LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class 
 
SCIENTIFIC PAPERS 
 
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 
 C. F. CLAY, MANAGER. 
 
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 50, WELLINGTON STREET. 
 
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 [All Rights reserved.] 
 
SCIENTIFIC PAPERS 
 
 BY 
 
 SIR GEORGE HOWARD DARWIN, 
 
 K.C.B., F.R.S. 
 
 FELLOW OF TRINITY COLLEGE 
 PLUMIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE 
 
 VOLUME I 
 OCEANIC TIDES 
 
 AND 
 
 LUNAR DISTURBANCE OF GRAVITY 
 
 OF THE 
 
 UNIVERSITY 
 
 Of 
 s4UFCKN 
 
 CAMBRIDGE : 
 
 AT THE UNIVERSITY PRESS 
 
 1907 
 
(JTamtrtDgc 
 
 PRINTED BY JOHN CLAY, M.A. 
 AT THE UNIVERSITY PRESS. 
 
PBEFACE. 
 
 WHEN the Syndics of the Cambridge University Press did me the 
 honour of offering to publish a collection of my mathematical papers, 
 I had to consider the method of arrangement which would be most con- 
 venient. A simple chronological order has been adopted in various collections 
 of this kind, and this plan certainly has advantages ; but an arrangement of 
 papers according to subject may be more convenient. In the case of my own 
 work the separation into well-defined groups of subjects was easy, and I have 
 therefore adopted this latter method. I shall, however, give at the beginning 
 of each volume a chronological list with a statement as to the volume in 
 which each paper will be found. 
 
 This first volume contains papers on Oceanic Tides and on an attempt to 
 measure the Lunar Disturbance of Gravity ; the second will give my papers 
 on Tidal Friction and on the astronomical speculations arising therefrom; the 
 third will be devoted to papers on Figures of Equilibrium of Rotating Liquid 
 and on cognate subjects; and the fourth will be on Periodic Orbits and on 
 various miscellaneous subjects. 
 
 Throughout corrections and additions will be marked by inclusion in 
 square parentheses. 
 
 The whole of my work on oceanic tides and the attempt made by my 
 brother Horace and me to measure the attraction of the moon sprang from 
 ideas initiated by Lord Kelvin, and I should wish to regard this present 
 volume as being, in a special sense, a tribute to him. 
 
 Early in my scientific career it was my good fortune to be brought into 
 close personal relationship with Lord Kelvin. Many visits to Glasgow and 
 to Largs have taught me to look up to him as my master, and I cannot find 
 words to express how much I owe to his friendship and to his inspiration. 
 
 181576 
 
VI PREFACE. 
 
 The following statement gives in a few words an explanation of how the 
 several investigations originated, and how they are connected together. 
 
 Part I is devoted to the consideration of OCEANIC TIDES. The advisability 
 of applying to the tides a method of analysis similar to that used in the 
 Lunar and Planetary Theories was first suggested by Lord Kelvin. Reports 
 on the Harmonic Analysis of tidal observations were drawn up by him and 
 presented to the British Association in 1868, 1870, 1871, 1872 and 1876. 
 As the analysis employed in those reports needed coordination and revision, 
 a new committee consisting of Professor Adams and myself was appointed in 
 1882. Professor Adams left the matter very much in my hands, although I 
 enjoyed the great benefit of his advice from time to time. The first three 
 papers below form the first, third and fourth reports of this Committee. The 
 second report being merely formal is not reproduced. Some paragraphs of a 
 merely temporary interest are omitted, and a few corrections and alterations 
 have been made by the light of subsequent experience. 
 
 These papers explain the method of harmonic analysis, and its connection 
 with the old method of hour-angles, declinations and parallaxes. The third 
 paper shows how a tide-table may be computed from the harmonic tidal 
 constants. 
 
 The fourth paper is an Article on the Tides written for the Admiralty 
 Scientific Manual, and is reprinted by permission of the Admiralty. It 
 explains the method of analysing a short series of observations and the 
 computation of an approximate tide-table. A small mistake in the Manual 
 has been corrected. The Article would have been too long for its place of 
 publication if it had contained the analytical reasoning on which the several 
 rules of procedure are based, but this analysis is contained in the three 
 preceding papers. 
 
 In the fifth paper a method is devised for evaluating the harmonic 
 constants from observations of high and low-water. 
 
 The sixth paper explains the use of a sort of abacus for the harmonic 
 analysis of hourly observations. 
 
 In the seventh paper I return to the subject of the third paper, and 
 show how a tide-table may be formed from the harmonic constants with any 
 desirable degree of accuracy. 
 
PREFACE. Vll 
 
 Next in logical order would have come two papers in which Colonel 
 Baird and I began a systematic collection of tidal constants for several 
 ports, but these papers are not reproduced. We did not continue our 
 attempt, and fortunately Mr Rollin Harris has made a large collection of 
 such results in his Manual of Tides published by the U. S. Coast and 
 Geodetic Survey. 
 
 The rest of the papers in Part I are devoted to the theory of the tides. 
 
 In Thomson and Tait's Natural Philosophy Lord Kelvin discussed the 
 correction required to make the equilibrium theory of tides true of an ocean 
 interrupted by continents. The eighth paper gives the numerical evaluation 
 of this correction by Professor H. H. Turner and me. 
 
 The ninth paper is an extract of paragraphs contributed to the second 
 edition of Thomson and Tait's Natural Philosophy, in which an attempt is 
 made to estimate the amount of tidal yielding to which the solid earth is 
 subject. 
 
 Laplace was of opinion that the tidal oscillations of long period might 
 be adequately represented by the equilibrium theory. But Lord Kelvin 
 showed that a yielding of the solid earth would produce a calculable reduction 
 of the oceanic tide according to the equilibrium theory. Accordingly it 
 appeared that a measurement of the actual values of the tides of long 
 period would afford a measure of the elastic yielding of the earth's mass. 
 In this ninth paper an evaluation is made of the ranges of these tides at a 
 number of ports; and thence an estimate is obtained of the tidal yielding of 
 the earth. 
 
 The tenth paper is on the dynamical theory of the tides, being an 
 extract from an unpublished article on Tides for a new edition of the 
 Encyclopaedia Britannica; it is reproduced by the kind permission of the 
 proprietors of that work. My article in the old edition forms the basis of 
 this new article, but Mr Hough's important addition to that theory is now 
 included. 
 
 In the eleventh paper arguments are adduced controverting Laplace's 
 opinion as to the adequacy of the equilibrium theory for discussing the tides 
 of long period, and it is shown that the problem is in fact a dynamical one. 
 Hence the evaluation of the amount of elastic yielding of the earth by 
 means of the observed heights of these tides cannot be accepted as exact. 
 
 The last paper in the first part contains the reduction of the tidal 
 observations made by the officers of the ' Discovery ' in the course of their 
 celebrated antarctic expedition. The volumes of scientific results of the 
 ' Discovery ' are not yet published. 
 
VU1 PREFACE. 
 
 Part II contains two papers on the LUNAR DISTURBANCE OP GRAVITY. 
 They were reports of a Committee to the British Association on an attempt 
 to make direct measurement of the attraction of the moon. The work was 
 carried on at Cambridge in 1880 by my brother Horace and me, with a form 
 of instrument of which the original idea is due to Lord Kelvin. Although 
 the report is written in my name, it should be explained that my brother 
 took the leading part in our experiments and designed all the mechanical 
 appliances used. 
 
 The second of these papers, the fourteenth in the volume, is principally 
 historical, but it contains an investigation of the results which may be 
 expected to result from the varying elastic flexure of the soil under varying 
 superincumbent weights. 
 
 The matters considered in these papers have been the subject of very 
 many experimental investigations during the last twenty-five years, and a 
 large bibliography of seismology would be required merely to enumerate all 
 that has been written on them. I therefore make no attempt to furnish 
 references, and leave the work in the form in which it appeared originally. 
 
 G. H. DARWIN. 
 
 September, 1907. 
 
CONTENTS. 
 
 PAGE 
 
 Chronological List of Papers with References to the Volumes in 
 
 which they probably will be contained ..... xi 
 
 PART I. 
 OCEANIC TIDES. 
 
 ART. 
 
 1. The Harmonic Analysis of Tidal Observations .... 1 
 
 [Report of a Committee, consisting of Professors G. H. DARWIN and 
 J. C. ADAMS, for the Harmonic Analysis of Tidal Observations. British 
 Association Report for 1883, pp. 49118.] 
 
 2. On the Periods Chosen for Harmonic Analysis, and a Com- 
 
 parison with the Older Methods by means of Hour-angles 
 
 and Declinations 70 
 
 [Third Report of the Committee, consisting of Professors G. H. DARWIN 
 and J. C. ADAMS, for the Harmonic Analysis of Tidal Observations. 
 Drawn up by Professor G. H. DARWIN. British Association Report for 
 1885, pp. 3560.] 
 
 3. Datum Levels: the Treatment of a Short Series of Tidal 
 
 Observations and on Tidal Prediction 97 
 
 [Report of the Committee consisting of Professor G. H. DARWIN, Sir W. 
 THOMSON, and Major BAIRD, for the purpose of preparing instructions 
 for the practical work of Tidal Observation ; and Fourth Report of the 
 Committee consisting of Professors G. H. DARWIN and J. C. ADAMS, 
 for the Harmonic Analysis of Tidal Observations. Drawn up by 
 G. H. DARWIN. British Association Report for 1886, pp. 4058.] 
 
 4. A General Article on the Tides 119 
 
 [Article "Tides," Admiralty Scientific Manual (1886), pp. 5391.] 
 
 5. On the Harmonic Analysis of Tidal Observations of High and 
 
 Low Water 157 
 
 [Proceedings of the Royal Society, XLVIII. (1890), pp. 278 340.] 
 
 6. On an Apparatus for Facilitating the Reduction of Tidal 
 
 Observations 216 
 
 [Proceedings of the Royal Society, LII. (1892), pp. 345389.] 
 
 7. On Tidal Prediction 258 
 
 [Bakerian Lecture, Philosophical Transactions of the Royal Society of London, 
 CLXXXII. (1891), A, pp. 159229.] 
 
 8. On the Correction to the Equilibrium Theory of Tides for the 
 
 Continents. I. By G. H. DARWIN. II. By H. H. TURNER . 328 
 
 [Proceedings of the Royal Society of London, XL. (1886), pp. 303315.] 
 
X CONTENTS. 
 
 ART. PAGE 
 
 9. Attempted Evaluation of the Rigidity of the Earth from the 
 
 Tides of Long Period 340 
 
 [This is 848 of the second edition of Thomson and Tait's Natural 
 Philosophy (1883). There have been some changes of notation, so as to 
 make the investigation consistent with the other papers in this volume. 
 Some portions have been omitted, where omissions could be made 
 without any interference with the main result.] 
 
 10. Dynamical Theory of the Tides 347 
 
 [This contains certain sections from the article " TIDES " written in 1906 
 for the new edition of the Encyclopaedia Britannica, being based on the 
 corresponding paragraphs in the original edition and on the article 
 " TIDES " in the supplementary volumes. Reproduced by special per- 
 mission of the Proprietors of the Enc. Brit.~\ 
 
 11. On the Dynamical Theory of the Tides of Long Period . . 366 
 
 [Proceedings of the Royal Society of London, XLI. (1886), pp. 337 342.] 
 
 12. On the Antarctic Tidal Observations of the 'Discovery' . 372 
 
 [To be published hereafter as a contribution to the scientific results of 
 the voyage of the ' Discovery.'] 
 
 PART II. 
 THE LUNAR DISTURBANCE OF GRAVITY. 
 
 13. On an Instrument for Detecting and Measuring Small Changes 
 
 in the Direction of the Force of Gravity. By G. H. DARWIN 
 
 arid HORACE DARWIN 389 
 
 [Report of the Committee, consisting of Mr G. H. DARWIN, Professor 
 Sir WILLIAM THOMSON, Professor TAIT, Professor GRANT, Dr SIEMENS, 
 Professor PURSER, Professor G. FORBES, and Mr HORACE DARWIN, 
 appointed for the Measurement of the Lunar Disturbance of Gravity. 
 This Report is written in the name of G. H. DARWIN merely for the 
 sake of verbal convenience. British Association Report for 1881, 
 pp. 93126.] 
 
 14. The Lunar Disturbance of Gravity; Variations in the Vertical 
 
 due to Elasticity of the Earth's Surface .... 430 
 
 [Second Report of the Committee, consisting of Mr G. H. DARWIN, 
 Professor Sir WILLIAM THOMSON, Professor TAIT, Professor GRANT, 
 Dr SIEMENS, Professor PURSER, Professor G. FORBES, and Mr HORACE 
 DARWIN, appointed for the Measurement of the Lunar Disturbance of 
 Gravity. Written by Mr G. H. DARWIN. British Association Report 
 for 1882, pp. 95119.] 
 
 INDEX 461 
 
 PLATE. Abacus for the harmonic analysis of tidal observations To face 219 
 
CHRONOLOGICAL LIST OF PAPERS WITH REFERENCES TO 
 THE VOLUMES IN WHICH THEY WILL PROBABLY BE 
 CONTAINED. 
 
 Probable volume 
 
 in collected 
 YEAR TITLE AND REFERENCE papers 
 
 1875 On two applications of Peaucellier's cells. London Math. Soc. Proc., IV 
 6, 1875, pp. 113, 114. 
 
 1875 On some proposed forms of slide-rule. London Math. Soc. Proc., 6, IV 
 
 1875, p. 113. 
 1875 The mechanical description of equipotential lines. London Math. Soc. IV 
 
 Proc., 6, 1875, pp. 115117. 
 
 1875 On a mechanical representation of the second elliptic integral. Mes- IV 
 senger of Math., 4, 1875, pp. 113115. 
 
 1875 On maps of the World. Phil. Mag., 50, 1875, pp. 431-444. IV 
 
 1876 On graphical interpolation and integration. Brit. Assoc. Rep., 1876, IV 
 
 p. 13. 
 
 1876 On the influence of geological changes on the Earth's axis of rotation. Ill 
 Roy. Soc. Proc., 25, 1877, pp. 328332 ; Phil. Trans., 167, 1877, 
 pp. 271312. 
 
 1876 On an oversight in the Mecanique Celeste, and on the internal densities III 
 
 of the planets. Astron. Soc. Month. Not., 37, 1877, pp. 77 89. 
 
 1877 A geometrical puzzle. Messenger of Math., 6, 1877, p. 87. IV ^ 
 
 1877 A geometrical illustration of the potential of a distant centre of force. IV 
 Messenger of Math., 6, 1877, pp. 9798. 
 
 1877 Note on the ellipticity of the Earth's strata. Messenger of Math., 6, III 
 1877, pp. 109, 110. 
 
 1877 On graphical interpolation and integration. Messenger of Math., 6, IV 
 1877, pp. 134136. 
 
 1877 On a theorem in spherical harmonic analysis. Messenger of Math., 6, IV 
 1877, pp. 165168. 
 
 1877 On a suggested explanation of the obliquity of planets to their orbits. II 
 
 Phil. Mag., 3, 1877, pp. 188192. 
 
 1877 On fallible measures of variable quantities, and on the treatment of IV 
 
 meteorological observations. Phil. Mag., 4, 1877, pp. 114. 
 
 1878 On Professor Haughton's estimate of geological time. Roy. Soc. Proc., IV 
 
 27, 1878, pp. 179183. 
 
 1878 On the bodily tides of viscous and semi-elastic spheroids, and on the II 
 
 Ocean tides on a yielding nucleus. Roy. Soc. Proc., 27, 1878, 
 pp. 419424; Phil. Trans., 170, 1879, pp. 135. 
 
Xll 
 
 CHRONOLOGICAL LIST OF PAPERS. 
 
 Probable volume 
 
 YEAR TITLE AND REFERENCE ^pel 
 
 1878 On the precession of a viscous spheroid. Brit. Assoc. Rep., 1878, II 
 
 pp. 482485. 
 
 1879 On the precession of a viscous spheroid, and on the remote history of II 
 
 the Earth. Roy. Soc. Proc., 28, 1879, pp. 184194 ; Phil. Trans., 
 
 170, 1879, pp. 447538. 
 1879 Problems connected with the tides of a viscous spheroid. Roy. Soc. II 
 
 Proc., 28, 1879, pp. 194199 ; Phil. Trans., 170, 1879, pp. 539593. 
 1879 Note on Thomson's theory of the tides of an elastic sphere. Messenger II 
 
 of Math., 8, 1879, pp. 2326. 
 
 1879 The determination of the secular effects of tidal friction by a graphical II 
 
 method. Roy. Soc. Proc., 29, 1879, pp. 168181. 
 
 1880 On the secular changes in the elements of the orbit of a satellite II 
 
 revolving about a tidally distorted planet. Roy. Soc. Proc., 30, 
 1880, pp. 110; Phil. Trans., 171, 1880, pp. 713891. 
 
 1880 On the analytical expressions which give the history of a fluid planet of II 
 small viscosity, attended by a single satellite. Roy. Soc. Proc., 30, 
 1880, pp. 255278. 
 
 1880 On the secular effects of tidal friction. Astr. Nachr., 96, 1880, omitted 
 
 col. 217222. 
 
 1881 On the tidal friction of a planet attended by several satellites, and on II 
 
 the evolution of the solar system. Roy. Soc. Proc., 31, 1881, 
 pp. 322325; Phil. Trans., 172, 1881, pp. 491535. 
 
 1881 On the stresses caused in the interior of the Earth by the weight of II 
 continents and mountains. Phil. Trans., 173, 1883, pp. 187 230; 
 Amer. Journ. Sci., 24, 1882, pp. 256269. 
 
 1881 (Together with Horace Darwin.) On an instrument for detecting and I 
 
 measuring small changes in the direction of the force of gravity. 
 Brit. Assoc. Rep., 1881, pp. 93126; Annal. Phys. Chem., Beibl. 6, 
 1882, pp. 5962. 
 
 1882 On variations in the vertical due to elasticity of the Earth's surface. I 
 
 Brit. Assoc. Rep., 1882, pp. 106119; Phil. Mag., 14, 1882, 
 pp. 409427. 
 
 1882 On the method of harmonic analysis used in deducing the numerical omitted 
 values of the tides of long period, and on a misprint in the Tidal 
 Report for 1872. Brit. Assoc. Rep., 1882, pp. 319327. 
 
 1882 A numerical estimate of the rigidity of the. Earth. Brit. Assoc. Rep., I 
 
 1882, pp. 472-474 ; 848, Thomson and Tait's Nat. Phil, second 
 edition. 
 
 1883 Report on the Harmonic analysis of tidal observations. Brit. Assoc. I 
 
 Rep., 1883, pp. 49117. 
 
 1883 On the figure of equilibrium of a planet of heterogeneous density. Ill 
 Roy. Soc. Proc., 36, pp. 158166. 
 
 1883 On the horizontal thrust of a mass of sand. Instit. Civ. Engin. Proc., IV 
 
 71, 1883, pp. 350378. 
 
 1884 On the formation of ripple-mark in sand. Roy. Soc. Proc., 36, 1884, IV 
 
 pp. 1843. 
 
CHRONOLOGICAL LIST OF PAPERS. 
 
 Xlll 
 
 YEAR 
 
 1884 
 
 1885 
 1885 
 
 1885 
 
 1886 
 
 1886 
 1886 
 
 1886 
 1886 
 
 1886 
 1887 
 
 1887 
 
 1888 
 1888 
 
 1889 
 1889 
 1890 
 1891 
 
 TITLE AND REFERENCE 
 
 Second Keport of the Committee, consisting of Professors G. H. Darwin 
 and J. C. Adams, for the harmonic analysis of tidal observations. 
 Drawn up by Professor G. H. Darwin. Brit. Assoc. Rep., 1884, 
 pp. 3335. 
 
 Note on a previous paper. Roy. Soc. Proc., 38, pp. 322 328. 
 
 (Jointly with 
 
 Probable volume 
 
 in collected 
 
 papers 
 
 omitted 
 
 II 
 
 omitted 
 
 Results of the harmonic analysis of tidal observations. 
 A. W. Baird.) Roy. Soc. Proc., 39, pp. 135207. 
 
 Third Report of the Committee, consisting of Professors G. H. Darwin I 
 
 and J. C. Adams, for the harmonic analysis of tidal observations. 
 Drawn up by Professor G. H. Darwin. Brit. Assoc. Rep., 1885, 
 pp. 3560. 
 
 Report of the Committee, consisting of Professor G. H. Darwin, I 
 
 Sir W. Thomson, and Major Baird, for preparing instructions for 
 the practical work of tidal observation ; and Fourth Report of the 
 Committee, consisting of Professors G. H. Darwin and J. C. Adams, 
 for the harmonic analysis of tidal observations. Drawn up by 
 Professor G. H. Darwin. Brit. Assoc. Rep., 1886, pp. 4058. 
 
 Presidential Address. Section A, Mathematical and Physical Science. IV 
 Brit. Assoc. Rep., 1886, pp. 511518. 
 
 On the correction to the equilibrium theory of tides for the continents. I 
 
 i. By G. H. Darwin, n. By H. H. Turner. Roy. Soc. Proc., 40, 
 pp. 303315. 
 
 On Jacobi's figure of equilibrium for a rotating mass of fluid. Roy. Ill 
 Soc. Proc., 41, pp. 319336. 
 
 On the dynamical theory of the tides of long period. Roy. Soc. Proc., I 
 
 41, pp. 337342. 
 
 Article 'Tides.' (Admiralty) Manual of Scientific Inquiry. I 
 
 On figures of equilibrium of rotating masses of fluid. Roy. Soc. Proc., 42, III 
 pp. 359362 ; Phil. Trans., 178A, pp. 379428. 
 
 Note on Mr Davison's Paper on the straining of the Earth's crust in IV 
 cooling. Phil. Trans., 178A, pp. 242249. 
 
 Article ' Tides.' Encyclopaedia Britannica. Certain sections in I 
 
 On the mechanical conditions of a swarm of meteorites, and on theories IV 
 of cosmogony. Roy. Soc. Proc., 45, pp. 3 16; Phil. Trans., 180A, 
 pp. 169. 
 
 Second series of results of the harmonic analysis of tidal observations, omitted 
 Roy. Soc. Proc., 45, pp. 556611. 
 
 Meteorites and the history of Stellar systems. Roy. Inst. Rep., Friday, omitted 
 Jan. 25, 1889. 
 
 On the harmonic analysis of tidal observations of high and low water. I 
 
 Roy. Soc. Proc., 48, pp. 278340. 
 
 On tidal prediction. Bakerian Lecture. Roy. Soc. Proc., 49, pp. 130 I 
 
 133; Phil. Trans., 182A, pp. 159229. 
 
XIV CHRONOLOGICAL LIST OF PAPERS. 
 
 Probable volume 
 
 in collected 
 YEAK TITLE AND REFERENCE papers 
 
 1892 On an apparatus for facilitating the reduction of tidal observations. 1 
 
 Roy. Soc. Proc., 52, pp. 345389. 
 
 1896 On periodic orbits. Brit. Assoc. Rep., 1896, pp. 708, 709. omitted 
 
 1897 Periodic orbits. Acta Mathematica, 21, pp. 101 242, also (with IV 
 
 omission of certain tables of results) Mathem. Annalen, 51, 
 pp. 523583. 
 
 [by S. S. Hough. On certain discontinuities connected with periodic IV 
 orbits. Acta Math., 24 (1901), pp. 257288.] 
 
 1899 The theory of the figure of the Earth carried to the second order of III 
 
 small quantities. Roy. Astron. Soc. Month. Not., 60, pp. 82 124. 
 
 1900 Address delivered by the President, Professor G. H. Darwin, on IV 
 
 presenting the Gold Medal of the Society to M. H. Poincare. 
 Roy. Astron. Soc. Month. Not., 60, pp. 406415. 
 
 1901 Ellipsoidal harmonic analysis. Roy. Soc. Proc., 68, pp. 248252 ; III 
 
 Phil. Trans., 197A, pp. 461557. 
 
 1901 On the pear-shaped figure of equilibrium of a rotating mass of liquid. Ill 
 
 Roy. Soc. Proc., 69, pp. 147, 148; Phil. Trans., 198A, pp. 301331. 
 
 1902 Article ' Tides.' Encyclopaedia Britannica, supplementary volumes. 
 
 Certain sections in I 
 
 1902 The stability of the pear-shaped figure of equilibrium of a rotating mass III 
 
 of liquid. Roy. Soc. Proc., 71, pp. 178183; Phil. Trans., 200A, 
 pp. 251314. 
 
 1903 On the integrals of the squares of ellipsoidal surface harmonic functions. Ill 
 
 Roy. Soc. Proc., 72, p. 492; Phil. Trans., 203A, pp. 111137. 
 
 1903 The approximate determination of the form of Maclaurin's spheroid. Ill 
 Trans. Amer. Math. Soc., 4, pp. 113 133. 
 
 1903 The Eulerian nutation of the Earth's axis. Bull. Acad. Roy. de IV 
 Belgique (Sciences), pp. 147 161. 
 
 1905 The analogy between Lesage's theory of gravitation and the repulsion IV 
 of light. Roy. Soc. Proc., 76A, pp. 387410. 
 
 1905 Address by Professor G. H. Darwin, President. Brit. Assoc. Rep., IV 
 
 1905, pp. 332. 
 
 1906 On the figure and stability of a liquid satellite. Roy. Soc. Proc., 77A, III 
 
 pp. 422425 ; Phil. Trans., 206A, pp. 161248. 
 
 Unpublished Article ' Tides.' Encyclopaedia Britannica, new edition to 
 be published hereafter (by permission of the proprietors). 
 
 Certain sections in I 
 
 Unpublished Article ' Bewegung der Hydrosphare ' (The Tides). Encyklo- IV 
 padie der Mathematischen Wissenschaften, vi. 
 
PART I 
 
 OCEANIC TIDES 
 
1. 
 
 THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [Report of a Committee, consisting of Professors G. H. DARWIN 
 and J. C. ADAMS, for the Harmonic Analysis of Tidal Obser- 
 vations. British Association Report for 1883, pp. 49 118.] 
 
 CONTENTS. 
 
 SECT. PAGE 
 
 PREFACE Account of Operations 2 
 
 1. Notation adopted in the Tidal Reports [A] Schedule of Notation . . 4 
 
 2. Development of the Equilibrium Theory of Tides, with reference to Tidal 
 
 Observations Tide-generating Potential Evection Variation Equili- 
 brium Tide [B i.] [B ii.] [B iii.] Schedules of Lunar Tides [C] Schedule 
 of Solar Tides [D] Schedule of Speeds in Degrees per m. s. hour 
 [E] Schedule of Theoretical Importance ....... 6 
 
 3. Tides depending on the Fourth Power of the Moon's Parallax ... 26 
 
 4. Meteorological Tides, Over-tides, and Compound Tides [F] Schedule of Over- 
 
 tides [G] Schedule of Speeds arising out of Combinations [H] Schedule 
 
 of Compound Tides 28 
 
 5. The Method of Reduction of Tidal Observations Definitions of A, B, R, , 
 
 f, H, V+u, K Treatment of Sidereal Diurnal, and Semi-diurnal Tides, 
 
 K 1} K 2 The Tide L The Tide M x 34 
 
 6. The Method of Computing the Arguments and Coefficients Formulae for 
 
 Computing /, i/, | Mean Values of the Coefficients in Schedules [B] 
 Formulae for Computing f Formulae for s, p, h, pi, N . . . . 40 
 
 7. Summary of Initial Arguments and Factors of Reduction Schedule [I] . 45 
 
 8. On the Reductions of the Published Results of Tidal Analysis [Schedule [J] 
 
 omitted] 48 
 
 9. Description of the Numerical Harmonic Analysis for the Tides of Short 
 
 Period Incidence of Special Hours amongst m. s. hours Schedule [K] 
 Form for entry of Tidal Observations Choice of Special Periods ; 
 [Schedule [L] omitted] Augmenting Factors, Schedule [M] Arrange- 
 ment of Harmonic Analysis Schedule [N], Form for Analysis Schedule 
 [0], Form for Evaluation of C, R, K, H 48 
 
 10. On the Harmonic Analysis for Tides of Long Period Methods of taking 
 
 Daily Means Clearance from Effect of Tides of Short Period Adams' use 
 of Tide Predictor for this end Ten final Equations for Components of 
 Tides Schedule [P] of the Coefficients in the Ten Equations Clearance 
 of Daily Means effected in the Final Equations, Schedule [Q] of Clearance 
 Coefficients Treatment for Gaps in the Series of Observations . . 57 
 
 11. Method of Equivalent Multipliers for the Harmonic Analysis for the Tides 
 
 of Long Period Schedule [R], Form for Reduction 66 
 
 12. Auxiliary Tables drawn up under the superintendence of Major A. W. Baird, 
 
 R.E. [largely abridged, and replaced by formulae] 68 
 
 D. I. 1 
 
HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 Preface. Account of Opwations. 
 
 A COMMITTEE appointed for the examination of the question of the 
 Harmonic Analysis of Tidal Observations practically finds itself engaged 
 in the question of the reduction of Indian Tidal Observations; since it 
 is only in that country that any extensive system of observation with 
 systematic publication of results* exists. 
 
 [Early in 1883] I proceeded to draw up a considerable part of this Report, 
 had it printed, and submitted it to Major Baird f. I was not at that time 
 aware of the extent to which Mr Roberts, of the Nautical Almanac office, 
 co-operated in England in the tidal operations, nor did I know that he was 
 not unfrequently taking the advice of Professor Adams. It was not until 
 Major Baird had read what I had written, and expressed his approval of the 
 methods suggested, that these facts came to my knowledge ; but it must be 
 admitted that it was through my own carelessness that this was so. I then 
 found that Professor Adams decidedly disapproved of the notation adopted, 
 and would have preferred to throw over the notation of the old Reports \ 
 and take a new departure. The notation of the old Reports seems to me 
 also to be unsatisfactory, but, seeing that Major Baird and his staff were 
 already familiar with that notation, I considered that an entire change 
 would be impolitic, and that it was better to allow the greater part of the 
 existing notation to stand, but to introduce modifications. The fact that 
 Major Baird, who was actually to work the method, approved of what 
 had been written, and had already mastered it, went far to prejudge the 
 question, and Professor Adams agreed, after discussion, that it would on the 
 whole be best to allow the work to go on in the lines in which it had been 
 started. 
 
 It has seemed proper to give this account of our operations in order 
 that Professor Adams may be relieved from responsibility for the analytical 
 methods and notation here adopted. I may state, however, that although 
 the Report is drawn up in a form probably differing widely from that which 
 it would have had if Professor Adams had been the author, yet he agrees 
 with the correctness of the methods pursued. I have been in constant 
 communication with him for the past eight months, and have received many 
 valuable criticisms and suggestions. 
 
 Mr Roberts has been supervising the printing of a new edition of the 
 computation forms; they have undergone some modification in accordance 
 
 * [This refers to the year 1883.] 
 
 f [Now Colonel Baird, R.E., F.R.S., at that time the officer in charge at Poona of the Tidal 
 Department of the Survey of India.] 
 
 J [These are the Reports to the British Association for the years 1868, 1870, 1871, 1872, 1876.] 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 3 
 
 with this Report. He has also computed certain new coefficients [Schedule Q] 
 which are required in the reductions. 
 
 The general scope of this paper is to form a manual for the reduction 
 of tidal observations by the Harmonic Analysis inaugurated by Sir William 
 Thomson, and carried out by the previous Committee of the British 
 Association *. 
 
 In the present Report the method of mathematical treatment differs 
 considerably from that of Sir William Thomson f. In particular, he has 
 followed, and extended to the diurnal tides, Laplace's method of referring 
 each tide to the motion of an astre fictif 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 fictif s, 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. 
 
 The Report of 1876 was not intended to be a final production, and it 
 did not contain any complete explanation of a considerable portion of the 
 numerical operations of the Harmonic Analysis. The present Report is 
 intended to systematise the exposition of the theory of the harmonic analysis, 
 to complete the methods of reduction, and to explain the whole process. 
 
 A careful survey of the methods hitherto in use has brought to light a 
 good many minor points in which improvements may be introduced, but it 
 has seemed desirable not to disturb the system, which is in working order, 
 more than can be helped. It has also appeared that the published results 
 have not been arranged in a form which lends itself to a satisfactory exami- 
 nation of the whole method. This defect will, we hope, now be remedied. 
 
 The first section refers to the notation, and contains a schedule of nomen- 
 clature by initials of the several tides under examination. The schedule is 
 not, strictly speaking, in its proper position at the beginning, because it 
 involves the results of subsequent analysis, but the advantage gained by 
 having this list in a position of easy reference seems to outweigh the want 
 of logic. 
 
 * See especially the Reports for 1872 and 1876. 
 
 t The present method of development is that pursued in a paper in the Phil. Trans. R.S., 
 Part II. 1880, p. 713. [To be included in Vol. n. of these collected papers.] 
 
 12 
 
4 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 The forms for computation are privately printed for the India Office, 
 and are therefore inaccessible to the public*. The type has been broken up, 
 and very few copies remain, but we shall send copies to the Libraries of 
 the following Societies, viz. : Royal Societies, London and Edinburgh ; the 
 Academies of Science of Dublin, Paris, Berlin, and Vienna, the Coast Survey 
 of the United States at Washington, and the Cambridge Philosophical 
 Society. 
 
 G. H. DARWIN. 
 
 1. The Notation adopted in the Tidal Reports. 
 
 In considering the notation to be adopted, much weight should be given 
 to the fact that a large mass of analysis and computation already exists in 
 a certain form. We have not thus got a tabula rasa to work on, but had 
 better accept a good deal that has grown up by a process of accretion. It 
 is certainly unfortunate that a dual system should have been adopted, in 
 which one set of letters are derived from the Greek and another from the 
 English. 
 
 The letters 7, <r, t), -GT are appropriated respectively to the earth's angular 
 velocity of rotation, to the mean motions of the moon, sun, and lunar perigee. 
 They form the initial letters of the words 777, O-\^VTJ, 77X109, and perigee. 
 There is also &>, derived from the obliquity of the ecliptic. In another 
 category we have M, S, E, for the masses of the moon, sun, and earth. It 
 is unfortunate that the letter 8 should thus be connected with the moon 
 in a- ; but it has not been thought advisable to change the notation in this 
 matter. In this Report the already existing notation is adhered to, as far 
 as might be without inconvenience ; but it must be admitted that the 
 notation is by no means satisfactory. 
 
 It is a matter of great practical utility to have a symbol for indicating 
 special tides. In the endeavour to meet this want initial letters were 
 assigned in the former Reports to each kind of tide ; but, except in the case 
 of M and S, for the principal ' moon ' and ' sun ' tides, the initials had no 
 connection with the tide. Although a new system of initials might be 
 devised which would have a direct connection with the tides to which they 
 refer, yet it has appeared best to adhere to the old initials and to introduce 
 certain new initials for the tides of long period and for some tides now 
 considered for the first time. 
 
 * [Other methods of effecting the computations have been devised of which one is described 
 in a subsequent paper in this volume " On an Apparatus, Ac." The original method is still in 
 use in India. ] 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [A.] Schedule of Notation. 
 
 Initials 
 
 Speed 
 
 Name of Tide 
 
 M, 
 
 M 2 
 M 3 
 &c. 
 
 y <r w, and 
 
 2(y"-o-T 
 3(y-<r) 
 &C. 
 
 Principal lunar series 
 
 K 2 
 
 2y 
 
 Luni-solar semi-diurnal 
 
 N 
 
 2y-3o- + or 
 
 Larger lunar elliptic 
 
 L 
 
 2y tr w and 
 
 Smaller lunar elliptic 
 
 2N 
 
 2y 4<r + 2 or 
 
 Lunar elliptic, second order 
 
 V 
 
 2y-3o--ar + 2^ 
 
 Larger lunar evectional 
 
 X 
 
 2y <r + <zr 2?7 
 
 Smaller lunar evectional 
 
 
 
 y-2o- 
 
 Lunar diurnal 
 
 00 
 
 y + 2o- 
 
 
 K, 
 
 7 
 
 Luni-solar diurnal 
 
 Q 
 
 y-Str + or 
 
 Larger lunar elliptic diurnal 
 
 
 y tr or 
 included in M t 
 
 Smaller lunar elliptic diurnal 
 
 J 
 
 y + o--or 
 
 
 
 y 4ir + 2 or 
 
 Lunar elliptic diurnal, second order 
 
 
 y 3<r zzr + 27 
 
 Larger lunar evectional diurnal 
 
 8, 
 
 y-ij 
 
 
 & 
 
 2 (y 77) 
 
 Principal solar series 
 
 &c. 
 
 &c. 
 
 1 
 
 T 
 
 2y-3 v 
 
 Larger solar elliptic 
 
 R 
 
 2y- 9 
 
 Smaller solar elliptic 
 
 P 
 
 y 2^ 
 
 Solar diurnal 
 
 Mm 
 
 o" OT 
 
 Lunar monthly 
 
 Mf 
 
 2o- 
 
 Lunar fortnightly 
 
 Sa 
 
 7; 
 
 Solar annual 
 
 Ssa 
 
 >5 
 
 Solar semi-annual 
 
 MSf 
 
 2(0-^) 
 
 Luni-solar synodic fortnightly 
 
 MS 
 
 4y-2o--2? 
 
 
 p. or 2MS 
 
 2y- 4<r + 2,7 
 
 
 2SM 
 MK 
 
 3y - 2<r 
 
 Compound tides 
 
 2MK 
 
 3y-4<r 
 
 
 MN 
 
 4y 5o~ + 07 
 
 ' 
 
6 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 In the old notation the L tide was simply the. tide of speed 2y a- in. 
 The values of this tide have probably been perturbed by another tide of 
 speed 2y a- + &, and this tide is supposed also to be included in L. 
 
 Where it is necessary to refer to any other tides -than those contained 
 in this schedule, it will be best to use the scientific nomenclature simply 
 by speed. For example, there may be a compound tide 87 2ij ; and 
 though this tide might be called SK, since 87 2rj = 2 (7 77) + 7, yet 
 reference to such a tide will be so infrequent as not to make the short 
 notation desirable. 
 
 Both the old and the new initials are given in the preceding schedule [A]. 
 
 2. Development of the Equilibrium Theory of Tides with reference 
 
 to Tidal Observations. 
 
 The first step is the formation of the tide-generating potential of the 
 moon ; that for the sun may then be written down by symmetry. 
 
 For the purpose we require to find certain spherical harmonic functions 
 of the moon's coordinates, with reference to axes fixed in the earth. 
 
 Let A, B, C (Fig. 1) be such axes, C being the 
 north pole and AB the equator. 
 
 Let X, Y, Z be a second set of axes, XY being 
 the plane of the moon's orbit. 
 
 Let M be the projection of the moon in her 
 orbit. 
 
 Let / = ZC, the obliquity of the lunar orbit to 
 the equator. 
 
 Let = AX = BOY. 
 
 FIG. 1. 
 
 Let I = MX, the moon's longitude in her orbit, measured from X. 
 
 Let 
 
 Then 
 
 = cos 
 
 ^i ^, .. 
 
 the moon s direction-cosines 
 
 M 2 = cos MB with reference to ABC 
 M 3 = cos MCJ 
 
 .(1) 
 
 of 
 
 M l = cos I cos % + sin I sin % cos / 
 
 M o = cos sin y + sin cos y cos J> (2) 
 
 * * 
 
 Jl/3 = sin I sin / ] 
 
 We may observe that JHf 2 is derivable from M 1 by putting ^ + |TT in place 
 
 % 
 Now for brevity let 
 
 p = cos 
 
 q = sin 
 
 .(3) 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 Then (2) may be written 
 
 J/j = jp 2 cos (% + <? cos (% + 0" 
 
 J/2 = p 2 sin (x 1) q 2 sin (% + I) 
 M.., = 2pq sin I 
 Whence 
 
 ! _ Mi* = p* cos 2 (% + 2p 2 5 2 cos 2^; + <? 4 cos 2 (% + I) 
 IM^M^ = the same with sines in place of cosines 
 M 2 M 3 = - p 3 q cos (x - 2/) + pq(p*- q 2 ) cos ^ + pq 3 cos (% + 20 
 = the same with sines in place of cosines 
 
 .(4) 
 
 _ - cos 
 
 y (5) 
 
 These are the required spherical harmonic functions of M l} M 2 , M s . 
 
 Let M denote the projection of the moon on the celestial sphere concentric 
 with the earth, and P that of any other point. 
 
 Let r, p be the radius- vectors of the moon and of P respectively, and let 
 f, rj, be the direction-cosines of P, with reference to the axes A, B, C. 
 
 Then p%, prj, p are the coordinates of P, and rM lt rM*, rM 3 those of M. 
 
 If M be the moon's mass, and p, the attraction between unit masses at 
 unit distance apart, then by the usual theory the tide-generating potential 
 V, due to the moon, of the second order of harmonics, at the point P, is 
 given by 
 
 (6) 
 
 But since cos PM = 
 
 ml M 2 M 
 
 2 
 
 cos 
 
 ' PM - 1 = 2fr Jf A + 2 
 
 
 Now let c be the moon's mean distance, e the eccentricity of the moon's 
 orbit, and let 
 
 Then putting 
 
 we have 
 
8 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 A simple tide may be defined as a spherical harmonic deformation of 
 the waters of the ocean which executes a simple harmonic motion in time. 
 Corresponding to this definition the expression for each term of the tide- 
 generating potential should consist of a solid spherical harmonic, multiplied 
 by a simple time-harmonic. 
 
 In (10) p 2 ?7, /a 2 ( 2 i? 2 ), &c., are solid spherical harmonics, and in order 
 to complete the expression for V it is necessary to develop the five functions 
 of X, Y, Z in a series of simple time-harmonics. 
 
 It will be now convenient to introduce certain auxiliary functions, 
 namely 
 
 [/-j 2\~1 ' 
 
 H^ J [ 
 
 r ,, rwi-^T (U> 
 
 W (a) = cos a, R = 
 
 L r J L r J 
 
 Then from (5) and (9) we have 
 
 X 2 F 2 = p*3> ( 2v) + 2p 2 <2 2> J r (2y) 
 2XY= the same with Y + Z-TT for ^ 
 
 (12) 
 
 XZ = the same with ^ ^TT for ^ 
 ^ (X 2 + F 2 2 2 ) = | (p 4 4p 2 g 2 + ^ 4 ) R + 2p 2 g 2 O (0) 
 
 Thus when the functions <E>, W, R are developed as a series of time- 
 harmonics, the further development of the X-Y-Z functions consists in 
 substitution in (12). 
 
 It will now be supposed that the moon moves in an elliptic orbit, un- 
 disturbed by the sun. The tides which arise from the lunar inequalities 
 of the Evection and Variation will be the subject of separate treatment 
 below. 
 
 The descending node of the equator on the lunar orbit will henceforth 
 be called ' the Intersection.' 
 
 Let <7, be the moon's mean longitude measured in her orbit from the 
 intersection, and r y the longitude of the perigee measured in the same 
 way. It has been already defined that I is the moon's longitude in her orbit 
 measured from the intersection. 
 
 The equation of the ellipse described by the moon is 
 
 r n _ (#\ 
 
 -^=l+ e cos(J-O (13) 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 Hence 
 
 R = 1 + | e 2 _j_ gg cos (i _ CT ^ _|_ | 6 2 cos 2(1 - OT X ) + . . . 
 3> (a) = R cos (21 + a) 
 
 = (1 + f e 2 ) cos (21 + a) + f e [cos (3Z + a - *7,) + cos (Z + a + *7,)] } (14) 
 
 + f e 2 [cos (4 + a - 2*7,) + cos (a + 2*7,)] + . . . 
 (a) = R cos a 
 
 By the theory of elliptic motion 
 
 I = o-, + 2e sin (o-, - *r,) + f e 2 sin 2 (<7, - *7,) + (15) 
 
 In order to expand <, ^ R in terms of <7, (which increases uniformly 
 with the time), we require cos (21 + a) developed as far as e-; cos (3/ + a 7,), 
 and cos (I + a + *r,), as far as e ; and only the first term of cos (4>l + a 2*7,). 
 
 Substituting for I its value (15) in terms of a;, it is easy to show that 
 cos (21 + a) = (1 - 4e 2 ) cos (2^ + a) - 2e cos (<r t + a + OT,) + 2e cos (3o-, + a - cr / ) 
 
 + f e 2 cos (a + 2*0 + J^ 2 cos (4^ + a - 2r,) + . . . 
 cos (31 + a- tff) = cos (So-, + a - -BT^ 
 
 - 3e cos (2<r y + a) + 3<? cos (4<r x + a - 2*7^ + . . . 
 
 cos (I + a + tzrj = cos (cr + a + -srj + e cos (2^ + a) e cos (a + 2-cO + . . . 
 cos (4 + a 2-57,) = cos (4^ + a 2*0 + . . . 
 
 Substituting these values in (14) we find, 
 <l> (a) = (1 -^-e 2 ) cos (2o- / + a) \e cos (<r / + a + *r,) 
 
 ' cos (3cr / + a *r x ) + J^-e 2 cos (4o- 7 + a 207^ + . . . 
 
 (a) = (1 fe 2 ) cos a + \e [cos (^ + a ta-^ + cos (<r, a *rj 
 
 + |e 2 [cos (2o- 7 + a - 2r,) + cos (2^ - a - 2*7,)] + . . . 
 Now substituting from (16) in (12), giving to a its appropriate value, 
 
 (16) 
 
 we have 
 
 X 2 - F 2 = (1 - 
 
 cos 2 ( x - <r,) + ^ cos 2 ( % + 
 
 \e [p* cos 
 
 cos 
 
 - e [p 4 cos (2% - o-, - *7,) + 2 4 cos (2^ + a, + **,)] [ (17) 
 
 + f e 2p 2 2 [cos (2% + <r / CT,) + cos (2^ cr / + *7 y )] 
 
 ' cos (2^ 4(7, + 2*7,)+ q 4 cos (2^ -f 4*7, 2*7j 
 [cos (2^; + 2(7, 2*7,) + cos (2^ 2<7 7 + 2*7,) 
 
 Clearly 2JTFis the same as (17) with sines in place of cosines. Also 
 since YZ is the same as X* F 2 when ^ replaces 2%, p s q replaces p 4 , 
 
10 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 (18) 
 
 pq (p 2 g 2 ) replaces 2p*q*, and pq 3 replaces q*, and since XZ is the same as 
 YZ with sines in place of cosines, we have from (17) 
 
 XZ = -(l- - l e*) [p s q sin (% - 2o-,) -pq 3 sin ( x + 2<r,)] 
 
 + (1 - fe 2 ) ><? (p 2 g 2 ) sin ^ 
 
 f e [p 3 q sin (% 3(7, + CT,) pg 3 sin (^ + 3cr, GJ-,)] 
 -f- \Q [p 3 Q sin (v o- -BT ) o :1 sin (v + o- + CT)] 
 
 - I_t i \/V / / ' * -* ' V / // J 
 
 + f epg (p 2 - q 2 ) [sin (% -f o-, - r y ) + sin ( X - *, + tsr,)] 
 - JgV [p 3 ? sin (% - 4o-, + 2ra-,) -pq 3 sin (% + 4o-, - 2w,)] 
 + ^e 2 </ ( 2 o 2 ) [sin ( v + 2o- 2-57 ) 4- sin ( v 2o- 7 + 2tn- ) 
 
 4i3.\* i/L \ /V / * ' * /V / / ' - 
 
 Lastly, 
 
 cos 20-, + 
 
 = HP 4 - V? 2 
 [(1 - 
 
 F 2 - 2^ 2 = 4 - 2 + 4 ) [(1 - I* 2 ) + 3e cos (^ - r,) 
 
 + f e 2 cos 2 (o-, - -or,)] 
 
 cos (So-, - -cr ( ) - \e cos (o- x + r y ) 
 (4<r / -2O] ............... (19) 
 
 Hitherto no approximation has been admitted with regard to /, the 
 obliquity of the lunar orbit to the equator. 
 
 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 
 / is 23 27''3, is "203, and its square is '041, and its cube '0084. The 
 eccentricity of the lunar orbit e = '0549 ; hence q 2 is a little smaller than e. 
 
 The preceding developments have been carried as far as e 2 , principally 
 on account of the terms involving -^-e 2 , which, as e is about T ^, have nearly 
 the same magnitude as if the coefficient had been \e. 
 
 It is proposed, then, to regard q 2 and q 3 as of the same order as e, and 
 to drop all terms of the order e 2 , 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 j ust sufficiently large to make it worth retaining. 
 
 Adopting this approximation, we may write (17), (18), (19), thus, 
 
 X 2 - F 8 = (1 - 
 
 e 2 )^ 4 cos 2 ( x - o-,) + (1 - fe 2 ) 
 (2^-3(7, + 1:7,) 
 
 cos 2% 
 
 cos 
 
 - r - r - 2 cos - 
 
 XZ=-(l- 
 + (1 - 
 
 cos (2^ - 4(7, + 2^,) 
 e 2 ) [p 3 q sin ( x - 20-,) -pq 3 sin 
 
 - q 2 ) sin % f 
 
 - <r, - -ar,) + 
 
 sn 
 
 sn 
 
 ~ sn 
 
 ^ + 2<r,)] 
 % 3<7, + -BT,) 
 sin (% - (7, + 
 in - 4<r 
 
 V (20) 
 
 -f F 2 - 2Z 2 = 
 
 + 9*) [(1 - |e 2 ) + 3e cos (o-, - r y )] 
 [(1 - -V-e 2 ) cos 2o- / + \e cos (3(7, - -cr,)] 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 11 
 
 The terms which have been retained in violation of the rule of approxi- 
 mation are that in X' 2 Y* with argument 2^ <r t -\-vf t , and that in 
 ( X' 2 + Y* - 2Z' 2 ) with argument 3<r, - r, . 
 
 The only other term which could have any importance is 
 f e 2p*q 2 cos (2 X + <7, - ,) in X 2 - F 2 
 
 Before proceeding to consider the tides due to lunar inequalities it will 
 be well to consider two pairs of terms in the expressions (20). 
 
 First, in X z F 2 we have the terms 
 
 - \ef- [p 2 cos (2% - (T t - 1*) - 6^ 2 cos (2^ - <r, + &,)] 
 The expression within [ ] may be put in the form 
 (p 2 - 6<? 2 cos 2r,) cos (2% - <r, - -or,) + Gq* sin 2^ sin (2^ - <r, - -ar,) 
 
 If then we write tan fl = . ., ?> ^ .................... (20*) 
 
 cot 2 / cos 2cr / 
 
 this pair of terms becomes 
 
 - \ep* V{1 - 12 tan 2 7 cos 2r,+ 36 tan 4 f/} cos (2 X - <r, - r, - #) . . .(20")* 
 
 [If we attribute to / its mean value 23 27''3 the square root becomes 
 V{1'066 - 12 tan 2 7 cos 2wJ.] 
 
 Secondly, in XZ we have the terms 
 
 + te [P 2 sin (% - 0-, - OT/ ) + 3 ( p* - q*) sin ( x - <r t + &,)] 
 [This is equal to 
 
 + ^ep s q [(4 3 tan 2 /) cos Ts t sin (^ a-) + (2 3 tan 2 7) sin or, cos (^ o-J] 
 
 1 _ s ^ an 2 1 j 
 
 If then we write tan Q = A - - 1 - -^-y tan is, 
 
 1 f tan 2 1/ 
 
 and if we attribute to I its mean value 23 27''3 the terms become 
 
 ep 3 q V }2'311 + 1-436 cos 2w 7 } sin ( x - <r, + Q) ......... (20 m ) 
 
 here tan Q = -483 tan r / ........................... (20 iv ) 
 
 w 
 
 In the original paper tan 2 ^/ was neglected, and the numbers which 
 appear here as 2'311, T436 and '483 were taken respectively as f, | and .] 
 
 The object of the transformations (20"), (20 iv ), 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 2^ a- / w / , from one whose argument is 
 2p o^ + -or,, nor one whose argument is ^ <j t -s^, from one whose 
 
 * [The term 3fi tan 4 \ I ranges between \ and T V> an< i ^ i 8 neglected in the original paper.] 
 
12 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 argument is % <r t + -CT,. In fact, the tide with argument 2^ <r, + -BT, (for 
 which no analysis has been as yet carried out) will only produce an 
 irregularity in that of argument 2^ <r / -!jr / , 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 ^ a t TV, will appear in the 
 harmonic analysis with the single argument % o",, and the resulting 
 numbers will necessarily appear very irregular, unless compared with the 
 theoretical expression (20 m ). 
 
 We will now consider the terms introduced by the two principal lunar 
 inequalities due to the disturbing action of the sun. 
 
 The Evection. 
 
 Let 6 be the moon's longitude in the ecliptic. 
 s the moon's mean longitude. 
 p the mean longitude of the perigee *. 
 h the sun's mean longitude. 
 
 m the ratio of the sun's to the moon's mean motion. 
 Then that inequality in longitude and radius vector is represented by 
 
 ..................... (21) 
 
 (22) 
 
 If we neglect the distinction between longitudes in the orbit and in the 
 ecliptic [which is in effect neglecting a term with coefficient sin 2 ( x 5 9')], 
 we have from (21), 
 
 I = (r/ + ^-me sin (s 2h+p) 
 whence 
 
 cos (21 + a) = cos (2(7, + a) 4- -\?-me [cos (2<r, + s - 2h+ p + a.) 
 
 - cos (2(7-, - s + 2h - p + a)] 
 
 And from (22) and the definitions of R, , <J> in (11), 
 
 (23) 
 
 (a) = cos a + $%me [cos (s - 2h + p + a) + cos (s - 2h + p - a)] ...... (24) 
 
 3> (a) = cos (2<r y 4- a) + ^f-me cos (2<r, + s - 2h + p + a) 
 
 - fflne cos (2o-, - s + 2h - p + a) ...... (25) 
 
 * p in this sense will easily be distinguished from the p used to denote cos \I which latter 
 will, moreover, be shortly discarded. 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 13 
 
 Then substituting from (23), (24), (25), in (12), and dropping the terms 
 which are merely a reproduction of those already obtained, and neglecting 
 terms in ( and q 3 , we have 
 
 X 2 - F 2 = ijgmep* cos (2 X - 20-, - s + 2h - p) 
 
 cos (2^; 2o- 7 + s 2h + p) 
 
 q 2 ) [sin (^ + s 
 
 F 2 - 
 
 I sin (<% 2c-, + s 2h+p) 
 + p) + sin (x s + 2h -p)] 
 p-me cos (s 2 
 
 It must be noticed that ^-me arises by the addition of the coefficient 
 of the Evection in longitude to three halves of that in the reciprocal of 
 the radius vector ; that yf we is the difference of the same two quantities ; 
 and that 4g-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 c/r '010022. 
 
 The ratio m of the mean motions is about y 1 ^, and is therefore a little 
 greater than e, hence me is somewhat greater than e 2 . Thus we may 
 abridge (25), and write the expressions thus: 
 
 X*-Y*= &j>-mep* cos ( 2 X - 2<r, - s + 2h - p) 
 
 - |f mep* cos (2 X - 2<7, + s - 2A + p) 
 XZ = - - I {g-mep 3 q sin (x - 2<r, - s + 2h - p) 
 F 2 - 2 2 ) = ! (P 4 ~ V? 2 + 2 4 ) * me cos - 2/i + P) 
 
 ...(26) 
 
 The equations (26) contain the terms to be added to (20) on account of 
 the Evection. 
 
 The Variation. 
 
 Treating this inequality in the same way as the Evection, we have 
 I = <r, + Jg'-ra 2 sin 2 (s - h) 
 
 2 
 
 R = 1 + 3m 2 cos 2(s-h) 
 
 V (a) = cos a + fm 2 [cos {2 (s - h) + a} + cos (2 (* - h) - a}] 
 
 <J> (a) = cos (20-, + a) + ^m 2 cos (2*, + 2s - 2h + a) + m 2 cos (2<r t ~ 
 
14 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 Whence we have to a sufficient degree of approximation, 
 
 X s - F 2 = -f-my cos (2 X - 2(7, - 2s + 2/0, XZ = 
 
 + F 2 - 2 2 ) = i (;> 4 - 4py + g 4 ) 3m 2 cos (2s - 2h) 
 
 In this case also the values of the coefficients are actually considerably 
 greater than the amounts as computed from the first terms; and regard 
 must be paid to this, as in the case of the Evection, when the values of 
 the coefficients in the tidal expressions are computed. According to Professor 
 Adams, the full values of the coefficients are, in longitude '011489, and in 
 c/r -008249. 
 
 We have now obtained in (20), (26), (27), the complete expressions for 
 the X-Y-Z functions in the shape of a series of simple time-harmonics; but 
 they are not yet in a form in which the ordinary astronomical formulae are- 
 applicable. 
 
 Further substitutions will now be made, and we shall pass from the 
 potential to the height of tide generated by the forces corresponding to that 
 potential. 
 
 The axes fixed in the earth may be taken to have their extremities as 
 follows : 
 
 The axis A on the equator in the meridian of the place of observation 
 of the tides ; the axis B in the equator 90 east of A ; the axis C at the 
 north pole. 
 
 Now , r), % are the direction-cosines of the place of observation, and if X 
 be the latitude of that place, we have 
 
 = cos X, rj = 0, = sin X 
 Thus 
 > _ if = cos 2 X, 
 
 Then writing a for the earth's radius, the expression (10) for V at the 
 place of observation becomes 
 
 V = ra \ ft cos 2 X (Z 2 - 7") + sin 2\XZ + f ( - sin 2 X) %(X* + F 2 - 2 2 )] 
 (1 e ) 
 
 The X-Y-Z functions being simple time-harmonics, the principle of 
 forced vibrations allows us to conclude that the forces corresponding to V 
 will generate oscillations in the ocean of the same periods and types as the 
 terms in V, but of unknown amplitudes and phases. 
 
 Now let X' 2 -|f, %:, H* 2 +IF- 2 ^ 2 ) be three functions, having 
 respectively similar forms to those of 
 
 Z 2 -F a XZ , (Z*+F'-2ff t ) 
 
 (1 - e 2 ) 3 ' (1 - e 2 ) 3 (1 - e 2 ) 3 
 
 but differing from them in that the argument of each of the simple time- 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 15 
 
 harmonics has some angle subtracted from it, and that the term is multiplied 
 by a numerical factor. 
 
 Then if g be gravity, and h the height of tide at the place of observation, 
 we must have 
 
 h = [ cos 2 X ( 2 - W] + sin 2XX5S + f ( - sin 2 X) | ( 2 + B 2 
 
 ......... (28) 
 
 The factor - - may be more conveniently written $ -s ( - ) a. where E is 
 g ' JS \C/ 
 
 the earth's mass. It has been so chosen that if the equilibrium theory of 
 tides were fulfilled, with water covering the whole earth, the numerical 
 factors in the 36-|9-5& functions would be each unity. The alterations of 
 phase would also be zero, or, with land and sea as in reality, they might be 
 computed by means of the five definite integrals involved in Sir William 
 Thomson's amended equilibrium theory of tides *. 
 
 The actual results of tidal analysis at any place are intended (see below, 
 5) to be presented in a series of terms of the form fH cos (V+ u K), 
 where dV/dt or n, 'the speed,' is the rate of increase of the argument per 
 unit time (say degrees per mean solar hour), and u is a constant. We 
 require, therefore, to present all the terms of the X-|9-i2j functions as cosines 
 with a positive sign. When, then, in these functions we meet with a 
 negative cosine we must change its sign and add TT to the argument; as 
 the X5S functions involve sines, we must add ^TT to arguments of the 
 negative sines, and subtract TT from the arguments of the positive sines, 
 and replace sines by cosines. The terms in the J ( 2 + |9 2 25%*) function 
 require special consideration. The function of the latitude being ^ sin 2 X, 
 it follows that when in the northern hemisphere it is high-water north of a 
 certain critical latitude, it is low water on the opposite side of that parallel ; 
 and the same is true of the southern hemisphere. The critical latitude is 
 that in which sin 2 X=^, or in Thomson's amended equilibrium* theory, 
 where sin 2 X = (1 + IE). An approximate evaluation of 3E, which depends 
 on the distribution of land and sea, given in 848 of the second edition of 
 Thomson and Tait's Natural Philosophy, shows that the critical latitudes 
 are 35 N. and S. It will be best to adopt a uniform system for the whole 
 earth, and to regard high-tide and high-water as consentaneous in the 
 equatorial belt, and of opposite meanings outside the critical latitudes. 
 In this Report we conceive the function always to be written J - sin 2 X, so 
 that outside the critical latitudes high-tide is low-water. Accordingly we 
 must add TT to the arguments of the negative cosines (if any) which occur in 
 the function J ( 2 + ff - 2$- 2 ). 
 
 * Thomson and Tait's Nat. Phil., or the Report on Tides for 1876. [See also Paper 9 in this 
 volume.] 
 
16 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 In continuing the development, the X-TB-jZs functions will be written 
 in the form appropriate to the equilibrium theory, with water covering 
 the whole earth ; for the actual case it is only necessary to multiply by the 
 reducing factor, and to subtract the phase alteration K. As these are un- 
 known constants for each place, they would only occur in the development 
 as symbols of quantities to be deduced from observation. It will be under- 
 stood, therefore, that in the following schedules ' the argument ' is that part 
 of the argument which is derived from theory, the true complete argument 
 being 'the argument' K, where K is derived from observation. 
 
 Following the plan suggested, and collecting results from (20), (26), (27), 
 we have 
 
 cos 2(x~ <O + (1 + |e 2 ) 2j3 2 ? 2 cos 2% 
 
 + fep* cos (2% - So-, + r,) 
 
 \ep V{1-066 - 12 tan 2 7 cos 2rJ cos (2 X - o-, - -sr, - .R + TT) 
 cos (2% - la, + 2r,) 
 
 cos (2% - 2o-, - s + 2h -p) 
 * cos (2 X - 2o- 7 + s - 2h + p + TT) 
 
 o- / -25 + 2A) ................................. (29) 
 
 = (!- &*) [p s q cos ( X - 2<r, + ^) +pq* cos ( x + 2<r, - |TT)] 
 + (1 + %e*)pq (p 2 - q*) cos ( x - ITT) 
 + \efq COS ( X - 3<r, + tr y + ^TT) 
 
 1'44 cos 2r,} cos (% - ^ + Q - ^?r) 
 
 2 - 2 s ) cos (% + "/ - "/ - i*^) 
 cos (x - 4o- 7 + 2r, + ^TT) 
 
 cos (%- 2^-^ + 2/1-^ + ^) ........................ (30) 
 
 pY + 5 4 ) [1 + |er' + 3e cos (<r, - *,) 
 -me cos (s 2A +p) + '3m 2 cos (2s 2/i)] 
 
 [(1 - f e 2 ) cos 2^ + 1 cos (3^ -or,)] ...... (31) 
 
 In these expressions 
 
 sin 2w 
 tan 7t = i - . , . r = , tan Q = '483 tan ta 
 
 COt 2 l COS 2-57 
 
 The next step is to express the angles X , a,, is,, each of which increases 
 uniformly with the time, in terms of the sidereal hour-angle "or of the local 
 mean time, and of the mean longitudes of the moon, and of the perigee. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 17 
 
 Let M be the moon in the orbit. A the extremity of the A-axis fixed 
 in the earth. 
 
 g be the sidereal hour-angle. 
 N the longitude of the node S3 . 
 v the right ascension of the intersection I. 
 the longitude 'in the moon's orbit' of the intersection. 
 * the inclination of the moon's orbit to the ecliptic, 
 w the obliquity of the ecliptic. 
 s the moon's mean longitude. 
 p the mean longitude of the perigee *. 
 
 Ecliptic 
 
 FIG. 2. 
 
 Then (Fig. 2) g = AT, i; = Tl, = V8 - 8l, N=V& 
 
 Now o-, and v l have been defined above as the moon's mean longitude 
 and the longitude of the perigee, both measured in the orbit from the 
 intersection I. 
 
 Since o- / Ts t is the moon's mean anomaly, we have 
 
 S p = (T t TS t 
 
 Let p' be the longitude of the perigee, measured from V in the ecliptic. 
 
 If P in Fig. 2 be perigee, we have by the ordinary formula for reduction 
 to the ecliptic, 
 
 QP = p' -N+ sin 2 i sin 2 (p - N) 
 
 But r i = IP = l8 + 8P = T8 -^+ 8P 
 
 P ~ % + \ sin 2 i si* 1 2 (p N} 
 
 Now p = p' + s in 2 i sin 2 (p' - JV) 
 
 and therefore ^,~P ~%\ 
 
 whence a- / = s g / 
 
 Again ^ = IA = AT - IT = g - v (33) 
 
 In this formula we suppose g to increase uniformly from the time when 
 the tidal observations begin. 
 
 * This p will easily be distinguished from the p used above to denote cos 1. 
 D. I. 9 
 
18 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 Since in all the tidal observations local mean solar time is used, it will 
 be better to substitute for g in terms of local mean solar time and the sun's 
 mean longitude. Let t be local mean solar time reduced to angle, so that 
 at noon t = 0. Let h be the sun's mean longitude ; hereafter we shall write 
 p l for the longitude of the sun's perigee. 
 
 Then we have ^ = t + h v (34) 
 
 We shall now substitute from (32) and (34) in the K-'jj)-j& functions 
 (29), (30), (31) ; substitute from them in (28), and express the final result 
 in the form of three schedules (pp. 20, 21, 22). 
 
 The schedules are arranged thus. First, there is the general coefficient 
 
 M fa\ s 
 % ~F ( ~ I a w hi cn multiplies every term of all the schedules. Secondly, there 
 
 are general coefficients one for each schedule, viz. cos 2 A. for the semi-diurnal 
 terms, sin 2A, for the diurnal, and J f sin 2 A, for the terms of long period. 
 These three functions of the latitude of the place of observation are the 
 values at that place of three surface spherical harmonic functions, which 
 functions have the maximum value unity, at the equator for the semi- 
 diurnal, in latitude 45 for the diurnal, and at the pole for the terms of long 
 period. 
 
 First, in each schedule there is a column of coefficients, functions of / and 
 e (and in two cases also of p). 
 
 In the second column is given the mean semi -range of the corresponding 
 term. This is approximately the value of the coefficient in the first column 
 when / = &>. We forestall results given below so far as to state that the 
 mean value is to be found by putting I = &> in the ' coefficient,' and when 
 the function of / is cos 4 ^7, sin / cos 2 |7, sin 7 sin 2 |7, sin 2 / (in B, iii.) 
 multiplying further by cos 4 t; and where the function of 7 is sin 2 7 (in B, i.) 
 sin 7 cos 7, 1 | sin 2 7 multiplying by 1 f sin 2 i. 
 
 Thirdly, there is a column of arguments, linear functions of t, h, s, p, 
 v, . In B, i. 2t + (2A 2i/), and in B, ii. t + (h v\ are common to all the 
 arguments, and they are written at the top of the column of arguments. 
 The arguments are grouped in a manner convenient for subsequent 
 computations. 
 
 Fourthly, there is a column of speeds, being the hourly increase of the 
 arguments in the preceding column, the numerical values of which are added 
 in a last column. 
 
 Every term is indicated by the initial letter (see 1) adopted for the 
 tide to which it corresponds, except in the case of certain unimportant terms 
 to which no initials have been appropriated. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 19 
 
 To write down any term : take the general coefficient ; the coefficient 
 for the class of tides; the special coefficient, and multiply by the cosine 
 of the argument. The result is a term in the equilibrium tide, with water 
 covering the whole earth. The transition to the actual case by the intro- 
 duction of a factor and a delay of phase (to be derived from observation) has 
 been already explained. 
 
 The solar tides. 
 
 The expression for the tides depending on the sun may be written down 
 at once by symmetry. The eccentricity of the solar orbit is so small, being 
 '01679, that the elliptic tides may be omitted, excepting the larger elliptic 
 semi-diurnal tide. 
 
 The lunar schedule is to be transformed by putting s = h, p=pi, 
 = v = 0, a- = 77, I=a), e = e l} CT = CTI . In order that the comparison of the 
 importance of the solar tides with the lunar may be complete, the same 
 
 M /a\ 3 
 
 general coefficient f-^ (-) a will be retained, and the special coefficient for 
 M \c/ 
 
 each term will be made to involve the factor TJ/T. Here T 1 = f^-g-, S being 
 the sun's mass. 
 
 With EIM = 81-5, ^ = -46035 = ^^ 
 
 The schedule [C] of solar tides is given on page 23. 
 
 The subsequent schedules [D] and [E] give all the tides of purely astro- 
 nomical origin contained in the previous developments, arranged first in 
 order of speed, and secondly in order of the magnitude of the coefficient. 
 In schedule [E] the tides K l5 K 2 originate both from the moon and sun, but 
 the lunar and solar parts are also entered separately. 
 
 The coefficients of the evectional and variational tides are computed from 
 the full values to those inequalities. 
 
 In the schedule [E] the tides are marked which occur in the 'Tide- 
 predicter' of the Indian Government in its present condition. 
 
 22 
 
20 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 'S 
 
 nip* 
 
 o 
 
 
 0) 
 
 
 
 o 
 
 QQ 
 
 00 1 
 
 
 g S oS i S 
 1* S 2 5 g S S 
 
 a TjcMOJ oo in CM m 
 r coooco CM o~i r-i in 
 >r * rt o^o^J* *n ooo 
 
 9682084 
 
 |i 
 
 CO O 00 O5 I> 00 O5 
 CM CO CM CM CM CM CM 
 
 CM 
 
 V 
 9) 
 
 5 f - 
 
 ^-. ^ t3 <M T 1 
 b + | + & & 
 
 1 ^ co b b ' + 
 
 C- 
 CM 
 
 b 
 
 & 
 
 do 
 
 i i f t 
 
 <^ ^ ' ^ 
 
 CM 
 
 1 
 
 
 e . i 
 
 
 "S T 
 
 V ' 
 
 B* 
 
 ^Y^ ^ a 
 
 + ]* x- J 
 
 ^^ a^ ** o o t4g -i- 
 
 i l^T 1 * 
 
 T T Mi f i 
 
 50 
 
 CM 
 1 
 g 
 
 o5 
 
 T i S 8 I - ^ 
 
 1 
 
 
 CM 1 ^ T | ^ 
 
 CM 
 
 
 
 1 
 
 
 w i ^ A 
 
 1 * 1 CM 
 
 
 
 1 
 
 
 
 O 
 
 * 
 
 
 Mean 
 Value of 
 Coefl&oient 
 
 -H -t- 
 
 CDO5CD !> COTfCOCOO 
 CMCM05 O ^JSPiZJS 
 T^ Oi J> CM rH (M !> f" 1 CO 
 
 T^OO O OOOOO 
 
 * 
 
 CO Tj< 
 
 CO Oi 
 
 8^~ t-H 
 9 
 
 
 1 
 
 
 
 <yj 
 
 
 
 HN ^ HM ^, 
 
 HM 
 
 .2 
 'S 
 
 g 'g *** ^ ID "* 
 
 O 
 
 B 
 o 
 O 
 
 * 5? *> ^ ^ ^ S 
 
 7 + n r g >, |o j 
 
 ! 
 
 
 IB 2 
 
 o i^ 
 
 
 
 2 x 
 
 
 H 
 
 N "^ h^ T S5 
 
 a. 
 
 S-s 
 
 * S 
 * S c 
 
 CO -- C 
 
 w 
 
1883] 
 
 * 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 
 to . 
 
 
 
 <u < 
 
 
 
 
 CO CO CD 
 O i i 00 
 
 OS ii TO ^l ^f 
 
 0^1 co co ^i* 
 
 ^rU9 
 
 TO O CO 
 
 >O ^* 00 i""^ 
 
 73 . 
 
 O f i O 
 TO OS i i 
 
 Tt* TO ^ 
 
 CO O ** <M 3 
 
 oo (N m TJI r-( 
 
 OS OS 00 tfS !> 
 
 ^ a 
 
 OS r-i O 
 
 
 <a ,_ 
 
 TO CD "b 
 
 CO ^ *O C'l CO 
 
 CO a 
 
 
 
 
 
 cr- 
 
 
 
 fe <M 
 
 i 
 
 b b 
 
 + b ^ JL 6 
 
 qj 
 
 (M <M 
 
 1 t~ 
 
 L 
 
 
 W Jv. b ^ ' 
 
 00 
 
 
 ^ ^ ?v 1 
 
 - 
 
 b fc 
 
 ^^ fc 
 
 + ^ i i 
 
 -^ H" ^ b a, ^ 
 
 *<rf 1 -1^ r" [ I G^ 
 
 s IT 
 
 + 1 
 
 1 OM | 4 + 
 
 Ij 
 
 Q ^ Jjj, 
 
 ^. +5. 'g- ^ ^ 
 
 &T 
 
 i ' 1 
 
 X ^'n i i i 
 
 < 
 
 (N <N 
 
 | -|- 
 
 7 A? + ? + 
 
 
 
 ^ ^ I ^ GO <y^J 
 
 
 
 ? i 
 
 
 
 S | s2- 
 
 
 
 -S <N 
 
 
 
 1 
 
 *" O 
 
 a 1 
 
 C8 01 
 
 co o^ o 
 
 O i I i I 
 00 00 I-H 
 
 # -1- 
 
 f-i <N in >n r- <M oo 
 
 O CN 00 00 00 r-i O 
 CO *O >O ^ ^ iO t~ 
 
 Ij 
 
 00 Q 00 
 
 i i O i i 
 
 o o o o o o o 
 
 
 N ^ s 
 
 ri Ci f- 
 
 <s ^^ 
 
 k^ ^^ ^^ 
 
 Coefficient 
 
 5 8 
 
 S 5 K, 
 
 III, 
 
 H^ Qq *^ 04 OS 
 
 "s 885 
 
 K| K, 
 
 S K.S fl -S 
 
 1 H^- ' ! HN 
 c -(- HN Hw ^ 
 
 
 u$ w|5) _L 
 
 1 1 J 
 
 A 8 ^ '** 3 3 
 
 s^i, 2F 
 
 | 
 
 | 
 
 g M 
 
 or a - , . . 
 
 
 21 
 
 a + 
 
 .2 
 
HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 CO 
 
 3 
 
 
 3 
 
 2 
 
 2 r~ > ' QO O r~ 
 
 %-G 
 
 3 T* i l O CO t- 
 c3 |^ QCJ 3^ j*^ (^) 
 
 go 
 
 fl . 
 
 '- S 
 
 CC >C CO O "^ 
 Q Tf i i O CO CM 
 ^* 1"~* f"* O5 ^^ 
 *^ kQ '^ ^^ ^^ CO 
 
 11 
 
 e<3 O O r i i i i i 
 i i 
 
 r3 
 B 
 O> 
 
 t3 ^ 
 
 ^5 ^2 l c- 1 b | 
 
 Pi 
 
 OQ 
 
 ^ b ^ ^ j; 
 
 
 b ^ 
 
 
 0) 
 
 
 +J G 5^ *j5 
 
 
 $H 1 . 
 
 a 
 a} 
 
 P , O 30 x^ -^-^ OQ 
 
 CM !-< *JJ* ^-^ 
 
 - g 
 
 O tin ^H -i- i I CM 
 
 3 
 
 |3 c I A % + 
 
 < 
 
 >j A i 
 
 
 .;- 
 
 "3 'S 
 
 -( Vt- H-4- 
 -* CO O >O CM r It CO 
 (M COCOiOCMCMCMr-i 
 
 cl 
 
 c3 O 
 
 (M i l 1C r ^t* ?D CO O 
 iC '^OOOOl^''' 
 CM OOOOOOO 
 
 S o 
 
 
 
 c~^ ^ ^r 
 
 
 <"* ^w <^ ^^ 
 
 "r^ 71 ^"^ 
 
 
 '* "G - o 'in K^ 
 
 "a 
 
 M(0 ' 'So -^, '* - " "^ 
 
 .2 
 
 1 frt cc|d 
 
 '5 
 
 1 1 '"''^ 'cc 
 
 
 " ' ' l ~' ^^ M^M 
 
 0) 
 
 ^j^ I 1 ^ ^ ^ ^ "<^j 
 
 O 
 
 ^ f-*iCC , t^"l 
 
 
 
 * J 
 
 Is 
 <u -43 
 
 1'! 
 
 o 
 
 3 
 
 - -ij 75 55 
 
 fl -g =_ G 
 
 4 S 9 * ^ o 
 
 S 3 
 EH 
 
 , 
 
 3 .2 
 
 + a 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 23 
 
 O 
 
 COfN 
 
 
 
 to 
 
 
 
 
 
 
 
 D 
 
 
 
 
 Tt< CO 
 
 
 (M 
 
 *"* O 
 
 
 ^^ 
 
 
 rH 00 
 
 
 t^ 
 
 CO 
 
 _n . 
 
 
 S <N 00 
 O O OS 
 
 
 CO 5O 
 
 O O 
 
 00 rH 
 
 o * 
 
 O5 O 
 
 
 CO 
 
 rH 
 (N 
 
 * 
 
 
 b b en 
 
 
 Th iO 
 
 
 b 
 
 OJ S 
 
 
 CO CC <M 
 
 
 rH rH 
 
 
 
 02 * 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 c- 
 
 !M 
 
 
 
 o> 
 
 
 ^* 1 
 
 
 1 <** 
 
 
 f*q 
 
 & 
 
 
 ?* >^ 
 
 
 1 
 
 
 
 00 
 
 
 ^ ^ 
 
 
 ^ 
 
 ^j 
 
 
 
 . 
 
 
 
 
 ."a 
 
 
 
 
 
 
 
 02 
 
 
 
 O 
 
 
 ^ 
 
 
 CC+M 
 
 
 
 o 
 II 
 
 
 <N 
 
 .S 
 
 
 Hn 
 
 
 
 to 
 
 
 '53 
 
 
 II 
 
 
 
 S 
 
 s~ ^ 
 
 H 
 
 
 to 
 
 
 "fl 
 
 o 
 
 &* 
 
 II 
 
 b fe 
 
 ss 
 
 
 1 
 
 a 
 
 <5? 
 
 n5 1 
 
 . 
 
 + 1 
 
 i 
 
 CM 
 
 u 
 
 ^ s -' 
 
 ^ i 
 
 ff^> 
 
 1 -j- 
 
 ^"^ 
 
 
 E 
 
 5 
 
 
 IN 
 
 o 
 
 
 S 
 
 
 
 ~e 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 8 
 
 
 
 s: 
 
 
 's 
 
 
 ^ 
 
 
 
 
 
 
 1 
 
 
 03 
 
 
 Value of 
 Coefficien 
 
 1 
 
 t^ CO CO 
 rH 00 M 
 
 rH i 1 rH 
 
 <jq 
 
 ^i 
 
 1^ cO 
 
 00 X 
 
 o 9 
 
 =0 
 
 3 
 
 "3 
 g 
 
 CO 
 
 2 
 
 CO 
 
 9 
 
 
 "e 
 
 
 EH 
 
 
 ^s 
 
 
 
 1 
 
 
 1 
 
 
 O 
 
 g 
 
 
 
 1 
 
 
 5 
 
 3 3 
 
 1 
 
 
 
 
 3 ^1 
 
 , , 
 
 HJ" 
 
 S 
 
 3 
 
 
 i 
 
 i-t|W o 
 
 r-i 
 
 O 
 
 
 
 
 rH 
 
 ^02 "^ H<N 
 
 
 
 3 
 
 l 
 
 
 
 a 
 .S 
 
 
 8 ' s 'So 
 
 
 3 .a 
 
 G co 
 
 ^H 
 
 L__J 
 
 "m 
 HCN 
 
 e 
 
 
 ^ ** " 
 
 
 r* ^" 
 
 
 ^ 
 
 tD 
 
 
 
 
 
 
 
 O 
 
 O 
 
 
 V + -*! 
 
 
 ,^ + 
 
 
 1 
 
 rH 
 
 
 
 ^ ^ f! i- 
 
 
 1 rH_ 
 
 
 fU- 
 
 
 
 
 
 ^T' ** ' *~ 
 
 
 
 'S 
 
 'S 
 
 hH 
 
 
 ^q W r i 
 
 
 
 
 1 
 
24 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 I 
 
 I 
 
 i 
 1 
 
 
 T3 
 
 t- o oo r~- i i <N 
 
 t~ CO O T}< r-H t~ 
 
 O co a> r^ sq co 
 
 
 
 
 
 
 M< O oo co 10 T < 
 
 
 
 
 s 
 
 (N 00 fJ rji i i (jq 
 
 
 
 a 
 
 A 
 
 CO 
 
 D O C 10 ^? O 
 
 
 
 'H 
 
 
 
 
 
 H 
 
 
 rH f-H -H O O O 
 
 
 
 OH 
 
 
 
 
 
 bO 
 
 a 
 
 
 
 
 
 o 
 
 
 
 
 
 
 a 
 
 * ^ f g + g 
 
 
 
 
 i 
 
 ^ ^ ^ S 
 
 
 
 
 
 b 
 
 
 
 
 
 CD CO to Tf i I CO TJ 
 
 (M 
 
 
 
 
 i ' CO 00 i i <M O r 
 O * CD CO iO CO r- 
 
 CD 
 
 CD OO 
 
 
 
 1 
 
 i i <* o as o o ic 
 
 Oi O i i 00 (M CO T 
 CO 00 Tt* lO Oi "^ l> 
 
 CD CN 
 00 "^ 
 O5 'O 
 
 
 
 CO 
 
 rH O O O5 ^ Oi Ti 
 
 CO 00 
 
 
 1 
 
 
 CD O O "^ T}< CO CC 
 
 CO C^ 
 
 
 H 
 
 
 
 
 
 "3 
 
 
 
 
 
 B 
 
 
 
 
 
 Q 
 
 
 
 
 
 
 
 C 
 
 
 
 
 
 <M 
 i 
 
 t3 
 
 
 
 "a 
 
 T 
 fc 
 
 (N 
 
 
 
 i-t 
 
 a 
 
 HH 
 
 8 - rf * rf ; 
 
 O> b 
 
 
 
 
 ct 
 
 1 
 
 ? 
 
 1 
 
 s 
 
 
 
 
 CM O Tj< 00 Tt< CM C 
 
 > CD -* Ot 
 
 3 
 
 
 
 t O < i 00 lO rj< ff 
 
 co o co r- <N o a 
 
 > O5 00 TI 
 ) CM O *f 
 
 i 
 i 
 
 
 r C 
 
 rH O CJi Tf CO f 1 If 
 
 ) 1" CM f 
 
 5 
 
 
 |j 
 
 CM O 00 00 >O -^ (> 
 
 1 O5 OO if 
 
 5 
 
 
 QJ 
 
 oo o *o cq iC 1 oo t 
 
 n CO CO O 
 
 j 
 
 8 
 
 CO 
 
 CO CO O5 &O M^ O5 if 
 
 > -^ 95 a 
 
 ) 
 
 H 
 
 
 O O O5 C3i 35 00 OC 
 CO CO OJ CM O^l G^ O 
 
 ) oo t^ r- 
 
 1 (M <N C- 
 
 i 
 
 -3 
 
 
 
 
 
 e 
 
 
 
 
 
 3 
 
 
 
 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 
 
 i i 
 
 
 
 
 
 
 '-3 
 
 S<^ ^ C^ hJ ^ ^ ' 
 
 k ^ a. ^ 
 
 I 
 
 
 'c 
 
 hH 
 
 
 
 1 
 
 CO 
 00 
 CO 
 
 o 
 
 o 
 
 CO 
 
 ^2 
 
 -1-2 
 '^ 
 
 CU 
 
 Td 
 
 d 
 
 * 
 CO 
 
 g 
 I 
 
 j> 
 
 0) 
 
 CD 
 
 ^ a 
 I 8 
 
 b cu 
 
 bXD 
 
 s 
 
 cu .2 o 
 
 - .13 
 
 co ! fe 
 
 cu cS ^ 
 
 'co cu 2 
 
 CU J3 rH 
 
 m H ^ 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 25 
 
 * 
 < 
 o 
 8 
 
 8 
 o 
 ' 
 
 I 
 
 3 
 
 8 m i-H 
 
 
 > CD 
 
 IS I 
 
 coi<r2gSco!oooa 
 
 ) t^ 
 
 1 
 
 ppppppppppppc 
 
 > O 
 
 
 
 
 ) O 
 
 3 
 
 sfi 
 
 t- CD <* (N (N i I O 00 1^ t^ CD "* f 
 
 5 g 
 
 
 OOOOOOOOOOOOC 
 
 > o 
 
 o 
 
 
 
 fH 
 
 1 
 
 
 
 
 
 
 fin 
 
 - ^ 1 1 1 i 1 
 
 i< 
 
 a 
 
 
 
 1 
 
 
 
 3 
 
 i 
 
 -3 + ^ JM 
 
 g & B <" + H 
 
 * J|? ? - ? < ?/ j 4 J , jM : 
 
 3 
 
 r -< 
 
 
 ^ " 3 ^ l 
 
 9 
 
 
 cS 
 
 
 OH 
 
 a 
 
 
 
 a; to i i 
 1 " 
 
 c5 00 CO O? t^ ^O i* CD ^O "^ I~H CO *^ 
 
 1 ^ rH 
 
 3 I-H i-l 
 
 2 O O 
 
 -2 S 
 
 OO^T^COi-Hi Ir-HrHOOOC 
 
 5 O O 
 
 ' S 
 
 i 1 
 
 
 B 
 
 
 
 i 
 
 CO G"? t^- CO ^O CO *O ^ G^-l O^ ^* i~^ Q 
 
 3 CC <M 
 
 3 
 
 ^SSoo^Sr^^SOTOiCDC 
 
 :> <M CM 
 
 5 00 00 
 
 ^E 
 
 kO CD i i 00 00 00 00 00 ^O C^ CO CO Cf 
 
 q i i i i 
 
 
 
 T^tT^CJ^rHj-HpOOOOOOC 
 
 3 O 
 
 o 
 
 
 
 1 
 
 * 
 
 1 
 
 'S 
 
 . 
 
 ' c 
 
 
 SM^O fc P4 ^ 1* 
 
 
 1 
 
 . 
 
 i 
 
 a 
 
 
 
 
 H 
 
 . 
 
 
 -3 
 
 M M W 
 
 Mi 
 
 ce 
 
 'a 
 
 SMai0 i^i a i sc?s 
 
 5 b aj 
 
 3 rS 02 
 
26 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 A tide of greater importance than some of those retained here is that 
 referred to where the approximation with regard to / was introduced, viz. 
 with speed 2j + a- - vr ; the value of its coefficient is '00323. There is 
 also the larger variational diurnal tide, which has been omitted: it 
 would have a coefficient '00450; also an evectional termensual tide, 
 *jtfme$Bm*lcos(38-2h+p), with coefficient of magnitude '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 development has been made by Professor J. C. Adams, and the values 
 of all the coefficients computed therefrom, in comparison with the above. 
 
 Besides the tides above enumerated, the predicter of the India Office 
 also has the over-tides M 4 and M 6 , of speeds 4(7 0-), 6(7 0-), and the 
 compound tides 2MS, 2SM, MS, of speeds 2y - 4o- + 2ij, 27 + 2cr - 4*;, 
 47 2o- 2?7, and the meteorological tides S 1} Sa, of speeds 7 - 77, r). 
 
 If this schedule is worth anything, it seems probable that the India Office 
 predicter would do better with some other term substituted for \. 
 
 If further examination of the tidal records should show that the tide M. l 
 is in reality regular, it should be introduced. 
 
 3. Tides Depending on the Fourth Power of the Moon's Parallax. 
 The potential corresponding to these tides is 
 
 7 = p* (f cos 8 PM - f cos PM) 
 
 We may obviously neglect the eccentricity of the lunar orbit, and it will 
 appear below, when the principal terms are evaluated, that the declinational 
 tides may be safely omitted. 
 
 By these approximations we may put r = c, and M 3 = 0, and neglect the 
 terms in M l , M 2 which involve q 2 . Following the same plan as in the 
 previous development of 2, we have, when M 3 = 0, 
 
 = ft (F - 3 ^ 2 ) W - 3 JW) + 1 tf - 3V (M? - 
 
 + f ( + T - 4 2 ) ( + 
 
 + 1 (pi? + if - 4i) WJT, + M*} 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 27 
 
 The four functions of , 77, , in this expression are surface spherical harmonics 
 of the third order, and therefore, corresponding to these four terms, there 
 will be four tides of the types determined by those functions. 
 
 Now, we have approximately 
 
 Mj, = p z cos (x - 1), M 2 = -p*sm(x-l) 
 From which we have 
 
 M, 3 - SMJf* = p s cos 3 (x - Z) 
 
 Mf+ M^M? = p 6 cos (x-l) 
 When 77 = ; 3 - 3^ 2 = cos 3 X, | + %<rf - 4>% 2 = cos X ( 1 - 5 sin" X) 
 
 Then, following the same procedure as before, we have for the height of 
 tide 
 
 M fa\ s a 2 
 #= - \-A> cos 3 X . p 6 cos 3 (v I) + cos X (1 5 sm 2 X) . p 6 cos (v l)\ 
 
 * H. \f 1 r L1 ^ * V " > 
 
 J-/ w / L- 
 
 (35) 
 
 i / 
 
 Now, cos X (5 sin 2 X 1) has its maximum value when cos X = ^ V15 : 
 
 o Y 15 
 
 that is to say, when X = 58 54'; thus we may write (35) 
 
 h = i ^ (~T a l~ cos3 x iV (-) cos6 K cos [3* + 3 (h - v) - 3 (s - )] 
 Is \c/ |_ \c/ 
 
 + ^* cos x (1 - 5 sin 2 X) ^* (") cos 6 \I cos [ + (/i -v)-(s- )] 1 
 
 (36) 
 
 In this expression observe that there is the same 'general coefficient' 
 outside [ ] as in the previous development ; that the spherical harmonics 
 
 cos 3 X, '-^r- cosX(5 sin 2 X 1) have the maximum values unity, the first 
 
 at the equator and the second in latitude 58 54'. The 'speeds' of these 
 two tides are respectively 3(7 0-) or 43 0< 4761563 per mean solar hour, and 
 7 0-, or 14'4920521 per mean solar hour. 
 
 The coefficient of the tide 3 (7 cr), which is comparable with those in 
 the previous schedules [B], [C], [E], is 
 
 and the mean value of this function multiplied by cos 3 (v ) is '00599 ; 
 
28 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 also the coefficient of the tide (7 <r), likewise comparable with previous 
 coefficients, is 
 
 45 
 
 and the mean value of this function multiplied by cos (v ) is '0050. 
 
 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 = 1^/5). 
 Outside this belt, what may be called high tide, will correspond with low 
 water. The distribution of land on the earth may, however, perhaps some- 
 what disturb the latitude of evanescent tide. 
 
 It must be noticed that the 7 0- tide is comparatively small in the 
 equatorial belt, having at the equator only about f of its value in latitude 
 58 54'. 
 
 Referring to the schedule [E] of theoretical importance, we see that the 
 ter-diurnal tide M 3 would come in last but four on the list, and the diurnal 
 tide M! (with rigorous speed 7-0-) would only be about a half as great again 
 as 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 M 3 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. 
 
 4. Meteorological Tides, Over-tides, and Compound Tides. 
 
 Meteorological Tides. 
 
 A rise and fall of water due to regular day and night breezes, prevalent 
 winds, rainfall and evaporation, is called a meteorological tide. 
 
 All tides whose period is an exact multiple or sub-multiple of a mean 
 solar day, or of a tropical year, are affected by meteorological conditions. 
 Thus all the tides of the principal solar astronomical series S, with speeds 
 7 rj, 2 (7 - 77), 3 (7 i]}, &c., are subject to more or less meteorological 
 perturbation. Although the diurnal elliptic tide, S x or 7 77, the semi- 
 annual and annual tides of speeds 2r) and 77, are all probably quite insensible 
 as arising from astronomical causes, yet they have been found of sufficient 
 importance to be included on the tide-predicter. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 29 
 
 The annual and semi-annual tides are of enormous importance in some 
 rivers; in such cases the ter-annual tide (3?/) is probably also important, 
 although no harmonic analysis has been as yet made for it. 
 
 In the reduction of these tides the arguments of the S series are t, 2t, 3t, 
 &c., and of the annual, semi-annual, ter-annual tides are h, 2h, 3h. As far 
 as can be foreseen, the magnitudes of these tides will be constant from year 
 to year. 
 
 Over-tides. 
 
 When a wave runs into shallow water its form undergoes a progressive 
 change as it advances ; the front slope generally becomes steeper and the 
 back slope less steep. The most striking example of such a change is when 
 the tide runs up a river in the form of a 'bore.' 
 
 A wave which in deep water presented an approximately simple harmonic 
 contour departs largely from that form when it has run into shallow water. 
 Thus in rivers the rise and fall of the water is not even approximately a 
 simple harmonic motion. From the nature of harmonic analysis we are, 
 however, only able to represent the motion by simple harmonic oscillations, 
 and thus to give the non -harmonic rise and fall of tide in shallow water it 
 is necessary to introduce a series of over-tides whose speeds are double, triple, 
 quadruple the speed of the fundamental astronomical tide. 
 
 The only tides, in which it has hitherto been thought necessary to 
 represent this change of form in shallow water, belong to the principal lunar 
 and principal solar series. Thus, besides the fundamental astronomical tides 
 M 2 and S 2 , the over-tides M 4 , M 6 , M 8 , and S 4 , S 6 have been deduced by 
 harmonic analysis. 
 
 The height of the fundamental tide M 2 varies from year to year, according 
 to the variation in the obliquity of the lunar orbit, and this variability is 
 represented by the coefficient cos 4 \I. It is probable that the variability of 
 M 4 , M 6 , M 8 , will be represented by the square, cube and fourth power of that 
 coefficient. 
 
 The law connecting the phase of an over-tide with the height of the 
 fundamental tide is unknown, and under these circumstances it is only 
 possible to make the argument of the over-tide a multiple of the argument 
 of the fundamental, with a constant subtracted. If that constant is found 
 to be the same from year to year, then it will be known that the phase of an 
 over-tide is independent of the height of the fundamental tide. 
 
 The following schedule gives the over-tides which must be taken into 
 consideration, the notation being the same as before : 
 
30 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 [F.] 
 
 Schedule of Over-tides. 
 
 Tide 
 
 Coefficient 
 
 Argument 
 
 Speed 
 
 Speed in degrees 
 per m. s. hour 
 
 M 4 
 
 (cos*i/) 2 
 
 4* + 4(A-v)-4(*-) 
 
 4y 4<r 
 
 57-9682084 
 
 M 6 
 
 (cos 4 ^/) 3 
 
 6* + 6(A-iO-6(-|) 
 
 6y 6<r 
 
 86 -9523126 
 
 M 8 
 
 (cosH/) 4 
 
 8< + 8(A-v)-8(-) 
 
 8y-8cr 
 
 115'9364168 
 
 S 4 
 
 1 
 
 4C 
 
 4y-4>7 
 
 60-0000000 
 
 S 6 
 
 1 
 
 & 
 
 6y-67 
 
 90-0000000 
 
 It will be understood that here, as elsewhere, the column of arguments 
 only gives that part of the argument which is derived from theory, and the 
 constant to be subtracted from the argument is derivable from observation. 
 It is necessary to have recourse also to observation to determine whether 
 the suggested law of variability in the magnitude of the M over-tides holds 
 good. 
 
 Compound Tides. 
 
 When two waves of different speeds are propagated in the same water 
 the vertical displacement at the surface is generally determined with sufficient 
 accuracy by summing the displacements due to each wave separately. If, 
 however, the height of the waves is not a small fraction of the depth of the 
 water, the principle of superposition leads to inaccuracy, and it becomes 
 necessary to take into consideration the squares and products of the dis- 
 placements. 
 
 It may be shown that the result of the interaction of two waves is repre- 
 sented by introducing two simple harmonic waves, whose speeds are the sum 
 and the difference of those of the interacting waves. When the interacting 
 waves are tidal these two resultant waves may be called compound tides. 
 They are found to be of considerable importance in estuaries. 
 
 A compound tide being derived from the consideration of the product 
 of displacements, we may form an index number, indicative of the probable 
 importance of each compound tide, by multiplying together the semi-ranges 
 of the component tides. 
 
 Probably the best way of searching at any station for the compound 
 tides, which are likely to be important, would be to take the semi-ranges 
 of the five or six largest tides at that station and to form index numbers 
 of importance by multiplying the semi-ranges together two and two. Since 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 31 
 
 O 
 
 -s 
 
 w 
 
 I I C- C- 
 
 b b T T 
 
 CN <M 
 I + 
 
 I I 
 
 b b 
 
 CM CM 
 
 C- C- 
 I I 
 
 + 1 
 b b 
 
 tO CO 
 
 <0 CM 
 
 I + CO CO 
 
 I + 
 
 sr- c- 
 (N CM 
 
 1 + 
 
 b b 
 
 * Tj< 
 1 1 
 
 
 ts 
 
 t 
 
 (M 
 
 &T 
 + & 
 
 ?i 
 
 I CO 
 
 b 
 
 cc 
 
32 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 these index numbers have no absolute magnitudes, we may omit the decimal 
 point in forming them. Having selected as many of these combinations, in 
 order of importance as may be thought expedient, the arguments of the 
 compound tides are to be found by adding and subtracting the arguments 
 of the components taken in pairs. 
 
 [Theory would, however, seem to indicate (see Encyc. Brit. Art. Tides) 
 that a better index of importance would be the product of the theoretical 
 importance of the combining tides multiplied by the sum or difference, as 
 the case may be, of their respective speeds.] 
 
 In the general case it is only possible to take the tides which the previous 
 schedules have shown usually to be large, and to form a list of compound 
 speeds, with index numbers derived from the multiplication together of the 
 mean values of the coefficients of the astronomical tides. The tides selected 
 here will be M 2 , K 1} S 2 , 0, and N; but to these we shall add M 4 and S 4 , although 
 it will not be possible to affix index numbers to combinations involving 
 them. 
 
 The schedule [G] gives the speeds of the compound tides. In many 
 cases it will be observed that the compound tide has itself a speed identical 
 with that of an astronomical or meteorological tide. These cases are in- 
 dicated by the addition of the initial after the speed in question. We thus 
 learn that the tides O, K,, Mm, P, M 2 , Mf, Q, M,, L will be liable to 
 perturbation in shallow water. 
 
 The schedule [G] contains 36 speeds of compound tides : 9 of these fall 
 into the category of astronomical or meteorological tides, 2 are repeated 
 twice, and of the remaining 25 we need only consider, say, the twelve most 
 important. 
 
 If either or both the component tides are of lunar origin, the height 
 of the compound tide will change from year to year, and will probably vary 
 proportionally to the product of the coefficients of the component tides. 
 
 For the purpose of properly reducing the numerical value of the compound 
 tides, we require not merely the speed, but also the argument. 
 
 The following schedule [H] gives the index of importance, argument and 
 speed of the compound tides. The index of importance is the product of the 
 coefficients of the two tides to be compounded. 
 
 [But the numbers given in square brackets immediately after these 
 coefficients are proportional to the product of the semi-ranges of the com- 
 ponent tides, multiplied by the sum or difference of their speeds. In 
 accordance with the considerations adduced above as to the theoretical 
 height of a compound tide, these numbers ought to give a better index of 
 importance than the simple product of the semi-ranges.] 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 33 
 
 As in the case of the over-tides, the law of variability of the amplitudes 
 of compound tides in various years is only to be tested by observation. 
 
 [HJ 
 
 Schedule of Compound Tides. 
 
 Index of 
 Importance 
 
 Initials 
 
 Arguments 
 combined 
 
 Speed 
 
 Speed in degrees 
 per m. s. hour 
 
 1205 [53] 
 
 MK 
 
 M^-0 1 
 
 3y-2o- 
 
 44-0251728 
 
 960 [57] 
 
 MS 
 
 M 2 + S 2 
 
 4y-2o--27 
 
 58-9841042 
 
 960 [1] 
 
 MSf 
 
 S 2 -M 2 
 
 2o fcj 
 
 1 -01 58958 
 
 857 [37] 
 
 2MK 
 
 M 2 +0 
 
 3y 4(r 
 
 42-9271398 
 
 561 [25] 
 
 
 
 S 2 + K! 
 
 3y-2i, 
 
 45 -0410686 
 
 400 [23] 
 
 MN 
 
 M 2 + N 
 
 4y 5<r + w 
 
 57'4238338 
 
 399 [18] 
 
 
 
 S 2 +0 
 
 3y-2o--27 
 
 43 -9430356 
 
 399 [6] 
 
 
 
 S 2 -0 
 
 y + 2o--27 
 
 16 -0569644 
 
 
 
 2SM 
 
 S 4 -M 2 
 
 2y + 2o--4,7 
 
 31-0158958 
 
 
 
 
 
 M 2 + S 4 
 
 6y-2o--47 
 
 88-9841042 
 
 
 
 2MS 
 
 M 4 -S 2 
 
 2y-4 f r + 2 V 
 
 27'9682084 
 
 
 
 
 
 M 4 + S 2 
 
 6y-4o--277 87 '9682084 
 
 It will be noticed that in two cases an over-tide of one speed arises in 
 more than one way, and accordingly different parts of it have different 
 arguments and coefficients. In these cases the utilisation of the results of 
 one year for prediction in future years can only be made by dividing up 
 the compound tide into several parts, according to its theoretical origin. 
 In order to do this it is necessary that the law should be known which 
 connects the heights of a summation and a difference compound tide. A 
 like difficulty arises from the fact that MSf and 2SM are also variational 
 tides. 
 
 In practice, however, the compound tide will generally be so small that 
 we may probably treat it as though it arose entirely in one way: and 
 accordingly it is proposed to treat the tides 3y 2<r or MK, and 3y 4><r or 
 2MK, as though they arose entirely from M 2 + Kj, M 4 Kj respectively, and 
 MSf and 2SM as though they were entirely compound tides. 
 
 D. I. 
 
34 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 5. The Method of Reduction of Tidal Observations. 
 
 The printed tabular forms on which the numerical harmonic analysis 
 of the tides is carried out are arranged so that the series of observations 
 to be analysed is supposed to begin at noon, or O h , of the first day, and 
 to extend for a year from that time. It has not been found practicable to 
 arrange that the first day shall be the same at all the ports of observation. 
 
 Supposing n to be the speed of any tide in degrees per mean solar hour, 
 and t to be mean solar time elapsing since O h of the first day; then the 
 immediate result of the harmonic analysis is to obtain A and B, two heights 
 (estimated in feet and tenths) such that the height of this tide at the time t 
 is given by 
 
 A cos nt + B sin nt 
 
 The question then arises as to what further reductions it will be convenient 
 to make, in order to present the results in the most convenient form. 
 
 T> 
 
 First, let us put R = V(A 2 + B 2 ), and tan = -r- > then the tide is repre- 
 
 A 
 
 sented by 
 
 R cos (nt ) 
 
 In this form R is the semi-range of the tide in British feet, and is an 
 angle such that /n is the time elapsing after O h of the first day until it is 
 high -water of this particular tide. 
 
 It is obvious that may have any value from O u to 360, and that the 
 results of the analysis of successive years of observation will not be com- 
 parable with one another, when presented in this form. 
 
 Secondly, let us suppose that the results of the analysis are to be presented 
 in a number of terms of the form 
 
 f H cos ( V + u - K) 
 
 Here F is a linear function of the moon's and sun's mean longitudes, the 
 mean longitude of the moon's and sun's perigees, and the local mean solar 
 time at the place of observation, reduced to angle at 15 per hour. F increases 
 uniformly with the time, and its rate of increase per mean solar hour is the 
 n of the first method, and is called the ' speed ' of the tide. 
 
 It is supposed that u stands for a certain function of the longitude of 
 the node of the lunar orbit at an epoch half a year later than O h of the 
 first day. Strictly speaking, u should be taken as the same function of the 
 longitude of the moon's node, varying as the node moves; but as the 
 variation is but small in the course of a year, u may be treated as a constant 
 and put equal to an average value for the year, which average value is taken 
 as the true value of u at exactly mid -year. Together V+u constitutes that 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 35 
 
 function which has been tabulated as ' the argument ' in the schedules 
 B, C, F, H. 
 
 Since V+ u are together the whole argument according to the equilibrium 
 theory of tides, with sea covering the whole earth, it follows that x/n is the 
 lagging of the tide which arises from kinetic action, friction of the water, 
 imperfect elasticity of the earth, and the distribution of land. 
 
 It is supposed that H is the mean value in British feet of the semi-range 
 of the particular tide in question. 
 
 f is a numerical factor of augmentation or diminution, due to the 
 variability of the obliquity of the lunar orbit. The value of f is the 
 ratio of ' the coefficient ' in the column of coefficients of the preceding 
 schedules to the mean value of the same term. For example, for all the 
 solar tides f is unity, and for the principal lunar tide M 2 , f is equal to 
 cos 4 ^//cos 4 leo cos 4 \i ; for as we shall see below, the mean value of this term 
 has a coefficient cos 4 ^co cos 4 %i. 
 
 It is obvious, then, that, if the tidal observations are consistent from year 
 to year, H and K should come out the same from each year's reductions. 
 It is only when the results are presented in such a form as this that it will 
 be possible to judge whether the harmonic analysis is presenting us with 
 satisfactory results. This mode of giving the tidal results is also essential 
 for the use of the tide-predicting machine. 
 
 We must now show how to determine H and K from R and f, 
 
 It is clear that H = R/f, and the mode of determination of f from the 
 schedules has been explained above, although the proof has been deferred. 
 
 If F be the value of V at O h of the first day, then clearly 
 
 --?; + - 
 
 So that K = % + F + u 
 
 Thus the rule for the determination of K is : Add to the value of % the 
 value of the argument at O h of the first day. 
 
 It is suggested that it will henceforth be advisable to tabulate R and f, 
 so as to give the results of harmonic analysis in the form R cos (nt ) ; and 
 also H and K, so as to give it in the form fH cos (V+u K), when the results 
 will be comparable from year to year. 
 
 A third method of presenting tidal results will be very valuable for the 
 discussion of the theory of tidal oscillations, although it is doubtful whether 
 it will at present be worth while to tabulate the results in this proposed 
 form. This method is to substitute for the H of the second method FK, 
 where F is the mean value of the coefficient as tabulated in the column 
 of coefficients in the schedules for example, in the case of M 2 we should 
 
 32 
 
36 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 have F = -| (1 |e 2 ) cos 4 ^w cos 4 ^i, and in the case of S 2 we should have 
 F = . | cos 4 |w. When this process is carried out it will enable us to 
 
 compare together the several K's corresponding to each of the three classes 
 of tides, but not the several classes inter se. 
 
 It might perhaps be advisable to proceed still further and to purify 
 
 M fa\ s 
 K of the coefficient f -^ I - j a, and of the function of the latitude, viz. 
 
 cos 2 X, sin 2X, ^ f sin 2 X, as the case may be. Then we should simply 
 be left with a numerical factor as a residuum, which would represent the 
 augmentation above or diminution below the equilibrium value of the tide. 
 This further reduction may, however, be left out of consideration for the 
 present, since it is superfluous for the proper presentation of the results of 
 harmonic analysis. 
 
 For the purpose of using the tide-predicting machine the process of 
 determining H and K from R and has simply to be reversed, with the 
 difference that the instant of time to which the argument is to refer is O h 
 of the first day of the new year, and we must take note of the different 
 value of u and f for the new year. Thus supposing V l to be the value of 
 V at O h of the first day of the year to which the predictions are to apply, 
 and u 1} fj, the values of u and f half a year after that O h , we have 
 
 R = fjM 
 
 This value of R will give the proper throw of the crank of the tide-predicter, 
 and will give the angle at which the crank is to be set. Mr Roberts states, 
 however, that the subtraction, in the predicter of the India Office, of V l + w a 
 from K is actually performed on the machine, one index being set at and 
 the other at V 1 + i^. 
 
 We learn also from him that one portion of the term u-^ has been 
 systematically neglected up to the present time : namely, that part which 
 arises in the form v or its multiples. If in the schedules above we were 
 to write f = v throughout we should arrive at the rule by which the tide- 
 predicter has hitherto been used. 
 
 The above statement of procedure is applicable to nearly all the tides, 
 but there are certain tides, viz. K 1? K 2 , which have their origins jointly 
 in the tide -generating forces of the moon and sun; also the tides L and 
 M! which are rendered complex from the fact that the tidal analysis only 
 extends over a year. 
 
 Treatment of the Sidereal Diurnal and Semi-diurnal Tides K 1} K 2 . The 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 37 
 
 expression for the whole Kj tide of luni-solar origin must, as we see from the 
 schedules B and C, 3, be of the form 
 
 M cos (t + h - far - v - K) + S cos (t + h - \TT - K) (39) 
 
 f /SV S ) 
 If now we put R = M \l + { ^ } + 2 ^ cos v\ 
 
 M I \ (40) 
 
 , sinv 
 
 tan v ~~ 
 
 cos v + S/M 
 
 these two terms may be written 
 
 R cos (t + h \TT v K) 
 
 If // be the sun's mean longitude at O h of the first day, t + h h is equal 
 to yt, where t is now mean solar time measured from that O h and not reduced 
 to angle. 
 
 Hence if we write = K + \ir h a + v ' (41) 
 
 the two terms become R cos (7^ ) 
 
 But this is the form in which the results of harmonic analysis for the total 
 K! tide is expressed in the first method. 
 
 From (41) we have 
 
 In this formula h \TT is V for the solar K x tide, and v is a complex 
 function of the longitude of the moon's node, to be computed (as explained 
 below) from the second of (40). 
 
 We must now consider the coefficient f. 
 
 If M be the mean value of the lunar Kj tide, then we know that its ratio 
 to M should according to theory be given by 
 
 M _ sin / cos / 
 
 M sin co cos co (1 f sin 2 *') 
 
 The ratio of M to S should also according to theory be given by 
 
 M _ T (1 + f e 2 ) sin I cos / 
 S r t (1 + fe^sin co cos co 
 
 We must therefore put the coefficient 
 
 sv .s 
 
 Mo n , S 
 
 .(43) 
 
 wnpi*p _ ? r ' 
 
 M ~ Ya+fe 2 ) '(1-fsin 2 ;) 
 S _ S sin co cos co (1 f sin 2 i) 
 
 M M sin / cos / 
 
38 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 f is clearly a complex function of the longitude of the moon's node to be 
 computed as shown below. 
 
 The reversal of the process of reduction for the use of the instrument for 
 prediction is obvious. 
 
 In the case of the K 2 semi-diurnal tide, if we follow exactly the same 
 process, and put 
 
 where 
 
 >./' 
 
 
 
 
 f 
 
 cos 
 M 
 
 2v + S/M 
 /S\ 2 S j* 
 
 
 
 M 
 
 -.So 
 
 \. 
 
 So 
 
 T I( 
 
 M 
 1 _i_ 3 2\ 1 
 
 r 
 
 M 
 
 s 
 
 " T( 
 
 S 
 
 sin 2 co (1 f sin 2 i) 
 
 
 M 
 
 ~M 
 
 sin 2 / / 
 
 
 (44) 
 
 the argument of the K 2 tide is Zt + 2k 2z/', and f is the factor for reduction. 
 
 S 
 The numerical value of ^ both for Kj and K 2 is '46407. 
 
 The Tide L. 
 
 Reference to the theoretical development in 3 shows that this tide 
 requires special treatment. 
 
 In schedule B (i.) it appears that it must be proportional to 
 cos 4 / Vl-066 - 12 tan 2 / cos 2 (p - ) 
 
 v)-2(s-%) + (s-p)-R + 7r] ...... (51) 
 
 where 
 
 n 
 
 tan E = 
 
 sin 2 (p - ) 
 
 
 
 co 
 
 cos 2 (p ) 
 
 In this expression we must deem R to form a part of the function u, for 
 which a mean value is to be taken. This is, it must be admitted, not very 
 satisfactory, since p increases by nearly 41 per annum. 
 
 Suppose, then, that P be the longitude of the perigee at mid-year, 
 measured from the intersection, and that we compute R from the formula 
 
 T> Sm 2 ^* /K0\ 
 
 tan# = i , , ^p ..................... ( 52 ) 
 
 cot 2 / cos 2P 
 
 Then the treatment will be the same as in all the other cases, if the 
 argument V + u be taken as 2t + 2 (h v) 2 (s f) + (s p) R + TT. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 39 
 
 The factor f in this case is equal to 
 
 _ 12 tan 2 ^Icos 2Pj 
 
 COS 
 
 The Tide M,. 
 
 Reference to schedule B (ii.) shows that this tide must be proportional to 
 sin 7cos4/ V{2'307 + V435 cos 2 (p - )} 
 
 v )-(s-) + Q-)>7r] ...... (52') 
 
 where tan Q = '483 tan (p ). 
 
 We must here deem Q to form a part of the function u, for which a mean 
 value is to be taken ; but as in the case of the L tide, this course is not very 
 satisfactory. 
 
 If P as before denotes the longitude of the perigee at mid-year, measured 
 from the intersection, and Q be computed from 
 
 tan Q = -483 tan P ........................ (52") 
 
 then the argument V + u will be 
 
 , 
 
 And the factor f is 
 
 sin / cos 2 \I y (2-307 + T435 cos 2P| 
 sin CD cos 2 <B cos 4 \i ~ V2'307 
 
 [In the computation forms the '483 of (52") has been taken as , and this 
 value of f has been taken as being 
 
 sin / cos 2 \I 
 
 (f +fcos2P) 
 
 sin o> sin 2 <w cos 4 $ i 
 
 because \/(f) which should have been in the denominator was accidentally 
 omitted, and tan 2 ^-/ was treated as zero, instead of having a mean value. It 
 is clearly a matter of indifference for practical work whether or not this 
 factor V(f) stands in the denominator or not. All the Indian work has 
 been carried out in the form just shown, and the practice may well be 
 adhered to. In so small a tide it cannot be of much importance whether or 
 not tan 2 ^/ is treated as zero*.] 
 
 ' It has been shown that the tide Mj , in as far as it depends on the fourth 
 power of the moon's parallax, is too small to be worth including in the 
 numerical analysis. 
 
 * [See Eccles, Vol. xvi., Great Trig. Survey of India, p. 57.] 
 
40 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 6. On the Method of Computing the Arguments and Coefficients. 
 
 In performing the reductions of the preceding sections a number of 
 numerical quantities are required, which are to be derived from the position 
 of the heavenly bodies. 
 
 Formula} for Computing I, v, . 
 From Fig. 2, 3, we see that 
 
 cot (N ) sin N = cos N cos i + sin i cot w 
 cot v sin N = cos N cos CD + sin o cot i 
 
 cos / = cos i cos a> sin i sin a> cos N 
 
 If ft be an auxiliary angle defined by 
 
 tan/3 = tani cos N 
 
 then cos / = cos i sec /3 cos (&> + /3) 
 
 sin v sin i cosec / sin N 
 sin (N ) = sin a> cosec / sin N 
 
 The formula? (53) also lead to the rigorous formula? 
 
 ...... (53) 
 
 (54) 
 (55) 
 
 sin i cot <u sin N (1 tan \ i tan &> cos N)} 
 
 tan r _ - _ - _ - 
 
 cos 2 \ i + sin i cot &> cos N - sin 2 i i cos 2 
 
 tan V = 
 
 . ,, 
 
 tan r cosec <w sin iv 
 
 (53') 
 
 -= - : -- ; - - TT 
 
 1 + tan i cot CD cos JV 
 But, if we treat i as small, (53') may be reduced to 
 
 1 \ sin 2 &) 
 
 tan = i cot &> sin N 
 
 sn 
 
 tan v = i cosec &> sin N i i 2 sin 2N 
 
 sin-* ft) 
 
 COS ft) 
 
 sm 2 co 
 
 cos m sn &> cos 
 
 .(53") 
 
 cos J = (1 - 
 
 A table of values of , y, /, for different values of N, with &> = 23 27 /- 3, 
 i = 5 8'*8, may be computed either directly from (53) or from (55). 
 
 The approximate formulae (53") will be of service hereafter. 
 
 On the Mean Values of the Coefficients in Schedules [B]. 
 
 In the three schedules [B] of lunar tides, ' the coefficients ' are certain 
 functions of /, and there are certain terms in the arguments which are 
 functions of v and . We may typify all the terms by Jcos(T + u), where 
 J is a function of /, and u of v and . If we substitute for J and u in terms 
 of to, i, N, and develop the result, we shall obtain a series of terms of which 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 41 
 
 the one independent of N is, say, J cos T. Then J^ is the mean value of the 
 semi-range of the tide in question. Such a development may be carried out 
 rigorously, but it involves a good deal of analysis to do so ; we shall therefore 
 confine ourselves to an approximate treatment of the question, using the 
 formulae (53") for and v. 
 
 It may be proved that in no case does J involve a term with a sine of an 
 odd multiple of N, and the formulae (54) or (55) show that in every term 
 of sin u there will occur a sine of an odd multiple of N ; whence it follows 
 that J sin u has mean value zero, and J l is the term independent of N in 
 J cos u. 
 
 It may also be proved that in no case does cos u involve a term in cos N, 
 and that the terms in cos 2N are all of order i 2 ; also it appears that /always 
 involves a term in cos N, and also terms in cos 2N of order ft. 
 
 Hence to the degree of approximation adopted, Jj is equal to J cosu , 
 where ./ is the mean value of J, and cos the mean value of cos u. 
 
 In evaluating cos w from the formulae (53"), we may observe that where- 
 ever sin 2 JV occurs it may be replaced by ^; for sin 2 JV = | cos 2N, and 
 the cos 2N has mean value zero. 
 
 The following are the values of cos u n thus determined from (53") : 
 
 .. /I cos 
 
 () 
 
 cos 2y = 1 i 2 -T 
 sin- G) 
 
 - 2 cos ft)\ 2 
 
 (S) 
 
 (e) cos v = I- $i* -v 
 
 1 
 
 sin 2 &> 
 
 (f) cos 2 =1 -i 2 cot 2 
 
 The suffix indicating the mean value. 
 Similarly the following are the J 's or mean values of J : 
 
 . sin 2 Aw costul 
 lf- l^^ J 
 
 (ff) & (O sin" /. = sin' m l + f -"-f 
 
 [~, . /cos 2&) 2 cos a)\~| 
 (7 ) sm / cos 2 iy o = sin to cos 2 *&> 1 + ft'- . - --- r^ 
 
 \ sin 2 a) cos-^ft>/J 
 
42 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 / */\ T o i r o i /COS 2<W 2 COS ft) 
 
 (') sm7 sm 2 7 =sma>sm 2 ift> \l + $i* - - + -^n;- 
 
 V sin 2 to sin 2 1 w 
 
 (e') sin 7 cos 7 = sin &> cos to [1 + ^i 2 (cot 2 <w 5)] 
 
 On referring to schedules [B], it appears that (a) multiplied by (a') is the 
 mean value of the cos 4 ^7 cos 2 (v ) which occurs in the semidiurnal terms ; 
 and so on with the other letters, two and two. Performing these multipli- 
 cations, and putting 1 ^fi in the results as equal to cos 4 ^-i, and 1 ft 2 as 
 equal to 1 f sin 2 i, we find that the mean values are all unity for the 
 following functions, viz.: 
 
 cos 4 ^ /cos 2 (v g) sin 2 7 cos 2v sin / cos 2 \ I cos (2 v) 
 
 cos 4 &) cos 4 t sin 2 o> (1 f sin 2 t) ' sin o> cos 2 &> cos 4 t 
 
 sin / sin 2 \I cos (2f + v) sin 7 cos / cos v sin 2 J cos 2 
 
 sin a) sin 2 ^ o> cos 4 t sin o> cos &> (1 | sin 2 t) ' sin 2 &> cos 4 \i 
 
 Lastly, it is easy to show rigorously that the mean value of 
 
 1-| sin 2 / 
 
 (1 - f sin 2 ) (1 - f sin 2 i) 
 is also unity. 
 
 If we write w = cos \ w cos \% - sin \u> sin ^' e iVt 
 
 K = sin ^w cos \i-\- cos | w sin \i e Nl 
 
 where i stands for V 1 ', and let w-j , /Cj denote the same functions with the 
 sign of N changed, then it may be proved rigorously that 
 
 cos 4 7 cos 2 (v - ) = \ (or* + nrf) 
 
 sin 2 7 cos 2v = 2 (w^ 2 + TiV) 
 sin 7 cos 2 7 cos (2 v} = ^K + HT^K! 
 sin 7 sin 2 ^7 cos (2 + i/) = cr/c 3 
 sin 7 cos 7 cos i> = 
 
 sin 2 7 cos 2| = 2 (r 2 2 + w, V) 
 
 1 Sm 2 7 = 
 
 The proof of these formula, and the subsequent development of the 
 functions of the -sr's and KB, constitute the rigorous proof of the formulae, of 
 which the approximate proof has been indicated above. The analogy between 
 the sr's and KS, and the p, q of the earlier developments of this Report, is 
 that if i vanishes or = ro-j =p, ic = tc l = q, 
 
 (See a paper in the Phil. Trans. R. 8., Part II. 1880, p. 713; to be 
 reproduced also in Vol. II. of the present work.) 
 
 This investigation justifies the statements preceding the schedules [B] as 
 to the mean values of the coefficients. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 43 
 
 Formula? for computing f. 
 
 In the original reduction of tidal observations we want 1/f; in the use 
 of the tide-predicter f is required. 
 
 On looking through the schedules [B], we see that the following values 
 of 1/f are required. 
 
 cos 4 ^o) cos 4 %i , . sin 2 w (1 f sin 2 i) sin <w cos 2 cos 4 \i 
 
 cos 4 i/ "sin 2 / sin/ cos 2 \I ' 
 
 sin to sin 2 \ to cos 4 ^i _, sin <o cos o> (1 f sin 2 i) 
 ~ sin 7 sin 2 1 7 sin 7 cos 7 
 
 sin 2 to cos 4 1 1 ,^ (1 f sin 2 eo) (1 f sin 2 i) 
 sin 2 / (l-fsin 2 7) 
 
 And in the case of the over-tides and compound tides (schedules [F], [H]), 
 powers and products of these quantities. 
 
 A table of values of these functions for various values of 7 is given in 12. 
 
 The functions (2) and (5) are required for computing f for the Kj and K 2 
 tides. 
 
 In this list of functions let us call that numbered (2) & 2 , and that 
 numbered (5) ^; k z and fa being the values of the reciprocal of f which 
 would have to be applied in the cases of the K 2 and Kj tides, if the sun did 
 
 not exist. 
 
 On referring back to the paragraph in 5 in which the treatment of the 
 K 2 and Kj tides is explained we see that for K 2 
 
 H = -46407 x k* 
 M 
 
 and therefore from (44) we see that for K 2 
 
 1 1-46407 x k z 
 
 {1 + (0-46407 x & 2 ) 2 + 0-92814& 2 cos 
 
 . 
 
 sm2i/ 
 
 tan2v = - ^ . _ ..^j 
 cos 2z; + -46407^2 
 
 And for Kj the similar formulae hold with &j in place of k z , and v in place 
 of 2v*. 
 
 Tables of 1/f and v', 2v" for the Kj and K 2 tides have been formed from 
 (56), and are given in Col. Baird's Manual of Tidal Observations. 
 
 The angle 7 ranges from 18 18''5, when it is w-i, to 28 36'% when it 
 is to + i. 
 
 * This method of treating these tides is due to Professor Adams. I had proposed to divide 
 the K tides into their lunar and solar parts. G. H. D. 
 
V (57) 
 
 44 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 Then in using these tables, we first extract / for any value of N, and 
 afterwards find the coefficients from the subsequent tables. 
 
 The coefficients for the over-tides and compound tides may be found 
 from tables of squares and cubes and by multiplication. 
 
 [Algebraic formulae for the several f s are given at the end of the present 
 paper ; they are so simple that the auxiliary tables may be dispensed with, 
 without much loss.] 
 
 Formulae for s, p, h, p lf N. 
 
 The numerical values may be deduced from the formulae given in Hansen's 
 Tables de la Lune. The following are reduced to a more convenient epoch, 
 and to forms appropriate to the present investigation. 
 
 s = 150-0419 + [13 x 360 + 132'67900] T + 13'1764 D \ 
 
 + 0-5490165# 
 
 p = 240-6322 + 40-69035 T + 0-1114 D + 0-0046418 H 
 h = 280-5287 + 360'00769 T + 0'9856 D + 0'0410686 H 
 p, = 280-8748 + 0-01711 T + 0-000047 D 
 N = 285-9569 - 19-34146 T - 0-0529540 D 
 Where 
 
 T is the number of Julian years of 365^ mean solar days, 
 D the number of mean solar days, 
 H the number of mean solar hours, 
 after O h Greenwich mean time, January 1, 1880. 
 From the coefficients of H we see that 
 
 a- = 0-5490165, OT = 0-0046418, 77 = 0-0410686, (58) 
 
 whence 7 = 15'0410686. 
 
 For the purposes of using the forms for harmonic analysis of the tidal 
 observations, these formulae may be reduced to more convenient and simpler 
 forms. 
 
 The mean values of N and p 1 are required, and for the treatment of the 
 L and MI tides the mean value of p g, denoted by P. For determining 
 these three quantities, we may therefore add half the coefficient of T once 
 for all, and write 
 
 N = 276'2861 - 0'05295 D - 19'34146 T] 
 
 p, = 280-8833 + 0-00005 D + 0'0171ir> (59) 
 
 P+=261-0 +0-111D +40-69T J 
 
 where T is simply the number of years, whether there be leap-years or not 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 45 
 
 amongst them, since 1880, and D the number of days from Jan. 1, numbered 
 as zero up to the first day of the year to be analysed. 
 
 Now, suppose d to denote the number of quarter days either one, two, or 
 three in excess of the Julian years which have elapsed since O h Jan. 1, 1880, 
 up to O h Jan. 1 of the jear in question ; let D denote the same as before ; 
 and let L be the East Longitude of the place of observation in hours and 
 decimals of hours. 
 
 Then for s 0) p , h , the values of s, p, h at O h of the first day, we have 
 
 * = 150-0419 + 132-67900 T + 3 C> 29410 d + 13-1764 7) - 0'54902 L} 
 
 p (1 = 240-6322 + 40-690357 T + 0-02785d + 0-1114 D-0'00464 L\ (60) 
 
 7/o = 280-5287 + 0'00769 T + 0-24641 d + 0-9856 D - 0'04107 L\ 
 
 In these formulae T is an integer, being the excess of the year in question 
 above 1880, and d is to be determined thus : if the excess of the year above 
 ] 880 divided by 4 leaves remainder 3, d is 1 ; if remainder 2, it is 2 ; if 
 remainder 1, it is 3 ; and if remainder zero, it is zero. For example for 1895, 
 T= 15, d=l; because from O h Jan. 1, 1880 to O h Jan. 1, 1895, is 15 Julian 
 years and a quarter day. 
 
 For all dates after Feb. 28, 1900, one day's motion must be subtracted from 
 *'o. po> h , pi, P + 1~, and one days motion added to N. 
 
 The terms in L may be described as corrections for longitude. 
 
 The 13 x 360 and 360 which occurred in the previous formulae for s 
 and h are now omitted, because T is essentially an integer. 
 
 If it be preferred, the values of s and N may be extracted from the 
 Nautical Almanac, and fi is (neglecting nutation) the sidereal time reduced 
 to angle. We may take p from a formula given by Hansen at p. 300 of the 
 Tables de la Lune. This latter course is that which is followed in the forms 
 for computation. 
 
 7. Summary of Initial Arguments and Factors of Reduction. 
 
 The results for the various kinds of tide are scattered in various parts of the 
 above, and it will therefore be convenient to collect them together. In order 
 to present the results in a form convenient for computation, each argument 
 is given by reference to any previous argument which contains the same 
 element. In the following schedule Arg. M 2 and Fac. M 2 (for example) mean 
 the argument and factor computed for the tide M 2 . 
 
46 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 [1 
 
 [I.] 
 
 Schedule of Arguments at O h of the first day, and Factors 
 for Ensuing Year. 
 
 
 Initial Arguments. 
 
 V ft + M 
 
 Factors for Reduction. 
 1 
 F 
 
 Si 
 
 
 
 83 
 
 S 4 
 
 zero 
 
 unity 
 
 p 
 
 -Ao + iTr 
 
 unity 
 
 T 
 
 -(Ao-ft) 
 
 unity 
 
 If, 
 
 where tan Q= -483 tan P 
 
 Fac. 0-7-v/d+f COS2P}* 
 
 M 2 
 
 
 ^COS ^0) COS ^i"\ 4 
 
 2(Ao i,) 2(* ) 
 
 \ COS ^/ / 
 
 M 3 
 
 f Arg. M 2 
 
 (Fac. M 2 )^ 
 
 M 4 
 
 2 Arg. M 2 
 
 (Fac. M 2 ) 2 
 
 M 6 
 
 3 Arg. M 2 
 
 (Fac. M 2 ) 3 
 
 M 8 
 
 4 Arg. M 2 
 
 (Fac. M 2 ) 4 
 
 K 2 
 
 2Ao-2i>" 
 a sin 2i/ 
 
 1 -46407 
 
 V{1 + ('464 x k) 2 + -928k cos 2i/} 
 
 cos 2v + '464x k 
 
 sin 2 / 
 
 K, 
 
 u , sin i/ 
 
 1 -46407 k 
 
 V{1-K'464 x /fc) 2 + -928 cos v} 
 ^ 7 sin 2<u (1 | sin 2 i) 
 
 cos / + '464 x k 
 
 sin 27 
 
 N 
 
 Arg. M 2 -(* -po) 
 
 Fac. M 2 
 
 2N 
 
 Arg. N-(SO-^O) 
 
 Fac. M 2 
 
 L 
 
 V 
 
 Arg. M 2 + (SQ ~PO) ~ R> + T 
 
 
 Fac. M 2 4-\ / l'066-12tan 2 ^7cos2P 
 Fac. M 2 
 
 Arg. M 2 + ( - Pa) + 2A - 2s 
 
 
 
 (Ao-,)-2 ( o-|)^ 
 
 sin w cos 2 ^<o cos 4 ^t 
 
 sin 7 cos 2 ^7 
 
 00 
 
 1 r 
 
 sin w sin 2 |cu cos 4 \i 
 
 " 2Tr 
 
 sin / sin 2 7 
 
 Q 
 
 Arg. O-(* -po) 
 
 Fac. O 
 
 J 
 
 ^t**"?? 
 
 sin2o>(l-| sin 2 -*) 
 
 sin 27 
 
 * [See 5 above as to a more correct value of f in this case, aud the reasons which have led 
 to the use of the value here given.] 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 Schedule [I.] continued. 
 
 47 
 
 
 Initial Arguments. 
 V + 
 
 _.. 
 
 Factors for Reduction. 
 1 
 f 
 
 MS 
 
 Arg. M 2 
 
 Fac. M 2 
 
 SMS 
 
 Arg. M 4 
 
 Fac. M 4 
 
 2SM 
 
 Sir Arg. M 2 
 
 Fac. M 2 
 
 MK 
 
 Arg. M 2 +Arg. Kj 
 
 Fac. M 2 x Fac. Kj 
 
 2MK 
 
 Arg. M 4 -Arg. K x 
 
 Fac. M 4 x Fac. K t 
 
 MN 
 
 Arg. M 2 +Arg. N 
 
 Fac. M 2 x Fac. N 
 
 MSf 
 
 2n--Arg. M 2 
 
 Fac. M 2 
 
 Mm 
 
 Mf 
 
 Sa 
 
 . (*o~Po) 
 
 2 (*-) 
 Ao 
 
 (1 - 1 sin 2 a) (1 - 1 sin 2 i) 
 
 1-f sin 2 / 
 sin 2 to cos 4 \i 
 
 sin 2 / 
 unity 
 
 Ssa 
 
 2/> 
 
 unity 
 
 There are two tables, numbered I. and II., given at pp. 304 and 305 of 
 the Report for 1876 of the Committee of the British Association on Tidal 
 Observations. The columns headed e give functions which, when their signs 
 are reversed, are the arguments at the epoch. To show the identity of these 
 expressions with those in the above schedule [I], we must put 
 
 f=-h , g = k , } = s + v- , 0=/i , *r' =p + v j-, *r=pi 
 
 For the sake of symmetry these tables contain several entries which we have 
 omitted from our schedule, because of the smallness of the tides to which 
 they refer. The entries of the tides of long period, Nos. 3 and 4, are .given 
 with the opposite sign from that here adopted*; thus those entries require 
 alteration by 180 to bring them into accordance with our schedule. 
 
 The following corrections have to be made in Table II. : No. 8, for 2i> 
 read 3v, No. 15, add 4i/; Nos. 17 and 19, add 2(i/-f); Nos. 18 and 20, 
 subtract 2 (v ). 
 
 The K x , K 2 tides, Nos. 9 and 16 of both tables, are entered separately 
 as to their lunar and solar parts. The two parts of the M a tide, Nos. 7 and 
 11, are entered separately. Also No. 14 only gives one part of the tide here 
 entered as L. 
 
 The reader is warned that the definition of e on p. 293 is incomplete, 
 and incorrect for proper reference to the equilibrium theory of tides. The 
 definition of BT' on p. 302 is incorrect. 
 
 * See the passage in 2 between equations (28) and (29). 
 
48 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 8. On the Reductions of the Published Results of Tidal Analysis. 
 
 In the Tide Tables published by the Indian Government, it is stated 
 that each tide is expressed in the form R cos (nt e), where R is the semi- 
 range in feet, n the speed of the tide, and e//?- is the time in mean solar 
 hours which elapses, after an epoch appropriate to the tide, until the next 
 high-water of that tide. Tables are then given for R and e at each station 
 for each year. 
 
 The mode of tabulation is the same as that followed in the Tidal Reports 
 of the British Association for 1872 and 1876. 
 
 It is advisable that all the results should be reduced according to one 
 system, such that the observations of the several years and the values for the 
 several speeds of tide may be comparable inter se. 
 
 In 5 it has been proposed that the tide should be recorded in the form 
 
 fHcos(F+w-*) 
 
 It appears from the statements in the Reports for 1872 and 1876 and 
 from an examination of the reductions of the published results that the e 
 of the tables is equal to K u, and that the R of the tables is equal to f H. 
 Thus in order to reduce the published results to proper forms, comparable 
 inter se, it is necessary to add to e the appropriate u, and to divide R by the 
 proper f. Following this process we obtain certain corrections to the e's to 
 obtain the 's. The values of 1/f by which the R's are to be multiplied to 
 obtain the H's, are those given in the preceding schedule [I]. [But it does 
 not seem worth while to reproduce the schedule of instructions for correcting 
 these old results.] 
 
 | 9. Description of the Numerical Harmonic Analysis for the Tides of 
 
 Short Period. 
 
