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Full text of "An introduction to celestial mechanics"

R. TRACY CRAWFORD 




Astronomy Library 



AN INTRODUCTION TO 
CELESTIAL MECHANICS 



THE MACMILLAN COMPANY 

NEW YORK BOSTON CHICAGO 
DALLAS SAN FRANCISCO 

MACMILLAN & CO., LIMITED 

LONDON BOMBAY CALCUTTA 
MELBOURNE 

THE MACMILLAN CO. OF CANADA, LTD. 

TORONTO 



AN INTRODUCTION 

TO 

CELESTIAL MECHANICS 



BY 



FOREST RAY MOULTON, PH.D. 

PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF CHICAGO 
RESEARCH ASSOCIATE OF THE CARNEGIE INSTITUTION OF WASHINGTON 



SECOND REVISED EDITION 



Neto 0rfc 
THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., LTD. 
1914 

All rights reserved 



COPYRIGHT, 1914 
BY THE MACMILLAN COMPANY 

Set up and electrotyped. Published April, 1914 



ASTRONOMY DEFT 



PREFACE TO FIRST EDITION. 

AN attempt has been made in this volume to give a somewhat 
satisfactory account of many parts of Celestial Mechanics 
rather than an exhaustive treatment of any special part. The aim 
has been to present the work so as to attain logical sequence, to 
make it progressively more difficult, and to give the various subjects 
the relative prominence which their scientific and educational 
importance deserves. In short, the aim has been to prepare such 
a book that one who has had the necessary mathematical training 
may obtain from it in a relatively short time and by the easiest 
steps a sufficiently broad and just view of the whole subject to 
enable him to stop with much of real value in his possession, or to 
pursue to the best advantage any particular portion he may choose. 

In carrying out the plan of this work it has been necessary to give 
an introduction to the Problem of Three Bodies. This is not only 
one of the justly celebrated problems of Celestial Mechanics, but it 
has become of special interest in recent times through the researches 
of Hill, Poincare, and Darwin. The theory of absolute pertur- 
bations is the central subject in mathematical Astronomy, and 
such a work as this would be inexcusably deficient if it did not 
give this theory a prominent place. A chapter has been devoted 
to geometrical considerations on perturbations. Although these 
methods are of almost no use in computing, yet they furnish in a 
simple manner a clear insight into the nature of the problem, and 
are of the highest value to beginners. The fundamental principles 
of the analytical methods have been given with considerable 
completeness, but many of the details in developing the formulas 
have been omitted in order that the size of the book might not 
defeat the object for which it has been prepared. The theory of 
orbits has not been given the unduly prominent position which it 
has occupied in this country, doubtless due to the influence of 
Watson's excellent treatise on this subject. 

The method of treatment has been to state all problems in 
advance and, where the transformations are long, to give an 
outline of the steps which are to be made. The expression "order 
of small quantities" has not been used except when applied to 
power series in explicit parameters, thus giving the work all the 

M572515 



VI PREFACE TO FIRST EDITION. 

definiteness and simplicity which are characteristic of operations 
with power series. This is exemplified particularly in the chapter 
on perturbations. Care has been taken to make note at all 
places where assumptions have been 'introduced or unjustified 
methods employed, for it is only by seeing where the points of 
possible weakness are that improvements can be made. The 
frequent references throughout the text and the bibliographies at 
the ends of the chapters, though by no means exhaustive, are 
sufficient to direct one in further reading to important sources 
of information. 

This volume is the outgrowth of a course of lectures given 
annually by the author at the University of Chicago during the 
last six years. These lectures have been open to senior college 
students and to graduate students who have not had the equivalent 
of this work. They have been taken by students of Astronomy, 
by many making Mathematics their major work, and by some who, 
though specializing in quite distinct lines, have desired to get an 
idea of the processes by means of which astronomers interpret 
and predict celestial phenomena. Thus they have served to 
give many an idea of the methods of investigation and the results 
attained in Celestial Mechanics, and have prepared some for a 
detailed study extending into the various branches of modern 
investigations. The object of the work, the subjects covered, and 
the methods of treatment seem to have been amply justified by 
this experience. 

Mr. A. C. Lunn, M.A., has read the entire manuscript with great 
care and a thorough insight into the subjects treated. His nu- 
merous corrections and suggestions have added greatly to the 
accuracy and the method of treatment in many places. Professor 
Ormond Stone has read the proofs of the first four chapters and 
the sixth. His experience as an investigator and as a teacher has 
made his criticisms and suggestions invaluable. Mr. W. O. Beal, 
M.A., has read the proofs of the whole book with great attention 
and he is responsible for many improvements. The author desires 
to express his sincerest thanks to all these gentlemen for the 
willingness and the effectiveness with which they have devoted so 
much of their time to this work. 

F. R. MOULTON. 
CHICAGO, July, 1902. 



PREFACE TO SECOND EDITION. 

THE necessity for a new edition of this work has given the 
opportunity of thoroughly revising it. The general plan 
which has been followed is the same as that of the first edition, 
because it was found that it satisfies a real need not only in this 
country, for whose students it was primarily written, but also in 
Europe. In spite of all temptations its elementary character has 
been preserved, and it has not been greatly enlarged. Very 
many improvements have been made, partly on the suggestion of 
numerous astronomers and mathematicians, and it is hoped that 
it will be found more worthy of the favor with which it has so far 
been received. 

The most important single change is in the discussion of the 
methods of determining orbits. This subject logically follows the 
Problem of Two Bodies, and it is much more elementary in char- 
acter than the Problem of Three Bodies and the Theory of Per- 
turbations. For these reasons it was placed in chapter VI. The 
subject matter has also been very much changed. The methods 
of Laplace and Gauss, on which all other methods of general applic- 
ability are more or less directly based, are both given. The 
standard modes of presentation have not been followed because, 
however well they may be adapted to practice, they are not noted 
for mathematical clarity. Besides, there is no lack of excellent 
works giving details in the original forms and models of com- 
putation. The other changes and additions of importance are 
in the chapters on the Problem of Two Bodies, the Problem of 
Three Bodies, and in that on Geometrical Consideration of Per- 
turbations. 

It is a pleasure to make special acknowledgment of assistance 
to my colleague Professor W. D. MacMillan and to Mr. L. A. 
Hopkins who have read the entire proofs not only once but several 
times, and who have made important suggestions and have pointed 
out many defects that otherwise would have escaped notice. 
They are largely responsible for whatever excellence of form the 
book may possess. 

F. R. MOULTON. 

CHICAGO, January, 1914. 



vn 



TABLE OF CONTENTS. 

CHAPTER I. 
FUNDAMENTAL PRINCIPLES AND DEFINITIONS. 

ART. PAGE 

1 . Elements and laws 

2. Problems treated 

3. Enumeration of the principal elements ...... 

4. Enumeration of principles and laws 3 

5. Nature of the laws of motion 

6. Remarks on the first law of motion . 4 

7. Remarks on the second law of motion 4 

8. Remarks on the third law of motion . . . . . . . 6 

DEFINITIONS AND GENERAL EQUATIONS 8 

9. Rectilinear motion, speed, velocity ....... 8 

10. Acceleration in rectilinear motion 9 

11. Speed and velocities in curvilinear motion . . ... . 10 

12. Acceleration in curvilinear motion 11 

13. Velocity along and perpendicular to the radius vector . . . 12 

14. The components of acceleration 13 

15. Application to a particle moving in a circle 14 

16. The areal velocity . 15 

17. Application to motion in an ellipse 16 

Problems on velocity and acceleration 17 

18. Center of mass of n equal particles 19 

19. Center of mass of unequal particles 20 

20. The center of gravity 22 

21. Center of mass of a continuous body 24 

22. Planes and axes of symmetry 26 

23. Application to a non-homogeneous cube 26 

24. Application to the octant of a sphere 27 

Problems on center of mass 28 

HISTORICAL SKETCH FROM ANCIENT TIMES TO NEWTON. 

25. The two divisions of the history 29 

26. Formal astronomy 30 

27. Dynamical astronomy 33 

Bibliography 35 

CHAPTER II. 

RECTILINEAR MOTION. 

THE MOTION OF FALLING PARTICLES. 

29. The differential equations of motion 36 

30. Case of constant force 37 

ix 



X TABLE OF CONTENTS. 

ART. PAGE 

31. Attractive force varying directly as the distance .... 38 
Problems on rectilinear motion . . 40 

32. Solution of linear equations by exponentials . . . . 41 

33. Attractive force varying inversely as the square of the distance . 43 

34. The height of projection 45 

35. The velocity from infinity 45 

36. Application to the escape of atmospheres 46 

37. The force proportional to the velocity 49 

38. The force proportional to the square of the velocity ... 53 
Problems on linear differential equations 55 

39. Parabolic motion .56 

Problems on parabolic motion 58 

THE HEAT OF THE SUN. 

40. Work and energy / 59 

41. Computation of work 59 

42. The temperature of meteors 61 

43. The meteoric theory of the sun's heat 62 

44. Helmholtz's contraction theory 63 

Problems on heat of sun 66 

Historical sketch and bibliography 67 

CHAPTER III. 

CENTRAL FORCES. 

45. Central force 69 

46. The law of areas 69 

47. Analytical demonstration of the law of areas 71 

48. Converse of the theorem of areas 73 

49. The laws of angular and linear velocity 73 

SIMULTANEOUS DIFFERENTIAL EQUATIONS. 

50. The order of a system of simultaneous differential equations . 74 

51. Reduction of order 77 

Problems on differential equations 78 

52. The vis viva integral 78 

EXAMPLES WHERE / is A FUNCTION OF THE COORDINATES ALONE. 

53. Force varying directly as the distance 79 

54. Differential equation of the orbit 80 

55. Newton's law of gravitation 82 

56. Examples of finding the law of force 84 

THE UNIVERSALITY OF NEWTON'S LAW. 

57. Double star orbits 85 

58. Law of force in binary stars 86 

59. Geometrical interpretation of the second law 88 

60. Examples of conic section motion 89 

Problems of finding law of force 89 

DETERMINATION OF THE ORBIT FROM THE LAW OF FORCE. 

61. Force varying as the distance '.90 

62. Force varying inversely as the square of the distance ... 92 



TABLE OF CONTENTS. XI 

ART. PAGE 

63. Force varying inversely as the fifth power of the distance . . 93 

Problems on determining orbits from law of force .... 95 

Historical sketch and bibliography 97 

CHAPTER IV. 

THE POTENTIAL AND ATTRACTIONS OF BODIES. 

65. Solid angles 98 

66. The attraction of a thin homogeneous spherical shell upon a 

particle in its interior 99 

67. The attraction of a thin homogeneous ellipsoidal shell upon a 

particle in its interior 100 

68. The attraction of a thin homogeneous spherical shell upon an 

exterior particle. Newton's method 101 

69. Comments upon Newton's method 103 

70. The attraction of a thin homogeneous spherical shell upon an 

exterior particle. Thomson and Tait's method .... 104 

71. Attraction upon a particle in a homogeneous spherical shell . 106 
Problems on attractions of simple solids 107 

72. The general equations for the components of attraction and 

for the potential when the attracted particle is not a part 

of the attracting mass 108 

73. Case where the attracted particle is a part of the attracting mass 110 

74. Level surfaces 113 

75. The potential and attraction of a thin homogeneous circular 

disc upon a particle in its axis . 113 

76. The potential and attraction of a thin homogeneous spherical 

shell upon an interior or an exterior particle 114 

77. Second method of computing the attraction of a homogeneous 

sphere 115 

Problems on the potential and attractions of simple bodies . .118 

78. The potential and attraction of a solid homogeneous oblate 

spheroid upon a distant particle 119 

79. The potential and attraction of a solid homogeneous ellipsoid 

upon a unit particle in its interior 122 

Problems on the potential and attractions of ellipsoids . . .126 

80. The attraction of a solid homogeneous ellipsoid upon an exterior 

particle. Ivory's method 127 

81. The attraction of spheroids 132 

82. The attraction at the surfaces of spheroids 133 

Problems on Ivory's method and level surfaces .... 137 

Historical sketch and bibliography 138 

CHAPTER V. 
THE PROBLEM OF TWO BODIES. 

83. Equations of motion 140 

84. The motion of the center of mass 141 



Xll TABLE OF CONTENTS. 

ART. PAGE 

85. The equations for relative motion 142 

86. The integrals of areas 144 

87. Problem in the plane 146 

88. The elements in terms of the constants of integration . . .148 

89. Properties of the motion 149 

90. Selection of units and determination of the constant k . . .153 
Problems on elements of orbits 154 

91. Position in parabolic orbits 155 

92. Equation involving two radii and their chord. Euler's equation 157 

93. Position in elliptic orbits 158 

94. Geometrical derivation of Kepler's equation 159 

95. Solution of Kepler's equation 160 

96. Differential corrections 162 

97. Graphical solution of Kepler's equation . . . . . .163 

98. Recapitulation of formulas .164 

99. The development of E in series . .165 

100. The development of r and v in series 169 

101. Direct computation of the polar coordinates 172 

102. Position in hyperbolic orbits 177 

103. Position in elliptic and hyperbolic orbits when e is near unity . 178 
Problems on expansions and positions in orbits . . . .181 

104. The heliocentric position in the ecliptic system . . . .182 

105. Transfer of the origin to the earth 185 

106. Transformation to geocentric equatorial coordinates . . .186 

107. Direct computation of the geocentric equatorial coordinates . 187 

Problems on transformations of coordinates 189 

Historical sketch and bibliography 190 

CHAPTER VI. 

THE DETERMINATION OF ORBITS. 

108. General considerations . 191 

109. Intermediate elements 192 

110. Preparation of the observations . . 194 

111. Outline of the Laplacian method of determining orbits . . .195 

112. Outline of the Gaussian method of determining orbits . . .199 
I. THE LAPLACIAN METHOD OF DETERMINING ORBITS. 

113. Determination of the first and second derivatives of the angular 

coordinates from three observations 202 

114. Determination of the derivatives from more than three ob- 

servations 205 

115. The approximations in the determination of the values of X, /*, v 

and their derivatives 206 

116. Choice of the origin of time 207 

117. The approximations when there are four observations . . . 208 

118. The fundamental equations 211 

119. The equations for the determination of r and p . . . .212 

120. The condition for a unique solution 215 



TABLE OF CONTENTS. Xlll 

ART. PAGE 

121. Use of a fourth observation in case of a double solution . . 218 

122. The limits on m and M 219 

123. Differential corrections 220 

124. Discussion of the determinant D 222 

125. Reduction of the determinants Z)i and D 2 224 

126. Correction for the time aberration 226 

127. Development of x, y, and z in series . 227 

128. Computation of the higher derivatives of X, /*, v . . . 229 

129. Improvement of the values of x, y, z, x', y', z' 230 

130. The modifications of Harzer and Leuschner 231 

II. THE GAUSSIAN METHOD OF DETERMINING ORBITS. 

131. The equation for P2 ... 232 

132. The equations for p\ and p 3 . 236 

133. Improvement of the solution 236 

134. The method of Gauss for computing the ratios of the triangles . 237 

135. The first equation of Gauss . . . - 238 

136. The second equation of Gauss . . 240 

137. Solution of (98) and (101) 241 

138. Determination of the elements a, e, and o> ..... 243 

139. Second method of determining a, e, and w 244 

140. Computation of the time of perihelion passage 248 

141. Direct derivation of equations defining orbits 249 

142. Formulas for computing an approximate orbit 250 

Problems on determining orbits 257 

Historical sketch and bibliography . 258 

CHAPTER VII. 
THE GENERAL INTEGRALS OF THE PROBLEM OF n BODIES. 

143. The differential equations of motion 261 

144. The six integrals of the motion of the center of mass . . . 262 

145. The three integrals of areas 264 

146. The energy integral 267 

147. The question of new integrals 268 

Problems on motion of center of mass and areas integrals . . 269 

148. Transfer of the origin to the sun 269 

149. Dynamical meaning of the equations 271 

150. The order of the system of equations 273 

Problems on differential equations for motion of n bodies . . 274 

Historical sketch and bibliography 275 

CHAPTER VIII. 
THE PROBLEM OF THREE BODIES. 

151. Problem considered 277 

MOTION OF THE INFINITESIMAL BODY. 

152. The differential equations of motion 278 



XIV TABLE OF CONTENTS. 

ART. PAGE 

153. Jacobi's integral 280 

154. The surfaces of zero relative velocity 281 

155. Approximate forms of the surfaces 282 

156. The regions of real and imaginary velocity 286 

157. Method of computing the surfaces 287 

158. Double points of the surfaces and particular solutions of the 

problem of three bodies 

Problems on surfaces of zero relative velocity 

159. Tisserand's criterion for the identity of comets .... 

160. Stability of particular solutions 

161. Application of the criterion for stability to the straight line 

solutions 300 

162. Particular values of the constants of integration .... 302 

163. Application to the gegenschein 305 

164. Application of the criterion for stability to the equilateral triangle 

solutions .... 306 

Problems on motion of infinitesimal body 308 

CASE OF THREE FINITE BODIES. 

165. Conditions for circular orbits 309 

166. Equilateral triangle solutions . . 310 

167. Straight line solutions 311 

168. Dynamical properties of the solutions 312 

169. General conic section solutions 313 

Problems on particular solutions of the problem of three bodies . 318 

Historical sketch and bibliography 319 

CHAPTER IX. 

PERTURBATIONS GEOMETRICAL CONSIDERATIONS. 

170. Meaning of perturbations 321 

171. Variation of coordinates 321 

172. Variation of the elements 322 

173. Derivation of the elements from a graphical construction . . 323 

174. Resolution of the disturbing force 324 

I. EFFECTS OF THE COMPONENTS OF THE DISTURBING FORCE. 

175. Disturbing effects of the orthogonal component ..... 325 

176. Effects of the tangential component upon the major axis . . 327 

177. Effects of the tangential component upon the line of apsides . 327 

178. Effects of the tangential component upon the eccentricity . . 328 

179. Effects of the normal component upon the major axis . . . 329 

180. Effects of the normal component upon the line of apsides . . 329 

181. Effects of the normal component upon the eccentricity . . .331 

182. Table of results 332 

183. Disturbing effects of a resisting medium 333 

184. Perturbations arising from oblateness of the central body . . 333 
Problems on perturbations 335 

II. THE LUNAR THEORY. 

185. Geometrical resolution of the disturbing effects of a third body . 337 



TABLE OF CONTENTS. XV 

ART. PAGE 

186. Analytical resolution of the disturbing effects of a third body . 338 

187. Perturbations of the node 342 

188. Perturbations of the inclination 343 

189. Precession of the equinoxes. Nutation 344 

190. Resolution of the disturbing acceleration in the plane of motion . 345 

191. Perturbations of the major axis 346 

192. Perturbation of the period . . . 348 

193. The annual equation 348 

194. The secular acceleration of the moon's mean motion . . . 348 

195. The variation 350 

196. The parallactic inequality 352 

197. The motion of the line of apsides 352 

198. Secondary effects 355 

199. Perturbations of the eccentricity 356 

200. The erection 359 

201. Gauss' method of computing secular variations .... 360 

202. The long period inequalities 361 

Problems on perturbations 362 

Historical sketch and bibliography 363 

CHAPTER X. 

PERTURBATIONS ANALYTICAL METHOD. 

203. Introductory remarks 366 

204. Illustrative example 367 

205. Equations in the problem of three bodies 372 

206. Transformation of variables 374 

207. Method of solution 377 

208. Determination of the constants of integration 381 

209. The terms of the first order 382 

210. The terms of the second order 383 

Problems on the method of computing perturbations . . . 386 

211. Choice of elements 387 

212. Lagrange's brackets 387 

213. Properties of Lagrange's brackets 388 

214. Transformation to the ordinary elements 390 

215. Method of direct computation of Lagrange's brackets . . .391 

216. Computation of [co, ft], [ft, i], [i, co] 395 

217. Computation of [K, P] 396 

218. Computation of [a, e], [e, a], [<r, a] 397 

219. Change from ft, co, and a to ft, TT, and e 400 

220. Introduction of rectangular components of the disturbing ac- 

celeration 402 

Problems on variation of elements 405 

221. Development of the perturbative function 406 

222. Development of Ri, 2 in the mutual inclination 407 

223. Development of the coefficients in powers of e\ and e* . . . 409 

224. Developments in Fourier series 410 



XVI TABLE OF CONTENTS. 

ART. PAGE 

225. Periodic variations . 413 

226. Long period variations . . . 416 

227. Secular variations 417 

228. Terms of the second order with respect to the masses . . . 419 

229. Lagrange's treatment of the secular variations 420 

230. Computation of perturbations by mechanical quadratures . . 425 

231. General reflections 429 

Problems on the perturbative function 430 

Historical sketch and bibliography . . . . , . .431 



INTRODUCTION TO CELESTIAL MECHANICS 



CHAPTER I. 

FUNDAMENTAL PRINCIPLES AND DEFINITIONS. 

1. Elements and Laws. The problems of every science are 
expressible in certain terms which will be designated as elements, 
and depend upon certain principles and laws for their solution. 
The elements arise from the very nature of the subject considered, 
and are expressed or implied in the formulation of the problems 
treated. The principles and laws are the relations which are 
known or are assumed to exist among the various elements. 
They are inductions from experiments, or deductions from previ- 
ously accepted principles and laws, or simply agreements. 

An explicit statement in the beginning of the type of problems 
which will be treated, and an enumeration of the elements which 
they involve, and of the principles and laws which relate to them, 
will lead to clearness of exposition. In order to obtain a com- 
plete understanding of the character of the conclusions which are 
reached, it would be necessary to make a philosophical discussion 
of the reality of the elements, and of the origin and character of 
the principles and laws. These questions cannot be entered into 
here because of the difficulty and complexity of metaphysical 
speculations. It is not to be understood that such investigations 
are not of value; they forever lead back to simpler and more 
undeniable assumptions upon which to base all reasoning. 

The method of procedure in this work will necessarily be to 
accept as true certain fundamental elements and laws without 
entering in detail into the questions of their reality or validity. 
It will be sufficient to consider whether they are definitions or 
have been inferred from experience, and to point out that they 
have been abundantly verified in their applications. They will be 
accepted with confidence, and their consequences will be derived, 
in the subjects treated, so far as the scope and limits of the work 
will allow. 



i^ATED. [2 



A PROBLEMS TH^ATED. 

2. Problems Treated. The motions of a material particle sub- 
ject to a central force of any sort whatever will be briefly con- 
sidered. It will be shown from the conclusions reached in this dis- 
cussion, and from the observed motions of the planets and their 
satellites, that Newton's law of gravitation holds true in the solar 
system. The character of the motion of the binary stars shows 
that the probabilities are very great that it operates in them also, 
and that it may well be termed "the law of universal gravita- 
tion." This conclusion is confirmed by the spectroscope, which 
proves that the familiar chemical elements of our solar system 
exist in the stars also. 

In particular, the motions of two free homogeneous spheres 
subject only to their mutual attractions and starting from arbi- 
trary initial conditions will be investigated, and then their motions 
will be discussed when they are subject to disturbing influences of 
various sorts. The essential features of perturbations arising from 
the action of a. third body will be developed, both from a geo- 
metrical and an analytical point of view. There are" two some- 
what different cases. One is that in which the motion of a satel- 
^*A- lite around a planet is perturbed by the sun; and the second is 
that in which the motion of one planet around the sun is per- 
1 turbed by another planet. 

Another class of problems which arises is the determination of 
the orbits of unknown bodies from the observations of their direc- 
tions at different epochs, made from a body whose motion is 
known. That is, the theories of the orbits of comets and plan- 
etoids will be based upon observations of their apparent positions 
made from the earth. This incomplete outline of the questions 
to be treated is sufficient for the enumeration of the elements 
employed. 

3. Enumeration of the Principal Elements. In the discussion 
of the problems considered in this work it will be necessary to 
employ the following elements: 

(a) Real numbers, and complex numbers incidentally in the 
solution of certain problems. 

(6) Space of three dimensions, possessing the same properties in 
every direction. 

(c) Time of one dimension, which will be taken as the inde- 
pendent variable. 

(d) Mass, having the ordinary properties of inertia, etc., which 
are postulated in elementary Physics. 



5] NATURE OF THE LAWS OF MOTION. 3 

(e) Force, with the content that the same term has in Physics. 

Positive numbers arise in Arithmetic, and positive, negative, 
and complex numbers, in Algebra. Space appears first as an 
essential element in Geometry. Time appears first as an essential 
element in Kinematics. Mass and force appear first and must be 
considered as essential elements in physical problems. No defini- 
tions of these familiar elements are necessary here. 

4. Enumeration of the Principles and Laws. In representing 
the various physical magnitudes by numbers, certain agreements 
must be made as to what shall be considered positive, and what 
negative. The axioms of ordinary Geometry will be considered 
as being true. 

The fundamental principles upon which all work in Theoretical 
Mechanics may be made to depend are Newton's three Axioms, or 
Laws of Motion. The first two laws were known by Galileo and 
Huyghens, although they were for the first time announced 
together in all their completeness by Newton in the Principia, 
in 1686. These laws are as follows:* 

LAW I. Every body continues in its state of rest, or of uniform 
motion in a straight line, unless it is compelled to change that state by 
a force impressed upon it. 

LAW II. The rate of change of motion is proportional to the force 
impressed, and takes place in the direction of the straight line in which 
the force acts. 

LAW III. To every action there is an equal and opposite reaction; 
or, the mutual actions of two bodies are always equal and oppositely 
directed. 

5. Nature of the Laws of Motion. Newton calls the Laws of 
Motion Axioms, and after giving each, makes a few remarks con- 
cerning its import. Later writers, among whom are Thomson and 
Tait,f regard them as inferences from experience, but accept New- 
ton's formulation of them as practically final, and adopt them 
in the precise form in which they were given in the Principia. A 
number of Continental writers, among whom is Dr. Ernest Mach, 
have given profound thought to the fundamental principles of 

* Other fundamental laws may be, and indeed have been, employed; but 
they involve more difficult mathematical principles at the very start. They 
are such as d'Alembert's principle, Hamilton's principle, and the systems of 
Kirchhoff, Mach, Hertz, Boltzmann, etc. 

t Natural Philosophy, vol. i., Art. 243. 



4: REMARKS ON THE FIRST LAW OF MOTION. [6 

Mechanics, and have concluded that they are not only inductions or 
simply conventions, but that Newton's statement of them is some- 
what redundant, and lacks scientific directness and simplicity. 
There is no suggestion, however, that Newton's Laws of Motion 
are not in harmony with ordinary astronomical experience, or that 
they cannot be made the basis for Celestial Mechanics. But in 
some branches of Physics, particularly in Electricity and Light, 
certain phenomena are not fully consistent with the Newtonian 
principles, and they have recently led Einstein and others to the 
development of the so-called Principle of Relativity. The astro- 
nomical consequences of this modification of the principles of 
Mechanics are very slight unless the time under consideration 
is very long, and, whether they are true or not, they cannot be 
considered in an introduction to the subject. 

6. Remarks on the First Law of Motion. In the first law the 
statement that a body subject to no forces moves with uniform 
motion, may be regarded as a definition of time. For, otherwise, 
it is implied that there exists some method of measuring time in 
which motion is not involved. Now it is a fact that in all the 
devices actually used for measuring time this part of the law is a 
fundamental assumption. For example, it is assumed that the 
earth rotates at a uniform rate because there is no force acting 
upon it which changes the rotation sensibly.* 

The second part of the law, which affirms that the motion is in 
a straight line when the body is subject to no forces, may be taken 
as defining a straight line, if it is assumed that it is possible to 
determine when a body is subject to no forces; or, it may be taken 
as showing, together with the first part, whether or not forces 
are acting, if it is assumed that it is possible to give an independent 
definition of a straight line. Either alternative leads to trouble- 
some difficulties when an attempt is made to employ strict and 
consistent definitions. 

7. Remarks on the Second Law of Motion. In the second law 
the statement that the rate of change of motion is proportional to 
the force impressed, may be regarded as a definition of the relation 
between force and matter by means of which the magnitude of a 
force, or the amount of matter in a body can be measured, accord- 
ing as one or the other is supposed to be independently known. 
By rate of change of motion is meant the rate of change of velocity 

* See memoir by R. S. Woodward, Astronomical Journal, vol. xxi. (1901). 



7] REMARKS ON THE SECOND LAW OF MOTION. 5 

multiplied by the mass of the body moved. This is usually called 
the rate of change of momentum, and the ideas of the second law 
may be expressed by saying, the rate of change of momentum is 
proportional to the force impressed and takes place in the direction 
of the straight line in which the force acts. Or, the acceleration of 
motion of a body is directly proportional to the force to which it is 
subject, and inversely proportional to its mass, and takes place in 
the direction in which the force acts. 

It may appear at first thought that force can be measured 
without reference to velocity generated, and it is true in a sense. 
For example, the force with which gravity draws a body downward 
is frequently measured by the stretching of a coiled spring, or the 
intensity of magnetic action can be measured by the torsion of a 
fiber. But it will be noticed in all cases of this kind that the 
law of reaction of the machine has been determined in some other 
way. This may not have been directly by velocities generated, 
but it ultimately leads back to it. It is worthy of note in this 
connection that all the units of absolute force, as the dyne, contain 
explicitly in their definitions the idea of velocity generated. 

In the statement of the second law it is implied that the effect 
of a force is exactly the same in whatever condition of rest or of 
motion the body may be, and to whatever other forces it may be 
subject. The change of motion of a body acted upon by a number 
of forces is the same at the end of an interval of time as if each 
force acted separately for the same time. Hence the implication 
in the second law is, if any number of forces act simultaneously on 
a body, whether it is at rest or in motion, each force produces the same 
total change of momentum that it would produce if it alone acted on 
the body at rest. It is apparent that this principle leads to great 
simplifications of mechanical problems, for in accordance with it 
the effects of the various forces can be considered separately. 

Newton derived the parallelogram of forces from the second 
law of motion.* He reasoned that as forces are measured by the 
accelerations which they produce, the resultant of, say, two forces 
should be measured by the resultant of their accelerations. Since 
an acceleration has magnitude and direction it can be represented 
by a directed line, or vector. The resultant of two forces will 
then be represented by the diagonal of a parallelogram, of which 
two adjacent sides represent the two forces. 

* Principia, Cor. i. to the Laws of Motion. 



6 REMARKS ON THE THIRD LAW OF MOTION. [8 

One of the most frequent applications of the parallelogram of 
forces is in the subject of Statics, which, in itself, does not involve 
the ideas of motion and time. In it the idea of mass can also be 
entirely eliminated. Newton's proof of the parallelogram of 
forces has been objected to on the ground that it requires the 
introduction of the fundamental conceptions of a much more 
complicated science than the one in which it is employed. Among 
the demonstrations which avoid this objectionable feature is one 
due to Poisson,* which has for its fundamental assumption the 
axiom that the resultant of two equal forces applied at a point is 
in the line of the bisector of the angle which they make with 
each other. Then the magnitude of the resultant is derived, and 
by simple processes the general law is obtained. 

8. Remarks on the Third Law of Motion. The first two of 
Newton's laws are sufficient for the determination of the motion 
of one body subject to any number of known forces; but another 
principle is needed when the investigation concerns the motion of 
a system of two or more bodies subject to their mutual interactions. 
The third law of motion expresses precisely this principle. It is 
that if one body presses against another the second resists the 
action of the first with the same force. And also, though it is 
not so easy to conceive of it, if one body acts upon another through 
any distance, the second reacts upon the first with an equal and 
oppositely directed force. 

Suppose one can exert a given force at will; then, by the second 
law of motion, the relative masses of bodies can be measured since 
they are inversely proportional to the accelerations which equal 
forces generate in them. When their relative masses have been 
found the third law can be tested by permitting the various bodies 
to act upon one another and measuring their relative accelera- 
tions. Newton made several experiments to verify the law, such 
as measuring the rebounds from the impacts of elastic bodies, and 
observing the accelerations of magnets floating in basins of water. f 
The chief difficulty in the experiments arises in eliminating forces 
external to the system under consideration, 'and evidently they 
cannot be completely removed. Newton also concluded from a 
certain course of reasoning that to deny the third law would be to 
contradict the first. f 

Mach points out that there is no accurate means of measuring 
* Traite de Mecanique, vol. I., p. 45 ei seq. 
f Principia, Scholium to the Laws of Motion. 



8] REMARKS ON THE THIRD LAW OF MOTION. 7 

forces except by the accelerations they produce in masses, and 
therefore that effectively the reasoning in the preceding paragraph 
is in a circle. He objects also to Newton's definition that mass 
is proportional to the product of the volume and the density of a 
body. He prefers to rely upon experience for the fact that two 
bodies which act upon each other produce oppositely directed 
accelerations, and to define the relative values of the masses as 
inversely proportional to these accelerations. Experience proves 
further that if the relative masses of two bodies are determined 
by their interactions with a third, the ratio is the same whatever 
the third mass may be. In this way, when one body is taken as 
the unit of mass, the masses of all other bodies can be uniquely 
determined. These views have much to commend them. 

In the Scholium appended to the Laws of Motion Newton made 
some remarks concerning an important feature of the third law. 
This was first stated in a manner in which it could actually be 
expressed in mathematical symbols by d'Alembert in 1742, and 
has ever since been known by his name.* It is essentially this: 
When a body is subject to an acceleration, it may be regarded as 
exerting a force which is equal and opposite to the force by which 
the acceleration is produced. This may be considered as being 
true whether the force arises from another body forming a system 
with the one under consideration, or has its source exterior to the 
system. In general, in a system of any number of bodies, the 
resultants of all the applied forces are equal and opposite to the 
reactions of the respective bodies. In other words, the impressed 
forces and the reactions, or the expressed forces, form systems 
which are in equilibrium for each body and for the whole system. 
This makes the whole science of Dynamics, in form, one of Statics, 
and formulates the conditions so that they are expressible in 
mathematical terms. This phrasing of the third law of motion 
has been made the starting point for the elegant and very general 
investigations of Lagrange in the subject of Dynamics.! 

The primary purpose of fundamental principles in a science is to 
coordinate the various phenomena by stating in what respects 
their modes of occurring are common; the value of fundamental 
principles depends upon the completeness of the coordination of 
the phenomena, and upon the readiness with which they lead to 
the discovery of unknown facts; the characteristics of funda- 

* See Appell's Mecanique, vol. n., chap. xxm. 
f Collected Works, vols. xi. and xn. 



8 SPEED AND VELOCITY IN RECTILINEAR MOTION. [9 

mental principles should be that they are self-consistent, that 
they are consistent with every observed phenomenon, and that 
they are simple and not redundant. 

Newton's laws coordinate the phenomena of the mechanical 
sciences in a remarkable manner, while their value in making 
discoveries is witnessed by the brilliant achievements in the 
physical sciences in the last two centuries compared to the slow 
and uncertain advances of all the ancients. They have not been 
found to be mutually contradictory, and they are consistent with 
nearly all the phenomena which have been so far observed; they 
are conspicuous for their simplicity, but it has been claimed by 
some that they are in certain respects redundant. One naturally 
wonders whether they are primary and fundamental laws of 
nature, even as modified by the principle of relativity. In view 
of the past evolution of scientific and philosophical ideas one 
should be slow in affirming that any statement represents ultimate 
and absolute truth. The fact that several other sets of funda- 
mental principles have been made the bases of systems of me- 
chanics, points to the possibility that perhaps some time the 
Newtonian system, or the Newtonian system as modified by the 
principle of relativity, even though it may not be found to be in 
error, will be supplanted by a simpler one even in elementary 
books. 

DEFINITIONS AND GENERAL EQUATIONS. 

9. Rectilinear Motion, Speed, Velocity. A particle is in 
rectilinear motion when it always lies in the same straight line, and 
when its distance from a point in that line varies with the time. 
It moves with uniform speed if it passes over equal distances in 
equal intervals of time, whatever their length. The speed is 
represented by a positive number, and is measured by the distance 
passed over in a unit of a time. The velocity of a particle is the 
directed speed with which it moves, and is positive or negative 
according to the direction of the motion. Hence in uniform motion 
the velocity is given by the equation 



Since s may be positive or negative, v may be positive or negative, 
and the speed is the numerical value of v. The same value of v is 
obtained whatever interval of time is taken so long as the corre- 
sponding value of s is used. 



10] ACCELERATION IN RECTILINEAR MOTION. 9 

The speed and velocity are variable when the particle does not 
describe equal distances in equal times; and it is necessary to define 
in this case what is meant by the speed and velocity at an in- 
stant. Suppose a particle passes over the distance As in the time 
At, and suppose the interval of time At approaches the limit zero in 
such a manner that it always contains the instant t. Suppose, 
further, that for every At the corresponding As is taken. Then 
the velocity at the instant t is defined as 

/o\ r / & s \ ds 

(2) v = lim [ - - ) = -=- , 

A ,=o \At J dt' 

ds 
and the speed is the numerical value of -77 . 

Uniform and variable velocity may be defined analytically in 
the following manner. The distance s, counted from a fixed point, 
is considered as a function of the time, and may be written 

s = 0(0. 
Then the velocity may be defined by the equation 



Tt ' 

where <j>'(t) is the derivative of 0(0 with respect to t. The velocity 
is said to be constant, or uniform, if 0'(0 does not vary with t', 
or, in other words, if 0(0 involves t linearly in the form 0(0 =+&, 
where a and b are constants. It is said to be variable if the value 
of 0'(0 changes with t. 

Some agreement must be made to denote the direction of 
motion. An arbitrary point on the line may be taken as the 
origin and the distances to the right counted as positive, and 
those to the left, negative. With this convention, if the value of s 
determining the position of the body increases algebraically with 
the time the velocity will be taken positive; if the value of s de- 
creases as the time increases the velocity will be taken negative. 
Then, when v is given in magnitude and sign, the speed and direc- 
tion of motion are determined. 

10. Acceleration in Rectilinear Motion. Acceleration is the 
rate of change of velocity, and may be constant or variable. Since 
the case when it is variable includes that when it is constant, it 
will be sufficient to consider the former. The definition of acceler- 



10 SPEED AND VELOCITIES IN CURVILINEAR MOTION. [11 

ation at an instant t is similar to that for velocity, and is, if the 
acceleration is denoted by a, 



/o\ 
(3) 



r f Av\ dv 
a = hm ( ) = -=- . 

A<=O \ At / at 



By means of (2) and (3) it follows that 

d /ds\ d 2 s 
a = dt(di) == dt*' 

There must be an agreement regarding the sign of the accelera- 
tion. When the velocity increases algebraically as the time 
increases, the acceleration will be taken positive ; when the velocity 
decreases algebraically as the time increases, the acceleration will 
be taken negative. 

11. Speed and Velocities in Curvilinear Motion. The speed 
with which a particle moves is the rate at which it describes a 
curve. If v represents the speed in this case, and s the arc of the 
curve, then 

ds 



< 6 > - - dt 



where 



ds 



represents the numerical value of -77 . As before, the 



velocity is the directed speed possessing the properties of vectors, 
and may be represented by a vector.* The vector can be resolved 
uniquely into three components parallel to any three coordinate 
axes; and conversely, the three components can be compounded 
uniquely into the vector. In other words, if the velocity is given, 
the components parallel to any coordinate axes are defined;. and 
the components parallel to any non-coplanar coordinate axes define 
the velocity. It is generally simplest to use rectangular axes and 
to employ the components of velocity parallel to them. Let 
X, M, v represent the angles between the line of motion and the 
x, y, and z-axes respectively. Then 

(ft \ = c s = c s v = 

Let v x , v y , v z represent the components of velocity along the three 
axes. That is, 

* Consult Appell's Mecanique, vol. I., p. 45 et seq. 



12] 



ACCELERATION IN CURVILINEAR MOTION. 



11 



(7) 



ds dx dx 

v x = v cos X = -77 -j- = -77 , 
dt ds dt 

ds dy dy 

Vy = V COS fJL = -TT -J- = -77 , 



dz dz 



From these equations it follows that 



(8) 



There must be an agreement as to a positive and a negative 
direction along each of the three coordinate axes. 

12. Acceleration in Curvilinear Motion. As in the case of 
velocities, it is simplest to resolve the acceleration into component 
accelerations parallel to the coordinate axes. On constructing a 
notation corresponding to that used in Art. 11, the following 
equations result: 

_ d 2 x _ d 2 y _ d 2 z 

OLx ~ ,7/2 a v ~ dp > a * ~ rffz ' 

Hi (Jili (Jiil/ 



(8) 



The numerical value of the whole acceleration is 



This is not, in general, equal to the component of acceleration 



d?s 



along the curve; that is, to -. For, from (8) it follows that 



ds 

V= dt 



whence, by differentiation, 

dx d 2 x 
d 2 s dt dt 2 



dz d 2 z 



en) 



It 2 " l/dx\ 2 idy\ 2 idz\ 
V\dt) \dt ) \dt) 

dx d 2 x dy d 2 y dz d 2 z 



Thus, when the components of acceleration are known, the 
whole acceleration is given by (10), and the acceleration along 



12 



POLAR COMPONENTS OF VELOCITY 



[13 



the curve by (11). The fact that the two are different, in general, 
may cause some surprise at first thought. But the matter becomes 
clear if a body moving in a circle with constant speed is con- 
sidered. The acceleration along the curve is zero because the 
speed is supposed not to change; but the acceleration is not zero 
because the body does not move in a straight line. 

13. The Components of Velocity Along and Perpendicular to 
the Radius Vector. Suppose the path of the particle is in the 
iC2/-plane, and let the polar coordinates be r and 6. Then 

(12) x = r cos 8, y = r sin 6. 

The components of velocity are therefore 

dx . dd . dr 

dt 
(13) 



-TT = v, 



Q dd 

r sin 6 -TT 



Let QP be an arc of the curve described by the moving particle. 
When the particle is at P, it is moving in the direction PV, and 
the velocity may be represented by the vector PV. Let v r and v & 




Fig. 1. 

represent the components of velocity along and perpendicular to 
the radius vector. The resultant of the vectors v r and v is equal 

(i'lr r/?y 

to the resultant of the vectors 3- and ~ , that is, to PV. The 

at at 

sum of the projections of v r and VQ upon any line equals the sum 

of the projections of 3- and -^ upon the same line. Therefore, 
at at 

projecting v r and v& upon the x and y-axes, it follows that 



(14) 



-j7 = v r cos 6 v^ sin 6, 
dt 

dii 

-r = v r sin + VQ cos 6. 

at 



14] POLAR COMPONENTS OF ACCELERATION. 13 

On comparing (13) and (14), the required components of velocity 

are found to be 

dr 



The square of the speed is 



The components of velocity, v r and vd, can be found in terms 
of the components parallel to the x and ?/-axes by multiplying 
equations (14) by cos 6 and sin 6 respectively and adding, and 
then by sin d and cos d and adding. The results are 

. dx . . n di/ 
v r = + cos 6 -=7 -f- sm 6 -, 
at at 

(16) 

. _ dx . n dy 
v e = sin 6 -=- -f cos 6 -37. 
at at 

14. The Components of Acceleration. The derivatives of 
equations (13) are 



(Px Y(Pr /< VI [ d*8 , n drd8l . 

a * = Hi? = [df- r (di) \ cose - [ r dt*+ 2 Jt Jt\ sm " 

<Pr /d\ 2 1 . 

de - T (jt) J sm " 



' *v_\_*9, n *d 



Let ct r and a& represent the components of acceleration along 
and perpendicular to the radius vector. As in Art. 13, it follows 
from the composition and resolution of vectors that 

{a x = OL T cos d O.Q sin 6, 
a y = a r sin 6 + a e cos 6. 
On comparing (17) and (18), it is found that 



The components of acceleration along and perpendicular to the 



14 



PARTICLE MOVING IN A CIRCLE. 



[15 



radius vector in terms of the components parallel to the x and 
2/-axes are found from (17) to be 



(20) 



a r = 



= - 







By similar processes the components of velocity and acceleration 
parallel to any lines can be found. 

15. Application to a Particle Moving in a Circle with Uniform 
Speed. Suppose the particle moves with uniform speed in a circle 
around the origin as center; it is required to determine the com- 



axis 




Fig. 2. 

ponents of velocity and acceleration parallel to the x and y-axes, 
and parallel and perpendicular to the radius. Let R represent 
the radius of the circle; then 

x = R cos 0, y = R sin 6. 

Since the speed is uniform the angle 6 is proportional to the time, 
or 6 = ct. The coordinates become 

(21) x = R cos (ct), y = R sin (ct). 

3 n /jj? 

Since -r = c and -r- = 0, the components of velocity parallel to 

CtL (juL 

the x and ?/-axes are found from (13) to be 

(22) v x = - Re sin (ct), v v = Re cos (ct). 
From (15) it is found that 

(23) v r = 0, v e = Re. 

The components of acceleration parallel to the x and i/-axes, 
which are given by (17), are 



16] 



AREAL VELOCITY. 



15 



(24) 



a x Rc 2 cos (ct), 
a.y Re 2 sin (ct). 



And from (19) it is found that 



(25) 



= Rc 2 , 



= 0. 



It will be observed that, although the speed is uniform in this 
case, the velocity with respect to fixed axes is not constant, and 
the acceleration is not zero. If it is assumed that an exterior 
force is the only cause of the change of motion, or of acceleration 
of a particle, then it follows that a particle cannot move in a 
circle with uniform speed unless it is subject to some force. It 
follows also from (25) and the second law of motion that the force 
continually acts in a line which passes through the center of the 
circle. 

16. The Areal Velocity. The rate at which the radius vector 
from a fixed point to the moving particle describes a surface is 



y - oris 




X-axis 



Fig. 3. 

called the areal velocity with respect to the point. Suppose the 
particle moves in the :n/-plane. Let AA represent the area of the 
triangle OPQ swept over by the radius vector in the interval of 
time A*. Then 



sin(A0); 



whence 
(26) 



r sin (A0) A0 



A* 2 A0 Ar 

As the angle A0 diminishes the ratio of the area of the triangle to 



16 MOTION OF A PARTICLE IN AN ELLIPSE. [17 

that of the sector approaches unity as a limit. The limit of 
r' is r, and the limit of - is unity. Equation (26) gives, on 

passing to the limit AZ = in both members, 

(27) = -r 2 

dt 2 T dt 

as the expression for the areal velocity. On changing to rect- 
angular coordinates by the substitution 

r = Vz 2 + 2/ 2 , tan = - , 

6 

equation (27) becomes 

(28) ^ = l/ x g_^ 

If the motion is not in the zt/-plane the projections of the areal 
velocity upon the three fundamental planes are used. They are 
respectively 

dA xy _ 1 / dy _ dx\ 

~dT ~2\ X dt~~ y dt)' 

dA yz 1 / dz dy 



(29) 



dt 

dA zx 
dt 



_ 1 / dx _ dz\ 
~2\ Z dt X dt)' 



In certain mechanical problems the body considered moves so 
that the areal velocity is constant if the origjn is properly chosen. 
In this case it is said that the body obeys the law of areas with 
respect to the origin. That is, 

r 2 -7- = constant. 
at 

It follows from this equation and (19) that in this case 

a e = 0; 
that is, the acceleration perpendicular to the radius vector is zero. 

17. Application to Motion in an Ellipse. Suppose a particle 
moves in an ellipse whose semi-axes are a and b in such a manner 
that it obeys the law of areas with respect to the center of the 
ellipse as origin; it is required to find the components of accelera- 






PROBLEMS. 17 

tion along and perpendicular to the radius vector. The equation 
of the ellipse may be written in the parametric form 

(30) x = a cos 0, y = b sin 0; 

for, if is eliminated, the ordinary equation 



is found. It follows from (30) that 

/0 -i\ dx . d<f> dy 

(31) _=-a S m0^, J- 

On substituting (30) and (31) in the expression for the law of areas, 



^ it is found that 



d$ = c_ 
dt ~ ab 



The integral of this equation is 



+ cz; 



and if < = when t = 0, then Cz = and < = c\t. 

On substituting the final expression for < in (30), it is found that 
'd 2 x 



- Ci'a cos = - 

Ci 2 6 sin cj) = c^y. 

If these values of the derivatives are substituted in (20) the 
components of acceleration along and perpendicular to the radius 
vector are found to be 

r a r = Ci 2 r, 

Oi& = 0. 

I. PROBLEMS. 

1. A particle moves with uniform speed along a helix traced on a circular 
cylinder whose radius is R; find the components of velocity and acceleration 
parallel to the x, y, and z-axes. The equations of the helix are 

x = R cos a), y = R sin o>, z = hu>. 
3 



18 



PROBLEMS. 



[ v x 

Ans. { 

I a x = - 



Re sin (d), v y = -\- Re cos (d), v z = he; 
Re 2 cos (d), a y = Re 2 sin (d), a z = 0. 



2. A particle moves in the ellipse whose parameter and eccentricity are 
p and e with uniform angular speed with respect to one of the foci as origin; 
find the components of velocity and acceleration along and perpendicular 
to the radius vector and parallel to the x and y-axes in terms of the radius 
vector and the time. 



Ans. 



v 6 = re; 



v x = - cr sin (d) + r 2 sin (2d), 

>C 

v y = cr cos (d) H r 2 sin 2 (d) ; 



2ec 2 



(d) + f- r 3 sin 2 (d) - c 2 r, 



*sin (ct); 



ot x = - c*r cos (d) + r 2 - -- r 2 sin 2 (d) 

2e 2 c 2 
H -- g- r 3 sin 2 (d) cos (d), 



sn 



sn 3 



3. A 'particle moves in an ellipse in such a manner that it obeys the law 
of areas with respect to one of the foci as an origin; it is required to find the 
components of velocity and acceleratio i along and perpendicular to the radius 
vector and parallel to the axes in terms of the coordinates. 



Ans. 



eA . 

v r = sin d, 



> 



eA . nn A sin 6 eA . , . A cos 6 

v, = ^ sin 20 j^, y, = - sm 2 + ; 



- 



A' 1 

'?' 



=0; 



i = 



A 2 cos 



p r 2 



A 2 sing 
p ' r 2 



4. The accelerations along the x and y-axes are the derivatives of the 
velocities along these axes; why are not the accelerations along and per- 
pendicular to the radius vector given by the derivatives of the velocities in 
these respective directions? Find the accelerations along axes rotating with 
the angular velocity unity in terms of the accelerations with respect to fixed 
axes. 



18] 



CENTER OF MASS OF SYSTEMS OF PARTICLES. 



19 



18. Center of Mass of n Equal Particles. The center of mass 
of a system of equal particles will be defined as that point whose 
distance from any plane is equal to the average distance of all 
of the particles from that plane. This must be true then for the 
three reference planes. Let (xi, yi, Zi), (xz, yz, Zz), etc., represent 
the rectangular coordinates of the various particles, and x, y, z 
the rectangular coordinates of their center of mass; then by the 
definition 



(32) 



X = 



y = 



+ X 2 



+ X, 







Zi 



Z 2 



i-l 

n 



Suppose the mass of each particle is m, and let M represent the 
mass of the whole system, or M = nm. On multiplying the 
numerators and denominators by m, equations (32) become 



(33) 



m 



x = 



y = 



z = 






nm 

n 



M 



nm 



M 



nm 



=1_ 
M 



It remains to show that the distance from the point (x, y, z) 
to any other plane is also the average distance of the particles 
from the plane. The equation of any other plane is 



ax 



by + cz + d = 0. 
The distance of the point (x, y, z) from this plane is 



(34) 



- _ ax + by + cz + d 
Va 2 + 6 2 + c 2 






20 



CENTER OF MASS OF SYSTEMS OF PARTICLES. 



[19 



and similarly, the distance of the point (Xi, y i} z) from the same 
plane is 

,*n j aXi + ^ + CZi + d 

(35) di = . 2 2 

It follows from equations (32), (34) ; and (35) that 



+ 6 2 + c 2 



n 



Therefore the point (x, y, z) denned by (32) satisfies the definition 
of center of mass with respect to all planes. 

19. Center of Mass of Unequal Particles. There are two 
cases, (a) that in which the masses are commensurable, and (fc) 
that in which the masses are incommensurable. 

(a) Select a unit m in terms of which all the n masses can be 
expressed integrally. Suppose the first mass is p\m, the second 
p 2 m, etc., and let pirn = mi, p 2 m = m 2 , etc. The system may be 
thought of as made up of p\ + p^ + particles each of mass m, 
and consequently, by Art. 18, 



(36) 



M 



M ' 



% 



M 



(b) Select an arbitrary unit m smaller than any one of the 
n masses. They will be expressible in terms of it plus certain 
remainders. If the remainders are neglected equations (36) give 
the center of mass. Take as a new unit any submultiple of m 
and the remainders will remain the same, or be diminished, 
depending on their magnitudes. The submultiple of m can be 
taken so small that every remainder is smaller than any assigned 



19] CENTER OF MASS OF SYSTEMS OF PARTICLES. 21 

quantity. Equations (36) continually hold where the m; are the 
masses of the bodies minus the remainders. As the submultiples 
of m approach zero as a limit, the sum of the remainders approaches 
zero as a limit, and the expressions (36) approach as limits the 
expressions in which the m t - are the actual masses of the particles. 
Therefore in all cases equations (36) give a point which satisfies 
the definition of center of mass. 

The fact that if the definition of center of mass is fulfilled for 
the three reference planes, it is also fulfilled for every other plane 
can easily be proved without recourse to the general formula for 
the distance from any point to any plane. It is to be observed 
that the i/z-plane, for example, may be brought into any position 
whatever by a change of origin and a succession of rotations of 
the coordinate system around the various axes. It will be neces- 
sary to show, then, that equations (36) are not changed in form 
(1) by a change of origin, and (2) by a rotation around one of the 
axes. 

(1) Transfer the origin along the #-axis through the distance a. 
The substitution which accomplishes the transfer is x = x' + a, 
and the first equation of (36) becomes 

n 

Sm,i(xi + a) 

~ _+_ n - 

whence 



~ _+_ n - _ - __ L = . 

M MM' 



n 



M 

which has the same form as before. 

(2) Rotate trie x and 2/-axes around the z-axis through the 
angle 6. The substitution which accomplishes the rotation is 

{x = x' cos 9 y f sin 0, 
y = x' sin 6 -\- y' cos 6. 
The first two equations of (36) become by this transformation 



x' cos 6 y' sin = cos 6 t= -- sin 



x' sin e + y' cos B = sin =_ (_ cos 6 



M M 



22 



CENTER OF GRAVITY. 



[20 



On solving these equations it is found that 



y' = 



M 



M 



Therefore the point (x, y, z) satisfies the definition of center of 
mass with respect to every plane. 

20. The Center of Gravity. The members of a system of 
particles which are near together at the surface of the earth are 
subject to forces downward which are sensibly parallel and pro- 
portional to their respective masses. The weight, or gravity, of a 
particle will be defined as the intensity of the vertical force /, 




Fig. 4. 

which is the product of the mass m of the particle and its accelera- 
tion g. The center of gravity of the system will be defined as the 
point such that, if the members of the system were rigidly con- 
nected and the sum of all the forces were applied at this point, 
then the effect on the motion of the system would be the same as 
that of the original forces for all orientations of the system. 

It will now be shown that the center of gravity coincides with 
the center of mass. Consider two parallel forces /i and / 2 acting 
upon the rigid system M at the points Pi and P 2 . Resolve these 
two forces into the components / and g\, and / and 2 respectively. 
The components /, being equal and opposite, destroy each other. 
Then the components g\ and # 2 may be regarded as acting at A. 
Resolve them again so that the oppositely directed components 



20] 



CENTER OF GRAVITY. 



23 



shall be equal and lie in a line parallel to PiP 2 ; then the other com- 
ponents will lie in the same line AB, which is parallel to the 
direction of the original forces /i and / 2 , and will be equal respec- 
tively to /i and / 2 . Therefore the resultant of /i and / 2 is equal to 
/! + / 2 in magnitude and direction. It is found from similar 

triangles that 

fi = AB_ f 

7~PiB> f 

whence, by division, 



The solution for x gives 



If the resultant of these two forces be united with a third force / 3 , 
the point where their sum may be applied with the same effects is 
found in a similar manner to be given by 



/1+/2+/3 

and so on for any number of forces. Similar equations are true 
for parallel forces acting in any other direction. 

Suppose there are n particles m subject to n parallel forces /,- 
due to the attraction of the earth. The coordinates of their 
center of gravity with respect to the origin are given by 



(37) 



M 



y = 



M 



M 



The center of gravity is thus seen to be coincident with the center 
of mass; nevertheless this would not in general be true if the sys- 
tem were not in such a position that the accelerations to which 



24 CENTER OF MASS OF A CONTINUOUS BODY. [21 

its various members are subject were both equal and parallel. 
Euler (1707-1783) proposed the designation of center of inertia for 
the center of mass. 

21. Center of Mass of a Continuous Body. As the particles 
of a system become more and more numerous and nearer together 
it approaches as a limit a continuous body. In the case of the 
ordinary bodies of mechanics the particles are innumerable and 
indistinguishably close together; on this account such bodies are 
treated as continuous masses. For continuous masses, therefore, 
the limits of expressions (37), as nii approaches zero, must be 
taken. At the limit m becomes dm and the sum becomes the defi- 
nite integral. The equations which give the center of mass are 
therefore 



(38) 



_ = fxdm 

fdm : 

_ Jydm 
~ fdm ' 



= 
' 



fdm ' 

where the integrals are to be extended throughout the whole body. 
When the body is homogeneous the density is the quotient of 
any portion of the mass divided by its volume. When the body 
is not homogeneous the mean density is the quotient of the whole 
mass divided by the whole volume. The density at any point is 
the limit of the mean density of a volume including the point in 
question when this volume approaches zero as a limit. If the 
density is represented by cr, the element of mass is, when expressed 
in rectangular coordinates, 

dm = (rdxdydz. 
Then equations (38) become 

fffffx dx dy dz 



(39) 



= fffvdxdydz ' 

= fffvydxdydz 
~ fffcrdxdydz ' 

- = 



fffadxdydz 

The limits of the integrals depend upon the shape of the body, 
and a must be expressed as a function of the coordinates. 



21] 



CENTER OF MASS OF A CONTINUOUS BODY. 



25 



In certain problems the integrations are performed more simply 
if polar coordinates are employed. The element of mass when 
expressed in polar coordinates is 

dm = a - ab be - cd. 




Fig. 5. 
It is seen from the figure that 

ab = dr, 
be = rd(f>, 



cd = r cos <f>dd. 
dm or 2 cos d<j> d& dr, 



Therefore 

(40) 
and 

Ix = r cos cos 6j 
y = r cos <f> sin 0, 
z = r sin <f>. 

Therefore equations (38) become 

///or 3 cos 2 <fr cos 6 d$ dB dr 
///or 2 cos d<j> dd dr 

fffvr 3 cos 2 </> sin d<f> d0 dr 



(42) 



y 



fffo-r 2 cos d(f> dd dr 



. _ ///or 3 sin 4> cos <ft d<f> dB dr 
fffar 2 cos <f>d<t>d8 dr 



26 



PLANES AND AXES OF SYMMETRY. 



[22 



The density a must be expressed as a function of the coordinates, 
and the limits must be so taken that the whole body is included. 
If the body is a line or a surface the equations admit of important 
simplifications. 

22. Planes and Axes of Symmetry. If a homogeneous body 
is symmetrical with respect to any plane, the center of mass is in 
that plane, because each element of mass on one side of the plane 
can be paired with a corresponding element of mass on the other 
side, and the whole body can be divided up into such paired ele- 
ments. This plane is called a plane of symmetry. If a homo- 
geneous body is symmetrical with respect to two planes, the center 
of mass is in the line of their intersection. This line is called an 
axis of symmetry. If a homogeneous body is symmetrical with 
respect to three planes, intersecting in a point, the center of mass 
is at their point of intersection. From the consideration of the 
planes and axes of symmetry the centers of mass of many of the 
simple figures can be inferred without employing the methods of 
integration. 

23. Application to a Non-Homogeneous Cube. Suppose the 
density varies directly as the square of the distance from one of the 
faces of the cube. Take the origin at one of the corners and let 
the i/2-plane be the face of zero density. Then a = kx 2 , where 
k is the density at unit distance. Suppose the edge of the cube 
equals a; then equations (39) become 



x = 



k I J I x 3 dx dy dz 



x 2 y dx dy dz 



x 2 dx dy dz 



z 



- 

These equations become, after integrating and substituting the 
limits, 



_ _a 
Z ~2' 



24] 



APPLICATION TO THE OCTANT OF A SPHERE. 



27 



If polar coordinates were used in this problem the upper limits 
of the integrals would be much more complicated than they are 
with rectangular coordinates, and the integration would be 
correspondingly more difficult. 

24. Application to the Octant of a Sphere. Suppose the sphere 
is homogeneous and that the density equals unity. It is preferable 
to use polar coordinates in this example, although it is by no 
means necessary. Either (39) or (42) can be used in any problem, 
and the choice should be determined by the form that the limits 
take in the two cases. It is desirable to have them all constants 
when they can be made so. If the origin is taken at the center 
of the sphere, and if the radius is a, equations (42) become 



x = 



IT IT 

Jo Jo Jo 



r 3 cos 2 ^ cos6d<f>dddr 



y = 



z = 



m 

Jo Jo Jo 

ITf 

Jo Jo Jo 



r 2 cos 4> d(f> d6 dr 



r 3 cos 2 sin 6 d(f> dd dr 



r 2 cos <f> d<j> dd dr 



r 3 sin cos < d(j> dd dr 



7T 7T 

m 



r 2 cos < d(l> dd dr 



Since the mass of a homogeneous sphere with radius a and density 
unity is fira 3 , each of the denominators of these expressions equals 
^wa 3 . This can be verified at once by integration. On integrating 
the numerators with respect to $ and substituting the limits, the 
equations become 



r 3 sin 6 d6 dr 



IT , 

6 



7T , 

r 



28 PROBLEMS. 

On integrating with respect to 6, these equations give 

7T /* 7T /" , 7T /" 

^ar r^dr 

^ = 4 Jo . = 4 J g = 4 Jo 

T 3 7T , 7T 3 

6 6 6 

and, finally, the integration with respect to r gives 
* = y = * = fa- 

The octant of a sphere has three planes of symmetry, viz., the 
planes defined by the center of the sphere, the vertices of the 
bounding spherical triangle, and the centers of their respective 
opposite sides. Since these three planes intersect not only in a 
point but also in a line, they do not fully determine the center of 
mass. 

As nearly all the masses occurring in astronomical problems are 
spheres or oblate spheroids with three planes of symmetry which 
intersect in a point but not in a line, the applications of the for- 
mulas just given are extremely simple, and no more examples 
need be solved. 



H. PROBLEMS. 

1. Find the center of mass of a fine straight wire of length R whose density 
varies directly as the nth power of the distance from one end. 

Ans. -. _ R from the end of zero density. 
n + 2 

2. Find the coordinates of the center of mass of a fine wire of uniform 
density bent into a quadrant of a circle of radius R. 

2R 

Ans. x = y = f 

where the origin is at the center of the circle. 

3. Find the coordinates of the center of mass of a thin plate of uniform 
density, having the form of a quadrant of an ellipse whose semi-axes are 
a and 6. 

f_ 4a 
5 -r' 
Ans. 1 

- 46 



25] HISTORICAL SKETCH TO NEWTON. 29 

4. Find the coordinates of the center of mass of a thin plate of uniform 
density, having the form of a complete loop of the lemniscate whose equation 
is r 2 = a 2 cos 20. 

Ans. J 2* ' 

I ?/ = 0. 

5. Find the coordinates of the center of mass of an octant of an ellipsoid 
of uniform density whose semi-axes are a, b, c. 

f - 3a 

x = . 



Ans. 



_ 36 



3c 

Z = TT 



6. Find the coordinates of the center of mass of an octant of a sphere of 
radius R whose density varies directly as the nth power of the distance from 
the center. 

_ _ _ n +3 R 
Ans. *- y ___.-. 

7. Find the coordinates of the center of mass of a paraboloid of revolution 
cut off by a plane perpendicular to its axis. 



(x = \h, 
Ans. < _ _ ' 

(y = z = 0, 



where h is the distance from the vertex of the paraboloid to the plane. 

8. Find the coordinates of the center of mass of a right circular cone whose 
height is h and whose radius is R. 

9. Find the coordinates of the center of mass of a double convex lens of 
homogeneous glass whose surfaces are spheres having the radii ri and r 2 = 2r t 

and whose thickness at the center is -^. - . 

4 

10. In a concave-convex lens the radius of curvature of the convex and con- 
cave surfaces are n and r 2 > n. Determine the thickness and diameter of the 
lens so that the center of mass shall be in the concave surface. 

HISTORICAL SKETCH FROM ANCIENT TIMES TO NEWTON. 
25. The Two Divisions of the History. The history of the 
development of Celestial Mechanics is naturally divided into two 
distinct parts. The one is concerned with the progress of knowl- 
edge about the purely formal aspect of the universe, the natural 
divisions of time, the configurations of the constellations, and 
the determination of the paths and periods of the planets in their 



30 HISTORICAL SKETCH. [26 

motions; the other treats of the efforts at, and the success in, attain- 
ing correct ideas regarding the physical aspects of natural phe- 
nomena, the fundamental properties of force, matter, space, and 
time, and, in particular, the way in which they are related. It is 
true that these two lines in the development of astrcnomical 
science have not always been kept distinct by those who have 
cultivated them; on the contrary, they have often been so 
intimately associated that the speculations in the latter have 
influenced unduly the conclusions in the former. While it is 
clear that the two kinds of investigation should.be kept distinct 
in the mind of the investigator, it is equally clear that they should 
be constantly employed as checks upon each other. The object 
of the next two articles will be to trace, in as few words as possible, 
the development of these two lines of progress of the science of 
Celestial Mechanics from the times of the early Greek Philosophers 
to the time when Newton applied his transcendent genius to the 
analysis of the elements involved, and to their synthesis into one 
of the sublimest products of the human mind. 

26. Formal Astronomy. The first division is concerned with 
phenomena altogether apart from their causes, and will be termed 
Formal Astronomy. The day, the month, and the year are such 
obvious natural divisions of time that they must have been 
noticed by the most primitive peoples. But the determination of 
the relations among these periods required something of the sci- 
entific spirit necessary for careful observations; yet, in the very 
dawn of Chaldean and Egyptian history they appear to have been 
known with a considerable degree of accuracy. The records left 
by these peoples of their earlier civilizations are so meager that 
little is known with certainty regarding their scientific achieve- 
ments. The authentic history of Astronomy actually begins with 
the Greeks, who, deriving their first knowledge and inspiration 
from the Egyptians, pursued the subject with the enthusiasm 
and acuteness which was characteristic of the Greek race. 

Thales (640-546 B.C.), of Miletus, went to Egypt for instruc- 
tion, and on his return founded the Ionian School of Astronomy 
and Philosophy. Some idea of the advancement made by the 
Egyptians can be gathered from the fact that he taught the 
sphericity of the earth, the obliquity of the ecliptic, the causes of 
eclipses, and, according to Herodotus, predicted the eclipse of the 
sun of 585 B.C. According to Laertius he was the first to deter- 
mine the length of the year. It is fair to assume that he borrowed 



26] FROM ANCIENT TIMES TO NEWTON. 31 

much of his information from Egypt, though the basis for pre- 
dicting eclipses rests on the period of 6585 days, known as the 
saros, discovered by the Chaldaeans. After the lapse of this 
period eclipses of the sun and moon recur under almost identical 
circumstances except that they are displaced about 120 westward 
on the earth. 

Anaximander (611-545 B.C.), a friend and probably a pupil of 
Thales, constructed geographical maps, and is credited with 
having invented the gnomon. 

Pythagoras (about 569-470 B.C.) travelled widely in Egypt and 
Chaldea, penetrating Asia even to the banks of the Ganges. On 
his return he went to Sicily and founded a School of Astronomy 
and Philosophy. He taught that the earth both rotates and 
revolves, and that the comets as well as the planets move in orbits 
around the sun. He is credited with being the first to maintain 
that the same planet, Venus, is both evening and morning star at 
different times. 

Meton (about 465-385 B.C.) brought to the notice of the 
learned men of Hellas the cycle of 19 years, nearly equalling 235 
synodic months, which has since been known as the Metonic cycle. 
After the lapse of this period the phases of the moon recur on the 
same days of the year, and almost at the same time of day. The 
still more accurate Callipic cycle consists of four Metonic cycles, 
less one day. 

Aristotle (384-322 B.C.) maintained the theory of the globular 
form of the earth and supported it with many of the arguments 
which are used at the present time. 

Aristarchus (310-250 B.C.) wrote an important work entitled 
Magnitudes and Distances. In it he calculated from the time at 
which the earth is in quadrature as seen from the moon that the 
latter is about one-nineteenth as distant from the earth as the sun. 
The time in question is determined by observing when the moon 
is at the first quarter. The practical difficulty of determining 
exactly when the moon has any particular phase is the only thing 
that prevents the method, which is theoretically sound, from 
being entirely successful. 

Eratosthenes (275-194 B.C.) made a catalogue of 475 of the 
brightest stars, and is famous for having determined the size of 
the earth from the measurement of the difference in latitude and 
the distance apart of Syene, in Upper Egypt, and Alexandria. 

Hipparchus (about 190-120 B.C.), a native of Bithynia, who 



32 HISTORICAL SKETCH. [26 

observed at Rhodes and possibly at Alexandria, was the greatest 
astronomer of antiquity. He added to zeal and skill as an ob- 
server the accomplishments of the mathematician. Following 
Euclid (about 330-275 B.C.) at Alexandria, he developed the 
important science of Spherical Trigonometry. He located places 
on the earth by their Longitudes and Latitudes, and the stars by 
their Right Ascensions and Declinations. He was led by the 
appearance of a temporary star to make a catalogue of 1080 fixed 
stars. He measured the length of the tropical year, the length 
of the month from eclipses, the motion of the moon's nodes and 
that of her apogee; he was the author of the first solar tables; he 
discovered the precession of the equinoxes, and made extensive 
observations of the planets. The works of Hipparchus are known 
only indirectly, his own writings having been lost at the time of 
the destruction of the great Alexandrian library by the Saracens 
under Omar, in 640 A.D. 

Ptolemy (100-170 A.D.) carried forward the work of Hipparchus 
faithfully and left the Almagest as the monument of his labors. 
Fortunately it has come down to modern times intact and contains 
much information of great value. Ptolemy's greatest discovery 
is the evection of the moon's motion, which he detected by fol- 
lowing the moon during the whole month, instead of confining his 
attention to certain phases as previous observers had done. He 
discovered refraction, but is particularly famous for the system of 
eccentrics and epicycles which he developed to explain the apparent 
motions of the planets. 

A stationary period followed Ptolemy during which science was 
not cultivated by any people except the Arabs, who were imitators 
and commentators rather more than original investigators. In 
the Ninth Century the greatest Arabian astronomer, Albategnius 
(850-929), flourished, and a more accurate measurement of the 
arc of a meridian than had before that time been executed was 
carried out by him in the plain of Singiar, in Mesopotamia. In 
the Tenth Century Al-Sufi made a catalogue of stars based on his 
own observations. Another catalogue was made by the direction 
of Ulugh Beigh (1393-1449), at Samarkand, in 1433. At this 
time Arabian astronomy practically ceased to exist. 

Astronomy began to revive in Europe toward the end of the 
Fifteenth Century in the labors of Peurbach (1423-1461), 
Waltherus (1430-1504), and Regiomontanus (1436-1476). It 
was given a great impetus by the celebrated German astronomer 



27] FROM ANCIENT TIMES TO NEWTON. 33 

Copernicus (1473-1543), and has been pursued with increasing 
zeal to the present time. Copernicus published his masterpiece, 
De Revolutionibus Orbium Coelestium, in 1543, in which he gave to 
the world the heliocentric theory of the solar system. The 
system of Copernicus was rejected by Tycho Brahe (1546-1601), 
who advanced a theory of his own, because he could not observe 
any parallax in the fixed stars. Tycho was of Norwegian birth, 
but did much of his astronomical work in Denmark under the 
patronage of King Frederick. After the death of Frederick he 
moved to Prague where he- was supported the remainder of his 
life by a liberal pension from Rudolph II. He was an indefatigable 
and most painstaking observer, and made very important contri- 
butions to Astronomy. In his later years Tycho Brahe had 
Kepler (1571-1630) for his disciple and assistant, and it was by 
discussing the observations of Tycho Brahe that Kepler was en- 
abled, in less than twenty years after the death of his preceptor, 
to announce the three laws of -planetary motion. It was from 
these laws that Newton (1642-1727) derived the law of gravitation. 
Galileo (1564-1642), an Italian astronomer, a contemporary of 
Kepler, and a man of greater genius and greater fame, first applied 
the telescope to celestial objects. He discovered four satellites 
revolving around Jupiter, the rings of Saturn, and spots on the 
sun. He, like Kepler, was an ardent supporter of the heliocentric 
theory. 

27. Dynamical Astronomy. By Dynamical Astronomy will be 
meant the connecting of mechanical and physical causes with 
observed phenomena. Formal Astronomy is so ancient that it is 
not possible to go back to its origin; Dynamical Astronomy, on 
the other hand, did not begin until after the time of Aristotle, and 
then real advances were made at only very rare intervals. 

Archimedes (287-212 B.C.), of Syracuse, is the author of the 
first sound ideas regarding mechanical laws. He stated correctly 
the principles of the lever and the meaning of the center of gravity 
of a body. The form and generality of his treatment were im- 
proved by Leonardo da Vinci (1452-1519) in his investigations 
of statical moments. The whole subject of Statics of a rigid body 
involves only the application of the proper mathematics to these 
principles. 

It is a remarkable fact that no single important advance was 
made in the discovery of mechanical laws for nearly two thousand 
years after Archimedes, or until the time of Stevinus (1548-1620), 
4 



34 HISTORICAL SKETCH TO NEWTON. [27 

who was the first, in 1586, to investigate the mechanics of the 
inclined plane, and of Galileo (1564-1642), who made the first 
important advance in Kinetics. Thus, the mechanical principles 
involved in the motions of bodies were not discovered until rela-. 
tively modern times. The fundamental error in the speculations 
of most of the investigators was that they supposed that it required 
a continually acting force to keep a body in motion. They thought 
it was natural for a body to have a position rather than a state of 
motion. This is the opposite of the law of inertia (Newton's first 
law). This law was discovered by Galileo quite incidentally in 
the study of the motion of bodies sliding down an inclined plane 
and out on a horizontal surface. Galileo took as his fundamental 
principle that the change of velocity, or acceleration, is deter- 
mined by the forces which act upon the body. This contains 
nearly all of Newton's first two laws. Galileo applied his principles 
with complete success to the discovery of the laws of falling bodies, 
and of the motion of projectiles. The value of his discoveries is 
such that he is justly considered to be the founder of Dynamics. 
He was the first to employ the pendulum for the measurement of 
time. 

Huyghens (1629-1695), a Dutch mathematician and scientist, 
published his Horologium Osdllatorium in 1675, containing the 
theory of the determination of the intensity of the earth's gravity 
from pendulum experiments, the theory of the center of oscil- 
lation, the theory of evolutes, and the theory of the cycloidal 
pendulum. 

Newton (1642-1727) completed the formulation of the funda- 
mental principles of Mechanics, and applied them with unparalleled 
success in the solution of mechanical and astronomical problems. 
He employed Geometry with such skill that his work has scarcely 
been added to by the use of his methods to the present day. 

After Newton's time, mathematicians soon turned to the more 
general and powerful methods of analysis. The subject of An- 
alytical Mechanics was founded by Euler (1707-1783) in his work, 
Mechanica sive Motus Scientia (Petersburg, 1736) ; it was improved 
by Maclaurin (1698-1746) in his work, A Complete System of 
Fluxions (Edinburgh, 1742), and was highly perfected by Lagrange 
(1736-1813) in his Mecanique Analytique (Paris, 1788). The 
Mecanique Celeste of Laplace (1749-1827) put Celestial Mechanics 
on a correspondingly high plane. 



BIBLIOGRAPHY. 35 



BIBLIOGRAPHY. 

For the fundamental principles of Mechanics consult Principien der 
Mechanik (a history and exposition of the various systems of Mechanics 
from Archimedes to the present time), by Dr. E. Diihring; Vorreden und 
Einleitungen der klassischen Werken der Mechanik, edited by the Phil. Soc. of 
the Univ. of Vienna; Die Principien der Mechanik, by Heinrich Hertz, Coll. 
Works, vol. in; The Science of Mechanics, by E. Mach, translated by T. J. 
McCormack; Principe der Mechanik, by Boltzmann; Newton's Laws of Motion, 
by P. G. Tait; Das Princip der Erhaltung der Energie, by Planck; Die geschicht- 
liche Entwickelung des Bewegungsbegriffes, by Lange. 

For the theory of Relativity consult Das Relativitdtsprincip, by M. Laue, 
and The Theory of Relativity, by R. D. Carmichael. 

For velocity and acceleration and their resolution and composition consult 
the first parts of Dynamics of a Particle, by Tait and Steele; Legons de Cine- 
matique, by G. Koenigs; Cinematique et Mecanismes, by Poincare"; and the 
works on Dynamics (Mechanics) by Routh, Love, Budde, and Appell. 

For the history of Celestial Mechanics and Astronomy consult Histoire de 
V Astronomic Ancienne (old work), by Delambre; Astronomische Beobachtungen 
der Alien (old work), by L. Ideler; Recherches sur V Histoire de V Astronomic 
Ancienne, by Paul Tannery; History of Astronomy, by Grant; Geschichte der 
Mathematik im Alterthum und Mittelalter, by H. Hankel; History of the Induc- 
tive Sciences (2 vols.), by Whewell; Geschichte der- Mathematischen Wissen- 
schaften (2 vols.), by H. Siiter; Geschichte der Mathematik (3 vols.), by M. 
Cantor; A Short History of Mathematics, by W. W. R. Ball; A History of 
Mathematics, by Florian Cajori; Histoire des Sciences Mathematiques et 
Physiques (12 vols.), by M. Maximilian Marie; Geschichte der Astronomic, by 
R. Wolf; A History of Astronomy, by Arthur Berry; Histoire Abregee de V Astron- 
omic, by Ernest Lebon. 



CHAPTER II. 

RECTILINEAR MOTION. 

28. A great part of the work in Celestial Mechanics consists of 
the solution of differential equations which, in most problems, are 
very complicated on account of the number of dependent variables 
involved. The ordinary Calculus is devoted, in a large part, to 
the treatment of equations in which there is but one independent 
variable and one dependent variable ; and the step to simultaneous 
equations in several variables, requiring interpretation in con- 
nection with physical problems and mechanical principles, is 
one which is usually made not without some difficulty. The 
present chapter will be devoted to the formulation and to the 
solution of certain classes of problems in which the mathematical 
processes are closely related to those which are expounded in the 
mathematical text-books. It will form the bridge leading from 
the methods which are familiar in works on Calculus and ele- 
mentary Differential Equations to those which are characteristic 
of mechanical and astronomical problems. 

The examples chosen to illustrate the principles are taken 
largely from astronomical problems. They are of sufficient 
interest to justify their insertion, even though they were not 
needed as a preparation for the more complicated discussions which 
will follow. They embrace the theory of falling bodies, the velocity 
of escape, parabolic motion, and the meteoric and contraction 
theories of the sun's heat. 

THE MOTION OF FALLING PARTICLES. 

29. The Differential Equation of Motion. Suppose the mass 
of the particle is m and let s represent the line in which it falls. 
Take the origin at the surface of the earth and let the positive 
end of the line be directed upward. By the second law of motion 
the rate of change of momentum, or the product of the mass and 
the acceleration, is proportional to the force. Let k 2 represent the 
factor of proportionality, the numerical value of which will depend 

36 



30] CASE OF CONSTANT FORCE. 37 



upon the units employed. Then, if / represents the force, the dif- 
ferential equation of motion is 

(1) m g = - ty. 

This is also the differential equation of motion for any case in 
which the resultant of all the forces is constantly in the same 
straight line and in which the body is not initially projected from 
that line. A more general treatment will therefore be given than 
would be required if / were simply the force arising from the 
earth's attraction for the particle m. 

The force / will depend in general upon various things, such as 
the position of m, the time t, and the velocity v. This may be 
indicated by writing equation (1) in the form 

(2) m^ = -Vf(8,t,v), 

in which / (s, t, v) simply means that the force may depend upon 
the quantities contained in the parenthesis. In order to solve 
this equation two integrations must be performed, and the char- 
acter of the integrals will depend upon the manner in which / 
depends upon s, t, and v. It is necessary to discuss the different 
cases separately. 

30. Case of Constant Force. This simplest case is nearly 
realized when particles fall through small distances near the earth's 
surface under the influence of gravity. If the second is taken 
as the unit of time and the foot as the unit of length then k 2 f = mg, 
where g is the acceleration of gravity at the surface of the earth. 
Its numerical value, which varies somewhat with the latitude, 
is a little greater than 32. Then equation (1) becomes 



This equation gives after one integration 

ds 

f --** + * 

where c\ is the constant of integration. Let the velocity of the 
particle at the time t = be v = VQ. Then the last equation 
becomes at t = 

v = ci; 



38 ATTRACTIVE FORCE VARYING AS DISTANCE. [31 

whence 

ds 

-dt = ~ gt + VQ ' 

The integral of this equation is 

2 

v Q t + c 2 . 



Suppose the particle is started at the distance SQ from the origin 
at the time t = 0; then this equation gives 

SQ = c 2 ; 
whence 

*2 

(4) 

When the initial conditions are given this equation determines the 
position of the particle at any time; or, it determines the time at 
which the body has any position by the solution of the quadratic 
equation in t. 

If the acceleration were any other positive or negative constant 
than mg, the equation for the space described would differ from 
(4) only in the coefficient of P. 

It is also possible to obtain an important relation between the 
speed and the position of the particle. Multiply both members 

ds f ds \ 2 

of equation (3) by 2 -=- . Then, since the derivative of ( -r ) is 

2 rfi ^7/2 ' ^ e m ^ e & ra l f ^ ne equation is 



1 = ~ 2gs + 

It follows from the initial conditions that 

c 3 = v Q 2 -f 
whence 

/CN /ds\ 2 

(5) (dt) 

31. Attractive Force Varying Directly as the Distance. Another 
simple case is that in which the force varies directly as the distance 
from the origin. Suppose it is always attractive toward the 
origin. This has been found by experiment to be very nearly the 
law of tension of an elastic string within certain limits of stretching. 
Then the velocity will decrease when the particle is on the positive 
side of the origin; therefore for these positions of the particle the 



31] ATTRACTIVE FORCE VARYING AS DISTANCE. 39 

acceleration must be taken with the negative sign, and the differ- 
ential equation for positive values of s is 

/7 2 Q 

(6) ^=-^ 

For positions of the particle in the negative direction from the 
origin the velocity increases with the time, and therefore the 
acceleration is positive. The right member of equation (6) must 
be taken with such a sign that it will be positive. Since s is 
negative in the region under consideration the negative sign must 
be prefixed, and the equation remains as before. Equation (6) is, 
therefore, the general differential equation of motion for a body 
subject to an attractive force varying directly as the distance. 

ds 
Multiply both members of equation (6) by 2 JT and integrate; 

the result is 



ds 

If s = SQ and -- = 0, at the time t = 0, then this equation 
at 

becomes 



which may be written, as is customary in separating the variables, 

ds kdt 



M - 
The integral of this equation is 

. s kt 

sin" 1 = = + 02. 
so \'m 

It is found from the initial conditions that c 2 = ; whence 

. , s kt . TT 

sin" 1 = + - . 

On taking the sine of both members, this equation becomes 

(7) 

From this equation it is seen that the motion is oscillatory and 
symmetrical with respect to the origin, with a period of ^ m . 

rC 



40 PROBLEMS. 

For this reason it is called the equation for harmonic motion. 

ds 
Obviously -r vanishes at some time during the motion for all 

initial conditions, and there was no real restriction of the gener- 
ality of the problem in supposing that it was zero at t = 0. 

Equation (6) is in form the differential equation for many physi- 
cal problems. When the initial conditions are assigned, it defines 
the motion of the simple pendulum, the oscillations of the tuning 
fork and most musical instruments, the vibrations of a radiating 
body, the small variations in the position of the earth's axis, etc. 
For this reason the method of finding its solution and the deter- 
mination of the constants of integration should be thoroughly 
mastered. 

m. PROBLEMS. 

1. A particle is started with an initial velocity of 20 meters per sec. and 
is subject to an acceleration of 20 meters per sec. What will be its velocity 
at the end of 4 sees., and how far will it have moved? 

= 100 meters per sec. 



r 

u 



Ans. 

240 meters. 

2. A particle starting with an initial velocity of 10 meters per sec. and 
moving with a constant acceleration describes 2050 meters hi 5 sees. What 
is the acceleration? 

Ans. a. 160 meters per sec. 

3. A particle is moving with an acceleration of 5 meters per sec. Through 
what space must it move in order that its velocity shall be increased from 
10 meters per sec. to 20 meters per sec.? 

Ans. 30 meters. 

4. A particle starting with a positive initial velocity of 10 meters per sec. 
and moving under a positive acceleration of 20 meters per sec. described a 
space of 420 meters. What time was required? 

Ans. t = 6 sees. 

5. Show that, if a particle starting from rest moves subject to an attractive 
force varying directly as the initial distance, the time of falling from any 
point to the origin is independent of the distance of the particle. 

6. Suppose a particle moves subject to an attractive force varying directly 
as the distance, and that the acceleration at a distance of 1 meter is 1 meter 
a sec. If the particle starts from rest how long will it take it to fall from a 
distance of 20 meters to 10 meters? 

Ans. 1.0472 sees. 

7. Suppose a particle moves subject to a force which is repulsive from 
the origin and which varies directly as the distance; show that if v = and 
s So when t = 0, then 



32] SOLUTION OF LINEAR EQUATIONS BY EXPONENTIALS. 41 



+ Vs 2 - so 2 \ K 



k 

whence, letting -=. = K, 
Vra 

s =~ (e Kt + e-**) = so cosh KL 
Observe that equation (7) may be written in the similar form 

s = ~ ( e */-LK _|- e -*/-i*<) = So C08 KI 

8. The surface gravity of the sun is 28 times that of the earth. If a solar 
prominence 100,000 miles high was produced by an explosion, what must have 
been the velocity of the material when it left the surface of the sun? 

Ans. 184 miles per sec. 

32. Solution of Linear Equations by Exponentials. The differ- 
ential equation (6) and the corresponding one for a repulsive force 
are linear in s with constant coefficients. There is a general 
theory which shows that all linear equations having constant 
coefficients can be solved in terms of exponentials; or, in certain 
special cases, in terms of exponentials multiplied by powers of the 
independent variable t. This method has not only the advantage 
of generality, but is also very simple, and it will be illustrated by 
solving (5). Consider the two forms 



Assume s = e and substitute in the differential equations; 
whence 

XV + fcV = 0, 

XV - fcV = 0. 

Since these equations must be satisfied by all values of t in order 
that e x< shall be a solution, it follows that 



(9) 

I X 2 - /c 2 = 0. 

Let the roots of the first equation be Xi and X 2 ; then the first 
equation of (8) is verified by both of the particular solutions 
e Al * and e Xs '. The general solution is the sum of these two particu- 
lar solutions, each multiplied by an arbitrary constant. Precisely 



42 SOLUTION OF LINEAR EQUATIONS BY EXPONENTIALS. [32 

similar results hold for the second equation of (8). On putting 
in the value of the roots, the respective general solutions are 






is = Cl e^ kt + c 2 e-^ kt , 
\ s = Cl 'e kt + c 2 'e- kt , 



where Ci, C 2 , c/, and c 2 ' are the constants of integration. 



ds 



s 
Suppose s = o, and -^ = so' when t '. = 0; therefore 



I Sn = 



= Ci -\- C 2 , 



The derivatives of (10) are 



On substituting f = and -r- = SQ, it follows that 



V^Hfc-CzV- lfc = o', 



Therefore 



fciV-lJfc 

I Ci'k c 2 ' 

H( 



so' \ 



Then the general solutions become 



(ID 



Or these expressions can be written in the form 



33] FORCE VARYING INVERSELY AS SQUARE OF DISTANCE. 43 

s = SQ cos kt + -|- sin kt, 

s = SQ cosh kt + y- sinh Atf . 

This method of treatment shows the close relation between the two 
problems much more clearly than the other methods of obtaining 
the solutions. 

33. Attractive Force Varying Inversely as the Square of the 
Distance. For positions in the positive direction from the origin 
the velocity decreases algebraically as the time increases whether 
the motion is toward or from the origin; therefore in this region 
the acceleration is negative. Similarly, on the negative side of 

k 2 
the origin the acceleration is positive. Since -r is always positive 

s 

the right member has different signs in the two cases. For 
simplicity suppose the mass of the attracted particle is unity. 
Then the differential equation of motion for all positions of the 
particle in the positive direction from the origin is 



ds 
On multiplying both members of this equation by 2 -r and inte- 

grating, it is found that 



/ds\ 2 2k 2 

(dt) = T 



Suppose v = V Q and s = s when t = 0; then 

2k 2 

C\ = V 2 -- . 
SQ 

On substituting this expression for Ci in (13), it is found that 



ds j2/<^ 2 2k 2 
= -v/ ^ v 2 . 

dt * S SQ 

2k 2 ds 

If v Q 2 - '- < there will be some finite distance Si at which - 
s dt 

will vanish; if the direction of motion of the particle is such that 
it reaches that point it will turn there and move in the opposite 

2k 2 ds 

direction. If v Q 2 = 0, 37 will vanish at s = co and if the 

s dt 

particle moves out from the origin toward infinity its distance will 



44 FORCE VARYING INVERSELY AS SQUARE OF DISTANCE. [33 

become indefinitely great as the velocity approaches zero. If 

2k 2 ds 

v Q 2 > 0, 37 never vanishes, and if the particle moves out 

So dt 

from the origin toward infinity its distance will become indefi- 
nitely great and its velocity will not approach zero. 

