* — « ) tanv = a ^ ^ ^ ^ x ' (*^) Aaf + ^ A, (iaf cosi f^' -a\ and tan w = a ~ Ai ai sin i f tt ■^^— a) Aa^-\- ^ Ai {iaf cos t (' ;r + . where i = 2,3, etc. (46) The respective values of each maximum and minimum of a wave composed of A and its harmonic only are y = ACOSV+ ^ AiCosi (^'^ - a-v^ (47) and y=—Acosw+ ^ A,cosi (^n + ^— u — w\ (48) These individual values are equivalent to average values, because they recur at each period of A. If there are also very small components of the form Aj cos U at + aj) (49) where j is a vulgar fraction, the effects of one such term upon the average value of the maxima and minima are . n = m —1 A, ^ m ^ and ^ w — m — 1 2 cosj f2n 7T+ ^ -a\ (50) ^ Yi -= m — 1 -^ 2 cosjf^nr+l7r+^-a\ (51) M = ^ J y where m is the denominator of the fraction j. Special case : When j = J, f, f .... each of these expressions reduces to zero. 10, Conditions for a predominating component. In order to have a simple wave determine the number of maxima and minima when combined with a series of other simple waves, it is necessary that its greatest slope be greater than the greatest possible slope of the series. For the two steepest parts of the predominating wave fall somewhere upon the serial wave; and the upward and downward slopes at these points cause the resultant curve to slope upward or downward at these points. But a maximum or a minimum implies the existence of both an upward and a downward slope. Since the greatest slope of any simple wave is proportional to amplitude -r- wave-length, the necessary condition for a predominating wave is Aa> Bb+ Gc+ • (52) This condition is also sufiBcient provided the greatest curvature of the predominating com- ponent be not less than the greatest possible curvature of the serial wave. Now, the greatest curvature of any component is at a maximum or minimum point; consequently the greatest possible 138 UNITED STATES COAST AND GEODETIC SURVEY. curvature of a series of waves would be the curvature at a time when all the maxima or the minima fall together. Writing the serial wave, we have y = B cos {bt + fi) + G cos {ct + y) + . . . , (53) ~f^^ = +Bb sin (6^ + /3) + Co sin {ct + y) + . . . , ~dt' ^ = + ^fe' cos [U + ft) + Gc^ cos {ct + y) + . . . . Let the origin of time be taken at this common maximum, say; then — dy d t These values substituted in the expression for the radius of curvature, gives at this point _ 1 . ^^ - Bb'+Gc'+ . . . ' or curvature = Bb^ + Gc^ + . . . . In like manner the greatest curvature of the predominating component is Aa'^ . .-. the remaining and, with (5U), sufiScient condition for a predominating component is Aa'^sBb^-'r Gc^+ .... (54) CHAPTER II. COMPUTATION OF NONHARMONIC QUANTITIES FROM HARMONIC TIDAL CONSTANTS. 11. In the preceding chapter on waves nothing has been said concerning the cause producing them. It will now be assumed that the maximum of a component follows its apparent cause, or fictitious moon, by a certain angle called the epoch of the component, and whose value is such that if it be divided by the speed of the component, the quotient is the time elapsing between the transit of the fictitious moon and the occurrence of the "maximum value of the component. For this reason the terms which have been written in the form A cos (at + a) will now be written in the form A cos (at + argo A — A°). A may be used to denote either the component A itself or its amplitude; a is the speed of the component expressed in degrees per mean solar hour; a is the initial argument or phase of the component A; i. e. its phase when t = 0; argo -A denotes the initial argument or hour angle of the fictitious moon producing J.; for brevity it is usually called the initial argument or initial equilibrium argument of the compo- nent A. To avoid misunderstanding at+a will be spoken of as the phase rather than as the argument of the component. A° denotes the epoch or lag of the component J. behind its fictitious moon; .-. a = argo A — A°. A°ya is the epoch expressed in time and may then be called the interval of the component. For harmonics of A we always have a, —- iai, and usually argo A, = i argo ^1. (55) The mean range, the spring range, etc., and the high or low water inequality in height. For some purposes it is convenient to give names to various ranges of tide. Mean range is the average value of the semidaily range of tide. Spring range is the greatest periodic semidaily range, occurring usually one or two days after new and full moon. Neap range is the smallest periodic semidaily range, occurring usually one or two days after the moon is in quadrature, that is, after the first and third quarters. Perigean range is the greatest periodic semidaily range of tide, occurring usually from one to three days after the moon is in perigee. Apogean range is the smallest periodic semidaily range, occurring usually from one to three days after the moon is in apogee. Great diurnal range is the difference between the mean of all the higher high waters and the mean of all the lower low waters of each day during one or more half tropical or declinational months. 139 140 UNITED STATES COAST AND GEODETIC SURVEY. Small diurnal range is the difference between the mean of all the lower high waters and the mean of all the higher low waters of each day during one or more half tropical or declinational months. Oreat tropic range is the greatest periodic daily range of tide, usually occurring soon after the moon is farthest north or south from the equator and therefore near one of the tropics. Small tropic range is the smallest periodic daily range of tide, usually occurring soon after the moon is farthest north or south from the equator and therefore near one of the tropics. Tides determining these ranges, or of simultaneous occurrence, may be referred to as spring, neap, perigean, tropic, etc. ; a like remark is applicable to lunitidal intervals, and occasionally to other quantities. Diurnal inequality as here used generally denotes the greatest periodic difference in height between two consecutive high waters or low waters, usually occurring soon after the moon is farthest north or south from the equator ; this inequality is therefore determined by the tropic tides. 12. To compute the mean lunitidal interval or corrected establishment. When the tide contains no components whose speeds are commensurable with that of M2, one can assume that the high water establishment is, in hours, 280-9841' °'" m2 ' ^^^> and the low-water establishment is the same increased, say, by one-fourth of a mean lunar day, that is -^+ 6''-2103 (= 6" 12"'-62). (57) When M2 has harmonics, which are denoted by M4, Me, etc., the high water establishment is ^^^^ (58) and the low water ^4^ + 6-2103 (59) where from § 9, since «, = t argo Ai — Ap, 2 M4 sin (2 M2° - M4°) + 3 Me sin ( 3 M^o - Ms°) + ■ • „„. **^^-rM2 + 22M4C0S(2M2O-M4°)+32M6cbs(3M2O-M6°)+ • ". • ' ^ ' 2 M4 sin (2 Mz" - M4O) - 3 Me sin (3 Ma" - M^o) A- . ._^ tan w - _ p jvi^ ^ 2^ M4 cos (2 M^o - M4O) - 3^ Me cos (3 M2O - MeO) + . . . ' '^^^> Less accurate formulae for v and w are tan V = 2 M:4 sin (2 M2O - M4°) + 3 Me sin (3 Mz" - Me°), ,q^. M2 2 M4 sin (2 M2° - M4°) -3 Me sin (3 M^o - Me^). „„, tan w = ^ — TTj 00) 13. To compute the mean range of tide from given harmonic constants. In formula (18) substitute M2 for A, and other components whose speeds are incommensurable with m2 for B, C, etc. Multiply tne result by 2. This should be increased by the excess of (47) — (48) over 3 M2, which is M2 (cos V + cos w) + M4 [cos (2 M20 -M,o-2v)- cos (2 M^o - M4° - 2 w)] + Me [cos (3 M20 -Me°-3v) + cos (3 MjO - MeO - 3 w)] + ....■ (64) -2M2 where v and w have the values given in § 12. REPORT FOR 1894— PART II. 141 Since v and w are assumed to be comparatively small, and as v is often approximately equal to — w, the above may be written Mj (cos V + cos w) + 0-035 M4 {v — w) sin (2 MjO - M4O) + 2 Me cos (3 M^o - MeO) - 2 Mj where v and w are expressed in degrees. The complete formula now becomes ^^•^+ 2M!m,^ [^''''' + ^''°''+ • • • +K,^V+0>o,^+. . .1, + M2 (cos V + COS w) + 0-035 M^{v-w) sin (2 MP - M4O) + 2 Mg cos (3 MjO - MeO) - 2 M3 (65) where the second part is due to the hartoonics of M2 and is usually small except at river stations. Table 21 has been prepared for the purpose of avoiding the labor of each time computing the residual effects of S2, N2, etc., upon the range of tide, as indicated in (65). See also Table 22. The above formula for computing the range of tide is not vitiated because of certain fixed relations between the speeds of some of the components. Let the parallax wave, where Ua + I2 = 2 mj, mz + 2 n = 2 nj, be first considered. Its amplitude at any particular time is approximately of the form N2 — Lj cos 2a; + 2 N cos x, (66) the average value of which is Nz- But in reality the residual effect of Lj and 2 N upon N2 is W V , (2N)^(2n )' 4 N2 n2^ "•" 4 N2 n2^ instead of zero. Therefore in finding the residual effect of the parallax wave upon M2, the amplitude to be used would be approximately L-,^ h^ (2 N)2 (2 n)2 ^^ + 4N75? + 4 N2 n2^ - L2 cos 2 a; + 2 N cos ( - x) The average value of the square of this expression (see § 7) multiplied by 112^ is, very nearly, ISTz^ ni" + W h' + (2 ^f (2 nf. The theoretical values of N2, L2; 2 K, give, for each value of x, an amplitude with which to enter Table 16. From Table 15 a correction to the phase ( — a;) is obtained. The mean of the correspond- ing tabular values shows that the fixed speed relations do not sensibly increase the range of tide. For the declinational wave- the fixed relations are ki + O; = m2, ki + pi = S2; the amplitude at any given time is approximately of the form Ki + O, cos 2/ + Pi cos z. (67) Now increase Ki by the residual effects of Oj and Pi upon it, and (§ 16, remark based upon Table 18) multiply the average value of the square of the amplitude so altered by the square of the speed. When the amplitude of the declinational wave is great, the speed is a trifle greater than mi ; when the amplitude is small, the speed exceeds kj.* For the present purpose the speed should be between mi and ki, but nearer to mi. The result indicated above becomes nearly equal to Ki^ ki^ + Oi^ oi^ + P,^ pi^. Having regard to the derivation of (15) and (16), it is not difficult to convince one's self that small components like K2 and Si, S4, Se, require no special treatment because of their speed relations to Ki and S2, respectively. •By means of Table 27, the speed of the Ki Oi wave may be ascertained for any given time; it is approximately equal to • Where 6 is the number of days before or after the nearest extreme declination of the moon, or rather maximum declinational effect. See also 5 8. 142 UNITED STATES COAST AND GEODETIC SURVEY. 14. The mean range of tide as determined from the tidal components is invariably less than the mean range of tide determined from observed high and low waters. The difference betiveen the two determinations is constant from year to year at a given station. In summing for any component, the curve is read at fixed times (which are determined in advance) depending in no way upon the height of the sea. In summing for high and low waters the tidal curve is actually read at times which could be only approximately determined in advance, and which times depend to a considerable extent upon the height of the sea. In the former case the elevations and depressions other than those due to the component (with its harmonics) sought will eventually become eliminated. In the latter case we have the liberty of selecting readings at about the fixed times of maxima of the wave got by combining all the tidal components. The mean range of the observed tide will always be greater than that of the theoretical tide. For, in order to get this observed range of tide to agree with that of the theoretical tide, we would have to observe the height of the sea at the theoretical or predictable times of high and low water. The magnitude of the discrepancy just pointed out will depend largely upon who observes the tides, and whether the high and low waters are observed upon a staff or are obtained from the record of an automatic tide gauge. Let it be assumed that what is meant by high and low water heights are heights obtained from a continuous curve by a tabulator who reads, as nearly as his judgment will permit, a curve which will eliminate such irregularities as are obviously not predictable. With this definition of high and low waters, and in want of definite information, the discrepancy may be assumed to be perhaps 1 per cent of the computed range. For high and low waters as ordinarily tabulated this discrepancy probably amounts to about 2 per cent. 15. To compute the hourly heights of the M tide. The height of the M tide h hours after high water is M2C0SJ28O69';i-'!)|+M4C0SJ57o58'A+2M2O-M4°-2ijj +MeCOS{86°5Th+3MiO-M6°-3v\; (68) and h hours after low water -M2C0si28o59'/i-wS+M4C0s|57O58';i+2M20-M4O-2w} -MeCOs\86° 5Th+SMiO-Me°-3w\.. (69) 16. To compute the speed and range of the diurnal wave when the moon is far from the equator; also the mean range of tide at these times. At the conspiring of Ki and O, the speed of the combined wave is, by § 8, k, Oioak,^_0i)._ g (70) when the value of K,/Oi is assumed to be 1-4066. In other words, it is very nearly Qo-l greater than mi. Calling this speed d,, it follows that the mean semirange of the diurnal wave when the moon is far from the equator is K, + O. + 4-(Kr+ OT)li? ^^' P'^ + ^'^ ^'^ + • -1; (71) = (1-0236) (K, + 0,) (72) when Pi and Qi are given their theoretical values. Since K, and Oi separate 13° 32' in one-half of a Di day (i. e. in 12»'-33), Ki + d should be multiplied by 0-9977, formula (25), thus making the required range 2 (1-021) (Ki + Oi). (73) EBPOET FOE 1894— PAET II. 143' In computing the mean range of tides at tliese times it should be noted: First, that the Kj moon is approximately 180° from the M2 moon ; second, ^he large wave whose amplitude is Ki + Oi has a fixed phase with respect to M2; third, from Table 18 it is seen that the mean range of a wave composed of a semidiurnal and a diurnal component is not sensibly affected by the phase of the latter with respect to the former. The mean range of tide at these times, which it is convenient to denote by Mc, differs from Mn, formula (65), by the substitution of (Kj + O,)^ di^ in place of K,^ k^'' + Oi' Oi^ and by the subtraction of 2 K2 and the residual effect of K2. That is Mc = M n + fch^lll^gM^i^O^..^^^ _ 2 K2, (74) =Mn + 0-127 (Ki + df - 04 35 K,^ - 0.116 O,^ - 0.540 K ,' _^ ^ iVl2 Mc will exceed Mn whenever (K, + O,)^' d,^ - K,2 k,^ - O,^ o,^ > 4 M2 K2 m2^- K^' k^^; (75) or, roughly, whenever Ki O, > 8 M2 K2; (76) which is about equivalent to saying whenever D, >2A2, (77) Di being the tropic amplitude of the diurnal wave and A 2 the mean amplitude of the semidiurnal wave. 17. Given the epochs of two components whose speeds are nearly equal, to find how much the time of their conspiring lags behind the time of conjunction of their fictitious moons. When the moons are in conjunction the angle between the crests of the component tides is The time required for the crests to come together will be BO—A° (78) where a mean solar hour is taken for the time unit. Whenever A° and B° appear to differ by more than 180°, zt 360° should be applied to one of them. T also represents the amount of lagging of the interference behind the time of opposition of the fictitious moons. For the components M2 and Sj 82° — M20 1-0159 : 0-984 (82° -M2°). (79) This may be written r (82; M2), meaning the age, or retard, of 82 relative to M2, expressed in hours. Assuming that ^^Il'^f tides occur when M2 and 82 have oppQgjte Phases, this is the interval between °«^ ?^ full moon ^ spring .^^^ moon m quadrature neap In like manner ^ (^-= ^^) =^i:kff = -^-5"ii?° =1.837 (M20-^20). (80) Assuming that the parallax wave f^J^^feres ^^^^ ^^ when M2 and N2 have oppogj^g phases, this is the interval between ^^oj^^'^P^^JI^^e^nd the greatest ^^^^^^^ of range of tide due to parallax. Similarly 144 UNITED STATES COAST AND GEODETIC SURVEY. is the number of hours between the time °^ ^"^^ moon's extreme north or south declination , when the moon crosses the equator the time of the ^'^i^^^^^ range of the diurnal wave, r (S^; Mj), r (N^; M^), and r (O, ; K,) may be referred to as the ages of the phase, parallax, and diurnal inequalities, respectively. 18. To compute spring and neap ranges. The phase wave consists of Su and /^j. Sj conspires with M3 at spring tide and interferes at neap tide. The argument of the /a^ moon exceeds that of the S2 moon by four times the mean longitude of the sun less that of the moon. Therefore, whenever the moon is new, full, or in quadrature, the S2 moon and that of //j are in conjunction. For this reason, and assuming that the epochs of S2 and Mi are equal, S2 and /a^ conspire at the times of spring or neap tides. The mean range of tide with the residual effect of S2 excluded is Disregarding the time perturbations other than those due to phase, also the slight separation of M2 and S2 during a quarter lunar day, the ^^^^^ range ought to be expression (82) ^°*'^®*®®^ by 2 {S2 + yU2). The perturbations just referred to diminish the direct effect of S2 and /J2 by about 0-02 + 0-04 ( ^' + ^' V M2 part of their value; see formula (18). The diminution in S2 due to the separation in a quarter lunar day is by formulae (25) and (26) very small. Ap^proximate values of the required ranges are Sg = Mn + 1-96 (S2 + /.2) - 0-08 (^L+^J (S2 + f^,) - ^^, (83) Np = Mn - 1-96 (S2 + M2) + O'OS (^5l^Y (S2 + M2) - ^J (84) 82^ Mo' .-. Sp + Np = 2 Mn - ^ . (85) 19. To compute perigean and apogean ranges. These terms are here used to denote ranges analogous to the spring and neap, but referring to perigee and apogee instead of syzygy and quadrature. The parallax wave consists chiefly of N2, L2, and 2N. The argument of M2 (i. e. of its fictitious moon) exceeds that of 2N bv^wice^the ™®'^'^ longitude of the moon from its perigee; in other words, , twice*the ^^''^''^ mesm anomaly. The argument of L2 (simple) is greater than that of M2 by the mean longitude of the moon from its perigee + 180°. From this it follows that when the moon is in mean perigee the fictitious JSTj moon is in conjunction with the fictitious M2 moon while that of Lj is in opposition. If con- spiring and interference took place immediately, the amplitude of the parallax wave when the moon is in mean perigee or apogee would be, approximately, ]sr2 - L2 ± 2 N. Proceeding as in the case of spring and neap ranges, but remembering that true and mean perigee differ considerably, we obtain for the required ranges the approximate values Pn = Mn + [2-1 - J C-^)'] (N2-L2 + 2 N) - 0-08 ^^-^^~ (N2 - L2 + 2 N) - ^^, (86) An = Mn-[2-l-j(|jy](N2-L2-2N) + 0-08^^5i^^'(N2-L2-2N)-2^|^; (87) .•.Pn + An = 2Mn-^+ 4[2N]. (88\ EEPOET FOE 1894— PAET II, 145 20. To find the heights and lunitidal intervals of the tropic tides which have a fixed order of occurrence. Compute the semirange of the tide by formula (65), omitting the diurnal components, and diminish the result by Ka. The theoretical value of this is 1-064 Ma -K2, or, more accurately, 1-006 M2 + 0-27 ^-Ka. (89) Where K2 is not known it may be taken equal to 0-272 S2. The semirange of the diurnal wave when Ki and Oi conspire is, formula (73), 1-021 (Ki + Oi). (90) (90) divided by (89) gives the ratio with which to enter Tables 17 and 18. By § 2 we see that for finding the acceleration in the times of maxima or minima, or the resultant heights, certain angles are involved which are of the forms (^2 nTT + I /3 - ayi(2 n + 1 7t + ^^- ay (91) & a It will be convenient to refer to these as the high water and low water phases, respectively ; and when no value of n is specified it is assumed to be zero. The argument of the Oi moon, added to the argument of the Ki moon is equal to that of the M2 moon. When Ki and Oi conspire, the argument of either component or the half sum of these arguments is the argument of the diurnal wave. Putting 6 = ^ a, § 16, the high and low water phases become ^ (2 « TT + M20 _ KjO - OiO), i (2w + 1 TT + M^o- K,o - OiO). (92) Making n = and 1, or any other even and odd integers, these phases become, after rejecting multiples of 360o, High water phase. Low water phase, i (M20 - K,o _ 0,0), i (M.,0 _ KiO - 0,o) + 900, (93) J (M2O - KjO - OjO) + 180°, J (M20 - KjO - OjO) + 270°. These are the phases with which to enter Tables 17 and 18. The heights of Table 18 are given in terms of the amplitude of the semidiurnal wave (89). The tropic lunitidal intervals are obtained from the mean values of the high and low water intervals, § 12, by subtracting from them the tabular values. Table 17, divided by 28-984. When the harmonics of M2 are large, M2° should be replaced by M2° — v in the abo^e expressions which refer to the maxima of M2, and by M2O — w in those referring to its minima. Having now determined the amplitude and phase arguments, the required quantities are readily obtained from the values in Tables 17 and 18.* 21. To find the height inequalities. When the order of the tides determining a height inequality is fixed its value becomes known by the preceding paragraph. Here it is proposed to find an expression for a height inequality when the sequence of the tides upon which it depends is subject to alteration. Suppose that we are here concerned with the high water inequality, HWQ, and let the value from the table be denoted by HWQ. For that portion of the time during which the change of order is not possible because the effect of the Pi wave in producing inequality in the heights of the high waters is less than HWQ, we have HWQ = HWQ. (94) 'Having thus obtained the heights of the tropic tides, each should he diminished by Mf cos Mf°. This is of importance when tronic LLW is required for the plane of reference. See also § 14. S. Ex. 8, pt. 2 10 146 UNITED STATES COAST AND GEODETIC SURVEY. Let X denote the distance in degrees of the maximum or minimum of Pi from the maximum of M2; then 2 P, cos x is the inequality in heights of the high waters which P, tends to introduce. Suppose that for the other portion of the time 2 P, cos x >HWQ. During this time one half of the inequalities will be in their normal condition and the other half will have their condition altered. The values of the former are represented by 2 Pi cos X + HWQ, (95) and of the latter by 2 P, cos X - HWQ, (96) where 15cos.^^=^say. (9^) The average value of cos x between the limits cos a; = 1 and cos x = I, or x between and cos ~' I, is V 1 - l\ COS"' V this substituted for cos x in (95) and (96) will give the required average values. The respective weights to be given to (94), (95), and (96), are 7C 1 1 -^ — cos-' I, - cos-' /, and ^ cos"' I. z z z The value of the required inequality is HWQ =(1 - ^'^^) HWQ + ^' ^/T^^ _r^_(cos:^-|^^ 4P^^ (98) ~L 90° J ^^'^ +3-1416 ^'• The fraction representing the percentage of (tropic) high water inequalities which have the order of their tides changed is cos-' I = icos-^^WQ)^X,say. (99) 71 Zi XI (If quite limited periods be considered, for example, particular years. Pi will remain constant from year to year, but HWQ will vary; see § 48. Consequently values must be given to HWQ suited to the particular years in order to ascertain the number of high water inequalities such that the high waters upon which they depend will have their order reversed.) 22. To compute the great and small tropic ranges, and the heights of the tides between which they occur. These are the greatest and least ranges of the tide upon a day when Ki and Oi conspire. Denoting them by Gc and Sc, respectively, we have Gc + Sc = 2 Mo, (100) Gc - Sc = HWQ + LWQ ; (101) Gc = Mc + I (HWQ + LWQ), (102) Sc = Mc - J (HWQ + LWQ). (103) Suppose the inequality in low water to be the more pronounced; Table 18 gives the values of tropic low waters, § 20. The tropic high waters are then obtained by the equations Tropic HHW = tropic LLW + Gc, (104) Tropic LHW = tropic HLW + Sc. (105) KEPOET FOK 189A— PAKT II. 147 23. To find the average range of the diurnal wave; also, roughly, the great and small diurnal ranges of tide. The average semirange of the diurnal wave is, formula (18), The theoretical value of this is 1-14 K], or 1'61 Oi. If this wave had a fixed position with respect to M2, the difference between the great and small diurnal ranges of tide would be almost directly proportional to the range of the diurnal wave. This can be seen by referring to Table 18. When the moon is far from the equator — i. e. when the diurnal wave is large — the position is nearly fixed. Assuming the fixed position to hold true for all declinations of the moon, the above s^tatement may be written Gt — SI _ Gt — Mn _ range of diurnal wave aoT) Go — So "" Gc — Mc ~ tropic range of diurnal wave since Gt + SI = 2 Mn, (108) Gc + Sc = 2 Mc. (109) (107) is theoretically equal to 1^7^ = «*^- (ii«) Since the above assumption is not correct when the declination of the moon is small, and more especially since oue is supposed to choose the greater range regardless of the sequence of the tides (see § 21), the above ratio ought to be increased by the positive quantity C representing these effects. For stations where there are usually two tides daily, C may be taken equal to 0-10, and for the present purpose Mc may be taken equal to Mn. Instead of (110) one may write (111) (112) If the tide had no semidiurnal components Mn would be zero, and the sequence of tides would not enter into the question. In such a case I* = 0-65. (113) For stations where there is usually but one tide a day, C must be very small. Mn can not be obtained from observation ; but it may be replaced by a rough value 2 (1-2) M2, while the ratio Gt - Mn Gc- Mn may be put equal to §; consequently Gt = I Gc + i M2. (114) At stations having tides of this character, the diurnal range, Gt, is often called the mean range of tide. 24. To compute roughly the average values of the height inequalities; application to planes of reference. If the sequence of tides remained fixed for all declinations of the moon, the average \4alues of the height inequalities would be to their maximum values (HWQ, LWQ) as the amplitude of the diurnal wave is to its tropic amplitude, or as 0-65 is to 1-00, §§ 21, 23. Considering now the more pronounced inequality, low water, say, the average value ought to be 0-65 LWQ + a fraction of (Gc - Mn), say. Gt- -Mn 3 Gc- -Mn 4' Gt = = !0c + |Mn. 148 UNITED STATES COAST AND GEODETIC SUEVEY. This assumed, the fraction must be such that it vary directly as Gc — Mn and inversely as LWQ. The expression now becomes Cro — Mn 0-65 LWQ + e j^^q (Gc - Mn) (115) where e is supposed to be constant. The special case LWQ = HWQ gives, making use of (111), £ = 040. Assuming that the mean of the two tropic low waters is the same as mean low water, it follows that the depression of the lower low waters below mean low water is LWQ 1 [-Gc - Mn O 1 1 rGc-Mn-]^ ,-,,„> LWQ The average value of the other and less pronounced inequality is obtained by subtracting the one already obtained from the quantity Gt — Mn. The following equations are obtained from Table 18 and show how the sum of the two tropic high or low waters differs from Mc, the mean tropic range of tide : Tropic HHW + tropic LHW = Mc - 0-126 ''^'^ '^ cos (M2O - KiO _ OjO); (117) Tropic LLW + tropic HLW = - Mc - 0-126 ''—^ — ^ cos {MP - K,o - OjO). (118) Less accurately we have (Ki + o,r Mean LLW + meau HLW = Mn - 0-08 ^-^^ — ^ cos (MjO - K,o - OjO) Mean HHW + mean LHW = Mn - 0-08 ^ 'j^ " cos (M2O - K,o - 0,o); (119) M, (120) The mean of both high and low water heights is the height of mean sea level, or rather half tide level (HTL), above the line about which the components are supposed to oscillate; that is, above mean sea level as determined from hourly ordinates (MSL). From § 13 the mean of high and low water gives the elevation of mean sea level due to the harmonics of M2. (Ki + O, .-. HTL = MSL + M4 cos (2 M^o - M^o) - O-Oi ' '^ " cos (M^o - K,o - MOiO). (121) 25. Stations where the tide is usually diurnal. The tropic lunitidal intervals may be found by means of formulae (58)-(61) after replacing 6''-2103 by 12''-33, M2, M2°, m2 by Ki + 0„ i (Ki° + OiO), dj; and M4, M4O, mt by M2, M2° m2, respec- tively, omitting the higher harmonics. The (great) tropic range of tide is by formulae (65) and (73) 2 (1-021) (K, + 0,) + (Ki+0,) (cos i;+ cos w) -0-035 M2 {v-io) sin (M20-K,o-OiO)-2 (Ki+Oi), (122) the corresponding high and low water heights being (cos V + 0.021) (K: + Oi) + M2 cos (M2° - K,° - 0,o) _ 0.035 M^ v sin [M^o - K,o _ o^o), and -(cos w + 0.021) (Ki + O,) + M2 cos (MP - K,o - Oi^) - 0.035 M2 w sin (M2O - K,o _ 0,c), where v and w are expressed in degrees. When M2 is small, this range is nearly equal to 2 (1-021) (K,+Oi). (123) Mean daily low water is then about 2 3 (K, + O0 below MSL. OHAPTBE III. REDUCTIONS OF OBSERVATIONS MADE UPON HIGH AND LOW WATERS. 26. The maiu object of this chapter is to determine, from a series of observed high and low waters, certain harmonic and nonharmonic quantities which are of special importance in deter- mining the best port of reference for a given station. In so far as the amplitudes and epochs of various components are determined, the reductions constitute a rude harmonic analysis.* The nonharmonic quantities obtained are of interest because of their obvious connection with observed tidal phenomena. The astronomical items used (transits, phases, etc.) are taken directly from the Greenwich ephemeris; the only change consists in using civil time. The tables given at the close of this paper would, by some alterations and extensions, enable one to use mean motions throughout, thus doing away with some of the inaccuracies which attend the process of reductiiDii as carried out below. Lt is believed, however, that such errors are too small to be of much consequence in an analysis of the kind proposed. There are some advantages, especially when the series is short, in using the real motions of the bodies. For instance, the epoch of S2 obtained from observing a few tides and referring them (without correction) to the mean sun will be much more in error than would have been the case had the true sun been used. But it will be seen from Table 31 that the correction to be applied to the epoch of S2 because of T2 and the solar part of K2 is practically the equation of time. FIKST EBDUCTION.+ 27. In this reduction the first process consists in tabulating the times and heights of high and low water, together with the times of transit of the moon across some known meridian. The next step is the subtraction of the transits from the times of the tides, usually selecting such transits as will give the smallest possible interval without introducing too many negative values. Before taking means of the intervals and heights, the length of series to be used must be decided upon. Length of series suitable for finding the semidiurnal range of tide and the mean lunitidal inter- 'cal. — Since the variation in range, from day to day depends chiefly upon the manner in which the semidiurnal components fall upon M2, the required length of series should be, as nearly as possible, a multiple of the synodic periods of S2, N2, L2, K2, ... with M2 ; i. e. of the form a Sz 14-765 + & (N; + L2) 27-555 + c K2 13-661 + . . . S2 + N2 + L2 + K2 + . . . ^^"*^ where a, b, c, . . . are integers. The weights S2, N2 + L2, K2, . . . are given to the synodic periods because the direct effect of any component in altering the range of the tide is nearly proportional to its amplitude. For the purpose of eliminating the diurnal inequality as far as possible, the j^°^ water which ends the series should be of the opposite kind from the j"^, water beginning it. As a length of * For a more elaborate mode of analysis, see a paper by Prof. G. H. Darwin entitled " On the harmonic analysis of tidal observations of high and low water," Roy. Soc. Proc, Vol. 48 (1890), pages 278-340, see also a correction to this paper, ibid.. Vol. 52 (1892), pages 388, 389. t For an example see 5 30. 149 /^ 150 UNITED STATES COAST AND GEODETIC SUEVBY. series suited to the high waters is geuerally also suited to the low waters, one should solve the above equation to the nearest lunar half day. This having been done, the results expressed in mean solar days are as follows : 29, 58, 87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, 384. It is often convenient to divide the> series, whatever its length, into periods of 29 days each. Since the transits of the true (not the mean) moon are used, the lunar semidiurnal components Nz, L2, lunar K2, . . . have little effect upon the interval. Consequently the synodic month of 29J days might seem to be a better length of series for the determination of the interval. But on account of the diurnal inequality in interval it is probable that even for this purpose 29 days would usually be preferable to 29J. Having fixed upon the length of series to be used, the sums and means of the intervals and the heights on staff are then taken. The resulting intervals are reduced to a value which would have been obtained had local transits been used by adding S-L + 0-035 {U - L) (125) where E, S, and L are the west longitude in time of the meridian of the transits, i. e. of the ephemeris used, of the (standard) time meridian, and of the meridian of the station, respectively. Using now the corrected intervals or establishments (HWI, LWI) expressed in hours, the epoch of M2 is (unless Me be large) 14-4921 (HWI + LWI =p 6-2103) = M2O. (126) The V^p„ sign is used when the low water interval is taken ^^^^ ^^ than the high water. The resulting LW height subtracted from the HW height gives the range of tide for the period of the observations. The intervals and ranges just considered refer to tides occurring twice daily. For this reason when evanescent tides occur such days' observations should be omitted in the summations. If only a few occur they may be interpolated, or if a continuous record be available the points of maximum curvature may be selected for the missing high and low waters.* For some purposes the average height of the higher high and lower low water may be useful, particularly for determining the great diurnal range of tide and the plane of reference defined by lower low water. These heights are always obtainable, as one high water and one low water occur each day, even when there are evanescent tides. In the first reduction, mark such heights and take the mean of the quantities so marked. The length of series used for this purpose should be an integral number of tropical months, or, less accurately, of half tropical months. The following numbers represent such periods expressed in mean solar days: (14), 27, (41), 55, (68), 82, (96), 109, (123), 137, (150), 164, (178), 191, (205), 219, (232), 246, (260), 273, (287), 301, (314), 328, (342), 355, (369), 382. 28. Determination of mean sea level and long-period tides. Where hourly heights of the tide are given, it is obvious that a suitable period of time for determining mean sea level would be one in which each of the principal tidal components makes an integral number of oscillations. By taking multiples of the hourly speed of each component, it is readily seen that the period of 29 mean solar days and zero hours is the best value obtainable unless the given heights of the tide are taken at intervals of less than one hour. The same period may be adopted for the determination of mean sea level (half-tide level, § 24) from observations of high and low waters. This mean sea level is the half sum of the heights of mean high and low water for the 29-day period or group. For long series of observations, it is desirable to have the periods conform to certain dates made out in advance. These dates should be the same for all stations in order that the fluctuations * This implies, as it should, that the small tropic range becomes negative. X REPORT FOR 1894— PART 11. 151 of mean sea level at various places may be more readily compared. The following dates are supposed to refer to any year; the dates marking: the middle of each group divide the average year into twelve equal parts ; the length of group is 29 days, as is shown by the terminal dates : Groups for mean sea level. Beginning Middle (noon). End (midnight pre- ceding). (midnight fol- lovfing). Jan. 2 Jan. I 6 Jan. 30 Feb. I Feb. 15 Feb. 29 Mar. 3 Mar. 17 Mar. 31 Apr. 3 Apr. 17 May I May 3 May 17 May 31 June 3 June 17 July I July 3 July 17 July 31 Aug. 3 Aug. 17 Aug. 31 Sept. 2 Sept. 16 Sept. 30 Oct. 2 Oct. 16 Oct. 30 Nov. 2 Nov. 16 Nov. 30 Dec. 2 Dec. 16 Dec. 30 For all spring and summer months the groups begin on the 3d of each month, while for all fall and winter months (except February) they begin on the 2d. In harmonically analyzing the twelve ordinates thus obtained, it should be borne in mind that the first ordinate is 15-218 days after 0'' (midnight preceding) January 1; this increases the Fo 4- M of Sa, Table 3, by 15°. If F„ + m be found for date of beginning of series used in analyzing short-period tides, then it must be increased by the number of days from this date to the middle of the first 29-day group used, multiplied by 0-9856. Supposing the groups to be so taken as to conform to the above scheme, then it is clear that the values of Table 3 are simply increased by the amount 300 X m — 15° where m is the number of the calendar month approximately covered by the group. Besides the yearly change in sea level, a small change having a period of about 428 days may be considered here. This depends upon the 428-day component of the variation of latitude. The annual and 428-day tidal components maybe nearly separated by combining 7 years' observations. If the annual inequality in sea level has been determined from analyzing the monthly heights just considered for 7 years, it may be taken away from the sea-level curve and the residual curve read for the 428-day tide, using the true lengths of period for the particular years. The fictitious moon has the longitude of the instantaneous minimum north latitude of the place so far as the latitude depends upon the 428-day component ; this moon goes around the earth once in 428 days from west to east. Suppose the time of minimum north latitude to be given for Greenwich, i. e. the time when the north pole of the earth's figure passes the instantaneous meridian of G-reenwich ; the time for a place L hours west is 15 i Time of Greenwich min. lat. daily speed days. 152 UNITED STATES COAST AND GEODETIC SURVEY. The average value of the daily speed is 0o-840. FormuliB.for more accurate values, together with a table of times of minimum north latitude or of high water of 428-day tide, are given below : No, of Time of high water Amplitude of periods from t86s. E at Greenwich of 428-day com- Amplitude of 428-day tide for 428-day component ponent of the latitude tide. latitude 45°. y-i variation. Civil date. // Feet. —40 1818, June 16 0-I-84 / 0062 -3S 1824, Apr. 13 •170 ■057 —30 1830, Jan. 28 •146 •049 -25 1835, Nov. II ■118 •040 — 20 1841, Aug. 29 ■096 •032 -15 1847, July I •085 •029 — 10 1853, May 18 ■090 ■030 — 5 1859, Apr. 20 ■108 •037 1865, Mar. 31 ■135 •046 + 5 1871, Mar. 13 ■162 •055 + 10 1877, Feb. 12 ■180 ■061 + 'S. 1882, Dec. 31 ■185 ■063 +20' 1888, Nov. I •174 ' ■°59 +25 1894, Aug. 20 ■152 ■051 +30 1900, June 3 0T24 0-042 Amplitude = 0"-135 -f 0"-05 sin 'V, Period = 428'i-6 + 5''-26 cos W, T, = 1865-25 + 428-6 ^ -f 55 sin W, where W={t- 1865-25) 50-48, t denoting the year. [These values are taken from Dr. S. 0. Chandler's paper on latitude variation in No. 322 (July 10, 1894) of the Astronomical Journal.] At the times of local upper transit of the fictitious moon, Height of 4281 ^ide =^ X change in latitude* (127) where the height is reckoned in feet from mean sea level, the (north) latitude, A, is expressed in degrees, and the change in latitude, column 3, is expressed in seconds.t This fluctuation or tide is of little or no practical importance. It is noticed here because, like the annual tide, it may be determined from a sufficient number of observations upon high and low water. In the special reductions which follow, the length of the series will usually be about one month. Certain modifications will be found necessary if, for any reason, other lengths be used. * The rounded value \ is written for 0'338. tA ready means of comparing theory and observation is to tabulate the times of local transits of the fictitious moon, as the times of high and low waters of the 428-day tide, and to find the reading of mean sea level, at such times using a 29-day group on each occasion. The sea level is then corrected for the annual component, and finally 6 high water or 6 low water groups (approximately 7 years) are combined and the mean taken. From hourly readings (1870-1887) at Pulpit Harbor, Maine, (lat. 44° 09' N., long. 68° 53' W.), the mean for the component high water was 10'495 feet, and for low water 10-357, showing, upon application of the group factor 1-007, an amplitude of 0-069 feet. The theoretical value of this, formula (127), is about 0.059 feet. From hourly readings (1869-1887) at San Francisco, Cal., (lat. 37° 50' N., long. 122° 24' W.), the corresponding observed quantities were 8'285, 8-206, giving an amplitude of 0-040 feet, the theoretical value being 0-057. EEPOET FOE 1894— PAET II. 153 REDUCTION OP SPRING AND NEAP TIDES. 29. In one column of the sheet headed " Ephemeris " will be found the times of the moon's phases expressed in the kind of time used in making the observations. Select from the first reduction, groups of observations, each group beginning about a day and a half before the times of new moon, full moon, or quadrature; then fill out the forms for spring and neap tides. The correction for parallax is taken from Table 25, using for the amplitude of ¥2 the value i^f Mn, and for its relative age the value at a neighboring station taken as 1-837 (M2° — ISTjO). For a series about 220 days in length the parallax correction becomes zero. Since transits of the true (not the mean) moon are used, the luni tidal intervals need not be corrected for parallax, especially when the springs or neaps are taken in pairs. The age of the solar tide. — The luni tidal intervals found in the column headed "Average" are plotted one-half lunar day apart, at times found in the average time column. The plotted values of each group are then connected, as nearly as possible, with a straight line. The values most distant from the time of spring or neap tide, as the case may be, will, theoretically, depart most widely from the straight line when the latter is properly drawn. The time of spring or neap tide is found by noting where the mean value of the interval (as deteimined from the first reduction before (126) has been applied) falls upon the straight line. Each such time is then diminished by the time of the appropriate phase of the moon. The mean of these differences is the age of the solar tide relative to the lunar. Spring and neap ranges. — One range is obtained from the reduction of spring tides, and one from that of neap tides. The age applied to the times of one or more phases of the moon will show what group factors must be used in reducing the amplitude of the solar wave as brought out from the spring and neap ranges. The group used in determining the ranges need not be as extensive as the group used in determining the age. According to § 18, the amplitude of the solar wave should be further increased by the factor 1.0. + 0.0. aw^+/^Q- , (m) HWQ and LWQ may be taken from the first reduction with sufficient accuracy. This will be referred to as the inequality factor. 