 It forms no part of the plan of this Report to give an account of the instru- 
 ments with which the tidal observations are made, or of the tide-predicting 
 instrument. A description of the tide-gauge, which is now in general use 
 in India and elsewhere, and of the tide-predicter, which is at the India Store 
 Department in Lambeth, and of designs for modifications of those instruments, 
 has been given in a paper by Sir William Thomson, read before the Institution 
 of Civil Engineers on March 1, 1881*, and to this paper we refer the reader. 
 Our present object is to place on record the manner in which the observations 
 have been or are to be henceforth treated, and to give the requisite information 
 for the subsequent use of the tide-predicting instrument. 
 
 * " The Tide Gauge, Tidal Harmonic Analyser, and Tide Predicter," Proc. Inst. C. E., Vol. 45, 
 Part in. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 49 
 
 The tide-gauge furnishes us with a continuous graphical record of the 
 height of the water above some known datum mark for every instant of 
 time. 
 
 It is probable that at some future time the Harmonic Analyser of 
 Professors James and Sir William Thomson may be applied to the tide- 
 curves. The instrument is nearly completed, and now lies in the Physical 
 Laboratory of the University of Glasgow, but it has not yet been put into 
 use*. The treatment of the observations which we shall describe is the 
 numerical process used at the office of the Indian Survey at Poona, under 
 the immediate superintendence of Major A. W. Baird, R.E. The printed 
 forms for computation were admirably drawn up by Mr Edward Roberts, 
 of the ' Nautical Almanac ' Office ; but they have now undergone certain 
 small modifications in accordance with this Report. The work of computation 
 is to a great extent carried out by native Indian computers. The results of 
 the harmonic analysis are afterwards sent to Mr Roberts, who works out the 
 instrumental tide-predictions for the several ports for the ensuing year. 
 The use of that instrument requires great skill and care. The results of 
 the tidal reductions have hitherto been presented in a somewhat chaotic 
 form, and we believe that it is only due to Mr Roberts' knowledge of the 
 manner in which the tidal results have been treated that they have been 
 correctly used for prediction. It may be hoped that the use of the methods 
 recommended in the present Report will remove some of the factitious 
 difficulties in the use of the instrument -f. 
 
 The first operation performed on the tidal record is the measurement in 
 feet and decimals of the height of water above the datum at every mean 
 solar hour. The period chosen for analysis is about one year, and the first 
 measurement corresponds to noon. It has been found impracticable to make 
 the initial noon belong to the same day at the several ports. It would seem, 
 at first sight, preferable to take the measurements at every mean lunar hour; 
 but the whole of the actual process in use is based on measurements taken 
 at the mean solar hours, and a change to lunar time would involve a great 
 deal of fresh labour and expense. 
 
 If T be the period of any one of the diurnal tides, or twice the period 
 of any one of the semi-diurnal tides, it approximates more or less nearly to 
 24 m. s. hours, and if we divide it into 24 equal parts, we may speak of each 
 as a T-hour. We shall for brevity refer to mean solar time as $-time. 
 
 Suppose, now, that we have two clocks, each marked with 360, or 
 24 hours, and that the hand of the first, or $-clock, goes round once in 24 
 
 * See Appendix, Thomson and Tait's Nat. Phil., 2nd ed. 1883. [It has not been found 
 expedient to use this interesting instrument, and it is deposited in the Museum at South 
 Kensington. ] 
 
 t [The tide-predicter was transferred to the National Physical Laboratory in 1904. 
 
 D. I. 4 
 
50 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 $-hours, and that of the second, or T-clock, goes round once in 24 T-hours, 
 and suppose that the two clocks are started at or O h at noon of the initial 
 day. For the sake of distinctness, let us imagine that a T-hour is longer 
 than an $-hour, so that the T-clock goes slower than the $-clock. The 
 measurements of the tide-curve give us the height of water exactly at each 
 $-hour ; and it is required from these data to determine the height of water 
 at each T-hour. 
 
 For this end we are, in fact, instructed to count T-time, but are only 
 allowed to do so by reference to $-time, and, moreover, the time is always to 
 be specified as an integral number of hours. 
 
 Beginning, then, with O h of the first day, we shall begin counting 0, 1, 
 2, &c., as the T-hand comes up to its hour-marks. But as the $-hand gains 
 on the T-hand, there will come a time when the T-hand, being exactly at 
 the p hour-mark, the $-hand is nearly as far as p + . When, however, the 
 T-hand has advanced to the p + 1 hour-mark, the $-hand will be a little 
 beyond p + 1 + ^ : that is to say, a little less than half an hour before p + 2. 
 Counting, then, in T-time by reference to $-time, we shall jump from p to 
 p 4- 2. The counting will go on continuously for a number of hours nearly 
 equal to %p, and then another number will be dropped, and so on throughout 
 the whole year. If it had been the T-hand which went faster than the 
 $-hand, it is obvious that one number would be repeated at two successive 
 hours instead of one being dropped. We may describe each such process as 
 a ' change.' 
 
 Now, if we have a sheet marked for entry of heights of water according 
 to T-hours from results measured at $-hours, we must enter the /^-measure- 
 ments continuously up to p, and we then come to a ' change,' and dropping 
 one of the ^-series, we go on again continuously until another ' change,' when 
 another is dropped, and so on. 
 
 Since a ' change ' occurs at the time when a T-hour falls almost exactly 
 half way between two $-hours, it will be more accurate at a ' change ' to 
 insert the two ^-entries which fall on each side of the truth. If this be 
 done the whole of the $-series of measurements is entered on the T'-sheet. 
 Similarly, if it be the T-hand which goes faster than the $-hand, we may 
 leave a gap in the T-series instead of duplicating an entry. For the analysis 
 of the T-tide there is therefore prepared a sheet arranged in rows and 
 columns ; each row corresponds to one T-day, and the columns are marked 
 O h , l h , ... 23 h ; the O h 's may be called T-noons. A dot is put in each space for 
 entry, and where there is a change two dots are put if there is to be a 
 double entry, and a bar if there is to be no entry. Black vertical lines mark 
 the end of each $-day. These black lines will of course fall into slightly 
 irregular diagonal lines across the page, and such lines are steeper and 
 steeper the more nearly T-tirne approaches to $-time. They slope downwards 
 
1883] 
 
 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 
 
 51 
 
 ^S 
 
 s 
 
 
 
 f* 
 
 3 
 O 
 
 V; 
 
 A 
 
 a 
 
 
 CO 
 
 ^^ 
 
 si 
 
 
 <M 
 
 ^_ ^^ 
 
 Si 
 
 
 O 
 <M 
 
 
 
 i 1 
 
 
 
 jg 
 
 
 r- 1 
 
 
 F- 
 
 i i 
 
 
 ^ 
 
 
 1-1 
 
 
 
 
 i-H 
 
 
 . 
 
 
 I 1 
 
 
 
 
 
 !) 
 
 
 
 
 
 
 ^H 
 i 1 
 
 
 
 
 H 
 
 
 O 
 
 
 00 
 
 
 Jl 
 
 
 t- 
 
 
 _ 
 
 
 CD 
 
 
 jd 
 
 
 
 
 x 
 
 
 % 
 
 
 
 
 03 
 
 
 - 
 
 
 .* 
 
 
 
 
 i-H 
 
 
 s 
 
 
 
 i i (N CO -* O 
 
 1- 00 OS CS .-H 
 
 o ^H <M * 
 i> t* t^ t^. i> 
 
 00 
 
 |- 
 
 P ^N 
 
 'o^ 
 
 M 
 0^ 
 
 
 
 
 * 
 
 
 t^ 
 
 
 cc 
 
 
 l^ 
 
 
 ^ 
 
 
 l^ 
 
 
 
 cc 
 
 L^ 
 
 
 ec 
 
 
 b- 
 
 
 
 * 
 i^ 
 
 
 
 o 
 
 
 M 
 
 
 
 l^ 
 
 
 m 
 
 _ 
 
 
 
 *t 
 
 
 
 t^ 
 
 
 ^ 
 
 _ 
 
 i> 
 
 
 it 
 
 
 B 
 
 
 n 
 
 
 t^ 
 
 
 ec 
 
 
 t- 
 
 
 ec 
 
 
 p* 
 
 
 iO 
 
 
 t~ 
 
 
 cc 
 
 m 
 
 t- 
 
 
 n 
 
 
 i^ 
 
 
 n 
 
 
 t^ 
 
 
 CO 
 
 
 t- 
 
 
 * 
 
 
 s 
 
 
 
 "* 
 1 
 
 
 n 
 
 
 
 
 
 n 
 
 
 i* 
 
 I 00 O5 O -H 
 CD CO CD t^ t- 
 
 a d 
 I* 
 
 42 
 
52 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 from right to left if the ^-hour is longer than the $-hour, and the other way 
 in the opposite case. The 'changes' also run diagonally, with a slope in the 
 opposite direction to that of the black lines. 
 
 We annex a diminished sample of a part of a page drawn up for the 
 entry of the M-series of tides, in which T-time is mean lunar time. 
 
 The incidence of the hours in the computation forms for the several series 
 was determined by Mr Roberts. 
 
 Since the first day is numbered 1, and the first hour O h , it follows that 
 the hourly observation numbered 74 d ll h is the observation which completes 
 a period of 73 d 12 h of mean solar time since the beginning; in fact, to find 
 the period elapsed since O h of the first day we must subtract 1 from the 
 number of the day and add one to the number of the hour. The 73 d 12 h of 
 m. s. time, inserted at the foot of the form, is very nearly equal to 71 days 
 of mean lunar or M-time. For each class of tide there are five pages, giving 
 in all about 370 values for the height of the water at each of the 24 special 
 hours ; the number of values for each hour varies slightly according as more 
 or less ' changes ' fall into each column. 
 
 The numbers entered in each column are summed on each of the five 
 pages ; the five sets of results being summed, the results are then divided 
 each by the proper divisor for its column, and thus is obtained the mean 
 value for that column. In this way 24 numbers are found which give the 
 mean height of water at each of the 24 special hours. 
 
 It is obvious that if this process were continued over a very long time 
 we should in the end extract the tide under analysis from amongst all the 
 others, but as the process only extends over about a year, the elimination 
 of the others is not quite complete. 
 
 [The choice of appropriate periods is considered in the next paper in 
 this volume, and I therefore omit the consideration of the forms which were 
 in use in India in 1883.] 
 
 Let us now return to our general notation, and consider the 24 mean 
 values, each pertaining to the 24 T-hours. We suppose that all the tides 
 excepting the T-tide are adequately eliminated, and, in fact, a computation 
 of the necessary corrections for the absence of complete elimination, which 
 is given in the Tidal Report of 1872, shows that this is the case. 
 
 It is obvious that any one of the 24 values does not give the true height 
 of the T-tide at that T-hour, but gives the average height of the water, as 
 due to the T-iide, estimated over half a T-hour before and half a T-hour 
 after that hour. We must now consider the correction necessary on this 
 account. 
 
1883] HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 53 
 
 Suppose we have a function 
 h = AI cos 6 + Bj sin + A 2 cos 28 + B 2 sin 20 + . . . + A,. cos rd + B r sin r0 + . . . 
 
 Then we see by integration that the function 
 
 h'= A/ cos 6 + B/sin + A 2 'cos 20 + B 2 'sin 26 + . . . + A,.' cos rd + B/ sin r0 + . . . 
 where 
 
 A/ B/ _ sin a A 2 ' B 2 ' sin 2a A/ B/ sin |ra 
 
 is derivable from h by substituting for the h, corresponding to any value 
 of 6, the mean value of h estimated over the interval from 6 + a to 6 ^a. 
 
 Thus when harmonic analysis is applied to the 24 jT-hourly values, the 
 coefficients which express that oscillation which goes through its period r 
 times in the 24 T-hours must be augmented by the factor ^rat/sin ^ra. Thus 
 we get the following expressions for the augmenting factors for the diurnal, 
 semi-diurnal, ter-diurnal oscillations, &c., viz. : 
 
 7'5?r . ^ OA , 15-7T / . 22"57r / . 000 OA/ p 
 
 ' ISO/ ' sm 3 ' &c ....... 
 
 Computing from these we find the following augmenting factors. 
 
 [M.] 
 Augmenting Factors. 
 
 For A a , Bj . . . 1-00286 
 
 A, B 2 1-01152 
 
 Aj, B 3 . . . 1-02617 
 
 A 4; B 4 1-04720 
 
 A 6 , B 1-11072 
 
 A 8 , B 8 1-20920 
 
 In the reduction of the S-series of tides, the numbers treated are the 
 actual heights of the water exactly at the $-hours, and therefore no aug- 
 menting factor is requisite. 
 
 We must now explain how the harmonic analysis, which the use of these 
 factors presupposes, is carried out. 
 
 If t denotes jf-time expressed in hours, and n is 15, we express the 
 height h, as given by the averaging process above explained, by the formula 
 
 h = A + A! cos nt + Bj sin nt + A 2 cos 2?? + B 2 sin 2nt + ... 
 where t is 0, 1, 2... 23. 
 
54 
 
 SCHEDULE FOR HARMONIC ANALYSIS. 
 
 [1 
 
 M 
 
 s '8 
 
 s - 
 
 1-3 
 
 .so g 
 
 X) ^a 
 
 rB* % 
 
 55 -2 
 
 ^ 
 
 X x 
 
 00 CO ^ Q 
 
 9999 
 
 -fill 
 
 Ti 
 
 tt O 
 
 O5 ^ iC rH CO iC CM 
 rH CO CM rH O 9 9 
 
 -* 00 O 
 
 999 
 
 I I 
 
 m 
 
 O5 1C 00 I^ CO . I CM 
 T-H O O C O O O 
 
 K ^ 
 
 f-3M C 
 
 > o> 
 
 9 J =- > 
 
 I I I 
 
 O2T-HCMCMCO^CMOi 
 
 ppppp999 
 
 + l 
 
 a + 
 
 O ^ "^ 1C CO CD rH CD 
 ^cboCO5C51-~iCCM 
 
 ire op cp 
 
 O5 05 rH 
 
 O5rH,-Hi--cpapcpcpcpr-pO5 
 
 CDOOOSOSOSOOI^CDiC^fiCiC 
 
 'COcpopcpoccpcocoi>'O5t- 
 
 I-- CO O5 O5 O5 00 ! CD b ^f 1 -f O 
 
 1C ^> IT- 00 O5 O rH 
 
1883] 
 
 SCHEDULE FOR HARMONIC ANALYSIS. 
 
 55 
 
 .> 
 
 *s 
 
 rH O 00 O "^ O 
 rH I"~ OS 
 1 1 + 
 
 i 
 ip p 
 
 + + 
 
 
 
 
 ^ z 
 
 to JH 
 O) . 
 
 (N O r- 
 
 (M iO t- 
 
 C5 05 P 
 CD O Q 
 CD CD O 
 00 O5 O 
 
 
 
 
 
 
 o 
 
 rH 
 
 a 
 
 rH O rH O rH 
 1 
 
 n 
 
 2 ^ 
 
 
 ^. ^ 
 
 II II II II 
 
 t~*. n M e< 
 
 O2 02 02 
 
 1 "o " 
 
 GO" 02 *"* 
 
 ~ s' 
 
 O 00 O O O O 
 
 eo ; . o ........ w 
 
 rH 
 + 1 + 
 
 i i 
 
 
 >3 
 S X 
 
 rH rH rH 
 
 00 OS OS 
 kO CM CM 
 
 + 1 1 
 
 9 8 
 
 s 
 
 O rH O rH O rH 
 1 
 
 n 
 
 S M 
 
 
 91 
 
 - 1 O2* 0? 
 1 1 
 
 'I 
 
 rH *^ 
 
 
 
 
 
 
 
 
 ** 
 s 
 
 1 + + 1 f 
 
 (M 
 
 00 00 
 
 1 1 
 
 
 rH K>( 
 
 > X 
 
 a i 
 
 CO "^ 
 
 o 6 6 
 
 o c 
 
 + 1 
 
 CO C^ 
 1 1 
 
 
 
 
 
 
 
 II 
 
 
 
 'L 
 
 (N <^ 
 
 
 y 
 
 O 02 02 
 1 
 
 2 
 
 
 1 1 
 
 rH ^ 
 
 
 .x 
 
 
 
 >* 
 
 O O5 00 O (M O 
 
 rH CD 6 00 M 
 
 a 
 
 00 Tj< 
 
 a> y\ 
 
 (M 
 
 
 ti 
 
 ob ob ob 
 
 
 *s 
 
 
 + + 
 
 
 I-H 
 
 00 S 
 
 
 * 
 
 O 02 02 rH O2 O2 
 
 II 
 CM ^ 
 
 
 t^4 fl 
 $* 
 
 + 1 + 
 
 
 
 
 
 
 
 
 ll 
 
 rH 
 
 
 
 
 ^ 
 
 '- ( O2 02 
 1 
 
 
 ^x 
 X '. 
 
 I-H 
 
 (1 
 
 X 
 
 rH 00 00 O -Jt* O 
 rH CO I- 6 OS CD 
 
 1 + + + + + 
 
 
 
 s'3 
 
 'l 1 
 
 rH O 
 'l 'l 
 
 
 
 
 
 *j 
 
 
 
 rH 
 I-H 
 . I-H 
 
 rH rH CM 
 
 O5 Ol O3 Oi O> O3 
 <N CN CM (N (M CM 
 
 
 
 M 
 
 O 02" 02* 
 
 II 
 
 2 
 
 S 
 
 
 
 
 5 v 
 
 00 W CD 
 
 
 "i 
 
 H^E 
 
 rH CD CD >C 00 5O 
 
 
 
 X 
 
 rH O rH 
 + 1 1 
 
 
 *C 
 
 OJ 
 
 02 
 
 >O (M O <35 O5 rH 
 
 rH rH rH rH 
 
 
 
 ^-^x 
 
 KJ( 
 
 O f-H (N 
 
 ~ Ct G5 
 
 
 O 
 
 
 
 
 02 
 
 
 
 <*-. 
 
 % I S 11 
 
 
 
 x^"' 
 
 T 1 9 9 
 
 
 n 
 
 _H 
 
 
 
 
 ^|o 
 
 _..._. 
 
 CM CM CM 
 
 
56 HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. [1 
 
 Then if S denotes summation of the series of 24 terms found by attributing 
 to t its 24 values, it is obvious that 
 
 A = ^Zh ; A! = -^Ji cos nt ; Bj = ^SA sin nt ; 
 
 A 2 = yj^A cos 2w ; B 2 = -^^h sin 2nt ; &c., &c. 
 
 Since n is 15 and is an integer, it follows that all the cosines and sines 
 involved in these series are equal to one of the following: viz.: 0, + sin 15, 
 + sin 30, + sin 45 C , + sin 60, + sin 75, 1. It is found convenient to denote 
 these sines, as 0, S 1} $ 2 , S 3 , S 4 , S s , I. The multiplication of the 
 24 A's by the various S's, and the subsequent additions may be arranged in a 
 very neat tabular form. 
 
 We give on the last two pages the form for the reduction of the M-tides, 
 filled in for Karachi 1880-81, but abridged by the omission of some of the 
 decimals. The columns marked M are the multipliers appropriate for each 
 series. 
 
 The columns I. and II. contain the 24 hourly values to be submitted to 
 analysis. The subsequent operations are sufficiently indicated by the headings 
 to the columns, and it will be found on examination that the results are in 
 reality the sums of the several series indicated above. We believe that this 
 mode of arranging the harmonic analysis is due to Archibald Smith, who 
 gives it in the Admiralty manual on the Compass. The arrangement seems 
 to be very nearly the same as that adopted by Everett (Trans. Roy. Soc. Edin. 
 1860) in his reductions of observations on underground temperature. 
 
 In most cases it is not necessary to deduce more than the tide of the 
 speed indicated by astronomical theory, but we give the full form by which 
 the over-tides are deducible. If we want only a diurnal tide, then the only 
 columns necessary are I. to VII. and IX. and X. ; if only a semi-diurnal tide, 
 the columns to be retained are I., II, III, XII, XIII, XV, XVI, XVII. 
 
 The A's and B's having been thus deduced, we have R = V(A 2 -I- B 2 ). 
 R must then be multiplied by the augmenting factors which we have already 
 evaluated (Schedule [M]). We thus have the augmented R. Next the angle 
 whose tangent is B/A gives f. The addition to of the appropriate V + u 
 (see Schedule [I]) gives K, and the multiplication of R by the appropriate 1/f 
 (see Schedule [I]) gives H. The reduction is then complete. 
 
 The following is a sample of the form used. 
 
1883] TIDES OF LONG PERIOD. 57 
 
 [O.] 
 
 Form for Evaluation of , R, K, H. 
 logB = 
 
 log tan = 
 
 v __ 
 
 K = 
 
 B 2 =~ 
 
 R 3 = 
 
 R = 
 
 Augtn. = 
 Augd. R = 
 
 H = 
 
 A form similar to [O] serves for the same purpose in the treatment of 
 the tides of long period, to the consideration of which we now pass ; it will 
 be seen, however, that for these tides there is no augmenting factor, and that 
 the increase of n for 11 hours has to be added to . 
 
 10. On the Harmonic Analysis for the Tides of Long Period. 
 
 For the purpose of determining these tides we have to eliminate the 
 oscillations of water-level arising from the tides of short period. As the 
 quickest of these tides has a period of many days, the height of mean water 
 at one instant for each day gives sufficient data. Thus there will in a year's 
 observations be 365 heights to be submitted to harmonic analysis. In leap- 
 years the last day's observation must be dropped, because the treatment is 
 adapted for analysing 365 values. 
 
 To find the daily mean for any day it has hitherto been usual to take 
 the arithmetic mean of 24 consecutive hourly values, beginning with the 
 height at noon. This height will then apply to the middle instant of the 
 period from O h to 23 h : that is to say, to ll h 30 m at night. We shall propose 
 some new modes of treating the observations, and in the first of them it will 
 probably be more convenient that the mean for the day should apply to 
 midnight instead of to ll h 30 m . For finding a mean applicable to midnight 
 we take the 25 consecutive heights for O h to 24 h , and add the half of the 
 first" value to the 23 intermediate and to the half of the last and divide by 
 
58 TIDES OF LONG PERIOD. [1 
 
 24. It would probably be sufficiently accurate if we took ^ of the sum of 
 the 25 consecutive values, if it is found that the division of every 24th 
 hourly value into two halves materially increases the labour of computing 
 the daily means. The three plans for finding the daily mean are then 
 
 (64) 
 
 (iii)] 
 
 And they will be denoted as methods (i), (ii), (iii) respectively. It does not, 
 however, seem very desirable to use the third method. Major Baird considers 
 that the use of method (i) is most convenient for the computers. 
 
 The formation of a daily mean does not obliterate the tidal oscillations 
 of short period, because none of the tides, excepting those of the principal 
 solar series, have commensurable periods in mean solar time. 
 
 A correction, or ' clearance of the daily mean,' has therefore to be applied 
 for all the important tides of short period, excepting for the solar tides. 
 
 Let R cos (nt ) be the expression for one of the tides of short period 
 as evaluated by the harmonic analysis for the same year, and let a be the 
 value of nt at any noon. Then the 25 consecutive hourly heights of 
 water, beginning with that noon, are 
 
 R cos a, R cos (n + a), R cos (2/i + a) ... R cos (23n + a), R cos (24n + a) 
 
 In the method (i) of taking the daily mean it is obvious that the 
 'clearance' is 
 
 ,, sin I2n . - , 
 
 ^VR - i cos (a 4- HA? 
 
 sin \ n 
 
 In the method (ii) it is easily proved to be 
 
 .(65) 
 tan#/i 
 
 and in method (iii) it is 
 
 Gin -SJi-ij 
 
 i T Olll i)~fv / , 1 \ \ 
 
 ^R : j cos (a + I2n) 
 
 The clearance, as written here, is additive. 
 
 It was found practically in the computation for these tides that only 
 three tides of short period exercise an appreciable effect, so that clearances 
 for them have to be applied. These tides are the M 2 , N, tides. It was 
 usual to compute these three clearances for every day in the year, and to 
 correct the daily values accordingly. But in following this plan a great deal 
 of unnecessary labour has been incurred, and when a simpler plan is followed 
 it may perhaps be worth while to include more of the short-period tides in 
 the clearances. 
 
1883] TIDES OF LONG PERIOD. 59 
 
 Professor J. C. Adams suggests the use of the tide-predicting machine 
 for the evaluation of the sum of the clearances, and if this plan is not found 
 to inconveniently delay operations in India, it may perhaps be tried. 
 
 In explaining the process we will suppose that method (i) has been 
 followed ; if either of the other plans be adopted it will be easy to change 
 the formulae accordingly. 
 
 It is clear that Rcos(a + lljra) is the height of the tide n at ll h 30 m ; 
 and the same is true for each such tide. Hence if we use the tide-predicter 
 to run off a year of fictitious tides with the semi-range of each tide equal to 
 ^j sin 12n/sm^n of its true semi-range, and with all the solar series and the 
 annual and semi-annual tides put at zero, the height given at each ll h 30 m 
 in the year is the sum for each day of all the clearances to be subtracted. 
 The scale to which the ranges are set may of course be chosen so as to give 
 the clearances to a high degree of accuracy. 
 
 In the other process of clearance, which will be explained below, a single 
 correction for each short-period tide is applied to each of the final equations, 
 instead of to each daily mean. 
 
 We next take the 365 daily means, and find their mean value. This 
 gives the mean height of water for the year. If the daily means be un- 
 cleared, the result cannot be sensibly vitiated. 
 
 We next subtract the mean height from each of the 365 values, and find 
 365 quantities &h giving the daily height of water above the mean height. 
 
 These quantities are to be the subject of the harmonic analysis ; and the 
 tides chosen for evaluation are those which have been denoted above as Mm, 
 Mf, MSf, Sa, Ssa. 
 
 Let 8h A cos (a- BJ) t + B sin (cr ar) i 
 
 -f C cos 2at + D sin 2crt 
 
 + C' cos 2 (a- - 17) t + D' sin 2 (a - 77) t- (66) 
 
 + E cos rjt + F sin rjt 
 
 + G cos 2i)t + H sin fyt 
 
 where tis time measured from the first ll h 30 m . 
 
 Now suppose 1 1} 1 2 are the increments in 24 m. s. hours of any two of 
 the five arguments (<r iff)t, 2(rt, 2 (<r ;), rjt, 2r}t, and that A 1} B x ; A 2 , B 2 , 
 are the corresponding coefficients of the cosine and sine in the expression 
 for Bh. 
 
 Then if Bh { be the value of Bh at the (i + l) th ll h 30 m in the year, we 
 may write 
 
 Bhi = Aj cos l-ii + B! sin lj, + A 2 cos Li + B 2 sin Li + (67) 
 
60 TIDES OF LONG PERIOD. [1 
 
 And therefore 
 
 Shi cos Iji = ^A 2 (cos (^ + 4) i + cos (^ Z 2 ) i\ 
 
 + B 2 {sin (^ + Z a ) i - sin (^ - Z 2 ) *'} + ... 
 />i sin /!* = A 2 {sin (^ + 2 ) i + sin (^ 2 ) 1} 
 
 + B 2 { cos (7j + >) i + cos (7j 2 ) 1} + . . . 
 
 m n -2 / f* 
 
 Now let 6 (#) = A ^ 
 
 ' 
 
 ,1 
 so that 
 
 We may observe that 
 
 <j>(x) = (j> (- x\ and </> (0) = 
 
 If therefore 2 denotes summation for the 365 values from * = to i = 364, 
 we have 
 
 cos l,i = [<^> (^ + L 2 ) cos 182 (^ + I,) + <f> (I, - I,) cos 182 (I, - I,)] A 2 
 
 + [<f> (li + k} sin 182 (I, + 4) - <^ (4 - 4) sin 182 (I, - 1. 2 )] B 2 + . . . 
 
 sin /,t = [<^> (^ + 4) sin 182 (^ + 4) + <^> (4 - / 2 ) sin 182 (^ - i a )] A 2 
 
 + [- <t> (4 + 4) cos 182 (^ + Z s ) + (j> (4 - 4) cos 182 (^ - Z 2 )] B a + . . . ] 
 
 ......... (68) 
 
 In these equations there is always one pair of terms in which 1 2 is 
 identical with 1 1} and since <f>(l 1 -l 1 ) = 182, and cos 182 (^ - ^) = 1, it follows 
 that there is one term in each equation in which there is a coefficient nearly 
 equal to 182*5. In the cosine series it will be a coefficient of an A ; in the 
 sine series, of a B. 
 
 , The following are the equations (copied from the Report for 1872*) 
 with the coefficients inserted, as computed from these formulae, or their 
 equivalents : 
 
 * [Some small corrections have been introduced.] 
 
1883] 
 
 TIDES OF LONG PERIOD. 
 
 61 
 
 .O 
 
 <*> 
 
 RH 
 
 I 
 
 "8 
 
 
 
 00 
 
 co 
 
 oo 
 
 i i 
 
 5 
 
 i i 
 CM 
 
 
 
 
 
 8 
 
 g 
 
 (M 
 
 CO 
 
 *" T! 
 
 O 
 
 O 
 
 O 
 
 O 
 
 
 
 o 
 
 o 
 
 o 
 
 
 
 CM 
 
 ^t* 
 
 
 
 
 
 
 
 
 
 
 oo 
 
 a 
 
 
 
 
 
 
 
 
 
 
 I-H 
 
 o 
 
 
 1 
 
 + 
 
 1 
 
 1 
 
 1 
 
 1 
 
 + 
 
 + 
 
 + 
 
 
 0-1 
 
 O 
 
 CD 
 
 00 
 
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 O 
 
 10 
 
 * 
 
 
 00 
 
 
 
 OS 
 
 X 
 
 Ip 
 
 O 
 
 
 CM 
 
 CM 
 
 o 
 
 cc 
 
 <~> 
 
 '" c*3 
 
 Tf 
 
 cc 
 
 1 1 
 
 CC 
 
 1 1 
 
 M 
 
 6 
 
 6 
 
 f>l 
 
 6 
 
 G) 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
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 + 
 
 + 
 
 1 
 
 + 
 
 1 
 
 * 
 
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 + 
 
 + 
 
 + 
 
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 O5 
 
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 1 1 
 
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 CM 
 
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 "* a 5ii 
 
 6 
 
 O 
 
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 6 
 
 6 
 
 6 
 
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 6 
 
 
 
 
 
 
 
 
 
 oo 
 
 
 
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 1 
 
 + 
 
 1 
 
 1 
 
 1 
 
 I 
 
 + 
 
 + 
 
 + 
 
 + 
 
 o-i 
 o 
 
 oo 
 op 
 
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 cp 
 
 CM 
 
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 -u , , 
 
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 cc 
 
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 cc 
 
 
 
 CC 
 
 01 
 
 6 
 
 6 
 
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 ^H 
 
 
 
 
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 + 
 
 + 
 
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 + 
 
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 + 
 
 + 
 
 + 
 
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 6 
 
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 3 
 
 
 
 
 
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 6 
 
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 00 
 
 
 
 
 
 
 
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 + 
 
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 1 
 
 1 
 
 ^J 
 
 tO 
 
 ^ 
 
 b 
 
 
 
 ^ 
 
 .^ 
 
 
 
 
 ^b. 
 
 b 
 
 b 
 
 CM 
 
 
 
 ". 
 
 B- 
 
 c- 
 
 01 
 
 c- 
 CM 
 
 
 
 
 _C 
 
 
 
 
 
 
 
 
 
 00 
 
 o 
 
 _fl 
 
 c 
 
 _g 
 
 
 o 
 
 "3 
 
 o 
 
 '& 
 
 o 
 
 02 
 
 
 
 'm 
 
 
 
 OS 
 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 
 | 
 
 
 
 
 
 
 
 
 
 
62 CLEARANCE OF DAILY MEANS. [1 
 
 If the daily means have been cleared by the use of the tide-predicter as 
 above described, these ten equations are to be solved by successive approxi- 
 mation, and we are then furnished with the two component semi-amplitudes, 
 say AU Bj of the five long-period tides. But the initial instant of time is 
 the first ll h 30 m in the year instead of the first noon. Hence if as before 
 we put R 2 = Aj 2 + B x 2 , and tan ^ l = ^B 1 /A l , we must, in order to reduce the 
 results to the normal form in which noon of the first day is the initial instant 
 of time, add to the increment of the corresponding argument for ll h 30 m , 
 according to method (i), or for 12 hours according to methods (ii) or (iii). 
 
 If, however, the daily means have not been cleared, then before solution 
 of the final equations corrections for clearance Avill have to be applied, which 
 we shall now proceed to evaluate. 
 
 For this process we still suppose method (i) to be adopted. 
 
 Let n be the speed of a short-period tide in degrees per m. s. hour, and 
 
 m n 1 2 ?? 
 
 let K) = 7T 1 T r- Then we have already seen that the clearance to 8/</, 
 4 sm$n 
 
 the mean height of water at ll h 30 m of the (i + l) th day, will be 
 - ^ (n) R cos [n {24i + 11$}- ] 
 
 If we write m = 24w (so that m is the daily increase of argument of the 
 tide of short period), and ft = n x 11| , this becomes 
 
 ^r (n) R cos (mi + /3) 
 Hence the clearance for Shi cos li is 
 
 - ii/r (TO) R {cos [(m + l){ + 0] + cos [(in -l)i + /3]} 
 and for Bhi sin li is 
 
 - ty (n) R (sin [(m + l)i + @]- sin [(m -l)i + ]} 
 
 Summing the series of 365 terms we find that the additive clearance for 
 25A cos li is 
 
 (ro) {</> (m + I) cos [182 (m + l) + ft] + <j> (m - I) cos [182 (m-l) + &]} 
 
 Sill OC 
 
 where as before <6(a;) = A -- ........................... (69) 
 
 1 pn$a 
 
 If A?i denotes the increase of the argument nt in 182 d ll h 30 m , this may 
 now be written 
 
 (n) {< (m + 1) cos [An + 1821- ?] + <f> (m - 1} cos [Aw - 1822 - ]} 
 
 If therefore R cos = A, R sin = B, so that A and B are the component semi- 
 ranges of the tide n as immediately deduced from the harmonic analysis for 
 the tides of short period, we have for the clearance to 2$A cos li 
 
 - [^ (n) <f>(m + l) cos (Aw + 1821) + ^ (n) <f> (m - 1) cos (Aw - 1821)] A 
 ) <f>(m + l) sin (An + 1821) + ^ (n) (m - 1) sin (An - 1821)] B 
 
1883] CLEARANCE OF DAILY MEANS. 63 
 
 In precisely the same manner we find the clearance for 28/t sin li to be 
 - O (n) </> (m + sin (A/i + 1820 - ^ (0 <f> (M ~ sin (& n ~ 182J)] A 
 
 + fy (0 < (* + cos ( Aw + 182 ^) - ^ 00 < ( ~ cos ( An ~ 182 OJ B 
 These coefficients may be written in a form more convenient for com- 
 putation. For 
 
 , " sin ap (m 
 
 6 (m + = o i / r~7\ 
 2 sin | (m + l) 
 
 = l cos 182 (m + + i sin 182 (m I) cot (m I) ......... (70) 
 
 Then let K (n, 1} = <j>(m + l) + <f>(m - 1)] 
 
 Z ( n , l) = <f> (in + I) - (m - 1}] 
 
 . sn . r< / \ 
 
 Also let & (n) cos A?i = Jr. r cos Aw = C (n) 
 
 4 sin fit ............ (72) 
 
 T/T (n) sin Aw = S (n)) 
 
 The functions K (n, I), Z (/<, 0> C (n), S (n) may be easily computed from 
 (70), (71), (72). 
 
 Then if we denote the additive clearance for SSA cos li by 
 
 [A, n, I, cos] A + [B, n, I, cos] B 
 and that for 2,8k sin li by 
 
 [A, ?i, I, sin] A + [B, n, I, sin] B 
 We have 
 
 [A, n, I, cos] = - C On) K (n, 1} cos 182Z + S (n) Z (n, 1) sin 182^ 
 
 [B, ?i, I, cos] = - S (n) K (n, cos 182Z - C (n) Z (n, 1} sin 182J 
 
 [A, n, I, sin] = - S (ro) Z (n, cos 182J - C (n) K (n, I) sin 182J I ...... < ~ 
 
 [B, n, I, sin] = C (w) Z (n, cos 182* - S (n) K (n, 1} sin 1821) 
 We must remark that if \ (m + I) = 360, (??i + is equal to 182'5. 
 
 This case arises when I is the tide MSf of speed 2 (<r 77), and m the tide 
 M 2 of speed 2 (7 - o-), for m + i is then 24 x 2 (7 - rj) = 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 
 M.,, 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 
 
64 
 
 CLEARANCE OF DAILY MEANS. 
 
 [1 
 
 epoch chosen for the tides of long period ; to determine them it is necessary 
 to multiply R by the cosine and sine of V+u K at the epoch. 
 
 too 
 
 Schedule of Coefficients for Clearance of Daily Means in the Final Equations*. 
 
 I 
 
 0" 7ZT 
 
 2ff 
 
 8(-9) 
 
 n 
 
 2i? 
 
 (M a ) 71 = 2(7-0-) 
 
 [A, n, I, cos] 
 
 -0-05557 
 
 +0-00302 
 
 + 5-7393 
 
 -0-10410 -0-10465 
 
 [B, n, I, cos] 
 
 -0-17036 
 
 -0-03773 
 
 -2-9228 
 
 - 0-07525 
 
 - 0-07546 
 
 [A, n, I, sin] 
 
 -0-17075 
 
 + 0-04170 
 
 - 2-8400 
 
 -0-00176 
 
 - 0-00353 
 
 [B, , I, sin] 
 
 + 0-04410 
 
 + 0-01052 
 
 -5-7271 
 
 + 0-00476 
 
 + 0-00958 
 
 (N) n = 2y - 3<r + OT 
 
 [A, n, , cos] 
 
 - 0-05884 
 
 + 0-03680 
 
 + 0-02938 
 
 -0-01760 
 
 -0-01760 
 
 [B, n, I, cos] 
 
 -0-07758 
 
 -0-22357 
 
 -0-19384 
 
 + 0-00254 
 
 + 0-00254 
 
 [A, w, I, sin] 
 
 - 0-02059 
 
 -0-15257 
 
 -0-12210 
 
 + 0-00020 
 
 + 0-00041 
 
 [B, , , sin] 
 
 +0-11381 
 
 - 0-08544 
 
 - 0-08081 
 
 + 0-00007 
 
 + 0-00015 
 
 (0) n * 7 - 2o- 
 
 [A, n, , cos] 
 
 -0-06485 
 
 +0-01662 
 
 + 0-01571 -0-19240 
 
 - 0-19340 
 
 [B, n, I, cos] 
 
 - 0-34765 
 
 -0-07775 
 
 -0-08158 -0-18260 
 
 -0-18311 
 
 [A, n, , sin] 
 
 - 0-34523 
 
 + 0-08411 
 
 +0-08754 
 
 - 0-00460 
 
 - 0-00926 
 
 [B, n, 1, sin] + 0-04052 
 
 +0-03384 
 
 + 0-03306 
 
 + 0-00897 
 
 +0-01802 
 
 It may happen from time to time that 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 Bh. 
 Now, the whole process employed depends on the existence of 365 continuous 
 values of Bh. 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 Bh graphically on each 
 side of the hiatus, and filling in the gap with a curve drawn by hand, use the 
 values of Bh given 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. 
 
 * [A few small corrections have been introduced.; 
 
1883] TREATMENT OF GAPS IN THE RECORD. 65 
 
 If there is another station of observation in the neighbourhood, the values 
 of Bh for that station may be inserted. 
 
 The values of Bh for 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 "%8h cos It and S8A 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., 2A cos (<r r) t 149 A approxi- 
 mately. After the approximate values of A, B, C, D, &c., 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 as 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 Il h 30 m of the year, instead of from the initial Il h 30 m 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. 
 
 D. i. 5 
 
66 METHOD OF EQUIVALENT MULTIPLIERS. [1 
 
 11. Method of Equivalent Multipliers for the Harmonic Analysis 
 for the Tides of Long Period. 
 
 Up to the present time (1883) 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 five 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 t = 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 Bh's 
 corresponding to each value of t, and the summations gave XSA cos It and 
 "ZSh sin It, the numerical results being the left-hand sides of one pair of the 
 ten final equations explained in 10. 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 TO, '9, '8, '7... '2, '1, 0. Then, as all the values of Bh 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 8/1 which belong to one 
 of these groups, sum them, and multiply the sum by the corresponding 
 multiplier, TO, '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 Stis. This changing of sign may be effected mechanically as 
 follows : In the spaces for entry of the Bh's, those Bh's 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 8h's 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 SS/t sin It is precisely similar. 
 
 The annexed form [Schedule R] is designed for entry for determination 
 of SSA cos (a- ?/) t. 
 
 The entries of Bh 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 (a 77). In the middle pair 
 for each period the + values of Bh 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 
 
1883] 
 
 SCHEDULE FOR TIDES OF LONG PERIOD. 
 
 67 
 
 tide of long period ; they are very easily prepared from the existing forms, 
 in which the values of the multipliers are already computed. 
 
 [In the form as finally prepared for the Indian Government the rows marked 
 ' change ' are treated in a schedule by themselves separated from the others.] 
 
 [R.] 
 Form for Reduction of the Tide MSf. 
 
 1 
 
 2 4 
 
 5 4 
 
 Total + . 
 
 Total - . 
 
 Total . . 
 
 Multiply . 
 
 Kesults . 
 
 + - 
 1-0 
 
 + - 
 9 
 
 1 
 
 8 
 
 | 
 
 7 
 
 + - 
 6 
 
 + - 
 5 
 
 + - 
 
 4 
 
 + - 
 3 
 
 + - 
 2 
 
 + - 
 1 
 
 CO 
 
 
 'E 
 
 *s 
 
 
 
 
 
 l 
 
 
 2 
 
 
 
 
 3 
 
 
 
 
 7 
 
 
 6 
 
 
 
 5 
 
 
 
 
 4 
 
 
 8 
 
 
 9 
 
 
 
 
 10 
 
 
 
 
 11 
 
 14 
 
 
 
 13 
 
 
 
 12 
 
 
 
 
 
 15 
 
 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. 
 
 
 
 
 *1 
 
 0> 
 
 xl-0 
 
 x-9 
 
 X'8 
 
 x-7 
 
 x-6 
 
 x '5 
 
 X'4 
 
 X'3 
 
 x-2 
 
 x-1 
 
 x-0 
 
 
 
 
 
 
 
 
 
 
 
 
 Sum laterally 
 
 . Sum of + = . 
 cos 2 (a- 77) t = . 
 
 . Sum of - 
 
 change 
 change 
 
 change 
 change 
 
 change 
 change 
 
 change 
 change 
 
 change 
 change 
 
 52 
 
68 
 
 AUXILIARY TABLES. 
 
 [1 
 
 12. AUXILIARY TABLES DRAWN UP UNDER THE SUPERINTENDENCE OF 
 
 MAJOR BAIRD, R.E. 
 
 [Largely abridged.] 
 
 Values of N (Long. Moons Ascending Node) for O h Jan. 1, G.M.T. 
 
 Value at O h G.M.T. Jan. 1, 1880 = 285-956863 
 
 Motion per Julian year in 1880 = 19 "34146248 
 
 Motion for 365 days = 19 -32822387, and for 1 e% = -052954 
 
 Year 
 
 N 
 
 Year 
 
 N 
 
 1900 
 
 259 ? 1276 
 
 1910 
 
 657395 
 
 1 
 
 239-7994 
 
 1 
 
 46-4112 
 
 2 
 
 220-4712 
 
 2 
 
 27-0830 
 
 3 
 
 201-1429 
 
 3 
 
 7-7018 
 
 4 
 
 181-8147 
 
 4 
 
 348-3736 
 
 1905 
 
 162-4335 
 
 1915 
 
 329-0454 
 
 6 
 
 143-1053 
 
 6 
 
 309-7172 
 
 7 
 
 123-7771 
 
 7 
 
 290-3360 
 
 8 
 
 104-4489 
 
 8 
 
 271-0078 
 
 9 
 
 85-0677 
 
 9 
 
 251-6795 
 
 Decrement of N since O h Jan. 1 up to midnight of certain days of the year. 
 
 [Omitted.] 
 
 Values of p l (Mean Long, of Solar Perigee) for O h Jan. 1. 
 
 [Abridged.] 
 
 
 
 Value at O h Jan. 1, 1880 = 280 '874802 
 Notion per Julian year =0-01710693 
 Motion for 365 days =0 -01709295 
 Motion for 1 day =0 -00004683 
 
 Year 
 
 ft 
 
 Year 
 
 ft 
 
 1900 
 
 281 ? 2169 
 
 1910 
 
 281 ? 3879 
 
 1 
 
 2340 
 
 1 
 
 4050 
 
 2 
 
 2511 
 
 2 
 
 4221 
 
 3 
 
 2682 
 
 3 
 
 4393 
 
 4 
 
 2853 
 
 4 
 
 4564 
 
 1905 
 
 3024 
 
 5 
 
 4735 
 
 6 
 
 3195 
 
 6 
 
 4906 
 
 7 
 
 3366 
 
 7 
 
 5078 
 
 8 
 
 3537 
 
 8 
 
 5249 
 
 9 
 
 3708 
 
 9 
 
 5420 
 
1883] 
 
 AUXILIARY FORMULAE. 
 
 69 
 
 [Tables, computed by Colonel Baird, for /, v, and the factors f in terms 
 of N the longitude of the node are here replaced by algebraic formulae, which 
 are sufficiently accurate for all practical purposes. The formulae are derived 
 from the tables in Baird's Manual by the method of special values. 
 
 The angles v, |, v', 2//' are expressible in the form 
 A! sin N + A 2 sin 2N + A 3 sin 3N 
 
 The values of the A's for the several angles are given in the following 
 schedule : 
 
 The angle 
 
 A! 
 
 A, 
 
 A, 
 
 V 
 
 12 '94 
 
 - 1-34 
 
 + 0-19 
 
 
 
 ll-87 
 
 - 1-34 
 
 +0-19 
 
 v 
 
 8-86 
 
 -0 3 -68 
 
 + 0-07 
 
 Zv" 
 
 17'74 
 
 -0-68 
 
 + 0-04 
 
 The factors f are expressible in the form 
 
 f = B + Bj cos N + E, cos 2N + B 3 cos 
 
 Tides for which 
 f is applicable 
 
 B 
 
 B i 
 
 B 2 
 
 BS 
 
 M 2 , N, 2N, v, i-oofm 
 MS, 2SM, MSf 
 
 - -03733 
 
 + 00017 
 
 + -ooooi 
 
 K 2 
 Ki 
 
 1-0241 
 1-0060 
 
 + 2863 
 + 1150 
 
 + -0083 
 - -0088 
 
 -0015 
 + 0006 
 
 0, Q 10089 
 
 + 1871 
 
 -0147 
 
 + 0014 
 
 OO 
 
 1-1027 
 
 + 6504 
 
 + 0317 
 
 -0014 
 
 J 
 
 1-0129 
 
 + 1676 
 
 -0170 
 
 + 0016 
 
 Mf 
 
 1-0429 
 
 + 4135 
 
 -0040 
 
 0000 
 
 Mm 
 
 1-0000 --1300 
 
 + 0013 
 
 0000 
 
 May 1906.] 
 
2. 
 
 ON THE PERIODS CHOSEN FOR HARMONIC ANALYSIS, AND 
 A COMPARISON WITH THE OLDER METHODS BY MEANS 
 OF HOUR-ANGLES AND DECLINATIONS. 
 
 [Third Report of the Committee, consisting of Professors 
 G. H. DARWIN and J. C. ADAMS, for the Harmonic Analysis 
 of Tidal Observations. Drawn up by Professor G. H. DARWIN. 
 British Association Report for 1885, pp. 35 60.] 
 
 I. RECORD OF WORK DURING THE PAST YEAR. 
 
 A LARGE number of tidal results have been obtained by the United States 
 Coast Survey, and reduced under the superintendence of Professor Ferrel. 
 Although the method pursued by him has been slightly different from that 
 of the British Association, it appears that the American results should be 
 comparable with those at the Indian and European ports. Professor Ferrel 
 has given an assurance that this is the case ; nevertheless, there appears to 
 be strong internal evidence that, at some of the ports, some of the phases 
 should be altered by 180. 
 
 II. CERTAIN FACTORS AND ANGLES USED IN THE REDUCTION OF 
 TIDAL OBSERVATIONS. 
 
 [These are given at the end of the last Paper.] 
 
1885] CHOICE OF PERIODS FOR ANALYSIS. 71 
 
 III. ON THE PERIODS CHOSEN FOR HARMONIC ANALYSIS IN THE 
 COMPUTATION FORMS. 
 
 Before proceeding to the subject of this section, it may be remarked that 
 it is unfortunate that the days of the year in the computation forms should 
 have been numbered from unity upwards, instead of from zero, as in the case 
 of the hours. It would have been preferable that the first entry should have 
 been numbered Day 0, Hour 0, instead of Day 1, Hour 0. This may be 
 rectified with advantage if ever a new issue of the forms is required, but the 
 existing notation is adhered to in this section. 
 
 The computation form for each tide consists of pages for entry of the 
 hourly tide-heights, in which the entries are grouped according to rules 
 appropriate to that tide. The forms terminate with a broken number of 
 hours. This, as we shall now show, is erroneous, although this error may not 
 be of much practical importance. 
 
 In 9 of the Report for 1883 the following passage [omitted in the pre- 
 ceding paper] occurs : 
 
 ' The elimination of the effects of the other tides may be improved by 
 choosing the period for analysis not exactly equal to one year. For suppose 
 that the expression for the height of water is 
 
 AjCOSMjtf + Bj sin n^t + A 2 cos w 2 + B 2 sin n 2 t (61) 
 
 'where n. 2 is nearly equal to n^, and that we wish to eliminate the r^-tide, so 
 as to be left only with the ^-tide. 
 ' Now, this expression is equal to 
 
 [A! 4- AJ, cos (X w 2 ) t B 2 sin (% n. 2 ) t] cos nj, ' 
 
 (62) 
 
 -f {Bj + A 2 sin (r?i n 2 ) t + B 2 cos (^ n 2 ) t} sin n^t j 
 
 'That is to say, we may regard the tide as oscillating with a speed w 1} 
 but with slowly varying range.' 
 
 Although this is thus far correct, yet the subsequent justification of the 
 plan according to which the computation forms have been compiled is wrong. 
 
 In the column appertaining to any hour in the form we have nj a multiple 
 of 15, if W! be a diurnal, and of 30, if n Y be a semidiurnal tide. 
 
 Consider the column headed '_p-hours ' ; then nj = 15^ for diurnals, and 
 30* p for semidiurnals. 
 
 Hence (62), quoted above, shows us that, for diurnal tides, the sum of 
 all the entries (of which suppose there are q) in the column numbered 
 j9-hours, is 
 
 , KO f A r x is r, N /2-Tr i5\i 
 
 cos 15 p {A^ + 4i cos(w! n 2 ) - +cos (n^ n 2 ) I 1 I 
 
 + cos (/?! - no) ( 2 + -^ j M- ... + B [&c.] [ + sin 15^ {&c.} . . .(a) 
 
 -' \ n, n, J] } 
 
72 CHOICE OF PERIODS FOR ANALYSIS. [2 
 
 And for semidiurnal tides the arguments of all the circular functions in (a) 
 are to be doubled. 
 
 Now, we want to choose such a number of terms that the series by which 
 A 2 and B 2 are multiplied may vanish. This is the case if the series is exactly 
 re-entrant, and is nearly the case if nearly re-entrant. 
 
 The condition is exactly satisfied for diurnal tides, if 
 
 where r is either a positive or negative integer. And for semidiurnal tides, if 
 
 fa w 2 ) q = 2jrr 
 HI 
 
 That is to say, fa n^)q = n^, for diurnal tides 
 
 or fa nz)q ^n : r, for semidiurnal tides 
 
 It is not worth while attempting to eliminate the effect of the semi- 
 diurnal tides on the diurnal tides, and vice versa, because we cannot be more 
 than a fraction of a day out, and on account of the incommensurability of the 
 speeds we cannot help being wrong to that amount. 
 
 S Series. 
 
 Now suppose we are analysing for the S 2 tide, and wish to minimise the 
 effect of the M 2 tide. 
 
 Then ^ = 2 (7 - 77) = 2 x 15 per hour 
 
 n z = 2 (7 - a) 
 
 n, - w 2 = 2 (a- - 77) = 1'0158958 per hour 
 The equation is 1-0158958? = 15r 
 
 If r= 25, 9 = 369-1 3 
 
 Thus 25 periods of 2 (cr - 77) is 369'1 3 mean solar days. It follows, there- 
 fore, that we must sum the series over 369 days in order to be as near right 
 as possible. 
 
 Now this is equally true of all the columns, and each should have 369 
 entries. 
 
 Hence, in order to have 369 entries in each column, the S 2 computation 
 form (as used in India) should be corrected accordingly. 
 
 M Series. 
 
 Now consider that we are analysing for M 2 , and wish to minimise the 
 effect of the S 2 tide. Hence 
 
 n, = 2 (7 - <r) = 2 x 14 C '4920521 per hour 
 ?i 2 = 2 (7 - 77) 
 Wj n 2 = 1 4 01 58958 per hour 
 
1885] CHOICE OF PERIODS FOR ANALYSIS. 73 
 
 Hence, taking r negative, the equation is 
 
 1-0158958 = 14-4920521r 
 If r= 25, 2 = 356-63 
 
 Thus 25 periods of 2 (<r 77) is 356'63 of mean lunar time. 
 It follows, therefore, that we must have 357 entries in each column. 
 
 The M 2 computation form in use should be also corrected by adding 
 9 entries amongst which there are no 'changes.' 
 
 K Series. 
 
 To minimise the effect of M 2 on K 2 , we have 
 
 n, = 2 7 = 2 x 15 0410686 per hour 
 n 2 = 2 (7 a-) 
 
 n, - n 2 = 2<r = 1'0980330 per hour 
 1-09803300 = 15-0410686r 
 If r = 27, g = 369'85 
 
 Hence we should complete the row numbered 370 ; and correct the form 
 accordingly. 
 
 To minimise the effect of O on K 1} we have 
 
 n, = 7 = 15-0410686 per hour 
 n z 7 2<r 
 
 n, - n a =2o-= 1 -0980330 per hour 
 1-0980330^= 15-0410686r 
 
 This is the same equation again, and it confirms the conclusion that the 
 row numbered 370 should be completed. 
 
 The N Series. 
 
 Here n^ = 2y - 3<r 4- or = 2 x 14'2 198648 per hour 
 
 To minimise the effect of M 2 , 
 
 n z = 27 2<7 
 Wl - w 2 = - (a - w) = - 0'5443747 per hour 
 
 0-5443747^= 14-2198648r 
 If r = 13, q = 339-58 
 
 Hence in the computation form we should complete the row numbered 
 340. 
 
 There is no justification for the alternative offered in the computation 
 forms of continuing the entries up to 369 d 3 b of mean solar time. 
 
74 CHOICE OF PERIODS FOR ANALYSIS. [2 
 
 The L Series. 
 
 Here ^ = 2 7 - a- - = 2 x 14 C> 7 642394 per hour 
 
 To minimise the effect of M 2 , 
 
 w 2 = 2720- 
 ?ij - w 2 = o- CT = 0< 5443747 per hour 
 
 0-5443747? = 14'7642394r 
 Ifr=13, q = 352-58 
 Hence we should complete the row numbered 353. 
 
 There is no justification for the alternative offered in the computation 
 forms of continuing the entries up to 369 d 3 h of mean solar time. 
 
 The v Series. 
 
 Here ^ = 2 7 - 3<r - w + 2^ = 2 x 14'2562915 per hour 
 
 To minimise the effect of M 2 , 
 
 w 2 = 2 7 - 2o- 
 Wj Wg = a- is + 2r) 0'47l5211 per hour 
 
 0-4715211 9 =14-2562915?- 
 If r= 11, 9 = 332-6 
 Hence we should complete the row numbered 333. 
 
 There is no justification for the alternative offered in the computation 
 forms of continuing the entries up to 369 d 3 h of mean solar time. 
 
 The X Series. 
 
 Here n, = 2 7 - a + vr - 2rj = 2 x 14' J '7278127 per hour 
 
 To minimise the effect of M 2 , 
 
 n 2 = 272o- 
 ??i ?i 2 = a- + TV 2rj = 0< 4715211 per hour 
 
 0-4715211^=14-7278127r 
 If r= 11, 9 = 343-58 
 Hence we should complete the row numbered 344. 
 
 There is no justification for the alternative offered in the computation 
 forms of continuing the entries up to 369 d 3 h of mean solar time. 
 
 The 2N Series. 
 
 Here ^ = 2? - 4<r + 2w = 2 x 13"9476774 per hour 
 
1885] CHOICE OF PERIODS FOR ANALYSIS. 75 
 
 To minimise the effect of M 2 , 
 
 2 = 27 - 2o- 
 Wl - W2 = - 2 (<r - r) = 1 -0887494 per hour 
 
 1-08874949 = 13'9476774r 
 If r = 26, 9 = 333-08 
 Hence we must in the computation form complete the row numbered 333. 
 
 The T Series. 
 
 Here n, = 2y - 3?? = 2 x 14'9794657 per hour 
 
 To minimise the effect of M 2 , 
 
 n z = 27 2<r 
 nj - a = 2<r - 377 = 0-9748272 per hour 
 
 0-97482729 = 14-9794657r 
 If r = 24, 7=368-79 
 Hence we must in the form complete the row numbered 369. 
 
 The R Series. 
 
 Here n 1 = 2y-r ) = 2x 15'0205343 per hour 
 
 To minimise the effect of M 2 , 
 
 n 2 = 27-2o- 
 w, - w s = 2o- - 77 = 1-0569644 per hour 
 
 1-05696449 = 15-0205343r 
 If r = 25, 9 = 355-28, arid r = 26, 9 = 369'49 
 Hence we should either complete the row numbered 355 or that numbered 
 
 369. 
 
 The 2MS Series. 
 
 Here n, = 2 7 - 4o- + 2iy = 2 x 13'9841042 per hour 
 
 To minimise the effect of M 2 , 
 
 w, = 27 2o- 
 Hl - n, = - 2 (or - T?) = - 1'01 58958 per hour 
 
 1-01589589= 13'9841042r 
 If r = 24, 9 = 330-37, and r = 25, g = 344'13 
 Hence we should either complete the row numbered 330 or that numbered 
 
 344. 
 
76 CHOICE OF PERIODS FOR ANALYSIS. [2 
 
 The 2SM Series. 
 
 Here w, = 2 7 + 2o- - 4>rj = 2 x 15'5079479 per hour 
 
 To minimise the effect of M 2 , 
 
 n 2 = 27 2<r 
 Wl - n 2 = 4 (a- - 77) = 2'03l7916 per hour 
 
 2-03179165 = 15-5079479r 
 If r = 48, ^ = 366-37 
 Hence we should complete the row numbered 366. 
 
 The O Series. 
 
 Here n, = 7 - 2o- = 13'9430356 per hour 
 
 To minimise the effect of Kj , 
 
 n 2 = 7 
 Wl - w 2 = - 2o- = - 1'0980330 per hour 
 
 1-0980330^= 13-9430356r 
 Ifr=27, 5 = 342-85 
 
 Hence we should complete the row numbered 343, cutting off the last 
 three entries in the forms as in use. 
 
 The P Series. 
 
 Here n l = y-2ij= 14'9589314 per hour 
 
 It is open to question whether it is best to minimise the effect of K! or 
 of O. 
 
 For Kj take n 2 = 7 
 
 n 1 -n 2 = -'2r) = - 0'0821372 per hour 
 
 0-08213725 = 14'9589314r 
 If r = 2, 5 = 364-24 
 
 Hence we should complete the row numbered 364. 
 For O, take W 2 = 7 2cr 
 
 n, - n 2 = 2 (a- - rj) = 1'0158958 per hour 
 
 1-01589585 = 14'9589314r 
 If r = 25, 5 = 36812 
 
 Hence we should complete the row numbered 368. 
 
 It is better to abide by this, for in the former case n^ n 2 varies very 
 slowly ; and we may be satisfied that on stopping with row 368 the effects 
 of O and Kj will both be adequately eliminated. 
 
1885] CHOICE OF PERIODS FOR ANALYSIS. 77 
 
 The J Series. 
 
 Here Wj = 7 + a nr = 15'5854433 per hour 
 
 To minimise the effect of K 1} 
 
 a = 7 
 OTI - w a = a - tar = 0'5443747 per hour 
 
 0-5443747?= 15-5854433?- 
 If r = 12, q = 343-56, and r = 13, = 37219 
 To minimise the effect of 0, 
 
 7i 2 = <y 2<r 
 ri x w, 2 = 3o- w = 1 -6424077 per hour 
 
 1-64240770 = 15-5854433r 
 If r = 36, q = 341'6, and r = 39, 5 = 370'09 
 
 Since in the latter case Wj w 2 varies three times as fast as in the former, 
 it will be better to abide by this, and stop either with the row numbered 
 342 or that numbered 370. 
 
 The Q Series. 
 
 Here n, = 7 - 3<r + vr = 13'3986609 per hour 
 
 To minimise the effect of K x , 
 
 2 = 7 
 Ml -n., = - (3o- - w) = - 1-6424077 per hour 
 
 1-6424077^ = 13'3986609r 
 If?- = 38, = 310-00 
 
 To minimise the effect of 0, 
 
 n. 2 = y2(7 
 TOJ - n 2 = - (o- - w) = - 0'5443747 per hour 
 
 0-54437470 = 13'3986609r 
 If r =12, = 307-36 
 
 Since in the former case n^ n. 2 varies about three times as fast as in 
 the latter, it will be better to abide by the former, and stop with the row 
 numbered 310. 
 
 With regard to the quaterdiurnal and terdiurnal tides, it does not 
 signify where we stop ; but it seems more reasonable to stop with the exact 
 year of 365 mean solar days. These tides are called MS, MN, MK, 2MK. 
 
78 
 
 CHOICE OF PERIODS FOR ANALYSIS. 
 
 Schedule II. 
 Periods over which Hie Harmonic Analysis should extend. 
 
 Initial of series 
 
 Number of day 
 last entry in 
 
 and hour of 
 special time 
 
 Period elapsing 
 cial day 1 to 23 h 
 day in mean 
 
 from O h of spe- 
 of last special 
 solar hours 
 
 S 
 
 369 d 
 
 23 h 
 
 368 J 
 
 23 h 
 
 
 M 
 
 357 
 
 23 
 
 369 
 
 11 
 
 
 K 
 
 370 
 
 23 
 
 368 
 
 23 
 
 
 N 
 
 340 
 
 23 
 
 358 
 
 15 
 
 
 L 
 
 353 
 
 23 
 
 358 
 
 14 
 
 
 V 
 
 333 
 
 23 
 
 350 
 
 8 
 
 
 \ 
 
 344 
 
 23 
 
 350 
 
 8 
 
 
 2N 
 
 333 
 
 23 
 
 358 
 
 2 
 
 
 T 
 
 369 
 
 23 
 
 369 
 
 11 
 
 
 R 
 
 355 
 or 370 
 
 23 
 23 
 
 354 
 or 369 
 
 11 
 11 
 
 
 2MS 
 
 330 
 or 344 
 
 23 
 23 
 
 353 
 
 or 368 
 
 22 
 23 
 
 
 2SM 
 
 366 
 
 23 
 
 353 
 
 23 
 
 
 
 
 343 
 
 23 
 
 368 
 
 23 
 
 
 P 
 
 368 
 
 23 
 
 368 
 
 23 
 
 
 J 
 
 342 
 or 370 
 
 23 
 23 
 
 329 
 or 356 
 
 3 
 
 1 
 
 
 Q 
 
 310 
 
 23 
 
 347 
 
 
 
 
 In the second column the numbers are given to the nearest mean solar 
 hour. 
 
 IV. A COMPARISON OF THE HARMONIC TREATMENT OF TIDAL 
 OBSERVATIONS WITH THE OLDER METHODS. 
 
 1. On the Method of Computing Tide-tables. 
 
 There is nothing in the harmonic reduction of tidal observations which 
 necessitates recourse to mechanical prediction of the tides. It may happen 
 that it is desirable to produce a tide-table by arithmetical processes, and that 
 the computers prefer to use the older methods of corrections, or it may be 
 desired to obtain the tidal constants in the harmonic notation from older 
 observations. For either of these purposes it is necessary to show how the 
 harmonically expressed results may be converted into the older form, so that 
 the constants for the fortnightly inequality in time and height, and the 
 
1885] 
 
 SYNTHETIC METHOD. 
 
 79 
 
 corrections for parallax and declination may be obtained from those of the 
 harmonic analysis, and conversely. 
 
 In the following sections I propose, therefore, first to reduce the harmonic 
 presentment of the resultant tide into the synthetic form, where we have 
 a single harmonic term depending on the local mean solar time of moon's 
 transit, and on corrections depending on the R.A., declination, and parallax 
 of the perturbing bodies. Subsequently it will be shown how a synthesis 
 may be carried out more simply by retaining the mean longitudes and 
 elements of the orbits. 
 
 2. Notation for Mean Heights and Retardations derived from the 
 
 Harmonic Method. 
 
 The notation of the Report of 1883 [Paper 1] is adopted; and I shall 
 carry the approximation to about the same degree as has been adopted by 
 the older writers. Closer approximation may, of course, be easily obtained. 
 
 In the Report of 1883 the mean height* of a tide is denoted by H, and 
 the retardation or lag by K. In the present note it will be necessary to refer 
 to several of the H's and K'S at the same time, and therefore it is expedient 
 to introduce the following notation : 
 
 Schedule III. 
 
 Initial of 
 tide 
 
 Mean height 
 (H) 
 
 Eetardation 
 
 w 
 
 Initial of 
 tide 
 
 Mean height 
 (H) 
 
 Retardation 
 
 W 
 
 M s 
 
 M 
 
 2fJL 
 
 L 
 
 L 
 
 2X 
 
 S, 
 
 S 
 
 K 
 
 T 
 
 T 
 
 K 
 
 Lunar K 2 
 
 K" 
 
 2 K 
 
 R 
 
 R 
 
 K 
 
 Solar K 2 
 
 K t " 
 
 2* 
 
 
 
 M' 
 
 ? 
 
 K 2 
 
 K- 2 
 
 2* 
 
 P 
 
 S' 
 
 C 
 
 N 
 
 N 
 
 2v 
 
 K! 
 
 K, 
 
 KI 
 
 In this schedule we assume T and R (of speeds 2y 3rj and 2y 17) to 
 have the same lag as S 2 ; and we use v in a new sense, the old v, the R.A. of 
 the intersection of the equator with the lunar orbit, being denoted by v . 
 The initials of each tide are used to denote its height at any time. 
 
 * I use height to denote semi-range. All references to this Report will simply be by the 
 date 1883. 
 
80 HOUR-ANGLES, DECLINATIONS, AND PARALLAXES. [2 
 
 3. Introduction of Hour-angles, Declinations, and Parallaxes. 
 
 We must now get rid of the elements of the orbit and of the mean 
 longitudes, and introduce hour-angles, declinations, and parallaxes. 
 
 At the time t let a, 8, ty be })'s R.A., and declination, and hour-angle, 
 and a t , &,, "fy t , 's R.A., and declination, and hour-angle. 
 
 Let I be ])'s longitude in her orbit measured from ' the intersection,' arid 
 a i/ (i/ being the v of 1883) be ])'s R.A. measured from the intersection. 
 
 The annexed figure exhibits the relation of the several angles to one 
 another. 
 
 The spherical triangle affords the relations 
 
 tan (a v ) = cos 7 tan I, sin 8 = sin 7 sin / (1) 
 
 From the first of (1) we have, approximately, 
 
 a = / + i/ -tan 2 J7sin2Z (2) 
 
 Now, s % is the moon's mean longitude measured from /, and s p is 
 the mean anomaly. Hence, approximately, 
 
 J = s-f+2esin(s-jp) (3) 
 
 And therefore, approximately, 
 
 a = s + i/o | + 2e sin (s - p) tan' 2 \I sin 2 (s ) (4) 
 
 Now, t + h being the sidereal hour-angle, 
 
 ty = t + h-a (5) 
 
 Therefore, from (4) and (5), 
 
 t + k s (v ) = ^ + 2e sin (s p) tan 2 ^7 sin 2 (s ) . . .(6) 
 By the second of (1) we have, approximately, 
 
 cos 2 8 = 1-1 sin 2 / + \ sin 2 7 cos 2 (s - % ) (7 ) 
 
 Hence, if A be such a declination that cos 2 A is the mean value of cos 2 S, we 
 have 
 
 cos 2 A = 1 i sin 2 7) 
 
 j- (<$) 
 
 and cos 2 A, = 1 sin 2 o>J 
 
 From this we have (neglecting terms in sin 4 A) the following relations : 
 cos 4 \I = cos 2 A, sin 7 cos 2 \1 = V2 sin A cos A, sin 2 7 = 2sin 2 A 
 cos 4 \ a) = cos 2 A /; sin &> cos 2 o> = \/2 sin &> cos CD, sin 2 o> = 2 sin 2 A, 
 
1885] HOUR- ANGLES, DECLINATIONS, AND PARALLAXES. 81 
 
 Thus we may put 
 
 cos 4 / _ cos 2 A sin 1 cos 2 \ I sin 2A \ 
 
 cos 4 \ o> cos 4 \ i cos 2 A, ' sin G>cos 2 |o> cos 4 i ~ siiT2 A, 
 sin 2 / sin 2 A 
 
 An approximate formula for A and the value of A are 
 
 A = 16-51 + 3H4 cos N - 0'19 cos 2N, A / = 16'36 ...... (10) 
 
 The introduction of A and A, in place of / and <w entails a loss of 
 accuracy, and it is only here made because former writers have followed that 
 plan. It may easily be dispensed with. 
 
 Now let us write 
 
 D = cos 2 - ), D' = sin 2 (s - )) 
 
 (11) 
 II = cos(s p), IT = sm(s p) ) 
 
 From (7) and (8), 
 
 ~ _ cos 2 8 - cos 2 A _ sin 8 cos 8 dS 
 
 sin'A : o-sin 2 A dt ............ ( } 
 
 Then, if we write for the ratio of the moon's parallax to her mean parallax P, 
 we have 
 
 P I =ecos(s p) 
 and II = iP-l H' 
 
 , - ~~ 
 
 e^ e (a- - vr) dt 
 
 Hence D, D', IT, II' are functions of declinations and parallaxes. The 
 similar symbols with subscript accents are to apply to the sun. 
 
 Now (6) may be written by aid of (9) and (11), 
 
 2[ + /*-s-( I / -f)]=2T/r + 4eir-Z> / tan 2 A ......... (14) 
 
 The left-hand side of (14) is the argument of M 2 (see Sched. B. i. 1883), 
 and from (9) the factor of M 2 is cos 2 A/cos 2 A,. Hence, subtracting the 
 retardation 2/i from (14) we have 
 
 (>/-!* A 
 
 (M 2 ) = >~ M cos [(2i/r + 4eH' - D' tan 2 A) - 2/a] 
 
 COS" tA 
 
 expanding approximately, 
 
 cos 2 
 
 -ri .................. (15) 
 
 D. i. 6 
 
82 HOUR-ANGLES, DECLINATIONS, AND PARALLAXES. [2 
 
 We shall see later that the two latter terms of (15) are nearly annulled 
 by terms arising from other tides, and as in the case of the sun the rates of 
 change of parallax and declination are small, we may write by symmetry, 
 
 (S 2 ) = S cos 2(^-0 ........................ (16) 
 
 In all the smaller tides we may write 
 
 t + h -S -(!/- |) = i|r 
 
 A general formula of transformation will be required below. Thus, if 
 cos 2x = X, sin '2x = X', 
 
 cos 2 ty + a; - a) = {X + tan 2 (a - p) X'} cos 2 (-f - a) 
 
 The lunar K 2 tide. 
 
 From Sched. B. i, 1883, we have 
 
 Lunar K, = *" cos 2 [t + A - . - *] 
 
 Applying (17) with X = D, X' = D', a = K, and taking the lower sign, 
 
 ci n 2 A |~ 
 
 Lunar K 2 = -^ K" {D + tan 2 (K - p) D'} cos 2 (^ - ) 
 sm -A 
 
 - sin 2 (^ - /*)] ...(18) 
 
 COS 2(K fJi) 
 
 In the case of the sun we neglect the terms in D', for the same reasons 
 as were assigned for the similar neglect in (16), and have 
 
 /:) .................. (19) 
 
 The tide N. 
 
 From Schedule B. i., Report 1883, 
 
 cos^/ ^^ 2 _ & _ 
 
 44 - 
 
 Then (N) = tfeoB 2 [f - - M* -/>)] 
 
 L-Ub ii-i 
 
 Then applying (17) with X=Yl, X'=H',a v, and taking the upper 
 sign, but writing //, v instead of v p, because this tide being slower than 
 M 2 suffers less retardation, 
 
 Pn ej2 A F 
 
 (N) = - ?-N m + tan 2 (p. - v) IT} cos 2 (^ - ) 
 cos' 2 a, 
 
 cos 
 
 sn 
 
1885] HOUR- ANGLES, DECLINATIONS, AND PARALLAXES. 83 
 
 The tide L. 
 
 We shall here omit the small tide of speed 2y <r + r, by which the 
 true elliptic tide is perturbed. Thus the R in the column of arguments in 
 Sched. B. i., 1883 is neglected, and we have 
 
 COS 4 ft) 
 
 cos 2 A 
 
 L COS 2 Nr \ + A (s 
 
 Applying (17) with X = II, X' = II', a=X, and taking the lower sign, 
 and changing the sign of the whole, because of the initial negative sign, 
 
 on<5 2 A r 
 
 (L) = ^^ L {- II - tan 2(X-/*) II'} cos 2 (f -X) 
 
 (OS jlA 
 
 swra o/ N and L. 
 
 In order to fuse these terms an approximation will be adopted. The L 
 tide is just as much faster than M 2 as N is slower, but the N tide should be 
 nearly 7 times as great as the L tide; hence the tan 2 (X /i) in (21) will be 
 put equal to tan 2 (p v). We then have 
 
 (N) + (L) = -[{n + tan 2(ji-v) II'} {Ncos 2 (f - ) - L cos 2 ty - X)} 
 
 COS LA 
 
 + IT {N sec 2 (/* - i/) + L sec 2 (X - /i)} sin 2 (^ - /A)] 
 But 
 
 7V COS 2 (l|r - i>) - L COS 2 (-\Jr X) = COS 2A/r (JV" COS 2v L COS 2X) 
 
 + sin 2i/r (JV sir 2v - L sin 2X) 
 
 . . JV sin 2i> - L sin 2X _ 
 
 Then writing tan 2e = -^ - ^ - ^ -- ^ ..................... (22) 
 
 
 
 ^ -- ^ 
 Lcos 2X 
 
 so that e is nearly equal to v, we have 
 
 /AT\ , /T\ cos 2 A ^Vcos2i/-Zcos2X rf , ,, 
 
 (IS ) + (L) = r- [IT + tan 2 (/i - i/) II cos 2 (^ - e)] 
 
 cos 2 A x cos 2e 
 
 .-2 
 
 sec 2 (^ - i/) + Z sec 2 (X - /*)} sin 2 (^ - ^)] . . .(23) 
 
 COS i- 
 
 In the symmetrical term for the sun, with approximation as in (16), we 
 get 
 
 (T) + (R) = (r- J R)n / cos2(^ / -r) .... ........... (24) 
 
 This terminates the semidiurnal tides which we are considering ; but before 
 proceeding to collect the results some further transformations must be 
 exhibited, 
 
 62 
 
84 EXPRESSION FOR SEMIDIURNAL TIDE. [2 
 
 Let us consider the function D +xD', where x is small. From (12) we 
 
 see that 
 
 cos 2 8 cos 2 A 2 sin 8 cos 8 d8 
 
 I) I rfl\ J. JL T 
 
 sin 2 A <rsin 2 A dt 
 
 Hence, if 8' be the moon's declination at a time earlier than the time of 
 
 observation by #/2cr, then 
 
 ,,, cos 2 8' cos 2 A 
 D + xD = - . . 
 sm 2 A 
 
 Hence, in (17), 
 
 D + tan2(.- / .)Z)^ COs2 ^-^ 0s2A (25) 
 
 when 8' is the moon's declination at time -^t 57'3 tan 2 (K ^)/2<7. The 
 period 57'3 tan 2 ( yu.)/2er may be called ' the age of the declinational 
 inequality.' 
 
 wT/Hlfn X dP] 
 
 Again, 11 + #11 = - \Jr 1 rr 
 
 e ( (r vrdt 
 
 Hence, if (P'-l)/e denotes the value of (P-l)fe at a time a?/(r - w) earlier 
 than that of observation, then 
 
 1 
 
 e 
 Hence, in (23), H + tan 2 (/A- i/) H' = -(P'~ 1) (26) 
 
 where P' is the ratio of the moon's parallax to her mean parallax at a time 
 fat - 57'3 tan 2 (/A - i/)/(r - CT). The period 57'3tan2(/t i/)/(o- r) may 
 be called 'the age of the parallactic inequality.' 
 In collecting results we shall write the sum 
 
 M 2 + S 2 + K. 2 + N + L + R + T = h. 2 
 
 For reasons explained below we omit terms depending on the rate of 
 change of solar parallax and declination. 
 
 Then, from (15), (16), (18), (19), (23), (24), (25), (26), we have 
 
 cos 2 8' cos 2 A cos 2 8. cos 2 A. 
 
 + -- i-j-r - K "cos 2 (i/r - ) + -- i- -- 'K"cos 2 (^ - ) 
 sm^ A sin 2 A 
 
 sin 8 
 
 cos 
 
 cr sm 2 
 
 cos 2 A . 7V n x -Af cos 2v L cos 2X, 
 + - r (P 7 - 1) - - cos 2 (-/r - e) 
 
 cos 2 A e cos 2e 
 
 cos 2 A 1 dP ( . , , ./Vsec 2 (tt - v) + Z sec 2 (X - /tt)\ . , , x 
 H --- T --- --n-fiJf -- - sin 2 (^r- /*) 
 
 cos 2 A 7 o- - r d* V e / 
 
 ......... (27) 
 
1885] APPROXIMATION FOR SEMIDIURNAL TIDE. 85 
 
 It may easily be shown, from Schedule B. i., 1883, that in the equilibrium 
 theory K"-Miatf A,= 0, and 4>M - (N + L)/e = ; hence the terms de- 
 pending on rates of change of declination and parallax are small. This 
 also shows that we were justified in neglecting the corresponding terms in 
 the case of the sun. Also, since the faster tides are more augmented by 
 kinetic action that the slow ones, the two functions, written above, which 
 vanish in the equilibrium theory are normally actually positive. The formula 
 (27) gives the complete expression for the semidiurnal tide in terms of hour- 
 angles, declinations, and parallaxes, with the constants of the harmonic 
 analysis. 
 
 We shall now show that with rougher approximation (27) is reducible to 
 a much simpler form. 
 
 The retardation of each tide should be approximately a constant, plus a 
 term varying with the speed. Hence all the retardations may be expressed 
 in terms of and fi, and 
 
 (T-M 
 
 /e = /*+ b ^ a- 
 
 <T 7) 
 
 _ 
 P-JI- -}(*.**) 
 
 It is clear that * differs very little from , and that 
 
 K fJi 2 (fM v) % ' fJb 
 
 a a is a 77 
 
 The time ( /A)/(O- ?) is called ' the age of the tide,' for reasons explained 
 below, and K /z, /* v, not being large angles, do not differ much from their 
 tangents. Hence the ages of the declinational and parallactic inequalities 
 are both approximately equal to the age of the tide. 
 
 Let ce, then, denote ( /i)/(o- 77), the age of the tide. 
 
 Now, as an approximation, we may suppose that heights of the lunar 
 K 2 tide, the N and L tides bear the same ratio to the M 2 tide as in the 
 equilibrium theory; and that the solar K 2 , the T and R tides bear the same 
 ratio to the S 2 tide as in that theory. Then reverting to the notation with 
 7, D, i in place of A, A /} and writing 
 
 cos 17 
 
 s ^w cos \i 
 we have 
 
 ^,, = s cos^ cos fj/ 
 
 sm 2 A cos 4 / cos 2 A cos 5 
 
 , T=7 8 
 cos 4 &> 
 
86 APPROXIMATION FOR SEMIDIURNAL TIDE. [2 
 
 Also, since (22) may be written 
 
 _ N sin 2 Q - v) + L sin 2 (A. - /*) 
 ( ^~ ~ r 
 
 we have, treating /u, i>, X ft, /i e as small, approximately, 
 e = p - \ffi (<r - 57) = fju - f (X - i;) 
 
 A , cos 2 A N cos 2i> L cos 2X 
 
 Also - = 3efM 
 
 cos 2 A, cos 2e 
 
 Then reverting to mean longitudes, and substituting the age of tide 
 where required, we find, on neglecting the difference between K and , 
 
 For the lunar declinational term, 
 
 2 tan 2 \I fM cos 2 [s - wo- - ] cos 
 For the solar declinational term, 
 
 2 tan 2 o> S cos 2/i cos 
 For the lunar parallactic term, 
 
 BefMcos [s p ce(<r cr)] cos 2 [t/r - //, + |o? (cr CT)] 
 For the solar parallactic term, 
 
 3^5 cos (A -p,) cos 2 fy, - f ] 
 
 Then omitting the terms depending on changes of declination and 
 parallax, we have as an approximation, 
 
 = fM {cos 2 (i/r - p) + 2 tan 2 |/cos 2 [s - ojo- - ] cos 2 (^r - ) 
 
 + 3e cos [s -p - CB (o- - r)] cos 2 [i/r - yu, + |cB (o- - )]} 
 
 ^) .................. (28) 
 
 In the equilibrium theory we have the lunar semidiurnal tide depending 
 on r~ 3 cos 2 8 cos 2-v/r. Now it is obvious that cos 2 S introduces a factor 
 1 + 2 tan 2 \I cos 2 (s - ), and r~ 3 a factor 1+ 3e cos (s -p). Thus, if we could 
 have foreseen the exact disturbance introduced by friction and other causes 
 in the various angles, the formula (28) might have been established at once ; 
 but it seems to have been necessary to have recourse to the complete 
 development in order to find how the age of the tide will enter. 
 
1885] REFERENCE TO MOON'S TRANSIT. 87 
 
 4. Reference to Time of Moons Transit. 
 
 It has been usual to refer the tide to the time of moon's transit, and we 
 shall now proceed to the transformations necessary to do so. 
 
 cos 2 A/ cos 2 A, goes through its oscillation about the value unity in 
 19 years; it is therefore convenient to write for, say, a whole year, 
 
 cos 2 A ,/ 
 M == M 
 
 cos 2 A, 
 
 and similarly. N = . N\ (29) 
 
 cos 2 A / 
 
 T _ cos 2 A , 
 
 * % "a?5. 
 
 We also observe that K " and K", being the lunar and solar parts of the 
 mean K 2 tide, and their ratio being *464 (Report, 1883), 
 
 #" = 68303^0, K," = -31697^ (30) 
 
 It will also be seen that in all the terms arising from the sun, excepting that 
 in K'', the argument of the cosine is 2 (^ ). It will be convenient, and 
 sufficiently accurate for all practical purposes, to replace K by in this solar 
 declinational term K". 
 
 We shall now proceed to refer the tide to the moon's transit at the place 
 of observation. 
 
 Let ot , h be ])'s R.A. and O's mean longitude at })'s transit say upper 
 transit, for distinctness. Then the local time of transit is given by the 
 vanishing of ty, and since ty = t + h a., it follows that the time-angle of ))'s 
 transit (at 15 to the hour) is a. h . 
 
 Now let r (mean solar hours) be the interval after transit to which the 
 time-angle t refers; then, since 
 
 dh _ da. _ /da 
 
 CLv (AJV \ CLv 
 
 //"//v \ 
 
 = [(7 - 77) T + ot - //] + [A + 777-] - + O-T + (- - <r T 
 
 L \ af / J 
 
 fda. _ 
 \dt 
 
 For the sake of brevity, put 
 so that T is T converted to angle at the rate of 14'49 per hour. Then we have 
 
88 . . REFERENCE TO MOON'S TRANSIT. [2 
 
 Similarly putting a, for 's R.A. at D's transit, we have 
 ^ = t + h - a, 
 
 = [(7 - ^) r + *o -/o] + [/*(, + T?T] - a, + O-T - Ur - ^J r 
 so that 1r = -o' 
 
 Thenlet A=a -ct t ........ ...................... (32)* 
 
 So that A is the apparent time of ])'s transit, reduced to angle at 15 
 per hour, and we have 
 
 It is only in the two principal tides that we need regard the changes of 
 R.A. since ])'s transit, and in all the smaller terms we may simply put 
 
 The first pair of terms of (27) now become 
 
 M cos 2 TT - (j*- o-) T- J +Scos 2 [T + A + (a- - ^ r - (f 
 
 and these are equal to 
 
 M cos 2 (T - fi) + S cos 2 (T + A - f ) 
 
 (34) 
 
 We may now collect together all the results, and write them in the form 
 of a schedule [Schedule IV.]. 
 
 Definition of symbols : 
 
 a, 8, a. t , 8, J)'s and 's R.A. and declination at moon's transit; A = a a t , 
 apparent time of ])'s transit at the port. 
 
 57'3 
 8' ])'s decl. at the time (generally earlier than transit) T ~ tan 2 (K /u,). 
 
 P, P t the ratio of ])'s and 's parallax to mean parallaxes. 
 P' the ratio for ]) at the time (generally earlier than transit) 
 
 T - tan 2 (//, v) 
 
 * It would be better to put 
 
 ff ri 
 
 A = a -a.-{ n 
 
 y-ff 
 
 If this be used the correction (40) for o 's change of R.A. becomes small. 
 
1885] 
 
 SCHEDULE OF SEMIDIURNAL TIDE. 
 
 89 
 
 I + 
 
 EH H 
 
 XX 
 
 + 
 EH 
 
 + 
 H_ 
 
 02 
 O 
 
 o 
 X 
 
 *^ 
 I 
 
 + 
 EH^ 
 
 (M 
 
 
 "a 
 
 + + 
 
 1 
 
 9 
 
 
 a 
 
 eg 
 
 
 
 
 
 
 
90 SYNTHESIS OF MEAN SEMIDIURNAL TIDE. [2 
 
 r the time elapsed since })'s transit in m. s. hours; T the same time 
 reduced to angle at 14'49 per hour. 
 
 A such a declination that cos' 2 A is the mean value of cos 2 8 ; A has a 
 19-y early period. 
 
 A y such a declination that cos 2 A, is the mean value of cos 2 B r 
 
 e, e / eccentricities of lunar and solar orbits ; a the ])'s mean motion ; 
 GT the mean motion of the ))'s perigee. 
 
 Mo_Ni_Lo_ cos 2 A 
 
 ~M~~N~J,~ cos 2 A, 
 
 M, 8, K 2 , N, L, T, R the mean semi-ranges H of the tides of those 
 denominations in the harmonic method. The retardations found by harmonic 
 analysis are 2/u, for M 2 , 2 for S 2 , 2* for K 2 , 2v for N, 2\ for L, and 2 for T 
 and R. 
 
 T , N sin 2i/ L sin 2X n , . ,, 
 
 .Lastly tan ze = -^ TT- = . 2e to be taken in the same quadrant 
 
 Ncos2v-Lcos2\ 
 
 as 2v. 
 
 5. Synthesis of the Several Terms. 
 Consider the two principal terms in Schedule IV., 
 
 M cos 2 (T - p) + S cos 2 (T + A - f ) 
 They may be written in the form 
 
 H cos 2 (T - $) 
 
 where Tif cos 2 (yt* 0) = M + S cos 2 ( J. + /A) 
 
 H sin 2 (p - <}>) = S sin 2 (4 - + /x) 
 
 If we compute <f> corresponding to the time of moon's transit from the 
 formula 
 
 S sin 2 (A - 
 
 tan 2 (a d)} = -= .. 
 
 JB, + o cos 2 ( J. 
 
 then < reduced to time at the rate of 14'49 per hour is the interval after 
 moon's transit to high water, to a first approximation. The angle </> + 90, 
 similarly reduced, gives the low waters before and after the high water, and 
 </> + 180 gives another high water. The high waters and low waters are to 
 be referred to the nearest transit of the moon. 
 
 The height or depression is given to a first approximation by 
 H = */{M * + S 2 + 2M S cos 2 O - <)} 
 
 This variability in the time and height of high water, due to variability 
 of <f>, is called the fortnightly or semi-menstrual inequality in the height and 
 interval. The period ( /*)/(o- rj) is called 'the age of the tide/ because 
 
1885] CORRECTIONS TO MEAN SEMIDIURNAL TIDE. 91 
 
 this is the mean period after new and full moon before the occurrence of 
 spring tide. 
 
 6. Corrections. 
 
 The smaller terms in Schedule IV. may be regarded as inequalities in the 
 principal terms. They are of several types. Consider a term B cos 2 (T - /3). 
 
 Then 
 
 5cos 2 (T -) = 5cos 2 ( - <)cos 2 (T - 0) + sin 2 (/3 - <)sin 2(T- <) 
 Hence the addition of such a term to H cos 2 (T - <) gives us 
 (# + S#) cos 2 (T - - 8<f>) 
 
 where SH = B cos 2 (/3 <f>), 2.ffS</> = 5 sin 2 (/3 <) (35) 
 
 Next consider a term G sin 2 (T - /A). Putting j3 = p + \ r rr, we have 
 
 i TTJ" ^f " O / -JL \ O f I ^L f^ O / ,^A \ / Q^? \ 
 
 Next consider a term .fe'cos 2 (T + J. - f). Putting /3 = ^- J., we have 
 85 = ^ cos 2 (4 - + </>), 2#S(/> = -#sin 2(^1 -+<) ...(37) 
 
 Lastly, consider a term ^sin2(T + .4- ^). Putting /3= -4 -f-^Tr, we 
 have 
 
 In writing down the corrections we substitute 14'498 for 8$, and 
 introduce a factor so that the times may be given in mean solar hours and 
 the angular velocities in degrees per hour. 
 
 Change of Moon's R.A., Sched. IV. 
 This is of type (36), and gives 
 2-7r fda 
 
 IftQ I ,7* " I ' "'"0 ""* * V* 
 
 This correction to the height is very small. 
 
 Change of Sun s R.A., Sched. IV.* 
 This is of type (38), and gives 
 
 (40) 
 
 * With the value of A suggested in footnote to (32) 
 
 (<r - dajdt) r becomes [(0 - p.) a - (<f>da t /dt - A"?)] / (7 ~ <*) 
 at high water. This is obviously very small. 
 
92 CORRECTIONS TO MEAN SEMIDIURNAL TIDE. [2 
 
 Moons Declination, Sched. IV. 
 This is of type (35), and gives 
 
 m . egPy-oM-A . 68Jj cos 
 
 sm A ' ......... (41) 
 
 U . xh-977 COS '*:- OS ' A -683 sin 2 ( - *) 
 
 $tm's Declination, Sched. IV. 
 This is of type (37), and gives 
 
 ,,. cos 2 8, -cos 2 A 01 Krr 
 
 "StfAT - 
 
 ft = - 1-977 C S ' g s : n "7' A ' '317 sin 
 
 Change of Moons Declination, Sched. IV. 
 This is of type (36), and gives 
 
 .> sin8cosSd8/ '683^2 M*. A "\ o/ j.\^ 
 g^T= r-rr- jTl- ^ - - tan 2 A / sm 2 (/i - </>) 
 
 o-sm 2 A / rfi \cos2(-/i) 
 
 K sin 8 cos 8 dS / '683^2 *,>. \\ n/ 
 S<= l h '977 . -- : ^rt J/tan 2 A y cos2(/i 
 
 o-.ff sm 2 A 7 rfi Vcos 2 (/c - /*) V 
 
 Moon's Parallax, Sched. IV. 
 This is of type (35), and gives 
 
 (44) 
 
 I's Parallax, Sched. IV. 
 This is of type (37), and gives 
 
 rn ~p 
 
 BH = (P - 1) - - cos 2 (A - + 4 
 
 \ (45) 
 
 Bt = - l h> 977 (P t - 1) -^g- sin 2 (A - %+ <\ 
 
 / 
 
 Change of Moons Parallax, Sched. IV. 
 This is of type (36), and gives 
 
 (46) 
 
1885] DIURNAL TIDES. 93 
 
 The lunar corrections involving sines are small compared with those 
 involving cosines. 
 
 To evaluate these corrections we must compute r from <j> reduced to time 
 at 14'49 per hour. 
 
 In the right ascensional terms, da/dt and a are to be expressed in degrees 
 per hour, da/dt is the hourly change of })'s R.A. at time of ))'s transit, and 
 dajdt is the hourly change of 0's R.A. at time of ])'s transit. 
 
 Similarly, d8/dt is to be expressed in degrees, if a be in degrees. 
 
 8', P' can be found for the antecedent moments, 57'3 tan 2(* 
 and 57 0< 3 tan 2 (/LI v)/(<r CT), before the time r. 
 
 7. The Diurnal Tides. 
 
 I shall not consider these tides so completely as the semidiurnal ones, 
 although the method indicated would serve for an accurate discussion, if it 
 be desired to make one. 
 
 The important diurnal tides are K 1; 0, P. 
 From Schedule B. ii., 1883, we have 
 
 (Q) = . sm J <**' * f , M' cos [t + h - Vo - 2 (s - f ) + iTr - /*'] 
 sin to cos 2 eo cos 4 ft 
 
 By (9) the coefficient is sin 2 A/sin 2 A,, and we shall put, as in the case of 
 the semidiurnal tides, 
 
 ,,, sin2A ,,, 
 M = -T-^T- M ' 
 
 sm 2A, 
 
 Then, since t + h = ^ + a. 
 
 (0) = M ' cos ty + (a + iv)- 2( S -) + |7r-/i'] 
 
 = J/ 'cosO, for brevity (47) 
 
 Again, from Schedule C., 1883, 
 
 (P) = S'cos[-/i + i7r-'] 
 Then let % = 2 (s - h) + v - 2% - + /*' 
 
 and we have (P) = 8' cos (O + x) ( 48 ) 
 
 Whence (0) + (P) = [M f + S' cos X ] cos n - S' sin X sin fl , 
 
 If we put H' cos (/*' - <^>') = Jf ' -H S' cos ^ 
 
 ff' sin (X - f ) = S' sin X 
 (O) + (P) = H' cos (H + f* - $ ) 
 
 = #' cos [^ + (a - i; ) - 2 (s - |) + TT - f ] (49) 
 
DIURNAL TIDES. 
 
 Where H' = \/{M '' 2 + S'' 2 + 2M 'S' cos xY 
 
 n( '-6'}= ' Sf/sin % 
 
 The rate of increase of the angle ^ is twice the difference of the mean 
 motions of the moon and sun, but it would be more correct to substitute for 
 s and h the true longitudes of the bodies. It follows from (50) that <' has a 
 fortnightly inequality like that of <. 
 
 fy is very nearly equal to T, and where the diurnal tide is not very large 
 we may with sufficient approximation put 
 
 So that with fair approximation 
 
 The synthesis of the two parts of the Kj tide has been performed in the 
 harmonic method (Report, 1883), and we have 
 
 (K x ) = i^Ki COS (t + h V^TT /Cj) 
 
 Then, writing H 1 K l = K , we have 
 
 (Kj) = K Q cos (T + a v \-n tfj) (52) 
 
 We have next to consider what corrections to the time and height of 
 high and low water are necessary on account of these diurnal tides. 
 
 If we have a function 
 
 COS i 
 
 where n is nearly equal to unity, and H l is small compared with H ; its 
 maxima and minima are determined b}^ 
 
 H n 
 sin 2 (T -<) = - -- sin (nT - j3) 
 
 If T = TO be the approximate time of maximum, and T + ST the true 
 time, then, since the mean lunar day is 24'84 hours, and the quotient when 
 this is divided by 8?r is O h> 988, we have in mean solar hours, 
 
 T = - O h -988 ^~ - sin (T 
 And the correction to the maximum is 
 
 .(53) 
 
 Again if T = Tj be the approximate time of minimum, and 1\ + STj the 
 true time, then 
 
 (54) 
 And the correction to the minimum is 
 
 SH=H 1 cos(nT 1 -/3) 
 
1885] SYNTHESIS IN HARMONIC NOTATION. 95 
 
 In the case of the correction due to (0) + (P), n is approximately 
 
 1 , and for the correction due to Kj , n is approximately 1 -f 
 
 7 0- j a 
 
 8. Direct Synthesis of the Harmonic Expression for the Tide. 
 
 The scope of the preceding investigation is the establishment of the 
 nature of the connection between the older treatment of tidal observation 
 and the harmonic method. It appears, however, that if the results of 
 harmonic analysis are to be applied to the numerical computation of a tide- 
 table, then a direct synthesis of the harmonic form may be preferable to a 
 transformation to moon's transit, declinations, and parallaxes. 
 
 Semidiurnal Tides. 
 
 We shall now suppose that M is the height of the M 2 tide, augmented 
 or diminished by the factor for the particular year of observation, according 
 to the longitude of the moon's node, and similarly K generically for the 
 augmented or diminished height of any of the smaller tides. As before, let 
 2/z, 2 be the lags of M 2 , S 2 ; and 2, generically, the lag of the K tide. 
 
 Let 6 = t + h-s-v + t; 
 
 Then might be defined as the mean moon's hour-angle, the mean moon 
 coinciding with the true, not at Aries, but at the intersection. 
 
 Let the argument of the K tide be written generically 2 [6 + a K\. 
 Then 
 h. 2 = J/" cos 2 (0 - fj,) + S cos 2 [0 + s - h + v, - % - f] + K Q cos 2[0 + u-ic] 
 
 ......... (55) 
 
 If we write = v + 
 
 and H cos 2 (/* - $) = Jf + 8 cos 2 [s - h - + /*] 
 
 Hsiu 2 O - (/>) = 8 sin 2 [s - h - + p] 
 the first two terms of (55) are united into 
 
 # cos 2 (0-0) ...................... ........ (56) 
 
 with fortnightly inequality of time and height defined by 
 
 o/ A\_ <Si sin 2 (s A ^o + /i) ) 
 
 '- (> - ......... (57) 
 
 The amount of the fortnightly inequality depends to a small extent on 
 the longitude of the moon's node, since and M are both functions of that 
 longitude. 
 
96 SYNTHESIS IN HARMONIC NOTATION. [2 
 
 For the K tide we have 
 
 KQ cos 2 (6 -f u - K) = K cos 2 (u - K + </>) cos 2 (B - 0) 
 
 - K sin 2 (tt - K + <) sin 2 (0 - </>) 
 
 Hence S# = /f cos 2 (it K + <f>) ] 
 
 K \ (58) 
 
 S< = ~j sin 2 (?< - K + <) 
 
 It is easy to find from the Nautical Almanac the exact time of mean 
 moon's transit on any day, and then the successive additions of 12 h< 420601 
 or 12 h 25 14 8- 16 give the successive upper and lower transits. The 
 successive values of 2 (s h) may be easily found by successively adding 
 12'618036 to the initial value at the time of the first transit of the 
 mean moon, and (f> may be obtained from the table of the fortnightly 
 inequality for each value of 2 (s h). 
 
 The function u is slowly varying, e.g., for the K 2 tide 
 Z*2(f-) + i(iV-i>") 
 
 and the increment of argument for each 12 h< 420601 may be easily computed 
 once for all, and added to the initial value. 
 
 In the case of the diurnal tides it will probably be most convenient to 
 apply corrections for each independently, following the same lines as those 
 sketched out in 5. 
 
 The corrections for the over- tides M 4 , S 4 , &c., and for the terdiurnal and 
 quaterdiurnal compound tides, would also require special treatment, which 
 may easily be devised. 
 
 At ports, where the diurnal tide is nearly as large or larger than the 
 semidiurnal, special methods will be necessary. 
 
 Although the treatment in terms of mean longitudes makes the correc- 
 tions larger than in the other method, yet it appears that the computation of 
 a tide-table may thus be made easier, with less reference to ephemerides, and 
 with amply sufficient accuracy. 
 
 [This subject is considered hereafter in the paper (7) on 'Tidal Prediction.'] 
 
3. 
 
 DATUM LEVELS; THE TREATMENT OF A SHORT SERIES OF 
 TIDAL OBSERVATIONS AND ON TIDAL PREDICTION. 
 
 [Report of the Committee consisting of Professor G. H. DARWIN, 
 Sir W. THOMSON, and Major BAIRD, for the purpose of prepar- 
 ing instructions for the practical work of Tidal Observation; 
 and Fourth Report of the Committee consisting of Professors 
 G. H. DARWIN and J. C. ADAMS, for the Harmonic Analysis of 
 Tidal Observations. Drawn up by G. H. DARWIN. British 
 Association Report for 1886, pp. 40 58.] 
 
 I. RECORD OF WORK DURING THE PAST YEAR. DATUM LEVELS. 
 
 IN the course of the Indian tidal operations a discussion has arisen as 
 to the determination of a datum level for tide-tables. The custom of the 
 Admiralty is to refer the tides to ' the mean low- water mark of ordinary 
 spring tides.' This datum has not a precise scientific meaning, but, at 
 ports where there are but few observations, has been derived from a mean 
 of the spring-tides available. At some of the Indian ports this datum has 
 been found by taking the mean of all spring-tides on the tide diagram for 
 a year, with the exception of those which occur when the moon is near 
 perigee. The diurnal tides enter into the determination of the datum in 
 an undefined manner. It follows that two determinations of this datum 
 level, both equally defensible, might differ sensibly from one another. 
 
 A datum level should be sufficiently low to obviate the frequent occur- 
 rence of negative entries in a tide-table, and it should be rigorously 
 determinable from tidal theory. It is now proposed to adopt as the datum 
 level at any new ports in India, for which tide-tables are to be issued, a 
 datum to be called ' the Indian spring low- water mark,' and which is to 
 
 i>. i. 7 
 
98 INDIAN LOW-WATER MARK. [3 
 
 be below mean sea-level by the sum of the mean semi-ranges of the tides 
 M 2 , S 2 , Kj, O; or, in the notation used below, 
 
 H m + H s H- H' + H 
 
 below mean water mark. 
 
 This datum is found to agree pretty nearly with the Admiralty datum, 
 but is usually a few inches lower. The definition is not founded on any 
 precise theoretical considerations, but it satisfies the conditions of a good 
 datum, and is precisely referable to tidal theory. 
 
 If, when further observations are made, it is found that the values of 
 the several H's require correction, it is not proposed that the datum level 
 shall be altered accordingly, but when once fixed it is to be always ad- 
 hered to. 
 
 II. ON THE TREATMENT OF A SHORT SERIES OF TIDAL OBSERVATIONS 
 AND ON TIDAL PREDICTION. 
 
 1. Harmonic Analysis, 
 
 Having been asked to write an article on the tides in a new edition 
 of the Admiralty Scientific Manual [see Paper 4 below], now in 
 the press, I thought it would be useful to show how harmonic analysis 
 might be applied to the reduction of a short series of tidal observations, 
 such as might be made when a ship lies for a fortnight or a month in 
 a port. 
 
 The process of harmonic analysis, as applicable to a year of continuous 
 observation, needs some modification for a short series, and as it was not 
 possible to explain the reasons for the rules laid down within the limits 
 of the article, it seems desirable to place on record an explanation of the 
 instructions given. 
 
 The observations to be treated are supposed to consist of hourly obser- 
 vations extending over a fortnight or a month. In the reduction of a long 
 series of observations the various tides are disentangled from one another 
 by means of an appropriate grouping of the hourly observations. When, 
 however, the series is short, the method of grouping is not sufficient in 
 all cases. 
 
 With the amount of observation supposed to be available, a determi- 
 nation of the elliptic tides was not possible, and it was therefore proposed 
 to consider only the tides M 2 , S 2 , K 2 , K 1} O, P that is to say, the principal 
 lunar, solar, and luni-solar semidiurnal tides, and the luni-solar, lunar, and 
 solar diurnal tides. The luni-solar and solar semidiurnal tides have, how- 
 ever, so nearly the same speed that we cannot hope for a direct separation 
 
1886] SHORT SERIES OF OBSERVATIONS. 99 
 
 of them by the grouping of the hourly values, and we must have recourse 
 to theory for completing the process; and the like is true of the luni- 
 solar and solar diurnal tides. 
 
 Also, the tides Kj and P have very nearly half the speed of S 2 ; hence 
 the diurnal tides K a and P will appear together as the diurnal constituent, 
 whilst S 2 and K 2 will appear as the semidiurnal constituent, from the 
 harmonic analysis of the same table of entries. 
 
 It thus appears that three different harmonic analyses will suffice to 
 determine the six tides, viz. : 
 
 First, an analysis for M 2 ; second, an analysis for O ; third, an analysis 
 for S 2 , K 2 , K!, P. 
 
 The rules therefore begin with instructions for drawing up three 
 schedules, to be called M, O, S, for the entry of hourly tide-heights. Each 
 schedule consists of twenty-four hour columns, and a number of rows for 
 the successive days. In M and O certain squares are marked, in which 
 two successive hourly entries are to be put. The instructions for drawing 
 up the schedules are simply rules for preparing part of the first page of 
 the series M, O, S of the computation forms for a year of observation. 
 
 In order to minimise the vitiation of the results derived from the M 
 sheet by the S 2 tide, and vice-versa, and similarly to minimise the vitiation 
 of the results from the sheet by the Kj tide, it is important to choose 
 the proper number of entries in each of the three sheets. 
 
 It was shown in Section III. of the Tidal Report to the British 
 Association for 1885 [Paper 2] how these periods were to be determined. 
 The equation by which we find how many rows to take to minimise the 
 effect of the S 2 tide on the M 2 tide is there shown to be 
 
 1-01589580 = 14-4920521r 
 If r = 1, q = 14-26 ; and if r = 2, q = 28'5. 
 
 For a reason similar to that given in 1885 we conclude that, in 
 analysing about a fortnight of observation we must have 14 rows of values 
 on the M sheet, and for a month's observation 29 rows of values. 
 
 Similarly, to minimise the effect of the M 2 tide on the S 2 tide the 
 equation is 
 
 1-0158958^ =15r 
 
 If r=l, = 14-76; and if r = 2, = 29'5. 
 
 Whence we must have 15 rows of values on the S sheet for a fort- 
 night's observation, and 30 rows of values for a month's observation. 
 
 These two rules are simply a statement that on the M and S sheets 
 we are to take a period equal to the interval from spring-tide to spring- 
 tide, or twice that period, 
 
 72 
 
100 SHORT SERIES OF OBSERVATIONS. [3 
 
 Similarly, to minimise the effect of the Kj tide on the tide, the 
 equation is 
 
 r0980330g = 13'9430356r 
 
 If r=l, 2=12-69; and if r = 2, q = 25'38. 
 
 Whence we must have 13 rows of values on the O sheet for a fort- 
 night's observation, and 25 rows for a month's observation. 
 
 Lastly, to minimise the effect of the O tide on the Kj tide, the 
 equation is 
 
 l-0980330g = 15-0410686r 
 
 If r = 1, q = 13-70 ; and if r = 2, q = 26'4. 
 
 Hence, in using the numbers on the S sheet for determining the 
 diurnal tides, we must use 14 rows of values for a fortnight's observation, 
 and 26 rows for a month's observation. 
 
 Thus, on the S sheet we use more rows for the semidiurnal tides than 
 for the diurnal namely, one more for a fortnight and three more for a 
 month. 
 
 The rules for drawing up the computation forms then specify, in 
 accordance with the above results, where the entries are to stop on the 
 three sheets, and give directions for the dual use of the S sheet, according 
 as it is for finding semidiurnal or diurnal tides. 
 
 When the entries have been made, the twenty-four columns on each 
 sheet are summed, and each is divided by the number of entries in the 
 column. On the S sheet there are two sets of sums and divisions, one 
 with and the other without the additional row or rows. 
 
 The three sheets thus provide us with four sets of twenty-four mean 
 hourly values; the M sheet corresponds with mean lunar time, the hour 
 being 15 -=- 14'49 of a mean solar hour; both the means on the S sheet 
 correspond with mean solar time; and the O sheet corresponds with a 
 special time, in which the hour is 15 -=- 13'94 of a mean solar hour. 
 
 The four sets of means are then submitted to harmonic analysis : the 
 semidiurnal components are only evaluated on the M sheet; the diurnal 
 components are evaluated from the shorter series on S, and the semidiurnal 
 from the longer series; and the diurnal components from the O sheet. 
 We may also evaluate the quaterdiurnal components from the M and S 
 sheets. 
 
 It might, perhaps, be useful to evaluate the diurnal component on the 
 M sheet, for if it does not come out small it is certain that the amount 
 of observations analysed is not sufficient to give satisfactory results. 
 
1886] NOTATION. 101 
 
 In the article the harmonic analysis is arranged according to a rule 
 devised by General [Sir Richard] Strachey*, which is less laborious than 
 that usually employed, and which is sufficiently accurate for the purpose. 
 
 2. On the Notation employed. 
 
 It will be convenient to collect together the definitions of the principal 
 symbols employed in this paper. 
 
 The mean semi-range and angle of lagging of each of the harmonic 
 constituent tides have, in the Tidal Report for 1883, been denoted gene- 
 rically by H, K ; but when several of the H's and KS occur in the same 
 algebraic expression it is necessary to distinguish between them. The tides 
 to which we shall refer are M 2 , S 2 , N, L, T, R, O, P, and K 2 , K x ; the 
 H and K for the first eight of these will be distinguished by writing the 
 suffix letters m , g> n , &c., e.g., H OT , tc m for the M 2 tide. With regard to 
 the K tides, we may put H", K", and H', K'. 
 
 Again, the factors of augmentation f (functions of longitude of moon's 
 node), as applicable to the several tides, will be denoted thus : for M 2 , 
 N, L, simply f; for K 2 , K 1? f", f respectively; for O, f . 
 
 The K 2 , K! tides take their origin jointly from the moon and sun, and 
 it will be necessary in computing the tide-table to separate the lunar 
 from the solar portion of K 2 . Now, the ratio of the lunar to the solar 
 tide-generating force is such that "683H" is the lunar portion and '31 7H" 
 is the solar portion of H". 
 
 In the Report of 1885 [Paper 2] a slightly different notation was 
 employed for the H's and K'S, but it is easy to see how the results of 
 that Report are to be transformed into the present notation. 
 
 As in the Report of 1883 [Paper 1] we write t, h, s for local mean solar 
 hour-angle, sun's and moon's mean longitude, and v, %, v, Zv" for functions of 
 the longitude of moon's node depending on the intersection of the equator 
 with the lunar orbit ; also 7 77, rj, <r, & are the hourly increments of 
 t, h, s and longitude of moon's perigee, and e, e, the eccentricities of lunar 
 and solar orbits. 
 
 Let p, p, denote the cubes of the ratios of the moon's and sun's paral- 
 laxes to their mean parallaxes ; 8, S, the moon's and sun's declinations ; 
 p' the value of p at a time tan (tc m - n )/(o- -or), or 105 h< 3 tan (/c m tc n ) 
 earlier than t ; 8' the moon's declination at a time tan (K" /e m )/2o- or 
 52 h '2 tan (K" x m ) earlier than t. 
 
 Let P, P,, P' be the cube roots of p, p,, p'. 
 
 * Proc. Roy. Soc. Vol. XLII. (1887), p. 01. 
 
102 REDUCTION OF RESULTS. [3 
 
 Let A, A, be declinations such that cos 2 A, cos 2 A, are respectively the 
 mean values of cos 2 B, cos 2 S, : obviously A has a small inequality with the 
 longitude of the moon's node. 
 
 Let e be an auxiliary angle denned by 
 
 H n sin K n - HI sin KI 
 tan e = ^j - rj - 
 
 il n COS K n tii cos KI 
 
 Lastly, let -\Jr, -fy, be the moon's and sun's local hour angles. 
 
 3. The Reduction of the Results of Harmonic Analysis*. 
 
 We now suppose the harmonic analysis of the hourly means on the three 
 sheets M, O, S completed. 
 
 The deduction of H, w , tc m and H , K O from the M and sheets follows 
 exactly the same rules as in a long series of observations, and the reader is 
 referred to the Report of 1883 [Paper 1] for an explanation. 
 
 With regard to the S sheet, the results of the harmonic analysis do not 
 separate the S 2 tide from the K 2 tide, nor the KI tide from the P tide, and 
 we have to employ theoretical considerations for effecting the separation. 
 
 The semidiurnal tides will be taken first. 
 
 The solar tide, as derived from a short series of observations, is of course 
 affected by the sun's parallax, and as the sun changes his parallax slowly, the 
 solar tide will follow the equilibrium law and vary as the cube of tKe sun's 
 parallax. Thus the height of the purely solar semidiurnal tide as derived 
 from our short series of observations will be p,H s instead of H,,, and this will 
 be fused with the luni-solar tide K 2 . 
 
 The schedules of the Report of 1883 thus show that we shall have as 
 the expression for this tide, compounded of S 2 (with parallactic inequality) 
 
 and K 2 , 
 
 h 2 = p,H, cos (2* - *,) + f"H" cos (2 + 2A-2i/'-/c") ......... (1) 
 
 The theoretical ratio of H" to R g is (see Schedule E, 1883) that of 
 12662 to '46531, or 1 to 3'67 ; and the tides having nearly the same speed, 
 we may assume K"'= K S . 
 
 Hence : 
 
 k, = H, |~p, cos (2* - *,) + -^ cos (2* + 2/i - 2v" - *.) 1 
 
 Again, the schedules of the Report of 1883 show that we shall have as 
 the expression for the tide which is compounded of K! and P 
 
 ^ = f ' H' cos (t + h - v - \ TT - *' ) -I- H p cos (t - h + % TT - Kp ) ...... (5) 
 
 * [Certain mistakes in this section have been corrected. This has involved the omission 
 of certain equations originally numbered (2), (3), (4). The portion of this section which has 
 been rewritten is enclosed in square brackets on pp. 103 4.] 
 
1886] REDUCTION OF RESULTS. 103 
 
 The theoretical ratio of R p to H' is (see Sched. E, 1883) that of '19317 
 to '58385, or 1 to 3, and the tides having nearly the same speed, we may 
 assume K P = K S . Hence : 
 
 A! = H' (f ' cos (t + h - v - \-rr - K) - | cos [t + h - v' - $ TT - K - (2h - v')]} 
 
 (6) 
 
 [The processes employed in the harmonic analysis are nearly the same for 
 the semidiurnal and for the diurnal tides. We find the mean height of 
 water at each of the 24 hours of mean solar time, as estimated over a succes- 
 sion of days, and then submit the 24 mean heights to harmonic analysis. 
 The algebraic details of the process will be found in Paper 6, 'On 
 an apparatus for facilitating the reduction of tidal observations,' 5 ; and 
 I shall here only give the results. It must, however, be remarked that the 
 present case differs slightly from the case treated in the paper referred to, 
 because we now suppose the analysis for the diurnal tides to embrace 27 days 
 or 14 days, instead of 30 or 15, which are the periods adopted for the semi- 
 diurnal tides. 
 
 For both classes of tide the sun's mean longitude at O h of the first day of 
 observation will be denoted by A . 
 
 The formulae for semidiurnal tides for 30 or for 15 days may be written 
 down together by means of a simple alternative notation ; and the formulae 
 for diurnal tides for 27 or for 14 days may be written in a similar manner. 
 
 For semidiurnal tides the harmonic analysis of the 24 hourly means 
 gives us the two components A 2 and B 2 , and we may arrange the result in 
 terms of R s and where 
 
 A 2 = n s cos % s , B 2 = R g sin s 
 
 w f"sin(2A -2i/' + a) 
 
 We now put tan -Jr = tT7i ,= -__ }^. -^ --,-, . 
 
 3-67P, Jp 2 + f " cos (2Ao - 2v + a) 
 
 f 30 ) A 
 
 where, for n _ > days, 
 15) 
 
 29'53{ - -01945) ,_,_- 3- 
 
 a= id,.7r and 1 S-lF2 == .AAASQf' also 3 ' 67 JF2 = o. 
 
 Ifr I O ) V/U^oO 1 O 
 
 / / 
 
 The investigation then shows that 
 
 K = 
 
 H = 3-67 jp 2 R g cos ^ H" = - H 
 
 r cos (2ho-2v fi + a) ' 3'67 
 
 Turning now to the diurnal tides, the harmonic analysis gives us 
 
104 COMPUTATION OF A TIDE-TABLE. [3 
 
 AIT ^ 
 
 We then put tan d> = ^7 
 
 sm 
 
 - cos 
 
 f 271 , 26-571 
 
 where, for days, = ^ 
 
 The investigation then shows that 
 
 13-29) 
 - 2 ^ - "') + + .QO k 
 
 D oo 
 
 nrl W , 
 
 ~ 3f - cos (2/i, -' + ) H * =iH ............... (7) 
 
 - 27} -00391] 3-027 
 where, for ^ days, log ^ = ^ and 
 
 These are the formula? which should have been used in the Admiralty 
 Scientific Manual, and are used in Paper 4, as given in the present volume. 
 There was a mistake in the Manual as published, and although pains 
 have been taken to insert errata in as many copies as possible, it is certain 
 that several uncorrected copies must be in circulation. It is fortunate that 
 the mistake was such as not to make a large difference in the result.] 
 
 4. Computation of a Tide-table. Semidiurnal Tides. 
 
 The computation of a tide-table from tidal constants which do not contain 
 the elliptic tides N and L presents some difficulty, because the total neglect 
 of these tides would make the results very considerably in error. On this 
 account it was found necessary to use the moon's hour-angle, declination, and 
 parallax in making the computations. 
 
 We shall begin by considering only the semidiurnal tide. 
 
 In the Tidal Report of 1885 [Paper 2] it was shown how the expression 
 for this tide in the harmonic notation may be transformed so as to involve 
 hour-angles, declinations and parallaxes, instead of mean longitudes and 
 eccentricities of orbits. 
 
 The formula (27) of the Report of 1885 for the total semidiurnal tide, 
 when written in the notation of 2, is 
 
 cos 2 
 
 ^^A" H ' cos ~ * + * cos 
 
 COS tA 
 
 cos 2 8' cos 2 A , 
 
 + -- ^ - -683H' cos (2ilr - ") 
 sin* A 
 
1886] TIDE-TABLE; SEMIDIURNAL TIDE. 105 
 
 sin 3 cos 8 dS f -683H" 1 
 
 - Vsin^7 * [COB (*-.) - H - tan A 'J S 
 
 g _ 
 
 cos- e cos e 
 
 cos 2 A dP/dzr w H M sec (/c m - ic n ) + 
 cos 2 A ~a~ 4Mm " 
 
 \ . 
 
 ............ (8) 
 
 H n sin /c n - H; sin 
 where tan e = 
 
 ^= 
 
 ti n COS K n tii COS /Cj 
 
 We shall now proceed to simplify this. 
 
 In the first place, the terms depending on d8/dt and dP/dt are certainly 
 small, and may be neglected. 
 
 Then let 
 
 ,, cos 2 A TT cos 2 B' cos 2 A , 
 
 M = co^ Rm + - sin^A, ' <683H C S ( * ~ ^ 
 
 , cos 2 A /T) , . H w cos K H HI cos KI 
 H -- TA (* ~ 1) -cos(e K m ) 
 
 cos 2 A/ ecose 
 
 cos 2 8' cos 2 A H" . . 
 
 COS 2 A /T)/ H n COS n H; COS KI . , , 
 
 (I - 
 
 M/ = H s + cos -317H" + (P - 
 
 2 
 
 sm 
 
 /*, = *. ........................................................................ (9) 
 
 Now observation and theory agree in showing that K" is very nearly 
 equal to K S \ hence we are justified in substituting K S for K" in the small solar 
 declinational term of (8) involving -317H". 
 
 This being so, (8) becomes 
 
 h a = M cos (2^- - A*) + M y cos (2^-^) ............... (10) 
 
 In the equilibrium theory each H is proportional to the corresponding 
 term in the harmonically developed potential. This proportionality holds 
 nearly between tides of nearly the same speed; hence in the solar tides 
 we may assume (see Sched. B, 1883, and note that cot 2 A, = cot 2 a>) that, 
 
106 SEMIDIURNAL TIDE. [3 
 
 and M / reduces to 
 
 M - = H * + 3 (P ' - 1} H * = H * [1 + 3 (P ' - 1)] nearly 
 
 Now, A, = 16'36 = 16 22', sec 2 A,= 1-086, also P, = p,, and therefore 
 M, = r086p,cos 2 8,H g ........................... (12) 
 
 In a similar way, according to the equilibrium theory, we should have 
 
 ^ (H n - Hf) = H OT 
 
 Although this proportionality is probably not actually very exact, yet in 
 our supposed ignorance of the lunar elliptic tides we have to assume its 
 truth. Also, we must assume that the two elliptic tides N and L suffer the 
 same retardation, and therefore x n = te t = e. 
 
 With these assumptions, 
 
 H m + (P'-l) -cos(e- K m } = H m [l + 3(P'- 1)] = H W P 
 
 e cos e 
 
 mu cos 2 A , ,, , 
 
 1 hen, since . = f, and P 3 = p 
 
 cos 2 A / 
 
 we have M = fp'H m + ^^?^ -683H" cos ( K " - *,) 
 
 nao? X P02 A U" 
 
 \S\JlJ \J \J\J75 L^ S^flt-t: AJ- X // \ /I O \ 
 
 683 , sin (*" - K m ) (13) 
 
 sin 2 A 
 
 / 
 
 Tf '683 '683 Kh _ o 
 
 If we put C, = ^ = C = - . - x 57 '3 
 
 2 sin 2 A, ' 2 sm 2 A / 
 
 then log d = '6344, logC 2 =2'3925 
 
 and GI, C 2 are absolute constants for all times and places. 
 Next, if we put 
 
 TT// 
 
 a = CjH" cos (K" K m ), /3 = C 2 vr- sin (K" K m ) 
 
 A-acos2A, B=y9cos2A (14) 
 
 then obviously a, $ are absolute constants for the port, and A and B are 
 nearly constant, for their small variability only depends on the longitude 
 of the moon's node entering through A. 
 
 Thus we have, from (9), (12), (13), (14), 
 
 M = fH m + (p' - 1) f H m + (a cos 28' - A) 
 
 /< = *m + (@ cos 28' B), expressed in degrees 
 
 M / = l'086p / cos 2 8 / H 8 
 
 ^ =*, (15) 
 
1886] SEMIDIURNAL TIDE. J07 
 
 where p', 8' are the values of p and 8 at a time earlier than that corresponding 
 to -^ by 'the age' 52 h< 2 tan (K" - K m ). 
 
 In the article [Paper 4 in this volume] fH TO is called R OT ; (p' 1) fH m , 
 the parallactic correction, is called 8 t R m ; (2 cos 2S' A), the declinational 
 correction, is called S 2 R W . Similarly, /3 cos 28' B, the declinational 
 correction to K W , is called S^K m . Also, M, is called S. 
 
 Thus, with this notation the whole semidiurnal tide is 
 
 /Is = (R, H + SjRm + S 2 R, n ) COS (2>/r - K m - S 2 m) + S COS (2^, - K g ) . . .(16) 
 
 The mean rate of increase of i|r is 7 <r, or 14'49 per hour; hence the 
 interval from moon's transit to lunar high water is approximately -^(Km+S^Km) 
 hours, when x m is expressed in degrees. If i be the mean interval, and S 2 i its 
 declinational correction, 
 
 Now, let A be twice the apparent time of moon's transit reduced to angle 
 at 15 per hour, or the apparent time reduced at 30 per hour. 
 
 Then the excess of the moon's over the sun's R.A. at lunar high water 
 is %A plus the increase of the difference of R.A.'s in the interval i. This 
 
 increase is approximately K m , and at lunar high water the sun's hour- 
 angle is given by 
 
 ^* TO ........................ (18) 
 
 Since the difference of time between lunar high water and actual high 
 water never exceeds about an hour and a half, if we neglect the separation 
 of the moon from the sun in that time, this relationship also holds at actual 
 hmi-solar high water. 
 
 Now, let 
 
 H cos (u <>) = M -f S cos A -\ -- - x m K* + K m + 8^K m 
 
 I 7-0- J 
 
 = M + S cos ( A Kg + f g/c m + S 2 /c w ) 
 Hsin(/t 0)= S sin ( A K S + f f K m + S 2 /c, rt ) ............ (19) 
 
 and we have for the whole luni-solar semidiurnal tide 
 
 ^ = Hcos(2^-</>) ........................... (20) 
 
 If We put 7 + 807 = K s l$K m + 
 
 we have, from (19), tan (u, d>) = ^ 
 
 M + S 
 
 cosx \ ..................... (21) 
 
 High water occurs approximately ^-, , or Jg</> after moon's transit. 
 
108 SEMIDIURNAL TIDE. [3 
 
 The determination of </> and H may be conveniently carried out by a 
 graphical construction. If we take O as a fixed centre, OS as an initial line, 
 and S a point in it such that OS = S, and set off the angle AOM equal to x, 
 and OM equal to M ; then OMS is the angle p. - <f>, and SM is the height H. 
 
 The angle x increases by 360 from spring-tide to spring-tide, and there- 
 fore one revolution in the figure corresponds to 15 days. 
 
 As a very rough approximation, M lies on a circle, but the parallactic 
 and declinational corrections 3jR m and S 2 R m cause a considerable departure 
 from the circle. 
 
 The angle <> and the height H are also easily computed numerically. 
 If cos x is positive, let 6 be an auxiliary angle determined by 
 
 tan 2 6 = ^ r cos x 
 M 
 
 and we have 
 
 tan (fju <) = sin 2 tan x, H = S cosec (/u, </>) sin x 
 If cos x is negative, let 6 be an auxiliary angle determined by 
 
 sin 2 = ^f cos x 
 M 
 
 and we have 
 
 tan (fj, <f>) = tan 2 6 tan x, H = S cosec (//- </>) sin x 
 These formulae are adapted for logarithmic computation. 
 
 5. Correction for Diurnal Tides. 
 
 The tide-table has to be corrected for the effect of three diurnal tides, 
 designated 0, K 1} P. 
 
 If we write V = t -f h 2s v + 2 4- |TT 
 
 V' = t + h- V '-^7T 
 
 then, in accordance with Schedules B of the Report of 1883, the expres- 
 sions for the three tides are 
 
 O =f H cos(V -* ) 
 K^f'H'cosCV'-*') 
 P =- Hp cos [V'-*'-- (2/i -i/) + ('-*)] (22) 
 
1886] CORRECTIONS FOR DIURNAL TIDE. 109 
 
 [The tides K x and P change their phases relatively to one another in 
 
 half a year, and since P is considerably smaller than K n we may without 
 
 serious error attribute to 2h a mean value for a short period such as a 
 
 month. Also we may assume tc p = K. 
 
 Hence if denotes the sun's mean longitude at the middle of the 
 month (or other short period), we may with rough approximation write 
 the expression for the P tide in the form ^ 
 
 P = - H p cos [V - ' - (2 - v')] 
 Therefore if we put 
 
 sin (20 -i/) , 3f'-coB(20-iQ , 
 
 n * == 3f- cos (20-0' 3cos4> 
 
 and if further we write f H = R the diurnal tides, reduced to two, are] 
 
 = R cos (V - K O ) 
 K 1 + P = R'cos(V'-'-0) ..................... (23) 
 
 <f> and R', having a semi-annual inequality, must be recomputed for each 
 month. 
 
 Now, suppose that we compute V and V at the epoch, that is, at the 
 initial noon of the period during which we wish to predict the tides, and 
 with these values put 
 
 = K O V at epoch 
 ' = K </> V at epoch 
 
 then the speed of V is 7-2<r, or 13'94 per hour, or 360-25 0> 37 per 
 day; and the speed of V is 7, or 15'04 per hour, or 360 0> 986 per day. 
 Hence, if t be the mean solar time in hours on the (n + l)th day since 
 the epoch, 
 
 V - KO = 360rc + 13-94t - - 2 
 
 V + (f> - K = 36Qn + 15-04t - f + 0'986w 
 
 Therefore the diurnal tide at the time t hours on the (n + l)th day is 
 given approximately by 
 
 = R cos [14t - - 25 J c x n] 
 K 1 +P = R / cos[15t-' + l xn] ........................ (24) 
 
 If we substitute for t the time of high or low water as computed simply 
 from the semidiurnal tide, it is clear that the sum of these two expres- 
 sions will give us the diurnal correction for height of tide at high or low 
 water. 
 
110 CORRECTIONS FOR DIURNAL TIDE. [3 
 
 If we consider the maximum of a function, 
 
 A cos 2n (t ot) + B cos n' (t /8) 
 
 where n is nearly equal to n', we see that the time of maximum is given 
 approximately by t = a, with a correction St determined from 
 
 - 2An sin (2/iSt) - n'B sin n (t - p 1 ) = 
 
 180 n'B . 
 or ot = -; -r sin ?i ( t /3) 
 
 47T71 71.0. 
 
 In this way we find the corrections to the time of high water from and 
 
 Ki + P : and since ti = 7 cr, and -: = O h> 988. and = 1 for 0, 
 
 47rw n 7 0- 
 
 and 1 -I for K x , we have 
 
 7 0" 
 
 8t = - O h '988 ( 1 -) sin [14t - , - 25 J x n] 
 
 \ 7-07 H 
 
 8t/ = - O h '988 ( 1 + - N ) sin [15t- ?' + 1 x n] (25) 
 
 \ 7 <7/ rl 
 
 where H is the height of the semidiurnal high water. 
 
 With sufficient approximation we may write these corrections : 
 
 Bt = - l h x ? s i n [] 4t - & - 254 x n] 
 ti 
 
 H 
 
 The computations are easily carried out, although the arithmetic is neces- 
 sarily tedious. Since two places of decimals are generally sufficient for R 
 and R', the multiplications by the sines and cosines are very easily made 
 with a Traverse Table. 
 
 The successive high and low waters follow one another on the average 
 at 6 h 12 m ; now, 14 x 6'2 - 87, and 15 x 6'2 = 93. Hence, if we compute 
 14t ^ 25^ x n for the first tide on any day, the remaining values are 
 found with sufficient approximation by adding once, twice, thrice 87; and 
 similarly, in the case of 15t ' + 1 x n we add once, twice, thrice 93. 
 
 [If the diurnal tide is the predominant one, as occurs at some places, 
 this method of correction would of course be insufficient.] 
 
1886] DETAILS AS TO COMPUTATION. Ill 
 
 6. Certain Details in the Computation of the Tide-table. 
 
 It will be well to give some explanatory details concerning the manner 
 of carrying out the computations. 
 
 The angle A is given by 16'51 + 3 '44 cos 8 -0'19cos2S3, where S3 is 
 the longitude of the moon's node. It is clear that A varies so slowly that 
 it may be regarded as constant for many months, and the same is true of 
 the factors f, f", f, f , and the small angles v, f, v, 2v". Approximate 
 formulae for these quantities in terms of S were given at the end of the 
 first paper in this volume, and are used in the article in the Manual 
 [Paper 4]. 
 
 To find the cube of the ratio of the sun's parallax to his mean parallax, 
 the following rule is given : Subtract the mean parallax from the parallax, 
 multiply the difference by 19J, read as degrees instead of seconds, look out 
 the sine, and add 1. This rule is founded on the fact that a mean parallax 
 8"'85 multiplied by 19 J gives 3 x 57", and 57 is the unit angle or radian, 
 whilst the sine of a small angle is equal to the angle in radians*. Similarly, 
 the cube of the ratio of the moon's parallax to her mean parallax is 
 
 1+3 sin [60 (parx mean parx)] 
 
 That is to say, for the moon : Subtract the mean parallax from the parallax, 
 read as degrees instead of minutes, look out the sine, multiply by 3, and 
 add 1. This rule depends on the fact that the moon's mean parallax in 
 radians is ^. 
 
 For the purpose of applying the corrections 8]R W , &jR m , 8 2 K m , 8 2 i, S^y, it 
 is most convenient to compute auxiliary tables for each degree of declination 
 of the moon and minute of her parallax, and then the actual corrections are 
 easily applied by interpolation. 
 
 These tables serve for the port as long as the longitude of the moon's 
 node is nearly constant, or with rougher approximation for all time. 
 
 The declinational and parallactic corrections to high water depend on 
 the moon's declination and parallax at a time anterior to high water by 
 ' the age.' Hence, in order to find these corrections we have to know the 
 time of high water in round numbers. Each high water follows a moon's 
 transit at the port approximately by the interval i. The Greenwich time 
 of the moon's transit at the port is the G.M.T. of moon's transit at 
 Greenwich, less 2 minutes for each hour of E. longitude, less the E. longi- 
 tude in hours. Then, if we subtract from this ' the age ' and add the 
 interval i, we find the G.M.T.'s at which we want the moon's declination and 
 parallax. 
 
 * [The mean solar parallax is now taken, in the Nautical Almanac, as 8" -80, but the 
 rule still remains sufficiently exact.] 
 
112 EXAMPLE OF TIDE-TABLE. [3 
 
 Thus, at Port Blair"! rn ]yr T f VA 
 
 the G.M.T. at which wel = -T ' S \ - long. corr. for transit (O h '2) 
 
 , , (transit at Gr.j 
 
 want parx. and decl. J 
 
 - E. long, of port (6 h< 2) - age of tide (32 h> 6) 
 + mean interval (9 h> 6) 
 = G.M.T. of Ys tr. at Gr. - 29 h> 4. 
 
 Thus at Greenwich, on Feb. 1st, 1885, the moon's lower transit was 
 at 2 h , and hence, corresponding to the lower transit at Port Blair of Feb. 1, 
 we require the moon's parallax and declination at 21 h Jan. 30, G.M.T. The 
 parallax at the nearest Greenwich noon or midnight is sufficiently near the 
 truth, and therefore we take the parallax at O h Jan. 31, which is 60 /g O, and 
 the excess above the mean is 3''0, and 1 + 3 sin 3 is T157, which is the factor 
 p'. Actually, however, we read off the correction ^Rm and the other correc- 
 tions S 2 R/i> & 2 i, 8 2 y straight from the auxiliary tables. 
 