2k 2 ds 

Suppose v 2 < and that 37 = when s = si. Then 

SQ dt 

equation (13) gives 

^ p | 

Si S 



The positive or negative sign is to be taken according as the 
particle is receding from, or approaching toward, the origin. 
This equation can be written in the form 

sds /o" 

-kdt, 



VSiS - S 2 \Si 

and the integral is therefore 



Since s = s when t = 0, it follows that 

: . Si . . /2 - *1\ 

C2 = \SiSo SQ ~h TT sin L I 

2 \ si / 

whence 

4 

4 

(15) ' 



s kt ' 

This equation determines the time at which the particle has any 
position at the right of the origin whose distance from it is less 
than si. For values of s greater than Si, and for all negative values 
of s, the second term becomes imaginary. That means that the 
equation does not hold for these values of the variables; this was 
indeed certain because the differential equations (13) and (14) 
were valid only for 

< s ^ si. 

Suppose the particle is approaching the origin; then the negative 
sign must be used in the right member of (15). The time at 
which the particle was at rest is obtained by putting s = si in 
(15), and is 



35] THE VELOCITY FROM INFINITY. 45 



The time required for the particle to fall from s to the origin 
is obtained by putting s = in (15), and is 



1 s / - o 1/siVF TT 

r 2 = --^-v,,.,-.^--^ [""a * 

The time required for the particle to fall from rest at s = si to 
the origin is 



34. The Height of Projection. When the constant Ci has been 
determined by the initial conditions, equation (13) becomes 



It follows from this equation that the speed depends only on the 
distance of the particle from the center of force and not on the 
direction of its motion. The greatest distance to which the particle 
recedes from the origin, or the height of projection from the origin, 
is obtained by putting v = 0, which gives 

1 -- 1 ._! 

Si s 2k 2 ' 

But if the height of projection is measured from the point of 
projection s , as would be natural in considering the projection of 
a body away from the surface of the earth, the formula becomes 



$2 = Si So = 



2/c 2 - 



35. The Velocity from Infinity. When the particle starts 
from s with zero velocity, equation (13) becomes 



If the particle falls from an infinite distance, So is infinite and the 
equation reduces to 



From the investigations of Art. 34 it follows that, if the par- 
ticle is projected from any point s in the positive direction with 



46 THE ESCAPE OF ATMOSPHERES. [36 

the velocity defined by (18), it will recede to infinity. The law 
of attraction in deriving (18) is Newton's law of gravitation; 
therefore this equation can be used for the computation of the 
velocity which a particle starting from infinity would acquire in 
falling to the surfaces of the various planets, satellites, and the 
sun. Then, if the particle were projected from the surfaces of 
the various members of the solar system with these respective 
velocities, it would recede to an infinite distance if it were not 
acted on by other forces. But if its velocity were only enough 
to take it away from a satellite or a planet, it would still be subject 
to the attraction of the remaining members of the solar system^ 
chief of which is of course the sun, and it would not in general 
recede to infinity and be entirely lost to the system. 

36. Application to the Escape of Atmospheres. The kinetic 
theory of gases is that the volumes and pressures of gases are 
maintained by the mutual impacts of the individual molecules, 
which are, on the average, in a state of very rapid motion. The 
rarity of the earth's atmosphere and the fact that the pressure is 
about fifteen pounds to the square inch, serve to give some idea 
of the high speed with which the molecules move and of the great 
frequency of their impacts. The different molecules do not all 
move with the same speed in any given gas under fixed conditions ; 
but the number of those which have a rate of motion different from 
the mean of the squares becomes less very rapidly as the differ- 
ences increase. Theoretically, in all gases the range of the values 
of the velocities is from zero to infinity, although the extreme 
cases occur at infinitely rare intervals compared to the others. 
Under constant pressure the velocities are directly proportional 
to the square root of the absolute temperature, and inversely pro- 
portional to the square root of the molecular weight. 

Since in all gases all velocities exist, some of the molecules of 
the gaseous envelopes of the heavenly bodies must be moving 
with velocities greater than the velocity from infinity, as defined in 
Art. 35. If the molecules are near the upper limits of the atmos- 
phere, and start away from the body to which they belong, they 
may miss collisions with other molecules and escape never to 
return*. Since the kinetic theory of gases is supported by very 
strong observational evidence, and since if it is true it is probable 
that some molecules move with velocities greater than the velocity 

* This theory is due to Johnstone Stoney, Trans. Royal Dublin Soc., 1892. 



36J THE ESCAPE OF ATMOSPHERES. 47 

from infinity, it is probable that the atmospheres of all celestial 
bodies are being depleted by this process; but in most cases it is 
an excessively slow one, and is compensated, to some extent at 
least, by the accretion of meteoric matter and atmospheric mole- 
cules from other bodies. In the upper regions of the gaseous 
envelopes, from which alone the molecules escape, the temperatures 
are low, at least for planetary bodies like the earth, and high 
velocities are of rare occurrence. If the mean square velocity 
were as great as, or exceeded, the velocity from infinity the deple- 
tion would be a relatively rapid process. In any case the elements 
and compounds with low molecular weights would be lost most 
.rapidly, and thus certain ones might freely escape and others be 
largely retained. 

The manner in which the velocity from infinity with respect 
to the surface of an attracting sphere varies with its mass and 
radius will now be investigated. The mass of a body is propor- 
tional to the product of its density and cube of its radius. Then 
k 2 , which is the attraction at unit distance, varies directly as the 
mass, and therefore . directly as the product of the density and 
the cube of the radius. Hence it follows from (18) that the 
velocity from infinity at the surface of a body varies as the product 
of its radius and the square root of its density. The densities 
and the radii of the members of the solar system are usually ex- 
pressed in terms of the density and the radius of the earth; hence 
the velocity from infinity can be easily computed for each of them 
after it has been determined for the earth. 

Let R represent the radius of the earth, and g the acceleration 
of gravitation at its surface. Then it follows that 

(19) ,-|. 

On substituting the value of k 2 determined from this equation 
into (18), the expression for the square of the velocity becomes 



(dt) s 

ds 
Let V = -37 when s = R\ whence 

7 2 = 2flffl. 
Let a second be taken as the unit of time, and a meter as the unit 



48 



THE ESCAPE OF ATMOSPHERES. 



[36 



of length. Then R = 6,371,000, and g = 9.8094*. On substi- 
tuting in the last equation and carrying out the computation, it 
is found that V = 11,180 meters, or about 6.95 miles, per sec. 
On taking the values of the relative densities and radii of the 
planets as given in Chapter XI of Moulton's Introduction to 
Astronomy, the following results are found : 



Body 


Velocity of Escape 


Earth 


11,180 meters, or 6.95 miles, per sec. 


Moon 


2,396 ' 




1.49 ' 


(i 


Sun 


618,200 ' 




384.30 ' 


U (( 


Mercury 


3,196 ' 




1.99 ' 


U (( 


Venus 


10,475 ' 




6.51 * 


(( 11 


Mars 


5,180 ' 




3.22 ' 


l( 11 


Jupiter 


61,120 ' 




38.04 ' 


It 


Saturn 


37,850 ' 




23.53 ' 


1C 11 


Uranus 


23,160 ' 


14.40 ' 


11 u 


Neptune 


20,830 " 


12.95 ' " " 



The velocity from infinity decreases as the distance from the 
center of a planet increases, and the necessary velocity of pro- 
jection in order that a molecule may escape decreases corre- 
spondingly; and the centrifugal acceleration of the planet's rotation 
adds to the velocities of certain molecules. 

The question arises whether, under the conditions prevailing 
at the surfaces of the various bodies enumerated, the average 
molecular velocities of the atmospheric elements do not equal or 
surpass the corresponding velocity from infinity. 

It is possible to find experimentally the pressure exerted by a 
gas having a given density and temperature upon a unit surface, 
from which the mean square velocity can be computed. It is 
shown in the kinetic theory of gases that the square root of the 
mean square velocity of hydrogen molecules at the temperature 
Centigrade under atmospheric pressure is about 1,700 meters per 
second. Under the same conditions the velocities of oxygen and 
nitrogen molecules are about one-fourth as great. 

The surface temperatures of the inferior planets are certainly 
much greater than zero degrees Centigrade in the parts where 
they receive the rays of the sun most directly, even if all the heat 
which may ever have been received from their interiors is neglected. 
It seems probable from the geological evidences of igneous action 

* Annuaire du Bureau des Long, g is given for the lat. of Paris, 48 50'. 



37] FORCE PROPORTIONAL TO THE VELOCITY. 49 

upon the earth that in the remote past they were at a much higher 
temperature, and the superior planets have not yet cooled down 
to the solid state. There is evidence that the most refractory 
substances have been in a molten state at some time, which implies 
that they have had a temperature of 3000 or 4000 degrees Centi- 
grade. Therefore the square root of the mean square velocity 
may have been much greater than the approximate mile a second 
for hydrogen given above, and it probably continued much greater 
for a long period of time. On comparing these results with the 
table of velocities from infinity, it is seen that the moon and 
inferior planets, according to this theory, could not possibly have 
retained free hydrogen and other elements of very small molecular 
weights, such as helium, in their envelopes; in the case of the 
moon, Mercury, and Mars, the escape of heavier molecules as 
nitrogen and oxygen must have been frequent. This is especially 
probable if the heated atmospheres extended out to great dis- 
tances. The superior planets, and especially the sun, could have 
retained all of the familiar terrestrial elements, and from this theory 
it should be expected that these bodies would be surrounded with 
extensive gaseous envelopes. 

The observed facts are that the moon has no appreciable 
atmosphere whatever; Mercury an extremely rare one, if any at 
all; Mars, one much less dense than that of the earth; Venus, one 
perhaps somewhat more dense than that of the earth; on the 
other hand the superior planets are surrounded by extensive 
gaseous envelopes. 

37. The Force Proportional to the Velocity. When a particle 
moves in a resisting medium the forces to which it is subject 
depend upon its velocity. Experiments have shown that in the 
earth's atmosphere when the velocity is less than 3 meters per 
second the resistance varies nearly as the first power of the velocity; 
for velocities from 3 to 300 meters per second it varies nearly as 
the second power of the velocity; and for velocities about 400 
meters per second, nearly as the third power of the velocity. 

(a) Consider first the case where the resistance varies as the 
first power of the velocity, and suppose the motion is on the 
earth's surface in a horizontal direction with no force acting except 
that arising from the resistance. Then the differential equation 
of motion is 

(20) +*- - 



50 FORCE PROPORTIONAL TO THE VELOCITY. [37 

where k 2 is a positive constant which depends upon the units 
employed, the nature of the body, and the character of the resisting 
medium. Equation (20) is linear in the dependent variable s, and 
the general method of solving linear equations can be applied. 
Assume the particular solution 

s = e". 
Substitute in (20) and divide by e xt ; then 

X 2 + k 2 \ = 0. 
The roots of this equation are 

Xi = 0, 

X 2 = - & 2 , 
and the general solution is 






ds 
Suppose -j- = v and s = s when t 0. Then the constants 

Ci and 02 can be determined in terms of VQ and s , and the solution 
becomes 

'(22) ' = .+^-^>. . 

(&) Consider the case where the resistance varies as the first 
power of the velocity and suppose the motion is in the vertical line. 
Take the positive end of the axis upward. When the motion is 
upward the velocity is positive and the resistance diminishes the 
velocity. Therefore when the motion is upward the resistance 
produces a negative acceleration, and the differential equation of 
motion is 



When the motion is downward the resistance algebraically in- 
creases the velocity; therefore in this case the resistance produces 
a positive acceleration. But since the velocity is opposite in 
sign in the two cases, equation (23) holds whether the particle is 
ascending or descending. 

Equation (23) is linear, but not homogeneous, and it can easily 
be solved by the method known as the Variation of Parameters. 



37] FORCE PROPORTIONAL TO THE VELOCITY. 51 

This method is so important in astronomical problems that it 
will be introduced in the present simple connection, though it is 
not at all necessary in order to obtain the solution of (23). In 
order to apply the method consider first the equation 

(<>A\ ^+fc2^_() 

d? H ^ dt ~ U ' 

which is obtained from (23) simply by omitting the term which 
does not involve s. The general solution of this equation is 
the first of (21). The method of the variation of parameters, or 
constants, consists in so determining Ci and c 2 as functions of t 
that the differential equation shall be satisfied when the right 
member is included. This imposes only one condition upon the 
two quantities Ci and c 2 , and another can therefore be added. 

If the coefficients Ci and c 2 are regarded as functions of t, it 
is found on differentiating the first of (21) that 



__ 9 
dt dt dt 



As the supplementary condition on Ci and c 2 these quantities will 
be made subject to the relation 



which simplifies the expression for -=- . Then it is found that 



(26) 

and equation (23) gives 

(27) % = - ,. 

It follows from this equation and (25) that 

dci _g_ dct 

dt " W dt 

(28) 



c 2 , 



where c/ and c 2 r are new constants of integration. When these 
values of Ci and c 2 are substituted in (21), it is found that 



(29) s = Cl ' + . 



52 FORCE PROPORTIONAL TO THE VELOCITY. [37 

Since c\ is arbitrary it can of course be supposed to include the 
constant p. 

The expression (29) is the general solution of (23) because it 
contains two arbitrary constants, c/ and c 2 ', and when substituted 
in (23) satisfies it identically in t. It will be observed that the 
part of the solution depending on c\ and c 2 ' has the same form 
as the solution of (20). It is clear that the general solution could 
have been found by the same method if the right member of (23) 
had been a known function of t, instead of the constant g. 

The velocity of the particle is found from (29) to be given by 
the equation 

(30) = ' 



ds 
f Suppose s = o, -57 = ^o at t = 0. On putting these values in 

equations (29) and (30), it is found that 

So = Ci' + Cs'+, 



whence 



, _ v g 

~P""F' 

Consequently, when, the constants are determined by the initial 
conditions, the general solution (29) becomes 



ds 
The particle reaches its highest point when -^ is zero. Let T 

represent the time it reaches this point and S s the height of 
this point; then it is found from equations (31) that 

k z T 1 |_ "*P(| 

+ T' 



38] FORCE PROPORTIONAL TO SQUARE OF VELOCITY. 53 

38. The Force Proportional to the Square of the Velocity. At 

the velocity of a strong wind, or of a body falling any considerable 
distance, or of a ball thrown, the resistance varies very nearly as 
the square of the velocity. An investigation will now be made 
of the character of the motion of a particle when projected upward 
against gravity, and subject to a resistance from the atmosphere 
varying as the square of the velocity. For simplicity in writing, 
the acceleration due to resistance at unit velocity will be taken as 
k z g. Then the differential equation of motion for a unit particle is 

< S- --*(*)' 

This equation may be written in the form 
d 



of which the integral is 

(33) tan-i = - k gt 



ds 
If -j7 = VQ and SQ = when t = 0, then 

at , 

Ci = tan" 1 (kvo). 

On substituting in (33) and taking the tangent of both members, 
it is found that 

, . ds _ 1 v k tan (kgt) 

dt kl + v k tan (kgt) ' 

This equation expresses the velocity in terms of the time. On 
multiplying both numerator and denominator of the right member 
of (34) by cos (kgt)j the numerator becomes the derivative of the 
denominator with respect to the time. Then integrating, the 
final solution becomes 

(35) s = T^- log [vok sin (kgt) + cos (kgt)] -f c 2 . 
K g 

It follows from the initial conditions that c 2 = 0. This equation 
expresses the distance passed over in terms of the time. 

The equations can be so treated that the velocity will be ex- 
pressed in terms of the distance. Equation (32) can be written 



54 FORCE PROPORTIONAL TO SQUARE OF VELOCITY. [38 

5*_ 

of which the integral is 



From the initial conditions it follows that 

ci' = log (1 + &W 
Therefore 

(36) 

The maximum height, which is reached when the velocity becomes 
zero, is found from (36) to be 



The time of reaching the highest point, which is found by putting 

ds 

-JT equal to zero in (34), is given by 

(it 



T = -tan- 1 (vjc). 
kg 

When the particle falls the resistance acts in the opposite 
direction and the sign of the last term in (32) is changed. This 
may be accomplished by writing k V 1 instead of k, and if this 
change is made throughout the solution the results will be valid. 
Of course the results should be written in the exponential form, 
instead of the trigonometric as they were in (34) and (35), in order 
to avoid the appearance of imaginary expressions. If the initial 
velocity is zero, VQ = Q and the equations corresponding to (34) , 
(35), and (36) are repectively 



(37) 



r ds 


1 


e kgt _ 


g-A: * 


dt 

(dsV 
I (dt) 


k 

e kgt . 


e kgt _|_ 
f e -kgt 


g-* ' 
j 


= AT 2(1 


2 



PROBLEMS. 



55 



1. Show that 



IV. PROBLEMS. 



dp - V? 

where the positive square root of s 6 is always taken, holds for the problem of 
Art. 33 whichever side of the origin the particle may be. Integrate this 
equation. 

2. Let s = s' p< in equation (23); integrate directly and show that the 
result is the same as that found by the variation of parameters. 

3. Find equations (37) by direct integration of the differential equations. 

4. Suppose a particle starts from rest and moves subject to a repulsive 
force varying inversely as the square of the distance; find the velocity and 
time elapsed in terms of the space described. 



Ans. 



So 



k\-t =>/s 2 - 



log 



5. What is the velocity from infinity with respect to the sun at the earth's 
distance from the sun? 

Ans. 42,220 meters, or 26.2 miles, per sec. 

6. Suppose a particle moves subject to an attractive force varying directly 
as the distance, and to a resistance which is proportional to the speed; solve 
the differential equation by the general method for linear equations. 

Ans. Let k 2 be the factor of proportionality for the velocity and Z 2 for the 
distance. Then the solutions are 



where 



Xi = 



- k* + V/c 4 - 4/ 2 



2 

Discuss more in detail the form of the solution and its physical meaning 
when (a) /c 4 - 4Z 2 < 0, (6) fc 4 - 4Z 2 = 0, (c) /c 4 - 4Z 2 > 0. 

7. Suppose that in addition to the forces of problem 6 there is a force jue"'; 
derive the solution by the method of the variation of parameters and discuss 
the motion of the particle. 

8. Develop the method of the variation of parameters for a linear differ- 
ential equation of the third order. 

9. If fc 2 = equation (23) becomes that which defines the motion of a 
freely falling body. Show that the limit of the solution (32) as /c 2 approaches 
zero is 

s = SQ + Vot %gP. 



56 



PARABOLIC MOTION. 



[39 



39. Parabolic Motion. There is a class of problems involving 
for their solution mathematical processes which are similar to 
those employed thus far in this chapter, although the motion is 
not in a straight line. On account of the similarity in the analysis 
a short discussion of these problems will be inserted here. 

Suppose the particle is subject to a constant acceleration down- 
ward; the problem is to find the character of the curve described 
when the particle is projected in any manner. The orbit will be 
in a plane which will be taken as the ^-plane. Let the y-axis be 
vertical with the positive end directed upward. Then the differ- 
ential equations of motion are 



(38) 



= n 
dt* 

tfy 



Since these equations are independent of each other, they can 
be integrated separately, and give 



x = 



--y 



dx dy 

0, 37 = ^0 COS o:, 37 
at at 



sin a when i = 0, 



Suppose x = y = U, 37 = v cos a, -77 = ^o 

where a is the angle between the line of initial projection and the 
plane of the horizon, and ^o is the speed of the projection. Then 




Fig. 6. 

the constants of integration are found to be 
ai = #o cos a, 0,2 = 0, 

61 = VQ sin a, b z = 0; 
and therefore 



\ 



39] 



PARABOLIC MOTION. 



57 



(39) 



X = VQ COS a t, 

at 2 
y = ~- + v Q sin a t. 



The equation of the curve described, which is found by elimi- 
nating t between these two equations, is 

(40) y-xt***- 1 "^"*. 

This is the equation of a parabola whose axis is vertical with its 
vertex upward. It can be written in the form 



x -- sin a cos a = 
Q 



y 
y 



20 



The equation of a parabola with its vertex at the origin has the 
form 

x 2 = 2py, 

where 2p is the parameter. On comparing this equation with the 
equation of the curve described by the particle, the coordinates 
of the vertex, or highest point, are seen to be 

x = sin a. cos a, 
9 



The distance of the directrix from the vertex is one-fourth of 
the parameter; therefore the equation of the directrix is 



p 



sin 2 a , v<? cos 2 a 

~ ~~ 



The square of the velocity is found to be 

"=(!)' +(*)-->* 

To find the place where the particle will strike the horizontal 
plane put y = in (40). The solutions for x are x = and 

2v 2 . v 2 . 

x = -- sin a cos a = sin 2a. 

g g 

From this it follows that the range is greatest for a given initial 
velocity if a = 45. From (39) the horizontal velocity is seen to 



68 PROBLEMS. 

be v cos a; hence the time of flight is -sin a. Therefore, if the 

y 

other initial conditions are kept fixed, the whole time of flight 
varies directly as the sine of the angle of elevation. 

The angle of elevation to attain a given range is found by 
solving 

V Q 2 

x = a = sin 2a 
Q 

for a. If a > there is no solution. If a < - there are two 

9 9 

solutions differing from the value for maximum range (a = 45) by 
equal amounts. 

If the variation in gravity at different heights above the earth's 
surface, the curvature of the earth, and the resistance of the air 
are neglected, the investigation above applies to projectiles near the 
earth's surface. For bodies of great density the results given by 
this theory are tolerably accurate for short ranges. When the 
acceleration is taken toward the center of the earth, and gravity 
is supposed to vary inversely as the square of the distance, the 
path described by the particle is an ellipse with the center of the 
earth as one of the foci. This will be proved in the next chapter. 



V. PROBLEMS. 

1. Prove that, if the accelerations parallel to the x and y-axes are any 
constant quantities, the path described by the particle is a parabola for 
general initial conditions. 

2. Find the direction of the major axis, obtained in problem 1, in terms of 
the constant components of acceleration. 

3. Under the assumptions of Art. 39 find the range on a line making an 
angle with the z-axis. 

4. Show that the direction of projection for the greatest range on a given 
line passing through the point of projection is in a line bisecting the angle 
between the given line and the ?/-axis. 

i 5. Show that the locus of the highest points of the parabolas as a takes 

all values is an ellipse whose major axis is , and minor axis, -. 

6. Prove that the ^elocityi) at any point equals that which the particle 
would have at the poinfl if it fell from the directrix of the parabola. 



40] . WORK AND ENERGY. 69 

THE HEAT OF THE SUN. 

40. Work and Energy. When a force moves a particle against 
any resistance it is said to do work. The amount of the work is 
proportional to the product of the resistance and the distance 
through which the particle is moved. In the case of a free particle 
the resistance comes entirely from the inertia of the mass; if there 
is friction this is also resistance. 

Energy is the power of doing work. If a given amount of work 
is done upon a particle free to move, the particle acquires a motion 
that will enable it to do exactly the same amount of work. The 
energy of motion is called kinetic energy. If the particle is retarded 
by friction part of the original work expended will be used in over- 
coming the friction, and the particle will be capable of doing only 
as much work as has been done in giving it motion. Until about 
1850 it was generally supposed that work done in overcoming fric- 
tion is partly, or perhaps entirely, lost. In other words, it was be- 
lieved that the total amount of energy in an isolated system might 
continually decrease. It was observed, however, that friction 
generates heat, sound, light, and electricity, depending upon the 
circumstances, and that these manifestations of energy are of 
the same nature as the original, but in a different form. It was 
then proved that these modified forms of energy are in every 
case quantitatively equivalent to the waste of that originally 
considered. The brilliant experiments of Joule and others, made 
in the middle of the nineteenth century, have established with 
great certainty the fact that the total amount of energy remains 
the same whatever changes it may undergo. This principle, 
known as the conservation of energy, when stated as holding 
throughout the universe, is one of the most far-reaching general- 
izations that has been made in the natural sciences in a hundred 
years.* 

41. Computation of Work. The amount of work done by a 
Newtonian force in moving a free particle any distance will now be 
computed. Let m equal the mass of the particle moved, and k 2 
a constant depending upon the mass of the attracting body and 
the units employed. Then 



* Herbert Spencer regards the principle as being axiomatic, and states his 
views in regard to it in his First Principles, part n., chap. vi. 



60 COMPUTATION OF WORK. [41 

The right member is the force to which the particle is subject. 
By Newton's third law it is numerically equal to the reaction, or 
the resistance due to inertia. Then the work done in moving 
the particle through the element of distance ds is 



d?s , , , TJ7 

m-jpds = -- as = dW. 

The work done in moving the particle through the interval from 
so to Si is found by taking the definite integral of this expression 
between the limits s and si. On performing the integrations and 
substituting the limits, it is found that 

1 



midsA* m 
2\~dt) "~2 



_ 

n 



Suppose the initial velocity is zero; then the kinetic energy equals 
the work W done upon the particle, and 



2 dt I Vi 

By hypothesis, the particle has no kinetic energy on the start, 
and therefore the power of doing work equals the product of one 
half the mass and the square of the velocity. If the particle falls 
from infinity, s is infinite, and the formula for the kinetic energy 
becomes 



, . m i dsi\ 2 _ 

2 \ dt / : 



If the particle is stopped by striking a body when it reaches the 
point si, its kinetic energy is changed into some other form of 
energy such as heat. It has been found by experiment that a 
body weighing one kilogram falling 425 meters* in the vicinity of 
the earth's surface, under the influence of the earth's attraction, 
generates enough heat to raise the temperature of one kilogram 
of water one degree Centigrade. This quantity of heat is called 
the calory. f The amount of heat generated is proportional to the 
product of the square of the velocity and the mass of the moving 
particle. Then, letting Q represent the number of calories, it 
follows that 

(44) Q = Cmv 2 . 

* Joule found 423.5; Rowland 427.8. For results of experiments and 
references see Preston's Theory of Heat, p. 594. 

t One-thousandth of this unit, denned in using the gram instead of the kilo- 
gram, is also called a calory. 



42] THE TEMPERATURE OF METEORS. 61 

Let m be expressed in kilograms and v in meters per second. 
In order to determine the constant C, take Q and m each equal to 
unity; then the velocity is that acquired by the body falling 
through 425 meters. It was shown in Art. 30 that, if the body 
falls from rest, 

J = - i<7* 2 , 

I v= -gt. 
On eliminating t between these equations, it is found that 



In the units employed g = 9.8094, and since s = 425 and 
v 2 = 8338, it follows from (44) that 

C- 



8338' 

Then the general formula (44) becomes 

mv 2 



(45) Q = 



8338 ' 



where Q is expressed in calories if the kilogram, meter, and second 
are taken as the units of mass, distance, and time. 

42. The Temperature of Meteors. The increase of temperature 
of a body, when the proper units are employed, is equal to the 
number of calories of heat acquired divided by the product of the 
mass and the specific heat of the substance. Suppose a meteor 
whose specific heat is unity (in fact it would probably be much 
less than unity) should strike the earth with any given velocity; it 
is required to compute its increase of temperature if it took up all 
the heat generated. The specific heat has been taken so that the 
increase of temperature is numerically equal to the number of 
calories generated per unit mass. Meteors usually strike the earth 
with a velocity of about 25 miles, or 40,233 meters, per second. 
On substituting 40,233 for v and unity for m in (45), it is found 
that T = Q = 194,134, the number of calories generated per unit 
mass, or the number of degrees through which the temperature of 
the meteor would be raised. As a matter of fact a large part of 
the heat would be imparted to the atmosphere; but the quantity 
of heat generated is so enormous that it could not be expected that 
any but the largest meteors would last long enough to reach the 
earth's surface. 

A meteor falling into the sun from an infinite distance would 



62 METEORIC THEORY OF THE SUN'S HEAT. [43 

strike its surface, as has been seen in Art. 36, with a velocity of 
about 384 miles per second. The heat generated would be there- 
fore (-V/) 2 , or 236, times as great as that produced in striking the 
earth. Thus it follows that a kilogram would generate, in falling 
into the sun, 45,815,624 calories. 

43. The Meteoric Theory of the Sun's Heat. When it is 
remembered what an enormous number of meteors (estimated by 
H. A. Newton* as being as many as 8,000,000 daily) strike the 
earth, it is easily conceivable that enough strike the sun to main- 
tain its temperature. Indeed, this has been advanced as a theory 
to account for the replenishment of the vast amount of heat which 
the sun radiates. There can be no question of its qualitative 
correctness, and it only remains to examine it quantitatively. 

Let it be assumed that the sun radiates heat equally in every 
direction, and that meteors fall upon it in equal numbers from 
every direction. Under this assumption, the amount of heat radi- 
ated by any portion of the surface will equal that generated by the 
impact of meteors upon that portion. The amount of heat 
received by the earth will be to the whole amount radiated from 
the sun as the surface which the earth subtends as seen from the 
sun is to the surface of the whole sphere whose radius is the 
distance from the earth to the sun. The portion of the sun's 
surface which is within the cone whose base is the earth and vertex 
the center of the sun, is to the whole surface of the sun as the 
surface subtended by the earth is to the surface of the whole 
sphere whose radius is the distance to the sun. Therefore, the 
earth receives as much heat as is radiated by, and consequently 
generated upon, the surface cut out by this cone. But the earth 
would intercept precisely as many meteors as would fall upon this 
small area, and would, therefore, receive heat from the impact of a 
certain number of meteors upon itself, and as much heat from the 
sun as would be generated by the impact of an equal number 
upon the sun. 

The heat derived by the earth from the two sources would be as 
the squares of the velocities with which the meteors strike the 
earth and sun. It was seen in Art. 42 that this number is ^i^- 
Therefore, if this meteoric hypothesis of the maintenance of the 
sun's heat is correct, the earth should receive -^^ as much heat 
from the impact of meteors as from the sun. This is certainly 

* Mem. Nat. Acad. of Sd., vol. i. 



44] HELMHOLTZ'S CONTRACTION THEORY. 63 

millions of times more heat than the earth receives from meteors, 
and consequently the theory that the sun's heat is maintained by 
the impact of meteors is not tenable. 

44. Helmholtz's Contraction Theory. The amount of work 
done upon a particle is proportional to the product of the resistance 
overcome by the distance moved. There is nothing whatever said 
about how long the motion shall take, and if the work is converted 
into heat the total amount is the same whether the particle falls 
the entire distance at once, or covers the same distance by a suc- 
cession of any number of shorter falls. When a body contracts 
it is equivalent to a succession of very short movements of all its 
particles in straight lines toward the center, and it is evident that, 
knowing the law of density, the amount of heat which will be 
generated can be computed. 

In 1854 Helmholtz applied this idea to the computation of the 
heat of the sun in an attempt to explain its source of supply. He 
made the supposition that the sun contracts in such a manner that 
it always remains homogeneous. While this assumption is 
certainly incorrect, nevertheless the results obtained are of great 
value and give a good idea of what doubtless actually takes place 
under contraction. The mathematical part of the theory is given 
in the Philosophical Magazine for 1856, p. 516. 

Consider a homogeneous gaseous sphere whose radius is R Q and 
density a. Let M represent its mass. Let dM represent an 
element of mass taken anywhere in the interior or at the surface 
of the sphere. Let R be the distance of dM from the center of 
the sphere, and let M represent the mass of the sphere whose radius 
is R. The element of mass in polar coordinates is (Art. 21) 

(46) dM = vR 2 cos (j>d(j>d8dR. 

The element is subject to the attraction of the whole sphere 
within it. As will be shown in Chapter IV, the attraction of the 
spherical shell outside of it balances in opposite directions so that 
it need not be considered in discussing the forces acting upon dM. 
Every element in the infinitesimal shell whose radius is R is 
attracted toward the center by a force equal to that acting on dM', 
therefore the whole shell can be treated at once. Let dM 8 repre- 
sent the mass of the elementary shell whose radius is R. It is 
found by integrating (46) with respect to 8 and 0. Thus 



(47) dM s = aR 2 dR 2 " f 2 cos 0d0 dO = 7raR 2 dR. 



64 HELMHOLTZ'S CONTRACTION THEORY. [44 



The force to which dM 8 is subject is --- ^ - . The element 

xc 

of work done in moving dM s through the element of distance dR is 
dW s = - 



The work done in moving the shell from the distance CR to R is 
the integral of this expression between the limits CR and R, or 

W - - 



r>2 r> I r^ 

K JK \ U 

But M = ^ircrR 3 ; hence, substituting the value of dM s from (47) 
and representing the work done on the elementary shell by 
W 8 = dW, it follows that 

dVr-J^MP(^^l*tt. 



= y TrW ( 



The integral of this expression from to R gives the total amount 
of work done in the contraction of the homogeneous sphere from 
radius CRo to RQ. That is, 



which may be written 

r = 

If C equals infinity, then 

(49) 

* Q 

If the second is taken as the unit of time, the kilogram as the 
unit of mass, and the meter as the unit of distance, and if k 2 is 
computed from the value of g for the earth, then, after dividing 
W by j , the result will be numerically equal to the amount of 
heat in calories that will be generated if the work is all trans- 
formed into this kind of energy. The temperature to which the 
body will be raised, which is this quantity divided by the product 
of the mass and the specific heat, is 

T - 



where 77 is the specific heat of the substance. Or, substituting 
(48) in (50), it is found that 



44] HELMHOLTZ'S CONTRACTION THEORY. 65 

T _W C-l M 2 
"5^" ~~C~ ~R~ Q '8338* 

By definition, k 2 is the attraction of the unit of mass at unit 
distance; therefore, if m is the mass of the earth and r its radius, 
it follows that 

k 2 m 



On solving for k z and substituting in (51), the expression for T 
becomes 

T - 3(C ~ 1} r * M O 2 ? 

5r?C ' #o ' m ' 8338 ' 

I f the body contracted from infinity (C = ), the amount of 
lieat generated would be sufficient to raise its temperature T 
degrees Centigrade, where T is given by the equation 



5 ' 77 ' R ' m ' 8338 ' 

Suppose the specific heat is taken as unity, which is that of water.* 
The value of g is 9.8094 and 

= 116,356, 

xt/o 

= 332,000. 

m 

On substituting these numbers in (53) and reducing, it is found that 
T = 27,268,000 Centigrade. 

Therefore, the sun contracting from infinity in such a way as to 
always remain homogeneous would generate enough heat to raise the 
temperature of an equal mass of water more than twenty-seven millions 
of degrees Centigrade. 

If it is supposed that the sun has contracted only from Neptune's 
orbit equation (52) can be used, which will give a value of T 
about -grAnr less. In any case it is not intended to imply that it 
did ever contract from such great dimensions in the particular 
manner assumed; the results given are nevertheless significant 
and throw much light on the question of evolution of the solar 
system from a vastly extended nebula. If the contraction of the 

* No other ordinary terrestrial substance has a specific heat so great as 
unity except hydrogen gas, whose specific heat is 3.409. But the lighter gases 
of the solar atmosphere may also have high values. 
6 



66 HELMHOLTZ'S CONTRACTION THEORY. [44 

sun were the only source of its energy, this discussion would give a 
rather definite idea as to the upper limit of the age of the earth. 
But the limit is so small that it is not compatible with the con- 
clusions reached by several lines of reasoning from geological 
evidence, and it is utterly at variance with the age of certain 
uranium ores computed from the percentage of lead which they 
contain. The recent discovery of enormous sub-atomic energies 
which become manifest in the disintegration of radium and several 
other substances prove the existence of sources of energy not 
heretofore considered, and suggest that the sun's heat may be 
supplied partly, if not largely, from these sources. It is certainly 
unsafe at present to put any limits on the age of the sun. 

The experiments of Abbott have shown that, under the assump- 
tion that the sun radiates heat equally in every direction, the 
amount of heat emitted yearly would raise the temperature of a 
mass of water equal to that of the sun 1.44 degrees Centigrade. In 
order to find how great a shrinkage in the present radius would 
be required to generate enough heat to maintain the present radi- 
ation 10,000 years, substitute 14,400 for T in (52) and solve for C. 
On carrying out the computation, it is found that 

C = 1.000528. 

Therefore, the sun would generate enough heat in shrinking about 
one four-thousandth of its present diameter to maintain its present 
radiation 10,000 years. 

The sun's mean apparent diameter is 1924", so a contraction of 
its diameter of .000528 would make an apparent change of only 
l."0, a quantity far too small to be observed on such an object by 
the methods now in use. On reducing the shrinkage to other 
units, it is found that a contraction of the sun's radius of 36.8 
meters annually would account for all the heat that is being radi- 
ated at present. 

VI. PROBLEMS. 

1. According to the recent work of Abbott, of the Smithsonian Institution, 
a square meter exposed perpendicularly to the sun's rays at the earth's distance 
would receive 19.5 calories per minute. The average amount received per 
square meter on the earth's surface is to this quantity as the area of a circle 
is to the surface of a sphere of the same radius, or 1 to 4. The earth's surface 
receives, therefore, on the average 5 calories per square meter per minute. 
How many kilograms of meteoric matter would have to strike the earth 
with a velocity of 25 miles (40,233 meters) per sec', to generate ^^ this amount 
of heat? 

Ans. .000,000,1115 kilograms. 



HISTORICAL SKETCH. 67 

2. How many pounds would have to fall per day on every square mile on 
the average? Tons on the whole earth? 

. (917 pounds. 

(90,300,000 tons. 

3. Find the amount of work done in the contraction of any fraction of 
the radius of a sphere when the law of density is <r = - . 



Ans. W = \Wm* R = kZ - i or \ -of the work 

done when the sphere is homogeneous. 

4. Laplace assumed that the resistance of a fluid against compression is 
directly proportional to its density, and on the basis of this assumption he 
found that the law of density of a spherical body would be 



Gsin 



(^) 



where G and ^ are constants depending on the material of which the body is 
composed, and where a is the radius of the sphere. This law of density is in 
harmony, when applied to the earth, with a number of phenomena, such as 
the precession of the equinoxes. Find the amount of heat generated by 
contraction from infinite dimensions to radius R Q of a body having the Lapla- 
cian law of density. 

5. Find how much the heat generated by the contraction of the earth 
from the density of meteorites, 3.5, to the present density of 5.6 would raise 
the temperature of the whole earth, assuming that the specific heat is 0.2. 

Ans. T = 6520.5 degrees Centigrade. 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The laws of falling bodies under constant acceleration were investigated 
by Galileo and Stevinus, and for many cases of variable acceleration by 
Newton. Such problems are comparatively simple when treated by the 
analytical processes which have come into use since the time of Newton. 
Parabolic motion was discussed by Galileo and Newton. 

The kinetic theory of gases seems to have been first suggested by J. Ber- 
nouilli about the middle of the 18th century, but it was first developed mathe- 
matically by Clausius. Maxwell, Boltzmann, and 0. E. Meyer have made 
important contributions to the subject, and more recently Burbury, Jeans, 
and Hilbert. Some of the principal books on the subject are: Risteen's 
Molecules and the Molecular Theory (descriptive work); L. Boltzmann's 
Gastheorie; H. W. Watson's Kinetic Theory of Gases; O. E. Meyer's Die Kine- 
tische Theorie der Gase; S. H. Burbury's Kinetic Theory of Gases; J. H. Jean's 
Kinetic Theory of Gases. 



68 HISTORICAL SKETCH. 

The meteoric theory of the sun's heat was first suggested by R. Mayer. 
The contraction theory was first announced in a public lecture by Helmholtz 
at Konigsberg Feb. 7, 1854, and was published later in Phil. Mag. 1856. 
An important paper by J. Homer Lane appeared in the Am. Jour, of Sri. 
July, 1870. The amount of heat generated depends upon the law of density 
of the gaseous sphere. Investigations covering this point are 16 papers by 
Ritter in Wiedemann's Annalen, vol. v., 1878, to vol. xx., 1883; by G. W. Hill, 
Annals of Math., vol. iv., 1888; and by G. H. Darwin, Phil. Trans., 1888. The 
original papers must be read for an exposition of the subject of the heat of 
the sun. Sub-atomic energies are discussed in E. Rutherford's Radioactive 
Substances and their Radiations. 

For evidences of the great age of the earth consult Chamberlin and Salis- 
bury's Geology, vol. n., and vol. in., p. 413 et seq.; for a general discussion of 
the age of the earth see Arthur Holmes' The Age of the Earth. 



CHAPTER III. 

CENTRAL FORCES. 

45. Central Force. This chapter will be devoted to the dis- 
cussion of the motion of a material particle when subject to an 
attractive or repelling force whose line of action always passes 
through a fixed point. This fixed point is called the center of force. 
It is not implied that the force emanates from the center or that 
there is but one force, but simply that the resultant of all the forces 
acting on the particle always passes through this point. The 
force may be directed toward the point or from it, or part of the 
time toward and part of the time from it. It may be zero at any 
time, but if the particle passes through a point where the force to 
which it is subject becomes infinite, a special investigation, which 
cannot be taken up here, is required to follow it farther. Since 
attractive forces are of most frequent occurrence in astronomical 
and physical problems, the formulas developed will be for this case; 
a change of sign of the coefficient of intensity of the force for unit 
distance will make the formulas valid for the case of repulsion. 

The origin of coordinates will be taken at the center of force, 
and the line from the origin to the moving particle is called the 
radius vector. The path described by the particle is called the 
orbit. The orbits of this chapter are plane curves. The planes 
are defined by the position of the center of force and the line of 
initial projection. The xy-plsme will be taken as the plane of the 
orbit. 

46. The Law of Areas. The first problem will be to derive the 
general properties of motion which hold for all central forces. The 
first of these, which is of great importance, is the law of areas, and 
constitutes the first Proposition of Newton's Prindpia. It is, 
if a particle is subject to a central force, the areas which are swept 
over by the radius vector are proportional to the intervals of time in 
which they are described. The following is Newton's demonstration 
of it* 

Let be the center of force, and let the particle be projected 
from A in the direction of B with the velocity AB. Then, by the 
first law of motion, it would pass to C' in the first two units of 



70 THE LAW OF AREAS. [46 

time if there were no external forces acting upon it. But suppose 
that when it arrives at B an instantaneous force acts upon it in 
the direction of the origin with such intensity that it would move 




Fig. 7. 

to b in a unit of time if it had no previous motion. Then, by the 
second law of motion, it will move along the diagonal of the 
parallelogram BbCC' to C. If no other force were applied it 
would move with uniform velocity to D' in the next unit of time. 
But suppose that when it arrives at C another instantaneous force 
acts upon it in the direction of the origin with such intensity 
that it would move to c in a unit of time if it had no previous 
motion. Then, as before, it will move along the diagonal of the 
parallelogram and arrive at D at the end of the unit of time. This 
process can be repeated indefinitely. 

The following equalities among the areas of the triangles in- 
volved hold, since they have sequentially equal bases and altitudes : 

OAB = OBC' = OBC = OCD' = OCD = etc. 

Therefore, it follows that OAB = OBC = OCD = ODE, etc. 
That is, the areas of the triangles swept over in the succeeding 
units of time are equal ; and, therefore, the sums of the areas of the 
triangles described in any intervals of time are proportional to 
the intervals. 

The reasoning is true without any changes however small the 
intervals of time may be. Let the path be considered for some 
fixed finite period of time. Let the intervals into which it is divided 
be taken shorter and shorter; the impulses will become closer and 
closer together. Suppose the ratio of the magnitudes of the impulses 
to the values of the intervals between them remains finite; then the 
broken line will become more and more nearly a smooth curve. 
Suppose the intervals of time approach zero as a limit; the suc- 
cession of impulses will approach a continuous force as a limit, and 



47] ANALYTICAL DEMONSTRATION OF LAW OF AREAS. 71 

the broken line will approach a smooth curve as a limit. The areas 
swept over by the radius vector in any finite intervals of time are 
proportional to these intervals during the whole limiting process. 
Therefore, the proportionality of areas holds at the limit, and the 
theorem is true for a continuous central force. 

It will be observed that it is not necessary that the central force 
shall vary continuously. It may be attractive and instantaneously 
change to repulsion, or become zero, and the law will still hold; 
but it is necessary to exclude the case where it becomes infinite 
unless a special investigation is made. 

The linear velocity varies inversely as the perpendicular from 
the origin upon the tangent to the curve at the point of the moving 
particle; for, the area described in a unit of time is equal to the 
product of the velocity and the perpendicular upon the tangent. 
Since the area described in a unit of time is always the same, it 
follows that the linear velocity of the particle varies inversely as 
the perpendicular from the origin to the tangent of its orbit. 

47. Analytical Demonstration of the Law of Areas. Although 
the language of Geometry was employed in the demonstration 
of Art. 46, yet the essential elements of the methods of the 
Differential and Integral Calculus were used. Thus, in passing 
to the limit, the process was essentially that of expressing the 
problem in differential equations; and, in insisting upon com- 
paring intervals of finite size when the units of measurement were 
indefinitely decreased, the process of integration was really em- 
ployed. It will be found that in the treatment of all problems 
involving variable forces and motions the methods are in essence 
those of the Calculus, even though the demonstrations be couched 
in geometrical language. It is perhaps easier to follow the reason- 
ing in geometrical form when one encounters it for the first time; 
but the processes are all special and involve fundamental difficulties 
which are often troublesome. On the other hand, the develop- 
ment of the Calculus is of the precise form to adapt it to the 
treatment of these problems, and after its principles have been 
once mastered, the application of it is characterized by comparative 
simplicity and great generality. A few problems will be treated 
by both methods to show their essential sameness, and to illustrate 
the advantages of analysis. 

Let / represent the acceleration to which the particle is subject. 
By hypothesis, the line of force always passes through a fixed 
point, which will be taken as the origin of coordinates. 



72 



ANALYTICAL DEMONSTRATION OF LAW OF AREAS. 



[47 



Let be the center of force, and P any position of the particle 
whose rectangular coordinates are x and y, and whose polar 
coordinates are r and 6. Then the components of acceleration 




Fig. 8. 

along the x and i/-axes are respectively =F / cos 8 and =F / sin 8 } 
and the differential equations of motion are 

d 2 x , ,x 

= ^ / cos 8 = ** f- 

(1) 



The negative sign must be taken before the right members of these 
equations if the force is attractive, and the positive if it is repulsive. 
Multiply the first equation of (1) by y and the second one 
by + x and add. The result is 

d?y d?x A 



On integrating this expression by parts, it is found that 
(2) x^-y=h, 

where h is the constant of integration. 

The integrals of differential equations generally lead to im- 
portant theorems even though the whole problem has not been 
solved, and in what follows they will be discussed as they are 
obtained. 

On referring to Art. 16, it is seen that (2) may be written 

dy _ dx _ s d8 _ ^ dA _ , 
X dt~ y dt~ r dt~ ~dt ~ ' 

where A is the area swept over by the radius vector. The integral 
of this equation is 

A = M + c. 



49] THE LAWS OF ANGULAR AND LINEAR VELOCITY. 73 

which shows that the area is directly proportional to the time. 
This is the theorem which was to be proved. 

48. Converse of the Theorem of Areas. By hypothesis 

A = Cit + c 2 . 

On taking the derivative with respect to t, it is found that 

dA 



This equation becomes in polar coordinates 

m -*-. I 

and in rectangular coordinates 

dy dx 

x ctt~ y dl = 2ci - 

The derivative of this expression with respect to t is 

x *y &* - . 
x w~ y ~d ' u ' 

or 

^.^_ a .. |/ 
dP 'd? ~ ' y ' 

That is, the components of acceleration are proportional to the 
coordinates; therefore, if the law of areas is true with respect to a 
point, the resultant of the accelerations passes through that point. 

Or. since r 2 -7- = 2ci, it follows that -n ( r 2 -=- J = 0. Hence, by 

(19), Art. 14, the acceleration perpendicular to the radius vector is 
zero; that is, the acceleration is in the line passing through the 
origin. 

49. The Laws of Angular and Linear Velocity. From the 
expression for the law of areas in polar coordinates, it follows that 

m ^-- 

dt ~ r*' 

therefore, the angular velocity is inversely proportional to the square 
of the radius vector. 
The linear velocity is 



74 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [50 

ds _ ds dd _ ds h 

dt == dedi == de7 2 ' 

Let p represent the perpendicular from the origin upon the tangent ; 
then it is known from Differential Calculus that 

ds = r* 

^sLa^-j^^ 
Hence the expression for the linear velocity becomes 

(4 ) ^ _ h . 

A p 9 

therefore, the linear velocity is inversely proportional to the per- 
pendicular from the origin upon the tangent. 

SIMULTANEOUS DIFFERENTIAL EQUATIONS. 

50. The Order of a System of Simultaneous Differential 
Equations. One integral, equation (2), of the differential equations 
(1) has been found which the motion of the particle must fulfill. 
The question is how many more integrals must be found in order 
to have the complete solution of the problem. 

The number of integrals which must be found to completely 
solve a system of differential equations is called the order of the 
system. Thus, the equation 



is of the nth order, because it must be integrated n times to be 
reduced to an integral form. Similarly, the general equation 

(6) / .g + / _ 1 *^ + ... +/1 *? + / ..o, 

where /, , /o are functions of x and t, must be integrated 

n times in order to express x as a function of t, and is of the nth 
order. 

An equation of the nth order can be reduced to an equivalent system 
of n simultaneous equations each of the first order. Thus, to reduce 
(6) to a simultaneous system, let 

_ dx _ dxi _ dXn-z 

Xl ~~dt' Xz ~~dt' "' ~dT' 

whence 



50] 



SIMULTANEOUS DIFFERENTIAL EQUATIONS. 



75 



(7) 



dx 
dt 



~dt 



dt f n f n f n ' 

Therefore, these n simultaneous equations, each of the first order, 
constitute a system of the nth order. An equation, or a system 
of equations, reduced to the form (7) is said to be reduced to the 
normal form, and the system is called a normal system. 

Two simultaneous equations of order m and n can be reduced 
to a normal system of order m + n. Consider the equations 



(8) 



fm dt m + 



dt' 



^ + / = 0, 
dy , 



0, 



where the fi and the < t are functions of x, y, and t. By a sub- 
stitution similar to that used in reducing (6), it follows that they 
are equivalent to the normal system 

dx 



(9) 



i-i 



dt 



x m -i 



dy 



which is of the order m + n. Evidently a similar reduction is 
possible when each equation contains derivatives with respect to 
both of the variables, either separately or as products. 

Conversely, a normal system of order n can in general be trans- 
formed into a single equation of order n with one dependent variable. 
To fix the ideas, consider the system of the second order 



76 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [50 

^ _ f ( r ,, A 

j t ~ j \ x ) y> i )i 



(10) 



= 0(x, y, t). 



In addition to these two equations form the derivative of one of 
them, for example the first, with respect to t. The result is 

d?x = df_dx dj_dy df 
dt 2 dx dt ^ dydt^ df 

dii 
If y and -j- are eliminated between (10) and (11) the result will 

be an equation of the form 

dx 



where F is a function of both x and -rr . Of course, / and of 

CLL 

equations (10) may have such properties that the elimination of y 
and -JJ- is very difficult. 

If the normal system were of the third order in the dependent 
variables x, y, and z, the first and second derivatives of the first 
equation would be taken, and the first derivative of the second and 
third equations. These four new equations with the original 

f/?7 (1% (1 77 {1% 

three make seven from which y, z, ~ , -j- , -~ , and -^ can in 

dt dt dt dt 

general be eliminated, giving an equation of the third order in x 

alone. This process can be extended to a system of any order. 

The differential equations (1) can be reduced by the substitution 

, dx . dy , 
x = -77 , y = -77 to the normal system 

dx _ , dxf_ .x 

dt~ X ' ~dT = =p; r > 



dt~ dt~ J r> 

which is of the fourth order. Therefore four integrals must be 
found in order to have the complete solution of the problem. 
The components of velocity, x' and y', play roles similar to the 
coordinates in (12), and, for brevity, they will be spoken of fre- 
quently in the future as being coordinates. 



51] REDUCTION OF ORDER OF DIFFERENTIAL EQUATIONS. 77 



51. Reduction of Order. When an integral of a system of 
differential equations has been found two methods can be followed 
in completing the solution. The remaining integrals can be found 
from the original differential equations as though none was already 
known; or, by means of the known integral, the order of the system 
of differential equations can be reduced by one. That the order 
of the system can be reduced by means of the known integrals 
will be shown in the general case. Consider the system of differ- 
ential equations 

d-r. 

\(Xi, , X n , t), 



dt 

dXji 

dt 



I X n , t), 



(13) 



Suppose an integral 

F(XI, x z , -', %n, t) = constant =c, 

has been 'found. Suppose this equation is solved for x n in terms 
of xi, , x n -i, c, and t. The result may be written 

X n = t(Xi, Xz, ', X n -l, C, t) . 

Substitute this expression for x n in the first n - 1 equations of (13) ; 
they become 



(14) 



This is a simultaneous system of order n 1, and is independent 
of the variable x n . 

It is apparent from these theorems and remarks that the order 
of a simultaneous system of differential equations is equal to the 
sum of the orders of the individual equations; that the equations 
can be written in several ways, e. g., as one equation of the nth 
order, or n equations of the first order; and that the integrals may 
all be derived from the original system, or that the order may be 



t -**... 


, Xn-l, C, t), 
', X n -i, C, 0, 


dXn-1 




dt < P"~ 1 ^ 1 





78 THE VIS VIVA INTEGRAL. [52 

reduced after each integral is found. In mechanical and physical 
problems the intuitions are important in suggesting methods of 
treatment, so it is generally advantageous to use such variables 
that their geometrical and physical meanings shall be easily 
perceived. For this reason, it is generally simpler not to reduce 
the order of the problem after each integral is found. 

VII. PROBLEMS. 

1. Prove the converse of the law of areas by the geometrical method, and 
show that the steps agree with the analysis of Art. 48. 

2. Prove the law of angular velocity by the geometrical method. 

3. Why cannot equations (1) be integrated separately? 

4. Derive the law of areas directly from equation (2) without passing 
to polar coordinates. 

5. Show in detail that a normal system of order four can be reduced to 
a single equation of order four, and the converse. 

6. Reduce the system of equations (12) to one of the third order by means 
of the integral of areas. 

52. The Vis Viva Integral. Suppose the acceleration is toward 
the origin; then the negative sign must be taken before the right 

members of equations (1). Multiply the first of (1) by 2 -5- , 

//7/ 

the second by 2 -3- , and add. The result is 

ffixdx ffiydy_ _2f( fa , ^A 
2 d? dt V dt = ~ r ( X dt ^~ y dt )' 

It follows from r 2 = x 2 + y 2 that 

dx dy _ dr 
X dt~^ y dt = r ~dt' 
therefore 

9 d^xdx d^ydy = _ f dr 
dP dt 1 dt 2 dt ~ J dt' 

Suppose / depends upon r alone, as it does in most astronomical 
and physical problems. Then/ = </>(r), and 

d 2 x dx d 2 y dy . dr 



The integral of this equation is 



53] FORCE VARYING DIRECTLY AS THE DISTANCE. 79 

When the form of the function <f>(r) is given the integral on the 
right can be found. Suppose the integral is $(r) ; then 

(16) v z = - 2$(r) + c. 

If $(r) is a single-valued function of r, as it is in physical prob- 
lems, it follows from (16) that, if the central force is a function of 
the distance alone, the speed is the same at all points equally 
distant from the origin. Its magnitude at any point depends upon 
the initial distance and speed, and not upon the path described. 
Since the force of gravitation varies inversely as the square of the 
distance between the attracting bodies, it follows that a body, for 
exaniple a comet, has the same speed at a given distance from the 
sun whether it is approaching or receding. 

EXAMPLES WHERE / is A FUNCTION OF THE COORDINATES ALONE. 
53. Force Varying Directly as the Distance. In order to find 
integrals of equations (1) other than that of areas, the value of 
f in terms of the coordinates must be known. In the case in which 
the intensity of the force varies directly as the distance the inte- 
gration becomes particularly simple. Let k 2 represent the acceler- 
ation at unit distance. Then / = k z r, and, in case the force is 
attractive, equations (1) become 



dt 2 



.dt* - 

An important property of these equations is that each is inde- 
pendent of the other, as the first one contains the dependent 
variable x alone and the second one y alone. It is observed, more- 
over, that they are linear and the solution can be found by the 

method given in Art. 32. If x = XQ, -rr = XQ , y= y Q , IT = yd at 

t = 0, then the solutions expressed in the trigonometrical form are 

/ 
x = -f- XQ cos kt + ~Y~ sin kt, 



(18) 



-57 = kxQ sin kt + XQ cos kt, 

f 

y = + 2/o cos kt + ^r- sin kt, 
-jr = ky Q sin kt -f- y Q ' cos Atf. 



80 DIFFERENTIAL EQUATION OF THE ORBIT. [54 

The equation of the orbit is obtained by eliminating t between the 
first and third equations of (18). On multiplying by the appro- 
priate factors and adding, it is found that 

(zo2/o' yoXo) sin kt = k(x y y Q x), 



\ (x Q yo f - yox r ) cos kt = y Q 'x - x 'y. 



The result of squaring and adding these equations is 

W + 2/o' V + (kV + Zo'V - 2(& 2 zo2/o + x*'y Q ')xy 



= x yo 2/o XQ. 

This is the equation of an ellipse with the origin at the center 
unless x Q y f 2/o#o' = 0, when the orbit degenerates to two straight 
lines which must be coincident; for, then 

#o 2/o 

-. = .= constant = c; 
XQ 2/0 
from which 

XQ = cxo, 2/0 = cy Q '. 

In this case equation (20) becomes 

(21) (k*<* + I)(y 'x - xo'y? = Q, 

and the motion is rectilinear and oscillatory. In every case both 
the coordinates and the components of velocity are periodic with 

the period ~r , whatever the initial conditions may be. 

K 

54. Differential Equation of the Orbit. The curve described 
by the moving particle, independently of the manner in which it 
may move along this curve, is of much interest. A general method 
of finding the orbit is to integrate the differential equations and then 
to eliminate the time. This is often a complicated process, and the 
question arises whether the time cannot be eliminated before the 
integration is performed, so that the integration will lead directly 
to the orbit. It will be shown that this is the case when the force 
does not depend upon the time. 

The differential equations of motion are [Art. 47] 



(22) 



d?x _ f x 
dt 2 = } r ' 



t 2 " r 

Since / does not involve the time t enters only in the derivatives. 



54] DIFFERENTIAL EQUATION OF THE ORBIT. 81 

But a second differential quotient cannot be separated as though 
it were an ordinary fraction; therefore, the order of the derivatives 
must be reduced before the direct elimination of t can be made. 
In order to do this most conveniently polar coordinates will be 
employed. Equations (22) become in these variables 



#0 



^ r ( de \- 

dt*~ r \Tt) 



__ 

dtdt 



The integral of the second of these equations is 

dO L 

r 2 -77 = h.~ 
di > rr 

On eliminating -7- from the first of (23) by means of this equation, 
dt 

the result is found to be 

(24) = ?-> 

Now let r = - ; therefore 
u 



dr 


1 du 


I du d0 , dw 


dt 


u 2 dt 
7 d (du\ 


w 2 dO dt dO' 

, d 2 u dO 



_ , 2 2 

~ l dp' 

When this value of the second derivative of r is equated to the 
one given in (24), it is found that 

(25) / 

This differential equation is of the second order, but one integral 
has been used in determining it; therefore the problem of finding 
the path of the body is of the third order. The complete problem 
was of the fourth order; the fourth integral expresses the relation 
between the coordinates and the time, or defines the position of 
the particle in its orbit. 

Since the integral of (25) expresses u, and therefore r, in terms 
of 6, the equation 

"$=> 

when integrated, gives the relation between 6 and t. 

7 



82 NEWTON'S LAW OF GRAVITATION. [55 

Conversely, equation (25) can be used to find the law of central 
force which will cause a particle to describe a given curve. It is 
only necessary to write the equation of the curve in polar coordi- 
nates and to compute the right member of (25) . This is generally 
a simpler process than the reverse one of finding the orbit when 
the law of force is given. 

55. Newton's Law of Gravitation* In the early part of the 
seventeenth century Kepler announced three laws of planetary 
motion, which he had derived from a most laborious discussion 
of a long series of observations of the planets, especially of Mars. 
They are the following: 

LAW I. The radius vector of each planet with respect to the sun 
as the origin sweeps over equal areas in equal times. 

LAW II. The orbit of each planet is an ellipse with the sun at one 
of its foci. 

LAW III. The squares of the periods of the planets are to each 
other as the cubes of the major semi-axes of their respective orbits. 

It was on these laws that Newton based his demonstration that 
the planets move subject to forces directed toward the sun, and 
varying inversely as the squares of their distances from the sun. 
The Newtonian law will be derived here by employing the analyti- 
cal method instead of the geometrical methods of the Principia* 

From the converse of the theorem of areas and Kepler's first law, 
it follows that the planets move subject to central forces directed 
toward the sun. The curves described are given by the second 
law, and equation (25) can, therefore, be used to find the expression 
for the acceleration in terms of the coordinates. Let a represent 
the major semi-axis of the ellipse, and e its eccentricity; then its 
equation in polar coordinates with origin at a focus is 



1 -+- e cos 6 
Therefore 

tfu = 1 

w ~T 7 /\O 



dd 2 a(l - e 2 ) ' 

On substituting this expression in (25), it is found that the equation 
for the acceleration is 

h 2 I _ *L 2 
* ~ o(l - e 2 ) ' r 2 ~ r 2 ' 

* Book i., Proposition xi. 



55] NEWTON'S LAW OF GRAVITATION. 83 

Therefore, the acceleration to which any planet is subject varies 
inversely as the square of its distance from the sun. 

If the distance r is eliminated by the polar equation of the conic 
the expression for / has the form 

/ = fci 2 (l + 6 cos 0) 2 , 

which depends only upon the direction of the attracted body and 
not upon its distance. Now for points on the ellipse the two 
expressions for / give the same value, but elsewhere they give 
different values. It is clear that many other laws of force, all 
agreeing in giving the same numerical values of / for points on the 
ellipse, can be obtained by making other uses of the equation of 
the conic to eliminate r. For example, since it follows from the 
polar equation of the ellipse for points on its circumference that 
(1 + e cos 0)r = 1 

a(l - e 2 ) 
one such law is 

+ e cos 



~ a(\- e 2 ) 

and this value of /, which depends both upon the direction and 
distance of the attracted body, differs from both of the preceding 
for points not on the ellipse. All of these laws are equally con- 
sistent with the motion of the planet in question as expressed by 
Kepler's laws. But the laws of Kepler hold for each of the eight 
planets and the twenty-six known satellites of the solar system, 
besides for more than seven hundred small planets which have so 
far been discovered. It is natural to impose the condition, if pos- 
sible, that the force shall vary according to the same law for each 
body. Since the eccentricities and longitudes of the perihelia of 
their orbits are all different, the law of force is the same for all 
these bodies only when it has the form 

W 
J ~ r f 

Another reason for adopting this expression for / is that in case of 
all the others the attraction would depend upon the direction of 
the attracted body, and this seems improbable. This conclusion 
is further supported by the fact that the forces to which comets 
are subject when they move through the entire system of planets 
vary according to this law. And finally, as will be shown in Art. 
89, the accelerations to which the various planets are subject vary 
from one to another according to this law. 



84 EXAMPLES OF FINDING THE LAW OF FORCE. [56 

From the consideration of Kepler's laws, the gravity at the 
earth's surface, and the motion of the moon around the earth, 
Newton was led to the enunciation of the Law of Universal 
Gravitation, which is, every two particles of matter in the universe 
attract each other with a force which acts in the line joining them, and 
whose intensity varies as the product of their masses and inversely as 
the squares of their distance apart. 

It will be observed that the law of gravitation involves con- 
siderably more than can be derived from Kepler's laws of planetary 
motion; and it was by a master stroke of genius that Newton 
grasped it in its immense generality, and stated it so exactly that 
it has stood without change for more than 200 years. When 
contemplated in its entirety it is one of the grandest conceptions 
in the physical sciences. 

56. Examples of Finding the Law of Force, (a) If a particle 
describes a circle passing through the origin, the law of force 
(depending on the distance alone) under which it moves is a very 
simple expression. Let a represent the radius; then the polar 
equation of the circle is 

r = 2a cos 6, u ^ '- - . 
2a cos B 

Therefore 



On substituting this expression in (25), it is found that 

8a 2 h 2 _ k 2 
J '- r 5 - r s 

(6) Suppose the particle describes an ellipse with the origin at 
the center. The polar equation of an ellipse with the center as 
origin is 



_ _ 

1 - e 2 cos 2 B 
From this it follows that 



bu = Vl - e 2 cos 2 6, 
d 2 u _ e 2 cos 2 e - e 2 sin 2 e 4 sin 2 6 cos 2 6 



U 



dB 2 ' A/1 - e 2 cos 2 (1 ~ g2 cog2 *) f ' 

d 2 u I - e 2 1 



57] DOUBLE STAK ORBITS. 85 

On substituting in (25), the expression for/ is found to be 



THE UNIVERSALITY OF NEWTON'S LAW. 

57. Double Star Orbits. The law of gravitation is proved 
from Kepler's laws and certain assumptions as to its uniqueness 
to hold in the solar system; the question whether it is actually 
'universal naturally presents itself. The fixed stars are so remote 
that it is not possible to observe planets revolving around them, 
if indeed they have such attendants. The only observations 
thus far obtained which throw any light upon the subject are 
those of the motions of the double stars. 

Double star astronomy started about 1780 with the search for 
close stars by Sir William Herschel for the purpose of determining 
parallax by the differential method. A few years were sufficient 
to show him, to his great surprise, that in some cases the two com- 
ponents of a pair were revolving around each other, and that, 
therefore, they were physically connected as well as being appar- 
ently in the same part of the sky. The discovery and measure- 
ment of these systems has been pursued with increasing interest 
and zeal by astronomers. Burnham's great catalogue of double 
stars contains about 13,000 of these objects. The relative motions 
are so slow in most cases that only a few have yet completed 
one revolution, or enough of one revolution so that the shapes of 
their orbits are known with certainty. There are now about thirty 
pairs whose observed angular motions have been sufficiently great 
to prove, within the errors of the observations, that they move 
in ellipses with respect to each other in such a manner that the 
law of areas is fulfilled. In no case is the primary at the focus, 
or at the center, of the relative ellipse described by the companion, 
but it occupies some other place within the ellipse, the position 
varying greatly in different systems. 

From the observations and the converse of the law of areas it 
follows that the resultant of the forces acting upon one star of a 
pair is always directed toward the other. The law of variation 
of the intensity of the force depends upon the position in the 
ellipse which the center of force occupies. It must not be over- 
looked at this point that the orbits of the stars are not observed 
directly, but that it is their projections upon the planes tangent 



86 LAW OF FORCE IN BINARY STARS. [58 

to the celestial sphere at their respective places which are seen. 
The effect of this sort of projection is to change the true ellipse 
into a different apparent ellipse whose major axis has a different 
direction, and one that is differently situated with respect to the 
central star; indeed, it might happen that if one of the stars was 
really in the focus of the true ellipse described by the other, the 
projection would be such as to make it lie on the minor axis of 
the apparent ellipse. 

Astronomers have assumed that the orbits are plane curves and 
that the apparent departure of the central star from the focus of 
the ellipse described by the companion is due to projection, and 
have then computed the angle of the line of nodes and the inclina- 
tion. No inconsistencies are introduced in this way, but the 



S ^\ Line of 




Nodes 



Fig. 9. 

possibility remains that the assumptions are not true. The 
question of what the law of force must be if it is not Newton's law 
of gravitation will now be investigated. 

58. Law of Force in Binary Stars. If the force varied directly 
as the distance the primary star would be at the center of the 
ellipse described by the secondary (Art. 53). No projection would 
change this relative position, and since such a condition has never 
been observed, it is inferred that the force does not vary directly 
as the distance. 

The condition will now be imposed that the curve shall be a 
conic with general position for the origin, and the expression for 
the central force will be found. The equation of the general 
conic is 

(26) ax 2 + 2bxy + cy 2 + 2dx + 2fy = 0. 



58] LAW OF FORCE IN BINARY STARS. 87 

On transforming to polar coordinates and putting r = -, this 
equation gives 



(27) u = A sin 6 + B cos VC sin 26 + D cos 26 + H, 
where 



^ _ d 2 + ag - / 2 - eg u _ d 2 + ag + J 2 + eg 
' 



On differentiating (27) twice, it is found that 

(28) 



d?U /> D Q 

-r^ = A sm B B cos B 



-C 2 -Z) 2 -(Csin20+Dcos20) 2 -2#(Csin20+Dcos20) 



(C sin 20 + D cos 20 + 
On substituting (27) and (28) in (25), it follows that 



r 2 (C sin 20 + D cos 20 + H)* ' 
This becomes as a consequence of (27) 



(30) 



/I \ 3 ' 

( A sin B cos j 



There are also infinitely many other laws, all giving the same 
values of / for points on the ellipse in question, which are obtained 
by multiplying these expressions by any functions of u and 
which are unity on the ellipse in virtue of equation (27). 

It does not seem reasonable to suppose that the attraction of 
two stars for each other depends upon their orientation in space. 
Equation (29) becomes independent of if C = D = 0, and (30), 
if A = B = 0. The first gives 

f - constant 

I = 2 > 

and the second, 

f = constant r. 

The first is Newton's law, and the second is excluded by the 
fact that no primary star has been found in the center of the orbit 
described by the companion. It is clear that can be eliminated 
from (29) and (30) by means of (27) without imposing the con- 



88 GEOMETRICAL INTERPRETATION OF LAW OF FORCE. [59 

ditions that A=B = C = D = 0. But Griffin has shown* 
that for all such laws, except the Newtonian, the force either 
vanishes when the distance between the bodi'es vanishes, or 
becomes imaginary for certain values of r. Clearly both of these 
laws are improbable from a physical point of view. Hence it is 
extremely probable that the law of gravitation holds throughout 
the stellar systems; and this conclusion is supported by the fact 
that the spectroscope shows the stars are composed of familiar 
terrestrial elements. 

59. Geometrical Interpretation of the Second Law. The 

expression for the central force given in (30) may be put in a very 
simple and interesting form. Let g 3 h 2 (H 2 - C 2 - D 2 ) = N, and 

transform A sin 6 B cos 6 into rectangular coordinates and 

the original constants; then (30) becomes 

ran f= ^ Nr 

(dx+fy-g)*' 

The equation of the polar of the point (x f , y') with respect to 
the general conic (26) isf 

ax,x f + b(x iy ' + y&') + cy<y' + d(xi + x') + f( yi + y') - g = Q, 

where x\ and y\ are the running variables. When (x 1 ', y'} is the 
origin this equation becomes 

(32) dxi +fyi-g = 0, 

and has the same form as the denominator of (31). The values 
of x and y in (31) are such that they satisfy the equation of the 
conic, while x\ and y\ of (32) satisfy the equation of the polar line. 
They are, therefore, in general numerically different from x and y. 
But the distance from any point (x, y) of the conic to the polar 
line with respect to the origin is given by the equation 

= dx + fy - g 
Vd 2 + f 2 

where x and y are the coordinates of points on the conic. Let 

N'- 
(# + W 
then (31) becomes 

* American Journal of Mathematics, vol. 31 (1909), pp. 62-85. 
t Salmon's Conic Sections, Art. 89. 



PROBLEMS. 89 






Therefore, if a particle moving subject to a central force describes any 
conic, the intensity of the force varies directly as the distance of the 
particle from the origin, and inversely as the cube of its distance from 
the polar of the origin with respect to the conic. 

60. Examples of Conic Section Motion, (a) When the orbit is 
a central conic 'with the origin at the center, the polar line recedes 

N' 
to infinity, and -j must be regarded as a constant. Then the 

force varies directly as the distance, as was shown in Art. 56 (7>). 
(6) When the origin is at one of the foci of the conic the polar 

line is the directrix, and p = - , where e is the eccentricity. Then 

e 

(33) becomes 

iv 



This is Newton's law which was derived from the same conditions 
in Art. 55. 

VTII. PROBLEMS. 

C C 

1. Find the vis viva integral when / = -^,/ = cr, / = . 

2. Suppose that in Art. 53 the particle is projected orthogonally from 
the z-axis; find the equations corresponding to (19) and (20). Suppose still 
further that k = 1, XQ = 1; find the initial velocity such that the eccentricity 
of the ellipse may be 1/2. 

or 
Am. 



3. Find the central force as a function of the distance under which a 
particle may describe the spiral r = ; the spiral r = e . 

h* 2h* 

Ans. / = ^- , / = j- . 

4. Find the central force as a function of the distance under which a 
particle may describe the lemniscate r 2 = a 2 cos 20. 

Ans - f = 



5. Find the central force as a function of the distance under which a 
particle may describe the cardioid r = a(l + cos 6). 

Ans. f = 



90 ORBIT FOR FORCE VARYING AS DISTANCE. [61 

6. Suppose the particle describes an ellipse with the origin in its interior, 
at a distance n from the x-axis and m from the ?/-axis. (a) Show that two of 
the laws of force are 

r , = W (oc)i 

r 2 [2mn sin cos + (a - c - n 2 + m 2 ) cos 2 + c - m 2 ]* ' 



L [ac am 2 en 2 cny amx] 3 ' 

where a and c have the same meaning as in (26), and where the polar axis 
is parallel to the major axis of the ellipse. (6) If the origin is between the 
center and the focus show that the force at unit distance is a maximum for 

= and is a minimum for = ; that if the origin is between a focus and 

JB 

the nearest apse the maximum is for = and the minimum for = 0; and 
that if the origin is on the minor axis the maximum is for = 0, and the 
minimum for = -~ . 

7. Interpret equation (29) geometrically. 
Hint. C sin 20 + D cos 20 + H = (dx + /y)2 + g(fl ? 9 + Cy * 



The numerator of this expression set equal to zero is the equation of the 
tangents (real or imaginary) from the origin to the conic. (Salmon's Conic 
Sections, Art. 92.) 

8. Find expressions for the central force when the orbit is an ellipse 
with the origin at an end of the major and minor axes respectively. Show 

k 2 
that they reduce to -^ when the ellipse becomes a circle. 




ar z cos 3 ' 



cr 2 sin 3 0' 



DETERMINATION OF THE ORBIT FROM THE LAW OF FORCE. 

61. Force Varying as the Distance. The problem of finding 
the orbit when the law of force is given is generally more difficult 
than the converse, since it involves the integration of (25). The 
method of integration varies with the different laws of force, and 
the character of the integrals depends upon the initial conditions. 
The process will be illustrated first in the case in which the force 
varies as the distance, a problem treated by another method in 
Art. 53. 

If / = k 2 r, equation (25) becomes 



61] ORBIT FOR FORCE VARYING AS DISTANCE. 91 



or 

d?u = k 2 I 
d6 2 ~ h* u 

The first integral of this equation is 

(du\ 2 = _Wl_ 
\dd) h 2 u 2 

whence 

(34) de = udu 



Let 



_ _ A 2 

4 Jj? 4 * 



The constant A 2 must be positive in order that -TT may be real, as 

du 

it is if the particle is started with real initial conditions. 
If the upper sign is used, equation (34) becomes 

(35) 2dd = 



It is easily verified that the same equation (36) would be reached, 
when the initial conditions are substituted, if the lower sign were 
used. The integral of (35) is 



or 

z = A cos 2(0 + c 2 ). 

On going back to the variable r, this equation becomes 

2 
= ci - 2A cos 2(0 + c 2 ) ' 

This is the polar equation of an ellipse with the origin at the center. 
Hence, a particle moving subject to an attractive force varying 
directly as the distance describes an ellipse with the origin at the 
center. The only exceptions are when the particle passes through 
the origin, and when it describes a circle. In the first h = 0, 
and equation (25) ceases to be valid; in the second, c\ has such a 
value that it satisfies the equation 



92 FORCE VARYING INVERSELY AS SQUARE [62 

(du\* k 2 1 

U)o = -/^- WO+C1 = ' 

and the equation of the orbit becomes u = U Q . In this case 
equation (34) fails. 

62. Force Varying Inversely as the Square of the Distance. 

Suppose a particle moves under the influence of a central attraction 
the intensity of which varies inversely as the square of the distance ; 
it is required to determine its orbit when it is projected in any 
manner. Equation (25) is in this case 

, Q7 x d 2 u k 2 

(37) de 2 = h 2 ~ u ' 

This equation can be written in the form 

d?u . k 2 



This is a linear non-homogeneous differential equation and can 
be integrated by the method of variation of parameters, which 
was explained in Art. 37. When its right member is neglected 
the general solution is 

u = Ci cos 8 H- c 2 sin 0. 

k 2 

It is clear that if -^ is added to this value of u the differential 
h* 

equation will be identically satisfied. Consequently the general 
solution of (37), which is the same as that found by the variation 
of parameters, is 

k 2 

u T5 + c i cos + C 2 sm ^' 
h 2 

On taking the reciprocal of this equation, it is found that 

1 



r = 



k 2 

-^ + GI cos 6 + C2 sin 

h 2 



Now let Ci = A cos 0o, c 2 = A sin , where A and are constants. 
It is clear that A can always be taken positive and equal to 
Vci 2 + c 2 2 and a real can be determined so that these equations 
will be satisfied whatever real values Ci and C 2 may have. Then 
the equation for the orbit becomes 



(38) 



5 + A cos (0 - 



63] AND INVERSELY AS FIFTH POWER OF DISTANCE. 93 

This is the polar equation of a conic with the origin at one of the 
foci. 

From this investigation and that of Art. 55 it follows that if the 
orbit is a conic section with the origin at one of the foci, and the 
force depends on the distance alone, then the body moves subject 
to a central force varying inversely as the square of the distance; 
and conversely, if the force varies inversely as the square of the 
distance, then the body will describe a conic section with the 
origin at one of the foci. 

Let p represent the parameter of the conic and e its eccentricity. 
Then, comparing (38) with the ordinary polar equation of the 

f) 

conic, r = ^ - , it is found that 

1 + e cos 



(39) 



h* 
P =17* 



and 0o is the angle between the polar axis and the end of the 
major axis directed to the farthest apse. The constants h 2 and A 
are determined by the initial conditions, and they in turn define 
p and e by (39). If e < 1, the conic is an ellipse; if e = 1, the 
conic is a parabola; lie > 1, the conic is a hyperbola; and if e = 0, 
the conic is a circle. 

63. Force Varying Inversely as the Fifth Power of the Distance. 

k 2 

In this case / = -g , and (25) becomes 



(40) 

u 
On solving for -r and integrating, it is found that 



Therefore 

(42) de - du 



The right member of this equation cannot in general be integrated 
in terms of the elementary functions, but it can be transformed 
into an elliptic integral of the first kind. Then u, and conse- 
quently r, is expressible in terms of by elliptic functions, and the 



94 FORCE VARYING INVERSELY AS [63 

orbits in general either wind into the origin or pass out to infinity, 
their particular character depending upon the initial conditions. 

There are certain special cases which are integrable in terms of 
elementary functions. 

(a) If such a constant value of u is taken that it fulfills (41) 
when its right member is set equal to zero, then r is a constant 
and the orbit is a circle with the origin at the center. It is easily 
seen that a similar special case will occur for a central force vary- 
ing as any power of the distance. 

(6) Another special case is that in which the initial conditions 
are such that c x =h and the right member of (41) is a perfect 

h 2 
square. That is, c\ = -. Then equation (41) becomes 



_ . _ u >__ 

u ~ A 



The integral of this equation is 

, 1 + A*U 

lo *l-A*u 
whence 



_ 

where coth ^- ( =t 6 c 2 ) is the hyperbolic cotangent of 

iV2( 0-c 2 ). 

(c) If the initial conditions are such that c\ = 0, equation (41) 
gives 



* a . *' 




the integral of which is 



On taking the cosines of both members and solving for r, the polar 
equation of the orbit is found to be 

k 
(44) r = -j=- cos (c 2 =F 0) , 

which is the equation of a circle with the origin on the circum- 
ference. 



63] THE FIFTH POWER OF THE DISTANCE. 95 

(d) If none of these conditions is fulfilled the right member of 
(41) is a biquadratic, and equation (42) can be written in the form 

, A r\ 7 fl Cdu 

(45) == d& = 7 



where C, a 2 , and /3 2 are constants which depend upon the coefficients 
of (41) in a simple manner. Equation (45) leads to an elliptic 
integral which expresses in terms of u. On taking the inverse 
functions and the reciprocals, r is expressed as an elliptic function 
of 0. The curves are spirals of which the circle through the origin, 
and the one around the origin as center, are limiting cases. 

If the curve is a circle through the origin the force varies in- 
versely as the fifth power of the distance (Art. 56); but if the 
force varies inversely as the fifth power of the distance, the orbits 
which the particle will describe are curves of which the circle is a 
particular limiting case. On the other hand, if the orbit is a 
conic with the origin at the center or at one of the foci, the force 
varies directly as the distance, or inversely as the square of the 
distance ; and conversely, if the force varies directly as the distance, 
or inversely as the square of the distance, the orbits are always 
conies with the origin at the center, or at one of the foci respectively 
[Arts. 53, 55, 56 (6)]. A complete investigation is necessary for 
every law to show this converse relationship. 

IX. PROBLEMS. 

1. Discuss the motion of the particle by the general method for linear 
equations when the force varies inversely as the cube of the distance. Trace 
the curves in the various special cases. 

2. Express C, a 2 , and /9 2 of equation (45) in terms of the initial conditions. 
For original projections at right angles to the radius vector investigate all the 
possible cases, reducing the integrals to the normal form, and expressing r as 
elliptic functions of 6. Draw the curves in each case. 

3. Suppose the law of force is that given in (29); i. e. 

M M 



e 



r z (C sin 26 + D cos 26 + 



Integrate the differential equation of the orbit, (25), by the method of vari- 
ation of parameters, and show that the general solution has the form 

1 

- = Ci cos 6 + Cz sin d + "V0(0), 

where c\ and Cn are constants of integration. Show that the curve is a conic. 



96 PROBLEMS. 



4. When the force is / = -^ + -^ show that, if v < A 2 , the general equa- 
tion of the orbit described has the form 

a 



I - e cos(^) ' 

where a, e, and k are the constants depending upon the initial conditions and 
fj. and v. Observe that this may be regarded as being a conic section whose 
major axis revolves around the focus with the mean angular velocity 

n = (l-fc)Y' 
where T is the period of revolution. 

5. In the case of a central force the motion along the radius vector is 
denned by the equation 

^ = _ / + ^! 

dP J ^ r 3 ' 

Discuss the integration of this equation when 



6. Suppose the law of force is that given by (30) ; i. e., 

N 



r 2 ( - A sin e B cos Y 



Substitute in (25) and derive the general equation of the orbit described. 
Hint. Let u = v + Asm&-\-B cos 6', then (25) becomes 



Ans. - = A sin 6 + B cos 6 + Vci cos 2 6 + c 2 sin 20 + c 3 sin 2 9, 
r 

which is the equation of a conic section. 

7. Suppose the law of force is 

_ ci + c 2 cos 26 

/ r 2 

show that, for all initial projections, the orbit is an algebraic curve of the 
fourth degree unless c 2 = 0, when it reduces to a conic. 



HISTORICAL SKETCH. 97 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The subject of central forces was first discussed by Newton. In Sections 
ii. and in. of the First Book of the Principia he gave a splendid geometrical 
treatment of the subject, and arrived at some very general theorems. These 
portions of the Principia especially deserve careful study. 

All the simpler cases were worked out in the eighteenth century by analyti- 
cal methods. A few examples are given in detail in Legendre's Traite des 
Fonctions Elliptiques. An exposition of principles and a list of examples 
are given in nearly every work on analytical mechanics; among the best of 
these treatments are the Fifth Chapter in Tait and Steele's Dynamics of a 
Particle, and the Tenth Chapter, vol. i., of AppelFs Mecanique Rationelle. 
Stader's memoir, vol. XLVI., Journal fur Mathematik, treats the subject in 
great detail. The special problem where the force varies inversely as the 
fifth power of the distance has been given a complete and elegant treatment 
by MacMillan in The American Journal of Mathematics, vol. xxx, pp. 282-306. 

The problem of finding the general expression for the possible laws of force 
operating in the binary star systems was proposed by M. Bertrand in vol. 
LXXXIV. of the Comptes Rendus, and was immediately solved by MM. Darboux 
and Halphen, and published in the same volume. The treatment given above 
in the text is similar to that given by M. Darboux, which has also been repro- 
duced in a note at the end of the Mecanique of M. Despeyrous. The method 
of M. Halphen is given in Tisserand's Me anique Celeste, vol. i., p. 36, and in 
Appell's Mecanique Rationelle, vol. i., p. 372. It seems to have been generally 
overlooked that Newton had reated the same problem in the Principia, 
Book i., Scholium to Proposition xvn. This was reproduced and shown to 
be equivalent to the work of MM. Darboux and Halphen by Professo: Glaisher 
in the Monthly Notices f R.A.S., vol. xxxix. 

M. Bertrand has shown (Comptes Rendus, vol. LXXVII.) that the only laws 
of central force under the action of which a particle will describe a conic 

fc 2 
section for all initial conditions are/ = =*= -^ and/ = k*r. M. Koenigs has 

proved (Bulletin de la Societe Mathematique, vol. xvn.) that the only laws of 
central force depending upon the distance alone, for which the curves de- 
ft 2 
scribed are algebraic for all initial conditions are / = =*= -5 and / = =*= & 2 r. 

Griffin has shown (American Journal of Mathematics, vol. xxxi.) that the 
only law, where the force is a function of the distance alone, where it does not 
vanish at the center of force, and where it is real throughout the plane, giving 
an elliptical orbit is the Newtonian law. 



CHAPTER IV. 

THE POTENTIAL AND ATTRACTIONS OF BODIES. 

64. THE previous chapters have been concerned with problems 
in which the law of force was given, or with the discovery of the 
law of force when the orbits were given. All the investigations 
were made as though the masses were mere points instead of being 
of finite size. When forces exist between every two particles of 
all the masses involved, bodies of finite size cannot be assumed to 
attract one another according to the same laws. Hence it is neces- 
sary to take up the problem of determining the way in which 
finite bodies of different shapes attract one another. 

It follows from Kepler's laws and the principles of central forces 
that, if the planets are regarded as being of infinitesimal dimen- 
sions compared to their distances from the sun, they move under 
the influence of forces which are directed toward the center of the 
sun and which vary inversely as the squares of their distances 
from it. This suggests the idea that the law of inverse squares 
may account for the motions still more exactly if the bodies are 
regarded as being of finite size, with every particle attracting every 
other particle in the system. The appropriate investigation shows 
that this is true. 

This chapter will be devoted to an exposition of general methods 
of finding the attractions of bodies of any shape on unit particles 
in any position, exterior or interior, when the forces vary inversely 
as the squares of the distances. The astronomical applications 
will be to the attractions of spheres and oblate spheroids, to the 
variations in the surface gravity of the planets, and to the per- 
turbations of the motions of the satellites due to the oblateness of 
the planets. 

65. Solid Angles. If a straight line constantly passing through 
a fixed point is moved until it retakes its original position, it gener- 
ates a conical surface of two sheets whose vertices are at the given 
point. The area which one end of this double cone cuts out of the 
surface of the unit sphere whose center is at the given point is 
called the solid angle of the cone; or, the area cut out of any con- 

98 



65] SOLID ANGLES. 99 

centric sphere divided by the square of its radius measures the 
solid angle. 

Since the area of a spherical surface equals the product of 4?r 
and the square of its radius, it follows that the sum of all the solid 
angles about a point is 4vr. The sum of the solid angles of one-half 
of all the double cones which can be constructed about a point 
without intersecting one another is 2?r. 

The volume contained within an infinitesimal cone whose solid 
angle is co and between two spherical surfaces whose centers are 
at the vertex of the cone, approaches as a limit, as the surfaces 
approach each other, the product of the solid angle, the square of 
the distance of the spherical surfaces from the vertex, and the 
distance between them. If the centers of the spherical surfaces 
are at a point not in the axis of the cone, the Volume approaches 
as a limit the product of the solid angle, the square of the distance 




Fig. 10. 

from the vertex, the distance between the spherical surfaces, and 
the reciprocal of the cosine of the angle between the axis of the 
cone and the radius from the center of the sphere; or, it is the 
product of the solid angle, the square of the distance from the 
vertex, and the intercept on the cone between the spherical 
surfaces. Thus, the volume of abdc, Fig. 10, is V = o>a0 2 ab. 
The volume of a'b'd'c' is 

T7 , coa'Q 2 b'e' -^ ,,, 

* ~ fr\ tr\t\ = uaO -ab. 

cos (Oa'O) 

Sometimes it will be convenient to use one of these expressions and 
sometimes the other. 

66. The Attraction of a Thin Homogeneous Spherical Shell 
upon a Particle in its Interior. The attractions of spheres and 
other simple figures were treated by Newton in the Principia, 



100 ATTRACTION OF ELLIPSOIDAL SHELLS [67 

Book i., Section 12. The following demonstration is essentially 
as given by him. 

Consider the spherical shell contained between the infinitely 
near spherical surfaces S and S', and let P be a particle of unit 
mass situated within it. Construct an infinitesimal cone whose 




A' 

Fig. 11. 

solid angle is w with its vertex at P. Let a be the density of 
the shell. Then the mass of the element of the shell at A is 
m = aABuAP ; likewise the mass of the element of A' is 
m' vA'B'uA'P . The attractions of m and m! upon P are 
respectively 

k 2 m f 



a. = =3 , a = 
AP 



Since A'B' = AB, a = WABuv = a'. This holds for every infini- 
tesimal solid angle with vertex at P; therefore a thin homogeneous 
spherical shell attracts particles within it equally in opposite directions. 
This holds for any number of thin spherical shells and, therefore, 
for shells of finite thickness. 

67. The Attraction of a Thin Homogeneous Ellipsoidal Shell 
upon a Particle in its Interior. The theorem of this article is 
given in the Principia, Book i., Prop, xci., Cor. 3. 

Let a homoeoid be defined as a thin shell contained between two 
similar surfaces similarly placed. Thus, an elliptic homoeoid is a 
thin shell contained between two similar ellipsoidal surfaces simi- 
larly placed. 

Consider the attraction of the elliptic homoeoid whose surfaces 
are the similar ellipsoids E and E f upon the interior unit particle P. 
Construct an infinitesimal cone whose solid angle is co with vertex 



68] 



UPON AN INTERIOR PARTICLE. 



101 



at P. The masses of the_two_infinitesimal elements at A and A' 
are respectively m = aABuAP and m r = aA'B'uA'P*. The 

k 2 m k^m' 

attractions are a = =5 and a = -==5 . Construct a diameter 
AP A P 

CC' parallel to A A' in the elliptical section of a plane throughjhe 
cone and the center of the ellipsoids, and draw its conjugate DD' . 
They are conjugate diameters in both elliptical sections, E and 
E'\ therefore DD' bisects every chord parallel to CC' , and hence 
AB = A'B'. The attractions of the elements at A and A' upon 




P are therefore equal. This holds for every infinitesimal solid 
angle whose vertex is at P; therefore the attractions of a thin elliptic 
homoeoid upon an interior particle are equal in opposite directions. 
This holds for any number of thin shells and, therefore, for 
shells of finite thickness. 