30. Example. 154 UNITED STATES COAST A]S"D GEODETIC SUEVEY. The following tabulations, relating to a month's observations at Sitka, Alaska, show how these reductions are carried out : Sifka, Alaska. First reduction. Date. Moon's transits. Time of— Lunitidal interval. Height of— Remarlts. H W L w H W L w H W L W Lat. = 57° 03' N. Long. = 125 i8 = 9'>oi»W. 1893- A. »«. A. w. A. 7?Z. h. ni. h. m. Feet. Feet. (13 08) July I I 34 I 08 8 09 (12 00) 6 35 14-6 3-4 (13 59) 14 51 19 54 13 17 (5 55) 12-8 8-4 2 2 24 2 01 8 42 (12 02) 6 18 14-4 3 '9 (H 48) •5 25 20 SO 13 01 (6 02) 130 8-3 3 , 3 II 2 3? 9 25 (" 42) 6 14 14-0 4-0 (IS 34) 15 58 21 14 12 47 (5 40) 12-8 7-8 4 3 37 3 12 9 54 (" 38) 5 57 13-2 4-3 Observations in mean local civil time. (16 19) 16 37 22 10 12 40 (5 51) 12-9 7-6 5 4 41 4 05 10 27 (" 46) 5 46 12-5 S'o ('7 04) 17 23 23 23 12 42 (6 '9) 132 7-2 Greenwich transits. 6 , 5 26 S 09 II 14 (12 05) 5 48 120 5-9 (17 48) 18 °3 12 37 34) 136 II-4 To convert Greenwich transit*; to local trrjn'?itt; 7 6 II 6 22 32 (12 (5" 44) 6"6 observation time,add Z—^+ 0-035 (Z — £) (18 36) 18 46 12 18 12 35 6 07 14-0 67 = l8m-9. 8 7 00 7 49 I 49 (13 13) (7 13) 1 1 -2 5 '8 (19 26) 19 44 13 03 12 44 6 03 14-6 7-6 9 7 53 9 02 2 52 (13 36) (7 26) "S 5-0 (20 22) 20 37 14 15 12 44 6 22 150 8-3 Greenwich civil time of perigee, P, and lO 8 5> 10 28 3 50 (H 06) (I 28) 11-9 3-9 apogee, A. (21 23) 21 30 15 25 12 39 6 34 15-5 8-6 II 9 55 II 31 4 50 (14 08) (7 27) 12-5 2-9 [II''23^s] (22 28) 22 29 16 36 12 34 6 41 157 8-4 p 12 II 01 12 17 5 40 (13 49) (7 12) 12-9 2-1 (23 34) 23 26 17 29 12 25 6 28 i6-i 8-0 13 6 28 (6 54) 1-9 12 07 13 09 18 21 (•3 "35) 6 14 135 .7-8 14 (° 38) 22 7 18 13 15 (6 40) ,6-3 1-8 13 09 13 55 19 19 (13 J7) 6 10 13-9 7-6 15 ( ' 38) I II 8 01 12 02 (6 23) i6-2 2-4 14 06 14 50 20 13 (13 12) 6 07 14-6 7-8 16 (2 32) I 56 8 46 11 5° ^\ 14) i6-3 3-6 14 57 15 19 21 04 (12 47) 6 07 151 7-9 P a.nd A hour of transit = 9J. 17 ( 3 20) 2 5° 9 31 II S3 (6 ") 15-7 4-3 IS 43 15 59 21 47 (12 39) 6 04 147 7-3 HWQ LWQ 18 (4 °S) 3 47 9 56 12 04 (5 51) 14-0 5-0 Feet. Feet. 16 27 16 38 22 50 (12 33) 6 23 14-2 7-0 July II, 12 3-2 6-3 19 (4 48) 4 40 10 35 12 13 (5 47) 127 5-9 " 26 2-6 5-0 17 09 17 22 (12 34) 13-8 20 (5 30) 5 40 00 12 31 6' SI II-6 6'8 Mean 2-9 5-6 17 51 18 08 II 21 (.2 38) (5 SI) 137 7-1 Sequence HHW to LLW. 21 (6 12) 6 52 48 13 01 6 57 1 1 -2 6-9 18 33 18 46 12 10 (12 34) (5 58) 13-8 8-3 July 1-29, 1893. 22 (6 55) 8 12 2 00 13 39 7 27 no 6-8 HWI LWI HW LW 19 17 19 34 13 08 (12 39) (f 13) 13-4 9-0 S6 56 56 56 23 ( 7 40) 9 30 3 18 14 13 8 01 II-O 6-0 ;4. m. h. m. Feet Feet. [24^ 2h-2] 20 04 20 28 14 41 (.2 48) (I 01) 13-4 9-2 Sum 722 04 369 38 760-2 351-2 A 24 ( 8 28) II II 4 07 15 07 8 03 II-3 5-3 Mean 12 53-6 6 36-0 13-58 6-27 20 53 21 34 15 21 ('3 06) (6 53) 13-5 II-8 9-3 Interval = 91= 44" -8. Mn = 7 -3 1 feet. To correct for transits subtract 18™ -9. HWI = l2i' 34™-7. LWI=6i i7'o-i. HWI=I2'' 34"'7 — I2'i 25"n-2=oi' 9"'-S. HWI — LWI=6'' 26"'-6=6''-443. 6''-443 — 6''-2io=o'>-233. i4°-492 xo-233 = 3°-4=Ms°. Duration of fall = 6" of^-d. 25 ( 9 '9) II 40 4 47 14 47 7 54 IS) 4-8 21 45 22 20 16 34 ^'^ 01) (7 141 9-2 26 27 (10 22 (" ") 37 03) 12 23 12 08 02 45 5 17 6 27 04 II (12 14 23 08 7 (6 7 42 34 127 iS-3 13-2 4-6 9-6 47 23 29 23 43 17 54 (12 40) (6 SI) 151 9-1 28 (" 54) 6 40 7 II * 4'0 13 14 18 30 13 45 ^^e 36) 5^ 132 8-S 29 20 25 7 16 (12 30 151 3-8 July 1-27, 1893. 30 '" 44) 08 13 44 57 19 7 00 46 13 24 (6 16) 13-5 8-2 HHW LLW Gt 26 2S (12 n) 6 38 15-4 3-8 (13 32) 14 16 19 47 13 08 (6 15) 14-0 7-9 Feet. Feet. Feet. 31 I 55 I 38 8 22 (12 06) 6 27 15-3 4'i 376-5 1 10-2 (14 18) 14 38 20 28 12 43 (6 10) 15-0 7-3 14-48 4-41 10-07 BEPOET FOE 1894— PART II. 155 Sitka, Alaska, 1893. Ephemeris. h. Longitude of ephemeris {£) o Longitude of time meridian (i') 9 01 E — .5'= — 9 01 Observation time. Observation time Observation time Moon. Gr. civ. time increased increased by -\-E-S. by assumed age. uncorrected age. d. h. d. h. d h. d. h. Full moon June 29 6-4 28 21-4 29 239 (J Last quarter July 6 22-1 6 13-1 7 15-6 • New moon " 13 12-8 13 3-8 14 6-3 ]) First quarter " 20 17-0 20 8-0 21 10-5 Full moon <• 28 20 -2 28 1 1 -2 29 137 Apogee June 26 137 26 47 28 37 28 17 Mid-time) July 4 6-6 3 21-6 5 20-6 5 i8-6 Perigee " u 23-5 II I4'S 13 13-5 13 "S (Mid-time) " 18 0-8 17 15-8 19 14-8 19 12-8 Apogee " 24 2-2 23 17-2 25 l6-2 25 14-2 (Mid-time) Aug. I o-o 31 is-o 2 140 2 12-0 Perigee " 8 217 8 127 10 117 10 97 On equator July 6 2-9 5 17-9 6 11-4 Farthest N. " 12 8-Q II 23-9 12 17-4 On equator " 18 18-3 18 9-3 19 2-8 Farthest S. " 26 S-2 25 20-2 • 26 137 m. 00 W. Sitka, Jlask., 1893. • h. m. Springtides. Longitude of ephemeris {E) o 00 W. Longitude of time meridian {S) 9 01 " E—S= — <) 01 From first reduction, July 1-29, 1893: Mean (uncorrected) interval^g'' 44"'8; Mn:=7-3i ft. Assumed: N2=-iVMn = 0-665 ft.; t (N^; M.j) = 4711, from analysis at St. Paul, Kadiak Id. Moon new and lvX\-\-E—S. Time. Average. Lunitidal interval. H W LW Average. Height. H W L W July 13 3-i July 28 d. II 12 12 12 13 14 27 29 29 30 30 h. m, 22 29 17 23 26 13 09 o 22 14 13 55 15 III 15 14 50 d. II 12 12 13 13 14 7 14 19 15 8 d. h. m. 16 36 5 40 17 29 6 28 18 21 18 19 13 3 08 Group cA 2" after perigean tides. 26 23 02 27 23 28 13 12 45 43 14 o 25 13 44 27 6 II 27 17 54 28 6 40 28 18 30 29 7 16 29 19 00 3° 7 46 30 19 47 28 9 32 Group 3* 6'! after apogean tides. k. m. 12 34 «3 49 12 25 13 35 12 15 13 17 12 02 13 12 h. m. 6 41 7 12 6 28 6 54 6 14 6 40 6 10 6 23 h. 04-0 58-5 50-5 44-5 36-5 32-2 26-8 Feet. 157 129 i6-i 13-5 16-3 13-9 i6-2 14-6 Correction in terms of Nj " " feet 12 51 14 08 12 40 13 45 12 31 13 24 12 13 7 34 6 SI 7 II 6 3& 6 S6 6 lb 6 38 6 15 08 Correction in terms of Nj " " feet 21 'O 12-5 03 "O 9 57-0 9 46-8 9 37-8 9 33 -S 119-2 14-90 — -90 — -60 14-30 15-3 13-2 I5-I 13-2 151 13-5 154 14-0 Feet. 8-4 2-1 8-0 1-9 7-8 1-8 7-6 2-4 40 o S'oo + -90 -f -60 5-60 47 9-1 4-0 8-5 3-8 82 3-8 7-9 114-8 14-35 -f -60 + -40 1475 Age = J (12 + 29): Spring range .-= 8-80 3 50-0 6-25 ■60 •40 5-85 = 20-5 156 UNITED STATES COAST AND aEODETIC SURVEY. h'eap tides. Sitka, Alaska, 1893. h. m. Longitude of ephemeris (£) o oo W. Longitude of time meridian (5) 9 01 " E — S=—<) 01 Moon in quadrature -\-E- S. d. July 6 13-1 Time. H W d. h. S 4 6 18 7 6 7 18 8 7 8 19 LW 10 27 23 23 11 14 32 12 18 1 49 13 03 2 52 Average. a^m^ + b^ \ / If the speed of the diurnal wave is less than mi, prefix the minus sign to the minimum range. The minimum ranges should be taken in pairs in order to eliminate Qi from Oi. 37. Moon far north or south. The uncorrected age of the diurnal inequality having been found, increase the times of extreme declination thereby, thus obtaining the times when the amplitude of the diurnal wave becomes a maximum. At these times tropic tides occur ; in the reduction, one or two or more high waters and as many low waters are taken on either side of each time. It may be noted that for any one determination of the quantities directly connected with the tropic tides, the lunitidal intervals having approximately equal values should be written in the same column. From a reduction of the high waters and of the low waters, according to the accompanying form, the following uncor- rected quantities are obtained : Intervals, heights, Gc, Sc, Mc, HWQ*, LWQ. Both heights and intervals of the great tropic tides can always be obtained from the reductions; but the small tropic tides can not be so obtained where evanescent tides occur. At such places Sc may be roughly inferred from the equation Gc + Sc = 2 Mn. The tropic range of the diurnal wave for a given time becomes approximately known by the formula 2 D, = VHWQ^ -f LWQ\ (137) The amplitude D^ of the semidiurnal wave when the moon is far from the equator is approxi- mately equal to 4Mc-i|L (138) * It is here supposed that LWQ > HWQ. HWQ indicates merely that the order of taking the tropic higher high and lower high water, has remained the same throughout the series. At stations wbere the order of the high waters, say, is sometimes reversed, even when the moon is far from the equator, a period of six months, or some multiple thereof, should he used in finding HWQ and X, § 21. These quantities are reduced to their mean values by Chapter IV. When a short series is used, X and HWQ should be computed from Pi =0-331 K„ §5 89, 40, and the corrected HWQ, by means of the formulsB Cos X 180° = ^^ , (135) 2 if, 2 HWQ =2 Pi [(1 — 2 X) cos X 180° + - sin X 180°] (136) EEPOET FOE 1894— PAET II. 161 where Mc is the mean of Gc and Sc. If Mc is poorly determined, the formula D2 = 0-44 Mn - 0-06 Mn (139) should be used. Having found Dj, two ratios are obtained by dividing HWQ and LWQ by 2 D2. The inter- section of two curves upon Plate III which correspond to these ratios gives an amplitude (in terms of Dj) and a HW phase. Stations where the tide is usually diurnal. — The quantities to be obtained from the reduction are the intervals (properly distinguished) and the heights of the tropic tides; also, approximately, mean sea level and the mean semidaily range of tide. For most stations, the two latter quantities can be found whenever the moon is near the equator, and especially about the times of the equinoxes. From these results are determined the ratio Gc -h Mn, the duration of the great tropic range (always taken to be less than a half lunar day), and whether the tropic high or tropic low water, departs farther from mean sea level. By means of Table 20 an amplitude and HW phase become known. Rule for distinguishing between the qurdrants : Interval (HW or LW) marked (a) HW phase falls in 1st quadrant. Sequence HHW to LLW u a Sequence LLW to HHW < ^ HW interval marked (a) LW (6) {&) HW interval marked (6) LW (a) u It " 3d it u " 4th u it it a li tt " 2d u it it li It is assumed that the HW interval is taken approximately equal to M2O/29 and the LW interval, to M2°/29 + 6'=; also that the mean intervals have the marks {a, b) belonging to the intervals of the great tropic tides, § 53. 38. Example. Declinational reductions for a month's observations at Sitka, Alaska, are given below: Sitka, Alaska, 189S. Minimum diurnal tide. h. m. Longitude of ephemeris (E) o oo W. Longitude of time meridian (.?) 9 01 " E — ^= — 9 01 Time. Height. Inequality. Moon on equator Time and value of minimum range. H W Average. L W Average. H W L W H W L.W d. h. d. h. m. d. h. m. d. h. m. d. h. m. Feet. Feet. 3 15 5« 3 9 25 12-8 4-0 7-8 4-3 7-6 +0-4 +0-3 —0-4 -0-7 — I -2 —1-6 — 2-2 —2-6 —2-8 -3-8 —3-5 —3-3 —2-6 4 4 5 3 16 4 12 37 05 3 4 4 21 14 9 54 22 10 132 12-9 12-5 Julys 17-9 S 6 6 7 7 8 17 5 18 6 18 7 23 09 03 22 46 49 5 23 16 5 5 6 7 7 8 10 27 23 23 n 14 32 12 18 1 49 5 16 55 13-2 120 13-6 114 14-0 II-2 S-o 7-2 S-9 6-6 6-7 S-8 — 2-2 -1-3 -07 +0-1 +0-9 July 6, 8" " 5. 18 14 = age. Range ==2-2o 16 16 I IS 56 19 IS 16 20 13 8 46 16-3 151 7-8 3-6 + 1-2 -i-o-6 + 10 —07 — 02 -I'S —4-2 -4 '3 -3-6 _3-o -2-3 — 20 17 2 50 lb 21 04 lyy 7-9 17 15 59 17 9 31 '47 4-3 July 18 9-3 18 18 TO , 3 16 4 47 38 40 18 10 12 17 18 18 21 47 9 56 22 50 18 3 52 14-0 142 127 7-3 5-0 7-0 July 19, 6l> " 18, 9 iq 17 22 IQ 10 35 138 5-9 — 09 +0-3 .21= age. 20 20 S 18 40 08 20 20 00 II 21 1 1 -6 137 5-8 7-1 — 2-1 Range = i-6o Uncorrected age^-i(l4+2l) ■ =17-5 Average date of moon on equator + age = July 12 Minimu m range -= I •90 S. Ex. 8, pt. 2 11 162 Tropic high waters. UNITED STATES COAST AND GEODETIC SUEVEY. Sifka, Alaska, 189S. From first reduction, July 1-29, 1893: Mean interval = gh 45m. From phase reduction: S,, = (S) — o-oi Mn = 0781 foot; r (Sj; M.2) = 26'^-',. h. m. Longitude of ephemeris {E) 00 W. Longitude of time meridian (5') 9 ol " E — •S'= 9 01 Moon. Time of liigh water. Lunitidal interval. He ight. Extreme declination. + £ - 5- + ir'-s. Transit. u n, 1 s, or 1 n, u s. For H H W. Corr'n. For LH W. Corr'n. Time. Kind. d. h. d. h. in. d. h. m. h. m. h. m. Feet. Feet. II 9 55 u n II 22 29 12 34 157 — -60 S2 July 12 N. 17-4 II 12 22 II 28 01 In u n 12 12 17 12 23 26 12 25 13 49 l6-I - -83 12-9 - 75 3, 12 23 34 In 13 13 09 13 35 13-5 — -90 25 9 19 Is 25 22 20 13 01 14-1 — 15 July 26 S. 137 25 26 21 ID 45 II u s Is 26 12 08 26 23 02 12 51 14 23 15-3 - -52 127 - 36 2b 22 37 u s 27 12 45 14 08 13-2 — -68 4 4 4 4 4 4 49 III 54 "5 61 -2 — 210 52-3 — 2-69 Mean Corrected for phase 12 43 12 15 13 59 13 31 15-33 14-89 — -52 13-08 12-56 - -67 EEPOET FOE 1894— PAET II. 163 Sitka, Alaska, 189S. Tropic low waters. h. m. Longitude of ephemeris (-£) o oo W Longitude of time meridian (i') 9 01 " ^ — 5' = — 9 01 Moon. Time of low water. Lunitldal interval. H eight. Transit. Extreme declination u n, 1 s, or 1 n,u s. For LL W. Corr'n. For H LW. Corr'n. Time Kind. d. h. d. h. »«. d. h. m. h. m. k. m. Feet. Feet. II 9 55 u n II 16 36 6 41 8-4 + -5382 July 12 17-4 N. II 22 28 In 12 5 40 7 12 2-1 + -6982 12 II 01 u n 12 17 29 6 28 8-0 + "79 12 23 26 In 13 6 28 6 54 I '9 + -87 25 9 19 Is 25 16 34 7 IS 9-2 + -05 July 26 137 s. 25 26 21 10 45 II u s Is 26 5 27 26 17 04 7 42 6 53 4-6 + -27 9-6 + -44 26 22 37 u s 27 611 7 34 47 + -60 4 4 4 4 4 4 27 142 25 137 133 +2-43 35-2 + I-8I Mean 7 20 6 49 3-32 + -61 8 -So + -45 Corrected for phase 6 52 6 21 3-80 9-15 Mean of tropic ints. ^= 10 13 " int. first red'n = Diff. = 9 45 28 Average date of extreme declination + age = July 20 Great tropic range = 1 1 -09 Small tropic " = 3-41 High water inequality = 2-33 Low " " = 5 -35 39. Treatment of declinational reduction results. 164 UNITED STATES COAST AND GEODETIC SURVEY. Station. Declnational tides. Amplitude relations. (1) Minimum semirange of diurnal wave from reduction. (2) i''(K|), Table 10; c\. Table 31. For av. time of min. diurnal tides. (3) HWQ from reduction. (4) LWQ (5) (3) X group factor. (6) (4) X " " (6') Gc from reduction + \ [(5) + (6) - (3) - (4)]. (6") Sc " " -H(5) + (6)-(3)-(4)]. (6'") Tropic LLW from reduction — \ [(6) — (4)]. (7) V (S)« + (6)» = approx. 2 D,. (8) o-88Mn-.o-03^ = 2D,. (9) (10) (5) HWQ (8)— 2D2 (6) LWQ (8) - 2 Ds ■ (11 J Amplitude and HW phase from Table 19. (12) i^(Ki), i^(Oi), Table 10; c-^^, Table 31. For av. time of tropic tides. (-) ^^-^^^^^ = 0. (15) K/xi^(K,) = K,. (16) 0/ X /'(Oi) = 0,. (17) /and J'"(Mn) for middle of series, Tables 6, 14. (18) Mn from first reduction. (19) (18) Xi^(Mn)==Mn {20) . ^^ , ?£l±°i. ^ ' i Mn i Mn (21) Tabular value, Table 23. (22) i(l9)X(2i) = (i+£)M,. Nonharmonic quantities (see Chapter IV). I -02 (Ki + OQ ^ , I -,/ = 1-02/^1 (5) X io2A = HWQ . (6) X io2i^, = LWQ [tropic HHWI from reduction — HWI *] X i 02 Fx + HWI f = tropic HHWI. [tropic LHWI from reduction — HWI] " " = tropic LHWI. [tropic LLWI from reduction — LWI] " LWI = tropic LLWI. [tropic HLWI from reduction — LWI] " " = tropic HLWI. [(60 — (i8)]X 1-02/1 + (19) = Gc. [(6'0-('8)] " " =Sc. [Gt from first reduction — (18)] X i -02 Fy + (19) = Gt. 2(i9)_Gt = SI. Mean LW from first reduction — -J [(19) — (i8)] = LW. [(6"/) — mean LW from first reduction] X i '02 F^ + LW — tropic LLW. [LLW from first reduction — mean LW from first reduction] X i 02 /", + L W = LLW. Epoch relations. (23) Age from reduction (hours). (24) (23) X I -098 = approx. (Ki° — O,"). (25) (24) + ace. in Ki due to P„ Table 31, = K,° — 0,°. o-gii (Ki° — Oi° = r(Oi; K,). (26) Mean lunitidal ihterval exceeds mean tropic interval (minutes). (27) (26) X 0-483. (28) 2 HW phase from amplitude relations. (29) (27) + (28) — ace. in K, due to P„ Table 31, = Mj" - K,° — 0,°. (30) M.° + (25)-(29)^^.^„^ (30 y-(25)-(29 )^^^„ ' Not corrected by (125). t Corrected by (125). KEPOET FOR 1894— PAET 11. 165 Sitka, Alaska, 1893. Declinational tides, July, tSgS- Amplitude relations. (0 0-95- (2) ii'(K,) = o-905,r,= 1.243; July 12. (3) 2-33- (4) S-3S- (5) 2-33 X 101= 2'353. (6) 5-35 X ''o' =5 ■404- (6') 1109 + i [0-077] = !i-i3. (6") 3-41— i [0077]= 3-37. (6"0 3-8i-i [0-054]= 3-783. (7) V 2-3532 + 5-404" = V 34740 = 5-894. (8) 0-88 X 7-31— 0-03 i^-Zi? = 6-433 — 0-143 = 6-290= 2 Dj. 7'3i (9) 6-i5 = °-374. (,o) |:^ = 0-859. 6-290 •'^ (11) Di = 0-945 03 = 2-972, HW phase = 246-1. (12) i^(K,) = 0-905,^^(0,) = 0-854, f„ = 1-196; July 20. (13) 3-9"^ 1.608= K/ 2-439 (14) ?^=, -049 = 0/ •*' 2-439 ^^ (15) 1-608x0-905 = 1-455 = ^1. (16) 1-049X0-854 = 0-896 = 01. (17) 7=28-23, i^(Mn) = 1-022; July 15. (18) 7-31- ('9) 7-31 X- 1'022 = 7-471 =Mn. <-) PS=--- PS-'* (21) 0949. (22) 3-736 X 0-949 = 3-545 =(' + «) Ms =1-02 M2; .-. 3-476 = M2. Nonharmonic quantities. 1-02(1-455 + 0-896) / 2-351 \ ,-.96x1601+1049 .= '-"H 2-^ ) = ' "^ X °-79' = °-8o7. 2353 X 0-807 = I -899 = HWQ. 5-404 X 0-807 = 4'36i = LWQ. [12 15 — 12 54] X 0-807 = — 31- — 31 + 12 35 ^ 12 04 ^ tropic HHWI. [13 31 — 12 54] X 0-S07 = + 30. + 30 + 12 35 = 13 05 = tropic LHWI. [652— 6 36] X 0-807 = + 13. +13+ 617= 6 30 = tropic LL-WI. [621 — 6 36] X 0-807^ — 12. —12+ 617^ 6 05 = tropic HLWI. [II -13— 7-31] X 0-807 = + 3-083. +3-083 + 7-471 = 10-55 = 00. [3-37- 7-31] X 0-807 = —3-180. —3-180 + 7-471= 4-29 = Sc. [10-07 — 7-31] X 0-807 = + 2-227. +2-227 + 7-471^ 9-7o = Gt. 14-94 — 9-70 = 5-24 = 51. 6-27 — i [7-47 — 7-31] = 6-19 = LW. [3-783 — 6-27] X 0-807 := — 2-007. — 2-01 + 6-i9^4-l8 = tropic LLW. [4-41 — 6-27] X 0-807 = ^ 1-501. — 1-50+ 6-19 = 4-69 = LLW. Epoch relations. (23) 17-5- (24) 17-5 X 1-098=19-2. (25) 19-2 + ( — 8-6) = IO-6, July 12. 0-91 1 X 10-6 = 9-7 = r (O, ; Ki). (26) —28. (27) —28x0-483 = — 13-5. (28) 2x246-0 = 492-0. (29) —13-5 + 492-0 — (— 1 1 -6) = 490-1, July 20. (3„) 3-4+.o-6-490i ^_^^g.^^,^^.^^^_, (3.) ^•^-'°t~'^°"' =-^48-6=.i.-4 = 0.°. 166 HOTTED STATES COAST AND GEODETIC SUEVEY. 40. Inferred amplitudes and epochs of components. Component. Amplitude. K. 0-272 Ss u 0-145 Nj Vi 0-194 Nj 2N 0-133 Nj p. 0-33IK, Q. o-:94 0i T, 0-0S9 S.2 M, i M2 (duration fall -^ (MS)4 2 Sa ,61>-2l) Epoch, M2°+i-098r(S2; Mj) M,° + o-544r(N,; Mj) Mj°-o-472r{N2; M^) M^°—i-oSgT{N^; M^) Ki" — 0-082 t(Oi; K,) K,"— I -6427(0,; K,) M.°+ 97Si-(S,; M,) 2 Mj" =F 90° * Kj°= Ma° + i-o8i (82° — M,°) L2° = 2M2° — N2° V2°= M2°— o-868(M2° — N3°) (2N)° = 2Ni,° — Ms° • P,°= Ki° — o-o75(Ki°— 0,°) Q,°= K,°— 1-495 (K,°—Oi°) Tj° = Ms° + o -960 ( Sj" — M2°) M4''=2M,j°=F9o°» 41. Collection of results, Sitka, Alaska. From 29 days* high and low water, July 1-29, 1893. From harmonic analysis of hourly ordinates i year, 1893-94. Amplitude. Epoch. Amplitude. Epoch. Feet. Feet. K, 1-46 122 I -51 125 K, 0-29 32 0-32 20 U 0-12 28 0-31 35 U, 348 3-4 3 -58 2-6 N, o-8o 339 0-69 338 2N o-n 3«4 0, 0-90 III 0-91 106 Pi 0-48 121 0-46 123 Qt 0-I7 106 0-14 106 s. I -05 30 I -14 34 V; 0-16 342 o-o6 295 From 29 days' high and low water, July 1-29, 1893. h. m. Feet. HWI 10 Mn 7-47 LWI 6 17 Gc 10-55 Tropic HHWI — 21^ So 4-29 Tropic LHWI 403 Gt 970 Tropic LLWI 6 30* SI 5-24 Tropic HLWI 6 05 a Sg 9-59 Age of phase inequality 26 Np 5-52 Age of parallax inequality 45 Pn 8-67 Age of diurnal inequality 10 An 5-83 HWQ 1-90 LWQ 436 LW (on staff) 6-19 Tropic LLW (on staff) 4-18 Mean LLW (on staff) 4-69 * Use iJJ^yg/ sign when duration of fall ^ 6''-2i. The determination of M4 m this manner is necessarily rough. EEPOKT FOE 1894— PART II. 167 DATUM PLANES. 42. To determine a plane of reference, having a given definition icith respect to the tide, when only a few observations are available. It is here supposed that observations, or more likely predictions, simultaneous with the observations in question are given for a neighboring principal station having a similar type of tide; it is also supposed that the observations or predictions at the principal station have been reduced to a plane of reference having the required definition. Determine from the observed heights at the subordinate station a plane approaching the one which definition would require. Call this the approximate plane and let h denote its height above the required plane. Find a range of tide by taking the mean of all observed low waters from the mean of all observed high waters. Let H denote the height of the tides at the principal station which correspond to the tides used in determining the approximate plane at the subordinate station. Let ^ _ o bserved r ange at subo rdina te station ,. ,q, corresponding range at principal station ' ' then h = r H. (141) The more nearly r remains constant, the better .the selection of the neighboring principal station. As a general rule field parties are supplied with tide tables giving predictions for the time in question. I^ow, unless the predicted heights are referred to a plane of reference having the same tidal definition as the one to be determined, they must be reduced to such a plane by adding a constant which we must suppose to be given in the tide tables. To dispense with this labor, predictions and soundings should be referred to planes having uniform definitions over a con- siderable area. To avoid confusion, which is sure to result near the limits of each such area, one definition should be used the world over. 43.* To determine mean sea level from one month's observations. — Use the period of 29 solar days. Take the mean of average high and average low water, or the mean of the hourly ordinates if a continuous record be available. To determine mean low water from one month's observations. — Use the period of 29 solar days. Find the mean range of tide and mean low water for this period. Eeduce this range by the factor F (Mn), Table 14. Depress the observed mean low water by ^ (mean range — observed mean range). (143) The value of (Ki + Oi)/M2 which is needed in Table 14 may be inferred from some neighboring station with sufficient accuracy for the present purpose, or it may be found by means of § 44. To determine mean lower low water from one month's observations. — Use the period of 27 solar days, i. e. a declinational month. Mark the lower low water of each day, and take the mean of the values so marked. Subtract this value from mean low water as determined from the 29-day period, and multiply the result by the factor 1-02 Fi, Table 32. Depress the cori-ected mean low water by the quantity just obtained, and the result will be mean lower low water. This plane of reference is unsatisfactory in accurate work, because no one reducing factor, like Fi, applies well to all stations. This is especially true where the low water inequality (LWQ) is small in comparison with the high water inequality (HWQ). 44. To determine the harmonic or Indian tide plane. This plane is, by definition, M2 -f Sa -1- Ki + O, (144) *To correct the planes of reference mentioned in this paragraph for the annual and semiannual components, depress the observation result by Sa cos ( /i — Sa° ) + Ssa cos ( 2 /i — Ssa° ) (142) where h is the mean longitude of the sun, Table 29. The amplitudes and epochs of these components may be taken from a neighboring station. 168 UNITED STATES COAST AND GEODETIC SUEVEY. feet below mean sea level as determined from hourly ordinates. Usually it nearly coincides with what might be called the tropic lower low water springs. At stations where harmonic analyses are available the depression of this plane below mean sea level becomes known at once. Where harmonic analyses are not available a month's observations upon high and low water may be used, as indicated below, for determining the required plane : Select a time near each extreme declination of the moon when the sum of the high and low water inequalities is a maximum. Denoting these inequalities by HWQ and LWQ, the tropic range of the diurnal wave for the series is, approximately, 2 D, = VHWQ^ -+rLWQ* ; (145) .-. K, + O, = Di X J', , (146) Table 32. From the observations determine spring and neap ranges, denoting them by Sg and Np; then S2 = 4 (Sg - Np) A.02 + 0-04 ^^^'^^J"^^' ) F, - 0-01 Mn . (147) M2 is obtained directly from Mn; i. e. the mean range from first reduction x F (Mn), Table 23. The quantities M2, S2, and K, + Oi, taken in connection with mean sea level as determined by the preceding paragraph, determine the required plane. This plane is in quite extensive use under the name of low water springs or Indian low water springs. It has the convenience of being so low that comparatively few low water heights are negative. 45. Stations where the tides are usually diurnal. When possible, mean sea level should be determined from hourly ordinates. If determined from high and low waters, they should be taken in pairs, so that each high water taken shall be accompanied by an adjacent low water having approximately the same displacement from mean sea level. Mean low water (semidiurnal) is here of no consequence as a plane of reference. The correction for annual and semiannual components is made as in § 43. The semidaily amplitude of tide when the moon is near the equator is, approximately, 1-1 M2 + correction for phase. Table 24. (148 S2 may be assumed to be 82 at neighboring station M2 at neighboring station ' From the observed tropic range we have [obtained by aid of formulae (65), (73), and (122)]. Having in this manner found M2, S2, and K, + O, from, say, one month's observations, the Indian tide plane becomes approximately known. feerel's expressions for inequalities in the tide. 46. If the lunitidal intervals and heights be classified according to an argument x whose period is that of some tidal inequality, the resulting interval and amplitude may, according to Fourier's theorem, be written B„ + M/ sin X + N't' cos x + M,/ sin 2 a? + JV,/ cos 2 as + . . . (7) (151) J Mn + -M( cos a; + JVi sin a; + Mu cos 2 a; + JT"., sin 2 a; + . . . (4) (152) where B^ denotes the mean lunitidal interval, Mn the mean range of tide, and i the characteristic of the inequality. These expressions may be written in the form 5„+5,sin(a7-f..) + -B„sin(2a?-e«)+ ... (7) (153) J Mn [1 + B, cos (X - a,) + E„ cos (2x-a,,)+ . . .] (1) (154) .^2 "•« x.c.s^^w.. . .s ^.c,.. ^^ where REPORT FOR 1894— PART II. 169 Jf.' M I 5. = ± VM,'' + Nr = —~, i?, = ± VM,/' + N,/' = cos f;' '■ -^ " « -r '« cose,i' • ■ ■ ' ,a\ MKf:\ jyr.' jyr / («) (155) taiifi = -,p, tanf„=-^, . . . ; M- M i Mn J?, = ± V.W + ^r =5i^ , J Mn i2„ = ± ^ilf„^ + N,? =^.— ,„ ■ ■ • , jVr ' AT (5) (156) tan«,=^ taix«„ = ^ .... In reducing tides, the observations are taken in groups. For this reason the coefficients -B,, Bf, and Ba, Bi, as determined above should be multiplied by the factors (a little greater than unity,) '"'-'^- and *"-*" 2 sin ^ (X, - x^) sin {x, - x^) where x^ and x^ are the values of x at the two limits of the group of observations. fj and ff, denote the epoch or lag of the principal term in the expression for any inequality. These divided by the " speed " of the inequality (i. e. the change in x per mean solar hour, say) give the "age of the tide" from times and heights, respectively. In all cases we shall assume that x corresponds to the hour of (local) transit of the moon, or to the moon's anomaly, longitude, etc., as the case may be, and not to the time of high water, low water, or a mean of these times; otherwise the value of x would correspond to a time a constant amount in advance or in retard of the moon's transit or other astronomical argument. Ferrel adopted this assumption in his "Discussion of tides in New York Harbor."* In earlier papers, notably in his "Discussion of tides in Boston Harbor"! and in his "Tidal researches,"! he has not corrected the e's and a's for the lunitidal interval. To make this correction, add the value of the lunitidal interval multiplied by the speed of the inequality when the f's and a's are affected with one subscript; use twice this speed when there are two subscripts, and so on. 47. To express FerreVs constants in the harmonic notation. For most inequalities this is not an easy matter, especially if much accuracy is required. The chief difficulties are, first, the inequality may be due to several components ; second, the argument X may not vary uniformly with the time. The work given below will illustrate some simple cases. Inequality due to a single component B. — From § 2 we have ^ ^^,sm{x-d) b :r~riB¥ ; r (is?) l + __co.s(.,-^) where e = BO-A°, X = (b—a) t, t = ^ ^ — - = the time of a high water of A reckoned a a from the conjunction of the fictitious moons of J. and jB, expressed in hours. If tan^'=/"^^(^-\ 1 + e cos (a?— (9) where e is a constant less than unity, then v' = e sin (x - d) - i ^ sin 2 {X - 6) + 'i e^ sin 3 (x - 6) - • ■ ■ p^ e^ . ^ . ^ (158) = e cos (9 sin a; — e sin (9 cos a? — 1^ cos 2 ^ sm 2 a? + ^ sm 2 (9 cos 2 a; + • • • • * United States Coast Survey Report for 1875. This discussion, because of its comparative clearness and consist- ency, will be referred to in preference to his other -works. The left-hand numbers in the above expressions refer to corresponding expressions in the "Discussion of tides in New York Harbor." t United States Coast Survey Report, 1868. t Ibid., 1874, Appendix. tan V = 170 UNITED STATES COAST AND GEODETIC SUEVEY. when V is small, and especially when a is nearly equal to b, u _ 1 r g2 2 - - J- |_ e cos (9 sin ,r - e sin 61 cos a? - 2 cos 2 ^ sin 2 a^ + I sin 2 ^ cos 2 a; + . . . 1 (159) = - [M/ sin X + N/ cos x + M-.' sin 2 a; + N,/ cos 2 a; + . . . ] (160) ^liere M>=-^cose=-^ cos (Bo _ ^o) A or " M,/ :^ ^ cos 2 ^ = ^^ cos 2 (£o _ ^o), i^./ = -|lsin2^=-^^sin2(£o_^o), Assuming the decrease of interval to be I = - [A sin {x - £,) + B,, sin (2 « - e„) + . . . ], (161) then -B; = ± V M,'- + if/2 = J^ ^ _ 57.3 ^ hours, cos «, J. a^ ' -B„ = ± V JW ,/2 + Jvr,/2 = ^^ = 57-3 ^:^JL hours, " ^ " cos &,• 2 J.^ a* ' Tyr/ tan fi = - ' ; 6;= (9 = 50 _ j^o^ a small angle, M tan e„ = - -^ii.'] f^, = 2 (9 = 2 (£o - J.o), a small angle, (162) (163) For a case as simple as this, the auxiliary quantities ilf/, N^'^ ... are introduced merely for the purpose of illustration. From § 3 we have y = A + B co^ {X — e) + B- t&n V sin {x - 6) — ^ A tan^ v + • • . (Xi + residual effects of components not involved in the inequality. (164) This may be written, without introducing if;., i^T;, . . . , 2/ = J Mn [1 + i?i cos {X - a.,) + i2„ cos (2 a? - a,>i + . , .] (i65) where 2 B B^ W ^' = Mi' ^" = ~ 2 A a? Mn' ' ' '' (1^^) «, = (9 = 50 _ ^o^ «„ r= 2 e = 2 [BO - AO), ... ' (167) When an inequality is due-to more than one component, the harmonic expressions for 5„ e^, Bi, «(, Bii, . . . become more complicated than those just given. Forms (151) and (152) are, however, sufficiently general to include such cases. REPORT FOR 1894— PART II. 171 The phase inequality. If A, B denote the components Mj, 82, and if there be no tide whose speed is a linear function of m2 and S2, we have ^2 ^2 Virvnr.c » _ K-T.Q 82^* (168) (169) £, = 82° - M20 , fii = 2 (82° - M2O), . . . ; i?, = ?A ' ij = 82' 82^ Mn " aMjma^Mn' " ' ' '. ai = 82° - M20 ■ an = 2 (82° - MjO), . . . When yU2 is taken into account, Bi becomes 2 (82 + Mi) Mn This should be divided by the inequality factor belonging to the phase reduction, § 29. The paraUax inequality. Whatever parallax inequality in time there may be is due chiefly to the transit selected in taking the intervals. The time coefficients B^, B^, ... of this inequality ought to be small, and so will not be considered here. In regard to the constants B2, 0-2 it may be noted that an approximate value for y is, § 3, 2/ = ^ + -B cos (a; — ^) + C cos (iT — «) + . . . , (170) and so M2* =Bcos6+ Gcosx JVj* = £ sin ^ + 6^ sin h a Mnf B2^ = Mi* + N/* = B'+G' + 2BG cos (0 ~ h) tan «2 =^J^^1±^-^^. B cos 6 + Gcos'k (171) When 0=MOTe=K^ 180°, i22 = 2 (S + 0) / Mn or £2 = 2 (B - 0)/ Mn, respectively. Here N2, L2 replace B, G, and differs from h by about 180°. B.= m^, (172) Mn ^ ' e = M.p - 1^2°, K = L20 - M20 ± 180°. Ui is found from the above expression for tan a^. The value of Bz should be divided by an inequality factor belonging to a parallax reduction where the mean, not the true, perigee is used. The declinational inequality. The time inequality whose period is a half tropical month, is due to the transit selected and to the sun's declination; it is usually small. The height constants B21 a-i have for their approximate values (see § 16) P_2X2_KiO, txz = K20 - M20. Diurnal tides. — In what has preceded no distinction has been made between the two high waters of a day, or between the two low waters. Let it now be supposed that the differences ' This is an auxiliary quantity, not a harmonic component. 172 UNITED STATES COAST AND GEODETIC SURVEY. between the high water heights and also between the low water heights have been found for each day. The amplitude of the diurnal wave for any particular day is approximately equal to J [ (HW inequality) ^ + (LW inequality) ^ ]4 (174) Having found this amplitude for each value of the moon's longitude (A), the constants involved in its expression may be determined. Let us assume the amplitude to be (see Ferrel's New York tides, § 22, 1. c.) Ki + Oi sin (2 A. - «/) + 0,i sin (4 A - «i,'). (175) The number of hours by which the maximum amplitude of the diurnal wave follows an extreme declination of the moon is ai' - 90°, = KiO - OjO, divided by ki — Oi or 1'098. This is the age of the diurnal inequality. CHAPTBE IV. TO REDUCE RESULTS TO THEIR MEAN VALUES. 48. All tidal components which depend upon the moon are assumed to have slightly variable amplitudes in order to better adapt them to each year of the luni solar eycle. The reason for this is that if they had fixed amplitudes many additional components would be necessary to take into account this irregularity of the tide; moreover, these additional components would have periods differing so little from the periods of the components with which they are now combined that a long series of observations would be necessary for their independent determination. Besides accounting for the small components due to the regression of the moon's node, the variation in the amplitudes may be made to take into account still other components whose speeds are nearly equal to the speeds of those components with which they are combined. The factors F and /, for reducing the amplitudes of the components obtained from a particular series to their mean values and vice versa, are given in Tables 10 and 13, which are based upon, or copied from, the tables given in Baird's Manual for Tidal Observations. In Table 14 are given F for the mean range of tide, for Kj + Oj, and X, based upon the factors of Baird's manual and the mean values of the coefficients as given in Darwin's report (1883). Table 14, for reducing the mean range of tide, may be tested by means of the observed ranges at Boston for the years 1848 to 1865, United States Coast Survey Eeport, 1868, page 81, Table VIII, last column ; also atlfew York for the years 1856 to 1874, United States Coast Survey Eeport, 1875, page 198, Table V, eighth column. The ratio (Ki + Oi)/M2 is about 0-2 at Boston and 0-3 at Few York. 49. The following approximate relations will suffice for reducing reduction results to their mean values. In fact, a consideration of the nature of these relations, together with the nature of the ^'s of the components, of Mn,Ki + d, and X, shows that the left-hand part of these relations lies very near to its mean value when the quantities composing the right-hand part have their respective mean values. To reduce a quantity represented by the left-hand part to its mean value, observe the corresponding increase, say, in the right-hand part and apply it to the given quaiitity, adding or multiplying as the case may be. Sg = Mn + 2 S2. (176) Ifp = Mn - 2 S2. (177) Pn = Mn + 2 (N^ - L^). (178) An = Mn - 2 (N^ - U). (179) HWQ, LWQ, proportional to K] + Oj. (180) H"WQ*= 2Pi [(1 - 2 X) cos X 180° +- sin X I8O0] (I8I) where cosX180o = l^Q. Mean duration of rise or fall ~ 6''.21, proportional to M4. (182) Lunitidal interval for the greit or small tropic tides -^ mean lunitidal interval, proportional to K, + Oi. (183) Duration of rise or fall for the great or small tropic tides ~ mean duration of rise or fall, propor- tional to K] -f Oi. • (184) Gc = Mn + i HWQ + i LWQ. (185) Sc = 2 Mn - J HWQ - J LWQ. (186) Gt = I Gc + i Mn. • (187) SI = 1 Mn - I Gc. (188) Depression of tropic LLW or mean LLW below mean LW, proportional to LWQ. (189) ' This supposes HWQ < LWQ ; when LWQ < HWQ, the low water values should replace the high water. 173 174 • UNITED STATES COAST AND GEODETIC SUEVEY. If K] and Oi are each known, the variation in K, + d should be found by means of Table 10 instead of Table 14. Quantities proportional to K, + Oi are obviously reduced to their mean values by the factor Ki' + Oi' (^'^"^ where the primes indicate values at a particular time as distinguished from mean values. For a short series Pi combines with Kj, and the factor becomes Cn K/ +Oi' (^^^> to be computed by aid of Tables 10 and 31, when Ki and Oi are known ; otherwise it may be taken equal to 1-02 F^, Table 32. CHAPTER V. ON THE CLASSIFICATION OF TIDES. 50. For places where continuous records of the tides have been obtained for any considerable length of time, it is usually possible to make predictions based upon the analysis of such observa- tions. But any tide table would soon become too bulky and expensive if predictions were to be given in detail for every place where forecasts are desired. It therefore becomes important to find a classification of tides, with the object of referring all stations having similar tides to one principal station where full predictions are given. The following assumptions implied in the proposed classification hold true, as regards the principal components, for nearly every station where an analysis has been made: First. The intervals of the ^^^/ypn™^ components are approximately equal to one another at any particular station. Second. The amplitudes of the ^^^d^^rnal^^^^^ semidiurnal components bear approximately the same ratios to one another at all stations. Granting the truth of these assumptions, the form of the tidal wave depends almost wholly upon (a) The form of the wave composed of the tropic diurnal wave and the principal lunar semi- diurnal component; that is, upon (Ki -f d) / M2 and M2° — K]0 — OiO. (b) The ratio of the amplitudes of the principal solar and lunar semidiurnal component; that is, upon S2/M2. (c) The form of the lunar semidiurnal wave; that is, upon M4/M2, 2 M2° — M4O, Me/Mj, 3 M2° — MgO, where, as is nearly always the case, M4 and Mg are the principal harmonics of M2. These quantities being equal at two stations, the tides at the one may be inferred from those at the other. Not only are the times and heights of high and low water comparable, but also the height of the sea or tide at any intermediate hour. When only high and low waters are to be compared, the quantities in (c) need not be alike at both stations unless (K + Oi)/M2 is a com- paratively large quantity; in which case the form of the lunar semidiurnal wave should be approximately the same at both stations. 51. If nonharmonic constants are used the quantities upon which the form of the tide wave depends are (a') The ratios of the high and low water inequalities in height to the mean range of tide, together with the sequence of the four tides of a day; that is, upon HWQ/Mn, LWQ/Mn, and whether the order of the tides is from higher high to lower low, or from lower low to higher high, the moon being far from the equator. (b') The ratio of the semimensual height inequality to the mean range of tide; that is, (Sg-]srp)/2Mn. (c') The form of the lunar semidiurnal wave. The quantities here selected for determining this are the duration of fall and the heights of the sea referred to high and low waters expressed in terms of the mean range ; that is, LWI — HWI, and heights of the sea or tide at various hours before and after high and low water expressed in terms of Mn, when the range has its mean value. It will be observed that (a'), (b'), and (c') are nearly equivalent to (a), (b), and (c), respectively, in fixing the form of the tide wave. Whenever the quantities in (a'), (b'), and (c') are equal at two stations the tides are comparable throughout. If only high and low waters are to be com- 175 176 UNITED STATES COAST AND GEODETIC SURVEY. pared, the quantities in (c') need not to be alike at both stations unless HWQ/Mn or LWQ/Mn is a comparatively large quantity; in which case the form of the lunar semidiurnal wave should be approximately the same at both stations. At stations where the tide is usually diurnal the height inequalities are not easily determined. The form of tide may be determined from the duration of the (great) tropic range, together with the rise of tropic HHW and the fall of tropic LLW reckoned from mean sea level. The amount of rise compared with the amount of fall shows whether HWQ > LWQ. 52. The sequence of the tides is determined by the following rule : If the HW phase, i. e. J (M20_KiO — 0,0), lie in the first or third quadrant, higher high precedes lower low ; if in the second or fourth quadrant, liigher high follows lower low. If twice the HW phase is used, it will lie in the first or second semicircle according as higher high precedes or follows lower low. Having found that the tides at two stations are comparable, the next step is to give rules for referriug the one to the other. The times. — The quantities which must be applied to the times of the tides at the first station to obtain the times of the tides at the second station may be expressed thus : Difference for HW= (HW)„- (HW),= (HWI),, - (HWI), + S,- 8„ + l^(L„-L,)+n(12^ 25"), Difference for LW = (LW)„- (LW), = (LWI)„ - (LWI), + 8,- S„ ^^^^^ where L„ L,, are the longitudes of the stations and S,, 8,, the longitudes of the time meridians, all expressed in time and reckoned westward from Greenwich, n (12'^ 25™) is such a multiple of a half lunar day as will cause the diurnal inequality at the first station to properly reappear at the second. As the form of the tide wave changes slowly from one day to another, n can be so taken as to allow for the change in date which occurs in crossing the Pacific Ocean, at the same time keeping the tidal differences between the limits — 12'' 25"° and + 12^ 25™. When the tidal waves at the two stations are of the same form, the time "difference for HW" is equal to the "difference for LW;" in fact, all like phases of the tide at the two stations are directly comparable by means of this time difference. The transit. — Rules for computing the high and low water establishments and for determining the sequence of the tides from harmonic constants alone have already been given. The question which now arises is, To which transit must a highwater interval (whose value is nearer M2°/28-98 than to M20/28-98 Jr 12'' 25™) be applied in order to obtain a higher high water when the moon's declination is "°''+|!^ The answer is, ^^Im^ev *'"^°^''' ™"®* ^^ ^®^*^ if the HW phase lies in the first or fourth quadrant, and ^ ower ^^ ^^ ^j^^ second or third quadrant. In order to obtain a lower low water when the moon's declination is g°^+jj' ^ lower *''^'^^''i* must be used if the HW phase lies in the first or second quadrant, and ^^ ^T,p^gj. if in t^e third or fourth quadrant (supposing the low water interval to be nearer M20/28-98 + 6" 13™ than to M20/28-98 + 6" 13™ ± 12^ 25™). These statements enable one to decide whether n in the above equations is odd or even. If the intervals at the two stations refer to the same transit^ then n = o or some even integer, as — 2 or + 3; if to opposite transits, n = some odd integer, as — 1 or + 1. 53. The following rules are-here collected together for convenience of reference: Sequence of tides. 