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 nonhomogeneous 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 socalled 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 selfconsistent, 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 noncoplanar 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 zaxes 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 yaxes, 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 yaxes,
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
Xaxis
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 semiaxes 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 zaxes. 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 yaxes 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 yaxes 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
il
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/zplane, 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 zaxis 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 (17071783) 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
fffor 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 NonHomogeneous 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/2plane 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 semiaxes 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 semiaxes 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 concaveconvex 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 (640546 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 (611545 B.C.), a friend and probably a pupil of
Thales, constructed geographical maps, and is credited with
having invented the gnomon.
Pythagoras (about 569470 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 465385 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 (384322 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 (310250 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 onenineteenth 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 (275194 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 190120 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 330275 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 (100170 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
(850929), 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 AlSufi made a catalogue of stars based on his
own observations. Another catalogue was made by the direction
of Ulugh Beigh (13931449), 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 (14231461),
Waltherus (14301504), and Regiomontanus (14361476). It
was given a great impetus by the celebrated German astronomer
27] FROM ANCIENT TIMES TO NEWTON. 33
Copernicus (14731543), 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 (15461601),
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 (15711630) 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 (16421727) derived the law of gravitation.
Galileo (15641642), 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 (287212 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 (14521519) 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 (15481620),
4
34 HISTORICAL SKETCH TO NEWTON. [27
who was the first, in 1586, to investigate the mechanics of the
inclined plane, and of Galileo (15641642), 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 (16291695), 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 (16421727) 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 (17071783) in his work,
Mechanica sive Motus Scientia (Petersburg, 1736) ; it was improved
by Maclaurin (16981746) in his work, A Complete System of
Fluxions (Edinburgh, 1742), and was highly perfected by Lagrange
(17361813) in his Mecanique Analytique (Paris, 1788). The
Mecanique Celeste of Laplace (17491827) 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 textbooks. 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^HfcCzV lfc = o',
Therefore
fciVlJfc
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 onefourth 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) tani =  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 _
gA: *
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 yaxis 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) yxt*** 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 onefourth 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 yaxes 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 zaxis.
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 farreaching 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
mjpds =  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 Onethousandth 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
dVrJ^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 Cl 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 twentyseven 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 subatomic 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 fourthousandth 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. Subatomic 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 xyplsme 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 _ dXnz
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)
ii
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
dr.
\(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 **...
, Xnl, C, t),
', X n i, C, 0,
dXn1
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 singlevalued 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 semiaxes 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 semiaxis 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 twentysix 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+fyg)*'
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 +fyig = 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. 6285.
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 zaxis; 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 xaxis 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 nonhomogeneous 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 *lA*u
whence
_
where coth ^ ( =t 6 c 2 ) is the hyperbolic cotangent of
iV2( 0c 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 = (lfc)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. 282306.
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 onehalf
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. 6669 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 ocoAP  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 oo 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' = STTOQ 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 zcomponents 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 onetoone 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 Xaxis, 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 plumbline 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 7TO2/C f o\r waiR 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 2axis 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 semiaxes
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~*, Cr*, 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 xcomponent of
attraction is proportional to the xcoordinate 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 semiaxes 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
Kaxts
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 onetoone correspondence between the points upon the two
80J
IVORYS METHOD.
129
ellipsoids will now be established by the equations (compare
Art. 77)
(47) rj'fc *v* rr.
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 recomponents 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 semiaxes
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 semiaxes 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 semiaxes 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 HK
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 (1e 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 zaxis, 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
semiaxes 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 fs)
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
2irafjLabc 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, 18091828, 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, 18371846, Jour. I'Ecole
Poly, and Mem. des Savants Strangers, vol. ix.; MacCullagh, 1847, Dublin
Proc., vol. in.; LejeuneDirichlet, 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 startingpoint 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 lastmentioned 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 pearshaped figure. Later computations by Sir George
Darwin (Philosophical Transactions, vol. 198) have shown it is so elongated
that it might well be called a cucumbershaped 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 zzplanes 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 zaxis 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
righthand focus is
P
r =
1 + e cos (6  co) '
89]
PROPERTIES OF THE MOTION.
149
where p is the semiparameter, 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 semiaxis; 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 semiaxis 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 selfevident 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 semiaxis 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 =  rZ = 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)
.(tanltan
whence
/ Q7 N + yyj 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 # = MM,
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
EA&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, 184950, p. 169.
164
RECAPITULATION OF FORMULAS.
