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Full text of "An Introductory Treatise On The Lunar Theory"

,CC. i''A 



, 



u 



AN 



INTBODUCTCXRY TREATISE 



ON THE 



LUNAE THEOEY 



SontiOtt: C. J. CLAY AND SONS, 

CAMBRIDGE UNIYEESITY PBBSS WAEEHOUSE, 

AYE MARIA. LANE. 

268, ARGYLE STREET. 




P. A. BROCKHATTS. 

larft: MACMILLAN AND CO. 
jn GEORGE BELL AND SONS. 



INTRODUCTORY TREATISE 



ON THE 



LUNAR THEORY 




EENEST W. BROWN _M.A. 

PROPESSQB OP APPLIED MATHEMATICS IN HAVBBFORD OOLLBG-B PA. U.S.A. 
SOMETIME PELLOW OF CHRIST'S COLLEGE 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 
1896 

[^tZZ Rights reserved] 



II/V l_lfc>., 




PBINTBD BY J. & C. F. CLAY, 
A.T THE BNIVEBSITY PRESS. 



PEEFACE. 

rpHE researches made during the last twenty years into the particular case 
-*- of the Problem of Three Bodies, known as the Lunar Theory, have had 
the effect of creating a wider interest in a subject which had been somewhat 
neglected by the majority of mathematicians. Enquiry has been made, not 
only into the value of the various methods from a practical point of view, but 
also into questions which have an equal theoretical importance but which, 
until just lately, have been almost entirely neglected. The existence of 
integrals and of periodic solutions, and the representation of the solutions 
by infinite series, may be cited as instances. 

In order to understand the bearing of these investigations on the lunar 
theory, some acquaintance with the older methods is desirable, if not necessary. 
In the following pages, an attempt has been made to supply a want in this 
direction, by giving the general principles underlying the methods of treat- 
ment, together with an account of the manner in which they have been 
applied in the theories of Laplace, de Pont^coulant, Hansen, Delaunay, and 
in the new method with rectangular coordinates. The explanation of these 
methods, and not the actual results obtained from them, having been my chief 
aim, only those portions of the developments and expansions, required for the 
fulfilment of this object, have been given. 

The use of infinite series requires that investigations into their 
convergency should take an important place in any treatise on the Lunar 
Theory, and it is with regret that I have been obliged to leave it almost 
entirely aside, owing to the lack of any certain knowledge on the subject. 
The applications of the results to the formation of tables have shown that 
the series are of practical use, but the right to represent the solutions by 



VI PREFACE. 

means of them has been discussed only for a few of the simpler cases, and 
the radius of convergence, when the series are arranged according to powers 
of any parameter, has been determined for elliptic motion only. It has also 
been found necessary to omit many other theoretical investigations which, in 
a more extended treatise, might have been included; but it is hoped that 
the references will cause the volume to be of service to those who desire 
to proceed to the study of these higher branches, as well to those who wish 
merely to obtain information concerning the older methods, 

The difficulties of the subject are, perhaps, less inherent to it than due to 
the manner in which it has been presented to a student approaching it for 
the first time. The classical treatises are, almost invariably, original memoirs, 
and, as such, do not contain the details which are essential for a clear 
understanding of the scope and limitations of the problem in the form in 
which it is usually considered. Moreover, the authors generally confine 
themselves to their own methods, and the discovery of the relations which 
exist between the various forms of expression for the same function, is often 
troublesome. I have therefore given special attention to this point and 
also to another, closely associated with it, namely, the definitions and 
significations of the constants in disturbed motion. 

A selection of one of the five methods, mentioned above, being necessary 
as a basis for the elucidation of the properties common to all, I have had 
no hesitation in adopting that of de Pontdcoulant. Laplace takes the true 
longitude, instead of the time, as the independent variable a method which 
renders the interpretation of the results difficult, until the reversion of series 
has been made while the theories of Hansen and Delaunay are not well 
adapted to the end in view. De Pont^coulant's method of approximation 
being similar to that of Laplace, it then seemed to be sufficient, for the 
explanation of Laplace's treatment, to give the equations of motion, the first 
approximation, and a brief account of the manner in which the second and 
higher approximations are obtained. 

In the Chapters on the methods of de Pont6coulant, Hansen and 
Delaunay, I have made some alterations in the form of presentation and 
the methods of proof, whenever these seemed to tend towards greater 
simplicity; where the differences from the original memoirs are important, 
they are noted. In order to facilitate references to the latter, the original 
notations are adhered to as far as possible, and this has necessitated the 



PREFACE. VII 

employment of three distinct notations. The tables given at the end of the 
volume will show the Chapters in which they are severally employed, and will 
enable the reader to find, without difficulty, the meaning of any frequently 
recurring symbol. 

The first four Chapters respectively contain investigations of the form 
of the disturbing function, of the equations of motion, of the expansions 
relating to elliptic motion, and of the methods adopted in order to approxi- 
mate to the solution. The term 'intermediary 7 is used to signify any 
orbit which may be adopted as a first approximation to the path actually 
described a definition somewhat different from that given by Prof. Gyldfen, 
In Chap. v. the equations for the variations of the elements in disturbed 
motion are obtained in an elementary way and also by the more elegant 
and symmetrical method of Jacobi. The properties and methods of de- 
velopment of the disturbing function are collected in Chap. vi. 

The details of the second and of parts of the third approximation to the 
solution of de Pont^couknt's equations will be found in Chap. vii. ; the 
inequalities are divided into classes in order to show their origin more 
clearly. Chap. VIIL, devoted to the arbitrary constants, is made intentionally 
simple and explicit. 

Chaps. IX. and x. contain the theories of Delaunay and Hanson, 
respectively. A special effort has been made to free the methods of the 
latter from the difficulties and obscurities which surround them in the 
Fundaments and the Darlegung. In Chap. XI. I have attempted to give 
a complete method for the treatment of the solar inequalities in the Moon's 
motion, based on that initiated by Dr Hill for those parts of them which 
depend only on the ratios of the mean solar and lunar motions. The 
infinite determinant, which arises in the calculation of the principal parts of 
the mean motions of the perigee and the node, is considered at some length, 
the conditions for its convergency and its development in series being 
included. Chap. xil. contains an account, necessarily brief, of methods 
other than those previously discussed. 

In Chap. xin. the inequalities arising from planetary action and from 
the ellipticity of the Earth are considered. It being impossible to give an 
adequate account of these in the space at my disposal, they are treated by 
Dr Hill's modification of Delaunay's method only. An exception is made 
in favour of the inequalities due to the motion of the ecliptic, and of 



VI11 PREFACE. 

the inequality known as the secular acceleration, since the effects of these 
appeared to be more simply explained by other methods. 

The various memoirs and treatises of which I have made use are 
referred to in the text. In particular, the excellent collection of methods, 
contained in the first and third volumes of the Mdcanique Celeste of 
M. F. Tisserand, has been frequently consulted. 

I take this opportunity of acknowledging a deep debt of gratitude to 
Professor G. H. Darwin and Dr E. W. Hobson, not only for their 
valuable criticisms and suggestions made while reading the proof-sheets of 
this work, but also for their advice and assistance rendered freely at all 
times during the last eight years. My thanks are also due to Mr P. H. 
Cowell, Fellow of Trinity College, Cambridge, for much help in the correc- 
tion of all the proof-sheets and in the verification of the formulae and 
results. 

I may add that the cooperation of the officers of the University Press 
has made it possible for me to see the printing almost completed during 
my temporary residence in Cambridge, and thus to avoid the delays and 
difficulties which would otherwise have arisen. 



ERNEST W. BROWN. 



HA.VKRFORD COLLEGE, 

1895, December 13. 



CONTENTS. 



CHAPTER I. 



FOKCE-FUNCTIONS. 

ARTS. PAGE 

1. Units 1 

2. Problem of three bodies 25 

3, 4. (i) Forces relative to the Earth 2 

5, 6. (ii) Forces on the Moon relative to the Earth, and those on the Sun 

relative to the centre of mass of the Earth and Moon . . 4 

7, 8. Force-function and Disturbing function usually used .... 6 

9. Distinction between the lunar and the planetary theories ... 8 

10. Force-function for p bodies 10 



CHAPTER II 

THE EQUATION'S OF MOTION. 

11. Methods of treatment i% 

12-15. (i) De Ponte'coulant's equations 13 

16, 17. (ii) Laplace's equations ^7 

18-21. (iii) Equations of motion referred to moving rectangular axes . , 19 

22. Particular case : the solar parallax neglected .... 23 

23. the solar parallax and eccentricity and the 
lunar inclination neglected .24 

24. The Jacobian integral 35 

25-30. (iv) Equations in the general problem of three bodies. Tho ten 

known integrals. The Invariable Plane. Special cases , . 25 



CHAPTER III 



UNDISTURBED ELLIPTIC MOTION. 

31. Method of procedure 

32-47. (i) Formula, expansions and theorems connected with the elliptic 



C CONTENTS. 

AETS. PAGE 

32. General formulae referring to an ellipse 29 

33-36, . Series connecting the radius vector, the true anomaly and 

the mean anomaly 31 

37-42, Similar expansions in terms of Bessel's Functions. . . 33 

43. Theorem of Hansen 36 

44-47. Ellipse inclined to the plane of reference .... 38 

48-53. (ii) Elliptic motion 40 

48-52. Undisturbed elliptic motion of the Moon .... 40 

53. Sun 42 

54. Convergence of elliptic series 43 

CHAPTER IV. 

FOKM OF SOLUTION. THE FIRST APPROXIMATION. 

55. The two principal methods of approaching the solution ... 44 

56. Form to be given to the expressions for the coordinates ... 44 
57-60. Intermediate orbits 45 

61. Solution by continued approximation 47 

62. the method of the variation of arbitrary constants . . 47 

63. The instantaneous ellipse 48 

64-66. Application of the method of solution by continued approximation to 

de Pont<3coulant's equations 49 

67, 68. Modification of the intermediary 51 

69. Convergency of the series obtained for the coordinates . . . 53 

70. Modification of the intermediary for Laplace's equations ... 53 



CHAPTER V. 



VARIATION OF ARBITRARY CONSTANTS. 

71. The two methods of development to be employed .... 54 

72-92. (i) Elementary methods of treatment 54 

73, 74. Change of position due to changes in the elements . . 55 

75. Expression of the derivatives of the disturbing function in 

terms of the forces 56 

76. Meanings to be attached to the symbols , 9 . . . . 58 
77-82. Differential equations for the elements in terms of the 

forces. Departure points 58 

83. Differential equations for the elements in terms of the 

derivatives of the disturbing function .... 63 

84-86. Delaunay's canonical system of equations deduced. . . 64 

87-92. Observations on the previous results 66 

93-105. (ii) The methods of Jacobi and Lagrange 67 

94. The dynamical methods of Hamilton and Jacobi ... 68 

95-97. Elliptic motion by Jacobi's method 69 

98. Variation of arbitrary constants by Jacobi's method . . 71 

99, 100. Lagrange's method 73 



CONTENTS. 



XI 



ARTS. 

101, 102. 
103. 
104. 

. 105. 



Pseudo-elements and ideal coordinates 

Lagrange's canonical system 

Hansen's extension to the method of the variation of arbitrary 

constants 

Beferences to memoirs and treatises ..... 



PAGE 
74 
76 

76 

77 



CHAPTER VI. 

THE DISTURBING FUNCTION. 

106, 107. Development in powers of the ratio of the distances of the Sun and 

the Moon 78 

108-121. (i) Development of R necessary for de Ponte'coulant's equations. 

Properties of R 79 

108. Development in terms of polar coordinates .... 79 

109, 110. the elliptic elements and the time 80 

111, 112. Form of the development 81 

113. Connection between arguments and coefficients . . . 81 

114. De Pont^coulant's expansion 82 

115, 116. Deduction of the disturbing forces ...... 83 

117-120. Eelations between the orders of the coefficients in. the dis- 
turbing function and those in the coordinates ... 84 

121. The second approximation to the disturbing function and to 

the forces 87 

122, 123. (ii) Development for Delaunay's theory 88 

124-126. (iii) Development for Hanson's theory 89 

127. (iv) Development for Laplace's theory 92 

128. (v) Development for the method with moving rectangular coordinates 92 



129. 

130-133. 
134-138. 
134-136. 

137. 

138. 

139-141. 
139, 140. 

141. 

142. 
143, 144. 

143. 

144. 
145-147. 

148. 



CHAPTER VII 

DE PONTlSCOULANT'S METHOD. 

Summary of the previous results required .... 
Preparation of the equations of motion. Order of procedure 
(i) Variational inequalities 

Second approximation 

Third approximation 

Results 

(ii) Elliptic inequalities. Motion of the perigee . 

Second approximation ...... 

Third approximation to the motion of the perigee 
(iii) Mean-period inequalities ....... 

(iv) Parallactic inequalities 

Second approximation 

Third approximation and results .... 
(v) Principal inequalities in latitude. Motion of the node . 
(vi) Inequalities of higher orders 



93 

93 

96 

96 

98 

99 

100 

100 

102 

103 

104 

104 

105 

106 

109 



xii CONTENTS. 

ABTS. 

149. Summary of results 110 

150. Direct deduction of the terms in the disturbing function required for 

special cases Ill 

151-153. Be Ponte'coulant's System du Monde 112 

154. Slow convergence of the series for the coefficients . . . .113 



CHAPTER VIII. 

THE CONSTANTS AND THEIR INTEKPRETATION. 

155. The questions to be considered 115 

156-161, Signification of the constants present in the final expressions for the 

coordinates 116 

162-164. Determination of the numerical values of the constants by observa- 
tion. The values of the solar constants 121 

165. Mean Period and Mean Distance 124 

166. The Variation, variational inequalities and the variational curve . 124 

167. The Parallactic Inequality, parallactic inequalities and the parallactic 

curve 125 

168. Methods for the determination of the ratio of the masses of the Earth 

and the Moon 127 

169. The Principal Elliptic Term, elliptic inequalities, the Evection and 

the motion of the perigee 127 

170. Bepresentation by means of variable arbitraries 128 

171. The Annual Equation and mean-period inequalities . . . .129 

172. Inequalities in latitude and the motion of the node . . . .130 

173. Magnitudes of the principal inequalities. References to memoirs in 

which the numerical values of the constants are obtained . . 131 

174. Motions of the perigee and the node when the true longitude is the 

independent variable 131 



CHAPTER IX. 



THE THEORY OF DELAUNAY. 

175. Method used. Limitations imposed on the problem . . . .133 

176. Defect in the canonical equations previously obtained, due to the 

presence of the time as a factor when the equations are in- 
tegrated .134 

177. Method used for transforming to a new set of variables . . .134 

178. Transformation to avoid the occurrence of terms containing the time 

as a factor 135 

179. Change of notation. Signification of the symbols . . . .136 

180. Form of the development of the disturbing function. Relations be- 

tween the two sets of elements used to represent the coefficients. 137 

181. Elliptic expressions for the coordinates in Delaunay's notation . . 138 

182. The method of integration 139 



CONTENTS, 



Xlll 



ARTS. PAGE 

183, 184. Integration when the disturbing function is limited to one periodic 
term and to its non-periodic portion. The new constants of 
integration 139 

185. The omitted portion of R is included by considering the new con- 

stants variable. The resulting equations are canonical . , 142 

186. Nature of the solution obtained in Arts. 183, 184 .... 144 

187. Form of the new disturbing function 145 

188. Lemma necessary for the next transformation 146 

189. First transformation of the new equations to new variables, in order 

to avoid the occurrence of terms containing the time as a factor. 147 

190. Second transformation to new variables, so that, when the coefficient 

of the periodic term previously considered is neglected, the new 
equations shall reduce to the old ones . . . . . .148 

191. Relations connecting the old and new variables, and the old and new 

disturbing functions . . . . . . . . . .149 

192, 193. Application to the calculation of an operation 150 

194-196. Particular classes of operations . . . . . . . .15.3 

197. Delaunay's general plan of procedure . . . . . . .155 

198. Integration when the disturbing function is reduced to a non-periodic 

term 156 

199, 200. Final expressions for the coordinates. Change of arbitraries. The 

meanings to be attached to the new arbitraries . . .157 

201. The results obtained by Delaunay .... ... 159 

CHAPTEE X. 



THE METHOD OF HANSEN. 

202, 203. Features of the method. History of its development . . . .160 
204. Change of notation 161 

205, 206. The instantaneous ellipse. The equations for the functions of the 

instantaneous elements required later 10& 

207. Eeasons for the use of the method ....... 164 

208, 209. The auxiliary ellipse. Its relation to the actual orbit . , . ,164 
210. Method of procedure 100 

211-213. The equations for , v 166 

214, 215. The equation for W. Certain parts of the expression may be con- 
sidered constant in the differentiations and integrations . . 160 

216. The constants arising in the integrations 171 

217. Motion of the plane of the orbit. Definitions. Mean motions of the 

perigee and the node 172 

218, 219. Equations for P, , K in terms of the force perpendicular to the 

plane of the orbit, First approximation to P, Q, K . . .174 
220, 221. Form of the development of the disturbing function . . . .177 
222, 223. Expression of the derivatives of the disturbing function with respect 

to P, Q, K 3 in terms of the force perpendicular to tho plane of 

the orbit 179 

224. First approximation to the disturbing function and to the disturbing 

forces X81 



XIV 



CONTENTS. 



ARTS. PAGE 

225. Expression of the first approximation to the disturbing forces in the 
plane of the orbit, in terms of the derivatives of the disturbing 
function in its developed form 182 

226-228. First approximation to the equation for W, Method for the calcula- 
tion of the equation 182 

229, 230. Integration of the equation for W. Determination of y and of the 

form of the arbitrary constant. First approximation to W . . 185 

231, 232. Integration of the equation for z. Signification to be attached to the 

new arbitraries and to the elements of the auxiliary ellipse . . 187 

233, 234. Integration of the equation for v. Determination of the arbitrary 

constant in terms of the other arbitraries 188 

235. Equations for P, , K 190 

236. Effect of the motion of the ecliptic 191 

237, 238. The first and second approximations to P, , K. Determination of 

a, rj and of the new arbitraries 191 

239, 240. Method of procedure for the higher approximations . . . .192 
241. * Beduction to the instantaneous ecliptic 194 



CHAPTER XL 



METHOD WITH RECTANGULAR, COOKDINATES. 

242. General remarks on the method. Limitations imposed . . .195 

243. History of its introduction 196 

244-246. The intermediate orbit. Equations for finding it 196 

247-256. (i) Determination of the intermediary, or of the inequalities depend- 
ing on m only 198 

247, 248. Form of solution required 198 

249-252, Equations of condition for the coefficients .... 200 
253, 254. Method of finding the coefficients from the equations of con- 
dition. The parameter in terms of which the series 
converge most quickly 203 

255. Determination of the linear constant 205 

256. Transformation to polar coordinates 206 

257-274. (ii) Determination of the terms whose coefficients depend only on m, e. 

Complete solution of the equations (1). Motion of the perigee . 200 
257, 258. Form of the general solution of equations (1). Equations of 

condition between the coefficients ..... 206 
259-261. Coefficients depending on m and on the first power of e. 

Equations of condition. Signification to be attached to 
the new arbitrary constant of eccentricity . . . 209 
262-264. Determination of the principal part of the motion of the 

perigee. Equation for the normal displacement . .211 
265" Calculation of the known quantities present in this equation 215 

266, 267. Determinantal equation for c. Properties of the infinite 

determinant. Deduction of a simplified equation for c 216 

268. Convergency of an infinite determinant . . . . .219 

269, 270, Development of an infinite determinant 220 



CONTENTS. 



XV 



ARTS. 

271, 272. Application to the determinant A(0) ..... 

273, 274 Determination of the coefficients depending on powers of e 

higher than the first. New part of the motion of the 
perigee. Further definitions of the linear constant and 
of the constant of eccentricity ...... 

275, 276. (iii) Determination of the terms whose coefficients depend only on 
w> e ' ............. 

277. (iv) Determination of the terms whose coefficients depend only on 

m, I/a' ............ 

278-286. (v) Determination of the terms whose coefficients depend only on 
>y ............. 

278. The equations of motion ........ 

279, 280. Terms dependent on the first power of y only. Principal 

part of the motion of the node ...... 

281. Signification to be attached to the constant of latitude . . 

282. Terms of order y 2 ......... 

283-285. Terms of order y\ Determination of the new part of the 

motion of the node ........ 

286. Further definition of the constant of latitude .... 

287. (vi) Terms of higher orders ......... 

288, 289. Connections of the higher parts of the motions of the perigee and the 

node with the non-periodic part of the parallax. Adams 3 theorems . 



PAGE 

222 



223 
225 

227 

228 
228 

229 
231 
231 

231 
233 

234 

234 



CHAPTER XII. 



THE PRINCIPAL METHODS. 

290. Newton's works ........... 237 

291. Clairaut's theory . .......... 238 

292. D'Alenibert's theory ........... 239 

293. Euler's first theory ........... 239 

294. Euler's second theory ........... 240 

295. Laplace's theory ..... ...... 242 

296. Discoveries of Laplace. The secular acceleration ..... 243 

297. Damoiseau's theory ...... ..... 244 

298. Plana's theory ............ 244 

299. Poisson's method ...... ..... 245 

300. Lubbock's method. Other theories. Airy's method of verification . 245 

301. [References to tables of the Moon's motion ...... 246 

302. Bemarks on the various methods ... ..... 246 



CHAPTER XIII. 



PLANETARY AND OTHER DISTURBING INFLUENCES. 

303. The method to be adopted 

304-307. General method of integration founded on Delaunay's theory 
308-313. Method for obtaining the planetary inequalities . 



248 
248 
252 



XVI 



CONTENTS. 



ARTS. PAGE 

308. The disturbing functions 252 

309. Separation of the terms in the disturbing functions. Their 

expressions in polar coordinates 253 

310, 311. Development of the disturbing functions. Connections be- 

tween the arguments and the coefficients . . . 255 

312. Example of an inequality due to the direct action of Yenus . 258 

313. Case of the indirect action of a planet. Example of an in- 

equality due to the indirect action of Venus . . . 259 
314-316. Inequalities arising from the figure of the Earth. Determination of 

the principal inequalities 260 

317, 318. Inequalities arising from the motion of the Ecliptic. Determination 

of the principal inequality in latitude 263 

319-322. Secular acceleration of the Moon's mean motion. Determination of 

the first approximation to its value. Effect of the variation of 

the solar eccentricity on the motions of the perigee and the node 265 



I. REFERENCE TABLE OF NOTATION . 
II. GENERAL SCHEME OF NOTATION 
III. COMPARATIVE TABLE OF NOTATION 
INDEX OF AUTHORS QUOTED 
GENERAL INDEX 



270 

272 
273 

274 
275 



ERRATA. 



Page 5. Head-line. For MOON'S read SUN'S. 
60. Line 1. a a 2 . 
108. 35. 
109. 8. 
10. 



CHAPTER I. 

FORCE-FUNCTIONS. 

1. THE Newtonian law of attraction states that either of two particles 
exerts on the other a force which is proportional directly to the product of 
their masses and inversely to the square of the distance between them. Let 
5 be the force between two particles of masses m, m f placed at a distance r 
from one another, then 

ft k m ^L 

r a ' 

where k is a constant depending only on the units employed. It is known 
as the Gaussian constant of attraction. 

If we use any terrestrial unit of mass, k will vary directly as the unit of 
mass and the square of the unit of time and inversely as the cube of the 
unit of length. 

The accelerative effect of the force exerted by w on 'm' is 



The Astronomical unit of mass corresponding to given units of length and 
time, is obtained by so choosing this unit that /<?= 1. Since SI is an accelera- 
tion, when the units of mass, length and time vary, the astronomical unit of 
mass varies directly as the cube of the unit of length and inversely as the 
square of the unit of time. It is used very largely in theoretical investiga- 
tions in astronomy and the frequent repetition of the constant k is thereby 
avoided. 

For further remarks on this unit see E. J. Routh, Analytical Statics^ Yol. II. pp. 1, 2, 3. 

For the sake of brevity the term ' force ' is often used to indicate ' accele- 
rative effect of a force/ In general, no confusion will arise in this use of the 
word. With an astronomical unit of mass, an acceleration may appear either 
as a mass divided by the square of a length or, in its usual form, as a length 
divided by the square of a time. 

* <^f _ 

^ B. L. T. 1 



2 FOECE-FUNCTIONS. [CHAP. I 

2. The general problem of Celestial Mechanics consists in the determina- 
tion of the relative motions of p bodies attracting one another according to 
the Newtonian law. This problem is not able to be solved directly : in order 
to deal with it, certain limitations must be made. 

The first simplification which we shall introduce, is made by eliminating 
from consideration the effects of the size, shape and internal distribution of 
mass of the bodies. A well-known theorem in attractions states, * that a 
spherical body, of which the density is the same at the same distance from 
its centre, attracts a similarly constituted spherical body as if the mass of 
each were condensed into a particle situated at its centre of figure/ The 
larger bodies of our solar system differ but little from spheres in shape 
and their centres of mass are certainly but little distant from their centres 
of figure if at all ; the shapes of very small bodies when under the attraction 
of a large one play little part in their motions. We assume therefore that 
the bodies are spherical and attract one another in the same way as particles 
of equal mass situated at their centres of figure, With these limitations 
the problem is known as that of p bodies. 

Again, owing to the conditions under which the bodies of our solar system 
move, we are farther able to divide the problem of p bodies into several others, 
each of which may be treated as a case of the problem of three particles, or, as 
it is generally called, the Problem of Three Bodies. 

The greater part of the Lunar Theory is a particular case of the Problem 
of Three Bodies ; it involves the determination of the motion of the Moon 
relative to the Earth, when the mutual attractions of the Earth, Moon and 
Sun, considered as particles, are the only forces under consideration. When 
this has been found, the effects produced by the actions of the planets, the 
non-spherical forms of the bodies eta, can be exhibited as small corrections to 
the coordinates. 

3. We proceed to consider the impressed forces in the problem of three 
particles. There are two methods of treating them, from the combination 
of which a suitable form of force-function can be obtained. In the first 
method, we find the accelerations due to the forces acting on the Sun and 
Moon relatively to the Earth, and in the second method, those acting on the 
Moon relatively to the Earth and on the Sun relatively to the centre of 
mass of the Earth and Moon. 

(i) The forces relative to the Earth. 

In figure 1, let J& T , M t m f , G respectively denote the places of the Earth, 
Moon, Sun and the centre of mass of the Earth and Moon. 

Let the masses of the Earth, Moon and Sun measured ia astronomical 



2-3] 



FORCES RELATIVE TO THE EARTH. 
M 




E 



m 1 

r' 2 



Fig. 1. ' 

units be E, M, m'. Let the distances, ME, m'E, m'M be r, /, A : these must 
"always he considered as positive quantities. 

We shall only deal here with the accelerations due to the forces and, in 
accordance with the remark made in Art. 1, speak of them as/or<m 

The forces acting on E are M/r 2 in the direction EM and m'/r' in the 
direction Em'. Similar expressions hold for the forces acting on M and w', an 
shown in the figure. Hence the forces acting, relatively to the Earth, are, 
On the Moon, On the Sun, 



in the direction of 



Mnf, 



m'ti; 



K7 JL ,/ 

- in the direction of m'E, 

M 



M 



ME. 



Take the Earth as the origin of a set of rectangular axes. Let #, y, z> and 
of, y', z' } be the coordinates of M and m', and 3c , g), 3, F, g) x , 3 ; , the forces 
acting on M and m' relatively to E, referred to these axes. We have then, 



r 2 r A s A 
__ / M of so 

t' /a T' A 2 A T* T 

with corresponding expressions for ), 3 2)'; 3^ 

If we put 

vM <m!_ wf + yy' + , 



we shall have 



, E + m f M T,^ aw' 4- vv 7 4- ##' 

^..,.., TO 1...- j ^^ ,.,, /i/r .___j..y^ 

^_ L.., . T m,,nr,j -.., _^~ I'.. - JJJ^ ' 

r A r tt 



3'- 



The expressions denoted by F, I' are the force-functions for the motions 
of the Moon and the Sun relative to the Earth. 

12 



4 FORCE-FUNCTIONS. [CHAP. I 

Let the cosine of the angle MEmf be denoted by S. Then 

QGOC' + yy' H- zz' = rr'$, 

and therefore F, F f are independent of the particular set of axes fixed or 
moving which may be used. 

The expressions for F, F' are quite general, and when they shall have been 
substituted in the six differential equations which represent the motion of M 9 
m' relative to E> we shall have the general equations of motion for the problem 
of three particles. (It has been tacitly assumed that the system is free from 
any external influence having .a tendency to disturb the relative motion 
of the three particles.) It is not possible to integrate these equations 
except in special cases. In order to obtain F in the most suitable form, 
certain observations must be made before proceeding further, 

4. In finding the motion of the Moon, the first assumption usually made 
consists in considering the motion of the Earth about the Sun or, what is the 
same thing, of the Sun about the Earth as previously known and therefore 
the coordinates #', ?/, z f as known functions of the time. Now the function 
F' which is used to find the motion of the Sun contains the unknown 
coordinates of the Moon. It becomes necessary to see what effect those terms 
in F 1 which have If as a factor produce in the relative motion of the Sun. 

If we limit F' to its first term (E + w')/r', the resulting motion of m' 
about E will be elliptic. Now in the lunar theory, r is very small compared 
to r f or A while r 1 and A are of nearly equal magnitude. Hence it appear** 
that the third term in F' is that which will cause most deviation from elliptic 
motion. This term is due to the force if/r 2 , which is one of the forces acting 
on E. If we had referred the motion of m' to G this term would not have 
entered. We shall therefore find the motion of m! relative to and 
the other terms in the new force-functions for m' and M. 



(ii) The forces acting on m 1 relatively to G and on M relatively to R. 

5. We have EQ=**EM 

& + M 

and therefore the accelerations of G relative to E are M/(E+M) times those 
of M relative to E. Hence the force on m 1 relative to G, parallel to the 
axis of x, is X' - $M /(E + if ) which, by Art. 3, is equal to 



E + M 



3-5] MOON'S FORCE-JFUNCTION RELATIVE TO (?. 5 

Let #i,yi,#i,be bhe coordinates of m' referred to parallel axes through G: 
let m'G-r-L. Then 

M M M 



Jp 

and therefore x 3) = ^ -- - x, etc. 



Also 

M V / H \* f M V 

a* 



E V / E \* f E 

+ 

If therefore we put 



a .___ 

where /, A are now expressed in terras of w, ?/, z, ^, y^ z lt the differentials 

_^ ! 1 w in fc e the forces acting on m! relatively to 0. 
8^ dy l '8^ b J 

Again replacing in Art. 3, a/, y' t / by their values above, we have 
m f / E \ m r ( M 

/*_. - */' 

1 E'+M 



The forces jf, ^, 3 can therefore be derived, by partial differentiation with 
respect to $, y, z, from the force-function 



m 

_ + _ _^.. .......... . + .. 



where A, / are expressed as before in terms of* 1 , y> #, ^, y l} ^. 

It is not difficult to show by expanding J?V in powers of r/r^ that it 
differs little from (E+M+m')/^. We have, putting cos M Gwf = & , 



whence 

Similarly 



11 E r f E \ 2 r a 

A = n + ^TS n 2 Sl + U+Jf J n 5 (l^ 2 - i) + 



6 FORCE-FUNCTIONS. [CHAP. I 

and therefore 

E.M r* 



Now r/rj differs little from ^ at any time and M/E is approximately 
^. Hence the order of the second term of FI relatively to the first is 
roughly 

JL l 1 

80 400 2 ~~ 12,800,000 ' 

a quantity which may be neglected here. We can then neglect the second and 
higher terms of F^ and consequently assume that the motion oftri about is 
an ellipse. 

6. The Moon thus produces little effect on the motion of the centre of mass of the 
Earth and Moon and consequently we can consider this point as moving in an ellipse in 
accordance with Kepler's laws. The actions of the planets, however, produce effects which 
become very marked after long periods of time. These effects, being exhibited by terms 
in the expressions of x lf y t , z l9 are transmitted to the Moon through I*. Wo ought 
properly to have considered the problem of p bodies or even less generally, to have 
included in the force-functions F. F r the forces produced by all the planets. But the 
action, both direct and indirect, of the other planets on the Moon is so small that in most 
cases it may be neglected : where it should be considered, it is always possible to do so in 
the form of small corrections. The variations produced in the motion of the Earth or of G 
by the other planets belong properly to the planetary theory and need not be considered 
here. All we require here concerning the motion of the Earth is, that it should be 
supposed known. The only reason for considering its force-function at all, is to see how 
the unknown coordinates of the Moon enter therein and to eliminate them as far as 
possible. 

It is a remarkable fact, and one which shows the extraordinary care required in the 
treatment of the lunar and the planetary theories, that the direct actions of the planets on 
the Moon are in general much less marked than their indirect actions as transmitted through 
the ^Earth. These indirect effects, though sometimes too small to be detected in the 
motion of the Earth, may become sufficiently large to be observed in the motion of the 
Moon. 

7. With the assumption that the motion of & is elliptic, we ought 
properly to use the force-function F l for finding the motion of the Moon. 
But as it is generally found more convenient to use F, we shall expand both 
functions in order to see in what way the latter must be corrected to give 
the former. 

The expansions of I// and I/A given in Art. 5, furnish 
1 1 lrl 1 1 r 2 



5-7] CORRECTIONS TO BE MADE TO F. 

When this is substituted in F l9 the term 



will contribute nothing to the forces since it is independent of the coordinates 
of the Moon : it may therefore be omitted. We shall then have 



Tr> , 1X , E-M r 3 ~ 3 ~ N 1 

^^^ 



Again, we find by expanding F in powers of r/r' and omitting the useless 
term m'/r', 



Now a/, y', z' ', r' refer to the motion of the Sun about the Earth, and 
*i, l/i, %> n to that about G, and these are contained in F, F l respectively in 
the same way. If therefore in F we consider o/ 9 y', /, r' to refer to the motion 
about ff, which motion is supposed known in terms of the time, F, F l will be 
the same except as regards the ratio M/E. 

Let now a, a' be the mean values of r, r x (or r'). It will be found later 
that expansions will be made in powers of a/a' and that the parts resulting 
from the term in F, 

771 l! f3.Sf2_l.\ 

f 72 (< 4J> 
do riot contain a/a'. The parts resulting from the term in F, 



contain a/a in its first power. Comparing F, ^ we see that if in the results 
produced by using F l} in addition to the change noted above, we replace the 
ratio a/a' wherever it occurs by 

E-Ma^ 
E'+Ma" 

we shall be able to use F instead of F^ The terms containing (a/a') 2 we 
should multiply by (E - M)*/(E + M ) 3 , and so on. 

This does not quite account for all the differences between F, F^ The 
terms of next highest order in F, F l are respectively 



E*-EM+M*m' 
and + . 



8 FORCE-FUNCTIONS. [CHAP. I 

The first expression must be multiplied by 



Hence, after the changes previously noted, there will still remain to be added 

to F, the term 

EM m! 



' + ) 

Now the order of this remainder, compared to the first of those terms 
in F which depend on the action of the Sun, is 

M o 2 ^ 1 

E a' 2 """ 12,800,000 

approximately. The largest periodic term produced by the Sun has a 
coefficient in longitude of less than 5000", so that it is very improbable 
that such a small quantity can produce any appreciable effect. The effect of 
the differences in the higher terms will be still smaller. We may therefore 
conclude that the replacing of a/a' by a (E - M )/a' (E 4- M) will sufficiently 
account for the remaining differences. 

8. The problem of the Moon's motion is therefore reduced to the deter- 
mination of the motion of a particle of mass M, under the action of a 
true force-function MF, where 



JL m ' _ 



in which af 9 y f , / are the known functions of the time obtained from purely 
elliptic motion. We shall now consider E instead of as the origin. 

Let# + Jf = /&, and put F=^ + R. 

The quantity E depends on the action of the Sun and is known as the 
Disturbing Function. 

There is now no further need to consider the functions F lf F', .F/. 

9. The distinction between the planetary and the lunar theories is one of analysis only, 
based on certain facts deduced from observations on the nature of the motion of the bodies 
forming the solar system, Both theories are particular cases of the problem of three 
bodies which, owing to the deficiencies of our methods of analysis, is at present only 
capable of being solved by tedious expansions, even when the bodies are so favourably 
placed relatively to one another as those which come within the range of observation* 
Almost nothing is at present known of the possible curves which bodies of any masses and 
placed at any distances from one another may describe. In the case of the planetary 
theory, where it is required to investigate the perturbations of one planet by another, or 
more properly, the mutual perturbations of two planets, we can use the functions F ly /\' 



^-9] THE LUNAR AND THE PLANETARY THEORIES. 9 

where E stands for the mass of the Sun and M y m f for the masses of the planets. For in 
this case the ratios (E~~M}I(E+M\ (E-m')l(E+m f ) are so near unity owing to the great 
mass of the Sun compared to that of any of the planets and the actual perturbations are 
so small, that the differences of these ratios from unity can in general he neglected. 

Bough observations extending over a sufficient interval of time show very quickly that, 
during that interval at least, the planets describe curves which approximate more or less 
closely to circles of which the Sun occupies the centres. A more exact representation of 
their motion is given by Kepler's well-known laws. The known satellites, and in 
particular the Moon, also approximately satisfy these laws with reference to the planets 
about which they move, but for a shorter time; they also exhibit larger deviations from 
them. Observation too has shown that the eccentricities of their ellipses and the mutual 
inclinations of the planes of motion of all the principal planets oscillate about mean values 
which are in no case very great. The same is true of the Moon and of most of the 
satellites with reference to the orbits of their primaries. 

It is assumed then that expansions may be made in powers of the eccentricities and of 
suitable functions of the inclinations. But when we are considering the perturbations of 
one body produced by another, it has just been seen that expansion will also be made 
in powers of the ratio of the distances of the disturbed and disturbing body from the 
primary. 

It is at this point that the first separation of the lunar and the planetary theories takes 
place. In the lunar theory, the distance from the primary the Earth of the disturbing 
body the Sun is very great compared to that of the disturbed body the Moon, and 
we naturally expand first in powers of this ratio in order that we may start with as few 
terms as possible. In the planetary theory, the distances of the disturbed and the disturbing 
bodies two planets from the Sun which is the primary, may be a large fraction. For 
example the mean distances of Venus and the Earth from the Sun are approximately in the 
ratio 7 : 10, and in order to secure sufficient accuracy when expanding in powers of this 
ratio, a very large number of terms would have to be taken. In the case of the planetary 
theory then, we delay expansion in powers of the ratio of the distances as long as possible 
and form the series first in powers of the eccentricities and inclinations. 

Again, in the lunar theory the mass of the disturbing body is very great compared to 
that of the primary, a ratio on which it is evident the magnitude of the perturbations 
greatly depends. On the other hand, in the planetary theory the disturbing body has a 
very small mass compared to that of the primary. From these facts we are led to expect 
that large terms will bo present in the expressions for the motion of the Moon due to the 
action of the Sun and, from the remarks made above, that the later terms in the expansions 
will decrease rapidly ; and in the planetary theory we expect large numbers of terms of 
nearly the same magnitude, none of them being very great. This expectation however is 
largely modified by somo further remarks about to be made. 

In the integrations performed in both theories the coefficients of the periodic terms by 
moans of which the coordinates are expressed, become frequently much greater than might 
have been expected a priori. In the lunar theory, before this can happen in such a way 
as to cause much trouble, the coefficients have previously become so small that it is not 
necessary to consider such terms beyond a certain limit. Suppose in the planetary theory 
that n, ri are the mean motions of the two planets round the primary. Then coefficients 
will, for example, continually be having multipliers of the form ri/(in i'ri) and n' t *j(ini'n > }'* 
produced by integration (i, i! positive integers). In general, the greater i, i' are, the 
smaller will be the corresponding coefficient. But owing to the two facts that the ratio 



10 FORCE-FUNCTIONS. [CHAP. I 

a : a! may be nearly unity and that the ratio n : n' may very nearly approach that of two 
small whole numbers, a coefficient may become very great. For example, five revolutions of 
Jupiter are very nearly equal in time to two of Saturn, while the ratio of their mean 
distances is roughly 6 : 11. One result is a periodic inequality which has a coefficient of 
28' in the motion of Jupiter and 48' in that of Saturn. Such inequalities take a long 
period to run through all their values, the one in question having a periodic time equal to 
about 76 revolutions of Jupiter or 913 years, so that the variation due to this term in one 
revolution is small, The periods of the principal terms in the motion of the Moon 
are generally short but some of them have large coefficients, so that the deviations from 
elliptic motion are well marked. 

One of the greatest difficulties in the planetary theory, perhaps owing chiefly to our 
methods of expression, is the presence, in the values of the coordinates (when the latter 
are obtained as functions of the time), of terms which increase continually with the time, 
and thus, after the lapse, of a certain interval, render the expressions for the coordinates 
useless as a representation of the motion. Whether such terms can be eliminated by the 
use of suitable functions is not at present certain. Kecently the work of Gylden* has 
gone far in the direction of achieving this object. In the lunar theory, the difficulty also 
occurs, but, as regards the perturbations produced by the Sun, is easily bridged by 
means of a slight artifice. 

It will readily be conceived, from the few statements made here, that in general, 
different methods will be required for treating the two problems of a satellite disturbed by 
the Sun and of a planet disturbed by another planet. When the disturbing planet is an 
inferior one, we use a function corresponding to F, but we have then to develope in powers 
of a' /a instead of a/a'. In the cases of some minor planets again special methods are 
required owing to the great eccentricity of their orbits. All the problems are essentially 
the same : the analytical difficulties alone compel us to treat them differently. 

As concerns questions of purely mathematical interest, the planetary theory has in the 
past opened out a larger field for the investigator than the lunar theory. In the last few 
years however the researches of Hillt, Adams t, Poincar^t and others have brought the 
latter problem forward again and given it a new stimulus. 



10. Let (fla^i), (tt*y*Zz), (#sya#s) be the coordinates, referred to any 
origin and any rectangular axes, of three bodies of masses m ly m 2 , m 3 which 
attract one another according to the Newtonian law. Let r ia , r<%, r u "be the 
distances between them. The force acting on m^ resolved parallel to the axis 
of a?, is 

m i m 2 #2 ~~ #1 Willis X S X l 

^" ?ia r 13 2 r 13 ' 

the forces acting on m z are therefore derivable from the function 



* Acta Mathematica> Vols. vn., XL, xv. etc., also Trait& analytique des Orbites absolues^ etc. 
Stockholm, 1893. 

t See Chapter xi. 

$ Les Mgthodes nouvelles de la Mecanique Celeste, Paris, Vol. i., 1892, Vol. n., 1893. These 
researches are outside the scope of this book. 



9-10] FORCE-FUNCTION FOR p BODIES. 11 

or, since ?~ 23 does not contain the coordinates a^ 9 y lt z ly from the function 



, 



The symmetry of this expression shows that it is also the force-function for 
the motions of m 2 and m 3 , 

Generally, if there be p bodies, it is easily seen that 



, 
where i,j receive values 1, 2... p, the terms for which i = j being excluded. 



CHAPTER II. 

THE EQUATIONS OF MOTION. 

11. THE methods used in the solution of the lunar problem may be 
roughly divided into four classes. In the first class we may place those 
methods in which the time is taken as the independent variable, the radius 
vector (or its inverse), the true longitude measured on a fixed plane, and 
the tangent of the latitude above this plane, as dependent variables; the 
equations of motion are expressed in terms of these quantities and solved by 
continued approximation with elliptic motion as a basis, so as to exhibit 
these coordinates as functions of the time and the arbitrary constants intro- 
duced by integration. Under this heading we may include the theories of 
Lubbock and de Pont&mdant. In the methods of the second class four 
similar variables are used, but the true longitude is taken to be the 
independent variable and the other three variables are expressed in terms 
of it. A reversion of series is finally necessary in order to obtain the 
coordinates in terms of the time. Clairaut, d'Alembert, Laplace, Damoiseau 
and Plana followed this plan. 

A third class will embrace those methods in which the Moon is supposed to 
be moving at any time in an ellipse of variable size, shape and position ; this 
is known as the method of the Variation of Arbitrary Constants and it was 
used in different ways by Euler in his first theory, by Poisson and with great 
success by Hansen and Delaunay. In the fourth class may be placed those 
theories in which rectangular coordinates referred to moving axes are used, 
with the time as independent variable. Euler's second theory and the general 
methods resulting from the works of Hill and Adams may be included under 
this heading. 

We shall give here the equations of motion used by de Pont^coulant 
and Laplace and the generalised form of Hill's equations, to illustrate the 
methods of the first, second and fourth classes respectively: the principles 
which form the basis of the methods of the third class will be given in 
Chapter v. We shall also include here some considerations on the general 
problem of three bodies with special reference to the known integrals. 



11-12] 



DE PONTICOULANT'S EQUATIONS. 



18 



The methods used by de Pont^conlant, Delaunay and Hansen will be found in 
Chapters VIL, ix. and x. respectively; the methods of Hill and Adams with the 
extensions to the complete problem are given in Chapter xr. A short summary of the 
methods employed by other lunar theorists is made in Chapter xn. 

(i) De Pontdcoulanfs equations of motion, 

12. Let x, y, z be the coordinates of the Moon referred to three 
rectangular axes, fixed in direction and passing through the centre of the 
Earth. The equations of motion of the Moon will be, according to Newton's 

laws of motion, since =- , ^- , ^- are, by Art. 8, the forces parallel to the 
da dy 08 



axes, 



MX M TT- 

ox 



or by the same article, 

/ _j _, = ._,., ^ 



$ + ' 



dF 

: dy' 



dy' 



where R is supposed expressed in terms of at, y, z, x', y', z'. 

Let the plane of (xy) be the plane of the Sun's motion, supposed fixed, 
and let the axis of x be a fixed line in this plane. Let x, y, z, M, M' be the 




Fig. 2. 

points where a sphere of unit radius cuts the axes, the radius vector and its 
projection on the (asy) plane, respectively. Let 

r be the radius vector of the Moon, 

TI its projection, 

v the longitude of this projection reckoned from the axis of a?, and 

s the tangent of the latitude of the Moon above the plane of (ay). 



14 THE EQUATIONS OF MOTION. [CHAP. II 

Hence <v = xM' 9 s = tan M 'M, 
and therefore 

rcosv rsinv rs r 



If we change the variables from x, y, z to r l9 v, z> the equations of motion 
become 

ju,n . dR 



(1), 



/P)7?\ 
where f-^ j denotes partial differentiation of R when R is expressed as a 

/-) 7? 

function of r lt v, z, in contradistinction to ^ which will denote partial differ- 
entiation when R is expressed as a function of r. v } s ; the two expressions 
are easily seen to be equal though differently expressed. 

Multiplying these equations by 2ri, 2^^), 2^, respectively, and adding, we 
obtain 



, , /r> dR 7 9JR 7 dR 7 

where a It = -5 a^ + -^- r x av + -^~ ^. 

3n r^v 3,s: 

Whence, if a be a constant, we have on integration 

V + ^ + J.^-f + 2/^c : (2), 

the expression for the square of the velocity. 

The integral on the right-hand side of this equation requires explanation. 
From the way in which d'R was formed, it is evident that d'R/dt denotes 
differentiation of R with respect to t> only in so far as t is present through the 
variability of the coordinates r 1} v, z and not through its presence explicitly in 
the coordinates of the Sun, We suppose then that d'R/dt has been formed 
in this way and we can put 

/f ^i** 

an equation which defines the meaning of f d'R. 

In order that the integration may be actually performed, d'Rjdt ought to be expressed 
in general as a function of the time only. It will, however, contain the unknown coordinates 
r l9 v, z, and its value can only be obtained by a process of gradual approximation to the 

values of these coordinates. 



12-13] DE PONT&COULANT'S EQUATIONS. 15 

Multiplying the first of equations (1) by r 1} the third by z and adding the 
sum to equation (2), we obtain 






But since we have rf + tf^r* and therefore r 1 r 1 H-r 1 2 + ^ + ^ 2 = 
the equation becomes 



........... ..-.(3). 

t r a 3r a dz 

Also from the second of equations (1), if h be a constant, we obtain 






Finally, from the third of equations (1) we have, substituting the value of 
in terms of r, s, 

P TS ^ dR (5) 

........................ 



dR fdR\ dR . dR dR dR , 

13. We must now express , ^J , ^ in terms of ^ , ^, ^ where, 

in the former three JR is considered a function of r lt v, 2, and in the latter 
three a function of r, v, s. Let Sr, 8v, S^ be any independent variations 
of n, v, z and Sr, 8v, &, BR the corresponding variations of r, u, 5, J2. 
We have 



Also, since r 2 = n 2 + ^ 2 , s = -^/n, and therefore 

8s = Sr x 



we obtain by substituting for Sr, Ss in the previous equation and equating the 
corresponding coefficients of the independent variations 8r x , 80, 8#, 



= 

3v ' dz r dr n 85 " 

These furnish immediately 



dR m __ s_ dR Vr+7 2 dR 
dr' * 9* ~V * 



; 16 THE EQUATIONS OF MOTION. [CHAP. II 

1 

I Substituting these results in equations (3), (4), (5), we obtain 



-(A), 



2/7/2V / ' ^ 
Uu / Ci 

i _i. 02 r^ P 

'dft 



! +_^ [3-R 
- "7F"J 9"^ 



*"* v 

the equations of motion sought. 

14. The first of these three equations contains no differentials, with 
respect to the time, of $, v, while the second has none of r, s : the third 
equation has differentials of both r, s. But since it is found by observation 
that the arbitrary constants introduced by integration are such that r differs 
from a constant by a small quantity only and that s is itself always a small 
quantity, we shall see that the three equations are respectively useful in 
determining the radius vector, the longitude of the projection of the radius 
vector on the fixed plane and the latitude above this plane. They will 
therefore be referred to as the radius-, longitude- and latitude- equations 
respectively. The equations (A) having been first used by de Pontdcoulant*" 
as a basis for the lunar theory, are referred to under his name. Equations 
of an almost identical form were, however, obtained by Laplace in Chap, vi., 
Book II. of the Mecanique Celeste. 

15, One of the chief difficulties of the lunar theory is the interpretation 
of the arbitrary constants arising from the integration of the differential 
equations. It is necessary, in order that we may be able to find them 
accurately from observation, to have them exhibited in such a way that their 
physical signification can be exactly fixed. Special stress is laid on this 
point. The same care is necessary also when it is desired to compare the 
results of one mode of development with another, in order that the relations 
between the constants used in the two sets of results may be determined. 

It will be noticed that the equations of motion have been reduced to two 
of the second order and one of the first ; the integration of them will there- 
fore give rise to five arbitrary constants. These five constants with the two 
A, a, already introduced, will make seven, while our original equations of 
motion three of the second order only demand six. There must therefore 
be some relation between these seven constants after the integrations have 
been performed and, to determine it, use must be made of some other 
combination of the original equations of motion. For this purpose we have 

* Systems du Monde, vol. iv. No. 1. 



13-16] LAPLACE'S EQUATIONS. 17 

from the sum of the first and third of equations (1) multiplied respectively 
t>y n, *, 



But since r 1 2 + ^ 2 = r 2 , s = tanZ7 (where, for a moment, v denotes the 
latitude) we obtain 



and therefore 

zz = rr + r 2 - f i 2 - # 2 = rr - r 2 ^ 2 = rr - r 2 s 



. . 9J? 312 9J2 

Also T! x -- h # 15- r =- . 

3rj 3# 3r 
Whence, after division by r 2 we obtain, since r x 2 = r 2 /(l + s 2 ), 



When the motion is elliptic, that is, when we neglect the right-hand 
sides of equations (A), the relation between the seven constants is very easily 
found (see Chapter in.). When the general equations of motion are treated, 
the equation (6) may be used to find the required relation. Six of the 
constants are determined in the disturbed motion so as to simplify the 
interpretation of the final results as much as possible (Chapter viil.). The 
presence of a seventh constant greatly assists us in this respect. 



(ii) Laplace's Equations of Motion. 

16. Let %, X ly 3x be the forces acting on the Moon, resolved parallel to 
the direction of the projection of the radius vector on the fixed plane, 
perpendicular to this direction in the fixed plane and perpendicular to the 
fixed plane respectively. The equations of motion will be 



^- 2 

Let M! = 1/n and r^v = H. Transforming so as to make the independent 
variable v, we obtain* from the first two equations, since dt^ 



dfv 

Tait and Steele, Dynamics of a Particle, 4th Ed., Art. 136. 



B. L. T. 



18 THE EQUATIONS OF MOTION. [CHAP. II 

Also, since H X^, we have 



__ 

JLL ~ jjt ~ t 

dv v Uj 3 

Integrating, we obtain, if h be an arbitrary constant, 



u 



and equation (7) may be written, 



Since J? = A when ^ is zero, 7i will have the same meaning as in (i) when E = 0. 
Again from the first and third equations of motion we obtain 



_ -= -- 

dv* dv dv r x 



where the values = 1 ~,ff= Vi have been substituted. Since 



~ 
there results, 



Also, since J 1 (Art. 8) is a function of , y, z and therefore of ,, , s, 
have, by the principle of virtual displacements, 



3$ 



and since Sn=-S Ml /v, 82= Ss/i^-sSuJu,*, we obtain by equating the 
coefficients of the independent displacements K, Sv, Ss, 



u, 



16-18] * EQUATIONS REFERRED TO MOVING AXES. 19 

Substituting for ty lt X l9 3i in equations (9), (10), (8), we obtain, 



= __ __ _ _ 

+ Ul " h* du, + Mut ds h*u? dv dv M \d& Ul j dv uf 



-. . 

+ " 



^.-- JL (l 4.^- 

dv hu^ \ /i 2 
which are the equations found and used by Laplace*. 

17. We have just obtained the equations of motion in the form of two 
equations of the second order and one of the first, furnishing five arbitrary 
constants on integration ; these with the constant h will form the six 
constants necessary. The form of these equations renders them very useful 
in certain departments of the lunar theory. For a complete development of 
the perturbations produced by the Sun, with the accuracy demanded by 
observation to-day, they are, nevertheless, almost excluded by the fact that, 
after u, s, t have been found in terms of v, a reversion of series is necessary 
to get v the most important coordinate in terms of t. This last process 
would probably demand as much labour as that necessary to find the other 
coordinates and the time in terms of v. 



(Hi) Equations of Motion referred to moving rectangular axes']-. 

18. Take axes EX, ET moving in the fixed plane of (ooy) with angular 
velocity ri round the axis of z (Fig. 2, p. 13). Let X, F, z be the coordinates 
of the Moon and JX 7 , F', those of the Sun referred to these axes, so that, as 
before, we assume the fixed plane of (XT) or (ooy) to be the ecliptic. The 
equations of motion of the Moon will be, according to the usual formulae for 
accelerations referred to moving axes, 



.*7-l 
n r- dY 

W 
If we define the function F', by the equation, 



.(12). 



(13), 



* Mgcanique Celeste, Books n. 15 ; vn, 1. 

*(- See remarks on Euler's second theory in Chapter XH. 

22 



20 THE EQUATIONS OF MOTION. * [CHAP. II 

the three equations of motion may be written, 



Let now v = X+7 V^~l, <r = -3T - F V -1. 

We have then 



Multiply the second equation of motion by V 1 : the first two equations of 
motion become, by addition and subtraction, 



-, -- -. 

ocr ov 

Again put = 



and change the independent variable from t to f; ??., ^ are two constants at 
present not defined. Let 



- -, 

at; 

We have 

~ 



according to the usual notation for operators. Substituting these values in 
the equations for v, a-, they become, 



19. We must now develope F'. From the expression of F given in 
Art. 7, we have, since E 



Let now ', 2o' be the mean motion and major axis of the ellipse 
described by the Sun. It will be shown in Chapter m. that wo -can put 
m - n o 3 . Let = ^. With these substitutions, we obtain from equation 
(13), since m = n'/v, X 2 + P = v<r, r* = 






.(15), 



18-20] EQUATIONS REFERKED TO MOVING AXES- 

where 

^ {6,- 



In this last expression ft^ stands for the terms of degree) J> "^ l^wtw niul 
products of u, <r, #, and therefore of degree - p + 2 in power* of ' (r I* of thw 
same degree as a!). 

Substituting the value of F given by the second of cquatinH (15) in 
equations (14), the latter become, since r 2 = vcr + # 2 , 

icv 911 \ 

>(i;+o-)-^ = - ^ 

V ..*.***+ % * ' i 

aO 

fin' (w + r) - ^ = - g-- J 
and the third equation of motion, after changing the iiidependont, vnrirtbt\ w 



-- 

r 3 2 9^r 

The equations (17), (18) form the basis for a general t,ro.atnumt of llw 
lunar theory. We shall now give Hill's transformation in HH intmt. K^ii^iiit 
form. 



20. Multiply the first of equations (17) by cr, the Hoeowl by n atttl 

subtract. We obtain 

- 



|m 9 (i/ a - <r 2 ) = 



Again, adding equations (17) with the same multiplier** to CHjtuil ,iot* (1H) 
multiplied by 2#, we have 

- 2m (vD<r - crDv) + f m 2 (v + <r) a 2m 81 * 9 *? 

80 



The last result is arrived at by applying Euler's theorem for 
functions to equation (16). 

Further, multiplying equations (17) and (18) by D<r, Dv t 2I) riK|**:*tIviV|| f 
and adding, the result may be put into the form * 



D \Dv . Do- + (D*) 2 + f m 2 (v -f cr) 2 - mW + - 1 = - ( | Do* + !" /> w 4, ? ? * /It 'I 

L r J v^ " <#/ " && ' ' / 

............... (21). ' 

* The termDu.Dtr is the product of Du and Dcr and must not be oonfrmrxcUkl wlili I> 
Similarly (Dz) 



22 THE EQUATIONS OF MOTION. [CHAP. II 

But since O is expressible in terms of the coordinates of the Moon and 
the Sun and since those of the latter are supposed known functions of the 
time or of f, we may suppose fi expressed as a function of v, &, z, f. We 
therefore have 

an an an , 



But ^=?=Afi 

where D t denotes the operation D performed on il with reference to the 
portions which contain t (or f ) explicitly, and D" 1 is the inverse operation 
to D. Substituting these results in equation (21) and integrating, we have 

Dv . Da + (Lzf + f m 2 (v -f a-) 2 - mV + = C - ft + D"" 1 (A ft). . -(22), 
where (7 is a constant. 

Adding this to equation (20), we obtain (since fl = ft a + ft 8 +...) an 
equation which may be put into the form, 

Z> 2 (v<r + 2*) -Dv.Do*- (jRgr)' - 2m (vD<r - <rDv) + m 8 (v + <r) 3 - 



Aft) ...... (23). 

2 

The three equations (18), (19), (23) are the generalised form of Hill's 
equations (see Chapter XL). 

21. It will be noticed that these three differential equations are each of 
the second order and therefore on integration will furnish six arbitrary 
constants. A constant of integration G has already been introduced, while tc 
or fi has disappeared from (19), (23), the equations which furnish the motion 
in the fixed plane. There is therefore a relation containing tc, between these 
seven arbitrary constants. This relation will be determined from one of 
our original equations of motion. The constants n, t Q introduced into the 
equations will be defined in Chapter XL as two of the arbitraries of the 
solution. 

The advantage possessed by the equations (19), (23), which are of principal 
importance for the determination of v, <r, arises chiefly from the fact that 
their left-hand members are homogeneous quadratic functions of v, <r, z. 
When we neglect the parallax of the Sun, that is, when we consider the 
Sun to be at an infinite distance, the right-hand members of the equations 
are also of the same form except as regards the constant C. Even when 
terms depending on the distance of the Sun are included, since it is not 
generally necessary to take them beyond the order l/a /s , the terms thus 
added will only be of the third and fourth degrees in v, er, z. Equation (18) 



20-22] PARTICULAR CASES. 23 

has not this form, but it is not difficult to obtain an equation of a form similar 
to (19), free from the divisor r 3 . 

The remarks of this last paragraph apply equally to equations (19), (23), 
when they are expressed in terms of the real variables X, Y } z, t. The use of 
the conjugate complexes v, <r enables us however to put our solution in an 
algebraic form, It will be seen later that X, Fare expressible respectively by 
means of cosines and sines of the same multiples of t. As a consequence of 
this, v, cr are expressible in series, with " as the variable and with real 
coefficients. Also, cr can be derived from v by putting I/? for so that 
it is only necessary to calculate either v or or. The advantage of algebraic 
over trigonometric series, when the multiplication of two series is in 
question, will be easily understood. 



Since Dv.J)<r + (Dzf = - (I 2 + 

the equation (22) is the Jacobian integral referred to moving axes. When the solar 
eccentricity is neglected the term D~~ l (D t &) vanishes. We may therefore look upon this 
term as the variation of the constant of Energy due to the eccentricity of the Sun's 
orbit. 

Also since v<r + %*> = r 2 , vDv vJDv = ( IfX- JLT)/j>, 

we can express immediately equations (19) and (23) in a real form. 

Although, either of the equations (1*7) is, since v, cr are complex quantities, a complete 
substitute for the first two of equations (12) the same cannot "be said of equations (19), (23). 
The reason of this is easily seen. If we give to v, o-, f their values in terms of JT, F, t, each 
of the equations (17) furnishes a real and an imaginary part. On the other hand when the 
same substitutions are made in (19), (23), the former givos an imaginary part only and the 
latter a real part only. 

22. There are two particular cases of equations (12) which require notice 
and, in order to treat them, we must know something further about the 
disturbing function. 

We have from Art. 7, putting m' = w /s a /8 (see Art. 19), 



Let v' be the true longitude of the Sun supposed to move in an elliptic orbit, 
and let the axis of X, which is rotating with the mean angular velocity of the 
Sun, point towards the Sun's mean place. If e be the angle, at time = 0, 
which this axis makes with the fixed line from which v' is reckoned, we shall 
have 

X ' = / cos (v' - n't - e'), F = r' sin (t/ - n't - e'). 

If now we neglect the solar eccentricity, these equations give 

Z' = r' = a', T = 0, rS = (XX' + FF')/r' = X, 
and the first term of R becomes n /2 (f Z 2 - 



24 THE EQUATIONS OF MOTION. [CHAP. II 

If therefore we neglect the terms beyond the first in J?, that is, if we 

neglect terms which depend on the parallax of the Sun (retaining those 
dependent on the solar eccentricity), we shall have 



The second term of this expression then vanishes with the solar eccentricity. 
Moreover since r 2 $ 2 is always a quadratic function of X, F, we can put 



in which A, 2?', C' ', K' are simple functions of the time depending on the 
solar elliptic motion. 

Substituting, equations (12) become 
!-2n'r-37i /2 jr~f AX 



r 8 



F+ 2'X + -B'Z + <7F= - 



,(24), 



in which those terms depending on the distance of the Sun are the only ones 
neglected. These equations form the basis of Adams' researches* into the 
connection between certain parts of the motions of the perigee and node and 
the constant part of l/r. 

23. A further simplification is introduced by supposing the solar eccen- 
tricity and the latitude of the Moon neglected. Giving therefore A ', J3', 0', K', z> 
zero values, the equations are reduced to the two 

X - 2fl/F~ Sn*X = 



r 8 



These play an important part in Hill's method of treating the lunar theory f. They 
are the equations of motion of a satellite disturbed by a body supposed to be of very great 
mass m', at a very great distance ', such that m'/a'****'* is a finite quantity. The 
disturbing body whose distance has just been supposed to be so great that I/a' is 
negligible, is moving with uniform velocity in a circular orbit in the plane of motion of the 
satellite and is placed on the positive half of the moving Z-axis. 

The equations admit of a particular solution 



r= 0, Jf = r = const. = ( 
The Moon is then always on the aris of X, or in other words, is constantly in con- 

^^ ftU RoyalAstronomical S eiet l/> l. ***vnr., pp. 460-472. 



22-25] GENEKAL PROBLEM OF THREE BODIES. 25 

junction with the Sun. The motion is however unstable, a fact which can easily be 
obtained from the equations (25). 

This is a particular case of a more general theorem mentioned below. (See Art. 30.) 

24. It is not difficult to find expressions for the velocity when we neglect the 
solar eccentricity only. Since in this case, r$= Jf, /=', we have, from Art. 22, 



Hence R does not contain the time explicitly. 

Multiply the equations (12) by A 7 , F, z and, after adding the results, integrate. We 
obtain 

(26), 



giving the velocity referred to moving axes. (If we include the solar eccentricity, 

/"/} 7? 
the term - 2 I -g- dt must be added to the right-hand side of (26)). 

But we have A r2 + P+z^r^+r^ (0- n')*+& 

= x 2 +$ 2 + & - 2w/ (yx - Ssy) 4- 7i' 2 ( 
Hence, since I 2 + F 2 = 



we obtain # 2 4- # 2 4- 2 = + 2 w' ($# - ^y) + 2 72 -f- const . , 

giving the velocity in space, when the solar eccentricity is the only quantity neglected. 
This expression was first obtained by Jacobi*. 



(iv) The general Problem of Three Bodies. 

25. The developments given above refer to the motion of one body about 
a second when that of the third body about the second is supposed known. 
The problem of three bodies or rather of three particles, considered without 
any limitations, admits of a much more general treatment and moreover, 
when looked at from this point of view, is seen to possess certain properties 
in the form of first integrals which do not appear in the more restricted 
problem. 

Let m^ m 2 , m 3 be the masses of the three bodies, r^, r 18 , r la their mutual 
distances and (^y^), (^yA), (^2/3^3) their coordinates referred to rect- 
angular axes, fixed in direction, through any origin. The nine equations of 
motion are 

., dF .. dl dF / , o o\ 

m^ = g^ , JIKK = ^ , mm = ^> ( = 1, 2, 3), 

where, according to Art. 10, Chap. L, 

v m l m z n^ms m 2 m 3 

ji ~ 1 1 m 

r^ r 13 r 23 
* Comptes JRendus, vol. in. pp. 5961. Collected Wwfa, vol. iv. pp. 37, 38. 



26 THE EQUATIONS OF MOTION. [CHAP. II 

Since n/ = fa - ^-) 2 4- (fa - ytf 4- fa - %jf an( i therefore 

we get immediately, by addition of the equations of motion, 

5m$#$ = 0, 2^2^ = 0, Smjj = 0. 
From these we obtain by integration 

where a, b, c are three constants. These constitute three first integrals of 
the equations of motion. 

On integrating again, we have three further equations of the form 

2m 7 : $i = at + const., 
or, eliminating a, 6, c, three equations of the form ^ 

Xm^ t (2<miXi) = const., (28), 

which are also three first integrals of the equation of motion. 
Again, since 

9 3 \ 1 yittt-tciyt f 3 9 \ 1 



we obtain from the equations of motion, 

S (<B$i - y&) = 0, 2 (y^i - Ziy^) = 0, 2 (^^ - ^) 0, 
and therefore three more first integrals of the equations, of the form 

S* (## y^i) = const (29). 

Finally, since F is a function of t only in so far as t occurs implicitly 
through the presence of the nine coordinates in F t we shall obtain, after 
multiplying the nine equations by their respective velocities, adding and 
integrating, 

m* (#i 2 + & 3 + zf) H- m 2 (<y + yj + 4 2 ) 4- m 3 (^ 2 4- 2/ 3 2 4- ^a 2 ) = %F 4- const. . . . (30), 
a tenth first integral of the equations of motion. 

26. At first sight the general integral of Energy (30) seems inconsistent with the ex- 
pressions obtained in Arts. 12, 20, 24 which, when the solar eccentricity was not neglected, 
contained an unknown integral. It is to be remembered, however, that we have earlier 
supposed the motion of the Sun round the Earth to be known and to be expressible 
by known functions of the time. In so doing we have neglected a portion of the effect 
produced by the Moon on the motion o"f the Earth, a portion which would produce an 
unknown integral in the expression for the relative Energy of the Sun. We have then 
divided the Energy, relative to the Earth, of the two bodies into two parts, one part being 



25-28] THE TEN KNOWN INTEGRALS. 27 

that of the Moon and the other that of the Sun. In the motion of the Sun, the portion 
depending on the Moon is so small compared to its own great mass that we have 
neglected it, while, in the motion of the Moon, the same portion, being great compared to 
the mass of the Moon, cannot be neglected. In fact, if we denote, in the expression for 
the square of the Moon's velocity, this portion by m'<, where <j> is a function of the 
coordinates and velocities, there will be, in the expression for the square of the Sun's 
velocity, a term :M<jf>. When we add the Energy of the Sun to that of the Moon, 
(j> will vanish identically. 

27. The ten integrals found above are the only known integrals for the 
general problem of three bodies. It has been demonstrated farther by 
M. Bruns*, that no other algebraic uniform integral can exist for any values 
of the masses. M. Poincar^-f has extended this result, from a practical 
standpoint, by proving that if the ratios of two of the masses to the third 
are sufficiently small quantities, there does not exist any other transcendental 
or algebraic uniform integral. For the proofs of these theorems, which are 
based on considerations altogether outside the scope of this book, the reader 
is referred to the original memoirs. 

It is evident that these ten integrals exist and are of the same form for 
any number of bodies attracting one another according to the Newtonian 
Law. The extension to this general case is made immediately, if we suppose 
i, j to receive the values 1, 2,...p, there being p bodies under consideration. 

The ten integrals might have been written down from purely dynamical 
considerations. The first six integrals (27), (28) are known as those of the 
Centre of Mass and they express the facts that the linear momentum in any 
direction is constant and that the motion of the -Centre of Mass is uniform 
and rectilinear. The three equations (29) are known as the integrals of areas. 
The dynamical equivalent is expressed by saying that the angular momentum 
round any line, fixed in direction, is constant. Equation (30) is that of 
Energy and expresses the fact that the sum of the Kinetic and Potential 
Energies is constant. 

28. Let the three constants of angular momentum be h l} h 2 , h s . The 
straight line whose direction cosines are proportional to h lt h<i,h 3 is invariable 
in direction and consequently the plane perpendicular to it is so also. If we 
consider all the bodies of the Solar System without any reference to those 
outside, the plane determined in this way is known as the Invariable Plane 
of the Solar System. Laplace suggested that this plane might be used as a 
plane of reference to which the motions of the bodies might be referred. 
There are however several difficulties in the way. 

* Acta Mathematica, vol. xi. p. 59. 

f Acta Mathematica, vol. xin. p. 264. Also M6canique Ctleste, vol. i. p. 253. 



28 THE EQUATIONS OF MOTION. [CHAP. II 

Por further remarks on the Invariable Plane, see 

E. J. Routh, Rigid Dynamics, Vol. i. Arts. 301-305. 

Laplace, Mec. C4l. Book VL Nos. 45, 46. 

De Pontdcoulant, System du Monde, Vols. I. p. 455 ; n. p. 501 ; in. p. 528, p. 555, 
Other references are given by Tisserand, Ne'e. 041. Vol. i. p. 15.8. 

29. We may consider any one of the equations of motion as replaced by two others, 
each of the first order. Let x be any coordinate, x its velocity. The equation 



, . , . 
may be replaced by 



fa ' 
dx 



so that if there be p bodies we shall have 6p equations to determine 6p variables, namely, 
the coordinates and the velocities. By means of the ten integrals, it is theoretically 
possible to eliminate ten of the 6p variables and the resulting equations will contain 
6p - 10 variables, or, in the case of three bodies, 8 variables. In general, it is found better 
to eliminate only six by means of the equations (27), (28), leaving in the case of three 
bodies, 12 variables between which four relations are known. 

The literature on the general problem of three bodies dates chiefly from the researches 
of Lagrange. An account of these is given by E. Tisserand, Mdcamque Celeste, Vol. I. 
Chap. viii. and by 0. Dziobek, Die mathematischen Theorien der Planeten-Bewegungen, pp. 
80-82. 

30. There are two special cases in which it is possible to integrate rigorously the 
equations of motion. They are obtained by supposing that the mutual distances of the 
three bodies remain constantly in the same ratio, In the first case, the three bodies are 
constantly in a straight line and each describes an ellipse with the common Centre of Mass 
as one focus. This motion is unstable. The result of Art. (23) is a special case of this. 
In the second case, the bodies always remain at the corners of an equilateral triangle 
of varying size. 

These two problems are discussed by 
F. Tisserand, Mfa Ml. Vol. i. Chap. vin. 
E. J. Routh, Rigid Dynamics^ Vols. I. Art. 286, n. Arts. 108, 109. 

In the former will be found further references to the papers on this subject. In the 
latter the stability of the second case is considered. 



CHAPTEK III. 



(JNDISTUBBED ELLIPTIC MOTION. 

31. THE subject of elliptic motion belongs properly to the problem of two 
bodies, the solution of which presents no difficulties : it is treated in most of 
the text-books on Dynamics. By the introduction of a certain angle (the 
eccentric anomaly), the relations between the coordinates and the time can 
be put into finite forms which, though useful for some purposes, are not 
convenient when we proceed to the problem of three bodies, this latter problem 
being generally treated, so far as the solar system is concerned, from the 
point of view of disturbed elliptic motion. In this case it becomes necessary 
to express most of the relations by means of series. These series, investigated 
mainly by means of Bessel's functions, will be given briefly in this Chapter. 
The subject will be divided into two parts, the first referring only to the 
properties of the elliptic curve, and the second containing applications of 
the results obtained in the first part, to elliptic motion about a centre 
of force in the focus. 



(i) Formulas, Expansions and Theorems connected with the 
elliptic curve. 

32. Let G be the centre of an ellipse, E one focus, A the apse nearer 
to E, P any point on the ellipse, Q the corresponding point on the auxiliary 
circle and QPN the ordinate drawn perpendicular to CA, 

Let EP-r and let the angle AGQ = M, the angle AEP =/, and let w be 
defined by the equation 

w _ area AEP 
%TT area of ellipse " 



30 



Then 



UNDISTURBED ELLIPTIC MOTION. 

/ is called the true anomaly, 
j& eccentric 



[CHAP, in 



w 



mean 




Fig, 3. 

Let the major axis of the ellipse be 2a, its eccentricity e and latus 
rectum I, From the well-known properties of corresponding points on an 
ellipse and its auxiliary circle, and from fig. 3, it is evident that the 
following relations hold : 

r = a (I - e cos B) = l/(l + e cos/) = a (1 - e 2 )/(l + e cos/) (1), 

rcos/ = a (cos J5? e), r sin/=aVl ~<s 2 sin fl (2), 



w = M e sin J, 



vl 



(3). 



Also, since a, e, I remain constant and r, / j, w vary with the position of 
P, we obtain easily 

*v 



There are two problems to be considered. The first consists in expressing 
certain functions of w and r in series proceeding according to sines and 
cosines of multiples of /and powers of e] the second consists in expressing 
certain functions of r and / by similar series in terms of w and e. These series 
will be investigated first in an elementary way and then by means of BesseFs 
functions. 



32-33] ELEMEKTTABY METHODS. 31 

33. To obtain series for r, w in terms off. 
Define X by the equation 

e = 2X/(l+X 2 ), or X = (1 - Vf^)/0 = e/(l + VI^7). 

We then have by (1) - = Q^* ~^^ . 

J ^ ' a 1+X 2 l+X 2 + 2Xcos/ 

Putting for a moment 2 cos/= x l + or 1 , and therefore 2 cos if= 0* + ar*, we 
easily obtain 

r^l-XV 1 Xar 1 \ 

a~l + X 2 U+Xa? 1+XtfT 1 /' 



and thence by expansion, since x is a complex quantity whose modulus 
is unity, 

r 

From this equation, after expanding X in powers of e, we obtain the required 
expansion of r. 

Again, from (5) we have the identity 
(1 + 6 cos/)- 1 = (1 - e^ {I - 2X cos/+ ... + (-!)< 2X* cos #*+...}. 

Since the series on the right is supposed to be convergent, we may differentiate 
this equation with respect to e ; we then get, after multiplying by e, 

+ 22(-iyx* 



(1 - 



Adding this to the previous equation we obtain, since dX/cfe = X/eVl "e a ", 



But we have, by (4), 



a 



^. 
$ cos/) 2 * 



Substituting for (1 + e cos/)~ 2 its expansion just found and integrating, we 
obtain, since w and / are zero together, 



w=/ 2X(1 + vl- e 2 )sin/+... +T( IVX^I + ivl e*)$iuif+ (6). 

* 7 ' ./ ^ v/v /./ \ / 



32 UNDISTUKBED ELLIPTIC MOTION. [CHAP. Ill 

Putting for X its value and expanding in powers of e, we deduce the required 
expansion of w. This is, as far as the order e*, 

w =/- 2e sin/+ (f e 2 + %e*) sin 2/- Je 3 sin 3/+ {^ sin 4/- 



34. To obtain f y r in terms of w. 

Since we have supposed that convergent series are possible, we can obtain 
/in terms of w by reversing the series (6') using f-w as a first approxima- 
tion. By this proceeding we shall get, as far as the order e 4 , 

/= w + O - ie 3 ) sin w + (f e 2 - ^e 4 ) sin 2w -f- jf e a sin 3w+ ^e 4 sin4 w +. . .(7). 



Also as r = a (I - e cos E), w-E e sin ., we can apply Lagrange's 
theorem * to the expansion of cos E = cos (w + e sin js) and thus obtain 



cos E = cos w - 35 -, 



From this we have, after replacing powers of sin w by sines of multiples of w, 
as far as the order e 4 , 



Corollary. Since a*/r*= df/dw, we can immediately obtain from (7) the 
expansion of a 2 /r 2 . 

35. From these expansions for/, r we can deduce those of any functions of/, r in term** 
of w. As however all these expansions (except perhaps that of/) are obtained much more 
easily by the use of other methods, those outlined in this Article will not be further 
developed. For a fuller discussion of the methods of Arts. 33, 34, aeo Tait and Stoele, 
Dynamics of a Particle , Arts. 162-167. 

36. An expansion for / in terms of w, exhibiting the general term of the series by 
means of a simple expression, has been given by S. S. Greatheedf. The process may be 
shortly sketched as follows. Let 



then by Lagrange's theorem applied to equations (3) we have 



Also, differentiating the expression for/ we have 



and} . 



* Williainson, Diff. Gal. Chap. vn. 

t Comb. Math. Jour., 1st Ed. Vol. i. (1839), pp. 208211. 

1 : B. W. Hotoon, Trigonometry, Art. 52. The formula there numbered (44), (45) can be 
combined into this form. . ; v ' 



BESSEL'S FUNCTIONS. 33 

giving 



2q - i)-i sin (p - Zq - i) ie}. 
Substituting this in the equation for / and finding the coefficient of gin/v in the 
resulting expression, Greatheed finally arrives at the symbolic formula for f: 



^ 



* / 

In order to ohtain the coefficient of sin/w, the expression for it given by this formula must 
he first developed in powers of X as it stands, all negative powers of X rejected and the 
terms of the order X divided by 2*. 

Cayley f has extended this result in a general manner to the expansion of any function 

f flnrl f 



off and/. 



Expansions by means of Bessel's functions. 



37. The following formula will be found}: in any treatise on Bessel's 
functions, i being a real integer : 



1 f 71 
(x) = - I 



cos icj> - so sin < 
(1 



o - a? 2 (1)0 0*0 1 



= L jr. 



where /<(*?) is the Bessel's function of the first kind In the applications to 
be made here, i is a positive integer and x a real quantity sufficiently small 
for series in powers of x to be convergent. If i be negative we have imme- 
diately from (9), Ji (a?) = Jlf (-. x). 

38. To expand COSJM, sinjjs in terms of w. 

From (3) we may assume that cosjX sin JK will be respectively expansible 
in cosines and sines of multiples of w. Let 



f = f.f cos iw, sin jjs = 2^1^ sin iw, 
where j, i are positive integers. Then, by Fourier's theorem, 

A * r 

-& H = cos JA T cos ^wdw. 

^ Jo 

* See note by Cayley on the expansion of this formula in Quart. Math. Jour. Vol. n 
pp. 229232. Coll Works, Vol. in. pp. 139142. ' "' 

t Camb. Math. Jour. 1st Ed., Vol. in. pp. 162167. Coll. Works, Vol. r. pp. 19-24. 