68. The Attraction of a Thin Homogeneous Spherical Shell 
upon an Exterior Particle. Newton's Method. Let AH KB 

and ahkb be two equal thin spherical shells with centers at and o 





Fig. 13. 



respectively. Let two unit particles be placed at P and p, unequal 
distances from the centers of the shells. Draw any secants from p 
cutting off the arcs il and hk, and let the angle kpl approach zero 
as a limit. Draw from P the secants PL and PK, cutting off the 



102 ATTRACTION OF THIN SPHERICAL SHELLS [68 

arcs IL and HK equal respectively to il and hk. Draw oe per- 
pendicular to pi, od perpendicular to pk, iq perpendicular to pb, 
and ir perpendicular to pk. Draw the corresponding lines in the 
other figure. 

Rotate the figures around the diameters PB and pb, and call 
the masses of the circular rings generated by HI and hi, M and m 
respectively; then 

(1) HI X IQ : hi X iq = M : m. 

The attractions of unit masses situated at / and i are respectively 
proportional to the inverse squares of PI and pi. The com- 
ponents of these attractions in the directions PO and po are the 

respective attractions multiplied by ~p and . respectively. If 

T J. jUv 

A' and a' represent the components of attraction toward and o, 
then 

(21 A >. a > --PQ.n 

~ PI' PI V Pi' 

Now consider the attractions of the rings upon P and p. Their 
resultants are in the directions of and o respectively because 
of the symmetry of the figures with respect to the lines PO and po, 
and they are respectively M and m times those of the unit particles. 
Let A and a represent the attractions of M and m; then 

M. PQ ^L 2 = HI x 7 # ?E hi x i( j vf_ 

~PI*PI : ^pi = Pf ~ PC*' pf po' 

In order to reduce the right member of (3) consider the similar 
triangles PIR and PFD and the corresponding triangles in the 
other figure. At the limit as the angles KPL and kpl approach 
zero, DF : df = 1 because the secants IL and HK respectively 
equal il and hk. Therefore 

PI : PF = RI : DF, 

pf:pi = DF(= df) :ri, 
and the product of these proportions is 

(4) PI X pf : PF X pi = RI : ri = HI : hi. 

From the similar triangles PIQ and POE, it follows that 

PI : PO = IQ : OE, 
and similarly 

po : pi = OE( = oe) : iq. 



69] UPON AN EXTERIOR PARTICLE. 103 

The product of these two proportions is 

(5) PI X po : PO X pi = IQ : iq. 
The product of (4) and (5) is 

Pl'xpfXpo: pi X PF X PO = HI X IQ : hi X iq. 
Consequently equation (3) becomes 

(6) A : a = po* : PO*. 

Therefore, the circular rings attract the exterior particles toward 
the centers of the shells with forces which are inversely propor- 
tional to the squares of the respective distances of the particles 
from these centers. In a similar manner the same can be proved 
for the rings KL and kl. 

Now let the lines PK and pk vary from coincidence with the 
diameters PB and pb to tangency with the spherical shells. The 
results are true at every position separately, and hence for all at 
once. Therefore, the resultants of the attractions of thin spherical 
shells upon exterior particles are directed toward their centers, and 
the intensities of the forces vary inversely as the squares of the distances 
of the particles from the centers. 

If the body is a homogeneous sphere, or is made up of homo- 
geneous spherical layers, the theorem holds for each layer sepa- 
rately, and consequently for all of them combined. 

69. Comments upon Newton's Method. While the demon- 
stration above is given in the language of Geometry, it really 
depends upon the principles which are fundamental in the Calculus. 
Letting the angle kpl approach zero as a limit is equivalent to 
taking a differential element; the rotation around the diameters is 
equivalent to an integration with respect to one of the polar angles; 
the variation of the line pk from coincidence with the diameter to 
tangency with the shell is equivalent to an integration with respect 
to the other polar angle; and the summation of the infinitely thin 
shells to form a solid sphere is equivalent to an integration with 
respect to the radius. 

Since Newton's method gives only the ratios of the attraction of 
equal spherical shells at different distances, it does not give the 
manner in which the attraction depends upon the masses of the 
finite bodies. This is of scarcely less importance than a knowl- 
edge of the manner in which it varies with the distance. 

In order to find the manner in which the attraction depends upon 
the mass of the attracting body, take two equally dense spherical 



104 THOMSON AND TAIT's METHOD. [70 

shells, Si and $ 2 , internally tangent to the cone C. Let POi = ai, 
P0 2 = 2, and MI and M 2 be the masses of Si and *S 2 respectively. 
The two shells attract the particle P equally; for, any solid angle 
which includes part of one shell also includes a similar part of the 
other. The masses of these included parts are as the squares of 




Fig. 14. 



their distances, and their attractions are inversely as the squares 
of their distances, whence the equality of their attractions upon 
P. Let A represent the common attraction; then remove Si so 
that its center is also at 2 . Let A' represent the intensity of the 
attraction of Si in the new position; then, by the theorem of 
Art. 68, 

A!_ = oi 2 = Mi 

A a 2 2 M 2 ' 

Therefore, the two shells attract a particle at the same distance with 
forces directly proportional to their masses. From this and the 
previous theorem, it follows that a particle exterior to a sphere which 
is homogeneous in concentric layers is attracted toward its center 
with a force which is directly proportional to the mass of the sphere 
and inversely as the square of the distance from its center-, or, as 
though the mass of the sphere were all at its center. 

Since the heavenly bodies are nearly homogeneous in concentric 
spherical layers they can be regarded as material points in the dis- 
cussion of their mutual interactions except when they are relatively 
near- each other as in the case of the planets and their respective 
satellites. 

70. The Attraction of a Thin Homogeneous Spherical Shell 
upon an Exterior Particle. Thomson and Tait's Method. Let 
be the center of the spherical shell whose radius is a and whose 
thickness is Ac, P the position of the attracted particle and PO a 
line from the attracted particle to the center cutting the spherical 
surface in C. Take the point A so that PO : OC = OC : OA, and 
construct the infinitesimal cone whose solid angle is co with its 
vertex at A. Let a be the density of the shell. Then the elements 
of mass at B and B' are respectively 



70] 



m = 



THOMSON AND TAIT*S METHOD. 

-r^i Aa , -r-^-72 Aa 



105 



m' = AB' 



cos (OBA) > cos (OB'A) ' 

The attractions of the two masses upon P are respectively 

Aa 



(7) 



a = 



a' = k 2 (ra> 



* cos (OBA) ' 

AB'* Aa 

&p* ' cos (OB' A) 




Fig. 15. 

From the construction of A it follows that 
PO : OB = OB : OA. 

Hence the triangles FOB and BOA, having a common angle in- 
cluded between proportional sides, are similar. Therefore 



AB 



Similarly 



BP 

AB' 
B'P 



OB 
OP 



a 
OP' 



a 
OP' 



The angle OBA equals the angle OB'A. Then equations (7) 

become 

i 2 Aa 



(8) 



OP 2 ' cos (OBA) ' 

a 2 Aa 

<5p ' cos (OBA) ~~ 



The angles BPO and B'PO are respectively equal to OBA and 
OB' A', therefore they are equal to each other. The resultant of 
the two equal attractions a and a' is in the line bisecting the angle 
between them, or in the direction of 0, and is given in magnitude 
by the equation 



106 ATTRACTION UPON PARTICLE IN THIN SHELL. [71 

Afl = a cos (BPO) + a' cos (B'PO) = 2a cos (OB A). 
This becomes, as a consequence of (8), 



OP 

This equation is true for every solid angle with vertex at A, and 
consequently for their sum. Therefore the attraction of the whole 
spherical shell upon the exterior particle is, on summing with 
respect to o>, 

D . I2 a 2 Aa k 2 M 

R = 47rA> 2 (7 ==- = ==- : 
OP OP 

or, the attraction varies directly as the mass of the shell and in- 
versely as the square of the distance of the particle from its center. 

71. The Attraction upon a Particle in a Homogeneous Spherical 
Shell. In Arts. 66-69 the attractions of a thin homogeneous 
spherical shell upon an interior and an exterior particle, respec- 
tively, have been discussed; the problem is now completed by 
treating the case where the attracted particle is a part of the shell 
itself. 

Let be the center of the spherical shell of thickness Aa, 
and P the position of the attracted particle. Construct a cone 
whose solid angle is co with its vertex at P. Let a- be the den- 




Fig. 16. 

sity of the shell; then the mass of the section cut out at A by 

the cone is o-coAP - fr . . n . . The attraction of the element 
cos (OAP) 

along AP is a = ArW ==5 (r . A m . The resultant attraction 

AP cos (OAP) 

of the shell is in the direction PO since the mass is symmetrically 
situated with respect to this line. The component in the direction 
POis 

AR = a cos (APO) = a cos (OAP) = 



PROBLEMS. 107 

The attraction of the whole shell is 
R = 



It follows from this equation and the results obtained in Arts. 
66 and 69 that the attraction on an interior particle infinitely near 



the shell is zero, on a particle in the shell, -- , and on an exterior 



particle infinitely near the shell, 5- .* The discontinuity in the 

attraction is due to the fact that the mass of any finite area of the 
shell is assumed to be finite although it is supposed to be infinitely 
thin. There is no such discontinuity at the surface of a solid sphere 
because an infinitely thin shell taken from it has only an infini- 
tesimal mass. 

X. PROBLEMS. 

1. Suppose any two similar bodies are similarly placed in perspective. 
Show that a particle at their center of perspectivity is attracted inversely as 
their linear dimensions if they are thin rods of equal density; equally, if they 
are thin shells of equal density; and directly as their linear dimensions if 
they are solids of equal density. Consider a nebula which is apparently as 
large as the sun. Suppose its distance is one million times that of the sun 
and that its density is one millionth that of the sun. Compare its attraction 
for the earth with that of the sun. 

2. Prove that the attractions of two homogeneous spheres of equal density 
for particles upon their surfaces are to each other as their radii. 

3. Prove that the attraction of a homogeneous sphere upon a particle in 
its interior varies directly as the distance of the particle from the center. 

4. Prove that all the frustums of equal height of any homogeneous cone 
attract a particle at its vertex equally. 

5. Find the law of density such that the attraction of a sphere for a particle 
upon its surface shall be independent of the size of the sphere. 

6. Prove. that the attraction of a uniform thin rod, bent in the form of 
an arc of a circle, upon a particle at the center of the circle is the same as 
that which the mass of a similar rod equal to the chord joining the extremities 
would exert if it were concentrated at the middle point of the arc. 

7. Prove that the attraction of a thin uniform straight rod on an exterior 
particle is the same in magnitude and direction as that of a circular arc of the 
same density, with its center at the particle and subtending the same angle 
as the rod, and which is tangent to the rod. 

* See note on the attraction of spherical shells, Lagrange, Collected Works, 
vol. vii., p. 591. 



108 



EQUATIONS FOR COMPONENTS OF ATTRACTION. 



[72 



8. Prove that if straight uniform rods form a polygon all of whose sides 
are tangent to a circle, a particle at the center of the circle is attracted equally 
in opposite directions by the rods. 

9. Prove that two spheres, homogeneous in concentric spherical layers, 
attract each other as though their masses were all at their respective centers. 

72. The General Equations for the Components of Attraction 
and for the Potential when the Attracted Particle is not a Part 
of the Attracting Mass. The geometrical methods of the pre- 
ceding articles are special, being efficient only in the particular 
cases to which they are applied; the analytical methods which 
follow are characterized by their uniformity and generality, and 
illustrate again the advantages of processes of this nature. 

Consider the attraction of the finite mass M whose density is a 
upon the unit particle P, which is not a part of it. That is, P is 
exterior to M or within some cavity in it. Let the coordinates 
of P be x, y, z. Let the coordinates of any element of mass dm 




Fig. 17. 

be , y, f , and the distance from dm to P be p. Then the com- 
ponents of attraction parallel to the coordinate axes are respectively 



(9) 1 



where 



--f **. 

J(M) p 2 



X 

Y = - 

Z = - 



M) p 



p 3 



dm, 



72] 



EQUATIONS FOR COMPONENTS OF ATTRACTION. 



109 



dm = <rddiidt, 

P 2 = (x - ) 2 + (y - r;) 2 + (z - 



The integral sign J signifies that the integral must be extended 

over the whole mass M. Then, if a is a finite continuous function 
of the coordinates, as will always be the case in what follows, X, 
Y, and Z are finite definite quantities. In practice dm is expressed 
in terms of <r and the ordinary rectangular or polar coordinates, 
and X, Y, and Z are found by triple integrations. 

The three integrals (9) can be made to depend upon a single 
integral in a very simple manner. Let 

dm 



do) 



-L 



V is called the potential function, the term having been introduced 
by Green in 1828. It is a function of x, y, and z and will be 
spoken of as the potential of M upon P at the point (x, y, z). 

Since P is not a part of the mass M, p does not vanish in the 
region of integration. The limits of the integral are independent 
of the position of the attracted particle; therefore the function 
under the integral sign can be differentiated with respect to 
x, y, z which are treated as constants in computing the definite 
integrals. The partial derivatives of V with respect to x, y 
and z are 



dx 



dy 

dv = r 

dz ~ ~ J(A 



dm, 
dm. 



(M) p 

On comparing these equations with (9), it is found that 



(11) 



110 EQUATIONS FOR COMPONENTS OF ATTRACTION [73 

Therefore, in the case in which P is not a part of M, the solution 
of the problem of finding the components of attraction depends 
upon the computation of the single function V. 

73. Case where the Attracted Particle is a Part of the Attracting 
Mass. It will now be proved that the components of attraction 
and the potential have finite, definite values when the particle is 
a part of the attracting mass, and that equations (11) also hold in 
this case. 

In order to show first that X, Y, Z, and V have finite, determi- 
nate values in this case, let dm and its position be expressed in 
polar coordinates with the origin at the attracted particle P. The 
equations expressing the rectangular coordinates in terms of the 
polar with the origin at P are 

x = p cos (p cos 6, 

y = P cos (p sin 0, 
- z = p sin (p, 

dm = ap 2 cos (pd<pdddp. 

Then the expressions for the components of attraction and the 
potential become 

' X = - Wfffo- cos 2 (p cosed<pd6dp, 
Y = - k 2 fffa cos 2 <p smed<pd6dp, 
Z = k 2 fff a sin <p cos (p d<p dd dp, 
V = + /// ap cos <p d(p dd dp, 

where the limits are to be so determined that the integration 
shall be extended throughout the whole body M. The integrands 
are all finite for all points in M, and therefore the integrals have 
finite, determinate values. 

The simplest method of proving that equations (11) hold when 
P is in the attracting mass M is to start from the definition of the 
derivative of V with respect to x. By definition 

ay = lim v - v 

dx A *=o Ax 

where V is the potential at the point P 1 whose coordinates are 
(x -f Ax, y, z). Construct a small sphere of radius e enclosing 
both P and P'. Let the mass contained within the sphere c be 



73] 



WHEN PARTICLE IS PART OF ATTRACTING MASS. 



Ill 



represented by M i and that outside of it by 7kf 2 . Let the corre- 
sponding parts of the components of attraction and the potential 
be distinguished by the subscripts 1 and 2. Then 




Fig. 18. 



(12) 



because all of these quantities are uniquely defined. Moreover, 
it follows from Art. 72 that 



X* = k* 



dV, 



= k< 



<97 2 



d7s 



dx ' dy ' dz ' 

Now consider the derivative of V with respect to x. It becomes 



(13) 



dV .. 
= hm 
dx A *=o 



i , ,. 
+ hm 



TV - 7 2 



Let the distance from P to dm be p, and from P f to dm be p f . 
Then 



Ax 



_ 

p f p I Ax * 



It follows from the triangle P dm P' that | Ax | ^ | p' - p |, 
where the vertical lines on a quantity indicate that its numerical 
value is taken. Hence it follows that 



Therefore 



Ax 



= PP '=2 

< l f dm,,\C dm 
-2 JIM P 2 + 2j(j/, )/> ' 2 ' 



112 EQUATIONS FOR COMPONENTS OF ATTRACTION. [73 

When dm is expressed in polar coordinates this inequality becomes 



Ax 



i ri r Zir f p 

^ - I I a cos <p dtp dd dp 

* J_ E J Jo 



cos 



Let o-o be the maximum value of <r in e. The result of integrating 
with respect to p and p' is 



TV - 



P cos 



cos 



Since P and P' are in the sphere e the distances p and p' cannot 
exceed 2c. Then 



and 



IT 
/^T /2T 

I I cos <p'd<p'dd' = STTO-Q e, 



lim 

Aa:=0 



V l f -V l 



ft 

07T(7o . 



It follows from this inequality and (13) that 



fc 2 - 
ox 



< /c 2 
dx 



Now pass to the limit e = 0. The limit of Xi, for e = 0, is 
easily proved to be zero by using polar coordinates. Hence it 
follows from (12) that 

lim X 2 = X, 



e=0 



and consequently, from the last inequalities, 



The corresponding relations for derivatives with respect to y 
and z are proved similarly, and therefore equations (11) hold 
whether or not P is a part of M. 



75] POTENTIAL AND ATTRACTION OF CIRCULAR DISC. 113 

74. Level Surfaces. The equation V = c, where c takes con- 
stant values, defines what are called level surfaces or equipotential 
surfaces. 

Any displacement bx, dy, 5z, of the particle from the point 
(XQ, i/o, 2o) in a level surface must fulfill the equation 



which is the condition that the points (XQ, T/O, 2o) and (XQ -f dx, 
2/o + 8y, ZQ + 8z) shall both be in the same level surface. This 
equation becomes as a consequence of (11) 

(14) Xdx + Ydy + Zdz = 0. 

The direction cosines of the resultant attraction to which the 
particle is subject are proportional to X, Y, Z, and the direction 
cosines of the line of the displacement are proportional to 8x } 
8y, 8z. Since the sum of the products of these direction cosines 
in corresponding pairs is zero, it follows that the resultant attrac- 
tion is perpendicular to the level surfaces. Consequently, if the 
particle starts from rest it will begin to move perpendicularly to 
the level surface through- its initial position; but after it has 
acquired an appreciable velocity it will not in general move 
perpendicularly to the level surfaces because the motion depends 
not only upon the forces, which have been shown to be orthogonal 
to the level surfaces, but also upon the velocity. 

75. The Potential and Attraction of a Thin Homogeneous 
Circular Disc upon a Particle in its Axis. Take the origin at th e 
center of the disc whose radius is R. Let the coordinates of P be 
x, 0, 0. Then 

Cdm C R C 2 rdrdd 

V = I = a I -. _ 

J P Jo Jo Ate 2 + r 2 

Upon integrating, it is found that 

[ V = 27r<r[ Vz 2 + R 2 ~ Vz 2 ], 

(15) ^ ^ ^dV 1 ,. T x x_-\ 




If x is kept constant and R is made to approach infinity as a limit, 
the attraction becomes 

(16) X = =t= 27rA; 2 (7, 

9 



POTENTIAL AND ATTRACTION OF 



[76 



according as the particle is on the positive or negative side of the yz- 
plane. The right member of this equation does not depend upon x; 
therefore a thin circular disc of infinite extent attracts a particle 
above it with a force which is independent of its altitude. Any 




Fig. 19. 

number of superposed discs would act jointly in the same manner. 
Hence, if the earth were a plane of infinite extent, as the ancients 
commonly supposed, bodies would gravitate toward it with 
constant forces at all altitudes, and the laws of falling bodies 
derived under the hypothesis of constant acceleration would be 
rigorously true. 

76. The Potential and Attraction of a Thin Homogeneous 
Spherical Shell upon an Interior or an Exterior Particle. Let 




(ff.O.O) 



Fig. 20. 

represent the angle between OP and the radius, and 6* the angle 
between the fundamental plane and the plane OAP. Then 



(17) V = f = a- C C 



* It must be noticed that the and 6 here are not the ordinary polar angles 
used elsewhere. 



76] THIN HOMOGENEOUS SPHERICAL SHELL. 115 

One of the three variables 0, 6, p must be expressed in terms of 
the remaining two. From the figure it is seen that 

p 2 = x 2 + R 2 - 2xR cos 0; 
whence 

(18) pdp = xR sin 

Then (17) becomes, if P is exterior, 



rtir 

-, 1 dpde ' 



and if P is interior, 



The integrals of these equations are respectively 

M 



M 
R' 

The z-components of attraction are respectively 

i^U^...-** 

(22) 



which agree with the results obtained in Arts. 66 and 70. 

The attraction of a solid homogeneous sphere also can be found 
at once. Considering the shell as an element of the sphere, the 
M of (22) is given by the equation 

M = 



Let X represent the attraction of the whole sphere M\ then 



. 

3 z 2 x 2 

Consider the mutual attraction of two spheres. In accordance 
with the results which have just been obtained, each one attracts 
every particle of the other as it would if its mass were all at its 
center. Hence the two spheres attract each other as they would 
if their masses were all at their respective centers. 

77. Second Method of Computing the Attraction of a Homo- 
geneous Sphere. A very simple method will now be given of 
finding the attraction of a solid homogeneous sphere upon an 



116 



SECOND METHOD OF COMPUTING 



[77 



exterior particle when it is known for interior particles. It is a 
trivial matter in this case and is introduced only because the 
corresponding device in the much more difficult case of the attrac- 
tions of ellipsoids is of the greatest value, and constitutes Ivory's 
celebrated method. 

Let it be required to find the attraction of the sphere S upon 
the exterior particle P', supposing it is known how to find the 
attraction upon interior particles. Construct the concentric 




Fig. 21. 

sphere S' through P' and suppose it has the same density as S. 
A one-to-one correspondence between the points on the surfaces 
of the two spheres is established by the relations 



/nn\ 

(23) 



x = 7 



z = - 7 z. 



The corresponding points are in lines passing through the common 
center of the spheres, and P corresponds to P'. Let X and X' 
represent the attractions of S' and S upon P and P' respectively. 
They are given by the equations 



(24) 



X = - k 2 C ^^- dm' = -k*<r C C C^^-dx'dy'dz', 
Joso p J J J P 



Jw P 3 
But it follows from the definition of p and p' that 



(25)- 



77] ATTRACTION OF HOMOGENEOUS SPHERE. 117 

where p 2 and pi are the extreme values of p obtained by integrating 
with respect to x. That is, the first integration gives the attrac- 
tion of an elementary column extending through the sphere parallel 
to the X-axis, and pi and p 2 are the distances from the attracted 
particle P' to the ends of this column. In completing the inte- 
gration the sum of all of these elementary columns is taken. The 
corresponding statements with respect to the first equation of (25) 
are true. 

Suppose the integrals (25) are computed in such a manner that 
corresponding columns of the two spheres are always taken at 
the same time. Consider any two pairs of corresponding elements, 
as those at A and A' '. For these p = p', and this relation holds 
throughout the integration as arranged above. Hence it follows 
from equations (24) and (25) that 

rr/i i \ c c i i i \ 

X f = k 2 v I ) dydz = k 2 a I I . -. } dydz. 

J J \P2 PI/ J J \P2 PI / 

But, from (23), 

R R 

therefore 

R*JJ \Pz Pi'/ R' 2 

Let M represent the mass of the sphere S, and M' that of '. 
The attraction of S f upon the interior particle P is given by 

X = - 



therefore it follows from the relation R'*X f = R 2 X that the 
attraction of S upon the exterior particle P' is 



agreeing with results previously obtained (Arts. 69, 70). 



118 PROBLEMS. 



XI. PROBLEMS. 

1. Prove by the limiting process that the potential and components of 
attraction have finite, determinate values, and that equations (11) hold when 
the particle is on the surface of the attracting mass. 

2. Find the expression for the potential function for a particle exterior to 
the attracting body when the force varies inversely as the nth power of the 

distance. 

, r 1 C dm 

Y - 



3. Find by the limiting process for what values of n the potential in the 
last problem is finite and determinate when the particle is a part of the at- 
tracting mass. 

4. Show that the level surfaces for a straight homogeneous rod are prolate 
spheroids whose foci are the extremities of the rod. 

5. Find the components of attraction of a uniform hemisphere, whose 
radius is R, upon a particle on its edge: (a) in the direction of the center of 
its base; (6) perpendicular to this direction in the plane of the base; (c) per- 
pendicular to these two directions. 

Am. (a) X = liraWR', (&) 7 = 0; (c) Z = &k*R. 

6. Find the deviation of the plumb-line due to a hemispherical hill of 
radius r and density a\. Let R represent the radius of the earth, assumed 
to be spherical, and <r 2 its mean density. 

Ans. If X is the angle of deviation, 



- . D . - - , 

f 7TO-2/C f o\r wa-iR air 
or 

tan X = - -^ approximately. 

7. Prove that if the attraction varies directly as the distance, a body of 
any shape attracts a particle as though its whole mass were concentrated at 
its center of mass. 



78] POTENTIAL AND ATTRACTION OF OBLATE SPHEROID. 119 



78. The Potential and Attraction of a Solid Homogeneous 
Oblate Spheroid upon a Distant Unit Particle. The planets are 
very nearly oblate spheroids, and they are so nearly homogeneous 
that the results obtained in this article will represent the actual facts 
with sufficient approximation for most astronomical applications. 

Suppose the attracted particle is remote compared to the 
dimensions of the attracting spheroid. Take the origin of co- 



P (x, y, z) 




Fig. 22. 

ordinates at the center of the spheroid with the 2-axis coinciding 
with the axis of revolution. Let R represent the distance from 
to P, and r the distance from to the element of mass. Then 

dm 



(26) 



= C d JH 

J(S) p 



p = 
R = 



- ) 2 + (y - r?) 2 +(z- 



+ y 2 4- 



I r = V? + r? 2 + r 2 - 
It follows from these equations that 

1 = 1 = 

P V# 2 + r 2 



1 



1 + 



r 2 - 2(sg + yjj + 
R 2 



-5 , 15 , and j^ be taken as small quantities of the first 
K K K 

order; then, on expanding the expression for p" 1 by the binomial 
theorem, it is found that, up to small quantities of the third order, 

+ vn + zf r 2 






P = < 



120 POTENTIAL AND ATTRACTION OF OBLATE SPHEROID. [78 

Therefore 



(27) 



+5/1* *+;-' 

Let M represent the mass of the spheroid; then 
J dm = M, 

and, since the origin is at the center of gravity, 

/r r 

dm = 0, I rjdm = 0, I f dm = 0. 
J J 

Let 0- represent the density; then 

dm = or 2 cos (f)d<f>d0dr, 
% = r cos cos 0, 
77 = r cos sin 0, 

and (27) becomes 



cos 3 sin (9 cos Bd^dBdr 
f T Tr 4 sin cos 2 smdd<t>dddr 

sin * cos2 ^ cos 



where the limits of integration are: for r, and r; for </> ; - and 

~ ; and for 0, and 2?r. Since r and are independent of 0, the 
2i 

integration can be performed with respect to first, giving 



78] POTENTIAL AND ATTRACTION OF OBLATE SPHEROID. 121 



(28) 



+ 



TT 

r 4 si 
J_JL Jo 



sin 2 4> cos 4>d(f>dr -f 



the last three integrals being zero. 

The next integration must be made with respect to r, as this 
variable depends upon <f>. Let the major and minor semi-axes 
of a meridian section of the spheroid be a and b respectively, and 
let e be the eccentricity. Then 



1 - e 2 cos 2 </> ' 

Upon integrating (28) with respect to r and expanding in powers 
of e, it is found that, up to terms of the second order inclusive, 



= M 
= R 



IT 

I * (1 + fe 2 cos 2 < + ) cos </>d< 



COS 3 0C?0 



cos 2 + ) sin 2 cos<j>d<t> 



On integrating with respect to <f> and arranging in powers of 
the expression for V becomes 



But 



M = 



122 POTENTIAL AND ATTRACTION OF [79 

therefore 

o '- 



The components of attraction are found from equations (11) and 
(29) to be 



(30) 

Z= - ^ft [" 1 + ^V 3(X * + | 2 4 } ~ 2 * 2 e 2 + 1 . 

If the spheroid should become a sphere of the same mass, the 
expressions for the components of attraction would reduce to the 
first terms of the right members of equations (30) . If the attracted 
particle is in the plane of the equator of the attracting spheroid, 
2 = 0; and if it is in the polar line, x = y = 0. Hence it follows 
from (30) that the attraction of an oblate spheroid upon a particle 
at a given distance from the center in the plane of its equator is greater 
than that of a sphere of equal mass; and in the polar line, less than that 
of a sphere of equal mass. As the particle recedes from the at- 
tracting body the attraction approaches that of a sphere of equal 
mass. Therefore, as the particle recedes in the plane of the equator 
the attraction decreases more rapidly than the square of the distance 
increases; and as it approaches, the attraction increases more rapidly 
than the square of the distance decreases. The opposite results are 
true when the particle is in the polar line. 

79. The Potential and Attraction of a Solid Homogeneous 
Ellipsoid upon a Unit Particle in its Interior. Let the equation 
of the surface of the ellipsoid be 



and let the attracted particle be situated at the interior point 
(x, y, z). Take this point for the origin of the polar coordinates 
p, 6, and <f>. On taking the fundamental planes of this system 
parallel to those of the first system, these variables are related to 
the rectangular coordinates by the equations 



79] 



A SOLID HOMOGENEOUS ELLIPSOID. 



123 



( = x + p cos <f> cos 6, 
TI = y + p cos sin 0, 
f = z + p sin 0. 
The potential of the ellipsoid upon the unit particle P is 

C dm fT f 2jr p 1 

7 = = o- I I p cos d<j> d6 dp. 

J(M) P J~Jo Jo 

Since the value of p depends upon the polar angles the integration 
must be made first with respect to this variable. The integration 
gives 



(33) 



= I f * J 2ff 



V = 



Pl 2 cos d<j> de. 



To express pi in terms of the polar angles substitute (32) in (31); 
whence it is found that 



(34) 
where 



(35) 



A Pl 2 + 25 Pl + C = 0, 
. _ cos 2 cos 2 cos 2 sin 2 sin 2 

-^ O 7~0 I o 



R 



a; cos cos ^ y cos sin z sin 
~~ ~~ ~~ 



From (34) it is found that 



Pl= 



-B 



The only pi having a meaning in this problem is positive; A is 
essentially positive, and C is negative because (a;, y, z) is within 
the ellipsoidal surface. Therefore the positive sign must be 
taken before the radical. On substituting this value of pi in (33), 
it is found that 



* n f 
= 2Jn) 



foa\ 

(36) 



Consider the integral 



- AC) 



124 



POTENTIAL AND ATTRACTION OF 



[79 



It follows from the expression for B that the differential elements 
corresponding to 6 = , <t> = 4>o and to 6 = TT + , </> = <o are 
equal in numerical value but opposite in sign. Since all the 
elements entering in the integral can be paired in this way, it 
follows that Vi = 0, after which (36) becomes 



(37) H 



iiun 



COS 2 COS 2 



?~ c ) 






X ( -nr - 



cos 2 sin 2 



cos <f> d<j> dd 



s (f> sin cos B yz sin cos sin B 

zx sin cos cos 1 cos <ft d</> c?0 
c 2 a 2 ^^ ' 



By comparing the elements properly paired, it is seen that the 
second integral is zero. 
Let 



(38) TF=. f 2 f 2 " cos<t>d<j>dd 

2 J_5_ Jo cos 2 cos 2 cos 2 4> sh 

2 - | 



sn 2 sin 2 <j> ' 
~~ 



then (37) can be written in the form 
(39) V = - 



a da b db 



dc 



For a given ellipsoid W is a constant, and the equation of the 
level surfaces has the form 

Ciz 2 + C 2 y 2 + C 3 z 2 = constant, 

which is the equation of concentric similar ellipsoids, whose axes 
are proportional to Ci~*, C-r*, and Cs~*. 

In order to reduce W to an integrable form, let 



(40) 
then (38) becomes 



,, _ cos 2 
~~ 



cos 2 (/> 



79] A SOLID HOMOGENEOUS ELLIPSOID. 125 

w _ <L f^ f 2ir cos<j>d<f>de 

~ 2 J_2L Jo M COs2 + N Sin ' 



-J 1 / 

/o /o 



cos 4>d(j>d0 



M cos 2 + N sin 2 ' 



M and N are independent of 6] hence, on integrating with respect 
to this variable, it is found that* 



(41) 



f 



cos d<f> 



V (a 2 sin 2 + c 2 cos 2 0) (b 2 sin 2 + c 2 cos 2 0) 

To return to the symmetry in a, 6, and c which existed in (38), 
Jacobi introduced the transformation 



Vc 2 + s ' 
whence 



C-= 

Jo V(a 2 



TF = 



On forming the derivatives with respect to a, 6, and c, and substi- 
tuting in (39), it follows that 



V = iraabc C ( 1 - - -- r 
Jo \ a 2 -f s 6 2 



c 2 + 

<42> 



^/( a 2 _{_ g )(2 _j_ S )( C 2 ^_ s ) 

The components of attraction are 

2ir<rabcxk 2 ds 



= /c 2 = C 
daT' Jo a 



F ^2 ___ _ 
~~" A/ . 



(a 2 + s) V(a 2 - 

ds 



(43) 



Equation (41) is homogeneous of the second degree in a, b, 
* Letting tan = x, the integral reduces to one of the standard forms. 



(6 2 + s) V(a 2 + s) (6 2 + s) (c 2 + s) ' 

z = k 2 ?. = C 

dz ' Jo C 2 + s 



126 PROBLEMS. 

and c; and therefore , , , computed from (39), are ho- 
mogeneous of degree. zero in the same quantities. It follows, 
therefore, that if a, 6, and c are increased by any factor v the 
components of attraction X, F, and Z, will not be changed; or, 
the elliptic homoeoid contained between the ellipsoidal surfaces whose 
axes are a, b, c and va, vb, and vc attracts the interior particle P 
equally in opposite directions. (Compare Art. 67.) 

The component of attraction, X, is independent of y and z 
and involves x to the first degree; therefore the x-component of 
attraction is proportional to the x-coordinate of the particle and is 
constant everywhere within the ellipsoid in the plane = x. Similar 
results are true for the two other coordinates. 

Suppose the notation has been chosen so that a > b > c. 
Then (41) can be put in the normal form for an elliptic integral 
of the first kind by the substitution 

c u 

sm <p = 



2 a 2 -& 
K "STT-? 

which gives 

(44) w = 27r(ra&c r~~ au 



W - c 2 

This integral can be readily computed, when K 2 is small, by ex- 
panding the integrand as a power series in /c 2 and integrating 
term by term. 

XII. PROBLEMS. 

1. Discuss the level surfaces given by equation (29). 

2. Set up the expressions for the components of attraction instead of that 
for the potential as in Art. 79. Determine what parts of the integrals vanish, 
integrate with respect to 0, and show that the results are 

X = - lirvbcxk 2 



Q 
Z = - 



>l (ft 2 sin 2 + a 2 cos 2 0) (c 2 sin 2 -f a 2 cos 2 0) ' 

sin 2 cos d<f> 
V (c 2 sin 2 + 6 2 cos 2 0) (a 2 ~sin 2 + 6 2 "co& 2 0) ' 



F = - ivacayk* C - -= sin 2 cos 

J Q 



C 

J 



sin2 * cos 



V (a 2 sin 2 <f> + c 2 cos 2 0) (6 2 sin 2 tf> + c 2 cos 2 <) 



80] 



IVORY'S METHOD. 



127 



Hint. Derive the results for Z, and since it is immaterial in what order 
the axes are chosen, derive the others by a permutation of the letters a, b, c. 
3. Transform the equations of problem 2 by 

b 



sin <f> 



Va 2 + 



Sin 



sin = = 



respectively, and show that equations (43) result. 

4. Show that the potential of an ellipsoid upon a particle at its center is 

s 



Vo = icaabc 



= W. 



5. From the value of Vo and equations (43) derive the value of the po- 
tential (42). 

6. Transform the equations of problem 2 so that they take the form 

u?du 



f 



7. Integrate equations (28) without expanding the expression for r 2 as a 
power series in e 2 . 

80. The Attraction of a Solid Homogeneous Ellipsoid upon an 
Exterior Particle. Ivory's Method. The integrals become so 
complicated in the case of an exterior particle that the components 
of attraction have not been found by direct integration except in 
series. They are computed indirectly by expressing them in 




Fig. 23. 

terms of the components of attraction of a related ellipsoid upon 

particles in its interior. This artifice constitutes Ivory's method.* 

Let it be required to find the attraction of the ellipsoid E upon 

the exterior particle P' at the point (x', y', z'). Let the semi- 

* Philosophical Transactions, 1809. 



128 



ATTRACTION OF A SOLID ELLIPSOID. 



[80 



axes of E be a, b, and c. Construct through P' an ellipsoid E f , 
confocal with E, with the semi-axes a', b', c', and suppose it has 
the same density as E. The axes of the two ellipsoids are related 
by the equations 



(45) 



a = 



V = 




where K is defined by the equation 
(46) 



a 2 + 



b 2 + K ' c 2 + 



-1 = 0. 



The only value of K admissible in this problem is real and positive. 
Equation (46) is a cubic in K and has one positive and two negative 
roots; for, the left member considered as a function of K is negative 



/(/C) axis 




K-axts 



Fig. 24. 



when K = + ; positive, when K = (because (x' } y', z') is 
exterior to the ellipsoid E) ; positive, when K = c 2 + e (where t 
is a very small positive quantity); negative, when K = c 2 e; 
positive, when K = 6 2 + e; negative, when K = 6 2 e; posi- 
tive, when K = a 2 + e; negative, when K = a 2 e; and nega- 
tive when K = oo . The graph of the function is given in Fig. 24. 
When the positive root is taken, a', b', and c' are determined 
uniquely. 

A one-to-one correspondence between the points upon the two 



80J 



IVORYS METHOD. 



129 



ellipsoids will now be established by the equations (compare 
Art. 77) 

(47) r-j'fc *v-|* r-|r. 

Let P be the point corresponding to P f . It will be shown that 
the attraction of E upon P r is related in a very simple manner to 
that of E' upon P. 

Let X, F, and Z represent the components of attraction of E' 
upon the interior particle P at the point (x, y, z). They can be 
computed by the methods of Art. 79, and will be supposed known. 
Let X', y, and Z' be the components of attraction of E upon P', 
which are required. The expressions for the re-components are 



X = - 




On performing the integration with respect to , it is found that 



(49) 



where P2 and pi are the distances from P' to the ends of the ele- 
mentary column obtained by integrating with respect to . The 
solution is completed by integrating over the whole surface of E. 
The first equation of (49) is interpreted similarly. 

Now X' will be related to X in a simple manner by the aid of 
the following lemma: 

// P and A are any two points on the surface of E, and if P' and 
A' are the respective corresponding points on the surface of E', then 
the distances PA' and P'A are equal. 

Let ~PA f = p' and ~AP' = p. Then p = p'. For, let the 
coordinates of P and A be respectively 1,771, Ti and 2 , r?2, 2; and 
of P' and A', fc', Tj/, f / and fr', 172', ft'. Then 



+ (i?! - 172')* + (ri - 



P 2 = tta - 



~ f i') 



10 



130 ATTRACTION OF SOLID ELLIPSOID. [80 

On making use of equations (45) and (47), it is found that 



Since P and A are on the surface of the ellipsoid whose semi-axes 
are a, 6, and c, each parenthesis equals unity. Therefore p' 2 p 2 = 0, 
or p = p'. 

Suppose the integrals (49) are computed so that the elements at 
corresponding points of the two surfaces are always taken simul- 
taneously. Then pi = p/ and p 2 = p 2 ' throughout the integration. 

b c 

Moreover, it follows from (47) that dy = r? dy' and d = -, d$'. 

Therefore 



(60) 

and similarly 

v , _ ca v 

c'n' ' 

(51) 

Z' = Z 

The letters a, b, c, and s of equations (43) should be given accents 
to agree with the notations of this article; and, since P and P' 

are corresponding points, x = , x', y = r- f y', z = -, z'. After 

CL \J C 

making these changes in equations (43) and substituting them 
in (50) and (51), it is found that 



X' = - 



(a* 
Y' = - 



c /2 



Z' = - 2<jraabck 2 z 



Jo (r' 2 



(c /2 + 8') V( a ' 2 + ') (6 /2 + (c /2 
It follows from equations (45) that 

a /2 = a 2 + K , b' 2 = 6 2 + K, c' 2 = c 2 + K; 



80] IVORY'S METHOD. 131 

hence, on letting s = s f + K, it follows that 
X' =- 



(52) 



(a 2 + s) V(a 2 + s) (6 2 + s) (c 2 + s) 



Y' = - 2jr<rabck 2 y 



f 

f 



(6 2 + s) V(a 2 + s) (6 2 + s) (c 2 + 



It follows from equations (40) and (41) that the components 
of attraction for interior particles are homogeneous of degree zero 
in a, b, and c, and that they are proportional to the respective 
coordinates of the attracted particle. Let X } as above, represent 
the attraction of the ellipsoid E' , whose semi-axes are a', b', c f , 
upon the interior particle at (x, y, z) ; let X" represent the attrac- 
tion of E' upon an interior particle at (x", y", z"), which will be 
supposed to be related to (x, y, z) by equations of the same form 
as (47). Then it follows that 

~Y''~~~^' ~T~'~~~y' ~Z == 7* 

Let the point (x", y", z"}, always corresponding to (x } y, z), 
approach the surface of E' as a limit. Then at the limit 

T'' = ~a^' T := 6"' ~Z~ == c* 
On combining these equations with (50) and (51), it is found that 

^ = II = ?1 = L^- = ML 
X' := Y' ~ Z' " abc '' = M' 

That is, the attraction of a solid ellipsoid upon an exterior particle 
is to the attraction of a confocal ellipsoid passing through the particle, 
as the mass of the first ellipsoid is to that of the second ellipsoid. 

Consider another ellipsoid confocal with the one passing through 
the particle and interior to it; by the same reasoning the ratios 
of the components of attraction of these two ellipsoids are as 
their masses. Let X'", Y"', Z'" be the components of attraction 
of the new ellipsoid whose semi-axes are a'", b'", c'". Then 



X'" ~ Y'" ~ Z'" ~ a" f b'"c f " ~ M'" ' 
On combining this proportion with (53), it is found that 



132 



THE ATTRACTION OF SPHEROIDS. 



[81 



X' 

X"' 



Z' 

Z'" 



__ 
M'"' 



Therefore, two confocal ellipsoids attract particles which are exterior 
to both of them in the same direction and with forces which are pro- 
portional to their masses. This theorem was found by Maclaurin 
and Lagrange for ellipsoids of revolution, and was extended by 
Laplace to the general case where the three axes are unequal. 
It is established most easily, however, by Ivory's method as above, 
and it is frequently called Ivory's theorem. 

The right members of equations (52) can be transformed to 
forms which are more convenient for computation by putting, in 

the first, . = u', in the second, = u; 



and in the 



third, 



= u. 



The results of the substitutions are 



(54) 



X'=-Trabck 2 x' 



a 

'*!**+*. 



u z du 



V[a 2 - (a 2 - 



v?du 



o V[c 2 - (c 2 - a 2 )^ 2 ][c 2 - (c 2 -6 2 K] 

When the attracted particle is in the interior of the ellipsoid the 
forms of the integrals are the same except that the upper limits are 
unity. 

81. The Attraction of Spheroids. The components of attraction 
will be obtained from (54), which hold for exterior particles. 
Suppose the attracting body is an oblate spheroid in which a = b>c 
and let e represent the eccentricity of a meridian section. Then 

c 2 = a 2 (l - e 2 ), 
and equations (54) become 



(55) 



The integrals of these equations are 




82] ATTRACTION AT THE SURFACES OF SPHEROIDS. 



133 



[X 1 Y' 


7.2 * ^ /1 tt ^ 


x' '- y' ~ 

z r _ 4?r 


ZTTffK ~ - | 2 , A / 1 g , 


| sin i . 
W\ ce vr ; 


? 


r .; I . \i o 

^LVc 2 H-K 
v f 1 / ce 



(56) , 



(l_ e 2 )(c 2 + K) 

The components of attraction for interior particles are obtained 
from equations (56) by putting K = 0. 

Now suppose the attracting body is a prolate spheroid and 
that a = b < c. Then a 2 = 6 2 = c 2 (l - e 2 ), and equations (64) 
become 



(57) 



J 



^ = - 4T<rfc 2 (l - 




The integrals of these equations are 

^ = 7 = 



(58) H 



c 3 L 

(1 - e) 

^ J_ -" ^ 


a, r 


2 1 


a 2 e 2 


A^T-.V 1 


^ H- , 

a 
ae 


+. 


log (v(r: 

*il 

,/ -2 ^ 


- e 2 )(a 2 + 


^ 


i i 


oV 


f (l- 
1+- 


e 2 )(a 2 + ic) 

C6 V 




* 




1 - 


' 



) 



When the particle is interior to the spheroid the equations for 
the components of attraction are the same except that K = 0. 

82. The Attraction at the Surfaces of Spheroids. The com- 
ponents of attraction for an interior particle, which are obtained 
in the case of an oblate spheroid from (56) by putting K = 0, 
are, omitting the accents, 



134 



ATTRACTION AT THE SURFACES OF SPHEROIDS. 



[82 



(59) H 



- = = - 27r<r/b 2 



- 

x 

f 



- e 



- e 2 + sin- 1 e], 



The limits of these expressions as the attracted particle approaches 
the surface of the spheroid are the components of attraction for a 
particle at the surface. As the attracted particle passes outward 
through the surface, K, in equations (56), starts with the value 
zero and increases continuously in such a manner that it always 
fulfills equation (46). Therefore equations (59), having no 
discontinuity as the attracted particle reaches the surface, hold 
when x, y, z fulfill the equation of the ellipsoid. 

When e is small, as in the case of the planets, equations (59) 
are convenient when expanded as power series in e. On substi- 
tuting the expansions 



sm" 1 e = 



in equations (59), it is found that 
X = Y 

(60) 



The mass of the spheroid is 

M = %iraa?c = 



^ .. v *, v -.. if A c/ 

The radius of a sphere having equal mass is defined by the equation 

M = ^TTffR 3 = -7rcra 3 Vl e 2 ; 
whence 

R = a(l - e 2 )*. 

The attraction of this sphere for a particle upon its surface is 
given by the equation 

(61) F = - 



82] ATTRACTION AT THE SURFACES OF SPHEROIDS. 135 

When the attracted particle is at the equator of the spheroid 
Vz 2 + y 2 = a; hence the ratio of the attraction of the spheroid 
for a particle at its equator to that of an equal sphere for a particle 
upon its surface is 



VZ 2 + F 2 _(l ~ &* ..-) <?_ 

p- (1-e 2 ) 1 30 "^ 

This is less than unity when e is small; therefore the attraction of 
the spheroid for a particle on its surface at its equator is less 
than that of a sphere having equal mass and volume for a particle 
on its surface. 

When the attracted particle is at the pole of the spheroid 
z = c = a Vl e 2 ; hence in this case 



= , 

(I - e 2 )* r 15 ^ 

This is greater than unity when e is small; therefore the attraction 
of the spheroid for a particle on its surface at its pole is greater 
than that of a sphere having equal mass and volume for a particle 
on its surface. 

There is some place between the equator and pole at which the 
attractions are just equal. The latitude of this place will now 
be found. The coordinates of the particle must fulfill the equa- 
tion of the spheroid; therefore 

(62) f(x,y,z)^^^ + -l=0. 



The direction cosines of the normal to the surface at the point 
(a?, y, z) are 

*L 

dx dy 



dz 



/#Y 

\dz) 



The last is the cosine of the angle between the normal at the 
point (x, y, z) and the z-axis, and is, therefore, the sine of the 



136 



ATTRACTION AT THE SURFACES OF SPHEROIDS. 



[82 



geographical latitude, which will be represented by <. Hence, 
it follows from (62) that 



(63) 



dz 



sin 



From (62) and (63) it is found that 
a 2 cos 2 <b 



I - e 2 sin 2 



z2 = a 2 (l - e 2 ) 2 sin 2 4> 



(64) < 



Let G represent the whole attraction of the spheroid; then it is 
found from (60) and (64) that 



G = - 



+ Y 2 + Z 2 



- cos 



The ratio of this expression to that for the attraction of a sphere 
of equal mass and volume, given by (61), is 



(65) = 



= 1 



- 3 sin 2 



30 



This becomes equal to unity up to terms of the fourth order in e 
when 3 sin 2 = 1, from which it is found that 

= 35 15' 52". 
Let r represent the radius of the spheroid; then 

2 = a 2 (l - e 2 ) 
~ 1 - e 2 cos 2 ^' 

where $ is the angle between the radius and the plane of the 
equator. Since this angle differs from <J> only by terms of the 
second and higher orders in e, it follows that, with the degree of 
approximation employed, 



PROBLEMS. 137 



When = 35 15' 52 



The radius of a sphere of equal volume has been found to be 
given by the equation 



which is seen to be equal to the radius of the spheroid up to terms 
of the second order inclusive in the eccentricity. Therefore, in 
the case of an oblate spheroid of small ecentricity, the intensity 
of the attraction is sensibly the same for a particle on its surface 
in latitude 35 15' 52" as that of a sphere having equal mass and 
volume for a particle on its surface; or, because of the equality 
of R and r, a spheroid of small eccentricity attracts a particle on 
its surface in latitude 35 15' 52" with sensibly the same force it 
would exert if its mass were all at its center. 

Xm. PROBLEMS. 

1. Show that Ivory's method can be applied when the attraction varies 
as any power of the distance. 

2. Show why Ivory's method cannot be used to find the potential of a 
solid ellipsoid upon an exterior particle when it is known for an interior particle. 

3. Find the potential of a thin ellipsoidal shell contained between two 
similar ellipsoids upon an interior particle. Hint. It has been proved 
(Art. 79) that the resultant attraction is zero at all interior points; therefore 
the potential is constant and it is sufficient to find it for the center. Let the 
semi-axes of the two surfaces be a, 6, c and (1 + n}a, (1 + fj,)b, (1 + ju)c; then 
the distance between the two surfaces measured along the radius from the 
center will be pp. Therefore 



-lif 



r 



cos (t>d<f>dd 



cos 2 <ft cos 2 cos 2 <f> sin 2 6 sin 2 
~ + ~ 



V(a 2 + s)(6 2 + s)(c 2 -f-s) 



138 HISTORICAL SKETCH. 

4. Show that in the case of two thin confocal shells similar elements of 
mass at points which correspond according to the definition (47) are propor- 
tional to the products of the three axes of the respective ellipsoids. Then 
show, using problem 3 and Ivory's method, that the potential of an ellipsoidal 
shell upon an exterior particle is 

J ds' 

. , , , ==- 

Va' 2 + s'&' +c' ' 



/ ds 

2ira-fjLabc I , 

J< V(a 2 



5. Prove that the level surfaces of thin homogeneous ellipsoids are confocal 
ellipsoids. What are the lines of force which are orthogonal to these surfaces? 

6. Discuss the form of level surfaces when they are entirely exterior to 
homogeneous solid ellipsoids. 

HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The attractions of bodies were first investigated by Newton. His results 
are given in the Principia, Book i., Sees. xn. and xin., and are derived by 
synthetic processes similar to those used in the first part of this chapter. 
The problem of the attraction of ellipsoids has been the subject of many 
memoirs, and the case in which they are homogeneous was completely solved 
early in the nineteenth century. Among the important papers are those 
by Stirling, 1735, Phil. Trans.; by Euler, 1738, Petersburg; by Lagrange, 
1773 and 1775, Coll Works, vol. in., p. 619; by Laplace, 1782, Mec. Cel., 
vol. ii.; by Ivory, 1809-1828, Phil. Trans.; by Legendre, 1811, Mem. de 
VInst. de France, vol. XL; by Gauss, Coll. Works, vol. v.; by Rodriguez, 1816, 
Corres. sur I'Ecole Poly., vol. in.; by Poisson, 1829, Conn, des Tem,ps; by 
Green, 1835, Math. Papers, vol. vin.; Chasles, 1837-1846, Jour. I'Ecole 
Poly, and Mem. des Savants Strangers, vol. ix.; MacCullagh, 1847, Dublin 
Proc., vol. in.; Lejeune-Dirichlet, Journal de Liouville, vol. iv., and Crelle, 
vol. xxxii. 

The earlier papers were devoted for the most part to the attractions of 
homogeneous ellipsoids of revolution upon particles in particular positions, 
as on the axis. Lagrange gave the general solution for the attractions of 
general homogeneous ellipsoids upon interior particles. This was extended 
by Ivory and Maclaurin (with Laplace's generalizations) to exterior particles. 
Ivory's theorem has been extended in a most interesting manner by Darboux 
in Note xvi. to the second volume of the Mecanique of Despeyrous. Chasles 
gave a synthetic proof of the theorems regarding the attractions of homo- 
geneous ellipsoids in Memoir es des Savants Strangers, vol. ix., and Lejeune- 
Dirichlet embraced in a most elegant manner in one discussion the case of 
both interior and exterior points by using a discontinuous factor (Liouville's 
Journal, vol. iv.). 

Laplace proved that the potential for an exterior particle fulfills the partial 
differential equation 

Sr+*r.r- 

ar 2 + d* + a? "' u ' 



HISTORICAL SKETCH. 



139 



and determined V by the condition that it must be a function satisfying this 
equation. This is a process of great generality, and is relatively simple 
except in the trivial cases. This has been made the starting-point of most 
of the investigations of the latter part of the last century, especially where 
the attracting bodies are not homogeneous. In a paper on Electricity and 
Magnetism, in 1828, Green introduced the term potential function for V, and 
discussed many of its mathematical properties. Green's memoir remained 
nearly unknown until about 1846, and in the meantime many of his theo- 
rems had been rediscovered by Chasles, Gauss, Sturm, and Thomson. One 
of Green's theorems has found an extremely useful application, when the 
independent variables are two in number, in the Theory of Functions. 

Poisson showed that the potential function for an interior particle fulfills 
the partial differential equation 



Among the books treating the subject of attractions and potential may be 
mentioned Thomson and Tait's Natural Philosophy, part u., Neumann's 
Potential, Poincar6's Potential, Routh's Analytical Statics, vol. n., and Tisser- 
and's Mecanique Celeste, vol. n. The last-mentioned develops most fully 
the astronomical applications and should be used in further reading. 

The attractions of spheroids and ellipsoids has been fundamental in the 
discussions of possible figures of equilibrium of rotating fluids. The reason 
is, of course, that the conditions for equilibrium involve the components of at- 
traction. Maclaurin proved in 1742 that for slow rotation an oblate spheroid, 
whose eccentricity is a function of the rate of rotation and the density of the 
fluid, ia a figure of equilibrium. There are, indeed, two such figures; for slow 
rotation one is nearly spherical and the other is very much flattened. For 
faster rotation the figures are more nearly of the same shape; for a certain 
greater rate of rotation they are identical; and for still faster rotation no 
spheroid is a figure of equilibrium. In 1834 Jacobi proved that when the rate 
of rotation is not too great there is an ellipsoid of three unequal axes which is a 
figure of equilibrium, which for a certain rate of rotation coincides with the 
more nearly spherical of the Maclaurin spheroids. For this work Tisserand's 
Mecanique Celeste, vol. n., should be consulted. In a very important memoir 
(Acta Mathematics, vol. vn.) Poincare" proved that there are infinitely many 
other figures of equilibrium which, for certain values of the rate of rotation, 
coincide with the corresponding Jacobian ellipsoid, as it, for a certain rate 
of rotation, coincides with the Maclaurin spheroid. The least elongated of 
these figures is larger at one end than it is at the other, and was called the 
apioid, that is, the pear-shaped figure. Later computations by Sir George 
Darwin (Philosophical Transactions, vol. 198) have shown it is so elongated 
that it might well be called a cucumber-shaped figure. 



CHAPTER V. 



THE PROBLEM OF TWO BODIES. 

83. Equations of Motion. It will be assumed in this chapter 
that the two bodies are spheres and homogeneous in concentric 
layers. Then, in accordance with the results obtained in Art. 69, 
they will attract each other with a force which is proportional ,to 
the product of their masses and which varies inversely as the 
square of the distance between their centers. 

Let mi and m 2 represent the masses of the two bodies, and 
mi + m z = M. Choose an arbitrary system of rectangular axes 
in space and let the coordinates of mi and w 2 referred to it be 
respectively (1, 171, fi) and ( 2 , ??2, 2). Let the distance between 
mi and ra 2 be denoted by r; then it follows from the laws of motion 
and the law of gravitation that the differential equations which the 
coordinates of the bodies satisfy are 



(1) 



dt 2 

d 2 rji 
dt* 









dt* 



r 6 



In order to solve these six simultaneous equations of the second 
order twelve integrals must be found. They will introduce twelve 
arbitrary constants of integration which can be determined in any 
particular case by the three initial coordinates and the three com- 
ponents of the initial velocity of each of the bodies. 

140 



84] 



THE MOTION OF THE CENTER OF MASS. 



141 



84. The Motion of the Center of Mass. On adding the first 
and fourth, the second and fifth, and the third and sixth equations 
of the system (1), it is found that 



These equations are immediately integrable, and give 



(2) 



dt 



On integrating again they become 






dt ~ 

= ai t 

= Pit 



182, 
72. 



Thus, six of the twelve integrals are found, the arbitrary constants 
of integration be_ing on, o% |8i, j8 2 , 71, 72- 

Let I", ^ i and p be the coordinates of the center of mass of the 
system; then it follows from Art. 19 and "equations (3) that 



(4) 



+ 



= ait 

= Pit 



/3 2 , 



72. 



From these equations it follows that the coordinates increase 
directly as the time, and, therefore, that the center of mass moves 
with uniform velocity. Or, taking their derivatives, squaring, 
and adding, it is found that 



'KfHfHfn 



7i 2 ; 



whence 



142 



THE EQUATIONS FOR RELATIVE MOTION. 



[85 



= _ 



M 



where V represents the speed with which the center of mass 
moves. The speed is therefore constant. 
On eliminating t from (4), it is found that 



on Pi 7i 

The coordinates of the center of mass fulfill these relations which 
are the symmetrical equations of a straight line in space ; therefore, 
the center of mass moves in a straight line with constant speed. 

85. The Equations for Relative Motion. Take a new system 
of axes parallel to the old, but with the origin at the center of mass 
of the two bodies. Let the coordinates of mi and m 2 referred to 
this new system be xi, y\, z\ and x*, yi, 22 respectively. They 
are related to the old coordinates by the equations 



(5) 



= rji rj, 



2 = 2 - , 
2/2 = rjz rj, 



On substituting in (1), the differential equations of motion in the 
new variables are found to be 



(6) 



2 2 ) 



dt 2 



r 3 

(22 - 



which are of the same form as the equations for absolute motion. 

The coordinates of the center of mass are given by equations (4) ; 

therefore if x\ t y\, >>>>, z 2 were known, and if the constants 



85] 



THE EQUATIONS FOR RELATIVE MOTION. 



143 



i, "2, j8i, 02, 7i> and 72 were known, the absolute positions in 
space could be found. But, since there is no way of determining 
these constants, the problem of relative motion, as expressed 
in (6), is all that can be solved. 

Since the new origin is at the center of mass, the coordinates 
are related by the equations 



(7) 



= 0, 
= 0, 

= 0. 



Therefore, when the coordinates of one body with respect to the 
center of mass of the two are known the coordinates of the second 
body are given by equations (7) . 

Equations (7) can be used to eliminate x*, 2/2, and z z from the 
first three equations of (6), and x\, y\, and z\ from the last three. 
The results of the elimination are 



(8) 



df 









dt 2 

In the first three equations the r which appears in the right 
member must be expressed in terms of x\, yi, and z\\ and in the 
second three it must be expressed in terms of x 2 , yz, and z 2 . It 
follows from equations (7) that 



M 



M 



M 



r. 



The equations in Xi, y\, z\ are now independent of those in z 2 , 2/2, 22, 
and conversely. But what is really desired in practice is the 



144 



THE INTEGRALS OF AREAS. 



motion of one body with respect to the other. Let x, y, and 2 
represent the coordinates of ra 2 with respect to mi, then 



x z - 



z = z z - 



Hence if the first, second, and third equations of (8) are sub- 
tracted from the fourth, fifth, and sixth equations respectively, the 
results are, as a consequence of these relations, 



(9) 






The problem is now of the sixth order, having been reduced 
from the twelfth by means of the six integrals (2) and (3). The 
six new constants of integration which will be introduced in 
integrating equations (9) will be determined by the three initial 
coordinates, and the three projections of the initial velocity of mi 
with respect to m 2 . 

86. The Integrals of Areas. Multiply the first equation of (9) 
by y, and the second by + x } and add; the result is 



d?z d?y 

-'5? 

d z x d 2 z 



The integrals of these equations are 

dy dx 



(10) 



dz dy 

di- z di = a *' 

dx dz 



_ 
Z dt X dt 



a s . 



It follows from Art. 16 that a\, a 2 , a 3 are the projections of 
twice the areal velocity upon the xy, yz, and zz-planes respectively. 



86] 



THE INTEGRALS OF AREAS. 



145 



Upon multiplying equations (10) by 2, x, and y respectively, and 
adding, it is found that 

(11) aiz + a z x + a^y = 0. 

This is the equation of a plane passing through the origin, and it 
follows from its derivation that the coordinates of mi always 
fulfill it ; therefore, the motion of one body with respect to the other is 
in a plane which passes through the center of the other. 

The constants 01, a 2 , and a 3 determine the position of the 
plane of the orbit with respect to the axes of reference. In polar 
coordinates equation (11) becomes 

(12) ai sin <p + a% cos <p cos + a 3 cos <p sin 6 0. 

The x7/-plane and the plane of the orbit intersect in a line L 
(Fig. 25). Suppose OL is that half line which passes through 




Fig. 25. 

the point at which the body mi goes from the negative to the 
positive side of the zi/-plane. Let &> represent the angle between 
the positive end of the z-axis and the line OL counted in the 
positive direction from Ox. This angle may have any value from 
to 360. Let i represent the inclination between the two 
planes counted in the direction of positive rotation around OL. 
The angle i may have any value from to 180. It is less or 
greater than 90 according as a x is positive or negative. Then, 
11 



146 PROBLEM IN THE PLANE. [87 

when <p = the value of 6 is ft or ft + IT. When 6 = ft + \ir 
the value of <p is i or TT i according as i is less than or greater 
than 90. In these cases equation (12) becomes respectively 

Ja 2 cos ft + a 3 sin ft = 0, 
[ai sin i =F a 2 cos t sin & =*= as cos i cos ,0, = 0, 

where the signs of the second equation are the upper if i is less 
than 90, and the lower if it is greater than 90. 

Since the projections of the areal velocity upon the three funda- 
mental planes are constants (viz., Jai, fa 2 , and ^a 3 ), the areal veloc- 
ity in the plane of the orbit is also constant. Let this constant 
be represented by Jcij then 



(14) ci = Vai 2 + a 2 2 + a 3 2 , 

where the positive value of the square root is taken. On solving 
(13) and (14) for a 1} a 2 , and a 3 , it is found that 

a\ = + Ci cos i, 

(15) - a 2 = =*= Ci sin i sin ft, 

. ct 3 = =F Ci sin i cos ft , 

where the upper or lower signs are to be taken in the last two 
equations according as i is less than or greater than 90; that is, 
according as ai is positive or negative. With this understanding 
equations (15) uniquely determine i and ft, which uniquely 
determine the position of the plane of the orbit. 

87. Problem in the Plane. Since the orbit lies in a known 
plane, the coordinate axes may be chosen so that the x and ?/-axes 
lie in this plane. If the coordinates are represented by x and y 
as before, the differential equations of motion are 

d z x 



The problem is now of the fourth order instead of the sixth as 
it was in (9), having been reduced by means of the integrals (10). 
It will be observed that, since the position of the plane is defined 
by the two elements ft and i, or by the ratios of ai, a 2 , and a 3 in 
(11), only two of the arbitrary constants were involved in the 
reduction. This problem might be solved by deriving the differ- 



87] PROBLEM IN THE PLANE. 147 

ential equation of the orbit as in Art. 54 and integrating as in 
Art. 62, the last integral being derived from the integral of areas; 
but, it is preferable to obtain the results directly by the method 
which is usually employed in Celestial Mechanics. 

Equations (16) give 

d?y d*x _ 
X W~ y dt* ~ 

The integral of this equation is 

dy dx 

*"*?"/" 

which becomes in polar coordinates 

on *S-* 

Let A represent the area swept over by the radius vector r; then 

o dA * de 
*-& = +* = *' 
whence 

(18) 2A= Cl * + c 2 , 

from which it follows that the areas swept over by the radius 
vector are proportional to the times in which they are described. 

On multiplying (16) by 2 -=- and 2 -j- respectively, and adding, 
the result is 

2k 2 M dr 



^xdx ^ydy_ k?M / dx dy\_ 

Z dt 2 dt ^ ' dt 2 dt = r 3 \ X dt y dt ) ~~ r 2 dt ' 

The integral of this equation is 



This equation, which involves only the square of the velocity 
and the distance, is known as the vis viva integral (Art. 52). On 
transforming the left member to polar coordinates, this equation 
becomes 



dd\ z 
U) = 



But- 

dr dr dO 

Tt = Tedt 

therefore 



148 ELEMENTS AND CONSTANTS OF INTEGRATION. [88 

- M. _L c 

r 
On eliminating -j- by means of (17), this equation gives 

de = 



V- d 2 + 2/cWr + c 3 r 2 ' 
which may be written in the form 



(20) de = 



Let B 2 and M be defined by 



in which B 2 must be positive for a real orbit; then (20) becomes 

du 

dB = 



, B 2 - u 2 
The integral of this equation is 

6 = cos- 1 ^ + c 4 . 

On changing from u, B, and C4 to r and the original constants, it is 
found that 



(21) r - 



k 2 M I 

-^ -V/ 



Cl V*^!?-' 



which is the polar equation of a conic section with the origin at 
one of its foci. 

88. The Elements in Terms of the Constants of Integration. 

The node and inclination are expressed in terms of the constants 
of integration by (15). 

The ordinary equation of a conic section with the origin at the 
right-hand focus is 

P 



r = 



1 + e cos (6 - co) ' 



89] 



PROPERTIES OF THE MOTION. 



149 



where p is the semi-parameter, and o> is the angle between the 
polar axis and the major axis of the conic. On comparing this 
equation with (21), it is found that 



(22) 



P = 



Ci 2 C 3 



e 2 = 1 + 
co = 04 T; 
d = k^Mp, 



P 



M. 



When e 2 < 1, the orbit is an ellipse and p = a(l e 2 ), where 
a is the major semi-axis; when e 2 = 1, the orbit is a parabola and 
p = 2q, where q is the distance from the origin to the vertex of 
the parabola; and when e 2 > 1, the orbit is an hyperbola and 
p = a(e 2 - 1). 

Let AQ represent the area described at the time the body passes 
perihelion;* then the time of perihelion passage is found from 
equation (18) to be 

2A - c 2 



(23) 



T = 



Ci 



This completes the determination of the elements in terms of 
the constants of integration. They are denned in terms of the 
initial coordinates and components of velocity by the equations 
where they first occur, viz., (10), (17), (18), (19), and (21). 

89. Properties of the Motion. Suppose the orbit is an ellipse. 
Then, when the values of the constants of integration given in 
(22) are substituted in (17) and (19), these equations become 




(24) 



where V is the speed in the orbit at the distance r from the origin. 
When the orbit is a circle, r = a and 

* Unless Wa is specified to be some body other than the sun the nearest apse 
will be called the perihelion point. 



150 



PROPERTIES OF THE MOTION. 



[89 



When the orbit is a parabola, a = oo and 



V ' 



Therefore, at a given distance from the origin the ratio of the 
speed in a parabolic orbit to that in a circular orbit is 

(25) 7 p :7 e = V2:l. 

Thus, in the motion of comets around the sun they cross the 
planets' orbits with velocities about 1.414 times those with which 
the respective planets move. 

The speed that a body will acquire in falling from the distance 
s to the distance r toward the center of force k 2 M is given by 
(see Art. 35) 

V 2 = 2k z M (---}. 
\r sj 

If s is determined by the condition that this shall equal the speed 
in the orbit, it is found, after equating the right member of this 




Fig. 26. 

expression to the right member of the second of (24), that s = 2a 
and 

(26) F* 

Therefore, the speed of a body moving in an ellipse is at every 



89] PROPERTIES OF THE MOTION. 151 

point equal to that which it would acquire in falling from the circum- 
ference of a circle, with center at the origin and radius equal to the 
major axis of the conic f to the ellipse. 

The speed at P in the ellipse is equal to that which would be 
acquired in falling from P' to P. 

Equation (26) gives an interesting conclusion about the possible 
motion of m\ on the basis of this equation alone, and without 
making any use of the detailed properties of motion in a conic 
section. Since the left member is necessarily positive (or zero) 
r can take only such values that the right member shall be positive 
(or zero). Consequently r ^ 2a in all the motion whatever it 
may be. This result is trivial in this simple case in which all 
the circumstances of motion are fully known, but the corresponding 
discussion in the Problem of Three Bodies (Chap, vm.) gives valu- 
able information which has not been otherwise obtained. 

Consider the second equation of (24) and suppose the body 
mi is projected from a point which is distant r from the body w 2 . 
It follows at once that the major axis of the conic depends upon the 
initial distance from the origin and the initial speed, but not upon 



the direction of projection. If V 2 < - = U 2 , which is the veloc- 

ity the body mi would acquire in falling from infinity, a is positive 
and the orbit is an ellipse; if V 2 = U 2 , a is infinite and the orbit 
is a parabola ; if V 2 > U 2 , a is negative and the orbit is an hyperbola. 
Let ti and t z be two epochs, and AI and A z the corresponding 
values of the area described by the radius vector. Then equation 

(18) gives 

2(A 2 - Ai) = (t 2 - Zi)ci. 

Suppose t z h = P, the period of revolution; then 2(A 2 AI) 
equals twice the area of the' ellipse, which equals 2irab. The 
expression for the period, found by substituting the value of Ci 
given in (22) and solving, is 

' 



From this equation it follows that the period is independent of 
every element except the major axis; or, because of (26), the period 
depends only upon the initial distance from the origin and the 
initial speed, and not upon the direction of projection. The 
major semi-axis will be called the mean distance, although it must 
be understood that it is not the average distance when the time is 



152 



PROPERTIES OF THE MOTION. 



[89 



used as the independent variable. (See Probs. 4 and 5, p. 154.) 
The three orbits drawn in Fig. 27 have the same length of 
major axis and are consequently described in the same time. 
The speed of projection from A is the same in each case, the 
differences in the shapes and positions resulting from the different 
directions of projection. 




Fig. 27. 

If the two systems mi, m z , and m z , m s are considered, and the 
ratio of their periods is taken, it is found that 



P 2 i, 



_ U* 1, 2 
3, 2 a 3 3 , 2 



M 



3, 2 



1,2 



If the two systems are composed of the sun and two planets 
respectively, then MI, 2 and Ma, 2 are very nearly equal because 
the masses of the planets are exceedingly small compared to that 
of the sun. Therefore, this equation becomes very nearly 

P\ 2 a\ 2 



or, the squares of the periodic times of the planets are proportional to 
the cubes of their mean distances. This is Kepler's third law. 

It is to be observed that, in taking the ratios of the periods, it 
was assumed that k has the same value for the different planets; 
that is, that the sun's acceleration of the two planets would be 
the same at unit distance. On the other hand, it follows from the 
last equation, which Kepler established directly by observations, 
that k has the same value for the various planets. This means 
that the force of gravitation between the sun and the several 



90] SELECTION OF UNITS. 153 

planets is proportional to their respective masses, as measured 
by their inertias. This result is not self-evident for the force of 
gravitation conceivably might depend upon the chemical con- 
stitution or physical condition of a body, just as chemical affinity, 
magnetism and all other known forces depend upon one or both 
of these things. In fact, it is remarkable that gravitation is 
proportional to inertia and independent of everything else. 

90. Selection of Units and the Determination of the Constant k. 

When the units of time, mass, and distance are chosen k can be 
determined from (27). It is evident that they can all be taken 
arbitrarily, but it will be convenient to employ those units in 
which astronomical problems are most frequently treated. The 
mean solar day will be taken as the unit of time; the mass of the 
sun will be taken as the unit of mass; and the major semi-axis of 
the earth's orbit will be taken as the unit of distance. When these 
units are employed the k determined by them is called the Gaussian 
constant, having been defined in this way by Gauss in the Theoria 
Motus, Art. 1. 

Let ra 2 represent the mass of the sun and mi that of the earth 
and moon together; then it has been found from observation that 
in these units 



(28) 



mi " 



354710 "354710' 
. P = 365.2563835. 

On substituting these numbers in (27), it is found that 

f k = . 2?r = 0.01720209895, 

(29) PVl + m! 

I log k = 8.2355814414 - 10. 

Since mi is very small k = -p- nearly, and is, therefore, nearly 
the mean daily motion of the earth in its orbit, or about -fa. The 

mean daily motion of a planet whose mass is m^ is -=- , and is 

* 

usually designated by n t -. This is found from (27) to be 

(30) ' 

The period of the earth's revolution around the sun and its 
mean distance were not known with perfect exactness at the 



154 PROBLEMS. 

time of Gauss, nor are they yet, and it is clear that the value of 
k varies with the different determinations of these quantities. 
If astronomers held strictly to the definitions of the units given 
above it would be necessary to recompute those tables which 
depend upon k every time an improvement in the values of the 
constants is made. These inconveniences are avoided by keeping 
the numerical value of k that which Gauss determined, and 
choosing the unit of distance so that (27) will always be fulfilled. 
If the mean distance between two bodies is taken as the unit of 
distance and the sum of their masses as the unit of mass, and if the 
unit of time is taken so that k equals unity, then the units form 
what is called a canonical system. Since M = 1 and k 2 = 1 in 
this system, and from (30) n = 1, the equations become some- 
what simplified and are advantageous in purely theoretical 
investigations. 



XIV. PROBLEMS. 

1. Find the differential equations for the problem of the relative motion of 
two bodies in polar coordinates. 

Ans. T r ( ) -j- 

dt 2 \ at / i* at 

2. Integrate the equations of problem 1 and interpret the constants of 
integration. 

3. The earth moves in its orbit, which may be assumed to be circular, with 
a speed of 18.5 miles per second. Suppose the meteors approach the sun in 
parabolas; between what limits will be their relative speed when they strike 
into the earth's atmosphere? 

Ans. 7.66 to 44.66 miles per second. (The Nov. 14 meteors meet the 
earth and have a relative speed near the upper limit; the Nov. 27 meteors 
overtake the earth and have a relative speed near the lower limit.) 

4. Find the average length of the radius vector of an ellipse in terms of 
a and e, taking the time as the independent variable. 

j*rdt 
Ans. Average r = = 



5. Find the average length of the radius vector of an ellipse, taking the 
angle as the independent variable. 

f rdd 2*aVT=l* 
Ans. Average r = ~= = - = b. 

fa 



91] POSITION IN PARABOLIC ORBITS. 155 

6. Prove that the amount of heat received from the sun by the planets 
per unit area is on the average proportional to the reciprocals of the products 
of the major and miner axes of their orbits. For a fixed major axis how does 
the total amount of heat received in a revolution depend upon the eccentricity 
of the orbit? 

7. If particles are projected from a given point with a given velocity but 
in different directions, find the locus of (a) perihelion points; (6) aphelion 
points; (c) centers of ellipses; (d) ends of minor axes. 

8. If particles are projected from a given point in a given direction but 
with different speeds, find the loci of the same points as in problem 7, and 
express the coordinates of these points in terms of the initial values of the 
coordinates and the components of velocity. 

9. Suppose a comet moving in a parabolic orbit with perihelion distance q : 
collides with and combines with an equal mass which is at rest before the 
collision. Find the eccentricity and the perihelion distance of the orbit of 
the combined mass. 

10. Suppose the mass of Jupiter is 1/1047 when expressed in terms of the 
mass of the sun, and that its mean distance from the sun is 483,300,000 miles 
(the mean distance from the earth to the sun is .92,900,000 miles). Find 
Jupiter's period of revolution around the sun, and the size of the orbit which 
the sun describes with respect to the center of gravity of itself and Jupiter. 

91. Position in Parabolic Orbits. Having found the curves in 
which the bodies move, it remains to find their positions in their 
orbits at any given epoch. The case of the parabolic orbit being 
the simplest will be considered first, and it will be supposed, to 
fix the ideas, that the motion is that of a comet with respect to 
the sun. Since the masses of the comets are negligible, M = 1 
and equation (17) becomes 

When the polar angle in the orbit is counted from the vertex of 
the parabola it is denoted by v, and is called the true anomaly. 
Then 

'd^dr 

r = - -r-Z = q sec 2 . 

1 + cos v 2 



Hence, equation (31) gives 
| 



- dt = sec 4 dv ( sec 2 ^ + sec 2 tan 2 1 dv. 



156 POSITION IN PARABOLIC ORBITS. [91 

The integral of this expression is 
(3 2 ) 



where T is the time of perihelion passage. This is a cubic equation 

in tan ~ . On taking the right member to the left side it is seen 
& 

that for t T > 0, the function is negative when v = 0, and that 
it increases continually with v until it equals infinity for v = 180. 

Therefore, there is but one real solution of (32) for tan , and it 

ft 

is positive. For t T < it is seen in a similar manner that 
there is one real negative solution. 
Equation (32) may be written 



Tables have been constructed giving the value of the right member 
of this equation for different values of v. From these tables v can 
be found by interpolation when t T is given; or, conversely, 
t T can be found when v is given. These tables are known as 
Barker's, and are VI. in Watson's Theoretical Astronomy, and IV. 
in Oppolzer's Bahnbestimmung. * 

In order to find the direct solution of the cubic equation let 



whence 



tan = 2 cot 2w = cot w tan w; 



tan 3 = 3 tan + cot 3 w tan 3 w. 



This substitution reduces (32) to 

3k(t - T) 
cot 3 w tan 3 w = 

Let 



whence 

cots=3fc( 2^ 7 ' ) - 
Therefore the formulas for the computation of tan ^ are, in the 

* In Oppolzer's Bahnbestimmung the factor 75 is not introduced. 



92] EULER^S EQUATION. 

order of their application, 



157 



(33) 



COt S 



cot w 



3k(t - T) 



3/ , S 

= \ cot 2' 



tan = 2 cot 2w. 



After v has been found r is determined by the polar equation of 



the parabola, r = 



cos v 



q sec 2 - 



2' 



92. Equation involving Two Radii and their Chord. Euler's 
Equation. Consider the positions of the comet at the instants 
ti and Z 2 . Let the corresponding radii be r\ and r 2 , and the chord 
joining their extremities s. Let the corresponding true anomalies 
be Vi and v z . Then it follows that 

k(ti - T) i . 

J 



- T} 



The difference of these equations is 



01, 



(M) 

The equation for the chord is 

s 2 = ri 2 + r 2 2 2rir 2 cos 



From this equation it is found that 
(35) 2V^c 



s)(ri 



- s). 



The + sign is to be taken before the radical if v z Vi < IT, and 
the sign if v 2 Vi > TT. 



158 POSITION IN ELLIPTIC ORBITS. [93 

It follows from the polar equation of the parabola that 

, r 2 = gsec 2 ^. 
These expressions for ri and r 2 , substituted in (35), give 



(36) 1 + tan tan = r ' r ' 



It also follows from the expressions for r\ and r 2 that 

ri + r 2 = (2 + tan 2 + 
The last two equations give 



(ri + r 2 + s) + (ri + r 2 - s) =F 2 V(ri + r 2 + s)(ri + r 2 - s) 

.(tanl-tan 

whence 



/ Q7 N + yy-j- s =F ri + r 2 - s v 2 vi 

--- =tan-- tan-. 



Equation (34) becomes, as a consequence of (36) and (37), 
(38) 6fc(* 2 - i) = (ri + r 2 + s)* ^ (ri + r 2 - s)i. 

This equation is remarkable in that it does not involve q. It was 
discovered by Euler and bears his name. It is of the first im- 
portance in some methods of determining the elements of a para- 
bolic orbit from geocentric observations. 

There is a corresponding equation, due to Lambert, for elliptic 
orbits. The right member is developed as a power series in I/a, 
the first term constituting the right member of Euler's equation. 

93. Position in Elliptic Orbits. The integral of areas and the 
vis viva integral are respectively 



dv 



/dr\ 
(dt) 



The result of eliminating -=- from the second of these equations 



94] GEOMETRICAL DERIVATION OF KEPLER' S EQUATION. 159 

by means of the first is 

(39) 



Let n represent the mean angular motion of the body in its orbit ; 
then _ 

2ir k-^l+m 
n= T = ~^~ 

On introducing n in (39) and solving, it is found that 
(40) ndt= T - 



(41) 



a Va 2 e 2 - (a - r) 2 

In order to normalize the integral which appears in the right 
member of (40), let the auxiliary E be introduced by the equation 

a r = ae cos E, whence 

r = a(l e cosE). 
This angle E is called the eccentric anomaly. Then (40) becomes 

ndt = (1 - ecosE)dE, 
the integral of which is 

n(t- T) = E -esmE. 

The quantity n(t T) is the angle which would have been de- 
scribed by the radius vector if it had moved uniformly with the 
average rate. It is usually denoted by M and is called the mean 
anomaly. Therefore 

(42) n(t - T) = M = E - e sin E. 

The M can at once be found when (t T) is given, after which 
equation (42) must be solved for E. Then r and v can be found 
from (41) and the polar equation of the ellipse. Equation (42), 
known as Kepler's equation, is transcendental in E, and the solution 
for this quantity cannot be expressed in a finite number of terms. 
Since it is very desirable to have the solution as short as possible 
astronomers have devoted much attention to this equation, and 
several hundred methods of solving it have been discovered. 

94. Geometrical Derivation of Kepler's Equation. Construct 
the ellipse in which the body moves, and also its auxiliary circle 
AQB. The angle AFP equals the true anomaly, v; the angle 



160 



SOLUTION OF KEPLER S EQUATION. 



[95 



ACQ will be defined as the eccentric anomaly, E, and it will be 
shown that the relation between M and E is given by Kepler's 
equation. 

.3 




Fig. 28. 

From the law of areas and the properties of the auxiliary circle, 
it follows that 

M __ area AFP = area AFQ 
2ir area ellipse area circle ' 

2 Tji n 

Area AFQ = area ACQ - area FCQ = -= ae sin E. 

2 2i 

Therefore 

M_ _ a 2 (E - e sin E) 
27r ~ 2 ~ Tra 2 ' : 



or, 



M = E e sin E, 



FP = r 



e cos 



FD = al- e cos 



which is the definition of the eccentric anomaly given in (41). 

95. Solution of Kepler's Equation. It will be shown first that 
Kepler's equation always has one, and only one, real solution for 
every value of M and for every e such that ^ e < 1. Write 
the equation in the form 



4>(E) = E - e sin E - M = 0. 

Suppose M has some given value between rnr and (n + l)?r, 
where n is any integer; then there is but one real value of E satis- 
fying this equation, and it lies between mr and (n + I)TT. For, 
the function <f>(E) when E = mr is 

<j>(nir) = mr M < 0. 



95] SOLUTION OF KEPLER'S EQUATION. 161 

And 4>(E) when E (n + I)TT is 

0[(w + !)TT] = (n + I)TT - M > 0. 

Consequently there is an odd number of real solutions for E which 
lie between mr and (n + I)TT. But the derivative 

4>'(E) = 1 - e cos E 

is always positive; therefore 0(#) increases continually with E 
and takes the value zero but once. 

A convenient method of practically solving the equation is by 
means of an expansion due to Lagrange. Suppose z is defined as 
a function of w by the equation 

(43) Z = W + 00(2), 

where a is a parameter. Lagrange has shown that any function 
of z can be expressed in a power series in a, which converges for 
sufficiently small values of a, of the form* 

( F(z) = F(w) * N ' a * d 
(44) 



This expansion can be applied to the solution of Kepler's equation 

by writing it 

E = M + e sin E, 

which is of the same form as (43). The expansion of E as a series 
in e can be taken from (44) by putting F(z) = E, 0(z) = sinJ, 
w = M, and a = e. The result is 

fc*^ 

(45) E = M + r sinM + - ^sin2M + . 

1 1 Z 

All the terms on the right except the first are expressed in radians 
and must be reduced to degrees by multiplying each of them by 
the number of degrees in a radian. The higher terms are con- 
siderably more complicated than those written, and the work of 
computing them increases very rapidly. In the planetary and 
satellite orbits the eccentricity is very small, and the series (45) 
converges with great rapidity, the first three terms giving quite 
an accurate value of E. 

* Williamson's Diff. Calc., p. 151. 
12 



162 DIFFERENTIAL CORRECTIONS. [96 

96. Differential Corrections. A method will now be explained 
in one of its simplest applications, which is of great importance 
in many astronomical problems. Suppose an approximate value 
of E is determined by the first three terms of (45). Call it E ', 
it is required to find the correct value of E. 

Kepler's equation gives 

MQ = EQ e sin E Q . 

For a particular value of M, viz., M , the corresponding value of 
E, viz., EQ, is known. It is required to find the value of E corre- 
sponding to M, which differs only a little from M . The angle M 
is a function of E and may be written 



M = M Q + AM = f(E Q + A# ). 

On expanding the right member by Taylor's formula, this equation 
becomes 

M = Mo + AM = f(E Q ) + /'(#o)A# + 

By the definitions of the quantities, M Q f(E Q )- therefore this 
equation becomes 

(46) M - M Q = f(Eo)AE Q + - = (1 - e cos #o)A#o + . 

Since &E is very small the squares and higher powers may be 
neglected, * and then equation (46) gives for the correction to be 
applied to EQ 



(47) 



1 e cos 



With the more nearly correct value of E, EI = E Q + AE , and 
MI can be computed from Kepler's equation, and a second correc- 
tion will be 

A # = M-M, 

I e cos EI ' 

This process can be repeated until the value of E is found as near 
as may be desired.f In the planetary orbits two applications of 

* If the higher terms in AEo were not neglected AE could be expressed as a 
power series in M M , of which the first term would be the right member 
of (47). 

t For the proof of the convergence of a similar, but somewhat more laborious, 
process see Appell's Mecaniqtte vol. i., p. 391. 



97] 



GRAPHICAL SOLUTION OF KEPLER'S EQUATION. 



163 



the formulas will nearly always give results which are sufficiently 
accurate, and usually one correction will suffice. 

97. Graphical Solution of Kepler's Equation. When the 
eccentricity is more than 0.2 the method of solving Kepler's 
equation given above is laborious because the first approximation 
will be very inexact. These high eccentricities occur in binary 
star and comet orbits, and are sometimes even so great as 0.9. 
In the case of binary star orbits it is usually sufficient to have a 



y 

90 
80 
70 
60 
50 
W 
SO 
20 
10 



Axis 

zp 40 


60 


80 


JOO 


120 


140 


160 , 180 2<jO 














































| 


J. 


^-j 


06 


e 




























E-A&s^s 


























































































































/ 














































































, 






























































































































































/ 






























































































































































/ 












































































^. 


-if- 








































































iX x- 


-^ 


















^ 


h^ 








L 


*W 


J, 





^y 


170 


L 





1< 



























. 


^ 


























V 


. 












5 


Or 


E 


r 






























x 










2 
























X 






































/ 


s 










/ 




























N 






































/ 














































\ 
































/ 














2 




































\ 




























/ 






















































\ 
























/ 
















' 










































\ 




















/ 
















7 














































\ 
















/ 
















j 


















































\ 












2 




































































\ 












T 20 


*~ lM 60 


^ SOB J00 1^0 . lk) 

Fig. 29. 


jft" 


3^0 ' ^OV 



solution to within a tenth of one degree. In this work a rapid 
graphical method is of great practical value. 
Consider Kepler's equation 

E - e sin E - M = 0, 

where M is given and E is required. Take a rectangular system 
of axes and construct the sine curve and the straight line whose 
equations are 

y = sin E, 



The abscissa of their point of intersection is the value of E satis- 
fying the equation;* for, eliminating y, Kepler's equation results. 
The first curve is the familiar sine curve which can be constructed 



* Due to J. J. Waterson, Monthly Notices, 1849-50, p. 169. 



164 



RECAPITULATION OF FORMULAS. 



[98 



once for all; the second is a straight line making with the E-axis 
an angle whose tangent is l/e. Instead of drawing the straight 
line a straight-edge can be laid down making the proper slope 
with the axis. To facilitate the determination of its position 
construct a line with the degrees marked on it at an altitude of 
100;* then place the bottom of the straight-edge at M and the 
top at M + lOOe, and it follows that it will have the proper slope. 
If M is so near 180 that the straight-edge runs off from the 
diagram, the top can be placed at M + 50e on the 50-line. As M 
becomes very near 180 the mean and eccentric anomalies become 
very nearly equal, exactly coinciding at M = 180. 

98. Recapitulation of Formulas. The equations for the com- 
putation of the polar coordinates, when, the time is given, will 
now be given in the order in which they are used. 



n 






(48) 




EQ = M + e sin M + sin 2M, 

MQ = EQ e sin E Q , 

Ag = l-e"co^k' 

#1 = #o + A#o, 

r = a(l e cos E) = 



whence 



(49) 



cos v = 

sin v = 

1 + cos v = 

1 cos v = 



1 + e cos z; ' 

cos E e 
1 - e cos # ' 

Vl e 2 sin # 
1 - e cos # ' 

(1 -e)(l + cosE) 

16 COS # 

(1 +e)(l - cosE') 



1 e cos E 
* This device is due to C. A. Young. 



99] DEVELOPMENT OF E IN SERIES. 165 

The square root of the quotient of the last two equations gives a 
very convenient formula for the computation of v, viz., 

(50) 

The last equation of (48) and equation (50) give the polar co- 
ordinates when E is known. 

99. The Development of E in Series. The equations which 
have been given are sufficient to enable one to compute the polar, 
and consequently the rectangular, coordinates at any epoch; 
yet, in some kinds of investigations, as in the theory of perturba- 
tions, it is necessary to have the developments of not only E, but 
also the polar coordinates, carried so far that the functions are 
represented by the series with the desired degree of accuracy. 