 7. On Tide-tables Computed by the above Method. 
 
 A great deal of arithmetical work was necessary in making trial of the 
 rules devised above and in various modifications of them, and I must record 
 my thanks to Mr Allnutt, who has been indefatigable in working out tide- 
 tables for various ports, and in comparing them with official tables. The 
 whole of the results, to which I now refer, are due to him. The following 
 table exhibits the amount of agreement between a computed table and one 
 obtained by the tide-predicting instrument. It must be borne in mind that 
 the instrument is rigorous in principle, and makes use of far more ample 
 data than are supposed to be available in our computations. The columns 
 headed 'Indian tables' are taken from the official Indian tide-tables. The 
 datum level, however, in those tables is 3'13 ft. below mean water mark, 
 whereas ' Indian spring low-water mark ' is 3'55 ft. below the mean. Thus, to 
 convert the heights given in the Indian tables to our datum 0'42 ft. or 5 ins. 
 have been added to all the heights in the official table. 
 
 A tide-table was computed for Aden for a fortnight, and the results were 
 found to be somewhat less satisfactory than those in the following table. It 
 must be remarked, however, that the sum of the semi-ranges of the three 
 diurnal tides Kj , O, P is 2'340 ft. and is actually greater than the sum of the 
 semi-ranges of the tides M 2 and S 2 , which is 2*265 ft. Thus, at some parts 
 of some lunations the semidiurnal tide is obliterated by the diurnal tide, and 
 there is only one high water and one low water in the day. In this case it is 
 obvious that the approximation, by which we determine semidiurnal high 
 and low water and apply a correction for the diurnal tides, becomes inapplic- 
 able. In the greater part of our computed table the concordance is fairly 
 good; but the tide-predicting instrument shows that on each of the days, 
 
1886] 
 
 EXAMPLE OF TIDE-TABLE. 
 
 113 
 
 7th and 8th February, 1885, there was only one high and low water, whereas 
 our table, of course, gives a double tide as usual. Again, on the 9th February 
 there is an error of 68 minutes in a high water. These discrepancies are to 
 be expected, since the approximate method is here pushed beyond its due 
 limits ; and for such a port as Aden special methods of numerical approxima- 
 tion would have to be devised. 
 
 TIDE-TABLE FOR PORT BLAIR, 1885. 
 
 
 Calculated 
 Times 
 
 Indian tables 
 Times 
 
 Calculated 
 Heights 
 
 Indian tables 
 Heights 
 
 
 h. m. 
 
 h. m. 
 
 ft. 
 
 ft. in. 
 
 Feb. 1, H.W. 
 
 11 3 p.m. 
 
 11 4 p.m. 
 
 7'4 
 
 7 2 
 
 Feb. 2, L.W. 
 
 5 21 a.m. 
 
 518 a.m. 
 
 o-o 
 
 -0 2 
 
 H.W. 
 
 11 26 a.m. 
 
 11 31 a.m. 
 
 6-6 
 
 6 5 
 
 L.W. 
 
 5 28 p.m. 
 
 5 25 p.m. 
 
 0-4 
 
 
 
 H.W. 
 
 11 39 p.m. 
 
 11 43 p.m. 
 
 7-1 
 
 6 11 
 
 Feb. 3, L.W. 
 
 5 56 a.m. 
 
 5 56 a.m. 
 
 0-2 
 
 1 
 
 H.W. 
 
 3 p.m. 
 
 9 p.m. 
 
 6-4 
 
 6 3 
 
 L.W. 
 
 6 4 p.m. 
 
 6 5 p.m. 
 
 07 
 
 7 
 
 Feb. 4, H.W. 
 
 14 a.m. 
 
 20 a.m. 
 
 6-7 
 
 6 6 
 
 L.W. 
 
 6 31 a.m. 
 
 6 33 a.m. 
 
 0-5 
 
 5 
 
 H.W. 
 
 40 p.m. 
 
 48 p.m. 
 
 6-1 
 
 6 
 
 L.W. 
 
 6 42 p.m. 
 
 6 44 p.m. 
 
 1-2 
 
 1 
 
 Feb. 5, H.W. 
 
 48 a.m. 
 
 56 a.m. 
 
 6-1 
 
 5 11 
 
 L.W. 
 
 7 5 a.m. 
 
 7 9 a.m. 
 
 1-0 
 
 10 
 
 H.W. 
 
 1 18 p.m. 
 
 1 28 p.m. 
 
 5-7 
 
 5 7 
 
 L.W. 
 
 7 20 p.m. 
 
 7 25 p.m. 
 
 1-7 
 
 1 7 
 
 Feb. 6, H.W. 
 
 1 24 a.m. 
 
 1 33 a. in. 
 
 5-5 
 
 5 4 
 
 L.W. 
 
 7 41 a.m. 
 
 7 45 a.m. 
 
 1-5 
 
 1 4 
 
 H.W. 
 
 2 1 p.m. 
 
 2 10 p.m. 
 
 5-3 
 
 5 2 
 
 L.W. 
 
 8 6 p.m. 
 
 8 12 p.m. 
 
 2-2 
 
 2 1 
 
 Feb. 7, H.W. 
 
 2 4 a.m. 
 
 2 13 a.rn. 
 
 4'9 
 
 4 9 
 
 L.W. 
 
 8 23 a.m. 
 
 8 25 a.m. 
 
 1-9 
 
 1 10 
 
 H.W. 
 
 2 53 p.m. 
 
 2 57 p.m. 
 
 4-9 
 
 4 10 
 
 L.W. 
 
 9 7 p.m. 
 
 9 8 p.m. 
 
 2-7 
 
 2 6 
 
 Feb. 8, H.W. 
 
 2 58 a.m. 
 
 3 8 a.rn. 
 
 4-4 
 
 4 3 
 
 L.W. 
 
 9 20 a.m. 
 
 9 24 a.m. 
 
 2-4 
 
 2 2 
 
 H.W. 
 
 4 10p.m. 
 
 4 14 p.m. 
 
 4-7 
 
 4 7 
 
 L.W. 
 
 10 42 p.m. 
 
 10 40 p.m. 
 
 3-0 
 
 2 4 
 
 Feb. 9, H.W. 
 
 4 29 a.m. 
 
 4 40 a.m. 
 
 4-0 
 
 3 10 
 
 L.W. 
 
 10 46 a.m. 
 
 10 57 a.m. 
 
 2-6 
 
 2 6 
 
 H.W. 
 
 5 47 p.m. 
 
 5 48 p.m. 
 
 4-7 
 
 4 7 
 
 In a table computed for Amherst the agreement is not quite so good as 
 was to be hoped ; the error in heights amounts in two cases in fifteen days 
 to nearly a foot, and in two other cases to three-quarters of an hour in time. 
 It may be remarked, however, that the tides are large at Amherst, having a 
 spring range of 20 ft. and a neap range of 6 ft., that the diurnal tide is con- 
 siderable, and that the sum of the semi-ranges of the over-tides M 4 , S 4 (which 
 
 D. I. 
 
114 EXAMPLE OF TIDE-TABLE. [3 
 
 we neglect entirely) amounts to 6 inches. It appears also that the tidal 
 constants are somewhat abnormal, for H" = ^H S instead of H" = jj7g_H s , and 
 
 further Hp = ^:gH / instead of H^jjH'. Under these circumstances it is 
 perhaps not surprising that the discrepancies are as great as they are. 
 
 Tables were also computed for Liverpool and West Hartlepool, but no 
 correction was here applied for the diurnal tides. The results were compared 
 with the Admiralty tide-tables for Liverpool and Sunderland. In the case 
 of Liverpool there were four tides in a fortnight in which there was a dis- 
 crepancy in the times amounting to 12 minutes, and four other tides in 
 which there was a discrepancy of a foot, and one with a discrepancy of 
 1 ft. 2 ins. It was obvious, however, that the agreement would have been 
 better if the correction for the diurnal tides had been applied. The spring 
 rise of tide at Liverpool is 26 ft. 
 
 In the case of Sunderland there were in a fortnight two discrepancies 
 of 15 m., two of 14 m., two of 13 m., two of 12 m., &c. in the times, and in 
 the heights one discrepancy of 3 ins., and four of 2 ins., &c. The spring rise 
 at West Hartlepool is 14 ft. 
 
 These two tables are quite as satisfactory as could be expected considering 
 the approximate nature of the methods employed. 
 
 Finally, in order to test the methods both of reduction and of prediction, 
 Mr Allnutt took the harmonic constants derived from our analysis of a 
 fortnight of hourly observation at Port Blair, from April 19 to May 2, 1880, 
 and computed therefrom a tide-table for that same fortnight. He then, by 
 interpolation in the observed hourly heights, determined the actual high 
 waters and low waters during that period. 
 
 The results of the comparison are exhibited in the table on next page. 
 
 If our method had been perfect, of course, the errors should be every- 
 where zero. 
 
 It must be admitted that the agreement is less perfect than might have 
 been hoped. If, however, the calculated and observed tide curves are 
 plotted down graphically side by side, it will be seen that the errors are 
 inconsiderable fractions of the whole intervals of time and heights under 
 consideration. 
 
 When we consider the extreme complication of tidal phenomena, together 
 with meteorological perturbation, it is, perhaps, not reasonable to expect any 
 better results from an admittedly approximate method, adapted for all ports, 
 and making use of a very limited number of tidal constants. In devising 
 these rules for reduction and prediction I could find no model to work from, 
 
1886] 
 
 EXAMPLE OF TIDE-TABLE. 
 
 115 
 
 
 
 
 K9$wVa92*asaasaVaaMaMS 
 
 
 o 
 
 - , , , ,+ ++ +++ ++++++1 , , , , , , , , , 
 
 
 ll 
 
 ^SSg8Sit3!S883?S833SgS3SS:3SS3ffS 
 
 
 ^w 
 
 (M W M ^ ^ ^ ^ rH ^4 ^4P- OI^^^CCrNCCOqw 
 
 
 H 
 
 d SSSSSSS&SSS8SSS9S8a98SSS8a 
 
 S3 
 
 I? 
 
 ..fOCM..CN(N (N r-i r-Hr-i^H(M<yi(Moi{<JcO..MfC 
 
 H 
 
 H 
 
 KT 
 
 o 
 
 (M-fl^OOOO'-HCOODOOr^l>rMi^WfMOCr-iOi lOCD 
 
 O 
 
 o 
 
 "* 1 1 1 1 1 I 1 1 1 1 + + + + + + V + + 4- + + 4-+ 1 l i 
 
 M 
 
 d 2 
 
 OCOCi i O O <M i ' ! ^ ifM'MX-fOOl'-O'^SOOOOiOCMOSOOTt' 
 
 
 O EH 
 
 ^HfO(>5 P9 i-4 t * *3 Cb CO * O f^ > OO t- 6i 00 O OS f^-O (N 
 
 
 
 <CN 
 
 
 T3 
 
 * 
 
 i a 
 
 81N 00 CC Q <N Oi 1 s - 00 O CO 00 00 Q 1^- <N TO O 00 00 O W O O i 
 CiOr iTt^QOi *1^ l^- ^lf>lCiOCOt^-COCOfM r M r MQOf>lO'*^CDOCf3 v l 
 
 
 jH 
 
 >^AieoGiA4M-i->i^i(N^'C | 9>oncD^i<oibrDdbt>dbaooai-<ocQ 
 
 
 o 
 
 
 
 -2 
 
 "3 
 
 Q 00 
 tl 00 
 
 -3 i 
 
 X 
 
 <! 
 
 Oi ^5 >"^ ^1 CO ^^ 'O CO t^* 00 O5 <O r-n GQ 
 i i'M(M(N(MCN(Maq<5q(MtNCi 
 
 1: ^ 
 
 
 O 
 
 .jsgsgsssssssggssssgsiisfeassgsjsisas 
 
 
 o 
 
 ^ + + + + + +1 Mll M. + + 4 + 4- + + + 4- + 4- + + + 4- 
 
 
 ll 
 
 ^^Sg^^S'ggfiig^^SS^^SSES^ilSSS^S 
 
 
 * 
 
 W iO P W W t W 1> t^ <0 t' CO 1> l> 00 t- 00 W t- ^ CD ^ CO W W 
 
 
 ^ ^ 
 &I 
 
 MG5(MCOOCOt^OO5'MOCO'-<Or- lOOOOSOW^fOSCO-^fOiD^O 
 ^JOO>OfM'MCOOiO3CDfCfMO-*Tt<T)<ffOi iOOOOOr-i(NXl^CDfOOC-^f 
 
 03 
 
 |w 
 
 tM ih ib cb i cb cb i> i^ t^ ob t~ oc i> do i-- ao i~- t- cb t- cb i^ ib cb o cb o cb 
 
 H 
 ^ 
 [ 
 
 o 
 
 .8^S^S8^^SSS822^2^??^^^8 
 
 a 
 o 
 
 o 
 
 1 1 1 1 1 1 1 i 1 1 1 1 + 1 1 4- +4-4-4- 4-4-4-4-1 l 
 
 a 
 
 41 
 
 CO vO ii T-l GC O^ f^ t^* CO 1^ G^l 00 GO *~^ CD l"^~ ^^ 00 O* CO !> CO CO GO CO G^I GO ^~^ 
 
 
 oS 
 
 ^rfi^occcbo^t^ocboa>i lOi^oco^-icodq co- t-^(>iibcocbib 
 
 
 
 
 
 T3 
 
 ll 
 
 .^^gcss^g^&s^^ifegigg^g^s^^ss^^ 
 
 
 is 
 
 ^4ficb6ii^6:i^odoi iO5i icsfNOfC' icos<iOMi iTHiyiibcct^ic 
 
 
 o 
 
 
 
 fil 
 
 
 
 ^ 
 
 O5O' ifMCO^iOCCt^OOOiOi i(NCC 
 ittq(MCN<M<M<M<MCN(NCNffC 
 
 ^ % 
 
 ^ 
 
116 VARIATIONS OF MEAN SEA-LEVEL. [3 
 
 and it seems probable that advantageous modifications may be introduced. 
 I spared, however, no pains to reduce the labour of computation. Nearly 
 half the work in forming a short tide-table is preparatory, and would serve 
 for a systematic computation of tables for all time. 
 
 III. AN ATTEMPT TO DETECT THE 19- YEARLY TIDE. 
 
 If M, E be the moon's and earth's masses ; a the earth's mean radius ; 
 c the moon's mean distance ; &> the obliquity of the ecliptic ; i the inclination 
 of the lunar orbit ; e the eccentricity of the lunar orbit ; & the longitude 
 of the moon's node ; and A, the latitude of the port of observation ; then the 
 term in the equilibrium tidal theory which is independent of the moon's 
 longitude (see Schedule B, iii., Paper 1, p. 22) is 
 
 M /ct\ 3 
 f ~v (~ I a (i ~ f si 11 " V) (1 + f^ 2 ) sm * cos * sm w cos m 
 
 Jit \C I 
 
 [ cos 3 -f j tan i tan co cos '2 & ] 
 Since tan t tan = '00975, the second term is negligeable compared with 
 
 the first. 
 If we take 
 M 1 
 
 a 1 
 
 E~81'5' 
 
 c ~ 60-27 
 
 a = 21 x 10 6 feet, i = 5 8', G> = 23 28' 
 
 the expression for this tide is, in British feet, 
 
 - 0-0579 (^ - | sin 2 X) cos a 
 
 Thus, at the poles this tide gives an oscillation of sea-level of 0"695 of an 
 inch, or a total range of If of an inch, and at the equator it is half as great. 
 
 In the Mecanique Celeste Laplace argues that all the tides of long 
 period (such as the fortnightly tide) must conform nearly to the equilibrium 
 law. I shall adduce arguments elsewhere* which seem to invalidate his 
 conclusion, and to show that in these tides inertia still plays the principal 
 part, so that the oscillations must take place nearly as though the sea were a 
 frictionless fluid. 
 
 With a tide, however, of as long a period as nineteen years Laplace's 
 argument must hold good, and hence the equilibrium tide of which the 
 above is the expression must represent an actual oscillation of sea-level, 
 provided that the earth is absolutely rigid. The actual observation of the 
 19-yearly tide would therefore be a result of the greatest interest for deter- 
 mining the elasticity of the earth's mass. 
 
 * See Paper 11 below " On the Tides of Long Period." 
 
1886] 
 
 VARIATIONS OF MEAN SEA-LEVEL. 
 
 117 
 
 g-g 
 
 T3 9 
 
 o 
 
 s * 
 
 I 
 
 II 
 
 
 ' 
 
 s.s 
 
 60 
 0) *"> 
 
 ' 
 
 
 
 C8 
 
 a e 
 o 
 
 > w 
 
 'S -2 
 
 3 S 
 
 o> "- 1 
 
 ^ a 
 
 
 
 H =2 eg 
 
 23 
 
118 VARIATIONS OF MEAN SEA-LEVEL. [3 
 
 A reduction of the observed tides of long period at a number of ports 
 was carried out in Thomson and Tait's Natural Philosophy, Part II.; 1883 
 [Paper 9 below], in the belief in the soundness of Laplace's argument with 
 regard to those tides, and the conclusion was drawn that the earth must have 
 an effective rigidity about as great as that of steel. The failure of Laplace's 
 argument, however, condemns this conclusion, and precludes us from making 
 any numerical conclusions with regard to the rigidity of the earth's mass, 
 excepting by means of the 19-yearly tide. The results given in the 
 Natural Philosophy merely remain, then, as generally confirmatory of 
 Thomson's conclusion as to the great effective rigidity of the earth's mass. 
 
 There are but few ports for which a sufficient mass of accurate tidal 
 observations are accumulated to make the detection of the 19-yearly tide 
 a possibility. 
 
 Major Baird has, however, kindly supplied me with the values of the 
 mean sea-level at Karachi for fifteen years. They are plotted out in the 
 figure on the preceding page. The horizontal line represents the mean 
 sea-level for the period from 1869-1883, and the sinuous curve gives the 
 variations of mean sea-level during that period. The dotted sinuous curve 
 gives the annual variations for a portion of the same period for Bombay. 
 The full-line sweeping curve has ordinates proportional to cos Q , and shows 
 the kind of curve which we ought to find if the alternations of sea-level were 
 due to the 19-yearly tide. 
 
 It is obvious at a glance that the oscillations of sea-level are not due to 
 astronomical causes. 
 
 At Karachi (lat. 24 47') the 19-yearly tide is 
 
 - O ft '0138 cos ga 
 
 The figure shows that the actual change of sea-level between 1870 and 
 1873 was nearly 0'25 feet, and this is just about nine times the range of 
 the 19-yearly tide, viz., 0'028 feet. 
 
 It is thus obvious that this tide must be entirely masked by changes 
 of sea-level arising from meteorological causes. 
 
 It seems unlikely that what is true of Karachi and Bombay is untrue 
 at other ports, and therefore we must regard it as extremely improbable that 
 the 19-yearly tide will ever be detected. 
 
4 
 
 A GENERAL ARTICLE ON THE TIDES. 
 
 [Article 'Tides/ Admiralty Scientific Manual (1886), pp. 53 91.] 
 
 I. INTRODUCTION. 
 
 THE object of the present article is to show how the best use may be 
 made, for scientific purposes, of a short visit to any port. 
 
 We refer to the article " Hydrography " [Admiralty Scientific Manual] 
 for an account of the method of observing the tides, and shall here assume 
 that the height of the water above some zero mark may be measured, in feet 
 and decimals of a foot, at any time, and that the zero of the tide gauge may 
 be referred by levelling to a bench-mark ashore. 
 
 Something of the law of the tide might be discovered from hourly or 
 half-hourly observations even through a single day and night, but to discover 
 the law at all adequately it is necessary that the observations should embrace 
 at least one spring tide and one neap tide. For the full use of the methods 
 given below, the observations should be taken each hour for 360 hours, or 
 720 hours. A longer series must be regarded as a new set of observations, 
 and the means must be taken of the results of the several sets. 
 
 It has been usual to recommend observations of the times and heights of 
 high and low water, but hourly observations are far preferable, the hours 
 being reckoned according to mean time of the port. 
 
 We shall, however, begin by a sketch of the treatment of .observations of 
 high and low water, and shall then give more detailed instructions for hourly 
 observations and the formation of a tide table. 
 
 The height of the water is subject to considerable perturbation from the 
 weather, and the most perfect tide table is one which gives the height of the 
 water, when abstraction is made of the disturbing causes. Such a table can 
 only be made from observations of such extent as to eliminate irregularities 
 by averages. 
 
120 OBSERVATIONS OF HIGH AND LOW-WATER. [4 
 
 No general rule can be given for wind disturbance, but it is often con- 
 siderable in bays and estuaries. The water stands higher with low, and lower 
 with high, barometer ; the amount of the effect appears to be very uncertain, 
 the estimates varying from 7 inches to 20 inches rise of water for an inch fall 
 of the mercury. It appears probable that the rule differs in different ports, 
 and even in the same port with different winds. To make the most, however, 
 of a short series of observations, it might perhaps be best to reduce each 
 hourly tide height to a standard height of barometer at the rate of a foot of 
 water to an inch of mercury, before undertaking the tidal reductions. 
 
 In order to discover the general run of the tide in any part of the world, 
 observations should be taken at several stations separated by 50 to 100 miles; 
 and this is the more important if some of the stations have to be chosen in 
 estuaries, since the tide wave takes a considerable time to run up from the 
 open sea and changes its form in doing so. 
 
 In estuaries and rivers it is important not to confuse flood and ebb with 
 high and low water, for the water often still runs up-stream for long after 
 the tide has turned and when the water-level is falling ; and the converse is 
 true of ebb and low water. We refer to " Hydrography " for remarks on tidal 
 currents and streams. 
 
 II. TIDAL OBSERVATIONS OF HIGH AND Low WATER*. 
 
 The immediate object is to connect the times and heights of high and 
 low water (H. W. and L.W.) with the time of the moon's transit. About high 
 and low tide the water often rises and falls irregularly, and the critical 
 moment cannot be found from a single observation. Observations are, there- 
 fore, to be taken every 5 or 10 minutes for half-an-hour or an hour about 
 H.W. and L.W. The time and height of H.W. or L.W. are then to be found 
 by graphical interpolation, i.e., take a straight line to represent time, and at 
 the points corresponding to the observations erect perpendiculars or ordinates 
 corresponding to the observed heights, draw a sweeping curve nearly through 
 the tops of the ordinates, so as to obliterate minor irregularities and measure 
 the height of the maximum or minimum ordinate, and note its incidence in 
 the time scale ^. 
 
 9 
 
 Dr Whewell recommends that the observation should begin with half- 
 hourly observations during 24 hours, for if there should be found to be 
 double H.W. or L.W., or only a single tide in the 24 hours, this method will 
 fail ; he also advises that tidal observations be referred to the moon's transit 
 
 * Founded on Dr Whewell's article in a former edition of the Admiralty Scientific Manual. 
 t A similar but less elaborate process would render hourly observations more perfect. The 
 readings might be every 2J minutes, from five minutes before to five minutes after the hour. 
 
1886] OBSERVATIONS OF HIGH AND LOW- WATER. 121 
 
 during their course, in order to detect irregularities in " the interval " from 
 transit to H.W., which might cause the observations to prove useless. 
 
 The object of the observations is to find " the establishment," or time of 
 high water, on days of full and change of moon, the heights of tide at spring 
 and neap, and " the fortnightly or semi-mensual irregularity " in the time 
 and height. The reference of the tide to the " establishment " is not, how- 
 ever, scientifically desirable, and it is better to determine the mean or 
 corrected establishment, being the average interval from moon's transit to 
 H.W. at spring tide, and "the age of the tide," being the average interval 
 from full and change to spring tide. For these purposes the observations 
 are conveniently treated graphically*. 
 
 An equally divided horizontal scale is taken to represent the 12 hours of 
 the clock of civil time, regulated to the time of the port or more accurately 
 arranged always to show apparent time by being fast or slow by the equation 
 of time ; this time scale represents the time of the clock of the moon's transit, 
 either upper or lower. The scale is perhaps most conveniently arranged in 
 the order V, VI, ..., XII, I, ..., IIII. Then each "interval" of time from 
 transit to H.W. is set off as an ordinate above the corresponding time-of- 
 clock of moon's transit. A sweeping curve is then drawn so as to pass nearly 
 through the tops of the ordinates, cutting off minor irregularities. Next 
 along the same ordinates are set off lengths corresponding to the height of 
 water at each H.W. 
 
 A second similar figure may also be made for the interval and height 
 at L.W. 
 
 In the curve of H.W. intervals the ordinate corresponding to XII is the 
 vulgar " establishment," since it gives the time of H.W. at full and change of 
 moon. That ordinate of H.W. intervals which is coincident with the greatest 
 ordinate of H.W. heights gives the " mean establishment f." 
 
 Since the moon's transit falls about 50 minutes later on each day, in 
 setting off a fortnight's observation there will be about five days for every 
 four hours-of-clock of moon's upper transit. Hence in these figures we may 
 regard each division of the time scale I to II, II to III, &c. as representing 
 25 hours instead of one hour. Then the distance from the maximum ordinate 
 of H.W. heights to XII, each division being estimated as 25 hours, is called 
 " the age of the tide." 
 
 From these two figures the times and heights of H.W. and L.W. may in 
 general be predicted with fair approximation ; we find the time-of-clock of 
 moon's upper or lower transit on the day, correct by the equation of time, 
 
 * For numerical treatment, see Directions for reducing Tidal Observation*, By Staff Com- 
 mander John Burdwood, R.N. London, 1876. J. D. Potter. Price Gd. 
 
 t See Section IV. for the numerical computation of mean from vulgar establishment. 
 
122 INSTRUCTIONS FOR HARMONIC ANALYSIS. [4 
 
 read off the corresponding heights of H.W. and L.W. from the figures ; and 
 the intervals to H.W. and L.W. being also read off are added to the time of 
 moon's transit, and give the times of H.W. and L.W. 
 
 We shall show below how a tide table may be otherwise computed from 
 establishment and spring rise and neap rise. 
 
 At all ports, however, there is an irregularity of intervals and heights 
 between successive tides, and in consequence of this our curves will present 
 more or less of a zig-zag appearance. Where the zig-zag is perceptible to the 
 eye, the curves must be smoothed by drawing them so as to bisect the 
 zig-zags, because these "diurnal inequalities" will not present themselves 
 similarly in the future. When, as in many equatorial ports, the diurnal tides 
 are large, this method of tidal prediction fails, but we shall show below how 
 the observations may then be treated scientifically. 
 
 III. INSTRUCTIONS FOR THE REDUCTION OF HOURLY TIDAL 
 OBSERVATIONS, WITH AN EXAMPLE. 
 
 We now suppose that the observations of the tides are taken at each hour. 
 If the observations are only taken every two hours, or if there are gaps in the 
 series, the hourly numbers must be filled in as indicated below. 
 
 All the measurements should be positive, and if the zero of the tide gauge 
 has been fixed too high it will be well to refer the measurement to an ideal 
 zero 10 feet lower. The following instructions for reduction should be read 
 along with the example. 
 
 COMPUTATION FORMS. 
 
 Mark three large sheets of paper with the letters M, 0, S ; divide them 
 into squares, with 24 columns; head the columns, O h , l h , 2 h ...23 h for the 
 several hours. On the left margin write the numbers of the days, O d , l d , 2 d , 
 &c. Each square may be specified by its day and hour. 
 
 M Sheet. 
 
 Place dots in the squares of each row, as follows : O d , 14 h ; l d , 18 h ; 2 d , 
 23 h ; 3 d , none; 4 d , 3 h ; 5 d , 8 h ; 6 d , 12 h ; 7 d , I7 h ; 8 d , 21 h ; 9 d , none; 10 d , 2 h ; 
 ll d , 7 h ; 12 d , ll h ; 13 d , 16 h ; (13 d is the last row required for a fortnight's 
 observation); 14 d , 20 h ; 15 d , none; 16 d , l h ; I7 d , 5 h ; 18 d , 10 h ; 19 d , 14 h ; 20 d , 
 19 h ; 21 d , 23 h ; 22 d , none ; 23 d , 4 h ; 24 d , 8 h ; 25 d , 13 h ; 26 d , 17 h ; 27 d , 22 h . (27 d 
 is the last row required for a month's observation.) 
 
 Sheet. 
 
 Place dots in the following squares: O d , 6 h and 19 b ; l d , 8 h and 22" ; 2 d , 
 ll h 3 d , O h and 13 h ; 4 d , 2 h and 16 h ; 5 d , 5 h and 18 h ; 6 d , 7 h and 20 h ; 7 d , 10 h 
 
1886] INSTRUCTIONS FOR HARMONIC ANALYSIS. 123 
 
 and 23"; 8 a , 12 h ; 9 d , l h and 14 h ; 10 a , 4 h and 17 h ; ll d , 6 h and 19 h ; 12 d , 8 h 
 and 22 h (12 <l is the last row required for a fortnight's observation); 13 d , ll h ; 
 14 d , O h and 13 h ; 15 d , 2 h and 15 h ; 16 d , 5 h and 18 h ; 17 d , 7 h and 20 h ; 18 d , 9 h 
 and 23 h ; 19 d , 12 h ; 20 d , 1" and 14 h ; 21 d , 3" and 17 h ; 22 d , 6 h and 19 h ; 23 d , 8 h 
 and 21 h ; 24 d , ll h . (24 d is the last row required for a month's observation.) 
 
 S Sheet. 
 
 There are no dots. For a fortnight's observation, let row 13 d be the last ; 
 then leave three rows blank, and add another row numbered 14 d . 
 
 For a month's observation, let row 26 d be the last ; then leave three rows 
 blank, and add three more rows numbered 27 d , 28 d , 29 (1 . 
 
 The shorter series (13 d or 26 d ) is for diurnal tides, the longer (14 d or 29 d ) 
 for semi-diurnal tides. 
 
 Entry in S. 
 
 The first hourly observation is supposed to be at noon or O d , O h ; enter it 
 in that square ; enter the second in O d , l h ; the third in O d , 2 1 ', and so on. 
 
 The O h 's of the rows are the noons of each day ; write the day of the 
 month opposite each row. 
 
 If any hourly observations are wanting or obviously vitiated through 
 accident or weather, fill in the blanks thus. Consider a column in which a 
 blank occurs ; take a number of equidistant points along a line, and let each 
 point correspond to one of the days before and after the blank ; then draw 
 lines perpendicular to the first line proportional to the height of water on 
 each of the days. Draw a sweeping curve through the extremities of the 
 lines, and measure off the height of the curve where it passes above the 
 blank. The resulting number is to be used for filling in the blank. 
 
 If the observations at night are taken at rarer intervals, say every two or 
 three hours, the same process of filling in blanks must be employed, by con- 
 sidering the hourly observations before and after the blank. 
 
 Every deficiency of data of course weakens the strength of the result. 
 
 The entry of observations is to stop as indicated in the instructions for 
 forming the S sheet. 
 
 Entry in M. 
 
 Copy the numbers on the S sheet ; begin entering as in S, but when a 
 dot is reached, enter two successive hourly values, the first above, the second 
 below ; then continue copying from S (the entries falling of course in a 
 different hour column) until a second dot is reached, where again make a 
 double entry ; and so on. 
 
124 INSTRUCTIONS FOR HARMONIC ANALYSIS. [4 
 
 The entries stop as indicated in the instructions for forming the M sheet. 
 
 For a fortnight's observation the last entry (which is in square 13 d 23 h ) is 
 that which is copied from square 14 d ll h of S. 
 
 For a month's observation the last entry (which is in square 27 d 23 h ) is 
 that which is copied from square 28 d 23 h of S. 
 
 Entry in O. 
 Follow the same rules for entry as in M. 
 
 For a fortnight's observation the last entry (which is in square 12 d 23 h ) is 
 that which is copied from square 13 d 23 h of S. 
 
 For a month's observation the last entry (which is in square 24 d 23 h ) is 
 that which is copied from square 26 d 20 h of S. 
 
 Rules of reduction on all three Sheets. 
 
 Add up the numbers in each column (in S there will be two sums, one 
 without rows 14 d or 27 d , 28 d , 29 d , as the case may be, and the other with 
 those additional rows). Divide the sum in each column by the number of 
 entries of which it is the sum. In S the divisors for each of the two sets of 
 sums are all the same, since there are not duplicate entries. 
 
 The results are 24 hourly mean values, and there are four of them, viz. 
 two from S, one from M, one from O. 
 
 The set from M, and the second set (longer series) on S are to be har- 
 monically analysed for semi-diurnal inequality ; the first set (shorter series) 
 on S, and the set from O are to be harmonically analysed for diurnal 
 inequality. 
 
 Thus where it is known that the diurnal tides are small, the sheet and 
 the shorter series on S may be omitted for rough results. 
 
 The height of mean water with reference to the zero of the tide gauge 
 has also to be determined from the second set on S. 
 
 Harmonic Analysis. 
 
 If we have any quantity which is variable during the day and night, such 
 as temperature or the height of the barometer, it is often desirable to express 
 it by the formula 
 
 A + AI cos 6 + Bj sin 6 + A 2 cos 20 + B 2 sin 26 + A s cos SO + B 3 sin 30 
 
 + A 4 cos 40 + B 4 sin 40 
 
 where is an angle which increases at the rate of 15 per hour, and is zero 
 at noon. 
 
1886] INSTRUCTIONS FOR HARMONIC ANALYSIS. 125 
 
 If we put 
 
 i> BI ., B 2 B 3 v B 4 
 
 tan ci = IT- , tan C. = -r- , tan C, = -r- , tan C 4 = -r- 
 
 A A A A 
 
 A.] xl-2 -".3 -"-4 
 
 and R t = A! sec ^ 1 = B 1 cosec^ 1 , R 2 = A 2 sec 2 = B a cosec f 2 
 
 and so on, the formula may be written 
 
 A + Rj cos (0 - &) + R 2 cos (20 - &) + R 3 cos (30 - &) + R 4 cos (40 - &) 
 
 The term in R! is diurnal, that in R 2 semi-diurnal, that in R 3 ter-diurnal, 
 that in R 4 quater-diurnal ; that is to say, they go through their changes once, 
 twice, thrice, and four times a day. 
 
 The term A gives the mean value for the day. 
 
 The A's and B's are the numbers which are derived from harmonic 
 analysis, as explained below. 
 
 The same process is applicable to the tides, but with the difference that 
 there are several kinds of days, viz. : first, the ordinary or mean solar or 
 S day ; second, the mean lunar or M day ; and a third kind the day for 
 which there is no name. 
 
 Thus in application to the tides there are to be four harmonic analyses, 
 two performed on the means on the S sheet, one on the means on the 
 M sheet, and one on the means on the O sheet. The matter is simplified, 
 however, by the fact that from the first means on S we only want A 1} B x ; 
 from the second means on S we only want A 2 , B 2 , and A ; from the means 
 on M we only want A 2 , B 2 ; and from the means on O we only want 
 A,, BL 
 
 The following schedule gives General [Sir Richard] Strachey's rules for 
 harmonic analysis. Columns I. and II. contain the 24 hourly values to be 
 analysed, and the headings to each successive column give the rules for its 
 derivation from the preceding ones. 
 
 If we only want A 1} B x the columns I. to VIII. inclusive are required ; if 
 we only want A,, B 2 the columns I., II., and IX. to XIV. inclusive are required, 
 and for A we require also column XV. 
 
 A comparison of this complete schedule with the numerical example 
 below will render the process intelligible. 
 
FORM FOR HARMONIC ANALYSIS. 
 
 h- 1 
 
 S 
 
 TAX - 'AX 
 
 3s s 
 
 s 
 i 
 
 1 
 
 X 
 
 'AX J JI l l puooag 
 
 . . . 
 
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 *' < i * 
 
 ^ 
 
 
 
 
 
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 z 
 
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 TIX + TX 
 
 . . . 
 
 
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 I-H 
 
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 t:|ua do 1 } Sut^juio 
 
 TIX-TX 
 
 
 
 
 H 
 X 
 
 TX J Jf 81 ! puooag 
 
 . . . 
 
 
 li II 
 
 3 ^ ' S ' 
 
 rS O II r9 O 1 
 
 X 
 
 'X -'XI 
 
 
 
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 S S 
 
 ^ 3 
 
 
 
 
 
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 X 
 
 XI J Jl^ 1 ! puooag 
 
 
 * 
 
 
 . 
 
 
 
 
 
 1 1 
 
 
 
 
 
 1-H 
 
 I-H 
 
 1 1 
 
 TA P UB Til "S 
 
 S fc * S 
 f 
 
 . oo 
 >o 
 
 1 P IL 
 
 co x- PQ 
 
 
 5 
 
 TA P UB Til ea S 
 
 ? 55 ^ c 
 1 
 
 28 
 
 CO X <1 
 
 
 h4 
 
 'A + 'AI + TII 
 
 ss'sS 
 
 3 
 
 s 
 
 > 
 
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 T H 9 4' 
 -? x 
 
 4- li O -T 5 
 
 II PQ 
 
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 2- 
 
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 c 
 
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 33. 
 
 
 
 
 
 
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 II - T 
 
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 1! ^ ^ 
 
 ^ II 
 
 tuns tjf 
 
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 HH 
 
 ~ %S* 
 
 . . . . 
 
 
 
 
 
 sjnojj 
 
 04 n -* o < 
 
 or- oc a 
 
 
 
 
 
 
 
 
 TT o? o 
 
 
 
 
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 seniBA AOH 
 
 
 
 
 
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 O I-H IN W 
 
 * o ;o i-- 
 
 X <S> O t-i 
 
 I-H t-H 
 
1886] INSTRUCTIONS FOR HARMONIC ANALYSIS. 127 
 
 General Rule for the Determination of and R from A and B. 
 
 If A is + and B is +, is less than 90, or in 1st quadrant. 
 
 If A is and B is +, lies between 90 and 180, in 2nd quadrant. 
 
 If A is and B is , lies between 180 and 270, in 3rd quadrant. 
 
 If A is 4- and B is , lies between 270 and 360, in 4th quadrant. 
 
 If tan is numerically less than 1, compute from R = A sec ; if greater 
 than 1, compute from R = B cosec . 
 
 In certain cases mentioned below, we shall have also to augment the 
 result R by a factor which is nearly equal to unity, as there explained. 
 
 General Rule as to Angles. 
 
 All angles are to be written as positive angles less than 360 ; if an angle 
 is greater than 360, subtract 360. 
 
 Certain small angles, however, determined below are to be estimated as 
 either positive or negative (e.y., 355 will in this case be written as 5), and 
 do not fall under this rule. The occurrence of the exception will always be 
 noted at the time. 
 
 It will often be convenient to write angles in degrees and decimals of 
 a degree. 
 
 Harmonic analysis of M. 
 Analyse the hourly means for A 2 , B 2 . 
 
 T> 
 
 Find m from tan m = ^ . 
 
 MH 
 
 Find R m from R m = A 2 sec % m x T0115 or B. 2 cosec f m x T0115, according 
 as tan m is numerically less or greater than 1. 
 
 N.B. Log 1-01 15 = '0050. 
 
 Harmonic analysis of S. 
 Analyse the second hourly means (the longer series) for A 2 , B 2 , and for A . 
 
 T> 
 
 Find f g from tan f g = . 2 . 
 
 -A-2 
 
 Find R 8 from R s = A, sec . or B a cosec &, according as tan & is numerically 
 less or greater than 1. 
 
 Analyse the first hourly means (the shorter series) for A 1} B!. 
 
 Find ' from tan ' = ir- 
 
 -n-l 
 
 Find R' from R'= A x sec f ', or B! cosec ', according as tan ' is numerically 
 less or greater than 1. 
 
128 INSTRUCTIONS FOR HARMONIC ANALYSIS. [4 
 
 Harmonic analysis of O. 
 
 Analyse the hourly means for A. 1} B x . 
 
 T> 
 
 Find from tan , = j 1 . 
 **i 
 
 Find R from R = A t sec x 1'0029 or B t cosec , x 1'0029, according as 
 tan , is numerically less or greater than 1. 
 
 N.B. Log 1-0029 = '001 3. 
 
 ANGLES AND FACTORS FOR REDUCTION. 
 N.A. stands for Nautical Almanac. 
 
 Call local mean noon of day of the series of observations to be reduced, 
 or of the tide table to be computed the Epoch. N.A., p. 1 : Find S3 the 
 mean longitude of the ascending node of ])'s orbit at epoch. Find sin S3, 
 cos S3, sin*2S3, cos 2 S3, and compute the following small angles (+ when & lies 
 between and 180, when S3 lies between 180 and 360), and numerical 
 factors 
 
 Angles to be determined 
 
 as + or 
 
 v = 12-9 sin S3 - 1'3 sin 2 S3 
 
 =ll-8sinS3-l-3sm2S3 
 i/= 8'9 sin S3 - 0'7 sin 2 S3 
 2v" = 17'7 sin S3 - 0'7 sin 2 S3 
 
 f = 1 _ -037 cos S3 
 
 f = 1-006 + -115 cos S3 - '009 cos 2 S3 
 r actors V 
 
 j f" = 1-024 + -286 cos S3 + '008 cos 2 S3 
 
 If = 1-009 + '187 cos S3 - -015 cos 2 S3 
 
 In the N.A. find the 's parx. at the middle of the fortnight or month of 
 observation, subtract from it the 's mean parx. (see Preface to N.A.). 
 Multiply the result by 19, and considering the product as degrees, look out 
 the sine of the angle and add 1, the result is p t ; e.g., if 's parx. be 8"'85, 
 and if the mean parx. be 8"'95*, we get 
 
 diff. - 0"-10, and - 19' x -10 = - 1'93 = - 1 56' 
 sin (- 1 56') = - -034, p, = 1 - '034 = -966 
 
 This is a short way of finding the cube of the ratio of 's parx. to 
 's mean parx. 
 
 * [The sun's mean parallax is now taken as 8"'bO, but the rule remains sufficiently exact.] 
 
1886] INSTRUCTIONS FOR HARMONIC ANALYSIS. 129 
 
 Arguments at Epoch. 
 
 From N.A., find )), the moon's mean longitude at epoch. From N.A. 
 find , the sun's mean longitude at epoch, by converting sidereal time to 
 angle at 15 per hour. The diurnal increase of J) = 13 11'. The corrections 
 for longitude of port are subtracted for E. long, and added for W. long. The 
 correction to J) is 0'549 for each hour of longitude, and 0'041 to for 
 each hour of longitude. 
 
 Find 2(]) ), - v, 2(0 v), i/, and compute the following 
 "arguments at epoch," 
 
 r=2(0-iO-2(})-f); V = -i/+ 270; V = - i/-2() - ) + 90 
 
 Find a mean value for for the period under reduction, by adding to 
 seven days' motion for a fortnight's observation, or 15 days' motion for 
 a month's observation. The motion for a day may be taken as 1, and thus 
 we add 7 or 15 ; with this mean compute, 2 - z/; V" = 2 - 2i/'. 
 
 FINAL REDUCTION. 
 Principal Lunar Tide called M a . 
 
 Let mean semi-range = H TO , and constant angle of retardation or lag = K n 
 The angle of retardation is hereafter called the lag. 
 Then from M sheet take R m , m , and compute 
 
 H * 4- V 
 
 L - L m f > ""in bwi ' ' 
 
 Principal Solar Tide called S 2 , and Lunisolar Semi-diurnal Tide 
 
 called K 2 *. 
 
 For S 2 , let mean semi-range = H s , lag = /c s . 
 For K 2 , let mean semi-range = H", lag = K". 
 Find t/r, as a positive or negative angle, for a fortnight, from 
 
 f'sinF" 
 
 Then from S sheet take R g , ,, and compute 
 
 TT 3-71 COST/r 1 TT _ // _ *. , 
 
 Hg = 371^ + r cos V* = 3 : 67 Hs ' KS ~ 
 For a month replace the 3'7l in these formulae by 3'84. 
 * [An error in the Manual has been corrected.] 
 
 
130 INSTRUCTIONS FOR HARMONIC ANALYSIS. [4 
 
 Lunisolar Diurnal Tide called K n and Solar Diurnal Tide called P*. 
 For Kj, let mean semi-range = H', lag = '. 
 For P, let mean semi-range = H p , \a,g = K p . 
 Find <j>, as a positive or negative angle, from 
 
 sin (2 - i/) 
 
 tan d> = ,5^7 , 0< ~ --- K 
 3f cos (2 - v) 
 
 Then from S sheet take R', ", and for a fortnight compute 
 
 , 3-007 cos $ I , , 
 
 = 3f'-cos(2-/) R ' H * = 3 H ' K = K *=t- +< ^ H 
 
 For a month replace 3'007 by 3'027 and 6'9 by 13'3. 
 
 Lunar Diurnal Tide called 0. 
 For O, let mean semi-range = H , lag = K U . 
 Then from sheet take R , <,, and compute 
 
 H- K t 4-V 
 
 J-^o f t "-o boi'o 
 Io 
 
 For rough results all the diurnal tides may be omitted, unless the diurnal 
 inequality is known to be large. 
 
 Collect Results. 
 
 These constants express six of the most important tides, and A gives the 
 height of mean water mark from the zero of the tide gauge. 
 
 * [An error in the Manual has been corrected.] 
 
1886] 
 
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136 EXAMPLE OF REDUCTION. [4 
 
 PORT BLAIR, ANDAMAN ISLANDS; lat. 11 41' N., long. 92 45' E. 
 
 Observation commences at epoch, O h mean time, April 19, 1880. Long. 
 92 45' E. = 6 h 183 E. 
 
 N.A. p. 1 S3 = 280 ; sin S3 = - "985 ; sin 2 S3 =- '34 
 cos 8 = + '174; cos 28 =-'94 
 
 Angles. 
 
 v = 12-9 sin S3 - 1'3 sin 2 S3 = - 12'26 
 
 = ll-8 sin S3 - l-3 sin 2 S3 = - 11'18 
 
 v' = 8'9 sin S3 - 0'7 sin 2 S3 = - 8'46 
 
 2//' = 17-7 sin S3 - 0'7 sin 2 S3 = - 17'36 
 
 Factors. 
 
 f= 1-000- -037 cos S3 ='994 
 P = 1-006 + -115 cos S3 - '009 cos 2 S3 = T035 
 f" = 1-024 + -286 cos S3 + '008 cos 2 S3 = T066 
 f = 1-009 + 187 cos S3 - '015 cos 2 S3 = T056 
 
 The mean value of 's parx. for the fortnight commencing April 19 is 
 's parx. on April 25 = 8"'89 (N.A. 1880); mean parx. from Preface to 
 N.A. = 8"-95 ; difference = - 0"'06 ; multiply by 19 = - 1"-16 ; change " to 
 = - 1 10' ; sin (- 1 10') = - '020 ; p t = 1 - '020 = "980. 
 
 Arguments at Epoch. 
 
 D's daily motion 13 11' ; hourly O c< 55 ; ))'s motion in 6 h< 18 = 3'40 
 's hourly motion 0'041 ; 's motion in 6 h '18 = 0-25 
 N.A. (Moon's Libration) 
 
 O h G.M.T. Ap. 20 D = 159 27' 
 one day's motion = 13 11' 
 
 O h G.M.T. Ap. 19 D = 146 16' 
 
 = 146'27 
 
 - 6 h -18 E. long. = - 3-40 
 
 D = 142 -87 
 - = + 11-18 
 
 D-= 154-05 
 
 2 (}) _ f ) = 308-10 
 
 - 2 (D - ) = 51-90 
 
1886] EXAMPLE OF REDUCTION. 137 
 
 Sid 1 time O h G.M.T. Ap. 19 = l h 51 m 51 s = 27'7 
 
 = l h 51 m '85 - v = 8-5 
 
 = l h -864 270 = 270 
 
 4 sid 1 time = 932 
 
 Sum V = 306-2 
 
 O h G.M.T. Ap. 19 = 27'96 
 
 - 6 h -18 E. long. = - '25 
 
 = 27-7l - v = 39-97 
 
 -i/ = + 12-26 _2 (])-) = 51-90 
 
 + 90= 90 
 _ j, = 39 -97 
 
 2 ( - ) = 79 -94 
 -2(-)= 51-90 
 
 Sum V = 181-87 
 
 Sum V= 131-84 
 
 Compute 2 v and V" with mean values of . The reduction is to 
 cover a fortnight, and therefore a week later than April 19 is the middle of 
 the period, and has increased by seven days' motion or 7. 
 
 April 19, = 28 2 = 70 2 = 70 
 
 a week's increase =7 i/ = + 8 2i/" = 17 
 
 mean =~35~ 2 - v = 78~ Sum V" = 87 
 
 REDUCTIONS. 
 M 2 , Principal Lunar Semi-diurnal. 
 
 From harmon. anal. f m = 147'93 _ R w , _ 2178 _ _ ft - __ 
 
 + F=131-82 ^ = ^994" 
 
 Sum Km = 280 
 
 K 2 , Lunisolar Semi-diurnal, and S 2 Principal Solar Semi-diurnal. 
 f' = 1-066; ^ = -980; R s = -731 ; C. = 297-9; F'^87 
 
 3-71 cos yr 
 
 '" 
 
 Compute log f" = '0278 
 
 log cos F"= 8-7188 
 log f" cos F"= ^8-7466 
 
 f"cosF' / = + 0-056 
 = + 3-636 
 
 rcosF'^ 3-692 
 log 3-692 = -5673 
 
138 EXAMPLE OF REDUCTION. [4 
 
 logf"= -0278 log cos i/r = 9-9827 
 
 log sin V" = 9-9994 log 3'7l = -5694 
 
 colog 3-692 = 9-4327 log R s = 9-8639 
 
 colog 3-692 = 9-4327 
 log tan ^ = 94599 
 
 sin V" is +, therefore ^r is +, and log H g = 9'8487 
 
 ^ = + 16 5' = + 16-1 H s = -706 
 
 C.= 
 
 Sum *. = 314 = K" 
 
 Kj, Lunisolar Diurnal, and P >SWr Diurnal. 
 
 2 - ' = 78 ; cos (20 - i/) = + '208 ; sin (2 - i/) = + '978 
 f = 1-035; R' = -468; ' = 2'2; F' = 306'2 
 H , 3-007 cos <ft 
 
 3f - cos (2 - z/) 
 
 Compute 3f = 3-1 05 
 
 - cos (2 - v) = - -208 
 
 3f - cos (2 - i/) = 2-897 
 
 sin (2 - i/') 
 
 <ft is +,and <ft = + 1840'= 18'7 
 V = 306 -2 
 C' = 2-2 
 6-9 = 6-9 
 Sum K = 334 = Kp 
 log cos (f> = 9-9765 
 log 3-007= -4781 
 colog 2-897 = 9-5381 
 log R' = 9-6702 
 
 log H' = 9-6629 
 
 H'= -460 ; H^ = iH / = 
 
 0, Lunar Diurnal. 
 T 7 = 181-9; f = 1-056; R = 146 ; 
 
 Compute 
 
 F = 
 Co - 
 
 TT = _ _ 
 
 " f 1-056 ~ 
 
1886] EXAMPLE OF REDUCTION. 139 
 
 RESULTS OF HARMONIC ANALYSIS of 15 days' hourly observations 
 at Port Blair, commencing O h , April 19, 1880. 
 
 Mean of Three Years' 
 Hourly Observation. 
 
 A = 474 ft 4-740 ft. 
 
 (H m = 219 ft 2-022 ft. 
 
 2 \ Km =280 278 
 
 (H s = 0-71 ft 0-968 ft. 
 
 2 j* s =314 315 
 
 (H" = 0-19 ft 0-282 ft. 
 
 2 V' =314 311 
 
 (H'-= 0-46 ft 0-397 ft. 
 
 1 K' = 334 327 
 
 H^ = 0-15 ft 0-134 ft. 
 
 K P =334 326 
 
 H = 0-14 ft 0-160 ft. 
 
 K =299 302 
 
 The second column is inserted for the sake of comparison, and gives the 
 results of three years of continuous hourly observation by the Tidal Depart- 
 ment of the Survey of India. The concordance between the two affords 
 evidence of the utility of even so short a series of observations as a fortnight. 
 
 IV. THE CONSTANTS TO BE USED IN COMPUTING 
 A TIDE TABLE. 
 
 The possibility of computing a tide table depends on the knowledge of 
 certain tidal constants appropriate to the port. In the preceding example 
 we have shown how these constants are derivable from a short series of 
 observations. The constants are there presented in what is called the 
 harmonic method, and an example is worked out below for Port Blair, with 
 such constants as have been derived above from a fortnight of observation. 
 The values used, however, are taken from the extended series of observations 
 made by the Indian Survey*. 
 
 The harmonic notation is, however, rather recent, and is not adopted 
 in the tide tables of the Admiralty. We must, therefore, show how the 
 principal constants of the harmonic method are derivable from the other 
 
 * The incompleteness of the data, with which we are supposed to be working, necessitates 
 the use of certain approximations which would not have been used if " the elliptic tides " had 
 been evaluated. 
 
140 TIDAL CONSTANTS FOR TIDE-TABLE. [4 
 
 notation, and thus the present method of computation will be made available, 
 wherever anything is known of the tides. 
 
 In the Admiralty tide tables the tides are specified by giving the time 
 of high water at full and change of moon, and the rise at spring and neap. 
 The semidiurnal constants of the harmonic method are derivable from these 
 very easily. Spring rise is the average height between low and high water 
 marks at spring tide ; neap rise the average height between high water-mark 
 at neap tide and low water-mark at spring tide ; neap range is the average 
 height between high and low water marks at neap tide. The average should 
 be taken from a great many springs and neaps. 
 
 Then H m + H s = \ spring rise 
 
 H m H s = neap range 
 H m = \ neap rise 
 
 If a the age of the tide be known, it may be expressed in hours. Then 
 reading the hours as degrees may be treated as an angle ; and if D be the 
 ratio of the neap rise to the excess of spring rise above neap rise, we have 
 
 TT 
 
 D= 
 H, 
 
 If T be the time of H.W. at full and change expressed in hours, 
 K m (in degrees) = 29 T tan" 1 yr , and K S = K m + a 
 
 AJ ~r COS Ct 
 
 If the age be unknown, we may take a as 36, and 
 
 sin a 3 
 
 tan" 1 yc = tan l ^^ -. 
 D + cos a 5D + 4i 
 
 For example, at Dungeness, Straits of Magellan (Adm. Tide Table) H.W. 
 at full and change is 8h. 30m. = 8 h '5; spring rise is 36 ft. to 44 ft., or say, 
 40 ft., neap rise is 30 ft 
 
 Hence H, n + H 8 = 20 ; H, H = 15 ; therefore H s = 5, and D = ^~ = 3. 
 
 The age of the tide being unknown, we assume 36 h. as a likely value, 
 so that a = 36, and 
 
 3 j3 1 
 
 ~5 + 4~ ~19~ 6 T 33~ 
 
 Again multiplying the time of H.W. at full and change by 29, we have 
 8-5 x 29 = 247, so that K m = 247 - 8 = 239, and Kg = 239 + 36 = 275. 
 
 The diurnal inequality is complex, and it seems unnecessary to enter into 
 details excepting in the harmonic notation. 
 
 Where it is stated that the tides are " affected by diurnal inequality," 
 it is not possible to predict the tides from the information contained in the 
 so-called tide table. 
 
1886] COMPUTATION OF A TIDE-TABLE. 141 
 
 A tide table is first computed with reference to mean water-mark, but it 
 is usual in navigational works to refer to " the mean level of low water of 
 ordinary spring tides." The datum level may be taken as H m + H g + H' + H 
 below mean water-mark, and hence to refer to the datum level we must add 
 H, n -f H s + H' + H to both H. W. and L. W. heights. 
 
 This datum has been defined for the first time in the prefaces to the 
 Indian Tide Tables for 1887, and is called "Indian spring low-water mark." 
 It has been chosen so as to agree as a general rule with " Low water of 
 ordinary spring tides." Accurate agreement was out of the question, since 
 the Admiralty datum does not appear susceptible of an exact scientific 
 definition. 
 
 In many estuaries and rivers the water rises much more rapidly than 
 it falls, and we sometimes find a double H.W. To take account of these 
 phenomena we should have to include, according to the schedule for Harmonic 
 Analysis, the terms A 4 , B 4 , both from the M sheet and S sheet. It is not 
 possible, without devoting too much space to the subject, to show how these 
 "over-tides" are to be included in the computation of the table. It is proper 
 to remark that in such an estuary either the H.W. or L.W., as computed by 
 the method below, may be found considerably in error. 
 
 V. THE COMPUTATION OF A TIDE TABLE. 
 
 The method of computation will be explained most easily by an actual 
 numerical example*. The computation is divided into a number of sections 
 and schedules, each line of each schedule is independent of all the others, 
 and thus a single tide may be computed as easily as a complete table. The 
 numerical value of any quantity required in the computation of any column 
 of a schedule is written at the top of the column, but outside the boundary 
 line of the schedule. In several cases explanatory headings are also put 
 outside the boundary line, but the process of derivation of each column from 
 what goes before is accurately stated inside the boundary line. It will be 
 stated below in VI. how the computations may be abridged where accuracy 
 is not desired. 
 
 TIDE TABLE FOR PORT BLAIR E. long. 6 h< 183, commencing Feb. 1, 1885. 
 Tidal constants serving as basis of table 
 
 H w = 2-022) H, = 0-968) H" = 0'282) 
 
 Km = 278 j K S = 315 j K" = 310 j 
 
 H' = 0-397) R p = 0-134) H = 0160) 
 
 K' = 327 j KJ, = 326 } KO = 302 j 
 
 * The reasoning on which the following processes are based is given in a report to the British 
 Association, 1886. [The preceding paper in this volume.] 
 
142 COMPUTATION OF A TIDE-TABLE. [4 
 
 A. Computation of Constants for a Fortnight, commencing Feb. I, 1885. 
 N.A. S3 = 187; sin S3 = -'122; sin 2 S3 = + '24 
 cos S3 = - '993 ; cos 2 S3 = + '97 
 Compute from the formula 
 
 A = 16*51 + 3'44 cos S3 - 019 cos 2 S3 
 Therefore 
 
 A = 16'51 - 3'416 - 0184 = 12*91 ; 2A = 25'82 = 25 49' 
 
 log cos 2A = 9-9543 
 
 By the formulae in reduction III. with above value of S3 , we find 
 y = -l-89; = -l-75; f=T037; f'=-882; f = '807 
 i/ = -l*22 
 Find mean value of 's parx. and decl. for a fortnight, beginning Feb. 1. 
 
 Parx. on Feb. 10 is (N.A.) 8"'96 ; mean parx. (Pref. N.A.) 8"'85 ; 
 diff. + 0"*11; multiply by 19J-, and read as degrees = + 213 = 2 8'; 
 sin 2 8' = -037 ; p t = 1'037. 
 
 Decl. on Feb. 8 : 
 
 S,= 1450'; 2^=29 40'; cos- 3 = (1 + cos 28,)= x 1-8689 = '935 
 r086# cos 2 S, = -970 ; H s . = 0'968 ; r086p, cos 2 ^H, = l ft '020 = S 
 
 Note that the 1*086 which occurs here is an absolute constant for all 
 times and places. 
 
 Compute " age of declinational inequality " as below : 
 
 ' Age ' = 52 h *2 tan ( K " - K m \ and K" - n m = 310 - 278 = 32 
 Therefore ' Age ' = 52 h '2 tan 32 = 32 h< 6 
 
 The 52 h *2 which occurs here is an absolute constant for all ports. 
 With constants, absolute for all ports, C l} C 2 (whose logarithms are given 
 
 TT// 
 
 below) compute a = C l H" cos (K" K m ) ; A = a cos2A ; ft = C 2 ^- sin (K" * m ) ; 
 
 -H-rn. 
 
 B = yS cos 2A, as follows : 
 
 logC 1= *6344 log C 2 = 2-3925 
 
 log cos (K" - Km ) = 9-9284 log sin (" - Km ) = 9'7242 
 
 log H" = 9-4502 log H" = 9-4502 
 
 colog H m = 9-6942 
 loga= -0130 
 
 log cos 2A = 9-9543 log = 1*2611 
 
 log cos 2A = 9-9543 
 
 log A = 9-9673 
 
 A = O ft -927 log B = 1-2154 
 
 B = 16'42 
 
1886] COMPUTATION OF A TIDE-TABLE. 143 
 
 Computation of Angles determining the Position of and ]) at O h Feb. 1, 
 
 1885, Port Blair M.T. 
 
 N.A. (Moon's Libration)* 
 
 D Jan. 31 = 138-6 E. long. 6 h 18 
 
 1 day's motion = + 13'2 Hourly in-) 
 
 > '55 
 crease of }) J 
 
 D Feb. 1 = 151-8 
 
 D = 148-4 
 - = +1-8 
 
 D - = 150-2 
 2 (D - ) = 300-4 
 2D-= 59-6 
 
 - i; = 314 
 
 })-) = 60 
 + 90 = 90 
 
 Corr" for E. long. = - 3'4 Corr" for E. long. - 3'399 
 
 N.A. Sid 1 time at O ll j = At epoch = 312 
 
 G.M.T. Feb. 1 -*/ = +! 
 
 | sid 1 time = 10'4 + 270 = - 90 
 
 Sum, O h Feb. 1=312 V = 223 C 
 
 -v = +2 
 
 F = 104 
 
 2 v is to be computed with the mean value of for a fortnight ; 
 add therefore 7 to at epoch: 
 
 at epoch = 312 mean 2 = 278 
 
 motion for 1 week = 4-7 v' = + I 
 
 mean = 319 2 - v' = 279 
 
 * [The moon's mean longitude is no longer to be found under the heading of ' Moon's 
 Libration ' but is on p. 1 of the Nautical Almanac.] 
 
144 COMPUTATION OF A TIDE-TABLE. [4 
 
 Compute as follows : 
 
 cos (2 -i/') = + '156; 3f = 2'646; 3f - cos (2 - v) = 2-490 
 sin (2 -i/) = -'988 
 
 sin (2 -i/') "988 _ 7 
 
 = ~~~ 
 
 </> is -, and </> = - 21 40' = - 21'7 
 
 3 cos</> 
 Compute 
 
 ' = 0-397 
 
 K '= 327 logsec< = "0318 
 
 _< = +22 colog 3 = 9-5229 
 
 log 2-490= -3962 
 
 *'-</>= 349 log H' = 9-5988 
 
 - V = - 223 
 
 lug R' = 9-5497 
 K > - - V' = ' = 126 R' = 0-355 
 
 = 302 logf= -0158 
 
 logH TO = -3058 
 
 = 198 logR m = logfH OT = "3216 
 
 log 3= -4771 
 
 log3R m = -7987 
 R m = 2-097 
 
 Collecting constants. 
 
 R m = 2 ffc -097; S = l ft '020 
 
 log3R m = '7987; 'Age' = 32 h -6 
 
 A = O ft -927; log a ='0130; R' = ft '355; R = ft '129 
 
 B = 16-42; log /3= 1-2611; ^'=126; ? = 198 
 
 Compute also 
 
 30 
 the mean interval i = ~ = 9 ll> 59 ; and 7 = K S - ^ x m = 27'4 
 
1886] 
 
 COMPUTATION OF A TIDE-TABLE. 
 
 145 
 
 S, R', ' must be recomputed for each month, the remaining constants 
 would serve for six months, or perhaps a year of continuous tidal com- 
 putation. The value of R would serve for six months, but care must be 
 taken in computing each month that be computed by reference to the 
 first noon of the month as a new epoch. 
 
 B. Parallactic Correction of Lunar Semi-diurnal Tide. 
 The semi-range of the lunar semi-diurnal tide is R m (=2 rt- 097) and it has 
 to be corrected for the D's parx. The parallactic correction is found by multi- 
 plying R m by the factor p, where 
 
 P = (})' s parx.) 3 -7- (])'s mean parx.) 3 
 
 The J)'s mean parx. is 57' 2", but the ))'s parx. in question must be taken 
 at a time anterior to H.W. by the "age" (- 32 h< 6). 
 
 To find p (approximately) subtract 57' 2" from the ])'s parx., substitute 
 for ' ", look out the sine of the angle, multiply it by 3, and add 1 
 to the result; then p is less than 1 if the J)'s parx. is below its mean value, 
 because the sine of a angle is , and vice versa. 
 
 We begin by making a table of SiR. m (the parallactic corrections to R m ) 
 for each 0''5 of parx. above or below the mean, to be applied + when the parx. 
 is greater than 57', and when it is below. 
 
 Tables B and C serve for all time, so long as the same tidal constants 
 are used. 
 
 B. Auxiliary Table for Parallactic Corrections, denoted by 
 
 SiRm,' 
 
 log3R m =-7987 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 Minutes of Parx. 
 in excess or 
 defect above or 
 below 57' 2" 
 
 Bead Degrees 
 for Minutes in I., 
 and enter 
 log sin (I.) 
 
 II. + lo g 3B w 
 
 Natural Number 
 of III. 
 5 t E m 
 
 0-5 
 
 7-9408 
 
 8-7395 
 
 05 
 
 1 
 
 8-2419 
 
 9-0406 
 
 11 
 
 1-5 
 
 8-4179 
 
 9-2166 
 
 16 
 
 2 
 
 8-5428 
 
 9-3415 
 
 22 
 
 2-5 
 
 8-6397 
 
 9-4384 
 
 27 
 
 3 
 
 8-7188 
 
 9-5175 
 
 33 
 
 3-5 
 
 8-7857 
 
 9-5844 
 
 38 
 
 4 
 
 8-8436 
 
 9-6423 
 
 44 
 
 4-5 
 
 8-8946 
 
 9-6933 
 
 49 
 
 5 
 
 8-9403 
 
 9-7390 
 
 55 
 
 D. I. 
 
 10 
 
146 COMPUTATION OF A TIDE-TABLE. [4 
 
 C. Declinational Correction to Lunar Semi-diurnal Tide. 
 
 A declinational correction has also to be applied to R m , say 8 2 Rm5 to K m , 
 say S 2 m ; to i, say & 2 i; to 7, say &. 2 <y. 
 
 If B be the ))'s decl. at a time anterior to H.W. by the ' age '; 
 5 2 Rwi = a cos 28 A ; & 2 K m = /3 cos 2S B ; 8 2 i = i + $ B. 2 K m ; B 2 y = 7 S 2 /c m 
 
 We begin by forming a table of declinational corrections for each degree 
 of decl. either N. or S. 
 
 w 
 
 'S 
 
 "-& <S> 
 
 V II 
 
 1 
 
 Q5 CO 
 
 ^jt I- 
 <*> li 
 
 a -i 
 
 O '-' 
 
 .3 CO 
 
 -K w 
 
 o 
 
 o 
 
 o 
 9 
 
 p 
 
 s-> o 
 1^ ? 
 
 H 
 
 ** . 
 
 d 
 
 pppppppppppppp 
 
 + 'l 
 il II II II II II 
 
 ' + 
 
 (MCNCNOlfMtNtNr^ 
 
 O O O O O O 
 
 ^^ ^^ ^^ 
 
 o oo 
 
 O rH ^H r-H <?q (M 
 
 oo 
 
 + + + + + + + + + + + + +I 1 1 1 1 1 1 
 
 t iOO>I>''OC^l~-COt^'-i'^ ( l>-OOi I O i ifMfOOQO 
 COeOiOOOiO-^-^MCCG^i IOO^^<MCO^>OCO 
 
 oooppopooopppppppopp 
 
 + + + + + + + + + + + + -f "l 'l 'l 'l 1 1 1 
 
 M IH 00 <N 10 . ifi 
 
 M > M; W 
 
 P* 
 
 O 
 
 + 
 
 +1 
 
 
 O 
 
 + 
 
 PPPppPPPPPPPP 
 
 + ' 'l 
 
 t35a01>'COTj<C y 3T ( 
 
 O 
 
 
 O 
 S 
 
 n (ft r lOOOCDMOSiOi iCOOfOCOQOOi i i i i i OS 
 PpOppOpOSO5O5O5O5OSOJOiOSOSO5O5O3 
 
 O5OS35O5OSOSO5O5OSOSO5OSO5O5O5O5O5O5QO 
 
1886] COMPUTATION OF A TIDE-TABLE. 147 
 
 D. Parallactic and Dedinational Corrections to Lunar 
 Semi-diurnal Tide. 
 
 Each H.W. follows a J)'s transit at the port, approximately, by the interval 
 i (9 h '6), and we require the J)'s parx. and decl. at a moment anterior to H.W. 
 by age of tide. Times are to be reduced to G.M.T. and round numbers used. 
 To reduce to G.M.T. subtract E. long., = 6 h '2 for Port Blair. (Add for W. 
 long.) The local time of })'s transit is G.M.T. of transit less 2 m for each hour 
 of E. long. ; for Port Blair less 12 m = O h '2 (for W. long, add this corrn.) 
 
 ( = G.M.T. of J)'s transit long, corrn. for transit 
 G.M.T. at which we want ,-, _. . ,,* ~^ 
 
 Vs mrx d decl ( 2) ~ K long " m time (6 2) + mean m ~ 
 
 [ ' terval i (9'6) - age of tide (32 h '6) 
 
 = G.M.T. of D's transit - 29 h '4 
 
 In the following table we determine roughly this moment of time, in 
 correspondence with each transit of ]), look out parx. and decl. and find 
 corrected R m from auxiliary tables B and C. These corrected values we 
 shall call m , M ; m,, Mj, &c., large letters being associated with upper, and 
 small with lower transits and the subscript numbers being the numbers of 
 the days of the tide table, the first day being numbered zero. 
 
 The corrected intervals, also from tables B and C, we call i , I ; i^ 
 /,; &c., and the corrected values of K S f| m (or 7) we call 70, F ; 7t, 
 I\; &c. 
 
 102 
 
148 
 
 COMPUTATION OF A TIDE-TABLE. 
 
 [4 
 
 X 
 
 O 
 
 -il2 
 
 7* &"~ 
 
 jd 
 
 ', P o 
 
 '6*dv6tOTd4ei<N9tOTMQ9r4iH 
 
 'L 'I " 'L I 'L 'L 'L " I " 'L " 
 
 > asasasa^fai^asasa.. 
 
 i~~*OOOOOOOOO 
 
 5"c>: "^ +11 
 
 r p 
 
 3 
 
 ^ J^ 02 02 
 
 ? q 7 H T'P < ?? : ' < ? :il ? )Q P ( ;! C( ? ) 
 
 + ' 'l 'l 
 
 rt M 
 
 ^ 
 
 I ^W (jv|(Nr^-lriHOCOOOl II 11 I 04 94 O4 
 
 + +1 I 
 
 X! 
 
 Co 
 nte 
 
 M 
 
 . 
 
 CD 01 
 O > 
 X 
 
 0) j 
 
 * * 
 
 a 
 
 *1 
 
 m * 
 
 <M (Mi ((Mr 1 
 
 cT r-T i-T of 
 
 M CO 
 
 COCNTflMO 
 
 i 1 i 1 r- 1 
 
 50" ^ 
 
 G 
 
 cS 
 
 -s 
 
1886] 
 
 COMPUTATION OF A TIDE-TABLE. 
 
 149 
 
 E. Determination of Local Mean and apparent Times of Moon's Transit, 
 and of Angles for Computing the Fortnightly Inequality. 
 
 It is convenient to treat the upper and lower transits separately in 
 schedules of similar forms. The angle x , X , x lf X 1} &c., on which the 
 fortnightly inequality of time and height depends, is twice the apparent 
 time of transit converted to angle at 15 per hour, and with the corresponding 
 angles j , F , j l} Fj, &c. subtracted. 
 
 The corrections for long, of port are for E., + for W. long. 
 E. Lower Transits. 
 
 
 
 Port Blair 
 
 
 
 
 
 Nautical 
 Almanac 
 
 Port Blair 
 M.T. of 
 
 appt. time 
 of transit 
 
 Equation 
 
 
 Twice 
 appt. time 
 converted 
 
 
 Required 
 angles 
 
 
 
 of time 
 
 
 to angle 
 
 
 
 
 
 = +14 m 
 
 
 
 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 Date 
 
 G.M.T. 
 
 of ])'s 
 transit 
 
 II. - 2 
 per hour 
 of 
 E. long. 
 
 III.- 
 equation 
 of time 
 
 IV. in 
 decimals 
 
 V. x30 
 
 Enter y 
 from 
 XII. of 
 D 
 
 VI. - VII. 
 
 1885 
 
 h. m. 
 
 h. m. 
 
 h. m. 
 
 h. 
 
 
 
 
 Feb. 1 
 
 1 50 
 
 1 38 
 
 1 24 
 
 1-40 
 
 42-0 
 
 26-9 
 
 x = 15-1 
 
 , 2 
 
 2 41 
 
 2 29 
 
 2 15 
 
 2-25 
 
 67-5 
 
 26-1 
 
 x t = 41-4 
 
 , 3 
 
 3 30 
 
 3 18 
 
 3 4 
 
 3-07 
 
 92-1 
 
 25-6 
 
 x 2 = 66-5 
 
 , 4 
 
 4 17 
 
 4 5 
 
 3 51 
 
 3-85 
 
 115-5 
 
 25-6 
 
 x 3 = 89-9 
 
 , 5 
 
 5 4 
 
 4 52 
 
 4 38 
 
 4-63 
 
 138-9 
 
 26-0 
 
 x 4 = 112-9 
 
 , 
 
 5 50 
 
 5 38 
 
 5 24 
 
 5-40 
 
 162-0 
 
 26-7 
 
 x 6 = 135-3 
 
 , 7 
 
 6 36 
 
 6 24 
 
 6 10 
 
 6-17 
 
 185-1 
 
 27-4 
 
 x 6 = 157-7 
 
 , 8 
 
 7 23 
 
 7 11 
 
 6 57 
 
 6-95 
 
 208-5 
 
 28-2 
 
 x 7 = 180-3 
 
 E. Upper Transits. 
 
 
 
 Port Blair 
 
 
 
 
 
 Nautical 
 Almanac 
 
 Port Blair 
 M.T. of 
 
 appt. time 
 of transit 
 
 Equation 
 
 
 Twice 
 appt. time 
 converted 
 
 
 Required 
 angles 
 
 
 
 of time 
 
 
 to angle 
 
 
 
 
 
 = + 14 m 
 
 
 
 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 Date 
 
 G.M.T. 
 
 of D'S 
 transit 
 
 II. - 2 m 
 per hour 
 of 
 E. long. 
 
 III.- 
 equation 
 of time 
 
 IV. in 
 decimals 
 
 V. x30 
 
 Enter T 
 from 
 XII. of 
 D 
 
 VI. - VII. 
 
 -360 
 
 1885 
 
 h. m. 
 
 h. m. 
 
 h. m. 
 
 h. 
 
 
 
 
 Feb.l 
 
 14 15 
 
 14 3 
 
 13 49 
 
 13-82 
 
 414-6 
 
 26-5 
 
 X = 28-1 
 
 2 
 
 15 5 
 
 14 53 
 
 14 39 
 
 14-65 
 
 439-5 
 
 25-8 
 
 X t = 53-7 
 
 3 
 
 15 53 
 
 15 41 
 
 15 27 
 
 15-45 
 
 463-5 
 
 25-6 
 
 X 2 = 77-9 
 
 4 
 
 16 40 
 
 16 28 
 
 16 14 
 
 16-23 
 
 486-9 
 
 25-7 
 
 X 3 = 101-2 
 
 5 
 
 17 27 
 
 17 15 
 
 17 1 
 
 17-02 
 
 510-6 
 
 26-3 
 
 X 4 = 124-3 
 
 
 
 18 13 
 
 18 1 
 
 17 47 
 
 17-78 
 
 533-4 
 
 27-0 
 
 X 5 = 146-4 
 
 ,, 7 
 
 18 59 
 
 18 47 
 
 18 33 
 
 18-55 
 
 556-5 
 
 27-8 
 
 X 6 = 168-7 
 
 
 
 19 46 
 
 19 34 
 
 19 20 
 
 19-33 
 
 579-9 
 
 28-6 
 
 X 7 = 191-3 
 
150 
 
 COMPUTATION OF A TIDE-TABLE. 
 
 [4 
 
 F. Figure for Times and Heights of Semi-diurnal Tide. 
 
 The next step is to find the resultant of the lunar and solar tides. Draw 
 the straight line OA. Produce AO to S, and take OS = S (for Port Blair 
 S = l ft "020) on any convenient scale. With OA as initial line set off the 
 angles x , X , x a , X 1? found in VIII. of E, for a fortnight; a new figure is 
 desirable for the next fortnight. On Om , OM , Onii, OM^ &c. set off to 
 adopted scale the heights m , M , m 1} MI, &c. found in X. of D. 
 
 v 
 
 Join m S, M S, nijS, MjS, &c., and measure all the lengths m S, M S, n^S, 
 MjS, &c. on adopted scale. These are the successive heights of H.W. 
 
 Measure all the angles Om S, OM S, &c. and count them as -f in the 
 upper half of the figure, and in the lower half. Each height and each 
 angle is associated with one upper or lower transit of }). 
 
 G. Formation of Tide Table for Semi-diurnal Tides. 
 
 The successive heights Om , OM , Orn 1? OMj, &c., of H.W. have been 
 found graphically in F. The angles Om S, OM S, OnijS, OMjS, &c. found in 
 the figure F must be reduced to time at the rate of 29 per hour, and 
 subtracted from the mean time of ))'s transit found in III. of E, to these have 
 to be added the corrected intervals i , /, i 1} I ly &c. from XI. of D. Next a 
 time half-way between consecutive H.W.'s is taken as the time of L.W. ; and 
 the height of L.W. is taken as the mean of the H.W. before and after with a 
 prefixed. 
 
1886] 
 
 COMPUTATION OF A TIDE-TABLE. 
 
 151 
 
 These processes are carried out in the following schedule. If the time 
 comes out greater than 24 h , the tide in question really belongs to the 
 next day. 
 
 H.W. L.W. H.W. L.W. 
 
 times times heights heights 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 
 
 
 Mean 
 
 
 
 
 
 
 
 Day, and 
 upper 
 or lower 
 transit 
 
 Angles 
 OMS 
 from 
 figure 
 
 11.^29 
 
 time of 
 transit 
 from 
 III. of 
 E. in 
 
 IV.- 
 III. 
 
 Cor- 
 rected 
 i from 
 XI. 
 of D 
 
 V. + VI. 
 
 Inter- 
 polate 
 in VII. 
 
 Hts. 
 
 from 
 figure 
 
 Inter- 
 polate 
 in IX. 
 
 
 
 
 decimals 
 
 
 
 
 
 
 
 1885 
 
 
 h. 
 
 h. 
 
 h. 
 
 h. 
 
 h. 
 
 h. 
 
 ft. 
 
 ft. 
 
 Feb. 1. L 
 
 + 4-4 
 
 + 0-15 
 
 1-63 
 
 1-48 
 
 9-61 
 
 11-09 
 
 
 3-45 
 
 
 
 
 
 
 
 
 
 17-24 
 
 
 -3-41 
 
 U 
 
 8-2 
 
 0-28 
 
 14-05 
 
 13-77 
 
 9-62 
 
 23-39 
 
 
 3-36 
 
 
 
 
 
 
 
 
 
 5-55 
 
 
 -3-31 
 
 2. L 
 
 12-0 
 
 0-41 
 
 2-48 
 
 2-07 
 
 9-63 
 
 11-70 
 
 
 3-25 
 
 
 
 
 
 
 
 
 
 17-85 
 
 
 -3-17 
 
 U 
 
 15-4 
 
 0-53 
 
 14-88 
 
 14-35 
 
 9-64 
 
 24-00 
 
 
 3-09 
 
 
 
 
 
 
 
 
 
 6-15 
 
 
 -3-00 
 
 3. L 
 
 18-7 
 
 0-64 
 
 3-30 
 
 2-66 
 
 9-65 
 
 12-31 
 
 
 2-91 
 
 
 
 
 
 
 
 
 
 18-46 
 
 
 -2-81 
 
 U 
 
 21-6 
 
 074 
 
 15-69 
 
 14-95 
 
 9-65 
 
 24-60 
 
 
 2-71 
 
 
 
 
 
 
 
 
 
 6-75 
 
 
 -2-59 
 
 4. L 
 
 24-4 
 
 0-84 
 
 4-08 
 
 3-24 
 
 9-65 
 
 12-89 
 
 
 2-47 
 
 
 
 
 
 
 
 
 
 19-05 
 
 
 -2-35 
 
 U 
 
 26-7 
 
 0-92 
 
 16-47 
 
 15-55 
 
 9-65 
 
 25-20 
 
 
 2-23 
 
 
 
 
 
 
 
 
 
 7-36 
 
 
 -2-10 
 
 5. L 
 
 28-3 
 
 0-98 
 
 4-86 
 
 3-88 
 
 9-64 
 
 13-52 
 
 
 1-98 
 
 
 
 
 
 
 
 
 
 19-70 
 
 
 -1-85 
 
 U 
 
 29-4 
 
 1-01 
 
 17-25 
 
 16-24 
 
 9-63 
 
 25-87 
 
 
 1-72 
 
 
 
 
 
 
 
 
 
 8-06 
 
 
 -1-59 
 
 6. L 
 
 29-2 
 
 1-01 
 
 5-63 
 
 4-62 
 
 9-62 
 
 14-24 
 
 
 1-47 
 
 
 
 
 
 
 
 
 
 20-47 
 
 
 -1-35 
 
 U 
 
 27-2 
 
 0-94 
 
 18-02 
 
 17-08 
 
 9-61 
 
 26-69 
 
 
 1-24 
 
 
 
 
 
 
 
 
 
 8-96 
 
 
 -1-13 
 
 7. L 
 
 22-3 
 
 0-77 
 
 6-40 
 
 5-63 
 
 9-59 
 
 15-22 
 
 
 1-02 
 
 
 
 
 
 
 
 
 
 21-56 
 
 
 - -94 
 
 U 
 
 + 13-5 
 
 + 0-47 
 
 8-78 
 
 18-31 
 
 9-58 
 
 27-89 
 
 
 85 
 
 
 
 
 
 
 
 
 
 10-33 
 
 
 - -81 
 
 8. L 
 
 - 0-4 
 
 -0-01 
 
 7-19 
 
 7-20 
 
 9-57 
 
 16-77 
 
 
 77 
 
 
 
 
 
 
 
 
 
 23-20 
 
 
 - -77 
 
 U 
 
 -14-9 
 
 -0-51 
 
 19-57 
 
 20-08 
 
 9-55 
 
 29-63 
 
 
 78 
 
 
 H. Correction for Diurnal Inequality. 
 
 We first enter the tide table as found in the final columns of G, bringing, 
 however, the times greater than 24 h into the succeeding day. 
 
 The successive processes are then adequately explained in the following 
 schedule. 
 
 Columns VII., VIII., and XIII., XIV. are conveniently taken from a 
 traverse table, for disregarding the decimal points, IT or R may be considered 
 as distances run, the angle in VI. or XI., as the " course," and R' cos VI. or 
 R cos XI. is the " Diff. lat.," whilst R' sin VI. or R sin XI. is the " Dep." 
 
152 
 
 CORRECTION FOR DIURNAL INEQUALITIES. 
 
 
 
 
 
 First cor- 
 
 
 
 
 
 
 Angles on 
 
 rection to 
 
 
 
 Tide Table without diurnal 
 
 
 
 which first 
 
 heights 
 
 
 
 corrections 
 
 
 o 
 
 diurnal tide 
 
 
 
 
 
 
 i i 
 II 
 
 depends 
 
 R' = 
 
 
 
 
 
 
 
 0-36 ft. 
 
 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 
 H 
 
 HH O 
 
 a 3 
 
 
 1. 2. 
 
 
 
 ah 
 3 
 
 
 11 
 
 f>- 
 
 r" 1 s 10 
 
 
 r-T oT oT 
 
 
 
 ~ 
 
 No. of 
 
 B ^ 
 
 a 
 
 O >>'~ l 
 
 
 <~ 
 
 a -2 
 
 
 
 * H ^ 
 
 Day, Date, 
 
 2*S 
 
 1~ 
 
 cfl 
 
 
 
 'm ^-g 
 
 
 
 SH 
 
 and 
 
 
 CH 
 
 *" rfl 
 
 
 6 
 
 O fl 
 
 S 
 
 t-5 
 
 -M * 
 
 H.W. or L.W. 
 
 Jq 
 
 *>> 
 
 O rrt 
 
 * 3 2 
 
 ^L, 
 
 * 
 
 ,2^ - 
 
 00 
 
 ^ 
 
 ** HH 
 
 
 "3 g 
 
 H 
 
 i^i 
 
 i 
 
 + Q 
 
 ^ t*> 5 
 
 O 
 
 o 
 
 'S 
 
 *^2 
 
 
 w * 
 
 H * 
 
 
 w 
 
 > 
 
 (^^-05 
 
 0H 
 
 PH 
 
 M 1 "" 
 
 
 ft. 
 
 h. 
 
 
 
 
 
 ft. 
 
 
 
 Feb. 1. H.W. 
 
 + 3-45 
 
 11-09 
 
 166 
 
 40 
 
 40 
 
 
 + 27 
 
 + 23 
 
 155 
 
 0. L.W. 
 
 -3-41 
 
 17-24 
 
 
 
 
 133 
 
 -25 
 
 + 27 
 
 
 H.W. 
 
 + 3-36 
 
 23-39 
 
 
 
 
 226 
 
 -25 
 
 -26 
 
 
 Feb. 2. L.W. 
 
 -3-31 
 
 5-55 
 
 83 
 
 317 
 
 318 
 
 
 + '27 
 
 -24 
 
 77 
 
 1. H.W. 
 
 + 3-25 
 
 11-70 
 
 
 
 
 51 
 
 + 23 
 
 + 28 
 
 
 L.W. 
 
 -3-17 
 
 17-85 
 
 
 
 
 144 
 
 -29 
 
 + 21 
 
 
 Feb. 3. H.W. 
 
 + 3-09 
 
 o-oo 
 
 
 
 234 
 
 236 
 
 
 -20 
 
 -30 
 
 
 
 L.W. 
 
 -3-00 
 
 6-15 
 
 
 
 
 329 
 
 + 31 
 
 -19 
 
 
 2. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + 2-91 
 
 12-31 
 
 
 
 
 62 
 
 + '17 
 
 + 32 
 
 
 L.W. 
 
 -2-81 
 
 18-46 
 
 
 
 
 155 
 
 -33 
 
 + 15 
 
 
 Feb. 4. H.W. 
 
 + 2-71 
 
 60 
 
 9 
 
 243 
 
 246 
 
 
 -15 
 
 -33 
 
 8 
 
 L.W. 
 
 -2-59 
 
 6-75 
 
 
 
 
 339 
 
 + 34 
 
 -12 
 
 
 3. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + 2-47 
 
 12-89 
 
 
 
 
 72 
 
 + 11 
 
 + 34 
 
 
 L.W. 
 
 -2-35 
 
 19-05 
 
 
 
 
 165 
 
 -35 
 
 + 09 
 
 
 Feb. 5. H.W. 
 
 + 2-23 
 
 1-20 
 
 18 
 
 252 
 
 256 
 
 
 -09 
 
 -35 
 
 17 
 
 L.W. 
 
 -2-10 
 
 7-36 
 
 
 
 
 349 
 
 + 35 
 
 -07 
 
 
 4. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + 1-98 
 
 13-52 
 
 
 
 
 82 
 
 + 05 
 
 + 36 
 
 
 L.W. 
 
 -1-85 
 
 19-70 
 
 
 
 
 175 
 
 -36 
 
 + 03 
 
 
 Feb. 6. H.W. 
 
 + 1-72 
 
 1-87 
 
 20 
 
 262 
 
 267 
 
 
 -02 
 
 -36 
 
 26 
 
 L.W. 
 
 -1-59 
 
 8-06 
 
 
 
 
 
 
 + 36 
 
 + 00 
 
 
 5. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + 1-47 
 
 14-24 
 
 
 
 
 93 
 
 -02 
 
 + 36 
 
 
 L.W. 
 
 -1-35 
 
 20-47 
 
 
 
 
 186 
 
 -30 
 
 -04 
 
 
 Feb. 7. H.W. 
 
 + 1-24 
 
 2-69 
 
 40 
 
 274 
 
 280 
 
 
 + 06 
 
 -35 
 
 37 
 
 L.W. 
 
 -1-13 
 
 8-96 
 
 
 
 
 13 
 
 + 35 
 
 + 08 
 
 
 6. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + 1-02 
 
 15-22 
 
 
 
 
 106 
 
 -10 
 
 + 35 
 
 
 L.W. 
 
 - -94 
 
 21-56 
 
 
 
 
 199 
 
 -34 
 
 -12 
 
 
 Feb. 8. H.W. 
 
 + '85 
 
 3-89 
 
 58 
 
 294 
 
 299 
 
 
 + 17 
 
 -32 
 
 54 
 
 L.W. 
 
 - -81 
 
 10-33 
 
 
 
 
 32 
 
 + 31 
 
 + 19 
 
 
 7. 
 
 
 
 
 
 
 
 
 
 
 H.W. 
 
 + -77 
 
 16-77 
 
 
 
 
 125 
 
 -21 
 
 + 29 
 
 
 L.W. 
 
 - -77 
 
 23-20 
 
 
 
 
 218 
 
 -28 
 
 -22 
 
 
 8. Feb. 9. H.W. 
 
 + -78 
 
 5-63 
 
 84 
 
 318 
 
 326 
 
 
 + 30 
 
 -20 
 
 78 
 
1886] 
 
 CORRECTION FOR DIURNAL INEQUALITIES. 
 
 153 
 
 
 
 
 
 
 
 
 Reduction 
 
 
 
 
 Angles on 
 
 Second 
 
 
 
 
 
 to "Indian 
 
 
 
 
 which second 
 diurnal tide 
 depends 
 
 correc- 
 tion to 
 heights 
 
 
 
 Total 
 correc- 
 tion to 
 heights 
 
 Cor- 
 rected 
 heights 
 
 Spring Low 
 Water." 
 
 Total 
 cor- 
 rection 
 to time 
 
 Cor- 
 rected 
 times 
 
 
 ,=198 
 
 0-13 ft. 
 
 
 
 
 
 = 3-55 ft. 
 
 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 XIV. 
 
 XV. 
 
 XVI. 
 
 XVII. 
 
 XVIII. 
 
 XIX. 
 
 M 
 
 1. 
 
 2. 
 
 
 
 
 
 
 rf 
 
 
 
 *"O 
 
 
 rH oT oT 
 
 
 
 
 
 
 
 
 
 SB 
 ^ >v 
 
 
 3 H 
 
 
 
 _; 
 
 
 
 s 
 
 
 j 
 
 +* ^ 
 
 O 2 <4-t 
 
 ce g o 
 J-. c 
 
 i 
 
 O.I" 
 
 2^0 "^ 
 
 H- 5 
 
 X 
 
 00 
 
 O 
 
 hH 
 
 X 
 a 
 
 1 
 
 i i 
 X 
 
 M 
 
 
 i5 
 i i 
 
 1; 
 
 i 
 
 X 
 1 
 
 5 3?6 
 
 i 
 
 i-^ "-" O 
 
 
 
 "01 
 
 Pj 
 
 . * 
 
 
 
 h^ 
 
 ^. 
 
 
 * 
 
 H 
 
 J fl - 00 
 
 rf 
 
 pf 
 
 H 
 
 " 
 
 X 
 
 X + 
 
 >t 
 X 
 
 h- 1 
 HH 
 
 
 
 
 ft. 
 
 
 h. 
 
 ft. 
 
 ft. 
 
 ft. 
 
 h. 
 
 h. 
 
 155 
 
 317 
 
 
 + 10 
 
 -09 
 
 + 14 
 
 + 37 
 
 + 3-82 
 
 7-37 
 
 + 04 
 
 11-05 
 
 
 
 44 
 
 + 09 
 
 + 09 
 
 + 36 
 
 -16 
 
 -3-57 
 
 -02 
 
 -11 
 
 17-35 
 
 
 
 131 
 
 -09 
 
 + 10 
 
 -16 
 
 -34 
 
 + 3-02 
 
 6-57 
 
 -05 
 
 23-44 
 
 52 
 
 214 
 
 
 -11 
 
 -07 
 
 -31 
 
 + 16 
 
 -3-15 
 
 40 
 
 + 09 
 
 5-46 
 
 
 
 301 
 
 + '07 
 
 -11 
 
 + 17 
 
 + 30 
 
 + 3-55 
 
 7-10 
 
 + 05 
 
 11-65 
 
 
 
 28 
 
 + 12 
 
 + 06 
 
 + 27 
 
 -17 
 
 -3-34 
 
 21 
 
 -09 
 
 17-94 
 
 309 
 
 111 
 
 
 -05 
 
 + 12 
 
 -18 
 
 -25 
 
 + 2-84 
 
 6-39 
 
 -06 
 
 05 
 
 
 
 198 
 
 -12 
 
 -04 
 
 -23 
 
 + 19 
 
 -2-81 
 
 74 
 
 + 08 
 
 6-07 
 
 
 
 285 
 
 + 03 
 
 -13 
 
 + 19 
 
 + 20 
 
 + 3-11 
 
 6-66 
 
 + 07 
 
 12-24 
 
 
 
 12 
 
 + 13 
 
 + 03 
 
 + 18 
 
 -20 
 
 -3-01 
 
 54 
 
 -06 
 
 18-52 
 
 292 
 
 94 
 
 
 -01 
 
 + 13 
 
 -20 
 
 -16 
 
 + 2-55 
 
 6-10 
 
 -07 
 
 67 
 
 
 
 181 
 
 -13 
 
 -00 
 
 -12 
 
 + 21 
 
 -2-38 
 
 1-17 
 
 + 05 
 
 6-70 
 
 
 
 268 
 
 -00 
 
 -13 
 
 + 21 
 
 + '11 
 
 + 2-58 
 
 6-13 
 
 + 09 
 
 12-80 
 
 
 
 355 
 
 + 13 
 
 -01 
 
 + 08 
 
 -22 
 
 -2-57 
 
 98 
 
 -03 
 
 19-08 
 
 276 
 
 78 
 
 
 + 03 
 
 + 13 
 
 -22 
 
 -06 
 
 + 2-17 
 
 5-72 
 
 -10 
 
 1-30 
 
 
 
 165 
 
 -13 
 
 + 03 
 
 -04 
 
 + 22 
 
 -1-88 
 
 1-67 
 
 + 02 
 
 7-34 
 
 
 
 252 
 
 -04 
 
 -12 
 
 + 24 
 
 + 01 
 
 + 1-99 
 
 5-54 
 
 + 12 
 
 13-40 
 
 
 
 339 
 
 + 12 
 
 -02 
 
 + 01 
 
 -24 
 
 -2-09 
 
 1-46 
 
 + 01 
 
 19-69 
 
 259 
 
 61 
 
 
 + 06 
 
 + '11 
 
 -25 
 
 + 04 
 
 + 1-76 
 
 5-31 
 
 -14 
 
 2-01 
 
 
 
 148 
 
 -11 
 
 + 07 
 
 + 07 
 
 + 25 
 
 -1-34 
 
 2-21 
 
 -04 
 
 8-10 
 
 
 
 235 
 
 -08 
 
 -11 
 
 + 25 
 
 -10 
 
 + 1-37 
 
 4-92 
 
 +17 
 
 14-07 
 
 
 
 322 
 
 + 10 
 
 -08 
 
 -12 
 
 -26 
 
 -1-61 
 
 1-94 
 
 + 08 
 
 20-39 
 
 245 
 
 47 
 
 
 + 09 
 
 + 10 
 
 -25 
 
 + 15 
 
 + 1-39 
 
 4-94 
 
 -20 
 
 2-89 
 
 
 
 134 
 
 -09 
 
 + 09 
 
 + 17 
 
 + 26 
 
 - -87 
 
 2-68 
 
 -15 
 
 9-11 
 
 
 
 221 
 
 -10 
 
 -09 
 
 + 26 
 
 -20 
 
 + -82 
 
 4-37 
 
 + 25 
 
 14-97 
 
 
 
 308 
 
 + 10 
 
 -09 
 
 -'21 
 
 -24 
 
 -1-18 
 
 2-37 
 
 + 22 
 
 21-34 
 
 237 
 
 39 
 
 
 + 10 
 
 + 08 
 
 -24 
 
 + 27 
 
 + 1-12 
 
 4-67 
 
 -28 
 
 4-17 
 
 
 
 126 
 
 -08 
 
 + 11 
 
 + 30 
 
 + 23 
 
 - -58 
 
 2-97 
 
 -37 
 
 10-70 
 
 
 
 213 
 
 -11 
 
 -07 
 
 + 22 
 
 -32 
 
 + -45 
 
 4-00 
 
 + 29 
 
 16-48 
 
 
 
 300 
 
 + 07 
 
 -11 
 
 -33 
 
 -21 
 
 - -98 
 
 2-57 
 
 + 43 
 
 22-77 
 
 235 
 
 37 
 
 
 + 10 
 
 + 08 
 
 -12 
 
 + 40 
 
 + 1-18 
 
 4-73 
 
 -15 
 
 5-78 
 
154 
 
 TIDE-TABLE. 
 
 [4 
 
 K. Final Tide Table. 
 
 It remains to reduce the decimals of an hour to minutes, and to change 
 from the astronomical to the civil date at the port. It will be found more 
 convenient to keep the heights in decimals of a foot, and not reduce to inches. 
 The times and heights are given in XIX. and XVII. of table H. 
 
 K. TIDE TABLE for Port Blair, the Heights being referred to " Indian 
 Spring Low Water Mark." 
 
 Civil Date 
 
 Times of 
 H.W. and 
 L.W. 
 
 Heights of 
 H.W. and 
 L.W. 
 
 Civil Date 
 
 Times of 
 H.W. and 
 L.W. 
 
 Heights of 
 H.W. and 
 L.W. 
 
 1885 
 
 h. m. 
 
 ft. 
 
 1885 
 
 h. m. 
 
 ft. 
 
 Feb. 1. p.m. 
 
 11 3 
 
 H.W. 7'4 
 
 Feb. 6. a.m. 
 
 1 24 
 
 H.W. 5-5 
 
 2. a.m. 
 
 5 21 
 
 L.W. -0 
 
 a.m. 
 
 7 41 
 
 L.W. 1-5 
 
 a.m. 
 
 11 26 
 
 H.W. 6-6 
 
 p.m. 
 
 2 1 
 
 H.W. 5-3 
 
 p.m. 
 
 5 28 
 
 L.W. -4 
 
 p.m. 
 
 8 6 
 
 L.W. 2-2 
 
 p.m. 
 
 11 39 
 
 H.W. 7-1 
 
 7. a.m. 
 
 2 4 
 
 H.W. 4-9 
 
 3. a.m. 
 
 5 56 
 
 L.W. -2 
 
 a.m. 
 
 8 23 
 
 L.W. 1-9 
 
 p.m. 
 
 3 
 
 H.W. 6-4 
 
 p.m. 
 
 2 53 
 
 H.W. 4-9 
 
 p.m. 
 
 6 4 
 
 L.W. -7 
 
 p.m. 
 
 9 7 
 
 L.W. 2-7 
 
 4. a.m. 
 
 14 
 
 H.W. 6-7 
 
 ,, 8. a.m. 
 
 2 58 
 
 H.W. 4-4 
 
 a.m. 
 
 6 31 
 
 L.W. -5 
 
 a.m. 
 
 9 20 
 
 L.W. 2-4 
 
 p.m. 
 
 40 
 
 H.W. 6-1 
 
 p.m. 
 
 4 10 
 
 H.W. 4-7 
 
 p.m. 
 
 6 42 
 
 L.W. 1-2 
 
 p.m. 
 
 10 42 
 
 L.W. 3-0 
 
 5. a.m. 
 
 48 
 
 H.W. 6-1 
 
 9. a.m. 
 
 4 29 
 
 H.W. 4-0 
 
 a.m. 
 
 7 5 
 
 L.W. 1-0 
 
 a.m. 
 
 10 46 
 
 L.W. 2-6 
 
 p.m. 
 
 1 18 
 
 H.W. 5-7 
 
 p.m. 
 
 5 47 
 
 H.W. 4-7 
 
 p.m. 
 
 7 20 
 
 L.W. 1-7 
 
 
 
 
 In the official Indian Tide Tables the tides of Port Blair are referred to a 
 datum 3'13 ft. below mean water, that is to say 0'42 ft. higher than the 
 datum here used. To effect a comparison then subtract 0'42 ft. from all 
 these heights, and the concordance will be found fairly satisfactory. 
 
 The Indian Tide Tables are formed by the tide-predicting instrument, by 
 which the approximations here used are avoided, and are based on much 
 wider data than those supposed to be here available. 
 
1886] ABRIDGEMENT OF COMPUTATION. 155 
 
 VI. ON ABRIDGEMENTS WHICH MAY BE ADOPTED IN COMPUTING 
 
 A TIDE TABLE. 
 
 For navigational purposes a very rough tide table will often suffice. Such 
 a table may be computed as follows : 
 
 H OT , H s , /c m , K S , and mean establishment may be derived from spring and 
 neap rise, age, and establishment as shown in IV. If the " age " be unknown 
 it may be assumed as 36 h., and K S K m may be taken as 36. 
 
 Then let A be the apparent time of any ))'s transit reduced to angle at 30 
 per hour, and we have for the height of H.W. from spring L.W. mark 
 
 2H m + H s [1 + cos (A - Kg + Km )] 
 and for the height of L.W. from same level 
 
 H s [1 COS (A Kg + ,,,,)] 
 
 The time of H.W. is 
 
 TJ" 
 
 M.T. of ])'s tr. + mean estab. 2 h ^- sin (A K S + tc m ) 
 
 H-m 
 
 And the time of L.W. is 6 h. 12m. later, or half-way between two con- 
 secutive H.W.'s computed by above rule. 
 
 For example : 
 
 At Port Blair H m = 2'0 ft., H g = TO ft., /c s K m = 37, mean establishment 
 = 9 h "6; and we found M.T. of J's lower transit on Feb. 5, 1885 = 4 h. 52m. 
 = 4 h- 9, and appt. time of transit reduced to angle at 30 per hour is 139, so 
 that A =139. 
 
 Then A-* s + * m =102; cos 102 = - 0'2 ; sin 102 = + TO 
 H s [l + cos 102] = 0-8; 2H m = 4'0 
 
 H g [1 - cos 102] = T2 ; 2 h ^ sin 102 = 2 x x 1 = l h '0 
 
 ti m 
 
 Time of H.W. = 4 h '9 + 9 h '6 - l h '0 = 13 h '5 
 Time of L.W. = 13 h '5 + 6 h '2 = 19 h '7 
 Hence H.W., Feb. 6, at 1 h. 30 m. a.m., height 4'8 ft. 
 
 L.W., Feb. 6, at 7 h. 42 m. a.m., height T2 ft. 
 
 It must be noticed that we are here supposed to know nothing of the 
 diurnal tides, and the datum level being H, n + H s or 3'0 ft. below mean water 
 is considerably higher than that used above. 
 
 The results are more nearly in accordance with the complete value as 
 found in the preceding section than would usually be the case. 
 
 A graphical method of using the same data would be more accurate. The 
 figure would be the same as that of the last section, but the m's and M's 
 would be determined by sweeping a circle with radius H m about as centre, 
 and OS would be taken as equal to H s 
 
156 WORKS OF REFERENCE. [4 
 
 The further step in accuracy would be to proceed as in computation of 
 V., but to compute auxiliary tables B. and C. for each minute of parx. and 
 each 2 of decl. only. Table D. may be abridged by computing corrections 
 for parx. and decl. for upper transits only, and columns XI., XII. may be 
 omitted entirely. Table E. for lower transit may be omitted. Figure F. may 
 be drawn for upper transits only, and the entries in G. for lower transits may 
 be filled in by interpolation. In Table H. for diurnal tides only the first 
 entry for each day in VI. (2) and XI. (2) need be made, and only the first two 
 entries for each day of VII., VIII., XII., XIII., XIV., XVIII. computed. 
 The third and fourth entries for each day of XV. and XIX. may be taken as 
 respectively numerically equal to the first and second ones, but with the 
 opposite signs. These abridgements would reduce the computation by nearly 
 a half. Other abridgements will doubtless occur to the computer, but they 
 will all involve loss of accuracy. 
 
 VII. WORKS OF REFERENCE. 
 
 A general account of the theory of tides will be found in most Popular 
 Astronomies, but we are not aware of any book which gives a complete 
 exposition of tidal theory and practice. Airy's well known article on " Tides 
 and Waves " in the Encyclopedia Metropolitana may be referred to, but as 
 great advances have been made since the time of its publication, it would 
 seem preferable to refer to the article by the present writer, which is about to 
 be contributed to the Encyclopedia Britannica*. 
 
 A complete list of all papers on the tides published since the time of 
 Newton will be found in the Bibliographie Astronomique, Houzeau and 
 Lancaster, Brussels, 1882. For an account of the harmonic method and its 
 connexion with the method of hour angles, &c., see the Reports to the British 
 Association for 1883 and 1885, and 1886, for an explanation of the methods 
 here used [Papers 1, 2, 3 above]. 
 
 Tables of the harmonic tidal constants at a considerable number of ports 
 are given in a paper by A. W. Baird and G. H. Darwin in the Proceedings of 
 the Royal Society, 1885. 
 
 Computation forms for the reduction of a long series of tidal observations, 
 and copies of the British Association Report, 1883, may be purchased of the 
 Cambridge Scientific Instrument Company. 
 
 A manual of practical tidal observation by Major A. W. Baird, R.E., will 
 shortly be on sale by Messrs Taylor and Francis, Red Lion Court, Fleet 
 Street. 
 
 * [Since this time my own Tides and Kindred Phenomena in the Solar System (Murray) has 
 been published.] 
 
5. 
 
 ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS 
 OF HIGH AND LOW WATER. 
 
 [Proceedings of the Royal Society, XLVIII. (1890), pp. 278 340.] 
 
 1. Introduction. 
 
 EXTENSIVE use of the tide-gauge has only been made in recent years, and 
 by far the largest number of tidal records consist only of observations of high 
 and low water (H. and L.W.). Such observations have usually been reduced 
 by determining the law governing the relationship between the times and 
 heights of H. and L. W. and the positions of the moon and sun. This method 
 is satisfactory so long as the diurnal inequalities are small, but it becomes 
 both complex and unsatisfactory when the diurnal inequality is large. In 
 such cases the harmonic notation for the tide is advantageous, and as, except 
 in the North Atlantic Ocean, the diurnal inequality is generally considerable, 
 a proper method of evaluating the harmonic constants from H. and L.W. 
 observations is desirable. 
 
 The essential difference between the method here proposed and that 
 followed by Laplace and his successors is that they introduced astronomical 
 considerations from the first and applied them to each H. and L.W., whereas 
 the positions of the sun and moon will only be required here at a single 
 instant of time. In their method, the time of moon's transit, and hence the 
 interval, was found for each tide ; the age of the moon, and the moon's and 
 sun's parallaxes and declinations were also required. An extensive table 
 from the astronomical ephemeris was thus necessary, and there still remained 
 the classification of heights and intervals according to the age of moon, and 
 two parallaxes, and two declinations. The classification could hardly be less 
 laborious, and was probably less mechanical, than the sorting processes 
 employed below. There is probably, therefore, a considerable saving of labour 
 in the present method, and, besides, I conceive that the results are more 
 satisfactory when expressed in the harmonic notation. 
 
158 NOTATION. [5 
 
 My object has been to make the whole process a purely mechanical one, 
 and, although nothing can render the reduction of tidal observations a light 
 piece of work, I believe that it is here presented in a form which is nearly 
 as short as possible. 
 
 The analytical difficulties to be encountered in such a task are small, but 
 the arrangement of a heavy mass of arithmetic, so as to involve a minimum 
 of labour and therefore of expense, is by no means easy. How far I have 
 succeeded must be left to the decision of those who will, I hope, use the 
 methods here devised. 
 
 When a question of this kind is attacked, the solution cannot be deemed 
 complete unless the investigation is left in such a state that an ordinary 
 trained computer is able to use it as a code of instructions by which to reduce 
 a series of observations, without any knowledge of tidal theory. 
 
 An actual numerical example is thus essential, both to test the method 
 and to serve as instructions to a computer. The Appendix contains so much 
 of the reduction of three months of observation at Bombay as will serve as 
 such a code. If the series be longer than three months, or in such cases as 
 the proper treatment of gaps in the series, it is necessary to refer back to the 
 body of the paper for instructions. 
 
 I now pass to the theoretical reasons for the rules for reduction. 
 
 2. Notation. 
 
 The notation of the Report to the British Association for 1883 [Paper 1], 
 and in use in the Indian tidal work and elsewhere, is here followed. 
 
 The earth's angular velocity is denoted by 7; the hourly mean motions of 
 the moon, sun, and lunar perigee by a, 77, nr (777, (Te\ijvtj, rpuo?) ; the mean 
 longitudes of moon, sun, and lunar perigee by s, h, p, and the mean solar 
 hour angle by t. The R.A. and longitude in the lunar orbit of the inter- 
 section of the equator with the lunar orbit are v, % ; and N is the longitude 
 of the moon's node. 
 
 The several harmonic tides are denoted by arbitrarily chosen initial letters. 
 Those with which we shall principally have to deal are 
 
 Semi-diurnal. 
 
 Name Initial Speed Equilibrium argument 
 
 Principal lunar ... M 2 2 (7 - a) . 2t + 2 (h - v) - 2 (6- - ) 
 
 solar ... S 2 2(7-77) 2t 
 
 Luni-solar K 2 2y 2t+2(h-v") 
 
 Larger elliptic ... N 27-80- + ^ 2t + 2(/i - v} - 2(s - ) - (s-p) 
 
 Smaller ... L 27-0--^ 2t + 2(h - v) - 2 (s- ) + (s-p)+ TT 
 
1890] GENERAL METHOD OF TREATMENT. 159 
 
 Diurnal. 
 
 Name Initial Speed Equilibrium argument 
 
 Luni-solar K x 7 t + (h v') \TT 
 
 Lunar O y-2<r t + (h -v) - 2 (s - Z) + %TT 
 
 Solar P 7-277 i-h + ^TT 
 
 The symbol H denotes the mean semi-range of any one of the tides, and 
 K its retardation of phase behind what it would be according to the equi- 
 librium theory ; f denotes a certain factor of augmentation of the lunar and 
 luni-solar tides depending on the value of N. 
 
 The particular tide to which H, K, f refer will in general be indicated by 
 a subscript small letter, the same as the letter constituting the initial of the 
 tide. Thus, for example, the M 2 tide is expressed by 
 
 f m H w cos {2t + 2 (h - v) - 2 (s - |) - *} 
 
 I have allowed a departure from this notation in the case of the tides 
 K 2 and K 1} where I write H", K", f" for the first, and H', K', f ' for the second. 
 The angles 2z/' and v (which, like v and , are functions of N) are also 
 involved in the arguments* (or angle under the cosine in the expression for 
 the height of the particular tide) of these two tides. 
 
 It is obviously necessary to suppose the reader to have some acquaintance 
 with the harmonic notation, or it would be necessary to repeat the Report on 
 Tides above referred to. 
 
 3. The General Method of Treating H. and L.W. Observations. 
 
 Noon of the day on which the observations begin is to be taken as the 
 epoch, and the mean solar time elapsed since epoch is denoted by t. V with 
 the proper subscript letter denotes the increase of argument since epoch ; 
 for example, V m = 2 (7 <r) t. 
 
 Then the height of the water h, estimated from mean sea-level, is 
 expressed by a number of terms of the form A cos V 4- B sin V, or, in an 
 alternative form, Rcos(F ). 
 
 In order to explain the principle of the method proposed, let us take two 
 typical terms involving V p and V q , and let the rates of increase of V p be p, 
 and of V q be q. 
 
 Then we have 
 
 h = A p cos V p + B p sin Vj, + A q cos V q + B q sin V q (1) 
 
 Since at H. or L.W. h is a maximum or a minimum, we must have 
 
 = A p sin Vp- B p cos V p + 3. A q sin V q -^B q cos V q (2) 
 
 * It is well to explain that I have sometimes elsewhere used argument to denote the argu- 
 ment according to the equilibrium theory, that is to say, with K equal to zero. In this paper 
 I call the latter the equilibrium argument. 
 
.(4) 
 
 160 GENERAL METHOD OF TREATMENT. [5 
 
 Let us write - = k g .................................... (3) 
 
 P 
 
 Then multiply (1) by cos V p and (2) by sin V p , and add; and again 
 multiply (1) by sin V p and (2) by cos V p , and subtract, and we have 
 
 h cos V p = A p + A q (cos V p cos V q + k q sin V p sin V q ) 
 + B q (cos V p sin V q k q sin V p cos V q ) 
 
 h sin V p = B p + A q (sin V p cos V q - k q cos V p sin F 9 ) 
 + B q (sin F p sin F 9 + k q cos F^ cos V q ) 
 
 Let S = icos(F i ,-F g )+|cos(F p +F 9 )= cosF p cosF g \ 
 
 A = -| cos(F p - F ? ) - | cos (F p + F 9 ) = sin V p sin F 9 I 
 <r = % sin (V p - V q ) + sin (F p + F ? ) = sin F p cos 
 
 Also let F = 2 + fc g A, f=o- + & g S ) 
 
 G = -S-V> g = A + jyJ 
 
 Then our equations are 
 
 h cos F p = ^ + FA q + GB q ) 
 
 I ..................... (7) 
 
 h sin Vp = B p + fA q + gB q } 
 
 A similar pair of equations will result from each H. and L.W. When a 
 series of tides is considered, we may take the mean of the equations and 
 substitute a mean F, G, f, g. 
 
 The general principle here adopted is to take the means over such periods 
 that the mean F, G, f, g become very small. In fact, we shall, in several 
 cases, be able to reduce them so far that these terms are negligible, and get 
 
 COS 
 
 simply - , 2/i . V v = J^ : but in other cases, where what is typified as 
 r J n + 1 sin B p 
 
 the p tide is a small one, whilst one or more of the tides typified as q is large, 
 it will be necessary to find F, G, f, g. The finding of these coefficients is 
 
 f*OS 
 
 clearly reducible to the finding of the mean values of (V p V q ). 
 
 Another useful principle may be illustrated thus : if the q tide does not 
 differ much in speed from the p tide, we may put V q = V p + vt, where v is a 
 small speed. Then we write 
 
 h = R p cos ( V p p) + R q cos ( V p + vt g ) 
 
 = COS V p {Rp COS p + R q COS (vt %q}} 
 
 + sin V p {Rp sin p - R q sin (vt - q )} 
 If we neglect v/p, the condition for maximum and minimum in conjunction 
 
 with this gives 
 
 h cos V p = R p cos p + R q cos (vt q ) 
 
 h sin V p = R p sin % p R q sin (vt % q ) 
 
1890] GENERAL METHOD OF TREATMENT. 161 
 
 Then taking the mean of these equations over a period beginning 
 with = and ending when t = 'jr/v > we have (writing A p = R p cos % p , 
 
 B p = R p sin 
 
 cos V p = A v + \R Q cos (a 
 
 n+l 
 
 2A sin V p = B p \R q sin (a 9 ) 
 
 iv ~T~ J_ 
 
 where X and a are certain constants, depending on the sum of a trigonometrical 
 series. 
 
 Again, if we take means from t = TTJV to t = ZTT/V, the second terms have 
 their signs changed. 
 
 Hence the difference between these two successive sums will give 
 \R q cos(a% q ) and \R q sin (a 9 ). There will be usually two terms such 
 as those typified by q, and we shall then have to take two other means, viz., 
 one beginning at 7r/2v and ending at 37r/2i/, and the other beginning at 
 37r/2i> and ending at 57r/2z>. From the difference of these sums we get 
 \R q sin (a % q ) and \R q cos(a g ). From these four equations the two 
 Rq's and the two q 's are found. The solution is a little complicated in 
 reality by the fact that it is not possible to take t = exactly at the beginning 
 of the series, because the first tide does not occur exactly at noon, but this is 
 a detail which will become clear below. 
 
 When all the A' a and B's or R's and 's have been found, the position 
 of the sun and moon at the epoch, found from the Nautical Almanac, and 
 certain constants found from the Auxiliary Tables in Baird's Manual of Tidal 
 Observations *, are required to complete the evaluation of the H's and KS. 
 
 The details of the processes will become clear when we consider the various 
 tides. 
 
 It may be worth mentioning that I have almost completely evaluated 
 the F's and G's, which give the perturbation of one tide on another, in the 
 case considered in the Appendix. Without giving any of the details of the 
 laborious arithmetic involved, it may suffice to say that the conclusion fully 
 justifies the omission of all those terms, which are neglected in the computation 
 as presented below. 
 
 4. The tides N and L. 
 These are the two lunar elliptic tides. 
 
 For the sake of brevity all the tides excepting M 2 , N, L are omitted from 
 the analytical expressions. 
 
 Since V n = V m - (a- - <sr) t, V t = V m + (<r-<er)t 
 
 * Taylor and Francis, Fleet Street, 1886. 
 D. I. 11 
 
162 ELLIPTIC SEMI-DIURNAL TIDES. [5 
 
 the expression becomes 
 
 h = A m cos V m + B rn sin V m + R n cos [ V m (a- - r) t - n ] 
 
 + ^ cos [V m + (cr - w) * - &] 
 
 = cos F m {.A,,, + ^ 7l cos [(cr -) + ?] + -Rz cos [(o- -) gj} 
 + sin F m [B m + R n sin [(o- vr)t + f,J - ^ sin [(o- - w) - i]{ 
 
 Hence, taking into account the equation which expresses that h is a 
 maximum or minimum, and neglecting the variation of s p compared with 
 that of V m , we have 
 
 h cos V m = A m + R n cos [(a - BT) + fn] + RI cos [(o- -vr)t- 
 
 /o\ 
 
 - sin V m =B m + R n sin [(o- - r) + ] ^ g i n K " - CT ) ^ - ft] ' 
 
 The mean interval between each tide and the next is 6'210 hours. Then 
 if e be the increment of s p in that period (so that with cr -57 equal to 
 '54437 per hour, e is equal to 3 -3807), and if a, b be the values of 
 (cr -or) t + M and (cr -ar) t j at the time of the first tide under considera- 
 tion, the equations corresponding to the (r + l) th tide are approximately 
 
 h cos V m = A m + R n cos (a + re) + RI cos (6 + re) 
 
 ......... (9) 
 
 h sin V m = B m -f R n sin (a + re) RI sin (b + re) 
 
 If we take the mean of n + 1 successive tides, the two latter terms on the 
 
 sin ft ( / yi ~4~ 1)6 
 
 right of (9) will be multiplied bv 7 ^. r , and the r in the arguments 
 
 * (rc + l)sme' 
 
 a + re,b + re, will be equal to \n. If the (n + 2) th tide falls exactly a semi- 
 lunar-anomalistic period later than the first, (n + 1) e = TT. On account of 
 the incommensurability of the angular velocity cr -^ this condition cannot 
 be rigorously satisfied, but if the whole series of observations be broken up 
 into such semi-periods, then on the average of many such summations it may 
 be taken as true. 
 
 Then, since \e is a small angle, 
 
 (n + 1) sin \e = TT, and sin \ (n + 1) e 1 ; 
 hence the factor is 2/vr. 
 
 Again \ne = \TT \e ; thus, if n + 1 is the mean number of tides in a semi- 
 anomalistic period, our mean equations are 
 
 7T \ 
 
 , , T v (2A cos V m - A m ] = - R n sin (a - \e) - R t sin (6 - %e) 
 
 - ...(10) 
 
 C S ( a ~ ^ ~ Rl C S ( b ~ 
 
 where the summations S are carried out over the first semi-lunar-anomalistic 
 period, which may be designated as 1. 
 
1890] ELLIPTIC SEMI-DIURNAL TIDES. 163 
 
 In applying these equations to the next semi-period 2, the result is got 
 by writing a + (n + 1) e or a + TT for a, and b + TT for b. 
 
 Thus the equations are simply the same as (10), with the signs on the 
 left changed. 
 
 The equations for semi-periods 3, 4, &c., will be all identical on the right, 
 with alternately + and signs on the left. 
 
 Let the observations run over m semi-lunar-anomalistic periods; then 
 double the equations appertaining to periods 2, 3, ... (m - 1), and add all the 
 m equations together, and divide by 2 (m 1), and we have 
 
 7T 
 
 -^-r- 2A cos V m = - R n sin (a - ^e) - R, sin (b - 1 
 
 : ...(n) 
 
 sin V m = R n cos (a ^e) RI cos (b \e) 
 
 rr \Jli ~r J ) yftl 1 
 
 where 2 now denotes summation of the following kind : 
 
 {2 (1) - 2 (2)} + {2 (3) - 2 (2)} + (2 (3) - 2 (4)} + {2 (5) - 2 (4)} + &c. 
 
 the numbers (1), (2), &c., indicating the number of the semi-lunar-anomalistic 
 periods over which the partial sums are taken. 
 
 Suppose the whole series of observations to be reduced covers 2m + 1 
 gwarfer-lunar-anornalistic periods, which we denote by i, ii, iii, &c. 
 
 First suppose that the semi-period denoted previously by 1 consists of 
 i + ii, that 2 consists of iii + iv, and so on. 
 
 Let t be the time of the first tide of the series, and since we take noon 
 of the first day as epoch, t cannot be more than a few hours. 
 
 Let j = \e (a- sr) t = 1'6903 (a- or) t , a small angle 
 
 Then a- \e = (a- - *r) t + n - \e = % n -j } 
 
 &-i e = (<r- CT K--^ = -(+j)j 
 
 7T 
 
 Then denoting the operation -T-. r . S by S (the mark c 
 
 4(w + l)(m- 1) 
 
 indicating that the first tide included is nearly at epoch, when (a -or) t = 0), 
 we have from (11) and (12) 
 
 Sh cos V m = - R n sin ( n - j) + R t sin (& + j) | 
 Sh sin V m = R n cos (f n -j) - RI cos (^ + j) j 
 
 Secondly, suppose the semi-lunar-anomalistic period indicated by 1 consists 
 of ii + iii, that 2 consists of iv + v, and so on. 
 
 Obviously the result is got by writing t + %7r/(cr -or) for t , or, what 
 amounts to the same thing, by putting j \TT in place oi'j ; but we must also 
 
 112 
 
164 ELLIPTIC SEMI-DIURNAL TIDES. [5 
 
 write S i?r for S, so as to show that the summation begins when (a- cr) t is 
 nearly equal to ^TT. Then 
 
 S**A cos V m = - R n cos (f n - j) - RI cos (ft + j) ) 
 
 L (14) 
 
 S*-A sin F m = - n sin ( B -j) - fl, sin (ft + j) j 
 Hence jR n sin (ft n j) = - Sh cos F m S*"7t sin F TO | 
 
 n cos ( H - j) = SA sin F m - S*-A cos F m [ 
 
 ......... (15) 
 
 R t sin (ft + j) = Sh cos F m - S**A sin F m 
 
 #, cos (ft + j) = - S% sin V m - S*"h cos F m J 
 
 These four equations give the four unknowns R n , n , RI, ft, and J is equal 
 to l -69-(<7--5r) . 
 
 Then if u n , u t denote the equilibrium arguments of the tides N and L at 
 epoch, we have 
 
 I*, = 2 (h - V) - 2 ( So - f) + (S - Po ) + 7T 
 
 where h , s , p are the mean longitudes of moon, sun, and lunar perigee at 
 epoch, and v and are small angles, functions of the longitude of the moon's 
 node (tabulated in Baird's Manual)*. 
 
 Then if f m is the factor of reduction (also tabulated by Baird) for the 
 tides M 2 , N, L, 
 
 K n = ?n + U n , K t = ft + U t 
 
 TT _ Rn TT _ RI 
 
 J^n f } J^-i f 
 
 'm v -fm 
 
 In this investigation the interferences of the solar and diurnal tides are 
 neglected, on the assumption that they are completely eliminated. 
 
 The difference between a lunar period and an anomalistic period is so 
 small that the elimination of the diurnal tides will be satisfactory, but the 
 effect of the solar tide will probably be sensible, unless we have under 
 reduction 13 quarter-lunar-anomalistic periods, which only exceed 6 semi- 
 lunations by about 25 hours. 
 
 The evaluation of the elliptic tides N and L from a series of observations 
 shorter than a quarter year would be very unsatisfactory, and it is not likely 
 that such an evaluation will be attempted. But if such a case is undertaken, 
 the solar disturbance may be found by a plan strictly analogous to that 
 pursued below in the case of the tides K n O, P. The reader may be left to 
 deduce the requisite formulae from the theory in 3. 
 
 In the case of a long series of observations, each quarter year should be 
 reduced independently, and the mean values of H, t cos x n and H n sin K H should 
 
 * [Given algo in algebraic form at the end of the first paper in this volume.] 
 
1890] PRINCIPAL LUNAR SEMI-DIURNAL TIDE. 165 
 
 be adopted as the values of the functions; whence H n and K n are easily found. 
 The L tide is, of course, to be treated similarly. 
 
 5. The Tide M 2 . 
 This is the principal lunar tide. 
 
 If we take the mean of n + 1 successive tides, the equations (9) give us 
 approximately 
 
 2A cos V m = A mt 2hsmV m = B m ...... (16) 
 
 We here assume that in taking this mean over an exact number of semi- 
 lunations, the lunar elliptic tides, the solar tides, and the diurnal tides are 
 eliminated. 
 
 With respect to the elliptic tides, this condition can only be approximately 
 satisfied, because no small number of semi-lunations is equal to a number of 
 anomalistic periods, and the like is true of the diurnal tides. In the example 
 given below the diurnal tides are much larger than the elliptic tides, and 
 I have found by actual computation (the details of which are not, however, 
 given) that the disturbance in the value of the M 2 tide arising from the 
 diurnal tides is quite insensible, and it may be safely accepted that the same 
 is true of the disturbance from the elliptic tides. 
 
 With respect to the disturbance arising from the principal solar tide S 2 , 
 I find that it is adequately, although not completely, eliminated by making 
 the number n + l of tides under summation S cover an exact number of 
 semi-lunations. 
 
 If the whole series of observations be short, it would be pedantic to 
 attempt a close accuracy in results, and we may accept these formula ; if the 
 series be long, the residual errors will be gradually completely eliminated. 
 
 We have then 
 
 R m cos f m = A m , R m sin m = B m 
 
 If u m be the equilibrium argument at epoch, we have 
 
 n 
 
 Whence Km - m + u rn , and H m = -~ 
 
 lm 
 
 The meanings of h 0> s , v, , f m , have been explained in the last section. 
 
166 SOLAR AND LUN1-SOLAR SEMI-DIURNAL TIDES. [5 
 
 6. The Tides S 2 and K 2 . 
 These are the principal solar and luni-solar semi-diurnal tides. 
 
 If the tide S 2 is in the same phase as K 2 at any time, three months later 
 they are in opposite phases. Hence, for a short series of observations, the 
 two tides cannot be separated, and both must be considered together. It is 
 proposed to treat a long series of observations as made up of a succession of 
 short series ; hence I begin with a short series. 
 
 For the sake of brevity all the tides excepting S 2 and K 2 are omitted 
 from the analytical expressions. 
 
 Since V"=V g + t 
 
 = cos Vg {R s cos % s + R" cos i 
 + sin V g {R s sin g - R" sin 
 
 Hence, taking into account the equation which expresses that h is a 
 maximum or minimum, and neglecting the variation of 2A or 2rjt compared 
 with that of V 8) we have 
 
 h cos V s = R s cos f g + R" cos (2^ - f ") 
 h sin F s = R s sin g - 5" sin (2rjt - ") 
 
 The mean interval between each tide and the next is 6 h> 210. Then if g 
 be the increment of 2h in that period (so that with 2ij equal to 0'082 per 
 hour, g is equal to 0'510), the equations corresponding to the (r + l) th tide 
 are approximately 
 
 h cos V s = R s cos g + R" cos (rg ") 
 
 h sin V s = R s sin s R" sin (rgr ") 
 
 Now, if P be the cube of the ratio of the sun's parallax to its mean 
 parallax, the expression for S 2 , together with its parallactic inequality (the 
 tides T, R of harmonic notation), is PH g cos (2t Kg). 
 
 Since t is the mean solar hour angle, 2t is the same thing as V s . 
 Hence R s = PH g , g = K S 
 
 Also if P be the value of P at epoch, then for a period of two or three 
 months we may take P = P (l + pt), where P p is equal to dP/dt. 
 
 Again, if we put 7 = ^- , we have 
 
 Also since the argument of the K 2 tide is 2t + 2A, - 2i/' - K", where 2i/" is 
 a certain function of the longitude of the moon's node (tabulated by Baird), 
 and since t = 0, h = h at epoch, it follows that 
 
1890] SOLAR AND LUNI-SOLAR SEMI-DIURNAL TIDES. 167 
 
 Now, when the means of the equations (17) are taken for n + 1 successive 
 
 tides, the latter terms become R" . (^nq "), where 
 
 7 sm v - 
 
 Also, if we write 
 
 .(19) 
 
 ~D q - ~* ^ii oiii v g 
 n -f 1 
 
 our equations become 
 
 A s = UH S cos K a + f"XH, cos (a) - K") | 
 B g = HH, sin K, - f'XH. sin ( - *") J ' 
 
 It may be observed that O is the mean value of P during the interval 
 embraced by the n + 1 tides. 
 
 In reducing a short series of observations we have to assume what is 
 usually nearly true, viz., that K" K S and 7 = 0'272, as would be the case in 
 the equilibrium theory of tides. 
 
 With this hypothesis, put 
 
 U cos <f) = II + X n f " cos &> 
 U sin </> = \ n f" sin <w 
 
 from which to find U and 0. Then 
 
 -4 g = HgfT" cos (Kg </>) 
 B, a* HfCr sin (, ) 
 from which to find H g and tc s . 
 Lastly, *" = Kt , H" = 7 H g = 0'272 H 8 
 
 In order to minimise the disturbance due to the lunar tide M 2 , we have 
 to make the n + 1 tides cover an exact number of semi-lunations, namely, 
 the same period as that involved in the evaluation of M 2 . The elimination 
 of the M 2 tide is adequate, although not so complete as the elimination 
 of the effect of the S 2 tide on M 2 , because M 2 is nearly three times as large 
 as S 2 . 
 
 A Long Series of Observations. Suppose that there is a half year of 
 observation, or two periods of six semi-lunations, each of which periods 
 contains exactly the same number of tides. 
 
168 SOLAR AND LUNI-SOLAR SEMI-DIURNAL TIDES. [5 
 
 Then each of these periods is to be reduced independently with the 
 assumption that 7 = 0*272 and K S = K". If this assumption is found sub- 
 sequently to be very incorrect, it might be necessary to amend these reduc- 
 tions by multiplying \ n by H"-r- 0'272H g , and by adding K^ K" to to; but 
 such repetition will not usually be necessary. From these reductions we 
 get independent values of Hgcos/e g , H s sin/^ from each quarter year, and 
 the mean of these is to be adopted, from which to compute H s and K S . It 
 remains to evaluate H" and K". 
 
 The factor f" and the angle 2i/' vary so slowly that the change may be 
 neglected from one quarter to the next, although each quarter is supposed to 
 have been reduced with its proper values. 
 
 Let h and h ' be the sun's mean longitude at the two epochs ; they will 
 clearly differ by nearly 90, and we put 2/? ' = 2h + TT + 28h. Hence it is 
 clear that the value of &> in the second quarter is w + 2Bh + TT. 
 
 Thus the four equations, such as (20), appertaining to the two quarters, 
 may be written 
 
 A g = HH 8 cos K, + . f "H" cos (a, - *") 
 7 
 
 B s = IIH 8 sin Ks - . f "H" sin ( - *") 
 
 7 
 
 A 8 '= ITH 8 cos K, - ~ . f "H" cos (co + 2Bh - K") 
 7 
 
 B s ' = ITH, sin K, + . f"H" sin (to + 2Sh - ") 
 7 
 
 where the accented symbols apply to the second quarter, and where 
 
 \ n sin(n + l)<7 ACK/3 
 = -. ^ -. f^ = 0'656, a constant. 
 7 (n + l)sm^g 
 
 From (21), 
 
 A g -A s '-(U- IT) H g cos *. = 2 ^- l . f "H" cos Sh cos Co> + Bh - K") 
 
 7 
 
 - B g + B g ' + (TI - H') H g sin Kg = 2 ^ . f "H" cos Bh sin (ta + Bh- K") 
 
 7 
 
 From these two equations, H" and K" may be computed, and since II II' 
 is very small, approximate values of H s cos /c s , H s sin K S suffice. 
 
 7. The Diurnal Tides K,, O, P. 
 
 Amongst the diurnal tides I shall only consider Kj the luni-solar diurnal, 
 the principal lunar diurnal, and P the principal solar diurnal tides. 
 
 There is the same difficulty in separating P from Kj as in the case of K 2 
 and S 2 , and therefore in a short series of observations P and Kj have to 
 be treated together. It is proposed to treat a long series of observations as 
 made up of a succession of short series ; hence I begin with a short series. 
 
 .(21) 
 
1890] DIURNAL TIDES. 169 
 
 For the sake of brevity all the tides excepting K x , 0, P are omitted from 
 the analytical expressions. 
 
 If i Vm denotes (7 a-) t, we have 
 
 V'=^7 m + at, V = \V m -at, V p = ^V m + (a-2 V )t, and 
 h = R cos (%V m + <rt- O + R cos (%V m -at- ,) 
 
 + R p cos { V m + (a- 2rj) t - ,} 
 
 = cos | V m {R cos (<rt - O + R cos (<rt + ?) + ^ cos [(<r -2rj)t- ,]} 
 + sin F m {- JR' sin (at - ') + sin (at + ,) - J2p sin [(<r -2i))t- ,]} 
 
 Hence, taking account of the equation which expresses that A is a 
 maximum or minimum, and neglecting the variation of at compared with 
 that of ^V rn *, we have 
 
 h cos \ V m = R cos (<rf - ') + cos (<rf + ?) + fl p cos {(<r -2rj)t- p ] 
 h sin F m = - ^' sin (at - ') + E sin (at + ,) - # p sin {(<r -2rj)t- p } 
 The mean interval between each tide and the next is 6 h> 210. 
 
 Then if e be the increment of s, and z the increment of s 2h in that 
 period (so that with a equal to 0'5490 per hour and a 2ij equal to 
 0-4669 per hour, e is equal to 3'4095 and z equal to 2'8994) ; and if a, b, c 
 denote the values of at ', at + % , (a 2rj)t- % p at the time of the first 
 tide under consideration, the equations corresponding to the (r + l)th tide 
 are approximately 
 
 h cos ^V m = R' cos (a + re) + R cos (b + re) + R p cos (c + rz) ) 
 h sin | V m = R sin (a + re) + R sin (b + re) R p sin (c + rz) ) 
 If we take the mean of n + 1 successive tides, the first pair of terms will 
 
 be multiplied by , \- and the last term by the similar function with 
 
 J (n + l)sm^e 
 
 z in place of e; also the r in the arguments must be put equal to ^n. 
 