E.g. Todhunter, Chapters xxx. xxxi. It will not here be necessary to suppose any 
knowledge of Bessel's functions, beyond the assumptions that the series are possible and that they 
converge. If we define J t (x) by (9) the results (10), (11) can be found by a few simple operations. 
B.L.T. 3 



g 4 UNDISTURBED ELLIPTIC MOTION. [CHAP. HI 

Integrating by parts, 

A = - F- cos fa sin iw] * + ^ I * sin w sin jj&cta 

^ -77 [_i ^ Jw=0 MT JO 

= -f f * sin (is - ie sin X) smjEdJS, by (3), 

iTTJo 

= 1 f * C o S {(i - j) *- 16 sin %} dE - J* cos {(- i -j) M - (- w) sin JB} <Z^ 

ITT J 

-|j^(te)+XjlM(-w), by(9) 3 
except when i = 0. For the determination of A Q we have 

Ao7r = f * cQzjEdw = f * (1 - e cos X) cos j^^ = - ^ TT or 0, 

Jo ^o 

according as j is equal or unequal to unity. 

If therefore we allow i to receive negative as well as positive values, wo 
obtain 

COSJJ= 2 -Ji-j^QOBlW ..................... (12), 



in which i/^(0)=- or 0, 

according as j is equal or unequal to unity. 
In an exactly similar way we may find 

J5 4 = -7 Ji-j (ie) -- -. JLv~y (- ie), 

- 



and 



there being no constant term. From the results (12), (13) we can get most 
of the expansions required. 

39. To expand r, r cos/, rsinf, r~ l , r~ 2 in terms of 10. 
Putting j = l in the two results just obtained and substituting for cos x 
and sin E in (1), (2), we deduce 

T e 

- = 1 2 -Ji^i(ie)coBiw (14), 

a -oo'fc 

(y* 00 1 

- cos/" = e + 2 t/i i (ie) cos iw, 

Ct ( t> 

M ._______ 00 "J 

- sin f = v 1 6 2 2 - e/i-i (ie) sin iw 
a J .. M i ^ ' 



(15). 



38-39] EXPANSIONS BY BESSEL/S FUNCTIONS. 35 

Since J^ ( ie) = Ji (ie) and since for i = we have 

we deduce, after the application of the formulae (1 1), 

r _ i g2 y ^ e ^*- 
a~~ + 2""fF "~ 



rcos^a^f. + Il^^osiJ 

L l c J 



. ___ Foo 2 1 

r sin/ = aVl - e 2 2 -r- Ji (ie) sin w 

LI ^ J 

Again, by (4), (3), (2), .we have 

a dJS _, d , . . d f er sin f 

=1 + - r=^= 



.(150- 



_ _. = T- r= 

r aw aw J aw \avl 

and therefore from (15 X ), 



- = l + 2Se/i(ic)oosiw ............ ......... ... (16). 



Further, since r/a = 1 e cos JB, we have 
d /r 2 \ r d 



d /r 2 \ ^r d fr\ ^rdjs . . , /1N ... 

^ ..... -= =2-^- (- =2- -r-esmJSf=2esm A T , by (1), (4); 
dw\a*J a dw \aj a, dw j \ / \ / j 

and since (15 X ) gives the expansion of rsin/=aVl e 2 sinj?, we obtain 



dw\ti 2 

T 2 4 

Integrating, -- = const. - S -^ /i (ie) cos iw. 

Cfc 1 ^^ 

But since r 2 /a 2 1 - 2e cos M + J e 2 + \ e* cos 2^ ; 

and since it was shown in the last Article that the constant part of cos /, 
when expanded in terms of w, is 6/2 and that the similar part of cos 2J0 is 
0, the above equation shows that the constant part of r 2 /a 2 , when so expanded, 
is 1 H- 3e 2 /2. Hence 

~ = l+f#-S*/i(w)cosiw .................. (17). 

a 1 1/ 

Corollary i. From (1) and (4) we have 



, 1 - e 2 a I . , \/l - 6-' dr 

cosf = , sm/= -7-. 

J ere J ae dw 

Whence, by using the developments (16), (14 r ), we can immediately deduce 
those of cos/, sin/ 

The difference between the true and mean anomalies is called the 
Equation of the Centre. Denote it by Eq. 

32 



36 UNDISTURBED ELLIPTIC MOTION. [CHAP. Ill 

Corollary ii. If a be any angle, we have, since /= w + Eq., 
sin (a + Eq.) = sin/ cos (a - w) + cos/sin (a - w). 
By means of Cor. i. we can then get the development of sin (a 4- Eq.). 

40. It will be noticed that the coefficient of sin w or cos iw (i positive) 
is always of the form **# + **** + o,^ 4 + . . . , (o, Oi, a* - - numerical quanti- 
ties). That this must be so in the expansions of all functions of r } f of the 
forms treated here, is sufficiently evident from Art. 32. Hence if we are 
considering any term whose argument is iw, we know immediately that the 
lowest power of e contained in the coefficient is not less than e\ This fact 
has an important bearing when we come to develope the disturbing function. 

41. In the development of the disturbing function it is important to obtain 
expansions for ^cosg/, 9*awqf (p being any positive or negative integer and q any 
integer including zero) in terms of w. These could be obtained from the expansions given 
in Art. 39 by multiplication of series. Such a process would be' somewhat tedious when 
many terms are required. On pp. 163-179 of the Fundamental Hansen obtains the 
expansions by finding the finite expressions corresponding to each value of p and q 
required for r cos qf in terms of positive or negative powers of r, and for r* sin qf in 
terms of the differentials of the same with respect to w. That this is possible is evident 
from the expressions for cos/ and sin/ given in Cor. i. of Art. 39. He then obtains a 
general formula giving the coefficients of the development of ** in terms of those of the 
developments of r 2 and r~ 2 . The coefficients of r 2 are obtained as in Art. 39 and those of 
r~ 2 as in the Cor. of Art. 34. 

In a later workt, he has considered them in a much more general manner and has 
obtained expressions for the coefficients of the development of r ?) exp. g/V - 1 in powers of 
exp. W~-l, by means of Fourier's theorem. The definite integral corresponding to the 
coefficient of exp. zW-Xis evaluated and a general expression which is the leant 
cumbrous to expand of any given up to the present time, is obtained for the coefficient. 
This is even true of the case jp=-2, =0, which by a simple integration gives the 
development of/. 

42. The literature on elliptic expansions up to 1862 has been collected by Cayley in a 
report On the progress of the solution of certain problems in Dynamics J. Later developments 
and references are to be found in Tisserand, Mfc. C4L 9 Vol. I. chaps, xiv., xv. 

43. The following Theorem and Corollary will be required later. 

Let F, (?, H be three functions of which F } are developable in cosines (or 
sines) and H in sines (or cosines) of a series of angles of the general form 
The function 



a 

* Fundamenta Nova Investigation Orbitae verae quam Luna perlustrat. Auctore P. A. 
Hansen. Gotha, 1838. This work will be referred to throughout as the Fundamenta. 

t 'Entwicklung des Products einer Potenz des Eadius-Vectors mit dem Sinus oder Cosirms 
eines Vielfachen der wahren Anomalie etc.' A bh. d. K. Sachs. Ges. zu Leipzig, Vol. n. pp. 183 281. 

Brit. Assoc. Reports, 1862, Coll Works, Vol. iv. pp. 513593, 



39-43] A THEOREM OF HANSEN. 3*7 

can be developed into a series of the form 



Mil 



also, when the coefficients <,, a x , cc* haw been found, all the other coefficients a?, 
can be obtained by a simple process. 

Suppose that F } 6- are developable in cosines and H in sines of angles 
of the form {It + &. Let 

- -J- & = 2.5 cos (/8$ + '), J? ---ir^ - 2-B' sin (/8tf -f '), 



in which 5, J3' are the typical coefficients corresponding to the argument 
J3t + ft. Let 

2 
- 7( >Ji(ie)=*Ri. 

it 

The formulae (15') may be written 



rcos/ dRi . rsmf Vl e 2 ^.^ . . ,, rtN 

+ |^ = _ 2, -j cos iw, ^ =- z i/tj smvw;. . .(18); 

a 2 i de a e i ^ 

also from (17) we have 

^,2 oo 

This last equation will be required in Chap. X. 
Substituting in the expression for T we obtain 

oo ///?. o 

r - J* = 2S5 cos (/3t + ^80 ^ --7-" cos w - 2S' sin (/Sit + ft') . S i J2< sin iw 

oo r/ ^jj^. \ 

= 2) -S ( JD 5 h -*5 'z/JKi 1 cos \w# H" ^^ ~\~ ]3 ) 
i=i [\ de J 

+ ( B -j ~ B f iRi\ cos ( iw -f 
If now we put R^ = J2$, jR = 0, this may be written 

00 

r F 2 2 i cos (w + /9^ 4- /3 r ), (i = excluded) 

JT? 

where ct^ 

If we put also 



we obtain T = S S i cos ( 

*S=! 00 

for all values of i. 



38 UNDISTURBED ELLIPTIC MOTION. 

By the definition of a^ we have, since R$ = . 



[CHAP, in 



whence B = (CL I + a_i)/ 2 -j- ' %' = ( a i ~" ( 

Substituting these values of B, B f in the expressions for a , a_i, we obtain 



de 



. 

4" 1> " 



de 



de 
d<3 



*JB, 



_de 

^r l 

C?6 



3; 



When therefore a 1} a_ x have been found for all the different arguments 
Pt-\r$ r , the coefficients a^ OL^ can be found without any trouble. This 
simple method of obtaining the coefficients in the product of two series, saves 
Hansen much labour in performing his developments. 

Corollary. From the last two equations we deduce immediately 



44. When the plane of the ellipse is inclined to the plane of reference, 
expansions for the longitude in this latter plane and for the latitude above it 
will be required. 

Let M (fig. 4) be the position on the unit sphere corresponding to the 
point P (fig. 3) whose true anomaly is /. Let MQ, be the position of the 




Fig. 4. 

plane of the orbit and draw MM* perpendicular to yjff the plane of reference. 
Then, according to the notation of Chapter n., v = a>M', s = tan M'M. Let 
& = 3 JW = L, &M' = L I} MQM = <* and therefore z* = v - 0. 

* The letter i is frequently used, as in the previous articles, to denote any integer : the addi- 
tional use of the letter to denote this angle will cause no confusion. 



43-45] PLANE OF ELLIPSE INCLINED. 39 

45. From the right-angled triangle MM'l we have 

tan Ly = cos i tan L. 
Putting L == V ii we can write this 



and therefore e lL ^ = e^ Ll 

l + tan 2 ^e 2 ^ 



Taking logarithms and expanding, we obtain 

2^ = Zpin +- 2u - tan 2 (e 2Lt - e~~ 2Xl ) + i^au 4 

ju 

where p is an integer. Since LI L when i is zero, we have p = 0. Hence 

LI = - tan 2 ^ sin 2^ + ^ tan 4 ~ sin 4^ tan 6 ^ sin 6i/ + .... 
z ^ A 

Let now the angular distance from the apse to the node O be -or 0. We 
have then 

L =/+ 5r-^ = ^ + OT-^4-Eq. = '? 7o + Eq., 

where r) Q = w + m- 0. Substituting for L this value and for L its value v6, 
we obtain 

-y ss f+ & - tan 2 sin (2t; + 2 Eq.) + J tan 4 sin (4^ H- 4 Eq.)- . - . (20). 

Zl -i 

The terms involving i and constituting the difference between the longitude 
in the orbit and that in the plane of reference are known as the Reduction. 

We can expand 

sin (2?7 + 2 Eq.) = sin 2^ cos 2 Eq. + cos 27/0 sin 2 Eq. 
by means of the formulas given in Cor. ii,, Art. 39. Let 

tan i = 7. 

Then tan 2 ~ = (2 4- 7 2 - 2 VTT 7 a )/7 2 = i 7 2 - i 7 4 + A7 6 ~ - - 

2t 

In the case of our Moon, 7 is a small quantity of the same order as e : it 
will, therefore, not be necessary to calculate a large number of terms in 
the Eeduction, when the latter is expanded in powers of and 7. 



40 UNDISTURBED ELLIPTIC MOTION. [CHAR III 

46. To obtain $, the tangent of the latitude, we have 

sin M'M = sin i sin L, 
sin i sin L 



. , 
and therefore 



-.~ ............ - - ~-,~ --- , 

vl - sm 2 i sin 2 L V 1 -f 7 s cos 2 L 

giving s = 7 sin L \<f sin z cos 2 + 7* sin L cos 4 ..., 

or 5 = 7 sin (^ ~f Eq.) - ^7* (sin (3?7o + 3 Eq.) + sin (r; + Eq.)] + . - ( 21 )- 

Of these terms in s, the first is the most important and the method of 
finding it has been given in Cor. ii., Art. 39. The other terms can be easily 
calculated ; since they are multiplied by 7 3 at least, it will not be necessary 
to take many of them. 

47. It will be noticed that there is a ^connection between the index 
of 7 and the multiple of T/Q similar to that between the index of e and 
the multiple of w. In longitude we have even multiples of T? O and even 
powers of 7 ; in latitude, odd multiples of 770 and odd powers of 7. In both 
cases, the lowest power of 7 which occurs in the coefficient of sini^o or cos irj^ 
is 7*. 

(ii) Elliptic Motion. 

48. When we neglect the disturbing action of the Sun, the forces on the 
Moon, relative to the Earth, are reduced to /i/r 2 acting inwards along the 
radius vector. Such a force is known to produce motion in an ellipse of 
which one focus is occupied by the Earth. We shall not here solve the 
problem which is merely that of two bodies, but assume that the solution 
has been completed ; all that then remains is to fix the constants to be used. 

Let (fig. 3) E<y be a fixed line from which we may reckon angles. Let 
7j??.4==t*r, and let e be the angle which a uniformly revolving radius vector 
Ep, makes with Ey at time = 0. Let the time of a complete revolution 
of this vector be Zir/n. Then fyEfju = nt + and AEp = nt + r. But since 
equal areas are described in equal times, we have, by the definition of w 
in Art. 32, AE/j, = w. Hence 

Mean anomaly = w = nt + e OT. 

Let 2a be the major axis and e the eccentricity. Then we have the 
following well-known results: 

^^^a 8 , (Velocity) 2 = ?-, 



Twice the area described in a unit of time =na 2 Vl e 2 , 
and n (or a), e, e, w may be taken as the four constants of integration. 



46-51] MOTION IN AN ELLIPSE. 41 

49. When the plane of motion is inclined to the plane of reference we 
require two more constants. Let them be those defined in Art. 44, namely, i 
the inclination and 6 the angular distance of the line of intersection of the 
two planes from the fixed line Ex. In this case the line E<y is taken to 
coincide with Ex, so that CT, e are reckoned from x along the fixed plane to 
O (the ascending node), and then along the plane of the orbit. Let if 
(fig. 4) be the position, on the unit sphere, of Ep (fig. 3). Then 

tar = 0fl4{L!i, 

and flM = r) Q 

The last angle is known as the mean argument of the latitude. The 
constants introduced by the three equations which determine undisturbed 
elliptic motion in space, are a (or n), e, e, nr, 6, i (or 7). 

These six constants are called the Elements of the ellipse. The meaning 
to be attached to. the word ' Element ' will be extended in Chapter v. 

50. From the results of Arts. 34, 45, 46, we obtain the following values 
of v, r, s in terms of the time, for the solution of the equations (A) of 
Chapter n. when we neglect R, as far as the 3rd order of the small quantities 
e,<y: 

i) = nt 4 e 4 (Ze -J0 3 ) sin w + \& sin %w 4 |f 3 sm ^ w + 

|7 a sin 2% -|e7 a sin(w- 2?? ) %e<fsm(w+ 2^ )4 ..., 
r/a = 1 4 \& (e |e s ) cos w \& cos %w fe 8 cos 3w - . . ., ^ -(22), 

s = (1 s -J7 2 ) 7 sin % 4 ey sin (w 770) -H #7 sin 
4 10 2 7 sin (2w ^ ) + 1^ 2 7 sin (2w 4 770) 
where w = nt 4 6 r, ^ = w# 4 0. 

51. It only remains to connect the constants a, A of Art. 13 with those 
used here. We have found ia equation (2), Chapter II., neglecting J?, 

Square of velocity in orbit = 2/^/r p/a. 

This being the same expression as that given in Art. 48, a has the same 
meaning in both cases, namely, the semi-major axis of the orbit. We have 
also by Art. 12, 

^h = Jr x 9 v = rate of description of areas in the plane of reference, 
whence 

1 -JL_=: ra te of description of areas in the plane of the orbit, 

2 cos i r 



= ^na'Vl ~- e\ by Art. 48. 
Hence A = 



This value refers to undisturbed motion only. 



42 UNDISTUKBED ELLIPTIC MOTION. [CHAP. Ill 

52. The solution of equations (1 1) of Chapter n. when F /*//, may, since u^ 
be put into the form 



where tan (^ - 6) = y 1 + y 2 tan ('<x 6), e 1 = e v 1 4- y 2 sin 2 (ID* - 0). 

For we have, by the figure of Art. 44, since M'M= u, 

e\/l+$ G osf^ e -^^^^^=ecos (vr-ff) ?5^+esin (w-^)-?!- 

^ ^nsa rr x ' rA.ct rr x ' cnri 



cos ir x / sin u 

sin L, 



=<2 cos (or - 0) cos Li+e sin (or - 0) 

v z K ' 

whence, after substituting for e and w in terms of e 1 and ra^ and putting ^ = v 0, wo get 
the required result. If AA^ drawn perpendicular to the plane of the orbit, intersect say 
in A i, we .easily obtain t ur l =-iisA- L . 

To obtain % in terms of v, we expand \fl+s* by the binomial theorem and, after 
substituting for s its value ysin(v 0), express these terms in cosines of multiples of 
%(v-6). To obtain tf in terms of v, we can expand l/% 2 by means of the formulae of 
Art. 33. For we have 



which can be expanded in powers of e x (l-hs 2 )" and cosines of multiples of v < zzr 1 . 
Expanding next the various powers of Vl-f s 2 in powers of s 2 and substituting for s its 
value, we shall obtain dt/dv expressed by means of cosines of multiples of v w^ and 
2 (o 6}. An integration will then give t in terms of v. 

It will be noticed that m-^ e t differ respectively from -or, e by quantities of the order y 2 . 

53. The expansions given in Art. 50 will apply equally to the motion of 
the Sun, but become simpler since we suppose the plane of its orbit to be the 
plane of reference. Only four constants will be required ; these will be 
called a!j e', e', tzr 7 . The mean motion n* is defined properly by the equation 

m' + p = m' + E + M = ri* a'* , 

We have, in Arts. 19, 22, put m' = n' 2 a' 3 . The ratio /^ : m' is approximately 
1 : 330,000, so that the error caused will be very small. 

We should properly have put in the disturbing function, 

mWW 3 - ^etfW 2 ^ (1 - wV/rc/V 3 ). 

The slight correction necessary is therefore obtained very easily after all the expansions, 
giving the motion of the Moon as disturbed by the Sun, have been made. The correction 
to be made to the largest coefficient in the expression of the longitude will not be so great 
as 0"-02. 

It is to be remembered that a', ri, e', e', tzr' are the constants of the ellipse which the 
Sun describes about the Centre of Mass of the Earth and Moon, in accordance with the 
principles laid down in Chapter i. 



52-54] CONVEKGENCE OF SERIES. 43 

54. An important question in connection with the series given in this Chapter is their 
convergence. Each coefficient of sin iw or cos iw is represented by a convergent series, 
and the series of coefficients thus arranged forms a convergent series as long as e is less 
than unity. But if we arrange the series according to powers of the eccentricity, this 
is no longer necessarily the case. We get in fact a double series, proceeding according 
to powers of e and sines or cosines of multiples of w, the convergency of which, for a 
certain range of values of e and w, depends on the manner of its arrangement. The 
problem is to find the greatest value of e for which the series is absolutely convergent. 
Laplace* has shown that if e be less than 0-6627432..., the series which have been 
discussed, together with those of the form r*> cos qf t r^ sin #/, will be absolutely convergent 
for all values of w. Full references will be found in Cayley's Report already referred to, 
and in Tisserand, Vol. i., Chap. xvi. In the latter are given some of the more important 
theorems on the subject. 

* Mtm. de VInst. de France, Vol. vi. (1823), pp. 6180. 



CHAPTER IV. 

FORM OF SOLUTION. THE FIRST APPROXIMATION. 

55. IT has already been pointed out in Chapter n. that there is no 
known method of obtaining directly a general solution of the differential 
equations which express the motion of the Moon as disturbed by the 
Sun. In consequence, we are obliged to resort to indirect methods. 
There are two well-recognized devices used, both of which depend on the 
right to neglect certain parts of the equations of motion, in the first 
instance, so as to reduce them to forms which are capable of integration, 
either by means of known functions or by the use of series, the coefficients 
of which can be found according to a definite law. 

The Form to be given to the expressions of the Coordinates. 

56. In discussing these methods of obtaining a solution of the general 
equations, it is necessary to keep certain physical considerations in view. It is 
not sufficient to obtain mere expressions for the coordinates ; they must 
be put into such a form that practical applications may be possible and 
sufficiently simple. Since infinite series will be used, this point becomes 
of the greatest importance. 

Now all records, ancient and modern, containing any mention of 
lunar observations whether made in a scientific way or not go to 
prove that, for a long period of time, the Moon has been circulating 
round the Earth in an orbit which is confined between limits not very far 
removed from one another. From this fact we infer that the motion is 
of such a nature that, at any rate during a considerable interval, its 
deviations from some mean state of motion (which, to fix ideas, we may 
think of as circular) are never very great. We ought then to try and 
express its coordinates in terms of the time, in a form which will give 
the position after any interval of time, whether it be short or long. 



55-57] INTERMEDIATE ORBITS. 45 

In order to be able to do this conveniently, the deviations from some 
mean state of motion ought to be expressible by functions which oscillate 
between finite and not very distant limits. The most convenient functions 
of this nature are the real periodic functions* sine and cosine. Should 
the variable, which is generally taken to be the time, occur, for example, 
in the form of a real exponential in the expression of a coordinate, such a 
term would cause the coordinate to increase indefinitely with t, for either 
positive or negative values of t. Again, should a term of the form tfsmnt 
be present, the same result would follow, provided the term has any indepen- 
dent existence. It may happen that such a term is present as one of a series 
which, in some other method of proceeding, would only have appeared through 
the expansion of some periodic function an expansion only permissible for 
small values of t. As it is desired to obtain expressions holding also for large 
values of t, one object to be sought after is to try and obtain a solution in 
which such terms are not present. If they should arise, they must, if possible, 
be eliminated by some alteration in the/orm of the solution. In the case of 
the Moon's motion as disturbed by the Sun only, when the latter is moving 
in an elliptic orbit, it will be seen that the coordinates can be expressed by 
periodic terms only. See Art. 69. 

A discussion of the limits, upper and lower, of the Moon's radius vector is given by 
G. W. Hill, Researches in the Lunar Theory -, Amer. Journ. Mat/L, VoL i., pp. 6-26. See also 
H. Gyld<5n, Traitti analytique des Orbits absblm, etc. VoL i., Chap. i. 



Intermediate Orbits. 

57. It has been stated in Art. 55, that the first step usually taken 
towards the determination of the Moon's path, is a simplification of the equa- 
tions of motion, made by neglecting certain portions of them, to forms which 
can be readily integrated. A solution of the equations, thus limited; should 
form an approximation to the true path of the Moon if it is to be of assistance 
in obtaining the general solution of the complete equations of motion. This 
approximate path is called the Intermediate^ Orbit, or more shortly the 
Intermediary. The intermediary need not necessarily be a general solution 
of the limited equations of motion ; it may not contain the full number of 
arbitrary constants, but should be such that, when the general solution of the 
complete equations is required, it forms in some way an approximation to the 
path described by the Moon. It is not even necessary that the intermediary 
shall exactly satisfy the limited equations of motion. We may, after having 
obtained the exact solution of the latter, modify it in any way which experi- 

* A periodic function is one which, after the addition of a definite and finite quantity to the 
variable, returns to its previous value. We consider periodic functions of a real variable only, 
t This term was introduced by Gylden. (German, intermediate ; Fr., interme'diaire.) 



46 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV 

ence may suggest, provided that the modified solution differs from the exact 
solution by quantities of an order not less than the lowest order of the 
portions neglected in the original equations. The intermediary may then 
be indefinite to a certain extent, until we have found the general solution 
of the complete equations. Such indefiniteness will, however, be only 
allowed for the purpose of facilitating the analysis and in order to put the 
expressions for the coordinates into a suitable form, in accordance with the 
remarks made in Art. 56. 

58. The next consideration is the choice of an intermediary. In this 
matter there is some freedom; it must partly depend on the particular 
method we intend to follow for the solution of the general equations. The 
usual plan is to choose an intermediate orbit which is such that, after a 
certain finite interval of time, the coordinates of any point on it, referred 
to axes which may be fixed or moving, return to the values they had at the 
beginning of the interval (an angular coordinate will have had its value 
increased by 2?r). An orbit which possesses this property is called periodic*. 
With reference to the axes used, the curve described will be closed. 

59. In most of the older methods, the first limitation of the equations 
of motion is made by neglecting the action of the Sun, so that the inter- 
mediary is a fixed ellipse. A certain indefiniteness is then given by 
supposing the apsidal line and the nodal line of its orbit on a fixed plane 
to be moving with uniform angular velocities; these motions are determined 
on proceeding to the second and higher approximations. In other words, 
the intermediary is periodic with respect to moving axes. By this choice 
we begin by considering the problem of two bodies rather than the problem 
of three bodies. 

Dr Hill starts from a different standpoint. He begins by neglecting, in 
the equations of motion, certain parts but not the whole, of the Sun's action, 
and he is able to obtain for the intermediary a solution, periodic with reference 
to axes moving in a definite manner; this is really a particular case of the 
problem of three bodies and the advantage of the orbit as a first approxima- 
tion arises from this fact. This intermediary is not indeed a general solution 
of his limited equations (which were given in Art. 23), in that it does not 
possess the full number of arbitrary constants; nevertheless it serves as 
a useful first approximation owing to the fact that one of the arbitrary 
constants (the . so-called eccentricity ' of the Moon's orbit), which has been 
tacitly put equal to zero to get the intermediary, appears to be small enough 
to permit of expansions in ascending power series. 

* On periodic solutions, see Pomcare*, Mec. Cl. Vol. i. Chap. iv. 



57-62] METHODS OF SOLUTION. 4*7 

60. The subject of intermediate orbits has been, treated by Gyldifo, Andoyer, Hill and 
others. The usual plan is to express the disturbing function by powers of the ratio 
of the distances of the Sun and the Moon and by cosines of multiples of their angular 
distances ; the coefficients and the term independent of this angle are then functions of the 
radii vectores of the Sun and Moon (or, in the planetary theory, of the two planets) only. 
All the terms containing this angle are neglected, so that the disturbing function involves 
only the radii vectores. A further simplification can be introduced by supposing the 
motion of the disturbing body to be circular. The case treated in Art. 67 is a simple 
illustration of the method followed. Gylden* uses a method akin to Hansen's to solve 
the resulting equations ; Andoyer f follows Laplace in taking the true longitude as in- 
dependent variable j HillJ uses a direct method, by finding equations for the small 
differences r a, r' a' (where a, a' are constants) and expanding in powers of them. 
The principal part of the motion of the perigee is determined without much difficulty. 

When the intermediary has been obtained, there are two methods of 
proceeding to the solution of the general equations : (i) by continued 
approximation, (ii) by allowing the arbitrary constants introduced into the 
intermediary to vary. 

(i) Solution by continued Approximation. 

61. We suppose that, by means of the intermediary, the four variables, 
namely, the three coordinates and the time, have been expressed in terms of 
one of them ; in these expressions there will be a certain number of the 
necessary six arbitrary constants present. With this solution or with a 
modified form of it, we then proceed to find what small corrections must be 
made to the variables when we include the omitted portions of the equations 
of motion. If the motion be stable, these corrections should take the form of 
small periodic terms. The method is then nothing else than that of small 
oscillations about a state of steady motion that in the intermediate orbit. 
In the case of the ]JIoon we shall generally have to proceed to the third and 
higher approximations in order to obtain the oscillations with sufficient 
accuracy. It is necessary to consider the amplitude, the period and the phase 
of each term. 

(ii) Solution by the Variation of Arbitrary Constants. 

62. The method is sufficiently well-known not to need explanation 
here . In the case of the Moon we have three differential equations of the 
second order and therefore six arbitrary constants in the solution. We assume 
that an intermediate orbit has been found and that the resulting relations 

* "Die intermediates Bahn des Mondes," Ada Math., Yol. vtt. pp. 125-172 (1885). 
t "Contribution & la Th6orie des orbites intermddiaires, " Annales de la Fac. des Sc. de 
Toulouse, Vol. i. M., pp. 1-72 (1887). 

J " On Intermediate Orbits," Annals of Math, (U. S. A.), Vol. vni. pp. 1-20 (1893), 
See A. B. Forsyth, Differential Equations, Chapter iv. 



48 FOBM OF SOLUTION. THE FIEST APPROXIMATION. [CHAP. IV 

between the coordinates and the time contain all the six arbitraries ; it is 
required to find what variable values the arbitraries must have in order that 
the same relations may satisfy the general equations of motion. The 
coordinates expressed in terms of the arbitraries and the time will thus have 
the same form for the intermediate orbit and the true orbit. There are three 
relations, which may be chosen at will, between the first and second 
differentials of the arbitraries. These are always taken such that the first 
differentials of the coordinates have the same form whether the arbitraries he 
constant or variable. Hence, the velocities, when expressed in terms of the 
arbitraries and the time, have the same form whether the arbitraries be constant 
or variable. This way of stating the relations enables us to change from 
one system of coordinates to another without trouble. The six arbitraries 
and any function of them not involving the coordinates, the velocities or the 
time, are named elements. It is usual to take the undisturbed ellipse as the 
intermediary. The method will be treated in the following chapter and an 
important extension will be given to the meaning of the term ' element/ 

The Instantaneous Ellipse. 

63. We assume that the intermediary is an ellipse obtained when the 
action of the Sun is neglected. It is evident that if at any instant during 
the Moon's actual motion, the disturbing forces were to suddenly cease to 
act and the Moon were to continue its motion from that point tinder the 
mutual action of the Moon and the Earth only, it would describe an ellipse. 
This orbit is called the Instantaneous Ellipse. 

Now when a particle is describing an ellipse under the Newtonian Law, 
if we are given the coordinates and the velocities* at any point, one ellipse 
can be constructed which satisfies the given conditions, and its six elements 
can be expressed uniquely in terms of the given coordinates and velocities* 
Conversely, the coordinates and velocities of the point considered can be 
determined uniquely in terms of the six elements. But since the coordinates 
and velocities of this point on the Instantaneous Ellipse arc the same as 
those in the actual orbit, and since in the actual orbit the coordinates and 
velocities, when expressed by means of the arbitraries and the time, have the 
same form whether the arbitraries be constant or variable, the Instantaneous 
Ellipse is the Intermediate Orbit at the time when, in the expressions for 
the arbitraries, we have given to t the value which corresponds to the Moon's 
position at that instant. Hence, the elements of the Instantaneous Ellipse 
at any time ^ can be obtained, after the solution by the method of the 
Variation of Arbitrary Constants has been carried out, by giving to t the 
value 4 in the expressions which determine the arbitraries in terms of the 
time. 

That is, the magnitude and direction of the velocity. 



62-65] NATURE OF EQUATIONS (A). 49 

Application of the Solution by continued Approximation. 

64. Let us now return to the first method and see how it is to be applied 
to the solution of equations (A), Chapter n. We may begin by neglecting 
their right-hand members, that is, the terras dependent on the action of the 
Sun. The equations so limited will give the intermediate orbit an ellipse 
of period Zir/ni and we have seen in Chapter III. that, in this case, the 
coordinates can be expressed by sums of periodic functions of the time* 
which are sines and cosines of multiples of angles of the form nt+ t a, 
where a is a constant. 

To obtain the second approximation, we substitute these values of the 
coordinates in the right-hand members. But the disturbing function also 
depends on the coordinates of the Sun, which is supposed to move in an 
elliptic orbit of period 2-Tr/ft', and these coordinates will be expressed by 
sines and cosines of multiples of angles of the form n't + a'. Since the time 
cannot enter into the right-hand members except through the coordinates, all 
the portions which depend on the action of the Sun will be periodic functions 
of the time and the arguments will be all of the form int + i'rit + A (i, i! 
integers, positive, negative or zero and A a constant depending on the integers 
i } i f and on the longitudes of perigee, node and epoch of the two orbits). 
The equations being integrated, we obtain for our second approximation new 
values of the coordinates which, when substituted in the right-hand sides of 
equations (A), will, after a new integration, furnish a third approximation, 
and so on. This process is repeated until the desired accuracy is obtained. 

65. Let us consider the nature of the equations which we obtain for the 
determination of the second approximation. The two integrals fd'R and 
fdtdR/dv must first be treated. It will be seen in Chapter VI. Art. 116, that 
the two expressions under the integral sign can contain no constant term when 
elliptic values in terms of the time have been substituted for the coordinates ; 
therefore, unless n, n f arc in the ratio of two whole numbers, no term directly 
proportional to the time can be introduced by these integrals. We assume 
that n, n f are incommensurable. 

Hence the right-hand sides of equations (A) consist entirely of periodic 
terms, whose arguments are of the form int + i'n't + A. When therefore the 
first of these equations has been prepared for the second approximation, we 
may write it 

1 S O" 2 ) - - + - = - I&B cos (int + i'rit + 4), 



where B is the constant coefficient corresponding to the argument int+i'n f t+ A. 

* An angular coordinate will be considered to be periodic, if its rate of increase with respect 
to the time can be expressed by periodic functions only. 

B. L. T. 4 



50 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV 

Let the elliptic value of r be r , and put 



1 1 , . 

- = h ou. 
r T Q 



Then Su is a small quantity of the order of the disturbing forces. Since r 
satisfies the equation when the right-hand member is put zero, we obtain for 
the left-hand member, by substituting the above value of r and neglecting 
powers of 8u above the first, 



For the purposes here, since the eccentricity e is a small quantity, we shall 
neglect the product e$u and therefore put r<?8u, = a?&u. Dividing by a a and 
giving to ^ its value w s a 3 , the equation becomes 

~ Bu + n^u = n^B cos (int 4- i'rit -f A). 
af 

66. This is a linear differential equation of well-known form*. Its 
solution consists of two parts the Complementary Function, containing two 
arbitrary constants, and the Particular Integral. The former may be con- 
sidered to be included in the first approximation, which already contains 
a similar expression with the requisite number of arbitrary constants. We 
are only concerned here with the Particular Integral. The latter is given by 



n* (m + in f 

There are two classes of terms included under the sign of summation 
(a) those in which n is different from w + iW, (6) those in which n=in + i'n' 
for some values of i, i!. 

Case (a) is simple. The resulting terms in Su are of the same period as 
those of R and they constitute forced vibrations of which the periods are the 
same as those of the disturbing forces. 

Case (6). Since n, n' are supposed incommensurable, this equality can 
only hold when t' is zero and therefore when i = 1. The corresponding 
particular integral is then of the form 



Proceeding to the third and higher approximations, it is evident that terms 
involving t\ t s ... in the coefficients will appear. As such forms are contrary 
to the assumption of stable motion when only a finite number of them are 

. * A. R. Forsyth, Differential Equations, Chapter in. This particular form is given on 
pp. 61, 62, 



65-67] MODIFICATION OF INTERMEDIARY. 51 

taken, the question arises as to whether all these powers of t are not in 
reality the expansion of some periodic function an expansion which cannot 
be convergent unless t be small and whether it is not possible, by including 
certain portions of the Sun's action, to get a solution which shall consist 
of periodic terms only. 



Modification of the Intermediate Orbit. 

67. For this purpose we shall examine more closely the first of equations 
(A) and see how terms of period %7r/n may arise through the Sun's action. 
Neglect 6' the tangent of the latitude of the Moon, that is, suppose the 
motion to be in one plane; neglect also the ratio of the distances of the 
Moon and Sun. Putting m' = n'^a'*, the value of R given in Art. 7 will become 

'*r* [fees' (v - v') - J], 



where, as before, v, v' are the true longitudes of the Sun and the Moon. As 
we substitute elliptic values in the first approximation, we may still further 
limit the expression by neglecting e f the solar eccentricity. Then 



and we have K = 7& /2 r 2 [^ + f cos 2 (v n't e')]. 

Also, since r } v do not contain the angle n't + e', when we substitute their 
elliptic values the second term of this expression will give no portion free 
from the angle %n't + 2e'. As only those terms which produce arguments of 
the form nt -f A are sought, we limit R to its first term. Hence 



the equations (A) and (6) of Chap. II. now become 

const., * = 



When the second and higher powers of n 7 - are neglected, the second and 
third of these equations give the steady motion 

(n />2 \ Ji f 7i /a \ 

1 -*^J' v = n + e, if n~^(l+%~^)> /* = ^ 2 a s : 

by a suitable determination of the arbitrary constant in the first of the three 
equations of motion, this value of r will also satisfy it. It is required to find 
the small oscillations about this motion. 



Let r = a(l + # ) . Neglecting powers and products of 7^ A2 /7^ 2 , x 
" 

42 



52 FORM OF SOLUTION. THE FIRST APPROXIMATION. [CHAP. IV 

beyond the first, we obtain from the substitution of this value in the equation 

for'/-, 

x + On 3 - 1??/ 2 ) = 0. 

The solution of this is given by 

x = G cos (cwi + D\ 
where c 2 n 2 =n 2 f n /2 ; and therefore 



Hence the period of the oscillation differs from that of the original motion by 
a small quantity of the same order as the small term introduced. 

If in finding the oscillation about the state of circular motion we had 
neglected n\ the solution would have been 

x = G cos (nt 4- jD), 

which is nothing else than the second term of the elliptic expansion for r in 
powers of the eccentricity. If we expand the previous value of & in powers 
of ?i /2 /^ 3 , we get 

OB = C cos (nt + D) + 1 (n,' a /w 3 ) ntC sin (nt + D), 

an expression which immediately shows how the occurrence of t in a coefficient 
took place. 

68. In order then to make the equations (22) of Art, 50 available as a 
suitable first approximation we shall, in the terms dependent on the 
eccentricity, put w = cnt -f e -or, where c is a definite constant which differ** 
from unity by quantities of the order of the disturbing forces and which is to 
be determined in the process of finding the second and higher approximations. 

Exactly the same difficulty occurs in the equation for s, which will 
evidently give a form similar to that for 8u when we proceed to a second' 
approximation. The same artifice will serve. We put 9? instead of % in the 
expressions, where rj = gnt + e - 0, g being a constant of the same nature as c. 

The term nt+einv requires no modification since the difficulty docs riot 
arise in the longitude equation. 

Hence, the assumed first approximation to the solution of equations (A) 
of Chapter IL will be obtained by giving to v, r, s the values (22) of Art 50, 
after we have substituted, </>, Tjfor w, VJ Q respectively, where 

<f> = cnt + e -GT, T? =gnt + e 6. 

It is evident that the same change would have been effected if we 
had substituted (1 - c) nt 4- -sr for r and (l-g) n t+Q for 6. A physical 



67-70] MOTIONS OF APSE AND NODE. 53 

meaning can therefore be given to these substitutions. Since w and 8 are the 
longitudes of the apse and node, the action of the Sun .not only produces 
periodic oscillations about elliptic motion but also causes the apse and node 
to revolve. (Fuller explanations of the physical interpretation will be given 
in Chapter vm.) The intermediary chosen may therefore be considered to 
be referred to moving axes. 

For the subject of oscillations about a state of steady motion, E. J. Eouth, Rigid 
Dynamics, Vol. n. Chap. vn. may bo consulted ; in particular, see Arts. 355-363 of the 
samp Chapter. 

69. Although by this modification of the intermediary wo have succeeded in avoiding 
the occurrence of secular terms, there is no security that the expressions for the coordinates, 
consisting as they do of sums of periodic terms, will actually represent the values of the coor- 
dinates at any time. The periodic terms are infinite in number and, in order that they 
may give the true values of the coordinates at any time, they must form converging series 
for any value of t. At the present time little is known concerning the convergoncy of these 
series. In Poincard's M&c(wique Celeste, certain groups of the terms are shown to converge 
for sufficiently small values of the quantity in powers of which expansion is made, but no 
definite numerical results have been obtained except in the case of purely elliptic motion 
(Art. 54). 

The comparison of theory with observation seems to indicate that we are justified in 
assuming that these series will represent the motion. Nevertheless it must be stated that 
Poincard's investigations just referred to, show that a limited number of terms of a divergent 
series may, under particular circumstances, give with great accuracy the numerical values 
of the function which the series was intended to represent. 

70. The remarks of the previous articles apply also to equations (11) of Chapter n. in 
which v is the independent variable. Before we can proceed to a second approximation it 
is necessary to express the coordinates of the Sun, which are given in terms of tf, in terms 
of v ; this causes no difficulty since wo have found tho elliptic value of t in terms of ?; in 
Chapter m. When this has been done, we substitute the elliptic values of M X , 8 (Art. 52) 
in the terms depending on the action of the Sun. Putting Wj, = (^i)o+^%j 9=,<? -f &?, tho 
equations for $%, 8s immediately take the linear form obtained for bu in Art. 65, with tho 
difference that v is now the independent variable. A device similar to that used for 
equations (A) can bo employed to avoid the presence of v in the coefficients of the periodic 
terms. We substitute in the first approximation av - tar for ?;- -zar, #v;~ 6 for ?- 6. It will 
be seen in Chap. vm. that the constants c, g so defined are the same as those introduced in 
Art. 68. 



CHAPTEE V. 

VARIATION OF ARBITRARY CONSTANTS, 

71. THERE are several ways of applying the method of the variation 
of arbitrary constants (as outlined in Arts. 62, 63) to the problem of disturbed 
motion. The assumption that the coordinates and velocities, when expressed 
in terms of the arbitraries and the time, have the same form in the disturbed 
and undisturbed orbits, lies at the basis of all these investigations. The 
intermediate orbit is, in all cases, an ellipse obtained by neglecting the 
action of the Sun, and the six elements of this ellipse or functions of 
them are the arbitraries used. 

The chapter is divided into two parts. The first part contains an elemen- 
tary investigation of the differential equations which express the arbitraries in 
terms of the time when the action of the Sun is taken into account. In 
the second part, the equations for elliptic motion and for the arbitraries in 
disturbed motion are treated by the more powerful method of Jacobi. 
Certain results which will be required in later chapters follow. 

(i) Elementary ethods. 

72. We suppose that the equations for elliptic motion have been solved 
and that the coordinates and the velocities have been expressed in terms of the 
elements and of the time by means of the formula given in Chapter in. After 
proving certain preliminary propositions, the equations which give the varia- 
tions of the six elements a, e, or, e, > i in terms of the resolved parts of the 
disturbing forces in three directions, will be obtained. These equations will 
be then expressed in terms of the partial differential coefficients of II with 
respect to the elements. Finally we shall deduce the so-called ' canonical ' 
system of equations used by Delaunay. 



71-73] 



GEOMETRICAL RELATIONS. 



55 



To find the change of position due to small arbitrary variations 
given to the elements*. 

73. Consider a set of moving axes defined in fig. 5 by the points where 
they cut the unit sphere, the axis of Y being along the radius vector, the axis 
of X being 90 behind that of Y in the plane of the orbit and the axis of Z 
being perpendicular to this plane. Since the coordinates are supposed* to be 
expressed in terms of the time and of the elements (Chap, in.), small changes 
in the latter will produce a change in the position of the Moon which may 
be defined by Br and by small rotations S0j, S0 2 , S0 3 of the axes of X, F, Z 
about themselves. The point F coincides with the point M of fig. 4, Art. 44. 

Let zZ meet yx in G and let (as in Chap, in.) flt be the node, xfl = 0, 



^ ,. 




Fig. 5. 

hence Cte = - o?C = - (90 - 0). By Baler's 
geometrical equations *f, we have then 

S0 1 = sin LBi sin i cos LB0 ") 

W% = cos iM -f sin i sin iBd > ........................ (1). 

J 



Recurring to the notations of Art. 82, let &/, Sfi, Sw, expressed in terms 
of the elements and of the time, denote the changes in f, /, w due to the 
variations Sa, Be, S-sr, 8e, &n. These last arc not all independent, owing 
to the equation o, 3 n 2 = /it. But since n, only occur in the coordinates 
in the form nt + e (Art. 50), we can replace Sn, 8e by the single variation 
BI = t8n + Be ; the four variations Ba, Be, 8-sr, Be L are then independent. 

* The assumption laid down in Art. 71 is not introduced until Art. 77. 
f B, J. Routh, Riyid Dynamic*, Vol. i. Art. 250, 



56 VARIATION OF ARBITRARY CONSTANTS. [OHAP. V 

74. We have then the six arbitrary variations Sa, Se, SOT, B { , 80, Si and 
these will produce changes -rS0 3 , Sr, rSff^ in the position of the Moon (whoso 
coordinates referred to the axes of X, Y 9 Z are 0, r, 0) towards the positive 
directions of the axes. It is required to express the latter variations in terms 
of the former. 

From Art. 32, we have r = a (1 - e cos //), w =.E-e sin K Hence 

Sr = - Sa (a cos E) Be + (ae sin A T ) 8/, 



- Sw + 
r 



[ - sin M 1 Se. 
\r ) 

Therefore 

Sr = - So. H Sw + a ( cos E + -e sin 2 R j Se 

a r \ r J 

r os ae sin f ^ cos 7 + e ^ 

= - da 4- O'W; 4- a 00, 

a v i 2 1 e cos .A 1 

by equations (2), (1) of Art, 32. But w = ra5 4- e OT and therefore 

$(; = jfS?l -J- O6 ~~ OOT = Oj OOT. 

Hence, transforming the coefficient of Se by means of the relations of Art. 32, 



Sr = - Sa+ ---(fc! - SOT) - O cos/) S^ (2), 

a VI - e 2 

Again = flJf=arg. of lat. =/+tsr- 6; then Sx -= S/+ SOT- Sft We 
have (equation (3), Art. 32) 

S/ Se , Sjs (I a\ , a , 

~ 7 ^__ i= + __ = . ^ gg ^ . , . g w 

sin / 1 6 2 KSin ^ v 1 e 2 r / r sin /<? 
after the substitution of the value of SE given above. Hence, since 



sin //sin M = a Vl 6 a /r, 
we obtain 



n/ r (^^ 

^ yi e u r y r y ^ /' 

and therefore, from (1), 



S0 3 = sin/ + Sa + VT3# Se x + l - Vl = # SOT - (1 - cos <) S(? 

.................. (8). 

Since z is immediately expressible in. terms of /and of the elements, the 
rotation S^ is immediately given by equations (1) ; S0 a will not be required 

75. To express the partial differential coefficients of R with respect to 
the elements in terms of the disturbing forces. 



74-75] 



VARIATIONS OF R IN TERMS OF THE FORCES. 



We suppose that the values of the coordinates of the Moon, given in 
Art. 50, have been, substituted in R ; R will then be a function of a, e, nt + e, 
r, 6, i and of the coordinates of the Sun : the latter, being expressed by 
elliptic formulae, are considered known functions of the time and of definite 
constants and they can therefore be left out of consideration. 

The changes in the elements of the Moon denoted by the symbol S, have 
produced changes r80 3 , 8r, r80 l in the position, towards the positive direc- 
tions of the axes of X, F, Z. Let the disturbing forces in these three directions 
be X, *|$, 3- Then s $ acts along the radius vector, perpendicular to it in 
the direction of motion and 3 perpendicular to the plane of the orbit. 

The Virtual Work done by the forces is 



Let the corresponding change in 11 be SJft. Since the change in position is 
produced by variations of the elements only, the Virtual Work is SR and 



da 



8e + d~(wt+7) l ~ 



Substitute in this equation the values of Sr, Sd 3) Sffi previously obtained; 
since the variations of a, e, nt + e, r, 0, i are independent, we can equate to 
zero their coefficients. The six resulting equations will give the values of 

O TT) *") 7? 

-...>. , in terms of $, 5E, 3- Before writing them down we notice that 
act (fa 

since e never occurs except in the form nt + e, 

_dR_ ^ dR 
3 (nt + e) 97 ' 

Also, dR/d(t is taken with reference to a, only as a occurs explicitly and not 
as it occurs through n. 

The resulting equations arc easily found to be 



da 

9JJ 



in r er 

..- = - su a cos/+ !Ea 
^ y 



1 sin/, 



e / cv ^ /- 
~ sin/+ 3^ V 1 - e j , 
- 6 2 ^ 

a<3 /- a- & 2 /T" " > , o- 
7 .^.- r .. sin/- S V 1 - e a 4- Zr, 
i- y r 



r sin i cos L, 



= r sin 



.(4). 



58 VARIATION OF ARBITRARY CONSTANTS. [CHAP, v 

Corollary. We deduce immediately 

oR SR />. 

_ + _ = <&r 

ovr 9e 

dR dR 9jR j ~ . . 

j. + 7T5- = r cos & or sin z cos L. 

013 Q QV 

If r 1 ! be the projected radius vector and t; the longitude in the fixed plane, 
dR/r^v is the disturbing force perpendicular to the projection of the radius 
vector in the fixed plane. Hence, resolving in this direction, we have by 
fig. 5, if M. M ' be perpendicular to asy, 

~ = r, (% sin QMM' - 3 cos IMM') 

= r (% cos MM' sin flMM / - 3 cos M M ' cos flMM') 
= r (S cos i 3 sin i cos x), 

t 9JS 9jB 9JJ 9JB 

whence _ .+ +_.=-._ 

OOT de 96 1 ov 



The expression 9J?/3w implicitly supposes that R is expressed in terms of 
Ti 9 v t z. (See Art. 13.) 

76. Let < be any function of the elements and of the time. The symbol 
Srf> denotes the change in < arising from the changes in the elements only 
and therefore tyfdt denotes differentiation of < with respect to t, only in 
so far as t occurs through the variability of the elements and not through its 
presence explicitly in $. If <j> is a function of the elements only, 



dn de 



Also, we denote by d<f>/dt the differential coefficient of <p with respect to t, 
only in so far as t occurs explicitly in <. Then 



As dr/dt occurs frequently in the following articles we shall denote it by r. 

To find the differential equations required in order to express the elements 
in terms of the time in disturbed motion. 

77. According to the principles laid down for forming these equations, 
the coordinates and velocities, when expressed in terms of the elements and 
of the time, are to have the same form whether the motion be undisturbed or 



75-77] 



THE DIFFERENTIAL EQUATIONS. 



59 



disturbed. Hence the part of the change in position, due to the variability of 
the elements alone, is zero. 

Let the variations 8a, ... Si of the elements be now the changes which 
actually take place in time dt, owing to the disturbing forces (see Art. 91). 
Then rS9$, Sr, r*80i, become the changes in position in time dt, due to 
the variability of the elements only. We therefore have 






dt~"> ' <ft- v 

Similarly Sv/dt = 0. 

The thi'ee equations of motion of the Moon may be replaced by 



.(5). 



dt* dt* r z 




rdt\ dt ) ' 

I d / a <M_l&B 
ridt\ l dt) n dv 





(6). 



For, by definition, dQ s is the angle, reckoned in the plane of the orbit, 
between two consecutive positions of the radius vector ; instead of the 
equation for the motion perpendicular to the plane of the orbit, we use 
the third of the above equations which, by the Corollary to Art. 75, introduces 
the force 3- The first two equations may also be deduced from the general 
formulae* for the motion of a point whose coordinates are 0, r, 0, referred to 
the moving axes used here, by putting S6 l = () = dO^ 

When the motion is undisturbed, we have 



3V_ Wf p 

ft* "~ r "g^a "" r a ' 



11 

rdt 

i d 



W- 



But since dejdt = 3ft/3* + 80 t /dt and since by (5) S0 t fdt = 0, etc. we have 

W$ d9$ . dr dr dv __ dv 
dt dt ' fit dt dt dt 

From the second of these we get 



_ 

dt* ~W dt' 

* E. J. Eoutli, Rigid Dynamics, Vol. i. Art. 238. 



60 VARIATION OF AEBITEARY CONSTANTS. [CHAP. V 



Let r 2 ~ = h = a Vl^F, r, = A = A, cos i (Art. 51). 

d c?c 

We have then, by the subtraction of equations (7) from (6), 



We shall deduce the required formulae from (5), (8). 

78. If we refer to fig. 5, we see that since ML, # 3 are, by Art. 77, both zero, the moving 
axes, as far as their rotation is due to the disturbing forces only, have the single rotation 
80 2 ; the instantaneous axis is therefore the radius vector. Hence, to get from one point to a 
consecutive point in the actual orbit when the values of the elements at the moment under 
consideration are given, wo calculate the displacement in the plane of the orbit by means of 
the elliptic formulas and then give the orbit a rotation 80 2 about the radiuw vector. The 
effect of this rotation on the position will be of the second order of small quantities, 

To obtain the rate of rotation of the orbit, we have from equations (1), 
Hence, since ^ = 0, 



79. Any line in the plane of the orbit which has no rotation about the axis of Z, is said to 
be fixed in the plane of the orbit. Such a line will be absolutely fixed when the motion is 
undisturbed ; when the motion is disturbed it will move only with the plane of the orbit. 
The point where such a line cuts the unit sphere is also fixed in the plane of the orbit and 
has been termed by Oayley * a Departure Point. When the plane of the orbit is in motion, 
the line joining any two consecutive positions of a departure point is perpendicular to the 
intersection of the orbit with the unit sphere. Hence the curves described by departure 
points cut the plane of the orbit at any time orthogonally. 

Since the variation in the position of the radius vector, due to the disturbing forces only, 
is zero, longitudes and angular velocities reckoned in the plane of the orbit from the depar- 
ture point have the same form whether the motion be disturbed or undisturbed. 

80. We shall first find the equations for the variations of the elements in 
terms of the forces $p, J, 3 : these will be required in Chap. X. It will 
be then easy to deduce their values in terms of the partial differential 
coefficients of R with respect to the elements by means of equations (4). 

The Inclination and the Longitude of the Node. 

We have, from equations (8) since /i is a function of the elements only, 

dh^ _ ~ d (h Q cos i) ___ dM 
~di~~ ' di 8tT" 

* On Hansen's Lunar Theory. Quart. Journ. Math. Vol. i. pp. 112-125, Coll. Works, Vol. in. 
p. 19. 



7*7-80] EQUATIONS FOR VARIABLE ELEMENTS. 61 

T r 9jR dhv . j . . di fr . 7 . . di 

Hence ^ = ir cos i A sm i -77 = Zr cos * A SHU -r- . 

ov (it at dt 

Substituting for dR/dv in terms of X, 3 (Cor. Art. 75) we obtain, after 
division by sin i, 



Whence, from the first of equations (1), since Sd^dt = 0, 



The Major Axis. 
We have in the ellipse, 



r' 2 r 



Submitting this to the operation <5/<&, we get, since Br/dt = ^, Sr/ctt = 
8A /(fe Sr, Sa/dt-da/dt, 



Whence, inserting the value off obtained by putting dr/dw = r/nin Art. 32, 
rfct _ 2r6a n e sin/^ u 2//.. a 3 ^ 

'-~'" "" 



Eccentricity. 

We have li ~ IJM (I ~ &). 

Differentiating ami putting 3/r for dh (} /dt, we obtain 



by f*[uatior) (11). Whence, since na?tj\ e 2 = /( , 






As a(l - #)/r 1 -f e cos/, r/a = 1 - cos /^ this may be also put into 
the form 

(12')- 



dt n /* 



62 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V 

The Longitude of Perigee. 
Since A 2 = id> we have f/sin/= pe/h^ and therefore 



Q \r r , 

Applying to this the operation $/dt and substituting as before, we obtain 



Whence, since A ' J = pi, na = pfna?, we have, after inserting the value of r, 

e |= * Vn?- $ cos/ _ 5* Vl ^ ( 1 + ^k sin / 

dt fji ^ J \ U 



But since S0 3 /dt = 0, the third of equations (1) gives 

n dO . $L 

= cos i -h ^j 

cfa & 

Therefore, substituting for Sf/dt, 



dO . $L dO . S/ efe ci<9 
= cos i -h ^j = -7-7 cos i + -- 4- -IT ------ v; 

cfa & ci^ dt dt dt 



The Epoch. 
We have, from the equations (4) and (2) of Art, 32, 



victe , ^ yc/u/ o 

r= ,..' _ sin / = 

Y I __. g2 ^ 7* 

and therefore, since fi = ?^ 2 a 8 , 

r 2 = -~j~===^ e sin ^ sin/ 

Taking logarithms and applying the operation 8/dt, we obtain 
2^] ___ 1 rfe 1 S (e sin js) .S/ 



r e(l-0 9 )& 

But, from equations (3) of Art. 32, we deduce 

1 rfg 1 



Substituting for 8(ednjs)fdt and then for S^/d^ in the previous equation, we 
shall find that de/dt disappears and that the equation becomes 



_ ^ 

f 6 sin jsr dif sin/ sin/ dt"" er sm/ d^ ' 



80-83] 

Putting f r = 
reduces to 

2r 



EQUATIONS FOR VARIABLE ELEMENTS. 63 

jsia.0, rsin/=aVF^sinJ0 (Art. 32), this equation 






= Vl e 2 ( 2 sin 2 |i -=- - -^ ) , by equation (13). 



' dt dt 



Finally, since Sw/dt-d^/dt - dvrjdt (Art. 74), we obtain 



(15), 



which, by the help of equations (10), (14), gives the value of Sejdt. 



81. Wo might of course immediately deduce tlio value of de/dt from this by obtaining 
the valuo of dn/dt from that of dajdt in (11). But its value introduces the time in the 
form tdn/dt, which posBQsacs the inconvenience mentioned in Art. 66. 

From the definition of 1? wo have 



or 



So that by Hubntituting $ndt for nt, wo change into c v Since n only occurs explicitly in 
tho form nt -f e, we shall cou&idor the substitution to have been made. With this understand- 
ing the Buffix of ex is very generally omitted. Tho integral $ndt is called the mean motion 
in tk& disturbed orbit* 

82, The results obtained may be written, after a few small changes : 



di' 

dw 



dd 



u// . / /fj w /.; t j f>w v JL ( 

i -r r~ sm/+ Z V 1 - e j( r 

Vl e a r ( " 



i/H- 5/a (1 H- TT jv ) sin/I + 2 sin 2 

Y C&(X"~"(3 )/ \ 



.d$ 
* l ~dt> 



nar 



nar ^ sn z 
eft "" ^ VT a Bin i ' 



*i- 
dt 



83. Finally, we desire to express the terms on the right hand of 
equations (1G) by means of the partial differentials of R with respect to the 



I. ..(16). 



VAKIATION OF ARBITRARY CONSTANTS. 



[CHAP, v 



elements and, if possible, with coefficients which are functions of the elements 
only. This can be done by means of the equations (4). None of the substitu- 
tions present any difficulty. The results are as follows : 



dt /JL de 

de _ na(l e- 
~dt "" u,e 



na\lle" 



na 



dT 
cfei 



9e 

: + 



*/M?+ 8 ^ 
" \3e 8W' 

, ..as 

4 tan^,-7vr, 



.(l-Vl-#)+-^te 



1 3JK 



^ 

dt "" /A VI "e 2 sm * 3" 1 ' 



d 

These equations are obtciined without the intervention of $p, S, 3 by 0. H. H. Choyne, 
Planetary Theory, Chap. u. 



The ordinary Canonical System of Equations. 

84. The system of equations just obtained is by no means the simplest 
in form ; by taking certain functions of the elements xused above, to form a new 
set of elements, we can reduce the equations to a very convenient form. Let 

/ \; A f\ 

l ~~^ i' 3 2 ~~^ ' 3 ~ 1 (18). 

The equations for the new elements a, /3 take the form, known as canonical, 

1 a \ a * > I (19), 

where R is now supposed expressed in terms of a lt a 23 a 3 , /3 1? /S 2 , /3 3 , t. 



dt 

da, 



85. To prove these we first notice that since the second three equations in (18) do not 
contain *, or or 0, we can immediately deduce from the first three* 



._4. rr:__ 

3a 2 9a 3 ' 9w 71 9a x 9a 2 



n 9a x * 



* It is to be noticed that the expressions 3J2/90... suppose that J2 is expressed in terms of 
a, e..,i and dJR/da,^., that Jft is expressed in terms of a lt a 2 .../8 3 . 



83-86] DEDUCTION OF A CANONICAL SYSTEM. 65 

For, since & is contained only in a 2 , a 3 , we have 



8<9 8a 2 8<9 803 
and so on. Hence, by the Cor., Art. 75, 



= , _- _ 

9o x 8e ' 8a 2 8e 8sr ~ </ '' 8a 3 ~ 8e 

We therefore have, by equations (17), (8), 



9/2 



giving the firwt tliroe of equations (19). 

86. Again, by the fifth of equations (17), we have 

A^^^ \ ?1^_ ?J? 
dt "~ 5* ' ~" h () sin f 9i ~~ 7 

for i only enterw into /^ through the clement j3 3 . 
Also, by the third and fifth of the same equations, 

da% __ d'tff dO _ 7i 8jfi5 
^ "" di " 5? "" /Liccft 8a 

A 9/i ,8/e 

aa^W. ^ -- |-COB'i!-57r . 
jbi&a 06 8)83 

But since o enters into R only through /3 2 , ^ 3 , we have 



wince A =?wVl "' a . Subntitutiiig this value of 8//3e in the previous equation we obtain, 
after putting n*sn*<& 9 the equation dajdt*= "~d 



Finally, since dr/dt, dOJdtj $d^/dt 9 are all zero and since the variations S are now those 
which actually take place, wo have from Art. 75, 



Therefore '' 4- 4- "* 4- 4- 2 4- -0 

i. nueiore 9 ^ -^ + ^ .^- + ^ ^ + ^ ^ + ^ .^- + g ^ d< -u. 

Substituting for ^% v\ ^, ^ 2 , -f 3 the valuos just found, the second and fifth 

C(it Cut Cut wt (M , 

and the third and sixth terms respectively cancel one another ; after division by ^R/d^ the 
equation becomes dajdt** -8 
B. L. T. 



66 VABIATION OF AKBITKARY CONSTANTS. [CHAP. V 

87. All systems of elements which satisfy equations of the form (19) are said to be 
canonical. 

Other canonical systems of elements and the conditions which must be satisfied in 
order to transform from one canonical system to another, will be found in the works of 
Jacobi, Dziobek and Poincare" referred to below. The general form of this transformation 
is that known as tangential (Beruhrungstransformation). 

88. The method of treatment given in this Chapterthat of causing the elements to 
vary in order to include the disturbing forces is more generally useful in its applications to 
the planetary than to the lunar theory. The equations for the variations do not admit, any 
more than the equations of motion examined in Chap, n., of a direct solution and we are 
obliged here also to use some method of approximation. This proceeds according to the 
plan explained in Chap. iv. We first find the values of dRfia... so that the right-hand 
sides of the equations (17) become functions of the time and of the elements. To solve, in 
general we may first consider the elements on the right-hand side to be constant or 
we may combine the equations in any suitable manner to make them integrable ; we thus 
obtain the values of the elements in terms of the time and of six new arbitrages. Using 
these new values in the terms on the right-hand sides, we again get the latter expressed as 
functions of the time and of absolute constants and we can proceed in this way until the 
desired accuracy is obtained ; the new arbitraries introduced at each stej) can be determined 
so as to simplify the final expressions as much as possible. 

In the lunar theory, the necessity for a large number of terms and for many 
approximations causes the process to become very tedious. Dekunay's theory (Chap, ix.) 

the only one worked out on these lines is very fully expanded, but the labour of 

obtaining the expressions was enormous and the results leave much to be desired. It 
is also to be remembered that we cannot start by giving the constants their numerical 

values a literal development is usually essential. Hanson's theory (Chap, x.) is not really 

treated after this method. He uses the variable arbitrary constants in order to obtain 
certain functions for the motion in the instantaneous plane but, having done so, he IB able 
to use numerical values for his constants from the outset. 

In the planetary theory, secular terms that is, terms increasing in proportion with the 
time appear, and also terms with large coefficients and of long period : thewe are very 
much more easily managed by considering them as attached to the elements than by con- 
sidering them as corrections to the coordinates. 

89. One of the most important properties of the equations and of the corresponding 
equations for all sets of elements which may be used is the fact that the coefficients of 
the partials 3./2/3X (where X is any element) are independent of the time explicitly, that 
is, they are functions of the elements alone. The time only occurs explicitly on the right- 
hand sides through the presence of the coordinates of the Sun in It. See Art. 99. 

It will be noticed that the method practically replaces three differential equations of the 
second order by six of the first order. For obtaining literal developments of the coordinates 
this is of doubtful advantage, but for theoretical investigations it is of tho highest importance. 
Canonical systems of elements, as used by Poincar6 and others, have been shewn to bo of 
great value in this respect. 

90. It is necessary to notice very carefully the meaning attached to dR/da in equations 
(17). By means of the equations of Art. 50, R is expressed in terms of a, n, e, *, or, 0, i and 



87-98] OBSERVATIONS ON THE PREVIOUS RESULTS. 67 



there exists between a, n the relation n z a^=jj,. It will be noticed that a only occurs as a 
coefficient and that n only occurs in the form nt+e. Hence we must not use the relation 
% 2 a 3 =/i before forming dR/da but differentiate with respect to a only as it occurs in E ex- 
plicitly. In the canonical system of equations (19) this difficulty is not present, 

The replacing of nt+e by lndt+c l does not cause any trouble, since 



91. Attention must be drawn to the meaning of the symbol d as used in Arts. 7375 
and as used later. In the first case the variations for each element were quite arbitrary 
and it was therefore permissible to equate the coefficients of each of them to zero. Later 
they were the variations actually taking place, owing to the disturbing forces. Thus, when 
the variations were arbitrary, dlt had a certain value depending on the arbitrary variations 
of the elements only ; when the variations were the actual ones it was seen (Art. 86) that 

ss n 

H - c&=0. This last equation is merely a direct consequence of the fact that R is a 

at 
function of the coordinates only and not of the velocities and therefore that 



__- 

dt ~~ 30 dt 3y dt ds dt' 

this expression is zero, since the velocities have the same form in disturbed and in un- 
disturbed motion. This fact is used in Art. 86 to obtain the sixth equation when the other 
five have been found. It might equally have been used in Art, 80 to obtain de^dt. The 
process would however have been somewhat longer. 

92. The canonical system (19) is much more easily found by the method of Jacobi. In 
fact the natural way is to obtain these equations first and then to deduce the results of 
Arts. 83, 82, With the transformations given in Arts. 85, 86, it will be quite simple to 
reverse the process. 

The equations (19) have been obtained by E. B. Hay ward* in a direct manner. 

The constants used may be defined geometrically and dynamically as follows : - 
-a 1 =Timo of passage through the nearer apse, 

a 2 = Distance from node to perigee, 

ag~ Longitude of node ; 

ft =5 Constant of Energy, 

^2= Twice area described in a unit of time in orbit, 



(ii) The methods of Jacobi and Lagrange. 

93. We shall now give a short account of the applications of the general 
dynamical methods of Hamilton, Jacobi and Lagrange to the problem of 
disturbed elliptic motion. In chronological order those of Lagrange should 
come first; their application to the discovery of equations (17) is however 
long and therefore his results will be stated only in so far as they are necessary 

* "A direct demonstration of Jacobi's Canonical Formula," etc., Quart. Jour. Math. Vol. in. 
pp. 2236, 

52 



68 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V 

for the explanation of Hansen's methods. The results of Jacobi's dynamical 
methods, which were based on those of Hamilton, will also be merely stated ; 
references will be given to the more advanced treatises on Mechanics in which 
the proofs may be found. 

94. The Methods of Hamilton and Jacobi. * 

Let T be the kinetic energy and F the force-function of a dynamical 
system. Suppose that there are n degrees of freedom and let q l} q% ... q n be 
the coordinates defining the position at time t. We suppose that there is no 
geometrical equation connecting the coordinates, that F is expressible in 
terms of the coordinates and of the time only, and that T does not contain 
the time explicitly. 

Let the velocities be q l9 g 2 <][n J then T is a function of </^, j$ (i = 1, 2 . . . n). 
Let 

*- ......................... ........ (2o): 



the quantities p$ are called the generalised component momenta of the 
system or, more simply, the momenta. 

Since I 7 is a quadratic function of the velocities it can be also expressed 
as a quadratic function of the momenta in the form 



p 1 jp 3 +... + -d. 3 flp a a + (21), 

where Ay is a function of the coordinates only. 

Theorem I. The equations of motion may be put into the form 

. _m ,._<*# 

where H=T~-I*. 

The principal function S is defined by the equation 



Suppose that the dynamical equations have been solved and that 8 has 
been expressed in terms of the coordinates, of the 2n necessary arbitrary 
constants (exclusive of the constant to be added to S by definition) and 
of the time. We then have 



Theorem II. 



. 

7T7 + jt---/ / =U, Pi = 7$ - 

dt ' ^ dqi 

E. J. Routh, Rigid Dynamics, Vol. i. Art. 414. 



93-95] ELLIPTIC MOTION BY JACOBl'S METHOD. 69 

Theorem III. 8 satisfies the partial differential equation . 



(This follows immediately from Theorem n. by the use of equation (21).) 

Theorem IV. If, knowing only F and the coefficients Ay, we can discover 
any integral of this partial differential equation, invoking n independent 
arbitrary constants &, /3 2 ... /3 n (exclusive of that additive to 8), of the form 



the n complete integrals of the dynamical system will be given by the equations 

> = !!., (i=l,2,.. v ) 

the i being n new independent arbitrary constants*. 

Solution of the Equations for Elliptic Motion by Jacobi's method. 

95. We shall first apply these theorems to the problem of simple elliptic 
motion. There being three degrees of freedom, choose as coordinates the 
radius vector r, the longitude v reckoned on the fixed plane and the latitude 
u above this plane. We take the mass of the Moon for simplicity to be 
unity, so that F** ^r. The velocities fa are r , v, u and 
22 T = r 2 + (r 2 cos 2 U) v z + 

Hence from equation (20) the momenta will be 

_ 3 >S rr 2 cos 2 */H== 

and therefore 



z 



The partial differential equation satisfied by S is then (Theorems n., in.) 

3S irvasy i_(^V + I 1 "l_e=o. 

+ 2 |_ W r 2 cos 2 u \dv) ^ r 2 ^^7 J r 

We first require some integral of this involving three independent arbitrary 
constants (Theorem IV.)- 

* Id Vol n Chap. x. Theorems n., m. are given fully. The result similar to Theorem 
. for the cMrLc function only is proved, but the proof for the primal function 
is almost identical and may be easily reproduced. 



70 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V 

96. To find one, assume 



where Si contains neither v nor t and /3 X , & are arbitrary constants. Sub- 
stituting, we obtain 

Y 4- ft 1 + - f^Y - 2 -^ = 2/3 

drj r^cos^U ^(.duj r Hl< 



Let \ 7-7- j -r -"^i ^2 > 

, being a constant, so that 



and assume $1 = $ 2 + $s> 

where S 3 is independent of r. Then, from the equation for S lt we have 



r*7 ___ 

, = I c??r V^ 2 2 - /3 3 2 / cosii cr. 



whence 

Substituting the values of fif a , >S 3 , S x in the assumed expression for S, an 
integral of the equation for S ia therefore given by 



This contains three independent arbitraries ft, /3 2 , /3 3 . The constant additive 
to S may be fixed by inserting any lower limits to the integrals. Let that of 
the second integral be and that of the first r a> where r a/ is the smaller root 
of the equation 



By Theorem iv. the integrals of the equations of motion are given by 
^ = 3^/3/3^. Whence 



cos 2 



96-98] VABIATION OF ARBITRAEIES BY JACOBl'S METHOD. 71 

The parts of a j? a 2 due to the differentiation with respect to the limit r a , are 



and they therefore vanish by the definition of r a . 

97. It only remains now to connect the six constants 1; a 2 , ot 3 , /3 X , /3 2 , /3 3 
with those ordinarily used in elliptic motion. 

Let the two roots of the equation defining r a be the greatest and least 
distances in the ellipse, that is a (1 4- e), a (1 e). We then have 



whence & = 

Again, as /3 2 2 fif/cos 2 u must be always a positive quantity, we give to u 
its greatest value i and to /3 3 a value such that the expression is then zero. 
Hence 

/9 3 = /3 2 cos i = ho cos i. 

Further, a z is the value of $ when r = r ffl , that is, at perigee where the 
mean anomaly nt + e -or is zero. Hence ! = ( i*r)/n. 

Also, a 8 is the value of v when *7= 0, that is, at the node. Hence ot 3 = 6. 
Finally, we have 



Let sin r/=sinisinx. Then (Fig. 6, Art. 73) since V=M'M, L is the 
angular distance from the node to the radius vector as in Art. 44 ; the value 
of the integral becomes L by the substitution. Hence 2 is the value of L 
when r = r a , that is, at perigee : therefore 2 = ur 6. 

The system of constants is then the same as that given by equations (18). 
It is now very easy to obtain the canonical system (19). 

98. Variation of Arbitrary Constants ly Jacobi's method. 

We have by Theorems n., iv. 

dS dS 

^^> * = W 

Since 8 is a function of the independent quantities $, <^, these may be 

written in one equation, 

(22), 



72 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V 

where 'S t pSq=p l Sq I + ... +p n tyn> etc.; the operator S may denote any varia- 
tion whatever. 

The Hamiltonian equations are 



Suppose we put F = p,jr + R. Let the values of gy, j^, already obtained 
for the case .R = 0, be made to satisfy the equations when R is not zero, by 
considering o$, ft variable. (This is merely another example of coordinates 
and velocities having the same form in two problems when they shall have 
been expressed in terms of the arbitraries and of the time.) 

Let A$, Api be the small increments to be added to q^ t pi in time dt, 
due to the presence of JS. Then from the Hamiltonian equations we have, 
since T is unaltered, 

. dR i, A dR 7 



* 



These, expressed in one equation, give 

dtSR = %(&pSq-~&qSp) ..................... (23), 

Again, as S denotes any increment, it may have the value A so that, from 
equation (22), 

AS = S (j>Ag + A/3). 

Whence 2S (pkq + aA^S) = SA/Sf = ASS = S A (pSq + aS/3). 

Therefore, as S ( j)Ag) = SpAg + pASg, etc., this equation gives 

S (SaA/3 - SySAcc) = S (Sq&p - Sp&q) = ^8E, 
by (23). 

Finally, as A denotes the actual increment due to the presence of R, 
we have 

A.-** *.-* 

Therefore, substituting in the equation just obtained, 



or 



where E is now supposed expressed in terms of t, a lt 2 , ... fa, /9 2 .... 
* Here Aj>Sg denotes the product of Ap and 85, and so elsewhere, 



I 

f 

98-99] LAGRANGE'S METHOD. 73 

From this result it is evident that Jacobi's method of solution produces a 
system of canonical constants. In the case of disturbed elliptic motion, we 
shall therefore have as one system the values of $, {$$ given in Art. 97. From 
the equations just found we can deduce (17) by reversing the processes of Arts. 

8486. 

i 

Lac/range's Method. 

99. Suppose for the sake of simplicity that in a dynamical problem there 
, ! are three degrees of freedom and that the complete integrals are 

i where q, p are defined as before and the system of constants of solution 71... 7 8 

{ is quite arbitrary. Since the constants are independent, we may suppose 

I them determined in terms of the coordinates and momenta by equations of 

5 the form 

i 

Let now R be added to the force function and let the solution be made to 
retain the same form by considering the arbitraries 7$ as variable. Lagrange 
has shown that the sice equations which determine the 7$ are 

* dt 872 l9 3 cfys ""* 9 r 



It is evident that after the differentiations have been curried out, the 
coefficients (7$, %) can be expressed in terms of the arbitraries and of the time. 
Lagrange has shown however that, when so expressed, t is not present ex- 
plicitly in any of these coefficients, so that the equations (24) only contain the 
time explicitly through its presence in dR/d%*. The equations (17) are a 
particular case of these results and were so obtained by Lagrange f; the 
original problem is that of undisturbed elliptic motion, R is the disturbing 
function and to the arbitraries y i} are given the values a, e...0. 

It is easy to see that any function X of the 7$ which does not contain 
t explicitly, may replace one of the 7$, say y l9 and that the equations 

* Proofs of these results are given by Bouth, Rigid Dynamics, Vol. n. Arts. 477, 478 and 
by Oheyne, Planetary Theory, Appendix. 

t M6c, Anal., Pt. n. Section vn. Chap. ii. See also Tisserand, M&c, Cil. Vol. i. Chap. x. 
and 0, Dziobek, Math. TJw. d<?r PlanetenBewegung<?n, 10, 11, 



74 VARIATION OF ARBITRARY CONSTANTS. [CHAP. V 

(24) will be still true if in them we replace 71 by X: for the system of 
constants 7$ was arbitrary. The disturbing function is then supposed to be 
expressed in terms of X, 72, 7s %> t - 

100. The transformation of the equations for 7^ 72 to those for X, 73 
might also have been made directly by means of the assumed relation between 
X and the y<. This way of looking at the problem enables us to give an 
extension to the meaning of X. If we define X by equations of the form 

X = %JAid<yi or d\ = SAicfry*, 

where the A* are functions of the 74 only although the expression SJU^y* 
may not be a perfect differential, the equations corresponding to (24) for 
X, 7a> will still hold because the direct transformation only involves the 
differentials of the arbitraries. 

"When R is not expressible explicitly in terms of X, %, ..., the expression 
9J?/3X may be defined by the equation 

dR * dR 



that is, 8JK/3X has the same meaning as if R had been expressible in terms 
of the new arbitraries. With this convention it will be unnecessary to make 
a direct transformation. 

101. Elements defined in this latter way have been called by Jacob! pseudo-elements*. 

Hansen defines ideal coordinates to be such that they and their first differentials with 
respect to the time have the same form whether the motion be disturbed or undisturbed*. 
Such are r, r l9 v, x, y, g, etc., these being all referred to fixed axes. We can however have 
ideal coordinates referred to moving axes. 

Consider a set of rectangular axes of which those of I 7 , Fare in the plane of the orbit 
and that of Z is perpendicular to it. Let the axis of X be placed at a departure point. Let 




Fig. 6. 
* See the letters of Hansen and Jacobi referred to at the end of Art, 102. 



99-102] PSEUDO-ELEMENTS AND IDEAL COORDINATES. 75 



the longitude of the Moon reckoned from the departure point. Then (Art. 79) 
v l and d\ have the same form whether the motion be disturbed or undisturbed ; %is there- 
fore an ideal coordinate. And yet v l is not expressible in terms of the elements and of 
the time unless one of these be a pseudo-element. For if XG = <r, we have from the figure, 
since the line joining two consecutive positions of X is perpendicular to OX, 

da-= cos i d&, v=f+ is B 4- cr. 

If then Q! be one of the coordinates in terms of which R is expressed, there will be present 
in R the pseudo-element cr. 

102. Let us see how R will be expressed when these axes are used. Euler's formulae 
of transformation of the coordinates of a point (xyz) to (XYZ) are* 

% = &jX 4- &i Y+ c\Z, X = a-^x 
y = a 2 X+ 6 a Y + c 2 Zj F 
3 F+ c 3 #, # 



where (ajjbft), (a a 6 a c 2 ), (% 6 3 c 3 ) are the direction cosines of the axes of X, F, ^, referred to 
those of #, T/, : they are trigonometrical functions of or, 6, i. If ( X YZ) be the coordinates 
of the Moon, we have Z*=Q. Hence R is expressible in terms of Jf, F, cr, 0, i, or in terms 
of ?, ly cr, ^, t. 

Here the differentials of R with respect to # 13 o- have a meaning without further 
definition. For d^ is the angle between two consecutive positions of the radius vector 
reckoned in the plane of the orbit and therefore dRIrdv^S,, S^/3r=$ ; the force perpen- 
dicular to the plane of the orbit will now depend on the differentials of R with respect* 
to or, 0, i (see Chap. x.). 

The use of the pseudo-element or introduces another arbitrary constant, namely, the 
value of or at the origin of time. 

The general condition that X, F, Z may be ideal coordinates when the new rectangular 
axes are any whatever, is 

x da-^ + y da^ + z da> 3 = 0, 
.1? db l -\~y db% +zdb 3 = 0, 
x dc-L -\-y dc 2 +z dc$ =0 ; 

for then cUT, o?F, dZ will have the same form whether a l9 6 X ... be constant or variable. 
These three conditions, involving only the differentials of a x , & 2 ,... are available whether 
the elements be true elements or pseudo-elements. 

On the subjects of Arts. 100-102, see 
Lagrange, Mc~.Anal. Pt. n. Sec. VIL No. 70. 

Binet, " Sur la Variation des Constantes Arbitrages," Journal de V$cole Polytcch- 
nique, Vol. xvn. p. 76. 