The application of Lagrange's method of Art. 95 to Kepler's 
equation gives E as a power series in e whose coefficients are 
functions of M. This method has been used to get the first terms 
of the series and it can be continued as far as may be desired. 
It is very elegant in practice and is subject only to the difficulty 
of proving its legitimacy. But a direct treatment of Kepler's 
equation based on more elementary considerations is not difficult. 
The solution of 

M = E e sin E 

for E is j?r when M jir, where j = 0, 1, 2, , whatever value e 
may have. Moreover, it has been shown that when e is less than 
unity the solution is unique for all values of M. When e = the 
solution is E = M for all values of M . If u is defined by the 
equation 

E - M = u 

Kepler's equation becomes 

(51) u = e sin (M + u), 

which defines u in terms of M and e. For every value of M different 
from jir, for which the solution is already known, the right member 
of 



sin (M + w) 

can be expanded as a converging power series in u. When this 
series is inverted u will be given as a power series in e whose 



166 DEVELOPMENT OF E IN SERIES. [99 

coefficients are functions of M. Since u vanishes with e, it will 
have the form 

(52) u = ^ e + u 2 e 1 + u 3 e 3 + 

Instead of forming the series in u and then inverting, it is 
simpler to substitute (52) in (51) and to determine u i} u%, - -. by 
the condition that the result shall be an identity in e. The result 
of the substitution is 

+ u 2 e 2 + Us & + = e sin M cos u + e cos M sin u 

TI/T f (u\e -\- u<2,^ ' *) 2 (u\ e -f- ) 4 

= e sin Mil =. H T-; 

2 ! 4 ! 



+ e cos Ml (HI e + u z e 2 + ) - -f- 

On equating coefficients of corresponding powers of e, it is found 

that 

,i = sin M, 



= Ui cos M = jr sin 2M , 



cos Af = 5 sin 3M - sin M, 

O O 



Some general properties of the solutions easily follow from 
these equations. It follows from (51) that if for any M = MQ 
the solution for u, which is known to exist uniquely, is u =- UQ, 
then the solution for M = M Q + 2jV (j any integer) is also u = U Q . 
Therefore u is a periodic function of M with the period 2?r. Since 
this is true for all values of e, each Uj is separately periodic with the 
period 2?r. If any M and UQ satisfy (51), then M and UQ 
also satisfy (51); therefore u is an odd function of M and the Uj 
are sines of multiples of M. If the sign of e is changed and TT is 
added to M in (51), the equation is unchanged; therefore the 
Uj with odd subscripts involve only sines of odd multiples of M , 
and those with even subscripts only sines of even multiples of M. 

It will be shown that the highest multiple of M appearing in 
Uj is jM. The general term of (53) is 

Uj = sinM PJ(UI, u 2 , -, Uj-i) + cos M QJ(UI, u z , , w/_i), 

where P 3 - and Qj are polynomials in u\, u z , , w/_i. These 
quantities must enter in such powers that they are multiplied 
by e*- 1 . Suppose the general terms of the polynomials P, and Qj 



99] DEVELOPMENT OF E IN SERIES. 167 

are, except for numerical coefficients which do not enter into the 
present argument, respectively 



j = u 1 ? 



The exponents of u\, , Uj-\ are subject to the condition that 
PJ and qj shall be multiplied by e*~ l . The term u m carries with it 
the factor e m , and therefore u^ carries the factor mn. Hence the 
exponents .of Ui, , Uj-.\ in PJ and q 3 - must satisfy 



(54) 

' 2k. 

Now suppose that the highest multiples of M in u m is mM for 
m = 1, , j 1. It follows from the properties of powers of the 
sines that the highest multiple in u^ is mnM. Since the highest 
multiple of the product of two or more sines is the sum of their 
highest multiples, the highest multiples in PJ and q, are respec- 
tively 

which are j 1 by (54) . But it follows from (53) that p } - is multi- 
plied by sinTlf and q, by cos M ; therefore the highest multiple 
appearing in u 3 - is jM. That is, u, has the form 

( U2k = + 42*) s in 2M + + ogf ) sin 2kM, 

(55) 

"smM -+ ' + a&*;V ; sin 



according as j is even or odd. 

It is easy to develop a check on the accuracy of the compu- 
tations. Since E = M + u, it follows that 

M = 1+-^= l +^! e 4.^2 , +^i e ; 

dM ^aM ^(9M e+ c)M e ^ 6 

But it follows from Kepler's equation that 



Suppose Af = 0; therefore E = and for this value of M 

r^T? 1 



Therefore, since the coefficient of e' in this series is unity, for M 



168 



DEVELOPMENT OF E IN SERIES. 



[99 



(57) 



dM 



(2k 



These equations constitute a valuable check on all the compu- 
tations. 

It is found from (56) that 

- e sin E dE - e sin E 



dM 2 (I - e cos E) 2 dM (1 - e cos E)* ' 
- e cos E 3e 2 sin 2 E 



(I e cos J5/) 4 (1 e cos ') 5 

For M = 0, the first of these equations is identically zero, but the 
second one becomes 



d 3 E 



-<r^"t 



1-2 



4-5 (n + 2) 
1-2 (n- 1) 



] 



Then the conditions similar to (57) are 



(58) 



4.5 ... (2k + 2) 
1-2 ..- (2/b - 1) 



12 2A; 

These equations constitute a check which is independent of that 
given in (57). In a similar way check formulas can be found 
from a consideration of all odd derivatives of E with respect to M. 

Equations (57), (58), and similar ones for higher derivatives 
of E } are linear in the coefficients af\ which it is desired to find; 
consequently, when the number of equations equals the number 
of unknowns, the latter are uniquely determined, at least if the 
determinant of the coefficients is not zero. It can be shown that 
the determinant is not zero. 

For the purposes of illustration suppose k = 0. Then the 
second equation of (57) gives a^ = 1, whence u\ = sin M 



100] 



DEVELOPMENT OF T AND V IN SERIES. 



169 



agreeing with the result in (53). Suppose k = 1; then the first 
equation of (57) gives 2a ( 2 2) = 1, whence u 2 J sin 2M. As an 
illustration involving both (57) and (58), suppose k = 1 and 
consider the second equations of (57) and (58). They become 
in this case 



3a ( 3 3) = 



,(3) 



ii5 

1-2' 



whence a ( ^ = f , a ( 3 3) = + f , agreeing with the results given 
in (53). 

When the expansion is carried out by the method of Lagrange, M 
or by that which has just been explained, the value of E to terms 
of the sixth order in e is found to be 



(59) 



E = M + e sin M + |- sin 2M 

(3 2 sin3M-3sinM) 






3!2 2 

e 4 
4!2 3 

e 5 
5!2 4 

/j6 



(4 3 sin 4M - 4 - 2 3 sin 2M) 
(5 4 sin 5M - 5 3 4 sin 3M + 10 sin M) 
6 4 5 sin4Af + 15 2 5 sin 



100. The Development of r and v in Series. The value of r in 
terms of e and M can be obtained by the method of Lagrange by 
letting F(z) = cos# and making use of the last equation of (48). 
This method has the disadvantage of being rather laborious. 

It follows from Kepler's equation that * 

BE e sin E 



Therefore 



de I e cos E ' 
dM=(l - e cosE)dE. 



-dM = esmEdE. 
de 



The method employed in this Art. is due to MacMillan. 



170 DEVELOPMENT OF T AND V IN SERIES. [100 

The integral of this equation gives 

f*M f\~j? 

(60) e I -dM = - ecosE + c, 

Jo de 

which expresses e cos E in terms of M very simply by sub- 
stituting in the left member the explicit value of E given in (59). 
For example, the first terms are 



e cos E = c + e I sin M + e sin 2M 

+ |e 2 (3 sin 3M - sinM) + -1 dM 

1 3 

= c ecos M ~e 2 cos2M -^e 3 (cos3M cosM) - . 

Z o 

The last equation of (48) and (60) give for r the series 

(61) - = 1 e cos E = 1 c e cos M - e 2 cos 2M 
a 2i 

It remains to determine the constant c. Since r is measured 
from the focus of the ellipse, it follows that r = a(l e) at 
M = 0; whence 



where 6,- is the coefficient of e 1 ' in the series for e cos E at M = 0. 
The two sides of this equation must be the same for all values 
of e for which (61) converges; therefore c must have the form 

c = c 2 e 2 + c 3 e 3 + , 

where C2, c?,, are determined so that the right member will 
contain no terms in e 2 , e 3 , ; that is, c/ + 6,- = 0, j = 2, 3, . 
Since e cos E, as defined by (60), is the integral of a sine series 
it contains no constant terms; therefore the 6,- are the sums of 
the coefficients of the cosine terms. Now consider 



= J[ 2 ' I 1 - c - e cos M - ^cos 2M + 



dM. 



It was shown in Problem 4, p. 154, that the value of this integral 
is 2ir(l + ^e 2 ). Therefore the coefficients of e 3 , e*, contain no 

constant terms and the exact value of c is 4-e 2 . 

' 

T 

The series for - up to the sixth power of e is 

"- 



100] 



DEVELOPMENT OF r AND V IN SERIES. 



171 



(62) 



- = 1 - 6 cosM - (cos 2M - 1) 
a * 

- 2^22 (3 cos 3M - 3 cosM) 

- 5^5 (4 2 cos 4M - 4 2 2 cos 2M) 

o ! ^5 

- -^~ (5 3 cos 5M - 5 - 3 3 cos 3M + 10 cos M) 

(6 4 cos QM - 6 4 4 cos 4Af + 15 - 2 4 cos 2M) 



5!2 5 



The computation of the series for v will now be considered. It 
follows from the first two equations of (49) that 



dv = 



ll - e 2 



dM, 



(l-e cos E) 2 
which becomes as a consequence of Kepler's equation 

, *( dE \rlM 

The quantity -=?-=. is found at once from (59), and the result squared 



and integrated gives, after Vl e 2 has been expanded as a 
power series in e 2 , 

v = M + 2e smM + fe 2 sin 2M 



(64) 



+ ~ (103 sin 4M - 44 sin 2M) 
9b 

H- jr^r (1097 sin 5M - 645 sin 3M + 50 sin M) 



jrr (1223 sin 6M - 902 sin 4M + 85 sin 2M) 



When e is small, as in the planetary orbits, these series are very 
rapidly convergent; if e exceeds 0.6627 they diverge, as 



172 DIRECT COMPUTATION OF POLAR COORDINATES. [101 

Laplace first showed, for some values of M. This value of e is 
exceeded in the solar system only in the case of some of the comets 7 
orbits, but developments of this sort are not employed in com- 
puting the perturbations of the comets. 

101. Direct Computation of the Polar Coordinates.* It has 
been observed that there is considerable labor involved in finding 
the coordinates at any time in the case of elliptic motion. The 
question arises whether it may not be due partly to the fact that 
the final result is obtained by determining E as an intermediary 
function from Kepler's equation. The question also arises 
whether the coordinates cannot conveniently be found directly 
from the differential equations. It will be shown that the answer 
to the latter question is in the affirmative. 

Equations (16) become in polar coordinates 



.<B 

On integrating the second of these equations and eliminating 

dv 

dt 



-77 from the first by means of the integral, the result is found to be 



d?r h* 

dt 2 r 3 ~* r 2 



After eliminating /b 2 (l + m) by the first equation of (48) and 
changing from the independent variable t to M by means of the 
second equation of (48), these equations become 



(65) 




The first equation of (65) is independent of the second and 
can be integrated separately. It is satisfied by r = a and e = 0, in 
which case the orbit is a circle. In order to get the elliptic orbit 
let 

*This method was first published by the author in the Astronomical 
Journal, vol. 25 (1907). 



101] DIRECT COMPUTATION OF POLAR COORDINATES. 173 

(66) r = a(l - pe), 

where ape is the deviation from a circle. When the planet is at 
perihelion, r = a(l - e). Therefore p = 1 for M = 0. When the 
planet is at aphelion, r = a(l + e). Therefore p = 1 for 

M = TT, and p varies between 1 and + 1. Since -j is zero 



for M equal to and TT, it follows that -7^ is zero for M equal to 

and TT. 

When (66) is substituted in (65), these equations become 




= 

Since e is less than unity and p varies from 1 to + 1, the 
second terms of these equations can be expanded as converging 
power series in e, giving 



(67) 



It has been shown that r, and hence p, is expansible as a power 
series in e. This fact also follows from the form of the first equa- 
tion of (67) and the general principles of Differential Equations. 
Hence p can be written in the form 

(68) p = po -f PI e + p 2 e 2 + , 

where po, PI, pz, are functions of M which remain to be deter- 
mined. Since p is periodic with the period 2w for all e less than 
unity, each py separately is a sum of trigonometric terms. Since 
the motion is symmetrical with respect to the major axis of the 
orbit, and since M = when the planet is at its perihelion, p is 
an even function of M. This is true for all values of e for which 
the series converges, and therefore each p/ is a sum of cosine 
terms. 

A change in the sign of e is equivalent to changing the origin to 
the other focus of the ellipse. Hence if the sign of e is changed 
and TT is added to M the value of r is unchanged; from (66) it fol- 



174 



DIRECT COMPUTATION OF POLAR COORDINATES. 



[101 



lows that the sign of p is changed. Since this is true for all values 
of e for which the series converges 

Pi (M)e* = - Pi (M + *)(- e)t. 

Therefore if j is even p/ is a sum of cosines of odd multiples of M, 
and if j is odd p/ is a sum of cosines of even multiples of M. It 
is seen on referring to equations (68) and (66) that this is the same 
property as that which was established Art. 100. 

It can easily be proved from the properties of the p/ and the 
second equation of (67) that v is expressible as a series of the form 



(69) v = VQ + vie + the 2 + , 

and that each Vj (j > 1) is a sum of sines of integral multiples of M. 
A more detailed discussion shows that if j is even v 3 - is a sum of 
sines of even multiples of M, and if j is odd v 3 - is a sum of sines of 
odd multiples of M. 

The solution can be directly constructed without any difficulty. 
The result of substituting (68) in the first of (67) is 

\d 2 p, j 



dM* 



d 2 ^ 
^dM 2 



+ [PO + Pie + p 2 e 2 
>o]e + [3po 6p pi 



On equating coefficients of corresponding powers of e in the left 
and right members of this equation, it is found that 

-A d *Po i 



(70) 



(&) dM~ 2+pl = l ~ 3p 2 > 



(c) 



P2 = 3 PO (1 - 2 P i - 2p 2 ), 



Equations (70) can be integrated in the order in which they 
are written. Two constants of integration arise at each step 

and they are to be determined so that p = 1 and -~f = for 

M = whatever may be the value of ' e. It follows from (68) 
that these conditions are 

p(0) = po(0) + pi(0)e + p 2 (0)e 2 + - - 



dp_ = dpo .dpi dpz 
dM dM^ dM r dM 



101] DIRECT COMPUTATION OF POLAR COORDINATES. 175 

where M is given the value after the derivatives of the second 
equation have been formed. Since these equations hold for all 
values of e, it follows that 

fpo(O) = i, PI(O) =o, P2 (0) = o, 

(71) 1 dp_o = n ^_i = n dpz = 

[ dM~ dM~ dM~ 

The general solution of equation (a) of (70) is (Art. 32) 
Po = do cos M + 60 sin M, 

where a and b Q are the constants of integration. It follows 
from (71) that a = 1, &o = C, and therefore that 



Po 



= cosM. 



The fact that 6 is zero also follows from the general property 
that the pj involve only cosines. 

On substituting the value of po in the right member of (6) of (70) , 
this equation becomes 

+ Pi = | f cos 2M. 



dM 2 

This equation can be solved by the method of the variation of 
parameters (Art. 37). But since the part of the solution which 
comes from the right member will contain terms of the same 
form as the right member, it is simpler to substitute the expression 

Pi = a\ cos M + 61 sin M + c\ + di cos 2M 

in the differential equation and to determine Ci and di so that it 
will be satisfied. This leads to the solution 

Pi = i cos M + 61 sin M \ + \ cos 2M, 

which is the general solution since it satisfies the differential 
equation and has the two arbitrary constants ai and 61. On 
determining ai and 61 by (71), the expression for pi becomes 

pi = i + | cos 2M. 

With the values of po and pi which have been found equation 
(c) of (70) becomes 



of which the general solution is 



176 DIRECT COMPUTATION OF POLAR COORDINATES. [101 

p 2 = a 2 cos M + 62 sin M + f cos 3M. 

If a 2 and 6 2 are determined by (71), the final expression for p 2 
becomes 

p 2 = f ( cos M + cos 3M). 

This process of integration can be continued as far as may be 
desired. It follows from the results which have been found that 

^ = 1 - pe = 1 - (po + pie + p 2 e 2 + -)e 

= 1 - e cos M -Je^cos 2M - 1) - |e 3 (cos 3M - cos M ) , 

which agrees with those given in (62). 

When the values for p , pi, are substituted in the second 
equation of (67), the result is 

1 + 2e cos M + fe 2 cos 2M + , 

and the integral of this equation is 

v = c + M + 2e sin M + f e 2 sin 2M + 

Since v = when M = 0, the arbitrary constant c is zero, and 
the result agrees with that given in (64). 

The method which has just been developed is, for this special 
problem, perhaps not superior to that depending upon the solu- 
tion of Kepler's equation. But if the conditions of the problem 
are modified a little, for example by adding the terms which 
would come from the oblateness of a planet when the body moves 
in the plane of its equator [equations (30), Chapter IV], Kepler's 
equation no longer holds and the method depending on it fails, 
while the one under consideration here can be applied without 
any modification except in the numerical values of the coefficients 
which depend upon the terms added to the differential equations. 
But additional terms in the differential equations change the 
period of the motion, if indeed it remains periodic, and in order 
to exhibit the periodicity explicitly some modifications of the 
methods of determining the constants of integration are in gen- 
eral necessary. This method of integrating in series is typical of 
those which are employed in the theories of perturbations and the 
more difficult parts of Celestial Mechanics, and for this reason 
it should be thoroughly mastered. 



102] POSITION IN HYPERBOLIC ORBITS. 177 

102. Position in Hyperbolic Orbits. There are close analogies 
between this problem and that of finding the position of a body 
in an elliptic orbit. But it follows from the polar equation of 
the hyperbola, 



r = 



1 + COS V ' 

where a is its major semi-axis and e its eccentricity, that in this 
case v can vary only from TT + cos" 1 ( - J to -f TT cos" 1 ( - J . 

The integrals of areas and vis viva are respectively in the case 
of hyperbolic orbits 



l_ \ / 2 ~\ \ 

U,l 

(72) \ 

(jt) +r \Jt) =A;2(1 + m) U + / 

On eliminating v from the second of these equations by means 
of the first and solving, it is found that 

,. rdr 

avdt = 

where 




This equation can be integrated at once in terms of hyperbolic 
functions, but it is preferable to introduce first an auxiliary 
quantity F corresponding to the eccentric anomaly in elliptic 
orbits. Let 

(73) a + r = ~ (e F + er f ) = ae cosh F; 
then 

vdt= I - 1 + - (e F + e~ F )}dF = [- 1 + e cosh F]dF. 
[ z j 

The integral of this equation is 

(74) M = v(t - T} = - F + I (e F - e~ F ) = - F + e sinh F, 

which gives t when F is known. The inverse problem of finding F 
when v(t T) is given is one of more difficulty. The most 
expeditious method would be, in general, to find an approximate 
value of F by some graphical process, and then a more exact 
13 



178 POSITION IN ELLIPTIC AND HYPERBOLIC [103 

value by differential corrections. The value of F satisfying (74) 
is the abscissa of the point of intersection of the line 

y = i (F + M), 

and the hyperbolic sine curve 

e f e -f 
y = -- -- 



The differential corrections could be computed in a manner 
analogous to that developed in the case of the elliptic orbits. 

From (73) and the polar equation of the hyperbola, it follows 
that 



r = = a _ 

1 -f- COS V 



and from this equation, 



f ~ F ) / + 1, 

tan = -v/ -- - =\ - -tann 

-' >/ - 1 



- 
- 
2 6 - 1 V+ 1 + i(e* + e-') - 1 



which is a convenient formula for computing v when F has been 
found. 

103. Position in Elliptic and Hyperbolic Orbits when e is Nearly 
Equal to Unity. The analytical solutions heretofore given have 
depended upon expansions in powers of e. If e is large, as in 
the case of some of the periodic comets' orbits, the convergence 
ceases or is so slow that the methods become impracticable. 
The graphical process, however, avoids this difficulty. 

In order to obtain a workable analytical solution, the develop- 
ments for elliptical orbits will be made in powers of y - . The 

start is made from the equation of areas and the polar equation 
of the orbit which will be assumed to be an ellipse. 
Let 

w = tan-, 
1 - e 



then the equation of areas becomes 



(1 + w 2 ) 



When X is very small the right member of this equation can be 



103] ORBITS WHEN 6 IS NEARLY EQUAL TO UNITY. 179 

developed into a rapidly converging series in X for all values of v 
not too near 180. Since the periodic comets are always invisible 
when near aphelion, there will seldom be occasion to consider the 
solution in this region. On expanding the right member and 
integrating, the result is found to be 



2(1 - 
(75) 



When the orbit is a parabola e = I and X = 0, and this equation 
reduces to (32), which is a cubic in w. Since the perihelion 

k 
distance in an ellipse is q = a(l e) and n = -j , it follows that 

n Vl + e _k Vl + e 
2(1 - e}* ~ 2q* 

It is desired to find the value of w for any value of t. If the 
eccentricity should become equal to unity, the left member keeping 
the same value, equation (75) would have the form 

(76) fc(1 2 + e)i ( t _ D = w + W 3 , 

where W would be the tangent of half the true anomaly in the 
resulting parabolic orbit. From this equation W can be deter- 
mined by means of Barker's tables, or from equations (33). 
Suppose W has been found; then w can be expressed as a series in 
X of which the coefficients are functions of W. For, assume the 
development 

(77) w = a + aiX + a 2 X 2 + a 3 X 3 + ; 

substitute it in the right member of (75), which is equal to the 
right member of (76). The result of the substitution is 

W + ^ = a + ^ + [a, + ao 2 a! - fa 3 - |a 5 ]X 
+ [a 2 + a 2 a 2 + a ai 2 

+ [a 3 + ao 2 a 3 + 7f 

- 4a W + 3a 4 ai + 3a 6 ai - fa 7 - |a 9 ] X 3 



180 POSITION IN ELLIPTIC AND HYPERBOLIC ORBITS. [103 

Since this equation is an identity in X, the coefficients of corre- 
sponding powers of X are equal. Hence 



ai(l + ao 2 ) = tao 3 + 
a 2 (l + ao 2 ) = - 



4a W 
3a 6 ai + ^a 7 



There are three solutions for a , only one of which is real. On 
taking the real root of the first equation, it is found that 



_ H^ 5 + tffTF 7 + j|TF 9 + 

(1 + Tf 2 ) 3 

= w*+ 



(1 + TF 2 ) 5 



When the values of these coefficients are substituted in (77) the 
tangent of one-half the true anomaly is determined. The first 
term gives that which would come from a parabolic orbit, the 
remaining terms vanishing for e = 1. In the series (64) the first 
term in the right member would be the true anomaly if the orbit 
were a circle, the higher terms being the corrections to circular 
motion. In the series (77) the first term in the right member would 
give the tangent of one-half the true anomaly if the orbit were a 
parabola, the higher terms being the corrections to parabolic 
motion. 

These equations apply equally to hyperbolic orbits in which the 
eccentricity is near unity if 1 e and \-\-e are changed to e 1 
and e H- 1 throughout, where is the eccentricity of the hyperbola. 



PROBLEMS. 181 

XV. PROBLEMS. 

1. Show how the cubic equation (32) can be solved approximately for 
tan | with great rapidity by the aid of a graphical construction. 

2. Develop the equations for differential corrections to the approximate 
values found by the graphical method. Apply to a particular problem and 
verify the result. 

3. If e = 0.2 and M = 214, find E , M , E l} M 1} E 2 , and M 2 . 

Ans. E = 208 39' 16".6, M = 214 8' 58".6; Ei = 208 31' 38".4, 
Mi = 213 59' 59".8; E 2 = 208 31' 38".6, M 2 = 214 00' 00". 

4. Show from the curves employed in solving Kepler's equation that the 
solution is unique for all values of e < 1 and M. 

5. In (50) the quadrant is not determined by the equation; show that 
corresponding values of \v and \E always lie in the same quadrant. 

6. Express the rectangular coordinates x = r cos v, y r sin v in terms 
of the eccentric anomaly, and then, by means of the Lagrange expansion 
formula, in terms of M. 

2 = cos M + I (cos 2M - 3) + ^ (3 cos 3Af - 3 cos M) 



^7^ (4 2 cos 4M - 4 2 2 cos 2Af ) + 
Ans. 

"' = sin M + ^sin 2M + ^ (3 2 sin 3M - 15 sin M) 

(4 3 sin 4M - 10 2 3 sin 2M) + 

7. Show that the properties of E as a power series in e, which were 
established in Art. 99, follow from the Lagrange expansion. 

8. Derive the first three terms of the series for r by the Lagrange formula. 

9. Give a geometrical interpretation of F (Art. 102) corresponding to that 
of E in an elliptic orbit. 

10. Express v as a power series in e by a method analogous to that used in 
Art. 103. 

11. Show that the branch of the hyperbola which is convex to the sun is 
described by the body in purely imaginary time. 

12. Add to the right members of equations (16) the terms TQ (1 +w)6 2 ei 2 ^ 

o 

and (1 + m)6 2 ei 2 ^ , which come from the oblateness of the central body 

[equations (30), Chap, iv.], where e\ is the eccentricity of a meridian section, 
and integrate by the method of Art. 101. 



182 THE HELIOCENTRIC POSITION [104 

104. The Heliocentric Position in the Ecliptic System. Methods 
have been given for finding the positions in the orbits in the 
various cases which arise. The formulas will now be derived 
for determining the position referred to different systems of axes. 
The origin will first be kept fixed at the body with respect to 
which the motion of the second is given. Since most of the appli- 
cations are in the solar system where the origin is at the center of 
the sun, the coordinates will be called heliocentric. 

Positions of bodies in the solar system are usually referred to 
one of two systems of coordinates, the ecliptic system, or the 
equatorial system. The fundamental plane in the ecliptic system 
is the plane of the earth's orbit; in the equatorial system it is the 
plane of the earth's equator. The zero point of the fundamental 
circles in both systems is the vernal equinox, or the point at which 
the ecliptic cuts the equator from south to north, and is denoted 
by V. The polar coordinates in the ecliptic system are called 
longitude and latitude; and in the equatorial, right ascension and 
declination. When the origin is at the sun Roman letters are 
used to represent the coordinates, and when at the earth, Greek. 
Thus 

Origin at sun. Origin at earth. 

longitude I . X measured eastward, 

latitude b + if north; if south, 

right ascension a a measured eastward, 

declination d d + if north; if south, 

distance r p 

In practice a and d are very seldom used. Absolute positions of 
fundamental stars are given in the equatorial system, and the 
observed positions of comets are determined by comparison with 
them. In some theories relating to planets and comets, especially 
in considering the mutual perturbation of planets and their per- 
turbations of comets, it is more convenient to use the ecliptic 
system; hence it is necessary to be able to transform the equations 
from one system to the other. 

The ascending node is the projection on the ecliptic, from the 
sun, of the place at which the body crosses the plane of the ecliptic 
from south to north. It is measured from a fixed point in the 
ecliptic, the vernal equinox, and is denoted by <&. The projection 
of the point where the body crosses the plane of the ecliptic from 
north to south is called the descending node, and is denoted by t3> 



104] IN THE ECLIPTIC SYSTEM. 183 

The inclination is the angle between the plane of the orbit and 
the plane of the ecliptic, and is denoted by i. It has been the 
custom of some writers to take the inclination always less than 
90, and to define the direction of motion as direct or retrograde, 
according as it is the same as that of the earth or the opposite. 
Another method that has been used is to consider all motion direct 
and the inclination as varying from to 180. The latter method 
avoids the use of double signs in the formulas and is adopted here. 
[See Art. 86.] The node and inclination define the position of 
the plane of the orbit in space. 

The distance from the ascending node to the perihelion point 
counted in the direction of the motion of the body in its orbit is w, 
and defines the orientation of the orbit in its plane. The longitude 
of the perihelion is denoted by TT, and is given by the equation 



This element is not a longitude in the ordinary sense because it 
is counted in two different planes. 

The problem of relative motion of two bodies was of the sixth 
order (Art. 85) , and in the integration six arbitrary constants were 
introduced. There are six elements, therefore, which are inde- 
pendent functions of these constants. They are 

a = major semi-axis, which defines the size of the orbit and 

the period of revolution. 

e = the eccentricity, which defines the shape of the orbit. 
&> = longitude of ascending node, and 
i = inclination to plane of the ecliptic, which together define 

the position of the plane of the orbit. 

a) = longitude of the perihelion point measured from the node, 
or TT = longitude of the perihelion, either defining the 
orientation of the orbit in its plane. 

T = time of perihelion passage, defining, with the other ele- 

ments, the position of the body in its orbit at any time. 

The polar coordinates have been computed; hence the rect- 

angular coordinates with the positive end of the re-axis directed to 

the perihelion point and the i/-axis in the plane of the orbit are 

given by the equations 



(78) 




184 



HELIOCENTRIC POSITION IN ECLIPTIC SYSTEM. 



[104 



If the x-axis is rotated backward to the line of nodes, the coordinates 
in the new system are 

fx = r cos (v + co) = r cos (v + TT ft), 
y = r sin (v + o>) = r sin (t; + TT ft), 
2=0. 

The longitude of the body in its orbit counted from the ascending 
node is called the argument of the latitude and is denoted by u. 
It is given by the equation 

u = v -\- w, 
hence u is known when v has been found. 




Fig. 30. 



Let S represent the sun and Sxy the plane of the ecliptic; S&A, 
the plane of the orbit; ft, the ascending node; n, the perihelion 
point; A, the projection of the position of the body; and angle 
USA = v. Then ftA = co + v = u. 

Let the position of the body now be referred to a rectangular 
system of axes with the origin at the sun, the x-axis in the line of 
the nodes, and the i/-axis in the plane of the ecliptic. Then equa- 
tions (79) become 

x' = r cos (v + w) = r cos u, 
(80) { y' = r sin (v + ) cos i = r sin u cos i, 
z' = r sin (v + o>) sin i = r sin u sin i. 



105] TRANSFER OF ORIGIN TO THE EARTH. 185 

But, in terms of the heliocentric latitude and longitude, 
-x' = r cos b cos (I &), 

(81) - y' = r cos b sin (I -&), 

. z' r sin b. 

Therefore, comparing (80) and (81), it is found that 
r cos b cos (I (fi> ) = cos u, 

(82) -j cos b sin (I &) = sin u cos t, 

I sin b = sin it sin i\ 

whence 

ftan (Z &) = tan it cos i, 

(83) -i 

[ tan 6 = tan i sin (I &). 

Since cos 6 is always positive, equations (82) and (83) determine 
the heliocentric longitude and latitude, I and 6, uniquely when 
&>, i, and u are known. 

105. Transfer of the Origin to the Earth. Let E, H, Z be the 
geocentric coordinates of the center of the sun referred to a system 
of axes with the x-axis directed to the vernal equinox, and the 
?/-axis in the plane of the ecliptic. Let P, A, and B* represent the 
geocentric distance, longitude, and latitude of tl*8 sun respectively. 
These quantities are given in the Nautical Almanac for every day 
in the year. The rectangular coordinates are expressed in terms 
of them by 

f*A = P cos B cos A, 
H = P cos B sin A, 
Z = P sin B. 

The angle B is generally less than a second of arc, and unless great 
accuracy is required these equations may be replaced by 

H = P cos A, 
H = P sin A, 
Z = 0. 

Let ", 77", and r" be the geocentric, and x", y", and z" the 
heliocentric, coordinates of the body with the o^axis directed 
toward the vernal equinox and the i/-axis in the plane of the eclip- 
tic. Therefore 

* P, A, B = capital p, X, ft. 



186 TRANSFORMATION TO GEOCENTRIC COORDINATES. [106 

" - x" + H, 
n" - y" + H, 

r" = z" + z. 

In polar coordinates these equations are 

p cos |8 cos X = r cos 6 cos I + P cos B cos A, 
p cos jS sin X = r cos 6 sin I + P cos B sin A, 
p sin j8 = r sin 6 + P sin B. 

From these equations X and ft can be found; but this system may 
be transformed into one which is more convenient by multiplying 
the first equation by cos A, the second by sin A, and adding the 
products; and then multiplying the first by sin A and the 
second by cos A, and adding the products. The results are 

fp cos ft cos (X A) = r cos b cos (I A) -f- P cos B, 
p cos ft sin (X A) = r cos b sin (I A), 
P sin ft = r sin b + P sin B. 

These equations give the geocentric distance, longitude, and 
latitude, p, X, and ft. 

106. Transformation to Geocentric Equatorial Coordinates. 

Let e represent the inclination of the plane of the ecliptic to the 
plane of the equator. Let ", rj", and f" be the geocentric co- 
ordinates of the body referred to the ecliptic system with the 
x-axis directed toward the vernal equinox. Then, the equatorial 
system can be obtained by rotating the ecliptic system around the 
x-axis in the negative direction through the angle e, the relations 
between the coordinates in the two systems being 

-r, 

]'" = 77" cos - f" sin c, 
'" = i?"sine + r" cose; 
or, in polar coordinates, 

cos 8 cos a = cos ft cos X, 

(86) - cos 6 sin a = cos ft sin X cos e sin ft sin e, 
. sin 5 = cos ft sin X sin e + sin ft cos e. 

In order to solve these equations conveniently for 5 and a the 
auxiliaries n and N will be introduced by the equations 



107] 
(87) 



COMPUTATION OF GEOCENTRIC COORDINATES. 



187 



n sin AT = sin 0, 
n cos N = cos j8 sin X, 
in which n is a positive quantity. Then equations (86) become 
' cos 8 cos a = cos |8 cos X, 
cos 5 sin a = n cos (N -f- e) , 
^sin 6 = n sin (JV + e) ; 

n sin A 7 " = sin 0, 

n cos TV = cos j8 sin X, 

cos (TV + e) tan X 
tan a. = 



whence 



(88) 



cos N 
tan 6 = tan (N + e) sin a. 

These equations, together with the first of (86), which is used in 
determining the quadrant in which a lies, give a and 8 without 
ambiguity when X and are known. 

If a and 8 are given and X and /3 are required, the equations from 
which they can be computed are found by interchanging a and 5 
with X and /?, and changing e to e in (88). They are* 

m sin M = sin 8, 

m cos M = cos 8 sin a, 

cos (M e) tan a 
tan X = - ,:. - , 

cos M 

tan j8 = tan (M e) sin X. 

107. Direct Computation of the Geocentric Equatorial Co- 
ordinates. The geocentric equatorial coordinates, a and 8, can 
be found directly from the elements, i and &, and the argument 
of the latitude u } without first finding the ecliptic coordinates, 
X and |S. 

In a system of axes with the z-axis directed to the node and the 
7/-axis in the plane of the ecliptic, the equations for the heliocentric 
coordinates are 

x' = r cos u, 

y' = r sin u cos i, 



z = r sin u sin i. 



* m and M are new auxiliaries, not being related to any of the quantities 
which these letters previously have represented. 



188 



COMPUTATION OF GEOCENTRIC COORDINATES. 



[107 



If the system is rotated around the z-axis until the z-axis is directed 
toward the vernal equinox, the coordinates are 

'x" = x' cos ft y' sin ft, 
y" = x' sin ft + y' cos ft, 



or, 



(90) 



= 



' x" = r (cos u cos ft sin i cos i sin ft), 
" = r (cos w sin ft -f- sin u cos i cos ft), 



r sin w sin i. 



If the system is rotated now around the z-axis through the angle 
e, the coordinates become 



== y cos e z sin e, 
X" = y" sin + z" cos e; 
or, in polar coordinates, 

;'" = rjcos u cos ft sin u cos i sin ft }, 
'"' = r { ( cos w sm ft 4~ sin it cos i cos ft) cos e 
(91) { sin w sin i sin e}, 

z/ " = r{(cos u sin ft + sin u cos i cos ft) sin e 

+ sin u sin i cos e } . 

In order to facilitate the computation Gauss introduced the new 
auxiliaries A, a, B, 6, C, and c by the equations 

sin a sin A = cos ft , 

sin a cos A = sin ft cos i t sin a > 0, 

sin b sin B = sin ft cos e, sin 6 > 0, 

sin b cos B = cos ft cos i cos e sin i sin e, 
sin c sin C = sin ft sin e, sin c > 0, 

L sin c cos C = cos ft cos i sin e + sin i cos c. 

These constants depend upon the elements alone, so they need be 
computed but once for a given orbit. They are of particular 
advantage when the coordinates are to be computed for a large 
number of epochs, as in constructing an ephemeris. When these 



(92) 



PROBLEMS. 189 

constants are substituted in (91), these equations for the helio- 
centric coordinates take the simple form 

x'" r sin a sin (A+ u), 



(93) 



y'" = r sin b sin (B + u), 
z'" = r sin c sin (C + w), 



from which x"', ?/'", and 2'" can be found. 

Then finally, the geocentric equatorial coordinates are defined 
by 

p cos 5 cos a = x'" + X', 
(94) - p cos 5 sin a = y'" + Y 7 , 



where X', Y', and Z' are the rectangular geocentric coordinates of 
the sun referred to the equatorial system. They are given in the 
Nautical Almanac for every day in the year, and, therefore, these 
equations define p, a, and 5. 

This completes the theory of the determination of the helio- 
centric and geocentric coordinates of a body, moving in any orbit, 
when either the ecliptic or the equatorial system is used. 



XVI. PROBLEMS. 

1. Interpret the angle N, equation (87), geometrically and show that n is 
simply a factor of proportionality. 

2. Suppose the ascending node is taken always as that one which is less 
than 180, and that the inclination varies from 90 to + 90; discuss the 
changes which will be made in the equations (78), , (93), and in particular 

write the definitions of the Gaussian constants a, A, , C f or this method 

of defining the elements. 

3. Interpret the Gaussian constants, defined by (92), geometrically. 



190 HISTORICAL SKETCH. 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The Problem of Two Bodies for spheres of finite size was first solved by 
Newton about 1685, and is given in the Principia, Book i., Section 11. The 
demonstration is geometrical. The methods of the Calculus were cultivated 
with ardor in continental Europe at the beginning of the 18th century, but 
Newton's system of Mechanics did not find immediate acceptance; indeed, 
the French clung to the vortex theory of Descartes (1596-1650) until Vol- 
taire, after his visit to London 1727, vigorously supported the Newtonian 
theory, 1728-1738. This, with the fact that the English continued to 
employ the geometrical methods of the Principia, delayed the analytical 
solution of the problem. It was probably accomplished by Daniel Bernouilli 
in the memoir for which he received the prize from the French Academy in 
1734, and it was certainly solved in detail by Euler in 1744 in his Theoria 
motuum planetarum et cometarum. Since that time the modifications have 
been chiefly in the choice of variables in which the problem has been expressed. 

The solution of Kepler's equation naturally was first made by Kepler 
himself. The next was by Newton in the Principia. From a graphical 
construction involving the cycloid he was able to find very easily the approxi- 
mate solution for the eccentric anomaly. A very large number of analytical 
and graphical solutions have been discovered, nearly every prominent mathe- 
matician from Newton until the middle of the last century having given the 
subject more or less attention. A bibliography containing references to 123 
papers on Kepler's equation is given in the Bulletin Astronomique, Jan. 1900, 
and even this extended list is incomplete. 

The transformations of coordinates involve merely the solutions of spherical 
triangles, the treatment of which in a perfectly general form the mathematical 
world owes to Gauss (1777-1855), and which was introduced into American 
Trigonometries by Chauvenet. 

The Problem of Two Bodies is treated in every work on Analytical Me- 
chanics. The reader will do well to consult further Tisserand's Mec. CeL, 
vol. i., chapters vi. and VH. 



CHAPTER VI. 

THE DETERMINATION OF ORBITS. 

108. General Consideration. In discussing the problem of 
two bodies [Arts. 86-88] it was shown how the constants of inte- 
gration which arise when the differential equations are solved can 
be determined in terms of the original values of the coordinates 
and of the components of velocity; and then it was shown how 
the elements of the conic section orbit can be determined in terms 
of these constants. Consequently, it is natural to seek to deter- 
mine the position and components of the observed body at some 
epoch. The difficulty arises from the fact that the observations, 
which are made from the moving earth, give only the direction of 
the object as seen by the observer, and furnish no direct informa- 
tion respecting its distance. An observation of apparent position 
simply determines the fact that the body is somewhere on one 
half of a defined line passing through the observer. The position 
of the body in space is therefore not given, and, of course, its 
components of velocity are not determined. It becomes necessary 
on this account to secure additional observations at other times. 
In the interval of time before the second observation is made the 
earth will have moved and the observed body will have gone to 
another place in its orbit. The second observation simply deter- 
mines another line on which the body is located at another date. 
It is clear that the problem of finding the position of the body and 
the elements of its orbit from such data presents some difficulties. 

The first question to settle is naturally the number of obser- 
vations which are necessary in order that it shall be possible to 
determine the elements of the orbit. Since an orbit is defined by 
six elements, it follows that six independent quantities must be 
given by the observations in order that the elements may be de- 
termined. A single complete observation gives two quantities, the 
angular coordinates of the body. Therefore three complete obser- 
vations are just sufficient, so far as these considerations are con- 
cerned, to define its orbit. It is at least certain that no smaller 
number will suffice. If the observed body is a comet whose 

191 



192 INTERMEDIATE ELEMENTS. [109 

orbit is a parabola, the eccentricity is unity and only five elements 
are to be found. In this case two complete observations and one 
observation giving one of the two angular coordinates are enough. 

109. Intermediate Elements. The apparent positions of the 
observed body are usually obtained by measuring its angular 
distances and directions from neighboring fixed stars. Since the 
stars are catalogued in right ascension and declination the results 
come out in these coordinates, but they can, of course, be changed 
to the ecliptic system, or any other, if it is desired. 

Suppose the observations are made at the times ti, t 2 , and 3 , 
and let the corresponding coordinates be denoted by their usual 
symbols having the subscripts 1, 2, and 3 respectively. The right 
ascensions and declinations are functions of the elements of the 
orbit and the dates of observation. These relations may be 
represented by 

'on = <?(&, i, co, a, 6, T] ti), 

2 = <?(&, i, co, a, e, T; 2 ), 
, i, co, a, e, T; t 8 ), 



, i, co, a, 6, T; J 2 ), 
<5 3 = <K&, i, co, a, e, T; U). 



The problem consists in solving these six equations for the six 
unknown elements. The functions $ and \f/ are highly transcen- 
dental and involve the elements in a very complicated fashion. 
In the case of an ellipse the position in the orbit is found by passing 
through Kepler's equation, in the hyperbola the process is similar, 
and in the parabola a cubic equation must be solved; and in all 
three cases the coordinates with respect to the earth are obtained 
by a number of trigonometrical transformations. Hence it is 
clear that there is no direct solution of equations (1) by ordinary 
processes. 

Although the ultimate object is to determine the elements of 
the orbit, the problem of finding other quantities which define the 
elements may be treated first. These quantities may be con- 
sidered as being intermediate elements. It has been remarked 
that if the coordinates and the components of velocity are known 
at any epoch, the elements can be found. Suppose it is desired 
to find the polar coordinates and their derivatives, which deter- 



109] 



INTERMEDIATE ELEMENTS. 



193 



mine uniquely the rectangular coordinates and their derivatives, 
at the time of the second observation tz. The equations corre- 
sponding to (1) become for this problem 

i = / (az, 8z, pz, oiz ', 82', pz'', t\, tz), 



(2) 



where 



as = f (az, 8z, P2, OLZ, 8z', pz', tz, ts), 

81 = g(az, 8z, P2, Wi 5 2 ', p/; ti, t z ), 
62 = 82, 

83 = 0(2, 5 2 , P2, 2', V, P 2 '; *2, 3), 



,_da 



dS , dp 

P2 ' = _ a t t = U. 



Since 2 and 6 2 are observed quantities only the first, third, fourth, 
and sixth equations are to be solved for the four unknowns P2, ctz, 
62', and P2 7 . The problem is therefore reduced to the solution of 
four simultaneous equations, and they are moreover much simpler 
than (1). These equations can be put in a manageable form, and 
this is, in fact, one of the methods of treating the problem. It was 
first developed and applied to the actual determination of orbits 
by Laplace in 1780, and it has been somewhat extended and 
modified as to details by many later writers. 

As another set of intermediate elements the three coordinates at 
two epochs may be taken. Suppose the times t\ and 3 are chosen 
for this purpose. Then the fundamental equations corresponding 
to (1) can be written in the form 



(3) 



F(ai, di, pi, 0:3, 5 3 , 



6 2 = G(ai, 
8s = 8s. 



pi, 0:3, 8 S , 



In this case the equations are reduced to two in the two unknowns 
Pi and p 3 , and they also can be solved. This is the line of attack 
on the problem laid out by Lagrange in 1778, taken up inde- 
pendently and carried out differently by Gauss in 1801, and fol- 
lowed more or less closely by many later writers. In spite of the 
14 



.194 PREPARATION OF THE OBSERVATIONS. [110 

hundreds of papers which have been written on the theory of the 
determination of orbits, very little that is really new or theoreti- 
cally important has been added to the work of Laplace and Gauss 
unless more than three observations are used. 

110. Preparation of the Observations. Whatever method it 
may be proposed to follow, the observations as obtained by the 
practical astronomer require certain slight corrections which should 
be made before the computation of the orbit is undertaken. 

The attractions of the moon and the sun upon the equatorial 
bulge of the earth cause a small periodic oscillation and a slow 
secular change in the position of the plane of its equator. Since 
the equinoxes are the places where the equator and ecliptic inter- 
sect, the vernal equinox undergoes small periodic oscillations 
(the nutation) and slowly changes its position along the ecliptic 
(the precession). It is obviously necessary to have all the obser- 
vations referred to the same coordinate system, and it is customary 
to use the mean equinox and position of the equator at the begin- 
ning of the year in which the observations are made. 

The observed places are also affected by the aberration of light 
due to the revolution of the earth around the sun and to its rota- 
tion on its axis. Since the rotation is very slow compared to the 
revolution, the aberration due to the former is relatively small 
and generally may be neglected, especially if the observations 
are not very precise. 

Suppose do and 5 are the observed right ascension and declina- 
tion of the body at any time. Then the right ascension and 
declination referred to the mean equinox of the beginning of the 
year, and corrected for the annual aberration, are 

(a = ao 15/ g sin (G-\-a ) tan BQ h sin (H +0:0) sec do, 
d = 5 i cos 5 g cos (G + o) h cos (H + a ) sin 5 , 

where /, g, h, G, and H are auxiliary quantities, called the Inde- 
pendent Star-Numbers, which are given in the American Ephem- 
eris and Nautical Almanac for every day of the year. In 
practice these numbers are to be taken from the Ephemeris. 
They depend upon the motions of the earth, but their derivation 
belongs to the domain of Spherical and Practical Astronomy, 
and cannot be taken up here.* The corrections to ao and 6 
furnished by equations (4) are expressed in seconds of arc. 

* Chauvenet, Spherical and Practical Astronomy, vol. i., chap. xi. 



Ill] OUTLINE OF THE LAPLACIAN METHOD. 195 

The corrections for the diurnal aberration are 

a = - 0".322 cos <p cos (6 - a ) sec 5 , 



I A5 = 0".322 cos <p sin (B a ) sin 5 , 

where <p is the latitude of the observer, and B 0:0 is the hour 
angle of the object at the time of the observation. The second 
of these corrections cannot exceed the small quantity 0".322, 
and the first is also small unless 5 is near =*= 90. 

111. Outline of the Laplacian Method of Determining an Orbit. 
Before entering on the details which are necessary for the deter- 
mination of the elements of an orbit by either of the two methods 
which are in common use, a brief exposition of the general lines of 
argument used in them will be given. From these outlines the 
plan of attack can be understood, and then the bearings of the 
detailed investigations will be fully appreciated. 

In order to keep to the central thought suppose only three com- 
plete observations are available for the determination of the orbit. 
Let the dates of the observations be ti, fa, and ts, and hence at 
these times the right ascensions and declinations of the observed 
body as seen from the earth are known. For the sake of definite- 
ness in the terminology let C represent the observed body revolv- 
ing around the sun, $, and observed from the earth E] , 77, the 
rectangular coordinates of C with respect to E; x, y, z the rectan- 
gular coordinates of C with respect to S; X, Y, Z the rectangular 
coordinates of S with respect to E; p the distance from E to C; 
r the distance from S to C; R the distance from E to S. Then 

i = p cos 5 cos a = p X, 
r] = p cos 5 sin a = PM, 
= p sin 6 = p v. 

The quantities X, ju, and v, which are the direction cosines of the 
line from E to C, are known at t\, fa, and fa. The distance p is 
entirely unknown. 

First Step. The first step is to determine the values of the 
first and second derivatives of X, /*, v, X, Y, and Z at some time 
near the dates of observation, say at fa. It will be sufficient at 
present to show that it can be done with considerable approxi- 
mation without discussing the best method of doing it. The 
value of the first derivative of X during the interval fa to fa averages 

x ' X2 Xi 

Al2 - ~ , 



196 OUTLINE OF THE LAPLACIAN METHOD. [Ill 

and this is very nearly the value of X' at the middle of the interval 
unless X' happens to be changing very rapidly. The approxima- 
tion is better the shorter the interval. In a similar manner X^ 
is formed. When the interval t-> ti equals the interval 3 t% 
the value of X' at t z is very nearly 

Xz ~ 2lXi2 ~T X 2 3J. 

If the intervals are not equal, adjustment for the disparity can of 
course be made. 

In a similar manner it follows from the definition of a derivative 
that the second derivative of X at 2 , in case the two intervals are 
equal, is approximately 



The first and second derivatives of M and v are given approximately 
by similar formulas, and it is to be understood that when the 
intervals are as short as they generally are in practice the approxi- 
mations, especially as obtained by the more refined methods 
which will be considered in the detailed discussion, are very close. 
The American Ephemeris gives the values of X, Y, and Z for every 
day in the year, and from these data the values of their first and 
second derivatives can be found. As a matter of fact only the 
first derivatives of these coordinates will be required. 

Second Step. The second step is to impose the condition that C 
moves around S in accordance with the law of gravitation. It 
will be assumed that C is not sensibly disturbed by the attractions 
of other bodies. Hence its coordinates satisfy the differential 
equations 

'd z x k 2 x 

dt 2 ~ r 2 ' 

fJ2ni Tf^ll 

(<-r\ \ & y K y 

*W m '.7' 

Off = "r 3 "' 
But it also follows from the relations of C, E, and S that 

x = P X - X, 

(8) -{ y = PM - Y, 

z = pv Z. 



Ill] 



OUTLINE OF THE LAPLACIAN METHOD. 



197 



On substituting these expressions for x, y, and z in equations (7), 
they become 



(9) 



(PM)" - Y 



(p.)" - Z" - 



r 3 

v - Z) 



But since E also revolves around S in accordance with the law 
of .gravitation, it follows that 

y _ * X 
R* > 

1.2V 

"V" 

' & ' 



7 n 

VBT- 



Therefore equations (9) become 
Xp" + 2X'p'+ |\" + 

(10) - 



p = - VX 



The unknown quantities in these equations are p", p', p, and r, 
the first three of which enter linearly. 

Third Step. The third step is to determine the distance of C 
from E and S by means of equations (10) and a geometrical 
condition which the three bodies must satisfy. In order to solve 
equations (10) for p, let 



(ID D = 



\ \' \" _i_ 

A, A , AT ~3~ 

/ \ /J I y 

M, M, x + ^ 



X, X', X" 



"', "" 



The second form of the determinant D is obtained by multiplying 



198 



OUTLINE OF THE LAPLACIAN METHOD. 



[Ill 



k 2 
the first column by -5 and subtracting the product from the third 

column. The determinant which is obtained by replacing the 
elements of the third column of D by the right member of (10) 



will also be needed. If the common factor I -^ 
this determinant is 



-^ -- - 



is omitted, 



(12) 



X 



v , 



The determinants D and DI involve only known quantities. 
The solution of equations (10) for p is 

(13) P = \ 

To this equation in the two unknown quantities p and r must be 
added the equation 

(14) r 2 = p 2 + R 2 - 2 P R cos ^, 

which expresses the fact that the three bodies C, S, and E form a 
triangle. The angle ^ is the angle at E between R and p, and 
this equation also has only the unknowns p and r. The problem 
of solving (13) and (14) for p and r is that which constitutes the 
third step. The solution of this problem gives the coordinates 
of C by means of equations (8) which involve only p as an unknown. 
Fourth Step. The fourth step is the determination of the 
components of velocity of C. It follows from (8) that 



= 



(15) 



y' = pV + P/*' - Y', 
= p'v + P v r - Z'. 

The only unknown in the right members of these equations is p' 
which can be determined from (10). The expression for it is 



(16) 



,'- -Mi. 11 

r 2D[R s r 3 ]' 



Z> = - 



X, X, X" 
M, Y, IL" 

V, Z, 

Therefore x f , y', and z' become known. 



v" 



112] 



OUTLINE OF THE GAUSSIAN METHOD. 



199 



Fifth Step. The fifth and last step is to determine the elements 
of the orbit from the position and components of velocity of the 
body. This is the problem which was solved in chap. v. 

112. Outline of the Gaussian Method of Determining an Orbit. 

First Step. The first step in the Gaussian method is to impose 
the condition that C moves in a plane passing through S. Since 
S is the origin for the coordinates x, y, and z, this condition is 

Axi + %i + Czi = 0, 
Ax 2 + By* + Cz 2 = 0, 
Axs + By s + Cz 3 = 0, 

where A, B, C are constants which depend upon the position of 
the plane of motion. The result of eliminating the unknown 
constants A } B, and C is the equation 



(17) 



Xz, 



2/i, 



2/3, 



= 0. 



The determinant (17) can be expanded with respect to the 
elements of the three columns giving the three equations 

' (2/223 Z 2 2/s)Xi (2/123 Zi2/s)x 2 + (2/l2 2 Zi2/ 2 )X3 = 0, 

(18) - (X 2 Z 3 - 22X3)2/1 - (XiZ 3 - 21X3)2/2 + (Xi2 2 - 2iX 2 )2/ 3 = 0, 

(x 2 2/ 3 - 2/2^3)21 - (xi2/ 3 - 2/1^3)22 + (xii/2 - 2/1^2)23 = 0. 

Evidently these three equations are but different forms of the same 
one; but when the nine parentheses are determined from additional 
principles and xi, x 2 , are expressed in terms of the geocentric 
coordinates by (8), they become independent in the unknowns 
pi, Pz, and p3. The parentheses are the projections of twice 
the triangles formed by S and the positions of C taken in twos 
upon the three fundamental planes. Since in each equation the 
three areas are projected upon the same plane the triangles 
themselves can be used instead of their projections. If [1, 2], 
[1, 3], and [2, 3] represent the triangles formed by S and C ai the 
times tit*, Ms, and Ms respectively, equations (18) become 

[2, 3]xi [1, 3]x 2 + [1, 2]x 3 = 0, 

(19) -( [2, 3] 2/1 - [1, 3] i/ 2 + [1, 2] 2/3 = 0, 

[2, 3] 2l - [1,3] 2 2 + [1,2] 2, =0. 



200 



OUTLINE OF THE GAUSSIAN METHOD. 



[112 



Second Step. The second step consists in developing the ratios 
of the triangles as power series in the time-intervals. This is 
done by integrating equations (7) as power series in the time- 
intervals, and then substituting the results for t = ti, t z , t 3 in the 
coefficients of (18) or (19). Inasmuch as these series are based 
upon equations (7) the condition that C shall move about S in 
accordance with the law of gravitation has been imposed. In 
order not to prolong the discussion at this point (for the details 
see Art. 127) the results will be given at once. For the purpose 
of simplifying the writing, let 



(20) 



In this notation the ratios of the triangles [2, 3] and [1, 2] to [1, 3] 
are found to be 



- t,) = 3 , 
k(t s - t 2 ) = Oi, 



(21) 



ro Ql 
l^> "J _ 



_ l - 

- f ~ 



[i, 

Uoi a r 1/32 /j 2 

> *J _ I 3 1 _|_ i ^2 #3 

Jl, 3] 2 L r 6 r 2 3 



I 
J' 



Third Step. The third step consists in developing equations 
for the determination of pi, p 2 , and p 3 . The results of substituting 
equations (8) and (21) in (19) are 



(22) 



0i 



01 



r, i i^ 2 -0i ; 

L 1 + 6~7^ 



+ 

102 2 - 



+ 



(x 3P3 - x t ) = o, 



['+1*5* 



( M ipi - Fi) - 



6 r 2 3 



(M3P3 - 



- 0i 5 



10 2 2 - 
6 r 2 3 



0, 



("3P3 - Z 8 ) = 0. 



112] OUTLINE OF THE GAUSSIAN METHOD. 201 

These equations involve the unknowns pi, p 2 , PS, and r 2 , the first 
three of which enter linearly. Since r 2 enters only as it is multi- 
plied by the small quantities 0i 2 , 2 2 , or 3 2 , it might be supposed 
that in a first approximation these terms could be neglected, after 
which pi, p 2 , and p 3 would be determined by linear equations. 
A detailed discussion of the determinants which are involved 
shows, however, that it is necessary to retain the terms in r 2 even 
in the first approximation. 

The solution of equations (22) for p 2 has the form 

(23) A P2 = P + J, 

where A is the determinant of the coefficients of pi, p 2 , and p 3 , 
and P and Q are functions of the known quantities \i, X 2 , , 
Xi, YI, 

Since 8, E, and C form a triangle at t z the quantities p 2 and r 2 
satisfy the equation 

(24) 7- 2 2 = p 2 2 + # 2 2 - 2p 2 E 2 cos fa. 

The solution of any two equations of (22) for pi and p 3 in terms 
of p 2 and r 2 has the form 






where M, PI, P 3 are functions of known quantities, and Q\ and Q 3 
involve only r 2 as an unknown. 

Fourth Step. The fourth step consists in determining pi and ps. 
The quantities p 2 and r 2 are found first by solving (23) and (24), 
which is exactly the same as the third step of the Laplacian 
method, and then pi and p 3 are given by (25). 

Fifth Step. The fifth step consists in determining the elements 
from the known positions of C at the times t\ and 3 . These two 
positions and that of C define the plane of the orbit without 
further work. Gauss solved the problem of determining the 
remaining elements by developing two equations involving only 
two unknowns. One equation was derived from the ratio of 
the triangle formed by S and C at ti and t 3 to the area of the 
sector contained between r\, r 3 , and the arc of the orbit described 
in the interval tit*. The other equation was derived from Kepler's 



202 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[113 



equation at the epochs ti and t^ The formulas are complex, 
but the method of solving the two equations is a rapid process 
of successive approximations. After the equations are solved the 
elements are uniquely determined without any trouble. Later 
methods have been devised which avoid many of the complexities 
of that due to Gauss. 

I. THE LAPLACIAN METHOD OF DETERMINING ORBITS. 

113. Determination of the First and Second Derivatives of the 
Angular Coordinates from Three Observations. It was found in 
the outline [Art. Ill] of this method of determining orbits that 
the first and second derivatives of the angular coordinates, or 
of the direction cosines X, n, and v will be required. 

Let k(t o) = T and then equations (7) become 



(26) 



_ _ 

~d?~ ~^' 

&y = _g. 

dr 2 r 3 ' 



tfz 
dr* 



^ 

r 3 * 



a 

Suppose x = x 0) y = y Q , z 



z , 



dx 



rfi/ 



$ = *,' at 



n\dr n 



n\\dr n 



r = 0. The solution of equations (26) can be expanded as power 
series in r which will converge if the value of r is not too great.* 
They will have the form 



(27) 



where the subscript on the parentheses indicates that the deriva- 
tives are taken for r = 0. The second derivatives can be replaced 
by the right members of (26) for r = 0; the third derivatives can 
be replaced by the first derivatives of the right members of (26), 
and so on. All the derivatives in this way will be expressed in 
terms of x , 2/0, ZD, XQ', yo', and z '. 

* For the determination of the exact realm of convergence see a paper by 
F. R. Moulton in The Astronomical Journal, vol. 23 (1903). 



113] FIRST AND SECOND DERIVATIVES OF X, /z, V. 203 

It is important to know for how great intervals the series (27) 
are of practical value. The limits are smaller the smaller the peri- 
helion distance and the greater the eccentricity, and moreover 
they depend upon the position of the body in its orbit at r = 0. 
For a small planet whose mean distance is 2.65, which is about 
the average for these bodies, and the eccentricity of whose orbit 
does not exceed 0.4, which is much greater than that of most of 
them, the series (27) always converge for an interval of less than 
160 days. If the orbit is a parabola whose perihelion distance is 
unity the series (27) converge if the interval of time does not 
exceed 54 days. Of course, the series are not of practical value 
in their whole range of convergence. In practice in the case of 
small planets an interval of 90 days is nearly always small enough 
to secure rapid convergence of (27), and in the case of the orbits 
of comets 20 days is rarely too great an interval. 

The coordinates of the earth also are expansible as series of the 
form of (27), and the rapid convergence holds for very long 
intervals because of the small eccentricity of the earth's orbit. 
Hence it follows from equations (8) that p, X, ju> and v can be 
expanded as power series of the type of (27). The range of 
usefulness of these expansions is the same as that of the series 
for x, y, and z. 

It will be sufficient to consider the series for X because those 
in fjL and v are symmetrically similar. The series for X for a 
general value of r and for n, r 2 , and r 3 , which correspond to 
h, U, and 3 respectively, are 

X = C + CIT + C 2 r 2 + 
(28) Xi^o + cxn + c^ 2 -!- 



X 2 CQ | ClT 2 -f- C 2 T 2 -p * * *, 
X 3 = C + CiT 3 + C 2 T 3 2 -H '."I 

where Co, Ci, c 2 , are constants. If these equations are termi- 
nated after the terms of the second degree the coefficients Co, Ci, 
and c 2 are determined in terms of the observed quantities Xi, X 2 , 
and X 3 , and the time-intervals n, r 2 , and r 3 . If more observations 
are available more coefficients can be determined; the number 
which can be determined equals the number of observations. 

The simplest way of expressing X in terms of r with known 
coefficients is to set equal to zero the eliminant of 1, c , Ci, and c 2 
in (28), which is 



204 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[113 



(29) 



X, 1, 

Xi, 1, 

X 2 , 1, 

X 3 , 1, 



= 0. 



The expansion of this determinant with respect to the elements of 
the first column is 



= (r 2 ri)(r 3 TO)(TI r 3 ), 



and where A\, A 2 , and A 3 are obtained from A Q by permuting r 
with TI, T 2 , and r 3 respectively. The determinant Ao is distinct 
from zero if TI, T 2 , and T 3 are distinct. Hence equation (30) 
becomes 

(r - r 2 )(r - T8 ) , (r - r 3 )(r - x 



(30) A X - Ai 


where 






1, 


Tl, Ti 2 


Ao = 


1, 


T2, Tl 2 




1, 


T 3 , T 3 2 



X = 



(31) 



(TI TZ)(TI r 3 ) (TZ - 

(r - TI )(T - T 2 ) 



X 2 



, 



(r 3 TI)(TS 



It follows from the form of (31) that this equation gives X 
exactly at n, TZ, and r 3 ; for other small values of T it gives X ap- 
proximately. The exact value of X is given by an infinite series, 





l^SS- 


V 




~^ 

1 


"\ 




I 


\^ 




1 

1 


\ 




! 


t 


1 


<* / 






Fig. 31. 





the first equation of (28), within the range of its convergence. 
Geometrically considered this series defines a curve, marked C in 
Fig. 31. The second degree polynomial (31) defines another 



114] 



DERIVATIVES FROM FOUR OBSERVATIONS. 



205 



curve C 2 . These two curves intersect at TI, T 2 , and T 3 , but in 
general do not intersect elsewhere. For small values of T the 
two curves nearly coincide, and the approximate value of X can 
be found from the polynomial near the origin. 

The first and second derivatives of X are found from (31) to be 
given approximately by 

2T - ( T2 + T 8 ) 2T - 



(32) 



T 2 )(Tl T 3 ) 
2T- I 






(r 2 r 3 )(r 2 TI) 
r 2 ) 



TI)(TS r 2 ) 



X 8| 



T 2 )(Tl T 3 ) 



(T 2 T 3 )(T 2 Tl) 



X 2 



X 2 



X 3 . 



T S TI)(TS r 2 ) 
There are similar expressions in ju and v. 

114. Determination of the Derivatives from more than Three 
Observations. If the observations were perfectly exact and 
near together, the more there were available the more exactly 
could X be determined for small values of T, and the more of its 
derivatives could be determined. Suppose there are four obser- 
vations. Then X is defined by a third degree polynomial analogous 
to (31) which reduces to Xi, X 2 , X 3 , and X 4 for r = TI, r 2; r 3 , and r 4 
respectively. The explicit expression for X is 

(r r 2 )(r r 3 )(r r 4 ) ^ 
TI r 2 )(ri TS)(TI T 4 ) 

(r TS)(T r 4 )(r 



(33) 



X= + 



(TZ T 3 )(T 2 T 4 )(T 2 TI 

(T T 4 )(r TI)(T T 2 ) 



TS T 4 )(T 3 

(T TI)(T 



S T2) 

T 3 ) 



X 2 



X 3 



X 4 , 



(T4 Ti)(T 4 T 2 )(T 4 T 3 ) 

from which the first, second, and third, but not higher, derivatives 
can be found. 

It is obvious from this how to proceed for any number of obser- 
vations. The process is unique and does not become excessively 
laborious unless the number of observations is considerable. The 
number of derivatives which can be determined, at least approxi- 
mately, is one less than the number of observations, but no 



206 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[115 



derivative higher than the third will in any case be used. If the 
observations extend over a long period so that the convergence 
of (28) fails or becomes slow for the largest values of r, it is neces- 
sary to omit some of them in the discussion. Usually, owing to 
the errors in the observations, four or five will give X and its 
first two derivatives as accurately as any greater number. 

115. The Approximations in the Determination of the Values 
of X, M> v and their Derivatives. In the applications it is im- 
portant to know the character of the approximations which are 
made, and whether all the quantities employed are determined 
with the same degree of accuracy. It is obvious no exact numerical 
answers can be given to these questions because the orbits under 
consideration are undetermined. But it has been insisted that 
the values of r must not be too great in order that the series (28) 
shall converge rapidly. Consequently, the values of r at the 
times of the observations can be considered as small quantities, 
and the degree of the approximation can be described in terms 
of the lowest powers of the T, which occur in the neglected terms. 
This gives a definite meaning to the order of approximation, and 
experience shows that it is a satisfactory measure of the accuracy 
of the results when the time-intervals are limited as described 
in Art. 113. 

Suppose first that only three observations have been made. 
The approximations in the determination of X and its derivatives 
arise from the fact that the higher terms of (28) are neglected. 
The coefficients c , Ci, and c 2 are determined by 

Co + Cm H- C 2 r i 2 = Xi C 3 Ti 3 C 4 Ti 4 , 






ClT 2 
CiT 3 



C 2 r i 

C 2 T 2 2 = X 2 - C 3 T 2 3 - C 4 T 2 4 - 



C 2 T 3 2 = X 3 C 3 T 3 



C 4 T 3 



4 



The errors of lowest degree in the TJ come from neglecting the 
terms in the right members which are multiplied by the unknown 
constant c 3 . Let the errors be denoted by Aco, Aci, and Ac 2 . 
Then 

C 4 Tl 4 + 



T 3 , 



T! 2 




C 3 Ti 3 


T 2 2 


Aco = 


C 3 T 2 3 


T 3 2 




C 3 T 3 3 




Tl 3 


= C.3 


T 2 3 




T 3 3 



C 4 T 2 4 + 
C 4 T 3 4 + 
Tl, Tl 2 
T 2 , T 2 2 
T3, T 3 2 



C 4 



Tl, 


Tl 2 











T 2 , 


T 2 2 






o 




T3, 


T 3 2 




Tl 4 , 


Tl, Ti 2 


T2 4 , 


T 2 , T 2 2 


T3 4 , 


T3, T 3 2 



116] CHOICE OF THE OKIGIN OF TIME. 207 

and similar expressions for Aci and Ac 2 . These determinants are 
easily reduced by the elementary rules for simplifying deter- 
minants, and it is found that 

^Co = C 3 TlT 2 T 3 C4TiT 2 T 3 (ri + T 2 + T 3 ) + ', 
^Cl = + C 3 (TlT 2 + T 2 T 3 + T 3 Tl) 

(35) - + C 4 (ri + r 2 )(r 2 + r 3 )(r 3 + n) + *, 

Ac 2 = C 3 (ri + r 2 + r 3 ) 

C 4 (Tl 2 ~h T 2 2 ~|- T 3 2 -{~ TlT 2 -f- T 2 T 3 -f- T 3 Ti) + 

It follows from these equations that c , Ci, and c 2 are determined 
up to the third, second, and first orders respectively. 

Now consider the first equation of (28). Since Ci is multiplied 
by r and c 2 by r 2 , each of ^he first three terms in the series for X is 
determined up to the third order in the r/. On taking the first and 
second derivatives, it is seen that X' and X" are determined up to 
the second and first orders respectively. Consequently, X'in 
general is determined by the first terms of (28) more accurately 
than its first derivative, and its first derivative in general is 
determined more accurately than its second derivative. These 
facts must be remembered in the applications. 

116. Choice of the Origin of Time. The origin of time has 
not been specified as yet except that it has been supposed that it is 
near the dates of the observations so that n, r 2 , and r 3 will be 
small. Any epoch fa which satisfies this condition can be used 
as an origin, and the problem at once arises of determining what 
one is most advantageous. 

The choice of the origin of time which has been universally made 
is the date of the second observation. That is, fa = fa and there- 
fore r 2 = 0. The value of X is exactly known at r = r 2 = 0, and 
the derivative of X at t = fa is 

X 2 ' = Ci -j- 2c 2 r 2 -}- = Ci t 

which is subject to the error Aci, which, by (35), is in this case 
c 3 T 3 ri. And similarly, the error in X 2 " is Ac 2 = c 3 [ri + r 3 ]. 
The error in X 2 ' is of the second order while that in X 2 " is of the 
first order. In general, an error of the first order is more serious 
than one of the second order. But it should be noticed that 
when o = fa the quantities n and r 3 are opposite in sign; and if 
the intervals between the successive observations are equal, 
TI H- r 3 = and the error in X 2 " is also of the second order. Con- 



208 LAPLACIAN METHOD OF DETERMINING ORBITS. [117 

sequently, when to = 2 it is advantageous to have the successive 
observations separated by as nearly equal time-intervals as 
possible. But unfavorable weather and other circumstances 
generally cause the observations to be unequally spaced. 

Suppose the epoch of the first observation is taken as the origin 
of time. The quantity Xi is exactly known* The error in X/ is 
Aci = c 3 T 2 T3, which is of the second order as before, but is approxi- 
mately twice as great numerically as that in X 2 ' because r 3 now 
represents k times the whole interval between the first and third 
observations. The error in X/' is Ac 2 = C 3 (r 2 + r 3 ) which 
is much larger than before because r 3 now depends on the whole 
interval covered by the observations, and because r 2 and r 3 in 
this case are both positive. It follows from this that it is not 
advantageous to use the time of the first observation as the origin 
of time; and for similar reasons the epoch of the third observation 
is to be rejected. 

The question now arises what should be taken for the origin 
of time when the epoch of the second observation is not midway 
between those of the other two. Since in general the error in X is 
only of the third order and that in X' is only of the second, while 
X" is subject to an error of tne first order, it is clear that the origin 
of time should be so chosen, if possible, as to make the first order 
error in X" vanish. It follows from the second equation of (35) 
that this result will be secured if 

fri + r 2 + r 3 = k(ti - to) + kfa - t ).+ k(t s - t ) = 0, 
(36) 4 

[ whence to = ^ (ti + 2 + 3). 

The best choice of the origin of time is therefore given by the 
second of (36), and this value of t becomes the date of the second 
observation when the successive observations are equally distant 
from one another. With this choice of to the errors in X' and X" 
are of the second order, while X is known up to the third order. 

117. The Approximations when there are Four Observations. 

When there are four observations the equations which correspond 
to the last three of (28) are 

CO + CiTi + C 2 Ti 2 + C 3 Ti 3 = Xl - C 4 Ti 4 H , 

J C ~*~ ClT2 + C2T22 + C3T 2 3 = ^2 C 4 T 2 4 H , 

(o7) 

O + CiT 3 + C 2 T 3 2 + C 3 T 3 3 = X 3 C 4 T 3 4 -I , 

I C + CiT 4 -f C 2 T 4 2 + C 3 T 4 3 = X 4 C 3 T 4 4 ~\ . 



117] APPROXIMATIONS WHEN THERE ARE FOUR OBSERVATIONS. 209 



The determinant of the coefficients of Co, Ci, c 2 , and c 3 is 



5 = 



T3 2 , 
T4 2 , 



T 2 
T3 3 

T4 3 



= (TZ TI)(TS Ti)(r 4 Ti)(r 8 r 2 ) 
X (r 4 T 2 )(T 4 T 3 ), 



which is not zero since the dates of the observations are distinct. 
The errors of lowest order in c , Ci, c 2 , and c 3 are determined 
from (37); when only the first terms in the right members are 
known they contain c 4 as a factor. Let these errors be represented 
by Ac , Aci, Ac 2 , and Ac 3 ; their orders in the TJ are required. The 
expression for Aco is 



Ac = 



C 4 



Tl 4 , 
T2 4 , 
T3 4 , 
T4 4 , 



T3, 



IV, 

T3 2 , 
T4 2 , 



Tl 



T2 



T3 



T 4 



When the factors TI, TZ, r 3 , and r 4 are removed from this deter- 
minant it is identical with 5 except the columns are permuted. 
Three permutations of columns bring it to the form of 6; hence 

(38) 



AC = 



C 4 TiT 2 T 3 T 4 . 





1, 


Tl 4 , 


Tl 2 , 


Tl 3 


- c 4 


1, 


T2 4 , 


T2 2 , 


T 2 3 





1, 


T3 4 , 


T3 2 , 


T 3 3 




1, 


T4 4 , 


T4 2 , 


3 



The expression for Aci is 



Ac, = 



If TZ is put equal to TI in this determinant it vanishes because then 
two lines become the same. Therefore it is divisible by TZ TI. 
Similarly, it is divisible by r 3 TI, T 4 TI, T 3 T 2 , T 4 T 2 , 
and T 4 T 3 ; that is, it is divisible by 6. All the elements of each 
column are of the same degree; and since every term of the ex- 
pansion of a determinant has a factor from each column, the terms 
of the expansion are all of the same degree. The degree of this 
determinant is nine, because this is the sum of the degrees of its 
columns. Hence Aci is of the third degree because d is of the 
sixth degree. Moreover, it is symmetrical in TI, , T 4 because 
both 8 and the numerator determinant are symmetrical in these 
quantities. Each term of the expansion contains TJ only to the 
15 



210 LAPLACIAN METHOD OF DETERMINING ORBITS. [117 

first degree because 77 occurs in the numerator determinant to 
the fourth degree as the highest, and in d to the third degree. The 
numerical coefficient of each term in the expansion is the same, 
because of the symmetry, and it can be determined by the con- 
sideration of a single term. It is found by considering the product 
of the main diagonal elements that it is -}- 1. Analogous dis- 
cussions can be made for Ac 2 and Ac 3 , and it is found in this way 
that 

f ACl = C 4 [TiT2T 3 + T2T3T 4 + T 3 T4Tl + T 4 TiT 2 ], 
(39) -I AC 2 = + C 4 [TiT 2 + TiT 3 + TiT 4 + T 2 T 3 + T 2 T 4 + T 3 T 4 ], 

[Ac 3 = - C 4 [ri + r 2 + r 3 + rj. 

It follows from (38) and (39) that when there are four obser- 
vations X, X', X", and X"' are determined up to small quantities 
of the fourth, third, second, and first order respectively. Ordi- 
narily X'" is not needed, though it becomes useful when the solution 
is double, as it may be, in determining which of them belongs to 
the physical problem. In this latter case it is advantageous to 
make Ac 3 vanish by determining t Q so that 

f TI + r 2 + T 3 + r 4 = 0, whence 
(40) 4 

1*0 = i01 + *2 + *3 + 

If the solution of the problem is made to depend only on X, X', 
and X", it is most advantageous to choose t so that Ac 2 shall 
vanish, for then all the quantities are determined up to the third 
order. This condition becomes 



(41) 



TlT 2 + TiT 3 + TiT 4 + T 2 T 3 + T 2 T 4 + T 3 T 4 = 0, 

W - 30! + t* + t, + *o + W 2 

+ Ms + tit* + * 2 Z 3 + * 2 * 4 + t,U = 0. 



The values of Zo determined by this quadratic equation are of 
no practical value unless they are real. The discriminant of the 
quadratic is 



= H - 30! - ttf + 3(*i - t.Y + 30i - UY 

+ 30 2 - * 3 ) 2 + 30 2 - ttf + 30 3 - * 4 ) 2 > 0. 
Therefore the solutions are always real, and are explicitly 



= 



118] 



THE FUNDAMENTAL EQUATIONS. 



211 



In order to get a concrete idea of the nature of the results 
suppose the intervals between the successive observations are 
equal to T. Then (42) gives 

(43) t Q = i(i + U + t, + U) t Vl5 T. 

The first term on the right is the mean epoch of the observations, 
and the two values of t Q are at the distance J Vl5 T either side of 
this time. Since the interval between the mean epoch and 
t z or 3 is f 7 1 , it follows that t is between ti and t z and distant 
($Vl5 %)T = %T approximately from 2 , or symmetrically situ- 
ated between t 3 and U. In practice it will be most convenient 
to choose IQ = fa or t t s , for then X is given exactly, the coef- 
ficients of (33) are as simple as possible, and (41) is nearly satisfied. 
The discussion when there are five or more observations can be 
carried out in a similar manner. For each additional observation 
one additional coefficient in the series (28) can be determined, 
and those which were determined previously become known to 
one order higher in the T/. In each case one additional order of 
accuracy in the determination of X" can be secured by properly 
selecting Z , but it is simplest to let U equal the date of the obser- 
vation which is nearest the mean epoch of all of the observations. 

118. The Fundamental Equations. The fundamental equations 
of the method of Laplace are (10), where X, n, v, X', M'> v', X", /*", /' 
are given by (31) and (32) and corresponding equations in p and v. 
The solution of equations (10) for p, p', and p" is 



(44) 



where 



(45) - 



2D R 3 r 3 



-I[n -^ 
~I>L r 8 



X" 



v" 



D l = - 



D - - 



X, X, X" 

M, y, /" 

JV // 

V ^v J^ 



x, 


X', Z 


M, 


M', Y , 


*i 


/, z 


X", 


X', X" 


r, 


M', M" 


ry 


/ // 


1 





212 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[119 



These determinants are subject to small errors because of the 
fact that the higher terms of equations (28) have been neglected. 
After p and p have been approximately determined corrections 
can be made for these omissions. The determinants are also sub- 
ject to small errors because they have been developed under the 
tacit assumption that the observations were made from the 
position of the center of the earth instead of from one or more 
points on its surface. After the approximate distances have 
been determined the observations can be corrected for the effects 
of the observer's position on the surface of the earth. 

119. The Equations for the Determination of r and p. Con- 
sider the triangle formed by S, E, and C. Let ^ represent the 
angle at E and <p that at C. Then it follows that 




(46) 



R 



p = 



JR+Tn.+ Sr, 

D sin (^ + <p) i 

ti ; / 

sin & 



sin 



When equations (46) are substituted in the first equation of (44) 
the result is 



R sin * cos v + fi cos * - 



[fi 



sn = 



In order to simplify this expression let 

'N sin m = R sin 



(47) 



N cos m = R cos i/" 
M 



DR 3 ' 
- NDR* sin 3 



119] EQUATIONS FOR THE DETERMINATION OF r AND p. 213 



where the sign of N will be so cnosen that M shall be positive. 
With this determination of the sign of N the first two equations 
of (47) uniquely determine N and m, and the equation in <p becomes 
simply 

(48) sin 4 <p = M sin (<p -f m). 

The quantities M and m are known and M is positive. 

Now consider the solution of (48) for <p. Since p = 0, r = R 
is a solution of the problem, it follows from (48) that <p = TT \f/ 
is a solution of (48). This solution belongs to the position of the 
observer and is to be rejected. It follows from Fig. 32 that the <p 
belonging to the physical problem, which must exist if the compu- 
tation is made from good observations, satisfies the inequality 

(49) <p < 7T - ^. 

The solutions of (48) are the intersections of the curves defined by 
the equations 



(50) 



2/2 = M sin(<? + m). 



For m negative and near zero and M somewhat less than unity 
these curves have the relation shown in Fig. 33. 
y 




Fig. 33. 

Consider first the case where -=^ is positive. Since both p and r 

must be positive, it follows from the first of (44) that in this case 
r > R. Since ^ is less than 180, it follows from (47) that N is 
negative, and that m is in the third or fourth quadrant. 

In case m is in the fourth quadrant the ascending branch of 
the curve y z crosses the <p-axis in the first quadrant, and, if M < 1, 
the relations of the curves are as indicated in Fig. 33. If m is 



214 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[119 



near 180 there are three solutions, <pi, <p z , and <p 3 , one of which 
is TT \f/ and belongs to the position of the observer. If ( p 3 = 7r ^ } 
both <pi and (pz fulfill all the conditions of the problem and it can 
not be determined-which belongs to the orbit of the observed body 
without additional information. However, it might happen that 
(Pi would give so great values of r and p that it would be known 
from practical observational considerations that the body would 
be invisible; it would be known in this case that <p 2 , which would 
give a smaller r, belongs to the physical problem. If <p 2 = TT \l/, 
it follows from (49) that <pi belongs to the problem. The case 
(Pi = TT \f/ cannot occur for then the physical problem could 
have no solution. If, for a fixed M, the ascending branch of the 
curve 2/2 moves to the right the roots <pi and <p2 approach coinci- 
dence; and as it moves farther to the right <p 3 alone remains real. 
This case, which corresponds to m far from 180 in the fourth 
quadrant or in the third quadrant, cannot arise, for then the 

problem would have no solution. Therefore, if -= is positive, then 

r > R, m is in the fourth quadrant, and there are one or two possible 
solutions of the physical problem according as <p 2 or <p 3 equals TT \J/. 

Now suppose - ^ is negative. In this case r < R and m is in 

the first or second quadrant. If m is in the first quadrant the 
descending branch of the curve y z crosses the ^>-axis in the second 




<t> t TT_ <j> 2 $ 3 * 

Fig. 34. 

quadrant, and for a small m and M < 1 the relations are as shown 
in Fig 34. In this case the solution of the problem is unique or 
double according as <? 2 or <p z equals TT \f/. If m is in the second 
quadrant the descending branch of the curve y z crosses the <p-axis 



120] THE CONDITION FOR A UNIQUE SOLUTION. 215 

in the first quadrant, <? 2 and <p a are not real, and the problem has 
no solution. Therefore, if -~ is negative, then r < R, misin the 
first quadrant, and there are one or two possible solutions of the 
physical problem according as #2 or <p 3 equals TT ^. 

120. The Condition for a Unique Solution. The solution of 
the physical problem is unique whether -^ is positive or negative 
if <pz = TT \j/j and otherwise it is double. Suppose p = TT \f/ + e, 
where e is a small positive number. When is positive, it is 
seen from Fig. 33 that if <? 2 = TT - ^ the difference yi - y z is 
positive for p = <?% + e; and, when ^ is negative, it is seen from 

Fig. 34 that yi y z is negative for v = ^ + e = TT $ + e. 

It follows from (50) that yi and y 2 can be expanded as power 
series in e when <p = TT $ + e. The first two terms of the 
difference are 

yi 2/2 = [sin 4 (TT ^) M sin (TT \j/ + m)] 
(51) + [4 sin 3 (TT - ^) COS (TT - ^) 

M cos (TT ^ + m)]<= + . 

The term independent of e is zero because <p = TT $ is a solution 
of (48). A reduction of the coefficient of e by equations (47) 

and (48) gives 

MR 3Di 



Therefore the condition that the solution of the physical problem 
shall be unique is 



f J-S1 

(52) 

' < if ^ 



This function is completely determined by the observations, and 
consequently it is known without solving (48) whether the solution 
of the problem is unique or double. 
The limit of the inequalities (52) is 

r 

(53) 1 



216 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[120 



On eliminating cos \f/ and ^ by the first equations of (44) and 
(46), it is found that 
(54) P 2 = 1 



2 R* 5 



The minimum value of the right member of this equation, con- 
sidered as a function of r, is zero; therefore for each value of r 
there is a unique positive value of p. All points defined by pairs 
of values of r and p which satisfy (54) are on the boundary of the 
regions where the inequalities (52) are satisfied. These boundary 
surfaces are evidently surfaces of revolution around the line 
joining the earth and the sun. The section of these surfaces by a 
plane through the line SE is shown in Fig. 35.* 




Fig. 35. 

The surfaces defined by (54) divide space into four parts, two 
of which in the diagram are shaded, and two of which are plain. 
The function (52) has the same sign throughout each of these 
regions and changes sign when the boundary surface is crossed 



* This figure was first given by Charlier, Meddelande fran Lunds Observa- 
torium, No. 45. 



120] THE CONDITION FOR A UNIQUE SOLUTION. 217 

at any ordinary point. This is a special case of a general propo- 
sition which will be proved. 

Suppose XQ, 2/0, ZQ is an ordinary point on the surface defined by 
F(x, y, z) = 0. Consider the value of F at XQ + Ax, y Q + Ay, 
ZQ -f Az, where Ax, Ay, and Az are small. The value of the function 
at this point is 

F(XQ + Ax, 2/0 + Ay, Z + Az) 

^+ -4- A + A 4- 
' dx dy dz 

The first term in the right member of this equation is zero because 
XQ } 2/0, ZQ is on the surface. Now suppose the point XQ + Ax, - 
is on the perpendicular to the surface at XQ, y , ZQ. Then 

dF 



Ax 



dF 



Ay = 

I / . -n \ n / n T-f \ n * *v Try \ O 



dF 

Az = 



dFV dF\*' 



where p is the distance from XQ, 2/0, ZQ to XQ + Ax, 2/0 + Ai/, ZQ + Az, 
because the factors by which p is multiplied are the direction 
cosines of the normal to the surface. On one side of the surface p 
is positive, and on the other side it is negative. The expression 
for the value of the function F at the point XQ + Ax, becomes 

F(x Q + Ax, 2/0 + Ay, ZQ + Az) 



For p very small the sign of the function is determined by the 
sign of the first term on the right whose coefficient is not zero. 
Since XQ, 2/0, ZQ is by hypothesis an ordinary point of the surface, 
not all of the first partial derivatives of F are zero, and conse- 
quently the sign of the function changes with the change of sign 



218 LAPLACIAN METHOD OF DETERMINING ORBITS. [121 

of p. That is, the function changes sign when the surface for 
which it is zero is crossed; and it does not change sign at any 
other finite point because the function is continuous. 

In order to find in which of the four regions of Fig. 35 the solu- 
tion is unique, and in which it is double, consider a point on the 
line SE to the left of E. At such a point r = p + R, \I/ = TT, and 
it follows that 



1 DR^ DR* 

which is clearly negative for p very large. Since in this case 
r > R it follows that -~ > 0, N < and the first inequality 

of (52) is the one under consideration. Since the inequality is 
satisfied the solution of the problem is unique if the observed 
body is in the unshaded area to the left of E. If the surface is 
crossed into the larger shaded area at a point for which r > R 
the function changes sign while the sign of N is unchanged. Then 
the first inequality of (52) is not satisfied and the solution of the 
physical problem is double. In this region the function (53) is 
positive and N is negative. If the surface is crossed into the 
smaller unshaded area the function (53) becomes negative, N 
becomes positive, and the second inequality of (52), which is now 
in question, is satisfied. Therefore the solution is unique in this 
unshaded area. It is shown similarly that it is double in the 
smaller shaded area. 

121. Use of a Fourth Observation in Case of a Double Solution. 

Suppose <p 3 = TT i/' so that there are two solutions of (48) which 
correspond to the conditions of the physical problem. One 
method of determining which solution actually belongs to the 
physical problem, in case there are four observations, is obviously 
to develop (48), using the fourth observation instead of one of 
the original three. In general, this will make the result unique. 
A better method of resolving the ambiguous case can be devel- 
oped from equations (44). Eliminate r from the second and 
third equations of (44) by means of the first. The results are 




122] THE LIMITS ON m AND M. 219 

The derivative of the first of these equations is 

P" r P'p + P P f = (P f + P 2 ) P , 

which equated to the right member of the second equation gives 
(55) D, - |i- + Dp = Z),(P' + P). 

Since this equation is linear p is uniquely determined unless D is 
zero. The determinant D will be examined in Art. 124. Equa- 
tion (55) must be based upon not less than four observations, for 
P' involves X"', /JL'", and v'" which cannot be determined, even 
approximately, from three observations. 

122. The Limits on m and M. In an actual problem of the 
determination of an orbit the constants m and M are subject to 
the condition that equation (48) shall have three real roots between 
and TT. The limits imposed by this condition can be determined 
from the conditions that it shall have double roots; for, suppose 
M is fixed and that m varies. In the first case, represented in 
Fig. 33, there are three real solutions of (48) until, the curve y z 
moving to the right, <pi and <? 2 become equal; and in the second 
case, represented in Fig. 34, there are three real solutions of (48) 
until, the curve y z moving to the left, <p 2 and <p 3 become equal. 
The two cases are not essentially different for <p\ in the first case 
corresponds exactly to <p 3 in the second. Similarly, if m remains 
fixed and M, starting from a small value, increases there are three 
real solutions of (56) until either <p 2 and <p$ or <pi and <p 2 , in the 
first and second cases respectively, become equal. When the 
limits are passed for which two values of <p which satisfy (48) 
are equal, there is only one real solution between and TT. 

The conditions that (48) shall have a double root are 

j sin 4 <p = M sin (<p + ra), 

(56) 

I 4 sin 3 <p cos <p = M cos (<p -f m). 