 If the (n + 2)th tide falls exactly a semi-lunar period later than the first, 
 (n + 1) e = TT. On account of the incommensurability of the angular velocity 
 a, this condition cannot be rigorously satisfied, but if the whole series of 
 observations be broken up into such semi-periods, then, on the average of 
 many such summations, it may be taken as true. 
 
 Since \e is a small angle, (n+ 1) sin^e = TT, and sin(n + \)e = 1 ; 
 hence the first factor is equal to 2/Tr. 
 
 Again, 
 
 O + l).z = |(w + l)e.- = i7r. ^ = 76 32' in degrees 
 
 CT 
 
 and (n + 1) sin^z |TT. - 
 
 a 
 
 * I have satisfied myself by analysis, which I do not reproduce, that on taking means this 
 error becomes very small. 
 
170 
 
 DIURNAL TIDES. 
 
 [5 
 
 Therefore 
 
 (n + 1) 
 Again 
 
 Now let 
 
 and we have 
 
 2^- c 
 
 (j w /-if) / ^ 
 
 7T CT 2?; 7T 7T 
 
 = i 7r _ 1 i e= i 7 r-l -7048 
 = TT - 13'4647 - l-4497 = TT - 14'9144 
 a = a- 1'7048 ' 
 
 6 = b- 1-7048 
 
 7 = c-14-9144 j 
 
 " ne = ^7r + a. 
 
 gu 
 
 .(23) 
 
 .(24) 
 
 Thus, if ?i + 1 is the mean number of tides in a semi-lunar period, the 
 means of equations (22) become 
 
 7T 
 
 7T 
 
 1) 
 
 S/i cos | V m = R' sin a R sin /3 X^ sin 7 
 2/i sin V m = - R' cos a + R cos /8 X^ cos 7 
 
 ...(25) 
 
 where the summations are carried out over the first semi-lunar period, which 
 may be designated as 1. 
 
 In applying these equations to the next semi-period 2, the result is 
 
 obtained by writing a + (n + l)e for a, b + (n + l)e for b, and c + (n+l)z 
 
 for c; that is to say, + 7r for a, b + ir for b, and c+153'070G or 
 c + 7r-26-9294 for c. 
 
 If, therefore, we put e = 26'9294, we obtain the result from (25) by 
 changing the signs on the left and writing 7 6 for 7. 
 
 The equations for semi-periods 3, 4, 5, &c., will be alternately 4- and - on 
 the left, and identical as regards the terms in a and ft, but with 7 2e, 7 3e, 
 y 4e, &c., successively in place of 7. 
 
 Let the observations run over m semi-lunar periods; then double the 
 equations appertaining to periods 2, 3...(w 1), add all the m equations 
 together, and divide by 2(m 1). 
 
 The terms in R p will involve the series 
 
 sin sin N sin sin , 
 
 7 + 2 (7 - e) + 2 (7 - 2e) + . . . + {7 - (m - 1) e 
 cos ' cos v ' cos v ' cos l ' 
 
 This is equal to 2 
 Then if we put 
 
 sin \(fn 1 ) e sin 
 - 
 
 tan|e cos 
 
 A, sin A (m 1) e 
 
 it = . ^ T -^- , where X = 11436 
 
 (m 1) tan^e 
 
1890] DIURNAL TIDES. 
 
 our equations (25) become 
 
 171 
 
 = R' sin a R sin @ fiR p sin {7 | (ra 1) e} 
 
 = R' cos a + R cos y# /u.Kp cos {7 -| (m 1) e} 
 where 2, now denotes summation of the following kind : 
 
 ...(26) 
 
 Suppose the whole series of observations to be reduced covers exactly 
 2m + 1 quarter-lunar periods, which we denote by I, II, III, &c. 
 
 First suppose that the semi-period denoted previously by 1 consists of 
 I + II, that 2 consists of III + IV, and so on. 
 
 .(27) 
 
 Let t denote the time of the first tide of the series, and since noon of the 
 first day is epoch, t cannot be more than a few hours. 
 
 Let i = \e- <rt = 1'7048 - 
 
 and k = \z - (a- - 27?) t = l -4497 - (a- - 2^) t \ 
 
 i and k are clearly small angles. 
 
 Since e = 26'9294, \e + 1'4497 = 14'9144 ; and from (23) 
 a = at - ?' - 1'7048 = -(' + i) 
 
 .(28) 
 
 If the same notation be adopted as that explained in 4 (the only 
 difference being that we now deal with quarter-lunar instead of quarter- 
 anomalistic periods), we have 
 
 Sh cos J V m = R' sin (" + i) R sin ( i) 4- /j,R p sin ( p + k + ^ we )) 
 S% sin ^ F m = R' cos (" + 1) + jR cos (^ i) /j,R p cos (f p + k + ^me) ) 
 
 Secondly, suppose the semi-lunar period indicated by 1 consists of 
 II + III, that 2 consists of IV + V, and so on. Then, obviously, the result 
 is got by writing to + ^ir/tr for t ; that is to say, write i fyrr for i and 
 k i (a- 2rj) TT/O-, or k \TT + T^TT/O- for k. But IJTT/O- is equal to e, and we 
 write k - JTT + ^e for A;. Therefore, following the notation used in 4 for 
 N and L, 
 
 ^h cos 
 
 i'/i sin 
 
 = -K cos (g + i} - R cos ( - i) 
 
 flR p COS {^ + A; + | (W + 1) 6; 
 
 i) - R sin (^ - i) 
 - pRp sin {f p + A + ^ (TO + 1) e} 
 
 = - sn 
 
 (30) 
 
DIURNAL TIDES. [5 
 
 These four S's require correction for the disturbance due to the semi- 
 diurnal terms M 2 and S 2 , and I shall return to this point later. In the 
 
 meantime write 
 
 and we have 
 
 (W + Z)= -R sm(^-i)-fj,R p sin ^ecos {, + & + (2m + l 
 
 i (X - Y) = # cos (& - i) - fiR p sin e sin {f p + jfc + (2m + l " 
 
 | ( W - Z) = ' sin (f ' + 1) + MJ R P cos ie sin {, + Ar + (2m + l 
 
 , 
 
 " 
 If we put L = { (X + Y) + R cos (f + i)} tan e ) 
 
 M = ft ( W - Z) - ' sin (' + i)} tan e J ' 
 the equations (33) may be written 
 
 - - 
 
 The four equations (32), (33) involve six unknown quantities, R', ", 
 RO, o, Rp, Zp, and are insufficient for their determination. 
 
 In reducing a short series of observations it is necessary to assume what 
 is usually nearly true, viz., that K P = K, and H^/H' = 0'3309, as would be the 
 case in the equilibrium theory of tides. 
 
 Then, writing q for 0'3309, we have approximately R p = H p = qW. The 
 argument of the Kj tide is t + (h - v') - TT - K, where v is a certain function 
 of the longitude of the moon's node (tabulated by Baird); and the argument 
 of the P tide is t h + \TT K P . 
 
 At the noon which is taken as epoch t = 0, h = h , and the two arguments 
 are equal to ' and % p . 
 
 Hence - ' = h -v -\ir- K 
 
 Therefore p = ? +2h n - v' -IT + (K P - K') 
 
 Putting K P = K as explained above, 
 
 p + k = % + i + 2h - v - TT + I 
 where l = k-i = [1'450 - (a- - 2iy) t ] - [1'705 - <rt ] 
 
 = -0'255 + 2^ , a small angle (36) 
 
 Then,if = 2h -v' + l + l(2m + I)e) 
 
 . f (P 1 ) 
 
 Pm = qp cos |e } 
 
 we have 
 
 |(W - Z) = f'H' sin (' + i) - p m R' sin (' + i + ff)} 
 
 > ("^/ 
 
 i (X + Y) = - f H' cos ( + *') + p m H' cos (' + i + 0) } 
 
1890] DIURNAL TIDES. 173 
 
 Let T cos </r = f - p m cos } 
 
 T sin A/T = p m sin 6 } 
 whence T and -fy may be computed ; and 
 
 (W - Z) = H'T sin (f + i - 
 
 .(40) 
 
 From these we compute H' and ' and ' + *. 
 
 Then if u h v ^TT, the equilibrium argument at epoch of K x , 
 
 '=' + u' 
 
 We have also H p = gH' = 0'3309H', K P = K. 
 Returning to equations (34) and (35), we compute R' = f'H', and hence 
 
 R' C S (f + i), and then L and M. 
 sin 
 
 Having these, we compute R and from (35). 
 
 Then, if u = h v 2(s f) + |TT, the equilibrium argument at epoch 
 
 ofO, 
 
 TT 
 
 K O = O + U O> arid RO = -T 
 
 r o 
 
 where f is a certain function of the longitude of the moon's node, tabulated 
 in Baird's Manual. 
 
 A Long Series of Observations. Suppose that there is a half year of 
 observation, or two periods of thirteen quarter-lunar periods, each of which 
 contains exactly the same number of tides. 
 
 Then each of these periods is to be reduced independently with the 
 assumption that q = 0'3309 and K P = K. If this assumption be found 
 subsequently to be very incorrect, it might be necessary to amend these 
 reductions by adding K P K' to the value of 6, and by multiplying p m by 
 Hp -r 0'3309H', but such repetition will not usually be necessary. 
 
 From these reductions we get independent values of H' cos ', H' sin K, 
 H cos K O , H sin K O from each quarter year, and the means of these are to be 
 adopted from which to compute H', K, H , K O . 
 
 It remains to evaluate H p and K P . 
 
 The factor f ' and the angle v vary so slowly that the change from one 
 quarter to the next may be neglected, although each quarter is supposed to 
 have been reduced with its proper values. 
 
 Let h , h ' be the values of the sun's mean longitude at the two epochs; 
 then since the second epoch is nearly a quarter year later than the first, 
 h ' will exceed h by about 90. 
 
 Let h ' = h + ^TT + Sh, so that Bh is small. 
 
174 DIURNAL TIDES. [5 
 
 If ' -I- 8%', p + 8 p be the values of ', p at the second epoch, we 
 have ' + ' = - /' + v + TT + *', ' = - A + i/ + ^TT + ', and therefore 
 
 Again, ^, + 8 p = /<' - \TT + K P , ^ p = h - 
 
 and therefore 8 p = |TT + 8h 
 
 Let * + Si, k + Bk be the values of i and & corresponding to the second 
 epoch, and let W, X', Y', Z' be the values of those quantities in the second 
 quarter. Then, replacing (2ra + l)e by 87'5, since that is its value when 
 2m + 1 is 13, we have from (33) 
 
 ( W - Z') = - R cos (? + i + Si - Bh) 
 
 + pRp cos e cos (p + k + 8k + Sh + 87'5) 
 ^ (X x + Y') = - R' sin (' + { + Si - 5A) 
 
 + /i,Rp cos e sin (? p + A; + SA; + 8h + 87'5) 
 
 ^ (W - Z) = -R' sin (f ' + i) + /i.Rp cos Je sin (^ + k + 87'o) 
 ^ (X + Y) =-R cos (f + i) - fjiR p cos |e cos ( p + k + 87'5) 
 
 Hence 
 
 i (W - Z') - i (X + Y) = R' sin I (Si - M) sin {? + i + ^ (Si - Sh)} 
 + }j,R p cos Je cos \ (Bk + Bh) cos { p + k + k (&k + Bh) + 87-5} 
 
 ...(41) 
 
 cos Je cos \ (8k + Bh) sin { p + k + ^ (8k + Bh) + 87*5} 
 In these equations R is equal to f'H' and R p is equal to H p . 
 
 The terms involving R' are clearly small, and approximate values of R' 
 and ', as derived from the first quarter, will be sufficient to compute them. 
 Afterwards we can compute R p or H^ and % p ; then if u p denotes h + ^TT, 
 the equilibrium argument of P at the first epoch, K P = P + u p . 
 
 The values of H^, K P thus deduced ought not to differ very largely from 
 those assumed in the two independent reductions. 
 
 The same investigation serves for the evaluation of the P tide from any 
 two sets of observations, each consisting of thirteen quarter-lunar periods, 
 and with a small change in the analysis we need not suppose each to consist 
 of thirteen such periods. But the two epochs must be such that sinS/i is 
 small and cos Bh is large, or the formulas, although analytically correct, will 
 fail in their object. 
 
1890] DISTURBANCE DUE TO SEMI-DIURNAL TIDES. 175 
 
 8. The Disturbance of K 1} O, P due to M 2 and S 2 . 
 
 It has been remarked in 7 that the diurnal tides are perturbed by the 
 semi-diurnal. The general method has been given in 3, by which to 
 calculate the effect on any one tide, whose increment of argument since 
 epoch is V p and speed is p, due to a tide whose increment is V q and speed q. 
 
 Since in the present instance all the diurnal tides have been consolidated 
 into one of speed 7 0-, we have to calculate the effect of the tides whose 
 speeds are 2(70-) and 2(7-77) on the tide whose speed is 7 0-. It 
 follows, therefore, that the factor q/p or k q of (3) is in the first case equal to 
 2 (7 er)/(7 - a-) or 2, and in the second case is 2 (7 77)7(7 ~ or 2'070 ; or 
 k m = 2, k s = 2-070. 
 
 The coefficients F, G, f, g, as due to the tide M 2 of speed 2 (7 <r), will be 
 written with suffix ra, and as due to the tide S 2 of speed 2 (7 77), with 
 suffix s. The sums and means have also to be taken in the two ways 
 denoted by S and SK Hence we have altogether to compute sixteen 
 coefficients, which by an easily intelligible notation may be written 
 
 F (0) G (0) f e (i7r) 2- (i7r) 
 
 J- m > **! > * > &s 
 
 In order to compute the sixteen coefficients, it is necessary to find the 
 mean cosines and sines of the four following angles, viz.: 
 
 ^V m V m , ^V m V s , and the means have to be taken in the two ways 
 denoted S and SK 
 
 These means are exactly the same in form as what the means of h cos and 
 h sin (which had to be evaluated in S and S**) would be if all the heights 
 were regarded as positive unity, irrespective of whether they are H.W. or 
 L.W. Hence the same plan of computation serves here as elsewhere ; the 
 plan is explained in the following section. 
 
 By comparison of equation (7) and the definitions (31) of W, X, Y, Z in 
 
 the last section, we have 
 
 ...(42) 
 
 The four quantities A m , B m , A s , B s are known from the evaluations of the 
 tides M 2 and S 2 ; whence the corrections referred to in 7 are calculable. 
 
 Y = &h cos \V m - {A m F m ^ + B m G m ^ + A g F s ^ + B.G.to] f 
 
176 THE METHOD OF SUMMATION. [5 
 
 9. On the Summations. 
 
 It will be seen from the preceding sections that sums have to be found of 
 the following functions : 
 
 and also of 
 
 , cos T7 , cos jr . cos 
 h . V m , h . K s , h . 
 sin sin sin 
 
 cos lF cos cos 
 
 Bin* "' sin^' Bin 1 * "V* ' ; 
 
 It is necessary to calculate the five angles \V m , V s> \V m V g , and V m , 
 for each tide, and the reader will easily see, by the example in the Appendix, 
 how they may be computed with considerable rapidity, by aid of an auxiliary 
 table A. 
 
 The computation of sines and cosines and multiplication by heights, may, 
 with sufficient accuracy, be abridged, by regarding the cosine or sine of any 
 angle lying within a given 5 of the circumference as equal to the cosine or 
 sine of the middle of that 5. 
 
 The process then consists in the grouping of the heights according to the 
 values of their F's (V m , V s , ^V m , as the case may be). The heights in each 
 group are then summed. Since the L.W. heights are all negative, they are 
 treated in a separate table, and are considered as positive until their combi- 
 nation with the H.W. at a later stage. We shall, for the present, only speak 
 of one of these groupings, taking it as a type of both. 
 
 OOS f*OS 
 
 Since . (a + 180) = . a, the eighteen groups forming the 3 rd quadrant 
 
 may be thrown in with the 1 st quadrant by a mere change of sign ; and the 
 like is true of the 4 th and 2 nd quadrants. 
 
 Since cos (180 a) = cos a and sin (180 a) = sin a, it follows that we 
 have to go through the 2 nd quadrant in reversed order, in order to fall in with 
 the succession which holds in the 1 st quadrant, and, moreover, the cosine 
 changes its sign, whilst the sine does not do so. Hence the following 
 schemes will give us the eighteen groups which all have the same cosines 
 and sines: 
 
 for cosines (1 st - 3 rd ) - (2 nd - 4 th ) reversed 
 
 for sines (1 st - 3 rd ) + (2 nd - 4 th ) reversed 
 
 Thus, one grouping of the heights serves for both cosines and sines, and, 
 save for the last step, the additions are the same. 
 
 The combination of the H.W. and L.W. results is best made at the stage 
 where 1 st - 3 rd and 2 nd - 4 th have been formed. 
 
1890] THE METHOD OF SUMMATION. 177 
 
 The negative signs for the L.W. results are introduced before addition to 
 the H.W. results, and total 1 st - 3 rd and 2 nd - 4 th are thus formed. 
 
 After the eighteen cosine and sine total numbers are thus formed, they 
 are to be multiplied by the cosines or sines of 2 30', 7 30', 12 30', ... 87 30'. 
 
 COS 
 
 The products are then summed so as to give "Zh , . 
 
 It was noted at the beginning of this section that we also have sums of 
 
 cos 
 
 the form S . These sums are obviously made by entering unity in place 
 
 of each height, and, of course, not treating the L.W. as negative. Thus, 
 where the H.W. and L.W. are combined it is not necessary to change the 
 sign of the L.W., as was done in the combination of H.W. and L.W. for 
 
 These summations are considerably less laborious than the others. 
 
 cos 
 sin 
 
 cos 
 
 In the case of the tides M 2 and So, the division of the sums *h . by the 
 
 sm 
 
 total number of entries gives the required results. But for N, L, and 
 similarly for the diurnal tides K 1} O, P, the grouping and summations have 
 to be broken into a number of subordinate periods, which are to be operated 
 on to form S and SK The multiplication by the eighteen mean cosines and 
 sines is best deferred to a late stage in the computation. 
 
 Thus, for example for N and L, the quarter-lunar-anomalistic periods, 
 i, ii, iii, &c., are treated independently, and we find (1 st 3 rd ) (2 nd 4 th 
 reversed) for each. There are thus eighteen cosine numbers and eighteen 
 sine numbers for each of i, ii, iii, &c. 
 
 We next form the sums two and two, i + ii, iii + iv, &c. ; next find the 
 differences (i 4- ii) - (iii + iv), (v + vi) (iii + iv), &c. ; add the differences 
 together ; then multiply by the eighteen cosines or sines of 2^, 7|, &c., and 
 
 finally multiply by T-. ^. z^-, and so find Sh . 
 
 J 4>(n+ l)(ra-l) sm 
 
 We next go through exactly the same process, but beginning with 
 
 COS 
 
 ii instead of i, and so find S^ w h . . 
 
 sin 
 
 COS 
 
 The same process applies, mutatis mutandis, for finding S and S*"" . 
 
 There are two cases which merit attention in particular. The sorting of 
 heights in quarter-lunar-anomalistic periods, according to values of V m , 
 serves, in the first instance, for the evaluation of N and L, but it serves, 
 secondly, to evaluate M 2 , for we then simply neglect the subdivision into 
 quarter periods and treat the whole as one series, but stop at the end of a 
 semi-lunation. 
 
 D. i. 12 
 
178 THE METHOD OF SUMMATION. [5 
 
 The sorting of heights in quarter-lunar periods, according to the values 
 f ^Vm> also serves several purposes. 
 
 COS 
 
 We first find from it S and S^h . \V m , and secondly, by merely 
 
 bill 
 
 counting the entries in each group for each quarter period, instead of adding 
 up the heights, we arrive at S and S^cos \V m , (It may be noted in 
 passing that what is wanted, according to preceding analysis, is the sum 
 
 COS 
 
 of (^ Vm V m ), so that there will be a change of sign in the sine sum to 
 sin 
 
 get the desired result.) 
 
 cos 
 
 But, besides these, S and S* 71 ' , (%V m +V m ) can be obtained with 
 
 sin 
 
 sufficient accuracy from the same sorting. 
 
 The angles %V m were sorted in four times eighteen groups, for each 
 quarter-lunar period. If each angle were multiplied by three, the eighteen 
 entries of the 1 st quadrant would be converted into three groups of six, lying 
 in three quadrants, viz., I st , II nd , III rd ; the 2 nd quadrant is changed to IV th , 
 I st , II nd ; the 3 rd to III rd , IV th , I st ; and the 4 th to II nd , III rd , IV th . Hence 
 eighteen entries of 1 st 3 rd are converted into three sixes, I st III rd , 
 II nd - IV th , - (I st - III rd ] ; and eighteen entries of 2 nd - 4 th are converted 
 into three sixes, - {II nd - IV th }, I st - III rd , II nd - IV th . 
 
 Hence a new I st III rd of six entries is made up thus : 
 first six of former 1 st 3 rd 
 
 + second six of former 2 nd 4 th 
 
 third six of former 1 st 3 rd 
 And a new II nd IV th of six entries is made up of 
 
 - first six of former 2 nd - 4 th 
 
 + second six of former 1 st 3 rd 
 
 + third six of former 2 nd - 4 th 
 
 These I st III rd and II nd -I V th may now be treated just like the other 
 ones. Thus, without calculating |F m , we have from the former 1 st 3 rd and 
 2 nd 4 th the results of a fresh grouping according to values of f V m . 
 
 It is true that there is a considerable loss of accuracy, because all angles 
 within 15 are now treated as having the same sine and cosine. 
 
1890] PARTITION INTO GROUPS. 179 
 
 10. Rules for the Partition of the Observations into Groups. 
 
 It appears from the preceding investigations that it is required to divide 
 up the observations into groups. This may be done, with all necessary 
 accuracy, and with great convenience, by dividing the tides just as they 
 would be divided if every H.W. followed L.W., and vice versa, at the mean 
 interval of 6 h> 2103. 
 
 Now a quarter-lunar-anomalistic period is 165 h '3272, a quarter-lunar 
 period is 163 h< 9295, and semi-lunation is 354 h> 3670. Hence, dividing these 
 numbers by 6 h "2103, we find that there are 26'62145 tides in a quarter- 
 anomalistic period, 26'3964 in a quarter period, and 57'0612 in a semi- 
 lunation. 
 
 It may be remarked in passing that these results show that the n + 1 of 
 (10), 4, is 53-243, and the n + 1 of (25), 7, is 52'793. 
 
 It is, of course, impossible to have a fractional number of tides, and, 
 therefore, we make a small multiplication table of these numbers, and take 
 the nearest integer in each case. For example, in the case of the semi- 
 lunations, we have 
 
 1. 57-0612 57 4. 228*2448 228 
 
 2. 114-1224 114 5. 285-3060 285 
 
 3. 171-1836 171 6. 342-3672 342 
 
 These have to be divided between H. and L.W. For the sake of con- 
 venience, I suppose that we always begin the series with a H.W., then when 
 the integer is odd we put in one more H.W. than L.W., and thus have the 
 following rule : 
 
 No. of serni-lunation ... 1 2 3 4 5 6 
 
 No. of last H.W. in the 
 
 semi-lunation 29 57 86 114 143 171 
 
 No. of last L.W. in the 
 
 semi-lunation 28 57 85 114 142 171 
 
 The H.W. and L.W. are here supposed to be numbered consecutively from 
 1 onwards in separate tables. 
 
 The other rules of partition given in Appendix E are found in the same 
 way. 
 
 
 
 11. On the Over- Tides. 
 
 Observations of H. and L.W. are very inappropriate for the determination 
 of these tides (of which the most important are M 4 , M 6 , S 4 , S 6 ), because they 
 express the departure of the wave from the simple harmonic shape, and we 
 are supposed to have no information as to what occurs between two tides. 
 
 12 2 
 
180 ANNUAL AND SEMI-ANNUAL TIDES. [5 
 
 These tides make the interval from H. to L.W. longer than from L. to H.W., 
 and there is no doubt that, assuming the existence in the expression for h 
 of a term of the form A 2m cos2F 7n + B 2w sin 2F W , we shall get an approxima- 
 tion to A 2wt and B^ by finding the mean of hcos2V m and A,sin2F m . But 
 the computation of the F, G, f, g, coefficients for the perturbation of M 4 by M 2 
 would be essential, and thus the amount of additional computation would be 
 very great, whereas in the analysis of continuous observation the over-tides 
 are found almost without any additional work. I am inclined to think that 
 it would be best to obtain hourly observations for several days at several 
 parts of a lunation, and by some methods of interpolation to construct a 
 typical semi-diurnal tide-wave, from which, by the ordinary methods of 
 harmonic analysis, we could find the ratio of the heights of the over-tides to 
 the fundamental, and the relationship of their phases. 
 
 I make no attempt at such an investigation in this place. 
 
 12. On the Annual and Semi-annual Tides. 
 
 These tides are frequently of much importance, so that they ought not 
 to be neglected from a navigational point of view. It is obviously impossible 
 to obtain any results from a series of observations of less than a year's 
 duration. 
 
 Rules for the partition of tides into months or 12 th parts of a year are 
 given in the Appendix E. The mean of all the H. and L.W. observations 
 for each month may be taken as the height of mean water at the middle of 
 the month, and the 12 values for the year may be submitted to the ordinary 
 processes of harmonic analysis for the evaluation of these two tides. 
 
 We have supposed in the previous investigation that the tide heights are 
 measured from mean sea-level, and although it is not necessary that this 
 condition should be rigorously satisfied, it might be well, where there is a 
 large annual tide, to refer the heights to different datum levels in the different 
 quarters of the year. 
 
 13. On Gaps in the Series of Observations. 
 
 It often happens in actual observations that a few tides are missing 
 through some accident, or are obviously vitiated by heavy weather. Now 
 the present method depends for its applicability on the evanescence of terms 
 in the averages. It is true that it is rigorously applicable even for scattered 
 observations, but if applied to such a case all the F, G, f, g coefficients have 
 to be calculated, and, as every tide reacts on every other, the computation 
 would be so extensive as to make the method almost impracticable. Thus, 
 where there is a gap, observations must be fabricated (of course noting that 
 they are fabrications) by some sort of interpolation, and even values which 
 
1890] TABLES AND RULES FOR REDUCTION. 181 
 
 are very incorrect are better than none*. If the interpolation is extensive, 
 it might be well to test its correctness in a few places when the reduction 
 is done. If a whole week or fortnight be missing, and if the computer 
 cannot find a plausible method of interpolation, I can only suggest a pre- 
 liminary reduction from the continuous parts, and the computation of a tide 
 table for the hiatus. Each, such case must be treated on its merits, and it is 
 hardly possible to formulate general rules. 
 
 APPENDIX. 
 
 Tables and Rules of General Applicability. 
 A. To find 
 
 The following table is for finding what would be the mean moon's hour- 
 angle, if the moon had been on the meridian at the epoch. This angle is 
 denoted by |- V m or (7 <r) t, and is equal to the angle through which the 
 earth has turned relatively to the mean moon (at 14'4920521 per mean solar 
 hour) since epoch. 
 
 It would be advantageous to extend the table up to 90 days, but it can 
 be used as it is for periods greater than 30 days by the division of the time 
 into sets of 30 days. In the second period of 30 days 5'7 must be subtracted 
 from the tabular entry, for the third period 11'4, and so onf. 
 
 For example : Find \V m for 78 d ll h 23 m . The day is 18 of the third 
 30, and the tabular entry for 18 d ll h is 113'9, and subtracting 11'4 we 
 have 102-5; 23 m gives 5'6, so that F m = 108 l. The correct result is 
 107'99, and it is obvious that an error of 0'l may easily be incurred by the 
 use of the table. 
 
 The row for day \ is given because it may be necessary to use one tide 
 before epoch ; this row is used in the example below. 
 
 * Fabricated times and heights would very likely be no worse than real observations during 
 a few days of rough weather. A perfect tide table only claims to predict the tide apart from the 
 influence of wind and atmospheric pressure ; and, conversely, tidal observations must be 
 sufficiently numerous to eliminate these influences by averages. 
 
 t Observe that the decimals run thus, -0, -5, -0, &c., then -4, -9, '4, &c., then -8, -3, -8, &c., 
 and so on. The first entry in which the sequence alters I call a " change." The incidence of 
 "changes" may be found thus: 7-0- is 14|- -00795; take Crelle's multiplication table for 795, 
 and note where the last digit but three, having been 4, becomes 5 ; I say that this is a "change." 
 For example, 559 x 795 = 444405, and 560 x 795 = 445200; then a change occurs at the 560 th hour, 
 or at the (12 x 46 + 8) th hour, or at 23 d 8 h . If the table be continued to 60 and 90 days, &c., by 
 subtracting 5 0> 7, 11 0> 4, &c., the changes will fall a little wrong, but they may easily be corrected 
 by means of Crelle's table, as here shown. 
 
182 
 
 TABLE OF (7 cr) t. 
 
 lOOCOOOr-iOCCDOOOCOirsOOO 
 
 ,_,, i, ,, ii ii I P H'-H I i^CMCMCMCMCMCMCMCNCMCM 
 
 ip-^OOCMrHpgSGCI^Cp 
 
 CM CM CM CM 
 
 ! i M rH CO --H CO r-H 
 
 c 
 
 i i 
 oo 
 
 i i CO t-i CM I-H 
 
 iO ^ 00 CM I-H O OS OS 00 !> 
 OSWf-i iiCCSCMCbO^ OO CM CD O ^J* 00 > i iO OS CO 
 
 CO "^ iO OO Tt* i t 00 O CM OS OOiOI>-~J 1 OOO'^i-HOO 
 
 CM CM CM CM CM CM I-H r-HOOi-HOOi i 00 i i 00 I-H 
 
 i i CM i i CM CM CM CM CM CM CM I-H I-H 
 
 -* t^ r-i i(5 OS 00 1^ I-H >(5 OS 
 >OCDTtiiCCM^i-HCOOi ' 
 j"5 1-1 CO I-H 00 r- 1 00 I-H 00 I-H 
 
 ^ 
 
 "^ 00 CM CO O "^ -_ - . _ 
 
 iO CM "^ I-H CM O I-H _ 
 CM ~ CM CM CM CM CM CM I-H I-H CO 
 
 ip-^MWCMr^pOlCp 
 
 i i 
 
 OOK5CN-*i-HCNOi iO>O 
 
 OOr-H?O-HOOl 1 00 i- 1 CM I- 1 
 
 - .. CM 00 I-H CMOS,-HOO .... 
 
 CM CM CM CM CM ri r-c i I 00 I-H OO I-H 00 -H OO I-H O5 i iCMi ICM 
 
 O i i CM ec 
 
 I 
 
 i lOOSOOt- 
 
 CM CM CM 
 
 CN CM CM I-H ^H OOi lOOi lOOr-HOOi lOOi i CM i i CM CM CM CM 
 
 iO-*OOCMi 'OOSOSOOI^- 
 
 CM CM -< i i JO 
 
 (M CM I-H i i 
 
 pOSOpt~CpCpipTtiOOCM r--ipOSqCl^Cpip-rJ<WCM 
 
 gOr-iib6aaoti^4Osn r-^H-^oocMcbo-^obcM 
 
 .T^-^i irMOi iO5pl^-OS COOOiOCO^iOOO^i 100 
 
 OOi lOOr-iCCi iCMr-HCM CM CM CM CM CM 
 
 poscpj>-cp ipT^oooocMrHposoor-- cpip^J<oocMiipOsoor~ 
 
 i lOO^HOJ^-i OO i i 0? rH CM i i CM CM CM CM CM CM CM 
 
 r-H pOC7SOJCl^CpipTt<O?CM 7-ipOSOpl^CpOTfOJCM 
 O OS OO CO O^OOCMCOO^ OOCMOOSOO t^- I-H iC OS OO 
 i i OJ ^H 00 r i CM I-H CM i i CM CM CM CM fM*CM CM I-H 
 
 KSCOt^OOOS Oi-HCMOO-<t 
 
1890] 
 
 TABLE OF (7 - <r) t. 
 
 183 
 
 t^t^t>-QOQOOOQCQOO5O5O5O5OOOOi ir-ii i ICMCNCNCMCOCOCOCO'*-'* 
 
 i i i I r-i CO 
 
 CN CO O5 CO t^ i iiCOlCOh- i 
 
 CO i i CO r^-i CM i I CN "CM CM CM ~ fM 
 
 CO i i CN r-H CM CM <N CM fN fM fM 
 
 CM I-H CN tM CM CN 
 
 -^iOCO 
 
 COr-ICO 
 
 COr-ICOl ICOr-<COr-l(M 
 
 p o Th co CM i i p 
 
 CM CM CM CM 
 
 I-H i i CO i I CO . icOi-HCOi i CO I-H CM 
 
 CM CN CN C<1 
 
 (M i i t-H C*3 i i CO i i P5 
 
 ieor-i<Mi I<M CM 
 
 -i H^i ~i "< "i ~ "i ~" ' ^ " ' ~ "' ," i ~"< "' "' ~ "' ~ "' 
 
 ICO1T-OOO5 Oi-HtMCOTjt iOCOJ>-OOC35 
 
 ii i rH I-H I-H CMCMCMCMCM CMCMCMCMCM 
 
 cp o> oc 
 
 
 
 <M oq sq i-i 
 
 i i CM r-i <y CM (M CM 
 
 O Oi CC 
 CO O5 CO 
 CM'CMCN'tM CM 
 
 CO i i CO (-H CO I-H CO I-H CO f 1 CM 
 
 CM CM CN CM 
 
 CO i i CO i i CO I-H CO ,-H CO i I CM r-i CM CM CM CM CM CM CM i i 
 
 t^ * CM <35 CO 
 
 CM CM CM T i i i 
 
 O CD l^ 00 Oi 
 
184 RULES FOR REDUCTION. [5 
 
 To find V s . 
 
 No table is necessary for the conversion of time into angle at 30 per 
 hour to find V s , or 2(7 v))t, since we multiply the hours by 30, and add 
 half the number of minutes. This rule is the same for every day. 
 
 B. The tides S 2 and K 2 . 
 It is required to compute U and < from 
 
 U COS <f> = II + X^f " COS ft) 
 
 U sin <J> = \nf" sin &> 
 where to = 2h 2v" + o^ 
 
 /sun's parx. mean parxA 
 and II = 1 + 3 
 
 \ mean parx. / 
 
 the sun's parallax referred to being its value at the middle of the period 
 under reduction. 
 
 If, for example, February 14 is the middle of the period, IT is found 
 thus: 
 
 Sun's parx. Feb. 14 = 8"-95, mean parx. = 8"-85, diff. = + 0"10 
 Then n = l+ 
 
 The period under reduction consists in this case of an exact number of 
 semi-lunations. The following table gives \n and a n , according to the number 
 of semi-lunations : 
 
 No. of semi- 
 
 lunations ... 1. 2. 3. 4. 5. 6. 
 
 log\ n 9-4300 9-4159 9'3920 9'3575 9'3113 9'2517 
 0^ 14-28 28-82 43'36 57'90 72-43 86'97 
 
 h is the sun's mean longitude at epoch, found from Nautical Almanac; 
 and 2 1/", f" are found from Baird's Manual in the tables applicable to the 
 tide K 2 [or from the formulae at the end of Paper 1]. 
 
 C. The Tides N and L. 
 
 Summations are carried out over quarter -lunar -anomalistic periods, 
 numbered i, ii, iii, &c. Grand totals are then made in two different ways, 
 viz. 
 
 [2 (i + ii) - 2 (iii + iv)] + [2 (v + vi) - 2 (iii + iv)] 
 
 + [2 (v + vi) - 2 (vii + viii)] + &c., to find S 
 and [2 (ii + iii) - 2 (iv + v)] + [2 (vi + vii) - 2 (iv + v)] 
 
 + [2 (vi + vii) - 2 (viii + ix)] -I- &c., to find S^ 
 
1890] RULES FOR REDUCTION. 185 
 
 where, for example, 2 (i -f ii) denotes summation carried over the half period 
 made up of i and ii. These totals are multiplied by certain mean cosines 
 and sines (whose values are given in F), and are summed. The next process 
 is multiplication by a factor <E> (?r/4 (n + l)(ra 1) of 4), of which the value 
 depends on the number of quarter-lunar-anomalistic periods under treatment. 
 The following table gives the value of this factor : 
 
 No. of 4:-lunar- 
 anom. periods ... iii. v. vii. ix. xi. xiii. 
 
 <I> 0-02950 0-01475 0-00738 0-00492 0-00369 0-00295 
 
 The angle j is also required ; it depends on the time of the first tide 
 under reduction. If t be the time in hours since epoch to the first tide, 
 
 j=l-690-0-5444 
 
 For instance, in the example below the first tide is at 3 h 14 m of day ; 
 this is 8 h 46 m , or 8 h -77, before epoch, so that t = - 8 h> 77 ; then 
 
 j = 1-690 + 0-5444 x 877 = + 6'46 
 
 D. The Tides K,, O, P. 
 
 Summations are carried out over quarter-lunar periods numbered I, II, III, 
 &c., and totals are formed like those mentioned in C, and a factor ^ (which 
 differs slightly from <f>) is required in the formation of S and SK This 
 factor depends on the number of quarter-lunar periods under treatment, and 
 the following table gives its value : 
 
 No. of |-lunar 
 
 periods ... III. V. VII. IX. XI. XIII. 
 
 0-02976 0-01488 0-00744 0-00496 0-00372 0-00298 
 
 The angles i and I are required ; they depend on the time of the first tide 
 under reduction. If t be the time in hours of the first tide since epoch, 
 
 i= 1-705-0-549 
 I = - 0-255 + 0-082 t 
 
 For instance, in the example below we have, as shown in C, t = 8 h> 77, 
 and 
 
 i = + 6-52 
 
 I = - 0-97 
 It is required to compute T and ty from 
 
 T cos >|r = f ' p n cos 6 
 T sin T/T = p n sin 6 
 where = 2h - v + I + 
 
186 RULES FOR REDUCTION. [5 
 
 The period under reduction consists in this case of an exact number of 
 quarter-lunar periods, and the following table gives the values of p n and ft n , 
 according to the number of quarter-lunar periods : 
 
 No. of J-lunar 
 periods ... 
 
 III. 
 
 V. 
 
 VII. 
 
 IX. 
 
 XI. 
 
 XIII. 
 
 log p n 
 
 9-5749 
 
 9-5628 
 
 9-5508 
 
 9-5303 
 
 9-5009 
 
 9-4618 
 
 fin 
 
 20-20 
 
 33'66 
 
 47-13 
 
 60-59 
 
 74-06 
 
 87-52 
 
 h is the sun's mean longitude at epoch, the formula for I is given above, 
 and v, f ' are found from Baird's Manual in the tables applicable to the tide 
 K! [or from the formulas at the end of Paper 1]. 
 
 E. Rules for the Partition of the Observations into Groups. 
 
 If the first event after epoch is a L.W., either omit it from the reductions, 
 or let the first tide be the H.W. which precedes epoch. Thus we are to begin 
 with a H.W.* 
 
 The H.W. and L.W. are treated apart in separate tables. 
 
 Each tide (H.W. or L.W., as the case may be) is numbered consecutively, 
 from 1 onwards. 
 
 The following are rules for partitfons : 
 
 * This is not necessary, but it makes the statements of the subsequent rules simpler, as they 
 have not to be given in an alternative form. 
 
1890] 
 
 RULES FOR REDUCTION. 
 
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188 
 
 RULES FOR REDUCTION. 
 
 For M 2 and S 2 . 
 
 Semi-lunations numbered 1, 2, 3, &c. 
 
 No. of semi-lunation 1234 
 
 No. of last H.W. in the semi- 
 lunation 29 57 86 114 
 
 No. of last L.W. in the semi- 
 lunation 28 57 85 114 
 
 5 
 
 143 
 142 
 
 171 
 171 
 
 Total No. of tides up to end of 
 each semi-lunation... 57 
 
 114 
 
 171 
 
 228 
 
 285 342 
 
 For Annual and Semi-annual Tides. 
 Months, or -jL th parts of a year, numbered 1, 2, 3. 
 
 No. of month 123 
 
 No. of last H.W. in month 59 118 176 
 
 No. of last L.W. in month 59 117 176 
 
 Total No. of tides up to the end of each month 118 235 352 
 
 No. of tides in each month 118 117 117 
 
 The epoch for the second quarter year should be 91 days after first epoch, 
 that for the third 92 days after the second, for the fourth 91 days after the 
 third, except in leap year, when the last should also be 92 days. 
 
 There are six tides (or about thirty-seven hours) more in a quarter year 
 than in xiii quarter-lunar-anomalistic periods; the times of these six tides 
 (or ten tides in one of the quarters) are to be omitted from the reduction, 
 and their heights are only required when the annual or semi-annual tides are 
 to be found. 
 
 F. Cosine and Sine Factors for all the Tides. 
 
 These are the cosines and sines of 2 30', 7 30', 12 30', &c. They are 
 as follows : 
 
 Cosine and Sine Factors. 
 
 Read downwards for cosines, upwards for sines. 
 
 1. 
 
 0-999 
 
 2. 
 
 0-991* 
 
 3. 
 
 0-976 
 
 4. 
 
 0-954 
 
 5. 
 
 0-924* 
 
 6. 
 
 0-887 
 
 7. 
 
 0-843 
 
 8. 
 
 0-793* 
 
 9. 
 
 0-737 
 
 r>n 
 
 of S C S 
 sin 
 
 10. 
 
 0-676 
 
 11. 
 
 0-609* 
 
 12. 
 
 0-537 
 
 13. 
 
 0-462 
 
 14. 
 
 0-383* 
 
 15. 
 
 0-301 
 
 16. 
 
 0-216 
 
 17. 
 
 0-130* 
 
 18. 
 
 0-044 
 
 COS 
 
 V m and S* 77 . $V m , only the factors marked 
 
 are required. 
 
1890] 
 
 REDUCTION OF BOMBAY TIDES. 
 
 189 
 
 G. Increments of Arguments in Various Times. 
 
 The following table gives the increments of arguments of the several 
 tides in various periods, multiples of 360 being subtracted. This table 
 facilitates verification of the calculation of the harmonic constants. 
 
 1 hour 
 
 M 2 
 28-984104 
 
 S 2 
 30-00000 
 
 K 2 
 
 30'08214 
 
 N 
 28 -43973 
 
 1 day 
 
 - 24 -3815 
 
 
 
 + 1 -9713 
 
 - 37 '4465 
 
 10 days 
 
 + 116 '185 
 
 
 
 + 19 -713 
 
 - 14 '465 
 
 100 days 
 
 + 81 -85 
 
 
 
 - 162 -87 
 
 - 144 -65 
 
 1 hour 
 
 L 
 
 29'52848 
 
 K, 
 
 15-04107 
 
 O 
 
 13-94304 
 
 P 
 14'95893 
 
 1 day . 
 
 - 11 -3165 
 
 + -9856 
 
 - 25 -3671 
 
 - -9856 
 
 10 days 
 
 -113 -165 
 
 + 9 '856 
 
 + 106 '329 
 
 - 9 '856 
 
 100 days 
 
 - 51 -65 
 
 + 98 '56 
 
 - 16 -71 
 
 - 98 -56 
 
 
 
 
 
 
 EXAMPLE. 
 (a.) Place, Time, Datum Level, and Unit of Length. 
 
 The case chosen is three months of observation (in reality the tidal 
 predictions of the Indian Government) at Bombay, and the epoch is O h , 
 January 1, 1887. 
 
 A datum at or very near mean water-mark is taken, so that all the H.W. 
 are positive and the L.W. negative. This datum is found by taking the 
 mean of all the H.W. and L.W. of the original observations. In this case 
 99 inches was subtracted from all the tide heights. I might more advantage- 
 ously have subtracted 102 or 103 inches, but 99 inches was chosen from 
 considerations applicable to my earlier attempts, but which do not apply to 
 the computation in its present form. 
 
 At places where there is a large annual inequality in the height of water, 
 it would be advisable to use a different datum for each quarter of a year. 
 It is not, however, important that the datum should conform rigorously to 
 mean water-mark, for even the discrepancy of 3 inches, which occurs in my 
 example, does not materially affect the result. 
 
 In recording the heights, a convenient unit of length is to be used, and 
 it is advantageous that the H.W. and the L.W. should be expressible by two 
 figures, so that the larger H.W. and L.W. shall fall into the eighties and 
 nineties. The unit of length is here the inch. 
 
190 REDUCTION OF BOMBAY TIDES. [5 
 
 (b.) Times and Angles. 
 
 The times of H.W., numbered consecutively, are entered in a table, as 
 shown on p. 191. Since O h astronomical time is the epoch, the P.M. tides will 
 come in the half days which are numbered with integrals, and the A.M. tides 
 in the half days which fall between the integral numbers. 
 
 From time to time there will be a half day with no H.W. ; this row in 
 the table should be left blank, but there happens to be no such row in the 
 sample shown. A computation form for times and angles might be printed, 
 for, although the exigencies of the printer have not allowed the entries to be 
 equally spaced in the sample below, yet the computation form might be 
 printed with equal spaces, and the dividing lines are to be filled in by hand. 
 
 The L.W. table is similar. 
 
 Both H. and L.W. are to be divided into quarter-lunar-anomalistic and 
 quarter-lunar periods, and semi-lunations, according to the rules given in E. 
 These partitions and the numbering of the entries could not be printed, 
 because of the occasional blank rows. 
 
 The formation of %V m and of V s , by means of Table A and the rule 
 following it, is obvious. In the subtractions and additions under the headings 
 i V m ~ V s an d V m + V s , 360 is added or subtracted where necessary. V m is 
 found by doubling V m . 
 
 (c.) The Heights. 
 
 The H.W. heights are written in columns, as shown in the column of 
 figures on the margin of the table on the next page (with the same blanks as 
 in the table of times and angles), and are so arranged, either on strips of 
 paper, or by folding the paper, that the heights may be pinned to the times, 
 bringing each height opposite to an angle on the same row with the time 
 corresponding to that height. The heights will on one occasion have to be 
 pinned opposite the V m column, on a second occasion opposite the V s column, 
 and on a third occasion opposite the V m column. 
 
 The L.W. heights are written in similar columns, but the minus signs 
 should be omitted. 
 
 It is well to divide the columns, or to put fiducial marks in the table 
 for easy verification of the proper allocation of the heights with the times. 
 Any marks suffice, but the division into quarter-anomalistic periods, as shown 
 in the column of heights printed at the margin of the table on the next page, 
 seems to be as good as any other. 
 
 If it is proposed to evaluate the annual and semi-annual tides, it is 
 necessary to carry on the heights beyond the times by 3 (or 5) H.W. and 
 
 (Continued on p. 195) 
 
1890] 
 
 REDUCTION OF BOMBAY TIDES. 
 
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192 
 
 REDUCTION OF BOMBAY TIDES. 
 
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1890] 
 
 REDUCTION OF BOMBAY TIDES. 
 
 193 
 
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194 
 
 REDUCTION OF BOMBAY TIDES. 
 
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1890] REDUCTION OF BOMBAY TIDES. 195 
 
 3 (or 5) L.W., and to partition them into months. The mean for each month 
 is evaluated, and if the successive quarters of the year are referred to different 
 data, the mean monthly heights must all be referred to a common datum. 
 This process is not carried out in my example, because it is useless to 
 attempt the evaluation of these tides of long period from three months of 
 observation. 
 
 The sum of all the H.W. entries to the end of xiii is 9791. The sum of 
 all the L.W. entries to the end of xiii is 8577. There are 173 H.W. and 
 173 L.W. Hence mean sea-level is 3^ (9791 8577), which is equal to 
 + 3'51. It would have been better, therefore, to have subtracted 103 inches 
 instead of 99 inches from all the heights. This would have given mean sea- 
 level at 0'49 from the datum adopted. 
 
 (d.) Sorting the Heights according to the Values of the Angles. 
 
 In the tables on pp. 192, 193, 194 the column belongs to all angles 
 between and 5 ; 5 belongs to 5 to 10 ; and so on. 
 
 Where an angle fells exactly on a multiple of 5, an arbitrary rule of 
 classification is required, and it is easiest to deem it to belong to the next 
 succeeding 5, rather than to the preceding 5. 
 
 (e.) Sorting according to Values of V m . (See pp. 192, 193, 194.) 
 
 The H. and L.W. are treated in separate tables, similar in form save that 
 the - signs of the L.W. heights are omitted. 
 
 The sheets of heights (c) are pinned opposite to the V m 's on the tables 
 of angles (b), and the heights are then entered successively into the columns 
 corresponding to their V m 's in a table like that on pp. 192, 193, 194. 
 
 The division into quarter-lunar-anomalistic periods is maintained, but as 
 this sorting is to serve a double purpose, it is necessary to mark the end of 
 the last semi-lunation. In these tables there are two H.W. and two L.W., 
 which fall after the end of 6 semi-lunations, and before the end of xiii quarter- 
 lunar-anomalistic periods. 
 
 Nearly all the entries fall into one quadrant for H.W., and into another 
 for L.W. Thus there are no H.W. entries in the 4 th quadrant, and no L.W. 
 entries in the 2 nd quadrant; there are altogether only 10 H.W. entries in 
 1 st quadrant, and 3 in 3 rd quadrant ; and there are only 4 L.W. entries in the 
 1 st quadrant, and 17 in 3 rd quadrant. Something like this would hold true 
 at all ports. 
 
 132 
 
196 REDUCTION OF BOMBAY TIDES [5 
 
 (f.) Table of Sums for N and L. (See p. 197.) 
 
 Maintaining the divisions i, ii, iii, &c., it is now necessary to sum each 
 of the four times 18 vertical columns in each of the xiii divisions, to subtract 
 the 18 columns of the 3 rd quadrant from the 18 columns of the 1 st , and to 
 subtract the 18 columns of the 4 th quadrant from the 18 of the 2 nd . The 
 2 nd 4 th columns have then to be reversed. 
 
 Since in this case nearly all the H.W. fall in the 2 nd quadrant, and nearly 
 all the L.W. fall in the 3 rd quadrant, it is easy to write down at once 2 nd 4 th , 
 and 1 st 3 rd , as shown on the next page. 
 
 In this the signs of the L.W. entries have to be reintroduced, but as 
 the L.W. lie mostly in 3 rd quadrant, which enters with negative sign, they 
 become positive again. It thus happens that nearly all the columns come 
 out + ; there are, however, a few in xii. 
 
 (g.) Table of Sums for M a . (See p. 198.) 
 
 We now disregard the sub-divisions i, ii, iii, &c., and sum the 4 times 
 18 columns into grand totals, stopping the summations, however, at the end 
 of 6 semi-lunations (i.e. at 171 H.W. and 171 L.W.). 
 
 It would hardly be wise to attempt in this case the subtractions 1 st 3 rd , 
 2 nd 4 th , without the intermediate steps. 
 
 The following table (p. 198) gives the results. 
 
 (h.) General Rule for Cosine and Sine Summations. 
 
 For 'cosines' the 18 numbers required are derived from (1 st 3 rd ) 
 _ (2 nd -4 th , reversed). 
 
 For 'sines' the 18 numbers required are derived from (1 st 3 rd ) +( 2 nd -4 th , 
 reversed). 
 
 (i.) Evaluations of N and L (continued). (See p. 199.) 
 
 Cosines. From Table (f) of Sums enter the 18 ' cosine ' numbers, in 
 accordance with (h) in xiii vertical columns, and perform the operations 
 indicated in the example on page 199. 
 
 The column of ' cosine factors ' are those given in F, and <!> is given in C 
 for xiii ^-lunar-anomalistic periods. 
 
1890] 
 
 REDUCTION OF BOMBAY TIDES. 
 
 197 
 
 
 
 
 ^ 
 
 
 
 I-H 
 
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 00 
 
 CD iO 
 CM CO 
 
 
 
 
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 ^^ 
 
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 S- 
 
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 33 
 
 CD 
 00 
 
 
 
 
 
 
 
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 H-l. 
 
 
 
 
 Tl 
 
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 ) Total.... 
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 11 
 
 M 
 
 
 
 
 N 
 
 
 
 
198 
 
 REDUCTION OF BOMBAY TIDES. 
 
 oo . 
 
 oo 
 
 CO 
 CM ' 
 
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 1 
 
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1890] 
 
 REDUCTION OF BOMBAY TIDES. 
 
 199 
 
 6 J 
 
 . 
 
 cc 'o P, "8 
 
 (M 1-1 00 
 
 TO . iO . i i 
 (N O i I 
 
 O cc CM >c m ;o 
 
 OS <N ^f CM 00 Tf 
 -4- I> CO iO O ^t< TO 
 
 L-- i i o o 
 
 . i i iQ !> . <N 
 
 TO i I 
 
 + 1 
 
 CO OS 
 GSJ 
 
 CO O 
 
 +9 
 
 ' ifOO5CMOiOOi(NCD 
 CN^tt^CMQOiCCCr-iCC CCC^ 
 
 1 + i +7 + + + I + 1 I 
 
 + 
 
 .3 3. 
 
 1 
 
 !> O CO 
 
 _l_ 
 "r 
 
 . 
 
 t I 1 r-t -t (M iO 
 
 +1 + 1 + 1 
 
 CCfO^CO 
 1 1 1 1 1 1 1 1 1 1 1 1 1 
 
 o c 
 
 eg =y 
 
 >fNCOiO(M 
 
 -I- 
 
 lOCCr-HCMCMOOTOOSCD TOCO 
 
 I I I I I I I I I I I I 
 
 ?O i iCOCM>-HCMOaOCO-<tCN 
 
 i-H i-H r I 1-H 1-H 
 
 I I I ! I I I I + I I 
 
 CM * (M TO T^ CD CD 
 
 CM ' ' CM r-H i-H r-H 
 
 I I I I I I I 
 
 
200 REDUCTION OF BOMBAY TIDES. [5 
 
 Cosines (continued). Form columns ii + iii, iv + v, vi + vii, viii 4- ix, 
 x + xi, xii + xiii. Form difference columns (ii + iii) (iv + v), (vi + vii) 
 - (iv + v), (vi -f vii) - (viii + ix), (x + xi) - (viii + ix), (x + xi) - (xii + xiii) ; 
 add the 5 difference columns together ; multiply by cosine factors ; sum and 
 multiply by 3> or 0'00295. 
 
 The result is SWi cos V m = - 8'38 
 
 Sines. From Table (f) of Sums, enter 18 "sine" numbers, in accordance 
 with (h) in xiii vertical columns. 
 
 Perform all the same operations as those on "cosine" numbers, save that 
 we use sine factors, which are the same as cosine factors in inverse order, viz., 
 beginning with 0'044 and ending with 0'999. 
 
 The two results are 
 
 SA sin V m = + 5-94, S*h sin V m = + 10-06 
 
 Collecting results, proceed thus : 
 
 S% cos V m = + 11-38. SWi cos V m = - 8'38 
 
 S*'A sin V m = + 10-06. S% sin V m = + 5'94 
 
 Sum = + 21-44. Sum =- 2'44 
 
 Diff. = + 1-32. Diff. =-14-32 
 
 P=|sum = + 1072. R = |sum = - 1'22 
 
 Q = diff. =+ 0-66. S = i cliff. =- M6 
 
 (j.) Evaluation of M 2 (continued). 
 
 From the Table (g) of Sums for M 2 enter in one vertical column 18 cosine 
 numbers, in accordance with (h) ; multiply them by cosine factors ; add up 
 and divide by the total number of entries for 6 semi-lunations, viz., 342. 
 
 The result is A m = ^2A cos V m = - 30'58 
 
 Then enter in vertical column 18 sine numbers, in accordance with (h); 
 multiply them by sine factors, add up, and divide by 342. 
 
 The result is B, rt = ^S,h sin V m = + 38*47 
 
 (k.) Sorting according to Values of V s , and Evaluation of S 2 , K 2 . 
 
 The H. and L.W. are treated in separate tables, similar in form save that 
 the signs of the L.W. heights are omitted. 
 
 The sheets of heights (c) are pinned opposite to the F s 's on the Tables 
 of Angles (b), and the heights are entered successively into the columns 
 corresponding to their V s 's in a table like (e), which was used for sorting 
 according to values of V m . The sorting is carried as far as the end of 
 
1890] REDUCTION OF BOMBAY TIDES. 201 
 
 an exact multiple of a semi-lunation, in this case to the end of 6 semi- 
 lunations. No sub-division is necessary, but for the purpose of verification 
 it is useful to break the entries into groups of about 40. This is conveniently 
 done by a division after each third ^-lunar-anomalistic period, so that i, ii, iii 
 would be the first group; iv, v, vi the second; vii, viii, ix the third; and 
 x, xi, xii, and all but the end of xiii, the last. 
 
 In this case the entries fall into all the four quadrants with about equal 
 frequency. 
 
 We next sum the four times 18 columns, just as with M 2 in (g), and form 
 1 st - 3 rd and 2 nd 4 th , reversed, in the same way. 
 
 Next we write the 18 cosine numbers, (1 st 3 rd ) (2 nd 4 th , reversed) in 
 vertical column, multiply by cosine factors, add, and divide by the total 
 number of entries, which is 342. Afterwards write the sine-numbers 
 (1 st 3 rd ) + (2 nd 4 th , reversed), multiply by sine factors, add, and divide by 
 342. 
 
 The results are 
 
 A s = 2/4 cos V s = + 21-08. B s = TZh sin V s = + 3'62 
 
 (1.) Sorting according to Values of 
 
 The whole process is precisely parallel to the sorting according to values 
 of V m in (e) ; the thirteen divisions are, however, given by the quarter-lunar- 
 periods I, II, ... XIII. The only difference lies in the substitution of the 
 factor ^F (for XIII equal to 0'00298) for <. It is unnecessary to give an 
 example. 
 
 The results are 
 
 S%cosF m =-10-50, S/4sinF m = + 8-04 
 
 sV m = + 0-40, S**A sin F m = + 3-74 
 
 (m.) Sorting o 
 
 It is required to find what the sums in (1) would be if every H.W. height 
 had been unity, and every L.W. the same both in magnitude and sign ; in 
 fact to find S cos iF m , S^ cos F m , &c. 
 
 This is done by counting the entries in the preceding sorting in (1) 
 without regard to magnitude, taking the L.W. entries as actually positive, 
 instead of being (as they are) negative quantities with the negative sign 
 suppressed. 
 
 Since in this case we have simply to count entries which are all treated 
 as positive, the table of sums of H. and L.W. may be written together. The 
 following example gives part of the work : 
 
202 
 
 REDUCTION OF BOMBAY TIDES. 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I-H . 
 
 T 1 
 
 CO 
 
 CN . 
 
 rH 
 
 rH 
 
 T 1 
 
 rH 
 1 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 CM -^ 
 
 CM 
 
 co . 
 .1 
 
 
 
 rH CO 
 
 I 
 
 rH i-H 
 
 CN <M 
 
 . 
 
 o ^ 
 
 . CO 
 
 CO 
 
 1 
 
 CM 
 
 1 
 
 CO 
 
 CO . 
 
 1 
 
 
 
 rH 
 
 
 
 1 + 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 + 
 
 
 (JCJ 
 
 
 
 
 1 
 
 . CO 
 
 
 d 
 
 
 
 
 1 
 
 
 
 
 1 
 
 
 . 
 
 
 
 
 
 
 
 
 ^ 
 
 
 Pd 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 + 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "2 c< 
 CN ^ 
 
 A 
 
 "* 
 1 
 
 1* 
 
 rH CO 
 
 E 
 
 to 
 r-H (M 
 
 CN ^ 
 
 1 I 
 
 r-H 
 
 1 
 
 rill 
 
 : 5 
 
 HH 
 I-H 
 I-H 
 
1890] REDUCTION OF BOMBAY TIDES. 203 
 
 We next proceed to form XIII columns of cosine numbers, and generally 
 to operate exactly as though these numbers were heights ; and then proceed 
 with XIII columns of sine numbers in the same way. 
 
 The results are 
 
 S cos iF m = - 0-0522, S sin \V m = + 0'0117 
 
 S*" cos iF w = - 0-0168, S** sin $V m = - 0'0129 
 
 (n.) Formation of the Mean Sums of cos|F m and sin|F TO . 
 
 These may be found with sufficient accuracy from the last Table (m) of 
 Sums, part of which is given. In that table lines are drawn dividing the 
 columns into three divisions of six each. These are treated in the way shown 
 in the following example : 
 
 H. and L.W. f F m . 
 
 
 
 
 Refer to preceding 
 
 
 
 
 sorting (rn) 
 
 I. 
 
 
 
 -1 st six of 2 nd -4 th 
 
 
 . -3 
 
 
 
 -f2 nd six of 1 st -3 rd 
 
 -3 
 
 + 1 
 
 
 + 3 rd six of 2 nd - 4 th 
 
 2 nd - 4 th -3 
 
 + 1 . -3 
 
 
 
 
 
 
 
 
 
 
 
 + I 8t six of lt-3 rd 
 
 
 + 1 +1 +1 
 
 -2 
 
 +2 nd six of 2 nd -4 th 
 
 + 1 
 
 -1 -2 
 
 
 -3 rd six of 1 st -3 rd 
 
 pt_3rd _j_l 
 
 -1 +1 
 
 -2 
 
 
 2 nd -4 th , rev. ... 
 
 0-3 
 
 + 1 -3 
 
 
 II. 
 
 
 
 -1 st six of 2 nd -4 th 
 
 
 
 -3 
 
 + 2 nd six of 1 st -3 rd 
 
 + 3 
 
 + 3 
 
 
 + 3 rd six of 2 nd -4 th 
 
 2 nd -4 th +3 
 
 + 3 
 
 -3 
 
 
 
 
 
 
 
 
 
 -f 1 st six of 1 st -3 rd 
 
 
 
 -2 -3 
 
 + 2 nd six of 2 nd -4 th 
 
 + 2 
 
 -3-2 
 
 
 -3 rd six of 1 st -3 rd 
 
 lst_3rd _|_ 2 
 
 -3 2 
 
 2 -3 
 
 
 2 nd -4 th , rev. ... 3 
 
 
 +3 +3 
 
 
 III. 
 
 &c. 
 
 
 &c. 
 
 We have now only 6 instead of 18 sub-divisions of the quadrant, but the 
 cosine and sine numbers are found in exactly the same way as before. 
 
 The following example shows part of the treatment, and the cosine factors 
 are those marked * in F. 
 
204 
 
 REDUCTION OF BOMBAY TIDES. 
 
 
 (M 00 rH 
 
 
 
 O5 OO rH 
 
 OS 
 
 1 
 
 . CO CD . >p . 
 
 
 
 !> CM SO 
 
 S 
 
 
 
 oo 
 
 
 co to ^? 
 
 i>- 00 CN 
 
 CO rH CM CM Tt< 
 
 1^ C5 00 O CO 
 
 + 
 
 OS . . r-H . rH 
 
 CM >p CD O O 
 
 
 * CM rH 
 
 OC CD rH O O 
 
 
 
 + x + 
 
 O> 00 
 
 .S S 
 
 OJ -W 
 
 i i TJI CO O5 CO O 
 O5 CM O5 O OC CO 
 
 05 05 r^* CD co rH 
 
 : " 
 
 < 
 
 O O O O 
 
 ^^ 
 
 
 
 -2 6 
 
 
 
 O 
 
 
 X 
 
 
 10 OQ 
 
 
 fe CC 
 
 o " 
 
 CO GC CO *O t^~ ^D 
 
 
 9 
 
 (M rH CM rH (M 
 
 
 s.3 
 
 + 1 1 + 1 + 
 
 
 C 
 
 
 
 02 
 
 
 
 
 ,lOrH^rH-Hr^ 
 
 TT + I i ++ 
 
 
 o 
 
 HH rS 
 
 
 S 
 
 i 
 
 
 1 
 
 p d 
 
 
 5 
 
 
 
 
 
 j^I 
 
 
 S 
 
 ^hH 
 
 
 
 rH +OOCDOCOOO-* 
 
 
 
 +H-: + 1 +11 
 
 
 
 rH 
 
 ><j CO rH (M rH CO CO 
 + +11+11 
 
 
 00 
 
 M 
 
 
 M 
 
 
 
 "5 
 
 ^ = 
 
 
 g 
 
 ^ 
 
 
 1 
 
 s 
 
 *3l CM CO O >O CM rH 
 
 T I + I + I 
 
 
 CO 
 
 hH 
 
 rH 
 HH 
 
 
 
 HH ^ CO O rH CD lO 
 
 
 
 + + 1 +11 
 
 rH 
 
 
 
 ^ O rH O rH rH i 1 
 P 1 | -(- | -|- 
 
 
 CO 
 
 
 
 
 o> 
 
 
 
 a 
 
 ** * 
 
 
 'tc 
 
 
 
 o 
 
 O CO CM >C CD 
 S + 1 1 II 
 
 
 
 rH O CM 1 rH rH 
 
 
1890] REDUCTION OF BOMBAY TIDES. 205 
 
 The remaining process is exactly like that pursued before, and the four 
 results are 
 
 Scos|F m = + 0-0347, Ssin|F m =-01830 
 
 S^ cos f V m = - 0-0479, S** sin f V m = - 0'0173 
 
 (o.) The Sorting of V m + V s and of %V m - V s . 
 
 These angles have to be sorted without reference to the heights, or just 
 as though all the heights were unity. Every entry is to be regarded as 
 unity. The following example shows part of the sorting of $V m + V s , and 
 1 denotes a H.W., f a L.W. ; by this device H. and L.W. may be sorted on 
 the same paper. 
 
 We may also, if it is found convenient, put on it the sorting of V m V s 
 by adopting, say 0, to denote a H.W. and * a L.W., each one of these four 
 signs denoting simply unity. 
 
206 
 
 REDUCTION OF BOMBAY TIDES. 
 
 
 
 ij 
 
 T3 
 
 o; 
 
 w 
 
 t>o 
 
 a 
 
 c 
 
 1 
 s 
 
 I, 
 
 tip 
 
1890] REDUCTION OF BOMBAY TIDES. 207 
 
 We then proceed to count these 1's and f's just as was done with the 
 number of entries in the sorting of ^V m , and to operate on them in the same 
 way. 
 
 The results are 
 
 
 
 S cos (i V m - F s ) = - -0078, S sin ft V m - F.) = - "0060 . 
 
 S*" cos (4 V m - F,) - + '0280, S*" sin (i V m - V g ) = 4- "0078 
 
 S cos(iF m + V.) = + "1244, S sin(iF m + F s ) = + '0094 
 
 S^ cos (i V m + V.} = + -0147, S*" sin ( F m + F s ) = + '0834 
 
 of F < 0) G (0 > f (f>) T (0) F ( i w) G <*"' f ( i' T) 0" ( i w) 
 ly i 7ft , vJ m , i wl , g m , J- m , \J m , i m , ^ m , 
 
 F (0) O (0) f (0) p. (0) ^ (JIT) Q (iT) f (J7T) (ijT) 
 
 - 1 - A- 5 v -"i> > 1 s ' &s > * a ' A' > s ' os ' 
 
 c*os 
 
 These 16 coefficients are required to correct the four sums S% . ^V m , 
 
 sin 
 
 SHi. ^F m , for the influence of the tides M 2 and S 2 . 
 
 olil 
 
 -I call S% cos ^F m + corrn., W, S/t sin |F wi , + corrn., X, and the other two 
 Y and Z. 
 
 The correction to be applied to S% cos | V m to get W is 
 TF <>A -i-G < 0) B -1- F < 0) A 4- G (0) B 1 
 
 L- 1 - m -"-m T *J W (, JJ m ' - 1 - s *! ' ** AJ((J 
 
 and the correction to be applied to S% sin |- F m to get X is 
 
 _ f'f (o)A , p- <)E 4-f(>A 4- P- <>B 1 
 L 1 *^ *** T gm JJm i J s -^s T^ &s -"sj 
 
 and the two other corrections are given by symmetrical formulae with (|TT) 
 in place of (o). 
 
 COS 
 
 These coefficients are computed from S and S^ of . (^ F m + F m ) and 
 
 sin 
 
 cos 
 
 of (y F m J^g), as given in (m) (n) (o). It must be especially noticed that 
 
 sin 
 
 we have above in (in) computed S and S i7r of sin -|-F WI ; but -F m F m = |-F m , 
 so that the signs of our previous results must be changed in these two cases. 
 
 If we remark that k m and k s are constants found by theoretical con- 
 siderations, that A m , B m , A s , B s , are already found, and that in the first 
 column we are compelled to omit the affixes to the letters S, k, and the F's 
 and G's, because they indicate various sorts of S's and k's and F's and G's 
 in the different columns, the computations in the following table are easily 
 followed : 
 
208 
 
 REDUCTION OF BOMBAY TIDES. 
 
 
 QO 
 
 S 
 
 s** 
 
 S 57r 
 
 
 4 = 2 
 
 4 = 2-07 
 
 4n = 2 
 
 4 = 2-07 
 
 
 F F 
 
 r p ' m 
 
 v p = r. 
 
 F F 
 
 ' p ' m 
 
 F F 
 
 ' P~ 'a 
 
 iScos(iF, (l -F p ) 
 
 - -0261 
 
 - -0039 
 
 -0084 
 
 + 0140 
 
 |Scos(|F m + F p ) 
 
 + 0174 
 
 + 0622 
 
 + -0240 
 
 + 0074 
 
 Sum 2 
 
 -0087 
 
 + '0583 
 
 - '0324 
 
 + 0214 
 
 Diff. A 
 
 - -0435 
 
 - -0661 
 
 + -0156 
 
 + 0066 
 
 fa 
 
 -0174 
 
 + 1207 
 
 - -0648 
 
 + '0443 
 
 A 
 
 - -0870 
 
 -1368 
 
 + 0312 
 
 + 0137 
 
 2 + A=F 
 
 - -0957 
 
 - '0785 
 
 -0012 
 
 + '0351 
 
 A + 2=g 
 
 - '0609 
 
 + '0546 
 
 - "0492 
 
 + 0509 
 
 *Ssin(*r m - F p ) 
 
 - '0059 
 
 -0030 
 
 + -0065 
 
 + -0039 
 
 |Ssin(|F m +F p ) 
 
 - -0915 
 
 + 0047 
 
 -0087 
 
 + 0417 
 
 Sum a 
 
 - -0974 
 
 + 0017 
 
 -0022 
 
 + '0456 
 
 Difl 1 . 8 
 
 + -0856 
 
 - '0077 
 
 + '0152 
 
 - -0378 
 
 k(T 
 
 -1948 
 
 + -0035 
 
 - '0044 
 
 + -0944 
 
 t 
 
 + '1712 
 
 -0159 
 
 + -0304 
 
 - '0782 
 
 <r+*d*f 
 
 + -0738 
 
 -0142 
 
 + '0282 
 
 - '0326 
 
 
 + '1092 
 
 + '0042 
 
 - '0108 
 
 - '0566 
 
 oefficients 
 
 
 B OT 
 
 -A. 
 
 -B 8 
 
 V m ) - 10-50 (F m ) 
 
 - '0957 
 
 (G m ) +-1092 
 
 (F g ) --0785 
 
 (G 8 ) +-0042 = 
 
 V m ) + 8-04 (f m ) 
 
 + -0738 
 
 (g)) - '0609 
 
 (f,) - '0142 
 
 (g.) +'0546 = 
 
 (~/i sin 4 K m ) + 8-04 (t m ) +-0738 ( gm ) v _ o/ .___ w/ . _ 
 
 (S iir AcosF m )+ 0-40 (F m i>r )--0012 (G,,, iir ) - '0108 (*".**) + -0351 (G.**) - '0326 = Y 
 
 (S iir Asii4F m )+ 3-74 (f m *) +'0282 (g **) - -0492 (f,**) - -0566 (g.**) +'0509 = Z 
 
 Multiply by -A m = +30-58, -B m =- 38-47, -A g =-21'08, -B g =-3'53 
 
 -10-50 -2-91 -4-23 +1-66 -O'Ol =W 
 
 + 8-04 +2-24 +2-36 +0'30 -0'19 
 
 + 0-40 -0-04 +0-42 -0-74 +0'12 =Y 
 
 + 3-74 +0-86 +1-91 +T20 -0'18 =Z 
 
 W 
 
 X 
 
 Y 
 
 Z 
 
 
 10-50 
 
 8-04 
 
 0-40 
 
 3-74 
 
 
 2-91 
 
 2-24 
 
 0-04 
 
 0-86 
 
 
 4-23 
 
 2-36 
 
 042 
 
 1-91 
 
 
 1-66 
 
 0-30 
 
 074 
 
 1-20 
 
 
 o-oi 
 
 0'19 
 
 0-12 
 
 0'18 
 
 
 17-65 
 
 12-94 
 
 0-94 0-78 
 
 7-71 
 
 
 1-66 
 
 0-19 
 
 0-78 
 
 0-18 
 
 
 - 15-99 
 
 + 12-75 
 
 + 0-16 
 
 + 7-53 
 
 
1890] REDUCTION OF BOMBAY TIDES. 209 
 
 W = - 15-99 X = + 1275 
 
 Z = + 7-53 Y = + 016 
 
 W + Z = - 8-46 X + Y = -H 12-91 
 
 W-Z = - 23-52 X-Y = + 12-59 
 
 = - 4-23 (X + Y) = + 6-46 
 
 Y = + 6-30 
 
 (q.) Computation of Astronomical and other Constants. 
 
 Find s , the moon's mean longitude (see Nautical Almanac), and h the 
 sun's mean longitude (sidereal time reduced to angle) from the Nautical 
 Almanac, and p f> the longitude of moon's perigee, from Baird's Manual*, 
 Appendix Table XII (there called TT), at the epoch O h , January 1, 1887, 
 Bombay mean time, in E. Longitude 4 h '855. 
 
 From Baird, Tables XIV, XV, XVIII, find N the longitude of Moon's 
 node, and /, v, % at mid-period, February 14, 1887 f [or see the Nautical 
 Almanac and the formulae at end of Paper 1]. 
 
 With the value of / find f m from XIX (1) for the tides M 2 , N, L; from 
 XIX (3) find f for the tide O ; from XIX (8) find f ' for the tide K x ; from 
 XIX (9) find f" for the tide K 2 ; from XX find v for the tide K x ; and from 
 XXI find 2i/' for the tide K 2 ; [or use the formulae at the end of Paper 1]. 
 The results are 
 
 s = 359-43, h = 280'63, p = 165'36 
 
 v = 9'60, = 9-00 
 
 l/f m = 0-9709, l/f = 1-161, f' = 0-915, f " = 0-802 
 
 v = 6-30, 2i/' = ll-75 
 
 Then compute initial equilibrium arguments, in the symbol for which 
 the subscript letters indicate the tides referred to, 
 
 Um = 2(h - v )-2(s -%), Uo = (h - v )-2(s -^ + ^, u s = 
 = 201-20, = 20-17 
 
 for K li u' = h -v t - far, for K 2 , u" = 2A - W 
 
 = 184-33, = 189'51 
 
 U n = U m - (S - p ), U { = U m + (S -po) + 7T 
 
 = 7-13, = 215 27 
 
 u p = -h + ^7r = 169'37 
 We have already shown in B the way of computing II, and II = 1'034J. 
 
 * Manual for Tidal Observations, by Major Baird. Taylor and Francis, Fleet Street, 1886. 
 
 f In making these reductions I have really used the value of N for July 1, 1887, because 
 I am operating on tidal predictions made for the whole year 1887, which were doubtless made 
 with mean N for that year. The difference is almost insensible. 
 
 J As the Indian tide-predicting instrument takes no account of solar parallax, I should in 
 reality have done better to take II as unity. But of course this consideration does not apply to 
 real observations. 
 
 D. I. 14 
 
210 REDUCTION OF BOMBAY TIDES. [5 
 
 In C and D we have shown how to compute j, i, I, and j = + 6'46, 
 t'= + 6-52, = -0'97. 
 
 By the formula in B, with ot n = 86'97 for 6 semi-lunations, 
 tw = 2h - 2v" + a n = u" + n 
 = 276'48 
 = - 83-52 
 
 By the formula in D, with @ n = 87'52 for XIII quarter-lunar periods, 
 
 = 2h -v' + l + 
 
 = 281-51 = - 78-49 
 By the formula in B, viz. : 
 
 U cos = II + X ?l f " cos w 
 U sin </> = \nf " sin &> 
 With log \ n 9'2517 for 6 semi-lunations, and with the above values of 
 
 n,f",: 
 
 </> = - 7-72, (+) log U = 0-0251 
 
 By the formula in D, viz. : 
 
 T cos \|r = f ' p n cos 6 
 T sin T/T = p n sin 
 
 with logp n = 9'4618 for XIII quarter-lunar periods, and with the above 
 values of f and 6: 
 
 ^ = - 18-32, (+) log T = 9-9557 
 
 (r.) Final Evaluation of M 3 . 
 From(j) B m = + 38-47, A m = -30'58, tari*f* 
 
 **m 
 
 B m is + and A m is , so that m lies in second quadrant ; whence 
 
 m = TT - 51-51 = 128'49 
 
 Then H m = ^ . B m cosec f m 
 
 ^m 
 
 whence, on reducing from inches to feet, 
 
 H m = 3-98ft. 
 
 Also K m = ? m + u m = 128-49 + 201'20 = 329'69 
 
 where the value of u m is taken from (q). 
 
1890] REDUCTION OF BOMBAY TIDES. 211 
 
 (s.) Final Evaluation of N and L. 
 Taking the values of P, Q, R, S from (i), 
 
 f m H n sin ( B - f) = - P = - 10-72, f m H ? sin ( + j) = + Q = + 0'66 
 
 f m H n cos( n -j) = ~S = + 7-16, f m Hj cos (^+j) = -R = + 1-22 
 
 n - j lies in 4 th quad., & + j lies in 1 st quad. 
 
 whence n -j = -56 0< 27 
 
 Then H B = ^ cosec (f n - j) x (- P) 
 
 ITO 
 
 whence, on reducing from inches to feet, 
 
 H w = 1-04 ft. 
 
 Again, since from (q) j = + 6'54, we have & = - 49'73 = 310'27, and 
 *=? + w n = 310-27 + 7 c -13 = 317-40, where the value of u n is taken 
 from (q). 
 
 Turning to the second pair of equations, 
 
 Si + j - 28-4 
 
 whence, on reducing from inches to feet, 
 
 H; = Oil ft. 
 
 Again, since j = + 6'5, we have & = 21'9, and 
 
 KI=& + UI = 21-9 + 215-3 = 237'2 
 where the value of u t is taken from (q). 
 
 (t.) Final Evaluation of S 2 and K 2 . 
 From(k) B s = + 3-62, A s = + 21*08; twif, 
 
 B s and A s are +, so that s lies in 1 st quadrant ; 
 whence & = 9'7l 
 
 Then H = AgSC ^ 
 
 whence, with log U already found in (q) as 0'0251, and, reducing inches to 
 feet, 
 
 H s = 1-68 ft. 
 
 Again KS = % s + = 9'7l - 7'72 = l-99 
 
 where the value of < is taken from (q). 
 
 Lastly, H" = 0'272H S = 0'46 ft., and *" = *, = 2 
 
 The factor - 272 is an absolute constant. 
 
 142 
 
212 REDUCTION OF BOMBAY TIDES. [5 
 
 (u.) Final Evaluation of K 1} O, P. 
 Taking the values of ( W - Z), (X + Y) from (p), 
 
 TH'sin(f' + ;-t) = i(W-Z):=- 11-76 
 TH' cos (f + i-^r) = -$(X + Y) = - 6-47 
 f ' + i i|r lies in third quadrant, and 
 
 ' + i-^ = 7r + 61'2 = 241'2 
 
 Then since, from (q), i/r = - 18'32, we have f + i = 222'9 ; and since from (q) 
 i = 6-52, therefore ' = 216'4; whence 
 
 K ' = % + u' = 216'4 + 184-3 = 40-7 
 where the value of u is taken from (q). 
 
 Then H' = ? ( ^~ Z) cosec (? + i - ^) 
 
 whence, with log T already found in (q) as 9'9557, and reducing from inches 
 to feet, 
 
 H' = l'24ft. 
 
 Also H P = 0-331H' = 0-41, and J , = * / =41 
 
 The factor 0*331 is an absolute constant. 
 We now have to compute 
 
 L = $ (X + Y) tan |e + fH' cos (' + i) tan ^e 
 M = | ( W - Z) tan e - f'H' sin (f + i) tan Je 
 
 where logtan^e = 9'0677, an absolute constant for all times and places. 
 With the values of f and (X + Y) and ( W - Z) given above in (q) and (p), 
 and with the values of H' and ' + i just found, there results 
 
 L = -0-410 M = --281 
 
 Now f H sin(r o -t) = 
 
 We have found in (p) 
 
 so that f H sin (, - i) = + 3'82 f H cos (? - i) = + 6'02 
 
 Whence i lies in the first quadrant, and 
 
 o _; = 32-4 ! 
 
 Then H = i [i (X - Y) + M] sec (, - 1) 
 
 ^o 
 
 whence, reducing from inches to feet, 
 
 H, = 0-69 ft. 
 
1890] REDUCTION OF BOMBAY TIDES. 213 
 
 Again, 
 
 KO = & + u = (, - *) + i + u = 32-40 + 6'52 + 20'17 = 59'09 
 where the value of u is taken from (q). 
 
 (v.) Final Reduction of Mean Water Mark. 
 
 We subtracted 99 inches from all the heights before using them, and the 
 mean of the heights was then + 3*51 inches. Hence mean water is 102*51 
 inches, or 8'54 feet above the datum of the original tidal observations. 
 
 (w.) Results of Reduction. 
 
 Error of present calc. 
 in inches and 
 
 
 Mean of 9 yrs. obs. 
 
 minutes 
 
 Mean water, 8'54 ft. 
 
 8-223 
 
 4 in. 
 
 . (H = 3-98 ft. 
 H-330- 
 
 4-043 
 330 
 
 f in. too small 
 nil 
 
 fH = 1-68 ft. 
 
 82 V = 2 
 
 1-625 
 3 
 
 | in. too large 
 2 m too slow 
 
 (H = 0-46 ft. 
 
 IK =2 
 
 0-405 
 352 
 
 | in. too large 
 20 m too fast 
 
 JH- 1-04 ft. 
 
 U = 317 
 
 0-997 
 313 
 
 ^ in. too large 
 8 m too fast 
 
 jH = Oil ft. 
 {* =237 
 
 0-088 
 308 
 
 \ in. too large 
 2 h 21 m too slow 
 
 (H = 1-24 ft. 
 1 \K =41 
 
 1-396 
 
 45 
 
 If in. too small 
 16 m too slow 
 
 Q (H = 0-69 ft. 
 (K =59 
 
 0-658 
 
 48 
 
 ^ in. too large 
 44 m too fast 
 
 (H = 0-41 ft. 
 1 =41 
 
 0-404 
 
 43 
 
 j 1 ^ in. too large 
 8 m too slow 
 
 The second column is given because, if the calculation had been con- 
 ducted by rigorous methods instead of approximately, my results should have 
 agreed very nearly* with these. The causes of several of the discrepancies 
 are explicable. The error of mean water mark is due to the necessity for 
 neglecting the annual and semi-annual tides in a short series of observations. 
 The error in phase in K 2 is a necessary incident of the shortness of the series 
 of observations. The tide L is only about an inch in height, and accuracy 
 of result could not be expected. 
 
 * I do not know the exact values of the constants used in the Bombay Tide Table, which has 
 been used as representing observation. 
 
214 REDUCTION OF BOMBAY TIDES. [5 
 
 The magnitude of the error in time in the diurnal tides is rather dis- 
 appointing, but it is clear that the length of observation has not been 
 sufficient to disentangle the O tide from the Kj tide. It may be remarked 
 also that an error of 1 in phase makes twice as much difference in time with 
 the diurnal tides as with the semi-diurnal. 
 
 Lastly it is probable that all these errors would have been sensibly 
 diminished if I had subtracted 103 inches from the heights all through 
 instead of 99, and I know that this is to some extent the case. 
 
 (x.) Verification. 
 
 In a calculation of this kind some gross error of principle may have been 
 committed, such, for example, as imputing to some of the K'S a wrong sign ; 
 and this is the kind of mistake which is easily overlooked in a mere verification 
 of arithmetical processes. It is well, therefore, to test whether the tide 
 heights and times are actually given by the computed constants. This is 
 conveniently done by selecting some three or four tides from amongst those 
 from which the reductions have been made, and it makes the calculation 
 much shorter if we pick out cases in which it is H. or L.W. within a few 
 minutes of noon. 
 
 For example, in the present case it was L.W. on February 16 (day 46) 
 at O h 7 m P.M., and the height was 4 ft. in. 
 
 Now, if U denotes the value of any equilibrium argument whose value 
 at the epoch, O h , January 1, was denoted in (q) by u, and if A denotes the 
 height of mean sea-level above datum, the expression for the height of 
 water is 
 
 h = A + f m R m cos ( U m - * m ) + R g cos ( U. - *.) + f'H" cos ( U" - *") 
 + f w H n cos ( Z7 - Kn ) + f w H, cos ( U t - Kl ) + f'H' cos ( U' - *') 
 + f H cos (U - K O ) + H p cos (Up Kp) 
 
 The time of H.W. depends on a formula involving the sines of the same 
 angles in place of cosines. 
 
 Since we have chosen cases where it is H. or L.W. at noon, the C7's exceed 
 the us by an exact number of days' motion. 
 
 The evaluation of the separate terms may be conveniently made by means 
 of an ordinary nautical traverse table, where (neglecting the decimal point) 
 f H is represented by the " Distance," and f H cos ( U K) is given by 
 " Latitude," and f H sin (U K) by " Departure." 
 
 If we know the time of H. or L.W. within 20 m or so, the following 
 calculation will give the true time and height. In this case we know that 
 there should be a L.W. at about O h of day 46. The increments of argument 
 
1890] 
 
 VERIFICATION. 
 
 215 
 
 are computed from the Table G, and the K'S are subtracted either by actual 
 subtraction or by addition of 2?r K. 
 
 M 2 
 
 Increment in 4O 1 464'7 
 Ditto 6 d - 146-3 
 
 K 2 N L 
 
 78-9 - 57'9 -452-7 
 11-8 -224-7 - 67-9 
 
 39-4 425-3 
 5-9 - 152-2 (see 
 
 Ditto 46 d 
 u 
 
 U= 
 
 U K = 
 U K = 
 
 318-4 
 201-2 
 
 90-7 
 189-5 
 
 - 282-6 
 7'1 
 
 - 520-6 
 215-3 
 
 45-3 
 196-9 
 
 273-1 
 3-4 
 
 - 45-3 
 169-4 
 
 519-6 
 - 329-7 
 
 280-2 
 -2-0 - 2-0 
 
 -275-5 
 + 42-6 
 
 -305-3 
 + 122-8 
 
 242-2 
 - 40-7 
 
 276-5 
 - 59-1 
 
 124-1 
 - 40-7 
 
 189-9 
 
 7T + 10 
 
 - 2-0 278-2 
 -2 - 82 
 
 -232-9 
 
 77-53 
 
 -182-5 
 
 7T-2 
 
 201-5 
 
 7T + 22 
 
 217-4 
 
 7T+37 
 
 83-4 
 83 
 
 II 
 
 fH 
 
 V- 
 
 fHsin(Ef-/c) 
 
 Is 
 
 
 'S 
 
 'a 
 
 <D 
 
 cc 
 
 ? 
 B 
 
 s 
 
 fM 2 
 
 S 2 
 
 IK, 
 
 N 
 
 U 
 
 ( KI 
 
 
 IP 
 
 1-03 
 1-0 
 0-80 
 1-03 
 1-03 
 
 0-915 
 0-86 
 1-00 
 
 3-98 
 1-68 
 0-44 
 1-04 
 O'll 
 
 1-24 
 
 0-69 
 0-41 
 
 4-10 
 1-68 
 0-37 
 1-07 
 0-11 
 
 1-14 
 0-59 
 0-41 
 
 -2 
 
 -82 
 
 7T-53 
 
 7T-2 
 
 1-68 
 0-05 
 
 4-04 
 
 0-64 
 0-11 
 
 0-86 
 
 o-oo 
 
 0-71 
 0-06 
 0-37 
 
 + 1-73 
 A 2 = 
 
 -4-79 
 + 1-73 
 
 + 0-86 
 
 -1-14 
 
 + 0-86 
 
 -3-06 
 
 B 2 = 
 
 -0-28 
 
 7T + 22 
 7T + 37 
 
 83 
 
 0-05 
 
 1-06 
 0-47 
 
 0-41 
 
 0-43 
 0-36 
 
 + 0-05 
 
 -1-53 
 + 0-05 
 
 + 0-41 
 
 -0-79 
 + 0-41 
 
 ! =-1-48 
 
 B 1= -0-38 
 
 Time = - 120 m 
 
 = - 120 m x = - 16 m = 1 l h 44 m A.M. 
 
 Tabular time=0 h 7 m P.M. 
 Error = -23 m 
 
 A 2 +A t =-4-54 
 Mean water +8-54 
 
 Height= 4-00 
 
 Height^L.W 4ft. in. 
 
 Tabular height 4 
 
 Error .. 
 
 This result is as good as might be expected, and, considered as a prediction, 
 would be amply sufficient for navigational purposes. 
 
6. 
 
 ON AN APPARATUS FOR FACILITATING THE REDUCTION 
 OF TIDAL OBSERVATIONS. 
 
 [Proceedings of the Royal Society, LII. (1892), pp. 345389.] 
 
 1. Introduction. 
 
 THE tidal oscillation of the ocean may be represented as the sum of a 
 number of simple harmonic waves which go through their periods approxi- 
 mately once, twice, thrice, four times in a mean solar day. But these simple 
 harmonic waves may be regarded as being rigorously diurnal, semi-diurnal, 
 ter-diurnal, and so forth, if the length of the day referred to be adapted to 
 suit the particular wave under consideration. The idea of a series of special 
 scales of time is thus introduced, each time-scale being appropriate to a 
 special tide. For example, the mean interval between successive culminations 
 of the moon is 24 11 50 m , and this interval may be described as the mean lunar 
 day. Now there is a series of tides, bearing the initials M,, M 2 , M 3 , M 4 , &c., 
 which go through their periods rigorously once, twice, thrice, four times, &c., 
 in a mean lunar day. The solar tides, S, proceed according to mean solar 
 time ; but, besides mean lunar and mean solar times, there are special time 
 scales appropriate to the larger (N) and smaller (L) lunar elliptic tides, to 
 the evectional (y), to the diurnal (Kj) and semi-diurnal (K^) luni-solar tides, 
 to the lunar diurnal (O), &c. 
 
 The process of reduction consists of the determination of the mean height 
 of the water at each of 24 special hours, and subsequent harmonic analysis. 
 The means are taken over such periods of time that the influence of all the 
 tides governed by other special times is eliminated. 
 
 The process by which the special hourly heights have hitherto been 
 obtained is the entry of the heights observed at the mean solar hours in a 
 schedule so arranged that each entry falls into a column appropriate to the 
 nearest special hour. Schedules of this kind were prepared by Mr Roberts 
 
1892] METHOD OF SORTING THE OBSERVATIONS. 217 
 
 for the Indian Government*. The successive rearrangements for each sort 
 of special time were made by recopying the whole of the observations time 
 after time into a series of appropriate schedules. The mere clerical labour 
 of this work is enormous, and great care is required to avoid mistakes. 
 
 All this copying might be avoided if the observed heights were written 
 on movable pieces. But a year of observation gives 8,760 hourly heights, and 
 the orderly sorting and re-sorting of nearly 9,000 pieces of paper or tablets 
 might prove more laborious and more treacherous than recopying the figures. 
 
 It occurred to me, however, that the marshalling of movable pieces might 
 be reduced to manageable limits if all the 24 observations pertaining to a 
 single mean solar day were moved together, for the movable pieces would be at 
 once reduced to 365, and each piece might be of a size convenient to handle. 
 
 The realisation of this plan affords the subject of this paper, and it will 
 appear that not only is all desirable accuracy attainable, but that the other 
 requisite of such a scheme is satisfied, namely, that the whole computing 
 apparatus shall serve any number of times and for any number of places. 
 
 The first idea which naturally occurred was to have narrow sliding tablets 
 which should be thrown into their places by a number of templates. It is 
 unnecessary to recount all my trials and failures, but it will suffice to say 
 that the slides and templates require the precision of a mathematical 
 instrument if they are to work satisfactorily, and that the manufacture would 
 be so expensive as to make the price of the instrument prohibitive. 
 
 The idea of making the tablets or strips to slide into their places was 
 then abandoned, and the strips are now made with short pins on their under 
 sides, so that they can be stuck on to a drawing board in any desired position. 
 The templates, which were also troublesome to make, are replaced by large 
 sheets of paper with numbered marks on them to show how the strips are to 
 be set. The guide sheet is laid on a drawing board, and the pins on the 
 strips pierce the paper and fix them in their proper positions. 
 
 The shifting of the strips from one arrangement to the next is certainly 
 slower than when they slid into their places automatically, but I find that 
 even without practice it only takes about 7 or 8 minutes to shift 74 of them 
 from any one arrangement to a new one, 
 
 * An edition of these computation forms was reprinted by aid of a grant from the Royal 
 Ooctotv. and is sold by the Cambridge Scientific Instrument Company, hot only about a dona 
 copies now remain. In the course of the preparation of the "guide sheets*' of die method 
 proposed in this paper, I found that there are many small migtahm in these Indian forms, bat 
 they are fortunately not of such a kind as to produce a sensible vitiation of results. I learn that 
 the mistakes arose from a misunderstanding on the part of a computer employed to draw up 
 the forms. [The whole of this edition of computation forms has, now in 1907, bean sold.] 
 
 The accuracy of my guide sheets was controlled by aid of Mr Roberta's forms, and it was the 
 occasional discrepancy between my results and the forms which led to the detection of the errors 
 lYtt'rrtM 50. 
 
218 DESCRIPTION OF THE APPARATUS. [6 
 
 The strip belonging to each mean solar day is divided by black lines into 
 24 equal spaces, intended for the entry of the hourly heights of water. The 
 strip is 9 in. long by 1 in. wide and the divisions (f by 1) are of convenient 
 size for the entries. There was much difficulty in discovering a good material, 
 but after various trials artificial ivory, or xylonite, was found to serve the 
 purpose. Xylonite is white, will take writing with Indian ink or pencil, and 
 can easily be cleaned with a damp cloth. It is just as easy to write with 
 liquid Indian ink as with ordinary ink, which must not be used, because it 
 stains the surface. 
 
 The strips have a great tendency to warp, and I have two methods of 
 overcoming this. A veneer of xylonite on hard wood serves well, or solid 
 xylonite may be stiffened by sheet brass let into a slot on the under side. 
 In the first plan the pins are fixed in the wood, and in the second the brass 
 is filed to a spike at each end*. Whichever plan is adopted, the strips are 
 expensive, costing about 7 for a set, and I do not at present see any way of 
 making them cheaper. 
 
 The observations are to be treated in groups of two and a half lunations 
 or 74 days. A set of strips, therefore, consists of 74, numbered from to 73 
 in small figures on their flat ends. 
 
 If a set be pinned horizontally on a drawing board in vertical column, 
 we have a form consisting of rows for each mean solar day and columns for 
 each hour. The observed heights of the water are then written on the strips. 
 
 When the 24 columns are summed and divided by the number of entries 
 we obtain the mean solar hourly mean heights. The harmonic analysis of 
 these means gives the mean solar tides. But for evaluating the other tides 
 the strips must be rearranged, and to this point we turn our attention. 
 
 Let us consider a special case, that of mean lunar time. A mean lunar 
 hour is about l h 2 m m.s. time ; hence the 12 h of each m.s. day must lie within 
 31 m m.s. time of a mean lunar hour. The following sample gives the incidence 
 to the nearest lunar hour of the first few days in a year : 
 
 Mean solar Mean lunar 
 
 time time 
 
 O d 12 h O d 12 h 
 
 1 12 1 11 
 
 2 12 2 10 
 
 3 12 39 
 
 4 12 48 
 
 5 12 58 
 
 6 12 67 
 
 7 12 76 
 
 &c. &c. 
 
 * [The first of these two plans was found to be the more convenient one.] 
 
G. H. Darwin's Scientific Papers. 
 
 M 
 
 222 to 222 + 73 
 or 295 
 
 M 
 
 222 to 222 t 73 
 or 295 
 
 Abacus for the harmonic analysis of tidal observations. 
 
1892] DESCRIPTION OF THE APPARATUS. 219 
 
 The successive 12 h of mean solar time will march retrogressively through 
 all the 24 hours of mean lunar time. 
 
 Now, if starting from strip 0, we push strip 1 one division to the left, 
 strip 2 two divisions to the left, and so on, the entries on the strips will be 
 arranged in columns of approximately lunar time. 
 
 The rule for this arrangement is given by marks on a sheet of paper 
 18 in. broad; these marks consist of parallel numbered steps or zigzags 
 showing where the ends of each strip are to be placed so as to bring the 
 hourly values into their proper places. 
 
 At the end of a lunation mean solar time has gained a whole day over 
 mean lunar time and the 12 h solar again agrees with the 12 h lunar. On the 
 guide sheet we see that the zigzag which takes its origin at the left end of 
 strip has descended diagonally from right to left until it has reached the 
 left margin of the paper, and a new zigzag then begins on the right margin. 
 
 When the strips are pinned out following the zigzags on the sheet marked 
 M, the entries are arranged in 48 columns, but the number of entries in each 
 column is different. The 48 columns are to be regarded as appertaining to 
 O h , l h , ..., 22 h , 23 h , O h , l h , ..., 22 h , 23 h . Thus, the number of entries in the 
 left-hand column of any hour added to the number of entries in the right- 
 hand column of the same hours is, in each case, 74. The 48 incomplete 
 columns may, in fact, be regarded as 24 complete ones. 
 
 The 24 complete columns are then summed; the 24 sums would, if 
 divided by the total number of strips, give the 24 mean lunar hourly heights. 
 The harmonic analysis of these sums gives certain constants, which, when 
 divided by the number of strips, are the required tidal constants. It must 
 be remarked, however, that, as the incidence of the entries is not exact in 
 lunar time, investigation must be made of the corrections arising out of this 
 inexactness. 
 
 The explanation of the guide sheet for lunar time will serve, mutatis 
 mutandis, for all the others. 
 
 The zigzags have to be placed so as to bring the columns into exact 
 alignment, and printers' types provide all the accuracy requisite. Accordingly, 
 the computing strips are made to suit a chosen type. The standard length 
 for one of the 24 divisions on the strips was chosen as that of a "2-em English 
 quadrate"; 24 of these come to 9 inches, which is the length of a strip. 
 I found the English quadrate a little too narrow, and accordingly between 
 each line of quadrates there is a "blind rule," of 42 to the inch. The depth 
 of the guide sheet is that of 74 quadrates and 74 rules, making 15 in. The 
 computing strips are 1 in. broad, and 74 of them occupy 14| in. The excess 
 of 15 above 14f , or -?-$ in., is necessary to permit the easy arrangement of 
 the strips. 
 
220 DESCRIPTION OF THE APPARATUS. [6 
 
 [The plate opposite p. 219 * shows portion of one of the guide-sheets 
 (that applicable to 222 d to 295 d ) for M or mean-lunar time. Thirty-seven of 
 the strips are shown in position, and the lower part of the sheet is still bare.] 
 
 To guard against the risk of the computer accidentally using the wrong 
 sheet, the sheets are printed on coloured paper, the sequence of colours being 
 that of the rainbow. The sheets for days to 73 are all red ; those for days 
 74 to 74 + 73, or 147, are all yellow; those for days 148 to 148 + 73, or 221, 
 are green ; those for days 222 to 222 + 73, or 295, are blue ; and those for 
 days 296 to 296 + 73, or 369, are violet. 
 
 Thus, when the observations for the first 74 days of the year are written 
 on the strips all the sheets will be red ; the strips will then be cleaned, and 
 the observations for the second 74 days written in, when all the guide sheets 
 will be yellow, and so on. 
 
 I must now refer to another considerable abridgment of the process of har- 
 monic analysis. It is independent of the method of arrangement just sketched. 
 
 In the Indian computation forms the mean solar hourly heights have 
 been found for the whole year, and the observations have been rearranged 
 for the evaluation of certain other tides governed by a time scale which 
 differs but little from the mean solar scale. I now propose to break the 
 mean solar heights into sets of 30 days, and to analyse them, and next to 
 harmonically analyse the 12 sets of harmonic constituents for annual and 
 semi-annual inequalities. By this plan the harmonic constants for 1 1 different 
 tides are obtained by one set of additions. In fact, we now get the annual, 
 semi-annual, and solar elliptic tides, which formerly demanded much trouble- 
 some extra computation. A great saving is secured by this alone, and the 
 results are in close agreement with those derived from the old method. 
 
 The guide sheets marked S and the computation forms are arranged so 
 that the observations are broken up into the proper groups of 30 days, and 
 they show the computer how to make the subsequent calculations. 
 
 I have also devised an abridged method of evaluating the tides of long 
 period MSf, Mf, Mm. The method is less accurate than that followed hitherto, 
 but it appears to give fairly good results, and reduces the work to very small 
 dimensions. 
 
 Before entering on the details of my plan it is proper to mention that 
 Dr Borgen has devised and used a method for attaining the same end. He 
 has prepared sheets of tracing paper with diagonal lines on them, so arranged 
 that when any sheet is laid on the copy of the observations written in daily 
 rows and hourly columns, the numbers to be summed are found written between 
 a pair of lines. This plan is excellent, but I fear that the difficulty of adding 
 correctly in diagonal lines is considerable, and the comparative faintness of 
 figures seen through tracing paper may be fatiguing to the eyes. Dr Bb'rgen's 
 
 * [From my Tides and Kindred Phenomena in the Solar System.] 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 221 
 
 plan* is simple and inexpensive, and had I not thought that the plan now 
 proposed has considerable advantages I should not have brought it forward. 
 
 In the investigations which follow the notation of the Report of 1883 
 to the British Association on Harmonic Analysis [Paper 1, above] is used 
 without further explanation. 
 
 2. Evaluation of A , Sa, Ssa, S 15 S 2 , S 4 , S 6 , T, R, K 2 , K 1} P. 
 
 The 24 mean solar hourly heights of water are entered in a schedule of 
 24 columns, with one row for each day, extending to n days ; the 24 columns 
 are summed, and the sums divided by n; the 24 means are harmonically 
 analysed ; it is required to find from the results the values of the harmonic 
 constituents. 
 
 The speed of any one of the tides differs from a multiple of 15 per hour 
 by a small angle ; thus, any one of the tides is expressible in the form 
 H cos [(15^ /3) t ], where q is 0, 1, 2, 3, &c., and ft is small. 
 
 When t lies between O h and 24 h this formula expresses the oscillation of 
 level due to this tide on the day of the series of days. 
 
 If multiples of 24 h from 1 to n 1 be added to t, the expression gives the 
 height at the same hour, t, of mean solar time on each of the succession of 
 days. 
 
 Then if !) denotes the mean height of water, as due to this tide alone, at 
 the hour t, we have 
 
 (1) 
 
 When t is put successively equal to O h , l h , ..., 23 h we get the 24 values of f) 
 which are to be submitted to harmonic analysis. 
 
 The mean value of f), say A (not to be confused with A as written at the 
 head of this section, where is denoted the mean sea level above datum) is 
 found by taking the mean of the 24 values of 1). 
 
 By the formula for the summation of a series of cosines it is easy to 
 prove that 
 
 We will now find the p th harmonic constituents A p> B p . By the ordinary 
 rules 
 
 A 1 23 
 
 sin 
 * Annalen der Hydrographie und Maritimen Meteorologie. June, July, Aug., 1894. 
 
222 THE TIDES GOVERNED BY MEAN SOLAR TIME. [6 
 
 f*f\Q 
 
 Now 2 15 pt cos [(15g - ) $ - f - 12/8 (w - 1)] 
 
 sin 
 
 +/>) - 0} - - 120 (n - 1)] 
 
 
 
 - sin - 1)] 
 
 and -^ of the sum of the 24 values corresponding to = 0, 1, ..., 23 is 
 
 A ^gSl'^4^!}- C S [V 115 (7 + J.) - /8) - f- 12/3 ( - D] 
 
 4 sin [15 (q + p) ft] sm L 
 
 + the same with sign of p changed. 
 
 This expression admits of simplification, because 12 x 15 = 180 ; making 
 this simplification, and introducing the result into (3), we obtain 
 
 In the particular case where p = q, we have 
 
 **</ 1 - TT ' -ict /a 
 
 //sin 12n/8 
 
 sin (15 g- 
 
 ............ (5) 
 
 sn / 
 
 If the number of days w be large, A. p> B p will be small unless the de- 
 nominator of one of the two terms in (4) be very small. This last case can 
 only occur when p = q and when /3 is small. Hence, in the analysis of a term 
 of the form under consideration, we may neglect all the harmonics except 
 the 5 th one. Accordingly (2) and (5) are the only formula required. 
 
 A case, however, which there will be occasion to use hereafter is when 
 n = 30, q = 2, when (4) becomes 
 
 A ) {sin (r + 359 **> + sin ( ^ + 3 + 359 H- 
 
 "21 1 TT o/->rvO 1 h111 T BID f \UJ 
 
 For the present we have to apply (5) in the two cases q I, ft = 0> 0410686 
 and q = 2, ft = 0'0821372 ; now the ratios of cosec to cosec (15j - i/8) in 
 these two cases are 722 to 1 and 697 to 1. In both cases the first term of 
 (5) is negligible compared with the second. 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 223 
 
 ^ 24>n sin i/3 
 
 Nowwrite S = ~ To^ .............................. ( 7 ) 
 
 sin I2np 
 
 arid (5) becomes, with sufficient exactness, 
 
 If this be compared with (2), we see that when q = this formula also 
 comprises (2). 
 
 In the applications to be made ft is very small, so that _jp is approximately 
 a function of the form cosec 6. This function increases very rapidly when 
 6 passes 90, but for considerable values less than 90 it only slightly exceeds 
 unity ; for example, when 6 = 60, jp = 1'2, but when 6 = 180, jp = infinity. 
 
 It follows, therefore, that if the number n of days in the series is such 
 that 12^/3 is less than say 60, the magnitudes of A q , B q are but little 
 diminished by division by jp; but if 12n/3 is nearly 180, A q , B q become 
 vanishingly small. 
 
 If the typical tide here considered be the principal lunar tide M a , and if 
 the number of days be as nearly as possible an exact multiple of a semi- 
 lunation, 12n/3 is nearly 180, and the corresponding A 2 , B 2 become very 
 small. No number of whole days can be an exact multiple of a semi-lunation, 
 so that A 2 , B. 2 corresponding to M 2 cannot be made to vanish completely. For 
 the present they may be treated as negligible, and we return to this point in 
 the next section. 
 
 The above investigation shows that in the expression for the whole 
 oscillation of sea level upon which the proposed analysis is performed all 
 those tides may be omitted from which /3 is not very small, and also all those 
 whose frequencies are such that the period under consideration 12n/3 is nearly 
 180. 
 
 Since the period under consideration will be a lunation, it follows that, 
 as far as is now material, the general expression for sea level may be written 
 as follows, t denoting mean solar hour angle equal to 15 t : 
 
 m. w., annual ............... A + H sa cos (h K sa ) 
 
 semi-annual ...... + H ssa cos (2h tc ssa ) 
 
 Solar tides, S 1} S 2 ......... + H$ g cos (t K$ S ) + H s cos (2t Kg) 
 
 S 3 , S 4 ......... + H^g cos (4t K 2S ) + HM cos ( 6t /f^) 
 
 Solar elliptic, T ............ 4- H t cos (2t h + p t K t ) 
 
 R ............ + H r cos(2t + hp t + ir- K T ) 
 
 Luni-solar, K 2 ............ + f"ff" cos (2t + 2k - 2v" - ") 
 
 K! ............ + f'H' cos (t + h - v' - TT - *') 
 
 Solar diurnal, P ......... + H p cos (t h + ^TT K P ) ............... (9) 
 
224 THE TIDES GOVERNED BY MEAN SOLAR TIME. [6 
 
 This includes all the tides whose initials are written at the head of this 
 section. . 
 
 It is now necessary to break up the year into 12 equidistant lunations of 
 30 days. This can be done by the omission of 5 days in ordinary years, and 
 of 6 days in leap years. 
 
 If the days of the year are numbered to 364 (365 in leap year), the 
 twelve months are as follows : 
 
 0, O d to 29 d ; 1, 30 d to 59 d ; omit 60 d ; 2, 61 d to 90 d ; 3, 91 d to 120 d ; omit 
 121 d ; 4, 122 d to 151 d ; 5, 152 d to 181 d ; omit 182 d ; 6, 183 d to 212 d ; 
 7, 213 d to 242 d ; 8, 243 d to 272 d ; omit 273 d ; 9, 274 d to 303 d ; 10, 304 d 
 to 333 d ; omit 334 d ; 11, 335 d to 364 d ; in leap year omit 365 d . 
 
 The increments of sun's mean longitude from O d O h of month up to O h 
 of the day numbered of each group of days or month are as follows : 
 
 0,0; 1, 30-0-431; 2, 60'124; 3, 90-0'306; 4, 120'249; 5, 150-0-182; 
 6, 180'373; 7, 210-0-057 ; 8, 240-0'488 ; 9, 270'068 ; 
 10, 300-0-364; 11, 330'191. 
 
 Thus if h be the sun's mean longitude at O d O h of month 0, the sun's 
 mean longitude at O d O h of month r is h + 30 T, with sufficient approxi- 
 mation. 
 
 Now let V with appropriate suffix denote the initial " equilibrium argu- 
 ment " at O d O h of month 0, so that 
 
 then the general expression (9) for the tide in the month r becomes 
 A + H sa cos (ijt + V ga + 30T - K sa ) + H gsa cos (fyt + ~V ssa + 60r - Kssa ) 
 
 + H s cos (15 - /c. s ) + H s cos (30 - ,) + H u cos (60 - K U ) 
 
 + H 3S cos (90 - K 3S ) 
 + H t cos [(30 - 77) t + V t - 30r - K t ] 
 
 + H r COS [(30 + T;) t + V r + 30T - Kr\ 
 
 + f"H" cos [(30 + 277) t + V" + 60r - K "] 
 
 + i"H' cos [(15 + n)t + V' + 30r - '] 
 
 + H p coa[(l5 -ti)t + V p -30r-K p ] ................................. (10) 
 
 Each of these terms falls into the type cos [(I5q ft}t ^], and ft is in 
 every case either 77, 2rj, or 0. 
 
 Now, when harmonic analysis of the mean of 30 days is carried out, 
 coefficients are introduced. 
 
1892] 
 
 THE TIDES GOVERNED BY MEAN SOLAR TIME. 
 
 225 
 
 Write therefore 
 
 24 x 30 sin 77 
 
 24 x 30 sin 77 
 sin 72077 
 
 sin 36077 
 With the known value of 77, 
 
 log jpj = 0-00483, log jp 2 = 0-01945 
 
 In applying the method investigated above, it will be observed that a 
 term of any frequency 15q /3 only contributes to the harmonic constituent 
 of order q. 
 
 Then applying our general rule (8) term by term*, and observing that 
 359^77 = 14'76, and 719?? = 29'53, the result may be written as follows : 
 
 = A + cos ( Kga - V m - 30r - 14-76) 
 
 cos 
 
 - 60r - 29'53) 
 
 = Hi 
 
 cos 
 sin 
 
 + ^ cos ( K ' _ V' - 30r - 14-76) 
 ffi sm v 
 
 -lip COS , -*Y OA i 1 A-*7G\ 
 
 -;=r ( V w + 30 T + 14 7b) 
 
 cos H t cos , v 1/t o.*.N. 
 
 . r g + -fir . (K t - V t + 30 T + 14 -76) 
 
 sm j sin 
 
 cos 
 sm 
 
 " _ V" - 60r - 29'53) 
 
 cos 
 sin 
 
 COS 
 
 .(11) 
 
 With the meaning of the term month in the present context, the sun has 
 a mean motion of 30 per month, and each of the first five it's and iU's is a 
 function with a constant part and with annual and semi-annual inequalities. 
 
 When T has successively the 12 values 0, 1, ..., 11, we have 12 equidistant 
 values of the ^I's and ifl's. These may be harmonically analysed for annual 
 and semi-annual inequalities. 
 
 * [The formulae in the text are devised for the case when a whole year's observations are 
 under reduction. When a short period is being treated, such as a fortnight or a month (as in 
 Papers 3 and 4 above), it is advisable to treat 15 or 30 days for semi-diurnal tides, but only 14 or 
 27 days for diurnal tides. I therefore give here, in the form of a schedule, certain numerical 
 values which have been used in the previous papers: 
 
 n 
 
 (12w-)2r) 
 
 logjT 2 
 
 n 
 
 0*-i)l 
 
 logj, 
 
 15 
 30 
 
 14 -76 
 29 -53 
 
 00483 
 01945 
 
 14 
 27 
 
 6'8-i 
 13 '29 
 
 00105 
 00391 
 
 D. I. 1, 
 
226 
 
 THE TIDES GOVERNED BY MEAN SOLAR TIME. 
 
 Suppose that the several coefficients to be determined by harmonic analysis 
 are defined by the following equations : 
 
 & (T > = A + A l cos 30V + B l sin 30r + A a cos 60r + B 2 sin 60r 
 
 C 
 
 Ep 
 
 e 
 
 cos 30r + 
 
 1 ! 
 
 - ci'n 2ft<V _L \ 
 
 sin 30V + T, f cos 60r + 
 E%] 
 
 e 2 ) 
 
 Then on comparing (12) with (11) we see that : 
 
 sin 60r 
 
 Co 
 
 D 
 
 E 
 
 e rt 
 
 H sa COS 
 
 f sin 
 
 COS 
 
 cos / _y _ 9Q'M\ 
 
 <*" 
 
 2 sn 
 
 cos 
 v 
 
 sn 
 
 cos 
 
 - dpi cos 
 
 jr COS 
 
 = a. 8 Kg 
 sm 
 
 _ + H t cos , 
 
 ~-3Fi sin 
 
 <7 p M 
 
 y -jp lS m v 
 
 -76) + 5 P Sm (p - V p + 14-76) 
 
 ' 
 
 1 s 
 
 K 
 
 e 2 
 
 cos (^" - V" - 29'53) 
 sm v 
 
 sm 
 cos 
 
 From these equations we get 
 
 t'H' cos , v/ _ 
 
 cos 40 . 7 
 
 sm 
 
 os _i4-76) 
 
 ( ' 
 
 I (E 2 - fa) ) f// ^ 7/ cos i'' _ v" - 29'53) 
 " v 
 
 .(13) 
 
 .(14) 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 227 
 
 The tidal constants of the several tides enumerated at the head of this 
 section are determinable from these equations. 
 
 Our rule is accordingly to analyse in twelve groups of 30 days, and then 
 to analyse the resulting harmonic constituents for annual and semi-annual 
 inequalities, combining the final results according to the formula just found. 
 
 The edition of "guide sheets" and computation forms which I have drawn 
 up are so arranged as to facilitate the whole process and to render it quite 
 straightforward. By this single set of additions it is thus possible to evaluate 
 eleven tides and mean water. 
 
 3. Clearance of T, R from perturbation by M 2 . 
 
 The method of the last section was designed to render all the tides 
 insensible excepting those enumerated. But M 2 is so much larger than any 
 other tide that there is a small residual disturbance which ought to be 
 corrected. 
 
 I have made computations, which I do not give, but which show that the 
 disturbance of all the harmonic constituents except i?l 2 , 33 2 is insensible. It 
 is required then to determine the correction to be applied to & 2 , 23 2 , and 
 thence those for E, F, e, f. 
 
 Suppose that, when time t is counted from O d O h of month r, the M 2 tide 
 is expressed by if cos [(30 -0)t- }, where j3 = 1'01 58958. 
 
 When means taken over 30 days are harmonically analysed the formula (6) 
 gives the contributions to & 2 , i3 2 . As it is now required to obliterate these 
 contributions, the signs must be changed, and the corrections are 
 
 For reasons which will appear below I now write 
 
 f=f TO M-0'5258 
 Then introducing the value of ft into (15), I find 
 
 fe(6. w + 4-689) 
 in 6' 43-35 fj^J^C 
 
 S (U"+ 34-689) 
 
 sin 29 29''52 
 
 Let TO denote the value of m < T > at O d O h of month 0, and let m (T) = m + 6 M , 
 and let -jp 2 denote a certain factor whose logarithm is 0'00849, and let 
 
 152 
 
228 
 
 THE TIDES GOVERNED BY MEAN SOLAR TIME. 
 
 In the harmonic analysis for the M 2 tide, considered below in 6, we 
 shall have 
 
 f ff fff 
 
 U-J- vn. t* Tl IJ.J. * t* 
 
 Accordingly 
 
 A _ 
 
 -"2 
 
 ,,cos 
 M . 
 sin 
 
 (T) = -TPo 
 
 Jj 2 
 
 COS ,,, , 
 
 sn 
 
 + 
 
 cos 
 
 These values of M . 
 sm 
 
 must now be introduced into (16), but the 
 algebraic process need not be given in detail. If we write 
 
 sin 5 43'-35 C S (4 41''32) 
 sin 
 
 sin 30' 28"'6 
 
 sin 5 43'-35 C S (34 41''32) 
 sin v 
 
 (R - Q) j ~ ^ 2 sm29 29'31"-4 
 
 it follows that 
 
 P = 0-01564, Q = 0-00114, R = 0-00147, S = 0-01611 
 
 SI Tl 
 
 Then, when the substitution of the values of M W (T) is carried out, 
 
 COS 
 
 we find 
 
 = cos 6 M 
 
 t 
 
 v io 
 
 Q 
 
 By the definition of 6 (r} it appears that 6 M is the increment of twice 
 the mean moon's hour angle during the time from O d O h of month up to 
 O d O h of month r, that is to say M = 2 (7 a-} t for the time specified. The 
 following table gives the values of (T} and of its cosine and sine for each 
 month : 
 
 Month 
 (r) 
 
 No. of days 
 from epoch 
 to epoch T 
 
 
 ** 
 
 
 cos (r) 
 
 sin 0< T > 
 
 
 
 
 
 
 
 
 0' 
 
 1-000 
 
 o-ooo 
 
 1 
 
 30 
 
 
 11 
 
 27 
 
 0-980 
 
 0-199 
 
 2 
 
 61 
 
 
 47 
 
 16 
 
 0-678 
 
 0-735 
 
 3 
 
 91 
 
 
 58 
 
 43 
 
 0-519 
 
 0-855 
 
 4 
 
 122 
 
 7T 
 
 -85 
 
 28 
 
 - 0-079 
 
 0-997 
 
 5 
 
 152 
 
 7T 
 
 -74 
 
 1 
 
 - 0-275 
 
 0-961 
 
 6 
 
 183 
 
 7T 
 
 -38 
 
 11 
 
 - 0-786 
 
 0-618 
 
 7 
 
 213 
 
 7T 
 
 -26 
 
 44 
 
 - 0-893 
 
 0-450 
 
 8 
 
 243 
 
 7T 
 
 -15 
 
 18 
 
 - 0-965 
 
 0-264 
 
 9 
 
 274 
 
 7T 
 
 + 20 
 
 32 
 
 - 0-937 
 
 - 0-351 
 
 10 
 
 304 
 
 7T 
 
 + 31 
 
 59 
 
 - 0-848 
 
 - 0-530 
 
 11 
 
 335 
 
 7T 
 
 + 67 
 
 49 
 
 - 0-378 
 
 - 0-926 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 229 
 
 If cos 6 (r} , sin (T} are regarded as quantities having annual and semi- 
 annual inequalities, we may write 
 
 cos 0v = + cd cos 30V + ft sin 30V + a 2 cos 60V + & sin 60V + . . . 
 sin 0< T > = 70 + 71 cos 30V + Si sin 30V + 72 cos 60V + S 2 sin 60V + . . . 
 
 On analysing the numerical values of cos (T] , sin (T) by the ordinary 
 processes, I find 
 
 = - 0165, 70 = + 0-273 
 
 ! = + 0-626, 7l = - 0-500 
 
 & = + 0-756, S, = + 0-642 
 
 2 =+0-159, 7 2 = -0-046 
 
 /9 2 = + 0-199, S 2 = + 0166 
 
 But in 2 the harmonic constituents of ^ 2 when analysed for annual and 
 semi-annual inequality were denoted by E , Ej, F 1? E 2 , F 2 , and the con- 
 stituents of 33 2 were denoted by e , e u f 1} e 2 , f 2 . Hence the ten corrections to 
 the E's and F's are (with an easily intelligible alternative notation) 
 
 SE , i, 2 = (- Pa fl , i, 2 + Q7o, i, 2) -4 2 + (Qo, i, 2 + P7o, i, 2) # 2 
 SF,, 2 = (- PA, , + QS:, 2 ) A, + (QA, , + PS,, 2 ) 5 2 
 Se , i, 2 = ( Ro, i, 2 870, j, 2 ) -4 2 + ( Sa , i, 2 + R7o, i, 2) -B 2 
 
 st;, 2 = (- R&, a - S8 lt a ) ^ 2 + (- SA, 2 + R^, 2 ) B 2 
 
 On substituting the numerical values of a, & 7, B, P, Q, R, S, I find 
 
 Coeffit. of A 2 Coeffit. of B 2 
 
 SE = + 0-0029 +0-0041 
 Se = - 0-0042 + 0-0031 
 
 S| (E, + fj) = - 0-0109 - 0-0091 
 81 (E, - fj) = + 0-0006 + 0-0020 
 
 l + Fj) = - 0-0020 + 0-0000 
 1 -F 1 ) = + 0-0091 -0-0108 
 
 = - 0-0028 -0-0018 
 8 (E. 2 - f 2 ) = + 0-0002 +0-0012 
 
 S \ (e 2 + F 2 ) = - 0-0012 + 0-0001 
 
 -0-0027 
 
 Most of these corrections are negligible, but the four which affect the 
 solar elliptic tides T, R must be included, because those tides are so small 
 that a small error affects them sensibly. Hence we may take, with sufficient 
 accuracy, 
 
 S ( 6l - Fj)= + 0-009^ 2 - 0-0115,, S| (e, + Fj)= - 0'0024 2 
 Si (E, + Q = - 0-011^1 2 - 0-009&, S| (E! - f,) = + 0-0006^1, + 0-0025 2 
 
 ...... (16*) 
 
230 THE TIDES GOVERNED BY MEAN SOLAR TIME. [6 
 
 where A 2 , B 2 are the components of the M 2 derived from the reduction of 
 that tide by the process of 6. 
 
 Provision for these corrections is made in the computation forms. 
 
 4. Evaluation of A , Sa, Sj, S 2 , S 4 , S 6 , K 2 , K 15 P, when a complete 
 year of observation is not available. 
 
 It is now proposed to consider the case where the period of observation 
 is as much as six complete months and less than a complete year. 
 
 The method of the last section apparently depends on the completeness 
 of the year, yet, with certain modifications, it may be rendered available 
 for shorter periods. 
 
 We suppose that so much of the year as is available is broken into sets 
 of 30 days by the rules of the last section, and that the means are har- 
 monically analysed. The results of such harmonic analysis for month (T) are 
 given in (11) of 3, but for the purpose in hand they now admit of some 
 simplification. It is clear that it is not worth while to evaluate the very 
 small solar elliptic tides T and R from a short period of observation. If 
 then, we denote by P (T) the ratio of the cube of the sun's parallax to its mean 
 parallax at the middle of the month (T), the first three terms of the third of 
 
 cos 
 
 (11) may be included in the expression P (T} H S . K S . The last term of this 
 
 equation really does involve the solar parallax to some extent and we may, 
 with sufficient approximation, write the third pair of equations 
 
 <a (r) . p( T )\ f"PJ" r>ns 
 
 2 I TT x I ( K " V" fiOV SQ -^ 
 
 'Vrt / \ TM,\ I **l ' K ' rt V K ' U" *"* OOI 
 
 l3 2 (T) -=-P (T) ) sm j^ 2 sm 
 
 Let us now consider the value of P (T) . The longitude of the solar perigee 
 is 281 or 79, and the ratio of the sun's parallax to its mean parallax is 
 approximately 1 + ^008(^+79), and the cube of that ratio is 1 + 3^008(^ + 79) 
 or 1 + 00504 cos (A + 79). Now h, the sun's longitude at the middle of 
 month (T), is h + 15 + 30r ; hence 
 
 P< T > = 1 + 0-0504 cos (A, + 30r + 94) 
 and ^rrc = 1 
 
 Thus it is easy to compute the values of 1/P (T) for the successive months, 
 when we know h the sun's mean longitude at O d O h of the month 0. 
 
 The semi-annual tide, being usually small, may be neglected in these 
 incomplete observations, and the equations (11) now become 
 
 ia o (T) = A + 5? cos ( Kga - N sa - 30r - 14-76) 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 231 
 
 . cos K + l " T" (*' - V - 30r - 14-76) 
 sin - jp! sin 
 
 XZ COS , TT on i i/l - h 7C\ 
 
 + =r , (r^ Vp + oU T + 14 7o) 
 
 COS 
 
 - o j 
 
 (17) 
 
 When the series of successive values of the $Ts and 33's are harmonically 
 analysed (by processes which we shall consider shortly) the several coefficients 
 resulting from such analysis will be denned by 
 
 & < T > = A + A, cos 30r + B l sin 30V 
 
 C< ^ ' Cl | cos 30V + ?'! sin 30V 
 E, 
 
 fif'2 " -*- I 
 
 Mean E 4 (T) = A t , Mean 23 4 < T) = 5 4 
 
 Mean & 6 < T > = A 6) Mean 23 6 < T ' = B 6 (18) 
 
 Then the subsequent procedure as given in (13) and (14) holds good, 
 the only difference being that we do not obtain the semi-annual and solar 
 elliptic tides. 
 
 We shall now consider the harmonic analysis of an imperfect series of 
 values. 
 
 It must be premised that each monthly value of g| 2 (T) , i3 2 <T) is to be 
 divided by its corresponding P (T) before the analysis is made. 
 
 Suppose that C (T) denotes a function which is subject to semi-annual 
 inequality, and that 
 
 CM = A + A 2 cos 60V + B 2 sin 60V 
 Then it is clear that C (u) = A + A 2 
 
 = A O - A 2 + yz B 2 
 
 &c. &c. 
 
232 THE TIDES GOVERNED BY MEAN SOLAR TIME. [6 
 
 I now define D , D l} D 2 thus : 
 
 If there be n equations and if they be treated by the method of least 
 squares, we get 
 
 These are the three equations from which AQ, A 2 , B 2 are to be found. 
 
 A schedule is given below for the formation of D Q , D l} D z , and a table 
 of the solutions of these equations according to the number of months 
 available. 
 
 Next, suppose C (T} denotes a function which is subject to annual inequality, 
 and that 
 
 CM = A 4- A! cos 30r + Bj sin 30r 
 
 Then C<> =A + A 
 
 &c., &c. 
 
 In this case the method of least squares gives 
 
 A = . 
 
 Tables are given below for the formation of D , D 1} D 2 , and of the 
 solutions of the equations according to the number of months available. 
 
1892] 
 
 THE TIDES GOVERNED BY MEAN SOLAR TIME. 
 
 233 
 
 y. 
 
 q 
 
 . 
 
 Or 
 
 
 
 1 
 
 02 1 
 
 I I 
 
 I I 
 
 V? 
 
 8 <> 
 
 02 * 
 
 QD 
 
 I 
 
 GO 
 
 6 
 
 II 
 
 02* 
 
 -tw 
 
 I I 1 i 
 
 I I 
 
 V* 
 
 . i I I I 
 
 * o^ 
 
 GO 
 
 r 
 
 I 
 
 x 
 
 ^ . 
 
 S o 
 
 -o S 
 
 .| 
 
 ><J a 
 
 , 
 
 08:3 
 
 *& t-- CO O5 O 
 
 O r-t (M CO -* O 
 
 8 -o 
 
 8 > 
 
 02 
 
 !"' 
 
 ym 
 
 3 
 
 -3 <r> t^ oo os o 
 
 8-0 
 
 O i i CM CO * C 
 
234 
 
 THE TIDES GOVERNED BY MEAN SOLAR TIME. 
 
 Rule for finding semi-annual inequality from an incomplete series. 
 
 Number of 
 months 
 available Coefft. of D 
 
 Coefft. of Dj 
 
 Coefft. of D 2 
 
 6 A = +0-167 
 
 
 
 A,j = 
 
 + 0-333 
 
 
 B 2 = 
 
 
 + 0-333 
 
 7 A = +0148 
 
 - 0-037 
 
 
 A 2 = -0-037 
 
 + 0-259 
 
 
 B 2 = 
 
 
 + 0-333 
 
 8 A = +0-136 
 
 - 0-045 
 
 - 0-026 
 
 A 2 = - 0-045 
 
 + 0-253 
 
 -0-019 
 
 B 2 = -0-026 
 
 -0-019 
 
 + 0-275 
 
 9 AO= +0-123 
 
 - 0-027 
 
 - 0-047 
 
 A 2 = -0-027 
 
 + 0-228 
 
 + 0-011 
 
 B 2 = - 0-047 
 
 + 0-011 
 
 + 0-241 
 
 10 AO= +0-107 
 
 
 - 0-041 
 
 A 2 = 
 
 + 0-182 
 
 
 B 2 = - 0-041 
 
 
 + 0-238 
 
 Rule for finding annual inequality from 
 
 an incomplete series. 
 
 Number of 
 months 
 available Coefft. of D 
 
 Coefft. of Dj 
 
 Coefft. of Z> 2 
 
 6 A = +0-977 
 
 - 0-326 
 
 -1-215 
 
 A! = - 0-326 
 
 + 0-442 
 
 + 0-405 
 
 B t = -1-215 
 
 + 0-405 
 
 + 1-845 
 
 7 A = +0-424 
 
 
 - 0-528 
 
 Ax = +0-250 
 
 
 
 B! = - 0-528 
 
 
 + 0-990 
 
 8 A = +0-226 
 
 + 0-062 
 
 - 0-233 
 
 A, = + 0-062 
 
 + 0-230 
 
 - 0-093 
 
 B! = - 0-233 
 
 - 0-093 
 
 + 0-552 
 
 9 A = +0-146 
 
 + 0-057 
 
 - 0-098 
 
 A, = + 0-057 
 
 + 0-230 
 
 - 0-083 
 
 B, = - 0-098 
 
 - 0-083 
 
 + 0-326 
 
 10 A = +0-110 
 
 + 0-036 
 
 - 0-036 
 
 A, = + 0-036 
 
 + 0-218 
 
 - 0-048 
 
 B, = - 0-036 
 
 - 0-048 
 
 + 0-218 
 
 We thus get the following rule for the evaluation of A , Ssa, S 1} S 2 , S 4 , 
 S 6 , KZ, Kj, P from 6, 7, 8, 9, or 10 months of observation : 
 
1892] THE TIDES GOVERNED BY MEAN SOLAR TIME. 235 
 
 Proceed as though the year were complete and find the &'s and 33's for as 
 many months as are available. Reduce the jH 2 > i$2 by multiplication by 1/P (T) 
 or 1 - 0-0504 cos (A + 30 r + 94). 
 
 Analyse a o (T) , &, (T) , 3$^, for annual inequality, and & 2 < T >/P (T >, t3 2 (T) /P (T) 
 for semi-annual inequality according to the rules for reduction of incomplete 
 series just given. 
 
 Complete the reduction as in 3. 
 
 These rules for reduction do not include the case of 11 months, nor the 
 case where any month in the series is incomplete (e.g., if a fortnight's obser- 
 vation were wanting in one of the months), because these cases may be 
 treated thus : the ^H's and 33's return to the same value at the end of a 
 year, and therefore the case of eleven months is the same as that of a missing 
 month at any other part of the year. In both these cases we may interpolate 
 the missing jTs and 33's and treat the year as complete. 
 
 If three or more weeks of observation were missing they might fall so as 
 to spoil two months, and in this case we should have an incomplete series. 
 It is then to be recommended that the equations of least squares be formed 
 and the equations solved. So many similar cases may arise that it does not 
 seem worth while to solve the equations until the case arises. 
 
 5. Evaluation of A,,, S 2 , S 4 , K 2 , K,, P from a short period of 
 
 observation*. 
 
 If the available tidal observations only extend over a few months, it is 
 useless to attempt the independent evaluation of those tides which we have 
 hitherto found by means of annual and semi-annual inequalities in the 
 monthly harmonic constants. We will suppose that 30 days of observations 
 are available. Then when we neglect the annual tide, and the solar 
 (meteorological) tide S 1} we have from (11) or (17), which give the analysis 
 of 30 days, 
 
 MQ = AO 
 
 , fl eos ( _ V - 14-76) + C 8 <* - V y + M-76) 
 
 x 
 
 = PH S KS + 1 -- u a ( K " - V" - 29-53) 
 sm JJ- 2 sm 
 
 |S 4 ! = H a C S *, P = 1 + 0-0504 cos (A + 15) 
 
 *J 4 j sm ' 
 
 It is now necessary to assume that the P tide has the same amount of 
 retardation as the K 1} and that the ratio of their amplitudes is the same as 
 
 * [See footnote to 2 above for changes to be made in this section when only 15 days are 
 available, and when the diurnal tides are analysed over either 27 or 14 days.] 
 
236 THE TIDES GOVERNED BY MEAN SOLAR TIME. [6 
 
 in the equilibrium theory. We also make the like assumption with respect 
 to the K 2 and S 2 tides. 
 
