ASTRONOMY DEFT PERIODIC ORBITS HY F. R. MOULTON IN COLLABOKATION WITH DANIEL BUCHANAN, THOMAS BUCK, FRANK L. GRIFFIN, WILLIAM R. LONGLEY AND WILLIAM D. MxcMILLAN PUBLISHED BY THE CARNEGIE INSTITUTION OP WASHINGTON WASHINGTON, 1920 M B 2 6-5- M<. i/tt. CARNEGIE INSTlftlTION OF WASHINGTON / ;;/':>:P^BLWTi-6rN'No. 161 ASTRONOMY PRESS OF GIBSON BROTHERS, INC. WASHINGTON, It. C. INTRODUCTION. Tlir problem of three bodies received a great impetus ill 1S7S, \\hen Hill published his celebrated researches upon the lunar theory. His inves- tigations \\ere carried out with practical objects in mind, and compara- tively little attention was given io ihe underlying logic of the processes which he invented. For example, the legitimacy of the use of infinite determinants was assumed, the validity of the solution of infinite systems of non-linear equations was not questioned, and the conditions for the conver- gence of the infinite series which he used were stated to be quite unknown. These deficiencies in the logic of his work do not detract from the brilliancy and value of his ideas, and his skill in carrying them out excites only the highest admiration. The work of Hill was followed in the early nineties by the epoch-making (' Poincare. which were published in detail in his Les Mithodes Xiiurfllfx ill In MI'CII >u'i/m (''/*/. Poincar<5 brought to boar on the prob- lem all the resources of modern analysis. The new methods of treating the difficult problem of three bodies which he invented were so numerous and powerful as to be positively bewildering. They opened so many new fields that a generation will lx> required for their complete exploration. On the one hand, the results were in the direction of purely theoretical considera- tions, in which Birkoff has recently made noteworthy extensions; on the other hand, they foreshadowed somewhat dimly methods which will doubt- less be of great importance in practical applications in celestial mechanics. The researches of Poincare are scarcely less revolutionary in character than wen- those of Newton when he discovered the law of gravitation and laid the foundations of celestial mechanics. In 1890 Sir ( Icorge Darwin published an extensive paper on the problem of three bodies in Adu MnUu-mntica. In mathematical spirit it was similar to the work of Hill; indeed, the methods used were essentially those of Hill, but the problem treated was considerably different. For a ratio of the finite masses of ten to one, Darwin undertook to discover by numerical processes all the j>eriodic orbits of certain types and to follow their changes with \ arying values of the Jacobian constant of integration. This program was excellently carried out at the cost of a great amount of labor. It gave specific numerical results for many orbits in a particular example. The investigations contained in this volume were begun in 1900 and. with the exception of the last chapter, they were completed by 1912. Those not made by myself were carried out by students who made their doctorates under my direction. in r/39883 iv INTRODUCTION. The following chapters have been heretofore published in substance : I. Sections III and IV American Journal of Mathematics, vol. xxxm (1911). II. Astronomical Journal, vol. xxv (1907). III. Rendiconti Matematico di Palermo, vol. xxxn (1911). IV. Transactions of the American Mathematical Society, vol. xi (1910). VII. Mathematische Annalen, vol. LXXIII (1913). VIII. Annals of Mathematics, 2d Series, vol. 12 (1910) XI. Transactions of the American Mathematical Society, vol. vn (190(i). XII. Transactions of the American Mathematical Society, vol. xni (1912). XIII. Transactions of the American Mathematical Society, vol. vin (1907). XIV. Transactions of the American Mathematical Society, vol. ix (1908). XV. Proceedings of the London Mathematical Society, Series II, vol. 2 (1912). The investigations and computations contained in the last chapter were completed in 1917. It was originally intended to publish only the first fifteen chapters, and if that program had been carried out they would have appeared in 1912. But as the work of printing progressed the ideas contained in the last chapter were being developed and the computations were begun. It was thought that an even more nearly complete and certain idea of the evolution of peri- odic orbits with changing parameters could be obtained in a year than were obtained in five years. The difficulties and enormous amount of labor involved were not foreseen. No one can now read with better appreciation than I the following words from Darwin's introduction to his paper : "As far as I can see, the search resolves itself into the discussion of particular cases by numerical processes, and such a search necessarily involves a prodigious amount of work. It is not for me to say whether the enormous amount of labor I have undertaken was justifiable in the first instance ; but I may remark that I have been led on by the interest of my results, step by step, to investigate more, and again more, cases." The results which now appear had all been obtained when service in the army made it necessary to lay them aside before the final chapter could be put into form for publication. After they had been laid aside for about two years it was not easy to gather up the details again and to arrange them in a systematic order. This explains the long delay in the appearance of this volume. It is clear that it was in no wise due to the Carnegie Institu- tion of Washington. Indeed, the patience of President Woodward with long and expensive delays has been far beyond what could reasonably have been expected. In the greater part of this work complete mathematical rigor has been insisted upon. On the other hand, the developments have been in a form applicable to practical problems in celestial mechanics. For example, sections III and IV of Chapter I treat non-homogeneous equations of the types which arise in practical problems; Chapter II is devoted to questions which have long been classic in celestial mechanics; Chapter III contains, among other things, a new and rigorous treatment of Hill's differential equation with periodic coefficients; Chapter IV treats a problem that arises, at least approximately, in the solar system; ( 'liaptcr V is developed in a form suitable for numerical applications; Chapter VI is an alternative treatment of the same problem, and in ( 'hapter VII an extension of the problem in\olving entirely new types of difficulties is found and more powerful methods of treatment are required; ('hapter XI covers the same ground as Hill's work on the moon's variational orbit and Brown's work on the paral- lactic terms; ( 'hapter XII contains the corresponding discussion for superior planet-; and ( 'hapter X I V treats a problem similar to that presented by the satellite systems of Jupiter and Saturn or by the planetary system. Chapter XV contains a discussion of limiting cases of jxriodic orbits, namely, closed orbits of ejection. It forms a basis for part of the work of the last chapter, and it may be found to have practical applications in the escape of atmos- pheres. The last chapter is an attempt to trace out the evolution of periodic orbits as the parameters on which they depend are varied. In spite of the fact that infinitely many families of periodic orbits were found, whereas only a few such families were previously known, the discussion remains in cer- tain respects incomplete. It >hould be stated that many results were found which have not been included because they did not contribute to the solu- tion of the particular question under consideration. For example, a series of orbits asymptotic to the collinear equilibrium points was computed. The amount of labor the last chapter cost can scarcely be overestimated. I should not be true to my own feelings if I did not express the appre- ciation of the assistance of my collaborators. The association with them has been a deep source of satisfaction and inspiration. They are to be held accountable only for those chapters which appear under their names. Most of the computations on which many of the results of the last chapter are based were made by Dr. W. L. Hart and Dr. I. A. Barnett. Without assistance of such a lugh order the laborious computations could not have been carried out. F. R. MOULTON. CONTENTS. C'llU'tKIl I. ('Kit IAIN TllK.iKKM- o\ 1\1I'I.|.I| I I NOTIONS AND DlFFEHENTIAL Kqr \IIM\-. (H\ K. H. Moi i.rox AND W. D. McMiLLAN.) I. Solution f Im/ilicit Functions. PAGE. 1. Formal solution of simultaneous equal ion-; when (lie functional determinant i-* distinct from zero at the origin 1 2. Proof of convergence of the solution- 2 :J. < ienerali/.ation to many parameters "i 4. 'l'h functional determinant /ero hut not all its first minors zero at origin 5 .V ( 'ase where P (a, M) -a* P, (o, M) ~M X PI (a, M) 6 6. A second simple case (i 7. ( Icneral case of power series in two variables 6 I 1. Solution.* <>f l)i]fi r< nlinl Ki/Hiilions as Power Series in Parameters. 8. The ty |H>S of equations treated 10 9. Formal solution of differential equations of Type 1 10 10. Determination of the constants of integration in Tyjie I 11 1 1 . Proof of the convergence of the solutions of Type 1 11 12. (iencralization to many parameters 14 1U. ( Jenerali/ation of the parameter 14 1 I. Formal solution of differential equations of Type II 15 !.">. Determination of the constants of integration in Type II Hi lii. Pnxf of the convergence of the solutions of Type II Hi 17. Case of homogeneous linear equations 19 III. Homogeneous Linear Differential Equations with Periodic Coefficients. 18. The determinant of a fundamental set of solutions 21 19. The character of the solutions of a set of linear homogeneous differential equations with uniform periodic coefficients 22 20. Solutions when the roots of the fundamental equation are all distinct ... 24 21. Solutions when the fundamental equation has multiple roots I'll 22. The characteristic equation when the coefficients of the differential equa- tions are expansible as power series in a parameter n 30 23. Solutions when a ( , 0) , a", . . . , a are di-tinct and their differences are not congruent to zero mod. V 1 32 24. Solutions when no two aj' are equal but when a!}" o| l " is congruent to zero mod. V 1 34 25. Solutions when, for n = 0, the characteristic equation has a multiple root. . 35 26. Construction of the solutions when, for M = 0, the roots of the character- istic equation are di-iim-t and their differences are not congruent to zero mod. V 1 36 27. Construction of the solutions when the difference of two roots of the characteristic equation is congruent to zero mod. \/ 1 40 28. Construction of the solutions when two roots of the characteristic equation are equal 43 IV. Non-Homogeneous Linear Differential Equations. 29. Case where the right members are periodic with the period 2r and the a, are distinct 40 30. Case where the right members are periodic terms multiplied Hy an exponen- tial, and the a, are distinct 47 31. Case where two characteristic exponents are equal and the right members are periodic t^ V. Equations of Variation and their Characteristic Exponents. 32. The equations of variations 50 33. Theorems on the characteristic exponents ' ">1 VII VIII CONTENTS. CHAPTER II. ELLIPTIC MOTION. PAGE. 34. The differential equations of motion 55 35. Form of the solution 36. Direct construction of the solution 58 37. Additional properties of the solution 38. Problem of the rotating ellipse GO 39. The circular solution 40. Existence of the non-circular solutions 41. Direct construction of the non-circular solution 64 42. Properties of the solution 66 CHAPTER III. THE SPHERICAL PENDULUM. I. Solution of the Z-Equation. 43. The differential equations 67 44. Transformation of the 2-equation 68 45. First demonstration that the solution of (12) is periodic and that n and the period are expansible as power series in n 69 46. Second demonstration that the solution of equation (12) is periodic 71 47. Third proof that the solution of (12) is periodic 73 48. Direct construction of the solution 73 49. Construction of the solution from the integral 76 II. Digression on Hill's Equation. 50. The x-equation 51. The characteristic equation 52. The form of the solution 81 53. Direct construction of the solutions in Case 1 83 54. Direct construction of the solutions in Case II 85 55. Direct construction of the solutions in Case III 87 III. Solution of the X- and Y-Equations for the Spherical Pendulum. 56. Application to the spherical pendulum 57. Application to the simple pendulum 90 58. Application of the integral relations 93 CHAPTER IV. PERIODIC ORBITS ABOUT AN OBLATE SPHEROID. (By WILLIAM DUNCAN MACMILLAN.) 59. Introduction 99 60. The differential equations 100 61. Surfaces of zero velocity 101 I. Orbits Re-entrant after one Revolution. 62. Symmetry 102 63. Existence of periodic orbits in the equatorial plane 103 64. Existence of periodic orbits which are inclined to the equatorial plane. . . 104 65. Existence of orbits in a meridian plane 106 66. Construction of periodic solutions in the equatorial plane 107 67. Construction of periodic solutions for orbits inclined to the equatorial plane. Ill 68. Construction of periodic solutions in a meridian plane 114 II. Orbits Re-entrant after many Revolutions. 69. The differential equations 116 70. The equations of variation 118 71. Special theorems for the non-homogeneous equations 127 72. Integration of the differential equations 132 73. Existence of periodic orbits having the period 2i. Statement of problem 7li. 'I'lie differential e<|iiat ions ol' niolioii 77. Regions of convergence of the -cries P,, P t , P, !>! 7v Introduction <>f the parameterM t and 5 155 Til. .lacobi's integral 156 sn The -yimnetry theorem 156 si. Outline of -tcps for proving the existence of |>eriodic solutions of equa- tions 157 sj i leneral M.lulions of conations (12) for i- = 83. I'erio<> 84. Noimal form for the different iiil equations K>1 s.'i. 1 \i-ieiice of |>eriodie orbits of Class A " ; f Stl. Some properties of solutions of Class A 166 s7. Din-it construction of the solutions for Class A 168 88. Additional pro|>erties of orliits of Class A 89. Application of .Jacolii's integral to the orbits of Class A 17ii 90. Numerical examples of orbits of Class A 179 '.M. onstruction of a prescribed orbit of Class A 180 ii'J. Kxistenee of orbits of Class B !M. Direct construction of the solutions for Class B . 183 1U. Additional properties of the orbits of Class B ii.V Numerical example of orbits of Class B 190 96. On the existence of orbits of Class C 191 ( 'IIAPTER VI. OSCILLATING SATELLITES. (SECOND METHOD.) 97. Outline of method 199 98. The differential equations 199 99. Integration of the differential equations . . 201 100. Kxistenre of periodic solutions 202 101. Direct construction of the solutions for Class A 210 102. Direct construction of the solutions for Class B 213 CHAPTER VII. < >s< ILLATING SATELLITES WHEN THE FINITE MASSES DESCRIBE ELLIPTICAL ORBITS. 103. The differential equations of motion 104. The elliptical solution 218 105. Equations for the oscillations 106. The symmetry theorem 107. Integration of equations (13) 108. The terms of the second degree 109. The terms of the third degree 1 10. General properties of the solutions 111. Conditions for the existence of symmetrical periodic orbits 230 112. The existence of three-dimensional symmetrical periodic orbits. . . 113. Case I, n even. .V odd - : < 114. Case II, n odd 1 1 ."). Convergence 116. The existence of two-dimensional symmetrical periodic orbits 239 117. Case I, n even 240 118. Case II, n aid 241 X CONTENTS. CHAPTER VII Continued. Construction of Three-Dimensional Periodic Solutions. 119. Defining properties of the solutions 241 120. Coefficients of Xi 241 121. Coefficients of X 243 122. Coefficients of X? 245 123. Coefficients of X 2 248 124. Coefficients of X? 248 125. The general step for the x and ^-equations 250 126. The general step for the z-equation 251 Application of the Integral. 127. Form of the integral 253 128. The integral in case of the periodic solutions 254 129. Determination of the coefficients of Z%+i when e = 257 Determination of the Coefficients of Z 2 j+i when e = 0. 130. Case when e 7* 0. General equations for Z : 260 131. Coefficients of e 262 132. Coefficients of e 263 133. Coefficients of e 2 and e 4 264 134. General equation for v = 1 265 135. Terjns independent of e 265 136. Coefficients of e 266 Direct Construction of the Two-Dimensional Symmetrical Periodic Solutions. 137. Terms in X* 267 138. Coefficients of X 269 139. Coefficients of X* 271 140. General proof that the constant parts of the right members of the first two equations of (148) are identical 278 141. Form of the periodic solution of the coefficients of X? 280 142. Coefficients of X 2 280 143. Induction to the general term of the solution 282 CHAPTER VIII. THE STRAIGHT-LINE SOLUTIONS or THE PROBLEM OF n BODIES. 144. Statement of problem 285 I. Determination of the Positions when the Masses are given. 145. The equations defining the solutions 286 146. Outline of the method of solution 288 147. Proof of theorem (B) 289 148. Proof of theorem (C) 290 149. Computation of the solutions of equations (6) 294 II. Determination of the Masses when the Positions are given. 150. Determination of the masses when n is even 296 151. Determination of the masses when n is odd 296 152. Discussion of case n = 3 297 CHAPTER IX. OSCILLATING SATELLITES NEAR THE LAGRANGIAN EQUILATERAL TRIANGLE POINTS. (By THOMAS BUCK.) 153. Introduction 299 154. The differential equations 299 155. The characteristic exponents 301 156. The generating solutions 303 157. General periodicity equations 305 CONTENTS. XI CIIAITER IX Continued. PAGE. 158. The first generating solution ........................................ 307 159. The third pciioniting solution ....................................... 309 160. The fourth Kcncniting solution ...................................... 309 161. Thr fifth generating solution ........................................ 311 \<\'2. The seventh generating solution ..................................... 312 163. ( 'oust ruction of the solutions in the plane ............................ 313 164. ( 'onstriK'tion of the solution with period 2r ........................... 319 ( 'IIAITKU X. I.-HM KI.KS-TRIANGLE SOLUTIONS OF THE PROBLEM OK THREE BODIES. (Br DANIEL BUCHANAN.) in:., introdui-tiun ...................................................... :v_>:. I. /'///( Orbits when the l-'iintt liodiea move in a Ciri-li i KH XI. PERIODIC ORBITS OF INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 179. Introduction ...................................................... 357 180. The differential equations ................ 181. Proof of the existence of the periodic solutions ........................ 360 182. Properties of the periodic solutions .................................. 363 Direct Construction of the Periodic Solutions. 183. General considerations ............................................. 364 184. Coefficients of m* .................................................. 365 185. Coefficients of m* .................................................. 365 186. Coefficients of m 4 .................................................. 366 187. Induction to the general step of the integration ........................ 369 188. Application of Jacobi's integral ...................................... 372 189. The solutions as functions of the Jacobian constant .................... 375 190. Applications to the lunar theory ..................................... 376 191. Applications to Darwin's periodic orbite .............................. 377 XII CONTENTS. CHAPTER XII. PERIODIC ORBITS or SUPERIOR PLANETS. PAGE. 192. Introduction 193. The differential equation 194. Proof of the existence of periodic solutions 195. Practical construction of the periodic solutions 385 196. Application of the integral CHAPTER XIII. A CLASS OF PERIODIC ORBITS OF A PARTICLE SUBJECT TO THE ATTRACTION OF n SPHERES HAVING PRESCRIBED MOTION. (Bv W. R. LONGLEY. ) 197. Introduction 389 198. Existence of periodic orbits having no line of symmetry 390 199. Construction of periodic orbits having no line of symmetry 396 200. Numerical Example 1 402 201 . Some particular solutions of the problem of n bodies 404 202. Existence of symmetrical periodic orbits 410 203. Construction of symmetrical periodic orbits 414 204. Numerical Example 2 415 205. Numerical Example 3 417 206. The undisturbed orbit must be circular 419 207. More general types of motion for the finite bodies 420 208. Numerical Example 4 422 CHAPTER XIV. CERTAIN PERIODIC ORBITS OF k FINITE BODIES REVOLVING ABOUT A RELATIVELY LARGE CENTRAL MASS. (By F. L. GRIFFIN.) 209. The problem 425 210. The differential equations 427 211. Symmetry theorem 427 212. Conditions for periodic solutions 428 213. Nature of the periodicity conditions 429 214. Integration of the differential equations as power series in parameters. . . . 430 215. Existence of periodic orbits of Type 1 434 216. Method of construction of solutions. Type 1 438 217. Concerning orbits of Type II 442 218. Concerning orbits of Type III 447 219. Concerning lacunary spaces 450 220. Jupiter's satellites I, II, and III 452 221. Orbits about an oblate central body 455 CHAPTER XV. CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 222. Introduction 457 223. Ejectional orbits in the two-body problem 457 224. The integral 461 225. Orbits of ejection in rotating axes 461 226. Ejectional orbits in the problem of three bodies 462 227. Construction of the solutions of ejection 463 228. Recursion formulas for solutions 465 229. The conditions for existence of closed orbits of ejection 469 230. Proof of the existence of closed orbits of ejection 471 231. Conditions at an arbitrary point for an orbit of ejection 473 232. Closed orbits of ejection for large values of n 477 233. Periodic orbits related to closed orbits of ejection 479 234. Periodic orbits having many near approaches 483 COM K\ I'S. Mil ( iiMiKu XVI. -StMiiK-i- .>K I'KUIODIC OHHII- IN IHI l.'i -i iu< i KI> PKOBLEM op THREE BODIES. PAGE. 2:r>. Statement of problem 485 j:5t). Periodic satellite ami planetary orbits ... 486 '2'17. The non-cxi>icnce nf isolated (icriodic orbits 487 _':;*. The persistence of double orbits with changing mavratio of the finite bolie> 4.M) J.'W. < 'iisps on ]>criodic orbits 495 210. Periodic orbits having l(M>ps which arc related to cusps 497 '-Ml. The (MTsistcnce of cusps with changing ina-ratio of the finite bodies. .. . 499 J IJ Some pro|x-rties of the jM-riodie oscillating stitcllitcs near the e<{iiilateral triangular jwiints 501 LM::. The analytic continuity of the orbits about the equilateral triangular points 504 '-Ml. The existence of jteriodie orbits about the equilateral triangular points for large values of /i 505 _M."i. Numerical periodic orbits about equilateral triangular |M>ints 507 Jlti. ( 'loscd orbit-, of ejection for large values of ^ 510 J 17. i >rbits of ejection from 1 n and collision with n 515 _Mv Proof of the existence of an infinite mimtter of closed orbits of ejection and of orbits of ejection and collision when n = ) > 517 '-M'.t. ( )n the evolution of jx-riodic orbits about equilibrium points 519 250. < >n the evolution of direct periodic satellite orbits 520 _'">1 . ( >n the evolution of retrograde periodic satollit* orbits 522 252. On the evolution of periodic orbits of superior planets 524 PERIODIC ORBITS BY F. R. MOULTON IN COLI.ABOHATION WITH DANIEL BUCHANAN, THOMAS BUCK, FRANK L. GRIFFIN, WILLIAM R. LONGLEY AND WILLIAM D. M A cMILLAN CHAPTER I. CERTAIN THEOREMS ON IMPLICIT FUNCTIONS AND DIFFERENTIAL EQUATIONS. BY F. R. MOULTON AND W. D. MACMlLLAN. I. SOLUTION OF IMPLICIT FUNCTIONS. I. Formal Solution of Simultaneous Equations when the Functional Determinant is Distinct from Zero at the Origin. -In applying the condi- tions for periodicity of the solutions of differential eq nations after the method of Poincare.* there will he frequent occasion to consider the solution of P,(a,, . . . ,a. ;M)=O (i-1, ...,), (1) for a, , . ... a. in terms of /i, where the P, are power series in the a, and n, vanishing with o, = 0, M = 0, but not with a, = 0, M^O, and converging for |tt;|0, \n\

0. We are interested in only those solutions which vanish with M: that is, if we regard a,, . . . , a,, M as coordinates in (H + l)-space, in those curves satisfying (1) which pass through the origin. Equations (1) can be satisfied formally by the series o, = S^V, (2) where the af are functions of the coefficients of the P, which are to be deter- mined. Upon substituting (2) in (1), expanding and arranging as power series in M, it is found that rW l> ..... a<*-")] M * (3) where the f *' are polynomials in a*,", . . . , a'/ ". Upon assuming for the moment that the series (2) converge and satisfy (1), it follows that (3) are identities in p. Hence the coefficients of each power of M separately are zero. Let Mtlhodct NoveeUe* de la Mctaniquc Ctletie. vol. I, chap. 3. i 2 PERIODIC ORBITS. From the coefficients of the first power of /x we get SAP. a o) = _ dPj da; ' dji i"j Since by hypothesis the functional determinant (t = (4) dPi d.Pi a ' ' ' da. da n y otti da n is distinct from zero for a,= =a n = M = 0, equations (4) can be solved uniquely for the a"'. If not all of the dP t /dn are zero, then not all of the a ( " are zero; but if all of the dP t /dn are zero, then all of the a ( " are zero. Equating the coefficients of the second power of M in (3) to zero, we get i = \, the right members of which are completely known. The determinant of the coefficients of the af is A, and the af are therefore uniquely determined, being all zero or not all zero according as the Ff are all zero or not all zero. And it is seen from (3) that the treatment of the general term is entirely simi- lar and depends upon the same determinant A. Hence, under the hypothesis as to A, a formal solution is possible, and it is unique. 2. Proof of Convergence of the Solutions. In order to prove the con- vergence of the series (2) , consider the solution of the comparison equations (t = .n) (10 for /3i , . . . , /8 in terms of M> where the Q t are power series in the /8j and n, vanishing for (8 ; = 0, M = 0, and convergent for | ft, \ < r > 0, /x | < p' > 0; and where also the coefficients of all terms beyond the linear in each Q, are real, positive, independent of i, and greater than the moduli of the corresponding coefficients* in the expansions of any of the P t . The character of the coeffi- cients of the linear terms will be specified when they are used. Suppose the solutions of (!') have the form (-!,. ..,*). (2 r ) *For proof of t.he possibility of satisfying these conditions see Picard's Traite d' Analyse, edition of 1905, vol. II, pp. 255-260. IMIM.lt IT H \< flOXS. On substituting ij'i in (!') and arranging in powers of n, then- results a -y-t.-iu of equations similar to (3). The cf," an- determined by 'V dfi 0=1 (4') It is necessary now to >peeify the properiie- of the linear terms of the (/. It will he supposed first that the MJ,/dn are real and positive, and that dQ t /dn= =dQ*/dn = dQ/dfji. It follows that for fixed value- ol the dQt/dfi, the values of the .r sati>fying I'i are proportional to dQ/d. The dQ,/(iJ, will now be so determined that when dQ,/dn is replaced by the greatest \dP,/dn\ the #'," determined by i4'i shall be equal, positive, and at least as great as the greatest ja"'! for all values of dP,/dn such that This must be done in such a way that the determinant dQ. a/3, dQ. shall be distinct from zero for #, = ^ = 0. These conditions can be satisfied in infinitely many ways. A simple way is to choose dQ,/dfij= 1 if j^i and ay,/a/3,= -(1 + r). Then the determinant of (4') is A' = (-l)"c"-' (c+n), wliich can vanish only if c = or c = n, and the solutions of (4') are (5) (c + n)' For any n we can give c such a value that the /3*, 1 ' shall be positive and at least as great as the greatest | a"' |. The &? are determined by equations of the form 2 *& = ~ ?w> ----- ^ I , (i (6) It follows from the properties of the Q, , together with the explicit structure of the G? and the values of the #", that G? ^ | F ? (t = l, . . . , n), and therefore that ?() a<" 4 PERIODIC CEBITS. It is very easily shown by induction that for every value of the index k ,...,, ^ |af>| Therefore if (2') converge for | /* | g p', then also (2) converge for | /* | ^ p'. All the conditions imposed upon the Q t can be satisfied by functions of the form* ft- -cff,- (8) where M is a real positive constant. Adding these n equations and solving for ft + + ft,, it is found that (9) Since each p, , and therefore the sum 18!+ +&, , is zero for M = 0, the negative sign must be taken before the radical in (9). The right member of (9) can be expanded as a converging power series in M if M I is taken so small that M I

where conditions which can always be satisfied, whatever may be the values of n, r, p, c, and M . Moreover, the coefficients of all powers of M in the expansion of (9) are real and positive. Hence it follows that 0i+ ' ' ' +/3 B = M #(M), (10) where #(M) is a power series in n whose coefficients are all positive. It follows from (8) that all the ft are equal. Hence f ----- -A-- (11) For |ju| sufficiently small the right members of these equations are converging power series in /z; moreover, they identically satisfy (8). It follows from this result and the second set of (7) that p" > exists such that the series (2) converge for | M I < p"- *Picard's Traitt d'Analyse, loc. cit. IMPLICIT FUNCTIONS. 5 3. Generalization to Many Parameters. Suppose the equations to l>e solved arc P, (a, ,..., a. ; MI ,...,/*) =0 (-!,..., n), (12) and that the functional determinant with respect to the a, is distinct from /en i for a, = =O. = MI= =M* = 0. Then the problem can be reduced to that discussed in 1 and 2 by letting n, = c,n. After the solutions have Keen obtained the c, M everywhere can be replaced by M>, for the c, M will occur only in integral powers. 4. The Functional Determinant Zero, but not All of its First Minors Zero at the Origin.* Consider the equations P,(a,,. . . , O,;M)=O (t = i ,), (13) where the P, have the properties imposed upon the P, of 1. Suppose that the determinant of the linear terms in the a, is zero for a, = = O,, = M =0, but that not all of its first minors are zero. It may be supposed without any loss of generality that the determinant of the terms remaining after deleting the last row and column of the linear terms is distinct from zero. Hence, as a consequence of the theorems proved in 1, 2, 3, the first nl equations can be solved uniquely for a, , . . . , a_i as power series in a. and n, vanishing for o = M = 0. Suppose the solutions of the first nl equations for a, , . . . , a,_, are substituted in the last equation. It will become a function of a, and M which may be written, omitting the now useless subscript on the a, , P(a, M) = S 2c w aV = (* + j>0). (14) t-O 1-0 Since the determinant of the linear terms of (13) is zero, this equation carries no linear term in a. Suppose the term of lowest degree in a alone is c w a*. Then, for each value of M whose modulus is sufficiently small there are k values of a satisfying (14) and, moreover, the modulus of M can be taken so small that the moduli of the solutions for a shall be as small as one pleases. f Also, for each set of values of a = a, and M whose moduli are sufficiently small there is one set of values a, , . . . , cu-i satisfying the first nl equations of (13). Consequently, for each M whose modulus is sufficiently small there are precisely k sets of values of a ( , . . . , a, satisfying (13). A special discussion is required to determine the character of these solutions and the method of finding them. These questions are taken up in the immediately following articles. The more difficult case, in which all the first minors of the functional determinant vanish, does not arise in this work. It has only recently (in 1911) been completely solved by MacMillan, in a paper which will appear in Malhrmatwche Annaltn. tWeientrmos, Abhandlungtn out der Functionenlehre, p. 107. Picard, TraiU d'Analyie, vol. II, chap. 9, |7, and chap. 13. Hardness and Morley, Trcotue on the Theory of Function*, chap. 4. 6 PERIODIC ORBITS. 5. Case where P(a, M ) = a*P,(o. M)~M X P s (tL, /*). Suppose the P(a, M) of (14) has the form a*P,(a, M ) - M A ^ 2 (a,M) =0, where P! and P 2 are power series in a and n which do not vanish for a = /u = and which converge for | a | < r > 0, M I < P > 0. Upon extracting the k th root, this equation gives a Q l (a, fji) - t\ M* Q 2 (a, ju) = 0, where Qi and Q 2 are power series in a and p which do not vanish for a = ^ = and which converge for a < r' > 0, M < P' > 0, and where 77 is a fc' A root of unity. If we let ju = v", this equation takes the form of those treated in 1 and 2 and can be solved uniquely for a in terms of v for each rj. The k solutions are obtained by taking for r? the A- roots of unity. 6. A Second Simple Case. Suppose P (a, /x) has the form P(o,/i) = Sc,a*-V+Q(a,M) =0, (15) J=0 where c ^0 and Q contains no term of degree less than k + 1 in a and ju. It can be supposed without loss of generality that C H = 1 . Then Sc ( a*-y= P (a, M) = (o-6 lM )(o-6, M ) ' ' ' (a-6 t /i). (16) i=0 Suppose (a bjfj,) is a simple factor of the homogeneous polynomial Po (a, ;u)> and exclude the trivial case in which it is also a factor of Q (a, ju). Then dPJda ^ for a = b, /t. Now let a = 6, M + j8M. (17) After this transformation both P and Q are divisible by M*- After // is divided out, P carries a term in /3 to the first degree whose coefficient is not zero, and Q carries no term independent of ju, but has at least one term in fj. alone, for otherwise P (a, /*) would be divisible by (a bjfj.). Conse- quently by 1 and 2 the equation in /3 and M can be solved for /3 as a con- verging power series in /z, vanishing for ju = 0. Therefore a can be expanded as a converging power series in /*, vanishing with /x, for each of the simple roots of the polynomial P (a, jj.) = 0. If 61 , . . . , 6* are all distinct, the expansions for the k branches of the function P(a, n) which pass through the origin can be found by this process. The actual determination of the coeffi- cients is by the method of 1 in the simple case n = 1 . 7. General Case of Power Series in two Variables.* The method of treatment consists in reducing the equation, by suitable transformations, to forms of a standard type from which the solutions can be found. In cer- tain special cases successive transformations are required. The analysis in *This problem has been treated by Puiseux, Nother, etc. For references and discussion see Harkness and Morley's Treatise on the Theory of Functions, chap. 4, and Crystal's Algebra, part 2, chap. 30. IMPLICIT FUNCTIONS. general, as well as tin- particular transformation required in any >pecial ca>e to reduce the equation to the standard forms, is indicated must simply by \t-\vt oil's Parallelogram. In constructing Newton's parallelogram it is sufficient to consider only those terms c u a'n' of P(a, /it) for which / ^ k, j ^ X, r^a' being the term of lowest degree in a alone and c,x M v the term of lowest degree in n alone. Take a set of rectangular axes and for each of the trim- > aV> 0,7*0, lay down a degrti point whose coordinates are i and j. Then the line passing through t lie origin and the point A-.O) is rotated around (k.O) as a pole so that it moves along the /-axis in the positive direction until it strikes at least one other degree point (it may, of course, strike several simultaneously). Ix i t the one of those which it first strikes having the greatest j he I/,,./,). Then the line is rotated around (ii,ji) in the same direction until it strikes at least one other degree point. Letting the one of these having the greatest j be (i 3 ,j), the line is rotated around (i' 2) jj) until another is encountered. This process is continued until the point (0, X) is reached. The number of steps in the process evidently can not be greater than /,- or X. The part of Newton's parallelo- gram needed in discussing the character of the function near the origin is made up of the segments (k, 0) to (t, j,), (t, j,) to (i t , j t ), . . . , (i,,j,) to (0, X). For the terms of P(a, n) corresponding to each one of these segments there is, as will be shown, a transformation which throws P(a, n) into a standard form. In order to illustrate Newton's parallelogram, consider the example P(a, M )=i where Q(a, M) contains only terms of the seventh and higher degrees in a and M- The coordinates of the points in Newton's Parallelogram are (5, 0), (3,1), (2,2), (1,3), and (0,6), and it consists of three segments which are shown in Fig. 1. Consider the segment (t, , j t ) to (i,j) and make the transformation x \(3jlj_ (5.0) where m and n are relatively prime inte- gers. The terms c t , h a' // and c hh a V* be- come respectively c^ h /3' 1 M* and c* h /3' 1 /**', where V- 7 ' 1 - (19) (2,2) Fio. 1. If there is another degree point (i f , f) on this segment its coordinates satisfy the equation Hence after the transformation (18) the term c ff a? p.' becomes c vf /3 1 ' 8 PERIODIC ORBITS. It follows from the position of the segment (ii,ji) to (i 2 , J 2 ) with reference to every other degree point, that in the case of any term c tj a M' of P (a, M) whose degree point is not on this segment the exponents i and j satisfy the inequality i (J2ji) +j (i\ ii) + i-iji iji > 0. Consequently, after the transformation (18) the term c aV becomes c 0V ", where This discussion proves that after the transformation (18) the terms belonging to the segment (ii,ji) to (12,32) contain y."' as a factor, and that every other term contains /z to a higher power than a '. Since a' is not necessarily an integer the series will not be, in general, a series in integral powers of n, but it will be in integral powers of // = M"- Hence, dividing out fj.' the series becomes = c hh 0'' + + c hh 0" + M ' P, (0, M')- (20) For ^' = equation (20) becomes c,, A 1 ' Pis'-'' + +*!- cu,0 (i 03 -c,) (0- *,_,) = 0. (21) L Gfc*J The solution 0' ! = is not to be considered for this transformation because it belongs to the solutions obtained from the segments having smaller values of i. Suppose j3 c,, is a simple factor of (21) and let = C, + T, . (22) Then the right member of (20) becomes a power series in 7, and n', van- ishing with y, = n' = 0, and the coefficient of 7, to the first power is distinct from zero. Therefore, by 1, the equation can be solved for y r uniquely as a converging power series in //, vanishing for // = 0. Then, on substituting back in (22) and (18), a is expressed in integral powers of /*'. This is an integral series in M only if a is an integer. If c, , .... c (l _,, are all distinct we obtain at this step i\ i- 2 solutions, and if a is an integer the number of them is precisely i l i 2 . Since when a is not an integer /*' has more than one determination, and since the series obtained by the transformation (18) after removing the factor n"' is not in integral powers of n, it would seem that the number of solutions for the segment is greater than i l ? 2 . But it will now be shown that the number of distinct solutions is {, i. 2 , whether a is an integer or not. Now a= (J2 ji)/(ii-i2)=m/n, where m and n are relatively prime integers. It is clear that ii-i 2 equals n, or is greater than n, according asj 2 -ji and ii i 2 do not have, or have, a common integral divisor greater than unity. Consider first the case where ii i 2 = n. There can be no degree point IM1M.HTI Kl \< I IONS. i/',/) on the segment between (/',,./,) and ii-..j-.), for its coordinate- would have to satisfy the relation \j. -j')/(i' i-j) = '//'. which is impossible when i' i 1 olution gives /, - /'.. values of J, differing only hy the /', 1 roots of unity. Hence the >aine linal results are obtained by using the principal value of n' in (18) and all of these i, 1 values of as are obtained if any other determination of // is used. Suppose now i', i-,= qn. where q is an integer. There can not be more than q 1 degree points on the segment /, ,j,) to (it,j), i. e. satisfying the relation (ji-j')/(i'-i,)=m/n. Therefore in this case (21) is a polynomial in/3* of degree q, and for each of the solutions for 0" there are n values of obtained one from the other by multiplying by the n M roots of unity. Hence in this case also the final results are the same as are obtained by using any other of the N determinations of n" ' . It follows that in every case all the distinct solutions are obtained by taking the principal value of n' , and that the number of them for the segment (;', , j,) to (ij , j 3 ) is precisely i, 1' 2 . In a similar manner the solutions associated with each of the other segments can be obtained. The whole number of solutions found in this way is N = (fc-t,) + (t, - 1,) + + (t, - 0) = *, (23) which is the number of solutions of (P a, n)=Q for a which vanish with n = 0. In this case the problem is completely solved. Suppose, however, that in treating the terms belonging to the segment (i'i, ji) to (it,j) it is found that c, = = c, . The analysis above fails to give the solutions for these roots. In this case the transformation ft = c, + 7 (24) is made, after which the right member of (20) is a power series in y and n'; and for /*' = 0, 7' = is a solution. That is, the equation is of the same form as P(a, M)=O only in place of having k zero roots for /* = there are now only p such roots. This number p is always less than k except when i, = k, ji = Q, i:, = 0, ;'j = X, and c, = Cj= =c, l _ i ,. But whatever the value of p a new Newton's parallelogram for the -y^'-equation is to be constructed. It will depend upon terms of higher degree in the original a/x-equation l>ecause the terms which gave rise to the p equal roots, c, , . . . , c,, have been concentrated, so to speak, by the transformations into the single one /, and the parallelogram depends upon the term in M' alone of lowest degree. By this step, or some succeeding one, the solutions will all become distinct unless, indeed, the original P(a, M)=O has two or more solutions for a which are identical in /* 10 PERIODIC ORBITS. II. SOLUTIONS OF DIFFERENTIAL EQUATIONS AS POWER SERIES IN PARAMETERS. 8. The Types of Equations Treated. In the course of this work certain types of differential equations will arise and they will be solved by processes adapted to attaining their solutions in convenient forms. It will tend to clearness and brevity of exposition of the actual dynamical problems to set down in advance those methods of solving differential equations which will be used, and to state the conditions under which the results obtained by them are valid. Consequently, this section will be devoted to these ques- tions without making here any applications to physical problems. The equations which will be treated are characterized chiefly by being analytic in the independent and dependent variables and in certain parameters upon which they depend; and the solutions are considered only for those values of the variables and parameters for which the equations are all regular. In the case where the differential equations are linear, their coefficients are either constants or periodic functions of the independent variable. 9. Formal Solution of Differential Equations of Type I.* The dif- ferential equations dx -~- = nf t (ar, , . . . , x u , M; (i=l , . . . , n) (25) will be said to be of the Type I when the right members have n as a factor and when all the f t are analytic in x v , . . .,x n ,n and t, and are regular at the point x t = a t , n = Q , for all t ^.t^ T. Then the /, (x l , . . . , x n , M; ') can be expanded as power series in (x t a,) and M which will converge if \x t -a i \Q and |Ai|

Ofor t^t^T. Suppose Xt = a t at t = to, whatever be the value of n. That is, suppose x t (Q = a t , (26) in which the letter under the identity sign = indicates the parameter in which the identity is denned. Equations (25) can be solved formally as power series in /* which have the form x i = S x\*> n*. (27) J=0 where the x ( ? are functions of t. On substituting (27) in (25) and arranging in powers of n, it is found that (28) v+ t _ i _, . a/i*j *See Moulton's Introduction to Celestial Mechanics, pp. 264-272. \i-\-I.INI-. \K DIFFERENTIAL EQUATIONS. 11 If tin - scries MIC convergent the coefficients of corresponding powers of n in the- right and left members are equal. On assuming for the moment that they are convenient, the identity relations become .." ;/, - i, . . . , n), (29) ./.(* ( ' 0; 0'ii . . . , (30) .-* ^ f af Qjt ^d) i ojt fii^ is, r).r, ' a M " wAJ '/' . y 9 / v + ! V V d */ - V 3'/ _(l) I 1 & ft /qO\ '" ... ' fcfc * r 2 ^ ^/ a^fcr, " x ' + W ; * + 2 a M ! ' Th of equations can be integrated sequentially. From (29) we get xs the primitive of/, (a^, 0; 0- Then (31), (32), give in order, similarly, *, *? = ^(0 + a", (35) In this manner the process can be continued as far as may be desired. 10. Determination of the Constants of Integration in Type I. At each step n additive constants of integration are obtained, and they must be determined in terms of the initial values of the x,. From (26), (27), (33), (34), (35) . . . , it follows that Therefore a l , 0) = a, , a? = - **,) (j-1, ... oo). (36) By these equations all of the constants of integration are uniquely determined in terms of the constants of the differential equations and of the initial values of the dependent variables. II. Proof of the Convergence of the Solutions of Type I. The method of integrating differential equations as power series in parameters has been in use in more or less explicit form since almost the beginnings of celestial mechanics. For example, in the year 1772 Euler published his 12 PERIODIC ORBITS. second Lunar Theory, in which he used a process quite analogous to this;* and the method of computing the absolute perturbations of the elements of the planetary orbits is virtually that of developing the solutions as power series in the masses. But the actual determination of the conditions for the validity of the process was not made until Cauchy published his celebrated memoirs on differential equations in 1842.f The results of Cauchy were extended by Poincare in his prize memoir on the Problem of Three Bodies,! and were proved again, following Cauchy 's Calcul des Limites, in LesMethodes Nouvelles de la Mecanique Celeste, vol. I, pp. 58-63. The theorem will be needed in this work in the form given by Poincare, viz.: // the f t (xi , . . . ,x n ,n; t) of equations (25) are analytic \\mxt,. . .,x tt ,fj., and t, and regular at Xi = a f , /u = 0, for all t can be taken so small that the series (27) mil converge for all t ^t^T provided \n\ j-5 dt dt for / ^/ g 7'. From this it follows similarly that yT^\af?\. This process can be continued indefinitely, giving by induction for the general term it? 5 I af? I (37) for Iv^t^T. Consequently. if the right members of (27') arc convergent .-cries when |/x|

0 for J ^/;S7', then likewise arc the right members of (27) convergent when |M|

0 for the same range in t. It is a simple matter to find the explicit expression for (27') by a direct integration of (25"). Since t he right members are the same for all i, we have yi c l = y t c t = = y.-c,, where Ci , . . . , c, are constants of integration. By the initial conditions it follows that c, c, = 6, b, and y l b, = y, b i . Let this common value of 3/1 61 be y b. Then each equation of (25") becomes d|/ dt On integrating this equation and determining the constant of integration by the condition that (y 6)=0 at t = t^, it is found that _ _ (l - H) (l - n fr~ 6 )) ' The solution of this equation for (y b) is r , r *-/o). ax > Since (y 6) = at t = t<>, the negative sign must be taken before the radical. It follows directly from equation (38) that whatever finite values n, M,r,p, and T is satisfied. This condition imposes the explicit limitation 1 - 2nM(T-tJ 1 (39) 14 PERIODIC ORBITS. upon n, which can always be satisfied by ^| < Mo> for r>0, p>0, and for M and T 1 finite. If these conditions are satisfied, the resulting expression for (y b) substituted in (25"') leads to convergent series. Moreover, the series for (y 6) satisfies (25") and is identical with (27') since (27') is unique. Consequently (27') converges, and therefore also (27) if M < Mo , where //o is the limiting value of M satisfying the inequality (39) , for all t in the range t ^t^ T. The theorem is thus established. 12. Generalization to Many Parameters. The differential equations may involve many parameters, /* 1? . . . , p k , instead of a single parameter n. The /, are supposed to be regular for ^ = ^ = = ju* = for t ^t^T. The discussion can be thrown upon the preceding case by letting After the solutions have been found /3, M can be everywhere replaced by //, . This groups the terms of the same degree in /i,, . . . , fj. t together. The equations can also be integrated as multiple series in the parameters Hi, . . . , p.t without the use of this artifice, and then the constants of inte- gration can be determined and the convergence proved. But the method is not essentially distinct from the other, and the details may be omitted. 13. Generalization of the Parameter. Suppose the differential equations depend upon a single parameter M- It may happen that this parameter enters in two distinct ways. For example, it may enter in one way so that, so far as this way alone is concerned, the f t can be expanded very simply as power series in ju. It may enter in another way so that, so far as this way alone is concerned, the expansions of the f t as power series in n are very complex, or even impossible without throwing the equations into an unde- sirable form. Under the circumstances thus described it is sometimes of the highest importance to generalize the parameter. Where it enters in the first way it is left simply as the parameter /* Where it enters in the second way it is replaced by m to preserve the distinction. In forming the solutions H is regarded as a variable parameter in terms of which identity arguments are made, while m is regarded simply as a fixed number. The solutions obtained are valid mathematically for any value of /j, whose modulus is sufficiently small, but they belong to the original (physical) problem for only one particular value of M, viz., for H = m. But it will be observed that when the differential equations are regular for a continuous range of values of m this restriction is of no importance, provided the solutions converge for fi = m, if the literal value of m has been retained in the solutions.* *For a practical application of this artifice see Moulton's Introduction to Celestial Mechanics, pp. 264-5. NON-LINEAR DIKr KKKNTIAL EQUATIONS. 15 14. Formal Solution of Differential Equations of Type II. The dif- ferent i;tl equations ^ = 0, (x, , . . . , z. ; + M/. (*. , . . , x, , M ; <) (,'-i ), (40) will he said to be of Ty|>e II if (a) the g,(x t , . . . , x.;<) are independent of n and not identically /TO; (6) the g,(x t , ., x.; <) and /,(*,, . . ., *., M ; are analytic* in X|, . . ., x, n, and I; (c) the 0,(:r, , . . . , x m ; /) and f,(x t , . . . x. , ; /) are regular at i, = x^(t), M=0, for 1*3. tT, where the x'? are the solutions of equations (40) for n = 0, and a;! 01 = a t at I = t . It follows from these conditions that the g, and /, can be expanded as power scries in (x,-xf) and /i, which converge if \x t x\ 0 and |M| < p >0 for all I in the range t^t^T. Equations (40) can be solved formally as power series in /* having the form x. = Sx ( ,V, (41) 1-0 where the x ( f are functions of t, and where *i(Wfo,. (42) Upon substituting (41) in (40) and equating coefficients of correspond- ing powers of n, it is found that (43) (44) = SA<'V, ; ) + F< (0, (48) where the A are the constants of integration. After these solutions are found, equations (45) can be integrated, and this process can be continued to the k th step, which gives The (f> fj (t) belonging to the complementary function are the same for each step, but the Ff (t), which depend upon the right members of the differential equation, are in general all different. The problem of finding the FJ 0> , the ip w and the Ff depends upon the explicit form of the differential equations, and can not be given a general treatment. 15. Determination of the Constants of Integration in Type II. At each step there are n constants of integration introduced which can be determined in terms of the initial values of the x t . It follows from equa- tions (41), (42), (47), (48), . . . , that ri(0) /_ ., . t \ I ^' v A '*) ^ / ~\ _j_ p'*' /"/ ^ I * /i ^fi\ r 4 ^Ci , . . . , CB > "oy T" ^^ I ** "i "w v / i ^/ I /* v ' \""J Hence *./' ' '^"_^ (t) ' ^ ' " ""' (51) Suppose the constants Ci , . . . , c n are uniquely determined in terms of a, by the first set of equations of (51) . Then the F"' become completely defined, and from the second set of (51) the A ( f are uniquely determined since the determinant A = |

>y< M; f), (40') where the conditions corresponding to (a), (6), (c), and (42) of 14 are satisfied, and where, in addition, the coefficients of the expansions of the

\a?\. The z'" an d y ( " are defined 1 >y dt dt a ) , (54) It follows from (48) and (51) that x = at t = / . Similarly y'," = at t = I* . Equations (54) can be solved by Picard's approximation process.* Let x ( and y'i' be the k" approximations to x and y'"- Then (t-1, . . . , w), (55) (5 ' "'' 0; (56) ' "'' y "' 0; (57) TraiU tTAnalyte, vol. II, edition of 1906, p. 340. 18 PERIODIC ORBITS. It follows from (55) and the relations between the /, and the ^ that y ( tl > |z"il for t \ x k for t ^ t ^ T, for all k. Now Picard has shown* that Urn x% = z ( " for a sufficiently restricted range of values of t. But equations (44) being linear, the range of values for t is precisely that for which the differential equations are valid. t There- fore we conclude that y? > \ x for t ^.t^T. The corresponding relation between y and x ( f can be proved in the same manner, and the process can be continued step by step indefinitely. Consequently the inequalities (52) are established. Hence, if the series (42') converge when |/*| < p', then the series (42) also converge when | n \ < p' f or t <: t ^ T. Since it follows from the reference given in 1 1 that the conditions imposed upon (40') can be satisfied by equations of the formt As a consequence of these equations (y f y ( f) = (y s yf) + c, , where the Cj are constants. Since y t = y^ at t = t , it follows that c } = 0. Now let (59) Then, upon taking the sum of equations (58) with respect to i, we get dz (60) On integrating this equation and determining the constant of integration by the condition that z = p. at t = t , it is found that Mn(t-t ) *Loc. tit. \Traite d' Analyse, vol. Ill, edition of 1894, p. 91. JSee Les Methodes Nouvelles de la Mecanique Celeste, vol. I, p. 60. NON-LINEAK IHKKKKKM I At. KQUATION8. 19 Solving this equation a no! determining the sign of the radical so that z = n at t = I,,, the expression for r heroine- (62) where Mn A = It follows from (62) that if |M! < P, |M| < 1, and T^r^i < 1. then z can be expanded as a converging power series in M for t^^t^T, and that in this range for / the values of z are such that the expansion of the right mem- ber of (60) as a power series in z also converges. ( 'onsequently, the y, and x, satisfying (40') and (40) respectively can also be expanded as converging series in M for all t in the interval 1 ^. t ^ T. The point to be noted in these results is that when the differential equa- tions are of the Types I or II, as defined above,and when the interval T 1<> has Keen chosen in advance and kept fixed, then the parameter M. in which the solutions are developed, can be taken so small in absolute value that the series in which the solutions are expressed will all converge in the whole interval > ^ t T. As in equations of Type I , there may be many parameters, MI , M* >M* instead of a single one. The treatment can be reduced to the case of the single one, just as in the preceding case. The parameter can be generalized precisely as was explained in 13. It is obvious that if there are many parameters they may all be generalized. Since the generalization can be made in an infinite number of ways, a great variety of possible expansions for these solutions is secured. 17. Case of Homogeneous Linear Equations. While the linear equa- tions are included in those already treated, they deserve some special attention for the reason that in their solutions the values of M are not restricted by so many conditions. Consider the equations w ill- i-i where the 0,, are expansible as power series in n of the form e tl = ztfV, t-o 20 PERIODIC ORBITS. which converge if | M| < P for t ^. t ^ T . Suppose o;, = o j at t = / . Then the solutions can be developed as power series of the form x t = 2 x? /, (64) t = precisely as in 14. To find the realm of convergence in /u of (64) , consider a comparison set of differential equations dn " -g! = 2 *()!/,, (63') Ult ,/=! where the \f/ tj are expansible as power series in n of the form which converge provided | /j, \ < p for t < t^ T. Suppose also that ^ ( * > \6*}\ for t ^ t^ T. Develop the solutions of (630 m the form 00 jfr-^itfV. (640 It can be shown by the method used in proving the inequalities given in (52) that if yfa) = |a,| = b t , then yf > xN -. 21 III. HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS. 18. The Determinant of a Fundamental Set of Solutions. Suppose where x[ is the derivative of x, with respect to t, is the set of linear homo- geneous differential equations under consideration, and let . /j\ / j\ _.. * ft\ I It be a fundamental set of its solutions. The determinant of this set of solutions may be denoted by A-|*,|. (67) It will be shown that A can not vanish for any t for which the 0,, are all regular. In the applications which follow, the V are analytic in t and in general regular for all finite values of t. The result of differentiating A with respect to t is A' where the index k denotes that in the k* column the

u > * 68) , V- The n determinants (68) can be expanded according to the elements 0,,. The result is * (_!/-! t-1 22 PERIODIC ORBITS. where A u is the minor of the element

Tor their actual cuiisi ruction, principally when the differential equation lias the form ^jj+ (ao + a,cos< + a,cos2<+ -)x = 0. (70) Different methods for constructing solutions of this equation have been proposed by Lindomunn.* Lindstedt,t Bruns,J Callandreau. Stieltjcs, and Harzer.* In what follows there will arise only equations with simply periodic coefficients having the form e li x J 0-1 ), (71) where the 6 lt are periodic functions of t with the period '2r. It will be assumed that the O u are uniform analytic functions of t and are regular forO^<^2T. Let *u = oury,\o\. XXI, No.4. lAttronomudu Nadirichlen, No. 2533, 2553 (1884). \\Attnmomitche Xachrichtcn, No. 2547 (1884). iWwiomwete Kachnchlm, No. 2602(1884). \AttrmomM* ffachriehtcn, Nos. 2860 and 2851 (1888). 24 PERIODIC ORBITS. where