0° < M2O — KjO — OiO <180o, HHW precedes LLW; I8O0 < M2° — KiO — 0°! <360o, LLW precedes HHW. Greater height inequality. - 90° < M20 - KjO - Oi° < 900, HWQ > LWQ; 90° < M20 - K,° - OjO < 270°, LWQ > HWQ. EEPOET FOR 1894— PAET II. 177 HWI takeu ap- proximately equal to MaO/29 hours. LWI taken ap- proximately equal to HWI + 6^ 13°-. The transit. -90o< J(M2O-K,0-0iO)< 90O 90° < i (M^o - KjO - OiO) < 270° 0° < iCM^o - KjO - OjO) < 180° 180° < i(M.2° - KjO - 0,o) < 360° For HHW use upper north, or lower south; lower north, or upper south. For LLW use upper north, or lower south; lower north, or upper south. For LHW use lower north, or upper south; upper north, or lower south. or HLW use lower north, or upper south; upper north, or lower south. Intervals applied to riorr'norTh'^;: upTer Ztht--*' ^^^ ^^ marked ^. At stations where J wQ 5 9 p" t^® order of ^^^ waters may be reversed even when the moon is farthest from the equator. See § 21. The heights. — If the planes of reference at the two stations have the same definition with respect to the tide, corresponding heights at the two stations differ only by a constant factor. This factor is usually taken as the ratio of the mean ranges. Where evanescent tides frequently occur, the ratio of the (great) tropic ranges or of the great diurnal ranges, should be used instead. If the planes of reference have different definitions at the two stations, the heights at one of the stations must be properly reduced before applying the factor. The height differences are Difference for HW= C j^^j" - D + (Mn),, - (Mn)„ Difference for LW= c |^°j" - D, (193) where D is the depression of the plane of reference below mean low water at the principal station and G the depression below mean low water at the principal station of a plane there having the same definition as the plane of reference at the subordinate station. When the two stations are far apart, it will generally be necessary to allow for the long period components if absolute heights of the tide are required, 64, Tides comparable by inversion. When the tide at a subordinate station is similar to the one at the principal station inverted, HW LW the J ™- at the subordinate station is proportional to the ^-^ at the principal station, the plane of reference being mean sea level. To ascertain if such a comparison can be made, find the quantities in (a), (b), and (c). The procedure is as in the case of tides directly comparable, except that M2° — KjO — OjO and 2 M2O — M4O should no longer have the same values at the two stations, but each should be 180° greater or less at the one station than at the other. Using nonharmonic criteria, LWQ at the one station should be proportional to HWQ at the other. The duration of fall at the one should equal the duration of rise at the other. 55. To infer harmonic constants through the comparison of nonharmonic constants, and con- versely. Supposing the tides to be simUar at two stations, we have for any short period component G {C%, = {GO), + c (HWI),, + (LWI)„ - (HWI), - (LWI), , (194) (Mn)„ (195) S. Ex. 8, pt. 2 12 178 m^ITED STATES COAST AND GEODETIC SUE VET. If tlie tides are not exactly similar, i. e. if the quantities (a'), (b'), and (c'), § 51, differ slightly at the two stations, then, instead of equation (195), we have (G-^). = (C^), X ^(5J^^^'; (196) where G^ denotes any lunar semidiurnal component; where G^ denotes any solar semidiurnal component, including lunisolar K2; (G) -(G) -.- (H WQ)„ + (LWQ)„ where d denotes any diurnal component; (GA =iGA _^ (LWI),,-(HWI),,^6''13°- ^ ''" ^ *'' ^ (LWl),-(HWI),=p6''13- (199) where C4 denotes any quarter diurnal component; and where the upper or lower sign is used according as LWI is taken greater or less than HWI. For inferring nonharmonie constants through the harmonic, we have (HWI)„ = (HWI), + (^^°)// - (M^°)/, (200) IB 2 (LWI)„= (LWI), + (M^°)// - (^a°)/, (201) ni2 (Mn)„ = (Mn), x <^'. (202) If the tides are exactly similar, Mn in the equation last written may be replaced by any other quantity measurable in feet, as HWQ, Sg, etc. If, on the other hand, (a), (b), and (c), § 50, differ somewhat at the two stations, then the following equations should be used: (HWQ)„ = (HWQ), X ^^^'^+^'^>^^ (203) (LWQ)„ = (LWQ), X J^^^. . (204.) (I^P)„-(Np).x|-|j^g-". (206) The deviation of a tropic interval from the mean is obtained by multiplying the value of this deviation at the first station byl-f i^-?^ \ • ^^^ ^^^^ connection see § 49. (IS-l + Uij, Long period components are taken from a neighboring station and without alteration. If the nonharmonie quantities at the second station are not properly corrected for the effect of tidal (not accidental) inequalities, they should be used along with the same quantities obtained simultaneously and in like manner, at the first station, using either observations or predictions. If the nonharmonie quantities are affected with accidental inequalities, it is important that the two stations be situated near each other, and that simultaneous observations be used in making inferences. EEPOET FOR 1894— PART II. 179 56. Gotidal lines. A cotidal line is an assemblage of points on the earth's surface where tides occur at the same absolute time. The number of each such line is usually taken as the lunar time (i. e. the lunar hour after upper or lower transit) at Greenwich when HW occurs at stations along the cotidal line. If solar hours are used, reckoned, of course, from the time of the moon's transits, each period of cotidal lines will consist of 12'42 hour lines instead of 12. The cotidal lunar hour of a place whose west longitude in time equals L is 0-966 HWI + L, or more exactly, 0-483 (HWI + LWI =p 6-210) + L; (207) while the cotidal solar hour is HWI + 1-035 L, or more exactly, HWI^+_™T6-"210^j.,3,^_ (208) The j™„j, sign is used when the low water interval is taken ^ , ^ than the high water. The cotidal lines relating to the semidiurnal portion of the tide are quite distinct from those relating to the diurnal. The tropic high and low water intervals of the diurnal wave are obtained thus (see § 16) : HWI(«)=5l^+2^ (209) LWI (a) = HWI (a) ± 12i^ 20°". (210) To change the transits add or subtract 121^ 25™. The cotidal lunar and solar hours for the diurnal wave become 0-966 HWI (a) + L (211) and HWI (a) + 1-035 L, (212) respectively. CHAPTEE YI. PREDICTION OF TIDES. TIDE-PREBICTING MACHINES. 57. British tide-predicting machines.* The object of these machines is to continuously sum the series S, +A cos {at + a) + B cos {bt + /?)+ . . . , (213) or to trace the curve y = S„ + A cos {at + a) + B cos (6« + /S) + . . . , (214\ thus giving the height of the sea or tide at any particular time. Among the earlier mechanical devices for summing a series of cosine or sine terms, where the angles vary uniformly with the time t, or with the angle 0, may be mentioned the following :t In the British Association for the Advancement of Science Eeport for 1845, Cambridge Meet- ing, Rev. P. Bashforth mentions a machine for describing the curve p = A cos {a d + a) + B COS {b d+ ^) -{■ . . . (215) where A, B, G, . . . a,b,c,... are real quantities, integral or fractional. He applies it to the fludiug of real roots of the algebraic equation jp„r' + ^, a;"-i + . . '. +p^ = (216) where « is a positive integer. By writing sn = cos d this equation becomes of the form q, cosn 6 + q^ cos n — 1 9+ . . . +g„=0. (217) The values of cos 6 at the intersections of p = lc + q^cosn (f + q^ cos n^^ 6 + . . + q^ and p=]c are, obviously, the roots required. In the Proceedings of the Eoyal Society of London, Vol. 18 (1869), pages 72, 73, Mr. W. H. L. Eussell shows how the curve y = Acos {at + a) + B cos {bt + /3) +.. . (218) may be described mechanically. From his diagram it will be noticed that the mechanism is " sub-, stantially the same as that of the tide predictor" (No. 2 or 3). In the Minutes of Proceedings of the Institution of Civil Engineers (London), Vol. 65, page 16, a desiTiption is giveu of a wooden model designed by Sir William Thomson and constructed for him in 1872-73. This model made provision for eight components, and its plan was very like tide ])redictor No. 1. *For an account of these machines see the Minutes of Proceedings of the Institution of Civil Engineers (I^ondon), Vol. 65 (1881), pages 15-72. For "predictor No. 1" see Thomson andTait's Natural Philosophy. For the Indian predictor see also Proceedings of the Eoyal Society of London, Vol. 29 (1879), pages 198-201, acd especially The Engineer (London) of December 19, 1879, where drawings of the machine are given. tSee Popular Lectures and Addresses by Sir William Thomson, Vol. Ill, page 185. 180 REPOET FOE 1894— PAET II. 181 Tide predictor N'o. 1. — This predictor, designed by Thomson, was constructed in about 1876. It contains ten components and is described in Thomson and Tait's Natural Philosophy, Part I. It differs from Noa. 2 and 3 in the number of components represented and in having cranks carrying pulleys Instead of pins working in slots. This machine is now kept at the South Kensington Museum. Tide predictor No. 2. — This machine was designed by Mr. E. Roberts and is now worked under his direction in preparing the " Tide tables for Indian ports." The number of components is twenty. Tide predictor No. 3. — This predictor, designed by Thomson, differs from No. 2 in the number of components, which is fifteen or sixteen instead of twenty, and in the direct character of the gearing. It was constructed in about 1881. General description. — Upon one or more shafts, driven by hand or by clockwork, are fixed a number of wheels which mesh into other wheels, causing the latter (or wheels moved by them) to revolve with angular velocities having given ratios to the angular velocities of the shafts. These ratios are taken, as nearly as possible, proportional to the speeds of particular tidal components. Eigidly connected to these wheels are cranks carrying pulleys, or pins working in slots, imparting to rods carrying pulleys rectilineal harmonic motions. At one end of the predictor a chain or flexible wire is made fast; thence it is laid alternately over and under the pulleys. To the free end of the wire is attached a marking point, which when moved transversely to the line of motion of the paper roll traces the tidal curve. This machine evidently sums the series (213) or traces the curve (214). 58. Maxima and minima tide-predicting machine. This machine, designed by the late Prof. William Perrel, of the United States Coast and Geodetic Survey, was constructed in the years 1881-82. A description is given in the Report for 1883, pages 253-272. Most of the predictions published by the Survey since 1885 have been made upon it. The machine provides for nineteen components. The amplitude of the principal lunar component is introduced into the predictor by shortening one of the summation chains about to be mentioned. Its argument is indicated by a hand on the face of the machine making two revolu- tions for each lunar day. Generally speaking, the speed ratios in the machine are not those of the components themselves, but are the differences between such ratios and the speed ratio of the principal lunar component. The smaller short-period components are each provided with two cranks fixed at right angles to each other, one for cosine terms and one for sine terms. Each set of cranks carries pulleys, and the resultant effects are obtained by means of two summation chains- One chain is made to impart a vertical motion to a slotted horizontal strip of steel, while the other imparts a horizontal motion to a slotted vertical strip. These slots intersect at the center of the machine when all short-period components are set at zero amplitude. In general, two settings of the machine are required, one for times and one for heights. When set for times, the direction of the intersection of the slots from the center of the machine shows (by aid of an oscillating needle) how the lunar hand must be. directed* at the time of a high or low water. When set for heights the distance of the intersection from the center t is, when multiplied by the cosine of the angle between the lunar hand and the oscillating needle,| the elevation or depression of the sea or tide from mea.n sea level. The machine gives the times of high and low water directly; no tidal curve is traced, nor is any computation required. Approximate values of the heights are also given directly, but the determination of their exact values or of the height of the sea at any intermediate time requires a little additional labor, viz., the setting and reading of an extensible cosine scale. 59. A proposed tide predictor. The machine briefly described below will carry on simultaneously four operations, viz. : 1. The tracing of the tidal curve. 2. The marking upon the axis of the curve the times of the maxima and minima. ■ * This angle reckoned counterclockwise from the vertical is the i* of § 60. t This distance is the iJ of ^ 60. (This angle is at -\- a -\- s oi § 60; at times of high or low water it becomes s — u. 2 is always the angle made by the oscillating needle with the vertical reckoned counterclockwise, the machine being set for heights; at -f q: is the argument or angle of the lunar hand reckoned clockwise, as on the face of the machine. 182 UNITED STATES COAST AND GEODETIC SUE VET. 3. The exhibiting upon the face of the machine the times of high and low water. 4. The exhibiting upon the face of the machine the heights of high and low water, and the height of the sea at any given time. (1) In regard- to the accomplishment of the first of these objects nothing farther need be said, for a mechanism similar to that used in the British machines seems quite satisfactory. (2) Now let there be a set of cranks at right angles to the set used in producing the tide curve; for distinction the former may be called the time cranks and the latter height cranks. Each set has its resultant effect shown by the movement of the free end (or of a mark a constant distance therefrom) of a wire or slender chain. In each case the result might be shown in the form of a continuous curve. This, however, is not ordinarily wanted in the case of the time cranks. A projection upon the wire uniting their effects comes into contact with a pencil placed at mean sea level, causing a short mark to be made every time high or low water occurs. (3) A mark on the wire at a fixed distance from the projection just referred to passes before an opening in the face of the machine, and is so adjusted that when it crosses a given line or mark the time shown on the face is the time of high or low water according to the direction of the wire's movement. (4) The heights may be conveniently shown by means of a pointer moving along a circular row of figures outside the time dial. A clear mode of indicating the time would be to have the day of the month anc^ the hour of the day show through two small openings, while a single hand points out the minutes. In order to warn the opei'ator that a high or low water is close at hand, a bell may be made to ring just before each tide.* Imaginary roofs of equations, etcA — A purely mathematical use for a machine of this kind is the finding of imaginary roots of equations. Let Z=Az''+B0''+ . . . =0 (219) where the coefficients are general but the exponents are real quantities. Now, as z describes a circle whose radius is Ic, Z will for each value of arg 2: or ^ have a certain affix mod A Ic' [cos {a 6 -\- a) + i sin {a 6 + a)] + mod B y [cos (6 6* + yS) + i sin (& (9 + /?)] + (220) The locus of Z will be mechanically described by a point whose ordinate and abscissa represent the resultant effects of the height and time cranks each set to the same amplitudes. Such values of h and d as will make both sums zero give roots of the equation. 60. Equations relating to tide-predicting machines. The height of the sea or tide at any given time t may be expressed in various ways : y = Ha + A(iO% {at + a) + B cos {ht + /i) + . . . (221) = Hq + A co% at + E^cosht + . . . + lwa.at + Bs.va.U+ . . . (222) where = A = Ac.09. a , A= —Asm a;^ .-. A^{A^ + I-)^ tan a = - J . (223) y = So+M COS {at + a) - N sin {at + a) (224) where M = A + B cos {b — at + ^ — a) + G cos {c — at + y —a) + . . . , (225) N=0 + B sin (b^^lit + /3 — a) + sin {c - at -\- y - a) + . . . ; (226) y = Ho+ B cos {at +a + z) (227) ^^®^® AT r99S> B = {M^ + N')i, tan z = ^. ^^^"f z is evidently the angle of a right-angled triangle, whose sides are JIf, JT, lying opposite JV. [* This survey has undertaken the desv.ning and construction of a machine which will carry on the above operations. When compl°ted, a full description of the machine will he published.— July, 1895.] +The notation here employed is independent of the notation used elsewhere in this paper. where EEPORT FOR 1894— PART II. 183 The times of high and low water are roots of the derived equation — ^ = Aasin{at+a)+Bb sin {bt + p) + . . . =0 (229) = A a sin at + E b sin bt + . . . — {lacosat + Ebcosbt+ . . .) (230) where where A = A cos a, A — — A sin a ; — = 3 .: A = {A^ + A)i,ta.na= - ^--. (231) -^y :=a M' sin (at+ a) + aN' cos at + a) = (232) M' =A +— cos (iT^at + p — a) + -^cos{c - at+ y - a)+ ■ ■ ■ , (233) N'' = + ^ sin (6 - at + (i — a) + -~ sin (c - at + ^ _ «) + . . . ; (234) -^y = R' sin (a^ + a + M) = (235) i2 ' = (i¥'2 + JVr'2)i, tan « = C • (236) The angle m shows how much (in A degrees) the high or low water is accelerated because of com- ponents other than A. The times of the tide are therefore t' = nn-a-u ^237) where n is any integer, even for a high and odd for a low water, v and w of § 1 are values of u for these two cases. A GRAPHIC METHOB OP PREDICTING TIDES PROM HARMONIC CONSTANTS. 61. The prediction of tides, as already stated, implies the combining of a series of waves into (toe for the purpose of ascertaining the height of the tide at a given time or of determining the times and heights of high and low water. The computation of these quantities, while not difficult, is extremely laborious ; on this account mechanical tide predictors have been constructed. As these instruments are too expensive for general use, a method is here proposed whereby predictions may be made, without computation, from the constants used in setting a machine — in other words, from the harmonic tidal constants, the initial equilibrium arguments, and certain factors (/of Table 10). The apparatus consists, first, of a smooth board (about 3 by 4 feet), one section of which is divided into twenty-four equal spaces, representing the hours of the day; second, of a set of pairs of movable curves, drawn once for all for a given station, each pair provided with a "calendar" good for all stations and all time; and third, of two stationary cleats for holding the curves in place, and a number of movable cleats for indices — one for each pair of curves. 62. The equation of any component with its harmonics may be written y = J., cos [a^t 4- argo A; — A-p) + A2 cos (2 a{t + argo A2 — AjO) + A3 cos (3 a^t + argo M - ^3°) + . . . (238) where A, is the oscillation of longest period, or the fundamental, not necessarily a diurnal component. Of course, a curve representing this compound wave can always be replaced by as many simple cosine curves as there are terms in y. In fact, the only instances where it is generally 184 UNITED STATES COAST AND GEODETIC SUEVEY. advantageous to use other than simple curves are M2 and S„ with their respective harmonics. Strictly speaking, the amplitude of the harmonics of M2 vary faster on account of the change in the inclination of the lunar orbit to the earth's equator than does the amplitude of M2; but as the amount of this change is small in small quantities, no great inaccuracy will be introduced by making the whole M wave vary as M2 varies. It is customary to put argo A2 = 2 argo J., argo A3 =3 argo ^1 (239) The above equation assumes that one has the value of the initial arguments at the time ^ = 0, local time, midnight, say. In practice it is convenient to take the origin of time at the instant of midnight on a certain standard time meridian, 8, and to have the initial arguments made out for the meridian of Greenwich ; then one might alter the arguments in such a way as to adapt them to the longitude of the locality, L, and to the standard time midnight. But if, instead of altering the Greenwich arguments, one alter the epochs by the same amounts, with opposite signs, the same results will be attained and with the great advantage of being done once for all. The epoch of any component Aj, thus modified is %o = Ao + (15 i) - *,) L + a^{L- S), ^ Ao + 15pL- apS (240) where J) is such an integer that 15 jp is nearly equal to «,. When A^ denotes a diurnal component, then p = 1, 2, . . . according as A^ is diurnal, semidiurnal, et«. 63. Construction of a height curve. — For convenience in drawing the curve, suppose argo A^ to be zero ; then its equation with modified epochs becomes y = A, cos {a,t - Sl,°) + A^ cos {2 ait - ^2°) + A3 cos (Sa^t- %o) + . . . . (241) The position of the maxima of A, are given by the equation Sl^ 2_^ «! ^ «! ^ ' where n is zero or a positive or negative integer. At this time the phases of the harmonics are 2 SliO - SI20, 3 SljO - 8(3°, . . . , (243) while the positions of their respective maxima are given by the equations _ %° ,2n7r X _ ?l3° , 2«5 ^244^ 2^1 "^ ^^' 3 ffl, "^ 3 «! ' ^ ' The beginning of the curve should be %° degrees or Sti°/ai hours before the first maximum. It should extend over a trifle more than two or three days according as Ai is a semidiurnal or a diurnal component. Prom the beginning and along the axis, generally lay off two periods, leaving the first blank and dividing the second into degrees from zero to 360. The number equivalent to argo Ai when brought over the stationary zero hour line fixes the position of the curve for the day. Construction of a time curve.— The equation of the derived curve corresponding to (241) is -2/ = ^ia,sin(ai*-SliO) + 2A2a,sin(2a, -Sl2°) + 3A3a,sin (3ai-8l3°)+ • . . . (245) It will be noticed that the aSend^ng^ ""'^^^ ^^ *^^ derived curve will coincide in time with the ^^^i"^^ of the height curve. This is of course equally true for each element or component of minima " the curve. In ascertaining the magnitude of the coefilcients Ai «i, 2 Ai a„ 3 J.3 a, . . . relative to Ai A2 A3 . . . , it is usug,lly convenient to take the speed of M2 as unity, expressing other speeds in term of this unit. The derived curve is used in ascertaining the times of tide, and is drawn in a broken Une upon the movable strip upon which the height curve has already been drawn. Construction of a '^ calendar. "—It will now be supposed that arg„ A^ is given for only one day of each year, January 1. The change in the given arg„ A^ necessary to adapt it to any other REPORT FOR 1894— PART II. 185 ilay of tlie year may be assumed to be uniform, and the same for all years. This change is intro- duced by means of a " calendar " peculiar to the component and its harmonics. Anywhere at the left of the curves, a section of the strip one period in length is set apart for the days of the year. The positions of the dates are given by Table 4. January 1 will be located at the right or left edge of the period according as the speed of the component is less or greater than that of the solar component of approximately the same speed. It is sometimes more convenient to set apart two or more periods for the calendar. This will necessitate two or more blank periods of the curve, instead of one, before the period used in laying down argo A,. This means, of course, an increase in the length of the curve. Lunar nodal components, etc.— In order to avoid numerous small components, especially those dependent upon the longitude of the moon's node, the lunar tides are made to undergo small changes in argument which are not directly proportional to the time, and also small changes of amplitude. The former have been accounted for in Table 3 by the m of F„ + u, or arg„, where u has an irregular variation, but so slow that for the purpose of prediction it may be regarded as constant during a year. The variations of amplitude are given by the/'s of Table 10, and can be shown upon the curves by means of a scale of proportional parts of the mean values of the amplitude and ordinates of each component. But as the/'s vary quite uniformly with I (the inclination of the lunar orbit to the equator. Tables 1 and 13) it is generally sufficient to mark upon certain ordinates of the curve their increased or decreased values corresponding to changes in I. The time scale. — This scale consists of twenty- five equidistant lines drawn upon the stationary board underneath all of the curves. The strips upon which the curves are drawn should be sepa- rated from one another by narrow spaces, so that the time scale, or rather the hour lines, may be seen between them. 64. Directions for predicting. Find in Table 3 the value of Vo + u of each component for the given year and bring this number of the degree scale over the zero hour line. Set the various indices over the date Janu- ary 1. For any other day move the scales until the indices, as already set, come over the required date upon each scale. Suppose, in the first instance, that the height of the sea or tide is required at any particular hour of this day. Along the edge of a piece of paper made to coincide with the given hour line, mark the positive ordinates of the full curves (slightly modified to account for the varying obliquity of the moon's orbit to the equator) so that all such ordinates shall be added together. Along the same edge, but on the opposite surface of the paper, mark the negative ordinates, beginning at the height upon the paper slip reached by the positive ordinates and going in an opposite direction. The resulting difference is the height of the tide above or below mean sea level at the stated hour. If a lower plane of reference be preferred, its depression below mean sea level should be laid off upon the paper slip before marking the ordinates upon it. To find the time of high or low icater. — By glancing at the dotted curves of the principal components one can usually tell roughly where the algebraic sum of their ordinates approach zero. If a high water time is sought, select a descending node ; if a low water, an ascending. At two adjacent hours near the high or low water, lay off' upon a slip of paper the positive and negative ordinates of the dotted curves, properly modified for the varying obliquity of the moon's orbit to the equator. Now take any scale about three hours in length, divided to single minutes. From this lay off the two resulting heights, one hour apart. A straightedge will show where a line joining the two points just laid down will cut the time scale. The time thus determined is the time of high or low water with reference to either hour line. 65. It is obvious that the foregoing method 'is applicable to periodic phenomena, although the constituent periodic curves may have no resemblance to simple cosine curves. Such curves are constructed by means of component hourly ordinates obtained from the observations. These remarks have special reference to the prediction of tidal currents. 186 UNITED STATES COAST AKD GEODETIC SUEVEY. APPROXIMATE PREDICTION OF HIGH AND LOW WATER.* »)6. By means of tlie foUowing method predictions can be made without resorting to graphic processes, and without access to a tide-predicting machine. Moon's transits, phases, etc.— It the transits used are for the meridian of Greenwich (H) and expressed in civil time, they may be adapted to another meridian (L) and expressed in standard or prediction time {S) by adding L- S + 0-035 (L - E) (246) § 27. The times of the moon's phases, distances, and declinations are expressed in standard time by adding E - 8 to their Greenwich civil times. A further increase of t hours, the age of the particular inequality, gives the standard or prediction time of the conspiring or interfering to which the inequality is due. See sheet headed " Ephemeris." The tidal constants. — The constants are as follows : HWI, LWI, Aj, M2, S2, T (S2; M2), N2, T (N2; Ma) Ki, ^p, 0„ r (d; K,), _ 12r ffl,; M2) + 5r (Sa; M^) ^' 17 X 24-84 (=422-3) (247) A 2 is the mean amplitude of the tide with the effects of the diurnal components excluded. ^1° is the modified form of KjO, defined in § 62. HWI and LWI are so taken that their sum is zero, as nearly as possible. A 2, Mu, 83^ N2, Ki, Oi are reduced to their values for the year of the predictions by the factors/ (M2), / (M2), 1, / (M2), / (Ki), / (Oi), respectively, found in Table 10. 67. The "Ephemeris" and the sheet following it indicate the work preparatory to making predictions. The process of prediction indicated below consists of three steps, the determination of the semidiurnal wave, of the diurnal, and of combining the two thus determined. Port Townsend, Washington, is taken as an illustration because of its large diurnal components. Ephemeris. Port Townsend, Wash., 1895. h. Longitude of ephemeris (E) c Longitude of time meridian [S) I £—S=~i 00 00 00 Moon. Greenwich civil time. £-S + r Prediction time of conspiring or interfering. d. A. h. h. h. d. h. ([ Last quarter H New moon ]) First quarter February it March 16 24 4 13 17 -8+21=13 ii Feb. 17 2 " 25 6 Mar. 5 2 Apogee (Midtime) Perigee February March n 22 2 10 19 10 I -8+51=43 Feb. 24 14 Mar. 4 5 (for ?,.) h. d. h. Farthest S. On equator Farthest N. February March 19 26 S I 14 IS -8+16= 8 Feb. 19 9+15=20 " 26 22+14=27 12 Mar. 5 23+12= 6 11 * For other modes of predicting high and low water from harmonic constants, see an article entitled "On tidal prediction," by Prof. G. H. Darwin, Phil. Trans., Vol. 182 (1891), A, pages 159-229; and, by the same author, a portion of the article on " Tides " in the Manual of Scientific Enquiry (fifth edition), pages 75-90. See also Gezeiten- tafeln (Berlin), 1894. The eq^uations for passing from the A's of the components in these tables to their epochs are B^° = M2° — A -Bs° + (62 — mj) (0-0345 Ms,° + i). K,° = — A K,° + (ki — mi) (0-345 MjO + X), ^1° = Ki° — A B^° + (61 — ki) (0-0345 M,° + L), where the small letters denote speeds, and A SP, A Ki^ are written instead of A *, A k, of the Gezeitentafeln. EBPOET FOE 1894— PAET II. 187 The standard time (120° W.) of the moon's transits across the meridian of Port Townsend (122° 45') can be obtained from the table of transits, Pacific Coast Tide Tables, by adding ll". Port Townsend, Washington, 1895. constants. HWI LWI Ai!, M„ S„ r(Ss,; M,), N„ r(N,;M,), K„ ^,°, 0„ r (O,; K,). t^ Z iZW —244(a) 2-40 2-24 0-55 21'' of'-46 51I1 2f'-47 ISI°-S l''-4l 16 1 70 lunar days. Factors for the year i8gs- /•(K,), /(M,), F{Y.{), /(Ki), /(O,). 1-308 0-964 0-901 i-iio 1-179. CONSTANTS. Amplitudes for the year i8gS- A J, M2, Si,, Nj, K„ O,. 2-31 2-16 0-55 0-44 2-74 1-66 Amplitudes for obtaining height corrections. February 20 Sj (with Kj and T2) 1-25 xo-5S=o-69,Table3l or 33 March 2 " 1-32x0-55 = 0-73 Febi-uary 22 N2 (with vi) i - 1 6 X o -44 = o -5 1 , Table 34 March 10 " 1-17x0-44=0-51 February 20 K, (with Pj) 0-90 x 2-74=2-47, Table 31 March 2 " 0-80x2-74=2-19 February 21 Oi(withQi) 0-81 x 1-66 =1-34, Table 26 22 " 0-81 X " =1-34 23 " 0-81 X " =i'34 24 " 0-83 X " =1-38 25 " 0-86 X " =1-43 26 " 0-88 X " =1-46 27 " 0-89 X " =1-48 28 " 0-93 X " =1-54 Amplitude for obtaining time corrections. February 20 SjCwith Tj and solar Kj) 1-09 X 0-55 =0-60. Sa/ M2 = o-27, Table 31 March 2 " I'lox " =0-61. " =0-28 Acceleration. T (Oi; Ki) increased. February 20 K, by Pi -17° = — 68". — 17(— !?) = + is\ Table 31 March 2 " —14 = — 56. — 14(— W) = + i3 Time of 1^1 tides. ArgoKi (Feb. 21)=- 62°-S5= 4'' 09" Tables «i°=iSi -5 = 10 04 -[argoKi-ti°]= 5 55 Feb. 2i=K, HW JofaKiday = ii 58 17 53 Feb. 2i=KiLW AUXILIARY TABLES FOB THE EEDUCTIOiT AND PEEDIOTION" OF TIDES. 189 AUXILIARY TABLES FOE THE EEDTJCTIOl^ AND PEEDICTION OF TIDES. 189 190 UNITED STATES COAST AND GEODETIC SUEVEY. Table 1. — The principal harmonic componenU, Sym- bol. [U] M, [M,] M2 Ms Ml Me Ms Nj 2 N Oi 00 s, S2 S3 s, A, Name of compoaent, etc. Smaller lunar elliptic diumal Lunar elliptic diumal, second order Larger lunar evectional diurnal Lunar diumal Solar diurnal Luni-solar diurnal Lunar semidiurnal Solar semidiurnal Luni-solar semidiurnal Smaller lunar elliptic semidiurnal [Ferrer s L2] Smaller lunar elliptic diurnal [Ferrel's wj or Qi'] ■Principal lunar series Larger lunar elliptic, semidiurnal Lunar elliptic semidiurnal, second order Lunar diumal Lunar diurnal, second order Solar diurnal Larger lunar elliptic diumal Smaller solar elliptic ^Principal solar series Larger solar elliptic Smaller lunar evectional Variational Larger lunar evectional Speed per mean solar hour. Speed ratios. _ Synodic period.* i5/(m~c) Formula. r c Si _5i c c mj c k7 d. y+ir-m 15-5854433 1*03903 0*96244 0*53772 1-03619 13*71879 T-4 iy-tr+nft J 14*4920521 0*96614 I '03505 0*50000 0*96350 y-cr+a 14*4966939 0*96645 1*03472 0*50016 0*96381 2 (y---|-2 7, 27*9682084 1*86455 0*53632 0-96495 1*85946 14-76529 2y-3"'-ra'+2i) 28-5125830 1-90084 0*52608 0*98373 1*89565 31*81193 *The diumal components are synodic -with Mj, and the semidiurnal components -with M2. tThe first of these components is Ferrel's ki ; the second his m\ or Qi', and is the same as [Mj]. TT. S. C. and G. S. Report, 1878, App. H. EEPORT FOE 1894— PART II. 191 with their speeds, coefficients, ete. Sym- bol. Coefficients. CoeflE. ratios. Factors for reduction. Equilibrium arguments. Formula. Mean value. C M, C F V u J. J « sin z / •01485 ©'03269 ©■05600 Vf^^ _ ©■72147 /+/5+j-/-|-9o° y f=sin/cos'i/ W mc sin / CDS' i I ■00487 5-00512* <• 00708 O"OI072 o'onz7 o'oi559 ©■ 01836 o- 01930? 0*026695 F F = F(0,) = F(0,) /-^-;4 — 4J+2/ — 90° '+3/»-3-r->-9o° 2i-y 2I-V [K,] [KM K, (i+S «^ sin 2 / (i+J^i) ^ sin so. (i+i <^ sin 2 / + (i+* «fl -' sin 2 0, •18115 ■08407 0-39878 o" 18507 o* 68302 0*31698 ■?^([K,]) ^(Ki) = _o-72i47_^y) Sin 2/ = unity I -05628 /+/.+90° ^-f/S-fgo- — V Zero ■26522 0-58385 I 00000 (3in2 2/4-0-66962 COS v%\n2.I-\-Q.-Li2\6)\ [KJ (i+l^sin' / ■03929 o' 08649 o'i48i4 -'^([K^]) _ 0-15652 sin*/ ^t\^h —2 V [KJ (H-l ^i') -' sin! „ ■01823 o" 0401 3 0^06874 "F'^i" = unity. 2i+2A Zero K, (i-B ^') sin" /+(i+4 ^)") -^' sitf 0, ■°575" 0*12662 0-21688 2t+2A -2V" (sin* /+©■ 14527 0052 1/ sin" /+o^oo528)4 U ie ji-t2tanH/coS2i'Pcosi4/t •01257 '02767 ©■04739 ^(1-2) = /?'(M2) XH' 2t+2/ts—p-\-iZo° ^^ — 2V—R [.U] i e cos« J / •01257 0-02767 ©■04739 F(i\^\i = ?5^/=^(^"> 22f-(-2;4-J— /-fiSo" 2^ 21* § e sin" / ■00323 o'oo7ii ©■oi2i8 F ©■ 15652 sin"/ 2t-\-2k+S-/' — 2 C M, i« Jf4-3coS2/>|4sin/coi=i/t (■00522} j 01649 0*01149 0-03630 ©■olg68) ©■062175 Fm-i = F(,Oi) X t" t+A-s+go" j-^+e [M,] § e sin 2 / ■01485 0*03269 ©■©5599 /■([M,]) -0.72x47 _^g,) sm2/ t+h-s^pJt-go° — V Jti (i-f^cos«J/ ■45426 1*00000 1*71277 ^(M2) _ 0-91538 cos* 4/ Zt-\-zk — 2S 21-2V Ms t*^ ^ = »/=^(^^ 2^-j4-/+i8o'> 2^ — 2f "2 f i »!" cos* 4 / I^OT«COS*4/ (■00736* ^01094 5 ■01234* V01706 0*01620 0-02408 0-02717 0-03756 ©■02775? 004125S ©•©4653? ©■©64325 F(j^d 2/+44-4J 2f — 2J' 2^ — 2V * The lower of these two figures gives the value when the coefficients in the evection and variation have their full values as derived from ^"Tthe^cSScients of U and Mi are aporoJEimately expressed by the given formulae ; the true mean values are 0-01278 and 0*01531, jThlfirltof thesetwo numbers is tlie mean value of the coefficients of the tidey-^-^; the second applies to the Ude Mi, from y—tr—xs and y~-a-\-^. respectively, compounded 192 UITITED STATES COAST AND GEODETIC SUEVEY. Table 1. — The principal harmonic components, Sym- bol. Name of component, etc. Speed per mean solar hour. Speed ratios. Synod c period.* is/im'^c) Formula; c c si c c mj c k, o d. MK 31'-2 Mean value. "07827 '00422^ '00621 04136 '03643 Coeflf. ratios. C_ Mj '00929 '01367 _C Ki o'2g5ii o'oi59i( o'0234ii o '15595 o' 13736 Factors for reduction. .P(MK) = j''(2MK) = y''(MN) = .F(MS) = FiiMS) = F (2 SM) = /■ (Mf) = .P(MSf) = F(Mm) = 7?(Ssa) = F (Mj) X F (K,) F(M4) X ^•(K,) COS" 4/ :^(M,) ^ 0-91538 . COS* J/ " o' 83792 cos" 4 / cos-" i / - °'IS779 sin"/ COS*^/ o'753'6 I — I '5 sin"/ = unity = unity Equilibrium arj^uments. 3<-|-3/,_2:t. f9o° 2^ — 2*' — 3/+3/:-4i -90° 4f-4''+ 4/+4/'-Si+> 4^-4" ^i-\-2/i—as zf — 21/ 2/+4A-+r 4^-4'' 2^ — 2A-J-2J -2f+2l. 2..- -2f -2/4+21 -2(+2'' i-> Zero h Zero ih Zero C = Coefficient or theoretical amplitude of any component. M2= Coefficient or theoretical amplitude of the component Mg. K] = Coefficient or theoretical amplitude of the component Kp mean motion of sun mean motion of moon : c 07480 : i? is such that tan R -<,. 12 tan- 4 /coS2 P^ 4 cot^J /— cos 2P' ,V; see Table 11. 13 '369 see Table 8. v' is such that tan v' sin v sin 2 / 0*334811 -f cos V sin 2 / ; see Table 7. ^ = eccentricity of moon's orbit = 0*0549. ei = eccentricity of earth's orbit = o"oi68. 0) = obliquity of the ecliptic = 23° 27' "3, . mass ( mass c i of sun ^/mean dist. of moon\* . *: „_ i —^ X{ -rr- 7 ) =046035=-; ... of moon \ mean dist. of sun / 2 1722 j . Q is such that tan Q = ^ tan P; see Table 9. Q' = ( ; 1 ")*; see Table 12. \2 ."i + I ■ S cos 2 Py sin 2 V sin* / ^^2 5 + I ■ 5 cos 2 p.- v" is such that tan 2 v' o "072634+005 2 V sin^/ ; see Table 7. / = inclination of lunar orbit to the plane of the earth's equator; it varies between u— 5° 8''8 and' 10+5" 8' '8, i. e. from 18° i8''5 to 28°36'*i. See Tables 6 and 7. o t. . P= mean longitude of lunar perigee measured from the intersection of moon's orbit with the plane of the earth's equator. See Table 6. rf= earth's radius divided by moon's mean distance, i. e. the lunar parallax expressed in radians =o' 01659 = 60*27 \ = longitude in moon's orbit of the intersection of the lunar orbit with the plane of the earth's equator. See Table 7, V = right ascension of the intersection of the lunar orbit with the plane of the earth's equator. See Table 7. The semidiurnals have a general coefficient cos* A ; the diurnals, sin2 A; those of long period, J— § sin^A; andMs,cos'X. Adenotesthe latitude. *The lower of these two figures gives the value when the coefficients in the evection and variation have their full values as derived from. Lunar Theory. S. Ex. 8, pt. 2 18 194 UNITED STATES COAST AlTD GEODETIC SUEVEY. Table 2. — Dependence of component speeds upon certain astronomical quantities {epoch 1900). h- ° 2 X -^60 2 X ^60 Solar day =: 24-000 0000 = — ClA — ; Sj = — —^3 — ^ 30-000 0000 Sj solar day Lunar day =24-841 2024= ?_>) of the particular component (C) as -will adapt it to the tabular V^ + u- 196 UNITED STATES COAST AND GEODETIC SURVEY. Table 3. — Equilitrium arguments (Fo + u) at the midnight precedinfi January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with the elements used in computing them — Continued. Component. i860 1861 1862 1863 1864 i86s 1866 1867 1868 1869 J. K, 0. 242-65 1670 21398 347°3S 18-88 218-19 76°95 19-28 218-62 . 165-17 18-69 216-95 25171 16-99 213-19 35o°55 15-22 209-68 74-00 11-71 203-16 i57°o3 7-97 196-37 . 240-76 4-62 190-05 340-06 3-11 186-95 L. %' 229 00 22492 178-86 26-04 43-36 61-04 228-97 232-88 340-98 72-20 62-16 280-05 259-39 251-23 179-71 62-53 68-81 56-79 249-41 257-60 311-75 86-02 86-37 235-51 283-89 275-18 162-59 100-40 92-79 47-19 M3 »56-44 181-82 272-72 57-57 258-47 207-70 329-72 359-27 178-90 240-49 99-83 149-74 149-59 200-17 120-25 44-86 275-96 53-94 310-86 1603 24-05 216-44 116-07 354-11 122-73 216-16 324-25 18-46 291 -99 257-99 M, Ms 3-63 185-45 7-26 156-94 55-40 313-87 358-53 357-80 357-06 199-65 299-48 39-30 40-33 240-50 80-66 191-92 107-88 23-84 32-06 48-10 64-13 232-15 348-22 104-30 ^^33 288-49 144-66 223-99 155-98 87-98 2 N 0, 318-71 95-61 162-27 293-58 328-68 236-01 305-65 252-04 335-93 317-49 175-15 77-00 329-11 98-05 179-52 303-11 330-27 258-25 3f4-46 252-89 3-60 325-78 175-49 109-34 337-15 98-14 214-42 311-19 330-39 292-74 00 P, Q. 56-81 350-12 299-17 348-91 349-37 27I-I2 250-74 349-61 282-32 148-66 349-85 294-66 41-77 350-09 308-46 317-16 349-34 285-41 201-26 349-58 302-03 84-13 349-82 319-05 329-07 350-06 335-40 245-78 349-31 311-93 Si, 3 359-34 180-00 0-07 180-00 359-82 180-00 359-56 180-00 359-30 1 80 -00 0-03 180-00 359-78 180-00 359-52 180-00 359-27 180-00 359-99 180-00 Ssj 4, 6 oco 0-00 0-00 0-00 0-00 0-00 0-00 0-00 0-00 0-00 T. ft. 0-66 318-69 1-83 359-93 217-19 154-86 0-18 128-51 356-41 0-44 39-59 197-72 0-70 310-46 38-81 359-97 208-10 190-98 0-22 118-69 31-80' 0-48 29-26 232-60 0-73 299-87 73-44 0-01 197-55 225-64 MK 2MK 224-94 198-51 346-93 119-75 277-35 138-06 50-02 18-54 339-25 340-06 118-51 180-96 269-87 217-15 23-34 163-82 291-18 176-70 93-37 27-74 20-36 22-89 124-04 224-18 312-45 220-78 67-71 206-44 295-10 220-88 MN MS 2 MS 140-53 181-82 3-63 192-04 258.-47 156-94 304-92 359-27 358-53 57-31 99-83 199-65 169-27 200-17 40-33 219-07 275-96 191-92 330-49 16-03 32-06 81-86 116-07 232-15 193-31 216-16 72-33 243-19 291-99 223-99 2SM Mf MSt 178-18 37-27 178-18 101-53 326-45 101-53 0-73 227-40 0-73 260-17 125-83 260-17 159-83 21-13 159-83 84-04 299-45 84-04 343-97 188-83 343-97 243-93 77-40 243-93 143-84 327-33 143-84 68-01 246-52 68-01 Mm Sa Ssa Elements. 223-11 279-88 199-76 324-";i 280-63 201-26 53-61 280-39 200-78 Val 142-34 280-15 200-30 ues at Gree 231-06 279-91 199-83 ;nwich, mi( 332-85 280-66 201 -32 Inight begi 61-57 280-42 200-84 ining each 150-29 280-18 200-37 year. 239-01 279-94 199-89 340-80 280 69 201 38 h s P N 279-88 9-87 146-77 ■Values a 280-54 175-91 303-12 280-63 152-43 187-54 the middlf 280-56 218-66 283-77 280-39 281-82 228-21 : of each y 280-57 260-42 264-44 280-15 51-21 268-87 ;ar, or for ^ in 280-59 300-91 245-11 279-91 180-59 309-53 uly 2, at G g midnight 280-61 339-89 225-76 280-66 323-15 350-31 reenwich m for leap ye 280-63 17-21 206-40 280-42 92-54 30-97 ean noon f ars. 280-64 53-18 187-07 280-18 221-92 71-63 or common 280-66 88-74 167-75 279-94 351-31 112-29 years and 280-68 125-04 148-39 280-69 13387 153-07 It preced- 280-69 162-79 129-03 I Q R 26-60 177-95 355-92 25-15 201 -80 17-32 23-49 251-35 3-91 21-77 320-13 349-96 20-19 349-62 351-83 18-98 8-81 6-27 18-36 33-73 8-19 18-46 87-48 0-34 19-25 144-51 351-29 20-58 171-19 352-39 f V v' 2V" 351-24 350-34 353-19 345-78 349-21 348-17 351-75 343-07 348-12 347-06 351-11 .342-16 348-29 . 347-32 351-46 343-35 350-03 349-27 352-92 346-64 353-43 352-95 355-44 35 '-64 358-12 357-99 35871 357-68 3-22 3-45 2-21 4-00 7-64 8-20 5-33 9-84 10-61 11-43 7-58 "4-43 From §62 we have for the modified epoch ((£") go = C»+l5/X-f 5 where * is to be put equal 1,2, . . . according as C is diurnal, semidiurnal, e'c; c is the hourly speed of C; L, S denote the west longitude in hours of the station and of the time meridian used. The values of L and 5 should always accompany the work of analysis or prediction, thus enabling one to pass from (1° to C" or C» to (£» as the case may be. EEPOET FOE 1894— PAET II. 197 Tablk 3.— Equilibrium arguments ( F„+u) at the midnight preceding January 1 of each year, from 1850 to 1950, for the mendian of Gi-eenwich, together toith the elements used in computing tAem— Continued. Component. 1870 1 1871 1872 1873 1874 187s 1876 1877 1878 1879 t 67-07 iS5°-69 24S°-6i 35o°54 82-08 174-02 266-11 1219 103-89 '95 03 171 I -41 2-04 4-40 6-29 8-5. 