 Accordingly we put 
 
 H P = \H', K P = K; H" = -2-H g , K" = K S 
 Now since 
 
 V = h - TT - i/, V p = -h +%7r, V" = 2h n - 2i/" 
 we have 
 
 K P -V P + 14-76 = ' - V - 14-76 + (2/* - i/ + 29'53) + TT 
 K " _ V" - 29'53 = KS - (2h n - 2i/' + 29'53) 
 Therefore 
 
 cos ,_ 
 
 v 
 
 T (*' - V - 14-76 + 2^ - v' + 29'53) 
 
 I prrCOS Af"fT s COS , 00KO\ 
 
 f = PH ' sin *' + ^T sin ( " s - 2/< + 2y 
 
 Let us put tan * = 
 
 sn 
 
 . . sn - 
 
 Q - . _ ," ' ..... 
 
 r cos (2flo _ 2l ," + 29'53) 
 Then 
 
 a,) ^ 3f / - cos (2A - ' + 29-53) cos , x 
 
 CM I*"* -- o~" ~~i - ' (/C V 14 /O 9) 
 
 a3j ) jp! 3 cos <j> sm v 
 
 cos (2A - 2i/ 7 + 29'53) cos 
 
 If therefore 
 
 | _ -R COS ^ 2 { _ -p COS 
 
 I "1 5 1 > r fl I -"2 S2 
 
 sm * 23 2 sm 
 
 we have ' = ^ + V + 14'76 + 9 = K 
 
 P 
 
 TT, _ ll COS <> _ IT = 
 
 3f / -cos(2A - J / / +29 -53) > * 
 
 Sr 77"__3_/f 
 
 cos (2A - 2i/" + 29'53) ' 
 
 If there be several months available it is recommended that each 30 days 
 be treated quite independently, so that from each group of days we shall get 
 H', K and H s , K S . Then the mean value of H'COSK' is to be taken as the 
 final value of that function, and H ' sin K is to be treated similarly ; finally 
 H', K are to be found. The several values of H s , K S may be treated in the 
 
1892] THE METHOD OF GROUPING THE MEAN SOLAR DAYS. 237 
 
 same way. Of course we assume throughout that te p = K, H p = ^ 
 H" =^H S . assumptions which are usually nearly correct. 
 
 The mean value of & must be taken as giving A , but at places with a 
 considerable annual tide it is impossible to obtain a good value of mean 
 water mark from a short series of observations. 
 
 6. On the evaluation of the several tides by grouping of mean 
 
 solar days. 
 
 Let n(y %) denote the speed in degrees per m.s. hour of any one tide, 
 n being equal to 1, 2, 3, 4, 5, or 6. Then 15/(7 - %) may be called one 
 "special hour." Since 15/(y w) is one m.s. hour, the ratio of the m.s. to 
 the special hour is (7 %)/(7 t}). 
 
 Let one m.s. hour be equal to 1 /3 special hour, then 
 
 ry 'y 
 
 8 = 1 - , special hours 
 
 y-rj 
 
 Let it be required to express the 12 h of any m.s. day of a series of days 
 by reference to special time. It is clear that 12 h m.s. time will be specified 
 by one of the 24 special hours, with something less than half a special hour 
 added or subtracted. 
 
 Having fixed the 12 h of m.s. time of a particular m.s. day in the special 
 time scale, let us treat that m.s. day as a whole, and consider the incidence 
 of the other 23 m.s. hours in special time. It is clear that in m.s. time we 
 work backwards and forwards from 12 h by subtracting or adding unity, and 
 that in special time we subtract or add 1 /3. 
 
 If 12 h m.s. time be aP + a, where a lies between + special time, the 
 following is a schedule of equivalence: 
 
 Mean solar 
 
 time Special time 
 
 O h = (a* 
 l h = (a* 
 2 h = tf h 
 
 12 h = a? + a 
 
 13 h = (**+ l h ) + (a-) 
 
 22 h = (x h + 10 h ) -f (a - 10/9) 
 
 23 h = # h 
 
 In the column of special time it is supposed that 24 h is added or sub- 
 tracted, so that the result is less than 24 h . For example, if a; is 10, the hour 
 column of special time will run 22 h , 23 h , O h , ..., 9 h , 10 h , ll h , ..., 20 h , 21 h . 
 
238 
 
 THE METHOD OF GROUPING THE MEAN SOLAR DAYS. 
 
 If the series of days be long x will have all integral values between and 
 23 with equal frequency, and since a has all values between + ^ and ^ with 
 equal frequency, the excess of the solar hour above the nearest exact special 
 hour (which may be called the error) will have all its possible values with 
 equal frequency. If the mean solar hours be arranged in a schedule of 
 columns headed O h , l h , ..., 23 h of special time, each column will be subject to 
 errors which follow the same law of frequency. 
 
 A' 
 
 FIG. 1. 
 
 Let abscissae (fig. 1) measured from O along A'OA represent magnitude 
 of a. 
 
 Since a lies between ^, the limit of the figure is given by OA = OA' = ^. 
 
 If magnitude of error (i.e. m.s. special hour), measured in special time, 
 be represented by ordinates, a line BOB' at 45 to AOA' represents all the 
 errors which can arise in the incidence of the m.s. 12 h in the schedule of 
 special time. 
 
 If a line 66' be drawn parallel to and above BB' by a distance ft, we have 
 a representation of all the errors of incidence of the m.s. ll h . If a series of 
 equidistant parallel lines be drawn above and below BB' until there are 12 
 above and 11 below, then the errors of all the m.s. hours are represented, the 
 top one showing the errors of the m.s. O h and the bottom one the errors of 
 the m.s. 23 h . 
 
 Any special hour corresponds with equal frequency with each solar hour, 
 and hence each mode of error occurs with equal frequency. 
 
 It is now necessary to consider in how many ways an error of given mag- 
 nitude can occur. If in the figure AM represents an error of given magnitude, 
 
THE METHOD OF GROUPING THE MEAN SOLAR DAYS. 
 
 239 
 
 then wherever MN cuts a diagonal line, it shows that an error may arise in 
 one way. 
 
 It is thus clear that there are no + errors greater than ^- + 12/8, and no 
 errors greater than ^ + 11/3, and 
 
 Errors of magnitude 
 
 ^ + 12/3 io ^ + 11/8 may arise in 1 way 
 
 I + 11/3 to | + 10/8 2 ways 
 
 \ + 10/8 to I + 9/3 3 ways 
 
 10/3 to 
 11/8 to -( 
 12/3) to - 
 
 11/3 
 
 12/8) 
 11/8) 
 
 23 ways 
 
 24 ways 
 23 ways 
 
 9/8) to - 
 10/3) to - 
 
 + 10/8) 
 + H/3) 
 
 2 ways 
 1 way 
 
 The frequency of error is represented graphically in fig. 2. The slope of 
 the two staircases is drawn at 45, but any other slope would have done 
 equally well. 
 
 A frequency curve of this form is not very convenient, and, as there are 
 many steps in the ascending and descending slopes, I substitute the frequency 
 curve shown in fig. 3. This is clearly equivalent to the former one. In fig. 3 
 all the times shown in fig. 2 are converted to angle at 15 to the hour; 
 e accordingly denotes 15/3. 
 
 FIG. 2. 
 
 B r Q B 
 
 T30'~" ~7 6 36~' 
 FIG. 3. 
 
240 THE METHOD OF GROUPING THE MEAN SOLAR DAYS. [6 
 
 Now let cos n (6 x) be the observed value of a function whose true 
 value is cos nd, and suppose that x, the error of 6, has a frequency f(x) ; 
 then the mean value of the function deduced from many observations 
 will be 
 
 i-+oo ,-+ao 
 
 I /(a;) cos n(6 x)dx-r- I / (x) dx 
 
 J CD J 00 
 
 In our case f(x) is the ordinate of the frequency curve whose abscissa 
 is x. 
 
 Let OQ=A, QB = a, QB'=6, OA = a + h, OA.'=b + h; then 
 
 r+oo 
 
 f(x) dx = (a + b+h)h 
 
 J 00 
 
 /+ 
 
 I /(#) cos n (6 x) dx 
 
 J oo 
 
 fa ra+h 
 
 = I h cos n (0 x)dx+ I (a + h x) cos n (6 x) dx 
 
 JO J a 
 
 fb rb+h 
 
 + I h cos n (6 + x) dx + I (b + h x) cos n (6 + x) dx 
 
 Jo J b 
 
 4 
 
 = cos n \9 \ (a b)] sin \nh sin \n (a + b + h) 
 
 VL 
 
 The algebraical steps involved in the evaluation of these four integrals 
 and subsequent simplification are omitted. 
 
 Hence the result is 
 
 sin \nh sin %n (a + b + h) 
 
 . , , 7 x cos n [6 - % (a - b)} 
 (a + b + h) 
 
 By reference to the figure it is clear that 
 
 a + b + h=15, A = 24e, = 7^-11^6, 6 = 7^-12^, a-b = e 
 
 ^ 12ne lf.n 
 
 Write, then 4p n = ^ ^ --- . \. 
 
 sin 12we sin ^ n 
 
 and we obtain as the mean value of cos n0, when found in this way, 
 
 js- cos n (6 - |e) 
 
 Jfn 
 
 It is obvious that if we had begun with sin nd, the argument in the result 
 
 and the factor Jp w would have been the same. Accordingly, a function 
 
 p/ 
 R' cos (nd % ') would yield the result -j=- cos [n (0 e) "]. If 24 equi- 
 
 Jfn 
 
 distant results of this sort are submitted to harmonic analysis to find A M , B n , 
 
 we shall get 
 
 R' 
 
 An = j=- cos (' + %ne) = R cos suppose 
 
 Jjn 
 
 p' 
 B n = j=- sin (^" + ^we) = R sin suppose 
 
1892] THE METHOD OF GROUPING THE MEAN SOLAR DAYS. 
 
 p' 
 
 Accordingly R = -=- , = ' + ^ne 
 
 3fn 
 
 But it is required to find R', ', so that 
 
 241 
 
 Thus when the 24 observed hourly tide heights on any m.s. day are re- 
 grouped so that the observed height at 12 h m.s. time is reputed to appertain 
 to an exact special hour, and each of the previous and subsequent hourly 
 values of that m.s. day are reputed to belong to previous and subsequent 
 exact special hours; and when a long series of m.s. days are treated similarly, 
 and when the mean heights of water at each of the 24 special hours are 
 harmonically analysed, we shall obtain the required result by augmenting R 
 by a factor Jp n , and by subtracting \ne from 
 
 The values of jp n and of Jne will be different for each kind of tide, and 
 the following table gives their numerical values. 
 
 Table of n and ^ne. 
 
 Initial of tide 
 
 n 
 
 log jf n 
 
 IP* 
 
 M, 
 
 1 
 
 0-00212 
 
 0-26 
 
 M 2 
 
 2 
 
 0-00849 
 
 0'53 
 
 M 3 
 
 3 
 
 0-01915 
 
 0-79 
 
 M 4 
 
 4 
 
 0-03416 
 
 1-05 
 
 M 6 
 
 6 
 
 0-07767 
 
 l-57 
 
 N 
 
 2 
 
 0-01361 
 
 0-82 
 
 L 
 
 2 
 
 0-00570 
 
 0-24 
 
 V 
 
 2 
 
 0-01278 
 
 0-78 
 
 O 
 
 1 
 
 0-00535 
 
 0-57 
 
 J 
 
 1 
 
 0-00225 
 
 - 0-28 
 
 Q 
 
 1 
 
 0-01149 
 
 0-90 
 
 fj, 
 
 2 
 
 0-02016 
 
 1-09 
 
 2SM 
 
 2 
 
 0-00805 
 
 - 0-49 
 
 MS 
 
 4 
 
 0-02342 
 
 0-52 
 
 X 
 
 2 
 
 0-00595 
 
 0-28 
 
 2N 
 
 2 
 
 0-02136 
 
 1-13 
 
 00 
 
 1 
 
 0-00481 
 
 - 0'53 
 
 MK 
 
 3 
 
 0-01438 
 
 0'50 
 
 2MK 
 
 3 
 
 0-02632 
 
 1-09 
 
 MN 
 
 4 
 
 0-04328 
 
 l-35 
 
 As it does not appear worth while to evaluate the tides written below 
 the line, no use will be made of the last six results given in this table. 
 D. i. 16 
 
242 THE PERIODS OVER WHICH THE MEANS ARE TAKEN. [6 
 
 7. On the periods over which the means are to be taken in evaluating 
 
 the tidal constants. 
 
 We have considered in previous sections the treatment of the group of 
 bides which are associated with solar time, when the period of observation 
 is less than a year, and we have now to consider the other tides. 
 
 It is important that the means be taken over such a number of days that 
 the perturbation arising from other tides shall be minimised. 
 
 The perturbation between semi-diurnal and diurnal tides is always 
 negligible. It is therefore only necessary to consider the action of the tides 
 M. 2 , S 2 in the case of semi-diurnal tides, and that of Kj and O for diurnal 
 tides. 
 
 It is easy to see that the influence of a disturbing tide is evanescent 
 when the means are taken over a period such that the excess of the argument 
 of the disturbed over that of the disturbing tide has increased through a 
 multiple of 360. As, however, we are working with integral numbers of 
 days, and as the speeds of tides are incommensurable, this condition cannot 
 be exactly satisfied. 
 
 From this consideration it appears that to minimise the perturbation of 
 S 2 , 2SM, fj, by M 2 (and vice versa) we must stop at an exact multiple of a 
 semi-lunation. To minimise the effect of M 2 on N and L, and of Kj on J 
 and Q, we must stop at an exact multiple of a lunar anomalistic period. To 
 minimise the effect of M 2 on v, we must stop at a multiple of the period 
 2?r/(o- + w 2r)). To minimise the effect of KI on 0, we must stop at an 
 exact multiple of a semi-lunar period. 
 
 For the quater-diurnal tide, MS, it is immaterial where we stop, and so 
 it may as well be taken at a multiple of a semi-lunation. 
 
 The following table (p. 243) gives the rules derived from these con- 
 siderations. 
 
 8. On the tides of long period. 
 
 The annual (Sa) and semi-annual (Ssa) tides are evaluated in the course 
 of the work by which other important tides are found. These are the only 
 two tides of long period which have a practical importance in respect to 
 tidal prediction, but the luni-solar fortnightly (MSf), the lunar fortnightly (Mf), 
 and the lunar monthly (Mm) tides have a theoretical interest. 
 
 It will therefore be well to show how they may be found. The process 
 is short, and, although it is less accurate than the laborious plan followed in 
 the Indian reductions, it appears to give fairly good results. 
 
 (Continued at foot of p. 243) 
 
1892] 
 
 THE PERIODS OVER WHICH THE MEANS ARE TAKEN. 
 
 243 
 
 Number of the last day to be included in the evaluation of the several 
 tides for observations extending over any period up to a year. 
 
 For M 2 , p, 2SM, MS. 
 Stop with one 
 of the following 
 days 
 (semi -lunations) 
 
 For 0. 
 
 Stop with one 
 of the following 
 days 
 (semi-lunar periods) 
 
 For N, L, J, Q. 
 Stop with one 
 of the following 
 days 
 (anom. periods) 
 
 For v. 
 Stop with one 
 of the following 
 days 
 (periods 
 
 14 
 
 13 
 
 27 
 
 31 
 
 29 
 43 
 
 58 
 
 26 
 40 
 54 
 
 54 
 
 63 
 
 74+8 
 
 74 + 20 
 
 73 
 
 67 
 
 + 35 
 + 63 
 
 + 52 
 
 74 + 14 
 
 + 28 
 + 43 
 + 58 
 + 73 
 
 74+7 
 + 21 
 + 34 
 + 48 
 + 62 
 
 148 + 10 
 + 42 
 
 148 + 16 
 
 + 44 
 
 + 71 
 
 222+ 
 + 31 
 + 63 
 
 222 + 25 
 
 + 53 
 
 148 + 13 
 + 28 
 + 43 
 
 + 58 
 + 72 
 
 148+ 1 
 + 15 
 + 29 
 + 42 
 + 56 
 + 70 
 
 296 + 21 
 + 53 
 
 296+ 6 
 + 34 
 + 61 
 
 
 222 + 13 
 
 + 28 
 
 
 222+ 9 
 
 + 43 
 
 + 23 
 
 
 
 + 58 
 
 + 37 
 
 
 
 + 72 
 
 + 50 
 
 + 64 
 
 
 
 296 + 13 
 
 + 28 
 
 296+ 4 
 
 + 43 
 
 + 17 
 
 
 
 + 57 
 
 + 32 
 
 
 
 + 72 
 
 + 45 
 + 58 
 + 72 
 
 
 
 
 
 For the sake of simplicity, let us consider the tide MSf. Its period is 
 about 14 days, and therefore a day does not differ very largely from a twelfth 
 part of the period. Accordingly, if about two days in a fortnight are rejected 
 
 162 
 
244 THE TIDES OF LONG PERIOD. [6 
 
 by proper rules, the mean heights of water on the remaining days may be 
 taken as representatives of twelve equidistant values of water height. 
 
 I therefore go through the whole year and reject, according to proper 
 rules, the daily sums of the 24 hourly heights corresponding to certain 69 of 
 the days out of 369. The remaining 300 values are written consecutively 
 into a schedule of 12 columns and 25 rows, of which each corresponds to a 
 half lunation. The 12 columns are summed, and the sums are harmonically 
 analysed for the first pair of harmonic components. These components have 
 to be divided by 24 times 25, or by 600, because the daily mean water height 
 is gV 11 f the daily sum, and there are 25 semi-lunations. 
 
 In the same way the semi-lunar period is about 13 \ days, and if we 
 erase by proper rules 45 daily sums out of 369, we are left with 324, which 
 may be written consecutively in a schedule of 12 columns and 27 rows, of 
 which each corresponds to a semi-lunar period. The summing and analysis 
 is the same as in the last case, but the final division is by 24 times 27, or by 
 648. 
 
 In this way we evaluate the luni-solar fortnightly and lunar fortnightly 
 inequalities in the height of the water. 
 
 The period of the moon is between 27 and 28 days, and if we erase 
 appropriately about one day in eight we are left with sets of 24 values which 
 may be taken as 24 equidistant values of the daily sums. Accordingly we 
 erase 46 daily sums out of 358, and write the 312 which remain consecutively 
 into a schedule of 24 columns and 13 rows, of which each corresponds to a 
 lunar anomalistic period. 
 
 The 24 columns are summed and the sums analysed for the first com- 
 ponents. Finally, the components are to be divided by 24 times 13, or by 
 312. In this way the lunar monthly tide is evaluated. 
 
 But the result obtained in this way is, as far as concerns the tide MSf, 
 to some, and it may be to a large, extent fictitious. It represents, in fact, a 
 residuum of the principal lunar tide M 2 . That this is the case will now be 
 proved. 
 
 Suppose that t Q is an integral number of days since epoch, being the time 
 of noon on a certain day ; then the principal lunar tide M 2 on that day may 
 be written H m cos [2 (7 - a-) (t + r) - OT ], where r is less than 24 hours. 
 Then the daily sum for that day will be 
 
 Hm Tin ( 7 - COS & ( ^ -*)* + 23 & ~ *> ~ 
 
 Now since t is an integral number of days 2 (7 <r) t only differs from 
 - 2 (o- rj) t by an exact multiple of 360 ; hence the argument of the cosine 
 may be written 2 (<r - 77) t Q 23 (7 cr) + f m . 
 
1892] THE TIDES OF LONG PERIOD. 245 
 
 But the true luni-solar fortnightly tide, which we may denote 
 
 varies so slowly in the course of a day that the daily sum is sensibly equal to 
 24f#cos [2 (a- -r))t + 23 (cr - 17) - ] 
 
 It thus appears that the residual effect of M 2 is of exactly the same form 
 as that of MSf. It becomes, therefore, necessary to clear the harmonic 
 components, determined as described above, from the effects of M 2 . 
 
 In order to determine the values of these clearances, I found the values 
 of cos2(cr fj)t and sin2(o- v))t for every noon in a year of 369 days. 
 I then erased the values selected for the treatment of MSf and analysed the 
 remaining values. In this way it was easy to find the effect of the known 
 M 2 tide. 
 
 Suppose that A 1} B x are the first harmonic components determined by 
 the treatment of a series of daily sums, and that BA 1} SBj are the corrections 
 to be applied to them to eliminate the effects of M 2 , then I find that if 
 Am, B m are the two components of M 2 as determined by the previous method 
 ( 6) of analysis, 
 
 SAi = + 0-0304A m - 0'0171B m 
 
 SB, = - 0-Ol71A m - 0-0304B m 
 
 C = A 1 + 8A 1 , D = B 1 + 8B 1 
 
 C - 0-047D = 0-992 fH cos 
 
 D + 0-047C = 0-992 fff sin 
 
 Whence f being known from Baird's Manual (being a function of the 
 longitude of moon's node), H and are determinable. We have also 
 
 In the set of computation forms which I have prepared for use on the 
 present plan, it is shown what days are to be erased for each of the three 
 analyses, and how they are to be entered in schedules, summed, and analysed. 
 
 9. On, abridgment in the computations. 
 
 It seemed probable that one decimal of a foot would suffice to express 
 the hourly tide heights. In order to test this, I have taken several individual 
 days of observation at Port Blair, and have found, by harmonic analysis, the 
 time and amplitude of the diurnal and semi-diurnal H.W., first, when the 
 hourly heights are expressed to two decimal places of a foot, and secondly, 
 when they are only entered to the nearest tenth of a foot. I find that the 
 times of H.W. agree within less than a minute of time, and that the 
 amplitudes agree within a fraction of an inch. If this much be true of 
 
246 
 
 ABRIDGEMENT OF COMPUTATION. 
 
 individual days, the difference of results arising from two or one place of 
 decimals will clearly entirely disappear when a series of days is considered. 
 Hence, by taking as unit the tenth of a foot, or the inch, or even two inches 
 at places with large tides, we may always express all, or nearly all, the 
 heights on which we are to operate by two significant figures. The adoption 
 of this rule not only saves the writing of a large number of figures, but also 
 enormously diminishes the labour of the additions which have to be made. 
 
 It also seemed probable that substantial accuracy might be attained from 
 the harmonic analysis of only 12 hourly values instead of 24. In order to 
 test this I took the tidal reductions for Port Blair, Andaman Islands (kindly 
 lent me by the Survey of India), and have compared the results which would 
 have been derived from 12 values with those actually obtained from 24 values 
 by the computers of the Indian Survey. The following tables give the 
 results : 
 
 Semi-diurnal tides. 
 
 Initial 
 
 Eesults from 12 
 two-hourly values 
 
 Results from 24 
 hourly values 
 
 Error, (12) -(24) 
 
 *{: 
 
 ft. 
 +0-6883 
 + 0-6775 
 
 ft. 
 + 0-6890 
 + 0-6768 
 
 ft. 
 -0-0007 
 +0-0008 
 
 {: 
 
 - 1-7032 
 
 + 1-0883 
 
 -1-7005 
 + 1-0872 
 
 -0-0027 
 + 0-0011 
 
 it: 
 
 -0-1437 
 -0-2527 
 
 -0-1407 
 -0-2515 
 
 -0-0030 
 -0-0012 
 
 L U: = 
 
 + 0-0357 
 + 0-0612 
 
 + 0-0347 
 + 0-0610 
 
 + 0-0010 
 +0-0002 
 
 *{*: 
 
 +0-3422 
 + 0-2192 
 
 + 0-3486 
 +0-2124 
 
 - 0-0064 
 +0-0068 
 
 {*: 
 
 -0-1217 
 +0-0867 
 
 -0-1165 
 
 + 0-0887 
 
 -0-0052 
 -0-0020 
 
 / A 2 = 
 ^IB 2 = 
 
 +0-0857 
 + 0-0383 
 
 + 0-0849 
 - 0-0388 
 
 - 0-0008 
 -0-0005 
 
 2SM { 2 = 
 
 I % 
 
 +0-0055 
 -0-0198 
 
 + 0-0037 
 -0-0200 
 
 +0-0018 
 +0-0002 
 
1892] 
 
 ABRIDGEMENT OF COMPUTATION. 
 
 247 
 
 Diurnal tides. 
 
 Initial 
 
 Besults from 12 
 two-hourly values 
 
 Besults from 24 
 hourly values 
 
 Error, (12) - (24) 
 
 S R' = 
 
 ft. 
 +0-0175 . 
 + 0-0223 
 
 ft. 
 
 + 0-0185 
 +0-0216 
 
 ft. 
 
 -o-ooio 
 
 + 0-0007 
 
 "{*: 
 
 + 0-0120 
 - 0-0168 
 
 +0-0059 
 -0-0173 
 
 +0-0061 
 + 0-0005 
 
 *{: 
 
 +0-3815 
 + 0-1398 
 
 +0-3847 
 + 0-1396 
 
 -0-0032 
 +0-0002 
 
 {fc: 
 
 -0-0818 
 +0-1335 
 
 - 0-0729 
 +0-1386 
 
 -0-0089 
 -0-0051 
 
 P U' = 
 
 -0-0167 
 -0-1287 
 
 -0-0178 
 -0-1280 
 
 +0-0011 
 
 -0-0007 
 
 J U-: 
 
 -0-0193 
 + 0-0315 
 
 -0-0167 
 +0-0347 
 
 -0-0026 
 - 0-0032 
 
 i{l i : 
 
 + 0-0140 
 - 0-0170 
 
 + 0-0136 
 -0-0194 
 
 +0-0004 
 + 0-0024 
 
 The mean discrepancy in the case of the semi-diurnal tides is 0'0022 ft., 
 and the greatest is + 0-0068 ; in the case of the diurnal tides the mean 
 discrepancy is 0'0026 ft., and the greatest is 0'0089. 
 
 In tidal work results derived from different years of observation differ far 
 more than do these two sets of results, and hence the analysis of 12 two- 
 hourly values for diurnal and semi-diurnal tides gives adequate results. 
 
 I find that this abbreviation does not give satisfactory results for quater- 
 diurnal tides, and the sixth harmonic is not derivable from 12 values. 
 Therefore, when these tides are to be evaluated the 24 hourly values must 
 be used. 
 
 It will still be necessary to write all the 24 hourly heights on each 
 computing strip, but when the strips are put into any one of the arrange- 
 ments, except where quater-diurnal tides are required, we need only add up 
 the columns 0, 2, 4, ..., 22, and may omit the columns 1, 3, ..., 23. 
 
 10. On a trial of the proposed method of reduction. 
 
 As already mentioned, I have the tidal reductions for one year (beginning 
 April 19, 1880) for Port Blair, Andaman Islands. I am thus able to make a 
 comparison between the results of the old method and of the new. The 
 computation was, in large part, done for me by Mr Wright. 
 
248 
 
 A TEST OF THE METHOD. 
 
 Port Blair. 1880-81. 
 
 
 I 
 
 Indian 
 calculation 
 
 N 
 New 
 method 
 
 I-N 
 (height) 
 
 I-N 
 (phase) 
 
 A = 
 
 ft. 
 
 4-792 
 
 ft. 
 4-795 
 
 ft. 
 - 0-003 
 
 
 Saj H = 
 l = 
 
 299 
 163 
 
 299 
 162 
 
 000 
 
 + 1 
 
 Ssajf I 
 
 106 
 165 
 
 111 
 164 
 
 - -005 
 
 + 1 
 
 T jH = 
 
 ( K = 
 
 099* 
 313* 
 
 094 
 339 
 
 
 
 RJ H = 
 
 (* = 
 
 020* 
 326* 
 
 004 
 312 
 
 
 
 s,S H = 
 
 1 (K = 
 
 028 
 49 
 
 026 
 53 
 
 + -002 
 
 - 4 
 
 s,J H - 
 
 j = 
 
 966 
 316 
 
 973 
 315 
 
 - -007 
 
 + 1 
 
 s JH = 
 
 S *J K = 
 
 003 
 107 
 
 003 
 105 
 
 000 
 
 + 2 
 
 K J H = 
 
 Kl i* = 
 
 403 
 326 
 
 401 
 326 
 
 + -002 
 
 
 
 K i H - 
 
 K2 JK = 
 
 286 
 314 
 
 268 
 311 
 
 + -018 
 
 + 3 
 
 P JH = 
 
 | K = 
 
 130 
 324 
 
 139 
 323 
 
 - -009 
 
 + 1 
 
 *{?: 
 
 014 
 23 
 
 013 
 34 
 
 + -001 
 
 -11 
 
 !?: 
 
 2-042 
 279 
 
 2-043 
 279 
 
 - -ooi 
 
 
 
 *; 
 
 004 
 20 
 
 004 
 54 
 
 000 
 
 -34 
 
 M-i?: 
 
 003 
 167 
 
 006 
 264 
 
 - -003 
 
 -97 
 
 .ffb 
 
 004 
 342 
 
 005 
 315 
 
 - -ooi 
 
 + 27 
 
 (24 values) Q j H = 
 
 ( K = 
 
 023 
 236 
 
 023 
 233 
 
 000 
 
 + 3 
 
 (12 values) Q \ H = 
 ( K = 
 
 023 
 236 
 
 022 
 234 
 
 + -001 
 
 + 2 
 
 * These are derived from 1880-82. 
 
1892] 
 
 A TEST OF THE METHOD. 
 
 Tides of Long Period. 
 
 249 
 
 
 T 
 
 
 
 
 
 Indian 
 calculation 
 
 New 
 method 
 
 I-N 
 
 (height) 
 
 I-N 
 
 (phase) 
 
 
 ft. 
 
 ft. 
 
 ft. 
 
 
 MSfj H = 
 (K = 
 
 045 
 163 
 
 019 
 168 
 
 + -026 
 
 -5 
 
 Mf| H = 
 (R = 
 
 056 
 356 
 
 056 
 356 
 
 000 
 
 
 
 Mmj H = 
 [K = 
 
 016 
 12 
 
 020 
 13 
 
 - '004 
 
 -, 
 
 It appeared sufficient to evaluate the tides of the S series and those 
 allied with them, the tides of the M series, and the tide Q ; also the tides of 
 long period MSf, Mf, Mm. 
 
 The S series test the new process of harmonic analysis of monthly 
 harmonic components for annual and semi-annual inequalities. I chose M 
 because it is the most important tide, and Q because it puts the proposed 
 method of grouping to a severe test, and is very small in amplitude. 
 
 In the Q time scale the day is 26 h 52 m of mean solar time, from which 
 it follows that one of the 24 mean solar hourly observations may fall as much 
 as 2 h O m away from the exact Q hour to which it is reputed to belong. Thus 
 the hourly observations are arranged in wide groups round the Q hours, and 
 the hypothesis involved in the method is put to a severe strain. 
 
 Lastly, the results for tides of long period test my proposed abridgment. 
 
 It will be seen in the table on p. 248 that the two methods give results 
 in close agreement. There is, however, a sensible discrepancy in the K 2 tide, 
 but in this case I am inclined to accept the new value as better than the old 
 one. This tide is governed by sidereal time, which differs but little from 
 mean solar time. Hence, in the Indian method of grouping, considerable 
 errors of incidence of the S hours in the K time scale prevail for many 
 days together, and the method seems of doubtful propriety. The same is 
 true of the P tide, and here also the two methods give somewhat different 
 results. 
 
 The accuracy with which the very small Q tide comes out, whether from 
 24 values or only from 12, is surprising, and may perhaps be, to some extent, 
 due to accident. It shows, however, that the present method may be safely 
 applied, even when the special time scale differs considerably from mean solar 
 time. 
 
 The results for the tides of long period are quite as close to the old values 
 as could be expected. 
 
250 TREATMENT OF GAPS IN THE OBSERVATIONS. [6 
 
 11. A comparison of the work involved in the new and old 
 methods of reduction. 
 
 It has been usual in the Indian reductions to use three digits in expressing 
 the height of water, and there have been 15 series, or even more. _Now 
 3 x 24 x 365 x 15 is 394,000 ; hence the computer has had to write that 
 number of figures in reducing a year of observation. This does not include 
 the evaluation of the annual and semi-annual tides, so that we may say that 
 there have been about 400,000 figures to write. 
 
 I propose to express the heights by two digits, and they only have to be 
 written once. Thus, in the present plan, the number of figures to write is 
 2 x 24 x 365, or' 17,500. Thus the writing of 382,000 figures is saved. 
 
 In the old method the computer had to add together all the digits written, 
 say, 394,000 additions of digit to digit. 
 
 I propose to use 24 hourly values in three series, viz., S, M, and MS, 
 and 12 two-hourly values in eight others. Therefore, the number of additions 
 will be 3 x 2 x 24 x 365 + 8 x 2 x 12 x 365 or 123,000. Thus 270,000 
 additions are saved. 
 
 We may say that formerly there were about 800,000 operations (writing 
 and addition), and that in the present method there will be about 140,000. 
 This estimate does not include a saving of several thousands of operations in 
 obtaining the tides of long period. I am therefore within the mark when 
 I claim that the work formerly bestowed on one year of observation will now 
 reduce at least five years. 
 
 It has been found that the manufacture of my computing strips of xylonite 
 is rather expensive, but as it formerly cost in England rather more than 20 
 to reduce a year of observation, the cost of the apparatus will be covered by 
 the saving in the reduction of a single year, and it will serve for any length 
 of time. 
 
 12. On the completion of the record for short gaps and long gaps. 
 
 In any long series of tidal observations there are usually some breaks in 
 the record in consequence of the stoppage of the clock of the tide gauge, or 
 from some other cause. Now the process of elimination by grouping depends 
 essentially on the completeness of the record, and it is therefore necessary to 
 fill in blanks by interpolation. 
 
 Such interpolation has not been usual in the operations of the Indian 
 Survey, and it might be thought that the complete omission of the missing 
 entries is the proper course to take ; but it is easily shown that this treat- 
 ment is exactly equivalent to the assumption that the water remained stagnant 
 
1892] 
 
 TREATMENT OF GAPS IN THE OBSERVATIONS. 
 
 251 
 
 at mean sea level during the whole time of stoppage of the gauge. It is 
 obvious, therefore, that any conjectural values are better than none. 
 
 The process by which it is proposed to interpolate is best shown by an 
 example. 
 
 At Port Blair (beginning April 19, 1880) the column of 6 h from days 99 to 
 112 gives the heights shown in the first column of the table below. I suppose 
 that the tide gauge broke down on day 103, and only came into action again 
 on day 110*. There was really no breakdown, and the actuality during the 
 supposed hiatus is shown in the last column but one. 
 
 Now if we look back about a month we find that the water stood about 
 the same height at the same hour of the day (viz., 6 h ). Then the " previous 
 record " (which is complete) beginning at 69 d is entered in the next column. 
 Similarly a "subsequent record" is found about a month later, and is entered 
 in a third column. The mean of the previous and subsequent records is then 
 taken as giving the values to be interpolated. 
 
 Table of Interpolation. 
 
 Defective 
 record 
 
 Previous 
 record 
 
 Subsequent 
 record 
 
 Mean of 
 previous and 
 subsequent 
 
 Actuality 
 
 Error 
 
 Day 6 h 
 99 2-54 
 
 Day 6 h 
 69 2-02 
 
 Day 6 h 
 129 2-56 
 
 2-29 
 
 
 
 
 100 3-13 
 
 70 2-83 
 
 130 3-07 
 
 2-95 
 
 
 
 
 101 3-86 
 
 71 3-70 
 
 131 3-70 
 
 3-70 
 
 
 
 
 102 4-40 
 
 72 4-55 
 
 132 4-27 
 
 4-41 
 
 
 
 
 103 .. 
 
 104 .. 
 
 73 5-10 
 74 5-60 
 
 133 4-88 
 134 5-27 
 
 4-99 \ 
 5-44 
 
 a 
 .2 
 
 4-83 
 5-24 
 
 + 0-16 
 + 0-20 
 
 105 .. 
 
 75 5-72 
 
 135 5-35 
 
 5-54 
 
 31 
 
 5-39 
 
 + 0-15 
 
 106 .. 
 
 76 5-67 
 
 136 5-28 
 
 5-48 
 
 K , 
 
 5-18 
 
 + 0-30 
 
 107 .. 
 
 77 5-59 
 
 137 4-92 
 
 5-26 
 
 
 
 4-90 
 
 +0-36 
 
 108 .. 
 
 78 5-04 
 
 138 4-44 
 
 4-74 
 
 "2 
 
 4-53 
 
 +0-21 
 
 109 .. 
 
 79 4-62 
 
 139 3-57 
 
 4-10 
 
 *~ 
 
 4-05 
 
 -1-0-05 
 
 110 3-32 
 
 80 3-81 
 
 140 2-95 
 
 3-38 
 
 
 
 
 111 2-64 
 
 81 3-26 
 
 141 2-50 
 
 2-38 
 
 
 
 
 112 2-17 
 
 82 2-73 
 
 142 1-99 
 
 2-36 
 
 
 
 
 The last two columns contain a comparison between the interpolation and 
 what in the present case we know to have been actuality. There is a mean 
 error of 0'20 ft. Thus it is clear that a fair record may be interpolated even 
 with so long a break as a week. . 
 
 In this example I have only shown the interpolation for one column, but 
 of course all the other twenty-three columns would really have to be treated 
 similarly. 
 
 * The days are here numbered from 1, instead of from 0. This has been the usage in India 
 hitherto. 
 
252 TREATMENT OF GAPS IN THE OBSERVATIONS. [6 
 
 I find by trial that the result would be a little improved by a graphical 
 method, but that process is slightly more troublesome than the numerical 
 one. 
 
 It may happen that the hiatus is too long for treatment in this way. 
 I do not think it would be safe to treat much more than a fortnight by 
 interpolation. 
 
 It has been shown in 4 how the tides associated with S are to be treated 
 where the record is deficient, and it remains to consider the other tides. 
 
 In 7 are given the days with which we must stop in the analysis of an 
 incomplete year, and this table affords us the means of treating a long hiatus 
 in the observation. 
 
 We may in fact omit all the entries between any two of the numbers 
 given in the table without seriously affecting the result. 
 
 Let us suppose, as an example, that the tide gauge broke down on day 
 210 and was only repaired and in operation again on day 226. Now 210 is 
 148 + 62, and 225 is 222 + 3. 
 
 Then we see by the table in 7 that in finding the means for M, 2SM, 
 MS, when the computing strips are written for the third time, we must 
 remove strips 59, 60, 61 (which have numbers written on them) and may 
 leave the remaining strips of that writing which are blank. When the strips 
 are written for the fourth time strips 0, 1, 2, 3 will be blank, but we must 
 remove strips 4 to 13 inclusive. When all the strips are used in a complete 
 year there are 369, and this is the divisor used in obtaining the harmonic 
 constants, but when there is this supposed hiatus we do not use 15 strips 
 of the third writing and 14 strips of the fourth writing, so that the divisor 
 will be 340. 
 
 Again, when we are evaluating O in the third writing, strips 57, 58, 59, 
 60, 61 must be removed, and in the fourth writing strips 4 to 9 inclusive. 
 In a complete year the divisor is 369, but we now do not use 17 strips of the 
 third writing and 10 of the fourth writing, so that the divisor becomes 342. 
 
 Again, in evaluating N, L, J, Q, in the third writing we remove strips 
 45 to 61 inclusive, and in the fourth writing strips 4 to 25 inclusive. The 
 divisor is reduced from 358 to 303. 
 
 Lastly in evaluating v, in the third writing strips 43 to 61 inclusive are 
 removed, and in the fourth writing strips 4 to 31 inclusive. The divisor is 
 reduced from 350 to 287. 
 
 Any hiatus, be it long or short, may be treated in this way, but it is clear 
 that if it be short enough to treat by interpolation, it is best to adopt that 
 method. 
 
1892] INSTRUCTIONS FOR USE OF APPARATUS. 253 
 
 INSTRUCTIONS FOR USING THE COMPUTING APPARATUS. 
 
 The apparatus for the reduction of tidal observations, together with com- 
 putation forms, can be purchased from the Cambridge Scientific Instrument 
 Company at a price (as far as can be now foreseen) of about 8*. 
 
 In case of any insufficiency in the following instructions recourse must be 
 taken to the preceding paper. 
 
 On the degree of accuracy requisite in the hourly heights. 
 
 It will usually be sufficient if the heights be measured to within one- 
 tenth of a foot, and the decimal point may, of course, be omitted in 
 computation. 
 
 This gives amply sufficient accuracy at a place where the semi-range of 
 the principal lunar tide is 2 ft., and where spring range is from 6 ft. to 7 ft. 
 
 At some places with small tides a smaller unit might be necessary, and 
 at others with very large tides a unit of 2 in., or of a fifth of a foot, might 
 suffice. 
 
 Whatever unit of length be taken it is important, for the saving of work, 
 and it is sufficient, that all or nearly all the heights should be expressed by 
 two digits. 
 
 Completion of record. 
 
 If there is an accidental break in the record, it is very important that it 
 should be completed according to the method shown in 12, or by some other 
 equivalent plan. 
 
 The computation forms are drawn up on the supposition that the year 
 of observation is complete, but with proper alterations, which will now be 
 indicated, they may be used in other cases. 
 
 In 4 it is shown how to treat the tides of the S group when the obser- 
 vations have been subject to a long stoppage in the course of the year, and 
 also when the observations extend over any period from six months to a year. 
 
 In 5 it is shown how to treat the S group for a short period of obser- 
 vation. 
 
 If the stoppage be a long one, the method explained in 12 must be 
 adopted for all the other tides. The same section also shows the treatment 
 for observations extending over any period, long or short, less than a year. 
 
 * [The price is, I believe, about 10. Each strip has to be carefully made by hand, and the 
 demand for the apparatus is of course very small. (1907).] 
 
254 INSTRUCTIONS FOR USE OF APPARATUS. [6 
 
 Entries and summations. 
 
 The computing strips are intended to take writing in pencil or liquid 
 Indian ink, but not in common ink. 
 
 They are to be cleaned with a damp cloth, and a little soda may be put 
 in the water if they become greasy. 
 
 Lay the red S sheet on one drawing board and set up the strips with their 
 ends abutting against the corresponding numbers. The strip numbered 60 
 is also to be put on the board. 
 
 Write the hourly heights for each day on the strip bearing the corre- 
 sponding number, strip for day 0, strip 1 for day 1, and so on up to strip 
 73 for day 73. The 24 hourly heights are to be written in the 24 divisions 
 of each strip, beginning on the left with O h and ending on the right with 23 h . 
 
 Remove strip 60. 
 
 Sum the 24 columns formed by the divisional marks on consecutive sets 
 of 30 strips. Thus, days to 29 afford 24 sums ; days 30 to 59 afford the 
 second set of 24 sums; days 61 to 73 afford 24 sums, which are the beginning 
 of a third 30, to be completed when the second set of 74 days shall have been 
 written on the strips. 
 
 The numbers 0, 1, 2, ..., 23, 0, 1, ..., 23 at the head and foot of the guide 
 sheet indicate the hours corresponding to the columns. 
 
 The sums of the columns on the board are to be entered in the corre- 
 sponding columns of the form " Hourly sums of S series in twelve months." 
 
 Lay the red M guide sheet on the other drawing board, and transfer the 
 strips from the first board to the new arrangement shown by the zigzag lines, 
 strip 60 being now reintroduced. 
 
 There will now be 48 columns (more exactly 47, since one of the columns 
 will be found to have nothing in it), numbered at top and bottom 0, 1, ..., 
 23, 0, 1,..., 23. Each of the 48 columns is to be summed from bottom to 
 top (not as for S in groups of 30), and the sums are to be entered in the form 
 " Sums of series M." The 24 sums which come from the left half of the 
 board will be entered in the row marked " red left," those from the right in 
 the row marked " red right." 
 
 Lay the red N sheet on the other board, and transfer the strips. 
 
 In accordance with 9, it will now usually suffice to sum .only the columns 
 appertaining to the even hours 0, 2, 4, . . . , 22 ; as these hours are repeated 
 twice, there will now be 24 columns to sum. 
 
 The sums are then to be entered on the form " Sums of series N," in the 
 alternate columns. The complete form is provided, so that all the 24 hourly 
 values may be used if it be thought desirable, but this labour seems un- 
 necessary, at least in a long series of observations. 
 
1892] INSTRUCTIONS FOR USE OF APPARATUS. 255 
 
 Lay the successive red sheets on the vacant board, transfer, sum, and enter, 
 until all the red sheets are exhausted. 
 
 In the case of S, M, MS the sums of all the columns are necessary, but 
 in the other eight arrangements only the sums of the alternate columns, 
 those of the even hours, are usually necessary. For a short series of obser- 
 vations it may be best to use all the columns, but in this case it will certainly 
 not be worth while to attempt the evaluation of v, J, Q, /*, 2SM, which are 
 all small in amount. 
 
 If the tides of long period MSf, Mf, Mm are required, the 24 numbers 
 written on each strip must be added together, and the sum entered in the 
 form "Long period tides daily sums." 
 
 Clean the strips. 
 
 In exactly the same way work through the next 74 days, from 74 d to 
 74 d + 73 d , with yellow guide sheets. Then clean the strips, and take another 
 74 days with green guide sheets, and so on with the blue and violet. 
 
 In the last (violet) set attention must be paid to the rules as to the places 
 where the analysis is to stop in each arrangement. 
 
 If the year of observation is so incomplete that the hiatus cannot be 
 made good by interpolation, or if the series does not run over the complete 
 year, the series must stop with one of the days specified in the table at the 
 end of 7, and a note must be made of the number of days used in each series. 
 
 The strips marked for omission on the violet sheets, or those selected for 
 omission under the rules of 7, may be hidden by a sheet of paper when the 
 summations are being made. 
 
 The additions of S, M, MS may be verified by proving that the grand 
 total of all the numbers (inclusive of omitted strips in S) written in each of 
 the sets of 74 days is the same in whatever way they are arranged. Thus, 
 the sum of the 48 columns should be equal to the sum of the daily sums. 
 An incomplete verification in the other arrangements, when only half the 
 columns are summed, is found by showing that the sum of all the hourly 
 sums of each 74 days is nearly equal to half the grand total of all the numbers 
 written in that period of 74 days. 
 
 When the guide sheets become worn with many pin pricks they may 
 easily be patched with adhesive paper. There seems no reason why this 
 patching should not go on almost indefinitely*. 
 
 * It is possible that it may be desired to evaluate the tides 00 and 2N, for which no guide 
 sheets are provided. I therefore give instructions for the preparation of guide sheets for these 
 cases. They will be understood by any one who has the set of guide sheets before him. With 
 the instructions given below, the computer might indeed set up the strips without a guide sheet. 
 
 I describe the staircase as descending from left to right or from right to left, and I define a 
 short step as being one space down and one space to the left or right, as the case may be, and a 
 
256 INSTRUCTIONS FOR USE OF APPARATUS. [6 
 
 Hourly sums and harmonic analysis. 
 
 Complete the summations in the forms for hourly sums, and copy into 
 the forms for harmonic analysis. In this copying it will generally suffice if 
 the last figure in the hourly sums be omitted ; for example, if the observa- 
 tions are entered to the nearest tenth of a foot the hourly sums will be given 
 in the same unit, and it will suffice if the hourly sums analysed be written to 
 the nearest foot. 
 
 There are 12 analyses (one for each month of 30 days) for the hourly 
 sums in S, and one analysis for each of the other 10 arrangements. All the 
 forms are provided with spaces for 24 hourly sums, but in the eight series 
 N, L, v, O, Q, J, yu., 2SM, where only 12 values will commonly be used, the 
 entries will only be made on the alternate rows of O h , 2 h , . . . , 22 h . In these 
 cases the divisor 12, which occurs in the penultimate stage of finding the A's 
 and B's, must be replaced by 6. 
 
 The large divisors (viz., 369 for M, /*, 2SM, MS ; 369 for O ; 358 for N, 
 L, J, Q ; and 350 for v) represent the number of days under reduction, and 
 must be altered appropriately (see table, 7) if there be long gaps in the 
 observations, or if the year be incomplete, or if the series be a short one. 
 
 If some one of the monthly analyses of S is deficient the missing ^'s and 
 23 's are to be made good by interpolation. 
 
 long step as one space down and two to the left or right, as the case may be. When I say, for 
 example, that a short follows 2, I mean that 2 to 3 is a short step. The first mark on each sheet 
 is specified by its incidence in the row of hours at the top. 
 
 00 : descending from left to right. 
 
 The sequence is long several times repeated and then short. 
 
 Red; between O h and l h ; shorts follow 2, 7, 13, 19, 24, 30, 36, 41, 47, 53, 58, 64, 69. 
 Yellow; between 15 h and 16 h ; shorts follow 1, 7, 12, 18, 24, 29, 35, 41, 46, 52, 57, 63, 69. 
 Green ; between 6 h and 7 h ; shorts follow 0, 6, 12, 17, 23, 29, 34, 40, 45, 51, 57, 62, 68. 
 Blue; between 21 h and 22 h ; shorts follow 0, 5, 11, 17, 22, 28, 33, 39, 45, 50, 56, 62, 67. 
 Violet; between ll h and 12 h ; shorts follow 5, 10, 16, 21, 27, 33, 38, 44, 50, 55, 61, 67, 72. 
 The last strip used for a year is 72. 
 
 2N : descending from right to left. 
 
 The sequence is long, long, short, long, long, short, and at intervals three longs and a short. 
 
 Bed; between 22 h and 23 h ; shorts follow 1, 4, , 16; 20, 23, , 35; 39, 42, , 57; 
 
 61, 64, , 70. 
 
 Yellow; between 18 h and 19 h ; shorts follow 2 ; 6,9, ,21; 25,28, 40; 44,47, , 
 
 59; 63, 66, , 72. 
 
 Green; between 13 h and 14 h ; shorts follow 1, 4; 8, 11, , 26; 30, 33 , 45; 49, 
 
 52, , 64; 68, 71. 
 
 Blue; between 8 h and 9 h ; shorts follow 0, 3, 6, 9; 13, 16, , 28; 32, 35, , 50; 54, 
 
 57, , 69; 73. 
 
 Violet; between 4 h and 5 h ; shorts follow 2, 5, ,14; 18,21, ,33; 37,40, ,52; 
 
 56, 59, 71. 
 
 The last strip used for a year is 61. 
 
1892] INSTRUCTIONS FOR USE OF APPARATUS 257 
 
 It is then necessary to analyse the monthly values of the ^Ts and 33's 
 derived from the 12 analyses of S. We thus obtain A , A u B 1? A 2 , B 2 , C , 
 c , d, D n Cj, dj, E , e , Ej, Fj, e^ f 1} E 2 , F 2 , e 2 , f 2 . The rules for these analyses 
 when the year is incomplete are given in 4, and the computation forms only 
 apply to the case of the complete year: 
 
 Astronomical data and final reduction. 
 
 Determine from the Nautical Almanac and Major Baird's Manual of 
 Tidal Observations* the astronomical data at O h local M.T. on day 0, and 
 proceed according to the form to find the initial arguments and factors for 
 reduction. The astronomical data are then to be used in the forms for final 
 reduction. 
 
 We have generally B = R sin f, A = R cos ; the forms are arranged so 
 that colog A is to be added to log B to find log tan , and thence . If 
 lies between 45 and 45 or between 135 and 225, log sec f is added to 
 log A to find R; if lies between 45 and 135 or between 225 and 315, 
 log cosec is added to log B to find R. Accordingly the computation form 
 has log sec , room being left for the syllable "co" if necessary; underneath 
 this is written log , and the computer will insert A or B as the case 
 may be. 
 
 There is usually required also a numerical factor 1/f or jp, or both ; the 
 logarithms of the 1/f are found amongst the astronomical data, and the 
 logarithms of the constant jjp's are printed in the forms in their proper 
 places. 
 
 The treatment of the 21 harmonic components derived from the harmonic 
 analysis of the five ^Ts and 23's is shown in the forms. 
 
 The tides of long period. 
 
 The processes involved in the evaluation of these tides are sufficiently 
 shown in the forms. 
 
 * Taylor and Francis, London, 1886, price 7s. Gd. 
 
 D. I. 17 
 
7. 
 
 ON TIDAL PREDICTION. 
 
 [Philosophical Transactions of the Royal Society of London, CLXXXII. 
 (1891), A, pp. 159229.] 
 
 TABLE OF CONTENTS. 
 
 PAGE 
 
 Introduction 258 
 
 I. Analysis 261 
 
 II. Computation 290 
 
 III. Comparison and Discussion ....... 323 
 
 INTRODUCTION. 
 
 AT most places on the North Atlantic the prediction of high and low 
 water is fairly easy, because there is hardly any diurnal tide. This abnormality 
 makes it sufficient to have a table of the mean fortnightly inequality in the 
 height and interval after lunar transit, supplemented by tables of corrections 
 for the declinations and parallaxes of the disturbing bodies. But when there 
 is a large diurnal inequality, as is commonly the case in other seas, the 
 heights and intervals after the upper and lower lunar transits are widely 
 different; the two halves of each lunation differ much in their characters, 
 and the season of the year has great influence. Thus simple tables, such as 
 are applicable in the absence of diurnal tide, are of no avail. 
 
 The tidal information supplied by the Admiralty for such places, consists 
 of rough means of the rise and interval at spring and neap, modified by the 
 important warning that the tide is affected by diurnal inequality. Information 
 of this kind affords scarcely any indication of the time and height of high 
 and low water on any given day, and must, I should think, be almost useless. 
 
 This is the present state of affairs at many ports of some importance, 
 but at others a specially constructed tide-table for each day of each year is 
 
1891] TIDE-TABLE TO BE DERIVED FROM HARMONIC CONSTANTS. 259 
 
 published in advance. A special tide-table is clearly the best sort of 
 information for the sailor, but the heavy expense of prediction and publication 
 is rarely incurred, except at ports of first rate commercial importance. 
 
 There is not, to my knowledge, any arithmetical method in use of com- 
 puting a special tide-table, which does not involve much work and expense. 
 The admirable tide-predicting instrument of the Indian Government renders 
 the prediction comparatively cheap, yet the instrument can hardly be deemed 
 available for the whole world, and the cost of publication is so considerable, 
 that the instrument cannot, or at least will not, be used for many minor 
 ports at remote places. It is not impossible, too, that national pride may 
 deter the naval authorities of other nations from sending to London for their 
 predictions *. 
 
 The object then, of the present paper, is to show how a general tide-table, 
 applicable for all time, may be given in such a form that any one, with an 
 elementary knowledge of the Nautical Almanac, may, in a few minutes, 
 compute two or three tides for the days on which they are required. The 
 tables will also be such that a special tide-table for any year may be computed 
 with comparatively little trouble. 
 
 Any tide-table necessarily depends on the tidal constants of the particular 
 port for which it is designed, and it is here supposed that the constants are 
 given in the harmonic system, and are derived from the reduction of tidal 
 observations. Where the observation has been by tide-gauge, the process of 
 reduction is that explained in the Report to the British Association for 1883, 
 but where the observations were only taken at high and low water a different 
 process becomes necessary. I have given, in a previous paper, a scheme of 
 reduction in these cases f. 
 
 At ports not of first rate commercial importance, tidal observation has 
 rarely been by tide-gauge, and thus it is exactly at those ports where the 
 method of this paper may prove most useful, that we are deprived of the 
 ordinary methods of harmonic analysis. On this account I regard my previous 
 paper as preliminary to the present one, although the two are logically 
 independent of one another. 
 
 In the harmonic method the complete analytical expression for the height 
 of water at any time consists of the sum of a number of heights, each 
 multiplied by the cosine of an angle. Each of these angles or arguments 
 involves some or all of the mean longitudes of moon, sun, lunar perigee, and 
 solar perigee ; there are, also, certain corrections depending on the longitude 
 of the moon's nodes. The variability of height of water depends principally 
 on the mean longitudes of moon and sun, and, to a subordinate degree, on 
 
 * The instrument may be used, I believe, on the payment of certain fees, 
 t " On the Harmonic Analysis of Tidal Observations of High and Low Water," Roy. Soc. 
 Proc., Vol. XLVIII., 1890, p. 278. [Paper 5, in this volume.] 
 
 172 
 
260 SKETCH OF METHOD. [7 
 
 the longitudes of lunar perigee and node for the solar perigee is sensibly 
 fixed. There are, therefore, two principal variables and two subordinate ones 
 besides the time. This statement suggests the construction of a table of 
 double entry for the variability of water due to the principal variables, and 
 of correctional tables for the subordinate ones; and this is the plan developed 
 below. 
 
 It would not be appropriate to retain the mean longitudes of moon and 
 sun as principal variables, nor the mean longitude of lunar perigee as one of 
 the subordinate variables. In the final table the principal variables are the 
 time of moon's transit, and the time of year, or strictly speaking the sun's 
 true longitude. As to the subordinate variables, the moon's parallax is 
 taken as one, whilst the longitude of the node is retained as the other. 
 
 Throughout the first part of the analytical development the moon's true 
 longitude, measured in the orbit, is taken as one of the principal variables. 
 Transition is then made to the time of transit of a fictitious satellite whose 
 right ascension is equal to the moon's true longitude, and the final transition 
 is to the transits of the real moon. This transposition of variables has 
 necessitated a complete development of the tide -generating forces from the 
 beginning, and thus the analysis of the paper is almost complete in itself, 
 although reference is necessarily made to the harmonic method. 
 
 Such a piece oi work as the present can only be deemed complete when 
 an example has been worked out to test the accuracy of the tidal prediction, 
 and when rules have been drawn up for the arithmetical processes, forming a 
 complete code of instructions to the computer. The example below is 
 intended to carry out these requirements. 
 
 I chose the port of Aden for the example, because its tides are more 
 complex and apparently irregular than those of any other place, which, as far 
 as I know, has been thoroughly treated. The tidal constants for Aden are 
 well determined, and the annual tide-tables of the Indian Government afford 
 the means of comparison between my predictions and those of the tide- 
 predicting instrument. 
 
 The arithmetic of the example has been long, and the plan of marshalling 
 the work has been rearranged many times. An ordinary computer is said to 
 work best when he is ignorant of the meaning of his work, but in this kind 
 of tentative work a satisfactory arrangement cannot be attained without a 
 full comprehension of the reason of the method. I was, therefore, very 
 fortunate in securing the enthusiastic assistance of Mr J. W. F. Ailnutt, 
 and I owe him my warm thanks for the laborious computations he has carried 
 out. After computing fully half of the original table, he made a comparison 
 for the whole of 1889 between our predictions and those of the Indian 
 Government. 
 
1891] SKETCH OF METHOD. 261 
 
 I had hoped that tables, less elaborate than those exhibited below, might 
 have sufficed. It appeared, however, that the changes during the lunation 
 and with the time of year, which affect the height and interval, are so abrupt 
 and so great, that short tables would give very inaccurate results, unless used 
 with elaborate interpolation. We can clearly never expect sailors to use 
 tables of double entry, in which interpolation to second and third differences 
 is required ; and indeed any interpolation is objectionable. It is proposed, 
 therefore, that the tables shall be made so full that interpolation will be un- 
 necessary. This plan, of course, throws the whole of the interpolation on 
 to the computer, and although extensive interpolation even when done 
 graphically is tedious, yet it is obviously best to have it done once for all, 
 rather than piecemeal by the user of the tables. 
 
 The first part of the paper gives the analytical development, the second 
 contains the numerical example and rules of computation, and in the third 
 I give some account of the comparison with other predictions and with 
 actuality and suggestions for abridgements in cases where less accuracy shall 
 be thought sufficient. 
 
 The analytical formulae of the first part are much scattered, and it may 
 not always be easy to see whither they tend ; but the second part virtually 
 contains a summary of the first, and a comparison between these two parts 
 will show both the reasons for the analysis and those for the arithmetical 
 processes. 
 
 I have tried to make the instructions to the computer so complete that 
 he need not be troubled with the reason for his operations ; but if there is, 
 through inadvertence, any incompleteness, reference must be made to the 
 analytical development for interpretation and instruction. 
 
 PART I. ANALYSIS. 
 
 1. Development of the Tide-Generating Potential. 
 
 Let A, B, C (fig. 1) be axes fixed in the earth, AB being the equator and 
 C the north pole. 
 
 Let r, p be respectively the radius-vectors of M the moon, and of P any 
 other point. Let M 1 , M 2 , M 3 be the direction cosines of the moon, and 
 i> a> s those of P, both referred to the axes A, B, C. 
 
 Then by the usual theory the potential V, of the second order of 
 harmonics, is 
 
 ) ........................ (1) 
 
 where M is the moon's mass, and p. the attractional constant. 
 
262 DEVELOPMENT OF THE TIDE-GENERATING POTENTIAL. 
 
 But since cos PM = ^M^ + % 2 M 2 + 
 
 cos 2 PM - 1 = 2 1 f 2 . M^ + 2 . ^-~ . ^ 
 
 (2) 
 
 (3) 
 
 FIG. 1. 
 
 Now let X, Y, Z, fig. 1, be a second set of rectangular axes, fixed in space ; 
 let M be the projection of the moon in her orbit XY. Let 7 = ZC, the 
 obliquity of the lunar orbit to the equator ; let x = AX = BOY, and -\Jr = MX, 
 the moon's longitude measured in her orbit from the point X. 
 
 From the figure it is clear that 
 
 M l = cos ty cos x + sin -^ sin x cos 7 
 
 = cos 2 \I cos (x ~ A / r ) + sin 2 \I cos (x ^~ 1 / r ) 
 M z = cos \|r sin x + sin -^ cos x cos f 
 
 = cos 2 \I sin (x ~ ^) ~ sm2 \J sm (x ~^~ ^ 
 M s = sin 7 sin -^r 
 
 Hence from (3) 
 
 M^ M 2 - = cos 4 ^7 cos 2 (x ~ JV / r ) 
 
 + | sin 2 7 cos %x + s i n * \I cos 2 (^ + ^) 
 ! = cos 4 \I sin 2 (x ^) 
 
 + ^ sin 2 7 sin 2^ + sin 4 17 sin 2 (^ + -\|r) 
 T, = sin -^7 cos 3 ^/ cos (^ 2-v/r) 
 
 + ^- sin 7 cos 7 cos ^ + sin 3 7 cos |7 cos (^ + 
 3/jMa = sin ^7 cos 3 ^7 cos (x 2^) 
 
 + ^ sin 7 cos 7 sin ^ + sin 3 ^ 7 cos \I sin (x + 
 l-M* = - -i sin 2 7 + 1 sin 2 7 cos 2i/r 
 
 In order to simplify the result of the substitutions from (4) into (2) and 
 (1), the axes fixed in the earth may be taken as follows : 
 
1891] DEVELOPMENT OF THE TIDE-GENERATING POTENTIAL. 263 
 
 The axis A on the equator in the meridian of the place where the tides 
 are observed, B 90 east of A, and C as already stated at the north pole. 
 
 Then if X be the latitude of the place of observation 
 
 = cos X, > = 0, 3 = sin X 
 
 Hence (2) becomes 
 cos 2 PM - = i (M* - MS) cos 2 X + M,M 3 sin 2X + f ( - M./) ( J - sin 2 X). . .(5) 
 
 We shall therefore only require the first and the last two of (4), and these 
 may be considerably simplified. 
 
 The angle 7 ranges between 23 27' 5 9' ; hence the term in Mi>-M 2 * 
 which involves sin 4 \I is negligible, and similarly that which involves 
 sin 3 \I cos \I in M^M^ may be omitted. 
 
 Further, it is only necessary to develop V in as far as it is variable ; now 
 as / ranges from 18 18' to 28 36', and back again in the course of 19 years, 
 sin 2 7 oscillates slightly in value about a mean. The " tides of long period " 
 corresponding to the term sin 2 7 cos 2^r in ^ M./, are, although sensible, 
 so small that we shall take no account of them ; a fortiori the variability of 
 J ^ sin 2 7 is negligible. It would be easy, however, to take account of these 
 terms if it were desirable to do so. 
 
 We need therefore only pay attention to the first and fourth of (4), and 
 the last term in each may be omitted. 
 
 In fig. 2 let M be the moon in her orbit, and let A be the axis A of 
 fig. 1 fixed in the earth. 
 
 FIG. 2. 
 
 Then let v be the right ascension of the point I, and its longitude 
 measured in the moon's orbit. 
 
 Let a) be the obliquity of the ecliptic, i the inclination of the moon's orbit, 
 S3 the longitude of the moon's node. 
 
 Let t be the mean solar hour angle at the place and time of observation, 
 h the sun's mean longitude, and I the moon's longitude measured in the orbit. 
 
264 DEVELOPMENT OF THE TIDE-GENERATING POTENTIAL. 
 
 Then, by the definition of ^ and \fr, we have 
 
 [7 
 
 Since v = Vl, = T S3 - S3 1, V S3 = S3 , and t + h the sidereal hour angle, 
 
 ...(6) 
 
 Thus to the degree of approximation adopted, (4) becomes 
 Mf - M* = cos 4 7 cos 2 (< + A - I - v + f) + sin 2 /cos 2 ( + A - 1) 
 cos 3 7 sin 7 cos ( + h - 21 + |TT - v + 2f) 
 
 + 1 sin 7 cos 7 cos ( 4- h |TT - v) 
 
 We must now obtain approximate formulae to express the functions of 
 7, i/, and in terms of &>, *', and S3 . The inclination i may be treated as so 
 small that its square may be neglected. 
 
 From fig. 2 we have 
 
 cos 7 = cos i cos &) sin i sin &> cos S3 
 
 cos 7 = cos &> i sin &> cos S3 
 cos 2 7 = cos 2 ^o (1 * tan ^ &> cos S3 ) 
 sin 2 ^7 = sin 2 |o> (1 + i cot ^<w cos S3 ) 
 cos 4 7 = cos 4 ^<w (1 2i tan o> cos S3 ) 
 
 sin 2 7 = sin 2 o> (1 4- 2i cot &> cos S3 ) 
 
 whence 
 
 cos 3 
 
 sn = cos 3 
 
 sn 
 
 . 2 cos &) 1 
 
 1 + I ; COS 
 
 sin &) 
 
 sin 7 cos 7 = sin &) cos G> (1 + 2i cot 2&> cos S3 ) 
 Again from the figure 
 
 sin v _ sin S3 _ sin ( S3 
 sin i sin 7 sin o> 
 
 Since is small, 
 
 Therefore 
 
 -. f. sin &) 
 
 1 E cot b6 = . f =li cot ft) cos S3 
 sin 7 
 
 sm i sin S3 
 sin 7 
 
 .sin S3 
 sin &) 
 
 = i cot to sin S3 
 v % = i tan |&) sin S3 
 . 2 cos &) 1 
 
 sm 
 
 sm S3 
 
 ,(8) 
 
1891] 
 
 DEVELOPMENT OF THE TIDE-GENERATING POTENTIAL. 
 
 265 
 
 Hence from (7) and (8) 
 
 cos 4 |7 cos 2 (v - ) = cos 4 ift) (1 2t tan \u> cos & ) 
 cos 4 |7 sin 2 (y - |) = cos 4 i&> . 2t tan i sin & 
 
 i sin 2 7 cos 2v = sin 2 &> (1 + 2i cot w cos a ) 
 sin 2 7 sin 2v = i sin 2 ft> . 2i cosec &> sin S3 
 
 cos 3 ^7 sin ^7 cos (2 ;;) = cos 3 &> sin i&> [1 + i (2 cos <u 1) cosec to cos S3 ] 
 cos 3 17 sin ^7 sin (2 z>) = cos 3 |to sin i&> . t (2 cos <w 1) cosec CD sin S3 
 1 sin 7 cos 7 cos ^ = sin &> cos &> (1 + 2i cot 2&> cos S3 ) 
 ^ sin 7 cos 7 sin i = ^ sin <a cos <w . i cosec co sin S3 
 
 (9) 
 
 Now < = 23 27S3, i = 5 8''8 ; whence 
 
 2i tan |<w = -0372, 2t cot w = 
 
 283 
 
 t cot 2<o = 
 
 683' 
 115 
 
 2i cosec co 
 
 683' 
 
 308 
 683' 
 
 154 
 
 I COSeC ft) = -vj^ 
 
 'boo 
 
 . 2 cos to 1 
 
 sm <u 
 
 Four of these are written with *683 in the denominator for reasons given 
 below. Then from (9) and (6) we have 
 
 M* - M? = cos 4 la) cos 2 (t + h - I) + i sin 2 <w cos 2 (t + h) 
 
 + j- cos 4 ift) . -0372 cos 2 (t + h - 1) 
 
 '283 
 
 + i sin 2 a) . -7? -- cos 2(t + h)\ cos 
 
 "Doo 
 
 . '0372 sin 2(t + h-l) 
 "308 
 
 sn w . 
 
 2 (^ + h)\ sin S3 
 
 . 
 
 "DoO 
 
 = cos 3 i&) sin |&) cos (^ + h 21 + ITT) 
 
 + ^ sin w cos ft> cos ( + ^ ^TT) 
 
 + | cos 3 G> sin ^o> . 188 cos (t + h - 21 + TT 
 
 '115 
 
 + A sin &) cos w . - ^ cos (t + h 
 "boo 
 
 + - cos 3 ^0) sin |ft) . '188 sin (t + h - 21 + 
 
 '154 
 
 + A sin &) cos &) . -^ r^ sin (^ + ^ 4)f sin S3 
 
 
 3 , and 
 
 cos S3 
 
 Now let c be the moon's mean distance, and let T f 
 
 r 3 
 1+JP-? 
 
266 DEVELOPMENT OF THE TIDE-GENERATING POTENTIAL. [7 
 
 Then as far as concerns the moon 
 
 V=T?(I+P){$cQS*\(M 1 *-M a a ) + sm2\.M,M 3 } ......... (11) 
 
 where M-f Mf, M^M^ are given by (10). 
 
 In writing down the corresponding functions for the sun, we shall write 
 a subscript accent to all the symbols, and accordingly V for the sun is 
 given by 
 
 V, = r,p n - (1 + P,) { cos 2 X (MJ - M/} + sin 2X . M tl M l3 ] ...... (12) 
 
 where, by symmetry with (10) 
 
 MJ - M/ = cos 4 ift> cos 2 (t + h - Z,) + 1 sin 2 <u cos 2(t + Ji)"\ 
 
 M tl M, a = cos 3 o> sin i<y cos (t + k - 2/, + ITT) I . . .(13) 
 
 + sin to cos to cos (t + h ^TT) 
 The masses and mean distances of the moon and sun are such that 
 
 ,..(14) 
 
 T + T, T + T, 
 
 Strictly speaking, T denotes f//.Mc~ 3 (l +fe 2 ), and T, denotes 
 
 where e, e, are the eccentricities of the lunar and solar orbits. A comparison 
 with the Report on Tides to the British Association for 1883 [Paper 1], will 
 show that, for the purpose in hand, it is correct to introduce these factors 
 involving e 2 and ef. 
 
 The whole potential may be collected from (10), (11), (12), (13), (14). 
 
 Since P is small the factor (1 + P) may be treated as unity in the small 
 terms which involve cos Q and sin 8, and we shall find it convenient to 
 distinguish the several parts of V, denoting by V the principal part, by Vp 
 the part involving P, by V& the part involving S3, and by V Pl the part 
 involving P y . 
 
 For brevity write 
 
 K = p" 2 . ^ cos 2 \ . cos 4 ^&> ; K tl = p".^ cos 2 X . sin 2 &> 
 
 K = p 2 . sin 2X . cos 3 ^<w sin ^to ; K t = p- . sin 2X. . i sin co cos &> 
 
 Then remarking that 
 
 approximately, we have 
 
 V= rKcos 2(t + h-l) + (T + T,) K tl cos 2(t + h) + r,Kcos 2(t + h - 
 + rK cos (t + h-2l + ITT) + (T + T,) K, cos (Z + h - TT) 
 
1891] FORMULA FOR THE HEIGHT. 267 
 
 + rK cos (t + h-2l + + '683 (r + T,) K, cos (t + h- $TT)J 
 
 V& f= cos S3 {- -rK "0372 cos 2(t + h-l) + (r + T ) K tl "283 cos 2 (S + A) 
 
 + rK '188 cos(t 4 h - 21 + ITT) + (T + T,) K, "115 cos ( + h - |TT)} 
 4 sin S3 {+ rK "0372 sin 2(t + h-l) + (r + T,) # "308 sin 2 ( + A) 
 - r^o '188 sin (< + A - 21 4 TT) 4 (T 4 T,) /f, 154 sin (< -f h - JT)| 
 
 F P/ = P, {'317 (T + T/ ) # cos 2 (Z 4 A) 4- r,K cos 2 (< + A - 1,) 
 
 + -317 (T 4 T,) ^ cos (t 4 A - ITT) 4 r t K cos ( 4 A - 2Z, 4 fri)} . . .(15) 
 The whole tide-generating potential is then V+ V P + F a 4 Fp^ 
 
 2. Formula for the Height of Tide. 
 
 If we had been going to consider the complete expression for the tide 
 in terms of a series of simple harmonic functions of the time, it would have 
 been necessary to substitute for I, I,, P, P t their values in terms of the mean 
 longitudes s and A of the two bodies, and of the eccentricities e, e t of their 
 orbits. When the potential is so expanded the principle of forced oscillations 
 allows us to conclude that the oscillations of the sea will be of the same 
 periods and types as the several terms of the potential, but with amplitudes 
 and phases which can only be deduced from observation. The oscillations 
 of the sea will not however be necessarily of the simple harmonic form, and 
 accordingly "over-tides" of double and triple frequency have to be introduced 
 in order to represent the motion according to Fourier's method. 
 
 This is the plan pursued in the "harmonic analysis of tidal observations," 
 and each simple harmonic oscillation is known by an arbitrarily chosen initial 
 letter. 
 
 It is found in fact, as is suggested by theory, that tides of approximately 
 the same frequency or "speed" have amplitudes approximately proportional 
 to their corresponding terms in the potential, and have their phases retarded 
 by approximately the same amount. 
 
 The notation of harmonic analysis will be adopted here, because it is 
 proposed to compute the tide-table from the harmonic constants. 
 
 The mean longitudes of the sun, moon, and lunar perigee are denoted by 
 A, s, p, and their hourly changes by <r, T/, -57 (from o-eXijvrj, rpuo?, perigee); 
 the angular velocity of the earth's rotation is 7 (from yrj), so that y rj is 15 
 per hour. 
 
 The eccentricities of the lunar and solar orbits are e, e t . 
 
 In the harmonic system the tides are denominated by the arbitrarily 
 chosen initials M,, M 4 , S 2 , S 4 , K 2 , K 1} N, L, v, \, p, T, R, O, P, Q, J, Sa, Ssa, 
 
268 FORMULA FOR THE HEIGHT. [7 
 
 and the semi-ranges of these tides will be here denoted by the suffixes 
 m, 2m, s, 2s, n, I, v, X, p, t, r, o, p, q, j, sa, ssa to the symbol H, and the 
 retardations of phase by the same suffixes to the symbol K. But the semi- 
 ranges and retardations of the tides Kg, K^ are denoted by H //; x it , H,, K, 
 instead of conforming to the rest of the notation*. 
 
 Then all the H's and AC'S are the immediate results of harmonic analysis, 
 and are supposed to be given when the construction of a tide-table is 
 contemplated. 
 
 In the development (15) of V, the first term corresponds with the 
 principal lunar tide M 2 , its over- tide M 4 , parts of the elliptic tides N, L, 
 parts of the evectional tides v, \, and part of the variational tide /t. 
 
 The first term of Vp contributes the remainder of N, L, v, \, p. 
 
 The second term of V corresponds with the luni-solar semi-diurnal tide 
 K 2 , and the second term of Vp corresponds with small inequalities in K 2 , 
 which are neglected in the harmonic analysis. 
 
 The third term of V corresponds with the principal solar tide S 2 , its over- 
 tide S 4 , and parts of the elliptic tides T, R. 
 
 The fourth term of V corresponds with the lunar diurnal tide O, and 
 parts of the elliptic tides Mj and Q ; the third term of V f corresponds with 
 another part of M t and the rest of Q. 
 
 The fifth term of V corresponds with the luni-solar diurnal tide K 1} and 
 with part of the elliptic tides Mj and J; its over-tide fuses with K 2 . The 
 fourth term of Vp corresponds with part of the elliptic tide Mj, and the rest 
 of J. 
 
 The last term of V corresponds with the solar diurnal tide P, and with 
 small inequalities neglected in the harmonic analysis. 
 
 The whole of V& corresponds with those factors of augmentation and 
 small alterations of phase, which are denoted by f and functions of v and in 
 harmonic analysis. 
 
 Vp t corresponds with the remainder of the solar elliptic tides T, R and 
 with other small inequalities usually neglected in the harmonic analysis. 
 
 The semi-annual (Ssa) and annual (Sa) tides are due to meteorological 
 causes, and have to be introduced after we have done with astronomical 
 considerations, so that they do not enter through the F's. 
 
 Since the terms V, Vp, VQ, Vp t are approximately simple harmonics, it 
 follows that each term will correspond to an approximately simple harmonic 
 
 * I adopted this originally for convenience in writing, and having got used to the notation 
 hall retain it. 
 
1891] FORMULA FOR THE HEIGHT. 269 
 
 tide, with amplitude and phase the same as that given by harmonic analysis, 
 and to its over-tides. If then we write 
 
 -2l- Km ........................... (16) 
 
 the height of water corresponding to V is 
 
 h = H m cos A + H,, cos (A + K m + 21- K tt ) + H s cos (A + tc m + 21 21, K S ) 
 
 + H COS [^ ( A + Km) - I + fa ~ Ko\ + H, COS [| (A + K m ) + l-$ir-K,] 
 
 p cos [ (A + K m ) + l-2l, + %7r- Kp] + H aia cos [2 (A + ,) - K 2m ] 
 
 We shall, in the developments immediately following, leave Vp t out of 
 consideration, and shall resume the effects of the sun's parallax in 11. 
 
 In order to represent, as far as possible, the elliptic tides, it is found to 
 be necessary to introduce certain numerical factors a, /3, and certain angles 
 m K, O K, as shown in the next formula. It is also practically convenient that 
 P should denote the value of c 3 /^, riot at the time corresponding to t, but 
 some hours earlier. 
 
 Then the height of water corresponding to Vp is taken as 
 h P = P {H, n cos (A + K m - mic) + '683H,, cos ( A + Km + 21 - ) 
 
 + /3H COS [ ( A + K m ) - I + ^7T - K] 
 
 + -Q8SH f coa[^ (& + K m ) + l-frr-K ( ]} ............... (18) 
 
 The height corresponding to V& is 
 
 h a = cos S3 {- H OT '0372 cos A + H,, '283 cos (A + Km + 21- ) 
 + H -188 cos [ (A + K m ) - 1 + TT - K ] 
 + H, -115 cos [| (A + K m ) + l-$-ir- *J} 
 + sin ga {H m '0372 sin A + H,, '308 sin (A + m + 21 - KII ) 
 - H '188 sin [i (A + ,) - 1 + %TT - K O ~\ 
 + H, -154 sin [(A + K m ) + 1 - far - K,]} ............... (19) 
 
 3. The Elliptic Tides. 
 
 The constants a, y8, m , O K, are insufficient to represent fully the harmonic 
 tides N, L, v, X, /x,, Q, J, and it would render the proposed method of 
 computing a tide-table too cumbrous for practical use if additional constants 
 were introduced. It is, therefore, necessary to adopt a compromise. 
 
 The tides N and L of the harmonic system are given by 
 H 7l cos [2t + 2h-2s-(s-p)- K n ] + Hj cos [2t + 2h - 2s + (s - p) + TT - KI] 
 
 In the present development the mean lunar tide M 2 and the elliptic tides 
 are to be represented by 
 
 H m cos [2t + 2h-2l- K m ] + H m P cos [2t + 2h-2l- m tc] 
 
270 THE ELLIPTIC TIDES. [7 
 
 It has been already remarked that it will be convenient that P should 
 refer to a time earlier than t, and the time referred to is that of the moon's 
 transit. Now H.W. occurs later than moon's transit by an interval whose 
 mean value is K m /2 (7 a-), and the mean value of the interval to the 
 
 preceding L.W. is (ic m ?r)/2 (7 <r). If, therefore, we put = 
 for H.W., and ^~ r (ic m TT) for L. W., we have approximately 
 
 also 21 = 2s + 4>e sin (s p) 
 
 Hence the elliptic tides are represented in the present development by 
 2eH w cos (2t + 2/t -3s+p- *,) + \ aeR in cos (2t + 2h,-3s+p- ,* + ) 
 + 2eH w cos (2t + 2/t s p + TT tc m ) - f eH m cos (2t + 2k-s p + TT - m * ) 
 
 In order that the first pair of these terms may give the N tide correctly 
 we must have 
 
 H n = 2eH, n cos ( - *,) + |eH m cos (K U - M K + ) 
 = 2eH m sin (* n - /c m ) + f aeH m sin ( - M K + f ) 
 And the condition that the second pair may give the L tide is 
 Hj = 2eH m cos (K L - K m ) - |aeH m cos (K L - m <c - ) 
 = 2eH m sin (K L - K m ) - |aeH m sin ( KI - m tc - ) 
 The condition for N may be written 
 
 2eK m sm(K m -fCn) 
 tan (, - TO A; + =- _ 2 H . _ x 
 
 ri n ^en TO cos ^ m K H ) 
 
 f aeH m = (H n - 2eH m cos (/c w - *)) sec (x n - m /c + 
 
 Then, since both /c n K m and /c ?l m ic + are small angles, we have 
 approximately 
 
 , u 2e 
 
 Similarly the condition for L gives 
 
 (ie t - /c in ) 
 
 _ ,, 
 
 These conditions cannot be satisfied simultaneously and a compromise 
 must be adopted. 
 
1891] THE ELLIPTIC TIDES. 271 
 
 If H; were zero we should have a = . Hence if a be taken as greater 
 than f we arc virtually making the L tide negative. Now, it is unadvisable 
 to take the L tide as negative even if small, because, if so, we are representing 
 the elliptic tide compounded from N and L as greatest when it should be 
 smallest and vice versa. 
 
 I therefore propose to use the condition for the L tide only for the 
 purpose of putting a superior limit to a, and to the constants to be derived 
 from the condition for the -more important N tide. 
 
 In order to express the result in the form which will ultimately be 
 required I write 
 
 _27-3<r + r _ 27 - 2o- _ 1 
 
 Pn ~ 27-20- ' Pm ~2y^2^~ 
 
 so that p n is the ratio of the "speed" of the N tide to that of M 2 , and p m is 
 merely introduced for the sake of algebraic symmetry. 
 
 Then 
 
 = (p m - p n ) Km for H. W. and (p m - p n ) (x m - TT) for L.W. 
 We thus have approximately, when referring to H.W. 
 
 Zeti m \K m K n ) 
 
 Pm K m Jft K Pn K m ^n ~i TT f) a fl 
 
 " *0***l 
 
 2 
 
 and H m is equal to the smaller of ^- (H n 2eH m ) and |H TO 
 
 oe 
 
 When referring to L.W. we have (rc m TT) in the above formula in the 
 place of K m wherever x m occurs multiplied by a p. 
 
 A similar argument may be followed with regard to the diurnal tides, 
 save that the smaller tide corresponding to L, has not usually been evaluated 
 in the harmonic analysis. The tide corresponding to N bears the initial Q, 
 and 
 
 7 3<r + or 7 2<r 
 
 Pq= 27-2^"' P ^fy^- 
 
 so that p q and p are the ratios of the " speeds " of the Q and O tides to that 
 of M. 2 . Then 
 
 cr TIT 
 
 7 \ == PO Pq 
 
 (7 <j) 
 
 and we find for H.W. 
 
 2eH (K O - Kg) 
 
 PO K m O K Pq K m K q ' TT 
 
 2 
 
 and /3H is equal to the smaller of ^- (H g - 2eH ) and |H 
 
 2 
 
 Since e = '0549, =- = 12144 = 19 x '639 (the factor being introduced 
 06 
 
272 THE ELLIPTIC TIDES. [7 
 
 because we shall hereafter have to divide by 19) ; also f=19x ? 4 T = 19x '070, 
 
 and J- 911. 
 
 2e 
 
 If we put then a' = -fay, /3' = fa/3, our formulae for H. W. become 
 
 \ tn ' *^7i) 
 Pm^rn m^ == Pn^m K n + Q.I i TT /TT 
 
 ^^" ............... (20) 
 
 (Kg Kg) 
 
 where p m = l,p n = '981, p = '481, p q = '462, and where for L.W. K m is to be 
 replaced by K m IT wherever it is multiplied by a p. 
 
 Also, if 36'4H n is less than 8H m 
 
 a = 12-14|p-f 
 
 U--070H, 
 
 If 36-4 H fl is less than 8H fl 
 
 = 12-14=;?- 
 
 .(21) 
 
 lg --070H / 
 
 If 36'4H n is greater than 8H m , = , and a'H m = '070H m . 
 If 36'4H 9 is greater than 8H , /3 = |, and j3'H = -070H . 
 Lastly, if the elliptic tides are unknown, a = /3 = 1, a' = /8' = yg. 
 
 4. Reference to Transit of Fictitious Moon. 
 
 Suppose that there is a fictitious satellite whose R.A. is equal to the 
 Moon's longitude, and let l (0) , h (0) be the R.A. of the fictitious moon and of 
 the mean sun at noon of the day under consideration, and let T be the mean 
 solar time of the fictitious moon's transit. 
 
 Then at transit t + h I = qjr 
 
 where q is even for upper and odd for lower transit. 
 
 The rate of change of t + h is 7, and if I be the rate of change of /, T is 
 given by 
 
 l(o) fi(o) -f qrTr 
 
 Then at time r + T 
 
 A 9# _L 9/j _ 97 u- 
 ^L T~ <U/i A\J Knt 
 
 and | A = q-rr + (y - V) T - %ic 
 
1891] REFERENCE TO TRANSIT OF FICTITIOUS MOON. 273 
 
 Now A occurs in the arguments of all the semi-diurnal tides, ^A in those 
 of the diurnal, and 2A in those of the quater-diurnal. 
 
 Hence we may omit the 2qir in the expression for A, and write 
 
 provided that when q is odd we change the signs of the diurnal terms. 
 
 Therefore we may write the diurnal terms with alternative signs + , and 
 the upper sign will be appropriate when we refer to an upper transit, and the 
 lower sign to a lower transit. 
 
 In the application of the formulas T + T will be the time for which the 
 mean solar hour angle is t, and we shall have 
 
 K A 
 
 T^m, I /OO\ 
 
 ; "T" \^LI&) 
 
 The first of these terms is obviously the interval from fictitious transit to 
 the mean lunar H.W., and the second is the interval from lunar H.W. to the 
 time t. 
 
 When we wish to discuss low waters it is convenient to put A equal to 
 8 TT, and in this case we shall have 
 
 T= Km ~^ + ^ ...(23) 
 
 2 (7 - 2 (7 - 
 
 The first of these terms is the interval from transit to the mean lunar 
 L.W., which precedes the H.W. given by (22), and the second is the interval 
 after lunar L.W. to the time t. 
 
 We shall proceed to consider H.W., and deduce therefrom the formulas 
 for L.W. 
 
 5. Reduction of Longitudes of Moon and Sun to time of 
 Fictitious Transit. 
 
 W T e have seen in (22) that the interval, from fictitious moon's transit to 
 the time t, is 
 
 = 2( 7 -/) 
 Now the equation of conservation of areas for the moon's motion gives 
 
 rH = o-c 2 V(l - e 2 ) 
 
 but since c'/r 3 = 1 + P, and since P is small, we have 
 
 I = a (1 -f |p) } very nearly 
 
 Therefore T= ^ + Km \ (\ + 
 
 2( 7 -<r)V 3 7 - 
 
 IT = ^-+^ i l" + |p ??_ (A + *j 
 
 D. I. 18 
 
274 REFERENCE TO TRANSIT OF FICTITIOUS MOON. 
 
 2 7 - 2o- 2 7 ^ - ' 
 
 Let Pi-SrroL. ^ = 977-9^ P = 
 
 2y-2<r' 2 7 -2o-' 2 7 - 2<r 
 
 JT o. O^. ' /^/ oil o^- ' -"* 97! 9^- 
 
 and let e = = - 1 = '01311 
 
 Then if 1 , 1 0/ be the moon's and sun's longitudes at the time of fictitious 
 transit, we have 
 
 1 = l + - (A + Km) + eP ( A + K ^ 
 
 l ' = lo/ + 2^2a ( A + * w) + e 7 P ( A + * m) 
 
 Then neglecting the term in P when multiplied by the small fraction 77/7, 
 and introducing the p's, we have 
 
 21 = 21 + ( Pli - 1) (A + O + 2eP (A + /O \ 
 
 2 (/ - = 2 (J - Z 0/ ) + (p. - 1) (A + * m ) + 2eP (A + m ) 
 
 _ / = - 1 + ( Po - 1) (A + m ) - eP (A + O ...(24) 
 
 1-21, = 1 - 21 0/ + (p p - i) (A + /c m ) + eP (A + *) 
 
 We must now take the several terms of the expressions for the tide in 
 (17) (18) (19), and rearrange them by aid of (24). 
 
 (i.) cos (A + Km + 21 - *) + -683P cos (A + m + 21 - ) 
 
 = cos [p tl (A + K m ) + 21 - K/l ] + -683P cos [ tl p (A + m ) + 21 - J 
 
 where ^ = p + ^- = p + '0384 = 1'0379 + '0384 = 1-0763 
 'boo 
 
 (ii.) cos (A + Km + 2(1-1,)- *,) = cos [p 8 (A + m ) + 2 (1 - 1 0/ ) - ,] 
 
 - 2eP (A + J sin [p s (A + w ) + 2 (Z - / 0/ ) - *J 
 
 Now the maximum value of P is about ^, and A + m will not differ 
 largely from te m , because A will oscillate about the value zero, and /c m might 
 always be taken as less than 180, either positively or negatively. Hence 
 the coefficient of this second term cannot exceed 2 x '013 x ^ x TT, or about 
 013. 
 
 Thus the second term cannot be more than about an eightieth of the 
 first, and may be neglected. This term may also be safely neglected, even 
 when /c m is greater than 180. 
 
 (iii.) cos [ (A + m ) - 1 + TT - *] + P/3 cos [ (A + *J - I + \TT - O K] 
 
 = cos [p (A + K m ) -IO + ITT- K O ] + P cos [ O p (A + K in ) - 1 + TT - O K] 
 
 e -0131 
 
 where p = p o -- = -4811 -- -g- 
 
1891] REFERENCE TO TRANSIT OF FICTITIOUS MOON. 275 
 
 The reader who verifies the above formula will perceive the nature of the 
 approximation adopted. 
 
 (iv.) cos [| (A + K M ) + l-^7r- K/ ] + -683P cos [ (A + K J + 1 - ^ - *,] 
 = cos [p, (A + K m ) + I -^7r- K,] + '683P cos O (A + x m ) +I -far- *,] 
 
 where . =p, + -^=p t + '0192 = "5189 + '0192 = '5381 
 
 'boo 
 
 (v.) The P term may be written 
 
 cos [p p (A + K m ) + 1 - 21 0/ + |TT - Kp \ 
 with a neglect similar to that involved in (ii.), but, of course, less important. 
 
 Before using these transformations the following notation must be 
 introduced 
 
 -'- (25) 
 
 = *,+ 5 
 So that 1 = J) + - 5, 1 0/ = Q- 5. We then write 
 
 ** in Pm *m m 
 
 m *=P m *m ~ m K ( see ( 2 )> 3) 
 
 = 2J +p 8 fc m -K 8 
 
 = -) + - 5 + K - K + 90 
 
 ^ = - () + - 5) + #,* w - O K - m + 90 (see (20), 3) 
 
 P 
 
 ^ = (D + Q-5) + p /Km - K/ -W 
 
 > = ^ + CP - ^) * = ^ + '0192^ 
 
 ^ = (D- + 5) +ft,* M -* p + 90 .............................. (26) 
 
 We also require for the quater-diurnal tides 
 
 ~2m == ^Pm^m ~~ KWH 
 ;T M = Zp s K m K^g 
 
 and for the annual and semi-annual tides 
 
 ^ = 2(0-5)-^ ........................ (26) 
 
 The parts of the ^'s which are independent of }) and will, in the 
 example, be written K, with appropriate suffix. 
 
 The moon's mean parallax is 57', then when its parallax is 57' + FT, 
 
 so that P = j'-gll = '052611 approximately. 
 
 182 
 
276 REFERENCE TO TRANSIT OF FICTITIOUS MOON. [7 
 
 II will be substituted for P, and the numerical coefficients in h P will be 
 divided by 19 or multiplied by '0526. 
 
 We observe then that ^ x "683 = "0360, and ^ = a', ^ = /3'. 
 
 We can, by aid of the transformations (i.) to (v.) and the formulae (26), 
 now re-write (17) (18) (19), but some part of what was originally included in 
 h is transferred to h P . The result is as follows : 
 
 h = H m cos (p m A + X) + H y/ cos (p tl A + ) + H s cos (p, A + ,) 
 + Ho cos (p A + * ) H, cos (p, A + $,) H p cos (p p A + * p ) 
 + Hcos(2# n A+* m ) + H M cos(2^A+Sr M ) (27) 
 
 A P = H {'H TO cos (p m A + m *) + -0360H,, cos ( /|P A + 
 
 /ra o cos(^A + *)-0360H,cos( / pA + >)} (28) 
 
 A a = cos s {- H m -037 cos (^ m A + * m ) + H,, "283 cos (p, A + *) 
 
 H 188 cos ( Po A +^ ) + H, -115 cos (jp,A + *,)) 
 + sin 3 {H ro '037 sin(p m A+^ TO )+ ^'3088^^^+^) 
 
 + H -188sin(p A+^)H/154sin(p / A + ^ / )} (29) 
 
 Annual and semi-annual tides = H ga cos *b s a + H ssa cos ^ ssa (30) 
 
 Except for the solar parallactic portion to be considered later, this is the 
 final expression for the tide. Apart from the slow variability of H and 83 
 the only variable is A, for the ^'s are not continuous variables, but change 
 per saltum from one lunar transit to the next. Thus, in finding the maximum 
 or H.W., we may treat A as the variable. 
 
 6. The Time and Height of High Water. 
 
 Apart from parallactic and nodal corrections, H.W. occurs when h in (27) 
 is a maximum, and the value of A which satisfies that condition will give 
 the time estimated from the epoch when A is zero. The value of A, when 
 reduced to time in the manner shown below, gives the inequality in the 
 interval after moon's transit, and will be called /. The time from moon's 
 transit until A is zero is the mean interval, and will be called i. Then the 
 interval from moon's transit is i + I. There is indeed a small parallactic 
 correction to i, which is omitted in this statement, but is included below. 
 
 I begin by considering the mean interval. 
 
 A vanishes when 2 (t + h I) x m = 0, that is to say, at a time /c m /2 (7 I) 
 after moon's transit. 
 
 Now, as shown in 5, 
 
 therefore =5-- = L- + 
 
1891] TIME AND HEIGHT OF HIGH-WATER. 277 
 
 The first of these terms is denoted by i, so that 
 
 *m /QI \ 
 
 l = 2Tf-V) (3 
 
 The second term is the parallactic correction to the mean interval, and 
 will be considered in 8, together with the other parallactic corrections. 
 
 If we denote by the suffixes 2, 1, 4 the semi-diurnal, diurnal, and quater- 
 diurnal terms, (27) may be written 
 
 li = SH 2 cos fa A + *) + SHi cos (p,A + ^,) + 2H 4 cos (^ 4 A + ^ 4 ) 
 
 The condition for maximum is that dh/d& should vanish, and the equation 
 consists of the sum of three pairs of terms of the form 
 
 2 [tip cos S- sin >A + Hp sin 
 equated to zero. 
 
 It is well known that 
 
 sin ka. 1 k z . 
 
 = 1 + s-r -sm a a+ ... 
 
 k sin a 3 ! 
 
 cos ka. 1 k z . 
 
 - =1+ s~r sm a + 
 
 cos a 2 ! 
 
 Now all the p 2 's are nearly unity, the p/s nearly |, the p 4 's nearly 2. 
 Hence to a near approximation 
 
 sin pA = -*-= sin fna - * I 
 
 cos p n A = cos ^??A 1 ^ ( -- 1 j sin 2 
 
 \ IV / 
 
 L_ x 1 
 
 where w is 2, 1, or 4. 
 
 The p's are so nearly equal to 1, |, 2 respectively, that the second terms 
 are small. In the case of the second term of sin^A, cospjA we have the 
 factor sin 2 A, which is equal to even when A is 90 ; also the height of the 
 quater-diurnal tide is small. Thus, both for the diurnal and quater-diurnal 
 tides, the second term may be neglected. 
 
 In the case of the semi-diurnal terms, indeed, the correction is so small 
 that it may certainly be neglected if A be less than 45 or 50, and without 
 much loss of accuracy even for larger values of A. We shall in the first place 
 neglect these second terms entirely. 
 
 Now let F = SHp sin & ) 
 
 r (32) 
 
 G = SH 2 cos * 
 
 with suffixes 2, 1, 4 for semi-diurnal, diurnal, quater-diurnal terms. 
 
 The constituent terms of F, G will be written separately below, for 
 example F s = H g p s sin ^ s . 
 
278 TIME AND HEIGHT OF HIGH-WATER. [7 
 
 Then the equation dh/dA. = 0, leads to 
 F 2 cos A + G 2 sin A (F 1 cos A + 2Gj sin A) + F 4 cos 2 A + G 4 sin 2 A = 
 
 (33) 
 
 With regard to the additional terms referred to above, the values of the 
 semi-diurnal p's are, p m =l, p /f =l'Q38, ^=1'035, and hence p,, 2 l and 
 p s 2 1 are both nearly equal to '072. Therefore if (33) be regarded as the 
 fundamental equation, the additional terms may be taken into account by 
 supposing that there are corrections to F 2 and G 2 given by 
 
 SF 2 = - -012 sin 2 A . 3F 2 ) 
 
 (34) 
 
 8G 2 = - -012 sin 2 A (G,, + G,) J 
 
 The equation (33) may be solved thus : 
 
 TJ! | Tji 
 
 take tan A = T^-^-p 1 (35) 
 
 Then if we put 
 D = F 2 sin A - G 2 cos A (Fj sin \ A - 2G X cos A ) 
 
 + 2 (F 4 sin 2A - G 4 cos 2A ) 
 N = F 2 cos A + G 2 sin A + (F a cos ^A + 2Gj sin A ) 
 
 + (F 4 cos 2A + ^G 4 sin 2 A ) . . .(36) 
 
 and &A-* (37) 
 
 " 
 
 the solution of the equation (33) is A = A + SA ; and if D is the value of D 
 when A replaces A , it is clear that 
 
 3D = D - D = SA {(F 2 cos A + G 2 sin A ) + (F, cos | A + 2G X sin |A ) 
 
 os2A + iG 4 sin2A )} (38) 
 
 The angle A has to be reduced to time by division by 2 (7 I). As in 
 the case of the reduction of the mean interval 
 
 i..J_^_/i + ,;^L 
 
 2 (7 - /) 2 (7 - <r) V 7-0" 
 
 Now the greatest value of P is about \, and a-/(y a-) is *038, hence 
 f Po-/(7 a) is at greatest '005 ; but the inequality in the interval is rarely 
 more than 2 h , and even if A/2 (7 <r) amounted to 3 h , the second term would 
 only be about l m . Therefore I neglect the parallactic inequality in the 
 reduction of A to time, and simply divide A by 2 (7 <r). 
 
 The value of 2 (7 or) is 28 0> 98 per hour, and the reciprocal of this is 
 
 345 >4+!4- 
 
 Hence, if A be given in degrees, we have to multiply by '0345, or its 
 equivalent, to find its value in hours. 
 
1891] TIME AND HEIGHT OF HIGH-WATER. 279 
 
 If any correction be given in circular measure the reduction to time is 
 also very simple, for 57'296 x "0345 = l h '977 = 119 m , or 
 
 . (circ. meas.) = 119 m . .......... (39) 
 
 2 (7 - o-) 
 
 Thus, we have for H.W., 
 
 A = A + SA 
 
 (40) 
 
 Turning now to the expression for the height : it is expressible as the 
 sum of three terms of the form 
 
 S (H cos ^ cos />A H sin ^ sin 
 Let 
 
 A = SH cos S- 
 then 
 h = A 2 cos A - F 2 sin A + (A, cos A - 2F, sin A) + A 4 cos 2A - F 4 sin 2A . 
 
 (41) 
 
 with a correction to the height corresponding with 
 
 8 A, = - -012 sin 2 A . 3 (A,, + A,) 
 
 (42) 
 = --012sin 2 A.F 2 J 
 
 In 8, 9 we shall have to consider the variations of the interval and 
 height due to variations of semi-diurnal and diurnal A, F, G. It is clear 
 from the preceding formulae and from (39) that 
 
 SI = g- {SF 2 cos A + SG 2 sin A (SF l cos |A + 28^ sin | A)} 
 
 Bh = 8A 2 cos A - SF 2 sin A + (SA l cos ^A - 2SF, sin 
 
 A particular case of the application of (43) is to the computation of the 
 corrections referred to in (34) and (42). 
 
 7. On Evanescent Tides. 
 
 At certain parts of the lunation, the diurnal tides sometimes suffice to 
 annul one H.W. and one L. W., so that there is only one tide a day, perhaps 
 for several days running. 
 
 If the inferior H.W. be watched as the condition of evanescence approaches, 
 it will be seen to become smaller and smaller, and to occur later and later 
 or earlier and earlier, and the adjacent L.W. undergoes similar changes. In 
 the limit H.W. and L.W. coalesce, and in a tide diagram the coalescence 
 appears as a point of contrary reflex ure with horizontal tangent. Beyond 
 
280 EVANESCENT TIDES. [7 
 
 this point, the reflexure is still maintained, although the tangent is not 
 horizontal ; finally, the tangent again becomes horizontal, and the double 
 H.W. and L.W. again reappear. 
 
 Now in the use of the method of this paper, the loss of the double tide 
 is very inconvenient, and I therefore propose to take the point of reflexure 
 as representing both H.W. and L.W. during evanescence. 
 
 If N cannot vanish there is evanescence, and the point of reflexure is 
 given by D = 0. The limit of the H.W. is given by N = 0, D = simul- 
 taneously, and beyond this only D can vanish. The vanishing of D is taken 
 to represent H.W. 
 
 Accordingly, when N cannot vanish, we proceed to make D vanish thus : 
 = F 2 sin A - G 2 cos A \ (F, sin |A - 2Gi cos |A) 
 
 + 2 (F 4 sin 2 A - G 4 cos 2A) ...... (44) 
 
 As a first approximation put 
 
 . G 2 + G! 
 
 tan A = -nr^TTp, 
 
 F,- t*i 
 
 Then writing 
 
 E = - {F 2 cos A + G 2 sin A \ (F, cos |A + 2G, sin A ) 
 
 os2A + iG 4 sin2A )} ......... (45) 
 
 we have A = A 4- 8 A, where 8A = ~ .................. (46) 
 
 &0 
 
 In the rest of the calculation this value of A is to be treated exactly as 
 though it had been determined by the former method. 
 
 The corrections M, N, P, Q, R, S considered in succeeding sections, 
 however, present a difficulty. In this case, A will always be very nearly 
 90, and I propose to compute P, Q, S (see 8, 9) as though that were 
 the true value of A. 
 
 But the correctional terms M', N', R, defined in 8, 9, become theoreti- 
 cally infinite, and we are therefore compelled not to compute them, and to fill 
 up the hiatus in the manner shown in the example below. 
 
 The process here suggested is a makeshift, but it is sufficient for the 
 construction of a trustworthy tide-table, since the real occurrence at these 
 times is a long period of nearly slack water, with or without a small maximum 
 and minimum. 
 
1891] PARALLACTIC CORRECTIONS. 281 
 
 8. Parallactic Corrections to Time and Height. 
 
 We will first consider the parallactic correction to the mean interval ; we 
 saw at the beginning of 6 that there is a correction to the interval due to 
 the moon's unequal motion in longitude of 
 
 * m - Pa- 
 
 2 (7 - a-) ' * 7 - a- 
 
 Now according to (31) TO /2 (7 - a-) is i, the mean interval, which we will 
 suppose expressed in hours ; also P is y^II and a/(y <r) is '03788, and | of 
 60 m is 40 m ; hence the correction in minutes is 
 
 i . 40 m . -03788 . T V . U = i x O m '0797II (47) 
 
 Now turning to the other parallactic terms : 
 
 Let us introduce the following notation, which follows the same plan as 
 that used before, viz. : 
 
 cos 
 ,,F J. = -036H,,,,p sin I < 
 
 cos ,A cos 
 
 F = /3'H O p sin 1 A ,F I = '036 H, f p sin j- 
 
 Gj O p"- cos) ft) yc 
 
 A comparison of the method of 6 with (28) 5 shows that the corrections 
 to the time and height are to be found by applying corrections to the A's, 
 F's, G's, as follows : 
 
 SF 2 = n ( m F + JF> = m 2 , 8G 2 = n ( m a + ,G) = um, 
 
 SF l = U( F+ J) = n^, 8G, = n ( G + ,G) = Urn, 
 
 SA 2 = n ( m A + = n^ 2 
 
 8A 1 = n( A+ 7 A) = n^ ........................ (48) 
 
 Then comparing (47) (48) with (39) and (43) we see that if 
 
 R = 119 m ^ {I, cos A + m 2 sin A + (I, cos A + 2m, sin |A)| + O m> 08 . i 
 
 S= [^co6A-i t flaaA(,ooe4A-2Z 1 sin4A)j ............ (49) 
 
 the corrections to the interval and height are 
 
 (50) 
 = US 
 
282 CORRECTIONS FOR LONGITUDE OF MOON'S NODE. [7 
 
 9. Corrections to Time and Height for Longitude of Moons Node. 
 Here again, we treat the terms as corrections to A, F, G. 
 Let 
 
 c 2 = -283 Hjo,, sin^,,= -283 F,, 
 
 d 2 = - -037 R m p m cos * m - -308 R aPtt cos * = - -037 G w - -296 G,, 
 e, = - -037 H m p m 2 cos * m + '283 H,^,, 2 cos = - -037 G m + "283G ; , 
 / 2 = -308 H,^, 2 8^,= -319F, 
 
 a, = - -037 H m cos * m + -283 H,, cos = - -037 A m + -283 A,, 
 & 2 = '308 H,, sin ^,, = -296 F,, 
 
 c,= -ISSHoposin^, -f llSHj^sin*, = -188F +-115F/ 
 
 = -391 Go -'297G, 
 = '188G + -115G, 
 
 /, = - -188H p 2 sin^ + 154H / ^ / 2 sin ^ = - -0905 F + '080 F, 
 
 = -188 A +115A, 
 
 =--391F + -297F 7 
 
 (51) 
 
 A comparison with (29) then shows that 
 
 SF 2 = c 2 cos 8 f d 2 sin 8 , SF t = c 1 cos S3 + d l sin & 
 
 &G. 2 = e 2 cos 88 -f / 2 sin 8 , 3Gj = e l cos S3 +fi sin Q 
 
 SA 2 = a 2 cos Q + b 2 sin Q, 8A X = a x cos S3 + &! sin s . . .(52) 
 We now put 
 
 AT - SM = 119 m =r {c 2 cos A + e z sin A + (c, cos A + 2c, sin 
 
 N' - SN = 1 19 m i {d 2 cos A +/ 2 sin A (d, cos | A + 2/, sin 
 
 P' = a 2 cos A c 2 sin A (aj cos A 2^ sin 
 
 Q' = 6 2 cos A d z sin A + (h cos ^A - 2^ sin 
 
 (53) 
 
 The corrections to interval and height, as far as concerns the investigation 
 up to the present point, are therefore 
 
 S7 a = (M ; - SM) cos a + (N x - SN) sin a 
 
 ^ Q = P'cos & +Q'sin Q (54) 
 
1891] REFERENCE TO MOON'S TRUE TRANSIT. 283 
 
 | 10. Reference to the Moons True Transit. 
 
 The intervals have been referred to the transits of a fictitious satellite 
 whose R.A. is equal to I, and we now require corrections so as to refer to the 
 moon's true transit. 
 
 Let T, T' be the times of true and fictitious transits, and let a be the 
 moon's R.A. 
 
 Then if t denotes the mean solar hour angle at time T, 
 
 t + k OL = qtr 
 and, dropping the suffix o to I, at time T' 
 
 t + h I = qjr 
 where q is an even integer at upper, and an odd one at lower transit. 
 
 If a be the value of the moon's R.A. at time T', then its value at T is 
 a + a (T T') ; also the t + h of the first equation is equal to the t + h of the 
 second corrected by 7 (T T'). 
 
 Hence the two equations may be written 
 
 t + h + <y(T-T)-a - d(T - T') - qTr 
 
 t + h - I = qir 
 
 Hence T - T' = *. 
 
 <y O. 
 
 It will afford a sufficiently close approximation if we replace a. by the 
 moon's mean motion cr, so that 
 
 T ' = T + l ~ a 
 <y cr 
 
 We have, therefore, to find the excess of the moon's longitude above her 
 R.A. at the time of fictitious transit. 
 
 It will be seen from fig. 2 that we have to determine the relationship 
 between the R.A. and longitude, both measured from the intersection of the 
 orbit and equator. The formula is exactly analogous with that which gives 
 the reduction of longitudes to the ecliptic, so that 
 
 l-% = a- v + tan 2 -!/ sin 2 ( ]l ~ ) ~ I tan4 K sin 4 (^ ~ ) + 
 The last of these terms is very small and may be neglected, so that 
 
 I - a = - (i/ - ) + tan 2 / cos 2 sin 2 J - tan 2 \I sin 2 cos 2Z 
 By the formula (7) 
 
 tan 2 \I = tan 2 ^ [1 + 2i cosec o> cos Q ] 
 sin 2 = 2t cot to sin Q 
 cos 2 = 1 
 v = i tan &) sin 3 
 
284 REFERENCE TO MOON'S TRUE TRANSIT. [7 
 
 Hence I - a = tan 2 f <u sin 21 + cos a |2 H?*!^ s i n 2/1 
 
 sin &> 
 
 + sin ffi [ 2z tan 2 |&> cot a> cos 21 i tan |w] 
 But 21 = 2]) + 2 - 10 = <H), suppose ; and 
 
 2t tan 2 jo, cot a, 
 
 (7 cr) sin o) 7 0- 
 
 and l^i 6 ^^ 
 
 7-0- 
 
 Let us put then 
 
 @ = 2)) + 2 - 10 
 
 = 4m 
 
 SM = 4 m -60 sin 
 
 .(55) 
 
 SN = - [4 m -22 cos + 4 m -41] 
 and we have T' = T + ST + SM cos Q 4- 8N sin 
 
 It is now clear that, when the tide is referred to the moon's true transit, 
 BT must be added to I, and SM, SN must be added to M' - SM, N' - 8N as 
 given in (53) to find M', N'. 
 
 Let be the height of mean sea level above the datum adopted for the 
 tide-table, and let 
 
 B = <a + H wa cos^ a (56) 
 
 Then i denotes the mean interval from moon's transit to H.W., and the 
 interval is 
 
 *' + / + 8T + M / cos sa H-N'sin a +8I P 
 and the height is 
 
 B + h + F cos Q + Q' sin Q +8h P + R sa cos ^ sa 
 
 In these formulae all the quantities are functions of }) ; now although }) 
 can easily be found from the Nautical Almanac, it is not tabulated, and the 
 difficulty of using the tide-table would be considerably augmented if it were 
 necessary to compute J) for each tide. We shall proceed, then, to convert our 
 formulae into others, in which there is direct reference to moon's true transit, 
 although there is some loss of accuracy in the process, and the amount of 
 computation required to form the table is increased. 
 
 We have seen that the time T' of fictitious transit is given by 
 T' = T + ST + SM cos a +SN sin ga 
 
 and that (7 17) T' + h I = qir where h and I correspond to the time of 
 fictitious transit. 
 
 Now l, = h + 2e, sin (I, - 281) 
 
 where e, is the eccentricity of the sun's orbit, and 281 the longitude of 
 perigee. 
 
1891] REFERENCE TO MOON'S TRUE TRANSIT. 285 
 
 Also I = ]) + I, 
 
 Hence I - h = ]) + 2e, sin (l t - 281) 
 
 But 
 
 -^- = 7 m -69 = O h 128 ; also I, = - 5, and - 286 = + 74 
 
 Therefore T' = ?- ^ + 7 m '69 sin ( + 74), 
 
 and ^ - J = T + ST + SMcos 83 +8Nsin 83 - 7 m '69 sin( + 74) 
 This formula connects J> with T, the time of moon's transit. 
 
 Now / is a function of ]) or of (qir + ^)/(y t]) ', hence if /(T) be the value 
 of / for which J) = (7 77) T qn, and if / ())) be the true value of /, 
 
 /()>) = / (T) -f ^ (8T + 8M cos 83 + 8N sin 83 - 7 m '69 sin ( + 74)} 
 and similarly 
 
 h($) = h (T) + J| {ST + SM cos 3 + 8N sin &3 - O h '128 sin ( + 74)} 
 
 These expressions have now to be substituted in those for the height and 
 interval ; but in the small terms 8T, W, N x , &c., we may regard }) as denoting 
 (7 - 77) T - qir. 
 
 In carrying out the substitutions preparations will be made for 
 computation. 
 
 First put SI = ^ ST = ~ x 10 m '32 sin 
 
 CL J. QJ \. 
 
 = (2^V(2-34)(2 m -2sine) (57) 
 
 Similarly, since 2 m> 2 is 3^ hrs., we put 
 
 300 / V / 
 
 Next put M" = ^ 8M = ^ x 4 m '60 sin 
 
 A) (59) 
 
 \ /j-/ 
 
 and N" = ^= 8N 
 
 = (2 ^ (2 ra -2) + (2 ^ (2 m -2 cos ) (1 - ^) (60) 
 
286 REFERENCE TO MOON'S TRUE TRANSIT. [7 
 
 Similarly put 
 
 -^) ......... (62) 
 
 Lastly put i = - ( 2 ^) {3 m '85 sin (0 + 74)} .................. (63) 
 
 \ CL \. 1 
 
 ft' = - ( 2 %} (O h '064 sin ( + 74)} ... . . .(64) 
 
 \ CtJ./ 
 
 With this notation 
 
 /(D) = 7(T) + 87 + M" cos S3 + N" sin 8 4- 1 
 = A(T) 4- 8A + P" cos 8 + Q" sin 8 + ft 
 
 These are now to be substituted in the expressions for interval and height, 
 and in doing so we may drop the (T) after the 7 and h. 
 
 I write X = 7 + 8T + i + 87 
 
 ^ = B + h + 8h 
 ft = ft' + H S(l cos ^ sa 
 M = M' + M", N = N' + N" 
 
 P* = P' + P", Q = Q' + Q" .................. (65) 
 
 Our formulae are designed to serve for any time of year and its opposite ; 
 now since t and ft involve the sun's longitude they change their signs in six 
 months, and we write + i, 4 ft, and it is to be understood that the upper sign 
 is to be used for the time of year under computation and the lower for its 
 opposite. 
 
 The interval is then 
 
 JE t + Mcos s +Nsin s + RII ..................... (66) 
 
 and the height is 
 
 |^ ft 4- P cos &3 4-Qsin Q 4- SH ..................... (67) 
 
 One part of M, N, P, Q arises from a true change in the tide when the 
 longitude of the moon's node changes, and the remainder (nearly equally 
 large) merely depends on the reference to true instead of fictitious transit. 
 The quantities t and ft depend partly on a portion of the equation of time 
 and partly on the annual tide. 
 
 We must now explain the computation of 2d//dT and of 2dh/dT. 
 
 * This P will not be confused with the P denned in 1, which has been replaced by II. 
 
1891] CORRECTION FOR SOLAR PARALLAX. 287 
 
 If u , MJ,...^ are cyclical values of a function, the symmetrical inter- 
 polation formula in the neighbourhood of u m is 
 
 U x+m = U m + \X (AE- 1 + A) U 
 
 ( A'E 
 
 where Eu m = u m+1 and AM,,, = u m+1 - u m 
 
 Then when x = 0, 
 
 2 = ( AE- 1 + A) w m - 
 
 or ^ _ M 
 
 CtCO 
 
 In the present case the first term will usually suffice, but the second term 
 may be easily computed. 
 
 In order then to compute the required differential coefficients we arrange 
 the /'s and h's in two columns, the even entries in one column and the odd 
 in another, and take the differences of the two columns independently of one 
 another. 
 
 11. The Correction for Solar Parallax. 
 
 The terms depending on solar parallax arise from the potential V P/ in (15), 
 but the correction is so small that I shall omit it from the example below. 
 It is well, however, to show how it may be computed. The only terms of 
 importance are those in H g , H, corresponding to the tides S 2 , Kj. 
 
 The variability of P / enters into the calculation in the form of corrections 
 to A 2 , G 2 , F 2 , A n G], Fj. 
 
 The sun's parallax is approximately 
 
 1 + e / cos (l t - 281 ), and hence P, = 3e, cos ( + 74) 
 Now 3e, is '0504, and 119 m x -0504 = 6 m '0 
 
 Then, since 8F 2 
 
 it follows that 
 
 gm.Q 
 
 HP, = -jj- cos ( + 74) (F 8 cos A + G sin A "317 (F, cos A + 2G, sin 
 
 Sh P , = -050 cos ( + 74) {A g cos A - F s sin A '317 (A, cos A - 2F ( sin 
 
 These must be deemed to be corrections to t and I), since they change 
 signs in six months. 
 
288 LOW-WATER. [7 
 
 12. The Formulce for a Low Water Table. 
 
 The formula (27), (28), (29), (30), for the height of water would (except 
 in one detail) be applicable to L.W., but they would not be convenient, 
 because A oscillates about TT at the L.W. which precedes the H.W. for 
 which mean A is zero. 
 
 Hence put A = 8 TT. 
 
 The L.W. formulae may be made exactly similar to those for H.W. by 
 making the heights negative, and this condition is satisfied by adding TT to 
 all the arguments. Thus the formulae for the height may be written 
 
 - 2H cos (pS + 0} 
 where 6 = ^ + (1 - p) TT 
 
 A similar change may be made in the nodal terms, but the parallactic 
 terms require further consideration. The term cos(p OT A + m ^) changes, not 
 only because its sign is to be changed and A is to be replaced by 8 TT, but 
 also because, as appears in 3, m ^ changes. 
 
 For H.W. 
 
 - - 
 
 m rj Pm*"m m m / / n /t m . * > Q.-I -i TT /TT _ i 
 
 but for L.W. K m must be replaced by /c w TT wherever it is multiplied by p; 
 hence, for L.W., p m K m - m /c must have (p m -p n )Tr or (lp n ) TT added to its 
 previous value. 
 
 Hence the term in cos (p m A 4- m ^) for H.W. corresponds in the expression 
 for h p for L.W., to a term involving 
 
 COS [p m 8 + (I - pj 7T + (I - p n ) TT + >] 
 
 Since p m = 1, the required term is cos (p m 8 + m Q), where 
 
 m 8 = m *>+(I-p n )7T 
 
 Again the parallactic term for involves cos ( pA + ^) for H. W., and 
 treating it in the same way, the corresponding term in - h p for L.W. is found 
 to be 
 
 cos ( pB + 0) 
 
 where Q = (1 - O p) TT + (p - p q ) -rr + fi 
 
 But oP=Po-g 
 
 Hence 6> = ^ + (l +fi~ 
 
 We have also 
 
1891] 
 
 LOW-WATER. 
 
 289 
 
 Then the L.W. formulae for depths below mean sea level are similar in 
 form to those for H.W. for elevation above mean sea level, with 8 in place of 
 A, and with 0's in the place of ^'s. 
 
 The connection of 0's with ^'s is given in the following table. 
 
 Initial 
 
 Principal and nodal terms 
 
 Parallactic terms 
 
 M 2 
 
 K 2 
 
 S 2 
 
 o 
 
 p 
 
 M 4 
 
 S 4 
 
 &m &m 
 
 Q H =5 // + (l-jt?J 7 r 
 
 (assume /3= 
 
 -6 
 
 6 = 
 
 All the #'s differ from the ^'s by a small angle, or by an angle nearly 
 equal to 90 or 180. Where the difference is small the A, G, F for L.W. 
 will be nearly equal to those for H.W. ; where the difference is nearly 90 
 the A, G, F for L.W. will be nearly the same as those for H.W., a quarter 
 year earlier or later; and where the difference is nearly 180 the A, G, F for 
 L.W. will be nearly equal and opposite to those for H.W. Hence we may set 
 aside the change of 90 and 180 to be satisfied by a shift of a quarter year, 
 or by change of sign. Suppose then, that = $ + a, where a is small, and let 
 [A], [G], [F] denote the values of L.W. A, G, F ; then, remembering that 
 
 A = H cos ^, G = Hj9 2 cos ^, F = Hp sin ^ 
 
 and that [A], [G], [F] are represented by similar formulae with 6 in place of 
 ^, we have 
 
 P 
 
 [F] = F + p sin a A 
 
 D. i. 
 
 19 
 
290 COMPUTATION OF TIDE-TABLE. [7 
 
 The values of a are given above, and those of the ps are known ; hence 
 it is easy to compute formulae of transition from L.W. to H.W. 
 
 The rules given below in the example are derived from these formulae, 
 but the coefficients sin a/p and p sin a are given in round numbers appropriate 
 for computation, and are sometimes treated as zero. 
 
 PART II. COMPUTATION. 
 Remarks on the Computations. 
 
 The multiplications are supposed to be done with Crelle's Bremiker's 
 multiplication table *. 
 
 The other tables required are tables of squares, natural tangents, circular 
 measure, and a traverse table. Bottom ley's tables f are convenient for the 
 purpose, because they give no more than is required. The nautical traverse 
 table, such as that in Inman's tables, or in Chambers' logarithms, is used for 
 finding such quantities as H cos^ and H sin^, for if H is "Distance," H cos^ 
 is "Lat." and Hsin^ is "Dep.," and the position of the decimal point is 
 determined by inspection. For the use of this table it is advantageous to 
 have only angles with a whole number of degrees, so as to avoid cross 
 interpolations J ; the whole calculation is therefore conducted so as to avoid 
 broken degrees. A traverse table is commonly given for "Distances" from 
 to 300, hence, if the "Distance" involves three digits and lies between 300 
 and 999, an interpolation is required, so as to use the entries between 30 and 
 99 ; this interpolation can easily be made by inspection. 
 
 All the angles are entered so that the significant part is less than 90, 
 by treating them as + or , with TT or 180 added where necessary. This 
 facilitates the use of the traverse table. 
 
 It is best to determine the signs of the cosines and sines independently 
 from their numerical values, and, accordingly, in the example where "000 is 
 entered as the value of a sine or cosine, it has a sign attached to it. 
 
 I suppose the computer to be able to add up a short column of figures, 
 where some of the entries are + and others . This is an arithmetical process 
 not much practised, but easily acquired. 
 
 The sequences of angles and of cosines and sines, which occur frequently 
 below, appertain (except in the cases of Sa and Ssa) to values of the excess 
 of moon's longitude over sun's (for which the symbol used is D) at intervals 
 of 15, beginning with }) = 0, and ending with }) = 345, 24 values in all. 
 But in the earlier part of the computation, the beginning of the sequence 
 occurs at a different part of the column at different times of the year. Thus, 
 
 * Eechsntafeln, Berlin, Georg Beimer. 
 
 t Four-figure Mathematical Tables, by J. T. Bottomley, F.E.S. Macmillan. 
 J Inman's Traverse Table is arranged so that the interpolation for a fraction of a degree is 
 not very awkward. 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 291 
 
 a list of months is written in the margin, to show where we are to begin at 
 any specified time of year. Strictly speaking these months are the times 
 when the sun's longitude + 5 (for which the symbol used is ) is equal to 
 a multiple of 30 ; thus, when is 0, we have March 15th, when is 30, 
 April 14th, and so on, as shown in Table VI. If the number of degrees in }) 
 be reduced to time, at the rate of 15 per hour, we have, approximately, the 
 time of the moon's transit, and in the later stages of the computation the 
 time of Moon's transit is made to replace }). The sequences of angles are 
 found by adding multiples of 30, adding or subtracting multiples of 15, or 
 adding multiples of 60 (see Table II.) to certain initial angles (see Table I.). 
 When the sequence has been carried so far that the next addition would 
 reproduce the first angle with TT added to it, it is unnecessary to proceed 
 further. In the sequences of cosines and sines of such angles, when we have 
 got to this same point, it is unnecessary to proceed further, since the remainder 
 is the same as the beginning, with the sign changed. 
 
 In subsequent stages where a constant has to be added to a sequence, 
 the new sequence will have double as many entries as the old, the first half 
 being formed by addition, and the second half by subtraction ; but in repeating 
 the new sequence the signs are not to be changed. This follows immediately 
 from what has been said of the signs of sequences. 
 
 Before proceeding with the computations I give some tables and rules of 
 general applicability to all ports. It will be best for the computer who is 
 learning the process to pass straight to the example, and to refer back to 
 these tables as they are required ; but I give them in the first place, because 
 they will be wanted in the case of any other port. 
 
 Tables and Rules applicable to all Ports. 
 
 TABLE I. For finding the K's, the initial entries of the several 
 sequences, for H.W. (See (26), 5.) 
 
 Initials Principal and nodal terms 
 
 Parallactic terms 
 
 M 2 
 K 2 
 
 
 
 K m =P m Km-K m = Q 
 
 Kg =P,K m -K, 
 
 .K = i 
 
 9'llH B /H m -l 
 
 13 , , *0-*q |05 o 
 
 /n K m'r.ii IT /TT i i " 
 
 K = 
 
 - 10 
 
 192 
 
292 
 
 COMPUTATION OF TIDE-TABLE. 
 
 TABLE II. For finding the sequences of the S-'s by putting n successively 
 equal to 0, 1, 2, 3, &c. (See (26), 5.) 
 
 Initials 
 
 Principal and nodal terms 
 
 * 
 
 Parallactic terms 
 
 a- 
 
 M 2 
 
 K 
 
 mK 
 
 K 2 
 
 K,, +30n 
 
 /t K + 3Qn 
 
 S 2 
 
 K s + 30n 
 
 
 
 
 K -I5n 
 
 gK \5n 
 
 Kx 
 
 K, + lbn 
 
 ^ + 15% 
 
 P 
 
 K p +lbn 
 
 
 M 4 
 
 K 2nt 
 
 
 S 4 
 
 K 2 , + 60n 
 
 
 Ssa 
 
 K^ + GO^ 
 
 
 Sa 
 
 K M + I0n 
 
 
 N.B. The sequence for Sa is required under conditions which differ from 
 those of the other S-'s. 
 
 TABLE III. The numerical values of the p's. (See 5.) 
 
 Initials 
 
 Principal and 
 
 nodal terms 
 
 Parallactic terms 
 
 M 2 
 
 fb-1, 
 
 ^ 2 =1 
 
 
 K 2 
 
 /> =1-038, 
 
 p,,s= 1-078 
 
 ,,p = 1-076, // /? 2 = 1-158 
 
 S 2 
 
 p t =1-035, 
 
 p, 2 =1-071 
 
 
 N 
 
 p n = -981 
 
 
 
 O 
 
 Po = '481, 
 
 2> 2 = -231 
 
 jo = -481--013/3- 1 , ^ 2 =-231--012^- 1 
 
 K! 
 
 P, = "519, 
 
 p? = -269 
 
 ,^ = 538, ^=-289 
 
 P 
 
 p p = -516, 
 
 jo p 2 = -266 
 
 
 Q 
 
 p q = -462 
 
 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 293 
 
 TABLE IV. For computing corrections for reference to moon's transit, 
 viz., 8T, SM, SN, and the sequence @. (See (55) 10.) 
 
 Sequence 
 
 e 
 
 March - 10 
 + 20 
 
 April + 50 
 + 80 
 
 May TT - 70 
 TT -40 
 
 June TT 10 
 
 7T+20 
 
 July 7T + 50 
 
 7T + 80 
 
 August - 70 
 -40 
 
 Bepeat the sequence. 
 
 5T 
 = h -172sm0 
 
 5M 
 = 4 m -60 sin 
 
 -4 ra -22cos6 
 
 8N = 
 - 4 m -22 cos - 4 m -41 
 
 h. 
 
 m. 
 
 rn. 
 
 m. 
 
 March - -030 
 
 March - 0'80 
 
 March -4'16 
 
 March - 8-57 
 
 + 059 
 
 + 1-57 
 
 -3-97 
 
 -8-38 
 
 April + -132 
 
 April +3-52 
 
 April -2-71 
 
 April - 7-12 
 
 + 169 
 
 +4-53 
 
 -0-73 
 
 -5-14 
 
 May + -162 
 
 May -t- 4-32 
 
 May +1-45 
 
 May -2-96 
 
 + 111 
 
 + 2-96 
 
 + 3-23 
 
 -1-18 
 
 June +-030 
 
 June +0-80 
 
 Rep. and eh. 
 
 June -0-25 
 
 -059 
 
 -1-57 
 
 
 -0-44 
 
 July - -132 
 
 July -3-52 
 
 
 July -1-70 
 
 -169 
 
 -4-53 
 
 
 -3-68 
 
 August --162 
 
 August -4-32 
 
 
 August -5-86 
 
 -111 
 
 -2-96 
 
 
 -7-64 
 
 The sequences for ST, SM, 8N are to be repeated without change of sign. 
 To find the succession of values for any month we begin with the entry 
 opposite to that month, read on down to the bottom, and then begin again at 
 the top. For example, ST for July begins with 132, and then, after going 
 on down to '111, it begins again at the top with '030. 
 
294 
 
 COMPUTATION OF TIDE-TABLE. 
 
 TABLE V. For corrections due to part of the equation of time. 
 (See (63) (64) 10.) 
 
 The following is a table of - 3 m '85 sin (0 + 74) (which I call c), and of 
 the same when the hour is unit of time (which I call d). 
 
 N.B. 286 is the longitude of sun's perigee + 5, and 74 is its supple- 
 ment to 360. 
 
 
 c 
 
 d 
 
 - 
 
 -3 m -85 sin (O+74) 
 
 -O h -064 sin (0+74) 
 
 m. 
 
 h. 
 
 March . . . 
 
 -37 
 
 -062 
 
 April .... 
 May .... 
 June .... 
 
 -3-7 
 
 -2-8 
 -1-1 
 
 -062 
 -046 
 -018 
 
 July .... 
 August . . . 
 
 -1-0-9 
 + 2-7 
 
 + 016 
 + 045 
 
 TABLE VI. Dates and Limits of Applicability of the Tide-Tables. 
 
 
 
 Applicability by 
 
 
 
 Applicability by 
 
 
 
 reference to Sun's 
 
 
 
 reference to Sun's 
 
 Heading for 
 
 G 
 
 long, at Moon's 
 
 Heading for 
 
 
 long, at Moon's 
 
 tide- table 
 
 
 transit 
 
 tide-table 
 
 
 transit 
 
 
 
 Sun's longitude 
 
 
 
 Sun's longitude 
 
 March 15 
 
 
 
 from 350 to 
 
 Sept. 17 
 
 180 
 
 from 170 to 180 
 
 25 
 
 10 
 
 10 
 
 28 
 
 190 
 
 180 190 
 
 April 4 
 
 20 
 
 10 20 
 
 Oct. 8 
 
 200 
 
 190 200 
 
 14 
 
 30 
 
 20 30 
 
 18 
 
 210 
 
 200 210 
 
 25 
 
 40 
 
 30 40 
 
 28 
 
 220 
 
 210 220 
 
 May 5 
 
 50 
 
 40 50 
 
 Nov. 7 
 
 230 
 
 220 230 
 
 15 
 
 60 
 
 50 60 
 
 17 
 
 240 
 
 230 240 
 
 26 
 
 70 
 
 60 70 
 
 27 
 
 250 
 
 240 250 
 
 June 5 
 
 80 
 
 70 80 
 
 Dec. 7 
 
 260 
 
 250 260 
 
 16 
 
 90 
 
 80 90 
 
 16 
 
 270 
 
 260 270 
 
 26 
 
 100 
 
 90 100 
 
 26 
 
 280 
 
 270 280 
 
 July 7 
 
 110 
 
 100 110 
 
 Jan. 5 
 
 290 
 
 280 290 
 
 ,, 17 
 
 120 
 
 110 120 
 
 15 
 
 300 
 
 290 300 
 
 28 
 
 130 
 
 120 , 130 
 
 25 
 
 310 
 
 300 310 
 
 Aug. 7 
 
 140 
 
 130 , 140 
 
 Feb. 4 
 
 320 
 
 310 320 
 
 17 
 
 150 
 
 140 , 150 
 
 13 
 
 330 
 
 320 330 
 
 28 
 
 160 
 
 150 , 160 
 
 23 
 
 340 
 
 330 340 
 
 Sept. 7 
 
 170 
 
 160 , 170 
 
 March 5 
 
 350 
 
 340 350 
 
 This table gives the days of the year on which 0, or Sun's longitude + 5, 
 is nearly equal to a multiple of 10. These days are used as headings to the 
 several tide-tables. It is intended that the tables shall be used without an 
 interpolation for the time of year, which ought strictly to be made. When 
 
1891] COMPUTATION OF TIDE-TABLE. 295 
 
 the time of a particular moon's transit, with reference to which a tide is to be 
 calculated, falls nearly halfway between any two of the specified days, it 
 becomes uncertain which of the two adjoining tables should be used, and the 
 question can only be decided by reference to the Sun's longitude. A column 
 is therefore given of the limits of applicability of the table. 
 
 It would be easy, by means of a table of four columns referring to leap 
 year, to give the Greenwich times at which the Sun's longitude is 0, 10, 
 20, &c., which would be accurate enough for the present purpose during 
 some twenty-five years. 
 
 VII. The Choice of a Unit of Length. 
 
 In a calculation of this kind it is advantageous to reduce the number of 
 digits as far as possible, consistently with due accuracy, and it is convenient 
 to omit the decimal point when we deal with heights. 
 
 The diurnal tides are so various at different places that no general rule 
 can be made to depend on them. 
 
 It is required to express as many of the heights as possible by two digits, 
 and it will be best to take such a unit that much of the work shall be con- 
 ducted with 70's and 80's, but to allow a margin and not to try to bring 
 them into the 90's. 
 
 After consideration I think it is best to take such a unit that a'H m (see 
 below) shall be expressed by 70 or 80. Since a' is usually about 3 4 T , or say 
 yL, then when H m is given in feet and decimals (or any other unit), we are 
 to multiply the heights by a simple factor lying between 14x70-=- H TO and 
 14 x 80 -r H m . 
 
 The rule therefore is: 
 
 Multiply the heights by a factor lying between 1000 -=- H m and 1200 -r- H m , 
 and omit decimals. 
 
 In the example below it would have been best to multiply all the heights 
 by 700. We should then have H m =1098, H s = 488, &c., and this would 
 have made a'H m equal to 77. The final step in the calculation would then 
 have been to divide all the heights by 700. 
 
 In my example I have not followed this plan, and accordingly the decimal 
 point is retained, and an unnecessary number of digits has been written. 
 
 VIII. Rules for the Calculation of a L. W. Table. 
 
 It will be more convenient to state these rules as part of the example for 
 the Port of Aden, although they are of course generally applicable to all 
 ports. 
 
296 
 
 COMPUTATION OF TIDE-TABLE. 
 
 EXAMPLE OF FOKMATION OF A TIDE-TABLE. 
 TABLE of Constants for the Port of Aden. 
 
 
 
 ft. 
 
 
 ft. 
 