21 3\ \ i3 l 33 8 = 0. (76) This is an equation of the n th degree in s, the constant term of which can not vanish since it is the determinant of a fundamental set of solutions. It is known as the fundamental equation for the period 2?r. Its roots can be neither zero nor infinite, because the coefficient of s" is unity and the term independent of s is A. 20. Solutions when the Roots of the Fundamental Equation are all Distinct. Suppose the roots s,, s 2 , . . . , s n of (76) are all distinct. Then at least one of the first minors of (76) is distinct from zero when s is put equal tos t , and therefore the ratios of the A,- are uniquely determined by (75). For each s* a set of y^ is determined by (74) involving one arbitrary constant. Since this solution depends upon s* it will be designated by y tk , and the corresponding A, by A jk . Since s = e mr , the a is uniquely determined in terms of s except for the additive constant v V l, where v is an integer. In every case the principal value of a can be taken, for its other values simply remove periodic factors from the y t . Consequently, for the n values of s there are n values a, and from equation (74) there are n solutions, one for each k from 1 to n, y ik = e- a ' S A jk <(> (t) (1 = 1, . , . , n), (77) 3=1 where the ratios of the A Jk are determined by (75). From (72), n solutions of equations (71) are thus found, one for each k, (t = l, . . . , n), (78) x ik = e " y it where the y ik are periodic in t with the period 2ir. These solutions (78) form a fundamental set, for, if they did not, there would exist linear relations among the x lk of the form S C t x (k (t) = 1=1 (79) where not all the C k = 0. Increasing t by 2-rr, it follows from the conditions imposed upon the x t and y t that C t x lt = S C t s t x it (t) = 0, LINEAR DIFFERENTIAL EQUATIONS. 25 and similarly that C t x tt (t + 4x) = 2 C* si z u (<) e 0, (80) *-l t-1 Since the C* are not all zero it follows that the determinant of these equations must vanish; that is, 1 ,1 ..1 , Si , Sj , Sj , ' e* * 01 , St , 5j , n v (*-*> -o (81) Since, by hypothesis, the s, are distinct this relation can not be satisfied unless some XU = Q. For the sake of definiteness take first i=l, and suppose that x^=Q, where &, is some particular value of the second subscript. But equation (79) becomes for t = 1 (?.* (0-0 and there is corresponding to (81) an identity of the form n From this it is inferred that another x it , say x u , , is identically zero. Repeating the process n times, the final conclusion is Upon starting from (79) for i = 2, the conclusion is reached in a similar way that On repeating the process, corresponding identities are obtained for all values of i from 1 to n. Now from the identities z,*=0 (i=l , . . . , n) and from (77) and (78), it follows that A.21 (fill ^T * * ' ^~ *T.nt <^in ^ U) " ' (82) m = 0. These identities can not all be satisfied unless each A, t = Q, for, at t = 0, *> ,'i). The if/a also constitute a fundamental set of solutions provided no A kt the determinant of the \{/ ik is n where \A jk \ and \

U , A.-II , .".32 , ,0 , ^33 , ,0 ,0 (85) LINKAIt DIKFKHKNTI \l IJ.'I \II()N8. 27 which is distinct from zero unless some A,, is zero. A special case, which will be used first, is that when- all the elements except those in the first line and in the main diagonal are zero. Now return to the point under discussion. By hypothesis not all the lii>t minors of 7ii v:ini>h for ,s = .s, . Let the notation be chosen so that one of those which is distinct from zero is formed from the elements of the last n 1 columns. Then .1, must be distinct from zero in order not to get the trivial case in which all the 4 , are zero. Now in placeof

'yl (y, t + ty n ); whence y,, = - ty n + B, y n e (a '- a - >4 + r^B,*,, . Since by hypothesis the x tl = e a>t y n satisfy (71), it is found by substitution that, if (83) are to constitute a solution, the y,, must satisfy the equations y( t + a, y tt = e tj y,*-yn (t = i n). (87) Since t enters only in the ( , and the y tl , which are periodic with the period 2ir, sufficient conditions that the y tl shall be periodic with the period 2v are y it (2v) - y tt (0) = - 2r y tt (0) + ^B, (

t y a , x i3 = e ttl( y a can be determined, where the y n , y f2 , and y i3 are periodic in t with the period 2ir. If all of the minors of the first order of the fundamental determinant vanish, but not all of those of the second order, then two of the A, can be taken arbitrarily, and two linearly distinct solutions of the form will be obtained, where the yn and y i2 are again periodic. In order to obtain a third solution associated with the root s t take as a new fundamental set of solutions e Bl< y,i, e a ' ( ?/ J2 , ViJ (i=i, . . . , n - j=3, . . . , n), so that any solution can be written in the form x t = B, e a >' y n + B, e a '' y a + S Bj

t S In a manner similar to that in the case just treated the periodicity conditions on the y i3 lead to the equations 0= -27T *! y n (0) - 27r s, y a (0) + S B, [

< [y a + t ( yil + yj], (89) where the y n , y a , and y i3 are periodic in t with the period 2-n-. Suppose now that not all of the first minors of the fundamental determi- nant vanish for s=Si. Then there will be one solution x tt = e"' y n and another x {2 e a '' (y l2 + t yn). It will be shown that in this case the third solution belonging to St is of the form x a = e tt1 ' [y a + t y a + f t y n }. (90) LINEAR DIFFERENTIAL EQUATIONS. 29 Take as a new fundamental set of solutions e a ''t/u , e' ( (y a + ly tl ),

n'->ion> for y a arc y= - ty a - If the a-,3 constitute a solution of the original equations (71), the y lt must satisfy the equations !/,', + a, t/,3 = 26 tl yK - y, t , since the y ( , satisfy (73) and the y, t satisfy (87). Hence sufficient conditions that the y n shall be periodic are that y,,(2r)-y,,(0) = (i-i , ). These conditions lead to the equations "5 B, [ ) t D t . But by hypothesis D admits (s = ,) as a triple root. Therefore D t ( t ) = and the equations are consistent. Since s = i is a simple root of D t , not all of its first minors are zero. Therefore the B t , . . . , B n are uniquely determined, and the z, 3 have the form (90). Suppose = s, is a root of multiplicity I. There is then a group of solu- tions, / in number, attached to this root. In general this group of solutions will have the following form = e a ''y lt (t-1, . . . ,n), (91) 30 PERIODIC ORBITS. If all the minors of the fundamental equation D = up to the order k l(k^l), but not all of order k, vanish for S=S T , then there are k solutions of the first form, i. e. of the form Xu = 6 ' t/il , Xi2 = 6 ' 2/f2 ) J %ik = & 2/ifc If now the fundamental set e a >'y tl , . . . ,e ait y ik , ?,.*+,, . . . , ? fc , be taken and the equation in s formed, it is found that ~r~\ / \ t T~\ f\ I J I o ~ o i ) K ~~~ ^ Since the roots of the fundamental equation are not changed by adopting the new fundamental set of solutions, D k = has s = Si as a root of multiplicity I k. Suppose all the minors of D k of order g 1, but not all of order g, vanish; then there are g solutions of the second form, viz., atT " 1 a,< f " 1 If k+g is one of the roots of the characteristic fln-a"", a,, a,, , <;., .(0) = 0. (93) This equation, which is of the n" degree in a <0) , has n roots, a, . . . , a. If these roots are all distinct there exists a fundamental set of solutions of the form xt? = c t) e*> (-i, n ; j = l, n). (94) If two of the roots are equal, say a{ 0) = aJ 0) , a fundamental set of solutions is obtained by taking r <*>- r A T *ia nt e > ) * ' vfc Suppose the roots of (93) are all distinct and that the fundamental set of solutions is (94); then, for n distinct from zero, the complete solutions of (92) are x, = 2 A, [c ( ,e a >"'+ S (f-i, . . . , n), (95) where, by 17, the series Zx* M U) converge for any preassigned finite range for t if | M I < P Without loss of generality the initial conditions can be taken so that the determinant of the c,, is unity and x(Q) =0. As before, the transformation = e at y t is made, and the equations corresponding to (92) and (95) are respectively (96) The conditions that the j/, shall be periodic with the period 2ir, viz., l/,(2T)-j/,(0) = 0, give = (97) 32 PERIODIC ORBITS. A = M Since the Aj must not all be zero, the determinant of their coefficients must vanish, whence *] | - 0. (98) This equation has an infinite number of solutions, for if a = a.j is a solution, then also is a = a i + v V 1, v any integer. The fundamental equation corresponding to (76) is obtained by the transformation e 2air = s. If the values of s satisfying the fundamental equation are distinct, the corresponding values of a are distinct but not the converse, for if two values of a differ by an imaginary integer the corresponding values of ,s are equal. Only those values of a will be taken which reduce to the a c ; 0) for M = 0, the a ( f being uniquely determined by (93) . Suppose now that two of the roots of the characteristic equation, say a? and a?, are equal. Then the solutions of (92) have in general the form x t = J=3 (99) The exception to this general form is that tc a may be absent from the second term for i = 1 , . . . , n, and this possibility must be considered at those places where it makes differences in the discussion. After making the transformation x ( =e at y t , the solutions for the y t are _ t .7=3 (100) The conditions for the periodicity of the y t , viz., y i (2ir') y t (0) =0, lead to the determinant A = ' S = 0, (101) where the elements which are not written are of the same form as those in (98) . If, for M = 0, the characteristic equation has a root of higher order of multiplicity, the fundamental equation is formed in a similar manner. 23. Solutions when af, o , af are Distinct and their Differ- ences are not Congruent to Zero mod. v/^T. The part of (98) independent of M is (102) and the determinant c t} \ is unity. LINKAU UIKKKKKVriAI, KlJt'ATIOXS. 33 If. in any particular case, (98) wen- an identity in n its n solutions would be simply o = o ( j". In <"i.-e it is not an identity, let (103) and A beoomei where f'(&,/i) is a -erie> in /3 and /i, converging for |0| finite and |/ Since, by hypothec-, i,i> i> an imaginary integer, the expan- sion of (104) as a power series in /3* and yu contains a term in ^ of the first degree and no term independent of both & and //. Therefore (see 1 and 2) it can be solved uniquely for & as a i>ower series in M of the form M). (105) Substituting this value of & in (103) and the resulting a in (97), n homo- geneous linear relations among A l , . . . , A n are obtained whose determi- nant vanishes, but for n sufficiently small not all of its first minors vanish, since the roots of the determinant set equal to zero were all distinct for M = . Therefore the ratios of the ^4 ; are uniquely determined as power series in n, converging for |/u| sufficiently small. When the ratios of the A, have been determined, the y u are determined as power series in //, and the coefficient of each power of p. separately is periodic in t. A solution is found similarly for each of. The origin of the singularities which determine the radii of convergence of the final solution series is known. If p is the smallest true radius of convergence of the original solutions (95) as t varies from to 2*, then, in general, the final solutions will converge only if | n \ < p. Consider the funda- mental equation, A(s, n)=Q, which is a polynomial in s of degree n and a power series in n converging if \p\ T e t?r . If s has a branch-point for /j. = /*, then ft also has a branch-point at the same place since dj3/ds = 1/2x8 is distinct from zero for all finite values of s. Therefore the series for ft converges only if \/j, < \JJL O \ . The root s t iss, = sf+ 2 4V and a< 0) + ft = ( 1/2*-) log [a + 2 s /*']. (=1 i=i If for any MI such that | MI I < P we have | sf \ = \ S s M! I , then ft has an essential singularity at yu = /xj , and the series for it converges only if |/x < /ii | . The zeros determining these singularities can also be found in a special numerical case by Picard's method. When \/j. satisfies the inequalities imposed by these various possible singularities, the solutions are convergent for all finite values of t. 24. Solutions when no two af are equal but when af a is Congruent to Zero mod. Vl. Suppose two roots of the characteristic equation for p. = 0, say dj 0> and af, differ only by an imaginary integer, and that there is no other such congruence among them. Then the equation corresponding to (104) becomes - e 1 - ') +ft M F t (ft , M ) +M 2 F 2 (ft , M) = 0. (107) /=3 The term of lowest degree in ft alone is 4.ir*P\. The term of lowest degree in M alone is at least of the second degree, and will in general be precisely of the second degree. This follows from the fact that every term in every element of the first two columns of the determinant (98) contains in this special case either ft or /x as a factor. In order to get the terms in fj, alone, those involving ft are suppressed, and then the conclusion follows from the fact that every term in the expansion of the determinant contains one term from each of the first two columns. In a similar way if p of the a ( " } are congruent to zero mod. V l, then the term of lowest degree in ft alone is exactly of degree p, and in /x alone it is at least of degree p. Consider the expansion of (107), which may be written in the form M 2 + ' =0, where y n , 702, are constants. The quadratic terms can be factored, giving (ft &!/x)(ft 6 2 M) + terms of higher degree =0. If 61 and 6 2 are distinct, as will in general be the case, the two solutions of (107) are then (see 6), /9n = 6iM+M'Pi(M), ft 2 = &2M + M 2 J P 2 (M), (108) where PI and P 2 are power series in n. If bi = b 2 the solutions are power series in VjZ or p,, depending upon the terms of higher degree. If 702 is UNKAK DlKKKHKVn.VL EQUATIONS. 35 zero at lea.-t mic the solutions of (101) are where e is any p th root of unity. Another case is that in which A = has a double root identically in n, the conditions for which are A(a, M )=0, |(, M )=0 for all |M| sufficiently small. Suppose a 2 = a, . If, for a = 0.1 , all the first minors of A are zero, the solutions of (97) for the ratios of the A t will carry two arbi- traries, and the two solutions associated with a, will be obtained. If not all the first minors of A vanish for a = a lt then in this way only one solution is found. But it is known from the general theory of 21 that the second solution has the form On substituting these expressions in the differential equations and making use of the fact that e a>t y n are a solution, it is found that n tfu+tiV*- 2 p*l If the left members of these equations are set equal to zero, they become precisely of the form of the equations satisfied by the i/ n - Consequently yn=yn plus such particular integrals that the differential equations shall be satisfied when the right members are retained. The method of finding the particular integrals will be taken up in 29-31. 26. Construction of the Solutions when, for M = 0, the Roots of the Char- acteristic Equation are Distinct and their Differences are not Congruent to Zero mod. V 1. The knowledge of the properties of the solutions and their expansibility as power series in M leads to convenient methods for constructing them. Under the conditions that for ju = the roots of the characteristic equation are distinct and that the difference of no two of them is congruent to zero mod. V i, it has been shown that there are exactly n distinct values of a expansible as converging power series in n, such that x ft = e att y a: (i = l, . . . ,ri), where the ?/ are purely periodic, constitute a fundamental set of solutions. It will be assumed that for M = no two o ( " are equal and that the difference of no two of them is congruent to zero mod. \< r ^l, and it will be shown that the coefficients of the expansions of the a t and ?/ are determined, except for a constant factor, by the conditions that the differential equations shall be satisfied and that the y tlc shall be periodic in t with the period 2w. LINEAR DIFKKKKXTIAL EQUATIONS. 37 r the value of // u is // = 2 y\>n', where the series converge for l-o all \n\ sufficiently small. It follows from the periodicity condition that 2 y* (2ir)M' = 2 /-O * J.O Since this relation is an identity, it follows that Therefore each y u separately is periodic with the period 2*. Now the original differential equations (92) after making the transfor- mation x l = e a 'y, become For /u = the roots of the characteristic equation belonging to these equations are a*', a', 0) , . . . , OL\ Consider any one of them, as a'". It has been shown that for M 7* , but sufficiently small in absolute value, a t and the y u are expansible in converging series of the form Z r-O (in) On substituting (111) in (110), arranging as power series in p, and equating coefficients of corresponding powers in n , there results a series of sets of equations from which a and the y u can be determined so that the y u shall be periodic with the period 2ir. The determination is unique except for an arbitrary constant factor of the y* . For simplicity of notation this constant factor will be determined so that j/" (0) = c u , provided c u ^ 0, and it can be restored in the final results by multiplying this particular solution by an arbitrary constant. Terms independent of p. The terms of the solution independent of n are defined by the differential equations ^-1 the general solution of which is 0-1 ..... ), (113) where the V" are the constants of integration. Since the y are periodic with the period 2*-, and since, by hypothesis, a^-Cf^O mod. V^T, except when j = k, every ^ = Q if j?*k. The initial value of y is c u ; therefore iju-1. 38 PERIODIC ORBITS. If Cu were zero the initial condition would be imposed upon another y ( , not all of which can be zero at / = 0. The solution satisfying the conditions laid down is then = Coefficients of ju. The differential equations for the terms in the first power of n are (tmX 8 !/!?- 2 ,,!# = -a^+ZflJ"^ (-l, . . . ,). (115) The general solution for the terms homogeneous in ?/|" is n f W W\I V -> r *A 1 * " /I 1ft\ * n tk c ij e > V AJL Oj ^=1 where the rj ( ^ are the as yet undetermined constants of integration, and the c i} are the same as in (113). Using the method of variation of parameters, we find where the (j\"(t) are periodic in Z with the period 2w. The determinant of the coefficients of the (rjlY is which can not vanish for any finite value of t. Therefore the solutions of equations (117) for (77"*)' are where the A ( ^ are periodic functions of t with the period 2 IT. The solutions of (118) forjVA; have the form ,a, = ,--<>> p= -' + , (120) where 8^ is A after the terms - a^y^ have been omitted from the k th column. It is a periodic function of t with the period 2-ir, and has in general a term independent of t. It can be written in the form where 4" is a constant and Q (t) is a periodic function whose mean value is zero. Then LINKAU DIFKKUKMIAL Kql'ATIONB. 39 It is clear that if V is to be periodic ihr right mcnilicr (if this c(]iiatiun must not contain any constant terms. Thrn-foiv ol"-*", (121) and C-PSI + B2, (122) where P is periodic with the period 2ir and B% is the constant of integration. Upon substituting (119) and (122) in (116), the general solution with the value of a"' determined by (121) becomes t* w w\i iff-ZIJQcfciiC +2 i ^P'(0. (123) In order that the y u shall be periodic with the period 2r, all the B^ must vanish except B u . From the condition that y u (0) = c u for all |M| sufficiently small, it follows that y"(0) = c tt and ^ (0) = (j = 1, . . . oo ). From the con- ditiiin that y\" = at t = it follows that Therefore the solution satisfying all the conditions is 1C = 2 [*, P% (0 - ^Cu P (0)] . (124) It remains to be shown that the integration of the coefficients of the higher powers of n can be effected in a similar manner. Let it be supposed that oi", of, . . . , a*-" and the j&\ y , . . . , t/*-" have been uniquely determined so that the y ( (t) are periodic with the period 2* and that y',' = 0, /=!, ...,m 1. It will be shown that the y can be determined so as to satisfy the same conditions. From equations (96) it is found that (125) Omitting the terms included under the sign of summation with respect to p, these equations are identical in form with equations (115) except for the superscripts (1) and (m). Obviously the integrations proceed with the index (m) just as with the index (1), and the character of the process is no wise altered by the inclusion of the terms under the sign of summation with respect to p, for they are all periodic with the period 2* and do not change the essential character of the $?(() . Therefore a and the y J" can be uniquely determined so that the y shall satisfy the differential equations and be periodic in t with the period 2r, and so that at the same time y (0) = 0. The induction is complete and the process can be indefinitely continued. The solutions associated with the other a' are found in the same way. 40 PERIODIC ORBITS. 27. Construction of the Solutions when the Difference of two Roots of the Characteristic Equation is Congruent to Zero mod. Vi. It will be supposed now that a a is congruent to zero mod. V 1 and that this relation is not satisfied by any other pair of af. The solutions associated with af , . . . , a ( n 0> are computed by the method of 26 without modifica- tion. It has been shown that in general a : , a 2 and the y tt , y i2 can be developed as converging series in integral powers of /z. It will be assumed further that the case under consideration is not an exceptional one. The general solutions of (96) for the terms independent of n is in this case " /V> r , w \i Iff- */! ->>'. j=i Imposing the conditions that y shall be periodic with the period 2ir and that y (0) = c n , these equations become, since af oi 0) is an imaginary integer, where TJ^ is so far arbitrary. Coefficients of n- It follows from (96) that the coefficients of /* must satisfy the equations The general solution of these equations when their right members are zero is y ( n = Sijjfc/* ' "' (i=l, . . . , n). (128) On considering the coefficients rj as functions of t and imposing the conditions that (127) shall be satisfied, it is found that On substituting the values of the yf from (126) and solving, there result (129) where the A and Z) are periodic functions of / with the period 2ir /()_ (0)\ depending upon the fl and e^ ; . In the first two equations the unde- termined constants a ( ," and y enter only as they are exhibited explicitly. I.INKAK DIFKKHKMI \\. VI!"V-. 41 Kquations i l_".i :nv ID he integrated ;intl tin- results Mihstituted in (128). In order that tin- // -hall he periodic the conditions must be imposed that 0=-( 0=-( = n), (130) whore 6', 1 ,', &', ef,", ef, 1 ,' are the constant terms of A'/,', A,", />,',', and D*," respec- tively, and where the /?",' are the constants of integration obtained with the last n 2 equations. These equations determine two solutions for the arbitraries a ( ," and 17 except in those special cases where the existence shows the solutions are expansible in other forms. Upon eliminating Vi between the first two equations of (130), it is found that a," must satisfy the equation - 0. (131) If the discriminant of this quadratic is not zero the case is that in the exist- ence proof, equations (108), where 6, and b 3 are distinct. In this case, which may be regarded as the general one, the solutions proceed according to integral powers of n. If the discriminant is zero the character of the solutions depends upon the coefficients of terms of higher degree, and they may proceed according to powers of n or =*= Vji. It will be supposed that the discrimi- nant is distinct from zero, and the method of constructing the solutions will be developed. Choosing one of the pairs of values of a"' and ijJJ which satisfy (130), it will be shown that henceforth the solution is unique. Upon imposing the condition that j/', 1 ,' (0) = , integrating (129), substituting the results in (128), and determining the constants of integration so that the solution shall be periodic, it is found that (132) where #*,', is an undetermined constant, and the P^' are entirely known periodic functions of t, having the jxriod 2r. Coefficients of p. The coefficients of ft 2 are defined by * !_-/*> v n i, (t > /. (Z) i, <0) _/, o >/ 1> _u y Iff ii m 4-flf l) ra, y tl 2, a (> j/,, a, j/ ( , a, y n -\- L \v\, y n -rv t , (133) 42 PEEIODIC ORBITS. The general solution of these equations when the right members are neglected is the same as (128) except that the superscripts are (2) instead of (1). On varying the if, the equations corresponding to (129) are (134) The undetermined constants af and fi^i are exhibited explicitly in the first two equations, and it is to be noted that A",* and A^' are precisely the same functions of t as those which appeared in (129). In order that these equations shall lead to periodic values of the y the undetermined constants must satisfy the conditions (135) 0= - af> l - 0= -aft-a (j=3, . . . , ri), The first two equations are linear in af and &, and they determine these quantities uniquely unless their determinant is zero. The determinant is A=- - (!) On eliminating 77^ and a by means of (130), it is found that where D is the discriminant of (131). Since by hypothesis D is not zero, the determinant A is not zero. Hence af and B% are uniquely determined by (135). Having determined B ( and af, equations (134) are integrated and the results are substituted in the equations corresponding to (128). Then the conditions that y^ (0) = are imposed and the final solution at this step becomes (136) where B% is undetermined until the next step of the integration. LINKAIt IHKI KKKVtUL EQUATIONS. -M The next step is similar to the preceding and all the equations are the same except the superscripts are (3) and (2) in place of (2) and (1) respec- tively. The determinant of the equations corresponding to (135) is precisely the same. In fact all succeeding steps are the same, and the whole process can he repeated as many times as is desired. The solutions associated with ajj", .... a||" are found as they were in 2i>. For congruence- of higher order similar methods can be used, and in the cases which are exceptions to this mode of treatment the existence discussion furnishes a sure guide for the construction of the solutions. 28. Construction of the Solutions when two Roots of the Characteristic Equation are Equal. It will he supposed aij" = aj" and that all the remaining a'' are mutually distinct and distinct from a. The solutions depending upon aJ 0> , . . . , a'. ' can be computed by the method of 26. It has been shown that the two solutions proceeding from a', 0) are in general expansible as power series in -N/M. The detailed discussion will be made only for the general case, where (137) Terms independent oj n. The terms independent of n are defined by W'+tt-*fdff-Q (,=!,..., n). (138) The general solution of these equations is -"'")'. (139) In order that the (/;"' shall be i>eriodic with the period 2r and the initial value Cn of if shall be obtained, the V* must satisfy the conditions The solution satisfying all the conditions is then t/ ( ,? = c n (i-l ---- ,). (HO) Coefficients of n* . The coefficients of ^ are defined by the equations 44 PERIODIC ORBITS. On neglecting the right members, the general solution of these equations is l = i^i+i?$?fc,,+fcn) + 2 tfc^-^'. (142) j=3 The method of variation of parameters leads to the conditions (Wc n + (r,%Y(c t2 +tc n )+ 2 WA/**"^' = -a'X 0-1, . . . , n). j = 3 On solving these equations for the (r?)', it is found that OlS)'=-a? ) , G$)' = (j = 2, ...,n). Consequently ,=*S-aS% = (j = 2, ...,n). (143) On substituting the values from (143) in (142), the result becomes To satisfy the conditions for periodicity and to make T/," (0) = 0, the B n must fulfill the relations B ( tl = a, B ( ilc n -\-B ( 2lc 12 = 0, -B ( /i=0 C?=3, . . . ,n). (144) Then the solutions satisfying all the conditions become Vn = (- fc tl +c^ ai", (145) where the constant a"' remains as yet undetermined. It is to be observed that not all the coefficients of a"' can vanish, for otherwise the determinant |cj itself would vanish. Coefficients of /z. The coefficients of /i are determined by the equations n n (2) (0) (1) (1) I ^ /l(l) (0) / -t \ /I Ad\ = ~ a \ Mi\ ~ a i 2/n+ 2 ^uUn (!,..., n). (14b; The solution of the homogeneous terms is of the same form as (142), and by varying the constants of integration, it is found that 2 (147) LINEAR DIKKKUKNTIAL EQUATIONS. The solutions of these equations for the (a'/,')' arc 45 (148) where the &%(i) and />(0 an- known jwriodic functions of t. The first equation gives rise to integrals of the type a,( t *^jtdl= + ^ gfajt+^^jt' J 3 The second equation gives rise to the corresponding integral When these results are substituted in equations (142) the terms of the type t ^jt destroy each other. Hence at this step , (149) where the P%(t) are periodic functions of /, and 6"' and n the particular integrals are also periodic with the period 2r. But if some of the characteristic exponents are congruent to zero mod. V\, then the par- ticular integrals in general contain, in addition to periodic terms, the corre- sponding part* of the complementary function multiplied by a constant limes t. 30. Case where the Right Members are Periodic Terms Multiplied by an Exponential, and the a, are Distinct. Consider the case where the 0,(0 have the form the / 4 (0 being periodic with the period 2r. When X = I \/~ i is a pure imaginary, this form includes such cases as g t (t) = X[ If the differential equations, which are now of the form l-jtf&i-ffM, (160) are transformed by x t = e*z { , they become (161) and have the same character as those treated in 29. If the character- istic exponents a., of (160) are distinct, then the characteristic exponents of (161) are a,-X, and are also distinct. Applying the results of the pre- ceding case, it is seen that if no a, X is congruent to zero mod. v' = T, then the solutions of (161) are where the Q t (t) are periodic with the period 2*. Therefore the solutions of (160) are ('-i ---- ,) (162) 48 PERIODIC ORBITS. But if one of the a,, say a t , is congruent to X mod. V- 7 ! , then the z, have the form and therefore the expressions for the x t become t (f) +c t t e a "'y it . (163) These results may be stated as follows: // the g t (t) have the form 0i (0 " **/(*) where f t (t) =f t (t + 2?r), and if none of the characteristic exponents is congruent to X mod. Vl, then the particular solution has the form x t = e*Q t (t) (i=l,...,n), where the Q ( (f) are periodic with the period 2-w; but if one of the characteristic exponents, a t , is congruent to \ mod. Vl, then the particular solution has the form z, = e w Q,(0+c**e a *V (t = l, . . . ,), where the Q t (() are periodic with the period 2 IT. 3 1 . Case where two Characteristic Exponents are Equal and the Right Members are Periodic. Suppose a 2 = a i . Then the solutions of xl-ZOuX^O (*-!,..., n), in general have the form y it (i= 1, . . . , n). (164) For the associated non-homogeneous equations *t-,Z4,*,-ft(<) (165) it is found by applying the method of the variation of parameters that e a >'y n r,{+e a > 1 (y a +tyM+ S e a y t ,-n', = g t (t) (i=l, ...,). ^=3 On solving these equations for the ^ , the results are found to be tyn), ya , . . . , y t n\e- a>t , g f (t), y t3 , . . . , y tn \e- ati , where A is the determinant \y tj \ . The expansions of these determinants have the form = - w (0-e r,' 2 = e- a 'T,(t), } O (j-3, ..., n), I where the P f (t) (j=l, . . . , n) are periodic with the period 2ir. I'Alilli I I.\K SOLUTIONS OF LINEAR EQUATIONS. 49 Suppose no a, is congruent to zero mod. \/^T. Then it is easily found that (167) where /?,(0, , #(0 are periodic with the period 2r. On substituting these values in (164), the solution become- x, = , e a "y (1 +# a e a % w +< j/ n ) + 5, e a >/ ( , + 2 fl/<) y,,. The 1' A*//, are periodic with the period 2r. Hence. if tiro of the characteristic j-i exponent* HIT eijinil hut mine of them is congruent to zero mod. V^l, then the /Hirtieiilnr .Dilution also /.s />< riutlic irith the period 2ir. The ease where one a, (j = 3 , . . . , /<) is congruent to zero mod. Vl is a combination of the present ease with the second part of that treated in S'J'.i. and that where a, = a, is congruent to zero mod. Vi does not differ in any essentials from that where a, = a, = 0. \<>w suppose a, = a l = 0. Then the equations which correspond to (166) become The Pj(t) are periodic with the period 2r and can be written in the form 00 Pj = a, + 2 [a tj cos kt+b u sin kt\. Hence the TJ, are found, by integrating, to have the form (j=3 where These values substituted in (164) give for the complete solutions [a l t+la 1 ('] y ll +a i ty a + Hence, irh< n the g t (t) are periodic with the, period 2*, and when two of the char- acteristic exponent* arc not only equal, but are zero, then the particular intei/rul int'olris nut null/ t hut. in + higher degree terms (t-1, ...,*), (172) the x i being replaced by

t (t) are periodic in t, so also are all the coefficients of (172). The & are expansible as power series in theft which, by the Cauchy-Poincare" theorem, 16, converge for any preassigned interval of time provided the |ft| are sufficiently small. The differential equations for the linear terms in the ft are the linear terms of (172), or These equations are known as the equations of variation. Suppose the solution (169) contains an arbitrary constant c, that is, one not contained in the differential equations (168). If c = c -\-y, the

\ AM) lH\K\i IKKISTlr KXl'iiNKNTO. 51 is a solution of equation- I 7_' and consequently ,= (i-1, ) (174) is a solution of equations t 1~:{). One such constant i- always present when the A, do not contain t explicitly and the v ~, \h are not mere constants, for then the origin of time is arbitrary. Hence in this case d

i* i a// '^ , dl< \. ? = a i + T~~ r " +T~ I {+ higher degree terms. The constant 7 is a power series in theft, and therefore the linear terms are - 1 / / 1 1.' ' 1 1.' *f J^ ' I- J^ _L 1 t flTfi 1 ^ which is therefore an integral of equations (173). The coefficients db\/dx t are I>eriodic functions of /, the x, having been replaced by

eriodic with the period 2ir. The solutions (175) have the form (177), but since the tp t are periodic so also are their derivatives, and the characteristic exponent of this solution is zero. There is an exception only if the ^>, are constants, in which case the solution (175) disappears. The solution obtained by differentiating with respect to the scale constant, which will be denoted by a, will, in general, have the form ^, and \jf t being periodic. The characteristic exponent of this solution is zero. If the generating solutions have p distinct arbitrary constants, the equa- tions of variation will have at least p characteristic exponents equal to zero. 52 PERIODIC ORBITS. From the existence of the integral (176) it follows also that at least one of the characteristic exponents is zero; for all solutions have the form and substituting them successively with respect to the index j in (176), we get -^V," (j = i,...,n). (178) The left members of these equations are periodic with the period 2ir, except, perhaps, for coefficients which are polynomials in t. It follows, therefore, that either the a_,=0 mod V^l, or all the 7 < / ) = 0. In this connection a congruence has the same properties as an equality, and they need not be distinguished from each other. If all the a } are distinct from zero, then < ) = = 7 = = , . . . , ri) and (178) becomes (179) Since the determinant | f tj \ T* 0, these equations can be satisfied only if Therefore, unless the integral (176) vanishes identically at least one of the characteristic exponents is zero. Suppose that a, = and that 7"' ^ 0. It is possible then to solve the equations corresponding to (178) uniquely for the dFj/dz, in terms of 7"' and the f (j . Suppose now there is a second integral F 2 (x l , . . . , x n ) = c 2 . Then = const. t On substituting in this equation successively the n fundamental solutions for the , it follows, since a t = 0, that 0' = 2, ...,n). (180) If 0,^0, and therefore 5^ = (j=2, . . . , n), these equations can be solved uniquely for dF 2 /do;, in terms of / and 5. It results that, aside from a constant factor, dF,_dF 1 dx i ~ dx t ' and the second integral is identical with the first. But if F l and F 2 are distinct, then there must be at least two characteristic exponents, say c^ and Oj , which are zero. Proceeding in this manner it follows that if the equa- tions of variation admit of p linearly distinct integrals not identically zero, then there are p characteristic exponents equal to zero. EQUATIONS OF VARIATION AM) CHARACTERISE' KXI'i INKNT8. 53 If the original differential equations have the form He ~~ dx, ( '" 1 n)> which is the case usually in celestial mechanics, they may be reduced to equations involving only first derivatives by writing If the generating solution is and the equations of variation are formed by putting there will result &. <" (181) - r rf/ dz, di t dx t d* ' 3x, dx. ' The main diagonal of the right members of these equations (considered as a determinant matrix) contains only zero elements. Therefore, by 18, the determinant of any fundamental set of solutions of these equations is a constant. But the determinant of the fundamental set of solutions a/i / has the form A = e'~ P (t) . This must therefore be a constant, from which it follows that the sum of the characteristic exponents is zero since P(t) has the period 2*. Suppose "', TjJ" and #?, jjf (t = l, ...,n) are any two solutions of equations (181). Then "5 and also < <" i un . " " ^' dj? d'v ^ . ^ . ^ ""^' f$ -- From these equations it follows that S(j d $> m dtf\_ Q *? (di?_p4i? k* dt * l ~dt)~ ' 2*\*> dt * dt The sum of these two equations is s[(^f+^D-( which can be written . . 54 PERIODIC ORBITS. Consequently 2 (W-SfY/0^ const. (186) 1=1 The relation (186) between any two solutions leads to important conclu- sions respecting the characteristic exponents. Suppose the ^ J) and ^' are " = e ai 'f ti (t), ^ = e ait g tj (t} (i=i, . . . ,n;/-l, . . . ,2/0, where f i} and g tj are polynomials in i with periodic coefficients, and that they constitute a fundamental set of solutions. On substituting any two of these solutions in (186) and dividing through by the exponential, there results n S( -f f . / n \ n~ (Cij-{-O,k)t /I Q f 7\ \J ij ,fik Jikaij) * Ijk " * \ ^ ' / It follows from the character of the left member of this equation that either ei^+aj. = 0, or 7^ = 0. It will be shown, however, that 7 ;t can not be zero for every k. Suppose j is kept fixed and give to k all the values from 1, . . . , 2n. Suppose 7,4 = (k = l, . . . , In). Then one equation of (187) is an identity and the others are linear in the f tj and the g tj . The determinant of this set of linear equations is the determinant of the fundamental set and is not zero. Hence they can be satisfied only by f tj =g t j=0- But this also is impossible since f tj and g tj are a solution of the fundamental set. Therefore not all the y jf can be zero. Hence for some k ^j~r & ~ 0, 7j*7 2 ^0. (18o) But since a, is any one of the characteristic exponents, it follows that corre- sponding to each characteristic exponent there is another one which differs from it only in sign. If two of the a, are equal but not equal to zero, then there are two others which are also equal and which differ from the first two only in sign. In order to show this suppose a_, = a j+ , = a m . Then a M = a m+I , because from (188) it follows that o,+a m = 0, 7 jm ^0. If 7 ;t = (k = l, . . . , 2n, k^m), then (187) can be solved for f tj and g tj uniquely in terms of f, k and g, k (k = l . . . , In). Now the corresponding equations for / w+1 and g t , j+1 will differ from (187) only in that j is replaced by j+1. They can be solved uniquely for / 4iJ+1 and olar coordinates, In writing these equations the origin has been placed at one of the bodies and the variables r and v are measured in the plane of motion. Equations (1) are easily integrated, and the integrals show that the relative motion is in a conic section for any initial conditions. If the initial velocity is not too great the orbit is an ellipse, and the discussion will be limited to this case. While the ordinary integration of (1) shows that under certain conditions the orbits are ellipses, it does not express the coordinates explicitly in terms of the time. The explicit developments are obtained through solving Kepler's equation, generally by Lagrange's method or by means of Bessel's functions. In treating elliptic motion as periodic motion the expressions for r and v in terms of t will be derived directly from the differential equations. On integrating the second equation of (1), and by means of this integral eliminating dv/dt from the first, it is found that ,dv p^ 1. Upon substituting (5) in (4), the latter becomes , p-e _ Q ( 6 ) 2 (1 pe) 3 The second term in this equation can be expanded as a power series in e for all the values of p if e\ < 1, as is explicitly assumed, giving (t + l)[t-(i+2) P V-V, (7) and the first equation of (2) becomes by the same substitutions (8) <=0 35. Form of the Solution. The solution of equation (7) will first be considered. After it has been found, v is determined from (8) by a simple quadrature. Equation (7) belongs to the type treated in 14-16, and therefore can be integrated as a power series in e, and \e\ can be taken so small that the series will converge for ^ r ^ 2ir. Since the body moves so that the law of areas is satisfied and completes a revolution in 2ir, p is periodic with the period 2ir. Consequently, if the series converges for O^T 5=271-, it converges for all real values of T. It is, indeed, possible to find the precise limits for \e\ within which the scries will converge for all values of T, and outside of which they will diverge for some values of r. The problem was first solved by Laplace,* who found that the series converge for all T if e< 0.6627 . . . , which is far above the eccentricity of the orbit of any planet or satellite in the solar system. *Mecaniqtie Celeste, vol. V, Supplement; see also Tisserand's Mecanique Celeste, vol. I, chapter 16, and a demonstration by Hermite, Courts a la Fac. des Sa. de Paris, 3d edition (1886), p. 167. Kl. Ml'TIC MOTION. 57 The solution of (7.) can !>< written in the form P=2 P> (ry, (9) J-O where the p,( T ) are functions of T. According to 1;~> and the initial condi- tions, the constants of integration which arise are to be determined by tlie conditions = l,p,(0)=0 (f-i, .00); ^(0)=0 (i-1, . . . oo). (10) As p is ]M-riodic with the period 2r, it follows that p(r+2ir)=p(T); whence =p,(r)e'. (11) Since (11) is an identity in e it follows that p,(T+2T)ssp,( T ). But this is simply the definition of periodicity. Therefore each p, separately is periodic. The body is at its nearest apse when T = 0, and the orbit is symmetrical with respect to the line of apses. Therefore it follows that p is an even function of T. Since p is periodic in r identically with respect to e, each P,(T) is expressible as a sum of cosines of integral multiples of r. If the sign of e in (6) is changed, then the body is at its farthest apse when T = 0. Consequently changing the sign of e and increasing T by T does not change the value of r. Since r = a(l pe) , it follows that -e)'- 1 . (12) Therefore when./ is even, P,(T) involves cosines of only odd multiples of T; and when j is odd, p,(r) involves cosines of only even multiples of r. If we substitute (9) in (8), we get (13) ;-o d ,. d'r =Vl - > where * C, t ..... ,. -.-.,, (t,+H+ +i t -i). (14) (| . ij .... i, . Suppose ij t + +ij. + i is even; then there are two cases to be considered, viz. (a) when i is even, and (6) when j is odd: (a) When i is even an even number of t, , . . . , i. must be odd, and the number of odd t\ multiplied by odd j x must be even. Therefore the number of odd i x multiplied by even j x must be even. All those factors p^ in (13) for which t x is even involve only even multiples of T, and those for which j\ is even and t x odd involve only odd multiples of T. Since there must be an even number of these terms involving only odd multiples of T, their product involves only even multiples of T. 58 PERIODIC ORBITS. (6) When i is odd it follows from (14) that an odd number of t\ , ...,, are odd, and from the hypothesis that iJt+ . . . +ij,+i is even it follows that an odd number of t\ j l , . . . , ij s are odd. A term t x j x can be odd only if both *' x and j^ are odd. When ,; x is odd the term involves only even multiples of r whether raised to an odd or even power. Since the whole number of odd i x is odd, and an odd number of them are multiplied by an odd j x , it follows that there is an even number of terms p , where j x is even and i x is odd. Therefore their product will be cosines of even mul- tiples of T. That is, in the right member of (13) the coefficients of even powers of e involve only even multiples of T. It is easily proved in a similar way that the coefficients of odd powers of e in (13) are odd multiples of T. Upon integrating (13), it is found that v is expansible as a power series in e of the form CO v = cr+2v t e i , (15) <=0 where v t is a sum of sines of even multiples of T when i is even, and of odd multiples of T when i is odd. The coefficient c is unity because, the ellipse being fixed in space, v increases by precisely 2ir in a period. 36. Direct Construction of the Solution. Upon substituting equation (9) in (7) and arranging in powers of e, we obtain S J=0 (16) + [-6p p 2 +3p 1 (l- Pl -6p*)+2pS(3-5p*)K + [-6p (p 3 +3pf+3p p 2 -2p 1 )+3p 2 (l-2p 1 )-5p?(8 Pl -2-3pD]e 4 . . . , ] where p'' is the second derivative of p } with respect to T. Upon equating coefficients of like powers of e, the differential equations which define the several coefficients become (a) P:+P O =O, (6) pf+ Pl = l-3 P *, Ps"+P3 = - The only solution of (a) satisfying (10) is P = cosr. (17) Then equation (6) becomes pr+ft=-{-f cos2r. The solution of this equation satisfying (10) is KI.UPTIO ~M<>TI<>\. .V.) In a similar MIMIUKT |ii;itions (c), (d), . . . can be integrated in order, and their solutions are found to In P 3 = (-eos T+cos3T), p 3 = -(co8 2T-cos 4r), (19) The general term of die solution is defined by an equation of the form C08T+ + A? CMltr, (20) where the A ( ^ are known constants. Since p, is periodic, A^ is zero for all values of j, and since p involves only even or odd multiples of T according as j is odd or even, all the A ( S with even subscripts are zero if j is odd, and all the A with odd subscripts are zero if / is even. On putting A\" equal to zero, the solution of (20) satisfying the initial conditions (10) is p J = A?+ \ -A?+ 2 . A '" 1 COST- 2 xi 1 * cosXT ( 21 ) L f^ x ~ 1J ; A - 1 On substituting (17), (18), (19), . . . in (9) and (5), the final expression for r becomes r = a \ 1 - [ cos T] e-f ^ [ 1 cos 2r ] e'+ - [ cos T cos 3 T ] e* i r ~i |[cos2T-cos4r] e*+ -\- (22. On making the same substitutions in (8) and integrating, the explicit value of v is found to he v = T + [2 sin T~\e+ |~4 sin 2rle J 4- ["- 7 sin T+ JJ sin 3rlc 3 L J L 4 J L 4 1 - J (23) 37. Additional Properties of the Solution. It will be proved that no p, carries a higher multiple of T than j+l. It has been seen that it is true for j = 0, 1, 2, 3. It will be assumed that it is true up to j - 1, and then it will be proved that it is true for the next step. The general term in the right member of (7) is, apart from its numerical coefficient, p'" l e'. After substituting the series (9) for p, any term of degree j in e arising from this term has the form Pop^pj 1 p^e', where X+X,+X,+ +X. = i*l, X,+2X J + After eliminating i from equations, it is found that X+2X,+3X,+ +(+l)X.-j*l, (24) where obviously a by the second, and 5 is a parameter to be determined later.* There are so far three arbitrary constants of integration a, e, and T; the fourth is introduced in integrating equation (26). With these substitutions equation (27) becomes p- (33) Poincan* introduces a parameter r somewhat analogous to ibis. Let Mfiltodet NowtUe* dt la Mfcanupu COetle, vol. I, p. 61. 62 PERIODIC ORBITS. Equation (33) admits a periodic solution, as is known from the fact that the orbit is a rotating ellipse. However, it will be shown directly by forming the first integral of (27), viz., aM= ^ (r) - (34) Suppose the initial conditions are real; then 0. For /x = the equa- tion r' 2 . Consequently it follows from dr/dt = V the orbit becomes a fixed ellipse and 5 = satisfies the periodicity condition. Hence (42) can be solved uniquely for 8 as a power series in n and e , vanishing with M = . When the value of 5 obtained from (42) is substituted in (37), p becomes periodic in T with the period 2ir , and is expanded as a power series in n and e which converges provided |M) and \e\ are sufficiently small. It can be written P =Z I P( , M V, 5= S 2 5 . M V. (43) t=0 j=0 1=1 1=0 From the reasoning of 35 it follows that each p (J separately is periodic. The range of convergence of (43) is limited in two ways. In the first place the inequalities 1 6 1 < 5 , fj.\ l e l< e i must be satisfied in order that the solution of (42) shall converge and give for 1 5 1 a value less than 5 . When | /z and | e \ satisfy both of these sets of inequalities the convergence of (43) is assured for all T. After the explicit development of equations (43) has been made the results can be substituted in (26), when v will be determined by a quadra- ture. The final form of v is v= 2 1 Vij^ei (44) i=0 j=0 The constant parts of the v t) are independent of e since f or n = it was found that v = T + periodic terms. 41. Direct Construction of the Non-Circular Solution. In carrying out the practical construction of the solution, we shall make use of the facts that (a) p^l, p'^0 at r = 0, (6) p is expansible in the form (43), and (c) each p tj separately is periodic with the period 2ir. On substituting (43) in (36) and equating coefficients of equal powers of /JL and e, it is found that the several coefficients must satisfy (A) (B) (D) Po2 3p 01 +3(l "00) llli: UOTATINc; KI.Ul'SE. <).") The solution of 1 -atisfying (a) is p w = cos r. The solution of (B) is given in equation (18). The solution of ((') is not periodic unless the coeffi- cient of POO is zero. Imposing also the condition (a), i, =- 3, p, = 0. The term p M is given in equation (1!)). Equation (E) becomes explicitly PM + PII = ~ 3 <''*~ S,,COST 3/2 cos2r. Upon imposing conditions (a) and (c), the solution of this equation is found to be 5,, = 0, PlI =-f+co8T+}co82r. (45) The explicit form of (/') now becomes P+P = ( $jo+3) coe T, whose solution satisfying the conditions (a) and (c) is * = 3, p = 0. (47) Hence the final expressions for p and 6 as power series in c and M are p = cosT-f [-| + { cos27-]e+ [- f + C08T + I co82r]ne + f[-cosT+cos3T]e'+ , where, from (32), T is to be replaced throughout by u(t- r)/\/l-j-j. The differential equation defining the general term is Suppose all the p.* and 6^ for which I when z' = 0; consequently in this case c t c 2 >0. Hence in all cases of the physical problem c, c 2 ^0. Now consider equation (5). If the initial conditions are real,2o 2 =/(z ) is zero or positive and >) = +00. Therefore the equation /(z) =0 has always three real roots. Let them be ttj , o 2 , and a 3 , where the notation is chosen so that e^ ^ a 2 ^ o 3 . Then equation (5) becomes On comparing this equation with (5), it is seen that 2(a l + o 2 +a 3 )=c 1 , a l a a +a a a 3 +a 3 a l = -f, 2 a t a 2 a 3 = -C 2 f . (7) It will now be shown that a 3 satisfies the inequalities / ^a 3 ^ 0. It follows from the last of (7) that c 2 is negative if a 3 is positive. On putting z' = Q and z = a 3 in (5), we get By hypothesis the first term on the right is positive, and by (2) the second can not be negative. Therefore c 2 must be positive, which contradicts the implication from the last of (7). Therefore a 3 ^0. Some special cases may be indicated : (1) It follows from (5), (6), and (7) that a 3 = -/ implies that a t = +1, 2a2 = c i = c 2 , a t a 3 = 2L The constant a 2 is not determined, and we shall suppose it is less than +1. This case is that of the ordinary simple pendu- lum making finite oscillations. In the sub-case where a 2 = I, we have c, = c a = 2f and the solutions of (4) are x = y = Q, z= I. THK M'llKRICAL PENDULUM. fil (2) Ifdj (/, it follows from the same (Munition- that cij- /, 'Jc^-C^C,. The constant a, is not determined. If a t >l, we have the case of the simple |)eii2/. In the sub-case where a, = /, we have c,=c,= +2/, and the solutions of (4) are x-y-0, z+l. (3) If a, = 0, it follows that o,^0. Therefore the second of (7) can not be satisfied except by o a = 0, a, = oo . Then, from the first equation we net <-, = <. This is the case of revolution in the xt/-plane with infinite >l>erd, and of course can not be n-ali/ed physically. Excluding this case and that of the simple pendulum, the constants o, , a,, a s satisfy the inequalities /-f/- Now make the transformation 2-o,= (o 1 -o,)tt t . (8) Then equation (5) becomes (9) fll Also let _ ot-o. yd p *-w where t is an arbitrary initial time and 5 is a constant as yet undefined. The constant n satisfies the inequalities O^M^!- Then (9) becomes u*= (!+)(! -u*) (l- M u f )=F(tt), (11) where it is the first derivative of u with respect to the new independent variable r. The first derivative of (11) is ']. (12) 45. First Demonstration that the Solution of (12) is Periodic, and that u and the Period are Expansible as Power Series in p. It will first be shown that, for any initial conditions belonging to the physical problem, except when /* = !, the solution of (12) is periodic. By the fundamental existence theorem of the solutions of differential equations* the solutions of (12) are regular in T for all finite values of r and u. For real initial condi- tions the coefficients of u and its derivatives expanded as power series in r are real, and by analytic continuation they remain real for all finite real values of T provided u does not become infinite. Now consider the curve F = F(u). Suppose u = u at r = Q and that is positive. Then u is increasing at a rate which is proportional to the square root of F(u ), and it continues to increase until u=l. It can not increase beyond +1 for then u would become a pure imaginary, and it has just been shown that it always remains real. It can not remain constantly equal to 1 unless M=l; for otherwise u = 1 does not satisfy (12). Therefore, unless n=l,u will increase Picard's Trait* fAnalyte, vol. II, chap. 11, fill. 70 PERIODIC ORBITS. to 1 and then decrease to 1 ; then, in a similar way, it changes at u = 1 from a decreasing to an increasing function. That is, at u= 1 the func- tion F(u) changes sign and u varies periodically between +1 and 1. This result follows, of course, from the fact that in the present problem u is the sine amplitude of T, one of whose properties is that of having a real period, but the argument given above applies to much more general cases, and the result can be read from the diagram for F = F(u). It may be mentioned in passing that the imaginary period of the elliptic function is associated in a similar way with the portions of the curve between + 1 and + 1/Vju, and between 1/V/x and 1 . The period of a complete oscillation is found from (11) to be P= _2_ f +1 du ../i r~s I -.//i ..2\/i _.2\' \ iO J which is finite unless /z = 1 . We shall exclude this exceptional case. It is well known that D = 1) i T In t the period is i-sV , 1 ) M + -J- (is) That is, the period is expansible as a power series in ju, and in the present simple case the series converges provided n\-0 from which it follows that u (> (0) = (t,j-0, oo), Woo(0)=a, u,,(0)=0 (i+j>0). (18) On substituting (17) in (12) and equating coefficients of corresjwnding powers of 5 and M, we get = -">, (19) The solution of the first of these equations satisfying (18) is MM = a sin T. (20) Upon substituting this result in the right member of the second equation of (19), integrating, and imposing the conditions (18), we find w 10 =- |sinT+ IT-COST. (21) Hence we have u = asinT+ % [_ 8 j nT + T cos T ] 5 + higher powers of 6 and . (22) m Since the right member of (12) does not contain T explicitly, sufficient conditions that u shall be periodic with the period 2 T in T are (2T)-u(0)=0, M(2r)-M(0)=0. (23) 72 PERIODIC ORBITS. It will now be shown that the second one of these equations is necessarily satisfied when the first is fulfilled. Let u = Q+v, u = a-\-v, where v and v vanish at r = 0. Then (11) becomes On making use of the fact that 1 + d = a, we have There are two solutions of this equation for v, but the one which vanishes at r = must be used. It has the form i>=vp(v), (24) where p(v) is a power series in v. Since u (0) = v (0) = 0, it follows from the first equation of (23) that w(27r)=0. Then, from (24) we have v (2ir) =u(2ii) M(0)=0. That is, by virtue of the existence of the integral (11), the second equation of (23) is a consequence of the first. Now let us consider the solution of the first equation of (23). Upon substituting u from (22), we get = Trad + terms of higher degree in 5 and //. (25) It follows from the theorems of 1-3 that this equation can be solved for 5 uniquely in the form (26) where P,(M) is a power series in n, which converges if |M| is sufficiently small. On substituting this result in (17), we have J=0 (27) which converges for all 0 >ymnic(ric:il in r with ropect to the value = 0, which, by (8), corresponds to z = a 3 , or the lowest point reached by the pendulum. Suppose // = (), it=n at T = and that the solution of (12) for these initial conditions is = /,, /,(0)=0, M-/ f (r), /,(0)-o. (28) Now make the transformation u= v, u i), T= u i = ~^f [sinr+ sin3r]. (40) Upon substituting the results already obtained in the third equation of (38), we find sinr+ sinr ^ sin3r j- sin5r. (41) IMF. SPHERICAL PKNIH I.UM. 75 Upon setting the coefficient of sinr equal to xero. as In-fore, and integrating subject to the condition- :>7), \ve find 5 > = "w ' "- = -i- (7 sin r + 8 sin 3r + sin ST]. (42) The induction to the general term can now he made. We n>.-ume that u , . . . , ,_, ; 5, , . . . , 6,_, have been determined and that it has been found that i/ is a sum of sines of odd multiplies of r, of which the highest is 2./ + 1. The differential equation for the coefficient of p \e W,+M,= -S,w +F,(,, u*) (k, X-0 ..... i-l), (43) where the F t are linear in the 5, and of the third degree in the u x . The general term of /', is (44) where m = or 1, i ++*.= lor 3, THK+J', X, + i' 1 X,+i' I X > = i or t-1 (vi+Vt+Vt**! or 3). It follows from the second of these equations that there are an odd number of odd v, . Consequently T, is a sum of sines of odd multiplies of r. The highest multiple of r is On reducing this expression by the third of equation (44), it is found that N,= 2mK+2i + l, the greatest value of which is, by (44), JV,-2t+l. (45) Hence (43) may be written l sin(2i+l)r, (46) where the A$ +l are known constants. In order that the solution of (46) shall be periodic, the condition ,-X (47) must be satisfied, which uniquely determines 5,. Then the solution of equation (46) satisfying the conditions (37) is sin(2i+l)r, l> hi (\ = \ >}. (48) 4T(x+T) i (0 = 7P C-l) x i & \ 4X(X+l The solution at this step has the same form as that which was assumed for , . . . , u,_, , and the induction is therefore complete. 76 PERIODIC ORBITS. On collecting results, we have for the first terms of the solution u = [sinr]+ [si (49) And substituting these results in (8), we get for the final result \ Oi-o 3 ) [1 ~ C0s2r] ft + -5 (o,- a s ) [1 -cos4r] 0i-a 3 ) [16+3 cos2r-16cos4T-3cos6r]M 3 + ; (50) the expression for 7 1 agreeing with that found in (15). 49. Construction of the Solution from the Integral. In the direct construction of the solution we have made no explicit use of the existence of the integral (11). We shall show that it can be used to check the computa- tions, or to furnish the solution itself. Equation (11) can be written in the form Since u, u, and 8 can be expanded as converging power series in n, we have V = 2H 2 + = 0- (51) Since this equation is an identity in /x, it follows that i = f t (u ,u ,d 2 , . . . ,u ( ,u ( ,d t ) = Q (i=o, ...), (52) where