Hansen, "Auszug eines Schreibens," etc.; Jacobi, "Auszug zweier Schreiben," etc. 
Crelky Vol. XLII. pp. 1-31. 

Hansen, " Auseinandersetzung einer zweckmassigen Methode zur Berechnung der 
absoluten Storungen der kleinen Planeten," Abh. d. K. Sticks. @es. d. Wtssensch. 
Vol. v. pp. 41218. 

* P. Frost, Solid Geometry (1875), Art. 146. 



76 VARIATION OF ARBITRABY CONSTANTS. [CHAP. V 

Cayley, "A. Memoir on Disturbed Elliptic Motion," Mem. R. A. S. Vol. xxvu. pp. 
1-29 ; Coll. Works, Vol. in. pp. 270-292. 

Donkin, "On the Differential Equations of Dynamics," Phil Trans. R. S. 1855, 
Pt. II. pp. 352-354. 

103. We can very quickly deduce a set of canonical equations from the formulae (24). 
Let yi, ygj Vs ke defined as the values of q lt q%, q z at time t~r and y 4 , y 6 , y 6 those of 

^15 Pzi ^3 a * ^ e same ti 1 * 16 - Since, in the process of forming the partials ^ , ~ etc., it 

JPi Q\ 

makes no difference whether t be constant or variable, and since t ultimately disappears 
from the coefficients (y, i} yy), we can give to t the value r, that is, we can put ^ 1 =y l etc., 
before forming these coefficients. We shall then have 



Whence, since all the arbitraries are independent, we obtain 



and all the other coefficients (yj, y/) will be zero. 

Denote the values of the coordinates and momenta at time r by Q , p im We have in 
the present case yj. <2i ..., y 4 =P 1 ... and therefore equations (24) become 

Hence the values of the coordinates and momenta at a given time form a system of 
canonical constants ; p^, Q$ are considered here as the arbitraries of the original solution 
and R is supposed to be expressed in terms of them and of the time. This system of 
canonical constants was first given by Lagrange. 

104. The disturbed values of Q it i\ are given by 

CdR , 



the lower limits of the integrals giving new arbitraries which are absolute constants. 
Suppose the integrations. have been performed by any process so as to give the disturbed 
values of p$, ^ ; these latter will then be functions of t, r and of absolute constants. 
But since the results must hold for every value of r, that is, at any point in the orbit, we 
shall get the disturbed values of the coordinates and of the momenta at time t by putting 
T t in these equations. Whence 



where the bar denotes that r has been changed into t after the integrations have been per- 
formed. 

This extension is due to Hansen*. It may also be written 



* -/* 

* " Commentatio de corporum coelestium'perttirbationibus," Atr, Nach. Vol. xi. Col. 322. 



102-105] REFERENCES. 77 

where, under the integral sign, we suppose disturbed values substituted. From the former 
of these results the general theorem, which lies at the basis of all Hansen's researches into 
the lunar and the planetary theories, can bo deduced. As however the form in which he 
uses it can be exhibited as an elementary result of the integral calculus, it will not be 
proved here. 

The theorem in question is constructed to prove that any function of the elements and 
of the time may be differentiated, the disturbed values "of the elements substituted and the 
result integrated, with the time as far as it occurred explicitly in the function constant 
during the whole process*. 

105. The earlier literature on the general dynamical principles of Lagrange, Hamilton 
and Jacobi and on their applications to the subject of this chapter, is very large. It has 
been collected and a summary of the results is given by 

Cayley, " Report on the recent Progress of Theoretical Dynamics," B. Ass. Rep. 1857 ; 
Coll. Works, Vol. in. pp. 156-204. See also "Report on the Progress of the 
solution of certain problems in Dynamics," B, Ass. Itep. 1862 ; Coll. Works, Vol. iv. 
pp. 514, 515. 

The chief original memoirs are to be found as follows : 

Lagrange, M4c. AnaL 

Poisson, "Mtki. sur la variation," etc., Jour, de I'j&c, Poly. Vol. vm. pp. 266-344. 

Hamilton, "On a general method in Dynamics," etc., Phil. Trans.. R. S. 1834, 
pp. 247-308 ; 1835, pp. 95-144. 

Jacobi, Vorlesungen ilber DynamiJc. 
The following treatises may also be consulted with great advantage : 

Thomson and Tait, Natural Philosophy, Vol. I. Chap. n. 

Tisserand, H4c. G41. Vol. I. Intro, and Chap. IX. 

Dziobek, Math. The., etc., Abschnitt n. 

Poincard, M&. G4L Vol. i. Chap, i. 

* F'wdamenta, pp. 22-25. A proof by Taylor's Theorem is given in the Commentatio, etc. 
Cols. 323-326. 



u 
I 



CHAPTER VI. 



THE DISTUEBING FUNCTION. 

106. IN the equations of motion obtained in Chapter n. we have 
expressed the forces in terms of the partial differential coefficients of M or F. 
In order to obtain the forces in terms of the variables, jR must be suitably 
expressed The object of this Chapter is to find expressions of such a 
form that the labour of making the developments may be as small as 
possible. 

We have seen that with the methods of procedure usually adopted in 
the lunar theory, the second approximation to the values of the coordinates 
is obtained by substituting the results of the first approximation in the 
terms previously neglected. In general, the first approximation being an 
ellipse, this amounts to expressing the disturbing forces in terms of the 
elliptic elements and of the time. 

Now the determination of motion in space requires a knowledge of three 
component forces. If we form these forces directly from the general expres- 
sion of R in terms of the coordinates a process easily performed and then 
develope the results in terms of the elliptic elements and of the time, there 
will be three developments to be made. To save this labour we develope jR 
in terms of the elements and of the time ; the forces can then be deduced 
by transforming their differentials with respect to the coordinates into differ- 
entials with respect to certain functions of the elements and of the time 
which occur explicitly in the development of R. 

The principal object is then to develope JR in terms of the time and 
of the elliptic elements of the orbits of the Moon and the Sun. According to 
the different functions of the elements used, there will be slightly different 
forms of expression. They can, however, be all deduced from those given in 
Section (iii) which contains Hansen's method. In connection with de Pont6- 
coulant's method, some general properties of the disturbing function will be 
given. The variety of forms by which R can be expressed arises from the 
fact that R depends only on r, r' and on the cosine of the angle between r, r'. 



106-108] INITIAL EXPANSION OF R. 79 

107. In the lunar theory, as already stated in Art. 9, we always begin 
by expanding the disturbing function in powers of r/r'. We have (Art. 8), 

m' , ccx' + yy' 4- zsf 

" 



"-{(, - fl O' + (y-/ 

Let S be the cosine of the angle between the radii vectores of the Sun 
and the Moon. Then xx + yy' + zz r = rr'S, 

and 

Expand the first term of this expression in powers of r/r' by means of the 
Binomial Theorem or by the use of Legendre's coefficients *. The first term, 
which is m'/r', may be omitted since it does not contain the coordinates 
of the Moon ; the second term m'rS/r'*, will be cancelled by the term 

m'rvSf/r' 2 . 

We therefore obtain 

JR = ' 



(i) Development of R necessary for the solution of Equation (A), 
Chapter n. The Properties of R. 

108. The first process is to develope R in terms of r, v, s, r, v'. Let mf 
be the place on xy (fig. 4, Art. 44) where the radius vector of the Sun cuts 
the unit sphere. According to the notation previously used, we denote by 
V) 1/ the true longitudes of the Moon and the Sun reckoned from w, and by s 
the latitude of the Moon above the plane of aty. 

From the light-angled triangle MM'm' we have, 

cos (v - v f ) = cos M 'm' = cos Mm'/cos M'M = SV(1 4- s 2 ). 



Hence ff. = (i- i#+ ^-...) C oB( V -^) ..... . ...... (2). 

Substituting this value of 8 in (1) we have, neglecting 4 , mV 4 /r /5 and 
* Todhunter, Functions of Laplace, LamS and Bessel, Chap. i. 



SO THE DISTURBING FUNCTION. [CHAP. VI 

higher powers of s 2 , r/r', and replacing powers of cosines by cosines of 
multiples of (v v')> 

+ 1 (1 - ") cos 2 (t - tO - M 

(i 0- - *") cos (*/-) + f (1 - |s 2 ) cos 3 (v - v')} 
+ ................................................................... (3). 

109. We have now to express R in terms of the time and of the elements 
of the elliptic orbits of the Moon and the Sun, according to the principles laid 
down in Chapter IV. Before doing so it is necessary to know something 
further about the numerical values of e, e', % a/of (in powers of which 
the expansions will be made), in order that we may have some idea of 
the number of terms necessary to secure a given degree of accuracy in 
the results. We ought strictly to know the meanings to be attached to 
these constants when the motion is disturbed ; but since in any of the 
systems used to fix their meaning, the numerical values only vary to a slight 
extent, for the purposes in view here it is sufficient to give a general idea of 
their magnitude in the case of the Moon. 

The most important ratio is that of the mean motions n', n. It does not 
occur directly in the expansion of R ; it will be seen, however, in Art. 114, 
that n'*lri* is a factor of R. The numerical values are approximately, 



n 1/1 1^1 

= & 0=^V> *-&> 7 = -ft ^" = ?*(T- 

We consider n'/n to be a small quantity of the first order. Consequently 
n'/n, e, e', 7 are small quantities of the first order and a/a' is one of the 
second order. 

On the basis that 1/13 is of the first order, e' 2 = ^^ would be of the tliird order, 
e f a/a'= ^jfa-Q of the fourth, order, and so on. But for simplicity we shall consider them of 
the order denoted "by the index. Hence (7&7ft) Pl ^ a ^ 3 y p *( a AO p * w ^l ^ sa> id to be of the 
order 



110, The equations of Art. 50 in which w, y Q are, by Art. 68, replaced by 
cnt -h e w = <, gnt 4- e 6 = y, give the values of r, v, s in terms of the ele- 
ments a, n, e, e, <&, 0, y and of the time. If, in the same equations, we accent 
the letters and put c 7 = 1, g' - 1, they will give the values of /, v', s f in. terms 
of the time and of the elliptic elements a', n' 9 e', e 7 , <&' ', 0', 7'. But since the 
Sun's orbit is in the plane of reference, 7' = 0, s' = : with these values & 
disappears. Substituting in R, the disturbing function will be found ex- 
pressed in terms of t and of the elements a, n, e, e, r, 0, % a', ri, e', e 7 , w'. 



108-113] PROPERTIES OF THE DISTURBING FUNCTION, 81 

The form of the development of JR. 

HI. Let ? = (w-w')* + e-'. 

Since v, v' only occur in jffi in the form cospO v'\ (p any integer) 
and as (Art. 50), 

i) = nt -f e + A , v' ^rit 



where A and A' consist only of periodic terms depending on the arguments 
<, rj and <'(= n't+ e'-zxr 7 ) respectively, we have 

cos p (v - v') = cos jpf Go&p (A - J/) - sinpf sin jp ( J. - A'). 

Also, jl, J/ being small quantities of the first order at least, we suppose 
that expansions in powers of A, A' are possible. Hence 



Therefore, all the terms arising from v, v' can be expressed by means of 
cosines of sums of multiples of the angles , $, 2??, </>'. 

Finally, r, r' and ,s 2 being expressible in terms of <, $ and of <, 2^, 
respectively, R can be expressed by a series of cosines of sums of multiples of 
the four angles f, <, <', 277, with coefficients depending on wf, a, e, 7 2 , a', e'. 

112, Owing to the introduction of e and <;, the coefficients of t in tlieso arguments 
and in all the arguments which are present in R, will never vanish unless tho argument itself 
vanishes. For these coefficients of t will all be linear functions with integral coefficients, 
of n - n\ m, n' y gn, that is, of n> n' 9 en, gn ; it will be seen later that c, g are not in general 
commensurable with an integer or with one another, and n'/n was assumed to be an 
incommensurable ratio. Hence no linear relation with integral coefficients will exist. 

113. The connection between the arguments and the coefficients. 

The constant y enters into M only through its presence in -y, s. Since 
only even powers of s are present in Ji, a glance at the equations of 
Art. 50 will show that only even powers of 7 are present in 11. 



Also, if we leave aside the factor m'a*/a' 3 which arises from 
equation (1) shows that even powers of a/a' in the coefficient of any term will 
accompany even multiples of v v' and therefore of in the argument of 
that term ; similarly, odd powers of a/a' accompany odd multiples of f , 

Combining these results with those of Arts. 40, 47, 111, we see 
(a) that the arguments of all terms in 11 are of the form 

ttP4>f'$ 

B. L. T. 



82 THE DISTUBBINO FUNCTION. [CHAP, VI 

(6) that the coefficient of the term having this argument is at least of 
the order 



f q or 

according as j is even or odd ; 

(c) that any term in the coefficient is of the order 

where p l , p/, q l are respectively equal top,p', q or are greater than them by 
even integers, and j, ji are odd or even together. 

The factor ePe'&ff 9 (or ePeftfyWaJaf) which occurs in the coefficient of the 
term with the above argument may be called the characteristic part of the 
coefficient or, more simply, the characteristic. 

De PonUcoulants expansion for R, 
114. Since m' =n'V 3 , p = n^a\ we have 

772* Q/ 2 It 71 2 /,</ , VI 

___ = c = c 0j, a , w here m = . 
a a n a n 

It will be found convenient in de Pont^coulanb's theory to choose the units 
of mass, length and time so that //, = !. With these units, m' is the ratio 
of the mass of the Sun to the sum of the masses of the Earth and the Moon. 
We can now put n 2 c# = 1, and 

m'a 2 m 2 ( *. 

7g-~ W- 

u/ a 

The development of R } complete as far as the first order in e, e f , </, a/a', 
is given below ; for the sake of illustration, some terms of higher orders are 
included. The shortest method of actually performing the expansions will be 
explained in Arts. 124 126. 



cos c/> - !* cos (2 - <) + f e cos 
cos $ + ^V cos (2f - $') - fe x cos (2 + <') 
fy 2 ~ fy 2 cos 2^ + f 7 2 cos 27; -f |7 2 cos (2f - 2iy) 



- & cos 2<> -f . . . + e 2 + e 2 cos < + e 2 cos - 



(5). 



113-116] PEOPEBTIES OF DERIVATIVES OF ft. 83 

To deduce the disturbing forces. 

O JD O "D T> 

115. "We have now to form ,,_. Since the disturbing function 

has been expressed in terms of the elliptic elements, these partial differentials 
must be transformed so that we can deduce the functions which they re- 
present directly from (5). For the purposes of this and of the next article, 
the factor m*/a must be supposed to be replaced in (5) by its value 



In the first place, since a only enters into R explicitly through r, and 
since r is of the form a (1 + p), where p is independent of a, we have 

dR dR 

r =a 

dr da 

Here dB/da has the meaning assigned in Art. 90. Whence, if we consider 
only the terms which have a 2 as a factor, 

r|? = a |? = 2JJ .............................. (6). 

dr da ^ J 

Similarly for those which have a 8 as a factor, 



and so on. 



Secondly, since v occurs only in the form v v' } and since only arises in 
R through the first term of the substitution of + A - A' for v - /, we have 



Thirdly, it is found to be simpler to deduce dR/ds directly from the 
equation (3) and then to substitute elliptic values for the coordinates. No 
transformation will therefore be necessary. 

116. Some further properties may be noted. We have from the defini- 
tion of ffR (Art. 12), since R is a function only of the coordinates of the Sun 
and of the Moon, 



dt dt 8r' dt W dt ' 
If we regard the first term of R only, 

,92? 



62 



84 THE DISTURBING FUNCTION. [CHAP. VI 

and generally, 

^ ^? _ d^ /c'\ 

8? "&y~~" "8f .............................. ^ ; ' 

Hence d'R = R + sj8 + cZi/ ..................... (9), 



a form which is frequently of value. If we are considering the term of R 
which contains the factor m'r^ 2 /^" 1 ^ instead of 3 we must put p -f 3. 



By means of this result we only need to form the single differential dR/dj~ 
in the radius and longitude equations (A), Chap, n., when R has been found. 



It is easy to see the truth of the statement made in Art. 65, that 
d'R/dt will contain no constant terms. For R contains only constant terms 
and cosines and therefore 3JB/9?J = 3B/3f only sines of angles without any 
constant term, Also in (9), E, r', M are expressible by means of cosines and 
constant terms while dr', dR/d% consist of sines only, whence d'R/dt contains 
no constant term. All the functions we have to deal with are expressible 
either by means of cosines and constant terms or by means of sines with or 
without a term of the form t x const. + const. 

The effect produced on the orders of the coefficients ly the integration 

of the equations (A). 

117. The substitution of m?/a for m'a a /a/ s shows that the coefficient of 
every term in R is at least of the second order of small quantities. It does 
not however follow that the corresponding terms in the expressions for the 
coordinates are of the same orders as the terms in R from which they arise. 
The integrations will, in certain cases, cause small divisors to appear which 
will lower the orders of the coefficients to which they are attached. 

We have seen in Art. 66, that a term of the form 

A cos (kt 4- a) 

present in the right-hand members of any of the three equations (A) will 
produce terms in r } s of the form 



and it is evident that it will produce in v a term of the form 



A 

~ sin 



There are three cases to be considered, depending on the magnitude 
offe 



116-117] ORDERS OF COEFFICIENTS AFTER INTEGRATION. 85 

(a) If k be a small quantity of the first order, the terms in r } s will be of 
the same order as A and the term in v will have its coefficient lowered one 
order. 

(&) If k* ri 2 be a small quantity of the first order, the terms in r, s will 
have the corresponding coefficients lowered one order, while the order of the 
coefficient of the term in v remains unaltered. 

f r) 7? 

Further, in the longitude equation there occurs the integral I -=- dt, and 
in the radius-vector equation the integral Jd'M. 

If a term of the form A sin (kt + a) is present in dR/dv and if k be of the 
first order, the coefficient of the term will be lowered two orders by the 
integration of the longitude equation. If the term occur in d'Rjdt, its 
coefficient will be lowered one order by the integration of the radius vector 
equation. 

(c) There is one term in R for which k = n, namely, the term with 
argument 4- <' ; its coefficient is of the order mVa/a'. Contrary to what 
might have been expected from the remarks of Art. 66, this term does not 
cause t to appear as a factor of the coefficient. The argument, expressed in 
terms of the elements, is 

nt + e - n't - e' + n't + e' - or' = nt -h e - <&'. 

To understand this, it is necessary to refer to Art. 67 where it was seen 
that the first approximation could only be obtained in a suitable form by 
supposing certain terms of the disturbing function (which should, by the 
method of continued approximation, have been neglected) to be included. It 
was seen that instead of the equation x + rp = Q, the more correct equation 
to deal with is #+ (n 2 + b^ sc = Q, where ^ and Q are small quantities arising 
from the disturbing function. The first approximation (that is, the Comple- 
mentary Function) then consisted of terms of period 2?r/cn. If, with this first 
approximation, Q be expressed in terms of the time and if a term A cos (nt + a) 
arises from Q, we see that no modification is necessary, since its period is 
27T/n and not %7r/cn: further, no terms proportional to the time will arise. 
Finally a term A' cos (en + a! ) in Q will cause no difficulty owing to the defini- 
tion of c, 

The terms for which k is small are known as long-period inequalities. 
Their effect is in general most marked on the longitude. The terms for 
which k is numerically nearly equal to n } are those whose periods nearly 
coincide with the mean period. They produce marked effects on the radius 
vector and latitude and thence on the longitude. 



86 THE DISTURBING FUNCTION. [CHAP. VI 

118. Let us examine the case (c) of the last article a little more closely and see in what 
way the ordinary method of approximation may be applied to a term of the form considered. 

The equation for the second approximation to r can be pxit into the form (see Art. 130), 



where b^ 6 2 Q are ^ ne portions arising from the action of the Sun which, when the 
results of the first approximation are substituted, consist entirely of known terms. 

In dealing with the second approximation we neglect a? 2 , & 3 , . . . and substitute the results 
of the first approximation in b^, , so that a term of the form A GOB (nt 4- a) in <J} & r v 
appears to give an infinite value to the coefficient of the corresponding term in a?. But 
we have seen that this is not really so and that the coefficient can only be found by in- 
cluding in the second approximation, terms of higher orders. It is the simplest plan, in 
actual calculation, to leave this coefficient indeterminate until the third approximation is 
reached : it can then be found because, in the third approximation, the results of the 
second approximation, substituted in # 2 , # 3 ,,.. will produce terms of this form and these 
can be equated to the corresponding terms in a?, Q. 

It is not difficult to see how a term with argument +$' and with a known coefficient 
may arise in so* in the third approximation. In the next chapter we shall see that the 
second approximation will produce in r or a, the terms ^we'cos^', A%maja'co&, 
(A ly A z numerical coefficients). On proceeding to a third approximation we should 
substitute the results of the second approximation in, for instance, aP. We thus get 
amongst others a term of the form A/n?e f (a/a')cos(f 4-$')> that is, in the equations for 
finding the third approximation we have a term of the same order as that in the disturbing 
function and therefore of the same order as that which would be used to find the second 
approximation. Hence, as far as this term is concerned, it is necessary, not only for the 
form of the solution to be correct, but also that the method of continued approximation 
may be applicable, to include certain parts of the equations which, in the second approxi- 
mation, would ordinarily be neglected. 

There are other terms for which the third approximation appears to produce coefficients 
of the same order as those given by the second approximation ; this peculiarity is chiefly 
due to the direct and indirect effects of small divisors. De Ponte'coulant* treats them 
by leaving the coefficients indeterminate until the higher approximations have been 
completed. Such terms illustrate the necessity, mentioned in Chap. iv. and insisted on 
here, of continually bearing in mind the effects produced by the higher approximations and 
the impossibility of obtaining the first and second approximations in the correct form, 
without considering them. 

119. There are certain results which we cannot stop to prove here but the statement 
of which may perhaps prevent misconceptions. They are : (a) The coefficients resulting 
from the action of the Sun in the coordinates are never of an order lower than the second ; 
(j8) The coefficients can always be represented by series of positive powers of m, e, e', y, a/a', 
with numerical coefficients ; (y) The characteristic of a term in radius vector or longitude 
is the same as that of the term in R from which it arose : in latitude, it is always less by 
one power of y ; (d) Nearly, but not quite, all the terms in the coordinates arising from the 
action of the Sun, have the factor m in its first power at least ; (e) the constant portions of 
the expansions of the functions considered contain only even powers of e, e', y, a/of. 

* Sy steme du Monde, Vol. iv. pp. 103, 145, 151, etc. The terms in longitude and radius vector 
of the form considered above, are those numbered 74. 



118-121] COEFFICIENTS INDEPENDENT OF w. 87 

With reference to the statement (y), it may be remarked that the divisors arising from 
integration are, in de Pontecoulant's method, linear or quadratic functions, with integral 
coefficients, of %, n'> en, gn. The constants 1 - c, l-g will be found to be represented by 
infinite series in powers of w, e 2 , y 2 , e' 2 , (a/a') 2 . Their principal parts begin with the power 
m 2 , so that the divisors involving c, g always contain powers of m. Hence none of these will 
have 0, y, e', a/of, as a factor. 

The exceptions to (d) as given by Delaunay*, are the terms in radius vector and longitude 
with arguments D -f V +pl + 2^, and those in latitude with arguments D + 1' -t-pl + (%q -f 1 ) F, 
or, with the notation used here, the terms with arguments 



respectively ( p, q - oo . . . + oo ). 

120. Since the expression of R has the factor m 2 , when we put m=0 all terms 
dependent on the action of the Sun should vanish in the expressions of the coordinates. 
The apparent exception of the terms just mentioned has been explained by Gogouf. It 
depends on the definitions of the constants in disturbed motion. When m is put zero the 
motions of the perigee and node vanish and the arguments of those periodic terms which 
remain, contain t only in the form jpwf + const. After suitable changes of the arbitraries have 
been made, Delaunay's expressions with m zero reduce to those for purely elliptic motion. 

On the subjects of Arts. 117 118, see de Ponte'coulant, TMorie du Systime du Monde. 
Vol. iv. Nos. 914, 90, 91, 100 ; Laplace, M&anique Celeste, Book vn., 5. On the short 
period terms whose coefficients are large in comparison with their characteristics, see also 
Part in. of a paper by the author, Investigations in the Lunar Theory $. 

121. The Second Approximation to R. 

It must be remembered that the substitution in R of elliptic values for the 
coordinates of the Moon, is only a means of finding a first approximation to 
jR. Suppose that with these elliptic values for r, v, s substituted in the right- 
hand members of the equations (J.), Art. 13, we have solved the equations 
and have found the new values r -f Sr, v + 8v, s + &, of the coordinates. 
According to the principles of the method, these new values must be sub- 
stituted in the right-hand members of the equations in order to find the 
third approximation. 

Let Q be a function of the coordinates of the Moon which may contain 
also the time. Put 



where 8Q is the new part of Q arising from the additions Sr, 8-y, 8s to the 
elliptic values of the coordinates. Expanding by Taylor's theorem, 

* Mem. de VAcad. des &c., Vol. xxix., Chap. xi. 

t Ann. de VObs. de Paris, Mttn., Vol. xvra. E. pp. 126. 

J Amer. Journ. Math., Vol. xvn. pp. 318 358. 



THE DISTURBING FUNCTION. [CHAP. VI 



. + .................................................................. (10). 

By putting R, dR/dl; ... successively for Q we can find the new values of 
these functions. In the partials dQ/dr, 8 2 Q/9r 2 , etc. we substitute the initial 
values of the elements, 



Care must be taken when we are proceeding to form such expressions as tfdtBll/dv. 
Omitting, for the sake of illustration, powers of dr, dv higher than the first and all terms 
dependent on the latitude, we have 



This, if we regard only the terms independent of a/a', gives 

.............................. <"> 

in which we substitute for &&[%&> Bjft/3 the values obtained from (5). 

(ii) Expansion of R for Delaunay's Theory. 

122. Let 3> ct 2 be the angular distances 0ft, XLA (fig. 4, Art. 44), so that 
3 = 5 a 2 = 'cr - as in Art. 84. Suppose for a moment that the Sun's orbit 
is inclined to the plane of ocy and let a/, a 2 ' be the distances #!', Of A', where 
A' is the solar perigee and XI' the intersection of the Sun's orbit with the plane 
of xy. When, the inclination of the Sun's orbit vanishes, XI' will become 
an indeterminate point on xy, but a/ H- a/ =* r' will be determinate. For 
symmetry, we use a/, a 3 f although they can only occur in the form a/ -f a/. 

We have from the spherical triangle Mlmf, 

S = cos Mm = cos SIM cos Xlm'+ sin X1M" sin fltm'cos i ...... (12), 

or, since XXM =/+ a 3) fim'sa/' + a 2 ' + / - 8 , 

S = (1 - 7l 2 ) cos (/+ a 2 + a 3 -/ - a,' - a/) + 7l 2 cos (/+,- a $ +/' + / + a/) 

......... (13), 

in which we have put sin %i = ^. 

From this we may form S 2 , S* . . . and, after expressing them as sums of 
cosines of multiples of angles, substitute them in (1). 

The first part of R will consist of five separate terms of the form 



121-124] DELATJNAY'S DEVELOPMENT. 89 

p, p taking the pairs of values 0, ; 0, 1 ; 1, ; 1, 1 ; 1, 1 ; K "being a 
function of 7^, and a, a' depending on the angles a a , a s , a%, ot 3 '. 

Delaunay proceeds by expanding 

r'cos . a' 3 ct' 3 cos 



(where a, a' may be any angles) in powers of e, e' and cosines or sines of 
multiples of w, w'. These may be obtained by means of the formulae given 

} in Chap. in. above. By the direct multiplication of series he is then able to 

form all the terms required. In a similar manner the rest of the terms in JK 

* may be found. 

| The arguments of all the terms will evidently be composed of the four 

I angles, 

w, w f , w-fot 2 , a 3 ~(i 



I 

I or of w, w', w -f ot 2 , w 4- a + a s "" (w' + a/ + a 3')- 

I 

123. It is not difficult to see that, after one or two small changes, this method of 

development will produce the same result as that obtained in Art. 114. In both cases we 

shall have expanded in terms of the mean anomalies and elements of the two orbits. 

| In the former case the inclination of the Moon's orbit was introduced through v and s, 

| while in Delaunay's method it is introduced directly through S. For simplicity and 

^ ease of calculation the latter method has a great advantage over the former, and more- 

i * over, it admits of a much more general treatment. 

Since y = tan z, y l = sin ^", if we put 



and for w, w 

the symbols <, $', ;, , respectively, we shall immediately obtain the development (5). 

Delaunay has performed the expansion so as to include in It all quantities up to the 
% 8th order inclusive ; in addition certain terms are carried to the 9th, 10th and even higher 
I orders where it appears to be necessary for accuracy. His development of R consists of a 

7 constant term and 320 periodic terms. See Mm. de VAcad. des /&., Vol. XXVIIL, 

Chap. ii. 

f * (iii) Hansens development 

124. Hansen's method is a more general one than either of those out- 
lined above since it is adapted to the case in which the Sun's orbit is in 
motion. It will give, after a few small changes, the expressions both of 
de Pont^coulant and Delaunay. 

Let CD, w' be the angular distances of the apses of the instantaneous 
orbits of the Moon and the Sun from the line of intersection of the planes of 
the orbits, that is, from their common node, and let J be the angle between 



90 THE DISTURBING FUNCTION. [CHAP. VI 

the planes (see Arts, 217, 220). The angular distances of the two bodies 
from this node will be eo +f, co' +/', and therefore 

S = cos (/+ o>) cos (/' 4- ') + sin (/+ ) sin (/' + /) cos J 
= (1 ~ sin 2 |/) cos (/-/' 4- o> - G>') + sin 2 y cos (/+/' + &> + o>') ...(14). 

Let Rfc = BW + E< 2 > + . . . , where pRP> is the term in (1) with coefficient 

wi! 9 . Then 



m? (~Y (~i) 3 (A + & cos (2/- 2/' 4- 2a> - 2') + A cos (2/+ 2a>) 



where ySj . . . /9 5 are definite functions of sin 3 \ J which it is not necessary 
to specify here. It is required to replace rja f offr',/,/' by series involving 
the mean anomalies w, w' and the eccentricities e, e'. 

The symbol % is here used instead of m because Hansen puts m'4-^=w' 2 a A3 and so 
does not neglect the small ratio jut : m'. We have then 



The ratio of the difference between m l and m to either, is the very small quantity 1/660,000. 
See Art. 53. 

r 2 . a /s 

125. Let ~ = S PJ cos jw, -73 = 2 K? cos j W, 

tt T 

~ COS /o /.x _ 5 Q/ COS / -A ^ COS /O.P/V _ ^ @j' C COS / / /x 

" - 



a 2 sn / sn i sn f * sn 

in which j t j' receive all integral values from + 00 to - oo *. The coefficients 
P, Q will be functions of e only and K, Q functions of e' only ; these may be 
calculated after the methods explained in Chap. in. Since, in each case, the 
coefficients for positive or for negative values of j, / are superfluous, it is 
supposed that 

P^ JL y , QL/, GL/, Q_/, 6L,P,, K f , Qf, G?*, -Qf, - (V 
respectively. 

Consider trigonometrical series of the forms 



* The letters c * placed above the coefficients are simply marks to distinguish between the 
coefficients of the cosine and sine. 



124-126] HANSEN'S DEVELOPMENT. 91 

and of the same nature as those just given. Their products may be expressed 
in the forms, 



j cos (jw) x 2E/ c g ? ( JW) = SSJSJ E/ c g ?* (jw + jV), 
j siii jw; x 2Ef sin JW = - SSJSj j/ cos ( jw + jW). 
Applying these results to the term of -R (1) with coefficient & we obtain, 

2 ()' = SPA " cos 

Also for the term with coefficient /3 2 , 



= cos (2 - 2 cos 2/. cos 



+ sin (2 - 20 cos 2/. sin 2/' - 2 sin 2/. 3 cos 

= cos (2< - 20 S2 (Q/Gy 6 - Q/<?/) cos (jw + JV) 

+ sin (2 - 2o)') 2S (Q/ G=/ - Of <?/) sin (> + jV) 

?/ + Q/G/) cos O +/ w/ + 2(B - 2eo ') 

- Of G,' a + Qf s s ~ Qf ff /) cos ( J + J' w ' ~ 2a) + 

Since j,j' receive negative as well as positive values, we can in the second 
line of this expression put - j, -/ for j, j'. Whence from the relations (16) 
the expansion becomes 

- ') cos (J w +J' W ' + 2<B 



This form of the product of two series is a sufficiently simple one to 
calculate, when we have obtained the values of Q in terms of 6 and of &m 
terms of e'. We can express by similar formula the terms in H whose coeffi- 
cients are /8 g , $41 & 

The terms in R are treated in like manner. The terms in JZ, being at 

j^ ta B t"Jtt^ ^*^^ ^ 

sensible coefficients and to find the latter directly. 



AttmO. i. I. 



92 THE DISTURBING FUNCTION. [CHAP. VI 

contain methods for the complete development of the disturbing function both in the 
lunar and the planetary theories. A very clear and concise account of Hansen's method 
and results has been given by Cayley in his first Memoir On the Development of the Dis- 
turbing Function in the Lunar Theory*. Eeference may also be made to two other papers 
by the same writer on the development of the disturbing function f. 

In order to deduce de Pont6coulant's developments from those of Hansen we put J=i 
and 0, <', i?, for w, w\ *+, w+a>- w'-<*' respectively. To deduce Delaunay's results 
we put sin 4/=yi, also a 2 for a>, and a 2 '+a 3 ' - s ^ or (k) '- 

(iv) Laplace's Equations. 

127. In order to develope Ffor the purpose of treating Laplace's equations (Chap, n.), 
we have by Arts. 8, 107, 



And since 



*e obtain 

Laplace forms the forces 3^/9^, 3^/9v, 9.F/3S directly from this expression. Since the 
independent variable is i>, it is then necessary to expand the results in terms of v and of 
the elements. 

By means of the results of Art. 52 the coordinates u ly s are immediately put into the 
required form. The coordinates u r , v' being given as functions of t must be expressed by 
means of the results of Arts. 52, 70 in terms of v. See Laplace, Mfc. G41., Book vii. 
Chap. I. 

(v) Equations referred to Rectangular Coordinates. 

128. The expansion of the disturbing function for the equations of Arts. 1820, has 
been there performed as far as it is necessary. It is a feature of the method that we do 
not substitute elliptic values for the coordinates of the Moon in the terms dependent on 
the action of the Sun, 

The part % of equation (16), Chap. IL, is that portion of & which is independent of the 
parallax of the Sun and which vanishes when e' is zero. As already pointed out in Art. 22, 
we can put 

. ......... (18), 




in which A, B, C, K depend only on the motion of the Sun and are at least of the order 
mV ; they can be easily expanded in powers of e' by the known elliptic formulae. 

* Mem, ofR. Astr. Soc, t Vol. xxvu. (1859). Coll Works, Vol. m. pp. 293318. 

t Mem. R. Astr. Soc., Vols. XXVIIL, xxix. Coll. Works, Vol. m. pp. 319343, 360474. 



CHAPTER VII. 
BE PONT^COULANT'S METHOD. 

129. WE have, in Section (i), Chapter II., obtained the equations (A) on 
which de Pont6ooulant has based his method. In Chapter in. Art. 50, are 
to be found the elliptic values of the coordinates which serve as a first 
approximation after the modification, formulated in Chapter IV. Art. 68, has 
been made. In Chapter VI. Art. 114, we have given a development of R 
obtained by using these modified elliptic values ; and, in the same division of 
Chapter VL, certain theorems which tend to simplify the algebraical processes 
of the second and higher approximations, are proved. The object of this 
chapter is to explain the manner of carrying out the various approximations, 
by applying the principles already discussed to the discovery of some of the 
larger inequalities in the Moon's motion. The arrangement of the inequalities 
into classes, although a natural one, is not essential to the method (Art. 151). 
It enables us, however, to explain the origin of the various periodic terms^in 
the expressions for the coordinates and to carry out portions of the third 
approximation with greater ease and security. 

Preparation of the equations (A) /or the second and higher approximations. 

130. Let r , U Q be the modified elliptic values of r, u (= 1/r). Let 

1/r = l/r + 8w, 

and therefore r 2 r 2 - 2r 8 ^ + 3 (r 2 &*) 2 - . - - ; 

$u is then the part of 1/r depending directly on the disturbing action of the 
Sun. 

Substituting for r on the left-hand side of the first of equations (A), 
Art. 13, we obtain, since p is put equal to unity, 



94 BE PONT^COTJLANT'S METHOD. [CHAP, vn 

Since R contains only even powers of 7, the equations for radius vector 
and longitude and therefore those coordinates contain only even powers of 
7. We shall neglect powers of 7 "beyond the first and consequently, in the 
first two of equations (A), neglect 7, 5 entirely. 

We can obtain other forms for P by means of the results of Arts. 115, 
116. Neglecting s we have, in the second approximation, 



(8), 



where p = for the terms independent of the solar parallax and p = 1 for the 
terms dependent on the first power of the ratio a/of. 

Also from equations (9) of Art. 116, in the second approximation, 



with the same definition of p. An arbitrary constant is considered to be 
present in the expression for P. The value of SP, necessary for the approxi- 
mations beyond the second, is found by Taylor's theorem as in Art. 121. 

131. Let V Q be the elliptic* value of the longitude and let h Q be the 
elliptic value of h. Put 

h = A + Sh. 



Neglecting s 2 and substituting, the second of equations (A) may be 
written, 



3JR ,A/1 2. /5 , ,A h Q 

w dt )^ 

We have also, since 7 is neglected and p = 1, 






or, when e 2 is neglected, 

fe = Va = na 2 ........... . ..................... (6). 

Finally, the equation to be used to find the latitude is not the third of 
equations (A) but 



__. 

% I n - "7\ "~~ /"\ I 

r 8 GZ T os T 



from Art. 13, neglecting s 8 . 

The equation (6) of Chapter II, for the determination of the constant part 
of 1/r is, when we neglect T 2 , a/ of, 



- .............................. < 8 >- 

That is, modified elliptic. This abbreviation will be used throughout the chapter, 



130-133] PRELIMINABIES. 95 

When we are considering terms independent of e, we have 



and not otherwise. For the introduction of c will cause terms of the order 1 c to appear 
if we substitute the modified elliptic values of r , # in these equations. They are only 
satisfied by the purely elliptic values of r , . 

The constant &h and that considered to be present in P are theoretically superfluous, 
but the presence of dh is of great assistance in determining the meanings to be attached to 
the arbitraries in disturbed motion. 

132. We have seen in Chapter VI. that the characteristic of a term in R 
is unaltered by the integration of the radius vector and longitude equations. 
All terms in It depending on the latitude are at least of the order y z : when 
introduced into the latitude equation they will be at least of the order 7. 
The order of the characteristic is not further lowered by the integration of 
this equation. 

Hence we can divide up the terms of the disturbing function and, instead 
of finding the complete first approximation with all the terms of R, we can 
separate them out according to the composition of their characteristics. 

The order in which the terms will be taken is as follows: the terms 
whose coefficients depend only on (i) m ; (ii) m, e ; (iii) m, e' ; (iv) m, a/a,' ; 
(v) m, 7 ; (vi) m and any combinations of e, e', 7, a/a' and of their powers. 
In the second, third, fourth and fifth classes we shall here develope only the 
terms depending on the first powers of e, e', a/a', 7 respectively ; the terms in 
the sixth class will not be developed. 

The approximations, which will in certain cases be carried to the third 
order*, are made according to powers of the disturbing forces, that is, of m 2 . 
In the first approximation we neglected the disturbing forces ; in the second 
approximation all the new parts added should be at least of the order m 2 ; in 
the third approximation of the order m 4 , and so on. But, owing to the small 
divisors introduced by integration, as already explained in the previous 
chapter, some of the terms in the second and third approximations contain m 
in its first power. It is this fact which causes the great labour necessary to 
produce expressions for the coordinates with an accuracy comparable with 
that of the best lunar observations of the present day. 

133, It is necessary to make mention here of the two constants n, a. In 

undisturbed motion we have 

wW=l. 

In disturbed motion, n will be defined (Art. 135) as the observed mean 
motion; n, a are two of the arbitraries of the solution. Since we cannot 
have seven independent arbitraries, the relation n*a? = 1 will be supposed to 

* The details of the third approximation are printed in small type. 



96 DE PONT^COULANT'S METHOD. [CHAP, vn 

hold between the symbols n t a in disturbed motion, whatever may be the meaning 
attached to n. When n has been defined, a definite meaning is thereby given 
to a. 

The necessary changes in the meanings to be attached to n, e, 7 when the 
motion is disturbed, are defined in the course of the chapter. Fuller expla- 
nations will be given in Chapter vni. 



(i) The terms whose coefficients depend only on m, 

134. From equation (5), Art. 114 we have, since all terms dependent on 
e } e', 7, a/a' are neglected, 

s2) ........................... (9); 



r = a, v Q nt + e, r' = a', v' = n't 4- e'. 

Also, as 2= 2 (n - n') t -f 2e 2e', we see by the results of Art. 117, that 
none of the terms here considered will have the orders of their coefficients 
lowered by integration. Hence, in the results of the second approximation 
all terms will be of the order m 2 at least, in those of the third approximation 
of the order w 4 , and so on. 

In the second approximation we neglect powers of Su higher than the 
first, The equation (1) therefore gives, since n*a* = 1, 

1 d* 

--- -2372^- &M=P. 

7i 2 dt* 

And from equation (4), since we neglect a/a' and therefore have p = 0, we 
obtain 



From the value of R given above, 

8 a 

/O D -__2 /yj' 

therefore 2w' I TT^ d = 4 r cos 2. 

J 9f A a n n " 

The equation for w then becomes, since %'/w = m, 



a 
where a/a is the constant attached to the integral in P. 

To find the particular integral assume 

1 

&u = - (6 + 6 2 cos 2|). 
a 



133-135] (i) VARIAT10NAL INEQUALITIES. 97 

Substituting and equating the constant term and the coefficient of cos 2 
to zero, we have 

-& = m 2 -ha (10), 



n 
The second of these gives 

6 a (4 (1 - m)* - 1} = 3m 2 + f m s /(l - m), 
or, expanding in powers of m, 



The third approximation will produce terms of the order m 4 and this 
value of Z> 2 is therefore only correct to m 3 ; it is given here to the order m 4 
in preparation for the next approximation. Since a is an arbitrary constant, 
the constant term 6 is at present undetermined. 

135. The equation (5) for the longitude is, with the same substitutions 
and to the same degree of approximation, since h Q = na 2 , 

d , , 1 [dR ,. &h 
TJ ov = ZnaSu 4- ~ I -^ dt + . 
dt <z 2 J of a 2 

f^ P 

This, from the values of $u, I -^dt found above, becomes as far as the 

J v% 
order m 3 , 

| fit; = 2w {6 + (m 4- ^m) cos 2f } + f ^ cos 2f + ^ . 
Therefore integrating, 



3 +...)sin2f ..................... (12), 

where we have put n/(n ^') = (1 m)" 1 == 1 + m -f- m 2 -f . , .. 

The longitude is v + S^ = ^ + e + Su This shows that the introduction 
of B is useless, for it merely adds an arbitrary part to e which was itself 
arbitrary, 

The coefficient of t is n -f 2n6 -f Sh/a z . Since n was an arbitrary of the 
original solution and since Bh is an arbitrary introduced into the second 
approximation we can determine the latter at will. We shall always give 
this arbitrary a value such that the coefficient of t in the expression for the 
true longitude is always denoted by n ; this statement defines the meaning of 
n in disturbed motion. Thus 2w& 4- S/i/<x 2 = and 

$v = (i^m* + f f m 8 ) sin 2 ........................ (13), 

as far as the order m 3 . 

B. L. T. 7 



98 DE PONT^COULANT'S METHOD. [CHAP, vn 

136. The constant term & in &u is found by substituting the values 
I/a 4 8w, nt + e + Sv for 1/r, v in (8). That is to say, we put 



~ 



therefore, as b Q is of the order m 2 at least, 
v 2 = 7i 2 4- n (n TZ,') (-V-m 2 4- 4 9 -m 3 ) cos 2, -~ = 1 4- 3& 4- (3m 2 4- -^ w 8 ) cos ! 



and similarly for r/a, r/a. 

The equation gives, since 2jR/a 2 can be put for 
8) cos 2f } 4 (w - ^ x ) 2 (^ 2 + V^ 3 ) cos 
?i 2 - n (n - TZ X ) (-^m 2 -f ^-m 3 ) cos 2 



+ {1 + 86 + (3 2 + ^ 3 ) cos 2 f ) = (| + f cos 2). 

Put 7i n' = ft (1 m), I/a 8 = w 2 and expand in powers of m, neglecting those 
beyond m 3 . On equating the coefficient of cos 2 and the constant term to 
zero, it will be found that the former vanishes identically, thus giving a 
verification of the previous work, while the latter furnishes to the order m 8 , 



- n 2 4- w 5 (1 4- 36 ) = ; 
or, correctly to the order m 8 , there being no term of that order, 

*>o = ^ (14). 

Thence Su = - {^m 2 4- (m 2 4- -^m 3 ) cos 2} (15), 

to the order m 3 . 

137. In order to exhibit the method of finding the third approximation, 
we shall calculate the coefficients of the periodic terms in 1/r up to the order 
m 4 . They can be found to the order m 5 in this approximation, but for the 
purpose of explanation it is not necessary to include the terms of this order 
as they merely involve further expansions. 

To find the third approximation we must include the first term of the right-hand 
member of (1) ; this term being of the order w* at the lowest, we can use the result (15) of 
the second approximation. We have therefore 

i 

torn 4 , 



= 2 cos2|+12 cos 4^, 

(Xf Ct> 

since (n-riym^n^m^cP to the order m 4 . 



136-138] (i) VAEIATIONAL INEQUALITIES. 99 

Also from (4), 



' f^ 
J vV 



The terms being all of the order m* at least, we can here put r=*a. Substituting the 
values of JB, 8jB/3|, 9 2 jft/9 2 obtained from (9) and those of dv and $r=~r^= -< 
from (13) and (15), we obtain to the order m 4 , 



nyj 2 ij M 2 

2 cos 2) - 4 - 



!...(16). 

On integration, the second line of this expression will be seen to be of the order m 5 . 
Hence 

.^4 

to w 4 . 
Let the new part of 1/r be ^u. The equation for d^u therefore becomes 

5 -j- g u _ $ ^ cos 2 4- ^fr* cos 4 -f const, part. 

n*dt* L1 a b ** a 

We do not put the constant terms into evidence since, as in the first approximation, 
they are determined by means of (8). 

1 
Assuming fail = - (6 + $^2 cos ^ + ^4 cos ^Sh 

CL 

we easily obtain d^ == - (56 - % m 4 cos 2 + 1 m 4 cos 4). 

Adding this to the value of dn which in (11) was carried to the order m 4 for the coefficient 
of cos 2, we have t 

138. In a similar manner the third approximation to the longitude may 
be calculated. In doing this we can omit all the constant terms which 
appear in the equation for dv/dt, since these merely add a known part and an 
arbitrary part to the coefficient of t in the expression for v> and these new 
parts, by the definition of w, always vanish. After the value of v has been 
found we can obtain the constant part of I/r by means of (8) and at the 
same time verify the results previously obtained. 

When these calculations have been performed we shall get as far as m 4 
for the terms whose coefficients depend on m only, 



100 DE PONTriCOXJIANT'S METHOD. [CHAP. VII 

From the remarks just made, it is evident that in calculating the right- 
hand member of the radius equation we can always omit the constant portions. 
This evidently applies to the calculation of all inequalities. 

Also, in calculating the terms in the equation for t), we always equate the 
constant portions to zero. This rule also applies to all inequalities. 

(ii) The terms whose coefficients depend on m and the first 
power of e only. 

139. The part of E required is* 



a 
Also, from Art, 50, since 7, e 2 are neglected, 

r Q = a (1 e cos </>), v = nt + e + 2e sin <, s = 0, 
r = a', v r = n't + e'. 

We first neglect all powers of m beyond m 2 . In forming P for the second 
approximation, it is evident that we do not need the first two terms of jR. 
Hence by (4), remembering that % = (n n') t -I- e e', <f> = cnt -f e -cr, 

pj?ie{- 2 cos < - 9 cos (2f- <) + 3 cos(2 + <)} 
a 



*)l 

J ' 

Since 1 c is of the order m at least, the second line of this expression is 
of the order m 9 and therefore, to the order w 2 , the value of P is given by the 
first line. 

The particular integral will be 
aSu = |<m 2 + m 2 cos 2 + ec Q cos <j6 + ec^i cos (2f <) + eci cos (2 + <), 

The first two terms are those obtained in (i); c , C-i, Ci are the coefficients to 
be found. We must substitute this expression for Su in (1). 

We have ^ (rf Su) = ~ {a 8 (1 - 3e cos <j>) Su}. 

Cbv Cut 

Whence, retaining only terms with the characteristic e, 

j-^ (TQ^SU) = a*e -r-^ { -|m 2 cos <jfr 3w 2 cos 2^ cos <|) -j- c cos ^ 

+ C i cos (2f <A) + GI cos 

* 

|(c - J-wi 2 ) cos <^ + (c-x f m 2 ) cos (2f <j>) + (c x - f m 2 ) cos 



* It is necessary to include the first two terms since, in combination with other terms of the 
order e, they may produce coefficients of the order considered. There is the same necessity for 

all inequalities. 



138-140] (ii) ELLIPTIC INEQUALITIES. 101 

Substituting this and the value (19) of P in equation (1) and neglecting 
all terms but those which have the first power of e in their coefficients, we 
obtain 

cWa 2 e cos <f> - ~ cos <f> - - {c cos < 4 c_x cos (2 - <) 4 ^ cos (2 4 <)} 

4 n?a*e {(c ^m 2 ) c 2 cos < 4 (c_ L |m 2 ) (2 - 2m - c) 2 cos (2f - </>) 

4 (c a - f ??i 2 ) (2 - 2m 4 c) 2 cos (2f 4 ^>)} 
= ~- e {- 2 cos </> - 9 cos (2f - <f>) 4 3 cos (2? 4 ^)}. 

Putting I/a for ?z- 2 a 2 and equating the coefficients of cos A, cos(2|-^), 
cos (2^ 4 ) to zero we have, after multiplication by a/e, the three equations 
of condition 

c 2 - 1 - c 4 c 2 (c - i w a ) = - 2m 2 , 

- c_! 4 (c-i - f m 2 ) (2 - 2m - c) 2 = - 9m 2 , 



The first of these may be written 

(c 2 -l)(l4Co) = -2m 2 4|-m 2 c 2 ............. . ....... (20). 

As c , c 1 are known to be of the order m at least, this shows that c 1 is 
of the order m 2 at least. Hence, neglecting all powers of m beyond the 
second, 

c 2 - 1 = - 2m 2 4 im a = -f m 2 , 

or c = l fm 3 

to the order m 2 . This value agrees with that found in Art. 67. 

The other two equations of condition then give, neglecting all powers 
of 7/4 but the lowest present, 

c_ 1 = J|fm ) c 1 =f|m 2 ; 

showing that the coefficient of cos (2 <j>) has been lowered one order by the 
integration a fact which might have been predicted, since the coefficient of 
t in 2 - (f> differs from n by a quantity of the first order. 

It is evident that c is a new arbitrary constant, for the term c cos $ 
might be considered to be included in the elliptic value of a/r. We shall 
not, however, neglect it here, but leave it arbitrary until the longitude 
has been found. 

140. To calculate the corresponding terms in longitude, the equation (5) 
gives to the order required, namely to m 2 e, 



______ _ - 



102 BE PONTECOULANT'S METHOD. [CHAP, vit 

Substituting the values of the various terms we have, since here A = wa, 
2 (c- 1) ne cos< -f $v=2nl+e cos <){|m 2 + m 2 cos 2 + ec, cos tf> 





+ Jme cos (2f? - <) -h f$w fl * cos 



j J " 

After expanding these expressions, we omit those periodic terms inde- 
pendent of e\ we also put l/a 3 = ^ 2 and c = l in those coefficients which are 
of the orders em or em 2 . We then have 

i^&;=|m 2 + ^ 2 + e(~2c + 2 + 

+ ^-me cos (2f - <^>) + ^m a e cos (2f + <^>), 
in which terms of a higher order than those required are neglected. 

The constant term is to be put zero. Hence 

g/ l=: -|7ia 2 m 2 . 

Since c is arbitrary we can determine it at will. Let it be such that the 
coefficient of sin </> in longitude is the same as in elliptic motion. This gives 
the definition of e in disturbed motion. 

We have therefore, 

!r / _2o + 2+2c + Jm + 2^)0 ......... . ........ (21), 

c \ no/ 1 

or, giving to c,.8A their values, 

C = - -&W 9 . 

Integrating the longitude equation, we finally obtain 

aSu = - &m*e cos </> + *me cos (2f -<j>) + $m*e cos (2f + ^), ) , 22 ^ 
Sv = ^me sin (2| - ^>) + ^-m^ sin (2? 4- ^), j " " V ;> 

in which the terms with argument 2f < are correct to the order w0 and 
those with arguments c/>, 2f -f ^ to the order 



141. We can, by paying attention to the orders of the terms, obtain c to the order m 3 
without much further labour. Its value was obtained by equating the coefficient of cos 
in the radius- vector equation to zero. The result (20) may be written 

Cc 2 -!) ........................... (23)- 



It was seen that c 2 - 1, c are of the order w 2 at least, so that c (c 2 - 1), w 2 (c 2 - 1) are 
of the order m 4 at least. It is not then necessary to further approximate to c . The new 
portions of the coefficient of cos in (1) can therefore only arise from the terms 



140-142] (iii) MEAN PERIOD INEQUALITIES. 103 

Now all terms in R are of the order m 2 at least, while only those terms in u, $v with 
argument 2-$ are of the order me. Hence a term of the order m%, with the argument 
<, can only arise in the parb 5P of (24) from the term in R of argument 2 combined 
with the term in 8it or dv of argument 2 <. The last term of the expression (24) 
furnishes only portions of the order mte, and it can therefore be neglected. As (Art. 121) 



we can, in this equation, put 



=^ cos (2-< 

C/b 



Whence 4&R= - ^ cos 2| cos (2 - $) - 4^ sin 2 sin (2 - <), 
of which the part depending on the argument < gives 

SP=4&ft= - if* cos 0. 
c& 

In the term -f d 2 (r<?$u)*/dt z we can, by similar reasoning, put 
r =a, a$^=m 2 cos 2f + ^ me cos (2 - </>) ; 
whence (r 2 Sw) 2 contributes Jg^wi 3 a 2 ecos<^, 

and - f ^ (r 2 S^) 2 contributes f *fma?ec 2 n 2 cos = f f ^ cos qf>, 

to the order required. 

Adding this to the value of 8.P, the new term of argument $ in the right-band member 
0f(l)is 

_ w 215 COB*. 

The equation of condition (23) therefore becomes, neglecting quantities of the order m\ 

c 2 -l=-fm 2 -^m 3 ; 
or, taking the square root, 

c = l-|m a -^m 8 ........................... (25). 

(iii) The terms whose coefficients depend on m and the first 
power of e f only. 

142. These are very easy to calculate and we shall only indicate the 
steps. There are no indeterminate coefficients to find as in cases (i), (ii). 
We put 



cos + -J cos (2f - - cos 

Cb 

T = C1D, V = n $ + 5=0. 

We use here the equation (3) to calculate P. The equation (1) becomes 



Tl O&C CK/ 

giving, as far as the order m?e', 

aSu = - fmV cos f + \m?e' cos (2 - <^.') - \m?d cos (2f + f ). ..(26). 



104 BE PONTECOUL ANT'S METHOD. [CHAP. VII 

Thence we obtain 



= nmV {- 3 cos f + ^ cos (2f - f ) - # cos (2 + <f>% 
and by integration 

to = - 3m</ sin f + ||mV sin (2 - f ) - iim 2 e' sin (2f + <') . . .(27). 

One term in Sv has been lowered to the order me' by integration. This 
will cause terms of the order wV to appear in the third approximation to 
I IT and therefore, we should suppose, terms of the order mV in the third 
approximation to v. But the only new terms in the radius-equation and in 
fdtS(dR/d^) which are of the order wV will be easily seen to be those with 
arguments 2 <' and no terms are lowered in order by the integration of 
the radius-equation ; hence, the only new terms which are of order mV in 
the longitude-equation are those with arguments 2 $'. Therefore, as the 
only terms in the longitude-equation which can have their orders lowered by 
integration are those of argument <' and as such terms are at least of the 
order m 4 e', the values (26), (27) for Su, Sv are correct to the order mV. 

The third approximation may be carried out as in cases (i), (ii). 

(iv) The terms whose coefficients depend on m and the first 
power of a/a' only. 

143. For these terms, we have 



andfrom(4), P = (p + 4)JB + 2n / * ..................... (28). 

In the second approximation to 1/r we shall not require the first two 
terms of R : p=*l for the third and fourth terms of J2. Since the coefficient 
of t in the argument of differs from n by a quantity of the order mn, 
the coefficient of cos f will be lowered one order by the integration of the 
radius equation. "We shall therefore, in preparation for the third approxima- 
tion, carry this term to the order m & a/af in the radiu equation so that, after 
the third approximation is complete, we may have it correct to the order 
m 2 a/o/. The calculations are very similar to those necessary for case (iii). 

We therefore have, to the order w 8 a/a' in the coefficient of cos | and to 
the order m z a/a' in that of cos 3, 

D K r> , 2n' m 3 a ^ K ^ , ^ 3 # t* 

P = 5J? + 7 f -7 cos = 5E 4 | - -- T cos f ; 
n n a * a 4 a a 



142-144] (iv) PARALLACTIC INEQUALITIES. 105 

and equation (1) becomes, 



The particular integral is 



3 ) COS 
in which the coefficient of cos is only correct to the order ma/ a'. 

The equation for the longitude becomes, after neglecting superfluous 
terms, 



a 

4 {- ( m + f I 2 ) cos + m " cos 



a cos + w? cos 



- (Jjf m + ^m 2 ) sin f + ifm 2 sin 3|, 



gvng v = - j m m sn 

where the coefficient of sin is only correct to the order ma/a'. 

144. Zfo jf/wrd: approximation. We are only going to find dt> dw correctly to the order 
m%/a'. In order to do so, we require the new terms of order m?a/a' in 1/r, that is, the 
new terms of order m*a/a! and of argument in equation (1). The only way in which such 
terms can be produced is from the term in du or in dv of argument fe order ma/ a', combined 
with terms of arguments 0, 2, order m 2 . 

We have from (28), since the second term, being multiplied by n' t may be omitted, 

aP=48/ or 55/2, 

and 8^=H 8v - 2 ^ 8% or If 8y "" 3 ^ dUi 

according as we take terms in R independent of a/a' or dependent on its first power. The 
only way in which we can get a term of argument fc order m%K from this expression is 
by putting 



Whence, as the terms considered in R are independent of a/a', 



106 DE PONT^COULANT'S METHOD. [CHAP, vn 

and therefore the terms of argument in 8P are given by 

w-W^jW 

For the first term on the right-hand side of (1) we take 






- -J-f m cos ; 



therefore the term of argument in (abuf is 






, cos J|-M 3 , cos = - 1 m 3 cos f ; 

so that - f ^ (r 2 Sw) 2 contributes - ^ ~, cos . 

Let the new part of 1/r be $ x w. We then have 



.. , nr ., 

-- ; =W -- ,cos A 
a a, * 1Q a a' * 

T , , . . T OK w 8 a '^ 2 cos _ ... m 2 ct , 

Integratmg 8 jW = W _ ? ( -^^= - \tf - -, cos 

Adding this to the value of 6& previously found we obtain, 

aSw - (}f m + fjm 9 ) -, cos + f f m 2 ~ cos 3f , correct to m 2 4 (29). 

tt 1 Ct (^ 

We shall now obtain the corresponding term in longitude. Since no coefficient has its 
order lowered by the integration, the new portion is simply that arising from the term djU 
just found and it is of the order w 2 ct/a'. Let it be denoted by d^. 

In the equation (5) the new portion Jc&9//9 is at least of the order m?a/a f and so 
contributes nothing. The same is true of (du)' 2 , dk x dw, 



The only new portion of the order m 2 a/a r is therefore given by 

$ d i vss "j~*i usss - W *^ ~' cos fc 

whence 8$ = - Iff- -2L- m 2 ~ sin ^ == - \^ m 2 -, sin f . 

Adding this to the value of &v previously found, we have 

8u = - (J^ m 4- ^p-m a ) -^ sin + J| m 2 4 sin 30, correct to m 2 ~ -(30). 



(v) The terms whose coefficients depend on m and the first 
power of 7. 

145. We have s = tang, of lat. = z V(l -f s z )/r } by Art. 12. 

Since the inclination of the plane of the Moon's orbit is always a small 
quantity oscillating about a mean value of 5, we can consider s and conse- 
quently z as small quantities of the first order and, in the first instance, 
neglect their squares and higher powers. Also, we have seen that the radius 



144-145] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 107 

vector and the longitude will contain only even powers of the small quantity 
7 (which is always a factor of s), and that the latitude will contain it in odd 
powers only. Hence in the equation for s or in the equation for z we are 
only neglecting quantities of the order s 3 or 3 . 

As stated in Art. 131, the equation (7) will be used to find z and thence 
to obtain the latitude. In order to express its right-hand member in terms 
of the coordinates, we use the development (3) of Chapter vi., neglecting 
the parallax of the Sun, that is, neglecting the terms beyond the first. We 
thus obtain, 



mrs mz , , , , j * 

-a -- _ -- -. to the order required*. 
r 8 r 8 

The equation of motion therefore takes the simple form, 



or, neglecting e' and dividing by n\ the form, 



an equation which is sufficient to determine all terms in latitude whose 

coefficients depend on m, e and the first power of 7 when those portions of r 

I which depend on m, e, are known. We shall, however, neglect e and therefore 

I give to a/r the part of its value dependent on m only that given by the 

first of equations (17) as far as the order m 4 . 

Wo are, in this method of finding the latitude, apparently departing from the principles 
i laid down in Chapter iv. concerning the method of solution by continued approximation. 

That is to say, instead of considering the first or elliptic approximation to s as known and 
then proceeding to find the new part due to the action of the Sun, we are including both in 
the equation of motion, so that we shall find a portion of the first and second approximations 
at one step. For the purposes here we do not need the value of s given in Art. 50. 

Put a/r = 1 4- o&fc We have 

a?lv* = 1 + SaSu + 3 (aSw) 3 + (a$u)\ 
The value (17) of Bu gives (afiw) 8 = to the order w 4 , and to the same order 

3 (afe) 2 = 3 (^-m 4 -i- Jm 4 cos 2| 4- m 4 cos 3 2g) 

\ = j|m 4 + m 4 cos 2 + f m 4 cos 4 

s The equation for z therefore becomes, 



I z + z {1 + f m 2 - ^m 4 + (3m 2 + tf m + ^mf) cos 2 + f-m 4 cos 4f } = 0. 
^fc 

* This result is nevertheless true when quantities of the orders 7 s , V- " taken ^ 
account. The only quantity actually neglected, when r^+j/o + s 3 , is a/a'. See Art. 150. 



108 DE PONT^COULANT'S METHOD. [CHAP, vn 

146. This equation is a case of the general form, 

g4.%{l + ^oos2^}=0, (/= 1,2,.. .00), 

which frequently occurs in physical problems. It is of principal importance in celestial 
mechanics for the determination of the mean motions of the perigee and the node. 

The solution is of the form 

=%&' cos (gnt+%jt +a), (/= - co ... + co ), 

the arbitrary constants being a and one of the q/. The chief part of the problem, which 
we cannot stop here to investigate generally, is the determination of g. Hill and Adams, 
as we shall see in Chapter XL, find g from the equation by means of an infinite determi- 
nant. For the purposes of Case (v) we shall assume the solution to be of the form given 
above and find the value of g to the required order by continued approximation. 

An investigation of the differential equation and of its solutions will be found, together 
with a large number of references to the literature on the subject, in Tisserand, M4c. CtfL 
Vol. in. Chaps, i. n. 

147. Assume as the solution, 

nt 

-^yteosin^+^sin^f--^)^ 

(JU 

where yg is considered to be one arbitrary constant. The symbol 77 stands 
for gnt + e 6 } where 9 is the other arbitrary constant : g is a constant as yet 
undetermined. 

If we put m = the motion is undisturbed and we shall have g = 1 ; also 

z = a<yg$ sin ^. 

In undisturbed motion, 7 = tan i. If therefore we put g Q = 1, we shall be able 
to define 7 in disturbed motion as being such that the coefficient of the 
principal term in z is the same as in undisturbed motion. It is evident that, 
in undisturbed motion, 0, e are the same as in Chapter in. 

We put then # = 1 &ftd substitute the assumed value of z. It will 
appear, when we write down the equations of condition, that since g differs 
from unity by a quantity of the order m at least, g^, g l} g^ 2) g% are of the 
orders m, m 2 , m 3 , m 4 respectively. Hence, omitting terms beyond the order 
m 4 , the equations of condition become, 



(31), 



Neglecting powers of m beyond the second, we obtain from the first of 
these, 

1-02 = -^ or = 1+^. 



{1 - (2 - 2m - gf] g^ + f m 2 ^ - (f m 2 + J^-m 3 + ^-m 4 ) = 
{1 - (4 - 4m - g}*} g^ - ff m 4 + (f m 3 + J^m) ^ = 

n.,7, =0 



146-148] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 109 

Therefore from the second and third, 

0_! (4m - m 2 ) = f m 2 + ^ w 3 to m\ 

g l (- 8 4- 12m - 7m 2 ) = - f m 2 - J^m 3 - ^fm 4 to m 4 ; 

or 0L. a = |m + f 4-m 2 to m 3 , 

^ = -^m 2 4-m 3 -ffffm 4 torn 4 , 
With these values we obtain from the fourth and fifth of the equations, 

g^ (- 8 + 24m) = f f m 4 - (f m? + J^m 3 ) (|m -f fjm 2 ) to m 4 , 

#, (-24) -fm 4 + tm a -&m a tow 4 : 

giving #- 2 = TfF m * + lHf 4 to m * 

#2 = iMr 4 torn 4 . 

Substituting the values of g^ ly g l in the first equation. we obtain for the 
further approximation to g, 



or ^l+f^-^ws-^w 4 torn 4 , 

or ^ =l+fm 2 -^m 3 -f||m 4 torn 4 ............ (32). 

With this value of g we can, from the second of equations (31), obtain a 
more approximate value of ff- l9 namely, 

g ^ (4 m - m - f| m 8 ) = f m 2 + ^m 3 4- \ 3 f m 4 w to m 4 , 

3 to m. 



or 

The value of z is therefore obtained to the order 7m 4 in all coefficients 
except in that of sin (2 - if) which is found to the order 7m 3 . As far as the 
| order 7m 2 , we have, 



- = 7 sin 77 + (|m + M^ 2 ) 7 sin ( 2 ^ "" 
a 

Also to this order, from the first of equations (17), 

s = 0[ r = z (1+ i^ 2 4- m 2 cos 2f)/a. 
Whence, to the order 7W 2 , 

si ^ 



(vi) The terms whose coefficients are dependent on m and the 
products and higher powers of e, e', 7, a/a. 

148. It is not intended to develope the algebraical expressions of the 
coordinates for terms other than those just given. To find the terms 
included in this class the developments become of great length, but there is 



110 BE PONT^COULANT'S METHOD. [CHAP, vn 

no change in the general method of finding them. When we wish to find 
any particular inequality defined either hy its argument or by the order of 
its coefficient relative to e, e' y 7, a/a', it is in general sufficient for the first 
approximation to the terms in l/r, to choose the corresponding terms present 
in R ; to find the terms in longitude and latitude, we must consider those 
of lower characteristics in l/r, s (these are supposed to have been previously 
found) and in 2? which, by their combinations, may produce terms of the 
required order. We shall then, as far as powers of m are concerned, have a 
first approximation to the term. It is generally necessary to proceed to 
further approximations. 

If, after the third approximation to the coordinates has been completed, the order of the 
new portion in the coefficient of any particular term is not higher than that of the portion 
with the same characteristic obtained in the second approximation, we must, in computing 
the second approximation, leave the corresponding coefficient indeterminate until the third 
approximation is reached : the coefficient may then be found. In this case the second 
approximation is not capable of giving the first term of the series for the coefficient in 
powers of m. See Arts. 118-120. 



Summary of the results. 

149. We shall now collect all the results of the first and second 
approximations and those portions of the third which have been determined 
in the, previous articles. The numerical coefficient of each term in the co- 
efficients is correct to the order given. The elliptic parts of the values of 
the coordinates will not be written down ; they are to be found in Art. 50. 

The various portions of afr are given by equations (17), (22), (26), (29). 
Whence, 



= Elliptic value -1- ^m 2 - |^|m 4 + (m 2 + ^m 8 + ^m 4 ) cos 2f + |m 4 cos 
- -farrtfe cos </> -f ^me cos (2 - <) + f |m 2 6 cos (2 + <f>) 
cos + my cos 2f - <') - 



- (tf m + f *m') J cos f + If m 2 J cos 8f . 

a LI 

The various portions of v given in equations (1*7), (22), (27), (30), furnish, 
-y = Elliptic value -f (Y^ 2 -f f| 3 + ^m 4 ) sin 2 + f^w 4 sin 4< 

+ 3me sin (2? - ) + ^-m*e sin (2f + <) 

- (3m + Om 2 ) e' sin $ 4- f^mV sin (2f - <') - ^-mV sin (2 f -f <f>') 



148-150] SUMMARY OF THE RESULTS. Ill 

The parts of these which were found by the third approximation are the 

terms of order m 4 and those of order m 2 ? in the coefficients of . : all the 

a! sin * ' 

other coefficients were found from the second approximation. 

The value of s, as given by (34), furnishes, 
5 = Elliptic value + %m*y sin 77 H- (m + f f m 2 ) 7 sin (2f T?) 

We have also from equations (25), (32), 



# = 1+ f w 2 ~ ^m 3 - f|f m 4 , 

the former being only obtained correctly to the order m 8 while the latter is 
found correctly to the order m 4 . 

Finally, n is the coefficient of t in the non-periodic part of ^ ; e, 7 are 
such that the coefficient of sin $ in longitude and that of sin 17 in z are the 
same as in undisturbed elliptic motion. 

150. For the cases (i) to (v) we have simply chosen out of the development of -ft, 
given in Art. 114, the terms required. It is easy and often useful to deduce each particular 
case directly from the disturbing function. 



txr v 
We have 



____.__, 

where $ is the cosine of the angle between the radii to the Sun and the Moon. 

Oases (i), (iv). Here, 0=0, e'=0, y=0; therefore r=a, r'=a', vnt+e, tf*=n f 
S sss cos (v - v 1 ) a* cos f . Whence 



(35). 

v 



Expanding as in Art. 107 and putting rc/ 2 a 2 = 



Case (v). Here r 2 =. < ' 2 4-y 2 4*^ 2 , $~xx' -^yy') and therefore 



If we neglect the ratio a : a f or r : r' this gives 



furnishing a2? inequalities in latitude independent of the solar parallax. 
Cases (ii), (iii). Here 7=0, a/a' = 0. "We obtain by expanding R, 



112 DE PONT^COUL ANT'S METHOD. [CHAP. VII 

For case (ii), we have /=&', v' = n't+c f and r, -y take their elliptic values. 
For case (iii), we have r=a, v=nt-\- e and /, v f take their elliptic values. 

By proceeding in this way we can without much trouble obtain any class of inequality 
denned by its characteristic. 

De Pontfaoulant's method as contained in his Systfane du Monde, Vol. iv. 

151. We have already mentioned in Art. 129 that the plan used here of dividing up 
the various terms into classes defined by the characteristic, is not essential for the 
development of the method nor is it used by de Pont^coulant. Further, if a complete 
development of the expressions were required, it would hardly be a saving of labour to 
proceed in this way. It will be readily seen that after R has been found in terms of the 
time to the required degree of accuracy by using the elliptic values of the radius vector, the 
longitude and the latitude, we should, in order to obtain the complete second approximation, 
use the complete elliptic value of R in the equation (1) which serves to find u. Certain 
coefficients would be indeterminate and they would be left so until, with this value of bu 9 
we had found the value of dv from equation (5), when they would be determined as in 
cases (i), (ii). Be Pont6coulant does not give full details of the method of procedure he 
adopted, but it is not difficult to see his general plan which is somewhat as follows. 

The first step is to neglect the latitude and with it all terms in the radius-vector 
equation, the longitude equation and the disturbing function, which depend on y or -s. In 
order to obtain a second approximation to 1/r, v of the terms independent of y such terms 
forming by far the greater part of R the parts on the right-hand side of (1) dependent on 
(Sw) 2 , (frw) 3 ... are neglected, the value of P* is calculated from the expression for R and the 
complete elliptic value of r is substituted on the left-hand side of the equation. In order 
to solve the resulting equation for du t we assume a value for this quantity which consists 
of a constant term and periodic terms of the same arguments as those occurring in the 
equation all the coefficients being indeterminate. When this value for du is substituted, 
by equating the coefficients of the various terms to zero we obtain definite values for all 
the indeterminate coefficients except for that of cos$ and for the constant term. The 
coefficients of cos <, however, give a first approximation to the value of c. Leaving these 
two coefficients indeterminate we proceed with the value of &u so obtained to find that of 
8v from (5). We then determine the new arbitraries after the manner explained in cases 
(i), (ii) and these, together with the coefficient of cos <, become definite. The constant 
part of I/r is found from an equation corresponding to that numbered (8) above. 

With these values of $u, fry we proceed to a third approximation by finding 8R 
according to the method of Art. 121 and we thence obtain the whole of the new terms on 
the right-hand side of (1). The resulting equations for the new parts of 1/r, v are solved 
as before* and a third approximation to these coordinates is deduced. Proceeding in this 
way by successive stages, all the coefficients are ultimately obtained accurately to quantities 
of the fifth order inclusive. In addition, certain coefficients which are either expressed by 
slowly converging series of powers of m, e, etc. or which, owing to the presence of small 
divisors, have their orders raised, are calculated to higher orders by choosing out the 
particular combinations required to obtain them. 

The latitude equation is then treated. Neglecting powers of y (and therefore of z and s) 
higher than the first and using the values of 1/r, v already found, we can obtain z by means 

* De Ponte"coulant uses the letter P to denote the terms on the right-hand side of (1) which 
depend on (5tt) 2 , (Suf.... 



150-154] THE STSTiSME DU MONDE. 113 

of equation (7) ; in doing so, since all terms in z have the factor y, it will only be necessary, 
except for certain coefficients, to use the previously found values of l/r, v as far as the 
fourth order, if we merely require z accurately to quantities of the fifth order. When z 
has been obtained the value of s is easily deduced. The terms of the order y 2 in 1/r, v can 
then be calculated by methods quite similar to those used before to approximate to these 
coordinates. Returning to the latitude equation we find the portions of z and thence those 
of s which are of the order y 3 ; and from the new parts of s so found we can obtain the parts 
in 1/r, v which are of the order y 4 . De Pont^coulant, neglecting quantities above the fifth 
order, stops at this point ; the terms of order y 6 in * are simply those given by the elliptic 
formulae. In the course of the approximations, the value of g is found and the meaning of 
y defined in disturbed motion according to the principles explained in (v). When all these 
operations have been performed we shall have found complete expressions for the coordi- 
nates, accurately to the fifth order at least, as far as the action of the Sun is concerned. 

152. Two important differences between the expressions obtained above and those 
found by de Ponte'coulant, must be noted : they refer to the meanings of e, y in disturbed 
motion. We have defined them to be such that the coefficient of sin < in longitude and 
that of sin rj in z shall be the same as in undisturbed motion. De Ponte'coulant, having 
before him the earlier results obtained by those who followed Laplace's method and 
desiring to compare his expressions with theirs, so determined these constants that, in the 
final expressions of the coordinates in terms of the time, the results, if correctly worked 
out, should agree. This point will be further explained in Art. 159 below. 

153. The successive approximations are not given in detail by de Pont^coulant. He 
merely states that the labour of performing them was very great, and then proceeds to 
write out the complete value of 11 obtained by substituting the results of the various 
approximations in the expression given in Art. 121 above ; he gives also the value of du 
furnished by the previous approximations. With these expressions for jR, Sw, he goes on 
to find the complete value of l/r and thence that of #, finally obtaining the value of s. The 
labour of performing this last approximation is divided into several portions : first, we are 
given those portions independent of y, whose coefficients include only the powers of e, e' t 
a/a' contained in their characteristics; secondly, those terms in the latitude whose coeffi- 
cients depend only on the first power of y and on those powers of e, e', a/a' present in 
their characteristics; thirdly, the omitted portions of the coefficients in all the three 
coordinates are found. 

He then proceeds to find the coefficients of certain long-period inequalities more 
accurately; and also to obtain the inequalities due to the actions of the planets, the 
non-spherical shape of the earth, etc. The results are worked out algebraically all through 
and the final expressions for the parallax, the longitude and the latitude are given in 
Chapter vn. of the volume. Numerical results are then obtained by substituting the 
numerical values of m, e, <?', y, a/a', I/a, in the coefficients, for the particular case of the 
Earth's Moon. These are followed by tables comparing his results with those obtained 
by Plana and by Danioiseau and with those obtained directly from observation. 

154. A brief examination of the literal expressions for the coordinates given by 
de PonttScoulant in Chap. vn. of his volume will show that the series for certain 
coefficients, when arranged according to powers of m, appear to converge very slowly owing 
to the large numerical multipliers; it is doubtful whether all these series are really 
convergent even for the small value which m has in the case of the Moon. If we arrange 
them according to powers of e 2 , e'\ y 2 , (a/a') 2 , such slow convergence does not seem to take 
place on account of the very small values of these four constants. It becomes important 

. L. T, 8 



114 DE PONT^COULANT'S METHOD. [CHAP, vii 

then to consider whether we can improve the slow convergence by substituting another 
parameter instead of m and also how we may estimate the remainders of such series on 
the assumption that they do really converge (see Art. 69). 

One of the series which converges very slowly is that which represents the coefficient 
of the term with argument 2 $ 4- <' and characteristic ee f . The portion of this coefficient 
in longitude, of the form ee'f(m), as given by de Ponte"coulant, is* 

^(^ m ^J^ m 3 + J^ 

The last term calculated has therefore a numerical multiplier of nearly 90,000 and its 
ratio to the first term is about 24,000 m 6 = 1/1 8 approximately. The numerical value of 
the terms given is about 30", so that six terms of the series are not sufficient to give the 
number of seconds accurately, although the complete coefficient is small It is such series 
as these which make the literal development tedious and difficult. 

It has been shewn by Hillf that if we expand in powers of m=#i/(l m) instead of in 
powers of m, most of the series will be rendered more convergent. Further, a careful 
inspection will often enable us to estimate the remainders with some exactness, owing to a 
certain regularity which these series appear to display $. 

* Sys. du Monde, Vol. iv. p, 577. It must be remembered that the definition of e is that 
used by de Ponte*coulant ; this coefficient will therefore not be the same as that obtained by 
Delaunay. See Art. 159 below, 

t "^Researches in the Lunar Theory,'* Amer. Journ. Math. Vol. i. p. 141. 

$ See two notes by the author in the Monthly Not. R. A. S. Vol. LII. pp. 71 80 ; uv. pp. 3 6. 
In the latter, the complete numerical value of the above term is given. 



CHAPTER VIII. 

THE CONSTANTS AND THEIR INTERPRETATION. 

155. ONE of the most important departments of the lunar theory is the 
interpretation of the arbitrary constants which arise when the equations of 
motion are integrated. It has already been mentioned that in obtaining 
expressions to represent the coordinates at any time, we must keep in view 
the necessity of putting the expressions into forms which enable us to readily 
give a physical interpretation of the results. In one direction the elimina- 
tion of all terms which increase in proportion with the time this object has 
been achieved: the difficulty was only of an analytical nature. Another 
question the convergency of the series obtained we have been obliged to 
leave aside owing to the lack of any certain knowledge on the subject. Further, 
assuming the convergency of the series which are obtained when the problem 
has been formally solved and when the coordinates have been expressed in 
terms of the time and of certain constants, it is necessary to so determine these 
constants that the initial conditions of the problem may be satisfied. In many 
physical problems it is sufficient to know the initial, coordinates and velocities 
in order to determine the constants easily. But the peculiar nature of the 
problems of celestial mechanics makes it impossible to find them with any 
approach to accuracy in this way ; this is owing partly to the difficulty of 
measuring distances, partly to the inaccuracy of a single observation and 
partly to the complicated nature of the equations which it would be necessary 
to solve. Consequently, indirect methods must be resorted to. 