The solution of the quotient of these equations for tan <p is 



(57) tan 



- 3 V9 - 16 tan 2 m 



2 tan m 
It follows at once that m is subject to the condition 

9-16 tan 2 m ^ 

in order that the double root shall be real. Hence 
(58) 323 8' 5 m g 360, ^ m ^ 36 52', 



220 LAPLACIAN METHOD OF DETERMINING ORBITS. [123 

the first range for m belonging to the first case, represented in 
Fig. 33, and the second to the second case, represented in Fig. 34. 

For each m there are two values of <p defined by (57) between 
and TT. In the first case, in which tan m is negative, tan <p is 
positive whether the upper or the lower sign is used before the 
radical, and it is smallest when the upper sign is used. Therefore 
the value of <p defined by (57) when the upper sign is used is that 
one for which <pi = <p z in Fig. 33, and the one determined when 
the lower sign is used is that one for which <p z = <p*. When m has 
the limiting value for which the radical vanishes tpi = <p% = <p 3 . 
The discussion is analogous in the second case in which tan m 
is positive. 

The limiting values of <p, defined by (57), which correspond to 
the limiting values of m as given in (58), are respectively 

(59) <p = 116 34', <p = 63 26', 

and for both of these values of <p the value of M defined by (56) 
is M = 1.431. This is the maximum M for which (48) can have 
three real roots between and IT. In order that the three roots 
shall be real for this M the value of m must be 36 52' or 323 8', 
and the three roots are then equal. 

Consider the first case and suppose m starts from 323 8' and 
increases to 360. The two values of <p defined by (57) start 
from 63 26'. One goes to and the other to 90. The two 
corresponding values of M start from 1.431, and one goes to 
and the other to unity. For each value of m between the limits 
(58) there are two limits between which M must lie in order that 
(48) shall have three real solutions. In constructing a table of 
the solutions of (48) depending on the two independent parameters, 
M and m, these limits should be observed in order to reduce the 
work as much as possible. 

123. Differential Corrections. Suppose the approximate solu- 
tion of (48) has been found from the graphs of y\ and y z , or by 
numerical trials, or from the tables of the roots of this equation. 
Let (po represent the approximate solution and <p + A<p the exact 
solution. The problem is to find A<p. 

Let 

(60) sin 4 <PQ M sin (<p + m) = rj, 

where rj will be a small quantity if <p is an approximate solution 
of (48). If po + A<p is substituted in (48) in place of <p, the result 
expanded as a power series in A<p becomes 



123] DIFFERENTIAL CORRECTIONS. 221 

77 = [4 sin 3 <po cos <p M cos (<p Q + ra)]A<p + [ ] (A<p) 2 + . 

This power series can be inverted, giving A<p as a power series in rj. 
The result is 

/ai\ A ~~ "n I n 2 _i_ 

(bl ) A<0 = -3 r ; T-= 7 ; r + I I 7? H~ 

4 sin 3 <PQ cos <po M cos (<po + w) 

The only exception is when the coefficient of A<p in the power 
series in A<p is zero. This is the second of equations (56), the 
conditions for a double root. In this case the expression for A<p 
proceeds in powers of == Vry. In practice difficulty arises if the 
coefficient of A<p is small without being zero, for then <p must be 
very close to the true value of <p before the method of differential 
corrections can be applied. 

The higher terms of (61) can be computed without any difficulty, 
but they rapidly become more complex. It is simpler in practice 
to neglect them and to repeat the process with successive improved 
values of (pe- 
lt is possible to develop a more convenient method for com- 
puting the differential corrections by making use of the fact that 
the work is done with logarithms. After m and M have been 
computed from the observational data the approximate solution 
of (48) can be determined from the diagram. The curve yi can 
be drawn accurately once for all. The better known sine curve, 
in this case flattened or stretched vertically by the factor M, can 
be drawn free hand with sufficient accuracy to enable one to get a 
very approximate estimate of the value of <p. Let it be ^o. The 
logarithms of the right and left members of (48) will be computed 
and they will of course be found to be unequal. Let 
4 log sin <p log M log sin (<p -\- m) = e. 

In the successive approximations only the first and third of these 
logarithms will be changed. The tables give the logarithms of 
the trigonometric functions. Let the tabular difference for the 
logarithm of sin <p and sin (<p + 5<p) be ci, where d<p is some 
convenient increment to <po, and let e 2 be the corresponding tab- 
ular difference for sin (<p + ni). These quantities are taken down 
from the margins of the tables when the logarithms of sin <p and 
sin (<PQ + m) are taken out. Then the correction A^? is given by 
the equation 



where the result is expressed in the units used for d<p. This 



222 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[123 



method is so convenient in practice that a very few minutes suf- 
fices in any case to find the solution of (48) with all the accuracy 
which may be desired. In the first approximation, where the 
error is in general large, one degree could be taken for d<p. In the 
later approximations 10" is a convenient increment because the 
tabular differences of the logarithms for differences of 10" are 
given on the margins of the tables.* 

124. Discussion of the Determinant D. The determinant D, 
equation (45), enters into the determination of the constants M 
and m, and the solution becomes indeterminate in form if it is zero. 
Consequently it is important to find under what circumstances 
it vanishes. 

Suppose the determination of the orbit is being based on only 
three observations. Then the values of X, X', and X", which occur 
in D, are given by (31) and (32). There are corresponding 
expressions for n, /, M"; v, v' , "" After they are substituted in 
(45) the determinant D can be factored into the product of two 
determinants. In order to simplify the notation let 



P l = 



, , 



r 2 ) (TI T 3 ) ' 



TI) (r 2 r 3 ) ' 



p = 



TI)(T TZ) 



TI) (T S TZ) ' 

and denote the derivatives of these functions with respect to r 
by accents. Then 

" D = AiAi 



(64) 





PI, 


Pi, 


p'; 


Ai = 


PI, 


p;, 


pj 




p., 


PL 


pi 




Xi, 


x fc 


X 3 


A 2 = 


Ml, 


M2, 


M3 


. 


''i. 


''2, 


V3 



Consequently D can vanish only if A i or A 2 is zero. 

*The solution of (48) depends on the two parameters M and m', if there were 
but one the relations between it and <p could easily be tabulated. In spite of 
the two parameters Leuschner has extended a table originally due to Oppolzer 
from which the solution can be read directly with considerable approximation. 
It is table xvi. in the third (Buchholz) edition of Klinkerfues' Theoretische 
Astronomie. 



124] 



DISCUSSION OF THE DETERMINANT D. 



223 



It will be shown first that AI is a constant which is distinct from 
zero. Since it is formally of the third degree in T, necessary and 
sufficient conditions that it shall be independent of r are that 
A/ = for all values of r. The derivative of a determinant is 
the sum of the determinants which are obtained by replacing suc- 
cessively the columns of the original determinant by their deriva- 
tives. Hence A/ is a sum of three determinants. Since the 
derivative of the first column is identical with the second column, 
the first of these determinants is zero for all values of r. Since 
the derivative of the second column is identical with the third, 
the second determinant is zero. The derivative of the third 
column is zero, and therefore the third determinant is zero. 
Hence A/ is identically zero and AI is a constant. Its value, 
which is easily found for r = 0, is 



(65) 



^1 





T 2 T 3 , T 2 - 


hT 3 , 1 




2 


T,Ti, T 3 J 


h-Ti, 1 






TiT 2 , Tl - 


hT 2 , 1 





T 2 ) 2 (T 2 T 3 ) 2 (T 3 Tl) 2 

2 



(TZ ri)(r 3 T 2 )(r 3 TI) " 

This determinant is distinct from zero and independent of the 
choice of the epoch t Q . 

In order to interpret A 2 multiply the first, second, and third 
columns by pi, p 2 , and p 3 respectively. Then, in the notation of 
equations (6), the determinant A 2 becomes 

, a 3 



The right member of this equation is numerically the expres- 
sion for six times the volume of the tetrahedron formed by the 
earth and the three positions of C with respect to E. The volume 
of this tetrahedron is zero only if the three positions of C lie in a 
plane passing through the fourth point E. This, of course, is 
referring the position of C to E as an origin. A simpler way of 
expressing the same result is, the determinant A 2 (and therefore D) 
is zero only if the three apparent positions of C as observed from E 
lie on an arc of a great circle. 

It follows from (44) that if D is zero, DI and D 2 are also zero 



224 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[125 



unless R = r. In general, the expressions for p and p' become 
indeterminate when D is zero, and they are poorly determined 
when D is small. One case in which A 2 and D are always zero is 
that in which C moves in the plane of the earth's orbit. But in 
this case there are only four elements to be determined, and since 
each observation gives a single coordinate (the longitude) four 
observations are required. 

An expression for A 2 can be obtained by means of equations (6). 
After some simple reductions it is found that 

A 2 = cos 5i cos 62 cos 63 [sin (0:2 ai) tan 3 

/ (**\ 

+ sin (0:3 at) tan 61 + sin (ai 3 ) tan 62]. 

125. Reduction of the Determinants DI and D 2 . The expres- 
sions for DI and D 2 , equations (45), become as a consequence of 
equations (31) and (32) and corresponding expressions for /*, //, v, 
and / 



D 1 = - 



Z), = + 



+ P 2 X 2 + P 3 X 3 , 
+ P 2M2 + PSMS, 

P\V\ + P2^2 + Ps^S, 
PiXi rj" P 2 X 2 ~~h P 3 X 3 , 

+ P 2 M2 + P 3 M 3 > 
+ P 2 I'2 + 



P/-V _|_ p /\ _|_ p /\ V" 

i AI ~(- *s A 2 ~i~ i 3 A 3 , -A 

Pl'/il + P2M2 + Ps'/iB, I" 

PiVi + P 2 V 2 + P 3 V 3 , Z 

P//-V I p //\ I p //\ V 

1 AI ~t~ -t 2 A 2 "p 3 A 3 , -A 

P// | p // i p // ^ 

i v\ H- r^ 2 ^ 2 T i 3 PS, Z 



If the first column of DI is multiplied by - and subtracted 
from the second column, the result is 

PA, (P/P 3 ~ PiP 3 ')Xi + (P 2 T 3 - P 2 P 3 r )X 2 , X 
(Pi'P, - PiPsOMi + (P.'Ps - P 2 P 3 OM2, Y 



P 3 



where 



P v , 



- PlP 3 ')"l + (P2 r P 3 - 

x = PiXi + P 2 X 2 + P 3 X 3 , 



P v - 

This determinant is the sum of the two determinants 

+ P-V /P/P PP /> \\_1_/'P / P P 

-t 2 A 2 , \JL i J 3 i IJL 3 ^Ai "I" \-t 2 * 8 * ! 

+ P 2 M 2 , (Pi'P, - PiPs')/*! + (P 2 r P 3 - P 



Pivi + P 2 ^ 2 , (Pi'P, - 



+ 



126] REDUCTION OF THE DETERMINANTS >i AND D 2 . 

and 



225 



Xs, (Pi'Pj - PiPsOXi + (P 2 'P 3 - P 2 P 3 ')X 2 , X 
Ms, (Pi'Pg - PiP 3 ')Mi + (P 2 'P 3 - P 2 P 3 ')M2, Y 

( ~D I ~D ~D ~D F\ I / ~D I D ~D D /\ ^7 

Vs, \i i r 3 r \r 3 )v\ -f- ^.r 2 /^s -r 2 -t 3 j^ 2 , Z 

The terms in X 2 , M 2 , and v z can be eliminated in a similar manner 
from the second column of the first of these determinants. Then 
each of the determinants is the sum of two others, and the reduced 
expression for DI becomes 

Xi, X 2 , X 



= - (PfJ - P/P 2 ) 



- (P 2 P 3 ' - P 2 'P,) 



- (PsP/ - P.'PO 



Mi, M 2 , Y 

vi, v 2 , Z 

\2) A 3 , J\. 

M 2 , Ms, Y 

\ \ y 

A 3 , Al, ^\. 

Ms, Mi, Y 



The coefficients of these determinants are needed for T = 0. It 
is found from (63) that 



1 f 2 



(r 2 



P 2 P/ - P,'P, = 



P 3 P/ - Ps'Px = 



+ 



(r 2 TI)(TS r 2 )(r 3 TI) J 



(r 2 ri)(r 3 r 2 )(r 3 TI) ' 
Then the expression for DI reduces to 



(67) 



Xi, X 2 , X 

Vl, J>2, Z 

\3, Xl, X 

Ms, MI, Y 
PS, v\) Z 

P = (r 2 TI)(T S T 2 )(T 3 TI). 



X 2 , X 3 , X 

M2, Ms, Y 

V*, J/3, Z 



16 



226 



LAPLACIAN METHOD OF DETERMINING ORBITS. 



[126 



In a similar manner the expression for D 2 reduces to 



2r 3 



(68) 



Xi, X 2 , X 
Mi, M2, Y 



Vz, 



2T2 



Ma, Ms, 

P2, *% 



X 



M3, Ml; 



Each of the determinants in the expressions for DI and Da can 
be developed in a form similar to (66) . 

126. Correction for the Time Aberration. Since the velocity 
of light is finite, the body C at any instant is apparently where it 
was at some preceding instant. This introduces a slight error in 
the data which must be corrected, if accurate results are desired, 
after the approximate distances have been determined. Since the 
velocity of light is very great and the apparent motions of the 
heavenly bodies are in general slow, it will not be necessary to 
know the distance of C with a high degree of accuracy in order to 
correct for the finite velocity of light. 

Let EI, Ez, and E 3 be the positions of the observer at ti, Z 2 , 
and 3 respectively. Let the observed directions of C at these 




Fig. 36. 

epochs be EiCi, E 2 Cz, and E S C 3 . In the time required for the 
light to go from C to E the former will have moved forward in its 
orbit to the positions pi, p z , and p 3 , which are its true places at 
the epochs t\ t t z , and t$. If the distances are known the observed 



127] DEVELOPMENT OF X 



227 



coordinates can easily be corrected for these slight motions; but 
this changes all the observed data of the problem and makes it 
necessary to recompute all the determinants. 

A second method, which is more convenient in practice, is to 
correct the times of the observations. The body C passed through 
the points Ci, C 2 , and C 3 , not at t 1} t z , and t 3 , but at these epochs 
diminished by the time required for light to move from Ci, C 2 , 
and C 3 to EI, E 2 , and E 3 respectively. In order to make these 
corrections to the epochs it is necessary to know EiCi = PI, 
EzCz = p 2 , E 3 Cz = p 3 . It will be supposed that (48), (46), and 
(44) have been solved and that p and p' are known. Then the 
values of pi, p 2 , and p 3 are given with sufficient approximations for 
present purposes by 

(Pi = P + P'TI, 
P2 = P + P'T 2 , 
Ps = P + P'TS. 

Let V represent the velocity of light. Then the epochs at 
which C was at Ci, C 2 , and C 3 are 



(70) 



A Pi (p -f- P'TI) 

Tl ATI = Tl y = Tl - y , 



P2 



Ar 2 = r 2 = r 2 - 



(p + P' 



A P 3 (p + P' 

T 3 Ar 3 = T 3 = r 3 - 



Now consider the correction to D, DI, and D 2 . In D only the 
factor AI is altered. But in the applications only the ratios of 
D to DI and D 2 are used, and the latter contain AI as a factor. 
Therefore the only change required is to replace TI, r 2 , and r 3 
by TI ATI, T 2 AT 2 , and T 3 ATS respectively in the numerators 
of the coefficients of the determinants in (67) and (68) . 

127. Development of z, y, and z in Series. In order to deter- 
mine the corrections which should be added to X' and X", so as 
to determine the elements of the orbit with greater accuracy, it is 
necessary to have x, y, and z developed as power series in T. These 
quantities satisfy the differential equations 



228 



(71) 



LAPLACIAN METHOD OF DETERMINING ORBITS. 
Z X X 



d?z z 

-=--=-uz. 



[127 



It is shown in the theory of differential equations that the solu- 
tions of differential equations of this type are expansible as power 
series of the form 



X = X XQT r X Q T 

y =y + y'r + ^ 'V + ^ "V 

So = 2 + Zc'r + ^o'V 2 + W'r 3 + ^o iv r 4 + yin ZoV + 
It is found from (71) and its successive derivatives that 



(72) 



The coefficients of the series for y and z differ only in that y Q , y Q ' 
and z , z</ appear in place of XQ, XQ respectively. Therefore 

x = fx Q + gxo, 
y =fy<> 



(73) 



g = r JwoT 3 TV^o'f 4 Ts-5-(3lo" UQ 2 )r 5 + . 
In order to have/ and g in a form for practical use the derivatives 
of u must be expressed in terms of XQ, yo, ZQ, XQ', y<>', and z r . La- 
grange has done this very elegantly by introducing p and q by the 
equations 



(74) 



, 
P = 2 fc = 






Then it is found that 



128] 



THE HIGHER DERIVATIVES OF X, fJL } v. 



229 



3 dr 

I3- 

r 4 dr 



3 1 dr 2 

i ?T ~^~ 

r 4 2r dr 



P 



By means of these equations and their successive derivatives the 
coefficients in the series for / and g can be expressed as polynomials 
in u, p, and q. The expressions for / and g become 



(75) 



The derivatives of x, y, and z can be determined from equations 
(73) and (75). For example 

" = f'"x + </"V, 
(76) -j z iv = f^xo + g^xo', 



128. Computation of the Higher Derivatives of X, M> v. The 

values of X, X', and X" determined by equations (31) and (32) are 
only approximate because c 3 , c 4 , were unknown. But after 
the higher derivatives become known these coefficients are obtain- 
able, and the approximate values can be corrected. 
The third derivatives of equations (8) are 

"X + 3p"X' + 3p'X" + P X'" = x'" + X"', 
(77) J p"' M + 3p'V + 3pV + PM'" = y" r + Y" f , 



3p'V + 3 P 



The left members of these equations involve the four unknowns 
p'", X'", IJL"', and /", the first and second derivatives having 
been determined approximately by equations (31), (32), and (44); 
but the unknowns are not independent because X, n, v, and their 
derivatives satisfy the relations 



XX r + MM' + w' = 0, 

XX" + MM" + vv" + X' 2 + M' 2 + v* = 0, 

XX'" + MM"' + vv'" + 3(X r X" + MM" + v'v") = 0. 



230 LAPLACTAN METHOD OF DETERMINING ORBITS. [129 

Consequently if equations (77) are multiplied by X, n, and v 
respectively and added, the result is 

p'" = 3 P '(X /2 + M ' 2 + "' 2 ) + 3p(X'X" + M V' + v'v") 

+ (*'" + X'")\ + (y" f + F'")M + (*'" + Z">, 

which uniquely defines p rrr . Then X'", /*'", and /" are deter- 
mined by (77) because x" f , y" f , z"' are given by (76) and X'", 
Y'", and Z"' can be found from the Ephemeris. 

The quantities X iv , ju iv , and v iv can be computed in a similar 
way by taking the derivatives of (77) and reducing by means of 
the relations among X, n, and v. 

129. Improvement of the Values of x, y, 2, x', y f , z'. After 
D, Z)i, and D 2 have been found from (65), (66), (67), and (68) 
equation (48) can be solved, and then x, y, z and their first deriv- 
atives can be determined from (8) and their first derivatives. 
These results are only approximate because of the errors to 
which X, fi, v, X', IJL', and v' are subject, and the problem is to 
correct them after X'", //'", have been determined. 

It follows from the first equation of (28) that 

C 3 = JX'", c 4 = AX iv , ...... . 

Then equations (35) give 

ACo = iX /// TlT2T3 -2 J X iv TiT2T 3 (Tl + T 2 + T 3 ) + ' ' ' , 

ACi = + iX'"(riT 2 + T2T3 + ran) 

+ 1&V V (T1 + T 2 )(T 2 + T 3 )(T 3 + Ti) + ' ' , 

AC 2 = - JX"'(TI + r 2 + TJ) 

- T^W + T 2 2 + T 3 2 + TlT 2 + T 2 T 3 + T,Ti) + , 

and the expression for X becomes 

X = C + Ac + (ci + Aci)r + (c 2 + Ac 2 )r 2 



where Co, Ci, and c 2 are the approximate values of the coefficients 
of the series which are obtained from (31) and (32) by putting 
r equal to zero. There are corresponding equations for /z and v. 
With these more nearly correct values of X, X', X", , the de- 
terminants D, D], and Z> 2 are computed from (45), <p is determined 
from (48), p and p' from (44), and x, y, z, x', y', z' from (8) and 
their first derivatives. Then still higher derivatives of X, /*, v can 



130] THE MODIFICATIONS OF HARZER AND LEUSCHNER. 231 

be computed and still more nearly exact values of X, X', and X" 
determined, or the elements can be determined from x, y, z, x', 
y'j z' by the methods of chap. v. 

There are two principal objections to the method of Laplace. 
One is that it is necessary to recompute all determinants and 
auxiliaries at each stage of the approximation, each of which 
costs a very considerable amount of labor. The other is that 
the method depends upon the motion of the observer through the 
equations by means of which X" ', Y", and Z" were eliminated 
from (9) . Obviously all that is really fundamental in the problem 
is that C shall have been observed from definite known places 
and that it shall move about the sun in accordance with the law 
of gravitation. 

130. The Modifications of Harzer and Leuschner. The 

method of Laplace for determining orbits has not been found 
very satisfactory in practice. The reason seems to be that the 
conditions that the first and third observations shall be exactly 
satisfied are not directly imposed as they are, for example, in the 
method of Gauss. To remedy this defect Harzer proposed* the 
plan of so determining x, y, z, x', y', z' by differential corrections, 
after their approximate values have been found, that the three 
observations shall be exactly fulfilled. If more than three obser- 
vations are under consideration, they cannot in general be exactly 
satisfied, and the adjustments are then made by the method of 
least squares. 

It will be sufficient here to sketch the method of making the dif- 
ferential corrections. The right ascensions and declinations are 
expressed in terms of the coordinates and components of velocity 
at t Q by 

pX = fx + gxo' + X, 

PM = fyo + gyo' + Y, 

.pv = fz + gz ' + Z, 

which are obtained by substituting equations (73) in equations (8). 
The right ascension and declination enter through X, /*, and v of 
equations (6). The result can be indicated 

| a = F(x , ?/o, Zo, XQ'J y ', z</), 

I 5 = G(x Q , 2/0, Zo, XQ', 2/0', z '). 

* Astronomische Nachrichten, Nos. 3371-2 (1896). 



232 GAUSSIAN METHOD OF DETERMINING ORBITS. [131* 

From these equations the variations in a and 5, which are the 
known differences between the observations and the approximate 
theory, are expressed in terms of the variations in x , - - , z ', which 
are required. The relations are 

dF . . dF . . dF . dF . , dF . . dF . 



dG . . dG . . dG . . dG . , , dG . , . dG . , 

^ A * + W Ayo + ^ A0 + to? AXQ + w* + s?^ 

In forming the partial derivatives it must be remembered that 
XQ, , 2o' enter through / and gf as well as explicitly. When these 
equations are written for three dates they become equal to the 
number of arbitraries, viz., A# , , A2 ', and consequently deter- 
mine them uniquely provided the determinant of their coefficients 
is distinct from zero. The circumstances under which it vanishes 
have not been investigated. If there are more than three obser- 
vations the number of equations exceeds the number of arbitraries 
and the method of least squares is employed. 

When the date of the second observation is taken as the origin 
of time and the number of observations is only three, the number 
of equations of condition reduces to four which in general can be 
satisfied by suitably determining Ap , Azo', AT/O', and Az '. This 
is the procedure adopted by Leuschner* to abbreviate the method 
of Harzer. In its simplified form the method has been found very 
convenient in practice and has led to highly satisfactory results. 

II. THE GAUSSIAN METHOD OF DETERMINING ORBITS. 
131. The Equation for p 2 . Equations (19) are fundamental in 
the method of Gauss. If the geocentric coordinates are intro- 
duced by equations (8), equations (19) become 

[2, 3]piXi - [1, 3] P2 X 2 + [1, 2]p 3 X 3 

= [2, 3]Zi - [1, 3]X 2 + [1, 2]X,, 

[2, 3]plMl - [1, 3]p 2M 2 + [1, 2JP3M3 

= [2,3]ri-[l, 3]F 2 + [1,2]F 3 , 
[2, 3]pin - [1, 3W 2 + [1, 2] P8 , 

= [2, 3]Z, - [1, 3]Z 2 + [1, 2]Z 8 . 

The left members of these equations are linear in the three un- 
knowns pi, p 2 , and p 3 . Their solution for p 2 is 

* Publications of the Lick Observatory, vol. vn., Part 1 (1902). 



(80) 



131] 



THE EQUATION FOR p 2 . 



233 



(81) - 



D 



Mb M2, M3 

Vl, *2t V* 



? X 3 



Mi, [2, 3]F t -[l, 3]F 2 +[1, 2]F 3 , 



= [2, 3][1, 2] 

-[1, 3]Z 2 +[1, 2]Z 8 , 
The determinant D is the sum of three determinants 
D = 



(82) < 



Mb 



Ma 



a, Xs 



M3 



7) (2) = 



Mb 
"b 



Consequently the first equation of (81) becomes 
(83) A 2P2 = - 



Suppose t z is taken as the origin of time. Then it follows from 
equations (73) that 



The expressions for the ratios of triangles then become 

[2, 3] _ 
[1, 3] " 
[1,2] 
[1, 3] ~ 



(84) 



The numerators and denominators of the expressions for the right 
members of these equations are found from (75) to be expansible 
as power series in TI and r 3 . But in order to simplify (83) it 
is convenient to let 



234 



GAUSSIAN METHOD OF DETERMINING ORBITS. 



[131 



(85) 



k(t s ti) = T 3 TI = 2r, 



Ck(t t - ti) 

[TI = - r 



+ e, r 3 = + r + e, 



where e is in general small compared to r, and will be supposed to 
be of the order of r 2 . Then the expressions for the ratios of the 
triangles become 

[2,3] _ +ga _1 1 , re 



(86) ^ 



[1, 2] _ - flfi = l__l , 1 
[1,3] /i03-/30i 2 2r" t "4 



re 
12 



where all terms up to the sixth order have been written. The 
quantity u is defined by u = and p and q are defined in (74). 
On making use of equations (86) , equation (83) becomes 

r 2 



A2P2 = K 



PK,+ 



re 



QK,, 



where 





AI, -^-i, ^3 Aij ./L 2, AS 


1 


1 


*--2 


Mi, ^i, Ms + Mi, Y Zt M3 2 

Vi, Zi } Vs. Vi. Z*>. Vz 




X 


1, *1| A3 




*y / 

Xl, ^L3, Xs 




^1 = 


Ml, Y lt M3 


. 


Mi, Y 3 , Ms 


t 




vi, Zi, */3 




Vl, Z 8 , *>3 






AI, .A. i, AS 




^ 1? *^- 3j *^3 




K.% = 


Mi, YI, Ms 


+ 


Ml, Y 3 , M3 


. 




V 


Li ^1, "3 




vi, Z 8 , v, 





Xi, 



M3 



The right members of the expressions for K, 
giving the simpler expressions 



I, and K 2 add, 



131] 



THE EQUATION FOR p 2 



235 



(88) 



K =~2 



Mi, 



+ ^3 - 2F 2 , 

+ Z 3 - 2Z 2 , 



\i 

- Ml 



+ 



Mi, Fi + Y 9 , Ms - Mi 

I/I, Zi + ^3, ?3 Vi 

Xi, Xa ^i, Xs Xi 

Mi, ^3 Yi, Ms Mi 



Consider equation (87). The determinant A 2 by which the left 
member is multiplied is given in terms of the on and 5; by (66), 
which appeared in 'the method of Laplace. It can also be written, 
by properly combining columns, in the form 



A 2 = 



Xi, X 2 , Xs 



Mi, M2, Ms 



Xi, 



Mi, 



Xi + Xs 2X2, 
Mi + Ms 2/z 2 , 
v\ + J>3 2i' 2 , 



Xs 



Ms Mi 



If Xf, MI, ^i are replaced by the series (28), taking r 2 = 0, the 
second column is of the second order and the third column is of 
the first order in the time-intervals. Therefore A 2 is of the third 
order. 

Since the left member of (87) is of the third order the right 
member also must be of the third order. The second column of 
the expression for K, the first equation of (88), is of the second 
order, and the third column is of the first order. Therefore K is 
of the third order. The determinant KI is of the first order and 
Kz is of the second order. The former is multiplied by r 2 , which 
is of the second order, and the latter by re, which is of the third 
order. In a preliminary determination of an orbit the terms of 
higher order may be omitted, after which (87) becomes 



= K 



4r 2 3 



This equation is of the same form as the first of (44) , and involves 
the two unknowns p 2 and r 2 . They are expressible in terms of 
a single unknown <p by means of equations (46) affected with the 



236 



GAUSSIAN METHOD OF DETERMINING ORBITS. 



[132 



subscript 2. The resulting equation has exactly the same form 
as (48), and its solution gives approximate values of p 2 and r 2 . 

132. The Equations for p t and p 3 . Equations (80) are linear 
in pi and p 3 , and these quantities can be determined from any 
two of the three equations. The two to be used in practice are 
those for which the determinant of the coefficients of pi and p 3 is 
the greatest, for they will best determine these quantities. 

The solution of the first two equations of (80) for pi and p 2 if 
they are written first in determinant form, and if they are then 
expanded as a sum of determinants, is 



Pi = 



(89) H 



' X t , 


X 3 


Mi, 


M3 


Xi, 


X 3 


Mi, 


M3 



P3 = 



Xi, X 3 [i 


,3] 


X 2 , X 3 




Y lt M3 [2 


,3] 


F 2 , Ms 




, [1, 2] 


F 3 , 


X 3 

Ms 


+ P2 [2 


,3] X 


h [2, 3] 


,3] p 


2, 3] Xi, Xi 




[1,3] Xi, 


X 2 


U> 2 1 MI, Fi 




U,2] Ml , 


F 2 




Xi, 


x, 


[1 


,3] X 




Mi, 


F 3 


^^U^] 



The solution of the first and third equations of (80) differs from this 
only in that the MI are replaced by the v i} and the F; by the Zi\ 
and the solution of the second and third equations of (80) can be 
obtained from (89) by changing the X t , jj, i} Xi, and F to m, Vi, 
Yi, and Zi respectively. 

After pi, p 2 , and p 3 have been computed the correction of the 
time for the time-aberration can be computed. The method was 
explained in Art. 126. 

133. Improvement of the Solution. The results so far obtained 
are only approximate because only the first term of P was retained 
while the term in Q was entirely neglected. Having found an 
approximate solution it is easy to correct it. The values of pi, p 2 , 
and p 3 are known, and the corresponding values of r can be found 
at each of the three epochs from 

r 2 = p 2 + R 2 - 2 P R cos ^, 

which expresses the fact that S, E, and C form triangles at the 
dates of the three observations. After r i} r 2 , and r 3 have been 



134] 



RATIOS OF TRIANGLES BY METHOD OF GAUSS. 



237 



found the first and second derivatives of r at t = t z can be found 
by the method of Art. 113. Then equations (74) define p and q 
after which more approximate values of P and Q can be determined. 

134. The Method of Gauss for Computing the Ratios of the 
Triangles. Equation (83), which is fundamental in determining 
p 2 and r 2 , involves two ratios of triangles. It follows from (86) 
that they can be written in the form 



(90) 



[2,3] 1 _e_ Pi 

[1,3] 2~ h 2r~ h r 2 3 ' 

[1,2] = 1 e_ P* 

.[1,3] '2 2r~ t ~r 2 3 * 



Consequently, if the ratios of the triangles can be determined 
PI and P 2 can be found from these equations. One of the im- 
portant features of the method of Gauss is a convenient means of 
determining the ratios of the triangles. In order to apply this 
method it is necessary to find the inclination and node of the orbit 
and the argument of the latitude at the dates of the observations. 
Since the geocentric coordinates are all known after pi, p 2 , p 3 
have been determined, the heliocentric coordinates can be com- 
puted. Suppose ecliptic coordinates are used and that the 




Fig. 37. 

longitudes and latitudes, as well as the distances, are known 
at ti, t z , and 3 . The inclination is less or greater than 90 according 
as Z 3 is greater or less than li. Then it follows from the spherical 
triangles Ci&li and C 3 &Z 3 that 

{tan i sin (li &) = tan 61, 
tan i sin (h &) = tan 6 3 . 

But ? 3 & = (h li) + (h &); therefore these equations 
become 



238 GAUSSIAN METHOD OF DETERMINING ORBITS. [135 



tan i sin (li Q>) = tan 6 



1, 



tan i cos (Z ; - ft) = tan 6. -tan 6 t cos (I.- t.) 

sm (3 li) 

which determine i and & uniquely since the quadrant of i is al- 
ready known from the sign of l s l\. 

The longitude of C from the node is called the argument of the 
latitude. It follows from Fig. 37 that 

(cos (lj- &) cos bj = cos Ujy (j = 1, 2, 3), 
sin (7/ &) cos 6,- = sin u } - cos i, 
sin 6y = sin Uj sin i, 

which uniquely define HI, u*, and u s . 

Let A equal the area of the sector contained between the 
radii r\ and r 2 and the orbit. Then the ratio of the area of the 
sector to the area of the triangle contained between r\ and r 2 is 



(93) = 



r 2 ri r 2 sn u z Ui 

where p now represents the parameter of the conic. Suppose the 
corresponding ratios for t s ti and Z 3 t z have been found; then 
the ratios of the triangles are known. The method of Gauss 
depends upon the determination of these ratios. Each of these 
quantities is denned by two simultaneous equations in two un- 
known quantities. 

135. The First Equation of Gauss. The polar equation of the 
conic gives 

= 1 + e cos 0i, 



whence 
(94) 



= 1 + e cos v 2 ' f 

17*2 



p - = 2 + e(cos vi + cos 



Since v% v\ = u z u\ is known, the only unknown in the right 
member of this equation is e cos ( -^= - ) , which will now be 
eliminated. From the equations of Art. 98 it follows that 



135] 



THE FIRST EQUATION OF GAUSS. 



239 



(95) 



r i l~77~ I 

Vri cos 2* = Va(l - e) cos , 

Vn sin ^ = Va(l + e) sin y , 

/ ^2 r~7^ - \ **J 
Vr- 2 cos = Va(l e) cos -^ , 

Vr^ sin ^ = Va(l + e) sin 2 . 



From these equations it is found that 

(E*- 
= acos( -- ^ -- ~ ae cos 



ir 2 cos 



= a cos 



~ ae cos 



(F -\- F 1 \ / I v \ 

2 - } and solving for e cos ( 2 - j , 

it is found that 
e cos 



As a consequence of this equation (94) reduces to 



rt i - fvz vi\ / E 
- 2Vr 1 r 2 cos( -^ 1 cos ( 



On eliminating p from this equation and (93) the equation 



(96) 



O/ -- /^2-^l\ 

| ri + r 2 -2Vr 1 r 2 cos( ) cos 



is obtained. In order to simplify it let 

t>2 vi = u<z HI = 2f, 
s- E! = 20, 



(97) 



m 



4 Vr 



r J_ 1 
f 2" 



cos 



240 



GAUSSIAN METHOD OF DETERMINING ORBITS. 



[136 



Then the expression for rj 2 reduces to 

m 2 



in which rj and g are the unknowns. This is the first equation in 
the method of Gauss. 

136. The Second Equation of Gauss. An independent equation 
involving 77 and g will now be derived. It will be made to 
depend upon Kepler's equation, thus insuring its independence 
of (98) which was derived without reference to Kepler's equation. 
The first equations are 



M l = 



= 2 -e sn 



whence 



*i) o 

' =2g -2e sin g cos 



(ET i TfJ \ 
-2^ l j must be eliminated in order 

to reduce this equation to the required type. On making use of 
the first equation following (95), it is found that 



(99) 






2 g - sin 2g + 2 



sin g cos /. 



It remains to eliminate a. By Art. 98 



whence 



= 1 e cos 



= 1 e cos E z ; 
a 



r-2 



= 2 2e cos Q cos 

#2 + 



(rr I F \ 
2~ - ) by the first equation following 



(95) this equation becomes 

1 2 sin 2 g 



r 2 



cosgr cos/ 



137] SOLUTION OF EQUATIONS (98) AND (101). 241 

which becomes as a consequence of the expression for if 



On eliminating a between (99) and (100), it is found that 

(101) 4_4=^_TLn^. i 

m? m 2 sin 3 g 

which is the second equation in 77 and g. There are similar 
equations for the time-intervals t s ti and t s 2 2 . 

137. Solution of (98) and (101). It follows from the definition 
of 77 that it is positive if the heliocentric motion in the orbit is 
less than 180 in the interval t z t\. It will be supposed in what 
follows that the observations are so close together that this con- 
dition is fulfilled. 

Let 



(102) 



2g - sin 2g _ v 

: a -A. 

sin 3 g 



Oh eliminating t\ from (98) and (101) and making use of (102), 
it is found that 

(103) m=-(l + a)* + X(l + x)*. 

The quantity X must now be expressed in terms of x, after which 
(103) will involve this quantity alone as an unknown. This will 
be done by first expressing X in terms of gr, and then g in terms of x. 
The following are well-known expansions of the trigonometrical 
functions : 

f sin 2g = 20 - f<7 3 + T V - " 



whence 

* ~ 



nrvn 



From the first of (102) it follows that 
g = 2 sin-K**) = 2x* + ^ + 



17 



242 GAUSSIAN METHOD OF DETERMINING ORBITS. [137 

Then (104) becomes 

J = 4[\ , 6 , 6-8 . , -6-8-10 

or 

X = 



Let 



6 6-8 6-8- 10 



3 _JLF JL 2 _ 52 ^ , 
4 10 [ X 35 x 1575^ 



If \g is a small quantity of the first order, x is of the second order 
and is of the fourth order. 
From (98) it is found that 

(106) x = ^ - I 
Let 

(107) ft = _"-_. 



then (101) may be written 

1 _m 2 X _ 
*- ^jT= 

from which it is found that 

(108) n 3 ~ f ~ hr, - g = 0. 

If were known h would be determined by (107) and f\ by (108), 
which has but one real positive root. In the first approximation 
compute h assuming that the small quantity is zero ; then find the 
real positive root of (108). Or, instead of computing the root, 
use may be made of the tables which have been constructed by 
Gauss, giving the real positive values of T\ for values of h ranging 
from to 0.6. * The value of x is then computed by (106) and the 
value of by (105) . f With this value of , h, and rj are recomputed, 
and the process is repeated until the desired degree of precision 
is attained. Experience has shown that this method of computing 

*This table is XIII. in Watson's Theoretical Astronomy, and VIII. in 
Oppolzer's Bahnbestimmung . 

t The value of with argument x is given in Watson's Theoretical Astronomy, 
Table XIV., and in Oppolzer's Bahnbestimmung, Table IX. 



138] DETERMINATION OF THE ELEMENTS a, 6, AND a>. 243 

the ratio of the sector to the triangle converges very rapidly, even 
when the time-interval is considerable. 

The species of conic section is decided at this point, the orbit 
being an ellipse, parabola, or hyperbola according as x is positive, 

zero, or negative; for, x = sin 2 -|- = sin 2 - (E 2 EJ, and E 2 and 

EI are real in ellipses, zero in parabolas, and imaginary in hyper- 
bolas. 

Gauss has introduced a transformation which facilitates the 
computation of I which was denned in the last equation of (97)4 
Let 

fe = tan (45 + '), ^ a/ ^ 45; 
whence 

Tj -^* = ^ + J^ = tan 2 (45 + co') + cot 2 (45 + a/), 
or 

fl i=^= 2 + 4tan 2 2a/. 



Then the last equation of (97) becomes 

sin 2 + tan 2 2o/ 



cos/ 

138. Determination of the Elements a, e, and o>. After g has 
been found by the method of Art. 137 it is easy to obtain the ele- 
ments a, e, and co. The major semi-axis a is defined by the last 
equation on page 240, or by the preceding equation for the longer 
time-interval 2 3 h, 

(109) a 



2 sin 2 g 

The parameter of the orbit p is determined by equation (93) . 
Since 

(110) p = a(l - e 2 ) or p = a(e* - 1) 

according as the orbit is an ellipse or hyperbola, e is determined 
when a and p are known. 

If the angle v is computed from the perihelion point it is related 
to the heliocentric distances and e and p by the polar equation of 
the conic, 

t Theoria Motus, Art. 86. 



244 SECOND METHOD OF DETERMINING a, 6, AND CO. [139 



Either of these equations determines a value of v since r is known 
at fc, t z , and 3 , and then co is determined by 

(112) co = Ui - v { . 

139. Second Method of Determining a, e, and to. The method 
of Gauss depends upon the complicated formulas of Arts. 135 and 
136. If the higher terms of P and Q, equations (86), give suf- 
ficiently accurate values of the ratios of the triangles, there is 
another method * which is simpler and especially advantageous 
when the intervals between the observations are not very great. 
The data which will be used in the solution are ri, u\] r z , u z ] r 3 , w 3 , 
the heliocentric coordinates at ti, t, and t s . 

The elements i and & can be computed by equations (91), 
which are valid for any orbit. The difficulties all arise in finding 
a, e, co. Let the parameter p be adopted as an element in place 
of the major semi-axis a. It is more convenient in that it does not 
become infinite when e equals unity, and it is involved alone in 
the equation of areas, 

k -Jpdt = r*dv = r*du. 
The integral of this equation is 

(113) kJp(t 3 - ti) = C*r*du. 



If r 2 were expressed in terms of u the integral in the right member 
could be found, when the value of p would be given. It will be 
shown from the knowledge of the value of r 2 when u = HI, Uz, u 3) 
viz., r 2 = ri 2 , r 2 2 , r 3 2 , that r 2 can be expressed in terms of u with 
sufficient accuracy to give a very close approximation to the 
value of p. 

For values of u not too remote from u% the function r 2 can be 
expanded in a converging series of the form 

(114) r 2 = r 2 2 + ci(u - u 2 ) + c^(u - u z ) 2 + c 3 (u - w 2 ) 3 + . 

In an unknown orbit the coefficients of the series (114) are 
unknown, but it will now be shown how a sufficient number to 
define p with the desired degree of accuracy can be easily found. 
By hypothesis, the radii and arguments of latitude are known at 
the epochs t\, t 2) and t s . Hence (114) becomes at ti and t 3 

* F. R. Moulton; The Astronomical Journal, vol. xxn., No. 510 (1901). 



139] SECOND METHOD OF DETERMINING a, 6, AND 03. 245 



r 2 



(115) 



For abbreviation let 



(116) 



0-3 

l 



Then equations (115) can be written 



+ c 2 o- 3 2 = r i 2 r 2 2 1, 
= r 3 2 r 2 2 e 3 . 



On solving for c\ and c 3 , it is found that 

- 6 1 )(7 1 2 + (f3 2 ~ 6 3 )(7 3 2 ~ 



Ci = 



- 3)0-3 - 



and, on substituting the values of ei and c 3 , 



(117) 



(7i(7 2 (7 3 



7V0- 2 



<7 3 ) 



Having obtained these expressions for the coefficients of the 
second and third terms of (114), let this series be substituted for 
r 2 in (113) and the result integrated. On making use of (116), it is 
easily found that 



<7 3 2 ) 



(73 3 ) 



246 



SECOND METHOD OF DETERMINING , 6, AND CO. 



[139 



On substituting the values of c\ and c 2 given in (117), this equation 
becomes 



(118) 



, 2/0 

r- -E -- ( 2o "3 - 





00-10-3 00-3 



60-1 



x 

- 0-3) 



12 



30 



{4(0-3 o-i) 2 + 0-10-3} 



If the second observation divides the whole interval into two 
nearly equal parts, as generally will be the case in practice, o-i 
and 0-3 will be nearly equal. Let 

(TI 0-3 = e, and o-i + 0-3 = 0"2j 



whence 



0-2 



0-3 = 



2 

0"2 



where e is in general a very small quantity. On substituting 
these expressions in the last terms of (118) this equation becomes 



(119) 



k 



/0 
( 2o "3 - 



60-10-3 60-3 

+ -& (2o-i - 0-3) - 



It is found in a similar way on integrating between the limits 
corresponding to t z and t\ that 

-2<rs) 



(120) ^ 



, 



.x 
" 






For the intervals of time which are used in determining an 
orbit these series converge very rapidly, and an approximate value 
of p, which is generally as accurate as is desired, can be obtained 



139] SECOND METHOD OF DETERMINING a, 6 AND CO. 247 

by taking only the first three terms* in the right member of (119). 
By considering equations (119) and (120) simultaneously and 
neglecting terms in c 4 and of higher order, it is possible to deter- 
mine both p and c 3 . But not much increase in accuracy is ob- 
tained because the term in c 3 in (119) is multiplied by the small 
quantity e, while that in c 4 does not carry this factor. Suppose 
the value of p has been computed; it will be shown how co and e 
can be found. 
The polar equation of the conic gives 

f f >. P 7*1 
e cos (HI co) = , 
n 
t \ P ~ r 3 
e cos (u s co) = . 
7*3 

Now u$ co (u z HI) + (ui co). On substituting this ex- 
pression for u s co in the second equation of (121), expanding, 
and reducing by the first, it is found that 



6 sin ( Ul - co) = ~ n COS U * - U " 



(122) 

e cos (HI co) = - 1 . 

Since e is positive these equations define e and co uniquely. When 
p and e are known, a is defined by p = a(l e 2 ) or p = a(e 2 1) 
according as the orbit is an ellipse or an hyperbola. 

If the elements a, e, and co have not been found with sufficient 
approximation it is now possible to correct them. It follows from 
(114) that 



1 d 3 (r 2 ) 1 

6 du 2 3 ' ~ 24 



and since 



[1 + e cos (u co)] 2 ' 
it is found that 

* For conditions and rapidity of convergence see the original paper in the 
Astronomical Journal, No. 510. It is shown there that the elements of asteroid 
orbits will be given by the first three terms of (119) correct to the sixth decimal 
place if the whole interval covered by the observations is not more -than 
40 days, and in the case of comets' orbits, if the interval is not more than 10 
days. When the two corrective terms defined by (123) are added the corre- 
sponding intervals are 100 days and 20 days. 



248 COMPUTATION OF THE TIME. OF PERIHELION PASSAGE. [140 



(123) * 



e sin (u co) 3e 2 sin (u co) cos (u co) 



3[1 + e cos (u co)] ; 



e cos 



co) 



[1 + e cos (u co)] 4 

4e 3 sin 3 (^ co) 
[1 + ecos (w - co)] 5 ' 

e 2 sin 2 (w co) 
12[1 + 6 cos (w - co)] 3 ~~ [1 + e cos (w - co)] 4 

3e 2 cos 2 (M -co) 6e 3 sin 2 (^- co) cos (^ - co) 
*" 4[1 + e cos (u - co)] 4 * [1 + e cos (u - co)] 5 

, 5e 4 sin 4 (u co) 

[1 + e cos (w co)] 6 ' 

With the values of Cs and c computed from these equations the 
higher terms of (119) can be added, thus determining a more 
accurate value of p, after which e and co can be recomputed by 
(122). Besides being very brief this method has the advantage of 
being the same for all conies. 

140. Computation of the Time of Perihelion Passage. The 

methods of computing the time of perihelion passage depend upon 
whether the body moves in a parabola, ellipse, or hyperbola, and 
are based on the formulas of chap. v. 

Parabolic Case. Equation (32), of chap, v., is 



(124) 



k(t - D - 



where 2q = p. Since u v + co, and HI, w 2 , and u s are known, 
this equation determines T. 

Elliptic Case. The first two equations of (49), chap, v., give 



(125) 



which uniquely define E. Then Kepler's equation 
(126) M = n(t - T) = E - e sin E 

determines T by using v and the corresponding E at h, t 2 , or t 3 . 
Hyperbolic Case. The quantity F is defined by 



sin E = 


Vl e 2 sin v 


\-\-e cos v 
e + cos v 




\-\-e cos v ' 



(127) 



- 1 



141] DIRECT DERIVATION OF EQUATIONS. 

after which T is given by 
k Vl -f- m 



249 



(128) 



(t - T) = - F + e sinh F. 



141. Direct Derivation of Equations Defining Orbits. The 
motion of an observed body must satisfy both geometrical and 
dynamical conditions. Altogether the simplest mode of pro- 
cedure is to write out at once these conditions. They will involve 
directly or indirectly many of the equations of the methods of 
Laplace and Gauss, for these methods both rest in the end on the 
essentials of the problem. 

Let the notation of Art. Ill be adopted. Think of the sun as 
an origin. Then obviously the ^-coordinate of C equals the 
^-coordinate of the observer plus the ^-coordinate of C with respect 
to the observer. Similar equations are of course true in the two 
other coordinates. These relations are explicitly 



- \ iPi 



a = i, 2, 3), 



(129) 



-f- y* = - 



These equations are subject to no errors of parallax because the 
coordinates of the observer have been used. Moreover, they 
contain all the geometrical relations which exist among the bodies 
S, E, and C at h, t z , and 3 . 

The next condition to be applied is that C shall move about S 
according to the law of gravitation. This is equivalent to stating 
that its coordinates can be developed in series of the form of (73). 
On making use of this notation, equations (129) become 



(130) 



+ fiX + 



+ f&o -f- 

+ /i2/o + 

/z 2 p 2 + /22/o + 22/0' 



= X 2 , 



= Y 



+ fiZ + g&o' = - 



V3P3 + /320 + QsZo' = Z 3 . 



250 FOBMULAS FOE COMPUTING AN APPROXIMATE ORBIT. [142 



If the date of the second observation is taken as the origin of 
time, as is convenient in practice, / 2 = 1 and g z = 0. 

Equations (130) contain fully the geometrical and dynamical 
conditions of the problem and are valid for all classes of conies. 
Since they are only the necessary conditions no artificial diffi- 
culties or exceptional cases have been introduced; and if in a 
special case they should fail no other mode of approach could 
succeed. 

The right members of equations (130) are entirely known; the 
unknowns in the left members are pi, p 2 , p 3 , X Q , X Q ', y , y Q ', z , and 
z</. That is, the number of unknowns exactly equals the number 
of equations. The quantities pi, p 2 , and p 3 enter linearly, but 
XQ, , 2</ occur not only explicitly but also in the higher terms 
of the/* and the gi. The solution of (130) for pi, p 2 , and p 3 is 



(131) 



where 



(132) 



A 2 pi = + Ai 



A 2 p 2 = 



+ 



r 1 + wi 



A 2 = 



*1; A2; AS 

Mi, M2, Ms 

^1, ^2, J'S 

Xi, X, Xs 



Ml, 



J> 3 



C 2 + C,, 
Xi, X 2 , 

Xl, X2, 

Mi, M2, 

Vl, V Z , 



M3 



In order to complete the discussion the coefficients of the deter- 
minants in the right members of these equations must be developed, 
as they were in (86) ; and since A 2 is of the third order, terms of 
the right member of the third order must be retained even in the 
first approximation. When applied to the second of (131), this 
leads to an equation of the form of the first of (44). The details 
of this and the completion of the solution of equations (130) will 
be called out in the questions which follow Art. 142. 

142. Formulas for Computing an Approximate Orbit. For con- 
venience in use the formulas for the computation of an approxi- 



142] FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. 251 



mate orbit are collected here in the order in which they are used. 
The numbers attached are those occurring in the text. 

Preparation of the data. The observed right ascensions and 
declinations, <*o and 6 , are corrected for precession, aberration, 
etc., by 

r a = ao 15/ g sin (G + o) tan 5 h sin (H -f a ) sec 6 , 

(4) 4 

[ 6 = 6 i cos 6 g cos (G + ao) h cos (# + a ) sin 6 . 
The direction cosines are given by 

= cos dj cos a/, (j = 1, 2, 3), 




(ta-t b )(t a ~ t e ) 



sn a 



The Method of Laplace. Take to = fa unless the intervals 
between the successive observations are very unequal, when 
to = K^i + ^2 -\-t 3 ). It will be supposed that t Q = fa- Suppose X, 
y, and Z are tabulated in the Ephemeris for t a , t b) t c where t b is 
near t Q . Then compute X, Y, and Z at to from formulas of the 
type* 



(31) 

(26) 
(67) 

(64, 65) 

(67) 

(68) 



_i_ (^0 tg)(to t b ) y, 

*" (tc - t a )(t e ~ t b r 

k(ti - fa) = T,; (j = 1, 2, 3; T 2 = 0). 
P = nrz(rz TI). 
Xi, X2, Xs 
Mi, M2, Ms 



T3 



Xi, 



MI, M2, 



X 



*% 



= + 2 -? 



Xi, 



MI, M2, 



M2, M3, 



X2, 



M2, M3, 



X 
Y 
Z 
X 
Y 
Z 



* These equations are very simple because t a , t b , and t c differ by intervals of 
one day, but there are other methods of interpolation which are even simpler. 



252 FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. [142 

(46) R cos } = X\ + YH + Zv, (0 < ^ <L TT). 





TV sin m = it sin ^, 


(47) 
(48) 


N cosm = Rcost - ^ , 


sin 4 ^> = Tkf sin ((p + m). 


Mtt = P sin * =/? sin(^+rf 



sm 



(44) 
(8) 



=- 

p * r 3 



P X - 



z = pv Z 



Compute X', //, ^' from equations of the type 

^, ^(r 2 + r 3 )Xi 

(32) (T1 " 



r s ) (r 2 T 3 )(r 2 
(TI + r 2 )X 3 



(r 3 TI) (T S T 2 ) 
Compute X', Y', and Z' from equations of the type 

(32) (ta ~ 









fe) 



y 



(8) 



P X r - 



i i i / /7/ 

2! = p j/ -j- pj; // . 



At this point the correction for the time aberration may be 
made by equations (70), and the approximate values of x, y, z, 
x'j y', and z' may be improved by the methods of Arts. 128 and 
129; or, the elements may be computed at once from the formulas 
given in chap. v. The formulas for the determination of the 
elements will be given and the numbers of the equations refer to 
chap, v 

The integrals of areas in the equator system are 



142] FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. 253 

)xy' - yx' = &i, 
yz' - zy f = & 2 , 
zx' - xz f = 6 3 . 

If e represents the obliquity of the ecliptic, the corresponding 
constants in the ecliptic system are 

1 = 61 cos e 63 sin e, 

2 = &2, 

3 = bi sin e + 6 3 cos e, 
and i and ft are defined by (chap, v.) 



(15) 



= Vai 2 + a 2 2 + a 3 2 cos i, 

= =*= Vai 2 + a<> 2 + as 2 sin i sin 



a 3 = q= Vai 2 + a 2 2 + a 3 2 sin i cos ft. 
The major axis and parameter are defined by 

(24) x ' 2 + y' 2 + z' 2 = 1 

(22) k*p = k 2 a(l - e 2 ) = a x 2 + a 2 2 + a 3 2 . 

It follows from Fig. 37, p. 237, that 

sin i sin u = sin b = - , 



11 x 
cosi sin u = cos 6 sin (I ft) = -cos ft sin ft, 

77 /v 

cos u = cos b cos (I ft) = -sin ft H cos ft, 



which define u. The angle v is given by 



and 



I + e cos v ' 

0) = U V. 

If the orbit is a parabola, T is defined by 
(32) k(t - T) = ip 



254 FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. [142 

If the orbit is an ellipse, E, n, and T are denned by 

\\ - e, v 



(50) 



E 
tan = 



(30) -j, 

(42) n(t - T) = E - e sin E. 

The corresponding equations for hyperbolic orbits are 

(73) . a + r ae cosh F, 

(74) n(t - T) = - F + e sinh F. 

The Method of Gauss. The observed data are corrected by (4) 
and the direction cosines are given by (6). The coordinates of the 
sun at ti f hi an d h can be computed from equations of the type 



~ t b )(tj - t c ) (tj - 

* *" 



i ~ t c ) 



(t a ~ t b )(t a -t e ) 



(t b ~ t a )(tb ~ t e ) 



~ <a)fo ~ fe) y 

C 



(31) 



where X a , ,Xb, -X" c are taken from the Ephemeris and t b is the time 
nearest to ti for which X is given. Then 



(64) 



X 2 , 



(88) 



K =- 



\i, 

Mi, 
^i, 

Xi, 

Mi, 



Xs 2X2, 
7 3 - 2F 2 , 



+ ^3, Xs 
+ Fa, Ms 



On neglecting the last term of (87), which is very small, and 
comparing the result with the first of (44), it is seen that the 
explicit formulas for determining r 2 and p 2 are 



(46) 



COS 



X,\t + ^2^2 + Z,vt, (0 < fa ^ IT), 



142] FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. 255 

C N sm m = Rz sin ^ 2 , 



(47) 

(48) 
(46) 



N cos m = Rz cos \f/z T~ , 
M = 2 J* m > 0. 

sin 4 <p = M sin (<p + m). 
7 2 sin 1^2 



sin <p 

= p sin (fa + <?) 

sin ^ 
Then pi and p 3 are given by 

_ [1, 3] 
Yi, 

_^ [1, 2] 
(89) 



Ml, Ms 



PI = 



+ 



[2,3] 



[2,3] 
-Xs, X 3 



Ms 



+ P2 



X 3 
M3 

[1,3] 

[2,3] 



Xi, X 3 

Mi, Ms 



P3 = 



[2,3] 



[1,2] 



X, 



Mi, 



[1,3] 
[1,2] 



+ 



Xi, 
Ml, 



+ P2 



Xi, 
Mi, 



[1, 3] 



[2, 3] 



X 2 , X 3 

M2, Ms 



Xi, X 2 

Mi, M2 



(or by formulas obtained from these by cyclical permutation of 
the letters X, /*, v and X, Y, Z), where 

(85) 2r = r 3 - 

and 



(86) 



[1,3] 

[2, 3] 1 



2 = T S + TI, 
1 



[1,2] 
[2,3] 



[2,3] 
[1,2] 



[1,3] 
[1,2] 



2 T 2T ^ 4r 2 3 

'7 + 27? 
1 + 7 + 2r? 
1 + -+^-: 



2r 2 3 



+ ^- 
2 2r "*" 4r 2 3 



256 FORMULAS FOR COMPUTING AN APPROXIMATE ORBIT. [142 

= P/Xy - X it (j = 1, 2, 3), 
(8) 



At this point the correction for the time aberration may be 
made; the first two derivatives of r 2 2 may be computed from the 
values ri 2 , r 2 2 , and r 3 2 by applying the formulas (32) to this case; 
p and q may be computed from (74) and more approximate 
values of P and Q may be determined from (86); and then the 
computation may be repeated beginning with equations (46); 
or, the method of Gauss of Art. 134 may be used to improve the 
accuracy of the expressions for the ratios of the triangles; or, the 
elements may be computed without further approximation of the 
intermediate quantities. The formulas for the computation of 
the elements will be given. Let the rectangular coordinates in 
the ecliptic system be Xi, y^ Zi, and the obliquity of the ecliptic e, 
which will not be confused with the e defined in (85). Then 

Xj, (j = 1, 2, 3), 

+ yj cos e + z i sm e > 
j = y } - sin e + Zj cos e. 

' Ax, + By, + Czi = 0, 
(17) - Ax, + By, + Cz 2 = 0, 

- Ax 3 + By, + Cz 3 = 0, 



from which 

A :B :C = 



Vi, 



Then, from equations corresponding to (11), (14), and (15) of 
chap, v., 

A 



(15) 



sin 



cost = 



sn i 



cos ft sin i = 



VA 2 + B 2 + C 2 ' 
=F C 



which define ft and i. 

It follows from Fig. 37 that the arguments of the latitude are 
defined by 



PROBLEMS. 257 



sin i sin Uj = - , (j = 1, 2, 3), 



v X' 

cos i sin Uj = cos ft sin 
TV TV 



. . 

= sin ft H - cos ft . 
TV 




define e and co. Hence a can be determined from p and e. 

Since w/ = u,- co (j = 1, 2, 3), the time of perihelion passage 
is determined precisely as in the method of Laplace by equations 
(of chap, v.) (32), [(50), (30), (42)], [(73), (74)] in the parabolic, 
elliptic, and hyperbolic cases respectively. 



XVII. PROBLEMS. 

1. Take three observations of an asteroid not separated from one another 
by more than 15 days, or three of a comet not separated from one another by 
more than 6 days, and compute the elements of the orbit by both the method 
of Laplace and also that of Gauss. 

2. Prove that the apparent motion of C cannot be permanently along a 
great circle unless it moves in the plane of the ecliptic. 

3. Apply formulas (31) and (32) on a definite closed function, as for ex- 
ample x = sin t. 

4. By means of the equation 

7-2 = #2 _j_ p2 _ 2R r cos $ 

eliminate p from the first equation of (44) and discuss the result by the methods 
of the Theory of Algebraic Equations, and show that the solutions agree 
qualitatively with those obtained in Art. 119. 

5. Discuss the determinants D, D\, and D 2 when there are four observations. 

6. Express A 2 when there are three observations in terms of the on and the 5 
in such a manner that the fact it is of the third order will be explicitly exhibited. 

18 



258 HISTORICAL SKETCH. 

7. Develop the explicit formulas, using the X;, /*-, and vi and the determi- 
nant notation, for the differential corrections of the method of Harzer and 
Leuschner. 

8. Give a geometrical interpretation of the vanishing of the coefficients 
of pi and ps in equations (89). 

9. Suppose three positions of C are known as in Art. 139. Show (a) that 
the three equations 

P (1 = 1,2,3), 



1 + e cos (in co) > 

define p, e, and co without using the intervals of time in which the arcs are 
described; (6) write out the explicit formulas for computing p, e, and co; 
(c) compare then- length with that of (119) and (122); and (d) show that p is 
not well determined as it depends upon ratios of small quantities of the third 
order. 

10. Suppose / 2 = 1, 02 = and regard (130) as linear equations in pi, p 2 , p 3 , 
Xo, XQ', 2/o, 2/o', z , z '. Show that the determinant of the coefficients is 



A = - 



Al, 
Ml, 
V\, 



11. Show that on using the expansions of equations (86) the second equa- 
tion of (131) becomes (87). 

% 12. Having found p 2 from the equation corresponding to (87), and p\ and p 3 
from (131), show that X O ,'X Q ', yt, y ', 2 , z</ can be determined from equations 
(130). (Then the elements can be determined as in the Laplacian Method.) 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The first method of finding the orbit of a body (comet moving in a parab- 
ola) from three observations was devised by Newton, and is given in the 
Principia, Book in., Prop. XLI. The solution depends upon a graphical con- 
struction, which, by successive approximations, leads to the elements. One 
of the earliest applications of the method was by Halley to the comet which 
has since borne his name. Newton seems to have had trouble with the 
problem of determining orbits, for he said, " This being a problem of very 
great difficulty, I tried many methods of resolving it." Newton's success in 
basing his discussion on the fundamental elements of the problem was fully 
explained by Laplace in his memoir on the subject 

The first complete solution which did not depend upon a graphical con- 
struction was given by Euler in 1744 in his Theoria Motuum Planetarum et 
Cometarum. Some important advances were made by Lambert in 1761. 
Up to this time the methods were for the most part based upon one or the 
other of two assumptions, which are only approximately true, viz., that in 
the interval t 3 ti the observed body describes a straight line with uniform 
speed, or that the radius at the time of the second observation divides the 



HISTORICAL SKETCH. 259 

chord joining the end positions into segments which are proportional to the 
intervals between the observations. In attempting to improve on the second 
of these assumptions Lambert made the discovery of the relation among the 
radii, chord, time-interval, and major axis mentioned in Art. 92. He later 
made the determination depend upon the curvature of the apparent orbit, 
which is closely related to the determinant A 2 , and in this direction approached 
the best modern methods. He had an unusual grasp of the physics and 
geometry of the problem, and really anticipated many of the ideas which 
were carried out by his successors in better and more convenient ways. 

Lagrange wrote three memoirs on the theory of orbits, two in 1778 and 
one in 1783. They are printed together in his collected works, vol. iv., pp. 439- 
532. As one would expect, with Lagrange came generality, precision, and 
mathematical elegance. He determined the geocentric distance of C at the 
time of the second observation by an equation of the eighth degree, which 
is nothing else than (87) with r-i eliminated by means of the equation which 
expresses the fact that S, E, and C form a triangle at t-i. He developed the 
expressions for the heliocentric coordinates as power series in the time-intervals 
[eqs. (73)], and laid the foundation for the development of expressions for 
intermediate elements in power series. These developments have been com- 
pleted and put in form for numerical applications by Charlier, Meddelande 
frdn Lunds A stronomiska Observatorium, No. 46. The original work of 
Lagrange was not put in a form adapted to the needs of the computer, and 
has not been used in practice. 

In 1780 Laplace published an entirely new method in Memoires de V Acad- 
emic Royale des Sciences de Paris (Collected Works, vol. x., pp. 93-146). This 
method, the fundamental ideas of which have been given in this chapter, has 
been the basis for a great deal of later work. Among the developments in 
this line may be mentioned a memoir by Villarceau (Annales de I'Observa- 
toire de Paris, vol. in.), the work of Harzer (Astronomische Nachrichten, vol. 
141), and its simplification by Leuschner (Publications of the Lick Observa- 
tory, vol. vii., Part i.). The approximations beyond the first are not con- 
veniently carried out in the original method of Laplace, but the method of 
differential corrections devised by Harzer and simplified by Leuschner has 
proved very satisfactory in practice. 

Olbers published his classical Abhandlung uber die leichteste und bequemste 
Methode, die Bahn eines Kometen zu berechnen, in 1797. This method has not 
been surpassed for computing parabolic orbits and is in very general use even 
at the present time. It is given in nearly every treatise on the theory of 
determining orbits. 

The discovery of Ceres in 1801 and its loss after having been observed only 
a short time drew the attention of a brilliant young German mathematician, 
Gauss, to the problem of determining the elements of the orbit of a heavenly 
body from observations made from the earth. The problem was quickly 
solved, and an application of the method led to the recovery of Ceres. Gauss 
elaborated and perfected his work, and in 1809 brought it out in his Theoria 
Motus Corporum Coeleslium. This work, written by a man at once a master 
of mathematics and highly skilled as a computer, is so filled with valuable 
ideas and is so exhaustive that it remains a classic treatise on the subject to 
this day. The later treatises all are under the greatest obligations to the work 
of Gauss. 



260 HISTORICAL SKETCH. 

In the Memoirs of the National Academy of Science, vol. iv. (1888), Gibbs 
published a method of considerable originality in which the first approximation 
to the ratios of the triangles was obtained more exactly by including all three 
geocentric distances as unknown from the beginning. The method is also 
distinguished by the fact that it was developed by the calculus of vector 
analysis. 

The works to be consulted are: 

The Theoria Motus of Gauss. 

Watson's Theoretical Astronomy (now out of print). 

Oppolzer's Bahnbestimmung, an exhaustive treatise. 

Tisserand's Legons sur la Determination des Orbites, written in the char- 
acteristically clear French style. 

Bauschinger's Bahnbestimmung, a recent book of great excellence by one 
of the best authorities on the subject of the theory of orbits. 

Klinkerfues' Theoretische Astronomie (third edition by Buchholz), an 
excellent work and the most exhaustive one yet issued. 



CHAPTER VII. 

THE GENERAL INTEGRALS OF THE PROBLEM OF n BODIES. 

143. The Differential Equations of Motion. Suppose the 
bodies are homogeneous in spherical layers; then they will attract 
each other as though their masses were at their centers. Let mi, 
^2, , m n represent their masses. Let the coordinates of m< 
'referred to a fixed system of axes be Xi, yi, Z{ (i = 1, , n). Let 
r t -, / represent the distance between the centers of m t - and m/. 
Let k 2 represent a constant depending upon the units employed. 
Then the components of force on mi parallel to the z-axis are 

(xi x n ) 



r\ 2 ri, 2 r\ n 

and the total force is their sum. Therefore 

(xi x,) 



and there are corresponding equations in y and z. 

There are similar equations for each body, and the whole system 
of equations is 

d 2 Xi n f ~ " x 



(i) 



m 

m,- 



Each of these equations involves all of the 3n variables re,-, 7/i, 
and Zi, and the system must, therefore, be solved simultaneously. 
There are 3n equations each of the second order, so that the 
problem is of order 6n. 

Equations (1) can be put in a simple and elegant form by the 
introduction of the potential function, which in this problem will 
be denoted by U instead of V. The constant k 2 will be included 
in the potential. In chap, iv the potential, 7, was defined by 

261 ' 



262 SIX INTEGRALS OF MOTION OF CENTER OF MASS. [144 

the integral V = I . In this case the system is composed of 
J p 



p 
discrete masses, and the potential is 

(2) v -&?* 



The partial derivative of U with respect to xt is 

dU 79 d A mj 79 A (xt xj) 

~~ ~ - - 



and there are similar equations in ?/ t - and 2,- . Therefore equations 
(1) can be written in the form 



dU 

(3) """ dU 

dU 



(i = 1, , n). 

144. The Six Integrals of the Motion of the Center of Mass. 

The function U is independent of the choice of the coordinate 
axes since it depends upon the mutual distances of the bodies 
alone. Therefore, if the origin is displaced parallel to the #-axis 
in the negative direction through a distance a, the x-coordinate 
of every body will be increased by the quantity a, but the potential 
function will not be changed. Let the fact that U is a function 
of all the x-coordinates be indicated by writing 

U = U(X 19 X 2 , -.., X n ). 

After the origin is displaced the ^-coordinates become 

Xi' = Xi + a, (i = 1, , n). 
The partial derivative of U with respect to a is 

dU = dU^ dxi' d7 dxj dU dx n f 

da dx\ da dxz' da dx n r da 

But -r- 1 - = 1, (i = 1, n), and -r- = 0, because U does not 
ua da 

involve a explicitly. Therefore, on dropping the accents and 



144] SIX INTEGRALS OF MOTION OF CENTER OF MASS. 263 



writing the corresponding equations in ?/ and z for displacements 
j8 and 7, it is found that 



da 

dl L 
d(3 



Therefore equations (3) give 



* = 

dp u - 



These equations are at once integrable, and the result of inte- x 
gration is 



(4) 



where i, j8i, 71 are the constants of integration. On integrating 
again, it follows that 



(5) 



Let J^ m { = M, and 5, y, and 2 represent the coordinates of 
the center of mass of the n bodies; then, by Art. 19, 



264 THE THREE INTEGRALS OF AREAS. [145 

i Xi = MX, 

n 

(6) 



iZi = Mz. 

Therefore, equations (5) become 

IMx = out + 012, 
Aff-jM-hft, 
Mz = 7i* + 7 2 ; 

that is, the coordinates of the center of mass vary directly as the 
time. From this it can be inferred that the center of mass moves 
with uniform speed in a straight line. Or otherwise, the velocity 
of the center of mass is 



which is a constant; and on eliminating t from equations (7), it 
is found that 

MX a* My 02 Mz 72 



(9) 



01 7i 



which are the symmetrical equations of a straight line in space of 
three dimensions. Equations (8) and (9) give the theorem: 

// n bodies are subject to no forces except their mutual attractions, 
their center of mass moves in a straight line with uniform speed. 
The special case V = will arise if i = 0i = 71 = 0. Since it 
is impossible to know any fixed point in space it is impossible 
to determine the six constants. 

The origin might now be transferred to the center of mass of 
the system, as it was in the Problem of Two Bodies, or, to the 
center of one of the bodies, as it will be in Art. 148, and the order 
of the problem reduced six units. 

145. The Three Integrals of Areas. The potential function is 
noli, changed by a rotation of the axes. Suppose the system of 
coordinates is rotated around the z-axis through the angle </>, 
and call the new coordinates x/, ?//, and z/. They are related to 
the old by the equations 



145] 
(10) 



THE THREE INTEGRALS OF AREAS. 



i = Xi cos 7/t sin 



265 



/ = Xi sin < + i/i cos $, 
/ = Zi, (i = 1, ,.*) 

Since the function U is not changed by the rotation it does not 
contain explicitly; therefore 

ATT Av ' w AT7 %, ' n ATT A S 

OU OXi 



(ID = 



But from (10) it follows that 
therefore (11) becomes 



= 
* 



(t = 1, , ri); 



On dropping the accents, which are of no further use, it is found 
as a consequence of (3) that 



F d 2 ?/i d 2 X{"| _ 
111 * I * r -^"3^ = 



, . ., , 
similarly, 



Each term of these sums can be integrated separately, giving 



(12) 



The parentheses are the projections of the areal velocities of the 
various bodies upon the three fundamental planes (Art. 16). 
As it is impossible to determine any fixed point in space, so also 
it is impossible to determine any fixed direction in space; conse- 
quently it is impossible to determine practically the constants 
Ci, c 2 , c 3 . Yet, in this case it is customary to assume that the 



266 



THE THREE INTEGRALS OF AREAS. 



[145 



fixed stars, on the average, do not revolve in space, so that, by 
observing them, these constants can be determined. It is evident, 
however, that there is no more reason for assuming that the stars 
do not revolve than there is for assuming that they are not drifting 
through space, each being a pure assumption without any possi- 
bility of proof or disproof. But it is to be noted that, if these 
assumptions are granted, the constants ci, c 2 , and c 3 can be deter- 
mined easily with a high degree of precision, while in the present 
state of observational Astronomy the constants of equations (4) 
cannot be found with any considerable accuracy. 

Let Ai, Bi, and d represent the projections of the areas de- 
scribed by the line from the origin to the body mi upon the xy, yz, 
and 20>planes respectively; then (12) can be written 



dBi 



dd 
dt 



= C 3 , 



the integrals of which are 



(13) 



6f 

n 



Ci', 



c 3 '. 



^ 



Hence the theorem: 

The sums of the products of the masses and the projections of the 
areas described by the corresponding radii are proportional to the 
time; or, from (12), the sums of the products of the masses and 
the rates of the projections of the areas are constants. 

It is possible, as was first shown by Laplace, to direct the axes 
so that two of the constants in equations (12) shall be zero, while 
the third becomes Vci 2 + c 2 2 -+- c 3 2 . This is the plane of maxi- 
mum sum of the products of the masses and the rates of the pro- 
jections of areas. Its relations to the original fixed axes are 
defined by the constants ci, c 2 , c 3 , and its position is, therefore, 
always the same. On this account it was called the invariable 



146] THE ENERGY INTEGRAL. 267 

plane by Laplace. At present the invariable plane of the solar 
system is inclined to the ecliptic by about 2, and the longitude 
of its ascending node is about 286. These figures are subject to 
some uncertainty because of our imperfect knowledge regarding 
the masses of some of the planets. If the position of the plane 
were known with exactness it would possess some practical ad- 
vantages over the ecliptic, which undergoes considerable vari- 
ations, as a fundamental plane of reference. It has been of great 
value in certain theoretical investigations.* 

146. The Energy Integral.f On multiplying equations (3) by 
-~ , -jf , -jj respectively, adding, and summing with respect to i, 
it is found that 



li { dt 2 dt + dt 2 dt " h dt 2 dt J 



/i . acy aY/t ou azi i 
t 4i I dxi dt dyt dt dzt dt )' 

The potential U is a function of the 3n variables a;*-, ?/;, z i} alone; 
therefore the right member of (14) is the total derivative of U 
with respect to t. Upon integrating both members of this equa- 
tion, it is found that 

(15) \ 

The left member of this equation is the kinetic energy of the whole 
system, and the right member is the potential function plus a 
constant. 

Let the potential energy of one configuration of a system with 
respect to another configuration be defined as the amount of work 
required to change it from the one to the other. If two bodies 
attract each other according to the law of the inverse squares, the 

force existing between them is r- 5 ' . The amount of work done 

ft. i 

in changing their distance apart from r ( 9^ to r t -, / is 
(16) 



* See memoirs by Jacobi, Journal de Math., vol. ix.; Tisserand, M6c. Cbl. 
vol. i., chap, xxv.; Poincare", Les Methodes Nouvelles de la Mec. Cel., vol. i., 
p. 39. 

t This is very frequently called the Vis Viva integral. 



268 THE QUESTION OF NEW INTEGRALS. [147 

If the bodies are at an infinite distance from one another at the 
start, then K 0) , = o, and (16) becomes 



hence 



Therefore, 17 is the negative of the potential energy of the whole 
system with respect to the infinite separation of the bodies as the 
original configuration. Hence (15) gives the theorem: 

In a system of n bodies subject to no forces except their mutual 
attractions the sum of the kinetic and potential energies is a constant. 

147. The Question of New Integrals. Ten of the whole 6n 
integrals which are required in order to solve the problem com- 
pletely have been found. These ten integrals are the only ones 
known, and the question arises whether any more of certain types 
exist. 

In a profound memoir in the Ada Mathematica, vol. xi., Bruns 
has demonstrated that, when the rectangular coordinates are 
chosen as dependent variables, there are no new algebraic integrals. 
This does not, of course, exclude the possibility of algebraic inte- 
grals when other variables are used. Poincare' has demonstrated 
in his prize memoir in the Ada Mathematica, vol. xin., and again 
with some additions in Les Methodes Nouvelles de la Mecanique 
Celeste, chap, v., that the Problem of Three Bodies admits no new 
uniform transcendental integrals, even when the masses of two 
of the bodies are very small compared to that of the third. In this 
theorem the dependent variables are the elements of the orbits 
of the bodies, which continually change under their mutual 
attractions. It does not follow that integrals of the class con- 
sidered by Poincare* do not exist when other dependent variables 
are employed. In fact, Levi-Civita has shown the existence of 
this class of integrals in a special problem, which comes under 
Poincare* 's theorem, when suitable variables are used (Ada 
Mathematica, vol. xxx.). The practical importance of the 
theorems of Bruns and Poincare* have often been overrated by 
those who have forgotten the conditions under which they have 
been proved to hold true. 



148] TRANSFER OF ORIGIN TO THE SUN. 269 



XVIH. PROBLEMS. 

1. Write equations (1) when the force varies inversely as the nth power 
of the distance. For what values of n do the equations all become inde- 
pendent? The Problem of n Bodies can be completely solved for this law 
of force; show that the orbits with respect to the center of mass of the system 
are all ellipses with this point as center. Show that the orbit of any body 
with respect to any other is also a central ellipse, and that the same is true 
for the motion of any body with respect to the center of mass of any sub- 
group of the whole system. Show that the periods are all equal. 

2. What will be the definition of the potential function when the force 
varies inversely as the nth power of the distance? 

3. Derive the equations immediately preceding (4) directly from equa- 
tions (1). 

4. Prove that the theorem regarding the motion of the center of mass holds 
when the force varies as any power of the distance. 

5. Derive the equations immediately preceding (12) directly from equa- 
tions (1), and show that they hold when the force varies as any power of the 
distance. 

6. Any plane through the origin can be changed into any other plane 
through the origin by a rotation around each of two of the coordinate axes. 
Transform equations (12) by successive rotations around two of the axes, and 
show that the angles of rotation can be so chosen that two of the constants, 
to which the functions of the new coordinates similar to (12) are equal, are 
zero, and that the third is V Ci 2 + c 2 2 + c 3 2 . (This is the method used by 
Laplace to prove the existence of the invariable plane.) 

7. Why are equations (13) not to be regarded as integrals of the differ- 
ential equations (1), thus making the whole number of integrals thirteen? 

148. Transfer of the Origin to the Sun. Nothing is known of 
the absolute motions of the planets because the observations 
furnish information regarding only their relative positions, or 
their positions with respect to the sun. It is true that it is known 
that the solar system is moving toward the constellation Hercules, 
but it must be remembered that this motion is only with respect 
to certain of the stars. The problem for the student of Celestial 
Mechanics is to determine the relative positions of the members 
of the solar system; or, in particular, to determine the positions 
of the planets with respect to the sun. To do this it is advanta- 
geous to transfer the origin to the sun, and to employ the resulting 
differential equations. 



270 



TRANSFER OF ORIGIN TO THE SUN. 



[148 



Suppose m n is the sun and take its center as the origin, and let 
the coordinates of the body mi referred to the new system be 
Xi, i//, Zi. Then the old coordinates are expressed in terms of 
the new by the equations 

Xi = Xi + x n , yi = y*' + y n , Zi = Zi + z, (i = l, , n-1). 

Since the differences of the old variables are equal to the corre- 
sponding differences of the new, it follows that 



3UdU 






dxi dxS' d yi dyr dZi dzi" 

As a consequence of these transformations equations (3) become 



(17) 



d 2 Xj f d 2 x n _ J_ dU 
~W ' ~ ~<W ~ mi 3x7 ' 



^ + 



d 2 y rt 
dt 2 



dt 2 



I- + 



dt 2 



1 dU 

mi dy t ' ' 

1 dU 

mi dz^ 



= 1, -., n- 1). 



Since the origin is at x n ' = y n ' = z n ' = 0, the first equation of 
(1) gives, on putting i = n, 



/ 10 \ 
(lo) 



d?x n = 

rift ' r3, " 
' 1, n 



. 





2, n 



n 1, n 



This equation, with the corresponding ones in y and z, substituted 
in (17) completes the transformation to the new variables; but 
it will be advantageous to combine the terms in another manner 
so that those which come from the attraction of the sun shall be 
separate from the others. The differential equations will be 
written for the body m\, from which the others can be formed by 
permuting the subscripts. 

The potential function can be broken up into the sum 



U 



or. 



si r t -, 



n 1 n 1 



i * j); 



(19) 



U 



U'. 



149] 



DYNAMICAL MEANING OF THE EQUATIONS. 



271 



On substituting equations (18) and (19) in equations (17), the 
latter become 



(20) < 



Let 






xi 



dt 2 



m) 



1 dU' 



mi 






n,,- 



then, equations (20) can be written in the form 



(21) 



'1, 



Let the accents, which have become useless, be dropped, and, 
in order to derive the general equations corresponding to (21), let 



(22) 



Then, the general equations for relative motion are 



(23) 



dt* 



. 
i + m,) 



x t 



= > my 



dt* nj i*i, n - 

in which i = 1, , n 1. 

149. Dynamical Meaning of the Equations. In order to under- 
stand easily the meaning of the equations, suppose that there are 
but three bodies, mi, m 2 , and m n Suppose m n is the sun, let its 
mass equal unity, and let the distances from it to mi and m 2 be 
r\ and r 2 respectively. Then equations (23) are, in full, 



272 



DYNAMICAL MEANING OF THE EQUATIONS. 







x\ 
- 3 



A.JJL 

dxi \ ri, 2 



[149 

}. 






r 2 , 






dt 2 

If m 2 were zero the first three equations would be independent 
of the second three, and they would then be the equations for the 
relative motion of the body mi with respect to m n = 1, and could 
be integrated. All the variations from the purely elliptical 
motion arise from the presence of the right members, which are, 
in the first three equations, the partial derivatives of RI, 2 with 
respect to the variables xi, yi, and z\ respectively. On this account 
mzRi, 2 is called the perturbative function. 

The partial derivatives of the first terms of the right members 
of the first three equations are respectively 

(zi - z 2 ) 



1, 2 



which are the components of acceleration of nil due to the attrac 
tion of w 2 . The partial derivatives of the second terms are 



which are the negatives of the components of the acceleration of 
the sun due to the attraction of w 2 . Therefore the right members 
of the first three equations of (24) are the differences of the com- 
ponents of acceleration of mi and of the sun due to the attraction 
of m 2 . Similarly, the right members of the last three equations 
are the differences of the components of the acceleration of w 2 
and of the sun due to the attraction of mi. If two bodies are 
subject to equal parallel accelerations their relative positions will 
not be changed. The differences of their accelerations are due to 



150] THE ORDER OF THE SYSTEM OF EQUATIONS. 273 

the disturbing forces, and measure these disturbances. The right 
members of (24) are, therefore, exactly those parts of the accelera- 
tions due to the disturbing forces. 

If there are n 2 disturbing bodies the right members are the 
sums of terms depending upon the bodies w 2 , , m n _i similar to 
the right members of (24), which depend upon ra 2 alone; or, in 
other words, the whole resultants of the disturbing accelerations 
are equal to the sums of the parts arising from the action of the 
separate disturbing bodies. 

150. The Order of the System of Equations. The order of the 
system of equations (23) is 6n 6, instead of 6n as (1) was in 
the case of absolute motion. In the absolute motion ten integrals 
were found which reduced the problem to order Qn 10. Six of 
these related to the motion of the center of mass, three to the 
areal velocities, and one to the energy of the system. In the 
present case but four integrals, the three integrals of areas and the 
energy integral, can be found, which leaves the problem of order 
6n 10 also. 

The problem can be reduced to the order 6n 6 by using the 
integrals for the center of mass directly. In particular, consider 
the differential equations for the bodies mi, w 2 , , w n _i. In the 
original equations they involve the coordinates of m n , but these 
quantities can be eliminated by means of equations (5). 

If the origin is taken at the center of mass 

n n n 

t -2/ = 0, ra^Zi = 0, 



and the elimination becomes particularly simple. Or, because of 
these linear homogeneous relations, the n variables of each set 
can be expressed linearly and homogeneously in terms of n 1 
new variables. Thus 



-ln- 



CL n 22 O, n , B -ln-l, 

and similar sets of equations for y and z. The coefficients a# are 
arbitrary constants except that they must be so chosen that every 
determinant of the matrix of the substitutions shall be distinct 
from zero; for, otherwise, a linear relation would exist among the &. 
These constants can be so chosen that the transformed equations 
19 



274 PROBLEMS. 

preserve a symmetrical form. This method was employed by 
Jacobi in an important memoir entitled, Sur I'elimination des 
noeuds dans le probleme des trois corps (Journal de Math. vol. ix., 
1844), and by Radau in a memoir entitled, Sur une transformation 
des equations differentielles de la Dynamique (Annales de I' E cole 
Normale, 1st series, vol. v.). 



XIX. PROBLEMS. 

1. Make the transformation xt = Xi + x n in the integrals (12) and (15), 
and eliminate x n , y n , z n , -V^, -J^, and -~ by means of equations (4) and (5). 
Prove that the resulting expressions are four integrals of equations (23). 

2. Derive equations (23) directly by taking the origin at m n , without first 
making use of the fixed axes. 

3. The equations (23) are not symmetrical, since each body requires a 
different perturbative function ./?,, in the right members. Construct the 
corresponding system of differential equations where the motion of m n -\ is 
referred to a rectangular system of axes with the origin at m n ; the motion of 
m n _2 to a parallel system of axes with origin at the center of mass of m n and 
ra n _i; the motion of m n -3 to a parallel system of axes with the origin at the 
center of mass of m n , m n -\, and m n _ 2 , and continue in this way. Show that 
the results are the symmetrical equations 

Mn ffiXn-l _ dU 

m n -l ,.o V , Mn Win, Mn-1 Wln-l T n, 



Mn-1 fe-2 dU 

Mn-2 



- 2 ~W - d^7 2 ' ^- 2 = m - 2 + mn ~ l + *"*> 



3U 

Mi=m 1 +m 2 +... +m n , 



and similar equations in y and z, where 
V -* 



. 

T n -l, n-3 



(These equations are the same as found by Radau from a different standpoint 
in the memoir cited in Art. 150. They have been employed by Tisserand in 
a very elegant demonstration of Poisson's theorem of the invariability of the 
major axes of the planets' orbits up to perturbations of the second order 
inclusive with respect to the masses. Poincare* has generally used this system 
in his researches in the Problem of Three Bodies.) 



HISTORICAL SKETCH. 275 

4. Derive the differential equations corresponding to (23) in polar co- 
ordinates. 



(J = 1, 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The investigations in the Problem of n Bodies are of two classes; first, 
those which lead to general theorems holding in every system; and second, 
those which give good approximations for a certain length of time in particular 
systems, such as the solar system. Investigations of the second class are 
known as theories of perturbations, the discussion of which will be given in 
another chapter. 

The first general theorems are regarding the motion of the center of mass, 
and were given by Newton in the Principia. The ten integrals and the 
theorems to which they lead were known by Euler. The next general result 
was the proof of the existence and the discussion of the properties of the 
invariable plane by Laplace in 1784. In the winter semester of 1842-43 
Jacobi gave a course of lectures in the University of Konigsberg on Dynamics. 
In this course he gave the results of some very important investigations on 
the integration of the differential equations which arise in Mechanics. In all 
cases where the forces depend upon the coordinates alone, and where a po- 
tential function exists, conditions which are fulfilled in the Problem of n 
Bodies, he proved that if all the integrals except two have been found the last 
two can always be found. He also showed, in extending some investigations 
of Sir William Rowan Hamilton, that the problem is reducible to that of 
solving a partial differential equation whose order is one-half as great as 
that of the original system. Jacobi's lectures are published in the supple- 
mentary volume to his collected works. They are of great importance in 
themselves, as well as being an absolutely necessary prerequisite to the reading 
of the epoch-making memoirs of Poincare, and they should be accessible to 
every student of Celestial Mechanics. 

It is a question of the highest interest whether the motions of the members 
of such a system as the sun and planets are purely periodic. Newcomb has 
shown in an important memoir published in the Smithsonian Contributions to 
Knowledge, December 1874, that the differential equations can be formally 
satisfied by purely periodic series. He did not, however, prove the convergence 
of these series; and, indeed, Poincare has shown in Les Methodes Nouvelles, 
chaps, ix. and xu., that they are in general divergent. 



276 HISTORICAL SKETCH. 

As was stated in Art. 147, Bruns has proved in the Acta Mathematica, 
vol. XL, that, using rectangular coordinates, there are no new algebraic inte- 
grals; and Poincare, in the Acta Mathematica, vol. xin., that, using the elements 
as variables, there are no new uniform transcendental integrals, even when 
the masses of all the bodies except one are very small. 

For further reading regarding the general differential equations in different 
sets of variables the student will do well to consult Tisserand's Mecanique 
Celeste, vol. i. chapters HI., iv., and v. 



CHAPTER VIII. 

THE PROBLEM OF THREE BODIES. 

151. Problem Considered. There are a number of important 
results in the Problem of Three Bodies which have been established 
with mathematical rigor if the initial coordinates and the com- 
ponents of velocity fulfill certain special conditions. While these 
special cases have not been found in nature, there are nevertheless 
some applications of the results obtained, and the processes 
employed are mathematically elegant and lead to most interesting 
conclusions. This chapter will contain such of these results as 
fall within the scope of this work, reserving the theories of per- 
turbations, by means of which the positions of the heavenly bodies 
are predicted, to subsequent chapters. 