 M 2 
 
 \*m 
 
 = 1-568 
 = 229 
 
 | H 
 
 = -653 
 
 = 38 
 
 i, 
 
 {TT 
 rig 
 K 8 
 
 = -697 
 = 248 
 
 *'{*' 
 
 = 1-299 
 = 36 
 
 K 
 
 JH,, 
 
 = -201 
 
 P H 
 
 = -388 
 
 
 I*// 
 
 = 244 
 
 I K P 
 
 = 33 
 
 N 
 
 \Kn 
 
 = -427 
 = 225 
 
 Q{ H - 
 
 u 
 
 = -151 
 = 42 
 
 L 
 
 e 
 
 = '046 
 = 230 
 
 Sa | H>a 
 
 = -390 
 = 357 
 
 
 fH 2 , 
 
 = -007 
 
 TH 
 
 ,= -095 
 
 M 4 
 
 J 
 
 
 Ssa] 
 
 
 
 l*2 
 
 = 314 
 
 L K 8a 
 
 = 126 
 
 
 {H 2j 
 
 = -006 
 
 a 
 
 = 3-859 
 
 S 4 
 
 
 
 
 
 K^4 
 
 <2 
 
 = 271 
 
 
 
 These are the results of four years of observation, and are the constants 
 from which the tide-table is to be computed. 
 
 Formation of Sequences (see Tables I., II., III., and 6). 
 
 K, 
 
 s, 
 
 
 
 K, 
 
 p t ,K m = 238 
 
 ^ gKra = 237 
 
 p K m = 110 
 
 JO /KTO = 119 
 
 AC , = 244 
 
 - Kg =-248 
 
 -K O =- 38 
 
 - K/ =_ 36 
 
 -10=- 10 
 
 
 + 95= 95 
 
 -95=- 95 
 
 
 
 167 
 
 
 JT,,= - 16 
 
 2T.= - 11 
 
 T7" -| o 
 
 K,= - 12 
 
 P 
 
 M 4 
 
 B 4 
 
 p p K m = 118 
 
 2p m K m = 458 
 
 2p 8 K m = 474 
 
 -KP-- 33 
 
 _ K _ _314 
 
 K2 = 271 
 
 + 95= 95 
 
 
 
 180 
 
 144 
 
 203 
 
 if i_ n 
 
 ./Ip == 7T "t~ U 
 
 IT og 
 
 Jf | OQ 
 
 Jl 2g 7T "t" ^O 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 Sequences of Angles. 
 
 297 
 
 M 2 
 
 K 2 
 
 S 2 
 
 
 
 *j 
 
 P 
 
 M 4 
 
 S 4 
 
 
 (K^ + SO ^) 
 
 (K, + 30n) 
 
 (K -15) 
 
 (K, + 15n) 
 
 (K p + 15n) 
 
 (K*J 
 
 (K 2g + 60) 
 
 *m 
 
 */, 
 
 *, 
 
 *0 
 
 *i 
 
 *, 
 
 ^2TO 
 
 ^"28 
 
 
 
 Mar. - 16 
 
 -11 
 
 Mar. 77-13 
 
 Mar. -12 
 
 Mar.7r+ 
 
 7T-36 
 
 77+23 
 
 (a const.) 
 
 + 14 
 
 +19 
 
 7T-28 
 
 + 3 
 
 7T + 15 
 
 (const.) 
 
 "S 77+83 
 
 
 April + 44 
 
 +49 
 
 April 7T - 43 
 
 April + 18 
 
 Feb. 7T+30 
 
 
 -37 
 
 
 + 74 
 
 a +79 
 
 7T 58 
 
 + 33 
 
 7T + 45 
 
 
 D 
 
 
 May 77 76 
 
 53 7T-71 
 
 May 7T-73 
 
 May +48 
 
 Jan. 77 + 60 
 
 
 " 
 
 
 7T-46 
 
 ^ IT -41 
 
 OQ 
 7T OO 
 
 + 63 
 
 7T+75 
 
 
 
 
 
 
 June +77 
 
 June +78 
 
 Dec. 77 + 90 
 
 
 
 
 
 
 + 62 
 
 -87 
 
 -75 
 
 
 
 
 
 
 July +47 
 
 July -72 
 
 Nov. - 60 
 
 
 
 
 
 
 + 32 
 
 -57 
 
 -45 
 
 
 
 
 
 
 Aug. +17 
 
 Aug. -42 
 
 Oct. -30 
 
 
 
 
 
 
 + 2 
 
 -27 
 
 -15 
 
 
 
 Semi-diurnal : A = H cos ^, G = Hp 2 cos ^, F = Up sin ^. 
 
 M 2 
 
 h 
 
 K, 
 
 
 (H. = ? 697) 
 
 <-.?=.> 
 
 <H.^, 21) 
 
 ( H // =''201) 
 
 (H,,^,n, 
 
 <H^, 
 
 H m= =H TO/ v> = l-568 
 
 +'684 
 
 + 732 
 
 -137 
 
 Mar. +-193 
 
 + 209 
 
 -058 
 
 A TO = G m = 1-568 
 
 3 +-659 
 
 + 705 
 
 + 234 
 
 + 195 
 
 + 211 
 
 + 051 
 
 F m =0 
 
 +'457 
 
 + 489 
 
 + 544 
 
 April + -145 
 
 + 156 
 
 + 145 
 
 (constants) 
 
 3 +-133 
 
 + 142 
 
 + 708 
 
 + 055 
 
 + 060 
 
 + 201 
 
 
 rt --227 
 
 -243 
 
 + 682 
 
 May -'049 
 
 - '053 
 
 + 203 
 
 
 *! --526 
 
 -'563 
 
 + 473 
 
 -140 
 
 -151 
 
 + 150 
 
 Repeat the sequences for K 2 and S 2 changing the signs. 
 Add M 2 to S 2 . 
 
 A, + A. 
 
 o + . 
 
 F M + F. 
 
 
 2-252 
 
 2-300 
 
 
 
 2-227 
 
 2-273 
 
 
 
 2-025 
 
 2-057 
 
 
 03 
 
 1-701 
 
 1-710 
 
 m 
 
 ^ 
 
 1-341 
 
 1-325 
 
 " 
 
 fl 
 
 o 
 
 1-042 
 
 1-005 
 
 9 
 
 Q 
 
 884 
 
 836 
 
 0) 
 
 ^H 
 
 909 
 
 863 
 
 a 
 
 eg 
 
 51 
 
 1-111 
 
 1-079 
 
 QQ 
 
 
 1-435 
 
 1-426 
 
 
 
 1-795 
 
 1-811 
 
 
 
 2-094 
 
 2-131 
 
 
 Repeat without change. 
 
298 
 
 COMPUTATION OF TIDE-TABLE. 
 
 Then write out the three K 2 sequences, with the months, in extenso, each 
 on a separate strip of paper ; place the A /7 strip opposite to the A TO + A g 
 table, so that the month for which the sequence is required falls in the first 
 place ; e.g., for March put + '193, for April put + '145, for May put '049, 
 for June put '193, &c., in the first place opposite 2'252 of A TO + A. Then 
 add the A. m + A s and A /7 tables together in a different way for each of the six 
 months from March to August. 
 
 Proceed with the G //} F 7/ strips in the same way. The next following 
 table is formed in this way. 
 
 Semi-diurnal : A 2 = A w + A,, + A,, G. 2 = G m + G,, + G g , F, = F m + F,, + F. 
 
 March 
 
 April 
 
 &c. 
 
 A-, 
 
 G 2 
 
 F 2 
 
 AI 
 
 G 2 
 
 F 2 
 
 o> 
 
 2-445 
 
 2-509 
 
 -195 
 
 2-397 
 
 2-456 
 
 + 008 
 
 "aj 
 
 a 
 
 2-422 
 
 2-484 
 
 + 285 
 
 2-282 2-333 
 
 + 435 
 
 43 
 
 2-170 
 
 2-213 
 
 + 689 
 
 1-976 
 
 2-004 
 
 + 747 
 
 93 
 
 1-756 
 
 1-770 
 
 + 909 
 
 1-561 
 
 1-559 
 
 + 858 
 
 |l 
 
 1-292 
 
 1-272 
 
 + 885 
 
 1-148 
 
 1-116 
 
 + 740 
 
 ^ 
 
 -902 
 
 854 
 
 + 623 
 
 847 
 
 794 
 
 + 422 
 
 oo 
 
 ce 
 
 691 
 
 627 
 
 + 195 
 
 739 
 
 680 
 
 -008 
 
 3 
 
 714 
 
 652 
 
 -285 
 
 854 
 
 803 
 
 -435 
 
 M t 
 
 
 
 966 
 
 923 
 
 -689 
 
 1-160 
 
 1-132 
 
 -747 
 
 cS 
 
 2 
 
 1-380 
 
 1-366 
 
 -909 
 
 1-575 
 
 1-577 
 
 -858 
 
 c 
 
 1-844 
 
 1-864 
 
 -885 
 
 1-988 
 
 2-020 
 
 -740 
 
 '+3 
 
 C 
 
 2-234 
 
 2-282 
 
 -623 
 
 2-289 2-342 
 
 -422 
 
 o 
 O 
 
 Repeat without change. 
 
 Diurnal: A = Hcos^, G=Hp 2 cos^, F = 
 
 
 
 K, 
 
 P 
 
 
 A 
 
 G 
 
 F 
 
 
 A 
 
 G, 
 
 F 
 
 
 A 
 
 G p 
 
 F 
 
 
 (H 
 
 (H p 2 
 
 (R p 
 
 
 /TT 
 \ / 
 
 
 (H/P/ 
 
 
 (Hp 
 
 
 (H p pp 
 
 
 = 653) 
 
 = 151) 
 
 = 314) 
 
 
 = 1-299) 
 
 = 350) 
 
 = 674) 
 
 
 = 388) 
 
 = 103) 
 
 = 200) 
 
 Mar. 
 
 -636 
 
 -147 
 
 + 071 
 
 Mar. 
 
 + 1-271 
 
 + 342 
 
 -140 
 
 Mar. 
 
 -388 
 
 -103 
 
 ooo 
 
 
 -577 
 
 - -133 
 
 + 148 
 
 
 + 1-297 
 
 + 350 
 
 + -035 
 
 
 -375 
 
 -100 
 
 -052 
 
 April 
 
 -477 
 
 -110 
 
 + 214 
 
 April 
 
 + 1-235 
 
 + "333 
 
 + 208 
 
 Feb. 
 
 -336 
 
 -089 
 
 -100 
 
 
 -346 
 
 -080 
 
 + 266 
 
 
 + 1-089 
 
 + 294 
 
 + 367 
 
 
 -275 
 
 - -073 
 
 -141 
 
 May 
 
 -191 
 
 -044 
 
 + 300 
 
 May 
 
 + -869 
 
 + 234 
 
 + 501 
 
 Jan. 
 
 -194 
 
 -052 
 
 -173 
 
 
 -023 
 
 -005 
 
 + 314 
 
 
 + -590 
 
 + 159 
 
 + 601 
 
 
 -100 
 
 -027 
 
 -193 
 
 June 
 
 + 147 
 
 + 034 
 
 + 306 
 
 June 
 
 + -270 
 
 + 073 
 
 + 659 
 
 Dec. 
 
 000 
 
 000 
 
 -200 
 
 
 + 307 
 
 + 071 
 
 + 278 
 
 
 - -068 
 
 -018 
 
 + 673 
 
 
 + 100 
 
 + 027 
 
 - -193 
 
 July 
 
 + 445 
 
 + 103 
 
 + 230 
 
 July 
 
 - -402 
 
 -108 
 
 + 641 
 
 Nov. 
 
 + 194 
 
 + 052 
 
 -173 
 
 
 + 554 
 
 + 128 
 
 + 166 
 
 
 - -708 
 
 -191 
 
 + 565 
 
 
 + 275 
 
 + 073 
 
 -141 
 
 Aug. 
 
 + 625 
 
 + 144 
 
 + 092 
 
 Aug. 
 
 - -965 
 
 -260 
 
 + -451 
 
 Oct. 
 
 + 336 
 
 + 089 
 
 -100 
 
 
 + 653 
 
 + 151 
 
 + 011 
 
 
 -1-157 
 
 -312 
 
 + 306 
 
 
 + 375 
 
 + 100 
 
 -052 
 
 Repeat changing signs. 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 299 
 
 Add together the and K x sequences as they stand above : 
 
 + K,. 
 
 
 A 4-A, 
 
 G fG, 
 
 F o+F, 
 
 March 
 
 + 635 
 
 + 195 
 
 -069 
 
 
 4- 720 
 
 + 217 
 
 + 183 
 
 April 
 
 + 758 
 
 + 223 
 
 + 422 
 
 
 + 743 
 
 + 214 
 
 + 633 
 
 May 
 
 + 678 
 
 + 190 
 
 + 801 
 
 
 + 567 
 
 + 154 
 
 + 915 
 
 June 
 
 + 417 
 
 + 107 
 
 + 965 
 
 
 + 239 
 
 + 053 
 
 + 951 
 
 July 
 
 + 043 
 
 -005 
 
 + 871 
 
 
 -154 
 
 -063 
 
 + 731 
 
 August 
 
 -340 
 
 -116 
 
 + 543 
 
 
 -504 
 
 -161 
 
 + 317 
 
 Repeat changing signs. 
 
 Write out the P sequences in extenso with the months on the margin 
 (24 entries), each on a separate strip of paper ; place the A p strip opposite 
 the A + A, table so that any chosen month in one agrees with that month 
 on the other ; add the two tables together, making the first entry in the new 
 sequence that opposite which the chosen month is written. For example the 
 April entry in the A p sequence (completed) is '336, and this added to 
 + '758, the April entry of A + A 7 , gives + '422, which is the initial entry in 
 the diurnal sequence for A x corresponding to April in the following table. 
 G p and F p are operated on in the same way. 
 
 The first time the computer does this sort of work he may find it con- 
 venient to write out the O + K x sequences in extenso, so as to see exactly how 
 the computation runs, but it will be found with a little practice that this is 
 unnecessary. 
 
 Diurnal: A 1 = A + A, + A,,, G x =G + G, + G,, F^Fo+^ + F,. 
 
 March 
 
 April 
 
 &c. 
 
 A! 
 
 GI 
 
 FI 
 
 A: 
 
 o, 
 
 F, 
 
 CD 
 
 + 247 
 
 + 092 
 
 -069 
 
 + 422 
 
 + 134 
 
 + 522 
 
 1 ' 
 
 'u 
 
 + 345 
 
 + 117 
 
 + 131 
 
 + 368 
 
 + 114 
 
 + 685 
 
 t g 
 
 + 422 
 
 + 134 
 
 + 322 
 
 + 290 
 
 + 087 
 
 + 801 
 
 -fj 
 
 + 468 
 
 + 141 
 
 + 492 
 
 + 192 
 
 + 054 
 
 + 863 
 
 1 
 
 + 484 
 
 + 138 
 
 + 628 
 
 + 081 
 
 + 018 
 
 + 865 
 
 gr 
 
 + 467 
 
 + 127 
 
 + 722 
 
 -036 
 
 -020 
 
 + 810 
 
 *fl 
 
 o 
 
 + 417 
 
 + 107 
 
 + 765 
 
 -151 
 
 -057 
 
 + 698 
 
 4B 
 
 + 339 
 
 + 080 
 
 + 758 
 
 -254 
 
 -090 
 
 + 538 
 
 
 + 237 
 
 + 047 
 
 + 698 
 
 -340 
 
 -116 
 
 + 343 
 
 CD 
 
 3 
 
 + 121 
 
 + 010 
 
 + 590 
 
 - -404 
 
 -134 
 
 + 124 
 
 .9 
 
 -004 
 
 -027 
 
 + 443 
 
 -441 
 
 -143 
 
 -104 
 
 g 
 
 -129 
 
 -061 
 
 + 265 
 
 -445 
 
 -144 
 
 -324 
 
 o 
 
300 COMPUTATION OF TIDE-TABLE. [7 
 
 Quater-diurnal : A = Hcos&, G = H (Ipf cos ^, F = H (2/>) cos ^. 
 
 M 4 
 
 4 
 
 A 2OT 
 (H 2m =-007) 
 
 <*! 
 
 (4H W = -028) 
 
 ^2m 
 
 (2H M =-014) 
 
 ^28 
 
 (H^-006) 
 
 G 2 
 
 (4H 2 .p.=-026) 
 
 TJ1 
 
 (2H 2A 2 4-012) 
 
 -006 
 const. 
 
 -023 
 const. 
 
 + 008 
 const. 
 
 j --006 
 
 a -'001 
 | +-005 
 _ Repeat 
 << and change 
 
 -024 
 -003 
 + 021 
 Repeat 
 and change 
 
 -005 
 -012 
 -007 
 Repeat 
 and change 
 
 Quater-diurnal: A 4 = 
 
 A. 
 
 G 4 
 
 F, 
 
 
 -012 
 
 -047 
 
 + 003 
 
 ^ 
 
 -007 
 
 -026 
 
 -004 
 
 a 
 o 
 
 -001 
 
 -002 
 
 + 001 
 
 a 
 
 ooo 
 
 + 001 
 
 + 013 
 
 a 
 
 -005 
 
 -020 
 
 + 020 
 
 <5 
 
 -on 
 
 -044 
 
 + 015 
 
 Repeat without change of sign. 
 
 Semi-annual and Mean Water, (See (56), 10.) 
 
 -10 = - 10 
 
 224 
 
 K gga = TT + 44 
 
 Sequence. 
 
 Htr 
 
 fc= 8 ^ 
 
 (^ = 3 8 -859) 
 
 March w+44 
 
 -068 
 
 March 3-791 
 
 April -76 
 
 + 023 
 
 April 3-882 
 
 May -17 
 
 + 091 
 
 May 3-950 
 
 
 Repeat and change 
 
 June 3-927 
 
 
 
 July 3-836 
 
 
 
 Aug. 3-768 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 301 
 
 Annual. 
 
 K sa = ~~ 
 
 5 = - 5 
 
 K, a = - 2 
 
 Sequence. 
 
 H.T 
 
 March 
 
 _ 2 
 
 June 
 
 + 88 
 
 
 + 8 
 
 77 
 
 -82 
 
 
 + 18 
 
 77 
 
 -72 
 
 April 
 
 + 28 
 
 July 77 
 
 -62 
 
 
 + 38 
 
 77 
 
 -52 
 
 
 +48 
 
 77 
 
 -42 
 
 May 
 
 + 58 
 
 Aug. 77 
 
 -32 
 
 
 + 68 
 
 77 
 
 -22 
 
 
 +78 
 
 77 
 
 -12 
 
 Annual Tide. 
 
 H 8a cos* ga 
 (H^-390) 
 
 
 ft. 
 
 ft. 
 
 March 
 
 + 390 
 
 June + -014 
 
 
 + 386 
 
 -054 
 
 
 + 371 
 
 -121 
 
 April 
 
 + 344 
 
 July - -183 
 
 
 + 307 
 
 -240 
 
 
 + 261 
 
 -290 
 
 May 
 
 + 207 
 
 Aug. - -331 
 
 
 + 146 
 
 - -362 
 
 
 + 081 
 
 -381 
 
 Mean Interval i, and Parallactic Correction to i. (See (31), 6 ; and (47), 8.) 
 
 30 K m= 7 '633 
 
 1 21 
 
 3Q2 ' 20 
 
 i= 7 h -900 
 Retaining i in hours, parallactic correction to i = + O m> 08 x i 
 
 = + O m -63 
 
 N.B. O m< 08 is an absolute constant. 
 
COMPUTATION OF TIDE-TABLE. 
 
 Parallactic Corrections. (See SS 3, 8.) 
 
 OO / 
 
 36-4 H n is greater than 8H m ; therefore a = f, a'H w = '07 H m = '110 
 36-4 H 9 is greater than 8H ; therefore ft = f, 'H = -07 H = -046 
 
 (N.B. When either of these inequalities is less instead of greater, put 
 a = 1214H n /H m - 1, a'H w = '639H n - -07 H m 
 /3=12-14H/E -|, 'H =-639H ? --07H 
 
 If the N tide is unknown take a = 1, a' = ^ ; if the Q tide is unknown 
 take = 1, /3' = T V) 
 
 (Table III) O p = '481 - -013/3-' = "481 - '010 = "471 
 
 O p* = -231 - -012/8- 1 = -231 - -009 = "222 
 (Table I) Let 
 
 7 = 
 
 the denominator is 911 x '272 1 = T48 ; the numerator is 4. 
 Hence 7 = + 4 -r T48 = + 3 
 
 , _ Ko - Kq -013 
 
 -911H,/H -1" -?"" 
 
 the denominator of the first term is 911 x '232 - 1 =111 ; the numerator 
 is 4. Hence the first term is 4. The second term is 
 
 -OlOx 229 =-2 
 Hence B = - 6 
 
 (See Tables I, II., III.) 
 
 M 2 
 
 K, 
 
 
 
 K: 
 
 p n < m = 225 
 - K n = - 225 
 + y 3 
 
 0384c m = 9 
 K /y =-16 
 
 P,*m= 106 
 
 - K, = - 42 
 + 8 =- 6 
 + 95= 95 
 
 0192c TO = 4 
 K ( =-12 
 
 *=+ 3 
 
 ,,K=- 7 
 
 153 
 K=n- 27 
 
 ,"=- 8 
 
 N.B. My calculations were made on a principle, now abandoned, which 
 led to slightly different values. I therefore now continue the calculation with 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 Sequences of Angles. 
 
 303 
 
 M 2 
 
 LK) 
 m * 
 
 K* 
 (,,K + 30n) 
 
 /> 
 
 
 
 ( K-15'n) 
 
 0* 
 
 KX 
 (,K + 15n) 
 A 
 
 + 5 
 
 March - 8 
 
 March 19 
 
 March - 8 
 
 const. 
 
 + 22 
 
 7T-34 
 
 + 7 
 
 
 April +52 
 
 April TT 49 
 
 April +22 
 
 
 + 82 
 
 7T-64 
 
 +37 
 
 
 May IT - 68 
 
 May TT - 79 
 
 May +52 
 
 
 IT -88 
 
 + 86 
 
 + 67 
 
 
 
 June +71 
 
 June +82 
 
 
 
 + 56 
 
 7T-83 
 
 
 
 July +41 
 
 July - 68 
 
 
 
 + 26 
 
 7T 53 
 
 
 
 Aug. +11 
 
 Aug. TT - 38 
 
 
 
 - 4 
 
 r-23 
 
 Semi-diurnal. 
 
 = a'H m cos ,A m G = a' 
 
 cos 
 
 a m _p m sn m S- 
 '036 H sin 
 
 ,- H ;= 4 , 22) 
 
 -A 
 
 = 1227 
 
 ( ,3 6 H;,%.00 8) 
 
 (.H^.m, 
 
 (036 H,",2 = -007) 
 
 + 122 
 
 March + -007 
 
 + 122 
 
 March + -008 
 
 + 011 
 
 March- -001 
 
 const. 
 
 + 006 
 
 const. 
 
 + 007 
 
 const 
 
 + 003 
 
 
 April +-004 
 
 
 April +-005 
 
 
 April +-006 
 
 
 + 001 
 
 
 + 001 
 
 
 + 007 
 
 
 May - -003 
 
 
 May - -003 
 
 
 May + -006 
 
 
 -006 
 
 
 -006 
 
 
 + 004 
 
 j* 
 
 March + -129 
 
 March + -130 
 
 Y 
 
 March + '010 
 
 + 128 
 
 + 129 
 
 + 014 
 
 April +-126 
 
 April +-127 
 
 April +-01 7 
 
 + 123 
 
 + 123 
 
 + 018 
 
 May +-119 
 
 May +-119 
 
 May + -017 
 
 + 116 
 
 + 116 
 
 + 015 
 
 June +'115 
 
 June +'114 
 
 June +-01 2 
 
 + 116 
 
 + 115 
 
 + O08 
 
 July +-118 
 
 July +'11 7 
 
 July + -005 
 
 + 121 
 
 + 121 
 
 + 004 
 
 Aug. +-125 
 
 Aug. +-125 
 
 Aug. +-005 
 
 + 128 
 
 + 128 
 
 + 007 
 
 Repeat without change. 
 
 It might suffice if the parallactic correction to K^ were neglected, in 
 which case z^ = m^ = TO A, 1 2 = m F. The labour of making the correct table is, 
 however, inconsiderable. 
 
304 
 
 COMPUTATION OF TIDE-TABLE. 
 
 Diurnal : A = /3'H cos >, G = /3'H O p 2 cos *t, F = /3'H O p sin ^ 
 ,A = '036 H, cos >, ,G = "036 H, ,p 2 cos >, ,F = "036 H, ^ sin 
 
 
 <A 
 (/S'H^-041) 
 
 A 
 
 (-036 H, 
 = 047) 
 
 G 
 
 (/3'H 00 p2 
 
 = 009) 
 
 ft 
 
 (036H, /P 2 
 = 013) 
 
 F 
 (P'tt ooP 
 = 019) 
 
 If 
 
 (036 H x t p 
 = -025) 
 
 March 
 
 -039 
 
 + 047 
 
 -009 
 
 + 013 
 
 + 006 
 
 -003 
 
 
 -034 
 
 + 047 
 
 -007 
 
 + 013 
 
 + 011 
 
 + 003 
 
 April 
 
 -027 
 
 + 044 
 
 -006 
 
 + 012 
 
 + 014 
 
 + 009 
 
 
 -018 
 
 + 038 
 
 -004 
 
 + 010 
 
 + 017 
 
 + 015 
 
 May 
 
 -008 
 
 + 029 
 
 -002 
 
 + 008 
 
 + 019 
 
 + 020 
 
 
 + 003 
 
 + 018 
 
 + 001 
 
 + 005 
 
 + 019 
 
 + 023 
 
 June 
 
 + 013 
 
 + 007 
 
 + 003 
 
 + 002 
 
 + 018 
 
 + 025 
 
 
 + 023 
 
 -'006 
 
 + 005 
 
 -002 
 
 + 016 
 
 + 025 
 
 July 
 
 + 031 
 
 -018 
 
 + 007 
 
 -005 
 
 + 012 
 
 + 023 
 
 
 + 037 
 
 -028 
 
 + 008 
 
 -'008 
 
 + 008 
 
 + 020 
 
 Aug. 
 
 + 040 
 
 -037 
 
 + 009 
 
 -010 
 
 + 004 
 
 + 015 
 
 
 + 041 
 
 -043 
 
 ** 
 
 + 009 
 
 -012 
 
 -001 
 
 + 010 
 
 
 ^ -v 
 *i 
 
 " ' >r 
 
 mi 
 
 ~v 
 h 
 
 
 March + -008 
 
 March + -004 
 
 March + -003 
 
 
 + 013 
 
 + 006 
 
 + 014 
 
 
 April +-01 7 
 
 April +'006 
 
 April +-023 
 
 
 + 020 
 
 + 006 
 
 + 032 
 
 
 May +-021 
 
 May +-006 
 
 May +-039 
 
 
 + 021 
 
 + 006 
 
 + 042 
 
 
 June +-020 
 
 June + -005 
 
 June +-043 
 
 
 + 017 
 
 + 003 
 
 + 041 
 
 
 July + -013 
 
 July +-002 
 
 July + -035 
 
 
 + 009 
 
 000 
 
 + 028 
 
 
 Aug. + -003 
 
 Aug. - -001 
 
 Aug. +-019 
 
 
 -002 
 
 -003 
 
 + 009 
 
 Repeat these sequences, changing the signs. 
 
 Nodal Corrections. (See (51) (52) 9.) 
 
 Find by reference to preceding sequences the following nine sequences : 
 (i.)-'0372A m , (ii.)+-283A // , (iii.) + -296F,,, (iv.) + -283F,,, (v.)-'0372G m , 
 (vi.) - -296G,,, (vii.) - -0372G m , (viii.) + -283 G /7 , (ix.) + -319 F y/ 
 
 Then the semi-diurnal sequences are as follows : 
 a 2 is (i.) + (ii.) ; b 2 is (iii.) ; c 2 is (iv.) ; d 2 is (v.) + (vi.) ; 
 
 e z is (vii.) + (viii.) ; / 2 is (ix.) 
 
1891] COMPUTATION OF TIDE-TABLE. 305 
 
 For example : 
 
 2 2- 
 
 March -'003 June -'113 
 
 -003 --113 
 
 April --017 July -'099 
 
 - -042 - -074 
 May --072 August -'044 
 
 - '098 - -018 
 
 This, and the other semi-diurnal sequences are repeated without change 
 of sign, and in all six of them the months run just as in this example, and 
 denote the places at which to begin reading the sequence for the month in 
 question. 
 
 The diurnal sequences are obtained thus : 
 Find, from preceding sequences, the twelve following 
 (i.) + -188A , (ii.) + -115A,, (iii.)--391F , (iv.) + "297F,, (v.)+188F , 
 (vi.) + 115F,, (vii.) + '391 G , (viii.) - -297G,, (ix.) + -188G , 
 
 (x.) + -115G,, (xi.) - -0905 F , (xii.) + -080 F, 
 Then 
 
 ! is (i.) + (ii.) ; h is (iii.) -I- (iv.) ; c, is (v.) + (vi.) ; 
 
 c?j is (vii.) + (viii.) ; ^ is (ix.) + (x.) ; /j is (xi.) + (xii.) 
 
 For example : 
 
 a,. a 1 . 
 
 March +'026 June +'059 
 
 + -041 + '050 
 
 April +-052 July +'038 
 
 + '060 + -023 
 
 May +'064 August +'007 
 
 + -064 - -010 
 
 This, and the other diurnal sequences, are repeated with change of sign, 
 and the months in all six of them run just as in this example, and denote 
 the places at which to begin reading the sequence for the month in question. 
 
 Calculation of Height, Interval, and Corrections for each Month. 
 (See 7, 8, 9, 10.) 
 
 Remarks. Each column in the following computation is arranged exactly 
 like the first, so that it is unnecessary to repeat the letters in the successive 
 columns. 
 
 For the month of March, which serves as an example, we refer to the 
 
 March sequences, and enter the twelve values of G 2 successively, in the top 
 
 left-hand corners of twelve columns ; below these are entered the twelve 
 
 values of G x , and the twelve values of G 4 , and on the right of the columns 
 
 D. i. 20 
 
306 COMPUTATION OF TIDE-TABLE. [7 
 
 are put the twelve values of F 2 , F 1} F 4 . A similar statement is true of all 
 the other symbols all the way down, and all the sequences are utilised up to 
 twelve entries in each. 
 
 The divisions and multiplications may be done by Crelle's table ; A is 
 found by a table of natural tangents, and 8A is converted into degrees by a 
 table of circular measure or radians. It is necessary to take as an approxi- 
 mate value of A the nearest even number of degrees. 
 
 From the places where the values of A are found, the left-hand side of 
 each column corresponds to the time of moon's transit written at the head of 
 the column, and the right-hand side to a time of moon's transit 12 h greater 
 than the time specified. But the whole table for any month serves for its 
 opposite (e.g., September opposite to March), by transposing the words right 
 and left in the preceding statement. Thus the whole computation has only 
 to be made for six months (up to August inclusive), instead of for twelve. 
 The diurnal terms with suffix 1, are written in the margin, with alternative 
 signs, and the upper sign is to be used on the left, and the lower on the right 
 of each column. 
 
 Thus, in finding, for example, FjCos|A on the right we deem the Fj 
 written at the head to have its sign changed. Thus, in the column of O h we 
 have Fj = '069 and on the right-hand, A = -f 4 ; then the required entry, 
 on the right-hand, for F a cos \ A is + '069 cos (+ 2) = + '069. 
 
 The values of 8T, 8M, SN are extracted from the sequence of those func- 
 tions in Table IV., and they are the same on each side of the column. The 
 value of B is taken from the sequence of the semi-annual tide and mean 
 water, and changes only with the month. 
 
 The parallactic correction to the mean interval, i, is introduced in com- 
 puting R. This is a constant of the port and is the same in all months. 
 
 In computing the height p^, and its corrections, an approximate value of 
 A is used, namely, the nearest even number of degrees ; this approximate A 
 will often be the same as A . 
 
 In this table it appears to me specially important that the signs of the 
 sines and cosines should be determined independently of their numerical 
 values. 
 
 Whereas in the right-hand of column 6 h we get, as a result of a second 
 approximation, no value of A, the conjectural value A = + 90 is adopted for 
 the computation of P, Q, S, and values of M, N, R are not computed. 
 
 The table has rows marked 81, Sh, M", N", P", Q" ; all these are derived 
 from a subsequent table of " Corrections for reference to the Moons transit." 
 But it appears convenient to finish off the computation on this sheet, although 
 we have to pause in the computation in order to calculate the said table of 
 corrections. 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 MARCH. 
 
 Interval or O h 
 
 15 or I 1 ' 
 
 
 90 or 6 h 
 
 
 G 2 +2-509 F 2 - -195 
 G! + -092 FI - -069 
 G 4 - -047 F 4 + -003 
 G 2 +G X +2-601 FJJ + F! - '264 
 G.J-G! +2-417 FJJ-F! - '126 
 
 + 2-484 + -285 
 + -117 + -131 
 - -026 - -004 
 + 2-601 + -416 
 + 2-367 + -154 
 
 &c. 
 
 + -627 + -195 
 + '107 + -765 
 - '047 + -003 
 + '734 + -960 
 + -520 - -570 
 
 
 tan A - F + F i , . 102 F *~ F i + .()r )2 
 
 
 
 
 
 tan a ~ i |uz n _, i uo/i 
 Lr 2 + IT ! 1*2 ^1 
 
 A +6 +4 
 
 - 10 - 4 
 
 
 -52 +48 
 
 
 a FgcnsAo - -194 - '195 
 F 2 sinA - -020 - "014 
 y G<,cosA +2-495 +2-503 
 S G 2 sinA + -262 + -175 
 
 + -281 + -284 
 - -050 - -020 
 + 2-446 +2-478 
 - -431 - -173 
 
 
 + -120 + -131 
 - -154 + -145 
 + -386 + -420 
 - -493 + -466 
 
 
 f + F 1 cosiA - -069 + -069 
 +F 1 sin|A - -004 + -002 
 i) +2G,cos^A + -184 - -184 
 e 2G 1 sin|A + -010 - '006 
 
 + -131 - -131 
 
 - Oil + -005 
 + -233 - -234 
 - -020 + -008 
 
 &c. 
 
 + -688 - -699 
 - -336 - -311 
 + -192 - -196 
 - -094 - -087 
 
 13 
 
 _g 
 
 CO 
 
 C 
 I 
 
 X F 4 cos2A + '003 + '003 
 in F 4 sin2A + -001 ^ + -000 
 v iG 4 cos2A - '023 .| - -023 
 p |G 4 sin2A 8 - -005 - -003 
 
 - -004 - -004 
 
 + -ooi + -ooi 
 
 - -012 - -013 
 + -004 + -002 
 
 
 - -ooi -ooo 
 
 - -003 + -003 
 + -006 + -002 
 + -023 - -023 
 
 "o 
 o 
 
 0) 
 
 1 
 
 a+8 + -068 g - -020 
 (+0 - -059 & + -063 
 \ + p - -002 -2 -000 
 
 - -150 + -111 
 + -111 - '123 
 000 - -002 
 
 
 373 + -597 
 + -594 - -786 
 + -022 - -023 
 
 ,a~ 
 r < 
 
 rH 
 
 s 
 
 Sum N + -007 + -043 
 
 - -039 - -014 
 
 
 + -243 - -212 
 
 lO 
 CD 
 
 -7; - '188 + '186 
 
 - '244 + -239 
 
 
 - -528 - -115 
 
 i i 
 
 o 
 -1-3 
 
 Hf-f) - ' 94 s - + ' 93 
 
 "/3-y -2-515 -2-517 
 2(/i-v) + -048 ,_, + -046 
 
 - -122 + -120 
 - 2-496 - 2-498 
 + -026 + -028 
 
 &c. 
 
 - -264 - -058 
 - -540 - -275 
 - -018 + -002 
 
 SL, 
 
 3 
 
 a> 
 3 
 
 a 
 
 Sum D - 2-561 - 2%378 
 
 -2-592 -2-350 
 
 
 - '822 - -331 
 
 -*^ 
 
 o 
 
 8A = N /D - -003 ^ - '018 
 (In degrees) SA -0-2 -1-0 
 
 + -015 + -006 
 +0-9 +0'3 
 
 
 - '296 + -640 
 -17-0 +36'7 
 
 O 
 
 A = A +SA +5-8 -| C +3-0 
 
 -9-1 -3-7 
 
 
 - 68-6* No H.W.* 
 
 
 ^A -193 * -100 
 
 1 21 A 7 & A 
 302 . 20 A / S 4 
 
 303 -123 
 
 11 4 
 
 
 2-287 | 
 80 2 
 
 
 (In hours)/ + -200 r| + -104 
 8T - -030 H - -030 
 
 - -314 - -127 
 + -059 + -059 
 
 &c. 
 
 -2-367 $ 
 
 + -030 fl 3 
 
 C fi 
 
 
 + '170 + -074 
 (Mean int.) 4 7 '900 7-900 
 
 - -255 - -068 
 7-900 7-900 
 
 
 -2-337 |-g 
 7-900 | 
 
 XI .= 
 
 
 8-070 7-974 
 
 7-645 7-832 
 
 
 5-563 U 
 
 
 8 h 4 m -2 7 h 58 m> 4 
 (See below) 81 +0'9 +0*4 
 
 7 h 38 ra -7 7 h 49 m '9 
 -1-8 -0-8 
 
 
 ^.s 
 
 5 h 33-8 ^ 
 +0-6 
 
 
 B 8 h 5 m 7h 5gm 
 
 711 37m 7h 4 9 m 
 
 
 5 h 34 m 02 
 * 
 
 
 202 
 
308 
 
 COMPUTATION OF TIDE-TABLE. 
 
 MARCH (continued). 
 
 Correction of D or O h 
 
 15 or l h 
 
 &c. 
 
 90 or 6 h 
 
 &c. 
 &c. 
 
 &c. 
 
 a + 8 
 
 *(+) 
 
 4(X+p) 
 
 Sum This correction need only be made when 
 (In circ. meas.) 8A D is small and 8A considerable 
 
 Product 8D 
 D 
 
 D = D +8D -2-56 -2*38 -2'59 -2'35 
 
 - -531* 
 + -126* 
 + -044* 
 
 - -361* 
 
 + -022* 
 
 - -008* 
 - '744* 
 
 - -752* 
 
 Height 
 
 A 2 2-445 -F a + '195 
 Aj + -247 -Fj + "069 
 A 4 - -012 -F 4 - -003 
 Approx. A +6 +4 
 
 2-422 - -285 
 + -345 - -131 
 - -007 + -004 
 - 10 - 4 
 
 &c. 
 
 &c. 
 
 &c. 
 
 &c. 
 
 691 - -195 
 + -417 - -765 
 - -012 - -003 
 - 70* None 
 
 A 2 cos A 2-432 2-439 
 -F 2 sinA + -020 + '014 
 + AjCOsiA + '247 - - '247 
 + 2F!sm|A + -007 <~ ^ - '005 
 A 4 cos2A - -012 ~-^ - -012 
 -|F 4 sin2A - -000 *C - '000 
 
 2-385 2-416 
 + '050 + -020 
 4- -344 - -345 
 + -023 - -009 
 - -007 - -007 
 
 - -ooi - -ooo 
 
 236 
 + -183 -g 
 + -342 | 
 + -878 g 
 + -009 i 
 
 + -ooi 
 
 V 
 
 Sum h 2-694 .? 2'189 
 B 3-791 * | 3-791 
 
 2-794 2-075 
 3-791 3-791 
 
 1-649 
 
 3-791 
 
 c 
 
 6-485 S" 2 5-980 
 (See below) 8k - -004 p + -001 
 
 6-585 5-866 
 + -004 - -012 
 
 5-440 J 
 - '014 o 
 
 
 $ 6-48 5-98 
 
 6-59 5-85 
 
 5-43 
 
 Nodal correction 
 
 A +6 +4 
 119 m -rD -46 m -50 m 
 
 -10 -4 
 -46 m -51 m 
 
 -70* +90* 
 - 158'"* 
 
 c 2 - '016 e z + -001 
 cj - -003 ei + -Oil 
 
 + -014 + -002 
 + -032 + -015 
 
 + -016 - -117 
 + -134 + -014 
 
 c 2 cos A - '016 - -016 
 e 2 sinA +0 +0 
 C!COSA - 3 +3 
 
 + 2e,sinAA + 1 - 1 
 
 <D -u> 
 
 + -014 + -014 
 0-0 
 + 32 32 
 3 + 1 
 
 + -005 
 
 + -no 
 + -no 
 
 - -16 
 
 Sum - -018 e.SP - '014 
 o *-< 
 
 + -043 - -017 
 
 + -209 
 
 Mult.byll9/D +0'8 g, +0'7 
 8M -0-8 'S fe -0-8 
 
 -2-0 +0-9 
 + 1-6 +1-6 
 
 - 33-0 
 
 + 0-8 
 
 SumM' 0-0 |jJ2 -0-1 
 (See below) M" +0-4 p +0-2 
 
 -0-4 +2-5 
 -0-8 -0-4 
 
 - 32-2 
 + 0-3 
 
 Sum M +0-4 +0'1 
 
 -1-2 +2-1 
 
 - 31-9 None 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 MARCH (continued). 
 
 309 
 
 Nodal correction 
 (continued) or " 
 
 15 or I 1 ' 
 
 
 90 or 6 h 
 
 
 d 2 --120 / 2 --019 
 o?! --159 /! --017 
 
 &c. &c. 
 
 &c. 
 
 &c. &c. 
 
 &c. 
 
 o? 2 cos A 
 / 2 sin A &c. 
 
 C?!COS A A 
 
 2/isin|A Compute like M 
 Sum 
 Mult. byl!9/D &c. 
 SN 
 
 Sum N' &c. 
 (See below) N" 
 SumN 
 
 &c. &c. 
 
 &c. 
 
 &c. &c. 
 None 
 
 &c. 
 
 a 2 --003 -c 2 +'01 6 
 a t + -026 -Cj + -003 
 
 - -003 - -014 
 + 041 -O32 
 
 &c. 
 
 -113 -'016 
 + 059 -'134 
 
 &c. 
 
 2 cos A - -003 - -003 
 -c 2 sinA + 2 ^ - + 1 
 ,cosiA +26 ^S.&p -26 
 
 + 2c,sin|A + 3 ^ - 
 
 o fl 
 
 - -003 - '003 
 + 2+1 
 + 41 - 41 
 + 6 2 
 
 
 - -039 -000 
 + 15 - 16 
 + 48 - 42 
 + 154 +-190 
 
 
 
 
 Sum F + -025 ft * - '028 
 
 (See below) P" - '002 g | -000 
 
 i . 
 
 + -046 - '045 
 + -002 - -005 
 
 
 + 178 +-132 
 -006 +-005 
 
 
 H- 1 -5 
 
 Sum P + -023 - -028 
 
 + 048 -'050 
 
 
 + 172 +-137 
 
 
 & 2 -'017 -c? 2 +-120 
 &! --070 -rf, +-159 
 
 &c. &c. 
 
 
 &c. &c. 
 
 
 6 2 cos A 
 - d 2 sin A &c. 
 bi cos ^ A 
 + 2o?isin^A Compute like P 
 Sum Q' &c. 
 (See below) Q" 
 Sum Q 
 
 &c. &c. 
 
 &c. 
 
 &c. &c. 
 
 &c. 
 
 Parallactic corrections 
 
 k +-010 ?., +-130 
 li +-003 wi +-004 
 
 + 014 +-129 
 + 014 +-006 
 
 &c. 
 
 + 012 +-114 
 + -043 + -005 
 
 &c. 
 
 ? 2 cosA +-010 +-010 
 w 2 sinA + 14 + 9 
 ^co.s|A + 3 4J.2? 3 
 2wi 1 sin|A +0 g " - 
 
 + 014 +-014 
 - 22 - 9 
 + 14 - 14 
 -1+0 
 
 &c. 
 
 + 004 
 -107 
 + 35 
 - 6 
 
 &c. 
 
 Sum +-027 Sjfl +-016 
 
 + 005 --009 
 
 
 -074 
 
 
 Mult. byl!9/D -1-2 *H * -0-8 
 Par. corr. to i +0'6 gi +0'6 
 
 -0-2 +0-5 
 + 0-6 +0-6 
 
 
 + 11-7 
 + 0-6 
 
 
 O 
 
 Sum R -0-6 -0-2 
 
 + 0-4 +1-1 
 
 
 + 12-3 None 
 
 
310 
 
 COMPUTATION OF TIDE-TABLE. 
 
 MARCH (continued). 
 
 Parallactic corrections Q C Q h 
 (continued) 
 
 15 or l h 
 
 &c. 
 &c. 
 
 90 or 6 h 
 
 &c. 
 &c. 
 
 z 2 + -129 -Z 2 -'010 
 2l + -008 -^ -'003 
 
 &c. &c. 
 
 &c. &c. 
 
 z 2 cos A 
 - 2 sin A &c. Compute like 
 ZiCos^ A 
 + 2^ sin | A &c. 
 
 P and Q 
 
 &c. &c. 
 
 Sum S + -135 + -120 
 
 + 143 + -115 
 
 + 115 +-035 
 
 SECOND approximation. 
 
 When the correction SA is large, as in the case of column 6 h , this is 
 
 necessary. 
 
 
 Column 
 
 of 90 or 6 h 
 
 
 Assume A 
 
 -70 
 
 + 86 
 
 
 a 
 
 ft 
 I 
 
 + 067 
 -183 
 + 214 
 -598 
 
 + 014 
 + 195 
 + 044 
 + 625 
 
 
 f 
 
 c 
 
 n 
 
 6 
 
 + 627 
 - -439 
 + 175 
 -123 
 
 -560 
 -529 
 -157 
 -146 
 
 
 \ 
 M 
 
 V 
 
 P 
 
 -002 
 -002 
 + 018 
 + 015 
 
 - -003 
 + 000 
 + -023 
 -003 
 
 
 a+S 
 
 + 
 
 X + p 
 
 - '531 
 + 504 
 + 011 
 
 + 639 
 -706 
 -006 
 
 Sum N 
 
 -016 
 
 -073 
 
 
 C-1 
 
 -614 
 
 -377 
 
 lff-9) 
 
 l-y 
 iQi-*) 
 
 -307 
 -397 
 -040 
 
 -189 
 + 151 
 
 -046 
 
 Sum D 
 
 -'744 
 
 -084 
 
 8A = N /D 
 (In degrees) SA 
 
 + 022 
 + l-4 
 
 + 87 
 + 50 
 
 A 
 
 = A + 8A 
 
 -68-6 
 
 No H. W. 
 
1891] COMPUTATION OF TIDE-TABLE. 311 
 
 It is concluded that as the correction in the second column is 50 there 
 is no H.W. A conjectural value of A = + 90 is used above in computing 
 P, Q, S * 
 
 * There ought in strictness to be further corrections to E and $J, but they are of little 
 importance. Thus : 
 
 Further correction to K and J|J. 
 When A is greater than (say) 50 there are further corrections [51], [Sh] computed from 
 
 012 
 [51] = - 2" (1 - fa) [3F 2 sin 2 A cos A + (G,, + G 8 ) sin 3 A] 
 
 [dh] = - -012 [3 (A,, + A,) sin 2 A cos A -- F 2 sin 3 A] 
 
 Thus in the column of 6 h on the left A= -70; then compute thus: 
 
 F 2 = + -195, G 2 = + -627, A,,= + '691 
 
 3F 2 = + -585, -H m =- 1-568, -H w =- 1-568 
 
 G 2 -H m =- -941 A 2 -H m =- -877 
 
 G /y + G.= - -941 3(A 2 -H OT )=-2-631 
 
 3(A /y + A 8 ) = -2-631 
 
 By successive use of Traverse table, 
 
 (G,, + G g ) sin A = + -884, 3F 2 sin A = - -550 
 
 (G,, + Gg) sin 2 A = - -831, 3F 2 sin 2 A= + -517 
 
 (G // + Gg)gin 3 A= + -781 
 + 3F 2 sin 2 AcosA= + -177 
 
 + -958 
 x - -012 
 
 Divide by D or - "752 - -0115 
 x2 h + -0153 
 
 + -0306 
 -*_ _4 
 
 [51 ]= + -030 
 
 = + l ra -8 
 
 Previous 5 = 5 h 33 m -8 
 
 Correct I = 5 h 36 m 
 Again, by successive use of Traverse table, 
 
 3 (A,, + A,) sin A = + 2-472, - F 2 sin A = + -183 
 
 3 (A,, + A,) sin 2 A = - 2-323, - F 2 sin 2 A = - -172 
 
 3(A // + Ag)sin 2 AcosA= - -795 
 -F.,sin 3 A=+ -162 
 
 633 
 x - -012 
 
 [Sh]=+ -008 
 Previous ffi = 5-426 
 
 Correct = 5-43 
 
312 
 
 COMPUTATION OF TIDE-TABLE. 
 
 Evanescent Tide. (See 7.) 
 
 The right-hand column of 6 h leads to no H.W., and the tables of I and 
 P^ must be completed by other formulae. 
 
 The following calculation is very like the preceding one. The value A 
 will always be nearly + 90, and in our example it is exactly + 90. 
 
 The computation of M, N, R is to be omitted, and that of P, Q, S has 
 been included in the general calculation with a conjectural A = + 90. 
 
 MARCH. 
 
 Evanescent 
 
 Tide 
 
 
 Evanescent Tide 
 
 (continued) 
 
 90 or 6 h 
 
 90 
 
 or 6 h 
 
 G 2 + -627 
 G! +-107 
 G 4 - -047 
 
 Gjj-Gj +.520 
 
 F 2 
 i F * 
 
 Ff^fF, 1 
 
 G 2 -G t 
 
 + 195 
 + 765 
 + 003 
 + 191 
 
 + 004 
 + 130-0 
 
 + 90 
 
 (In degrees) SA 
 7!\>A 
 
 A 
 * ) 
 
 8T 
 Mean int. i 
 
 (See below) 87 
 ^ 
 
 + 110 
 + 6'3 
 
 +96-3 
 
 3-210 
 112 
 
 tanA " A F : +iFi I 
 
 F 2 -Fi 
 
 3-322 
 
 + 030 
 
 a F 2 cos A 
 /3 F 2 sin A 
 y G 2 cos A 
 8 G 2 sin A 
 
 
 ooo 
 
 + 195 
 
 ooo 
 
 + 627 
 
 3-352 
 7-900 
 
 e F] cos ^A 
 f F 1 sin|A 
 T} 2G!COsiA ,, 
 6 2G! sin |A O 
 
 ja 
 
 -541 
 -541 
 -151 
 -151 
 
 11-252 
 ll h 15 m -l 
 + 2-5 
 
 ll h 18 
 
 X F 4 cos 2A 
 p. F 4 sin 2A 
 v iG 4 cos 2A 
 p |G 4 sin 2A 
 
 S 
 
 o 
 
 
 ooo 
 
 + 024 
 
 ooo 
 
 A 2 -691 
 A t +-417 
 A 4 -'012 
 
 A 
 
 F 2 - -195 
 F! - -765 
 F 4 - -003 
 96 or TT - 84 
 
 I ~"9 5) 
 
 i(f-*) 
 SO*-*) '! 
 Sum D 
 
 ^ 
 
 -390 
 
 A 2 cos A 
 - F 2 sin A 
 + A! cos iA 
 + 2FJSU1IA 
 A 4 cos 2 A 
 - |F 4 sin 2A 
 
 Sum h 
 B 
 
 (See below) 8h 
 
 5 
 
 - -072 
 .SP - -194 
 - -279 
 + 1-137 
 - -012 
 J + -001 
 
 o 
 5b 
 
 -195 
 + 195 
 -048 
 
 
 -048 
 
 ('+*) 
 a + 8 " 
 Sum - E 
 
 1 
 P 
 
 -692 
 
 % + -581 
 ^ 3-791 
 
 
 -173 
 
 + 627 
 - -012 
 
 So 4-372 
 
 + -oil 
 
 
 + 442 
 -442 
 
 J3 4-38 
 
1891] COMPUTATION OF TIDE-TABLE. 313 
 
 Corrections for Reference to Moon's Transit. (See (57) (64), 10.) 
 
 Of these corrections 81, Sh, M", N", P", Q" have already been used in the 
 preceding calculation, and we have to show how they are to be computed ; 
 we also have to compute f and ft'. 
 
 From March " intervals " and " heights " we extract / and h, and arrange 
 them in double columns the even entries in one column and the odd in 
 another. The columns O h to ll h afford the 12 values for O h to ll h of/ and h 
 by means of their left hand entries, and they afford the 12 values for 12 h to 
 23 h by means of their right hand entries. The entry for 23 h is repeated 
 at the top and that for O h at the bottom, so that each column has 13 entries, 
 and thus each provides 12 first differences. After finding these differences, 
 the distinction of odd and even entries is unnecessary. 
 
 The numerical factors 2'2, $, 2'34, ^, -fa, 2'34 in the legends at the 
 top of the columns are absolute constants. 
 
 The @'s are derived from the sequence in Table IV., beginning the 
 sequence with the month treated. 
 
 The values of c and d are derived from Table V., for the month named ; 
 thus for March c is - 3 m> 7 and d is - O h> 062. 
 
 The values of M", N", SJ are found in columns vii., xi., xii. of the first 
 table, and P", Q", Bh in vii., xi., xii. in the second table. The entries 
 opposite O h to ll h were used above on the left-hand side of columns O h to 
 ll h , and the entries opposite 12 h to 23 h were used above on the right-hand 
 side of the columns O h to ll h . 
 
 The quantity ft' is not a final result, but after interpolated values of ft' 
 (see below on Interpolation) have been found, we shall add to it computed 
 values of the annual tide, so as to form ft. 
 
 The arithmetical processes involved in these tables are sufficiently ex- 
 plained by the instructions at the head of each column. 
 
314 
 
 COMPUTATION OF TIDE-TABLE. 
 
 nj 
 
 a o'x 
 *** >J 
 
 00 00 IT- CO . 
 
 O 
 
 S < 
 
 *! 
 
 KS .' X 
 
 40 'S -* 
 
 CO 
 
 OS 00 O Oi . 
 
 CO ^C 
 
 g 6 AH -^ 4t< a 
 + 1 1 1 ^ 
 
 + + 
 
 ? M 
 
 * '* .i 
 
 rH 
 
 -f CM iO i~ CM rH 
 
 - TTTT^ 
 
 '\ 1 
 
 1 
 
 |H 
 
 iC O ^ iO 00 i i 
 i i O CO CO -J t> 1^- 
 CN (N AH 6 og 6 AH 
 1 1 1 1 II 
 
 X '> 
 
 Jo 
 
 77 i i ^ 
 
 1 1 
 
 00 
 
 :a o 
 
 cq I-H Tt< co j 
 
 CM O 
 
 oo oo 
 
 (M (N rH O cW 
 
 1 1 1 1 
 
 O rH 
 
 1 1 
 
 ir<-j 
 
 -f 00 00 (N . 
 
 -* CO 
 
 + : ?7T^ 
 
 +7 
 
 ' ? * 4 
 
 CM Tt* 00 O 
 
 + i i 7^ 
 
 rH 00 
 rH 
 
 + + 
 
 rH 
 
 .S 
 
 *" '53 
 
 O 1> O5 C3J CO i l 
 
 O O AH oq j* CM AH 
 + 111 + + 
 
 SH 
 
 >r ;g h 
 
 ^* ^H 
 
 O O O O 
 i l CM C 00 O 
 
 o o 
 
 rH 
 
 ^ x 
 := w 
 
 Ot) ^^ (O I~H 
 CM CM CM I-H . 
 
 o *c 
 
 TTTT^ 
 
 1 1 
 
 1 .- ti 
 
 CN S 
 
 J rH rH AH O o 
 
 lilt 
 
 Oi 1~- 
 
 1 1 I-H 
 
 j 1 
 
 
 CO i i-i 
 t~- CO CO 
 
 O O AH 
 
 + ' 1 . 
 
 O 
 
 O CM 
 
 ^ CM 00 
 
 6 6 
 
 + 1 
 
 CO 
 
 6 
 
 + 
 
 t^* O> 
 CM CM 
 
 AH 6 
 + + 
 
 
 rH CO O i < CM CO OCMCOO 
 'CM og CM CM 
 
 i :fl S^ 
 
 CO l^~- "C O LC 1C 
 
 * 9 ^j" 
 1 1 ++ 1 1 
 
 " :g i 
 
 *^t* ^t i?t (^* ^O CC 
 O 1 G^l ? "^ G^l 
 
 'i + i i J i i 
 
 x 
 O" + 
 
 1 x .j 
 
 X 00 O Tt* t^ CO 
 
 + + 1 1 ^ + + 
 
 .d 
 
 X 
 
 O5 C5 00 O5 !> O5 
 
 00 CO rH rH . CD O 
 
 + + 1 1 + + 
 
 .a 1 
 
 ,fj CM rH rH . Tjl CD 
 + + 1 1 ** + + 
 
 iH 
 
 'I 1 
 
 * 1 ' O5 O r-H 1C 
 
 OS -^f I-H CM . I>- i i 
 
 O O i 1 
 
 9 =8 
 ++ 1 1 ++ 
 
 rH : ? + 
 
 CM CM CN CM O O 
 
 O i i CM I-H 
 jg O ^ 
 
 1 + 1 1 II 
 
 ^ i 
 
 rH rH r-H CD J O O 
 
 i + i i ^7 i 
 
 = i 
 
 I> >C CM 1C M* CO 
 
 r-H rH CM I-H . C35 O5 
 S I-H O I-H 
 ^3 
 
 'l + 1 1 II 
 
 *'ll 
 
 O O O O 
 
 rHCMlCOO^t^-* 
 
 a x 
 
 iC "^ C5 1>- CO O 
 
 O5 Tt* CM rH .OO 
 O rH O CM I-H 
 
 o eg 
 
 + + 1 1 + + 
 
 H .j 
 
 CO (M 00 CM . CD rH 
 
 .j CM rH O CO O 1C "^ 
 
 CN S 
 
 "H , J, . eg , , 
 
 + + 1 1 + + 
 
 
 CO O5 i l CO 
 *" CM CM CM CM 
 
 O 
 eg 
 OS rH 00 OS 
 jg CD 00 CM CO 
 
 
 -a *aa 
 
1891] 
 
 COMPUTATION OF TIDE-TABLE. 
 
 315 
 
 Additional Values of Intervals and Heights. 
 
 Where the intervals change largely between one column and the next, it 
 would add much to the accuracy if additional values were calculated. Thus, 
 in the calculation for March further values between 75 or 5 h and 90 or 6 h , 
 and again, between 90 or 6 h and 105 or 7 h , would be desirable. The like is 
 true for August, where a column between 105 or 7 h and 120 or 8 h would be 
 useful. I choose this last case for my example. 
 
 If Inman's traverse table be used, interpolation may be made for 112^ 
 without much difficulty, but I think it is better to interpolate for an even 
 number of additional degrees, and to compute a column for 113, found by 
 adding 8 D to 105. 
 
 It is proposed then to add a new column between the 8th and 9th. 
 
 We begin by interpolating in the sequences of angles. In each sequence 
 we have to find the 8th entry for August ; then, if it is semi-diurnal, add 16 ; 
 if diurnal add 8 for Kj and P, and subtract 8 for O ; and, if quater-diurnal, 
 add 32. In the sequence for (*) we add 16, since it is similar to a semi- 
 diurnal term. The calculation runs thus : 
 
 
 K 2 
 
 s, 
 
 
 
 K, 
 
 P 
 
 S 4 
 
 8th entry . . 
 Add .... 
 
 7T 46 
 
 + 16 
 
 7T + 19 
 
 + 16 
 
 -88 
 - 8 
 
 77 + 63 
 
 + 8 
 
 7T-45 
 
 + 8 
 
 7T+83 
 
 + 32 
 
 3 
 
 7T-30 
 
 7T+35 
 
 . 7T + 84 
 
 if + 71 
 
 77-37 
 
 -75 
 
 
 
 
 
 
 
 
 Also Xi = 0, ^2, n = TT 36, as before. 
 
 The 8th entry of is TT - 40, to which we add 16, and find = TT - 24. 
 With this value of 8, compute ST, SM, 8N. The interpolation amongst the 
 sequences of angles for the parallactic terms is done in the same way. With 
 these new values, and with the former H, Hp 2 , Hp we now compute new A's, 
 F's, G's, and are then in a position to compute a new column corresponding 
 to 113 or 7 h 32'". 
 
 In computing 81, 8h, M", N", P", Q", t, j)', column ii., for intervals, or 
 2dI/dT, must be put equal to 2 (/ 120 7 105 ) ; and similarly, column ii. for 
 heights, or 2dh/dT, must be put equal to 2 (h m h 105 ). 
 
 This interpolation would be especially valuable in the case of M, N, P, Q, 
 which change abruptly. 
 
 Some interpolation of the kind has been done in my example, but I do 
 not reproduce the work. 
 
316 COMPUTATION OF TIDE-TABLE. [7 
 
 The Calculation of a Low Water Table (referred to in general rule VII.). 
 
 This may be done almost independently of the H.W. table by replacing 
 the Sy's by B's and using the rules given in 12. The calculation may, 
 however, be materially abridged, and I will now go over the several steps 
 of the calculation noting the mode of transition from one case to the other. 
 
 The new A, G, F will be distinguished from the old by enclosing the new 
 ones in square parentheses. 
 
 Semi-diurnal [A 2 ], [G 3 ], [F 2 ]. 
 
 These sequences are derivable directly from the old ones by the rule 
 
 [AJ = A 2 + F 2 , [GJ = G 2 + F 2 , [FJ = F 2 - $ (A,, + A.) 
 Diurnal [AJ, [GJ, [FJ. 
 
 The rule is here more complex : 
 
 [AJ = - {A - F }, [G ] = - (G - s'_F }, [FJ = - {F + ^ A } 
 [AJ= A,+ *F,, [GJ= G 
 
 and in all these sequences shift the list of months in the margin six places 
 downwards, so that in the O and Kj sequences March stands where June 
 stood, and in the P sequence March stands where December stood. 
 
 If it be agreed to neglect the terms involving , -fa, &c., the rule is 
 simply to shift the months and change the signs of the O sequences; but 
 at Aden where the diurnal tide is very large, this would lead to a sensible 
 error. 
 
 After the new sequences for 0, K 1} P have been found, they are combined 
 to find [AJ, [GJ, [FJ just as for H.W. 
 
 Quater-diurnal 
 
 [A 4 ] = -A 4 , [GJ = -G 4 , [FJ=-F 4 
 that is to say, simply change signs. 
 
 Semi-annual and Mean Water and Annual Tide, 
 The old calculation serves again. 
 
 Mean Interval and its Parallactic Correction. 
 Here we subtract 180 from K m ; thus 
 
 K m 7T = 49 
 
 &(*.-)= 1-633 
 
 2-1 1 / \ K.^7 
 
 20'3on*m~ 1T)= O/ 
 
 i= 1-690 
 Parallactic correction = + O m< 08i = + O m< 14 
 
1891] COMPUTATION OF TIDE-TABLE. 317 
 
 It may be well to warn the computer that i may be negative, that is to 
 say L.W. may occur on the average earlier than moon's transit. 
 
 Parallactic Corrections. 
 
 a, a', /3, /3' are unchanged. 
 
 The rules are 
 
 [ W A] = m A - T V m F, LA] 
 [mG] = m G -fr m F, [,,G] 
 
 We then compute [/ 2 ], [w 2 ], M by the same rules as before. 
 In the diurnal terms compute by the following rules : 
 
 F 
 
 the list of months in the margin being shifted six places downwards, so that 
 March stands where June stood. The values of [z^. [wj], [/J are then com- 
 puted by the same rule as before. 
 
 Nodal Corrections. 
 
 We may, with sufficient accuracy, take 2 , 6 2 , c 2 , d 2 , e 2 ,/ 2 to be unchanged. 
 
 Referring to the instructions for the computation of a 1} b 1} &c., the new rule 
 
 may be stated thus : 
 
 [oj is (ii.) - (i.), [6,] is (iv.) - (iil), [c,] is (vi.) - (v.) 
 
 [dj is (viii.) - (vii.), |>i] is (x.) - (ix.), and [/,] is (xii.) - (xi.) 
 
 and the list of months in the margin is pushed down six places, so that 
 March stands where June stood. 
 
 The corrections 8T, 8M, 8N remain unchanged. 
 
 When the L.W. sequences have been formed the calculation follows the 
 lines of H.W. calculation precisely, save in three respects first, in the 
 "heights" h is to be subtracted from B , and Bh is to be then subtracted 
 from B h, instead of the corresponding additions in the H.W. calculation ; 
 secondly, the signs of P, Q, S are to be changed as a last step in the 
 calculation of those quantities, in order that the corrections to the heights 
 may be additive instead of subtractive as they would be if we left off exactly 
 as in the case of H.W. ; thirdly, after the final table for J)' has been made, 
 its values must be subtracted from the annual tide. 
 
 The reader will easily understand the necessity for these changes when it 
 is remarked that h, 8h, f)' have been estimated as depressions below mean 
 water, whereas B arid the annual tide are estimated as elevations above the 
 adopted datum ; in the result we require, of course, to estimate heights with 
 reference to the datum. 
 
318 COMPUTATION OF TIDE-TABLE. [7 
 
 It may be well to warn the computer that i + 1 may often be negative. 
 
 It will be unnecessary to refer henceforth to L.W., since the instructions 
 for H.W. serve also for L.W. 
 
 Interpolation. 
 
 The sun's longitude increased by 5 is indicated by , and the months 
 March, April, May, &c., really mean the dates when is 0, 30, 60, &c. 
 that is to say, about the middle of the months. The dates which, on the 
 average, fall the nearest to these times, are given in Table VI? 
 
 The 12 columns for any month, headed O h , l h , 2 h , . . . ll h contain on the 
 left the 12 values of 5, p^, M, N, &c., corresponding to moon's transit at 
 O h , l h , 2 h , . . . ll h , and they contain on the right the 12 values for moon's 
 transit at 12 h , 13 h , . . . 23 h . This applies to the month named at the head 
 of the table. But these values also appertain to the opposite month (i.e., 
 September opposite to March, October to April, and so on) by reading the 
 right hand entries as appertaining to O h , l h , . . . ll h , and the left to 12 h , 13 h , 
 . . . 23 h . The same is true of l and f)' (see Corrections for reference to moon's 
 transit), except that here the values change sign in the opposite month ; 
 thus the values of t and ])' which we have computed for March must be taken 
 with the opposite sign when applied for September. 
 
 Now it is required to form interpolated tables for every 10 of , and 
 in all the 18 tables (of which 6 will be originally computed and 12 inter- 
 polated) to interpolate for every 20 m of moon's transit. 
 
 These interpolations may be done graphically, and I find with millimetre- 
 square paper a convenient scale for is 1 mm. to 1, and for time of moon's 
 transit 15 mm. to l h . These will be set out horizontally as abscissae, and the 
 ordinates will be time in treating 5, and height in treating f^. 
 
 A convenient time scale for 3t is 30 mm. to the hour. In the case of ^ 
 the scale must depend on the range of tide at the place for Aden (with a 
 small range) I have found 50 mm. to the foot convenient. 
 
 I will begin with interpolation for (i.e., for time of year), and will 
 only refer to I, since |^ follows the same plan. Write March, April, May, 
 . . . January, February, March, at 0, 30, . . . 360 mm. along a horizontal line, 
 corresponding to the same number of degrees of . It may be well to 
 repeat February before the first March, and April after the last March. Set 
 off as an ordinate the left-hand entries from column O h for the six months 
 March, April, . . . August, and from the right-hand entries of O h for the same 
 six months set off ordinates for their opposite months, e.g., the right-hand I 
 of O h for March, affords 5 of O h for September. Through the tops of these 
 
1891] COMPUTATION OF TIDE-TABLE. 319 
 
 ordinates draw a smooth curve of O h .* Proceed similarly to form curves of 
 l h , 2 h , &c., twelve in all. If the figures get confused we may have two or 
 more, and confusion may often be avoided by drawing parts of the curve with 
 upward or downward shift, so as to make things clear where a number of 
 curves go through nearly the same point. 
 
 We now start a fresh figure with time of moon's transit as horizontal 
 line, and 31 as ordinate corresponding to March (or = 0). These 24 
 (computed) values of I, joined by a smooth curve, enable us to read off 
 the values of J for = for every 20 m of moon's transit, i.e., on the adopted 
 scale, at every 5 mm. of horizontal space. 
 
 We now set off from the previous figure the 24 (interpolated) values of I 
 corresponding to =10, of which the first 12 are found at "March + 10 mm." 
 and the last 12 at " September + 10 mm." These 24 values being joined by 
 a curve, give 5 for = 10, and for every 20 m of moon's transit. 
 
 We next set off 24 values of J corresponding to = 20, of which the 
 first 12 are found in the preceding figure at " March -f 20 mm.," and the last 
 12 at " September + 20 mm." These are treated the same way. 
 
 The next in the series are the computed April ( = 30) 3Ps which are 
 set off like the March ones, joined by a curve and read off to each 20 m 
 of moon's transit. 
 
 We then take 24 (interpolated) values of $ from " April + 10 mm." and 
 " October + 10 mm.," to give 5 for = 40. The 24 from " April + 20 mm." 
 and " October + 20 mm." afford 1 for = 50. The next is the computed 
 May ( = 60) series, and we so pass on through six months, the last in 
 the set being derived from "August + 20 mm." and "February + 20 mm." 
 corresponding to = 170. 
 
 * The following rule is probably known, but I do not know where it has been stated, except 
 in a note of my own in the Messenger of Mathematics. I have found it very useful in drawing 
 good curves. 
 
 Rule for Graphical Interpolation half-way between Computed Ordinates. 
 
 Draw the polygon (A) joining the tops of a number of equidistant ordinates, and draw the 
 two polygons (B) joining the tops of alternate ordinates. Then every ordinate has marked on it 
 an intercept or sagitta where a side of polygon (B) cuts it. On the half-way ordinates next on 
 each side of a sagitta, set off one-fourth of the sagitta from the points where the two sides of 
 polygon (A) cut those half-way ordiuates; the set-off is to be in the direction in which the 
 sagitta would shoot if it were an arrow. When all the quarter sagittas are thus set off, every 
 half-way ordinate (except the first or last of the series) has two points marked on it. 
 
 The interpolated curve passes half-way between the pairs of marked points, except in the 
 case of the first and last half-way ordinate, when it passes through the single marked points. 
 
 This rule is correct to fourth differences, except in the case of the first and last ordinate, 
 when it is correct only to third differences. In the cases in the text the computed values are 
 cyclical, and there are therefore no first or last. 
 
 By means of proportional compasses set to 4, the quarter sagittas may be set off rapidly, and 
 the bisection of the pairs of marked points may be made by eye. 
 
320 COMPUTATION OF TIDE-TABLE. [7 
 
 If one person reads the numbers from the figure, whilst another writes, 
 the tabulation may be done very rapidly. 
 
 The same process is applied for tabulation of the p^'s. We should, in 
 strictness, do the same by M, N, P, Q, R, S, t, ft', but it appears unnecessary 
 to work with so much accuracy. 
 
 I have done much of the interpolation by simply writing out the com- 
 puted values of the quantity to be tabulated in a chess-board table with 
 blanks for the interpolated values. If sixteen squares be considered, a 
 computed value will stand at each corner. Then a great many of the in- 
 terpolated values may be put in by inspection of the march of the quantity 
 in the two directions. In other parts I make a pencil curve, on millimetre- 
 square paper, of four or five adjacent values, and pass a freehand curve 
 through them to fill in the interpolated values ; I rub out the curve when 
 used. It must be remembered that close accuracy in these terms would be 
 mere pedantry. 
 
 M, N are computed to the decimal of a minute, and the decimal part 
 may be useful for drawing the pencil interpolation curves, but the result 
 should be tabulated only to the nearest minute. Similarly the third decimal 
 in P and Q may be dropped. 
 
 The interpolation of t and f)' follows the same plan, but it must be borne 
 in mind that these functions change sign in opposite months, and this con- 
 sideration is important when we come to interpolate for = August -f 10, 
 and + 20. 
 
 When there is an evanescent tide (as in the case of March, 18 h ) the 
 corrections M, N, R become infinite. As a practical solution this is absurd, 
 and the fact is that there may or may not be a H.W. according to the 
 values of 8 and II. Again, in other parts of this and other lunations there 
 may be no H.W., although the tide-table predicts one. In all such cases 
 there is a long period of four or five hours' duration of nearly slack water, 
 and it is accordingly almost a matter of indifference whether or not a small 
 H.W. is predicted. It would necessitate very laborious computations to make 
 correct predictions in these cases, and the result would not be worth the 
 labour. I have adopted, therefore, a makeshift, and have replaced H.W. by the 
 height of water and time when the rate of change of water level is a minimum. 
 
 It has been proposed above that P, Q, S shall be computed with a con- 
 jectural A = + 90, and this is better than the plan which I actually adopted 
 in my experimental table for Aden, of which a sample is given below*. The 
 practical point to consider in the present instructions is the manner of treat- 
 ment of M, N, R about the time of evanescence. I propose then that the 
 gap in the values shall be bridged by a conjectural curve, and that the values 
 
 * Thus if any one seeks to verify my table, he will not get exactly my values for P, Q, S in 
 the neighbourhood of 18 h . 
 
1891] 
 
 TIDE-TABLE. 
 
 321 
 
 be only given in round numbers. For example, for March we have the 
 following values of M : (16 h ) + 9 m> 3, (17 h ) + 28 m '5, (18 h ) blank, (19 h ) + 45 m -0, 
 (20 h ) + 17 m '7, &c. By drawing a curve I conjecture + 50 m for the missing 
 value at 18 h . 
 
 A comparison with the corresponding complete curves for February and 
 April helps us in filling the gap. 
 
 After the complete table for !)' is formed we proceed to add to it the 
 values computed in the table of the annual tide for every 10 of , and so 
 form a table of |). For example, the first five values of f)' are '02*, - *01, 
 '01, '01*, '00 (of which those marked * are computed), and to these 
 we add '390, the computed annual tide for March, and obtain '37, '38, '38, 
 38, -39. 
 
 The final results are then arranged in a table f. If the L.W. were also 
 computed I should propose that the L.W. and H.W. should be given 
 alternately. The following is a sample of the table computed only for 
 H.W. at Aden: 
 
 PORT of Aden ; High Water Tide-Table. 
 
 Times of moon's 
 transit for 
 
 For March 15th (i.e., from sun's long. 350 to 0), and for September 17th 
 (i.e., from sun's long. 170 to 180). The upper signs of i and f) apply 
 to March, the lower to September 
 
 March 
 
 Sept. 
 
 I 
 
 i 
 
 & 
 
 *> 
 
 M 
 
 N 
 
 B 
 
 P 
 
 Q 
 
 S 
 
 h. m. 
 
 h. m. 
 
 h. m 
 
 m. 
 
 ft. 
 
 ft. 
 
 m. 
 
 m. 
 
 m. 
 
 ft. 
 
 ft. 
 
 ft. 
 
 
 
 12 
 
 8 5 
 
 + 4 
 
 6-49 
 
 + 37 
 
 
 
 + 9 
 
 - 1'3 
 
 + 02 
 
 -08 
 
 + 14 
 
 20 
 
 20 
 
 7 55 
 
 + 4 
 
 54 
 
 + 38 
 
 
 
 + 9 
 
 - 1-0 
 
 + 03 
 
 -08 
 
 14 
 
 40 
 
 40 
 
 46 
 
 4 
 
 57 
 
 38 
 
 - 1 
 
 + 9 
 
 - 0-6 
 
 + 04 
 
 -09 
 
 14 
 
 1 
 
 13 
 
 7 37 
 
 + 4 
 
 6-59 
 
 + '38 
 
 - 1 
 
 + 9 
 
 - 0-2 
 
 + '05 
 
 -09 
 
 14 
 
 20 
 
 20 
 
 28 
 
 + 4 
 
 60 
 
 + 39 
 
 2 
 
 + 9 
 
 + O'l 
 
 + '06 
 
 -09 
 
 14 
 
 40 
 
 40 
 
 18 
 
 + 4 
 
 60 
 
 + 39 
 
 - 2 
 
 + 9 
 
 + 0-4 
 
 + 07 
 
 -09 
 
 15 
 
 &c. &c. &c. &c. &c- 
 
 17 
 
 5 
 
 8 29 
 
 + 14 
 
 4-21 
 
 + 37 
 
 + 28 
 
 -11 
 
 + 2-3 
 
 -16 
 
 -01 
 
 09 
 
 20 
 
 20 
 
 9 52 
 
 + 15 
 
 21 
 
 + 36 
 
 + 35 
 
 - 2 
 
 - 7'3 
 
 -10 
 
 -03 
 
 10 
 
 40 
 
 40 
 
 11 2 
 
 + 14 
 
 27 
 
 35 
 
 + 43 
 
 + 20 
 
 -16 
 
 -05 
 
 -05 
 
 10 
 
 18 
 
 6 
 
 11 17 
 
 + 10 
 
 4-38 
 
 + 35 
 
 + 50 
 
 + 30 
 
 -20 
 
 00 
 
 -06 
 
 09 
 
 20 
 
 20 
 
 15 
 
 + 6 
 
 53 
 
 + 34 
 
 + 54 
 
 + 25 
 
 -30 
 
 + '05 
 
 -05 
 
 08 
 
 40 
 
 40 
 
 7 
 
 + 1 
 
 71 
 
 + 33 
 
 + 50 
 
 + 20 
 
 -30 
 
 + 10 
 
 -04 
 
 06 
 
 &c. &c. &c. &c. &c. 
 
 t I have found that it is convenient to cut the constituent tables into strips and paste them 
 together again, so as to save much copying and verification. 
 
 D. I. 21 
 
322 METHOD OF .USING THE TIDE-TABLE. [7 
 
 If -or be the moon's parallax at moon's transit in minutes of arc, and S3 be 
 the longitude of moon's node, the interval is 
 
 + t + M cos 3 + N sin S3 + O - 57') R 
 and the height is 
 
 ffi + ft + P cos S3 + Q sin & + (tsr - 57') S 
 
 After the table has been completed the computer should test the correct- 
 ness of the prediction by computing two or three tides in each month, and 
 comparing the results either with the observations from which the harmonic 
 constants were originally derived, or with other known values of high and 
 low water. 
 
 Examples of the Use of the Table. 
 
 From and after the year 1887, the datum for the tide-tables of the Indian 
 Government for Aden has been 0'37 ft. lower than that used in my table ; as 
 I am going to compare my results for 1889 with those of the Indian Govern- 
 ment, 0'37 ft. will be added to my heights to make the two comparable. 
 
 (A) The moon crossed the meridian at Aden on March 17th, 1889, at 
 O h ll m , Aden M.T. Aden is in 3 h O m E. long., and therefore this is about 21 h , 
 March 16th, G.M.T. ; whence from the Nautical Almanac we find or the moon's 
 parallax at Aden transit was 58''2, and OT - 57' = + 1'2. The longitude of 
 moon's node was 108, and cos 83 = '3, sin S3 = + I'O. 
 
 Then referring to our table and interpolating between O h O m and O h 20 m , 
 and taking the upper signs of t and ft, we find 
 
 I + i = 8 h 4 m , f^ + ft = 6-89 ft., M = 0, N = + 9 m , R = -l m '2, 
 
 P = + -03ft., Q = -'08ft., S = + '14ft. 
 Hence M cos S3 + N sin 3 + R (r - 57) = + 8 m 
 
 and P cos S3 + Q sin S3 + S (w - 57) + 0*37 = + O45 ft. 
 
 Therefore the interval is 8 h 12 m , and the time of H.W. 8 h 12 m + O h ll m or 
 8 h 23 m p.m., March 17th ; and the height is 7'34 ft. or 7 ft. 4 in. 
 
 The Indian tide-table gives as time 8 h 12 m p.m., and as height 7 ft. 4 in. 
 
 (B) On September 17th, 1889, the moon crossed the meridian at 
 18 h 36 m , w was 54'-2 or -m - 57' = - 2'8, S3 = 98, cos &3 = - '14, sin S3 = + I'D. 
 
 Interpolating in our table between 18 h 20 m and 18 h 40 m , and taking the 
 lower signs of t and ft, we find 
 
 E-i = ll h 22 m , f$- ft = 4-34, M = + 50 m , N = + 20 m , R=-30 m , 
 
 P = + '10, Q = -'04, S = + '06 
 
 Hence M cos &3 + N sin 83 + R ( - 57) = + 97 m = + l h 37 m 
 and PcosS3 + sinS3 + S<r-57) + (K37= + 015 ft. 
 
1891] COMPARISON WITH MECHANICAL PREDICTION. 323 
 
 Therefore the interval is 12 h 59 m , and the time of H.W. 12 h 59 m + 18 h 36 m 
 or 31 h 35 m September 17th, i.e., 7 h 35 m p.m. September 18th, and the height 
 is 4'49 or 4 ft. 6 in. 
 
 The Indian tide-table gives " no inferior H.W." 
 
 This example shows our table at its worst, for it is clear that a nominally 
 small correction to the time which amounts to an hour and forty minutes 
 must give unsatisfactory results. At this time of year mean water has a 
 height of 3 ft. 10 in., hence our prediction only shows a rise of 8 inches. 
 Although there was probably in reality no maximum (as predicted in the 
 Indian table), I should expect that the water stood at about 4 ft. 6 in. at 
 half-past seven of the evening of September 18th, 1889. 
 
 PART III. COMPARISON AND DISCUSSION. 
 
 Comparison. 
 
 As stated in the Introduction, Mr Allnutt computed a complete H.W. 
 tide-table for 1889 for Aden in order to compare the results with the Indian 
 tide-tables, which are made with the tide-predicting instrument. When this 
 comparison was made our tables had not been brought into exactly the form 
 given above, arid Mr Allnutt's work was considerably more laborious than it 
 would have been if undertaken later. 
 
 The mechanical predictions were probably made with constants which are 
 the means of the results derived from eight years of observation, whereas the 
 constants used in our tables are derived from only four years. Mr Roberts 
 has supplied me with six weeks of prediction for the year 1887, worked 
 mechanically in duplicate, namely, with the eight-year and the four-year 
 constants. In the latter case the times of H.W. seem to run about 6 m later 
 than in the former, but the difference often rises to 10 m , occasionally to 15 m , 
 and at rare intervals to 20 m . The two sets of constants also give a systematic 
 difference in the heights amounting to about 2 inches, but the difference 
 often rises to 3 inches or falls to 1 inch, and occasionally reaches 4 inches and 
 zero. It follows, therefore, that a sensible part of the discrepancy, or error as 
 it may be called, of our computation is due to the difference of constants. 
 But not nearly all of the " error " can be set down to this cause ; it is due to 
 a combination of causes, namely, flaws in the interpolation, imperfect repre- 
 sentation of the elliptic tides, and partial inclusion in our method of all the 
 lunar inequalities which are totally neglected in the machine. The principal 
 cause of " error," however, is the imperfect correction for longitude of moon's 
 node and parallax about the time in each lunation when there is partial or 
 total evanescence of the inferior H.W. 
 
 212 
 
324 COMPARISON WITH MECHANICAL PREDICTION. [7 
 
 Omitting, as we must do, the cases of evanescence, there were 689 H.W. 
 computed by Mr Allnutt, and he finds " the probable errors " in the time and 
 height to be 9 m and 1'2 inches. 
 
 In the course of the year there were thirty-two occasions on which the 
 time " error " amounted to 30 m or more. All of these occur in the inferior 
 H.W. at the times of approximate evanescence, where the nodal and parallactic 
 corrections are large. At these times there is always a period of four or five 
 hours of nearly slack water, and the time at which a small maximum occurs 
 is of no importance from a navigational point of view. If these thirty-two 
 cases be taken away the probable error falls to 7 m . A cursory inspection of 
 the table shows also that nearly all the " errors " of 25 m to 30 m fall about the 
 time of evanescence and are therefore unimportant. We are accordingly 
 justified in saying that our predictions do not differ materially from the 
 Indian tables. 
 
 It has been already mentioned that I have six weeks of mechanical 
 prediction for 1887, made with the identical constants used in our table. 
 
 1 have, therefore, taken a month, or 58 H.W., out of these six weeks, and 
 compared them with my predictions, made without cross interpolation for 
 date. I find that the errors of time are 27 from O m to 5 m , 13 from 5 m to 10 m , 
 10 from 10 m to 15 m , 4 from 15 m to 20 m , one error of 26 m , and one of 34 m . 
 These give a probable error of 7 m . All the large errors fall on the inferior 
 H.W., at the time when it is very small, and they are, therefore, practically 
 unimportant. 
 
 In the heights there are 16 cases of agreement. 22 errors of 1 in., 15 of 
 
 2 in., and 5 of 3 in. These give a probable error of I'O inch. 
 
 The errors of 3 inches all fall about the time when the moon's parallax 
 was small, and I also observe that this was commonly a time when the height 
 errors rose to their greatest in 1889. This is probably due to the imperfect 
 representation of the elliptic tides, which, as shown in 3, is inherent to our 
 method. 
 
 The concordance between the two is good enough, but less perfect in the 
 heights than I expected. 
 
 The last comparison is between our predictions and actuality. The 
 observed times and heights for part of 1884 have been furnished me by 
 Colonel Hill, R.E., by the direction of Colonel Strahan, R.E., Deputy Surveyor- 
 General in India. For the purpose of testing the present method, I have 
 computed a H.W. tide-table from 10th March to 9th April, and again from 
 12th November to 12th December, 1884. In these periods, there are 117 
 actual and one evanescent H.W. The observation of one H.W. is missing 
 through an accident to the tide-gauge; there are, therefore, 117 H.W. for the 
 purpose of comparison. 
 
1891] COMPARISON WITH ACTUALITY. 
 
 The following is a table of errors, regardless of signs : 
 
 325 
 
 Time 
 
 
 Heigh 
 
 t 
 
 Magnitude of error 
 
 No. of H.W. 
 
 Magnitude of error 
 
 No. of H.W. 
 
 m. m. 
 
 
 Inches 
 
 
 to 5 
 
 35 
 
 
 
 15 
 
 5 , 10 
 
 32 
 
 1 
 
 48 
 
 10 , 15 
 
 19 
 
 2 
 
 28 
 
 15 , 20 
 
 19 
 
 3 
 
 14 
 
 20 , 25 
 
 5 
 
 4 
 
 11 
 
 26 28 
 
 2 
 
 
 
 33 36 
 
 2 
 
 
 
 56 57 
 
 2 
 
 
 
 Evanescence 
 
 1 
 
 Evanescence 
 
 1 
 
 
 117 
 
 
 117 
 
 Omitting the case of evanescence and assuming the errors to conform in 
 distribution to the normal exponential law (which is not accurately the case), 
 the probable error is about 9 m in time, and 1*4 inch in height. 
 
 When the rise from the higher L.W. to the lower H.W. is nil there is 
 evanescence, and when that rise is small there is approximate evanescence. 
 1 have accordingly examined the 12 cases in which the error of time is equal 
 to or greater than 20 ra , and the following table gives the result. 
 
 Eise from L.W. 
 to H.W. 
 
 
 Time errors 
 
 Nil 
 
 
 
 6 in to 8 in 
 
 22, 
 
 26, 28, 56, 57 minutes 
 
 13 in 
 
 36 
 
 minutes 
 
 17 in 
 
 22 
 
 > 
 
 19 in 
 
 33 
 
 j 
 
 2 ft 10 in 
 
 22 
 
 5 
 
 3ft gin 
 
 23 
 
 ) 
 
 3" 11 in 
 
 20 
 
 5 
 
 It appears that where the errors of time were 56 m and 57 m the tide was 
 very nearly evanescent, and that the two other considerable errors of time, 
 viz., 36 m and 33 m , pertain to very small tides. 
 
 It has been already pointed out that in such cases a considerable error in 
 the time is of no importance, and it is justifiable, in testing the calculations, 
 to set aside the nine tides in which the rise is less or equal to 19 inches. 
 
 There remain 108 H.W., and the greatest error in the times amounts to 
 only 23 minutes. In 58 cases the error is 7 minutes or less, and in 51 cases 
 
326 COMPARISON WITH ACTUALITY. [7 
 
 it is 6 minutes or less ; as the half of 108 is 54, it follows that the probable 
 error is a little over 6 minutes. 
 
 The Indian predictions maintain their standard of excellence fairly well 
 through the periods of approximate evanescence, but out of 116 tides there 
 are 59 cases with time errors of 10 minutes or less; as the half of 116 is 58, 
 the probable error is about 10 minutes. 
 
 Turning now to the heights I find that both mine and the Indian 
 predictions present 63 out of 116 H.W. with zero errors or errors of 1 inch. 
 We may take it then that the probable error for both modes of prediction is 
 about 1 inch. The Indian predictions have, however, the disadvantage that 
 several errors of 5, 6, 7 inches occur. On the other hand, the 11 cases of 
 4 inches of error which occur in mine have a systematic character ; they are 
 all positive (actuality the greater), and all but one affect the higher H.W. 
 about the time when the moon's parallax is small. This defect is doubtless 
 due to the imperfect representation of the elliptic, evectional, and variational 
 tides inherent to my method. 
 
 The slight superiority shown over the mechanical prediction must be 
 attributed to the fact that I have used better values of the tidal constants 
 than were available in 1883, when the Indian predictions must have been 
 made. 
 
 I learn from Colonel Hill that two independent observers reading the 
 same tide-curve will frequently differ by 5 m and sometimes by 10 m in their 
 estimate of the time, and by 1 and sometimes by 2 inches in the height. 
 Accordingly, predictions which agree with a reading of a tide-curve with 
 probable errors of 6^ m in time and 1 inch in height may claim to possess a 
 high order of accuracy. 
 
 I conclude from the preceding discussion that with good values of the 
 tidal constants the present method leads to excellent predictions, and that 
 they are even better than are required for nautical purposes. 
 
 Discussion. 
 
 It is probable that methods may be invented by which some abridgement 
 of the computations may be made, but I am, of course, unable to suggest such 
 improvements. 
 
 The last-mentioned comparison seemed to show that but little accuracy 
 would be lost if P, Q, were entirely omitted, and if |)' were treated as zero, so 
 that 1) would consist simply of the annual tide. Indeed, the only advantage 
 gained by the retention of P, Q, f)' appeared to be the avoidance of a few 
 considerable errors in the inferior H.W. about the time of approximate 
 evanescence. Experience must decide whether the computation and the 
 tables may be lightened by the omission of these quantities. 
 
1891] DISCUSSION OF METHOD. 327 
 
 The advantage gained from M and N is marked, but as these quantities 
 arise almost entirely from the diurnal tides, I am inclined to think that, at 
 places where the diurnal tide is not extremely large, a very fair tide-table 
 might be made without them. 
 
 The present method will probably be applied to ports of second-rate 
 importance, where there are not sufficient data for very accurate determination 
 of the tidal constants. In such cases it will be best to omit the computation 
 of P, Q, \)', and to postpone that of M, N, and perhaps also of R and t, until 
 the simple tide-table has been tested as to its adequacy for navigational 
 purposes. At most places the annual tide is so large that f) cannot be 
 omitted, and it is impossible to dispense with the value of S. But it is 
 possible that it might suffice to attribute to S a constant value*, although 
 this would certainly cause very perceptible error in the heights of the lower 
 H.W. and higher L.W. A tide-table which only gave 5, f% ]), and a constant 
 S would be fairly short, even if computed for every ten days in the year ; and 
 this would be a great gain. 
 
 The question of how far to go in each case must depend on a variety of 
 circumstances. The most important consideration is, I fear, likely to be the 
 amount of money which can be spent on computation and printing ; and after 
 this will come the trustworthiness of the tidal constants and the degree of 
 desirability of an accurate tide-table. 
 
 My aim has been to reduce the tables to a simple form, and if, as I 
 imagine, the mathematical capacity of an ordinary ship's captain will suffice 
 for the use of the tables, whether in full or abridged, I have attained the 
 principal object in view. 
 
 * I may suppose the elliptic tides unknown, and I should then take S= I VH OT + -036H,,. For 
 Aden this would give S = '129, or say ^. 
 
8. 
 
 ON THE CORRECTION TO THE EQUILIBRIUM THEORY OF 
 TIDES FOR THE CONTINENTS. 
 
 I. By G. H. DARWIN. II. By H. H. TURNER. 
 [Proceedings of the Royal Society of London, XL. (1886), pp. 303 315.] 
 
 I. 
 
 IN the equilibrium theory of the tides, as worked out by Newton and 
 Bernoulli, it is assumed that the figure of the ocean is at each instant one of 
 equilibrium. 
 
 But Sir William Thomson has pointed out that, when portions of the 
 globe are occupied by land, the law of rise and fall of water given in the 
 usual solution cannot be satisfied by a constant volume of water*. 
 
 In Part I. of this paper Sir William Thomson's work is placed in a new 
 light, which renders the conclusions more easily intelligible, and Part II. 
 contains the numerical calculations necessary to apply the results to the case 
 of the earth. 
 
 If m, r, z be the moon's mass, radius vector, and zenith distance ; g mean 
 gravity ; p the earth's mean density ; or the density of water ; a the earth's 
 radius ; and J) the height of tide ; then, considering only the lunar influence, 
 the solution of the equilibrium theory for an ocean-covered globe is 
 
 This equilibrium law would still hold good when the ocean is interrupted 
 by continents, if water were appropriately supplied to or exhausted from the 
 sea as the earth rotates. 
 
 * Thomson and Tait's Natural Philosophy, 1883, 808. 
 
1886] DARWIN, CORRECTION TO THE EQUILIBRIUM THEORY. 329 
 
 Since when water is supplied or exhausted the height of water will rise 
 or fall everywhere to the same extent, it follows that the rise and fall of tide, 
 according to the revised equilibrium theory, must be given by 
 
 ft 3raa 1 ,,. ,n\ 
 
 - = - r (cos 2 z 4) a (2) 
 
 a 2 5 rr 3 l-f(r// 
 
 where a is a constant all over the earth for each position of the moon 
 relatively to the earth, but varies for different positions. 
 
 Let Q be the fraction of the earth's surface which is occupied by sea ; let 
 X be the latitude and I the longitude of any point; and let ds stand for 
 cos \d\dl, an element of solid angle. Then we have 
 
 integrated all over the oceanic area. 
 
 The quantity of water which must be subtracted from the sea, so as to 
 depress the sea level everywhere by era, is 4<7ra 3 aQ ; and the quantity required 
 
 to raise it by the variable height -^- -^- is the integral of this function, 
 
 taken all over the ocean. But since the volume of water must be constant, 
 continuity demands that 
 
 dWld 1 I I / o 1 \ J /Q\ 
 
 a = s r=- , x . IT* I (cos 2 - A) as (6) 
 
 integrated all over the ocean. 
 
 On substituting this value of a in (2) we shall obtain the law of rise 
 and fall. 
 
 Now if X, I be the latitude and W. longitude of the place of observation ; 
 h the Greenwich westward hour-angle of the moon at the time and place of 
 observation ; and 8 the moon's declination, it is well known that 
 
 cos 2 z \ = \ cos 2 A, cos 2 S cos 2 (h l) + sin 2\ sin 8 cos 8 cos (h I) 
 
 + f(i-sin 2 S)(i-sm 2 \) (4) 
 
 We have next to introduce (4) under the double integral sign of (3), and 
 integrate over the ocean. 
 
 To express the result conveniently, let 
 
 ~ 1 1 cos 2 X cos 21 ds = cos 2 \2 cos 21 2 , ^^ 1 1 cos 2 \ sin 2lds = cos 2 \ 2 sin 21 2 
 
 ~ I sin 2\ cos Ids = sin 2\ cos 1 1} -7^ 1 1 sin 2X sin Ids = sin 2\ sin h 
 
 ^VJJ ^TT^JJ 
 
 f' o-v 1 /C\ 
 
 sm-A - (5) 
 
 the integrals being taken over the oceanic area. 
 
330 DARWIN, CORRECTION TO THE EQUILIBRIUM THEORY. [8 
 
 These five integrals are called by Sir William Thomson gl, 33, <, 19, IE, 
 but by introducing the five auxiliary latitudes and longitudes, X 2 , ^> ^-i, ^i. X , 
 we shall find for the conclusions an easily intelligible physical interpretation. 
 
 It may be well to observe that (5) necessarily give real values to the 
 auxiliaries. For consider the first integral as a sample : 
 
 Every element of JJ cos 2 \cos2lds is, whether positive or negative, neces- 
 sarily numerically less than the corresponding element of 4-TrQ, and therefore, 
 even if all the elements of the former integral were taken with the same sign, 
 (47rQ)~ 1 //cos 2 Xcos 21 ds would be numerically less than unity, and d fortiori 
 in the actual case it is numerically less than unity. 
 
 Now using (5) in obtaining the value of //(cos 2 ^ i) ds, and substituting 
 in (3), we have 
 
 - -T- ~ 77- ^ ,-^ - = i cos 2 S [cos 2 A, cos 2 (/*. 1} cos 2 X n cos 2 (h L}] 
 a 2^(1 - fay/a)?- 3 
 
 + sin 2S [sin X cos X cos (h 1) sin Xj cos Xj cos (h ^)] 
 + f (-sm 2 S)(sin 2 X -sm 2 X) (6) 
 
 The first term of (6) gives the semi-diurnal tide, the second the diurnal, and 
 the third the tide of long period. 
 
 The meaning of the result is clear. The latitude and longitude X 2 , L give 
 a certain definite spot on the earth's surface which has reference to the semi- 
 diurnal tide. Similarly \ lt ^ give another definite spot which has reference 
 to the diurnal tide ; and X gives a definite parallel of latitude which has 
 reference to the tide of long period. 
 
 From inspection we see that at the point X 2 , 1 2 the semi-diurnal tide is 
 evanescent, and that at the point Xa, l. 2 + 90 there is doubled tide, as com- 
 pared with the uncorrected equilibrium theory. At the place X 1( , the 
 diurnal tide is evanescent, and at Xj , l there is doubled diurnal tide. 
 
 In the latitude X the long period tide is evanescent, and in latitude 
 (sometimes imaginary) arc sin \/{f sin 2 X } there is doubled long period tide. 
 
 Many or all of these points may fall on continents, so that the evanescence 
 or doubling may only apply to the algebraical expressions, which are, unlike 
 the sea, continuous over the whole globe. But now let us consider more 
 precisely what the points are. 
 
 It is obvious that the latitude and longitude X,, and L, being derived from 
 expressions for cos 2 X2Cos2 2 and cos^sin 2/ 2 , really correspond with four 
 points whose latitudes and longitudes are 
 
 X,,/ 2 ; -X 2 , Z 2 ; X 2 , / 2 +180; - X,, Z 2 + 180 
 
 Thus there are four points of evanescent semi-diurnal tide, situated on a 
 single great circle or meridian, in equal latitudes N. and S., and antipodal 
 
1886] DARWIN, CORRECTION TO THE EQUILIBRIUM THEORY. 331 
 
 two and two. Corresponding to these four, there are four points of doubled 
 semi-diurnal tide, whose latitudes and longitudes are 
 
 A 2 , / 2 + 90; -X 2 , Z a + 90; X 2 , J 2 + 270; - X 2 , l a + 270 
 
 and these also are on a single great circle or meridian, at right angles to the 
 former great circle, and are in the same latitudes N. and S. as are the places 
 of evanescence, and are antipodal two and two. 
 
 Passing now to the case of the diurnal tide we see that \ l} l t , being 
 derived from expressions for sin2\icos^ and sin 2A 1 sini, really correspond 
 with four points whose latitudes and longitudes are 
 
 *!, li ; - \, li + 180 ; 90 - X,, I, ; - 90 + X,, I, + 180 
 
 Thus there are four points of evanescent diurnal tide, situated on a single 
 great circle or meridian, two of them are in one quadrant in complemental 
 latitudes, and antipodal to them are the two others. Corresponding to these 
 four there are four points of doubled diurnal tide lying in the same great 
 circle or meridian, and situated similarly with regard to the S. pole as are 
 the points of evanescence with regard to the N. pole ; their latitudes and 
 longitudes are 
 
 -X 1} I,; X lf ^ + 180; - 90 +X 1} J,; 90 - X,, I, + 180 
 
 Lastly, in the case of the long period tide, it is obvious that the latitude X 
 is either N. or S., and that there are two parallels of latitude of evanescent tide. 
 In case siri 2 X n is less than f , or X less than 54 44', there are two parallels of 
 latitude of doubled tide of long period in latitude arc sin \/{f ~ sin 2 X }. 
 
 From a consideration of the integrals, it appears that as the continents 
 diminish towards vanishing, the four points of evanescent and the four points 
 of doubled semi-diurnal tide close in to the pole, two of each going to the 
 N. pole, and two going to the S. pole ; also one of the points of evanescent 
 and one of doubled diurnal tide go to the N. pole, a second pair of points of 
 evanescence and of doubling go to the S. pole, a third pair of points of evan- 
 escence and of doubling coalesce on the equator, and a fourth pair coalesce at 
 the antipodes of the third pair ; lastly, in the case of the tides of long period 
 the circles of evanescent tide tend to coalesce with the circles of doubled 
 tide, in latitudes 35 16' N. and S. 
 
 We are now in a position to state the results of Thomson's corrected 
 theory by comparison with Bernoulli's theory. 
 
 Consider the semi-diurnal tide on an ocean-covered globe, then at the 
 four points on a single meridian great circle which correspond to the points 
 of evanescence on the partially covered globe, the tide has the same height ; 
 and at any point on the partially covered globe the semi-diurnal tide is the 
 excess (interpreted algebraically) of the tide at the corresponding point on 
 the ocean-covered globe above that at the four points. 
 
332 TURNER, NUMERICAL EVALUATION. [8 
 
 A similar statement holds good for the diurnal and tides of long period. 
 
 By laborious quadratures Mr Turner has evaluated in Part II. the five 
 definite integrals on which the corrections to the equilibrium theory, as 
 applied to the earth, depend. 
 
 The values found show that the points of evanescent semi-diurnal tide 
 are only distant about 9 from the N. and S. poles ; and that of the four 
 points of evanescent diurnal tide two are close to the equator, one close to 
 the N. pole, and the other close to the S. pole ; lastly, that the latitudes of 
 evanescent tide of long period are 34 N. and S., and are thus but little 
 affected by the land. 
 
 Thus in all cases the points of evanescence are situated near the places 
 where the tides vanish when there is no land. It follows, therefore, that the 
 correction to the equilibrium theory for land is of no importance. 
 
 G. H. D. 
 
 II. 
 
 For the evaluation of the five definite integrals, called by Sir William 
 Thomson 1, 33, <2T, 19, IE, and represented in the present paper by functions 
 of the latitudes and longitudes A , \ l} X 2 , and l l} 1 2 , respectively similar in 
 form to the functions of the "running" latitude and longitude to be integrated, 
 it is necessary to assume some redistribution of the land on the earth's 
 surface, differing as little as possible from the real distribution, and yet with 
 a coast line amenable to mathematical treatment. The integrals are to be 
 taken over the whole ocean, but since the value of any of them taken over 
 the whole sphere is zero, the part of any due to the sea is equal to the part 
 due to the land with its sign changed ; and since there is less land than sea, 
 it will be more convenient to integrate over the land, and then change the 
 sign. 
 
 Unless specially mentioned, we shall hereafter assume that the integra- 
 tion is taken over the land. 
 
 The last of the integrals has already been evaluated by Professor Darwin*, 
 with an approximate coast line, which follows parallels of latitude and longi- 
 tude alternately. 
 
 * Thomson and Tait's Natural Philosophy, 1883, 808 [not reproduced in this volume]. 
 
1886] TURNER, NUMERICAL EVALUATION. 
 
 His distribution of land is given in the following table : 
 
 333 
 
 N. lat. 
 
 W. long. 
 
 E. long. 
 
 Lat. 80 to 90 
 
 20 to 50. 
 
 
 70 80 
 
 22 to 55: 85 to 115. 
 
 55 to 60: 90 to 110. 
 
 60 70 
 
 35 to 52 : 65 to 80 : 90 to 165. 
 
 10 to 180. 
 
 50 60 
 
 to 6 60 to 78 : 90 to 130. 
 
 10 to 140 : 155 to 160. 
 
 40 50 
 
 to 5 65 to 123. 
 
 to 135. 
 
 30 40 
 
 to 8 78 to 1 20. 
 
 to 120: 135 to 138. 
 
 20 30 
 
 to 15 : 80 to 82 : 97 to 110. 
 
 to 118. 
 
 10 20 
 
 to 17 : 87 to 95. 
 
 to 50 : 75 to 85 : 
 
 
 
 95 to 108: 122 to 125. 
 
 10 
 
 53 to 78. 
 
 to 48 : 98 to 105 : 
 
 
 
 112 to 117. 
 
 S. lat. 
 
 W. long. 
 
 E. long. 
 
 to 10 
 
 37 to 80 
 
 12 to 40: 110 to 130. 
 
 10 20 
 
 37 74 
 
 12 to 38 : 45 to 50 : 
 
 
 
 126 to 144'. 
 
 20 30 
 
 45 71 
 
 15 to 33: 115 to 151. 
 
 30 40 
 
 55 73 
 
 20 to 23 : 132 to 140. 
 
 40 50 
 
 65 73 
 
 170 to 172. 
 
 &(\ fin 
 
 fi7 72 
 
 
 <J\J DU 
 
 60 70 
 
 < M * *J 
 
 55 65 
 
 120 to 130. 
 
 70 80 
 
 about 20 of longitude. 
 
 80 90 
 
 180 
 
 N.B. The Mediterranean, being approximately a lake, is treated as land. 
 
 The limits of the 20 and 180 of longitude between S. latitudes 70 and 
 90 are not specified. For the evaluation of the last integral this is not 
 necessary, for restricting 
 
 (3 sin'X - 1) cos \d\dl 
 
 to a representative portion of the land bounded by parallels X x and X 2 , x and 1 2 , 
 
 r ~|A, TJ- 
 
 we get \(1 2 l\) sin X + sin 3X x - ; and similarly for Q ; so that if 
 
 j and t 2 be the number of degrees of longitude N. and S. of the equator 
 respectively between latitudes X 3 and Xj, the last of the integrals becomes 
 
 2 i (1 + Q sin X + sin 3X 
 
 sin X + sin 3X 
 
 But for (e.g.) 
 
 A* i t 
 
 720 - 2 (t, + 4) sin X 
 
 /A* fi t 1 r ~|A 2 r -p, 
 
 cos 2 Xcos2cosXc?XeW = Try 9sinX +sin 3X sin 21 \ 
 hji, 24[_ _U|_ _k 
 
 the actual limits 2 and h must be given, and not merely their difference. 
 
334 
 
 TURNER, NUMERICAL EVALUATION. 
 
 It is, however, obvious, on inspection of these integrals, that the land in 
 high latitudes affects them but little ; and we shall not lose much by neglect- 
 ing entirely the Antarctic continent in their evaluation. 
 
 This evaluation is reduced by the above process to a series of multiplica- 
 tions, and on performing them the following values of 0, 23, (>, HJ, 15, and Q 
 are obtained on the two hypotheses : 
 
 (1) That there is as much Antarctic land as is given in the schedule, 
 which is, however, only taken into account in the last integral IE, and the 
 common denominator 4-TrQ of each. 
 
 (2) That there is no land between S. latitude 80 and the pole. 
 
 The value of Q is given in terms of the whole surface, and represents the 
 fraction of that surface occupied by land ; it must be remembered that the 
 Mediterranean Sea is treated as land. Professor Darwin quotes Rigaud's 
 estimate* as 0'266 : 
 
 
 1st hypothesis 
 
 2nd hypothesis 
 
 <a 
 
 4-0-03023 
 
 +0-03008 
 
 ia 
 
 + 0-00539 
 
 +0-00537 
 
 ( 
 
 0-01975 
 
 G'01965 
 
 u 
 
 + 0-02910 
 
 + 0-02895 
 
 IE 
 
 -0-01520 
 
 -0-00486 
 
 Q ... 
 
 0-283 
 
 0-278 
 
 
 
 
 These results for 15 and Q have already been given by Professor Darwin 
 in Thomson and Tait's Natural Philosophy, and I have found them correct. 
 
 We then find for the set of latitudes and longitudes of evanescent tide : 
 
 Nature of tide 
 
 
 1st hypothesis 
 
 2nd hypothesis 
 
 Long period 
 
 lat. A 
 
 34 39' N. 
 
 35 4' N. 
 
 Diurnal \ 
 
 lat. A! 
 
 1 S. 
 
 1 S. 
 
 Semi-diurnal \ 
 
 long, li 
 lat. A 2 
 
 55 50 E. 
 
 79 54 N. 
 
 55 50 E. 
 
 79 56 N. 
 
 
 long. 1 2 
 
 5 3 W. 
 
 5 4 W. 
 
 The other points of evanescence are of course easily derivable from these, 
 as shown in the first part of this paper. 
 
 As a slightly closer approximation to truth, I have calculated these 
 integrals on another supposition. There are cases where lines satisfying the 
 
 equations 
 
 I = const, or X = const. 
 
 * Transactions of the Cambridge Philosophical Society, Vol. vi. 
 
1886] TURNER, NUMERICAL EVALUATION. 335 
 
 diverge somewhat widely from the actual coast line, but a line 
 
 al &X = const. 
 
 (where a and b are small integers) can be found following it more faithfully. 
 An approximate coast line of the land on the earth is defined in the following 
 schedule, west longitudes and north latitudes being considered positive. 
 
 Limits 
 longitud 
 
 + 20 to 
 
 of 
 
 le (0 
 
 + 10 ... 
 
 Equation 
 
 -X-/-40 
 
 Limits 
 latitude 
 
 + 20 to 
 
 1 Of 
 
 ,(X) 
 
 + 30' 
 + 40 
 + 73 
 
 
 I 10 
 
 4- 30 
 
 + 10 
 
 - 23 
 
 - 23 
 
 - X - 1 - 50 
 
 T O\J 
 
 + 40 
 
 + 120 
 
 X- 73 
 
 
 
 I 120 
 
 4- 73 
 
 + 80 
 
 + 120 
 
 + 20 
 
 X 80 
 
 -f ' O 
 
 
 / 20 
 
 4- 80 
 
 + 70 
 + 60 
 + 80 
 
 + 20 
 + 50 
 + 70 
 + 80 
 + 50 
 + 90 
 + 100 
 + 80 
 + 70 
 + 30 
 
 + 50 
 
 ... 3X 1 - 230 
 
 T o\J 
 
 + 70 
 
 + 70 
 
 X- + 10 
 
 + 60 
 
 + 80 
 
 X- 80 
 
 
 + 50 
 
 \ = l 
 
 + 80 
 
 + 50 
 + 30 
 
 + 90 
 
 ... -2X-Z-150 
 
 + 50 
 
 + 100 
 
 X 30 
 
 
 + 80 
 
 X--70 
 
 + 30 
 
 + 10 
 
 + 70 
 
 X 10 
 
 
 + 30 
 
 2X-Z-50 
 
 + 10 
 
 -10 
 -53 
 -14 
 
 + 60 
 
 + 73 
 
 -X-J-20 
 
 - 10 
 
 
 I 73 
 
 KQ 
 
 + 73 
 + 80 
 + 140 
 -150 
 -100 
 - 90 
 - 80 
 - 65 
 
 + 80 
 
 X- 2^-160 ... 
 
 oo 
 
 - 14 
 
 + 140 
 
 X-/-80 
 
 o 
 
 -150 
 
 X 60 
 
 v 
 
 -100 
 
 - X - / + 90 
 
 + 60 
 
 + 10 
 + 20 
 + 10 
 + 25 
 
 90 
 
 - 90 
 
 + A.-/+110 
 
 + 10 
 
 - 80 
 
 -X-j + fO 
 
 + 20 
 
 - 65 
 
 X - I + 90 
 
 + 10 
 
 - 40 
 
 -X-Z + 40 
 
 + 25 
 
 
 I 40 
 
 o 
 
 - 40 
 - 20 
 8| 
 
 - 20 
 
 X- 1 + 60 
 
 v 
 
 20 
 
 -40 
 + 5 
 
 - 8| 
 
 X 41 + 40 
 
 *v 
 
 -40 
 
 + 121 
 
 X 5 
 
 
 + 12*,, 
 
 -130 to 
 -150 
 -140 , 
 
 + 20 
 
 X-2J-20 
 
 + 5 
 
 + 20 
 -10 
 
 -150 
 
 New Guinea. 
 2X = + 130 
 
 to 
 
 - 140 
 
 X 10 
 
 
 -130 
 
 X - I + 130 
 
 -10 . 
 
 
 
336 TURNER, NUMERICAL EVALUATION. [8 
 
 Australia. 
 
 Limits of Limits of 
 
 longitude (1) Equation latitude (X) 
 
 -140to-150 A, = Z + 130 -10to-20 
 
 I = - 150 - 20 - 35 
 
 -150 -115 X = -35 
 
 - 35 - 221 
 
 -115 ,,-140 -2X = + 160 -22|,,-10 
 
 It will be seen that it is only rarely necessary to depart from the forms of 
 equation + X = I + x and the two original forms A, = const., I = const, to repre- 
 sent the coast line with considerable accuracy. There are still left one or two 
 outlying portions, of which mention will be made later. 
 
 Now supposing we are to find the value of the first integral for the 
 portion of land indicated by the shaded portion of the diagram, EQ being 
 the equator: 
 
 \=c 
 
 the equations to its boundaries being written at the side of each. 
 We have 
 
 cos 3 Xo5X cos 2ldl = ^ I I 9 sin X -f sin 3X I cos 2ldl 
 
 ~12 
 
 r/3 
 + I (9 sin c 4- sin 3c) cos 2ldl 
 
 J & 
 
 fa 
 
 4- - [9 sin ^ (I + y} -h sin %(l + y}} cos 2ldl 
 
 J ft 
 
 We may thus simply travel round the boundary omitting the places 
 where A, = constant : being careful to go round all the pieces of land in the 
 same direction. If we suppose I = a to be the meridian of Greenwich, and 
 the land to be in the northern hemisphere, the direction indicated above is 
 the wrong one for obtaining the value of the integrals over the land, for the 
 longitudes increase to the left; but by following this direction we shall 
 obtain the values over the sea as is in reality required. 
 
 The result of integration has, of course, a different form for each form of 
 the relation between I and X representing the boundary. In computing the 
 numerical values of the integrals, it is convenient to consider together all the 
 parts of the boundary represented by similar equations. 
 
TURNER, NUMERICAL EVALUATION. 
 
 337 
 
 Below are given as representative the forms which the numerator of the 
 first integral & assumes for different forms of the boundary, the quantities 
 within square brackets being taken within limits. 
 
 Form Value of Integral 
 
 + \ = l + x + J ? [^ cos (51 + 3#) + 3 cos (3 + x) + cos (I + 3#) 
 
 - 9 cos (- I + x)] 
 
 \ = i + J ? (9 sin x + sin 3#) [sin 21] 
 
 1 = x Zero 
 
 X = 21 + a; ^V [f cos (4>l + x) - 1 sin x + 1 cos (81 + 3#) 
 
 + 1 cos (4>l + 3#)] 
 A, = 4>l + x ^ [f cos ( til + x) + | cos (21 + x) + -Jj cos (14 + 3#) 
 
 + 2X = I + x i A [-^- c s (5 + a:) 6 cos ^ ( 3 + x) 
 
 + f cos ^ (71 + 3ar) - 2 sin ^ (- ^ + 3#)] 
 
 + ^ cos (3^ + x) cos ( I + ac)] 
 Evaluating these integrals on this supposition, we obtain 
 
 1st hypothesis 2nd hypothesis 
 
 + 0-02119 +0-02110 
 
 + 0-00778 +0-00775 
 
 -0-01890 -0-01882 
 
 + 0-03159 +0-03128 
 
 IE -0-04364 -0-03319 
 
 0-283 0-278 
 
 It will be noticed that the values of Q are exactly the same as before. 
 From these we deduce 
 
 Nature of tide 1st hypothesis 2nd hypothesis 
 
 Long period lat. \, 33 29' N. 33 55' N. 
 
 Diurnal I lat - X i 1 3 S. 1 3 S. 
 
 long. I, 59 7 E. 58 58 E. 
 
 Semi-diurnal. ,.. } j at '^ 81 22 N 81 23 N 
 
 long. l z 10 5 W. 10 5 W. 
 
 The agreement of these values of the quantities with the values calculated 
 on the previous supposition is not quite so close as I anticipated, but it 
 should be remarked that the numerators of the quantities &, i3, <ZD, 19, IE 
 D. i. 22 
 
338 TURNER, NUMERICAL EVALUATION. [8 
 
 are the differences of positive and negative quantities of very much greater 
 magnitude, as becomes obvious on proceeding to the numerical calculation ; 
 and thus a comparatively small change in one of the large compensating 
 quantities, due to large tracts of land in different portions of the globe, affects 
 the integrals to a considerable extent. 
 
 In this connexion I was led to investigate the effect of counting various 
 small islands and promontories as sea, or small bays and straits as land. For 
 instance, a portion of sea in the neighbourhood of Behring's Straits is 
 included as land, and a corresponding correction must be applied to the 
 integrals. This correction I have estimated as follows : The area of the sea 
 is estimated in square degrees, by drawing lines on a large map corresponding 
 to each degree of latitude and longitude and counting the squares covered by 
 sea, fractions of a square to one decimal place being included, though the 
 tenths have been neglected in the concluded sum. This area has then been 
 multiplied by the value of (say) cos 3 \ cos *2l for the approximate centre of 
 gravity of the portion, to find an approximate value of the integral 
 
 J/cos'X cos 21 d\dl 
 over its surface. 
 
 By drawing the assumed coast line on a map, it will become obvious that 
 such corrections may be applied for the following portions, defined by the 
 latitude and longitude of their centres of gravity; remarking that when there 
 is a portion of land which may be fairly considered to compensate a 
 portion of sea in the immediate neighbourhood, no correction has been 
 applied. For instance, it would be seen that part of the Kamschatkan 
 Peninsula is excluded from the coast line, and part of the Sea of Okhotsk is 
 included; but these will produce nearly equal effects on the integrals in 
 opposite directions, and are thus left out of consideration. 
 
 Area in 
 square degrees Longitude Latitude 
 
 -1-160 +172 +64 
 
 + 240 +150 + 71 
 
 + 166 + 85 +60 
 
 + 80 + 60 +52 
 
 + 68 + 85 + 9 
 
 20 + 75 +21 
 
 69 + 55 + 4 
 
 + 43 + 34 -11 
 
 + 22 - 37 -20 
 
 48 47 -19 
 
 + 65 53 +18 
 
 - 16 -107 +13 
 
 34 -102 + 2 
 
 - 49 -114 +1 
 
1886] 
 
 TURNER, NUMERICAL EVALUATION. 
 
 339 
 
 Area in 
 square degrees 
 
 - 27 
 11 
 43 
 
 - 39 
 
 Longitude 
 
 -123 
 -118 
 -138 
 -173 
 
 Latitude 
 + 12 
 - 5 
 + 36 
 
 -42 
 
 N.B. Land-areas are considered positive, sea negative. 
 
 We then find the following corrected values of the integrals : 
 
 
 1st hypothesis 
 
 2nd hypothesis 
 
 a 
 
 + 0-02237 
 
 + 0'02247 
 
 J3 
 
 + 0-00230 
 
 + 0-00231 
 
 <& 
 
 Q-01952 
 
 0'01961 
 
 IB 
 
 + 0-02665 
 
 +0-02676 
 
 
 
 -0-01775 
 
 0'02810 
 
 O .. 
 
 0-279 
 
 0-274 
 
 
 
 
 and finally the following values of the latitudes and longitudes of evanescent 
 tides : 
 
 Nature of tide 
 
 
 1st hypothesis 
 
 2nd hypothesis 
 
 Long period 
 
 lat. X (J 
 
 34 33' N. 
 
 34 7' N. 
 
 Diurnal < 
 
 lat. X t 
 
 57 S. 
 
 57 S. 
 
 Serai-diurnal < 
 
 long. ?j 
 lat. X 2 
 
 53 47 E. 
 
 81 23 N. 
 
 53 46 E. 
 
 81 21 N. 
 
 
 long. J 2 
 
 2 56 W. 
 
 2 56 W. 
 
 The estimation of corrections due to these supplementary portions has 
 been checked in two cases by a detailed extension of the method of square 
 blocks of land used previously for evaluation of the whole integrals ; that is 
 to say, two of these portions were separately divided into square degrees 
 (instead of squares whose sides were each ten degrees), and the integral 
 evaluated in a similar manner to that previously described. The agreement 
 of the values so calculated with those obtained by the above method of 
 estimation was sufficiently exact to justify a certain confidence in the close 
 agreement of the finally corrected values of the integrals with their theoreti- 
 cally perfect values. 
 
 H. H. T. 
 
 222 
 
9. 
 
 ATTEMPTED EVALUATION OF THE RIGIDITY OF THE EARTH 
 FROM THE TIDES OF LONG PERIOD. 
 
 [This is 848 of the second edition of Thomson and Tait's Natural 
 Philosophy (1883). There have been some changes of notation, so as to make 
 the investigation consistent with the other papers in this volume. Some 
 portions have been omitted, where omissions could be made without any 
 interference with the main result.] 
 
 It appears from Schedule B iii. of the Report to the British Association 
 of 1883 (p. 22 of this volume) that the fortnightly tide Mf is expressed by 
 
 . (i - 1 sin 2 X) (1 - f e>) . * sin 2 / cos 2 (, - f ) 
 and the lunar monthly tide Mm by 
 
 M" //7\3 
 
 f|;(-U(-fsm 2 X)e(l-fsin 2 /)cos(s-p) 
 
 But in the paper on the correction to the equilibrium theory for the 
 continents (p. 330 of this volume) it is shown that the factor 
 
 - $ sin 2 X = f ( - sin 2 X) = f (sin 2 35 16' - sin 2 X) 
 
 should be replaced by f (sin 2 X sin- X). It appears further from that paper 
 that X is 34 39' according to one hypothesis and 35 4' according to another 
 as to the distribution of land on the earth. Hence we may with sufficient 
 accuracy replace this factor by f sin (35 X) sin (35 + X). 
 
 M /a\ 3 
 Thus if we write T = f - ( - , and let <f> denote the equilibrium fortnightly 
 
 -^ - 
 tide, and /* the equilibrium monthly tide, we have 
 
 <j> = f ra (1 - f e 2 ) sin 2 / sin (35 - X) sin (35 + X) cos 2 (s - ) 
 fj, = T ae (1 - f sin- /) sin (35 - X) sin (35 + X) cos (s - p) 
 
1883] METHOD OF REDUCTION. 341 
 
 Thus the actual fortnightly and monthly tides must be expressed in the 
 forms 
 
 <f> = f ra (1 - f e 2 ) sin 2 / sin (35 - X) sin (35 + X) {x cos 2 (s - % ) + y sin 2(* - f )} 
 p = * T ae (1 - f sin 2 /) sin (35 - X) sin (35 + X) [u cos (s -p) + v sin (s -p}} 
 
 09) 
 
 where ac, y, u, v are numerical coefficients. If the equilibrium theory be 
 nearly true for the fortnightly and monthly tides, y and v will be small ; 
 and x and u will be fractions approaching unity, in proportion as the 
 rigidity of the earth's mass approaches infinity. 
 
 If we now put 
 
 a = f ra (1 - fe 2 ) sin 2 /sin (35 - X) sin (35 + X) 
 
 (20) 
 c = f roe (1 - f sin 2 /) sin (35 - X) sin (35 + \) ' 
 
 and for the fortnightly tide write 
 
 a * =A | ...(21) 
 
 ay=B j 
 
 and for the monthly tide write 
 
 CM = C l ...(22) 
 
 cv =D) 
 
 we have <f> = A cos 2 (s ) + B sin 2 (s f) 
 
 /t = C cos (s p) + D sin (s - p) 
 
 Every set of tidal observations will give equations for x, y, u,v', and the 
 most probable values of these quantities must be determined by the method 
 of least squares. 
 
 For places north of 35 N. lat., or south of 35 S. lat. the coefficients a 
 and c become negative. This would be inconvenient for the arithmetical 
 operations of reduction, and therefore it is convenient to regard the co- 
 efficients a and c as being in all cases positive, for we may suppose (X 35) 
 to be taken for places in the northern hemisphere North of 35, and 35 X 
 for places in the same hemisphere to the South of 35 ; and similarly for the 
 southern hemisphere. 
 
 [This is merely an artifice for avoiding the insertion of many negative 
 signs, and it makes no difference in the result.] 
 
 In collecting the results of tidal observation I have to thank Sir William 
 Thomson, General Strachey, and Major Baird* for placing all the materials 
 in my hands, and for giving me every facility. The observations are to 
 be found in the British Association Reports for 1872 and 1876, and in the 
 Tide-tables of the Indian Government. 
 
 * [Now Lord Kelvin, Sir Richard Strachej', and Colonel Baird.] 
 
342 OMISSION OF CERTAIN STATIONS. [9 
 
 The results of the harmonic analysis of the tidal observations are given 
 altogether for 22 different ports, but of these only 14 are used here. The 
 following are the reasons for rejecting those made at 8 out of the 22 ports. 
 
 One of these stations is Cat Island in the Gulf of Mexico ; this place, in 
 latitude 30 14' N., lies so near to the critical latitude of evanescent fortnightly 
 and monthly tides, that considering the uncertainty in the exact value of 
 that latitude, it is impossible to determine the proper weight which should 
 be assigned to the observation. The result only refers to a single year, viz. 
 1848, and as its weight must in any case be very small, the omission can 
 exercise scarcely any effect on the result. 
 
 Another omitted station is Toulon ; this being in the Mediterranean Sea 
 cannot exhibit the true tide of the open ocean. 
 
 Another is Hanstal in the Gulf of Cutch. The result is given in an 
 Indian Blue Book. I do not know the latitude, and General Strachey informs 
 me that he believes the observations were only made during a few months 
 for the purpose of determining the mean level of the sea, for the levelling 
 operations of the great survey of India. 
 
 The other omitted stations are Diamond Harbour, Fort Gloster and 
 Kidderpore in the Hooghly estuary, and Rangoon, and Moulmein. All these 
 are river stations, and they all exhibit long period tides of such abnormal 
 height as to make it nearly certain that the shallowness of the water has 
 exercised a large influence on the results. The observations higher up the 
 Hooghly seem more abnormal than those lower down. I also learn that the 
 tidal predictions are not found to be satisfactory at these stations. 
 
 The following tables exhibit the results for the 14 remaining ports. 
 
 No attempt has been made to assign weight to each year's observations 
 according to the exact number of months over which the tidal records extend. 
 The data for such weighting are in many cases wanting. In computing the 
 value for a the factor 1 |e 2 was omitted, but it has been introduced finally 
 as explained below. 
 
1883] 
 
 OBSERVATIONS AT THE SEVERAL STATIONS. 
 
 343 
 
 BRITISH AND FRENCH PORTS, NORTH OF LATITUDE 35. 
 
 [Tidal Reports of Brit. Asssoc. 1872 and 1876.] 
 
 o 
 K f 
 
 "^32 J \ 
 
 H 
 
 PLACE 
 
 RAMSGATE 
 51 21' 
 
 LIVERPOOL 
 53 40' 
 
 WEST HARTLEPOOL 
 54 41' 
 
 BREST 
 48 23' 
 
 N. Latitude... 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 YEAR 
 
 1864 
 
 1857-8 
 
 1858-9 
 
 1859-60 
 
 1866-7 
 
 1858-9 
 
 1859-60 
 
 1860-1 
 
 1875 
 
 
 A 
 B 
 
 a 
 
 -0112 
 + 0311 
 0439 
 
 + 0917 
 - '0156 
 0955 
 
 + O337 
 -0153 
 0946 
 
 -0015 
 - '0240 
 0905 
 
 - O346 
 + 0095 
 0416 
 
 + 0497 
 + 0153 
 0996 
 
 + '0297 
 + 0438 
 0952 
 
 + 0729 
 -0027 
 0884 
 
 - -0244 
 - -0959 
 0684 
 
 C - -0223 
 D - '0224 
 c -0332 
 
 -0153 
 + 0434 
 0302 
 
 - -1687 
 - -1038 
 O304 
 
 + 1508 
 -0190 
 0312 
 
 + 0127 
 + -0708 
 0392 
 
 - -0688 
 - -0299 
 0320 
 
 + 1347 
 
 - -0100 
 
 0328 
 
 - -0261 
 - -1365 
 0339 
 
 -0279 
 + 0178 
 0218 
 
 INDIAN PORTS. 
 
 [Indian Tide Tables for 1881.] 
 
 f J 
 
 PLACE 
 Latitude . . . 
 
 ADEN 
 12 47' 
 
 KURRACHEK 
 
 24 47' 
 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 YEAR 
 
 1877-8 
 
 1879-80 
 
 1868-9 
 
 1869-70 
 
 1870-1 
 
 1873-4 
 
 1874-5 
 
 1875-6 
 
 1876-7 
 
 1877-8 
 
 A 
 
 + 0606 
 
 + 0597 
 
 + 0287 
 
 + 0421 
 
 -'0076 
 
 -0123 
 
 + 0429 
 
 + 0131 
 
 + 0460 
 
 + 0508 
 
 .6073 ] 
 
 B 
 
 + 0131 
 
 + 0167 
 
 - -0249 
 
 -0482 
 
 - '0342 
 
 - -0102 
 
 + -0328 
 
 + 0050 
 
 -0012 
 
 + 0406 
 
 11 
 
 a 
 
 0818 
 
 0706 
 
 0218 
 
 0248 
 
 0287 
 
 0411 
 
 0440 
 
 0456 -0459 
 
 0448 
 
 C 
 
 + 0194 
 
 + 0329 
 
 - -0288 
 
 -0429 --0140 
 
 + 0281 
 
 + -0522 
 
 - -0133 
 
 + 0631 
 
 + -0722 
 
 H~ 
 
 D 
 
 - -0158 
 
 + 0027 
 
 - -0703 
 
 + 0035 +-0288 
 
 + 0414 
 
 + -0229 
 
 + -0565 
 
 + -0570 
 
 + 0831 
 
 a 
 
 c 
 
 0268 
 
 0286 
 
 0185 
 
 0180 '0173 
 
 0153 
 
 0148 
 
 0145 
 
 0145 
 
 0147 
 
344 
 
 OBSERVATIONS AT THE SEVERAL STATIONS. 
 
 INDIAN PORTS (continued). 
 
 [Indian Tide Tables for 1881-2.] 
 
 
 PLACE 
 
 OKHAPOINT 
 AND BEYT 
 HARBOUR 
 
 BOMBAY APOLLO BUNDER 
 
 KARWAR 
 
 BEYPORE 
 
 
 
 Latitude.. 
 
 22 28' 
 
 18 55' 
 
 14 48' 
 
 11 10' 
 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 YEAR 
 
 1874-5 
 
 1876-7 
 
 1878-9 
 
 1879-80 
 
 1878-9 
 
 1879-80 
 
 1878-9 
 
 1879-80 
 
 A 
 
 + '0529 
 
 + 0699 
 
 + O888 
 
 + 0653 
 
 + 0649 
 
 + 0699 
 
 + -1040 
 
 + O907 
 
 '18 1 
 
 B 
 
 + '0459 
 
 + O121 
 
 - Ol 99 
 
 + OIOO 
 
 -O168 
 
 + 0025 
 
 + O208 
 
 + O281 
 
 ^ / 
 
 3 
 
 -gT3 J 
 
 |HJ 
 
 a 
 
 0525 
 
 0671 
 
 0617 
 
 0565 
 
 0726 
 
 0665 
 
 0803 
 
 0735 
 
 C 
 D 
 
 + -0331 
 -0375 
 
 -O048 
 + O266 
 
 + O372 
 
 - O378 
 
 + O459 
 -0036 
 
 + O415 
 -0069 
 
 + O562 
 + O143 
 
 + O687 
 + 0072 
 
 + 0057 
 + O708 
 
 
 c 
 
 0177 
 
 0212 
 
 0220 
 
 0229 
 
 O260 
 
 0270 
 
 0287 
 
 0298 
 
 
 PLACE 
 
 PAUMBEN PASS, 
 Island of 
 Ramesweram 
 
 VlZAGAPATAM 
 
 MADRAS 
 
 PORT BLAIR, 
 Ross ISLAND 
 
 
 Latitude ... 
 
 9 16' 
 
 17 41' 
 
 13 4' 
 
 11 40 J' 
 
 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 
 YEAR 
 
 1878-9 
 
 1879-80 
 
 1879-80 
 
 1880-1 
 
 1880-1 
 
 1880-1 
 
 t f 
 
 A 
 
 0560 
 
 0448 
 
 + O328 
 
 + O524 
 
 + O318 
 
 + O589 
 
 *^2 ** ~\ 
 
 B 
 
 -0010 
 
 -0043 
 
 + O1 48 
 
 -0167 
 
 + 0037 
 
 -0029 
 
 II 
 
 a 
 
 O835 
 
 0764 
 
 0597 
 
 0534 
 
 0626 
 
 O649 
 
 3 & 
 
 C 
 
 + 0579 +0279 
 
 + O194 
 
 + O468 
 
 + O304 
 
 + O195 
 
 12 
 
 D 
 
 -01 13; + -0439 
 
 + 0080 
 
 + 0611 
 
 + 0260 
 
 + 0045 
 
 S 
 
 c 
 
 0298 O310 
 
 O242 
 
 0253 
 
 0296 
 
 O307 
 
 Gauss's notation is adopted for the reductions*. That is to say, [A A] 
 denotes the sum of the squares of the A's, and [Aa] the sum of the products 
 of each A into its corresponding a. 
 
 * See Gauss's works, or the Appendix to Chauvenet's Astronomy. 
 
1883] LEAST SQUARES. 345 
 
 In computing the value of a for the fortnightly tide the factor (1 fe 2 ) 
 which occurs therein was treated as being equal to unity; since fe 2 = '00754, 
 it follows that the [aa], which would be found from the numbers given in 
 the table, must be multiplied by (1 - '01508), and the [Aa] and [Ba] by 
 (1 '00754). After introducing these correcting factors the following results 
 were found : 
 
 [aa] = '14573, [AA] = '09831, [BB] = '02576, [Aa] = '09836, [Ba] = '00291 
 [cc] = -02253, [CC] = '11588, [DD] = '07552, [Cc] = '01533, [Dc] = '00202 
 
 Then according to the method of least squares, the following are the most 
 probable values of x, y, u, v. 
 