There are three questions to be considered in the interpretation of the 
results. In the first place, we must give .definite physical meanings to all the 
constants involved in order that we may be able to apply the results to the 
case of any satellite which moves under the general conditions initially 
assumed. Secondly, we must be able to determine the numerical values 
of the constants from observation, as accurately as the observations permit, 
in the case of any such satellite and more particularly of the Moon so 

.82 



116 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

that tables may be formed which will give the place of the satellite at any 
time, with an error not exceeding that of a single observation. The discovery 
of new causes of disturbance, and of their magnitude, is rendered far easier 
when tables, calculated from theory alone and including the effects of all known 
causes of disturbance, have been formed. It is the small differences between 
the tables and the observations which will be most likely to lead to an 
advance in our knowledge of the peculiarities in the motions of the bodies 
forming the solar system provided these differences cannot be wholly 
accounted for either by errors of observation or by inaccurate values of the 
constants used in forming the tables. Thirdly, when the constants have been 
determined, the magnitude of the effects produced by the various terms 
present in the expressions for the coordinates, will be inquired into. 

The Signification of the Constants. 

156. We have, in the previous Chapter, formed certain developments for 
the coordinates and, in so doing, we have introduced new arbitrary constants 
or defined those introduced into the first approximation in a new way. It is 
immediately obvious that as soon as we begin to use the results of the second 
approximation, the constants can no longer have their former significations. 
They wer6 specially defined for the case of elliptic motion, that is, motion 
in a curve of known properties. When this orbit was modified by the 
introduction of c and g, it was still possible to interpret the results 
geometrically, namely, by the use of an elliptic orbit of fixed size and shape, 
"with its apse and its node moving with uniform velocities in given directions 
(Art. 68). When, however, we go further and approximate to the path of the 
Moon by the methods of Chap. TIL, no such easy interpretation of the 
results is possible : the curve described is not one with whose properties we 
are familiar. In order to use the results, it will be necessary to consider 
the constants separately; we must also give them such meanings that the 
determination of them by observation shall be as easy as possible and that 
the results of any other method, in which the arbitrary constants of the 
solution may be introduced in a different manner, can be compared with 
those just obtained. 

Besides the arbitrary constants of solution there are present certain 
constants which have been supposed known, namely, those referring to the 
elliptic solar orbit. These are m', n\ of, e', e', r', of which the first three are 
connected by the relation m' = n /2 a' 3 . When the orbit of the Sun is supposed 
not to be elliptic and not to be in one plane, two new constants depending on 
the position of the plane of the orbit will be introduced, and all the 
constants must be defined again. The problem of the determination of the 
solar constants, although it does not differ much from that of the lunar 
constants, belongs mainly to the planetary theory and we shall 



155-158] THE NEW ARBITRARIES INTRODUCED. 

leave it aside. In the following we retain the former suppositions that the 
orbit of the Sun is elliptic and that it lies in the fixed plane of reference. 

In attaching meanings to the arbitrary constants used in disturbed 
motion, the principal object which will be kept in view is the consideration of 
definitions which depend in no way on the method of integration adopted. 
We shall thus be able to compare the results obtained by any of the methods 
used for treating the lunar theory. 

157. In the first approximation we had seven arbitrary constants, 
a, n } e, e, <cr, 6, i (or y) and one constant JUL present in the differential equations : 
p was eliminated from the results by means of the necessary relation 
^_ n 2 a 3 rpi^g re i a tion is not used to eliminate a or n because ^ is a 
much more difficult constant to determine observationally than either a or n : 

"< /JL is, in fact, considered as an unknown, although definite, constant. When 

: we proceed to the second approximation which, in de Pont&oulant's method, 

is really the discovery of particular integrals of the equations, it is theo- 
retically unnecessary to introduce new arbitrary constants, the requisite 
number being already present in the complementary function. The relation 
p = n 2 a 8 would, however, have been replaced by one much more complicated 
and certain important terms in the expressions for the coordinates would 
have been much less simple. It is possible to make the required changes in 
the arbitrary constants, after the expressions have been obtained, but we gain 
much by defining them as we proceed in the way finally necessary. To do 
so, the new constants must be retained and suitable meanings must be given 
to them. 

In order to simplify the explanations as much as possible, we shall first 

neglect y (or i) and therefore 6 ; we then merely treat the constants present in 

cases (i) (iv) of Chap. vil. In cases (iii), (iv) no new arbitraries were 

introduced : we were only finding particular integrals. It is therefore only 

, necessary to pay attention to those present in (i), (ii). As already stated in 

Art. 151, we might have taken the four cases at one step : the parts of 1/r, v, 

due to the second approximation and collected in Art. 149, would then have 

j been found together. Had we done so, the terms considered in (iii), (iv) 

* would have appeared as parts of the solution which it is not necessary to 

[ consider here. We therefore treat cases (i), (ii) only and suppose that the 

|ji second approximations, there separated, have been made together. 

"i 

158. In these two cases there were five constants a, 6 , Sh, B, c which 
" were either new arbitraries introduced or parts of the assumed solutions 
I which were not directly determinate. Between a, 60 there was one relation, 
V namely, equation (10) of Art. 134. But since p was not present in the 

solution, another relation involving p* existed between a, & , Sh. Hence only 



118 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

three new independent arbitraries, which may be taken to be Sh, JJ, c , were 
introduced. It is required to find what connection there is between these 
and the old arbitraries a, n, e, e, -cr. 

The constant B arose exactly in the same way that e did, namely, as the 
additive constant in the integration of the longitude equation. The constant 
additive to the longitude therefore becomes J9+ e instead of e. As both 5, e 
are arbitrary, the introduction of B is unnecessary and we can put it zero. 
The longitude is then expressible in the form 

t x const, -f e -f periodic terms. 

Hence, in disturbed (or undisturbed) motion, e is the value of the mean* 
longitude at time tf=0. 

The constant Sh is determined in a similar way so that the meaning 
of n may be fixed, but a complication arises owing to the way in 
which h occurs. The expression for dv/dt consists of a constant term and 
periodic terms. When the motion is undisturbed there is only one constant 
term and it was denoted by n. (The arbitrary actually introduced was h ; n 
is the constant term in the development of hjr\) When the motion is 
disturbed, definite constant terms are present as well as the arbitrary constant 
&h. The expression for the mean longitude appears in the form 

nt (1 + powers of m* etc.) + e. 

The presence of Sh enables us to get rid of all these other terms, so that the 
longitude is expressible in the form 

nt 4- e -f periodic terms. 

Hence n is the mean angular velocity of the Moon or, as it is generally 
called, the mean motion> whether the orbit be disturbed or undisturbed. 
Since the new definite terms which appear in the expression of dv/dt can 
always be eliminated in any stage of the approximations, n will be the mean 
motion at any stage and therefore in the final results. 

It will be noticed tlxat in case (i), bh was simply used to get rid of these new terms. In 
case (ii), the result obtained by equating to zero the new terms multiplied by t, was required 
in order to determine the new value of e. The introduction of th in cases (iii), (iv) is 
unnecessary, for if we stop at the first powers of e", a/a', no definite constant terms arise in 
the new part of dv/dt ; the value of d/i would therefore have been found to be zero and it 
was neglected *at the outset. 

Next, to find 6 , we substituted in equation (8), Art. 131, assuming that 
the relation /i-nW still held. Had we simply used equation (1) without 
introducing a, this would have been unnecessary for the purpose of finding 

* The mean value of any quantity expressed in this form is defined, in celestial mechanics, as 
its value when all- the periodic terms are neglected. 



158-159] THE CONSTANT OF ECCENTRICITY. HD 

&o which would have been determined by equation (10), Art. 134. But then 
the new relation between //,, a, n and the other constants would have to be 
determined from some equation of motion similar to (8) involving /u, ; a simple 
definition for a in disturbed motion is also required. It is found best to 
define a by the equation ^ = w 2 a 3 , where n has the meaning just defined. This 
we have done in case (i). The relation /JL = % 2 a 3 having been assumed to hold 
and & having been found from equation (8), the introduction of a is un- 
necessary as far as the discovery of the solution is concerned, but its presence 
is necessary in order to make it evident that the equation for &u, p. 96, is 
satisfied. 

159. The constant ec Q in case (ii), was the coefficient of cos $ in the value 
assumed for aSw. Now the first approximation gave the value of ajr to be 

1 + e cos <, 

neglecting powers of e higher than, the first. Therefore, when we include in 
the value of ajr the results of the second approximation, the coefficient of 
cos 6 becomes 



But e was an arbitrary of the first approximation to 1/r and ec is an 
arbitrary introduced exactly in the same way in the second approximation to 
1/r. We can therefore determine c at will. Instead of putting c zero, it is 
found best to determine it so that the coefficient of sin c/> in longitude is the 
same whether the motion be disturbed or undisturbed. Certain definite terms 
will also occur with C in this coefficient (p. 102) ; these can always be 
eliminated, by the proper use of c , at any stage of the approximations, in the 
same way that Sh was used to cancel those occurring in the coefficient of t in 
the expression for the longitude, 

Other methods for fixing the meaning of e have been used. The older lunar theorists, 
taking the true longitude as independent variable and expressing the time or the mean 
longitude and the other coordinates in terms of it, fixed the meaning of e so that the 
coefficient of the principal elliptic term, in the expression of the parallax in terms of the true 
longitude, was the same in disturbed and undisturbed motion. After the true longitude 
has been expressed in terms of the mean longitude (or of the time), the principal elliptic 
term in longitude contains powers of m, e' 2 ,. .. in its coefficient. The characteristic is, in any 
case, e. De Pont<$coulant, when working out his theory, wished to compare his results with 
those of the earlier investigators. He therefore determined c and e so that the coefficient 
of sin< in longitude was the same as with them. We have not followed him in this 
detail because the more complete theory of Delaunay has the definition of e used here and 
because e so defined is obtained observationally with much greater ease. 
. In working out with rectangular coordinates the inequalities considered in case (ii)* I 
have defined the constant of eccentricity by the coefficient of sin <f> in the expression of the 
coordinate Y (Section iii, Chap. n.). This appeared to be the simplest plan, in view of 
the later approximations necessary -to form a complete development by this method, bee 
Section (ii), Chap. XL 



120 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

160. It is not necessary to introduce a new constant for the determination 
of tzr in disturbed motion. The reason for this will be seen more clearly when 
we come to the method of the Variation of Arbitrary Constants as exemplified 
by Delaunay's theory. It will then be seen that the variable longitude of 
perigee, in disturbed motion, is expressed in the form 

(1 c) nt + CT -f .periodic terms. 

The constant or is therefore the value of the mean longitude of perigee at 
time t = and it keeps this name in any theory ; also (1 c) n is the mean 
motion of the perigee, As the longitude of perigee only occurs in the 
elliptic expressions for the coordinates under the signs, sine and cosine, and 
as the periodic terms which occur in the above expression for the longitude of 
perigee have coefficients of the first order at least, we can, when substituting 
in the elliptic expressions for the coordinates, expand the sines or cosines so 
that no periodic terms appear in the arguments : the coordinates are therefore 
ultimately expressed as in Chap. vii. Since -er only occurs in the principal 
elliptic term of de Pont^coulant's method in the form cut + e w, and since e 
may be supposed to be known, <& may be defined by means of the value of 
this argument at time t = 0. 

It is usual to retain the term ' eccentricity ' for e in disturbed motion 
whatever be the plan used to fix its meaning. The reason for giving to 
(1 c) nt + tar the term ' mean longitude of perigee ' will be evident from the 
remarks just made. The constants e, -or are called the ' epoch of the mean 
longitude ' and the ' epoch of the mean longitude of perigee/ respectively. 
The constant a, defined by the equation p = ?i 2 a 3 , is often called the c mean 
distance ' or the ' semi-major axis of the orbit.' 

The term rnean distance' thus applied to a is inexact according to the usual definition 
of the word 'mean 7 (see footnote, p. 118), With the above definition of a, we have 
determined r in the form 

a/r = 1 4- ft 4- periodic terms, 

which would give r/a=l+0' + periodic terms (); 

3, ff being constants depending on m, e' 2 .... According to the definition of the word 
* mean/ the mean distance should be a (I +/3'), while a/(l +0) is the distance corresponding 
to the mean "value of the sine of the parallax. The latter is the quantity determined 
observationally and therefore of most importance in this connection. The terms in ft are 
small and are easily found when the values of the other constants have been determined by 
observation. 

161. Eemarks quite similar to those made concerning e, r, apply to 7, 6. 
We have determined 7 in Art. 147 so that the coefficient of the principal 
term in z is the same as in undisturbed motion. It is better to define it so 
that the coefficient of the corresponding term in the latitude u, is the same 
as in undisturbed motion. The transformation from the old constant to the? 



160-162} DETERMINATION BY OBSERVATION. 121 

new one is easily made when u has been found. The coefficient of sin T? in 
V will be Y(! + 0"), where /3" depends on m, e 2 , e' 2 ,.... We replace 7 by 
7/(l 4- /3") wherever the former constant occurs ; 7 is called ' the tangent 
of the mean inclination.' The longitude of the node, when found as in 
Delaunay's theory, will be expressed in the form 

(1 g) nt 4- 4- periodic terms, 

Hence (l g)n is the mean motion of the Node, is the epoch of the mean 
longitude of the Node; the latter is determined in de Pont6coulant's method 
by finding the value of the angle gnt + e at time t = 0. 

Determination of the Constants by Observation. 

162. The three coordinates of the Moon which are observed directly, 
are the longitude, the latitude and the parallax. Of these, an expression for 
the longitude has already been obtained; the expressions for the parallax 
and the latitude are deducible immediately. 

Take the Earth's equatoreal radius as the unit of distance. Then 1/r 
will be the ratio of the Earth's equatoreal radius to the distance of the Moon, 
that is, the sine of the equatoreal horizontal parallax of the Moon. Let II 
denote this parallax. We have approximately 



The average value of sin II is about ^. The error caused by neglecting the 
term | sin 8 II is about ,^00 of the whole, corresponding to an error of 0"'2 : 
this is within the limits of error of a single observation. To this degree of 
accuracy we can therefore put II = a]r, 

To find the latitude we have, since tan Z7 = s, 



As a is a small quantity of the order 7, we can quickly find u when s is 
known. 

These three angular coordinates are therefore expressible in the form 

(1). 



sin 



When Q denotes the longitude, the periodic terms are sines; when Q denotes 
the latitude, they are also sines and A, B are both zero ; when Q denotes the 
parallax, we have B = and the periodic terms are cosines. 

In all cases Q, and therefore A, G, are the circular measures of angles. To 
express them in degrees we multiply by 180/ir, or in seconds of arc by 
180 x 60 x 60 -r 77 = 206,264-8. 



122 THE CONSTANTS AND THEIR ItfTEKPKETATIGN. [CHAP. VIII 

The number of seconds of arc in any coefficient is therefore obtained by 
inserting the numerical values of the constants and multiplying the result by 
206,265. 

163. Suppose that in the expressions for the coordinates, represented by 
the general form (1), we stop at a given order ; they will then be reduced to 
a finite number of terms. If a number of values of Q, equal to the number 
of constants A, B, C, /3, /3' present, be given, each of tfrese constants could be 
determined independently. But our expressions have shown that only six or, 
if we suppose /JL unknown, only seven of these constants are independent. 
(We consider the solar constants known.) Hence, if the observations and the 
theory were both correct, exact relations ought to exist between the various 
constants thus found when the number of observed values of Q is greater 
than seven. But these conditions are not quite fulfilled. In the first place, 
each observation is only approximate and must be regarded as subject to 
error. In the second place, the coefficients of the periodic series, being 
each of them formally represented by an infinite series about the convergency 
of which we have no information, can only be considered at the best as 
approximate, apart from the question as to whether the infinite series is a 
correct representation of the coefficient. Assuming that the infinite series 
are possible and convergent, in order to determine the numerical values 
of the seven arbitraries, it is still necessary to choose the particular terms 
which are best adapted to our needs. 

Now the methods used to find the constants present in an equation of the 
form (1) enable us, in general, to obtain with a high degree of accuracy the 
coefficient, period and argument of any term when the number of observed 
values of Q is very great. 

Suppose that it be required to determine the constant e. We naturally 
choose out of one of the coordinates the term or terms in which a given 
alteration to e will produce the greatest effect on the value of that coordinate. 
This term is immediately recognised as being that with argument <f> in the 
longitude. All the other terms in longitude containing e have either powers 
of e higher than the first in their characteristics, or e is multiplied by some 
small quantity such as m, y 2 ; in parallax, all terms are multiplied by the 
small quantity I/a. Again, the number of available trustworthy observations 
of the longitude is far greater than those of the other coordinates. Finally, 
since we" have chosen that the coefficient of sin< in longitude shall be the 
same as in elliptic motion, this coefficient can be obtained theoretically to 
any degree of accuracy we desire. For all these reasons the determination 
of e by observation from the term with argument <f> in the longitude is the 
most suitable. The advantages of the definition of this constant, adopted in 
Art. 159, now become very evident. 



162-164] NUMERICAL VALUES. 123 

It will easily be seen how the numerical values of all the seven arbi- 
trary constants are determined. The values of n, e are obtained from 
the non-periodic terms nt, e in v. The principal elliptic term then furnishes 
0, txr and also en if we wish to find the period by observation. The principal 
term in latitude that with argument 77 (= gnt + e 0) gives 7, 6 and also gn. 
Finally, the constant part of I/r furnishes the value of a ; this constant part 
contains also a few terms depending on w, e /a , etc. which are known, since 
their numerical values were previously found. 

164. At the present day the numerical values of most of the constants are known with 
a very high degree of accuracy. Tables have thus been formed of the motion of the Moon 
from theory alone. Notwithstanding the great care bestowed by various investigators in 
including in, them the results of all known causes, small differences between the tables and 
the observations are continually to be found. Some of these can be put down to errors of 
observation but many of them, especially when they exceed a certain limit and appear 
to be either periodic or secular, are due to imperfections either in the theory or in the 
numerical values of the constants used in the tables. Even when all the corrections due 
to known causes have been made, certain empirical corrections, not indicated by theory, 
have to be applied to these tables in order that they may agree with the observations. 
The tables published in 1857 by Hansen, together with certain corrections investigated by 
Newcomb (see Art. 173), are still used to obtain the places of the Moon given in the 
Nautical Almanac on pages iv to xii of each month. 

A comparison of the values of en, gn, as determined from theory and directly from 
observation, furnishes a valuable test of the sufficiency of the known causes to completely 
account for the motion of the Moon. The recorded observations of eclipses have enabled 
astronomers to obtain ?i, (l-c)n, (lg)n with a high degree of accuracy, and the values 
of c, g deduced therefrom agree very closely with those calculated by theory. Nevertheless, 
the small differences between theory and observation still leave something to be desired. 
As far as may be judged, the results deduced from observation appear to be rather more 
trustworthy than those deduced from theory. 

In order to reduce the results of the preceding Chapter to numbers it is 
necessary to know beforehand the numerical values of certain of the constants. 
We take the units of time and length to be the sidereal day and the Earth's 
equatoreal radius, respectively: the numerical values of the constants used 
below are those by which Delaunay* reduced his theory to numbers (see 
Art. 173). He takes 

2ir/n'= 365-25637 days, I/a' = 8"'T5, d = <M)16 77106 (2). 

The object of the following articles being merely to find the extent by 
which the principal inequalities affect the place of the Moon, we shall not 
here require to know the values of e', -cr' or of e, tsr, 0. The parts of chief 
importance are the coefficients and periods. In other words, we consider 
mainly the amplitudes and periods of the periodic oscillations and not their 
phases. 

* Mem. de VAcad. des Sc. Vol. xxix. Chap. xi. 



124 THE CONSTANTS AND THEIB, INTERPRETATION. [CHAP. VIII 

The Mean Period and the Mean Distance. 

165. The longitude is expressed in the form 

v = nt 4. 4. periodic terms. 

The mean longitude is therefore nt 4- e and the Mean Period I 7 is the 
time in which the mean longitude increases its value by ZTT. Hence T = 2?r/n. 
We find T directly frorn observation to be about 2T-J days or more exactly 

277/71 = T =27*321661 ........................... (3). 

With the Yalue of n r given in equations (2) of the previous Article, we 
deduce 

w = ny = -07480133 = 1/13J, approximately ............ (4). 

The parallax is given (Art, 138) by 

- =-(1 -f ?n 8 -||w/ 4 ) -4- periodic terms, 

The mean value of the sine of the equatoreal parallax is found directly 
from observation to be 3422"* t / r . To obtain I/a we have therefore 



Cv 

which, with the value of w just found, gives 



This value is very little altered by including the terms which have not been 
calculated here for the constant part of 1/r. 

The distance of the Moon corresponding- to this value is 206,265/341 9"6 
equatoreal radii of the Earth or about 239,950 miles, taking the Earth's 
equatoreal radius as 3,978 miles. The real mean distance, calculated by the 
formula (a) of Art. 160, is 238,840 miles. 

From the value (2) of of and (5) of a, we deduce 

~> = '002559 = -^ T approximately .................... (6). 



The VwrioMan* 

166. The term with argument 2f in longitude or parallax is known 
by the name of the Variation. Let T be the mean periodic time, The 
Variation runs through all its values in time 

27T/2 - nf) = T/2 (1 - m) = H| days, by (8), (4), 
or iu half the mean synodic period of revolution. 



THE VARIATIONAL CURVE. 

The coefficient of this term in longitude was (Art. 138) seen to be 



which, when the value of m has been substituted and the result multiplied by 
2Q6\265, gives 34' 51". When the portions in the coefficient depending on 
higher powers of m and on e\ if* etc. are taken into consideration, the value 
of the coefficient is found by Delaunay to be 39' 30". 

The investigations of case (i) have shown that if the approximations 
to the coefficients be continued, the result for the terms dependent on m 
only, will be 



a 
r 



cos 2jfc v - n't - e' = f + 26' 2g sin 2yf , (q = 0, 1, 2, . . .), 



where 6 ag , l' m depend only on m and are each of the order m?*. The terms 
considered in case (i) therefore constitute a curve, periodic with reference to 
axes moving in their own plane with uniform angular velocity ri. The curve, 
relatively to these axes, will be closed and symmetrical ; the time of revo- 
lution round it will be 2?r/(n - n'), or a mean synodic month. Since, in the 
case of our Moon, all the coefficients b^ are positive, the maxittmm and 
minimum values of afr are given by f = and f = 7r/2 respectively. Hence 
the shortest diameter of the oval is directed towards or away from the mean 
place of the Sun. We shall call this line the X-axis of the oval. 

All the inequalities with arguments 2# may be called 'Variational 
Inequalities 1 and the curve just defined the 'Variational Curve. 1 This 
curve has been calculated and drawn by Dr Hill for satellites of periods 
differing from that of the Moon. 

| He has shown* that as the value of m is increased, the ratio between the 

< f lengths of the two axes increases while the velocities at the ends of the 

i longer axes, that is, in quadratures, diminish. For the value m = l/2 78 the 

' velocities in quadratures vanish and the curve has cusps at those two points. 

M. Poincar^t has shown that if the value of m be still further increased the 

cusps are replaced by loops, so that a satellite whose mean period relative to 

, that of the Sun is greater than 1/2*78, would appear in quadrature six times 

I during one revolution. 

The Parallactic Inequality. 



167. The terms considered in case (iv) of the previous Chapter are 
closely related to the Variational inequalities by their arguments, the latter 
being simply odd instead of even multiples of f. The principal term 
that of argument % is called the Parallactic Inequality. 

* Amer. Jour. Hath. Vol. i. pp. 259, 260, 
t Mc, Ga. Vol. i. p. 10$, 



126 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

The coefficient of this term in longitude was found in Art. 144 to be 
_ (j^ m + s$ m *) ~ = - r 48", 

by the values of m, a/a' given in (4) and (6) respectively. Delaunay's value 
of the whole coefficient is 2' 7". According to the results of Chap. I. these 
numbers are to be multiplied by (E - M )I(E -I- M ) = 39/40 approximately 
(Art. 168), 

The period of the term is 2ir/(n - n') t that is, one mean synodic month. 

Suppose that these inequalities be included with the Variational Inequali- 
ties in the expressions for a/r and v. We shall have 

a/r = %b q cos g, v - n't - e' = f 4- %b' q sin q% ; 

where the terms for which q is even are functions of m only and those for 
which q is odd, besides being functions of w, have the factor 



When we put for , r is unaltered and v n't e' changes sign. The 
curve, referred to the same moving axes as before, is therefore symmetrical 
about the line directed to the mean place of the Sun. In this case, however, 
= and = TT do not give the same values for r. Let r c , r v denote the values 
of r when = D, TT respectively. We then have 



'0 'IT 

== a negative quantity 

in the case of our Moon, for 6 X is then greater than the sxim of the quantities 
6 3 , & 5 ... and it is negative (see equation (29), Art. 144). Hence 



The longer JT-axis is therefore directed towards the Sun and the shorter 




. 7. 



PARALIACTIC TERMS. 127 

X-axis ' away from it. The effect of the inequalities depending on a/a' is 
to slightly distort the Variational curve in the direction of the Sun. The 
general character of the remarks just made, is not altered by the introduction 
of the. squares and higher powers of a/ a!. 

All the inequalities which have arguments of the form (2# -f 1) f may be 
called Paralktctic Inequalities. The principal Parallactic Inequality has 
been used to determine the parallax of the Sun, that of the Moon being 
supposed known. It will be immediately seen that if we find the coefficient 
by observation and also by theory, we can, knowing m with great accuracy, 
deduce the value of I/a'. This method is, however, defective since it involves 
the accurate knowledge of the ratio (E - M)/(E + M). 

The variation as well as the other principal inequalities in the Moon's motion and 
the motions of the Apse and the Node, were first explained by Newton on the theory 
of gravity only. The values of their coefficients were obtained by him more or less approxi- 
mately. The oval curve, called above the Variational Curve, was also recognised by him 
and the ratio of its two diameters was shown to be approximately as 69 : 70, corresponding 
to a coefficient S5' 10" of sin 2 in longitude. The results of Newton's investigations 
in the lunar theory are contained chiefly in Props, xxn. xxv. xxxv. Book in. of the 



168. The determination of the ratio MjE is a matter of some difficulty. There is no 
direct way of obtaining it. Probably the best method consists in finding the inequality in 
the motion of the Earth due to the action of the Moon. It will be readily seen from 
Chap. i. that E will describe a circle of radius Ma/(JS+M) about G, if we suppose that S 
deaeribes a circle about G and that M describes a circle of radius a about E. As the 
Moon performs a synodic revolution, the apparent place of the Sun as seen from the Earth 
will therefore oscillate to and fro about its mean position. By observing the extent of this 
oscillation we can, knowing the other constants with considerable accuracy, deduce the value 



The constant may be determined by comparing the tides produced by the Moon 
with those produced by the Sun and also by comparing the observed nutation of the 
Earth's axis with the value deduced from theory. See de Ponte'coulant, Sytttom du Monde, 
Vol. iv., p. 651. The differences between the values so obtained, indicate that the 
numerical value of the ratio MjE is not certainly known within five per cent, of its true 
value. 

The Principal Elliptic term, the Evection and the 
Mean Motion of the Perigee. 

169. The Principal Elliptic term in longitude, having the same coefficient 
as in undisturbed motion, is, as far as quantities of the third order (Art. 50), 



The coefficient is found directly from observation to be 



128 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

Dividing this by 206,265 and equating the result to Ze - Je 3 , we obtain 

e = -0548993. 

The argument of the term is cnt + w and its period is therefore Tjc. 
The value of c given by equation (25) of the last Chapter, is 



Whence 1 - c = '0041964 -h '0029428 = '0071392. 

Delaunay finds the complete value of 1 c to be '00845. The period of 
the term is therefore about 27-J- days. 

The term with argument 2 <f> is called the Evection. The value of the 
coefficient in longitude was found (Art. 140) to be -}-^-me which, with the 
values of m, e deduced above, gives 52' 56". The complete value of this 
coefficient is 116'26". 

The period of the term is 

2w/(2n - 2n' - en} = T/(l - 2m + 1 - c) 

= 31| days approximately. 

In Art. 1 60 we have seen that (1 c) n is the mean motion of the 
Perigee. The Perigee will therefore make a complete mean revolution in 
time Tf(l c) = 3827 days with the value of c found above. The period as 
calculated by Delaunay* is 3232'38 days or about 9 years. 

The class of inequalities defined by the arguments 2q% p<, (p, q integers) 
may be termed Elliptic Inequalities. The characteristic of the terms with 
arguments 2qg p$ is e*. 



170. We can combine the principal elliptic term and the Evection in a 
manner which enables us to illustrate the connection between the results 
obtained by the method of Chapter vn. and those obtained by causing the 
arbitrary constants to vary. 

The principal elliptic terms in longitude and parallax are, neglecting 
quantities of the orders e 2 and m 2 , 

?;=... -f 2e sin </> + ^me sin (2 <) -f . . . , 

- = ... 4- e cos < + ^f-me cos (2 $) + .... 

20J sin <>! = 2e sin -f ^f-me sin (2f <), 
61 cos ! = e cos -f i<nie cos 2 - <. 



* " Note sur les mouvements du p6rige et du noeud $e la Jjune," Comptes Rendus^ Vol. 
pp. 1724, 



169-171] THE ELLIPTIC INEQUALITIES. 129 

From these we obtain, neglecting quantities of the order w 2 , 

e* = e* + if-me? cos (2f - 20), 
or e l = e + *-me cos (2| 

We also deduce 

BT. sin (0! 0) = - 

Since e 1} e are of the same order, fa must be a small quantity of the order 
m at least. To the order required we therefore have 

a - = Jm sin (2f - 20). 
The transformations give 

v = . . . 4- 20! sin 0! -f . . ., a/r = . . . + e l cos X , 
where ^ = e 4- J^we cos (2f 20), 

0, = + J^me sin (2f - 20) = w + e - {(1 - c) n$ + tsr - ^w sin (2? - 20)}. 

The effect of the action of the Sun as far as it produces the Evection, is 
to cause periodic variations of the eccentricity and of the longitude of perigee 
of the Moon. Had we solved by causing the arbitrary constants to vary, the 
variable values of the eccentricity and longitude of perigee would have been 
found to contain terms of this form. 

In a similar manner, the other terms due to the action of the Sun may be 
included by assuming variable values for the constants. In order to perform 
the process completely, it would be necessary to assume that the velocities 
have the same form as in elliptic motion, in accordance with the principles 
of Chapter v. To the order considered in the above example this is easily 
seen to be true. We should of course include the terms depending on higher 
powers of e, e lt This method of expressing the coordinates is not generally 
useful in itself: it is merely given here as an illustration. In fact, when we 
use the method of the variation of arbitrary constants, the reverse process 
has to be gone through to achieve the object in view, namely, the expression 
of each coordinate as a sum of periodic terms. 

The Annual Equation. 

171. The terms in longitude and parallax with argument 0' = n't + e' -or' 
are known as the Annual Equation ; their period is one year. The coefficient 
in longitude was found (Art. 142) to be 



the complete value being II 7 10". 
B. L. T. 



130 THE CONSTANTS AND THEIR INTERPRETATION. [CHAP. VIII 

The coefficient of the Annual Equation in parallax is, from the same 
Article, 



which is the value correct to one hundredth of a second of arc. 

The terms with arguments 2gf p$ in longitude and parallax may be 
called the Mean Period Inequalities. The variable parts of their arguments 
depend only on the mean motions of the Sun and the Moon ; they are in- 
dependent of the longitudes of the perigee and the node of the Moon's orbit. 

The Latitude and the Mean Motion of the Node. 

172. We have found (Arts. 147, 162) as far as the order 7m 2 , 
u = s = (1 + &O 7 sin 97 + (f m + f f m 2 ) 7 sin (2 - 17) + # w 2 7 sin (2f 4- 17). 
We shall replace the constant 7 by that used in Delaunay's theory. In 
undisturbed motion let 7! be the sine of half the inclination ; therefore, to the 
order considered here, 7 = 27!. In disturbed motion, ^ is defined so that the 
coefficient of the principal term in U is the same as in undisturbed motion. 
Hence, in disturbed motion, we have to the order 7m 2 , 

7(1 + m 2 ) = 2 7l , or, 7 = 2^ (1 - %m z ). 
The value of u expressed in terms of ^ is then given by 

u= 27! sin T? + (f m + f$m 2 ) 7! sin (2f - 17) + V^i sin (2? + 17). 
The coefficient of sin 17 is found from observation to be 

57'41"-26==lS461"-26 
corresponding to the value *04475136 of 7^ The correct value is 

7 X = -04488663 ........................ . ....... (7), 

the difference being due to the omission of the elliptic terms of higher orders 
in the coefficient of sin oj. 

The period of the term is Zir/gn = T/g, The value of g was found to be 
given (Art. 147) by 

1 - g = - (j m - ^ m s - f||m 4 ) = - -0040119, 

the complete value being* ' '0040212. The period is therefore about 27^ 
days. 

The mean motion of the node is (L g)n. This being negative, the node 
moves backward, that is, in the direction opposite to that in which the Moon 
moves. It completes a revolution in time 

T/(g - 1) = 6794'4 days or about 18 J years. 

* See footnote, p. 128, 



171-174] INEQUALITIES IN LATITUDE. 131 

It will be noticed that the first terms in the expressions for 1 - <?, g - 1 are the same. 
We have seen however that the apse moves forward twice as fast as the node moves back- 
ward. The difference is principally due to the near equality of the first two terms in the 
expression for I c : the second term in the expression for g 1 is quite small. 

The method of Art. 170 may be applied to the remaining terms in the 
expression for u, by assuming the inclination and the longitude of the node 
to be variable. The two equations necessary for their determination are 
furnished by the supposition that both U and dujdt have the same form as in 
undisturbed motion. 

173. The other terms present in the coordinates will not be examined here. Enough has 
been said to show how the effect of any particular term on the place of the Moon may be 
examined. The magnitudes of the coefficients can all be obtained in the manner explained 
above. These are best seen in a paper by Newcomb, Transformation of Hanseris Lunar 
Theory*. A reference to it will show that there are 2 coefficients in longitude (those of the 
principal elliptic term and of the evection) which surpass 1, 11 coefficients lying between 
1 and 1', 14 coefficients between 1' and 10" ; in latitude, 1 coefficient (that of the principal 
term) greater than 1, 7 coefficients between 1 and 1' and 6 between V and 10" j in 
parallax, one term (the mean value) of nearly 1 in magnitude, one coefficient (that of the 
principal elliptic term) of just over 3', and 7 coefficients lying between 35" and 1". The 
number of large coefficients is therefore not great, as far as the solar perturbations are 
concerned. 

The methods used for deducing the numerical values of -the constants from the recorded 
observations will be found in the various memoirs which contain determinations of these 
constanta The values of < 4 which have been employed in the preceding articles were 
obtained by Leverrier (Am. de I'Ols. de Paris, Mc'moires, Vol. iv.), those of e, y by Airy 
(Mem. of II. A. & Vol. xxix.) and that of I/a by Breen (Mem. of It. A. S. Vol. xxxn.). The 
values used by Hansen for the seven lunar constants will be found in the Darlegung (see 
Art. 202 below) and the Tables de la Lune. Later determinations have been made by 
Newcomb (Papers published l>y the Commission on the Transit of Venus, Pt ill. and various 
memoirs in the first *two volumes of the Papers published for the use of the Amer. JEph.). 
Further references will be found in the memoir mentioned in the previous paragraph, and 
in the Nautical Almanac, the Oonnaissance des Temps, the American JEphmms, etc. 

174 It is not difficult to verify the statement made in Art. 70, that c, g, found by 
Laplace'B method with * as the independent variable, are the same as the values obtained 
when t is the independent variable. 

In Laplace's method we modify the first approximation by substituting for or, 6 the 
values (!-*) + , (1-0)0+0, respectively. If * be the true longitude of the Sun, the 
disturbing function, which is expressed in terms of r, /, *, 0-tf, will contain the angles 



In order to express v-v', n't + '-&' in terms of v, we have 



* Astr Papers for the use of the Amer. Eph. Vol. i. pt. n. pp. 57107. 



92 



132 THE COUNTS AM) XNTBKPKKTATION. [CHAP. VHZ 

Hence . - ', ** - *' can be stressed in tern, of the four angles 



- 

t. 



- *- *- 



both eases, c, g must have the same values. 



CHAPTER IX. 

THE THEORY OE DELAUNAY. 

175. THIS Chapter will be devoted to an explanation of the manner 
in which Delaunay has applied the principles of the method of the Variation 
of Arbitrary Constants to the discovery of expressions for the coordinates which 
will represent the position of the Moon at any time. The principal object 
which Delaunay had in view and which he fully carried out, is stated in the 
preface to the two large volumes* containing his investigations, in the 
following words f : 

f Determiner, sous forme analytique, toutes les infyalites du mo'uvement de 
la Lune autour de la Terre, jusqu'aux quantitds du septikme ordre inclusive- 
rnent, en regardant ces deux corps comme de simple points wiatdriels, et tenant 
compte uniquement de fraction perturbatrice du Soleil, dont le mouvement 
apparent autour de la Terre est supposi sefaire suivant les lois du mouvement 
elliptique,' 

The limitations imposed on the problem are therefore the same as those 
made in the previous Chapters. The motion of the Sun is supposed to 
be elliptic and in the fixed plane of reference, the disturbing function is the 
same as that given in Art. 8, and the intermediate orbit is an ellipse 
obtained by neglecting the action of the Sun. No modification of the inter- 
mediate orbit, like that given in Chap. IV. and used in de Pont&oulant's 
and Laplace's theories, is necessary here. 

The use of canonical systems of elements being the basis of Delaunay's 
theory, we shall depart from the notation used above and, after Art. 179, adopt 
that of Delaunay. The latter has the advantage of retaining a certain 
symmetry in the formulae : it will also facilitate references to Delaunay's 

* Mm. de VAcad. des Sc. 4to Vols. xxvm. (1860) 883 pp., xxix. (1867) 931 pp. These will be 
referred to in this Chapter as * Delaunay, i., n.' 
f Delaunay, i. p. xxvi. 



134 THE THEORY OF DELAUNAY. [CHAP. IX 

expressions and to the further developments (e.g. those in Chap, xm.) which 
have been made according to his method by other investigators who have 
generally adopted his notation in their memoirs. 

The method by which the transformations contained in Arts. 178, 185 
189, 190 below, are carried out, is not the same as that of Delaunay ; the 
latter performed them by direct differentiation a process somewhat tedious. 
Tisserand, in his account of the theory*, uses Jacobi's general dynamical 
methods, stated in Art. 94 above, to perform the transformations. The 
method used here is short and it has the advantage of showing immediately 
the terms which are to be added to 



176. In Chapters IV., v. have been given the principles on which the 
method of the variation of arbitrary constants is based. When the motion is 
undisturbed it is elliptic, and the coordinates are expressible in terms of the 
elements and of the time. "When the action of the Sun is taken into 
account by considering the elements variable, it has been shown (Arts. 84-86 
or 98) that the equations which express them in. terms of the time are 

^fti _ ^^ ^ a ?" __ ^R 

In these, a*, fa have certain definite meanings with reference to the elliptic 
orbit : they are explained in Art. 92. 

The equations in this form possess a serious defect. It will be remembered 
that J? contains in its arguments, terms of the form nt + const. Now (Art. 84) 
n = rfa^~p~ l (-%&)*. When, therefore, we form 3JR/3&, the time t will 
appear outside the signs sine and cosine and thus produce terms in the value 
of %, which increase continually with the time. It has been seen in 
Chap, iv. that such terms are to be eliminated if possible. The artifice used 
by Delaunay consists in simply replacing the variables fa, a x by two others. 

Before changing the variables to effect this object, some remarks must be 
made on the method of performing this and similar transformations required 
later. 

177. Method for transforming from one set of variables to another. 

Let any arbitrary variations So*, 8fa be given to ^, fa, and lot 8R be the 
corresponding change in R. We have then 



* Mtc. C$1. Vol. m. Chaps. XL, am. Also, Jour, de Liouvilk, Vol. xxn. pp. 255 WB. 
t The method is used in a different way by Badau on pp. 336840 of a paper " Bemarques 
relatives a la Throne des Orbites." Bulletin Astronomiyue, Vol. ix. 



175-178] CHANGE OF VARIABLES. 135 

The six canonical equations (1) can therefore be written, as in Art. 98, 



f .-<)-** 



or 

where R is supposed to be expressed in terms of a$, &, t. The symbol cZ there- 
fore denotes the actual change taking place in time dt } while 8 denotes any 
arbitrary variation of the elements. If we wish to transform from oti, ft to 
another set of variables j lt 72, ... %, the process of finding the new equations 
is rendered very easy. Suppose 

i =fi (t> 7i> 72, - - 7)> ft = fi fc 7i> 7a , - 7e) 5 
we have, by the definitions of d, S, 



and similarly for the variables ft. 

The substitutions being made in the first member of (I 7 ), we suppose R 
expressed in terms of the new variables, so that 

&p aR. 312. ^ 3B. 

Sli == ^ - d7x 4- ^r- 072 + ... 4- 5 - Sy Q . 
dji ' 872 37 

Equating the coefficients of the independent arbitraries 87!, 873,... 870 to 
zero, we obtain the new set of equations satisfied by y lf 72, ... 7< ? . 

The transformations will in general be possible and definite if the Jacobian of c^, ft 
with respect to y 1? y a , ...y c does not vanish. For transformations in which, the system 
remains canonical (Art. 87), see the works of Jacobi, Dziobek and Poincard referred to in 
Art. 105. The formula) for these are not of groat value here because, in Delaunay's 
method, terms are added to R to keep the system canonical. 

178, Transformation to the variables w, 2 , 3 , L, /3 a , ft. 
Let ^^-^/ai 3 , w^nO + O. 

We have from Art. 84, &=*- p/Za and n* = pa~*. Hence 



The formute of transformation are 

^ = -/4V2^ Gti = 

Whence 8ft = Si, So, = ^ Sw H- 



136 THE THEOEY OF DELAUNAY. [CHAP. IX 

We have from these, 

d&Sc*! - dotifySi = dL&w - dwSL 4- p dtSL ; 

and therefore from (!'), since the other variables remain unaltered, 
a 2 + d3,Sa s - dw8L - dM&> - <M& = * *R - 



where R,= R+ i#fi& = R-& ........................ (3). 



If we now suppose jR to be expressed in terms of the new variables, 
the equations remain canonical. They are 





dt ~ dw ' ~dt "" 3ct 2 ' 




It will be noticed that w is the mean anomaly in the undisturbed orbit 
(Chap. in.). 

179. Change of notation. 

We now take up the notation adopted by Delaunay and replace 



by Z, jr, h, I, Q, If, B, % 

respectively. 

The six canonical equations (4) may be written 
4- AffSA - cZZSJD - dgSQ - 



where 



* 



It will not be necessai'y to change the signification of n', a/, ^ the 
constants referring to the solar orbit, Abo ?i, a/, e retain, for undisturbed 
motion, the meanings previously given. 

According to the definitions given in Art. 92, A, #, I are, in undisturbed motion, the 
angular distances x&, QA 9 AM (fig. 4, p. 38) ; also //, G 9 L arc double the rates of 
description of areas by the projection of the radius vector in the piano of reference, by the 
radius vector in the plane of the orbit, and by the uniformly revolving radius vector- 
supposed to be of length equal to the semi-major axis in the plane of the orbit, 



178-180] THE DISTURBING FUNCTION. 137 

respectively. The correlation of areas to angles will be readily noticed. We have also, in 
undisturbed motion, 

mean longitude of Moon, 
= longitude of perigee, 
h = longitude of node, 
I = mean anomaly, 
y = sine of half the inclination. 

In the notation previously used, these were respectively denoted by 



It must be remembered that in de Pontecoulant's theory the letter y was used to denote 
tan i, 

180. The form of the development of R. 

The results of Art. 113 show that, after changing the notation, the 
arguments of all terms in jR are expressible by means of the four angles 

l, I', l+g, h-l'-tf-K. 

The argument of any term is therefore of the form 



where i, i! , i", i f " may be any integers, positive, negative or zero, and 



Since we suppose the perigee of the Sun's orbit fixed, q is a constant which 
vanishes when i!' = 0. 

The coefficients of the development of R given in Art. 114 are functions 
of e, 7, e', a/a' with the factor m*/a ; since m = n'/w, u 2 a 8 = /^, n' 2 a' 8 = m' *, they 
may be expressed as functions of a, e, % a', e'* Hence if we put m'aP/a'*, 
I + g + h I' g f h', I, l r > l + g for m 2 , , <^>, <ft t 77, respectively and for <f the 
expression 4<<y* (1 ry 2 ) (1 2y 2 )^ we can, by expanding this latter quantity, 
deduce the form of the development required for this Chapter, This is, after 
adding the term /x 2 /2i 2 = / 






in which only the principal periodic terms have been written. Delaunay's 
development contains 320 periodic terms -[-. 

* In A.rt. 114 we have put ^ = 1 and ni' for the ratio of the mass of the Sun to the sum of the 
masses of the Earth and the Moon. 
t Delaunay, Vol. i. pp. 3354. 



138 THE THEORY OF DELAUNAY. [CHAP. IX 

If we neglect powers of 7 2 above the second, it is sufficient (remembering 
the change of notation) to put 4ty 2 for j z in the result of Art. 114. In the 
following, V = n't -f const, and a' } e f , n r , //,, w', g\ h' are absolute constants. 

From the relations given in Art. 84 we have, in the new notation, 

r / O/O \ J. 2 



H = A cos i = (1 27 2 ) A//AO- (1 e 2 ), 
and therefore 



(n 



by means of these relations the coefficients in R can be expressed in terms 
! of L, G, H. We shall see later that it is not necessary to actually perform 

the substitutions. 

Hence, as far as the elements of the Moon's orbit are concerned, the 
arguments are functions of I, g, h only and the coefficients are expressible 
as functions of L, (?, H only. 

It is to be noticed that R and the elliptic formulae for v, 1/r, u, given in 
the next Article, do not contain the time- explicitly. It is only present 
in the initial development of R through its presence in I, I'. 

\ 181. The elliptic expressions for the coordinates. 

i The longitude v is immediately obtained from the expression given 

j in- Art. 50 if we replace therein, nt + e, w, % by I + g + h, 1,1 + g, respectively, 

1 and to the order given, 7* by 47*. The value of 1/r is obtained from equation 

I (16) of Art. 39 by expanding Jt(ie) and replacing w by I The latitude v 

L can be deduced from the expression of Art 50 by means of the equation, 



cr=5~|-s 3 "4-..., 
In the expression for s } just referred to, we replace ^ "by I + g, w by I aod 

y 

2 7 (l-fy(1^2 7 ^ = 2 7 (l + |^+...). 



"We thus obtain 



- y" sin (2<7 + 21) - 2y a e sin (20 + 3Z) + 2^% sin (2g + l) + ..., 



1 1 f ) 

-=4l4-(e-46 8 )cosZ-h6 9 cos2Z+... L 



276 sin (gr + 21) - 



...(8). 



Delaunay gives the values of v, u correct to the sixth order and that of 
1/r correct to the fifth order*. 

*' Belaimay, Vol. i. pp. 5559. He uses V to denote the longitude. 



180-183] INTEGRATION OF CANONICAL EQUATIONS. 139 



Delaunay's method of Integration. 

182. The method of integration adopted by Delaunay consists, in 
the first instance, in choosing out of R the constant term and one of the 
periodic terms, neglecting the rest of R. It is then found that the canonical 
equations can be integrated by means of series, and that the variables 
Z, (?, Hj I, g, h can be expressed in terms of the time and of six new 
constants (7, (G), (H), c, (#), (h). By means of these values, jR is expressed 
in terms of the time and of the six new arbitraries. Having solved the 
equations by neglecting a portion of the disturbing function, enquiry is 
made in order to find what variable values these new arbitraries must have 
when this omitted portion of R is included. It is shown that by adding 
certain terms to the disturbing function, the equations which express G, ((?), 
(J5T), c, (g), (h) in terms of the time are canonical inform. The power of the 
method arises from the fact that when the change in jR from L, G, H, I, g, 
h to 0, ((?), (T), c, (g), (h) has been performed, the periodic term considered 
has disappeared. 

The new arbitraries not being suitable for the purposes in view, two 
transformations will be made. Finally, we shall have formula similar to 
those from which we started, that is to say, the equations which express the 
new arbitraries in terms of the time are canonical, their relation to the new 
disturbing function is the same as before and the new disturbing function is 
of the same general form and has the same general properties as R. 

Integration of the Canonical Equations when R is limited to one 
periodic term and the non-periodic portion. 

183. The canonical equations, given by (4f) Art. 179, are 

dL ^. <M *?= = 

W" dl' dt dg' dt dh> [ ............... (4 "). 

dl_ dR dg __ 9^ ^ = 

3$-"~aZ' dt" d&' dt 

Let R = - B - A cos (il + i'g + i"h + V/ T 

or, putting 



in which - B is the non-periodic part of R and - A cos is any one of the 
periodic terms of JR. We shall first integrate the equations (4 ) wth U, 



neglected. 



140 THE THEORY OF DELAUNAY. [CHAP. IX 

Substituting in (4/') we obtain, since A, B are independent of I, g, h and 
6 is independent of L, G, H, 

~ 

dl dA a dB dg dA 
= COsd *"" 




The first three equations give 

L = i, G = i' + (Q)> JSTWe + Off) ............ (11), 

where ($), (H) are arbitrary constants and where 



Differentiating (9) with regard to t, we obtain 

dff .dl .; dg , . dh ., , 

-TT = l-j7+t -77+* -Jl+* W 

d cfe dt dt 
which, by means of (10), becomes 

B v dB . dB\ 



, , 



Now L, Gr, PI and therefore A } B are, by (11), expressible as functions of 
one variable . Hence, putting 

B + i'V^ .............................. (12), 

there results, since i = 3L/3, etc., 

dd dA n dB l /1tn 

Tt = m co*e +m ........................... (13). 

We have therefore the two simultaneous equations (II'), (18) for the 
expression of > 6 in terms of t Multiply (13) by dB and (IT) by dd, 
subtract and integrate. We obtain 



where G is an arbitrary constant. Whence 
sin = 

M 
Substituting in (11') and integrating, we find, if c be an arbitrary constant, 



The lower limit of the integral is taken to be the value of & when 5 = 0, 
that is, when A =C-B l . Since A, B l are supposed to be known functions 
of L, (?, H 9 and therefore, by (11), of and of the constants (G), (If), 



183-184] INTEGBAT10N OF CANONICAL EQUATIONS. 141 

this integral gives the value of t + c in terms of and therefore of 6 in 
terms of t + c ; hence L, G, H can be expressed in terms of t + c and of the 
arbitraries C, ((?), (H). The equation (14) then gives in terms of t + c. 

184. The values of I, g, h are now to be found. 

da dA a , dB 



We have 

U/C/ L/V-f vvjf 

Substituting for <fe from (15) and for cos from (14), this becomes 

^ = M g-ff* + |Jl ^ v{^ 2 - (^ - ^i) 2 }- 

d IdG A dGr\ 



Also, from the way in which (G), (H) were introduced, 
dA dA dB dB I 



Hence, integrating, 

rdA C-B 1 



+ 



Similarly A = ] ^ {A ^ (C ^ B ^ 



.(16). 



Here (^), (A) are arbitrary constants ; the upper limit of each integral is 
and the lower limit is, as before, that value of @ for which A = G - B,. 

Finally, as 6, g, h, 1! are knowa in terms of t, we can find I from (9). 
The three equations (15), (16) can be put into a more convenient form by 
assuming 



the upper and lower limits of the integral being the same as before. The 
three equations then become 



for K may be considered to be a function of , C, (<?), (H). 

The complete solution is contained in equations (9), (11), (14), (17), (18) 
which, after the elimination of 0, 0, K, will give the values of L, G, H, I, g, h 
in terms of t and of the six new arbitraries C, (0), (J5T), c, (g), (h). The solu- 
tion in this form is not, however, convenient for actual calculation ; the method 
to be used will be outlined in Arts. 192, 193, 



THE THEOBY OF DELAUNAY. [CHAP. IX 

The Canonical Equations for 0, ((?), (H), c, (#), (A), when M 1) the 
portion of R previously omitted, is considered. 

185. The canonical equations having been thus solved when jRj is neglect- 
ed, it is required to find the solution when jR x is included. The solution 
just obtained contains six arbitraries (7, ((?), (H), c, (g), (A). We are going 
to inquire what variable values these six arbitraries must have when S^ is 
not neglected. The method is then a further application of the principles of 
Chapter v., for we suppose L } Q > H, I, g, h to have the same forms whether 
the new arbitraries be constant or variable. See Art. 98. 

The canonical equations are given by 

dUl 4- dGSg + dffih - mi - dgSG - dhSH 

) ......... (19). 



Into this equation we must substitute the values of L, G, H, I, g, h given 
by equations (11), (14), (18), the arbitraries C, (G), (H), c, (g), (h) being now 
considered variable. We suppose any arbitrary variation S given to the 
latter and consider what changes are produced in R ly L, etc. 

From (12), '(14) we deduce 



The second member of (19) is therefore 

i'VeftSe - dt&C + 

For the first member we have, from (11), 
SL = iS, 8G = i'&Q + S(ff), 
dL = id, dG = fd& + d(G), dH = i"d& + d(H). 
Substituting these and remembering that 

g0 igz + i'ty 4- i"$h 9 
d0 = idl + i'dg 4- i"dh + i'"n'dt 
the left-hand member of (19) becomes 



But from the equations (18) we have 



with corresponding expressions for dg, dh. Substituting these values of 
Sg, Sh, dg, dh in the last expression for the left-hand member of (19), and 



185] THE NEW CANONICAL EQUATIONS. 143 

cancelling out the term i"VcKS common to both members, we reduce 
equation (19) to 



d(H)S(K) - <%)S() - d(h)S(S) = dt(SR 1 - SO)... (20). 
Since the operators d, 8 are commutative, the part [...] may be written 



But by (17), K is a function of , 0, (0), (H) only, and each of these is 
now considered to be a function of t, owing to the variability of the six 
arbitraries. Hence 



air. a.sr 3jsr 
_ g + w sa 



and a corresponding expression of dfif. The portion [...] therefore becomes 



This expression, by reason of equations (17), (18), is equal to 
8 {- 6W + (* + c) ci(7) (2 {- 6>Se + (* + c) 8G}, 
or, since S = 0, 0Sd = ^d8, etc,, to 

- dS<9 + d(7Sc + d<9S - (^ + dc) BO. 

If this be substituted in (20), the terms d&0, d9S, dt&C disappear 
and the equation reduces to 

dOSc + d(G)S(g) + d(H)S(h) -dcSC- d(g)8(0) - d(h)S(H) = d^j^. . .(21). 

Supposing now that, by means of the values of L } (?, JET, I, g, h, the function 
J?i has been expressed in terms of t and of (7, ((?), (T), c, (</), (/t), we obtain 
from (21), in accordance with the remarks of Art. 177, the canonical 
equations 

^L aE j W = aft d(g) j^ \ 

3? 3c ' dt d(/y dt d(hy 

^w-f^ i 

~ 



A glance at the results obtained in Arts, 183, 184 will show that the variables in (2V) 
have meanings, with respect to the motion to which they refer, "bearing a close analogy to 
those of a i9 fa (Art. 176) with respect to elliptic motion. "We shall see in Art. 187 that a 
transformation, similar to that of Art. 178, must be made, 



144 THE THEOKY OF DELAUNAY. [CHAP. IX 



The nature of the solution obtained in Arts. 183, 184. 

186. Let us examine the nature of the solution obtained in Arts. 183, 
184 when the variables are expressed in terms of the time and of the 
six arbitraries 0, (G), (H), c, (g), (h) : as E : is there neglected, these arbi- 
traries are still considered to be constant. Since A is the coefficient 
of one of the periodic terms of the disturbing function, A, 9-4/3 are small 
quantities of the first order at least. Hence, the equations 

d . . a d9 dA a dS, 
~=~ = A sm 0, -vr = -JT=C cos -h -=7= 
dt dt d d 

show that, if we neglect quantities of the first order and remember that 
A, B l are supposed to be expressed as functions of and of the arbitraries 
($), (IT), a first approximation to the solution is given by 

= const. = o, 6 = (t + c), 

in which @ , c are arbitraries and is a definite function of @ 0> (G), (H). 
A second approximation furnishes 

= o + @i cos (t + c), = (t + c) + & 1 sin (* 4- c). 

Hence, assuming that developments in series are possible, the solution of 
these equations is given by 

@- 



2 sin20 (?5+c)4-... j ...... ( } ' 

Since 0=0 when t = - c, the arbitrary c is the same as that defined by (15). 
The arbitrary constant attached to is which, by (15), may be expressed 
as a function of (7, (), (J?); hence @ 0j 1? @ 2 , ..., , a , 2 ... are functions of 
0, (G), (H). 

For the sake of brevity, denote by a e , OL S the series 



The solutions may then be written 

@= + <B) C , 0=0 ( + c )4-0, .................. (22'). 

With these values, the equations (18) give 

# = (#) + <7o( + c)+^, , A = (A) + Ao($ + e) + fc. ......... (23), 

where g Q , ^i, ..., A , &i,... are functions of G, (G), (H). Therefore, from 
equation (9), we have 

-W 



186-187] NATUEE OF SOLUTION. 145 

where Z 1? Z 2 ,... are determined by 



Finally, the value of in (22'), substituted in equations (11), gives 

= i + c , 0=0o + 00, H= 
where 

i = i @o, A =*!, 2 = 

(?!=;',, <?, = ; e,, ......(24). 



The coefficients fy, $/, fy, /, 6^, 23} being quantities of the order j at 
least, the above investigation shows that the new values of the elements can 
be formally expressed by series of cosines or sines of multiples of one angle 
#o (t + c), with coefficients in descending order of magnitude. The values of 
L, G, H, I, g, h may then be substituted in the disturbing function and in the 
elliptic expressions for the coordinates and the results expressed as sums of 
cosines or sines; the arguments may be freed from periodic terms in the 
manner explained in Art. 111. 



187. The form of the disturbing function after the substitutions have 
been made. 

Let us now consider the effect of the substitution of these values of 
L, G, H, I, g, h in R^ Previously, R l consisted of periodic terms whose 
arguments were multiples of Z, g, h, I', g', h' and whose coefficients were 
functions of L, G } H. After the substitutions have been made, J^ will 
consist of periodic terms whose arguments, besides being multiples of 
^i $'> ^ contain multiples of 

0o (t + c), (g) + #, (t 4- c), (70 + Ao (t + c) ; 

, # , A , and the coefficients of these periodic terms, are functions of C, (0) } (H). 
Hence, when we commence to solve the equations (21') by differentiating B^ 
with respect to 0, ((?), (H), the time 1 4- c will appear as a factor of the 
periodic terms. In order to avoid this, instead of C, (6?), (JST), c, (gr), (h), 
a new set of variables can be chosen which are such that the equations 
expressing them in terms of the time are still canonical ; when R^ has been 
expressed in terms of the new variables, it will have a form similar to that 
which R had, that is to say, R 1 will consist of periodic terms whose arguments 
contain multiples of three only of the variables and whose coefficients are 
functions of the three conjugate variables only. In making the transformation, 
the following Lemma will be required. 

B. L. T, 10 



146 THE THEOBY OF DELAUNAY, [CHAP. IX 

188, Lemma. Let 



<f> = ^i 4- 20 2 
then -7r = ?r7v((! 



, 



Differentiating (22'), we have 

d = d Q dt [@! sin (t + c) + 2 2 sin 20 ( + c) + ...], 
and = (t + c) + 0i sin (tf + c) + 2 sin 20 ( + c) + . . . ; 

whence 0c?@ = \6 Q dt [i0i + 2 2 02 + 3 8 0s + . . . J 

+ (t + c) dt x periodic terms + dt x periodic terms. 
Therefore, since K~J0d and since K, t+c are zero together, we obtain 

X = -|0 < (t + c) + (t + c) x periodic terms + periodic terms. ..(26). 
Now K was a function of @, (7, ((?), (IT) and @ is a function of (t + c), 
C, (Q), (#). Denote by ( ^ J , (5775;) , (Ww\ ) t ' le P art i a l differentials of jfiT 

with respect to (7, (0), (H), after the value of has been substituted in /^. 
We then have 

Uc/ 80 "^aeao' 

and similarly for ((?), C^)- Since dKjd = 0, we obtain from these equations, 



idK\ .36 dK _(dK\ .30 8JT _ _ . 

^^ J """ ' "-^ 



...... (27). 

Now by (26), 



+ periodic terms having (t + c)', (*+ c) 1 , (< + c) 2 as factors. 
Also, from (22'), 



+ periodic terms having (t + c), (<H- c) 1 , (< + c) 2 as factors. 
Further, by (18), 8^/90 = - (* + c). 



188-189] FIRST CHANGE OF VARIABLES IN THE NEW EQUATIONS. 147 

Substituting these values for (JQ\ 6~, ^ in the first of equations 

(27) and equating those coefficients of - (t + c) which are independent of 
periodic terms, we find 



In a similar manner, by (18), (23), 



and therefore, equating coefficients of (t + c), 



' - 2 d(G) 
with a corresponding equation for h Q . 



189. First change of variables in Equations (21') to avoid the occurrence 
of terms increasing in proportion with the time. 

Put now 



X (* + C), A: = for) + <7o (* + c), *> = (A) + ^ (* + o> j " " 

and change the variables in (21') or (21) from (7, ($), (J?), c, (jr), (A) to A, 
, (ff), X, /c, 77. The Lemma just proved gives 

1 9A <7o 3A /d 3A 



We have also, from the relations (28), 



, dc = -- 



Substituting, the first member of (21) becomes 
d 



102 



148 THE THEORY OF DELATJNAY. [CHAP. IX 

1 g A ^ . , 8A J)A_ 3A 
Substitute everywhere for ^ , - g- , - ^ their values -^ > 9 () ' 9 (JJ)' 

The coefficient of 8X becomes dA and that of d\ becomes - SA. The 
coefficient of -f X is 



= coefficient of X, since 8, d are commutative. 
The equation (21) therefore reduces to 

d(G)$* + d (H)*i + dtSO - ^8 ((?) - ^ S (JI) + dASX 

...... (29). 

Hence, putting K = Ri-C .............................. ( 30 X 

and supposing K to be expressed in terms of A, (<?), (Jff), X, , rj } we obtain 



= 
dt 8X J cfo 9/c ' d* 3*? ' ............ (290- 

^ = 
d* 



The equations remain canonical and R has a form similar to that which R 
had, namely, it is expressed by cosines of sums of multiples of X, /c, 77, Z', </, A 7 , 
with coefficients depending on A, (G), (If). The partials of R with respect 
to the new variables will no longer introduce terms proportional to the time. 

Second change of variables. 

190. It is advisable to make a further transformation in order that when 
A (the coefficient of the periodic term previously considered) is put equal to 
zero, the new canonical equations shall reduce to the old ones (4"), Art. 183. 

When A vanishes, it is easily seen from (22), (28), (23) that 



189-191] SECOND CHANGE OF VARIABLES. 149 

Put therefore 



, , 

In 00 ft * 'I 

v 6 ) 

and transform from A, (0), (H), \, K, 77 to A', <?', H', \', K , rj. 

We obtain by substitution in (29), after the elimination of by means 
of (30), 



= i eZA' (tfix' + i'&K + t"8ij) + f d& - ^ d A' ) S + f dff' - C dA' 

t \ % J \ I 



- (tax' + i'djt + i"dr, + i'"n'df) 4 SA' - die (80' - -r SA' }-d>r,(SH'--^ SA' 

^ \ i ) \ 

= ciA'Sx' + dO'B,c + dH'Zr, - d\'SA' - d>c$G' - dij&ff' - C '. 

/// 

Hence, by putting R" = R' + -r- n'Af (310? 

i 

the equations can be written in the canonical form 

J A f Q TV >7 '/"' ^ T?" ^7 TJT f ^ D^ 

OuxX C/JU/ CtUT OjCt CyJLt, C/Xt 

OD^ dX Out u/C C6t d77 

^ = - _ , -~ = - ._ , = - , 

and these new variables will reduce to the old ones when A is put zero. It is 
evident that the remarks made at the end of the previous article will also 
apply to these equations, 

191. We will now see how the new variables are related to those found 
in the solution given in Art. 186 ; also, how the new disturbing function R" 
IB related to jR. 

by (11), (25), The equations (31), (28), (24) furnish 



, 17 = (A) + AO (* + c), 
where, by (880,. ., 



. 



T r-jJ. 

The new variables X', /c, 97 are therefore nothing else than those non- periodic 
parts of I, g> h which were obtained by the solution of the equations (4"). 



150 THE THEOUY OF DELAUNAY. [CHAP. IX 

Again, we have 

R = - A cos 6 - B + R 1 

by (30), 

^ / , by (31'). 



But since A' = ih. = i ( + 

we obtain JR" = R l - 4- i"W (6 

that is, JB" is the same as B^ except for some additional non-periodic terms. 
The periodic term A cos Q is therefore not present in the new disturbing 
function. Since jRi does not contain A cos 0, the last equation also shows 
that, when we make the change of variables in jR by giving to Z, G, H, I, g, h 
their values as functions of A', (7, J5P, X', K, 77, the periodic term A cos 9 in 
JB will identically vanish (See Art. 19*7). This furnishes a means of veri- 
fying the calculations. 

The application of the previous results to the calculation of an operation. 

192. It is necessary to see how the results obtained in the previous 
articles are applied in the actual calculations. In the first place, R was 
expressed in terms of a, e, 7, Z, g, h. Out of R the terms B A cos 9 were 
chosen : of these, -B is the whole of the non-periodic part of JB and A cos 9 
is any one periodic term. In the canonical equations (4") for L y 0, H y I, g, h, 
we insert this value of JR. 

Equations (7) give the values of L 3 (?, H in terms of a, e, 7. By means of 
them, we find from the integrals 



any two of the quantities a, e, 7 in terms of the third, say a, 7 in terms of e } 
and substitute in the expression in (7) for L, which then becomes a function 
of e } ((?), (H) only. From this, dL/dt is deduced as a function of de/dt, e> 

(Q), 



But dL/dt = dR/dl, where jR = - A cos 6 - B. Whence, after expressing 
A in terms of e, ((?), (JT), we find 

de/dt = sin x function of e, (Q), (H) ............... (33). 

Also 



191-193] THE CALCULATION OF AN OPERATION. 151 

Now ?, , may be expressed in terms of ^, -g^-> g> ^ e, 7, by 



means of the relations (7), (7'). We therefore obtain 

jf/3 

~~ = function of a, e, 7 + cos 5 x function of a, e, 7, 
at 

or, according to the remarks made above, 

^ = function of e, (G), (H) + cos x function of 6, (0), T . . . (34). 
at 

The equations (33), (34) involve only the two dependent variables e, 9 
and the independent variable t. They are equivalent to (11'), (13) and are 
integrated as in Art. 186 by continued approximation or by some similar 
method, giving 

6 2 = e * + cosines of multiples of (< + 

= ($ + c) + sines of multiples of ( + 

where e , c are the two arbitrary constants and is a function of e Q3 (CF), (H). 

In certain operations where e appears as a denominator in (34), it is found 
to be more convenient to solve (33), (34) by finding 

e cos 6 = const. + cosines of multiples of (t 4- c), } 735'} 
e sin 6 = sines of multiples of (t + c) j ....... 

Here the arbitrary e, is the coefficient of sin <9 (t + c) and it does not appear 
in the right-hand members as a denominator. See Art. 196. 

From these we can deduce the values of a, 7* (which were found in 
terms of e, (ff), (H)), expressed as functions of <? , (&), (J?) and cosines of 
multiples of (* + <0- In a11 cases the coefficients of the sitie8 and cosines 
are functions of * , (<?), (H) only. Let a , 7 o 2 be the non-periodic terms of a, 7* 
and eliminate (<?), (J5T). We have then 

a = ao + functions of a , ^ > 7o 8 multiplied by 

cosines of multiples of (* + c), 
e s -. e f 4. similar terms, 
ry2 =5 ry Q 2 + similar terms 

When the form (350 is used > the expression for e* contains some other 
non-periodic terms. 

^7 da dh , , r 8jR ^ 

193. Next, equate -| - J, - ^ to the values of ^ ^ , 



152 THE THEOBY D^ DELAUNAY. [CHAP. IX 

already found. Eliminating a, e, 7, from dR/dL, etc. by means of (35) or 
(35'), (36) and integrating, we obtain . 

I = X' + functions of a , e<>, % multiplied by 

sines of multiples of (t + c), 
. 
g = /c + similar terms 

^ = ^ + similar terms 

Here X', K, y are the non-periodic parts of I, g, h] also, by means of the 
relation 6 = il -f i'# -f i"& 4- i'"l' + q, we can express # (t -f c), which is simply 
the non-periodic part of 9, in terms of X', /c, 17, V, q. 

The results (36), (37) contain the required solution of the equations. 

In order to prepare for the next operation, we substitute these results in 
E (and also in the expressions for the longitude, latitude and parallax, 
previously obtained); J? will then become a function of a , e Q> y 9 , X', /e, ??, 

Our new variables will be A', (?', H', X', K, rj, 
where A', G', H f = functions of a , e , 70 .................. (38), 

found by means of the relations given in Art. 191. For since a, e, 7 are 
given by (36), the values of L y G, H can be deduced ; each of them will be 
expressed by a constant term and by cosines of multiples of (* + #)> fch 
former and the coefficients of the latter being furjctioris of <x , e , 70- We then 
have LQ, Q-Q, HQ-, the function <f>' is deduced from the series obtained for 6, L. 

Finally, since (Art. 191) 

n'Af 



= R - '"Ve + n'A', by (12), (14), 






by (11), 



we can obtain the new disturbing function. It will bd found that, when JR, 
and L have been expressed in terms of the new variables, the term A cos 6 
will not be present in !". 

Taking the new variables and the new disturbing function the equations 
for them are still canonical. Also, as (38) gives the values of A', (?', H 1 in 
terms of a , e , 70, we are in a position to go through the whole process again 
with another periodic term. Since the letters L, G, H t a, e, 7, I, g, h, E have 
disappeared completely, there is no further need of the symbols A', $', etc, : 
as soon as the operation is finished, they are replaced for simplicity by the 
letters L, G> etc. During the process, it is frequently more convenient to use 
n instead of a : the relation between n, a is always defined by the equation 
p, 5= ri*a*. 



193-195] PARTICULAR CASES. 153 

Particular cases. 

194. It is evident that if any or all of the integers i', i", i'" are zero, the 
same methods will hold ; the only difference will be a greater simplicity in 
the results. When i f or i n is zero, we get 



respectively. When i f " is zero, the new part to be added to R vanishes. 

When i is zero but either i f or i" not zero, the method requires a slight 
modification. Suppose i' unequal to zero. We assume (? = i'@ instead of 
L = i ; the solution evidently proceeds on exactly the same lines since, in 
the first instance, the equations were symmetrical with respect to i, G, fl". 

195. When i 9 i\ i" are all zero, we can modify the solution in a way 
which saves several operations. 

The angle is reduced to i"T, for q = when i" 0. We take out of R, 
instead of a single term, the terms 

- A l cos V - J.2 cos 21' - A 3 cos 3V . . . = - %A P cosjpi'. 
When this is substituted in the canonical equations (4") we obtain 

dL d&_ dH_ 

~dt =0 ' W~' dF~' 

dl ^dA p dg ~dA p ,, dh -9J 

jg-i^OOBJpJ', ^-S-g^OOBjrf, ^g-S 

Hence L, G-, H are constant and therefore 

l = (l) + ^~^8inpl' ........................ (39), 

x pn oL 

with similar expressions for g, h. 

It is necessary to see what the new canonical equations become when J^ 
is not neglected. They are expressed by the equation 

dti (& - 2A P cos pi') = dUl + dGSg + dH&h - dUL -dg$G- dh$H. . . (40). 

Since L, G-, H remain unaltered, take L, G, H, (I), (g), (h) as new variables. 
Substituting from (39) for I, g, h, the second member of (40) becomes 



154 THE THEORY OF DELAUNAY. [CHAP. IX 

The second line of this expression can be shown to vanish in the same way as 
a similar expression considered in Art. 189, p. 148 ; the third line is equal to 

- 2,$A P cospl'dt = - dtS [ZA P Gospl'l 

and it therefore disappears with the same term present in the first member 
of (40). Hence the first line is equal to dtSR l3 and when J^ has been 
expressed in terms of L, G, H, (I), (g), (h), the equations for these new 
variables will be canonical. 

The rule for transformation is simple. We solve and find 
I = (1) + %l p sin pl' 9 g = (g) + 2#p sin pi', h = (A) + ^h p sin pV, 

where l p , g p} h p are functions of a, e, 7, or of L, G, H. All that is necessary 
is to replace I by I + StpSinyZ', with similar changes for g, h, in jR, v, V, I/r. 
The new equations for L, G, H, I, g, h are canonical, they reduce to their 
former values when all the coefficients A p are zero and, after the change 
of variables has been made, the terms %A p cospl' will not be present in 
the new disturbing function. 

It will be noticed that the constant term is not included here. Since it 
is a function of L, $, If (which were shown to be constants) only, it can 
produce no new parts depending on I, g, h and it need not, in this case, be 
considered. 

196, It lias been mentioned in Art. 192 that, when the equations (33), (34) have been 
prepared for integration, they sometimes take the forms 



~ 

d* M K'-i.a.->; 





here M is of the second order at least and N> J/i, N^ Pi are of order zero ; these 
coefficients are supposed to be independent of e, , 0, so that, as far as the integration 
of the equations is concerned, they are constants. If we integrate in series by continued 
approximation, difficulties may arise owing to the presence of e as the denominator of a. 
fraction. This may be avoided as follows. 

We deduce from the equations just given, 



~ i - i (esin ff) (ecos 0)~ 

(M 

j t (e sin &)=M+M[(e sin 6}* 2jMi- + (e cos 0) 2 2.2V 

Since the least value of i is unity, e does not occur as a denominator in these expressions ; 
also, the second members are expressible by integral powers of e cos 0, e sin 9. 

To solve these equations assume 



e sin 6 = &[ sin (+c) 4-JS r 2 ' sin 20 (t + 



195-197] DELAUNAY'S METHOD OF PROCEDURE. 155 

If we put .&y=0 , it can be shown that E i9 E{ are of the order ej at least; also, 
E^E^ E%=E%) etc. The arbitrary constants are e , c : the quantity is a certain definite 
function of Jf, $", M^ 3 N"^ P^ c 2 and it does not contain e Q as a denominator. 

In the cases where e appears as a denominator, it is found, when we proceed to the 
substitution of the values of 0, e in the expressions for R and for the coordinates, that we 
only require to know e cos #, e sin 0, e 2 and powers and products of these quantities ; 
since their values do not contain e$ as a denominator, no difficulty will ensue. See 
Delaunay, i. pp. 107, 108, 878882 and Tisserand, Mfc. Gel Vol. in. pp. 216220. 



The general plan of procedure. 

197. A general view of the whole process will perhaps make the compre- 
hension of Delaunay 7 s method easier. 

We find first the elliptic values of the coordinates of the Sun and the Moon 
and, hy means of them, express R in terms of 1 9 g, h, a, e } 7, referring to the 
Moon, and of I', g' + h' 3 a', e', referring to the Sun. We have also definite 
relations between L, (?, H and a, e, 7. 

Choosing out of R the non-periodic part and one of the periodic terms, we 
solve the canonical equations for these portions of R and find the values of 
I, g } h, a, e, 7 in terms of the time and of X', K, 77, a , e , 70 ; of the latter, 
X', /c, t? contain three new arbitraries which are the constant parts of i, g, h : 
the other three new arbitraries are a , 70- These values are substituted 
in R } v, 1/r, u which then become functions of X', /c, ??, a , e Q , 70 Certain 
terms are added to R and, when the change of variables has been made, 
it is found that the periodic term considered has disappeared. As I, g, h 
occur in the alignments of the periodic terms of the expressions for R and for 
the coordinates, and as the periodic terms, introduced by the change of varia- 
bles, are small, we can expand each cosine or sine (as in Art. Ill) so as to 
free the arguments from periodic terms. The form of the new disturbing 
function is similar to that of the old, except that the new g, h now contain the 
time explicitly ; this is due to the fact that the action of the Sun causes the 
perigee and the node to revolve. We also find the relations between A', G', 
H' and a , e Q , 7 ; the equations for the new variables X', /e, 77, A', 6?', H' 
being 'canonical, we are ready for the next operation. 

We proceed in exactly the same manner. With the relations, just found, 
between the new L, G, H and the new a, e, 7 (that is, between the old 
A', (?', H' and the old a , e , 7o) w ^ take the non-periodic part of the new R and 
one of its periodic terms and with these we solve the canonical equations for 
the new arbitraries (now considered variable), introducing in the same way six 
other arbitraries. By this operation another periodic term of R is eliminated. 
Continuing in the same way, we eliminate one of the periodic terms in R at 
every operation xintil R is reduced to a non-periodic part only. 



156 THE THEORY OF DELAUNA.Y. [OHAP. IX 

Each change of variables may produce new terms in R and may cause 
a reappearance of terms whose arguments are the same combinations of Z, g, h, 
T, g f , h'> as those of terms previously eliminated or as that of the term under 
consideration; in the first case, they will evidently have the same general 
form as before, that is, they will be of the form il + i'g + i"h + i" r l -f q ; 
in the latter cases, the new coefficients will be of a higher order than the 
coefficients of the terms with the same arguments previously eliminated. 
The series of operations thus continually raises the order of the coefficients of 
the periodic terms in R. We go on with the operations until these co- 
efficients become sufficiently small to be neglected. Delaunay has continued 
them until he has found all terms in the longitude correctly to the seventh 
order inclusive ; in addition, some coefficients are calculated to higher orders 
when slow convergence indicated the necessity of carrying the approxima- 
tions further. 

The number of operations required is very large. Delaunay retains all terms in JK up 
to the eighth order inclusive. He first carries out 57 operations, by means of which he 
eliminates all periodic terms in II which are of an order less than the fourth. The 
first operation is that outlined in Art. 195 above ; then follows the elimination of the terms 
with arguments I, %k+%g-}>%l-%fi f ~-%g'~-%l f .l ) etc. those terms whose coefficients are 
lowered by the integrations being, in general, considered first. The expression for It (in 
which every, term produced by the successive changes of the variables is shown separately) 
together with the details of these operations occupy the greater part of Vol. i. ; the ex- 
pression for R alone occupies pages 119-256. 

Vol. ii. opens with the value of R which remains after the 5*7 operations have been 
carried out : it now contains no periodic term of an order less than the fourth and the 
great majority of the terms are of a higher order. He then makes 435 further operations 
in order to eliminate these remaining terms. In most of these operations it is not 
necessary to change the variables in R : the small changes produced ar made in the 
coordinates only. There are, however, five periodic terms, arising from changes in 
R, to be taken into account and these are eliminated by five further operations. Then 
follow the values of the longitude, latitude and parallax with the successive modifications, 
written out in full, which they have undergone owing to the 57+485+5=497 operations. 
The next chapter is devoted to the further researches into the longitude necessary to carry 
some of the coefficients to higher orders ; this demands a recalculation of some of the 
operations. In performing them, certain errors are detected and the necessary corrections 
are added. Finally he gives the reduced values of the coordinates after the change of 
arbitraries (explained in Art, 199 below) has been made, together with the numerical value 
of each term in every coefficient, for the case of the Moon. 

198. Finally, the disturbing function is reduced to a non-periodic term 
B. Since S does not contain I, g, h, the canonical equations give 



dt~~ ' eft ' <ft dt~dL' dt^dG' ete 

Hence L, &, H and therefore a, e, 7 are unchanged, while we have for I, g, h, 
respectively, the values 



FINAL EXPKESSIONS OF COORDINATES. 157 

where 1 Q , g Q) h Q are the values of dB/dL, dB/dG, dB/dH and (l) 9 (#), (h) are 
arhitraries. Since the previous operation has furnished the connection 
between L, (?, H and a, e, 7, we can obtain l Qt g Q , A &s a function of a, e, 7. 