The first part of the chapter will be devoted to a discussion of 
some of the properties of motion of an infinitesimal body when it 
is attracted by two finite bodies which revolve in circles around 
their center of mass, and will include the proof of the existence of 
certain particular solutions in which the distances of the infinitesi- 
mal body from the finite bodies are constants. The second part 
of the chapter will be devoted to an exposition of a method of 
finding particular solutions of the motion of three finite bodies such 
that the ratios of their mutual distances are constants. These 
solutions include the former, but the discoverable properties of 
motion are so much fewer, and are obtained with so much more 
difficulty, that it is advisable to divide the discussion into two 
parts. 

The particular solutions of the Problem of Three Bodies which 
will be discussed here were given for the first time by Lagrange in 
a prize memoir in 1772. The method adopted here is radically 
different from that employed by him, and lends itself much more 
readily to a generalization to the case where a larger number of 
bodies is involved. But, on the other hand, the reduction of the 
order of the problem by one unit, which was a very interesting 
feature of Lagrange's memoir, is not accomplished by this method. 
However, as it has not been possible to make any use of this 
reduction, it has not been of any practical importance. 

Mathematically speaking, an infinitesimal body is one that is 

277 



278 THE DIFFERENTIAL EQUATIONS OF MOTION. [152 

attracted by finite masses but does not attract them. Physically 
speaking, it is a body of such a small mass that it will disturb 
the motion of finite bodies less than an arbitrarily assigned amount, 
however small, during any arbitrarily assigned time, however long. 
To actually determine a small mass fulfilling these conditions it is 
only necessary to make it so small that its whole attraction, which 
is always greater than its disturbing force, on one of the large 
bodies, if placed at the minimum distance possible, would move the 
large body less than the assigned small distance in the assigned 
time. 

MOTION OF THE INFINITESIMAL BODY. 

152. The Differential Equations of Motion. Suppose the 
system consists of two finite bodies revolving in circles around their 
common center of mass, and of an infinitesimal body subject to 
their attraction. Let the unit of mass be so chosen that the sum 
of the masses of the finite bodies shall be unity; then they can be 
represented by 1 ^ and ju, where the notation is so chosen that 
fi ^ J. Let the unit of distance be so chosen that the constant 
distance between the finite bodies shall be unity. Let the unit of 
time be so chosen that k 2 shall equal unity. Let the origin of 
coordinates be taken at the center of mass of the finite bodies, 
and let the direction of the axes be so chosen that the ^-plane is 
the plane of their motion. Let the coordinates of 1 n, M> and the 
infinitesimal body be 1, rji, 0; 2, 172, 0; and , rj, f respectively, and 



ri = Vtt - i) 2 + (n - rn) 2 + r 2 , 

T2 = V( - 2 ) 2 + (r) - i7 2 ) 2 + r 2 - 

Then the differential equations of motion for the infinitesimal 
body are 



(1) 



i\ _ n 
= ~ 



y r? 

*7l) (l? ^2) 



<* 2 r v r r 

-=-(i-ti--- 



As a consequence of the way the units have been chosen the 
mean angular motion of the finite bodies is 



a* 



152] 



THE DIFFERENTIAL EQUATIONS OF MOTION. 



279 



Let the motion of the bodies be referred to a new system of 
axes having the same origin as the old, and rotating in the ??- 
plane in the direction in which the finite bodies move with the 
uniform angular velocity unity. The coordinates in the new 
system are defined by the equations 



= x cos t y sin t, 
= x sin t + y cos t, 







and similar equations for the letters with subscripts 1 and 2. On 
computing the second derivatives of (2) and substituting in (1), 
it is found that 



+ M 



(3) 



f d?x _ 9 dy _ 
[d? ~ dt~ 

~{a- 

+ 



r 2 



^) cos* 



- + 



J^!_2-^ 1 ' / 4- I 4- 
U^ 2 d ^ J S1 1 rf^ 2 " 

= -(d-M)^-^ + 



- (1- 



+ 



sin 



\ cos ^, 



df 



Multiply the first two equations by cos t and sin t respectively, 
then by sin t and cos t, and add; the results are 

(x - x,) (x- x z ) 



/-72/y //'?/ 

_ _2 =0; (1 
rf 2 ?/ . ^dx 



The position of the axes can be so taken at the origin of time 
that the z-axis will continually pass through the centers of the 



280 



JACOBI S INTEGRAL. 



[153 



finite bodies; then y\ = 0, ?/ 2 = 0, and the equations become 
d?x_ ( ,dy_ _ n _ ,(x-x 1 )_ ii (x_-x z ) 



(4) H 



d z y 



d z z 



dx 



y_ 

z 
fl 3 



z 
-i 



These are the differential equations of motion of the infinitesimal 
body referred to axes rotating so that the finite bodies always lie 
on the rr-axis. They have the important property that they do 
not involve explicitly the independent variable t because the 
coordinates of the finite bodies have become constants as a conse- 
quence of the particular manner in which the axes are rotated. 
On the other hand, in equations (1) the quantities 1, 2 , r?i, and r? 2 
are functions of t. 

The general problem of determining the motion of the in- 
finitesimal body is of the sixth order; if it moves in the plane of 
motion of the finite bodies, the problem is of the fourth order. 

153. Jacobi's Integral. Equations (4) admit an integral which 
was first given by Jacobi in Comptes Rendus de I' Academic des 
Sciences de Paris, vol. in., p. 59, arid which has been discussed by 
Hill in the first of his celebrated papers on the Lunar Theory, 
The American Journal of Mathematics, vol. i., p. 18, and again by 
Darwin in his memoir on Periodic Orbits in Acta Mathematica, 
vol. xxi., p. 102. Let 

a -:,>.. 



(5) 



U = i(z 2 + y 2 ) + 



then equations (4) can be written in the form 



(6) 



d*x dy _ dU 
dt* ~di~ dx' 



dt 2 



dt* 



= 
dt 



dy 



dz ' 



If these equations are multiplied by 2 , 2 -^ , and 2 -r re- 
spectively, and added, the resulting equation can be integrated, 



154] THE SURFACES OF ZERO RELATIVE VELOCITY. 281 

since U is a function of x, y, and z alone, and give 



Five integrals more are required in order completely to solve 
the problem. If the infinitesimal body moved in the xy-plane 
only three would remain to be found, the last two of which could 
be obtained by Jacobi's last multiplier,* if the first one were found. 
Thus it appears that only one new integral is needed for the com- 
plete solution of this special problem in the plane. f But Bruns 
has proved in Ada Mathematica, vol. xi., that, when rectangular 
coordinates are used, no new algebraic integrals exist; and Poin- 
care has proved in Les Methodes Nouvelles de la Mecanique Celeste, 
vol. i., chap, v., that when the elements of the orbits are used as 
variables, there are no new uniform transcendental integrals, 
even when the mass of one of the finite bodies is very small com- 
pared to that of the other (see Art. 147). These demonstrations 
are entirely outside the scope of this work and cannot be repro- 
duced here. 

154. The Surfaces of Zero Relative Velocity. J Equation (7) 
is a relation between the square of the velocity and the coordinates 
of the infinitesimal body referred to the rotating axes. Therefore, 
when the constant of integration C has been determined numeri- 
cally by the initial conditions, equation (7) determines the velocity 
with which the infinitesimal body will move, if at all, at all. points 
of the rotating space; and conversely, for a given velocity, equa- 
tion (7) gives the locus of those points of relative space where alone 
the infinitesimal body can be. In particular, if V is put equal to 
zero in this equation it will define the surfaces at which the velocity 
will be zero. On one side of these surfaces the velocity will be 
real and on the other side imaginary; or, in other words, it is 

* Developed in Vorlesungen uber Dynamik, supplementary volume to 
Jacobi's collected works. 

t Hill put his special equations in such a form that they would be reduced 
to quadratures if a single variable were expressed in terms of the time, American 
Journal of Mathematics, vol. i., p. 16. 

t First discussed by Hill in his Lunar Theory, The American Journal of 
Mathematics, vol. i.; and again, for motion in the xy-plane, by Darwin in his 
Periodic Orbits, in Ada Mathematica, vol. xxi. 



282 APPROXIMATE FORMS OF THE SURFACES. [155 

possible for the body to move on one side, and impossible for it 
to move on the other. The general proposition that a function 
changes sign as the surface at which it is zero is crossed (at least 
at a regular point of the surface) was proved in Art. 120. While 
it will not be possible to say in any except very particular cases 
what the orbit will be, yet this partition of relative space will 
show in what portions the infinitesimal body can move and in 
what portions it can not. 

The equation of the surfaces of zero relative velocity is 



(8) 






+ y 2 + z 2 , 



I r 2 = 4(x - z 2 ) 2 + y 2 + z 2 . 



Since only the squares of y and z occur the surfaces defined by (8) 
are symmetrical with respect to the xy and zz-planes, and, when 
/A = |, with respect to the j/z-plane also. The surfaces for JJL =j= J 
can be regarded as being deformations of those for /* = J. It 
follows from the way in which z enters that a line parallel to the 
z-axis pierces the surfaces in two (or no) real points. Moreover, 
the surfaces are contained within a cylinder whose axis is the 
z-axis and whose radius is VC, to which certain of the folds are 
asymptotic at z 2 = o ; for, as z 2 increases the equation approaches 
as a limit 

z 2 + y 2 = C. 

155. Approximate Forms of the Surfaces. From the properties 
of the surfaces given in the preceding article and from the shapes 
of the curves in which the surfaces intersect the reference planes, 
a general idea of their form can be obtained. The equation of 
the curves of intersection of the surfaces with the xy-pl&ne is 
obtained by putting z equal to zero in the first of (8), and is 



(x-xtf + y* A(z - * 2 ) 2 + 2/ 2 

For large values of x and y which satisfy this equation the third 
and fourth terms are relatively unimportant, and the equation 
may be written 



155] 



APPROXIMATE FORMS OF THE SURFACES. 



283 



where e is a small quantity. This is the equation of a circle whose 
radius is A/C e; therefore, one branch of the curve in the in/- 
plane is an approximately circular oval within the asymptotic 
cylinder. It is also to be noted that the larger C is, the larger 
are the values of x and y which satisfy the equation, the smaller 
is e, the more nearly circular is the curve, and the more nearly 
does it approach its asymptotic cylinder. 



x-Aocis 




Fig. 38. 

For small values of x and y satisfying (9) the first and second 
terms are relatively unimportant, and the equation may be 
written 



1 M , M 



This is the equation of the equipotential curves* for the two centers 
of force, 1 n and n> For large values of C they consist of 
closed ovals around each of the~bodies 1 M and M; for smaller 
values of C these ovals unite between the bodies forming a dumb- 

* Thomson and Tait's Natural Philosophy, Part II., Art. 508. 



284 



APPROXIMATE FORMS OF THE SURFACES. 



[155 



bell shaped figure in which the ends are of different size except 
when n = J; and for still smaller values of C the handle of the 
dumb-bell enlarges until the figure becomes an oval enclosing 
both of the bodies 

From the foregoing considerations it follows that the approxi- 
mate forms of the curves in which the surfaces intersect the in/- 
plane are as given in Fig. 38. The curves Ci, C 2 , Cs, C 4 , CB are 
in the order of decreasing values of the constant C. They were 
not drawn from numerical calculations and are intended to show 
only qualitatively the character of the curves. 



z -Axis 




Fig. 39. 

The equation of the curves of intersection of the surfaces and 
the rcz-plane is obtained by putting y equal to zero in equation 
(8), and is 

2(1 - M ) 2 M 



(10) 



= C. 



For large values of x and z satisfying this equation the second 



155] 



APPROXIMATE FORMS OF THE SURFACES. 



285 



and third terms are relatively unimportant, and it may be written 

x 2 = C - e, 

which is the equation of a symmetrical pair of straight lines 
parallel to the z-axis. The larger C is, the larger is the value of x 
which, for a given value of z, satisfies the equation, and, therefore, 
the smaller is e. Hence, the larger C the closer the lines are to the 
asymptotic cylinder. 

z -Axis 




yAxis 



Fig. 40. 

For small values of x and z satisfying equation (10) the first 
term is relatively unimportant, and the equation may be written 



fl = C 

r 2 2 



6. 



This is again the equation of the equipotential curves and has the 
same properties as before. Hence, the forms of the curves in the 
zz-plane are qualitatively like those given in Fig. 39. Again, 
the curves Ci, , CB are in the order of decreasing values of the 
constant C, and were not drawn from numerical calculations. 
The equation of the curves of intersection of the surfaces and 



286 THE REGIONS OF REAL AND IMAGINARY VELOCITY. [156 

the 2/z-plane is obtained by putting x equal to zero in equation 
(8), and is 
(11) y. 2(1 -M) 2 M 



> 

+ I/ 2 + 2 2 VZ 2 2 + ?/ 2 + Z 2 

For large values of y and 2 satisfying this equation the second and 
third terms are relatively unimportant, and it may be written 

2/ 2 = C - e, 

which is the equation of a pair of lines near the asymptotic cylinder, 
approaching it as C increases. 

If 1 ju is much greater than jj,, the numerical value of # 2 is 
much greater than that of x\\ hence, for small values of y and z 
satisfying (11), this equation may be written 



r l 

which is the equation of a circle which becomes larger as C de- 
creases. Hence, the forms of the curves in the t/2-plane are quali- 
tatively as given in Fig. 40. Again, the curves Ci, , CB are 
in the order of decreasing values of the constant C. 

From these three sections of the surfaces it is easy to infer their 
forms for the different values of C. They may be roughly de- 
scribed as consisting of, for large values of C, a closed fold approxi- 
mately spherical in form around each of the finite bodies, and of 
curtains hanging from the asymptotic cylinder symmetrically 
with respect to the xy-pl&ue; for smaller values of C, the folds 
expand and coalesce (Fig. 38, curve C 3 ); for still smaller values 
of C the united folds coalesce with the curtains, the first points of 
contact being in every case in the :n/-plane; and for sufficiently 
small values of C the surfaces consist of two parts symmetrical 
with respect to the :n/-plane but not intersecting it (Figs. 39, 
curve C&, and 40, curve Ce). 

156. The Regions of Real and Imaginary Velocity. Having 
determined the forms of the surfaces, it remains to find in what 
regions of relative space the motion is real and in what it is imagi- 
nary. The equation for the square of the velocity is 

e\/-t \ r 



Suppose C is so large that the ovals and curtains are all separate. 



157] METHOD OF COMPUTING THE SURFACES. 287 

The motion will be real in those portions of relative space for 
which the right member of this equation is positive. If it is 
positive in one point in a closed fold it will be positive in every 
other point within it, for the function changes sign only at a surface 
of zero relative velocity. 

It is evident from the equation that x and y can be taken so 
large that the right member will be positive, however great C may 
be; therefore, the motion is real outside of the curtains. It is also 
clear that a point can be chosen so near to either 1 JJL or /*, that 
is, either ri or r 2 may be taken so small, that the right member will 
be positive, however great C may be; therefore, the motion is real 
within the folds around the finite bodies. 

If the value of C were so large that the folds around the finite 
bodies were closed, and if the infinitesimal body should be within 
one of these folds at the origin of time, it would always remain 
there since it could not cross a surface of zero velocity. If the 
earth's orbit is supposed to be circular and the mass of the moon 
infinitesimal, it is found that the constant C, determined by the 
motion of the moon, is so large that the fold around the earth is 
closed with the moon within it. Therefore the moon cannot 
recede indefinitely from the earth. It was in this manner, and 
with these approximations, that Hill proved that the moon's 
distance from the earth has a superior limit.* 

157. Method of Computing the Surfaces. Actual points on 
the surfaces can be found most readily by first determining the 
curves in the :n/-plane, and then finding by methods of approxi- 
mation the values of z which satisfy (7). Besides, the curves in 
the rn/-plane are of most interest because the first points of contact 
as the various folds coalesce occur in this plane, and, indeed, on 
the x-axis, as can be seen from the symmetries of the surfaces. 

The equation of the curves in the xy-plane is 



\ tf \ " M \ 



z - *i y x - * 2 y 

If this equation is rationalized and cleared of fractions the result 
is a polynomial of the sixteenth degree in x and y. When the value 
of one of the variables is taken arbitrarily the corresponding 
values of the other can be found by solving this rationalized 
equation. This problem presents great practical difficulties 
* Lunar Theory, Am. Jour. Math., vol. i., p. 23. 



288 METHOD OF COMPUTING THE SURFACES. [157 

because of the high degree of the equation, and these troubles 
are supplemented by the presence of foreign solutions which are 
introduced by the processes of rationalization. 

The difficulty from foreign solutions can be avoided entirely, 
and the degree of the equation can be very much reduced by 
transforming to bi-polar coordinates. That is, points on the 
curves can be denned by giving their distances from two fixed 
points on the o>axis. This method could not be applied if the 
curves were not symmetrical with respect to the axis on which 
the poles lie. Let the centers of the bodies 1 M and /* be taken 
as the poles; the distances from these points are r\ and r 2 respec- 
tively. To complete the transformation it is only necessary to 
express x 2 + y 2 in terms of these quantities. 



y -\axia 




X-axis 
-4 

Fig. 41. 



Let P be a point on one of the curves; then OA =_x, AP = y, 
and, since is the center of mass of 1 /z and /*, OM = 1 ju, 
and 0(1 - /z) = - M- It follows that 



r jf = fl i _ ( X + M )2 
[ = r , 2 _ x _ 1 _ 



= ^2 _ X 2 + 2(1 - rfx ~ (I ~ M) 2 - 

On eliminating the first power of x from these equations and solv- 
ing for x 2 + y 2 , it is found that 

x 2 + y 2 = (1 - ju)n 2 + jur 2 2 - /*(! - /*) 
As a consequence of this equation, (9) becomes 

(12) (1 - M ) Tl 2 + + M r 2 2 + = C + /il-/=C". 



If an arbitrary value of r 2 is assumed n can be computed from 
this equation; the points of intersection of the circles around 
1 ;u and v as centers, with the computed and assumed values 
respectively of r\ and r 2 as radii, will be points on the curves. To 
follow out this plan, let equation (12) be written in the form 



157] 



METHOD OF COMPUTING THE SURFACES. 



289 



0, 



(13) 



= 2. 



Since b = 2 is positive there is at least one real negative root of 
the first of (13) whatever value a may have. But the only value 
of n which has a meaning in this problem is real and positive; 
hence the condition for real positive roots must be considered. 

It follows from (12) that C" is always greater than /* r 2 2 H 

for all real positive values of r\ and r 2 ; therefore a is always nega- 
tive. It is shown in the Theory of Equations that a cubic equa- 
tion of this form has three distinct real roots if 276 2 + 4a 3 < 0; 
or, since b = 2, if 

(14) a + 3 < 0. 



Suppose this inequality is satisfied. 
of solving the cubic is 



Then a convenient method 



(15) 




where rn, r*i 2 , r n are the three roots of the cubic. 

The limit of the inequality (14) is a + 3 = 0; or, in terms of 
the original quantities, 



(16) 



r 2 3 + aV 2 + &' = 0, 



b' = 2. 



The solution of this equation gives the extreme values of r 2 for 
which (13) has real roots. Therefore, in the actual computation 
equation (16) -should be solved first for r 2 i and r 22 . The values of 

20 



290 PARTICULAR SOLUTIONS OF [158 

r 2 to be substituted in (13) should be chosen at convenient inter- 
vals between these roots. 

Equation (16) will not have real positive roots for all values 
of a', the condition for real positive roots being 

a' + 3 ^ 0; 

the limiting value of which is, in the original quantities, 
C" 3(1-,.) 

-- T ' --- O, 

M M 

whence 

C' = 3. 

Therefore C' must be equal to, or greater than, 3 in order that the 
curves shall have real points in the xy-pl&ne. For C' = 3 the 
curves are just vanishing from the plane, and it follows at once 
\ that equation (12) is then satisfied by r\ = 1, r 2 = 1; that is, the 
surfaces vanish from the xy-pl&ne at the points which form equi- 
lateral triangles with 1 M and M- 

158. Double Points of the Surfaces and Particular Solutions 
of the Problem of Three Bodies. It follows from the general 
forms of the surfaces that the double points which appear as C 
diminishes are all in the rri/-plane. Therefore it is sufficient in 
this discussion to consider the equation of the curves in the 
zi/-plane. There are three double points on the z-axis which 
appear when the ovals around the finite bodies touch each other 
and when they touch the exterior curve enclosing them both. 
There are two more which appear, as the surfaces vanish from the 
zi/-plane, at the two points making equilateral triangles with the 
finite bodies. 

These double points are of interest as critical points of the 
curves, and it will now be shown that they are connected with 
important dynamical properties of the system. Let the equation 
of the curves be written 



The conditions for double points are 



(17) F(x, y) ma + 1 f 

ouble oints are 



1 dF . , (x Xi) (x Xz) n 

o T~ = x (1 M) 5^ M 3 5 i = 0; 

2 dz r x 3 r 2 3 

A " 

la^ 7 ? / 

2 dy ~ y ^ ri 3 M r 2 3 * 






158] THE PROBLEM OF THREE BODIES. 291 

The left members of these equations are the same as the right 

1 r^ff 

members of the equations (4) for z = 0. The expressions - - 

Z ox 

1 r) W 

and - are proportional to the direction cosines of the normal 

at all ordinary points of the curves; and since 3- and -jr are zero 

at at 

at the surfaces of zero velocity it follows from (4) that the directions 
of acceleration, or the lines of effective force, are orthogonal to the 
surfaces of zero relative velocity. Therefore, if the infinitesimal 
body is placed on a surface of zero relative velocity it will start 
in its motion in the direction of the normal. But at the double 
points the sense of the normal becomes ambiguous; hence, it might 
be surmised that if the infinitesimal body were placed at one of 
these points it would remain relatively at rest. 

The conditions imposed by (17) and (18) are also the conditions 

that -JTJ and -^ , or the components of acceleration, in equations 

(4) shall vanish. Hence, if the infinitesimal body is placed at a 
double point with zero relative velocity, its coordinates will identically 
fulfill the differential equations of motion and it will remain forever 
relatively at rest, unless disturbed by forces exterior to the system 
under consideration. These are particular solutions of the Problem 
of Three Bodies, and are special cases of the Lagrangian solutions. 
Consider equations (18), the second of which is satisfied by 
y = 0. The double points on the z-axis, and the straight line 
solutions of the problem are given by the conditions 

(x - xi) (x .- x t ) 



(19) 



-- 

y = o, 

z = 0. 



The left member of the first equation considered as a function 
of x is positive f or x = + oo ; it is negative for x = x 2 + e, where e 
is a very small positive quantity; it is positive for x = 2 e; 
it is negative for x = x\ + e; it is positive for x = Xi e; and it 
is negative for x = oo. Since the function is finite and con- 
tinuous except when x = + <*> , x*, x\, or -- oo, it follows that 
the function changes sign three times by passing through zero, 
(a) once between + oo and x z , (b) once between z 2 and Xi, and 
(c) once between xi and oo. Therefore, there are three posi- 



292 PARTICULAR SOLUTIONS OF [158 

tions on the line through 1 /* and M at which the infinitesimal 
body will remain when given proper initial projection. 

(a) Let the distance from ^ to the double point on the #-axis 
between + oo and x 2 be represented by p. Then x # 2 = P, 
x Xi = ri = 1 + p, x = I M + P; therefore the first equation 
of (19) becomes after clearing of fractions 

(20) p 6 + (3 - /x)p 4 + (3 - 2/z)p 3 - jup 2 - 2/zp - M = 0. 

This quintic equation has one variation in the sign of its coef- 
ficients, and hence only one real positive root. The value of this 
root depends upon /*. Consider the left member of the equation 
as a function of p and ju. For /* = the equation becomes 

P V + 3 P + 3) = 0, 

which has three roots p = 0, and two others, coming from the 
second factor, which are complex. It follows from the theory 
of the solution of algebraic equations that, for /JL different from 
zero but sufficiently small, three roots of the equation are ex- 
pressible as power series in /**, vanishing with this parameter.* 
The one of these three roots obtained by taking the real value of /** 
is real; the other two are complex. Therefore, the real root has 
the form 

On substituting this expression for p in (20) and equating to zero 
the coefficients of corresponding powers of M*> it is found that 

_ 3* _ 3* 1 

ai ~3~' a2 ~~9' ~27' 

Hence 

(21) 

P- 

The corresponding value of C" is found by substituting these 
values of r\ and r 2 in equation (12). 

(6) Let the distance from /* to the double point on the x- 
axis between x% and x\ be represented by p. Then in this case 
x x 2 = p, x x\ = r\ = 1 p, x = (1 M) P; therefore 
the first equation of (19) becomes 

p 5 - (3 - M)p 4 + (3 - 2^)p 3 - MP 2 + 2/zp - M = 0. 
* See Harkness and Morley's Theory of Functions, chapter iv. 



158] THE PROBLEM OF THREE BODIES. 293 

On solving as in (a), the values of r 2 and r\ are found to be 
r / M \* 

(22) r 2 = p = UJ " . 

In';- i-p- 

The corresponding value of C' is found by substituting these 
values of r\ and r 2 in equation (12). 

(c) Let the distance from 1 /* to the double point on the 
z-axis between x\ and oo be represented by 1 p. In this case 
z - z 2 = - 2 + p, z - Zi = - 1 + p, x=-fj.-l + p, and 
the first equation of (19) becomes 

P 6 - (7 + M )P 4 + (19 + 6 M )p 3 - (24 + 13 M )p 2 

( ^o ) 

+ (12 + 14 M ) P - 7 M = 0. 
When /* = this equation becomes 

P 8 - 7p 4 + 19p 3 - 24 P 2 + 12p = 0, 

which has but one root p = 0. Therefore p can be expressed as a 
power series in /* which converges for sufficiently small values of 
this parameter, and vanishes with it. This root will have the 
form 

P = Cin + c 2 ju 2 + c 3 M 3 + c 4 M 4 +' 

On substituting this expression for p in (23), and equating to zero 
the coefficients of the various powers of M, it is found that 

7 23 X 7 2 

Cl= l2' 2 = ' Cs = T2 4 ' 
Hence 

7 , 23 X 7 2 3 . 
P = 12 M+ 12 4 M+ B| 

(24) 

1 = 1 - p, 

2 = 1 + ri = 2 - p. 

The corresponding value of C' is found by substituting these 
values of r\ and r 2 in equation (12). 

If the values of r\ and r 2 given by the first three terms of the 
series (21), (22), and (24) are not sufficiently accurate, more 
nearly correct values should be found by differential corrections. 

In order to find the double points not on the z-axis consider 
equations (18) again. They, or any two independent functions 
of them, define the double points. Since y is distinct from zero 
in this case the second equation may be divided by it, giving 



294 



PROBLEMS. 



1 _ 



_ JL = o. 



Multiply this equation by x rc 2 , and x x i} and subtract the 
products separately from the first of (18). The results are 



But x 2 1 ju> X 
equations reduce to 



= ^ and 2 1 = 1; therefore these 






- 1 + -, \ = 0, 
r 2 3 

2 = 0. 

The only real solutions are r\ = 1, r 2 = 1, and the points form 
equilateral triangles with the finite bodies whatever their relative 
masses may be. As was shown in the last of Art. 157, they occur 
at the places where the surfaces vanish from the xy-plane. 



XX. PROBLEMS. 

1. The units defined in Art. 152 are called canonical units; what would 
the canonical unit of time be in days for the earth and sun? 

2. Show on d priori grounds that, when the niotion of the system is referred 
to axes rotating as in Art. 152, the differential equations should not involve 
the time explicitly. 

3. Why cannot an integral corresponding to (7) be derived from equations 
(1) at once without any transformations? Prove that there is an integral 
of (1). 

4. What are the surfaces of zero velocity for a body projected vertically 
upward against gravity? For a body moving subject to a central force 
varying inversely as the square of the distance? 

5. Show by direct reductions from (13) and (14) that 



rn)(ri ri 2 )(n - r 13 ) 



+ 



+ 6 = 0. 



6. Prove that the solution of (16) gives the extreme values of r 2 for which 
(14) has real roots, Hint. Consider the graph of y = r 2 3 + a'r 2 + b'. 



159] TISSERAND'S CRITERION FOR IDENTITY OF COMETS. 295 

7. Impose the conditions on (12) that C" shall be a minimum and show 
that it is satisfied only for r L = 1, r 2 = 1, and that the minimum value of C' 
is 3. 

8. Why are not the lines of effective force orthogonal to all of the surfaces 
of constant velocity? 

9. Prove that the double point between /j, and 1 ^ is nearer /* than is 
the one between ju and + . 

10. Prove that, as C' diminishes, the first double point to appear is the one 
between /j. and 1 /*; the second, the one between p. and + J the third, 
the one between I n and w ; and the last, those which make equilateral 
triangles with the finite bodies. 

11. If /i = TT> 1 - M = TT, find the values of n, r 2 , and C' from (21), (22), 
(24), and (12). 

1(21) r 2 = 0.340, ri = 1.340, C' = 3.535; 
(22) r 2 = 0.276, n = 0.724, C' = 3.653; 
(24) r 2 = 1.947, ri = 0.947, C" = 3.173. 

12. From the approximate values of the last example find by the method 
of differential corrections more accurate values. 

f (21) r 2 = 0.347, n = 1.347, C' = 3.534; 

Ans. J (22) r 2 = 0.282, n = 0.718, C' = 3.653; 

[ (23) r 2 = 1.947, n = 0.947, C' = 3.173. 

13. Considering the earth's orbit to be a circle, find the distance in miles 
from the earth to the double point which is opposite to the sun. Would an 
infinitesimal body at this point be eclipsed? 

Ans. 930,240 miles. 

159. Tisserand's Criterion for the Identity of Comets.* Comets 
sometimes pass near the planets in their revolutions around the 
sun, and then the elements of their orbits are greatly changed. 
The planet Jupiter is especially potent in producing these per- 
turbations because of its great mass and because at its distance 
the attraction of the sun is much less than it is at the distances of 
the earth-like planets. Since a comet has no characteristic 
features by which it may be recognized with certainty, its identity 
might be in question if it were not followed visually during the 
time of the perturbations. 

One way of testing the identity of two comets appearing at 
different epochs is to take the orbit of the earlier and to compute 
the perturbations which it undergoes, and then to compare the 
derived elements with those determined from the later obser- 

* Bulletin Astronomique, vol. vi., p. 289, and Mec. Cel., vol. iv., p. 203. 



296 TISSERAND'S CRITERION FOR IDENTITY OF COMETS. [159 

vations; or, the start may be made with the elements of the later 
comet, and by inverse processes the earlier elements may be com- 
puted and the comparison made. One or the other of these plans 
has been followed until recent years. 

But the question arises if there is not some relation among the 
elements which remains unaltered by the perturbations. This 
is the question which Tisserand has answered in the affirmative in 
one of his characteristically elegant and important papers on 
Celestial Mechanics. 

Let the eccentricity of Jupiter's orbit be supposed equal to zero, 
and the mass of the comet infinitesimal. While both of these 
assumptions are false they are very nearly fulfilled, and the error 
introduced will be inappreciable, especially as the comet will be 
near enough to Jupiter to suffer sensible disturbances only a very 
short time. Under these suppositions, and when the units are 
properly chosen, the integral 



holds true. This is an answer to the question; for, when the 
elements are known the velocity and coordinates can be computed 
at any time, and the motion referred to rotating axes by equations 
(2). Hence, to test the identity of two comets, compute the 
function (7) for each orbit and see if the constant C is the same 
for both. If the two values of C are the same, the probability is 
very strong that only one comet has been observed; if they are 
different, the two comets are certainly distinct bodies. 

The process just explained has the inconvenience of involving 
considerable computati9n. This can be largely avoided by ex- 
pressing (7) in terms of the ordinary elements of the orbit. The 
first step is to express (7) in terms of coordinates measured from 
fixed axes. The equations of transformation are the inverse of 
equations (2), viz., 

' x = + % cos t + 17 sin t, 
y = % sin t + 77 cos t, 
z = f. 

From these equations it is found that 



159] TISSERAND'S CRITERION FOR IDENTITY OF COMETS. 297 



Hence equation (7) becomes 



= 2 U ~ M) , 2^ _ ^ 
T\ TZ 

Let r represent the distance of the comet from the origin, and i 
the angle between the plane of its instantaneous orbit and the 
i7-plane. Then equations (24), Art. 89, give 



dt \dt / \dt 



Hence equation (25) becomes 
(26 ) ?_i_ 



ri r 2 

In the case of Jupiter and the sun ju is less than one-thousandth. 
Therefore the origin is very near the center of the sun, and TI is 
sensibly equal to r. In both instances the elements will be deter- 
mined when the comet is far from both Jupiter and the sun so that 

2u 2u 
-- - H - will be so small that it may be neglected without 

fl Tz oj 

important error; then (26) reduces to the simple expression 

cos i = C. 

It will be noticed that the elements of this formula are the 
instantaneous elements for motion around a unit mass situated 
at the center of mass of the finite bodies. The actual elements 
used in Astronomy are the elements referred to the center of the 
sun, with the sun as the attracting mass. Nevertheless, on 
account of the small relative mass of Jupiter the two sets of 
elements are very nearly the same, and if the two orbits are of 
the same body, the equation 



298 



STABILITY OF PARTICULAR SOLUTIONS. 



[160 



(27) -. + 2^(1- 



cos 



= + 2 a 2 (l - e 2 2 ) cos 



must be fulfilled, where the elements are those in actual use by 
astronomers. Such is the criterion developed by Tisserand, and 
employed later by Schulhof and others. 

160. Stability of Particular Solutions. Five particular solutions 
of the motion of the infinitesimal body have been found. If the 
infinitesimal body is displaced a very little from the exact points 
of the solutions and given a small velocity it will either oscillate 
around these respective points, at least for a considerable time, 
or it will rapidly depart from them. In the first case the particular 
solution from which the displacement is made is said to be stable; 
in the second case, it is said to be unstable. 

The question of stability must be formulated mathematically. 
Consider the equations 



(28) 



Suppose x = XQ, y = y^ where X Q and y Q are constants, is a par- 
ticular solution of (28). That is, 

/(zo, yo) = 0, g(x , yo) = 0. 

Give the body a small displacement and a small velocity so that 
its coordinates and components of velocity are 

x = x Q + x', 




(29) 



y = yo + 

dx = dx^ 
dt == dt ' 

dy = dy f 
dt " dt > 



y', 



where x', y', , and -^- are initially very small. On making 
these substitutions in (28), the differential equations become 



(30) 



160] STABILITY OF PARTICULAR SOLUTIONS. 299 

When the right members are developed by Taylor's formula, they 
take the form 



2/0 - g(xo, 2/o) + o/ + 



In the partial derivatives x = x and y = y . The first terms in 
the right members are respectively zero; hence equations (30) 
become 

r ^~' 9 dy' _ df , df , 

(31) 



If #' and 2/' are taken very small on the start the influence of 
the higher powers in the right members will be inappreciable, at 
least for a considerable time. If the parts which involve second 
and higher degree terms in x' and y r are neglected, the differential 
equations reduce to the linear system 



(32) 



_< 2 . = _ x > - tf 
* " d 



___ 
dt ~ dx' dy'' 



The solutions of a system of linear differential equations with con 
stant coefficients can in general be expressed in terms of exponen 
tials in the form 



where ai, , "<* 4 are the constants of integration, and 0i, , 4 
are constants depending upon them and the constants involved in 
the differential equations. If Xi, , X 4 are pure imaginary 
numbers, then x' and y f are expressible in periodic functions, and 
the solution from which the start was made is said to be stable; if 
any of Xi, , X 4 are real or complex numbers, then x' and y' 
change indefinitely with t, and the solution is said to be unstable. 
There are exceptional cases where the solution contains constant 

terms instead of exponentials; they are of course stable if all the 





300 



APPLICATION OF CRITERION FOR STABILITY 



[161 



exponentials are purely imaginary. There are other exceptional 
cases in which the solution contains exponentials multiplied 
by some power of t\ these solutions are usually regarded' as 
unstable. 

161. Application of the Criterion for Stability to the Straight 
Line Solutions. The definitions and general methods of the last 
article will now be applied to the special cases which have arisen 
in the discussion of the motion of the infinitesimal body. The 
original differential equations were (Art. 152) 



d?x 



(x 



(x 



dx 
dt 



dt 2 



- (1 - M) ~ M = h(x, y, z). 



The straight line solutions occur for 

x = x 0i , y = 0, z = 0, 

where i = 1, 2, 3 according as the point lies between + oo and /z, 
IJL and 1 M, or 1 /z and oo , and where these values of x, y, 
and z satisfy equation (19). Make the substitution 



_ 
dt' 



X = Xoi + X 1 , 


y = y' } 


z 


dx dx' 
dt == W' 


dy dy' 
dt " dt ' 


dz 
dt 



Then it is found that 



*. 4- . x 

* ^ 27 ^ 2 



u y i i u y t i u y / __ / _ 

dx' dy' dz' 

dh_ , dh_ , dh^ , = 

dx' X dy' y dz' Z 



(i - M 



Let 

(34) 



Then the equations corresponding to (32) become in this case 



161] 



TO THE STRAIGHT LINE SOLUTIONS. 



301 



(35) 



df 



The last equation is independent of the first two and can be 
treated separately. The solution is (Art. 32) 



(36) 



z > = 



Therefore the motion parallel to the z-axis, for small displace 
ments, is periodic with the period = . 

4Ai 

Consider now the simultaneous equations 



(37) 



To find the solutions let 

(38) 

where K and L are constants. On substituting these expressions 
in equations (37) and dividing out e^, it is found that 

[X 2 - (1 + 2Ai)]K - 2XL = 0, 
2\K + [X 2 - (1 - Ai)]L = 0. 



(39) 



In order that equations (38) shall be particular solutions of (37) 
equations (39) must be fulfilled. They are verified by K = 0, 
L = 0; but in this case x' = 0, y f = 0, and the solutions reduce 
to the straight line solutions. Equations (39) can be satisfied by 
values of K and L different from zero only if the determinant 
of the coefficients vanishes. This condition is 



(40) 



X 2 - (1 + 2A,), 
+ 2X 



- 2X 

X 2 - (1 - A,) 



0. 



This equation is the condition upon X that equations (38) may be 
a solution of (37). There are four roots of this biquadratic, each 



302 PARTICULAR VALUES OF THE CONSTANTS. [162 

giving a particular solution, and the general solution is the sum 
of the four particular solutions multiplied by arbitrary constants; 
that is, if the four roots of (40) are Xi, X 2 , X 3 , X 4 , the general solu- 
tion is 

I Ju J\~\\s | J-\- 2^ T~ -**- 3^ *" 1 ./V 4& 9 



where the K 3 - are the arbitrary constants of integration, and the 
LJ are denned in terms of them respectively by either of the 
equations (39). The X, depend of course upon the subscript i on 
A, but the notation need not be burdened with this fact since the 
equations all have the same form whether i is 1, 2, or 3. 

It remains to determine the character of the roots of the bi- 
quadratic (40). It follows from (34) and (21), (22), and (24) 
respectively that 



(42) 



- 1 "A* - M _ 4_o . S^V 
~(l+r 2 )^ + r 2 3~ Z 3 \3y 



4 -o^+^ = 4 + 2 - 3 (i V 

1 -M M 

A3 -(l-^+ (2^7)5- l ,** 



It follows from (42) that, for small values of M> the term of (40) 
which is independent of X satisfies the inequality 

1 + A.--2A; 2 <0, (i = 1, 2, 3); 

and, indeed, this relation is true for values of M up to the limit J, 
as can be verified easily.* Therefore the biquadratic has two real 
roots which are equal in numerical value and opposite in sign, and 
two conjugate pure imaginaries. It follows from the definitions 
given that the motion is unstable. If the infinitesimal body were 
displaced a very little from the points of solution it would in 
general depart to a comparatively great distance. 

162. Particular Values of the Constants of Integration. The 
constants of integration will now be expressed in terms of the 
initial conditions, and it will be shown that the latter can be 
selected so that the motion will be periodic. 

Suppose Xi and X 2 are the real roots of equation (40); then 
^i = X 2 . The imaginary roots are 

* H. C. Plummer gave a general proof in Monthly Not. of Roy. Astr. Soc., 
vol. LXII. (1901). 



162] 



PARTICULAR VALUES OF THE CONSTANTS. 



303 




where a is a real number. The Lj are expressed in terms of the 
KJ by equations (39), and are 



(43) 



[V - (1 + 2AJ] 

" -- 



' = 1,2, 3; 
1,2, 3, 



Since the X,- are equal in numerical value but opposite in sign in 
pairs, and the last two are imaginary, it follows that 



Ci = C 2 , 



(44) 

Af 1 c, 
where c is a real constant depending on i. 

Let XQ, 2/ ', ~^~ , and -~ be the initial coordinates and com- 
ponents of velocity; then equations (41) give at t = 



-K 2 )+ 



dxj 
dt 



dt 



The values of the constants of integration are found in terms of 
the initial coordinates and components of velocity by solving these 
equations. 

The values of x' and y r increase in general without limit with the 
time, but if the initial conditions are such that KI = K<> = they 
become purely periodic. This case will now be considered. The 
initial coordinates, XQ, y ', will determine K s and K^ by means 

of which ~~ and ~- are defined. Thus 
at dt 



whence 



304 



PARTICULAR VALUES OF THE CONSTANTS. 



[162 




2c 



The equations (41) become 



(45) 



2c 



yo 

= XQ cos at + sin at, 
c 



= coV sin at + 2/0' cos <r. 

The equation of the orbit is found by eliminating t from these 
equations. Solve for cos at and sin at; then square and add, and 
the result, after dividing out common factors, is 



(46) 



1. 



c 2 



This is the equation of an ellipse with the major and minor axes 
lying along the coordinate axes, and with the center at the origin. 
Since X 3 is imaginary it follows from (43) and (44) that c 2 > 1 ; 
therefore the major axis of the ellipse is parallel to the 7/-axis. 
The eccentricity is given by 






which, for large values of c, is. very near unity. The orbits have 
the remarkable property that their eccentricity is independent 
of the initial small displacements, depending only upon the dis- 
tribution of the mass between the finite bodies, and upon the one 
of the three straight line solutions from which they spring. 

It is obvious that this discussion is not completely rigorous 
because the terms of higher degree in the right members of the 
differential equations have been neglected. The linear terms 
alone do not give sufficient conditions for the existence of periodic 
orbits, and consequently when the discussion is thus restricted it 
answers only the question as to the stability of the solution. But 
in the present case periodic orbits actually exist about all three 



163] APPLICATION TO THE GEGENSCHEIN. 305 

points for all < M ^ ^. Some special examples for JJL = ^ were 
found by Darwin in his memoir in Ada Mathematica, vol. 21. 
The complete analysis for these orbits, including the much more 
difficult case in which the finite bodies describe elliptical orbits, 
was given by the author in the Mathematische Annalen, vol. 
LXXIII. (1912), pp. 441-479, and in the Publications of the Carnegie 
Institution of Washington, No. 161, Periodic Orbits, chapters v., 
vi., and vii. 

163. Application to the Gegenschein. If the constants KI 
and KI are zero the infinitesimal body will revolve in an ellipse 
around the point of equilibrium. If these constants are not zero 
but small in numerical value compared to K 3 and K 4 , the motion 
will be nearly in an ellipse for a considerable time, but will eventu- 
ally depart very far from it. It would be possible to have any 
number of infinitesimal bodies revolving around the same point 
without disturbing one another. 

Consider the motion of the earth around the sun. It is in a 
curve which is nearly a circle. One of the straight line solution 
points is exactly opposite to the sun, and if a meteor should pass 
near it with initial conditions approximately such as have been 
defined in the last article it would make one or more circuits around 
this point before pursuing its path into other regions. If a very 
great number were swarming around this point at one time they 
would appear from the earth as a hazy patch of light with its center 
at the anti-sun, and elongated along the ecliptic. This is the 
appearance of the gegenschein which was discovered independently 
by Brorsen, Backhouse, and Barnard in 1855, 1868, and 1875 
respectively. 

The crucial question seems to be whether or not there are enough 
meteors with the approximate initial conditions to explain the 
observed phenomena, but no certain answer can be given. How- 
ever, it is certain that the meteors are exceedingly numerous, as 
many as 8,000,000 striking into the earth's atmosphere daily 
according to H. A. Newton; and it is only reasonable to sup- 
pose that they cause the zodiacal light which is very bright com- 
pared to the gegenschein. The suggestion that this may be the 
cause of the gegenschein was first made by Gylden in the closing 
paragraph of a memoir in the Bulletin Astronomique, vol. i., en- 
titled, Sur un Cas Particulier du Probleme des Trois Corps.* 

* See also a paper by F. R. Moulton in The Astronomical Journal, No. 483. 
21 



306 



APPLICATION OF CRITERION FOR STABILITY 



[164 



164. Application of the Criterion for Stability to the Equilateral 
Triangle Solutions. The particular solutions of the original differ- 
ential equations in this case are r x = 1, r 2 = 1. The equations 
corresponding to (33) are 



bA 



dh f dh , dh f _ , 

x * v z ~ z ' 



and the differential equations up to terms of the second degree are 



(47) 



The last equation is independent of the first two, and its solution is 
z' = d sin t + Cz cos t. 

Therefore the motion parallel to the 2-axis, for small displace- 
ments, is periodic with period 2ir, the same as that of the revo- 
lution of the finite bodies. 
To find the solutions of the first two equations let 



(48) 



fx' = Ke", 
\y' = Le. 



On substituting these expressions in the first two equations of (47) 
and dividing out common factors, it is found that 



(49) 



[X 2 - 



- 2 M ) L = 0, 



2 \ - 



K 



f ]L = 0. 



In order that solutions may be obtained other than x f = 0, y' = 
the determinant of these equations must vanish. That is, 



164] 



TO EQUILATERAL TRIANGLE SOLUTIONS. 



307 



(50) 



X 2 - i, - 2X - 



(1 - 2/*), X 2 - f 



Let Xi, X 2 , X 3 , X 4 be the roots of this biquadratic. Then the 
general solutions of (47) are 



x = 
y > = 



where KI, 1^2, -^3, ^4 are the constants of integration, and LI, L 2 , 
L 3 , L 4 are constants related to them by either of equations (49) . 
It is found from (50) that 



Xi = - X 2 = 



- 1+A/l - 



-M). 



- 1 - Vl - 27^(1 - 



The number /i never exceeds > an< i if 1 27/x(l /z) ^ the 
roots are pure imaginaries in conjugate pairs; if this inequality 
is not fulfilled they are complex quantities. The inequality may 
be written 

1 - 27 M (1 - M) = e, 

where e is a positive quantity whose limit is zero. The solution of 
this equation is 



Since ju represents the mass which is less than one-half the negative 
sign must be taken. At the limit = 0, /i = .0385 . There- 
fore if ju < .0385 the roots of (50) are pure imaginaries and 
the equilateral triangle solutions are stable ; if ju > .0385 the 
roots of (50) are complex and the equilateral triangle solutions 
are unstable. 

XXI. PROBLEMS. 

1. If a comet approaching the sun in a parabola should be disturbed by 
Jupiter so that its orbit remained a parabola while its perihelion distance was 
doubled, what would be the relation between the new inclination and the old? 



Ans. 



COS 



V2 

= - COS 

2 



308 PROBLEMS. 

2. Prove that if a comet's orbit, whose inclination to Jupiter's orbit is 
zero, is changed by the perturbations of Jupiter from a parabola to an ellipse 
the parameter of the orbit is necessarily decreased. Investigate the changes 
in the parameters for changes in the major axes of the other species of conies. 

3. Suppose a comet is moving in an ellipse in the plane of Jupiter's orbit 
and that the perturbing action of Jupiter is inappreciable except for a short 
time when they are near each other. Prove that if the perturbation of Jupiter 
has increased the eccentricity, the period has been increased or decreased 
according as the product of the major semi-axis and the square root of the 
parameter in the original ellipse is greater or less than unity when expressed 
in the canonical units. 

4. A particle placed midway between two equal fixed masses is in equilib- 
rium. Investigate the character of the equilibrium by the method of Art. 161. 

5. Suppose 1 fj, and fj. are the sun and earth respectively; find the period 
of oscillation parallel to the z-axis for an infinitesimal body slightly displaced 
from the xy-plaue near the straight line solution point opposite to the sun 
with respect to the earth as an origin. 

Ans. 183.304 mean solar days. 

6. In the same case, find the period of oscillation in the xy-pl&ne. 
Ans. ISfaft mean solar days. 

1~7 k 

7. Prove that in general for small values of /JL the periods of oscillation 
both parallel to the z-axis and in the xy-plane, are longest for the point opposite 
to n with respect to 1 n as origin; next longest for the point opposite to 
1 n with respect to n as origin; and shortest for the point between 1 n 
and fj,. 

8. Find the eccentricity of the orbit in the xy-plane opposite to the sun in 
the case of the sun and earth. 

9. The differential equations (35) admit the integral 



discuss the meaning of this integral after the manner of articles 154-159. 

10. What can be said regarding the independence of equations (39) after 
the condition has been imposed that the determinant shall vanish? 

11. If the explanation of the gegenschein given in Art. 163 is true what 
should be its maximum parallax in celestial latitude for an observer in lati- 
tude 45? 

Ans. Roughly 15'. (Too small to be observed with certainty in such an 
indefinite object.) 

12. Suppose /z = \ and reduce the problem of finding the motion of the 
infinitesimal body through the origin along the z-axis to elliptic integrals. 



165] 



CONDITIONS FOR CIRCULAR ORBITS. 



309 



CASE OF THREE FINITE BODIES. 

165. Conditions for Circular Orbits. The theorem of Lagrange 
that it is possible to start three finite bodies in such a manner 
that their orbits will be similar ellipses, all described in the same 
time, will be proved in this section. It will be established first 
for the special case in which the orbits are circles. It will be 
assumed that the three bodies are projected in the same plane. 
Take the origin at their center of mass and the ^-plane as the 
plane of motion. Then the differential equations of motion are 
(Art. 143) 

= (i = 1 2 ^] 



(52) 



ldU 



dt 2 ~ 
U = 



The motion of the system is referred to axes rotating with the 
uniform angular velocity n by the substitution 



(53) 



i = Xi cos nt yi sin nt, 
rji = Xi sin nt + yi cos nt. 



(i = 1, 2, 3), 



On making the substitution, and reducing as in Art. 152, it is 
found that 



(54) 



d 2 yi . dxf 1 dU n 

-~ + 2?i -r n 2 t -- - =0 



dt 2 



j 
dt 



If the bodies are moving in circles around the origin with the 
angular velocity n } their coordinates with respect to the rotating 
axes are constants. Since the first and second derivatives are 
then zero, equations (54) become 



(55) 



I, 2 



L, 2 



I, 3 



2, 3 



2, 3 



310 



EQUILATERAL TRIANGLE SOLUTIONS. 



[166 



(55) 



+ 



+ 



^2, 



= 0, 
= 0, 
= 0. 



And conversely, if the masses and initial projections are such 
that these six equations are fulfilled the bodies move in circles 
around the origin with the uniform angular velocity n. 

Since the origin is at the center of mass the coordinates satisfy 



(56) 



+ 



+ 



= 0, 
= 0. 



If the first equation of (55) is multiplied by mi, the second by ra 2 , 
and the products added, the sum becomes, as a consequence of 
the first equation of (56), the third of (55). In a similar manner 
the last equation of (55) can be derived from the others in y and the 
last of (56). Therefore the third and sixth equations of (55) can 
be suppressed, and equations (56) used in place of them, giving a 
somewhat simpler system of equations. 

The units of time, space, and mass are so far arbitrary. It is 
possible, without loss of generality, to select them so that ri, 2 = 1 
and k 2 = 1. Then necessary and sufficient conditions for the 
existence of solutions in which the orbits are circles are 



(57) 




= 0. 



166. Equilateral Triangle Solutions. There is a solution of the 
problem for every set of real values of the variables satisfying 
equations (57). It is easy to show that the equations are fulfilled 



167] 



STRAIGHT LINE SOLUTIONS. 



311 



if the bodies lie at the vertices of an equilateral triangle. 
PI, 2 = TZ, 3 = ri, 3 = 1, and equations (57) become 

-f- m 3 # 3 = 0, 



Then 



(m 2 
(mi 

(m 2 



m 3 



m 3 - 



m 3 n 2 )y 



= 0, 
= 0, 

+ m 2 ?/ 2 + m 3 ?/ 3 = 0, 
- m 2 ?/ 2 - m s y 3 = 0, 
0. 



These equations are linear and homogeneous in x\, Xz, - , 2/3. 
In order that they may have a solution different from Xi = x z 
= - =2/3 = 0, which is incompatible with 7*1, 2 = r 2 , 3 = ri, 3 = 1, 
the determinant of their coefficients must vanish. On letting 
M = mi -\- mz -\- m 3 , it is easily found that this condition is 

m 3 2 (M - w 2 ) 4 = 0, 

from which ri 2 = M. Then two of the z and two of the y< are 
arbitrary, and hence the equations have a solution compatible 
with rt, / = 1. Therefore, the equilateral triangular configuration 
with proper initial components of velocity is a particular solution of 
the Problem of Three Bodies; and, if the units are such that the 
mutual distances and k 2 are unity, the square of the angular velocity 
of revolution is equal to the sum of the masses of the three bodies. 

167. Straight Line Solutions. The last three equations of (57) 
are fulfilled by y\ y z = y 3 = 0, that is, if the bodies are all on the 
x-axis. Suppose they lie in the order mi, m 2 , m 3 from the negative 
end of the axis toward the positive. Then x 3 > x 2 > x\ and 
TI, 2 = Xz x\ = 1, and the first three equations of (57) become 



(58) 



= 0, 



m 3 



1 (, - zi - I) 2 
On eliminating z 3 and n 2 , it is found that 

(59) m 2 -|- (mi + m z )xi + 7 




= 0. 



1 (Mzi + m 2 ) 2 
If this equation is cleared of fractions a quintic equation in x\ is 



312 DYNAMICAL PROPEETIES OF SOLUTIONS. [168 

obtained whose coefficients are all positive. Therefore there is 
no real positive root but there is at least one real negative root, 
and consequently at least one solution of the problem. 

Instead of adopting xi as the unknown, # 3 z 2 , which will be 
denoted by A, may be used. The distance Xi must be expressed 
in terms of this new variable. The relations among x\, x 2 , x z , 
and A are 

I rv 

Xz Xi = 1, 

_ _ A . 

3/3 "^ *C 2 ~~ ^i , 

whence 



M 

On substituting this expression for Xi in (59), clearing of fractions, 
and dividing out common factors, the condition for the collinear 
solutions becomes 

(mi + m 2 )A 5 + (3mi + 2m 2 )A 4 + (3mi - 



(60) 

- (ra 2 + 3m 3 )A 2 - (2m 2 + 3w 3 )A - (m 2 + m a ) = 0. 

This is precisely Lagrange's quintic equation in A,* and has but 
one real positive root since the coefficients change sign but once. 
The only A valid in the problem for the chosen order of the masses 
is positive; hence the solution of (60) is unique and defines the 
distribution of the bodies in the straight line solution of the 
Problem of Three Bodies. It is evident that two more distinct 
straight line solutions will be obtained by cyclically permuting 
the order of the three bodies. 

168. Dynamical Properties of the Solutions. Since the bodies 
revolve in circles with uniform angular velocity around the center 
of mass, the law of areas holds for each body separately; therefore 
the resultant of all the forces acting upon each body is constantly 
directed toward the center of mass (Art. 48). 

Let the distances of mi, w 2 , and ra 3 from their center of mass 
be ai, a 2 , and a 3 respectively. Then the centrifugal acceleration 

V 2 
to which m t - is subject i& = - , where V t - is the linear velocity 

Q/i 

of m. But this may be written on = tfai. The centripetal force 

* See Lagrange's Collected Works, vol. vi., p. 277, and Tisserand's Mec. Cel., 
vol. i.) p. 155. 



169] 



GENERAL CONIC SECTION SOLUTION. 



313 



exactly balances the centrifugal; therefore the acceleration toward 
the center of mass is 

cti = n 2 di', 

that is, the accelerations of the various bodies toward their common 
center of mass are directly proportional to their respective distances 
from this point. 

169. General Conic Section Solutions. The solutions of the 
problem of three bodies which have been discussed are char- 
acterized by the fact that their orbits are circles. It will be shown 
that corresponding to each of them there is a solution in which 
the orbits are conic sections of arbitrary eccentricity. These 
solutions are characterized by the fact that in them the ratios of 
the mutual distances of the bodies are constant, though the dis- 
tances themselves are variable. 

The differential equations of motion when the system is referred 
to fixed axes with the origin at the center of gravity of the system 
are 



(61) 



dP 







dt 2 
~W 
W 
~W 

w 



-m) 



-u) 



-10 



2,3 



Wi(r? 3 171) 



Suppose the coordinates of m\, m 2 , and w 3 at i = are respec- 
tively (a?i, 2/1), (a? 2 , 2/2), and (x 3 , 2/3), and let the respective distances 
from the origin be ri ( % r 2 (0) , and r 3 (0) . Suppose the angles that 
ri (0) , r 2 (0) , and r 3 (0) make with the ^-axis are <pi, <p 2 , and <p 3 . Then 



(62) 



<- 

[yi = 



= ri (0) cos 



r 2 (0) cos (pz, 



ri (0) sin 



= r 2 (0) sin 



= r* 3 (0) cos 
> (0 > sin 



2/3 = r 3 ^ 

Now let the coordinates of the bodies at any time t be (1, 771), 
(2, *? 2 ), and ( 3 , rjs). Suppose the ratios of the mutual distances 



314 GENERAL CONIC SECTION SOLUTIONS. 

are constants; then the mutual distances at t are 



[169 



, 3, 



where p is the factor of proportionality. Since the shape of the 
figure formed by the three bodies is unaltered, it follows that 

(63) ri = ri (0) p, 



= r 2 (0) p, 




Fig. 42. 

Moreover, the radii n, r 2 , and r 3 will have turned through the same 
angle 6. Hence 



(64) < 



1 = n (0) P cos (B + 

??i = ri (0) p sin (0 + 

2 = r 2 (0) p cos (0 + 

?? 2 = r 2 (0) p sin (0 + 

^3 = r 3 (0) p cos (0 + 

773 = r 3 (0) p sin (0 + 



= (zi cos 6 - 2/1 sin 0)p, 

= (0:1 sin + 2/1 cos 0)p, 

= (x 2 cos 6 2/2 sin 0)p, 

= (a; 2 sin + 2/2 cos 0)p, 

= (x 3 cos 2/3 sin 0)p, 

= (z 3 sin + 2/3 cos 0)p. 



If equations (61) are transformed by means of (64) they will 
involve only the two dependent variables p and 0, and they will 
be necessary conditions for the existence of solutions in which the 
ratios of the mutual distances are constants. It follows from 
the first two equations of (61) and (64) after multiplying the results 



169] 



GENERAL CONIC SECTION SOLUTIONS. 



315 



of the transformation by cos 6 and sin and adding, and then by 
sin and cos and adding, that 

dd\ 2 d*e 



(65) < 



Let 
(66) 
Then 



f 



. 



de 



de\ 2 



dp dS 
dt dt 



I m z (yi - yd m 3 (j/i - 2/3) 1 1_ 
" I " r\ 2 r\ 3 J P 2 ' 



,dO 



P dt' 



and equations (65) become 



(67) 



dt 2 



_ 
dt p 3 



x\ 



r 3 i, 



2/ip 



1 



~ 3/2) , 
, 2 



r 3 i, 3 
IT 1 " JP 1 ' 



And the equations which are similarly derived from the last four 
equations of (61) and of (65) are 



(68)^ 



__ 
dt 2 x 2 p dt p 3 



r 3 1> 



, 2 



dt 2 



eft p 3 



k{ 
i{ 






. 



+ 



r 3 



2 , 3 



\ 1 
Jp 2 ' 



1,3 



, 



-2/2)U 

~ I ~2 

J, 3 J P 



Equations (67) and (68) are necessary conditions for the exist- 
ence of solutions in which the ratios of the distances of the bodies 
are constants. There are but two variables, p and ^, to be de- 
termined. The first gives the dimensions of the system by means 
of (63) , and the second its orientation by means of (66) . In order 



316 



GENERAL CONIC SECTION SOLUTIONS. 



[169 



that the solutions in question may exist these equations must be 
consistent. In pairs of two they define p and ^ when the initial 
conditions are specified. In order that for given initial con- 
ditions the p and \J/ shall be identical as defined by each of the 
three pairs of differential equations, the coefficients of corre- 
sponding terms in p and ^ must be the same. This can be proved 
by considering the expansion of the solutions as power series in 
t to by the method of Art. 127. In order that the solutions 
shall be the same the coefficients of corresponding powers of 
t t must be identical; and in order that these conditions shall 
be satisfied the coefficients of corresponding terms in the differ- 
ential equations must be identical. Therefore the conditions for 
the consistency of equations (67) and (68) are either 



(69) 



or 



(70) 



dt 



2/3 



= 0, 



and the system of six equations 



(71) 



+ 



+ 



* 



,., 



+ 



2 (2/i - 2/2) 



+ 



= n 2 



i f , 



+ 



+ 



o 

^ 



2, 3 

usi-.rf 



where n 2 is the common constant value of the brackets in the right 
members of (67) and (68). And it follows from equations (71), 
as well as from the original definitions of the Xi and the y iy that 
the center of mass equations 

f m'lXi -f- m 2 z 2 -f ^30:3 = 0, 

I m l y l -f- m z y 2 + m 3 y 3 = 0, 
are fulfilled. 



169] GENERAL CONIC SECTION SOLUTIONS. 317 

Equations (69) are satisfied only if the three bodies are in a 
straight line at i t Q . Since, by hypothesis, the shape of the 
configuration is constant, they always remain in a straight line 
in this case. The position of the axes can be so chosen at t = t 
that 2/1 = 2/2 = 2/3 = and the conditions for the existence of the 
solutions reduce to the first three equations of (71). These 
equations are the same as (55) of Art. 165, and it was shown 
in Art. 167 that they have but three real solutions. 

Suppose equations (69) are satisfied and that the bodies remain 
collinear; therefore the resultant of all the forces to which each 
one is subject is directed constantly toward the center of gravity 
of the system, and consequently the law of areas with respect to 
this point holds. Hence 



where Ci, 0%, and Ca are constants. It follows from (63) that 
p 2 = . ( * , and then from (66) that 
(66), (67), and (68) become in this case 



p 2 = . ( * , and then from (66) that ~ = 0. Hence equations 



(72) 



\l/ = Co = constant, 



_ 
di 



These are the differential equations in polar coordinates for the 
Problem of Two Bodies. Except for differences of notation, they 
are the same as equations (65) of chap. v. Therefore p and 8 
satisfy the conditions of conic section motion under the law of 
gravitation, and it follows from (63) and the definition of 6 that the 
three bodies describe similar conic sections having an arbitrary 
eccentricity. These solutions include the straight line solutions 
in which the orbits are circles as a special case. 

Suppose equations (69) are not satisfied; then the bodies are 
not collinear. But if the bodies are not collinear equation (70) 
must hold in order that equations (67) and (68) may be com- 
patible. It follows from equations (66) and (63) that the law of 
areas with respect to the origin holds for each body separately. 
It was shown in Art. 166 that equations (71) are satisfied if the 



318 PROBLEMS. 

bodies are at the vertices of an equilateral triangle. It is easy to 
show that, unless they are collinear, there is no other solution. 
In the case of the equilateral triangle solution equations (67) and 
(68) also reduce to (72), and the orbits are similar conic sections 
of arbitrary eccentricity. 

XXII. PROBLEMS. 

1. Take as an hypothesis that a solution exists in which the three bodies 
are always collinear. Prove that the law of areas holds for each body sepa- 
rately with respect to the center of mass of the system, with respect to either 
of the other bodies, and with respect to the center of mass of any two of the 
bodies. 

2. Write the conditions that the accelerations to which the bodies are 
subject shall be directed toward their common center of mass and proportional 
to their respective distances. 

Ans. Equations (55). 

3. The resultant of the forces acting on each body always passes through 
a fixed point. Prove that the equilateral triangle configuration is the only 
solution of equations (55) unless the bodies lie in a straight line. 

4. Suppose nil = m 2 = m 3 = 1, and that the bodies move according to 
the equilateral triangular solution. Find the radius of the circle in which a 
particle would revolve around one of them in the period in which they revolve 
around their center of mass. 

Ans. R = 3 *. 

5. Prove that the equilateral triangular circular solutions hold when the 
mutual attractions of the bodies vary as any power of the distance. 

6. Find the number of collinear solutions when the force varies as any 
power of the distance. 

7. Prove that when the force varies inversely as the fifth power one solution 
is that each of the bodies moves in a circle through their center of mass in 
such a way that the three bodies are always at the vertices of an equilateral 
triangle. 

8. Prove that if the three bodies are placed at rest in any one of the con- 
figurations admitting circular solutions, they will fall to their center of mass 
in the same time in straight lines. 

9. Find the distribution of mass among the three bodies for which the time 
of falling to their center of mass will be the least; the greatest. 

10. Prove that if any four masses are placed at the vertices of a regular 
tetrahedron, the resultant of all the forces acting on each body passes through 
the center of mass of the four, and that the magnitudes of the accelerations are 
proportional to the respective distances of the bodies from their center of mass. 

11. Prove that there are no circular solutions in the Problem of Four 
Bodies in which the bodies do not all move in the same plane. 

12. Investigate the stability of the triangle and straight line solutions 
of the Problem of Three Bodies when all of the masses are finite. 



HISTORICAL SKETCH. 319 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The first particular solutions of the Problem of Three Bodies were found 
by Lagrange in his prize memoir, Essai sur le Probleme des Trois Corps, which 
was submitted to the Paris Academy in 1772 (Coll. Works, vol. vi., p. 229, 
Tisserand's Mec. Cel. vol. i., chap. vin.). The solutions which he found are 
precisely those given in the last part of this chapter. His method was to 
divide the problem into two parts; (a) the determination of the mutual dis- 
tances of the bodies, (6) having solved (a), the determination of the plane 
of the triangle in space and the orientation of the triangle in the plane. He 
proved that if the part (a) were solved the part (6) could also be solved. 
To solve (a) it was necessary to derive three differential equations involving 
the three mutual distances alone as dependent variables. He found three 
equations, one of which was of the third order, and the remaining two of the 
second order each, making the whole problem of the seventh order. The reduc- 
tion of the general problem of three bodies by the ten integrals leaves it of the 
eighth order; hence Lagrange's analysis reduced the problem by one unit. He 
found that he could integrate the differential equations completely by assuming 
that the ratios of the mutual distances were constants. The demonstration 
was repeated by Laplace in the Mecanique Celeste, vol. v., p. 310. In I'Expo- 
sition du Systeme du Monde he remarked that if the moon had been given to 
the earth by Providence to illuminate the night, as some have maintained, the 
end sought has been only imperfectly attained; for, if the moon were properly 
started in opposition to the sun it would always remain there relatively, and 
the whole earth would have either the full moon or the sun always in view. 
The demonstration upon which he based his remark was made under the 
assumption that there was no disturbing force. If there were disturbing 
forces the configuration would not be preserved unless the solution were stable, 
which it is not, as was proved by Liouville, Journal de Mathematiques, vol. vn., 
1845. 

A number of memoirs have appeared following more or less closely along 
the lines marked out by Lagrange. Among them may be mentioned one by 
Radau in the Bulletin Astronomique, vol. in., p. 113; by Lindstedt in the 
Annales de VEcole Normale, 3rd series, vol. i., p. 85; by Alle^ret in the Journal 
de Mathematiques, 1875, p. 277; by Bour in the Journal de I'Ecole Poly technique, 
vol. xxxvi.; and by Mathieu in the Journal de Mathematiques, 1876, p. 345. 

Jacobi, without a knowledge of the work of Lagrange, reduced the general 
Problem of Three Bodies to the seventh order in Crelle's Journal, 1843, p. 115 
(Coll. Works, vol. iv., p. 478). It has never been reduced further. 

Concerning the solutions of the problem of more than three bodies in which 
the ratios of the mutual distances are constants a number of papers have 
appeared, among which are one by Lehmann-Filhes in the Astronomische 
Nachrichten, vol. cxxvu., p. 137, one by F. R. Moulton in The Transactions of 
the American Mathematical Society, vol. i,, p. 17, and one by W. R. Longley in 
Bulletin of the American Mathematical Society, vol. xin., p. 324. 

No new periodic solutions of the problem of three bodies were discovered 
after those of Lagrange until Hill developed his Lunar Theory, The American 
Journal of Mathematics, vol. i. (1878). These solutions of Hill are of im- 
mensely greater practical value than those of the Lagrangian type. It should 



320 HISTORICAL SKETCH. 

be stated, however, that they are not strictly periodic solutions of any actual 
case, because a small part of the perturbing action of the sun was neglected. 

The next important advance was made by Poincare in a memoir in the 
Bulletin Astronomique, vol. i., in which he proved that when the masses of two 
of the bodies are small compared to that of the third, there is an infinite 
number of sets of initial conditions for which the motion is periodic. These 
ideas were elaborated and the results extended in a memoir crowned with 
the prize offered by the late King Oscar of Sweden. This memoir appeared 
in Ada Mathematica, vol. xm. The methods employed by Poincar6 are 
incomparably more profound and powerful than any previously used in 
Celestial Mechanics, and mark an epoch in the development of the science. 
The work of Poincare" was recast and extended in many directions, and pub- 
lished in three volumes entitled, Les Methodes Nouvelles de la Mecanique 
Celeste. It is written with admirable directness and clearness, and is given 
in sufficient detail to make so profound a work as easily read as possible. 

An important memoir on Periodic Orbits by Sir George Darwin appeared 
in Acta Mathematica, vol. xxi. (1899). In this investigation it was assumed 
that one of the three masses is infinitesimal and that the finite masses, hav- 
ing the ratio of ten to one, revolve in circles. A large number of periodic 
orbits, belonging to a number of families, were discovered by numerical ex- 
periments. The question of their stability was answered by using essen- 
tially the method employed by Hill in his discussion of the motion of the 
lunar perigee. 

A considerable number of investigations in the domain of periodic orbits, 
employing analytical processes based on the methods of Poincare, have been 
published by F. R. Moulton and his former students Daniel Buchanan, Thomas 
Buck, F. L. Griffin, Wm. R. Longley, and W. D. MacMillan. These papers 
have appeared in the Transactions of the American Mathematical Society, the 
Proceedings of the London Mathematical Society, the Mathematische Annalen, 
and the Proceedings of the Fifth International Congress of Mathematicians. 
Besides containing the analysis for a great variety of periodic orbits, they 
show the existence of infinite sets of closed orbits of ejection which form the 
boundaries between different classes of periodic orbits. These investigations 
are published under the title " Periodic Orbits " as Publication 161 of the 
Carnegie Institution of Washington. 



CHAPTER IX. 

PERTURBATIONS GEOMETRICAL CONSIDERATIONS. 

170. Meaning of Perturbations. It was shown in chapter v. 
that if two spherical bodies move under the influence of their 
mutual attractions each describes a conic section with respect to 
their center of mass as a focus, and that the path of each body 
with respect to the other is a conic. The converse theorem is 
also true; that is, if the law of areas holds and if the orbit of one 
body is a conic with respect to the other as a focus, then if the force 
depends only on the distance it varies inversely as the square of 
the distance (see also Art. 58). If there is a resisting medium, 
or if either of the bodies is oblate, or if there is a third body at- 
tracting the two under consideration, or if there is any force acting 
upon the bodies other than that of the mutual attractions of the 
two spheres, their orbits will cease to be exact conic sections. 
Suppose the coordinates and components of velocity are given at 
a definite instant t Q ; then, if the conditions of the two-body problem 
were precisely fulfilled, the orbits would be definite conies in 
which the bodies would move so as to fulfill the law of areas. 
The differences between the coordinates and the components of 
velocity in the actual orbits and those which the bodies would 
have had if the motion had been undisturbed are the perturbations. 
It is necessary to include the changes in the components of velocity 
as perturbations, for the paths described depend not only upon 
the relative positions of the bodies and the forces to which they 
are subject, but also upon the relative velocities with which they 
are moving. 

Several methods of computing perturbations have been devised 
depending upon the somewhat different points of view which may 
be taken. Of these the two following are the ones most frequently 
used. 

171. Variation of Coordinates. The simplest conception of 
perturbations is that the coordinates are directly perturbed. For 
example, if a planet is subject to the attraction of another planet 
the coordinates and components of velocity of the former at any 
time t differ by definite amounts from what they would have been 

22 321 



322 VARIATION OF THE ELEMENTS. [172 

if the sun had been the only source of attraction, and these differ- 
ences are computed by appropriate devices. No attempt is made 
to get the equations of the curve described, and usually no general 
information as to what will happen in the course of a long time is 
secured. This method is applied only to comets and small planets. 

172. Variation of the Elements. This method is variously 
called the Variation of the Elements, the Variation of Parameters, 
and the Variation of the Constants of Integration. According to 
this conception, a body subject to the law of gravitation is always 
moving in a conic section, but in one which changes at each instant. 
The variable conic is tangent to the actual orbit at every point 




Fig. 43. 

of it; and further, if the body were moving undisturbed in any 
one of the tangent conies it would have the same velocity at the 
point of tangency which it has in the actual orbit at that point. 
This conic is said to osculate with the actual orbit at the point of 
contact. The perturbations are the differences between the ele- 
ments of the orbit on the start, and those of the osculating conic 
at any time. An obvious advantage of this method is that the 
elements change very slowly, since in most of the cases which 
actually arise in the solar system the perturbing forces are small. 
But if the perturbations were very large, as they are in some of 
the multiple star systems, this method would lose its relative 
advantages. 



173] DERIVATION OF THE ELEMENTS. 323 

The conception of perturbations as being variations of the 
elements arises quite naturally in considering the factors which 
determine the elements of an orbit. It was shown in chap. v. 
that the initial positions of the two bodies and the directions of 
projection determine the plane of the orbit; that the initial posi- 
tions and the velocities of projection determine the length of the 
major axis; and that the initial conditions, including the direction 
of projection and the velocities, determine the eccentricity and 
the line of the apsides. 

Suppose a body m is projected from P , Fig. 43, in the direction 
Qo with the velocity V . Suppose there are no forces acting upon 
it except the attraction of S', then, in accordance with the results 
of the two-body problem, it follows that it will move in a conic 
section Co whose elements are uniquely determined. Suppose that 
when it arrives at PI it becomes subject to an instantaneous 
impulse of intensity /i in the direction PiQi; this position and the 
new velocity and direction of motion determine a new conic Ci in 
which the body will move until it is again disturbed by some 
external force. Suppose it becomes subject, to the impulse / 2 in 
the direction P 2 Q 2 when it arrives at P 2 ; it will move in the new 
conic C 2 . This may be supposed to continue indefinitely. The 
body will be moving in conic sections which change from time to 
time when it is subject to the disturbing impulses. Suppose the 
instantaneous impulses become very small, and that the intervals 
of time between them become shorter and shorter. The general 
characteristics of the motion will remain the same. At the limit 
the impulses become a continually disturbing force, and the orbit 
a conic section which continually changes. 

173. Derivation of the Elements from a Graphical Construction. 

It was shown in Art. 89 that the major semi-axis is given by the 
very simple equation 

(1) V* - 

where V is the initial velocity, & 2 'the Gaussian constant, S + m 
the sum of the masses, r the initial distance of the bodies from 
each other, and a the major semi-axis. Suppose the major semi- 
axis has been computed by (1) ; it will be shown how the remaining 
elements can be found by the aid of very simple geometrical 
constructions. The initial positions of S and m, and the direction 



324 



RESOLUTION OF THE DISTURBING FORCE. 



[174 



of projection of m, determine the position of the plane of the 
orbit, and therefore & and i. 

Suppose m is at the point P at the origin of time, and that it is 
projected in the direction PQ with the velocity V. The sun S is 
at one of the foci. It is known from the properties of conic 
sections that the lines from P to the two foci make equal angles 
with the tangent PQ. Draw the line PR making the same angle 
with the tangent that SP makes. Let ri represent the distance 




from S to P, and r 2 the distance from P to the second focus. 
Therefore r l + r 2 = 2a; or, r 2 = 2a - r lt which defines the 

SiO 
position of Si. Call the mid-point of SSi, 0; then e = --. 

Suppose S& is the line of nodes; then the angle &SA = , and 



CO 



The only element remaining to be found is the time of perihelion 
passage. The angle ASP, counted in the direction of motion, 
is v. The eccentric anomaly is given by the equation (Art. 98) 



(2) 



tan 



E_ /r 

2 \1 



e 



After E has been found the time of perihelion passage, T, is defined 
by the equation (Art. 93) 



(3) 



n(t - T) = E - e sin E. 



174. Resolution of the Disturbing Force. Whatever may be 
the source of the disturbing force it is convenient, in order to find 
its effects upon the elements, to resolve it into three rectangular 
components. It is possible to do this in several ways, each having 



175] DISTURBING EFFECTS OF ORTHOGONAL COMPONENT. 325 

advantages for particular purposes. The one will be adopted 
here which on the whole leads most simply to the determination 
of the manner in which the elements vary when the body under 
consideration is subject to any disturbing force. It would be 
possible without much difficulty to derive from geometrical con- 
siderations the expressions for the rates of change of the elements 
for any disturbing forces, but the object of this chapter is to 
explain the nature and causes of perturbations of various sorts, 
and the attention will not be divided by unnecessary digressions 
on methods of computation. This part falls naturally to the 
methods of analysis, which will be given in the next chapter. 

The disturbing force will be resolved into three rectangular 
components: (a) the orthogonal component,* S, which is per- 
pendicular to the plane of the orbit, and which is taken positive 
when directed toward the north pole of the ecliptic; (6) the 
tangential component, T, which is in the line of the tangent, and 
which is taken positive when it acts in the direction of motion; 
and (c) the normal component, 'N, which is perpendicular to the 
tangent, and which is taken positive when directed to the interior 
of the orbit. 

The instantaneous effects of these components upon the various 
elements will be discussed separately; and, unless it is otherwise 
stated, it always must be understood that the results refer to the 
way in which the elements are changing at given instants, and not 
to the cumulative effects of the disturbing forces. Although the 
effects of the different components are considered separately, yet 
when two or more act simultaneously it is sometimes necessary to 
estimate somewhat carefully the magnitude of their separate 
perturbations, in order to determine the character of their joint 
effects. 

I. EFFECTS OF THE COMPONENTS OF THE DISTURBING FORCE. 
175. Disturbing Effects of the Orthogonal Component. In 

order to fix the ideas and abbreviate the language it will be sup- 
posed that the disturbed body is the moon moving around the 
earth. The perturbations arising from the disturbing action of 
the sun are very great and present many features of exceptional 
interest. Besides, this is the case which Newton treated by 
methods essentially the same as those employed here.f The 

* A designation due to Sir John Herschel, Outlines of Astronomy, p. 420. 
t Prindpia, Book i., Section 11, and Book m., Props xxn.-xxxv. 



326 DISTURBING EFFECTS OF ORTHOGONAL COMPONENT. [175 

character of the perturbations arising from positive components 
alone will be investigated; in every case negative components 
change the elements in the opposite way. 

It is at once evident that the orthogonal component will not 
change a, e, T, and co, if co is counted from a fixed line in the plane 
of the orbit. But the co in ordinary use is counted from the 
ascending node of the orbit; hence if the negative of the rate of 
increase of ft be multiplied by cos i the result will be the rate 
of increase of co due to the change in the origin from which it is 
reckoned. Consequently it is sufficient to consider the changes 
in ft and i when discussing the perturbations due to the orthogonal 
component. 




Fig. 45. 

Let AB be in the plane of the ecliptic, PoQo in the plane of the 
undisturbed orbit, and ft and i the corresponding node and 
inclination. Suppose there is an instantaneous impulse PoS 
when the moon is at P ; it will then move in the direction PoP\, 
and the new node and inclination will be fti and i\. It is evident 
at once that i\ > i Q and fti < ft . Suppose a new instantaneous 
impulse PiSi acts when the moon arrives at PI. The new node 
and inclination are ft 2 and i z , and it is evident that i 2 < i\ and 
ft 2 < fti. If Pofti = ftiPi, P S Q = PA, and the velocity of 
the moon at P equals that at PI, then i = i 2 . The total result 
is a regression of the node and an unchanged inclination. 

From the corresponding figure at the descending node it is 
seen that a negative S before node passage and a symmetri- 
cally opposite positive S after node passage will produce the 
same results as those which were found at the ascending node. 
Therefore, a positive S causes the nodes to advance if the moon is 
in the first or second quadrant, and to regress if it is in the third 
or fourth quadrant; and a positive S causes the inclination to 
increase if the moon is in the first or fourth quadrant, and to 
decrease if it is in the second or third quadrant. 



177] EFFECTS OF TANGENTIAL COMPONENT. 327 

The following quantitative results may be noted: The rate of 
change of both & and i is proportional to S. The rate of change 
of & is greater the smaller i\ for i = evidently & is not defined, 
but in this case in such problems as the Lunar Theory S vanishes. 
For a given i the rate of change of & is greater the nearer the point 
at which disturbance occurs is to midway between the two nodes. 
The rate at which i changes is greater the nearer the point at which 
the disturbance occurs is to a node. 

176. Effects of the Tangential Component upon the Major Axis. 
Instead of deriving all the conclusions directly from geometrical 
constructions, it will be better to make use of some of the simple 
equations which have been found in chapter v. If it were desired 
the theorems contained in these equations could be derived from 
geometrical considerations, as was done by Newton in the Prin- 
cipia, but this would involve considerable labor and would add 
nothing to the understanding of the subject. 

The major semi-axis is given in terms of the initial distance and 
the initial velocity by equation (1); viz., 



V 2 = k 2 (E + m) - - - 
\r a 

In an elliptic orbit a is positive ; hence, since a positive T increases 
V 2 and does not instantaneously change r, a positive T increases 
the major semi-axis when the moon is in any part of its orbit. It 
also follows from this equation that a given T is most effective in 
changing a when V has its largest value, or when the moon is at 
the perigee, and that the rate of change is more rapid the larger a. 
Expressed in terms of partial derivatives, the dependence of a 
upon T is given by 

- 2a 2 V dV 

'dT~dV'dT~ k 2 (E + m) ~dT' 

177. Effects of the Tangential Component upon the Line of 
Apsides. The tangential component increases or decreases the 
speed, but does not instantaneously change the direction of 
motion. The focus E is of course not changed, n is unchanged, 
and, according to the results of the last article, a' is increased. 
Since r 2 = 2a r\ while the direction of r 2 remains the same, 
it follows that the focus EI is thrown forward to EI, Fig. 46. The 
line of apsides is rotated forward from AB to A'B'. Hence it is 
easily seen that a positive tangential component causes the line of 



328 



EFFECTS OF TANGENTIAL COMPONENT. 



[178 



apsides to rotate forward during the first half revolution, and back- 
ward during the second half revolution. 

The instantaneous effects are the same for points which are 
symmetrical with respect to the major axis. When the moon is 
at K or L the whole displacement of the second focus is per- 
pendicular to the line of apsides, and at these points the rate of 




Fig. 46. 

rotation of the apsides is a maximum for a given change in the 
major axis. But the major axis is changed most when the moon 
is at perigee; therefore the place at which the line of the apsides 
rotates most rapidly is near K and L and between these points 
and the perigee. The rate of rotation of the line of apsides 
becomes zero when the moon is at perigee or apogee. It should 
be remembered that the whole problem is complicated by the 
fact that the magnitude of T depends upon the distances of both 
moon and sun, and these distances continually vary. 

178. Effects of the Tangential Component upon the Eccentricity. 

Tjl Tjl 

The eccentricity is given by the equation e = -=, Fig. 46. 

When the moon is at the perigee EEi and 2a are increased by the 
same amount. Since EEi is less than 2a the eccentricity is 
increased at this point. When the moon is at apogee 2a is in- 
creased while EEi is decreased equally, hence the eccentricity is 
decreased. Consequently there is some place between perigee 
and apogee where the eccentricity is not changed, and it is easy 
to show that this place is at the end of the minor axis. Let 2Aa 
represent the instantaneous increase in 2a when the moon is at 
C or D, Fig. 47. Then r 2 will be increased by the quantity 2Aa, 

and EEi by A#. If 6 is the angle CEtE, cos e = ^ = 2= e , 



180] 



EFFECTS OF NORMAL COMPONENT. 



329 



and, moreover, AE = 2Aa cos 6 
EE, + AE 



2eAa. Therefore 
2ae + 2eAa 



2a + 2Aa ~ 2a + 2Aa 



e; 



or, the eccentricity is unchanged by the tangential component 
when the moon is at an end of the minor axis of its orbit. 

The changes in the time of perihelion passage depend upon the 
changes in the period and the direction of the major axis, as well 
as on the direct perturbations of the longitude in the orbit. Since 
the period depends upon the major axis alone, whose changes 




have been discussed, the foundations for an investigation of the 
changes in the time of perihelion passage have been laid, except 
in so far as they are direct perturbations in longitude; but further 
inquiry into this subject will be omitted because geometrical 
methods are not well suited to such an investigation, and because 
the time of perihelion passage is an element of little interest in 
the present connection. 

179. Effects of the Normal Component upon the Major Axis. 

It follows from (1) that the major axis depends upon the speed 
at a given point and not upon the direction of motion. Since 
the normal component acts at right angles to the tangent, it 
does not instantaneously change the speed and, therefore, leaves 
the major axis unchanged. 

180. Effects of the Normal Component upon the Line of Apsides. 

Consider the effect of an instantaneous normal component when 
the moon is at P, Fig. 48. Let PT represent the tangent to the orbit. 
The effect of the normal component will be to change it to PT'. 
Since the radii to the two foci make equal angles with the tangent 



330 



EFFECTS OF NORMAL COMPONENT. 



[180 



the radius r 2 will be changed to r 2 '; and, since the normal com- 
ponent does not affect the length of the major axis, r 2 and r/ 
will be of equal length. Consequently, when the moon is in the 
region LAK a positive normal component will rotate the line of 
apsides forward, and when it is in the region KBL, backward. At 




Fig. 48. 

the points K and L the normal component does not change the 
direction of the line of apsides. 

In the applications to the perturbations of the moon it will be 
important to determine the relative effectiveness of a given normal 
force in changing the line of apsides when the moon is at the two 
positions A and B. When the moon is at either of these two 
points the second focus EI is displaced along the line KL. The 
effectiveness of a force in changing the direction of motion of a 
body is inversely proportional to the speed with which it moves; 
but by the law of areas the velocities at A and B are inversely 
proportional to their distances from E. Let EA and E B represent 
the effectiveness of a given force in changing the direction of 
motion at A and B respectively, and let VA and VB represent the 
velocities at the same points. Then 



E A ' E B = V B : VA = a(l - e) 



e). 



The rotation of the line of apsides is directly proportional to 
the displacement of E\ along the line KL. The displacements 
along KL are directly proportional to the products of the lengths 
of the radii from A and B to E\ and the angles through which they 
are rotated. But the angles are proportional to EA and E B , and 
the lengths of the radii to EI to a(l + e) and a(l e}. There- 
fore, letting HA and R B represent the rotation of the line of apsides 
at the two points, it follows that 



181] EFFECTS OF NORMAL COMPONENT. 331 

RA : KB = a(l + e)E A : a(l - e)E B = 1:1; 

or, equal instantaneous normal forces produce equal, but oppositely 
directed, rotations of the line of apsides when the moon is at apogee 
and at perigee. 

Suppose the forces act continuously over small arcs. Since the 
linear velocities are inversely as the radii, the effectiveness, in 
changing the direction of the line of apsides, of a constant force acting 
through a small arc at A is to that of an equal force acting through 
an equal arc at B as a(l e) is to a(l + e). In practice the 
disturbing forces are not instantaneous but act continuously, 
their magnitudes depending upon the positions of the bodies; 
consequently, unless the normal component is smaller at apogee 
than at perigee the average rotation of the line of apsides due to a 
normal component always having the same sign is in the direction 
of the rotation when the moon is at apogee. 

181. Effects of the Normal Component upon the Eccentricity. 

If 2a represents the major axis, the eccentricity is given by 

EE l 

e " -25" 

After the action of the normal component the eccentricity is 



the major axis being unchanged. It is easily seen from Fig. 48 
that a positive normal force decreases the eccentricity during the first 
half revolution and increases it during the second half, EE\ being 
less than EEi in the first case, and greater in the second. The 
instantaneous change in the eccentricity vanishes when the moon 
is at A or B. 

It follows from Fig. 48 that a given change in the direction of r 2 
produces a greater change in the eccentricity when the moon is 
in the second or third quadrant than when the moon is in a 
corresponding part of the first or fourth quadrant. Besides this, 
the moon moves slower the farther it is from the earth, and conse- 
quently a given normal component is more effective in changing 
the direction of motion, and therefore of r- 2 , when the moon is near 
apogee than when it is near perigee. Hence a given normal com- 
ponent causes greater changes in the eccentricity if the moon is near 
apogee than it does if the moon is near perigee. 



332 



TABLE OF RESULTS. 



[182 



182. Table of Results. The various results obtained will be of 
constant use in the applications which follow, and they will be 
most convenient when condensed into a table. The results are 
given for only positive values of the disturbing components; for 
negative components they are the opposite in every case. 




The orthogonal component, S, is positive when directed toward 
the north pole of the ecliptic. 

The tangential component, T, is positive when directed in the 
direction of motion. 

The normal component, N, is positive when directed to the 
interior of the ellipse. 



Component . . . 


S 


T 


N 


Nodes 


Advance in first 








and second quad- 
rants; regress, in 
third and fourth 
quadrants. 








Inclination. . . . 


[ncreases in first 
and fourth quad- 
rants ; decreases 
in second and 
third quadrants. 








Major Axis . . . 





Always increases 





Line of Apsides 


No effect if <a is 
counted from a 
fixed point rather 
than from ft . 


In interval ACS, 
forward ; 
In interval BDA, 
backward 


In interval LAK, 
forward; 
In interval KBL, 
backward 


Eccentricity. . . 





In interval DAC, 
increases; 
In interval CBD, 
decreases 


In interval ACS, 
decreases; 
In interval BDA, 
increases 



184] 



PERTURBATIONS DUE TO OBLATE BODY. 