1 i 1 /~<(n o t ottj -(-^Jl/j , a, -- o, 'o^o > (56) which uniquely determine 6, and the Let us apply these equations to the computation of the first terms of u. Suppose u = sinT and take t = l. Then we find from (11) that whence Therefore _4.<'> 1 * _n H (| >_ o-'" * n , ^*i+g = "> =-4a, -^6 l = 0, a* =Za t - * U. agreeing with the results already found. The process can be continued as far as may be desired. Two different methods of computing the solutions have been developed. They are both reduced by the general discussion to the mere routine of handling trigonometric terms. When they are both used each serves as a check on the other. There is little advantage with either over the other, so far as their convenience is concerned; the integral has a slight advantage in that when using it the computations are made with cosines, and the disad- vantage of involving u to the fourth degree instead of only to the third. 78 PERIODIC ORBITS. II. DIGRESSION ON HILL'S EQUATION. 50. The x-Equation. The value of z was given explicitly in (50), and since it is periodic the second equation of (4), after transforming to the independent variable T, has the form 2 + ' ]z = 0, (57) where a is a constant independent of /z, and 6 1 , B 2 , . . . are periodic functions of T, having in this case the period TT. Since the period can always be made 2ir by a linear transformation on the independent variable, we shall suppose for the sake of uniformity that the period is 2ir. This problem belongs to the class which was treated in Chapter I, and can be transformed to the form considered there. But it is now in the form used first by Hill, and later by Bruns, Stieltjes, Harzer, Callandreau, etc., and because of its historical interest and for the sake of comparison with this earlier work, it will be treated directly in the form (57) . 51. The Characteristic Equation. Suppose that with the initial con- ditions z(0) = 1, i(0) = the solution of (57) is and that with the initial conditions z(0) =0, i(0) = 1, the solution of (57) is X = *(T), i = iKr), The determinant being equal to unity at T = 0, these solutions form a fundamental set. In fact, A is independent of T for an equation of the form (57), as was shown in 18. It follows from the initial conditions that its value is unity. Hence the general solution of (57) is ^(T). (58) where c t and c 2 are arbitrary constants. Now let us make the transformation x = e"t, (59) where a is an undetermined constant. Then equation (57) becomes +2a+a 2 +[a 2 +0 lM + 2 M 2 + - - - ] = 0. (60) The general solution of this equation is, by (58) and (59), M]. (61) HILL'S DIFFKHKMIAL EQUATION. 79 We now raise the question whether it is |>ossible to determine a in such a way that shall l>e periodic with the period 2ir. It follows from the form of (60) that sufficient conditions for the periodicity of $ with the period 2* arc On substituting from (61), we get, after making use of the initial values of + and /, In order that these equations may have a solution other than the trivial one r, =c, = 0, the determinant of the coefficients of c, and c, must vanish; or, * (63) Since A is equal to unity, equation (63) is a reciprocal equation, and becomes D=(e* a ') t -[(2ir)+v,(2ir)]e t * w +l=Q l (64) of which t he roots are e 2a '* and e- la< *. (This is not the o, of 44.) When the value of e 3a ' r which satisfies (64) is substituted in (62) the ratio of c, to c, is determined. Then equations (61) give '" and (l> periodic with the period 2r. We get a second solution <2) and <2) by using the other root e" 2 * 1 '. The (l) and (2) will each carry an arbitrary factor. We shall determine these factors so that ( " (0) = {2) (0) = 1, and multiply the solutions by arbitrary constants at the end. Consider equation (64). If \ 2 , a, is real ; and if eriod 2ir , viz. c tt ' r (1) and e~ a ' r (2) . The coefficients of (57) by hypothesis are all real, and the 9, are sums of cosines of multiples of r. Therefore, if the sign of v^T be changed in a solution the result will be a solution. Suppose 80 PERIODIC ORBITS. a and n have such values that a t is a pure imaginary. Then it follows that (1> (V i)= <2> ( V i). Similarly, since (57) is unchanged by changing the sign of r, it follows that cl> (r) = <2) ( - r) . And finally, since (57) is unchanged by changing the sign of both v^T and r, it follows that e\V=l,T-) = e\-^S=i,-T), r(v^7,T)= (2 >(-V^T,-r). Therefore in the expressions for (1) and <2) the coefficients of the cosine terms are real and of the sine terms pure imaginaries, and (1) and (2) differ only in the sign of Vi. Hence, writing them as Fourier series, we have U) = S [CIJCOSJT+ V 1 bjsmjr], (2) = S [a,; COSJT V 1 bj sinjr], where the aj and the bj are real constants. It follows from this that it is sufficient to compute (1) . Any solution of (57) can be expressed linearly in terms e a ' r (1> and e -a,T<2> jj. f o n ows f rom the initial values of , and (2) and the equations above that Since ^ a> and | (2) are periodic with the period 2 TT, and since their initial values are unity, we have I (e a .'+ e - 2ft ") = cosh 2a l v = But by (64), e 2a ' T +e- 2a " r = 7i that* can le expanded as ;i power serie- in n which will converge for 0^r^2r if |M| \8 sufficiently small. Hence (67) Therefore equation (6ii) cos2air-e' a '+ X-l ' " 2 *<2 X-l -asin2air + 2 ^(2ir V, cos2ar -** + 2 ^ x ( X-l x-l = 0. (68) This e(|ii:it ion expresses the condition that ~7 >liall ha\e a |>eriodic solution of the form (59), where $ is i>eriodic with the period 2r. If it is satisfied by a^a,, it is also satisfied by a = o t) +' V 1, where v is any integer. Tin - different values of o do not, however, lead to distinct values of x. \\ C -hall use only those values which reduce to a V 1 for /u = 0. 52. The Form of the Solution. For the principal solutions of (68) are a ( , 0) = +(i V -T and oa >= a V l. There are three cases depending upon the value of : Case I. ^0 and 2 not an integer. Case //. a 7^0 and 2o an integer. Case ///. a = and therefore oj 01 = a?' = 0. Case I. This may be regarded as being the general case, and is that actually found and discussed by Hill and later writers on the same subject. It follows from the form of (68) that # = P(a,M), (69) where P is a power series in a and M which vanishes for n = 0, a= a V i. It is also easily found for /* = and a = *a V 1 that = 4ir V= da which is distinct from zero under the conditions of Case I. Therefore it follows from the theory of implicit functions that (68) can be solved for a in the form where the series converge for > sufficiently small. Since the equation for a is a reciprocal equation in e 7 ", it follows that a\" = o^O'-l, 2, ...,). 82 PERIODIC ORBITS. If we substitute either of the series (70) in (62), we shall have the ratio of Cj and c 2 expressed as a power series in M- If this result and equations (67) are substituted in (61), will be expressed as a power series in n, converging for \p\ sufficiently small, and carrying one arbitrary constant factor. We shall omit the superfix and adopt the notation 2 + (71) Since the periodicity conditions have been satisfied, is periodic and it follows from its expansibility that each , separately is periodic. Hence it follows from this property and the initial condition (0) = 1 that < (2ir+T)- i (T)=0 (t-0, 1, . . . oo), j (0) = 1, ,(0) = (i = l,2, . . . oc). J It will be shown when the solutions are constructed that these properties uniquely define their coefficients. Case II. In this case we find from (68) f or n = 0, a = a V^Tf , that Hence if we let a a V 1+/8, the expression for D has the form 47r 2 /3 2 + Cll /3 M +c 02M 2 + -0, (73) where in general c u and c 02 are distinct from zero. In order not to multiply cases indefinitely we shall suppose that c u and c 02 are distinct from zero and that the discriminant of the quadratic terms of (73), viz. 8 = c 2 n 167r 2 c 02 , is distinct from zero. Suppose the quadratic terms factor into where now b^b 2 since d^O.* Then by the theory of implicit functions equation (73) is solvable for /8 as converging power series of the form + (74) Hence we get two solutions for /3, and consequently for a, as power series in M starting from the root a= +a V 1 for fj, 0. There are two similar ones obtained by starting from a = a Vi for n = 0, but they do not lead to distinct solutions since they differ from the former values by purely imaginary integers. Then, by means of (62) and (61), we obtain the final solutions as before. There are other sub-cases, for example &j = & 2 , all of which can be treated by the theory of implicit functions, but they will be omitted. *Since a = o, and a= c^ are the roots of (63), it follows that in this case 6i = b,, and that they are therefore distinct unless b t = b, = Q. 1 Ml KKKKMIAI. Kljl Ali<>\ M /. ///. I'nder the conditions of this case \ve have for /i = 0, instead of equation.- till and i7 . the solution Jo = a,, V.= l, ^o = ^- (75) Hence equation D = e~ i .-e ta '+ 2 < expanded into a converging |>ower series in a and n, and for n = Q the principal solutions of D = are o, o, = 0. We find from (76) for M = a = that In general ^,(2ir) is distinct from zero, and when it is we know from the theory of implieit functions that (76) can be solved for a in the form (77) Since a, and a s differ only in sign the a', 1 " are all zero. After a, and a, have been determined, we obtain the final solutions as before, except now the series proceed in powers of N/P instead of in powers of //. 53. Direct Construction of the Solutions in Case I. On substituting the first of (70) and (71) in (60) and equating the coefficients of the several powers of n to zero, we obtain ,7sj the left members of all the equations being the same except for the subscripts, and the first terms on the right being the same except for the superscripts on a, . There is a similar set of equations defining the other solution, which differ from these only in the sign of V\. Consider the solutions of (78) subject to the conditions (72). The general solution of the first equation is where b and 6^ are arbitrary constants of integration. Since in this case 2 a is not an integer, it follows from (72) that o=l. (79) 84 PERIODIC ORBITS. The right member of the second equation of (78) now becomes a known function of r. When the left member of this equation is set equal to zero, its general solution is e- 2aV - lT . (80) Now regarding &<" and &" as variables, according to the method of the variation of parameters, and imposing the conditions b + b e~ 2a " / ~~ 1T =0 and that the second equation of (78) shall be satisfied, we obtain fQ-i \ Let the constant part of B l be d^. Then in order that &"* shall not contain a term proportional to r, which would make (80) non-periodic, we must impose the condition (1) V 1 Q! (Q<)\ ft, ^o^^ The integrals of sines and cosines are cosines and sines respectively, and Therefore it follows, when (82) is satisfied, that -r P --r -? -' + 2 sin , (83) where P : and Qj are periodic functions of r having the period 2ir, and BI" and B" 5 are the constants of integration. Since 2 a is not an integer, / 4a 2 can not vanish and there are no terms with infinite coefficients. On substituting (83) in (80) and imposing the conditions (72), we get *i = P (r) + Q 1 (r} ~ P l (0) - Q, (0) . (84) It is easy to show that ah 1 succeeding steps of the integration are entirely similar. The differential equation for the coefficient of M* is where F t (r) is an entirely known periodic function of T after the preceding steps have been taken. The general solution of the left member of this equation set equal to zero is the same as (80) , except that has the subscript i, and 6, and b 2 have the superscript i. The equations analogous to (81) are />"> <*> ^ *~ 61 - "L" 1 ' "2^ If we represent the constant part of F t by d t , we must impose the condition in order that the solution shall be periodic. Then integrating, substituting in the equation analogous to (80), and imposing the conditions (72), we get where P 4 (r) and Q,(r) are periodic with the period 2?r. Thus the general step in the integration is in all essentials similar to the second step. HILL'S DIH i \i. EQUATION. 85 54. Direct Construction of the Solutions in Case II. Since in this case the solutions arc al.-u in ^rnrral oVvrlopahlr as power series in ft, we start from equations (78). The general solution of the first equation is Since 2o is an integer, is periodic for all values of 6[ w and b. In this case it is convenient in the computation to impose the initial condition (0) = 1 instead of {(0) = 1, whence (0) = 1, ,(0)=0 (i-1, ...). 1 It ace we have for the solution at the first step of the integration where b is a constant which will be determined at the next step. The second equation of (78) now becomes (85) (86) The equations analogous to (81) are in this case (87) In order that , shall be periodic the right members of these equations must contain no constant terms. Hence we must impose the conditions 6r = 0, (88) where d, is the constant part of 0, , and where 5,, and & M are the constant parts of \^~l e i e~ uv=lT /2a and v^T 1 e ta>/:r7r /2a respectively. If 6 t is an even function of T, then $, = $, and if it is an odd function, 5,,= 5 W . Equations (88) express the conditions that the right member of (86) shall contain no terms independent of T, or which involve T only in the form e -i%/=7T This is only an expression for the fact that in order that the solutions shall be periodic the right member of the differential equation must not contain terms of the type obtained by integrating the left member set equal to zero. Upon eliminating 6J ' from (88), we get [20 V^T C+d,] [2av^T a -d.1-40 1 *-(), of which the solutions are ^^ (89) The two values of a"' are distinct unless 5 M -|-d?/4a* = 0. They will not be equal to zero except for special values of the coefficients of the differential 86 PERIODIC ORBITS. equations, and we shall assume here that they are distinct. This was the case treated in 52, Case II, and when the two values of a"' are equal the solutions may be developable in a different form. After a"' has been found, &i 0) can be obtained at once from either of equations (88) . There is a difficulty only if 5 2l = 5 22 = 0, when one solution for b ( } becomes infinite; but in this case we impose a different initial condition on . After satisfying equation (88), the integrals of (87) have the form 2 > where P l (T) and Q t (T) are periodic functions of r, and Bf and B^ are unde- termined constants of integration. These results substituted in equation (80) give, after applying the condition ^(0) = 0, 2 ^-, (90) where B is so far undetermined. It is necessary to carry the integration one step further in order to prove that the general term satisfying the periodicity condition and the initial condition can be found. The differential equation for the coefficient of /i 2 is -2aV~l al" B^-6, B ( +F 2 (r), I where af and B ( " are undetermined constants, and where F 2 (r) is an entirely known periodic function of T. In the case under consideration the equations corresponding to (88) are -a + 2a V^ where d. 2 and 5 2 are known constants depending on F 2 . The unknowns aJ 2> and B enter (92) linearly, and, by means of (88), the determinant of their coefficients becomes 4aV^-7 a"', which, by hypothesis, is distinct from zero. Therefore a"' and B\" are uniquely determined by these equa- tions. When equations (92) are satisfied, the solution of (91) satisfying the initial condition is which has the same form as (90). Therefore the next step can be taken in the same manner. Thus it is seen that the process is unique after the choice of the sign of aj ]) is made, and in this way two solutions which satisfy the periodicity and initial conditions are obtained. III1. 1.'- miKKHKM I AI. K.I I \!ION. 87 55. Direct Construction of the Solutions in Case III. In this case t he- solutions were proved to have, in ^eiientl. the form = *+*, v^+&M+ , a = 0+a"VM+a M + (94) Substituting these equal ions in (60), we have for the term independent of Vji u=o, of which the solution satisfying the conditions (72) is .= !. (95) The differential equation which the coefficient of \//I must satisfy is and the solution of this equation which satisfies the conditions (72) is , = 0, o (l) = an undetermined constant. (96) The differential equation which defines the coefficient of M is then - -2a"> -.''-- -a mf - - When the left member of this equation is set equal to zero, its general solution i. found to be , (97) where 6 ( , 2) and 6 are the constants of integration. On making use of the variation of parameters and imposing the condition that the differential equation shall be satisfied, we find &}"= -[('")'+*,], 4?- +[(a n> m] r. (98) In order that, when the first of (98) is integrated and the result substituted in (97), there shall be no term proportional to T*, the condition (a m )'+d, = (99) must be imposed, where d t is the constant part of 0, . This equation deter- mines two values of a (l) which differ only in sign, and they are reals or pure imaginaries when the coefficients of 0, are real. Since there are no more arbitraries available in (98), no more conditions can be satisfied. The first equation of (98) gives rise to integrals of the type The second of (98) gives rise to the corresponding and associated integrals Hence, imposing the condition (99), integrating (98), and substituting the results in (97), we have *, = P,(r)-P,(0), (100) where P,(T) is periodic and a (1) is as yet undetermined. 88 PERIODIC ORBITS. In determining the coefficient of M V! , the equations for &J 3) and &" corre- sponding to (98) are found to be where 3 , we must impose the condition which uniquely determines a <2) if a a) is distinct from zero, as it is in general. If a (1) = the expansion may be of another form, for this is an exceptional case in the existence discussion, and it is necessary to go to higher terms of the differential equation to determine the character of the solution. But limiting ourselves here to the case where a u) is distinct from zero, the solution is carried out as in the preceding step. All the succeeding steps are the same except for the indices and the numerical values of the coefficients. III. SOLUTION OF THE X AND Y-EQUATIONS FOR THE SPHERICAL PENDULUM. 56. Application to the Spherical Pendulum. On transforming from t to T as the independent variable in the second equation of (4), and making use of (50), we get ,-,) cos 2r], (101) Ctl 0.3 fli 0.3 Obviously a will not in general be an integer. It will be shown that the only integral value it can have in the problem of the spherical pendulum is unity. Suppose a equals the integer n. In this case the second of (101) gives It was shown in 43 that in the problem of the spherical pendulum a t is positive and a 3 is negative. Therefore n- must be unity or zero. In the former case we have a,= a 3 , which, because of the inequalities satisfied by a, and a 3 , can be true only if a 1 = +1, a 3 = I. This is the special case of the simple pendulum. If n = we have 4a : = 2a 3 , which, because of the inequalities to which a, and a 3 are subject, can not be satisfied. Therefore a is not an integer and the equations can be integrated by the methods of 53. Upon emitting the subscript on the a in e ar , so as not to confuse it with a, , cu , a, defined in 43, it is found by actual computation that 4V(a I -a 3 )(2a 1 +a 3 ) ' . fl _ 1 t X ft \ ft _ I . , V--1 Sin2r, 1111. M'HKIUCAK 1'KMH I.I M. The other solution is found from this one simply l>y eh.-inninn the sign of \ 1. Hence I lie general solution is fV2v 2a, , 4V -1) "1 ' J V- (102) 1 1 can he shown from the properties of 0, , 0, , . . . and the method of constructing the solutions, that the coefficients of n, in (l) and (2) are cosim > and sines of even multiples of T, the highest multiple being 2j. It follows from the form of equations (102) that, for real initial condi- tions, A and B must be conjugate complex quantities, 2A = A t V I A t , and 2 B = A t + V^~iA s . Hence the solution takes the form (103) x, = l- -^r(cos2T-l)/i + (cosines) / $(a. t -\-a t ) V2 V(a -J (2 where /I, and ^4, are arbitrary constants. Since the second and third equations of (4) have the same form, the solution of the latter can differ from that of the former only in the constants of integration. Therefore y = B l (x l cosXr x t sin \r]-\-Bj [x, sinXr+x, cosXr]. The constants A l , A a , B, , and B t are subject to the conditions that equations (2), the first equation of (4), and the relation xx+yy+zz=Q shall be satisfied. This leaves one arbitrary which may be used to dispose of the orientation of the xy-&xcs at r = 0. Let the axes be chosen so that x = at T = 0. Then , since z also vanishes at T = 0, we have from this equation and the values of x and y above x,sinXT], j/ = /?,[x, sinXr+x, cosXr]. (104) 90 PERIODIC ORBITS. Making use of (104), it is found from (2) and the first of (4), that at r = 0, -(a,-a 3 )[X+X 2 (0)T 2 Therefore, the solution is completely determined when the positive directions on the x and ?/-axes are chosen. The well-known properties of the motion can easily be derived from equations (104). The variables x and y always oscillate around their initial values since they are made up of terms which are the product of two periodic functions that are always finite. Since the period of x l and x 2 is TT, the solutions are periodic and the curves described by the spherical pendulum are re-entrant provided X is a rational number. Let the period of x l and x 2 be P I = TT, and that of sinXr and cosXr be P 2 =27r/X. Then, when P l and P 2 are commensurable, P? = 2 = 2j P l X " p ' p and q being relatively prime integers. Hence the least common multiple of the two periods is ^ =2?7r. (105) In the period P the variables z, x l , and x 2 make 2q complete oscillations, and sin Xr and cos XT make p complete oscillations. In the independent variable T the period of z, x t , and x 2 is independent of M, but P 2 is a continuous function of /JL. In the original independent variable t both periods are continuous functions of t. But in either variable the period P is a discontinuous function of n, being finite only when the ratio of P l to P 2 is rational. It is seen from the solution expressed in terms of T, in which P, is constant with respect to n, that this ratio fills a portion of the linear continuum, and therefore that onhy exceptionally is it rational. 57. Application to the Simple Pendulum. Since the problem of the simple pendulum is a special case under that of the spherical pendulum, it can be treated by the same methods. Of course, it is not advisable to do so in practice, for x and z must satisfy the relation z 2 + z 2 =f, (106) from which x can be found when z has been determined. Before discussing the properties of x we must find the expression for z in this case. Since in the simple pendulum z always passes through the I Hi: -IMIKKH Al. I'KNDl I. I'M. 91 value /. it follows that a 3 = I. I'Voiii the fact that z' - II \\hcii z=l, we get, by (5), c, = f, and z' = for z=+l. Hence, in the case of the simple pendulum we have from (50) and (101) z = -1+1(1 -cos 2T)M+{(l-cos4T) M *+ a, 0,=J(l+2cos2T), ]x = 0. (107) In the expression for z the coefficient of each power of n separately vanishes at T = and is a sum of cosines of even multiples of r. Therefore contains sinV as a factor. The parameter n is also a factor. From the relation M(! cos2r) = 2/x sinV, it follows that x=VF^* (108) is expansible as a power series in VM, containing only odd powers of v//u. It is easy to show that the coefficient of (V/f) tl+l is a sum of sines of odd multiples of T, the highest multiple being 2t + l. We find directly from the second line of (107) and from (108) that (109) It follows from (109) that the last equation of (107) has a solution of the form (110) where the x w+1 are periodic with the period 2* instead of T. and where *n+i(0) = 0. It is not possible to determine completely the constants of integration from these conditions, for if (110) is a solution, then, since the last equation of (107) is linear, (110) multiplied by any power series in p having constant coefficients is also a solution. For example, we have for the determination of x, and x, i.+x^O, x l +x J =-|(l+2cos2T)x, ) the solutions of which, satisfying the conditions x,(0) =x,(0) =0, are n x, = CisinT, *t = c sinr+ c, sinSr, where r, and r, are undetermined. This indeterminateness continues as far as the solution is carried, unless additional conditions are imposed. 92 PERIODIC ORBITS. The value of x at T = is an infinite series in VM whose general term is not easily obtained; but, from the fact that z(ir/2) = a. 2 = Z+2/ju and from equation (108), we get (111) On determining c t and c 3 by these conditions, we find x = if [2 sinr] M V '+ |[-5sinT+3sin3T]M v '+ }' agreeing with the direct computation (109). We may also consider the last equation of (107) from the standpoint of the general theory of linear differential equations having periodic coefficients. From the fact that the part of the coefficient of x which is independent of /x is unity, it follows that the solution of this problem belongs to Case II. Since there is one solution which is periodic with the period 2ir, the values of a are independent of /j. and are simply V l. We have here the case in which the two values of a not only differ by an imaginary integer for /* = 0, but for all values of /*. It follows from 21 that in this case the second solution is either r times a periodic function or, for special values of the coefficients of the differential equation, a periodic function. In the problem of the simple pendulum the second solution is T times a periodic function, and is most simply found by integrating the last equation of (107) with the initial conditions If we make the transformation x = e aT , then the last equation of (107) becomes (112) -^(5+16cos2T+4cos4r) M 2 + We shall integrate this equation and determine a so that shall be periodic with the period 2w. The chief point of interest will be that a will be inde- pendent of n so far as the work is carried. The equations corresponding to (85) and (86) are t SO - f (1+2 cos2T) (26+ V- (113) THK Sl'IIKUICAI. I'KNOULUM. 93 The conditions that the solution of the second of these equations shall be periodic are - 2v /- i CC- f &1 0) - f v - 1 =0, -a',"- f &1 '- f V-l=(}, (114) of which the solutions are > = , &> = -v-i. (115) Hence, we see by direct computation from the differential equations that in this problem a is independent of /* up to ju 3 at least. 58. Application of the Integral Relations. We now return to the con- sideration of the problem of the spherical pendulum. Since z and x have been determined the value of y can be found from the first equation of (4). But it will be noticed that in this work no explicit use has been made of the integral (2). Now x, y, and z must satisfy the differential equations, given in the last three equations of (4), and the integral relations x> (116) Oi O xx+yy+zz=Q, xy-yx=c t , where the second equation is determined from (2) by changing the inde- pendent variable from t to T; the third equation expresses the fact that the motion must be along the surface of the sphere; and the fourth equation is obtained from the second and third equations of (4), and expresses the fact that the projection of the areal velocity on the zj/-plane is constant. The solutions of the second and third equations of (4) have been shown to have the form x = a, e^.+a, e~ ar ^ , y = b v e* T t t +b t e-* T !-, , (117) where a, , a 2 , 6, , and 6, are arbitrary constants, , and 2 are jwwer series in n and are periodic with the period T, and o is a pure imaginary which is also a power series in n, but is not an integer. Moreover, , and , are conjugate complex quantities. If we make use of (117), the first equation of (116) becomes (aH-6^?e iar +(^+6J)^e- w '-|-2(a l a,+6 1 6,)f l { 2 = r-z 1 . (118) Now it has been shown that z 2 is expansible as a power series in n and that it is periodic with the periodic T. Before proceeding further we shall prove a lemma. Suppose there is given Suppose the ) 2 , and the K,(j=\, . . . oo) involve the a u) linearly. On referring to (122), we see that the H t and the K s have the form K , = &+& cos2r+ +D$ cos2;r. j Since (131) are identities in T, it follows that %=& (i=o, ...,;; j=0, ... oo). (133) It will now be shown that equations (123), (129), and (133) determine uniquely the aV , 6",', a" 1 , a ' (j>0), in the order of increasing values of j when z and 8 are known. To do this it is necessary to develop the explicit forms of (129) and (133) by reference to equations (121) and (130). It is necessary to eliminate a 2 and I 2 from their right members by equations (124) and the first of (10). When j = 0, we get from (121) and (130) 4(a[ 0) ) 2 =-2a 3 (a 1 0=-(o (0) ) 2 [2o 3 (a 1 4 a3(2ai+a 3 )(ai -fas) (134) 0103 The first of these equations determines aJ 0> except as to sign. The sign of af depends upon which is taken as the positive end of the z-axis. The second equation gives (o (0) y 2 = - a 2 = - 2(2ai+a3) , (135) 01 as agreeing with the result in (101). I UK -IMIKHI. Al. I'KMM l.l \! . 97 When./= 1. we find from lL'1 i and l.SOi that I B(o -r-8 = /; I ^ fO 0-2a t c ( t " = B i t t \ ' = = - 4 a "' a, ( a, + a,) a'" - (a") 1 (aj - oj) - 2 (a (0) ) s a, ri" 13li Oi n C?->-16(af A -\b=-2(* m ttl Ol !< which we must add the lir-t equation of ( 123) for>= 1. The unknown.- in the>e five c |iiations are <", a 1 ,", a"', a 1 ", and &i", which enter linearly. The second equation determine- <;_.' ; then f^" is found from the first of (123); then the first of (136) defines ;", while a ( " and &"' are jjiven ly the last two equation.- <>l" 136). We shall apply (136) in computing the first terms of the solutions. We find from (49) and (50) that *,-, ci"=-C= !(,-,). (137) Upon substituting in (136) and solving these equations and the first of (123) for oj", ending upon t lie coefficients having smaller numbers for the su|M-rscripts. The j+2 equa- tions of the first two and the last lines define uniquely the j + 2 quantitie.- aml iV (i = j). The equation of the third line determine- a ', and the equations of the fourth line, the coefficients ///, (t"l, . . . J). 98 PERIODIC ORBITS. Therefore we have the interesting result that in this problem the coefficients of the general solution can all be determined from the integral relations alone, the solution of the z-equation having been previously obtained from another integral in 49. The last two equations of (116) are unused integrals. Let us consider the last equation, which is the more complicated. By means of (117), we get 4 a\ a cos2r+ +E ( 2j cos2jr, " } cos2r+ -\-F 2 j cos2jr, from which it follows that ETO') ft-') ( ' A ,,'. f\ rf*\ ('I 4O^ The EM and the F 2 " (i = l, . . . , j) are known functions of the coefficients already computed, while the F ( J) involves the unknown coefficient of n 1 in the expansion of c a . Consequently equations (142) determine this constant for i = 0, and also furnish a check on the earlier computation of the coefficients for i=l, . . . , j. CHAPTER IV. PERIODIC ORBITS ABOUT AN OBLATE SPHEROID. BY WILLIAM DUNCAN MACMILLAN. 59. Introduction. The orbit of a particle about an oblate spheroid is not. in general, closed geometrically. The motion of the particle is not, therefore, in general, periodic from a geometric point of view. But if we consider the <>rbit as described by the particle in a revolving meridian plane which passes constantly through the particle, several classes of closed orbits can be found in which the motion is periodic. The failure of these orbits to close in space arises from the incommensurability of the period of rotation of the line of nodes with the period of motion in the revolving plane. When these periods happen to be commensurable the orbits are closed in space and the motion is therefore periodic, though the j>eriod may be very great. Indeed, it seems that much of the difficulty in giving mathematical expres- sions to the orbits about an oblate spheroid rests upon the incommensura- bility of periods. The difficulty arising from the node can be overcome in the manner just described, but elsewhere it is more troublesome. Orbits closed in the revolving plane are considered most conveniently in two general classes: I, Those which re-enter after one revolution; II, those which re-enter after many revolutions. The existence of both classes is established in this chapter and convenient methods for constructing the solutions are given. Orbits which re-enter after the first revolution are naturally the simpler and will be considered in the first part of the chapter. Those lying in the equatorial plane of the spheroid become straight lines in the revolving plane, and within the realm of convergence of the series employed all orbits in the equatorial plane are periodic. When the motion is not in the equatorial plane there exists one, and only one, orbit for assigned values of the inclination and the mean distance. These orbits reduce to circles with the vanishing of the oblateness of the spheroid. In considering orbits which re-enter only after many revolutions the differential equations are found to be very complex, and one would despair of ever finding any of these orbits by direct computation. However, a proof of their existence and a method for the constructions of the solutions are given by the aid of theorems on the character of the solutions of non- homogeneous linear differential equations with j>eriodic coefficients. 100 PERIODIC ORBITS. These periodic orbits of many revolutions involve five arbitrary con- stants. One, only, is lacking for a complete integration of the differential equations. The orbits are all symmetric with respect to the equatorial plane. 60. The Differential Equations. The differential equations of motion of a particle about an oblate spheroid are* d*x k^Mxr, 3 , 2 2 z 2 +2/ 2 -4z 2 ~\ dV j?, 2 2 x'-^y 4g* , -i _ d F R* ^ 10 M fl 4 = ^~' k*My r, 3,22 x 2 +t/ 2 -42 2 n ay 3,2 2 ir-f-y , w 6 M ~7F" 'J = '] = dt* R 3 10" fl dy' d?z tfMzr, 3,2 2 3(x 2 +2/ 2 )-2? 2 -i 5F df /? L- w " ^ J - a 2 ' (1) The symbols employed are defined as follows: The x, y, z arc rectangular coordinates, the origin being at the center of the spheroid and the xy-plane being the plane of the equator, k is the Gaussian constant, b is the polar radius of the spheroid, M is the mass of the spheroid, ^ is the eccentricity of the spheroid, , n -I Since - - , we obtain one integral of areas, namely x dx y dy That is, the projection of the area described by the radius vector upon the equatorial plane is proportional to the time. We have also the vis viva integral There are no other integrals which can be expressed in a finite number of terms, and for further integration we are compelled to resort to the use of infinite series. It will be advantageous to transform the differential equations to cylindrical coordinates by the substitutions z = arcosv, y = arsmv, z aq, (4) A &! 10 a 2 *Moulton's Introduction to Celestial Mechanics, p. 113. ORBITS ABOUT AN oHI.VIK >1'IIKH< HI). I'fll Vl'l.T llir-r -lll>-tittllion> << |ll:il ioli- i 1 . heroine (a) r'-r(v')'- =- ^^- r> ~ 4r0, there is but one positive value for p, which is c 2 If c 2 ?*Q, none of these curves cross the axis (f> = w/2. But if c 2 = 0, we have the circle p= 2/c 2 inside of which the motion is real when c 2 is negative. For values of //^O, but sufficiently small, we can put r=(p+p) cosy, q=(p+p)sin ei ies in a and n' of the form " p'(T)=0 = , (6) ff (7r ) =0 = ^(6,6+6, e'+^a'+^aHAMM- ' ' ' ), t = r , a'+f, M'+f, a'+c, aM*+c, M 4 + (19) On substituting (19) in (18a), we obtain a series of the form (a) = J 1 aM l +rf ! a 1 +rf jM 4 +rf 4 aV J +rf J a 4 + (20) If in this equation we make the substitution we obtain which can be solved uniquely for y as a power series in p. This solution substituted in (206) gives a as a power series in n*. This value of a sub- stituted in (19) gives t as a power sei ie- in p. We thus have a solution where P, and P, are power series in ft 1 . Newton's parallelogram shows that equation (20a) has two additional solutions, but as they are imaginary w we shall not develop them. 106 PERIODIC ORBITS. 65. Existence of Orbits in a Meridian Plane. If in equations (6) we put the area constant c equal to zero, the motion of the particle is in a meridian plane; that is, the plane has ceased to revolve, and the orbit in this plane is the true orbit. After changing to polar coordinates by the substitution r = p cos, the differential equations are - + | cos2 A Hil I AN M|t|. Ml. -l'IH.K(ill). 1(1? 66. Construction of Periodic Solutions in the Equatorial Plane. \\ e consider first orbits in the r<|ii;ttunal plane. We take the differential (-(illa- tions (8), and by means of the transformations there given we proceed at once to the integration of equation (10). It was shown in equation (13) that 6 can be expanded uniquely as a power M-ries in c in such a manner that the solution for p a.s a power series in < shall he periodic with the period 'Iv. Since the series is j)eriodic with the same period for all values of ( Miflicieiitly small, it follows that the coefficient of each power of e is it -elf periodic. Since the solution exists and is unique, it must be possible to determine the 6 uniquely by the condition that the solution shall be periodic. In the existence proof it was shown that 6 vanishes with e. Therefore p and 6 have the form P = P +P 1 e+p,e 5 +p,e'+ , -*, e+6 1 e 1 -|- J e > + (23) The p are to be determined by the integration of equation (10) and by the initial vah, P(0)=-l, -0. (24) The 5, are to be determined in such a manner that the p, shall be periodic. Upon substituting (23) in (10) and equating the coefficients, we find (6) (c) (25) These equations can be integrated in succession. The solution of (a) which satisfies the initial conditions is p =-cosT. 2(1. Since the initial conditions are independent of c, every p, except p must vanish at T = 0. On substituting (26) in (256) and integrating, we have p l = $ 1 (l-cosT) + [3-30y-^+6$M l + -][||co8T-|cos2T]- (27) The constants of integration in equation (27) have been determined so as to satisfy the initial conditions, but the constant 5, is, as yet, undetermined. 108 PERIODIC ORBITS. On substituting the values of p and p^ in (25 c), we find -)] COST 2 -)]cos2T + [3-40 2 M 2 -]cos3T. In order that the solution of this equation shall be periodic the coefficient of cos T must be zero. This is the condition that determines S l , and con- sequently (28) which agrees with (13) of the existence proof. With this value of d l equa- tion (28) becomes f^r+P 2 = [S 2 +30 2 M 2 -]-H-90 2 M 2 ]cos2T+[3-40 2 y . .]cos3T. The solution of this equation which satisfies the initial conditions is -f0 2 M 2 + -JcosT + |0 2 M 2 + -]cos3T. The constant 5 2 is as yet entirely arbitrary. It is determined by the periodicity condition on p 3 in the same manner that 6, was determined by the periodicity condition on p 2 . Without giving the details of the computation, its value is found to be 5 2 =-60 2 M 2 + This method of integration can be carried as far as is desired. In order to show this, let us suppose that p , . . . , p n _, have been computed and that all the constants are known except 5 n _! . From (25 d) we have ]2 PoPn _ 1 +/n(po, , P.-,), (30) where / (p , . . . , p B _ 2 ) is a polynomial in the p s and contains only known terms. It is easy to see that p n -\ depends upon 5 n _ t in the following way, p n _i = 5 n _i(l cosl)+ known terms. Equation (30) may therefore be written 2 // . . .]._, COST + [3-30 2 M 2 + ] cos2T+ known terms. ORBITS AHot'T AN OBLA I K >1 > HKIUD. 109 In order that the solution of this equation shall he period ic the coefficient of CO>T mu-t he /ero. This condition determines $_,. The e(|uation can then he integrated, and the constant> of integration will he determined by the conditions that, at T =0, dp, P* = -r=. =0. Kveryth'mn is then determined with the exception of 6,, and we have p.= (1 cosT)5,+ kimwn terms. On substituting the values of 5, and 5, in the solution as far as it has been computed, we find p = COST, COST [f -i*lM*+ -JcosT + [30V- -]cos2T From these expressions the series for r becomes (a) r=l-ecosT + j[f + {0y+(0t-4^) M 4 + -JcosT 4 ]cos2T]e' (31) ]cos3T }(?+ On suhstituting this value of r in the equation (86), transforming to the independent variable T, and integrating, we find (6) t,-r = -JsinTJ 110 PERIODIC ORBITS. Equations (31a) and (31&) are the periodic solutions sought. If we return to the symbols denned in the original differential equations (1) by means of equations (5), with the additional notation n VT-0?M 2 -302M 4 ' ' = ", we have the following expressions for the polar coordinates : (32) v v = (f (33) Equations (32) and (33) contain four arbitrary constants,* a, e, v a , and . Since the differential equations of motion in the equatorial plane are of the fourth order, these series, within the realm of their convergence, represent the general solution. The expression for the radius vector, R, is always periodic with the period 2-ir/v. At the expiration of this period v has increased by the quantity in excess of 2?r; that is, the line of apsides has rotated forward by this amount. If 6 is commensurable with unity the orbit is eventually closed geometrically. If 6 = I/J , where / and J are relatively prime integers, then v = 2(I + J)ir at t = 2Jir/v, and the particle is at its initial position with its initial components of velocity. The particle has completed I+J revolutions, and the line of apsides has completed / revolutions. The mean sidereal period is (35) Equation (34) for the rotation of the line of apsides has an interesting application in the case of Jupiter's fifth satellite. On the hypothesis that Jupiter is a homogeneous spheroid whose equatorial diameter is 90,190 miles and whose polar diameter is 84,570 miles, that the mean distance of the satellite is 112,500 miles, that the eccentricity of its orbit is .006, and that *The constant a is also contained implicitly in v through the constant n, and t can obviously be replaced by (l t,) since I does not occur explicitly in the differential equations (1). ultHllS AUtUT \\ OUI.AIK M'HKKOID. Ill its sidereal period is ll k O m 22!?, equation :\\ gives for the rotation of the lino of apside- 1 1 liT per year. Thr values derived from observations arc somewhat discordant, hut are in the neighborhood of 883 per year. If we still keep the hypothesis that Jupiter is homogeneous in density and of the same oblateness as before, we areconipelled to suppose that the value adopted for its polar radius was about 9,000 miles too great. In reality Jupiter must be much more dense at the center than at its surface, and therefore it is not necessary to suppose so large a reduction in making an allowance for its atmosphere. 67. Construction of Periodic Solutions for Orbits Inclined to the Equatorial Plane. By means of the area integral the problem has been reduced to the three equations (6), the first two of which are () >*=r (6) g*=- After the solution of these equations has been obtained, the third coordinate is found from the equation (c) "' = F'' \Ve have already proved, equations (14) to (20), the existence of periodic solutions of these equations of the following type: r = I+PJ M*+p 4 M 4 + , (36) q =g, M+gM r with the initial conditions r'(0)=g(0)=0, the constant /S being arbitrary. We can therefore integrate the equations so as to satisfy these initial conditions, and determine the c, in such a manner as to render the solution periodic. On substituting (36) in (6) there results <7i a, 4-6 p, a? (37) (38) Equation (37) contains only the even powers of n, and (38) only the odd powers. For the integration we have: f ri+stfgjM'+IX+g.-Sftg,- f 112 PERIODIC ORBITS. Coefficient of p. The coefficient of fj, is defined by q'i+qi = 0, and the solution of this equation satisfying the initial conditions is ^ = /3sinr. (39) Coefficient of p. The coefficient of //, from (37), is denned by The solution of this equation which satisfies the assigned initial conditions is The constant c 2 is determined by the periodicity condition on q 3 , where it is found that it must have the value c 2 = 20, /3 2 ; and a 2 , which is deter- mined by the periodicity condition on p 4 , is found to be zero. If we anticipate these determinations, we have Coefficient of p. 3 - The coefficient of (j?, from (38), is defined by &+? 3 = ?.(3p 2 +ftf-30 2 ) = (30 2 + 3c 2 -60 2 )sinr + Jo 2/ 3sin2r. In order that the solution shall be periodic it is necessary that the coefficient of sinr be zero. Therefore c 2 =26~ l p' 2 . On substituting this value and integrating, we find & = 0, sin r--~-a 2 /3 sin2r. From the initial conditions we must have q' 3 (Q) =0, and therefore But it will be shown in the next step that a 2 = 0, and consequently that ft = 0. (41) Coefficient of ^. It follows from (37) that the coefficient of /x 4 is defined by . (42) ORHITS \H<>lT AN OBLA1K H'HKKOlD. 113 Before expanding the right member of this equation we will examine the coefficient of cosr, which we know must he zero from the periodicity condition. It is noticed in the first place that terms in COST can arise only through terms involving p. and */, , and .secondly that all such terms carry a, as a factor. No other arbitrary enters the coefficient ; therefore we must take o, = 0. It can be shown by induction that the arbitrary con- stant a,(the coefficient of COST), which arises in the integration of p, , is determined 1>\ the periodicity condition on p {Jrt , and further that its value is zero. The proof is omitted for the sake of brevity. Upon substituting the value a, =0 in p, and q t and expanding the right member of (42), we find Since the constants of integration must both be zero, the solution is 1 1" we anticipate the value of c 4 which is found below, we have Coefficient of //. We find from (38) that the coefficient of /** is defined by From the periodicity condition we have ( )n integrating and imposing the initial conditions, we find at this step (43) This is sufficient to make evident the general character of the series. The r-equation contains only even multiples of T and the ^-equation con- tains only odd multiples. The r-equation contains only even powers of n and of T, while the (/-equation is odd in both these respects. The series are therefore triply even and odd. 114 PERIODIC ORBITS. On collecting the various coefficients, we have the following series: (a) r- (b) ~ " ' " ' ' (44) 64 r (d) c 2 =l + [20 2 -/3 2 ]M 2 +[-40 4 .-^0*0 2 ] / u 4 + In this solution the constants* a, /3, v a , and T Q are arbitrary. As is shown by equation (44c) the nodes regress, the measure of regression being The generating orbit of these solutions is a circle in the equatorial plane. A circle having any assigned inclination might have been used, e. g., r = Vl s 2 sin 2 T, g = ssinr, w = tan~ 1 [Vl s 2 tanr], (45) where s is the sine of the inclination. The solution thus obtained would have been identical with (44) . If we should expand (45) as power series in s and put s= /3/x, we should find that the terms thus obtained are identical with the terms independent of 6\ in the solution which has been worked out. It might therefore be of assistance in the physical interpretation of the constants to put /3/* = s in the series (44). 68. Construction of Periodic Solutions in a Meridian Plane. When the constant c is zero the motion is in a meridian plane. The equations of motion (21) are (46) We have proved in 65 the existence of periodic solutions of these equations as power series in tf, which, for M 2 = 0, reduce to the circle p = l, v = r. Let us therefore put *The constant a is contained implicitly through r and 9?; see equations (4). i "KBITS ABOUT AN OBLATK SI'HK1HD. 115 Upon substituting tlu-x- e\i>n->iuns in (40), expanding, and collecting the eoeilieients (if the vari">u> powers of p, we find -2 sin2r + sin4r) tf p, (47) The initial e<>ii(0) =0. On proceeding to the integration, we have: Coefficients of /i 1 . The coefficients of M' are defined by (a) p* 3p,-2v>j = - ^t (48) (b) On integrating (6) once, we have (c) ^=-2p,+^ cos 27- If we substitute this value of 3 M] and [^W+r&M] are obtained by differentiating ar and aq with respect to a. Thus two of the solutions of the fundamental set can be found without integration. We will consider first the two solutions in which the X, are not zero. Let us substitute in (51) the expressions ORBITS ABOUT AN OBLATE SPHEROID. 119 After dividing out the exponential, there remains (a) * , f>2 1 V I i~* * I / I r 1 * I 1 I t 1 . 1 . . r I 1 ft: (o) where a,= 3/3 sin T, 6 4 = a sum of cosines of even multiples of T. With respect to equations (52), it is known that

cos2T -\fiyT sin2T, ^a 0) -|-a 0> cos2T- ^ C sin2T. (56) 120 PERIODIC ORBITS. Coefficients of tf. The coefficients of // are defined by *+ft = -2tX 2 -ai'/'i , if'* +& = -2i\t' -b^ -a 1

COST. In order to satisfy the periodicity conditions we must have 2 I (59) 40i a< 0> - 2iX 2 a 2 0) = 0, +2t'X 2 7 2 0) = 0. The last two of these equations are satisfied by taking On solving the first two, we find Equations (59) can also be satisfied by X 2 = a <> = af = y f = 0, T[ O) arbitrary. This leads to the development of the solution in which the characteristic exponent is zero, which will be discussed, beginning with equations (80), by using the integral relations. It was known at the outset that the two values of X are equal but of opposite sign. We will choose the one with the positive sign. The solution for the negative X can be derived from it. The condition ai 0> ia 2 0) = still leaves us with an arbitrary constant. Since the equations are linear, this constant will enter the solution linearly, and may therefore be taken equal to unity, inasmuch as the solution after development is multiplied by an arbitrary constant. We will take then a[ 0> = l, which therefore makes o. 2 0) = i. Consequently i sinr, ^c = 0- (60) On integrating equations (58) with these values of ai 0> and a 2 0> , we get snr + -3/3 2 t sin3r, (61) ORBITS ABOUT AN OBLATE SPHEROID. 121 Coefficients of p*. The terms of the third decree in /* are defined by +10y? in 2r-f fty? cos2r, (62) From the periodicity conditions we must have The last two equations can be satisfied only if y" } = y ( t " = 0. The first two can be satisfied only if X, = (aj" - 1 oj") = 0. The condition a'," - 1 a<" = again leaves us an arbitrary constant; it gives us V>I = C(COST i sinr), but this is the same as multiplied by c/x. That is, the solution is repeating itself one degree higher in n, and this, of course, should be expected since the equations are linear, and a multiplied by any power of n must satisfy them. We may then choose the arbitrary multiplier equal to zero, which is the same as choosing a *, "' = " = = a = 0. On integrating (62) with these values, we find (63) It can be shown by induction at this point that

+ af) + 16 i ^ = 0. On solving these equations, we find In the last equation we can choose af* = 5/4 /3 2 and e4 2) = 0. This choice will make the coefficient of sinr in