The final expressions for i;, 27, 1/r are therefore obtained as a sum of 
periodic terms whose arguments are of the general form 

H + i'g + t"fc + i'l' - t" (^ 4. fc'), 

and whose coefficients are functions of the constants a, e, 7 introduced by the 
last operation ; also, I, g f h are each of the form, t x function of a, e, 7 -f const. 
Further, v contains the term t x function of a, e, 7 + const and 1/r contains a 
constant term which is a function of a, e, 7. We must now see how the final 
I, g, h, a, e, 7 are related to the quantities denoted by those letters in purely 
elliptic motion. 

The Arbitrary Constants and the Mean Motions of the Perigee 

and the Node. 

199. The result of any operation was to replace a by a -f periodic terms 
introduced by the operation : the periodic terms, depending on the action of 
the Sun, will be small Similar remarks apply to e, 7. Hence a, e, 7 at any 
stage will differ from their original values (which were arbitraries of the elliptic 
solution) by terms depending on the action of the Sun, and their principal 
parts will be their elliptic values. 

Again, after any operation we find for l y g or h expressions of the form, 
Arb. const, -f t x function of a, e, 7 + periodic terms. 

The new 1 9 g, h are the non-periodic parts of these, so that I is replaced 
by I + periodic terms ; similarly for g, h. 

At the outset we had I = nt 4- e -or, g = tar 9, h = (6 being here the 
longitude of the node). Hence the relations of the final I, g, h to their 
initial values are, when the whole series of operations is completed, 

Final l = nt+e~-vr + tx function of a, e, 7, 
Final # = -GT 6 -Mx function of a, e, 7, 
Final h = 6 + t x function of a, e, 7. 

The last terms of each of these expressions, depending on. the action of 
the Sun, are all small. 

Since a', e', n r are present and since at any stage n is defined by the 
relation ^ = ^ 2 a 8 , the coordinate v can be expressed in the form, 

Const. 4- 1 x const. + periodic terms with coefficients depending 

on n'/n, e, e', 7, afa' and arguments depending on I, l\ g> h I 1 g'~ h', 



158 THE THEOKY OF DELAtJNAY. [CHAP. IX 

The coordinates u, 1/r are expressed by periodic terms of similar form, 
the coordinate 1/r having further a constant term. We now change the 
arbitraries so that they may be defined as in Chap. vin. and consequently be 
independent of the method of solution adopted. 

200. Since the constant parts of I, g, h are arbitraries, we define e, 0, VT 
in disturbed (or undisturbed) motion as follows : 

e = the constant term in I + g -f h } that is, the constant part of the 
mean longitude, 

or = the constant term in g-\-h, 
= the constant term in h. 

Again, since n (or a), e, 7 are arbitraries, we take a new n, e, 7, a defined 
as follows : 

n = coefficient of in I + g + h, that is, n is the mean motion in longi- 
tude whether we consider disturbed or undisturbed motion ; 

e is such that the coefficient of sin I in longitude is the same in 
disturbed motion as in undisturbed motion; 

7 is such that the coefficient of sin (I + g) in latitude is the same 
as in undisturbed motion; 

a = (/4/n 2 )i, where n has the meaning just defined. 

In order to transform to these new arbitraries we equate n to the coeffi- 
cient of t in the final non-periodic part of v ; e, 7 are found by equating the 
coefficients of the principal elliptic term in longitude and the principal term 
in latitude, found from purely elliptic motion (with e,. 7 as the eccentricity 
and the sine of half the inclination, respectively), to the coefficients of the 
corresponding terms found by the theory. We have then sufficient equations 
to express the old arbitraries in terms of the new and thence all the 
i coefficients can be expressed in terms of the new arbitraries. 

Since I + g + h, 1,1 + g are respectively the mean longitude, the argument 

f ^Q principal elliptic term and the argument of the principal term in 

latitude, the mean motions of the perigee and the node are given by the 

j, coefficients of t in the final expressions for I -f g + h I = g + h and l + g + h 

> i^g--^ respectively; these coefficients of t must also be expressed in 

ir terms of the new a, e, 7, n. They were denoted in de Pont&oulant's theory 

\ 

* The arguments of all terms are combinations of the four angles I, l\ l + g, 

; l + g + h-l'-g'-h'. Delaunay puts D = l + g + h-l'-g' ~h', F*=*l+g,BQ 

that 



199-201] DELAUNAY'S BESULTS. 159 

2D = Argument of the Variation, 

I = Principal Elliptic term, 

V = Annual Equation, 

D = ,, Parallactic Inequality, 

F = Principal term in Latitude. 

These were respectively denoted by 2, <f>, </>', , v\ in de Pont&ioulant's 
theory. It is necessary, in Delaunay's final results, to replace a/a' by 



in order to take into account the correction obtained in Art. 7. 

201. The literal results obtained by Delaunay in using the methods explained above, 
far surpass any other complete developments in their general accuracy and the high order 
of approximation to which they are carried, although further terms of certain portions, such 
as the principal parts of the mean motions of the perigee and the node, have been found to 
a greater degree of approximation. The only results which can be at all compared with them 
are those of Hanson. The latter, however, confined his attention to numerical develop- 
ments by substituting the values of m, e, y, e', a/a r at the outset, while Delaunay gives complete 
literal results for the three coordinates this being necessary in his method of treatment. 
Although the disturbances produced by the Sun are alone treated, the method can be and 
indeed has been continued from the point where Delaunay stopped, so as to include the 
effects produced by the actions of the planets, the figure of the Earth, etc. (see Chap. XIIL). 
Had Delaunay lived, it was his intention to complete the lunar theory by a full examination 
of all these inequalities and so add a third volume to the two large ones already referred to. 

M. Tisserand graphically described Delaunay's work in the following terms* : ' Cette 
4 th^orie est trbs interessante au point du vue analytique ; dans la pratique, elle atteint le 

* but poursuivi, mais au prix de calculs algelmques effrayants. C'est comme une machine 
' aux rouages savamment combine's qu'on appliquerait presque inde'finimeirt pour broyer un 

* obstacle, fragments par fragments. On ne saurait trop admirer la patience de 1'auteur, 
' qui a consacre* plus de vingt anne'es de sa vie &, I'exdcution rnate'rielle des calculs algelbriques 
' qu'il a effectue's tout seul. J 

* Mfe. Gtl. vol.. in. p. 232. 



CHAPTEE X. 



THE METHOD OF HA.NSEN. 

* 202. THIS chapter contains an explanation of the methods adopted by 

J Hansen to solve the lunar problem. In the earlier portion of the chapter 

,1 to the end of Art. 223 the various equations to be used are formed in a 

perfectly general manner; the next portion from Arts. 224 to 238 contains 

an explanation of the manner in which the approximations, as far as the first 

!' order of the disturbing forces, are carried out. When these have been 

i ! grasped, the extensions necessary for the further approximations follow very 

1 , easily ; they will be outlined in Arts. 239, 240. 

i] For convenience of reference, the notation is based on that of the 

1; Darlegung*; in the few places where a different notation is adopted in order 

j to avoid confusion, the differences will be pointed out. 

! The distinguishing features of Hansen's method are: (i) the angular 

t perturbations in the plane of the orbit are added to the mean anomaly of an 

^ auxiliary ellipse placed in the plane of the instantaneous orbit, its major 

\, axis and eccentricity being constant and its perigee moving in a given 

,!( manner; (ii) the radial perturbations are determined by finding the ratio of 

*; the actual radius vector to the portion of it cut off by the auxiliary ellipsef; 

(iii) the reckoning of longitudes from a departure point (Art.. *79) in the 

' plane of the orbit ; (iv) the discovery and use of one function W to find all 

} the inequalities in the plane of the orbit; (v) the perfect generality of the 

] , method which permits, without difficulty, the inclusion of inequalities from 

; every source; (vi) the completeness with which the method is worked out 

1 numerically and the close agreement with observation of the tables which 

> were founded on the theory. 

* This title refers to Hansen's paper entitled Darlegung der theoretischen Berechnung der in 

i den Mondtafeln angewandten StQrungen. Abh. der. K. Sachs. Gesell. d. Wissensclxaften, Vol. YI. 

1 pp. 91-498, Vol. vn. pp. 1-399. The two parts will be referred to as I., II. 

'. t I* 1 the Fundamenta (see footnote, p. 36) Hansen finds the logarithm of this ratio. 



202-204] NOTATION. 161 

A general explanation of Hanson's method has been given in a note by Delaunay* 
and also in two papers by Hanson, Note sur la tMorie des perturbations plantftaires and 
Bemerkungen iiber die Behandlung der Theorie der Storungen des Mondesi. 

203. Hansen's theory is much the most difficxilt to understand of any of those given up 
to the present time, partly on account of the somewhat uncouth form in which it is given 
in the Fundainenta and partly on account of the very unusual way in which the perturba- 
tions are expressed. It was first published in a series of papers entitled, Disquisitiones circa 
th&oriwni perturbationwn quae motum corporwn coelestium afficiunt and Commentatio de GOT- 
porum coelestium perticrbationibus^. The methods, although they are in general there 
worked out with special reference to the planetary theory only, are, after a few changes, 
equally applicable to the lunar theory : the chief difference being that, in the former, 
terms increasing with the time are permitted to be present while, in the latter, they are 
eliminated by the introduction of a certain quantity y. In the Fundamenta, which 
was published in 1838, the methods, as far as they refer to the Moon's motion, are fully 
elaborated and detailed expansions are given in forms ready for calculation. A method 
for the solution, on the same lines, of the problem of four bodies is added. 

In 1857 Hansen began another series of papers in which the perturbations are 
expressed in a similar manner, but the methods of arriving at the equations are much 
simpler. The first paper 1 1 refers to the planetary theory : the method is the same as in the 
Dowlegung which followed a few years later. The latter was chiefly published in order 
to verify the 'Tables de la Lime 'II which had been previously formed by an application of 
the principles explained in the Fundamenta, As far as p. 212 of the first part, the Darleg- 
ung is, however, available for the general development and it will be used for that purpose 
here; when it is a question of forming the successive approximations, the Fundamenta 
must be referred to. The early parts of the Fwndammta and of the Da/rlcgung, though 
expressed in forms very different in appearance, can, with some trouble, be seen to be 
equivalent. 

204. Change of Notation, 

In order that the expressions obtained below may be the same as those 
of Harden, a few changes from the notation of Chaps. I VIII, are necessary ; 
these changes chiefly affect the results of Art. 82, which will be required 
directly. 

Replace *p, , 3, R 

by /A*P, /*$, ^ pR, 

respectively. The right-hand members of the equations of Art. 82 must 
therefore be all multiplied by yu; the results of Art. 75 will remain unaltered. 
Also, in Art, 124, we have put R//M = jR (l) 4- JK (2) -f .,.; we shall have now 



* Jour, des fSavmts, 1858, pp. 16, 17. 
t Axtr. Nach. Vol. xv. Cols. 201-216, Vol. xix. Cols. 33-92. 
$ These are contained in various numbers of the Astr. Nach. from 1829-1836. 
These were published in the early volumes of the Abh. d. K, Slicks. Ges> der Wissensch. 
II Auseinandersetzung einer zweclcm&sngen Methods zur Berechnuny der absoluten Sttftungen 
der kleinen Planeten. Abh. Vol. v. pp. 1-148. 

^] London, 1857. Published by the Government. 

B. L. T. 11 



162 THE METHOD OF HANSEN. [CHAP. X 

where R (l) + -R (2) 4- ... retain the same meanings as before*. The mean 
anomalies w, w' will be replaced later by g, g 1 , respectively. 

Hansen also uses h in a different sense. He puts 

f ............ a). 

The significations of the other quantities present in the equations of 
Art, 82, remain unaltered. 

The instantaneous elliptic orbit. 

205. The elements of the instantaneous orbit being denoted by a, n, e, e, 
j, 6, i, the disturbing forces by /^, //&, /*3, the true anomaly by/, the radius 
vector by r, the latus rectum by I, the distance of the Moon from the node 
by L, we have from the second, third, fifth and sixth of equations (16), Art. 82, 
after replacing therein % , 3 by /*$, pX, /*3 and using the expression for k 
just given, 



de 

_- - sin 



. ,. . ~ / I r\] 
in/4- 2, ---- ) k 
J \er Wj 




. . dO 7 ^ di 

sin. r -77 = fenr sin L, -r: 

at at 

We also have from Art. 77, after putting pX for S, 

pXr = d (?ia 2 VI -e^/dt ; 
whence by (1), 

* ........................... (3). 



The last equation replaces that for c?a/^ in Art. 82. The equation for d^/dt will not b 
required since those functions of the instantaneous elements, which are used in Hanson's 
particular method of treatment, do not directly involve . ,The method being to find the 
perturbations of the mean anomaly, the equation which would be obtained by making c 
vary, is really included in the equation giving the disturbed value of the mean anomaly, 

206. In Hansen's method the plane of the Sun's orbit is not necessarily 
a fixed one. "We take as a fixed plane of reference either the ecliptic at a 
given date or the Invariable plane (Art. 28); any fixed plane inclined at a 
small angle to the , ecliptic will serve at present. As before, we define the 
positions of all lines by means of their intersections with the unit sphere. 
Let so be a fixed point on x^ the plane of reference. 

* Hansen uses instead of R, 



204-206] THE INSTANTANEOUS ORBIT. 163 



Let X be a departure point (Art. 79) on Xflj the instantaneous orbit of 
the Moon. Let Oj be the node of the instantaneous orbit with the fixed 




Kg. 8. 

plane, TT the instantaneous position of its perigee and M the corresponding 
position of the Moon's radius vector. 

We have aeO^ = 0, fljTr = -or - 0, fl^M = L. 

Let JTfi^Gr, X7r = x , 

then = v <r, -cr ~ = 

Also, from Art. 101, 

do- = cos i d0, 
and therefore 



Whence, the second of equations (2) gives 



the third and fourth of the same equations become 

sin i -T7 = A3?* sin (v <r), -IT = A3^ cos (v cr) ............ (5). 

ctt dt 

These three, with the equations for de/dt, dh/dt, given in the previous 
article, are all we shall require. 

The new clement x> liko cr, is a pseudo-element and its presence is due to tho measuring 
of the coordinate v* from a departure point. It is not a complete substitute for m 
since the point X is not completely definite ; in order to make it so, it is necessary to 
define the initial position of X. The latter is assumed to be such that when i~ 0, X coincides 
with $ ; hence X is on that orthogonal to the orbit which passes through x (Art. 79). 

The equations for de/dt, dh/dt give 
d v , de dh 



ea 

* See Art. 101, where this coordinate is called i; r In the Jfundamenta, p. 87, it is denoted 
by v t and in the Darlegung, i. p, 102 by v. 

112 



164 THE METHOD OF HANSEN. [CHAP. X 

But since l/r = 1 + e cos/, we have 

I r r I r " ( r\ 

---- e2=^_ == i4_ e cos f 

r a I r I \ I/ J 
and therefore 



Let )8i be any function of t. Multiply (6) by cos (^ - &) and (4) by 
sin (% /3i) and subtract : we obtain 

{he cos (x- ft)} = A* $ sin (/+ % -&) + l + 1 COS 



the expression for that function of the instantaneous elements required by 
Hansen. 

207. The considerations which guided Hansen in his method of treating perturbations 
are set forth in a reply* to some ill-founded criticisms by de Pontdcoulant on the Funda- 
menta. Hansen remarks that the solution of the equations which give the elements in 
terms of the time is very troublesome, requiring that six integrations be performed. But the 
quantities really sought are not the variable values of the elements but only three definite 
functions of them, namely, the three coordinates. He therefore sought for functions of 
these elements, by means of which the coordinates could be found in a more direct manner. 
It is true that, in any case, six integrations must be performed and also that some method 
of continued approximation must be used, but the ease or difficulty of carrying them out 
varies enormously according to the plan of treatment. The most numerous of the in- 
equalities in the Moon's motion are those which occur in the plane of the orbit. Hansen 
succeeded in obtaining a function W 9 the equation for which was of the first degree ; when 
this function is Isnown in terms of the time, two very simple integrations furnish the in- 
equalities in the plane of the orbit. 

One point which differentiates Hansen's methods from all others consists in the addition of 
j the perturbations directly to the mean anomaly of a certain auxiliary ellipse in the plane of 
I the instantaneous orbit instead of to the true anomaly or to the true longitude on the fixed 
I plane. This fact is sometimes stated by saying that he uses a variable time. The 
I auxiliary ellipse will now be defined : it may be looked upon as the intermediate orbit 
adopted by Hansen. 

The Aucciliary Ellipse. 

208. Consider an auxiliary ellipse placed in the plane of the Moon's 
instantaneous orbit, with one focus at the origin, Let its mean anomaly be 
denoted by ^ its major axis by 2a , where w 2 a 3 = M O = sum of the masses 
of the Earth and the Moon), and its eccentricity by e . Throughout the whole 
of the theory n , a , e Q are absolutely constant. 

* 'Note stir la thtforie des perturbations planMaires,' Aatr. Nach, Vol, xv f Cols, 201-216. 



206-209] THE AUXILIARY ELLIPSE. 165 

In the auxiliary ellipse, let H be the eccentric anomaly*, / the true 
anomaly, r the radius vector. We have then (Art. 32) 

r cos/= a (cos s - e ), JB - e sin E = n^z 9 \ 



these, after the elimination of 1, will give r, /as functions of one variable z 
and of the constants a , n , e . 

Also let 

ritf-f ...... (9); 



so that //, is the same function of n Q9 a , # that A was of n, a, #. 

Let the perigee of this ellipse have a forward motion in the plane of the 
orbit equal to n (] y, and let TT O be the longitude of the perigee from the 
departure point X at time t = 0. The longitude from X of the point whose 
true anomaly is/ will be at time t, 



Thiw ellipse being used as an intermediate orbit, we shall have initially, z = t + const, or 
n Q z s= g. The plane of the orbit is then supposed to be fixed and X will be a fixed point on it. 
Also %, 2 {) , e will be the mean motion, major axis and eccentricity, while y is a constant, 
as yet indeterminate, depending on the Sun's action in the same manner as did the constant 
c introduced in Chap. iv. When the complete action of the Sun is taken into account, the 
value of % will be </+ periodic terms, 

209. So far the only relation between the auxiliary ellipse and the 
actual position of the Moon consists in the fact that the former is placed 
in the instantaneous plane of the orbit. The connecting link is made by 
allowing the point whose true anomaly is / to be on the actual radius vector 
of the Moon, This fact, expressed in symbols, is, by Arts. 206, 208, 

y> % = v =/+ n Q yt + TT O , 

BO that r, f are the radius vector and true anomaly of the point on the 
auxiliary ellipse where the actual radius vector of the Moon cuts this ellipse. 
When z and the actual position of the Moon are known in terms of the time 
and of the constants, the auxiliary ellipse is completely defined. 

Let the actual radius vector of the Moon be, as earlier, r and put 

r = r (1 + v), 

When $, v are known in terms of the time and of the constants, the position 
of the Moon in its orbit will be known. The problem of motion in the 
instantaneous plane therefore consists in the determination of z, v and in 

* Hanson in the Darlegung, i. p. 102, where these equations are given, denotes the eccentric 
anomaly by c. 



166 THE METHOD OF HANSEN. [CHAP. X 

the determination of the meanings to be attached to the constants n , a , e 
and to the arbitraries which arise when the equations for z, v are integrated. 

210. Some idea of Hansen's method can now be given. Suppose that the initial position 
of X has been defined and that z, v have been expressed in terms of the time. The equa- 
tions (8) will then give r,/; from these, by means of the equations 



we can calculate v,r; y is a certain quantity (which is constant when the solar perturba- 
tions only are considered) to be denned during the process of solution so that no terms, 
increasing continually in proportion with the time, shall be present in the expressions for 
n Q zg, v. The first object to be sought is therefore the determination of #, v. 

The second object in view is the determination of the motion of the plane of the orbit; 
this is given by the equations (5). But in order to reduce the longitude in the orbit to 
that along the plane of reference we must know <r. The latter is found, when i, are 
known in terms of the time, from the equation 

do- .dS 



Also, when o*, i, 6 are known, the latitude above the plane of reference will be obtainable. 

The determination of 2, v will be reduced to the consideration of a function W which 
will presently be constructed ; the variables o-, i, 6 will be replaced by three others. The 
integration of the equations for W 9 #, v will furnish four arbitrary constants which will be 
determined in Art. 231, and those for the variables <r, i, 6 three further arbitrary constants ; 
the latter three will furnish the initial position of JT and of the plane of tho orbit. 
All the equations considered are reduced to the first order. The equations for z, v aro 
really of the second order, since W is determined by an equation of the first order. The 
equations for P, Q, K the variables which ultimately replace o-, i, 6 are each of th first 
order, so that the seven arbitrary constants are necessary for the general solution of tho 
equations. Six constants only are necessary to define the position of the Moon : tho 
seventh constant is that which defines the initial position of X 

The Equations for z, v. 

211. Since v, the longitude in the orbit, is measured from a departure 
point, dv/dt has the same form, when expressed in terms of the instantaneous 
elements and of the time, in disturbed or in undisturbed motion (Art. 79) ; 
hence r z dv/dt = na?*/i~e* ) in disturbed motion. From this and from the 
equations of Art. 208 we have, as in Art. 32, 

<x(l~e 2 ) - , - a (l~0o 2 ) i 

'- - ' = 1 + e cos/, QV , Oy = l 



dr nae /.,./. dr n Q a Q e Q . ? , . ? 
-=- = . .......... T sin/= he sin/, -r- = ~=== sin/ = h Q e sin/ 

dt Vl~e 2 J J &* 2 ' J 




T = r (1 + v). 



209-212] THE EQUATIONS FOR Z, v. 167 

Of these, it is to be remembered that the portions to the left, involving 
the letters r, /, a, etc., refer to the instantaneous ellipse ; those to the right, 
involving r, / a , etc., refer to the auxiliary ellipse. The connection is fur- 
nished by the two values for v and by the fact that the two ellipses lie in the 
same plane. 

Since / is a function of one variable z which is itself supposed to 
be a function of t, we deduce immediately 

p __ dv __ df dz 
^^ 



Eliminating df/dz from this equation by means of the value given for it in 
(10), we obtain 

<k_*oP_ _y__(r\* an 

"VT - --" - - _, i i t VJLJLy. 

dt hr* \/l-e 3 \a / v x 

Also, from the last of the same equations, 

v Y 2 i . 0^ y l+ecos/ A ( v 



since 7i ^ = /i = /^L Substituting in (11) and putting /=/+ 7t y + TT O 
we find 



where F 1-? + 2 ^ ......... (13). 

h h Q a 1 - e<? v y 

These furnish the required equation for z. 

212. To find the equation for v we have, since r is a function of z only, 



Substituting from (11) for dz/dt and observing that 1 + z> r/r, we obtain 

^ = H4. \ ^ ^f 4. " 1^ 4. ?.(" ^ 

^ v ; fa ^ 

= _ h ^ 4. 1 ^ 4. 

^r d^ r d^ 

1 rr'y* 1 ^/v* 

Put for - , -5- , = , jr their values from (10). The first and second terms 
r cLz T cLt 

of the latter expression for dv/dt become 

/* l-fecos/, . -14-0Q cos/. . 
- - ........... ........ *- - 



. 

~ Q 2 a dz 

^. / \* 

? ^ \a / 



168 THE METHOD OF HANSEN. [CHAP. X 

or, since h^/h? = a (1 - e 2 )/a (1 2 )> they are 

^._ {- e sin/(l -f e cos/) + e sin/(l + e, cos/)} ....... (14). 

But, differentiating (13) partially with respect to #, we have (since z is 
only present explicitly in r, /), after inserting the values of drfdz, dffdz given 
in (10), 



sm 



which, since /+ ^ojrf 4- TT O % =/, n a = A Vl S , becomes 

-O ^x) 



+ * cos /) - (1 + o cos/) e sin/}. 

Comparing this with the expression (14) which contains the first two 
terms of dv/dt, we obtain 

<*"_ i 
a - t 

the required equation for v. 

213. It is easy to see that when the two sets of elements coincide, W, 
v } y vanish ; further, if the disturbing forces vanish, dz/dt = 1. The quantities 
TT, v, y are therefore at least of the first order of the disturbing forces. 
Hence, in the expression (12) for dzjdt, the third term is of the order of the 
square of the disturbing forces and it may be neglected in the first ap- 
proximation; in the fourth term we can, to the same degree of accuracy, 
put n^z = n(f-\- const. =g: this amounts to neglecting the disturbing forces 
in the coefficient of y. The same remark may be made concerning the 
second term in the expression (15) for dv/dt. Hence, the principal parts 
of the equations for z, v depend on W and this function must now be 
expressed in terms of the disturbing forces. 

So far the equations (12), (15) are purely algebraical results obtained by the com- 
bination of two sets of elliptic formulae and connected by the single fact that the longitude 
in each orbit, reckoned from one origin, is the same. One mean anomaly is therefore 
a function of the other, but no supposition, involving any relations between the two sots 
of elements, has been made and the results would be equally true for any two sots of 
elements in one of which the motion ^of the perigee is directly proportional to that of the 
mean anomaly*. 

* See Hansen, Weber die Anwetidung osculirender Elemente als Grundlage der Berechnung tier 
Storwigen eines Planeten, und iiber die unabhlingigen Elemente der " Fwidamenta nova etc" Astr. 
Nach. Vol. xvni. Cols. 287-288. 



212-215] THE DEFINITION OF W. 169 

The equation for W. 

214. The expression (13) for W is a function of the variable elements 
h, e, %; it also contains t through the term n Q yt and through r,fihe latter 
being functions of z and therefore of t. But since the equations which 
express A, e, % in terms of the disturbing forces are given by their differ- 
entials, it will be better to form dW/dt and then to find W by a single 
integration, instead of performing the three integrations necessary to find 
h, e, % directly and then substituting their values in the expression for W. 
It will now be shown that, in performing these processes, r > f, which 
are functions of z and therefore of t } may be considered constant. (See 
Art. 104.) 

Let T be a constant and let f, p, <j> denote the values of z, r, f when 
T replaces t: ", p, <j> are then the same functions of the constant T that #, r,f 
are of t. Let W denote the value of W when p, </> are put for r, f. 

Now the expression (13) for W may be put into the form 
W- LI + L 2 r + L s r cos/+ L^r sin/ 

where L l} L^ X 3 , 4 do not contain r, f, being functions of A, e } %, n & yt only. 
We may write this 



since T is a constant. After the integration, r must be put equal to t. 
Hence we need only consider the function If, in which p, ^ are constants. 

We have, substituting p, <, ? for r,/ ^ respectively, in equations (10), 



215. The definition of W gives 

T rr i h () ~h p 1 + e cos (^ + n,yt _ Try- %) 

|^/ = X -- -_ - ---- ..... - - * 



Therefore, considering p, <^ as constant and remembering that a (]) e^ n Q) h Q , 
TT O are always constant, 

AW _ A, dh 2_ p_ dh 2 .......... p _d f/ (v- 

^ " 77 ^ + //o Oo + - { V% 

where & = ^ + n ^ + TT O . 



170 THE METHOD OF HANSEN. [CHAP, X 

The equations (3), (7) are immediately applicable. By means of them 
we find, 

dW 



- A) + 2 in ( X - A). 



But since p = A 9 a (1 e 2 ) = A 2 a (1 S ), i = a (I - ei*), 

/+ % - A =/> Wojtf + 7r ~ - W y$ - 7T =/- $, 

we obtain 



sn - + 2,,, cos - < 



In order to get the last term of this expression into a suitable form, 
differentiate (17) with respect to This gives 



8? 

which, by means of (16), becomes 



Substituting for e sm ($ + n^ + ?T O - #) from this result in the Itwt 
expression for dW/dt and rearranging the other terms, we obtain 




which is the required equation for F. When F hasjboen found from this 
equation and thence, by putting r = i, tho value of IT, tho equation*! (12), 
(15) will give z> v. In the process of integrating (18), J5, $ are, by Art. 214, 
to be considered constant. 

Several methods of arriving at this expression for d W/dt have boon given. On pp. 41-43 
of the Fundamenta, Hansen arrives at It by a direct transformation from tho equatloriH of 
variations of the elements, but the form obtained is slightly different from (IB) above ; the 
latter becomes the equation given in the DarUgmg, I. p. 107, if Utfiv, d&fir IHJ flulwtitutod 
for %r 9 $. The method given above is based on one by Zech*. la tho J)arUffunff 1. 1^ 

* Neue Ableitung der Hawerfschen Fundmiental/ormeln fttr die Ikrechnww <kr Httintnatn 
Astr. Naoh. Vol. XLI. Cols. 129-142, 205-208. 



215-216] THE NUMBER AND SIGNIFICATION OF THE CONSTANTS. 171 

104-107 * is another investigation obtained directly from the fact that the coordinates and 
the velocities have the same form in disturbed and in undisturbed motion. In all these 
methods Hansen's theorem, enunciated in Art. 104, is used ; Brunnow f gave develop- 
ments of a different form which suggested that this theorem was not necessary (see M. N. 
R. A. S. 1895, No. 2). Earlier, Cay ley J had also given a method of obtaining the equations 
of the JFmdamenta which assisted in clearing up several difficult points in that work. 

216. Six constants have been introduced with the auxiliary ellipse, namely, a , # , n Qt y, 
7r and that attached to n Q z (which in undisturbed motion is of the form n Q t-^G ). These 
are not all independent and arbitrary. The two , n Q are connected by the equation 
w 2 ao 8=B /Aj while y will be seen to be a certain constant necessary (like c in Chaps, iv. vu.) 
to put the solution into a suitable form. The number of independent constants is therefore 
four ; the other three arise from o-, i, 6 (or from P, Q, K ). Hence, as far as the motion in 
the plane of the orbit is concerned, we have the necessary number (Art. 210). The four 
new constants, which will be introduced when the equations for #, v are integrated, can be 
determined at will, and they will be so determined that the meanings of ?i , e , TTQ, <? in 
disturbed motion may be rendered independent of the method of solution adopted. As 
numerical values are used by Hanson, it is necessary to know beforehand what signifi- 
cations are to be attached to n Q , e Q> h Q . These are, however, better explained when the 
equations for #, v have been integrated : the definitions will be found in Art. 231 below. 
It is only necessary to state here that 7* , e differ from h 9 e by quantities of the order of 
the disturbing forces. 

It will therefore be seen that the elements with suffix zero are not the purely elliptic 
values of the instantaneous elements, if we understand by ' purely elliptic values ' those with 
which we start. On any development with the latter as a basis, the observed mean 
motion, for example, would no longer be denoted by a single letter but would consist of the 
purely elliptic value together with a series of constant terms clue to the disturbing forces. 
This was seen in de Ponte'coulaut's theory where the new arbitrages arising during the 
integrations were used in such a way that the mean motion might be denoted by n. The 
same thing occurred in Delaunay's theory, but there it was necessary to make a direct 
transformation in the final results. Hansen, like de Pontdcoulant, keeps the arbitraries 
(denoted by &, below ) which arise in the integrations, for the purpose of defining %, e . 
These remarks are necessary for a clear understanding of the three sets of elements used in 
the J^undamenta. There Hansen denotes by (a), (n\ etc., the quantities denoted by & , ^ , 
etc., above and by a , w , etc., the purely elliptic or initial values of c&, %, etc. (the latter 
being the instantaneous elements), that is, the values of a, n, etc., when the disturbing 
forces vanish || . With the notation used in this chapter, and in the Darlegung^, a , %, 
etc. implicitly contain terms due to the disturbing forces. 

* It was also given by Hansen in the Astr. Naoh. Vol. LXII. Cols. 273-280, Neuc Ableitung 
meiner Fundamentalformeln flir die Berechnung der Storungen, 

t Saturn etc., nebst einer Ableitung der Hansen'schen Fundamentalformeln. Astr. Nach. Vol. 
LXIV. Cols. 259-266. 

$ On Hansen's Lunar Theory. Quar. Math. Jour. Vol. i. pp. 112-125 ; A Memoir on the 
Problem of Disturbed Elliptic Motion. Mem. E. A. S. Vol. xxvn. pp. 1-29 ; A Supplementary 
Memoir on the Problem of Disturbed Elliptic Motion. Mem. B. A. S. Vol. xxvin. pp. 217-234. 
These are also found in Ms collected works Vol. in. pp. 13-24, 270-292, 344-359. 

Art. 230. 

|| Fundamenta, pp. 62, 64. 

II Darlcgung, I. p. 102. 



172 THE METHOD OF HANSEN. [CHAP. X 

It is necessary to point out that n^y is not the mean motion of the Moon's perigee along 
the true ecliptic although it accounts for the greater part of this motion. It is the mean 
motion of the perigee in the orbit. A small correction, which depends on the mean motion 
of the Moon's node along the ecliptic and on the square of the inclination, has to be applied 
in order to obtain the mean motion of the perigee along the ecliptic. See Arts, 217, 237. 

One great advantage of Hansen's method of computing the longitude in the plane of 
the orbit is that the inequalities produced in z by the motion of the plane of the orbit are 
necessarily very small. Since the force 3 does not occur in the equations for 0, i/, the 
inequalities produced in these variables by the motion of the plane of the orbit must all 
be small quantities of the order of the square of the disturbing forces at least. 



The Motion of the Plane of the Moon's orbit. 

217. Definitions. It is necessary now to define the variables by means 
of which the motion of the plane of the Moon's instantaneous orbit is found. 
We suppose here that the Sun's orbit is not fixed but that it is moving in a 
known manner. 

On the unit sphere, let XQ.M, X'&m! be the orbits of the Moon and the 
Sun respectively. Let X' be a departure point on the Sun's orbit, defined in 
the same manner as X was. Let X1 1? Ii/ be the ascending nodes of the orbits 
on the fixed plane of reference. "With the notation used in Art. 206 we 
have, if accented letters refer to the Sun's orbit, the following old and new 
definitions*: 




Fig. 9. 



* The angles denoted here by \f/> f are called by Hansen 0, ^ respectively. Fundamenta, 
p. 84. Darlegung, i. p. 110. The change is made to prevent confusion with the letter used 
earlier. 



216-217] QUANTITIES DEFINING THE PLANE OF THE ORBIT. 173 

Hence (Art. 101) 

Ha- = cos i d0, dcr' = cos i' d6 f . \ 

Also, put _p =s sin i sin <r, p' = sin i' sin </, > .................. (19). 

q = sin i cos cr, q' = sin i' cos </ J 

All the quantities denoted by accented letters, except -\|/, are supposed 

known, Bince they refer solely to the Sun's orbit. 

Let N, K be defined by the equations 



where 7r ' denotes the distance of the Sun's apse from X' at time t = : a, 9? 
will be so defined that JV, J5T contain no terms directly proportional to the 

time, 

We have 



Since X'fl-^, and since JV", J5T contain no terms directly proportional to 
the time, the quantity - w (a + 77) represents the mean motion of the argu- 
ment ^ ', that is, - Mo(a-h^) is ^ mean motion of the Moons node along the 
true ecliptic. 

Again, if TT be the perigee of the auxiliary ellipse, the mean motion of 
the Moon's perigee, when reckoned along the true ecliptic and then along the 
orbit, will be the same as that of TT, when reckoned in the same way. Now 



X'Q. + IMr 

by Art, 208. The mean motion of ^ - ^ is - 2w^, by the second of equations 
(20). Hence, the mean motion of the Moon's perigee along the true ecliptic is 

NO (y - 2 ^)< 

In flonoral, y, , rj are constant quantities. The actions of the planets, however, produce 
mall aoooloratioiu* in tho mean motions of the perigee and of the node, that is, they 
produce terms dependent on , *> ... These can be taken into account by putting to y, 
U "o* the JegralH ^Jy* i^Ja* J,* respectively. The differentmls of these 
qmntiSB with respect to the time will then be still denoted by ^, n a, ^ respec- 
tively*. 

Hanson introduces the quantities <I>, * to denote the angles *-r, *'-^t. They are 
hwW merely intermediaries in his development of the equations obtained below : as 
Z 2 not neceBBary in the proof given here, they .ill not be used m this sens, 
He UBCB tho letter *, in another place, to denote an entirely different quantity. See 
Art. 230 below. 

* Fmdame,nta, pp. 51, 97, 98; VarUgung, i. pp. 103, 112. 
t Fundamenta, p. 82 ; Darlegung, i. p. 112. 



THE METHOD OF HANSEN. [CHAP. X 



The equations satisfied by P, Q, K. 

218. In any spherical triangle ABO whose sides, denoted by a, b } c and 
angles, denoted by A, B, (7, all vary, we have 

dC = dA cos b dB cos a -f dc sin A sin 6 , 
db = dc cos J. + da cos (7 + d.B sin (7 sin a, 
da = dc cos 5 + db cos (7 + dA sin (7 sin 6. 

To prove these, draw ED, AD 1 so that the angles CBD, CAD' are each equal to a right 
angle. "We have, in the triangle ABC, 

cos 0= -cos A cos B +sin A sin Jocose ......... (22), 

and therefore, when the sides and angles all vary, 
- dCsin C=dA (sin A cos j3-hcos A sin B cos c) 

4- cZ? (cos A sin 5 + sin ^i cos J5 cos c) - dc sin J. sin 5 sin c. 

The coefficient of dA in this equation is equal to 
-cos (90 --4) cos (180 -B)+ sin (90-4) sin (180-5) cose 

=cos AD'B=sm Ccos b } 
by the spherical triangles ABD', AD'C. m &' 10 ' 

Similarly, by considering the triangles ADB, CDB y we prove that the coefficient of dB 
is equal to sin C cos a. Also, since sin B sin o= sin b sin (7, the coefficient of c?o i equal 
to sin A sin 6 sin 0. Substituting and dividing by - sin C we obtain the expression given 
above for dC. 

Again, in the result for dC, put 4, *r-5, -# for a, b, c and TT - a, r-6, TT-C for 
4, jB, <7, respectively. We immediately deduce, from the known property of the polar 
triangle, the value of dc in terms of da, db dC, and thence, by interchanging the letters, 
the values of db, da, given above. 




For the triangle Qf^fl (fig. 9), put A = i', B = 180 - i, C = J, a = ^ - 
ifr'-^c^ff-ff. We obtain 

= - di' cos (^' - or') + di cos (^ - cr) + (cZ^ - d9 f ) sin i 7 sin (^ - o-'). . .(23), 
dty' - do- 7 = (d<9 - dff) cos i x + (dty - do-) cos / - di sin / sin (^ - a), 
d^r - do- - (eZtf - d0') cos i + (dty f - do- 7 ) cos /+ dA' sin J sin (>/r 7 - o- 7 ). 



Substituting for do- 3 da their values cosidO, cosi' df)', and transposing, 
the second and third of these equations become, 

dty' d\|r cos J=d6 (cos i 7 - cos i cos J) - di sin J sin (^ - <r) 

= d6 sin i sin J cos (-f - <r) - ^ di sin J" sin ty - o-), 

d->|r - <&|r 7 cos /= dff (cos i - cos i 7 cos /) + ^ 7 sin /sin (^ 7 - o- 7 ) 

= - d6 f sin i 7 sin /cos (f x - a- 7 ) + dA' sin /sin (^ 7 ~ </) : 



218] THE MOTION OF THE PLANE OF THE ORBIT. 175 

the second lime in each case being obtained by the successive application of 
the formula (22) to the triangle O'A& 

But we have, from (20), 

+ d^) (1 - cos J) = - 4 (dN -f n acfa) sin 2 |</, 
- top) (1 4 cos J) = - 4 ((ZJ? - noqdt) cos 2 J J. 



Therefore, substituting these values in the sum and difference of the two 
previous equations, we obtain 



dN 4- n Q adt = 4 cot \J {di sin (^|r <r) d# sin i cos (^ cr) 

- di f sin (i|r f - </) + dff sin i' cos (^' - <r')}, 
-o*7 di = i tan \J { cZi sin (^ cr) + rf^ sin i cos (^ cr) j 

di' sin (T// </) -f cZ^' sin i' cos (^ </)} J 



The equations (23), (24) for cZJ", dJV", dJf are purely geometrical results ; 

it IB necessary now to introduce the disturbing forces. 

We deduce immediately from equations (5), 

d% / . v d6 . , . , . x r , . . 

"^7 cos (Y <r) + -3- am i sin (y cr) = /^r cos (v y), 

-y: sin (ty a*) -j- sin i cos (^ <j) = ^3^ sin (v i|r). 

tttJ CtC 

Also, differentiating the expressions for p' t q' in (19) and remembering 
that dor' sse cos i' dQ', we obtain 

dp' = di' cos H sin cr -f d0' cos i' sin i! cos </, 
dg' SB di' cos i x cos <r' d^' cos i x sin i' sin cr'. 

Whence 







Dividing the equations (23), (24) by (ft and using the results just 
obtained, we find, since (fig. 9) d& sin i' sin (^ - </) = d^ sin i sin (ifr - o-), 



1*76 THE METHOD OF HANSEN. [CHAP. X 

219. The final transformation is made by changing from the variables 
/, N to P, Q, where 

P = 2 sin JBm(N-N<>) 9 6 = 2 sin^/cos (JT- JST ) ...... (26). 

In these, N denotes the constant part of N. We deduce 

dP = dJcoB | J sin (JV - JV ) + 2^3111 J J cos (F - JV ) 
= A/cos J /sin (JT - -ZV ) + Qdtf, 

dQ = dJ cos 4/cos ( jf - NO) - 2<iZV" sin | J" sin (AT - #) 



Substituting for dJ, dN their values just obtained, we find 

dP ^ ,^ ., T- - / . TIT- -XT N coslJ/d 

r 



(27), 

where /*' = ^' $ + JV"o* 

The equations (25), (27) for K, P, Q, are those required. The angle \ 
may be eliminated from (25) by means of the equation 

rt . T fdp ./ dq f . A / r\&p r -r>dQ f \ / 
2 sin * J ( -4- cos Y 1 - sin y 1 = ( y --=-- 4- JT -y - 1 cos ^ 
\ at at J \ at at J 



which follows immediately from the definitions of P, Q, /i, 7 . 

When all the disturbing forces are omitted, we have K, P, Q constant 
and therefore N, J constant, for a, 77 are of the order of the disturbing forces. 
Now, by definition, N Q is the constant part of .JT; let J" be the constant part 
of / and KQ that of K. Hence : 

The first approximation to P, Q, K is given by 

P = 0, Q2siniJo, J5T = J5To (23). 

These values correspond to fixed positions of the orbits of the Sun and 
the Moon. 

The quantities p> q, defined in Art. 217, have not been used. It is evident from the 
definitions that the equations for P, $, K should be symmetrical (except with regard to 
signs) with respect to the quantities referring to the Sun and the Moon ; the parts of these 
equations dependent on 3 can, in fact, Tbe exhibited in terms of dp/dt, dq/dt by expressions 
similar to those which contain dp'jdt, dq'/dt. The reason for not expressing them in this 
form is that the latter are known functions while the former are functions of the quantities 
we wish to find. It will be seen from fig. 9 that p 9 q are the sines of the latitudes of 
the points JT, F below the plane of reference and that p r , q' are those of Jf', Y' below the 
same plane. 



219-220] DEVELOPMENT OF THE DISTURBING FUNCTION. 177 

Since J is small and since NN Q contains only periodic terms dependent on the 
disturbing forces, equations (26) show that the principal part played by Q is to bear the 
periodic variations in the inclination of the Moon's orbit to the ecliptic. The quantity P 
is small and it carries chiefly the perturbations included in N\ they are multiplied by the 
small quantity sin J,/. Also, by equations (21), N - ^+/t -/t, N^N-K^+K contain 
the periodic parts of the motions of the Moon's node, along the Moon's orbit and along the 
ecliptic, respectively ; the difference between these is very small. 



The Form of the Development of the Disturbing Function. 
220. We have from Art. 107, after replacing R by pR, 



As before, 8 is the cosine of the angular distance between the Sun and Moon 
and therefore, by fig. 9, 

8 = cos HM cos Qm' + sin flM sin Om' cos J. 

In order to obtain a perfectly general development of R, the auociliary 
(not the instantaneous) ellipse, with its variable mean anomaly n c #, is used 
for the developments in the plane of the orbit, and the instantaneous values 
of i, 6, a (or of the variables replacing these) for those of the plane of the 
orbit. For symmetry, we suppose the Sun's motion to be defined also by an 
auxiliary ellipse with a mean anomaly w V, the perturbations of its radius 
vector being denoted by v and the mean motion of its auxiliary perigee in 
the plane of the orbit by n Q y'. 

To develope R we have 

+TT O ^, 



by Art. 209. Substituting, we obtain 



v ^ . 

R = _ __ ^ ( |,S- - J) + - 

where 

S = cos (/+ n,yt + TT O - ^) cos (/' + n Q y't 4- W - ^' 



TT O - ^) sn + w ^ + < - ^ ') cos J". 

Since the variables /', r', v 7 , referring to the Sun's orbit, are supposed to 
be known functions of the time, R is thus expressed as a function of t and of 
the unknown variables f, r, v, -^, ty', /. 

Let ft) = n ^ + 7To-'f, a>'**n<>tft + 7ro'-'ty' ............... (29), 



B. L. T. 



178 THE METHOD OF HANSEN. [CHAP. X 

so that, by fig. 9, CD, co' are the distances of the apses of the auxiliary orbits 
from the common node 1. We have then 

S = cos (/+ ) cos (/' + o>') + sin (/+ o>) sin (/' 4- CD') cos / ... (30). 

Now / r are, by equations (8), the elliptic true anomaly and radius vector 
corresponding to a mean anomaly w #, with constants a , n Q , ; in the same 
way, f', r f correspond to a mean anomaly H V with constants a Q ', w ', e Q '. 
Therefore, putting mfa^/fia^ = m*, we find 



where $ has the value (30). 

Comparing these values of It, S with those given in Art. 124, we see, 
by the remarks just made, that the method of development, given in Art. 125, 
will be available if we simply replace a, e, a' ', e', w, w f by a , > <V> e*> n^> 
TI^Z', respectively, and multiply 

Jjw by (1 + z/) 2 /(l + z/) 3 , R by (1 -f j/)/(l + v')\ etc. 

Finally, to take into account the correction noted in Art. 7, we must further 
multiply R by (E-M)I(E + M). 

If we look at the developments of 1/r', I/A given in Art. 5, it is not difficult to see that 
the general form of the multiplier of jRtO, necessary when the force function given in Art. 8 
is used, is 



This expression was first obtained in an indirect manner by Harzer*. 

221. We shall thus have the development of R in a perfectly general 
form : it will be expressed as an explicit function of the unknown variables 
#, v, o>, a/, J, and of t through the known variables /, v the rest of the 
symbols present being absolute constants. It will be shown later how 5$, % 
are obtained from this development of R. In order to find R in a form 
suitable for the determination of the motion of the plane of the orbit, we 
must transform from the variables o>, a/, J to P, Q, K. 

By equation (14) of Art. 124-, we see that o> and <*' will only occur in R 
in the form of multiples of &> + <</, and that a term containing in its argument 
ji (co + a/), where j l is a positive integer, will have its coefficient at least of 
the order sin# JJ, Hence, all terms in R are of the form 



where J. contains only integral powers of e Q , e Q ', a /a ', sin 2 \J, and j, /, j{ are 
positive or negative integers. 

* Ueler die Riickwerkung der von dem Monde in der Bewegung tier Sonne erzeugten Storungen 
auf die Bewegung de$ Monties, Astr. Nach. Vol. gx3fin. Cols, 193-200 t 



220-222] EXPRESSION OF R IN TERMS OF #, v, P } Q, K. 179 

But, from (29) and (20), we deduce 

o> + o>' = n*t (y 4- y' + 2a) + 2^, GO - a>' = n Q t (y-y f - 2??) 
and therefore the general term is of the form 

cos {/3 + 2j x (JV- 



where 

=W +/wV +j L nt(y + y 1 + 2a) 
Also, from (26), we have 

4 sin 2 y = P 2 + Q 2 , 2 sin 2 y sin 2 ( F - #,) = PQ, 

4 sin 2 y cos 2 ( N - JV ) = Q 2 - P 2 . 



The expression of R as a sum of periodic terms therefore contains the five 
unknowns n^s 9 v, P, Q, K ; of these, the variables ??- #, K occur in the argu- 
ments only and the variables z/, P, Q in the coefficients only. 

It is to be noticed that, since rc #, v enter into R only through v, r, we can 
express R as a function of the time and of the five variables r, v, P, Q, K. 
We then have 5)3 = 3jR/3r, Xr dR/dv, and the expressions for the disturbing 
forces may therefore be directly inserted into equation (18). But as the 
latter has to be solved by continued approximation, this process would 
necessitate the expansion, of the two expressions dR/dr, dR/dv. In the first 
approximation, the latter can be transformed into the differentials of Jti 
with respect to certain quantities present explicitly in the expansion of -R. 
We shall first find relations between 3 & n <l the partial derivatives of R with 
respect to P, Q, K, since the results for these are quite general. 

222. The general expression for R is 

R = ^L (1 ^ X %'JKZ' + ZZ '\ = (L-. r \ 
"~ fj, VA F" 3 " y fj, \A r'V ' 

where A 2 = (X - XJ + ( F- YJ + (Z- ZJ = r 8 + r" 2 - 2rr'$ ; 
(X, F, Z), (X', F', #') being the coordinates referred to any axes. 
From the first form of expression, we deduce 



when we take the axis of Z perpendicular to the plane of the Moon's 
instantaneous orbit. And, from the second form of expression for R } 



m 



122 



180 THE METHOD OF HANSEN. [CHAP. X 

Hence 3 = 3$ > 

or, since (fig. 9) Z' = r' sin (v' - ^') sin J", the relation becomes 

223. Again, from Art. 220, 

$ = cos (v ty) cos (y' ^') -h sin (-y ^) sin (v' -^r') cos J 
= cos 2 1/ cos (v t;' -^ + t^O + sin 2 i/cc 



and, by equations (20), 
-,^ + ^r'ss: 2^- 7r 
Therefore, as JV", JT enter into $ only through -\^, ^ 



Again, since / only occurs in >S in its explicit form, 

rW 

%~f = - sin / sin (v ^) sin (v i/r'). 
dt/ 

But we easily deduce from (26) by differentiation, 



Equating the two values of 9/S/9/, we obtain 

oa on 

J r sin(w-^)sin(v / -^ / ) ...... (33). 



Also, from the values of dS/dK, dS/dN, we find 

o<^ on 

^ cos 2 1/- ra sin 2 1 J = - sin 2 / cos (v - ^r) sin (v' - ^')- 

Multiply this equation by Q, the first equation for dS/dJ by P sin / and 
add. We obtain, after using the values (26) of P and Q in the right hand 
members, 

P l^sin J+ Q MOO* |/_ Q || sin , ^ 

= - 2 sin 2 / sin %J cos (v - ^ - 2V" + JV ) sin (/ - ^')- 
But by (32), 



222-224] THE FIKST APPROXIMATION TO R. 181 

and therefore we have, after dividing by P 2 + Q 2 = 4 sin 3 -JJ", 

|p cos 2 ^jr-JQ^ = - sin /cos,}/ cos (v-^-N+N*)&n(v' -tf)...(M). 

In a similar manner we can deduce 
^ cos 2 1/+ JP ^ = sin /cos f J" sin (v -^ - N 4- jZV^) sin (t/ - ^r 1 ).". .(35). 

Since P, Q, JT enter into E only through S 9 we have 

f\ T) ^ T> o CY ^k D Ci D d O d Z? Ci "?? ^1 Cf 

OJtti C/jti/ C/O (/jCw C/XL C/yO U-v C7-tb C/O 



Wheace, multiplying the equations (33), (34), (35) by dR/dS and substituting 
in their right hand members the value ofdR/dS given by (31), we find 



.(36). 



cos 2 |J+ JP = r3 cos |/sin (t; - ^ - 



These results, which are quite general, are put into the form in which 
they will be useful in Art. 235. 

The First Approximation to R and to the disturbing forces. 

224. A limitation of the general value of R will now be made by 
supposing the orbit of the Sun in its instantaneous plane to be an ellipse, so 
that nj z' = gf =* njt + c{ (where c ' is a constant) and z/ = 0, r' = r'. The 
perturbations of the solar orbit, thus neglected, only produce small effects on 
the motion of the Moon. 

The first approximation to R is obtained by substituting for the coordi- 
nates their elliptic values, that is, we put 

n^^g^n^t + CQ, i/ = 0, K = K* 9 N = N 0) J r =J r ; 
whence co 4- o>' = n g t (y -f y' + 2cc) 4- 

co-a> f = n Q t(y-y - 2*]) - 
in which y\ being a known constant, is retained : y, a, ^ are constants to be 
found. We have also, by (26), 



The first approximation to R is therefore expressed explicitly as a 
function of the time, the arguments being sum's of multiples of the four 
angles g, g', co, &/, and the coefficients being expanded in powers of e Q , '> 



182 THE METHOD OF HANSEN. [CHAP. X 



sin 2 4/o; fl&o/flo'- For the motion of the plane of the orbit when R is 
expressed in terms of n Q 2 } v, P, Q, K, we also put n Q z = g, v = Q. The partial 
derivatives of R with respect to P, Q must be formed before we give to P 
the value and to Q the value 2sin^J" . As regards J5T, we evidently have 



In the 'jfandamenta (p. 81), Hansen used the derivatives of jR, with regard to p, q as 
though R were a function of the/ozw variables r, v, p, q only. The difficulty is merely that 
Hansen has attached a meaning to dll/dp, dlt/dq which is unusual. The point was cleared 
up by Jacobi *. 

225. To express the disturbing forces 5)J , 3' w, terms of the partial 
derivatives of R. 

Denote by < , $ the values of the disturbing forces *p, X when the first 
approximation to R is used. In the terms multiplied by quantities of the 
order of the disturbing forces, we can put r ~r =r ,/=/=/ , where r ,/ are 
the values of r,/ when n*z = g. Since, in the first approximation, e 9> g enter 
into R only through r,/and since g enters here in the same way that nt -f e 
entered in the expressions of Art. 75, the second and third of equations (4) of 
that article are available. We therefore obtain, with the necessary changes 
in notation, 



...(37). 




(1- 

By means of these results we can express 5)J , $ in terms ofdM/de Q , 
Also, since o> enters into R only in the form v + w, wo have 

cr 9JK 9J? 



Approximation to W. 

226. The general process of solution adopted by Hansen is one of 
continued approximation. There has been found, in Art. 215 a general 
expression for dW/dt which contains 5)}, . Now W is of the order of the 
disturbing forces and all the terms present in the expression for dW/dt are 
implicitly functions of z, t . 

* Auszug easier Schreiben etc. Crelle, Vol. XMI. pp. 12-31. 



224-227] THE FIRST APPROXIMATION TO W. 183 

Put 

77 # = n Q t 4- C 4 S# = g 4- UQ&Z, n f = n^r 4 C 4- S = 7 4 '^oSf. , .(39), 

and, in finding the first approximation to TF, neglect 8#, 8f. Let p , <^o,/o, 
TP > etc. be the values of p, <|>,/ W, etc. when for n<>z 9 n ^ are put ^7,7, respec- 
tively. Also, as h, r differ from A 0} r by quantities of the order of the 
disturbing forces, we can, in the terms multiplied by quantities of that order, 

Let 

r () A J^^^ 

The equation for TF" may be written 



But since y is of the order of the disturbing forces, we can, in the coefficient 
of y, neglect F and put A = V Hence, the first approximation will be 
obtainable from 



227. We shall now transform 1\ so that the values of ^ , X , given 
by (37), (38), may be inserted. The suffix zero, which occurs in every symbol 
present in T Q) will, for the sake of brevity, be omitted until the end of this 
article. 

With this understanding we have, by the elliptic formula (16), 



Therefore 






> - ?(J cos/ 



] 



+ 



* - e- 

2 P. CO ?_* + ^ r acos / cos / +e 



a v 



a^E sin/1 
J 



2psin<f) r/asin/" sin A ~ 00> /-I 
- /,----> -- - H- i --- *^ ~ a =P C08 / 
a Vl - e 2 [\ r l ~ e "' -1 



184 THE METHOD OF HANSEN. [CHAP. X 

We deduce, from (37), 

ae 



On the left-hand side of this equation put Xr = dR/dco and substitute in 
the second line of the latter expression for T: for the first and third lines, 
the equations (37) are available. We obtain, on restoring the suffixes, 

T = - ^dflpJ! JL I (%P_ Q ?2? ^o 4. Q ^ f^bQ ___ * ___ 




This expression is now very easily calculated from the first approximation 
to the value of a Q R, for R has been expressed as a sum of periodic terms 
whose arguments contain g, co and whose coefficients are functions of e Q . 
The portions dependent on R are thus expressed by means of periodic series 
with constant coefficients and with arguments of the form fit 4- /3'. 

228. Let, for a moment, 

m _ v , /PQ cos ftp 8 \ , e posin <#> ^ 

JL JJ -- h w Co IT . ........................ =r JGt , 



in which the signification of F', G f , JJ' is evident. Since aJR is expressible 
by means of cosines, and since g, a> occur in the arguments only while # 
occurs in the coefficients only, J", &' will be developable in sines and H' in 
cosines of angles which are all of the form yStf-f/3' (ft, j3 f constant). 
Here, ftt + fi' is formed of multiples of the angles g, g, a), &>', all of which, 
owing to the introduction of a, ??, contain t ; also, /3' = when /3 = 0, for 
n Q) n ', etc. are supposed to be incommensurable with one another. 

Since /> , ^> are the radius vector and true anomaly corresponding to a 
mean anomaly 7, we have, by the theorem of Art. 43, 



The first sign of summation refers to the angles pt H- $' and the second to the 
integral values of j; a is the symbol for the general coefficient corresponding 
to the angle fit -f ft?. The extra labour, caused by the presence of the angle 
7 (which does not occur in pt +'), is compensated by the case with which 
the other coefficients can be obtained, when the values of C , 1} 0- l9 for all 
values of fit + /3', have been calculated. See Art. 43. 

When T Q has been thus found in terms of the time, we obtain from (40) 



227-229] DETERMINATION OF y. 185 

But, by equation (19) of Art. 43, we have 



<*y 
Hence 



-- -j t = S S a, sin 0' 7 + /a + /3') + S ~7=^JRj + c/ - c'-j sinj % 
n Q av -oo i Lvl . J 

where the terms for which /3 + /?' = are written separately, their coeffi- 
cients being denoted by c/. 

229. Integration of the Equation for W Q and Determination of y. 
We have on integration, since 7 is constant during the process, 



[-7=^ 

i LV1 e " J 



~oo p 

where the additive arbitrary constant, denoted by ^(7), may be a function 
of 7. 

Putting r = t and therefore 7 = g y we find 



'-c'-j 



- 

vl e - J 

......... (41). 

Now y was specially introduced in order that expressions of the form 
t x periodic term might be eliminated, if possible. Determine y so that the 
coefficient of t sing vanishes. This gives 



^V + d'-c^^O ..................... (42), 

The corollary to the theorem of Art. 43, applied to c/ - c'_^, then shows that 

2jo^/VT^li? + c/ - c%- = 0. 

Hence the coefficient of t sinjg vanishes, and therefore all the terms having 
the time as a factor disappear from W Q . 

The equation (42) gives a first approximation to y* That it should be 
capable of being determined so that all terms of the forbidden form may be 
eliminated, is sufficiently evident from what has been said in previous 
chapters. All that now remains is the determination of the form of F(g), 
the function which contains the arbitrary constants. 

* Fundamenta, p. 191. The determination of y, given in the DarUgimQi * P' S38 > is no * 
directly applicable here because Hansen is there simply performing a verification of his 
tables. ., 



186 THE METHOD OF HANSEN. [CHAP. X 

230. Determination of the Form of F(g). 

The constants are found by considering the initial form of W . Put 
p = a (1 e/) pe cos ^>, 



and express W as a linear function of jo cos <f>, p sin $ ; the coefficients of these 
quantities being of the order of the disturbing forces, we can put p , < for p, 0. 
Hence, the expression (17) may be written 



, 
where 



= E + T c? cos 0o + fo + ^ - sin , 

\Ct Q J QJQ 

ft A h e cos (y n$yt TT O ) e 
2 j- - 8* ^ - ^ x 5^ ; '. 



T - 9 A. 6 _52i(xj: 7l ?y ^"^z. ^ \i/ ~ 9 A 

..... " ...... ~' 



..(43). 



Since the arbitrary jp (7) is the only constant in the expression for TT given 
in the previous article, all that is required is to find the form of the constant 
part of W Q . Let the constant part of % (which, by Art. 206, is |;he distance 
be TT O ; then "^ will contain no constant term. 



The constant parts of h, e are as yet undefined : it was merely assumed 
(Art. 216) that they differed from h 0> e Q by quantities of the order of the 
disturbing forces. In Art. 226 these differences were multiplied by 
quantities of the order of the disturbing forces and they were therefore 
neglected ; here this does not take place. Let the differences be such that 
the constant parts of 5, T are denoted by 6, ; the approximate constant 
part of e - e, is then | and that of h/h Q - 1 is ^6 -h e (See Art. 234.) 

The constant part of Tf is therefore 



by equation (18), Art. 43. From this we deduce F(g) by putting T = t or 7 = g. 

Since the terms multiplied by t have been made to vanish, the equation 
(41) becomes, on the substitution of this value of F(g), 



On the subject of the determination of the constants &, and on their general meaning 
with respect to A, e and A , e 0) Hansen's paper, referred to in the foot-note of page 168 above, 
will be found of great assistance. See also Fundamenta, pp. 65, 66, 196, Darlegung, I 
pp. 332 337, etc. 



230-231] DETERMINATION OF Z. 187 

The Integration of the Equations for z, v, and the Signification to be 
attached to the Constants of the Auxiliary Ellipse. 

231. Substituting the first approximation W Q for W in (12) and neglecting 
the third term of the equation that term being of the order of the square 
of the disturbing forces, we find 



os^) ......... (44), 

where, in the last term, r has been put for r (since y is of the order of the 
disturbing forces) and for r has been inserted its value given by 
equation (19) of Art. 43. We can also put for y the value found in 
Art. 229. 

The non-periodic part of dzjdt will be 

1 + b + d, 

where b } d are of the order of the disturbing forces, b being arbitrary and d 
containing the rest of the known non-periodic terms in (44). On integration, 
it will produce in ?i the term 

n ( f (1 + 6 + d). 

There is now an opportunity of defining rz . Let it represent the mean 
motion of the mean anomaly n () z. With this definition we must determine the 

arbitrary b, so that 

6 + ^=0. 

Again, if d' represent the known coefficients of cos g in (44), the coefficient 
of this term will be 



where d', % are of the order of the disturbing forces. On integration the 
coefficient of sing in z will therefore be (d f - f 3JBi/3OMo- We define % so 
that the coefficient of sin g in z vanishes. This, as we shall see directly, 
amounts to a definition of e . 

The value ofz is therefore given by 

n* ~<7 + SB sin (# + '), 
where fit + j3 f is an angle of the form 

j ff +j'g' + j lfl > + jiV, ( j, /, j x , ji = + oo . . . - oo ) : 
the term for which j = 1, / =j x = ji = having its coefficient zero. 



188 THE METHOD OF HANSEN. [CHAP. X 

The arbitrary constant present in g is the value of the non-periodic part 
of the disturbed mean anomaly at time t = 0. 

232. The value of n Qy defined above, is the mean motion of the mean anomaly of the 
auxiliary ellipse, that is, the mean rate of separation of the Moon from the perigee of the 
auxiliary ellipse. To obtain the mean motion of the Moon it is necessary to add to n the 
term n Q (y -fy) which (Art. 217) represents the mean motion of the perigee of the auxiliary 
ellipse. Since the mean motion of the Moon is observed directly, in order to obtain n Q for 
purposes of computation we must also know the mean motion of the Moon's perigee a 
quantity which is found from theory. The latter is, however, capable of being observed 
with great accuracy and Hansen, in performing his computations, used a value of % obtained 
from these observed values. The computed value for the motion of the perigee agrees 
very nearly with the value obtained directly from observation : the small difference causes no 
sensible error in the coefficients of the periodic terms *. 

When the value of %s is inserted in the expansion of /in terms of the moan anomaly, 

we obtain 

sin n^z -f e< 2 sin 



where e l9 e 2 v are known functions of e given by equation (7) of Art. 34. Since 
ntfs = g + %, we find 



Although n$z contains no term with the argument </, terms with argument g, other 
than ^ sin ^, will arise in /, owing to combinations of terms in the powers of n$ts with 
those of the elliptic development. For instance, the term_in n$z with argument %g will 
combine with e 1 cos g to produce a term of argument g in /. These terms, as well as those 
of the same argument which arise from the reduction of the longitude in the orbit to that 
on the ecliptic, are very small. 

Hansen computes n^s with e Q = -05490079 ; this produces 22637"'15 as the coefficient of 
sin g in the expression for the ecliptic longitude. The observed value of this coefficient is, 
according to him, 22640"-15 and, in order to produce this coefficient, the value of <? should 
have been -05490807. The difference is very small and it is sufficiently taken into account 
by multiplying those terms whose characteristic is e by '05490807/"05490079. Very few 
terms need to be thus corrected : the principal one is the evection. (See the papers of 
Newcomb referred to in Art. 2t38 below.) 

233. The equation (15) will now serve for the calculation of v. But 
since dW/d^^dW/d^ (where the bar denotes that r is changed into t after 
the differentiation) and since dW/n d^=dW/dy in the first approximation, we 



can put dW/djs = n$WQ/dy. Also, to the same degree of accuracy, we can 
neglect the product yv and put d/dz~n Q d/dg. The equation therefore 
becomes 



dt 



Darlegung, i. pp, 173, 348. 



231-234] DETEKMINATION OF v. 189 

The value of 3 W /dy may be obtained from Art. 229 and thence, by putting 
r = t, that of the first term of this equation ; therefore, after inserting the 
values of y, r 2 /a 2 as before, we have 



whence, integrating, 

i/=0-s4o 

Here fit + fi' is of the same form as before and A is the corresponding 
coefficient. The constant G, owing to the relation n 2 a 3 = //-, is not arbitrary : 
we proceed to find it. 

234. We have, from equations (43), Art. 230, 



where 8 (h/h^ S (h Q /}i) denote the differences of h/h^ h Q ]h from unity. Also, 
in the same article, 6, f were defined to be the constant parts of S, T. 
Hence 

6 H- f e = constant part of 2 j S ~ 

/IQ II 



But, by definition (Art. 211), 

w~ 1 ^0,0 h r l + ecosf 

yy JL "~ "5- -p A 7" - z - 

h A O a 1 e ( ? 



since M = h<?I and r = r(l + y). Therefore 



J (TTTy + < 2 + "> { r+ 1 <J 

an equation which is true generally. 

Neglecting, in the first approximation, the terms which are of the order 
of the square of the disturbing forces, this equation gives 

const, part of 2z> = const, part of f S j- W J 

const, part of Wo. 



190 THE METHOD OF HABT8EN. [CHAP. X 

As b, % and the constant part of W are already known, we can find from 
this result. 

Cor, "When z, v have heen found, the first approximation to k () /h can be most easily 
calculated by means of equation (11) which, when r IUIH boon put oqiuil to v (1 4- v \ gives 

jfe^ o __y /fV 

Stf 7i(l-f*/) a Jl-~e$ \ a J ' 

^o -I 9 i a /I i \s ^ ^* j ? 7.(^ "t"") 2 /^ ** \ a 
^ ^ " Ji'-e^ \<ij 

This will be required in the next approximation to z, 

It can also be found hy integrating the equation tlhjdt = / 



The mhies of P t Q, K as far as the First Order of the 
Disturbing Forces. 

235. In Arts. 218, 219, the equations satisfied by 1\ Q, 1C Lave been 
given in terms of the disturbing force 3 and of certain known quantities p' t 
q'. Further, in Art. 223, expressions for the partial derivatives of It with 
respect to P, Q, K Lave been found in terms of 3- Substitute the results 
(36) in the second terms of the right-hand members of equations (25), (27), 
and substitute the result (2V) in the third term of (25). The equations for 
dP[dt, dQ/dt, dKfdt "become 



dK 



where 



Qf 



by (21). 



These are the general equations for />, ft A r ; they are given ou p. 93 of the ftmdcmmta 
and on p. 117 of the first volume of the Darlegnngi. Tlio second of equations (21) shows 
that tihe constant part f ' of y i ^ ~-JV Q + /f (> . therefore, if we neglect the periodic part 
(which is very small) of JT, we have/- ^ (+,)<- ^ '. Since -^(<H-,)*is the mean 
motion of the node (Art. 21 7), this result shows that-/ is the meim longitude of the ascend- 

* Darlegung, r. pp. 164, 165. See also G. W, Hill, Note on Xawen'* general Formula for 
ferturlatians. Amer, Jouru. Math. Tol. iv. pp. 256-259. 

f Hansen, in the Darlegwy, denotes the angle / b.y <9, The change is made to avoid confusion 
h the angle arty. 



234~23'7] DETERMINATION OF P, Q, K. 191 

ing node of the Moon's orbit on the ecliptic. It is to be remembered that a, 77 are, by 
definition, to be so determined that N, K or that P, K contain no terms proportional to 
the time, 

236. We can immediately show that the secular motion of the ecliptic will only produce 
periodic terms in P, Q. Let this motion be given by 

p' = frjtf cos i' + const. , q' b^t cos i' + const. , 

whore & 1} ft/ arc constants supposed known. Substituting in (45), we see that the parts 
ddP/dt, dSQ/dt, due to these terms, are periodic. The corresponding terms in ddE/dt, being 
multiplied by P or Q and therefore by sin I/, are much smaller and they maybe neglected. 
If wo put K = 7i in the expression for p, the periods of these terms will be seen to be the 
name as that of //, that is, they will be 2?r/w (a + 77) ; this quantity is the period of revolution 
of the Moon's node along the ecliptic* (Art. 217). 

Another method of finding the effect of the motion of the ecliptic will be given in 
Chap. xiii. 

237. It has been seen, in Art. 219, that when the disturbing forces 
are neglected, 

P = 0, Q = 2 sin J/o, K = K*. 

Neglect the motion of the ecliptic, that is, consider p' t tf as constants. 

Put 

P = + SP, Q = 2 sin l/o + SQ, K = K Q + &E. 

In the terms containing the disturbing forces, neglect SP, SQ, K and 
put A = //<<). 

Let WoBo = - h* ^|) cos* 4/ , 



where the zero suffix indicates that, after R has been differentiated, the 
constant values of P, Q, K are to be substituted. 

The equations (45) become 
|l SP = - 2woO sin 4/ + %B , ^ 8Q = ^oG , ^ SJ? = wo*? +oA. 

It is not difficult to see, from Art. 221, that JB , A will be expansible in series 
of cosines, and in series of sines of angles depending on the time ; hence 
Q will contain no constant term. 

Let A* be the constant term in (dR/dQ\ : the constant term in n Q B Q will 
be - Mo cos 2 4/o and that in n Q D Q will be 4*0 A, sin 4/ . Hence the constant 
term in dSP/dt, d$K/dt are respectively 

- 2w sin 4/0 - ^o^-o cos 2 4/o> 

* Fundamenta, p. 94 ; Darlegung, I. p. 



192 THE METHOD OF HANSEN. [CHAP. X 

"When the equations for SP, SK are integrated, these will be the terms 
multiplying the time. 

Now a, ?) were introduced so that N, K should contain no terms pro- 
portional to the time ; the condition demands that P, K contain no such 
terms. We therefore determine a, y so that the two expressions written 
above are zero. Hence 

w-o a = - Po A o cos 2 1 / /sin J Q , TC O T? = - //. 1 sin ^ J~ ; 
giving tj = a tan 2 Jt/o 

These equations determine the first approximations to a, 77. The remark- 
able relation between them is modified in the higher approximations. 

The integration of the equations (45) will furnish for P, Q values 
depending on sines, and for K a value depending on cosines of arguments 
of the form {ft + '. 

238. On integration we can add arbitrary constants to SP, SQ, S-ST : these 
arbitraries, since the necessary number has been already introduced, may be 
determined at will. That added to 8K merely adds to KQ a *id it may 
therefore be put zero ; K K Q is thus expressed as a series of sines. The 
constant additive to 8P will also be put zero, so that P is expressed as a 
series of sines. The constant part of Q was 2 sin |J" , where ,/ was arbitrary; 
to BQ we add a constant /e, so that Q is expressed in the form 

2 sin | JQ + K H- series of cosines. 

The constant K is used so that, when the latitude has been found, the 
coefficient of the principal term (which has as its argument the distance 
of the Moon from the node) is sin/ "this coefficient being determined 
directly from observation. 

A careful investigation of the meanings to be attached to these constants and to those 
denned in Art. 232, and a comparison with the constants used by Delaunay, is given by 
Newcomb, Transformation of Hansen's Lunar Theory*. Another paper, Investigation of 
Corrections to Hansen's Tables of the Moon with Tables for their Application^ > by the same 
author, may also be consulted with advantage. 

To find the newt Approximation to n Q z, v, P, Q, K. 

239. When the disturbing forces were neglected, we had z=*g, v = and P, Q, K 
constants. Let 8z, v, 5P, $g, dK be the parts, just found, depending on the first order of the 
disturbing forces. In order to find the next approximation to the values of the variables, it 
is necessary to substitute in R and in the various functions used, instead of the initial 
values of the variables, their initial values increased by the parts just found. This is easily 
done by Taylor's theorem in the following manner. 

* Astron. Papers for the Amer. Ephemeris, Vol. I. pp. 57-107. 

t Papers published by the Commission on the Transit of Venus, Pt. ni. pp. 1-51. 



237-240] THE THIRD APPROXIMATION. 193 

From the value of dz we deduce, by putting =1-, that of $f. Now the expression (18) 
for dW/dt is, owing to the general form in which the disturbing function has been 
expressed (Art. 220), a function of 2, f, i/, P, Q, K. Put n f=y-|-^ % and let W be the 
value of W when Sf=0. We have 



and so for any function containing . Expand the expression (18) for d W/dt, in this way. The 
factor of y is quite easy to calculate when f y. After putting y=y +% (where y is the 
first approximation to y\ the values of W , 7i/7i , y , furnished by the previous approximation, 
are inserted ; the value of by will be afterwards determined so that no terms proportional 
to the time shall be present in W. The only difficulty that remains is the calculation of 
the terms containing $, X, and this arises from the presence of i/. It is to be remem- 
bered that when f=y we have p=po, 0=<jf>o* 

240. Denote the first two terms of d W'/dt by T, so that 

755 ( cos (/- *o) ~ 1}1+ 2/l o~ $r sin (/- < ). 



Put for $, 5> their values dRjdr, dE/da, and let 



whore ^ denotes the value of R (Art. 220) when v is put zero. Let 



so that ^i 1 ), jS( 2 ), ... , as well as &, are independent of v explicitly. 
We have evidently, since r=*r(l + v) 9 

v aw .3^(2) a/aw 



, ^ etc . 
+,), etc., 

and therefore, if the values of 5, (7, 1 be inserted in 2 1 , we obtain 



.............. .......... ( 46 )' 

where J l ^&+ Z7+S ; the numbers in brackets denote that the term 72W of 72 is alone con- 
sidered. A similar expression may be obtained for OT. 

Hence T is a function of wt, P, , j^, and it is independent of v, 7i/\ 5 if ^o be its value 
when fte, P, <2, 5/^ are put zero, we have, by Taylor's theorem, 

M 0+ |%a*^ 

In this we substitute the values of d, 5P 5 dft- a/f furnished by the previous approxi- 
mation. Since, in (46), <7, 17", 2 are all multiplied by small quantities, we can substitute 
elliptic values for the quantities present in the expressions which they denote ; v, h/h Q 
receive the values furnished by the previous approximation. As T Q is the quantity called 
TO in Art. 226, we already have its value. Hence the terms in (18) can all be expressed 
in terms of the time when it is desired to obtain the second approximation to W. 

13 



194 THE METHOD OF EANSEN. [CHAP. X 

In the same way we can isolate v, h in the equations (45) for P, Q, K. For example, if 
be that portion of the first of these equations which depends on R, we have 



^- 

then J BW= s J(i)+ JW j~ - l + 

( 
which can be treated as before. 

All the equations are finally expressed in terms of the time and integrated, the 
determination of $#, &a, 77 being made as in the first approximation. 

Reduction to the Instantaneous Ecliptic. 

241. A final step is necessary to obtain the longitude and the latitude 
referred to the ecliptic ; the parallax is found from r as in Art. 162. 

Draw ME (fig. 9, Art. 217) perpendicular to Z'a Then 
X'H = longitude = v } HM = latitude = u. 
Also, from the right-angled triangle MHl, 

sin (F ^r) cos u = sin (v -^r) cos J, sin u = sin J sin (v ^) . . .(47). 
From the relations (10), (21), 

v - ^ /+ ^o yt + TT O + rc-o ( - ??) * -H -iV 4- .fir - TT O 



where we now take a> to denote only the mean, part of its value given by (29). 
Also, cos / = 1 - (P 2 + Q 2 )/2, and 

', by (21), 



where /^ denotes the .mean longitude of the node on the ecliptic. Thus 
g + (D + fjii denotes the mean longitude of the Moon on the ecliptic. Since 
fg, 8-ZV, SJT, P> Q contain only periodic terms which are at least of the first 
order of small quantities, we can, after substituting their values, obtain v and 
17 in the usual form. 

Hansen finds u from the equation 

sin cr= sin 7 sin (/+ &>) + s, 

so that s denotes the perturbations of the sine of the latitude; /may 
be expanded in powers of w &gr as in Art. 232. 

For the details of the transformations, the reader is referred to the Darlegung l. 9. 
They are also to be found in Tisserand, Mfa C4l. Vol. m., Arts. 153155 of Chap. XVIL : 
the chapter referred to consists of an account of Hansen's Darlegung. 



CHAPTER XL 

METHOD WITH RECTANGULAR COORDINATES. 

242. THE use of rectangular coordinates is an essential feature of the 
latest method for the treatment of the solar inequalities in the Moon's 
motion. The equations of motion, referred to axes of which two move in 
their own plane with uniform angular velocity while the third is fixed, have 
already been investigated in Section (iii) of Chapter II.; a plan for the 
complete solution of these equations by means of series will now be given. 
The method of obtaining it is, to a certain extent, one of continued approxi- 
mation. The approximations do not, however, proceed according to powers 
of the disturbing forces, that is of m, but according to powers of e, e' } 7, I/a' 
the constants which are naturally present in the coefficients but which, in 
the earlier approximations, do not occur explicitly in the arguments. The 
advantage of the method used here is due to the possibility of carrying a 
coefficient to any degree of accuracy, as far as m is concerned, without 
making the large number of calculations which the methods discussed earlier 
would entail ; a reference to Art. 154 will show the importance of this. 

The theory is adapted to a complete literal development, but the labour 
necessary to secure accurate expressions for tlie coordinates of the Moon will 
be best employed by giving to M its numerical value and by leaving the 
other constants arbitrary. The fact that the mean motions of the Sun and 
Moon are the two constants which have been obtained by observation with the 
greatest degree of accuracy, will justify this abbreviation of the work ; any 
alteration in their values which future observations might give, must neces- 
sarily be very minute, and its effect can therefore be deduced from the literal 
developments of Delaunay. 

A difficulty which caused some trouble in de Pontdcoulant's theory -does 
not arise here. In obtaining the approximations to a given order, it was 
frequently necessary to consider terms of orders higher than those actually 
required, owing to the presence of small divisors. Since the order of a term, 
as far as e, e', 7, I/a' are concerned, is never lowered by integration, this 

132 



196 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

proceeding will be unnecessary here, for the method enables us to include all 
powers of m when calculating terms of a given order with respect to the four 
constants just mentioned: those terms, whose orders with respect to m are 
lowered by the integrations, merely present themselves with larger coefficients 
than they might otherwise be expected to have. 

243. The introduction of this method is due to Dr G. W. Hill who in 1877 published 
two important papers entitled * Researches in the Lunar Theory* ' and ' On the Part of the 
Motion of the Lunar Perigee which is a function of the Mean Motions of the Sun and 
Moonf 3 . These papers, besides throwing a new light on the methods of celestial 
mechanics, contain entirely new analytical devices for treating the problem of three 
bodies ; M. Poincare", in the preface to the first volume of his Me'oanique Cdleste, remarks, 
'Dans cette 03uvre...il est permis d'apercevoir le germe de la plupart des progres que 
la Science a faits depuis.' 