[98
once for all; the second is a straight line making with the Eaxis
an angle whose tangent is l/e. Instead of drawing the straight
line a straightedge 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 straightedge 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 straightedge runs off from the
diagram, the top can be placed at M + 50e on the 50line. 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 = le"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 , , Uji) + 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
12
45 (n + 2)
12 (n 1)
]
Then the conditions similar to (57) are
(58)
4.5 ... (2k + 2)
12 .. (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
12'
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 sin3M3sinM)
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 4e 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,
(le 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 semiaxis 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 + jTF 9 +
(1 + Tf 2 ) 3
= w*+
(1 + TF 2 ) 5
When the values of these coefficients are substituted in (77) the
tangent of onehalf 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 onehalf 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 semiaxis, 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 reaxis 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 xaxis 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 xaxis 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 xaxis 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
xaxis directed toward the vernal equinox. Then, the equatorial
system can be obtained by rotating the ecliptic system around the
xaxis 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 zaxis 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 zaxis until the zaxis 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 zaxis 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 (15961650) until Vol
taire, after his visit to London 1727, vigorously supported the Newtonian
theory, 17281738. 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 (17771855), 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. 8688] 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 StarNumbers, 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 timeintervals. 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 timeintervals 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 timeintervals 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 timeintervals 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 <paxis 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 determinedwhich 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 <paxis
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 JS1
(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 Tf \ 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
916 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
hTi, 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
+ PV /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 Ts5(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. 33712 (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 timeintervals. 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 timeaberration 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
r2
= 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 timeintervals 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 wellknown 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 sinK**) = 2x* + ^ +
17
242 GAUSSIAN METHOD OF DETERMINING ORBITS. [137
Then (104) becomes
J = 4[\ , 6 , 68 . , 6810
or
X =
Let
6 68 68 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 timeinterval 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 semiaxis a is defined by the last
equation on page 240, or by the preceding equation for the longer
timeinterval 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 semiaxis 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)
03
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)03 
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 
00103 003
601
x
 03)
12
30
{4(03 oi) 2 + 0103}
If the second observation divides the whole interval into two
nearly equal parts, as generally will be the case in practice, oi
and 03 will be nearly equal. Let
(TI 03 = e, and oi + 03 = 0"2j
whence
02
03 =
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 
60103 603
+ & (2oi  03) 
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),
(tat 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
72 = #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, timeinterval, 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 ri eliminated by means of the equation which
expresses the fact that S, E, and C form a triangle at ti. He developed the
expressions for the heliocentric coordinates as power series in the timeintervals
[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. 93146). 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 zaxis 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 xcoordinate
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 xcoordinates 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,
AffjMhft,
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 zaxis 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, LeviCivita 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, , n1).
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 lnl,
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 ffiXnl _ dU
m n l ,.o V , Mn Win, Mn1 Wlnl T n,
Mn1 fe2 dU
Mn2
 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, n3
(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 184243
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 onehalf 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 epochmaking 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
=(iti
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 "
= (dM)^^ +
 (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 zaxis 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 _ ,(xx 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 rraxis. 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 xyplane
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 xyplane, 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 zzplanes, and, when
/A = , with respect to the j/zplane 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
zaxis pierces the surfaces in two (or no) real points. Moreover,
the surfaces are contained within a cylinder whose axis is the
zaxis 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 xypl&ne is
obtained by putting z equal to zero in the first of (8), and is
(xxtf + 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.
xAocis
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
dumbbell 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 rczplane 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 zaxis. 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
zzplane 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/zplane 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/2plane 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 xypl&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 xaxis, as can be seen from the symmetries of the surfaces.
The equation of the curves in the xyplane 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 bipolar 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
Xaxis
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 xypl&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 xypl&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 zaxis 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 zaxis, 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'; ip
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
zaxis 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 zaxis 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 xyplane.
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 earthlike 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
i7plane. 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 onethousandth.
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 zaxis, 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. 441479, 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 antisun, 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 2axis, 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 onehalf 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 semiaxis 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 zaxis for an infinitesimal body slightly displaced
from the xyplaue 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 xypl&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 zaxis and in the xyplane, 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 xyplane 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 154159.
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 zaxis 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
xaxis. 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 LehmannFilhes 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 twobody 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 twobody 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 semiaxis 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 semiaxis. 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 midpoint 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 semiaxis 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 semiaxis 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 xypl&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 zaxes 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 zaxis.
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 xypl&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 =  Df" 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 commonsense 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
79 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, itJs^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 semiaxis 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
SE
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 (17071783), 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 (17131765), born at Paris, and spending
nearly all his life in his native city; D'Alembert (17171783), also a native
and an inhabitant of Paris; Lagrange (17361813), 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 (17491827), 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 (17231762), 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 (17681846) carried out Laplace's method to a high degree of
approximation in 182428, and the tables which he constructed were used
quite generally until Hansen's tables were constructed in 1857. Plana
(17811869) published a theory in 1832, similar in most respects to that of
Laplace. An incomplete theory was worked out by Lubbock (18031865) in
18304. A great advance along new lines was made by Hansen (1795
1874) in 1838, and again in 18624. His tables published in 1857 were very
generally adopted for Nautical Almanacs. De Ponte"coulant (17951874)
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 (18161872) 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 (18191892)
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 (18011892) Gravitation, and Sir John Herschel's
(17921871) 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 frr = ,
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],
2ktl]
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
(i1, .., 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 nonhomogeneous, 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 dRi, 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 rjaxis 90
forward in the plane of the orbit, and the axis perpendicular to
the plane of the orbit. Let the direction cosines between the
xaxis and the , r;, and faxes be a, a', a"; between the ?/axis and
the , 77, and axes be ft t 0', #"; and between the zaxis and the
, TJ, and faxes 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
TT = 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
le j
dcr do
Then equation (57) gives
(59) [a, e] = 0, [e, <r] = 0,
de 16
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. 225227] 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 xaxis, 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/Li,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 1e 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>/
le 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 zaxes 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,
4r,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 zaxis [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 Vl _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
vl
(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)
12 (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 semiaxes. 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 fiftythree 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 twentyfour 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 semiaxis, 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(tnl)(tn ~ <nl).
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)
(tQ 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 subintervals. 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 subintervals 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 rajttja/ 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 farreaching
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 (17991805).
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
PlanetenBe 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 RadiusVectors 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
AlSufi, 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 starnumbers, 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
LehmannFilhes, 319
Leibnitz, 429
Leonardo da Vinci, 33
Leuschner, 222, 231, 232, 259
Level surfaces, 113
Leverrier, 361, 363, 400, 406, 413,
430, 432
LeviCivita, 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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