10-86 14-12 16-15 17-73 Kj '8373 182-64 183-58 188-21 192-12 196-81 201-81 208-64 212-86 216-00 Ml 27095 100-60 307-85 139-35 296-92 121-83 334-74 170-64 331-40 151-77 281 92 II I -26 300-85 "9-34 309-36 J39-50 329-70 148-55 338-61 168-50 29771 201 -00 145-32 47-99 301-76 202-48 145-46 55-40 309-99 207-85 ^m'^ 28803 199-20 1 1 1 -67 13-03 287-13 201-62 116-27 18-77 293-03 206-72 Ms 32-39 133-02 233-88 310-59 51-88 153-30 254-77 331-84 73-18 174-35 M, 228-59 199-52 170-81 105-88 77-82 49-95 22-16 317-76 289-77 261-52 M. 64-79 266-03 107-75 261-18 103-76 306-60 149-55 303-68 146-36 348-70 Ms 97-18 39-05 341 -63 211-76 155-63 99-90 44-32 275-53 219-55 163-04 129-58 172-06 215-50 162-35 207-51 253-20 299-10 247-37 292-73 337-39 N, ■ 322-87 334-77 346-91 321-87 334-40 347-10 359-85 335-13 347-75 019 2 N 253-35 176-52 99-94 333-08 256-92 180-90 104-93 • 338-42 262-32 186-04 o, 34-84 135-57 ' 235-24 308-80 47-27 145-48 243-58 316-37 54-72 '53-47 OO 140-26 39-33 302 -04 234-76 '41-93 50-16 318-80 254-59 162-20 68-28 Pi 349 55 349-79 350-03 349-28 349-52 349-75 349-99 349-25 349-49 349-73 Qi 325-32 337-32 348-27 320-05 329-80 339-28 348-66 319-65 329-29 339-32 R. 359-74 359-49 359-23 359-96 359-70 359-45 359-19 359-92 359-67 359-41 Si, 3 180-00 180-00 180-00 i8o-oo 180-00 180-00 180-00 180-00 180-00 180-00 S2, 4, 6 0-00 0-00 0-00 0-00 o-oo 0-00 0-00 0-00 000 o-oo T, 0-26 0-51 0-77 0-04 0-30 0-55 o-8i o-o8 0-33 0-59 Xs 108-47 19-62 291 -00 189-56 101-37 13-32 285-32 184-23 96-09 7-78 /is 66-79 26817 109-78 262-86 104-91 307-08 149-31 302-75 144-84 346-76 V2 136-32 66-42 356-75 251-62 182-39 113-28 44-23 299-46 230-27 160-91 MK 34-11 134-42 235-92 314-99 S8-I7 161-S1 265-63 345 -97 89-33 192-07 2MK 63-07 264-62 105-71 256-78 97-46 298-09 138-69 289-56 130-21 330-97 MN 355-27 107-79 220-78 272-42 26-28 140-40 254-63 306-97 60-93 '74-54 MS 32-39 133-02 233-88 310-59 51-88 153-30 254-77 331-84 73-18 174-35 2 MS 64-79 266-03 107-75 261-18 103-76 306-60 149-55 303-68 146-36 348-70 2SM 327-61 226-98 1 26 12 49-41 30812 206-70 105-23 28-16 286-82 185-65 Mf 142-71 41-88 303-40 232-98 137-33 42-34 307-61 239-11 143-74 47-41 MSf 327-61 226-98 126-12 49-41 308-12 206-70 105-23 28-16 286-82 185-65 Mm 69-52 158-25 246-97 348-75 77-48 166-20 254-92 356-71 85-43 '74-'5 Sa 280-45 280-21 279-97 280-72 280-48 280-25 280-01 280-75 280-51 280-27 Ssa 200-90 200-43 199-95 201-44 200-97 200-49 200-01 201-50 201 -03 200-55 Elements. Val ues at Grec nwich, rai( Inight begi iming each year. A 280-45 280-21 279-97 280-72 280-48 280-25 280-01 280-75 280-51 280-27 s 263-25 32-64 162-02 304-58 73-97 203-35 332-74 115-30 244-68 14-07 P 193-73 234-39 275-05 315-83 356-49 37-15 77-81 118-59 . 159-25 199-91 Values at the middle of each ye ■ar, or for ^ ins ulv 2, at G I midnight reenvvich n for leap ye. lean noon f irs. or common years and i It preced- i>. 280-71 280-73 280-75 280-76 280-78 280-80 280-81 280-83 280-85 280-87 P 202-16 243-03 285-12 328-07 11-52 55 -.30 99-27 143-18 186-77 229-88 N 109-71 90-38 71-03 51-67 32-34 13-01 353-66 334-30 314-97 295-65 T 1 22-22 11-at. 2e-i:7 26-Q7 27 -03 28 -AQ 28-1:7 28-18 27-52 ofi'C\1 198 UNITED STATES COAST AND GEODETIC SURVEY. Table 3. — Equilibrium arguments ( Fo + «) at the midnight preceding January 1 of each year, from 1850 to 19S0, for the meridian of Greenwich, together with the elements used in computing them — Continiled. Component. i88o 1881 1882 1883 1884 1883 1886 1887 188S 1889 o J. 285-32 28-47 1 1 6 -06 201-92 286-12 23-31 106-49 190-75 276-68 18-39 K, 18-64 19-66 18-65 16-53 13-43 10-74 7-07 4-01 1-94 1-96 1 K. 217-59 219-20 216-70 212-21 206-24 201-48 194-79 188-87 184-45 184-02 L, 359-19 189-85 8 -So 184-51 17-70 204-53 38-89 219-81 35-38 219-58 [L.] 358->7 176-28 5-47 194-47 23-31 200-77 29-54 218-40 47-40 225-28 i M, 142-24 58-16 312-82 203-44 108-15 31-14 301-37 192-56 83-72 344-54 [M,] 119-56 19-14 289-29 197-71 104-45 358-07 263-80 170-62 79-11 337-25 M^ 275-29 351-62 92-08 192-36 292.48 8-15 108-20 208-33 308-61 24-71 M, 232-94 167-42 138-13 108-54 78-72 12-22 342-30 312-50 282-91 217-06 ^^ 190-58 343-23 184-17 24-72 224-96 16-30 216-40 56-67 257-22 49-42 K 105-88 334-85 276-25 217-07 157-44 24-44 324-61 265 -00 205-83 72-12 Ma 21-17 326-46 8-34 49-43 89-92 32-59 72-81 113-34 154-44 98-83 N, 12-41 346-95 358-70 10-25 21-65 355-53 6-86 18-27 29-82 4-13 2 N 109-54 342-29 265-31 188-14 110-82 342-91 265-52 188-21 1 1 1 -04 343-56 o, 252-85 327-81 69-41 172-54 277-19 357-46 103-06 207-65 310-72 26-89 oo 332-03 259-80 155-95 47-08 293-37 203-93 87-23 333-72 225-07 148-75 p. 349-96 349-22 349-46 349-69 349-93 349-'9 349-43 349-66 349-90 349-16 Q. 349-97 323-15 336-02 350-43 6-36 344-84 1-72 17-59 31-93 6-31 Ra 359-15 359-88 359-63 359-37 359-11 359-84 359-59 359-33 359-08 359-81 y„3 180-00 180-00 180.00 180-00 180-00 180-00 180.00 180-00 180-00 180-00 Ss, 4> 6 0-00 0-00 0-00 0-00 0-00 0-00 0-00 0-00 0-00 0-00 T, 0-85 0-12 0-37 0-63 0-S9 0-16 0-41 0-67 0-92 0-19 X, 279-25 177-42 88-41 359-21 269-85 167-37 77-94 348-60 259-40 157-34 fii 188-46 341-15 182-37 23-40 224-27 16-31 217-12 58-00 259-03 51-50 v% 91-33 345-82 275-76 205-51 135-" 28-93 318-46 248-07 177-82 72-08 MK 293-93 11-28 110.73 208-89 305-91 18-88 115-28 212-34 310-55 26-67 2MK 171-95 323-57 165-52 8-18 211-53 5-56 209-33 52-66 255-28 47-46 MN 287-71 338-57 90-78 202-61 3i4->3 3-68 115-06 226-60 338-43 28-84 MS 275-29 351-62 92-08 192-36 292-48 8-15 io8-20 208-33 308-61 24-71 2MS 190-58 343-23 184-17 24-72 224-96 16-30 2 1 6 -40 56-67 257-22 49-42 2SM 84-71 8-38 267-92 167-64 67-52 351-85 251-80 151-67 5 '-39 335-29 Mf 309-59 236-00 133-27 27-27 278-09 193-23 82-09 333-04 227-17 150-93 MSf 84-71 '8-38 267-92 167-64 67-52 351-85 251-80 151-67 51-39 335-29 Mm 262-88 4-66 93-39 182-11 270-83 12-62 101-34 lgO-06 ■278-79 20-57 Sa 280-04 280-78 280-54 280-31 280-07 280-81 280-57 280-34 280-10 280-84 Ssa 260-07 2bi-57 201 -09 200-61 200-13 201 -63 201-15 200-67 200-20 201.69 Elements. Val ues at Gret nwich, mk .night begi aning each year. h 280-04 280-78 280-54 280-31 280-07 280-81 280-57 280-34 280-10 ^80-84 s '43-45 286-01 55-40 184-79 314-17 96-73 226-12 355-50 124-89 267-45 P 240-58 281-35 322-01 2-68 43-34 84-11 124-77 165-44 2o6-io 246-87 ■Values a the middl t of each 1 ear, or for ing :uly 2, at G midnight fc ireen-wich n r leap year tiean nbori s. bi: coiiimor years, anc . at preced- P. 280-88 280-90 280-92 280-93 280-95 280-97 280-99 281-00 281-02 281 -04 P 272-30 313-67 353-58 31-86 68-60 104-33 140-03 176-79 215-19 255-22 N 276-29 256-94 237-61 218-28 198-93 179-57 160-24 140-91 121-56 102-21 I 24-53 22-82 21-12 19-66 18-66 18-31 18-69 19-71 21-19 22-89 Q 274-60 332-35 356-78 17-26 51-91 117-06 157-27 178-39 199-42 242-19 R 358-98 346-43 356-67 9-96 5-61 356-24 350-65 358-58 12-01 5-71 f 348-66 348-02 348-77 35i->5 355-13 0-11 5-07 8-98 11-30 11-98 i> 347-60 346-98 347-87 350-49 354-78 0-12 5-43 9-65 12-21 13-03 v' 351-40 351-12 351-89 353-77 3^6-64 •oS 3-50 6-33 8-16 8-89 2V" 342-48 342-37 344-39 348-40 353-90 0-14 6-36 11-80 15-74 17-67 REPORT FOll 1894— PART II. 199 Table 3. — Equilihrium arguments (1^0 + '*) <"' "^^ midniglu preceding January 1 of each year, from 1850 to 1950, for ike meridian of Greenwich, together with the elements used in computing them — Continued. Component. J. K, K., M, [M,] M, M, M4 Me Ms N, 2N O, 00 Pi Q. Si, 3 S3, 4, 6 T2 ■ ^ Vi MK zMK MN MS 2MS 2SM Mf MSf "Mm Sa Ssa Elements. k s P N i8go •oyss 2-04 1837s 6575 5472 28773 248-96 125 42 188-14 250-85 16-27 141-70 16-13 266-83 127-17 49 35 349-39 17-87 3S9-SS 1 80 -GO o-oo 0-45 68-58 252-97 2-27 127-46 248-81 141-55 125-42 250-85 234-58 51-09 234-58 109-30 280-61 201-21 1891 1892 280-61 3683 287-54 ■Values 281 -05 296-58 82-88 197-88 2-98 185-40 260-29 244 -40 192-32 161-84 226-38 159-57 92-76 319-14 185-52 28-36 190-34 226-52 313-21 349-63 28-50 359-30 1 80 -GO 0-70 340-06 94-68 292-70 229-36 89-78 254-74 226-38 92-76 133-62 313-34 133-62 198-02 280-37 200-74 280-37 166-22 328-20 289-05 4-58 188-60 55-86 74-29 85-29 75-57 327-55 131-33 295-10 262-66 230-21 40-81 114-07 325-24 219-38 349-87 38-50 359-04 180-00 0-96 251-76 296-60 223-35 332-13 290-53 8-36 327-55 295-10 3^-45 217-07 32-45 286-74 280-13 200-26 1893 34-83 7-61 194-35 240-32 253-04 340-37 337-77 44-52 66-77 89-03 133-55 178-06 15-99 347-46 38-21 154-39 349-13 9-69 359-77 180-00 0-23 150-56 89-94 118-47 52-13 81-42 60-50 44-52 89-03 315-48 148-09 315-48 28-53 280-87 201 -75 126-86 9-90 199-71 9398 83-22 288-59 252-35 145-97 38-95 291-94 77-91 223-88 28-72 271-47 136-37 62-85 349-36 19-11 359-51 180-00 0-49 62-54 292-15 49-40 155-87 282-04 174-69 145-97 291-94 214-03 53-24 214-03 117-25 280-64 201 -27 189s 218-95 12-24 204-70 295-30 273-42 199-79 167-00 247-45 11-17 134-89 22-34 269-78 41-47 195-50 234-47 331-47 349-60 28-50 359-26 180-00 o-oo 0-74 334-54 13438 340-35 259-68 122-65 288 92 24745 134-89 112-55 318-50 112-55 205-97 280-40 200-80 1896 310-88 14-45 209-37 86-65 103-56 93-36 81-48 348-86 343-29 337-72 326-58 315-44 54-16 119-47 332-68 239-68 349-84 37-98 359-00 180-00 0-00 100 246-48 336-54 271-24 3'3' 323-27 43-02 348-86 337-72 11-14 223-50 11-14 294-70 280-16 200-32 1897 56-45 17-31 215-21 268-40 282-25 344-85 343-48 65-76 278-64 131-52 197-29 263-05 29-28 352-80 45-80 174-14 349-09 9-3« 359-73 180-00 0-27 145-23 129-82 166-30 83-07 114-21 95-04 65-76 131-52 294-24 154-17 294-24 36-48 280-91 201-82 1898 280-13 295-60 Values at Greenwich, midnight beginning each year. 280-87 78-16 49-63 280-64 207-55 90-30 280-40 336-93 130-96 280-16 106-32 171-62 280-91 248-88 212-40 at the middle of each year, or for July 2, at Greenwich mean noon for common ing midnight for leap years. 281 -07 338-99 63-55 281 -09 22-18 44-19 281-11 65-85 24-84 281-12 109-70 5-51 281-14 153-61 346-18 281-16 197-44 326-83 281-17 240-93 307-47 7,6-88 147-29 18-65 217-80 118-43 112-05 289-57 256-88 166-84 250-26 333-68 140-53' 307-37 41-64 276-43 144-76 79-41 349-33 19-55 359-48 180-00 0-00 0-52 56-83 331-65 96-85 185-50 315-03 208-48 166-84 333-68 193-16 57-33 193-16 125-21 280-67 201-33 280-67 18-26 253-06 years, and 281-19 28379 288-15 1899 237-16 19-25 21S-67 317-10 301 -63 205 -07 169-30 267-70 221-55 175-40 83-09 350-79 53-77 199-84 244-47 341 -99 349-57 30-54 359-22 180-00 0-78 328-21 173-26 27-19 286-95 156-14 321 -47 267 -70 175-40 92-30 318-76 92-30 213-93 280-43 200-86 280-43 147-65 293-72 at preced- 281-21 325-79 268-82 200 UNITED STATES COAST AND GEODETIC SUEVEY. Table 3. — Equilibrium arguments {Va -\- u) at the midnight preceding January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with tJie elements used in computing them — Continued. Component. igoo igoi igo2 1903 1904 i9°5 igo6 1907 1908 1909 K. 32572 18-90 217-49 52-66 17-46 214-18 137-84 14-93 209 -05 22°-53 11-56 202-71 304°-55 7-81 195-90 42-oS 5-30 191-37 i26°-95 2-59 185-96 213-56 0-93 182-27 301 -84 0-39 180-69 4'5-54 1-81 183-17 L, M, 127-09 130-96 97-46 309-02 320-07 351-64 147-11 149-00 271-67 345-00 337-81 203 -30 174-12 166-58 101-59 339-64 344-05 338-11 162-47 172-95 232-12 1-20 2-0^ 158-08 204-78 191-32 9059 17-54 9-53 336-18 ■ [M,] M^ M, • 80-41 8-31 192-46 349-91 108-70 163-04 257-65 208-90 133-35 163-90 309-00 103-49 69-47 49-04 73-55 323-42 124-72 7-09 230-85 224-90 337-36 140-02 325-26 307-89 50-85 65-83 278-74 310-97 142-25 213-37 i M^ : Me j Ms 16-62 24-92 33 23 217-39 326-09 74-78 57-80 266-71 115-61 257-99 206-99 155-98 98-07 147-11 196-14 249-45 14-17 138-90 89-81 314-71 179-62 290-52 255-77 221-03 131-65 197-48 263-30 284-50 66-74 208-99 ! N, 2N 65-66 123-01 345 '24 77-32 45-95 87-41 88-81 328-71 191-16 100-18 251-36 296-28 HI -50 173-96 42-03 85-40 46-07 121-97 96-85 328-81 226-01 108-49 251-72 328-47 1 20-33 I74.-84 69-48 94-97 47-69 143-99 OO P, Q. 240-89 349-81 42-59 135-15 350-05 56-04 24-33 350-29 71-06 269-15 350-53 87-46 151-99 350-76 104-49 63-53 350-02 82-65 311-79 350-26 97-96 205 11 350-49 111-70 103-20 350-73 123-99 32-52 349-99 96-71 Si, 3 358-97 180-00 358-71 180-00 358-45 1 80 -00 358-20 180-00 357-94 180-00 358-67 1 80 -00 358-41 180-00 358-16 180-00 357-90 180-00 358-63 180-00 S2, 4, H 0-00 0-00 o-oo 0-00 O-QO 0-00 0-00 o-oo 0-00 0-00 T, 1-03 239-35 14-62 1-29 150-26 215-76 I -55 60-99 56-72 1-80 331-61 257-57 2-06 242-17 98-36 1-33 139-70 250-42 1-59 50-41 91-35 1-84 321- 9 292-46 2-10 232-38 133-78 1-37 130-64 286-57 MK 2MK 317-27 27-21 357-71 247-13 126-16 199-93 176-81 223-83 42-87 106-39 320-55 246-44 35-90 56-84 90-26 289-75 130-02 244-15 219-40 227-49 87-22 149-23 326-19 289-58 79-27 66-21 131-27 333-85 144-06 282-69 MN MS 2 MS 73-97 8-31 16-62 186-02 108-70 217-39 297-71 208-90 57-80 49-17 309-00 257-99 160-53 49-04 98-07 210-12 124-72 249-45 321-76 224-90 89-81 73-75 325-26 290-52 186-16 65-83 131-65 237-21 142-25 284-50 2SM Mf MSf 351-69 217-82 351-69 251-30 113-87 251-30 151-10 6-59 151-10 51-00 256-44 51-00 310-96 144-98 310-96 235-28 60-78 235-28 135-10 312-89 135-10 34-74 208-32 34-74 294-17 106-86 294-17 217-75 34-26 217-75 Mm Sa Ssa Elements. 302-65 280-19 200-38 31-37 279-95 199-90 120-09 279-71 199-42 Va 208-82 279-47 198-95 ues at Gre 297-54 279-24 198-47 jn-wich, mi 39-33 279-98 199-96 dnight begi 128-05 279-74 199-49 nning each 216-77 279-51 199-01 year. 305-49 279-27 198-53 47.-28 280-01 200-03 h s P /. P N 280-19 277-03 334-38 ■Values a 281-23 6-59 249-49 279-95 46-42 15-05 the middl 281-24 45-89 230-16 279-71 175-80 55-71 e of each y 281-26 83-53 , 210-83 279-47 305-19 96-37 ;ar, or for \ ir 281-28 119-73 191-51 279-24 74-57 137-03 uly 2, at G g midnigh 281-29 155-34 172-15 279-98 217-13 177-81 reenwich n for leap y 281-31 191-39 152-80 279-74 346-52 218-47 lean noon : ;ars. 281-33 228-73 133-47 279-51 115-90 259-13 or common 281-35 267-72 114-14 279-27 245-29 299-80 years, and 281-36 308-32 94-79 280-01 27-85 340-57 at preced- 281-38 350-18 75-43 I Q R 22-15 3-87 20-52 27-29 11-05 19-21 77-22 1-89 18-44 138-80 352-82 18-37 167-07 352-45 19-01 185-75 4-41 20-24 209-67 10-48 21-83 265-45 0-83 23-56 327-68 346-54 25-22 355-06 351-99 V v' 1v" 348-12 347-13 351-29 342-89 349-48 348-67 352-49 345-72 352-51 351-97 354-78 350-37 356-97 356-76 357-92 356-24 2-08 2-23 1-43 2-57 6-75 7-23 4-68 8-59 10-07 10-85 7-16 ■3-53 11-74 12-72 8-57 16-73 11-86 12-92 8-88 17-84 10-72 : 11-76 8-21 16-85 EEPOET FOR 1894— PART II. 201 Table 3. — Eqnilibiium arguments ( Fo + i«) dt the midniglit preceding January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with the etementa used in computing them — Continued. Component. Ji Ki K, U [L,] M, [M,] M, M, Me Ms N, 2N . O, oo p, Q. Si, 3 5-2) 4» 6 MK 2MK MN MS 2 MS 2SM Mf MSf Mm . Sa Ssa Elements. h P N 136-22 3-03 185-47 179-69 199-29 229-57 224-21 243-29 184-94 126-58 9-88 253-'7 107-29 331-29 243-07 297-36 350-23 107-07 358-38 180-00 1-62 42-21 128-37 264-37 246-32 123-55 350-58 243-29 126-58 116-71 297-14' 116-71 136-00 279-77 199-SS 279-77 157-24 21-23 Values 281-40 _72-go 56-10 26-65 227-63 4-82 189-13 22-00 29-27 146-18 138-19 344-54 156-81 329-08 313-63 298-17 119-82 255-09 341-63 204-19 350-47 116-91 358-12 180-00 0-00 1-88 313-99 330-37 195-10 349-37 324-26 104-36 344-54 329-08 15-46 201-28 15-46 224-73 279-53 199-07 319-51 6-99 193-69 235-00 219-39 92-11 52-61 85-94 128-91 171-88 257-82 343-76 132-49 179-04 79-88 112-24 350-70 126-43 357-87 i8o-oo 0-00 2-13 225-91 172-52 125-97 92-93 164-89 218-43 85-94 171-88 274-06 106-18 274-06 313-45 279-30 198-59 65-64 10-30 200-64 55-42 38-26 343 35 315-17 1^3-03 64-54 326-06 129-08 292-11 107-79 52-56 152-62 48-19 349-96 97-39 358-60 180-00 1-40 124-84 325-98 21-22 ■73-33 315-75 270-82 160-03 326-06 196-97 37-79 196-97 55-23 280-04 200 -09 1916 157-69 12-61 205-56 206-55 228-44 237-05 229-78 264-49 36-73 168-97 73-46 337-94 120-53 336-57 250-75 316-73 350-20 106-79 358-34 180-00 1-66 36-83 168-19 312-15 277-10 156-36 25-01 264-49 168-97 95-51 302-99 95-51 143-96 279-80 199-61 249-49 14-72 209-97 48-99 58-54 149-20 144-14 5-86 8-79 11-72 17-59 23-45 133-18 260-50 349-05 224-58 350-43 116-37 358-08 180-00 1-92 308-72 10-32 243-00 20-58 357-01 139-05 5-86 11-72 354-14 207-77 354-14 232-68 279-57 199-13 340-79 16-41 213-40 260-18 248-48 96-33 57-99 107-07 340-61 214-15 321-22 68-30 145-67 184-27 87-69 131-09 350-67 126-28 357-83 18000 o-oo 2-17 220-46 212-29 173-69 123-49 197-73 252-75 107-07 214-15 352-93 II I -70 252-93 321-40 279-33 198-66 85-35 18-49 217-36 81-51 66-88 350-12 318-97 183-69 275-53 7-38 191-07 14-76 120-50 57-31 161-53 62-79 349-93 98-34 358-56 180-00 0-00 1-44 118-92 5-27 68-46 202 - 1 7 348-89 304-19 183-69 7-38 176-31 40-63 176-31 63-19 280-07 200-15 1918 Values at Greenwich, midnight beginning each year. 79-53 279-30 280-04 279*80 279-57 279-33 280-07 86-62 56-01 198-57 327-95 97-34 226-72 9-28 61-89 102-56 143-33 183-99 224-66 265-32 306-09 174-75 18-74 217-46 245-33 256-36 242-01 230-93 284-45 246-67 208-89 133-34 57-78 132-53 340-62 261-61 324-07 350-17 109-70 358-30 180-00 1-70 30-20 206 -7 8 358-69 303-18 190-15 56-98 284-45 208-89 75-55 301-23 75-55 151-91 279-83 199-67 279-83 138-67 346-76 262-72 17-98 215-46 75-84 85-61 142-58 141-45 24-97 217-46 49-94 74-92 99-89 144-34 263-71 2-89 221 -28 350-40 122-26 358-05 180-00 0-00 1-95 301-25 48-06 288-69 42-95 31-97 169-31 24-97 49-94 335-03 199-19 335-03 240-63 279-60 199-19 279-60 268-05 27-42 at the middle of each year, or for July 2, at Greenwich mean noon for common years and at preced- ing midnight for leap years. 281-41 76-25 36-77 27-74 281-43 120-03 17-42 28-40 281-45 163-99 358-06 28-60 281-47 207-87 338-74 28-31 281-48 251-54 319-41 27-55 281-50 294-84 300-05 26-39 281-52 337-46 280-70 24-90 281-53 19-03 261-37 23-21 281-55 59-29 242-04 21-50 202 UNITED STATES COAST AND GEODETIC SUEVBY. Table 3. — Equilibrium arguments F„ + «) at the midnight preceding Januari/ 1 of each year, from 1850 to 1950, for the meridian of Green wich, together with th e elements used in computing them— Continued^ Component. I, ig20 348-98 I92I 1922 1923 253-94 1924 1925 1926 1927 1928 1929 87°57 i7o°87 337-87 77-45 164-74 253-58 343-67 88-73 ki 161I 14-19 10 -60 6-88 3-63 2-29 1-08 0-93 1-70 4-15 K. 211-39 207-66 201-05 194-28 188-11 185-27 182-37 181-62 182-87 187-73 U 277-13 101-57 284-24 101-71 291-23 119-89 319-09 133-44 306-72 «4i-37 ^b'^ 274-64 92-19 280-98 109-75 298-58 116-22 305-37 13475 324-38 142-91 M, 78-36 344-03 235-99 124-59 25-11 309-86 226-49 119-97 15-31 292-00 [M,] 50-27 305-28 211-14 116-77 23-25 279-26 189-10 100-50 13-15 274-63 M, 125-28 201-05 301 -I I 41-16 141-27 217-12 3'7-55 58-21 159-11 235-86 M, 187-93 121-58 91-67 61-74 31-90 325-68 296-33 267-32 238-67 173-79 M. 250-57 42-10 242-22 82-32 282-53 74-24 275-10 116-42 318-23 111-72 Me 15-85 243-16 183-33 123-48 63-80 291-37 232-65 174-64 117-34 347-57 Ms 141-14 84-21 124-44 164-64 205 -06 148-49 190-20 232-85 276-46 223-43 N, >55-93 129-91 141-24 152-57 163-96 138-02 149-73 161-67 173-85 148-81 2N 186-57 58-77 341 -38 263-99 186-65 58-93 341-91 265-13 188-59 61-75 0, 105-66 184-63 290-10 35-81 140-70 218-77 320-63 61-17 160-71 234-18 00 113-58 28-22 271-91 154-90 40-43 317-94 213-20 112-91 16-11 309-18 Pi 350-64 349-89 35o-"3 350-37 350-61 349-86 350-10 350-34 350-58 349-83 Qi 136-31 "3-49 130-24 147-22 163-39 139-67 152-81 164-63 175-44 H7-13 R2 35779 358-52 358-26 358-01 357-75 358-48 358-23 357-97 357-72 358-45 s„. 180-00 1 80 -00 180-00 180-00 i8o-oo 180-00 180-00 180-00 i8o-oo 180-00 Sj, 4, 6 0-00 0-00 0-00 0-00 0-00 o-oo 0-00 0-00 o-oo o-oo T. 2-21 1-48 1-74 1-99 2-25 1-52 1-77 2-03 2-28 1-55 ?.2 212-08 109-70 20-28 290-85 201 -48 99-18 10-14 281-32 192-75 91-33 fl2 249-13 41-26 242-07 82-88 283-74 65-96 277-14 118-56 320-22 "3-33 Vl 218-48 112-41 41-94 331-47 261-05 155-06 84-96 15-10 305-48 200-38 MK 141-39 215-24 311-71 48-04 144-90 219-42 318-63 59-15 i6o-8i 240-01 2MK 234-46 27-91 231-62 75-44 278-90 71-95 274-02 115-49 316-53 107-56 MN 281-21 330-96 82-35 193-73 305-22 355-15 107-28 219-88- 332-96 24-66 MS 125-28 201 -05 301-11 41-16 141-27 217-12 317-55 58-21 I59-II 235-86 2 MS 250-57 42-10 242-22 82-32 282-53 74-24 275-10 116-42 318-23 111-72 2SM 234-72 158-95 58-89 318-84 218-73 142-8S 42-45 301-70 , 200 -89 124-14 Mf 93-96 ii-80 260-90 149-54 39-86 319-59 216-28 1.5-87 17-70 307-50 MSf 234-72 158-95 58-89 318-84 218-73 142-88 42-45 301-79 200-89 124-14 Mm 329-36 71-14 159-87 248-59 337-31 79-10 167-82 256-54 345-26 87-05 Sa 279-36 280 - 1 1 279-87 279-63 279-39 280-14 279-90 279-66 279-42 280-17 Ssa 198-72 200-2I 199-73 199-26 198-78 200-27 199-79 199-32 198-84 200-33 Elements. Va ues at Gre ;nwich, mil Inight begi nning each year. h 279-36 280-11 279-87 279-63 279-39 280-14 279-90 279-66 279-42 280-17 s 37-44 i8o-oo 309-38 78-77 208-15 35071 120-10 249-48 18-87 161-43 P 68-08 108-86 149-52 190-18 230-84 271-62 312-28 352-94 33-61 74-38 Values a t the middl e of each y ear, or for . in uly 2, at G g midnight reenwich, r for leap ye nean noon ars. or common years and at preced- P\ 281-57 281-59 281-60 281-62 281-64 281-65 281-67 281-69 281-70 281-72 P 9801 135-09 170-92 206-52 243-01 281-03 320-65 J -73 43-97 87-03 A 222-69 203-33. 1 84 -00 164-68 145-32 125-97 . 106-64 87-31 67-96 48-60 / 19-96 18-83 18-32 18-54 '9-43 20-83 22-50 24-22 25-81 27-12 Q 105-72 153-50 175-43 194-01 224-47 291-29 337-71 0-87 25-75 84-07 R 357-51 350-62 356-73 8-04 7-35 356-33 346-28 1-32 17-66 1-54 f 350-46 354-10 358-93 4-00 8-22 10-92 11-96 11-55 10-02 7-68 V 349-74 353-68 358-86 4-28 8-83 11-78 12-98 12-62 11-01 8-49 v' 3S3-25 355-92 359-27 2-75 5-75 7-84 8-82 8-73 772 6-01 2v" 347-33 352-55 358-68 4-98 10-67 15-01 17-42 17-69 15-97 12-6l REPORT FOR 1894— PART II. 203 Table 3. - JEguilibrium arguments ( F„ + w) at the midnight preceding January 1 of each year, from 1850 to 1950, for \ mei-idian of Greenwich, together with the elements used in computing them — Continued. Component. 1930 193 1 . 272°33 1932 1933 1934 193s 1936 1937 1938 1939 Ji 1 80-35 4-43 110-47 202-10 293° 12 23-24 I26°i8 2i3°-5i 299°09 K, 611 8-36 10-71 13-95 15-92 17-40 18-19 19-06 17-87 15-58 K, 19179 196-56 201-57 208-33 212-41 215-33 216-64 217-91 215-07 210-31 U 352-21 170-08 33°-77 167-92 18-80 201-72 5-95 193-11 35-89 227-40 [U] 332-94 163-10 353-30 172-15 2-18 192-04 21-67 199-74 28-90 217-87 M, 228-28 126-48 21-61 293-28 234-44 134-12 26-99 285-68 227-36 132-55 [Ml] 188-80 103-33 17-99 280-46 194-64 108-22 20-90 280-26 190-14 98-28 Mj 337-17 78-60 180-08 257-14 358-45 99-59 200-50 276-78 17-21 117-46 Ms 145-75 117-91 90-13 25-71 357-68 329-38 300-74 23517 265-82 176-19 M4 314-34 157-21 0-17 154-28 356-90 19918 40-99 193-56 34-43 234-92 M„ 291-51 235-81 18025 51-42 355-36 298-76 241 -49 110-34 51-64 352-39 M 268-68 314-42 0-34 308-56 353-81 38-35 81-98 27-12 68-86 109-85 N2 161-40 174-11 186-87 162*13 174-72 187-14 199-32 173-82 185-53 197-06 2N 345-62 269-61 193-65 67-13 351-00 274-69 108-15 76-86 353-85 276-65 0, 332-59 70-77 168-87 241-68 340-08 78-92 178-43 . 253-56 355-39 98-78 no 216-56 124-89 33-55 329--5 236-66 142-43 45-71' 332-87 228-26 118-58 P, 350-07 350-31 350-55 349-80 350-04 350-28 350-52 349-77 350-01 350-25 Ql 156-82 166-28 175-65 146-67 15635 166-47 177-25 150-60 163-71 178-38 Rs 358-19 357-93 357-68 358-41 358-15 357-90 357-64 358-37 358-11 357-87 Si, 180-00 i8o-oo 180-00 180-00 1 80 -00 1 80-00 180-00 180-00 180-00 180-00 S2, 4> 6 o-oo 0-00 0-00 0-00 o-oq o-oo 0-00 o-oo 0-00 6-60 T, I -81 2-07 2-32 1-59 1-85 2-10 2-36 1-63 1-89 2-14 h 3-17 275-13 187-13 86-03 357 -B7 269-53 180-96 79-09 350-05 260-82 lii ■315-39 157-58 359-82 153-24- 355-30 197-20 38-86 191-51 32-70 233-70 V2 I3I-I7 62-08 353-03 248-25 179-03 109-65 40-03 294-47 224-38 154-10 MK 343-28 86-96 190-79 27 1 -09 14-37 116-99 2i8-6y 295-84 35-09 133-04 2MK 30823 148-85 349-46 140-33 340-99 181-77 22-80 174-50 16-56 219-35 MN 138-57 252-71 6-95 59-27 173-18 286-73 39-82 90-60 202-7S 314-52 MS 337-17 78-60 180-08 257-14 358-45 99-59 200-50 276-78 17-21 117-46 2 MS 314-34 157-21 0-17 154-28 356-90 199-18 40-99 193-56 34-43 234-92 2SM 22-83 281 -40 179-92 102 -86 1-55 260-41 159-50 83-22 342-79 242-54 Mf 211-98 117-06 22-34 313-78 218-29 121-76 23-64 309-66 206 -43 99-90 MSf 22-83 281-40 179-92 102-86 1-55 260-41 159-50 83-22 342-79 242-54 Mm 175-77 264-49 353-22 95-01 183-73 272-45 1-17 102-96 191-68 280-40 Sa 279-93 279-69 279-45 280-20 279-96 279-72 279-48 280-23 279-99 199-98 279-75 Ssa 199-86 199-38 198-90 200-40 199-92 199-44 198-96 200 -46 199-50 Elements. Valu es at Greer i-wich, mid night begin ning each ; ^ear. k 279-93 279-69 279-45 280-20 279-96 279-72 279-48 280-23 279-99 279-75 41-63 s 290-82 60-20 189-59 332-15 101-53 230-92 0-30 142-86 272-25 p 115-04 155-70 196-37 237-14 277-80 318-47 359-13 39-90 80-57 121-23 ■Values a t the middl e of each y ear, or for ir July 2, at ( g midnigh Greenwich i for leap ye nean noon ars. or commor years and at preced- P N 281 -74 130-55 29-27 281-76 281-77 281-79 281-81 281 82 281 -84 281-86 281-88 281-89 174-37 9-94 218-33 350-59 262-22 331-23 305-75 311-91 348-76 292-58 31-03 273-22 72-20 253-87 111-87 234-54 149-88 215-21 . « . n - . _o.- . «(?.«., O.T.T r- ■jc-Xc 2/t-26 22-1; d. 20-87 IQ'4.7 2U4 UNITED STATES COAST AND GEODETIC SURVEY. Table S.— Equilibrium arguviente {V„ + u) at the midnight preceding January 1 of each year, from 1850 to 1950, for the meridian of Greenwich, together with the elements used in computing (/lem— Continued. Component. J. K, K, U M, [M,] Ma Mh Me Ms N, 2N O, OO p. Q. R2 S|, 3 S2> 4) 6 T. hi /"2 MK 2MK MN MS 2 MS 2SM Mf MSf Mm Sa Ssa Elements. h P N I Q R 23-06 12-35 204-17 44-29 46-69 21-99 4-80 217-57 146-35 75-14 292-70 150-27 208-44 199-31 203-64 4-22 350-49 194-51 357-60 180 -00 o-oo 2-40 171-45 74-56 83-69 229-92 62-78 66-01 217-57 75-14 142-43 350-29 142-43 9-'3 279-51 1 99 -02 279-51 171-02 161-89 Values at 120-18 9-62 199-37 215-18 224-15 260-90 258-35 293-23 79-85 226-46 159-70 92-93 182-32 71-40 283-97 274-55 349-74 173-06 358-33 i8o-oo 1-67 68-96 226-59 337-51 302-85 2x6-84 "5-55 293-23 226-46 66-77 265-29 66-77 i 10-91 280-26 200-52 203-46 6-02 192-74 49-56 52-93 173-39 164-19 33-29 49-93 66-58 99-87 133-16 193-65 354-02 29-47 158-19 349-98 189-83 358-08 180-00 1 -92 339-54 67-40 267 -04 39-31 60-56 226-94 33-29 66-58 326-71 154-36 326-71 199-64 280-02 200-04 287-96 3-09 187-02 248-99 241-80 109-54 71-25 133-44 20-16 266-88 40-33 17377 205 -08 276-73 133-84 45-38 350-22 205 -48 357-82 180-00 0-00 2-18 250-22 268-31 196-67 136-53 263-79 338-53 133-44 266-88 226-56 45-77 226-56 28S-36 279-78 199-56 14-18 1-19 182-90 82-00 70-83 340-01 233-75 350-62 107-50 34J -24 214-99 216-67 199-59 236-65 297-54 350-46 219-57 357-57 180-00 2-43 161-05 109-57 126-45 234-94 106-30 90-41 233-75 107-50 126-25 300-44 126-25 17-08 279-54 199-09 '945 1946 1947 116-15 1-39 182-80 205-52 1-62 182-85 296-00 2-68 184-79 240-19 248-75 251-95 64-35 78-23 151-20 271-82 267-94 91-30 238-41 309-88 284-82 150-34 50-63 255-95 63-37 151-63 227-44 259-76 209-65 159-53 101-27 151-90 202-54 303-25 94-88 246-50 191-01 72-14 312-60 203 -04 355-45 52-71 21531 279-00 151-94 221-96 349-71 193-73 123-15 349-95 205-12 27-42 350-19 215-63 358-29 180-00 358-04 180-00 357-78 180 -00 0-00 0-00 o-oo I-71 59-02 261.-87 1-96 330-30 103-38 2-22 241-82 305-12 20-74 311-27 258-38 310-97 52-25 99-65 241 -43 154-31 300-57 140-89 309-88 259-76 253-68 50-63 101 -27 6-94 151-63 303-25 50-12 224-68 50-12 309-37 125-22 309-37 208-37 27-74 208-37 118-87 280-29 200-58 207-59 280-05 200-10 296-31 279-81 199-62 1948 o 27-28 4-37 188-19 118-73 '97-87 14-50 337-20 252-83 199-25 145-67 38-50 291-34 227-80 202-76 25059 293-89 350-43 225-56 357-53 180-00 0-00 2-47 153-55 147-08 172-12 257-20 141-30 120-63 252-83 145-67 107-17 291-65 107-17 25-04 279-57 199-15 Values at Greenwich, midnight beginning each year. 280-26 280-02 279-78 279-54 280-29 280-05 279-81 313-58 82-96 212-35 341-73 124-29 253-68 23-06 202-66 243-33 283-99 324-65 5-43 46-09 86-75 279-57 152-45 127-41 133-12 7-45 194-55 270-72 276-64 257-36 239-47 329-82 134-73 299-64 269-46 239-28 203 -00 76-17 323-52 229-07 349-68 196-70 35826 180-00 . 0-00 1-74 52-38 300-44 67-26 337-27 292-19 272-82 329-82 299-64 30-18 222-77 30-18 126-82 280-32 200-64 280-32 295-01 168-19 the middle of each year, or for July 2, at Greenwich mean noon for common years and at ing midnight for leap years. 281-91 186-41 195-86 281-93 222-o6 176-50 281-94 257-87 157-17 281-96 294-86 I37-8S 281-98 333-53 118-49 281-99 13-80 99-14 282-01 55-35 79-81 282-03 97-89 60-48 18-55 1S3-2I i-41 18-32 204-29 8-96 ■ 18-81 246-75 3-36 19-92 312-82 352-81 21-45 346-02 348-82 23-17 7-01 8-56 24-86 35-89 13-88 26-35 105-49 35612 355-87- 355-58 357-16 354-85 0-93 1-00 0-64 I-15 5-78 619 4-00 7-30 9-46 10-17 6-69 12-54 II-51 12-45 8-35 16-19 "-95 13-01 8-90 17-78 11-07 12-12 8-43 17-25 9-19 10-13 7-13 14-84 282-05 141-17 41-13 27-53 158-08 339-14 6-63 7-33 5-21 10-96 282-06 184-89 21-77 28-29 182-45 5-92 3-63 4-03 2-87 6-09 225-17 9-75 199-4J 86-64 106-82 155-34 154-08 71-28 106-92 142-56 213-83 285-11 215-73 0-19 61-65 137-59 349-92 2o6-ii 358-00 i8o-oo 0-00 2-00 324-36 142-65 358-19 bl-03 132-80 287-01 71-28 142-56 288-72 127-97 288-72 215-55 280-08 200-16 280-08 64-40 208-85 preced- 282-08 228-77 2-44 28-60 209-71 20-18 0-41 0-46 0-33 0-70 where 7" is the numb.r of Julian years of 365^- mean solar days; D, the number of mean solar days; H, the number of mean solar hours after Greenwich mean noon, Januai-y 1 , 1 880. On account of the slowness of the secular changes in the coefficients o T, D, or If, the epoch of this table may be reg-nrded is 1900. See Hansen's Tables de la Lune, p. 15, from which these formulas may be obtained by putting / = 80. Newcomb's corrections (Washington Observations, Vol. 22 (1875), App. II, pp. 268, 274) are not of sufficient magnitude to affect the values in Table 3. EEPOET FOE 1894— PAET II. 205 Table ^.—For adapting the uniformly varying portion ( Vo) of the equilihrium arguments of Table S to Greenwich midnight, beginning any day throughout the year. Months. K, Kj L, Mj M3 M, . -^•« >:, N, 0, Pi Ui Jan. Feb. 30-56 6111 9-19 324-17 306-26 288-35 252-52 216-69 279-16 293-62 329-44 248-60 Mar. f Com. yr \ Leap yr 581S 116-31 52-33 1-49 2-24 2-98 4-48 5-97 310-66! 303-34 301-85 252-50 59-14 118-28 41-01 337-11 325-66 314-22 291-33 268-44 273-21 277-97 300-86 214-07 Apr. f Com. yr \ Leap yr 8871 177-42 61-54 325-66 308-50 291-33 257-00 222-66 229-82 236-96 271-29 141-11 89-69 179-39 50-20 301-28 271-93 242-57 183-85 125-13 192-37 211-59 270-31 102-68 May f Com. yr \ Leap yr 118-28 236-56 82-02 314-22 291-33 268-44 222-66 176-88 186-42 195-94 241-72 68-14 119-26 238-53 70-70 289-84 254-76 219-68 149-52 79-36 148-98 170-58 240-74 29-71 June f Com. yr \ Leap yr 148-83 297-66 91-21 278-39 237-59 196-79 115-18 33-58 105-58 129-56 211-17 316-75 149-82 299-64 79-89 254-01 201-02 148-02 42-04 296-05 68-13 104-19 210-18 278-32 July J Com. yr \ Leap yr 178-40 356-80 ni-71 266-95 220-42 173-90 80-85 347-80 62-18 88-55 181-60 243-78 179-39 358-78 100-40 242-57 183-85 125-14 7-70 250-27 24-74 63-18 i8o-6i 205-35 Aug. f Com. yr t Leap yr 208-96 57-91 120-90' 231-12 166-68 102-24 333-37 204-49 341-34 22-16 151-04 132-39 209 94 59-89 109-58 206-74 130-11 53-48 260-22 106-96 303-90 356-80 150-06 93-96 Sept. f Com. yr \ Leap yr 239-51 119-02 130-09 195-30 112-94 30-59 225-89 61-18 260-50 3>5-78 120-49 20-99 240-50 121-00 118-77 170-92 7637 341-83 152-74 323-66 223-06 290-42 119-50 342-56 Oct. f Com. yr \ Leap yr 269-08 178-16 150-59 183-85 95-78 7-70 191-55 15-40 217-11 274-77 90-92 308 -03 270-07 180-13 139-28 159-47 59-20 318-94 118-41 277-88 179-66 249-40 89-93 269-59 Nov. J Com. yr \ Leap yr 299-64 239-27 159-78 148-02 42-04 296-05 84-07 232-10 136-27 208-39 60-36 196-63 300-62 241-24 148-47 123-64 ■ 5-46 247-29 10-93 134-57 98-82 183-02 59-38 158-20 Dec. f Com. yr \ Leap yr 329-21 298-41 180-29 136-58 24-87 273-16 49-74 186-32 92-87 167-37 30-79 123-67 330-19 300-38 168-97 112-20 348-30 224 -40 336-59 88-79 55-43 142-01 29-81 85-23 Dec. ,2 / Com. yr ■' \ Leap yr 359-76 359-52 189-48 IOO-7S 331-13 201-51 302-26 43-01 12-03 100-99 0-24 12-27 0-75 1-49 178-16 76-37 294-56 152-74 229-11 305-49 334-58 75-62 359-25 333-84 Day of month I 2 0-99 1-97 348-68 335-62 323-43 311-24 286-86 262-47 322-55 334-63 — 0-99 321-57 3 1-97 3-94 337-37 311-24 286-86 262-47 213-71 164-95 285-11 309-27 — 1-97 283-14 4 2-96 5-91 326-05 286-86 250-28 213-71 140-57 67-42 247 -66 283 -90 — 2-96 244-70 5 3-94 7-88 314-73 262-47 213-71 164 95 67-42 329-90 210-21 258-53 — 3-94 206-27 6 4-93 9-86 303-42 238-09 177-14 116-18 354-28 232-37 172-77 233-16 — 4-93 i67-)<4 7 5-91 11-83 292-10 213-71 140-57 67-42 2S1-13 134-84 135-32 207-80 — 5-91 129-41 8 6-90 13-80 280-78 189-33 103-99 18-66 207-99 37-32 97-88 182-43 — 6-90 90-98 9 7-88 15-77 269-47 164-95 67-42 329-90 134-84 299-79 60-43 157-06 - 7-88 52-54 lO 8-87 17-74 258-15 140-57 30-85 281-13 61 70 202 27 22-98 131-70 — 8-87 14-1 1 II 9-86 19-71 246 -84 116-18 354-28 232-37 348-56 104-74 345-54 106-33 — 9-86 335-68 12 10-84 21-68 235-52 91-80 317-70 183-61 275-41 7-21 308-09 80-96 — 10-84 297-25 '3 11-83 23-66 224-20 67-42 281^13 134-84 202-27 269-69 270-64 55-59 — 11-83 258-81 14 I2-8l 25-63 212-88 43-04 244-56 86-08 129-12 172-16 233-20 30:23 — 12-81 220-38 IS 13-80 27-60 201 -57 18-66 207-99 37-32 55-98 74-64 195-75 4-86 —13-80 181-95 i6 14-78 29-57 190-25 354-28 171-42 348-56 342-83 337-11 158-30 339-49 -14-78 143-52 17 15-77 31-54 178-94 329-90 134-84 299-79 269-69 239-58 120-86 314-13 —15-77 105-09 i8 16-76 33-51 167-62 305-52 98-27 251-03 196-54 142-06 83-41 288-76 — 16-76 66-65 19 17-74 35-48 156-30 281-13 61-70 20227 123-40 44-53 45-96 263-39 — 17-74 28-22 20 18-73 37-46 144-99 256-75 25-13 153-50 50-26 307-01 8-52 238-02 -18-73 349-79 21 19-71 39-43 133-67 232-37 348-56 104-74 337-" 209-48 331-07 212-66 —19-71 311-36 22 20-70 41-40 122-35 207-99 311-98 55-98 263-97 111-95 293-62 187-29 — 20-70 272-92 23 21-68 43-37 iii-o.^ '83-61 275-41 7-21 190-82 14-43 256-18 161-92 —21-68 234-49 24 22-67 45-34 99-72 159-23 238-84 3'8-4S 117-68 276 -go 218-73 136-56 — 22-67 196-06 25 23-66 47-31 88-40 134-84 202-27 269-69 44-53 179-38 181-28 111-19 -23-66 157-63 26 24-64 49-28 77-09 1 10-46 165-69 220-92 331-39 81-85 143-84 85-82 —24-64 119-20 27 25-63 51-25 65-77 86-08 129-12 172-16 258-24 344-32 106-39 60-45 -25-63 80-76 28 26-61 53-22 54-45 61-70 92-55 123-40 185-10 246-80 68-94 35-09 —26-61 42-33 29 27-60 55-20 43-14 37-32 55-98 74-64 111-95 149-27 31-50 9-72 — 27-60 3-90 3° 28-58 57-17 31-82 12-94 19-40 25-87 38-81 51-75 354-05 344-35 -28-58 325-47 31 • 29-57 59-14 20-50 348-56 342-83 337-'i 325-66 314-22 316-60 318-99 —29-57 287-04 The upper line for each month is for common years, and the lower line for leap years. For longitude corrections see Table 5, and for the portion («) of the equilibrium arguments of Table 3, -which depend upon the longitude of the moon's node, see Tables 6 and 7. The changes for other components may be found from those above, as follows : J. = L, - O, S2, 4> 6 =0 [L2]=L5|M| =/i —s |[M,] = K, +;i2|2N = N.J + X.2 1 00 = K^ — O, I R2 = K, I S,, 3 o T2 =P, I MN = M, + N.J I MS =Mj |2MS = M, 1 2 SM = MSf | Sa = K, | Ssa = K, 206 UNITED STATES COAST AND GEODETIC SUEVEY. Table i.—For adapting the uniformly varying portion ( V„) of the equiKirium arguments of Table 3 to Greenwich midnight, beginning any day throughout the year — Continued. Months. Aj M2 "2 MK '2MK MSf Mf Mm A s > JV* Jan. I .0 r C.+9-66 IL.+9-69 Feb. I 314-98 288-35 333-36 354-73 257-79 35-83 96-94 45-02 30-56 48-47 3-45 +8-02 +8-05 Mar. I /Com.yr. \ Leap yr. 309-16 2-98 53-82 59-64 304-83 358-51 114-82 50-84 58-15 57-41 6-57 +6-54 296-10 314-22 18-12 36-25 255-08 22-89 141-17 63-90 59-14 70-58 6-68 +6-51 •^p^'JLrprr: 264-15 291-33 27-18 54-37 202-62 34-34 211-75 95-85 88-71 105 -88 10-03 +4-90 251-09 242-57 351-48 30-98 152-87 58-72 238-10 108-91 89-69 119-05 10-14 +4-87 ^^y'iSrpy^;; 232-20 268-44 36-24 72-50 150-16 45-78 282-34 127-80 118-28 141-17 '3-37 +3-31 219-14 219-68 0-54 49-10 100-41 70-16 308-69 140-86 119-26 154-34 13-48 +3-28 Junei/Com.yr. ■> \Leapyr. 187-19 196-79 9-60 67-23 47-96 81-61 19-27 172-81 148-83 189-64 16-82 +1-67 174-12 148-02 333-90 43-83 358-21 105-99 45-62 185-88 149-82 202-81 16-93 + 1-64 July I 1 9°'"- y"- J ' \Leapyr. I5S'24 173-90 18-66 85-35 355-50 93-05 89-86 204-76 178-40 224-93 20-16 +0-08 142-17 125-14 342-96 61-96 305-75 117-43 116-21 217-83 179-39 238-10 20-28 +0-05 ^' \ Leap yr. 110-22 102-24 352-02 80-08 253-29 128-88 186-79 249-78 208-96 273-40 23-62 -1-56 97-16 53-48 316-32 56-68 203-54 153-26 213-14 262-84 209-94 286-57 23-73 -1-59 sept-{Srp,^.^; 65-21 30-59 325-38 74-81 151-08 164-70 283-73 294-79 239-51 321-86 27-07 — 3-20 52-14 341-83 289-69 51-41 101-33 189-08 310-08 307-86 240-50 335-04 27-18 —3-23 o->{Srp];r: 33-26 7-70 334-44 92-93 98-62 176-15 354-31 326-74 269-08 357-16 30-41 —4-79 20-19 318-94 298-75 69-54 48-88 200-53 20-66 339-81 270-07 10-33 30-52 —4-82 Nov. i|^°'"-y- \ Leap yr- 348-24 296-05 307-81 87-66 356-41 211-98 91-25 11-76 299-64 45-62 33-87 -6-43 335-iS 247-29 272-11 64-26 306-66 236-36 117-60 24-82 300-62 58-80 33-98 —6-46 Dec. ,;Com.yr. \ Leap yr. 316-29 273-16 316-87 105-79 303-95 223-42 161-83 43-71 329-21 80-92 37-21 —8-02 303-23 224-40 281-17 82-39 254-20 247-80 188-19 56-77 330-19 94-09 37-32 -8-05 •^ \Leap yr. 271-28 201-51 290-23 100-51 201 -74 259-25 258-77 88-72 359-76 129-38 40-66 —9-66 258-21 152-74 254-53 77-12 152-00 283-63 285-12 101-79 0-75 142-56 40-77 —9-69 Day of month. I 2 346-93 311-24 324-30 336-60 310.25 24-38 26-35 13-07 0-99 13-18 o-ii — 0-05 3 333-87 262-47 288-60 313-21 260-50 48-76 52-71 26-13 1-97 26-35 0-22 — 0-11 4 320-80 213-71 252-91 289-81 210-75 73-14 79-06 39-20 2-96 39-53 0-33 — 0-16 5 307-74 164-95 217-21 266-42 161-00 97-53 105-41 52-26 3-94 52-71 0-45 — 0-21 6 294-68 Ii6-i8 181-51 243-02 111-26 121-91 131-76 65-32 4-93 65-88 0-56 — 0-26 7 281-61 67-42 145-81 219-62 61-51 146-29 158-12 78-39 5-91 79-06 0-67 —0-32 8 268-54' 18-66 110-11 196-23 11-76 170-67 184-47 91-46 6-90 92-23 0-78 —0-37 9 255-48 329-90 74-42 172-83 322-01 195-05 210-82 104-52 7-88 105-41 0-89 — 0-42 lO 242-42 281-13 38-72 149-44 272-26 219-43 237-18 117-58 8-87 118-59 1-00, —0-48 II 229-35 232-37 3-02 126-04 222-51 243-82 263-53 130-65 9-86 ■131-76 i-ii —0-53 12 216-28 183-61 327-32 102-65 172-76 268-20 289-88 143-72 10-84 144-94 1-23 -0-58 13 203-22 134-84 291-62 79-25 123-02 292-58 316-23 156-78 11-83 158-12 1-34 — 0-64 14 190-16 86 -08 255-93 55-85 73-27 316-96 342-59 169-84 12-81 171-29 1-45 — 0-69 IS 177-09 37-32 220-23 32-46 23-52 341 -34 8-94 182-91 13-80 184-47 1-56 —0-74 i6 164-02 348-56 184 '53 9-06 333-77 5-72 35-29 195-98 14-78 197-65 1-67 —0-79 17 150-96 299-79 148-83 345-67 284-02 30-10 61-64 209 -04 15-77 210-82 1-78 —0-85 i8 137-90 251-03 113-13 322-27 234-27 54-48 88 -oo 222-10 16-76 224-00 1-89 — 0-90 19 124-83 202-27 77-44 29S-88 184-52 78-87 II4-35 235-17 17-74 237-18 2-01 —0-95 20 I II -76 153-50 41-74 275-48 134-78 103-25 140-70 248-24 18-73 250-35 2-12 — 1-01 21 98-70 104-74 6-04 252-08 85-03 127-63 167-06 261-30 19-71 263-53 2-23 — 1-06 22 85-64 55-98 330-34 228-69 35-28 152-01 193-41 274-36 20-70 276-76 2-34 — i-ii 23 72-57 7-21 294-64 205-29 345-53 176-39 219-76 287-43 21-68 289-88 2-45 — 1-16 24 59-50 318-45 258-95 181-90 295-78 200-77 246-11 300-50 22-67 303 -06 2-56 — 1-22 25 46-44 269-69 223-25 158-50 246-03 225-16 272-47 313-56 23-66 316-23 2-67 —1-27 26 33-3>i 220-92 '87-55 135-10 196-28 249-54 298-82 326-62 24-64 329-41 2-79 —1-32 27 20-31 172-16 151-85 111-71 146-54 273-92 325-17 339-69 25-63 342-59 2-90 —1-38 28 7-24 123-40 116-15 88-31 96-79 298-30 351-52 352-76 26-61 355-76 3-01 —1-43 29 354-18 74-64 80-46 64-92 47-04 322-68 17-88 5'!? 27-60 8-94 3-12 — 1-48 30 341-12 25-87 44-76 41-52 357-29 347-06 44-23 18-88 28-58 22-12 3-23 —1-54 31 328-05 337-11 9-06 l8-I2 307-54 11-44 70-58 31-95 29-57 35-29 3-34 —1-59 * This column gives the longitude of the moon's ascending node for any day when applied to the N oi Table 3. The change in the dial reading of any component (except M2 and Sa) on the Ferrel machine = ± (tabular value for component — tabular value for Mj), the JJ^I^ sign being used when the speed of the component is ^jg^'*'' than that of Mj. The changes for Mj and Sa are given directly by this table. EEPOET FOR 1894— PART II. 207 Table 5. — For adapting the uniformly varying portion ( ¥„) of the equilibrium arguments of Table S to local midnight for any degree of west longitude. West longitude. i i I 2 3 4 5 6 lO 20 3° 40 50 60 70 80 90 100 1 10 120 130 140 ISO 160 170 180 I go 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 K, o-oo O'OO o-oo O'OO O'OI o-oi o-oi 0-01 0-02 0'02 0-02 0-02 o-oj 005 o-o8 o-ii 0-14 o-i6 0-19 0-22 0-25 0-27 0-30 0-33 o"36 038 0-41 0-44 0-47 0-49 0-52 0-S5 0-57 o-6o 0-63 0-66 0-68 071 074 077 079 0-82 0-85 0-88 o-go 0-93 0-96 099 o-oo o-oo 0-00 o-oi O-OI 0-02 0-02 0-03 0-03 0-04 0-04 0-05 0-05 O-II o-i6 0-22 0-27 0-33 0-38 0-44 0-49 o-SS 0-60 0-66 0-71 077 0-82 0-88 0-93 o-gg 1-04 i-io IIS I -20 1-26 I -31 1-37 1-42 1-48 1-53 1-59 1-64 1-70 17s I -81 1-86 I 92 1-97 M. — O-OI — 0-02 — 0-02 — 0-03 — 0-06 — o-og — 0-13 — o-i6 — 0-19 0-25 0-28 031 0-63 ■ 0-94 • 1-26 ■ 1-57 • 1-89 ■ 2 -20 . 251 • 2-83 - 3-14 - 3-46 • 377 - 4-09 - 4-40 - 472 - 5 'OS - 5-34 - 5-66 - 5-97 - 6-29 - 6 -60 - 6-92 - 7-23 - 7 -54 - 7-86 -8-17 - 8-49 - 8 -80 - 9-12 - 9-43 - 974 -10-06 -10*37 -io-6g -11-00 -11-32 - 0-02 - 0-03 - 0-05 - 007 - 0-14 - 0-20 - 0-27 - 0-34 - 0-41 - 0-47 - 0-S4 - 0-61 -'0-68 - 1-35 - 2 03 - 2-71 - 3-39 - 4-06 - 474 - 5 '42 - 6-10 - 6-77 - 7-45 -8-13 - 8 -So - 9-48 -10-16 -10-84 -11-51 -12-19 -12-87 -13-55 -14-22 -14-90 -15-58 -16-25 -16-93 -17-61 -i8-2g -18-96 -19-64 -20-32 —2 1 -00 —21-67 -22-35 -23-03 -23-70 -24-38 — 0-03 — 0-05 — 0-08 — o-io — 0-20 — 0-30 — 0-41 — 0-51 — o-6i — 0-71 — 0-81 — 0-91 - I -02 - 2-03 - 3-05 - 4-06 - 5-08 - 6-10 - 7-11 -8-13 - 9-14 -1016 -II -17 -l2-ig -13-21 -14-22 -15-24 -16-25 -17-27 -18-29 -19-30 -20-32 -21-33 -22-35 -23-37 -24-38 -25-40 -26-41 -27-43 -28-45 -29-46 -30-48 -31-49 -32-51 -33-52 -34-54 -35-56 -36-57 Mt 0-03 0-07 O-IO 0-14 0-27 0-41 0-54 0-68 o-8i 0-95 1-08 - '-35 - 2-71 - 4-06 - 5 -42 - 6-77 - 8-13 - 9-48 -10-84 -i2-ig -13-55 -i4-go -16-25 -17-61 -18-96 -20-32 -21-67 -23-03 -24-38 -25-74 -27-09 -28-45 -29-80 -3i-'5 -32-51 -33-86 -35-22 -36-57 -37-93 -39-28 -40-64 -41-99 -43-34 -44-70 -46-05 -47-41 -48-76 0-05 o-io 0-15 0-21 0-41 0-61 0-81 1-02 1-22 1-42 1-63 1-83 • 2-03 ■ 4-06 6-IO -10-16 -i2-ig -14-22 -16-25 -i8'29 -20-32 -22-35 -24-38 -26-41 -28-45 -30-48 -32-51 -34-54 -36-57 -38-60 -40-64 -42-67 -44-70 -46-73 -48-76 -50-79 -52-83 -54-86 -56-89 -58-92 -60-95 -62-99 —65-02 -67-05 -69-08 -71-11 -73-14 - 0-07 - 0-14 - 0-20 - 0-27 - 0-54 - o-8i - 1-08 - I -35 - 1-63 1-90 - 2-17 - 2-44 - 2-71 - 5-42 - 8-13 -10-84 -13-55 -16-25 -18-96 -21-67 -24-38 -27-09 -29-80 -32-51 -35-22 -37-93 -40-64 -43-34 -46-05 -4S-76 -51-47 -54-18 -56-S9 -59-60 -62-31 -65-02 -67-73 -70-44 -73-14 -75-85 -78-56 -81-27 -83-98 -86-69 -189-40 - g2-ii - g4-82 -97-53 ■ 0-03 ■ 0-05 • o-o8 010 • 0-21 ■ 0-31 - 0-42 - 0-52 - 0-62 - 0-73 - 0-83 - 0-94 - I -04 - 2-08 - 3-12 - 4-16 - 5-20 - 6-24 - 7-28 - 8-32 - 9-06 -10-40 -11-44 -12-48 -13-52 -14-56 -15-60 -16-64 -17-68 -18-72 -ig-76 -20-80 -21-84 -22-88 -23-92 -24-96 -26-00 -27-04 -28-08 -29-13 -30-17 -31-21 -32-25- -33"29 -34-33 -35-37 -36-41 -37-45 0-02 0-04 0-05 0-07 0-14 0-21 0-28 0-35 ; 0-42 0-49 0-56 0-63 ■ 0-70 I -41 211 • 2-82 - 3-52 - 4-23 - 4-93 - 5-64 - 6-34 - 7-05 - 7-75 - !-46 - g-i6 - 9-87 -10-57 -1 1-27 -11-98 -13-39 -14-og -14-80 -15-50 -16-21 -16-91 -17-62 -18-32 -19-03 -19-73 -20-43 -21-14 -22-55 -23-^5 -23-96 -24-66 -25-37 0-00 o-oo 0-00 0-00 O-OI O-OI O-OI O-OI 0-02 0-02 0-02 0-02 0-03 0-05 0-08 O-II 0-14 0-16 0-19 0-22 0-25 ■ 0-27 ■ 0-30 ■ 0-33 - 0-36 ■ 0-38 - 0-41 - 0-44 - 0-47 - 0-49 - 0-52 - 0-55 - 0-57 - o-6o - 0-63 - 0-66 - 0-68 - 0-71 - 0-74 - 0-77 - 0-79 - 0-82 - 0-85 - 0-88 - o-go - 0-93 - 0-96 - o-gg - 0-03 - 0-05 - o-o8 - O-II - 0-21 - 0-32 - 0-43 - 0-53 - 0-64 0-75 - 0-85 - o-g6 - 1-07 - 2-14 - 3-20 - 4-27 - 5-34 - 6-41 - 7-47 - 8-54 - g-6i -IO-68 -11-74 -12-81 -13-88 -14-95 -16-01 -17-08 -18-15 -19-22 -20-28 -21-35 -22-42 -23-49 -24-55 -25-62 -26-6g -27-76 -28-82 -2g-8g -30-96 -32-03 -33-09 -34-16 -35-23 -36-30 -37-36 -38-43 The changes for other components may be found from those above as follows : S2, = Ls-Oi . 6 = ni — T„IM, —h — s I rM,l = Ki4-Ai 1 2N = N2-fAj| OO =K2 — Oi I R2 =Ki I S„ 3 = o K''^ = ^|mN = M, + nJmS =M, |2MS = M. |2SM = MSf | Sa = Kj Ssa = K, 208 UNITED STATES COAST AND GEODETIC SURVEY. Table 5.