333 



183. Disturbing Effects of a Resisting Medium. The simplest 
disturbance of elliptic motion is that arising from a resisting 
medium. The only disturbing force is a negative tangential 
component, which has the same magnitude for points symmetri- 
cally situated with respect to the major axis. Therefore, it is 
seen from the Table that: (1) & and i are unchanged; (2) a is 
continually decreased; (3) the line of apsides undergoes periodic 
variations, rotating backward during the first half revolution, 
and rotating forward equally during the second half; (4) the 
eccentricity decreases while the body moves through the interval 
DAC, and increases during the remainder of the revolution. It 
takes the body longer to move through the arc CBD than through 
DAC\ but, on the other hand, if the resistance depends on a high 
power of the velocity, as experiment shows it does for high veloci- 
ties, the change is much greater at perigee than at apogee, and 
the whole effect in a revolution is a decrease in the eccentricity. 
The application of these results to a comet, planet, or satellite 
resisted by meteoric matter, or possibly the ether, is evident. 

184. Perturbations Arising from Oblateness of the Central 
Body. Consider the case of a satellite revolving around an oblate 
planet in the plane of its equator. It was shown in equations 
(30), p. 122, that the attraction under these circumstances is always 
greater than that of a concentric sphere of equal mass, but that 





D 
Fig. 50. 

the two attractions approach equality as the satellite recedes. 
The excess of the attraction of the spheroid over that of an equal 
sphere will be considered as being the disturbing force, which, 
it will be observed, acts in the line of the radius vector and is 
always directed toward the planet. Therefore the normal com- 



334 PERTURBATIONS DUE TO OBLATE BODY. [184 

ponent is always positive, and is equal in value at points which 
are symmetrically situated with respect to the major axis. If the 
eccentricity of the orbit is not large the tangential component is 
relatively small, being negative in the interval ACB, and positive 
in BDA. 

(a) Effect upon the period. This is most easily seen when the 
orbit is a circle. The attraction will be constant and greater 
than it would be if the planet were a sphere. This is equivalent 
to increasing k 2 , the acceleration per unit mass at unit distance; 
therefore it is seen from the equation 

P = 



that for a given orbit the period will be shorter, and for a given 
period the distance greater, than it would be if the planet were a 
sphere. 

(b) Effects upon the elements. On referring to the Table, it is 
seen that: (1) & and i are unchanged; (2) a decreases and in- 
creases equally in a revolution; (3) the line of apsides rotates 
forward during a little more than half a revolution, and that while 
the disturbing force is of greatest intensity ; and (4) the eccentricity 
is changed equally in opposite directions in a whole revolution. 
That is, & and i are absolutely unchanged; a and e undergo periodic 
variations which complete their period in a revolution; and the line 
of apsides oscillates, but advances on the whole. 

The effects will be the greater the more oblate the planet and 
the nearer the satellite. The oblateness of the earth is so small 
that it has very little effect in rotating the moon's line of apsides. 
The most striking example of perturbations of this sort in the 
solar system is in the orbit of the Fifth Satellite of Jupiter. This 
planet is so oblate and the satellite's orbit is so small that its 
line of apsides advances about 900 in a year. 



PROBLEMS. 



335 



XXIII. PROBLEMS. 

1. A body subject to no forces moves in a straight line with uniform speed. 
The elements of this orbit are the constants which define the position of the 
line, viz., the speed, the direction of motion in the line, and the position of 
the body at the time T. Show that they can be expressed in terms of six 
independent constants, and that it is permissible in the problem of two bodies 
to regard one body as always moving with respect to the other in a straight 
line whose position continually changes. Find the expression of these line 
elements in terms of the time in the case of elliptic motion. 

2. Show from general considerations based on problem 1 that the methods 
of the variation of coordinates and the variation of parameters are essentially 
the same, differing only in the variables used in denning the coordinates and 
velocities of the bodies. 

3. Suppose the sun moves through space in the line L, orthogonal to the 
plane II . Take n as the fundamental plane of reference. Let the point 
where the planet Pi passes through the plane n in the direction of the motion 
of the sun be the ascending node, and, beginning at this point, divide the 
orbit into quadrants with respect to the sun as center. Suppose the ether 
and scattered meteoric matter slightly retard the sun and the planets, but 
neglect the retardation arising from the motion of the planets in their orbits 
around the sun. 

(a) If the resistance is proportional to the masses of the respective bodies, 
show that the nodes and inclinations of their orbits are unchanged. 

(6) Let a and R represent the density and radius of the sun, and o-; and Ri 
the corresponding quantities for the planet P. Then, if the resistance is 
proportional to the surfaces of the respective bodies, show that with respect 
to the plane II the inclination and line of nodes undergo the following vari- 
ations: 

(1) If (nRi < aR. 



Quadrant 


1 


2 


3 


4 


Inclination 


decreases 


increases 


increases 


decreases 












Line of nodes 


regresses 


regresses 


advances 


advances 



(2) If info > <rR. 



Quadrant 


1 


2 


3 


4 


Inclination 


increases 


decreases 


decreases 














Line of nodes 


advances 


advances 


regresses 


regresses 



336 PROBLEMS. 

(c) If the orbits were circles the various changes in both cases would 
exactly balance each other in a whole revolution. How must the lines of 
apsides in the two cases lie with respect to the line of nodes in order that, for 
a few revolutions, (1) the inclination shall decrease the fastest, and (2) the 
line of nodes advance the fastest? 

(d) Is it possible to make the relation of the line of apsides to the line 
of nodes such that, for a few revolutions, the inclination shall decrease and 
the line of nodes advance? 

(e) If the line of apsides remains fixed in the plane of the orbit is it possible 
for the line of nodes to rotate indefinitely in one direction? 

4. Suppose the orbit of a comet passes near Jupiter's orbit at one of its 
nodes; under what conditions will the inclination of the orbit of the comet 
be decreased? Show that if the major axis remains constant while the in- 
clination is decreased the eccentricity is increased. (Use Art. 159.) 

5. What is the effect of the gradual accretion of meteoric matter by a 
planet upon the major axis of its orbit? 

6. Consider two viscous bodies revolving around their common center of 
mass, and rotating in the same direction with periods less than their period 
of revolution. They will generate tides in each other which will lag. The 
tidal protuberances of each body will exert a positive tangential and a positive 
normal component on the other, these components being greater the nearer 
the bodies are together. Moreover, the rotation of each body will be retarded 
by the action of the other on its protuberances. Suppose the bodies are 
initially near each other and that their orbits are slightly elliptic; follow out 
the evolution of all of the elements of their orbits. 



185] DISTURBING EFFECTS OF A THIRD BODY. 337 



II. THE LUNAR THEORY. 

185. Geometrical Resolution of the Disturbing Effects of a 
Third Body. The problem of the disturbance by a third body 
is much more difficult than those treated in Arts. 183 and 184, 
because the disturbing force varies in a very complicated manner. 




Fig. 51. 

Suppose the three bodies are S, E, and m, and consider S as 
disturbing the motion of m around E. Two positions of m are 
shown at mi and w 2 , and all the statements which are made apply 
for both subscripts. Let EN represent in magnitude and direction 
the acceleration of S on E. The order of the letters indicates the 
direction of the vector representing the force, and the magnitude 
of the vector depends upon the units employed. In the same 
units let mK represent in direction and amount the acceleration 
of S on m. The vector m\K\ is greater than EN because m\S is 
less than ES, and mzKz is less than EN because w 2 $ is greater 
than ES. By the law of gravitation they are proportional to the 
inverse squares of the respective distances. 

Now resolve mK into two components, mL and mP, such that 
mL shall be equal and parallel to EN. Since mL and EN are equal 
and parallel these components will not disturb the relative posi- 
tions of E and m. Therefore the disturbing acceleration is mP. 

One important result is evident from Fig. 51, viz., that the 
disturbing acceleration is always toward the line joining E and S } 
or toward this line extended beyond E in the direction opposite 
to S when mS is greater than ES. Similar considerations applied 
to movable particles on the surface of the earth show why there 
tends to be a tide both on the side of the earth toward the moon, 
and also on the opposite side. 
23 



338 



DISTURBING EFFECTS OF A THIRD BODY. 



[186 



186. Analytical Resolution of the Disturbing Effects of a Third 
Body. Take a system of rectangular axes with the origin at the 
earth and with the xy-pl&ne as the plane of the ecliptic. Let 
(x, y, z) and (X, Y, 0) be the coordinates of the moon and sun 
respectively referred to this system. Let r, p, and R represent 
the distances Em, mS, and ES respectively. Let F x , F y , and F z 
represent the components of the disturbing acceleration parallel 
to the x, y, and z-axes respectively. It follows from equations 
(24) of chapter vn., p. 272, that in the present notation 




Fig. 52. 



(4) < 



By 



R 3 



In order to get the components of the disturbing acceleration in 
any other directions it is sufficient to project these three com- 
ponents on lines having those directions and to take the respective 
sums. 

Let F r represent the component of the disturbing acceleration 
in the direction of the radius vector r; let F v represent the com- 
ponent in a line perpendicular to r in the plane of motion of m; 
and let Fy represent the component which is perpendicular to 



186] 



DISTURBING EFFECTS OF A THIRD BODY. 



339 



both F r and F v . The component F r will be taken as positive when 
it is directed from E\ the component F v will be taken positive when 
it makes with the direction of motion an angle less than 90; and 
the component F& will be taken positive when it is directed to 
the hemisphere which contains the positive end of the z-axis. 
The expression for F r is 

F r = F x cos (xEm) + F y cos (yEm) + F z cos (zEm). 

The expression for F v can be obtained from this one by replacing 
the angle &Em by &>Em + 90, because r will have the direction 
of the tangent at m after the body has moved forward 90 in its 
orbit. The expression for F N can be conveniently obtained by 
first projecting F x and F y on a line in the xy-pl&ne which is per- 
pendicular to lft, then projecting this result on the line perpen- 
dicular to the plane &>Em, and projecting F z directly on the 
same final line. Let the angle &>Em be represented by u; then 
it is found from Fig. 52 by spherical trigonometry that 

F r = + F x [cos u cos ft sin u sin ft cos i] 
+ F tf [cos u sin ft + sin u cos ft cos i] 
+ F z sin u sin i, 

(5) -j F v = + F x [ sin u cos ft cos u sin ft cos i] 

+ F y [ sin u sin ft + cos u cos ft cos i] 
-\- F z cos u sin i, 
= + F x sin ft sin i F y cos ft sin i + F z cos i. 

Let U represent the angle ft#*S; then, since the sun moves in 
the :n/-plane, 

x = r[cos u cos ft sin u sin ft cos i], 
y = r[cos u sin ft + sin u cos ft cos i], 

z = r sin u sin i\ 

(6) \ 

X = R[cos U cos ft sin U sin ft], 

F = R[cos U sin ft + sin U cos ft], 
Z = 0. 



On substituting the expressions for F T , 
use of (6), and reducing, it is found that 



and F z in (5), making 



340 



DISTURBING EFFECTS OF A THIRD BODY. 



[186 



(7) - 



F r = k*S - 



R cos U cos u 



r 



+ sin U sin u cos i\ -^ \\ , 



cos 17 sin u 
+ sin 7 cos u cos i 
R sin C7 sin t -= =- 



][?-*]} 



The geometry of equations (7) is important for a complete 
understanding of the problem. Consider a system of axes with 
origin at E, one axis directed toward m, another at right angles 
to it and 90 forward in the plane of the orbit of m, and the third 
perpendicular to the other two. Then it follows from the figure 

that the coefficients of k z SR -5 j^ in (7) are respectively the 

cosines of the angles between these axes and the line ES. There- 
fore F v vanishes if the line through E parallel to the perpendicular 
to the radius is also perpendicular to ES, and F N vanishes if m 
is in the plane of the orbit of S. They both vanish also if r = p } 

k 2 S 
and in this case F r becomes simply ^ . 

Let \f/ represent the angle between r and R', then 



(8) 



Therefore the expression for F r becomes 

(9) ^ r 



p 2 = 12 2 + r 2 - 2/^r cos ^, 

2r , r 2 



Consequently F r vanishes, if the terms of higher order are ne- 
glected, when 



(10) 



i; 



+ 3 cos 2\f/ = 0, whence 

= 54 44/ f 125 16', 234 44', 305 16 ; . 

Now consider the problem of finding the tangential and normal 
components of the disturbing acceleration. Let P represent a 



186] 



DISTURBING EFFECTS OF A THIRD BODY. 



341 



general point in the orbit, Fig. 53. Let PT be the tangent at P 
and PN the perpendicular to it. It follows from the elementary 
properties of ellipses that PN bisects the angle between n ancj r 2 . 




Fig. 53. 

Then the tangential and normal components of the disturbing 
acceleration are expressed in terms of F r and F v by 



(11) 



T = + F r sin 6 + F v cos e, 

N = - F r cos + F v sin 0. 



In order to complete the expressions for T and N the factors 
sin 6 and cos must be expressed in terms of v. It follows from 
the geometrical properties of the ellipse and from the triangle 
t that 



1 + e cos v 
+ r 2 = 2a, 

+ r 2 2 - 2rir 2 cos 20 = 4a 2 e 2 . 



When 7*1 and r 2 are eliminated from these three equations, it is 

found that 

e sin v 



. . . 
sin 6 



. 
VI + e 2 + 2e cos 

1 + e cos i; 



Therefore 



(12) 



T = 



S 



e sin # 
Vl + e 2 + 2e cos v 



l + e 2 + 2e cos v 



(1 + e cos t;) 



Vl + e 2 + 2e cos v 

e sin v 

Vl + e 2 + 2e cos v 



F v . 



342 



PERTURBATIONS OF THE NODE. 



[187 



On making use of (7) and the relation u = co + v t the final 
expressions for the tangential and normal components of the 
disturbing acceleration become 



(13) 



T = 



{ e sin v -= 
P 3 



Vl + e 2 + 26 cos 

+ cos U (sin u + e sin ) 

+ sin 7 cos i (cos w + e cos w) \R ^ | , 



Vl + e 2 + 26 cos v 

cos U (cos w + e cos w) 

+ sin U cos i (sin u + e sin co) \R \ -^ ^ | . 

All the circumstances of the variation of T and TV can be inferred 
from these equations. 

187. Perturbations of the Node. By definition, the orthogonal 
component S is identical with F# ; therefore by the last of (7) 



(14) Orthog. Comp. = S = - k 2 SR sin U sin i \ - 3 - -^ . 

The sign of the right member depends upon the signs of sin U and 
~s ~~ #s ' both of which can be either positive or negative. 

In order to determine which sign prevails in the long run so as 
to find whether on the whole there is an advance or retrogression 
of the line of nodes, it is necessary to expand the last factor of (14). 
On making use of the last equation of (8), it is found that 



(15) 



S = p- sin U sin i cos ^ + 

3/b 2 ST 
o?~ sm U sm ^I cos U cos u + sin U sin u cos i] + , 



where the S in the right member represents the mass of the sun. 

The angular velocity of the sun in its orbit is slow compared to 

that of the moon; hence, in order to simplify the discussion, it 

may be supposed to stand still while the moon makes a single 



188] PERTURBATIONS OF THE INCLINATION. 343 

revolution. Since the periods of the moon and sun have no simple 
relation the values of sin U and cos U in the long run will be as 
often decreasing as increasing, and hence the assumption will 
cause no important error. 

Suppose S is broken up into the sum of two parts, Si and $2, 
where 



(16) 



a . . . TT TT 

Si = -- D-f" sm l sm " cos " cos u > 



sin i cos i sin 2 U sin u. 



p 



In order to get the greatest degree of simplicity suppose the orbit 
of the moon is a circle so that r is a constant and u = nt. Suppose 
U has a definite value and consider the effects of Si during a revo- 
lution of the moon, starting with the ascending node. It follows 
from the table of Art. 182 that the effects of Si in the first and 
second quadrants are equal and opposite because cos u has equal 
numerical values and opposite signs in the two quadrants. It 
is the same in the third and fourth quadrants. Therefore Si 
produces only periodic perturbations in the line of nodes. 

Now consider the effects of S& In the first half revolution, 
starting with the node, $2 is negative because sin u is positive 
and all the other factors are positive. In the second half revo- 
lution 82 is positive because sin u is negative. Therefore, it 
follows from the table of Art. 182 that 82 causes a continuous, but 
irregular, regression (except when it is temporarily zero) of the line 
of nodes. The complete motion of the line of nodes is the resultant 
of the periodic oscillations due to Si and the periodic and con- 
tinuous changes produced by S 2 . 

The period of revolution of the moon's line of nodes is about 
nineteen years. Since eclipses of the sun and moon can occur 
only when the sun is near a node of the moon's orbit, the times of 
the year at which they take place are earlier year after year, the 
cycle being completed in about nineteen years. 

188. Perturbations of the Inclination. The expression for the 
orthogonal component is given in (15), which may again be broken 
up into the two parts Si and $ 2 . It follows from the table of Art. 
182 that a positive S increases the inclination in the first and fourth 
quadrants and decreases it in the second and third quadrants. 

Consider the effects of Si. If sin U cos U is positive the effect 



344 PEECESSION OF THE EQUINOXES. NUTATION. [189 

in each quadrant is to decrease the inclination. But this case 
can be paired with that in which sin U cos U is negative and of 
equal numerical value. Since all possible situations can be paired 
in this way, Si produces only periodic changes in the inclination. 

The case of $2 is even simpler than that of Si. Since sin u is 
positive in the first two quadrants, the effect in the second quad- 
rant offsets that in the first. Similarly, the effects in the third 
and fourth quadrants mutually destroy each other. Therefore the 
inclination undergoes only periodic variations. 

Some things have been neglected in this discussion to which 
attention should be called. No account has been taken of the 
eccentricities of the orbits of the moon and earth. When they 
are included the terms do not completely destroy one another in 
the simple fashion which has been described. Moreover, each 
perturbation has been considered independently of all other ones. 
As a matter of fact, each one depends en all the others. For 
example, if the node changes, the effects on the inclination are 
different from what they would otherwise have been, and con- 
versely. It is clear that a very refined analysis is necessary in 
order to get accurate numerical results. But this does not mean 
that common-sense geometrical and physical considerations are 
not of the highest importance, especially in first penetrating 
unexplored fields. 

189. Precession of the Equinoxes. Nutation. Suppose the 
largest sphere possible is cut out of the earth leaving an equatorial 
ring. Every particle in this ring may be considered as being a 
small satellite; then, from the principles explained in Arts. 185 
and 186, the attractions of the moon and sun will exercise dis- 
turbing accelerations upon them which will tend to shift them 
with respect to the spherical core. But the particles of the ring 
are fastened to the solid earth so that it partakes of any dis- 
turbance to which they may be subject. Since their combined 
mass is very small compared to that of the spherical body within 
them, and since the disturbing forces are very slight, the changes 
in the motion of the earth will take place very slowly. 

From the results of the last article it follows that the nodes of 
the orbit of every particle will have a tendency to regress on the 
plane of the disturbing body. The angle between the plane of 
the moon's orbit and that of the ecliptic may be neglected for the 
moment as it is small' compared to the inclination of the earth's 



190] RESOLUTION OF DISTURBING ACCELERATION. 345 

equator. They communicate this tendency to the whole earth 
so that the plane of the earth's equator turns in the retrograde 
direction on the plane of the ecliptic. On the other hand, it follows 
from the symmetry of the figure with respect to the nodes of the 
orbits of the particles of the equatorial ring that there will be no 
change in the inclination of the plane of the equator to that of 
the ecliptic or the moon's orbit. The mass moved is so great, 
and the forces acting are so small, that this retrograde motion, 
called the precession of the equinoxes, amounts to only about 
50".2 annually; or, the plane of the earth's equator makes a revo- 
lution in about 26,000 years. 

The moon is very near to the earth compared to the sun, and the 
orthogonal component arising from its attraction is greater than 
that coming from the sun's attraction. The main regression is, 
therefore, on the moon's orbit, which is inclined to the ecliptic 
about 5 9'. Since the line of the moon's nodes makes a revo- 
lution in about 19 years, the plane with respect to which the 
equator regresses performs a revolution in the same time. This 
produces a slight nodding in the motion of the pole of the equator 
around the pole of the ecliptic, and is called nutation. 

The quantitative agreement between theory and observation of 
the rate of precession proves that the equatorial bulge is solidly 
attached to the remainder of the earth. If the earth were a 
relatively thin solid crust floating on a liquid interior, as was once 
supposed, it would probably slide somewhat on the interior and 
give a more rapid precession. 

190. Resolution of the Disturbing Acceleration in the Plane of 
Motion. It follows from the table of Art. 182 that the orthogonal 
component does not produce perturbations in the major axis, 
longitude of perigee, and eccentricity, except indirectly as it 
shifts the line of nodes from which the longitude of the perigee is 
counted. Consequently an idea of the way these elements are 
perturbed can be obtained even if the inclination, with which the 
orthogonal component vanishes, is supposed to be zero. But it 
must be remembered the results obtained under these restrictions 
are not rigorous because T and N depend on the inclination. But 
the approximation is fully justified because it results in great 
simplifications which aid correspondingly in understanding the 
subject. 

On taking i = equations (13) become 



346 RESOLUTION OF DISTURBING ACCELERATION. [190 



(17) 



T = 



N = 



1 e sin v -= 
P 3 



VI + e 2 + 2e cos 

-fl[sin (t* - CO + esin ( - CO] [?-gi]}i 



Vl + e 2 -h 2e cos v 









Tangential Component. 




When i equals zero ^ = u U, and on using the last equation 
of (8), it is found that 

7-9 Of f 

T = 



(18) 




r . 



Seisin (o> U) cos (u U) 




In the orbit of the moon e is approximately equal to ^ 
consequently a good idea of the numerical magnitudes of T and N 
and the circumstances under which they change sign can be 



191] 



PERTURBATION OF THE MAJOR AXIS. 



347 



obtained by neglecting those terms which have e as a factor. If 
these terms are neglected it is found that T vanishes at u U = 

Q 

- , TT, and -- ; it is negative in the first and third quadrants, and 

& z 

positive in the second and fourth quadrants. Under the same 
circumstances N vanishes at 54 44', 125 16', 234 44', and 
305 16'; it is negative from - 54 44' to + 54 44' and from 
125 16' to 234 44', and is positive from 54 44' to 125 16' and 
from 234 44' to 305 16'. If the terms depending on e and the 

Normal Component. 
m. 




Fig. 55. 

higher terms in the expansion of p" 3 are retained, the points 
where T and N vanish are in general slightly different from those 
which have been found, but the differences are not important in 
a qualitative discussion whose aim is simply to exhibit the general 
characteristics of the results. 

The signs of T and N for the moon in different parts of its orbit 
are shown in Figs. 54 and 55. 

191. Perturbations of the Major Axis. If the perigee were 
at mi or m^ the tangential component, which alone changes a, 
would be equal and of opposite sign at points symmetrically 
situated with respect to the major axis. In this case a would be 
unchanged at the end of a complete revolution. But this con- 
dition of affairs is only realized instantaneously, for the disturbing 
body S is moving in its orbit; yet, in a very large number of revo- 
lutions, when the periods are incommensurable, an equal number 
of equal positive and negative tangential components will have 



348 PERTURBATION OF THE PERIOD. [192 

exerted a disturbing influence. The result is that in the long 
run a is unchanged, although it undergoes periodic variations. 

192. Perturbation of the Period. The normal component is 
not only negative more than half a revolution, but the negative 
values are greater numerically than the positive ones. If the terms 
involving e are neglected, it-Js^seen from the second equation of 
(18) that thejgeatest pSswevalue of N is twice its numerically 
greatest iregSfeTvalue. One effect of the whole result is equiva- 
lent to a diminution, on the average, of the attraction of E for m; 
that is, to a diminution of k 2 , the acceleration at unit distance. 
The relation of the period to the intensity of the attraction and 
the major axis is (Art. 89) 



Hence, for a given distance, P is increased if k is decreased. In 
this manner the sun's disturbing effect upon the orbit of the moon 
increases the length of the month by more than an hour. (Com- 
pare Art. 184 (a).) 

193. The Annual Equation. Since the orbit of the earth is an 
ellipse the distance of the sun undergoes considerable variations. 
The farther the sun is from the earth the feebler are its disturbing 
effects, and in particular, the power of lengthening the month 
considered in the preceding article. Therefore, as the earth moves 
from perihelion to aphelion the disturbance which increases the 
length of the month will become less and less; that is, the length 
of the month will become shorter, or the moon's angular motion 
will be accelerated. While the earth is moving from aphelion to 
perihelion the moon's motion will, for the opposite reason, be 
retarded. This is the Annual Equation amounting to a little 
more than 11', and was discovered from observations by Tycho 
Brahe about 1590. 

194. The Secular Acceleration of the Moon's Mean Motion. 
In the early part of the 18th century Halley found from a com- 
parison of ancient and modern eclipses that the mean motion of 
the moon is gradually increasing. Nearly 100 years later (1787) 
Laplace gave the explanation of it, showing that it is caused by the 
gradual average decrease of the eccentricity of the earth's orbit, 
which has been going on for many thousands of years because of 
perturbations by the other planets, and which will continue for a 
long time yet before it begins to increase. 



194] SECULAR ACCELERATION OF MOON'S MEAN MOTION. 349 

One effect of a change in the eccentricity of the earth's orbit is 
to change the average disturbing power of the sun on the orbit of 
the moon. It will now be shown that if the eccentricity decreases, 
the average disturbing power decreases. 

The effect upon the moon's period is due almost entirely to the 
normal component, because it alone acts nearly along the radius 
of the orbit, and therefore in this discussion consideration of the 
tangential component may be omitted. The average value of 
N in a revolution of the moon, for R and U constant and e placed 
equal to zero, is found from the second equation of (18) to be 



Average N = - ^Sll - 3 cos 2(nt - U)]dt 
(19) 



That is, the normal component of the disturbing acceleration on 
the average is very nearly proportional to the radius of the moon's 
orbit and the inverse third power of the radius of the earth's orbit. 
But if the earth's orbit is eccentric, the result for a whole year 
depends upon the eccentricity. When the nature of the depend- 
ence of the average N upon the eccentricity of the earth's orbit 
has been found, the effect of an increase or decrease in this ec- 
centricity can be determined. 

Let N represent the average N for a year. Then it follows 
from (19) that 

- 



where P is the earth's period of revolution. By the law of areas 
it follows that hdt = R 2 d6; hence equation (20) becomes 

- lWSrr 2n dB 1 k*Sr C 2 " (1 + e' cos 



r 2n dB = 1 k*Sr C 2 
J> R = 2 Ph J> 



2 Ph > R 2 Ph > O '(l - e") 



Pha'(l - e' 2 ) ' 

where a' and e' are the major semi-axis and eccentricity of the sun's 
orbit. But it follows from the problem of two bodies that 

, 9 _/! 

h = k V(l + m)o'(l ~ e' 2 ) i P = , - . 

fc\l + m 



350 THE VARIATION. [195 

Therefore the expression for N becomes 

k 2 Sr 



2a' 3 (l - ") r 

As e f decreases N numerically decreases; therefore, as the eccen- 
tricity of the earth's orbit decreases, the efficiency of the sun in 
decreasing the attraction of the earth for the moon gradually 
decreases, and the mean motion of the moon increases corre- 
spondingly. The changes are so small that the alteration in the 
orbit is almost inappreciable, but in the course of centuries the 
longitude of the moon is sensibly increased. The theoretical 
amount of the acceleration is about 6" in a century. The amount 
derived from a discussion of eclipses varies from 8" to 12" '. It 
has been suggested that tidal retardation, lengthening the day, 
has caused the unexplained part of the apparent change, but the 
subject seems to be open yet to some question. 

The very long periodic variations in the eccentricity of the 
earth's orbit, whose effects upon the motion of the moon have 
just been considered, are due to the perturbations of the other 
planets. Although their masses are so small and they are so 
remote that their direct perturbations of the moon's motion are 
almost insensible, yet they cause this and other important varia- 
tions indirectly through their disturbances of the orbit of the 
earth. This example of indirect action illustrates the great 
intricacy of the problem of the motions of the bodies of the solar 
system, and shows that methods of the greatest refinement must 
be employed in order to derive satisfactory numerical results. 

195. The Variation. There is another important perturbation 
in the motion of the moon which does not depend upon the eccen- 
tricity of its orbit. It was discovered by Tycho Brahe, from 
observation, about 1590. Newton explained the cause of it in the 
Prindpia by a direct and elegant method which elicited the praise 
of Laplace. 

It can be explained most readily by supposing that the undis- 
turbed motion of the moon is in a circle. As has been shown, the 
normal component of the sun's disturbing acceleration is negative 
in the intervals ra 8 Wim 2 and m^m^m^ with maximum values at 
mi and ra 6 . Suppose the undisturbed motion at mi is in a circle; 
that is, that the acceleration due to the attraction of the earth 
exactly balances the centrifugal acceleration. There is no tan- 



195] 



THE VARIATION. 



351 



gential component at this point but a large negative normal com- 
ponent. The result is that the force which tends toward E is 
diminished and the orbit is less curved at this point than the 
circle. Therefore the moon will recede to a greater distance 
from the earth in quadrature than in the circular orbit. At the 
point ms the tangential component is zero, the force which tends 
toward E is increased, and the curvature is greater than in the 
circle. The conditions vary continuously from those at mi to 




Fig. 56. 

those at w 3 in the interval m\m$. The corresponding changes in 
the remainder of the orbit are evident. The whole result is that 
the orbit is lengthened in the direction perpendicular to the line 
from the earth to the sun. If the sun is assumed to be so far dis- 
tant that its disturbing effects in the interval m 3 m 5 W7 are equal 
to those in the interval m 7 mim 3 , the orbit, under proper initial 
conditions, is symmetrical with respect to E as a center, and 
closely resembles an ellipse in form. This change of form of the 
orbit, and the auxiliary changes in the rate at which the radius 
vector sweeps over areas, give rise to an inequality in longitude 
between the mean position and the true position of the moon 
which amounts at times to about 39' 30", and is called the variation. 
The variation has an interesting and important connection 
with the modern methods in the Lunar Theory, which were 
founded by G. W. Hill in his celebrated memoirs in the first volume 
of the American Journal of Mathematics, and in the Acta Mathe- 
matica, vol. vm. A complete account of this method is given in 
Brown's Lunar Theory in the chapter entitled, Method with Reel- 



352 THE PARALLACTIC INEQUALITY. [196 

angular Coordinates. Hill neglected the solar parallax; that is, he 
assumed that the disturbing force is equal in corresponding points 
in conjunction with, and opposition to, the sun. Instead of 
taking an ellipse as a first approximation, he took as an inter- 
mediate orbit that variational orbit which is closed with respect to 
axes rotating with the mean angular velocity of the sun, with a 
synodic period equal to the synodic period of the moon. The 
conception is not only one of great value, but the analysis was 
made by Hill with rare ingenuity and elegance. 

196. The Parallactic Inequality. Since the sun is only a finite 
distance from the earth, its disturbing effects will not be exactly 
the same in points symmetrically situated with respect to the line 
m 3 w 7 , but will be greater on the side m 7 mim 3 . For example, if 
the expansion of p~ 3 in (17) is carried one order farther so as to 

r 2 

include the terms of the second order, that is in ^, the part of N 
which is independent of e is found to be 



(t*- E7)] 

(22) 

- ~ [3 cos (u - U) + 5 cos 3(t* - U)] ---- } . 

When u U = the term of the second order has the same sign 
as the first one, and when u U = ir it has the opposite sign. 
The effect of this term is relatively small because r -5- R = .0025 
nearly. The terms which are of the second order introduce a 
distortion in the variational orbit, which leads to an inequality 
of about 2' 7" in the longitude of the moon compared to the 
theoretical position in the variational orbit. Since it is due to 
the parallax of the sun it has been called the parallactic inequality. 
Laplace remarked that, when it has been determined with very 
great accuracy from a long series of observations, it will furnish a 
satisfactory method of obtaining the distance of the sun. The 
chief practical difficulty is that the troublesome problem of finding 
the relative masses of the earth and moon must be solved before 
the method can be applied.* 

197. The Motion of the Line of Apsides. On account of the 
more complicated manner in which the different components 
affect the motion of the line of apsides, the perturbations of this 

* See Brown's Lunar Theory, p. 127. 



197] 



MOTION OF THE LINE OF APSIDES. 



353 



element present greater difficulties than those heretofore con- 
sidered. Suppose first that the line of apsides coincides with the 
line ESj and that the perigee is at m\. The normal component 
at mi is negative, and therefore (Table, Art. 182) produces a 
retrogression of the line of apsides. On the other hand, when 
the moon is at m& the negative normal component causes the 
line of apsides to advance. It was shown in Art. 180 that the 
effectiveness of a normal component acting while the moon 
describes a short arc at apogee is to that of an equal normal 
component acting while an equal arc is described at perigee as 
a(l + e) is to a(l e). Moreover, the second equation of (18) 
shows that the normal component varies directly as the distance 
of the moon from the earth. Therefore the normal component 
is greater at apogee, and is more effective in proportion to its 
magnitude, than the corresponding acceleration at perigee. The 

Normal Component. 



m 




normal component is positive, though comparatively small, in the 
intervals mmtfn and memymg. These intervals are almost equally 
divided by K and L (Fig. 48) where the effect of the normal com- 
ponent on the line of apsides vanishes. Therefore it follows from 
the Table that the total effect in these intervals is very small. 
Hence when the perigee is at mi the result in a whole revolution is 
to rotate the line of apsides forward through a considerable angle. 
Similar reasoning leads to precisely the same results when the 
perigee is at m 5 . 

When the perigee is at mi the tangential component is equal in 
24 



354 



MOTION OF THE LINE OF APSIDES. 



[197 



numerical value and opposite in sign on opposite sides of the major 
axis. Hence it follows from the Table that the effects are in the 
same direction and equal in magnitude for points symmetrically 
situated on opposite sides of the major axis. But the effects in 

Tangential Component. 




the second and third quadrants are opposite in sign to those in 
the first and fourth quadrants; moreover, they are a little greater 
in the second and third quadrants because then r is greatest and 
the tangential component, by (18), is proportional to r. Hence 
when the perigee is at mi the total effect of the tangential compo- 
nent in a whole revolution is to rotate the apsides forward. Now 
pair this with the case where the perigee is at w 5 , a condition which 
will arise because of the motion of the sun even if the apsides were 
stationary. Under these circumstances the apsides are rotated 
backward, and the rotations in the two cases offset each other. 

Suppose now that the line of apsides is perpendicular to the line 
ES. It is immaterial in this discussion at which end of the line 
the perigee is, but, to fix the ideas, it will be taken at ra 3 . The 
normal component is positive in the interval ra 2 ra 3 W4, and, ac- 
cording to the Table, rotates the line of apsides forward. It is 
also positive in the interval m 6 m 7 m 8 and there rotates the line of 
apsides backward. In the latter case the disturbing acceleration 
is greater, and more effective for its magnitude, so that the whole 
result is a retrogression. The intervals m 8 Wim 2 and m^n^m^ in 
which the normal components are negative, are divided nearly 



198] SECONDARY EFFECTS. 355 

equally by L and K] hence it is seen from the Table that their 
results almost exactly balance each other in a whole revolution. 
Therefore, when the perigee is at m 3 , the result of the normal com- 
ponent on the line of apsides for a whole revolution is a consider- 
able retrogression. 

When the perigee is at m s the tangential component is positive 
in the interval w 3 ra 5 and negative in m 5 ra 7 . From the Table it is 
seen that a positive T in the interval W 3 w 5 w 7 causes the line of 
apsides to rotate forward, and a negative, backward. Since the 
sign of T is opposite in the two nearly equal parts of the interval 
the whole result upon the line of apsides is very small. The result 
is the same in the half revolution w 7 raiW 3 . Thus it is seen that 
the combined effects of the normal and tangential components in a 
whole revolution is to rotate the line of apsides backward when it 
is perpendicular to the line from the earth to the sun. 

It was found that the line of apsides rotates forward when it 
coincides with the line from the earth to the sun. The next 
question to be answered is whether the advance or the retro- 
gression is the greater. It is noticed that the total changes arising 
from the action of the tangential components are the differences 
of nearly equal tendencies, and therefore small. The same may be 
said of the normal components which act in the vicinity of the 
ends of the minor axis of the ellipse. Moreover, in the two 
positions considered they act in opposite directions so that their 
whole result is still smaller. The most important changes arise 
from the normal components which act in the vicinity of the ends 
of the major axis. It follows from the second equation of (18) 
that in the first case, in which the line of apsides advances, they 
are about twice as great as in the second, in which the line of apsides 
regresses. Therefore, the whole change for the two positions of 
the line of apsides is an advance. The results for the positions 
near the two considered will be similar, but less in amount up to 
some intermediate points, where the rotation of the line of apsides 
in a whole revolution of the moon will be zero. From the way in 
which the tangential components change sign (Fig. 58) it is evident 
that these points will be nearer to m 3 and ra? than to mi and ra 5 ; 
therefore the average results for all possible positions of the perigee 
is an advance in the line of apsides. 

198. Secondary Effects. The results thus far have been derived 
as though the sun were stationary. It moves, however, in the 
same direction as the moon. It has been shown that when the 



356 PERTURBATIONS OF THE ECCENTRICITY. [199 

moon is near apogee and the sun near the line of apsides, the 
normal component makes the apsides advance. This advance 
tends to preserve the relation of the orbit with reference to the 
position of the sun, and the advance of the apsides is prolonged and 
increased. On the other hand, when the moon is at perigee and 
the sun near the line of apsides the line of apsides moves back- 
ward; the sun moving one way and the line of apsides the other, 
this particular relation of the sun and the moon's orbit is quickly 
destroyed, and the retrogression is less than it would have been if 
the sun had remained stationary. In a similar manner, for every 
relative position of the line of apsides, the advance is increased 
and the retrogression is decreased. 

There is another important secondary effect which depends 
upon the tangential component and is independent of the motion 
of the sun. As an example, take the case in which the line of 
apsides passes through the sun with the perigee at m\. The 
tangential component in w 3 ra 5 is positive, and, according to the 
Table, rotates the line of apsides forward until the moon arrives 
at apogee. But, as the line of apsides advances, the moon will 
arrive at apogee later, and the effect of this component will be 
increased. When the motion of the sun is also included this 
secondary effect becomes of still greater importance. In this 
manner, perturbation exaggerates perturbation, and it is clear 
what astronomers mean when they say that nearly half the motion 
of the lunar perigee is due to the square of the disturbing force, 
or that it is obtained in a second approximation. 

The theoretical determination of the motion of the moon's line 
of apsides has been one of the most troublesome problems of 
Celestial Mechanics; the secondary effects Escaped Newton when 
he wrote the Principia* and were not explained until Clairaut 
developed his Lunar Theory in 1749. The most successful and 
masterful analysis of the subject yet made is undoubtedly that of 
G. W. Hill, in the Acta Mathematica, vol. vin., which, for the 
terms treated, leaves nothing to be desired. The line of apsides 
of the moon's orbit makes a complete reyolutionin about 9| years. 

199. Perturbations of the Eccentricity. Suppose the line of 
apsides passes through the sun and that the perigee is at mi. 

* In the manuscripts which Newton left, and which are now known as the 
Portsmouth Collection, having been published but recently, a correct explana- 
tion of the motion of the line of apsides is given, and nearly correct numerical 
results are obtained. 



199] PERTURBATIONS OF THE ECCENTRICITY. 357 

From the symmetry of the normal components with respect to 
the line ES and the results given in the Table, it follows that the 
increase and the decrease in the eccentricity in a complete revolu- 
tion due to this component, are exactly equal under these cir- 
cumstances. From the way in which the tangential component 
changes sign, and from the results given in the Table, it follows 
that the changes in the eccentricity, due to this component, also 
exactly balance. Therefore there is no change in the eccentricity 
in a complete revolution of the moon under the conditions pos- 
tulated. In a similar manner the same results are reached when 
the perigee is at ra 5 . 

Suppose the line of apsides has the direction ra 3 w 7 . It can be 
shown as before that neither the normal nor the tangential com- 
ponent makes any permanent change in the eccentricity. 

Now consider the case in which the line of apsides is in some 
intermediate position; for simplicity suppose it is in the line W 2 w 6 
with the perigee at m 2 . Consider simultaneously with this case 
that in which the perigee is at m 6 . First consider only the effects 
of the normal component. It follows from Fig. 57 and the Table 
of Art. 182 that when the perigee is at m 2 and the moon is in the 
region m 2 m 4 , the normal component decreases the eccentricity; 
and when the perigee is at ra 6 , increases the eccentricity. The two 
effects largely destroy each other. But it was shown in Art. 181 
that a given normal component is more effective in changing 
the eccentricity when the moon is near apogee than it is when the 
moon is correspondingly near perigee. Besides this, since N is 
proportional to r, as follows from the second equation of (18), the 
normal component is larger the greater the moon's distance. For 
both of these reasons, while the moon is in the arc ra 2 w 4 the 
increase of the eccentricity with the perigee at w 6 is greater than 
the decrease with the perigee at w 2 . The two cases combined 
give a small second order residual increase in the eccentricity 
which may be represented by + Ai6. Similarly, while the moon 
is in the region ra 4 w 6 the effects of the normal component on the 
eccentricity with the perigee at ra 2 and w 6 are respectively an 
increase and a decrease. On paying heed to the relative positions 
of the apogee, it is seen that the combined effect on the eccentricity 
is a second order residual increase + A 2 e. By analogous dis- 
cussions, the combined effects for the moon in the arcs W 6 w 8 and 
ra 8 ra 2 are the positive second order residuals + A 3 e and + A 4 e. 

The question arises whether the second order residuals are not 



358 PERTURBATIONS OF THE ECCENTRICITY. [199 

in some way destroyed. In order to show that they also vanish 
consider the case in which the line of apsides has a symmetrically 
opposite position with respect to the line ES, that is, the case in 
which the perigee is at w 8 or w 4 . When the perigee is at m 4 and 
the moon in the region ra 2 ra 4 the eccentricity is increased by the 
normal component; when the perigee is at ra 8 , the eccentricity is 
decreased. The decrease in the latter case is greater than the 
increase in the former because when the perigee is at ra 8 the 
region ra 2 w 4 is near the apogee. Therefore the combined effect 
is a second order decrease in the eccentricity; and, since the arc 
W 2 w 4 is not only situated the same relatively with respect to the 
earth and sun but also with respect to the moon's orbit as when 
the line of apsides was the line ra 2 w 6 , it follows that the second 
order decrease in the eccentricity is Aie. It is found similarly 
that when the moon is in the arcs w 4 w 6 , mtfns, and w 8 w 2 the sums 
of the changes of the eccentricity when the perigee is at w 4 and m% 
are respectively A 2 e, A 3 e, and A 4 e. When these second 
order residuals are added to those obtained when the line of 
apsides was the line W 2 ra 6 the result is zero. A corresponding 
discussion leads to the same results for any other position of the 
line of apsides, viz., it can be paired with another which is sym- 
metrically opposite with respect to the line ES so that when the 
perigee is taken in both directions on each line the total effect of 
the normal component on the eccentricity is zero. Therefore the 
normal component in the long run makes no permanent change in the 
eccentricity of the moon's orbit; and a somewhat similar discussion 
establishes the same result for the tangential component. 

The sun does not, however, stand still while the moon makes 
its revolution, and the conditions which have been assumed are 
never exactly fulfilled. Nevertheless, it is useful to show how the 
different configurations, even though changing from instant to 
instant, may be paired. In a very great number of revolutions the 
supplementary configurations will have occurred an equal number 
of times, and the eccentricity will have returned to its original 
value. The period required for this cycle of change depends in the 
first place upon the periods of the sun and the moon; in the second 
place, upon the eccentricity of the sun's orbit (the earth's orbit) ; 
and lastly, upon the manner in which the lines of apsides of the 
sun's and moon's orbits rotate. 

The present system, with abundant geological and biological 
evidence of a very long existence for the earth in at least approxi- 



200] THE EVECTION. 359 

mately its present condition, shows with reasonable certainty that 
the system is nearly stable, if not quite. It is an interesting fact, 
though, that those two elements, the line of nodes and the line of 
apsides, which may change continually in one direction without 
threatening the stability of the system do, on the average, re- 
spectively retrograde and advance forever. 

200. The Evection. It has just been shown that the eccentricity 
does not change in the long run; yet it undergoes periodic varia- 
tions of considerable magnitude which give rise to the largest lunar 
perturbation, known as the evection. At its maximum effect it 
displaces the moon in geocentric longitude through an angle of 
about 1 15' compared to its position in the undisturbed elliptic 
orbit. This variation was discovered by Hipparchus and was 
carefully observed by Ptolemy. 

The perturbations of the elements, and of the eccentricity in 
particular, depend upon two things, the position of the moon in 
its orbit, and the position of the moon with respect to the earth 
and sun. Suppose the moon and sun start in conjunction with 
the perigee at mi. Consider the motion throughout one synodic 
revolution. It follows from the Table of Art. 182 and Figs. 57 
and 58 that the eccentricity is not changing when the moon is at 
m\\ that it is decreasing, or zero, when the moon is at m 2 , Ws, 
and m 4 ; that it is not changing when the moon is at m 5 ; that it is 
increasing, or zero, when the moon is at m 6 , m 7 , and m 8 ; and that 
it ceases to change when the moon has returned to m t again. 
This is true only under the hypothesis that the perigee has re- 
mained at mi throughout the whole revolution; or, in other words, 
that the line of apsides advances as fast as the sun moves in its 
orbit. Now, the actual case is that the sun moves about 8.5 
times as fast as the line of apsides rotates. Since the synodic 
period of the moon is about 29.5 days while the sun moves about 
one degree daily, the moon will be about 26 past its perigee when 
it arrives at m\. What modification in the conclusions does this 
introduce? The normal component is negative and, in this part 
of the orbit, causes an increase in the eccentricity, while the 
tangential makes no change, since it is zero. As the moon pro- 
ceeds past mi the normal component becomes less in numerical 
value, while the tangential component becomes negative and tends 
to decrease the eccentricity. The tendencies of the two com- 
ponents to change the eccentricity in opposite directions balance 
when the moon is at some point between mi and m^ instead of 



360 SECULAR VARIATIONS. [201 

at Wi, after which the eccentricity decreases. There is a corre- 
sponding advance of the point ne.ar ra 5 at which the eccentricity 
ceases to decrease and begins to increase. Similar conclusions 
are reached starting from any other initial configuration. 

The results may be summarized thus: The perturbations of the 
sun decrease the eccentricity of the moon's orbit somewhat more 
than half of a synodical revolution, and then increase it for an 
equal time. These changes in the eccentricity cause deviations 
in the geocentric longitude from the ones given by the elliptic 
theory, which constitute the evection. The appropriate methods 
show that the period of this inequality is about 31.8 days. 

201. Gauss' Method of Computing Secular Variations. It has 
been shown in the preceding articles that some of the elements, 
such as the line of nodes and the line of apsides, vary in one 
direction without limit. This change is not at a uniform rate, for 
in addition to the general variations, there are many short period 
oscillations which are of such magnitude that the element fre- 
quently varies in the opposite direction. When the results are 
put into the symbols of analysis, the general average advance is 
represented by a term proportional to the time, called the secular 
variation, while the deviations from this uniform change are 
represented by a sum of periodic terms having various periods and 
phases. Thus it is seen that the secular variations are caused by a 
sort of average of the disturbing forces when the disturbing and 
disturbed bodies occupy every possible position with respect to 
each other. 

There are other elements, such as the inclination and the 
eccentricity which, though periodic in the long run, vary con- 
tinuously in one direction on the average for many thousands 
,'\ / of years. These changes may be regarded as secular varia- 
tions also, and they likewise result from a sort of average of 
perturbations. 

In 1818 Gauss published a memoir upon the theory of secular 
variations based upon the conceptions just outlined. His method 
has been applied especially in the computation of the secular 
variations of the elements of the planetary orbits. Instead of 
considering the motions of the bodies, Gauss supposed that the 
mass of each planet is spread out in an elliptical ring coinciding 
with its orbit in such a manner that the density at each point is 
inversely as the velocity with which the body moves at that point. 
He then showed how to compute the attraction of one ring upon 



202] LONG PERIOD INEQUALITIES. 361 

the other, and the rate at which their positions and shapes change 
under the influence of these forces. 

The method of Gauss has been the subject of quite a number of 
memoirs. Probably the most useful for practical purposes is by 
G. W. Hill in vol. i. of the Astronomical Papers of the American 
Ephemeris and Nautical Almanac. Hill's formulas have been 
applied by Professor Eric Doolittle with great success, the results 
which he obtained agreeing very closely with those found by 
Leverrier and Newcomb by entirely different methods. 

202. The Long Period Inequalities. In the theories of the 
mutual perturbations of the planets very large terms of long 
periods occur. They arise only when the periods of the two bodies 
considered are nearly commejisjjTahle, and it is easy to discover 
their cause from geometrical considerations. 

Since the most important variation occurs in the mutual per- 
turbations of Jupiter and Saturn the explanation will be adapted 
to that case. Five times the period of Jupiter is a little more 
than twice the period of Saturn. Suppose that the two planets 
are in conjunction at the origin of time on the line IQ. After five 




Fig. 59. 

revolutions of Jupiter and two of Saturn they will be in conjunction 
again on a line l\ very near 1 , but having a little greater longitude. 
This continues indefinitely, each conjunction occurring at a little 
greater longitude than the preceding. Conjunctions occurring 
frequently at about the same points in the orbits cause very large 
perturbations, and the Long Period is the time which it takes the 
point of conjunction to make a complete revolution. In the case 



362 PROBLEMS 

of Jupiter and Saturn it is about 918 years. This inequality, which 
is the greatest in the longitudes of the planets, displacing Jupiter 
21' and Saturn 49', long baffled astronomers in their attempts to 
explain it as a necessary consequence of the law of gravitation. 
Laplace finally made one of his many important contributions to 
Celestial Mechanics by pointing out its true cause, and showing 
that theory and observation agree 4 . 



XXIV. PROBLEMS. 
1. Prove that the locus of the point at which the attractions of the sun 

7? -\/ Q Ti 1 

and earth are equal is a sphere whose radius is ~ ^ , and whose center is 

o ~~~ Jbs 

on the line joining the sun and earth, at the distance ~ ^ from the center 

o -ft 

of the earth opposite to the sun, where S and E represent the mass of the sun 
and earth respectively, and R the distance from the sun to the earth. 

If R = 93,000,000 miles, and | = 330,000, then 

= 161,550 miles, 
= 281 miles. 



S -E 

RE 

S-E 



Since the moon's orbit has a radius of about 240,000 miles, it is always at- 
tracted more by the sun than by the earth. 

2. The moon may be regarded as revolving around the earth and disturbed 
by the sun, or as revolving around the sun and disturbed by the earth. As- 
sume that the moon's orbit is a circle, and find the position at which the 
disturbing effects of the sun will be a maximum; show that the disturbing 
effects due to the earth, regarding the moon as revolving around the sun, are 
a minimum for the same position. 

3. Find' the ratio of the greatest disturbing effect of the sun to the least 
disturbing effect of the earth. 

-4ns. Let R equal the distance from the sun to the earth, p the distance 
from the sun to the moon, and r the distance from the earth to the moon; 
then 

^ = r l &^P* = $ ? ?LJ> = 0114 
D E E p* ~K> -r* E' p 3 ' R + r 

4. Find the ratio of the sun's disturbing force at its maximum value to 
the attraction of the sun, and to the attraction of the earth. 



HISTORICAL SKETCH. 363 

5. Prove in detail the conclusion of Art. 199 that the tangential com- 
ponent produces no secular changes in the eccentricity of the moon's orbit. 

6. Suppose a planet disturbs the motion of another planet which is near to 
the sun. Find the way in which all the elements of the orbit of the inner 
planet are changed for all relative positions of the bodies in then* orbits. 

7. Show that, if the rates of change of the elements are known when the 
planet is in a particular position in its orbit, the intensity and direction of 
the disturbing force can be found. Show that, if it is assumed that the 
distance of the disturbing body from the sun is known, its direction and mass 
can be found. (This is part of the problem solved by Adams and Leverrier 
when they predicted the apparent position of Neptune from the knowledge of 
its perturbations of the motion of Uranus. There are troublesome practical 
difficulties which arise on account of the minuteness of the quantities involved 
which do not appear in the simple statement given here.) 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The first treatment of the Problem of Three Bodies, as well as of Two 
Bodies, was due to Newton. It was given in Book I., Section XL, of the 
Principia, and it was said by Airy to be " the most valuable chapter that 
was ever written on physical science." It contained a somewhat complete 
explanation of the variation, the parallactic inequality, the annual equation, 
the motion of the perigee, the perturbations of the eccentricity, the revolution 
of the nodes, and the perturbations of the inclination. The value of the motion 
of the lunar perigee found by Newton from theory was only half that given 
by observations. In 1872, in certain of Newton's unpublished manuscripts, 
known as the Portsmouth Collection, it was found that Newton had accounted 
for the entire motion of the perigee by including perturbations of the second 
order. (See Art. 198.) This work being unknown to astronomers, the motion 
of the lunar perigee was not otherwise derived from theory until the year 1749, 
when Clairaut found the true explanation, after being on the point of sub- 
stituting for Newton's law of attraction one of the form a = 2 + ^-. Newton 

regarded the Lunar Theory as being very difficult, and he is said to have told 
his friend Halley in despair that it " made his head ache and kept him awake 
so often that he would think of it no more." 

Since the days of Newton the methods of Analysis have succeeded those 
of Geometry, except in elementary explanations of the causes of different 
sorts of perturbations. In the eighteenth century the development of the 
Lunar Theory, and of Celestial Mechanics in general, was almost entirely the 
work of five men: Euler (1707-1783), a Swiss, born at Basle, living at St. 
Petersburg from 1727 to 1747, at Berlin from 1747 to 1766, and at St. Peters- 
burg from 1766 to 1783; Clairaut (1713-1765), born at Paris, and spending 
nearly all his life in his native city; D'Alembert (1717-1783), also a native 
and an inhabitant of Paris; Lagrange (1736-1813), born at Turin, Italy, 
but of French descent, Professor of Mathematics in a military school in Turin 



364 HISTORICAL SKETCH. 

from 1753 to 1766, succeeding Euler at Berlin and spending twenty years 
there, going to Paris and spending the remainder of his life in the French 
capital; and Laplace (1749-1827), son of a French peasant of Beaumont, in 
Normandy, Professor in 1'Ecole Militaire and in 1'Ecole Normale in Paris, 
where he spent most of his life after he was eighteen years of age. The only 
part of their work which will be mentioned here will be that relating to the 
Lunar Theory. The account of investigations in the general planetary theories 
comes more properly in the next chapter. 

There was a general demand for accurate lunar tables in the eighteenth 
century for the use of navigators in determining their positions at sea. This, 
together with the fact that the motions of the moon presented the best test 
of the Newtonian Theory, induced the English Government and a number of 
scientific societies to offer very substantial prizes for lunar tables agreeing 
with observations within certain narrow limits. Euler published some rather 
imperfect lunar tables in 1746. In 1747, Clairaut and d'Alembert presented 
to the Paris Academy on the same day memoirs on the Lunar Theory. Each 
had trouble in explaining the motion of the perigee. As has been stated, 
Clairaut found the source of the difficulty in 1749, and it was also discovered 
by both Euler and d'Alembert a little later. Clairaut won the prize offered 
by the St. Petersburg Academy in 1752 for his Theorie de la Lune. Both he 
and d'Alembert published theories and numerical tables in 1754. They were 
revised and extended later. Euler published a Lunar Theory in 1753, in the 
appendix of which the analytical method of the variation of the elements was 
partially worked out. Tobias Mayer (1723-1762), of Gottingen, compared 
Euler's tables with observations and corrected them so successfully that he 
and Euler were each granted a reward of 3000 by the English Government. 
In 1772 Euler published a second Lunar Theory which possessed many new 
features of great importance. 

Lagrange did little in the Lunar Theory except to point out general methods. 
On the other hand, Laplace gave much attention to this subject, and made 
one of his important contributions to Celestial Mechanics in 1787, when he 
explained the cause of the secular acceleration of the moon's mean motion. 
He also proposed to determine the distance of the sun from the parallactic 
inequality. Laplace's theory is contained in the third volume of his Mecanique 
Celeste. 

Damoiseau (1768-1846) carried out Laplace's method to a high degree of 
approximation in 1824-28, and the tables which he constructed were used 
quite generally until Hansen's tables were constructed in 1857. Plana 
(1781-1869) published a theory in 1832, similar in most respects to that of 
Laplace. An incomplete theory was worked out by Lubbock (1803-1865) in 
1830-4. A great advance along new lines was made by Hansen (1795- 
1874) in 1838, and again in 1862-4. His tables published in 1857 were very 
generally adopted for Nautical Almanacs. De Ponte"coulant (1795-1874) 
published his Theorie Analytique du Systeme du Monde in 1846. The fourth 
volume contains his Lunar Theory worked out in detail. It is in its essentials 
similar to that of Lubbock. A new theory of great mathematical elegance, 
and carried out to a very high degree of approximation, was published by 
Delaunay (1816-1872) in 1860 and 1867. 

A most remarkable new theory based on new conceptions, and developed 



HISTORICAL SKETCH. 365 

by new mathematical methods, was published by G. W. Hill in 1878 in the 
American Journal of Mathematics. The first fundamental idea was to take 
the variational orbit as an approximate solution instead of the ellipse. Ex- 
pressions for the coordinates of the variational orbit were developed with rare 
mathematical skill, and are noteworthy for the rapidity of their convergence. 
A second approximation giving part of the motion of the perigee was published 
in volume vm. of Acta Mathematica. This memoir contained the first solution 
of a linear differential equation having periodic coefficients, and introduced 
into mathematics the infinite determinant. Hill's researches have been 
extended to higher approximations, and completed, by a series of papers 
published by E. W. Brown in the American Journal of Mathematics, vols. 
xiv., xv., and xvn., and in the Monthly Notices of the R.A.S., LII., LIV., and 
LV. As it now stands the work of Brown is numerically the most perfect 
Lunar Theory in existence, and from this point of view leaves little to be 
desired. The motion of the moon's nodes was found by Adams (1819-1892) 
by methods similar to those used by Hill in determining the motion of the 
perigee. 

For the treatment of perturbations from geometrical considerations con- 
sult the Principia, Airy's (1801-1892) Gravitation, and Sir John Herschel's 
(1792-1871) Outlines of Astronomy. For the analytical treatment, aside 
from the original memoirs quoted, one cannot do better than to consult 
Tisserand's Mecanique Celeste, vol. in., and Brown's Lunar Theory. Both 
volumes are most excellent ones in both their contents and clearness of expo- 
sition. Brown's Lunar Theory especially is complete in those points, such 
as the meaning of the constants employed, which are apt to be somewhat 
obscure to one just entering this field. 






CHAPTER X. 

PERTURBATIONS ANALYTICAL METHOD. 



203. Introductory Remarks. The subject of the mutual 
perturbations of the motions of the heavenly bodies has been one 
to which many of the great mathematicians, from Newton's time 
on, have devoted a great deal of attention. It is needless to say 
that the problem is very difficult and that many methods of 
attacking it have been devised. Since the general solutions of 
the problem have not been obtained it has been necessary to treat 
special classes of perturbations by special methods. It has been 
found convenient to divide the cases which arise in the solar system 
into three general classes, (a) the Lunar Theory and satellite 
theories; (6) the mutual perturbations of the planets; and (c) the 
perturbations of comets by planets. The method which will be 
given in this chapter is applicable to the planetary theories, and 
it will be shown in the proper places why it is not applicable to the 
other cases. References were given in the last chapter to treatises 
on the Lunar Theory, especially to those of Tisserand and Brown. 
Some hints will be given in this chapter on the method of com- 
puting the perturbations of comets. 

The chief difficulties which arise in getting an understanding of 
the theories of perturbations come from the large number of 
variables which it is necessary to use, and the very long trans- 
formations which must be made, in order to put the equations in a 
form suitable for numerical computations. It is not possible, 
because of the lack of space, to develop here in detail the explicit 
expressions adapted to computation; and, indeed, it is not desired 
to emphasize this part, for it is much more important to get an 
accurate understanding of the nature of the problem, the^m^tlie- 
matical features of the methods employed, the limitations which 
arlfnecessary, the exact places where approximations are intro- 
duced, if at all, and their character, the origin of the various sorts 
of terms, and the foundations upon which the celebrated theorems 
regarding the stability of the solar system rest. 

There are two general methods of considering perturbations, 
(a) as the variations of the coordinates of the various bodies, 

366 



204] ILLUSTRATIVE EXAMPLE. 367 

and (6) as the variations of the elements of their orbits. These 
two conceptions were explained in the beginning of the preceding 
chapter. Their analytical development was begun by Euler and 
Clairaut and was carried to a high degree of perfection by La- 
grange and Laplace. Yet there were points at which pure as- 
sumptions were made, it having become possible to establish 
completely the legitimacy of the proceedings, under the proper 
restrictions, only during the latter half of the nineteenth century 
by the aid of the work in pure Mathematics of Cauchy, Weier- 
strass, and Poincare*. 

204. Illustrative Example. The mathematical basis for the 
theory of perturbations is often obscured by the large number 
of variables and the complicated formulas which must be used. 
Many of the essential features of the method of computing per- 
turbations can be illustrated by simpler examples which are not 
subject to the complexities of many variables and involved 
formulas. One will be selected in which the physical relations 
are also simple. 

Consider the solution of 




where k 2 , ju, *>, and I are positive constants. If /* and v were zero 
the differential, equation would be that which defines simple 
harmonic motion. It arises in many physical problems, such as 
that of the simple pendulum, and of all classes of musical instru- 
ments. In order to make the interpretation definite, suppose it 
belongs to the problem of the vibrations of a tuning fork. The 
first term in the right member may be interpreted as being due 
to the resistance of the medium in which the tuning fork vibrates. 
It is not asserted, of course, that the resistance to the vibrations 
of a tuning fork varies as the third power of the velocity. An 
odd power is taken so that the differential equation will have the 
same form whether the motion is in the positive or negative direc- 
tion, and the first power is not taken because then the differen- 
tial equation would be linear and could be completely integrated 
in finite terms without any difficulty. 

The left member of equation (1) will be considered as defining 
the undisturbed motion of the tuning fork. The first term on the 
right introduces a perturbation which depends upon the velocity 



368 ILLUSTRATIVE EXAMPLE. [204 

of the tuning fork; the second term on the right introduces a 
perturbation which is independent of the position and velocity 
of the tuning fork. The first is analogous to the mutual per- 
turbations of the planets, which depend upon their relative posi- 
tions; the second is more of the nature of the forces which produce 
the tides, for they are exterior to the earth. The tides are defined 
by equations analogous to (1). 

In order to have equation (1) in the form of the equations which 
arise in the theory of perturbations, let 

(2) x = xi, -j t = x z . 

Then (1) becomes 

(3) 

ttX'2 .in 91 7. 

-T- + k 2 Xi = ju 2 -h v cos It. 
at 

The corresponding differential equations for undisturbed motion 
are 



Equations (4) are easily integrated, and their general solution is 
i = + a cos kt + )8 sin kt, 



(5) 

ka sin kt + k(3 cos kt, 

where a and are the arbitrary constants of integration. In the 
terminology of Celestial Mechanics, a and /3 are the elements of 
the orbit of the tuning fork. 

Now consider the problem of finding the solutions of equations 
(3). Physically speaking, the elements a and /3 must be so varied 
that the equations shall be satisfied for all values of t. Mathe- 
matically considered, equations (5) are relations between the 
original dependent variables x\ and x z , and the new dependent 
variables a and which make it possible to transform the differ- 
ential equations (3) from one set of variables to the other. This 
would be true whether (5) were solutions of (4) or not, but since 
(5) are solutions of (4) and (4) are a part of (3), a number of terms 
drop out after the transformation has been made. On regarding 



204] ILLUSTRATIVE EXAMPLE. 369 

(5) as a set of equations relating the variables x\ and #2 to a and j3, 
and making direct substitution in (3), it is found that 

>s *-j7 + sin f-rr = , 

sin kt~ + cos kt-^ = Ma sin kt cos Atf? -f 7 cos ft. 
at at K 

These equations are linear in -,- and -j- and can be solved for these 

at at 

derivatives because the determinant of their coefficients is unity. 
The solution is 

-77 = fjik 2 [a sin kt cos kt] 3 sin kt j cos ft sin &Z , 

Ctl /C 

(7) ; 

-77 =+ nk 2 [a sin kt (3 cos fc(| 3 cos Atf + T cos ft cos Atf . 
etc /c 

The problem of solving (7) is as difficult as that of solving (3) 
because their right members involve the unknown quantities a 
and |8 in as complicated manner as x\ and x z enter the right mem- 
bers of (3). But suppose // and v are very small; then, since they 
enter as factors in the right members of equations (7), the depen- 
dent variables a and /3 change very slowly. Consequently, for a 
considerable time they will be given with sufficient approximation 
if equations (7) are integrated regarding them as constants in the 
right members. To assist in seeing this mathematically consider 
the simpler equation 

(8) ~ = /ia(l + k cos kt). 



The solution of this equation is 



where C is the constant of integration. If the right member of 
this equation is expanded, the expression for a becomes 

(9) a = C \ I + /*(< + sin kt) +^(t + sin kt)* +]. 

If fjL is very small and if t is not too great the right member of this 
equation is nearly equal to its first two terms. If it were not for 
the term t which is not in the trigonometric function no limitations 
on t would be necessary. But in general such limitations are 
necessary; and in most cases, though not in the present one, they 
are necessary in order to secure convergence of the series. 
25 



370 



ILLUSTRATIVE EXAMPLE. 



[204 



It is observed that the solution (9) is in reality a power series in 
the parameter /*, and the coefficients involve t. If it is desired 
equation (8) can be integrated directly as a power series in /*. 
The process is, in fact, a general one which can be used in solving 
(7), and equations (10), which follow, are the first terms of the 
solution. The conditions of validity of this method of integration 
are given in Art. 207. 

The fact that when n is very small a and may be regarded 
as constants in the right members of (7) for not too great values 
of t can be seen from a physical illustration. Consider the per- 
turbation theory. The changes in the elements of an orbit depend 
upon the elements of the orbits of the mutually disturbing bodies 
and upon the relative positions of the bodies in their orbits. It is 
intuitionally clear that only a slight error in the computation of 
the mutual disturbances of two planets would be committed if 
constant elements were used which differ a little, say a degree in 
the case of angular elements, from the true slowly changing ones. 

If equations (7) are integrated regarding a and /3 as constants 
in the right members, it is found that 



(10) 



a = a - 



(a 



(3a 2 + /3 2 )[cos 2kt - 



- sin 2kt + (a 2 - 



sn 



2k(l + k) 



2k(l - k) 



[cos (I + k)t - 1] 



[cos (I - k)t - 1], 



2kt-l] 



|T sin 2kt 

v 
2k(l + k) 

v 



- /3 2 ) sin 4to J 



2k(l - k) 



sin (I + k)t 



sin (I k)t, 



204] ILLUSTRATIVE EXAMPLE. 371 

where ao and /3 C are the values of a and respectively at t = 0. 
When these values of a and are substituted in (5) the values of 
Xi and #2 are determined approximately for all values of t which 
are not too remote from the initial time. 

Consider equations (10). The right member of each of them 
has a term which contains t only as a simple factor, while elsewhere 
t appears only in the sine and cosine terms. The terms which 
are proportional to t seem to indicate that a and /3 increase or 
decrease indefinitely with the time; but it must be remembered 
that equations (10) are only approximate expressions for a and j8, 
which are useful only for a limited time. It might be that the 
rigorous expressions would contain higher powers of t, and that 
the sums would have bounded values, just as 

t 3 t 6 



is an expression whose numerical value does not exceed unity, 
though a consideration of the first term alone would lead to the 
conclusion that it becomes indefinitely great with t. On the 
other hand the presence of terms which increase proportionally 
to the time may indicate an actual indefinite increase of the 
elements a and . For example, it was found in the preceding 
chapter that the line of nodes and the apsides of the moon's orbit 
respectively regress and advance continually. The terms which 
change proportionally to t i re called secular terms. 

The right members of equations (10) also contain periodic terms 

having the periods -r , ^r , , , , and , _ , . These are known 

as periodic terms. If Z and k are nearly equal the terms which in- 
volve sines or cosines of (lk)t have very long periods, and are called 
long period terms. Sometimes terms arise which are the products 
of t and periodic terms. These mixed terms are called Poisson 
terms because they were encountered by Poisson in the discussion 
of the variations of tfhe major axes of the planetary orbits. If (10) 
are substituted in (5) the resulting expressions for x\ and x 2 contain 
Poisson terms but no secular terms. 

The physical interpretation of equations (10) is simple. The 
elements a and /3 continually decrease because of the secular terms; 
that is, the amplitudes of the oscillations indicated in (5) con- 
tinually diminish. This reduction is entirely due to the resistance 
to the motion as is shown by the fact that these terms contain the 



372 



EQUATIONS IN THE PROBLEM OF THREE BODIES. 



[205 



coefficient /* as a factor. There are terms in x\ and x% of period 
three times and five times the undisturbed period which are also 
due to the resistance. And the periodic disturbing force intro- 
duces in a and terms whose periods depend both on the period 
of the disturbing force and also on the natural period of the tuning 
fork. But it is noticed that the periods of the terms which they 
introduce into the expressions for x\ and x 2 are the period of the 
disturbing force and the natural period of the tuning fork. 

205. Equations in the Problem of Three Bodies. Consider 
the motion of two planets, mi and m 2 , with respect to the sun, S. 
Take the center of the sun as origin and let the coordinates of mi 
be (xi, y\, Zi), and of m 2 , (x z , y Zj z 2 ). Let the distances of mi 
and m 2 from the sun be r\ and r 2 respectively, and let ri, 2 repre- 
sent the distance from mi to m 2 . Then the differential equations 
of motion, as derived in Art. 148, are 



(11) 







, 2 



N Zi Ofii 2 

mi ) ; = m 2 - : 
ri 3 dzi 



, 1 



\ Z 2 (7/1/2 1 

m 2 ) ~ = mi 

r 2 3 6z 2 



K 2 ,. = fc 2 



ri 1 



J- 



The right members of equations (11) are multiplied by the 
factors mi and m 2 which are very small compared to S', therefore 
they will be of slight importance in comparison with the terms 
on the left which come from the attraction of the sun, at least for 
a considerable time. If mi and m 2 are put equal to zero in the 
right members, the first three equations and the second three 



205] 



EQUATIONS IN THE PROBLEM OF THREE BODIES. 



373 



form two sets which are independent of each other, and the 
problem for each set of three equations reduces to that of two 
bodies, and can be completely solved. 

It will be advantageous to reduce the six equations (11) of the 
second order to twelve of the first order. Let 



dx 



then equations (11) become 



(12) 



W-*' = o> 

4 1 -*-- ' 



s-"" 1 = 



% / = (fe. 
# * d* * 



^- + /b 2 (>S + m 1 )^ = 
eft ri 3 



dt 



L + 



dt 



^ + 






+ ^- 



da:i 
dyi 
TzT 



and similar equations in which the subscript is 2. 

If the motions of m\ and m^ were not disturbed by each other 
equations (12) would become 



(13) 






p + k z (S + mi) 2l = 0, 



= 0, 



and an independent system of similar equations in which the 
subscript is 2. Let Q x = i(*i' 2 + y,' 2 + z/ 2 ) - fe (jSf " 
then equations (13) take the form 



(14) 



(ft d^i' 

rf^i _ dQi 
"dT ~ dW 



^i 
(ft 



dz, 



This form of the differential equations is convenient in connection 
with the problem of transforming equations so that the elliptic 



374 



TRANSFORMATION OF VARIABLES. 



[206 



elements become the dependent variables whose values in terms 
of t are required. 

206. Transformation of Variables. In order to avoid confusion 
in the analysis, and to be able to say where and how the approxi- 
mations are introduced, the method of the variation of param- 
eters must be regarded in the first instance as simply a trans- 
formation of variables, which is perfectly legitimate for all values 
of the time for which the equations of transformation are valid. 
From this point of view the whole process is mathematically simple 
and lucid, the only trouble arising from the number of variables 
involved and the complicated relations among them. 

In chapter v. it was shown how to express the coordinates in 
the Problem of Two Bodies in terms of the elements and the 
time. Let 01, , 6 represent the elements of the orbit m\, 
and 0i, , 6 those of m^. Then the equations for the coordinates 
in the Problem of Two Bodies may be written 



(15) 



= 0(oi, 



*2 = 



6 , I), 

oe, 

oe, t), 

06, 0, 

06, 0, 

06, 0, 



= 8(a lt 

= 4>(oi, 



6 , 0, 

"6, 0, 

6, 0, 

06, 0> 

06, 0, 

06, 0- 



A transformation of variables in equations (12) will now be 
made. Let it be forgotten for the moment that equations (15) 
are the solutions of the Problem of Two Bodies, and that the 
ai and are the elements of the two orbits; but let (15) be con- 
sidered as being the equations which transform equations (12) in 
the old variables, Xi, yi, z x , Xi, yi, zi', x z , y*, z 2 , xj, 2/2', z 2 ', into an 
equivalent system in the new variables, ai, -, o 6 , 0i, , 0r>. 
The transformations are effected by computing the derivatives 
occurring in (12) and making direct substitutions. The deriva- 
tives of equations (15) with respect to t are 



(16) 




206] 



TRANSFORMATION OF VARIABLES. 



375 



The direct substitution of (16) in (12) gives 

**1 -. x > + 
dt ^ 



dt 



(17) 1 



and similar equations in z 2 , , z*', and 



These equations 



are linear in the derivatives -^ and can be solved for them, ex- 

pressing them in terms of i, , 6 , 0i, , &, and <, provided 
the determinant of their coefficients is distinct from zero. 

But if equations (15) are the solution of the problem of un- 
disturbed elliptic motion equations (17) are greatly simplified, 
for it is seen from (13) that, when ai, , 0:5 are constant, 

1 xi = for all values of t . The partial derivative -rr , 
at dt 



when 



are regarded as variables, is identical with - 






when they are regarded as constants. Therefore -r-^ x\ = 0; 
and similarly ^- + k 2 (S -f- mi) 3 = 0, and similar equations in 

ut 7*1 

y and z. As a consequence of these relations equations (17) 
reduce to 

A dx\ dai 



(18) 



^ ox\ aoi _ ^ ox\ aa{ _ 

&d*< dt" U> &(~ ~~~ 



yi aa j _ n 

T, V"j 

oai dt 



dt 



dt 

V^- 
^i dai dt 

V* ^5_L ^ = 
t^ 6i ^ 



dx 



and similar equations in the &. These equations are linear in the 



376 TRANSFORMATION OF VARIABLES. [206 

derivatives r-* and can be solved for them unless the determinant 
dt 

of their coefficients is zero. But the determinant of the linear 
system (18) is the Jacobian of the first set of equations (15) with 
respect to i, , 6 , and cannot vanish if these functions are 
independent and give a simple and unique determination of the 
elements.* These functions are independent, and in general they 
give simple and unique values for the elements since they are the 
expressions for the coordinates in the Problem of Two Bodies. 
The problem of determining the elements from the values of the 
coordinates and components of velocity was solved in chap. v. 
If m z = equations (18) are linear and homogeneous, and since 

the determinant is not zero they can be satisfied only by -rf = 0, 

(i = 1, , 6). That is, the elements are constants, which, of 
course, is nothing new. 

On solving equations (18), it is found that 



1, * , 6i |8l> ' ' ' j 06) t}) C' ~ 1) ' ' ' > 6), 

(19) "C 

- ' ' a 0, 8 t) (i = 1 6) 



It will be remembered that in determining the coordinates in 
the Problem of Two Bodies the first step, viz., the computation of 
the mean anomaly, involved the mean motion, defined by the 
equation 

nya8 feVS + m, 0' = 1,2). 
a? 

Since the HJ involve the masses of the planets the right members of 
(15), and consequently of (19), involve m\ and w 2 implicitly. 

In order to justify mathematically the precise method of inte- 
grating equations (19) which is employed by astronomers, some 
remarks are necessary upon m\ and w 2 . In those places where 
they occur implicitly in the functions (pi and ^i they will be 
regarded as fixed numbers; as they appear as factors of the \f/i 
and ^i respectively they will be regarded as parameters in powers 
of which the solutions may be expanded. Such a generalization 
of parameters is clearly permissible because, if a function involves 
a parameter in two different ways, there is no reason why it may 

* See Baltzer's Determinanten, p. 141. 



207] METHOD OF SOLUTION. 377 

not be expanded with respect to the parameter so far as it is 
involved in one way and not with respect to it as it is involved in 
the other. If the function, instead of being given explicitly, is 
denned by a set of differential equations the same things regarding 
the expansions in terms of parameters are true. If the attractions 
of bodies depended on something besides their masses (measured 
by their inertias) and their distances, as for example, on their 
rates of rotation or temperatures, then mi and m 2 so far as they 
enter in the <pi and \f/i implicitly through n\ and n z , where they 
would be defined numerically by their individual mutual attrac- 
tions for the sun, would be different from their values where they 
occur as factors of the pi and i^ t -, for in the latter places they 
would be defined by their attractions for each other. 

Hence, the values of the masses m\ and ra 2 entering implicitly 
in equations (15) and (19) are treated as fixed numbers, given in 
advance, and do not need to be retained explicitly; on the other 
hand, the m\ and w 2 which are factors of the perturbing terms of 
the equations are retained explicitly, being supposed capable of 
taking any values not exceeding certain limits. 

207. Method of Solution. Equations (11) are the general 
differential equations of motion for the Problem of Three Bodies. 
Equations (12) are equally general. No approximations were 
introduced in making the transformation of variables by (15); 
therefore equations (19) are general and rigorous. The difference 
is that if (19) were integrated the elements would be found instead 
of the coordinates as in (11), but as the latter can always be 
found from the former this must be regarded as the solution of the 
problem. 

Instead of interrupting the course of mathematical reasoning by 
working out the explicit forms of (19), it will be preferable to show 
first by what methods they are solved. Explicit mention will be 
made at the appropriate times of all points at which assumptions 
or approximations are made. 

When mi and w 2 are very small compared to $, as they are in 
the solar system, the orbits are very nearly fixed ellipses, and 
therefore a { and /3 t - change very slowly. Consequently if they 
were regarded as constants in the right members of (19) and the 
equations integrated, approximate values of the <* t and the & 
would be obtained for values of t not too remote from the initial 
time. This is the method adopted in the illustrative example 



378 METHOD OF SOLUTION. [207 

of the preceding article, and has been the point of view often 
taken by astronomers, especially in the pioneer days of Celestial 
Mechanics. But any theory which is only approximate, even 
though it is numerically adequate, does not measure up to the 
ideals of science. 

Equations (19) are of the type which Cauchy and Poincare* have 
shown can be integrated as power series in mi and m 2 . Cauchy 
proved that m\, ra 2; and t can all be taken so small that the series 
converge. Poincare* proved the more general theorem* that if 
the orbits in which the bodies are instantaneously moving at the 
initial time do not intersect, then for any finite range of values 
of t, the mi and ra 2 can be taken so small that the solutions 
converge for every value of t in the interval. However, the 
masses cannot be chosen arbitrarily small but are given by 
Nature. Hence the practical importance of the additional the- 
orem that, whatever the values of mi and w 2 , there exists a range 
for t so restricted that the solutions of equations (19) as power 
series in the parameters mi and m z converge for every value of t in 
the range. In general, the larger the values of the parameters 
the more restricted the range. This is, of course, a special case of 
a general theorem respecting the expansion of solutions of differ- 
ential equations of the type to which (19) belong as power series 
in parameters.! 

It follows from the last theorem quoted that, if the range of t is 
not too great, the solutions of equations (19) can be expressed in 
convergent power series in mi and w, of the form 



(20) 

j=0 k=0 

where the superfixes on the ai and ft t simply indicate the order of 
the coefficient. The a^ k) and fr'' k) are functions of the time 
which are to be determined. It has been customary in the theory 
of perturbations to assume without proof that this expansion is 
valid for any desired length of time. As has been stated, it can be 
proved that it is valid for a sufficiently small interval of time; 
but as the method of demonstration gives only a limit within 
which the series certainly converge, and not the longest time 

* Les Methodes Nouvelles de la Mccanique Celeste, vol. I., p. 58. 
fSee Picard's Traite d' Analyse, vol. 11., chap. XL, and vol. HI. 



207] 



METHOD OF SOLUTION. 



379 



during which they converge, and as the limit is almost certainly far 
too small, it has never been computed. It is to be understood, 
therefore, that the method which is just to be explained, is valid 
for a certain interval of time, which in the planetary theories is 
doubtless several hundreds of years. 

On substituting (20) in (19) and developing with respect to 
MI and 7?i2, it is found that 



da t -<'> 



(21) 



dt 



efi 



dt 



dt 



dt 



; 



a,- 



d(f>i 



+ higher powers in m\ and w 2 , 

/7fl.(o,i) /7fl.(i,o) 



dt 



dt 



0) . Q (0 , 0) Q (0 , 0) . A 

; HI > > Mo > v 



+ higher powers in mi and w 2 , (i 1, , 6). 



In the partial derivatives it is to be understood that a t - and jS are 
replaced by t (0> 0) and jS/ - 0) respectively. If wi and m 2 were 
not regarded as fixed numbers in the left members of equations 

\ i _ r\ i _ 

(11), fa, \l/i, -^ , r^, etc., would have to be developed as power 

OCX] Opj 



380 



METHOD OF SOLUTION. 



[207 



series in Wi and ra 2 , thus adding greatly to the complexity of the 
work. 

Within the limits of convergence the coefficients of like powers 
of mi and w 2 on the two sides of the equations are equal. Hence, 
on equating them, it follows that 



(22) 




(23) 



dt 



(i-1, ..-, 6), 



>,0) ... ,^(0,0). /Q, (0,0) 
, , 6 , Pi , 



= 0, 



dt 



(24) 



dt fa 

^ = 0, 



eft 



On integrating equations (22) and substituting the values of 
a f (0 - 0) and fr (0 - 0) thus obtained in (23), the latter are reduced to 
quadratures and can be integrated; on integrating (23) and sub- 



stituting the expressions for a^ - 1 ), a t - (1 ' 0) , 



U, t (1 ' 0) in (24), 



the latter are reduced to quadratures and can be integrated; 
and this process can be continued indefinitely. In this manner 



208] 



CONSTANTS OF INTEGRATION. 



381 



the coefficients of the series (20) can be determined, and the 
values of on and &i can be found to any desired degree of precision 
for values of the time for which the series converge. 

208. Determination of the Constants of Integration. A new 

constant of integration is introduced when equations (22), (23), 
are integrated for each oti (3 '' k \ Pi ( i' k) . These constants will now 
be determined. 

Let the constant which is introduced with the a; - k) be denoted 
by - ai^'V and with the ft '-* , by &<<'*>. Since the first set 
of differential equations have m 2 as a factor in their right members, 
while the second set have mi as a factor, it follows that 
.</,< = , (/,<, y = 0, ->), 
0.(o,*> = 6.(o,*) > (fc = 0, oo). 

Since the a; '-* and j8 t ' fc) are defined by quadratures all the 
constants of integration are simply added to functions of t. That 
is, the oti ( i' fc) and /3i (/i fc) have the form 







</,*> 



. *> 



(0 - a t - 



Therefore equations (20) become 



(25) 



Let the values of on and ft at = be 
Then, at ^ = t Q , equations (25) become 



and /3i (0) respectively. 



Since these equations must be true for all values of mi and m 2 
below certain limits, the coefficients of corresponding powers of 
mi and m 2 in the right and left members are equal; whence 



(26)- 



^.(0,0) = 



a .(/.o> =0 , 

^.(O.t) = 0, 

= 0, 
=0, ( 



1, ...oo), 

1, . . 00), 



382 TERMS OF THE FIRST ORDER. [209 

Since all the terms of the right members of (25) except the first 
vanish at t = to, it follows that a; (0 ' 0) and /3i (0i0) are the osculating 
elements [Art. 172] of the orbits of mi and m z respectively at the 
time t = to, and that the other coefficients of (20) are the definite 
integrals of the differential equations which define them taken 
between the limits t = t Q and t = t. 

209. The Terms of the First Order. The terms of the first 
order with respect to the masses are defined by equations (23). 
Since the terms of order zero are the osculating elements at to, 
the differential equations become 

v (0) . a (0) , a (0) . f\ 

-*6 7 Pi j j P6 j ^jj 

(27) 

~dt 

The right members of these equations are proportional to the rates 
at which the several elements of the orbits of the two planets 
would vary at any time t, if the two planets were moving at that 
instant strictly in the original ellipses. The integrals of (27) are, 
therefore, the sums of the instantaneous effects; or, in other words, 
they are the sums of the changes which would be produced if the 
forces and their instantaneous results were always exactly equal 
to those in the undisturbed orbits. Of course the perturbations 
modify these conditions and produce secondary, tertiary, and 
higher order effects. They are included in the coefficients of 
higher powers of mi and ra 2 in (20). 

The quantities ai (0il) and /3i (li0) are usually called perturbations 
of the first order with respect to the masses. The reason is clearly 
because they are the coefficients of the first powers of the masses 
in the series (20). In the planetary theories it is not necessary to 
go to perturbations of higher orders except in the case of the 
larger planets which are near each other, and then comparatively 
few terms are great enough to be sensible. It is not necessary in 
the present state of the planetary theories to include terms of the 
third order except in the mutual perturbations of Jupiter and Saturn. 

Instead of there being but two planets and the sun there are 
eight planets and the sun, so that the actual theory is not quite 
so simple as that which has been outlined. Yet, as will be shown, 
the increased complexity comes chiefly in the perturbations of 
higher orders. If there were a third planet w 3 whose orbit had 
the elements 71, , 70, equations (23) would become 



210] 



TERMS OF THE SECOND ORDER. 



383 



(28) 



=TF ' 

da ( (0.1.< 

~~dT =<i>< 

/7/v- (o.o.i) 



ftt 

^.(1,0,0) 

~~dT~ 

^ t . (0,1,0) 
^.(0,0,1) _ 



. (o.o.i) 



= 0. 



If there were more planets more equations of the same type 
would be added. Consider the perturbations of the first order of 
the elements of the orbits m\\ they are composed of two distinct 
parts given by the second and third equations of (28), one coming 
from the attraction of ra 2 , and the other from the attraction of m 3 . 
Therefore, the statement of astronomers that the perturbing ef- 
fects of the various planets may be considered separately, is true 
for the perturbations of the first order with respect to the masses. 

210. The Terms of the Second Order. It has been shown that 
a .a.o) = a .(2,o) = ^.(o.i) = 0.(o.2) = o ; therefore it follows from 
(24) that the terms of the second order with respect to the masses 
are determined by the equations 



(29) 



; Q 



dt 



, 

dt f={ 



day 



( 

i ' J 






dt 



a, 



dt 



0) 
' 



384 TERMS OF THE SECOND ORDER. [210 

The perturbations of the first order are those which would result 
if the disturbing forces at every instant were the same as they 
would be if the bodies were moving in the original ellipses. If the 
bodies mi and ra 2 move in curves differing from the original ellipses 
the rates at which the elements change at every instant are dif- 
ferent from the values given by equations (27) . The perturbations 
of the elements of the orbit of mi due to the fact that w 2 departs 
from its original ellipse by perturbations of the first order are 
given by the equations of the type of the first of (29), for, if 
p.d.o) = 0, it follows that a/ 1 - 1 * = also. The perturbations of 
the elements of the orbit of mi due to the fact that mi departs from 
its original ellipse by perturbations of the first order are given by 
the equations of the type of the second of (29), for, if a/ - 1} = 0, 
it follows that i (0 ' 2) = also. The terms ft' 1 - 1 * and /3 t - (2 ' 0) in 
the elements of the orbit of m z arise from similar causes. Thus the 
perturbations of the second order correct the errors in the terms of 
the first order, and those of the third order correct the errors 
in the second, and so on. 

As has been said, the solutions expressed as power series in the 
masses converge if the interval of time is taken not too great. 
In a general way, the smaller the masses of the planets the longer 
the time during which the series converge. In the Lunar Theory 
the sun plays the r61e of the disturbing planet. Since its mass is 
very great compared to that of the central body, the earth, the 
series in powers of the masses as given above would converge for 
only a very short time, probably only a few months instead of 
years. Such a Lunar Theory would be entirely unsatisfactory. 
On this account the perturbations in the Lunar Theory are de- 
veloped in powers of the ratio of the distances of the moon and the 
sun from the earth, and special artifices are employed to avoid 
secular terms in all the elements except the nodes and perigee. 

If there is a third planet the perturbations of the second order 
are considerably more complicated. Let the planets be Wi, w 2 , 
and ra 3 , and consider the perturbations of the second order of the 
elements of the orbit of mi. From purely physical considerations 
it is seen that the following sorts of terms will arise: (a) terms 
arising from the disturbing action of w 2 and w 3 , due respectively 
to the perturbations of the first order of the elements of ra 2 and m s 
by mi; (b) terms arising from the disturbing action of ra 2 and w 3 , 
due to the perturbations of the first order of the elements of the 
orbit of mi by w 2 and w 3 ; (c) terms arising from the disturbing 



210] TERMS OF THE SECOND ORDER. 385 

action of ra 2 , due to the perturbations of the first order of the 
elements of the orbit of mi by ra 3 ; (d) terms arising from the 
disturbing action of w 2 , due to the perturbations of the first order 
of the elements of the orbit of w 2 by w 3 ; (e) terms arising from the 
disturbing action of ra 3 , due to the perturbations of the first order 
of the elements of the orbit of mi by ra 2 ; and (/) terms arising 
from the disturbing action of w 3 , due to the perturbations of the 
first order of the elements of w 3 by w 2 . 

Under the supposition that there are three planets, the terms of 
the second order with respect to the masses are found from equa- 
tions (19) and (20) to be 

Lll^xi.o.o^ 



(30) 



dt f=i . dft 

dT~ = S J ^y~ -^ 



dt 



dt U dctj 

Jo;. (0,1,1) 6 QJ ( (0) . . . (0). 0(0) . . 0(0). f) 

-dT = ^i~ -r ' 



dft 



.0X0,0,1) 



ddj 

^^ y , v ^ i ,-",a 6 ; 71 ,---,76 ; t) (n t _ n) 
i / j i /ii 



and similar equations for -^ and -JT 

at at 

The first two equations give the perturbations of the class (a), 
for, <f>i(a, |8) and #(, 7) are the portions of the perturbative 
function given by w 2 and m 3 respectively, while /3/ (1>0 ' 0) and 
y y d.0,0) are fa e perturbations of the first order of the elements of 
the orbits of w 2 and ra 3 by mi. Similarly, the third and fourth 
equations give the perturbations of the class (&); the first term 
of the fifth equation, those of class (c) ; the second term, of class 
(d) ; the third term, of class (e) ; and the fourth term, of the class (/). 
26 



386 PROBLEMS. 

It appears from this that the terms of the second order cannot be 
computed separately for each of the disturbing planets. 