The first paper is devoted partly to an examination of the equations most useful 
for the actual determination of the Moon's motion and of the limits between which the 
radius vector must lie, and partly to the determination of the principal parts of those 
inequalities in the motion which have been called, in Art. 166, the Variational Class. In 
the second paper, the determination of (1 - c) n, the principal part of the motion of the 
perigee, is shown to depend on the computation of an infinite determinant and the 
numerical value of this quantity is calculated with a high degree of accuracy. 

On the publication of these papers, the late Prof. J. C. Adams gave the results of an 
investigation which he had made several years before, in order to find the corresponding 
part of the motion of the nodej. This likewise depends on the calculation of an infinite 
determinant of similar form ; owing, however, to the simplicity of the equation from which 
it arises, no transformations, like those necessary in the case of the perigee, are required. 
The full details of his investigation have not been published. 

The further developments which have been made and which directly concern the lunar 
theory, will be referred to in the course of the Chapter. 

The limitations imposed on the problem are the same as those adopted 
in the methods of de Pont^coulant and Delaunay. The disturbing function 
used is that of Art. 8, and the orbit of the Sun is an ellipse situated in the 
plane of (XY) with the Earth occupying one focus. 

The preliminary Limitations imposed on the Equations of Motion. 
The Intermediate Orbit. 

244. The general equations of motion (17), (18) of Chap, n., referred 
to axes moving with uniform angular velocity n 1 in the plane of reference, 
have undergone certain transformations: the forms to be used are there 

* Amer. Journ. Math. Vol. i. pp. 5-26, 129-147, 245-260. 
t Cambridge U.S.A. 1877 and Acta Math. Yol. vm. pp. 1-36. 

These two papers will be respectively referred to below by the titles c [Researches ' and ' Motion 
of the Perigee ' and by the pages of the journals in which they appeared. 
$ See footnote, p. 2S(K 



242-245] DEFINITION OF THE INTERMEDIARY. 197 

numbered (23), (19), (18). Since it is not possible to solve these equations 
directly, it will be necessary to neglect certain terms which are known 
to be small. Connected with this limitation is the choice of an inter- 
mediate orbit : the intermediary will not be the ellipse or the modified ellipse 
chosen by previous lunar theorists but will be defined to be a certain periodic 
solution of the differential equations after some, but not all, of those parts 
of them due to the Sun's action have been neglected. It is assumed, as 
before, that expansions, in powers and products of the small quantities which 
will be initially neglected in the differential equations, are possible. 

Let the equations (23), (19), (18) of Chap. II. be limited by neglect- 
ing the small quantities e', I/a'. Then / = a', rS = JST and (Art. 19) 1 = 0. 
Further, neglect the latitude of the Moon, so that z = 0. The equation 
(18) disappears and the equations (23), (19) reduce to 

D 2 (va) - DvDo- - 2m (vD<7 - <rDv) + fm 2 (v + <r) 2 = C, \ ( . 

2 (u 2 -<T 2 ) = j ......... ( )m 



It is advisable to notice that the equations (1) are equivalent to equations (17) of 
Art. 19 with &=0 and to no others. The second of equations (1) may be written 

{D 2 v -f 2mDv + fni 2 (u + <r)}/v = {D V - 2nxZ)<r + f na 2 (v + o-)}/cr = x, 
suppose. The first of them is then 

C- DvDo- - |m 2 (v + of. 



Whence, by differentiation, 

|m 2 (v + <r) (Dv + D<r) = - 



Therefore + |=o or, Const. ^xW^X^ 

which proves the equivalence of the two forms. 

The constant /e, which has disappeared, must be determined in terms 
of the arbitrariea by a reference to one of the original equations of motion. 
See Art. 21. 

In order to see the connection between these equations and the ordinary 
forms by which- the lunar motion is expressed, reference is made to the 
expressions collected in Art. 149. "When e'=0, 1/^ = 0, # = () (or, in, the 
notation of Chap, vn., ef = 0, a/a' == 0, 7 = 0), the coefficients of the remaining 
periodic terms depend on m, e only, while the arguments depend only on the 
angles 2 = 2 (n ri) 1 4- 2 (e e'), <p = ont + e r. Hence, the equations (1) 
will furnish all inequalities which depend only on m and e and will besides 
give the value of c so far as this quantity depends on m, e. 

245. The Intermediate Orbit is defined to be the path described by the 
Moon when we neglect e', a/a', v, e; it therefore consists of those terms 
in the expression for the Moon's motion which depend only on m. Now e is 



198 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

an arbitrary of the solution of the equations which determine the Moon's 
motion, and since the equations (1) determine all inequalities which depend 
only on m, e, the intermediate orbit must be a solution of (1) not containing 
the full number of arbitrary constants. 

The inequalities which depend only on m have been considered in case i. 
of Chap. vn. In Chap. vin. (Art. 166) they have been more fully examined; 
it was seen that they correspond to a symmetrical closed curve referred 
to axes moving with angular velocity n f about the Earth : this closed curve 
was called the Yariational curve. Since the coordinates in equations (1) 
refer to axes moving in the same manner, the Intermediate Orbit is the 
Yariational curve. If we look at the constants present in the expressions for 
the coordinates of a point on the Variational curve, it is seen that only two 
of them are arbitrary, namely, n, e, for a was defined by the equation >u, = u 2 a 3 . 
Hence the Intermediate Orbit is a periodic solution of (I) involving only two 
arbitrary constants. The first object in view is the determination of this 
orbit in the most effective and accurate manner. 



246. The only small quantity not neglected in the equations (1) is now n' or m. 
The reason for retaining this small quantity at the outset is derived from a knowledge 
of previous results. When the full literal developments in a theory like that of Delammy 
are examined, it is seen directly that if the series which represents any coefficient converges 
slowly, the slow convergence takes place chiefly when the series is arranged according to 
powers of m and not when it is arranged according to powers of e, e', a/a' or y. It is 
therefore of the greatest importance to find an intermediate orbit in which the coefficients 
may be obtained with any accuracy desirable as far as m is concerned. 

It might have seemed better to consider the complete solution of the equations (1) as the 
intermediate orbit; this evidently involves a determination of e. Later developments 
have shown that there is a saving of labour if we find those terms which depend on w, &' 
immediately after obtaining the parts which depend on m only, the next step being the 
determination of the inequalities which depend on m, <?', e. This plan will not be followed 
here, because the developments necessary to show the reason for it are too long to be 
inserted in this Chapter (see Art. 288) ; the order of procedure will bo the same as that of 
Chap. vn. 

It is evident that if we neglect m in the equations (1), they must redxico to those for 
elliptic motion. Since vcr=r 2 , -(^-^) 2 Dui)cr=(veL) 2 , the first of them can, in this case, 
be deduced by eliminating /*/r between equations (2), (3) of Art. 12, after putting R, z 
zero. -The second equation expresses the fact that equal areas are described in equal times. 

(i) The Terms whose Coefficients depend only on m. 
The Determination of the Intermediate Orbit. 

247. In order to arrive at the forms of the expressions for v, <r when the 
Intermediate Orbit is under consideration, recourse may be had to Art, 166, 
It was there seen that when the terms dependent on m are alone retained, r, 



245-247] (i) INTERMEDIARY. FORM OF SOLUTION. 199 

the radius vector, and v, the longitude measured from a fixed line, are given by 
a/r = 26 2 i c s 2i , v n't e' = 4- 2&'ai sin 2i, (i = 0, 1, ... oo ). 

Here = (n n') + e e' arid 6 2l -, 6'^ depend only on m. But since X, T are 
the coordinates referred to axes, of which that of X points towards the mean 
i place of the Sun, we have 

f X = r cos (v - n't- e'), Y= r sin (v - n't - e') (2). 

Hence X, Y are of the forms 

00 00 

^ _ __ 

where i receives negative as well as positive values and where we suppose 
a-o = l; a Q will, however, be retained for symmetry. We have then 

\ X = aS^ cos (2i + 1) %, )/__._ , >, /Q/N 

Whence, since v = X + Y V 1 , cr = JT F V 1 , the forms to be given to 
f , or are 



ai exp. 2 + 1 l , cr = aiC^ exp. 1 1 . 

Two arbitrary constants n, t (Arts. 18, 21) have been introduced into 
equations (1). Let n have the same meaning as before, namely, the observed 
mean motion of the Moon, and let 

( n _ w ')t oBS _( _ e ') ; 

then (w, -n ; ) (* - tf ) = (n- n*) t+ e - e' = ^, 

m = n'Kn ri) = m/(l m). 

As e, n were arbitraries of the solution previously found, n, # axe thus 
defined to be the arbitraries of that solution of (1) which is considered here. 

Finally, since " = exp. (7^ n') ( < ) V 1, (Art. 18), the assumed values 
for u, cr may, after putting i 1 for i in the expression for cr, be written* 

v^aSiO,,^ 1 , craSia_^p +l (3); 

or, i/f-^aSiCfoP 1 ' , o-^aSiOMrfp* ..(3'); 

where i = oo . . . + . 

It is now only a question of so determining the coefficients a^ that these 
values for u, cr may satisfy the equations (1). 

* In the Researches p. 130, a< takes the place of the coefficient here called a%. The change 
is convenient, firstly, because the chief consideration is the determination of a(/a and secondly, 
hecause it will not now be necessary to introduce another letter for the coefficients of the 
parallactic terms (Art. 277). 



200 METHOD WITH KECTANGULAR COORDINATES. [CHAP. XI 

248. It must be remembered that the object in view is the determination of the 
coefficients <%. They have been obtained to the order m 4 in Chap, vii., but no general 
law was forthcoming by which they could be found to any degree of accuracy without 
great labour. We have only made use of the results of that Chapter in order to discover 
the form of the required solution of equations (1). From the point of view of the lunar 
theory, it is very important that the connections between the results of the various theories 
should be clearly set forth ; for this reason it has seemed preferable to deduce the forms of 
the solutions from our previous knowledge rather than to obtain them by a general consi- 
deration of the differential equations. 

The latter method has been followed by Poincare" in his Mfaanique Celeste. In 
Chap, in., Vol. i. of his treatise, he has considered the conditions under which the equations 
of dynamics admit of periodic solutions. In 39 41, he gives some applications of his 
results to the Problem of Three Bodies and, as a special case, he proves that the equations 
(1) or rather the equations (25) of Art. 23 above, from which the former were deduced, 
do admit of periodic solutions in general. Moreover, in the same Chapter, he shows 
that these solutions, for sufficiently small values of the parameter, are in general 
developable in series. 

The Determination of the Coefficients of the Variational Inequalities. 

249. Let j be an integer with the same range of values as i, namely, 
from +00 fco oo . Since D^ = i^, we have, from the equations (3), 

Dv = aS* 2i + 1 a' 2i+ \ Da = 

/- = 2 



As j and i extend independently from +00 to oo, we may put j i 1 for j. 
The expression for va then becomes 



Similarly, 



Dv.Da= - a 2 2^ (2i + 1) (2* - 
vD<r - crDv = - a 2 2jSi (4i - 2j + 2) a^a^^ g*. 

When these results have been substituted in equations (1), the coefficients 
of the several powers of must be equated to zero in order that the values 
assumed for v, a- may satisfy the equations. The coefficients of f^ give 

S . [4j* + ( + 1) (2< _ 2j + 1) + 4 (K - j + 1) m + fin 2 ] 0*0*4 ) 

+ |m 2 S^a 2l -(a^_ 2i _ 2 + 0^-3^) = 0,1... (4); 
m) 0*0*4 ~ f m^a* (a 2j _ 2i _ 2 - a_ 2 ^_ 2 ) = J 



except for the value j = 0, when the second equation is identically satisfied 
and the right hand side of the first equation is (7/a 2 . 

Multiply the second of these equations by (2m + l)/2j and subtract from 



248-251] (i) EQUATIONS OF CONDITION FOE THE COEFFICIENTS. 201 

the first ; also, divide the second equation by 4j. Excluding the value j = 0, 
the results are 



= 2< {4? 2 - 1 - 2m + |m 2 + 4i 2 - 4trij] 

+ | y (2m + 1)1 SiOrfO^^g + ifm 2 - f 5?- (2m -i- 1)1^0* 
J J v J ) 

m 2 
= 2* (1 - j + m + 2 ' 



1,,,. 
i V* ) 



For the case j = we have, from the first of equations (4), 

C/a 2 = 4Si {(2i + 1 + 2m) 3 + -Jm 2 } a. 2i 2 + f m 2 2^ 2i a_^_ 2 ......... (o) ; 

an equation which serves to determine when a, oy have been found. 

250. As $ =1, these equations show that the coefficients <% are functions of m only. 
Also, it is not difficult to see that 2i will bo of the order m' 2il *. For example, put/! in 
equations (4'), and write down a few terms given by small values of i. We obtain 



= (3 - 2m + ^m 2 ) ( 2 a -f- a a_ 2 ) + ( 1 1 - 2m + ^ 

2 -f^^^ 



= . . . -|- (4 + m) afy + (2 + m) a 2 a + nia a_ 2 + ( 2 -f m) aL. 2 a_ 

m 2 



If we suppose a 4 , a_. 4 of higher order than 2 , a_ 2 , these equations show that a 2 , _ 2 
are of the order m 2 at least. "We may similarly treat the equations for j = 2 and deduce 
the fact that % a- 4 are of the order m 4 , and so on. It will appear presently that, in 
finding the % from equations (4'), m cannot be a factor of any of the denominators. We 
see further that the equations always occur in pairs and that those of principal im- 
portance in finding <%, a^ 2j -, are obtained by equating the coefficients of 2jf to xero. 

251. It is now necessary to show how the coefficients may be most 
suitably obtained from equations (4'). Owing to the fact that a^ is of the 
order 1 2j | at least, they may be found by continued approximation ; but in 
order to explain how the approximations are carried out, some further remarks 
must be made. Suppose that it be desired to determine a^ correctly to the 
order 1 2j |, the values of the coefficients, for values of j numerically smaller 
than that considered, being supposed known. The equations (4 ; ) show that 
we shall have two simultaneous linear equations for a^a , c^c& from which 
these quantities may be determined; the same is true when we desire to 
obtain them to any degree of approximation whatever. When we use the 
numerical value of m at the outset, these are the simplest equations to deal 
with ; but when a literal development in powers of m is required, the labour 
will be lessened if, before commencing the calculations, we deduce from the 
two equations giving %-, a^ two new equations in which the coefficients of 
e respectively zero. 



* If a be any real quantity, | a | denotes that a is to be taken positively whichever sign it may 
have. 



202 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

The terms containing a^a Q) CL^OO in equations (4') are obtained by putting 
i=j j t = 0. They are, in the respective equations, 

(4js - 1 _ 2m -f |m 2 ) (a Q a. 2j 
(1 -f j -f m) a a 2 j + (1 - j + m) 
If, therefore, we multiply the two equations by 

j-l-m, 4j 2 -l-2m + im 2 ..................... (6), 

respectively, and add the results, the coefficient of a a^ will be zero and that 
of a a 2 j will become 

2 ) ........................... (6'). 



Further, if we divide the resulting equation by this expression, the 
coefficient of a Q a will be 1. 



Multiply then the equations (4') by the two expressions (6), respectively, 
add them and divide the resulting equation by (6 ; ). Let it be written 



__. 2 ,^ 3 } =0 ...... (7). 

The coefficients [2j, 2i], [2JJ, (2j,) * will be 



^ 



i 8j 2 - 2 -4m + m 2 + 4 (i-j)(j-l -m) 



(2/0 = ^^^^^ - 1 - 2m +| 

- 2 - 4m + m 3 ) 



It is evident that [2j, 0] = 0, [&, %] = -!, so that the coefficients of 
are respectively 0, 1. 



Equation (7) will also serve for the determination of a_ 2 . For j may 
receive both positzve and negative values and, if -j is put fo r j, the coeffi- 






251-253] (i) SOLUTION OF THE EQUATIONS OF CONDITION. 203 

cient of a Q a 2 j becomes [ 2j, 0] which is zero, while that of a Q a^ is 
[__2j, _2j] which is equal to 1. The value j=0 is excluded as before. 

252. The denominator (6') has its least value when j = 1 and it cannot then vanish 
for any real value of m. Also, since 8/ 2 - 2 is never zero, m cannot appear as a divisor : the 
statement made in Art. 250, with reference to the order of <%, will therefore be true for all 
these coefficients. 

It will be noticed that we have obtained one equation (7) instead of the two (4') : this 
is a necessary consequence of the use of imaginary variables. With such variables it is 
possible in general to replace two equations by one, as was seen in Art. 18, where either of 
the two equations (14) of that article was a complete substitute for the first two of those 
there numbered (12). The equations (4) of Art. 249 do not possess this property because 
they arise from the equations (1) which respectively contain real terms and imaginary 
terms only (see Art. 21). When, however, the equations (4) are combined in a suitable 
way, one equation can be made to replace the two. In either of the equations (4), positive 
values of j are sufficient; this can be seen by putting -.; for j and then changing i into 
i~j in the first summations: after the two changes the equations become the same as 
before. 

253. The determination of the coefficients a. 2 i from the equations (7). 

In order to show the method of finding the coefficients a^ from these 
equations, the latter will be written out in full with a number of terms 
sufficient to find a 2 , &_ 2 , Oj, a_ 6 to the ninth order inclusive and & 4> a_ 4 to 
the seventh order inclusive. In writing them down, it is only necessary to 
recall that [2j,], (2/,) are of the second order and that a$ is of the order | 2j |. 
In the parts multiplied by [2/J, (2j,), a term 0^0^ occurs twice when i, i' are 
different. We find 



2 = [2, - 2] cL+a-t + [2, 4] a,a, + [2,] (a 2 + 2a_ 2 a 2 ) + (2,) (a_ 2 3 

t-2, - 4] a^au*+[-2, 2]a 2 a 4 +[-2,](a^ 2 -f 2a a 
a a 4 =[4, 2] a*a>-*+ [4,] (2a a 2 ), 
a a 4 =h 4, - 2] a_ 2 a 2 + (- 4,) (2a a 2 ); 
a a 6 =[6, 2] a 2 GL. 4 + [6, 4] a 4 a^ 2 H- [6,] (a 2 2 + 2a a 4 ), 
a a_ 6 =h 6, - 4] a_ 4 a 2 + [- 6, - 2] a_ 2 a 4 + (- 6,) (of 4- 

These are easily solved by continued approximation. Since a = l, we 
have, neglecting terms of the sixth and higher orders, 

^=[2,], o^ = (-2,), torn 5 . 

Whence, neglecting terms of the eighth and higher orders, 
a. = [4, 2][2,](- 2,) + 2 [4,][2,], a^ = [-*,- 2](- 2,)[2,] + 2 (- 4 ; )[2J, to m 7 . 
Similarly, using these results, we can obtain c& 6 , a_ 6 correctly to the order m 9 . 



204 METHOD WITH RECTANGULAB COORDINATES. [CHAP. XI 

To obtain a 2 , a_ 3 correctly to the order m 9 , it is sufficient to add to the 
values of a 2 , a_ 2 just found, the terms previously neglected ; these latter are 
obtained to the required accuracy if we calculate them with the values of 
a 2 , a_2, 4 , a-4 previously found. For c&_ 2 , &- 4 , which are of the second and 
fourth orders, were respectively obtained correctly to the fifth and seventh 
orders: their product is therefore correct to the ninth order; the same pro- 
perty holds for the other terms. Should it be desired to obtain their values 
still more approximately, a similar process will serve. We proceed by con- 
tinued approximation, using the results of the previous approximation for the 
calculation of the terms previously neglected. 

If these results be utilised to obtain a numerical development, we can, 
at the outset, calculate the coefficients [2j, 2i], [2jJ, (Zj 9 ) for all values of j, i 
required ; the continued approximations are then very simple. For a literal 
development in powers of m, these coefficients may be first expanded in 
powers of m to the degree of accuracy ultimately demanded ; the approxi- 
mations then only entail multiplications of series of such powers. 

254. The rapidity with which the values approximate may be grasped from the fact that 
each new approximation carries the value of the coefficient under consideration four orders 
higher. Also, for each such increase of accuracy, it is only necessary to add four new 
terms to the equation for that coejBfioient, two with factors of the form [2/, 2t], one with 
the factor 2[2/,] and one with the factor 2(2;,). The advantage of the method, when 
we compare it with a laborious one like that of de Pont&soulant, is very striking. I)r Hill 
has calculated the values of % literally* to the order m and numerically f to fifteen places 
of decimals. To show the rapidity with which they approximate in the latter case, the value 
of a_ 2 the largest coefficient is given below. 

He takes ^/(w-^^m- -08084 89338 08312 and finds 

IstApprox. =--00869 58084 99634, 
additional terms in the 2nd ~ 4- "00000 00615 51932, 

" " " 3rd " I' . OOOC) . 0000 13838 
resulting value of a_ a = "~ T 66869 ""57469 61540. 

When literal developments in powers of m are considered, the convergenoy of the series 
obtained depends to a large extent on the system of divisors 2 (4j/ 2 - 1) - 4rn + rn 2 These 
divisors increase very rapidly with j. The smallest of them, given by j~l, is 
6-4m+m 2 ; slow convergence would then chiefly arise from this divisor. Dp Hill inquires 
what function of m, of the form m=m/(l+m), will make the expansion, in powers of m 
of the inverse of this divisor converge most quickly. It is easily found that the necessary 
value for ais-J; the divisor 6-4m+rn2 gives rise to the divisor 6+ Jm a , and new divisors 
which are powers of 1 + Jin, are also introduced. The expansion of the inverse of each of 
these converges quickly. 

^ It may be remarked that it is not necessary to repeat the whole series of approximations 
in order to have the results expressed in terms of m. In the results expressed in terms of 
m, we simply put m=m/(l- Jm) and then expand in powers of m, stopping at that power 

* Researches, pp. 142, 143. f id. pp. 247, 248. 



253-255] (i) DETERMINATION OF THE LINEAR CONSTANT. 205 

of in to which the expansions in m were carried. Any of the results, whether they 
have been expressed in polar or rectangular coordinates, may be treated thus. 

One of the most interesting parts of the Researches (pp. 250260) is that which 
contains the investigation of the forms of the variational curves for increasing values of m. 
When m is much greater than J, it is found to be no longer possible to use the expansion 
in powers of m and mechanical quadratures must be employed. See Art. 166 above. 

255. Determination of a. 

Since tc (or /*.) has disappeared from the equations used above, there must 
be relations connecting K, p, with n, a. It will now be shown that these 
relations are of the forms, K = a 3 (1 4- powers of in), p, = ft 2 a 3 (1 + powers of m). 
The first of equations (17) of Art. 19, with O = 0, will be used, This equation 
may be written 

if\] 
^ = (D 2 + 2mD + f m 2 ) v + f m 2 <r. 

Substitute the values (3) of v, a. Since the result must hold for all 
values of f, we can, after the substitution, put f = 1. We have then 

v = a = a^Ojrf, Dv = a2* (2i + 1) a*, D 2 u = aS* (2i + 1) 2 a zi ; 
and the result is 

tear* (StOrf)-* = 2* {(2i 4- 1-f m) 2 + 2m 2 } a*. 
But K = p/(n - nj- = p (I + m) 2 /n 2 , by Art. 19. Hence 

a = *(! + m) [S< {(2* + 1 + m) 2 + 2m 2 } a*]-* [Sid*]-* ... (8). 



When the values of the coefficients <% have been inserted, a will be ex- 
pressed in the required form. The quantity /^ is that* usually called 
a 8 in the lunar theory. Hence a/a differs from unity by powers of in only, 
and it is equal to unity when m = 0. 

Since the parallax is observed directly, it will not be generally necessary 
to make the transformation from a to a in the coefficients ; if we desire, for 
the sake of comparison, to have the results expressed in terms of a, the 
transformation can always be delayed till the end of the investigation. 

The value of a may be also found by substituting for v, or as before and equating the 
coefficients of f 1 to zero. The results, obtained by this method and by that just given in 
detail, will be entirely different in form and their agreement will therefore constitute a 
useful verification. 

The value of may be found (if it be required) from (5). Another method for obtaining 
it, is given by substituting the values of v, o- in the first of equations (1) and putting f = 1 
in the result. For this we have 

V =o-=a2 i %, JDv= - 2)<r=aS i (2z+l)o 2i , D 2 u=jDV=a2i(2^+l) 2 %. 

The two results for (7, being also of different forms, furnish another means of verification. 



206 METHOD WITH RECTANGULAR COORDINATES. [CHAP, XI 

256. Transformation to real rectangular and to polar coordinates. 
The coefficients aa 2 having been found, the coordinates may be expressed 



in the usual manner by the formulae, 

r cos (v nt e) = r cos (v n't e' f ) = | (uf""" 1 4- erf), 
r sin (v nt e) = r sin (v rc/ e' f ) = -| (f f"" 1 <r) V 1. 
Hence equations (3') give 

r cos (i> nt e) = a [1 4- (& 2 4- &~ 2 ) cos 2 + (& 4 + a__ 4 ) cos 4 + . . .], 
r sin (# nt e) = a [ (a 2 a_ 2 ) sin 2 -f (a 4 OL. 4 ) sin 4f + . . .]. 
If it is desired to express these in polar coordinates, we have 
v nt = tan (v nt e) ^tan 8 (v n e) + . . . . 



As tan (v nt e) is a small quantity of the second order at least, the 
terms in the right-hand member of this equation can be calculated from 
the above values of r cos (v nt e), r sin (v nt e), by expansion. Also, since 
an equation for C has been given, the parallax 1/r may be found from either 
of the equations (20), (22) of Art. 20, after we have put z and fl zero therein. 
When the numerical value of m has been used, it is simplest to make the 
transformations for the true longitude by the method of special values. 

(ii) The Terms whose Coefficients depend only on m, e. 

The General Solution of Equations (1). 

257. We shall deduce the form of the general solution of equations (1) 
in the manner of Art. 247- From the results of Chaps. VI., VII., it is evident 
that, when all the terms whose coefficients depend only on m, e are considered, 
we have 

- = S^Zte^ cos (2if +#</>), v - n't - e' = + 2^6^+^ sin (2if + jp</>), 

where i ranges from oo to + oo and p from to oo ; the coefficients depend 
only on m, e and btf+pe, Vzi +pc are of the order e p at least. 



Hence X, T will be of the forms 



where A^ +pCy A^ +PG depend only on in, e and are of the' order e* at least. 
If we allow p to receive negative as well as positive values, the accent of 
be omitted, 



256-257] (ii) ELLIPTIC INEQUALITIES. FORM OF SOLUTION. 207 

From these values of X y F, the corresponding expressions for u, a may 
be deduced. After putting i 1 for i, and p for p, in the expression for o-, 
we obtain 



v = a cp- exp. _ 

<r = a StSp-A-jfrHH-jK, ^p. {(2i + 1) f + jp^} V- 1,) 

Let <m = c (ft TI'), r e = c^ (ft ft') ; 

so that ti replaces the arbitrary or. We have then 

<f> = cnt 4* e -or = c (ft n') ( #1), c = c (1 + HI). 

If we put exp. (n ft') (b ti) V 1 = 5i, 

we have ff exp. {(Zi +l)%+p<l>} V-l = ^ +1 ^ c . 

Now in order that the values (10) may satisfy the equations (1), it 
is necessary to substitute them in the latter and to equate to zero the 
coefficients of the various powers of ", f^. In the process of doing so, it will 
be necessary to use the differentials, 

D (** &P C ) = (2i + 1 4- pc) ^ +1 ^ c , D 2 (Z* i+l 7 C ) = (2t + 1+ pc) 2 ^ f/ c . 

Since the value of c will not be substituted in the index of fi, if we put 
g L == f (which corresponds to making ^ = ^ ) the equations of condition 
will be perfectly general ; the only point to remember -is that, when we 
return to real coordinates, the part of the index of f which contains c 
corresponds to the argument c (n n') (t ^). The equations (10) may 
therefore be written 



v = a, 
or, in the more symmetrical forms, 

Substitute the values (11) in (1). For this purpose we have, as in 
Art. 249, 

/"),, oV.V ^9V J_ 1 JL. niF\ A . 5"24-l-hpC f*t.r> " 

JLs U - cl^-y^jiv'kj \~ J " iT J- "l fJ / '^Sl'HifCs J Uv> j 

1JCT sss Q, - ^* ^/ ^^ jfjp ''"' 2't~HWC "^"2'fc~~2? "j- WC~~ftC a / t>uC. j 

where j, g have the same range of values as i, p, namely, from +00 to oo , 
Equating the coefficients of 2 .? + # c to zero, we obtain 

)c + 1) (2i -fpc 2j gc 
4- 2 (4i 4 2pc 2J gc 4 2) m 4 f m 2 

. _ 2j gc 4 2 4 2m) .4 2 *4 



except for j = = g, when the right hand member of the first equation is (7/a 2 . 



208 METHOD WITH RECTANGULAB COOEDINATES. [CHAP. XI 

258. On comparing these equations with (4), we see that the latter would 
be the same as the former if the symbols 



were respectively replaced by 
A, 2 

In making the corresponding transformations, we can therefore use the results 
previously obtained. Indeed this fact was evident as soon as we had arrived 
at the equations (11). 

To get the equations corresponding to (4'), multiply the second of equa- 
tior^s (12) by (2m + l)/(2j+ gc) and subtract it from the first. The second 
equation is to be divided by 2 (2j + qc). It is not necessary to write down 
the results, for they are immediately deducible from (4'). 

A pair of corresponding coefficients will be A^ +PG) j4_ 2l ^ c . If, in order 
to isolate A Z { +pC) we make the transformations of Art. 251 after putting 
c, Zi +jpc for 2j, 2i respectively, we arrive at the equation 



= 

4- [2; + ffcj Ati+ pG A 2j _^+ qc ^ pc -f (2j + gc,) ^L 2Hpc ^.__ 2j ^_ 2 _ ac _ pc } . . . (13), 



the value ^ = = g being excluded. The coefficients in this equation are 
the values of (7') after the changes noted have been made. We evidently 
have 

[2/ + ffo, 2j + ? c] = -~l, [2j + gc, 0] = 0; 

so that the coefficient of the term A Q A^ +pc is - 1 and that of A^A^^Q is 0. 



The equations for G and a can be obtained in a similar manner. 

The equations of condition (13) are to be solved by continued approxima- 
tion, but since there is a double sign of summation involved, they are by no 
means so simple as those of Art. 251. We know that the term A^ +pG is 
of order e^ 1 at least; the simplest method therefore appears to consist in 
finding initially those terms which depend on the first power of e only, 
neglecting all higher powers of e, thus restricting q to the values 1. 

If we neglect all powers of e and put ^1 =1, the equations will reduce to (7). "We have 
then 



It must be remembered that when powers of e are not neglected, A 2i is no longer equal 
to a^ but differs from it by terms of the order e 2 at least. The A 2i are the coefficients of 
the variational terms when powers of e are not neglected, while the a 2i are the coefficients 
of the same terms when the parts dependent on e\ e 4 ... are neglected. 



258-260] (ii) ELLIPTIC INEQUALITIES OF THE FIRST ORDER. 209 

The Terms dependent on m and on the First Power of e. 

259. As powers of e above the first will be neglected and as the coeffi- 
cients of the variational terms contain only even powers of e, we put 



so that e$, e/ are of the form ef(m). 

In the equations of condition (13), q takes the values +1, 1 only. 
When q = 4- 1, p has the values 1, in the first two terms, and the values 0, 
1 in the third term ; any other values of p will give terms of the orders 
e 8 , e 5 , Similarly, when ^ = 1, we must give to p the values 1, in 
the first two terms and the values 0, 1 in the third term. The equations 
for obtaining e j} e/ are therefore 



= 2* {[2j + c, Zi + c] ia 2i . 2j H- [Zj -h c, 2 

+ [2? + c >] (**<ty-2-si + %i<=M-i) + (2? + c,) (0*6%^-.! -h 

= S< {[Zj - c, 2i - c] e'tOrt^ + [Zj - c, Zi] a^e^ 

+ (2j - C, 




Since j receives positive and negative values, the equations of condition 
may be put into a more symmetrical form by considering those for 6j, e'-_; 
which form a ' pair/ Also, since each term is summed for all values- of i, we 
can, in any term, put i integer for i. In the second term of the first equa- 
tion put i + j for i ; in the second part of the third term, j i 1 for i ; 
and so on. With like changes in the second equation after -j has been 
put for y, the equations of condition may be written, 



(14). 



260. There are no other relations satisfied by the unknown coefficients q, 
e^. (^* = 4- oo ... oo ), and therefore the equations (14) only suffice to deter- 
mine the ratios of the unknowns to one of them, say to e or </. Moreover, 
since the coefficients, as well as the equations (14), always occur in pairs, some 
relation must exist between the equations, and it is necessary to so determine 
the unknown constant c that the relation may be satisfied. When c has been 
found, the ratios of the unknowns e^, e~j may be calculated by the ordinary 
methods of approximation. Hence one of them is an arbitrary constant, and 
this corresponds to the arbitrary constant which, in other theories, is denoted 
by e. 

a L. T. 14 



c, 2i + c] 6*0^- + [2j + c, Zj - 2i] 

+ 2 [2; + c,] e %_ 2 ^ 2 4- 2 (2j 4- c,) e 
- c, - 2i - c] e'_^_ 2 i + [- 2jf - c, 2i - 

+ 2 [- 2j - c,] e^flto-tf-a + 2 (- 2j - c,) 



210 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

It is not difficult to see that the unknowns <?y, e'_y, like the coefficients a< 2 3> are > m 
general, in descending order of magnitude with ascending values of \j\. Assuming this 
fact, suppose that we neglect all powers of m above the second and also the unknowns beyond 
those given by the values j 1 ; we shall have six equations to find the ratios of e , e ', 
_ 15 c/, !, c'-!, namely, those given by jf=0, 1, and the relation determining c will be 
expressible by a determinant with six rows. If we include the values for which 
j 2, the determinant will have 10 rows, and so on. Finally, including all the equa- 
tions, the determinant will have an infinite number of rows and columns the infinite 
number being of the form 4/ +2. 

It is possible to approximate to c while the approximations to e,-, c'_y are being carried 
out, but this would be very troublesome owing to the way in which this constant is involved 
in the coefficients [2/+c, 2t], etc. Or it might be found by calculating the determinant 
after neglecting the smaller terms; this again would be very laborious owing to the large 
number of rows necessary to secure the accuracy required at the present day. In his 
paper ' The Motion of the Perigee'*, Dr Hill has shown that c may be made to depend on 
a symmetrical infinite determinant in which the number of rows or columns is of the 
form 2; + 1, and further, he has succeeded in representing the value of this determinant 
by means of a series whose terms diminish with great rapidity. See Arts. 262 et seq. 

The homogeneous form of the equations (14) with respect to the unknowns, shows that 
the value of c, when powers of e above the first are neglected, is independent of the 
particular definition to be assigned to the eccentricity constant. The remark is made 
because, when finding c by de Ponte'coulant's method (Art. 139), it appeared to be involved 
with the definition there given to e. We also notice that, when powers of e above the 
second are neglected, c will be a function of m only a fact which renders the previous 
remark obvious from another point of view. 

In working out a complete theory by this method, it will be found more convenient to 
put -^ 2 i +0 =e2-) -^-o^ij in order that the introduction of a new symbol for the coefficients 
of the terms dependent on e and on powers of a/a' may be avoided. These coefficients are 
of the form ^ + 1+0, ^ 2 i+i-c- 

261. Suppose that c has been obtained in some way, accurately to the 
order ultimately required, either as a series in powers of m or numerically. 
The coefficients in the equations (14) can then be found and the values of the 
unknowns calculated. 

When the numerical value of m is used at the outset, the best method of 
dealing with these equations is to neglect, in the first instance, the two equa- 
tions given by j = 0. The equations given by j = 1, 2, . . . will then furnish 
the rest of the unknowns in terms of e 0> e ', by the usual methods of approxi- 
mation. Substituting these in the equations given by the value j = 0, we 
shall have two equations to find the ratio e /e ', and if the value of c, 
previously obtained, is correct, these should agree. A formula of verification 
is thus available. The arbitrary constant may be taken to be e, where 

e = e Q. 
* See footnote, p. 196. 



260-263] (ii) THE MOTION OF THE PERIGEE. 211 

It is found that Je differs very little from the constant e used by Delaunay 
and defined in Arts. 159, 200 above. 

The method outlined above will be found more fully developed in Parts L IV. of a 
paper by the author, The Elliptic Inequalities in the Lunar Theory*. The notation used is 
slightly different : i and j, p and q are interchanged ; for c, m are put c, m j instead of e 
the symbol 7 is used; the other differences will be obvious. The values of <%, c, as 
obtained by Hill, are used, and the results for e y /e, e'^/Q are computed numerically to 10 
places of decimals. The ratio of e to e is found by transforming to polar coordinates and 
comparing the resulting coefficient of the principal elliptic inequality in longitude with 
that given by purely elliptic motion. 



The Determination of the Part of c which depends only on m. 

262. The problem to be considered here is the discovery of an equation 
which will give c : we are not concerned with the unknown coefficients $, 
e/. Now a transformation of coordinates will not affect the periods of the 
various terms and we are therefore at liberty to choose those coordinates 
which will put the problem into the simplest form. It was noticed that, when 
the method of Art. 259 was followed, the determinant giving c was of infinite 
order and that the number of rows or columns was of the form 4jf+2. It 
will be shown here that c may be made to depend on an infinite determinant 
with only half that number of rows and columns ; in other words, only half 
the number of rows or columns are necessary for a given degree of accuracy. 
When, however, the determinant has been found, instead of limiting the 
number of rows, we shall show how it may be expanded generally as a series, 
and we shall then see to what degree of approximation the series must 
be taken in order to secure a given accuracy in the results. 

The results of Arts. 244-246 show that the problem may be stated as 
follows : Given a periodic solution of a pair of differential equations, to find 
the periods of a solution differing but little from the given solution. 

263. The equations (1) being equivalent to (25) of Art. 23, let the latter 
be written 



r + 2n' = a, 

where I 2 + F> = F 2 = 2F f + const. = 2yu,/r + 3n /s JC s + const, j ...... ( ' ; 

F 1 is then independent of t explicitly, and V is the velocity with respect to 
the moving axes of X, Y. 

Let ty be the angle which the tangent at (X, Y) makes with the Jf-axis, 

and put 

8 ,8. .8 8 .8. ,8 

9T=cos ^ gir + sm ^_, ^=cost 9T -smtgj ...... (16); 

* 4-mer, Journ. Math. Vol. xv. pp. 244-263, 321-338. 

142 



212 



METHOD WITH RECTANGULAR COORDINATES. 



[CHAP. XI 



then dF'fiT, dF'/dN represent the resolved parts of the forces corresponding 
to the force-function F', along the tangent and the normal respectively*. 

Since X = Fcos^, F= Fsin-^, we deduce 



_ 

dt3N~~ dWdT dt dT 
where the meanings to be attached to &F'/dN* t 

Also, since tan^/r= Y/X and therefore 
easily from (15), (16), 

dV dF' 1 dF' 



are evident. 
X jf F)/F 2 , we obtain 



Let 8Z, SF be any small variations of X, F, which are such that X + $X , 
F+SF, as well as X, F, satisfy the differential equations, and let ST, SN 
be the corresponding variations along the tangent and normal to the orbit of 
X, Y, Neglecting squares of SX, $ F, we have 

...(18); 



sin'f 8F, 
and the equations (16) show that 



etc. 



All the equations may therefore be submitted to a variation S. We 
have, from (15), 



.(19). 



Now the left-hand members of these equations are the accelerations of a 
point referred to axes moving with angular velocity n'. Take the sum of 
these equations multiplied by cos^r, sin^ ; respectively, and also their 
sum multiplied by - sin -^, cos -^ respectively : the results will be the 
accelerations referred to the tangent and normal of the original orbit. 
The latter rotate with angular velocity n' 4- d^/dt. Since the coordinates, 



* It is necessary to define the partial differentials in this way because F' is not expressible in 
terms of T, N only ; they have the same meaning as if F' were so expressible, 



263] (ii) EQUATION FOE THE NORMAL DISPLACEMENT. 213 

referred to these axes, are S2 7 , SJV", we obtain, by the well-known formula for 
the acceleration in the direction of the normal*, and by (16), (18), 



- - - * 

Now 



$T d ( T ,Aty _ ,\] , 

F 3i F (1 + 2w )J + T W * by 



d^ /^ AdFST 

^? + (dt +n ) dt V 

By means of this result, equation (20) becomes 



Also, since SF is the variation of the velocity, relative to the moving 
axes, in the direction of the tangent, we have 

sr-isr-^stf. 

dt dt 
But, by the third of equations (15) and by (17), 



Eliminating SF, we find 

*t T - 

dt dt 
The equation (2()') therefore becomes 

........................ (21), 



Cut" 
where (ft _.e = 3 + ' 2 + n'- ............... (22). 



This method of arriving at the equation (21) is a modification of one given by Prof. 
Adams in his lectures on the Lunar Theory in the academic year 1885-6. A similar 
investigation was given independently by Prof. G. H. Darwin in his lectures of 1893-4 
The method used by Dr Hill demands several transformations t ; the variable there called 
w is equal to 2dN^/l> 

* Tait and Steele, Dynamics of a Particle, Art. 42. 
t Motion of the Perigee, pp. 6-9. 



214 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

264. Form of the Solution of the Equation for SN. 

It will be convenient to express in terms of the rectangular coordinates. 
For this purpose we have, as before, 

cty YX-XY , X . , F 

^r = - Y* - , cosf = T , smf = ^. 

Whence, substituting for d*F'/dN* from (16'), we find 



......... (22'), 

where F 2 = J 2 -f F 2 , F' = p (X 2 + F 2 )~^ + f n'* X 2 . 

Since the quantities present in refer to the given periodic solution 
(which is the intermediate orbit), it is evident from equations (2') that it is 
expressible by cosines of multiples of 2 = 2 (n n') (t ). We can therefore 
express @ in the form 

o + 2! cos 2 + 2@ 2 cos 4f + . . ., 
where , i} ... are functions of m only. If $ = $, equation (21) becomes 

2 O< cos 2if) 8^=0 .............. (23). 

; oo / 

This equation is of the form referred to in Art. 146, and it admits of a 
solution of a similar nature. In order to retain the connection with the 
previous methods, the form of the solution will be deduced from our previous 
knowledge of the forms of 8-3T, SY. 

We have, from (18), 



Since X, Y are respectively expressible by means of sines and cosines of odd 
multiples of , and SF, SJT, from (9), by sines and cosines of the angles 
(2i + !)+<, the function XSY- Y8X will be expressible by means of 
cosines of the angles 2if </>; also F and 1/F are expressible by means of 
cosines of the angles 2if . Hence 8N will be expressible by means of cosines 
of angles of the form 

2* (n - w') (* - fc) c (n - n') (t - fc) 



Introducing the operator /), defined as before by means of the relation 
= - d*f( n - n y dt\ equation (23) becomes 

... ..... , ............. (230; 



264-265] (ii) FORM OF SOLUTION OF THE EQUATION. 215 

where i = +oo ... oo, and _; = *. The general solution, according to 
the remarks just made, will be of the form 



B N = (S<b^ +c ) exp. c (n - ri) (t Q - tj ^_ 

- 1. 



On substituting this expression in (23') and equating the coefficients of 
the several powers of f to zero, we see that, since i receives negative as well 
as positive values, the equations of condition for the coefficients b; are the 
same as those for the coefficients b'_<. It is therefore only necessary for the 
determination of c to consider the integral 



This result, which has been deduced from our previous knowledge of the 
form of the solution, is a well known property of equations of the form (2:3'). 
All that now remains is the substitution of (24) in (230, and tbe deduction 
of c from the equations of condition obtained by equating the coefficients of 
the various powers of to zero. 

265. In order to bring 8 into a form suitable for calculation, it will be better to 
express (22) in terms of the complex variables v, 0% We have 



and 

v J= r*-i^=-K-7(-v^-/>wM, ^ 

therefore, from (22), 



where &i=W l(n-ri) 

the transformation used in Arts. 18, 19. 
But, from Art. 18, 



therefore ^ + ^ = - ^ -57 - ^ 9(r 

Hence 



g^-(l> (r ) J . 8, 
The last two terms, by the equations of motion, are equal to 



3 2 & , K. . 2 

and, since r 2 = -uer, g^. = f 55 + fc m 5 



216 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 



and therefore 
e=- 



The quantities present in this equation have to be expanded in powers of Assume 
D^v/Dv^^Ui^ K /r 3 -i-m 2 = 22^ i f 2i (25) ; 

When 17^ Mi have been found, the calculation of 8 requires only two operations of 
any length, namely, the squaring of two series. 

"We have, after the substitution of the values (3) of v, <r, 

from which, by equating coefficients, the coefficients % can bo obtained. 

The coefficients M can be found by treating, in the same way, the equation 
D z v 4- 2nxZ)v 4- f-mV + |?m 2 i; = v (m 2 + K/T) = 2u2^ M 2i . 

When the numerical value of m is used, it is easier to calculate the coefficients U^ M 
by the method of special values, after eliminating the second differentials by means of the 
equations of motion. It is evident that e$ will be of the order m m at least 

The Equation for c. 

266. When the value (24) is substituted in (23'), wo have, on 
equating the coefficient of ^' +c to zero, 



or, snce _$ = ^, 

... - 2 v 2 - e^ + {(c + 2/) - } b ; - e^ - e 9 b;^ - . . . = o ; 

by giving to j the values 0, 1, 2,. .., we obtain a set of linear homogeneous 
equations. Since the unknowns and the equations are infinite in number, 
some remarks concerning the treatment of them are necessary. 

Suppose that the two series of quantities b , b 1 , b a ,..., O , <H) I? . J4 .., are 
in descending order of magnitude, and let all the coefficients beyond b^, b-p, 
@ p (p a positive integer) be neglected; we shall have 2p + l unknowns and 
Zp + I equations. As the equations are linear and homogeneous with respect 
to the unknowns, a relation, which may be expressed by equating to zero a 
determinant of 2p -f 1 rows, must exist between the equations. We shall 
assume that the same results hold when p becomes infinitely great (see the 
note at the end of Art 267). 



265-267] (ii) INFINITE DETERMINANT AL EQUATION FOR c. 217 

We thus suppose that the ordinary rules for treating a set of linear 
homogeneous equations may be used when the unknowns and the equations 
are infinite in number. To every positive value of j there corresponds an 
equal negative value, and there is one equation for j = 0: we may therefore 
consider that the number of equations is odd and that the coefficient of b , in 
the equation given by j = 0, is the central constituent of the determinant 
formed by eliminating the unknowns. Let the two equations, obtained by 
equating to zero the coefficients of ^"-t- , be divided by 4j 2 . The 
condition that the new series of equations, thus formed, shall be consistent is 
expressed by the equation 

A(c) = 0, 
where A(c) represents the infinite symmetrical determinant, 













(c-4) 2 - 


X 


e 2 


3 


4 


4 a - ' 


4 2 - ' 


x 


4 2 -@ ' 


4 2 - ' " 

3 


2 2 -@ ' 


2 2 - ' 


2 2 - ' 

C 2 -@ 


2 2 - ' 


2 2 -@ ' '" 
, 


2 - ' 


2 -@ ' 
, 


2 -@ ' 

x ( 


2 - ' 

c + 2) 2 -@ 


2 - ' " 


2 2 -@ ' 
8, 


2 2 -@ ' 

3 


2 2 - ' 
, 


2 2 - ' 


2 2 - ' " 


4 3 - ' 


4'- ' 


4 2 -@ ' 


4 2 - ' 


4"- '" 



267. The equation A (c) = may be regarded as an equation for c with 
an infinite number of roots which have the following properties : 

(i) The roots ooowr in pairs. For when - c is put for c, A (c) remains 
unaltered. Hence, if c be a root of the equation, - c is another root. 

(ii) If c be a root, the expressions 



are also roots. Let c + 2 be put for c, and let the divisors 4f- (which 
are infinitely great at infinity) be all moved on one place ; the central line 
and column are merely altered in position, and the determinant therefore 
remains unaltered. The same result holds if c be increased or diminished by 
any even integer. 

Since these roots are also roots of the equation COSTTC = cos 77C 0? we have 

A (c) = k (COS 7TC - COS 7TC ) ..................... ( 26 )' 



218 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

(iii) All the roots of A (c) = are included in the expression c 4- 2;, that 
is, k is independent of c. Since , & 19 @ 2 ,... do not contain c, the number of 
roots cannot be altered by giving any special values to x , 2 , ... ; let them 
be put zero, and let the resulting value of A (c) be denoted by A (c). 
We have evidently 



The roots of the equation A (c) = are therefore the same as those of the 
equation 

cos TTC cos TT V = ; 

whence it follows that the roots of A (c) == are the same as those of the 
equation 

COS 7TC COS 7TC = 0. 

This result, combined with (26), shows that k is independent of c. 
(iv) Any root c satisfies the equation 

sin a J7rco A (0) sin^We^ ..................... (28), 

where A (0) is the determinant obtained by putting c= in A (c). 

Since (26) is true for all values of c, the value of k may be found by 
equating the coefficients of the highest power of c in the two members of the 
equation. But the highest power of c, contained in A (c), is obtained from 
the elements of the leading diagonal only; hence k is independent of j, 2 ,.... 
Let zeros be put for the latter quantities ; A (c) then reduces to A (c) and c 
to V . We therefore have 

A (c) = k (COS 7TC COS 7T \/ ) 

Putting c = in this identity, we obtain 

k (1 - cos TT V5J) = A (0) = 1, by (27). 

Finally, substituting this value of k and putting c = in (26), we deduce 
the required equation (28). 

Since the substitution of c 4- 2j for c leaves (28) unaltered, this equation 
gives the periods of all the terms in the solution. The period we require is 
known to be that of the principal term, and therefore, in obtaining c from the 
value of sin 2 |-7rc , we must choose that value nearest to unity. 

The only step that now remains, for the determination of c , is the 
calculation of A (0). Every element of the leading diagonal of this deter- 
minant is unity and the other elements are functions of m which may be 



267-268] (ii) CONVEKGENCY OF INFINITE DETERMINANT. 



219 



found according to the methods explained in Art. 266. Putting c = in 
A (c), we find that A (0) is equal to 









"1 * 


2 


3 4 


' 4 2 -/ 


' 4 2 -@ ' 


#_@ > 4, 3 _@ "" 


01 1 


i 


2 3 


* @ @0 ' , 


2 3 - ' 


2 2 - ! 2 2 - J '" 


2 - ' 2 -o' 


' 


2 - ' s - 5 " 


3 2 


i 


I 1 


2"- ' 2 2 -,' 


2'- ' 


2 2 - > "' 


1 3 


2 


@1 1 


4 a - ' 4 2 - ' 


4 3 - ' 


4 2 - ' ' "' 



The complete examination of the assumptions involved in the preceding results is 
outside the limits of this treatise ; they have been considered by Poincare*. We shall 
only give below the conditions which must be fulfilled in order that an infinite determinant 
may be expanded according to the ordinary methods. 

268. Convergency of an Infinite Determinant. 

Let a determinant of 2n + 1 rows or columns be denoted by A n ; the determinant A rt is 
said to be convergent if it continually approaches a finite and determinate limit as n 
approaches to infinity. 

Let the element in the ith row above the middle row and in the ji'th column to the right 
of the middle column be denoted by it y, and the element of the principal diagonal in the 
ith row by 1+&, _*, where i,j*=+n... -n; in other words, let /3 U be a non-diagonal 
element and 1 4- ft, -< a diagonal element, the central element being 1 +/3 , o- Consider the 
continued product f 



where, under the sign of summation, the value j= - i is excluded. This expression, which 
we shall for brevity denote by IL n , contains all the terms of the development of A* and 
other terms besides. Since, by definition, all the terms of Ii n have positive signs, A^ is less 
than n n and therefore, when n is infinite, 

Lim. A H < Lirn. U n . 

But nn2Sni(l+S,|&,y|), 

where the value j/= - i is not excluded from the second member. Whence, when n becomes 

infinite, 

Lim. 



The second expression is convergent if 2 u |/3j converges J. This condition includes 
the convergence of S, |ft, -_| which is the condition of convergence of the product n,(l +&, _,) 

* Bar les determinants d'ordre infini. JBtrfL de la Soc. Math, de France, Vol. X iv. pp. 77-90. 
, Pt. n. P. 1B7. B. W. Hobso,, Tn>no^, Art, 279-2BL 



220 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

Hence the limit of A n is finite if the sum of the non-diagonal elements be absolutely convergent 
and the product of the elements in the principal diagonal be absolutely convergent. 

Again, the expansion of the determinant A n can be deduced from that of A n+p (p a 
positive integer), by putting zeros for certain of the elements which are present in the latter 
but not present in the former. Hence A n+p -A n represents the terms which vanish 
when these elements are annulled. But n n+ 1J>-n n must contain all these terms (and 
possibly others) all affected with a positive sign, and H u+p contains every term present in 
n n . Hence 



But, if S^|ftJ *> e convergent, H n+p -n n and therefore A n+p -A n can be made less 
than any finite quantity by sufficiently increasing n. That is, the conditions given in 
italics are sufficient for the convergence of A u when n is infinite*. 

The condition sufficient for the convergence of the determinant A(0) can be easily 
deduced. The second of the conditions, given above, is evidently satisfied, for all the 
elements in the principal diagonal are unity. The first condition is satisfied if 

1 



be convergent. The latter series is well known to be convergent. Hence, in order that 
A(0) may be convergent, it suffices that 2<|eJ be convergent. We shall assume that 
this condition is satisfied. 

The Development of an. Infinite Determinant. 

269. It will be convenient to denote the non-diagonal element {3^ / by (i : j), and the 
diagonal element 1 +&, -i by (i : i) f. The central constituent is then (0 : 0), and the term 
in the development arising from the product of the elements of the leading diagonal is 



Any other term in the development may be obtained from this by interchanging any of 
the second numbers of the several symbols contained in the above expression, the first 
numbers remaining unchanged J. 

If one change be made between two of the second numbers, the other elements 
remaining the same, the term is said to be produced by one exchange j for example, if we 
interchange - i, 0, so that the elements (i : - ^) (0 : 0) become (i : 0) (0 : - i), the new term 
is said to be produced by one exchange. 

Two exchanges may be made in two ways. They may either be made amongst three 
elements (e.g. we may exchange i, and then 0, 1), or they may be made amongst four 
elements (e.g. by exchanging -a, and also 1, i). Similarly three exchanges may be made 

* The proof, when the elements of the principal diagonal are all unity, was given by Poincare" 
in the paper just referred to. The proof given here is from a paper by H. yon Koch, ' Sur les 
Determinants Infinis, &c.,' Acta. Math., Yol. xvi. pp. 217-295. 

t The explanation will be more easily followed by taking the principal row and principal 
column as axes : the positive direction of the ^-axis being to the right and that of the a?-axis 
upwards. The ' coordinates ' of any element are then i, j. 

$ Of course the development may be also made "by interchanging the first numbers, the 
second numbers remaining unchanged. 



268-270] (ii) DEVELOPMENT OF AN INFINITE DETERMINANT. 221 

amongst four, five, or six elements. Any term produced by n exchanges which might, by a 
different proceeding, have been produced by n-k exchanges, is excluded from the conside- 
ration of the terms produced by n exchanges, since it is considered in the n-k exchanges. 
Further, it is evident that the terms produced by n exchanges have a positive or 
negative sign according as n is even or odd. 

In the following, i is an integer which may have any value, including zero, between 
~oo and +00 ; k, V, &",... are integers having any values from 1 to +co . 

270. Let all the elements of the leading diagonal be unity. The term in the develop- 
ment obtained by using all the elements of the leading diagonal is therefore 1. 

Consider any two elements of the leading diagonal 

(i :-i\ (i + k : -i-). 

Owing to the fact that k can only be a positive integer greater than zero, these expressions 
will always denote different elements, and their product will never denote the same pan- ot 
elements for different pairs of values of t, k. 
One exchange between these gives 

(t : -i-*), (i+k : -*) 
Since all the other elements of the leading diagonal are unity, the expression 



gives the terms, in the development of the determinant, obtained by one exchange. 
Consider three elements of the leading diagonal 
(i : -t), (i+k : -i-k\ 



There are only two possible ways of making two exchanges amongst these three elements, 
so that none of them remain in the leading diagonal. They are 



(t :-*-*), (>* I-*'-*-*), 

and (i : --;&-#), (i+k : -i), (i+k+k' : -t-*). 

Hence, all the terms arising from two exchanges amongst three elements are 



: -i-*)} ...... (29')- 

Two exchanges amongst the/of elements 



so that none of them remain in the leading diagonal, are given by exchanging - J -t- J 
and exchanging -i-A-A', --*- A'-*^ ; the combinations obtained by exchangmg -s 
-ilk-l* or -t, -%'-*-#-*" are included. Hence the terms in the development, 
obtained by making two exchanges amongst four elements, are 



In a similar manner, we may consider the terms of the development, produced by three 
or more exchanges. 



222 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

Development of A(0). 
271. Let us now apply these results to the development of A(0). We have 



Let 4i 2 60=o$, so that d$ = a_.i. 

Now efc=e_fc is of the order m 2 * at least; it is therefore evident that the order of any 
element (i :j) is Z\i+j\- Since k, k' are always positive and greater than zero, we see 
immediately that the terms (29), produced by one exchange, will be of the fourth order at 
least ; the terms (29'), (29"), produced by two exchanges, will be of the eighth order at 
least; similarly, the terms produced by n exchanges will be of the order 4n at least. We 
shall here neglect terms of the 12th and higher orders. 

The terms (29) give 

torn". 



Since 6^=9^, etc., the terms obtained from (29') are 

-^ 9l * e2 , torn"; 



since values of &, ' greater than 1 give terms of an order higher than the eleventh. 
The terms (29") give 

**. O *+. 9 -I 

1 . to m 11 . 



Hence, as far as the eleventh order inclusive, we have 

- ...... ( 3 o). 



272. The final process consists in replacing the summations, in the expression (30), by 
finite terms. For this purpose we have, if k be a finite integer*, 

.=2$ - . S$ -; j =2$ - : j , ({=00 ... GO), 

a-i ^ - ~~' v 



Here a is never equal to an integer, and, in the method used below, the semi-con vergency 
of these forms will not affect the values of the functions which they are supposed to 
represent. 

Let & =4a 2 ; then ai=4i 2 e =4(^ 2 ~a 2 ). Hence, decomposing into partial fractions, 

s 16 ~ 1 __ vfAo. B 

^OiOi^r *\&~W~(i+l^^ 

where l/^ 



, ^ D 1 



Therefore, summing each of the four terms by means of the formulse given above, 

1 7T COt 1TQ, /I 1 \ ft- COt 7TCI 

By giving to Js the values 1, 2, successively, the second and third terms of (30) may be 
obtained from this result. 

* E, W. Hobson, Trigonometry, Art. 293, 



271-273] (ii) ELLIPTIC INEQUALITIES OF HIGHER ORDERS. 223 

In the same manner, l/oiOi^oi^ may be decomposed into partial fractions and 
summed for all values of i\ then, by giving to Tc, V the values 1, 2, respectively, the 
fourth term of (30) may be found. 

The last term may be obtained by decomposing l/a i a i+1 a i+fc ' +1 ai +fc ' +2 into partial 
fractions, as before, and summing for all values of i. The result is 



Replacing 2a by its value, this may be again decomposed and summed for all values of 
# from 1 to OD . The decomposition is best effected by putting k for # + 1 ; the expression 
can then be exhibited as a function of F. After summation for all values of k from to QO , 
the terms for &=0, &=1 must be subtracted. 

The results are given by Dr Hill so as to include all terms of an order less than the 
sixteenth* As far as the sixth (since 1 - 6 is of the first) order, we have found in equa- 
tions (30), (31), 



Taking that value of c , obtained from (28), which is nearest to unity, this expression for 
A (0) gives c =l-07158 28. The value obtained by Dr Hillf to 15 places of decimals, with 
m= -08084 89338 08312, is C =1'07158 3277^S|ffl2, giving 

25730 04864. 



He has also obtained c literally, to the order m 11 , from the equations of condition*. 

The Terms whose Coefficients depend on m and on the Second and 
Higher Powers of e. 

273 The Terms of Order e 2 . There are two classes of terms to be considered, namely, 
those terms whose characteristics are zero and those whose characteristics are A The 
former are the parts of the Yariational Inequalities, that is, of the terms with arguments 
ft* whose coefficients are of the order #; the latter are those Elliptic Inequalities whose 
coefficients are of the order a and whose arguments are of the form 2if2$. We shall 
only indicate the method of procedure. 

(a) The Parts of the Variational Inequalities which are of tJie Order e\ 
For these it is necessary to put ^=0 in equation (13), but, instead of neglecting in the 
results all terms which depend on a--a proceeding which gave the equations (7)-we must 
g ve to puch values that terms of order * are included. In the first two terms p therefore 
Ses the values 0, 1, and in the third term the values 1, 0. The equation ^ obtained 
from the coefficient of & becomes, on combining those terms multiplied by [2,J (2,,) and 
containing the suffix c, which are equal when /-- 1 is put for ^, etc., 



-^ 

except for j0. It will be noticed that those terms whose factors involve c, are at least 

* Motion of the Perigee, p. 32. 

I S W.ffll, "Literal Expression for the Motion of the Moon's Perigee," Annals 
(U. S. A.), Vol. ix. pp. 31-41, 



224 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

of the second order with respect to e j hence the known part of c, which depends only on 
m and which is denoted above by c , is sufficiently accurate. 

Let ^2i=<%+&%> 

where 2i is the coefficient found in Section (i) and 8a 2i is the new part of the order e 2 , 
Also, neglecting powers of e above the second, we have 



Substituting these in the equation just given and making use of equations (7), we 
obtain, to the order required, 



The quantities a 2i , ^, e/ having been already found, we have a series of linear equations 
for the determination of the coefficients ba>i ; but since the value y=0 is excluded, these 
equations are only sufficient to determine && 1 , Sa 2 , ... in terms of m, e 2 , 8a Q . It is 
necessary to see how a may be found. 

Owing to the introduction of a in the values of v, <r, one of the members of the product 
a-4 is arbitrary. When the inequalities dependent on m only were considered, we put 
^L =a =l (Art. 258) ; since we are now considering the terms dependent on e 2 as well as 
those dependent on in, the definition of A Q must be extended so as to cover these terms. 
It is found to be most convenient, in making the calculations after the method outlined 
here, to define A Q in the same way as before, so that 



The same definition holds when the terms of the orders e*, e' 2 , etc. are under consideration ; 
the new part of a, of the order e 2 , can then be found by an extension of the method of Art. 
255. The equations, which are linear with constant terms, are now sufficient to determine 
a 1 , Sa^g,.. . in terms of e 2 , m, and the method of continued approximation may be used 
to find the unknowns. 

It may be remarked that, when the method, referred to at the end of Art. 28*7 below, is 
used, it is more convenient to define a to be such, that its value obtained in Art. 255 is 
unaltered by any of the subsequent approximations ; thus Sa=0, but 8a Q is no longer j^ero. 
In this method, the requisite number of equations for finding & , Sa 1 ,,.. appear naturally. 

(6) The Terms whose Characteristics are e 2 . 

These terms have arguments of the form 2i 2<. We therefore put q = -h 2, q = - 2, 
successively, in the equation (13), such values being given to p that terms of a higher 
order shall not occur. All the terms are directly seen to be at least of the order e\ We 
then obtain, for the determination of the unknown coefficients, a series of linear equations 
which correspond in number to the unknowns, and which contain constant terms ; all the 
unknowns can therefore be found by continued approximation. Since all the terms are of 
the order e\ the value of c to be substituted in the coefficients [2/+2c, 2i], etc. is known 
with sufficient accuracy. 

274. The Terms of Order e\ These again are of two kinds: those with arguments 
$, 2i3<. For the latter we put #=+3 in the equations (13); all the terms 
present are of order e 3 , and the determination of the unknowns proceeds on exactly the 
same lines as that of the terms whose characteristics are e 2 , The terms of arguments 



273-275] (iii) MEAN PERIOD INEQUALITIES. 225 



2i<j> are troublesome because the value of c, already found, is not sufficiently accurate: 
it must be known to the order e 2 . 

Put =-fl, -1 successively in equations (13) and give to p such values that terms of 
the order e 3 may be included. The only coefficients present will be found to be A^ A zi+0 , 



-4 2 i-c SSB */ + &/, c=c +5c; 

Set, e/ are then of order e B and &a 2i , fie of order e 2 . The terms which are of the order e 
vanish by equations (14); the coefficients A^ i+ ^ -4 2 i-2< are known by Art. 273 (5). We 
have remaining a set of linear equations for the determination of Sc, &$, </, containing 
known terms independent of these quantities. 

Since one of the coefficients A 2 i a was arbitrary, the same must be true of one of the 
Se$, </ ; this fact is also dedxicible from the consideration that the number of unknowns 
is greater by one than the number of equations. It is most convenient to determine the 
arbitrary so that, when all powers of e are included, 



so that &? =e '. "With this assumption the values of 6^, */, c may be obtained by 
continued approximation. 

The method of carrying out the approximations outlined in this and in the previous 
article may be found in Pts. v. ix. of the paper referred to in Art. 261 above. The 
results are obtained numerically as far as m is concerned, the value of e being left arbitrary. 

The fact that one of the dc^ 5c/ is arbitrary implies that some relation free from these 
quantities and containing only dc can be obtained from the equations of condition. The 
author has given a definite form to this relation, with several extensions, on pp. 336 338 
of a paper entitled Investigations in the Lunar Theory*. The method of obtaining it 
will be outlined in connection with the latitude inequalities arid the determination of fig. 
See Arts. 284, 285, 288 below. 



(iii) The Terms whose Coefficients depend only on in, d. 

275. Since the terms dependent on the solar parallax and on the latitude 
are neglected, we put H = fl a , # = Q in the equations (23), (19) of Art. 20. 
Hence, it is necessary to add to the right-hand members of the equations (1) 

of Art. 245 the terms 



respectively. But when z is neglected, f) a is of the form (Art. 128), 



where A, B, C depend only on the coordinates of the Sun. The terms to be 
added to the right-hand members of equations (1) are therefore 

- 3 (Au 2 + mva* + Co- 2 ) 4- D- 1 (i/DA 4- 2ucrjDB + <r 2 DC), 2(V - 2AtA . .(32). 

* Amcr. Journ. Math. Vol. xvn. pp. 318-358. 
B. L. T. 15 



226 METHOD WITH BECTANGTJLAR COORDINATES. [CHAP. XI 

To find A, B, we have, by Art. 19, 

fl 2 = 3m' J^ffif - J ( + cr) j - m'v* gj - l) . 
But, by Art. 22, 



_ 
- (u + <r) cos <> 7 - n'tf - e) - 4 V^ - <r) sin (V - n' - e'). 

Let<y'~^-e 7 =F'. Then 

rflf = J-v exp. (- F 7 V^l) + J<r exp. (+ F 7 V^l), 
r 2$ 2 = j ^ ex p ( (- 2 F' V^l) + i a 2 exp. (+ 2 T 7 V^l) + 4 ucr - 
Substituting in 1 2 , we find 



C + A = |m 2 ('~ COS2F 7 -!) , * C - A = fm 2 V^l ~7 3 siri2F'. 



whence 

Since v' = i/ - *?' - (w' 4- e' - w 7 ) =/' - w' (Arts. 48, 53), the only functions 
which have to be calculated are 

a' 3 /r' 3 , (a /3 // 3 ) cos 2 (/' - /), (a /3 /r /3 ) sin 2 (/' - w f ). 

These are expanded in powers of e f and in cosines and sines of multiples of w f 
after the manner explained in Arts. 39 41. The results for them have been 
fully worked out by several investigators*. 

Assume -^ 1 = SjpOp' cos pw' 9 

a' 3 a' 3 

s cos 2 v 1 - I = Sjp^Sp 7 cos puf, -7 3 sin 2 F 7 = 2^/9/ sin pw/, 

where p = oo . . . + oo and a?^ p = o^ 7 ; then a/, /8p' are known functions of e'. 
Using these expressions, we have 



Since the coefficients of t in m and e^'^- 1 are the same, we can put m for 
e w'V_i ^f we remember that, when returning to real variables, e pw '* f ~~ l is to be 
put for P m ; the value of m will not be substituted in the index of ". Hence 
we can write 



* See the references given in Arts. 123 ? 126, 



275-277] (iv). PARALLACTIC INEQUALITIES. 227 

These values must be substituted in the terms (32), and the results added 
to the right-hand members of equations (1). 

276. It is evident that the required solution of the equations is a 
particular integral corresponding to the newly added terms. Since e is 
neglected, the solution will be of the form 



These values being substituted in, the equations, and the coefficients of 
equated to zero, we shall have a series of equations of condition to deter- 
mine the unknown quantities. 

The method is similar to that used in Case (ii). Neglecting, initially, 
powers of e f higher than the first, the values v = V Q> cr = CT O , given by equations 
(3), can be used in the right-hand members, since A, B, C are at least of the 
order e. We obtain a series of equations similar to (14) ; but since their 
right-hand members are no longer zero, there is no relation like that which 
was necessary to find c : this is otherwise evident, for the index of % in all 
these terms is quite known and no new arbitrary constant is required. The 
higher approximations proceed in the same way. The inverse operation D"" 1 
will introduce divisors of the form 2j 4 gm ; but as m is assumed to be 
incommensurable with any integer, none of these divisors will vanish. 

Some results, obtained by the author for inequalities dependent on e', will be found in 
two Notes in the Hon. Not. It, A. S. Vols. LIV. p. 471, LV. pp. 35. 

(iv) The Terms whose Coefficients depend only on m, I/a'. 
277. Since e f is neglected we have 



therefore the terms to be added to the right-hand members of equations (1) 
are respectively 



Also, by Arts. 19, 22, we have rS = X = % (v + <r), r' = a! ', and therefore, as 
z is neglected, 

^ 



The terms to be added to the equations are therefore of the third, 
fourth,... degrees in u, cr, corresponding to terms of the first, second,... degrees 
with respect to I/a'. 

152 



228 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

If, in the expressions (33), the values v , <r , which are odd power 
series in f and which correspond to the intermediate orbit, be substituted 
for v, a, the terms produced by fl p will be odd or even power series in * 
according as p is odd or even. The values of v, <r, when terms dependent on 
the solar parallax are included, will therefore contain even as well as odd 
powers of Hence we assume 



This solution includes the intermediate orbit. When I/a' is neglected, we 
have AX = a^ and A^^i = 0. 

The process is similar to that of Case (iii). First, neglecting I/a' 2 and 
higher powers, we find the odd coefficients A^^\ the parts of A& which are 
of the order l/c&' 2 are next found ; and so on. Since the equations of condition 
at any stage of the process are linear, with known terms in their right-hand 
members, no relation, independent of these coefficients, exists and the un- 
knowns can be calculated by continued approximation. 

For the details of the calculations, two papers by the writer-' On the Part of the 
Parallactic Inequalities in the Moon's Motion which is a Function of the Mean Motions 
of the Sun and Moon*,' and 'On the Determination of a Certain Class of Inequalities in 
the Moon's Motion t ' may be consulted. See also G. W. Hill, 'The Periodic Solution as 
a first approximation in the Lunar Theory J.' 



(v) The Terms whose Coefficients depend only on m, 7. 

278. When e\ I/a" are neglected, fl 0, and the equations (23), (19), (18) 
of Chap. ii. become 

D*(va + z 2 ) - DvD<r - (Lzf- 2m (vDa - <rDv) + f m 2 (v + a") 2 - 3mV = 9 



9 ) ( ~ . 
J 

(35). 

The form of the last equation shows that its integral contains a constant 
factor which is one arbitrary of the solution. In the case of the Moon this 
factor is small : it has been denoted in previous chapters by 7. If, in the first 
two equations, terms of the order 7 2 be neglected, they reduce to those of 
Art. 244 ; the new parts of v, a are therefore at least of the order 7 2 . We 
shall neglect the constant of eccentricity and consider the first approximation 
to the solution of equations (34), (35) to be the intermediary. 

The procedure is the same as in the previous cases. We suppose the 
intermediate orbit known and consider first the terms of the order j 1 ; these 

* Amer. Journ. Math. Vol. xiv. pp. 141-160. 
t Mon. Not. E. A.S. Vol. LII. pp. 71-80. 
Astron, Journ, Vol. xv. pp. 137-143, 



277-279] (v) PRINCIPAL INEQUALITIES IN LATITUDE. 229 

only occur in z and they are obtained from equations (35) by inserting therein 
that value of r which corresponds to the intermediary. The terms whose 
coefficients are of order <y 2 only occur in v, a and they are found, when those 
of order 7* have been obtained, from the equations (34); the terms of the 
order 7* only occur in z and they are obtained from (35) ; and so on. 

The solution may be represented by 



where u^-, cr^, z y ti+i represent the terms whose coefficients are of the orders 
denoted by the suffixes. Therefore, neglecting powers of 7 above the third 
and remembering that r 2 = u<r-M 2 , r 2 = v ^o> where r , v 0) <7 refer to the 
intermediary, we have 

1 1 

r* " {(1/0 + tv)(oo+ ay) 4- V}* 

Substituting, equation (35) becomes 



_ 1 /.. _ 3 l/oOr Y 2 + g- lV + V\ 
" ^ \ * r* ) 



the differential equation for the terms in z as far as the order rf. 

Instead of (35), we may use an equation free from the divisor r 3 . This equation 
deduced immediately from equations (17), (18) of Art. 19, after putting ii = 



D (zDv - vDz) + ZmzDv -f f m 2 (v + <r) + m 2 ^ = 

(or a similar equation in which o-, v, D replace v, <r, D). This new form will be found useful 
for a literal development in powers of m ; the method given in the text is less laborious 
when the numerical value of m is used at the outset. 



The Terms dependent on mi and on the First Power of 7. 
Principal Part of the Motion of the Node. 

279. "When terms above the order 7 1 are neglected, the equation (36) 
reduces to 



(37). 



Substituting for m 2 + */r 8 the series 22Jf<?* (Art. 265), in which Jf^ = M i9 

i = oo . . . -H oo , we obtain 



Since Mi is of the order m' 2 * 1 at least, this equation is of exactly the same 
form as (23 X ) and it may be treated in an exactly similar manner. The two 
independent integrals are given by 

0' = - oo . . . + oo ), 



230 METHOD WITH RECTANGULAR COORDINATES. [CHAP, XI 

and the substitution of either of these will give a series of linear homogeneous 
equations, from which the ratios of the unknown coefficients may be found; 
the infinite determinant, formed by eliminating the unknowns, will give g. 

280. In order to connect this solution with that obtained in Art. 147, 
let the latter be written 

z = 2a2 sin2 v = - oo ... + oo 



As with the elliptic inequalities (Art. 257) we put 

p = exp. 2f V^l = exp. 2 (n - ri) (t - ) V^OL, 

Sf = exp. TJ V^l = exp. (gut + e - 6) V~l = exp. g (n - ri) (t - 2 ) V - 1 ; 
so that g = gn/(n - w/) = g (1 -f m), g 2 - w') = 6 - e. 

If it be recollected that tf is to be replaced by 8 in the part of the index of 
which contains g, the solution may be written in the form 



where K'-j = K^ 

When this is substituted in equation (37') and the coefficient of f 
equated to zero, we have 

0, (i,j = -~ <*... + oo).. .(39). 



The equations for K^> obtained by equating to zero the coefficients of 
are of the same form. The elimination of the Kj or of the K f ^ gives an 
infinite determinant V (g). This determinant being of the same form as A (c), 
all the results proved in Arts. 266 272 will be available if we replace &i by 
i and c by g. 



The part of g which depends only on m, may therefore be found by taking 
the value of g , nearest to unity, given by the equation 

sin 2 ^TTgo = V (0) sin 2 ^ V2 ; 

where V (0) denotes the determinant A (0) of Art. 267, after < has been 
replaced by 2Jfi. 

The determination of g by this method was first made by Adams*. With the value 
m=n'/n= -0748013 (exactly) or m = '08084 89030 51852, he finds g=r085l7 13927 46869, 
giving #-1 = -00399 91618 46592. Mr P. H. Cowell has verified this value and he has ob- 
tained the literal and numerical values of g , K h as well as those of the terms of orders 



* " On the Motion of the Moon's Node in the case when the Orbits of the Sun and Moon are 
supposed to have no Eccentricities, and when their Mutual Inclination is supposed to be small/' 
Mon. Not. R. A. S. Vol. xxxvm. pp. 43-49 ; Coll. Works, pp. 181-188. 



279-283] (v) TERMS IN LATITUDE OF HIGHER ORDERS. 231 

281. When g has been found, the coefficients Kj can be determined in 
terms of one of them, say of K Q , by means of equations (39). We first leave 
aside the equation given by j = ; when the other coefficients have been 
found in terms of K^, their values should satisfy this equation: it is therefore 
useful for purposes of verification. 

The coefficient of sin 77, that is, of (*-^)/2V~i will be 2a J BT , since 
JST ' = J5T . In Art. 147 this was denoted by ay. Hence 

ay = 2a.K" . 

The ratio a : a being previously known, we have the new constant of 
latitude in terms of the one used in Chap. vn. The relation will be modified 
when terms of the order 7* are considered. 

The Terms of Order 7 2 . 

282. It is not necessary to give detailed explanations concerning the 
calculation of these terms. They are deduced from equations (34), when # Y 
has been found, in exactly the same manner as the terms in e 2 were deduced 
when those of order e had been found. We give to z the value # y just obtained 
and put v=si/ + Uy, <r = cr -fay, rejecting, after the substitution, all terms 
of an order higher than 7*. The solution divides into the parts which depend 
on f 3 * and f 2 ** 3 *, respectively, and the coefficients are calculated in the manner 
explained in Art. 273. The value g of g is sufficiently approximate. 

We suppose then that u, <r or ty, ay are known correctly to the order 7*. 

The Terms of Order 7*. 
The Part of the Motion of the Node which is of Order 7 a . 

283. Eeturning to the equation (36), we substitute in its right-hand 
member the values of $ Y , v y *> oy, v , <T O , r , previously obtained, since this 
portion is of the order 7 at least. Also, from its symmetry with respect to 
v, <r, we see that it will be expressible in terms of (* - -**), (?** - 

with known coefficients. With regard to the left-hand member, m 2 
is expressible as before in terms of f 2 ^ + f"" 2 *. We cannot, however, put 
D% - (m 2 + /c/r 8 ) z y = 0, for, when the value of # y is substituted, D% contains 
g in its coefficients. Denoting the value of g, to the order 7 2 , by g + 8g, we 
see that D% will produce terms of the order 7g, that is, of the order 7'. 



Assume as the solution 

^T = a 2, {(} + &ZJ) ^ + (E'-j + &' 

+ terms dependent on 



232 METHOD WITH KECTANGULAR COOKDINATES. [CHAP. XI 

where K'^ = - Kj and SK'^ = - BKj. The terms of the second line will be 
all of the order 7 3 , since the index of contains 3g : they can be equated to 
the terms of corresponding form on the right-hand side of (36), and the unknown 
coefficients found in the usual way. The value g of g is evidently sufficient 
for these terms. We shall therefore leave them aside and consider only the 
equations of condition obtained by equating the powers of ^ s to zero. 

284. With this understanding we have, since g = go + Sg, 

D*z V^l = a % (2/ + go 
a 2, (2j + g ) 

+ 2a Sg 2, (2; + go) (JZ} (*" + *%, (T^*) + a^ (2j + go )(6Z, p+* 
omitting terms of an order higher than </. 

When the latter expression is substituted in (36), the terms under the 
first sign of summation cancel with - (m a + */r 8 ) **> by (87 7 ); the remaining 
terms are all of order 7 s . We have therefore 

2aSg 2, (2j + go) (^ &** + JST^ r^*) + a2 j (2j -1- go ) 2 
- a (m 2 + /./r 3 ) 2,- (SJ^ &+* + SK'^ J^M) = - f ^ 

'0 

those terms on the right, depending on 2j8g > being left aside. Equating to 
zero the coefficient of ^+ 8 and putting m 2 + /c/r 3 = 22^^, we deduce 



. . .(40). 



The number of equations obtained from this by giving to j the values 
0, 1, 2,. . . is one less than the number of unknowns 8JS}, % ; but since Jf 
was arbitrary, SJT must be also arbitrary and it may be determined at will : 
when the arbitrary value, to be given to 8jBT , has been fixed, the number 
of unknowns will correspond to the number of equations. As Sg is of the 
order 7 2 , while Sj6T is of the order y, the value of Sg is independent of the 
value to be given to 8-2^ and therefore some relation, independent of the 
unknowns 8jK} but involving Sg, must exist between these equations. We 
shall now find this relation, 








285. The Equation for Sg. 
We have 

ToUya + # v 2 )/r 5 = terms of order rf in the expansion 

f z t {( V Q + U Y S ) (^o + ^-y 2 ) + Vl"* ^ n powers of y. 