- —For adapting the uniformly varying portion ( ¥„) of the equilibrium argumei any degree of west longitude — Contiuued. ts of Table S to local midnight for West ongitude. ^2 (*2 •■a MK 2MK MSf Mf Mm h s / M i — O-QI i — 0-03 — 0-02 — 0-02 — 0-03 0-02 0-02 o-oi 000 O-OI 0-00 o-oo i — 0-02 — 0-03 — 0-07 — o-io — 0-05 — 0-07 — 0-03 - 0-05 — 0-07 — O-IO 0-03 0-05 0-04 0-05 0-02 0-03 0-00 0-00 0'02 0-03 0-00 0-00 0-00 O'OO I 2 — 0-04 — 0-07 — o'i4 — 0-27 — O-IO — 0-20 — o-o6 — 0-13 — 0-14 — 0-28 0-07 0-14 0-07 0-15 004 0-07 o-oo O-OI 0-04 0-07 o-oo 0-00 0-00 O'OO 3 — o-ii — 0-41 — 0-30 — 0-I9 — 0-41 0-20 0-22 O-II o-oi O-II 0-00 0-00 4 5 - 0-I5 — o-i8 — 0-54 — 0-68 — 0-40 — 0-50 — 0-26 - 0-32 — 0-55 — 0-69 0-27 0-34 029 0-37 0-I5 0-18 o-oi o-oi 0-15 o-i8 0-00 0-00 O-OO 0-00 6 0'22 — 0-81 — 0-59 — 0-39 - 0-83 0-41 0-44 0-22 0-02 0-22 0-00 O-OO 7 8 9 — 0-25 — 0-29 — 0-33 — 0-95 — 1-08 — 1-22 — 0-69 — 0-79 — 0-89 — 0-45 — 0-52 — 0-58 — 0-97 — I-II — 1-24 0-47 0-54 0-61 0-51 0-59 0-66 0-25 0-29 0-33 0-02 0-02 0-02 0-26 0-29 0-33 0-00 0-00 o-oo 0-00 o-oo 0-00 10 20 30 — 0*36 — 0-73 — 1-09 — 1-35 — 2-71 — 4-06 — 0-99 — 1-98 — 2-97 - 0-65 - 1-30 - 1-95 - 1-38 - 2-76 - 4-15 0-68 1-35 2-03 0-73 1-46 2 20 0-36 0-73 1-09 0-03 005 o-o8 0-37 0-73 i-io o-oo O-OI O-OI 0-00 0-00 o-oo 40 so 60 — 1-45 — i-Si — 2-18 — S-42 -6-77 -8-13 — 3-97 — 4-96 — 5-95 — 2-60 — 3-25 — 3-90 — 5-53 — 6-91 — 8-29 2-71 3-39 4'o6 2-93 3-66 4-39 1-45 1-81 218 o-ii 0-I4 0-16 1-46 1-83 2 -20 O-OI 0-02 0-02 — O-OI — O-OI ^— 001 70 80 90 — 2-54 — 2-90 — 3-27 - 9-48 — 10-84 — 12-19 — 6-94 — 7-93 — 8-92 — 4-55 — 5-20 -5-85 -9-67 — 11-06 —12-44 4*74 5-42 6-10 512 5-86 6-59 2-54 2 90 3-27 0-19 0-22 0-25 2-56 2-93 3-29 0-02 0-02 0-03 — 001 — 0-01 — O-OI 100 no 120 -3-63 — 3-99 — 4-36 -13-55 —14-90 —16-25 — 9-92 — 10-91 — 11-90 — 6-50 — 7-15 — 7-80 —13-82 — 15-20 -16-58 6-77 7-45 8-13 7-32 8-05 - 8-78 3-63 3-99 4-36 0-27 0-30 0-33 3-66 4*03 4-39 0-03 0-03 0-04 — o-oi — 0-02 — 0-02 130 140 150 — 4-72 — 5-08 — 5 "44 — 17-61 —18-96 — 20-32 —12-89 — 13-88 -14-87 — 8-45 — 9-10 — 9-75 —17-96 —19-35 —20-73 8-8o 9-48 10-16 9-52 10-25 10-98 4-72 5 -08 5-44 0-36 0-38 0-41 4-76 5-12 5-49 0'O4 0-04 0-05 — 0-02 — 0-02 — 0-02 160 170 180 -5-81 - 6-17 -6-53 — 21-67 — 23-03 —24-38 -15-87 -16-86 -17-85 — 10-40 — 11-05 — 11-70 — 22-11 — 23-49 -24-87 10-84 11-51 12-19 11-71 12-44 13-18 5-81 6-17 5-53 0-44 0-47 0-49 5-86 6-22 6-59 005 005 0-06 — 0-02 — 0-03 — 0-03 190 200 210 — 6-90 — 7-26 — 7-62 -25-74 -27-09 -28-45 —18-84 -19-83 —20-82 -12-35 —13-00 -13-65 -26-26 —27-64 — 29-02 12-87 13-55 14-22 13-91 14-64 15-37 6-90 7-26 7-62 0-52 0-55 0-S7 6-95 7-32 7-69 0-06 0-06 006 — 0-03 — 003 — 0-03 220 230 240 - 7-98 -8-35 -8-71 — 29-80 -31-1S -32-51 -21-82 — 22-8l —23-80 -14-30 -14-95 — 15-60 —30-40 -31-78 -33-17 14-99 15-58 16-25 1610 16-84 17-57 7-98 8-35 8-71 0-60 0-63 0-66 8-05 8-42 8-78 0-07 0-07 0-07 — 0-03 — 0-03 — 0-04 250 260 270 — 9-°7 — 9-44 — 9-80 -33-86 —35-22 -36-57 —24-79 -25-78 —26-77 — 16-25 — i6-go —17-55 -34-55 —35-93 —37-31 16-93 17-61 18-29 18-30 19-03 19-76 9-07 9'44 9-80 0-68 0-71 0-74 9-15 9-52 9-88 o-o8 o-o8 0-08 — 0-04 — 0-04 — 0*04 280 290 300 —10-16 — 10-52 — 10-89 —37-93 -39-28 — 40-64 -27-77 -28-76 —29-75 — 18-20 —18-85 — 19-50 -38-69 —40-08 —41-46 18-96 19-64 2032 20-50 21-23 21-96 io-i6 10-52 10-89 0-77 0-79 0-82 10-25 10-61 10-98 0-09 0-09 0-09 — 0-04 — 0-04 — 0-04 310 320 330 -11-25 — if6i —11-98 —41 -99 —43-34 —44-70 —3074 — 3«-73 —32-72 —20-15 — 2o-8o —21-45 —42-84 —44-22 -45-60 2I-00 21-67 ■22-35 22-69 23-42 24-16 11-25 11 -61 11-98 0-85 •0-88 0-90 "■35 11-71 1 2 -08 O-IO O-IO O-IO — 0-05 — 0-05 — 0-05 340 350 360 -12-34 — 12-70 -13-07 -46-05 —47-41 -48-76 -3371 —34-71 -35-70 — 22-IO -22-75 —23-40 —46-98 -48-37 -49-75 23-03 23-70 24-38 24-89 25-62 26-35 12-34 12-70 13-07 0-93 0-96 0-99 12-44 1281 13-18 on 0-n O-II — 0-05 — 0-05 — 0-05 EEPOET FOR 1894— PAET II. 209 Table 6.— Values of N, J, and P for Greenwich midnight, beginning each month, from 1850 to 1949. Month. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June Tuly Aug. Sept. Oct. Nov. Dec. 146-201 144-559 [43-076 141-435 139-846 138-205 136-616 134-974 133333 131-744 130-103 128-514 126-872 125-231 123-748 122-107 120-518 118-876 117-288 115-646 114-005 112-416 1 10-774 109-186 107-544 105-903 104-367 102-726 101-137 99-495 97-907 96-265 94*624 53 -035 91-393 89-805 88-163 86-522 85-039 83-397 81-809 80-167 78-578 76-937 75-295 73707 72-065 70-476 68-834 67-193 65-710 64-069 62-480 60-838 59-250 57-608 55-967 54-378 52-736 51-148 1850 i9°379 19-479 19-572 19-679 19-785 19-899 20 -on 20-130 20-253 20-373 20-500 20-625 1851 20-755 20-889 21 -on 21-148 21-281 21-421 21-558 21-700 21-845 21-984 22-129 22-270 1852 22-416 22-563 22-701 22-848 22-991 23-137 23-279 23-426 23'573 23-714 23-859 23-999 1853 24-143 24-285 24-414 24-555 24-689 24-827 24-960 25-095 25-229 25-357 25-488 25-612 1854 25-746 25-864 25-974 26*096 26-210 26-328 26-438 26-550 26-659 26-763 26-867 26-966 91-861 95 015 97-875 loi -049 104-135 107-335 110-442 113-664 116-900 120-041 123-301 126-468 129-753 133-047 136-034 139-353 142-575 145-918 149-164 152-528 155-905 159-183 162-582 165-882 169-301 172-732 175-951 179-401 182-750 186-221 189-588 193-076 196-574 199-968 203-484 206-895 210-426 213-966 217-172 220-727 224-174 227-746 231-207 234-790 238-382 241-863 245-467 248-960 252*587 256-198 259-475 263-107 266-626 270-281 273-799 277-451 281-107 284-651 288-316 291-867 49-506 47-865 46-382 44-741 43-152 41-510 39-922 38-280 36-639 35-050 33-408 31-820 30-178 28-537 27-001 25 -360 23-771 22 129 20-541 18-899 17-258 15-669 14-027 12-439 10-797 9-156 7-673 6-032 4-443 2-801 1-213 359-571 357-930 356-341 354-699 353-I" 351-469 349-828 348-345 346*704 345-115 343*473 341*885 340-243 338-602 337-013 335-371 333-783 332-140 330-499 329-016 327-375 325-786 324-144 322-556 320*914 319-273 317-684 316*042 314-454 1855 o 27-066 27-162 27-247 27-339 27-424 27-509 27-589 27-669 27-746 27-817 27-888 27-952 1856 28-017 28-078 28-132 28-187 28-237 28-285 28-329 28-370 28-408 28-443 28-474 28-501 1857 28-526 28-547 28-563 28-578 28-589 28-597 28-601 28-602 28-599 28-593 28-583 28-571 1858 28-555 28-535' 28-514 28-487 28-458 28-424 28-389 28-349 28-306 28-260 28-211 28-159 1859 28-102 28-043 27-986 27-920 27-853 27-781 27*709 27-629 27-549 27-468 27*380 27-293 295-540 299-217 302-542 306-226 309-794 313-485 317-059 320-756 324-456 328-038 331-744 335-331 339-040 342-752 346-226 349"94i 353-539 357-258 0-858 4-580 8-303 1 1 -909 15-633 19-239 22-968 26-695 30-064 33-793 37-402 41-132 44-741 48-471 52-202 55-842 59-542 63-150 66-879 70-607 73-975 77-701 81-307 85-032 88-636 92-358 96-079 99-679 103-398 106-994 110-709 114-421 117*773 121-481 125*067 128-771 132-353 136*050 139-746 143-319 147*008 150-575 N 312-812 311-171 309-635 307-994 306-405 304-763 303-175 301 -533 299-892 298-303 296-661 295-073 293-431 291-790 290-307 288-666 287-0^7 285-435 283-847 282-205 280-564 278-975 277-333 275-745 274-103 272-462 270-979 269-338 267-749 266-107 264-519 262-877 261-236 259-647 258-005 256-417 254-774 253-133 251-650 250-009 248-420 246-778 245-190 243-548 241-907 240-318 238-676 237-088 235-446 233-805 232-269 230-628 229-039 227-397 225-809 224-167 222-526 220-937 219-295 217-707 i860 o 27-201 27-105 27-014 26-913 26-813 26-707 26-603 26-491 26-378 26-266 26-149 26-032 1861 25-911 25-787 25-673 25-545 25-419 25-288 25-159 25-025 24-888 24-755 24-617 24-481 1862 24-340 24-197 24-068 23-924 23-783 23-638 23-497 23-350 23-203 23 -060 22-914 22-771 22-624 22-477 22-345 22-199 22-058 21-913 21-774 21-631 21-489 21-353 21-214 21 -081 1864 20-945 20-810 20-687 20-556 20-433 20-307 20-189 20-068 19-951 19-840 19-728 19-623 154-257 157-936 161-374 165-045 168-594 172-258 175-797 179-452 183-102 186-629 190-269 193-787 197-415 201 -038 204-309 207-919 211-410 215-011 218-489 222-076 225-656 229-115 232-682 236-126 239-678 243-221 246-415 249-943 253-349 256-860 260-249 263-742 267-226 270-589 274-054 277-398 280-842 284-277 287-371 290-785 294-079 297-473 3OO-.746 304-117 307-476 310-716 314-052 317-270 320-582 323-883 326-960 330-237 333-397 336-650 339-786 343'oi5 346-232 349-332 352-526 355-605 1865 216-065 214-424 212-941 211-300 209-711 208-069 206-481 204-839 203-198 201 -609 99-967 98-379 96-737 95-096 93-613 91-972 90-383 88-741 87-153 85-511 83-870 82-281 80-639 79-051 77-409 75-768 74-285 72-644 71-054 69-413 67-825 66-183 64-542 62-953 61-311 59-723 58-080 56-439 54-903 53-262 51-673 50-031 48-443 46-801 45-160 43-571 41-929 40-341 i9°5i8 19-416 19-329 19-234 19-146 19-060 18-980 18-902 18-828 18-761 18-696 18-638 1866 18-583 18-532 18-490 18-449 18-415 18-383 18-359 18-338 18-323 18-314 18-309 18-309 1867 18-315 18-326 18-341 18-362 18-387 18-419 18-454 18-496 18-542 18-592 18-649 18-707 1868 18-774 18-844 18-914 18-992 19-073 19-160 19-248 19-343 19-442 19-541 19-647 19-752 1869 38-699 19-864 37-058 19-980 35-575 20-087 33-934 20-208 32-345 20-328 30-703 20-453 29-115 20-577 27-473 20-707 25-832 20-840 24-243 20-970 22-601 21-106 21-013 21-240 358-775 1-933 4-777 7'9i3 10-938 14-054 17-061 20-157 23-246 26-223 29-294 32-258 35-312 38-360 41-108 44-144 47-078 50106 53-031 56-051 59-070 61-989 65 -004 67-923 70-939 74-955 76-682 79-676 82-628 85-657 88-591 91-628 94-671 97-623 100-679 103.-644 106-715 109-795 112-686 115-783 118-791 121-908 124-935 128-074 131-224 134-283 137-456 140-538 143-733 146-940 149-848 153-078 156-216 159-472 162-633 165-911 169-205 172-399 175-715 178-934 A^^the mean longitude of the moon's ascending node. /=the inclination of the lunar orbit to the plane of the earth's equator. P = the mean longitude of the lunar perigee measured from the intersection of moon's orbit vpith the plane of the earth's equator. S. Ex. 8, pt. 2- -14 iilO UNITED STATES COAST AND GEODETIC SURVEY. Table 6. — Values of N, I, and P for Greenwich midnight, beginning each month, from 1850 to 1949 — Continued. Month. N / p N '^ P N / p N I p 1870 1875 1 1880 1885 t ] Jan. I H9-37I 21-379 „ 182-274 22-677 28-270 33-381 285°983 25-332 251-173 189-236 18-393 86-557 Feb. I 117730 21-520 185-623 21 -036 28-316 37-100 284-342 25-200 254-770 187-595 18-366 89-582 Mar. I 116-247 21 -649 188-660 19-553 28355 40-462 282*806 25-074 258-128 186-II2 18-345 92-312 Apr. I 114-606 21-793 192-032 17-912 28-394 44-184 281-165 24-939 261-710 184-471 18-327 95-329 May I 113-017 21 -930 195-306 16-323 28-429 47-788 279-576 24-806 265-171 182-882 18-317 98-249 June I m-375 22-076 198-703 14-681 28-462 51-514 277-934 24-667 268-741 181-240 18-309 101-265 July 1 109-787 22-217 201 -997 13-093 28-490 55-120 276-346 24-532 272-188 179-652 18-309 104-183 Aug. I Sept. I 108-145 22-363 205-414 11-451 28-517 58-846 274704 24-392 275-742 178-010 18-312 107-198 106-504 22-510 208-839 9-810 28-539 62-574 273-063 24-249 279-289 176-369 18-321 IIO-214 Oct. I 104-915 22-652 212-166 8-221 28-558 66-183 271-474 24'1II 282-711 174-780 18-335 "3-'3S Nov. I Dec. I 103-273 22-798 215-613 6-579 28-573 69-913 269-832 23-967 286-244 173-138 18-355 116-155 101-685 22-941 218-958 4-991 28-585 73-521 268-244 23-828 289-652 171-550 18-379 119-080 1871 1876 1881 1886 1 Jan. I 100-043 23-088 222-425 3-349 28-595 77-249 266-602 23-682 293-166 169-908 18-409 122-107 Feb. I 98-402 23-236 225-901 1-708 28-600 80-981 264-961 23*536 296-671 168-267 18-444 125-138 Mar. I 96:919 23-368 229-050 0-172 28-602 84-471 263-478 23-403 299-829 166-784 18-480 128-881 Apr. I 95-278 23-515 232-543 358-531 28-600 88-203 261-837 23-257 303 -3 '6 165-143 18-525 130-920 May I 93-689 23-656 235-934 356-942 28-596 91-811 260-248 23-114 306-682 163-554 18-574 133-867 June I 92-047 23-802 239-446 355-300 28-587 95-541 258-606 22-968 310-151 161-912 18-627 136-923 July I 90-459 23-942 242-853 353-712 28-576 99-149 257 018 22-823 313-498 160-324 18-685 139-886 Aug. I 88-817 24-086 246-382 352-070 . 28-561 102-878 255-376 22-678 316-946 158-682 18-749 142-954 Sept. I 87-176 24-229 249-920 350-429 28-542 106-608 253-735 22-531 320-386 157-041 18-817 146-031 Oct. I 85-587 24-367 253-350 348-840 28-521 110-215 252-146 22-389 323-704 155-452 18-888 149-017 Nov. I 83-945 24-508 256-904 347-198 28-495 "3-943 250-504 22-242 327-123 153-810 18-965 152113 Dec. I 82-357 24-643 260-349 345-610 28-467 117-549 248-916 22-102 330-419 152-222 19-044 155-116 1872 1877 1882 1887 Jan. I 80-715 24-782 263-916 343-967 28-435 121-273 247-273 21-956 333-815 150-579 19131 158-230 Feb. I 79-074 24-919 267-492 342-326 28-399 124-997 245-632 21-813 337-200 148-938 19-220 161-355 Mar. I 77-538 25 -046 271-843 340-843 28-364 128-361 244-149 21-683 340-249 147-455 19-305 164-187 Apr. I 75-897 25-180 274-431 339-202 28-322 132-082 242-508 21-541 343-612 145-814 19-463 167-332 May I 74-308 25-309 277-910 337-613 28-278 135-682 240-919 21-404 346-856 144-225 19-500 170-387 June I 72-666 25-440 281-512 335-971 28-229 139-401 239-277 21-264 350-197 142-583 19-604 173-554 July I 71-078 25-565 285 -003 334-383 28-179 142-998 237-689 21-131 353-418 140-995 19-708 176-631 Aug. 1 69-436 25-693 288-615 332-741 28-123 146-708 236-047 20-994 356-735 139-353 19-819 179-822 Sept. 1 67-795 25-819 292-236 331-100 28-065 150-426 234-406 20-859 0-040 137-712 19-934 183-025 Oct. I 66-206 25-938 295-744 329-511 28-005 154-017 232-817 20-731 3-227 136-123 20-047 186-136 Nov. I 64-564 26-059 299-375 327-869 27-940 157-727 231-175 20 -600 6-509 134-481 20-167 189-363 Dec. I 62-976 26-175 302-893 326-281 27-874 161-314 229-587 20-47S 9-672 132-893 20-307 192-461 1873' 1878 1883 1888 Jan. I 61-333 26-292 305 ■5.34 324-639 27-803 165-019 227-945 20-349 12-930 131-251 20-411 195-747 Feb. I 59-692 26-408 310-180 322-998 27-729 168-720 226-304 20-226 16-174 129-610 20-538 199-021 Mar. I 58-209 26-509 313-478 321-515 27-659 172-062 224-821 20-116 19-095 128-074 20 -660 202-075 Apr. I 56-568 26-620 3'7-i33 319-874 27-579 175-757 223-180 19-997 22-316 126-433 20-791 205-361 May I 54-979 26-724 321-674 318-285 27-501 179-331 221-591 19-884 25-422 124-844 20-921 208-553 June I 53-337 26-829 324-338 316-643 27-413 183-020 219-949 19-772 28-620 123-202 21-056 211-864 July I 51-749 26-929 327-889 315-055 27-327 186-590 218-361 19-666 31-704 121-614 21-189 215-081 Aug. I 50-107 27-030 331-560 313-413 27-235 190-273 216-719 19-559 34-878 119-972 21-328 218-413 Sept. I 48-466 27-127 335-236 311-772 27-141 193-954 215-078 19-457 38-041 118-331 21-468 221-760 Oct. I 46-877 27-219 338-7^6 310-183 27-047 197-511 213-489 19-360 41-091 116-742 21-606 225-009 Nov. I 45-235 27-311 342-480 308-541 26-947 201-185 211-847 19-264 44-231 115-100 21-748 228-378 Dec. I 43-647 27-397 346-047 306-953 26-848 204-735 210-259 19-176 47-261 113-512 21-887 231-649 1874 1879 1884 1889 Jan. I 42-005 27-484 349-736 305-3" 26-743 208-399 208-617 19-088 50-379 1 1 1 870 22-032 235-040 Feb. I 40-364 27-567 353-429 303-670 26-635 212-059 206-976 19-004 53-489 110-229 22-177 238-442 Mar. I 38-881 27-642 356-768 302-187 26-536 215-361 205-440 18-930 56-390 108-746 22-309 241-524 Apr. I 37-240 27-718 0-465 300-546 26 -424 219-012 203-799 18-854 59-480 107-105 22-455 244-947 May I 35-651 27-790 4-047 298-957 26-313 222-542 202-210 18-786 62-462 105-516 22-598 248-269 June I 34-009 27-862 7-752 297-315 26-196 226-184 200-568 18-719 65-535 103-874 22-744 251-713 July I 32-421 27-928 11-338 295-727 26-081 229-703 198-980 18-659 68-501 102-286 22-887 255-055 Aug. I 30-779 27-994 15-046 294-085 25-959 233-334 197-338 18-602 71-558 100-644 23-034 258-518 Sept. I 29-138 28-056 18-757 292-444 25-837 236-961 195-697 18-548 74-610 99-003 23-182 261-991 Oct. I 27-549 28-113 22-349 290-855 25-715 240-464 194-108 18-504 77-555 97-414 23-324 265-361 Nov. I 25-907 28-169 26-066 289-213 25-588 244-080 192-474 18-461 80-594 95-772 23-470 268-853 Dec. I 24-319 28-220 29-663 287-625 25-463 247-571 190-878 18-425 83-530 94-184 23-612 272-241 EEPOET FOE 1894— PAET II. 211 Table 6.~ Values of N, I, and P for Greenwich midnight, beginning each month, from 1850 to i949— Continued. Month. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jul e July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 1890 92-542 23-757 90-901 23-903 89-418 24-033 87777 24-176 86-188 24-314 84-546 24-456 82-958 24-593 81-316 24730 79-675 24-868 78-086 25-001 76-444 25-136 74-856 25 264 73-214 71-573 70-090 68-449 66-860 • 65-218 63-630 61-988 60-347 58-758 57-116 55-528 53-885 52-244 50-708 49-067 47-478 45-836 44-248 42-606 40-965 39*376 37734 36-146 34*504 32-863 3 1 '380 29-739 28-150 26-508 24-920 23-278 21-637 20-048 18-406 16-818 25-396 25-526 25 -642 25-769 25-889 26-011 26-128 26-246 26-362 26-471 26-583 26-688 26-794 26-898 26-993 27-091 27-185 27-277 27-365 27-453 27-537 27-616 27-695 27-768 1893 27-841 27 -909 27-970 28-034 28-093 28-149 28 -202 28-252 28-299 28-342 28-382 28-418 1894 15-176 28-452 13-535 28-482 12-052 28-507 10-41 1 28-531 8-822 28-551 7-180 28-568 5-592 28-582 3-950 28-591 2-309 28-598 0-720 28-602 359-078 28-601 357-490 28-598 275-751 279-267 282-454 285-988 289-415 292-966 296-409 299-974 303-547 307-010 310-597 3«3-o74 317-673 321-278 324-540 328-157 331-663 335-291 338-808 342-446 346-091 349-622 353-276 356-815 0-478 4-145 7-579 n-253 14-812 18-494 22-o6l 25-748 29-440 33-016 36-714 40-295 43-998 47-703 51-054 54-763 58-355 62-070 65-666 69-383 73' 103 76-703 80-426 84-030 87-754 91-479 94-846 98-573 102-181 105-915 109-520 113-248 116-979 120-589 124-320 127-929 355-848 354-207 352-724 351-083 349-494 347-852 346-264 344-622 342-981 341 -392 339-750 338-162 336-519 334-878 333-342 331-701 330-112 328-470 326-882 325 -240 323-599 322-010 320-368 318-780 317-138 315-497 314-014 312-373 310-784 309-142 307-554 305-912 304-271 302-682 301-040 299-452 297-810 296-169 294-686 293-045 291-456 289-814 288-226 286-584 284-943 283-354 281-712 280-124 278-482 276-841 275-358 273-717 272-128 270-486 268-898 267-256 265-615 264-026 262-384 260-796 1895 o 28-590 28-580 28-568 28-550 28-530 28-506 28-479 28-448 28-414 28-377 28-337 28-294 1896 28-246 28-195 28-144 28-086 28-028 28-963 27-899 27-830 27757 27-682 27-603 27-524 1897 27-439 27-351 27-269 27-176 27-081 26-984 26-885 26-782 26-675 26-569 26-458 26-348 1899 24713 24-574 24-448 24-306 24-168 24-025 23-885 23-740 23-594 23-452 23-306 23-164 26-231 26-113 25-004 25-881 25-762 25-634 25-511 25-381 25-238 25-119 24-984 24-851 131-659 135-388 138-754 142-486 146-104 149-822 153-428 157-153 160-878 164-482 168-204 171-805 175-523 179-240 182-717 186-430 190-023 193-733 197-321 201-026 204-728 208-309 212-007 215-582 219-273 222-960 226-290 229-970 233-529 237 -204 240-756 244-421 248-084 251-623 255-276 258-807 262-450 266-087 269-370 272-997 276-503 280-120 283-614 287-218 290-818 294-294 297-879 301-342 304-915 308-478 311-692 315-241 318-668 322-203 325-614 329-131 332-639 336-026 339-517 342-886 Jf 259-154 257-513 256-030 254-389 252-800 251-158 249-570 247-928 246-287 244-698 243-056 241 -468 239-825 238-184 236-701 235 060 233-471 231-829 230-241 228-599 226-958 225-369 223-727 222-139 220-497 218-856 217-373 215-732 214-143 212-501 210-913 209-271 207-630 206-041 204-399 202-811 201-169 199-528 198-045 196-404 194-815 193-173 191-585 189-943 188-302 186-713 185-071 183-483 1900 23-017 22-869 22-737 22-589 22-447 22-301 22-159 22 015 21-869 21-730 21-588 21-451 1901 21-311 21-172 21-048 20-913 20-784 20-652 20-527 20-400 20-275 20-156 20-036 19-923 1902 19-809 19-698 19-602 19-497 19-400 19-302 19-212 19-123 19-037 18-958 i8-88i 18-811 1903 18-743 18-679 18-626 18-572 18-524 18-479 18-441 18-406 18-377 18-353 18-334 18-320 1904 181-841 18-312 180-200 18-308 178-664 18-310 177-023 18-317 175-434 18-329 173-792 18-346 172-204 18-369 170-562 18-397 168-921 18-430 167-332 18-466 165-690 18-510 164-102 18-556 346357 349-819 352938 356-380 359-702 3-124 6-426 9-826 13-217 16-487 19-854 23-102 26-447 29-779 32-780 36-089 39-280 42-567 45-735 49-007 52-247 55-380 58-606 61-716 64-918 68-107 70-979 74-146 77-200 80-345 83-379 86-502 89-61J6 92-619 95-714 98-699 101-774 104-842 107-607 110-660 113-609 116-650 119-587 122-617 125-644 128 5 69 131-589 134-509 137-525 140-540 143-361 146-376 149-296 152-315 155-239 158*264 161 -296 164-229 167-269 170-215 62-460 60-819 59-336 57-695 56-106 54-464 52-876 51-234 49-593 48-004 46-362 44-774 43-131 41-490 40-007 38-366 36-777 35-135 33-547 31-905 30-264 28-675 27-033 25-445 23-803 22-162 20-679 19-038 17-449 15-807 14-219 12-577 10-936 09-347 07-705 06-117 04-475 02-834 01-298 99-657 98-068 96-426 94-838 93-196 91-555 89-966 88-324 86-736 85-094 83-453 81-970 80-329 78-740 77-098 75-510 73-868 72-227 70-638 68-996 67-408 1905 18-608 18-666 18723 18-790 18-858 18-935 19-011 19-096 19-184 19-273 19-369 19-466 1906 19-568 19-676 19-775 19-887 20-000 20- 119 20-237 20-360 20-497 20-6l2 20-743 20-872 1907 2 1 -007 21-143 21-268 21-408 21-544 21-687 21-826 21-969 22-115 22-256 22-402 22-544 1908 22-691 22-838 22-976 23-123 23-265 23-413 23-554 23-700 ^3-845 23-985 24-128 24-267 1909 24-409 24-549 24-676 24-814 24-946 25-082 25-212 25-345 25-475 25-599 25-726 25-848 173-267 176-328 179-055 182-168 185151 188-242 191-243 194-353 197-474 200-504 203-645 206-696 209-861 213-035 215-915 219-113 222-205 225-441 228-570 231-815 235-075 238-239 241-521 244-709 248-015 251-333 254-341 257-682 260-926 264-290 267-557 270-942 274-340 277-638 281-058 284-376 287-816 291-266 294-503 297.971 301-338 304-826 308-210 311-715 315-231 318-640 322-172 325-597 329-145 332-700 335-918 339-488 342-949 346-532 350-007 353-603 357-206 0-699 4-314 7-819 212 UNITED STATES COAST AKD GEODETIC SUEVEY. Table 6. — Values of N, I, and Pfor Greenwich midnight, leginning each month, from 1850 to 1949— Caatiaxiaa. Month. N / p iV I P N / p N / P 1910 I9I5 1920 1925 Jan. I 65-765 25-970 1 1 -446 329°o7i 27-988 229-741 232°377 2o-°695 78-837 135°630 20-083 26i°833 Feb. I 64- 1 24 26-092 15-077 327-430 27-922 233-449 230-736 20-565 82-115 133-989 20-204 265 -064 Mar. 1 62'64i 26-199 18-363 325-947 27-860 236-797 229-200 20-446 85-171 132-506 20-315 267-992 Apr. I 61 -000 26-316 22-004 324-306 27-788 240-500 227-559 20-318 88-425 130-865 20-441 271-245 May I 59-411 26-427 25-534 322-717 27-716 244-082 225-970 20-201 91-562 129-276 20-564 274-406 June I 57769 26-539 29-186 321-075 27-638 247-781 224-328 20-080 94-793 127-634 20-695 277-684 July I 56-181 26-645 32-723 319-487 27-559 251-357 222-740 19-966 97-907 126-046 20-822 280-868 Aug. I 54-539 26-753 36-384 317-845 27-476 255-049 221-098 19-851 101-112 124-404 20-957 284-169 Sept. I 52-898 26-865 40-049 316-204 27-389 258-739 219-457 19-739 104-307 122-763 21-093 287-483 Oct. I 51-309 26-956 43-600 314-615 27-302 262-307 217-868 19-634 107-387 121-174 21-226 290-701 Nov. I 49-667 27-056 47-273 312-973 27-210 265 -990 216-226 19-528 110-558 119-532 21-365 294-039 Dec. I 48-079 27-149 50-831 311-385 27-118 269-550 214-638 19-430 113-616 117-944 21-502 297-280 19U 1916 1921 1926 Jan. I 46-437 27-244 54-512 309-743 27-021 273-224 212-996 19-332 116-765 116-302 21-644 300-641 Feb. I 44-796 27-335 SS-I9S 308-102 26-919 276-897 211-355 19-237 119-902 114-661 21-787 304-012 Mar. I 43-313 27-415 61-526 306-566 26-823 280-328 209-872 19-155 122-727 113-178 21-916 307-068 Apr. I 41-672 27-501 65-216 304-925 26-718 283-991 208-231 19-068 125-843 111-537 22-062 310-461 May I 40-083 27-581 68-790 303-336 26-613 287-532 206-642 18-988 128-850 109-948 22-203 313-756 June I 38-441 27-661 72-487 301-694 26-502 291-187 205-000 18-909 131-948 108-306 22-348 317-171 July 1 36-853 27-736 76-066 300-106 26-393 294-719 203-412 18-837 134-937 106-718 22-490 320-486 Aug. I 3S-2H 27-810 79-768 298-464 26-277 298-365 201-770 18-767 138-016 105-076 22-637 323-922 Sept. I 33-570 27-881 83-473 296-823 26-160 302-005 200-129 18-702 141-087 103-435 22-784 327-368 Oct. I 31-981 27-946 87-060 295-234 26-044 305-523 198-540 18-643 144-052 101-846 22-927 330-712 Nov. I 30-339 28-011 90-770 293-592 25-923 309-153 196-898 18-588 147-107 100-204 23-074 334-179 Dec. I 28-751 28-070 94-362 292-004 25-803 312-661 195-310 18-538 150-058 98-616 23-217 337-541 1912 1917 1922 1927 1 Jan. I 27-109 28-128 98-074 290-362 25-677 316-280 193-668 18-492 153-099 96-974 23-363 341-028 Feb. I 25-468 28-184 101-790 288-721 25-549 319-892 192-027 18-451 156-136 95-333 23-510 344-521 Mar. I 23-932 28-232 105-268 287-238 25-432 323-151 190-544 18-418 158-875 93-850 23-641 347-686 Apr. I 22-291 28-281 108-986 285-597 25-301 326-751 188-903 18-386 161-903 92-209 23-787 351-196 May I 20-702 28-325 112-586 284-008 25-173 330-230 187-314 18-361 164-829 90-620 23-927 354-602 June I 19-060 28-367 116-308 282-366 25-038 333-819 185-672 18-340 167-850 88-978 24-072 358-131 July I 17-472 28-404 119-911 280-778 24-906 337-284 184-084 18-325 170-771 87-390 24-210 1-553 Aug. I 15-830 28-439 123-635 279-136 24-769 340-858 182-442 18-315 173-787 85-748 24-353 5-096 Sept. 1 14-189 28-471 127-361 277-495 24-631 344-426 180-801 18-309 176-803 84-107 24-494 8-649 Oct. I i2-6oo 28-498 130-966 275-906 24-495 347-871 179-212 18-309 179-721 82-518 24-629 12-094 Nov. I 10-958 28-524 134-695 274-264 24-354 351-424 177-570 18-315 182-737 80-876 24-768 15-662 Dec. 1 9-370 28-545 138-302 272-676 24-215 354-853 175-982 18-324 185-655 79-288 24-902 19-121 1913 1918 1923 1928 ' Jan. I 7-728 28-563 142-032 271-034 24-073 358-389 174-340 18-340 188-674 77-646 25-037 22-701 Feb. I 6-087 28-578 145 -.760 269-393 23-929 1-917 172-699 18-361 190-695 76-005 25-172 26-290 Mar. I 4-604 28-588 149-130 267-910 23-798 5-097 171-216 18-385 194-426 74-469 25-296 29-653 Apr. I 2-963 28-596 152-858 266-269 23-652 8-608 169-575 18-415 197-453 72-828 25-427 33-253 May I 1-374 28-600 156-468 264-680 23-510 11-999 167-986 18-450 200-387 71-239 25-552 36-744 June I 359-732 28-602 160-199 263-038 23-364 15-493 166-344 18-491 203-425 69-597 25-680 40-356 July I 358-144 28-599 163-809 261-450 23-223 18-866 164-756 18-534 206-368 68-009 25-802 43-859 Aug. I 356-502 28-594 167-539 259-808 23-075 22-341 163-114 18-587 209-417 66-367 25-926 47-483 Sept. I 354-861 28-584 171-268 258-167 22-929 25-807 161-473 18-643 212-473 64-726 26-047 51-113 Oct. I 353-272 28-572 174-877 256-578 22-785 29-152 159-884 18-701 215-437 63-137 26-163 54-631 Nov. I 351-630 28-557 178-607 254-936 22-638 32-599 158-242 18-772 218-508 61-495 26-280 58-272 Dec. I 350-042 28-537 182-215 453-348 22-496 35-924 156-654 18-834 221 -488 59-907 26-393 61-800 1914 1919 1 1924 1 1929 1 Jan. I 348-400 28-514 185-944 251-706 22-350 39-350 155-011 18-908 224-575 58-264 26-505 65-450 Feb. I 346-759 28-488 189-670 250-065 22-204 42-765 153-370 18-987 227-673 56-623 26-616 69-104 Mar. I 345-276 28-461 193-036 248-582 22-072 45-841 151-834 19-065 230-580 55-140 26-714 72-409 Apr. I 343-635 28-428 196-761 246-941 21-927 49-235 150-193 19-152 233-696 53-499 26-819 76-072 May I 342-046 28-393 200-364 245-352 21-788 52-508 148-604 19-239 236-714 51-910 26-919 79-622 June I 340-404 28-354 204-088 243-710 21-645 55-881 146-962 19-334 239-860 50-268 27-020 83-294 July I 338-816 28-312 207-689 242-122 21-507 59-133 145-374 19-429 242-907 48-680 27-114 86-851 Aug. I 337-174 28-265 211-408 240-480 21-366 62-482 143-732 19-531 246-067 47-038 27-210 90-529 Sept. I 335-533 28-216 215-127 238-839 21-228 65-820 142-091 19-636 249-238 4S'397 27-302 94-212 Oct. I 333-944 28-164 218-723 237-250 21-094 69-038 140-502 19-742 252-318 43-808 27-388 97-779 Nov. I 332-302 28-108 222-438 235-608 ,20-958 72-353 138-860 19-854 255-513 42-166 27-475 101-469 Dec. I 330-714 28-051 226-031 234-020 20-828 75-548 137-272 19-965 258-616 40-578 27-556 105-042 REPOET FOE 1894— PAET II. 213 Table 6. — Values of N, I, and P for Greenwich midnight, beginning each month, from 1850 to 1949 — Continued. Month. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June J«iy Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. N 1930 38936 37-295 35-812 34-171 32-582 30-940 29-352 27-710 26-069 24-480 22-838 21-250 19-608 17-967 16-484 I4'843 I3-2S4 11-612 10-024 8-382 6-741 5-152 3'Sio 1-922 0-280 358-639 357-103 355-462 353-873 352-231 350-643 349-001 347-360 345-771 344-129 342-541 340-899 339-258 337-775 336-134 334-545 332-903 331-315 329-673 328-032 326-443 324-801 323-213 321-570 319-929 318-446 316-805 315-216 313-574 311-986 3'o-344 308-703 307-114 305-472 303-884 27-637 27-715 27-783 27-855 27-922 28-003 28-048 28-107 28-164 28-216 28-265 28-310 1931 28-353 28-393 28-425 28-459 28-487 28-514 28-537 28-556 28-572 28-584 28-594 28-599 1932 28-602 28-600 28-596 28-589 28-577 28-562 28-545 28-523 28-498 28-470 28-439 28-404 1933 28-366 28-324 28-283 28-234 28-184 28-129 28-072 28-012 27-946 27-881 27-810 27-739 1934 27-661 27-581 27-507 27-421 27-336 27-244 27-153 27-057 26-957 26-858 26-753 26-649 108-738 112-436 115-780 119-483 123-069 126-778 130-368 134-081 137-796 141-392 145 1 1 1 148-710 152-432 156-154 159-519 163-243 166-849 170-576 174-184 177-912 181-642 185-250 188-980 192-674 196-319 200-050 203-539 207-268 210-877 214-607 218-216 221-943 225-671 229-277 233-003 236-607 240-330 244-052 247-412 25I-I3I 254-728 258-444 262-037 265-748 269-458 273-045 276-750 280-333 284-032 287-728 291-065 294-755 298-323 302-007 305-570 309-246 312-919 316-470 320-136 323-678 1935 ,0 302-242 26-539 300-601 26-428 299-118 26-324 297-477 26-207 295-888 26-093 294-246 25-971 292-658 25-853 291-016 25-727 289-375 25-600 287-786 25-476 286-144 25-346 284-556 25-217 282-914 281-273 279.737 278-096 276-507 274-865 273-277 271-635 269-994 268-405 266-763 265-175 263-533 261-892 260-409 258-768 257-179 255-537 253-949 252-307 250-666 249-077 247-435 245-847 244-205 242-564 241-081 239-440 237-851 236-209 234-621 232-979 231-338 229-749 228-107 226-519 224-876 223-235 221-752 220-111 218-522 216-880 215-292 213-650 212-009 210-420 208-778 207-190 1936 25-083 24-947 24-819 24-681 24-546 24-405 24-268 24-125 23-981 23-842 23-696 23-555 1937 23-409 23-261 23-130 22-983 22-839 22-692 22-550 22-403 22-257 22-116 21-970 21-832 1938 21-688 21-546 21-418 21-278 21-144 21-008 20-877 20-744 20-613 20-488 20-361 20-241 1939 20-120 20-001 19-896 19-783 19-676 19-569 19-470 19-370 19-274 19-185 19-095 19-015 327-333 330-985 334-280 337-922 341-441 345-074 348-584 352-204 355-820 359-313 2-916 6-396 9-987 13-570 16-916 20-486 23-934 27-489 30-922 34-462 37-992 41-402 44-917 48-310 51-807 55-294 58-437 61-906 65-254 68-704 72-033 75-463 78-882 82-180 85-579 88-855 92-231 95-594 98-623 101-965 105-187 108-506 111-705 115-000 118-283 121-447 124-706 127-847 131-082 134-303 137-203 140-402 143-487 146-662 149-725 152-877 156-019 159-049 162-179 165-180 1940 205-548 203-907 202-371 200-730 99-141 97-499 95-911 94-269 92-628 91-039 89-397 87-809 86-167 84-526 83-043 81-402 79-813 78-171 76-583 74-941 73-300 71-711 70-069 68-481 66-839 65-198 63-715 62-074 60-485 58-843 57-255 55-613 53-972 52-5S3 50-/41 49-153 47-510 45-869 44-386 42-745 41-156 39-514 37-926 36-284 34-643 33-054 31-412 29-824 28-182 26-541 25-005 23-364 21-775 20-133 18-545 16-903 15-262 13-673 12-031 10-443 18-935 18-859 18-793 18-725 18-665 18-607 18-556 18-509 18-465 18-429 18-396 18-369 I94I 18-346 18-328 18-317 I8-3II 18-308 18-312 18-320 18-333 18-353 18-377 18-406 18-440 1942 18-479 18-523 18-568 18-621 18-679 18-742 18-808 18-881 18-958 19-036 19-122 19-208 1943 19-302 19-399 19-490 19-594 19-698 19-808 19-920 20-036 20-155 20-274 20-399 20-522 1944 20-651 20-783 20-908 21-043 21-176 21-315 21-450 21-592 21-734 21-873 22-018 22-158 168-280 171-372 174-256 177-329 180-296 183-355 186-308 189-352 192-392 195-327 198-356 201 -284 204-306 207-324 210-050 213-066 215-983 218-998 221-917 224-934 227-955 230-879 233-906 236-839 239-873 242-914 245 -666 248-719 251-680 254-749 257-725 260-809 263-904 266-907 270-021 273-043 276-178 279-322 282-173 285-339 288-415 291-605 294-703 297-916 301 -141 304-275 307-524 310-681 313-946 317-241 320-325 323-634 326-849 330-182 333-419 336-775 340-143 343-413 346-803 350-094 108-801 107-160 105-677 104-036 102-447 100-805 99-217 97-575 95-934 94-345 92-703 91-115 1945 o 22-305 22-451 22-583 22-731 22-873 23-020 23-162 23-309 23-456 23-598 23-743 2^-884 1946 89-473 24-028 87-832 24-171 86-349 24-300 84-708 24-442 83-119 24-578 81-477 24-717 79-889 24-850 78-247 24-987 76-606 25-122 75-017 25-252 73-375 25-384 71-787 25-511 70-145 68-504 67-021 65-380 63-791 62-149 60-561 58-919 57-278 55-689 54-047 52-459 50-816 49-175 47-639 45-998 44-409 42-767 41-179 39-537 37-896 36-307 34-665 33-077 31 "435 29-794 28-311 26-670 25-081 23-439 21-851 20-209 18-568 16-979 15-337 13-749 1947 25-638 25-765 25-876 25-999 26-117 26-234 26-347 26.461 26-570 26-678 26-784 26-885 1948 26-986 27 085 27-175 27-268 27-356 27-444 27-526 27-608 27-687 27-761 27-834 27-901 1949 27-968 28-032 28-086 28-143 28-196 28-247 28-293 28-338 28-378 28-414 28-449 28-479 353-506 356-928 0-029 3-470 6-811 10-275 13-634 17-115 20-606 23-993 27-502 30-905 34-431 37-965 41-164 44-714 48-155 51-721 55-177 58-755 62-340 65-817 68-416 72-904 76-515 80132 83-404 87-031 90-547 94-186 97-712 101-360 105-013 108-552 112-215 115-762 119-433 123107 126-548 130-228 133-795 137-494 141-055 144-750 148-447 152-028 155-731 159-316 163-025 166-737 170-088 173-801 177-397 181-115 184-714 188-433 192-157 195-760 199-486 203-091 214 UNITED STATES COAST AND GEODETIC SURVEY. Table 7. — Values of I, v, ?, v' , and S v", corresponding to each half degree of N. N / V £ v' 2 v" N N / V r v' 2 v" N o „o „ ■ a o-o 28-602 0-000 o-ooo 0-000 o-ooo 360-0 30-0 28-024 S-478 4-939 3-903 8-249 330-0 o-S 28-602 0-094 0-084 0-067 0-I43 359-5 30-5 28-005 5-564 5-017 3-964 8-377 329-5 i-o 28-601 0-188 0-169 0-I34 0-285 3S9-0 31-0 27-985 5-651 5-095 4-025 8-504 329-0 i-S 28-600 0-281 0-253 0-201 0-427 358-5 31-5 27-965 5-736 5-173 4-085 8-631 328-5 2-0 28-599 0-37S 0-337 0-268 0-569 358-0 32-0 27-945 5-822 5-251 4-146 8-757 328-0 2-S 28-598 0-468 0-421 0-335 0-711 357-5 32-5 27-925 5-907 5-328 4-206 8-883 327-5 3-0 28-596 0-562 0-506 0-402 0-853 357-0 33-0 27-904 5-992 5-406 4-266 9-008 327-0 3-5 28-594 0-656 0-590 0-469 0-995 356-5 33-5 27-884 6-077 5-482 4-326 9-133 326-5 4-0 28-591 0-749 0-674 0-536 1-137 356-0 34-0 27-862 6-162 5-559 4.386 9-257 326-0 4-S 28-589 0-843 0-758 0-603 1-279 355-5 34.5 27-841 6-246 5-636 4-445 9-381 325-5 S-o 28-585 0-936 0-842 0-670 I -421 355-0 35 -o 27-819 6-330 5-712 4-504 9-504 325-0 S-5 28-582 1-030 0-926 0-737 1-563 3S4-S 35-5 27-797 6-414 5-788 4-563 9-626 324-5 6-0 28-578 1-123 i-oio 0-804 1-705 354-0 36-0 27-775 6-497 5-864 4-621 9-748 324-0 6-5 28-574 1-217 1-094 0-871 1-847 353-5 36-5 27-752 6-580 5-939 4-679 9-869 323-5' 7-0 28-570 1-310 I-I78 0-938 1-989 353-0 37-0 27-729 6-663 6-015 4-737 9-989 323-0 7-5 2S-565 1-403 1-262 1-004 2-130 352-5 37-5 27-706 6-745 6-090 4-797 10-109 322-5 8-0 28-560 1-496 1-346 1-070 2-271 352-0 38-0 27-682 6-828 6-164 4-855 10-228 322-0 8-S 28-555 1-590 1-430 1-137 2-413 351-5 38-5 27-658 6-909 6-239 4-912 10-347 321-5 9-0 28-549 1-683 I-5I4 1-203 2-554 351-0 39-0 27-634 6-991 6-313 4-969 10-465 321-0 9-5 28-543 1-776 1-598 1-270 2-695 350-5 39-5 27-610 7-072 6-387 5-026 10-583 320-5 lO'O 28-537 1-869 1-681 1-337 2-836 3500 40-0 27-585 7-153 6-461 5-083 10-700 320-0 IO-5 28-530 1-962 1-765 1-403 2-977 349-5 40-5 27-560 7-234 6-534 5-139 io-8i6 319-5 ii-o 28-523 2-054 1-848 1-469 3-117 349-0 41-0 27-535 ■ 7-314 6-608 5-195 10-932 319-0 "■5 28-516 2-147 1-932 1-535 3-257 348-5 41-5 27-510 7-394 6-680 5-251 11-047 318-5 I2-0 28-508 2-240 2-015 I -601 3-397 348-0 42-0 27-484 7-473 6-753 5-307 11-161 318-0 12-5 28-500 2-332 2-099 1-667 3-536 347-5 42-5 27-458 7-553 6-825 5-362 11-275 317-5 I3-0 28-492 2-424 2-182 1-733 3-676 347 -o 43-0 27-432 7-631 6-897 5-416 11-388 317-0 13-5 28-483 2-517 ?-265 1-799 3-816 346-5 43-5 27-405 7-710 6-969 5-471 11-500 316-5 14-0 28-475 2-609 2-348 1-864 3-9SS 346-0 44-0 27-378 7-788 7-040 5-526 11-612 316-0 14-5 28-465 2-701 2-431 1-930 4-094 345-5 44-5 27-351 7-866 7-111 5-580 11-723 315-5 15-0 28-456 2-793 2-514 1-996 4-233 345-0 45-0 27-324 7-943 7-182 5-634 11-833 315-0 15-5 28-446 2-885 2-596 2-061 4-372 344-5 45 -5 27-296 8-020 7-253 5-687 1 1 -942 314-5 i6'o 28-436 2-977 2-679 2-127 4-510 334-0 46-0 27-268 8-097 7-323 5-740 12-051 314-0 i6-5 28-425 3-068 2-762 2-192 4-648 343-5 46-5 27-240 8-173 7-392 5-793 12-159 313-S 17-0 28-414 3-160 2-844 2-257 4-786 343-0 47-0 27-212 8-249 7-462 5-846 12-266 313-0 17-5 28-403 3-251 2-927 2-322 4-923 342-5 47-5 27-183 8-324 7-531 5-898 12-372 312-5 i8-o 28-392 3-342 3-009 2-387 5-060 342-0 48-0 27-154 8-399 7-600 5-950 12-477 312-0 185 28-380 3-433 3-091 2-452 5-197 341-5 48-5 27-125 8-474 7-668 6-002 12-582 311-5 ig-o 28-368 3-524 3-173 2-517 5-334 341-0 49-0 27-095 8-548 7-736 6-053 12-686 311-0 19-5 28-356 3-615 3-255 2-581 5-471 340-5 49-5 27-066 8-622 7-go4 6-104 12-789 310-5 2tTO 28-343 3-705 3-337 2-646 5-607 340-0 50-0 27-036 8-695 7-871 6-154 12-892 310-0 20-5 28-330 3-796 3-418 2-710 5-743 339-5 50-5 27-006 8-768 7-938 6-205 12-994 309-5 21 'O 28-317 3-886 3-500 2-774 5-878 339-0 51-0 26-975 8-841 8-005 6-255 13-095 309-0 2T-S 28-103 3-976 3-581 2-838 6-013 338-5 51-5 26-944 8-913 8-071 6-305 13-195 308-5 22-0 28-289 4-066 3-662 2-902 6-148 338-0 52-0 26-913 8-985 8-137 6-354 13-294 308-0 22-5 28-275 4-156 3-743 2-966 6-282 337-5 52-5. 26-882 9-056 8-203 6-403 13-392 307-5 23-0 28-260 4-245 3-824 3-030 6-416 337 -o 53-0 26-851 9-127 8-268 6-452 13-489 307-0 2ys 28-245 4-335 3-905 3-093 6-550 336-5 53-5 26-819 9-197 8-333 6-500 13-586 306-5 24-0 28-230 4-424 3-985 3-156 6-683 336-0 54-0 26-787 9-267 8-397 6-547 13-682 306-0 24-5 28-215 4-513 4-066 3-219 6-816 335-5 54-S 26-755 9-336 8-461 6-594 13-777 305-5 25-0 28-199 4-602 4-146 3-282 6-948 335-0 S5-0 26-723 9-405 8-525 6-641 13-871 305-0 25 'S 28-183 4-690 4-226 3-345 7-080 334-5 55-5 26-690 9-474 8-588 6-688 13-964 304-5 26-0 28-166 4-779 4-306 3-408 7-212 334-0 56-0 26-657 9-542 8-651 6-735 14-057 304-0 26-5 28-149 4-867 4-386 3-471 7-343 333-5 56-5 26-624 9-609 8-713 6-781 14-148 303-5 27-0 28-132 4-955 4-466 3-534 7-474 333 -o S7-0 26-591 9-676 8-775 6-827 14-238 303-0 27-5 28-115 5-043 4-545 3-596 7-604 332-5 57-5 26-557 9-743 8-836 6-872 14-327 302-5 28-0 28-097 5-130 4-624 3-658 7.734 332-0 58-0 26-523 9-809 8-897 6-917 14-415 302-0 28-5 28-079 5-218 4-703 3-720 7-863 331-5 58-5 26-489 9-874 8-958 6-961 '^^■5°? 301-5 29-0 28-061 5-305 4-782 3-781 7-992 331-0 S9-0 26-455 9-939 9-01S 7-005 14-588 301-0 29-5 28-043 5-391 4-861 3-842 8-121 330-5 59-5 26-42 1 10-004 9-078 7-048 14-673 300-5 30-0 28-024 S-478 4-939 3-903 8-249 330-0 60-0 26-386 10-068 9-137 7-091 14-757 300-0 Se,e Tables 1 and 5 for explanation of symbols, /is always positive; v, f, v', and 2v" are positive when iVis between 0° and 180°, and negative when Nis between 180° and 360°. TV is the longitude of the moon's ascending node. «(],)=_,;; «(K,) = — v'; u{'K^) = — 2v"; «(La) = 2f — 2v — ^,- see Table 8 ior R ; « (M,) = f - v + 5/ see Table 9 lor Q; «( (Mj) = 2 f — 2 v; '« (M3) = 3 f — 3 "; «(M4)=4| — 4v; u(M^) = 6^ — 6v; « (Mg) = 8 f — 8 v; »(Na) = 2f — 2v = «(Ms.); « (2 N) = 2 f — 2 v = » (Mj); M(0i) = 2f — v; «(00)= — 2f — v; «(Pi) =0; « (Qi) = 2f — v = «(Oi); «(Rj) = o; a (S,,s,,3, 4) = o; a(T2)=0; « (A2) = «« (;Us) = «(i;,) = 2 | — 2 v = » (M,); »(MK) = 2f — 2v — v' = 2^(M.;) + «(K,); «(2MK) = 4f- 4v + 7/ = «(M4) — «(K,); « (MN} =4f — 4 v = « (M4); «(MS) = 2f — 2v = «(M.); «(2MS) = 4f — 47^ = «(M,); « (2 SM) = — 2 f + 2 v = — a (Mj); «(Mf)= — 2f; «(MSf) = — 2f + ^. ,, /'C„> . EEPOET FOE 1894— PAET II. 215 Table 7. — Values of I, v, f, v', and 2 v", oorreiponding to each half degree of N — Continued. N / V i v' zv" N N / V ( y' 2P" N o 6o'o 26-386 io-o68 9-137 7-091 14-757 300-0 90-0 23-982 12-751 11-681 8-797 17-783 270-0 6o-5 26-351 10-131 9-196 7-134 14-841 299-5 90-5 23-938 12-772 11-704 8-808 17-795 269-5 6i'o 26-316 10-194 9-255 7-177 14-924 299-0 91-0 23-894 12-793 11-725 8-819 17-805 269-0 6i-S 26-280 10-256 9-313 7-219 1 5 -006 298-5 91-5 23-850 12-814 11-745 8-829 17-814 268-5 62-0 26-245 10-318 9-370 7-261 15-087 298-0 92-0 23-806 12-833 11-765 8-839 17-822 268-0 62-5 26-209 10-379 9-427 7-302 15-167 297-5 92-5 23-761 12-851 11-784 ■8-848 17-829 267-5 63-0 26-173 10-440 9-484 7-343 15-246 297-0 93-0 23-717 12-869 1 1 -802 8-856 17-834 267-0 63-5 26-137 10-500 9-539 7-383 15-324 296-5 93-5 23-673 12-886 11-819 8-863 17-837 266.5 64-0 26-101 10-560 9-595 7-423 15-401 296-0 94-0 23-628 12-901 11-835 8-870 17-839 266-0 64-5 26-064 10-619 9-650 7-462 15-477 295-5 94-5 23-584 12-916 11-851 8-876 17-840 265-5 65-0 26-027 10-677 9-705 7-501 15-551 295-0 95-0 23-539 12-930 11-866 8-882 17-840 265-0 65 -S 25-990 IO-735 9-759 7-539 15-624 294-5 ■ 95-5 23-495 12-943 11-880 8-887 17-838 264-5 66-0 25-953 10-793 9-812 7-577 15-696 294-0 96-0 23-450 12-955 11-893 8-891 17-835 264-0 66-s 25-916 10-849 9-865 7-614 15-767 293-5 96-5 23-406 12-966 11-905 8-895 17-830 263-S 67-0 25-878 10-906 9-918 7-651 15-837 293-0 97-0 23-361 12-976 11-916 8-898 17-824 263-0 67-5 25-841 10-961 9-970 7-688 15-906 292-5 97 -5 23-316 12-985 11-927 8-900 17-816 262-5 68-0 25-803 11-016 10-021 7-724 15-974 292-0 98-0 23-271 12-994 11-936 8-902 17-807 262-0 68-5 25-765 1 1 -070 10-072 7-760 16-042 291-5 98-5 23-227 13-001 11-945 8-903 17-796 261-5 69-0 25-726 11-124 10-123 7-795 16-109 291-0 99-0 23-182 13-007 11-953 8-903 17-784 261-0 69-5 25-688 II-I77 10-173 7-830 16-174 290-5 99-5 23-137 13-013 11-960 8-902 17-770 260-5 70-0 25-649 1 1 -230 IO-222 7-864 16-238 290-0 loo-o 23-092 13-017 11-966 8-901 17-755 260-0 70-5 25-610 11-282 10-271 7-898 16-300 289-5 100-5 23-047 13-021 11-971 8-899 17-739 259-S 71*0 2S-57I 11-333 10-319 7-932 16-361 289-0 loi-o 23-003 13-023 11-975 8-896 17-721 259-0 7i'S 25-532 11-383 10-366 7-965 16-421 288-5 101-5 22-958 13-024 11-979 8-892 17-702 258-5 72-0 25-493 "■433 10-414 7-997 16-480 288-0 102-0 22-913 13-025 11-981 8-888 17-681 258-0 72-5 25-453 1 1 -482 10-460 8-029 16-538 287-5 102-5 22-868 13-024 11-983 8-883 17-659 257-5 73"o 25-413 11-531 10-506 8-061 16-594 287-0 103-0 22-823 13-023 11-983 8-878 17-636 257-0 73'S 25-374 11-579 10-551 8-092 16-649 286-5 103-5 22-778 13-020 11-983 8-872 17-611 256-5 74-0 25-334 11-626 10-596 8-122 16-703 286-0 104-0 22-734 13-017 11-982 8-865 17-584 256-0 74-5 25-293 11-673 10-640 8-152 16-756 285-5 104-5 22-689 13-012 11-979 8-858 17-556 255-5 75 -o 25-253 H.