= cos T i sin r, (65) The coefficient, a 4 , of COST in

j+. depends upon constants as yet undetermined, it is found from (-([nations (52) tliat f |8 I )+ f / (66) where A^, and B, +t are the known terms in the coefficients of COST and sinr respectively. From the periodicity condition we must have COST i =0. (67) The solution of those equations is . (68) As has already been pointed out, we can choose a"' = 0, and we have then In order to show that X H2 and a{ n are real, it will be sufficient to show that A i+t is real and that B J+1 is a pure imaginary. This is readily proved by induction, for, up to j = 4 inclusive, we have t+i t+i /-i /-i ipj= S W,COSICT+I S n^sinKT, tj = i 2 /COSKT+ 2 ^sinicr, K-l *-l *-0 -0 \\liere m, n, , /,, and g t are all real. From the form of the differential equations it follows at once that the same forms hold for j = 5, then 6, and so on. That is, A }+t is real while 5 /+ , is purely imaginary. Furthermore, it is to be noticed that A t+t and B l+t do not contain any terms in /3 independent of 6\ , and consequently the 0J , which appears in the denominator of a\'\ will divide out. This is proved as follows: If 0J be put equal to zero in the differential equations, then equations (52) become the equations of variation of a circular orbit in the ordinary two-body problem, the plane of the circle being inclined to the plane of reference by an angle whose sine is Pn = s. Equations (6) can then be written where the constant c 1 is given the form (1 -s*)(l e 1 ). For these equations we have the solution (1 - e 5 ) Vl-jPain'tf-'ft) (l-eOain(J-O) ,-, v - - where 124 PERIODIC ORBITS. If now we form the equations of variation by varying r, q, and e, that is by putting where r c , q , and e are the values in (71), we find dR n ,dR fr ,dR _dQ n ,dQ ,dQ p ~-'--fr p +^q- ff --~te ' -aF p+ aT "^ Three solutions of equations (72) are given by (1) (2) (3) 3r dr dr dq - ' If e 9^ these three solutions are distinct. The case in which we are interested is that for which e g = 0, but then these three solutions are not distinct, for the first two coincide, as is readily seen by putting e = in (71). Since the equations are linear, it follows that - f - - ' C ' - - drj' ' i dn dr' ' t dSl dr> is also a solution; but, since it vanishes with e , it carries e as a factor which we can divide out and absorb in the arbitrary constant c 4 . For e = this solution does not now vanish, and it is moreover distinct from the first solution. Thus we have three distinct solutions even when e = 0, but since dQ/de =0 and dR/de carries e as a factor, equations (72) pass over to the equations of variation of a circle when e = 0. For these equations we have therefore three solutions which are periodic with the period 2 TT. The fourth solution is not periodic for it involves a term of the form T times a periodic function. Let us return now to the solution which we have developed, (65), and consider only the terms which belong to the two-body problem, viz., the terms which are independent of B\ . This solution can be separated into two solutions, one of which is real, the other purely imaginary. The real solution is the third one of (73), and the purely imaginary solution is the second. Since both of these solutions are certainly periodic with the period 2ir, it follows that no terms in /3 alone can occur in the A }+t and B j+z , (67), because the presence of such terms would give rise to non-periodic terms in the two-body problem. Hence A J+Z and B J+2 carry 0* as a factor which can be divided out of equation (69). Furthermore, X J+2 , equation (68), carries 6\ as a factor, and therefore X vanishes with the oblateness of the spheroid. where ORBITS ABOUT AN OBLA 1 K Sl'll KU< HI). 125 The solution which we have obtained may be written (76) By putting e' Xr = cosXr+i sinXr, the solution takes the form , 1 We have thus one solution of the differential equations. A second solution can be derived from it by merely changing the sign of i, or By adding and subtracting these two solutions, we have finally A and B being arbitrary constants. As above developed, there is a certain arbitrariness in these solutions, owing to the manner in which the constants of integration were determined. They may be reduced to a normal form by multiplying each solution by the proper series in ^ with constant coefficients. By this process we can make p (1) (0)+P (2> (0) = 1, a ( "(0) - 4 and ^^ are periodic functions of r with the period 2w. As in the previous solutions, this can be normalized so that at r = The functions

uppo-e merely the conditions that the coefficients are periodic with the period 27r. Additional facts with regard to the solutions can lieestab- li.vhed when additional fact.- are specified with regard to the coefficients of the differential equations. The equations of variation, (51), may be written where the notation with respect to the 6's has the following significance: F.verv subscripts denote functions even in T, and odd subscripts denote functions odd in T; one dash indicates that only odd multiples of T are involved, and two dashes indicate that only even multiples of T are involved. The solution of equations (82) and (83) may be characterized in the same manner, and are then (85) where the notation is the same as for the 0's with the exception that in the first two solutions every integral multiple of T is increased by XT, e. g., cos (3+A)T. On these terms the dashes refer only to the integral part. Suppose now we have the non-homogeneous differential equations (86) where g(r) and/(r) are periodic with the period 2r. Since the character- istic exponents are 0, 0, == V^l\, by 31, the general solution has the form (87) 128 PERIODIC ORBITS. where the (p t ) and (o-,) are the complementary functions, and the , and rj t are the particular integrals of which the u t are the periodic parts, and where a and b are constants which depend upon the differential equations. Let us suppose that g(r) is an even function of T and that/(r) is an odd function of T, and let us seek the character of the solutions which satisfy the initial conditions On changing r into T in equations (86) , we get (88) From equations (86) and (88) we obtain, by eliminating g (T) and/(r), the differential equations j-[Pl(T) Pi( T)]= [P 2 (T)+P 2 ( T)], (89) These equations are the same as the original homogeneous set (84) . Hence their general solutions have the same form, viz., r) = A T, (T) + B ^ (r) + C T t (T) +>[; (r (90) Upon putting r = 0, we find from the first and the fourth of these equations that ORIUI- \UniT AN OBLATK s|'MKI HI). 129 Hither A = D = 0. or the determinant i^(0)J^(0)-~t(0)(0)Q. But it IB readily verified that this determinant is not zero. Therefore A=*D = Q. By virtue of the hypothec- made on the initial values, it follows from the second and third equations that and hence B = C = 0; consequently Pi(r)-p,(-r)=0, P,(r)+p,(-r)=0, Since tliese equations are identities in r, we have the following theorem : Theorem I. If g(r) is an even function of T and f(r) is an odd function of T, and if p,(0) = 3 (T)+a> 3 (T+7r) =0, a> 2 (r)-co 2 (r+7r) = 0, co 4 (r)+w 4 (r+7r) =0. Therefore c^ (r) and o> 2 (T) contain only even multiples of T, while w 3 (T) and w 4 (r) contain only odd multiples of T, and by carrying this result into equa- tion (87), we have -|T 2 a 3 +r a 4 ], % = c (97) These results may be expressed in Theorem II. If g(r) contains only even multiples of T andf(r} contains only odd multiples of T, then & and 2 contain only even multiples of T, and iji and ij 2 contain only odd multiples of T. If in addition to the above hypotheses we suppose that g(r) is an even function of r and/(r) is an odd function of T, then t and r? 2 are even functions and & and ^ are odd functions; therefore 6 = 0. But if Q(T) is an odd function and/(r) is an even function of T, then & and ij 2 are odd functions and s and ^ are even functions, and in this case a = 0. ciKHII- \H(H 1 AN OBLATK SI'HKKOID. 131 In the sunn- manner we can prove Theorem I II. // g(r) contains only odd multiples of r and f(r) contains unly even multiplf* f T. tli>n , and , contain only odd multiples of T, and t?, and j, contain only < re// multiples of T. Furthermore , , , , 17, , and ij t are periodic with th i>< riod 2w. If g(r) is of the form m > cosO'X)r and also if f(r) has the form -\n(j\)T, then, since i-v/^X arc th<> characteristic exponents of the homogeneous equations, the form of the solution is, by 30 and 31, ('.IS | luit, since g(r) is an even function and/(r) is an odd function, , and ij,arc t-vcn functions, and , and r;, are odd functions. Therefore all the coefficients in (98) which have the superfix (2) are zero. But if g(r) were an odd function of T and/(r) an even function, then all the coefficients in (98) which h:i\c the superfix (1) would be zero. Therefore Theorem IV. // g(r) has the form 2m y cosO'X)T, and if /(T) has the form 2n ; *in(.;'X)r, where V IX are the characteristic exponents of the homogeneous equation, then the particular solution has the form ij, = 2p, sin (cX) (00) From similar reasoning we have Theorem V. // g(r) has the form 2w,st'nO'X)r and if /(T) has the form 2n ; cos (j\)r, where V^T X are the characteristic exponents of the homogeneous equations, then the particular solution has the form It is understood that a, 6, c, d, and j arc integers in Theorems IV and V. 132 PERIODIC ORBITS. 72. Integration of the Differential Equations. It will be convenient hereafter to use as notation for the fundamental set of solutions, (82) and (83), of the equations of variation P = Aa 2 (r)+ J Ba 1 (r)+Ca 3 (r)+Z)[a 4 (r)+Ta 3 (T)], 1 where the a and y-i unctions are characterized as follows : UJ(T) involves only terms of the form cos [(2n+l)x] r, Tl (r) " " " " " sin[2n X]r, ttl (r) " " " " " sin[(27i+l)X]r, 7 2 (r) " " " " " cos[2n X] T , O,(T) " " " " " sin[2n +0] r, 74 W " " cos[(2n+l)+0]r, a 4 (r) " " " " " cos[2n +0]r, 7 3 (r) " " " " " sin[(2n+l)+OJT. It will be also convenient to write the differential equations (50) for p and a in the form where all the 0's are periodic with the period 2?r; 2 and 4 contain only cosines of even multiples of r; and 3 contains only sines of odd multiples of r. On the right side of the first equation the coefficients of terms carrying odd powers of a contain only sines of odd multiples of T, while all the other coefficients contain only cosines of even multiples of r. In the second equation odd powers of u have coefficients involving only cosines of even multiples of T, while all other coefficients contain only sines of odd multiples of T. The initial conditions are p(0) = a, p'(0)=0, cr(0)=0, Pioo = 0> 0"ioo = *ioo = From these conditions we find that The solution of these conditional equations is (105) A A = Hence the solution of equations (103) takes the form Coefficients of 5. The terms of the first degree in 6 must satisfy Pb,.+iA.+.o = 0, ffi u +0 t ff m +6 t p m = 0. (107) These equations are the same as (103), and from the initial conditions we must have, at r = 0, The solutions of equations (107) are therefore r) + ra 1 (r)], r) +T y<(r)}, where A m 4(0) A2. ^010= - ^010 134 PERIODIC ORBITS. Coefficients of e. The differential equations for these terms are Pan + 2 Pooi +#3 ff m = 0 cr ooi + ^4 ^ooi+^s Pooi = 0- The right member, 6 m , is a periodic function of r with the period 2ir. Furthermore, it involves only cosines of even multiples of T. Consequently, by Theorem II of 71, the solution has the form r), r),| where a is a constant depending on d m ; a 5 (T) is a cosine series involving only even multiples of r; and 7 5 (r) involves only sines of odd multiples of T. From the initial conditions it follows that p m , +04P.+0,P> =>, (H3) where the right members have the following expressions: By the initial conditions p^, a m , and their first derivatives vanish at r = 0. Since the equations (113) are linear, the solutions have the form = A O,(T) +B o,(r) + CO,(T) -f-Z> [o 4 (r) +TO,(T)] (114) 136 PERIODIC ORBITS. Upon imposing the initial conditions, we find . _ 4 - 4 3 .,, , 4 A ~ ^100 I A ~ 100 I A " -^ 100 -^ 100 , ^(0)a 4 (0)-fr(0)[74(0)+7i(0)] A 001 Ml > L 100 , I A L. -. , , A -"-100 ) where A-a t (0)[7 4 (0)+7l(0)I-7i<0)(0). On substituting these values in (141), we have for the solutions P^A^x^+A^A^x^+A^x^r), } o- = ^lS y.(r}+A^A?Mr}+A y,(r), J where Tl (r) and similar expressions for X 2 ,y 2 ,x 3 , and i/ 3 , the values of which we shall find we do not need. The properties of ^ and y^ are known with the exception of (p, and \f/i, which we will now investigate. The functions ^ and ^ are those portions of the solution of the differential equations which depend upon the coefficients of A^- These coefficients are homogeneous of the second degree in a 2 (r) and T^T). In R m and S 2W the expressions 6 200 , no , and 6^ contain only cosines of even multiples of r; &M, 6 m , and ~o~m contain only sines of odd multiples 00 of T; a 2 (r) has the form a 2 = 2 a n cos[(2n+l)X] T; y^r) has the form n=0 00 7i = 2 b n sin [2wX] T. Consequently, so far as the coefficient of ^[o 2 is n=0 concerned, R 200 and S 20B have the form R m = S a cos2wr+ 2 a cos [2n2X] T, n = n = >S 200 = 2 61" sin(2w+l)r+ 2 6 sin [(2n+l)2X] T. n=0 n=0 ORBITS ABOUT AN OBLATE Sl'HKHtUK 137 By :*<) terms involving multiple* of XT give rise only to periodic term- in the solution. By 31 those part* of /,'.. and >',,., which are independent of X give rise to irnn- in the solution which have the form P = Pi( r )+r'\T*i(T), ,( T ) and p,(r) are periodic with the period 2*. Consequently the function- J-,(T) and J/,(T) have the form /' r)+ Cl ra,(r), V.M =P,(T)+c,ry 4 (r), (116) where I\ (V) and P,(T) are periodic with the period 2*7r. of aS. The differential equations for these terms are ~5U (H7) where 7?] and S IIO is obtained from /? 110 by replacing 0,, by (? (J . The functions /?,, and S m differ from ^ and S,M only in the con- stants A tlk . The initial conditions impose the same conditional equations. Consequently the solutions differ only in the constants A ttt , so that we can express them at once without computation in the form ^ x l (T\ 1 j/ 1 (r),J where the x t (r) and ?/,(T) are the same functions of T as in (115). Coefficients of S 1 . By symmetry with the coefficient of a 1 , it is seen that A. = ^C' *, W +A A% x, (r) (119) <*<*> = A y l (T) +A A y, (r) + A y> (T). Coefficients of . Since the coefficients of the first powers of a and 6 are homogeneous of the first degree in the A's, the coefficients of a*, 06, and 6 1 are homogeneous of the second degree in the A's. The coefficients of the first power of are not homogeneous in the A's; hence the coefficients of the second power are not homogeneous. But if the functions a, and 7, were zero the coefficient of the first power of would be homogeneous, and therefore the second also. By symmetry, therefore, we can at once write down the terms involving the A 's to the second degree. To these must be added terms in the first degree in the A's, and one term independent of the A's. 138 PERIODIC ORBITS. The differential equations for these terms are // __ f- Po02 I ^2P()02 ~T~ "3*^002 = -*^002 > where Rm = 02ooPooi + flnoPooi^ooi + 0o20oo 2 i ~a I ~a 2 > = S m , (120) The terms involved in R m are shown in the following table, where the coefficients of the constants given in the first line are the products of the functions in their respective columns and the functions of the same line in the last column. Thus, one of the coefficients of A is a 2 yj 6 m , and this coefficient comes from the expansion of p^ , (T0,+o 4 ) a, S 101 For the S m it is necessary in the above table only to change the 6 i}t into B t]t in the last column. The solutions of equations (120) can be expressed in the form (r (121) where x^ x t , x 3 ,y l} y 2 , and ?/ 3 are the same functions as in (115). The coefficients of A^ in the differential equations (120) are homo- geneous of the first degree in a^ and 7j , every term of which involves the first multiple of XT. Hence the solutions for these terms, by Theorem IV, 71, involve non-periodic terms, and we can write ) +C 2 Ta 1 (r)+c 3 TO,(T), (122) where P 3 and P 4 are periodic with the period 2 KIT. It is seen from the table that x^(r) and y 6 (r) do not involve the X. They have, therefore, the form x 6 (T)=P 6 (T)+c 4 Ta 3 (T), y^T^PM+CiTy^T). (123) It will be verified at the bottom of page 141 that we do not need to know the character of x 6 (r) and (iKWTS ABOUT AN OBLATE Sl'HKKOID. 139 Cocffici< N/.S i if at. Those tcnn- >:itisfy tlic ri(!in of term ,|(1) ,.(1) 100^001 .(i) j< , "IOO ^001 T " lfM\ * IHt 1 *W WII 11 n A (2> 11 u ioo -'MOO w > u oio -^oic *i w noi -^ooi u > J, _ A (2) ,. h A <2> ,, }, A (2) "100 ""-^MOO V i "010 " A (2) -I- A (2> A (i n ^r 4-9 A -' 1 oio~-' :l ioo -^oioJ *t ' A m ^ _oj( /Id) ^"j.fJt 1 ) / d( 2 )_L/l(2) ,4 (DlTT in /l (2) /(2) 17 "no ^^100 -^010 i/iT^L^ioo -^OIOT-AIOO -^oioJ 4/2 i ^-^loo -^010 i/3 > _ A (1)2 r~ i j (i) x (2) ~ I A (2)2 ~r W o2o~-' l oio * / i~-^ 1 oio -^oio ''si ^010 -^s ? h -A (1)2 i7-4- A (1> xl (2) i74- A C2)2 77 "020 -^010 /l 1^-^010^010 i/2 I -^010 i/3 ) 9 A m A m r~-l-r/l (1) /1 (2) 4-4 (2) /l (I) lr"4-9 4 (2) A (2) r"-l- /4 (1) "loi^^-^-ioo-^ooi^i i L-^ 1 loo-^ooi i -^loo-^ooiJ-*-! i ^-^loo-^ooi^a i -^IOO _o/i(i) /(i)7 i r /id) j<2> id (2) /id) 177-4-9 /4 (2) xl <2> i74- /4 U) TT-)-/!' 2 ' ?7 ~^O /1 OOlJi/2 I ^^lOO-^OOl i/3 ^^lOO^T^-^ 1 100i/5 ,, _o /jo) / 4' 1 )_i_r / 4( /i<2) _i_ / 4(2) xdMio /i(2) 4(2) _i_ j "Oil ^-^OlO-^OOl^l I L-^OlO-^MlT-^olO-^OOlJ^ I "-' 1 010-' 1 001' t '3 I -' 1 __ _ 010-' 1 001' t '3 I -' 1 010' t '4T^-' 1 010- l '5 ! _ A (D2 ~_l ,4 (1) A (2) ~ I /( (2)2 7T_I_ yj ( ~ I A (2) "002 -"^ooi *i t **ew -^ooi *a i -"ooi *> "001*4 <***w 0(X)2 = -^001 2/1 I -^001 -^001 2/2~)~-4o01 2/3~t~-4o01?/4~f~-4oOl where U = KTT~ , V = Kirj t , Xt = ^ , yi = y { atT = K7T. (129) UT ' ar OKBIIS AUDI T AN OHLATK H'HKKOII). 141 Let us solve the first equation of 1 127) for as a power series in a and 6. \\ V obtain e = e 10 a+ Bl + w a t +,,a+< - * l + ... (130) where the coefficients t,, have the values _ a,o oJo,- a 100 a,, ai>i (131) f -- "10 > *> ' ": lo ,a M , a,,, <, Solving the second equation of (127) for e in terms of a and 6, we obtain = iia+^6+^a 1 +e^a5+^a,+ , (132) where the e^ have the same expressions in the 6 (>t as the / have in the a, Jt . Upon subtracting (130) from (132), we have = S- 1 o]a+[^- eoi ]5 + S;-e i( ,]a ! -r-S;- 1I la6 + [ < --e ( ,]5 1 +- (133) We must examine the coefficients of this series. The first two are QOIO Oaio ^ i. = jm Oooi Oooi ^001 QIOO _ OIOQ " IPO V -A IPO P. Oooi Oooi -"001 W /looi P Both of the linear terms therefore vanish. The computation of the second degree terms is somewhat more compli- cated. It will simplify matters somewhat if we observe that the e^ are the same expressions in v and j/j as the , are in u and x< . It will therefore be sufficient to compute one and derive the other from it. On substituting in the expression for e^ in (131) the values of the a ljt from (128), we get ,1 -.. r A wt l "3^1 ' " |- **ooi L A a) * ?? -4- A ' tt)t ' 100 -100 - 100 '^001 - <*> ?4 Xl -U ~ r A o) A < T ^100 -^00 4 _ 4 -^ 4- 1 r (1)1 J () tt)t a>* ?* _i. A tt>1 ^i\ u -' M loo uJ On forming the sum of these three expressions, there results (135) (fil 100 100001 u the coefficients of o^/u, x,/u, and xju being identically zero. 142 PERIODIC ORBITS. On changing the ^ into ^ and u into v, we get &T . Hence 1C 2 " |r/|CU AW _ AW J">1 fS_al 20 C 20 4 <2>3 1 ^100 ^OOl ^MOO^OOl ^001 L <* " ~ |_ ,1( 1 ^6 _ |/6~| 1 LU v \ \ But f- 1 - ^1 and f- 6 - 1 vanish since LM yJ LM yJ as is readily seen from (116) and (123). On referring to (122), it is also seen that |^ 4 fel does not vanish, but is equal to c,|- -^1 Hence LU VJ LU dr VJrmxr [ "1 _ , _ _ - 20 20 J \A m A (2) A 100 L-rt- 100 ^001 -^100 - ~ ln7 H7~\ J Xj _ #4 lu~v\ (136) Without repeating the details of the computation, we find similarly r/i a) 4 <2) 4 (2) 4 (1 Mr^ ,, L-^ L 010- a 001~' ft 010-^ 1 OOlJ *4_i/ Lw e t-l , _ _ 608 e j ( 4 a 1 ^010 **B "KM *981 1 a 5 (0)-aa 4 (0) a ^S J (2) J (2) -Aooi -"-001 Oe(O) ] = ^-a+..., = 0-a+ , (139) (140) ORBITS ABOUT AN OBLATE SPHEROID. 143 where a, and y, are the quantities defined in (111), and -yi(0) is the value of dy t /dr for r = 0. It was shown in (112) thata,(0) is distinct from zero, and in (79) that o,(0) i< equal to unity. Thus one solution for t begins with the first power of a, while the other certainly does not begin before the second, but in both solutions 5 begins with the first power of a. 74. Construction of the Solutions with the Period 2 KIT. We have proved the existence of series for p, a, and t proceeding in powers of the initial value of p, which we will now denote by e*. The series for p and a are periodic in r with the period 2KW, and since this condition holds for all values of e sufficiently small, each coefficient separately is periodic. The series for p is even in T, and the series for a is odd in T. These series have the form p = p,e+p,e 1 +p,e'+ , (141) i i _ i e = We shall substitute these series in equations (102) and integrate the coeffi- cients of the powers of e in order, and determine the constants in such a way that p and a shall be periodic and shall satisfy the initial conditions p(0)=e, Mini I \N OBL.VIK Sl'MKUiUD. 145 We have &]><> OL* CLt O It follows from (147) that R. contains only cosines of even multiples of T, and = 0. Anticipating this step, we have (151) so that Pt contains only cosines of even multiples of T, and l I \\ OBLATK SI'IIKK. UD. 147 Let us suppose we h:il started with a definite value of 0, for example , which gives a definite lienerating orbit with a definite initial distance r . Let us seek now the general ing orbit for which the initial distance is r +e. It . is sufficiently small we can evidently give an increment to , which will increase r by the amount c. We have '=/(&), r.+e-/(A+). Expanding the right member of the second equation in powers of e, we have df e " which gives, by inversion, a series of the form Then, by substituting /3 = /3 +c l e+c t e*+ in the generating orbit and arranging the solutions as a power series in e, we obtain the orbit in which the initial distance is r +e. As these are the same conditions that were imposed when we sought new orbits through the equations of variation, it was to have been expected that one of the class of generating orbits would satisfy them. SECOND SOLUTION. We return now to equation (146), and continue with the second solu- tion, in which , = and A (I) = l/o,. From (82) it is seen that a, = a,(0) = l, and therefore A m = 1. Hence in the second solution . (155) On using these values of A (l> and , , J?, and S t of (146) become 0,7, All of the terms in these expressions except e,^, are of the second degree in o, and 7, . Therefore they involve only terms carrying 2 XT and terms independent of XT, and m is independent of XT. In the solutions the terms depending upon 2 XT are periodic. As for the terms independent of X, R t contains only cosines of even multiples of r, and S t contains only sines of odd multiples of r. These terms give rise to non-periodic terms in the solution, which has the form where ^>,(X, T) and ^,(X, T) are the periodic terms involving X; TJ, and f, are the periodic terms with the period 2ir; a, is the constant belonging to the non-periodic part; and the coefficients of t, are the solutions depending 148 PERIODIC ORBITS. on the coefficient of e 2 in the differential equations. In order that this solution shall be periodic, it is necessary that n<2> _ r,. U Q 2 fl 2 j which reduces p 2 and 9f ^ W S If ^ 19 S It* (4) 2 - f - Moulton'a Introduction to Ctlettial Medutniet, p. 185. fSee Introduction to CeUttial Mechanict, Art. 121, and especially Charlier'i Die Mtchanik de* HimmeU, TO). II, pp. 102-111, for a detailed diacuaaion. 154 PERIODIC ORBITS. Suppose the coordinates of (a), (6), or (c) are =$ > *7 = 0, T=0, the value of depending upon which point is in question. It will not be necessary to distinguish among them except in numerical computation. Now give the infinitesimal body a small displacement from one of these points, and a small velocity with respect to the finite masses such that h z', (5) y, r=+ z, - dt " dt dt~ dt dt ~ ~ dt The differential equations (1) are transformed by these relations into __, p( , , 2 , dt ~dx'~ l( ' y ' - +2 - - v'P (x' y' 2 z' 2 ) df ' dt ~dy'~ y * ( >y ' )j where P l , P 2 , and P 3 are power series in x', y'\ and z' 2 . 77. Regions of Convergence of the Series P l , P 2 , P 3 . It follows from the form of U in equations (6) that P lt P t , and P 3 converge for the common region of convergence of the expansions of \/r l and l/r 2 . We are consider- ing only real values of x', y', and z', and consequently the conditions for the convergence of the expansions of l/r t and l/r 2 as power series in x', y', and z' are respectively _ 2x' x . * 2 The surfaces which bound the regions of convergence of the expansions are obtained by replacing these inequalities by equalities. For the convergence of the expansion of \/r l , the equations of the bounding surfaces are 0, i 0. j The first is the equation of the point occupied by the finite body 1 ju. The second is the equation of a sphere whose center is at 1 M and whose radius is V2(| +M) 2 - The convergence of the expansion of l/r : holds for the space between the point and this sphere. O8CILL.MIM. > \ I 1 II 1 I IS FIRST METHOD. 155 The o(|iiations of corresponding surfaces for the expansion of 1/r, arc 0, 1 0. These are respectively tin- equations of the point occupied )>\ the mass H and of a sphere whose center is at n and whose radius is V2( 1 +/*)' The convergence of the expansion of 1/r, holds for all points within this sphere except the center. The distances from I-M and n to the point (a) are respectively &+M and 1 +/*. The radii of the spheres which have been defined in (8) and (9) are \/2 times these distances. Since \/2 ( 9 +n)-l> \/2 (& I+M), the sphere around 1 M as a center is entirely outside of the one around M as a center. ( 'onseqiiently, the series P, , P, , and P, converge in the case of the t must".. i -niation to the point (a) for all points within the sphere whose center is at n and whose radius is \/2 ( O +M)> except the point M itself. The distances from 1 -n and n to the point (6) are +M and %/({ !+/*)*. The radii of the two spheres are \/2~ times these distances, and hence they both include the point (b) in their interiors. In this case the two spheres intersect unless n is small, when one will be entirely within the other. The distances from 1 -^ and n to the point (c) are , M and 1 & p. Since -v/2 (1 - - /*) - 1 > \/2 ( - - M), the sphere around n as a center includes in its interior the one around 1 -/* as a center. The latter includes (c) in its interior, and everywhere within it, except at 1 /*, the series P, , P,, and P, converge. 78. Introduction of the Parameters e and 6 . Let us now make the transformations x' = xt', y' = yt', z' = zt' (VO), *-/ e =(l + 5)r, (10) where ' and 5 are constant, but at present undetermined, parameters. Then equations (6) become 1, (11) where A'. , Y, , Z. are homogeneous functions of x, y, and z of degree n. These differential equations are valid for all values of x, y, z, and ' satis- fying the conditions for convergence which have been developed. 156 PERIODIC ORBITS. We shall now generalize the parameter e (see 13) by replacing it everywhere by e, where e may have the value zero or any value in its neighborhood. When e ^ e', the differential equations belong to a purely mathematical problem; but when e = e' they belong to the physical problem. Since the value of e' has not been specified except that it is distinct from zero, the generalization may appear trivial, but the same method can be used where the parameter corresponding to d does not have this arbitrary character, and where the device is of the highest importance. We have therefore to consider the differential equations - 2(1 + 6) = (12) 79. Jacobi's Integral. Equations (1) admit the integral where C is the constant of integration. This integral was first given by Jacobi in Comptes Rendus de I' Academic des Sciences de Paris, vol. Ill, p. 59. For equations (12) there is the corresponding integral where U is now a power series in e. 80. The Symmetry Theorem. Let us consider the solution of equations (12) and suppose that at r = we have dy _ dz _ dx _ n :ClJ dr~ dr~~ ~ dr~ that is, that the infinitesimal body crosses the .r-axis perpendicularly at T = 0. The solution will have the form Now transform equations (12) by the substitution x=+x', y=-y', z=-z', T=-T', dx _ _ dx' dy = i_dy^ dz _ . dz' dr ~ ~ dr' ' dr = h dr' ' dr r dr' ' OSCILLATING SATELLITES FIRST METHOD. 157 The equations in the new variables ;m- precisely the same as in the old; consequently, if the values of the dependent variables at T' = are -< -* -< "'-*'- 5? the solution is 57=/;, g-/;(o, 37 =/;(''). Now it follows from the relations between the two sets of variables that /,(T) = +/,(T') = +/,( - T), /; and therefore that D is positive. From (15) we have (21) At the points (a), (b), and (c) we have _ __ dx dr, dx dr t dx *See also Introduction to Celestial Mechanics, and Charlier's Mechanik des Himmels, vol. II, pp. 117-137. fFirst proved for all it by H. C. Plummer, Monthly Notices of Royal Astronomical Society, vol. LXII (1901). OSCILLATING SATELLITES FIRST METHOD. 159 It is found from the definition of U in (1) that For the point (a) the relation 1>H \\ccn r| 0) and r' is rj m = 1+rJ", and therefore dr _dr_ dx dx Hence, equations (22) and (23) give for this point m 11 r< 1 . -^J= -M[T. -^ Therefore, since the first two factors are positive while the third is negative, (21) becomes, for the equilibrium point (a), l-^r)<0. (24) Similarly, since only the second factor in the expression for 1 A is negative, for the point (6) we find < 25 ) For the point (c) we have the corresponding equations (26) Then 1^1 is negative because the third factor alone is negative. Since 1 A is negative in every case for O^M^O.5, it follows that two of the roots of (19) are real and equal numerically but opposite in sign, and that the other two are conjugate pure imaginaries. Let the real roots be p and the imaginary and q satis- fying these inequalities. .1 lies between +00 and 8/5. On taking the lower sign in ('.\\ ). we must have 0