283-286] (v) NEW PART OF THE MOTION OF THE NODE. 233 

Let JB 2 Y 2 = (i; + v y z) (<T O + cry) -1- Y 2 = mfce o/r 2 correct to ry 2 . 

Substituting, we see that the right-hand member of (40) is 
The part, of order 7*, of the coef. of ^ +s in the expansion of tcz y V l/aJJ%2. 

Multiply (40) by Kj and sum for all values of j. Since M^ = -MJM, the 
terms involving the unknowns SKj on the left-hand side will be 



As j, i have the same range of values, we may interchange them in the 
second term of this expression which then becomes 

2,. {(2j + go)' K } - 2S; M^ K t ] SK } . 

This is zero because the coefficient of SJZ} vanishes for all values of j by 
equations (39). Hence all the unknowns SKj disappear. 

The result of the summation of the equations (40) is therefore 

28g2j (2j + go) Kf = Sj \Kj x coef., order 7 3 , o 
In an exactly similar manner we may find 

2Sg2,- (2j + go) K^ = 2,- [K'-j x coa/., order f , of ?*-* in 

But since JfLj = - Kj, equation (38) may be written 

- Y V"l = a Sj (^ ? 2 ^ g + J2i r 2 ^ 8 ), 

and therefore &K jt &K f ^ are respectively the coefficients of ~ 2 >- g , ^' +g in 
z y ^~^l. Whence, adding the two previous equations and putting 
jBT%-jff/, we have 
4% 2j (2j + go) JST/ = Sj [(cofl/. o 



or, 4Sg 2j (2j + go) J5Tj 2 == Part, order y\ of const, term in KZ 

where ^/B?Sf^ is supposed to be expanded in powers of 7 and in cosines of 
multiples of the arguments contained therein. Since all the other quantities 
present in this equation have been previously found, it constitutes a simple 
equation for finding Sg. 



286. We have seen that SJ? niay be determined at will It may be 
fixed so that the coefficient of sin v\ in z is 2a r at any stage of the ap- 
proximations and therefore in the final results ; hence &5T = 0. When Sg has 
"been found, the equations obtained by putting j= 1, 2,... in (40) will 
enable us to find the coefficients SJ5T 1 , $K^ 3 ... by continued approximation. 
The equation given by j = is then superfluous : it may be used as an equa- 
tion of verification for the values of 8g, SK J9 when these have been calculated. 



234 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 



(vi) Terms dependent on m and on Combinations of e' y e, 7, I /a 

and their Powers. 

287. The developments which have been given above suffice to indicate 
the general method by which all the solar inequalities in the motion of the 
Moon may be found. The only real difficulties which occur are those arising 
when the motions of the Perigee and of the Node are required. The infinite 
determinant gives the principal parts of these with an accuracy which leaves 
nothing to be desired ; the other smaller portions can all be found by simple 
equations in the manner explained in Art. 285. 

The coefficients of the terms of any order in e, e', 7, !/&' will always be 
determinate by a set of linear equations when the terms of lower orders in 
these constants have been found. It will be noticed from what has gone 
before that, at every stage, all powers of m are included, and also that, when a 
term of argument i'l;p<l> qfi j*r] is being found, all the terms of argu- 
ments (2i + i') % p<j> q<j> f jrj (if = or 1 and i, p, q, j, positive integers or 
zeros) are determined at the same time. The process is therefore reduced 
to the approximate solution of a certain number of sets of linear equations 
the number of such sets depending on the order in e, e' 9 y, I/a' to which it 
is desired to take the solution. 

After finding the intermediate orbit and the principal parts of the motions of the perigee 
and of the node, the use of the equations (19), (23) of Art. 20 for obtaining the other 
inequalities is not essential. We may return to one of the original equations (17) of 
Art. 19 and develope the functions KV/T^ no-fr* in powers of Sv, da- by putting v = u 4-fru, 
<F= (T -Ho-, where v, So- denote any of the series of inequalities to be found. This plan of 
procedure, which has some advantages, especially in the terms of lower orders in e, <s', y, I/a', 
is outlined in the memoir referred to in Art. 274. The method, given above, for the deter- 
mination of &g, together with its application to the other parts of the motion of the node 
and to the motion of the perigee, will be found in the same place. 



Relations "between the Higher Parts of the Motions of the Perigee and the Node 
and the Non-periodic Part of the Moon's Parallax. 

288. The equation (41) is only one particular case of a much more general theorem by 
which all the parts of the motions of the perigee and of the node may be found when their 
principal partsthose depending on m only have been obtained. Modified forms of the 
equation will serve for the determination of those parts of g which depend on all powers 
and products of e' 2 , I/a' 2 , e\ y 2 , so that the motion of the node really depends only on the 
solution of an infinite determinant and of a series of simple linear equations with one 
unknown. ^ The same is true of the perigee. For example, the part of the motion of the 
perigee which is of the order e 2 may be shown to be the value of Sc given by 



Con$t. part, order et, in the expansion of K (Z e2 ^+ r e 



287-289] SOME GENERAL THEOREMS. 235 

where ^ , % e , %& and y , y^ y& denote the parts in X and Y whose coefficients are of 
orders e, e\ e 2 , respectively, and where 

TV 



An important extension of the results (41), (42) consists in the fact that they are true 
when all powers of e' 2 are included in the various terms. In other words, when we suppose 
that all parts of the functions z^ R yZj which depend on m, e' 2 and on y 1 , y 2 , have been found, 
and when g , Ay are replaced by their more accurate values g +e' 2 /(m, e' 2 ), Kj+ePyf (m, e' 2 ), 
the part of the value of g which depends on y 2 and on all powers and products of m, e' 2 is 
given by (41) ; a similar result holds for part of the motion of the perigee given by (42). It 
is for this reason that the determination of the parts of the solution which depend only 
on m, e' should be the step immediately following the calculation of the intermediate 
orbit ; the necessary parts of the motions of the perigee and of the node, which depend on 
m, e'\ are found by the same method. 

289. One general theorem is as follows : Let e, y be the constants of eccentricity and 
latitude, and let the coefficients e,-, e'_y, JTy be expressed in terms of them ; these coefficients 
are supposed to be of the form e/(m, e' 2 ), and c , g of the form < (m, e' 2 ). The constant 
part of I IT will contain the terms 



and the motions of the perigee and the node the terms 

He 2 +Ky 2 , M<2 2 +Ny 2 , 
respectively, where E, F, G, H, K, M, N are functions of m, e' 2 only. Put 



where powers of e' 2 are supposed to be included in c , ey, etc. It may then be shown that 
HT.6E, KT =6F=MT Y , NT V =6G ........................ (43). 

The theorem which lies at the basis of these results is as follows : If JT, F, z (Art. 18) 
have been fully calculated to the order &y**~*, where p=0, 1, ... 2? (that is, to the order 
20 in e, y), the constant part of 1/r can be obtained to the order e*y+ 2 -*>, where p=0, 
1, ... 4+2 (that is, to the order 2$r+2), without further reference to the equations of 
motion, by a purely algebraic formula involving only the values of Z, T, z to the order 
calculated. A different and more complete statement of this theorem is, that the terms of 
order 2# with respect to e, y, m the constant part of the expansion of 3//2 2 in powers of e, y, 
are equal to the corresponding terms in the expansion of 



where X m , Y m , Z n contain the terms as far as the order 2? ; ^ 2Q , y 2a , % are the terms in 
jf Y % , whose coefficients are of the order 20, and ^ M = X\ q + Y\+ AQ-I- Tms 
result also holds when all powers of the solar eccentricity are included ; the only quantity 
neglected is the solar parallax. 

The proofs of the theorems in this and in the previous Article are too long to be 
inserted here; they will be found in Part n. of the author's paper referred to in 
Art. 274 above. 



236 METHOD WITH RECTANGULAR COORDINATES. [CHAP. XI 

The first of Adams' two theorems* can be deduced from the result just enunciated, by 
putting q=l. It states that, in the constant part of the expression for a/r, there are no 
terms of the forms <? 2 /(m, <?' 2 ), y 2 /(m, e' 2 ) ; here, a/(/x/^ 2 )* may be a function of m, e'\ but 
it must not contain e, y. His second theorem is immediately obtainable from the equa- 
tions (43). It states that 

H/K=E/F, 



These interesting results may be also utilised as equations of verification. 



* J. C. Adams, " Note on a remarkable property of the analytical expression for the constant 
term in the reciprocal of the Moon's radius vector," M. N. JR. A. 8. Vol. xxxvm. pp. 460-472 ; 
Coll. Works, pp. 189-204. 



CHAPTEE XII. 



THE PRINCIPAL METHODS. 

290. A BRIEF account of the methods which have been used to attack 
the lunar problem, other than those considered above, will be given in this 
Chapter. In general, only those theories which have been tested by actual 
application to the discovery of the perturbations produced by the action 
of the Sun will be analysed. The historical order will be adhered to as 
far as possible, although the time elapsed between the first announcement 
of a plan of treatment and the full publication of the results, frequently 
makes it difficult to assign any exact date to a theory. 

The history of the lunar theory and of celestial mechanics generally (in 
the sense in which these terms are now understood) began in 168*7 with the 
publication of the Principia] the portion which especially refers to the 
motion of the Moon is contained in Props. 22, 25 35 of Book in. In this 
and in the later editions of the same work Newton succeeded in showing that 
all the principal periodic inequalities, as well as the mean motions of the 
perigee and the node, were due to the Sun's action, and he added some other 
inequalities which had not been previously deduced from the observations. 
The result which he obtains for the mean motion of the perigee is only half 
its observed value ; it appears, however, from the Manuscripts * of Newton 
which have come down to us, that he had later succeeded in obtaining its 
value within eight per cent, of the whole. The conciseness of the proofs, 
when they are given, makes his work very difficult to follow. It is now 
generally recognized that he used his method of fluxions to arrive at many 
of the results, afterwards covering up all traces of it by casting them into a 
geometrical form ; if this be so, the claim of Clairaut to be the first to apply 
analysis to the lunar theory must be somewhat modified. No substantial 
advance was made until the publication, more than sixty years later, of 
Clairaut' s TMorie de la Lune. 

* Catalogue of tlw Portsmouth Collection of Books and Papers written by or belonging to 
Sir Isaac Newton. Cambridge, 1888. 



238 THE PRINCIPAL METHODS. [CHAP. XII 

For the methods which Newton used, the reader is referred to the Principia and to the 
numerous commentaries which have been written on it. An analysis of the Prindpia 
and an account of Newton's life and works published or in manuscript together with 
many references, have been given by W. W. Rouse Ball*. 

The history of the lunar theory up to the publication of the third volume of the M&ani- 
que Celeste of Laplace and an account of the methods used, have been given by Gautierf. 
For further details concerning the theories of Newton, Clairaut, d'Alembert, the method 
given by Euler in the Appendix to his first theory, Euler's second theory, and concerning 
the theories of Laplace, Damoiseau and Plana, reference may be made to the Mdcanique 
Celeste of TisserandJ. 

291. Clairaut's Theory . 

Clairaut commences by finding the differential equations of motion when 
the latitude of the Moon is neglected. He takes the inverse of the radius 
vector and the time as dependent variables, the true longitude as the inde- 
pendent variable, and he finally arrives at equations which are equivalent 
to (8), (9) of Art. 16 ; since the latitude is neglected, u^ is, in his theory, the 
inverse of the radius vector. 

The process of solution is one of continued approximation. He considers 
the orbit of the Moon to be primarily an ellipse, but, recognizing that the 
apse revolves quickly, he introduces the quantity denoted above by c, 
so that the first approximation is the modified ellipse. The forces due to 
the Sun's action are expressed in terms of the radii vectores of the Sun and 
the Moon and the difference of their longitudes, and thence, by means of the 
elliptic formulse, in terms of the true longitude of the Moon somewhat 
in the manner explained in Art. 127 above. The calculations are, in certain 
cases, carried to the second order of the disturbing, forces, and, in particular, 
the motion of the perigee is obtained to this degree of accuracy. 

The motion of the Moon in latitude is obtained by considering the 
equations for the variations of the node and the inclination ; these correspond 
to the fifth and sixth of the equations of Art. 82. They are much less 
accurately worked out than the motion in the plane of the orbit. 

Clairaut was the first to publish a method for the treatment of the lunar theory founded 
on the integration of differential equations. The artifice of modifying the first approxima- 
tion by the introduction of c, in order to take the motion of the perigee into account from 
the outset, is also due to him. The determination of this motion is marked by an event 

* An Essay on Newton's Principia. Macmillan, 1893. 

f JEssai historique sur le Prolttme des Trois Corps. Paris, 1817. 

Vol. in. Chaps, in, vn. , ix. 

Announced in 1747. The first edition of the TUorie de la Lune was published in 1752, the 
second and most complete edition in 1765 at Paris. The latter is a quarto volume of 162 pages 
and it contains Tables for the calculation of the position of the Moon, founded on gravity only. 
The first set of tables was published separately in 1754. 



290-293] CLAIRAUT. D'ALEMBERT. EULEE. 239 

which shows what progress astronomers had made in admitting the sufficiency of Newton's 
laws to account for all the known phenomena of the celestial motions. It must be remem- 
bered that Newton's later determination, as shown in his manuscripts, was then unknown. 

It had long been a difficulty that the first approximation to the motion of the perigee 
gave only half the observed motion. Clairaut, thinking that the Newtonian law must be 
considerably in error, tried the effect of adding a term, of the form K/r 3 , to the forces 
resolved along the radius vector, and he succeeded in showing that it would account for the 
observed motion. It then occurred to him to carry his approximation a step further, with 
the law of the inverse square only, and to see what effect the evection had when introduced 
into the expressions for the disturbing forces. He soon found that the new term was nearly 
equal in absolute value to the second term and that the greater part of the motion was thus 
accounted for by the law of Newton (see Art. 169). 

292. D'Alembert' s Theory*. 

This is very similar in its general plan to that of Clairaut, but while the 
latter worked out his results numerically, d'Alembert considered a literal 
development and carried out his computations with more completeness. He 
gave the term 3ffi m 3 in the motion of the perigee : Clairaut had only 
obtained its numerical value. His method of approaching the modified orbit 
is much more logical ; he introduces a part of the Sun's action into the first 
approximation by proceeding in a manner analogous to that of Art. 67 above. 

D'Alembert made several contributions to the theory. He succeeded in showing that 
terms increasing continually with the time can be avoided, and he gave a direct method of 
approaching the first approximation. He also recognized the fact that the question of the 
convergence of the series obtained ought not to be neglected. He further considered the 
effects produced by small divisors and showed that the coordinates might be expressed by 
means of only four arguments which were necessarily related to the orders of the coefficients. 
Tables are added at the end of his researches. 

293. Eider's First Theory f. 

Euler commences by considering the equations of motion referred to 
cylindrical coordinates; these, translated into modern notations, are the 
equations given at the beginning of Art. 16. The equation for the latitude 
is immediately replaced by two others which are practically those for the 
variations of the node and inclination, and they are obtained under the 
assumption that the first differential of r tan 17, with respect to the time, 
has the same form in disturbed and undisttirbed motion. The second 
equation of motion is integrated and the resulting value of v is substituted 
in the first equation which then contains only the differentials of r l9 with 
respect to the time, and certain integrals depending on the forces. 

* Eecherches sur diff&rents Points importans du Systems du Monde. Pt. i. (1754) Thforie de la 
Zune, 8vo. LXVIII. -|- 260 pp. D'Alembert sent in his theory to the Secretary of the Academy 
in January, 1751. 

f Theoria Motus Lunae, etc. (with an Appendix) 4to. 347 pp. Petrop. 1753. 



240 THE PBINCIPAL METHODS. [CHAP. XII 

The independent variable is changed from t to the true anomaly/ and for 
r is put its elliptic value (in terms of/) multiplied by l + v, v is then a 
small quantity depending on the disturbing forces (cf. Hansen's method, 
Art. 209). Euler thus arrives at a differential equation of the second order 
in which v is the dependent and / the independent variable. 

After expanding the disturbing forces according to powers of e, etc., he 
divides them into classes: those independent of e, e',i\ those independent 
of e' 9 i\ and so on, after the manner explained in Chaps. VII., xi. In 
considering the inequalities of the second class (Art. 132), he finds 

v = const. + Ci/4- periodic terms ; 

so that the motion of the perigee depends on c x . This is not determined 
directly. Its observed value is used and Euler then compares the latter with 
the value deduced from theory in order to test the Newtonian law. He has 
previously assumed that the attraction between the Earth and the Moon is 
of the form ytt/r 2 const. ; the constant is shown to be very small and well 
within the limits of error caused by the neglect of higher terms in the 
approximations. 

The numerical values of the constants to be used in the theory which is 
numerical as far as m is concerned and algebraical in respect of the other 
constants are determined by the consideration of thirteen eclipses; the 
want of definiteness in the meanings to be assigned to the constants, which 
affected the results of Clairaut and d'Alembert, is avoided, for Euler uses the 
formulae of his own theory in the calculation of these eclipses. 

In the Appendix, an investigation, which practically amounts to the 
method of the variation of arbitrary constants, is given and worked out with 
some detail. Euler expresses himself as unsatisfied with both the theories he 
has explained. 

294. Euler s Second Theory*. 

The method which Euler has here set forth with much detail is interesting 
as the first attempt to employ rectangular coordinates referred to moving 
axes in the Lunar Problem. He considers an axis of x revolving with the 
mean angular velocity of the Moon in the Ecliptic, that of y being also in 
this plane and that of z perpendicular to the plane. Taking the mean distance 
of the Moon as a, its coordinates are assumed to be a (1 -f #), ay, az, so that 
x, y, z are small quantities depending on the solar action and on the lunar 
eccentricity and inclination ; a is defined to be such that x contains no constant 
term. The equations of motion are found in the usual way, the disturbing 

* Theoria Motuum Lunae, nova methodo pertractata una cum Tabulis astronomicis unde at 
quodvis tempus Loca Lunae expedite computaripossunt,,.. : J. A. Euler, W. L. Kraft, J. A. Lexell. 
Opus dirigente L. Eulero. 4to. 775 pp. Petrop. 1772. 



293-294] EULER'S SECOND THEOEY. 241 

forces being developed in powers of 1/r'. It is to be noticed that this 
method, like that of Chap. XI., has the advantage of allowing the disturbing 
forces to be expressed as homogeneous functions of x, y, z of the first, 
second, and higher degrees ; but the relation between the degrees of the 
homogeneous functions and those of I/a', which was observed in the 
equations of Section (iii), Chap. IL, does not hold in Euler's method: the 
coefficients of these functions, in Euler's theory, are expressible by means 
of the two arguments , w'* 

The independent variable is w' 9 and Euler puts 

n/n' = mj. + 1 ; 

so that 1 + m 1 is the ratio of the mean motions of the Moon and the Sun : 
observation gives m x = 12"36.... The forces a (I -f a?)/?* 3 , ay/r*, a#/r 8 , due only 
to the mutual actions of the Earth and the Moon, are expanded in powers 
of a?, y, z. 

The general solution is then supposed to be of the form 



with similar expressions for y, z. Here e, i are the two arbitrary constants of 
the sohition corresponding to the eccentricity and the inclination. Sub- 
stituting these values in the differential equations and equating the coefficients 
of the various powers and products of e, e', etc. to zero, he obtains a series of 
differential equations for the determination of A, B lr . . . . The parts dependent 
on A give the variational inequalities, those dependent on jB 1? J? 2> ... the 
elliptic inequalities, and so on. In the determination of JE?i, the motion of the 
perigee arises ; as in his earlier methods, he assumes its value from observa- 
tion and verifies his results by means of the calculated value. The motion of 
the node is treated in the same manner. The various differential equations 
are solved by the method of indeterminate coefficients. 

M. Tisserand remarks that Eulor's method of dividing up the inequalities into classes 
requires some modification when we proceed to terms of higher orders, owing to the fact 
that the motions of the perigee and the node contain powers of e\ e' 2 ,,.- ; the arguments 
depending on these motions, when expanded BO as to put the solution into Euler's form, 
would introduce into the coefficients terms depending on the time. A reference to Art. 283 
above, will show how this objection to Euler's method may be removed. 

Euler's main contributions to the lunar theory are : the application of moving rect- 
angular axes ; the method of the variation of arbitrary constants, as given in the appendix 
to his first theory ; the use of indeterminate coefficients in the solution of the differential 
equations ; a new method for the determination of the constants from observation ; the 
formation and solution of equations of condition to determine the constants from observa- 
tion when the number of unknowns is less than the number of equations ; the final ex- 
pression of the coordinates by means of angles of the form a + pt. He also added to the 
subject in many other directions, and much of the progress which has since been made, 
may be said to be founded on his results. 

B. L. T. 16 



242 THE PRINCIPAL METHODS. [CHAP. XII 

295. Laplace's Lunar Theory*. 

The publication of Laplace's Mecanique Cdleste marked a new epoch in the 
history of the lunar theory, owing to the general plan of treatment adopted 
and to the manner in which it was carried out. Some account of Laplace's 
method has already been given in previous chapters. In Section (ii) of 
Chap. ii. his general equations of motion with the true longitude as inde- 
pendent variable and with the time, the inverse of the projected radius vector 
and the tangent of the latitude as dependent variables have been obtained. 
The first approximation found by neglecting the action of the Sun has 
been given in Art. 52, and the manner in which this is modified to prevent 
the occurrence of terms proportional to the time, in Art. 70. By means of 
the modified ellipse, those parts of the equations of motion which are due to 
the action of the Sun are expressed in terms of the true longitude (Art. 127). 

The equations can then be integrated. Laplace's method is to assume 
the solution to be a sum of periodic terms whose coefficients are unknown, and 
to substitute it in the differential equations: in each unknown coefficient 
the characteristic is written separately ; he thus obtains, on equating the 
coefficients of the different periodic terms to zero, a series of equations of 
condition by means of which the coefficients can be calculated. The new 
values of %, t, s are then used to find the third approximation. The method 
of procedure is similar to that of Chap. VIL, except that, instead of 
the equation for #, the second of the equations (11), Art. 16, is used and 
solved in the same manner as the first of these equations ; the analysis is, 
however, rather more simple owing to the forms of the left-hand members of 
the equations for u l} s. Terms up to the second order in e } e', 7, a/of are 
considered and certain terms of higher orders whose coefficients become large, 
owing to small divisors, are also included. The approximations are, in general, 
taken to the second order of the disturbing forces. 

The coefficients are not developed in powers of m. As soon as the 
equations giving the values of c, g and the equations of condition between 
the coefficients have been obtained, Laplace substitutes the numerical values 
of the constants in all terms ; only the characteristics are left arbitrary, so that 
a small change in the numerical values of any of the constants, except m } will 
not sensibly affect the coefficients. The theory is therefore a semi-algebraical 
one. The value of the coefficient of the principal elliptic term in the expres- 
sion of the mean longitude in terms of the true being thus obtained, the 
value of e necessary to his theory is deduced from observation ; the constant 
7 is found in a similar manner. His constants e, 7 are such that the 

* MScanique Celeste, Pt. n. Book vn. pp. 169303, 4to. Paris, 1802. Several editions have 
since appeared ; the latest, now in the course of publication, is in a collection of all Laplace's 
works. Laplace's investigations cover a period of thirty years anterior to the publication of 
Vol. in. of the Mec. C61, 



295-296] LAPLACE. SECULAR ACCELERATION. 243 

coefficient of the principal elliptic term in the expression of %. in terms of v 
and that of the principal term of s in terms of v, are the same as in undis- 
turbed motion*. 

Finally, the numerical values of the constants are all substituted, and a 
'reversion of series gives u^ v, s and thence 1/r, v, u in terms of the time f. 

296. Besides giving a general treatment of the Lunar Theory, Laplace enriched the 
subject with several new discoveries. Of these, the most noted is his explanation of the 
cause of the secular acceleration of the Moon's mean motion t a phenomenon which had 
been observed many years before and which had been the subject of several prizes offered 
by various academies. Laplace, after an attempt to account for it by supposing that a finite 
time was necessary for the transmission of the force of gravity, announced in 1787 that it 
was due to a slow variation in the eccentricity of the Earth's orbit, and his theoretical deter- 
mination agreed almost exactly with the vahie deduced from the observations. He also 
showed that the same cause produced sensible accelerations in the motions of the node and 
the perigee ; his results were confirmed by a later examination of the observations. 

A curious fact concerning the discovery of the cause of the secular acceleration of the 
mean motion, the theoretical value of which remained unquestioned for over sixty years, 
was pointed out by J. 0. Adams . Laplace and his followers had integrated the equations 
of motion as if e f were constant, substituting its variable value in the results, and had then 
determined the acceleration to a high degree of approximation. Adams showed that, although 
this method of procedure is permissible in a first approximation, it is necessary to introduce 
the variability of e' into the differential equations themselves when proceeding to higher 
orders. He then found that the true theoretical value, which amounted to about 6" per 
century in a century, was only a little more than half of the value obtained by Laplace and 
Plana and therefore that theory was insufficient to account completely for the observed 
value. A controversy, which lasted for several years, arose concerning the validity of 
Adams' method ; his value was, however, confirm eel at various times by several investi- 
gators amongst whom maybe mentioned Delaunayll, Plana IT, Lubbook**, Cayleyft and 
Hansen{|. It must be stated, however, that doubts have been raised concerning the 
correctness of the value deduced from observation by the researches of Prof. Newconab 
into ancient eclipses. The question turns chiefly on the trustworthiness of the records. 
A full discussion of the points at issue is given by Tisserand|| ||. 

* See Art. 159 above. 

t A portion of Laplace's second approximation and the determinations of c, g to the order w :l 
are given by H. Godfray, Elementary Treatise on the Lunar Theory. 

$ See Arts. 819-322 below. 

" On the Secular Variation of the Moon's Mean Motion," Phil Trans. 1853, pp. 397406 ; 
M. N.R.A. S. 1858; Coll. Works, pp. 140157. 

|| " Sur I'acG&e'ration s<culaire du moyen mouvement de la Lune," Cowptes J&endits, Vol. XLVIII. 
pp. 137138, 817827. 

If "Me'moire sur liquation s6culaire de la lame," Mem. d. Accad. d. So. di Torino, Vol. xvm. 
pp. 157. 

** " On the Lunar Theory," Mem. E. A. S. Vol. xxx. pp. 4352. 

ft "On the Secular Acceleration of the Moon's Mean Motion," M, N. R. A: S. Vol. XXH. 
pp. 171231 ;"ColL Works, Vol. in. pp. 522561. 

t " Sur la controverse relative & liquation se*culaire de la Lune," par M. Delaunay, Comptes 
Rendus, Vol. LXII. pp. 704 707. 

"^Researches on the Motion of the Moon," Washington Observations, 1875, pp. 1 280. 

Illl MGccwique Ctileste, Vol. in. Chaps, xm, xix. 

162 



244 THE PRINCIPAL METHODS. [CHAP. XII 

297. The Theory of Damoiseau *. 

Damoiseau follows Laplace's method almost exactly. He assumes the 
final forms of the expressions for u I} nt } s in terms of v and substitutes them 
directly in the differential equations. A number of equations of condition, 
involving the unknown coefficients in a more or less complicated manner, are 
thus obtained and these are solved by continued approximation. Numerical 
values are used all through and the theory is therefore entirely numerical. 
When the coefficients have been obtained., a reversion of series is made in 
order to express the coordinates in terms of the time ; this is also done by 
the use of indeterminate coefficients a method always available when the 
arguments of the required series are known. 

The object of the theory appears to be the determination of the coefficients accurately 
to one-tenth of a second of arc. ITor this purpose he carries them to the hundredth of a 
second and includes certain sensible terms due to the actions of the planets and to the 
figure of the Earth. The results are given very concisely, but the work will be easily 
followed after a perusal of Laplace's theory as given in the Mfoanique Celeste. The labour 
of finding the values of the coefficients may be grasped from the fact that the mere writing 
down of the equations of condition occupies half the Memoir. The tables which he 
deduced t from the results of this theory were not entirely disused until those of Hansen 
appeared. 

298. The Theory of Plam$. 

This is an extension of a theory worked out by Plana and Carlini and 
sent in to compete for a prize offered by the Paris Academy of Sciences 
in 1820. A prize was awarded to them and also to Damoiseau for his theory. 
The results of Plana and Carlini were not printed, but later Plana issued 
the three large volumes referred to in the footnote. The method of Laplace is 
used ; Plana, however, instead of substituting numerical values, makes a literal 
development in powers of m, e, e f , % a/a'. The results are, in general, carried 
to the fifth order of small quantities ; certain coefficients, which are expressed 
by slowly converging series, are carried to the sixth, seventh and eighth orders. 
In point of accuracy, judged by Hansen's theory, it is about equal to that of 
de Pont^coulant and slightly inferior to the numerical theory of Damoiseau ; 
the inferiority is partly due to the slow convergence of the series arranged in, 
powers of m and partly to errors which have crept into the work errors 
unavoidable where the developments are of such length and complexity. As 
a literal development it has only been completely superseded by Delaunay's 
theory. 

* " Memoire sur la Theorie de la Lune," Mm. (par divers savants) de VImt. de France 
Vol. i. (1827), pp. 313598. 

t Tables de la Lune, formges par la seule th&orie de Vattraction et suivant la division de la 
circonfgrence en 400 degrfs, Paris, 1824. TaUes^en 360 degrto, Paris, 1828. 

J Theorie du Mouvement de la Lune, 8vo. Turin, 1832, Vol. i. 794 pp. : u. 865 t>r> - 
m, 856 pp. *** ' 



297-300] DAMOISEAU. PLANA. POISSON. LUBBOCK. AIRY. 245 

299. The Method of Poisson*. 

Poisson proposed to apply the method of the Variation of Arbitrary Con- 
stants to the solution of the lunar problem. For this purpose lie introduces the 
equations of Art. 83 above. The disturbing function is to be expanded by 
the purely elliptic values of the coordinates and the result substituted in the 
right-hand members of the equations. To obtain the second approximation 
to the values of the elements, they are regarded at first as constants in 
the right-hand members; the equations may then be solved and the resulting 
values of the elements, in terms of the time, are to be used as the basis of a 
second approximation by substituting them, instead of their constant values, 
in the same parts of the equations. To obtain the solar inequalities in the 
Moon's motion, the method in this form is almost useless on account of the 
enormous developments which it would entail, and it would not be considered 
here were it not for its value in investigating the inequalities arising from 
other sources and, in particular, for those inequalities known as ' long-period ' 
and * secular.' In fact, Poisson only gives a few calculations as illustrations 
of the method. It is chiefly of value in the planetary theory, 

300. The Method of Lubbock^. 

The publication of Lubbock's researches in the volumes of the Phil. 
Trans, between 1830 and 1834, places his method next in historical order ; 
they are collected and extended in the pamphlets referred to in the footnote. 
The method is the same as that of de Pontdcoulant, whose results were not 
published until 1846 ; but, from the remarks made by Lubbock and de Ponte- 
coulant in their prefaces, it is evident that they had adopted the same plan 
independently. Lubbock never carried out his method with any complete- 
ness; his published papers contain an explanation, of the method, a full 
development of the second approximation, and the calculation of the earlier 
approximations to the coefficients of certain classes of terms ; his results are 
compared with those of Plana. 

Next in order come the theories of Hansen and Delaunay which have been already 
treated. Finally, mention must be made of Airy's method J, which was rather a verification 
of previous results than a complete theory in itself. 

Airy proposed to take Delaunay's expressions after numerical values had been sub- 
stituted for the constants and, considering each coefficient to need a small unknown 
correction, to substitute the results, together with the unknown parts, in the equations of 

* "M&noire sur le mouvement de la Lime autour de la Terre," Mm. de VAcad. des Sc, de 
VInst. de France, Vol. xni. (1885) pp. 209335. (Bead in 1838.) 

t On the Theory of the Moon and on the Perturbations of the Planets, London, 8vo. Pfc. i. 
(1834), 115 pp., with an Appendix containing Plana's results; Pts. n, (1836), in. (1837), iv. (1840), 
417 pp. ; PL x. (1861), 94 pp., with tables. 

$ Numerical Lunar Theory, London, 1886, fol. 178 pp. 



246 THE PBINCIPAL METHODS. [CHAP. XII 

motion*. He had worked at this for several years but, after the volume containing his 
results was published, he discovered a serious omission which altogether invalidated them ; 
the large corrections which he had found were necessary to make Delaunay's results satisfy 
the equations of motion, were probably due to this unfortunate error. In a letter f to the 
Secretary of the Boyal Astronomical Society, he says, " I keep up my attention to the 
general subject, but with my advanced age (eighty-eight) and foiling strength I can 
scarcely hope to bring it to a satisfactory conclusion. I will only further remark that I 
believe the plan of action which I had taken up would, if properly used, have led to a 
comparatively easy process, and might in that respect be considered as not destitute of all 
value." 

301. Tables. 

The tables of the Moon's motion which have been formed from the results of theory 
alone, in order to calculate the position of the Moon at any time, have already been referred 
to, in connection with the theories from which they were deduced. In addition, we may 
mention those of Mayer (London, 1770) formed by a combination of theory and observation, 
of Mason (London, 1787), which were Mayer's tables improved, of Burg (Paris, 1806), of 
Burckhardt (Paris, 1812) and, for the Parallax of the Moon, of Adams (M. J\ r . R. A. $. 
Vol. xni. 1853; Nautical Almanac, 1856; Coll. Works, pp. 89107). Later efforts in 
this direction have been made chiefly for the purpose of correcting Hanson's tables (see 
Art. 238). 

302. In making a comparison of the various methods of treating the lunar problem, 
several considerations enter. There doen not appear to bo any method which is capable of 
furnishing the values of the coordinates with a degree of accuracy comparable with that of 
observation, without great labour ; and, in the present state of the lunar theory, looking 
only to a practical issue, what is required is rather a verification of the results of previous 
methods, say those of Hansen and Delavmay, than new developments. Again, some 
methods appear to be most effective for one class of inequalities while other methods give 
another class of inequalities moat accurately. The question to bo discusMod is mainly the 
relation between the accuracy obtained and the labour expended. 

As regards the inequalities produced by the action of the Bun, the methods may be 
divided into three classes, The first or algebraical class contains those iu which all the 
constants are left arbitrary ; the second or numerical, those in which the numerical values 
of the constants are substituted at the outset ; the third or semi-algebraical, those in which 
the numerical values of some of the constants are substituted at the outset, the others 
being left arbitrary : the most useful cawe of the last class appears to be that iu which the 
numerical value of the ratio of the mean motions is alone substitutod. The advantage of 
an algebraical development will be readily recognized. In a numerical development, slow 
convergence is to a great extent avoided, but the source of an error is traced with groat 
difficulty and any change in the values of the arbitraries can not be fully accounted for 
without an extended recalculation. The semi-algebraical class, in which the value of m is 
alone substituted, appears to possess an accuracy nearly equal to that of a numerical 
development, and it has the advantage of leaving those constants arbitrary whoso values 
are known with least accuracy. 

It is difficult to judge of the labour which any particular method will entail, without 
performing a considerable part of the calculations by that and by other methods. As far 

* An exhaustive analysis and criticism of Airy's method is given by M. Badau, Bull. 
Astronomique, Vol. iv. pp. 274286. 

t " The Numerical Lunar Theory," M. N. 11. A. 8. Vol. XLIX, p. 2. 



300-302] REMARKS ON THE METHODS. 247 

as it is possible to estimate, either by general considerations or by the amount of time previous 
lunar theorists have spent over their calculations, it may be stated that those methods 
which have the true longitude as the independent variable must be altogether excluded if the 
solar perturbations are required, owing to the necessary reversion of series. For a complete 
algebraical development carried to a greater accuracy than that of Delaunay, none of the 
methods given up to the present time seem available without the expenditure of enormous 
labour: Delaunay's calculations occupied him for twenty years. If we may judge from 
the inequalities computed up to the present time, the methods of Chap. xi. seem 
to be best suited to a numerical or semi-algebraic development. It is true that they give 
the results expressed in rectangular instead of in polar coordinates, but the labour of 
transformation is not excessive in comparison with that expended on the previous 
computations, while the accuracy obtained far surpasses that of any other method ; the 
transformation of the series, however, would not be necessary for the formation of tables. 
The disadvantage of de Ponte'coulant's method is the necessity of obtaining the parallax, 
with an accuracy much beyond that required for observation, before the longitude can be 
found ; this remark applies also to the methods of Chap. XL, but in rather a different way. 
Hansen's method labours under the disadvantage of putting the results under a form which 
makes comparison with those of other methods difficult. Another consideration which is a 
powerful factor, is the question as to how far the ordinary computer, who works by definite 
rules only, can be employed in the calculations ; and here the methods of Chap. XL appear 
to have an advantage not possessed by any of the earlier theories. 

With reference to the classical treatises on Celestial Mechanics, there is little doubt 
that the works of Euler and Laplace will best repay a careful study ; those of Lagrange in 
a different direction the general problem of three bodies must also be mentioned. The 
ideas upon which all the later investigations have been built, may be said to have originated 
from the works of one or other of these three writers. 



CHAPTER XIII. 

PLANETARY AND OTHER DISTURBING INFLUENCES. 

303. AJST explanation of the way in which the principal effects of 
planetary action and of the figure of the Earth may be included in. the 
lunar theory will be given in this chapter. A general plan of integration for 
the new terms introdxiced into the disturbing function will be first explained ; 
the discovery and development of the disturbing functions for the direct and 
indirect actions of the planets and for the direct effect of the ellipticity of 
the Earth then follow, the results being illustrated by applying them to a 
few of the principal inequalities. The perturbations produced by the motion 
of the ecliptic and by the secular variation of the solar eccentricity are, 
owing to their peculiar nature, treated by special methods. In all cases, 
the developments will be only given as far as they are necessary for the 
purpose of explanation; references are given to the memoirs in which more 
complete investigations may be found, As far as the end of Art. 318, Delau- 
nay's notation will be used; the determination of the secular acceleration 
being made by the use of de Pontteoulant's equations, we use the notation 
of Chap. vil. in Arts. 319322, 

The effect of the terrestrial Tides and of the figure of the Moon on the motion of the 
latter will not be treated here. The former is considered in the Memoirs of Prof. (1 1L 
Darwin* in detail ; the chief effect is on the Moon's mean period and mean distance, and 
the amomt of the correction, within the limits of time during which observations have 
been recorded, is -very small, As to the latter, it is very doubtful whether it produces 
any appreciable eft'ect : Hansea introduces an empirical periodic term supposed to Tbe due 
to the difference between the centre of mass and the centre of figure of the Moon f, 

G-eneral ethod of Integration. 

304. The expression of the disturbing causes which affect the motion of 
the Moon can, in nearly all cases, be made by inserting additional periodic and 
constant terms in the disturbing function. The periods and coefficients of 
these terms of the disturbing function the parts which involve the elements 

* Phil. Tram, 1879-1881. 

t Darlegmg, i. pp. 175, 474479. 



303-305] GENERAL METHOD OF INTEGRATION. 249 

of the Moon as well as those arising from other sources can generally he 
found with an accuracy sufficient for practical purposes ; for this reason, it is 
advisable to use a method of integration which shall be adaptable easily to 
any periodic term, and such a method, founded on Delaunay's formulae, has 
been devised by Dr Hill*. Its value chiefly depends on the fact that the 
coefficients of the new terms in the disturbing function are always small and 
that, in consequence, it is seldom necessary to consider the changes produced 
in the new terms of the disturbing function by those changes of the elements 
which occur when any one of the old or new periodic terms is eliminated 
by Delaunay's processes. The operations are therefore similar to the majority 
of those mentioned in the last paragraph of Art. 197, but it will be seen that 
we may use numerical values for the elements of the Moon's orbit and that, 
owing to this feet, the operations of Delaunay may be very considerably 
abridged. The numerical results given below are those obtained by 
Kadauf in a valuable Memoir to which frequent reference will be made. He 
introduces a slight modification of Hill's method and his numerical values for 
S,M differ to a small extent from those of Hill. Periodic terms only will be 
discussed; the changes produced by new non-periodic terms due to the 
inequalities considered below, are very small. 

It is supposed that the periodic terms, arising solely from the action of 
the Sun considered to be moving in an elliptic orbit, have been eliminated, 
and that the disturbing function contains only the remaining constant portion 
together with the new terms to be considered. It is further supposed that the 
operation of Art, 198 above has not been carried out, and that the final change 
of constants, which Delaunay makes in order to reduce his expressions to a 
suitable form (Art. 200), is as yet not performed. The results required here 
are all contained in Delaunay's volumes : the latter will be referred to as in 
Chap. ix. 

305. Delaunay's canonical equations are (Art. 183) 

dL^dR L-?^ 

W~"3T "" '"' dt dL 3 "" "" 

Let - B be the constant part of R which remains after the periodic terms due 
to the Sun have been eliminated; we then have, by Art. 198, 

9J5 ae as 



* It is contained in pt. m. of hie Memoir ' On certain Lunar Inequalities due to the action of 
Jupiter and discovered by Mr E. Nelson," Astron. Papers for Amer. Eph. Vol. in. pp. 373 - 393 - 

f "Becherches concernant les InegalitSs plantoires du Mouvement de la Lune, Ann. de 
I'Ofc. de Paris (Jtanoir*), Vol. ai, pp. 1- 114. See also, Eemarques sur certaines ^f 
& longue p&iodd du mouvement de la Lune," Bulletin Astronomique, Vol. ix. pp. 137 lib, 
185212, 245246. 



250 PLANETABY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

Let &R be the new part of JS, and let S, SG, SH, 8Z, %, Bh be the new 
parts of Lj G, H, I, g, h, due to 6.R; let SI Q , Sg , SJi G be the new parts of 
L 9o> ^o (which are functions of L, Q, H\ due to 8L } S0 9 SH. The canonical 
equations may be written, 



(1). 



We choose out one of the periodic terms of SR and put 

SJR = + A cos (il + i'g + i"h + <*t + ) = J. cos (9, 

where a* -f /3 is the part of the argument independent of the lunar elements, 
and where A is the coefficient, containing three of the six lunar elements, 
namely, L, G, H or a, e, 7. We have 

= (il, + i'g, + i"h, -f a) t + j3' = jrt + /3', 
suppose, so that JW is the motion of the argument 9. 

As A is always very small, we can, with a sufficient approximation, 
substitute the value of SJ? in (1) and integrate on the supposition that 
L 3 G, H t 1 Q , g Q , h Q are constant when multiplied by the small quantity A. 
In this way we find 



, ., 

M M M 

Whence, since a, e y 7, 1 , g Q , h Q are functions of L, G, H, 
.da ./da . r , da\A /i ^ 



The second three of equations (1) give 

d \j ^j dA * 
di Sl=Bl -dL coad > -' - 

Substituting the values of SZ , $ff 0} 8/^ a]Q d integrating on the supposition that 
the lunar elements are constant in the right-hand members, we find 

~ , 7 ,, 

8 ^---* 8 *- ...... (3) * 

The equations (2), (3) give the new terms to be added to the elements. 

306. As regards the calculation of the various quantities present in (2), 
(3), the partial differential coefficients of a, e, 7, 1 Q , <7 , A> with respect to 
L 9 G, H may be obtained from the expressions given by Delaunay*. Since 

* Delaunay, n. pp. 235- 238. 



305-306] EQUATIONS FOR THE VAEIATIONS OF THE ELEMENTS. 251 

we shall not consider the changes produced in E by the changes in the 
elements, the numerical values of n'ln, e, e, 7, a/a' may be substituted in the 
results : the numerical values of the lunar constants will not be quite the same 
as those used by Delaunay in his final results, because the final transformation 
which alters the meaning of n, e, 7, a (Art. 200) has not been made ; the 
necessary modifications can be obtained from the formulae given by Delaunay 
for the transformations*. The values thus obtained by Eadau are 

n'/tt = 0-0744, e = 0*0549, 7 = 0-0449, a/a' = 0'00257. 
To obtain the partial derivatives of A with respect to L, (?, JET, we have 

^A^A^ d A^ 4-<A^L 

dL~~ da 3 de 3 87 3' *'" ""' 

in which the partials da/dl,... may be numerically calculated in the manner 
just explained. These calculations being made once for all, we can obtain 
very simple formulae for the determination of the coefficient of any periodic 
term. 



Instead of g, the mean longitude M = l + g + his introduced, so that 

Mfo^W + fy 
and instead of M 9 the ratio p = n'/M. Put 



After inserting the numerical values of all the known terms, as explained 
above, Radau findsf that the equations (2), (3) become 

g a / a ^ (0-14901% - 0'000246 v - 0-000006*") A' cos (9, ^ 
$e = (1-4215 i - 1-4238 if + 0'00024 i") A' cos 0, 
S 7 =: (O'OOOlOi + (H1203 H - 0-41370 i ;/ ) A' cos 6, 

8Jf ={(- 8-0576 % + 0-05601 i'- 0-01124, i")p 

~ 01487 6j + 0-02551 / 4- 0*03492 j"} A' sin 0, 
SI ={(- 3'0826 % + 0-06142 i'- 0-03621 i")p 

- 0-14901J - 25-891 / - 0'00232 /} A' sin 0, 
8A {(-0-0364K + 0-02890 i'-O'OOSTS i")p 

+ 0-000006J - 0-00435 / H- 9'2169 /'} A' sin (9 

The simplicity of these equations enables us to calculate easily the first 
approximation (generally sufficient), according to powers of A or A', to the 
coefficient of any term. Examples will be found below. 






Delaunay, n. p, 800. t Recherches etc., p. 36. 



252 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

307. To calculate the corresponding terms in the coordinates, it will 
often be sufficient to limit their expressions to the principal elliptic and 
solar terms, and to find the small changes in the coordinates induced by the 
calculated changes in the elements, after inserting the numerical values of all 
parts of the coefficients except the characteristics. For example, it is in many 
cases sufficient to put 

v = M + 2e sin Z + 0'41e sin (2D - Z), 

where the numerical part 0*41 arises from the series in powers of n'/n etc. 
From this expression, Sv may be obtained by putting D = if n't e', 
and causing M Q) e, I to receive the small increments calculated above. 

If this he not sufficiently approximate, we can take the principal terms in 
the unreduced values of the coordinates in terms of the elements* and submit 
them to a variation , all the elements being supposed variable. 



General method for the Inequalities produced by the 
Direct and Indirect Actions of the Planets. 

308. The Disturbing Functions. 

Let #/, y, /, r' be the coordinates of the Sun, a?, y, z> r those of the Moon, 
referred to axes fixed in direction and passing through the Earth, and let S 
be the cosine of the angle between r, r'. The disturbing function for the 
motion of the Moon, due to the Sun, is, by Art. 107, 



In order that the coordinates a/ 9 y' } z' } r' may be considered to refer to the 
motion of the Sun about G (the centre of mass of the Earth and the Moon), 
it is necessary to multiply the second term of this expression by 



f , 97, D be the coordinates and distance of a planet P, referred to the 
same axes. The action of a planet of mass m", on the motion of the Moon, 
will evidently be expressed by a disturbing function of the same form as JB, 
namely R, where 



S' being the cosine of the angle between r, D. In order that f , 77, f, j> may 
be considered to refer to G as origin, it is necessary to multiply the second 
term by (E - M)/(E + if). 

* Delaimay, n. Chaps, vn, ix. 



307-309] BISTUEBING FUNCTIONS FOR PLANETARY ACTION. 253 

The ratios r//, r/D, m"/m' are always small and, in most cases, the effect 
of R f on the motion of the Moon will be sufficiently accounted for by 
considering only the first term of R' ; the other terms will therefore be neg- 
lected here. The inequalities produced by R f are said to be due to the direct 
action of the planets. To each planet will correspond a function R ; but 
since the terms produced by the combination of two terms, one from each 
such function, are generally negligible, it is only necessary to consider one of 
these functions, applying it to the case of each planet successively. 

The solar inequalities, as far as they arise from the purely elliptic motion 
of the Sun, are supposed to have been determined. The actions of the 
planets on the motion of the Earth produce small deviations from elliptic 
motion in the apparent motion of the Sun: these, being substituted in JS, 
may be considered as small corrections Sa/, %', / to the coordinates af, y', z\ 
As these corrections are never large, it will be sufficient, for the inequalities 
thus produced in the motion of the Moon, to limit R to its first term. The 
lunar inequalities arising in this way are said to be due to the indirect action 
of the planets. Since m" is very small compared with m', it will not be 
necessary to consider these variations of of, y', z f in R\ See Art. 310 (d). 

309. Separation of the terms in R, R', and their expressions in polar 
coordinates. 

Confining R, R to their first terms we have, on introducing rectangular 
coordinates, 

1 p^ 

-,^~ 



These may be written 

1 pa _ 8^/1 3/ 2 



/r") + 3y* (yY/r' 5 ) ...... (5), 



in which it will be noticed that the coordinates of the Moon are separated 
from those of the Sun or of the Planet. It is now necessary to express JZ, K 
by means of the polar coordinates of the planet and of the Earth (or of (?), 
referred to the Sun, and those of the Moon referred to the Earth. 

The notation of Chap. IX. will be used whenever it differs from that of 



254 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

previous chapters. We suppose the ecliptic to be a fixed plane perpendicular 
to the axis of #. As before, 

L = Distance of the Moon from its node, 

h = Longitude of the node, 

7 = Sine of half the inclination of the lunar orbit. 

We have then (fig. 5, Art. 73) 

x = r cos L cos h r cos i sin L sin h, 
y = r cos L sin h -f r cos i sin L cos h, 
z = r sin L sin i ; 

or # == (1 7 2 ) r cos (L + A) + y 2 r cos ( - 

(6). 




Let V = Longitude of the Earth as seen from the Sim. The longitude of 
the Sun, as seen from the Earth, will be V + 180, and therefore 

#' = - r' cos F', 2/' = -r'sin F, #' = (7). 

The coordinates f, 77, f, being those of P relative to E, are those of P 
relative to the Sun added to those of the Sun relative to E. The coordi- 
nates of P relative to the Sun may be deduced from (6) if we put 

7" = Sine of half the inclination of the orbit of P to the ecliptic, 
A" = Longitude of its node on the ecliptic, 
V" = Longitude of P as seen from the Sun, reckoned along the ecliptic 

to its node and then along its orbit, 
r" = Solar radius vector of P, 

for 7, A, L -f A, r, respectively ; the coordinates of the Sun are given by (7). 
We therefore obtain 

f = - r' cos F' + (1 - 7" 2 ) r" cos F" + 7' V cos ( F" - 2A");i 

<ri = - r' sin 7' + (1 - 7 //2 ) r" sin V" - */ V sin ( V" - 2A"), I (8). 

f = 27"Vl-7 / V / sin ( F" - A") J 

Whence D 2 = f 2 + ?? 2 + f 2 

- D 2 + 4 7 " W sin ( F - h") sin ( F 7 ' - h"), 
where D 2 = r' 2 + r //2 -2rV / cos(F / - F") (9). 

It will be unnecessary to consider powers of 7" beyond the second ; we shall 
therefore have 

" "*"'"" {cos (F 7 4- V" - 2A") - cos ( F - F")| (10). 



By means of the formulae (6) (10), -R, Rf can be expressed in terms of 
r, r, r", L, V, 7", A, A 7 , A", 7, 7". 



309-310] DEVELOPMENT OF THE DISTURBING FUNCTIONS. 255 

310. Development of the Disturbing Functions. 

We shall first expand the expressions obtained for the various parts of 
the disturbing functions, by substituting elliptic values for the coordinates of 
the three bodies, and then show how non-elliptic terms present in these 
coordinates may be taken into account. "When this has been done, the 
disturbing functions are to be expressed as sums of periodic terms. 

(a) The portions which depend only on the coordinates of the Moon. 
Neglecting powers of 7 beyond the fourth, we obtain, from (6), 

J (r 2 - 3O/V 2 = i (1 - 6 7 2 -h 6y 4 ) 4- f 7 2 (1 - 7 2 ) cos 2z, 
f (^ _ yu)/^ = (1 _ 7^)2 cos (ZL 4- 2A) 4- f 7 4 cos (2 - 
corresponding expressions may be obtained for f &y 

As in Chap. IX., let g be the distance of the lunar perigee from the node 
and /the true anomaly. Then 



and, from the expressions given in Art. 39, 



- cos (2/+ ) = (!- f e 2 ) cos (21! + a) 4- e cos (3Z + a) - 3e cos (I + a) 4- . . ., 
(it 

where a may be any angle. 

By giving to a suitable values, all the five functions JO" 2 - 3^ 2 ), l(tf-y*)> 
etc., present in R, R', can be expressed in series of cosines or sines involving 
Z, g, h in their arguments and e, 7 in their coefficients. Moreover, the orders 
of the coefficients can be associated with the multiples of I, g, h in the 
corresponding arguments, by the rules obtained in Chap. VI. Putting M for 
l^. g 4. Ji } it is easily seen that 

r2 _ 3^2 = 24 oe * cos Jd + 7 2 ^jB e fc cos (2Jf - 2A fcQ, 

where Jc is a positive integer and where A Q) Q are coefficients of zero order 
containing powers and products of e\ 7 l The other four functions depend- 
ing only on the coordinates of the Moon may be similarly treated. 

(6) The parts which involve the coordinates of the Sun and 
of the Planet in the second degree. 

By means of the formula (8), we find the values of f 2 , 2 -Y, ft, & rt 
expressed as sums of cosines or sines. The arguments of these terms will 
contain multiples of F, V", h", and the coefficients will contain 7' and will 



256 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP, XIII 

have r'*, r" 2 or r'r" as factors. They are expanded in terms of the elliptic 
elements of the Sun and of the planet by the formulae 

..., V = Mi ' + 2e' sin V +. 



where Jl/ ', Jf " denote the mean longitudes of the Earth and the Planet 
respectively. The portions of R which depend only on the coordinates of the 
Sun, present no difficulty. 

In the same manner as with the coordinates of the Moon, the composition 
of the argument of any term in the development of these expressions may be 
associated with the order of its coefficient, though the connection is by no 
means so simple. For instance, it may be shown that 



f'*' cos {2.M " -2*lfo" + j (M Q ' - M,") + 2&" TdV VI"}, 

where J. is a coefficient of zero order : k' t "k" are positive integers or zeros, 
and i, j have positive integral or zero values such that i + j< 3. 



(c) The parts of R' arising from the Divisors D^. 

The equation (10) shows that it is only necessary to consider the divisors 
D tf They are the functions which cause the great difficulty in finding the 
planetary inequalities in the Moon's motion; the difficulty is of the same 
nature as that encountered in the planetary theory and it arises from the 
near equality of r', r" or of < n" in the cases of those planets which are not 
far from the Earth (see Art. 9). 

We can expand l/7> ff , by means of Legendre's coefficients, in the form 

A-* - W } + # w cos ( F ^ F") + JV 2) cos 2 ( F 7 - F') + . . ., 
"\ 

where B^ is a homogeneous function of r', r" ; when r', r" are comparable 
with one another in magnitude, these coefficients dimmish very slowly and it 
becomes frequently necessary to consider terms in which j is a large number*. 
In the case of a superior planet, expansion must be made in powers of r'/r" 
and, in the case of an inferior planet, in powers of r"/r. 

Substituting the values of /, r", V, V", given by (11), it is easily seen 
that 



'*" cos { j (jf; - jf ") KV m"}, 

in which A* is a homogeneous function of a, a" and of zero order with respect 
to e' 3 e", and j, Itf, k" have positive integral or zero values. 

* Badau's method (Reclierches, pp. 1731), for abbreviating the calculations in such cases, 
should be consulted. 



310-311] TERMS DUE TO INDIRECT ACTION". 257 

(d) The terms arising in R from non-elliptic terms present in the 
coordinates of the Earth, the Planet and the Moon. 

Two methods may be used for these terms. We may either consider r, V' 
(and also u f , if the terms dependent on the motion of the ecliptic be not 
neglected) to receive small increments Sr', 8?/ and then expand the formula 
(3) of Art. 108 in powers of these increments by means of Taylor's theorem, 
substituting for r', v their elliptic values and for Sr', &v' the small terms 
given by the planetary theory. Or we may suppose the additional terms 
to be given as small corrections to the elements of the solar orbit, in which 
case the development (5) of Art. 114 will be available after the changes of 
notation, necessary to express the result in Delaunay's form (Arts. 123, 180), 
have been made. The same methods may also be employed to take into 
account any non-elliptic terms present in the coordinates of the planet. 

The solar terms present in the coordinates of the Moon cannot, in all 
cases, be neglected. In the process of eliminating, by Delaunay's method, 
the periodic terms of J2 which are due to the action of the Sun, the lunar 
elements present in R f will be changed at each operation. Instead of 
inserting the changes, thus produced, by adding them to the elements, it 
will generally be more convenient to suppose that the elliptic values of the 
coordinates receive small increments, these increments being the principal 
solar terms which occur in the unreduced values of the coordinates, as given 
by Delaunay. Numerical values may usually bo substituted in all parts of 
the coefficients of the new terms, except in the characteristics, 

311. After the various processes, outlined above, have been carried out, 
it is only necessary to multiply the series obtained for the various parts of 
R or R and to express them as sums of cosines of angles. To do this in any 
general manner, would involve enormous labour due chiefly to the divisors 
/#; and much of the labour would be without result, because the great 
majority of the terms have quite insensible coefficients in the coordinates 
of the Moon. The plan usually adopted consists in trying to discover the 
terms which have long periods and which, in consequence, may have coeffi- 
cients lai^ge in comparison with their order when the equations of motion 
are integrated. Certain short-period terms which are cither associated with 
these long-period terms, or which have an independent existence in the 
disturbing function, must also be" included when there is a possibility of a 
large coefficient in one of the coordinates. In every case, the methods by 
which R, R have been developed, give the order of the coefficient in the 
disturbing function in relation to the eccentricities and inclinations. No 
furthejr rules can be given to guide us in the choice of these terms. Many of 
them have been indicated by observation : others have been obtained directly 
from theory in the course of investigations into the effects of planetary 
action. 

B. L. T. 17 



258 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

The method here outlined for the treatment of the disturbing function was first given 
by Dr Hill* and was afterwards extended and applied to many planetary inequalities 
by M. Radaut Combined with Hill's method of integration (Arts. 304307), it forms 
the only complete and generally effective method known up to the present time for the 
investigation of the planetary inequalities. 

There are several ways in which the calculation of the coefficient of a term in R\ 
with a given argument, may be abridged ; to give an account of them would lead us 
outside the limits of this treatise. The reader is referred to Radau's memoir and also 
to Tisserand's M&anique Ce'lestel which contains an account of this memoir. 

We shall illustrate the methods of the previous articles by applying them to the 
calculation of two celebrated inequalities one due to the direct action of Venus and the 
other due to its indirect action. 

312. Example of an inequality due to the Direct action of Venus. 

There is a term in R' of period 27r/(Z + 16w'-18w"), where n" is the 
mean motion of Venus. Observation furnishes, for the daily motions, 

Z = 47033"-97, n'= 3548"-19, n" = 5767"-67. 

The daily motion of this inequality is therefore - 13"'0, giving a period 
of 273 years. 

It can be shown that the term in R', having this period, is 

- 0"-001337i /2 a 2 6 cos (I + 16Jf ' - ISM," + 2A"). 
We therefore have, on applying the formulae of Art. 306, 

p = - 273, A' = - 0"-001S3p6 = 0"'0199 ; 
i = l, ' = o = i"; j = 2, j' = l, j" = 0; 
and the equations (4) give 

SM Q = 16"'6 sin (I + 16 Jf ' - 18M " + 2A" ), 

which also gives the approximate value of the term in longitude, since $1 is 
nearly equal to 8M Q and since the equations for 8a, Be, 87, Bh only give small 
coefficients. The more accurate value of the coefficient, when terms of 
higher orders are included, is 14"'4||. 

This is the largest known periodic inequality in longitude, produced by the action of 
the planets. Indeed, according to the table given by Eadau at the end of his memoir, no 
other inequality in longitude has a coefficient so great as I'', although there are several 
greater than half a second ; the majority of the inequalities are of comparatively short 
period either approximating to the lunar month or having a period of a few years. 

* U 0n certain Possible Abbreviations in the Computation of the Long-Period Inequalities of 
the Moon's Motion due to the Direct Action of the Planets," Amer. Journ. Math. Vol. vi. pp, 
115130. 

t Recherches etc. 

$ Vol. ni. Chap. xvm. 

Tisserand, Mec. Gel. Vol. m. p. 396. 

11 Kadau, Recherches etc., p. 64. 



311-313] CASE OF INDIRECT ACTION. 259 

The inequality just calculated was discovered by Hansen*, who found by theory a 
coefficient of 27" -4. He also noticed another inequality with a mean motion 8ri'lZn' 
and a coefficient 23"*2. In both cases Hansen was in error ; the former coefficient has 
just been seen to be about 14"'4, while Delaunayt and others have shown that the 
coefficient of the latter term is less than 0"-004. The values of the coefficients, which 
Hansen obtained by a discussion of the observations and which he adopted in his tables, 
were 15" -34 and 21"-47, respectively, including the parts due to the indirect action (see 
Art. 313). 

The Indirect Action of a Planet. 

313. A very simple formula can be obtained for this in many cases. 
Neglecting the perturbations of the plane of the ecliptic and the ratio of 
the parallaxes, we have, by Art. 116, 

SR = - (3jR/r') or' - (dR/dv) 8 T. 

Let us confine our attention to the term mV 2 / 4r/3 f R (Art. !0 8 )> since 
most of the larger inequalities of long period will arise from this term. Sub- 
stituting in the expression for Sit and neglecting the solar eccentricity, 
we obtain immediately 



Suppose that the solar tables give an inequality oV = a' A cos 0, where A, 
are independent of the lunar elements. We obtain 

SE=-f n'' 2 a?A cos 0. 
Using the equations (4), we have i, f, i",j' t j" zero and j = 2. Whence 

&Jf = U = f x Q'UQpA sin = %pA sin 8, 
approximately. 

If we suppose further that the inequality Sr' is of long period and that it 
arises principally from a variation 8af of a', a direct approximate relation 
between Sv and SF' can be deduced. For (Art. 81) 



and therefore, owing to the various conditions assumed above, 

S/ = S< 
But since n'V 8 = m', we have 



r' = - Snfa'A cos 0. 
Therefore SF = - f sin 6 = - \$A sin 0. 



' Auszug aus einem Briefe," etc. A>lr. Nach. Vol. xxv. Cols. 325-332 ; " Lettre & M. Arago," 

lendus. Vol. xxrv. pp. 795799. 

T - aur 1'Inegalite Innaire & longue periode due h 1'action pertobatrice de Venus et ^pendant 
de {'argument W'-6l"," Conn, de, Terry,, 1863, Additions, pp. 1-66. The result m given 

^P-* 6 ' . ^ o 



260 PLANETAKY AND OTHEE DISTUEBING INFLUENCES. [CHAP. XIII 

On combining this result with the value just obtained for SJf , we find 



approximately. In this case we can therefore obtain an approximate idea of 
the magnitude of the coefficient in longitude, by dividing the corresponding 
inequality in the Earth's longitude by - 7. 

For example, the solar tables * give an inequality of period 

2*1(13*,'- 8n") = 239 years, 
due to the action of Venus. In longitude, this is 

87' = + l"-92 sin (13MJ - 8Jf " + 132). 

Multiplying the coefficient by 4/27, we obtain for the corresponding in- 
equality in the Moon's motion, due to the indirect action of Venus, 

Sv = - 0"'284 sin (18Jf ' - 8. " + 132) ; 
the correct value, as found by Delaunay t, being 

Sv = - 0"'272 sin (LW ' - 8Jf " + 1 38). 

The inequality having this period, due to the indirect action of Venus, 
is therefore much greater than that, with the same period, produced by the 
direct action (Art. 312). 

For a complete investigation of the inequalities produced by the direct and indirect 
actions of the planets, the reader is referred to Radaoa's memoir. A largo number of 
references to the labours of other investigators on the same subject is also given. To those 
may he added an important paper hy Newcombf, "Theory of the Inequalities in the 
Motion of the Moon produced by the Action of the Planets," in which the whole theory of 
the subject is treated in a very general manner. 

Inequalities arising firom the Figure of the Earth. 

314. Let A, B, be the moments of inertia of the Earth about three 
rectangular axes meeting in the centre of mass, and let / be the moment of 
inertia about the line connecting this point with the centre of mans of the 
Moon. The difference of the attractions on the Moon, of the Earth and of a 
spherical body of equal mass, produces a potential 



We suppose that one principal axis of the Earth is its polar axis and that 
the moments of inertia about the other two axes are equal. Let B**A, and 
* Ann. de Z'OZ*. de Paris (Mem.), Vol. IT. p. 35. The inequality is given in th* form 

- 1"'283 sin (13M ' - 8M ") + l"-425 cos (18tf ' - 8M/'). 
t See footnote, p. 259. 

t Astron. Papers for Amer. Eph. Vol. v. Pt. in, pp. 97-295. 
E. J. Bouth, Rigid Dynamics, Vol. n. Art. 481. 



313-315] THE FIGURE OF THE EARTH. 261 

let d be the declination of the Moon. We then have 

I = A cos 2 d + (7sin 2 d; 
and the new part to be added to the disturbing function is 

SB = ^ (2A + C- 3-4. cos 2 d - 3(7 sin 2 d) 

2iT 

= ^(l-3sin*d) = ^'a-sm=d) ............... (12); 

where /ut, is the sum of the masses of the Earth and the Moon, and 



It is proved in works on the figure of the Earth* that, if the Earth's 
surface be supposed to be an equi-potential surface, 



where E is the Earth's mass, j& its equatoreal radius, a its ellipticity and ft 
the ratio of the centrifugal force to gravity at the equator. 

' The numerical determination of pM may be made in several ways. It is possible to 

i find it by the reverse process of comparing the theoretical values of the coefficients of 

the principal terms produced by the figure of the Earth on the motion of the Moon, with 

those deduced from observation ; owing to the near equality of the periods of these terms 

with the periods of certain terms produced by planetary action terms whose coefficients 

are not known with a great degree of certainty this method is not capable of very great 

accuracy. The value may "be deduced from the latter of the formulae given above for /*#, 

! by obtaining a from geodetic measures and $ from pendulum observations ; this method 

involves an assumption concerning the interior constitution of the globe. Thirdly, it may 

be obtained from the first formulaand Hill so finds itt by a discussion of pendulum 

observations, made to find the intensity of gravity at various stations on the surface of the 

; Earth. A determination has also been made by comparing the observed and the calculated 

i values of the yearly precession of the Equinoxes. 

i 315. Lot as (Fig. 5, Art. 73) be the ascending node of the Ecliptic on the 

\ Equator the place from which longitudes are reckoned and let o^ be the 

inclination of these two planes. If / be the pole of the equatoreal plane, we 



have zz' = a) l} /Ar=90~d, *Jtf - 90 - tf, 
The triangle %'zM. therefore gives 

sin d = sin u cos o)j 4- cos u sin % sin v. 
We also have sin v sin i sin L 9 

cos u cos (v Ji) = cos I, cos u sin (v h) = sin L cos i ; 

* e.g. J. H. Pratt, Art. 85. 

f In chapter v. of his Memoir " Determination of the Inequalities of the Moon's Motion which 
are produced by the Figure of the Earth : a- supplement to Delaunay's Lunar Theory," Astron. 
Papers for Amer. Mph. Vol. in. pp, 201344. 



262 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

and therefore 

cos u sin v = cos 2 Ji sin (L + h)- sin 2 $i sin (L - h). 
Hence 

sin d = cos o^ sin i sin x -h cos 2 Ji sin c^ sin (z + /i) sin 3 Ji sin (#i sin ( h). 
Putting sin ^i~y and neglecting powers of 7 beyond the first, we obtain 
sin d = sin &>! sin (L -f h) + 27 cos caj sin i ; 



J sin 2 ^ -f sin 2 (! cos 2 (L -f A) - 7 sin 2^ {cos & cos (2z + /;/)}]< 

As the method of integration will be that of Arts. 304 307, it is neces- 
sary to substitute elliptic values for r and L. We put 

r = a (1 - e cos Z + ...)> L = g + l + Ze sin 1+ .... 

The terms in SR which will give the greatest coefficients are those of long 
period : it is easily seen that, after the substitution of elliptic values, there 
is only one such term that with argument h. We therefore take 

8.R = fiJsf (7/a 8 ) sin 2o> a cos h. 

All that now remains is the application of the formula* (4). Since tho 
diurnal motion of the node is - 190 //f 77, we find 

p = n'fa = - (3548"'2 + 190"'77) = - 1*8'60 ; 



Substituting and retaining only three places of decimals in the coefficient of 
A' cos h, we obtain 



81f t = + 0'69(U' cos h, 81 = + 1118^'sm h, $h = + 9'2864' sin h} ' ' ' 
Hill finds, by his discussion of pendulum observations *, 
(k'/a?) sin ^ = 0"'072854. 

Hence, with the values of 7, n'/n given in Art, 306, A' = 10"-99. Tho results 
(13) then give 



= + 7">58 sin h, & = + 12"-28 sin h, 8A = + 1()2"-0 sin 

alled 



315-317] INEQUALITIES DUE TO THE EAKTtl's ELLIPTIOITY. 263 

316. To find the corresponding inequalities in the coordinates, it is only 
necessary to subject Delaunay's results for the elliptic and solar terms to 
a variation & and to insert the above values. It is sufficient for our purposes 
to take, in longitude, 

v = M o 4- 2e sin I ^ sin 2 (g + 1), 
Sv = M" + 2eSl cos I - 2787 sin 2(g + 1) - 27 s (By + SI) cos 2 (# + J). 

When the values of 8Z, &y, Sg -f SZ == SJf - SA, are substituted, it will be found 
that the first term only gives an inequality so great as 1". Hence 

Sv = SM = + 7"-5S sin A. 
In latitude, we have 

sin u = sin i sin (0r + I). 

Putting 7 = sin i and neglecting quantities of the order 7 3 , we find 

8 u = 287 sin (gr + Z) + 27 (8M" - 8A) cos (# -f Z) 
= - 9"'10 sin (g + Z) cos A - 8"-48 cos (gr 4- Z) sin A 
= - 8"-79 sin (A + ^ + Z) - 0"'31 sin (^ 4- Z - A). 

The only inequalities, having coefficients greater than 1" in longitude 
and latitude, have therefore arguments equal to the longitude of the node and 
to the mean longitude, respectively ; the periods are 18"6 years and one mean 
sidereal month. The coefficients, as found by Hill* who followed Delaunay's 
method exactly, are + 7"*67 and 8"'73, so that the calculations made above 
give the values with considerable accuracy. The extensions necessary to find 
the coefficients of the other periodic terms can be easily made by the method 
used here : there are several of about half a second of arc in magnitude. 

Other determinations of the inequalities due to the figure of the Earth are to be found 
in the works of Laplace f, de PontdcoulantI, Hansen, Tisserand||, etc, 

The otion of the Ecliptic. 

317. Owing to planetary action, the plane of the Earth's orbit, which has 
been hitherto considered to be the plane of reference, is not fixed. If the 
plane of reference, e.g. the ecliptic at a given date, had been fixed, this 
motion of the ecliptic, being very small, would have produced but little effect 
on the motion of the Moon when introduced into R. But it is usual to use 
the instantaneous ecliptic as the plane for the measurement of longitudes and 

* Mem. cit. pp. 341, 342. 

t M&e. C6L Pt. u. Book vn. Chap. n. ; Book xvi. Chap. m. 

t Sys. du Monde, Vol. iv. Chap. iv. 

Darlegung, I. pp. 459474, 11. pp. 273322. 

|| N4c. CM. Vol. m. pp. 144149, 155160. 



264 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

latitudes. Hence the apparent place of the Moon will be affected to an 
extent which is comparable with the motion of the ecliptic. Since the 
inclination of the lunar orbit is a small quantity, and since the line of in- 
tersection of two consecutive positions of the ecliptic has a motion so small 
that it may be neglected in comparison with the rotation of the ecliptic, the 
principal effect produced by referring the Moon's place to the moving ecliptic 
will occur in the latitude of the Moon. The approximate fixture of the node 
of the ecliptic reduces the consideration of its motion to that of a small 
rotation about a fixed line. 




Fig. 11. 

318. Let flf^M be the position of the lunar orbit, and let JST'fl/Jlf', 
XiQ^Mi be the two ecliptics at times t, t + dt. The position of the Moon 
will have changed during the interval dt ; but since two small changes may 
be calculated by considering their effects separately and adding the results, 
we can consider the position of the Moon as unchanged in finding the 
apparent change in time dt due to the motion of the ecliptic. Let the 
rate of rotation of the ecliptic be denoted by /3ri. 



Draw fljQ and MM' perpendicular to X'M'. As before, we put 

X'l=h } nif=L, X'M' = v, M f M = u; 
and further 

x'n^h'^xso*', 

since X' 9 XJ are now departure points and iy is fixed with reference 
to them. 

We have 

dh = OQ = QjQ cot i = fa'dt cot i sin (h - /*'). 

Also, by considering a point on the ecliptic 90 in advance of fl, we obtain 

- di = pn'St sin (90 + A - h') = pn'dt cos (h - h'). 
The equations for h, i are therefore 

jj_ = /3ri cot i sin (i - #), * = - fa' COS (h - h'\ 

When the motion of the ecliptic is neglected, h = h,t + const. Since is 
a very small coefficient, we may integrate the equations on the supposition 



317-319] EFFECT OF THE MOTION OF THE ECLIPTIC. 265 

that h has this value and that i is constant in their right-hand members. 
The new parts of h, i are therefore given by 

8A = - G8n'/A ) cot i cos (h - h'), Si = - (/3?z'//i ) sin (h - h'). 

Further, S(y + *)= -QO, -^-.^ CQS ( A "*'). 

cos^ A sin i 

The latitude is given by the equation, sin /"= sini sin (# 4- 1). Hence 
Bu cos (7 = 8i cos i sin (# + 1) + 8 ((/ -f Z) sin i cos (</+) 

= (pn'lht) { cos i sin (# + ) sin (h /*/) -f cos (g 4- Z) cos (fe A')}, 
which, by considering the triangles Jf fliV> JfO/lf/, becomes 



The period of the inequality is therefore the same as that of the Moon. 
The annual motion of the ecliptic is 0"*48, and the node of the Moon's orbit 
makes a revolution in 18*6 years. Hence 



therefore /3n'lh = ;/ -48 x 18'6 -s- 618 = l /; -42, 

giving 8 u = 1"'42 cos (v - A 7 ). 

The corresponding inequality in longitude is much smaller. Its period is 
that of the mean motion of the node and its coefficient is less than one-third 
of a second. The calculation of it presents some difficulties and requires a 
more extended investigation. 

The above method of investigation was given by Adams in a " Note on the Inequality 
in the Moon's Latitude which is due to the secular change of the Plane of the Ecliptic *." 
A more complete investigation by Hill will be found in a paper "On the Lunar Inequalities 
produced by the Motion of the Ecliptic f. 33 Reference may also be made to Hanson (, 
Eadau , Tisseraud 1 1 . 

The Secular Acceleration of the oon's ean otion. 

319. The action of the planets produces a slow variation in the eccen- 
tricity of the Earth's orbit which is usually expressed in the form, 

e / = 6 / -a* + a'? + .... 
The coefficients a, a',.-- are yiitQ insensible in the motion of the Earth, 



* M. N. JR. A. 8. Vol XLI. pp. 385 403, Coll Works, pp. 231252. Godfray's Lunar 
Theory, Art. 113. 

t Annals of Math, (U. S. A.), Vol. i. pp. 510, 2531, 5258. 

J Darlegung, i. pp. 118120, 490491. 

" Influence du ^placement s6oulaire cle 1'Ecliptique," Bull Astron. Vol. ix. pp. 363373. 

|| me. 61. Vol. ni. pp. 136140, 160164. 



266 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

but the first of them produces in the longitude of the Moon an effect which 
is easily noticeable when observations, extending over a hundred years or 
more, are discussed. When this expression, instead of e', is introduced into 
the disturbing function, it is evident that terms of the forms & cos (at + b) 
will be introduced into R and therefore into the coordinates of the Moon. 

In order to make the investigation as brief as possible, we shall make two 
assumptions which have been verified by actual calculation. The first is that 
the only sensible coefficient in the expression for e' is a and that the squares 
and higher powers of a may be neglected ; the second is that terms of the 
form t sin (at + 6), which will arise in the final expressions for the coordinates, 
have coefficients so small that they may be neglected. It is required, therefore, 
to find what non-periodic terms are produced in the coordinates by the term 
at. The method of Chapter vn. will be used for the investigation. 



320. Let us consider how non-periodic terms were produced in the 
right-hand member of the equation (1), Art. 130. It was seen in Chap. 
vn. that the first approximation to the coefficient of any term in Su was 
obtained by simply considering the corresponding term in SR : terms of a 
lower characteristic could be neglected. As only non-periodic terms are re- 
quired here, and as it was shown in Art. 116 that d'R/dt contained no such 
terms, we have, from equation (2), Art. 130, neglecting a/a', 

P = r dR/dr + const. = 2 R + const. 

There is only one non-periodic term in R which need be considered, 
namely, that containing e' 2 . We have therefore, from Art. 



P = f mV 2 /a + const. 
Equation (1), Art. 130, then becomes, since e, y are neglected, 

, d% <v o m 2 

a ~j^ u ~ w = f e + const. 
at* 4 a 

Putting e' = e Q f at and neglecting squares of a, this equation furnishes 

aSu = - !?nV 2 + const., -or, a$u = f m 2 (e ' 2 - e /2 ) + const., 
giving the inequality produced in the parallax. 

321. Next, consider the equation (5), Art. 131, for the longitude. 
Neglecting e, 7, a/a', we obtain 

As.,-M_ L i_faR,*, 2*. fi .. 



319-322] THE SECULAR ACCELERATION. 267 

The integral in the right-hand member gives no term free from sines or 
cosines. We therefore get, on substituting for $u, and putting h Q = no?, 

J CS7 

$ v = + const. + f m*n (> ' 2 - e' 2 ). 

CtC d 

Now n was defined so that all constant terms in this equation should 
vanish: this definition will be retained. As e^e /z contains t as a factor, 
the term involving this quantity cannot be made to vanish. Hence, by a 
suitable determination of Sh, we find, on integration, 



The additive constant is put zero, according to the remarks of Art. 158. 
The general expression for the longitude therefore becomes 

v = nt + e -f f mPn, I (0 ' 2 O dt 4- periodic terms 

J 

= nt + + f mn'eQCitf + periodic terms. 

Let the unit of time be one Julian year ; n' will then be the angle de- 
scribed by the Sun in one year. The planetary theory gives, for the epoch 
1850-0*, 

e ' = 0-016771, a = 0*0000004245, ri = 1295977", m = 0'07480. 
The term in v 9 involving t* 9 is therefore "f 



The presence of this term in the longitude is usually expressed by saying 
that the mean angular velocity of the Moon is not quite constant but has 
a secular acceleration of 10""35 per century ; the more correct statement 
being that the mean motion is increasing at the rate of 2 x 10"*35 per century 
in a century. 

322. This is approximately the value found by Laplace and it requires 
considerable modification when we proceed to higher powers of m and to the 
terms dependent on e 2 , ry 2 , etc. It is in the second and higher approximations 
that the difficulty of the subject arises. To obtain the next approximation, 
it is necessary to consider not only the non-periodic part of R but also those 
periodic terms which, in combination with periodic terms of equal arguments, 
may produce non-periodic terms in the longitude equation : and it is to 
be remembered, when integrating the equations of motion, that &' is variable. 
For instance, to get the next approximation in powers of m by this method, 

* Ann. de VObs. de Paris (M#m.), Vol. iv. p. 102. 

t The result obtained for this term by Adams and Delaunay (see Art. 296) is -hlO"-66, owing 
to the use of a slightly different value for a. 



268 PLANETARY AND OTHER DISTURBING INFLUENCES. [CHAP. XIII 

it Is necessary to retain (i) the parts of II of characteristic zero, (ii) the 
non-periodic term of the order e' 2 , (iii) the periodic terms of characteristic 
e', (iv) the values of r', v' as far as the order e /2 in the non-periodic terms and 
to the order e' in the periodic terms. The details of the next approximation 
are too extended to be given here ; they may be found, calculated to the 
order m 4 after this method, in a memoir by Cayley (see Art. 296 above). 
The value to the order m 5 is 

- (f m fl - *L w 4 - Styfi m 5 ) n I (ej* - e' 2 ) dt. 