-719 10-684 8-181 16-808 285-0 105-0 22-644 13-006 • 11-976 8-850 17-526 255-0 75 "5 25-213 11-764 10-726 8-209 16-859 284-5 105-5 22-599 13-000 11-972 8-841 17-495 254-5 76-0 25-172 11-809 10-769 8-237 16-909 284-0 106-0 22-554 12-992 11-967 8-831 17-463 254-0 76-5 25-131 11-852 io-8n 8-264 16-958 283-5 106-5 22-510 12-983 11-961 8-821 17-430 253-S 77 -o 25-090 11-895 10-852 8-291 17-005 283-0 107-0 22-465 12-974 11-954 8-810 17-395 253-0 77'S 25-049 11-938 10-892 8-318 17-051 282-5 107-5 42-420 12-963 1 1 -946 8-799 17-358 252-5 78-0 25-008 11-980 10-932 8-344 17-096 282-0 108-0 22-376 12-951 11-937 8-787 17-320 252-0 78-5 24-966 12-021 10-971 8-370 17-140 281-5 108-5 22-331 12-9-38 n-927 8-774 17-281 251-5 79-0 24-925 12-061 1 1 -009 8-395 17-183 281-0 1090 22-287 12-924 11-916 8-760 17-240 251-0 79-5 24-883 12-100 11-047 8-420 17-225 280-5 109-5 22-242 12-909 1 1 -904 8-745 17-197 250-5 8o-o 24-841 12-139 1 1 -084 8-444 17-265 280-0 iio-o 22-198 12-892 11-891 8-729 17-153 250-0 80-5 24-800 12-177 11-121 8-467 17-303 279-5 110-5 22-153 12-875 11-877 8-713 17-107 249-5 81 -o 24-757 12-214 11-157 8-490 17-340 279-0 11 1-0 22-109 12-857 11-862 8-696 17-060 249-0 8i-5 24-715 12-251 11-192 8-512 17-376 278-5 111-5 22-065 12-837 11-846 8-678 I7OII 248-5 82-0 24-673 12-287 11-227 8-533 17-410 278-0 1120 22-021 12-817 11-829 8-659 16-961 248-0 82-5 24-631 12-322 11-261 8-554 17-443 277-5 112-5 21-976 12-795 11-811 8-639 16-910 247-5 83-0 24-588 12-356 1 1 -294 8-574 17-475 277-0 1130 21-932 12-772 11-792 8-619 16-858 247-0 83-5 24-545 12-389 11-326 8-594 17-505 276-5 1135 21-888 12-748 11-772 f'599 16-805 246-5 84-0 24-503 12-422 "-358 8-613 17-534 276-0 1140 21-845 12-723 11-750 8-578 16-750 246-0 84-5 24-460 12-454 11-389 8-631 17-562 275-5 114-5 21-801 12-697 11-728 8-556 16-693 245-5 85-0 24-417 12-485 11-419 8-649 17-589 275-0 115-0 21-757 12-670 11-705 8-533 16-635 245-0 85-5 24-374 12-515 u-449 8-666 17-614 274-5 115-5 21-713 12-642 11-681 8-509 16-576 244-5 86-0 24-33' 12-545 11-478 8-683 17-638 274-0 II6-0 21 670 12-612 11-655 8-484 16-515 244-0 86-5 24-287 12-573 11-506 8-700 17-661 273-S 116-5 21-627 12-581 1 1 -629 8-458 1.6-453 243-5 87-0 24-244 12-601 "-533 8-716 17-683 273-0 117-0 21-583 12550 11 -601 8-432 16.389 243-0 .87-5 24-200 12-628 11-560 8-731 17-703 272-5 117-5 21 -540 12-517 11-572 8-405 16-323 242-5 88-0 24-157 12-654 11-586 8-745 17-722 272-0 118-0 21-497 12-483 11-543 8-378 16-256 242-0 88-5 24-113 12-680 ii-6n 8-759 17-739 271-S 118-5 21-454 12-447 11-512 8-350 16-188 241-5 89-0 24-070 12-704 11-635 8-772 17-755 271-0 119-0 21-411 12-411 11 -480 8-321 16-118 241-0 89-5 24-026 12-728 11-659 8-785 17-770 270-5 119-5 21-368 12-373 11-447 8-291 16-047 240-5 90-0 23-982 12-751 11-681 8-797 17-783 270-0 I20-0 21-326 12-335 11-413 8-260 15-975 240-0 216 UNITED STATES COAST AND GEODETIC SUEVET. Table 7. — Values of I, v, f , v' , and 2 v" , coi-responding to each half degree of iV— Continued. N I V i v' 2^" N N / V f v' zv" N o I20-0 21-326 12-335 11-413 8-260 15-975 240-0 150-0 19-162 7-854 7-324 5-096 9-392 210-0 120-5 21-283 12-295 11-378 8-229 15-902 239-5 150-5 19-135 7-745 7-223 5-024 9-251 209-S I2I-0 21-241 12-254 11-342 8-197 15-827 239-0 151-0 19-108 7-635 7-120 4-951 9-109 209-0 121-5 21-199 12-211 11-304 8-165 15-751 218-5 151-5 19082 7-523 7-017 4-878 8-966 208-5 122-0 21-157 12-168 11-266 8-132 15-673 238-0 152-0 19-056 7-411 6-913 4-805 8-823 208-0 122-5 21-115 12-123 11-226 8-098 15-594 237-5 152-5 19-030 7-298 6-809 4-730 8-679 207-5 123-0 21-073 12-078 11-186 8-063 15-513 237-0 153-0 19-005 7-184 6-703 4-655 8-534 207-0 123-5 21-032 12-031 H-144 8-028 15-431 236-S 153-5 18-981 7-069 6-596 4-579 8-389 206-5 1 24-0 20-990 11-983 II-IOI 7-992 15-347 236-0 154-0 18-956 6-953 6-488 4-502 8-243 206-0 124-5 20-949 11-933 11-057 7-955 15-262 235-5 154-5 18-933 6-836 6-380 4-424 8-096 205-5 125-0 20-908 11-883 11-012 7-917 15-176 235-0 155-0 18-909 6-718 6-270 4-346 7-948 205-0 125-5 20-867 11-831 10-965 7-878 15-089 234-5 155-5 18-886 6-599 6- 160 4-268 7-800 204-5 126-0 20-826 11-778 10-918 7-839 15-001 234-0 156-0 18-863 6-480 6-049 4-189 7-651 204-0 126-5 20-786 11-724 10-869 7-799 14-911 233-S 156-5 18-841 6-359 S-937 4-110 7-502 203-5 127-0 20-746 11-669 10-820 7-758 14-820 233-0 157-0 18-819 6-238 5-824 4-030 7-352 203-0 127-5 20-705 11-612 10-769 7-716 14-728 232-5 157-5 18-798 6-116 5-710 3-950 7-201 202-5 128-0 20-666 11-555 10-717 7-674 14-635 232-0 158-0 18-777 5-993 5-596 3-869 7-050 202-0 128-5 20-626 1 1 -496 10-664 7-631 14-540 231-5 158-5 18-7-56 5-869 5-480 3-787 6-898 201-5 129-0 20-586 11-436 10-610 7-587 14-444 231-0 159-0 18-736 5 -744 5-364 3-705 6-745 201 -o 129-5 20-547 "•374 IO-S54 7-542 14-347 230-5 159-S 18-716 5-618 5-247 3-623 6-592 200-5 130-0 20-508 11-312 10-498 7-496 14-248 230-0 160-0 18-697 5-492 5-130 3-541 6-438 200 -o I30-S 20-469 11-248 10-440 7-449 14-148 229-5 160-5 18-678 5-365 5-011 3-458 6-284 199-5 I3I-O 20-430 11-184 10-382 7-401 14-048 229-0 161-0 18-660 S-237 4-892 3-375 6-130 199-0 I3IS 20-392 11-118 10-322 7-353 13-946 228-5 161-5 18-642 5-109 4-773 3-291 5-975 198-5 132-0 20-353 11-050 10-261 7-304 13-842 228-0 162-0 18-624 4-980 4-652 3-207 5-819 198-0 132-5 20-315 10-982 10-199 7-255 13-737 227-5 162-5 18-607 4-850 4-531 3-122 5-663 I97-S 133-0 20-278 10-912 10-135 7-205 13-631 227-0 163-0 18-591 4-719 4-409 3-037 5-506 197-0 I33-S 20-240 10-841 10-071 7-154 13-524 226-5 163-5 18-575 4-588 4-287 2-952 5-349 196-5 134-0 20-203 10-769 10-005 7-102 13-416 226-0 164-0 •8-559 4-456 4-164 2-866 5-192 196-0 134-5 20-166 10-696 9-939 7-050 13-306 225-5 164-5 18-544 4-323 4-040 2-780 5 -034 195-5 135-0 20-129 10-622 9-8*71 6-998 13-195 225-0 165-0 18-529 4-190 3-916 2-694 4-875 195-0 135-5 20-092 10-546 9-802 6-945 13-083 224-5 165-5 18-515 4-056 3-791 2-607 4-716 194-5 136-0 20-056 10-469 9-732 6-891 12-970 224-0 166-0 18-501 3-922 3-665 2-520 4-557 194-0 136-5 20-020 10-391 9-660 6-837 12-856 223-5 166-5 18-487 3-787 3-539 2-433 . 4-397 193-5 137-0 19-984 10-312 9-588 6-782 12-741 223-0 167-0 18-475 3-651 3-413 2-345 4-238 193-0 137-5 19-949 10-232 9-515 6-727 12-624 222-5 167-5 18-462 3-515 3-286 2-257 4-078 192-5 138-0 19-913 10-150 9-440 6-671 12-506 222-0 168-0 18-450 3-379 3-158 2-169 3-918 192-0 138-5 19-878 10-068 9-364 6-614 12-387 221-5 168-5 18-439 3-242 3-030 2-081 3-757 I9I-5 139-0 19-844 9-984 9-287 6-556 12-268 221-0 169-0 18-428 3-104 2-902 1-992 3-596 I9I-0 139-5 19-809 9-899 9-209 6-498 12-148 220-5 169-5 18-417 2-966 2-773 1-903 3-435 190-5 140-0 19-775 9-813 9-130 6-439 12-027 220-0 170-0 18-407 2-828 2-644: 1-814 3-273 190-0 140-5 19-742 9-725 9-050 6-379 11-904 219-5 170-5 18-398 2-689 2-514 1-725 3-110 189-5 141-0 19-708 9-637 8-969 6-318 11-780 219-0 171-O 18-388 2-550 2-384 1-635 2-948 189-0 141-5 19-675 9-547 8-887 6-256 11-655 218-5 171-5 18-380 2-410 2-254 1-546 2-786 188-5 142-0 19-642 9-457 8-803 6-193 11-529 218-0 172-0 18-372 2-270 2-123 1-456 2-623 i88-o 142-5 19-610 9-365 8-719 6-129 1 I -402 217-S 172-5 18-364 2-130 1-992 1-365 2-460 187-5 143-0 19-577 9-272 8-633 6-064 11-274 217-0 173-0 18-357 '■ft I -860 1-275 2-297 187-0 143-5 19-545 9-177 8-546 5-999 11-145 216-5 173-5 18-350 1-848 1-729 1-185 2-133 186-5 144-0 19-514 9-082 8-458 5-933 11-016 216-0 174-0 18-344 1-707 1-597 1-094 1-970 186-0 144-5 19-483 8-986 8-370 5-866 10-886 215-S 174-S 18-338 1-566 1-464 1-003 1-806 185-5 145-0 19-452 8-888 8-280 5-798 10-755 215-0 175-0 18-333 1-424 1-332 0-912 1-642 185-0 145-5 19-421 8-790 8-189 5-730 10-623 2I4-S 175-S 18-328 1-282 1-199 0-821 1-478 184-5 146-0 19-391 8-690 8-097 5-661 10-490 214-0 176-0 18-324 I -140 1-067 0-730 1-314 184-0 146-5 19-361 8-589 8-004 5-592 10-356 213-5 176-5 18-320 0-998 0-934 0-639 I -150 183-5 147-0 19-332 8-487 7-910 5-522 10-221 213-0 177-0 18-317 0-856 0-801 0-548 0-986 183-0 147-5 19-302 8-384 7-814 5-452 10-084 212-5 177-5 18-315 0-713 0-667 0-457 0-822 182-5 148-0 19-273 8-280 7-718 5-382 9-947 212-0 178-0 18-312 0-571 0-534 0-366 0-658 1S2-0 148-5 19-245 8-175 7-621 5-311 9-809 211-5 178-5 18-311 0-428 0-401 0-274 0-493 181-5 149-0 19-217 8-069 7-523 5-240 9-671 2II-0 179-0 18-309 0-286 0-267 0-183 0-329 181-0 149-5 19-189 7-962 7-424 5-168 9-532 210-5 179-5 18-309 0-143 0-133 0-091 0-165 180-5 150-0 19-162 7-854 7-324 5-096 9-392 210-0 180-0 18-308 0-000 o-ooo 0-000 o-ooo 180-0 EEPOET FOE 1894— PAET II. 217 Tablk 8. — Values of B for completing u for hi. p /= Inclination of moon's orbit. • 18° 19° 20" 21° 22° 23° 24° 25° 26" 27° 28° 29° O 5 lO O'OO 176 3 '43 0-00 2-00 3-90 0-00 2-27 4-42 o-oo 2-57 5-00 O-OO 2-90 5-63 0-00 3-27 6-32 0-00 3-67 7-09 0-00 4-14 7-94 o-oo 4-65 8-89 0-00 5-22 9-94 0-00 5-88 11-12 o-oo 6-57 12-43 15 20 25 4"94 6-24 7-28 5=61 7-06 8-21 6-39 7-97 9-22 7-15 8-94 10-32 8-03 10-00 11-49 8-99 ii-i6 12-76 10-04 12-40 14-12 11-20 13-76 15-57 12-47 15-23 17-14 13-86 16-82 i8-8i 1 5 "40 18-56 2o-6o 17-10 20-43 22-50 3° 35 40 8-02 8-48 8-66 9-02 9-51 9-69 10-10 io-6o 10-75 11-26 11-77 11-89 12-48 13-00 13-08 13-80 14-31 14-34 15-19 15-68 15-65 16-68 17-13 17-02 18-25 18-64 18-44 19-90 20-23 19-92 21-66 21-89 21-44 23-50 23-61 23-02 45 SO 55 8-56 8-22 7-66 9-54 9-13 8-49 10-57 lo-n • 9-36 11-64 11-09 10-26 12-77 12-12 n-18 13-95 13-20 1214 15-17 14-30 13-12 16-43 15-44 14-13 17-73 16-61 15-16 19-08 17-81 l6-20 20-46 19-03 17-27 21-87 20-28 18-34 6o 65 70 6-91 6-00 4-96 7-64 6-62 5-46 8-40 7-27 5-99 9-19 7-94 6-53 I o-oo 8-62 7-08 10-83 932 7-64 11-68 10-03 8-21 12-55 10-75 8-79 •3-43 11-49 9-38 14-32 12-23 9-97 15-23 12-98 10-56 16-14 13-73 ii-i6 75 80 85 3-80 2-58 1-30 4-19 2-84 1-43 4-59 3-11 1-57 5-00 3-38 1-70 3-66 1-84 5-84 3-94 1-98 6-26 4-23 2-13 6-70 4-52 2-27 7-14 4-81 2-42 7-58 5-10 2-56 8-02 5 -40 2-71 8-47 5-69 2-86 90 95 100 o-oo 358-24 356-57 0-00 358-00 356-10 0-00 357-73 355-58 o-oo 357-43 3SS-00 0-00 357-10 354-37 0-00 356-73 353-68 o-oo 356-33 352-91 o-oo 355-86 352-06 0-00 355-35 351-11 0-00 354-78 350-06 0-00 354-12 348-88 o-oo 353-43 347-57 105 no 115 355-06 353-76 352-72 354-39 352-94 351-79 353-61 352-03 350-78 352-85 351-06 349-68 351-97 350-00 348-51 351-01 348-84 347-24 349-96 347-60 345-88 348-80 346-24 344-43 347-53 344-77 342-86 346-14 343-18 341-19 344-60 341-44 339-40' 342-90 339-57 337-50 120 125 130 ■351-98 351-52 351-34 350-98 350-49 350-31 349-90 349-40 349-25 348-74 348-23 348-11 347-52 347-00 346-92 346-20 345-69 345-66 344-81 344-32 344-35 343-32 342-87 342-98 341-75 341-36 341-56 340-10 339-77 340-08 338-34 338-11 338-56 336-50 336-39 336-98 135 140 14s 351-44 351-78 352-34 350-46 350-87 351-51 349-43 349-89 350-64 348-36 348-91 349-74 347-23 347-88 348-82 346-05 346-80 347-86 344-83 345-70 346-88 343-57 344-56 345-87 342-27 343-39 344-84 340-92 342-19 343-80 339-54 340-97 342-73 338-13 339-72 341-66 ISO 155 160 353-09 354-00 355-04 352-36 353-38 354-54 351-60 352-73 354-01 350-81 352-06 353-47 350-00 351-38 352-92 349-17 350-68 352-36 548-32 349-97 351-79 347-45 349-25 351-21 346-57 348-51 350-62 345-68 347-77 350-03 344-77 347-02 349-44 343-86 346-27 348-84 165 170 175 180 356-20 357-42 358-70 360-00 355-81 357-16 358-57 360-00 355-41 356-89 358-43 360-00 355-00 356-62 358-30 360-00 354-64 356-34 358-16 360-00 354-16 356-06 358-02 360-00 353-74 355-77 357-87 360-00 353"3o 355-48 357-73 360-00 352-86 355-19 357-58 360-00 352-42 354-90 357-44 360-00 351-98 , 354-60 357-29 360-00 351-53 354-31 357-14 360-00 «for I^^2(f — v) — R. puted from the equation tan R The values of f and v and to be obtained from Table 7; and the above values of R were com- sin 2 /" \ cot -2 J /—cos 2 P' The values of /and P for the first day of every month are given in Table 6. When /" lies between 180° and 360°, subtract 1 80° from it and enter the table with the remainder. 218 UOTTED STATES COAST AND GEODETIC SUEVEY. Table 9. — Values of Q for completing ufor M,. p Q p Q P Q p Q p Q p Q o o o-oo 6c! 40-89 120 139-11 180 180-00 240 220-89 300 319-11 I 0-50 61 42-05 121 140-24 181 180-50 241 222-05 301 320-24 2 I'OO 62 43-24 122 141-34 182 181-00 242 223-24 302 321-34 3 1-50 63 44-46 123 142-41 183 181-50 243 224-46 303 322-41 4 2-00 64 45-71 124 143-45 184 182-00 244 225-71 304 323-45 5 2-50 65 47-00 125 144-47 185 182-50 245 227-00 305 324-47 6 3-01 66 48-32 126 145-46 186 183-01 246 228-32 306 325-46 7 3'5i 67 49-67 127 146-44 187 183-51 247 229-67 307 326-44 8 4-02 68 51-06 128 147-38 188 184-02 248 231-06 308 327-38 9 4-53 69 52-48 129 148-31 189 184-53 249 232-48 309 328-31 lO 5 '04 70 53-95 130 149-21 190 185-04 250 233-95 310 329-21 II S'55 71 55-45 131 150-09 191 185-55 251 235-45 3" 330-09 12 6-07 72 56-98 132 150-96 192 186-07 252 236-98 312 330-96 13 6-58 73 58-56 133 151-80 193 186-58 253 238-56 313 331-80 14 7-10 74 60-17 134 152-63 194 187-10 254 240-17 314 332-63 IS 7-63 75 6i-8i 135 153-44 195 187-63 255 ,241-81 315 333-44 16 8-i6 76 63-50 136 154-23 196 i88-i6 256 243-50 316 334-23 17 8-69 77 65-22 137 155-00 197 188-69 257 245-22 317 335-00 18 9-23 78 66-97 138 155-76 198 189-23 258 246-97 318 335-76 19 977 79 68-76 139 156-51 199 189-77 259 248-76 319 336-51 20 10-32 80 70-58 140 157-24 200 190-32 260 250-58 320 337-24 21 IO-86 81 72-42 141 157-96 201 190-86 261 252-42 321 337-96 22 1 1 -42 82 74-30 142 158-66 202 191-42 262 254-30 322 338-66 23 11-98 83 76-20 143 159-36 203 191-98 263 256-20 323 339-36 24 12-55 84 78-13 144 160-04 204 192-55 264 25813 324 340-04 25 13-12 85 8o-o8 145 160-70 205 193-12 265 260-08 32s 340-70 26 13-70 86 82-04 146 161-36 206 193-70 266 262-04 326 341-36 27 14-29 87 84-02 147 162-01 207 194-29 267 264-02 327 342-01 28 14-89 88 86 -oo 148 162-65 208 194-89 268 266-00 328 342-65 29 1 5 '49 89 88-00 149 163-28 209 195-49 269 268-00 329 343-28 30 16-10 90 90-00 150 163-90 210 196-10 270 270-00 330 343-90 3' 16-72 91 92-00 151 164-51 211 196-72 271 272-00 331 . 344-51 32 1 7 "35 92 94-00 152 165-11 212 197-35 272 274-00 332 345-" 33 17-99 93 95-98 153 165-71 213 197-99 273 275-98 333 345-71 34 18-64 94 97-96 154 166-30 214 198-64 274 277-96 334 346-30 35 19-30 95 99-92 •55 166-88 215 199-30 275 279-92 335 346-88 36 19-96 96 101-87 156 167-45 216 199-96 276 281-87 336 347-45 37 38 20-64 97 103-80 157 168-02 217 200-64 277 283-80 337 348-02 21-34 98 105-70 158 168-58 218 201 -34 278 285-70 338 348-58 39 22-04 99 107-58 159 169-14 219 202-04 279 287-58 339 349-14 40 22-76 100 109-42 160 169-68 220 202-76 280 ,289-42 340 349-68 41 23-49 lOI II I -24 161 170-23 221 203-49 281 291-24 341 350-23 42 24-24 102 113-03 162 170-77 222 204-24 282 293-03 342 350-77 43 25-00 103 114-78 163 171-31 223 205-00 283 294-78 343 351-31 44 25-77 104 116-50 164 171-84 224 205-77 284 296-50 344 351-84 46 26-56 105 118-19 165 172-37 225 206-56 285 298-19 345 352-37 27-37 106 119-83 166 172-90 226 207-37 286 299-83 346 352-90 47 28-20 107 121-44 167 173-42 227 208-20 287 301-44 347 353-42 48 29-04 108 123-02 168 173-93 228 209-04 288 303-02 348 353-93 49 29-91 109 124-55 169 174-45 229 209-91 289 304-55 349 354-45 SO SI 52 53 54 57 58 59 30-79 no 126-05 170 174-96 230 210-79 290 306-65 350 354-96 >- 31-69 32-62 III 127-52 171 175-47 231 211-69 291 307-52 351 335-47 112 128-94 172 175-98 232 212-62 292 308-94 352 355-98 33'56 "3 130-33 173 176-49 233 213-56 293 310-33 353 356-49 34'54 114 131-68 174 176-99 234 214-54 294 311-68 354 356-99 35-53 "5 133-00 175 177-50 235 215-53 295 313-00 355 357-50 36-55 116 134-29 176 178-00 236 216-55 296 314-29 356 358-00 37-59 38-66 117 135-54 177 178-50 237 217-59 297 315-54 357, 358-50 ii8 136-76 178 179-00 238 218-66 298 316-76 358' 359-00 39-76 119 137-95 179 179-50 239 219-76 299 317-95 359 359-50 f-Qj. ]yi 3_ c v+ Q- The values of f and v are to be obtained from Table 7 ; the above values of Q were computed from the equation tan g=i tan /". The value of P for the first day of every month is given in Table 6. EBPORT FOE 1894— PART II. 219 Table 10. — Factors F andf for rediicUon and prediction of tides; computed for the middle of each year, or for July 2, at Greenwich mean noon for common years, and at preceding midnight for leap years. Component. 1850 1851 1852 j ■853 1854 185s F logF logy F \ogF logy F logF logy F \agF logy F log F y logy F \ogF logy J..[M,]* I-I2I6 0'0498 0-8916 9-9502 1-0553 0-0234 0-9476 9-9766 0-9935 9-9972 I -0066 0-0028 0-9428 9-9744 I -0607 0-0256 0-9047 9-9565 1-1053 0-0435 0-8788 9-9439 I-I380 0-0561 K, I -0840 0-0350 0-9225 9-9650 I -0426 0-0181 0-9591 9-9819 I-OOll 0-0005 0-9989 9-9995 0-9647 9-9844 1 -0366 0-0156 0-9358 9-9712 1-0686 0-0288 0-9153 9-9616 1-0925 0-0384 K. I •2263 0-0886 0-8I5S 9-91 14 I-I276 0-0521 0-8869 9-9479 1-0238 0-0102 0-9767 9-9898 0-9305 9-9687 1-0747 0-0313 0-8559 9-9324 1-1684 0-0676 0-8026 9-9045 1-2459 0-0955 u 0-8599 9"9344 1-1629 0-0656 1-1050 0-0434 0-9050 9-9566 1-3801 0-1399 0-7246 9-8601 0-9476 9-9766 1-0553 0-0234 0-7920 9-8987 1-2626 0-IOI3 1-0594 0-0251 0-9439 9-9749 \u-\* _ 0-9733 9-9882 I -0274 0-0118 0-9830 9-9925 1-0173 0-0075 0-9947 9-9977 1-0053 0-0023 1-0073 0-0032 0-9928 9-9968 1-0192 0-0083 0-9811 9-9917 1-0291 0-0124 0-9717 9-9876 M. 0-9777 9-9902 1-0228 0-0098 0-5972 9-7761 1-6746 0-2239 0-5066 9-7046 1-9741 0-2954 0-6414 9-8072 1-5591 0-1929 0-8942 9-95I4 1-1183 0-0486 0-5378 9-7306 I -859s 0-2694 Ms, MS 0-9733 9-9882 1-0274 0-0118 0-9830 9-9925 1-OI73 0-0075 0-9947 9-9977 1-0053 0-0023 1-0073 0-0032 0-9928 9-9968 1-0192 0-0083 0-9811 9-9917 1-0291 0-0124 0-9717 9-9876 M, 0-9602 9-9824 1-0415 0-0174 0-9746 9-9888 1-0261 0-0112 0-9921 9-9966 1 -0080 0-0034 i-oiio 0-0047 0-9891 9-9953 1 -0289 0-0124 0-9719 9-9876 1-0440 0-0187 0-9579 9-9813 M4, MN 0-9473 9-9765 1-0557 0-0235 0-9662 9-9851 1-0350 0-0149 0-9895 9-9954 1-0106 0-0046 I-OI47 0-0063 0-9856 9-9937 I -0388 0-0165 0-9626 9-9835 1-0590 0-0249 0-9443 9-9751 Me 0-9220 9-9647 I -0846 0-0353 0-9498 9-9776 1-0529 0-0224 0-9843 9-9931 1-0159 0-0069 1-0221 0-0095 0-9784 9-9905 1-0588 0-0248 0-9445 9-9752 I -0898 0-0374 0-9176 9-9626 Ms 0-8973 9-953° I-II44 0-0470 0-9336 9-9702 1-0711 0-0298 0-9791 9-9908 1-0213 0-0092 1-0295 0-0126 0-9713 9-9874 1-0791 0-0331 0-9267 9-9669 I-1215 0-0498 0-8917 9-9502 Ns, 2N 0-9733 9-9882 1-0274 0-0118 0-9830 9-9925 1-0173 0-0075 0-9947 9-9977 1-0053 0-0023 1-0073 0-0032 0-9928 9-9968 I -0192 0-0083 0-9811 9-9917 1-0291 0-0124 0-9717 9-9876 o„q, I -1449 0-0588 0-8735 9-9412 1-0715 0-0300 0-9333 9-9700 1-0023 o-ooio 0-9978 9-9990 0-9446 9-9752 1-0587 0-0248 0-9006 9-9545 1-II04 0-0455 0-8700 9-9395 1-1495 0-0605 00 I -5841 0-1998 0-6313 9-8002 1-2732 0-1049 0-7855 9-8951 1-0175 0-0075 0-9829 9-9925 0-8305 9-9194 1-2041 0-0806 0-7031 9-8470 1-4223 0-1530 0-6218 9-7936 1 -6084 0-2064 Pll Rsj Tg I -0000 0-0000 I -0000 0-0000 1 -0000 o-oooo 1-0000 0-0000 1 -0000 0-0000 1-0000 o-oooo I-OOOO 0-0000 i-oooo o-oooo I -0000 o-oooo 1 -0000 o-ooco I-OOOO 0-0000 i-oooo 0-0000 Si, 3, 3» 4 I -0000 0-0000 I -0000 0-0000 1-0000 0-0000 1-0000 0-0000 1. 0000 0-0000 I-OOOO 0-0000 I -0000 o-oooo I-OOOO 0-0000 I-OOOO o-oooo I -0000 0-0000 I-OOOO 0-0000 1-0000 0-0000 \, fli, Vi 0-9733 9-9882 I -0274 0-0118 0-9830 9-9925 1-0173 0-0075 0-9947 9-9977 1-0053 0-0023 • I -0073 0-0032 0-9928 9-9968 1-0192 0-0083 0-9811 9-9917 1-0291 0-0124 0-9717 9-9876 MK :-o55o 0-0233 0-9479 9-9767 I -0249 0-0107 0-9757 9-9893 0-9958 9-9982 I -0042 0-0018 0-9717 9-9875 1-0291 0-0125 0-9538 9-9795 1 -0484 0-0205 0-9420 9-9740 1-0616 0-0260 2MK 1-0268 0-0115 0-9739 9-9885 1-0074 0-0032 0-9926 9-9968 0-9906 9-9959 1-0095 0-0041 0-9788 9-9907 1-0217 0-0093 0-9722 9-9877 1-0286 0-0123 0-9693 9-9865 1-0316 0-0135 2MS 0.9473 9-9765 I -0557 0-0235 0-9662 9-9851 1-0350 0-0149 0-9895 9-9954 1-0106 0-0046 I -0147 0-0063 0-9856 9-9937 1-0388 0-0165 0-9626 9-9835 1-0590 0-0249 0-9443 9-9751 Msf, 2SM 0-9733 9-9882 1-0274 o-oii8 0-9830 9-9925 1-0173 0-0075 0-9947 9-9977 1-0053 0-0023 1-0073 0-0032 0-9928 9-9968 1-0192 0-0083 0-9811 9-9917 1-0291 0-0124 0-9717 9-9876 Mf I -3467 0-1293 0-7426 9-8707 1-1680 0-0674 0-8562 9-9326 I -0098 0-0042 0-9903 9-9958 0-8857 9-9473 1-1290 0-0527 0-7957 9-9008 1-2567 0-0992 0-7355 9-8666 1-3597 0-1334 Mm 0-9138 9-9608 I -0944 0-0392 0-9446 9-9752 1-0587 0-0248 0-9837 9-9929 1-0165 0-0071 1-0278 0-0119 0-9729 9-9881 1-0720 0-0302 0-9328 9-9698 1-1106 0-0455 0-9005 9-9545 Sa, Ssa i-oooo 0-0000 I -0000 0-0000 I -0000 0-0000 I-OOOO o-oooo 1-0000 0-0000 I-OOOO 0-0000 1 -0000 0-0000 I-OOOO 0-0000 i-oooo 0-0000 I-OOOO 0-0000 I-OOOO 0-0000 I-OOOO 0-0000 , .. . . i : i„ 220 UNITED STATES COAST AND GEODETIC SURVEY. Table 10. — Factors F and f for reduction and prediction of tides; computed for the middle of each year, or for July S, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1856 1857 1858 1859 i860 I86I J' log F / logy F \0%F lo^y F log F lo^y F \ogF logy F \0gF lo^y F log F logy I -0665 0-0280 Ji. [M,] 0-8636 9-9363 I-I580 0-0637 0-8583 9-9336 1-1651 0.0664 0-8625 9-9357 1-1595 0-0643 0-8764 9-9427 1-1411 0-0573 0-9010 9-9547 1-1098 0-0453 0-9376 9-9720 K^ 0-9030 9-95S7 I -1074 0-0443 0-8987 9-9536 I-II28 0-0464 0-9021 9'9553 1-1085 0-0447 0-9134 9-9607 I -0948 0-0393 0-9330 9-9699 1-0718 0-0301 0-9609 9-9827 1-0407 0-0173 K, 0-7707 9-8869 1-2976 0-II3I 0-7593 9-8804 1-3170 0-1196 0-7683 9-8855 1-3017 0-1145 0-7977 9-9018 1-2537 0-0982 0-8484 9-9286 1-1786 0-0714 0-9207 9-9641 1 -0862 0-0359 L, 2-1324 0-3289 0-4690 9-6711 1-0393 0-0167 0-9622 9-9833 0-7796 9-8919 1-2827 o-io8i 0-9992 9-9996 ) -0008 0-0004 1-7598 0-2455 0-5683 9-7545 1-0822 0-0343 0-9241 9-9657 iU} I -0357 0-0152 0-9655 9-9848 1-0382 0-0163 0-9632 9-9837 1-0362 0-0155 0-9650 9-9S45 1-0301 0-0129 0-9708 9-9871 I -0205 0-0088 0-9799 9-9912 1 -0088 0-00*38 0-9913 9-9962 M, 0-4260 9-6294 2-3475 0-3706 0-5343 9-7278 I-87I6 0-2722 0-8499 9-9294 1-1766 0-0706 0-5631 9-7506 1-7759 0-2494 0-4490 9-6522 2-2274 0-3478 0-5581 9-7467 1-7919 0-2533 Ms, MS I •0357 0-0152 0-9655 9-9848 1-0382 0-0163 0-9632 9-9837 1-0362 0-0155 0-9650 9-9845 1-0301 0-0129 0-9708 9-9871 1-0205 0-0088 0-9799 9-9912 1 -0088 0-0038 0-9913 9-9962 Ms I -0540 0-0229 0-9487 9-9771 1-0578 0-0244 0-9453 9-9756 1-0548 0-0232 0-9480 9-9768 1-0454 0-0193 0-9566 9-9807 1 -0309 0-0132 0-9700 9-9868 1-0131 0-0057 0-9870 9-9943 M„MN 1-0727 0-0305 0-9322 9-9695 1-0779 0-0326 0-9278 9-9674 1-0738 0-0309 0-9313 9-9691 I -061 1 0-0257 0-9425 9-9743 1-0415 0-0177 0-9602 9-9823 1-0176 0-0076 0-9827 9-9924 M, I-IIIO 0-0457 0-9001 9-9543 1-1190 0-0488 0-8936 9-9512 1-1127 0-0464 0-8987 9-9536 1 -0930 0-0386 0-9149 9-9614 I -0629 0-0265 0-9408 9-9735 1-0265 0-0114 0-9742 9-9886 Ms 1-1507 0-0610 0-8690 9-9390 i-i6i8 0-0651 0-8608 9-9349 1-1530 0-0618 0-8673 9-9382 1-1258 0-0515 0-8882 9-9485 I -0847 0-0353 0-9219 9-9647 1-0355 0-0152 0-9657 9-9848 N2, 2N 1-0357 0-0152 0-9655 9-9848 I -0382 0-0163 0-9632 9-9837 1 -0362 0-0155 0-9650 9-9845 1-0301 0-0129 0-9708 9-9871 1-0205 0-0088 0-9799 9-9912 1 -0088 0-0038 0-9913 9-9962 0„Q, 0-8519 9-9304 1-1739 0-0696 0-8455 9-9271 I-I827 0-0729 0-8505 9-9297 1-1758 0-0703 0-8672 9-9381 1-1532 0-0619 0-8962 9-9524 1-1158 0-0476 0-9387 9-9725 1 -0654 0-0275 00 0-5763 9-7606 1-7352 0-2394 0-5607 9-7488 1-7834 0-2512 0-5730 9-7581 1-7453 0-2419 0-6145 9-7885 1-6273 0-2115 0-6912 9-8396 1-4467 0-1604 0-8127 9-9099 1-2305 0-0901 Pi. R2. Ti I -oooo I -oooo I -oooo I -oooo I -oooo 1 -oooo 1 -oooo 1-0000 I -OOOO I -oooo 1 -oooo I -oooo 0-0000 o-oooo o-oooo 0-0000 o-oooo 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 Si, 2. 3» 4 I -oooo I -oooo I -oooo 1-0000 I -oooo i-oooo 1 -oooo 1 -oooo 1 -OOOO I -oobo 1 -oooo 1 -oooo 0-0000 0-0000 6-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 ^ /i2 V2 I -0357 0-0152 0-9655 9-9848 1-0382 0-0163 0-9632 9-9837 1-0362 0-0155 0-9650 9-9845 1-0301 0-0129 0-9708 9-9871 1 -0205 0-0088 0-9799 9-9912 1-0088 0-0038 0-9913 9-9962 MK 0-93S3 9-9709 I -0692 0-0292 0-9330 9-9699 1-0718 r-0301 0-9348 9-9707 1-0697 0-0293 0-9409 9-9736 1 -0628 00264 0-9521 9-9787 1-0503 0-0213 0-9693 9-9864 I-03I7 0-0136 2MK 0-9687 9-9862 1-0323 0-0138 0-9686 9-9862 1-0324 0-0138 0-9687 9-9862 1-0323 0-0138 0-9692 9-9864 1-0318 0-0136 0-9717 9-9875 1-0291 0-0125 0-9778 9-9902 1-0227 0-0098 2 MS 1-0727 0-0305 0-9322 9-9695 1-0779 0-0326 0-9278 9-9674 1-0738 0-0309 0-9313 9-9691 1-0611 0-0257 0-9425 9-9743 1-0415 0-0177 0-9602 9-9823 1-0176 0-0076 0-9827 9-9924 Msf, 2 SM 1-0357 0-0152 0-9655 ' 9-9848 1-0382 0-0163 0-9632 9-9837 1 -0362 0-0155 0-9650 9-9845 I -0301 0-0129 0-9708 9-9871 1 -0205 0-0088 0-9799 9-9912 I -0088 0-0038 0-9913 9-9962 Mf 0-7007 9-8455 1-4272 0-1545 0-6886 9-8379 1-4523 0-1621 0-6981 9-8439 1-4325 0-I56I 0-7300 9-8633 1-3699 0-1367 0-7871 9-8960 1-2705 0-1040 0-8734 9-9412 I-I449 0-0588 Mm 1-1374 0-0559 0-8792 9.9441 I -1476 0-0598 0-8713 9-9402 I-J395 0-0567 0-8776 9-9433 1-1145 0-0471 0-8973 9-9529 1-0770 0-0322 0-9285 9-9678 1-0331 0-0142 0-9679 9-9858 Sa, Ssa I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo 1-0000 1-0000 ; 1-0000 I-oooo i-oooo I-oooo o-oooo 0-0000 o-oooo 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 o-oooo o-oooo 0-0000 REPORT FOR 1894— PART II. 221 Table 10. — Factors F and / for reduction and prediction of tides ; computed for the middle of each year, or for July ; Greenwich mean noon for common years, and at preceding midnigh' for leap years — Continued. at I J862 1863 1864 i86s 1S66 1867 Component J!- logii- log/ logi? log/ log^ lo^/ log^ log/ log.F / log/ logF log/ J., [M,] 0-9868 9-9942 I -0134 0-0058 I -0474 0-020I 0-9547 9-9799 I-II39 0-0468 0-8978 9-9532 1-1731 0-0693 0-8524 9-9307 1 -2068 0-0816 0-8287 9-9184 1-2013 0-0796 0-8325 9-9204 K. 0-9964 9-9984 I -0036 0-0016 1-0375 0-0160 0-9638 9-9840 1-0793 0-0331 0-9265 9-9669 I-1I40 0-0469 0-8977 9-953" 1-1327 0-0541 0-8828 9-9459 1-1297 0-0530 0-8852 9-9470 K„ I -0120 0-0052 0-9882 9-9948 1-1150 0-0473 0-8969 9-9527 I-2I54 0-0847 0-8228 9-9153 1-2941 0-1120 0-7727 9-8880 1-3344 0-125^ 0-7494 9-8747 1-3280 0-1232 0-7530 9-8768 L. 0-8162 9-9118 1-2252 0-0882 0-8950 9-9518 I-II73 0-0482 I -1564 0-0631 0-8647 9-9369 1-1371 0-0558 0-8794 9-9442 0-9239 9-9656 1 -0824 0-0344 0-8405 9-9245 1-1898 0-0755 [U], 0-9962 9-9984 I -0038 o-ooi6 0-9843 9-9931 I -0159 0-0069 0-9743 9-9887 1 -0264 0-0113 0-9672 9-9855 1-0339 0-0145 0-9638 9-9840 1 -0376 0-0160 0-9643 9-9842 1-0370 0-0158 Ml 0-9558 9-9804 I -0462 0-0196 0-7939 9-8998 1-2596 0-1002 0-5952 9-7746 1-6802 0-2254 0-6215 9-7934 1-6091 o-2o66 0-8590 9-9340 1-1641 0-0660 1-2312 0-0903 0-8122 9-9097 Mj, MS 0-9962 9-9984 1-0038 0-0016 0-9843 9-9931 I -0159 0-0069 0-9743 9-9887 I -0264 0-0113 0-9672 9-9855 1-0339 0-0145 0-9638 9-9840 1-0376 0-0160 0-9643 9-9842 1-0370 0-0158 M3 0-9944 9-9975 1-0057 0-0025 0-9766 9-9897 I -0240 0-0103 0-9617 9-9830 1 -0398 0-0170 0-9513 9-9783 1-0512 0-0217 0-9462 9-9760 1-0569 0-0240 0-9470 9-9763 1 -0560 0-0237 Ml, MN 0-9925 9-9967 I -0076 0-0033 0-9689 9-9863 I -0321 0-0137 0-9492 9-9774 1-0535 0-0226 0-9355 9-9711 I -0689 0-0289 0-9289 9-9680 1-0765 0-0320 0-9300 9-9685 1-0753 0-0315 Mfi 0-9887 9-9951 I-OI14 0-0049 0-9537 9-9794 I -0486 0-0206 0-9248 9-9661 1-0813 0-0339 0-9049 9-9566 1-1051 0-0434 0-8953 9-9520 1-1169 0-0480 0-8968 9-9527 1-1151 0-0473 Ms 0-9850 9"9934 I -0152 0-0066 0-9387 9-9725 1-0653 0-0275 0-9011 9-9548 1-1098 0-0452 0-8752 9-9421 1-1426 0-0579 0-8629 9-9360 11589 0-0640 0-8648 9-9369 1-1563 0-0631 N2, 2N 0-9962 9-9984 1-0038 0-0016 0-9843 9-9931 I-OI59 0-0069 0-9743 9-9887 1 -0264 0-0113 0-9672 9-9855 1-0339 0-0145 0-9638 9-9840 1-0376 0-0160 0-9643 9-9842 1-0370 0-0158 0„Q, 0-9947 9-9977 1-0053 0-0023 1-0627 0-0264 0-9410 9-9736 1-1363 0-0555 0-8800 9-9445 1-2014 0-0797 0-8324 9-9203 1-2382 0-0928 0-8076 9-9072 1-2322 0-0907 0-8II6 9-9093 00 0-9917 9-9964 I -0084 0-0036 1-2389 0-0930 0-8072 9-9070 "•5457 0-1891 0-6470 9-8109 1-8536 0-2680 0-5395 9-7320 2-0436 0-3104 0-4894 9-6896 2-oii8 0-3036 0-4971 9-6964 Pi) 1^2) T2 I -0000 o-oooo I -0000 o-oooo I -oooo o-oooo I -oooo 0-0000 I -oooo 0-0000 1 -oooo 0-0000 I -oooo o-oooo 1-0000 0-0000 1 -OOOO 0-0000 1-0000 o-oooo I-oooo 0-0000 1-0000 0-0000 Si, 2» 3j 4 I -0000 0-0000 I -0000 0-0000 I -oooo 0-0000 I -oooo 0-0000 I -oooo 0-0000 1-0000 0-0000 I -oooo 0-0000 1-0000 0-0000 i-oooo 0-0000 I-oooo 0-0000 1-0000 0-0000 1-0000 0-0060 As, fil, Vl 0-9962 9-9984 1-0038 0-0016 0-9843 9-9931 I -0159 0-0069 0-9743 9-9887 1 -0264 0-0113 0-9672 9-9855 1-0339 0-0145 0-9,638 9-9840 1-0376 o-oi6o 0-9643 9-9842 1-0370 0-0158 MK 0-9927 9-9968 1-0074 0-0032 I -021 2 0-0091 0-9792 9-9909 1-0516 0-0218 0-9510 9-9782 1-077S 0-0324 0-9281 9-9676 1-0917 0-0381 0-9160 9-9619 I -0894 0-0372 0-9179 9-9628 2MK 0-9889 9-9952 I-OII2 0-0048 1-0052 0-0022 0-9948 9-9978 1-0245 0-0105 0-9761 9-9895 I 0422 0-0179 0-9S95 9.9821 1-0522 0-0221 0-9504 9-9779- 1-0506 0-0214 0-9519 9-9786 2 MS 0-9925 9-9967 1-0076 0-0033 0-9689 9-9863 1-0321 0-0137 0-9492 9-9774 I -0535 0-0226 0-9355 9-9711 1-0689 0-0289 0-9289 9-9680 1-0765 0-0320 0-9300 9-9685 1-0753 0-0315 MSf, 2SM 0-9962 9-9984 [ -0038 0-0016 0-9843 9-9931 I -0159 0-0069 0-9743 9-9887 1 -0264 0-0113 0-9672 9-9855 1-0339 0-0145 0-9638 9-9840 1-0376 0-0160 0-9643 9-9842 1-0370 0-0158 Mf 0-9932 9-9970 1-0069 0-0030 I -1474 0-0597 0-8715 9-9403 1-3253 0-1223 0-7546 9-8777 1-4923 0-1738 0-6701 9-8262 1-5907 0-20l6 0-6287 9-7984 1-5744 0-1971 0-6352 9-8029 Mm 0-9888 9-9951 IOII3 0-0049 0-9489 9-9772 I -0539 0-0228 0-9169 9-9623 I -0906 0-0377 0-8951 9-9519 1-1172 0-0481 0-8848 9-9468 11302 0-0532 0-8864 9-9476 1-1281 0-0524 Sa, Ssa I -oooo 0-0000 I -0000 0-0000 I -oooo 0-0000 I -oooo 0-0000 1 -oooo 0-0000 1 -oooo 0-0000 I -oooo 0-0000 1-0000 0-0000 i-oooo 0-0000 I -oooo 0-0000 I-oooo 0-0000 I-oooo 0-0000 222 UNITED STATES COAST AND GEODETIC SURVEY. Tablk 10. — Factors F and f for reduetion and prediction of tides; computed for the middle of each year, or for July : Greenivich mean noon for common years, and at preceding midnight for leap years — Continued. j,t Component. J.. [M,] K, [L,] M, Ms, MS M3 Ml, MN Me Ms N2, 2N Oi.Qi 00 Pi I Rj) T2 Si, 2» 3» A MK 2MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa F \ogF I-1589 0-0640 I-1059 0-0437 1-2762 0-1059 0-9164 9-9621 o-( 9-9862 0-8409 9-9247 0-9688 9-9862 o'9S3S 9'9793 0-9386 9-9725 0-9093 9-9587 0-8809 9-9449 09688 9-9862 1-1858 0-0740 1-7768 0-2496 I -0000 0-0000 i-oooo 0-0000 o- 9-9862 I -07 14 0-0299 1-0379 0-0162 0-9386 9-9725 o- 9-9862 1-4515 0-I6I8 0-8999 9-9542 I-oooo 0-0000 logy 0-8629 9-9360 0-9043 9-9563 0-7836 9-8941 1-0912 0-0379 1-0322 0-0138 I -1892 0-0753 I -0322 0-0138 I -0487 0-0207 1-0655 0-0275 1-0998 0-0413 I-I352 0-0551 1-0322 0-0138 0-8433 9-9260 0-5628 9-7504 I-oooo 0-0000 I-oooo 0-0000 1-0322 0-0138 0.9334 9-9701 0-9635 9-9838 1-0655 0-0275 1-0322 0-0138 o- 9.8382 I-III2 0-0458 I-oooo 0-0000 F logF I -0961 0-0398 I -0684 0-0287 1-1898 0-0755 I-I90I 0-0756 0-9767 9-9898 0-5776 9-7616 0-9767 9-9898 0-9653 9-9847 0-9540 9-9795 0-9318 9-9693 0-9IOI 9-9591 0-9767 9-9898 I-II67 0-0479 1-4596 0-1642 I-oooo 0-0000 I-oooo 0-0000 0-9767 9-9898 1-0435 0-0185 I-OI92 0-0083 0-9540 9-9795 0-9767 9-9898 1-2766 o-io6i 0-9246 9-9659 I-oooo 0-0000 logy 0-9124 9-9602 0-9360 9-9713 0-8405 9-9245 0-8403 9-9244 I -0238 0-0102 I-73I3 0-2384 I -0238 0-0102 1-0359 0-0153 I -0482 0-0205 1-0732 0-0307 0-0409 1-0238 0-0102 0-8955 9-9521 0-6852 9-8358 I-oooo 0-0000 I-oooo 0-0000 1-0238 0-0102 0-9583 9-9815 0-981 1 9-9917 I -0482 0-0205 1-0238 0-0102 0-7833 9-8939 i-o8i6 0-0341 I-OOOO 0-0000 1870 F logF I -0303 0-0130 1-0262 0-0II2 I -0869 0-0362 1-2073 0-0818 0-9874 9-9945 0-5521 9-7420 0-9874 9-9945 0-981 1 9-9917 0-9749 9-9890 0-9626 9-9834 0-9504 9-9779 0-9874 9-9945 I -0436 0-0185 I-1658 0-0666 I-OOOO 0-0000 I-OOOO o-oooo 0-9874 9-9945 1-0132 0-0057 I -0004 0-0002 0-9749 9-9890 0-9874 9-9945 1-1030 0-0426 0-9589 9-9818 I-oooo o-oooo y logy 0-9706 9-9870 0-9745 9-9888 0-9201 9-9638 0-8283 9 9182 1-0128 0-0055 1-8113 0-2580 1-0128 0-0055 1-0192 0-0083 1-0257 0-0110 1-0389 0-0166 1-0522 0-022I 1-0128 0-0055 0-9582 9-9815 0-8578 9-9334 I -0000 0-0000 I-oooo 0-0000 1-0128 0-0055 0-9870 9-9943 0-9996 9-9998 1-0257 0-0110 I -01 28 0-0055 0-9066 9-9574 I -0428 0-0182 I-oooo 0-0000 I87I F logF lo^y 0-9724 9-9878 0-9862 9-9940 0-9859 9-9938 0-8708 9-9399 0-9996 9-9998 0-7693 9-8861 0-9996 9-9998 0-9994 9-9997 0-9992 9-9996 0-9988 9-9995 0-9983 9-9993 0-9996 9-9998 0-9784 9-9905 0-9373 9-9719 1 -0000 0-0009 1 -0000 0-0000 0-9996 9-9998 0-9858 9-9938 0-9854 9-9936 0-9992 9-9996 o- 9-9998 0-9576 9-9812 1-0004 0-0002 I -0000 0-0000 1 -0284 0-0122 1-0140 0-0060 1-0143 0-0062 1-1484 0-0601 1 -0004 O-0002 1-2999 0-1139 1-0004 0-0002 1 -0006 0-0003 1-0008 0-0004 1-0012 0-0005 1-0017 0-0007 1-0004 0-0002 1-0221 0-0095 l-( 0-0281 1 -0000 0-0000 I -0000 0-0000 I -0004 0-0002 1-0144 0-0062 1-0148 0-0064 1 -0008 0-0004 I -0004 0-0002 I -0443 0-0188 0-9996 9-9998 I -0000 o-oooO 1872 1873 F iogF y logy F \og F log/ 0-9266 9-9669 1-0793 0-0331 0-8933 9-9510 1-1195 0-0490 0-9526 9-9789 1 -0498 0-02H 0-9269 9-9670 1-0789 0-0330 0-8992 9-9539 I-1I21 0-0461 0-8327 9-9205 1-2010 0-0795 0-8172 9-9123 1-2237 0-0877 1-2260 0-0885 0-8157 9-9115 1-0121 0-0052 0-9881 9-9948 I -0234 o-oioo 0-9772 9-9900 0-8438 9-9262 1-1851 0-0738 0-4990 9-6981 2-0040 0-3019 1-0121 0-0052 0-9881 9-9948 1-0234 0-0100 0-9772 9-9900 I-01S2 0-0078 0-9821 9-9922 1-0352 0-0150 0-9660 9-9850 1-0243 0-0104 0-9763 9-9896 1-0473 O-O201 0-9549 9-9799 1-0366 0-0156 0-9647 9-9844 1-0717 0-0301 0-9331 9-9699 1 -0492 0-0208 0-9531 9-9792 1-0968 0-0401 o-gii8 9-9599 I-0I21 0-0052 0-9881 9-9948 1-0234 0-0 no 0-9772 9-9900 0-9259 9-9666 1 -0800 0-0334 0-8871 9-9480 1-1272 0-0520 0-7750 9-8893 1 -2904 0-1107 0-6667 9-8239 1-5000 0-1761 I -COOO 1 -0000 I-oooo I -0000 o-oooo 0-0000 0-0000 0-0000 I -0000 1 -0000 I -0000 1 -0000 o-oooo 0-0000 0-0000 0-0000 I-OI2I 0-0052 0-9881 9-9948 1-0234 O-OIOO 0-9772 9-9900 0-9641 9-9841 1-0373 0-0159 0-9485 9-9770 1-0543 0-0230 0-9757 9-9893 1 -0249 0-0107 0-9707 9-9871 1-0302 0-0129 1-0243 0-0104 0-9763 9-9896 1-0473 0-0201 0-9549 9-9799 1-012I 0-0052 0-9881 9-9948 1-0234 0-0100 0-9772 9-9900 0-8471 9-9279 i-i8o6 0-0721 0-7690 9-8860 1-3003 0-1140 1-0452 0-0192 0-9568 9-9808 1 -0880 0-0366 0-9191 9-9634 I-OOOO o-oooo I-oooo 0-0000 I -0000 0-0000 I -0000 o-oooo REPOET FOR 1894— PART II. 223 Tabie 10. — Factors F and f for reduction and prediction of tides ; computed for the middle of each year, or for July S, at Greenxvich mean noon for common years, and at preceding midnight for leap years — Continued. 1874 Component. J.. [M.] LU-] M. M2, MS M3 Mi, MN Me M„ Ng, 2N 0„Q, 00 Si, 2, 3, 4 X2, /l-i, V^ MK 2MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa logi^ logy 0-8717 9-9404 0-9096 9'9589 0-7878 9-8964 1-8337 0-2633 1-0321 0-0137 0-4374 9-6408 I -0321 0-0137 I -0486 0-0206 1-0653 0-0275 1-0995 0-0412 1-1348 0-0549 I -0321 0-0137 0-8615 9-9353 0-6003 9-7784 I -0000 o-oooo I -0000 0-0000 I -0321 0-0137 0-9388 9-9726 0-9690 9-9863 1-0653 0-0275 1-0321 0-0137 0-7192 9-8568 1-1226 0-0502 I -0000 0-0000 1875 \ogF 1-1472 0-0596 1-0994 0-04H 1-2694 0-1036 0-54S3 9-7367 0-9689 9-9863 2-2864 0-3592 0-9689 9-9863 0-9S37 9-9794 0-9387 9-9725 0-9095 9-9588 0-8812 9-9451 0-9689 9-9863 I -1607 0-0647 1-6658 0-2216 I -0000 0-0000 I -0000 0-0000 0-9689 9-9863 I -065 1 0-0274 1-0320 o;oi37 0-9387 9-9725 0-9689 9-9863 1-3905 0-1432 0-8908 9-9498 I -0000 0-0000 logy 0-8604 9-9347 0-9004 9-9544 0-7638 9-8830 0-9196 9-9636 1-0372 0-0159 0-6038 9-7809 1-0372 0-0159 1-0563 0-0238 1-0758 0-0317 1-1158 0-0476 I-IS73 0-0634 1-0372 0-0159 0-8480 9-9284 D-5669 9-7535 1-0000 0-0000 1-0000 0-0000 1-0372 0-0159 0-9339 9-9703 0-9686 9-9862 1-0758 0-0317 I -0372 0-0159 0-6934 9-8410 I -1435 0-0582 1-0000 0-0000 1-1622 0-0653 I -1 106 0-0456 1-3092 0-1170 I -0874 0-0364 0-9641 9-9841 1-6562 0-2191 0-9641 9-9841 0-9467 g-9762 0-9296 9-9683 0-8962 9-9524 0-8641 9-9366 0-9641 9-9841 1-1792 0-0716 1-7639 0-2465 1-0000 0-0000 i-oooo 0-0000 0-9641 9-9841 I -0708 0-0297 I -0324 0-0138 0-9296 9-9683 0-9641 9-9841 1 -4422 0-1590 0-8745 9-9418 1-0000 0-0000 1876 F logF f logy 1877 0-1 9-9339 0-8991 9-9538 0-7604 9-8810 0-7874 9-8962 1 -0380 0-0162 0-8150 9-9111 1 -0380 0-0162 I -0575 0-0243 1-0774 0-0324 I-1I83 0-0485 1-1607 0-0647 1-0380 0-0162 0-8461 9-9274 0-5622 9-7499 1-0000 0-0000 I-oooo 9-0000 1-1644 0-0661 1-1122 0-0462 I-3151 0-1190 1-2700 0-1038 0-9634 9-9838 1-2270 0-0889 0-9634 9-9838 0-9456 9-9757 0-9282 9-9676 0-8943 9-951S 0-8615 9-9353 0-9634 9-9838 1-1819 0-0726 1-7787 0-2501 I-OOOO 0-0000 I-OOOO 0-0000 F XogF 1-0380 0-9634 0-0162 9-9838 0-9332 9-9700 0-9686 9-9862 1-0774 0-0324 I -0380 0-0162 0-6897 9-8387 1-1466 0-0594 1 -0000 0-0000 1-0716 0-0300 1-0324 0-0138 0-9282 9-9676 0-9634 9-9838 1-4499 0-1613 0-8721 9-9406 I-oooo 0-0000 y logy 0-8667 9-9378 0-9056 9-9569 0-7772 9-8905 1-1657 0-0666 1-0343 0-0146 0-5005 9-6994 1-0343 0-0146 1-0519 0-0220 1 -0698 0-0293 1-1065 0-0440 I -1445 0-0586 1-0343 0-0146 0-8556 9-9323 0-5854 9-7675 I-oooo 0-0000 1-0000 0-0000 1-0343 0-0146 0-9366 9-9716 0-9687 9-9862 1-0698 0-0293 1-0343 0-0146 0-7077 9-8499 I-I3I6 0-0537 I-oooo 0-0000 I-I538 0-0622 1-1043 0-0431 1-2867 0-1095 0-8578 9-9334 0-9668 9-9854 1-9981 0-3006 0-9668 9-9854 0-9506 9-9780 0-9348 9-9707 0-9038 9-9561 0-8738 9-9414 0-9668 9-9854 I-I688 0-0677 1-7082 0-2325 1-0000 0-0000 I-oooo 0-0000 0-9668 9-9854 1-0677 0-0284 I -0323 0-0138 0-9348 9-9707 0-9668 9-9854 1-4130 0-I50I 0-8837 9-9463 I-oooo 0-0000 1878 F \o%F y logy 0-8846 9-9468 0-9200 9-9638 0-8148 9-9110 1-8420 0-2653 I -0267 0-OII5 0-4408 9-6442 I -0267 0-0115 I -0404 0-0172 1-0542 0-0229 I -0824 0-0344 I-III3 0-0458 1-0267 0-0115 0-8769 9-9430 0-6396 9-8059 I-oooo 0-0000 I-oooo 0-0000 1-0267 0-OII5 0-9446 9-9753 0-9699 9-9867 I -0542 0-0229 1-0267 0-OII5 0-7489 9-8744 I-I0I2 0-0418 1-0000 0-0000 1-1305 0-0532 1 -0869 0-0362 1-2273 0-0890 0-5429 9-7347 0-9740 9-9885 2-2688 0-3558 0-9740 9-9885 0-9612 9-9828 0-9486 9-9771 0-9239 9-9656 0-8998 9-9542 0-9740 9-9885 I -1404 0-0570 I-56.S4 0-1941 1-0000 o-oooo I-oooo 0-0000 0-9740 9-9885 I -0586 0-0248 I -031 1 0-0133 0-9486 9-9771 0-9740 9-9885 1-3352 0-1256 0-9081 9-9582 I-oooo 0-0000 1879 F \ozF 0-9137 9-9608 0-9428 9-9744 0-8738 9-9414 0-9649 9-9845 1-0162 0-0070 0-6079 9-7838 1-0162 0-0070 1-0244 0-0105 1 -0326 0-0139 I -0493 0-0209 I -0663 0-0279 1-0162 0-0070 0-91 10 9-9595 0-7322 9-8646 1 -0000 0-0000 i.-oooo 0-0000 1-0162 0-0070 0-9580 9-9814 0-9735 9-9883 1 -0326 0-0139 1-0162 0-0070 0-8167 9-9121 1 -0605 0-0255 i-oooo 0-0000 logy I -0945 0-0392 I -0607 0-0256 I-I444 0-0586 1 -0364 0-0155 0-9841 9-9930 1-6450 0-2162 0-9841 9-9930 0-9762 9-9895 0-9684 9-9861 0-9530 9-9791 0-9378 9-9721 0-9841 9-9930 1-0977 0-0405 1-3658 0-1354 1-0000 0-0000 1-0000 o-oooo 0-9841 9-9930 1 -0438 1-0272 0-0117 0-9684 g-9861 0-9841 9-9930 1-2244 0-0879 0-9430 9-9745 I-oooo o-oooo 224 UNITED STATES COAST AND GEODETIC SUEVEY. Table 10. — Factors F and f for reduction and prediction of tides ; computed for the middle of each year, or for July IS, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. i88o 1 i88i I8S2 1883 1884 I i85 1 F 1 log jS' f logr F logF loR/ F log F log/ F logF logy F logF logy logF logy J., [M,] 0-955I 9-9801 I -0470 0-0199 I -0092 0-0040 0-9909 9-9960 1-0731 0-0306 0-9319 9-9694 I-I386 0-0564 0-8783 9-9436 I -1902 0-0756 0-8402 9-9244 1 -2095 0-0826 0-8268 9-9174 K, 0-9738 9-9884 I -0269 0-OII6 I-OII9 0-0052 0-9882 9-9948 I -0540 0-0229 0-9488 9-9772 I -0940 0-0390 0-9140 9-9610 1-1236 0-0506 0-8900 9-9494 1-1342 0-0547 0-8817 9-9453 Kj 0-9540 9-9795 I -0483 0-0205 I -05 1 2 0-0217 0-9512 9-9783 0-0627 0-8656 9-9373 I -2494 0-0967 0-8004 9-9033 I -3149 0-1189 0-7605 9-8811 1-3376 0-1263 0-7476 9-8737 L, 0-8024 9-9044 1-2463 0-0956 0-9803 9-9914 1.0201 0-0086 1-2726 0-1047 0-7858 9-8953 1-0594 0-0250 0-9440 9-9750 0-8678 9-9384 1-1523 0-0616 0-8538 9-9314 1-1712 0.0686 [L.] I -0039 0-0017 0-9961 9-9983 0-9914 9-9963 1 -0086 0-0037 0-9801 9-9913 1 -0203 0-0087 0-9712 9-9873 1-0297 0-0127 0-9654 9-9847 1-0358 0-0153 0-9635 9-9839 1-0378 0-0161 M, 0-9564 9-9807 1-0456 0.0193 0-6543 9-8158 1-5284 0-1842 0-5482 97389 1 -8242 .0-2611 0-6541 9-8156 1-5289 0-1844 1-0314 0-0134 0-9696 9-9866 1.1408 0-0572 0-8766 9-9428 Mj, MS I -0039 0-0017 0-9961 9-9983 0-9914 9-9963 1-0086 0-0037 0-9801 9-9913 1 -0203 0-0087 0-9712 9-9873 1-0297 0-0127 0-9654 9-9847 1-0358 0-0153 0-9635 9-9839 1-0378 0-0161 Ms 1-0059 0-0026 0-9941 9-9974 0-9872 9-9944 1-0130 0-0056 0-9703 9-9869 1 -0306 0-0131 0-9571 9-9810 1 -0448 0-0190 0-9487 9-9771 1-0541 0-0229 0-9458 9-9758 1-0573 0-0242 M<, MN 1-0079 0-0034 0-9922 9-9966 0-9830 9-9925 1-0173 0-0075 0-9606 9-9826 1-0410 0-0174 0-9432 9-9746 I -0602 0-0254 0-9321 9-9695 1-0729 0-0305 0-9284 9-9677 1-0771 0-0323 Me 1-0118 0-0051 0-9883 9-9949 0-9746 9-9888 1-0261 0-0112 o'94i5 9-9738 1-0621 0-0262 0-9160 9-9619 1-0917 0-0381 0-8999 9-9542 1-1112 0-0458 0-8946 9-9516 1-1179 0-0484 Ms 1-0158 0-0068 0-9845 9-9932 0-9662 9-9851 1-0350 0-0149 0-9228 9-9651 1-0836 0-0349 0-8896 9-9492 I-I241 0-0508 0-8688 9-9389 1-I5IO 0-0611 0-8619 9-9355 I -1602 0-0645 Nj, 2N 1-0039 0-0017 0-9961 9-9983 0-9914 9-9963 1 -0086 0-0037 0-9801 9-9913 1 -0203 0-0087 0-9712 9-9873 1-0297 0-0127 0-9654 9-9847 1-0358 0-0153 0-9635 9-9839 1-0378 0-0161 0„Q, 0-9587 9-9817 I -0430 0-0183 I -0199 0-0086 0-9804 9-9914 1-0912 0-0379 0-9164 9-9621 1-1635 0-0658 0-8595 9-9342 1-2201 0-0864 0-8196 9-9136 1-2412 0-0938 0-8057 9-9062 OO 0-8744 9-9417 I '1437 0-0583 1-0794 0-0332 0-9264 9-9668 1-3526 0-1312 0-7393 9-8688 1-6700 0-2227 0-5988 9-7773 • 1 -9486 0-2897 0-5132 9-7103 2-0597 0-3138 0-4855 9-6862 Pi, Rs, T, I -0000 0-0000 I -0000 0-0000 I -OOOO 0-0000 1 -oooo o-oooo I -oooo 0-0000 I -oooo 0-0000 I -oooo o-oooo I-oooo 0-0000 1 -oooo o-oooo 1 -oooo 0-0000 1 -oooo 0-0000 1 -oooo 0-0000 Sl,2» 3» -t I -0000 o-oooo I -oooo o-oooo I -oooo o-oooo 1-0000 0-0000 1 -oooo 0-0000 1-0000 0-0000 1-0000 0-0000 1-0000 0-0000 1 -oooo 0-0000 I -oooo 0-0000 1 -oooo 0-0000 I -oooo O-OOOO' Aj, ^3, V2 1-0039 0-0017 0-9961 9-9983 0-9914 9-9963 I -0086 0-0037 0-9801 9-9913 1 -0203 0-0087 0-9712 9-9873 1-0297 0-0127 0-9654 9-9847 1-0358 0-0153 0-9635 9-9839 1-0378 o-oi6i MK 0-9776 9-9902 I -0229 0-0098 I -0033 0-0014 0-9967 9-9986 1-0331 0-0141 0-9680 9-9859 1 -0625 0-0263 0-9412 9-9737 1-0848 0-0353 0-9219 9-9647 1-0929 0-0386 0-9150 9-9614 2MK 0-9814 9-9918 I-0I89 0-0082 0-9947 9-9977 I -0053 0-0023 1-0125 0-0054 0-9876 9-9946 1-0319 0-0136 0-9691 9-9864 1-0473 0-0201^ 0-9S49 9-9799 1-0530 0-0224 0-9496 9-9776 2 MS 1-0079 0-0034 0-9922 9-9966 0-9830 9-9925 1-0173 0-0075 0-9606 9-9826 1-0410 0-0174 0-9432 9-9746 I -0602 0-0254 0-9321 9-9695 1-0729 0-0305 0-9284 9-9677 1-0771 0-0323 MSf, 2 SM I -0039 0-0017 0-9961 9-9983 0-9914 9-9963 i-oo86 0-0037 0-9801 9-9913 1 -0203 0-0087 0-9712 9-9873 1-0297 0-0127 0-9654 9-9847 1-0358 0-0153 0-9635 9-9839 1-0378 0-0161 Mf 0-9156 9-9617 I -0922 0-0383 I -0493 0-0209 0-9530 9-9791 1-2149 0-0845 0-8231 9-9155 1-3939 0-1442 0-7174 9-8558 1-5419 0-1880 0-6486 9-8120 1-5989 0-2038 0-6255 9-7962 Mm I-OI57 0-0068 0-9845 9-9932 0-9725 9-9879 I -0283 0-0121 0-9354 9-9710 I -0691 0-0290 0-9072 9-9577 1-1023 0-0423 0-8897 9-9493 1-1239 0-0507 0-8840 9-9465 1-I3I2 0-053S Sa, Ssa I -oooo 0-0000 I -oooo 0-0000 I -oooo 0-0000 i-oooo o-oooo 1-0000 0-0000 1-0000 0-0000 1-0000 0-0000 1 -oooo 0-0000 I-oooo 0-0000 I -oooo o-oooo 1-0000 i o-oooo 1-0000 o-oooo EEPOKT FOR 1894— PAET II, 225 Table 10. — Faetors F and f for reduction and prediction of tides; computed for the middle of each year, or for July ; Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. at Component. J.. [M,] K, . K, U M, M2, MS M3 M„MN Ms Ms N2, 2N Oi,Q, 00 Pi) Rj) T2 Si, 2, ii, 4 MK 2 MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa F log f / logr 1-1885 0-0750 I'I226 0-0502 I-3I29 0-1182 0-9943 9-9975 0-9556 9-9848 0-7330 9-8651 0-9656 9-9848 0-9489 9-9772 0-9324 9-9696 0-9003 9-9544 0-8694 9-9392 0-9656 9-9848 I 21 82 0-0857 1-9390 0-2876 I -oooo o-oooo I "OOOO O'OOOO 6-9656 9-9848 I -0840 0-0350 I -0467 0-0198 0-9324 9-9696 0-9656 9-9848 1-5369 0-1866 0-8903 9-9495 I -oooo 0-0000 1887 F logF 0-8414 9-9250 0-8908 9-9498 0-7617 9-8818 1-0058 0-0025 1-0356 0-0152 1-3643 0-1349 1-0356 0-0152 1-0539 0-0228 1-0725 0-0304 I-II07 0-0456 I-I503 0-0608 1-0356 0-0152 0-8209 9-9143 0-5157 9-7124 I -0000 o-oooo : -oooo o-oooo 1-0356 0-0152 0-9225 9-9650 0-9554 9-9802 1-0725 0-0304 1-0356 0-0152 0-6506 9-8134 I-I233 0-0505 i-i5ooo o-oooo log/ I-I360 0-0554 1-0925 0-0384 1-2460 0-0955 1-2143 0-0843 0-9715 9-9874 0-5810 9-7642 0-97IS 9-9874 0-9575 9-9811 0-9438 9-9749 0-9168 9-9623 0-8907 9-9497 0-9715 9-9874 I-I607 0-0647 1-6568 0-2192 I -oooo o-oooo 1 -oooo 0-0000 0-9715 9-9874 I -0614 0-0259 1-0311 0-0133 0-9438 9-9749 0-9715 9-9874 1-3867 0-1420 0-9082 9-9582 1-0000 0-0000 0-8803 9-9446 0-9153 9-9616 0-8026 9-9045 0-8235 9-9157 1-0294 0-0126 I-72II 0-2358 I -0294 0-0126 1-0444 0-0189 1-0596 0-0251 I -0907 0-0377 1-1227 0-0503 I -0294 0-0126 0-8616 9-9353 0-6036 9-7808 i-cooo 0-0000 I -oooo o-oooo I -0294 0-0126 0-9422 9-9741 0-9698 9-9867 1-0596 0-0251 I -0294 0-0126 0-7212 9-8580 I-lOIl 0-0418 1 -oooo o-oooo F log-F f log/ I -0702 0-0295 1-0522 0-0221 1-1508 o-o6io I -0581 0-0245 0-9806 9-9915 0-6278 9-7978 0-9806 9-9915 0-9710 9-9872 0-9615 9-9830 0-9428 9'9744 0-9245 9-9659 0-9806 9-9915 I -0880 J -0366 1-3395 0-1270 i-oooo 0-0000 I -oooo o-oooo 0-9806 9-9915 1-0317 0-0136 1-0117 0-0050 0-9615 9-9830 0-9806 9-9915 1-2072 o-o8i8 0-9368 9-9717 1 -oooo 0-0000 F \ozF 0-9344 9-9705 0-9504 9-9779 0-8689 9-9390 0-9451 9-9755 1-0198 0-0085 1-5929 0-2022 1-0198 0-0085 I -0299 0-0128 I -0400 0-0170 1 -0606 0-0256 i-o8i6 0-0341 1-0198 0-0085 0-9191 9-9634 0-7465 9-8730 I -oooo 0-0000 I -oooo o-oooo I -0198 0-0085 0-9692 9-9864 0-9884 9-9950 1 -0400 0-0170 1-0198 0-0085 0-8283 9-9182 1-0674 0-0283 I -oooo 0-0000 / log/ I -0066 0-0028 I'OIOI 0-0044 I -0467 0-0198 0-8301 9-9191 0-9920 9-9965 0-9302 9-9686 0-9920 9-9965 0-9880 9-9948 0-9840 9-9930 0-9761 9-9895 0-9683 9-9860 0-9920 9-9965 1-0170 0-0073 1 -0689 0-0289 I -oofto 0-0000 1 -oooo 0-0000 0-9920 9-9965 1 -0020 0-0009 0-9940 9-9974 0-9840 9-9930 0-9920 9-9965 I -0426 0-0181 0-9744 9-9887 I-oooo 0-0000 0-9935 9-9972 0-9900 9-9956 0-9554 9-9802 I -2047 0-0809 1-0081 0-0035 I -07^0 0-0314 I -0081 0-0035 1-0122 0-0052 1-0162 0-0070 I -0245 0-0105 I -0328 0-0140 I -0081 0-0035 0-9833 9-9927 0-9356 9-9711 1 -oooo o-oooo I'OOOO o-oooo i-oo8i 0-0035 0-9980 9-9991 1 0061 0-0026 1-0162 0-0070 i-oo8i 0-0035 0-9591 9-9819 1 -0263 0-0113 1-0000 o-oooo i8go F \0%F 0-9531 9-9792 0-9723 9-9878 0-9502 9-9778 0-8671 ■ 9-9380 1 -0045 0-0019 0-7560 9-8785 1-0045 0-0019 1-0067 0-0029 I -0089 0-0039 1-0134 0-0058 1-0179 0-0077 1-0045 0-0019 0-9564 9-9806 0-8671 9-9381 I -oooo o-oooo I -oooo o-oooo I -0045 0-0019 0-9766 9-9897 0-9810 9-9917 1 -0089 0-0039 I -0045 0-0019 0-9107 9-9594 1-0176 0-0076 1 -oooo o-oooo / log/ I89I 1 -0492 0-0208 1 -0285 0-OI22 1-0524 0-0222 I-I533 0-0620 0-9956 9-9981 1-3228 0-1215 0-9956 9-9981 0-9933 9-9971 0-9912 9-9961 o- 9-9912 0-9824 9-9923- 0-9956 9-9981 1-0456 0-0194 1-1532 0-0619 1 -oooo 0-0000 I -oooo O'OOOO 0-9956 9-9981 I -0239 0-0103 1-0194 0-0083 0-9912 9-9961 0-9956 9-9981 1-0981 0-0406 9827 9-9924 1 -oooo 0-0000 F log/^ 0-9123 9-9601 0-9417 9-9739 0-8710 9-9400 1-4101 0-1492 i-oi66 0-0072 0-4783 9-6797 I -0166 0-0072 I -025 1 0-0108 1-0336 0-0143 I -0508 0-0215 1-0683 0-0287 I -0166 0-0072 0-9093 9-9587 0-7275 9-8618 1-0000 0-0000 1 -oooo 0-0000 1-0166 0-0072 0-9573 9-9811 0-9733 9-9882 1-0336 0-0143 I -0166 0-0072 0-8133 9-9103 1-0623 0-0262 I-oooo 0-0000 / log/ 1 -0962 0-0399 1 -0620 0-0261 1-1481 0-0600 0-7092 9-8508 0-9836 9-9928 2-0907 0-3203 0-9836 9-9928 0-9755 9-9892 0-9675 9-9857 0-9517 9-9785 0-9361 9-97'3 0-9836 9-9928 I -0997 0-0413 1-3746 0-1382 1 -oooo 0-0000 I -oooo 0-0000 0-9836 9-9928 I -0446 0-0189 1-0275 0-OII8 0-9675 9-9857 0-9836 9-9928 1-2295 0-0897 0-9414 9-9738 I -oooo o-oooo 226 UNITED STATES COAST AND GEODETIC SUEVEY. Table 10. — Factors F and f for reduction and prediction of tides ; computed for the middle of each year, or for July 2, at Greenirich mean noon for common years, and at preceding midnight for leap years — Continued. Component. Ji. [M,] u [L.] Ml Ma, MS Ms M4, MN Ms Ma Na, 2N 0„Q, OO Pi> R2. Tj Sl-2-3-4 MK 2MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa 1S92 logF log/ 0-8836 9-9463 0-9192 9"9634 0-8128 9-9100 1-4651 0-1659 I -027 1 o-oii6 0-4633 9-6659 1-0271 0-0116 I -0410 0-0174 1-0550 0-0232 1-0836 0-0349 1-1130 0-0465 1,-0271 0-0116 0-8758 9-9424 0-6367 9-8039 I -oooo' 0-0000 I -oooo 0-0000 I-027J 0-OII6 0-9442 9-9750 0-9698 9-9867 1-0550 0-0232 1-0271 0-OII6 0-7467 9-8732 1-1027 0-0424 I -oooo 0-0000 1-1317 0-0537 1-0878 0-0366 1-2303 0-0900 0-6826 9'834i 0-9736 9-9884 2-1582 0-3341 0-9736 9-9884 0-9607 9-9826 0-9479 9-9768 0-9229 9-9651 0-8985 9-9535 0-9736 ■893 F log P 1-1419 0-0576 1-5707 0-1961 I -oooo o-oooo I -oooo o-oooo 0-9736 9-9884 I -0591 0-0250 1-0312 0-0133 0-9479 9-9768 0-9736 9-9884 1-3392 0-1268 0-9069 9-9576 I -oooo 0-0000 / log/ 0-8661 9-9376 0-9051 9-9567 0-7760 9-8899 0-8437 9-9262 I -0346 0-0148 o-6^5 9-«435 1-0346 0-0148 1-0523 0-022I I -0703 0-0295 1-1073 0-0443 I-I456 0-0590 I -0346 0-0148 0-8549 9-9319 0-5837 9-7662 I -OOOO 0-0000 I -oooo O'OOOO I -0346 0-0148 0-9364 9-9714 0-9687 9-9862 I -0703 0-0295 I -0346 0-0148 0-7064 9-8491 1-1326 0-0541 I -oooo 0-0000 1-1546 0-0624 1-1049 0-0433 1-2886 OlIOl 1-1853 0-0738 0-9666 9-9852 1-4337 0-1565 0-9666 9-9852 0-9503 9-9779 0-9343 9-9705 0-9031 9-9557 0-8729 9-9410 0-9666 9.9852 1-1697 0-0681 1-7131 0-2338 I -oooo o-oooo I -oooo 0-0000 0-9666 9-9852 1-0679 0-0286 1-0323 0-0138 0-9343 9-9705 0-9666 9-9852 1-4156 0-1509 0-8829 9-9459 1 -oooo o-oooo F logy? / lo?