1. Consequently, there arc infinitely many values of n between and 0.5, such that a and VA are commensurable. \Yhen the commensurability relation is satisfied we have the periodic solution z = c,cosvTr+c 1 8inv'jT. We can choose t, so that c, = 0, and let a represent twice the real part of A', and A',, and b twice the imaginary part of A\ and K t . Then this solution becomes y = nas\n at +n&cosar, z= r- sinVAr. (32) The period of this solution is P = 2irp/ff = 2irq/VA. In this period x and y make p complete oscillations, and z makes q complete oscillations. These will be called Orbits of Class C. 84. Normal Form for the Differential Equations. We are about to prove that when e^O the initial values of x, y, z, and their derivatives can be so determined, depending on , that periodic solutions having the periods of (29) and (30) exist for all values of e sufficiently small, and reduce to these solutions for <=0. In this discussion it is convenient to have the differential equations in a normal form, and it is necessary to compute the first terms of the solutions as power series in 5, , and the increments to the initial values of the dependent variables. The linear terms of (12) are found by (14) to be 162 PERIODIC ORBITS. The general solution of the first two of these equations is 2/ = nv /-l [K, Therefore we see that the transformation X = (33) changes (12) into ,N, - ( 2(m(r-np) h +n(l + 8)[]e ( w 4 - - h (34) where Ft - + i-** + JL , n - 1 ~ M -l- JL , -O + r 0, c+j, d, e) =0. (41) This equation is satisfied at r = 2ir/VA by v t = = f'= 0, and we find from the explicit form of F in (13) that for these values =: -i A -l]+tQ 3 , -v= 2L _ = 0, [e 0= It follows that (41) can be solved for f' as a power series in a,, 7, 5, e, !>!,..., t> 4 , f, which vanishes with i> 4 = f = 0. That is, if u 1} . . . , w 4 , and z retake their initial values at r = 2ir/VA, z' also retakes its initial value. Hence we can suppress the last equation of (40) and consider the solution of the first five equations. It follows from (38) that the explicit forms of the first terms of (40) are (42) where the Q 4 are power series in the cif, j, 8, and e. The coefficients of a 3 and a 4 are always distinct from zero, and the parts of the coefficients of a t and a 2 which are independent of 5 vanish only if o/VZ is an integer. We shall suppose at present that this ratio is not an integer, and that it is incommensurable. Part of the discussion becomes quite different when it is commensurable, and this case will be taken up when we discuss, in 96, the question of the existence of orbits of Class C. It follows from (34) that since [ ] and { \ involve terms in z 2 alone, and since z a does not vanish identically for a t = y = 5 = 0, the Q t carry terms in e alone. The determinant of the linear terms in a n . . . , a 4 of the first four equations of (42) is the product of the coefficients of 04 , . . . , a 4 , and is distinct from zero. Therefore these' equations can be solved for a : , . . . , a 4 in the form a, = 6^(7,8,6), (43) where the Rt are power series in 7, 5, and e. When these results are substituted in the last equation of (42), we have = e P(y, d, e). (44) OSCILLATING SATKI.I.I I l-:s MUST METHOD. 165 The solution of this equation gives us the periodic orbits in question. We have the two arbitrary parameters 7 and 6, and we shall show first that an not give 6 an arbitrary value and solve the equation for 7 as a power series in f , vanishing with . Suppose that <5 is neither /ero nor an integer. Then equation (44) is not satisfied by 7 = 4 = 0, and the solution can not be made. Now suppose 5 is zero or an integer. Then the left member of (44) vanishes, and the equation is divisible by t. It is, in fact, divisible by *. It is seen from (34) that x and y do not enter in the last equation except in terms involving as a factor, and since the a, defined in (43) enter only through x and y, the part of (P coining from the first four equations is divi.-ible by e*. It is seen also that the part of the right member of the last equation of (34) which is independent of x and y is multiplied by *. Therefore the right member of (44) is divisible by e 2 . We shall now prove that after e* has been divided out there is left a term which is independent of 7 and , and which is distinct from zero. Terms in i- in the right member of (44) are introduced both through the a, defined in (43), and directly in the integration of the last equation of (34). The terms obtained in the former way involve B as a factor and depend upon a and p; the terms entering in the latter way carry C as a factor. Hence, if the coefficient of ' is to be identically zero, the parts involving B and C as factors separately must be zero. We shall verify that the part involving C as a factor is distinct from zero. The coefficient of e 1 in (44), so far as it is independent of B and 7, is denned by the equation (46) The solution of this equation satisfying the conditions 2, = 0, z,' = 0, at r = 0, is 3 (46) Consequently the last equation of (42) becomes - - (47) Hence, after division by e 1 , there is a term independent of both 7 and , and the solution for 7 as a power series in , vanishing with , does not exist. That is, periodic solutions of the type in question do not exist. 166 PERIODIC ORBITS. Now let us give 7 an arbitrary value and attempt to solve equations (42) for d! , . . . , a 4 , and 5 as power series in e, vanishing with e. Since c is arbitrary, we may put 7 equal to zero without loss of generality. Or, more conveniently for the construction of the periodic solution, we may give 7 such a value that z' = c/VZ for T = 0, whatever e may be. But for simplicity in writing we shall suppose that 7 is included in c. The first four equations can be solved for a, , . . . , a 4 in terms of 5 and e, and the results substituted in the last. The result differs from that above only in the terms multiplied by e , and, as before, we find that the lowest term in e alone has e 2 as a factor. There is a linear term in d alone whose coefficient is 2-irc/VA. Therefore, after a, , . . . , a 4 have been eliminated by means of the first four equations, the last equation can be solved for 5 as power series in e, the term of lowest degree being e 2 . When this result is substituted in the solutions of the first four equations, we have o t , . . . , a 4 expressed as power series in e, vanishing with e. That is, when 7 = the solutions of (42) have the form 5 = 2 p(e), a f = ep,(e) (t = l, . . . , 4), (48) where p and the p f are power series in e. When these results are substituted in (39), we have z = z +e 2 g(e; T), u i = tq i (r,T) ({ = !,..., 4), (49) where q and the q t are power series in e and are periodic in T with the period 2-ir/VA. These series converge for kl sufficiently small. The circle of convergence is determined by the singularities which are present in the differential equations (34), which are introduced in forming (39), and which are introduced in the solution of (42). Since the right members of (49) converge and are periodic for all | e \ sufficiently small, the coefficient of each power of e separately is periodic. 86. Some Properties of Solutions of Class A. It will now be shown that the orbits under consideration are re-entrant after one revolution, and that they cross the x-axis perpendicularly. Let us find the orbits whose periods are 2,v^^/^/ r A, v being an integer. We form equations analogous to (42) simply by replacing 2 IT by 2i>ir. The determinant of the linear terms in e^ , . . . , a 4 is distinct from zero unless vv/\fA is an integer. We exclude this case here and treat it when we con- sider orbits of Class C. Therefore the first four equations can be solved for a L , . . . , a 4 as power series in 7, d, and e. On substituting the results in the last equation, we find, as before, that the solution can not be made for 7, taking 5 arbitrary, but that the solution for 6 as a power series in e is unique. That is, the solution for a t , . . . , a 4 , and 5 as power series in e, vanishing with e, is unique. Hence for a given value of e there is a single orbit of Class A having the period 2i>ir/VA. We have shown also OSCILLATING SATELLITES FIKST METHOD. 167 that for a given value of t there is one orbit of Class A having the period 2T/\~A. Since an orbit of period 2w/VA has also the period 2.vr/\fA t the latin- are included in the former. It follows, therefore, from the unique- ness of both orbits for a given , and from the fact that, for = 0, they re-enter after the period 2T/V/71", that all orbits of this class re-enter after a single revolution. Let us now suppose that, at T = 0, we have dx/dr = j/ = z = 0; that is, that the orbit crosses the x-axis perpendicularly at r = 0. It follows from equation (33) that i l (0)- 1 (0) = > tt,(0)-t 4 (0) = 0; (50) whence fl l = l > 1 = fl 4 The e< [nations corresponding to (39) will now be power series in o, , a, , y, 6, and t. \Ye may suppose -y = on the start, for orbits that may be found in this way will be included in those found from more general initial condi- tions, under which it was always permissible to put y equal to zero. The orbits obtained with these initial conditions will be symmetrical with respect to the x-axis. Therefore necessary and sufficient conditions for periodicity are that the infinitesimal body shall cross the z-axis perpendicularly at the half period. These conditions are that at r = T/ VA *-,-<-o. It follows from (33) that these conditions imply that, at T = T/VJ, M, w, = 0, , M 4 = 0, 2 = 0. (51) These conditions give us, in place of (42), the equations (52) where Q( , Q' 3 , and Q' & are power series in a, , a, , 8, and . It is easy to see that these equations are solvable uniquely for a, , a, , and d as power series in , vanishing with t. Therefore, for a given value of there is one, and but one, of these symmetrical periodic orbits of this class. Since for a given value of e there is but one periodic orbit for unrestricted initial con- ditions, it follows that all orbits of Class A cross the x-axis perpendicularly at every half period. " = a,e - 0= sin2T(l +6) + 168 PERIODIC ORBITS. 87. Direct Construction of the Solutions for Class A. In the practical construction of the solutions for the orbits of Class A it is most convenient to use equations (12). The explicit values of the right members are (53) the A, B, and C being constants which are defined in (15) and (35). The x, y, z, and 5 can be expanded uniquely as series of the form X 1 =(l+2A)x, X 2 =[ Y 2 =3Bxy, Z 2 =3Bxz, z= -sinVJ r + V^W, = .', (54) where the Z<(T), y ( (r), and z t (r) are each periodic with the period On substituting these expressions in (12) and making use of (53), we obtain a series of sets of equations for the determination of the x t , y t , z t , 6, . The coefficients of e in (54) must satisfy the equations 2^ -(1 + 2.1)*,= \B [-2xl+y 2 +z 2 ], (55) But x = y = 0, z c = c/VT sin y/~Ar. Therefore the solution of (55) which satisfies the conditions that x 1 , ?/, , and z 1 shall be periodic with the period 2ir/Vl, and that z = 0, z' = c, at r = 0, whence z l (0) = z[ (0) = 0, is (56) 9 /T z > = Since in all cases A>1, the coefficients are always finite. The coefficients of e 2 in (54) must satisfy the differential equations (57) Upon substituting the values of z and x l , the third equation becomes , . o /J 3 2 = S1 OSCILLATING SATELLITES FIRST METHOD. 169 The solution of this equation will not be periodic imli-< we impose the con- dition that the coefficient of sinx.tr shall vanish. This condition is satisfied l>v <-(}, hut this leads to the trivial solution z=t/=z = 0. If we reject this solution, we may use the condition for the determination of 5,. After i( has been satisfied, the periodic solution of (57), having the period 2ir/v'7f and fulfilling the conditions that Zj = z' t =0 at r = 0, IS r==n -27 '(1-34 + 144*) c* 9Cc| "*- ' (fig) In this manner the construction of the periodic solution can be continued as far as may be desired. We shall prove this statement by induction, and at the same time we shall derive certain general properties of the solution which are sati.-fied by the terms already computed. suppose x t , . . . , _! ; i/ , . . . , y n -\ ', z , . . . , 2,_i ; o, , . . . , S m -\ have been determined and that they have the following properties: 1. The x tj and y tj are identically zero, j an integer. 2. The z, ;+l are identically zero, j an integer. 3. The function x, /+1 is a sum of cosines of even multiples of \f~Ar, and the highest multiple is 2j+2. 4. The function y tj+l is a sum of sines of even multiples of V~AT, and the highest multiple is 2j+2. 5. The function 2,, is a sum of sines of odd multiples of vTr, and the highest multiple is 2j + l. 6. The $,,+, are zero. It will now be shown that these properties hold for x.,y u ,z u , and 5. . The terms z. , y, , z* , and 5, satisfy the differential equations (59) - 2A z, .+/?.(* y,, z t , *,), where P. , Q, and /? are polynomials in x, , . . . , &, (j = l, . . . , n-1). It is seen from (12) that P. and Q involve y, and x', only in the products y't&n-j and x',t n -,. If n is even, these terms are zero by properties 1 and 6, for then either j must be even or nj must be odd. But if n is odd, they are in general not zero. 170 PERIODIC ORBITS. We shall now prove that P n =0 if n is even. The general term of P n is rr _ r "i . . -^ . 7 X X' . /" X'" . sir. XJ2 /n\ ln ~ y d " di " ' where Xj , . . . , X K , . . . , \" ; MI , , M?" ; Pi , P 2 , ft , and g 2 are all integers. Since the 5, enter only through (1 + 5) 2 , the exponents q l and q 2 satisfy the relation 2. (61) The exponents and subscripts of (60) satisfy the following relations : (a) /*!+' +AV is an even integer because the right member of the first equation of (12) is a function of y z . (b) /*i'+ ' ' ' +MK" is an even integer for a similar reason. (c) \! , . . . , XK , X( , . . . , XK- are odd integers by property 1. (d) Xf , . . . | XK are even integers by property 2. (e) p l and p 2 are even integers by property 6. (f) MA+ + M-c because the term of degree MI+ +M+M[+ in x, ?/, and z has e as a factor to a degree one less than this sum, the terms x^J, . . . introduce e to the degree MiX 1; . . . , and the sum of the exponents of e must equal n. There are two sub-cases, according as MJ+ +M is an even integer or an odd integer. When ^4- +MK is an even integer, the following statements are true: (a) There is an even number of odd /^ , . . . , V, K . (/3) MJ X t + +MK \c is an even integer by (c) and (a). (T) Mi^i+ ' ' ' +/vX' is an even integer by (a) and (c). (5) juiX'+ +A&XK" is an even integer by (d). () Pi tfi+Pa & is an even integer by (e). It follows from the assumption that MI+ +M K is even, and from (a), (b), (a) , . . . , (e) that the left member of (f ) is odd. Therefore in this case T n is identically zero if n is even, and in general is not identically zero if n is odd. Suppose now that ju t + +M* is an odd integer. Then (a') There is an odd number of odd MI , , M* (/3') MI ^1+ ' ' ' +M* \ is an odd integer by (c) and (a'). The properties (y f ), (d'), and (') are the same as (7), (d), and (e) respectively. Therefore the left member of (f) is again odd, and hence every T n is identi- cally zero if n is even, and in general not identically zero if n is odd. It follows that P n is identically zero if n is even, and in general is not zero if n is odd. The treatment of the general term of Q n can be made in a similar way. The only differences are that in (a) and (7) the sums are individually odd instead of even. But since (f ) involves their sum, the result is that Q n is identically zero if n is even, and in general is not zero if n is odd. -i-ILLATINC SATKI.I.IIK- HUM MKTIIOD. 171 The nen.-ral term in lt n has the form of i()i. where tin- subscripts and (x]M)iiciits satisfy the relations: (a*) // + ' ' ' + /v is a even integer because the right member of the third equation of li' is a function of y*. (b*) n* + +* is an odd integer because the riht member of this equation involves only odd powers of z. (c*), (d*), (e*), and (P) are the same as (c), (d), (e), and (f) respectively. Suppose /i,+ +A* is an even integer. Then (a*), (0*), (/), (*") and (e*) are the same as (a), (/3), (7), (6), and () respectively. Therefore in t his case t he left member of (P) is an even integer. . It is shown similarly that the same result is true when /!,+ +/* is an odd integer. Therefore, # IN identically zero if n /s odd, and in general is not identically zero if n is even. The discussion now naturally divides into two cases, viz., where n is even, and where n is odd. We shall treat them separately. Case I. We shall prove that if n is even, R n is a sum of sines of odd multiples of VAT, the highest multiple being n+1. Consider the general term (60). The x^ are all cosines of even multiples of VAT; therefore the product x" 1 x"' is a sum of cosines of even multiples. Because \ x of property 3 and the properties of the products of cosines of multiples of an argument, it follows that the highest multiple which occurs is .. (62) Similarly, from properties 4 and (a*) , it follows that y"^. y^f is a sum of cosines of even multiples of VAT, the highest multiple being M i(x;+i)+ +/V(X:<+I)=M;X;+ + M ;X+M;+ +/*. (63) From properties 5 and (b*), it follows that/ 1 , " is a sum of sines *? " of odd multiples of VAT, the highest multiple being '-. (64) On taking the product of these three sets of terms, we find that R n is a sum of sines of odd multiples of VAT, the highest multiple being By (P), which is of the same form as (f), we have For those terms in which 9, = 9, = 0, we have, as the largest value of N, tf-n+1. (66) 172 PERIODIC ORBITS. Hence, for n even, equations (59) become (67) +C +1 sin(2j+l)VAT+ where the +, are known constants which depend upon the coefficients of the terms with lower subscripts. The solution of these equations which satisfies the periodicity conditions and z n = z' a = Q at r = 0, is (7 1 these denominators can not vanish for any integral j. It is obvious that in practice it is not necessary to refer to the differ- ential equations at each step. The most convenient method to follow is to substitute as many terms of (54) in the right members of (12) as will be required in carrying the computation to the desired order in e, and to arrange the results as power series in of the form and similar series for the other equations. From the P, Q n , and /? the A ( *\ B, and O can be computed sequentially with respect to n without explicit reference to the left members of the differential equations. The coefficients of the solutions are given by (68) and (70). The whole process is unique and can be continued as far as may be desired. 88. Additional Properties of Orbits of Class A. It will be observed that, so far as the computations have been carried, x,, y,, and z, carry c' +l as a factor and that 8, carries c' as a factor. We shall prove that this is a general property. Suppose it is true for j = 0, . . . , n l, and consider the question for.; = /i. The terms of order n are defined by equations (59). In P. there are terms y',8 m .j- It follows from the assumed properties of the y, and S, that this term carries c*+' as a factor. Similarly the x', $_, occurring in Q, carry c^ 1 " 1 as a factor. Now consider the general term (60). It follows from the assumed properties of x,, y,, z,, and S, that this term carries c as a factor to the power It follows from (f) that N = n+l, and therefore this property is general. 174 PERIODIC ORBITS. In order to obtain the coordinates in the physical problem, we must replace e by e and multiply x, y, and z by e' [equations (10)] . Then e' and c occur in every term in x', y', and z' to the same degree, and are equivalent to a single parameter. That is, without loss of generality we may put c equal to unity, and the value of t will determine the dimensions of the orbit. Or, if we put e' equal to unity, c will determine the dimensions of the orbit. It follows from these results that the coordinates and 8 are expansible as power series in c, and the solutions could have been derived in this way without the introduction of e and e', but the discussion would have been less simple. The explicit expressions for the periodic solution, so far as they have been worked out in (29), (56), and (58), are* sinVJr]e' _ n 9 r 3 2 (1-3A + 14A 2 ) -I , 2 , ~ 16 A 2 L(1+2A)(1-7A + 18A 2 ) " (71) Since x' and y' are sums of cosines and sines respectively of even multiples of VAr, and since z' is a sum of sines of odd multiples of VZr, it follows that x' and y' are periodic with half the period of z'. Since the relations X'(r) = x'(-r), y'(r) = -y'(-r), z'(r] = -z'(-r) are satisfied, the orbits are symmetrical with respect to the z-axis, as was shown in the existence proof. Let T equal half the period. Then x'(r) = o/(T+r), y'(r} = 2/'(T + r), z'(r} = -z'(T + r). Therefore the orbits are symmetrical with respect to the x'y'-pl&ne. Simi- larly, since z'(r)=z'(T-r), y'(r)=-y'(T-T), z'(r] =z'(T-r), the orbits are symmetrical with respect to the x'z'-plane. It follows from the form of (71) that a change of the sign of e' is equivalent to changing T by a half period. The period of the solutions in r is 2w/\ r A, but it follows from the last equation of (10) that in the time t the period is p _ T 2a- fi 9 f 3 2 (1-3 A + 14 A 2 ) ~-| " V4l6~^ 1(1+2^X1-7 A + 18 A 2 ) *The x', y', and z' are the actual coordinates and not their derivatives. OSCILLATING SATKM.H KS KIHST \IKIIHU). 175 It is found from (71) that the equation of the projection of the orbit on the j-y-pkuie is. up to terms of the fourth l^rcr in ', z'4 T , _ 9BV' -> h 4A(l+2A)J "(1-7A + 18A')'' _ OD-'J This is the equation of an ellipse whose center is at x'= ^~\ , y' = 0, and whose semi-axes are 3fl(l+3A)e" :Uh n ,- A . 4 A (1-7 A + 18 A')' vCT(i-7A + 18A ! )' The equation of the projection of the orbits on the yV-plane is approximately -&Bz' A:' /7KN - (1-7A+17AV V 1 -?- This is tin- equation of a figure-of-eight curve with its center at the origin, touching the i/'-axis at no other point, and having two other intersections with the z'-axis. The equation of the projection of the orbit on the z'z'-plane is approximately -1 (1+3,4) The orbit is a parabola whose axis is the z'-axis, and whose vertex is at X '= _ 93(1 -A)e" >=f = Q (l+2Ani-7A + 18A>) Only that part of the parabola for which z'* 0. It is seen from (35) that B is positive for the point (a). Hence these orbits intersect the z'-axis between (a) and the finite mass . It follows from (35) and the second equation of (4) that B is negative for the point (6), at least if n is small. Hence these orbits intersect the z'-axis between the point (6) and the finite mass M- Similarly, those orbits near (c) intersect the z'-axis between (c) and the finite body 1 n. The vertices of these parabolas are the ends of the ellipses whose axes are given in (74). It is seen from the expressions for the coordinates of the centers of the ellipses and the ends of the parabolas, that the distance from the vertices of the parabolas to the centers of the ellipses is 44(1 -74 + ISA') ' It follows from the signs of B that the orbits in all cases open out away from the points of equilibrium near which they lie. 176 PERIODIC ORBITS. From the properties which have been derived it is possible to infer the geometric character of these orbits. In a general way they have the shape of the handles of ice-tongs, one of the two handles being situated on one side of the r;-plane, and the other symmetrically on the other side of this plane. The place of the hinge is where they cross the -axis. In the case of the points (a) and (6) they open toward the finite mass n, and in the case of (c), toward the finite mass 1 /j.. In Fig. 3 an orbit of each class is shown. No attempt has been made to represent them to scale for any particular case, but the figures show their general positions and the directions of motion in them. The curves are drawn on elliptic cylinders to make the figures as clear as possible. (C) (I) fj- FIG. 3. 89. Application of Jacobi's Integral to the Orbits of Class A. The original differential equations admit the integral (13), which holds for all orbits, and therefore in particular for the periodic orbits which we are dis- cussing. It has already been seen that it plays an important role in the proof of the existence of the periodic solutions when we start from general initial conditions, and we shall now show that it is almost equally important in the construction of these solutions. We are illustrating, in a particularly simple problem, a new and valuable use to which integrals may be put. The explicit form of the integral (13) is +J5(-2z 2 +3?/ 2 +32 2 )ze+ .1 = constant. (77) OSCILLATING SATELLITES FIRST METHOD. 177 If we substitute the solutions i49),or rather the equivalent series for .r. //, z, and their derivatives, and the >cric- for 5 in (77), and then arrange as a l>ower series in c. we have 'W dr/ f "+ = constant. +F n ( ,7s, Since this is an identity in e, each coefficient is a constant. We shall now work out the form of the general term of (78). It is seen from (77) that the function F, will contain terms of the types $!*=/, i'^N, '^-' and (60). dr dr dr dr dr dr It follows from the properties of x, y, and z that the terms of the first three types are zero unless n is even. Now consider the term of ty{>e (60). All the properties of its exponents and subscripts are the same as when it belonged to P n except that in the relation (f) the 1 in the left member must be replaced by 2. Hence we see that this term is also identically zero unless n is even. Since /' involves only even powers of the y, and z,, it is a sum of cosines of even multiples of V^Ar. It follows from the relations (a), . . . , (f) (the last one modified as indicated) that the highest multiple is n+2. Hence we may write + +D ) ,cos(n+ 2) vTr = constant. Since F n is identically a constant, we have D;- constant, D'= =D'\=0. (80) The quantities D' t {m} , . . . , D'$ t depend upon a<;>, . . . , a';'; 0<", . . . , ft"; 7; , . . . , 7". Suppose all the a,, 0,, 7, up to <-, ft-*, y-'> have been computed and are known to be accurate. Equations (80) can then be used in two ways, as we shall show. First, they test the accuracy of the computation of the a? 1 "", ff~ l \ y ( "~ l \ for these quantities must have such values that the equations shall be satisfied. And secondly, we can compute the a7~", P*~" from equations of the type of (69), and then find the 75 and 6. directly from (80) without referring to the differential equations of the type of (67). The first use is obvious and we need to consider further only the second. We are working under the hypothesis that n is even. Therefore the a*' and the 0*' are identically zero. It is seen from (77) that the only terms which can introduce the 7^' and 6. are (81) 178 PERIODIC ORBITS. From the form of z n given in (68), we have cos(n+2) VTr j, (82) Therefore equations (80) become D; W = 2 c vT 7< B) +c 2 8.+J5J = constant, D' = 4 c < n) = 0, (83) where D^J i ^"+2 are known constants depending upon a^", . . . , of~"; 0(1) oW-1). (0) (n-2) Pj t ; Pj > I) } ' ) ij The last n/2 equations, beginning with the last one, can be solved for 7^+! , . . . , 7g B) in order. Then the second equation gives d n uniquely. The results of these solutions are W = (n) - : D? All the constants are uniquely determined except 7i n) , which is defined by the condition that z' n shall be zero at r = 0. This condition gives (85) Thus we see that in orbits of Class A we can suppress the z-equation, if we wish, and compute the 7^ from the integral; or, we may use the integral, step by step, as a check on the computations. OSCILLATING SATELLITES FIHST METHOD. 179 90. Numerical Examples of Orbits of Class A. No periodic orbits of this class have -<> far been published. It is clear that it is practically impossible to discover them by numerical experiment. We shall suppose the ratio of the finite ma.--e- i- ten to one, or 1 A = 10/1 1, n = 1/11. Then, in the computation of the coefficients of the series for the .solutions in the vicinity of the points (a), (6), and (c), the following results are found: Coefficient Point (a). I'oint (6). Point (c). rj" (EquatioiiH (4)] + 0.347 + 0.282 + 1.947 r,'" (Equations (4)] + 1.347 + 0.718 +0.947 A [Equation (15)] + 2.548 + 6.510 + 1.082 a* [Equation (19)] + 2.811 + 6.820 + 1.144 P ' [Equation (19)] + 3.359 + 11.330 +0.226 n [Equation (28)] + 2.657 + 3.990 +2.014 HI [Equation (28)] - 0.747 - 0.397 -3.091 K (Equation (35)] + 6.548 -10.961 -1.136 C [Equation (35)] + 18.283 +55.740 +1.196 -'US t(lM\\ - 0.316 + 0.090 +0.249 4A(1+2A) ' +3BU+3A) + 0.151 - 0.036 -0.230 A A 1 1 7 A t 1 Q A S \ \*^*/ J * /i ^ i ^ / **t -|~ o ** / -3B - 0.112 + 0.018 +0.226 V/A(1 7A+18A*) r3B(l+3A) -I - 0.037 - 0.020 -0.002 64A'/'Ll-7A+18A _P -9 T 3B(1-3A + 14A) ^ + 0.184 + 0.467 +0.001 16.*lL(I"i"*) (1 t A ~\~\oA) _} ^-period 3.936(1+0.184'+--) 2.463(1+0.467 +) 6.041(1+0.001 "+) From these results we find that the solutions in the neighborhood of the three points of equilibrium are x' = [-0.316+0.151 cos 2v/Ar]"+ , (a) "+ 7/' = [-0.112sin2v/Jr]' 1 + 2' = [+0.626 shWArlt'+f- 0.037 (3sinvGfr-i x' = [+0.090- 0.036 cos2 VA r] "+ , ' = [+0.018 sin2v/AT] e"+ z' = [ +0.392 sin VAr] '+ [ - 0.020 (3 sin VAr - sin 3 vdr)] "+ x' = [+0.249- 0.230 cos2VJT]'*+ , (6) \y' = [+0.018 sin2v/ZT]e"+ , (86) (c) , ' = [+0.961sinVXr]'+[-0.002(3sinVAT-8in3>/XT)]"+ If we regard the motion of the finite bodies as direct, and consider the projections of the motion of the infinitesimal body upon the zV-plane, we find that in all cases the motion in orbits of Class A is retrograde. 180 PERIODIC ORBITS. 91. Construction of a Prescribed Orbit of Class A. Suppose the masses of the finite bodies are given. Then a periodic orbit of Class A for the infinitesimal body may be defined (1) by the place it crosses the a/-axis, (2) by the y' or z'-component of velocity with which it crosses, (3) by the greatest value of the y' or z'-coordinate, (4) by the constant of the Jacobian integral, and (5) by the period. It is understood, of course, that these various quantities are arbitrary only within such limits that the series for the coordinates converge. For ' = the Jacobian constant C and the period have definite values depending only upon /JL. The increments to these values and all the other defining quantities enumerated above can be developed as power series in t, vanishing with e'. These series are odd or even, depending upon which other quantity is taken as defining the orbit. If s represents any of these quantities, we can write s = S 1 e'+S 2 e' 2 +s 3 e' 3 + where the coefficients s l , s 2 , s 3 , . . . are constants which depend upon /z alone. If s is assigned numerically the inversion of this series gives e. This value of e' substituted in (71) gives the desired orbit. Thus, the methods which have been developed not only prove the existence of the periodic orbits and give convenient processes for constructing them and testing the accu- racy of the computations, but they furnish a ready means of finding any particular orbit that may be desired. 92. Existence of Orbits of Class B. For e = the coordinates in these orbits are given by (30). Therefore a t = a 2 7^ 0, a 3 = a 4 = c= in (38). Sufficient conditions that the solutions (39) shall be periodic with the period 2ir/ati>!inl. at r = 2*/ff, by r,= f = f'=0; and we find from the explicit form of /' and the traii>formation (33) that, for these values of the variables and , = a 1 = s = = o, Hut from (-(iimtions (19) and (28) we have S+1+2A 2 4 , f, f', and this solution vanishes for r, = v t = v t = f = f ' = 0. Hence, if we impose the condition that r, = r, = v t = f = f = at T = 2rA, the equation r, = at r = 2r/a will be satisfied. Therefore the first equation is redundant, and it will be suppressed. It will be shown that the orbits of Cla.vs B liejn the ary-plane. It follows from the form of the last equation of (34) and the initial values of z and z' that I\ and P t contain 7 as a factor. Therefore the last two equations of (88) contain 7 as a factor. The explicit form of the next to the last one is ' ~ P ' (0) = VA 8in When VA/tr is not an integer the only solution of this equation, vanishing with the parameters in terms of which the solution is made, is 7 = 0. The case where vCT/a is commensurable will be considered in connection with the orbits of Class C. Therefore z = 0, and the orbits are plane curves. Necessary and sufficient conditions for the existence of the ixriodic solutions of Class B reduce to (87), where t = 2, 3, 4. The explicit forms of these equations are (90) We have three ecjuations to satisfy and five arbitrary parameters, besides , at our disposal. The parameter o, enters only in the combination a,+a,, and since a, is as yet subject only to the condition that it shall not vanish, we may let it absorb o, . \Ve may determine / , which enters in the 182 PERIODIC ORBITS. definition of T, so that at r = we shall have x' = 0, a condition which is fulfilled in all closed orbits in which the coordinates have continuous derivatives. By (33) this condition becomes ffV 1 o,+p(a 3 <0=0, which we may regard as eliminating a, . We now consider the solution of (90) for a, , a 4 , and 6 as power series in e, vanishing with . The determinant of the linear terms in a a , a 4 , and 6 is (91) which is not zero. Therefore equations (90) have a unique solution for a 3 , a 4 , and 5 as power series in , vanishing with . When these results are substituted in the first four equations of (39), the latter become power series in which are periodic in r with the period 2ir/(r. It will now be shown that all orbits of this class are symmetrical with respect to the or-axis. We choose t a so that we have y = x' = at r = 0. Therefore it follows from equations (33) and from the initial values of the w, that otj = 0, a 3 = a 4 . Necessary and sufficient conditions that these S3'mmetrical solutions shall be periodic are It follows from (33) that these equations are equivalent to u l = u t , u t = u 4 at T = ir/ff. The explicit expressions for the latter become (92) / t \ , (a,, , ), -i , n/ , N J + Q 3 (, 3, e). The determinant of the coefficients of the terms which are linear in a 3 and 5 is , - r Z -2r ~\ 1 v^T [e e "J> which is not zero. Therefore equations (92) can be solved uniquely for a, and d as power series in t, vanishing with e. Since, for a given value of e, there is but one unrestricted orbit of this class, and since there is also one which is symmetrical with respect to the x-axis, it follows that all orbits of this class are symmetrical with respect to the z-axis. The orbits of this class all re-enter after one revolution, for if we impose the conditions that they re-enter after v revolutions, we find the solution is unique. Since it includes those which re-enter after one revolution, it follows that all orbits of this class re-enter after precisely one revolution. OSCILLATING SATELLITES FIRST METHOD. ISM 93. Direct Construction of the Solutions for Class B. It has been shown that in the periodic orliits of ( 'la>s H the coordinates are uniquely developable in M-rir- of the form w,= 2 uJV i-o l, -.. ,4), 5=2 ,', (93) where the u\" are periodic functions of r with the period 2/ = _, 3 nBa? [ (n 2 - 2) - (n 2 + 2) cos 2 ar\ __ 3n.Ba 2 sin2 = i 8(mr-np)(r ' a ,, 6 n, = _ , 2(mp+na)_ a> U 10 "20 ) The solution of the third equation of (99) has the form cos 2 ffr + &S' sin 2 or, OSCILLATING SATELLITES FIRST METHOD. 185 where, because of the periodicity condition. ri" = 0. \\ e find by substitution in the differential equations that = + _3nfio 1 (n t -2) a = 8(m 2(wur-np)V-l 2(mp+n (108) _. dr 2 (ma np) 2 (rap + no) 2 (ma np) 2 (rap + no) where the &, , . . . , d y ^ are included in [ ] and { } . OSCILLATING SATELLITES- FIRST METHOD. 187 It follows from properties 1, 2, 4, and 5 that [ ] is a sum of cosines of multiples of ar, and that : I is a sum of sines of multiples of or. The general term of [ ] is r ' = x x' <' !' I/?'*;- < 109 ) I C I .' The exponents and subscripts of thi- expression satisfy the conditions: (a) n[+ -f/v is an even integer because of 4. (b) q is or 1 , since & enters (34) linearly, (c) The product x% z is a sum of cosines of multiples of ar, by 1. There is an even number of odd /uj , AV, by (a). Those factors $$ for j which /ij are odd are sums of sines of multiples of or. The product of an even number of such factors is a sum of cosines of ar. It follows that T, is a sum of cosines of multiples of or. The highest multiple of or in T, is It follows from (c) that when = 0. (110) Therefore [ ] is a sum of cosines of multiples of or, the highest multiple being v+1. It can be shown in a similar way that |- is a sum of sines of multiples of or, the highest multiple being v+1. Equations (108) can be written, therefore, in the form G+i ~i f'+' - , J () I . - / , L .4, cosiffT I i 2 B. . J L<-i M f_ dr aoVl .-vr ^ c [ r + l S ^{ 1.0 ,.() r'^ 1 ~\ u _ p M w = + ;, | 2 ^ , {r) cos tar ar Li-o J r'^ 1 n * = - n | 2 .4 i"cos Vr L-o J T f r +' 1-J21? J 1 1.1 f+i 1 2 ^'"sintffT [ (HI) where the A and ^<" arc all known real constants. 188 PERIODIC ORBITS. In order that the solution of equations (111) shall be periodic, we must impose the conditions that the coefficient of c av ' Tr in the first equation, and of e~" v=riT in the second equation, shall be zero. It is easily seen that the two conditions are identical, and they uniquely determine 5,, by the equation - a (112) The periodic solutions of (111) are of the form = v+1 +l 2 a COS jar V 1 S cos jar -V-l 2 smjffT, r+l r+l + 2 v+l S (113) where cj" and c^' are arbitrary constants of integration. Upon substituting (113) in (111) and equating coefficients of corre- sponding functions of T, we find v 1 _mAT + j_B_ JV+pP Then equations (33) give (3 = 2, (3 = 1, j = l, . . . ,v+l). (114) v+l y v = nv/ -T [clVv^' -c^V^'] + 2 2 i [n 6J7 + m Cl " (115) OSCILLATING SATELLITES FIRST METHOD. 189 The arbitraries cj" and c arc determined l>y the conditions that x, = Q and ?/ r = Oat r = 0. Upon applying these conditions, the final results are -(_,.()__ y fn c i c z * l a i-o x, = 2[<+S?l-2 S l [<+<>] cosar+2 j -o ( ' >+ y, =2n 2 [<+ ^ = constant. Since this equation is an identity in e, each F, separately is a constant. It is seen that F, is a sum of cosines of multiples of (101) + 0.905 - 0.605 -0.595 mB(n'+2) + 0.540 - 0.259 -0.583 Q , \\\}\.) nB + 1.682 - 0.922 -1.562 2(mp+n<7)<7 a', (101) - 1.142 + 0.663 +0.979 by (101) - 0.239 + 0.057 +0.385 a (103) - 2.944 + 4.691 +0.872 -3nfi(n'+2)p + 1.212 - 1.772 -0.121 8 (ma np) (4 y, 5, ), 0= + (c+7)cos2*g(l+*)+Q,(a 1> . . . , a 4 , 7, , e). Since the last equation of (34) carries 2 as a factor, it follows that the last two equations of (122) are divisible by c+y. It will be assumed that this factor is distinct from zero and is divided out. We shall let the unde- termined constant c absorb the arbitrary y. Since we assume that c is distinct from zero, it follows from the integral (13) that the last equation of (122) is redundant and can be suppressed. There remain five equations whose solutions for ^ , . . . , a 4 , and 5 as power series in , vanishing with t, will now be considered. It will appear in the course of the work that we shall need all of the terms of the first degree in , and all of those of the second degree which arc not periodic. We integrate equations (34) as power series in t, introducing 5 in the combination (l + S)r, and o, , . . . , a 4 by means of the initial conditions (120). It will be found that only the first power of 5 is needed, and then only in terms independent of ; elsewhere it will be omitted. Likewise only the first powers of a, and a 4 will be needed, and therefore the higher powers will be omitted. Since a, and o, enter only in the com- binations a,+o! and a,+o,, we may omit them for brevity until the end, using simply a, and a, , and then restore them where they are needed. 192 PERIODIC ORBITS. The terms of degree zero in e satisfying the initial conditions (120) are w< 0) = a 3 e+ p(1+5)T , < = (a 2 + a 2 )e- = a 4 e- pa + 5)T , c (123) where i = V 1. The terms of the first degree in e are defined by the equations -F\ , --/<-ffcB[ ]+*{ f, i i , (124) where 4(mT - (m'+m) a 2 a 4 e <-"-'"- } . > 325) The solutions of (1 24) are the respective complementary functions, #{V lT , K?e- aiT , K?e pT , K^c~ pr , plus terms of the same character as their right members. In the solution of the first equation, the coefficients of these terms are respectively the coefficients of the right members written in the order given in (1 25), omitting the term cos2vTr, divided by ffi, -\-tri, 3cri, +p, p, 2^< II.I,.VMN<; SATKI.LITKS HUM Ml; I MOD. 193 III tin 1 solution of tin- third and fourth equation- it i- uiinece.-sary to compute the terms which carry a and a, as factors. Omitting these term-. the divisors for the third equation arc respectively - p, '2t e<|Uation of i 124) is I 2(p*+2tpVT) r i I 3BC04J (126) The constants of integration K[ l} . . . . , A'J" are determinod by the conditions that 7/J , .... "' shall vanish at T = 0, and the constants LI" and /,;" by the conditions 2,(0)=*;(0) = 0. It is necessary to compute all non-periodic terms of , , it, , and 2 which are of the second degree in t and which are independent of a, , . . . , o 4 , and 6. The right members of the differential equations involve - (127) The quantities j - , and T/, are defined by X, = Mj" + Mi" + Mi" + Mi", I/, = Hi( Mi" - Mi") + m(U? - Ui"). In order to get all the non-periodic parts of the solutions at this >tep, the terms of the differential equations which are non-periodic, that is, which carry p in the exponential, must be retained; in the first and second equations the terms in r" T and c~" T respectively must be retained; and in the 2-equation the terms in cos V^T and sin \TA T must l)e retained, for the.-e |>eriodic terms give rise to terms in the solution which are multiplied by T, and which therefore are not periodic. 194 PERIODIC ORBITS. The conditions for a periodic solution with the period -& = -v* = T are a VA = u t (T] - u t (0) = [<> (T) - u (0)] + [u?(T) - M< n> (0)] e + (i = l ----- 4), 0= z(T)- + [2 2 (7 1 )-z 2 (0)]e 2 + By means of the steps explained on pages 191 to 193, we find explicitly tC(0) = (a 2 +a 2 )[e- 2T ' )(1+5)< - 1], ^"'(T 1 ) -<(0) = a 4 [ z (T)- 2 (0) = (129) M < (j 1 ) - w a) (0) = I [ 2mEi(2 mm') - F( m' + w) ] a t a 3 [ 2mEi(2 "> (T\ ,,M(C\\ - if [ 2mEi('2 + mni) + F (ni m)]a 1 a 4 I P [ 2mEi( 2 mni) F(ni + m)] a 2 a 4 "I r -2^ 1 ~\ 2 ff i + p ~\\- e 2mEi( 2 mni) F(ni + m)] a^ . 2mEi( 2 + mni) F(ni m)] a^ 2(rt-p _, [ -|- 2mEi(2 mni ) F ( m + w& ) ] Q 2 a 4 1 f - 2jrp T _ i "I u m (T) -w a> (0) = ( [-rc-ff^+n'Q-nFtK + 2n^(2-n 2 )a 1 a, [n#(2 + n 2 ) - nFt] a 2 2n^c 2 \ r +2^ _ 1 -i 2 " 4 2 L 6 2 o-{ + p p [-nE( 2+n*)-nFi]c tnEc* \ r -*rf _ , i 2cri-p " (4Z+7H L 6 OSCILLATING SATELLITES FIRST METHOD. 195 . 2 If a. _g *)pJlp 2 )-nft]a; 2n^ If a. a, 2 fa, . 2)+nFi]al , h2 (131) When these expressions for a, and a 4 are substituted in the first, second, and fifth equations of (128), we obtain = (a 1 + a 1 )[e +w5 '-l]+(a 1 + a 1 )[(a 1 + a 1 )(a 2 + a 2 )L 1 + c 2 M 1 ] 2 + -mo- 4A-a 2 3(7 ] (132) OSCILLATING SATELLITES FIRST METHOD. 197 After removing the factor < from the hist equation, solving for 6, and substituting tin- result in the first two equations. \ve liave where j L> "/>' 3C(4-n') ' ' ' ~~ rmp+nff_ i j/ mo np \~ _ 9C ap M - ff_ i / L ap ' = (a, + a,M(n, + a, )(, + a t )L+c'J/] '+ , (134) Kquation- i:>3) can not be solved for o, and a, as power .-erics in t, vanishing with t. unless ,U","^+cMf]=0, a t [a,o,L+c 1 M] = 0. (135) One solution of these equations is aj = a, = 0, and with this determination of a and o,, equations (133) are uniquely solvable fora, and a, as power series in (. vanishing with t. In this case the generating solution reduces to the form of that of Class A. But the orbits of Class A heretofore treated were those for which V A and a are incommensurable. Thi- restriction was not necessary in order to prove that orbits exist which re-enter after one revo- lution, but it was not certain that there are not others re-entering only alter many revolutions. The uniqueness of the solution of (133), for a, = r/, = 0, proves that all of the orbits of ('/e under consideration In /> . 1 1 -enter after a single revolution. At the beginning of the present discussion the assumption was made that c is distinct from zero, and this permitted the suppression of the last equation of (122). If we had assumed that a, is distinct from zero, we could have suppressed the first equation of (122); and solving in a different order, we should finally have arrived at two equations corresponding to (,133) containing (c-r"y) asa factor. The equations would have been found solvable after imposing the condition c = 0, and we should have arrived at the eon elusion that all orbits of Class B re-enter after one revolution. Equations (135) also have the solution a,a,L+c'M = 0. (136) This equation defines < when a, and a, have been given arbitrary values. If the orbits are to be real, a, and a, must be conjugate complex quantities. Under these circumstances their product is positive, and L and M must be opposite iii sign in order that c shall be real. After the condition (136) has been applied, equations (133) become [a 1 , a. , t]+ , -[o 1 , a, , e]+ 198 PERIODIC ORBITS. Since the terms of these equations which are independent of e are identical, except for the non-vanishing factors e^-fa, and a-j+Oj , it follows that if one is solved for a x and the result substituted in the other, the latter becomes divisible by e. After dividing out e, there is a term independent of a., and e which must be equal to zero in order that the equation may be solved for Oj as a power series in e, vanishing with e. This term involves the coefficients of e 3 in the original solutions (122), since e 3 has been divided out. Likewise, terms enter from lower powers of e through the elimination of a 3 , a 4 , and 5. It is not possible to construct these terms without an unreasonable amount of work. But we see from the way in which they originate that they are homogeneous of the fourth degree in a : and 2 . Unless one or the other of these constants is absent, their ratio is determined by this constant term set equal to zero. If one is absent, the only solution is the other set equal to zero, which throws us back on Class A, which has been already completely treated. Suppose both constants are present and that their ratio is determined. Since they must be conjugate in order that the orbit may be real, the solu- tion for the ratio has the form at " a-b V=l ' a 2 +6 2 It is clear that only for special values of the coefficients, which might never be possible in the problem, could the solution for the ratio have this form. The complexity of the problem is such that no further attempt will be made here to determine whether there exist solutions of Class C which are dis- tinct from those of Class A and Class B. If the attempt is made to construct the periodic solution of which (32) are the terms independent of e, no difficulty will be encountered until the terms in e 2 are reached. Then it will be found that equations (135) must be satisfied in order that the solutions at this step shall be periodic. That is, step by step, the construction agrees with the existence, though the computation is somewhat less laborious. CHAPTER VI. OSCILLATING SATELLITES. SECOND METHOD.* 97. Outline of Method. The problem treated in this chapter is the same as that considered in the preceding, but the method employed is quite different. In this particular question the preceding method is somewhat more convenient, but in other problems where the same general style of analysis can bo used it is much less so. There is a definite physical situation for which the analysis is to be develo|>ed. One of its principal features is that the periodic orbits form a continuous series from those of zero dimensions at the points of equilibrium, and as they vary in dimensions the periods undergo corresponding changes. In the analysis of Chapter V the dimensions were controlled by means of the scale factor ' and the varying periods were properly secured by the introduction of 5 and its subsequent determination in terms of ' As ' approached zero the orbits approached zero dimensions and the period approached the value which corresponds to 6 = 0. In the present treatment no parameters corresponding to e' and 6 are employed. Instead, we introduce a parameter X by means of /* = /* +X, where MO is kept fixed in numerical value while X is a parameter in terms of which the solutions are expressed. Periodic solutions are found for all X whose moduli are sufficiently small, but only those solutions belong to the physical problem for which X = M~MO- The dimensions of the physical orbit depends upon this value of X, and its period depends upon MO- That is, we find a family of periodic solutions having a constant period depending upon M, , but only one of them belongs to the physical problem. It is because of this fact that it is not necessary to make the period variable and dependent upon the parameter in terms of which the solutions are developed. 98. The Differential Equations. We shall start from equations (6) of Chapter V, omitting the accents which will not be needed. The right members of these equations involve the parameter n explicitly in the last two terms of U, and implicitly through r, and r, which depend upon r f M) de- fined in (4). We shall make the transformation M = M,+X, (1) but it is not necessary to do so in all places, both explicit and implicit, in which this parameter occurs in the differential equations. The problem of oscillating satellites was first treated by the author by the method* of this chapter. However, the two methods were reported on simultaneously in the paper referred to at the beginning of Chapirr V. 200 PERIODIC ORBITS. For simplicity the transformation will be made where it appears explicitly in U, and elsewhere M will be supposed to retain its original given value, which is regarded as a fixed constant. This particular generalization of the parameter ju is not the only possible one, and the series obtained differ according to the particular generalization made, but when the conditions for convergence are satisfied their sums are identical in t. After the transformation (1), equations (6) of Chapter V become x*-2y'-(l+2AJx= P,(x, if, z% X), 2 \ \ x , y > z ~, A;, 2 2 "\ "\ x i y > z ) xj, (2) y" -\- A y y P (v ifl y 2 \} r YI O z zr 2 \j., y , z , A^, where A . . Mo , _Mp_ ** r (0)3 I r (0)3 r ' it _ _i T) +1 Mo + _Mo r< i Mo i Mo " " r (0)3 I r (0)3 ' ** + 7-(0)4 r (O)4 ' ^8 T r (0)5 T ..(0)5 ' '1 '2 'l '! '2 '2 (3) the signs in B being the first, second, or third according as orbits in the vicinity of (a), (6), or (c) are in question. The regions of convergence of P! and P 2 are precisely the same as those found in 77. We shall need the differential equations in the normal form so far as the linear terms are concerned. When the right members of (2) are put equal to zero, their solutions are (4) where K l} . . . , 7v 4 , c, , and c 2 are arbitrary constants of integration, and where + a V i, )+A'(2x>-y t -z')\ (8) 1 | =constant, which holds for x, y, and z within the region for which the series converge. Since there is always a component of acceleration toward the xj/-plane, there can be no closed orbit entirely on one side of this plane. Therefore,, in all cases we can take the origin of time so that z = at t = 0. Suppose Uj = aj, z = 0, z' = y aU = 0. (9) We now integrate equations (7) as power series in the a n y, and X. The solutions arc . . . , a 4 , y, X; 0, *' +P*(I> , 4, y, *; 0, . . . , o 4 , y, X; 0, ,JQ, i > o 4 , y, X; 0, a -v X- /") i> > a 4 it A *y where p,, . . . , p, are power series in o,, . . . , o 4 , y, and X. The moduli of these parameters can be taken so small that the series converge for all O^J^T, where T (finite) is taken arbitrarily in advance (16). The j)j arc of the second and higher degrees in the a } , y, and X. It follows from the way in which X was introduced that the p, identically vanish for a,= =a 4 = -y = 0. Since the last equation of (7) contains 2 as a factor, p t = p, = for 7 = 0, whatever the other initial conditions may be. 202 PERIODIC ORBITS. 100. Existence of Periodic Solutions. Since the right members of equation (7) do not contain t explicitly, sufficient conditions that (10) shall be a periodic solution with the period T are '"" ij -TVs\i) /A,W> -"' T -l] +p 4 (T)-p 4 (0), 0= z(T)- z (0) = -^= sin VT T + p s (T)-p,(0), 0=z'(T)-z'(0)=7[cosVT T-l]+p 6 (T)-p 6 (0). The last two equations of (11) are satisfied by 7 = 0. Suppose 7^0. Then it follows from the form of the integral (8) that unless VZ T = (2n-}-l)ir/2, where n is an integer, the last equation is a consequence of the first five. We shall suppose T does not have one of these special values, and we shall suppress the last equation since it is a redundant condition. The first five equations are to be solved for Oj , . . . , a 4 , and 7 in terms of X, and we can use only those solutions which vanish with X . These equations are satisfied by e^ = = a 4 = 7 = 0. In order that this may be not the only solution vanishing with X, the determinant of the coefficients of the linear terms in a, , . . . , a 4 , and 7 must be zero. This condition is explicitly _ j-j |y.T _ ^ |- e -,.T _ j J This equation has the solutions T > = ' T ' = <" an inte g er ). (13) Consider first the solution T = Tj . For this value of T the determinant of the linear terms in ai , . . . , a 4 of the first four equations of (11) is distinct from zero unless v is an integer. This condition can not be fulfilled for all v unless = ) =o is (15) It follows from the non-periodic term of this equation that the fifth equation of (1 1) has a term in -yX, and therefore that the solutions exist. To get their character we must find the terms of lowest degree in y alone. The coefficients of 7* are defined by a.o) + =- 4 | ~ST ' 0.0) : where [ (16) l a - 4- ?/ -I- 2* 1 o T tfi.o i z i.oJ> y, . (17) ' as a factor. We shall need only the terms M" |W , . . . , u^ n carrying 7 Hence in the first four equations we may omit z,,, and y,, . From equa- tions (6) and (10) we have 2VA, V-i a*v 2VT t v^^T ^"Y t v /:r T 047 v /Tr r ^i "I (18) 204 PERIODIC ORBITS. Therefore, integrating (16) so far as the first four equations depend upon terms involving 7 2 as a factor, and determining the constants of integration so that w?' 0) , z,. , and z 2 . are zero at t = 0, we get # 0) Bm2VT t t, w . = a . . - ' 0> uf- 0) = a' 2 ' 0) + a,* 0) e +p " + a 0) cos 2 V3; < + 6- sin 2 v^ < , M . > = a * 0) 4- a* 0) e-"' + a- 0) cos 2 Vi; - 6< 2 2 ' 0) sin 2 V3; < , " 2 (o-o 2 V^ ) Vl)p VA V- a 4 7 2(p +2 V^" V-l)p where r/ (2,o>_ , 3 mo Bo 7 s , a .o. = + 3 n .Bo 7 2 a ' 8 (m, (^ -n p ) A a 8 (?/? o- -r? Po ) A Po (20 )_ 3 W BO a ' 2 ' h (19) (2 o> = 3m Bo 7 2 , _. o) = _ _3o Bo 7 2 (7 2 ) m p 4-w (r = - - -> 2ff po (25) Hence, after dividing by y, we have ' 2 + , (26) _ - 7 A + ISA?) which can be solved for y in terms of X in the form . (27) Upon substituting this result in the series for e^ , . . . , a 4 when they are expressed in terms of y and X from (24), we have a, = XP,(X) (i = l, ...,4). (28) After (27) and (28) are substituted in (10) the coordinates u t and z become power series in X*, vanishing with X 1 , and they are periodic, since the conditions for periodicity have been satisfied. The series converge for |X| sufficiently small. The radius of convergence depends on M and MO an d it is easy to see from the explicit forms of the equations that it remains finite as /z approaches /x. For X = /x fj^ the orbits belong to the physical problem, and /x c can be taken so near p that the series converge. That is, periodic solutions exist having the form x= S z^, y- 2 y^, z= i z,X% (29) 1-1 <- 1 <=i where the x t , y ( , and z, separately are periodic functions of t having the period 2ir/VA a . OSCILLATING SATELLITES SECOND METHOD. 207 I The last t\\o equations of (11) arc satisfied byy = 0. For T = 2ir/V/l the determinant of the linear terms of the first four equations is distinct from zero; therefore their only solution for a , a t as power scries in X, vanishing with X, is o,= =a 4 = 0. Hut then M, , . . . , u t , and z are identically zero. That is, the solutions having the period 2r/VA 9 are in three dimensions and not in two alone. In this respect they agree with the solutions of Class A of Chapter V. It will now be shown that for these solutions are those of Class A. The solutions of Class A were developed as power series in ' of the form z= S V, y= 2 y,t", z= S z _ I 7 , f 1(2) I I I (2) ~~jy~ TT Po M 4 i "n ] i i i > where 2A'\x l +lB.[-24+ri\ t ( m A'\ i0ii (35) It follows from the forms of the right members of these four equations that their solutions will contain Poisson terms,* whose coefficients involve Xttj , . . . , Xa 4 respectively as factors. The coefficients of all the other terms are of the second degree in a : , . . . , a 4 and linear in B , and those which are not periodic involve a 3 or a 4 at least to the first degree. Conse- quently, when we solve the third and fourth equations of (11) for a 3 and o,,, the results will start with terms of the second degree in o, , a 2 , and X. When these results are substituted in the second equation of (11), it will contain a term in X Oj and terms of the third degree as the lowest in a, and cu alone. If we now eliminate o t by means of (33), we have an equation whose terms of lowest degree are a,X and a 2 . We shall verify first that the coefficient of a 2 X is not zero. *In Celestial Mechanics terms which arc of the form of I multiplied by cosine or sine terms are called Poisson terms, from the results in Poisson's theorem on the invariability of the major axes of the planetary orbits. II. I. \ I IN', -\ll.l.l.lll.- -SECOND MKTHOD. L'O'.t It follows from .',1 and 36) that tin- Poisson terms in i/.,' are ^ _ > * " V-l Hence, so far a term- arc concerned, we find 5s- -1. o + HflffoJ which, I iy ('({nations (25), reduce.- tu - < - Therefore the coefficient of Xo, in (31) is not zero unless .!' = (). But except for tli" center of liliration it distinct from zero. Consequently, since all the equations are iden- tically satisfied by a, = =a, = 0, the equation obtained by eliminating a, between (31) and (33) is divisible by a,, after which there is a term in X alone whose coefficient is distinct from zero. Therefore the equation can be solved for a, in terms of X, vanishing with X, and the periodic solutions exist. The form of the solution depends upon the degree of the term of the lowest degree in a, alone in the final equation after a,, a,, and a, are elimi- nated. It is ea.-y to show that the coefficient of oj in this equation is not identically zero. It has been shown that the terms arising from the solu- tions of (34) involve B t linearly. There are also terms of the third degree in a,, . . . , a 4 arising from the terms of the third order. The terms of , of the third order are defined by H vA- dt where The Poisson terms in the solution of (37) involving (' as a factor are M o) = _ 3(7, w,(2 - np a.aite-''^^ 1 , aC^n. (4 - 3 np a. ajt e-*<^ ' (mtfftr^pt) Vl 4(wpo+n, a , p , m a , n , and (7 are the same function of ^ for both points, but B is different because of the change of sign in its second term [eq. (3)]. Consequently, the sum of the terms in B and C can not be identically zero in MO f r both the points of libration (a) and (6). Hence in this case the second, third, and fourth equations of (11) and (33) are solvable for Oj , . . . , a 4 as power series in X } , vanishing with X. Therefore the periodic solutions with the period 2ir/ff are expansible as power series in X } . In a manner similar to that used to prove that (29) are series which represent orbits of Class A, it can be shown that the orbits now under con- sideration belong to Class B. 101. Direct Construction of the Solutions for Class A. As in the method of Chapter V, the coordinates in the orbits of Class A are most con- veniently obtained from the x, y, and ^-equations. Consequently we start from equations (2) and (3). Since the solution is periodic for all [X 1 ] suf- ficiently small, each term of the expansion separately is periodic with the period 2ir; and since z = at t = 0, each term in the expansion of z separately vanishes at t = 0. The coefficients of X* are defined by x';-2y' l -(l+2A )x l = Q, y^+2x' 1 -(l-A )y i = 0, zl+A& = Q. (39) The solutions of these equations satisfying the periodicity and initial con- ditions are 2, = - sinv/To t, (40) where c t is so far undetermined. u;+2t' t -(i- .1 v. :;/; A 3 B* MOLLATHra SATELLITES SECOND METHOD. 211 The coefficients of ,\ :t re defined by (41) I 'pon making u>e of > J(>>. integrating. and applying the periodicity and initial conditions, \\e have 3fi.C? 3g.(l + 3;4.)c; 4(1+2,4 M H 4(l-7i.+ l*4SM. < whore r, is so far arbitrary. It will be necessary to carry tho computation two stops further in ordor to .-how how the general term is found. Tho coefficients of X 1 arc defined by y' t +2x' 3 -(l- .4 )/, = 0, - |81 .. . n (43) Consider first the solution of the third equation. In order that it shall be periodic tho coefficient of sinv^f must be zero, or Ai - (44) Thi.- <( illation, which is identical with (26) of the existence proof, has tho solutions . 2 V2A,A f C, = 0, C, = - / 1 e\ 3 212 PERIODIC ORBITS. The solution c i = leads to the trivial case x = y = z = 0, as can be shown easily by an induction to the general term. The double sign before the radical plays the same role as the double sign before X ! in the existence. If it is used in one place in the final solution it is superfluous in the other. With the value of c t determined from the second of (45), the solution of (43) satisfying the periodicity and initial conditions is *> AJQlsllsZ 1 ^3 0/1 I o A \ A i 2(1+ 2A )A ' 2(1 - 7A +isAl)A ~^==smVA t ^-f; Tjsm 64(1-7^+18^)^0 (46) where c 3 is so far undetermined. The equation for the determination of z 4 is z"<+A^ = -A'z 2 +3B Q (x z z,+x 3 z 1 } + f <7 z*z 2 . (47) In order that the solution of this equation shall be periodic the coefficient of sin V~A~ 1 in its right member must equal zero. This function of t arises from every term in the right member of (47), and it follows from (40), (42), and (46) that its coefficient carries c 2 linearly and homogeneously. Therefore this condition determines c 2 uniquely by the equation c 2 = 0, whence z 2 =.T 3 =i/ 3 =0. (48) After the sign of c, has been chosen all the other c t are determined uniquely by the conditions that all the z< separately shall be periodic. For, suppose that x lt . . . , x^; y lt . . . , y^; z lt . . . , z t ^ have been computed and that their coefficients are entirely known except the arbi- trary terms c^-j/VZ^ sin \/3^ t in z,_ 2 and c 4 _i/v^ sin VA~ in z t ^ , and the arbitrary constant c,_ 2 , which enters linearly in x^ and y t ^ . The z t is defined by z?+A z i =-A'z i _t+3B,(XtZ t _ t +x i _ 1 z 1 ) + ^C zlz < _ 2 + , (49) where the terms not written are completely known. The arbitrary c 4 _ 2 enters linearly in the coefficient of sin v^ t in the right member of this equation, which does not involve c ( - 1} and the constant c t ^ 3 is uniquely determined by the condition that this coefficient shall vanish. OSCILLATING SATELLITES SECOND METHOD. 213 It can !> -hown without difficulty thut the solutions have the following propertii 1. The x t , n , y tj+t , z^, are identically zero (j-1, 2, ...). 2. The x lt y,, z, involve r, homogeneously to the degree j. 3. The x tl are sums of COHIM-> of even multiples of vT.J, the highest multiple being 2j. 4. The y tl an >ums of sines of even multiples of VA~ t t, the highest multiple being 2j. 5. The z tt+t are sums of sines of odd multiples of \/~A~ t t, the highest multiple being 2j+l. (i. Changing the sine of c, is equivalent to changing the sign of X', which is equivalent to increasing t by r/^/~A t . Therefore the two values off, (or X*) belong to the same physical orbit, the origin of time being different by half a period in the two cases. 7. The orbits are symmetrical with respect to the x-axis, the xy-pl&ne, and the zz-plane. It is not necessary to go into the proofs of these properties, which are the same, so far as the comparison can be made, as those found in 87. 102. Direct Construction of the Solutions for Class B. For these orbits it is advantageous to use the first four equations of (7), the last one being identically zero. We have proved that the solutions are expansible as power series in X*, that the coefficients of each power of X* are periodic with the period 2T/a , and that z' = at t = for all X. The coefficients of X' are defined by the differential equations (50) O wi u = 0. The periodic solutions of these equations are seen to be From equations (6) and the initial value of x' it is found that /\ (1) (I) rt 4 "*) a, l = a t = a , Xi = a cosfficient a (l) is so far undetermined. 214 PERIODIC ORBITS. The coefficients of X are defined by .(2) *ii r i (2) \ \ < 2 > ti (<>) i *''(\\ jr 411^' I "I J _ dt (m ffo n po) V 1 (2) \ ff t -^(2) _ 3 _ ^(2) _ ___ o __ I dt (mo where a <2) is so far undetermined, '"" -)-Wopo)po ?) (a (1) ) 2 ' <2) rz p )o- "o <7o g a (a a) ) 2 < = , 00 - (54) (oo; OSCILLATING SATELLITES SECOND METHOD. 215 The arbitrary a"' is determined. except as to sign, by the periodicity condition in the next step of the integration; and the double sign is equivalent to the double sign on X*. After '" ha> been determined, an a" 1 is uniquely determined by the periodicity condition at each succeeding step of the integration. The coefficients of X 1 are defined by _ r tV - . *j dt * " i HI. a. II. n.\ v'_ 1 (m. a-4-n.a.} (57) ._, rt v- dt dt where [ ] U) =-f _ (m u i" " M4 ' - h (58) In order that the solution of the first equation shall be periodic it is necessary that in its right member the coefficient of e"^ 7 ' be equal to zero. That part of this coefficient which arises from A'z, involves a (1) linearly and homogeneously. Those parts which arise from x, y t , xj etc. carry (a (l> )' as a factor and involve it in no other way. Consequently, the condition that the coefficient of e* tvrrr ' shall vanish is satisfied by a (l) = 0, or by an equa- tion of the form P(a (U )'+Q = 0, (59) where P and Q are known constants. It is easily shown that they are identical with the coefficients of oj and aX which arise, in 100, in the demonstration of the existence of the solutions. The first determination of a t leads to the trivial solution x=y=0; equation (59) gives the double determination for a by an equation which is identical with (59). That is, the same value of a'" makes the solutions of both the first and the second equations periodic. The particular integrals of the third and fourth equations are periodic. In solving the third and fourth equations the constants of integration are always to be taken equal to zero. 216 PERIODIC ORBITS. The right members of the differential equations for the terms of the next order are (60) Before integrating the first equation the coefficient of e an/ ^' in its right member must be put equal to zero. It is found from an examination of the terms of (60) that a (2) is involved linearly but not homogeneously. Moreover, a (2) is the only unknown quantity in this coefficient. Therefore a (2) is uniquely determined by setting the coefficient of $""'=>' equal to zero. The condition that the solution of the second equation shall be periodic is identical with that imposed by the first. The particular integrals of the third and fourth equations are periodic. At the i lh step the right members of the differential equations involve (61) where the parts not explicitly written are independent of a" 2) . In order that the solution of the first equation shall be periodic it is necessary that the coefficient of e ff v=It in its right member be put equal to zero. This coefficient carries a (< ~ 2) linearly and in general non-homogeneously, and the known factor by which a (< ~ 2) is multiplied is precisely the same as that of a w in the equation by which the latter was determined. Therefore the arbi- trary a (i ~ 2) is uniquely determined by setting the coefficient of e'" v=li equal to zero, for it carries no other unknown. When this condition is satisfied the coefficient of e- ffov=Tl in the second equation is zero, and the entire solu- tion at this step is periodic. Therefore after the sign of a (1) has been chosen the process is unique, and it can be continued indefinitely. CHAPTER VII. OSCILLATING SATELLITES WHEN THE FINITE MASSES DESCRIBE ELLIPTICAL ORBITS. 103. The Differential Equations of Motion. Suppose the finite bodies describe ellipses whose eccentricity is c. Let 1 n and n (/j^0.5) represent their masses, and then determine the linear and time units so that their mean distance apart and the gravitational constant shall be unity. With these units their mean angular motion is unity. Now refer the system to a set of rectangular axes with the origin at the center of gravity, and let the direction of the axes be so chosen that the 7>-plane is the plane of motion of the finite bodies. Suppose the i?-axes rotate with the constant angular rate unity around the f-axis in the direction of motion of the finite masses, and suppose 1 M and /* are on the {-axis when they are at the apses of their orbits. Then the differential equations of motion for the infinitesimal body are (1) <# (1 M)(n-ih) dt i\ rj where and where , , ^,, 77, , and ij t are determined by the fact that the finite bodies move in ellipses. If we let p, and t>, be the polar coordinates of 1 n referred to fixed axes having their origin at the center of mass of the system, and p, and t' f the corresponding coordinates of /* we have , = -p, cos(,-0, & = J,= -p.sin^.-O, 77,= |l-ecos<+ j (1 -cos 20+ } (2) where the initial value of t has been so determined that the bodies are at their nearest apses at ( = 0. n 7 218 PERIODIC ORBITS. 104. The Elliptical Solution. In order to get the Lagrangian elliptical solution for the infinitesimal body we consider first the two-body problem. The equations of motion for the infinitesimal body subject to the attraction of a mass m are, when referred to the rotating axes, =~ k * m > (3) We shall consider the solution in the ^-plane. If the eccentricity of the orbit is e and if the mean motion with respect to the fixed axes is unity, then the solution with the same determination of the origin of time and apses as in (2) is (4) v f), r? = rsin(v f), (l-cos20+ It will now be shown that equations (1) will be satisfied if the infini- tesimal body moves so that the ratios of its coordinates to the corresponding coordinates of the finite masses have certain constant values. Let the coordinates in these special solutions be represented by , rj , and 0; then t - i _ 2 , ,rx Upon making use of (2) and (5), it is found that t i % + (6) SATKI.UTKS, KI.UPTICAL r\ 219 Then equation- 1 i heroine r* ., ft (7) The first two of these equations are of the same form as (3), and their solu- tions eorresjMmding to (4) are From these equations and (2), we find & _. 30 _ [(l l Til and from (6), On equating these two expressions for the ratio ,/, = JJ./T;, , and rationalizing, we have M = 0. (9) It is easily verified that starting from the expressions for the ratio the same quintic equation is obtained. Therefore, for those values of M satisfying (9), equations (2) and (8) are a particular solution of the thn-e- hody problem where one mass is infinitesimal. As is well known, there :ire three real solutions, one for each ordering of the three masses. As in Chapter V, we shall call them (a), (6), and (c) in the order of decreasing values of their x-coordinates. Equation (9) is Lagrange's quintic in case one mass is infinitesimal and the units are chosen so that the masses of the finite bodies are 1-p and . For example, if in equation (60), page 216, of Introduction to Crlextial s. we put /, = !-//, i,=0, and W,=M, we get equation (9). 220 PERIODIC ORBITS. 105. Equations for the Oscillations. We shall study the oscillations in the vicinity of the Lagrangian solutions. For this purpose we make the transformation (10) in equations (1), and expand as power series in x, y, and z. After this transformation and expansion, we let in those places where /z appears explicitly. This is not the only way in which the original p can be divided into the new ju and ju +^> and sometimes others are advisable. The coefficients of the various powers of x, y, and z are expansible as power series in e, the terms independent of e being constants, as is seen from (2) and (8). We find from (6) and (8) that +2esinH- si = (12) Consequently, after making use of these expansions and the transformations (10) and (11), equations (1) become 4 x 1* (13) ( ll.I.AIIN., SATBLUTBB, Kl.I.ll'TICAL C.\ _'_'! where I - L '"Mo, Mo ' "1 _ ,. y ' = ft, z = 0, z' = y. (15) Then the solutions of (12) are x=f(a, 0, 0, ft, 0, 7; 0, x'=f(a, 0, 0, ft, 0, 7; 0, y = g(a, 0, 0, ft, 0, 7; 0, y' = &'(<>, 0, 0, 0, 0, y; t), (16) z=h(a, 0, 0, 0, 0, 7; 0, ' = /i'(a, 0, 0, ft, 0, y; t). Now make the transformation x=+x, x'=-x', y=-y, y'=+y', z=-z, z'=+z', t=-t. (17) It follows from the properties (a) , . . . , (d) that the form of equations (13) is not changed by this transformation. Consequently the solutions with the initial conditions x = a, x' = 0, y = 0, y' = ft, = 0, z' = y, are identical with (16), and we have, making use of (17), (18) Therefore, with the initial conditions (15), x, y', and z' are even functions of t, while x', y, and z are odd functions of t. That is, if the infinitesimal body crosses the z-axis perpendicularly when the finite bodies are at an apse, its motion is symmetrical with respect to the it-axis. u.-< II.I.AIIM, >\ i u.u n..-. Ki.i.ii'in \i. i \-i. _'_':; 107. Integration of Equations (13). The Terms of lh< f-'irst It, ,,,,,. Suppose tlu- initial conditions are ' < = a,. *'(()) = a,, j/(0) = a lt J/'(0) = a 4) z(0) = a, z'(0) = a,. (19) \\ e shall now integrate c(|iiations (13) as power series in a,, a,, a,, a 4 , a> , a t , and X. Since there are no terms in the differential e<|iiation> inde- pendent of .r. //, and 2 and their derivatives, there will he no terms in the solutions independent of a, a,. The terms of the first degree in a , a, are defined liy the differ- ential equations 20) -,- [l-.l - z'+ [A +3 Accost + f 4e*(l+3cos20 + ] z, =0, subject to the initial conditions (19). The coefficients are power series in <. periodic with the period 2ir, and reduce to constants for r = 0. The first two equations are independent of the third, and conversely. For e = equations (20) become simply x' l -2y[-(].+2A}x l =Q, yi+2x f l -[l-A]y l =0, z' t +Az t =Q. (21) From the results obtained in 23-25 it follows that the properties of the solutions of (20) depend upon the character of the roots of the characteristic equation of (21). This equation for the first two of (21) is 4 +(2-^)s-*+(l-yl)(H-2^)=0. (22) Two roots of this equation can be equal only if ^4 has one of the values - 1/2, 0, S/9. or 1. The first two are excluded by the fact that A is necessarily po>itive. When M, is near n in value, as it will always be taken here, A is greater than unity and, consequently, the last two values are excluded. Equation (22) has two pairs of roots equal in numerical value but opposite in sign, and for A>\ two of them are pure imaginaries and two are real. Let us represent them by M II.I.MIM. -\ I F.I.I. II'KS, ELLIPTICAL CASE. '2'2~i According to tin- results obtained in 2:i, the solutions and character- istic exponent.- of I'D ;ire always expansible as power series in except when .1 has the special values noted above: and according to the results obtained in 24, the same result is true, in general, even if the roots of the character- i.-tic e(iuations differ by imaginary integers. However, in the latter case the construction of the solutions is quite different. It was proved in M that in equations of the type under consideration here the characteristic exponents occur in pairs which are equal numerically luit opposite in sign. Therefore 1 the solutions of (20) arc of the form +a t e pl [pu l +u' t ]+a (25) M i 0> > " w > u ' <0) &re constants. The initial values of the v, and w, can be taken equal to unity without loss of generality, and will be so chosen. Moreover, the t/,. v t , and u\ are periodic with the period 2*-, and since this property holds for all e for which the series converge, each ?/{", v\'\ and w\" separately is j>criodic with the period 2r. The coefficients of these series can be found by the methods set forth in 26. The a, and r, are uniquely expressible in terms of the initial values of x, x', y, y', z, and z', the a, of equations (19), because the solutions (24) constitute a fundamental set, by hypothesis, and the determinant of the coefficients of the a, and r, is therefore distinct from zero. We may use either the a, and c, or the a, as arbitrarics. The characteristic exponents es. There are terms which contain X as a factor, and a part of these are sums of periodic terms, period 2*-, multiplied by t times the fundamental exponentials; the remaining part lacks the factor t. These terms are homogeneous and linear in the (it and the c, . The corresponding parts of the solu- tions are sums of periodic terms, period 2*-, multiplied partly by ? , partly by t, and partly by f. They are all homogeneous and linear in the a, and the c,, the x and t/-terms being independent of the c t , and the z-terms of the a, . The terms of the solutions coming from the other part of (b) are sums of periodic terms, period 2r, multiplied by squares and second- degree products of the fundamental exponentials. They are homo- geneous of the second degree in the a, and the c, . [7] The terms of the type (c) are homogeneous of the third degree in the a, and the c ( . They consist of terms of two classes, the first of which are sums of periodic terms, period 2r, multiplied by the fundamental exponentials to the first degree; and the second of which are sums of periodic terms, period 2r, multiplied by cubes and non-canceling third- degree products of the fundamental exponentials. The corresponding parts of the solutions consist respectively of t times sums of periodic terms, period 2r, multiplied by the fundamental exponentials to the first degree, and sums of periodic terms, period 2*-, multiplied by cubes and non-canceling third-degree products of the fundamental exponentials. [8] The part of the solution coming from the terms (d) consists of terms of two kinds, the first depending upon those parts of x t , y t , and z, which contain X as a factor, and the second depending upon those parts of x,, y t , and z, which are independent of X. The parts of the solu- tions corresponding to the first are homogeneous of the second degree in the a, and the c, , and they are sums of periodic terms, period 2r, multiplied by squares and second-degree products of the fundamental exponentials, and some of these products contain t as a factor while others do not. The other parts of the solutions coming from the terms (d) have the proj>erties of those coming from (c). 230 PERIODIC ORBITS. 110. General Properties of the Solutions. It will be necessary to use the following general properties of the solutions : [9] Since the right members of the first two equations of (13) involve only even powers of z, it follows that x and y are even functions of Cj and c 2 taken together. [10] Since the right member of the third equation of (13) is an odd function of z, it follows that z is an odd function of C L and c, taken together, and that z identically vanishes for d = ci = 0. [11] Since the right members of the first two equations of (13) vanish identically for x = y = z = 0, but not f or x = y = 0, it follows that x and y vanish identically for a t = a 4 = c t = c 2 =0, but not for [12] Since the equations reduce to those having constant terms for e = Q, it follows that the sums of the periodic terms, period 2ir, reduce to constants for e = 0. 111. Conditions for the Existence of Symmetrical Periodic Orbits. The differential equations are periodic in t with the period 2ir. Consequently the period of the periodic solutions, if they exist, will be T = 2nir, where n is an integer. When the initial conditions are such that the orbit of the infinitesimal body is symmetrical, as defined in 106, then sufficient con- ditions for the existence of the periodic solutions are (31) These equations are power series in a,, . . . , a 4 , c lt c 2 , and X, and vanish identically with a t = = a 4 = c : = c 2 = 0. In order that they may have any solution for a : , . . . , a t , c lt and c 2 , vanishing with X, aside from this one, the determinant of the coefficients of the linear terms in a, , . . . , a 4 , c u and c 2 must vanish. It follows from (28) that , U.CO), -(0) =0, z -2(0) = (32) OSCILLATING SATELLITES, ELLIPTICAL CASE. 231 and therefore the determinant of the coefficients of the linear terras of (31) is found from (24) to be A-A.A,, (33) 1 o rvcn A t e -R l , A, -vrrr 1 ,(34) where U V | X (35) On reducing, we find r ,vrri I _r^ A,= [e 7 -e A,= w,(-|)[e uv -e"" It will now be shown that t0,(7Y2) and A, A t A, (36) are distinct from zero. Let D, represent the determinant of the fundamental set of solutions of the x and {/-equations, and D t that of the z-eq nation. On writing these determinants for the time t = T/2 and making use of the relations (32), we find ".(I). r '-'I *(f) -c, 232 where PERIODIC ORBITS. The determinants become, after some reductions, .(!). X /T\ V *\2) ' (37) Since A and D 2 are determinants of fundamental sets of solutions of linear differential equations for the regular point t = T/2, they are distinct from zero. Therefore their second factors are not zero, and equations (36) can be satisfied only by r \ e : *]=0 or [e uV - l *-e- uV -**]=0. (38) If either of these equations is satisfied, A is zero. In order that one of equations (38) shall be satisfied it is necessary that either aT = 2N 1 w, or wT = 2NsTr (N,, N, integers). (39) Since T = 2mr, where n is an integer, these conditions become a = - , or co = n n (40) Hence the conditions for the existence of the symmetrical periodic solutions in question can be satisfied only when a or co is rational. These quanti- ties, given in (25), are power series in e and they depend upon M and /z and the way /* is generalized when the transformation JU = M O +^ is made. Since a and to are continuous functions of /z, MO, and e, the rationality of at least one of them at a time can be assured. It should be noted further that erty [10], and therefore vanish identically for c, = c, = 0. The substitution of the values for a, , . . . , a 4 does not change the linear terms. for the first four equations were even in c, and c, alone. Let r, be eliminated by means of the fifth equation of (31). Then r, is a factor of the result, which has the form = c l [a M X+a x (^+ ]. (41) The solution c, = is trivial and we are interested only in those obtained by setting the other factor equal to zero. The second factor set equal to zero is satisfied by r, = X = 0. If a,, is distinct from zero, solutions for c, as power series in fractional powers of X certainly exist. If a /l0 is the first a lo which does not vanish, then the solutions are expansible as power series in X !//1 . In particular, if a*, is distinct from zero the solutions are expansible as power series in X*. If the number of solutions is odd, only one is real; and if even, only two are real, and these are real only for positive or negative values of X according as a,, and a /|0 are unlike or like in sign. It is clear a priori that the number of solutions will be even, for there is nothing of a dynamical nature by which to distinguish the two sides of the xi/-plane. Consequently, if any initial projection gives rise to a periodic orbit, a symmetrically opposite one with respect to the ary-plane will also produce a periodic orbit. It remains to show that a,, is distinct from zero. To prove this the terms of the second degree in the a,, c t , and X must be considered (108). It follows from the form of (29) and (30) that o^ depends only upon the z-equation, for it is not changed by the substitution of the solutions of the fir ; then let c 1 = cf ) (l+7). It is easily found from (41) that dy/de is a power series in e and 7 which is distinct from zero for 7 = e = provided X has such a value that cf'^0. Hence the solution of (41) can be written in the form (43) where p(\*) is a power series in X* whose coefficients are power series in e. On substituting this result in the solutions of the first four equations of (31) for the a t as power series in c t and X, we have c^ , . . . , a 4 expressed as power series in X*. But since a x , . . . , a 4 contain only even powers of Cj , they have X instead of X* as a factor after Cj is eliminated by (43). The expressions for the coordinates become, since x and y are functions of c\ , x = \P l (\*;t), y = XP,(X;0, 2 = X*P,(X*; 0, (44) where P, , P 2 , and P 3 are power series in X*. Since the problem is dynamically symmetrical with respect to the rc?/-plane, a solution symmetrically opposite with respect to the xy-pl&ne exists for all |X| sufficiently small. Therefore z must be an odd series in X J , and x and y even series in X*. Hence equations (44) become * = XQ 1 (X;0, y = XQ,(X;0, s = XQ,(X; 0, (45) where Q 1( Q 2 , and Q 3 are power series in X. We suppose that e>0 and therefore that the finite bodies are at their perifoci* at t=0. The infinitesimal body crosses the re-axis perpendicularly at t=0 and at t=T/2. It follows from the symmetry of the motion that it can cross the re-axis perpendicularly only at the end and middle of the true period. Therefore if n and N are relatively prime, T = 2mr = 2Nir/u is the true period. The two cases I, n even, and II, n odd, merit a little further discussion. In the first N is odd because n and N are relatively prime; in the second it may be either odd or even. 113. Case I, n even, N odd. The infinitesimal body crosses the re-axis perpendicularly at t = and also at t = T/2 = mr = 2n'ir, where n' is an integer. Since at both of these epochs the finite bodies are at their peri- foci, the infinitesimal body crosses the re-axis perpendicularly only when the finite bodies are at their perifoci. It follows from this that the infini- tesimal body crosses the re-axis perpendicularly at the same point but in the opposite direction with respect to the rn/-plane at t=0 and t = T/2. To *Perifoci will be used to denote the points at which the finite bodies are nearest each other, and apofoci those at which they are moat remote from each other. These points correspond to perihelia and aphelia in planetary motion. O8CILLATINC! SATKI I I I l>. KI.I.IIMICAL CA 235 prove it Mippo.M- tin- two points were different. Then we should li:i\c four solutions corresponding to X* at each of the points, and it is known that there are but two. The values of z'(0) and z\T/2) are opposite in sign because othriwi-r the period of the motion would be T/2. It can now be shown that in this case the orbits obtained by taking the two signs before X* are geometrically the same one. Consider the orbit defined by the positive sign before X*. At the time t = T/2 the infinitesimal body crosses the x-axis perpendicularly at the point at which it crossed at t = Q, but in the opposite z-direction. We may consider T/2 as an origin of time for defining orbits which cross the x-axis perpendicu- larly. It has been shown that there is but one with the given z-direction of motion, and at t= T/2 + T/2 = T the infinitesimal body will again cross the x-axis perpendicularly with the opposite z-direction, viz., with that which it had at J = 0. Since this orbit was unique, it follows that the two orbits which correspond to the double sign before X 1 are geometrically the same, but that in one the infinitesimal body is half a period ahead of its position in the other one. That is, changing the sign of A' in the solution is equivalent to adding T/2 to t. When the solutions are actually constructed and are reduced to the trigonometric form, they involve sines and cosines of (ju + k)t, where,; and A; are integers. Since x, y', and z' are even functions of t, they will involve only cosines, and since x', y, and z are odd functions of t, they will involve only sines. Since x and y are even series in X', it follows from the foregoing properties that in them These identities are satisfied if, and only if, cosO'w+A;) T/2 = cos(ju+k)2irn f = 1 (n f an integer); therefore N (ji -ILL. MINI; >Mi:i,i.m>. Ki.i.ii'neAL i:.\ 237 115. Convergence. The existenceof the >\ mmetrical three diniriisional periodic orbits has been proved except for a discussion of the convergence of the series which have been employed, a matter which must now he taken up. The nature of the difficulty will first be pointed out. Equations (13) were integrated a> POWIT -erics in the*/,, c,, and X. It was shown in 14-16 that for any prea-Mum-d T the moduli of these parameters can tie taken so small that the series converge for all 0^<^ T. The limits on the moduli of the a,, c t , and X are functions of /*, M 3 , and e. But T can not be taken arbitrarily in advance, for it is a discontinuous function of u, which is in turn a function of n, M O , and c. It is not evident a priori that values of M, MO, and e, satisfying the relations M MO = X, W=/(M, MO, e), exist such that all the series which are employed are convergent. The final parameters of the solutions are M, M> e, and the mode of generalizing M is arbitrary. Suppose the ratio of the finite masses is given, that, is, that M is a fixed number. Suppose also that the mode of general- izing M has been determined. It will be shown that values of MO and e exist such that the series all converge for X = M MO- In equations (25) it was shown that where \/A and the w, are functions of M. Moreover, a detailed examina- tion of the functional relation shows they are continuous functions of MO- Suppose for any MO such that |M Mo|^i>0 the series for w converges for M ^ ?i , where ?, depends on , . It is easy to show from the nature of the dependence of the series for w upon M and MO that n, >0. Take any particu- lar MB" such that |M Mi"l^i and suppose that, while e runs over the range to T;,. the value of w (M O , e) runs over the range (MO", 0) = VA to (MO", fi)- For brevity, let w <0) and w (U represent the smallest and largest values of w. An examination shows that ,, u> t , . . . are not all identically zero, and it follows from this that | co (l) - w <0) | > 0. Let MO" take all real values such that |M MO"!^*! and find the corresponding values of w <0) and w'". Let u be the greatest (0) , and wj" be the least w (l> . The value of , can be taken so small that w^-w^X), and hence from the continuity of w as a function of MO '' follows that u takes all values satisfying the inequalities "' ^w ^w, (l> . Take any rational value of w depending on M, Mo , an d e which satisfies these inequalities, say o=^-' (46) n where N t and n, are relatively prime integers. Then determine T= T t by the equation (47) 238 PERIODIC OKBITS. Now consider the series (31), in which we put T= T . Since they vanish identically for a, = c, = 0, the discussion in 14-16 shows that, for any values of X and e for which the differential equations are regular, r t > and p 4 >0 can be so determined that the series converge for all <: T ^ T provided |a 4 | \ and e. We may eliminate X by the relation ^ = /z +X, and we shall take |/u MO I = e i an d ^ e i= r] 1 , where e t and ^ are both distinct from zero and have such values that the differential equations are regular for M Mol = e i > e <^i Let r and p, <0) be the least values of r t arid p, as M O and e take all values satisfying the ine- qualities |M Mol = i> 2+M = 2+M +X, then the series in X are convergent if |X| <2+M where X and MO are subject to the condition M O + ^=M- For example, if M = l/3 in the first case the series converge if |X|<2/3, and in the second case if |MJ<7/6. Now it is clear from the variety of ways in which X can be introduced that convergence of the series can be secured in many, if not all, cases when M and e are given in advance. OBCII.I.UIV, -\ I 1 .1.1.111.-. Ll.l. 11MICAL CA 239 1 16. The Existence of Two-Dimensional Symmetrical Periodic Orbits. The last two equations of (31) are satisfied identically by c, = c, = 0, in which case the orbits become plane curves. We have to consider the solution of the first four equations for a, , . . . , a, in terms of X. The determinant A, equation (33), now becomes simply A = A,. It follows from i.'M) that the condition Ai=0 can be satisfied only by . KI.UPTICAL CASE. 241 1 18. Case II, n odd. If in the case where n is odd the infinitesimal body crosses the .r-axis perpendieuhirly at i = () and the finite bodies arc at (heir perifoci, then it also crosses the .r-axis perpendicularly at i^T/'l \\hen the finite bodies arc at their apofoci. If N is even, the infinitesimal body erocs the ./--axis in the same direction at t = () and J = 7'/2. Hence the orbits for + X* and X ! are in this case geometrically distinct. If N is odd, the orbits for +X' and X* are also distinct, because other- ui.-ethe mot ion of the infinitesimal body would be the -a me while the finite bodies are moving from perifoci to apofoci as while they are moving from apofoci to perifoci. This is impossible because / enters the differential equations differently in the two cases. Similarly, starting when the finite bodies are at their apofoci, two geo- metrically distinct orbits arc obtained, in both of which the infinitesimal body crosses the .r-axis perpendicularly when the finite bodies are at their apofoci. and also \\hen they are at their perifoci. These orbits, therefore, are identical with those obtained starting when t he finite bodies were at their perifoci. CONSTRUCTION OF THREE-DIMENSIONAL PERIODIC SOLUTIONS. 119. Defining Properties of the Solutions. It has been shown that the periodic solutions have the form x= 2 x tl \', y= 2 y,,\', z= 2 z^-.X" 2 *", (51) /-i /-i i-\ whore the x,, y n and z t are all periodic with the period 2T/w. It has been shown that symmetrical orbits exist, two for each value of X, and their coefficients are uniquely determined by the periodicity conditions and 2(0) =0. It follows from this last relation and from the fact that the series for x, y, and z converge for all | X | sufficiently small, that each z,(0) sepa- rately is zero. 120. Coefficient of X'. It follows from (13) that this term is defined by , = (>. (52) This equation is of the type of that treated in 53, and its general solution is of the form z, = c| u e"^'wi(; +cj l) e-"^' w,(e; t), (53) where while w t differs from w l only in the sign of N/^M, and where each ir{" is separately periodic with the period 2r. Since (53) is unchanged by changing the signs of both t and v^T, it follows that in the expressions for u>, and w t 242 PERIODIC ORBITS. the coefficients of the cosine terms are real, and those of the sine terms are purely imaginary. Since w^e; 0) = w 2 (e; 0) = 1 for all values of e, it follows from the condition z,(0) = that c[ l) +c? = Q. It will now be shown that w is a series in even powers of e. The coeffi- cient of Zj in (52) is derived from the expansion of r^ 3 and r^ 3 . Now the expressions for r, and r 2 are unchanged if in them e is replaced by e and t by t+w. Consequently, if ef aMvrrii w 1 (e', t) is a solution of (52), then also e w( ~ e)v/ ^ T( ' +T) t0 1 ( e; t+w) is a solution. Since any solution can be expressed linearly in terms of the two solutions in (53), and since the solutions now under consideration hold for all e sufficiently small, and, for e = 0, differ only by the factor e?'^^ 1 **, it follows that for any e they differ only by a constant factor. Therefore * Wi ( e; t) - Ce^^^w^-e; t+ir) = 0. This relation is satisfied identically in t and e. Since w l is periodic with the period 2^, it follows also that Hence we have two homogeneous linear equations in e^^^'w^e; t) and C r e w( ~" v ^ T( ' + ' r) w; 1 ( e; t+ir), and as these quantities are not identically zero, the determinant must vanish. Since by definition o> must reduce to VA for e = 0, we have w( e)=u(e); that is, w is a function of e 2 . Upon carrying out the computation by the method of 53, we find (54) _ 9A(1-3A-2A') 2 ,_ 8(1-A)(1-4A) C It will be noticed that the coefficients of cosjt and sinjt have e j as a factor so far as they are written. This is a general property of the differential equations, and an examination of the process of integration, as explained in 53, shows it is also a general property of the solutions. If we had solved the differential equations for x^ and y l , we should have found that these quantities are identically zero because of the periodicity conditions to which the solutions are subject. OSCILLATING SATELLITES, ELLIPTICAL CASE. 243 121. Coefficients of ,\. It follows from (13) and (14) that these term- are defined bv (66) ,- [l-,4-34ecos(+ where Jzj, the signs in B l)eing the first, second, or third pair according as the point (a), (6), or (c) is in question. By means of (53) and (54), we find \ A,- The character of the solutions of equations of the type to which (55) belong was discussed in 29. It was there shown that they consist of the complementary function plus terms of the same character as X t and F,. It follows from the hypothesis that the imaginary characteristic exponent irV 1, arising in the solution of (55), and wV I are incommensurable, that the constants of integration must all be put equal to zero. The par- ticular integrals can be found most conveniently by assuming their form with undetermined coefficients, and then defining them by the conditions that the equations shall be identically satisfied. 244 PERIODIC ORBITS. We shall need the following properties of the solutions of equations (55). They are homogeneous of the second degree in Ci 1) e wv/ ^ 7 ' and c a) e - Wv ^T( The terms in the solutions multiplied by (c 2 ") 2 differ from those multiplied by (c"') 2 only in the sign of V i, because this is a property of the right members of the differential equations. If throughout equations (55) we change y 2 into -y z , V^i into -v^T, and t into -t, the equations are unchanged. Therefore, in the expression for z 2 the coefficients of the cosine terms are real and the coefficients of the sine terms are purely imaginary. The opposite is true for ?/ 2 . The terms in x 2 having c"'^" as a factor involve only cosines and are independent of the exponentials e"^~ 11 and e~ a ^^ 1 , while those in 7/ 2 having c"X" as a factor involve only sines and are also indepen- dent of the exponentials e<" v/ = T( and e~ us/ ^ T( . It follows from these properties and c[ u = - c 2 " , that a(0) = &(0) = 0. Certain divisors are introduced in the integration of (55). When the right members of these equations are omitted, the solutions are of the form (24) . On using the method of the variation of parameters, we find fl| -r (i = l> > *), where F t (t) has the form of X t and F 2 , and where A is the determinant of the fundamental set of solutions (24). It follows from the principles of 18 applied to this case that A is constant. The expressions F t (t) contain terms of the types given in (56). Consequently the a t contain terms of the types Aa,= (O z je^^'lA, cosjt+ ^f=\B 1 s\njt~]dt or, performing the indicated integrations, @^e- 2 2 that their product involves only cosines and that the coefficients of their product are all real. Hence the constant parts of (60) and (61) are power series in e which, aside from the coefficients c[ l) and c"', differ only in sign, and the parts independent of e are respectively Cl J_\ if 1 / Consider now the terms coming from that part of Z 3 which contains as a factor. The constant parts of the coefficients of e Wv/=Tl and e~ wV=lt in the products and +w l x i z 1 [3B+12Becost } respectively are required. It follows from the properties of w l and u> 2 that w l [3B+12Beco8t+ ] and w 2 [3B+12Becost+ } differ only in the sign of V^-T, which is a factor of all the sine terms, while the coefficients of the cosine terms are all real. These products do not involve the exponentials e" v/=T ' and e~ wv ^ T( . In the product z 2 z l the coefficients of e uv ^^ 1 and e~ uv=lt are multiplied by (0*4" and c" } (c" 1 ) 2 respec- tively. It follows from the properties of x 2 and z r that in this product the coefficients of (c^Yc^e"^^' and c{ u (^ ) t e-"^ =T ' differ only in the sign of V 1, which is a factor of all the sine terms, while the coefficients of the cosine terms are all real. The typical terms of the products iv t [3B+ 12Be cost ] and that part of the product x 2 2, which contain e wv/ ~" as a factor are respectively The corresponding terms from +u>, [3B + 12 Be cost -\- } and the part of x^Zi containing g-^'^ 7 ' as a factor are respectively OSCILLATING SATELLITES, ELLIPTICAL CASE. 247 The constant parts of tin 1 products of these terms are respectively Since these properties hold each term individually, it follows that the con- stant parts of the coefficients of e uy '-~ il and e~ vt>=::Tl in w t x t z t [3B+ -]and are res Actively of the form + ], +cj B (O i [nding discussion shows that the same result is true for the third and fourth terms of Z t . Therefore, in order that the solutions of (57) shall be periodic, we must impose the conditions = A'c[" +L(c; i) )'ci 1) , - Kc?> +Lc; u (O', (63) where A' and L are constants and power series in e. It follows from the corresponding work in Chapter VI that both K and L have terms independent of e which are distinct from zero. The solutions of (63) are c is homogeneous of the third degree in ^ef v ~ 11 and c^e~ a ^' lt . In the right member of (57) the coefficients of all cosine terms are real, and the coeffi- cients of all sine terms are purely imaginary; and moreover those which are multiplied by c|", (c"')', (OM" differ from those respectively which are multiplied by ci, (O 1 , c|"(O' only in the sign of ^/ F =\. It follows that PI" and F? have these properties also. The u>, and w t are given in (53) and (54). The P[ 3) and P* are entirely known functions, while c, a) and cj" are subject to the relation 248 PERIODIC ORBITS. It follows from the properties of the expressions Pf } (t) and P (t) and c?=-c? that P| 3) (0)=Pf(0)=0. Hence cf and cf are subject to the condition 0. (67) Therefore but one undetermined constant remains in (66). The divisors introduced at this step can be found as they were in the preceding step. They are the determinant of the fundamental set of solutions of the z-equation, j 2 o> 2 , andj 2 9w 2 , the j- or appearing with terms which involve e* uV=It * jt, and the / 9w 2 with those which involve e -*ov=n u*jt. The coefficients of ^ jt contain e' as a factor. 123. Coefficients of X 2 . The differential equations at this step are x"-2y(-[l+2A+GAecost+ -~k- [~6 Seisin t + (68) y"+2x' t -[6Aesmt+ -]z 4 - [l-A-3Aecost+ where X 4 and F 4 are of even degree in e wv/rri( and e~ wV=It considered together. It follows, as in the case of x 2 and y 2 , that in the right member of the first equation the coefficients of all cosine terms are real, and those of all sine terms are purely imaginary ; the opposite is true in the right member of the second equation. The coefficients of c{ u c{ 8) and (c" } ) 4 differ from those of c u c and (c^) 4 respectively only in the sign of V i. The coefficients of WyC*?) 1 are rea l> independent of e u%/=T< and e~ Wv/=7( , and involve only cosines. The solutions of (68) have the same properties. It follows from these properties and cf = -of, that x( (0) = y t (0) = 0. The divisors introduced in the solution are the determinant of the fundamental set of solutions of the x and ^-equations, j 2 16u> 2 , j* 4co 2 , and j in connection with the terms involving e 4wv/rrr 'gf n 9 jt, e* wv ^~ 11 I jt, and C 8 jt respectively. The coefficients of these terms contain e' as a factor. 124. Coefficient of X' A . It is necessary to consider this step in order to be able to make the general discussion. The differential equation defin- ing z 6 is = Z ;> , (69) where it is found from (14) and properties (a) and (d) of 106 that Z & is made up of real cosine series multiplied into z 3 , .T 2 z 3 , x t z l , x\z^, y\ z, , ZjZ 3 , z\, and of real sine series multiplied into yz 3 and y 4 z lf It follows from the properties of z n z 3 , x 2 , y 2 , x t , and y t that Z 5 is of odd degree in e w%/ ~' and e~^^' taken together; that the coefficients of the cosine terms OSCILLATING SATELLITES, ELLIPTICAL CASE. arc real, and those of the sine terms purely imaginary; that cj" ami rj" outer linearly; that the coefficients of c*, (ci u )', (O'CO 1 differ from the coefficients of , Z, e-"*^ 1 ' and +>, Z e wv/ ^ r< must be zero. It follows from the properties of Z enumerated above that these conditions are of the form where A t , . . . , A< are power series in e. Upon reducing by means of the last equation of (64), we get ' (72) where P"(0 is linear and homogeneous in c^'e^^ 7 ' and cJ"e~ w%ArT/ taken together, and P"'(0 an d PJ"(0 are homogeneous of the third and fifth degrees respectively in e"*^ 1 ' and e~ a ^~' taken together. It follows from the facts noted above that Z t is unchanged if c|", c|", + Vi are interchanged with cj", cj", V 1, and that the coeffi- cients of the cosine terms are real; hence the solution (72) has the same properties. Consequently, since cj" = c" } , c, u> c,*, it follows that WO) = P, U) (0) = P*(0) = 0; and since z, (0) = 0, that O. (73) 250 PERIODIC ORBITS. 125. The General Step for the x and ^-Equations. Suppose z,, x t , j/,, . . . , z^-i, z 2( ,-2, 2/2X-2 have been computed and that they have the following properties: (A) The x 2J and y 2} are even functions of e Wv/ ^ T< and e'"'*^ 1 ' taken together, and the z 2 j+i are odd functions of e av ^ 7t and e~ uv ~' taken together. (B) In the x 2J and z 2;+ i the constant parts of the coefficients of all cosine terms are real, and those of all sine terms are purely imaginary. (c) In the y 2 , the constant parts of the coefficients of all cosine terms are purely imaginary, and those of all sine terms are real. (D) In the x 2J and y- is the highest powers of e uV ~' and e""^^ 1 ' are 2j, and in z+i they are 2j+\. (B) The coefficients of ( 2 A; 2 9 ._i taken together; that in X the coefficients of all terms which are of even degree in y 2 , y t , . , . , y }v _ taken together are sums of cosines of integral multiples of t; that in X tv the coefficients of all terms which are of odd degree in t/ 2 , y t , . . . , y tv _ t taken together are sums of sines of integral multiples of t; that the last two properties are reversed in the case of Y tv ; and that if in X tv or Y tt the general term has as a factor A . . -A then 2X.+ OSCILLATING SATELLITES, ELLIPTICAL CA 251 It follows fi om these properties and (A), . . . , (<;) that X^ and }', are even functions of e"^ 7 ' and e~" v ~" taken together; that in A' u the constant parts of the coefficients of all cosine terms are real, and those of all sine terms are purely imaginary; that in !' the last property is reversed; that in A'j, and }', the highest powers of e" v ~' and e-* v ' =71 are 2v; and that the coefficients of (O^O* : (OHO^c*)*' (ef 1 )*- differ from the coefficients of (c, ft) )*(O* (c 1 a> )''(c l U) )*'(O* f (c**)'- only in the sign of Vl. It follows from the form of (74) and these properties that the periodic solutions of equations (74) (i. e., the particular integrals) have the properties (A), . . . , (o) so far as they pertain to x^, and y tj ; and then from c, U) = -ci", that x^(0) = y u (0) = 0. 126. The General Step for the ^-Equation. The differential equation defining z jr M is zl+i+[A+3Aecost+ ] z, r+l = Z tf+l . (75) It follows from the properties (a), . . . , (g) of 106 that Z t ,+i is of odd degree in z,, z,, . . . , z-i taken together, and that it contains z,,_, linearly. In Z tf+l the coefficients of terms which are of even degree in t/,. . . . , //: taken together are cosines of integral multiples of /, and the coefficients of those terms which are odd in the same quantities are sines of integral multiples of t. It follows from these properties and (A), . . . ,(G) of 125 that Z tf +i is an odd function of e"*^ 1 ' and e~~ t Z, r+I and +w l Z tr+ i the constant parts of the coefficients of ^ /=lt and e'""^ 1 ' respectively shall be equal to zero. The z jr _i enters Z,,+, linearly and has the same coefficient that z, has in Z, . Hence, from the relations of the preceding paragraph and equations (70), we have I.I.J-MI-MAJ ^an.^-^p ,< ,^, ft ^ where P, r +, is a polynomial of odd degree in c" } and cj 1 ' taken together. It is supposed that c, a/+l) and c?' + " = 1 > "~ 2) have been eliminated at the successive steps by the equations corresponding to (71). If the general term in P*^ is (c"'X(O*, then j and k satisfy the relation (77) 252 PERIODIC ORBITS. On reducing (76) by means of (64), making use of (77) and (F) of 125, and the fact that c^ 1>= c[ l \ it is seen that equations (76) are equivalent, and that cf"~ l) = cf"~ ' is uniquely determined as a power series in e. Since in Z 2v+l the coefficient of (O' 1 (c<")'< (O* 1 (cf)*- differs from that of (O' 1 ' ' ' (OHO* 1 ' ' ' (c^)*" 1 only in the sign of V 1, it follows that the solution has the same property, and from this that z 2H -i(0) =0. For small values of v there will be no other terms than those considered above in w^Z 2v ^\ and -\-w l Z 2 i>+i ) which are constants multiplied by the exponentials e uV= *' and e- wv/=I < respectively. When there are no other terms and when equations (76) are satisfied, the solution of (75) is periodic and z 2t , +1 has all the properties of z 2J _i specified in 125. But since u=N/n, where n and N integers, there is a value of v for which other terms of the type in question can arise. It follows from the properties of Z w+ i that m,Z 2 x+i and +^ 1 Z 2 i/ + i contain respectively the terms +a t coskt-\- V lb t smkt ], \ (78) -{-a t coskt+ Now e<*+<''V=i< = e" v:= * t [cos2vut + V ism2vut], Consequently terms of the type in question will arise if Ivu = k, k being an integer. Upon substituting the value of co , this relation becomes 2 m = kN, which can be satisfied when 2i> becomes a multiple of N. Suppose the integer n is odd. In this case when 2v N, the smallest k satisfying the relation, viz. k = n, is obtained. The term in which this occurs is multiplied by X 2 " +1/1 e" = \ N+l/> e n . But if n is even and N odd, the relation is satisfied first for increasing values of v when 2v = 2N, and then k = In. The term in which this relation occurs is multiplied by X 2 " + ' A e 2 " = X 2Ar+ ' A e 2 ". After terms of this type once appear they in general occur similarly at all subsequent steps of the integration. The coefficients of e Wv/Tri( and e~ a ^ = ^' obtained from (78) are respec- tively -C 2 x + i(O 2 * +1 an d +C*2^+i(c2 1) ) 2 " +1 , where C 2v+i is a constant multiplied by e" or e 2 " according as N is even or odd. Therefore, when these terms arise we have in place of equations (76) Consequently cj 2 """ and Cj 2 """ are determined in this case as well as in that in which the terms multiplied by C tv+l do not arise. This completes the proof of the possibility of constructing the solutions. OSCILLATING SATELLM !>. KI.L1PT1CAL CASE. 253 APPLICATION OF THE INTEGRAL. 127. Form of the Integral. Kquations (1) can be written in the form eeond degree in x, y, and z. It contains terms independent of X and others, for the particular transformation (11), multiplied by X to the first degree only. The coefficients in the series for (7 are power series in c whose coefficients, in turn, arc periodic in t with the period 2r, and which reduce to constants for t = 0. The first terms of U are seen from equations (10) and (11) to be -~\xy *! + ,wi - 3 ( , - a LT, 7, V| 't i- r ri + 3 ( 4,), - r (b.i)ecosf+ ' t \' i 'i / I (82) The integral of equations (81), analogous to the Jacobi integral in the case where U does not involve t explicitly, is dt+C, (83) where (dU/df) is the partial derivative of U with respect to / so far as t occurs explicitly, and not as it enters through x, y, and z. This partial derivative is zero for e equal to zero, and therefore it contains e as a factor. 254 PERIODIC ORBITS. 128. The Integral in Case of the Periodic Solutions. In the periodic solutions which have been considered, x, x, y, and y' are expansible as power series in X, while z and z' are expansible in odd powers of A*. It follows from property (a) of 106 that U is a power series in z\ Therefore when the expansions of all these variables are substituted in (83), the result is a power series in integral powers of X of the form F = F l \+F t \*+ +F V \*+ = y' 2v _ 2 , z l ,z{, . . . , z 2 ,-i, z-i Equation (85) therefore has the form /r)O jw JWMJ y^n Z 2j+i> t)dt = C v) (86) where P v and Q v are polynomials in the indicated arguments with t entering the coefficients in sines and cosines. The subscript j runs from to v 1. The derivatives enter (86) only in the form x^^x'^.^ , y' t) -ty' tv - ti , and z-i 2 2K-2j Suppose the general term of P v or Q r is x, x x; X', x" x",, x x"*?y j y*'2 * * * z " 2M, 2^ J 2p.\ V^, 2^+1 2^',,+l ' The exponents and subscripts satisfy the relation The partial derivative of Q n with respect to t is taken only so far as t enters explicitly in the coefficients of x zj , . . . , z lj+1 , but the integral must be computed for t entering both explicitly and also implicitly through the x t} It was shown in 125 that the x zj , the y 2J! and the z v+1 have the form ~ y ~<2*)/,2ito-v/=n'( ., _ y 7/ <2*>2tu\/=T< - _ y x tj z, x tj e , y tj LI y tj e , z tj+1 z, t--j j tj MX II, LATIN". -AIKLI.II K>. ELLIPTICAL CASE. 255 where the j^\ the //", and the 2,%" are power series in < whose coefficients are periodic with the period 2r. In j" and 2,7+V the eorHieients of the cosine terms are real, and those of the sine terms are purely imaginary : the opposite is true in y { "\ It follows from these projKTties and those of F f enumerated on page 254 that P r and dQ f /dt can \w written in the form /',- 2 *= 2 (88) where P"' and /C' are periodic with the period 2r. Since in U the coefficient > of odd j>owers of y are multiplied by sine series, it follows from the properties of the x } , the y\ and the z ( "" that in P" } the coefficients of the cosine terms are real, and those of the sine terms are purely imaginary; the opposite is true in R*\ The integrals coming from j -*' dt are of the types .' ../ Binjt> e- Therefore we have (89) C.K)) where the S are i>eriodic with the period 2*. Moreover, the coefficients of the cosine terms are real, and those of the sine terms arc purely imaginary. It follows from these properties that (85) can be written in the form F,= 2 L (91) where the f'"' are j)eriodic with the period 2*-. The coefficients of the cosine terms are real, and those of the sine terms are purely imaginary. If we let <**"=i* = ff > (92) and make use of the fact that the F"' are periodic with the period 2*, we get from (91) 2 / --r 2 I tm-f 2 1 lmf These equations and (91) can be satisfied only if either F, W -C, FJ" = ( k--, -l.+l or TI j_/~i fai} o\ -o r . (.yo; (94) 1 , = ^F^e ] =C v = i C v .,e>, J**0 *=0 F = S F e'=0 (k=-v, ...,-!,+!,..., +). J=0 Since these equations are identically satisfied in e, we have F <0) = C F (l> = (95^) 1 y.J ^v,)1 L v,S V. \.V=0 F* = S [<), cos pt + V^-T &. sin pt] = 0. P=0 Since these equations are identities in t } we have finally (9b) "- II.I.VI1M, >\IKI.I.IIKS. KU.II'TICAL CA-I 257 129. Determination of the Coefficients of z,,^ when r =0. Equations (96) are relations anionn the coefficients of the solutions and may be used for checking tho computations. The control is very effective because at each step all tho preceding coefficients are in general involved. But equations (96) can also be used, step by step, for the determination of the coefficients of the expansion for z when the coefficients of the expansions for x and y are determined, alternately with those for z, from the first two equations of (13). Before taking up the general problem we shall treat the case of e = 0. \\hcn this condition is >ati.-fied the integral becomes ( 97 ) " o _) + _Wi ' 'i 'i In this case w = \fA and equations (87) become +> +/ +/ Zjj = o^ t e^ , y t) = 2 p ti e^ > ~ , z tl ^. l = 2 where the o, a /', /3^', and y^ +l> are constants. We shall show how to compute the 7*+i" for successive values of j. Terms for j = Q. In this case, since x =sy = 0, equation (97) becomes, as a consequence of the last of (98), Since this equation is an identity in t, we get 4Ay[~" y\" = (\. Since C, is unknown, this equation imposes no relation on y[~ l) and -y"'. But since 2, (0) = 0, it follows that y[~" = 7"', and there remains the single unde- termined constant y. Terms in x and y for j=l. It follows from (13) and (14) that x, and y t are defined by the equations xZ-2 y ',-(l+2A}x,= \Bz\= \ -, (flg) The solutions of these equations, which have the period 2r/\/A) are , . 3B(7;") i 2(1 - (100) 258 PERIODIC ORBITS. Terms in z for j=l. It follows from (97) that the terms for j=l are Since z, has the form equation (101) gives rise to the relations Y=-^Ayryr, There is another equation which is useless for present purposes because it involves the unknown constant (7 2 . Since z 3 (0) = 0, we have also . (104) It follows from equations (103) and (104) that yi-*=-y?, 73- l) =-7 3 1) . (105) In order that 7j~ 3) shall be the same as determined by both equations of (103), we must impose the condition (I , V , ft K,_ n ) ^3-(7.) 0, which determines 7"' except as to sign. Then 73~ 3) and 73" are uniquely determined in terms of 7"', but 73" remains so far arbitrary. The problem for e = was treated in Chapter VI. If we compare equations (100) of the present work with equations (42) of page 211, we find that 7i lJ and c t are related by the equation Upon making use of this relation, it is seen that equations (106) of this chapter and (44) of Chapter VI are identical. Terms in x and y for j = 2. It follows from (13) and (14) that we have in this case /Y," O / * _ f 1 I O A 1 /* Q Z? ty /y I P a .1 , O /) _ _ /a 1 /7/ f) I 1 07 1 vi '~~ * i/4 ^^ I J- 1^ ^ -^-1 ^d *-* *-* ^l *3 1^ 4 J c/4 ^^ ^ L ^T- J t/4 ~~" V^j^ ^ ^ XVI I j where P 4 and Q 4 are entirely known periodic functions of t having the period T. We wish the details of the solutions of these equations only so OSCILLATING 8ATELLITKS, KLLIPTICAL C.\ 259 far a* they depend upon the undetermined constant >i"= y' t ~ l \ So far as these terms arc involved, the right member of the first equation is '*-"]. Hence the solutions of (107) arc (108) U = -f /=7ll _l_ 7J Vi. where ~f\ and ^> 4 are known periodic functions of /. '(N in z forj = 2. It follows from (97) that these terms are defined by 2z(z' t +2Az l z k =-2x',i'<-2y' 1 y'<+2(\+2A)x t x<+2(l-A)y t y< (109) where /j 1 , is a known periodic function of t. The expression for z, has the form (110) On substituting this expression in (109) and equating the coefficients of the several powers of VJV=T ', beginning with e~' >:ivrrT ', we get 4 y\"ylr" = known function of 7!" =-8 A of [ "/^ There is also an equation, coming from the terms independent of e^*^ 1 ', which involves the unknown T 4 and need not be written. The first equation uniquely defines 7j~ B , which equals 7,"'. In order that the second and third equations shall be consistent, we must impose the relation ?-7A + UA* + C ]W)'7;" = known function of y[. (112) Therefore 7!" is uniquely determined, since its coefficient in this equation is positive; and then the second or third of (111) defines 7}"", which is the negative of 7,"'. Now, on imposing the condition that z,(0)=0, we get 7j~"= y'", and 7"' remains undetermined. All succeeding steps are precisely similar to the one which has just been explained. The parts of the equations which contain undetermined coefficients differ from those of (111) and (112) only in the subscripts. 260 PERIODIC ORBITS. 130. Case when e ^ 0. General Equations for z t . It will now be shown that when the coefficients of the series for x and y are determined from equa- tions (13) and (14), the coefficients of the expansion for z can be determined from the integral (83) . We shall need the partial derivative of U with respect to t so far as this variable occurs explicitly. We find from (82) that -~\xy + Jz 2 + [jj(-- 2 +---, +z 0+0 zi-z[ m =0+0 -"C = +z z > 2l "-'> = o + (120) Equating to zero the coefficients of the various exponentials of (116), we get 3Aesmt+ -J[ r - \[3Aesint+ where the coefficients under the integral sign / must be transformed by equations (117) instead of forming the ordinary integrals, and where (7 t is an undetermined constant. These equations are power series in e, and setting their coefficients equal to zero we have equations (119). 131. Coefficients of e\ On referring to (120), we find for these terms ,._ _ - Uo lt0 = .LQ ) , 01:fl - l<9 , 01>0 li0 Since 2,(0) =0, we must add to these equations z{^ l) + 2j" = 0. It follows from these equations that ul = A, 2, ( ro"=-C, 4A(C) = C- llQ , (122) and z"o = ai!o,o remains as yet undetermined since C",, is an unknown constant. OSCILLATING SATELLITES, ELLIPTICAL CASE. 263 132. Coefficients of e. On referring to (120), (117), and (115), we get at this step rj- [ _;- sin<+ v^Ttf:,!', cos<] = -3( 2 ;: e ") ,i Jr", CGS/+ v^T/S^ cos*] -{.[ -a^si -2 v^ T^:,", cos<]} =lft? t sini] J (123) Since these equations are identities in t, we find, after making use of equa- tions (122), that 0Ilfl . t , 1.1 ' +2 A i.T.V - 6 A (f)', 264 PERIODIC ORBITS. From the first two sets of these equations we get _ 4 2 > The first of the last three equations imposes no condition upon the unknown coefficients since it involves the undetermined C lfl , and the second and third equations of the last set become identities. The coefficients a^," and u,o> which are still undetermined, are not involved in these equations. The fact that , is zero was known in advance, for it was proved in 120 that w is a series in even powers of e. It has also been shown that z{~ l) and z", aside from constant factors, differ only in the sign of Vi. Since 2(0) =0 these constant factors differ only in sign, from which it follows that a^^ differs from a[" Jip only in sign, while /3^, l j, and $", are equal for all j and p. Applying the condition z(0) = to the terms under consideration at present, we have z u (0) + z"} (0) = 0. On making use of (124), this equation leads to the result olS =-&,, (125) and a{" t0 alone remains undetermined at this step. Of course, it should be noted that aj" fl and /3{" , t are expressed linearly in terms of the undetermined constant z"i = a"o, , whose value will be fixed when we treat the coefficient of X 2 in the integral. 133. Coefficients of e- and e*. From the consideration of this step we can infer the character of the process in general. Because of the relations between a^,", p[~" f and a{" t ,, $", it is sufficient to equate to zero the coefficient of &>^*' in (116). Since we are interested only in the possi- bility of determining the unknown coefficients, it will be sufficient to write out the equations explicitly only so far as they involve these unknowns. Upon equating to zero the terms independent of t and the coefficients of cos<, cos2<, sint, and sin2i in order in the coefficient of e 2 , we find =o - Pi,2,i J/1,2,U -'l.ol,I,Z ),2,*> /. no\ (1) n. (1) = f co> - 4 \/~A z a) a 11 ' f <0> yi a li2il j Ii2il , v A z li0 a 1>2 2 7 1,2,2 , J where / 0) is known and where {/{",, , . . . , /[ 22 are homogeneous functions of the second degree in z ( " = a"o, and aj" , and are linear in a"J alone. The first equation uniquely determines w 2 ; the remainder determine /3" 2 , , . . . , a" 2 , 2 uniquely when z and a['| become known, as they do when the coefficient of X 2 is considered. The coefficient a" 2 . is so far entirely arbitrary. The coefficients of e k lead to similar equations. The first has co t in place of co 2 , and the left members of the remainder involve j8"i,,, . . . , $",, o"i.!, . . . , a"i, t , the numerical coefficient of j8"i ip and a"i, p being 2p\/JzJ'o . The right members are homogeneous second-degree functions of a"o, , . . . , a^_ li0 . The a"i, remains arbitrary. OSCILLATING SATELLITES, ELLIPTICAL CASE. 265 134. General Equations for v= 1. The coefficients of z, are determined from F t = 0. From (82), (83), and (113) we find explicitly that 2z' l z' 3 +2[A+3Aecost+ ]z l z>=-(x' t ) t -(y',)'+[l+2A+GAecost+ *!+2j[ 3Aea\nl+ -2 -2 j [ J [34esin< + ]z l z,dt+ f [3(pi-p (127) The x t and j/, are determined from equations (55), and it is seen from these equations that they are homogeneous of the second degree in the coefficients of z,. Other properties of the solutions are given in 121, among which is that they are of even degree in e" N/=T< and e~ aV=ril . The expression for z, has the form z^zi^e-^^'+zi-V-^'+^e-^^'+zfe*^^ 1 '. (128) The coefficients of zj~" and zj~ differ from those of zf and z"' respectively, aside from constant factors, only in the sign of V 1, and these constant factors differ only in sign. Hence it is sufficient to determine the coeffi- cients of z"' and z*, which have the form t-o f-a m I y, o ^ (129) 135. Terms Independent of e. Equations (128) and (129) are to be substituted in (127) and the coefficients of g**^ 7 ' and e***^' set equal to zero. These terms are power series in e whose coefficients separately must be set equal to zero. We are now interested in the terms which are inde- pendent of e. These results were worked out in 129, where the parts of x, and y t independent of e were derived. The explicit results were given in equations (103), the relations in the present notations being .(1) _ ..(1) _0> _ /v < it > i.^ Ti 266 PERIODIC ORBITS. The condition for the consistency of the two expressions for 7"' is equation (106), which determines 7"' = Co except as to sign. Therefore, by 132, a"}.! and $",! are also determined except as to sign, and d, is defined. And by 133 it is seen that the C, , $" >p (p^O) are all determined except as to sign, while the aj", remain as yet undetermined. The equations corre- sponding to (103) determine 73 3) uniquely, but 7," remains so far arbitrary. 136. Coefficients of e. We shall write explicitly only the terms which involve those coefficients a ( ^, v , /8jJi. p , d,> and ^ tij> which, at the successive steps, are unknown. The quantity z s is involved in the right member of (127) under the integral sign, but since this term is multiplied by e, it introduces at this step only C. as an unknown coefficient, and this unde- termined constant enters linearly. It is determined from the terms which are independent of e when v = 2. Consider first the parts of the coefficients of e* aV=It and e 4a)N/=IT( which are independent of sint and cos<. We find from (127) that ~ 2A a}" a,*},,, = 2^4 C a3 3 00 +/3'J , (130) where f ( t ~$ and /}. are linear functions of C , which is the only un- known that they involve. Since C" = C, , the condition that equations (130) shall be consistent is a condition on their right members which a detailed discussion shows uniquely determines the coefficient C - Then equations (130) uniquely define a t0 . Now we set equal to zero the coefficients of e 2a)v/:rT ' sint, e* uV= *' sint, & uV=It cost, and e twv ^'cost. The explicit expressions are found from equa- tions (127) to be, respectively, VI [+C. > & + 4 VA C j VA [ - C & ~ 2 VJ C u , where ft , . . . ,