The types of terms which will arise in the perturbations of the 
third order can be similarly predicted from physical considera- 
tions, and the predictions can be verified by a detailed discus- 
sion of the equations. 



XXV. PROBLEMS. 

1. In equations (3) take the term v cos It to the left member before starting 
the integration, and include it in equations (4). Carry out the whole process 
of integration with this variation in the procedure. 

2. If equations (7) are integrated as power series in /* and v, what types of 
functions of t will arise in the terms of the second order? 

3. Write the equations defining the terms of order zero, one, and two in 
the masses when equations (11) are integrated as series in mi and ra 2 . Show 
that the terms of order zero are the coordinates that m\ and w 2 would have 
if they were particles moving around the sun in ellipses defined by their 
initial conditions. Show that the equations defining the terms of the first 
and higher orders are linear and non-homogeneous, instead of being reduced 
to quadratures as they are after the method of the variation of parameters 
has been used. 

4. Suppose there are four planets, m\, w 2 , w 3 , m' 4 ; write all the terms of 
the second order with respect to the masses according to (30) and interpret 
each. 

5. Suppose there are two planets m\ and w 2 ; write all of the terms of the 
third order with respect to the masses and interpret each. 

6. Suppose mi = ra 2 = w 3 and that the planets are arranged in the order 
mi, m 2 , m a with respect to their distance from the sun. Show that of the 
perturbations defined by equations (30) the most important are those given 
by the first and third equations and the second term of the fifth; that the 
perturbations next in importance are given by the first, third, and fourth 
terms of the fifth equation; and that the least important are given by the 
second and fourth equations. 



212] 



LAGRANGE S BRACKETS. 



387 



211. Choice of Elements. In order to exhibit the manner in 
which the various sorts of terms enter in the perturbations of the 
first_order, it will be necessary to develop equations (19) explicitly. 
This was deferred, on account of the length of the transformations 
which are necessary, until a general view of the mathematical 
principles involved could be given. 

If terms of the first order alone are considered the functions 
<f>i(a, j8) can be considered independently of $i(a, &). Any inde- 
pendent functions of the elements may be used in place of the 
ordinary elements. In fact, one of the elements already employed, 
TT = co + Q>, is the sum of two geometrically simpler elements. 
Now the form of 4>i(a, j8) will depend upon the elements chosen; 
with certain elements they are rather simple, and with others very 
complicated. They will be taken in the first example which 
follows so that those functions shall become as simple as possible. 

212. Lagrange's Brackets. Lagrange has made the following 
transformation which greatly facilitates the computation of (19). 



Multiply (18) by - 



da\ 



3T 5T IT respec- 



tively and add. The result is 



(31) 



daj I dxi dxi _ dxi dxi 
dt [dai daz dai daz 



dyi 
daz 



dai 



dzidzi' dzi'dzi} 
dai daz dai daz J 






das I dxi dxi _ dxi' dxi 1 

dt \ dai das dai day J 



da? f dxi dxi r _ dxi' dxi . ] 
dt \ dai da& dai da& 



* ' * t n 

dai 
Lagrange's brackets [a t -, a/] are defined by 



(32) 



[ a . a ] = dx ^ dx i dxi f dxi dyi dy^ dyi' dyi 
dai da } - dai da } - dai da } - dai da/ 



i daj 



388 



PROPERTIES OF LAGRANGE's BRACKETS. 



[213 



Form the equations corresponding to (31) in a2, , a 6 ; the result- 
ing system of equations is 

, dai dRi, 2 



(33) 



, 



r na; 

K,-]^- 



These equations are equivalent to the system (18) and will be used 
in place of them. 

213. Properties of Lagrange's Brackets. It follows at once 
from the definitions of Lagrange's brackets that 

[" I rv 

[a*, ay] = [ay, aj. 

A more important property is that they do not contain the time 
explicitly; that is, 

d[a { , ay] 



(34) 



(35) 



dt 



= 0, 



*~ 1, ...,6; j = 1, ,6), 



as will be proved immediately. 

Many complicated expressions will arise in the following dis- 
cussion which are symmetrical in x, y, and z. In order to abbrevi- 
ate the writing let S, standing before a function of x, indicate that 
the same functions of y and z are to be added. Thus, for example, 

In starting from the definitions of the brackets and omitting the 
subscripts of x, , z', which will not be of use in what follows, 
it is found that 

{dtx_ dx^ dx_ _dV_ &x?_ dx_ _dx^_ jPx_ 1 
daidt daj dai dajdt daidt daj dai dajdt J 

*?L + MJ!?L.\ 

idaj dt daidaj J 
I dx_ d z x' dx^_ 



dt 



A.sl -^-^\ i of ^ 

don" [dt daj dt da, J H 1 dt da 



^AQ/^^.^^I AQ/^^-^^I 

.da,-' [dt da } - dt da,) Bat [dt da> dt don]' 



213] PROPERTIES OF LAGRANGE*S BRACKETS. 389 

The partial derivatives of the coordinates with respect to the time 
are the same in disturbed motion as the total derivatives in un- 
disturbed motion. Therefore this equation becomes as a conse- 
quence of (14) 



<Mdx_ d^M\ d_nf<Mdx_ dQds_' 

dx da, M &*/ \ ddj { dx dai dx' dai 

d / dfl \ d / dti \ _ d 2 fl d 2 12 

dai\daj/ da } - \dai/ daidaj dajdai 



dt 



which proves the theorem that the brackets do not contain t 
explicitly. This would iiardlv be anticipated since each of the 
quantities which appears in the brackets^ an explicit function of t. 

Since the brackets do not contain the time explicitly they may 
be computed for any epoch whatever, and in particular for t = t Q . 
The equations become very simple if the coordinates at the time 
t = t Q are taken for the elements i, , a 6 . This is permissible 
since the ordinary elements are defined by these quantities, and 
conversely. It must not be supposed that they are constants; 
they are such quantities that if the elements are computed from 
them, and then if the coordinates at any time t are commuted using 
these elements, the correct results will be obtained. Since in 
disturbed motion the elements vary with the time, the values of 
the coordinates at t = t also vary. Otherwise considered, if the 
osculating elements at t are used and if the coordinates at the 
time t = t are computed, it will be found in the case of disturbed 
motion that the coordinates at t = t vary, and these values of 
the coordinates are the ones in question. 

Let the coordinates at the time t = t Q be X Q) ', z '; then 

o dx ' dx Q ' dx<> 



which equals zero because XQ is independent of yo and XQ. Simi- 
larly, 

= [20', 3</] = [so, 2/o] = 0, 



But 

(37) [s , so'] = [2/o, </o'] = [zo, zol = 1. 

Therefore equations (33) become in this case 




TRANSFORMATION TO ORDINARY ELEMENTS. 



[214 



dt " 

dyo = 
dt ~ 

dz Q 



dt 

dy ' _ 
dt 



dzp 1 
dt 



= - ra 2 



Any system of differential equations of the form (38) is known 
as a canonical system, and they possess properties which make them 
particularly valuable in theoretical investigations. There is a 
theorem that any dynamical problem in which the forces can be 
represented as partial derivatives of a potential function can be ex- 
pressed in this form; and if it is possible to put a problem in the 
canonical form it is possible to do so in infinitely many systems of 
dependent variables. 

If equations (38) were solved they would give the values of the 
coordinates at to which would have to be used to obtain the true 
coordinates at the time t, under the supposition that the planet 
moved in an undisturbed ellipse during t U. If the variables 
were the elliptic elements the solutions of the equations would 
give the elements which would have to be used to compute the 
coordinates at the time , when they are supposed to have been 
constant during the interval t t . Thus, when the elements 
have been found the remainder of the computation is that of 
undisturbed motion. 

214. Transformation to the Ordinary Elements. The elements 
used in Astronomy are not the coordinates at t = to, but &, i, a, 
e, TT, and T (or e = IT nT), which were expressed in terms of the 
initial conditions in Arts. 86, 87, and 88. It will be necessary, 
therefore, to transform equations (38) to the corresponding ones 
which involve only the elements which are actually in use by 
astronomers. 

Let s represent any one of the elements & , i, a, e, TT, e. It may 
be expressed symbolically in terms of the initial conditions by 

(39) s = f(x<>, y Q , z , XQ', y Q ', *</) 

Hence it follows that 



ds = f df dx df dx ' } 
dt *\ dx dt "*" dx ' dt J ' 



or, because of (38), 



COMPUTATION OF LAGRANGE's BRACKETS. 
ds ^ f df dRi t 2 df dR\, 



215] COMPUTATION OF LAGRANGE's BRACKETS. 391 

(40) 

The partial derivatives of Ri, 2 are expressed in terms of the 
partial derivatives with respect to the new variables by the 
equations 



(41) 



i d-Ri, 2 dw dRi, 2 de 
dir dx~Q de dx Q ' 



i, 2 



afli, 2 di dRi, 2 aa afli, 2 de 

-- ' 






, 2 






de 



a 



On carrying out the complicated computations of -r- , , 7 

OXQ OZQ 

by means of the equations given in Arts. 86, 87, and 88, and ex- 
pressing all the partial derivatives in terms of the new variables, 

\ p Af? 

the partial derivatives . 1; 2 , . . *', 2 are found in terms of 

OXQ OZo 



the elements and 



On substituting in (40) and - 



expressing 5*- , , ^77 in terms of the elements, -r. is found in 
dXQ OZQ at 

terms of the elements and the derivatives of the perturbative 
function, Ri t 2 , with respect to the elements. 

215. Method of Direct Computation of Lagrange's Brackets. 

The transformations required in the method of the preceding article 
are very laborious, and the direct computation of the brackets, 
though considerably involved, is to be preferred from a practical 
point of view. All of the computation in the transformations of this 
sort might be avoided by using canonical variables; but, in order to 
employ them, a lengthy digression upon the properties of canonical 
systems would be necessary, and such a discussion is outside the 
limits of this work. Still, the labor may be notably reduced by 
first taking elements somewhat different from those defined in 
chapter v., and then transforming to those in more ordinary use. 
The following is based on Tisserand's exposition of Lagrange's 

method.* 

* Tisserand's Mecanique Celeste, vol. i., p. 179. 



392 



COMPUTATION OF LAGRANGE S BRACKETS. 



[215 



Let the ^-plane be the plane of the ecliptic, &>P the projection 
of the orbit upon the celestial sphere, II the projection of the peri- 
helion point, and P the projection of the position of the planet at 
the time t. In place of TT and e, adopt the new elements o> and a 
defined by the equations 

(42) 




Fig. 60. 

The following equations are either given in Art. 98, or are ob 
tained from Fig. 60 by the fundamental formulas of Trigonometry 



(43) 



n = 






r = 



COS V = 

sin v = 
x = 

y = 

z = 



e sin # = nt -f- 
a(l e cos E), 



IA 



cos E e 
1 - e cos E ' 

Vl - e 2 sin E 
1 - e cos E ' 

r{cos (v + co) cos ft sin (v + co) sin ft cos i] , 
rjcos (v + w) sin & + sin (t; + co) cos ft cos i], 
r sin (v -f- co) sin i 



215] 



COMPUTATION OF LAGRANGE's BRACKETS. 



393 



From these equations and their derivatives with respect to the 
time the partial derivatives of the coordinates with respect to the 
elements can be computed. The elements have been chosen 
in such a manner that they are divided into two groups having 
distinct properties; , i, and co define the position of the plane of 
motion and the orientation of the orbit in the plane, and a, e, 
and ff define the dimensions and shape of the orbit and the position 
of the planet in its orbit. Therefore the coordinates in the orbit 
can be expressed in terms of the elements of the second group 
alone, and from them, the coordinates in space can be found by 
means of the first group alone. 

Take a new system of axes with the origin at the sun, the positive 
end of the -axis directed to the perihelion point, the rj-axis 90 
forward in the plane of the orbit, and the -axis perpendicular to 
the plane of the orbit. Let the direction cosines between the 
x-axis and the , r;, and f-axes be a, a', a"; between the ?/-axis and 
the , 77, and -axes be ft t 0', #"; and between the z-axis and the 
, TJ, and f-axes be 7, 7' ', 7". Then it follows from Fig. 60 that 



cos , 
cos i, 



(44) 



a 


= COS CO COS ii 


sin co sin 66 


ft 


= cos co sin + 


sin co cos 


7 


= sin co sin i, 




a' 


= sin co cos 


cos co sin 


0' 


= sin co sin 


+ COS CO COS 


7' 


= cos co sin i, 




a" 


= sin sin i, 




ff' 


= cos sin ^, 




v 


= cos i. 





cos , 
cos i, 



There exist among, these nine direction cosines, as can easily be 
verified, the relations 



(45) 



a 2 + 2 + 



a' 2 + 



+ 



aa' + 08' + 77' = 0, 
a' a" + ft' ft" + 7V = 0, 



a = - 7 0, a = / T - y, a = y - y, 
ft = 7 '<*" - a V, 0' = y"a - a"7, ft" = 7 ; - 7 ; , 
7 = a' ft" - VOL", 7' = a" ft - VOL, 7" = aft' - fta 1 '. 



394 



COMPUTATION OF LAGEANGE S BRACKETS. 



[215 



It follows from (43) and (44) and the definition of the new 
system of axes that 



= r cos v = a(cos E e), ij = a Vl e 2 sin E, 



(46) < 



dt I - e cos E ' 

r- 



, _ na sin E _ k V$ + mi sin E 
1 - e cos E ~ Va(l - e cos E) ' 



, _ ria Vl e 2 cos # _ fc VS -f mi Vl e 2 cos 
1 - e cos # Va(l - e cos #) 



y' = 



where the accents on x, y, z, , rj, and f indicate first derivatives 
with respect to t. 

The partial derivatives of a, , y" with respect to the elements 
may be computed once for all; they are found from (44) to be 



(47) 



da 



(48) 



da 

d& 

dp 






9ft" 



ir = - % - - 0; 

dco aco 



da 



r > n r fi 

30 "' a " " "' 



(49) 



^ - 0: 

dt = + sin &> cos t, 

^' = cos &, cos i, 
j // 
^- = y" sin co, -V = T " cos co, -~ = - sin i. 



da ,, . da 

T-T = a" sin co, = a' 

01 di 



= ,3" sin co, - = 






di 



216] 



COMPUTATION OF LAGRANGE J S BRACKETS. 



395 



There are as many brackets to be computed as there are combi- 

6! 
nations of the six elements taken two at a time, or 



2! 4! 



15. 



Three of them involve elements of only the first group; nine, one 
element of the first group and one of the second; and three, ele- 
ments of only the second group. Let K and L represent any of 
the elements of the first group, &>, i, co; and P and Q any of the 
elements of the second group, a, e, a. Then the Lagrangian 
brackets to be computed are 



/ i <v ri o f dx dx ' dx' dx ] 

(a) [X,I,] = S{ _-__}, (3 equations), 

f dx dx' dx' dx ] 
(6) [JC,P]-S{_ -__j, (9 equations), 

/ \ fD 01 C. f SX dX ' dX ' dX } 

(c) [P, Q] = S _-__, (3 equations). 



(50) 



It is found from (46) that 

da , da' 
I dK~ * 
(51) 



dtf 
dK 

dP 



,da' 



and similar equations in y and z. 

216. Computation of [co, &], [&, i], [i, ]. Let S indicate that 
the sum of the functions, symmetrical in a, j8, and 7, is to be 
taken. Then the first equation of (50) becomes as a consequence 
of (51) 



But the law of areas [Art. 89] gives 



Therefore 

(52) [K, L] 




On computing the right member of this equation by means of (47), 



396 



COMPUTATION OP LAGRANGE^S BRACKETS. 



[217 



(48), and (49), and reducing by means of (45), the brackets in- 
volving elements of only the first group are found to be 

[co, &] = na 2 Vl - e 2 (- ap - a'p' + ap + a'0') = 0, 
[&, i] = na 2 Vl - e 2 {(a$" - /3a") QOS co 

+ (/3V -a' ft") sin co } 
(53) -} = na 2 Vl e 2 ( 7' cos co 7 sin a>) 

= na 2 Vl e 2 sin i, 
[i, co] = - na 2 Vl - e 2 { (aV + jS'/S" + T'T") cos co 

+ (a" a + P"$ + y"y) sin co} = 0. 

217. Computation of [K, P]. The second equations of (50) 
become, as a consequence of (51), 



' + p H y 

^ ^ T a^ 

It follows from equations (45), (47), (48), and (49) that 



** AW ' P AW ' *Y ^v ^> 
oJ oJ\. oA 



>/ r / 



Therefore 



218] 



(54) 



COMPUTATION OF LAGRANGE's BRACKETS. 
, da , a , 80 , , d 



397 



a >* + *' * + y *x I ^!jL=jf 

" dtf + ^ d# +T d# J aP 



f da , 8(3 , dy 1 d V? 

+ mi [ a dK + ^dK +J dK\-dP 

Let P = a, e, a in succession. Then it is found that 



(55) 



Vo(l - 






n 

= 0. 



Let K = co, ^, i in turn in (54), and make use of (55); then it 
is found that 



(56) S 



r i na r, ^ 

[co,a] = y Vl^6 2 , 



, 6] = 



na?e 



[co, a] = 0, 



na 



Vl - e 2 ' 

j ] = ~2~ Vl e 2 cos t, [i, a] = 0, [i, e] = 0, 

na z e 



, e 



_ ^>2 



COS I, 



,*]= 0, [i, (r] = 0. 



218. Computation of [a, e], [e, a], [a, a]. The third equation 
of (50) becomes, as a consequence of (51), 



/ /2 



, /-, 

- J ) -' 



>\ r a * a '' a? 5 "' 4- ^' a ' ^'^ 

~ - 



COMPUTATION OF LAGRANGE'S BRACKETS. 



[218 



As a consequence of equations (45), the right member of this equa- 
tion reduces to 



m = __L_ 

dPdQ dQdP^dPdQ dQdP' 



Since the brackets do not contain the time explicitly t may be 
given any value after the partial derivatives have been formed. 
The partial derivatives become the simplest when t = T, the time 
of perihelion passage. For this value of t, E = 0, r = a(l e), 
and it is found from equations (46) that* 



(58) - 



d _ 1 dr; _ n d' _ n dr/' n 1 -\- e 

da" ~ e ' d^~ U ' d^ = ' d^ = "2Vl"-^ 



-^ _ ^7 _ Q d _ _ "! 



1 



na 



de 



de 



= 0, 



de 



l-e j 



dcr do- 

Then equation (57) gives 

(59) [a, e] = 0, [e, <r] = 0, 



de 1-6 
na 



na 



On making use of the fact that [, ,] = [a/, aj and equations 
(53), (56), and (59), equations (33) become 



(60) 



na 



da 



n ^ . . di . na n r . da 

na 2 VI e 2 sin t -3- + -p- VI e z cos 1 37 
at 2 at 



na 2 Vl e 2 sinf 



dt 



na 2 e .de dRi, 

, COS I -J7 = ^2 ^^ 

/I _ e 2 di d^ 

1, 2 



* It should be remembered that a and e enter explicitly and also implicitly 
through E and n, for # is denned by the equation 



E - e sin E = n(t - T) = 



\ |Tt 

Then, e. g., ~ = cos E e a sin E = 1 - e when = T 7 , etc, 



218] 



COMPUTATION OF LAGRANGE*S BRACKETS. 



399 



(60) 



na r- dco na r. . d&> na da 

~T A e 'dt~~2'* 1 ' l ~dt"~2dt 

na 2 e do> ntfe cos i d& _ dRi, 2 
^#~di^' ^T^~dt~'~ m2 ~de~' 



da ' 



2 (ft - da 

These equations are easily solved for the derivatives, and give 



(61) 



na 2 Vl - e 2 sin i 



^2 



, 2 



di m 2 cos i dRi t 2 _ 

dt na 2 Vl e 2 sin i do> na ? Vl e 2 sin ^ ^ ^ 




nae 



de 



rwi da 



The perturbative function Ri t 2 involves the element a explicitly, 
and also implicitly through n which enters only in the combi- 
nation nt + cr. Consequently the last equation of (61) becomes 



da 



m 2 (l - 



*\ 7") O/i/i/i / m I? \ 

o/ti^ _ ^niz i Q/II, 2 \ _ 
de na \ da ) 



6//l 2 C'-tt'l, 2 CF/t 

?ia dn da ' 



where the partial derivative in parenthesis indicates the derivative 
is taken only so far as the parameter appears explicitly. ^ 
It follows from the combination nt + a that 



(63) 



2m 



dn 



na 



da 



, 2 _ da 

" 



It will be shown [Arts. 225-227] that 



CCT 



*'. 

is a sum of periodic 



terms; therefore cr, as denned by (62), contains terms which are 
the products of t and trigonometric terms. It is obvious that such 
an element is inconvenient when large values of t are to be used. 



400 



COMPUTATION OF LAGRANGE's BRACKETS. 



[219 



In order to avoid this difficulty Leverrier used* in place of a the 
mean longitude from the perihelion as an element. It is defined by 

(64) I = fndt + cr, 
whence 

(65) -. + +*. 




Since n 



it follows that 



(66) 



dn _ _3n dn _ 3n da 

~da ~ ~2 a' ~dt ~ ~ 2adi' 

Therefore equation (65) becomes, on making use of (62), 



-r. = n 



na?e 



o/ti, 2 2wi2 / dR\, 2 \ 
de na \ da ) ' 



Since 



or) 

*' 



Off 



, the fourth and fifth equations, where alone 



the partial derivative of Ri t 2 with respect to a occurs, will not be 
changed in form. Hence, if Us used in place of cr throughout (61), 
the equations will be unchanged in form, and the partial deriva- 
tive of Ri, 2 with respect to a is to be taken only so far as a occurs 
explicitly. 

219. Change from ft, co, and cr to ft, TT, and . The trans- 
formation from the elements ft, co, and a to ft, TT, and e is 
readily made because the relations between the co and cr and the 
TT and e are very simple. It follows from the definitions of Arts 
214 and 215 that 

< V* 



V 



(68) 



whence 



(69) 



e - TT; 



dt 



dt ' 



da) _ dir 

~dt ~ ~di " ~dt '. 

da _ de dir 
dt ~ ~dt ~ ~dt' 



On solving (68) for & , TT, and e in terms of & , w, and cr, it is found 
that 

* Annales de VObservatoire de Paris, vol. i., p. 255. 



219] COMPUTATION OF LAGRANGE's BRACKETS. 401 

' ft= ft, 

(70) - 7T = CO + ft, 

Hence the transformations in the partial derivatives are given by 
the equations 

afli, 2 _ /a# ll2 \ aft 



(71) 



aft 



aft aft 



lt 2 \_gjr_ /afli. a \ _de 

Tr / aft "*" \ ae ) e& 



= \ aft 
-jjj = ( J^J j - _|_ ( k* | JL _|_ [ LJ j _i 

aco \ aft / dco \ dw / aco \ de / d<*3 



li. 2 = / 1?- 2 ) ^i _|_ / !_? j _J[ _|_ / _ k 2 J _1 

dff \ aft / dff \ dir / do" \ de / dff 

-(^): 

On substituting (69) and (71) in (61) and omitting the parentheses 
around the partial derivatives, and on solving for the derivatives 
of the elements with respect to t, it is found that 

dft ra 2 a#i, 2 



(72) - 



na 2 Vl e 2 sin i 



tan I 



dir 
dt 

da 

dt 

de 
^- 
at 



na 2 Vl e 2 sin i & ft na 2 Vl 

i 

L, 2 . m 2 Vl - e 2 a^i, 2 



i t 2 i 



H. 



ll - e 2 di 
dR\, 2 



^ + 



ae 



a 



02 



- Vl - 



, 2 



l - 



2 



de m2tan 2 
d^ 



na 2 e dir 



na^l 



nae 



de 



na da 



27 



402 RECTANGULAR COMPONENTS OF ACCELERATION. [220 

These equations,* together with the corresponding ones for the 
elements of the planet w 2 , constitute a rigorous system of differ- 
ential equations for the determination of the motion of the planets 
mi and w 2 with respect to the sun when there are no other forces 
than the mutual attractions of the three bodies. 

If Ri t 2 is expressed in terms of the time and the osculating 
elements at the epoch to, equations (72) become the explicit 
expressions for the tirst half of the system (27), and define the 
perturbations of the elements which are of the first order with 
respect to the masses. 

220. Introduction of Rectangular Components of the Disturbing 
Acceleration. Equations (72) require for their application that 
Ri t 2 shall be expressed first in terms of the elements, after which 
the partial derivatives must be formed. In some cases, especially 
in the orbits of comets, it is advantageous to have the rates of 
variation of the elements expressed in terms of three rectangular 
components of the disturbing acceleration. 

The disturbing acceleration will be resolved into three rect- 
angular components W, S, R, where W is the component of 
acceleration perpendicular to the plane of the orbit with the 
positive direction toward the north pole; S is the component in 
the plane of the orbit which acts at right angles to the radius 
vector with the positive direction making an angle less than 90 
with the direction of motion; R is the component acting along the 
radius vector with the positive direction away from the sun. 
The components used in the preceding chapter evidently might be 
employed here instead of these, but the resulting equations would 
be less simple. 

In order to obtain the desired equations it is only necessary to 
express the partial derivatives of JKi, 2 with respect to the ele- 
ments in terms of W, S, and R, and to substitute them in (61) 
or (72), depending upon the set of elements used. The trans- 
formation will be made for the elements used in equations (61). 

The quantities m 2 -^- 2 , w 2 -^, m 2 ^j-^ are the com- 
ox oy oz 

ponents of the disturbing acceleration parallel to the fixed axes of 
reference. It follows from the elementary properties of the 

* The subscript 1, which was omitted from the coordinates and elements in 
Art. 213, should be replaced when the equations for more than one planet are 
written. 



220] 



RECTANGULAR COMPONENTS OF ACCELERATION. 



403 



resolution and composition of accelerations that 



2 



i 

is equal 



to the sum of the projections of W , S, and R upon the x-axis, and 
similarly for the others. 

Let u represent the argument of the latitude, or the distance 
from the ascending node to the planet P, Fig. 61. Then it follows 




Fig. 61. 

from the fundamental formulas of Trigonometry that 

= + R(cos u cos ft sin u sin ft cos i) 
*S(sin u cos ft + cos u sin ft cos i) 
+ W sin ft sin i, 



(73) 



a/L-i,2 

dx 



ra 2 



/) 7? 

, 1>2 = + R(cos u sin ft + sin u cos ft cos i) 



, 
dy 



S(sm u sin ft cos u cos ft cos i) 

W cos ft sin i, 

A 7? 

ra 2 = + R sin u sin i + S cos u sin i -\- W cos i. 
Let s represent any of the elements ft, ,*; then 



(74) 



L, 2 



dx S 

as a?/ as 



, 



__. . . a/ti 2 a /LI 2 a/ti, 2 /wr\ i i. 

The derivatives . ' , ' , , are given in (73) and when 
ox dy dz 

, , and have been found, the transformation can be com- 
as as as 

pleted at once. 



404 RECTANGULAR COMPONENTS OF ACCELERATION. 

It follows from equations (51) that 

da' 



[220 



(75) 



dx_ 
dK 

dy_ 
dK 



da 



dP dp' 

n TT" I */ r\ 7^ ) 



dz dy 
dK ^dK 



dy' 



!/</ *-^S I f ^ I 

JP = a ~dP~* * 'dP ) 

dy_ _ R d^_, R , dr) 
dP~ PdP^P dP' 

dz _ d , df] 

dP~ T aP + T dP> 



where K is any of the elements &, i, , and P any of the ele- 
ments a, e, a. The quantities a, , y' are defined in (44), and 
their derivatives are given in (47), (48), and (49); the derivatives 

f\t, *\ 

ft and -r^ are to be computed from (46). 

or or 

It is found after some rather long but simple reductions that 

m z .,*' 2 = Sr cos i Wr cos u sin i, 
oii 

m z ?' 2 = Wr sin u, 



(76) 



da a 

dfli. 2 = 



-Ra cos v + *S 1 + - a sin v, 



/i, 2 _ /cae . i c^_ 

o" "Y^ g2 r 



Therefore equations (61) become 
^ r sin u 



(77) 



7W, 



r cos 



at no? Vl - e 2 

<fo _ - Vl - e 2 cos 
d< 



i^ + JZfft^llffln^ 

nae P .1 



r sin tt cot t ~- 

na 2 Vl - e 2 ' 



(77) 



da 
dt 



2e sin v 



de_ _ Vl e 2 sin v 
dt na 




dcr 1 [2r 1-e 2 1 D 

= CQSZ ; # 

d na |_ a e 



405 



(!_-*, 

nae 



1 + - sm 



XXVI. PROBLEMS. 

1. Find the components S and J? of this chapter in terms of T and N, 
which were used in chapter ix., Art. 174. 



(1 -)- e cos v) 



A.ns. 




e sin v 



T - 



+ e 2 + 2e cos v 
1 + e cos v 



+ e 2 + 2e cosv Vl + e 2 + 2e cos w 



N. J 



2. By means of the equations of problem 1 express the variations of the 
elements ft, , a in terms of T and A", and verify all the results contained in 
the Table of Art. 182. 

3. Explain why -^ contains a term depending upon W. 

4. Suppose the disturbed body moves in a resisting medium; find the 
equations for the variations of the elements. 



Ans. 



<fc 
Jt 

dt 


= 0, 
= 0, 




dot 


2 V 1 e 2 sin y ^ 


dt 


nae 


V 1 + e 2 + 2e cos v 


da 


2Vl +< 


? + 2e cos v T 


dt 


n>/ 


l-e 2 


de 
dt 


2Vl -6 2 (cosy + e) m 


naVTT 


e 2 + 2e cos y 


do- 


2(1 


e 2 )(l + e 2 + e cos y) sin y 


dt 


nae(l 


+ e cos y) V 1 -f e 2 + 2e cos y 



5. Discuss the way in which the elements vary in the last problem, including 
the values of v for which the maxima and minima in their rates of change 
occur, when T is a constant, and when it varies as the square of the velocity. 



406 DEVELOPMENT OF PERTURBATIVE FUNCTION. [221 

6. Derive the equations corresponding to (77) for the elements &, i, TT, 
a, e, and e. 

r sin u 



dt 
Ans. 



dt na V 1 e 2 sin i 
di r cos u 



j^ 
1 e cos v 




dt 

221. Development of the Perturbative Function. In order 
to apply equations (72) the perturbative function /2 lf 2 must be 
developed explicitly in terms of the elements and the time. From 
this point on only perturbations of the first order will be con- 
sidered; therefore, in accordance with the results of Art. 208, 
the elements which appear in R i, 2 are the osculating elements at 
the time t . 

In the notation of Art. 205 the perturbative function is 

fit, 

(78) 



+ (2/2 - 2/i) 2 + fe - zi) 



The perturbing forces evidently depend upon the mutual 
inclinations of the orbits, rather than upon their inclinations 
independently to the fixed plane of reference. It will be con- 
venient, therefore, to develop Ri t 2 in terms of the mutual inclina- 
tion. Since this angle is expressible in terms of ii, iz, &>i, and &2, 
the partial derivatives of Ri t 2 with respect to these elements will 
depend in part on their occurring implicitly in this angle. 

The development of the perturbative function consists of three 
steps:* 

* There are many more or less important variations of the method outlined 
here, which is based on the work of Leverrier in the Annales de VObservatoire 
de Paris, vol i. 



222] 



DEVELOPMENT IN THE MUTUAL INCLINATION. 



407 



(a) Development of RI, 2 as a power series in the square of the 
sine of half the mutual inclination of the orbits. 

(6) Development of the coefficients of the series obtained in 
(a) into power series in e\ and e 2 . 

(c) Development of the coefficients of the preceding series into 
Fourier series in the mean longitudes of the two planets and the 
angular variables in, 7r 2 , & i, and & 2 . 

In the little space available here it will not be possible to give 
more than a general outline of the operations which are necessary 
to effect the complete development. A detailed discussion is 
given in Tisserand's Mecanique Celeste, vol. I., chapters xn. to 
xvin. inclusive. 



222. (a) Development of Ri,z in the 
Let S represent the angle between the radii 



Mutual Inclination. 

and r 2 ; then 



(79) 



+r 2 2 - 2rir 2 cos S)~ 




Fig. 62. 

Let the angles between r\ and the x, y, and z-axes be i, 0i, 71 
respectively, and in the case of r 2 , 2 , j8 2 , and 7 2 . Then it follows 

that 

(80) Xi = ri cos i, 2/1 = r v cos 0i, Zi = n cos 71, etc., 
and 

i 2 + 2/i2/2 + ZiZ 2 = rir 2 (cos i cos 2 + cos /3i cos /3 2 

+ cos 71 cos 7 2 ) = r\r<i cos S. 



Let 7 represent the angle between the two orbits, and T\ and r 2 



408 



DEVELOPMENT IN THE MUTUAL INCLINATION. 



[222 



the distances from their ascending nodes to their point of inter- 
section. From the spherical triangle PiP 2 C the value of cos S is 
found to be 

cos S = cos (u\ TI) cos (u z T 2 ) 

+ sin (ui TI) sin (w 2 T 2 ) cos 7, or 
cos S = cos (ui Uz + T 2 TI) 



(82) 



2 sin (HI TI) sin (u z T 2 ) sin 2 > 



T 2 



7T 2 



T 2 . 



The quantities 7, TI, and T 2 are determined by the formulas of 
Gauss applied to the triangle & i &> 2 C : 



sin 7 sin TI 
sin I sin T 2 



sn 
sin 



sn 
sin 



sin / cos TI = sin ii cos t' 2 cos ii sin i z cos (h 
sin / cos T 2 = cos i\ sin i 2 + sin ii cos t 2 cos ( & i ^ 2 ), 
cos 7 = cos i'i cos iz + sin ii sin t' 2 cos (^i 



(83) 



For simplicity 7, TI, and T 2 will be retained, but it must be remem- 
bered when the partial derivatives of Ri, 2 are taken that they are 
functions of ii, i*, &>i, and ^ 2 . 

As a consequence of (79), (81), and (82), the perturbative 
function can be written in the form 



(84) 



, 2 = 



+ r 2 2 2rir 2 cos (HI u% + T 2 
4rir 2 sin (ui TI) sin 




T 2 ) sin 2 - 



+ r 2 2 - 2nr 2 cos (ui - U Z + T Z - TI) 
- ^ cos (ui - u z + T 2 - TI) 

2 sin (ui TI) sin (u^ T 2 ) sin 2 - . 

The radii ri and r 2 are independent of 7. The second factor of 
the first term of the right member of this equation can be expanded 
by the binomial theorem into an absolutely converging power 

series in sin 2 ~ so long as the numerical value of 



223] 



DEVELOPMENT IN POWERS OF 61 AND 



409 



(85) 



sn 



n) sin (u z r 2 ) sin 2 - 



2 



r 2 rir 2 cos ( 1 w 2 + r 2 n) 
is less than unity. This fraction is less than, or at most equal to, 

4-r,r sin 2 

(86) 



4rir 2 sin 2 - 



(ri - 



If this expression is less than unity for all the values which ri 
and r 2 can take in the given ellipses the expansion of (84) is valid 
for all values of the time. In the case of the major planets it is 

always very small, the greatest value of sin 2 - being for Mercury 

and Mars, 0.0118. In the perturbations of the planetoids by 
Jupiter it often fails, for I is sometimes of considerable magnitude 
while r 2 ri may become very small. In the case of Mars and 
Eros r 2 r\ may actually vanish and this mode of development 
consequently fails. It is needless to say that it is not generally 
applicable in the cometary orbits. 

In those cases in which the expansion of (84) does not fail, the 
expression for R it 2 becomes 



(87)- 



Ri, 2= + [n 2 + r 2 2 - 2nr 2 cos (MI - 
r 2 2 2/*ir 2 cos (u 



+ r z - n)] * 
u z + r 2 TI)]- 
X 2 sin (MI n) sin (u z r 2 ) sin 2 



cos MI - 



Tl 

rCOS 

r 2 2 



X 6 sin 2 (HI n) sin 2 (u z r 2 ) sin 4 - 



u 2 + r 2 






H -- sin (HI TI) sin (u z r 2 ) sin 2 - . 



7*2 



223. (6) Development of the Coefficients in powers of ei and e 2 . 
The radii ri and r 2 vary from ai(l d) and a 2 (l e 2 ) to ai(l + 
and a 2 (l + e 2 ) respectively. Let 



(88) 



f 

L 



p 2 ). 




410 DEVELOPMENTS IN FOURIER SERIES. [224 

The angles HI and u 2 are expressed in terms of the true anomalies, 
Vi and v 2 , and the elements by (82) . The true anomalies are equal 
to the mean anomalies plus the equations of the center, which 
may be denoted by Wi and w 2 . Let li and 1 2 represent the mean 
longitudes counted from the z-axis [Fig. (62)]; then 



(89) 

LW 2 T 2 = 1 2 &2 T 2 + W 2 . 

It follows from (811) that Ri, 2 can be written in the form 
Ri, 2 = F[ai(l + PI), a 2 (l -f p 2 )], 



where F is a homogeneous function of ai and a 2 of degree 1. 
Therefore 

(90) fi ltl = 



The right member of this equation can be developed by Taylor's 
formula, giving 



_ I v(n _ N , PI - P2 ai aF(oi, q a ) 

T- - S P (dl, dz) T T~L - "i -- 5 - 

+ P2 I 1 + p 2 1 dai 

( y i ) 



, /PI -P 2 \ 2 ai 2 a 2 F(oi, a 2 ) 1 

r \ 1+P2/ 1 2 " (to! 2 ' J ' 

The expressions ( ^ p2 J can be developed as power series in 

Pi and p 2 . But in Art. 100, equation (62) , p is given as a power series 
in e whose coefficients are cosines of multiples of the mean anomaly. 
On making these expansions and substitutions in (91), Ri, 2 can 
be arranged as a power series in e\ and e 2 . These operations are 
to be actually performed upon the separate terms of the series 
(87), so the resulting series is araanged according to powers of 

e\ t e 2 , and sin 2 - . The angles Wi and w 2 also depend upon e\ 

and e 2 respectively, but their developments will not be introduced 
until after the next step. 

224. (c) Developments in Fourier Series. The first term 
within the bracket of (91) is obtained by replacing rj. and r 2 by ai 
and a 2 respectively in (87). The higher terms involve the deriva- 
tives of the first with respect to a\. On referring to the explicit 
series in (87), it is seen that the development of the expressions of 
the type 



224] DEVELOPMENTS IN FOURIER SERIES. 411 



v-\ _ 

2 2 , 



2 [ai 2 + a 2 2 2aia 2 cos (ui u 2 + r 2 T 2 



where *> is an odd integer, must be considered. 

Let HI u 2 + r 2 TI = ^. It is known from the theory of 
Fourier series when a\ and a 2 are unequal, as is assumed, that 

_ V 

[a i 2 + 2 2 2ai& 2 cos \f/] 2 can be developed into a series of cosines 
of multiples of \f/ } which is convergent for all values of ^. That is, 



(92) (a!a 2 )[ ai 2 + a 2 2 - 2aia 2 cos fl"* = V < cos 



where B v = B F <-. 

The coefficients J3 v (i) are of course given by Fourier's integral 

1 /^7T V-l _V 

B/^ = - J o (aia 2 ) 2 [ Ol z + a 2 2 - 2aia 2 cos ^] cos 



but the difficulty of finding the integral makes it advisable in this 
particular problem to proceed otherwise. 

Let z = e**~^, where e represents the Napierian base. Then 

2 cos \j/ = z + z~ 1 J 2 cos i\l/ = z { + z~*'. 

Suppose a 2 > 0,1 and let = a; then (92) becomes 
a 2 

v-l 

(93) (1 + a 2 - 2a cos ^)"5 = i E ^v (i) cos i>. 

2 ^ t=^oo 

Let 

(1 + a 2 - 2a cos i0~5 = (1 - a^)~^ (1 " 

therefore 

v-\ 

(94) R =^-6/ 



Since the absolute values of az and az~ l are less than unity for 

n _ v 

all real values of ^, the factors (1 az) * and (1 az~ l ) * can 

be expanded by the binomial theorem into convergent power 
series in az and car 1 . The coefficient of z 1 in the product of these 
series is %b y (i \ after which B v (i) is obtained from (94). The 
general term of the product of the expansions is easily found to be 



412 DEVELOPMENTS IN FOURIER SERIES. [224 



(95) 



1-2 (i + !)(; + 2) 



In this manner the coefficients of p\ Jl pJ* ( sin 2 - ) are de- 

\ */ 

veloped in Fourier series in cos i(^i u z + r 2 TI). But these 
functions are multiplied by the factors sin (u\ TI) sin (u* T 2 ) 
raised to different powers [equation (87)]. These powers of 
sines are to be reduced to sines and cosines of multiples of the 
arguments, and the products formed with cos i(u\ u 2 -f r 2 TI), 
and the reduction again made to sines and cosines of multiples 
of arcs. The final trigonometrical terms will have the form 
cos ( jiUi + j z uz + kiTi + & 2 r 2 ), where ji, jz, ki, and & 2 are integers. 
As a consequence of (89) this expression can be developed into 



cos (jii 
= cos 

X JCOS (JiWi) COS (jzWz) ~ sm O'lWi) sm 

X {sin (jiWi) cos 0' 2 w 2 ) + cos (jiWi) sin 
Since 

<o) = n\t - 



the first factors of the terms in the right member of this equation 
are independent of e\ and e 2 . Cos (jiWi), etc., are to be expanded 
into power series in Wi and w z by the usual methods. Now 
Wi Vi MI, Wz = Vz MZ, and these quantities were developed 
into power series in e\ and e 2 [Art. 100, eq. (64)] whose coefficients 
were Fourier series with multiples of the mean anomaly as argu- 
ments. On substituting these series for w\ and w 2 in the expansions 
of the second factors of the terms of the right member of (96) , and 
reducing the powers of sines and cosines of the mean anomaly to 
sines and cosines of multiples of the mean anomaly, and multi- 
plying by the factors 

cos (jili -+ jzlz 
and 



225] PERIODIC VARIATIONS. 413 

Sin (jili + j z lz - jlftl T J2&2 + fclTl + fe 2 T 2 ), 



and again reducing to sines and cosines of multiples of the argu- 
ments, the expression (96) is developed as a power series in e\ 
and 62 whose coefficients are series in sines and cosines of sums of 
multiples of l iy l z , &i, & 2 , TI, r 2 , Mi, M 2 . But MI = Z x in,' 
M 2 = Z 2 ?T 2 ; therefore the arguments will be l iy 1 2 , fti, & 2 , 
TI, T2, TTi, 7r 2 , where TI and r 2 are functions of fti, & 2 , ii, and t' 2 
denned by (83). 

When the several expansions and reductions which have been 
described have all been made, R it 2 will be developed in a power 

series in e\, e 2 , and sin 2 = , the coefficients of which are series of 

sines and cosines of multiples of l\, Z 2 , &i, & 2 , n, r 2 , in, 7r 2 , the 
coefficient of each trigonometric term depending upon the ratio 
of the major semi-axes. If the signs of h, & 2 , in, 7r 2 , TI, T 2 , 
1, 2, and are changed the value of Ri, 2 , as denned in (84), 
obviously is unchanged; therefore the expansion in question 
contains only cosines of the argument. Hence 



(97) 



, z = 2C cos D, 

+ kiTi + i 
C = / ( 01, a 2 , ei, e 2 , sin 2 - j , 



in which ji, , A; 2 ' take all integral values, positive, negative, and 
zero, the summation being extended over all of these terms. 

It is clear from the foregoing that the series for #1, 2 is very 
complicated and that much labor is required to expand it in any 
particular case. Leverrier has carried out the literal development 

of all terms up to the seventh order inclusive in e\ t e z , sin 2 - , 

it 

and the length of the work is such that fifty-three quarto pages of 
the first volume of the Annales de I'Observatoire de Paris are 
required in order to write out the result. 

225. Periodic Variations. It follows from equations (72) and 
(97) that the rates of change of the elements of mi are given by 



414 



PERIODIC VARIATIONS. 



[225 



(98) 



dt 



VT= 



,, 



Sin 




dir\ 

~df 



j^ 7 n 

_ c062) _ l _i 



ac 



n 

cos D, 



l ei 2 v^ f 7 / , 7 5n , , ar 2 

o 2^1 *i +*13 ^^27 

i 2 ei ^^ I d?ri a?ri 




+ 



, 
w 2 tan 



2 ^ f ac r an,, dT 2 l 

V < TT- cos D - ki ^- + k 2 rr- 

- 6l 2 ^ I aii L #ii ^i J 



cos 



The perturbations of the elements of the orbit of mi of the first 
order with respect to the mass w 2 are the integrals of these equa- 
tions regarding the elements as constants in the right members. 
Similar terms must be added for each disturbing planet. 

There are terms in Ri t 2 of three classes: (a) those in which 
ji^i + jzn z is distinct from zero and not small; (b) those in which 
j\n\ + J 2 n 2 is very small, but distinct from zero; and (c) those in 
which jini + J 2 n 2 equals zero. Denote the fact that RI, 2 contains 
these three sorts of terms by writing 



225] PERIODIC VARIATIONS. 415 

Ri, 2 = 2Co cos DO + SCi cos DI + 2C 2 cos D 2 , 

where the three sums in the right member include these three 
classes of terms respectively. Hence the perturbations of the 
elements of mi by w 2 of the first order and of the first class are 



(99) 



l 



{H/^ 
C/U 
dij 



sin Do 



+ J2ft 2 



sin ii 

T dr z 1 Co cos Do 



6 1 2 sin ii 



D 



w 2 tan 



?7i 2 tan7r f _.~ . T. 

_____?__ v 1 1 - sm 

o 1^ ^ ^-^ I Qi' . A 

, dri | 7 dr 2 1 ^7o COS Do 



W 2 \l 



sin Z) 



cos -o 



nidi^ J jiWi +J 2 n 2 



W 2 Vl 6i 2 T-> f 7 i ^TI , T dr 2 1 Co COS D 

~ X J M * I A* L f* A _ _ \ 

9 / ^ i /vi ~r "'i i " / 2 f I 

nidi 2 6i " t OTTi OTTi J JiWi -f- J 2 ? 

!7 sin Do 



m 2 tan 

fo = mai 2 Vf^l 
, 7 5r 2 1 Co COS Do 



Do 1 

j 2 n 2 J 



, 1 - Vl - ei 2 ^ 5C sin Do 

+ m 2 VI - ei 2 - . 2. 

uiai z ei ^ 



sin Dp 



416 LONG PERIOD VARIATIONS. , [226 

These terms are purely periodic with periods - - ~ , and 



constitute the periodic variations. Every element is subject to 
them, depending upon an infinity of such terms whose periods 
are different. The larger jiWi + .7*2^2 is, the shorter is the period 
of the term and in general the smaller is its coefficient. 

The method of representing the motion of the planets by a series 
of periodic terms is somewhat analogous to the epicycloid theory 
of Ptolemy, for each term alone is equivalent to the adding of a 
small circular motion to that previously existing. This theory is 
more complex than that of Ptolemy in that it adds epicycloid 
upon epicycloid without limit ;' it is simpler than that of Ptolemy 
in that it flows from one simple principle, the law of gravitation. 

226. Long Period Variations. The letters ji and j z represent 
all positive and negative integers and zero. Therefore, unless 
HI and n z are incommensurable ji and j- 2 exist such that jini + 
jznz = 0, where ji and jz are not zero. But then D is a constant 
and the integral is not formed this way. However, whether n\ and 
nz are incommensurable or not, such a pair of numbers can be found 
that jiUi -f jznz is very small. The corresponding term will be 
large unless its C is very small. It is shown in a complete dis- 
cussion of the development of R it 2 that the order of C in e\, e z , 

sin 2 5 is at the least equal to the numerical value of ji + jz (see 

Tisserand's Mec. C6L, vol i., p. 308). Since n\ and n 2 are both 
positive, one of the numbers ji, jz must be positive and the other 
negative in order that the sum jiUi + j 2 nz shall be small. The 
more nearly equal ji and jz are numerically the smaller the numeri- 
cal value of ji + jz is, and consequently, the larger C will be. 
When the mean motions of the two planets are such that they are 
nearly commensurable with the ratio of n\ to n 2 expressible in 
small integers, then large terms in the perturbations will arise 
from the presence of these small divisors. The period of such a 

term is = - -r. , which is very great, whence the appellation 

long period. These terms are given by equations of the same 
form as (99), but with the restriction that jini + jznz shall be 
very small. 

Geometrically considered, the condition that the periods shall 
be nearly commensurable with the ratio expressible in small 
integers means that the points of conjunction occur at nearly the 



227] 



SECULAR VARIATIONS. 



417 



same part of the orbits with only a few other conjunctions inter- 
vening. The extreme case is that in which there are no con- 
junctions intervening, i. e., when ji and j 2 differ in numerical value 
by unity. 

The mean motions of Jupiter and Saturn are nearly in the ratio 
of five to two. Consequently ji = 2, j 2 5 gives a long 
period term, and the order of the coefficient C is the absolute 
value of 2 5, or 3. The cause of the long period inequality of 
Jupiter and Saturn was discovered by Laplace in 1784 in com- 
puting the perturbations of the third order in ei and e 2 . The 
length of the period in the case of these two planets is about 850 
years. 

227. Secular Variations. The expression D is independent 
of the time for all of those terms in which ji = j z = 0. The 
partial derivatives of D with respect to the elements are also 
independent of the time; hence, on taking these terms of (98) and 
integrating, it is found that 



(100) 



ra 2 



sn 



Z 2 



ra 2 



VI 



sn 



2 Vi - 



tan I 



nidi 



*vr^ 



-- cos L> 2 



[A:^+ /c 2 ^? J C 2 sin 



28 



418 



(100) 1 



SECULAR VARIATIONS. 

o, 



[227 



w 2 tan 

Zi 



dC, 



-j- m 2 Vl - 



2w 2 



1 - Vl - 



X 



It follows that there are no secular terms of this type of the first 
order with respect to the masses in the perturbations of a. This 
constitutes the first theorem on the stability of the solar "system. 
It was proved up to the second powers of the eccentricities by 
Laplace in 1773,* when he was but twenty-four years of age, in 
a memoir upon the mutual perturbations of Jupiter and Saturn; 
it was shown by Lagrange in 1776 that it is true for all powers of 
the eccentricities.f It was proved by Poisson in 1809 that there 
are no secular terms in a in the perturbations of the second order 
with respect to the masses, but that there are terms of the type 
t cos D, where D contains the time 4 Terms of this type are 
commonly called Poisson terms. 

All of the elements except a have secular terms. It appears 
to have been supposed that the secular terms, which apparently 
cause the elements to change without limit, alone prevent the use 
of equations (72) for computing the perturbations for any time 
however great. Many methods of computing perturbations have 
been devised in order to avoid the appearance of secular terms; 
yet it is clear that, whether or not terms proportional to the time 

* Memoir presented to the Paris Academy of Sciences. 
t Memoirs of the Berlin Academy, 1776. 
t Journal de I'Ecole Poly technique, vol. xv. 



228] TERMS OF THE SECOND ORDER. 419 

appear, the method is strictly valid for only those values of the 
time for which the series (20) of Art. 207 are convergent. 

Secular terms may enter in another way, usually not considered. 
If jini + J 2 n 2 = with ji =|= 0, j 2 =t= 0, D is independent of the 
time and the corresponding terms are secular. In this case D is 
not independent of ei and there will be secular terms in the per- 
turbations of a. As has been remarked, this condition will always 
be fulfilled by an infinity of values of j\ and j 2 if n\ and n 2 are not 
incommensurable. But it is impossible to determine from obser- 
vations whether or not ni and n 2 are incommensurable, for there 
is always a limit to the accuracy with which observations can be 
made, and within this limit there exist infinitely many com- 
mensurable and incommensurable numbers. There is as much 
reason, therefore, to say that secular terms in a of this type exist 
as that they do not. However, they are of no practical im- 
portance because the ratio of HI to n 2 cannot be expressed in small 
integers, and the coefficients of these terms, if they do exist, are 
so small that they are not sensible for such values of the time as are 
ordinarily used. 

228. Terms of the Second Order with Respect to the Masses. 

The terms of the second order are defined by equations (29), 
Art. 210. The right members of these equations are the products 
of the partial derivatives, with respect to the elements, of the right 
members which occur in the terms of the first order, and the 
perturbations of the first order of the corresponding elements. 
Thus, the second order perturbations of the node are determined 
by the equations 



dt Uiai 2 Vl 6i 2 sin 

(101) 

~!^< 



Vl - ei 2 sin ii i < 
where Si and s 2 represent the elements of the orbits of mi and m 2 

*\2 D 

respectively. The partial derivative . *' 2 is a sum of periodic 

C7 2/iC/u ]_ 

and constant terms; s/ -^ and s 2 (1 ' 0) are sums of periodic terms 
and terms containing the time to the first degree as a factor. The 

products 1; 2 si (0> 1} and . *' 2 s 2 (1 - 0) therefore contain terms of 



420 LAGRANGE'S TREATMENT OF SECULAR VARIATIONS. [229 

four types: (a) sm D, where D contains the time; (b) t s D] 
cos cos 

(c) ^ m Z> 2 , where D 2 is independent of the time; and (d) t ^ D 2 . 
The integrals of these four types are respectively: 

cos T} cos ft sin ^ 

sin sin cos 



+ j z n 



Therefore, the perturbations of the second order with respect to 
the masses have purely periodic terms; Poisson terms, or terms 
in which the trigonometric terms are multiplied by the time; 
secular terms where the time occurs to the first degree; and secular 
terms where the time occurs to the second degree. This is true 
for all of the elements except the major semi-axis, in the case of 
which the coefficients of the terms of the third and fourth types 
are zero, as Poisson first proved. 

In the terms of the third order with respect to the masses there 
are secular terms in the perturbations of all the elements except 
d, which are proportional to the third power of the time, and so on. 

229. Lagrange's Treatment of the Secular Variations. The 
presence of the secular terms in the expressions for the elements 
seems to indicate that, if it is assumed that the series represent 
the elements for all values of the time, then the elements change 
without limit with the time. But this conclusion is by no means 
necessarily true. For example, consider the function 



/3 

(102) sin (cmt) = cmt - ~. -\ ---- , 

o! 

where c is a constant and m a very small factor which may take the 
place of a mass. The series in the right member converges for 
all values of t. This function is never greater than unity for any 
value of the time; yet if its expansion in powers of m were given, 
and if the first few terms were considered without the law of the 
coefficients being known, it might seem that the series represents 
a function which increases indefinitely in numerical value with 
the time. 

On following out the idea that the secular terms may be ex- 



229] LAGRANGE'S TREATMENT OF SECULAR VARIATIONS. 421 



pansions of functions which are always finite, Lagrange has shown 
(see Collected Works, vols. v. and vi.), under certain assumptions 
which have not been logically justified, that the secular terms are 
in reality the expansions of periodic terms of very long period. 
These terms differ from the long period variations (Art. 226) in 
that they come from the small uncompensated parts of the periodic 
variations, instead of directly from special conditions of con- 
junctions. As a rule these terms are very small, and their periods 
are much longer than those of the sensible long period terms. It 
will not be possible to give here more than a very general idea of 
the method of Lagrange. 

The first step in the method of Lagrange is a transformation of 
variables by the equations 



(103) 
and 
(104) 



= j Sin 7T/, 
= 6j COS 7T/, 

Pi = tan ij sin < 
#/ = tan ij cos 



where e h TT/, etc., are the elements of the orbit of m h and lj is a 
new variable not to be confused with the mean longitude. These 
transformations are to be made simultaneously in the elements of 
the orbits of all of the planets. The elements a/ and e/ remain 
without transformation. On omitting the subscripts, it is found 
from (103) and (104) that 



(105) 



' dh 


dir . . de 


dl 


= - e sin 


w 


+ cosx (fe 


dR 

de 


_ dR dh 
~ dh de 


dR 


az _ . 

de ~ 


dR . dR 

7T -TT- + COS T -77 
6/1 6t 


[ ; 


dR 


_dRdh 


dR 


dl _ 


a^ 


dR 


dir 


~ dh dir 


dl 


dir 


18 ^aF" 


dl 


Tt 


= -f- tan i 


cos 


a d& 
1 rf< 


^<K 

sec 2 1 sin ft 37 , 
at 




dq 
_ dt 


= tan i 


sin 


n d& . 
d< " 


-.in 

sec 2 z cos & -T. , 
at 





422 LAGRANGE'S TREATMENT OF SECULAR VARIATIONS. [229 



dR = dR dp dR dq 
d& ~ dp d& dq d& 

^dR . . 

= tan i cos Q> - -- tan i sin 

dp 



= , 

di ~ dp di dq di 

dR 



dK 
dq' 



dR 



= sec 2 i sin Q> - \- sec 2 i cos & -r . 



dp 

Then it follows from (72) that 
e& m 2 Vl - h 2 - I 2 



dq 



(106) H 



na 



m 2 Vl - h 2 - I 2 



no 



mjtan- 



Vl -h z -l 2 



na 



- m 2 Vl - /t 2 - 



2 



m 2 Vl - A 2 - 



dR 



na 



m 2 A tan 



dR 




na 2 Vl - /i 2 - 



cos 



3 



2na 2 Vl - /i 2 - / 2 cos i cos 2 



+ 



1, 
ae J 



na 



2 Vl - /I 2 - Z 2 cos 3 i 3? 



ra# + aj 

t- L a?r 6e J 



2na 2 Vl h 2 I 2 cos i cos 2 - 

2i 

On developing the right members of these equations and neglecting 
all terms of degree higher than the first* in h, I, p, and q, these 

* The terms of order higher than the first are neglected throughout in a 
later step in the method. 



229] LAGRANGE'S TREATMENT OF SECULAR VARIATIONS. 423 

equations reduce to 

dh 



(107) 



~dt 



no? dh ' 
m z dR 



dp _ 

dt " *" no? dq ' 

dq _ m 2 dR 

~dt~ ~ na 2 ~dp ' 



The terms which involve the derivative of R with respect to e, i, 
and TT do not appear in these equations because they involve h, I, 
p, or q as a factor. This fact follows from the properties of C 
given in Art. 226 and the form of equations (103) and (104). 

Each perturbing planet contributes terms in the right members 
of equations (107) similar to the ones written which come from ra 2 . 
These differential equations are not strictly correct, since the 
first approximation has already been made in neglecting the higher 
powers of the variables. 

The second step is in the method of treating the differential 
equations. The expansions of the Ri, / contain certain terms 
which are independent of the time, which in the ordinary method 
give rise to the secular terms. Let R w i, / represent these terms. 
Lagrange then treated the differential equations by neglecting the 
periodic terms in R it ,-, and writing 



(i =1, -, n] j 4= i), 



(108) 




The values of hi, Z, p, and g determined from equations 
(108) are used instead of the secular terms obtained by the 
method of Art. 227. The process of breaking up a differential 
equation in this manner is not permissible except as a first approxi- 
mation, and any conclusions based on it are open to suspicion. 



424 



LAGEANGE S TREATMENT OF SECULAR VARIATIONS. 



[229 



In spite of the logical defects of the method and the fact that it 
cannot be generally applied, there is little doubt that in the 
present case it gives an accurate idea of the actual manner in which 
the elements vary. 

The right members of equations (108) are expanded in powers of 
hi, li, p^ and #, and all of the terms except those of the first degree 
are neglected; consequently the terms omitted in (107) would 
have disappeared here if they had been retained up to this point. 
The system becomes linear, and the detailed discussion of the 
R it j shows that it is homogeneous, giving equations of the form 



(109) 



dfe 

dt 



dh 



and a similar system of equations in the PJ and the <?/. 

The coefficients c t / depend only on the major axes (the e/ not 
appearing in the secular terms) which are considered as being 
constants, since the major axes have no secular terms in the 
perturbations of the first and second orders with respect to the 
masses. It is to be noted here that the assumption that the c^ 
are constants is not strictly true because the major axes have 
periodic perturbations which may be of considerable magnitude. 

When these linear equations are solved by the method used in 
Art. 160, the values of the variables are found in the form 



(110) 



Pi - 



230] PERTURBATIONS BY MECHANICAL QUADRATURES. 425 

where the H^, L^, P ih and Qij, are constants depending upon the 
initial conditions. A detailed discussion shows that the X/ and /*/ 
are all pure imaginaries with very small absolute values; there- 
fore the hi, It, pi, and g t - oscillate around mean values with very 
long periods. Or, since the e,- and tan t,- are expressible as the 
sums of squares of the h,, l h p h and q h it follows that they also 
perform small oscillations with long periods; for example, the 
eccentricity of the earth's orbit is now decreasing and will continue 
to decrease for about 24,000 years. 

Equations (109) admit integrals first found by Laplace in 1784, 
which lead practically to the same theorem. They are 

mjn,jaf(hf + If) = Constant = C, 
(111) 



or, because of (103) and (104), 

mjrijafef = C, 

_ fij^-. C ^ <5^^ 

jUjaj 2 tan 2 ty = C', 

where n,- is the mean motion of my. The constants C and C" as 
determined by the initial conditions are very small, and since the 
left members of (112) are made up of positive terms alone, no e, 
or ij can ever become very great. There might be an exception 
if the corresponding my were very small compared to the others. 

Equations (112) give the celebrated theorems of Laplace that 
the eccentricities and inclinations cannot vary except within very 
narrow limits. Although the demonstration lacks complete rigor, 
yet the results must be considered as remarkable and significant. 
Equations (112) do not give the periods and amplitudes of the 
oscillations as do equations (110). 

230. Computation of Perturbations by Mechanical Quadratures. 

If the second term of the second factor of (84) in absolute value is 
greater than unity, the series (87) does not converge and cannot 
be used in computing perturbations. The expansions may fail 
because r\ and r 2 are very nearly equal; or, sometimes when they 
are not nearly equal, because / is large. In the latter case 



426 PERTURBATIONS BY MECHANICAL QUADRATURES. [230 

another mode of expansion sometimes can be employed, * but there 
are cases in which neither method leads to valid results. They 
both fail if the two orbits placed in the same plane would intersect, 
for in this case 

r 2 i, 2 = ri 2 + r 2 2 - 2r l r 2 cos (ui - u 2 + r 2 - n), 

would vanish when the two bodies arrive at a point of inter- 
section of their orbits at the same time. Unless the periods are 
commensurable in a special way this would always happen. Of 
course, it is not necessary that ri, 2 should actually vanish in 
order that the expansion of (84) should fail to converge. 

Perturbations can be computed by the method of mechanical 
quadratures without expanding the perturbative function explicitly 
in terms of the time. Consequently, this method can be used in 
computing the disturbing effects of planets on comets and in other 
cases where the expansion of R i, 2 fails altogether or converges 
slowly. Let s represent an element of the orbit of Wi; then 
equations (77) can be written in the form 



dt 
and the perturbations of the first order in the interval t n t Q are 

(113) 8 = So + 



where s is the value of s at t = to. 

The only difficulty in computing perturbations is in forming the 
integrals indicated in (113). When the perturbative function can- 
not be expanded explicitly in terms of t the primitive of the 
function f a (t) cannot be found. But in any case the values of 
f(t) can be found for any values of t, and from the values of f s (t) 
for special values of t an approximation to the integral can be 
obtained. Geometrically considered, the integral (113) is the 
area comprised between the -axis and the curve / = f,(t) and the 
ordinates t and t n . An approximate value of the integral is 

8 = SO +/.( )(<1 ~ <o) +/.(l)(2 ~ *l) + ' "-\-fs(tn-l)(tn ~ <n-l). 

The intervals ti t , t z ti, - - , t n t n _i can be taken so small 
that the approximation will be as close as may be desired. 

Another method of obtaining an approximate value of the inte- 
* Tisserand, Mecanique Celeste, vol. i., chap, xxvin. 



230] PERTURBATIONS BY MECHANICAL QUADRATURES. 427 

gral is to replace the curve / 8 (0, whose explicit value in convenient 
form may not be obtainable, by a polynomial curve of the nth 
degree which agrees in value with f s (i) at t = to, ti, , t n . The 
equation of this polynomial is 

f. 



(-l)(*-*2) 


(t-Q ffn 


' (*o- *i)(o- fc) ' 
(-*o)(*- IV 


o - U /s(iu) 


' (tl ~ to)(t! - t 2 ) '" 


(4 4 \ Ja^l) 

(tl t n ) 



^ 



Since there is no trouble in forming the integral of a polynomial 
there is no trouble in computing the perturbation of s for the in- 
terval t n to. If the value of the function f s (t) is not changing 
very rapidly or irregularly, its representation by a polynomial is 
very exact provided the intervals ti to, - , t n t n -\ are not 
too great. 

However, the area between the polynomial, the -axis, and the 
limiting ordinates is not the best approximation to the value of 
the integral that can be obtained from the values of f a (t) at t , 
- , t^ The values of the function give information respecting 
the nature of the curvature of the curve between the ordinates 
(this being true, of course, only because the function f,(t) is a 
regular function of t), and corrections of the area due to these 
curvatures can easily be made. Ordinarily they would involve the 
derivatives of f s (t) at o, , t n , which would require a vast amount 
of labor to compute; but the derivatives can be expressed with 
sufficient approximation in terms of the successive differences of 
the function, and the differences are obtained directly from the 
tabular values by simple subtraction. The derivation of the 
most convenient explicit formulas is a lengthy matter and must 
be omitted.* 

Suppose the computation of the integrals from the values of 
f a (t) at t = to, , t n has not given results which are sufficiently 
exact. More exact ones can be obtained by dividing the interval 
t n to into a greater number of sub-intervals. A little experience 
usually makes it unnecessary to subdivide the intervals first chosen. 

* See Tisserand's Mecanique Ctleste, vol. iv., chaps, x. and XL; and Char 
lier's Mechanik des Himmels, vol. n., chap. 1. 



428 PERTURBATIONS BY MECHANICAL QUADRATURES. [230 

There is a second reason why the results obtained by mechanical 
quadratures may not be sufficiently exact. It has so far been 
assumed that/ s (0 is a function of t alone; or, in other words, that 
the elements of the orbits on which it depends are constants. 
This is*the assumption in computing perturbations of the first 
order. If it is not exact enough, new values of / 8 (i), , f s (t n ) 
can be computed, on using in them the respective values of 
the elements s which were found by the first integration. From 
the new values of f s (ti), , f s (tn) a more approximate value of 
the integral can be obtained. Unless the interval t n t is too 
great this process converges and the integral can be found with 
any desired degree of approximation, because this method is 
simply Picard's method of successive approximations whose 
validity has been established.* In practice it is always advisable 
to choose the interval t n t so short that no repetition of the 
computation with improved values of the function at the ends of 
the sub-intervals will be required. At each new stage of the inte- 
gration the values of the elements at the end of the preceding 
step are employed. It follows that the method, as just explained, 
enables one to compute not only the perturbations of the first order, 
but perturbations of all orders except for the limitations that 
the intervals cannot be taken indefinitely small and the compu- 
tation cannot be made with indefinitely many places. 

The process of computing perturbations by the method of 
mechanical quadratures, as compared with that of using the 
expanded form of the perturbative function, has its advantages 
and its disadvantages. It is an advantage that in employing 
mechanical quadratures it is not necessary to express the per- 
turbing forces explicitly in terms of the elements and the time. 
This is sometimes of great importance, for, in cases where the 
eccentricities and inclinations are large, as in some of the asteroid 
orbits, these expressions, which are series, are very slowly con- 
vergent; and in the case of orbits whose eccentricities exceed 
0.6627, or of orbits which have any radius of one equal to any 
radius of the other the series are divergent and cannot be used. 
The method of mechanical quadratures is equally applicable to 
all kinds of orbits, the only restriction being that the intervals 
shall be taken sufficiently short. It is the method actually em- 
ployed, in one of its many forms, in computing the perturbations 
of the orbits of comets. 

* Picard's Traite d' Analyse, vol. n., chap. XL, section 2. 



231] GENERAL REFLECTIONS. 429 

The disadvantages are that, in order to find by mechanical 
quadratures the values of the elements at any particular time, 
it is necessary to compute them at all of the intermediate epochs. 
Being purely numerical, it throws no light whatever on the general 
character of perturbations, and leads to no general theorems 
regarding the stability of a system. These are questions of 
great interest, and some of the most brilliant discoveries in Ce- 
lestial Mechanics have been made respecting them. 

231. General Reflections. Astronomy is the oldest science 
and in a certain sense the parent of all the others. The relatively 
simple and regularly recurring celestial phenomena first taught 
men, in the days of the ancient Greeks, that Nature is systematic 
and orderly. The importance of this lesson can be inferred from 
the fact that it is the foundation on which all science is based. 
For a long time progress was painfully slow. Centuries of obser- 
vations and attempts at theories for explaining them were neces- 
sary before it was finally possible for Kepler to derive the laws 
which are a first approximation to the description of the way in 
which the planets move. The wonder is that, in spite of the 
distractions of the constant struggles incident to an unstable 
social order, there should have been so many men who found their 
greatest pleasure in patiently making the laborious observations 
which were necessary to establish the laws of the celestial motions. 

The work of Kepler closed the preliminary epoch of two thousand 
years, or more, and the brilliant discoveries of Newton opened 
another. The invention of the Calculus by Newton and Leibnitz 
furnished for the first time mathematical machinery which was 
at all suitable for grappling with such difficult problems as the 
disturbing effects of the sun on the motion of the moon, or the 
mutual perturbations of the planets. It was fortunate that the 
telescope was invented about the same time; for, without its use, 
it would not have been possible to have made the accurate obser- 
vations which furnished the numerical data for the mathematical 
theories and by which they were tested. The history of Celestial 
Mechanics during the eighteenth century is one of a continuous 
series of triumphs. The analytical foundations laid by Clairaut, 
d'Alembert, and Euler formed the basis for the splendid achieve- 
ments of Lagrange and Laplace. Their successors in the nine- 
teenth century pushed forward, by the same methods on the 
whole, the theories of the motions of the moon and planets to 
higher orders of approximation and compared them with more 



430 PROBLEMS. 

and better observations. In this connection the names of Lever- 
rier, Delaunay, Hansen, and Newcomb will be especially remem- 
bered. Near the close of the nineteenth century a third epoch 
was entered. It is distinguished by new points of view and new 
methods which, in power and mathematical rigor, enormously 
surpass all those used before. It was inaugurated by Hill in his 
Researches on the Lunar Theory, but owes most to the brilliant con- 
tributions of Poincare to the Problem of Three Bodies. 

At the present time Celestial Mechanics is entitled to be regarded 
as the most perfect science and one of the most splendid achieve- 
ments of the human mind. No other science is based on so many 
observations extending over so long a time. In no other science 
is it possible to test so critically its conclusions, and in no other 
are theory and experience in so perfect accord. There are thou- 
sands of small deviations from conic section motion in the orbits 
of the planets, satellites, and comets where theory and the obser- 
vations exactly agree, while the only unexplained irregularities 
(probably due to unknown forces) are a very few small ones in 
the motion of the moon and the motion of the perihelion of the 
orbit of Mercury. Over and over again theory has outrun practise 
and indicated the existence of peculiarities of motion which had 
not yet been derived from observations. Its perfection during 
the time covered by experience inspires confidence in following it 
back into the past to a time before observations began, and into 
the future to a time when perhaps they shall have ceased. As 
the telescope has brought within the range of the eye of man the 
wonders of an enormous space, so Celestial Mechanics has brought 
within reach of his reason the no lesser wonders of a correspond- 
ingly enormous time. It is not to be marveled at that he finds 
profound satisfaction in a domain where he is largely freed from 
the restrictions of both space and time. 

XXVII. PROBLEMS. 

1. Suppose (a) that R i, 2 is large and nearly constant; (6) that R\,z is 
large and changing rapidly; (c) that Ri, 2 is small and nearly constant. If the 
perturbations are computed by mechanical quadratures how should the 
/ to be chosen relatively in the three cases, and how should the numbers of 
subdivisions of t n t compare? 

2. The perturbative function involves the reciprocal of the distance from 
the disturbing to the disturbed planets. This is called the principal part and 
gives the most difficulty in the development. How many separate reciprocal 



HISTORICAL SKETCH. 431 

distances must be developed in order to compute, in a system of one sun and 
n planets, (a) the perturbations of the first order of one planet; (6) the per- 
turbations of the first order of two planets; (c) the perturbations of the second 
order of one planet; and (d) the perturbations of the third order of one planet? 

3. What simplifications would there be in the development of the per- 
turbative function if the mutual inclinations of the orbits were zero, and if 
the orbits were circles? 

4. What sorts of terms will in general appear in perturbations of the third 
order with respect to the masses? 



HISTORICAL SKETCH AND BIBLIOGRAPHY. 

The theory of perturbations, as applied to the Lunar Theory, was developed 
from the geometrical standpoint by Newton. The memoirs of Clairaut and 
D'Alembert in 1747 contained important advances, making the solutions 
depend upon the integration of the differential equations in series. Clairaut 
soon had occasion to apply his processes of integration to the perturbations 
of Halley's comet by the planets Jupiter and Saturn. This comet had been 
observed in 1531, 1607, and 1682. If its period were constant it would pass 
the perihelion again about the middle of 1759. Clairaut computed the 
perturbations due to the attractions of Jupiter and Saturn, and predicted that 
the perihelion passage would be April 13, 1759. He remarked that the time 
was uncertain to the extent of a month because of the uncertainties in the 
masses of Jupiter and Saturn and the possibility of perturbations from un- 
known planets beyond these two. The comet passed the perihelion March 13, 
giving a striking proof of the value of Clairaut's methods. 

The theory of the perturbations of the planets was begun by Euler, whose 
memoirs on the mutual perturbations of Jupiter and Saturn gained the prizes 
of the French Academy in 1748 and 1752. In these memoirs was given the 
first analytical development of the method of the variation of parameters. 
The equations were not entirely general as he had not considered the elements 
as being all simultaneously variables. The first steps in the development of 
the perturbative function were also given by Euler. 

Lagrange, whose contributions to Celestial Mechanics were of the most 
brilliant character, wrote his first memoir in 1766 on the perturbations of 
Jupiter and Saturn. In this work he developed still further the method of 
the variation of parameters, leaving his final equations, however, still incorrect 
by regarding the major axes and the epochs of the perihelion passages as 
constants in deriving the equations for the variations. The equations for 
the inclination, node, and longitude of the perihelion from the node were 
perfectly correct. In the expressions for the mean longitudes of the planets 
there were terms proportional to the first and second powers of the time. 
These were entirely due to the imperfections of the method, their true form 
being that of the long period terms, as was shown by Laplace in 1784 by 
considering terms of the third order in the eccentricities. The method of the 
variation of parameters was completely developed for the first time in 1782 
by Lagrange in a prize memoir on the perturbations of comets moving in 



432 HISTORICAL SKETCH. 

elliptical orbits. By far the most extensive use of the method of variation of 
parameters is due to Delaunay, whose Lunar Theory is essentially a long 
succession of the applications of the process, each step of it removing a term 
from the perturbative function. 

In 1773 Laplace presented his first memoir to the French Academy of 
Sciences. In it he proved his celebrated theorem that, up to the second 
powers of the eccentricities, the major axes, and consequently the mean 
motions of the planets, have no secular terms. This theorem was extended 
by Lagrange in 1774 and 1776 to all powers of the eccentricities and of the sine 
of the angle of the mutual inclination, for perturbations of the first order with 
respect to the masses. Poisson proved in 1809 that the major axes have no 
purely secular terms in the perturbations of the second order with respect to 
the masses. Haretu proved in his Dissertation at Sorbonne in 1878 that 
there are secular variations in the expressions for the major axes in the terms 
of the third order with respect to the masses. In vol. xix. of Annales de 
I'Observatoire de Paris, Eginitis considered terms of still higher order with 
respect to the masses. 

Lagrange began the study of the secular terms in 1774, introducing the 
variables h, I, p, and q. The investigations were carried on by Lagrange 
and Laplace, each supplementing and extending the work of the other, until 
1784 when their work became complete by Laplace's discovery of his celebrated 
equations 

C, 



rf rajttj-a/ 2 tan 2 ij = C'. 
I 

These equations were derived by using only the linear terms in the differential 
equations. Leverrier, Hill, and others have extended the work by methods of 
successive approximations to terms of higher degree. Newcomb (Smithsonian 
Contributions to Science, vol. xxi., 1876) has established the more far-reaching 
results that it is possible, in the case of the planetary perturbations, to repre- 
sent the elements by purely periodic functions of the time which formally 
satisfy the differential equations of motion. If these series were convergent 
the stability of the solar system would be assured; but Poincare has shown 
that they are in general divergent (Les Methodes Nouvelles de la Mecanique 
Celeste, chap. ix.). Lindstedt and Gylden have also succeeded in integrating 
the equations of the motion of n bodies in periodic series, which, however, 
are in general divergent. 

Gauss, Airy, Adams, Leverrier, Hansen, and many others have made 
important contributions to the planetary theory in some of its many aspects. 
Adams and Leverrier are noteworthy for having predicted the existence and 
apparent position of Neptune from the unexplained irregularities in the motion 
of Uranus. More recently Poincare" turned his attention to Celestial Mechanics, 
publishing a prize memoir in the Ada Mathematica, vol. xm. This memoir 
was enlarged and published in book form with the title Les Methodes Nouvelles 
de la Mecanique Celeste. Poincare" applied to the problem all the resources 
of modern mathematics with unrivaled genius; he brought into the investiga- 
tion such a wealth of ideas, and he devised methods of such immense power 



HISTORICAL SKETCH. 433 

that the subject in its theoretical aspects has been entirely revolutionized in 
his hands. It cannot be doubted that much of the work of the next fifty 
years will be in amplifying and applying the processes which he explained. 

The following works should be consulted : 

Laplace's Mecanique Celeste, containing practically all that was known of 
Celestial Mechanics at the time it was written (1799-1805). 

On the variation of parameters Annales de V Observatoire de Paris, vol. i. ; 
Tisserand's Mecanique Celeste, vol. i.; Brown's Lunar Theory; Dziobek's 
Planeten-Be wegungen . 

On the development of the perturbative function Annales de I 'Observatoire 
de Paris, vol. i.; Tisserand's Mecanique Celeste, vol. i.; Hansen's Entwickelung 
des Products einer Potenz des Radius-Vectors mil dem Sinus oder Cosinus eines 
Vielfachen der wahren Anomalie, etc., Abh. d. K. Sachs. Ges. zu Leipzig, vol. n.; 
Newcomb's memoir on the General Integrals of Planetary Motion; Poincare, 
Les Methodes Nouvelles, vol. i., chap. vi. 

On the stability of the solar system Tisserand's Mecanique Celeste, vol. i., 
chaps. XL, xxv., xxvi., and vol. iv., chap, xxvi.; Gylden, Traite Analytique 
des Orbites absolues, vol. i.; Newcomb, Smithsonian Cont., vol. xxi.; Poincare, 
Les Methodes Nouvelles de la Mecanique Celeste, vol. n., chap. x. 

On the subject of Celestial Mechanics as a whole there is no better work 
available than that of Tisserand, which should be in the possession of every 
one giving special attention to this subject. Another noteworthy work is 
Charlier's Mechanik des Himmels, which, besides maintaining a high order of 
general excellence, is unequaled by other treatises in its discussion of periodic 
solutions of the Problem of Three Bodies. 



29 



INDEX. 



Abbott, 66 

Acceleration in rectilinear motion, 9 
curvilinear motion, 1 1 
Adams, 363, 432 
Airy, 363, 365, 432 
Albategnius, 32 
Allegret, 319 
Almagest, 32 
Al-Sufi, 32 
Anaximander, 31 
Annual equation, 348 
Anomaly, eccentric, 159 

mean, 159 

true, 155 

Appell, 7, 10, 35, 97, 162 
Archimedes, 33 
Areal velocity, 15 
Argument of latitude, 162 
Aristarchus, 31 
Aristotle, 31 

Atmospheres, escape of, 46 
Attraction of circular discs, 103 

ellipsoids 99, 122, 127 
spheres, 99, 101, 104, 

114 
spheroids, 119, 132 ; 133 

Backhouse, 305 
Ball, W. W. R., 35 
Baltzer, 376 
Barker's tables, 156 
Barnard, 305 
Bauschinger, 260 
Bernouilli, Daniel, 190 

J., 67 
Berry, 35 
Bertrand, 97 
Boltzmann, 3, 67 
Bour, 319 
Brorsen, 305 

Brown, 351, 352, 365, 433 
Bruns, 218, 276, 281 
Buchanan, Daniel, 320 
Buchholz, 222, 260 
Buck, 320 
Budde, 35 
Burbury, 67 
Burnham, 85 

Cajori, 35 

Calory, 60 

Canonical equations, 390 

Cantor, 35 



Carmichael, 35 
Cauchy, 367, 378 
Center of gravity, 22 

mass, 19, 20, 24 
Central force, 69 
Chaldaeans, 31 
Chamberlin and Salisbury, 68 
Charlier, 216, 259, 427, 433 
Chasles, 138, 139 
Chauvenet, 190, 197 
Circular orbits for three bodies, 309 
Clairaut, 356, 363, 364, 367, 429, 431 
Clausius, 67 

Contraction theory of sun's heat, 63 
Copernicus, 33 

d'Alembert, 3, 7, 363, 429, 431 
Damoiseau, 364 
Darboux, 97, 138 

Darwin, 68, 139, 280, 281, 305, 320 
Delambre, 35 
Delaunay, 364, 430, 432 
De Pontecoulant, 364 
Descartes, 190 
Despeyrous, 97, 138 
Differential corrections, 162, 220 
Differential equations of orbit, 80 
Dirichlet, 138 

Disturbing forces, resolution of, 324 
Doolittle, Eric, 361 
Double points of surfaces of zero ve- 
locity, 290 

Double star orbits, 85 
Duhring, 35 
Dziobek, 433 

Eccentric Anomaly, 159 
Eginitis, 432 
Egyptians, 30 

Elements of orbits, 146, 148, 183 
Elements, intermedisrte, 192 
Energy, kinetic, potential, 59 
Equations of relative motion, 142 
Equipotential curves, 283 

surfaces, 113 
Eratosthenes, 31 
Escape of atmospheres, 46 
Euclid, 32 
Euler, 24, 34, 138, 158, 190, 258, 363, 

364, 367, 429, 431 
Euler's equation, 157, 275 
Evection, the, 359 



434 



INDEX. 



435 



Falling bodies, 36 

Force varying as distance, 90 

inversely as square of 
distance, 92 
fifth power 
of dis- 
tance, 93 

Galileo, 3, 33, 34, 67 
Gauss, 138, 139, 153, 154, 188, 190, 
193, 194, 231, 238, 240, 242, 243, 
244, 249, 259, 260, 360, 361, 432 
Gauss' equations, 238, 240 
Gegenschein, 305 
Gibbs, 260 
Glaisher, 97 
Grant, 35 

Greek philosophers, 30, 429 
Green, 109, 138, 139 
Griffin, 88, 97, 320 
Gylden, 305, 432 

Halley, 258, 348, 363 

Halphen, 97 

Hamilton, 3, 275 

Hankel, 35 

Hansen, 364, 430, 432, 433 

Haretu, 432 

Harkness and Morley, 292 

Harzer, 231, 232, 259 

Heat of sun, 59 

Height of projection, 45 

Helmholtz, 63, 68 

Herodotus, 30 

Herschel, John, 325, 365 

William, 85 
Hertz, 3, 35 
Hilbert, 67 
Hill, 68, 280, 281, 287, 319, 351, 352, 

356, 361, 365, 430, 432 
Hipparchus, 31, 32, 359 
Holmes, 68 
Homoeoid, 100 
Huyghens, 34 

Ideler, 35 

Independent star-numbers, 194 
Infinitesimal body, 277 
Integrals of areas, 144, 264 

center of mass 141, 262 
Integral of energy, 267 
Integration in series, 172, 200, 202, 

227, 377 

Invariable plane, 266 
Ivory, 116, 127, 132, 138 

Jacobi, 139, 267, 274, 275, 280, 281, 

319 

Jacobi's integral, 280 
Jeans, 67 
Joule, 60 



Kepler, 33, 82, 83, 152, 190, 429 
Kepler's equation, 159, 160, 163, 165 

laws, 82 

third law, 152 
Kinetic theory of gases, 46 
Kirchhoff, 3 
Klinkerfues, 222, 260 
Koenigs, 35, 97 

Laertius, 30 

Lagrange, 7, 34, 107, 132, 138, 161, 

193, 227, 259, 277, 312, 319, 363, 

364, 387, 418, 421, 423, 429, 431, 

432 
Lagrange's brackets, 387 

quintic equation, 312 
Lagrangian solutions of the problem 

of three bodies, 277, 291, 309, 313 
Lambert, 158, 258, 259 
Lane, 68 
Laplace, 34, 132, 138, 172, 193, 194, 

231, 249, 258, 259, 266, 275, 319, 

348, 350, 352, 362, 364, 367, 418, 

425, 429, 431, 432, 433 
Laue, 35 
Law of areas, 69 

converse of, 73 
force in binary stars, 86 
Laws of angular and linear velocity, 73 
Kepler, 82 
motion, 3 
Lebon, 35 
Legendre, 97, 138 
Lehmann-Filhes, 319 
Leibnitz, 429 
Leonardo da Vinci, 33 
Leuschner, 222, 231, 232, 259 
Level surfaces, 113 
Leverrier, 361, 363, 400, 406, 413, 

430, 432 
Levi-Civita, 268 
Linstedt, 319, 432 
Liouville, 319 
Long period inequalities 361, 371, 

416 

Longley, 320 
Love, 35 
Lubbock, 364 
Lunar theory, 337 

MacCullagh, 138 
Mach, 3, 6, 35 
Maclaurin, 34, 132, 139 
MacMillan, 169, 320 
Marie, 35 
Mathieu, 319 
Maxwell, 67 
Mayer, Robert, 68 
Tobias, 364 
McCormaok, 35 
Mean anomaly, 159 



436 



INDEX. 



Mechanical quadratures, 425 

Meteoric theory of sun's heat, 62 

Meton, 31 

Metonic cycle, 31 

Meyer, O. E., 67 

Motion of apsides, 352 

center of mass, 141, 262 
falling particles, 36 

Neumann, 139 

Newcomb, 275, 361, 430, 432, 433 

Newton, H. A., 62, 305 

Newton, 3, 5, 6, 7, 29, 33, 34, 67, 82, 
84, 97, 99, 101, 138, 190, 258, 275, 
320, 327, 350, 356, 365, 429, 431 

Newton's law of gravitation, 82, 84 
laws of motion, 3 

Normal form of differential equations, 
75 

Node, ascending, descending, 182 

Nyren, 318 

Gibers, 259 

Omar, 32 

Oppolzer, 156, 222, 242, 260, 370 

Order of differential equations, 74 

Osculating conic, 322 

Parabolic motion, 56 

Parallactic inequality, 352 

Parallelogram of forces, 5 

Periodic variations, 371, 413 

Perturbations, meaning of, 321 

by oblate body, 333 
resisting medium, 333 
of apsides, 352 
elements, 322, 382 
first order, 382 
inclination, 343 
major axis, 346 
node, 342 
period, 348 

Perturbative function, 272 

resolution of, 337, 

338, 345, 402 
development of, 406 

Peurbach, 32 

Picard, 378, 428 

Plana, 364 

Planck, 35 

Plummer, 302 

Poincare, 35, 139, 267, 268, 275, 276, 
281, 320, 367, 378, 432, 433 

Poisson, 6, 138, 371, 418, 420, 432 

Poisspn terms, 371 

Position in elliptic orbits, 158 

hyperbolic orbits, 177 
parabolic orbits, 155 

Potential, 109, 261 

Precession of equinoxes, 344 

Preston, 60 



Problem of two bodies, 140 
three bodies, 277 
n bodies, 261 

Ptolemy, 32, 359 

Pythagoras, 31 

Question of new integrals, 268 

Radau, 274, 319 

Ratios of triangles, 233, 237 

Rectilinear motion, 36 

Regiomontanus, 32 

Regions of real and imaginary ve- 
locity, 286 

Relativity, principle of, 4 

Resolution of disturbing force, 337, 
338 

Risteen, 67 

Ritter, 68 

Rodriguez, 138 

Routh, 35, 139 

Rowland, 60 

Rutherford, 68 

Salmon, 88 

Saracens, 32 

Saros, 31 

Secular acceleration of moon's motion, 
348 

Secular variations, 360, 371, 417 

Solid angles, 98 

Solution of linear equations by ex- 
ponentials, 41 

Solutions of problem of three bodies, 
290, 309, 313 

Speed, 8 

Spencer, 59 

Stability of solutions, 298, 306 

Stader, 97 

Stevinus, 33, 67 

Stirling, 138 

Stoney, 46 

Sturm, 139 

Surfaces of zero relative velocity, 281 

Sliter, 35 

Tait, 35 

Tait and Steele, 35, 97 

Tannery, 35 

Temperature of meteors, 61 

Thales, 30, 31 

Thomson, 139 

Thomson and Tait, 3, 104, 139, 283 

Time aberration, 226 

Tisserand, 97, 139, 190, 260, 267, 276, 

295, 296, 312, 319, 365, 391, 407, 

426, 427, 433 
Tisserand's criterion for identity of 

comets, 295 
True anomaly, 155 
Tycho Brahe, 33, 348, 350 



INDEX. 437 

Uniform motion, 8 Vis viva integral, 78, 267 

Ulugh Beigh, 32 Voltaire, 190 
Units, 153 

canonical, 154 Waltherus, 32 

Variation, the, 350 Waterson, 162 

Variation of coordinates, 321 Watson, 156, 242, 260 

elements, 322 Weierstrass, 367 

parameters 50, 322 Whewell, 35 

Vector, 5 Williamson, 161 

Velocity, 8 Wolf, 35 

areal, 15 Woodward, 4 

from infinity, 45, 46 Work, 59 
of escape, 48 

Villarceau, 259 Young, 164 



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