/ 0-8587 9-9338 0-8990 9-9538 0-7601 9-8809 0-8202 9-9139 1 -0380 0-0162 0-7305 9-8636 1 -0380 0-0162 1-0576 0-0243 I -0775 0-0324 1-1185 0-0486 I-16IO 0-0648 1-0380 0-0162 0-8459 9-9273 0-5618 9-7496 I -oooo o-oooo 1-0000 0-0000 1-0380 0-0162 0-9332 g-9700 0-9687 9-9862 1-0775 0-0324 1-0380 0-0162 1-1646 0-0662 1-1124 0-0462 1-3156 0-1191 1-2192 0-0861 0-9634 9-9838 1-3689 0-1364 0-9634 9-9838 0-94S5 9-9757 0-9281 9-9676 0-8941 9-9514 0-8613 9-9352 0-9634 1895 0-1 9-8385 1-1469 0-0595 r -oooo o-oooo 1-1821 0-0727 1-7800 0-2504 I -oooo 0-0000 1-0000 0-0000 0-9634 9-9838 1-0716 0-0300 1 -0324 0-0138 0-9281 9-9676 0-9634 9.9838 I -4506 0-1615 0-8719 9-9405 1 -oooo 0-0000 F \o%F 0-8607 9-9348 0-9006 9-9546 0-7644 9-8833 1-4211 0-1526 1-0371 0-0158 0-4596 9-6624 1-0371 0-0158 1-0562 0-0237 I -0755 0-0316 I-IIS4 0-0474 1-1568 0-0632 1-0371 0-0158 0-8484 9-9286 0-5677 9-7541 1 -oooo 0-0000 1-0000 0-0000 1-0371 0-0158 0-9340 9-9704 0-9687 9-98,62 1-0755 0-0316 1-0371 0-0158 0-6940 9-8414 1-1430 0-0580 1 -oooo 0-0000 / log/ I-I619 0-0652 1-1103 0-0454 1-3082 0-1167 0-7037 9-8474 0-9643 9-9842 2-1758 0-3376 0-9643 9-9842 0-1 9-9763 0-9298 9-9684 0-8965 9-9526 0-8645 9-9368 0-9643 9-9842 I-I787 0-0714 1-7614 0-2459 I -oooo 0-0000 1 -oooo o-oooo 0-9643 9-9842 1-0706 0-0296 1-0324 0-0138 0-9298 9-9684 0-9643 9-9842 1-4409 0-1586 0-8749 9-9420 1 -oooo 0-0000 F \a%F 0-8724 9-9407 0-9102 9-9591 0-7892 9-8972 1-6465 0-2166 1-0318 0-0136 0-4465 9-6498 1-0318 0-0136 1-0481 0-0204 I -0646 0-0272 1-0985 0-0408 1-1334 0-0544 1-0318 0-0136 0-8624 9-9357 0-6024 9-7799 1 -oooo 0-0000 1 -oooo 0-0000 1-0318 0-0136 0-9391 9-9727 0-9690 9-9863 1 -0646 0-0272 1-0318 0-0136 0-7208 9-8578 1-1214 0-0498 1 -oooo* 0-0000 / log/ 1-1463 0-0593 1-0987 0-0409 1-2670 0-1028 0-6073 9-7834 0-9692 9-9864 2-2398 0-3502 0-9692 9-9864 0-9541 9-9796 0-9393 9-9728 0-9103 9-9592 0-8823 9-9456 9-9864 1-1596 0-0643 I -6600 0-2201 I -oooo 0-0000 1 -oooo 0-0000 0-9692 9-9864 I -0648 0-0273 1-0320 0-0137 0-9393 9-9728 0-9692 9-9864 1-3874 0-1422 0-8918 9-9502 1 -oooo 0-0000 1897 F \a%F 0-8945 9-9516 0-9278 9-9675 0-8352 9-9218 0-8766 9-9428 1 -0229 0-0098 0-6799 9-8324 1-0229 0-0098 1-0346 0-0148 I -0464 0-0197 1-0703 0-0295 1-0949 0-0394 1-0229 0-0098 0-8886 9-9487 0-6706 9-8264 1 -oooo 0-0000 1 -oooo 0-0000 ■ 1-0229 0-0098 0-9491 9-9773 0-9709 9-9872 I -0464 0-0197 1-0229 0-0098 0-7719 9-8876 I -0862 0-0359 1 -oooo 0.0000 / log/ 1-1179 0-0484 1-0778 0-0325 1-1974 0-0782 1-1408 0-0572 0-9776 9-9902 1-4708 0-1676 0-9776 9-9902 0-9666 9-9852 0-9557 9-9803 0-9343 9-9705 0-9134 9-9606 0-9776 9-9902 1-1254 0-0513 1-4914 0-1736 i-oooo 0-0000 1 -oooo. 0-0000 0-9776 9-9902 1-0536 0-0227 1 -0300 0-0128 0-9557 9-9803 0-9776 9-9902 1-2955 0-1124 0-9206 9-9641 1 -oooo 0-0000 EEPORT FOR 1894— PAET II. 227 Table 10. — Factors F and f for reduction and prediction of tides; computed for the middle of each year, or for July S, at Greenivich mean noon for common years, and at preceding midnight for leap years — Continued. Component. Ji. [Ml] K, M, Mj, MS M, M4,MN Me Ma N3,2N 0„Q, OO Si, 2j 3, 4 A21 /i2> V2 MK 2MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa i8g8 log^P log/ 0-9283 9-9677 0-9538 9'979S 0-9025 9-9SSS 0.8138 9-9105 I-OII6 0-0050 0-8576 9'9333 1-0116 0-0050 1-0173 0-0075 1-0232 o-oioo 1-0350 0-0149 I -0469 0-0199 I-0II6 0-0050 0-9279 9-9675 0-7807 9-8925 I -0000 0-0000 I -0000 0-0000 I-0II6 0-0050 0-9648 9-9844 0-9760 9-9894 I -0232 O-OIOO 1-0116 0-0050 0-85 1 1 9-9300 I '0433 0-0184 I -0000 0-0000 '•0773 0-0323 I -0484 0-0205 I -1080 0-0445 1-2288 0-0895 0-9886 9-9950 1. 1660 0-0667 0-9886 9-9950 0-9829 9-9925 0-9773 9-9900 0-9662 9-9851 0-9552 9-9801 0-9886 9-9950 1-0778 0-0325 1-2809 0-1075 I -0000 0-0000 1-0000 0-0000 0-9886 9-9950 1-0365 0-0155 I -0246 o-oio6 0-9773 9-9900 0-9886 9-9950 1-1750 0-0700 0-9585 9-9816 I -oooo 0-0000 1899 F log/? log/ 0-9746 9-9888 0-9878 9'9947 0-9899 9-9956 1-1150 0-0473 0-9991 9-9996 0-5615 97494 0-9991 9-9996 0-9986 9 "9994 0-9981 9-9992 0-9972 9-9988 0-9962 9-9984 0-9991 9-9996 0-9808 9-9916 0-9454 9-9756 I -0000 0-0000 I -0000 0-0000 0-9991 9-9996 0-9868 9-9942 0-9859 9-9938 0-9981 9-9992 0-9991 9-9996 0-9630 9-9836 0-9986 9-9994 I -oooo 0-0000 1-0261 0-0II2 I -0124 0-0053 I 0102 0-0044 0-8969 9-9527 I -0009 0-0004 I -7809 0-2506 1-0009 0-0004 I -001 5 0-0006 1-0019 0-0008 1-0028 0-0012 I -0038 0-0016 I -0009 0-0004 1-0195 0-0084 1-0577 0-0244 I -oooo o-oooo 1-0000 0-0000 1-0009 0-0004 1-0133 0-0058 1-0143 0-0062 1-0019 0-0008 1-0009 0-0004 1 0384 00164 1-0014 0-0006 I -oooo 0-0000 F \osF log/ I -032^ 0-OI4I 1-0279 0-0120 1-0913 0-0379 1-3279 0-1232 0-9869 9-9943 0-5259 9-7209 0-9869 9-9943 0-9804 9-99'3 0-9739 9-9885 0-961 1 9-9828 0-9485 9-9770 0-9869 9-9943 I -0465 0-0198 1-1769 0-0707 I -oooo 0-0000 I -oooo 0-0000 0-9869 9-9943 I -0144 0-0062 1-oon 0-0005 0-9739 9-9885 0-9869 9-9943 I -1098 0-0452 0-9573 9-9811 I -oooo 0-0000 0-9681 99859 0-9728 9-9880 0-9164 9-9621 0-7531 9-8768 I-OI33 0-0057 1-9016 0-2791 I-OI33 0-0057 I -0200 o-oo86 1-0268 0-0115 I -0404 0-0172 1-0543 0-0230 1-0133 0-0057 0-9555 9-9802 0-8497 9-9293 I -oooo 0-0000 1 -oooo 0-0000 1-0133 0-0057 0-9858 9-9938 0-9989 9-9995 1 -0268 0-0115 I-OI33 0-0057 0-9011 9-9548 I -0446 0-0189 I -oooo 0-0000 F logF 1-0988 0-0409 I -0701 0-0294 I-I938 0-0769 0-9704 9-9870 0-9764 9-9896 0-7149 9-8542 0-9764 9-9896 0-9648 9-9844 0-9533 9-9792 0-9307 9-9688 0-9087 9-9584 0-9764 9-9896 I-II97 0-0491 1-4725 0-I68I I -oooo o-oooo I -oooo 0-0000 0-9764 9-9896 I -0448 0-0190 I-020I 0-0086 0-9533 9-9792 0-9764 9-9896 I -2840 o-io86 0-9234 9-9654 I -oooo 0-0000 / log/ 0-9101 9-9591 0-9345 9-9706 0-8377 9-9231 1-0305 0-0130 1 -0242 00104 1-3988 0-1458 1-0242 0-0104 1-0365 0-0156 1 -0490 0-0208 1-0744 0-0312 1-1004 0-0416 1 -0242 0-0104 0-8931 9-9509 0-6791 9-8319 I -oooo o-oooo 1-0000 0-0000 1 -0242 0-0104 0-9571 9-9810 0-9803 9-9914 I -0490 0-0208 1 -0242 0-0104 0-7788 9.8914 I -0830 0-0346 I -oooo 0-0000 F log/? I-I6II 0-0649 I -1072 0-0442 1-2790 0-1069 0-8383 9-9234 0-9686 9-9861 1-1662 0-0668 0-9686 9-9861 0-9532 9-9792 0-9381 9-9722 0-9086 9-9584 0-8800 9-9445 0-9686 9-9861 1-1882 0-0749 1-7884 0-2525 I -oooo o-ooco 1 -oooo 0-0000 0-9686 9.9861 1-0723 0-0303 1-0386 0-0164 0-9381 9-9722 0-9686 9-9861 1-4578 0-1637 0-8991 9-9538 I -oooo 0-0000 / log/ 0-8612 9-935' 0-9032 9-9558 0-7819 9-8931 1-1929 0-0766 I -0325 0-0139 0-8575 9-9332 1-0325 0-0139 I -049 1 o-o2o8 1 -0660 0-0278 1-1006 0-0416 1-1364 0-0555 1-0325 0-0139 0-8416 9-9251 0-5592 9-7475 1-0000 0-0000 I -oooo 0-0000 1 -0325 0-0139 0-9326 9-9697 0-9629 9-9836 I -0660 0-0278 1-0325 0-0139 0-6860 9-8363 1-II22 0-0462 I -oooo o oooo F \o%F 1-2022 0-0800 1-1302 0-0532 1-3292 0-1236 0-8951 9-9518 0-9642 9-9842 0-9354 9-9710 0-9642 9-9842 0-1 9-9763 0-9298 9-9684 0-8965 9-9526 0-8645 9-9368 0-9642 9-9842 1-2332 0-0910 2-0172 0-3048 I -oooo o-oooo 1 -oooo 0-0000 0-9642 9-9842 / log/ 1-1 0-0374 I 0508 0-0215 0-9298 9-9684 0-9642 9-9842 1-5772 0-1979 0-8861 9-9475 I -oooo 0-0000 0-8318 9-9200 0-8848 9-9468 0-7524 9.8764 1-1172 0-0482 I 0371 0-0158 1 -0690 0-0290 1-0371 0-0158 1-0562 0.0237 1-0755 0-0316 1-1154 0-0474 1-1568 0-0632 1-0371 0-0158 0-8109 9-9090 0-4957 9-6952 I -oooo o-oooo 1 -oooo 0-0000 I -037 1 0-0158 0-9176 9-9626 0-9516 9-9785 I -0755 0-0316 I -037 1 0-0158 0-6340 9-8021 1-1285 0-0525 I -oooo 00000 228 UNITED STATES COAST AND GEODETIC SUEVB¥. Table 10. — Factors F and/ for reduction and prediction of tides; computed for the middle of each year, or for July 2, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1904 I9°S igo6 1907 1908 1 )°9 F log/? log/ F \osF log/ F log F log/ F \osF log/ F logF log/ \0gF log/ J.. [Mi] I -2061 0-0814 0-8291 9-9186 1-1711 0-0686 0-8539 9-9314 I-III3 0-0458 o-85r99 9-9542 1-0449 0-OI9I 0-9570 9-9809 0-9846 9-9933 I-OI56 0-0067 0-9358 9-9712 1-0686 0-0288 K, 1-1323 0-0540 0-8831 9-9460 I-II28 0-0464 0-8986 9-9536 1-0777 0-0325 0-9279 9-9675 1-0358 0-0153 0-9654 9-9847 0-9949 9-9978 1-0052 0-0023 0-9595 9-9821 1-0422 0-0179 K. 1-3336 0-1250 0-7499 9-8750 I-2916 o-iiii 0-7742 9-8889 I-2II7 0-0834 0-8253 9-9166 I -1 109 0-0457 0-9002 9-9543 1 -ooSo 0-0035 0-9920 9-9965 0-9172 9-9625 I -0902 0-0375 L. I -0807 0-0337 0-9254 9-9663 I-I650 00663 0-8584 9-9337 0-9514 9-9783 I -05 1 1 0-0217 0-8192 9-9134 1-2207 0-0866 0-9416 9-9739 I -0620 0-0261 i-S3«4 0-1851 0-6530 9-8149 [L..] 0-9639 9-9840 1-0375 o-oi6o 0-9674 9-9856 1-0336 0-0144 0-9746 9-9888 I 0260 0-0112 0-9848 9-9933 I-0IS5 0-0067 0-9967 9-9986 1-0033 0-0014 1-0093 00040 0-9908 9-9960 M, 0-6636 9-8219 1-5070 0-1781 0-6086 9-7843 1-6432 0-2I57 0-7465 9-8730 1-3396 0-1270 1-0574 0-0242 0-94S7 9-9758 0-6762 9-8300 1 -4789 0-1700 0-4735 9-6753 2-1120 0-3247 M,, MS 0-9639 9-9840 J -0375 0-0160 0-9674 9-9856 1-0336 0-0144 0-9746 9-9888 I 0260 0-0112 0-9848 9-9933 1-0155 0-0067 0-9967 9-9986 1-0033 0-0014 1 -0093 0-0040 0-9908 9-9960 M, 0-9463 9-9760 1-0567 0-0240 0-9515 9-9784 1-0509 0-0216 0-9623 9-9833 I -0392 0-0167 0-9773 9-9900 1-0233 o-oioo 0-9951 9-9979 1 -0049 0-0021 I -0140 0-0060 0-9862 9-9940 M,, MN 0-9291 9-9680 1-0764 0-0320 0-9360 9-9713 I -0684 0-0287 0-9499 9-9777 1-0527 0-0223 0-9698 9-9867 1-0312 0-0133 0-9934 9-9971 I -0066 0-0029 1-0186 o-oo8o 0-9817 9-9920 Ms 0-8955 9-9521 1-H67 0-0479 0-9055 9-9569 I -1044 0-0431 0-9259 9-9666 I -0801 0-0334 0-9550 9-9800 1-0472 0-0200 0-9902 9-9957 I -0099 0-0043 1-0281 0-0120 0-9727 9-9880 M, 0-8631 9-9361 1-1586 0-0639 0-8760 9-9425 I-I4IS 0-0575 0-9024 9-9554 1-1082 0-0446 0-9404 9-9733 I -0634 0-0267 0-9869 9-9943 1-0133 0-0057 1-0376 0-0160 0-9637 9-9840 N^, 2N 0-9639 9-9840 1-0375 0-0160 0-9674 9-9856 I -0336 0-0144 0-9746 9-9888 I -0260 0-0II2 0-9848 9-9933 1-0155 0-0067 0-9967 9-9986 1-0033 0-0014 1 -0093 0-0040 0-9908 9-9960 0„Qi 1-2375 0-0925 0-8081 9-9075 I -1992 0-0789 0-8339 9-921 1 I-I33S 0-0544 0-8823 9-9456 I -0599 0-0253 0-9435 9-9747 0-9922 9-9966 1 -0078 0-0034 0-9366 9-9716 I -0677 0-0284 OO 2-0396 0-3095 04903 9-6905 I -8424 0-2654 0-5428 9-7346 1-5329 0-1855 0-6524 9-8145 1*2279 0-0892 0-8144 9-9108 0-9833 9-9927 1-0170 0-0073 0-8066 9-9066 1-2398 0^0934 Pi J 1^2) Ta I -0000 I -0000 I -0000 I -0000 I -0000 I -0000 I -0000 I -oooo I -oooo 1 -oooo I -oooo 1 -oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 ^I) 2) 3) 4 I -0000 I -0000 I -0000 I -0000 I -0000 I -0000 I -OOOO I -oooo I -oooo 1 -oooo 1 -oooo 1 -oooo 0-0000 o-oooo 0-0000 o-oooo 00000 0-0000 0-0000 0-0000 0-0000 o-oooo o-oooo 0-0000 As, 1X2, Vl 0-9639 9-9840 1-0375 0-0160 0-9674 9-9856 1-0336 0-0144 0-9746 9-9888 I -0260 0-0II2 0-9848 9-9933 1-0155 0-0067 0-9967 9-9986 1-0033 0-0014 1-0093 0-0040 0-9908 9-9960 MK I -0914 0-0380 0-9162 9-9620 1-0766 0-0321 0-9288 9-9679 I -0504 0-0214 0-9520 9-9786 I -0201 0-0086 0-9803 9-9914 0-9916 9-9963 1 -0085 0-0037 0-9684 9-9861 1-0326 0-0139 2 MK 1-0520 0-0220 0-9506 9-9780 I -0416 0-0177 0-9601 9-9823 1-0238 0-0102 0-9768 9-9898 I -0045 0-0020 0-9955 9-9980 0-9883 9-9949 1-0118 0-0051 0-9774 9-9901 1-0231 0-(i099 2 MS 0-9291 9-9680 I -0764 0-0320 0-9360 9-9713 I -0684 0-0287 0-9499 9-9777 1-0527 0-0223 0-9698 9-9867 1-0312 0-0133 0-9934 9-9971 1 -0066 0-0029 I -0186 0-0080 0-9817 9-9920 MSf, 2 SM 0-9639 g-9840 1-0375 0-0160 0-9674 9-9856 1-0336 0-0144 0-9746 9-9888 I -0260 0-0II2 0-9848 9-9933 1-0155 0-0067 0-9967 9-9986 I -0033 0-0014 1-0093 0-0040 0-9908 9-9960 Mf 1-58S7 0-2010 0-6295 9-7990 I -4864 0-I72I 0-6728 9-8279 I-3181 0-I200 0-7586 9-8800 1-1408 0-0572 0-8766 9-9428 0-9878 9-9947 1-0124 0-0053 0-8692 9-9391 1-1506 0-0609 Mm 0-8850 9-9470 1-1299 0-0530 0-8958 9-9522 1-1163 0-0478 0-9180 9.9628 1-0893 0-0372 0-9503 9-9779 1-0523 0-0221 0-9905 9-9958 1 -0096 0-0042 1-0350 0-0149 0-9662 9-9851 Sa, Ssa I -0000 o-oooo I -0000 0-0000 I -0000 0-0000 I -oooo 0-0000 I -0000 0-0000 I -0000 0-0000 I -0000 0-0000 I -OOOO O-OOOO I -oooo 0-0000 1 -oooo 0-0000 I -oooo o-oooo I -oooo 0-0000 EEPOET FOR 1894— PART II. 229 Table 10. — Factors F and f for reduetion and predictioti of tides; computed for the middle of each year, or for July Z, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1910 ign 1912 I9I3 1914 19 IS F log F log/ \osF logy F log F io{y F \osF / log/ F log F log/ F log F log/ Ji, [M.] 0-8998 9-9542 I-III3 0-0458 0-8757 9-9423 I -1420 0-0577 0-8621 9-9356 I-I599 0-0644 0-8583 9-9336 1-1650 0-0664 0-8640 9-9365 1-1574 0-0635 0-8796 9-9443 1-1369 0-0557 K, 0-9320 9-9694 1-0729 0-0306 0-9128 9-9604 I-09SS 0-0396 0-9018 9-9551 1-1089 0-0449 0-8987 9-9536 I-I127 0-0464 0-9034 9-9559 1-1070 0-0414 0-9160 9-9619 r-0917 0-0381 K, 0-8460 9-9274 1-1820 0-0726 0-7962 9-9010 1-2560 0-0990 0-7675 9-8851 1-3030 0-II49 0-7S94 9-8805 1-3168 0-1195 0-7715 9-8873 1-2962 0-II27 0-8043 9-9054 1-2433 0-0946 L, I-I998 0-0791 0-8335 9-9209 0-8024 9-9044 1-2463 0-0956 0-881 1 9-9450 1-1349 0-0550 1-7834 0-2512 0-5607 9-7488 1-3712 0-1371 0-7293 9-8629 0-8193 9-9J34 1-2206 0-0866 \.U} I -0210 0-0090 0-9795 9-9910 1-0304 0-0130 0-9705 9-9870 1-0364 0-0155 0-9649 9-9845 1-0382 0-0163 0-9632 9-9837 1-0355 0-0152 0-9657 9-9848 1-0288 0-0123 0-9720 9-9877 Ml 0-5070 9-7050 1-9723 0-2950 0-8010 9-9036 1-2484 0-0964 0-6424 9-8078 1-5567 0-1922 0-43S4 9-6388 2-2969 0-3612 0-4661 9-6685 2-1455 0-3315 0-7636 9-8829 I -3096 0-II7I Ms, MS I-02I0 0-0090 0-9795 9-9910 I -0304 0-0130 0-9705 9-9870 I -0364 0-0155 0-9649 9-9845 1-0382 0-0163 0-9632 9-9837 1-0355 0-0152 0-9657 9-9848 1-0288 0-0123 0-9720 9-9877 M3 I -0316 0-0135 0-9693 9-9864 I -0459 0-0195 0-9561 9-9805 1-0551 0-0233 0-9477 9-9767 1-0578 0-0244 0-9454 9-9756 1-0538 0-0227 0-9490 9-977:! I -0435 0-0185 0-9583 9-9815 M<, MN I -0424 0-0180 0-9594 9-9820 I-0617 0-0260 0-9419 9-9740 1-0741 0-0311 0-9310 9-9689 1-0778 0-0325 0-9278 9-9675 1-0723 0-0303 0-9326 9-9697 1-0584 0-0246 0-9448 9-9754 Me I -0642 0-0270 0-9397 9-9730 I -0940 0-0390 0-9141 9-9610 1-1132 0-0466 0-8983 9-9534 1-1189 0-0488 0-8937 9-9512 1-1104 0-0455 0-9006 9-9545 1-0888 0-0370 0-9184 9-9630 Ms I -0865 0-0360 0-9204 9-9640 1-1272 0-0520 0-8872 9-9480 I-I538 0-0621 0-8667 9-9379 1-1617 0-0651 o-86o8 9-9349 1-1498 o-o6o6 0-8697 9-9394 I-1202 0-0493 0-8927 9-9507 N2, 2N I-02I0 0-0090 0-9795 9-9910 I -0304 0-0130 0-9705 9-9870 I -0364 0-OI55 0-9649 9-9845 1-0382 0-0163 0-9632 9-9837 1-0355 0-0152 0-9657 9-9848 1-0288 0-0123 0-9720 9-9877 0„Q, 0-8948 9-9517 1-1175 0-0483 0-8663 9-9377 I -1543 0-0623 0-8501 9-9295 1-1763 0-0705 0-8455 9-9271 1-1827 0-0729 0-8524 9-9306 1-1732 0-0694 0-8710 9-9400 I-I482 0-0600 00 0-6874 9-8372 1-4548 0-1628 0-6124 9-7870 1-6330 0-2130 0-5719 9-7574 1-7484 0-2426 0-5609 9-7488 1-7830 0-2512 0-577S 9-7615 1-7317 0-2385 0-6242 9-7953 1 -6020 0-2047 Pli Rsj T2 i-oooo I -oooo I -OOOO I -oooo I -oooo I-oooo I -oooo I -oooo 1 -oooo 1 -oooo I-OOOO 1 -oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 Oi, 2) 3, 4 I -oooo I -oooo I -OOOO I-oooo I -oooo I-oooo 1-0000 I -oooo I -oooo I -oooo I-oooo 1-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 X^, IMl, V2 I -0210 0-0090 0-9795 9-9910 I -0304 0-0130 0-9705 9-9870 I -0364 0-0155 0-9649 9-9845 I -0382 0-0163 0-9632 9-9837 1-0355 0-0152 0-9657 9-9848 1-0288 0-0123 0-9720 9-9877 MK 0-9516 9-9784 1-0509 0-0216 0-9406 9-9734 1-0632 0-0266 0-9347 9-9706 1-0699 0-0294 0-9330 9-9699 I-07I8 0-0301 0-9354 9-9710 I -0690 0-0290 0-9423 9-9742 I -0612 0-0258 2MK 0-9715 9-9874 I -0293 0-0126 0-9692 9-9864 I-03I8 0-0136 0-9687 9-9862 1-0323 0-0138 0-9686 9-9862 I -0324 0-0138 0-9687 9-9862 1-0323 0-0138 0-969S 9-9865 I -0315 0-0135 2 MS I -0424 0-0180 0-9594 9-9820 I -06 1 7 0-0260 0-9419 9-9740 I -0741 0-031 1 0-9310 9-9689 1-0778 0-0325 0-9278 9-9675 1-0723 0-0303 0-9326 9-9697 1-0584 0-0246 0-9448 9-9754 MSf, 2 SM I-02I0 0-0090 0-9795 9-9910 I -0304 0-0130 0-9705 9-9870 I -0364 0-0155 0-9649 9 -984s 1-0382 0-0163 0-9632 9-9837 I -0355 0-0152 0-9657 9-9848 1-0288 0-0123 0-9720 9-9877 Mf 0-7843 9-8945 1-2750 0-1055 0-7284 9-8623 1-3730 0-1377 0-6973 9-8434 I-434I 0-1566 0-6886 9-8380 I -4521 0-1620 0-7016 9-8461 1-4254 0-1539 0-7373 9-8677 1-3562 0-1323 Mm 1-0787 0-0329 0-9270 9-9671 I-II57 0-0476 0-8963 9-9524 I -1402 0-0570 0-8770 9-9430 1-1476 0-0598 0-8714 9-9402 1-1366 0-0556 0-8798 9-9444 I -1092 0-0450 0-9015 9-9550 Sa, Ssa I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo 1 -oooo 1-0000 I-oooo I-oooo I-oooo 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 230 UNITED STATES COAST AND GEODETIC SURVEY. Table 10. — Factors Fandf for reduction an d prediction of tides ; computed for the middle of each year, or for July S, at Gj'eenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1916 I9I7 1918 1819 1920 I92I \OgJ? log/ logF log/ logy? log/ J!' logy-- log/ log^ log/ 7? iogy=- log/ J.. [M>] o'go6o 9-9571 I -1038 0-0429 0-9446 9-9752 1-0587 0-0248 0-9958 9-9982 I -0042 0-0018 1-0578 0-0244 0-9453 9-9756 I-I242 0-0508 0-8895 9-9492 I -1806 0-0721 0-8470 9-9279 K. 0-9368 9-9717 1-0674 0-0283 0-9660 9-9850 1-0352 0-0150 1-0027 0-0012 0-9973 9-9988 I -0443 0-0188 0-9576 9-9812 1-0855 0-0356 0-9212 9-9644 III82 0-0485 0-8943 9-9515 K, 0-8585 9-9337 1-1649 0-0663 0-9340 9-9704 1-0707 0-0296 1-0278 0-OII9 0-9729 9-9881 1-1316 0-0537 0-8838 9-9463 1-2299 0-0899 0-8I3I 9-9101 1-3034 0-1151 0-7672 9-8849 U 0-8529 9-9309 1-1725 0-0691 1-3141 O-I186 0-7610 9-8814 I-282I 0-1079 0-7800 9-8921 0-8944 9-9515 I-II8I 0-0485 0-8351 9-9218 I-I975 0-0782 0-9669 9-9854 1-0342 0-0146 lui I -0188 0-0081 0-9816 9-9919 I -0068 0-0029 0-9933 9-9971 0-9942 9-9975 1-0058 0-0025 0-9826 9-9924 I-OI77 0-0076 0-9730 9-9881 1-0278 0-OII9 0-9664 9-9852 I -0347 0-0148 M, 0-7294 9-8630 I-37IO 0-1370 0-5018 9-7005 1-9929 0-2995 0-5237 9-7I9I 1-9094 0-2809 0-8047 9-9056 1-2427 0-0944 I-II57 0-0476 0-8963 9-9524 0-7644 9-8833 1-3083 0-1167 Mj, MS I -0188 o-ooSi 0-9816 9-9919 I -0068 0-0029 0-9933 9-9971 0-9942 9-9975 1-0058 0-0025 0-9826 9-9924 I-OI77 0-0076 0-9730 9-9881 1-0278 001 19 0-9664 9-9852 1-0347 0-0148 M3 1-0283 0-0121 0-9725 9-9879 I -0102 0-0044 0-9899 9-9956 0-9914 9-9962 1-0087 0-0038 0-9740 9-9885 1-0267 0-0115 0-9597 9-9822 1-0420 0-0179 0-9501 9-9778 1-0525 0-0222 M4, MN 1-0379 0-0162 0-9635 9-9838 1-0136 0-0059 0-9866 9-9941 0-9886 9-9950 I-0II6 0-0050 0-9654 9-9847 1-0358 0-0153 0-9466 9-6762 I -0564 0-0238 0-9340 9-9703 1-0707 0-0297 Me I -0574 0-0242 0-9457 9-9758 I -0205 0-0088 0-9799 9-9912 0-9829 9-9925 I-OI74 0-0075 0-9486 9-9771 1-0542 0-0229 0-9210 9-9643 1-0858 0-0357 0-9026 9-9555 I -1079 0-0445 Ms i-°773 0-0323 0-9283 9-9677 I -0274 O-OI18 0-9733 9-9882 0-9772 9-9900 1-0233 o-oioo 0-9321 9-9694 1-0729 0-0306 0-8961 9-9524 I-H59 0-0476 0-8723 9-9407 I -1464 0-0593 N,, 2 N I -0188 I -0081 0-9816 9-9919 1-0068 0-0029 0-9933 9-9971 0-9942 9-9975 1-0058 0-0025 0-9826 9-9924 I-OI77 0-0076 0-9730 9-9881 1-0278 0-0119 0-9664 9-9852 1-0347 0-0148 0,, Qi 0-9020 9-9552 1-1086 0-0448 0-9466 9-9762 1-0564 0-0238 I -0048 0-0021 0-9952 9-9979 I -0743 0-031 1 0-9308 9-9689 I-I477 0-0598 0-8713 9-9402 1-2096 0-0826 0-8267 9-9174 00 0-7072 9-8495 1-4141 0-1505 0-8369 9-9227 I -1948 0-0773 1-0263 0-OII3 0-9744 9-9887 1-2843 0-1086 0-7787 9-8914 1-5969 0-2033 0-6262 9-7967 1-8950 0-2776 0-5277 9-7224 P|» ^2) 1*2 I -0000 I-oooo I -0000 I -0000 I -0000 I-oooo I-oooo I -oooo I -oooo I-oooo I -oooo I-oooo 0-0000 o-oooo 0-0000 0-0000 0-0000 o-oooo o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 Si, 2) 3, 4 i-oooo I-oooo I-OOOO I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo 0-0000 o-oooo o-oooo o-oooo o-oooo 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 As, /i2, Vl I-OI88 o-ooSi 0-9816 9-9919 I -0068 0-0029 0-9933 9-9971 0-9942 9-9975 1-0058 0-0025 0-9826 9-9924 I-OI77 0-0076 0-9730 9-9881 1-0278 0-OII9 0-9664 9-9852 1-0347 0-0148 MK 0-9544 9-9797 1-0478 0-0203 0-9726 9-9879 1-0282 0-0I2I 0-9969 9-9987 1-0031 0-0013 I -0261 0-0II2 0-9746 9-9888 1-0562 0-0237 0-9468 9-9763 I -0807 0-0337 0-9253 9-9663 2MK 0-9723 9-9878 1-0284 0-0122 0-9792 9-9909 I-02I3 0-0091 0-9912 9-9962 1-0089 0-0038 I -0082 0-0035 0-9919 9-9965 1-0276 0-OII8 0-9732 9-9882 I -0444 0-0189 0-9575 9-981 1 2 MS 1-0379 0-0162 0-9635 9-9838 I -0136 0-0059 0-9866 9-9941 0-9S86 9-9950 I -01 16 0-0050 0-9654 9-9847 I -0358 0-0153 0-9466 9-9762 I -0564 0-0238 0-9340 9-9703 1-0707 0-0297 MSf, 2 SM I-OI88 o-oo8i 0-9816 9-9919. 1-0068 0-0029 0-9933 9-9971 0-9942 9-9975 I -0058 0-0025 0-9826 9-9924 I-OI77 0-0076 0-9730 9-9881 1-0278 0-OII9 0-9664 9-9852 1-0347 0-0148 Mf 0-7987 9-9024 I-252I 0-0976 0-8901 9-9494 I-I235 0-0506 I-OI55 0-0067 0-9847 9-9933 I -1746 0-0699 0-8514 9-9301 1-3538 0-I3I6 0-7387 9-8684 1-5140 0-I80I 0-6605 9-8199 Mm 1-0703 0-0295 0-9343 9-9705 I -0260 o-oiii 0-9747 9-9889 0-9820 9-9921 1-0183 0-0079 0-9432 9-9746 I -0602 0-0254 0-9127 9-9603 1-0956 0-0397 0-8927 9-9507 I-I202 0-0493 Sa, Ssa I -0000 I-oooo I -0000 I -0000 I -oooo I -oooo I-oooo I -oooo I -oooo I -oooo I -oooo I-oooo o-oooo o-oooo 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 REPOET FOE 1894— PAET II. 231 Table 10. — Factors F and f for reduction and prediction of tides; computed for the middle of each year, or for July Z, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. J.. [M,] K, 1^ [L,] M, Ms, MS Ms M4, MN M„ Ma N2, 2N 0„Q, 00 Pi, R-2, T2 Si, 2, 3, 4 MK 2MK 2 MS MSf, 2 SM Mf Mm Sa, Ssa log^F log/ 1-2087 o'o823 1-1338 0-0545 1-3366 0-1260 1-1489 0-0603 0-9636 9-9839 0-6260 9-7966 0-9636 9-9839 0-9459 9-9758 0-9285 9-9678 0-8947 9-9517 0-8622 9-9356 0-9636 9-9839 1-2403 0-0935 2-0547 0-3128 I -0000 0-0000 I -0000 0-0000 0-9636 9-9839 I -0925 0-0384 1-0527 0-0223 0-9285 9-9678 0-9636 9-9839 1-5964 0-2031 0-8843 9-9466 I -0000 0-0000 F log F 0-8273 9-9177 0-8820 9-9455 0-7482 9-8740 0-8704 9-9397 1-0378 O-OI61 I -5974 0-2034 1-0378 0-0161 I 0572 0-0242 1-0770 0-0322 1-1176 0-0483 1-1598 0-0644 1-0378 00161 0-8063 9-9065 0-4867 9-6872 I -0000 0-0000 I -0000 0-0000 1-0378 0-0161 0-9153 9-9616 0-9499 9-9777 1-0770 0-0322 1-0378 0-0161 0-6264 9-7969 1-1309 0-0534 I -0000 0-0000 / log/ I-I967 0-0780 I-I272 0-0520 1-3226 0-I2I4 I -0734 0-0308 0-9648 9-9844 0-6653 9-8230 0-9648 9-9844 0-9476 9-9766 0-9308 9-9689 0-8980 9-9533 0-8664 9-9377 0-9648 9-9844 1-2272 0-0889 1-9853 0-2978 I -0000 0-0000 I -0000 0-0000 0-9648 9-9844 1-0875 0-0364 I -0492 0-0208 0-9308 9-9689 0-9648 9-9844 1-5608 0-1933 0-8878 9-9483 I -0000 0-0000 F \0%F 0-8357 9-9220 0-8872 9-9480 0-7561 9-8786 0-9316 9-9692 1-0365 0-0156 1-5030 0-1770 1-0365 0-0156 1-0553 0-0234 1-0743 0-0311 1-1136 0-0467 1-1542 0-0623 1-0365 0-0156 0-8149 9-9111 0-5037 9-7022 1-0000 0-0000 I -0000 0-0000 1-0365 0-0156 0-9196 9-9636 0-9531 9-9792 1-0743 0-0311 1-0365 0-0156 0-6407 9-8066 I - 1 264 0-0517 I -0000 0-0000 / log/ 1-1498 0-0606 i-ioo5 0-0416 1-2644 0-1019 0-8828 9-9459 0-9698 9-9867 0-9244 9-9659 0-9698 9-9867 0-9551 9-9800 0-9406 99734 0-9122 9-9601 0-8847 9-9468 0-9698 99867 11758 0-0704 1-7284 0-2376 I -0000 o-cooo 1 -0000 0-0000 0-9698 9-9867 1-0674 0-0283 1-0352 0-0150 0-9406 9-9734 0-9698 9-9867 0-8697 9-9394 0-9086 9-9584 0-7909 9-8981 1-1328 0-0541 1-0311 0-0133 1-0818 0-0341 103U 0-0133 1-0470 0-0200 I -0632 0-0266 1 -0962 00399 I -1303 0-0532 1-0311 0-0133 0-8504 9-9296 0-5786 9-7624 I -0000 0-0000 i-oooo 0-0000 1-0311 0-0133 0-9368 9-9717 0-9660 9-9850 I -0632 0-0266 1-0311 0-0133 F \a%F 1-4256 0-7014 0-1540 9-8460 / log/ 0-9031 9-9557 I -0060 0-0000 I -1073 0-0443 I -0000 0-0000 1 -0854 0-0356 i-o6i8- 0-0260 1-1740 0-0697 0-8340 9-9212 0-9783 9-9905 I -0488 0-0207 0-9783 9-9905 9676 99857 0-9570 9-9809 0-9362 99714 0-9159 9-9618 0-9783 99905 1-1049 0-0433 1 -4095 0-1491 1 -0000 0-0000 I -0000 0-0000 0-9783 9-9905 1-0387 0-0165 1-0162 0-0070 0-9570 9-9809 1926 F \0%F 0-9213 9-9644 0-9418 9-9740 0-8518 9-9303 1-1990 0-0788 1-0222 O-OO9S 0-953S 9-9793 1-0222 0-0095 1-0335 0-0143 I 0449 00191 I -0681 0286 1 0918 0-0382 1-0222 0-0095 0-9050 9-9567 0-7095 9-8509 I -0000 00000 1 -0000 0-0000 1-0222 0-0095 0-9627 9-9835 0-9841 9-9930 1 -0449 o-oigi 0-9783 1-0222 9-9905 0-0095 1-2480 I 0-8013 0-0962 99038 / log/ 0-9294 9-9682 1 -oooo 0-0000 1-0759 0-0318 I -0000 0-0000 1 -0204 0-0088 1-0195 0-0084 1-0703 0-0295 1-0387 0-0165 0-9892 9-9953 0-6177 9-7908 0-9892 9-9953 0-9839 99930 0-9786 9-9906 09681 99859 0-9576 9-9812 0-9892 9 9953 1-0325 00139 1-1248 00510 10000 0-0000 1-0000 00000 0-9892 9-9953 I -0086 0-0037 0-9977 9-9990 0-9786 9-9906 09892 9'9953 1-0776 o-o;j25 0-9651 9-9846 I -0000 0-0000 0-9800 9-9912 0-1 9-9916 0-9343 9-9705 0-9627 9-9835 1-0109 0-0047 1-6190 0-2092 1-0109 0-0047 I -0164 00070 1-0219 00094 I 0330 00141 1-0442 00188 1-0109 0-0047 0-9685 9-9861 08891 9-9490 I -0000 0-0000 I -0000 0-0000 1-0109 0-0047 0-9915 9-9963 I -0023 0-0010 1-0219 0-0094 I -0109 0-0P47 0-9280 9-9675 I -0361 0-0154 1-0000 o-oooo F log/? 0-9643 9-9842 0-9804 9-9914 0-9710 9-9872 1-4953 0-1747 1-0016 0-0007 0-4847 9-6855 1-0016 0-0007 1 -0024 00010 1-0032 00014 I -0048 0-0021 1-0064 0-0028 1-0016 0-0007 09692 9-9864 09074 99578 1 0000 0-0000 I-OOOO 0-0000 I 0016 00007 0-9820 9-9921 09835 9-9928 1 -0032 0-0014 I 0016 0-0007 0-9378 9-9721 1-0074 0-0032 I-oooo 0-0000 / log/ 1-0370 0-0158 I -0200 0-0086 I -0299 0-0128 0-6688 9-8253 0-9984 9-9993 2-0630 0-3145 0-9984 9-9993 0-9976 9-9990 09968 9-9986 0-9952 9-9979 0-9937 9-9972 0-9984 9"9993 1-0318 0-0136 I -1020 0-0422 1-0000 0-0000 1-0000 0-0000 09984 9-9993 1-0184 0-0079 I -0168 o 0072 9968 9-9986 0-9984 9 9993 1 -0664 0-0279 0-9926 9-9968 1-0000 0-0000 232 [JISriTED STATES COAST AND GEODETIC SUEVEY. Table 10. — Factora F and f for reduction and prediction of tides; computed for the middle of each near, or for July's, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1928 1929 1930 I93I 1932 1 933 F log/; log/ F logF logy- F \0gF log/ F lOgF / log/ F log 7^ log/ F \ogF / log/ J.,[M,] 0-9204 9'9640 I -0864 00360 0-8891 9-9490 1-1247 0-0510 0-8692 9-9391 I-I505 0-0609 0-8595 9-9342 I -1634 0-0658 0-8594 9-9342 1-1636 0-0658 0-8688 9-9389 1-1510 O-061I K, 0-9479 9-9768 I 0549 0-0232 0-9236 9-9655 1-0827 0-0345 0-9076 9-9579 I-IOI8 0-0421 0-8997 9-9541 I-III5 0.0459 0-8996 9-9540 1-1116 0-0460 0-9073 9-9578 1-1022 0-0422 K, 0-8872 9-9480 II27I 0-0520 0-8241 9-9160 1-2135 0-0840 0-7826 9-8935 1-2778 0-1065 0-7619 9-8819 1-3125 0-II8I 0-7617 9-8818 1-3129 0-1182 0-7818 .9-8931 1-2791 0-1069 L, 1-0256 o-oiio 0-9750 9-9890 0-7874 9-8962 1-2700 0-1038 0-9781 9-9904 1-0224 0-0096 2-1230 0-3270 0-4710 9-6730 1-1452 0-0589 0-8732 9-941 1 0-7874 9-8962 1-2700 0-1038 \.w\ I -0140 00060 0-9862 9-9940 1-0250 0-0107 0-9756 9-9893 1-0332 0-0142 0-9679 9-9858 1-0376 0-0160 0-9637 9-9840 1-0377 0-0161 0-9637 9-9839 I -0334 0-0142 0-9677 9-9858 M. 0-5750 9-7597 1-7392 0-2403 0-8787 9-9438 I -1380 0-0562 0-7625 9-8823 I -31 15 0-III7 0-4250 9-6284 2-3529 0-3716 0-5020 9-7007 1-9921 0-2993 0-8354 9-9219 1-1970 0-0781 M,,MS I -0140 0-0060 0-9862 9-9940 I -0250 0-0107 0-9756 9-9893 1-0332 0-0142 0-9679 9-9858 1-0376 0-0160 0-9637 9-9840 1-0377 0-0161 0-9637 9-9839 1-0334 0-0142 0-9677 9-9858 Ma I -0210 0-0090 0-9794 9-9910 1-0376 0-0160 0-9637 9-9840 1-0502 0-0213 0-9522 9-9787 1-0570 0-0241 0-9461 9-9759 1-0570 0-0241 0-9460 9-9759 1-0505 0-0214 0-9519 9-9786 M4,MN 1-0282 0-0121 0-9726 9-9879 I -0506 0-0214 0-9519 9-9786 1-0675 0-0284 0-9368 9-9716 1-0767 0-0321 0-9288 9-9679 1 -0768 0-0321 0-9287 9-9679 1-0678 0-0285 0-9365 9-9715 Me 1-0425 0-0181 0-9592 9-9819 1-0768 0-0321 0-9287 9-9679 1-1029 0-0425 0-9067 9-9575 I-I772 0-0481 0-8951 9-9519 I-II73 0-0482 0-8950 9-9518 1-1034 00428 0-9063 9-9572 M, 1-0571 0-0241 0-9460 9-9759 I -1037 0-0428 0-9061 9-9572 I-I395 0-0567 0-8776 9-9433 I-I592 0-0642 0-8627 9-9358 1-1594 0-0642 0-8625 9-9358 1-1402 0-0570 0-8770 9-9430 N„2N I -0140 0-0060 0-9862 9-9940 1-0250 0-0107 0-9756 9-9893 1-0332 0-0142 0-9679 9-9858 1-0376 0-0160 0-9637 9-9840 1-0377 0-0161 0-9637 9-9839 1-0334 0-0142 0-9677 9-9858 0„Qi 0-9188 9-9632 I -0883 0-0368 0-8822 9-9456 I -1335 0-0544 0-8586 9-9338 I -1647 0-0662 0-8470 9-9279 I -1807 0-072 [ 0-8468 9-9278 1-1809 0-0722 0-8582 9-9336 I-1653 0-0664 OO 0-7545 9-8777 1-3254 0-1223 0-6536 9-8153 1-5300 0-1847 0-5930 9-7730 1-6865 0-2270 0-5643 9-7515 I-772I 0-2485 0-5640 9-7513 1-7732 0-2487 0-5919 9-7722 1-6896 0-2278 P„R2,T, I -0000 I -0000 I -OOOO I -oooo I -oooo I -oooo I -oooo I -oooo 1 -oooo I -oooo 1 -oooo I -OOOO 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 Si, It 3t 4 I -0000 I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo I -oooo 1-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 li, lii, V^ I -0140 0-0060 0-9862 9.9940 1-0250 0-0107 0-9756 9-9893 1-0332 0-0142 0-9679 9-9858 1-0376 o-oi5o 0-9637 9-9840 1-0377 o-oi6i 0-9637 9-9839 1-0334 0-0142 0-9677 9-985S MK 0-9612 9-9828 I -0404 0-0172 0-9467 9-9762 1-0564 0-0238 0-9377 9-9721 I -0664 0-0279 0-9335 9-9701 I-07I2 0-0299 0-9335 9-9701 1-0713 0-0299 0-9376 9-9720 1 -0666 0-0280 2MK 0-9746 9-9888 1-0260 0-0II2 0-9703 9-9869 I -0306 0-0131 0-96S9 9-9863 I -0321 0-0137 0-9686 9-9862 1-0324 0-0138 0-9686 9-9862 1-0324 0-0138 0-9689 9-9863 1-0321 0-0137 2 MS 1-0282 00121 0-9726 9-9879 I -0506 0-0214 0-9519 9-9786 1-0675 0-0284 0-9368 9-9716 1-0767 0-0321 0-9288 9.9679 1-0768 0-0321 0-9287 9-9679 1-0678 0-0285 0-9365 9-9715 MSf, 2 SM I -0140 o-oo6o 09862 9-9940 1-0250 0-0107 0-9756 9-9893 1-0332 0-0142 0-9679 9-9858 1-0376 00160 0-9637 9-9S40 1-0377 0-0161 0-9637 9-9839 1-0334 0-0142 0-9677 9.9858 Mf 08326 9-9204 1-2010 0-0796 0-7594 9-8804 I-3I69 0-II96 0-7135 9-8534 I -4015 0-1466 0-6913 9-8397 1-4465 0-1603 0-691 1 9-8395 1-4470 0-1605 0-7127 9-8529 1-4032 0-1471 Mm 1-0523 0-0221 0-9503 9-9779 I -0942 00391 0-9139 9-9609 I 1270 0-0519 0-8873 9-9481 I -1452 0-0589 0-8732 9-9411 I-I455 0-0590 0-8730 9-9410 1-1277 0-0522 0-8868 9-9478 Sa, Ssa I -oooo 0-0000 I -oooo 0-0000 r -oooo O'OOOO I -oooo 0-0000 I -oooo 0-oobo I -OOOO 0-0000. I -oooo 0-0000 I -oooo 0-0000 I -oooo 0-0000 I -oooo 0-0000 1 -oooo 0-0000 1-0000 0-0000 REPORT FOE 1894— PART II. 233 Table 10. Factors F and f for reduetion and prediction of tides; compuiedfor the middle of each year, or for July 2, at Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1934 ! .. 1 1935 i 1636 1937 1938 1939 F \ogF io^/ F logF log/ F \ogF log/ F logF logy F logF log/ F \QgF lo{/ Ji. [M,] 0-8884 9-9486 I-I256 0-0514 0-9194 9-9635 1-0877 0-0365 0-9629 9-9836 1-0385 0-0164 I -0188 0-0081 0-9815 9-9919 1-0837 0-0349 0-9228 9-9651 1-1482 0-0600 0-8710 9-9400 K. 0-9231 9-9652 I -0834 0-0348 0-9471 9-9764 1-0558 0-0236 0-9794 9-9910 1-0210 0-0090 1-0185 o-oo8o 0-9819 9-9920 I -0607 00256 0-9428 9-9744 1-0997 0-0413 0-9094 9-9587 K, 0-8227 9-9152 I-2I55 0-0848 0-8852 9-9470 1-1297 p-0530 0-9684 9-9861 I -0326 0-0139 1-0677 0-0284 0-9366 9-9716 1-I7I4 0-0687 0-8537 9-93'3 1-2622 o-ioii 0-7923 9-8989 L, 0-9274 9'9673 1-0783 0-0327 1-5720 0-1965 0-6361 9-8035 1-1645 0-0661 0-8588 9-9339 0-8401 9-9243 I -1903 0-0757 0-8602 9-9346 1-1625 0-0654 1-0681 0-0286 0-9363 9-9714 [I^] 1-0252 o-oio8 0-9754 9-9892 I -0143 0-0062 0-9S59 9-9938 1-0019 0-0008 0-9981 9-9992 0-9895 9-9954 1-0106 0-0046 0-9785 9-9906 I -0220 0-0094 0-9700 9-9868 I 0309 00132 M, 0-6196 9-7921 1-6139 0-2079 0-4655 9-6679 2- 1482 0-3321 0-5407 9-7329 1-8495 0-2671 0-9109 9-9595 1-0978 0-0405 0-9269 9-9670 1-0789 0-0330 0-6518 9-8141 1-5342 0-1859 M2, MS 1-0252 0-0108 0-9754 9.9892 I -0143 0-0062 0-9859 9-9938 1-0019 0-0008 0-9981 9-9992 0-9895 9-9954 I -0106 0-0046 0-9785 9-9906 1 -0220 0-0094 0-9700 9-9868 I 0309 0-0132 M3 I -038 1 0-0162 ^•9633 9-9838 I -0216 0-0093 0-9789 9-9907 I -0028 0-0O12 0-9972 9-9988 0-9843 9-9931 1-0159 0-0069 0-9680 9-9859 I -0331 0-0141 0-9554 9-9802 I -0467 0-0198 . M4, MN I -05 1 1 0-0216 0-9514 9-9784 1-0288 0-0123 0-9720 9-9877 1 -0038 0-0017 0-9962 9-9983 0-9792 9-9909 1-0213 0-009 1 0-9575 9-981 1 1 -0444 0-0189 0-9409 9-9736 1-0628 0-0264 Me 1-0776 0-0325 0-9280 9-9675 I -0435 0-0185 0-9513 9-9815 1-0057 0-0025 0-9943 9-9975 0-9689 9-9863 1-0321 0-0137 0-9369 9-9717 1-0673 0-0283 0-9127 9-9603 1-0956 0-0397 Ma 1-1048 0-0433 0-9051 9-9567 1-0585 00247 0-9448 9-9753 1-0077 0-0033 0-9924 9-9967 0-9588 9-9817 1 -0430 0-0183 0-9168 9-9623 1 -0907 0-0377 0-8854 9-9471 1-1295 0-0529 Ns, 2N 1-0252 o-oioS 0-9754 9-9S92 1-OI43 0-0062 0-9859 9-9938 1-0019 o-ooo8 0-9981 9-9992 0-9895 9-9954 1-0106 0-0046 0-9785 9-9906 1-0220 0-0094 0-9700 9-9868 1 -0309 0-0132 Oi,Q, 0-8814 9-9452 I -1345 0-0548 0-9176 9-9627 I -0898 0-0373 0-9676 9-9857 I -0335 0-0143 1-0307 0-0132 0-9702 9-9868 1-1030 0-0426 o-90d6 9-9574 1-1740 0-0697 0-8518 9-9303 00 0-6515 9'8i39 1-5348 0-1861 0-7510 9-8757 1-3315 0-1243 0-9024 j 9-9554 i-io8i 0-0446 1-1184 0-0486 0-8942 9-9514 1-4014 0-1466 0-7136 9-8534 1-7198 0-2355 0-5814 9-7645 Pi, R2, T, I -0000 0-0000 I -0000 0-0000 i-oooo 0-0000 1-0000 0-0000 1 -oooo 0-0000 I-oooo o-oooo 1 -oooo o-oooo 1 -oooo o-oooo i-oooo 0-0000 I -oooo o-oooo I-oooo 0-0000 I -oooo 0-0000 Si, 2, 3, 4 1-0000 0-0000 I -0000 0-0000 I -0000 0-0000 1 -oooo O'OOOO 1-0000 0-0000 I-oooo 0-0000 1 -oooo o-oooo I -oooo 0-0000 I-oooo o-oooo I -oooo 0-0000 1-0000 0-0000 I-oooo 0-0000 ^,M2, Vi 1-0252 o-oio8 0-9754 9-9892 1-0143 0-0062 0-9859 9-9938 1-0019 o-oooS 0-9981 9-9992 0-9895 9-9954 1-0106 0-0046 0-9785 9-9906 1-0220 0-0094 0-9700 9-9868 1-0309 0-0132 MK 0-9463 9-9760 1-0567 0-0240 0-9607 9-9826 I -0409 0-0174 0-9813 9-9918 1-0191 0-0082 1-0078 0-0034 0-9922 9-9966 I -0379 0-0162 0-9635 9-9838 1-0667 0-0280 0-9375 9-9720 2MK 0-9702 9-9iS69 1-0307 0-0131 0-9744 9-9888 1-0262 0-0112 0-9832 9-9926 1-0171 0-0074 0-9973 9-9988 I -0027 0-0012 1-0156 1 0-0067 0-9846 9-9933 1-0347 0-0148 0-9665 9-9852 2 MS I -05 1 1 0-0216 0-9514 9-9784 1 -0288 0-0123 0-9720 9-9877 1-0038 0-0017 0-9962 9-9983 0-9792 9-9909 10213 0-0091 0-9575 ; 9-981 1 1-C444 0-0189 0-9409 9-9736 I -0628 0-0264 MSf, 2 SM 1-0252 o-oio8 0-97S4 9-9892 I -0143 0-0062 0-9859 9-9938 1-0019 0-0008 0-9981 9-9992 0-9895 9-9954 1-0106 0-0046 0-9785 9-9906 1-0220 0-0094 0-9700 9-9868 1 -0309 0-0132 Mf 07578 9-8796 1-3196 0-1204 0-8302 9-9192 I -2046 o-o8o8 0-9344 9-9706 I -0702 0-0294 1-0736 0-0309 0-9314 99691 1-2433 0-0946 0-8043 9-9054 1-4210 c-1526 0-7038 9-8474 Mm 1-0952 0-0395 0-9131 9-9605 I -0535 0-0226 0-9492 9-9774 1 -0086 0-0037 0-9914 9-9963 0-9662 9-9850 1-0350 0-0150 ' 0-9302 i 9-9686 1-0750 0-0314 0-9037 9-9560 1-1066 0-0440 Sa, Ssa lOOOO O'OOOO I -oooo o-oooo I-oooo o-oooo I -oooo 0-0000 I'OOOO o-ooco I-oooo 0-0000 1 -oooo 0-0000 1 -oooo o-oooo I-oooo 0-0000 1 -oooo 0-0000 1-0000 o-oooo 1 -oooo 0-0000 234 UNITED STATES COAST AND GEODETIC SUEVET. Table 10. — Factors F and f for reduction and prediction of tides; computed for the middle of each year, or for July S, at Greemcicli mean noon for common years, and at preceding midnight for leap years — Continued. 1940 1941 1942 1943 1944 194s Component. logF log/ log F log/ F \ogF log/ F XogP \o{f F logy log/ F \0g F lo^/ Ji. [Ml] 1-1958 0-0776 0-8363 9-9224 1-2089 0-0824 0-8272 9-9176 T-18I8 0-0726 0-8462 9-9274 1-1260 0-0515 0-8881 9-9485 I -0597 0-0252 0-9437 9-9748 0-9973 9-9988 1 -0028 0-0012 K, 1-1267 0-0518 0-8876 9-9482 I-I339 0-0546 0-8819 9-9454 1-1189 0-0488 0-8937 9-9512 I -0866 0-0361 0-9203 9-9639 1-0454 0-0193 0-9565 9-9807 1-0037 0-0016 0-9963 9-9984 K, I-3216 0-I2II 0-7567 9-8789 1-3369 O-I261 0-7480 9-8739 1-3048 0-1156 0-7664 9-8844 1-2323 0-0907 0-8115 9-9093 I-I344 0-0548 0-8815 9-9452 1-0305 0-0130 0-9704 9-9870 U I-I635 0-0658 0-8595 9-9342 0-9794 9-9909 1-0211 0-0091 0-8474 9-9281 1-I80I 0.0719 0-8738 9-9414 I -1444 0-0586 1-1415 0-0575 0-8760 9-9425 1-3364 0-1259 0-7483 9-8741 [L.] 0-9649 9-9845 I -0364 o-oiss 0-9636 9-9839 1-0378 0-0161 0-9663 9-9851 1-0349 0-0149 0-9727 9-9880 I -0280 0-0120 0-9823 9-9922 i-oi8i 0-0078 0-9939 9-9974 1-0061 0-0026 M, 0-6160 9-7896 I -6234 0-2104 0-7616 9-8817 1-3I3I 0-1183 1-1380 0-0561 0-8787 9-9439 0-9294 9-9682 1-0760 0-0318 0-5834 9-7660 I -7142 0-2340 0-5144 9-7113 1-9442 0-2887 Mj, MS 0-9649 9-9845 1-0364 0-0155 0-9636 9-9839 1-0378 0-0161 0-9663 9-9851 1-0349 0-0149 0-9727 9-9880 1-0280 0-0120 0-9823 9-9922 i-oi8i 0-0078 0-9939 9-9974 1-0061 0-0026 M3 0-9477 9-9767 1-055 1 0-0233 0-9459 9-9758 1-0572 0-0242 0-9499 9-9777 1-0527 0-0223 0-9S93 9-9820 1 -0424 0-0180 0-9735 9-9883 1-0272 0-0117 0-9909 9-9960 1-0092 0-0040 M4, MN 0-9310 9-9689 I -0741 0-03 1 1 0-9285 9-9678 1-0770 0-0322 0-9338 9-9702 I -0709 0-0298 0-9462 9-9760 1-0569 0-0240 0-9649 9-9845 1-0364 0-0155 0-9879 9-9947 1-0122 0-0053 Me 0-8983 9-9534 1-1132 0-0466 0-8947 9-9517 1-1177 0-0483 0-9023 9-9554 1-1083 0-0446 0-9204 9-9640 1-0865 0-0360 0-9477 9-9767 1-0551 0-0233 0-9819 9-9921 1-0184 0-0079 Ms 0-8667 9-9379 1-1538 0-0621 0-8621 9-9356 1-1600 0-0644 0-8719 9-9405 1-1469 0-0595 0-8953 5-9520 I-I170 0-0480 0-9309 9-9689 1-0742 0-0311 0-9760 9-9894 1-0246 0-0106 N2, 2N 0-9649 9-9845 I -0364 0-0155 0-9636 9-9839 1-0378 0-0161 0-9663 9-9851 1-0349 0-0149 0-9727 9-9880 1 -0280 0-0120 0-9823 9-9922 1-0181 0-0078 0-9939 9-9974 1 -0061 0-0026 0„ Qi 1-2262 o-o886 0-8155 9-9114 1-2406 0-0936 0-8061 9-9064 1-2109 0-0831 0-8258 9-9169 1-1497 0-0606 0-8698 9-9394 1-0763 0-0319 0-9291 9-9681 1 -0065 0-0028 0-993S 9-9972 00 I -9802 0-2967 0-5050 9-7033 2-0561 0-3130 0-4864 9-6870 1-9016 0-2791 0-5259 9-7209 1-6059 0-2057 0-6227 9-7943 1-2923 01114 0-7738 9-8886 I -0321 0-0137 0-9689 9-9863 Pi) Rs) Ts I -oooo 1-0000 1 -OOOO 1 -oooo i-oooo 1 -oooo i-oooo 1 -oooo I -oooo I -oooo I -OOOO 1 -oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 o-oooo 0-0000 Si, 3j 3, 4 I -0000 1-0000 1-0000 1-0000 I-oooo 1-0000 1 -oooo I-oooo I -oooo I -oooo 1-0000 I-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 ^2, fh, "2 0-9649 9-9845 1 -0364 o-oiss 0-9636 9-9839 1-0378 0-OI6I 0-9663 9-9851 1-0349 0-0149 0-9727 9-9880 I -0280 0-0120 0-9823 9-9922 I -0181 0-0078 0-9939 9-9974 I -0061 0-0026 MK 1-0871 0-0363 0-9199 9-9637 I -0926 0-0385 0-9153 9-9615 1-0812 0-0339 0-9249 9-9661 1-0570 0-0241 0-9461 9-9759 1-0269 0-0115 0-9738 9-9885 0-9976 9-9990 I -0024 o-ooio 2MK 1-0489 0-0207 0-9534 9-9793 1-0528 0-0224 0-9498 9-9776 1-0448 0-0190 0-957' 9-9810 1-0281 0-0120 0-9726 9-9880 I -0087 0-0038 0-9914 9-9962 0-9916 9-9963 I -0085 0-0037 2 MS 0-9310 9-9689 1-0741 0-0311 0-9285 9-9678 1-0770 0-0322 0-9338 9-9702 I -0709 0-0298 0-9462 9-9760 1-0569 0-0240 0-9649 9-9845 I -0364 0-0155 0-9879 9-9947 1-0122 0-0053 MSf, 2 SM o-9'ii9 9-9845 I -0364 o-oiss 0-9636 9-9839 1-0378 0-0161 0-9663 9-9851 I -0349 0-0149 0-9727 9-9880 1 -0280 0-0120 0-9823 9-9922 1-0181 0-0078 0-9939 9-9974 1-0061 0-0026 Mf 1-5582 0-1926 0-6417 9-8074 1-5971 0-2033 0-6262 9-7967 1-5174 0-1811 0-6590 9-8189 1-3588 0-1331 0-7360 9-8669 1-1794 0-0717 0-8479 9-9283 1-0192 0-0083 0-981 1 9-9917 Mm O-83S0 9-9484 I -1261 0-0516 0-8842 9-9466 1-1310 0-0534 0-8923 9-9505 1-1206 0-0495 0-9120 9-9600 1-0965 0-0400 ■ 0-9422 9-9742 1-0613 0-0258 0-9810 9-9916 1-0194 0-0084 Sa, Ssa I -oooo 1-0000 1 -oooo I -oooo i-oooo 1 -oooo 1 -oooo 1-0000 I -oooo 1 -oooo 1 -oooo 1 -oooo 0-0000 0-0000 0-0000 00000 0-0000 0-0000 0-ooco 0-0000 0-0000 0-0000 0-0000 0-0000 EEPORT FOE 1894— PART II. 235 Table 10. Factors F and f for reduction and prediction of tides; corg,puted for the middle of each year, or for July : Greenwich mean noon for common years, and at preceding midnight for leap years — Continued. Component. 