which enters linearly. This constant remains undetermined until the equations are derived for v 1. The unknowns in the left members of (131) are af^, f}.u d> an d /3j" >u which enter linearly. On making use of the fact that C" = a"o, , the determinant of their coefficients becomes A=-4'(C) 4 [4A-1], (132 ) which is distinct from zero. Therefore these quantities are uniquely deter- mined as linear functions of the arbitrary C, This illustrates sufficiently the method of determining the coefficients from the integral. The complexity of the details makes it unprofitable to carry the explicit results further. OSCILLATING SATELLITES, ELLIPTICAL CASE. 267 DIRECT CONSTRUCTION OF THE TWO-DIMENSIONAL SYMMETRICAL PERIODIC SOLUTIONS. 137. Terms in X v> . It was shown in 116 that the periodic solutions i \i-t and are expansible as power series in X v> . It is found from equations (13) that the terms of the first degree in X 1 ' 1 are defined by x"-2y(-[l+2A+6Aecost+ ]x l -[6Aesmt+ y'+2x f l -[6Aeiant + }x l -[l-A-3Aecost+ (133) The general solutions of equations (133) are known from the general theory to have the form (134) x, = a y, =a a =a e + (t-1 ,4), where a"', . . . , a"' are arbitrary constants, and where the u t and the v t are periodic functions of t with the period 2r. In order that the solutions shall be periodic we must first impose the conditions ai u =ai u = 0. (136) The constant a is a continuous function of n, /u,, and e. It will be supposed that these parameters have such values that a is a rational number. Then, since u\" and t> 4 0) are periodic with the period 2*-, the solution at this step is periodic with the period T, where T is a multiple of 2r and 2* / ( = in case of periodic orbits), [e pt u, - i= +< [e'^'w,- e-"^' v t ~\+dS i [ef*v 1 -e'" 1 vj . (136) We shall suppose that the initial conditions are real as well as such as to give the symmetrical orbits. Then, since changing the sign of V 1 in (133) does not alter these equations, with the same initial conditions the solutions will be identical with (136). Therefore we have If we change the sign of both t and y l , equations (133) are unaltered. With the same initial conditions as before, which this transformation does not affect, since ?/i(0) = 0, we have an identical solution except that y l is changed in sign. Therefore Now if Vl, t, and y^ are changed in sign the differential equations are unchanged, and hence it follows that u 4 ( Vl, f) = u 3 (Vi,f); v 4 ( Vi, t) = +v 3 (Vi,f). It follows from the last three sets of relations that u 4 , . . . , u t v lt . . . , v t , when expressed as Fourier series, have the form M, = S [ + a, cos jt + V 1 bj sinjt] , u 3 = S [ + c } cosjt + dj sin jt] , u t = 2 [ a,j cos jt + V 1 bj sin jt] , u 4 = S [ c, cosjt + dj sin jt] , Vi = S [ + V 1 a y cosjt + ft sin jt] , v 3 = 2 [ + y, cosjt -f- 5, sin $] , v i = S [ + V 1 a ; cos.# ft sin jt] , v 4 = S [ + 7, cosjY 5j sin j<] , , (137) where the a/, fy, c^, d ; , a/, ft, 7,, and 5, are real constants and power series in e. OSCILLATING SATELLITKS, KLLIPT1CAL CASE. In the case of the |>eriodic orbits we have simply L'li'.t (138) In the ease of the periodic orbits it follows from equations (137) that the numerical coefficients of the cosine terms in j, are real, and that those of the sine terms are purely imaginary; and the opposite is true in y,. 138. Coefficients of X. It is found from equations (13) that these coefficients are defined by -]:z;J+[-24flesin<-f -]zJ+[3fi+12Becos<+ + [9Besmt+ .. ' (140) The character of the solutions of the equations of the type to which (139) belongs was determined in 30, where it was shown that they consist of the complementary function plus terms of the same character as X t and y, . Hence the periodic solution of (139) is x t = t/, = (141) where aj l> and o^*' are constants which are as yet undetermined, and where /, and 0, are the particular solutions. We shall need certain properties of /, and g t . It is evident from (139) and (140) that they are homogeneous of the second degree in aj" and in e*^^ 7 ' and e~ 9V=l1 . It follows from (137) and (138) that in xj and j/J the coefficients of those cosine terms which are multiplied by e**^ 7 ' and e~" v=:T ' are real and identical, while the coefficients of the sine terms are purely imaginary and differ only in sign. The terms which are independent of e*^^ 1 ' and e~ rv:rT ' consist only of cosines whose coefficients are real. In the product x, y l the coefficients of those cosine terms which are multiplied by e"*^' are purely imaginary and differ from the coefficients of those cosine terms which are multiplied by e~ tfV={t only in sign, while the coefficients of 270 PERIODIC ORBITS. the sine terms are real and identical. The terms which are independent of e W)' are the derivatives of a{ 3> , . . . , a"' with respect tot. The solutions of these equations for (af)'j , WY are A(a[ 3 ')' = [D n X t +D a Y 3 ] e'"^ , A(a')' = [A, e'" 1 , (148) where A = MI o pu 3 +u' 3 , V 3 , pV 3 + V' 3 , ~ o-xA^T Vt+v'z, pv 3 +v' 3 , , u 3 , V 3 , V t w^, pu 3 +u' 3 , pu,i+u( <>-' II. I. \ I IN', -\IKI.t.llO. Kl.l.ll'l li'AI. < \ 273 l> and l> are obtained from I) ., and I). . respectively, by changing the sub- script '_' to 1 and by changing the sign of \ ] and of the whole 1 expression: />j, and D K are obtained from l) n and ]),,, resjx>ctively, by changing the subscript .'{ to 1 . p to a \ - i . and by changing the sinn of the whole expre sion: and /> and />.,are obtained from I), and I),., respectively, by changing the subscript I to 1, ptoff\ I . and by changing the sign of the whole expression. It follows from the di.-cu-Mon of is that A is a constant, and in this case it is a power series in . In order that the solution -hall be periodic it is necessary that the right members of (148) shall contain no constant terms. \Ye shall show these conditions are sufficient. \Yhen they are satisfied the general term of the riuht member of either of the first two equations has the form [a,. cosjt+bjj slnjt] e**"^^', where j and /. are integers distinct from xero and where n, k and b JL are con- stant-. < '.-iiseijuently a? and a* are sums of terms of the t \ pe V^itAnjt] ^ijacf (149) The right member of the third equation of (148) never has any terms which are independent of I, but contains terms of the type .os/7 rAtriiyf [(fUeri<)dic with the jjeriod T. They may be written in the form \\here aj" and aj 11 arc constants which so far are undetermined. 274 PERIODIC ORBITS. It remains to show that a"' can be so determined that the right members of the first two equations of (148) shall have no constant terms. Let us consider the first of these equations. We are to set equal to zero the con- stant part of the coefficient of e" v ^' in D n X 3 +D a Y 3 . It follows, from the form of X 3 and Y 3 , equations (145), that the term which must be made to vanish does not depend on af\ It also follows that the conditional equa- tion which must be imposed has the form where P l and Qi are power series in e, the former coming from those terms of X 3 and F 3 which are linear in x^ and y^ and independent of x 2 and y t , and the latter coming from those terms which are of the third degree in x l and y l , or which involve x 2 and y 2 . The solutions of (152) are a{" = 0, which leads us to the trivial result x=y=Q, and In (153) The significance of the double sign was discussed in 116-118 in connection with the existence of the solutions. The expressions for P t and Qi are power series in e and both of them contain terms independent of e, as was shown in Chapter VI in the discussion of the corresponding problem for e = 0. Therefore crj l) is a power series in e having an absolute term. It remains to be shown that this value of a also satisfies the equation which is obtained when the constant term of the right member of the second equation of (148) is set equal to zero. We shall show that the constant part of the coefficient of e av ^' in D n X, + D 12 Y 3 is identical with the con- stant part of the coefficient of e~ av =~- 1 in Z) 21 X 3 +D M Y 3 . Let us first consider the term [ 2K QKecost-}- ]x l of X t which contributes to P l of equa- tion (152). So far as this term is concerned, we have (154) On referring to (138) and the values of D n and D 21 , we see that the constant parts of the right members of these two equations are respectively the constant parts of -a u. -< , v, , pV t + V f ,, -pt> 4 + t>4 -pv t +v' 4 [-2K-6Kecost-\ [-2K-6Kecost-\ (i>( II. I. \ IIM. >\IKI.I.m.- M.I.II'IK \L CASE. These expressions arc equivalent to 275 MI' t>,-t> 4 whore The parts of these expressions containing M,M, as a factor are identical ami need no further consideration. The parts multiplied by /,-, and u,w lf so far as they apjwar in the second lines of the determinants, arc respectively [-2A'-6A'ccosH J, _(!> ~2~ ?t,-U, [-2/C-6A'ecos,, so far as they come from the second lines of the determinants, are identical. The parts of the expressions which contain u,t>, and ,t>, , so far as they come from the third lines of the determinants, are respectively tt,-U 4 V, ~ The determinant is in this case a sum of cosine terms. Therefore we need only the cosine terms from -fttif, and u^r, . It is seen from (137) that 276 PERIODIC ORBITS. they are identical Therefore the constant parts of the two expressions, so far as they arise in this manner, are identical. It remains to consider only the constant parts of the two functions - *>4 U 3 -U 4 , [-2K-6Kecost+ [-2K-6Kecost+ It follows, as before, that we need only the cosine terms of wX an( i We see from (137) that the coefficients of the cosine terms in these products are identical. Therefore the constant parts of the right members of equa- tions (154) are identical. The discussion for the other terms of X 3 and Y 3 which are linear in x l and ?/! is made in a similar manner, and it is thus proved that the P l which is obtained from the second equation of (148) is identical with the one which depends on the first. It is now necessary to consider those terms of X 3 and Y 3 which are not linear in x l and y l . Let us treat in detail the term in X 3 which contains x l x t as a factor. So far as this term is concerned, the first two equations of (148) become ] -oV if (155) On referring to equations (138), (144), and the expressions forZ) n andD 21 , it is seen that the constant parts of the right members of these equations are respectively identical with the constant parts of U 3 v't, pV 3 +V 3 , - *>4 -K') 3 M! 1*4 4 pv 3 +v' a , - (-GB-24Becost+ Since f ( is a cosine series, and the product of it and [ 65 24 Be cos t-\- ] is also a cosine series, the discussion for the terms multiplied by Wj /2 0) and Ut /i 0) does not differ from that given above for the terms multiplied by x^ . OSCILLATING SATELLI'l 1>. KLLIPTICAL CASE. \\ 'c have now to find the constant parts of K')' a K')' 2 277 -v' t , p(v t -v t )+v' t +v' t i.-,<; where /',(*) = [-6B-24Becost+ ]. The factors by which ti^/f and MJ/^ C are multiplied in these respec- tive expressions are identical, and it follows from equation (137) that they are a sum of cosine terms having real coefficients. Consequently we need only the cosine terms of wj f* and wj/i" 1 ' in order to obtain the constant parts of (156). Now it follows from (137) and (143) that the cosine terms of the products wj/i" and w?/i~ B are identical. Therefore the constant parts of (156), which involve H| and u\ as factors, are identical. Now consider r, and v r in so far as they occur in the second lines of the determinants. It follows from (137) that the factors by which w,r, /" and J^fi/i"*' must be multiplied are sine series having real coefficients. Therefore we need only the sine terms in these products. It also follows from (137) that the expressions for n^v t and w,v, are respectively cosine terms with purely imaginary coefficients which differ only in sign, and sine terms with real coefficients which are identical. Therefore, in the products MjVj/J" and u^vj^ the coefficients of the sine terms are real and respec- tively equal. There remain only the terms coming from the third line and first column of the determinants. We have first aV 1 w,t; t /, U) and -f vV 1 u t vj^. These expressions are multiplied into the same cosine series having real coefficients. Consequently we need compare only the coefficients of their nine terms, which we find from (137) and (143) are real and respectively identical. Therefore the constant parts of the right members of (155) to which these terms give rise are identical. Finally, there remain only the terms multiplied by +M,t>i/J n and by + w X/i~ l) respectively. The term into which these factors are multiplied is a cosine series having real coefficients. It is seen from (137) and (143) that the coefficients of the cosine terms of +t/X/, a> and -fwX/~ c are real and respectively identical. Therefore the constant parts of the right members of (155) are altogether identical. The discussions for all the other terms of A', and 1', which involve x, or t/, are made in a similar manner and lead to the same result. There remain only terms in X t and Y t which are of the third degree in a;, and y t . 278 PERIODIC ORBITS. Let us consider, for example, the term of A' 3 which is multiplied by x^y}. Then, so far as this term alone is concerned, the first two equations of (148) become 3 '' = - . . .]x,rfr'^',l = The constant parts of the right members of these equations are respectively the constant parts of U 3 , v, pv 3 +v' 3> - u t pv 3 +v' 3 , pv t +v' t [-6CH (158) Since v^v t is a cosine series having real coefficients, the discussion for the the terms multiplied by 2M,?^ 2 and 2u 2 v 1 v t does not differ from that given above for that part of X 3 which is multiplied simply by #, . If we refer to equations (137) and (138), we see that v\ and v\ have the properties of ff and / 2 ~ 2> , as regards the relations existing between their respective coefficients. Therefore the discussion of these terms of (158) is identical with that of (156), for which the proposition was established. In a manner similar to this the identity of the constant parts of the right members of the first two equations of (148) can be established for all of the elements of which X 3 and F 3 are composed. 140. General Proof that the Constant Parts of the Right Members of the First two Equations of (148) are Identical. We shall treat first the parts which depend on X 3 . We shall need the following properties of X 3 : (1) It is a polynomial in x 1} y lt z 2 , y 2 . (2) Those terms which are of even degree in y l and ?/ 2 taken together are multiplied by cosine series having real coefficients. (3) Those terms which are of odd degree in y l and t/ 2 taken together are multiplied by sine series having real coefficients. (4) If the general term is XiX^y^y^-, then j i +2j 2 -}-k l -}-2k, is an odd integer (in the present case one or three). The parts of the first two equations of (148) which depend on X 3 a,re It is obvious from (137) and properties (2) and (3) that those parts of X 3 e~ av/ ~' and X 3 e ffv ~' which are independent of the exponentials e~ av ^' and e**'^ 7 ' are sums of cosines having real coefficients and of sines having purely imaginary coefficients, and that the real coefficients in the two ex- OSCILLATING SATELLITES, ELLIPTICAL CASE. 279 pn -sinus differ respectively only in sign, while the imaginary coefficients are respectively identical. Hence, referring to the expressions for /.),, :iml I) . we may write these parts of equations (159) in the form r, U, t>,-t> 4 ', P(v*+v 4 )+v',-v'< , 2[A,cosjt U,-U 4 v{, P(v t +v 4 )+v t -v' 4l It easily follows from these expressions, as in the discussion in 139, that their constant parts are real and identical. Now consider the terms depending on Y tl which has the properties (1) and (4) belonging to A", , and (2) Those terms which are of even degree in y l and y t taken together are multiplied by a sine series having real coefficients. (3) Those terms which are of odd degree in y l and y t taken together are multiplied by a cosine series having real coefficients. The parts of the first two equations of (148) which depend on F, are It follows from (137) and properties (2) and (3) that those parts of F, g-v=ri and F, e* yrr7 ' which are independent of the exponentials e~* vrrT ' and e 9 ^^' are sums of cosine terms having purely imaginary coefficients, and of sine terms having real coefficients, and that the purely imaginary coefficients are respectively identical while the real coefficients differ respectively only in sign. Hence, using the explicit values of D a and D a , these parts of the right members of (160) are found to have the form w,-u 4 v t -v 4 ^lAi cosjt +B,smjt], B,8injt]. It follows from (137) that the constant parts of these two expressions are real and identical. Therefore the constant parts of the right members of the firsl two equations of (148) are identical, and when one of them is made to vanish by a special determination o/a|" the other one also vanishes. 280 PERIODIC ORBITS. 141. Form of the Periodic Solution of the Coefficients of X Vl . It follows from the form of X 3 and Y 3 , given in equations (145), that /, and g 3 of equa- tion (151) have the form g? g ,,(0) (-3) -3 5 t/ (161) J where /"' , . . . , = 2 cosjt+0 sinjt], cosjt+ff smjt], sin^], / 3 - 1) =2[-v^T< ) cosj7+^ -,(0) (162) /i- B -2(ofcosj<-V=i /< =2 o'cos^, where the af , . . . , /3j" are real constants. It follows from equations (161) and (162) that \IKU.m>. Kl.l.IITK-AI, f\ where \' 4 =+[-2K-(\Kcco8t+ ]*,+[ -(iAYsin< + I'M [-24fiesin<+ ] [^ [--j/H-ttlfeco8H- [ZB+2Beco*t+ (165) \\ It- -re X 4 and Y 4 are independent of x, and j/, and linear in x, and j/, . In .V, tin in ins which are of even degree in y l and j/ t are multiplied by cosine M-rirs having real coefficients, while those which are of odd degree in y l and y, are multiplied by sine series having real coefficients. In the case of Y t tin- cosine series and sine series are interchanged. If z{' xj xj yj' j/J' yj 1 is tin- -i ncral term in X 4 or 1' 4 , then j t +2j t +3j, + k l + 2k t +3k t = 4 or 2. \Yhon the right members of (164) are set equal to zero, the general solution of the equations is where a\ *> aj 4 * are arbitrary constants. Now, on varying them and subjecting them to the conditions that (164), including the right members, >li:ill bo satisfied, we find A ) f -[fl,, A' 4 + D a 1'J A A whore D ; , are the same as in 139. \rrr--ury conditions that the solution shall be periodic at this stoj) are that the constant terms in the right members of the first two equations of ! KIT ' >hull IK> zero. It follows from the form of A' 4 and Y, , as given in (165), :iiid from their properties, that these constant terms are independent of .nd involve n\-' linearly. Therefore the condition that the constant trim of the right member of the first equation shall be zero has the form (168) 282 PERIODIC ORBITS. where P 2 and Q 2 are power series in e. It was shown in Chapter VI, in the treatment of the case where e = 0, that P 2 has a term independent of e which is distinct from zero. Therefore for \e\ sufficiently small a is uniquely determined by (168) as a power series in e. The equation obtained by setting the constant part of the right member of the second equation of (167) equal to zero is of the same form as (168); it is, in fact, identical with (168), as will now be shown. It follows from the properties of X t and Y 4 that the parts of the right members of the first two equations of (167) which are independent of the exponentials e - e -2 ffv /=l +a*f* + +4 -4 (170) where the /"' , g\" , /" } , and g[ 1} have properties exactly analogous to those of equations (162). 143. Induction to the General Term of the Solution. We shall suppose the x l _,; ?/, y n -i have been computed and that their coefficients have all been determined except a,"" 2 ' and aj"~", which enter in x n -2 , y n -2 , x n -i , and i/ n _, in the form =* v.-e^^^' v t ] (171) OSCILLATING SATELLITES, ELLIPTICAL C.\ 283 \\V shall suppose that x, and y, (p= 1, . . . , n-1) have the properties x, = 2 /y'e"^ 7 ', y, = 2' 0i" (172) 2[+v'=7a,*-' ) co8-<+#r-"8in>/i] The differential equations which define x n and y. are seen from (13) and (14) to be ]z.-[6v4esin<+ -]y. = X., y'+2x n -[6Ae smt+ where (173) +[-3B-12Becos(+ x n _ t +(K+3Kecost+ (174) - ' ' ][2y l y._ 1 +2y 1 y._,]+r 4 . The functions X, and Y n do not involve z._i or y._i , and are linear in z._ t and y._,. In X. the terms which are of even degree in y, , . . . , y,_, are multiplied by cosine series having real coefficients, while those which are of odd degree in y, , . . . , y._ are multiplied by sine series having real coefficients. In the case of Y m the cosine series and sine series are inter- changed. Ifz{' zi'-V yf 1 yi"_V is the general term of X, or Y u , then ^i+2j,+ +(n l)j,_,-}-A^4-2^+ : +(n !)&_, = n or n-2. (175) Necessary conditions that the solutions of (173) shall be periodic are that the right members of shall contain no constant terms. It follows from (174) that these constant terms are independent of a""" and involve aj"~" linearly. The coefficient of a|"~" is distinct from zero for |e| sufficiently small, for in Chapter VI it 284 PERIODIC ORBITS. was seen to be distinct from zero for e equal to zero. Therefore af" 2 ' is uniquely determined as a power series in e by setting the constant term of the right member of the first equation of (176) equal to zero. It can be shown, precisely as in the discussion when w = 4, that the constant parts of the right members of equations (176) are identical. There- fore af~ 2> is uniquely determined by the conditions that the solutions of (173) shall be periodic. It follows from the properties of x i} . . . , x n ^; y 1 , . . . , y n -i, and from (175), that when these conditions are satisfied the solution of (173) has the form = /,<) aV= {"- 1) [g o-Pt u + S j=-n + (177) (n) aj" are undetermined, and where the /"', /),~ 2> , /1 0) , where a[ B) , . . . ^. a) , On"", 9?,f", and ^' have the properties of (172). In order that (177) shall be periodic it is necessary and sufficient to impose the conditions a = a% t> = 0. Then it follows, from the properties of !, that o ,, y that y. (0) - of' - < = 0. Since y. (0) = 0, it follows = d?\ and equations (177) become x n = (-2) g-2 + + S f" I " J n + 2 (178) and equations (172) are satisfied for p = n. In picking out the constant part of the right member of the first equation of (176), in general only those terms in X n and Y n which contain e a ^ /=lt as a factor to the first degree will be used. But because a is a rational number there will eventually occur, in the higher powers of the exponentials, multiples of a which are integers, and constant terms in the right member of the first of (176) may occur from these terms, but their presence does not prevent the determination of the constants so that the solutions shall be periodic. After such terms once appear, they occur in general at each succeeding step of the integration. CHAPTER VIII. THE STRAIGHT- LINE SOLUTIONS OF THE PROBLEM OF H BODIES. 144. Statement of Problem. In his prize memoir* on the problem of time bodies, Lagrange showed that, for any three finite masses mutually attracting one another according to the Newtonian law, there are four distinct configurations such that, under proper initial projections, the radios of the mutual distances remain constant. The bodies describe similar conic sections with respect to the center of mass of the system, the simplest case being that in which the orbits are circular. In three of the four solutions the masses lie always in a straight line, and in the fourth they remain at the vertices of an equilateral triangle. This memoir is one of the most elegant written by Lagrange, and its mathematical form does not seem capable of improvement. But the method which he employed can not lie extended conveniently to the case of more than three bodies, and it has not led to practical results in applied celestial mechanics. This chapter is devoted to the solution of two closely related problems: I. The number of straight-line solutions is found for n arbitrary positive masses; that is, the ratios of the distances are determined so that under proper initial projections the bodies will always remain collinear. This is the generalization of Lagrange's straight-line solutions to the problem of n bodies. For each straight-line solution of n finite masses there aro n-fl points of libration near which there are oscillating satellite orbits of the types treated in Chapters V VII. Therefore the results of this chapter lead to generalizations of those of the preceding three chapters. II. The problem is solved of determining, when possible, n masses such that, if they are placed at n arbitrary collinear points, they will, under proper initial projection, always remain in a straight line. The first problem, in a somewhat different form, has been considered b\ I>ehmann-Filhes.t The method of treatment adopted heref, though originally developed independently, has considerable in common with that of Lehmann-Filhes, and the discussion completes in certain essential respects the demonstration of the German astronomer. It is shown that whatever real positive values the masses may have, and whatever the rate of their revolution, there are $n! real collinear solutions. Lagrange's Collected Work*, vol. VI, pp. 229-324. TuMrand'i Mttaniqve CtietU, vol. I, chap. 8. \Atlronomitche Nachrichten, vol. CXXVII (1891). No. 3033. JRead before the Chicago Section of the American .Mathematical Society, December 28, 1900; abstract in Bull. Am. Math. Nor., vol. VII (1900-1901), p. 249. tu lN(i I'KIIIODIC Ill MIC second problem it is proved that when the number of arbitrarily chosen real collinear points is even, the n masses are, in general, uniquely determined by the condition thai it, shall be possible to place them at these points and to give them initial projections so that they will always remain collinear and revolve in orbits of specified linear dimensions. Or, if it is preferred, the period of revolution can be taken as the arbitrary in place of the linear dimensions of the orbit. In general, the masses will not all be positive, and therefore the problem will not, always have a physical inter- pretation. When I he number of points is odd, it is not possible to determine the masses so as to satisfy the solution conditions unless the coordinates of the points themselves satisfy one algebraic equation. \Yhen they satisfy this condition, any one of the masses may be chosen arbitrarily, after which all of the others are, in general, uniquely determined. I. DETERMINATION OF THE POSITIONS WHEN THE MASSES ARE GIVEN. 145. The Equations Defining the Solutions. Let the origin of coordi- nates be taken a! the center of gravity of the system, which will be supposed to bo at rest. This point and the line of initial projection of any mass determine a plane. All the other masses must be projected in this plane, for otherwise they would not be collinear at the end of the lirst element of time. All the bodies being initially in a line and projected in the same plane, I hey will always remain in this plane. Consequently, if solutions exist in which the n masses arc always in a straight line, the orbits arc plane curves. Let the plane of the motion be the 7; plane. Let the masses be de- noted by m n m,, . . . , m, and their respective coordinates by (,, TJ,), (&> *?) (> '?) Then, choosing the unite so that the Gaussian constant is unity, the differential equations of motion are ? "jj J // '"*i ^ "jT c > r V d T >?( KM'.III LINK x.1.1 IK. N> H)l< BODIKB. 287 Iii ca-e tin- n bodie- remain collinear, the line of the n-ultaiit acceleration to which each one is -ubject alwav- pa--e- through the origin. 'I 'herefore. in collinear solution- it follows from the law of ureas that, for each body separately. Hut when the bodie- remain collinear we have also (z.-z > ) | >~\ i\* ** 288 PERIODIC ORBITS. Suppose the notation is so chosen that in any solution x l < < x n ; then the terms of the left member of the last equation are all positive. Since the origin is at the center of gravity, x n is positive, and therefore a" is positive in all real solutions. For every set of values of x 1 , . . . , x n satisfying equations (6) the solutions of (5) are the same for all values of i, and these solutions substituted in (4) give the coordinates in the collinear configurations. Since equations (5) have the same form as the differential equations in the two-body problem, it follows that in the collinear solutions the orbits are always similar conic sections. In case the orbits are ellipses, the coefficient of /r 3 and rj/r 3 is the product of the cube of the major semi-axis of the orbit and the square of the mean angular speed of revolution. If the undetermined scale factor be so chosen that x, is the major semi-axis of the orbit of m t , the mean angular velocity of revolution of the system is w. The hypothesis is made that or and m l , . . . , m n are real positive numbers, and the problem is to find the number of real solutions of (6) for any value of n. For each of these solutions there is a six-fold infinity of collinear configurations, the six arbitraries being the two which define the plane of motion, the one which defines the orientation of the orbits in their plane, the one which determines the epochs at which the bodies pass their apses, the one which determines the scale of the system, and, finally, the eccentricity of the orbits. 146. Outline of the Method of Solution. The method of solution involves a mathematical induction and consists of the following steps: Assumption (A). It is assumed that for n = v the number of real solutions of (6) for x^ , . . . , x v is N v , whatever real positive values or and m, , . . . , m v may have. It is known from the work of Lagrange that when v 3 the number is N s = 3 = f 3 ! . Theorem (B). If to the system w, ,...,? of positive masses an infinitesimal mass m v+l be added, then the whole number of real solutions Theorem (C). As the infinitesimal mass m p+l increases continuously to any finite positive value whatever, the total number of real solutions remains precisely (v+l)N v . Conclusion (D). From successive applications of theorems (B) and (C) it follows that the number of real solutions of (6) for n = v+n is Since ^3 = ^ 3! , it follows that N 3+lt = % Gu-j-3)! . Let /*+ 3==n and we have #. = }n!. (7) To complete the demonstration of this conclusion it remains only to prove theorems (B) and (C). -I K\K,H1-I,INK Sill. I | KINS FOR H BOD1KS. L'S'.t 147. Proof of Theorem (#). -\Yhcn there are v finite bodies m,, . . . , m, and the infinitesimal body j, M , equations (6) become M.r+l +0+ (8) r.r+l The last column of these equations is zero because m r+1 = (is infini- tesimal). Consequently the first v equations, which involve x, , . . . , x f alone as unknowns, are the equations defining the solutions when n = v. By (A), it is assumed that there are precisely N, real solutions of these equations. Let any one of these solutions be x, = xj 01 , . . . , x, = a; 1 . Then the last equation of (8) becomes t.H-1 *r+l 'r.r+I The number of real solutions of this equation is required. Consider r+1 is finite and continuous except at x f+l = xf\ . . . , x*\ +00, w, it follows that there is an odd number of real solutions in each of the intervals - oo to x, where x is the smallest x, xf to x{ 0> , where x 1 and x are any two x ( f which are adjacent, and x*' to +00, where x* is the largest x"'. But we find from (9) that ax r+l 'l.r+l which is negative except at x, +1 = x, , . . . , x^^x, , where it is infinite. Therefore +l of these intervals, there are, for each real solution of the first v equations of (8), precisely v + 1 real solutions of the last equation of (8). Since the first v equations have, by hypothesis (A), N p real solu- tions, equations (8) altogether have precisely (v-\-l)N v real solutions. This completes the demonstration of Theorem (B). 148. Proof of Theorem (C). Let Xj=xf (j=l, . . . , v + l] be any one of the (v-\-\)N v real solutions of equations (8) which are known to exist when m v+l = 0. It will be shown that as m, +l increases continuously to any finite positive quantity whatever, the xf can be made to change continu- ously so as always to satisfy equations (8), and that they remain distinct, finite, and real. From this it will follow that there are at least (v-\-l)N v real solutions of (8) for every set of finite positive values of m l , . . . , m v+l . It will also be shown that no new solutions can appear as r >+1 increases from zero to any finite value. Consequently, it will follow that the number of real solutions of (8) is exactly (v + 1)N V for all finite positive values of the masses m, , . . . , m v+l . The roots of algebraic equations are continuous functions of the coef- ficients of the equations so long as the roots are finite and the equations do not have indeterminate forms. Consequently, the xf are continuous functions of m v+l if no xf becomes infinite and if no xf = xf . The real roots of algebraic equations having real coefficients can disappear only by passing to infinitjr, or by an even number of real solutions becoming conjugate complex quantities in pairs. Therefore we have to determine (1) whether any finite xf can become equal to any xf, (2) whether any xf can become infinite, and (3) whether any two real solutions can become complex for any finite positive values of m l , . . . , m y+1 . (1). The masses m l , . . . , m v+l , by hypothesis, are all positive. Let the notation be so chosen that for any values of m 1 , . . . , m v+l for which the xf are all distinct the inequalities xf >u IIM\> HH n BODIES. 291 that some r. can :i|)|>rii:ich zero for finite values of j-', . . . , xJJ^., and finite positive values of w, .... m ; , leads to an impossibility, and it is therefore false. (2). On multiplying equations (8) by m, , w,, . . . , m r+1 , respectively, and adding, it is found that It follows from this etiuation that no x', 01 alone ran become infinite, and that if one of them becomes negatively infinite then some other one must become positively infinite. Suppose the notation is again so chosen that Then, if any .r. become- negatively infinite .r, 0) must also become negatively infinite, and from the equation above it follows that x, +1 must become positively infinite. Now suppose this occurs and consider ^ = 0. In order that this equation may remain satisfied, xJ 0) must also become negatively infinite in such a way that xj" x' shall approach zero. But now it follows from