 [Aa] [Ba] [Cc] [Dc] 
 
 ff, - L _ nl - L _ J . = L - J q, _ L - f 
 
 ~[aa]' * ISP M' M 
 
 And if m be the number of observations (which in the present case is 33) the 
 mean errors of x, y, u, v are respectively 
 
 J^ /[A A] [ [aa] -[Aaf J_ /[BB] [aa] - [Ba] 2 
 
 jaa]V m-l [aa]V m-l 
 
 J_ /[CC][cc]-[Ccp J_ / 
 
 ccV m-l ccV 
 
 _ 
 [cc] m-l [cc] m-l 
 
 The probable errors are found from the mean errors by multiplying by 
 6745. 
 
 I thus find that 
 x = -675 + '056, y = '020 '055, u = '680 '258, v = '090 '218 
 
 The smallness of the values of y and v is satisfactory ; for, as stated in 
 848 (d) of the Natural Philosophy, if the equilibrium theory were true for 
 the two tides under discussion, they should vanish. Moreover the signs 
 are in agreement with what they should be, if friction be a sensible cause 
 of tidal retardation. But considering the magnitude of the probable errors, 
 it is of course rather more likely that the non-evanescence of y and v is due 
 to errors of observation *. 
 
 If the solid earth does not yield tidally, and if the equilibrium theory is 
 
 * Shortly after these computations were completed Professor Adams happened to observe a 
 misprint in the Tidal Report for 1872. This Report gives the method employed in the reduction 
 by harmonic analysis of the tidal observations, and the erroneous formula relates to the reduction 
 of the tides of long period. On inquiring of Mr Roberts, who has superintended the harmonic 
 analysis, it appears that the erroneous formula has been used throughout in the reductions. 
 A discussion of this mistake and of its effects will be found in a paper communicated to the 
 British Association by me in 1882. It appears that the values of the fortnightly tide are not 
 seriously vitiated, but the monthly elliptic tide will have suffered much more. This will probably 
 account for the large probable error which I have found for the value of the monthly tide. If a 
 recomputatiou of all the long-period tides should be carried out, I think there is good hope that 
 the probable error of the value of the fortnightly tide may also be reduced. 
 
346 ELASTIC YIELDING OF THE SOLID EARTH. [9 
 
 fulfilled, x and u should each be approximately unity, and if it yields tidally 
 they should have equal values. The very close agreement between them is 
 probably somewhat due to chance. From this point of view it seems reason- 
 able to combine all the observations, resulting from 66 years of observation, 
 for both sorts of tides together. 
 
 Then writing X and Y for the numerical factors by which the equilibrium 
 values of the two components of either tide are to be multiplied in order to 
 give the actual results, I find 
 
 X = -676 -076, F= -029 "065 
 
 These results really seem to present evidence of a tidal yielding of the 
 earth's mass, showing that it has an effective rigidity about equal to that of 
 steel*. 
 
 But this result is open to some doubt for the following reason : 
 
 Taking only the Indian results (48 years in all), which are much more 
 consistent than the English ones, I find 
 
 X = -931 -056, Y = 155 "068 
 
 We thus see that the more consistent observations seem to bring out the 
 tides more nearly to their theoretical equilibrium-values with no elastic 
 yielding of the solid. 
 
 It is to be observed however that the Indian results being confined 
 within a narrow range of latitude give (especially when we consider the 
 absence of minute accuracy in the evaluation of the critical latitude X ) a less 
 searching test for the elastic yielding, than a combination of results from 
 all latitudes. 
 
 On the whole we may fairly conclude that, whilst there is some evidence 
 of a tidal yielding of the earth's mass, that yielding is certainly small, and 
 that the effective rigidity is at least as great as that of steel. 
 
 [Postscript. It is interesting to compare this conclusion with the results 
 obtained by Dr 0. Hecker by means of the horizontal pendulum (" Beobach- 
 tungen an Horizontalpendeln liber die Deformation des Erdkorpers," K. 
 Preuss. Geoddt. Inst., Neue Folge, No. 32, 1907), for he finds the deflections 
 of the pendulum to be two-thirds as great as they would be on a rigid earth.] 
 
 * It is remarkable that elastic yielding of the upper strata of the earth, in the case where the 
 sea does not cover the whole surface, may lead to an apparent augmentation of oceanic tides at 
 some places, situated on the coasts of continents. This subject is investigated in the Report for 
 1882 of the Committee of the British Association on " The Lunar Disturbance of Gravity." 
 (Paper 14 in this volume.) It is there, however, erroneously implied that this kind of elastic 
 yielding would cause an apparent augmentation of tide at all stations of observation. 
 
10. 
 
 DYNAMICAL THEORY OF THE TIDES. 
 
 [This contains certain sections from the article " TIDES " written in 1906 
 for the new edition of the Encyclopaedia Britannica, being based on the cor- 
 responding paragraphs in the original edition and on the article " TIDES " 
 in the supplementary volumes. Reproduced by special permission of the 
 Proprietors of the Enc. Brit.] 
 
 12. Form of Equilibrium. 
 
 Consider the shape assumed by an ocean of density o- on a planet 
 of mass M, density 8 and radius a, when acted on by disturbing forces whose 
 potential is a solid spherical harmonic of degree i, the planet not being 
 in rotation. 
 
 If Si denotes a surface spherical harmonic of order i, such a potential 
 is given at the point whose radius vector is p by 
 
 In the case considered in an earlier section m is the moon's mass and 
 r her distance, and i = 2, while Si becomes cos 2 (moon's z. d.) - ^ ; [in the 
 present instance the form of the coefficient is immaterial, save that its 
 dimensions shall be correct]. 
 
 The theory of harmonic analysis tells us that the form of the ocean, when 
 in equilibrium, must be given by the equation 
 
 P = a + e i S i ................................. (2) 
 
 Our problem is to evaluate 0i. 
 
 We know that the external potential of a layer of matter, of depth ;$; 
 and density <r, has the value 
 
348 EQUILIBRIUM TIDE. [10 
 
 Hence the whole potential externally to the planet and up to its surface is 
 M 
 
 eiSi 
 
 The first and most important term is the potential of the planet, the 
 second that of the disturbing force, and the third that of the departure from 
 sphericity. T 
 
 Since the surface of the ocean must be a level surface, the expression (3) 
 equated to a constant must be another form of (2). Hence if we put 
 p = a + BiSi in the first term of (3) and p = a in the second and third terms, 
 (3) must be constant; this can only be the case if the coefficient of S{ vanishes. 
 
 Hence on effecting these substitutions and equating that coefficient to 
 zero, we find 
 
 M 3ma? 4nra-a 
 
 ~^ ei + ~W + W^l ei ~ 
 
 But by definition of 8 and a we have M = TrSa 3 = gar, where g is gravity, 
 and therefore 
 
 3. 
 
 If <7 were very small compared with 8 the attraction of the water on 
 itself would be very small compared with that of the planet on the water; 
 
 // Q \ 
 
 (1 7^^-^} is the factor by which the mutual 
 V (2i + I)6J 
 
 gravitation of the ocean augments the deformation due to the external 
 forces. This factor will occur frequently hereafter, and therefore for brevity 
 we write 
 
 and (4) may be written 
 
 Comparison with (1) then shows that 
 
 is the potential of the disturbing forces under which 
 
 p = a+eiSi ( 8 ) 
 
 is a figure of equilibrium. 
 
1906] RECENT ADVANCES IN DYNAMICAL THEORY. 349 
 
 We are thus provided with a convenient method of specifying any dis- 
 turbing force by means of the figure of equilibrium which it is competent 
 to maintain. In considering the dynamical theory of the tides of an ocean- 
 covered planet, we shall specify the disturbing forces in the manner expressed 
 by (7) and (8). 
 
 This way of specifying a disturbing force is equally exact whether or 
 not we choose to include the effects of the mutual attraction of the ocean. 
 If the augmentation due to mutual attraction of the water is not included, 
 hi becomes equal to unity ; in this case there is no necessity to use spherical 
 harmonic analysis, and we see that if the equation to the surface of an 
 ocean be 
 
 p = a + S 
 
 where S is a function of latitude and longitude, it is in equilibrium under 
 forces due to a potential whose value at the surface of the sphere (where 
 
 p = a) is gS. 
 
 14. Recent Advances in the Dynamical Theory of the Tides. 
 
 The problem of the tidal oscillation of the sea is essentially dynamical. 
 In two papers in the second volume of Liouville's Journal (1896), M. Poincare 
 has considered the mathematical principles involved in the problem, where 
 the ocean is interrupted by land as in actuality. He has not sought to 
 obtain numerical results applicable to any given configuration of land and 
 sea, but he has aimed rather at pointing out methods by which it may some 
 day be possible to obtain such solutions. It would hardly be in place to 
 attempt to follow the details of an investigation of this character, but we 
 may say that it affords a conspicuous example of M. Poincare's power ol 
 reaching the very heart of a difficult problem. 
 
 Even when the ocean is taken as covering the whole earth, the problem 
 presents formidable difficulties, and this is the only case in which it has been 
 solved hitherto*. 
 
 Laplace gives the solution in Bks I. and iv. of the Me'canique Celeste ; but 
 his work is unnecessarily complicated. In the first edition of the present 
 work we gave Laplace's theory without these complications, but the theory 
 is now accessible in Lamb's Hydrodynamics, and perhaps in other works 
 
 of the kind. It will not, therefore, be reproduced here. 
 
 c 
 In 1897 and 1898 Mr S. S. Hough undertook an important revision of 
 
 Laplace's theory, and succeeded not only in introducing the effects of the 
 
 * Lord Kelvin's (Sir W. Thomson's) paper on the " Gravitational oscillations of rotating 
 water," Phil. Mag., Oct. 1880, bears on this subject. It is the only attempt to obtain numerical 
 results in respect to the effect of the earth's rotation on the oscillations of land-locked seas. 
 
350 EQUATIONS OF MOTION. [10 
 
 mutual gravitation of the ocean, but also in determining the nature and 
 periods of the free oscillations of the sea*. A dynamical problem of this 
 character cannot be regarded as fully solved unless we are able not only 
 to discuss the " forced " oscillations of the system but also the " free." 
 Hence we regard Mr Hough's work as the most important contribution 
 to the dynamical theory of the tides since the time of Laplace. We shall 
 accordingly present the theory in the form due to Mr Hough, although 
 limitations of space compel us to omit many points of interest. 
 
 The analysis is more complex than that of Laplace, where the mutual 
 attraction of the ocean was neglected, but this was perhaps inevitable. 
 
 Our first task is to form the equations of motion and continuity, which 
 will be equally applicable to all forms of the theory. 
 
 15. Equations of Motion. 
 
 Let r, 0, <f> be the radius vector, colatitude, and east longitude of a point 
 with reference to an origin, a polar axis, and a zero-meridian rotating with a 
 uniform angular velocity n from west to east. Then if R, H, 3 be the 
 radial, colatitudinal, and longitudinal accelerations of the point, we have 
 
 d 2 r fd0\* . , ,, /d<f> V 
 r ( -r.- 1 r sm- 01-^ + n ] 
 
 J 
 
 V 
 
 ~ Id/ ,d0\ (dj> , V 
 
 tt = - -r. (r- -r. }- r sm cos (-.- + n } 
 
 r dt \ dt) \dt I 
 
 1 d f , . , a fdtf> \~1 
 
 ' a ^ r ~ sm " 9 \J*+ n l\ 
 
 in dt |_ \dt j J 
 
 r sin dt 
 
 If the point were at rest with reference to the rotating meridian, we 
 should have R = ri*r sin 2 0, E = n' 2 r sin cos 0, H = 0. When these con- 
 siderations are applied to the motion of an ocean relative to a rotating 
 planet, it is clear that these accelerations, which still remain when the ocean 
 is at rest, are annulled by the permanent oblateness of the ocean. As then 
 they take no part in the oscillations of the ocean, and as we are not con- 
 sidering the figure of the planet, we may omit these terms from R and E. 
 
 / (j (j) \ '^ 
 
 This being so, we must replace ( -53 4- n } as it occurs in R and 5 by 
 
 \dt ) 
 
 Now suppose that the point whose accelerations are under consideration 
 never moves far from its zero position, and that its displacements f , 77 sin 
 in colatitude and longitude are very large compared with p, its radial dis- 
 placement. Suppose further that the velocities of the point are so small 
 
 * Phil. Trans. Roy. Soc., Vol. 189 A, pp. 201258, and Vol. 191 A, pp. 139185. 
 
1906] EQUATIONS OF MOTION. 351 
 
 that their squares and products are negligible compared with w 3 r 2 ; then 
 we have 
 
 -j- = -jjr- , a very small quantity 
 dt clt 
 
 = 
 r dt~ dt 
 
 Since the radial velocity always remains very small, it is not necessary 
 to concern ourselves further with the value of R, and we only require the 
 two other components which have the approximate forms 
 
 -- sn 006 
 
 (9) 
 
 We have now to consider the forces by which an element of the ocean 
 is urged in the direction of colatitude and longitude. These forces are those 
 due to the external disturbing forces, to the pressure of the water surrounding 
 an element of the ocean, and to the attraction of the ocean itself. 
 
 If denotes the equilibrium height of the tide, it is a function of Co- 
 latitude and longitude, and may be expanded in a series of spherical surface 
 harmonics t;. Thus we may write the equation to the equilibrium tide in 
 
 the form 
 
 r = a + = a + St; 
 
 Now it appears from (7) and (8) that the value of the potential, at the 
 surface of the sphere where p = a, under which this is a figure of equi- 
 librium, is 
 
 We may use this as specifying the external disturbing force due to 
 the known attractions of the moon and sun, so that t; may be regarded 
 as known. 
 
 But in our dynamical problem the ocean is not a figure of equilibrium. 
 and we may denote the elevation of the surface at any moment of time by f). 
 Then the equation to the surface may be written in the form 
 
 r = a + $ = a + 2f); 
 where f); denotes a spherical harmonic just as ; did before. 
 
 The surface value of the potential of the forces which would maintain the 
 ocean in equilibrium in the shape it has at any moment is 2<jr& t -|),. 
 
 Hence it follows that in the actual case the forces due to fluid pressure 
 and to the attraction of the ocean must be such as to balance the potential 
 
352 EQUATIONS OF MOTION AND CONTINUITY. [10 
 
 just determined. Therefore these forces are those due to a potential ^.gb^. 
 If we add to this the potential of the external forces, we have a potential 
 which will include all the forces, the expression for which is #26; (f){ tj). 
 If further we perform the operations d/ad0 and d/asin 0d(f> on this potential, 
 we obtain the colatitudinal and longitudinal forces which are equal to the 
 accelerations H and H. 
 
 It follows, then, from (9) that the equations of motion are 
 
 Q *, d . \ 
 
 - sm cos 
 
 d 
 
 -j-- a -^ 
 
 dt 2 dt a sin d<f> 
 
 It remains to find the equation of continuity. This may be deduced 
 geometrically from the consideration that the volume of an element of the 
 fluid remains constant ; but a shorter way is to derive it from the equation 
 of continuity as it occurs in ordinary hydrodynamical investigations. If $ 
 be a velocity potential, the equation of continuity for incompressible fluid is 
 
 s d f dQ . 0* as ,\ , * a d ( . a d<& . 5 \ 
 
 or j- (r 2 -j~ sin0808(b} + 80 ^Irsmtf ^ Sr8d>) 
 
 dr \ dr r j d0 \ rdd r / 
 
 , , , d ( 1 d<& 
 
 + 8 4>^a\ r -^a ^r 
 
 d<f> \ r sin d(f> 
 
 The element referred to in this equation is defined by r, 0, (f>, r + Sr, 
 + &0, <f> + S<. The colatitudinal and longitudinal velocities are the same 
 for all the elementary prism defined by 0, <, + 80, <f> + 8^>, and the sea 
 
 bottom. Then j^ = ~ , ; , = sin ~ ; and, since the radial velocity 
 rdd dt rsm0d(f> dt 
 
 is dfy/dt at the surface of the ocean, where r = a + y, and is zero at the sea 
 bottom, where r = a, we have -r- = ~A< Hence, integrating with respect 
 
 to r from r = a + j to r = a, and again with respect to t from the time t 
 to the time when f), , rj all vanish, and treating 7 and f) as small compared 
 with a, we have 
 
 f)asin0+ -^ ( 7 sin 0) + -^ (77; sin 0) = ........ '....(11) 
 
 du a<p 
 
 This is the equation of continuity, and, together with (10), it forms the 
 system which must be integrated in the general problem of the tides. The 
 difficulties in the way of a solution are so great that none has hitherto been 
 found, except on the supposition that 7, the depth of the ocean, is only a 
 function of latitude. In this case (11) becomes 
 
1906] FORCED OSCILLATIONS. 353 
 
 16. Adaptation to Forced Oscillations. 
 
 Since we may suppose that the free oscillations are annulled by friction, 
 the solution required is that corresponding to forced oscillations. Now , 
 which is proportional to V, has terms of three kinds, the first depending on 
 twice the moon's (or sun's) hour-angle, the second on the hour-angle, and 
 the third independent thereof. The coefficients of the first and second vary 
 slowly, and the whole of the third varies slowly. Hence has a semi-diurnal, 
 a diurnal, and a long-period term. We shall see later that these terms may 
 be expanded in a series of approximately semi-diurnal, diurnal, and slowly 
 varying terms, each of which is a strictly harmonic function of the time. 
 
 Thus, according to the usual method of treating oscillating systems, we 
 may make the following assumptions as to the form of the solution 
 
 = 20; = 2#j cos (Znft + s<J> + a)\ 
 
 = 2f)i = S/?i cos (2nft + S(f> + a) [ 
 
 >- (13) 
 
 = ^biXi cos (2nft + S(f) + a) 
 
 i) = "Zbtyi sin (Znft + S(f> + a) / 
 
 where ei, h i; x i} y t are functions of colatitude only, and e t -, h t are the 
 associated functions of colatitude corresponding to the harmonic of order i 
 and rank s. 
 
 For the semi-diurnal tides 5 = 2, and / approximately unity ; for the 
 diurnal tides 8=1, and/ approximately ; and for the tides of long period 
 s = 0, and / is a small fraction. 
 
 Substituting these values in (12), we have 
 
 Then if we write m for hi - e i} and put m = iPaJg, substitution from (13) 
 in (14) leads at once to 
 
 1 d 
 
 f'^biX; + /"sin 6 cos 6 2&i?/i = -r TZ 26ftt< 
 J J J 4m dv 
 
 .AI s a^i s 2M< 
 
 sin V zbiVi + T cos a 2,D<< = -. : ^ 
 
 y J 4m sin 6 
 
 (15) 
 
 , 7 * a*! s *M 
 
 >OiVi + /cos 6 Zbi^i = : a 
 
 iyi J 4m sin 6 
 
 Solving (15), we have 
 
 COS u) j, 1OT \ fjf) /'sin fi */ 
 
 r (16) 
 
 D. i. 23 
 
354 ZONAL OSCILLATIONS. [10 
 
 Then substituting from (16) in (14), we have 
 
 I d 
 sin dO 
 
 s 
 
 d0 '/ 
 
 / 2 - cos 2 
 
 f dd sine 
 sin (/ 2 - cos 2 0) 
 
 (17) 
 
 This is closely analogous to Laplace's equation for tidal oscillations in an 
 ocean whose depth is only a function of latitude ; indeed the only difference 
 is that we have followed Mr Hough in introducing the mutual attraction of 
 the water. 
 
 When Ui is found from this equation, its value substituted in (16) will 
 give iK{ and y t . 
 
 17. Zonal Oscillations. 
 
 We might treat the general harmonic oscillations first, and proceed to the 
 zonal oscillations by putting s = 0. These waves are, however, comparatively 
 simple, and it is well to begin with them. The zonal tides are those which 
 Laplace describes as of the first species, and are now more usually called the 
 tides of long period. 
 
 As we shall only consider the case of an ocean of uniform depth, 7 the 
 depth of the sea is constant. Then since in this case s = 0, our equation (17) 
 to be satisfied by U{ or hi ei becomes 
 
 a d 
 smt> -T7 
 
 c , . /. A 
 2% sin = 
 
 / 2 - cos 2 e 
 
 This may be written 
 
 d ^ 4<ma f 2 cos 
 
 f * . a , D 
 
 2/^-sm Ode + A = (18) 
 
 J 
 
 dO 7 sin 6 
 
 where A is a constant. 
 
 Let us assume hi = CiPi, e-i = EiPi 
 
 where PI denotes the iih zonal harmonic of cos 6. The coefficients Gi are 
 unknown, but the EI are known because the system oscillates under the 
 action of known forces. 
 
 If the term involving the integral in this equation were expressed in 
 terms of differentials of harmonics, we should be able to equate to zero the 
 coefficient of each dPi/d0 in the equation, and thus find the conditions for 
 determining the G's. 
 
 /2 cos 2 9 f 
 
 The task then is to express = -. Pi$\n@d0 in differentials of 
 
 sm J 
 
 zonal harmonics. 
 
1906] ZONAL OSCILLATIONS. 355 
 
 It is well known that Pi satisfies the differential equation 
 
 in0 = .................. (19) 
 
 Therefore f P, sin 0d0 = - * ^ sin 
 
 J i^ + I) 
 
 dd 
 
 and fl^f. [p. S i n Od6 = - ^- (/ - cos* 
 sm0 J i(i + 1) v ^ 
 
 dP t 
 
 Another well-known property of zonal harmonics is that 
 
 If we differentiate (20) and use (19) we have 
 
 Multiplying (20) by sin 6, and using (21) twice over 
 
 _ __ L [dPi^_dPi\. 
 
 " ' ' ' 
 
 Therefore 
 
 1 
 
 r . 
 
 Pis 
 
 J 
 
 . 
 
 ism0d0= - 
 sm 
 
 + 
 
 It (i + 1) ' (2i - 1) (2i + 3)] dtf (2t + l)(2t + 3) d0 
 
 This expression, when multiplied by 4ma/7 and by G t and summed, is the 
 second term of our equation. 
 
 The first term is 2& (d - E { ) ^ 
 
 In order that the equation may be satisfied, the coefficient of each d 
 must vanish identically. Accordingly we multiply the whole by <y/4>ma and 
 equate to zero- the coefficient in question, and obtain 
 
 / 2 - 1 
 
 4-ma v (2* - 1) (2i - 3) [i (i + 1) (2t - 1) (2i + 3) 
 
 C- 
 
 w 
 
 (22) 
 
 This equation (22) is applicable for all values of i from 1 to infinity, provided 
 that we take , E , C_i, ^_! as being zero. 
 
 We shall only consider in detail the case of greatest interest, namely that 
 of the most important of the tides generated by the attraction of the sun and 
 
 232 
 
356 ZONAL OSCILLATIONS. [10 
 
 moon. We know that in this case the equilibrium tide is expressed by a 
 zonal harmonic of the second order ; and therefore all the E i} excepting E z , 
 are zero. Thus the equation (22) will not involve EI in any case excepting 
 when i = 2. 
 
 If we write for brevity 
 
 j?2 "I 9 /)-iv 
 
 Li = T(j~+T) + (2i-l)(2i~+3) ~ 4^ 
 the equation (22) becomes 
 
 (2* + 3)(2i + 5) ~ LiGi + (~2z-3)(2i-l) = 
 save that when i 2, the right-hand side is -~- E 2 , a known quantity ex 
 
 hypothesi. 
 
 The equations naturally separate themselves into two groups, in one of 
 which all the suffixes are even and the other odd. Since our task is to 
 evaluate all the G's in terms of E 2 , it is obvious that all the C"s with odd 
 suffixes must be zero, and we are left to consider only the cases where 
 i=2, 4, 6, &c. 
 
 We have said that G must be regarded as being zero ; if however we take 
 
 G = -T-^- E 2 , so that C is essentially a known quantity 
 
 the equation (23) has complete applicability for all even values of i from 2 
 upwards. 
 
 The equations are 
 
 O-^'+O- - ^- i ' a ' + T03 = ' &c - 
 
 It would seem at first sight as if these equations would suffice to deter- 
 mine all the G's in terms of G 2 , and that G z would remain indeterminate ; but 
 we shall show that this is not the case. 
 
 For very large values of i the general equation of condition (23) tends to 
 assume the form 
 
 _J!^ !=? _) 
 
 Gi ma 
 
 By writing successively i +2,i + 4i,i + 6 in this equation and taking the 
 differences we obtain an equation from which we see that, unless Gi/C i+2 tends 
 to become infinitely small, the equations are satisfied by G{ = G i+2 in the limit 
 for very large values of i 
 
 Hence, if Gt does not tend to zero, the later portion of the series for h 
 tends to assume the form Gi(Pi + P{ +2 + P t - +4 ...) All the P's are equal to 
 unity at the pole ; hence the hypothesis that G t does not tend to zero leads to 
 
1906] ZONAL OSCILLATIONS. 357 
 
 the conclusion that the tide is of infinite height at the pole. The expansion 
 of the height of tide is essentially convergent, and thus this hypothesis is 
 negatived. Thus we are entitled to assume that C{ tends to zero for large 
 values of i. 
 
 Now writing for brevity 
 
 we may put (23) into the form 
 
 By successive applications of this formula, we may write the right-hand side 
 in the form of a continued fraction. 
 
 Jjf^U j\ . . 
 
 n. in. 
 Then we have 
 
 or --(t 
 
 ^i-2 
 
 Thus 
 
 C t = 3.5K a C ; C, = 3.5.7.9K 2 K,G - C e = 3.5 . 7 .9. 11 . 13K 2 K.K 6 C &c. 
 
 If we assume that any of the higher C's, such as C u or C 16 , is of negligible 
 smallness, all the continued fractions K%, K i} K Q &c. may be computed ; and 
 
 thus we find all the C's in terms of G , which is equal to -r-^ E^ 
 
 4>ma 
 
 The height of the tide is therefore given by 
 
 = - E, {3 . 5 K,P, + 3 . 9 K Z K,P, + . . . } cos (2nft + a) 
 
 It is however more instructive to express f) as a multiple of the equi- 
 librium tide , which is as we know equal to E 2 P 2 cos (Znft + a). Whence we 
 find 
 
 {3 . 5K,P 2 + 3 . 5 . 7 . 9K,K,P. + 3 . 5. 
 % 
 
 The number / is a fraction such that its reciprocal is twice the number of 
 
 sidereal days in the period of the tide. The greatest value of / is that 
 
 appertaining to the lunar fortnightly tide (Mf in notation of Harmonic 
 
 Analysis) and in this case / is in round numbers ^, or more exactly 
 
 2 = -00133. 
 
358 ZONAL OSCILLATIONS. [10 
 
 The ratio of the density a- of sea-water to 8 the mean density of the earth 
 is "18093 ; which value gives us 
 
 b 2 = l-~ = -89144 
 oo 
 
 The quantity m is the ratio of equatorial centrifugal force to gravity, and 
 is equal to ^-g. 
 
 Finally y/a is the depth of the ocean expressed as a fraction of the earth's 
 radius. 
 
 With these numerical values Mr Hough has applied the solution to 
 determine the lunar fortnightly tide for oceans of various depths. Of his 
 results we give two : 
 
 First, when 7 = 7260 ft. = 1210 fathoms, which makes -^ = J*. he finds 
 
 4ana 
 
 f) = ^ {'2669P 2 - *1678P 4 + '0485P 6 - '0081P 8 + '0009P 10 - -0001P 12 . . .] 
 * 
 
 If the equilibrium theory were true we should have 
 
 thus we see how widely the dynamical solution differs from the equilibrium 
 value. 
 
 Secondly, when 7 = 58080 ft. = 9680 fathoms, and - = |, he finds 
 
 i) = ~ {'7208P 2 - *0973P 4 + '0048P 6 - -0001P 8 ...} 
 * 
 
 From this we see that the equilibrium solution presents some sort of 
 approximation to the dynamical one; and it is clear that the equilibrium 
 solution would be fairly accurate for oceans which are still quite shallow 
 when expressed as fractions of the earth's radius, although far deeper than 
 the actual sea. 
 
 The tides of long-period were not investigated by Laplace in this manner, 
 for he was of opinion that a very small amount of friction would suffice to 
 make the ocean assume its form of equilibrium. In the arguments which he 
 adduced in support of this view the friction contemplated was such that the 
 integral effect was proportional to the velocity of the water relatively to the 
 bottom. It is probable the proportionality to the square of the velocity 
 would have been nearer the truth, but the distinction is unimportant. 
 
 The most rapid of the oscillations of this class is the lunar fortnightly 
 tide, and the water of the ocean moves northward for a week and then 
 southward for a week. In oscillating systems, where the resistances are 
 proportional to the velocities, it is usual to specify the resistance by a 
 ' modulus of decay,' namely the time in which a velocity is reduced by 
 friction to e~ l or 1/2*78 of its initial value. Now in order that the result 
 contemplated by Laplace may be true, the friction must be such that the 
 
1!J06] TESSERAL OSCILLATIONS. 359 
 
 modulus of decay is short compared with the semi-period of oscillation. It 
 seems practically certain that the friction of the ocean bed would not reduce 
 a slow ocean current to one-third of its primitive value in a day or two. 
 Hence we cannot accept Laplace's discussion as satisfactory, and the investi- 
 gation which has just been given becomes necessary*. 
 
 18. Tesseral Oscillations. 
 
 The oscillations which we now have to consider are those in which the 
 form of surface is expressible by the tesseral harmonics. The results will be 
 applicable to the diurnal and semidiurnal tides Laplace's second and third 
 species. 
 
 Q 
 
 If we write a- = > the equation (17) becomes 
 
 d 
 
 ( sin 6 -JQ + a- cos 6 j S^-M; 
 
 ( a d 
 
 <7 COS V -j^ 
 
 \ av 
 
 + s 2 cosec^j ^biUi 
 
 de 
 
 s 2 - cr- cos 2 d 
 
 4 
 
 s 2 
 
 4<ma ^ . 
 Zh; sin t 
 
 - a 2 cos 2 6 
 >-0.. 
 
 (24) 
 
 < 
 
 If D be written for the operation sin -^ , the middle term may be arranged 
 
 in the form 
 
 crcot 9 (D + a COS 6) 26;Mi v,, . 
 
 -- T ^- - - 2 biUi sin 
 
 s 2 cr cos- 07 
 
 Therefore on multiplying (24) by sin 6 it becomes 
 
 /r> a\ rCD + o-cos0)2&tH.i~| ^ i, , 4raa v , . ,, A /OK , 
 
 (D - a cos 0) - ^b t Ui H Zjli sin 2 - . . .(25) 
 
 s 2 a- cos 2 6 70- 2 
 
 We now introduce two auxiliary functions, such that 
 2&i(A;-e;) = 2&;^ 
 
 = (D - a- cos 6} + (s" - o-' 2 cos 2 0) <J> (26) 
 
 It is easy to prove that 
 
 (J5 + o- cos 0)(D-tr cos 6} = D*- s 2 + a- sin 2 + (s 2 - <r 2 cos 2 
 
 (D - o- cos 0) (D + o- cos 0) = D 2 - s 2 - a- sin 2 + (s 2 - a' 2 cos 2 0) 
 Also 
 
 (D + <r cos 0) (s 2 - o- 2 cos 2 0) 3> = (s 2 - a- 2 cos 2 0) (D + cr cos 0) <E> 
 
 + 2<r 2 sin 2 cos O (28) 
 
 Now perform D + a cos 6 on (26), and use the first of (27) and (28), and we 
 have 
 
 (D + a- cos 0) ZbiUi = (D' 2 -s+<r sin 2 + s 2 - a 2 cos 2 0} ^ 
 
 + (s 2 - o- 2 cos 2 6} (D + a- cos 0) <l> + 2o- 2 sin 2 cos 6 4> (29) 
 
 * [A fuller discussion of this subject is contained in Paper 11 below.] 
 
TESSERAL OSCILLATIONS. [10 
 
 The functions and <l> are as yet indeterminate, and we may impose 
 another condition on them. Let that condition be 
 
 (D 2 - s 2 + a sin 2 (9) ^ = - 2o- 2 sin- cos <f> (30) 
 
 Then (29) may be written 
 
 (D + r cos 0)26^ 
 
 Substituting from this in (25), and using the second of (27), the function 
 disappears and the equation reduces to 
 
 -#-* 8in 
 
 Since by (30) - <r cos* 6 * = * ^ (& - s* + a sin* 6} V, (26) may be 
 written 
 
 = D- 
 
 Sill 
 
 > ...... (32) 
 
 The equations (30), (31), and (32) define and <f>, and furnish the 
 equation which must be satisfied. 
 
 If we denote cos 6 by p the zonal harmonics are defined by 
 
 d 
 
 The following are three well-known properties of zonal harmonics : 
 
 1 ) P - .................. (38) 
 
 17^=0 .................. (34) 
 
 If ^ sin S< ^ are the two tesseral harmonics of order * and rank s, it is also 
 known that 
 
 P- =(I _^^ ........................... (86) 
 
 Let us now assume 
 
 hi = ClPl ei=ElPl, ^ = ^Pl, 4> = 20lPl 
 
 These must now be substituted in our three equations (30), (31), (32), 
 and the result must be expressed by series of the P* functions. It is clear 
 then that we have to transform into P^ functions the following functions of 
 P*, namely 
 
1906] TESSERAL OSCILLATIONS. 361 
 
 If we differentiate (33) s times, and express the result by means of the 
 operator D, we find 
 
 (7) 2 -s 2 )P* + t(* + l)P*sm 2 = .................. (37) 
 
 Again, differentiating (34) s times and using (35), we find 
 
 (i-s+l)P s i+1 -(2i+l)coseP s i + (i + s)P s i _ 1 = ......... (38) 
 
 Lastly, differentiating (36) once and using (33), (35), and (38) 
 
 s _ 
 
 s 
 
 
 s 
 
 l l l 
 
 (S 
 
 By means of (37), (38), and (39) we have 
 
 sin 2 
 
 r . 
 
 , 
 
 2(2t 
 
 Therefore the equations (30), (31), (32) give 
 
 O 8 f ' / ' , 1 \ , ) TlS , O n Q } * ~^~ & T)S , * ^ "t" 1 r> f\ 
 
 S ^ {-^(^+l) + a} P 4 + 2^/3, j^-j P,., + -^^^ P m j J = 
 
 2(2 - +1) 
 
 2(81-1-1) 
 
 P s 
 
 * i-\ 
 
 s -0 
 
 Since these equations must be true identically, the coefficients of P* in 
 each of them must vanish. Therefore 
 
 a! [a - i (i + 1)} 
 
 = 
 
 
 - 
 
 2(2t-l) 
 
 a 
 i 
 
 (40) 
 
362 TESSERAL OSCILLATIONS. [10 
 
 If we eliminate the a's and /S's from the third equation (40) by means of 
 the first two, we find 
 
 <Vf)- 
 
 j^& T C* -i- C 1 7r ^4il ^ 
 
 where 
 
 s 2 (-a)[> + (z-l)(i-2)] 
 
 L< ~ CT 2 [cr + I (i + D] + (4i 2 - 1) [cr - (l - 1) *'] [cr + i (i + 1)] 
 
 [( t - + l) 2 _ s2 ] [(r + ( t - + 2)(t + 3)] 
 
 [4 i (i + 1) 2 - l][cr - (i + l)(i + 2)][o- + i (t + 1)] 4ma 
 
 5) [<r - (i + 1) (i + 2)] 
 
 In the case of the luni-solar semi-diurnal tide (called K 2 in the notation 
 of harmonic analysis) we have i = 2, s = 2, a = 2. Hence it would appear that 
 these formulae for L\, |_ 2 fail by becoming indeterminate, but i and s are 
 rigorously integers whereas cr depends on the 'speed' of the tide; accordingly 
 in the case referred to we must regard terms involving i s as vanishing in 
 the limit when cr approaches to equality with i(il). For this particular 
 case then we find 
 
 and = 
 
 The equation (41) for the successive C's is available for all values of i 
 provided that G_ l} E_ l} C , E are regarded as being zero. 
 
 As in the case of the zonal oscillations, the equations with odd suffixes 
 separate themselves from those with even suffixes, so that the two series may 
 be treated independently of one another. Indeed, as we shall see immediately, 
 the series with odd suffixes are satisfied by putting all the C's with odd 
 suffixes zero for the case of such oscillations as may be generated by the 
 attractions of the moon or sun. 
 
 For the semi-diurnal tides i = 2, s = 2, and / is approximately equal to 
 unity. Hence the equilibrium tide is such that all the E\, excepting E" 2 , are 
 zero. 
 
 For the diurnal tides i = 2, s = l, and/ is approximately equal to . 
 Hence all the E\, excepting E\, are zero. 
 
 Since in neither case is there any E with an odd suffix, we need only 
 consider those with even suffixes. 
 
 In both cases the first equation among the C's is 
 
1906] TESSERAL OSCILLATIONS. 363 
 
 It follows that if we were to write 
 
 the equation of condition amongst the C's would be of general applicability 
 for all even values of i from 2 upwards. 
 
 The symbols jj, rf z do not occur in any of the equations, and therefore we 
 may arbitrarily define them as denoting unity, although the general formulae 
 of and T; would give them other values. Accordingly we shall take 
 
 ; ( ,= 2 orl) 
 
 With this definition the equation 
 
 _ 2 Cl_ 2 -LICI + r)l +z C s i+2 = (s = 2 or 1) 
 is applicable for i = 2, 4, 6, etc. 
 
 It may be proved as in the case of the tides of long period that we may 
 regard Cl/C s i+2 as tending to zero. 
 
 Then our equation may be written in the form 
 
 C s -~ fl 7 ?; 
 
 and by successive applications the right hand side may be expressed in the 
 form of a continued fraction. 
 
 Let us write H=& &* && ... 
 
 Hence our equation may be written 
 
 i - 2 G\ Hi 
 Whence c\ = ^ G\_ z 
 
 'i 
 
 It follows that 
 
 rrs rjs Tjs rjs 
 
 fl- X2/2 * 
 
 7 ?2 7 ?4 7 ?6 
 
 Then since we have defined 
 
 S 1 J /~S 
 
 ; 2 =1 and C = 
 
 7^2 
 
 - 2 
 
 4>ma 
 
 all the (7's are expressed in terms of known quantities. 
 
 Hence the height of tide f) is given by 
 J) = ^hi cos (2nft + s<f> + a) 
 
 \ I 
 
 ) 
 L 
 
 ^c, /0// _L oA .I \ 77 P _L 
 
 cos 2) * + + ^^ -I- 
 
364 SEMIDIURNAL TIDE. [10 
 
 But the equilibrium tide t is given by 
 
 e = E\Pl cos (2/i/tf + s<f> + a) 
 
 Hence we may write our result in the following form, which shows the 
 relationship between the true dynamical tide and the equilibrium tide : 
 
 A 
 
 4>ma P 2 ( 77 
 
 From a formula equivalent to this Mr Hough finds for the lunar semi- 
 diurnal tide (9 = 2), for a sea of 1210 fathoms ( 
 
 \4raa 
 
 f> = \ {-10396P1 + -57998P* - '19273P;! + -03054P'...) 
 Pg 
 
 This formula shows us that at the equator the tide is 'inverted/ and has 
 2 '41 87 times as great a range as the equilibrium tide. 
 
 For this same ocean he finds that the solar semi-diurnal tide is 'direct' at 
 the equator, and has a range 7'9548 as great as the equilibrium tide. 
 
 Now the lunar equilibrium tide is 2'2 times as great as the solar equili- 
 brium tide, and since 2'2 x 2'4187 is only 5'3 it follows that in such an ocean 
 the solar tides would have a range half as great again as the lunar. Further, 
 since the lunar tides are 'inverted' and the solar 'direct,' spring- tide would 
 occur at quarter-moon and neap-tide at full and change. 
 
 We give one more example from amongst those computed by Mr Hough. 
 
 In an ocean of 9680 fathoms ( -~- = 1 ) , he finds 
 
 \4>ma 5 / 
 
 ft = 2 {r7646PH - -06057PJ + "001447 Pj...} 
 
 * 
 
 At the equator the tides are 'direct' and have a range T9225 as great as 
 the equilibrium tide. In this case the tides approximate in type to those 
 of the equilibrium theory, although at the equator, at least, they have nearly 
 twice the range. 
 
 Our space will not permit us to give others of the remarkable conclusions 
 reached by Mr Hough. 
 
 We do not give any numerical results for the diurnal tides, for reasons 
 which will appear from the following section. 
 
1906] DIURNAL TIDE. 365 
 
 19. Diurnal tide approximately evanescent. 
 The equilibrium diurnal tide is given by 
 
 e = E\P\ cos (2nft + < + ) 
 where /is approximately | and the associated function for i = 2, s= 1 is 
 
 P\ = 3 sin 6 cos 6 
 Now the height of tide is given by 
 
 f) = ^,C\P i cos (2??/ + (j) + a) 
 and the problem is to evaluate the constants C\. 
 
 If possible suppose that f) is .also expressed by a single term like that 
 which represents , so that 
 
 j) = 3(/2 sin 6 cos cos (2n/55 + </> 4- a) 
 
 Then the differential equation (17) to be satisfied becomes 
 ew 1 A cos 6 du u 
 
 V-TX+ -^U COS 8 \ ;r- ^ + 
 
 !_ ! JJ^ d -- ^'/ 
 
 7l^ 2 ^Mgin^^X /-"-cos 
 
 where w is written for brevity in place of sin 6 cos 0. 
 
 Now when /is rigorously equal to , it may be proved by actual differen- 
 tiation that the expression inside { } vanishes identically, and the equation 
 reduces to G 2 = 0. 
 
 We thus find that in this case the differential equation is satisfied by 
 zero oscillation of water level. In other words we reach Laplace's remarkable 
 conclusion that there is no diurnal rise and fall of the tide ; there are, it is 
 true, diurnal tidal currents, but they are so arranged that the water level 
 remains unchanged. 
 
 In reality/ is not rigorously ^ (except for the tide called K x ) and there 
 will be a small diurnal tide. The lunar diurnal tide called has been 
 evaluated for various depths of ocean by Mr Hough and is found always to 
 be small. 
 
11. 
 
 ON THE DYNAMICAL THEORY OF THE TIDES OF 
 LONG PERIOD. 
 
 [Proceedings of the Royal Society of London, XLI. (1886), pp. 337 342.] 
 
 IN the following note an objection is raised against Laplace's method of 
 treating these tides, and a dynamical solution of the problem, founded on 
 a paper by Sir William Thomson, is offered. 
 
 Let 0, (f> be the colatitude and longitude of a point in the ocean, let and 
 i) sin be the displacements from its mean position of the water occupying 
 that point at the time t, let f) be the height of the tide, and let t be the 
 height of the tide according to the- equilibrium theory ; let n be the angular 
 velocity of the earth's rotation, g gravity, a the earth's radius, and 7 the 
 depth of the ocean at the point 0, <f>. 
 
 Then Laplace's equations of motion for tidal oscillations are 
 
 c 2 -di) q d /t N 
 
 -r| - 2n sm cos - = - ^ -^ (f) - *) 
 dtf dt ad0^ , 
 
 a d*-n a d q d ., 
 
 sm -T-' + 2n cos - = ----- " - a -- (fr- 
 dt 2 dt a sm d<f> ^ ' 
 
 And the equation of continuity is 
 
 The only case which will be considered here is where the depth of the 
 ocean is constant, and we shall only treat the oscillations of long period in 
 which the displacements are not functions of the longitude. 
 
 As the motion to be considered only involves steady oscillation, we assume 
 
 = e cos (2nft + a) 
 ^ = h cos (Znft + a) 
 = x cos (2nft + a) [ ........................... (3) 
 
 77 = y sin (%nft + a) 
 u = h e 
 
1886] EQUATIONS OF MOTION. 367 
 
 Hence, by substitution in (1), we have 
 
 f 2 + yfsin 9 cos 6 = -. -- , 
 4<md0 
 
 where m = - 
 
 9 
 
 Whence x (f 2 - cos 2 6) = -. -^ 
 
 4m dd 
 
 . a . , 1 cos 6 du 
 
 y sin 6 (f 2 cos 2 6) = -r j- ^ 
 
 4<m f dd 
 
 Then substituting for x and y in (2), which, when 7 is constant and rj is 
 not a function of </>, becomes 
 
 ' ^sM^"')- 
 
 7 d [sin0du/d0~\ 
 
 we get -^f. -r a ,,., - 1 T + 4<ma (u 
 
 sm 6 dd \_f - - cos 2 J 
 
 This is Laplace's equation for tidal oscillations of the first kind*. In 
 these tides / is a small fraction, being about -% in the case of the fortnightly 
 tide, and e the coefficient in the equilibrium tide is equal to E( cos*6), 
 where E is a known function of the elements of the orbit of the tide-gene- 
 rating body, and of the obliquity of the ecliptic. 
 
 If now we write /3 = 4<ma/y, and p cos 0, our equation becomes 
 
 In treating these oscillations Laplace does not use this equation, but 
 seeks to show that friction suffices to make the ocean assume at each 
 instant its form of equilibrium. His conclusion is no doubt true, but the 
 question remains as to what amount of friction is to be regarded as sufficing 
 to produce the result, and whether oceanic tidal friction can be great enough 
 to have the effect which he supposes it to have. 
 
 The friction here contemplated is such that the integral effect is repre- 
 sented by a retarding force proportional to the velocity of the fluid relatively 
 to the bottom. Although proportionality to the square of the velocity would 
 probably be nearer to the truth, yet Laplace's hypothesis suffices for the 
 present discussion. In oscillations of the class under consideration, the water 
 moves for half a period north, and then for half a period south. 
 
 Now in systems where the resistances are proportional to velocity, it is 
 usual to specify the resistance by a modulus of decay, namely, that period in 
 
 * Mecaniqite Celeste. 
 
368 SOLUTION OF THE EQUATIONS. [11 
 
 which a velocity is reduced by friction to e~ l or 1 -r- 2'783 of its initial value ; 
 and the friction contemplated by Laplace is such that the modulus of decay 
 is short compared with the semi-period of oscillation. 
 
 The quickest of the tides of long period is the fortnightly tide, hence for 
 the applicability of Laplace's conclusion, the modulus of decay must be short 
 compared with a week. Now it seems practically certain that the friction 
 of the ocean bed would not much affect the velocity of a slow ocean current 
 in a day or two. Hence we cannot accept Laplace's hypothesis as to the 
 effect of friction. 
 
 We now, therefore, proceed to the solution of the equation of motion 
 when friction is entirely neglected. 
 
 The solution here offered is indicated in a footnote to a paper by 
 Sir William Thomson (Phil. Mag., Vol. L., 1875, p. 280), but has never 
 been worked out before. 
 
 The symmetry of the motion demands that u, when expanded in a series 
 of powers of fj,, shall only contain even powers of /A. 
 
 Let us assume then 
 
 Then 
 
 ......... (5) 
 
 Again 
 
 ~ = - f*B lP , + (B, -/*,) ?+. 
 
 u = C-$f*B 1 *+$(B l -f*B 3 )^+... + l- i (B^ s -f*B^}v? i +'.. 
 
 ......... (6) 
 
 where G is a constant. 
 
 Then substituting from (5) and (6) in (4), and equating to zero the 
 successive coefficients of the powers of p, we find 
 
 i- 3 = 
 
 Thus the constants C and B 3 , B 5 , &c., are all expressible in terms of B l . 
 
1886] SOLUTION OF THE EQUATIONS. 369 
 
 We may remark that if 
 
 then the general equation of condition in (7) may be held to apply for all 
 values of i from 1 to infinity. 
 
 Let us now write it in the form 
 
 Bji+i _ T _ J _ fzQt _ ! __ Q -"2t-3 /n\ 
 
 . 2i2i+l J P "*" 2i2i + l P . 
 
 When i is large, 5 2i+1 /5 2 j_ 1 either tends to become infinitely small, or it 
 does not do so. 
 
 Let us suppose that it does not tend to become infinitely small. Then it 
 is obvious that the successive B's tend to become equal to one another, and 
 so also do the coefficients J5 2 i-i f*Bti +l in the expression for du/d/j,. 
 
 rJft fyT 
 
 Hence -j- = L + -= 2 , where L, M are finite, for all values of //,. Hence 
 
 -j^ = L Vl jj? + 7= , and therefore a; is infinite when //, = 1 at the pole, 
 
 Ct<C7 ^ j^ /^ 
 
 and d%jdt is infinite there also. 
 
 Hence the hypothesis, that 5 iM - +1 /5 !a -_ 1 does not tend to become infinitely 
 small, gives us infinite velocity at the pole. But with a globe covered with 
 water this is impossible, the hypothesis is negatived, and 5^ +1 /J5 2 t-i tends to 
 become infinitely small. 
 
 This being established let us write (8) in the form 
 o 1 R 
 
 -2i 1 2t(2i+l)^ /Q\ 
 
 ft = ^T i fto R 7^ w 
 
 2i(2i + l)/ " -Ozt+l/^i-l 
 
 By repeated applications of (9) we have in the form of a continued 
 fraction 
 
 (27+2) (27+ 3) (2^+1X27 + 5) 
 
 Zi ~ 3 
 
 
 ......... (10) 
 
 And we know that this is a continuous approximation, which must hold 
 in order to satisfy the condition that the water covers the whole globe. 
 
 Let us denote this continued fraction by Nt. 
 Then, if we remember that B^ = 2E, we have 
 
 v. i, 24 
 
370 SOLUTIONS FOR VARIOUS DEPTHS OF OCEAN. [11 
 
 so that 
 
 B s = - 2EN,N 2 , B 5 = - 2EN.N.N,, B 7 =- ZEN^N^, &c. 
 
 and G**-$X+*~k 
 
 pf 
 
 Then the height of tide j) is equal to h cos (Inft + a), the equilibrium 
 tide is equal to E (^ /i 2 ) cos (2?i/i5 4- a), and we have 
 
 h = u + E ($ - ^ 2 ) 
 
 Now when /3 = 40, we have 7 = ^ x 4raa = ^ M a = 7260 feet ; so that 
 /3 = 40 gives an ocean of 1200 fathoms. 
 
 With this value of /3, and with /= '036501 2, which is the value for the 
 fortnightly tide, I find 
 
 ^ = 3-040692, JV; = 1-20137, N 3 = "66744, N, = '42819, N 5 ='29819 
 N 6 = -21950, N 7 = '16814, ^ = 13287, N 9 
 These values give 
 
 N, = -15203, 1 +f'N l = 1-0041, ^N, (1 + f*N 2 } = 1-5228 
 
 = 1-2187, 
 = -2089, 
 
 7 ) = -0098, i ^ . . . N 7 (1 +/W 8 ) = '0014 
 = -00017, &c. 
 
 So that 
 
 4 = '1520 - 1-0041^ 2 + 1-5228/* 4 - 1'2187/A 6 + -6099/i 8 - -2089^ 10 
 A 
 
 + -0519/-1 12 - -0098/i 14 + -0014/A 16 - '0002^ 18 + ... 
 
 At the pole, where (j,= 1, the equilibrium tide is f E; at the equator it 
 is + E. 
 
 Now at the pole h = -Ex '1037 = - f E x -1556 
 and at the equator h = + E x '1520 = ^E x '4561 
 
 In a second case, namely, with an ocean four times as deep, so that 
 = 10, I find 
 
 4 = '2363 - 1'0016/i 2 + -5910/u 4 - '1627/i 6 + -0258/* 8 - -0026^ + '0002/* 12 - ... 
 .& 
 
 At the pole h = -E x '3137 = - f^ x -471 
 
 at the equator h = + Ex '2363 = + j 2? x '709 
 
1886] SOLUTIONS FOR VARIOUS DEPTHS OF OCEAN. 371 
 
 With a deeper ocean we should soon arrive at the equilibrium value for 
 the tide, for N 2 , N 3 , &c., become very small, and 2NJ/3 becomes equal to . 
 
 These two cases, /9 = 40, /3 = 10, are two of those for which Laplace has 
 given solutions in the cases of the semi-diurnal and diurnal tides. We notice 
 that, with such oceans as we have to deal with, the tide of long period is 
 certainly less than half its equilibrium amount. 
 
 In Thomson and Tait's Natural Philosophy (edition of 1883) [Paper 9, 
 p. 340 above] I have made a comparison of the observed tides of long period 
 with the equilibrium theory. The probable errors of the results are large, 
 but not such as to render them worthless, and in view of the present in- 
 vestigation it is surprising to find that on the average the tides of long 
 period amount to as much as two-thirds of their equilibrium value. 
 
 The investigation in the Natural Philosophy was undertaken in the belief 
 of the correctness of Laplace's view as to the tides of long period, and was 
 intended to evaluate the effective rigidity of the earth's mass. 
 
 The present result shows us that it is not possible to attain any estimate 
 of the earth's rigidity in this way, but as the tides of long period are dis- 
 tinctly sensible, we may accept the investigation in the Natural Philosophy 
 as generally confirmatory of Thomson's view as to the great effective rigidity 
 of the whole earth's mass. 
 
 There is one tide, however, of long period of which Laplace's argument 
 from friction must hold true. In consequence of the regression of the nodes 
 of the moon's orbit there is a minute tide with a period of nearly nineteen 
 years, and in this case friction must be far more important than inertia. 
 Unfortunately this tide is very minute, and as I have shown in a Report 
 for 1886 to the British Association on the tides [p. 166, above], it is entirely 
 masked by oscillations of sea level produced by meteorological or other causes. 
 
 Thus it does not seem likely that it will ever be possible to evaluate the 
 effective rigidity of the earth's mass by means of tidal observations. 
 
 242 
 
12. 
 
 ON THE ANTARCTIC TIDAL OBSERVATIONS OF THE 
 
 'DISCOVERY.' 
 
 [To be published hereafter as a contribution to the scientific 
 results of the voyage of the ' Discovery.'] 
 
 THE ' Discovery' wintered in 1902 and in 1903 at the south-eastern 
 extremity of Ross Island, on which Mount Erebus is situated, in south 
 latitude 78 49' and east longitude 166 20'. 
 
 The station is near the west coast of a great bay in the Antarctic 
 Continent, and the westerly coast line runs northward from the station for 
 about 9 of latitude. To the eastward of the bay, however, the coast only 
 attains a latitude of about 75 and follows approximately a small circle of 
 latitude. Since the tide-wave comes from the east and travels to the west 
 the station is not sheltered by the coast to the westward, and the continent 
 to the eastward can do but little to impede the full sweep of the tide-wave 
 in the Antarctic Ocean. It is true that Ross Island itself is partially to the 
 east of the anchorage, but it is so small that its influence cannot be important. 
 Of course the westward coast line must exercise an influence on the state of 
 tidal oscillation, for regarding the tide-wave as a free wave coming in from 
 the east, it is clear that it will run up to the end of the bay and then wheel 
 round northward along the westerly coast. It would seem then that the 
 situation is on the whole a good one for such observations. Of course their 
 value would have been much increased if it had been possible to obtain other 
 observations elsewhere. 
 
 The following account by Lieutenant Michael Barne, R.N., explains the 
 manner in which the tidal observations were made. 
 
 "On our arrival in the vicinity of our winter quarters on Feb. 8th, 1902, 
 a good deal of the previous year's ice remained attached to the land. As 
 there was no foreshore, and pieces of this ice were constantly moving out, it 
 was impossible to erect a tide-pole. With the final departure of the old ice, 
 
1907] 
 
 LIEUTENANT BARNE ON THE METHOD OF OBSERVATION. 
 
 373 
 
 the temperature fell, and young ice formed continually, only to be quickly 
 broken up by the almost incessant easterly winds. 
 
 " As this state of affairs promised to last for a considerable time, an effort 
 was made to obtain records of the tides. A stout graduated pole was erected 
 alongside the ice foot in about 10 feet of water, the lower end being heavily 
 weighted and the upper end securely guyed. Some intermittent observa- 
 tions were secured in this manner, but they are probably of little value, as 
 the ice was continually forming round the pole, which was only with difficulty 
 freed from it. Besides this, communication with the shore, and consequently 
 approach to the tide-pole, was constantly interrupted. 
 
 " On the ship being finally frozen in, a tide gauge of the following nature 
 was erected (Fig. 1). 
 
 Mark 
 
 Indicator 
 
 . sinkers 
 
 FIG. 1. 
 
 " A single length of pianoforte wire (sounding wire) was led through 
 a block, secured to the head of a tripod. One end of this length was attached 
 to eight 25 Ib. sinkers, which were lowered to the bottom. Four 25 Ib. 
 sinkers were secured to the other end in such a manner as to allow of their 
 free movement, between the ice and the block, as the ice, with the tripod, 
 rose and fell with the tide. An indicator was clamped to the wire, and a 
 suitable scale secured to the tripod. 
 
 "It was thought that the motive force supplied by the weight of four 
 sinkers, would be sufficient to draw the smooth surface of the wire through 
 the ice, as the water rose and fell ; whilst, in case it should fail to do so, the 
 weight of eight sinkers would not be sufficient to break the wire. 
 
 " As it was considered possible that the ice, owing to the proximity of the 
 land, might not maintain a uniform position relative to the surface of the 
 water, a small hole was occasionally opened close to one of the tripod legs, 
 
374 LIEUTENANT BARNE ON THE METHOD OF OBSERVATION. [12 
 
 to which was attached a mark, indicating the height to which the water 
 should rise. A few observations shewed that no error from this cause was to 
 be apprehended. 
 
 " This gauge was placed about 200 yards from the ship, and two-hourly 
 readings with but slight interruption were continued from April 12th to 
 April 28th. 
 
 " Some sluggishness in its movement which was eventually noticed, and 
 its final breakdown was possibly partly due to the thickening of the ice, but 
 principally I think to the fact that too small a block was employed at the 
 tripod head. The scale was, by accident, secured so that the readings 
 increased upwards; consequently they have to be inverted*. 
 
 " It was originally intended to place a tide gauge in the ship, owing to the 
 far greater convenience of position, but it was thought that the position 
 of the ship relatively to the water surface might alter and this might lead 
 to errors. It was hoped that, by placing one on the ice as well as one in 
 the ship, check observations might be obtained to determine if this source 
 of error existed. This was eventually accomplished on April 25th, but by 
 the time the ship gauge was erected, the outside gauge had ceased to be 
 entirely satisfactory for the reasons given. The observations however shew a 
 close approximation of movement. 
 
 " The ship gauge was arranged as shewn in Fig. 2. The supporting blocks 
 were secured rigidly, and, until May 10th, the wire was led directly through 
 the ice. As the friction was gradually increasing, a suggestion made by 
 Dr Wilson was adopted on that date, and the wire was taken through a tube, 
 filled with paraffin oil, and closed at the top and bottom with a hard wooden 
 plug through which the wire passed. A maximum and minimum arrange- 
 ment, with balanced weights, was added as shewn in the sketch f. Un- 
 fortunately, both on May 10th and May 12th, in re-fitting the gauge, the 
 indicator had to be fixed afresh, and therefore the observations cannot be 
 referred to a common zero. 
 
 "A mark was placed on the ship's side to ascertain any vertical move- 
 ment of the ship relatively to the water surface, and a long plummet was 
 secured in the engine-room, to shew any alteration in her inclination to 
 the vertical. 
 
 " On April 6th, 1903, the tide gauge was re-erected, and observations con- 
 tinued ; but, owing to the large number of observers employed, the maximum 
 and minimum arrangement was not fitted. 
 
 " The height of the mark (b) on the ship's side above the water, was 
 ascertained about once a month, in the same manner as during the winter 
 
 * This oversight was rectified before May 12. G. H. D. 
 t No use has been made of this arrangement. G. H. D. 
 
1907] 
 
 LIEUTENANT BARNE ON THE METHOD OF OBSERVATION. 
 
 375 
 
 of 1902, i.e. by digging a hole through the ice below the mark, and measuring 
 its height. On these occasions the difference between the heights of the 
 leading blocks was measured in the following manner. A wooden scale (c) 
 marked in half-inches was secured to the beam (a), in a vertical position, 
 
 Copper tube 
 filled with 
 paraffin 
 
 minimum 
 'Arrangement 
 
 on 
 scale 
 
 tide rising 
 
 FIG. 2 
 
 close to the outer block, with its zero mark on a level with the top of the 
 sheave. A wooden instrument, shaped like the letter T, and having a lead 
 weight attached to its lower end, and a hole A (see Fig. 3) in the centre of 
 its upper end, was hung, freely, on a nail, in such a manner that its upper 
 side was horizontal, and on a level with the top of the sheave of the inner 
 block. By bringing the eye on a level with the upper side of this T-piece, 
 and noting the position on the scale at which the upper side, if produced, 
 would cut it, the reading of the scale was obtained which gave the difference 
 in height of the blocks. 
 
376 
 
 LIEUTENANT BARNE ON THE METHOD OF OBSERVATION. 
 
 [12 
 
 " By taking periodical measurements of the height of the mark, and the 
 difference between the heights of the blocks, data were obtained, by which 
 readings could be corrected for alteration of the trim and the list of the ship 
 
 FIG. 3. 
 
 respectively and reduced to a common zero, namely that on April 6th, when 
 the tide gauge was erected for the winter of 1903. 
 
 " On Sept. 21st, 1903, the wire carried away close to the place where it 
 was secured to the weight resting on the bottom. On examination, the wire 
 was found to be greatly eaten away, from the point of attachment to a height 
 of several feet, presumably on account of an action between the cast-iron 
 sinkers, and the steel pianoforte wire." 
 
 The series of hourly observations was occasionally interrupted by accidents, 
 and the trim and list of the ship changed a little from time to time. 
 Accordingly it is not possible to treat the observations as a continuous whole. 
 The series was therefore broken into a succession of months, so chosen as to 
 avoid periods of manifest irregularity or of accidental interruption, and each 
 month was treated independently. 
 
 The choice of the method of harmonic analysis to be employed seemed 
 to lie between that explained in the Admiralty Scientific Manual and that 
 devised for the use of the tidal Abacus*. The method of the Manual is 
 considerably more laborious than the other, and it was highly desirable 
 that the Abacus should be used if it could be trusted for a short series of 
 observations. I therefore asked Mr Wright, who carried out the reductions 
 and was familiar with the use of the Abacus, to reduce the first month in 
 duplicate by the two methods. Curves were drawn through ordinates repre- 
 
 * Manual of Scientific Inquiry, Article Tides ; and " On an apparatus for facilitating the 
 reduction of tidal observations," Proc. Hoy. Soc., Vol. 52, p. 345. [Papers 4 and 6 in this volume.] 
 
1907] DIVISION OF OBSERVATIONS INTO MONTHLY SERIES. 377 
 
 senting the mean height of water at the 24 hours of mean lunar time, as 
 derived in the two ways. Although the whole range of height in the 24 
 mean lunar hours was only about six inches, the two curves shewed a 
 substantial agreement. The same process was then applied with O-time, 
 when the range was found to be about 15 inches, and the agreement of 
 the two curves was very close. The method of the Manual shewed several 
 sharp peaks or irregularities in the curves which were nearly smoothed out 
 by the use of the Abacus. Such peaks would not affect the values of semi- 
 diurnal or of diurnal components to a sensible amount, and as they are 
 clearly accidental, I concluded that the use of the Abacus was quite satis- 
 factory, and accordingly that method was adopted throughout. 
 
 In the use of harmonic analysis it is necessary that the month under 
 discussion should differ a little in length according to the tide which is being 
 evaluated. For finding the M 2 -tide months of 30 days or of 29 days would 
 be almost equally advantageous, but as 30 days gives us one more day of 
 observation that period was adopted. Similarly 30 days is appropriate for 
 the S 2 -tide. For a short period of observation it is necessary to regard this 
 tide as compounded of the S 2 and K 2 tides, and we must also suppose its 
 range to vary with the sun's parallax. The separation of these two tides 
 from one another depends on theoretical considerations, which appear to be 
 well founded. 
 
 Similarly in a short series of observations the K! and P tides must be 
 treated as fused together in a single tide, and they are separable by 
 theoretical considerations only. For these two tides a month of 27 days is 
 appropriate. 
 
 Lastly the analysis for the O-tide demands the use of a month of 
 28 days*. 
 
 I determined, then, to separate the months in such a way that the 
 shortest months (27 days) should follow one another as closely as possible, 
 while the longer months should overlap slightly. Whenever any event 
 occurred, whereby it seemed likely that the observations might be vitiated, 
 the months were chosen so as to omit the time of possible or actual ab- 
 normality. 
 
 It was clearly desirable that the largest possible number of independent 
 or nearly independent months should be discussed. This consideration led 
 in one case to an overlap of as much as six days ; thus the fifth month of 
 27 days ended on October 19, while the sixth month began on October 13. 
 
 In the few cases where hourly observations were missing, the defects were 
 made good by interpolation. Although the observations began in April, 1902, 
 the first satisfactory continuous period began on May 12. It will be well to 
 
 * This use of months of various lengths necessitates some small arithmetical changes in the 
 method as explained in the paper on the Apparatus referred to above. [See p. 225 above.] 
 
378 SPECIFICATION OF THE SEVERAL MONTHLY SERIES. [12 
 
 state the epochs for the succession of twelve months which it was possible 
 to obtain, and to add a few comments on the observations. 
 
 First month. This begins with O h May 12, 1902. The observations 
 really begin at 2 h , but extrapolated values were used for O h and l h . 
 
 Second month. This begins with O h June 5, 1902. 
 
 On the afternoon of July 5 the wire attached to the sinker parted and 
 the observations ceased. The apparatus was only reinstalled at 5 p.m. on 
 July 23rd. 
 
 Third month. This begins with O h July 24, 1902. 
 Fourth month. This begins with O h Aug. 23, 1902. 
 
 Fifth month. This begins with O h Sept. 23, 1902. The height for 6 h on 
 October 20 was interpolated. 
 
 On October 1 it was found that the ship had shifted so as to affect the 
 readings by one inch. The date at which the shift had occurred was un- 
 known, and moreover so small a change could not affect the results sensibly. 
 
 Sixth month. This begins with O h Oct. 13, 1902. On Nov. 9, the four 
 hourly values, l h to 4 h inclusive, were missing and were supplied by inter- 
 polation. 
 
 As already remarked this month considerably overlaps the one before 
 it. This was necessary if a seventh month was to be secured before the 
 observations ceased for the season, but the choice of the stage at which 
 the overlapping should be made to occur was more or less arbitrary. 
 
 Seventh month. This begins with O h Nov. 13, 1902. The observation for 
 22 h of Dec. 9 is missing and was supplied by interpolation. On Dec. 13 the 
 wire parted and the series ended for the year. 
 
 In the second winter, that of 1903, somewhat greater care seems to have 
 been taken to note the small shifting of the ship. 
 
 Eighth month. This begins with O h April 6, 1903. Between April 22 and 
 May 3 the ship shifted so as to make the readings too high by 3 inches, 
 compared with the earlier ones. As an arbitrary correction I deducted one 
 inch from all heights from O h April 24 to O h April 27 ; from l h April 27 to 
 O h April 30 I deducted two inches ; and for the rest of the month the full 
 three inches. These arbitrary corrections were submitted to independent 
 harmonic analysis, and it appeared that they afforded corrections so minute 
 as to leave the tidal constants virtually unchanged. 
 
 Ninth month. This begins with O h May 9, 1903. The ship shifted 
 considerably at some time about June 12, and as it is only possible to obtain 
 one month before that date, there is an unutilized gap of a few days between 
 this month and the one before it. 
 
1907] SPECIFICATION OF THE SEVERAL MONTHLY SERIES. 379 
 
 Tenth month. This begins with O h June 15, 1903. On July 10 a sensible 
 shift in the trim and height of the ship was discovered. This necessitates 
 the addition of 4^ inches to all heights, 2 inches being due to angular 
 movement and 2 inches to vertical movement. As an arbitrary correction 
 I added 2 inches to all heights from O h July 8 to O h July 9 ; and afterwards 
 I added the full 4| inches. 
 
 Eleventh month. This begins with O h July 14, 1903. 
 Twelfth month. This begins with O h August 14, 1903. 
 
 After Sept. 8 the observations were only taken every two hours, and for 
 the remainder of the month the values at the odd hours were interpolated. 
 
 The observations stop on Sept. 20, but are not used in the reductions 
 after Sept. 13. 
 
 No corrections have been applied for changes in the barometric pressure. 
 As the application of such a correction would have been very laborious and 
 moreover somewhat speculative, I have relied on the automatic elimination 
 of the inequalities produced by taking mean values. 
 
380 
 
 RESULTS OF HARMONIC ANALYSES. 
 
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1907] 
 
 VERIFICATION BY TIDE-PREDICTING INSTRUMENT. 
 
 381 
 
 The values of H and K are somewhat irregular from month to month, and 
 it is therefore not permissible to adopt the mean values of H and K as 
 representing the mean tide. I have therefore formed H cos K and H sin K, for 
 each month and have taken the mean of each as giving the mean values of 
 H cos K, and H sin K. It is easy to compute from these the proper mean 
 values of H and K for each tide. The results are given in the following 
 table : 
 
 Mean values of tidal constants. 
 
 Semidiurnal tides 
 
 M 2 H = 1-966 inches = O164 feet 
 * = 9-9 = 10 
 
 S 2 H = 1-142 inches = 0-095 feet 
 K = 272-l = 272 
 
 K 2 H = -311 inches = 0-026 feet 
 K = 272-l = 272 
 
 
 
 Diurnal tides 
 
 H = 9-245 inches = 0770 feet 
 K = 14-05 = 14 
 
 H = 3-082 inches = 0'257 feet 
 K = 14-05 = 14 
 
 H = 9-264 inches = 0-772 feet 
 K = 359-5 = 
 
 The sum of the semi-ranges of the three diurnal tides is 21'6 inches and 
 of the three semidiurnal tides is only 3'4 inches. This result corresponds 
 with the fact that little trace of the semidiurnal tide is to be discovered 
 from mere inspection of the tide curve. 
 
 When tidal observations have been reduced it is always important to 
 verify that the constants found do really represent the tidal oscillation, for in 
 computations of such complexity it is always possible that some gross mistake 
 of principle may have slipped in unnoticed. Such a verification is especially 
 important in a case where the tides are found to be very abnormal, as here, 
 and where the results from month to month are not closely consistent. 
 I accordingly asked Mr Glazebrook to run off curves for two periods with the 
 Indian tide-predicter at the National Physical Laboratory. The constants 
 used were the means for the tides evaluated. It is probable that a better 
 result would have been attained if a number of other tides, with constants 
 assigned by theoretical considerations from analogy with the constants actually 
 evaluated, had also been introduced ; but I did not think it was worth while 
 to do so. Evidence will be given hereafter to shew that the smaller elliptic 
 diurnal tides must exercise an appreciable influence. 
 
 The periods chosen for the comparison were about three weeks beginning 
 on May 12, 1902, and nearly the same time in November. It does not seem 
 worth while to reproduce the whole of the observed and computed curves 
 for these periods. The observed tide-curve has frequently sharp irregularities 
 presumably produced by weather or by unperceived shifts of the ship, and 
 the maxima are sometimes sharp peaks instead of flowing curves. However, 
 on the whole the computed and observed curves follow one another very well, 
 at least throughout all those portions where the diurnal tide is pronounced. 
 Where the diurnal inequality is nearly evanescent, and the semidiurnal tide 
 
382 PROBABLE EXPLANATION OF IRREGULARITY OF DIURNAL TIDE. [12 
 
 becomes perceptible the discordance is sometimes considerable, although even 
 in these cases every rise and fall of the water is traceable in the computed 
 curve. Such discordance was inevitable, for at this part of the curve all 
 those tidal oscillations which have any importance have disappeared, and 
 only those tides remain which are very small ; moreover most of these tides 
 are avowedly omitted from the computed curve. 
 
 I give two figures. The first (Fig. 4) shews the two curves where the 
 diurnal tide is large, viz. from O h May 24th to O h May 25th, 1902 ; it is a 
 rather favourable example of the general agreement referred to above. The 
 second figure (Fig. 5) is selected because it exhibits by far the worst discord- 
 ance which occurred in the six weeks under comparison. 
 
 I conclude that the reductions are quite as good as could be expected 
 from tide-curves which present as much irregularity as these do. It would 
 not be possible to make a very good tide-table from the constants, but no 
 one wants a tide-table for Ross Island. We only need sufficient accuracy to 
 obtain an insight into the nature of the Antarctic tides, and the constants 
 are quite sufficient for that end. 
 
 When the mean heights of water at the 24 hours of mean lunar time 
 were plotted in curves for each month, it became obvious that a pure semi- 
 diurnal inequality did not represent the facts very closely, and that there 
 remained also a sensible diurnal inequality. Such an inequality is given by 
 the tide M 1} and if we neglect the minute portion of the tide Mi which 
 depends on the terms in the tide-generating potential which vary as the 
 fourth power of the moon's parallax, such an inequality is found to depend 
 on the composition of two elliptic tides with speeds 7 a ta and 7 cr + nr. 
 The genesis of this compounded tide is explained in the .report to the British 
 Association for 1883 [Paper 1]. 
 
 I accordingly thought it worth while to evaluate the Mj tide for each of 
 the twelve months under reduction. The results come out sufficiently dis- 
 cordant to render it impossible to assign any definite value to the tide, yet 
 there appears to be some sort of method in the phases. Thus the phases for 
 the twelve months come out for 1902 9, - 3, - 45, 6, - 32, 70, 12, and 
 for 1903 6, - 159, - 179, - 42, - 10. 
 
 Two of the phases, those for the 9th an'd 10th months, are very discordant, 
 but for these months the amplitude of M a is small ; it is also very small for 
 the 6th month with phase 70. The mean of all the other phases is such 
 that K is pretty small, and this agrees with what is to be expected because K 
 for the tide is small. It thus appears probable that there has been 
 a sensible disturbance from the Mj tide of the values of the mean heights of 
 water as arranged in mean lunar time. It should be noted that the whole 
 amplitude of oscillation is so small that it is really surprising that this effect 
 should be traceable at all. 
 
1907] " ACTUALITY COMPARED WITH COMPUTED TIDE-CURVES. 
 
 383 
 
 '8 20 22 NOON 
 
 FIG. 4. 
 
 MIDNIGHT 14 16 IS 20 22 NOON 
 
 6 8 to MIDKIOK 
 
 MAY 29'-" 
 
 WAV 30' 
 
 FIG. 5. 
 
384 
 
 SEASONAL CHANGE IN LUNAR SEMIDIURNAL TIDE. 
 
 [12 
 
 There is one feature in the results which is so singular that it is well to 
 refer to it. If we look at the heights and phases of the M 2 -tide it will be 
 observed that there is a progressive change both in amplitude and phase as 
 the season of 1902 advances, and this "change is repeated in 1903. 
 
 Mere inspection does not convince one of the degree of regularity, and 
 I have therefore prepared a figure which exhibits the march of H cos K and 
 of H sin K. The values for each month may be taken to appertain to the 
 middle of the month, and the points surrounded by rings in Fig. 6 give the 
 
1907] SEASONAL CHANGE IN LUNAR SEMIDIURNAL TIDE. 385 
 
 values for the season of 1902, while those marked with crosses give the values 
 for 1903. The broken line shews conjectural curves which appear to satisfy 
 the observations. The conjectural curves are such that (in inches) 
 
 H cos K = T65 - '75 cos (rjt + 2) 
 H sin K = '23 + "53 cos (tjt + 79) 
 where 77 is 360 per annum and t is expressed in months. 
 
 There would thus be an annual inequality in H cos K and H sin K, and 
 their mean values, viz. 1'65 and 0'23 inches, would shew that the mean lunar 
 semidiurnal tide is expressed by H = 1| inches, K = 8. 
 
 The mean given previously as derived only from the observations was 
 H = 2 inches, K = 10. 
 
 It will be noticed that the greatest retardation occurs about midsummer, 
 and at the same season there is a considerable decrease of amplitude. It is 
 almost impossible to believe that the thawing of the sea could decrease the 
 amplitude of the tide, although it might possibly increase it. 
 
 It would be strange if this result, depending as it does on 12 independent 
 observations, should arise from mere chance. Yet there is no astronomical 
 tide which can give an annual inequality in the lunar semidiurnal tide. 
 I note that if the observations of 1903 were pushed backward one month 
 the whole of the observations would fall into a more perfect curve. Hence 
 an inequality of 13 months would satisfy the conditions more perfectly than 
 one of 12 months. There is theoretically a minute tidal inequality of long 
 period (Laplace's first species) with a period of 14 months due to the variation 
 of latitude ; but it is difficult to see how any perturbation of the lunar 
 semidiurnal tide could be produced in this way. 
 
 But if we have found a true physical phenomenon, the same kind of effect 
 ought probably to be produced on all the other tides. Yet when the 
 observations for the other tides are plotted out in the same way, the points 
 appear to be arranged almost chaotically. It is true that some slight 
 tendency may be perceived for an increase of amplitude towards mid- winter, 
 but the effect is too uncertain to justify reduction to numbers. 
 
 A much longer series of observations would be needed to throw a clear 
 light on the point raised, but the result is so curious that it would not have 
 been right to pass it by in silence. 
 
 Tidal observations were made at Ross Island (called Erebus Island on 
 the memorandum) by Dr Wilson from 2 h Jan. 11, 1904 to 8 h Jan. 13. The 
 place of observation was some 40 or 50 miles to the northward of the winter 
 D. i. 25 
 
386 COMPARISON WITH OBSERVATIONS OF JANUARY 1904. [12 
 
 
 
 station. As there seemed some reason to suspect a seasonal variability in 
 the tides, it seemed worth while to compare with actuality a tide-curve 
 computed with the constants derived from the winter observations. A curve 
 was therefore run off at the National Physical Laboratory for a few days 
 beginning with O h Jan. 11, 1904. Although the sites of the two sets of 
 observations are not identical, comparison with actuality shews a satis- 
 factory agreement. It is unfortunate that these observations were made just 
 after the time when the diurnal inequality has vanished and is beginning to 
 increase again ; for at these times the agreement is liable to be imperfect 
 between computed and observed curves. On these grounds no surprise need 
 be felt on account of the fact that the semidiurnal tide is somewhat more 
 clearly marked in the observed tide-curve than in the computed one, and 
 that the whole range of the diurnal tide on Jan. 11 was three inches greater, 
 and on Jan. 12 about six inches (out of 28 inches) greater than appears 
 from the computed curve. The computed and observed times of high and 
 low water agree closely with one another. We may, on the whole, accept 
 these summer observations as proving that our tidal constants are sub- 
 stantially correct. 
 
 The semidiurnal tides, although small, exhibit clearly another peculiarity; 
 it is that (K of S 2 ) (K of M 2 ) exhibits a seasonal change of roughly the same 
 character in both years. 
 
 In all cases ' the age of the tide ' is negative and its mean value is about 
 4 days ; in other words spring-tide occurs four days before or ten days after 
 full and change of moon. 
 
 If the phases of M 2 and S 2 differed by 180 we should have neaps at full 
 and change, and springs at half moon. This case corresponds to 'direct' 
 lunar tide and ' inverted ' solar tide. In the actual case 
 
 (K of M 2 ) - (K of S 2 ) = 370 - 272 = 98 
 
 thus the observations shew a re'sult a little nearer to this condition than to 
 the ordinary one where springs coincide with full and change of moon. 
 
 The unusual relationship between the M 2 and S 2 tides is such as to make 
 it worth while to examine what would be the condition of affairs in an ocean 
 of uniform depth covering the whole planet. From the few soundings which 
 have been made it would seem that the ocean may be about 600 fathoms in 
 depth, although further north the depth appears to be considerably greater. 
 I have therefore taken the formulae of Mr Hough (Phil. Trans. A, 191 (1878), 
 pp. 177, 180) and evaluated the lunar and solar semidiurnal tides for an 
 ocean of 7260 ft. in latitudes 60, 65, 70, 75 with the following 
 results : 
 
1907] 
 
 COMPARISON WITH TIDES OF AN OCEAN-COVERED EARTH. 
 
 387 
 
 Lunar semidiurnal tide. 
 
 Latitude 60 65 
 
 H of equilibrium tide 6*052 cm. 4*324 cm. 
 
 Factor of augmentation for dy- 
 namical tide 1-932 1-496 
 
 H of dynamical tide for ocean! 11*69 cm.) 6-47 cm. 1 
 
 of 7260 ft. (direct tide) / 4iinchesJ 2 inches] 
 
 Solar semidiurnal tide. 
 60 65 
 
 2-816 cm. 2*012 cm. 
 Factor of augmentation for dy- 
 namical tide . 
 
 Latitude 
 H of equilibrium tide 
 
 70 
 
 2-832 cm. 
 
 1-098 
 
 3-11 cm. 1 
 lj inches/ 
 
 70 
 1*318 cm. 
 
 75 
 1-622 cm. 
 
 755 
 
 1-22 cm. "I 
 i inch j 
 
 75 
 0*755 cm. 
 
 - 6-441 
 
 -4-390 
 
 H of dynamical tide for ocean) -18*14 cm.} -8*83 cm. 
 
 of 7260 ft. (inverted tide) J 7 inches J 3^ inches - Ij inches/ inch 
 
 -2-556 -1-003 
 
 -3-37 cm. ) -0-751 cm 
 
 '} 
 
 We thus find that in these high latitudes the solar tide is more magnified 
 than the lunar, and is inverted. Thus in latitude 60 the solar tide is much 
 larger than the lunar and is inverted, whereas in latitude 70 they are nearly 
 of equal magnitude and the inversion of the solar tide still continues. 
 
 For an ocean of twice the depth both the tides are direct, and they are 
 not so much magnified. 
 
 Although the Antarctic Ocean runs all round the globe it is of course 
 unjustifiable to apply these results directly to the oscillations of the actual 
 ocean, but they serve to shew that we have no reason to expect considerable 
 semidiurnal tides so near to the pole, and also that the great discrepancy 
 between the phases of M 2 and S 2 is not so surprising a fact as might appear 
 at first sight. 
 
 It is useless to carry out a similar investigation for the diurnal tides, 
 because the variations in the depth of ocean exercise so large an influence 
 on the result. We know in fact that for an ocean of uniform depth the Kj 
 tide vanishes completely, and the O-tide nearly vanishes. 
 
 I find that the equilibrium O-tide is 3 inches in lat. 60 and falls to 
 2 inches in lat. 75. Thus the amplitudes of the diurnal tides observed by 
 the ' Discovery ' are very much larger than the equilibrium values. 
 
 The Scottish Antarctic Expedition passed the winter of 1903 in S. lat. 
 60 44' and W. long. 44 39' at the South Orkney Islands ; they were thus 
 nearly opposite to the station of the 'Discovery.' Their station was well 
 adapted for determining the general character of the tides in the Antarctic 
 Ocean. The reduction of their observations was made by Mr Selby at the 
 National Physical Laboratory, and gave the following results : 
 
 H 
 
 M 2 
 
 S 2 
 
 K 2 
 
 K! 
 
 P 
 
 
 
 1*522 ft. 
 
 902 ft. 
 
 245 ft. 
 
 496 ft. 
 
 166 ft. 
 
 559 ft. 
 
 172 
 
 198 
 
 198 
 
 15 
 
 15 
 
 359 
 
 
 
 
 
 
 252 
 
388 COMPARISON WITH RESULTS OF SCOTTISH EXPEDITION. [12 
 
 It will be noticed that these results are quite normal, save that the S 2 -tide 
 is rather large compared with M 2 ; and there is a well-marked diurnal in- 
 equality. They acquire a special interest when considered in connection with 
 the ' Discovery's ' results. We see that the semidiurnal tides are ' inverted ' 
 but have little or no retardation ; whereas the M 2 -tide of the ' Discovery ' is 
 small, but ' direct ' also with little retardation. We are thus led to suspect 
 that to the northward of the latitude of the South Orkneys, where the 
 ' Scotia ' wintered, the semidiurnal tides are inverted with small retardation ; 
 that somewhere between the South Orkneys and near to the Antarctic 
 Continent there is a nodal line for the M 2 -tide. There must be also a 
 similar node for the S 2 -tide, and we may perhaps suppose that the node 
 of the S 2 -tide is nearer to Ross Island than that of the M 2 -tide. 
 
 When we turn to the diurnal tides we find an entirely different condition, 
 for at both places the phases are virtually identical, and there seems a primd 
 facie case for maintaining that the phase of the diurnal tide throughout the 
 whole Antarctic Ocean is approximately the same as in the equilibrium 
 theory. I cannot venture to offer any theory in explanation of the greater 
 magnitude of the diurnal tide at Ross Island than at the South Orkneys. 
 
PABT II 
 
 LUNAR DISTURBANCE OF GRAVITY 
 
13. 
 
 ON AN INSTRUMENT FOR DETECTING AND MEASURING 
 SMALL CHANGES IN THE DIRECTION OF THE FORCE 
 OF GRAVITY. 
 
 By G. H. DARWIN and HORACE DARWIN. 
 
 [Report of the Committee, consisting of Mr G. H. DARWIN, 
 Professor Sir WILLIAM THOMSON, Professor TAIT, Professor 
 GRANT, Dr SIEMENS, Professor PURSER, Professor G. FORBES, 
 and Mr HORACE DARWIN, appointed for the Measurement of 
 the Lunar Disturbance of Gravity. This Report is written 
 in the name of G. H. DARWIN merely for the sake of verbal 
 convenience. British Association Report for 1881, pp. 93 
 -126.] 
 
 I. Account of the experiments. 
 
 WE feel some difficulty as to the form which this report should take, 
 because we are still carrying on our experiments, and have, as yet, arrived 
 at no final results. As, however, we have done a good deal of work, and 
 have come to conclusions of some interest, we think it better to give at 
 once an account of our operations up to the present time, rather than to 
 defer it to the future. 
 
 In November, 1878, Sir William Thomson suggested to me that I should 
 endeavour to investigate experimentally the lunar disturbance of gravity, 
 and the question of the tidal yielding of the solid earth. In May, 1879, we 
 both visited him at Glasgow, and there saw an instrument, which, although 
 roughly put together, he believed to contain the principle by which success 
 might perhaps be attained. The instrument was erected in the Physical 
 Laboratory of the University of Glasgow. We are not in a position to 
 give an accurate description of it, but the following rough details are 
 quite sufficient. 
 
390 DESCRIPTION OF SIR WILLIAM THOMSON'S INSTRUMENT. [13 
 
 A solid lead cylinder, weighing perhaps a pound or two, was suspended 
 by a fine brass wire, about 5 feet in length, from the centre of the lintel or 
 cross-beam of the solid stone gallows, which is erected there for the purpose 
 of pendulum experiments. A spike projected a little way out of the bottom 
 of the cylindrical weight ; a single silk fibre, several inches in length, was 
 cemented to this spike, and the other end of the fibre was cemented to 
 the edge of an ordinary galvanometer-mirror. A second silk fibre, of equal 
 length, was cemented to the edge of the mirror at a point near to the 
 attachment of the former fibre. The other end of this second fibre was 
 then attached to a support, which was connected with the base of the stone 
 gallows. The support was so placed that it stood very near to the spike 
 at the bottom of the pendulum, and the mirror thus hung by the bifilar 
 suspension of two silks, which stood exceedingly near to one another in 
 their upper parts. The instrument was screened from draughts by paper 
 pasted across between the two pillars of the gallows ; but at the bottom, on 
 one side, a pane of glass was inserted, through which one could see the 
 pendulum bob and galvanometer-mirror. 
 
 It is obvious that a small displacement of the pendulum, in a direction 
 perpendicular to the two silks, will cause the mirror to turn about a vertical 
 axis. 
 
 A lamp and slit were arranged, as in a galvanometer, for exhibiting the 
 movement of the pendulum, by means of the beam of light reflected from 
 the mirror. 
 
 No systematic observations were made, but we looked at the instrument 
 at various hours of the day and night, and on Sunday also, when the street 
 and railway traffic is very small. 
 
 The reflected beam of light was found to be in incessant movement, of so 
 irregular a character that it was hardly possible to localise the mean position 
 of the spot of light on the screen, within 5 or 6 inches. On returning to 
 the instrument after several hours, we frequently found that the light had 
 wandered to quite a different part of the room, and we had sometimes to 
 search through nearly a semicircle before finding it again. 
 
 Sir William Thomson showed us that, by standing some 10 feet away 
 from the piers, and swaying from one foot to the other, in time with the 
 free oscillations of the pendulum, quite a large oscillation of the spot of 
 light could be produced. Subsequent experience has taught us that con- 
 siderable precautions are necessary to avoid effects of this kind, and the 
 stone piers at Glasgow did not seem to be well isolated from the floor, and 
 the top of the gallows was used as a junction for a number of electric 
 connections. 
 
 The cause of the extreme irregularity of the movements of the pendulum 
 was obscure ; and as Sir William Thomson was of opinion that the instru- 
 
1881] DESCRIPTION OF THE NEW INSTRUMENT. 391 
 
 ment was well worthy of careful study, we determined to undertake a series 
 of experiments at the Cavendish Laboratory at Cambridge. We take this 
 opportunity of recording our thanks to Lord Rayleigh* for his kindness in 
 placing rooms at our disposal, and for his constant readiness. to help us. 
 
 The pressure of other employments on both of us prevented our 
 beginning operations immediately, and the length of time which we have 
 now spent over these experiments is partly referable to this cause, although 
 it is principally due to the number of difficulties to be overcome, and to 
 the quantity of apparatus which has had to be manufactured. 
 
 In order to avoid the possibility of disturbance from terrestrial mag- 
 netism, we determined that our pendulum should be made of pure copper f. 
 Mr Hussey Vivian kindly gave me an introduction to Messrs Elkington, 
 of Birmingham; and, although it was quite out of their ordinary line of 
 business, they consented to make what we required. Accordingly, they 
 made a pair of electrolytically-deposited solid copper cylinders, 5^ inches 
 long, and 2f inches in diameter. From their appearance, we presume that 
 the deposition was made on to the inside of copper tubes, and we under- 
 stand that it occupied six weeks to take place. In November, 1879, they 
 sent us these two heavy masses of copper, and, declining any payment, 
 courteously begged our acceptance of them. Of these two cylinders we 
 have, as yet, only used one; but should our present endeavours lead to 
 results of interest, we shall ultimately require both of them. 
 
 Two months before the receipt of our weights, the British Association 
 had reappointed the Committee for the Lunar Disturbance of Gravity, and 
 had added our names thereto. Since that time, with the exception of com- 
 pulsory intermissions, we have continued to work at this subject. My brother 
 Horace and I have always discussed together the plan on which to proceed ; 
 but up to the present time much the larger part of the work has consisted 
 in devising mechanical expedients for overcoming difficulties. In this work 
 he has borne by very far the larger share ; and the apparatus has been 
 throughout constructed from his designs, and under his superintendence, by 
 the Cambridge Scientific Instrument Company. 
 
 Near the corner of a stone-paved ground-floor room in the Cavendish 
 Laboratory there stands a very solid stone gallows, similar to, but rather 
 more massive than, the one at Glasgow. As it did not appear thoroughly 
 free from rigid connection with the floor, we had the pavement raised all 
 round the piers, and the earth was excavated from round the brick base- 
 ment to the depth of about 2 feet 6 inches, until we were assured that 
 there was no connection with the floor or walls of the room, excepting 
 
 * Professor Maxwell had given us permission to use the ' pendulum room,' but we had not 
 yet begun our operations at the time of his death. 
 
 f We now think that this was probably a superfluity of precaution. 
 
392 DESCRIPTION OF INSTRUMENT. [13 
 
 % 
 
 through the earth. The ditch, which was left round the piers, was found 
 very useful for enabling us to carry out the somewhat delicate manipulations 
 involved in hanging the mirror by its two silk fibres. 
 
 Into the middle of the flat ends of one of our copper weights (which 
 weighed 4797 grammes, with spec. gr. 8'91) were screwed a pair of copper 
 plugs ; one plug was square-headed and the other pointed. Into the centre 
 of the square plug was soldered a thin copper wire, just capable of sustaining 
 the weight, and intended to hang the pendulum. 
 
 A stout cast-iron tripod was made for the support of the pendulum. 
 Through a hole in the centre of it there ran rather loosely a stout iron rod 
 with a screw cut on it. A nut ran on the screw and prevented the rod from 
 slipping through the hole. The other end of the copper wire was fixed into 
 the end of the rod. 
 
 The tripod was placed with its three legs resting near the margin of 
 the circular hole in the centre of the lintel of the gallows. The iron rod 
 was in the centre of the hole, and its lower end appeared about six inches 
 below the lower face of the lintel. The pendulum hung from the rod by 
 a wire of such length as to bring the spiked plug within a few inches 
 of the base of the gallows. This would of course be a very bad way of 
 hanging a pendulum which is intended to swing, but in our case the 
 displacements of the end of the pendulum were only likely to be of a 
 magnitude to be estimated in thousandths or even millionths of an inch, 
 and it is certain that for such small displacements the nut from which 
 the pendulum hung could not possibly rock on its bearings. However, 
 in subsequent experiments we improved the arrangement by giving the 
 nut a flange, from which there projected three small equidistant knobs, 
 on which the nut rested. 
 
 The length of the pendulum from the upper juncture with the iron rod 
 down to the tip of the spike in the bob was 148'2 cm. 
 
 An iron box was cast with three short legs, two in front and one 
 behind; its interior dimensions were 15 x 15 x 17^ cm.; it had a tap at 
 the back; the front face (15 x 17|) was left open, with arrangements for 
 fixing a plate-glass face thereon. The top face (15 x 17) was pierced by 
 a large round hole. On to this hole was cemented an ordinary earthenware 
 4-inch drain pipe, and on to the top of this first pipe there was cemented 
 a second. The box was thus provided with a chimney 144 cm. high. The 
 cubic contents of the box and chimney were about 3 gallons. 
 
 The box was placed standing on the base of the gallows, with the chimney 
 vertically underneath the round hole in the lintel. The top of the chimney 
 nearly reached the lower face of the lintel, and the iron rod of the pendulum 
 extended a few inches down into the chimney. The pendulum wire ran 
 down the middle of the chimney, and the lower half of the pendulum bob 
 
1881] DESCRIPTION OF INSTRUMENT. 393 
 
 was visible through the open face of the iron box. The stone gallows faces 
 towards the S.E., but we placed the box askew on the base, so that its open 
 face was directed towards the S. 
 
 The three legs of the box rested on little metal discs, each with a conical 
 hole in it, and these discs rested on three others of a somewhat larger size. 
 When the box was set approximately in position, we could by an arrangement 
 of screws cause the smaller discs to slide a fraction of an inch on the larger 
 ones, and thus exactly adjust the position of the box and chimney. 
 
 A small stand, something like a retort stand, about 4 inches high, stood 
 on a leaden base, with a short horizontal arm clamped by a screw on to the 
 thin vertical rod. This was the ' fixed ' support for the bifilar suspension 
 of the mirror. The stand was placed to the E. of the pendulum bob, and 
 the horizontal arm reached out until it came very close to the spike of the 
 pendulum. 
 
 The suspension and protection from tarnishing of our mirror gave us 
 much trouble, but it is useless to explain the various earlier methods 
 employed, because we have now overcome these difficulties in a manner 
 to be described later. The two cocoon fibres were fixed at a considerable 
 distance apart on the edge of the mirror, and as they were very short they 
 splayed out at nearly a right angle to one another. By means of this 
 arrangement the free period of oscillation of the mirror was made very short, 
 and we were easily able to separate the long free swing of the pendulum 
 from the short oscillations of the mirror. 
 
 The mirror was hung so that the upper ends of the silks stood within 
 an eighth of an inch of one another, but the tip of the spike stood or T ^ of 
 an inch higher than the fixed support. The plate-glass front of the box was 
 then fixed on with indiarubber packing. 
 
 It is obvious that a movement of the box parallel to the front from E. 
 to W. would bring the two fibres nearer together; this operation we shall 
 describe as sensitising the instrument. A movement of the box perpen- 
 dicular to the front would cause the mirror to show its face parallel to 
 the front of the box ; this operation we shall describe as centralising. As 
 sensitising will generally decentralise, both sets of screws had to be worked 
 alternately. 
 
 The adjusting screws for moving the box did not work very well; never- 
 theless, by a little trouble we managed to bring the two silks of the bifilar 
 suspension very close to one another. 
 
 After the instrument had been hung as above described, we tried a 
 preliminary sensitisation, and found the pendulum to respond to a slight 
 touch on either pier. The spot of light reflected from the mirror was very 
 unsteady, but not nearly so much so as in the Glasgow experiment; and 
 
394 SENSITIVENESS OF THE INSTRUMENT. [13 
 
 we were quite unable to produce any perceptible increase of agitation by 
 stamping or swaying to and fro on the stone floor. This showed that the 
 isolation of the pier was far more satisfactory than at Glasgow. 
 
 We then filled the box and pipes with water. We had much trouble 
 with slow leakage of the vessel, but the most serious difficulty arose from 
 the air-bubbles which adhered to the pendulum. By using boiled water 
 we obviated this fairly well, but we concluded that it was a great mistake 
 to have a flat bottom to the pendulum. This mistake we have remedied in 
 the final experiment described in the present paper. 
 
 The damping effect of the water on the oscillations of the pendulum 
 and of the mirror was very great, and although the incessant dance of the 
 light continued, it was of much smaller amplitude, and comparatively large 
 oscillations of the pendulum, caused by giving the piers a push, died out 
 after two or three swings. A very slight push on the stone piers displaced 
 the mean position of the light, but jumping and stamping on the pavement 
 of the room produced no perceptible effect. If, however, one of us stood on 
 the bare earth in the ditch behind, or before the massive stone pier, a very 
 sensible deflection of the light was caused ; this we now know was caused 
 by an elastic depression of the earth, which tilted the whole structure in 
 one or the other direction. A pull of a few ounces, delivered horizontally 
 on the centre of the lintel, produced a clear deflection, and when the pull 
 was 8 Ibs., the deflection of the spot of light amounted to 45 cm. We then 
 determined to make some rough systematic experiments. 
 
 The room was darkened by shutters over all the windows, and the doors 
 were kept closed. The paraffin lamp stood at three or four feet to the S.E. 
 of the easterly stone pier, but the light was screened from the pier. 
 
 We began our readings at 12 noon (March 15, 1880), and took eight 
 between that time and 10.30 P.M. From 12 noon until 4 P.M. the lamp was 
 left burning, but afterwards it was only lighted for about a minute to 
 take each reading. At 12 the reading was 595 mm., and at 4 P.M. it was 
 936 mm.*; these readings, together with the intermediate ones, showed 
 that the pendulum had been moving northwards with a nearly uniform 
 velocity. After the lamp was put out, the pendulum moved southward, 
 and by 10.30 P.M. was nearly in the same position as at noon. 
 
 During the whole of the two following days and a part of the next 
 we took a number of readings from 9 A.M. until 11 P.M. The observations 
 when graphically exhibited showed a fairly regular wave, the pendulum 
 being at the maximum of its northern excursion between 5 and 7 P.M., and 
 probably furthest south between the same hours in the morning. But 
 
 * I give the numbers as recorded in the note-book, but the readings would sometimes differ 
 by 2 or 3 mm. within half-a-minute. The light always waves to and fro in an uncertain sort of 
 way, so that it is impossible to assign a mean position with any certainty. 
 
1881] MODIFICATIONS OF THE INSTRUMENT. 395 
 
 besides this wave motion, the mean position for the day travelled a good 
 deal northward. We think that a part of this diurnal oscillation was due 
 to the warping of the stone columns from changes of temperature. An 
 increase of temperature on the south-east faces of the piers carried the 
 lintel towards the north-west, and of this displacement we observed only 
 the northerly component. The lamp produced a very rapid effect, and the 
 diurnal change lagged some two hours behind the change in the external 
 air. The difference between the temperatures of the S.E. and N.W. faces 
 of the pier must have been very slight indeed. At that time, and indeed 
 until quite recently, we attributed the whole of this diurnal oscillation to 
 the warping of the piers, but we now feel nearly certain that it was due in 
 great measure to a real change in the horizon. 
 
 We found that warming one of the legs of the iron tripod, even by contact 
 with the finger, produced a marked effect, and we concluded that the mode 
 of suspension was unsatisfactory. 
 
 Although we had thus learnt that changes of temperature formed the 
 great obstacle in the way of success, there were a good many things to be 
 learnt from the instrument as it existed at that time. 
 
 After the box and pipes had been filled for some days the plate-glass 
 front cracked quite across, and a slow leakage began to take place ; we were 
 thus compelled to dismount the whole apparatus and to make a fresh start. 
 
 It is obvious that to detect and measure displacements of the pendulum 
 in the N. and S. direction, the azimuth of the silks by which the mirror is 
 suspended must be E. and W., and that although any E. and W. displace- 
 ment of the pendulum will be invisible, still such displacement will alter 
 the sensitiveness of the instrument for the N. and S. displacements. In 
 order to obviate this we determined to constrain the pendulum to move 
 only in the N. and S. azimuth. 
 
 Accordingly we had a T-piece about 4 inches long fixed to the end of 
 the iron rod from which the pendulum hung. The two ends of a fine 
 copper wire were soldered into the ends of the T-piece ; a long loop of 
 wire was thus formed. The square-headed plug at the top of the pendulum 
 bob was replaced by another containing a small copper wheel, which could 
 revolve about a horizontal axis. The bearings of the wheel were open on 
 one side. 
 
 When the wheel was placed to ride on the bottom of the wire loop, and 
 the pendulum bob hooked on to the axle of the wheel by the open bearings, 
 we had our pendulum hanging by a bifilar suspension. The motion of the 
 pendulum was thus constrained to take place only perpendicular to the plane 
 of the wire loop. 
 
 The iron tripod was replaced by a slate slab large enough to entirely 
 cover the hole in the lintel of the gallows. Through the centre of the slab 
 
396 MEASUREMENT OF THE DEFLECTIONS OF THE VERTICAL. [13 
 
 was a round hole, of about one inch in diameter, through which passed the 
 iron rod with the T-piece at the lower end. The iron rod was supported 
 on the slate by means of the flanged nut above referred to. There was also 
 a straight slot, cut quite through the slab, running from the central hole to 
 the margin. The purpose of this slot will be explained presently. 
 
 In the preceding experiment we had no means of determining the abso- 
 lute amount of displacement of the pendulum, although, of course, we knew 
 that it must be very small. There are two methods by which the absolute 
 displacements are determinable ; one is to cause known small displacements 
 to the pendulum and to watch the effect on the mirror ; and the second is 
 to cause known small horizontal forces to act on the pendulum. We have 
 hitherto only employed the latter method, but we are rather inclined to 
 think that the former may give better results. 
 
 The following plan for producing small known horizontal forces was 
 suggested by my brother. 
 
 Suppose there be a very large and a very small pendulum hanging by 
 wires of equal length from neighbouring points in the same horizon; and 
 suppose the large and the small pendulum to be joined by a fibre which 
 is a very little shorter than the distance between the points of suspension. 
 Then each pendulum is obviously deflected a little from the vertical, but 
 the deflection of the small pendulum varies as the mass of the larger, and 
 that of the larger as the mass of the smaller. If ra be the mass of the 
 small pendulum, and M that of the large one, and if a be the distance 
 between the points of suspension, then it may be easily shown that if a be 
 increased by a small length 8a, the increase of the linear deflection of the 
 large pendulum is m8a/(m + M}. If I be the length of either pendulum, the 
 angular deflection of the larger one is mSa/l (m + M), and this is the deflec- 
 tion which would be produced by a horizontal force equal to m&a/l (m + M) 
 of gravity. It is clear, then, that by making the inequality between the two 
 weights m and M very great, and the displacement of the point of suspension 
 very small, we may deflect the large pendulum by as small a quantity as we 
 like. The theory is almost the same if the two pendulums are riot of exactly 
 the same length, or if the length of one of them be varied. 
 
 Now in our application of this principle we did not actually attach the 
 two pendulums together, but we made the little pendulum lean up against 
 the large one ; the theory is obviously just the same. 
 
 We call the small pendulum ' the disturber,' because its use is to disturb 
 the large pendulum by known forces. A small copper weight for the dis- 
 turber weighed '732 grammes, and the large pendulum bob, with its pulley, 
 weighed 4831 '5. Therefore the one was 6600 times as massive as the other. 
 The disturber was hung by a platinum wire about y^^th of an inch in 
 diameter, which is a good deal thinner than a fine human hair. 
 
1881] MEASUREMENT OF THE DEFLECTIONS OF THE VERTICAL. 397 
 
 We must now explain how the disturber was suspended, and the method 
 of moving its point of suspension. 
 
 Parallel to the sides of the slot in the slate slab there was riveted a pair 
 of brass rails, one being V-shaped and the other flat; on these rails there 
 slid a little carriage with three legs, one of which slid on one rail, and the 
 other two on the other. A brass rod with an eyelet-hole at the end was 
 fixed to the centre of the carriage, and was directed downwards so that it 
 passed through the centre of the slot. The slot was directed so that it was 
 perpendicular to the T-piece from which the pendulum hung, and the brass 
 rod of the little carriage was bent and of such length, that when the carriage 
 was pushed on its rails until it was as near the centre of the slab as it would 
 go, the eyelet-hole stood just below the T-piece, and half-way between the 
 two wires. A micrometer screw was clamped to the slab and was arranged 
 for making the carriage traverse known lengths on its rails, and as the wires 
 of the pendulum were in the E. and W. plane, the carriage was caused to 
 travel N. and S. by its micrometer screw. 
 
 One end of the fine platinum wire was fastened to the eyelet, and the 
 other (as above stated) to the small disturbing weight. The platinum wire 
 was of such length that the disturber just reached the pulley by which the 
 big pendulum hung. We found that by pushing the carriage up to the 
 centre, and very slightly tilting it off one rail, we could cause the disturber 
 weight to rest on either side of the pulley at will. If it was left on the side 
 of the pulley remote from the disturber-carriage, it was in gear, and the 
 traversing of the carriage on its rails would produce a small pressure of the 
 disturber on to the side of the pulley. If it was left on the same side of the 
 pulley as the disturber-carriage, the two pendulums were quite independent 
 and the disturber was out of gear. 
 
 On making allowance for the difference in length between the pendulum 
 and the disturber, and for the manner in which the thrust was delivered at 
 the top of the pendulum, but omitting the corrections for the weights of the 
 suspending wires and for the elasticity of the copper wire, we found that 
 one turn of the micrometer screw should displace the spike at the bottom 
 of the pendulum through O'OOOl mm. or ^-ijVocjtli of an inch. The same 
 displacement would be produced by an alteration in the direction of gravity 
 with reference to the earth's surface by ^th of a second of arc. 
 
 A rough computation showed that the to and fro motion of the pendulum 
 in the N.S. azimuth, due to lunar attraction, should, if the earth be rigid, be 
 the same as that produced by 2| turns of the micrometer screw. 
 
 We now return to the other arrangements made in re-erecting the 
 instrument. 
 
 A new mirror, silvered on the face, was used, and was hung in a slightly 
 different manner. 
 
398 METHOD OF DAMPING THE OSCILLATIONS. [13 
 
 The fluid in which the pendulum was hung was spirits and water. The 
 physical properties of such a mixture will be referred to later. In order to 
 avoid air-bubbles we boiled 3^ gallons of spirits and water for three hours in 
 vacuo, and the result appeared satisfactory in that respect. 
 
 After the mirror was hung, the plate-glass front to the box was fixed 
 and the vessel was filled by the tap in the back of the box. The disturber 
 was not introduced until afterwards, and we then found that the pendulum 
 responded properly to the disturbance. 
 
 As the heat of a lamp in the neighbourhood of the piers exercised a 
 large disturbance, we changed the method of observing, and read the reflec- 
 tion of a scale with a telescope. The scale was a levelling staff divided into 
 feet, and tenths and hundredths of a foot, laid horizontally at 15 feet from 
 the piers, with the telescope immediately over it. 
 
 Since the amount of fluid through which the light had to pass was 
 considerable, we were forced to place a gas-flame immediately in front of 
 the scale ; but the gas was only kept alight long enough to take a reading. 
 
 After sensitising the instrument we found that the incessant dance of 
 the image of the scale was markedly less than when the pendulum was hung 
 in water. A touch with the finger on either pier produced deflection by 
 bending the piers, and the instrument responded to the disturber. 
 
 The vessel had been filled with fluid for some days, and we had just 
 begun a series of readings, when the plate-glass front again cracked quite 
 across without any previous warning. Thus ended our second attempt. 
 
 In the third experiment (July and August, 1880) the arrangements were 
 so nearly the same as those just described that we need not refer to them. 
 The packing for the plate-glass front was formed of red lead, and this 
 proved perfectly successful, whereas the indiarubber packing had twice 
 failed. As we were troubled by invisible leakage and by the evaporation 
 of the fluid, we arranged an inverted bottle, so as always to keep the 
 chimney full. We thought that when the T-piece at the end of the shaft 
 became exposed to the air, the pendulum became much more unsteady, but 
 we now think it at least possible that there was merely a period of real 
 terrestrial disturbance. 
 
 From August 10 to 14 we took a series of observations from early morning 
 until late at night. We noted the same sort of diurnal oscillatory motion as 
 before, but the outline of the curve was far less regular. This, we think, may 
 perhaps be explained by the necessity we were under of leaving the doors 
 open a good deal, in order to permit the cord to pass by which Lord Rayleigh 
 was spinning the British Association coil. 
 
 Notwithstanding that the weather was sultry the warping of the stone 
 columns must have been very slight, for a thermometer hung close to the 
 
1881] TESTS OF SENSITIVENESS. 399 
 
 pier scarcely showed a degree of change between the day and night, and the 
 difference of temperature of the N. and S. faces must have been a very small 
 fraction of a degree. At that time, however, we still thought that the whole 
 of the diurnal oscillation was due to the warping of the columns. 
 
 We next tried a series of experiments to test the sensitiveness of the 
 instrument. 
 
 As above remarked the image of the scale was continually in motion, and 
 moreover the mean reading was always shifting in either one direction or 
 the other. At any one time it was possible to take a reading to within 
 Y^th of a foot with certainty, and to make an estimate of the y^th of a foot, 
 but the numbers given below are necessarily to be regarded as very rough 
 approximations. 
 
 As above stated, the gallows faced about to the S.E., and we may describe 
 the two square piers as the E. and W. piers, and the edges of each pier by 
 the points of the compass towards which they are directed. 
 
 On August 14, 1880, my brother stood on a plank supported by the 
 pavement of the room close to the S.W. edge of the W. pier, and, lighting 
 a spirit lamp, held the flame for ten seconds within an inch or two of 
 this edge of the pier. The effect was certainly produced of making the 
 pendulum bob move northwards, but as such an effect is fused in the 
 diurnal change then going on, the amount of effect was uncertain. He 
 then stood similarly near the N.E. edge of the E. pier, and held the spirit 
 flame actually licking the edge of the stone during one minute. The effect 
 should now be opposed to the diurnal change, and it was so. Before the 
 exposure to heat was over the reading had decreased '15 feet, and after the 
 heat was withdrawn the recovery began to take place almost immediately. 
 We concluded afterwards that the effect was equivalent to a change of 
 horizon of about 0"'15. 
 
 When the flame was held near but not touching the lintel for thirty 
 seconds, the effect was obvious but scarcely measurable, even in round 
 numbers, on account of the unsteadiness of the image. 
 
 When a heated lump of brass was pushed under the iron box no effect 
 whatever was perceived, and even when a spirit flame was held so as to lick 
 one side of the iron box during thirty seconds, we could not be sure that 
 there was any effect. We had expected a violent disturbance, but these 
 experiments seemed to show that convection currents in the fluid produce 
 remarkably little effect. 
 
 When a pull of 300 grammes was delivered on to the centre of the lintel 
 in a southward direction, we determined by several trials that the displace- 
 ment of the reading was about '30 feet, which may be equal to about 0"'3 
 change of horizon, 
 
400 TESTS OF SENSITIVENESS. [13 
 
 Two-thirds of a watering-can of water was poured into the ditch at the 
 back of the pier. In this experiment the swelling of the ground should 
 have an effect antagonistic to that produced by the cooling of the back 
 face of the pier, and also to the diurnal changes then going on. The 
 swelling of the ground certainly tilted the pier over, so that the reading 
 was altered by '10 feet. A further dose of water seemed to have the same 
 effect, and it took more than an hour for the piers to regain their former 
 position. As the normal diurnal change was going on simultaneously, we 
 do not know the length of time during which the water continued to 
 produce an effect. 
 
 On August 15 we tried a series of experiments with the disturber. When 
 the disturber was displaced on its rails, the pendulum took a very perceptible 
 time to take up its new position, on account of the viscosity of the fluid in 
 which it was immersed. 
 
 The diurnal changes which were going on prevented the readings from 
 being very accordant amongst themselves, but we concluded that twenty-five 
 turns of the screw gave between '4 and '3 feet alteration in the reading on 
 the scale. From the masses and dimensions of the pendulum and disturber, 
 we concluded that 1 foot of our scale corresponded with about 1" change 
 in horizon. Taking into account the length of the pendulum, it appeared 
 that 1 foot of our scale corresponded with y^Vtfth o f a mm displacement of 
 the spike at the bottom of the pendulum. Now as a tenth of a foot of 
 alteration of reading could be perceived with certainty, it followed that when 
 the pendulum point moved through y^^th of a mm. we could certainly 
 perceive it. 
 
 During the first ten days the mean of the diurnal readings gradually 
 increased, showing that the pendulum was moving northwards, until the 
 reading had actually shifted 8 feet on the scale. It then became necessary 
 to shift the scale. Between August 23 and 25 the reading had changed 
 another foot. We then left Cambridge. On returning in October we found 
 that this change had continued. The mirror had, however, become tarnished, 
 and it was no longer possible to take a reading, although one could just see 
 a gas-flame by reflection from the mirror. 
 
 Whilst erecting the pendulum we had to stand on, and in front of, the 
 piers, and to put them under various kinds of stress, and we always found 
 that after such stress some sort of apparently abnormal changes in the piers 
 continued for three or four hours afterwards. 
 
 We were at that time at a loss to understand the reason of this long- 
 continued change in the mean position of the pendulum, and were reluctant 
 to believe that it indicated any real change of horizon of the whole soil ; but 
 after having read the papers of MM. d'Abbadie and Plantamour, we now 
 believe that such a real change was taking place. 
 
1881] DESCRIPTION OF MODIFIED INSTRUMENT. 401 
 
 By this course of experiments it appeared that an instrument of the kind 
 described may be brought to almost any degree of sensitiveness. We had 
 seen, however, that a stone support is unfavourable, because the bad conduc- 
 tivity of stone prevents a rapid equalisation of temperature between different 
 parts, and even small inequalities, of temperature produce considerable warping 
 of the stone piers. But it now seems probable that we exaggerated the 
 amount of disturbance which may arise from this cause. 
 
 A. cellar would undoubtedly be the best site for such an experiment, but 
 unfortunately there is no such place available in the Cavendish Laboratory. 
 Lord Rayleigh, however, placed the ' balance room ' at our disposal, and this 
 room has a northerly aspect. There are two windows in it, high up on the 
 north wall, and these we keep boarded up. 
 
 The arrangements which we now intended to make were that the 
 pendulum and mirror should be hung in a very confined space, and should 
 be immersed in fluid of considerable viscosity. The boundary of that space 
 should be made of a heat-conducting material, which should itself form the 
 support for the pendulum. The whole instrument, including the basement, 
 was to be immersed in water, and the basement itself was to be carefully 
 detached from contact with the building in which it stands. By these means 
 we hoped to damp out the short oscillations due to local tremors, but to 
 allow the longer oscillations free to take place ; but above all we desired 
 that changes of temperature in the instrument should take place with great 
 slowness, and should be, as far as possible, equal all round. 
 
 We removed the pavement from the centre of the room, and had a 
 circular hole, about 3 feet 6 inches in diameter, excavated in the 'made 
 earth,' until we got down to the undisturbed gravel, at a depth of about 
 2 feet 6 inches. 
 
 We obtained a large cylindrical stone 2 feet 4 inches in diameter and 
 2 feet 6 inches in height, weighing about three-quarters of a ton. This we 
 had intended to place on the earth in the hole, so that its upper surface 
 should stand flush with the pavement of the room. But the excavation had 
 been carried down a little too deep, and therefore an ordinary flat paving 
 stone was placed on the earth, with a thin bedding of cement underneath 
 it. The cylindrical block was placed to stand upon the paving stone, with 
 a very thin bedding of lime and water between the two stones. The surface 
 of the stone was then flush with the floor. We do not think that any sacrifice 
 of stability has been made by this course. 
 
 An annular trench or ditch a little less than a foot across is left round 
 the stone. We have lately had the bottom of the ditch cemented, and the 
 vertical sides lined with brickwork, which is kept clear of any contact with 
 the pavement of the room. On the S. side the ditch is a little wider, and 
 this permits us to stand in it conveniently. The bricked ditch is watertight, 
 D. i. 26 
 
402 DESCRIPTION OF MODIFIED INSTRUMENT. [13 
 
 and has a small overflow pipe into the drains. The water in the ditch stands 
 slightly higher than the flat top of the cylindrical stone, and thus the whole 
 basement may be kept immersed in water, and it is, presumably, at a very 
 uniform temperature all round. 
 
 Before describing the instrument itself we will explain the remaining 
 precautions for equalisation of temperature. 
 
 On the flat top of the stone stands a large barrel or tub, 5 feet 6 inches 
 high and 1 foot 10 inches in diameter, open at both ends. The diameter of 
 the stone is about 2 inches greater than the outside measure of the diameter 
 of the tub, and the tub thus nearly covers the whole of the stone. The tub 
 is well payed with pitch inside, and stands on two felt rings soaked in tar. 
 Five large iron weights, weighing altogether nearly three-quarters of a ton, 
 are hooked on to the upper edge of the tub, in order to make the joint 
 between the tub and the stone watertight. Near the bottom is a plate-glass 
 window ; when it is in position, the window faces to the S. This tub is filled 
 with water and the instrument stands immersed therein. 
 
 We had at first much trouble from the leakage of the tub, and we have 
 to thank Mr Gordon, the assistant at the Laboratory, for his ready help in 
 overcoming this difficulty, as well as others which were perpetually recurring. 
 The mounting of the tub was one of the last things done before the instru- 
 ment was ready for observation, and we must now return to the description 
 of the instrument itself. 
 
 We used the same pendulum bob as before, but we had its shape altered 
 so that the ends both above and below were conical surfaces, whilst the 
 central part was left cylindrical. The upper plug with its pulley is replaced 
 by another plug bearing a short round horizontal rod, with a rounded groove 
 cut in it. The groove stands vertically over the centre of the weight, and is 
 designed for taking the wire of the bifilar suspension of the pendulum ; when 
 riding on the wire the pendulum bob hangs vertically. 
 
 Part of this upper plug consists of a short thin horizontal arm about an 
 inch long. This arm is perpendicular to the plane of the groove, and when 
 the pendulum is in position, projects northwards. Through the end of the 
 arm is bored a fine vertical hole. This part of the apparatus is for the 
 modified form of disturber, which we are now using. 
 
 The support for the pendulum consists of a stout copper tube 2| inches 
 in diameter inside measure, and it just admits the pendulum bob with 
 ^th inch play all round. The tube is 3 feet 6 inches in height, and is 
 closed at the lower end by a diaphragm, pierced in the centre by a round 
 hole, about inch in diameter. The upper end has a ring of brass soldered 
 on to it, and this ring has a flange to it. The upper part of the brass 
 ring forms a short continuation f of an inch in length of the copper tube. 
 
1881] DESCRIPTION OF MODIFIED INSTRUMENT. 403 
 
 The ring is only introduced as a means of fastening the flange to the 
 copper tube. 
 
 The upper edge of the brass continuation has three V notches in it at 
 120 apart on the circumference of the ring. A brass cap like the lid of a 
 pill-box has an inside measure \ inch greater than the outside measure of 
 the brass ring. The brass cap has three rods which project inwards from 
 its circumference, and which are placed at 120 apart thereon. When the 
 cap is placed on the brass continuation of the upper tube, the three rods 
 rest in the three V notches, and the cap is geometrically fixed with respect 
 to the tube. A fine screw works through the centre of the cap, and actuates 
 an apparatus, not easy to explain without drawings, by which the cap can be 
 slightly tilted in one azimuth. The object of tilting the cap is to enable us 
 to sensitise the instrument by bringing the silk fibres attached to the mirror 
 into close proximity. 
 
 Into the cap are soldered the two ends of a fine brass wire ; the junctures 
 are equidistant from the centre of the cap and on opposite sides of it ; they 
 lie on that diameter of the cap which is perpendicular to the axis about 
 which the tilting can be produced. 
 
 When the pendulum is hung on the brass wire loop by the groove in the 
 upper plug, the wires just clear the sides of the copper tube. 
 
 It is clear that the tilting of the cap is mechanically equivalent to a 
 shortening of one side of the wire loop and the lengthening of the other. 
 Hence the pendulum is susceptible of a small lateral adjustment by means 
 of the screw in the cap. 
 
 To the bottom of the tube is soldered a second stout brass ring ; this ring 
 bears on it three stout brass legs inclined at 120 to one another, all lying in 
 a plane perpendicular to the copper tube. From the extremity of each leg 
 to the centre of the tube is 8| inches. The last inch of each leg is hollowed 
 out on its under surface into the form of a radial V groove. 
 
 There are three detached short pieces of brass tube, each ending below 
 in a flange with three knobs on it, and at the upper end in a screw with a 
 rounded head. These three serve as feet for the instrument. These three 
 feet are placed on the upper surface of our basement stone at 120 apart, 
 estimated from the centre of the stone. The copper tube with its legs 
 attached is set down so that the inverted V grooves in the legs rest on the 
 rounded screw-head at the tops of the three feet, and each of the feet rests 
 on its three knobs on the stone. The bottom of the copper tube is thus 
 raised 5| inches above the stone. By this arrangement the copper tube is 
 retained in position with reference to the stone, and it will be observed that 
 no part of the apparatus is under any constraint except such as is just 
 necessary to determine its position geometrically. 
 
 262 
 
404 DESCRIPTION OF MODIFIED INSTRUMENT. [13 
 
 The screws with rounded heads which form the three feet are susceptible 
 of small adjustments in height, and one of the three heads is capable of more 
 delicate adjustment, for it is actuated by a fine screw, which is driven by 
 a toothed wheel and pinion. The pinion is turned by a wooden rod, made 
 flexible by the insertion of a Hook's joint, and the wooden rod reaches to the 
 top of the tub, when it is mounted surrounding the instrument. 
 
 The adjustable leg is to the N. of the instrument, and as the mirror 
 faces S. we call it the ' back-leg.' When the copper support is mounted on 
 its three legs, a rough adjustment for the verticality of the tube is made 
 with two of the legs, and final adjustment is made by the back-leg. 
 
 It is obvious that if the back-leg be raised or depressed the point of 
 the pendulum is carried southwards or northwards, and the mirror turns 
 accordingly. Thus the back-leg with its screw and rod affords the means 
 of centralising the mirror. The arrangements for suspending the mirror 
 must now be described. 
 
 The lower plug in the pendulum bob is rounded and has a small hori- 
 zontal hole through it. When the pendulum is hung this rounded plug 
 just appears through the hole in the diaphragm at the bottom of the 
 copper tube. 
 
 A small brass box, shaped like a disk, can be screwed on to the bottom 
 of the copper tube, in such a way that a diameter of the box forms a 
 straight line with the axis of the copper tube. One side of the box is 
 of plate glass, and when it is fastened in position the plate glass faces 
 to the S. This is the mirror-box ; it is of such a size as to permit the 
 mirror to swing about 15 in either direction from parallelism with the 
 plate-glass front. 
 
 The fixed support for the second fibre for the bifilar suspension of the 
 mirror may be described as a very small inverted retort-stand. The vertical 
 rod projects downwards from the underside of the diaphragm, a little to the 
 E. of the hole in the diaphragm ; and a small horizontal arm projects from 
 this rod, and is of such a length that its extremity reaches to near the centre 
 of the hole. This arm has a small eyelet-hole pierced through a projection 
 at its extremity. 
 
 The mirror itself is a little larger than a shilling and is of thin plate 
 glass; it has two holes drilled through the edge at about 60 from one 
 another. The mirror was silvered on both sides, and then dipped into 
 melted paraffin; the paraffin and silver were then cleaned off one side. 
 The paraffin protects the silver from tarnishing, and the silver film seen 
 through the glass has been found to remain perfectly bright for months, 
 after having been immersed in fluid during that time. A piece of platinum 
 wire about y^^th of an inch in diameter is threaded twice through each 
 hole in opposite directions, in such a manner that with a continuous piece 
 
1881] DESCRIPTION OF MODIFIED INSTRUMENT. 405 
 
 of wire (formed by tying the two ends together) a pair of short loops are 
 formed at the edge of the mirror, over each of the two holes. When the 
 mirror is hung from a silk fibre passing through both loops, the weight of 
 the mirror is sufficient to pull each loop taut. 
 
 A single silk fibre was threaded through the eyelet-hole at the end of 
 the blunt point of the pendulum bob, and tied in such a way that there was 
 no loose end projecting so as to foul the other side of the bifilar suspension. 
 The other end of the silk fibre was knotted to a piece of sewing silk on 
 which a needle was threaded. 
 
 The pendulum was then hung from the cap by its wire loop, outside 
 the copper tube, and the silk fibre with the sewing silk and needle 
 attached dangled down at the bottom. The cap, with the pendulum 
 attached thereto, was then hauled up and carefully let down into the 
 copper tube. The sewing silk, fibre, and blunt end came out through 
 the hole in the diaphragm. 
 
 We then sewed with the needle through the two loops on the margin 
 of the mirror, and then through the eyelet-hole in the little horizontal arm. 
 The silk was pulled taut, and the end fastened off on to the little vertical 
 rod, from which the horizontal arm projects. 
 
 The mirror then hangs with one part of the silk attached to the pendulum 
 bob and the other to the horizontal arm. 
 
 The two parts of the silk are inclined to one another at a considerable 
 angle, so that the free period of the mirror is short, but the upper parts of 
 the silk stand very close to one another. The mirror-box encloses the mirror 
 and makes the copper tube watertight. 
 
 There is another part of the apparatus which has not yet been explained, 
 namely, the disturber. This part of the instrument was in reality arranged 
 before the mirror was hung. 
 
 We shall not give a full account of the disturber, because it does not 
 seem to work very satisfactorily. 
 
 In the form of disturber which we now use the variation of horizontal 
 thrust is produced by variation in the length of the disturbing pendulum, 
 instead of by variation of the point of support as in the previous experiment. 
 It is not easy to vary the point of support when the pendulum is hung in 
 a tube which nearly fits it. 
 
 The disturber weight is a small lump of copper, and it hangs by fine 
 sewing silk. The silk is threaded through the eyelet in the horizontal arm 
 which forms part of the upper plug of the pendulum ; thus the disturber 
 weight is to the N. of the pendulum. The silk after passing between the 
 wires supporting the pendulum has its other end attached to the cap at the 
 top at a point to the S. of the centre of the cap. Thus the silk is slightly 
 
406 DESCRIPTION OF MODIFIED INSTRUMENT. [13 
 
 inclined to the plane through the wires. The arrangement for varying the 
 length of the disturbing pendulum will not be explained in detail, but it 
 may suffice to say that it is produced by a third weight, which we call the 
 'guide weight/ which may be hauled up or let down in an approximately 
 vertical line. This guide weight determines by its position how much of 
 the upper part of the silk of the disturber shall be cut off, so as not to form 
 a part of the free cord by which the disturbing weight hangs. 
 
 The guide weight may be raised or lowered by cords which pass through 
 the cap. If the apparatus were to work properly a given amount of displace- 
 ment of the guide weight should produce a calculable horizontal thrust on 
 the pendulum. The whole of the arrangements for the disturber could be 
 made outside the copper tube, so that the pendulum was lowered into the 
 tube with the disturber attached thereto. 
 
 After the mirror was hung and the mirror-box screwed on, a brass cap 
 was fixed by screws on to the flange at the top of the copper tube. This 
 cap has a tube or chimney attached to it, the top of which rises five inches 
 above the top of the cap or lid from which the pendulum hangs. From 
 this chimney emerges a rod attached to the screw by which the sensitising 
 apparatus is actuated, and also the silk by which the guide weight is raised 
 or depressed. 
 
 The copper tube, with its appendages, was then filled with a boiled 
 mixture of filtered water and spirits of wine by means of a small tap 
 in the back of the mirror-box. The mixture was made by taking equal 
 volumes of the two fluids; the boiling to which it was subjected will of 
 course have somewhat disturbed the proportions. Poiseuille has shown* 
 that a mixture of spirits and water has much greater viscosity than either 
 pure spirits or pure water. When the mixture is by weight in the propor- 
 tion of about seven of water to nine of spirits, the viscosity is nearly three 
 times as great as that of pure spirits or of pure water. As the specific 
 gravity of spirits is about '8, it follows that the mixture is to be made by 
 taking equal volumes of the two fluids. It was on account of this remarkable 
 fact that we chose this mixture in which to suspend the pendulum, and we 
 observed that the unsteadiness of the mirror was markedly less than when 
 the fluid used was simply water. 
 
 The level of the fluid stood in our tubular support quite up to the top 
 of the chimney, and thus the highest point of the pendulum itself was 
 5 inches below the surface. 
 
 The tub was then let down over the instrument, and the weights hooked 
 on to its edge. The plate-glass window in the tub stood on the S. opposite 
 to the mirror-box. The tub was filled with water up to nearly the top of 
 
 * Poggendorjf's Annalen, 1843, Vol. LVIII., p. 437. 
 
1881] SENSITIVENESS OF THE INSTRUMENT. 407 
 
 the chimney, and the ditch round the stone basement was also ultimately 
 filled with water. The whole instrument thus stood immersed from top to 
 bottom in water. 
 
 Even before the tub was filled we thought that we noticed a diminution 
 of unsteadiness in the image of a slit reflected from the mirror. The filling 
 of the tub exercised quite a striking effect in the increase of steadiness, and 
 the water in the ditch again operated favourably. 
 
 We met with much difficulty at first in preventing serious leakage of 
 the tub, and as it is still not absolutely watertight, we have arranged a 
 water-pipe to drip about once a minute into the tub. A small overflow 
 pipe from the tub to the ditch allows a very slow dripping to go into 
 the ditch, and thus both vessels are kept full to a constant level. We 
 had to take this course because we found that a rise of the water in the 
 ditch through half an inch produced a deflection of the pendulum. The 
 ditch, it must be remembered, was a little broader on the S. side than 
 elsewhere. 
 
 In May, 1881, we took a series of observations with the light, slit and 
 scale. The scale was about 7 feet from the tub, and in order to read it we 
 found it convenient to kneel behind the scale on the ground. I was one 
 day watching the light for nearly ten minutes, and being tired with kneeling 
 on the pavement, I supported part of my weight on my hands a few inches in 
 front of the scale. The place where my hands came was on the bare earth 
 from which one of the paving stones had been removed. I was surprised 
 to find quite a large change in the reading. After several trials I found 
 that the pressure of a few pounds with one hand only was quite sufficient 
 to produce an effect. 
 
 It must be remembered that this is not a case of a small pressure 
 delivered on the bare earth at say 7 feet distance, but it is the difference 
 of effect produced by this pressure at 7 feet and 8 feet ; for of course the 
 change only consisted in the change of distribution in the weight of a small 
 portion of my body. 
 
 We have, however, since shown that even this degree of sensitiveness 
 may be exceeded. 
 
 We had thought all along that it would ultimately be necessary to take 
 our observations from outside the room, but this observation impressed it 
 on us more than ever; for it would be impossible for an observer always 
 to stand in exactly the same position for taking readings, and my brother 
 and I could not take a set of readings together on account of the difference 
 between our weights. 
 
 In making preliminary arrangements for reading from outside the room 
 we found the most convenient way of bringing the reflected image into 
 
408 METHOD OF READING THE DEFLECTIONS. [13 
 
 the field of view of the telescope was by shifting a weight about the room. 
 My brother stood in the room and changed his position until the image 
 was in the field of view, and afterwards placed a heavy weight where he 
 had been standing; after he had left the room the image was in the field 
 of view. 
 
 On the S.W. wall of the room there is a trap-door or window which opens 
 into another room, and we determined to read from this. 
 
 In order to read with a telescope the light has to undergo two reflections 
 and twelve refractions, besides those in the telescope ; it has also to pass 
 twice through layers of water and of the fluid mixture. In consequence of 
 the loss of light we found it impossible to read the image of an illuminated 
 scale, and we had to make the scale self-luminous. 
 
 On the pavement to the S. of the instrument is placed a flat board on 
 to which are fixed a pair of rails ; a carriage with three legs slides on these 
 rails, and can be driven to and fro by a screw of ten threads to the inch. 
 Backlash in the nut which drives the carriage is avoided by means of a 
 spiral spring. A small gas-flame is attached to the carriage ; in front of it 
 is a piece of red glass, the vertical edge of which is very distinctly visible 
 in the telescope after reflection from the mirror. The red glass was intro- 
 duced to avoid prismatic effects, which had been troublesome before. The 
 edge of the glass was found to be a more convenient object than a line 
 which had been engraved on the glass as a fiducial mark. 
 
 The gas-flame is caused to traverse by pullies driven by cords. The cords 
 come to the observing window, and can be worked from there. A second 
 telescope is erected at the window, for reading certain scales attached to 
 the traversing gear of the carriage, and we find that we can read the 
 position of the gas-flame to within a tenth of an inch, or even less, with 
 certainty. 
 
 From the gas the ray of light enters the tub and mirror-box, is reflected 
 by the mirror, and emerges by the same route ; it then meets a looking- 
 glass which reflects it nearly at right angles and a little upwards, and 
 finally enters the object-glass of the reading telescope, fixed to the sill of 
 the observing window. 
 
 When the carriage is at the right part of the scale the edge of the red 
 glass coincides with the cross wire of the reading telescope, and the reading 
 is taken by means of the scale telescope. 
 
 Arrangements had also to be made for working the sensitiser, centraliser, 
 and disturber from outside the room. 
 
 A scaffolding was erected over the tub, but free of contact therewith, 
 and this supported a system of worm-wheels, tangent-screws, and pullies 
 by which the three requisite movements could be given. The junctures 
 
1881] SENSITIVENESS OF THE INSTRUMENT. 409 
 
 with the sensitising and centralising rods were purposely made loose, because 
 it was found at first that a slight shake to the scaffolding disturbed the 
 pendulum. 
 
 The pullies on the scaffolding are driven by cords which pass to the 
 observing window. 
 
 On the window-sill we now have two telescopes, four pullies, an arrange- 
 ment, with a scale attached, for raising and depressing the guide weight, and 
 a gas tap for governing the flame in the room. 
 