1946 1947 1948 1949 1950 Mean value of/ from the years 1850-1942. \ozF lo"^/ F \ogF log/ F \osF log/ F log^ log/ F logF / log/ Ju [Ml] 0-9458 9-9758 I '0573 0-0242 0-9069 9-9576 I -1027 0-0424 0-8801 9-9446 1-1362 0-0554 0-8643 9-9367 1-1570 0-0633 0-8584 9-9337 1-1650 0-0663 1 -00490 K, 0-9669 9-9854 1-0342 0-0146 o'9375 9-9720 I -0666 0-0280 0-9164 9-9621 1-0912 0-0379 0-9036 9-9560 1-1067 0-0440 0-8987 9-9536 1-1127 0-0464 I -00609 K. 0-9363 9-9714 I -0681 0-0286 o-86o2 9-9346 I -1624 0-0654 0-8055 9-9061 1-2415 0-0939 0-7721 9-8877 1-2951 0-1123 0-7594 9-8805 1-3168 0-1I95 1-02421 L, 0-9165 9-9621 1-0911 0-0379 0-7970 9-9015 1-2547 0-0985 I-I182 0-0485 0-8943 9-9515 2-0764 0-3173 0-4816 9-6827 0-9888 9-9951 1-OII3 0-0049 0-97803 [L.] I -0065 0-0028 0-9936 9-9972 1-0185 0-0080 0-9819 9-9920 1-0285 0-0122 0-9723 9-9878 1-0354 O-O151 0-9658 9-9849 1-0382 0-0163 0-9632 9-9837 1 -00033 Ml 0-6755 9-8296 I -4804 0-1704 0-8786 9-9438 1-1382 0-0562 0-5190 9-7151 1-9269 0-2849 0-4275 9-6310 2-3390 0-3690 0-5572 9-7460 1-7947 0-2540 1-55050 Mj, MS 1-0065 0-0028 0-9936 9-9972 I -0185 0-0080 0-9819 9-9920 1-0285 0-0122 0-9723 9-9878 1-0354 0-0151 0-9658 9-9849 1-0382 0-0163 0-9632 9-9837 I -00033 Ms I -0096 0-0042 0-9904 9-9958 1-0278 0-0H9 0-9729 9-9881 1-0431 0-0183 0-9587 9-9817 1-0535 0-0226 0-9492 9-9774 1-0578 0-0244 0-9454 9-9756 I -00074 M4, MN I -01 29 0-0056 0-9872 9-9944 1-0373 0-0159 0-9641 9-9841 1-0579 0-0244 0-9453 9-9756 1-0721 0-0302 0-9328 9-9698 1-0778 0-0325 0-9278 9-9675 1-00137 Me I -0195 0-0084 0-9809 9-9916 1-0565 0-0238 0-9466 9-9762 1-0881 0-0367 0-9191 9-9633 i-iioo 0-0453 0-9009 9-9547 1-II89 0-0488 0-8937 9-9512 1-00307 Ms 1-0260 0-0H2 0-9746 9-9888 1-0760 0-0318 0-9294 9-9682 i-ngi 0-0489 0-8936 9-9511 I -1493 0-0604 0-8701 9-9396 I-16I7 0-0651 0-8608 9-9349 1-00552 Nj, 2 N I -0065 0-0028 0-9936 9-9972 I -0185 0-0080 0-9819 9-9920 1 -0285 0-0122 0-9723 9-9878 1-0354 0-0151 0-9658 9-9849 1 -0382 0-0163 0-9632 9-9837 1-00033 0„Qi 0-9480 9-9768 I -0548 0-0232 0-9031 9-9557 1-1073 0-0443 0-8716 9-9403 I -1473 0-0597 0-8527 9-9308 1-1727 0-0692 0-8456 9-9272 1-1826 0-0728 1 -00905 00 0-8411 9-9249 I -1889 0-0751 0-7100 9-8513 I -4084 0-1487 0-6259 9-7965 1-5976 0-2035 0-5784 9-7622 1-7290 0-2378 0-5609 9-7489 1-7828 0-2511 1.10310 P„ Rs, T-2 I -oooo 0-0000 I -0000 0-0000 I -0000 0-0000 I -0000 0-0000 1 -0000 0-0000 I -0000 0-0000 1 -0000 0-0000 I -oooo 0-0000 I -oooo 0-0000 1-0000 0-0000 1 -00000 Si, 2, 3j -1 I -0000 0-0000 I -0000 0-0000 I -0000 o-oooo I -0000 0-0000 1-0000 0-0000 i-dooo 0-0000 I -oooo 0-0060 I -oooo 0-0000 I -oooo 0-0000 1 -oooo 0-0000 1 -00000 ^, f^2> Vl I -0065 0-0028 0-9936 9-9972 1-0185 0-0080 0-9819 9-9920 1-0285 0-0122 0-9723 9-9878 1-0354 0-0151 0-9658 9-9849 1 -0382 0-0163 0-9632 9-9837 1-00033 MK 0-9731 9-9882 1-0276 0-0118 0-9548 9-9799 1-0473 0-0201 0-9426 9-9743 1 -0609 0-0257 0-9356 9-9711 1 -0688 0-0289 0-9330 9-9699 I -0718 0-0301 1 -00428 2MK 0-9794 9-9910 1-0210 0-0090 0-9725 9-9879 1-0283 0-0121 0-9695 9-9865 1-0315 0-0135 0-9687 9-9862 1-0323 0-0138 0-9686 9-9862 1-0324 0-0138 1-00315 2 MS 1-0129 0-0056 0-9872 9-9944 1-0373 0-0IS9 0-9641 9-9841 1-0579 0-0244 0-9453 9-9756 1-0721 0-0302 0-9328 9-9698 1-0778 0-0325 0-9278 9-9675 1-00137 MSf, 2 SM I -0065 0-0028 0-9936 9-9972 I -0185 0-0080 0-9819 9-9920 1-0285 0-0122 0-9723 9-9878 1-0354 0-0151 0-9658 9-9849 1-0382 0-0163 0-9632 9-9837 I -00033 Mf 0-8930 9-9508 1-1198 0-0492 0-8008 9-9035 1-2488 0-0965 0-7386 9-8684 1-3539 0-1316 0-7023 9-8465 I -4240 0-1535 0-6887 9-8380 1-4520 0-1620 1-04317 Mm I -0248 0-0106 0-9758 9-9894 1-0692 0-0290 0-9353 9-9710 1-1083 0-0447 0-9023 9-9553 1-1360 0-0554 0-8802 9-9446 I-I475 0-0598 0-8714 9-9,02 0-99992 Sa, Ssa I -0000 I -0000 I -0000 I -0000 1 -0000 I -0000 I -oooo 1 -oooo 1 -oooo 1 -oooo I -00000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 0-0000 236 UNITED STATES COAST AND GEODETIC SURVEY. Table H. — Values of log Ifjor obtainitig the factor F of Lj from that of M2. /> ^= inclination- of moon's orbit. 18° 19° 20* 21* 22" 's" 24° 25° 26° 27° 28° 29» 5 10 0-0778 0-0764 0-0722 0-0889 0-0873 0-0824 0-IOI4 0-0994 0-0938 0-1154 0-1131 0-1064 0-I3I2 0-1284 0-1206 0-1491 0-1459 0-1365 0-1696 0-1658 0-1547 0-I935 0-1888 0-I754 0-2216 0-2158 0-1995 0-2555 0-2482 0-2279 0-2976 0-2881 0-2622 0-3523 0-3393 0-3047 IS 20 25 0-0656 0-0569 0-0467 0-0747 0-0646 0-0528 0-0847 0-0731 0-0595 0-0959 0-0824 0-0668 0-1083 0-0926 0-0748 0-1221 0-1040 0-0835 0-1377 0-II65 0-0930 0-1552 0-1305 0-1035 0-I753 0-1462 0-1150 0-1984 0-1639 0-1277 0-2255 0-1840 0-1418 0-2579 0-2072 0-1575 30 35 40 0-0354 0-0236 0-0117 0-0400 00265 0-0130 0-0448 0-0296 0-0146 0-0501 0-0330 0-0161 0-0558 0-0366 0-0178 0-0620 0-0404 0-0196 0-0687 0-0445 0-0215 0-0759 0-0489 0-0235 0-0837 0-0536 0-0256 0-0922 0-0586 0-0278 0-1014 0-0640 0-0301 0-III4 0-0697 0-0326 45 50 55 0-0000 99889 9-9787 o-oooo 9-9877 9-9764 0-0000 9-9864 9-9739 o-oooo 9-9850 9-9714 0-0000 9-9836 9-9687 0-0000 9-9820 9-9659 0-0000 9-9805 9-9631 0-0000 9-9788 9-9601 0-0000 9-9771 9-9570 0-0000 9-9754 9-9539 o-oooo 9-9736 9-9506 O'OOOO 9-9717 9-9473 60 65 70 9-9696 9-9616 9-9549 9-9663 9-9575 9-9503 9-9628 9-9533 9-9454 9-9593 9-9490 9-9404 9-9556 9-9445 9-9353 9-9518 9-9398 9-9300 9-9479 9-9351 9-9246 9-9439 9-9302 9-9190 9-9398 9-9252 9-9'34 9-9355 9-9201 9-9077 9-9312 9-9149 9-9018 9-9267 9-9097 9-8959 75 80 . 85 9-9497 9-9459 9-9436 9-9445 9-9404 9-9379 9-9392 9-9347 9-9320 9-9337 9-9289 9-9260 9-9281 9-9229 9-9198 9-9223 9-9168 9-9135 9-9164 9-9106 9-9071 9-9104 9-9042 9-9006 9-9043 9-8978 9-8939 9-8981 9-8913 9-8872 9-8918 9-8846 9-8804 9-8854 9-8780 9-8735 90 95 100 9-9429 9-9436 9-9459 9-9371 9-9379 9-9404 9-9312 9-9320 9-9347 9-9250 9-9260 9-9289 9-9188 9-9198 9-9229 9-9124 9-9135 9-9168 9-9059 9-9071 9-9106 9-8993 9-9006 9-9042 9-8926 9-8939 9-8978 9-8858 9-8872 9-8913 9-8790 9-8804 9-8846 9-8720 9-8735 9-8780 105 no "5 9-9497 9-9549 9-9616 9-9445 9-9503 9-9575 9-9392 9-9454 9-9533 9-9337 9-9404 9-9490 9-9281 9-9353 9-9445 9-9223 9-9300 9-9398 9-9164 9-9246 9-9351 9-9104 9-9190 9-9302 9-9043 9-9134 9-9252 9-8981 9-9077 9-9201 9-8918 9-9018 9-9149 9-8854 6-8959 9-9097 120 125 130 9-9696 9-9787 9-9889 9-9663 9-9764 9-9877 9-9628 9-9739 9 -9864 9-9593 9-9714 9-9850 9-9556 9-9687 9-9836 9-9518 9-9659 9-9820 9-9479 9-9631 9-9805 9-9439 9-9601 9-9788 9-9398 9-9570 9-9771 9-9355 9-9539 9-9754 9-9312 9-9506 9-9736 9-9267 9'9473 9-9717 135 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 0-0000 o-oooo 0-0000 0-0000 140 145 0-0117 0-0236 0-0130 0-0265 0-0146 0-0296 0-OI6I 0-0330 0-0178 0-0366 0-0196 0-0404 0-0215 0-0445 0-0235 0-0489 0-0256 0-0536 0-0278 0-0586 0-0301 0-0640 0-0326 0-0697 ISO 155 160 0-0354 0-0467 0-0569 0-0400 0-0528 0-0646 0-0448 0-0595 0-0731 0-0501 0-0668 0-0824 0-0558 0-0748 0-0926 0-0620 0-0835 0-1040 0-0687 0-0930 0-II65 0-0759 0-1035 0-1305 0-0837 0-1150 0-1462 0-0922 0-I277 0-1639 0-I0I4 0-1418 0-1840 0-III4 0-IS7S 0-2072 165 170 175 180 0-0656 0-0722 0-0764 0-0778 0-0747 0-0824 0-0873 0-0889 0-0847 0-0938 0-0994 0-1014 0-0959 0-1064 0-1I3I 0-II54 0-1083 0-1206 0-1284 0-1312 0-I22I 0-1365 0-1459 O-I49I 0-1377 0-1547 0-1658 0-1696 0-1552 0-I754 0-1888 0-1935 0-I753 0-1995 0-2158 0-2216 0-1984 0-2279 0-2482 0-2555 0-2255 0-2622 0-2881 0-2976 0-2579 0-3047 0-3393 0-3523 Log /^ (U) = log F (Mi) + log V?', where Ji' = ( i _ 12 tan^ ^ /cos 2 f)^' "^''^ ™'"^^ °^ ^ ^^^ ^ ^°'^ *-^^ ^^^^ '^^y of every month are given in Table 6. When P lies between 180° and 360° subtract i8o° from it, and enter the table with the remainder. EBPORT FOR 1894— PART II. Tabi,k 12. — Values of log Q' for obtaining the factor F of Mi from that of O,. 237 p Loge- 1 P LogC^ P 1.0% Q' p Log CI' p Logg- P LogC o o 9-6990 1 60 9-8785 120 9-8785 180 9-6990 240 9-8785 300 9-8785 I 9-6990 61 9-8841 121 9-8729 181 9-6990 241 9-8841 301 9-8729 2 9-6992 62 9-8898 122 9-8673 182 9-6992 242 9-8898 302 9-8673 3 9-6994 63 9-8955 123 9-8618 183 9-6994 243 9-8955 303 9-8618 4 9-6998 64 9-9012 124 9-8563 184 9-6998 244 9-9012 304 9-8563 5 9-7002 65 9-9068 125 9-8509 185 9-7002 245 9-9068 305 9-8509 6 9-7008 66 9-9125 126 9-8456 186 9-7008 246 9-9125 306 9-8456 7 9-7014 67 9-9181 127 9-8403 187 9-7014 247 9-9181 307 9-8403 8 9-7022 68 9-9237 128 9-8351 188 9-7022 248 9-9237 308 9-8351 9 9-7030 69 9-9292 129 9-8300 189 9-7030 249 9-9292 309 9-8300 lO 9-7039 70 9-9347 130 9-8249 190 9-7039 250 9-9347 310 9-8249 II 9-7050 71 9-9400 131 9-8200 191 9-7050 251 9-9400 3" 9 -8200 12 9-7061 72 9-9453 132 9-8151 192 9-7061 252 9-9453 312 9-8151 13 9-7074 73 9-9504 133 9-8103 193 9-7074 253 9-9504 313 9-8103 14 9-7087 74 9-9554 '34 9-8056 194 9-7087 254 9-9554 314 9-8056 15 9-7102 75 9-9602 135 9-8010 195 9-7102 255- 9-9602 315 9-8010 16 9-7117 76 9-9649 136 9-7965 196 9-7117 256 9-9649 316 9-7965 17 9-7134 77 9-9693 137 9-7921 197 9-7134 257 9-9693 317 97921 18 9-7151 78 9-9735 138 6-7878 198 9-7151 258 9-9735 318 9-7878 19 9-7170 79 9-9775 139 9-7836 199 9-7170 259 9-9775 319 9-7836 20 9-7190 80 9-9812 140 9-7795 200 9-7190 260 9-9812 320 9-7795 21 9-7210 81 9-9846 141 9-7755 201 9-7210 261 9-9846 321 9-7755 22 97231 82 9-9877 142 9-7716 202 9-7231 262 9-9877 322 9-7716 23 9-7254 83 9-9905 143 9-7678 203 9-7254 263 9-9905 323 9-7678 24 9-7277 84 9-9930 144 9-7641 204 9-7277 264 9-9930 324 9-7641 25 9-7302 85 9-9951 145 9-7605 20s 9-7302 26s 9-9951 325 9-7605 26 9-7328 86 9-9968 146 9-7570 206 9-7328 266 9-9968 326 97570 27 97354 87 9-9982 147 9-7536 207 9-7354 267 9-9982 327 9-7536 28 9-7382 88 9-9992 148 9-7503 208 9-7382 268 9-9992 328 9-7503 29 9-741 1 89 9-9998 149 9-7471 209 9-7411 269 9-9998 329 9-7471 3° 9-7441 90 10-0000 150 9-7441 210 9-7441 270 10.0000 330 9-7441 31 9-7471 91 9-9998 151 9-7411 211 9-7471 271 9-9998 331 9-7411 32 9-7503 92 9-9992 152 9-7382 212 9-7503 272 9-9992 332 9-7382 33 9-7536 93 9-9982 153 9-7354 213 9-7536 273 9-9982 333 9-7354 34 9-7570 94 9-9968 154 9-7328 214 9-7570 274 9-9968 334 9-7328 35 9-7605 95 9-9951 155 9-7302 215 9-7605 275 9-9951 335 97302 36 9-7641 96 9-9930 156 9-7277 216 9-7641 276 9-9930 336 9-7277 37 9-7678 97 9-9905 157 9-7254 217 9-7678 277 9-9905 337 9-7254 38 9-7716 98 9-9877 158 9-7231 218 9-7716 278 9-9877 338 9-7231 39 9-7755 99 9-9846 159 9-7210 219 9-7755 279 9-9846 339 9-7210 40 9-7795 100 9-9812 160 9-7190 220 9-7795 280 9-9812 340 9-7190 41 9-7836 lOI 9-9775 i6i 9-7170 221 9-7836 281 9-9775 341 9-7170 42 9-7878 102 9-9735 162 9-7151 222 97878 282 9-9735 342 97151 43 9-7921 103 9-9693 163 9-7134 223 9-7921 283 9-9693 343 9-7134 44 9-7965 104 9-9649 164 9-7117 224 9-7965 284 9-9649 344 97117 45 46 9-8010 J05 g-9602 165 9-7102 225 9-8010 285 9-9602 345 9-7102 9-8056 106 9-9554 166 9-7087 226 9-8056 286 9-9554 346 9-7087 4.7 9-8103 107 9-9504 167 9-7074 227 9-8103 287 9-9504 347 9-7074 48 9-8151 108 9-9453 168 9-7061 228 9-8151 288 9-9453 348 9-7061 49 9-8200 109 9-9400 169 9-7050 229 98200 289 9-9400 349 9-7050 50 SI 52 53 54 55 56 59 9-8240 no 9-9347 170 9-7039 230 9-8249 290 9-9347 350 9-7039 9-8300 III 9-9292 171 9-7030 231 9-8300 291 9-9292 351 9-7030 9-8351 112 9-9237 172 9-7022 232 9-8351 292 9-9237 352 9-7022 9-8403 113 9-9181 173 9-7014 233 9-8403 293 9-9181 353 9-7014 9-7008 9-8456 98509 9-8563 9-8618 9-8673 9-8729 114 9-9125 «74 9 -7008 234 9-8456 294 9-9125 354 115 116 9-9068 9-9012 17s 176 9-7002 9-6998 23s 236 9-8509 9-8563 295 296 9-9068 9-9012 355 356 9-7002 9-6998 117 118 119 9-8955 9-8898 9-8841 177 178 179 9-6994 9-6992 9-6990 237 238 239 9-8618 9-8673 9-8729 297 298 299 9-8955 9-8898 9-8841 357 358 359 9-6994 9-6992 9-6990 T ^« ZT /■^A_^ — ino- jp en .> -U Ino- (T w'hprp n' — . I — =,i (2-5 + 1-5 COS 2 . The value of P for ihe firsl day of every month is given in Table 6. 238 UNITED STATES COAST AND GEODETIC SURVEY. Table 13. — Factors F andf, corresponding to every tenth of a degree of I, for redubtion and prediction of tides. PQi) ■4 ■5 •6 •7 •8 i8-9 1-21005 ■20440 ■19882 •19330 ■18784 ■1824s •I77I2 19-0 ■I ■2 •3 ■4 ■I7I85 ■16665 ■I6I5I ■15642 •I5I39 19-5 6 •7 ■8 19-9 ■14642 •14I5I •13665 ■I3I85 ■I27IO 2O'0 ■I ■2 ■3 •4 ■12240 ■II776 ■II3I7 •10863 ■I04I4 20-S •6 •7 -8 20'9 •09970 •0953 1 •09096 •08667 •08242 210 •I •2 •3 •4 ■07822 ■07406 ■06995 ■06588 ■06185 21-5 ■6 ■7 ■8 21^9 ■05787 ■05393 ■05004 ■04618 ■04237 22 ■o •i •2 •3 •4 •03859 •03486 •03116 •02751 •02389 22-5 •6 •7 ■8 22^9 •02031 •01677 •01326 -00979 -00636 23^0 ■I •2 23'3 1-00296 0-99959 •99626 0-99297 Diff. S65 558 552 546 539 533 527 520 S14 509 503 497 491 486 480 475 470 464 459 454 449 444 439 435 429 425 420 ,416 411 407 403 394 389 386 381 378 373 370 365 362 358 354 351 347 343 340 337 333 329 326 /Oi) 0-82640 •83029 •83416 •83801 •84186 ■84570 ■84953 ■85335 •85716 ■86095 •86474 -87228 -87603 •87978 •88351 •88723 •89095 •89465 •89834 •90202 •90568 ■90934 •91299 •91662 •92024 -92386 •92746 •93'o5 •93463 •93819 •94175 •94529 •94883 •95235 •95586 •95936 •96284 •96632 •96978 •97323 •97667 •98010 •98351 ■98692 -99031 ■99369 0-99705 I -00041 •0037s 1 -00708 389 387 385 385 384 383 382 381 379 379 377 377 375 375 373 372 372 370 369 368 366 366 365 363 362 362 360 359 358 356 356 354 354 352 351 350 348 348 346 345 344 343 341 341 339 338 336 336 334 333 332 ^(Ji) 23^3 ■4 •5 -6 ■7 -8 23-9 0^99297 •98971 •98648 •98329 •98012 •97699 •97389 24-0 ■1 ■2 •3 •4 -97083 •96779 •96479 •96181 •95886 24-5 •6 •7 •8 24-9 •9559s ■95307 •95021 •94738 •94458 25^0 •I •2 ■3 •4 •94181 •93906 ■ ^93634 •93365 •93099 25^5 •6 •7 ■8 25^9 •92835 •92574 •92316 •92060 •91806 26-0 •-I ■2 ■3 •4 •91555 •51307 •91061 •90817 •90576 26-s •6 ■7 ■8 26-9 •90337 •90101 ■89867 •8963s •89405 27^0 •I -2 •3 •4 •89178 •88953 •88730 •88510 •88291 27^5 •6 ■7 ■8 27-9 ■88075 •87861 •87648 •87438 •87230 28-0 •I •2 ■3 •4 •5 28^6 •87025 •86821 •86619 •86419 •86221 •86025 0-85831 Diff. 326 323 319 317 313 310 306 304 300 298 295 291 288 286 283 280 277 275 272 269 266 264 261 258 256 254 251 248 246 244 241 239 236 234 232 230 227 225 223 220 219 216 214 213 210 208 205 204 202 200 198 196 194 fOi) I ^00708 •01040 •01370 •01700 •02028 •0235s •02681 •03005 •03328 •03650 •03970 •04290 •04608 •04924 •05240 •05554 •05867 •06179 •06489 •06798 •07106 •07412 •07717 •08021 •08324 •08625 •08925 •09223 •09521 •09817 •lom •10404 •10696 •10987 ■11276 ■11564 •11850 •12135 •12419 •12701 •12982 •13262 •13540 •13817 •14092 •14366 •14639 •14910 •15180 •15448 •1571S •15981 •16245 1^16508 Diff. 332 330 330 328 327 326 324 323 322 320 320 318 316 316 314 313 312 310 309 308 306 305 304 303 301 300 298 296 294 293 292 291 289 288 286 285 284 282 281 280 278 277 27s 274 273 271 270 268 267 266 264 263 sinMC0Su(i — I sm%') 0-72147 ^— ^If— sin /cos / /^(Ji) = i^ for lunar Ki. /■(Ji) =/ for lunar Ki. /is given for the first of each month in Table 6. sin 2/ REPORT FOR 1894— PART II. 239 Table 13. — Factors Fandf, corresponding to every tenth of a, degree of I, for reduction and prediction of tides — Continued. ^(K,) 18-3 ■4 •5 •6 •7 •8 i8-9 ig-o •I •2 ■3 ■4 I9-S •6 •7 •8 ig-g 20-0 •I •z ■3 •4 20-5 ■6 •7 ■8 20"9 2I-0 ■I •2 ■3 •4 2I-S •6 •7 •8 2 1 '9 22 -o •I ■2 •3 •4 22'5 •6 •7 ■8 22'9 23-0 ■I ■2 23'3 1-13450 •13142 •1283s ■12530 •12227 •11926 ■I 1627 •11329 •11033 •10739 •10447 ■10156 •09867 •09579 ■09293 •09009 ■08727 ■08446 ■08166 ■07889 ■07613 •07339 •07067 ■06796 •06527 ■06259 ■05992 •05727 ■05464 " ^05203 •04943 •04685 •04429 •04174 •03920 •03668 •03417 •03168 •02921 •02676 •02432 ■02189 ■01948 ■01709 •01471 •01234 ■00998 ■00764 ■00532 ■00302 I ^00073 Diif. /(K,) 308 307 305 303 30« 299 298 296 294 292 291 289 288 286 284 282 281 280 277 276 274 272 271 269 268 267 265 263 261 260 258 256 25s 254 252 251 249 247 24s 244 243 241 239 238 237 236 234 232 230 229 228 0-88145 •8S385 •88626 •89106 •89345 ■89585 •89824 •90064 ■90304 •90543 ■90781 •91019 •91258 ■91497 •91735 ■91974 ■92212 ■92451 ■92689 ■92927 •93164 •93400 •93637 •93874 •94111 •94347 •94583 •94819 •95054 •95289 •95524 •95759 •95994 •96228 ■96462 ■96696 •96929 ■97162 •97394 ■97626 •97858 ■98089 ■08320 •98551 ■98781 ■99011 ■99241 ■99470 ■99698 0^99927 240 241 240 240 239 240 239 240 240 339 238 238 239 239 238 239 238 239 238 238 237 236 237 237 237 236 236 236 235 235 235 235 235 234 234 234 233 233 232 232 232 231 231 231 230 230 230 229 228 229 228 •4 •s ■6 •7 -8 23-9 240 ■I -2 •3 •4 24-5 -6 •7 -8 24-9 25^0 •I •2 •3 •4 25-5 •6 •7 •8 25^9 z6'o -I ■2 ■3 ■4 26-5 ■6 ■7 ■8 26^9 27^0 •I ■2 ■3 ■4 27-5 -6 27-9 28-0 •1 -2 •3 •4 •5 28-6 ^(K,) Diff. /(Ki) 1 -00073 0^99845 ■99619 •99395 -99172 -98950 -98729 -98510 ■98293 ■98077 ■97862 •97648 •97435 •97225 •97016 ■96601 ■96396 ■96193 ■95991 •95790 •95590 •95391 •95194 ■94999 •94805 ■94612 ■94419 ■94228 •94039 •93851 ■93664 •93479 •93295 ■93112 -92930 •92749 ■92570 ■92392 ■92216 ■92041 ■91867 ■91693 ■91521 ■91351 •91181 ■91013 ■90846 ■90680 •90515 •90352 ■90190 ■90028 228 226 224 223 222 221 219 217 216 215 214 213 210 209 208 207 205 203 202 20 1 200 199 197 195 194 193 193 191 189 188 187 185 184 183 182 181 179 178 176 175 174 174 172 170 170 168 167 166 165 163 162 162 160 0-99927 1-00155 ■00383 •00610 •00836 •01062 •01287 ■01512 ■01737 •01961 •02185 •02409 •02632 •02854 •03076 •03297 •03518 •03738 •03958 •04177 ■04396 ■04614 ■04832 •05049 ■05265 ■05481 ■05696 •05911 ■06125 •06338 •06551 ■06764 •06976 •07187 •07398 •07608 •07817 •08026 •08234 •08441 •08647 •08853 •09059 •09264 •09468 •09672 •09875 ■10077 ■10278 ■10478 •10678 •10878 •11077 i^ii275 Diff. 228 228 227 226 226 225 225 225 224 224 224 223 222 222 221 221 220 220 219 219 218 218 217 216 216 215 215 214 213 213 213 212 211 211 210 209 209 208 207 206 206 206 205 204 204 203 202 201 200 200 200 199 ig8 F=ilf-. 1-05628 (sin2 2 7 + 0-66962 cos V sin 2 7-f o-il2lo)i I is given for the first of each month in Table 6. 240 UNITED STATES COAST AND GEODETIC SUEVEY. Table 13. — Factors F and f, corresponding to every tenth of a degree of J, for reduction and prediction of tides — Continued. 18-3 •4 •s •6 •7 •8 i8-9 ig-o •I •2 •3 •4 19-5 •6 19-9 20'0 •I •2 •3 ■4 20'5 •6 ■7 •8 20'9 2I-0 •I ■2 ■3 •4 21-5 ■6 ■7 •8 2 1 '9 22'0 •I ■2 ■3 •4 22-5 •6 •7 ■8 22 '9 23-0 •I ■2 23'3 /=-(Kj) 1-33821 ■33169 •32s '8 •31866 •31214 •30562 •29910 •29257 •28604 •27951 •27298 •26645 ■25993 ■25341 •24690 •24039 •23389 •22740 ■22092 ■21445 •20799 •20154 ■19510 ■18227 ■17588 ■16950 ' -16313 •15679 •15047 •14417 ■13788 •13161 ■12536 ■11914 ■I 1294 •10676 •10060 •09447 •08837 •08229 •07623 ■07019 •06418 •05820 •05225 •04633 •04043 •03456 •02872 I •02291 Diff. 652 651 652 652 652 652 653 653 653 653 653 652 652 651 651 650 649 648 647 646 645 644 642 641 639 638 637 634 632 630 629 627 625 622 620 618 616 613 610 608 606 604 601 598 595 592 590 587 584 581 578 r(K,) 0-74732 -7509s -75462 •75834 •76211 ■76592 •76977 -77365 -77757 ■78154 ■78555 •78960 •79369 •79781 •80198 ■80619 •81044 ■81473 ■81905 ■82341 ■82781 ■83226 ■8367s •84127 ■84583 ■85043 ■85507 •85975 •86446 ■86921 •87400 ■87883 ■88370 -89354 •89852 ■90354 ■90860 ■91369 •91882 ■92398 •92918 ■93442 ■93969 ■94500 ■95034 ■95572 •961 14 •96659 •97208 097760 363 367 372 377 381 38s 388 392 397 401 405 409 412 417 421 425 429 432 436 440 445 449 452 456 460 464 468 471 • 475 479 483 487 490 494 498 502 506 509 S13 516 520 524 527 531 534 538 542 545 549 552 556 23 '3 •4 -s ■6 -7 •8 23-9 24^0 •I •2 ■3 -4 245 ■6 24-9 25-0 •I •2 -3 ■4 25-5 ■6 25^9 26^o •I •2 ■3 -4 26^5 ■6 26 9 27'0 •I •2 ■3 27-5 •6 27-9 28^0 •I ■2 -3 -4 -5 28^6 F(Ki) I -02291 •01713 •01 137 •00565 0^99996 ■99429 ■98865 ■98304 •97746 ■97192 •96641 -96093 •95548 •95006 •94467 ■93932 ■93400 •92871 ■92345 •91822 •91302 ■90786 ■90273 ■89763 ■89256 ■88752 ■88252 •87755 •87261 ■86770 ■86282 ■85798 ■85317 ■84839 ■84364 ■83424 ■82959 •82497 •82038 •81582 •81 129 •80679 •80232 •79788 •79348 •78477 •78046 •77618 •77193 •76771 •76351 0^75934 Diff. 578 576 572 569 567 564 561 558 554 SSI 548 545 542 539 535 532 529 526 523 520 516 513 510 507 504 500 497 494 491 478 475 472 468 465 462 459 456 453 450 447 444 440 437 434 431 428 425 422 420 417 /(Kj) 0^97760 •98316 •98876 0-99439 I ^00005 •OOS7S •01 148 •01725 •02305 •02888 •0347s ■04065 •04659 •05256 •05856 •06460 •07067 •07677 •08290 ■08906 ■09526 ■10149 ■10775 ■I 1404 ■12037 ■12673 •13312 •13954 •14599 •15247 ■15898 ■16552 ■17210 ■17870 •18533 ■19199 •20542 •21217 •2189s •22576 •23260 •23948 •24638 •25331 •26026 •26724 •27425 •28129 •28836 •29546 •30259 •30974 K31692 556 560 563 566 570 573 577 580 583 587 590 594 597 600 604 607 610 613 616' 620 623 626 629 633 636 639 642 645 648 651 654 658 660 663 666 670 673 075 678 681 684 688 690 693 695 698 701 704 707 710 713 71S 718 F=j// = 0-22915 (sin ■■ /-j- 0^14527 cos 2 y sin 2 /-f- o-oo528)j' / is given for the first of each month in Table 6. EEPOET FOR 1894— PAET II. 241 Table IS.—Faotms Fandf, corresponding to every tenth of a degree of I, for reduoiion and prediction of 3 3'6 319 322 324 327 330 333 335 339 342 344 348 350 354 357 360 363 367 369 373 377 379 383 I. 01614 •01361 •01108 •00853 -00597 •00341 1-00084 0-99826 •99567 •99308 •99048 •98786 •98524 •98262 •97998 •97734 •97469 •97203 •96936 •96669 •96401 •96132 ■95862 •95591 -95320 -95048 ■94775 •94502 •94227 •93952 ■93677 •93400 ■93123 •92845 -92566 -92287 •92007 •91726 •91444 ■91162 ■90879 -9059s •90311 •90026 •89740 ■89453 •89166 ■88878 •88590 -88301 •88011 •87720 •87429 o^87i37 253 253 255 256 256 257 258 259 259 260 262 262 262 264 264 265 266 267 267 268 269 270 271 271 272 273 273 275 275 27s 277 277 278 279 279 280 281 282 282 283 284 284 285 286 287 287 288 288 289 290 291 291 292 F= 1//= (1 — I sin' m) (1 — f sin' i) o^753i6 /is given for the first of each month m Table 6. I — f sin» / "1— fsin"/' Table 14.- EEPOET FOE 1894— PAET II. -Factors (F) for reducing Mn, K,+Oi, and X to mean values. 247 i8J" '9° I, or Inclinaaon of orbit to equator. 20° 21" 22° 23° 24° 25° 26'' 27° 28" 28i° F{Un) =o-o 0-970 0-972 0-977 0-982 0-988 0-994 I-OOOi 1-006 1-012 1-019 1-026 1-029 0'2 0-971 0-973 0-978 0-982 0-988 0-994 I-ooo I -006 I-0I2 1-019 1-026 1-029 0-4 0-972 0-974 0-979 0-983 0-988 0-994 I-OOO I -006 I-OI2 I-018 1.025 1-028 0-6 0-974 0-976 0-980 0-984 0-989 0-994 I-ooo 1-006 I -01 1 I -017 1-023 1-026 0-8 0-977 0-979 0-982 0-986 0-990 0-995 I-OOO 1-005 I-OIO 1-015 1-020 I -022 I"0 9-980 0-982 0-985 0-989 0-992 0-996 I-OOO 1-004 I -008 I -013 I -017 I -019 1'2 0-984 0-985 0-988 0-991 0-994 0-997 I-OOO 1-003 1-007 I-OIO I -014 I -016 1-4 0-988 0-989 0-991 0-994 0-996 0-998 I-OOO 1-002 1-005 1-007 I-OIO I -012 1-6 0-994 0.994 0-995 0-997 0-998 0-999 I-OOO I-OOI 1-003 1-004 1-005 1-006 1-8 I -001 i-ooo i-ooo I-ooo I-OOO I-ooo I'OOO I-ooo I-OOO I-ooo I-OOO 0-999 2'0 1-008 1-007 I -006 1-004 1-003 I-OOI I-OOO 0-998 0-997 o'99S. 0-994 0-993 2-S 1-025 1-023 i-oig 1-014 1-009 1-004 0-999 0-994 0-989 0-984 0-979 0-977 i^(K,+O0 I-I68 1-148 1-109 I -073 1-040 i-oio 0-982 0-9S5 0-931 0-909 0-888 0-878 ^(X) X'= o-i 0-47 0-89 I-I6 1-38 ••54 1-68 I -80 1-86 0-2 0-54 0-6: 073 0-83 0-91 0-98 1-04 1-09 I-I4 1-19 1-23 1-24 0-3 0-86 0-88 0-91 0-94 0-97 0-99 I -01 1-04 1-05 107 1-08 1-09 0-4 0-96 0-96 0-97 0-98 0-99 i-oo i-oi I-OI 1-02 1-02 1-03 103 o-S I -co I -00 i-oo I -00 i-oo I -00 I-oo I-oo I-OO I-oo i-oo I-oo 2-4066 J- (K, -f O.) = ,.4o66/(K,) +/(0,)' This table is based upon Tables I, 13, 21, and ^(X)=J. cos (X- 180°) =^(Ki + 0,) cos (X' • 180°). 3. 21. SO- 248 UNITED STATES COAST AND GEODETIC SUEVBY, Table 15. — Acceleration in HW and LW of a [The amplitude of the principal wave is taken as unity.] HW phase." LW phase.* 0° 180 10° 190 20" 200 30° 210 40° 220 50° 230 60° 240 70° 250 80° 260 90° 270 1 °-° a o'x ■§ 0'2 1 °-3 S „• G-4 °i °1 ■g 07 S 0-8 ^ 0-9 1 i-o / GO 00 00 00 00 00 00 GO 00 00 GO / GO 54 1 40 2 18 2 51 3 20 3 45 4 07 4 27 4 44 5 00 / 00 1 47 3 18 4 35 5 41 6 38 728 853 9 28 10 00 / GO 238 452 6 47 827 9 54 11 IG 12 18 13 18 14 12 15 00 / GO 3 25 6 22 8 55 II g8 13 OS 14 48 16 19 17 41 iS 54 20 00 / O GO 4 07 7 44 10 54 13 42 16 10 18 21 20 18 22 02 23 36 25 GO / 00 4 43 857 12 44 16 06 19 06 21 47 24 II 26 2G 28 16 30 00 / G GO 5 12 9 59 14 20 18 18 21 52 25 04 27 57 30 33 32 53 35 00 / GO s 32 10 47 15 41 20 13 24 22 28 09 31 35 34 40 37 28 40 00 / 00 5 42 II 19 16 42 21 48 26 34 30 58 35 00 38 40 41 59 45 00 HW phase. LW phase. 360° z8o 3SO° 170 340° 160 330° ISO 320° 140 310° 130 300° 120 290° no 280° 100 270° 90 *i. e. the argument, or phase, of the subordinate component (S) at the time of HW and LW, respectivelv,of the principal component (A). By §2 5*2 .„HW phase , .■ ■ HW a T^' LW phase tan acceleration m y ~ = _ iiii . LW i Bf^^^ HW phase ± ' +^4^2 "^"^ LW phase (When 6 = a, this formula is exact.) If i denote the time after the conspiring of A and 5, § 17, then phase = {l) — a)i] b = — 2 « jrfor HW, ^b ^ (J 2 « + I IT for LW, n being an integer denoting the number of high waters of A since the coincidence of the maxima of A and S, Tabm Vo.—HeigU of HW anA IM for a [The amplitude of the principal wave is taken as unity.] HW phase. 0° 10° 20"* 30° 40° s°° 60° 70° 80" 90° LW phase. 180 190 200 210 220 230 240 2SO 260 270 & O-O I -0000 I-OGOO i-oooo I-oooo I-OOOO I-oooo I-OOOO I-OOOO I-OGOO I-OOOO C O'l I-IGOO 1-0986 1-0945 1-0877 1-0785 I -0670 1-0536 1-0385 I -0221 1-0050 •a 0-2 I -2000 I-I975 I -1899 I-I77S I-I603 1-1389 I-II35 I -0849 1-0532 I -0198 .§ 0-3 1-3000 1-2965 1-2860 1-2687 1-2448 I -2148 1-1790 I-I38I 1-0928 1-0440 S u 0-4 1-4000 1-3957 1-3827 1-3611 1-3315 1-2939 1-2490 I-I973 I-I397 1-0770 •S S o'S 1-5000 1-4949 1-4798 1-4546 1-4198 1-3758 1-3228 I-26I8 I-I93I I-II80 °^ 0-6 i-6ooo 1-5943 1-5772 1-5490 1-5097 1-4599 1-4000 1-3306 1-2523 1-1662 ■§ 07 1-7000 1-6937 1-6749 1-6439 1-6007 I -5459 1-4798 1-4032 I -3165 1-2207 1 0-8 1-8000 1-7932 1-7730 1-7395 1-6928 1-6336 1-5620 1-4789 1-3849 1-2806 % 0-9 1-9000 1-8928 I-87I2 1-8354 1-7858 1.7225 1-6463 1-5575 1-4569 1-3453 2-0000 1-9924 1-9696 i-93'8 1-8794 I -8126 1-7320 1-6384 1-5320 I -4142 HW phase. 360° 350° 340° 330° 320" 3.o» 300- 290° 380° 270° LW phase. i8a 170 160 ISO 140 130 no 100 90 For high waters use the tabular values as given; but for low waters alter their signs. EBPOET FOE 1894— PAET II. 251 Table 20. — Great tropic range and its duration. [The amplitude of the semidiurnal wave is taken as unity.] Slevation of great trop ic HW > depression of great tropic LW. Depressioi of gt. tropl LW > elevation of gt. tropic HW. HW phase. 5 i8o 10° 170 20° 160 , 30° 150 40° 140 50° 130 60° 120 70° 110 80° 100 90° 90 ■■'{ 3-921 7 SO 4-149 7 54 4-326 7 56 4-444 7 57 4-504 7 57 4-504 7 57 4-444 7 57 4-326 7 56 4-149 7 54 3-921 7 50 n{ 4-062 7 S7 4-291 7 59 4-487 8 01 4-610 8 02 4-674 8 03 4-674 8 03 4-610 8 02 4-487 8 01 4-291 7 59 4-062 7 57 '■'{ 4-205 8 03 4-456 8 05 4-649 8 06 4-777 8 07 4-844 8 08 4-844 8 08 4-777 8 07 4-649 8 06 4-456 8 05 4-205 8 03 '■"{ 4-3SI 8 10 4-612 8 II 4-813 8 12 4-947 8 12 S-015 8 12 5-015 8 12 4-947 8 12 4-813 8 12 4-612 8 II 4-351 8 10, '■'{ 4-500 8 17 4-772 8 17 4-978 8 17 5-117 8 17 5-187 8 17 5-187 8 17 S-117 8 17 4-978 8 17 4-772 8 17 4-500 8 17 "{ 4-651 8 24 4-932 8 23 S-144 8 22 5-288 8 21 5-360 8 21 5-360 8 21 5-288 8 21 5-144 8 22 4-932 8 23 4-651 8 24 '■'{ 4-805 8 31 5-093 8 28 S-3" 8 27 5 -460 8 26 5-535 8 26 5-535 8 26 5-460 8 26 S-311 8 27 Hi 4-805 8 31 'A 4-962 8 38 5 -256 8 34 S-487 8 31 5-634 8 31 S-710 8 30 5-710 8 30 5-634 8 31 5-487 8 31 5-256 8 34 4-962 8 38 3 .,{ 5-121 8 45 5-421 8 40 S-652 8 37 5-808 8 35 S-886 8 34 5-886 8 34 5-808 8 35 5-652 8 37 5-421 8 40 5-121 8 45 "o 1 .s{ 5-282 8 S3 5-589 8 45 5-824 8 41 5-984 8 39 6-063 8 38 6-063 8 38 5-984 8 39 5-824 8 41 5-589 8 45 5-282 8 53 ! '■«{ S-44S 9 01 S-757 8 51 5-997 8 46 6-160 8 44 6-241 8 42 6-241 8 42 6-i6o 8 44 5-997 8 46 5-757 8 51 5-445 9 01 .7{ 5-611 9 08 5-927 8 57 6-170 8 51 ^337 8 48 6-420 8 46 6-420 8 46 6-337 8 48 6-170 8 51 5-927 8 57 5-611 9 08 '■>{ 5-780 9 17 6-097 9 02 6-345 8 55 6-514 8 52 6-6oo 8 50 6-600 8 50 6-514 8 52 6-345 8 55 6-097 9 02 5-780 9 17 =•'{ S-9SI 9 2S 6-274 9 07 6-522 8 59 6-692 8 55 6-779 8 54 6-779 8 54 6-692 8 55 6-522 8 59 6-274 9 07 S-9SI 9 25 S-o{ 6-125 9 34 6-446 9 13 6-700 9 03 ■6-875 8 59 6-959 8 57 6-959 8 57 6-875 8 59 6-700 9 03 6-446 9 13 6-125 9 34 ,-.{ 8-000 12 25 8-261 10 03 8-517 9 42 8-701 9 32 8-795 9 28 8-795 9 28 8-701 9 32 8-S17 9 42 8-261 10 03 8.000 12 25 »•»{ 10-000 12 25 10-165 10 43 10-399 10 II IO-S79 9 58 10-675 9 52 10-675 9 52 10-579 9 58 10-399 10 II 10-165 10 43 lo 00 12 25 lo-o- 20-000 12 25 20-037 II, 48 20-146 II 18 20-289 II 01 20-387 10 53 20-387 10 53 20-289 II 01 20-146 11 18 20-037 II 48 20 00 12 25 1 HW phase. J i8o» 360 igo- 35° 200° 340 2IO* 330 220*" 320 230° 310 240° 300 2SO- 290 260° 280 270° 270 The first value of each pair is the value of the great tropic range ; the second, its duration in hours and minutes. This table assumes that dj = mj. See ?? 25, 37, and 53. 252 UNITED STATES COAST Al^TD GEODETIC SURVEY. Table 21. — Effects of various tidal components upon the mean semirange of tide. [The amplitude of M^ is taken as unity.] Ampli- Semidiurnal components. Ampli- Diurnal components. tude of subordi- tude of subordi- nate com- ponent. K, L, Nj s. A, Ms "1 4 nate com- ponent. Ki Oi P. Qi 16 0-02 ■0001 •0001 •0001 •0001 •0001 •000 1 •0001 •ooqi 0-04 •0001 •0001 •0001 ■0001 ■ooot 0'04 •0004 •0004 •0004 •0004 ■0004 •0004 •0004 •0004 o-o8 •0004 •0004 •0004 ■0003 •0004 o-o6 •0010 •0009 •0009 •0010 ■0009 •0008 •0009 •0009 0-12 •0010 •0008 •0010 •0008 •0009 o-o8 •0017 •0017 •0015 •0017 ■0017 •0015 •0015 •0016 9'i6 •0017 •0015 •0017 •0014 •0016 o-io •0027 •0026 •0024 •0027 •0026 •0023 ■0024 •0025 0-20 ■0027 •0023 •0027 •0021 ■0025 0'12 •0039 •0037 •0035 •0039 •0036 0'24 •0039 ■0033 •0038 ■0030 •0036 o'l4 •0053 ■0051 •0047 ■0052 •0049 0-28 •0053 ■0045 •0052 •0041 •0049 o'i6 •0069 •0061 •0069 •0064 0.32 •0069 •0059 •0068 •0064 o-i8 •0087 •0078 •0087 ■0081 0-36 •0087 •007s •0086 •0081 0-20 •0108 ■0096 •0107 ■0100 0-40 ■0108 •0093 •0106 •QIOO 0'22 •0130 •01 1 6 •0130 •0121 ©•44 •0130 ■01 12 •0127 •0I2I 0-24 •OIS5 •0138 •01 54 •0144 0-48 ■OIS5 •0133 ■0150 •0144 0'26 •0162 •0181 •0169 0*52 •0182 •0156 •0176 •0169 0-28 •0188 •0210 •0196 0-56 •02 1 1 ■01 8 1 •0196 0-30 •0216 •0241 •0225 o-6o •0242 ■0208 •0225 0-32 ■0274 •0256 0-64 •0276 •0237 •0256 0-34 •0310 •0289 0-68 ■0311 ■0268 •0289 0-36 •°347 •0324 0^72 •0349 •0300 •0324 0-38 •0387 •0361 0-76 ■0389 •0334 ■0361 0-40 •0428 •0400 o-8o •0431 •0370 •0400 0-42 •0472 ■0441 0^84 •°47S •0407 •0441 0-44 •0518 •0484 0-88 •0521 ■0447 •0484 0'46 •0567 •0529 0-92 •0570 ■0488 •0529 0-48 •0617 •0576 0-96 •0620 •0532 •0576 0-50 •0669 •0625 I"0O 1-04 •0673 •0728 •0577 ■0624 •0625 •0676 o'6o •0964 •0900 I -08 ■078s •0673 •0729 070 •1312 ■1225 I-I2 ■084s ■0723 •0784 o-8o •1715 ■1600 i-i6 •0906 •0776 •0841 0-90 •2169 •2025 I^20 ■0969 •0831 •0900 i-oo •2678 •2500 r24 r28 1-32 • •36 1-40 1-44 1-48 1-52 1-56 I -60 I '64 1-68 •103s •I 103 •"73 •I 24s •1320 •1396 •147s •isss •1637 •1720 •1805 •1892 •0887 •0945 •0961 •1024 •1089 •II56 •1225 •1296 ■1369 •1444 •152I •1600 ■i68i •1764 * Tabular value for component C ■■ --4M, EBPOKT FOE 1894— PART II. 253 Table 22.— Value ofi Mn when M2= i. K,+0, S^M, O'l 0-2 o"3 °'4 °-5 0-6 0-7 0-8 o-o I '0127 1-0213 I '0357 1-0558 1-0817 1-1134 1-1508 1-1940 0"2 I •0142 I -0228 1-0372 1-0573 1-0832 1-1149 1-1523 I-I95S 0-4 i-oiSs I -0271 1-0415 i-o6i6 1-0875 I-II92 1-1566 1-1998 0-6 1-0257 1-0343 1-0487 1-0688 1-0947 1-1264 1-1638 1-2070 0-8 I -0357 I -0443 1-0587 1-0788 1-1047 1-1364 1-1738 1-2170 i-o I -0486 1-0572 1-0716 I -0917 1-1176 I-I493 1-1867 1-2299 1-2 1-0643 1-0729 1-0873 I -1074 I-I333 1-1650 1-2024 1-2456 1-4 1-0830 I -0916 I -1060 1-1261 1-1520 1-1837 I-221I 1-2643 1-6 I-I045 1-1131 I-I275 1-1476 I-I73S 1-2052 1 -2426 1-2858 1-8 1-1289 I-I37S 1-1519 1-1720 1-1979 1-2296 1-2670 I -3102 2'0 '■ISS9 I -1645 1-1789 I -1990 1-2249 1-2566 1 -2940 1-3372 2-S 1-2360 1-2446 1-2590 1-2791 1-3050 1-3367 I -3741 I -41 73 This table, based upon Tables l and 21, is computed upon the assumption that the ratios between the diurnal components, the pure lunar semidiurnals, the solar semidiumals (including luni-solar Kj), are respectively constant for all stations; also that shallow water tides do not occur. On account df nonpredictable inequalities, the tabular values should be multiplied by about 1-02. Table 23. — Value 0/M3 when i Mn = l. Sj/iMn K,+0, iMn o.r 0*2 °'3 o'4 o'S 0-6 0-7 0-8 0-0 0-9873 0-9785 0-9634 0-9414 O-9I16 0-8725 0-8221 0-7574 0-2 0-9858 0-9770 0-9618 0-9396 0-9096 0-8702 0-8194 0-7542 0-4 0-9814 0-9725 0-9572 0-9348 0-9045 0-8648 0-8136 0-7479 0-6 0-9740 0-9650 0-9495 0-9269 0-8963 0-8562 0-8045 0-7382 0-8 0-9635 0-9S43 0-9386 0-9157 0-8847 0-8442 0-7919 0-7249 1-0 0-9497 0-9403 0-9243 0-901 1 0-8696 0-9286 0-7756 0-7077 1-2 0.9322 0-9225 0-9062 0-8826 0-8506 0-8090 0-7552 0-6863 1-4 0-9106 o-goo6 0-8839 0-8598 0-8272 0-7849 0-7302 0-6602 1-6 0-8844 0-8740 0-8568 0-8321 0-7988 0-75S7 0-7000 0-6288 1-8 0-8529 0-8421 0-8243 0-7989 0-7648 0-7207 0-6639 0-5913 2-0 0-8151 0-8038 0-7854 0-7592 0-7242 0-6789 0-6208 O-546S 2-S 0-7175 0-7049 0-6848 0-6566 0-6193 0-5710 0-5093 0-4304 This table is Table 22 reverted. On account of nonpredictable inequalities, the tabular values should be divided by about 1-02. 254 HtflTED STATES COAST AND GEODETIC SUEVEY. Table 24. — Variation in lunitidal interval and mean aemirange of tide, due to the phase wave composed of Sa and fH Increase n mean semirans^e of tide due to S2 and i^i- Time Increase iu lunitidal Length o£ half group interval due lo 02. tides. 4 tides 6 tides 8 tides . 10 tides. (/. A. zw. r o CX3 S-VM, +0-93 S2 +0-90 s. +0'88 Si, +0-84 S, +o'79 S2 o o6 — 9 -)-o-92 " +0-90 2 18 Ti 2 12 2 06 d 2 00 r? ' I 18 I 12 V I 06 •a I 00 pq 18 12 06 +0-94 Ng +0-94 " +0-93 " -|-0"92 " +0-91 " +0-89 " +0-87 " +0-8S « +0-82 " +079 " +07S " -j-072 " 4-0-68 " +0-64 " -|-o-6o " +0-6I " +O.S7 " 4-0-52 " +0-48 " +0-43 " +0-38 " +0-33 " +0-28 " 4-0-23 " +o-i8 « +0-12 " -|-o'o8 " -[-0-02 " — O-02 " — 0-07 " — 0-07 " — 0-12 " — 0-17 " — 0-21 " — 0-26 " — 0-29 " -0-33 " —0-37 " —0-41 " -0-4S " —0.47 " — 0-51 " -o-SS " — 0-S7 " —0-60 " —0-58 " —0-61 " —0-63 " — 0-66 " — 0-68 " — 0-69 " — 0-71 " —0-73 " —0-74 " — 0-7S " —0-75 " —0-76 " — 0-77 " — 0-77 " — •0-77 " +0-93 N2 +0-93 " -1-0-93 " -1-0-91 " -)-o-9i " -1-0-89 " +0-87 " -1-0-85 " -j-o-82 " -1-0-79 " +0-75 " -1-0-72 " -i-o-68 " -f 0-64 " -|-o-6o " -I-0-6I " +0-57 " -1-0-51 " +0-47 " 4-0-43 " 4-0-38 " +0-33 " -I-0-28 " +0-23 " +0-I8 " -)-o-i2 " -f-o-o8 " 4-0-02 " —0-02 " — 0-07 " —0-07 " — 0-I2 " — 0-17 " — 0-21 " —0-26 " — 0-29 " — 0-33 " —0-37 " —0-41 " -0-4S " —0-47 " —0-51 " -0-S4 " -0-S7 " — o-6o " —0-58 " —0-61 " —0-63 " —0-66 " —0-68 " —0-69 " — 0-70 " — 0-72 " —0-74 " —0-75 " — 0-7S " —0-76 " -077 " — 0-77 " —0-77 « -1-0-90 N2 4-o*90 " -1-0-89 " +0-88 " -I-0-87 " 4-0.85 " 4-0-84 " 4-0-82 " 4-0-80 " 4-0-77 " 4-0-74 " 4.0-72 " 4-0-67 « 4-0-63 « 4-0-59 " 4-0-59 " 4-0-5S " 4-0-50 " 4-0-4S " 4-0-40 " 4-0-36 " 4-0-31 " 4-0-27 " 4-0-23 " 4-0-I9 " 4-0-13 " 4-0-09 " 4-0-04 " 0-00 " —0-06 " — o-o6 " — o-io " —0-15 " — 0-19 " — 0-24 " — 0-27 " —0-32 " -0-35 " — 0-40 " —0-44 " —0-46 " — 0-50 " —0-52 " -0-5S " —0-58 " -0-S7 " — o-6o " —0-62 " —0-64 « —0-65 " —0-67 " —0-68 " —0-72 " — 0-73 " —0-74 " —074 " — 0-7S " —0-76 " —0-76 " — 0-76 " 4-0-85 Na 4-0-85 " 4-0-84 " 4-0-83 « 4-0-82 " -1-0-80 " 4-0-79 " 4-0-77 " -I-0-76 " 4-0-73 " 4-0-70 " 4-0-67 " -I-0-64 " 4-0-61 " 4-0-57 " 4-0-57 " 4-0-S3 " 4-0-49 " 4-0-45 " -I-0-40 " 4-0-36 " +0-31 " -1-0-28 " 4-0-24 " 4-0-20 " 4-0-I4 " 4-0-I0 " 4-O-04 " -)-o-oi " — 0-04 " — 0-04 " — 0-09 " — 0-13 " —0-16 " — 0-21 " — 0-24 " — 0-29 " — 0-32 " -0-37 " — 0-41 " —0-43 " —0-46 " —0-50 " —0-53 " -0-56 " -0-54 " -0-57 " —0-60 " -0-63 « —0-64 " —0-65 " —0-67 " —0-70 " —0-71 " — 0-72 " — 0-72 " -0-73 " —0-73 " —0-73 " —0-73 " d. h. o 00 06 12 18 00 06 12 18 2 00 2 06 2 2 3 3 13 3 3 3 2 2 2 2 I I I I o o o o o o o o I I I I 2 2 2 2 3 3 13 12 18 00 06 12 12 06 00 18 12 06 00 18 12 06 00 18 12 06 00 00 06 12 18 00 06 12 18 00 06 12 18 00 06 12 3 12 3 06 3 00 2 18 2 12 2 06 2 00 I 18 I 12 I 06 I 00 18 12 06 -0-77 Ns -0-77 " -0-77 « -0-76 " -0-75 " -0-75 " -0-74 " -0-73 " -0-71 " -0-69 " -0-68 " -0-66 " -0-63 " -o-6i " -0-58 " -o-6o " -0-57 " -0-5S " -0-51 " -0-47 " — 0-4S — 0-41 (i — o"37 t< — 0-33 it — 0-29 — 0-26 cc — 0-2I it — 0-I7 *' — 0-I2 a 0-07 tt — 0-07 (( — 0-02 tt 4-0-02 node. * i. e. Kx' or the Ki for the given year. For the acceleration in S3 due to lunar K2, use the tabular value multiplied by [/(Kj) — 0-317]. This table is based upon Tables i, 15, 16, and 29. 260 UNITED STATES COAST AND GEODETIC SURVEY. Table 32. — Factor Fi for clearing D| of the effects of the longitude of the moon's node and of Pi. Date. /, or inclination of orbit to equator. isr '9° 20° 21" 22° 0-874 230 24° 25° 0-814 26° , 27° 28° 281° Jan. I 0963 0-949 0-923 0-898 0-853 0-833 0-796 0-779 0-764 0-757 II 0-988 0-974 0-946 0919 0-895 0-872 0-851 0-831 0-813 0-795 0-779 0-772 21 1-030 I -014 0-983 0-955 0-929 0-905 0-882 0-861 0-841 0-822 0-805 0-797 3° I -091 1-073 1-038 1-008 0-978 0-951 0-926 0-903 0-881 0-861 0-842 0-833 Feb. lo 1172 I-I5I 1-112 1-077 1-043 I -012 0-984 0-958 0-933 0-911 0-889 0-879 20 1-270 I 245 1-200 1-158 I-II9 1-0^4 1-052 I -022 0-994 0-968 0-945 0933 Mar. 2 1-373 I -345 1-293 1-245 I -200 1-159 I -122 1-088 1-057 1-028 i-ooi 0-989 12 1-463 I -43 1 I -371 1-317 1-268 1-223 1-182 I 144 r-109 1-077 1-047 1034 22 1-499 1-466 1-404 1-347 1-296 1-248 1-205 I -166 1-130 1-097 1-067 1-052 Apr. I 1-468 1-436 1-376 I -321 1-272 1-226 1185 1-147 I-II2 1-080 1-050 1-037 II 1-384 1-356 1-302 1-253 1-208 I -167 1-130 1-095 1-063 1-034 1-007 0-994 21 1-280 1-256 1-209 1-167 I-I28 1-093 1-059 1-029 I -000 0-974 0-951 0-939 May I i-i8i 1-161 1-121 1-084 1-051 I 020 0-991 0-964 0-940 o-gi6 0-895 0-885 II i-ioo 1-082 1-048 1-015 0-986 0-959 0-933 0-909 0-887 0-866 0-848 0-839 21 1-036 I -019 0-989 0-961 0-934 0-910 0-886 0865 0-845 0-826 0-809 0-801 31 0-991 0-977 0-949 0922 0-898 0-875 0-854 0-833 0-815 0-798 0-782 0-774 June lo 0-965 0-951 0-925 0-900 0-876 0-854 0-834 0-815 0-797 0-780 0-765 0-758 20 0-9S4 0-941 0-915 0-890 0-867 0-846 0-826 0-808 0-790 0-773 0-758 0-751 3° 0-958 0-944 0-918 0-893 0-871 0-849 0-829 0-810 0-793 0-776 0-761 0-754 July lo 0-980 0-965 0-938 0912 0-888 0-866 0-845 0-825 0-807 0-790 0-774 0-767 20 1-017 1-002 0-972 0-945 0-920 0-895 0-873 0-852 0-833 0-815 0-798 0-790 30 1-072 I -05s 1-022 0-992 0-964 0-937 0-913 0-890 0-869 0-850 0-832 0-823 Aug. 9 1-146 1-127 1-090 1-055 1-024 0-994 0-966 0-941 0-917 0-895 ■ 0-875 0-866 19 1-243 1-219 1-176 1-135 1-099 I -065 . 1-033 1-005 0-977 0-953 0-930 0-919 29 1-346 1-319 I 268 1-222 1-179 I -140 1-104 I -07 1 1-040 1-013 0-987 0-974 Sept. 8 1-443 I -412 I -354 1-301 1-253 1-209 1-169 1-132 1-098 1-066 1-038 I 024 18 1-499 1-465 1403 1-346 1-295 1-248 1-205 1-166 1-130 1-097. 1-066 1-052 28 1-487 I -454 1-392 1-336 1-286 1-240 1-197 1-159 I-I23 I -091 1-060 1-046 Oct. 8 I -412 1-382 1-326 1-275 1-229 .1-187 I -148 I-II2 1-079 1-050 I 021 1-008 18 I 309 1-283 1-235 I-I9I 1150 "1-113 1-079 1-047 I 018 0-991 0-967 0-955 28 1-205 1-184 I 142 I -105 1-070 1-037 I -008 0-980 0-954 0-931 0-909 0-898 Nov. 7 1-119 i-ioo 1-065 1031 I -001 0-973 0-946 0-922 0-899 0-878 0-858 0849 17 1-050 1-033 • 1-002 0-973 0-946 0920 0-896 0-874 0-855 0-836 o-8i8 0-809 27 I-OOI 0986 0-957 0-930 0-906 0-883 0-861 0-841 0-822 0-804 0-788 0-780 Dec. 7 0-970 0-956 0-929 0-904 0-880 0-858 0-838 0-818 0-801 0-784 0-768 0761 17 0-9S5 0-941 0-915 0-891 0-868 0-846 0-826 0-808 0-790 0-774 0-759 0-752 27 0-956 0-942 0-916 0-892 0-869 0-847 0-827 0-809 0-791 0-775 0-760 0-752 Jan. 6 0-976 0-962 0-934 0-908 0-884 0-862 0-842 0-822 0-804 0-787 0-772 0-764 F, = . 2-4066 f„ i-4o66/(K0+/(O,) This table is based on Tables 13 and 31. J'\ X observed Di ^ Ki -|- Oi. 1-02 Fi X observed Di = Di. REPORT FOR 1894— PART II. 261 Table S3.— Factor Fifor clearing S2 o/Kj and Tj. Date. /, or inclination of orbit to equator. .8J° 19° =0° 21" 22° 23° 24° 25° 26° 27° 28° 281° Jan. I 114 115 116 1x8 119 I 20 I 22 1 24 f27 1-30 1-33 I 34 II I -08 1-09 IIO I^IO I^II 112 114 IIS I-I7 I-I9 I-2I 1-22 21 I 02 1-02 I -02 I ■03 1-03 1^04 1^04 • •OS I -06 I '06 1-07 I -08 31 ■96 ■96 •96 •96 ■96 •96 ■96 •96 •96 •96 •96 •96 Feb. 10 •90 •90 •90 •89 ■89 •89 •88 •88 •88 •87 ■87 •86 20 ■86 •86 •85 •85 •84 •84 •83 •83 ■82 ■81 •81 •80 " Mar. 2 •83 •83 ■83 •82 •82 •81 •80 ■79 •78 •77 •76 •76 12 •83 •82 •82 •81 ■80 •79 •78 •77 ■76 •75 •74 •74 22 •83 •82 •81 •80 •79 •79 •78 •77 •76 •75 •74 •73 Apr. I ■83 ■S3 •83 •82 •81 •80 ■79 ■78 •77 •76 •75 •75 II •86 •86 •85 •84 •83 •82 ■82 ■81 •80 •79 •78 •78 21 ■91 •91 •90 •89 •89 ■88 ■88 ■87 •86 •85 ■85 •84 May I •96 •96 •96 •95 •95 •95 •94 ■94 •93 •93 ■92 •92 II 1-04 1-04 1^04 1-04 1^04 f04 f04 1^04 1^04 1-04 1-04 1-04 21 I-I2 112 113 113 I 14 114 115 ri6 116 117 118 119 3> I 23 1^24 I 25 1-26 I 27 1-28 I 30 I 32 '•34 1-37 ••39 I -40 June 10 1-30 1-31 i^33 1^35 1^37 . 1^39 I 41 I 44 1-47 i-Si '•55 157 20 I -35 1-37 I 39 I 41 1^44 1-47 I ■SO I 54 158 1-63 1-68 1-70 30 1-34 I 35 1^37 1-39 1-42 "•45 1^48 I 51 1-55 1-59 1-64 f67 July 10 1-26 I 27 I 28 I •SO 1-32 1-34 136 • •38 141 • •43 I •46 1^47 20 117 i-i8 119 119 I -20 1*22 ^23 I 24 I -25 1^26 I 27 1^28 30 I -07 K07 I -08 i^o8 I -08 1^09 1^09 1^09 1-09 I^IO I^IO I^IO Aug. 9 I'OO i-oo i-oo i^oo I^OO •99 •99 •99 •98 ■98 •97 .97 •9 ■94 •93 •93 •93 •92 ■92 ■91 •90 •89 ■88 •88 •87 29 •88 •88 ■88 ■87 ■ ^86 •85 •84 •83 ■83 ■82 ■8i' •80 Sept. 8 •85 ■85 ■84 •84 ■83 ■82 ■8i ■80 •79 •78 •77 •76 18 •84 •83 •83 •82 ■81 ■80 •79 •78 •77 ■76 •75 •75 28 •83 ■83 •83 ■82 ■81 •80 •79 •78 •77 ■76 ■75 •74 Oct. 8 •85 •84 •83 •83 ■82 •81 • •80 •79 •78 •77 •77 •76 18 ■86 •86 •86 •85 •85 ■84 •83 •82 •81 ■80 •79 ■79 28 •90 •89 •89 ■89 ■88 ■88 ■88 •87 ■86 ■86 ■85 •85 Nov. 7 •95 •95 •95 •95 •95 •94 •94 •94 •94 •94 •93 •93 17 I-OI I^OI f02 I^02 I^02 1^03 ro3 1^04 1^04 1-05 I •OS i^o6 27 I 07 I -08 1^09 1^09 flO rii 1^12 114 115 116 118 119 Dec. 7 i^i3 ri4 '•15 ri6 118 r-i9 I-2I I 23 1-25 I 28 I 31 1^32 17 116 118 ri9 121 123 I 25 1^27 I 30 1-33 1-36 '•39 141 27 116 1-17 I 18 119 I-2f 1-23 1-25 I 28 I 31 I 34 1-37 ••39 Jan. 6 i-ii III 1^12 ri4 ••IS fi6 ri8 I^20 I 22 f24 1-27 1-28 This table is based upon Tables 13 and 31. Table 34. — Effect of v-i upon the amplitude of N.2. Apparent time of Resultant Apparent time of Resultant moon's upper or lower transit. amplitude N2 and V2. Nj— I. lower transit. amplitude N2 and ^a- J) in perigee or B in perigee or Nj=i. apogee. apogee. A. h. m. h. h. in. 0, 12 00 119 6, 18 00 o^8i 20 1^19 20 o'8i 40 118 40 0^82 I, 13 00 I-I7 7, 19 00 0-84 20 i^i6 20 0-86 40 114 40 089 2, 14 00 i^ii 8, 20 00 0^92 20 I -08 20 0^95 40 1-05 40 o^99 3, '5 00 1-02 9, 21 00 I -02 20 0-99 20 1^05 40 095 40 I 08 4, 16 00 0-92 10, 22 00 I'll 20 089 20 114 40 086 40 116 5. 17 00 0-84 II, 23 00 I-I7 20 082 20 118 40 o-8i 40 ri9 This table is based upon Tables i and 16. 262 UNITED STATES COAST AND GEODETIC SUEVEY. Table 35. — Gi-oiip factors. For phase reduction. Number of Number of tides after springs or neaps, tides before springs or neaps. o 2 4 6 8 10 12 • o I-OO i-oi I -03 1-07 113 1-22 1-34 2 lOI I-OI 1-02 I -06 III I-l8 1-29 II 4 103 I 02 1-03 I -OS IIO I-I6 1-25 6 I -07 I 06 I -OS 1-07 III 116 1-24 8 113 III I-IO i-ii 113 118 1-25 (J^ lO 1-22 ii8 116 i-i6 118 1-22 1-28 Ll2 1-34 129 1-25 1-24 1-25 1-28 1-34 • i-oo I 01 103. 1-07 I-I4 1-23 1-35 o 2 I'OI I 01 1-03 I -06 III I-I9 1-30 II 4 1-03 1-03 1-03 I -06 IIO 117 1-26 6 1-07 I 06 i-o6 I -08 I'll 117 1-25 IS 8 114 rii IIO l-ii 114 119 1-26 c^ lO 1-23 I-I9 117 117 1-19 1-23 1-29 .12 1-35 1-30 1-26 1-25 I 26 1-29 1-35 O I'OO I '01 1-03 I -08 I-I4 1-24 1-37 6 II 2 i-oi i-oi I 03 I '06 I'I2 1-20 1-31 4 1-03 1-03 1-03 I -06 I-II i-i8 1-27 6 I -08 I -06 I 06 I -08 I-I2 I-I7 1-26 S 8 114 I'I2 III I-I2 i-iS 1-19 1-27 C/j lO 1-24 I -20 i-i8 I-I7 I-I9 I 24 1-30 12 1-37 1-31 1-27 1-26 1-27 1-30 1-37 O I -00 I'd 1-04 I 09 i-i6 1-25 I '39 6 2 I-OI I 01 1-03 I '07 113 I-2I 1-33 t? ^ 4 1-04 1-03 1-04 1-07 I-I2 I-I9 1-29 6 1-09 1-07 I -07 1-09 I-I3 I-I9 1-27 § 8 i-i6 I-I3 I-I2 1-13 i-i6 I-2I 1-29 Cfi 10 1-25 I -21 119 119 I-2I I -25 1-32 12 1-39 1-33 1-29 1-27 1-29 1-32 1-39 For parallax reduction. For deoUnational reduction. Number of tides before or after greatest and least Factor. parallax effects. I-OO 4 I-OI 6 1-02 8 1-04 10 1-06 Number of tides before or after moon's extreme declination. Factor (for diurnal wave). 2 4 6 8 I -000 1-005 I-OI2 1-024 1-040 This table is based upon Tables i and 16.