shall approach zero. This reasoning continues until it is found that .rl' , . . . , x, must all become negatively infinite. But xj,j, at least must become positively infinite. Therefore x' can not become negatively infinite, and similarly x|, can not become positively infinite. Hence no x', ' can become infinite. In order to prove now that, as m, + , approaches zero, equations (8) and their solutions remain always determinate, and that there are accord- ingly no solutions besides those obtained in theorem (B), consider a solution x, , . . . , x r+1 , in which the x, are all distinct for a set of positive values of wi, , . . . , m r+ i , and then let m, approach zero as a limit. In the first place, if x, approaches neither x,+, nor x,_, as a limit as m, approaches zero as a limit, then by the reasoning of (1) and (2) above no x, can approach any x as a limit. In the second place, x, can not approach x, +1 as a limit as m, approaches zero as a limit unless x,_i approaches x i+t as a limit, for otherwise ,_, = and <(> l+t =0 can not be satisfied unless x,_ 2 and x,+, respectively approach x, as a limit. This shifts the difficulty to v(-"0 and f +i =0 are reached, which can not be satisfied under the hypotheses it has been necessary to make. In the third place, x, can not become positively infinite as m, approaches zero, for then <(>, = Q can not be satisfied unless Xi_i becomes infinite in such 292 PERIODIC ORBITS. a way that x, x ( _i approaches zero. Continuing through ,H-D, shall be a multiple solution of (8) are that these values shall satisfy (8) and also the equation dx, = 0. (11) Consider two solutions of a set of algebraic equations having real coefficients. As they change from real to conjugate complex quantities, or from conjugate complex to real quantities, for some value of a continuously varying parameter, then for this particular value of the parameter they are not only equal but they are real. Consequently, it is necessary to examine A only when all of its elements are real. It will now be shown that it can not vanish for any set of real values of the Xj when m l , . . . , m v+1 are positive, and consequently that it can not vanish for any particular set which satisfies equations (8). When this is established, it will have been proved that all the solutions of (8) which are real for m (/+1 = remain real when m v+l increases to any positive value, and that those which are complex remain complex. STRAKIHT-LINK SOLUTIONS FOK Tl BODIES. From equations (8) and (11), it follows that I'm, f*~~ ' r 3 ' I. r+l ' J. r+l where i M t , . . . , (12) A/, = -'- - p - M* 2ttli /\ , = -w' -j - i* 2m,,, t M. r+l 2m, .r+l 0. If WH-,= this determinant breaks up into the product of a determinant of the same type as (12) and a factor which is negative. Therefore, in examining \\het her or not A can vanish, it is sufficient to consider the general case in which all the m, are positive. Several properties of A are evident, (a) If the i lk row be multiplied by >n t (i=l, . . . , v + 1), the determinant becomes symmetrical. (6) The sum of the elements in each row is *, from which it follows that the expansion of the determinant contains w 1 as a factor, (c) The expansion >f the determinant contains ( l)' +1 o> 1( " +1> as one of its terms, and since all the m, are positive and all the x, are real the sign of all the terms coming from the product of the elements of the main diagonal is ( l)' +l . The fact is that, when A is completely expanded, all those terms not having the sign ( l)' +l are canceled by terms coming from the product of the main diagonal elements, and since the term (- l) r+ V" +l) is certainly present the determinant can in no case be zero. The following demon- stration of this fact was invented in 1907 by Dr. T. H. Hildebrandt, now of the University of Michigan, as a class exercise.* Since the determinant contains w' as a factor, every term in its expansion must depend upon at least one of the elements of the main diagonal. Fasten the attention upon any term of the expansion. It can be supposed without loss of generality that it depends upon the first main diagonal element. In the expansion of the determinant this element is multiplied by its minor; consequently we must see if the minor can vanish. The minor is of the An earlier proof WM devined by the author, and rtffl another jointly by Profor N. B. McLean, of the University of Manitoba, and Mr. E. J. Moulton, now of Northwestern University. 294 PERIODIC ORBITS. same form as the original determinant, and the sum of the elements of its i th row is co 2 m-i/rlf . Consequently every term in the expansion of the minor which does not vanish will contain at least one of the o> 2 m^/r^ as a factor. But these elements appear only in the main diagonal of the minor. Hence all terms in the expansion of the minor which do not vanish depend upon at least one element of the main diagonal. In considering our particular term it may be supposed, without loss of generality, to depend upon the first main diagonal element of the minor. In the expansion of the original determinant the product of these two diagonal elements will be multiplied by the co-factor of the minor of the second order of which they are the main diagonal. This co-factor has the same properties as the first minor just considered, and in the same way it is proved that at least one of its diagonal elements must be involved in the term in question; that is, the term under consideration depends upon at least three elements of the main diagonal. On continuing in this manner it is proved that any term in the final expansion depends upon all the elements of the main diagonal, which are all of the same sign in every one of their terms. Consequently, all the terms which do not cancel out in the expansion of the determinant have the sign ( 1)" +1 ; and it has been seen that there is at least one such term, viz. ( 1)" +1 co 2< " +1) . Therefore the determinant not only can never vanish, but it can never be less than a> 2 '" +u in numerical value. Since A can never vanish for real distinct x ( f when all the m, are real and either zero or positive, it follows that no real solutions can ever be lost or gained as the m, vary so as not to become negative, and therefore that the number of real solutions of (8) is (v+\}N v = %(v+\}\ for all positive finite values of m l , . . . , ?n v+l , and co 2 . 149. Computation of the Solutions of Equations (6). There are well- known methods of finding the roots of a single numerical algebraic equation of high degree, but they are not readily applicable to simultaneous equations of high degree. However, when the order of the masses has been chosen, equations (6) will become polynomials in x { , . . . , x n after they have been cleared of fractions. Then by rational processes n 1 of the x t can be eliminated from these equations, giving a single equation in the remaining unknown. The solutions of this equation can be found by the usual methods and the results can be used to eliminate one unknown. By repeated application of this process to the successively reduced equations, the solu- tions can all be found. The one satisfying the conditions of reality of x l , . . . , x n and their order relation is the one desired. The solutions of (6) can also be found by a method closely related to that by means of which their existence was proved above. Suppose for m t = mf (i=l, . . . , ri) a solution x f = xf of equations (6) is known. The m f are supposed to be zero or positive. Suppose it is desired to find the corresponding solution, that is, the one in which the masses are arranged 8TKAICHT-LINE SOLUTIONS FOR n BODIES. J'.t.') on the line in the same order, for m =///,+/,. Let the corresponding set of the *, satisfying >) be .r, = or', 01 + {, , where the , are functions of M,, . . . ,M. to be determined. On substituting*, = .rr + < and m t = m+n ( in (6), making use of the notation of (8), expanding tus power series in the {, and /j, (which is always poihle. since it has been shown that no a:', ' can become infinite and no .< ,' can e(|iial any .r'"'), and remembering that x, = x ( ?' -olution of (6) for m, = ///"'' , the resulting equations are found to be (13) S 1 4. v j r T ^ * T - v d *> u h "La*i j 2/a^/ S^!t_i_y>J_rv^.tT- y d *>* u ,_,*/ ^ZalZdx/ / where the are the symbolic powers used in connection with the power-series expan- sions of functions of several variables. The determinant of the terms of the first degree in the %, in equations i:: is the ^ of equation (11), which has been proved to be distinct from zero in this problem. Therefore equations (13) can be solved by the method explained in 1, and by 2 the solutions converge for |/x,|>0, but sufficiently small. Suppose they converge if |M<|^r. Keeping the MI within this limit, a solution x t = x\" is computed. Then this can be used as a starting-point for a second application of the process, which can be related as many times as may be desired. Hence, to find the solution in which the bodies ,,..., in m have any finite positive values and lie in a determined order on the line, we may start with /, , m, , and m, and solve the Lagrangian quintic* which defines their distribution on the line. Then an infinitesimal body m t is added and its position is found by solving the single equation (9), in which v = 3. This infinitesimal mass m t is made to increase, step by step, to the required finite value and the corresponding values of z, , . . . , z 4 are computed. It follows from the fact that the d > T 3 , + r 2 '2,n-l '2n 1 1 1 1 to ' r-j n ' r 2 , , '3n r .2 , + (15) is distinct from zero. This is a skew-symmetrical determinant, and when n is even it is the square of an associated Pfaffian, and therefore is not in general zero. Therefore if n is even the masses are, in general, uniquely determined when and x l , , . . , z n are given, though it should be noted that they are not necessarily all positive. 151. Determination of the Masses when n is Odd. In this case the skew-symmetric determinant is identically zero, but its first minors of the main diagonal elements, being skew-symmetrical determinants of even order, are in general all distinct from zero ; consequently the x t must satisfy one relation in order that equations (14) shall be consistent. To get this relation, take the right members to the left and add the equation STRAIGHT-LINE SOLUTIONS FOR n BODIES. 297 rn, x, + ?, x,+ + w. x, = 0, which is a consequence of (14), to the set of equations. There are then n + 1 linear homogeneous equations in w 7 m, In order that they shall be consistent their eliminant , + x, , + x, ,...,+ x. I 0, +i * 1 ~Xu, ~ -~ ~ .f , 'in 'In >!. must vanish. This is also a skew-symmetrical determinant and is the square of the Pfaffian F, where F=|*,, x,, . . ., *. l 1 '-I.* (16) Equation (16) can be found also by solving any n 1 equations of (14) for the corresponding m, and substituting the solutions in the remaining one. The result is a sum of determinants which can be shown to be the expansion of F multiplied by the square root of the determinant of the coefficients of the n 1 masses m, in the equations used. When F = is satisfied by x, , . . . , x., equations (14) are consistent. Then, after any w, has been chosen arbitrarily, the corresponding nl equations can in general be uniquely solved for the remaining m, , and the unused equation will be satisfied because F = Q. 1 52. Discussion of Case n = 3. When n = 3 the determinant D becomes and the Pfaffian F is (r u r a r u )' (17) It will now be shown that when any two of x, , x, , x, are so chosen as to satisfy the conditions x,r a , r u >r u , it follows that if x, is positive, then x t /r* a +x l /r t ll is positive, and therefore that x, must be negative in order that (17) may be satisfied. If x, is negative, x, , being less, must also be negative. That is, x, is necessarily negative; and similarly x, is necessarily positive. 298 PERIODIC ORBITS. Suppose x 2 and x 3 are chosen and consider F as a function of x l . Then it follows at once that From the inequalities x 2 x 2 when x 2 and a negative x l are chosen. If a;, is negative and x, is positive, but both otherwise arbitrary, F considered as a function of x 2 gives x t . Then, for a certain positive value of w 2 the mass m 3 vanishes while ra 2 is still positive. For a certain greater value of mj, the mass m 1 is zero and m, is negative. For still greater values of w 2 , both m 1 and m 3 are negative. From the fact that x t must be negative and x 3 positive, and from equations (20), it follows that not all three of the masses m 1} m t , and m s can be negative simultaneously. CHAPTER IX. OSCILLATING SATELLITES NEAR THE LAGRANGIAN EQUILATERAL-TRIANGLE POINTS. By THOMAS BUCK. 153. Introduction. This chapter is devoted to an investigation of certain periodic orbits which an infinitesimal body may describe when attracted according to the Newtonian law by two finite bodies revolving in circles about their center of mass. It has been shown by Lagrange that three bodies placed at the vertices of an equilateral triangle can be given such initial projections that they will retain always the same configuration. The orbits here considered are in the vicinity of the equilateral-triangle points defined by the two finite bodies. The infinitesimal body is displaced from the vertex of the equilateral triangle, and its initial projection is deter- mined so that its motion is periodic with respect to that of the finite bodies. The existence of the solution is established by the method of analytical continuation. The construction is made by the method of undetermined coefficients, using the properties obtained in the discussion of the existence. The solutions are given in the form of power series which converge for sufficiently small values of the parameter employed. 154. The Differential Equations. The motion of the infinitesimal body will be referred to a rotating system of axes, the origin being at the center of mass, the j-plane being the plane of the motion of the finite bodies, and the rate of rotation such that they remain on the -axis. The masses of the finite bodies will be represented by n and 1 n so taken that n ^$ , their distance apart will be taken as the unit of distance, and the unit of time will be so chosen that the proportionality factor k? is unity. Then the equations of motion for the infinitesimal body are where _ ___ -. I # # ' de dt " dr, ' de ' ar ' r, and r, being the distances from the infinitesimal body to the bodies 1 j and M respectively. 300 PERIODIC ORBITS. The Lagrangian equilateral triangle-solutions are ii. I O =-M, ?o=-V3, r=o. The two points in the rotating plane defined by these solutions will be referred to as point I and point II respectively. The question of the exist- ence of periodic solutions of equations (1) in the vicinity of these points is to be investigated. For this purpose the origin is transferred to the point in question by means of the transformation |=-M+z, v=^V3+y, r=*. (2) After the transformation is made the right members of the equations are expanded as power series in x, y, and z. The region of convergence of these series is determined by the singularities of the functions l/r l and l/r 2 . When point I is considered, the region of convergence is given by the values of x, y, and z satisfying the inequalities This region consists of the common portion of two spheres, excluding their centers which are at the finite bodies, each of radius \/2. For point II the region of convergence is defined by the inequalities -l<:c 2 +7/ 2 +2 2 -Hc-\/32/<+l, -l\u;i.U||->, Kgi 11. \1KHAL-TKIANGLE CASE. 301 where A", = + i [- 37z+75v/3(l - (4) For sufficiently small values of x, y, z, and e these series are all convergent. Equations (1) admit the integral The corresponding integral of equations (3) can be expressed as a power series in . The terms independent of e are (5') These terms will be found useful in the existence proofs which follow. 155. The Characteristic Exponents. For = 6 = equations (3) become (6) The last equation, being independent of the first two, can be integrated immediately, giving 302 PERIODIC ORBITS. To integrate the first two equations, let On substituting in the first two equations of (6) and dividing out e XT , we have (7) [2X- In order that these equations may be satisfied by values of K and L different from zero, the determinant of the system must vanish. This gives for the determination of X the characteristic equation X 4 +X 2 +f M (l- M )=0. (8) Each of the four values of X satisfying this equation gives a particular solu- tion of equations (6). The corresponding K and L must satisfy equations (7). Since these equations have a vanishing determinant the ratio only of the K and L is determined. In what follows K will be considered as arbi- trary, and L will be determined in the form L = bK. In order that a solution shall be periodic, the corresponding X must be a purely imaginary quantity. Upon solving (8), we have For small values of /* the roots of (8) are pure imaginaries; the limiting value of M for which this is true is given by the equation 1-27 M (1-M)=0. The root of this equation which is less than \ is /JL = .0385 .... For fj. ^ .0385 . . . the values of X are purely imaginary and the corresponding particular solutions are periodic. Let . KqUILATEKAL-TKIAM.I.K CASE. 156. The Generating Solutions. The general solution of (6) is r= ,/,,." -"4. a,e-" v -> T + o,e" v '- lr + a 4 c-" v -' r , ij = b, , <'- ' r +6, a, e~" '-' r + 6, a, e"^~ lT +6 4 a, c~" *-' ', 2=c,sinT-r-c,cosT. The quantities ",,,, a, , o 4 , c, , and c, are arbitrary, while 6, , 6, , 6, , and 6, are determined by equations (7) when the proper values of X are sub- stituted. Thus it is found that h - h . -<,->-M . Various periodic solutions are obtained from this general solution by assign- ing suitable values to the arbitrary constants and to the quantity p. For M<.0385 ... we have two distinct periodic solutions: y = b l a l c" v/ -' r + 6,0,6-"^'-' T , 2 = 0. II. x These equations represent ellipses in the .n/-planc with centers at the origin. The major axes of the ellipses coincide and make an angle with the positive z-axis defined by tan20=->/3(l-2 M ), with ros20 positive. The major and minor axes of the second are greater and less respectively than those of the first. The periods are 1-K/a^ and 2jr/\ II.I.I.I I l>. KijULATERAL-TUIANGLE CASE. Those periodic -ohition.- are tin- generating solutions for the general problem. \Ye sh;ill now suppose tlial is not zero and collider the question of the existence of the continuations of these .solutions with re>|)e<-t to the parameter t. The period in the variable r will in all cases be taken the same as that of the generating solution. The |x>riod in t of the solution is found from the relation l=(l + 6)r. 157. General Periodicity Equations. For = the general solution of equations uli is x = 2 =r,sn (9) Normal \ariables are introduced by the transformation x = ^1+9 x l y' = a, ( 1 + *) v^T (6, x, - b t x t ) + a t ( 1+ i) v^T (6, The differential equations then become (10) (11) where A being the determinant of the transformation (10), and A,, the minor of an element in this determinant. The first subscript indicates the row and the second one the column. 306 PERIODIC ORBITS. F or = 5 = the initial values of the variables x t , x 2 , x 3 , and z 4 are flj , a 2 , a, , and a 4 respectively. In the general problem we take as the initial conditions Since there is a component of force always directed toward the xy-plane, it is clear that at some time z must be zero. Hence we have supposed that z = 0at r = 0. According to 14 and 15 equations (11) can be integrated as power series in the parameters a lt Oj, a 3 , a 4 , 7, 5, and e, which converge for |a,| , . . . , | e| sufficiently small, and for O^r^T, the value of T, which in this case is the period, being given in advance. These solutions have the form 3 1 =(a 1 +a 1 )e +^'+ep 3 (a M a,, o,, a 4 , 7, 6, e; r), Z 4 =(a 4 +a 4 ) e-x'+'^'+ep, (a,, a,, o,, o 4 , 7, 5, ej T), 2 =(C+T) sin(l + S)T+ep,(a,, a 2 , a 3 , a 4 , 7, 5, e; T), Z' = (l + 3)(c+7)cos(l+5)r+ep 6 (a 1 , a 2 , a, , a 4 , 7, 5, e; T). The general periodicity equations for the period T are s.(0) = (t = l, ... ,4), (13) z(T)- (0)=0, ' / These equations are sufficient conditions for the periodicity of the solution. On solving them for the arbitraries a, , cu, , a 3 , a 4 , 7, 5 as power series in e, a determination of these quantities is obtained such that the corresponding solution is periodic. Hence, on substituting these series in (13), the resulting expressions for the x t , z, and z' are periodic. These expressions can be rearranged as power series in e which will converge for e sufficiently small, and for all ^ T econd equation is redundant and can be suppressed. On computing the necessary terms of (13), it is found that the remaining equations have the form '] 0, K t*n*/i - ~i r ~i e*-l)+ \+<[e ll a,+e n a i +e nt + -J=0, y ( sin ) -f '!+*! ]=o, (15) where the explicit computation shows that the e t , are constants different from zero. The right member of the 2-equation in (1 1) carries the factor z. Conse- quently the solution carries the factor y, and hence the last two equations of (15) have y as a factor. If y is not zero and is divided out, there remains a term in each equation which is independent of the arbitraries. These terms can vanish only if \l KI.UTES, EQ1 II.. \TEHAL-TRIANGLE CASE. The discussion of the existence of the continuation of the -croud gener- ating solution, which depend- upon a , as the tirst does on ut 1 it-cause of its complicated charaeter this possibility has not been considered. The question of the existence of these orbits is thus left open, but it seems improbable that the necessary conditions can be satisfied. 161. The Fifth Generating Solution. The values of n in this case are such that the period of the generating solution is 2wT = 2wi l T/ +& t$ (a l -|-a 1 )(c+T) 1 +&i,(a 1 +a t )(c+7) i -f ] + -0, 0, 0, (20) where the a t) , . . . , e,, are functions of /i which are readily determined. The last three equations are solved for a, , a 4 , and 5, and the results thus obtained are substituted in the first and second equations. After dividing by e*, these equations have the form a.) ( 1 +a 1 )'+D(a i +o,)(c+7) f +( )- 312 PERIODIC ORBITS. In order that solutions of (21) of the required form shall exist, it is necessary that O. (22) These equations are satisfied by a^ = o 2 = 0. With Oj and o 2 having this value, equations (21) then have a unique solution for a, and a 2 . But the generating solution has reduced to that considered in 159. The orbit obtained is, therefore, the continuation of the third generating solution re-entering after m revolutions, and moreover the value of n is such that ma l = m^ , where m 1 is an integer. Thus we have a proof of the existence of an orbit in one of the exceptional cases omitted in discussing the third solution. In order that equations (22) may have a solution for which c^ , a 2 , and c are different from zero, it is necessary that the determinant of the A, B, C, and D shall vanish. This determinant can be developed as a power series in V/I. If it is identically zero, each coefficient in this development sepa- rately must vanish. The coefficient of \/ju was computed and found to be different from zero. For special values of M it may be possible to satisfy (22) by values of a : , o 2 , and c which are distinct from zero. Because of the complicated character of the coefficients, this possibility has not been established. As in the preceding case, the existence of orbits of this type seems improbable, but complete proof is lacking. The discussion for the sixth generating solution differs only in notation from that just given. No new orbits are found, but a proof is obtained of the existence of the continuation of the third generating solution re-entering after m revolutions, when fj, is such that ma z = m 2 . This is another excep- tional case not treated in the discussion of the third generating solution. 162. The Seventh Generating Solution. The values of /j, for this case are such that the period is 2w7r = 2rn 1 7r/ H (a, + a ,) (aj + a,) +^ 62 (a, + a,) (a 4 + a 4 ) +*(c+7)'y+ =o,j where the

2 which, expressed in terms of the original variables by (10), becomes The coefficients of the first power of are given by the equations *i,-r,v'=7 x u = +^^^1 ,+ A, X?+ B, Y?, . Y?, where A'^ 01 and Y represent the expressions obtained by substituting x a and r/ for x and y in A', and Y t . In order that the solution of the first equation shall be periodic, the coefficient of e">^' r in the right member must vanish. Otherwise non-periodic terms of the type re'^^ ir will be introduced. For the same reason the coefficients of e~" v -' T , e" v '~ lr , and e - u ii ^ U i2 h , bl - 7 / _ SQ^OH 3V3(l 2M)u u " 4 (i=l, . . . , 4; j = 0, . . . , n 1) have been determined and that the x u are periodic. For the determination of x tn and 6 n we have equations of the following form : (30) where the 6 tJ and r? are known constants. Since .r so = x 40 = 0, no non- periodic terms can enter the solutions of the last two equations. In order that the solutions of the first two equations shall be periodic, it is necessary that the coefficients of e fflV - lr and e~ fflV=[T in the first and second respectively osriu, \TI.\I; >\n:i.i.iiKs. Kijrn. \ i KI; \i.-i i;i \\<.i.i: CASK. 317 shall vanish. Tliis gives for the determination of 6,, the only undetermined (mi-!. -int. two ('(illations n- a- power >eries in t and the terms are rearranged so that the integral remains a power serie< in t. Since the integral is an identity in r and t, it follows that the coeflicient of each power of t must reduce to a constant identically in T. \Ye will consider the coefficient of t*. When the expressions for the x ( , as functions of T arc substituted, this coefficient consists of a sum of linearly independent functions of T. The coefficients of each of these functions must then vanish. Let (f l and