 After the arrangements which have been described were completed we 
 sensitised the instrument from outside the room. The arrangements worked 
 so admirably that we could produce a quite extraordinary degree of sensi- 
 tiveness by the alternate working of the sensitising and centralising wheels, 
 without ever causing the image of the lamp to disappear from the field 
 of view. This is a great improvement on the old arrangement with the 
 stone gallows. 
 
 We now found that if one of us was in the room and stood at about 
 16 feet to the S. of the instrument with his feet about a foot apart, and 
 slowly shifted his weight from one foot to the other, then a distinct change 
 was produced in the position of the mirror. This is the most remarkable 
 proof of sensitiveness which we have yet seen, for the instrument can 
 detect the difference between the distortion of the soil caused by a weight 
 of 140 Ibs. placed at 16 feet and at 17 feet. We have not as yet taken 
 any great pains to make the instrument as sensitive as possible, and we 
 have little doubt but that we might exceed the present degree of delicacy, 
 if it were desirable to do so. 
 
 The sensitiveness now attained is, we think, only apparently greater 
 than it was with the stone gallows, and depends on the improved optical 
 arrangements, and the increase of steadiness due to the elimination of 
 changes of temperature in the support. 
 
 From July 21 to July 25 we took a series of readings. There was 
 evidence of a distinct diurnal period with a maximum about noon, when 
 the pendulum stood furthest northwards ; in the experiment with the stone 
 gallows in 1880 the maximum northern excursion took place between 
 5 and 7 P.M. 
 
 The path of the pendulum was interrupted by many minor zigzags, and 
 it would sometimes reverse its motion for nearly an hour together. During 
 the first four days the mean position of the pendulum travelled southward, 
 and the image went off the scale three times, so that we had to recentralise 
 it. In the night between the 24th and 25th it took an abrupt turn north- 
 ward, and the reading was found in the morning of the 25th at nearly the 
 opposite end of the scale. 
 
410 VALUE OF THE SCALE OF EEADINGS. [13 
 
 On the 25th the dance of the image was greater than we had seen it at 
 any time with the new instrument, so that we went into the room to see 
 whether the water had fallen in the tub and had left the top of the copper 
 tube exposed ; for on a previous occasion this had appeared to produce much 
 unsteadiness. There was, however, no change in the state of affairs. A few 
 days later the image was quite remarkable for its steadiness. 
 
 On July 25, and again on the 27th, we tried a series of observations with 
 the disturber, in order to determine the absolute value of the scale. 
 
 The guide weight being at a known altitude in the copper tube, we took 
 a series of six readings at intervals of a minute, and then shifting the guide 
 weight to another known altitude, took six more in a similar manner; and 
 so on backwards and forwards for an hour. 
 
 The first movement of the guide weight produced a considerable disturb- 
 ance of an irregular character, and the first set of readings were rejected. 
 Afterwards there was more or less concordance between the results, but it 
 was to be noticed there was a systematic difference between the change 
 from ' up ' to ' down ' and ' down ' to ' up.' This may perhaps be attributed 
 to friction between certain parts of the apparatus. We believe that on 
 another occasion we might erect the disturber under much more favourable 
 conditions, but we do not feel sure that it could ever be made to operate 
 very satisfactorily. 
 
 The series of readings before and after the change of the guide weight 
 were taken in order to determine the path of the pendulum at the critical 
 moment, but the behaviour of the pendulum is often so irregular, even 
 within a few minutes, that the discrepancy between the several results and 
 the apparent systematic error may be largely due to unknown changes, which 
 took place during the minute which necessarily elapsed between the last 
 of one set of readings and the first of the next. The image took up its 
 new position deliberately, and it was necessary to wait until it had come 
 to its normal position. 
 
 Between the first and second sets of observations with the disturber, it 
 had been necessary to enter the room and to recentralise the image. We 
 do not know whether something may not have disturbed the degree of 
 sensitiveness, but at any rate the results of the two sets of observations 
 are very discordant*. 
 
 The first set showed that one inch of movement of the gas-flame, which 
 formed the scale, corresponds with y^th of a second of arc of change of 
 horizon ; the second gave |th of a second to the inch. 
 
 As we can see a twentieth of an inch in the scale, it follows that a change 
 of horizon of about 0" - 005 should be distinctly visible. In this case the 
 
 * See, however, the postscript at the end of this part. 
 
1881] VALUE OF THE SCALE OF READINGS. 411 
 
 point of the pendulum moves through ^^^th of a millimetre. At present 
 we do not think that the disturber gives more than the order of the changes 
 of horizon which we note, but our estimate receives a general confirmation 
 from another circumstance. 
 
 From the delicacy of the gearing connected with the back-leg, we esti- 
 mate that it is by no means difficult to raise the back-leg by a millionth 
 of an inch. The looseness in the gearing was purposely kept so great that 
 it requires a turn or two of the external pulley on the window-sill before 
 the backlash is absorbed, but after this a very small fraction of a turn is 
 sufficient to move the image in the field. 
 
 We are now inclined to look to this process with the back-leg to enable 
 us to determine the actual value of our scale, but this will require a certain 
 amount of new apparatus, which we have not yet had time to arrange. In 
 erecting the instrument we omitted to take certain measurements which 
 it now appears will be necessary for the use of the back-leg as a means 
 of determining the absolute value of our scale, but we know these measure- 
 ments approximately from the working drawings of the instrument. Now 
 it appears that one complete revolution of a certain tangent-screw by which 
 the back-leg is raised should tilt the pendulum-stand through almost exactly 
 half a second of arc, and therefore this should produce a relative displace- 
 ment of the pendulum of the same amount. We have no doubt but that 
 a tenth of the turn of the tangent-screw produces quite a large deflection 
 of the image, and probably a hundredth of a turn would produce a sensible 
 deflection. Therefore, from mere consideration of the effect of the back-leg 
 we do not doubt but that a deflection of the pendulum through a ^th of 
 a second of arc is distinctly visible. This affords a kind of confirmation of 
 the somewhat unsatisfactory deductions which we draw from the operation 
 of the disturber. 
 
 Postscript. The account of our more recent experiments was written 
 during absence from Cambridge from July 29 to August 9. In this period 
 the gradual southerly progression of the pendulum bob, which was observed 
 up to July 28, seems to have continued; for on August 9 the pendulum 
 was much too far S. to permit the image of the gas-flame to come into 
 the field of view of the telescope. On August 9 the image was recen- 
 tralised, and on the 9th and 10th the southerly change continued ; on 
 the llth, however, a reversal northwards again occurred. During these 
 days the unsteadiness of the image was much greater than we have seen 
 it at any time with the new instrument. There was some heavy rain 
 and a good deal of wind at that time. We intend to arrange a scale for 
 giving a numerical value to the degree of unsteadiness, but at present it 
 is merely a matter of judgment. 
 
 It seems possible that earthquakes were the cause of unsteadiness on 
 
412 VALUE OF THE SCALE OF READINGS. [13 
 
 August 9, 10, and 11, and we shall no doubt hear whether any earthquakes 
 have taken place on those days. 
 
 After August 11 we were both again absent from Cambridge. On 
 August 16 my brother returned, and found that the southerly progression 
 of the pendulum bob had reasserted itself, so that the image was again 
 far out of the field of view. After recentralising he found the image to 
 be unusually steady. 
 
 This appeared a good opportunity of trying the effect of purely local 
 tremors. 
 
 One observer therefore went into the room and, standing near the instru- 
 ment, delivered some smart blows on the brickwork coping round the ditch, 
 the stone pavement, the tub, and the large stone basement underneath the 
 water. Little or no effect was produced by this. Very small movements 
 of the body, such as leaning forward while sitting in a chair, or a shift of 
 part of the weight from heels to toes, produced a sensible deflection, and it 
 was not very easy for the experimenter to avoid this kind of change whilst 
 delivering the blows. To show the sensitiveness of the instrument to steady 
 pressure we may mention that a pressure of three fingers on the brick coping 
 of the ditch produces a marked deflection. 
 
 On August 171 returned to Cambridge, and noted, with my brother, that 
 the image had never been nearly so steady before. The abnormal steadiness 
 continued on the 18th. There was much rain during those days. 
 
 On the afternoon of the 19th there was a high wind, and although the 
 abnormal steadiness had ceased, still the agitation of the image was rather 
 less than we usually observe it. 
 
 The image being so steady on the 17th, we thought that a good oppor- 
 tunity was afforded for testing the disturber. At 6.15 P.M. of that day we 
 began the readings. The changes from ' up ' to ' down ' were made as quickly 
 as we could, and in a quarter of an hour we secured five readings when the 
 guide weight was ' up,' and four when it was 'down.' 
 
 When a curve was drawn, with the time as abscissa, and the readings 
 as ordinates, through the ' up's,' and similarly through the ' down's,' the 
 curves presented similar features. This seems to show that movement of 
 the disturber does not cause irregularities or changes, except such as it is 
 designed to produce. 
 
 The displacement of the guide weight was through 5 cm. on each 
 occasion. 
 
 The four changes from ' up ' to ' down ' showed that an inch of scale cor- 
 responded with 0"'0897, with a mean error of 0"'0021 ; the four from 'down ' 
 to ' up ' gave 0"'0909 to the inch, with a mean error of 0"'0042. Thus the 
 systematic error on the previous occasions was probably only apparent. 
 
1881] THE WORK OF PREVIOUS OBSERVERS. 413 
 
 Including all the eight changes together, we find that the value of an 
 inch is 0"'0903 with a mean error of 0"'0030. 
 
 A change in the scale reading amounting to a tenth of an inch is 
 visible without any doubt, and even less is probably visible. Now it will 
 give an idea of the delicacy of the instrument when we say that a tenth 
 of an inch of our scale corresponds to a change of horizon* through an 
 angle equal to that subtended by an inch at 384 miles. 
 
 II. On the work of previous observers. 
 
 In the following section we propose to give an account of the various 
 experiments which have been made in order to detect small variations of 
 horizon, as far as they are known to us; but it is probable that other 
 papers of a similar kind may have escaped our notice. 
 
 In a report of this kind it is useful to have references collected together, 
 and therefore, besides giving an account of the papers which we have con- 
 sulted, we shall requote the references contained in these papers. 
 
 In Poggendorff's Annalen for 1873 there are papers by Prof. F. Zollner, 
 which had been previously read before the Royal Saxon Society, and which 
 are entitled " Ueber eine neue Methode zur Messung anziehender und 
 abstossender Krafte," Vol. 150, p. 131, " Beschreibung und Anwendung des 
 Horizon talpendels," Vol. 150, p. 134. A part of the second of these papers 
 is translated, and the figure is reproduced in the supplementary number of 
 the Philosophical Magazine for 1872, p. 491, in a paper " On the Origin of 
 the Earth's Magnetism." 
 
 The horizontal pendulum was independently invented by Prof. Zollner, 
 and, notwithstanding assertions to the contrary, was probably for the first 
 time actually realised by him ; it appears, however, that it had been twice 
 invented before. The history of the instrument contains a curious piece of 
 scientific fraud, of which we shall give an account below. 
 
 The instrument underwent some modifications under the hands of Pro- 
 fessor Zollner, and the two forms are described in the above papers. 
 
 The principle employed is as follows : There is a very stout vertical 
 stand, supported on three legs. At the top and bottom of the vertical 
 shaft are fixed two projections. Attached to each projection is a fine 
 straight steel clock spring; the springs are parallel to the vertical shaft 
 of the stand, the one attached to the lower projection running upwards, and 
 
 * We use the expression ' change of horizon ' to denote relative movement of the earth, at the 
 place of observation, and the plumb-line. Such changes may arise either from alteration in the 
 shape of the earth, or from displacement of the plumb-line ; our experiments do not determine 
 which of these two really takes place. 
 
414 ZOLLNER ON THE HORIZONTAL PENDULUM. [13 
 
 that attached to the upper one running downwards. The springs are of 
 equal length, each being equal to half the distance between their points 
 of attachment on the projections. 
 
 The springs terminate in a pair of rings, which stand exactly opposite 
 to one another, so that a rod may be thrust through both. 
 
 A glass rod has a heavy weight attached to one end of it, and the other 
 end is thrust through the two rings. The rings are a little separated from 
 one another, and the glass rod stands out horizontally, with its weight at 
 the end, and is supported by the tension of the two springs. It is obvious 
 that if the point of attachment of the upper spring were vertically over 
 that of the lower spring, and if the springs had no torsional elasticity, then 
 the glass rod would be in neutral equilibrium, and would stand equally well 
 in any azimuth. 
 
 The springs being thin have but little torsional elasticity, and Professor 
 Zollner arranges the instrument so that the one support is very nearly over 
 the other. In consequence of this the rod and weight have but a small 
 predilection for one azimuth more than another. The free oscillations of 
 the horizontal pendulum could thus be made extraordinarily slow; and even 
 a complete period of one minute could be easily attained. 
 
 A very small horizontal force of course produces a large deflection of 
 the pendulum, and a small deflection of the force of gravitation with 
 reference to the instrument must produce a like result. He considers 
 that by this instrument he could, in the first form of the instrument, 
 detect a displacement of the horizon through (T'OOOSS ; in the second his 
 estimate is 0"'001. 
 
 The observation was made by means of a mirror attached to the weight, 
 and scale and telescope. 
 
 The maximum change of level due to the moon's attraction is at 
 St Petersburg 0"'0l74, and from the sun 0"'0080 [C. A. F. Peters, Bull 
 Acad. Imp. St Petersbourg, 1844, Vol. in., No. 14] ; and thus the instrument 
 was amply sensitive enough to detect the lunar and solar disturbances of 
 gravity*. 
 
 * We are of opinion that M. Zollner has made a mistake in using at Leipzig Peters' results 
 for St Petersburg. Besides this he considers the changes of the vertical to be 0"'0174 on each 
 side of a mean position, and thus says the change is 0"-0348 altogether. Now a rough computa- 
 tion which I have made for Cambridge shows that the maximum meridional horizontal component 
 of gravitation, as due to lunar attraction, is 4-12 x 10~ 8 of pure gravity. This force will produce 
 a deflection of the plumb-line of 0" - 0085, and the total amplitude of meridional oscillation will be 
 0" - 0170. -The maximum deflection of the plumb-line occurs when the moon's hour-angle is 45 
 and 135 at the place of observation. The change at Cambridge when the moon is S.E. and 
 N.W. is 0"'0216. The deflection of the plumb-liue varies as the cosine of the latitude, and is 
 therefore greater at Cambridge than at St Petersburg. Multiplying -0216 by sec 51 43' cos 60 
 we get -0174, and thus my calculation agrees with that of Peters. 
 
1881] ZOLLNER ON THE HORIZONTAL PENDULUM. 415 
 
 Professor Zollner found, as we have done, that the readings were never 
 the same for two successive instants. The passing of trains on the railway 
 at a mile distant produced oscillations of the equilibrium position. He 
 seems to have failed to detect the laws governing the longer and wider 
 oscillations performed. Notwithstanding that he took a number of precau- 
 tions against the effects of changes of temperature, he remarks that " the 
 external circumstances under which the above experiments were carried out 
 must be characterised as extremely unfavourable for this object (measuring 
 the lunar attraction), so that the sensitiveness might be much increased in 
 pits in the ground, provided the reaction of the glowing molten interior 
 against the solid crust do not generate inequalities of the same order." 
 
 Further on he says that if the displacements of the pendulum should be 
 found not to agree in phase with the theoretical phase as given by the sun's 
 position, then it might be concluded that gravitation must take a finite time 
 to come from the sun. 
 
 It appears to me that such a result would afford strong grounds for 
 presuming the existence of frictional tides in the solid earth, and that 
 Professor Zollner's conclusion would be quite unjustifiable. 
 
 Earlier in the paper he states that he preferred to construct his 
 instrument on a large scale, in order to avoid the disturbing effects of 
 convection currents. We cannot but think, from our own experience, that 
 by this course Professor Zollner lost more than he gained, for the larger 
 the instrument the more it would necessarily be exposed in its various 
 parts to regions of different temperature, and we have found that the 
 warping of supports by inequalities of temperature is a most serious cause 
 of disturbance. 
 
 The instrument of which we have given a short account appears to us 
 very interesting from its ingenuity, and the account of the attempts to use 
 it is well worthy of attention, but we cannot think that it can ever be made 
 to give such good results as those which may perhaps be attained by our 
 plan or by others. The variation in the torsional elasticity of the suspending 
 springs, due to changes of temperature, would seem likely to produce serious 
 variations in the value of the displacements of the pendulum, and it does not 
 seem easy to suspend such an instrument in fluid in such a manner as to kill 
 out the effects of purely local tremors. 
 
 Moreover, the whole instrument is kept permanently in a condition of 
 great stress, and one would be inclined to suppose that the vertical stand 
 would be slightly warped by the variation of direction in which the tensions 
 of the springs are applied, when the pendulum bob varies its position. 
 
 In a further paper in the same volume, p. 140, "Zur Geschichte des 
 Horizontalpendels," Zollner gives the priority of invention to M. Perrot, who 
 
416 HENGLER'S SCIENTIFIC FRAUD. [13 
 
 had described a similar instrument on March 31, 1862 (Comptes Rendus, 
 Vol. 54, p. 728), but as far as he knows M. Perrot did not actually 
 construct it. 
 
 He also quotes an account of an "Astronomische Pendelwage," by 
 Lorenz Hengler, published in 1832, in Vol. 43 of Dingier s Polytechn. 
 Journ., pp. 81 92. In this paper it appears that Hengler gives the most 
 astonishing and vague accounts of the manner in which he detected the 
 lunar attraction with a horizontal pendulum, the points of support being 
 the ceiling and floor of a room 16 feet high. The terrestrial rotation was 
 also detected with a still more marvellous instrument. 
 
 Zollner obviously discredits these experiments, but hesitates to charac- 
 terise them, as they deserve, as mere fraud and invention. 
 
 The university authorities at Munich state that in the years 1830-1 
 there was a candidate in philosophy and theology named Lorenz Hengler, 
 of Reichenhofen, " der weder friiher noch spater zu linden ist." 
 
 V 
 
 At p. 150 of the same volume Professor Safafik contributes a "Beitrag 
 zur Geschichte des Horizontalpendels." He says that the instrument takes 
 its origin from Professor Gruithuisen, of Munich, whose name has " keinen 
 guten Klang" in the exact sciences. 
 
 This strange person, amongst other eccentricities, proposed to dig a 
 hole quite through the earth, and proposes a catachthonic observatory. 
 Gruithuisen says, in his Neuen Analekten fur Erd- und Himmelskunde 
 (Munich, 1832), Vol. I., Part I. : "I believe that the oscillating balance 
 (Schwung-wage) of a pupil of mine (named Hengeller), when constructed 
 on a large scale, will do the best service." 
 
 Some of the most interesting observations which have been made are 
 those of M. d'Abbadie. He gave an account of his experiments in a paper, 
 entitled " Etudes sur la verticale," Association Francaise pour I'avancement 
 des Sciences, Congres de Bordeaux, 1872, p. 159. As this work is not very 
 easily accessible to English readers, and as the paper itself has much 
 interest, we give a somewhat full abstract of it. He has also published 
 two short notes with reference to M. Plantamour's observations (noticed 
 below), in Vol. 86, p. 1528 (1878), and Vol. 89, p. 1016 (1879), of the 
 Comptes Rendus. We shall incorporate the substance of his remarks in 
 these notes in our account of the original paper. 
 
 When at Olinda, in Brazil, in 1837, M. d'Abbadie noticed the variations 
 of a delicate level which took place from day to day. At the end of the 
 two months of his stay there the changes in the E. and W. azimuth had 
 compensated themselves, and the level was in the same condition as at 
 first; but the change in the meridian was still progressing when he had 
 to leave. 
 
1881] D'ABBADIE, BOUQUET DE LA GRYE. 417 
 
 In 1842, at Gondar, in Ethiopia, and at Saqa, he noticed a similar 
 thing. In 1852 he gave an account to the French Academy (Comptes 
 Rendus, May, p. 712) of these observations, as well as of others, by means 
 of levels, which were carried out in a cellar in the old castle of Audaux, 
 Basses Pyrenees. 
 
 Leverrier, he says, speaks of sudden changes taking place in the level 
 of astronomical instruments, apparently without cause. Airy has proved 
 that the azimuth of an instrument may change, and Hough notes, in 
 America, capricious changes of the Nadir. 
 
 Henry has collected a series of levellings and azimuths observed at 
 Greenwich during ten years, and during eight of the same years at 
 Cambridge (Monthly Notices R. A. S., Vol. vin., p. 134). The results with 
 respect to these two places present a general agreement, and show that 
 from March to September the western Y of the transit instrument falls 
 through 2"'5, whilst it deviates at the same time 2" towards the north. 
 Ellis has made a comparison of curves applying to Greenwich, during 
 eight years, for level and azimuth. He shows that there is a general 
 correspondence with the curves of the external temperature (Memoirs of 
 the R. Ast. Soc., Vol. xxix., pp. 45 57). 
 
 In the later papers M. d'Abbadie says that M. Bouquet de la Grye 
 has observed similar disturbances of the vertical at Campbell Island, 
 S. lat. 52 34'. M. Bouquet used a heavy pendulum governing a vertical 
 lever, by which the angle was multiplied*. He found that the great 
 breakers on the shore at a distance of two miles caused a deviation of 
 the vertical of I"*!. On one occasion the vertical seems to have varied 
 through 3"'2 in 3 hours. 
 
 M. d'Abbadie also quotes Elkin, Yvon Villarceau, and Airy as having found, 
 from astronomical observations, notable variations in latitude, amounting to 
 from 7" to 8". 
 
 As M. d'Abbadie did not consider levels to afford a satisfactory method 
 of observation of the presumed changes of horizon, he determined to proceed 
 in a different manner. 
 
 The site of his experiments was Abbadia, in Subernoa, near Hendaye. 
 The Atlantic was 400 metres distant, and the sea-level 62 metres below 
 the place of observation. The subsoil was loamy rock (roche marneuse), 
 belonging to cretaceous deposits of the south of France. Notwithstanding 
 the steep slope of the soil, water was found at about 5 metres below the 
 surface. 
 
 * I do not find a reference to M. Bouquet in the R.S. catalogue of scientific papers. It 
 appears from what M. d'Abbadie says that certain observations have been made with pendulums 
 in Italy, but that it does not distinctly appear that the variations of level are simultaneous over 
 wide areas. No reference is given as to the observers. 
 
 D. i. 27 
 
418 D'ABBADIE'S NADIRANE. [13 
 
 In this situation he had built, in 1863, a steep concrete cone, of which 
 the external slope was ten in one (une inclinaison d'une dixieme). The 
 concrete cone is truncated, and the flat surface at the top is 2 metres in 
 diameter. It is pierced down the centre by a vertical hole or well 1 metre 
 in diameter. This well extends to within half a metre of the top, at which 
 point the concrete closes in, leaving only a hole of 12 centimetres up to the 
 flat upper surface. 
 
 From the top of the concrete down to the rock is 8 metres, and the 
 well is continued into the rock to a further depth of 2 metres: thus from 
 top to bottom is 10 metres. 
 
 A tunnel is made to the bottom of the well in order to drain away the 
 water, and access of the observer to the bottom is permitted by means of 
 an underground staircase. Access can also be obtained to a point half-way 
 between the top and bottom by means of a hole through the concrete. At 
 this point there is a diaphragm across the well, pierced by a hole 21 centi- 
 metres in diameter. The diaphragm seems to have been originally made 
 in order to support a lens, but the mode of observation was afterwards 
 changed. The diaphragm is still useful, however, for allowing the observer 
 to stand there and sweep away cobwebs. 
 
 The cone is enclosed in an external building, from the roof of which, as 
 I understand, there hangs a platform on which the observer may stand 
 without touching the cone ; and the two staircases leading up to the top 
 are also isolated*. 
 
 On the hole through the top of the cone is riveted a disk of brass 
 pierced through its centre by a circular hole 21 mm. in diameter. The 
 hole in the disk is traversed across two perpendicular diameters by fine 
 platinum wires ; at first there were only two wires, but afterwards there 
 were four, which were arranged so as to present the outline of a right- 
 angled cross. The parallel wires were very close together, so that the four 
 wires enclosed in the centre a very small square space. 
 
 At the bottom of the well is put a pool of mercury. The mercury 
 was at first in an iron basin, but the agitation of the mercury was found 
 sometimes to be so great that no reflection was visible for an hour together. 
 At the suggestion of Leverrier the iron basin was replaced by a shallow 
 wooden tray with a corrugated bottom, and a good reflection was then 
 generally obtainable. Immediately over the mercury pool there stood a 
 lens of 10 cm. diameter and 10 metres focal length, and over the brass 
 disk there stood a microscope with moveable micrometer wires in the eye- 
 piece, and a position circle. The platinum wires were illuminated, and on 
 looking through the microscope the observer saw the wires both directly and 
 
 * This passage appears to me a little obscure, and I cannot quite understand the arrange- 
 ment. 
 
1881] D'ABBADIE'S NADIRANE. 419 
 
 by reflection. The observations were taken by measuring the azimuth and 
 displacement of the image of the central square relatively to the real 
 square enclosed by the wires. 
 
 One division of the micrometer screw indicated a displacement of 
 vertical of 0"'03, so that the observations were susceptible of considerable 
 refinement. 
 
 The whole of the masonry was finished in 1863, and M. d'Abbadie then 
 allowed the structure five years to settle before he began taking observa- 
 tions. The arrangements for observing above described were made in 1868 
 and 1869. 
 
 In the course of a year he secured 2000 observations, and the results 
 appear to be very strange and capricious. 
 
 Throughout March, 1869, the perturbations of the mercury were so 
 incessant that observations (taken at that time with the iron basin) were 
 nearly impossible ; on the 29th he waited nearly an hour in vain in trying 
 to catch the image of the wires. Two days later the mercury was perfectly 
 tranquil. On April 6 it was much agitated, although the air and sea were 
 calm. A tranquil surface was a rare exception. 
 
 In 1870 the corrugated trough was substituted for the iron basin; and 
 M. d'Abbadie says : 
 
 "Cependant, ni le fond inegal du bain raine ni sa forme ne m'ont 
 empeche d'observer, ce que j'appelle des ombres fuy antes. Ce sont des 
 bandes sombres et paralleles qui traversent le champ du microscope avec 
 plus ou moins de vitesse, et qu'on explique en attribuant au mercure des 
 ondes tres tenues, causees par une oscillation du sol dans un seul sens. Le 
 plus souvent ces ombres semblent courir du S.E. au N.O., approximative- 
 ment selon 1'axe de la chaine des Pyrenees ; mais je les ai observees, le 
 15 Mars 1872, allant vers le S.O. A cette epoque, le mercure etait depuis 
 le 29 fevrier, dans une agitation continuelle, comme mon aide 1'avait constate 
 en 1869, aussi dans le mois de Mars*." 
 
 He observed also, from time to time, certain oscillations of the mercury 
 too rapid to be counted, which he calls ' tremoussements.' There were also 
 sudden jumpings of the image from one point to another, or 'fre'tillements,' 
 indicating a sudden change of vertical through 0"'49 to 0"*65. 
 
 He observed many microscopic earthquakes, and in some cases the image 
 was carried quite out of the field of view. 
 
 He also detected the difference of vertical according to the state of the 
 tide in the neighbouring sea ; but the change of level due to this cause was 
 often masked by others occurring contemporaneously. 
 
 * M. d'Abbadie writes to me that this phenomenon was ultimately found to result from air 
 currents (Nov. 5, 1881). 
 
 27 2 
 
420 PLANTAMOUR'S OBSERVATIONS WITH LEVELS. [13 
 
 From observations during the years 1867 to 1872 (with the exception 
 of 1870) he finds that in every year but one the plumb-line deviated 
 northwards during the latter months of the year, but in 1872 it deviated 
 to the south. 
 
 He does not give any theoretical views as to the causes of these 
 phenomena, but remarks that his observations tend to prove that the 
 causes of change are sometimes neither astronomical nor thermometrical. 
 
 The most sudden change which he noted was on October 27, 1872, 
 when the vertical changed by 2"'4 in six hours and a quarter. Between 
 January 30 and March 26 of the same year the plumb-line deviated 4"'5 
 towards the south. 
 
 We now come to the valuable observations of M. Plantamour, which 
 we believe are still being prosecuted by him. His papers are " Sur le 
 deplacement de la bulle des niveaux a bulle d'air," Comptes Rendus, 
 June 24, 1878, Vol. 86, p. 1522, and " Des mouvements periodiques du sol 
 accuses par des niveaux a bulle d'air," Comptes Rendus, December 1, 1879, 
 Vol. 89, p. 937. 
 
 The observations were made at Secheron, near Geneva, at first at the 
 Observatory, and afterwards at M. Plantamour's house. After some pre- 
 liminary observations, he obtained a very sensitive level and laid it on 
 the concrete floor of a room in which the variations of temperature were 
 very small. The azimuth of the level was E. and W., and the observations 
 were made every hour from 9 A.M. until midnight. Figures are given of 
 the displacement of the bubble during April 24, 25, and 26, 1878. The 
 results indicate a diurnal oscillation of level, the E. end of the level being 
 highest towards 5.30 P.M.; the amplitudes of the oscillations were 8"'4, 
 11"'2, 15"'75 during these three days. It also appeared that there was 
 a gradual rising of the mean diurnal position of the E. end during the 
 same time. 
 
 The level was then transported to a cellar in M. Plantamour's house, 
 when the temperature only varied by half a degree centigrade. The bubble 
 of the level often ran quite up to one end. A new and larger level was 
 obtained, together with the great 'chevalet de fer,' which is used by the 
 manufacturers in testing levels. Both levels were placed E. and W., at 
 about two metres apart. During May 3 and 4, 1878, the bubble travelled 
 eastward without much return, and it is interesting to learn that simul- 
 taneous observations by M. Turretini, at the Level Factory, three kilometres 
 distant, at Plainpalais, showed a similar change. 
 
 Between May 3 and 6 the level actually changed through 1 7". Up to 
 the 19th the level still showed the eastward change. 
 
 M. Plantamour remarks that the eastern pier of a transit instrument is 
 
1881] PLANTAMOUR'S OBSERVATIONS WITH LEVELS. 421 
 
 known to rise during a part of the year, but not by an amount comparable 
 with that observed by him, and that the diurnal variations are unknown. 
 
 After further observations of a similar kind, one of the levels was 
 arranged in the N. and S. azimuth. 
 
 The same sort of diurnal oscillations, although more irregular, were 
 observed, but the hours of maximum were not the same in the two 
 levels. During the four days, May 24 to 28, the maximum rising of the 
 north generally took place about noon. This is exactly the converse of 
 what we have recently observed. 
 
 In the second paper he remarks : 
 
 " Dans le sens du meridien, les mouvements diurnes sont tres rares 
 irreguliers et toujours tres faibles, le niveau en accuse parfois, quand il n'y 
 en a point de Test a 1'ouest, et inversement, quand ces derniers sont tres 
 prononces, on n'en aperyoit que tres rarement du sud au nord." 
 
 In our experiment of March 15 to 18, 1880, we found that the pendulum 
 stood furthest north about 6 P.M., so that at that time the S. was most 
 elevated; and in the short series of observations during the present summer 
 the maximum elevation of the S. took place about noon. 
 
 On October 1, 1878, M. Plantamour began a new series of observations, 
 which lasted until September 30, 1879. The levels were arranged in the 
 two azimuths as before, and the observations were taken five times a day, 
 namely, at 9 A.M., noon, 3, 6, and 9 P.M. The mean of these five readings 
 he takes as the diurnal value. 
 
 During October and November the eastern end of the level fell, which 
 is exactly the converse of what happened during the spring of the 
 same year; he concludes that the eastern end falls when the external 
 temperature falls. 
 
 When a curve of the external temperature was placed parallel with 
 that for the level, it appeared that there was a parallelism between the 
 two, but the curve for the level lagged behind that for temperature by 
 a period of from one to four days. 
 
 This parallelism was maintained until the end of June, 1879, when it 
 became disturbed. From then until the beginning of September the E. 
 rose, but in a much greater proportion than the rise of mean temperature. 
 It must be noted that July was a cold and wet month. 
 
 Although the external temperature began to fall on August 5, the E. 
 end continued to rise until September 8. This he attributes to an accumu- 
 lation of heat in the soil. The total amplitude of the annual oscillation 
 from E. to W. amounted to 28"'08. 
 
422 NYR^N'S OBSERVATIONS AT PULKOVA. [13 
 
 There was also a diurnal oscillation in this azimuth which amounted to 
 3"'2 on September 5. The east end appeared to be highest between 6 and 
 7.45 P.M., and lowest at the similar hour in the morning*. 
 
 The meridional oscillations were much smaller, the total annual amplitude 
 being only 4"'89. From December 23, 1878, until the end of April, 1879, 
 there was a correspondence between the external temperature curve and 
 that for N. and S. level. We have already quoted the remark on the diurnal 
 meridional oscillations. 
 
 M. Plantamour tells us that in 1856 Admiral Mouchez detected no 
 movement of the soil by means of the levels attached to astronomical 
 instruments. On the other hand, M. Hirsch established, by several years 
 of observation at Neuchatel, that there was an annual oscillation of a transit 
 instrument from E. to W., with an amplitude of 23", and an azimuthal 
 oscillation of 75". Similar observations with the transit instrument were 
 made at the observatory at Berne in the summer of 1879. 
 
 It is to be regretted that M. Plantamour does not give us more informa- 
 tion concerning the manner in which the iron support for the levels was 
 protected from small changes of temperature, nor with regard to the effect 
 of the observer's weight on the floor of the room. We have concluded that 
 both these sources of disturbance should be carefully eliminated. 
 
 Some interesting observations were made at Pulkova on a subject 
 cognate to that on which we are writing. M. Magnus Nyren contributed, 
 on February 28, 1878, an interesting note to the Imperial Academy of 
 St Petersburg, entitled " Erderschiitterung beobachtet an einem feinem 
 Niveau 1877 Mai 10+." On May 10 (April 28), 1877, at 4.16 A.M., a 
 striking disturbance of the level on the axis of the transit was observed 
 by M. Nyren in the observatory at Pulkova. The oscillations were watched 
 by him for three minutes; their complete period was about 20 seconds, 
 and their amplitude between 1"'5 and 2". At 4.35 A.M. there was no 
 longer any disturbance. He draws attention to the fact that it afterwards 
 appeared that one hour and fourteen minutes earlier there had been a 
 great earthquake at Iquique. The distance from Iquique to Pulkova is 
 10,600 kilometres in a straight line, and 12,540 kilometres along the arc 
 of a great circle. He does not positively connect the two phenomena 
 together; but he observes that if the wave came through the earth 
 from Iquique to Pulkova it must have travelled at the rate of about 
 
 * It seems that M. Plantamour sent a figure to the French Academy with the paper, but no 
 figure is given. This figure would doubtless have explained the meaning of some passages which 
 are somewhat obscure. Thus he speaks of the minimum occurring between 6 and 7.45, but it is 
 not clear whether minimum means E. highest or E. lowest. I interpret the passage as above, 
 because this was the state of things in the observations recorded in the first of the two papers. 
 There is a similar difficulty about the meridional oscillations. 
 
 t Bull. Acad. St Petersb., Vol. xxiv., p. 567. 
 
1881] OBSERVATIONS AT PULKOVA. 423 
 
 2 '4 kilometres per second. This is the speed of transmission through 
 platinum or silver. 
 
 M. Nyren thinks the wave-motion could not have been so regular as 
 it was, if the transmission had been through the solid, and suggests that 
 the transmission was through the fluid interior of the earth. 
 
 It appears to us that this argument is hardly sound, and that it would 
 be more just to conclude that the interior of the earth was a sensibly 
 perfectly elastic solid ; because oscillations in molten rock would surely be 
 more quickly killed out by internal friction than those in a solid. How- 
 ever, M. Nyren does not lay much stress on this argument. He also 
 draws attention to the fact that on September 20 (8), 1867, M. Wagner 
 observed at Pulkova an oscillation of the level, with an amplitude of 3", 
 and that seven minutes before the disturbance there had been an earth- 
 quake at Malta. On April 4 (March 23), 1868, M. Gromadzki observed an 
 agitation of the level, and it was afterwards found that there had been 
 an earthquake in Turkestan five minutes before. 
 
 Similar observations of disturbances had been made twice before, once 
 by M. Wagner and once by M. Romberg ; but they had not been connected 
 with any earthquakes at least with certainty. 
 
 Dr C. W. Siemens has invented an instrument of extraordinary delicacy, 
 which he calls an "Attraction-meter." An account of the instrument is 
 given in an addendum to his paper " On determining the depth of the sea 
 without the use of the sounding-line" (Phil. Trans., 1876, p. 659). We 
 shall not give any account of this instrument, because Dr Siemens is a 
 member of our committee, and will doubtless bring any observations he may 
 make with it before the British Association at some future time. 
 
 III. Remarks on the present state of the subject. 
 
 Although our experiments are not yet concluded, it may be well to make 
 a few remarks on the present aspects of the question, and to state shortly our 
 intentions as to future operations. 
 
 Our experiments, as far as they go, confirm the results of MM. d'Abbadie 
 and Plantamour, and we think that there can remain little doubt that the 
 surface of the earth is in incessant movement, with oscillations of periods 
 extending from a fraction of a second to a year. 
 
 Whether it be a purely superficial phenomenon or not, this consideration 
 should be of importance to astronomical observers, for their instruments are 
 necessarily placed at the surface of the earth. M. Plantamour and others 
 have shown that there is an intimate connection between the changes of 
 level and those of the temperature of the air; whence it follows that the 
 
424 VARIATIONS IN PIERS OF TRANSIT INSTRUMENTS. [13 
 
 principal part of the changes must be superficial. On the other hand, 
 M. d'Abbadie has shown that it is impossible to explain all the changes by 
 means of changes of temperature. It would be interesting to determine 
 whether changes of a similar kind penetrate to the bottom of mines, and 
 Gruithuisen's suggestion of a catachthonic observatory seems worthy of 
 attention, although he perhaps went rather far in the proposition that the 
 observatory should be ten or fifteen miles below the earth's surface. 
 
 It may appear not improbable that the surface of the soil becomes 
 wrinkled all over, when it is swollen by increase of temperature and by 
 rainfall. If this, however, were the case, then we should expect that instru- 
 ments erected at a short distance apart would show discordant results. 
 M. Plantamour, however, found that, at least during three days, there was 
 a nearly perfect accordance between the behaviour of two sets of levels at 
 three kilometres apart ; and during eight years there appeared to be general 
 agreement between the changes of level of the astronomical instruments at 
 Greenwich and Cambridge. It would be a matter of much interest to 
 determine how far this concordance would be maintained if the instrument 
 of observation had been as delicate as that used by M. d'Abbadie or as our 
 pendulum. 
 
 M. Plantamour speaks as though it were generally recognised that one 
 pier of a transit circle rises during one part of the year and falls at 
 another*. But if this be so throughout Europe, we must suppose that 
 there is a kind of tide in the solid earth, produced by climatic changes ; 
 the rise and fall of the central parts of continents must then amount to 
 something considerable in vertical height, and the changes of level on the 
 easterly and westerly coasts of a continent must be exactly opposite to one 
 another. We are not aware that any comparison of this kind has been 
 undertaken. The idea seems of course exceedingly improbable, but we 
 understand it to be alleged that it is the eastern pier of transit instru- 
 ments in Europe which rises during the warmer part of the year. Now 
 if this be generally true for Europe, which has no easterly coast, it is not 
 easy to see how the change can be brought about except by a swelling of 
 the whole continent. 
 
 We suggest that in the future it will be thought necessary to erect at 
 each station a delicate instrument for the continuous observation of changes 
 of level. Perhaps M. d'Abbadie's pool of mercury might be best for the 
 longer inequalities, and something like our pendulum for the shorter ones ; 
 
 * " Dans 1' operation au moyen de laquelle on verifie I'horizontalit6 de 1'axe d'une lunette 
 meridienne, il parait qu'on remarque bien un leger mouvement d'exhaussement de Test pendant 
 une partie de 1'annee, mais il n'est pas aussi considerable que celui qu'accuse mon niveau, et 1'on 
 n'a jamais remarque, que je sache, une oscillation diurne comme celle qu'a indiqu6e le niveau 
 dans le pavillou." Comptes Eendus, June 24, 1878, Vol. LXXXVI., p. 1525. 
 
1881] HORACE DARWIN'S EXPERIMENT. 425 
 
 or possibly the pendulum, when used in a manner which we intend to try, 
 might suffice for all the inequalities. 
 
 At present the errors introduced by unknown inequalities of level are 
 probably nearly eliminated by the number of observations taken; but it 
 could not fail to diminish the probable error of each observation if a correc- 
 tion were applied for this cause of disturbance from hour to hour, or even 
 from minute to minute. If the changes noted by M. Plantamour are not 
 entirely abnormal in amount, such corrections are certainly sufficient to 
 merit attention. 
 
 In our first set of experiments we found that stone piers are exceedingly 
 sensitive to changes of temperature and to small stresses. Might it not be 
 worth while to plate the piers of astronomical instruments with copper, and 
 to swathe them with flannel ? We are not aware as to the extent to 
 which care is taken as to the drainage of the soil round the piers, or as 
 to the effect of the weight of the observer's body; but we draw attention 
 to the effect produced by the percolation of water round the basement, and 
 to the impossibility we have found of taking our observations in the same 
 room with the instrument. 
 
 In connection with this subject we may notice an experiment which 
 was begun 3^ years ago by my brother Horace. The experiment was 
 undertaken in connection with my father's investigation of the geological 
 activity of earthworms, and the object was to determine the rate at which 
 stones are being buried in the ground in consequence of the excavations 
 of worms. 
 
 The experiment is going on at Down, in Kent. The soil is stiff red 
 clay, containing many flints lying over the chalk. There are two stout 
 metal rods, one of iron and the other of copper. The ends were sharpened 
 and they were hammered down vertically into the soil of an old grass field, 
 and they are in contact with one another, or nearly so. When they had 
 penetrated 8 feet 6 inches it was found very difficult to force them deeper, 
 and it is probable that the ends are resting on a flint. The ends were 
 then cut off about three inches above the ground. 
 
 A stone was obtained like a small grindstone, with a circular hole in 
 the middle. This stone was laid on the ground with the two metal rods 
 appearing through the hole. Three brass V grooves are leaded into the 
 upper surface of the stone, and a moveable tripod-stand with three rounded 
 legs can be placed on the stone, and is, of course, geometrically fixed by 
 the nature of its contact with the Vs. An arrangement with a micrometer 
 screw enables the observer to take contact measurements of the position 
 of the upper surface of the stone with regard to the rods. The stone 
 has always continued to fall, but during the first few months the rate of 
 fall was probably influenced by the decaying of the grass underneath it. 
 The general falling of the stone can only be gathered from observations 
 
426 VARIATION OF LATITUDE. [13 
 
 taken at many months apart, for it is found to be in a state of continual 
 vertical oscillation. 
 
 The measurements are so delicate that the raising of the stone produced 
 by one or two cans full of water poured on the ground can easily be per- 
 ceived. Between September 7 and 19, 1880, there was heavy rain, and 
 the stone stood 1'91 mm. higher at the latter date than at the former. 
 The effect of frost and the wet season combined is still more marked, for 
 on January 23, 1881, the stone was 4'12 mm. higher than it had been on 
 September 7, 1880. 
 
 The prolonged drought of the present summer has had a great effect, 
 for between May, 8 and June 29 the stone sank through 5'79 mm. The 
 opposite .effects of drought and frost are well shown by the fact that on 
 January 23 the stone stood 8'62 mm. higher than on June 29, 1881. The 
 observations are uncorrected for the effect of temperature on the metal 
 rods, but the fact that the readings from the two rods of different metals 
 always agree very closely inter se, shows that such a correction would amount 
 to very little. 
 
 The changes produced in the height of the stone are, of course, entirely 
 due to superficial causes ; but the amounts of the oscillations are certainly 
 surprising, and although the basements of astronomical instruments may be 
 very deep, they cannot entirely escape from similar oscillations*. 
 
 In his address to the mathematical section at the meeting of the British 
 Association at Glasgow in 1876, Sir William Thomson tells usf that Peters, 
 Maxwell, Nyren, and NewcombJ have examined the observations at Pulkova, 
 Greenwich, and Washington, in order to discover whether there is not an 
 inequality in the latitude of the observatories having a period of about 
 306 days. Such an inequality must exist on account of the motion in that 
 period of the instantaneous axis of rotation of the earth round the axis of 
 maximum moment of inertia. The inequality was detected in the results, 
 but the probable error was very large, and the epochs deduced by the 
 several investigators do not agree inter se. It remains, therefore, quite 
 uncertain whether the detection of the inequality is a reality or not. But 
 now we ask whether it is not an essential first step in such an enquiry 
 to make an elaborate investigation by a very delicate instrument of the 
 systematic changes of vertical at each station of observation ? 
 
 We will next attempt to analyse the merits and demerits of the various 
 methods which have been employed for detecting small changes in the 
 vertical. 
 
 * [An account of this experiment is given in Proc. Roy. Soc., Vol. LXVIII., 1901, pp. 253 261.] 
 
 t B. A. Report for 1876, p. 10. For " Nysen " read " Nyren." 
 
 I Peters' paper is in Bull. St Pet. Acad., 1844, p. 305, and Ast. Nach., Vol. xxn., 1845, pp. 71, 
 103, 119. Nyren's paper is in Mem. St Pet. Acad., Vol. xix., 1873, No. 13. With regard to 
 Maxwell, see Thomson and Tait's Nat. Phil., 2nd edit., Part i., Vol. i. An interesting letter 
 from Newcornb is quoted in Sir W. Thomson's address. 
 
1881] ON THE SEVERAL ADVANTAGES OF VARIOUS INSTRUMENTS. 427 
 
 The most sensitive instrument is probably the horizontal pendulum of 
 Professor Zollner, and its refinement might be almost indefinitely increased 
 by the addition of the bifilar suspension of a mirror as a means of exhibiting 
 the displacements of the pendulum bob. If this were done it might be 
 possible to construct the instrument on a very small scale and yet to 
 retain a very high degree of sensitiveness. We are inclined to think, 
 however, that the variation of the torsional elasticity of the suspending 
 springs under varying temperature presents an objection to the instrument 
 which it would be very difficult to remove. The state of stress under 
 which the instrument is of necessity permanently retained seems likely 
 to be prejudicial. 
 
 Next in order of sensitiveness is probably our own pendulum, embodying 
 the suggestion of Sir William Thomson. We are scarcely in a position as 
 yet to feel sure as to its merits, but it certainly seems to be capable of 
 all the requisite refinement. We shall give below the ideas which our 
 experience, up to the present time, suggest as to improvements and future 
 observations. 
 
 Although we know none of the details of M. Bouquet de la Grye's pendu- 
 lum actuating a lever, it may be presumed to be susceptible of considerable 
 delicacy, and it would be likely to possess the enormous advantage of giving 
 an automatic record of its behaviour. On the other hand the lever must 
 introduce a very unfavourable element in the friction between solids. 
 
 M. d'Abbadie's method of observation by means of the pool of mercury 
 seems on the whole to be the best which has been employed hitherto. But 
 it has faults which leave ample fields for the use of other instruments. The 
 construction of a well of the requisite depth must necessarily be very expen- 
 sive, and when the structure is made of a sufficient size to give the required 
 degree of accuracy, it is difficult to ensure the relative immobility of the 
 cross-wires and the bottom of the well. 
 
 Levels are exceedingly good from the point of view of cheapness and 
 transportability, but the observations must always be open to some doubt 
 on account of the possibility of the sticking of the bubble from the effects 
 of capillarity. The justice of this criticism is confirmed by the fact that 
 M. Plantamour found that two levels only two metres apart did not give 
 perfectly accordant results. Levels are moreover, perhaps, scarcely sensitive 
 enough for an examination of the smaller oscillations of level. Dr Siemens' 
 form of level possesses ample sensibility, but is probably open to the same 
 objections on the score of capillarity. 
 
 In the case of our own experiments we think that the immersion of 
 the whole instrument in water from top to bottom has proved an excellent 
 precaution against the effects of change of temperature, and our experience 
 leads us to think that much of the agitation of the pendulum in the earlier 
 
428 TREMORS OF THE SOIL OBSERVED AT GREENWICH. [13 
 
 set of experiments was due to small variations of temperature against which 
 we are now guarded. 
 
 The sensitiveness of the instrument leaves nothing to be desired, and 
 were such a thing as a firm foundation attainable, we could measure the 
 horizontal component of the lunar attraction to a considerable degree of 
 accuracy. We believe that this is the first instrument in which the 
 viscosity of fluids has been used as a means of eliminating the effects of 
 local tremors. In this respect we have been successful, for we find that 
 jumping or stamping in the room itself produces no agitation of the pendu- 
 lum, or at least none of which we can feel quite sure. We are inclined 
 to try the effect of fluids of greater viscosity, such as glycerine, syrup of 
 sugar, or paraffin oil. But along with such fluids we shall almost inevitably 
 introduce air-bubbles, which it may be hard to get rid of. If a fluid of 
 great viscosity were used, we should then only observe the oscillations of 
 level of periods extending over perhaps a quarter to half a minute. The 
 oscillations of shorter periods are, however, so inextricably mixed up with 
 those produced by carriages and railway trains, that nothing would be 
 lost by this. 
 
 In connection with this point Mr Christie writes to me, that "In the 
 old times of Greenwich Fair, some twenty years ago, when crowds of people 
 used to run down the hill, I find the observers could not take reflection 
 observations for two or three hours after the crowd had been turned 
 
 out We do not have anything like such crowds now, even on Bank 
 
 holidays, and I have not heard lately of any interference with the obser- 
 vations." If the observers attributed the agitation of the mercury to the 
 true cause, the elasticity of the soil must be far more perfect than is 
 generally supposed. It would be surprising to find a mass of glass or steel 
 continuing to vibrate for as long as two hours after the disturbance was 
 removed. May it not be suspected that times of agitation, such as those 
 noted by M. d'Abbadie, happened to coincide on two or three occasions 
 with Greenwich Fair ? 
 
 As the sensitiveness of our present instrument is very great, although 
 the sensitising process has never been pushed as far as possible, we think 
 that it will be advantageous to construct an instrument on half, or even 
 less than half, the present scale. The heavy weights which we now have 
 to employ will thus be reduced to one-eighth of the present amount. The 
 erection of the instrument may thus be made an easy matter, and an easily 
 portable and inexpensive instrument may be obtained. 
 
 Our present form of instrument has several serious flaws. The image is 
 continually travelling off the scale, the gearing both internal and external 
 to the room for observing is necessarily complex and troublesome to erect, 
 and lastly we have not yet succeeded in an accurate determination of the 
 value of the scale. 
 
1881] PLANS FOR MODIFYING THE BIFILAR PENDULUM. 429 
 
 We are in hopes of being able to overcome all these objections. We 
 propose to have a fixed light, which may be cast into the room from the 
 outside. This will free us from the obviously objectionable plan of having 
 a gas-flame in the room, and at the same time will abolish the gearing for 
 traversing the lamp on the scale. We should then abolish the disturbing 
 pendulum and thus greatly simplify the instrument. The readings would 
 be taken by the elevation or depression of the back-leg, until the image of 
 the fixed light was brought to the cross-wire of the observing telescope. 
 
 The ease with which the image may be governed with our present 
 arrangements leads us to be hopeful of the proposed plan. The use of the 
 back-leg will, of course, give all the displacements in absolute measure. 
 
 The only gearings which it will be necessary to bring outside the room 
 will be those for sensitising and for working the back-leg. The sensitising 
 gearing, when once in order, will not have to be touched again. 
 
 The objections to this plan are, that it is necessary to bring one of the 
 supports of the instrument under very slight stresses, and that it will not 
 be possible to take readings at small intervals of time, especially if a more 
 viscous fluid be used*. 
 
 Our intention is to proceed with our observations with the present instru- 
 ment for some time longer, and to note whether the general behaviour of the 
 pendulum has any intimate connection with the meteorological conditions. 
 We intend to observe whether there is a connection between the degree of 
 agitation of the pendulum and the occurrence of magnetic storms. M. Zollner 
 has thrown out a suggestion for this sort of observation, but we find no notice 
 of his having acted on it^f* 
 
 We shall also test how far the operation by means of the back-leg may be 
 made to satisfy our expectations. 
 
 We have no hope of being able to observe the lunar attraction in the 
 present site of observation, but we think it possible that we may devise 
 a portable instrument, which shall be amply sensitive enough for such a 
 purpose, if the bottom of a deep mine should be found to give a sufficiently 
 invariable support for the instrument. 
 
 The reader will understand that it is not easy to do justice to an 
 incomplete apparatus, or to give a very satisfactory account of experiments 
 still in progress; but as it is now two years since the Committee was 
 appointed, we have thought it best to give to the British Association such 
 an account as we can of our progress. 
 
 * [Mr Horace Darwin has designed a new form of bifilar pendulum, in which the mirror 
 itself is the bob of the pendulum. Such an instrument, with continuous photographic record, 
 has been used at Birmingham and at Edinburgh. See Committee on Earth Tremors, B.A. Reports 
 for 1893 and 1894.] 
 
 t Phil. Mag., Dec. 1872, p. 497. 
 
14 
 
 THE LUNAR DISTURBANCE OF GRAVITY; VARIATIONS IN 
 THE VERTICAL DUE TO ELASTICITY OF THE EARTH'S 
 SURFACE. 
 
 [Second Report of the Committee, consisting of Mr G. H. 
 DARWIN, Professor Sir WILLIAM THOMSON, Professor TAIT, 
 Professor GRANT, Dr SIEMENS, Professor PURSER, Professor 
 G. FORBES, and Mr HORACE DARWIN, appointed for the 
 Measurement of the Lunar Disturbance of Gravity. Written 
 by Mr G. H. DARWIN. British Association Report for 1882, 
 pp. 95119.] 
 
 SHORTLY after the meeting of the British Association last year (1881), 
 the instrument with which my brother and I were experimenting at the 
 Cavendish Laboratory, at Cambridge, broke down, through the snapping 
 of the wire which supported the pendulum. A succession of unforeseen 
 circumstances have prevented us, up to the present time, from resuming 
 our experiments. 
 
 The body of the present Report, therefore, will merely contain an account 
 of such observations by other observers as have come to our knowledge 
 within the past year, and it must be taken as supplementary to the second 
 part of the Report for 1881. The Appendix, however, contains certain 
 theoretical investigations, which appear to me to throw doubt on the utility 
 of very minute gravitational observations. 
 
 The readers of the Report for 1881 will remember that, in the course 
 of our experiments, we were led away from the primary object of the 
 Committee, namely, the measurement of the Lunar Disturbance of Gravity, 
 and found ourselves compelled to investigate the slower oscillations of the 
 soil. 
 
1882] SEISMOLOGICAL OBSERVATIONS IN ITALY. 431 
 
 It would be beyond the scope of the present Report to enter on the 
 literature of seismology. But, the slower changes in the vertical having 
 been found to be intimately connected with earthquakes, it would not have 
 been possible, even if desirable, to eliminate all reference to seismology from 
 the present Report. 
 
 The papers which are quoted below present evidence of a very mis- 
 cellaneous character, and therefore this Report must necessarily be rather 
 disjointed. It has seemed best in our account of work done rather to 
 classify together the observers than the subjects. This rule will, however, 
 be occasionally departed from, when it may seem desirable to do so. 
 
 The interesting researches in this field made during the last ten years 
 by the Italians, are, I believe, but little known in this country, and as the 
 accounts of their investigations are not easily accessible (there being, for 
 example, no copy of the Bulletino, referred to below, at Cambridge), it 
 will be well to give a tolerably full account of the results attained. I have 
 myself only seen the Transactions for four years. 
 
 The great extension which these investigations have attained in Italy 
 has been no doubt due to the fact of the presence of active volcanos and 
 of frequent sensible earthquakes in that country. But it is probable that 
 many of the same phenomena occur in all countries. 
 
 In 1874 the publication of the Bulletino del Vulcanismo Italia.no was 
 commenced at Rome under the editorship of Professor S. M. de Rossi, of 
 Rome*. As the title of this publication shows, it is principally occupied 
 with accounts of earthquakes, but the extracts made will refer almost entirely 
 to the slower oscillations of level. 
 
 I learn from the Bulletino that in 1873 Professor Timoteo Bertelli, 
 of Florence, had published an historical account of small spontaneous 
 movements of the pendulum, observed since the seventeenth century up 
 to that timef. 
 
 In 1874 (Anno 1 of the Bulletino) Rossi draws attention to the fact 
 that there are periods lasting from a few days to a week or more, in which 
 the soil is in incessant movement, followed by a comparative cessation of such 
 movement. This he calls a ' seismic period.' In the midst or at the end of 
 a seismic period there is frequently a sensible earthquake. 
 
 At page 51 he remarks, in a review of some observations of Professor 
 Pietro Monte (Director of the Observatory of Leghorn), that he was led 
 to suspect that the crust of the earth is in continuous and slow movement 
 
 * I am compelled to make this abstract from manuscript notes, but my papers having become 
 somewhat disarranged, I am not absolutely certain in one or two places of the year to which the 
 observations refer. 
 
 t Bulletino Boncampayni, t. vi., Gennaio, 1873. Reprinted Via Lata, No. 2114, Borne. 
 
432 SEISMOLOGICAL INVESTIGATIONS IN ITALY. [14 
 
 during the seismic period, and that this movement is influenced by varia- 
 tions of barometric pressure. This suspicion was, he says, confirmed by 
 finding, in his observations of a pendulum at Rocca di Papa (of which 
 we shall speak again below), that during the seismic period the excursions 
 of the pendulum were mostly in the S.W. and N.E. azimuth. This is 
 perpendicular to the volcanic fracture, which runs towards the Alban lake 
 and the sea. The lips of the. fracture rise and fall, and there result two sets 
 of waves along and perpendicular to the fracture. In an earthquake these 
 waves are propagated with great velocity (the phenomenon being in fact 
 dynamical), but during the seismic period the same class of changes takes 
 place slowly. This view accords with observations at Velletri made by 
 Professor D. G. Galli. 
 
 With regard to the influence of barometric pressure Rossi elsewhere 
 quotes M. Poey (October 15, 1857 ?) as having attributed the deviations 
 of the vertical to this cause, and remarks : 
 
 " Although he (Poey) gave too much weight to the baro-seismic action 
 of large variations of atmospheric pressure, yet after very numerous obser- 
 vations made by me in these last three years (I suppose 1871-4), I can 
 affirm that no marked barometric depression has occurred without having 
 been immediately preceded, accompanied, or followed by marked micro- 
 seismic movements; but besides these there are other irregular, often 
 considerable and instantaneous movements, which occur under high pressure. 
 To distinguish them, I have called the first baro-seismic, and the second 
 vulcano-seismic, movements." The reader will find a theoretical investiga- 
 tion on this subject in the Appendix to the present Report. 
 
 Rossi states (page 118, Anno 1 ?) that whilst Etna was in a condition 
 of activity his pendulums at Rocca di Papa were extraordinarily agitated 
 at the beginning of each barometric storm. 
 
 At page 90 of the second year are given graphical illustrations of the 
 simultaneous deflections of pendulums at Rome, Rocca di Papa, Florence, 
 Leghorn, and Bologna. There is some appearance of concordance between 
 them, and this shows that the agitations sometimes affect considerable tracts 
 of land, but that the minor deflections are purely local phenomena. 
 
 M. d'Abbadie, in presenting a memoir on micro-seismic movements by 
 Father Bertelli to the French Academy, relates (Comptes Rendus, 1875, 
 Vol. 81, p. 297) the following experiment made by Count Malvasia, as proving 
 the independence of the disturbances of the pendulum from the tremors 
 produced by traffic. Two batteries of artillery were marching through 
 Bologna, and it was arranged that at 30 metres from the Palazzo Malvasia 
 they should break into a trot. The pendulum, situated only 6 metres from 
 the street, was observed to be unaffected by this, and continued its oscilla- 
 tions in the E.W. azimuth. A pool of mercury was violently agitated, and it 
 
1882] SEISMOLOGICAL INVESTIGATIONS IN ITALY. 433 
 
 was concluded that the motion communicated to the ground by the artillery 
 was exclusively vertical. 
 
 At page 5 of the Bulletino for 1876 (January to May), Rossi writes 
 a " Guida pratica per le osservazioni sismiche." This article contains a 
 description of the instruments which have been used by the Italian 
 observers. 
 
 Bertelli used a pendulum protected from the air, with a microscope 
 and micrometer for evaluating the oscillations. The upper part of the 
 support of the pendulum consisted of a spiral spring, so that vertical 
 movements of the ground could be recorded. This instrument he calls a 
 tromo-seismometer. 
 
 Professor Egidi, of Anagni, proposed to use the reflection from mercury. 
 The object observed was to be a mark fixed on a wall, and the reflected 
 image of the mark was to be observed with a telescope. The deviation of 
 the vertical was to be evaluated by noting the amount of movement required 
 to bring the cross-wires of the telescope on to the mark. This instrument 
 has not, I think, the advantages of M. d'Abbadie's, because the light was 
 incident at about 45 on the mercury, and thus the mark and telescope were 
 remote from one another ; whereas in the arrangement of M. d'Abbadie the 
 mark and microscope are close together, and only a micrometer wire in the 
 microscope is movable. 
 
 Cavalleri used ten pendulums of graduated length, and found that some- 
 times one of the pendulums was agitated and sometimes another. Rossi 
 observed the same with his pendulums at Rocca di Papa. It thus appears 
 that the free period of oscillation of the pendulum is a disturbing element. 
 
 In order to obviate the discrepancies which must arise in the use of 
 various kinds of pendulums for simultaneous observations in different places, 
 Bertelli and Rossi propose a normal 'tromometer/ of which a drawing is 
 given. The length of the pendulum is 1^ metres, the weight 100 grammes, 
 and it makes forty-nine free oscillations in a minute. To the bottom of the 
 pendulum is attached a horizontal disk, on the underside of which are 
 engraved two fine lines at right-angles to one another. These lines are 
 observed, after total internal reflection in a glass prism placed immediately 
 below the disk, by a horizontal microscope, furnished with a micrometer. 
 The azimuth of the deflection of the vertical is observed by a position- 
 circle. 
 
 This paper also contains a description of the author's observatory at 
 Rocca di Papa. It is established in a cave at 700 metres above the sea, on 
 the external slope of the extinct Latian volcano. There is a large central 
 pendulum hanging from the roof, and there are four others with different 
 weights and lengths hanging in tubes cut in the native rock. Only the ends 
 of these pendulums are visible, and they are protected by glass at the visible 
 D. i. 28 
 
434 SEISMOLOGICAL INVESTIGATIONS IN ITALY. [14 
 
 parts. A great part of this paper is occupied with descriptions of seismo- 
 meters, and this is outside the scope of the present Report. 
 
 In presenting a pamphlet by Father Bertelli, entitled " Riassunto delle 
 osservazioni microsismiche, &c.," to the French Academy (Comptes Rendus, 
 1877, Vol. 84, p. 465), M. d'Abbadie summarises Bertelli's conclusions 
 somewhat as follows: 
 
 The oscillation of the pendulum is generally parallel to valleys or chains 
 of mountains in the neighbourhood. The oscillations are independent of 
 local tremors, velocity and direction of wind, rain, change of temperature 
 and atmospheric electricity. 
 
 Pendulums of different lengths betray the movements of the soil in 
 different manners, according to the agreement or disagreement of their 
 free-periods with the period of the terrestrial vibrations. 
 
 The disturbances are not strictly simultaneous in the different towns of 
 Italy, but succeed one another at short intervals. 
 
 After earthquakes the ' tromornetric ' or microseismic movements are 
 especially apt to be in a vertical direction. They are always so when the 
 earthquake is local, but the vertical movements are sometimes absent when 
 the shock occurs elsewhere. Sometimes there is no movement at all, even 
 when the shock occurs quite close at hand. 
 
 The positions of the sun and moon appear to have some influence on the 
 movements of the pendulum, but the disturbances are especially frequent 
 when the barometer is low. 
 
 The curves of ' the monthly means of the tromornetric movement ' exhibit 
 the same forms in the various towns of Italy, even those which are distant 
 from one another. 
 
 The maximum of disturbance occurs near the winter solstice and the 
 minimum near the summer solstice ; this agrees with Mallet's results about 
 earthquakes. 
 
 At Florence a period of earthquakes is presaged by the magnitude and 
 frequency of pendulous movements in a vertical direction. These movements 
 are observable at intervals and during several hours after each shock. 
 
 At page 103 of the first part of the Bulletino for 1878 ?, there is a 
 review of a work by Giulio Grablovitz, " Dell' attrazione luni-solare in rela- 
 zione coi fenomeni mareo-sismici," Milano, Tipografia degli Ingegneri, 1877. 
 
 In this work it appears that M. Grablovitz attributes a considerable part 
 of the deviations of the vertical to bodily tides in the earth, but as he 
 apparently enters into no computations to show the competency of this cause 
 to produce the observed effects, it does not seem necessary to make any 
 further comment on his views. 
 
SELSMOLOGICAL INVESTIGATIONS IN ITALY. 435 
 
 At page 99 of the volume for September December, 1878, Rossi writes 
 on the use of the microphone for the purpose of observing earthquakes 
 ("II microfono nella meteorologia endogena"). He begins by giving an 
 account of a correspondence, beginning in 1875, between himself and Count 
 Giovanni Mocenigo*, of Vicenza, who seems to have been very near to the 
 discovery of the microphone. When the invention of the microphone was 
 announced, Mocenigo and Armellini adopted it for their experiments, and. 
 came to the conclusion that the mysterious noises which they heard arose 
 from minute earthquakes or microsisms. 
 
 Rossi then determined to undertake observations in his cavern at Rocca 
 di Papa, with a microphone, made of silver instead of carbon, mounted on a 
 stone beam. The sensitiveness of the instrument could be regulated, and he 
 found that it was not much influenced by external noises. 
 
 The instrument was placed 20 metres underground, and remote from houses 
 and carriage-roads. It was protected against insects, and was wrapped up in 
 wool. Carpet was spread on the floor of the cave to deaden the noise from 
 particles of stone which might possibly fall. Having established his micro- 
 phone, he waited till night and then heard noises which he says revealed 
 'natural telluric phenomena.' The sounds which he heard he describes as 
 ' roarings, explosions occurring isolated or in volleys, and metallic or bell-like 
 sounds ' [fremiti, scopii isolati o di moschetteria, e suoni metallici o di 
 campana]. They all occurred mixed indiscriminately, and rose to maxima at 
 irregular intervals. By artificial means he was able to cause noises which he 
 calls ' rumbling (?) or crackling ' [rullo o crepito]. The roaring [fremito] was 
 the only noise which he could reproduce artificially, and then only for a 
 moment. It was done by rubbing together the conducting wires, in the 
 same manner as the rocks must rub against one another when there is an 
 earthquake. 
 
 A mine having been exploded in a quarry at some distance, the tremors 
 in the earth were audible in the microphone for some seconds subsequently. 
 
 There was some degree of coincidence between the agitation of the 
 pendulum-seismograph and the noises heard with the microphone. 
 
 At a time when Vesuvius became active, Rocca di Papa was agitated by 
 microsisms, and the shocks were found to be accompanied by the very same 
 microphonic noises as before. The noises sometimes became 'intolerably 
 loud ' ; on one occasion in the middle of the night, half an hour before a 
 sensible earthquake. The agitation of the microphone corresponded exactly 
 with the activity of Vesuvius. 
 
 Rossi then transported his microphone to Palmieri's Vesuvian observatory, 
 
 * Count Mocenigo has recently published at Vicenza a book on his observations. It is 
 reviewed in Nature for July 6, 1882. 
 
 28-2 
 
436 SEISMOLOGICAL INVESTIGATIONS IN ITALY. [14 
 
 and worked in conjunction with him. He there found that each class of 
 shock had its corresponding noise. The sussultorial shocks, in which I con- 
 ceive the movement of the ground is vertically up and down, gave the volleys 
 of musketry [i colpi di moschetteria], and the undulatory shocks gave the 
 roarings [i fremiti]. The two classes of noises were sometimes mixed up 
 together. 
 
 Rossi makes the following remarks : " On Vesuvius I was put in the way 
 of discovering that the simple fall and rise in the ticking which occurs with 
 the microphone [battito del orologio unito al microfono] (a phenomenon 
 observed by all, and remaining inexplicable to all) is a consequence of the 
 vibration of the ground." This passage alone might perhaps lead one to sup- 
 pose that clockwork was included in the circuit; but that this was not the 
 case, and that ' ticking ' is merely a mode of representing a natural noise, is 
 proved by the fact that he subsequently says that he considers the ticking to 
 be ' a telluric phenomenon.' 
 
 Rossi then took the microphone to the Solfatara of Pozzuoli, and here, 
 although no sensible tremors were felt, the noises were so loud as to be heard 
 simultaneously by all the people in the room. The ticking was quite masked 
 by other natural noises. The noises at the Solfatara were imitated by 
 placing the microphone on a vessel of boiling water. Other seismic noises 
 were then imitated by placing the microphone on a marble slab, and 
 scratching and tapping the under surface of it. 
 
 The observations on Vesuvius led him to the conclusion that the earth- 
 quake oscillations have sometimes fixed nodes and loops, for there were places 
 on the mountain where no effects were observed. Hence, as he remarks, 
 although there may sometimes be considerable agitation in an earthquake, 
 the true centre of disturbance may be very distant. 
 
 In conclusion Rossi gives a description of a good method of making a 
 microphone. A common nail has a short piece of copper wire wound round 
 it, and the other end of the wire is wound round a fixed metallic support 
 The nail thus stands at the end of a weak horizontal spring ; but the nail is 
 arranged so that it stands inclined to the horizon, instead of being vertical. 
 The point of the nail is then put to rest on the middle of the back of a silver 
 watch, which lies flat on a slab. The two electrodes are the handle of the 
 watch and the metallic support. He says that this is as good as any instru- 
 ment. The telephone is a seismological instrument, and therefore, strictly 
 speaking, beyond the scope of this Report ; but as some details of its use 
 have already been given, I will here quote portions of an interesting letter by 
 Mr John Milne, of the Imperial Engineering College of Tokio, which appeared 
 in Nature for June 8, 1882. Mr Milne writes : 
 
 " In order to determine the presence of these earth-tremors, at the end 
 of 1879 I commenced a series of experiments with a variety of apparatus, 
 
1882] SEISMOLOGICAL INVESTIGATIONS IN JAPAN. 437 
 
 amongst which were microphones and sets of pendulum apparatus, very 
 similar in general arrangement, but, unfortunately, not in refinement of 
 construction, to the arrangements now being used in the Cavendish 
 Laboratory. 
 
 " The microphones were screwed on to the heads of stakes driven in the 
 ground, at the bottom of boxed-in pits. In order to be certain that the 
 records which these microphones gave were not due to local actions, such as 
 birds or insects, two distinct sets of apparatus were used, one being in the 
 middle of the lawn in the front of my house, and the other in a pit at the 
 back of the house. The sensitiveness of these may be learnt from the fact 
 that if a small pebble was dropped on the grass within six feet of the pit, a 
 distinct sound was heard in the telephone, and a swing produced in the needle 
 of the galvanometer placed in connection with these microphones. A person 
 running or walking in the neighbourhood of the pits, had each of his steps so 
 definitely recorded, that a Japanese neighbour, Mr Masato, who assisted me 
 in the experiments, caused the swinging needle of his galvanometer to close 
 an electric circuit and ring a bell, which, it is needless to say, would alarm a 
 household. In the contrivance we have a hint as to how earth-tremors may 
 be employed as thief-detectors. 
 
 " The pendulum apparatus, one of which consisted of a 20-lb. bob of lead 
 at the end of 20 feet of pianoforte wire provided with small galvanometer 
 mirrors, and bifilar suspensions were also used in pairs. With this apparatus 
 a motion of the bob relatively to the earth was magnified 1000 times, that is 
 to say, if the spot of light which was reflected from the mirror moved a dis- 
 tance equal to the thickness of a sixpence, this indicated there had been a 
 relative motion of the bob to the extent of 1000th part that amount. 
 
 "The great evil which everyone has to contend with in Japan when 
 working with delicate apparatus is the actual earthquakes, which stop or 
 alter the rate of ordinary clocks. 
 
 " Another evil which had to be contended with was the wind, which shook 
 the house in which my pendulums were supported, and I imagine the ground 
 by the motion of some neighbouring trees. A shower of rain also was not 
 without its effects upon the microphones. After many months of tiresome 
 observation, and eliminating all motions which by any possibility have been 
 produced by local influence, the general result obtained was that there 
 were movements to be detected every day and sometimes many times 
 per day. . . . 
 
 " A great assistance to the interpretation of the various records which an 
 earthquake gives us on our seismographs is what I may call a barricade of 
 post-cards. At the present moment Yedo is barricaded, all the towns around 
 for a distance of 100 miles being provided with post-cards. Everyone of 
 them is posted with a statement of the shocks which have been felt. 
 
438 INVESTIGATIONS AT CAMPBELL ISLAND. [14 
 
 " For the months of October and November it was found from the records 
 of the post-cards that nearly all the shocks came from the north and passed 
 Yedo to the south-west. When coming in contact with a high range of 
 mountains, they were suddenly stopped, as was inferred from the fact that 
 the towns beyond this range did not perceive that an earthquake had occurred. 
 This fact having been obtained, the barricade of post-cards has been extended 
 to towns lying still farther north. The result of this has been that several 
 earthquake origins have, so to speak, been surrounded or corralled, whilst 
 others have been traced as far as the seashore. For the latter shocks, earth- 
 quake hunting with post-cards has had to cease, and we have solely to rely 
 upon our instruments. Having obtained our earthquake centres, at one or 
 more of these our tremor instruments might be erected, and it would soon be 
 known whether an observation of earth-tremors would tell us about the com- 
 ing of an earthquake as the cracklings of a bending do about its approaching 
 breakage. To render these experiments more complete, and to determine 
 the existence of a terrain tide, a gravitimeter might be established. I men- 
 tion this because if terrain tides exist, and they are sufficiently great from a 
 geological point of view, it would seem that they might be more pronounced 
 and therefore easier to measure in a country like Japan, resting in a heated 
 and perhaps plastic bed, than in a country like England, where volcanic 
 activity has so long ceased, and the rocks are, comparatively speaking, cold 
 and rigid, if an instrument, sufficiently delicate to detect differences in the 
 force of gravity, in consequence of our being lifted farther from the centre of 
 the earth every time by the terrain tide as it passed between (sic) our feet, 
 could be established in conjunction with the experiments on earth-tremors." 
 
 The only account which I have been able to find of M. Bouquet do la 
 Grye's observations (mentioned in the last report) is contained in the Comptes 
 Rendus for March 22, 1875, page 725. M. Bouquet writes : 
 
 "... The observation of the levels of our meridian telescopes put us on 
 the track of a curious fact. Not only is Campbell Island subject to earth- 
 quakes, but it also exhibits movements when the great swell falls in breakers 
 on the coast. I thought that it would be interesting to study this new 
 phenomenon. The instrument, which was quickly put together, consisted of 
 a steel wire supporting a weight, to which was soldered a needle ; the move- 
 ments of the weight were amplified 240 times by means of a lever; by 
 passing an electric current through this multiplying pendulum, which was 
 terminated at the bottom by a small cup of amalgamated tin, regular oscilla- 
 tions of T <^joth f a mm - cou ld be registered. I propose to repeat these 
 observations with a pendulum of much larger amplifying power, so as to try 
 to register the variations of the plumb-line." 
 
 In a letter to me, M. d'Abbadie mentions an attempt by Brunner to 
 improve M. Bouquet de la Grye's apparatus, but considers that the attempt 
 was a failure. 
 
1882] 
 
 D'ABBADIE'S NADIRANE. 
 
 439 
 
 He also tells me that Delaunay directed M. Wolf to devise an apparatus 
 for detecting small deviations of the vertical, and that the latter, without 
 M. d'Abbadie's knowledge, adopted his rejected idea of a pendulum, about 
 30 metres long, bearing a prism at the end by reflection from which a scale 
 was to be read by means of a distant small refractor. The pendulum was 
 actually set up, but the wire went on twisting and untwisting until Delaunay's 
 death, and no observations were made with it. Our own experience is enough 
 to show that nothing could have been made of such an instrument. 
 
 M. d'Abbadie gives further explanations of a passage in his own paper 
 about the arrangement of the staircases for access to and observation with 
 his Nadirane. In writing the Report of 1881 I had found the description of 
 the arrangements difficult to understand. 
 
 The woodcut below is a copy of the rough diagram that he sent me. 
 There were three staircases : 
 
 T cut in the rock ; CB to ascend from the cellar-flags CD ; and, lastly, AS 
 to mount from the boarded ground floor, AB, to the small floor SN, which 
 was hung from the roof. The two upper staircases did not touch the trun- 
 cated cone of concrete anywhere. 
 
 FIG. 1. 
 
 Judging from this figure, I imagine that the concrete cone has an 
 external slope of ten in one; the French expression was 'une inclinaison 
 d'une dixieme.' 
 
 M. d'Abbadie informs me that the apparently curious phenomenon of the 
 'ombres fuyantes,' which were observed in the reflection from the pool of 
 
440 D'ABBADIE'S NADIRANE. [14 
 
 mercury, to which we drew attention last year, was of no significance. It 
 arose from the currents of air caused by a candle left standing on the stair- 
 case T cut in the rock. The light was required for pouring out the mercury, 
 and it was left burning whilst the observation was being taken; but now that 
 this operation is done entirely from above, the phenomenon has disappeared. 
 
 In a paper entitled " Recherches sur la Verticale " (Ann. de la Soc. Sclent. 
 de Bruxelles, 1881), M. d'Abbadie continues the account of his observations 
 with his instrument, called by him a Nadirane. It was described in the last 
 year's Report, and some further details have been given above. A portion of 
 this paper refers to his old observations, and gives further important details 
 as to the exact method of making observations, and of various modifications 
 which have been introduced. 
 
 Each complete observation consists of the following processes: measure- 
 ment of the distance between the cross-wires and their image, (1) in the 
 meridian, (2) in the prime vertical, (3) in the N.W. azimuth, (4) observation 
 of barometer, (5) of thermometer, (6) of direction and force of the wind, 
 
 (7) condition and movement of the image estimated with the micrometer, 
 
 (8) condition of the heavens, (9) of the breakers called ' les Criquets,' which 
 can be observed from the neighbouring room. 
 
 This last is to determine whether it is possible to have a rough sea with a 
 calm image ; a condition which has not hitherto been observed. This state- 
 ment seems somewhat contradictory of the following : 
 
 " Aucunes des variations dans les circonstances concomitantcs n'a paru se 
 rattacher a 1'etat de 1'image qui, pendant des journees entieres, parait tantot 
 belle, tantdt faible, et parfois meme disparait entierement, bien que ce dernier 
 inconvenient ait ete evite en grande partie par 1'usage d'un recipient en bois 
 a fond raine pour contenir le mercure." I presume we are to understand that 
 the roughness of the sea and the badness of the image is the only congruence 
 hitherto observed. 
 
 M. d'Abbadie's observations on the effect of the tides will be referred to 
 in the Appendix to this Report. He then discusses the various causes which 
 may perhaps influence the vertical. 
 
 The variations of air temperature are insufficient, because the vertical has 
 been seen to vary 2"'4 in six hours. If the effects are to be attributed to 
 variations in the temperature of the rock, it would be necessary to suppose 
 that that temperature varies discontinuously, which it is difficult to admit. 
 
 If it be supposed that the changes take place in the instrument itself, the 
 like must be true of astronomical instruments. And there is no reason to 
 admit the reality of such strange variations. 
 
 Another cause, more convenient because more vague, is variation of a 
 chemical or mechanical nature in the crust of the earth. But if this be so, 
 
1882] D'ABBADIE'S NADIRANE. 441 
 
 why does the vertical ever return to its primitive position ? Another cause 
 may be variation in the position of the earth's axis of rotation. 
 
 The azimuthal variations in astronomical instruments, referred to by 
 M. d'Abbadie (see a paper by Mr Henry, Vol. vm. p. 134, Month. Not. R.A.S.), 
 are difficult to explain without having recourse to such variation in the axis 
 of rotation. 
 
 He also tells us that Ellis (Vol. xxix. 1861, p. 45, Mem. R.A.S.) has dis- 
 cussed the Greenwich observations from 1851 to 1858. A comparison of the 
 results obtained from two neighbouring meridian instruments seemed to 
 show that the azimuthal variations are partly purely instrumental. 
 
 M. d'Abbadie's paper contains diagrams illustrating the variations of the 
 vertical observed with the Nadirane during nearly two years. He sums up 
 the results as follows : 
 
 " En resume le maximum d'ecart du sud au nord entre le fil et son image 
 a ete egal a 49 U< 2 (this is 15"'94; it seems as though this should be twice the 
 deviation of the vertical) le 30 Novembre a 8 h. 43 m. du matin. Ce meme 
 jour, a 7h. 28m., on a lu 40 VV> 1, chiffre porte ici au tableau, et 37 V1 '6 seulement 
 a 1 h. 32 m. du soir. Dans 1'espace de six heures la verticale a done varie de 
 2 VV> 5 ou 0" - 81 (as this is the deviation of the image, should not the deviation 
 of the vertical be half as much ?). Le minimum de 1'annee, ou 3'06, fut 
 atteint le 19 Janvier a 3h. 3m. du matin, ainsi que le 21 du meme mois a 
 midi, bien qu'on eut observe 3*44 et 3'30 dans les matinees de ces deux jours, 
 ainsi qu'on le voit au tableau ci-apres .... Pendant 1'annee entiere la verti- 
 cale, consideree selon le plan du meridien, a done varie d'un angle de 12 VV> 45 
 ou 4"'034 .... On aura . . . . 8 XV> 3 ou 2"7 pour la plus grande variation dans 
 le sens Est-Ouest ou Ton nivelle les tourillons des lunettes meridiennes." 
 
 Towards the end M. d'Abbadie makes the excellent remark, that in dis- 
 cussing latitudes and declinations of stars, account should be taken of the 
 instantaneous position of the vertical at the moment of taking the observa- 
 tion. 
 
 In the Archives des Sciences, 1881, Vol. V. p. 97, M. P. Plantamour 
 continues the account of his observations on oscillations of the soil at Secheron, 
 near Geneva. The account of the earlier observations, which we quoted from 
 the Comptes Rendus in our previous Report, is also contained in Vol. II. 
 of the Archives, p. 641. The paper to which we are now referring contains a 
 graphical reproduction of the previous series of observations, as far as concerns 
 the daily means. 
 
 The new series extends from October 1, 1879, to December 31, 1880, the 
 disposition of the levels being the same as was described in our last Report. 
 The observations were taken at 9 A.M. and 6 P.M., which hours are respectively 
 a little before the diurnal minimum and maximum. The meanings of the 
 terms maximum and minimum were somewhat obscure in the Goinptes Rendus, 
 
442 PLANTAMOUR'S OBSERVATIONS WITH LEVELS. [14 
 
 but I now find that the right interpretation was placed on M. Plantarnour's 
 words, for maximum means for the two levels E. end highest and S. end 
 highest. 
 
 The N.S. level seems to have behaved very similarly in the two years of 
 observation ; the total annual amplitudes in the two years being 4"'89 and 
 4"'56 respectively. In both years this level followed, with some retardation, 
 the curve of external temperature, except between April and October, when 
 the curves appear to be inverted. The E.W. level behaved very differently 
 in the two years. In 1879 the E. end began to fall rapidly at the end of 
 November, and continued to fall until December 26, when the reading was 
 88"'7l; it rose a little early in January and then fell again, so that on 
 January 28, 1880, the reading was - 89"'95. The amplitude of the total fall 
 (viz. from October 4, 1879, to January 28, 1880) was 95"'80. In the preced- 
 ing year the amplitude was only 28"'08. The E. end has never recovered its 
 primitive position, and remains nearly 80" below its point of departure. 
 
 It is difficult to believe that so enormous a variation of level is normal, 
 and one is tempted to suspect that there is some systematic error in his mode 
 of observation. If such oscillations as these were to take place in an astro- 
 nomical observatory, accurate astronomical observations would be almost 
 impossible. 
 
 I have seen nothing which shows that M. Plantamour takes any special 
 precaution with regard to the weight of the observer's body, nor is it ex- 
 pressly stated that the observer always stands in exactly the same position, 
 although, of course, it is probable that this is the case. It would be interest- 
 ing, also, to learn whether any precautions have been taken for equalising the 
 temperature of the level itself. To hold the hand in the neighbourhood of a 
 delicate level is sufficient to quite alter the reading. In one of his letters to 
 me M. d'Abbadie also remarks on the slow molecular changes in glass, which 
 render levels untrustworthy for comparisons at considerable intervals of time. 
 Although we must admire M. Plantamour's indomitable perseverance, it is to 
 be regretted that his mode of observation is by means of levels ; and we are 
 compelled to regard, at least provisionally, these enormous changes of level 
 either as a local phenomenon, or as due to systematic error in his mode of 
 observation. 
 
 In the Report for 1881 we referred to some observations by Admiral 
 Mouchez, made in 1856, on changes of level. A short paper by Admiral 
 Mouchez on these observations will be found in the Comptes Rendus for 1878, 
 Vol. 87, p. 665. I now find that the observations were, in fact, discussed by 
 M. Gaillot, in a paper entitled " Sur la direction de la verticale a 1'observa- 
 toire de Paris," at p. 684 of the same volume. The paper consists of the 
 examination of 1077 determinations of latitude, made between 1856 and 1861, 
 with the Gambey circle. 
 
1882] CHANGES OF LEVEL AT THE OBSERVATORY OF PARIS. 443 
 
 M. Gaillot concludes that the variation from year to year is accidental, 
 and that the variation of latitude in the course of the year is represented by 
 
 ^n"on 
 = + Q 20sm 
 
 [360 (-95)1 
 [ 365-25 
 
 where t is the number of days since January 1. 
 
 By a comparison of day and night observations he concludes that there is 
 no trace of a diurnal variation. On this we may remark that, if the maximum 
 and minimum occur at 6 P.M. and 6 A.M. (which is, roughly speaking, what we 
 found to be the case), then the diurnal oscillation must necessarily disappear 
 by this method of treatment. 
 
 Individual observations ranged from 2"'48 above to 3"' 17 below the mean. 
 On this he remarks : 
 
 " Ceux qui savent combien 1'observation du nadir presente parfois de diffi- 
 culte dans un observatoire situe au milieu d'une grande ville, .... ceux-la ne 
 trouveront pas ces ecarts exageres, et ne croiront nullement avoir besoin de 
 faire intervenir une deviation de la vertical e pour les expliquer." 
 
 M. Gaillot concludes by remarks adverse to any sensible deviations of the 
 vertical. 
 
 It seems to me, however, that in the passage about the influence of the 
 traffic of a great town, M. Gaillot begs the whole question by setting down to 
 that disturbing influence all remarkable deviations of the vertical. Our obser- 
 vations, and those of many others, are entirely adverse to such a conclusion. 
 
 M. d'Abbadie, in a letter to me, also expresses himself as to the inconclu- 
 sivencss of M. Gaillot's discussion. 
 
 He also tells me that M. Tisserand, in his observations of latitude in 
 Japan, found variations amounting to nearly 7" ; and when asked " How he 
 could be so much in error," answered " That he was sure of his observations 
 and calculations, but could not explain the cause of such variations." 
 
 The following further references may perhaps be useful: Maxwell's paper 
 on the 306-day inequality in the earth's rotation, which was mentioned in 
 the Report of last year, is in the Trans. Roy. Soc. of Edinburgh, 1857, 
 Vol. xxi. pp. 559-70. See also Bessel's Abhandlungen, Vol. II. p. 42, Vol. in. 
 p. 304. In Nature for January 12, 1882 (p. 250), there is an account of 
 the work of the Swiss Seismological Commission. The original sources 
 appear to be a text-book on Seismology by Professor Heim, of Bern, the 
 Annuaire of the Physical Society of Bern, and the Archives des Sciences of 
 Geneva. I learn from M. d'Abbadie that Colonel Orff has been making 
 systematic observations twice a day with levels at the Observatory at Munich, 
 and that Colonel Goulier has been doing the same at Paris, with levels filled 
 with bisulphide of carbon. 
 
444 [14 
 
 APPENDIX. 
 
 ON VARIATIONS IN THE VERTICAL DUE TO ELASTICITY OF THE 
 EARTH'S SURFACE. 
 
 By G. H. DARWIN. 
 
 1. On the Mechanical Effects of Barometric Pressure on the 
 Earth's Surface. 
 
 The remarks of Signore de Rossi, on the observed connection between 
 barometric storms and the disturbance of the vertical, have led me to make 
 the following investigation of the mechanical effects which are caused by 
 variations of pressure acting on an elastic surface. The results seem to show 
 that the direct measurement of the lunar disturbance of gravity must for ever 
 remain impossible*. 
 
 The practical question is to estimate the amount of distortion to which 
 the upper strata of the earth's mass are subjected, when a wave of barometric 
 depression or elevation passes over the surface. The solution of the following 
 problem should give us such an estimate. 
 
 Let an elastic solid be infinite in one direction, and be bounded in the 
 other direction by an infinite plane. Let the surface of the plane be every- 
 where acted on by normal pressures and tractions, which are expressible as a 
 simple harmonic function of distances measured in some fixed direction along 
 the plane. It is required to find the form assumed by the surface, and 
 generally the condition of internal strain. 
 
 This is clearly equivalent to the problem of finding the distortion of the 
 earth's surface produced by parallel undulations of barometric elevation arid 
 depression. It is but a slight objection to the correctness of a rough estimate 
 of the kind required, that barometric disturbances do not actually occur in 
 parallel bands, but rather in circles. And when we consider the magnitude 
 of actual terrestrial storms, it is obvious that the curvature of the earth's 
 surface may be safely neglected. 
 
 This problem is mathematically identical with that of finding the state of 
 stress produced in the earth by the weight of a series of parallel mountains. 
 The solution of this problem has recently been published in a paper by me in 
 the Philosophical Transactions (Part II. 1882, pp. 187 230t), and the solution 
 there found may be adapted to the present case in a few lines. 
 
 The problem only involves two dimensions. If the origin be taken in the 
 mean horizontal surface, which equally divides the mountains arid valleys, 
 and if the axis of z be horizontal and perpendicular to the mountain chains, 
 
 * [This prevision has now been falsified; see p. 346.] 
 t [To be reproduced in Vol. n. of these collected papers.] 
 
1882] 
 
 STATE OF STRESS AND STRAIN OF AN ELASTIC SOIL. 
 
 445 
 
 and if the axis of x be drawn vertically downwards, then the equation to the 
 mountains and valleys is supposed to be 
 
 , z 
 
 x = h cos y 
 6 
 
 so that the wave-length from crest to crest of the mountain ranges is 2?r&. 
 
 The solution may easily be found from the analysis of section 7 of the 
 paper referred to. It is as follows : 
 
 Let a, 7 be the displacements at the point x, z vertically downwards and 
 horizontally (a has here the opposite sign to the a of (44)). Let w be the 
 density of the rocks of which the mountains are composed ; g gravity ; 
 v modulus of rigidity, then 
 
 dW 
 
 ry = 
 
 ' 
 
 dW 
 
 -j- 
 
 dz 
 
 where 
 
 W = gwh e~ xlb cos 
 
 (1) 
 
 From these we have at once 
 
 = 
 
 *)-,., 
 
 z\ 
 
 7 = 
 
 gwh ., . z 
 - xe~ xlb sm r 
 6 
 
 da. 
 dz 
 
 qwh /, x\ _, . z 
 
 = -if- 1 1 + T I e sm 7 
 
 2v \ b) bt 
 
 The first of these gives the vertical displacement, the second the horizontal, 
 and the third the inclination to the horizon of strata primitively plane. 
 
 At the surface 
 
 qwh , 
 a. = ^r 6 cos 
 
 .- , 
 b 
 
 = 
 
 .(3) 
 
 da. _ gwh . z 
 dz 2v b 
 
 * It is easy to verify that these values of a and 7, together with the value p=gwhe~ x / l 'coszlb 
 for the hydrostatic pressure, satisfy all the conditions of the problem, by giving normal pressure 
 gwhcoszjb at the free surface of the infinite plane, and satisfying the equations of internal 
 equilibrium throughout the solid. I take this opportunity of remarking that the paper from 
 which this investigation is taken contains an error, inasmuch as the hydrostatic pressure is 
 erroneously determined in section 1. The term - W should be added to the pressure as deter- 
 mined in (3). This adds W to the normal stresses P, Q, R throughout the paper, but leaves the 
 difference of stresses (which was the thing to be determined) unaffected. If the reader should 
 compare the stresses, as determined from the values of a, 7 in the text above, and from the value 
 of p given in this note, with (38) of the paper referred to, he is warned to remember the missing 
 term W. 
 
 [The mistake, referred to, will be corrected in the reproduction of the paper.] 
 
446 DEFLECTION OF VERTICAL DUE TO ATTRACTION. [14 
 
 Hence the maximum vertical displacement of the surface is + gwhb/Zv, and 
 the maximum inclination of the surface to the horizon is 
 
 + cosec 1" x gwh/Zv seconds of arc 
 
 Before proceeding further I shall prove a very remarkable relation be- 
 tween the slope of the surface of an elastic horizontal plane and the deflection 
 of the plumb-line caused by the direct attraction of the weight producing 
 that slope. This relation was pointed out to me by Sir William Thomson, 
 when I told him of the investigation on which I was engaged ; but I am 
 alone responsible for the proof as here given. He writes that he finds that it 
 is not confined simply to the case where the solid is incompressible, but in 
 this paper it will only be proved for that case. 
 
 Let there be positive and negative matter distributed over the horizontal 
 plane according to the law wh cos (z/b) ; this forms, in fact, harmonic moun- 
 tains and valleys on the infinite plane. We require to find the potential and 
 attraction of such a distribution of matter. 
 
 Now the potential of an infinite straight line, of line-density p, at a point 
 distant d from it, is well-known to be 2/*p log d, where p, is the attraction 
 between unit masses at unit distance apart. Hence the potential V of the 
 supposed distribution of matter at the point x, z, is given by 
 
 I cos | log </{# 2 + ( - zf] d% 
 
 V = - 
 
 It is not hard to show that the first term vanishes when taken between the 
 limits. 
 
 L ^~~ 2 tlT 2 /*/' 2 
 
 Now put t = - , so that sin ~* = sin -7- cos 7 + cos -7- sin T , and we have 
 x b b b b b 
 
 f + / iff ? 
 
 Fn 771 I ^i/* " 
 
 = ZfiWnO I sin ~r cos 7 + 
 /-, \. b b 
 
 toe . z\ tdt 
 
 cos T- sin T 
 
 b bjl 
 
 But it is known* that 
 
 +> / oir, rtjt /+ / cos c t 
 
 inctdt f + tc 
 
 F *- L 
 
 Therefore V = 27rfj,whbe~ xlb cos y 
 
 If f/ be gravity, a earth's radius, and B earth's mean density, 27T/A ~^ . 
 
 And 
 
 F= ^ 4e -^ cos f (4) 
 
 See Todhunter's Int. Gale.; Chapter on "Definite Integrals." 
 
1882] PROPORTIONALITY OF DEFLECTION AND SLOPE. 447 
 
 The deflection of the plumb-line at any point on the surface denoted by 
 x = 0, and z, is clearly d Vjgdz, when x = 0. Therefore, 
 
 , 1 Sqwh . z 
 
 the deflection = -- x -~~- sin 7 .............................. (5) 
 
 g 2a6 6 
 
 But from (2) the slope for -j- , when z is zero) , is ~_ s i n . 
 
 Therefore deflection bears to slope the same ratio as v/g to ^aB. This 
 ratio is independent of the wave-length %7rb of the undulating surface, of the 
 position of the origin, and of the azimuth in the plane of the line normal to 
 the ridges and valleys. Therefore the proposition is true of any combination 
 whatever of harmonic undulations, and as any inequality may be built .up of 
 harmonic undulations, it is generally true of inequalities of any shape what- 
 ever. 
 
 Now a = 6'37 x 10 8 cm., 8 = 5|; and ^aS = 12'03 x 10 8 grammes per square 
 centimetre. The rigidity of glass in gravitation units ranges from 1*5 x 10 8 
 to 2*4 x 10 8 . Therefore the slope of a very thick slab of the rigidity of glass, 
 due to a weight placed on its surface, ranges from 8 to 5 times as much as 
 the deflection of the plumb-line due to the attraction of that weight. Even 
 with rigidity as great as steel (viz., about 8 x 10 8 ), the slope is 1 times as 
 great as the deflection. 
 
 A practical conclusion from this is that in observations with an artificial 
 horizon the disturbance due to the weight of the observer's body is very far 
 greater than that due to the attraction of his mass. This is in perfect 
 accordance with the observations made by my brother and me with our 
 pendulum in 1881, when we concluded that the warping of the soil by our 
 weight when standing in the observing room was a very serious disturbance, 
 whilst we were unable to assert positively that the attraction of weights 
 placed near the pendulum was perceptible. It also gives emphasis to the 
 criticisms we have made on M. Plantamour's observations namely, that he 
 does not appear to take special precautions against the disturbance due to 
 the weight of the observer's body. 
 
 We must now consider the probable numerical values of the quantities 
 involved in the barometric problem, and the mode of transition from the 
 problem of the mountains to that of barometric inequalities. 
 
 The modulus of rigidity in gravitation units (say grammes weight per 
 square centimetre) is vfg. In the problem of the mountains, wh is the mass 
 of a column of rock of one square centimetre in section and of length equal 
 to the height of the crests of the mountains above the mean horizontal plane. 
 In the barometric problem, wh must be taken as the mass of a column of 
 mercury of a square centimetre in section and equal in height to a half of the 
 maximum range of the barometer. 
 
448 NUMERICAL VALUES TO BE USED IN THE SOLUTION. [14 
 
 This maximum range is, I believe, nearly two inches, or, let us say, 5 cm. 
 The specific gravity of mercury is 13'6, and therefore wh = 34 grammes. 
 
 The rigidity of glass is from 150 to 240 million grammes per square 
 centimetre ; that of copper 540, and of steel 843 millions. 
 
 I will take v/g = 3 x 10 8 , so that the superficial layers of the earth are 
 assumed to be more rigid than the most rigid glass. It will be easy to adjust 
 the results afterwards to any other assumed rigidity. 
 
 w .,, gwh 5-67 648,000 5'67 
 
 with these data we have ^ = - ; also - x - - = '0117. 
 
 2v 10 s TT 10 s 
 
 It seems not unreasonable to suppose that 1500 miles (2'4 x 10 s cm.) is 
 the distance from the place where the barometer is high (the centre of the 
 anti-cyclone) to that where it is low (the centre of the cyclone). Accord- 
 ingly the wave-length of the barometric undulation is 4'8 x 10 s cm., and 
 b = 4-8 x 10 8 -r- 6-28 cm., or, say, 6 = '8 x 10 8 cm. 
 
 Thus, with these data, ~- b = 4'5 cm. 
 2v 
 
 We thus see that the ground is 9 cm. higher under the barometric 
 depression than under the elevation. 
 
 If the sea had time to attain its equilibrium slope, it would stand 
 5 x 13'6, or 68 cm. lower under the high pressure than under the low. But 
 as the land is itself depressed 9 cm., the sea would apparently only be 
 depressed 59 cm. under the high barometer. 
 
 It is probable that, in reality, the larger barometric inequalities do not 
 linger quite long enough over particular areas to permit the sea to attain 
 everywhere its due slope, and therefore the full difference of water-level can 
 only be attained occasionally. 
 
 On the other hand, the elastic compression of the ground must take place 
 without any sensible delay. Thus it seems probable that the elastic com- 
 pression of the ground must exercise a very sensible effect in modifying the 
 apparent depression or elevation of the sea under high and low barometer. 
 
 It does not appear absolutely chimerical that, at some future time when 
 both tidal and barometric observations have attained to great accuracy, an 
 estimate might thus be made of the average modulus of rigidity of the upper 
 500 miles of the earth's mass. 
 
 Even in the present condition of barometric and tidal information, it 
 might be interesting to make a comparison between the computed height of 
 tide and the observed height, in connection with the distribution of baro- 
 metric pressure. It is probable that India would be the best field for such an 
 attempt, because the knowledge of Indian tides is more complete than that 
 for any other part of the world. On the, other hand we shall see in the 
 
1882] AMOUNT OF DISTORTION OF THE EARTH'S SURFACE. 449 
 
 following section that tidal observations on coast-lines of continents are liable 
 to disturbance, so that an oceanic island would be a more favourable site. 
 
 It has already been shown that the maximum apparent deflection of the 
 plumb-line, consequent on the elastic compression of the earth, amounts to 
 0"0117, and this is augmented to 0"'0146, when we include the true deflec- 
 tion due to the attraction of the air. It is worthy of remark that this result 
 is independent of the wave-length of the barometric inequality, and thus we 
 get rid of one of the conjectural data. 
 
 Thus if we consider the two cases of high pressure to right and low to 
 left, and of low pressure to right and high to left, we see that there will be a 
 difference in the position of the plumb-line relatively to the earth's surface of 
 0"'0292. Even if the rigidity of the upper strata of the earth were as great 
 as that of steel, there would still be a change of 0"'011. 
 
 A deflection of magnitude such as 0"'03 or 0"'01 would have been easily 
 observable with our instrument of last year, for we concluded that a change 
 of 2^5- th of a second could be detected, when the change occurred rapidly. 
 
 It was stated in our previous Report that at Cambridge the calculated 
 amplitude of oscillation of the plumb-line due directly to lunar disturbance 
 of gravity amounts to 0"'0216. Now as this is less than the amplitude 
 due jointly to elastic compression and attraction, with the assumed rigidity 
 (300 millions) of the earth's strata, and only twice the result if the rigidity 
 be as great as that of steel, it follows almost certainly that from this cause 
 alone the measurement of the lunar disturbance of gravity must be im- 
 possible with any instrument on the earth's surface. 
 
 Moreover the removal of the instrument to the bottom of the deepest 
 known mine would scarcely sensibly affect the result, because the flexure of 
 the strata at a depth so small, compared with the wave-length of barometric 
 inequalities, is scarcely different from the flexure of tne surface. 
 
 The diurnal and periodic oscillations of the vertical observed by us were 
 many times as great as those which have just been computed, and therefore 
 it must not be supposed that more than a fraction, say perhaps a tenth, of 
 those oscillations was due to elastic compression of the earth. 
 
 The Italian observers could scarcely, with their instruments, detect de- 
 flections amounting to T ^jth of a second, so that the observed connection 
 between barometric oscillation and seismic "disturbance must be of a different 
 kind. 
 
 It is not surprising that in a volcanic region the equalisation of pressure, 
 between imprisoned fluids and the external atmosphere, should lead to earth- 
 quakes. 
 
 If there is any place on the earth's surface free from seismic forces, it 
 might be possible (if the effect of tides as computed in the following section 
 ix i. 29 
 
450 RENDING OF THE COAST DUE TO THE TIDE. [14 
 
 could be eliminated) with some such instrument as ours, placed in a deep 
 mine, to detect the existence of barometric disturbance many hundreds 
 of miles away. It would of course for this purpose be necessary to note the 
 positions of the sun and moon at the times of observation, and to allow for 
 their attraction. 
 
 2. On the Disturbance of the Vertical near the Coasts of Continents due 
 to the Rise and Fall of the Tide. 
 
 Consider the folloAving problem : 
 
 On an infinite horizontal plane, which bounds in one direction an infinite 
 incompressible elastic solid, let there be drawn a series of parallel straight 
 lines, distant I apart. Let one of these be the axis of y, let the axis of z be 
 drawn in the plane perpendicular to the parallel lines, and let the axis of x 
 be drawn vertically downwards through the solid. 
 
 At every point of the surface of the solid, from z = to I, let a normal 
 pressure gwh(l 2z/l) be applied; and from ^ = to I let the surface be 
 free from forces. Let the same distribution of force be repeated over all the 
 pairs of strips into which the surface is divided by the system of parallel 
 straight lines. It is required to determine the strains caused by these forces. 
 
 Taking the average over the whole surface there is neither pressure nor 
 traction, since the total traction on the half-strips subject to traction is equal 
 to the total pressure on the half-strips subject to pressure. 
 
 The following is the analogy of this system with that which we wish to 
 discuss: the strips subject to no pressure are the continents, the alternate 
 ones are the oceans, g is gravity, w the density of water, and h the height of 
 tide above mean water on the coast-line. 
 
 We require to find the slope of the surface at every point, and the vertical 
 displacement. 
 
 It is now necessary to bring this problem within the range of the results 
 used in the last section. In the first place, it is convenient to consider the 
 pressures and tractions as caused by mountains and valleys whose outline is 
 given by x h (1 22/1) from z = to I, and # = from z = to I. To 
 utilise the analysis of the las^i section, it is necessary that the mountains and 
 valleys should present a simple-harmonic outline. Hence the discontinuous 
 function must be expanded by Fourier's method. Known results of that 
 method render it unnecessary to have recourse to the theorem itself. It is 
 known that 
 
 \TT - \Q = sin 6 + \ sin 20 + $ sin 3(9 + . . . 
 - 10 = - sin 6 + \ sin 20- % sin 30 + ... 
 4(1 1 
 
 ITT + B = - JC080 + S COS 30 + = COB 50+ 
 
 TT 6" o* 
 
1882] FOURIER EXPANSION FOR THE SOLUTION. 451 
 
 The upper sign being taken for values of 6 between the infinitely small 
 positive and + TT, and the lower for values between the infinitely small nega- 
 tive and TT. 
 
 Adding these three series together we have 
 
 2 ji sin 26 + i sin 40 + ...} + -{cos + i cos 30 + i cos 56 + ...I 
 
 7T ( O 2 5 2 j 
 
 equal to TT 20 from = to + TT, and equal to zero from = to TT. 
 Hence the required expansion of the discontinuous function is 
 
 (6) 
 
 where = "7~ .................................... 00 
 
 ( 
 
 For it vanishes from z= l to 0, and is equal to h (1 2z/l) from .z = 
 to +1. 
 
 Now looking back to the analysis of the preceding section we see that if 
 the equation to the mountains and valleys had been oc = h sin (z/b), a would 
 have had the same form as in (2) but of course with sine for cosine, and 7 
 would have changed its sign and a cosine would have stood for the sine. 
 Applying then the solution (2) to each term of our expansion separately, and 
 only writing down the solution for the surface at which x = 0, we have at 
 once that 7 = 0, and 
 
 qwh I (1 . n/1 1 . . n 1 . nn . 
 a = ^~ -\-= : sin 20 + Ta sin 40 + ^ sm 60 + ... 
 
 7TV 7T [2 2 4 2 D 2 
 
 qwh 21 ( /, 1 o/i.l eA } 
 + ^ .Jcos0+K.cos30+7-cos50 + ...^ ............... (8) 
 
 TTV 7T 2 [ 3 s O 3 ) 
 
 The slope of the surface is -7- or T -j^ ; thus 
 
 az L av 
 
 qwh 2f. /1 1. 0/1 1 r/i , /n\ 
 
 * - sin 30 + ^r sin 50+...^ ............ (9) 
 
 . , 
 
 TTf 7T { 3 2 5 2 
 
 The formulae (8) and (9) are the required expressions for the vertical 
 depression of the surface and for the slope. 
 
 It is interesting to determine the form of surface denoted by these equa- 
 tions. Let us suppose then that the units are so chosen that gwhlfir*v may 
 be equal to one. Then (8) becomes 
 
 292 
 
452 
 
 FORM OF THE DISTORTED SURFACE. 
 
 [14 
 
 ... ...(11) 
 
 da 2 
 -ja = * cos 26 + i cos 40 + . . . 
 
 (W 7T 
 
 When is zero or + TT, dafdO becomes infinite, which denotes that the 
 tangent to the warped horizontal surface is vertical at these points. The 
 verticality of these tangents will have no place in reality, because actual 
 shores shelve, and there is not a vertical wall of water when the tide rises, as 
 is supposed to be the case in the ideal problem. We shall, however, see that 
 in practical numerical application, the strip of sea-shore along which the 
 solution shows a slope of more than 1" is only a small fraction of a millimetre. 
 Thus this departure from reality is of no importance whatever. 
 
 When 9 = or + TT, 
 
 2 (1 1 1 
 
 = - x 1-052 = -670 
 
 7T 
 
 being + when = 0, and when = + TT. 
 
 When = + I-TT, a vanishes, and therefore midway in the ocean and on 
 the land there are nodal lines, which always remain in the undisturbed 
 surface, when the tide rises and falls. At these nodal lines, defined by 
 
 = H, 
 
 da._ 2 jl 11 
 
 rJf) ~ ^ & e "t~ ~ jTii 03 ' KB ' 
 
 \ 
 
 = - '3466 + -6168 = - -9634 and + -2702 
 
 Thus the slope is greater at mid-ocean than at mid-land. By assuming 
 successively as ^TT, ^TT, ^TT, and summing arithmetically the strange series 
 which arise, we can, on paying attention to the manner in which the signs of 
 the series occur, obtain the values of a corresponding to 0, ^TT, + JTT, f TT, 
 f T, ITT, ITT, |TT. The resulting values, together with the slopes as 
 obtained above, are amply sufficient for drawing a figure, as shown annexed. 
 
 The straight line is a section of the undisturbed level, the shaded part 
 being land, and the dotted sea. The curve shows the distortion, when warped 
 by high and low-tide as indicated. 
 
 The scale of the figure is a quarter of an inch to ITT for the abscissas, and 
 a quarter of an inch to unity for the ordinates ; it is of course an enormous 
 exaggeration of the flexure actually possibly due to tides. 
 
1882] FORM OF THE DISTORTED SURFACE. 453 
 
 It is interesting to note that the land regions remain very nearly flat, 
 rotating about the nodal line, but with slight curvature near the coasts. It 
 is this curvature, scarcely perceptible in the figure, which is of most interest 
 for practical application. 
 
 The series (8) and (9) are not convenient for practical calculation in the 
 neighbourhood of the coast, and they must be reduced to other forms. It is 
 easy, by writing the cosines in their exponential form, to show that 
 
 cos0 + cos20 + J cos 30 + ... =-log e ( 2sin|0) ......... (13) 
 
 cos0-cos20 + icos30-}-...= log e (2cos^0) ............ (14) 
 
 Where the upper sign in (13) is to be taken for positive values of 6 and the 
 lower for negative. 
 
 For the small values of 0, for which alone we are at present concerned, 
 the series (13) becomes log e ( + 6) and the lower log e 2. 
 
 Taking half the difference and half the sum of the two series we have 
 i cos 20 + 1 cos 40 +... = - i log (0)- 1 log 2 ......... (15) 
 
 ... =-|log( 0) + |log2 ......... (16) 
 
 Integrating (16) with regard to 0, and observing that the constant intro- 
 duced on integration is zero, we have 
 
 Sin0 + isin30 + ism50 = -i0[log(0)-l] + i01og2 ...(17) 
 Then from (15) and (17) 
 
 2 f 1 
 
 20 + i cos 40 + ... -- \ sin + sin 30 + . . . 
 
 * 
 
 A cos 
 
 7T 
 
 Integrating (15), and observing that the constant is zero, we have 
 
 2 ...... (19) 
 
 Integrating (17) and putting in the proper constant to make the left-hand 
 side vanish when 0=0, we have 
 
 ...... (20) 
 
 For purposes of practical calculation may be taken as so small that the 
 right-hand side of (18) reduces to log( 20), and the right-hand sides of 
 (19) and (20) to zero. 
 
454 FORM OF THE DISTORTED SURFACE. [14 
 
 Hence by (8) and (9), we have in the neighbourhood of the coast 
 
 gwh 21 fl 1 1 U 
 
 = x - \\ r 3 + oi + + 
 
 TTV 7T 2 [_1 3 3 3 5 J J 
 
 ...(21) 
 
 1x2-1037 
 
 TT 
 
 . gwh 
 
 I shall now proceed to compute from the formulse (21) the depression of 
 the surface and the slope, corresponding to such numerical data as seem most 
 appropriate to the terrestrial oceans and continents. 
 
 Considering that the tides are undoubtedly augmented by kinetic action, 
 we shall be within the mark in taking h as the semi-range of equilibrium 
 tide. At the equator the lunar tide has a range of about 53 cm., and the 
 solar tide is very nearly half as much. Therefore at the spring-tides we may 
 take h = 40 cm. It must be noticed that the highness of the tides, say 15 or 
 20 feet, near the coast is due to the shallowing of the water, and it would not 
 be just to take such values as representing the tides over large areas ; w, the 
 density of the water is, of course, unity. 
 
 If we suppose it is the Atlantic Ocean and the shores of Europe with 
 Africa, and of North and South America, which are under consideration, it is 
 not unreasonable to take I as 3,900 miles or 6*28 x 10 s cm. Then 
 
 =z x 10~ 8 
 
 Taking vjg as 3 x 10 8 , that is to say, assuming a rigidity greater than 
 that of glass, we have for the slope in seconds of arc, at a distance z from the 
 sea-shore 
 
 40 
 X - e (22) 
 
 = 0"-01008(8-log 10 *) 
 
 From this the following table may be computed by simple multiplica- 
 tion : 
 
 Distance from mean 
 
 water-mark Slope 
 
 ..... 0"-0806 
 
 ..... -0706 
 
 . . . . . -06U5 
 
 ..... -0504 
 
 ..... -0403 
 
 . . . . '0302 
 
 ..... -0202 
 
 2xl0 6 cm.= 20kilom ....... -0170 
 
 5xll) 6 cin.= 50 kilom ....... -0131 
 
 10 7 cm. = 100 kilom. -0101 
 
 1 Clll. 
 
 = 1 cm. 
 
 10 cm. 
 
 = 10 cm. 
 
 10* cm. 
 
 = 1 metre 
 
 10 3 crn. 
 
 = 10m. 
 
 10* cm. 
 
 = 100 m. 
 
 10 5 cm. 
 
 = 1 kilom. 
 
 10" cm. 
 
 = 10 kilom. 
 
1882] AMOUNT OF APPARENT OSCILLATION OF THE VERTICAL. 455 
 
 On considering the formula (22) it appears that z must be a very small 
 fraction of a millimetre before the slope becomes even as great as 1'. This 
 proves that the rounded nick in the surface, which arises from the discon- 
 tinuity of pressure at our ideal mean water-mark, is excessively small, and 
 the vertical displacement of the surface is sensibly the same, when measured 
 in centimetres, on each side of the nick, in accordance with the first of (21). 
 
 The result (5) of section 1 shows that, with rigidity 3 x 10 8 , the true 
 deflection of plumb-line due to attraction of the water is a quarter of the 
 slope. Hence an observer in a gravitational observatory at distance z from 
 mean water-mark, would note deflections from the mean position of the 
 vertical 1^ times as great as those computed above. And as high water 
 changes to low, there would be oscillations of the vertical 2 times as great. 
 We thus get the practical results in the following table : 
 
 Distance of obser- Amplitude of 
 
 vatory from apparent oscillation 
 
 mean water-mark of the vertical 
 
 10 metres 0"'126 
 
 100m -101 
 
 1 kilom -076 
 
 10 kilom -050 
 
 20 kilom '042 
 
 50 kilorn '035 
 
 100 kilom -025 
 
 It follows, from the calculations made for tracing the curve, that halfway 
 across the continent (that is to say, 3,142 kilometres from either coast) the 
 
 slope is - - x - x '2703 seconds of arc. = 0" - 00237 ; and the range of 
 
 7T TTV 
 
 apparent oscillation is 0"'006. 
 
 In these calculations the width of the sea is taken as 6,283 kilometres. If 
 the sea be narrower, then to obtain the same deflections of the plumb-line, 
 the observatory must be moved nearer the sea in the same proportion as the 
 sea is narrowed. If, for example, the sea were 3,142 kilometres wide, then at 
 10 kilometres from the coast the apparent amplitude of deflection is 0"'042. 
 If the range of tide is greater than that here assumed (viz., 80 cm.), the 
 results must be augmented in the same proportion. And, lastly, if the 
 rigidity of the rock be greater or less than the assumed value (viz., 3 x 10 8 ) 
 the part of the apparent deflection depending on slope must be diminished 
 or increased in the inverse proportion to the change in rigidity. 
 
 I think there can be little doubt that in narrow seas the tides are gene- 
 rally much greater than those here assumed ; and it is probable that at a 
 gravitational observatory actually on the sea-shore on the south-coast of 
 England, apart from seismic changes, perceptible oscillations of the vertical 
 would be noted. 
 
456 LORD KELVIN ON THE COMPUTED ATTRACTION OF THE TIDE. [14 
 
 Sir William Thomson has made an entirely independent estimate of the 
 probable deflection of the plumb-line at a sea-side gravitational observatory*. 
 He estimates the attraction of a slab of water, 10 feet thick (the range of tide), 
 50 miles broad perpendicular to the coast, and 100 miles long parallel with 
 the coast, on a plummet 100 yards from the low water mark, and opposite 
 the middle of the 100 miles of length. He thinks this estimate would very 
 roughly represent the state of things say at St Alban's Head. He finds then 
 that the deflection of the plumb-line as high-tide changes to low would be 
 ?oiyo<nmth of the unit angle or 0"'050. The general theorem proved above, as 
 to the proportionality of slope to attraction, shows that, with rigidity 3 x 10 8 
 for the rocks of which the earth is formed, the apparent deflection of the 
 plumb-line would amount to 0"'25. 
 
 It is just possible that a way may in this manner be opened for determin- 
 ing the modulus of rigidity of the upper 100 or 200 miles of the earth's 
 surface, although the process would be excessively laborious. The tides of 
 the English Channel are pretty well known, and therefore it would be possible 
 by very laborious quadratures to determine the deflection of the plumb-line 
 due to the attraction of the tide at any time at a chosen station. If then the 
 deflection of the plumb-line could be observed at that station (with corrections 
 applied for the positions of the sun and moon), the ratio of the calculated 
 to the observed and corrected deflection, together with the known value of 
 the earth's radius and mean density, form the materials for computing the 
 rigidity. But such a scheme would be probably rendered abortive by just 
 such comparatively large arid capricious oscillations of the vertical, as we, 
 M. d'Abbadie and others, have observed. 
 
 It is interesting to draw attention to some observations of M. d'Abbadie 
 on the deflections of the vertical due to tides. His observatory (of which an 
 account was given in the Report for 1881) is near Hendaye in the Pyrenees, 
 and stands 72 metres above, and 400 metres distant from, the sea. He 
 writes f : 
 
 " J'ai reuni 359 comparaisons d'observations speciales faites lors du maxi- 
 mum du flot et du jusant; 243 seulement sont favorables a la theorie de 
 1'attraction exercee par la masse des eaux, et 1'ensemble des resultats pour 
 une difference moyenne de marees egale a 2*9 metres donne un resultat 
 moyen de O vv '56 ou 0"'18 pour le double de 1'attraction angulaire vers le 
 Nord-Ouest. Ceci est conforme a la theorie, car les differences observees 
 doivent etre partage*es par moitie, selon la loi de la reflexion ; mais comme il 
 y a toujours de 1'inattendu dans les experiences nouvelles, on doit aj outer que 
 sur les 116 comparaisons restantes il y en a eu 57 ou le flot semble repousser 
 le mercure au lieu de 1'attirer. Mes resultats ont ete confirmes pendant 
 
 * Thomson and Tait's Nat. Phil. 818. 
 
 t " Recherches sur la Verticale." Ann. de la Soc. Scient. de Uruxelles, 1881. 
 
1882] D'ABBADIE ON THE OBSERVED ATTRACTION OF THE TIDE. 457 
 
 1'hiver dernier par M. 1'abbe Artus, qui a eu la patience de comparer ainsi 71 
 flots et 73 jusants consecutifs, de Janvier a mars 1880. Lui aussi a trouve un 
 tiers environ de cas defavorables a nos theories admises. On est done en 
 droit d'affirmer que si la mer haute attire le plus souvent le pied du fil a 
 plomb, il y a une, et peut-etre plusieurs, autres forces en jeu pour faire varier 
 sa position." 
 
 We must now consider the vertical displacement of the land near the 
 coast. In (21) it is shown to be = - '- x x 21037, where indicates 
 
 7TU 7T 2 
 
 the displacement corresponding to z 0. 
 
 With the assumed values, h = 40, v = 3 x 10 8 , I = 6'28 x 10 8 , I find 
 Of = 5'684cm. Hence the amplitude of vertical displacement is 11*37 cm. 
 As long as hi remains constant this vertical displacement remains the same ; 
 hence the high tides of 10 or 15 feet which are actually observed on the 
 coasts must probably produce vertical oscillations of quite the same order as 
 that computed. 
 
 If the land falls the tide of course rises higher on the coast-line than it 
 would do otherwise ; hence the apparent height of tide would be h + a . But 
 this shows there is more water resting on the earth than according to the 
 estimated value h ; hence the depression of the soil is greater in the propor- 
 tion 1 + a /h to unity ; this again causes more tide, which reacts and causes 
 more depression, and so on. Thus on the whole the augmentation of tide due 
 to elastic yielding is in the ratio of 
 
 H + + + . . . or to unity 
 
 h \hj \tij I. a ( ,/h 
 
 This investigation is conducted on the equilibrium theory, and it neglects 
 the curvature of the sea bed, assuming that there is a uniform slope from 
 mid-ocean to the sea-coast. The figure shows that this is not rigorously the 
 case, but it is quite near enough for a rough approximation. The phenomena 
 of the short period tides are so essentially kinetic that the value of this 
 augmentation must remain quite uncertain, but for the long-period tides (the 
 fortnightly and monthly elliptic) the augmentation must correspond approxi- 
 mately with the ratio 
 
 x 2 . 103 7 
 
 The augmentation in narrow seas will be small, but in the Atlantic Ocean the 
 augmenting factor must agree pretty well with that which I now compute. 
 
 With the previous numerical values we have a /h (which is independent 
 of h) equal to '1421, and 1 a /h = *8579 = very nearly. 
 
 Thus the long-period tides may probably undergo an augmentation at the 
 coasts of the Atlantic in some such ratio as 6 to 7. 
 
458 VALUES OF RIGIDITY OF SUPERFICIAL ROCKS. [14 
 
 The influence of this kind of elastic yielding is antagonistic to that 
 reduction of apparent tide, which must result from an elastic yielding of 
 the earth's mass as a whole. 
 
 The reader will probably find it difficult to estimate what degree of 
 probability of correctness there is in the conjectural value of the rigidity, 
 which has been used in making the numerical calculations in this paper. 
 The rigidity has not been experimentally determined for many substances, 
 but a great number of experiments have been made to find Young's modulus. 
 Now, in the stretching of a bar or wire the compressibility plays a much less 
 important part than the rigidity, and the formula for Young's modulus shows 
 that for an incompressible elastic solid the modulus is equal to three times 
 the rigidity*. Hence a third of Young's modulus will form a good standard 
 of comparison with the assumed rigidity, namely, 3 x 10 8 grammes weigh. 
 per square centimetre. The following are a few values of a third of Young's 
 modulus and of rigidity, taken from the tables in Sir William Thomson's 
 article on Elasticity! in the Encyclopcedia Britannica. 
 
 A third of Young's mod. and rigidity in 
 Material terms of 10 s grammes weight per sq. cm. 
 
 Stone About 1'2 
 
 Slate About 3 to 4 
 
 Glass Rigidity 1-5 to 2 -4 
 
 Ice 4-7 
 
 Copper ...... 4 and rigidity 4'6 to 5'4 
 
 Steel 7 to 10 and rigidity 8-4 
 
 It will be observed that the assumed rigidity 3 is probably a pretty high 
 estimate in comparison with that of the materials of which we know the 
 superficial strata to be formed. 
 
 It is shown, in another paper read before the Association at this meeting, 
 that the rigidity of the earth as a whole is probably as great as that of steel. 
 That result is not at all inconsistent with the probability of the assumption 
 that the upper strata have only a rigidity a little greater than that of glass J. 
 
 3. On Gravitational Observatories. 
 
 In the preceding sections estimates have been made of the amount of 
 distortion which the upper strata of the earth probably undergo, from the 
 shifting weights corresponding to barometric and tidal oscillations. These 
 results appear to me to have an important bearing on the probable utility of 
 gravitational observatories. 
 
 * Thomson and Tait's Nat. Phil. 683. 
 
 t Also published separately by Black, Edinburgh. 
 
 J [See Paper 9, p. 340 above.] 
 
1882] GRAVITATIONAL OBSERVATORIES. 459 
 
 It is not probable, at least for many years to come, that the state of tidal 
 and barometric pressure, for a radius of 500 miles round any spot on the 
 earth's surface, will be known with sufficient accuracy to make even a rough 
 approximation to the slope of the surface a possibility. And were these data 
 known, the heterogeneity of geological strata would form a serious obstacle to 
 the possibility of carrying out such a computation. It would do little in 
 relieving us from these difficulties to place the observatory at the bottom of 
 a mine. 
 
 Accordingly the prospect of determining experimentally the lunar disturb- 
 ance of gravity appears exceedingly remote, and I am compelled reluctantly 
 to conclude that continuous observations with gravitational instruments of 
 very great delicacy are not likely to lead to results of any great interest. It 
 appears likely that such an instrument, even in the most favourable site, would 
 record incessant variations of which no satisfactory account could be given. 
 Although I do not regard it as probable that such a delicate instrument 
 should be adopted for regular continuous observations, yet, by choosing a site 
 where the flexure of the earth's surface is likely to be great, it is conceivable 
 that a rough estimate might be made of the average modulus of elasticity of 
 the upper strata of the earth for one or two hundred miles from the surface. 
 
 These conclusions, which I express with much diffidence, are by no means 
 adverse to the utility of a coarser gravitational instrument, capable, let us 
 say, of recording variations of level amounting to 1" or 2". If barometric 
 pressure, tidal pressure, and the direct action of the sun and moon, combined 
 together to make apparent slope in one direction, then at an observatory 
 remote from the sea-shore, that slope might perhaps amount to a quarter of 
 a second of arc. Such a disturbance of level would not be important com- 
 pared with the minimum deviations which could be recorded by the supposed 
 instrument. 
 
 It would then be of much value to obtain continuous systematic observa- 
 tions, after the manner of the Italians, of the seismic and slower quasi-seismic 
 variations of level. 
 
 I venture to predict that at some future time practical astronomers will 
 no longer be content to eliminate variations of level merely by taking means 
 of results, but will regard corrections derived from a special instrument as 
 necessary to each astronomical observation. 
 
INDEX TO VOLUME I. 
 
 Abacus, tidal an apparatus for facilitating 
 harmonic analysis, 216 
 
 Abbadie, A. d' deflection of the vertical, 416, 
 420, 434, 439, 440 
 
 Adams, J. C. on notation for harmonic 
 analysis, 2; confirms numerical values of 
 coefficients, 26; fusion of lunar and solar 
 Kj, K 2 tides, 43; equivalent multipliers for 
 tides of long period, 66; B. A. Eeports, 
 70, 97; correction to B.A. Keport of 1872, 
 345 
 
 Admiralty Scientific Manual paper on Tides 
 from, 117 
 
 Airy, Sir G. B. Tides and Waves, 156; 
 changes in azimuth of transit, 417 
 
 Analysis see Harmonic 
 
 Antarctic tidal observations reduction of, 372 
 
 Armellini seismological observations, 435 
 
 B 
 
 Baird, Col. A. W. in charge of Indian Tidal 
 Survey, 2; auxiliary tables for harmonic 
 analysis, 68; Manual for tidal observations, 
 156 
 
 Barne, Lieut. M., B.N. method of observing 
 tides on the 'Discovery,' 372 
 
 Barometric pressure apparent deflection of 
 vertical due to, 444 
 
 Bertelli, Prof. seismological observations, 
 431, 434 
 
 Bouquet de la Grye deflection of vertical at 
 Campbell Island, 417, 438; form of seismo- 
 meter, 438 
 
 British Association Keports on harmonic 
 analysis, 1, 2, 3, 71, 341; on lunar dis- 
 turbance of gravity, 389, 430 
 
 Burdwood directions for reducing tidal ob- 
 servations, 121 
 
 C 
 
 Christie, Sir W. H. M. vibrations of the 
 ground at Greenwich, 428 
 
 D 
 
 d' Abbadie, see Abbadie 
 
 Darwin, Horace instrument for measuring 
 deflections of the vertical, 389; experiment 
 on variations of surface soil, 425 
 
 Deflection of vertical d' Abbadie, Plantamour, 
 416, 417; Nyren, 422, 456 
 
 ' Discovery ' antarctic tidal observations, 372 
 
 Dynamical theory of tides, 347, 366 
 
 E 
 
 Earth, rigidity of estimated from tides of 
 
 long period, 340; Hecker on, 346 
 Earth's surface changes of level due to 
 
 elasticity of, 444 
 
 Eccles, J. on treatment of M x tide, 39 
 Egidi, Prof. seismological observations, 433 
 Elasticity of earth's surface changes of level 
 
 due to, 444 
 Encyclopedia Britannica, extract from on 
 
 the dynamical theory of the tides, 347 
 Equilibrium theory of tides correction for 
 
 continents, 328 
 
 Ferrel, Prof. tidal results of U. S. Coast 
 Survey, 70 
 
 G 
 
 Gaillot changes of level at Paris, 443 
 Galli, D. G. seismological observations, 432 
 Glazebrook, R. T. antarctic tide-curves for 
 
 verification of antarctic reductions, 381, 
 
 386 
 
462 
 
 INDEX TO VOLUME I. 
 
 Grablovitz, G. seismological observations, 
 
 434 
 
 Gruithuisen, Prof. on horizontal pendulum, 
 
 416 
 
 H 
 
 Harmonic analysis see table of contents 
 Paper 1, 1; Indian computation forms, 4; 
 choice of periods for analysis, 71; com- 
 parison with method of hour-angles and 
 parallaxes, 78; treatment of a short series 
 of observations, 98, 122; of high and low- 
 water observations, 157; an instrument for 
 facilitating, 216 
 
 Harmonic analysing instrument, 49 
 Hecker, Dr 0. rigidity of earth determined 
 
 by horizontal pendulum, 346 
 Hengler fraudulent account of horizontal 
 
 pendulum, 416 
 
 Henry changes in transit at Greenwich, 417 
 Hough, S. S. introduction of mutual gravi- 
 tation of water in dynamical theory of tides, 
 349 
 
 M 
 
 Malvasia, Count observation as to local 
 
 tremors, 432 
 
 Microphone as seismological instrument, 435 
 Milne, Prof. J. seismological observations, 
 
 437 
 Mocenigo, Count seismological observations, 
 
 435 
 Monte, Pietro seismological observations, 431 
 
 N 
 Nadirane d'Abbadie's instrument, 418, 439 
 
 441, 456 
 National Physical Laboratory tide-curves for 
 
 verification of antarctic tidal reductions, 
 
 381, 386 
 Nyren distant earthquakes observed by, 422 
 
 
 
 Observations, tidal harmonic analysis, 1; of 
 short series, 98, 122 ; of high and low water, 
 120; Whewell's treatment of, 120 
 
 Indian tidal operations Colonel Baird in 
 charge of, 2 
 
 K 
 
 Kelvin, Lord use of astres fictifs in equi- 
 librium theory of tides, 3; on tidal instru- 
 ments, 48; harmonic analysing instrument, 
 49; correction of equilibrium theory for 
 land, 328; oscillations of rotating water, 
 349; tides of long period, 366; instrument 
 for measuring deflections of vertical, 389 ; 
 computation of deflection of vertical due to 
 the tide, 456; values of modulus of rigidity 
 of rock, 458 
 
 Laplace astres fictifs in equilibrium theory, 
 3 ; theory of tides in Mecanique Celeste, 349, 
 367 
 
 Latitude variations of, 422, 426, 443 
 Levels Plantamour's observations with, 420 
 Levels, Datum for tidal observations, 97 
 Long period tides equivalent multipliers in 
 reduction of, 66 ; dynamical theory of, 366 ; 
 rigidity of earth estimated from, 340, 371 
 Lunar deflection of vertical Hecker measure- 
 ment of, 346; attempted measurement of, 
 389; theoretical amount of, 414 
 
 Pendulum, horizontal Zollner on, 413; sug- 
 gestion by Perrot, 416 ; Safarik on, 416 ; 
 Heugler's fraudulent account of, 416 ; 
 Gruithuisen on, 416 
 
 Pendulum, form of devised by Bouquet de la 
 Grye for observing deflection of vertical, 
 417, 427, 438 
 
 Perrot suggestion of horizontal pendulum, 
 416 
 
 Peters amount of lunar deflection of gravity, 
 414 
 
 Plantamour variations of level, 416; obser- 
 vations with levels, 420 
 
 Poincare, H. dynamical theory of tides in 
 ocean interrupted by land, 349 
 
 Poiseuille viscosity of alcohol and water, 406 
 
 Prediction of tides Kelvin on instrument 
 for, 48; from approximate tidal constants, 
 104, 141; method of, 258; comparison with 
 actuality, 325 
 
 R 
 
 Eigidity of earth estimated by tides of long 
 period, 340; by horizontal pendulum, 346 
 
 Roberts, E. in charge of Indian tidal pre- 
 diction, 1, 36 ; computation forms, 49 ; 
 correction of the same, 345 
 
 Rossi, S. M. de seismological observations, 
 431, 435 
 
INDEX TO VOLUME I. 
 
 463 
 
 S 
 
 Safarik on horizontal pendulum, 416 
 Scottish antarctic tidal observations, 387 
 Sea-level, datum for tidal observations, 97 
 Sea-level slow oscillations of, 116 
 Seismological observations Italian, 431; Ja- 
 
 panese, 437 
 Selby reduction of Scottish antarctic tidal 
 
 observations, 387 
 Strachey, Sir B. rules for harmonic analysis, 
 
 101, 125, 132 
 
 Thomson, James harmonic analyser, 49 
 Thomson, Sir William also Thomson and 
 
 Tait, see Kelvin, Lord 
 Tides see several other titles 
 Tides distortion of coast due to weight of, 
 
 450 
 Tide- table see Prediction 
 
 Transit variation of piers of, 417, 422, 443 
 Turner, Prof. H. H. correction of equilibrium 
 theory for continents, 328 
 
 U 
 United States Coast Survey tidal results, 70 
 
 Vertical, deflection of an instrument for 
 measuring, 389 
 
 w 
 
 Whewell, Dr treatment of tidal observations, 
 120 
 
 Wilson, Dr antarctic observations in sum- 
 mer, 385 
 
 Zollner on the horizontal pendulum, 413 
 
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