The coefficients corresponding to the second and third terms are 2"*27, 
1"'54, causing the acceleration to be diminished to 6"*5. Delaunay * finds 
6"'ll to be the complete theoretical value. 

It is necessary to make one further remark. The value of e! does not 
always continue to dimmish ; after a period of about 24000 years it will have 
reached its minimum value and begin to increase again, attaining a maximum 
after the lapse of another period of similar magnitude. Were it not for this 
fact, the period of revolution of the Moon would go on increasing and its 
distance diminishing until it was brought within the limits of the terrestrial 
atmosphere. The periodic nature of the variation of e' prevents this small 
inequality from producing any great change in the relations between the 
Moon and the Earth. 

The observable effect on the mean distance is quite insensible. This will be readily 
understood when it is mentioned that the annual mean approach of the Moon to the 
Earth, due to this cause, is less than one inch ; in 200 years, the mean distance will be 
only fourteen feet smaller than at the present time, corresponding to a change in the 
parallax of less than one twenty-thousandth of a second of arc. 

Since the expressions of c, g contain terms dependent on e' 2 , direct observations of the 
motions of the Perigee and the Node will show similar secular changes. The values of 
c, g contain the terms -|mV 2 , +fwV 2 . The first approximations to the secular 
accelerations of these motions will therefore be 



respectively. The first of these is very much altered by the further approximations. 
Delaunay f finds -40"-0 and +6"'8 for their complete values. 

For further references on the subject of the secular acceleration, see Art. 296. 

* " Calcul de ^acceleration seculaire du moyen mouvement de la lune," Comptes Rendus, 
Vol. XLVIII. pp. 817 827. 

t '* Calcul des variations s6culaires des moyens mouvements du pe"rige*e et du nceud de 1'orbite 
de la lune," Comptes Rendus^ Vol. XLIX. pp. 309 314, 



TABLES OF NOTATION 



AND 



INDICES 



I EEFEEENCE TABLE OF NOTATION. 



The numbers following the symbols refer to the articles in which the symbols cure first 
used. Brackets denote tfiat the symbol is defined and used with that definition only, TO the 
articles which accompany it. Symbols which are defined and used in, one article only, or in 
two consecutive articles only, are not generally included. The new symbols occurring only 
in articles 308-313 are also omitted. The letters used in the figures are not included. 



General Notation i.-vin., XL, XIL 
(1-178, 242-302, 319-322). 



Delaunay ix., xm. 
179-201, 303-318). 



a 12, a' 19, a^ 247, A' 22, A t A t ' 247, 
(A 64-66), (A A' 111, 115), a 247, 
A 128. 
&* 134, &' 166, ff 22, (JJ 135, 158), 

bt V 264, B 128. 
c 68, Ci 139, 20, 0' 22, c 257, c 

267, C 128. 
d' 12, D 18, D t 20. 
e 32, e' 53, (<?' 319-322), E 3, E 32, 

e 261, Eq. 39. 
/ 32, /' 110, F 8, (F 3-8), (F 1 V{ 

5-9), F 18. 

g 68, g t 147, g 279, g, 280. 
h 12, h a 77, (If 94-98). 
{44. 

Ji ( ) 37. 
K' 22, Ki Ki 279, K 128. 

V fjMy JU JLf Ji X'. 

m 114, Wx 124, m' 3, IT 3, Jf 4 265, 

m 18. 

?i 18 and 48, w' 19, -^263. 
(Pi 94-104), P 130, $ a 16, $ 75. 
(ji 94-104). 



a, 192, J. 183, 

ji' 306, 
JS jBi 183. 
c G 183. 
D200. 
e fl 192. 
P200. 



183, (gr) 184, 
186, G' 190. 



183, (A) 184, 
hi h s Hi H c 
186, H' 190. 

i {' i" i'" 180. 

j/j" 306. 

Jfl84. 

/ T T T7Q 77 

t X/ & J- t/, v<l v$ 



195, L 309. 



Hansen x. 
(202-241). 



a Q 208, cto 7 220. 

6230. 

00^224,^^0/228, 

0233. * 

e<> 208, e<>' 220, i 
_208. 
/ 208, f 220, / 

225. 

^ 208, ^' 224. 
h 204, A 208. 
i' 217. 

/217, Jo 219. 
Z 217, K 219. 
Z 208. 
m, 220. 
w 208, < 220, N 

217, -ZVo 219. 
p p' 217, P 219, $ 

204, ?)3 225. 
5 ^ 217, Q 219. 
r 208, T 220, r 

225, R 204. 
5241. 



REFERENCE TABLE OF NOTATION. 



271 



r r 3, (r x 5-7), r, 12, r 130 3 R 8, Ri 

43, JR 124. 

5 12, 8 3, ($ 5-7), (8 94-98). 
i the time, 4 18, ^ 257, 4 280, (T 

94-98), (T 165-172), ^ 16, 75. 
w 130, ^ 16, Z7i 265, 7 15. 
v 12, t/ 22, <y 131, F 263. 
w 32, w' 110. 



xyzx'y'z 3, 



278, ZFZ'F 



18, (g)33e')'3' 3-5), 3, 16, 3 

75. 
(a 134, 158), a x a, a s 84, (a 319- 

322). 

fa A A 84. 
7 45, 7j 122. 

A 3, A (c) 266, V (g) 280. 
e 48, e' 22, * 73, e/ 259. 
18. 

9j 45, 17 68. 
6 44, (0! e, 6 3 73-80), 263, * 

264. 
*19. 
(\ 33, 36). 



| 111. 

45, w' 53. 

a 18, o- 276, <7 v2i 278. 

v 18, v 276, *| 278. 

^ 68, ^' 110. 

^ 263. 

a) a>' 124, O 19, flp 19. 



2 180. 


y o .226, 204, X, 


R 179,^183, 


225, 


R 189, JT 


v 206, v' 217, (F 


190, (R 308- 


241). 


312). 


F211, F214, F 


7 179, 70 192. 


226, F 229. 


77 189, (77 308- 


y 208, y' 220. 


310). 


z 208, / 220, 3 


6 @ 183, Oi 


204. 


e 8 t c 186. 


a 217. 


ic 189. 


(ft 206, 215), /3/3' 


\ A 189, X' A' 


228. 


190. 


7226. 


188, </>' 191. 


? 214. 




77217. 


, If, /A, m', 


0' 217, 


r, r', v, */, 5, 


^ 219, 0^ 241). 


$ 92/ tt' 6 X 00 


v 209, i/ 220. 


y, a, of, y', 


f & 230. 


z' retain the 


TT O 208, < 217. 


same mean- 


p 214, po 226. 


ings as in the 


<T 206, er 217. 


first column. 


r214. 


7i, a, e, (and 7) 


T 230. 


refer first to 


$ 214, ^ 226. 


the instanta- 


%206. 


neous orbit. 


,fr tf 217, 230. 


In Art. 200, 


w to' 220. 


they denote 




the arbitrary 


jE, AT, /^, m', r, r', 


constants. 


A, cr, fif, JS, Ri 


For change of 


retain the same 


notation, see 


meanings as in the 


Art. 179. 


first column. 




n, a, e, e, cr, 0, i, I, 




f, jg?, L refer here to 




the instantaneous 




orbit. 




For change of nota- 




tion, see Art. 204. 



II. GENERAL SCHEME OF NOTATION. 



The symbols connected with the lunar orbit refer to undisturbed elliptic motion in 
Chap. in. and to the Instantaneous Ellipse in Chap, v.; their meanings in Chaps, ra, FUJT., 
together with those of other symbols, will be found in Table III. Accented letters, in general, 
refer to the solar orbit. 



Symbols. 



E, M, m' 

V* 

r, r', A 

8 

x, y, z 

X, F, * 

v } a 

% 

v, v' 
u 
s 

F 

R 

$, .3 



n, 

a, a! 

e, e' 

e, e' 

-or, t& 

6 

i 

7> 7i 

// 

w, w' 

E 
L 
I 
h 

r 

D 



Significations. 



Masses of Earth, Moon and Sun. 

E + M. (p = 1 in de Pont6coulant's theory,) 

Distances of Moon and Sun from Earth, and of Moon from Sun. 

Cosine of angle between r, r'. 

Coordinates of Moon referred to Fixed axes through Earth. 

Coordinates of Moon referred to Moving axes through Earth. 



l/r. ^ 

Longitudes of Moon and Sun reckoned on Ecliptic. 

Latitude of Moon above Ecliptic. 

Tan u. 

Force Function for Motion of Moon. 

Disturbing Function arising from solar action. 

Solar Forces on Moon, along r, perpendicular to r in plane of 

orbit, and perpendicular to plane of orbit. 
Solar Forces on Moon, along the projection of r, perpendicular 

to projection of r in Ecliptic, and perpendicular to Ecliptic. 
Mean Motions of Moon and Sun. 
Defined by equations fju = n*a?, m! + p n'V 3 . 
Eccentricities of lunar and solar orbits. 
Longitudes of Epochs of Mean Motions. 
Longitudes of lunar and solar Perigees. 
Longitude of lunar Node. 
Inclination of lunar orbit to Ecliptic. 
Defined by equations 7 =. tan i, ^ = sin \i. 
Lunar and solar True Anomalies. 
Mean 

Eccentric Anomaly. 
Angular distance of Moon from its Node. 
Latus Rectum of Moon's orbit. 
Eate of description of areas by Moon in Ecliptic. 
Exp. {(n -vfyt + e- e'} V^l. 



Ill, COMPAEATIVE TABLE OF NOTATION. 



In the first column is the notation used ly de Pont&oulant, the second and third columns 
contain the corresponding symbols used ly Delaunay and Hansen. The fourth column 
contains the final definitions; those in square brackets refer to the methods of Chap. xi. 



Do Pont. 


Del. 


Hansen*. 


Definitions. 


n 


n 


no(l + y- 


Observed Mean Motion. 


en 


l * 


W 


Mean Motion of Mean Anomaly. 


(I - c) n 


#>+AO 


% (2/ - 29?) 


Perigee, [c = c/(l - m).] 


(1 - g) n 


A 


-w (a +17) 


Node. [g = ^/(l-m).] 


%' 


71' 


n ' + nj' 


Observed Mean Motion of Sun. 






y 


Mean Motion of Solar Mean Anomaly. 


<!> 


I 


5 1 


Arg. of Principal Ell. Term in Long. 


& 


l f 


^ 


Annual Equation. 


1 


I) 


</ H- ft) $' 6>' 


Half Arg. of Variation. 


* 


JP 


<7+w 


Arg. of Principal Term in Lat. 


m 


m 




m= ^!:^ 


a 


a 


a 


aW = p = afnf. 








[For a, see Arts. 255, 2*73.] 


a 


a' 


a ' 


a V 2 = m' } a /5 V 2 = m' -f yw. 


g-j- 


e 




Ecc., defined by Principal Ell. Term in 








Long. [For e, see Arts. 261, 274.] 






e 


Ecc., defined by Aux. Ellipse. 


0' 


</ 




Solar Eccentricity. 






^C/ 


Solar Ecc., defined by Solar Aux. Ellipse. 


ryf 






Tan i, defined by Principal Term in zfa. 


7tt 


7 




Sinji, L a ^ 






/ 


Inclin., sin?7. 








[For K Q> see Arts. 281, 286.] 






/n0 


True and Mean Anomalies of Aux. Ell. 






a,'wL 


Mean and Periodic parts of njs. 






5 


Sin V - sin J sin ( /+ ) 








Radius vector of Aux. Ellipse. 






1 + w 


r/f. 



* Hansen leaves out the zero suffix when the earlier developments have been completed 
t These are not the definitions actually assigned by de Ponteooulant. See Art. 159 



B, L. T, 



18 



INDEX OF AUTHORS QUOTED. 



(The numbers refer to the pages.} 

Adams 10, 24, 196, 213, 230, 236, 243, 246, Hansen 36, 74, 75, 76, 77, 91, 123, 131, 

265, 267. 160-194, 243, 246, 248, 259, 263, 265. 

Airy 131, 245, 246. Harzer 178. 

Andoyer 47. Hayward 67. 

Arago 259. Hill 10, 45, 47, 114, 125, 190, 196-213, 223, 

228, 249, 258, 261, 262, 263, 265. 
Hobson 32, 219, 222. 



Ball (W. W. B.) 238. 
Binet 75. 
Breen 131. 
Briinnow 171. 
Bruns 27. 
Burckhardt 246. 
Burg 246. 

Carlini 244. 

Cayley 33, 36, 43, 60, 76, 77, 92, 171, 243, 

268. 

Cheyne 64, 73. 
Chrystal 219. 
Clairaut 237, 238. 
Cowell 230. 



Jacobi 25, 66, 74, 75, 77, 135, 182. 

Koch, von 220. 
Krafft 240. 

Lagrange 73, 75, 77, 247. 

Laplace 16, 19, 28, 43, 87, 92, 238, 242, 243, 

244, 247, 263, 267. 
Leverrier 131, 260, 267. 
Lexell 240. 
Lubbock 243, 245. 



Mason 246. 
Mayer 246. 
Damoiseau 238, 244. 
Darwin 213, 248. 
D'Alembert 238, 239. 
Delaunay 87, 89, 123, 125, 126, 128, 130, 133, Newton 127, 237, 238, 239. 
137, 138, 155, 156, 161, 243, 249-252, 259, 

260, 267, 268. Plana 238, 243, 244. 

De PontScoulant 16, 28, 86, 87, 112, 113, 114, PoincarS 10, 27, 46, 53, 66, 77, 125, 135, 196, 



Neison 249. 

Newcomb 123, 131, 188, 192, 243, 246, 260. 



127, 245, 263. 
Donkin 76. 
Dziobek 28, 66, 73, 77, 135. 

Euler 238, 239, 240, 247. 

Forsyth 47, 50. 
Frost 75. 

G-autier 238. 
Godfray 243, 265. 
GogOTi 87. 
Greatheed 32. 
Gyld6n 10, 45, 47. 

Hamilton. 77, 



200, 219, 220. 
Poisson 77, 245. 
Pratt 261. 

Eadau 134, 246, 249, 251, 258, 260, 265. 
Bouth 1, 28, 53, 55, 59, 68, 69, 73, 260. 

Tait and Steele 17, 32, 213. 

Thomson and Tait 77. 

Tisserand 28, 36, 43, 73, 77, 108, 134, 155, 

159, 194, 238, 241, 243, 258, 263, 265. 
Todhunter 33, 79. 

Williamson 32. 
Zeeh 170, 



GENEEAL INDEX. 



(The numbers refer to the pages.) 

The following abbreviations are frequently used: P. for de Pont<coulant's method, D. for 
Delaunay's method, H. for Hansen's method, B. and rect. coor. method for the 
method of Chapter xi. These abbreviations refer only to the methods as set forth in 
the text. 



Acceleration, dimensions of, with astronomical 
units, 1 ; 

secular, 243, 265 (see secular). 
Accelerath e effect, term 'force 1 used for, 1. 
Action of the Sun, causes the perigee and node 
to revolve, 53 ; 

D'Alembert's method of introducing the, 

in the first approximation, 239. 
Action f the planets (see planetary). 
Adams, equations used in proving the theorems 
of, connecting the parallax with the motions 
of the perigee and the node, 24 ; 

statements of the theorems, 236; 
method of, for the motion of the node, 

230; 

for the secular acceleration, 243 ; 
for the motion of the ecliptic, 263 et seq. 
Airy, numerical lunar theory of, 245. 
Algebraic uniform integrals, number of, in the 

problem of three bodies, 27. 
Algebraic development, de Pont<coulant's, 113 ; 
d'Alembert's, 239 ; 
Delaunay's, 159 ; 
Lubbock's, 245 ; 
Plana's, 244; 

contrasted with semi- algebraic and nu- 
merical, 246. 

Analysis, claim of Clairaut in first using, 237. 
Angular coordinate, periodicity of, defined, 49. 
Angular coordinates, final forms of the three 

lunar, 121. 
Annual equation, defined, 129 ; 



order of coefficient of, lowered by inte- 
gration, 104 ; 
coefficients of, 129, 130; 
notation for the argument of (D.), 159. 
Anomaly, eccentric, defined, 30 ; 

expansion of functions of, in terms of 

the mean anomaly, 34 ; 
of Hanseri's auxiliary ellipse, 165. 
Anomaly, mean, defined, 30 ; 
in terms of the true, 31 ; 
in elliptic motion, 40 ; 
convergence of series in terms of, 43 ; 
development of disturbing function in 

terms of, 89 et seq.. ; 
used as a variable (D.), 136; 
of the auxiliary ellipse (H.), 165 ; 
perturbations added to, 160, 164 ; 
equation for, 167 ; 

integration of, 187 ; 
considered constant in the inte- 
grations, 169; 

arbitrary constant present in, 171 ; 
used in the development of the 
disturbing function, 177 et seq.. ; 
elliptic value of, 181 ; 
form of expression for the disturbed, 

187; 

definition of mean motion of," in 
disturbed motion, 187 ; 

remarks on, 188 ; 
third approximation to, 192; 
of the Sun, notation for, 177 ; 

182 



276 



GENEKAL INDEX. 



of the Sim, used by Euler as independent 

variable, 241. 
Anomaly, true, defined, 30 ; 

in terms of the mean, 32 ; 

symbolic formula for, 33 ; 
expansion of functions of, in terms of 
the mean, 34 et seq. ; 

Hansen's method for, 36 ; 
used by Euler as independent variable, 

240. 

Approximation, solution by continued, 47 (see 
continued) ; 

second, to the disturbing function, 87 ; 
second, and third (P.), 95 ; 
second (H.), 181 et seq..; 
third (H.), 192; 
method of (B.), 195; 
rapidity of (B.), 204. 
Apse (see perigee). 

Arbitrary constants in elliptic motion, 40, 41; 
connection between those present in 
de PontScoulant's equations and, 
41; 

any function of, called Elements, 48 ; 
meanings to be attached to the, in 

Jacobi's method, 71 ; 
in disturbed motion, difficulties in the 
interpretation of, 16 ; 
interpretation of, 115 et seq. ; 
numerical values of the solar, 123 ; 
of the lunar, 124 et seq. ; 

references to memoirs contain- 
ing the, 131; 

in de Ponte"coulant 3 s method, number of, 
introduced and necessary, 16; 

equation to determine the extra 

constant, 16; 
definitions of, used by de Pont- 

coulant, 113, 119; 

in Laplace's equations, number of, intro- 
duced and necessary, 19 ; 

method of defining the, 119, 243 ; 
in Delaunay's method, used as new 
variables, 139, 142; 

transformation of the final, 158 ; 

final definitions of, 158; 

a change of, explains an apparent 

error in the results, 87 ; 
in Hansen's method, seven necessary 
166; 

number introduced into the equa- 
tions, 171; 

their significations, 171 ; 
for the motion of the orbital plane, 
176; 



determination of, in W., 186 ; 
definitions of, 187, 188, 192; 

remarks on the, 188 ; 
in rect. coor. method, number introduced 
and necessary, 22 ; 
definitions of the, introduced, 199 ; 
final definitions of the new, intro- 
duced, 210, 225, 231, 233 ; 
numerical values of the, obtained by 
Euler, 240; 
his method for the determinations 

of the, 241;. 

(see also constants, elements) ; 
variation of the (see variation). 
Areas, integrals of, in the problem of three 
bodies, 27 ; 

rate of description of, in elliptic motion, 

40, 41, 67, 136. 
Argument, mean, of latitude defined, 41; 

in the disturbing function, whose motion 

is equal to the mean motion, 85. 
Arguments, connections between coefficients 
and, in elliptic expansions, 36 ; 

when the ellipse is inclined to the 

plane of reference, 40 ; 
in the disturbing function, 82 ; 

for planetary actions, 255, 256; 
the expressions for the coordinates con- 
tain multiples of only four, 84; 

discovered by d'Alembert, 239; 
form of, in de Pontdcoulant's method, 
in the disturbing function, 81 * 
in the final expressions for the co- 
ordinates, 110; 
connection of, withLaplace'smethod, 

131; 

for terms with coefficients not con- 
taining m as a factor, 87 ; 
in Delaunay's method, 137; 

in the disturbing function 89, 138; 
after any operation, unaltered, 156 ; 
final, 157 ; 

notation for, in the results, 159 ; 
in Hansen's method, in the disturbing 
function, 91, 178; 

in the disturbed mean anomaly, 187 ; 
in the disturbed radius vector, 189 ; 
in the motion of the orbital plane, 

192; 
in rect. coor. method, 199, 206, 227, 

228, 230, 231; 
used by Euler, 241; 

in Laplace's theory, functions of the 
true longitude, 132, 242 ; 
in the disturbing function, 92 ; 



GENERAL INDEX. 



277 



coefficients of the time in, incommen- 
surable, and will not vanish unless 
the arguments vanish, 49, 81, 184. 
Ascending node, denned, 41 (see node) ; 

of the ecliptic on the equator, the origin 

for reckoning longitudes, 261. 
Astronomical unit of mass, denned, 1. 
Attraction, Newton's law of, 1; 
G-aussian constant of, 1 ; 
law of, for spherical bodies, 2. 
Auxiliary ellipse in Hansen's method, defined, 
164; 

used as an intermediary, 164 ; 

relation of, to the actual position of the 

Moon, 165 ; 

formulae referring to, 166 ; 
coordinates of, considered constant, 169 ; 
constants of, 171 ; 

signification of, 187 ; 
used for development of disturbing func- 
tion, 177 et seq. ; 
the solar, 177. 
Axes, rectangular (see rectangular) ; 

Euler's formulas, for rotations of, 55 ; 
of the variational curve, 125, 127. 

Bessel's functions, defined, 83; 

used for elliptic expansions, 33 et seq. 
Bodies, the celestial, considered as particles, 2 ; 

problem of three, of p (see problem). 

Canonical constants, defined, 66 ; 
Delaunay's, 64; 

dynamical and geometrical meanings 

of, 67; 
produced by Jacobi's dynamical method, 

73; 
initial coordinates and velocities form a 

system of, 76. 

Canonical system of equations, Delaunay's, 
deduced, 65; 

obtained by Jacobi's method, 72 ; 
transformation from a, to a, tangential, 66; 
Lagrange's, 76; 

Hansen's extension of, 76; 
in Delaunay's method, defect of the 
first, 134; 

second system of, transformation to, 
to avoid the presence of the time 
as a factor, 136 ; 
integration of the, 139 et seq. ; 
nature of the solution, 144 ; 
the arbitrary constants of the solu- 
tion give a new, 143 ; 
second system, to avoid the 



presence of the time as a 
factor, 147; 

third system, to correspond 
with the previous second 
system, 149; 
Hill's method of using Delaunay's, for 

small disturbances, 249 et seq. 
Centre, equation of, defined, 35 ; 

expansions of functions of, 36. 
Centre of mass of the Earth and Moon, the 
Sun's force-function relative to, 5 ; 

motion of, considered an ellipse, 6 ; 
correction to the disturbing function 
when the motion is referred to the 
Earth instead of to the, 8 (see cor- 
rection). 

Change of position due to changes in the ele- 
ments, general formulae for, 56 ; 

zero in the motion, 59. 
Characteristic, defined, 82; 

connection with the argument, 82 ; 
unaltered by the integration of the 

radius- and longitude-equations, 86; 
diminished one order, by substitution 

in the latitude-equation, 86; 
left arbitrary in Laplace's method, 242 ; 
and in finding the variations of the co- 
ordinates in Hill's method for small 
disturbances, 252. 
Clairaut, lunar theory of, 238 ; 

showed that the observed and theoretical 

motions of the perigee agree, 239. 
Classes, division of the inequalities into, P., 95; 
not made by de Ponte'coulant, 112 ; 
in rect, coor. method, 198 ; 
by Euler, 240, 241. 
Coefficients, orders of, defined, 80; 

denoted by the index, 80 ; 
form of the, in the disturbing function, 

81, 82; 

connection between arguments and, 82 ; 
discovered by d'Alembert, 289 j 
in the planetary disturbing func- 
tions, 255, 256 ; 
characteristic parts of, defined, 82 (see 

characteristic) j 

of the time in the arguments, will not 
vanish unless the argument vanishes 
and assumed incommensurable, 49, 
81, 184; 

effect produced on the orders of, in the 
coordinates by the integrations, 84 
et seq. ; 

certain, to be left indeterminate until 
the third approximation, 86, 110; 



278 



GENERAL INDEX. 



of the same order in the second and 

third approximation, 86 ; 
some properties of, 86; 
orders of, in the successive approxima- 
tions, 95; 

division of the inequalities, into classes 
according to the orders of, 95 (see 



slow convergence of the series repre- 
senting the, 113; 
the particular, used to determine the 

arbitraries, 119 et seq.; 
conversion of the, into seconds of arc, 121 ; 
numerical values of certain lunar, 124 

et seq. ; 

magnitudes of the, 131 ; 
in Delaunay's theory, form of the, in the 
disturbing function, 138 ; 
in the solution of the canonical 

equations, 144; 
in the calculation of any operation, 

150 et seq. ; 
relations between the new and old, 

155; 
of the time, in the arguments, 157, 

158; 
in Hansen's theory, in the disturbing 

function, 181 et seq. ; 
of inequalities due, to Venus, 258, 260 ; 
to the figure of the Earth, 263 ; 
to the motion of the ecliptic, 265. 
Comparison, of theoretical and observed values, 
a test, 123; 

of the motion of the perigee, 188, 

239; 

of the secular acceleration, 243 ; 
of certain inequalities, to determine 

the figure of the Earth, 261; 
of the systems of notation used, 136, 

161, 273 ; 
of Delaunay's results with Hansen's, 

159; 

of the values of the various methods, 246. 
Complex variables, used in the lunar theory, 

20 (see rectangular coordinates). 
Condition, equations of, for the variational 
coefficients, 200 et seq.; 

method of solution, 203 ; 

rapidity of approximations in, 

204; 

for the elliptic inequalities, 207, 208 ; 
of the first order, 209; 

method of solution, 210 ; 
of the second order, 223, 224; 
for finding the motion of the perigee, 216 ; 



for inequalities in latitude, of the first 
order, 230; 

of the third order, 232 ; 
when the number of unknowns is greater 
than the number of equations, 122 ; 

first used by Euler, 241. 
Condition, that an infinite system of linear 
homogeneous equations be consistent, 217; 
of convergency, of elliptic series, 43 ; 
of an infinite determinant, 219. 
Connection between, arguments and coefficients, 
in elliptic expansions, 36, 40; 

in the disturbing function, 82 ; 

for planetary action, 255, 256; 
the developments of the disturbing func- 
tions of P., D., H., 89, 92; 
the auxiliary and instantaneous ellipses 

(H.),165; 
(see relations). 
Constant, Gaussian, of attraction, 1 ; 

of mean motion, in elliptic motion, 40; 
in disturbed motion, P., 97, 118; 

D., 158; H., 188; E., 199; 
numerical value, 124; 
when the secular acceleration is 

considered, 267; 
of epoch, in elliptic motion, 40 ; 

in disturbed motion, P., 97, 118; 

D., 158; E., 199; 
when the secular acceleration is 

considered, 267; 
the linear, in elliptic motion, 40; 

in disturbed motion, P., 95, 98, 
119; D., 158; H., 171, 189; E., 
205, 224; 

numerical value of, 124; 
remarks on, 120 ; 

of eccentricity, in elliptic motion, 40 ; 
in disturbed motion, P., 102, 119; 
D., 158; H., 187, 188; E., 210, 
225; 
used by Laplace and de Ponte"- 

coulant, 113, 119, 243 ; 
numerical value, 128 ; H., 188 ; 
of latitude, in elliptic motion, 41; 

in disturbed motion, P., 108, 120, 
130; D., 158; H,, 192, 194; E., 
231, 233; 

in Laplace's method, 243; 
numerical value, 130 ; 
of epoch of mean longitude of perigee, 
in elliptic motion, 40 ; 
in disturbed motion, P., 120; D., 

158; E., 207; 
of epoch of mean anomaly (H.), 181 ; 



GENERAL INDEX. 



279 



of epoch of mean longitude of node, in 
elliptic motion, 41 ; 
in disturbed motion, P., 121; D., 

158; H., 173, 192; H., 230; 
of energy (B.), determination, 205; 

used for verification, 205. 
Constant parts of the functions used, form of 

the, 86. 

Constants, introduced and necessary in P.'s 
equations, 16 ; 

in Laplace's equations, 19; 
in Hansen's method, 166; 
in rect. coor. method, 22 ; 

definitions of the, introduced, 

199; 

interpretation of, 16, 115 et seq. ; 
determination by observation of, 121 
et seq.. ; 

Euler's method, 240 ; 
the solar, 42 ; 

numerical values of, 123 ; 
numerical values of the lunar, 124 et seq. ; 
references to memoirs containing 

the determination of the, 131 ; 
(see also constant, arbitrary, elements) ; 
in Hansen's method, of the auxiliary 
ellipse, defined, 164; 

their significations, 187, 188; 
determination of the arbitrary, in 
W., 186; 

their significations, 187; 
for the motion of the orbital plane, 
176; 

their significations, 192 ; 
in the problem of three bodies, the ten, 

26; 
unreduced numerical values of De- 

launay's, 251; 

for the figure of the Earth, methods for 
the numerical determination of, 261 ; 
variation of arbitrary (see variation). 
Controversy concerning the secular acceleration, 

243. 

Continued approximation, solution by, method 
of, 47; 

applied to de Pont&coulant's equa* 
tions, 49; 

to Hansen's method, 181, 182; 
to rect. coor. method, 195. 
Convergence, of Bessel's functions assumed, 33 ; 
conditions of, for elliptic series, 43 ; 

for an infinite determinant, 219 ; 
slow, of series for the coordinates, 113 j 
a particular case of, 114 ; 
indicates the rect. coor. method, 198; 



change of parameter to improve, 

114, 204; 
avoided by using the numerical 

value of m, 246 ; 
question of, recognised by d'Alembert, 

239. 

Coordinates, referred to fixed axes, 13 ; 
to moving rectangular axes, 19 ; 

used by Euler, 240; 
elliptic expressions for, 41 ; 

forms of, used by Delaunay, 138 ; 
forms necessary for the, 45; 
modified elliptic expressions for, 52; 
and velocities have the same forms in 
disturbed and undisturbed motion, in 
the method of the variation of arbi- 
trary constants, 48, 58, 72 ; 
in Jacobi's method, 68; 
ideal, defined, 74; 

general conditions for the existence 

of, 75; 

Euler's formulas for transformation of, 75 ; 
initial velocities and, form a canonical 

system, 76; 

expression of disturbing function in 
terms of fixed rectangular, 8, 79, 136; 
in the case of planetary actions, 
253; 

transformation to polar, 254 ; 
of moving rectangular, 20 et seq., 

179; 
expressions for, in de Pontexioulant's 

method, 110; 
forms of the three angular, of the Moon, 

121; 
forms of expressions for, after Delaunay' s 

operations, 157 et seq. ; 
of the auxiliary ellipse, can be con- 
sidered constant (H.), 169; 
cylindrical, used by Euler, 239. 
Corrections, to be made to the force-functions 
for the solar and lunar motions, 5-8 ; 

to take into account the Moon's mass, 
to the disturbing function, 8, 178, 252 ; 

general form of, 178; 
to the parallactic inequalities, 126, 159 ; 
to Kepler's laws, to account for the 

Earth's mass, 42, 90; 
to be applied to the tables, 123; 
to Hansen's eccentricity, 188; 
to Newton's law found unnecessary, by 
Clairaut, 239; 

by Euler, 240; 

to Laplace's value for the secular accele- 
ration, 243. 



280 



GENEEAL INDEX. 



Curve, the elliptic, 29 et se<i. (see elliptic). 
Cylindrical coordinates, used by Euler, 239. 

D'Alembert, method and discoveries of, 239. 
Damoiseau, method of, 244. 
Darlegung, full title of the, 160; 
object of publication, 161 ; 
determination of y in, not available, 185. 
Definitions of the constants (see constant). 
Delaunay, theory of, canonical system of ele- 
ments and equations, 64, 134 ; 
signification of, 67 ; 
deduced by Jacobi's method, 72 ; 
problem solved in, 133 ; 
development of the disturbing function, 
88; 
comparison with other methods, 89, 

92; 

form of, in Delaunay's notation, 137 ; 
canonical equations used in, 136; 
notation used, 136; 

meanings of variables, 137 ; 
elliptic expressions for the coordi- 
nates in, 138; 

method of integration, 139 et seq. ; 
general procedure, 155 ; 
analysis of the theory, 156 ; 
changes of the arbitraries and final form 

of the results, 158; 
correction to, to account for the Moon's 

mass, 159; 
numerical values of solar and lunar 

constants used in, 123 et seq. ; 
Airy's method of verification of the re- 
sults of, 245 ; 

compared with other theories, 247; 
Hill's method of continuing the, for small 

disturbances, 248 et seq. 
Departure point, denned, 60 ; 

curve described by a, cuts the orbital 

plane orthogonally, 60 ; 
longitudes reckoned from a, have the 
same form in disturbed and undis- 
turbed motion, 60 ; 

use of a, introduces a pseudo-element, 
75; 

and an arbitrary constant, 163 ; 
when ecliptic is in motion, 264 ; 
used in Hansen's method, 160, 163. 
Dependent variables used in the various 

methods, 12 (see variables). 
De Ponteeoulant's method, variables used in, 12; 
equations of motion, 13 et seq. ; 

arbitrary constants introduced into, 
and necessary for, 16 ; 



solution of, when the solar action is 

neglected, 41 ; 
solved by continued approximation, 

49; 

modified intermediary for, 52 ; 
development of the disturbing function, 
82; 

deducible from Delaunay's, 89; 
and from Hansen's, 92 ; 
unit of mass used in, 82 ; 
the effects produced by the integrations, 

84 et seq. ; 
method for the higher approximations, 

87; 

preparation of the equations for the 
second and higher approximations, 
93 et seq. ; 

"details of the second, and of parts 
of the third, approximation, 96 et 
seq.; 

summary of the results, 110 ; 
analysis of, as contained in the Systeme 

du Monde, 112 et seq. ; 
slow convergence of the series for the 

coefficients, 113; 

meanings to be attached to the arbitraries 
in, 117 et seq. ; 
definition of the eccentricity used 

by de Ponte*cotilant, 119 ; 
comparison of the arguments in Laplace's 

method, with those in, 131 ; 
form of solutions for rect. coor. method 

deduced from, 199, 207, 230 ; 
similar to Lubbock's, 245; 
compared with other methods, 247 ; 
first approximation to the secular accele- 
ration by, 266 et seq. 
Derivatives of the disturbing function (see 

disturbing). 
Determinant, of a system of linear homogeneous 

equations, 210 et seq. (see infinite). 
Development, of the disturbing function (see 
disturbing) ; 

of an infinite determinant, 221 et seq. 
Differences between theory and observation 

(see comparison). 
Direct action of a planet, defined, 253 (see 

planetary). 

Discoveries of, Newton, 237 ; 
Clairaut, 238 ; 
d'Alembert, 239 ; 
Euler, 241 ; 
Laplace, 243 ; 

Adams, with reference to the secular 
acceleration, 243. 



OENERAL INDEX. 



281 



Distance, mean, defined, 120; 

method of finding the, from observation, 

123; 

numerical value of the, 124 ; 
effects of the secular acceleration on the, 

268. 

Distance, constant of mean, in disturbed 
motion, P., 119 ; D., 158; H., 171, 189; B., 
205,224; 

Euler's, 240. 

Distances, ratio of, the disturbing function first 
developed in powers of, 5 et seq. ; 
large in the planetary theory, 9; 
of the second order, 80 ; 
(see parallactic). 

Distribution of mass of the Moon, Hansen's 
empirical term to account for a supposed 
non-uniform, 248. 
Disturbances, Hill's method of integrating for 

small, 248 et seq. 
Disturbed body, defined, 9 ; 

mass of a, relative to the primary, 9. 
Disturbed elliptic orbit, coordinates and veloci- 
ties have the same form in the undisturbed 
and, 58 ; 

also, longitudes reckoned from a 

departure point, 60; 
(see variation of arbitrary constants). 
Disturbing body, defined, 9 ; 

mass of, relative to the primary, 9.* 
Disturbing forces, derivatives of the disturbing 
function in terms of, 57 ; 

rate of rotation of the orbital plane due 

to, 60 ; 
variations of the elements in terms of, 63 ; 

in Hansen's theory, 163 ; 
deduced from the development of the 
disturbing function (P.), 83; 

higher approximations to, 88 ; 
in Hansen's theory, 179 et seq., 182; 

higher approximations to, 193 ; 
deduced direct from the force function in 

Laplace's theory, 92 ; 
in Hansen's theory, notation for, 161 ; 
equation for W in terms of the, 170 ; 
for the motion of the orbital plane, 

in terms of the, 176. 

Disturbing function, due to the Sun's action, 
defined, 8 ; 

for planetary actions, 252 ; 

separation of the terms, 253 ; 
expression in polar coordinates, 254; 
for the figure of the Earth, 261. 
Disturbing function, derivatives of the, with 
respect to the polar coordinates, 14, 15, 58 ; 



to the elements in terms of the forces, 
57; 
in Hansen's method, 180 et seq., 

182; 

to the major axis, remarks on, 66 ; 
equations for the elements in terms of, 64 ; 
coefficients of, independent of the 

time, 73 ; 
properties of, 83 ; 
higher approximations to, 88. 
Disturbing function, development of, in powers 
of the ratio of the distances, 5, 20, 79 ; 

in the planetary and lunar theories, 

9; 

for planetary action, 252 ; 
in terms of the elliptic elements, 80 ; 

properties of, 81, 86 ; 
higher approximations to, 87 ; 
in de Ponte'coulant's theory, 80 ; 
result, 82; 

parts of, for certain inequalities, 
96, 100, 103, 104, 107 ; 
deduced directly, 111 ; 
in Delaunay's theory, method for, 88 ; 
form used, 137 ; 

one term of, used for integration, 139 ; 
form of, with the new variables, 145 ; 
effect of an operation on, 150 ; 
relations between the new and old, 

152; 

reduced to a non-periodic term, 156 ; 
in Hansen's theory, method for, 89 ; 
form of, 177 ; 

first approximation to, 181 ; 
higher approximations to, 192 ; 
in rect. coor. method, 20, 92 ; 

terms in, for certain inequalities, 

225, 227 ; 

in Laplace's theory, 92 ; 
for planetary action, 255 et seq. ; 
term in, due to Venus, 258 ; 
due to the variation of the solar 

eccentricity, 266 ; 
for the figure of the Earth, 262. 
Divergent series may represent functions, 53. 
Division of inequalities into classes (see classes),. 
Divisors, effect of, on the orders of coefficients, 
84 et seq., 95, 96 et seq.; 

on the variational inequalities, 204 ; 
on the mean-period inequalities, 227 ; 
on planetary inequalities, 257. 
Dynamical methods of Hamilton, Jacobi and 
Lagrange, 67 et seq. 

Eccentric anomaly (see anomaly). 



282 



GENEKAL INDEX. 



Eccentricities, considered of the first order, 80 ; 

used as parameters in expansions, 30, 80; 

connection between the arguments and 

powers of, in the disturbing functions, 

36, 82, 255, 256 ; 
forms in which the, occur, 86. 
Eccentricity, lunar, equation for the variation 
of, 61 ; H., 162 ; 

constant of, in disturbed motion, P., 
102, 119; D., 158; H., 187, 188; B., 
119, 210, 225 ; 

determined from the principal ellip- 
tic term, 123 ; 
relation of, to that used in 

rect. coor. method, 211 ; 
numerical value, 128 j H., 188; 
used by de Pont6coulant andLaplace, 

113, 119, 243 ; 

relation of, to Delaunay's variables, 138 ; 
presence of, as a denominator avoided 

(D.),154; 
inequalities dependent on (see elliptic 

inequalities). 
Eccentricity, solar, numerical value of, 123 ; 

how the, is included in the motions of 

the perigee and node, 234 ; 
variation of, the cause of the secular 
acceleration, 243, 265 ; 

periodic nature of, 268. 

Eclipses, used by Euler to determine the arbi- 
traries, 240 ; 

ancient, and the secular acceleration, 

243. 
Ecliptic, considered fixed, 13 ; 

in Hansen's method, movable, 162 ; 
quantities defining the motion of, 

172; 

reduction of expressions to, 194 ; 
effect of secular motion of (H. ), 191 ; 
Adams' method, 263 ; 
principal inequality due to, 265 ; 
ascending node of, on equator, the origin 

for reckoning longitudes, 261. 
Elements, definitions of, 41 ; 
extended, 48, 74; 
of the lunar orbit, 41 ; 
of the solar orbit, 42 ; 
of the instantaneous orbit, 48 ; 
coordinates of the Sun and Moon in 
terms of the, and of the time, 41, 42, 
138; 

of the true longitude, 42 ; 
development of the disturbing functions 
in terms of the (see disturbing func- 
tion) ; 



not used in the rect. coor. method, 

92; 

derivatives of the disturbing function 
with respect to, in terms of the dis- 
turbing forces, 57 ; 

change of position due to variability of, 
56; 

is zero in the actual motion, 59 ; 
equations for the variations of, in terms 
of the disturbing forces, 63 ; 

required in Hansen's method, 

162 et seq. ; 
in terms of the derivatives of the 

disturbing function, 64 ; 
Lagrange's, 73 j 
canonical systems of, 64 et seq. (see 

canonical) ; 
pseudo-, definition and properties of, 74, 

75; 

meanings of Delaunay's, 67, 136 ; 
purely elliptic values of, defined (H.) , 171 ; 
used in the Pundamenta, 171 ; 
relations of the final constants to the 

elliptic (D.), 158; 
Badau's numerical equations for the, 

for small disturbances, 251 ; 
variations of, due to motion of ecliptic, 

264. 

Ellipse, motion of the Sun considered an, 6 j 
formulae and expansions connected with 

the, 29 et seq. (see elliptic) ; 
used as an intermediary, 46 ; 
also, when modified, 52 j 

used by Clairaut, 238 ; 
instantaneous, 48 ; H., 162 ; 
auxiliary (H,) 164. 
Elliptic elements (see elements). 
Elliptic expansions, in terms of the true ano- 
maly, 31 ; 

in terms of the mean anomaly, 82 ; 
by Bessel's functions, 33 et seq, ; 
Hansen's theorem concerning, 36 ; 
when the plane is inclined, 88 et seq. ; 
for the coordinates, 41, 138 j 
convergence of, 43. 
Elliptic inequalities, defined, 128 ; 

determination of, de Pont^coulant's me- 
thod, 100 et seq. ; 
rect. coor. method, 206 ; 
of the first order, 209 ; 
of higher orders, 224 j 
terms in the disturbing function 

for, 112, 

Elliptic motion, formulas and expansions con- 
nected with, 40 et seq. j 



GENERAL INDEX. 



283 



method of including the effects of the 
solar and planetary deviations from, 
253, 257. 

Elliptic term, principal, in longitude, used to 

define the eccentricity, 102, 119, 123, 128,158; 

observed value of the coefficient of, 

127; 

period of, 128 ; 
combination of, with the evection, 

128. 
Ellipticity of the Earth, 260 (see figure of the 

Earth). 

Empirical term, Hansen's, supposed to be due 
to the non-uniform distribution of the Moon's 
mass, 248. 

Energy, integral of, in the problem of p bodies, 
26; 

in the lunar theory, 14, 18, 22 ; 

apparent inconsistency of, 26 ; 
used as a means of verification, 205. 
Epoch of the mean longitude, defined, 120; 
equations for the variation of, 62, 64 j 

not used by Hansen, 162 ; 
constant of, in disturbed motion, 118 

etc. (see constant). 

Epochs of mean longitudes of perigee and 
node, defined, 120, 121; 

constants of, 120 etc. (see constant). 
Equation of the centre, defined, 35 ; 

expansions containing, 36. 
Equation, determinantal, for the motion of the 
perigee, 217 ; 

of the node, 230. 

Equations of condition (see condition). 
Equations of motion, de Ponte"coulamVs, 13 
et seq. ; 

solution of, when the solar action is 

neglected, 41; 
effects produced by the integration 

of, 84 et seq. ; 

preparation of, for the second and 
higher approximations, 93 et seq. ; 
Laplace's, 17 et seq.; 

solution, neglecting the Sun's action, 

42; 

Hansen's, for radius and longitude, 167, 
168, 170 j 

for the plane of the orbit, 190 ; 
in reot. coor. method, 19 et seq. ; 

simplified forms of, for the inter- 
mediary and the elliptic inequali- 
ties, 24, 197,211; 
for mean-period inequalities, 225 ; 
for parallactic inequalities, 227 ; 
for inequalities in latitude, 228 ; 



for Adams' researches, 24 ; 
simplified to obtain a first approximation, 

44; 

referred to polar coordinates, 59 ; 
Hamilton's, 68; 

Jacobi's solution of elliptic, 69 et seq. ; 
Clairaut's, 238 ; 
Euler's, 239, 240 ; 

for the problem of three bodies, 25 et seq. ; 
the ten first integrals of, 26; 
cases when the, are integrable, 28. 
Equations for the variations of the elements, 

59 et seq. (see elements, variation). 
Equations, linear (see linear). 
Equator, ascending node of ecliptic on, origin 

for reckoning longitudes, 261. 
Equinoxes, precession of, used to determine the 

figure of the Earth, 261. 
Equivalence of the two forms of the equations 

for the intermediary (B,), 197. 
Error in Airy's theory, 246. 
Euler, methods of analysis of the, 239, 240 ; 
contributions of, 241 ; 
formulae of, for rotations, 55 ; 

for transformation of coordinates, 75. 
Evection, defined, 128 ; 

order lowered by integration, 101 j 
coefficient and period of, 128 ; 
combination of, with the principal ellip- 
tic term, 129 ; 
effect of, on the motion of the perigee, 

discovered by Clairaut, 239. 
Existence of integrals in the problem of three 

bodies, 27. 
Expansions (see elliptic, disturbing function, 

etc.). 

Expressions for the coordinates, in undisturbed 
motion, 41 ; 

form to be given to, 45; 
effects of small divisors on, 84 et seq, ; 
facts concerning, 86, 87 ;, 
obtained by de Pont^coulant's method, 
110; 

slow convergence of, 113 5 
form of, B., 158; H., 166, 194; 
Euler's, 240, 241 ; 
Laplace's, 243 ; 
(see coordinates). 

Figure of the Earth, disturbing function for, 
261; 

numerical determination of, 261 j 
principal inequalities due to, 263, 
Figure of the Moon, Hansen's empirical term, 
supposed to be due to, 248. 



284 



GENEKAL INDEX. 



Force, used instead of accelerative effect of, 1. 
Forces, relative to the Earth, 3 ; 

on the Moon relative to the Earth and 
on the Sun relative to the centre of 
mass of the Earth and Moon, 5 ; 
disturbing (see disturbing). 
Force-function used by Laplace, 92. 
Force-functions for the lunar and solar motions, 
3; 

second form of, 5; 

corrections to, 6-8. 

Form (see disturbing, expressions, etc.). 
Function, disturbing (see disturbing). 
Functions, Bessel's, defined, 33 ; 

used in elliptic expansions, 34 et seq. 
Fundamenta, full title of the, 36; 

contents of, 161; 

elements used in, 171. 

Gaussian constant of attraction, 1. 

Geodetic measures and pendulum observations 

used to determine the figure of the Earth, 

261. 

Hamilton's dynamical method, 67 et seq. 
Hansen, methods of, for elliptic expansions, 
36; 

theorem of, concerning elliptic expan- 
sions, 36 et seq.. ; 

extension of, of method for the variation 
of arbitrary constants, 76 ; 
theorem concerning, 77; 
method of, for the development of the 

disturbing function, 89 et seq. ; 
two inequalities of, due to Venus, 259. 
Hansen's theory, features of, 160, 164, 166 ; 
history of, 161 ; 
notation for, 161; 
instantaneous ellipse, 162 ; 
auxiliary ellipse, 164 et seq. ; 

relation of instantaneous to, 165 ; 
disturbing function, form of, 177 et seq.; 
first approximation to, 181 ; 
derivatives of, in terms of the forces, 

179 et seq., 182; 

motion in the orbital plane, equations 
for, 167 et seq. ; 
introduction of r, 169 ; 
the function W, 169 ; 

first approximation to, 182 et 

seq.; 
integration of the equations, 185 et 

seq.; 

the arbitrary constants, 171, 187, 
188; 



motion of the orbital plane, definitions 
for, 172 et seq. ; 
equations for, 174 et seq., 190; 
integration of the equations, 191 ; 
the arbitrary constants, 192; 
third and higher approximations, 192 et 

seq. ; 

reduction to true ecliptic, 194. 
Hill, equations used in the researches of, 24, 
197; 

particular solution of, 24 ; 
method of, for the variational inequali- 
ties, 196 et seq. ; 
for the motion of the perigee, 211 

et seq. ; 
for adapting Delaunay's theory to 

small disturbances, 248 et seq. ; 
for separating the terms in the 
planetary disturbing functions, 
253. 

History of the lunar theory since Newton, 237 
et seq. ; 

of Hansen's theory, 161. 

Ideal coordinates, defined, 74; 

general conditions for, 75. 
Inclination, equation for the variation of, 61 ; 
first used by Euler, 239 ; 
when the ecliptic is in motion, 264. 
Inclination, sine of half or tangent of, a 
parameter in elliptic motion, 39 ; 

considered of the first order, 80 ; 
a parameter in the development of 

the disturbing function, 80 ; 
connection between arguments and 

powers of, 82; 
properties of, in the coordinates, 

86; 

(see constant of latitude, latitude). 
Incommensurable, coefficients of the time in 

the arguments assumed, 49, 81, 184. 
Independent variable (see variable). 
Indeterminate coefficients in the second ap- 
proximation (P.), 86, 110; 

method of solution by, first used by 

Euler, 241. 

Index of a coefficient denotes the order, 80. 
Indirect action of a planet, 253 (see planetary). 
Inequalities, division into classes, 95, 198, 
241; 

variational, 96, 125, 198 (see varia- 
tional) ; 

elliptic, 100, 128, 209, 224 (see elliptic); 
mean-period, 103, 129, 225 (see mean 
period) ; 



GENERAL INDEX. 



285 



parallactic, 104, 127, 227 (see paral- 

lactic) ; 
principal, in latitude, 106, 130, 228 (see 

latitude) ; 

of higher orders, P., 109; B., 234; 
special, deduced directly from the dis- 
turbing function, 111; 
long- and short-period, denned, 85; 
method of calculating small, 248 et seq. ; 
planetary, 252 et seq.. ; 

due to Venus, 258, 260; 
to the motion of the ecliptic, 265 ; 
to the variation of the solar eccen- 
tricity, 266, 267; 

due to the figure of the Earth, 263 ; 
principal, obtained by Newton, 237. 
Infinite determinant, to find the motion of the 
perigee, 217; 

properties of, 217 et seq., ; 
to find the nodal motion, 230 ; 
convergency of, 219 ; 
development of, 220 ; 

application to the perigee, 222. 
Instantaneous axis, the radius vector, rate of 

rotation of the orbit about, 60. 
Instantaneous ellipse, defined, 48 ; 

the intermediary when the method of 

the variation of arbitraries is used, 48 ; 

relations between, and the auxiliary 

(H.), 165 ; 

(see variation, Hansen). 
Integrable, case when Hill's equations are, 24 ; 
cases when the equations for the pro- 
blem of three bodies are, 28. 
Integrals, the ten first, in the problem of three 
or p bodies, 26 et seq. ; 

Jacobian (see velocity, Jacobian). 
Integration by continued approximation (see 

continued). 

Integration, small divisors introduced by, 84 
et seq. ; 

effects of, on the orders of coefficients, 

86; 

of the prepared equations (P.), 96 et seq.; 
of canonical equations with one periodic 
term of the disturbing function (D.), 
139 et seq. ; 

in particular cases, 153, 156 ; 
mean anomaly constant in (H.), 169 ; 
of equations, for mean anomaly and 
radius vector (H.), 185 et seq,; 

for motion of orbital plane, 191 ; 
of equations of motion, by Clairaut, 
238; 
method of Laplace, 242 ; 



with a variable solar eccentricity, 

243, 267; 
Hill's general method of, for small dis 

turbances, 248 et seq. 
Intermediary, defined, 45; 

in the various methods, 46 et seq. ; P,, 
52 ; D,, 134 ; H., 165 ; B., 197 ; La- 
place, 53 ; 

modification of, 51, 238, 239. 
Intermediate orbit (see intermediary). 
Interpretation of arbitraries (see arbitrary). 
Invariable plane, defined, 27 ; 

as a fixed plane of reference, 27, 162. 

Jacobi, dynamical method of, 68 ; 
elliptic motion by, 69 ; 
produces a system of canonical con- 
stants, 73. 

Jacobian integral, when the solar eccentricity 
is neglected, 25 ; 

in the problem of three bodies, 26 ; 
(see velocity). 
Jupiter, large inequality in motion of, 10. 

Kepler's laws, approximate representation of 
motions of planets and satellites by, 9 j 

correction to, due to Earth's mass, 42, 90. 

Lagrange, equations for the variation of arbi- 
traries of, 73 ; 

canonical system of, 76. 
Laplace, condition of convergence of, for el- 
liptic series, 43; 

discoveries of, 243 ; 

value of the secular acceleration of, 267. 
Laplace's method, equations of motion for, 17 
et seq. ; 

solution of, when the solar action is 

neglected, 42; 
intermediary for, 53 ; 
development of force-function for, 92 ; 
form of solution compared with P., 131 ; 
definition of eccentricity, 119, 243 ; 
analysis, of, 242. 
Latitude, argument of, defined, 41 ; 

in disturbed motion, P., 52; D., 

159; H.,194; 

constant of, in disturbed motion, P., 108, 
120, 130; D., 158; H., 192, 194; B., 
231,233; Laplace, 243; 
used by de Pontecoulant, 113 ; 
niimerical value of, 130 ; 
of the first order, 80; 
development of the disturbing func- 
tion in powers of, 82 ; 



286 



GENERAL INDEX. 



connections between arguments and 

powers of, 81; 

form of expression for, 121; EL, 194; 
long-period inequalities in, 85 ; 
magnitudes of coefficients in, 131 ; 
perturbations of, Hansen's form for, 194 ; 
principal inequalities in, determination 
of, P., 106 et seq.; B., 228 et seq.; 
due to the figure of the Earth, 263 ; 
to the motion of the ecliptic, 265 ; 
principal term in, used for the determi- 
nation of the constant, 108, 120, 123, 
130, 192, 231j 233, 243; 

coefficients and period of, 130 ; 
tangent of, expression for, in elliptic 
motion, in terms of the time, 41 ; 
of the true longitude, 42 ; 
in disturbed motion (P.), Ill; 
terms not containing m in, 87. 
Latitude -equation, denned (P.), 16; 

effect of integration of, on the orders of 

coefficients, 84 et seq. ; 
not used in the calculations, 94; 
Euler replaces, by two equations, 239. 
Legendre's coefficients, expansions by, of the 

disturbing functions, 79, 256. 
Limitations of the lunar theory, 2; D., 133; 

B., 196. 

Line, fixed in the orbital plane, defined, 60. 
Linear constant (see distance, constant). 
Linear equations to find the motions of the 

perigee and node, 108, 213, 229. 
Linear equations arising in the second approxi- 
mation, 50, 53. 
Linear homogeneous equations, determinant of 

an infinite number of, 210, 216, 230. 
Longitude, derivatives of the disturbing func- 
tion with respect to, 14 et seq,, 58, 83, 179. 
Longitude of epoch, perigee, node (see epoch, 

perigee, node). 

Longitude, mean, in elliptic motion, 40; D. 
137; 

in disturbed motion, P., 97, 118; D., 

158; H., 194; (see mean motion). 
Longitude, true, expression for, in elliptic 
motion, 41; D,, 138; 

in disturbed motion, 110; 
form of expression for, P., 121; D., 157; 

H., 194; Euler, 240; 
independent variable the, theories using, 
12, 242, 244; 

remarks on, 247; 
equations with, 17 et seq, ; 
elliptic motion with, 42 ; 
intermediary, 53 ; 



motions of perigee and node with, 

131; 

magnitudes of coefficients in, 131 ; 
terms in, long- and short-period, 85 ; 
not containing m as factor, 87; 
due to figure of the Earth, 263; 
to motion of ecliptic, 265; 
to secular acceleration, 267; 
transformation to find (B.), 206. 
Longitudes, origin for reckoning, 261; 

reckoned from a departure point, pro- 
perty of, 60; 

introduce pseudo-elements, 75 ; 
used by Hansen, 160. 
Longitude-equation, defined, 16; 

effect of integration of, on the orders of 

coefficients, 84 et seq. ; 
prepared form of (P.), 94. 
Long-period inequalities, defined, 85 ; 

found best by the method of the varia- 
tion of arbitrages, 66, 245 ; 
due to planetary action, 257. 
Lubbock, method of, 245. 
Lunar theory, a particular case of the problem 
of three bodies, 2 ; 

limitations initially assigned, 2; 
distinction between the, and the planet- 
ary, 8-10, 66; 
variables used in the various methods, 

12; 

analysis of the methods given by 
Airy, 245; 
Clairaut, 238; 
Damoiseau, 244; 
d'Alembert, 239; 
de Ponte"coulant, 112 et seq.; 
Delaunay, 156; 
Euler, 239, 240; 
Hansen, 166; 
Laplace, 242; 
Lubbock, 245; 
Newton, 237 ; 
Plana, 244; 
Poisson, 245; 
Bectangular coordinates, 195 et seq.; 

another method, 234; 
comparison of the methods, 246; 
tables deduced from (see tables). 

Magnitudes of the coefficients, 131. 

Major axis, equation for the variation of, 61 ; 

derivative of the disturbing function 
with respect to, 66; 

relation of, to Delaunay 's elements, 138 ; 



GENERAL INDEX. 



287 



Mass, astronomical unit of, defined, 1 ; 
unit of, used (P.), 82; 
of the Moon, of the Earth, correction 
necessary to include the (see correc- 
tion); 

methods for determination of, 127. 
Mean anomaly (see anomaly). 
Mean argument of latitute, defined, 41 (see 

latitude). 

Mean distance (see distance). 
Mean motion, in an ellipse, 40 ; 

in Delaunay's notation, 138; 
in the disturbed orbit, defined, 63 ; 
in disturbed motion, P., 97, 118; D., 
158; H., 188; E., 199; 
obtained from observation, 123 ; 
numerical value of the, 124; 

of the solar, 123 ; 
term in the disturbing function having 

a period equal to that of the, 85, 86; 
secular acceleration of, 243, 265 et seq. 
Mean motions, ratio of the, assumed incom- 
mensurable, 49; 

considered of the first order, 80 ; 
square of, a factor of the disturbing 

function, 82; 

a factor of the terms in the coordi- 
nates due to the Sun, 86 ; 
cases of exception, 87 ; 
numerical value of, 124 ; 
inequalities dependent only on (see 

variational) ; 
of the perigee, node, mean anomaly (see 

perigee, node, anomaly); 
of two planets, nearly commensurable, 9. 
Mean period, obtained by observation, 123 ; 
numerical value of the, 124 ; 

of the solar, 123. 
Mean period inequalities, defined, 130; 

determination of, P., 103; K., 225; 
terms in the disturbing function for, 
112. 
Modification of intermediary, 51 et seq. (see 

intermediary). 

Motion, oscillation about a steady, 47, 52, 211 ; 
of the Moon, effect of, on the motion of 
the centre of mass of the Earth and 
Moon, 6 ; 

of the Sun, referred to the centre of 
mass of the Earth and Moon, as- 
sumed to be known, 4, 6; 
Kepler's laws an approximate repre- 
sentation of, 9 ; 

(see ecliptic, elliptic, equations, perigee, 
node, etc.). 



Newton, law of attraction of, 1 ; 

sufficiency of, to account for the 

motion of the perigee, 239 ; 
tested by Euler, 240 ; 
results and discoveries of, 127, 237. 
Node, ascending, defined, 41 ; 

of the ecliptic on the equator, the 
origin for reckoning longitudes, 
261; 
generally assumed to be in motion, in 

the intermediary, 46 ; 
made to revolve by the Sun's action, 53 ; 
period of revolution of, 130 ; 
distance of, from the perigee (H.), 177. 
Node, longitude of the, in elliptic motion, 41 ; 
equation for the variation of, 61, 64 ; 
used by Euler, 239 ; 
due to the motion of the ecliptic, 

264; 

notation for (D.), 137 ; 
mean, on the ecliptic (H.), 194 ; 

epoch of the, defined, 121. 
Node, mean motion of, determination of, P., 
109; D., 158; H., 192; E., 230, 233; by 
Newton, 237; 

notation for (H.), 173; 
numerical value of, 130 ; 

Adams', of the principal part of, 

230; 

a test of the theory, 123 ; 
higher parts of, equations for, 233, 234 ; 
in Euler's method, 241 ; 
connections of, with the constant 

part of the parallax, 235 ; 
in Laplace's method, 131; 
secular acceleration of, 243, 268. 
Notation, in Delaunay's theory, 136, 248 j 
in the operations, 152 ; 
for the arguments, in the final 

results, 159; 

in Hansen's theory, 161, 172 ; 
in the Fundamenta, 171 ; 
tables of, 270-273. 
Numerical orders, defined, 80. 
Numerical theories, Clairaut's, 238 ; 
Damoiseau's, 244 ; 
Hansen's, 171; 
Airy 3 s,245; 

contrasted with other methods, 246. 
Numerical values, of the lunar constants, 124 
et seq. ; 

difficulties in the determination of, 

115 et seq, ; 

determination of, by observation, 
121 et seq. ; 



288 



GENERAL INDEX. 



references to memoirs with, 131 ; 
of Hansen, 188 ; 
of Delaunay, unreduced, 251 ; 
of the principal coefficients and periods, 
124 et seq.; 
magnitudes of, 131 ; 
rapid approximation to (B.), 204 ; 
of the motion of the perigee, 128, 223 ; 

of the node, 130, 230 ; 
of the coefficients in Eadau's equations, 

for small disturbances, 251 ; 
of inequalities due, to Venus, 258-260 ; 
to the figure of the Earth, 262, 263 ; 
to the motion of the ecliptic, 265 ; 
of the secular acceleration, 267, 268 ; 
of the solar constants, 123. 
Nutation of the Earth's axis, used to determine 
the ratio of the masses of the Earth and 
Moon, 127. 

Observation, determination of the constants 
by, 121 et seq.; 

references to memoirs contain- 
ing, 131; 

method of Euler for, 240, 241 ; 
theoretical motion of the perigee recon- 
ciled with, 239 ; 
tables deduced from, 246 ; 
coefficients of Hansen' s inequalities as 

obtained from, 259 ; 
determination of the ellipticity constant 

from, 261; 
the secular change in parallax insensible 

to, 268. 
Observed mean motion (see mean motion, 



Operation (D.), method for the calculation of 
an, 150 et seq. ; 

particular cases of, 153 et seq. ; 
the final, 156 ; 

effect of an, on the variables, 157. 
Orders, of parameters, defined, 80 ; 

of coefficients in the disturbing function, 
connection between arguments and, 
82; 

for planetary action, 255 et seq. ; 
eff ect on, produced by the integra- 
tions, 84 et seq. ; 
highest, given by Delaunay, 89 ; 
least, of the solar terms in the coordi- 
nates, 86 ; 

of certain terms in the higher approxi- 
mations, 86 ; 

of the coefficients in the successive 
approximations (P.), 95 et seq.; 



to which the theories are carried, P., 113 ; 
D., 133; B., 204, 211, 223, 230 ; 

other methods, 237 et seq. 
Origin, of coordinates considered to be the 
Earth, 8 ; 

for reckoning longitudes, 261. 
Oscillations about steady motion, examples of, 
47, 52, 211. 

Parallactic inequalities, defined, 127; 

determination of, P., 104; B., 227 ; 
terms for, deduced directly from the 

disturbing function, 111 ; 
effect of, on the variational curve, 127 ; 
correction to, for the Moon's mass, 126, 

159. 
Parallactic inequality, denned, 125 ; 

order of, lowered by integration, 104 ; 

period and coefficient of, 126 ; 

used to determine the Sun's parallax, 

127; 

notation for argument of (D.), 159. 
Parallax of the Moon, determination of, from 
the inverse of the radius vector, 121 ; 
mean value of the, 124 ; 
magnitudes of coefficients in, 131 ; 
secular inequality in, 266 ; 

effect on the mean, 268 ; 
(see radius vector). 

Parallax of the Sun, determined by the paral- 
lactic inequality, 127 ; 

mean value of, 123. 
Parameter, change of, to improve convergence, 

114, 204. 

Parameters, orders of, used, 80 (see orders). 
Particles, Earth, Moon and Sun considered 

as, 2. 
Particular integrals, forms of, in the second 

approximation, 50, 227. 
Pendulum observations used to determine the 

figure of the Earth, 261. 
Perigee, generally assumed to be in motion in 
the intermediary, 46 ; 

made to revolve by the Sun's action, 58 ; 
period of revolution of, 128 ; 
distance of, from the node (H.), 177. 
Perigee, longitude of, in elliptic motion, 41 ; 
equations for the variation of, 62, 64 ; 
notation for (D.), 137 ; 
epoch of the mean, defined, 120. 
Perigee, mean motion of the, determined, P., 
101, 103; D., 158; H., 185, 192; B., 218, 
223, 234 ; 

by Laplace's method, 131 ; 
by Newton, 237 ; 



GENERAL INDEX. 



289 



to the second order, numerically by 
Clairaut, 238 ; 

incident connected with, 239 ; 
algebraically by d'Alembert, 

239; 

by Euler, 240 ; 
of the auxiliary ellipse (H.), 165 ; 

determined, 185 ; 

in rect. coor. method, determinantal 
equation for principal part of, 217; 
simple equation for, 218 ; 
the higher parts of, 225, 234 ; 

connections of, with the con- 
stant part of the parallax, 
235 ; 

higher parts of, in Euler's method, 241 ; 
numerical value of, 128 ; 

Hill's, for the principal part, 223 ; 
observed, used by Hansen, 188 ; 

by Euler, 240; 

a test of the theory, 123 ; 

secular acceleration of, 243, 268. 

Period, the mean, obtained from observation, 

123; 

numerical value of, 124 ; 

of the Sun, 123 ; 

of the mean motion of the perigee, 128; 
of the node, 130 ; 
of the variational curve, 125 ; 
mean-, inequalities (see mean period). 
Periods, of oscillations about a steady motion, 
47, 52, 211 ; 

case of an infinite number of, 217 ; 
of the principal inequalities, 124 et seq. ; 
due to planetary action, 257, 258 ; 
to Venus, 268, 260; 
to the figure of the Earth, 263 ; 
to the motion of the ecliptic, 265. 
Periodic functions, defined, 45 ; 

used to represent the coordinates, 45 ; 
time as a factor of, to be avoided, 45 

(see time). 
Periodic solution, defined, 46 ; 

used as an intermediary, in general, 46 ; 
Hill's (H.), 198 et seq. 
Periodicity, of an angular coordinate, defined, 

of the variation of the solar eccentricity, 

268. 

Perturbations, of the solar orbit included (H.), 
172 j 

in the disturbing function, 177 ; 
neglected, in the first approximation, 

181; 
of the ecliptic (H.), 191 ; 

B. L. T, 



Adams' method for, 264 ; 
of latitude, Hansen's method of ex- 
pressing, 194. 

Plana and Carlini, theory of, 244. 
Plane of orbit, line fixed in, defined, 60 ; 

properties of, 60 ; 
rate of rotation of, due to the disturbing 

forces, 60 ; 

quantities denning (H.), 172 ; 
equations of motion for, in terms of 'the 
disturbing forces (H.), 176 ; 
in terms of the derivatives of the 
disturbing function, 190. 
Plane of reference, the plane of the Sun's orbit 
supposed fixed, 13 ; 

the invariable plane as a, 27 ; 
Hansen's, 162. 

Planetary action, effect of, on the apparent 
solar motion, 6 ; 

direct and indirect, disturbing functions 
for, 252 ; 

separation of the terms in, 253 ; 
expressions by polar coordinates, 

254; 
developments in terms of the elliptic 

elements, 255, 256 ; 
nature of the terms in, 257 ; 
direct, inequality due to Venus, 258 ; 
indirect, methods of including, 253, 257 ; 
ease of, 259 ; 

inequality due to Venus, 260 ; 
motion of the ecliptic, 263 et seq.; 
secular acceleration, 265 et seq. 
Planetary theory, distinction between the lunar 
and the, 8-10; 

variation of arbitrary constants used in 

the, 66, 245; 
Hansen's, 161 ; 

theorem at the basis of, 77. 
Poisson, method proposed by, 245. 
Polar coordinates, transformation from rect- 
angular to (B.), 206 (see coordinates). 
Potential due to the figure of the Earth, 260 ; 

(see force-function). 
Precession of the Equinoxes, used to determine 

the figure of the Earth, 261. 
Primary, defined, 9 ; 

mass of, compared to that of the dis- 
turbed body, 9. 

Principal elliptic inequality (see elliptic). 
Principal function, defined, 68; ^ 

satisfies apartialdifferential equation, 69 ; 
used for equations of elliptic motion, 

69 et seq.; 
and in the variation of arbitranes, 71. 

19 



290 



GENERAL INDEX. 



Principal inequality in latitude (see latitude). 
Problem of $ bodies, denned, 2 ; 

force-function for, 11. 
Problem of three bodies, defined, 2 ; 
lunar theory, a case of, 2 ; 

limitations of, 2 ; 
equations of motion for, 25 ; 
the ten first integrals, 26; 

the number of variables reduced 

by, 28. 
Pseudo-element, defined, 74 ; 

derivative with respect to a, 74; 

occurs when longitudes are reckoned 

from a departure point, 75; 
introduces another arbitrary, 75 ; 
used by Hansen, 163. 

Eadau, numerical equations of, for small dis- 
turbances, 251; 

application to various inequalities, 

258, 259, 262. 
Radius-equation, defined, 16; 

effect produced on the orders of coeffi- 
cients by the integration of, 84 et seq. ; 
prepared form of (P.), 93 ; 
constant parts of, omitted, 100. 
Radius vector, elliptic expression for, 41 ; 

as an instantaneous axis, rate of , .rota- 
tion of the orbit about, 60 ; 
derivative of disturbing function with 

respect to the, 14 et seq., 83, 179; 
terms not containing m in, 87 ; 
constant part of, 120 (see distance) ; 
Hansen's method of computing, 160 ; 
relation of, to the, of the auxiliary 
ellipse (BL), 165 ; 
equation for, 168; 
solution, 189. 

Radius vector, inverse of the elliptic value of, 
35; D., 138; 

in disturbed motion (P.), 110 ; 
form of expression for (D.), 158; 
constant part of, 120 ; 

connections of, with the motions of 

perigee and node, 234 et seq. ; 
general theorem concerning, 235 ; 
determination of the parallax from the, 

121 ; 

transformation to find the (R.), 206; 
projection of the, theories using as a 
dependent variable, 17, 238 et seq.. ; 
(see parallax). 
Eadius vector of the Sun, perturbations of, H., 

177, 259. 
Eatio (see mean motions, distances, mass). 



Kectangular axes, moving with the mean solar 
angular velocity, 19 ; 

moving with the mean lunar angular 

velocity, used by Buler, 240. 
Bectangular coordinates, method with, equa- 
tions of motion, 19 et seq.; 

particular cases of, 24 ; 
development of the disturbing function, 
20, 92; 

elliptic series not used in, 92 ; 
origin of, and limitations imposed on, 

196; 
Intermediate orbit, 46, 197 ; 

determination of, 198 et seq.. ; 
transformation to polars, 206 ; 
elliptic inequalities, 206 et seq.; 
of the first order, 209 ; 
of higher orders, 223 ; 
motion of the perigee, method for, 211 
et seq. ; 

determinantal equation for, 217 ; 
simple equation for, 218 ; 

value obtained from, 223; 
parts of, of higher orders, 225, 234 ; 
mean period inequalities, 225; 
parallaetic inequalities, 227; 
inequalities in latitude, 228 et seq. ; 
of the first order, 229 ; 
of higher orders, 231 ; 
motion of the node, 230 ; 

equation for part of, of the second 

order, 233; 
of higher orders, 235 ; 
another mode of development, 234 ; 
theorems in connection with, 235, 236 ; 
compared with other methods, 247. 
Reduction, the, defined, 39 ; 

terms constituting the, in elliptic mo- 
tion, 39. 

Reference, plane of (see plane). 
Relations between the, developments of the 
disturbing functions, 89, 92 ; 

constants, in the various methods, 116 ; 

in R. and P., 205,211, 231; 
solutions, in R. and P., 199, 207, 230; 
elliptic elements and Delaunay's vari- 
ables, 138; 

old and new variables, after any opera- 
tion (D.), 149; 
disturbing functions, after any operation 

(D.), 152; 
motions of perigee and node and the 

constant part of the parallax, 235 ; 
coefficients of a solar and the resulting 
lunar inequality, 259. 



GENERAL INDEX. 



291 



Relative forces (see force, force-function). 
Results, summary of (P.), 110; 

Delaunay's, form of, after the operations, 
157; 
correction to, to account for the 

Moon's mass, 159; 
comparison of, with Hansen's, 159 ; 
Hansen's, form of, 188 ; 

reduction of, to the ecliptic, 194. 
Reversion of series, necessary in Laplace's 
method, 243; 

theories requiring, useless when great 

accuracy is needed, 247. 
Boots of an infinite determinantal equation, 

properties of, 217, 218. 
Kotation of orbits due to the disturbing forces, 
rate of, 60. 

Saturn, large inequality in the motion of, 10. 
Seconds of arc, reduction of coefficients to, 121. 
Secular acceleration, denned, 267 ; 
cause of, 265 ; 
first approximation to, determined, 266, 

267; 

controversy concerning, 243 ; 
of the perigee, of the node, 243, 268. 
Secular terms, avoided by a modification of the 
intermediary, 58 ; 

treated by the variation of arbitrages, 

66, 245. 

Semi-algebraical theories, rect. coor., 195; 
of Buler, Laplace, 240-242; 
value of, 246. 

Separation of terms in the planetary disturb- 
ing functions, 258. 

Scries (see coordinates, convergence). 
Short-period terms due to planetary action, 257. 
Signification of the constants, elements (see 

constants, elements). 
Small divisors introduced by integration, 85 et 

seq. 

Solution, when the solar action is neglected, 
of do Polite" coulant's equations, 41 ; 
of Laplace's equations, 42 ; 
form to be given to the general, 45 ; 
a periodic, defined, 46 ; 

used as an intermediary, 46 ; 
by continued approximation (see con- 
tinued approximation) ; 
of a certain linear equation, 108 ; 
nature of, of canonical equations (D.), 

144; 

methods of deducing the (B.), 200; 
deduction of the (B.), from de Ponte"- 
coulant's results, 199, 207, 230; 



of differential equations by indetermi- 
nate coefficients, first used by Euler, 
241. 

Summary of results (P.), 110. 
Symbolic formula for the true anomaly in 
terms of the mean, 33. 

Tables, of the Moon's motion, references to, 
123, 161, 192, 238 et seq., 246; 

of the notation used, 270-273. 
Tangential transformation from one canonical 

system to another, 66. 

Theorem, of Hansen, relative to elliptic ex- 
pansions, 36 ; 

application, 184, 185 ; 
relative to any function of the 

elements and the time, 77; 
concerning the Moon's parallax, 235 ; 
of Adams, 236. 

Theory, lunar, planetary (see lunar, planetary). 
Three bodies, problem of (see problem). 
Tides, used to determine the ratio of the 
masses of the Earth and Moon, 127; 
effect of, on the lunar motion, 248. 
Time, expression for, in terms of the true 
longitude, in undisturbed motion, 42; 

not present explicitly, in the equations 
for the variable elements, 73 ; 

or in Delaunay's formulae, 138 ; 
coefficient of, in the true longitude, 97; 
coefficients of, in the arguments, assumed 
incommensurable and will not vanish 
unless the arguments vanish, 49, 81, 
184; 

introduction of a constant (H.), 169; 
terras increasing with, produced by the 

secular acceleration, 266. 
Time as a factor of periodic terms, to be 
avoided if possible, 10, 45; 

how it may occur, 50; 

* modification of intermediary to avoid, 52 - r 

explanation of apparent occurrence of, in 

the second approximation, 85 et seq. ; 

removed from the equation for the 

epoch, 63; 
in Delaunay's method, 134; 

transformation to avoid, 135 ; 
appears again, 145; 
transformation to avoid, 147 ; 
in Hansen's method, how avoided, 166, 

173; 
neglected in the secular acceleration,. 

266. 

Transcendental uniform integrals, in the prob- 
lem of three bodies, limited number of, 27. 



292 



GENEEAL INDEX. 



Transformation, tangential, denned, 66 ; 

method of, used in Delaunay's theory, 
134; 

conditions of the possibility of, 135; 
applications of, 136, 142, 147, 149; 
to new constants (D.), 158; 
from rectangular to polar coordinates 

(B.), 206. 
Triangle, variations of the sides and angles of 

a spherical, .174. 
True anomaly (see anomaly). 
True longitude (see longitude). 

Undisturbed elliptic motion, 40 et seq. 
Unit of mass, astronomical, denned, 1 ; 

used in de Ponte*coulant's method, 82. 

Variables, used in the various methods, 12, 
238 et seq.; 

number of, in the problem of three 

bodies, 28; 
change of, in Delaunay's method, 135, 

136, H7, 148; 

used in Hansen's method, 166, 172, 179. 
Variation of arbitrary constants, method of, 
47; 

coordinates and velocities have the same 
form, for disturbed and undisturbed 
motion in, 48, 58, 72; 
application of, elementary, 54 et seq.. ; 
Jacobi's method, 71 ; 
Lagrange's method, 73; 
given by Euler, 241 ; 
suggested by Poisson, 245 ; 
remarks on use of, in the lunar and 

planetary theories, 66, 245; 
(see variations). 
Variation of the solar eccentricity, the cause of 

the secular acceleration, 243, 265. 
Variation, the, defined, 124; 

period and coefficient of, 124, 125 ; 

Newton's value, 127 ; 
notation for the argument of (D.), 159. 
Variations of the elements, change of position 
produced by, 56; 

zero in the motion, 59 ; 

equations for, found, 59 et seq. ; 

. in terms of the forces, 63 ; 

in terms of the derivatives of the 
1 disturbing function, 64; 

Delaunay's canonical equations for, 64; 



in the form used, 136 ; 
remarks on, 66; 

deduced by Jacobi's method, 72 ; 
use of pseudo-elements in, 74 ; 
Lagrange's canonical system, 76 ; 

Hansen's extension, 76 ; 
equations for, used by Hansen, 162; 
for small disturbances, Badau's numerical 

equations, 251; 

due to the motion of the ecliptic, 264. 
Variations of the sides and angles of a spherical 

triangle, 174. 
Variational curve, defined, 125; 

for different values of w, 125 ; 

effect of the parallactic inequalities on, 

127; 

Newton's ratio of the axes of, 127 ; 
used as an intermediary (B.), 198; 

a periodic solution of Hill's equa- 
tions, 198. 

Variational inequalities, defined, 125; 
determination of (P.), 96 et seq. ; 

terms for, deduced directly from the 

disturbing function, 111 ; 
in rect. coor. method, equations for, 
196; 

form of solution, 199 ; 
determination of coefficients, 200 et 

seq. ; 

rapidity of the approximations, 204 ; 
transformation to polar coordinates, 

205; 

parts of, of higher orders, 223, 231. 
Velocity, expression for the square of, with 
de Ponte"coulant's equations, 14 ; 
with Laplace's equations, 18 ; 
with rect. coor, method, 22, 25 ; 
in elliptic motion, 40. 

Velocities and coordinates, having the same 
form in disturbed and undisturbed motion, 
48, 58, 72; 

initial, a canonical system, 76. 
Venus, ratio of distance of, from the Sun to 
that of the Earth, 9; 

inequality due to the direct action of, 
258; 

to the indirect action, 260 ; 
Hansen's two inequalities due to, 259. 
Verification, equations for (B.), 205, 236; 

of Delaunay's results, Airy's method for, 
245. 



CAMBRIDGE I PBINTED BY J. AND C. P. CLAY, AT THE UNIVEBSITY PKESS.