\ , the coefficient of i does not vanish, and we have ^,+^ = 0. Both i and

y 2) I b *j cosjr-^-b'^ sinjV] , /-o /-o .= - _2u 2r CILLATING SATELLITES, EQUILATERAL-TRIANGLE CAM 321 For the general terms we proceed by induction. Suppose th&tx,, y,, z,, and S J (j = Q. 1 ..... n 1) have been determined, and that for j even it has been found that i *i = 0, y, = 0, z,= jj [c t cos(2*+l)T+cJsin(2*+l)T]; while for./ odd, it has been found that a cos2kr+a' t sin2&r], O+H/l *- z, = 0, 5, = 0. It can be readily shown that when n is even the coefficients of e" are given by the equations (44) In order that the last equation shall have a periodic solution we must impose the conditions -2c*.+C"=0, CT-0. The first relation serves for the determination of 5. . Since by the existence proof the periodic solution is known to exist it follows that the expression Ci" is zero. An additional proof that C is zero is obtained by considering the integral (5'). The series for x, y, z are substituted and the terms are re-arranged as a power series in . Each coefficient of this series must reduce to a constant identically in T. Consider the coefficient of e". The terms of this coefficient which carry z n are 2z' z' m +2z t z.. (45) Suppose T , . . . , x n ; y t , . . . , y n ; z, , . . . , 2.., ;,,..., ._, have been determined and that the x, , y,, and z, are periodic. The equation for the determination of z n has the form COST+ On integrating, the following non-periodic terms are obtained z n = IJT COST CJ" r sinr. 322 PERIODIC ORBITS. When the expressions for x } , y, , and z, as functions of r are substituted in the coefficients of e" in the integral, the only non-periodic terms obtained come from z n . They are of the form T, r sin2r, and r cos2r. Since the coefficient of e" is a constant identically in T, it follows that the coefficients of these non-periodic terms are zero. Those for T sin2r and r cos2r vanish identically. The coefficient of T gives the relation c C=Q. Since c^O, it follows that <7{ B) = 0. Upon integrating (44) and imposing conditions (35), we find z n = c nl cos r+c' nl sinr (46) The quantities C^ +l and C'^ are known from the differential equations. The constants of integration c Bl and c nl are found by (c) and (d) of (35) to have the values n/2 n/2 When n is odd the equations obtained from the coefficients of e" are i; (n) sin2jr], From the last equation it follows at once that 5 n = 0, for otherwise z n will not be periodic. Integrating these equations and imposing the periodicity equations, we find 1"- (l6j-'+9)" - 16J-B? +3 V3(l - 2M) _ _ , L 16f(4f-l)+27 M (l- M ) _ ( ^ /2 ri6jAr+3V3(l-2M)A^-(l6j 2 + 3)^ . . L" 16/(4f-l)+27 M (l- M ) , 2 =0, (48) OSCILLATING SATELLITES, EQUILATERAL-TRIANGLE CASE. 323 If we make use of the transformation by which the parameter c was introdiicr.l, we have for the final serir- the .r,, y,, z,, and 6, being given by (46) and (48). It is not difficult to show that x n , y n , and z n carry the factor c" +1 , and S n the factor c", and that c enters tin M e\pres>ions in no other way. Consequently in (49) the arbi- traries r and e occur always in the combination c. Therefore we may put c=l without loss of generality. The final series then contain only the arbitrary e. An approximate idea of the shape of the orbit can be obtained by considering the first two terms of (49). These terms \\rre computed for /u = 0.01 and = 0.5. The projection on the xy-p\a,ne is an ellipse of small eccentricity whose center is on the negative ?/-axis and whose major axis cuts the positive x-axis. This projection is shown in Fig. 5. The projections on the xz and 7/z-planes are shown in Fig. 6 and Fig. 7 respectively. The orbit thus con- ~ sists of two elongated loops, one above FIG. 5. Fio. 6. Fio. 7. and the other below the xj/-plane, the double point being in the fourth quadrant of the xy-plane. If T is replaced by T+T in the expressions for x }) y,, and z,, then x, and y, remain unchanged while it is seen that z, changes sign. It follows that the loops are symmetrical with respect to the xy-pl&ne, and that each loop is described in half the period. For positive values of c the upper loop is described first, and the motion is such that the projection of the infini- tesimal body on the xy-pl&nc moves in the clockwise direction. The period of the motion is given by the relation T = 324 PERIODIC ORBITS. Orbits about the Point II. To each of the orbits about point I there corresponds an orbit about point II. The proofs of the existence of these orbits were omitted, since they are similar to those for the first point. The series for these orbits can be obtained easily from the corresponding series for the first set of orbits. The differential equations for the orbits about point II are obtained from those for the orbits about point I by changing the sign of -\/3- The periodicity conditions to be imposed are the same in both cases. It follows, therefore, that the solutions for the one case can be obtained from those for the other by changing the sign of -v/3"- There- fore, in order to get the series for the orbits about the second point we make this change in the series already obtained for the first point. On referring to equations (1) and (2), it is seen that the differential equations for point I reduce to those for point II if the signs of y and T are changed. Hence this transformation can be made geometrically by a reflection in the zz-plane and a reversal of the direction of motion. Thus, it is easy to get an idea of the shape of these orbits from those already discussed. CHAPTER X. ISOSCELES-TRIANGLE SOLUTIONS OF THE PROBLEM OF THREE BODIES. BY DANIEL BUCHANAN. 165. Introduction. - This chapter treats of periodic solutions of the problem of three bodies, in which two of the masses are finite and equal. The thinl I >< uly is started at the initial time t a from the center of gravity of the equal masses, and the initial conditions are so chosen that it moves in a -traight line and remains equidistant from the other bodies. In I the third l)oi ly is assumed to he infinitesimal and the initial conditions are so choM-ii that the equal bodies move in a circle about the center of mass.* In II the third body is considered infinitesimal and the initial conditions an so chosen that the equal bodies move in ellipses with the center of mass a- the common focus. In III the third body is considered finite and the solutions derived have the same period as those obtained in I, and reduce to those solutions when the third body becomes infinitesimal. I. PERIODIC ORBITS WHEN THE FINITE BODIES MOVE IN A CIRCLE AND THE THIRD BODY IS INFINITESIMAL. 166. The Differential Equation of Motion. Let wi, and m, be two finite bodies of equal mass, and n an infinitesimal body. Let the unit of mass he so chosen that m l = m t = 1/2; the linear unit so that the distance between >, and m t shall be unity; and the unit of time so that the Gaussian con- stant shall be unity. Let the origin of coordinates be taken at the center of mass, and the to-plane as the plane of motion of the finite bodie-s. Let the coordinates of MI, , m t , and /z be ,, ;,, 0; ,, ij,, 0; and 0, 0, f respectively. If w, and m t are started at the points 1/2, 0, and 1/2, 0, 0, respectively. i hat they move in a circle, then \\ hen the finite bodies move in a circle, the motion of the infinitesimal body can be completely deter- mined by means of elliptic integrals. The problem was first solved in this way by Pavanini in a memoir, "Sopra una Nuova Categoria di Soluzioni Periodiche nel Problema dei Tre Corpi," Annati di Matematica, Series HI, vol. XIII (1907), pp. 179-202. The elliptic integrals obtained by Pavanini were later developed independently by MacMillan in an article, "An Integrable Case in the Restricted Problem of Three Bodies," Astronomical Journal, NOB. 625-626 (1911). MacMillan further developed the solution as a power series in a parameter, the coefficient* of which are periodic functions of I. The solution obtained in I baa a close relation to MnoMillnn'a solution. ais 326 PERIODIC ORBITS. The differential equation for the motion of the infinitesimal body is __ __ l+4f 2 ) 3/2 where the accents denote derivatives with respect to t. The integral of (1) is (2) where C is the constant of integration. If C is positive, the particle M recedes to infinity. If C is negative, the particle n does not pass beyond a finite distance from the origin. From a consideration of (2) it can be shown that, if C is negative, the particle crosses the ij-plane. Hence the initial time can be chosen, without loss of generality, as the time when the particle is in the ^-plane. It can also be shown from (2) that if C is negative, the motion of the particle is periodic, and that the period can be expressed as a power series in the initial velocity of n, whose limit is 27r/\/8 as the initial velocity approaches zero. We shall, however, prove the existence of a periodic solution of (1) by a different method. 167. Proof of Existence of a Periodic Solution of Equation (1). In this proof it is convenient to make the transformation , (3) where 5 is to be determined so that the solution of (1) shall be periodic in T with the period 2w. At T = let r = 0, f = a, (4) where f = d^/dr. Let us make the further transformation t = az; (5) then when (3) and (5) are substituted in equation (1), we obtain . _ (1 + 5)2 , 6) 22 where z is the second derivative of z with respect to r. The initial conditions for z become, as a consequence of (4) and (5), 2 (0) =0, 2(0) = 1. . (7) For \a\ sufficiently small the right member of (6) can be expanded into the series z=-(l + 5)z{l+(- i 3/2 )4aV+ . + (- 3/2 )(4aV)<+ }, (8) where /-3/ 2 \ V i ) ' ISOSCELES-TRIANGLE SOLUTIONS. Equation (8) can be integrated as a power series in a 1 and 6 which converges for O^T^2r, provided |a| and |6| are sufficiently small. Let us write this solution in the form z=2 2 z u 5'a". (9) 1.0 / The initial values of the z,, as determined from (7) are *u(0)=0, (i,j-0, ... oo), I J Upon substituting (9) in (8) and equating the coefficients of the various powers of 5 and a 1 , we obtain differential equations from which the z u can be determined so that the initial conditions (10) shall be satisfied. The differential equation for the term independent of 5 and o 1 is and the solution of it which satisfies (10) is z M =sinr. The differential equation for the term in & alone is *i*+i*= -sinr, and the solution of it which satisfies (10) is The solution of equation (8) is therefore z = sinT + S [|COST |sinr] + terms of higher degree in a 1 and 6. (11) With the initial conditions (7), the variable z is an odd function of T, and therefore a sufficient condition that it shall be periodic with the period 2x in T is Z(T) = 0. With the value of z given in (1 1), this condition becomes 0= - |a+termsof higher degree in a* and 6. (12) Since the coefficient of 5 is different from zero, this equation can be solved uniquely for 6 as a power series in a*, vanishing with a*. Let us denote this solution for 5 by 5=25,, a 1 '- (13) i-\ When (13) is substituted in (9), we obtain z=z f ,a", (14) 328 PERIODIC ORBITS. which converges for |a| sufficiently small. Since the periodicity condition has been satisfied, z is periodic in r with the period 2ir. Hence z 2J (T+2ir)=z 2j (T) (j = 0, ... QO). (15) The initial values of z 2i as determined from (7) are Z 2 ,(0)=0 (j = 0, .;.), ] i (lo) Z (0) = l, 2,,(0)=0 (] = !, ... oo). j 168. Direct Construction of the Periodic Solution of Equation (1). Let us substitute (13) and (14) in (8) and equate the coefficients of the various powers of a 2 . Since the result is an identity in a 2 , there is obtained a series of differential equations from which the coefficients of the solution (14) can be determined. The differential equation for the term independent of a 2 2 is and the solution of it which satisfies (15) and (16) is z = sinT. The differential equation for the term in a 2 is z 2 +z 2 =-5 2 z +6zJ=(-5 2 +|)sinr-|sin3T. The term sinr gives rise to a non-periodic term in the solution, and, in order that (15) shall be satisfied, its coefficient must be zero. Hence and the solution for z 2 satisfying (16) is z,= J(sin3r- ! The differential equation for the term in o 4 is 4+4= - (5 4 +^)sinr+6sin3r- |sin5r. In order that (15) shall be satisfied, 5 4 must have the value 5 = > and the solution for z 4 is found to be z 4 =r^[431sinT-192sin3T+29sin5T], 3EBQ where the constants of integration have been determined so as to satisfy (16). I8O8CEI.KS lltl VM.l.K SOLUTION- ii'J'.l - far as computed, it has been found that the S t) are uniquely deter- mined b\ -the periodicity and the initial conditions, and t hat each z,, is a sum of sines of odd multiples of T. the highest multiple being '2j+ 1. \\ . -liall now show by an induction to the general term that all the & tj are uniquely determined by the same conditions, and that all the z tl have the properties which have been stated. Let us assume that S t , . . . , $_,; z t , . . . , z,,_, have been uniquely determined, and that each z st (k = 0, . . . , j 1) is a sum of sines of odd multiples of T, the highest multiple being 2k+l. From these assumptions and the differential equations it will be shown that 6,, and z,, are uniquely determined, and that z,, is a sum of sines of odd multiples of r, the highest multiple being 2j+l. Let us consider the term in a". The differential equation is , (17) where Z t; isa known function of 2^(1 = 0, . . . ,j 1) and 5 tt (k = l, . . . ,j 1). The general term in Z tl has the form T ._-*', .A st lt >~\ \ 6 " where X, , . . . , X t ; ^, , . . . , M ', P, and q are positive integers (or zero) having the following properties: (a) /*,+ +M* is an odd integer, (6) Ml X,+ +/I.X.+P.+ +n t -l+qp = 2j, (c) q is or 1. Since each z , . . . , z,,_, is a sum of sines of odd multiples of T, it follows from (a) that Z tl is a sum of sines of odd multiples of T. The highest multiple is The highest value of N tJ is obtained when 9 = and is, therefore, 2J+1. Hence (17) has the form ^+z,/ = [-$,,+a| 1 >>nT+a J a/) sin3T+ +a t %sin(2>+l)r, (18) where n{ l/> , . . . , a+, are known constants. From (15) it follows that S -n' n 5,,-a, . The solution of (18) satisfying (16) is therefore z 1/ = /l l (1/) sinT+^ 1 a/) 8in3T+ -f X i where <*- * 330 PERIODIC ORBITS. Hence the periodic solution of (1) in terms of the variable T is It is a power series in odd powers of a with sums of sines of odd multiples of T in the coefficients. The highest multiple of T in the coefficient of a 2t+1 is 2&-fl. In the sequel we shall call such a series a triply odd power series. The period in T is 27r, and in t it is II. SYMMETRICAL PERIODIC ORBITS WHEN THE FINITE BODIES MOVE IN ELLIPSES AND THE THIRD BODY IS INFINITESIMAL. 169. The Differential Equation. Let m^ and w 2 represent the two finite bodies and ju the infinitesimal body. Let the system of coordinates be chosen as in 166. Let the unit of mass be so chosen that m l = m z = 1/2, and then let the linear and time units be so determined that the mean dis- tance from Wi to m 2 and the gravitational constant are each unity. With these units the mean angular motion of the bodies also is unity. Let M be started from the center of gravity of m t and m 2 perpendicu- larly to the plane of their motion when they are at apsides of their orbits, which can be assumed to lie on the -axis. From the symmetry of the motion with these initial conditions, it follows that 2 = ~~ 1 > ^2 = ~~ li Let the motion of the finite bodies be referred to a system of axes rotating about the f-axis with the uniform velocity unity. The coordinates referred to the rotating axes are defined by o;, = ,cos+??sin, !/=& sin <+!? cos J, z = r ft = 1,2). The x t and y t are determined by the conditions that m l and m t shall move in ellipses and be at apsides at t = t a , which in this case is put equal to zero. Then it follows, from the properties of elliptic motion, that -t), (20) where r =m[l v = 7n = m, = m.= ^> e = eccentricity of ellipses. ISOSCELES-TRIANGLE SOLUTIONS. 331 The differential equation for the motion of /i is // mz mi 2m z 2 =- -JT--T = - r- (21) M <* r \ where When we substitute the values of x, and y t from (20), equation (21) becomes .//_ 2mz (22) ' Win -re //i occurs in the denominator we shall substitute ita value 1/2, but in the numerator we shall make the substitution m = m +X, and con- sider X as a variable parameter while m and m t both remain fixed. In order to obtain the solution of the physical problem we must put X = m i in the final results. With these substitutions, (22) becomes " (mrHOSAw** 1 , (23) 1-9 where and where each # 2 /+i is a power series in e with cosines of integral multiples of / in the coefficients, the highest multiple being the same as the exponent of the eccentricity e. 170. Determination of the Period by a Necessary Condition for a Periodic Solution of (23). If the motion is periodic, let the period be denoted by T. Since the period of motion of the finite bodies is 2r, we must have T = 2yr, (24) where v is an integer which denotes the number of revolutions made by the finite bodies in the period T. Let us take the initial conditions 2(0) = 0, z'(0) = a. (25) With these initial conditions it can be shown from (23) that z is an odd function of t. Hence if M is started from the i?-plane when m, and wi, are at apsides of their orbits, a necessary and sufficient condition that z shall be periodic with the period T is z(772)=0. (26) 332 PERIODIC ORBITS. In order to determine the period T from the condition (26), we inte- grate equation (23) as a power series in a and X, but only in so far as the term of the first degree in a is concerned. The differential equation for this term is ii.o = 0. (27) This equation belongs to the class of differential equations with periodic coefficients which was treated in Chapter III, where it was found that the character of the solutions depends upon whether or not 4\/m^ is an integer. Since m depends upon the way in which m is separated into ra +X, and since |X| must be taken small in order that certain solutions appearing in the sequel shall be convergent, the value of w is in the vicinity of 1/2. We may therefore regard 4-v/m^ as not an integer, and when it is not an integer the general solution of (27) is 2 , (28) where w t and w 2 are conjugate complex functions of the form oo n ; fl) (coskt - 1)] e", i n) (coskt - 1)] e", The A ( and A are constants of integration; the a and 6*' are real con- stants which depend upon the coefficients of the various powers of e in E\ ; and a is a power series in e with real constant coefficients, determined by the condition that w t and M, shall be periodic with the period 2-rr. From (25) we have A? +A? = 0, [erie> in .I, 01 is r > \,e; I), where /' i- ;i power -erie- in .! .'. ,\. and e. Upon imposing the condition (26) that : >!ia!l he |)eriodic witli the period 7'. we obtain Tliis c(iuation is sati>fied l>y .11'" = 0, but this value of .4,"" leads to the trivial solution 2 = 0. In order, then, that (31) shall have a solution for A which i- ditTercnt from /er<>. the coefficient of -AJ ' must he zero. Xow , (j = 1, or 1 -fa power series in e, according as v in (24) is even or odd respectively. Hence u^T/ty^Q for \e\ sufficiently small, and we have . (32) In order that this condition for the existence of a periodic solution may be satisfied, T must have the value where N is an integer which denotes the number of oscillations made by the infinitesimal body in the period T. Then it follows from (24) that N = v )' and (a^)' is a constant, by 18, and is the same as (30), viz., A = V 1 A^O. Therefore (36) The integration of (36) gives non-periodic terms as well as periodic terms having the period T. We shall be concerned only with the non-periodic terms. Let the constant part of V 1 E^uJ^ be denoted by P t ; it is a power series in e with constant coefficients which are purely imaginary, the absolute term of which is found to be 2V i/Vrn^. Hence [P, J+periodic terms], a< = A? +X A ( [P, Aperiodic terms], where A and A are constants of integration which are to be so determined that 2 U (0) = z' ltl (0) = 0. Then z ia = X A? P.tie" V=T ' u l +e- aV=Tl u,] + periodic terms. (37) It is necessary to obtain in addition to this only the term in (A ( y. This term is obtained from the differential equation r t 7 - vn v / 3 ^QC^ J i z 3,o~ A ~ ^ j 3 z . (68) On forming the equations analogous to (36), we have )' = + (39) The terms not written in (39) carry the exponentials e 8/ ) i [P t <+ periodic terms], 1 af = A? + (A[ 0> ) 3 [P 2 Z+periodic terms], ] where A and Af are constants of integration which are to be determined so that z 3 , (0) = < (0) = . Hence z 3 , =(A? ) )*Ptt[e'' v=r[ 'u 1 +e-' rV = I 'Uv]+periodic terms. (41) ISO.-' 1 .1 .1 ;> r It I. \NULE SOLUTIONS. 335 Now imposing the condition (26) that z shall he periodic with the period T, we obtain from (37) and (41) (42) Th.- expremOQ [e'^^ l (r/2)+-" / = TrA u,(7y2)] is different from /on. for e = 0, and therefore remains different from zero for \e\ sufficiently small. Initiation (42) is satisfied by A = 0, and hence the right side carries A as a factor. In order to tind a solution of (23) other than 2=0, it is necessary to consider A{VO; therefore the factor A can be divided out of (42). There remains a power series in X and A and, since P, and P t are different from zero for \e\ sufficiently small, the terms of lowest degree are X and (A') 1 . There are no terms in Ae and e alone, and the coefficient of (^"O'hasa term independent of e, viz., 36 V 1 Vm 9 . Hence, after A is divided out, equation (42) can be solved for A as a power series in X', the coefficients hoi M; power series in integral powers of e. Two periodic solutions of (23) therefore exist having the period T. They have the form where Q is a power series in X* whose coefficients are power series in e. In the practical construction of the solutions it can be shown that 2 is a power series in odd powers of X'. This fact follows also from the dynamical nature of the problem, since the motion of n is obviously symmetrical with respect to the :ry-plane. The two solutions are therefore of the form =-s (43) where each Zj ]+ i is periodic with the period T. In 117-118 it is shown by a discussion, which is applicable in this problem, that if v is even in (24), the orbits obtained by taking the two signs before X' are geometrically the same, but in the one the infinitesimal body is half a period ahead of its position in the other. If v is odd, the orbits for -r-X' and X* are geometrically distinct. By an argument similar to that in 115, it can be shown that it is pos- sible to choose \>Q so that the solutions (43) will converge for all ^ t^ T. 172. Direct Construction of Symmetrical Periodic Solutions of (23). Let us substitute (43) in (23) and equate the coefficients of the various powers of X*. The constants of integration occurring at each step are determined by the conditions that the orbits shall be symmetrical and periodic with 336 PERIODIC ORBITS. the period T. The condition to be imposed in order that the orbits shall be symmetrical is 2(0) =0, from which it follows that w+ ,(0)=0 tf-0, ...). (44) It is necessary to consider the terms up to X 5/2 before the induction to the general term can be made. The differential equation for the term in X* is and the solution of this equation is (28). When (44) is imposed z, = A? [e V=I ' u, - e~' v=r ' ,], where A ( " is an undetermined constant. The differential equation for the term in X 3/2 is l z t = Z s =-E 1 z l -m 9 E l iS' l . (45) When expressed in terms of t, the right side of (45) has the form z 3 =A?e?+(Arre?\ (46) where the 0f l+u (i=0, 1) are homogeneous and of degree 2i+l in e +ffV= *' and e - are purely imaginary. In order that z 3 shall be periodic, the constant parts of the coefficients of e +av=l1 and e~ av=lt in u 2 Z 3 and M^ respective^ must be zero. From the form of u t , u t , and Z 3 it follows that, when we equate to zero the constant parts of these coefficients, we obtain only the one equation -A^nPH-WO'P.l-O, (47) where P t and P 2 are the power series which appear in (42). Equation (47) is satisfied by A"' = 0, but this value of A ( " leads to the solution 2 = and is excluded. The solutions of (47) for A ( " which are different from zero are (48) ISOSi i 1 Kj IKIANUI.E SOU'TIOV-v '.M7 where />, is a power series in * with constant coeflicients which are real since P, and /'.an- loth p-.irely imaginary and their absolute terms have the same sign. The absolute tcnn of /;, is found by computation to be Since .1," is purely imaginary the expression for z, is real. When the sign <>f v 7 ^! is chosen in Iv, the periodic solution of (23) which satisfies the initial condition 2(0) = is unique. The general solution of (45) is z^Arf^'^+Afe-'^'^+V^iW+v?}, (49) whprr .I* and A are the constants of integration. The particular integrals an d Z, it follows that, when we equate to zero the constant parts of these coefficients, we obtain only the one equation ^PJ B + V r ^TP 1 > (52) where P' and P, are power series in e with real constant coefficients which are unique after the sign of V 1 in (48) has been chosen. The absolute term of PJ" is found to be 32, and therefore the solution of (52) is Af-V^lp., (53) where p, is a power series in e with real constant coefficients. 338 PERIODIC ORBITS. With A determined as in (53), the general solution of (51) is periodic and has the form z t = A? er^'Ut+A? e-'^'Ui+V^TW+tf+v?], (54) where A and A are the constants of integration and where the f l+n (i = 0, 1, 2) are of the same form as the 6f i+a (i = 0, 1, 2), respectively. It follows from the form of ipf, ( +" in (56). Let us consider the term in x C2a+1> / 2 . The differential equation for this term is - (57) The part of Z 2n+1 not written explicitly involves z l , . . . , z 2n _ 3 to odd degrees when considered together. The only undetermined constant which enters Z 2n+i is Af ~ u , and it has the same coefficient in Z 2n+l that Af has in Z 5 . In order that z 2 +i shall be periodic, the constant parts of the coefficients of e +crv=lt and e~ av=lt in u 2 Z 2n+ ^ and M^an+i respectively must be zero. Now since Z 2n+l is similar in form to Z 6 , we obtain only one equation when the constant parts of these coefficients are equated to zero. The form of the equation is A*-P?+ V^lP^ = 0, (58) where P 2n _i is a power series in e with real constant coefficients. The solution of this equation for Af""" is Ar- 1) = V^Tp 2n _ 1 , (59) where p 2n _i is a power series in e with real constant coefficients. ISOSCELES-TRIANGLE SOLUTIONS. 339 In general, there are no other terms in M,Zj,, + , and w^j.+i which yield non-periodic terms in z,. fl . But since , N and v being integers, there are values of n for which other non-periodic terms than those already di-cussed can occur. It follows from the properties of Z 2 . + , that contain- the term [ (60) +v^T6, sin*+ + v/=T6. sinfa-f- ] J where K is a constant. Now e .+Dav=7i = e >^T[ c08 2 n /^T s\n2not]. Consequently these non-periodic terms arise if k = 2nv, k an integer, or if kv = 2nN. This relation is satisfied if 2n becomes a multiple of v. Suppose v is odd. Since v and N are taken relatively prime, the smallest values of n and A - for which the non-periodic terms in question can arise are n = v and A - = 2A r . If v is even, AT is odd; and the smallest values of n and /, are // = v/2 and k = N. The terms in which these non-periodic terms first arise are multiplied by X tt " +l)/t e 1Ar or X'+'e", according as v is odd or even. After these terms first appear they in general occur similarly at all subse- quent steps. When they are present, the equation analogous to (58), in so far as the terms in u^Z^+i are concerned, is (61) where #,._, is a constant multiplied by e w or tf according as v is odd or even. The terms in jZ 2l ,+, corresponding to (60) differ from (60) only in the sign of K and Vi. Non-periodic terms arise from these terms in the same way as from (60). The equation analogous to (58), in so far as the terms in w,Z ta+ , are concerned, is the same as (61). This equation can be solved uniquely for Af*~ ti and the solution is of the same form as (59). Hence in all cases A^~" can be determined by the symmetrical and the periodicity conditions. With A I* 1 "" determined as in (59), the solution of (57) is periodic. The general solution of (57) is From the form of ^2+" it follows that *C3?fl-P, and when the condition (44) is imposed, z,,+, becomes where A?* +0 remains undetermined at this step. This solution is real if AJ* 1 ^" is purely imaginary. This completes the induction. 340 PEEIODIC ORBITS. III. PERIODIC ORBITS WHEN THE THREE BODIES ARE FINITE. 173. The Differential Equations. We shall now consider the question of the existence of orbits which are periodic when n is finite, and which have the same period as those obtained in I. The question is one of deter- mining initial conditions for m l , m 2 , and /x so that the motion of the system shall be periodic when M is finite, and shall have the same period as when fj, is infinitesimal. The origin of coordinates will be taken at the center of mass of the system. The plane passing through the center of mass and perpendicular to the initial motion of n wih 1 be taken as the ^rj-plane. Let the coordinates of mi , w, , and M be , , 17, , f t ; 2 , ^ , f z ; and , 77, f respectively. Let the values of , 17, f, ', V ', ^ i? 2 , f 1; and f 2 be zero at t = t a . Further, let Under these symmetrical initial conditions &=-&, ?i=-%, fi=r,. (62) On making use of (62) in the center of gravity equations, which are we have 0. (63) Hence M always remains on the f-axis. With the units chosen as in 166, the differential equations are "_ -III- l ~ 8 -3 (64) where r s = ^+'??. Let us transform (64) by the substitutions , (65) where 5 has the value determined in I. Then equations (64) become [ (66) (1 + /*)(! + 8) f 2372 ' ISOSCBUBS-TKIAMiLE SOLUTIONS. .ill For n = these equations admit the solutions where i/ i> the function defined in (1 ( J) and is periodic in T with the period 2r. Now let (67) where p, u, and u> vanish with ^ = 0. When equations (67) are substituted in (66), the differential equations for p, u, and w are found to be - ~ ' ' liM The second equation of (68) admits the integral d (69) where d is an arbitrary constant. Since u and p vanish with n, we substitute where do = -v/V8(l + 5), and X is an undetermined constant. On substi- tuting (69) in (68), we obtain " (70) 174. Proof of Existence of Periodic Solutions of Equations (70). For M = equations (70) admit the periodic solutions p = p = w = w = Q. It will now be proved that if |M| is not zero, but sufficiently small, equations (70) admit solutions expansible as converging power series in /*, which vanish with jt and which are periodic in T with the period 2*-. Let us take the initial conditions a,, p(0)=0, tt(0) = 0, tb(0) = a,. (71) With these initial conditions it can be shown from the properties of (70), by the usual method, that p is even in T and that w is odd in T. Therefore P(T)=P(-T), >(*) = U>(-T), and if the conditions P(T)=U;(T) = O, (72) are satisfied, p and w will be periodic in T with the period 2r. 342 PERIODIC ORBITS. Equations (70) will now be integrated as power series in ^ , a, , and p in so far as the a! and a 2 enter linearly in the solutions. If the terms of the solutions in which the Oj and a 2 enter linearly are denoted by p l and w l , then the differential equations defining p l and w t are (73) The first equation of (73) is independent of the second equation. The complementary function of the first equation is where A^ and .B! are constants of integration. The function P t involves \J/ to even degrees, and is therefore a power series in a 2 with cosines of even multiples of ^ in the coefficients. The highest multiple of T in the coeffi- cient of a 2 * is 2k. In the sequel, such a power series is called a triply even power series. The particular integral arising from P t is a triply even power series unless WsU + S) is an even integer. If Wsd + S) is an even integer the left side of the first equation of (73) has the same period as certain terms of the right side, and the solution will therefore contain non-periodic terms. When V 1 /s(l + 5) is an even integer, the period of the motion of m 1 and m t is an even integral multiple of the period of the oscillations of /z. The mutual attractions of the three bodies will then have a cumulative effect and produce non-periodic motion. We therefore exclude from our consideration those values of a for which Ws(l + 5) is an even integer. With this restriction upon a, the solution of the p r equation satisfying the initial conditions (71) is T), (74) where C t (r) is a triply even power series, and it contains A as an undeter- mined constant. When (74) is substituted in the ^-equation, all the terms of W l are known. With the left side simplified and ^ = 0, the equation becomes >,+ [! + HX'Jw^O, (75) where 6, = - I- + 9 cos2r, 4 = ^ - 48 cos2r + ^ cos 4r, 6Z o and where each 2 * is a sum of cosines of even multiples of r, the highest multiple being 2k. ISOSCELES-TRIANGLE SOLUTIONS. 343 Equation (75) is one of the equations of t'uriulion , and the expression ^(r) or ^-,(t- O/WsU + 6) ! , obtained in (19), is the generating solution. Two jirl>itr:iry constants, viz., < and a, appear in its generating solution, and according to j^2 and M the two fundamental solutions of (75) are obtained liy taking the tir.-t partial derivatives of ^ with re-pcct to these constants. One solution is therefore and it is periodic in r with the period 2ir. This solution contains the factor a, and since it is multiplied later by an undetermined constant the factor a may be absorbed by the undetermined constant. This solu- tion can then be expressed as (see page 330 for ^) 2 ^a w = cosT + a f (cos3T cosr)+ , (76) 1-0 where titp = ty/dr. Therefore the ^ are sums of cosines of odd multiples of T, the highest multiple being 2./+1. The initial values of this solution are The other solution of (75) is obtained by differentiating the generating solution with respect to the constant a; hence this solution is (78) where f^) denotes that the differentiation is performed only in so far as a \da/ occurs explicitly. Now ?M kda and therefore the solution (78) is The initial values of this solution are u> u (0)=0, tb lf (0) = B 344 PERIODIC ORBITS. Since it is more convenient for computation to have a solution w l2 in which the initial values are w u (0)=0, w u (0) = l, (79) we take as the second solution of (75) (80) where x and A are found by computation to be 00 - X = 2xsX'=sinT + a 2 (5sinT+sin3r)+ , ^=-Wf + From the way in which x has been derived, viz., it follows that each x 2 > is a sum of sines of odd multiples of T and that the highest multiple is 2j+l. Further, since it follows from the character of \j/ that the coefficients of the cosines and sines of the highest multiples of r in

[x+ATd, (81) where n"' and n are constants of integration. When (74) is substituted in (73), W l becomes an odd power series in a with two types of terms in the coefficients. (1) There are terms not multiplied by cosV 1 /8(l + 5)r which enter through p t and they form a triply odd power series. They have ^ as a factor and will be denoted by fj. M 1 , (2) The remaining part of W t consists of terms which are multiplied by cos WsU + S) T. As we have already excluded those values of a for which WsU + S) is an even integer, and as we subsequently exclude those values of a for which WsU + S) is an odd integer, these terms in W l do not have the period 2ir. In the direct construction of the solutions such terms do not appear in the right members of the ^-equations. They appear as the complementary functions of the p,-equations and, since they do not have the period 2tr, they are excluded by assigning zero values to the con- stants of integration. We can, therefore, disregard the terms in W^ which do not have the period 2ir and consider W l to have the form ISOSCELES-TRIANGLE SOLUTIONS. 345 By varying the paraim-trr- nf and 7^' in (81), we obtain (82) The determinant of the coefficients of nj" and nj" in (82) is a constant [ IS], and from (77) and (79) it is seen that the value is unity. Equation.- (82) can therefore be solved for n^' and n, <0 , and the solutions are (83) integrating these equations, we obtain (84) The 7?"' and ij"' are the constants of integration. The A/J" is a power series in if with constant coefficients. The A/{ is a power series in odd powers of a with sines of even multiples of r in the coefficients, the highest multiple of T in the coefficient of a u+I being 2k+2. The R l has the same form as the M, u> except that it has cosines instead of sines. Since the coefficients of a w cos (2j + 1 ) T and a?sm(2j+l)T occurring in and \ respectively are equal, so also the coefficients of a w+l sin (2j + 2) T and a v+1 cos(2j + 2)T in A/, U) and R l respectively are equal. When (84) is substituted in (81) the solution of the second equation in (73) is found to be u> 1 = ,?V+inx+4Td-M(2*r)-M>(0) = 0. (90) These four conditions can not be satisfied by the three constants a, unless one condition is a consequence of the other three. We now show that the last condition can be suppressed when the first three have been imposed. The original differential equations (64) admit the integral |i +const. When the substitutions (65) and (67) are made and u is eliminated by means of (69), this integral takes the form UofTp) + [(i+p) 2 +4(i 2 +MW+) 2 ]' + const -J ' ISOSCELES-TRIANGLE SOLUTIONS. 347 Let us make in (91) the usual substitutions /> = p(0)+p, /> = p(0)+p w = 0+w, w = w(Q)+w, (92) where /'), p, w, and vanish at r = 0, and let us denote the resulting equation by (91a). By putting r = 0, we obtain from (91a) an equation (916) con- nect iim the terms in (91a) independent of p, p, w, and w. When this equa- tion (916) is substituted in (91a) there results an equation of the form F(p,p,w,w) = 0, (93) in which there are no terms independent of the arguments indicated. The linear term in w enters (93) with the coefficient 8/i(l +/*)[*+(<))], which, we shall show, is different from zero at r = 2f. Since the third body is assumed to be finite, 8^(1 +/*) is distinct from zero. The coefficient of w is therefore different from zero at r= 2ir unless u?(0) = ^(2ir) = a. Now the third body, when assumed to be infinitesimal, has the initial speed ), and ti;(0)/VV8(l+8) is the additional initial speed to be so determined that the orbits shall be periodic in r with the period 2ir when the t hinl body becomes finite. If this additional initial speed is a then the whole initial speed is zero, and the third body remains at the center of gravity since there is then no force component normal to the i;-plane. In order therefore to obtain solutions in which f is not identically zero, we must consider tb(0) ^ a. Hence the coefficient of the linear term w in (93) is dis- tinct from zero at T = 2r, and therefore (93) can be solved for J(2r) as a power series in p(2ir), p(2ir), and w(2*) which vanishes with p(2r), p(2ir), and uJ(2r) . This power series is unique if X, which appears in the coefficients, is assigned. Hence if the first three conditions of (90) are imposed, then p(2ir)=p(2ir)=uJ(2T)=0; and since w(2*) vanishes with p(2r), p(2ir), and w(2r), it follows that w(2v) =0 or w(2ir) -ti>(0) =0. Therefore the fourth equation of (90) is a consequence of the first three and may be suppressed. Equations (70) will be integrated as power series in n and o,(t = l, 2, 3), but only in so far as the a, enter linearly in the solutions. The differential equations from which these linear terms in o< are obtained are the same as (73). Their solutions satisfying (89), in so far as the linear terms are con- cerned, are (94) 348 PERIODIC ORBITS. When the first three conditions of (90) are imposed on p and w, we have as a consequence of (94) = 0, [cos Vi/8(l + 5) 27T - 1 ] + ^ 1/81 + g) sin VVsI 1 + 3) 27T + terms in ju, jua 3 , and higher degree terms, (95) + terms in ju, /ia 3 , and higher degree terms, Q = 2a 3 Air+ terms in o a.,, /z, and higher degree terms. The determinant of the coefficients of the linear terms in a t in (95) is 2r], This determinant does not vanish when a is not zero, and consequently (95) can be solved for a, as power series in /*, vanishing with /*. These solutions are therefore unique if X, which enters the coefficients, is assigned. Hence the periodic solutions of (70) for p and w, with the initial conditions (89), are of the same form as those obtained in (88) for the symmetrical orbits. The unrestricted and the symmetrical orbits are unique for a not zero and for any value of X, and therefore, since the unrestricted orbits include the symmetrical orbits, all the periodic orbits are symmetrical. 176. Direct Construction of the Periodic Solutions of (70). In order to construct the periodic solutions of (70), we substitute (88) in (70) and equate the coefficients of the various powers of /*. The arbitrary constants of integration are to be so determined that w(0) =0 and that each p f and w t shall be periodic in T with the period 2w. The differential equations for the terms in p are ( 96 ) The general solution of the p r equation is where C^T) is the same triply even power series as in (74). Since the complementary function does not have the period 2w when VVs(l + 5) is not an integer, the arbitrary constants A^ and B l must be zero in order that p l shall be periodic with the period 2ir . Hence the desired solution of the ^.-equation is Pi-C^r). (97) BWMI KI.K- IKIANGLE SOLUTIONS. When (<)7) is substituted in ]\\ all the terms of It", arc known. The general solution of the (^-equation is the same as (85) except that the particular integral does not carry the factor M . Hence (98) Since all the periodic orbits are syimnetrical we impose the condition 0, from which it follows that w t (Q) = (t-i, . . . oo). (99) A- a consequence of (99), the constant >?," is zero. In order that to, shall l>e periodic, the right member of (98) must contain no terms in r except those in which it occurs under the trigonometric symbols. The non-periodic terms in 1 98) disappear if the constant t;J l) is so determined that A-tf = aM( m , from which it follows that where P u> (a l ) is a power series in a* with constant coefficients. When these values of i?, in (100). The form of the w, is apparent from (100) and (106). The form of the P>0>2) is not apparent from (97) and (102), and, before the induction can be made, it is necessary to consider the term in /x 3 in so far as p 3 is concerned. The differential equation for p 3 is P 3 , (107) where all the terms of P 3 are known. That part of P 3 independent of the Wj is a triply even power series multiplied by I/a 2 . The w } enter P 3 multi- plied by power series which are triply odd or triply even according as the W], considered together, enter to odd or even degrees respectively. These terms form a triply even power series multiplied by I/a 4 . Since P 3 contains the term w t \l/, the lowest power of a 2 in a 4 P 3 is unity. The complementary function of (107) does not have the period 2ir, and the solution desired is the particular integral. This solution has the same form as P 3 and is denoted by where the Cf n have the same form as the C in (102). ISOSCELES-TRIANGLE SOLUTIONS. ii.'i 1 Let us suppose that the p t , w t , ijj(t-l, . . . , n-1; j-1, 2), have been computed and that (109) . /-I /-O ,r' " where the C, SJ M+I) I and P is odd or even respectively. The X,, X[, . . . , n,, n' t are positive integers (or zero) such that X.XI+ +XX+^ M ;+ + M X^n-l. (112) From the form of the general term it follows that P n is a triply even power -lies multiplied by I/a', where i = (2X,-2)x;+ +(2X.-2)x;+2( Ml M;+ +*$. (113) This expression is even and has its highest value when X| = = X^ = 0, i. e., in the terms of P. in which only the w, appear. Hence, from (112), the highest value of i is 2n 2. Since P. contains the term u>,_,^, the lowest power of o' in a w ~*P. is found to be unity. Therefore the form of P. is where the P have the same form as the CJ". The only solution of (llOa) which has the period IT is the particular integral, and it has the form where the C are of the same form as the 352 PERIODIC ORBITS. When p n has been determined, all the terms in W n are known since they arise from p^ (j and n we have as the complete solution - =i M?T . This completes the induction. iMiSCELES-TKIANciLE SOLUTIONS. 177. The Periodic Solution of Equation (69). When the solution for p is substituted in (li'.)i, u can be obtained by a single integration. Since p is a power series iii n with coeflicients which are triply even power series multiplied by l/r', the right side of (69) will contain terms independent of r. After the integration, u will therefore contain a term in T with X appearing in the coeliicient. \Ye shall now show that X can be so determined that this coefficient shall be zero. When X has been so determined u will be periodic in T with the period 2r. The solution for p, obtained in (97), in so far as the term in X is con- cerned, is \\ hen (116) is substituted in (69) and d is replaced by d,,+X/x, the constant terms appearing on the right side of (69) are 3 Xju+ higher degree terms in \n and n. (117) Since X carries the factor //, we may replace \n by ) and (120), and the solution already obtained for p, in which X is to be replaced by (119). The various X, are determined so that the right side of 354 PERIODIC ORBITS. (69) shall contain no constant terms in the coefficients of the n 1 The con- stant term appearing in the coefficient of n is cx> o r\ "v \ fly) j-i^l o LA O *~~ t AO d \ 9 the Xo 2 ^ being known constants. This term is zero if OO X = X a . It is necessary to consider the terms up to fj? before the induction to the general term can be made. The constant term appearing in the coefficient of M 2 has the form 00 o r\ v* \ (2/> rt 2.n O[A 1 2j A t 1 \, where the Xf^ are known constants. This term vanishes if \ has the value OO /=0 The constant term in the coefficient of ^' has a similar form; that is, it can be written J-O the Xj 2J) being known constants, and this term vanishes if \ is so determined that -9 2j ^ a Suppose Xo, \ , . . . , X B _i have been uniquely determined in the same way and that J-0 the \ being known constants. From the form of p t in (115) it follows that the constant term in the coefficient of M" +I has the form a ], J=0 where the X" are constants derived from the p } and X,(t = l, . . . , n-1), and are therefore known. This constant term is zero if ^--JM The same process of determining the X, obviously can be indefinitely con- tinued. I-" IUIAM.1.K SOLUTIONS. \\ ith X thus determined as a power scries in p, the integration of (69) yields periodic terms only, and from the form of the p, it follows that the u t have the form the f/ are sums of sines of even multiples of r, the highest multiple being 2j. The periodic solution of (69) is'therefore where t/ is the constant of integration. Since the mass m, is started from the point 0, 0, 1/2, at = < or at r = 0, it follows thatt> = at T = and there- fore, from (67), the value of u at r = is zero. Hence the constant U is zero. 178. The Character of the Periodic Solutions. When the periodic solu- t ions for p, u, and w have been determined, the solutions for , , 17, , f, (t = 1 , 2) and f are obtained by means of the equations (67), (65), (63), and (62). These solutions are all periodic in t with the period P = 2*-\/ 1 /8(l-r-5)- Three arbitrary constants appear in the solutions, viz., a, /*, and ^. The expression a/vVsU+di) represents the initial speed of the third body in I, n the mass of the third body, and ^ the epoch. The mass n is restricted in magnitude but can be increased step by step by making the analytic continuation of the solutions already obtained, provided the series do not pass through any singularities in the intervals. This can be done by the process already developed. The parameter a is restricted in mag- nitude and so that -y/VsU+S) is not an integer. As already stated in 166, it can be shown from equation (2) that the motion of the infinitesimal body will be periodic if the initial conditions are chosen so that the constant C is negative. With the initial conditions chosen as in (4), the constant C has the value 4 j2ay(l + S)-l}. Now if 2ay(l + 5) = l or >1, the infini- tesimal body recedes to infinity with a velocity which is zero or greater than zero respectively, and therefore the motion will not be periodic. Hence a must be restricted so that 2a*/(l + 6) fl U u / /\ I A/ it vi 79 I/ Ly // I f-\ t t 79 M" __ ,yi ( ai' \ ^ 1 * - J>* fV) ^^, V* 41 ^1^ x'V* 4) i 1 ^ 'VM / I \V I ~~ A/ //to_ /f | ^. / I/ "* IV I/ Vn N ' ' rt*Z * ,3** * * r 2 3r ' 3z; ' r/_ _ 1 rcosit^F) (1) [^ 2 -2r-^cos(t;-y)+r 2 ] } ^ 2 where r and t> are the polar coordinates of infinitesimal body, and R and y are the polar coordinates of m t . In the present discussion those orbits are considered in which r is small relatively to R. In this case U is expansible as a converging power series in r/R, and equations (1) become (2) It was assumed in the beginning that the relative motion of m^ and ra, is circular. Hence we have, from the two-body problem, Q, (3) where N is the angular velocity of the relative motion of the finite bodies. When the right members of the second and third equations of (2) are put equal to zero, that is, when the infinitesimal body is supposed to revolve about m l without being disturbed by ra 2 , they have the particular solution r = a = constant, v= ^ (t 4) =n(t Q, (4) U where n is the angular velocity of the infinitesimal body with respect to m t . In the periodic solutions of (2) when the right members are included, the mean angular velocity will be kept equal to n and a will be defined by the equation n 2 a' = A; 2 m 1 . (5) INHMIKSIMAL SATELLITES AND INFERIOR PLANETS. Since ///, is given, a has three determinations, two of which are complex. In the physical two-body problem there is interest only in the real value of a, and it is immaterial whether corn is regarded as given. It will be observed, however, from the purely astronomical point of view, that n can be determined by observation much more accurately than a, because the former is an angular variable and any error in its determination causes the theory to deviate secularly from the observations; while the latter is a linear variable and the discrepancies which arise from errors in its determination do not accumu- late. But in the three-body problem all three values of a must be included in determining orbits whose period is defined by n, because all the orbits may become real for certain values of the parameters which occur in the right members of the differential equations. In fact, Darwin found three real orbits for certain values of the Jacobian constant.* In the Lunar Theory, as developed by de Pont4coulant, for example, the solutions were forced into the trigonometric form by various artifices. But it will be noticed that the method of procedure was the opposite of that adopted here, so far as comparison can be made, in that the mean motion was continually modified by de Pont^coulant in order to preserve the trigonometric form; while here the mean motion is regarded as being given arbitrarily in advance by the observations or otherwise, and it is kept fixed. New quantities p, w, r, and m will be introduced by the equations (6) For brevity a/ A will be written in place of its expression as a power scries. As a result of these transformations, the last two equations of (2) become !j(l+p)[3cos(T+uO+5cos3(T+u>)]+ (7) where the dots over p and w indicate derivatives with respect to T. These equations are valid for the determination of the motion of the infinitesimal body as long as a(l+p) is less than A. Ada MaOiematica, vol. XXI (1807), pp. W-242. 360 PERIODIC ORBITS. It follows from equations (6) that ap and w are the deviations from uni- form circular motion due to the right members of the differential equations. They are functions of m, and the initial conditions are to be determined so that they shall be periodic in T with the period 2ir. Since the right members of (7) contain m 2 as a factor, the periodic expressions for p and w will contain m 2 as a factor. 181. Proof of the Existence of the Periodic Solutions. For m = Q equations (7) admit the periodic solution p = w = 0. It will now be proved that for m distinct from zero, but sufficiently small, equations (7) admit a periodic solution which has the period 2ir in T, and which is expansible as a power series in m 1/3 , vanishing with m. It is the analytic continuation of the solution p = w = m = Q with respect to m as the parameter. Since T enters explicitly in the right members of (7) in terms having the period 2ir, it follows that the period of the solution must be 2?r, or a multiple of 2ir. Equations (2) are not altered if we change the sign of v, V, and t. It easily follows from this that, if v (0 = V (0 = p' (<) = 0, the dependent vari- ables p and v are even and odd functions respectively of t 1 . An orbit in which these conditions are satisfied is symmetrical with respect to a line always passing through m 1 and w 2 . Such an orbit will be called symmetrical. Expressed in the variables of (7), p and w are respectively even and odd functions of T in the case of symmetrical orbits; and w(ty =p(0) =0. The existence of symmetrical periodic orbits will be established, and then it will be shown, in connection with the construction of the solutions, that the condition that the solutions are periodic implies that they are also sym- metrical. It will follow from this that all of the periodic orbits of the type under consideration are symmetrical. Suppose that the initial conditions are (8) If the right members of equations (7) are neglected and the transformation is made, then equations (7) reduce to the ordinary polar equations of the two-body problem for the motion of the infinitesimal body with respect to m l . In this two-body problem with the initial conditions (8), it is found that aa is the increment to the semi-axis a, and e is the eccentricity of the orbit of the infinitesimal body. Since the properties of the solution of the two-body problem in terms of the major semi-axis and eccentricity are known, we can at once write down the properties of the solution of (7) in terms of a and e, so far as they are independent of the right members of the equations. It was precisely for this reason that p and w were given the peculiar initial values defined in (8). INHMI1>1\IAL SATELLITES AND INFERIOR PLANETS. 361 Equations (7) arc now integrated as |>o\ver series in a, e, and m {/t . The results have the form p = p l (a,e,m^;r), w = p, (a, e, w 1 *; T), 1 P = p t (a, e, w"; T), w = p. (a, e, ro 1 *; T). The moduli of a, c, and m' ;i can be taken so small that the series converge while T runs through any finite range of values starting from zero. The period must be a multiple of 2*-, say 2kir, and consequently it will be supposed that these parameters have such small moduli that (9) converge for O^r^A'ir. It follows from the symmetry of the orbits under the initial conditions (8), that if, at r = kir, the three bodies are in a line, and if the infinitesimal body is crossing perpendicularly the rotating line which joins w r and m t , then the orbit necessarily re-enters at 2kr, and the motion is periodic with the period 2kir. Conversely, if, at r = 0, the orbit crosses perpendicularly the rotating line which joins m, and m t , and if the motion is periodic with the period 2ir, then the orbit at r = kir necessarily crosses perpendicularly the rotating line which joins i, and w,. Therefore, necessary and sufficient conditions for the existence of symmetrical periodic solutions having the period 2k* are p,(a, e, m"; *) =0, p,(o, e, m*; kw) = 0. (10) Since p t = p t = at r = 0, it follows that equations (10) are identically satis- fied by a = e = w l/1 = 0. The problem of solving them for a and e as power series in m l/l , vanishing for m = 0, will now be considered. All the terms of the solu- tions which do not carry w* as a factor are obtained from the solutions of the left members of (7) set equal to zero. Since these terms belong to the two- body problem equations (9) become, when these terms are explicitly exhibited, (11) where / 19 x "(1+a)" The series 9, , . . . , q t depend upon the right members of the differential equa- tions (7). All terms in the [ ] which are not written contain e 1 as a factor. The conditions, (10), for the existence of symmetrical periodic solutions become as a consequence of (11) - 1)+ - + J-l-m 1 &(a, e, m 1 ";^, (l+m)T+T-r-2esinjT-f +m t g,(o,e, m^; T), t (a,e,m tn ;kir) = 0, [(13) R(*) - L-(l-H)*r-h*r-f aena*4- 362 PERIODIC ORBITS. where all the unwritten terms in the [ ] carry e 2 as a factor and are linear functions of sines of multiples of vkir. On substituting the value of v from (12), it is found that r = - sn m-a where j is an integer. Every term in the [ ] contains either m or a as a factor; therefore every term of the second equation of (13) contains either m or a as a factor. The coefficient of a to the first power is 3/2 kir, which is distinct from zero; therefore the second equation of (13) can be solved for o as a power series in m and e, vanishing with m = 0. The term of lowest degree in m alone is the second, and its coefficient depends upon the right member of the differential equations (7). Therefore the solution of the second equation for a carries m 2 as a factor, and has the form a = m*p(m 1/3 , e). (14) Suppose a is eliminated from the first of (13) by means of (14). After the elimination, a factor m can be divided out. The result then contains a term in e alone to the first degree, and its coefficient is ( 1)*. Therefore the resulting equation can be solved for e as a power series in m t/3 , vanishing with m = 0, of the form l/3 ). (15) As a matter of fact, the expression for e contains m 2 as a factor, as can easily be shown. Suppose equations (7) are integrated as power series in m 1/3 , and let the initial conditions be p = P = w - w = 0, in order to get the terms which are independent of a and e. The series will have the form p = p 1 ra+p 2 ra 2 +p 3 ra 8/3 + > w = The explicit result of the integration is r-l+2cos-r-cos2r1, 1 - From these equations it follows that p 2 (kir) has m 2 as a factor, but that it does not have m 3 as a factor. Therefore the expression for e as a power series in w 1/3 contains m 2 as a factor. Then (15) and (14) together give o = m t P 1 (w 1A ), e = m 2 P 8 (m 1/3 ), (16) INFINITESIMAL SATELLITES AND INFEHIoK I'LANETB. where I\ and P t arc power series in m 1 ' 3 which converge for the modulus of m sufficiently small. Therefore t he symmetrical periodic orbits e\i>t.and equations (16) and (8) give the initial values of the dependent variables in terms of the parameter m 1 " which is defined, except for the cube root of unity, by the data of the problem and the third equation of (6). 182. Properties of the Periodic Solutions. The periodic orbits whose existence lias been proved re-enter after the period 2for, where k is any integer. Those orbits for which k is greater than unity include those for which k equals unity. Since, according to the discussion which has just been made, the number of periodic orbits is the same for all values of k, it follows that the period of the solutions is 2r. When w = the infinitesimal body makes a revolution in 2*-, and m can be taken so small that the orbit is as near this undisturbed orbit as may be desired. Therefore a synodic revolution is made in 2ir for all |m| sufficiently small. If the expressions for a and e given in (16) are substituted in (9), the result becomes p=p,(T)m'", w- Z^Wro'*, (17) where the summation starts with .7 = 6, because the expressions for a and e have m 1 as a factor, and p and w have no terms in m alone of degree less than the second. The p and w are periodic with the period 2* for \m\ sufficiently small because the conditions for periodicity have been satisfied. Therefore I Pj (T + 2 T)ro' /J = S p, (T)ro", ^ w, (r + 2 T)m' /J = 2 w, (r)m'" ; whence p,(T+2T) = p,(r), Wj(T+2^=Wj(r') (j-6, ...). (18) Since p(0) = u?(0) =0, it follows that 2 p, (0) m" 1 = 0, 2 w, (0)m'/' = ; ;- /- whence p/0)=0, t^(0)=0 tf-6, ...oo). (19) If the orbit of the infinitesimal body is retrograde, n is negative and m has the definition m= N/(n+N) for a given sidereal period. Therefore, for a given numerical value of n, the parameter m is smaller in retrograde motion than it is in direct motion. For a given sidereal period the deviations from circular motion are less in the retrograde orbits than they are in the direct. The physical reason is that the disturbance of the motion of the infinitesimal body by m, is greatest when the three bodies are in a line, as can be seen from (7) or by graphically resolving the disturbing acceleration; and in retrograde motion this approxi- mate condition lasts a shorter time than in direct motion. 364 PERIODIC ORBITS. DIRECT CONSTRUCTION OF THE PERIODIC SOLUTIONS. 183. General Considerations.- It has been proved that equations (7) have solutions of the form (17) which satisfy (18) and (19). The solutions are in m l/3 only because a/ A is a series in m 1/3 , given explicitly in (6). The expression for a/A can be modified by writing = Mm, (20) where M is to be regarded in the analysis as a constant independent of m. This amounts to a generalization of the m as it appears in certain places in the last equation of (6). The particular transformation (20) is made in order that the right members of (7) shall be in integral powers of m. The proof of the existence of the periodic solutions can be made precisely as before, because the transformation (20) affects only the higher terms which were not explicitly used. While there is nothing essential* in the trans- formation, it will be made for the sake of convenience, after which equations (7) become (21) where In the right member of the first equation of (21) the coefficient of m' is a sum of cosines of integral multiples of T+W, the highest multiple being j; the coefficient of m j in the right member of the second equation is a sum of sines of integral multiples of (T+W), the highest multiple being j. In a closed orbit around m l there are two points at which w is zero. The arbitrary will be so determined that w(0)=0. The first condition of (8) will not be imposed in advance, and it will be shown that it is a conse- quence of the others. It will follow from this that all of the periodic solutions of the type under consideration are symmetrical. Equations (21) will there- fore be integrated in the form CO 00 p = S Pi m', w=2 w } m! (22) ,/=2 subject to the conditions (18) and the second of (19). *A different transformation was made in Transactions of the American Mathematical Society, vol. VII, (1906), p. 542. INFINITESIMAL 8ATKI.I.ITKS AND INFERIOR PLANETS. 184. Coefficients of m-. 'l'\n-- term- an- y the (Munitions P 1 -3p J -2-, = |;(l+3cos2T), tP t +2^= -|rj8in2r, - A the general solution of which is P, = \ i\ + 2 c{" + c? cos T -f cf sin T - jj cos 2r , "': = cr- (7j+3c I tt) )r-2c I fI > 8 inT+2c i a) cosT+ TJ sin2r, (23) when- r,-', . . . , c are the constants of integration. By conditions (18) and tin- second of (19), it follows that (24) Therefore the solution (23) becomes p,= w t = - 2 cj 8 - 2c sin r + 2^ cos r + ij-ij sin 2r, O where ci" and cf are constants which remain so far undetermined. (25) 185. Coefficients of m'. The differential equations which define the t en i is of the third degree in m are iv t +2p t = -2p,- 1 = f 2cf - 1 Mrj) sin T - 2 c?> cos T - 4 17 sin 2r - ^ Mn sin 3r. \ o / 8 From the second of these equations it is found that which substituted in the first gives 366 PERIODIC ORBITS. In order that the solution of this equation shall be periodic the coefficients of COST and SUIT must be zero; whence _ J2 /->9\ Cj 77. v'J^/ With this value of cf it is found, upon integrating (31) and imposing the second condition of (19), that, so far as the computation has been made, 1 15 TI ,r 15 it f !! P 2 = -77 + M77COS7 7jcos27, w> 2 = A/?? sin 7+--77sm27, 6 P 3 = +-77 +cf cos7+cf sin7- -7JCOS27 - M77cos37, [(33) w 3 =- 2cf + 2cf cos 7 - ( 2 cf + 1 M-n ) sin 7 + ^ 77 sin 2 r + ^77 sin 37, where cf and cf are constants which are as yet undetermined. 186. Coefficients of m 4 . The integration will be carried one step further and then the induction to the general term of the solution will be made. The differential equations which define the coefficients of m 4 are - 3 w 2 sin 2r -f ^ Af 2 r? [9 + 20 cos 2r + 35 cos 4r] , (34) .^ ' 4 = p 2 w 2 ~ 2 p 3 2p 2 w 2 - p 2 sin 2r 3ir 2 cos 2r IMIMI1MMAL SATELLITES AND INFERIOR PLANETS. Upon developing the explicit values of the right members of (34) by means of (33.), it U found that -f { M , ] cos 2r + |M n cos 3r + [I ,'+ f 6 M V,] cos 4r, (35) The first integral of the second of these equations is (36) where c{ is an undetermined constant. Then, on substituting this value of w^ , the first equation of (35) becomes (37) In order that the solution of (37) shall be periodic, the conditions ci = fM,'-|M^ cr=0 (38) must be satisfied. Then its general solution becomes 675 M t t _i_ JL 512 M ' " h !6 J (39) where of, cJ 4) , and ci" are as yet undetermined constants. 368 PERIODIC ORBITS. If (39) is substituted in (36), it is found, by using (38), that (40) The periodicity condition determines cj* by the equation Then the integral of (40) satisfying the second condition of (19) is (42) The results so far obtained are P 2 = t =- 2c< 4) +2cf cos r - [2c< 4) + 1| Mr; 2 - ^ Af 1,] sin r AfV+|AP,]8in2r (43) where rj= - ^ - , and where Cj 4> and c"' are so far undetermined. ~~ IXFIXITF.SIM.M. S-ATF.I.I.MT.s AND INFERIOR PLANETS. 369 It is observed that, so far as the variables are completely determined, the p, and are sums df cosines ;iiid sine- respectively of integral multiples of r, the highest multiple being 7. At the j' h step of the intejiration one of the four arl lit rary constants which arise at that step is determined by the perio- dicity condition on the w,, and another by the initial condition on the w,. The other two constants remain undetermined until the next step, but two which arose at the preceding step are determined by the periodicity condition on p,. It will be shown that these properties are general. 187. Induction to the General Step of the Integration. Suppose P 5 , . . . , p.-,; M'J, . . . , w n ^i have been computed and have the properties expressed in the following equations: p, = C + a' l "cosT+a?cos2T+ +CL?COSJT (j=2, . . . , n-2), w,= /3 I (/) sinT+/3 1 0) sin2T+ +fftanjr (j=2, .... n-2), P.-,= +c;- 1) sinT+o;"- 1) +ci-"cosT+a;- l) cos2T+ +a ( :: I I ) cos(n-l)r, (44) where the a, 0J , and &{"~ u are known constants, and cj""' and cj"~" are undetermined constants. In writing the differential equations which define p, and IP, all unknown < luant it ies will be given explicitly. The terms involving these undetermined coefficients are the same at every step. It is found from equations (7) that the coefficients of ra" are defined by p.-3p.-2w.= +2r;-' > cosT+2cr l) sinT+P.(p,, w,, w,; r), } I (45) p / , p,, w,, w; T ).J where P. and Q n are polynomials in p,, p,, w,, and w, (j = 2, . . . , n-2) and the known parts of p._i , p._i , w.- t , and w u -\, and where T enters in the coefficients only in sines and cosines. It follows from (7) that, aside from numerical coefficients, P. has terms of the types Pi" = p h W h 0', +h = n, or n - 1 ) , PI* = p h W h tb A (Ji+h+h = ), Pi 1) = p* 1 ' p*' (kji+ ' ' ' +k r j r = n, n l,orn 2), Pi 4 ' = M'p*' t p};u' wJji'i'JTJj O'-0, 1, .... n-2; *;,+ +! t'^.7+2; j+kJt+ +k,j,+\pi+ +\p If X,+ +X M iseven, the term is multiplied by costV; and if X,+ is odd, the term is multiplied by sinir. The terms Pi", Pf, and P? come from the left member of (7), and P? comes from the right member. 370 PERIODIC ORBITS. It follows at once from (44) and the conditions on the j, and k t that P\ Pn\ an d P are sums of cosines of integral multiples r, the highest multiple being n at most. If X t + +X M is even, the product w% w^ is a sum of cosine terms, and it follows therefore that in this case P M 4) is a sum of cosines of integral multiples of r. The highest multiple of T is which becomes, as a consequence of the relations to which the exponents and subscripts are subject, If A!+ +X M is odd, the product w% wfy is a sum of sines of integral multiples of T. Therefore, in this case also, P is a sum of cosines of integral multiples of T; and it is shown, precisely as before, that the highest multiple is n. Hence the general conclusion is that P B is a sum of cosines of integral multiples of T, the highest multiple being n. By a similar discussion it can be proved that Q n is a sum of sines of integral multiples of T, the highest being n. Hence equations (45) can be written in the form (46) where the A and the B Q' = 0, . . . , ri) are known constants. The first integral of the second equation of (46) is w n = - 2 pn+c? - 2 c'"-" sin r - [2 c?- +B?] cos T - 1 B cos 2r j (47) _ ... $" C os nr, n where cJ B) is an undetermined constant. On substituting equation (47) in the first of (46), it is found that cosnr, w a +2 Pn =-2 cf- l> cos r+[2 c'"-" +B] sin r +5 B n) sinnr, 2 D(n) 1 cos nr. (48) In order that the solution of this equation shall be periodic, the conditions (49) must be imposed. They uniquely determine the constants cJ"~ J and cj""", which remained undetermined at the preceding step of the integration. INKIMCKSIMAL SATKLUTKS AND INKKUloK PLANETS. 371 After e(iuations (49) arc fulfilled, the general solution of (48) is of the form p. = ci" ) siiiT + ai" > +ci" ) cosT + ai" > co82r+ + ai," cos HT, (50) where , /' and c arc arbitrary constants, and where a <->- _L_r4<->- * <>-)- \JA?-2BT] < 51 > tt; -j . \_ I 4 ' J ~ 1(1* i i" U~*> i "/ If ein The results obtained at this step are +< ) cosjr + +a| > 11) cosnr, + + ft"' sin TIT, (55) where ci" and cj m) are as yet undetermined constants. 372 PERIODIC ORBITS. Since the results expressed in the first two equations of this set are identical in properties with the equations (44), with which the discussion of the general step was started, and since the properties of (44) were fulfilled for the subscripts 2, 3, and 4, it follows that the induction is complete. The process of integration can be carried as far as may be desired. The hypotheses under which the discussion has been made are that the solutions are periodic and that w(Q) = 0. Solutions satisfying these properties and p(0) =0 were known to exist from the existence discussion, and therefore they could certainly be found because the assumed properties are included in those of the symmetrical orbits. It appears in the construction that the hypotheses adopted imply also that p(0) =0. Therefore all periodic orbits of the type under discussion which are expansible as power series in m are symmetrical orbits. It can be shown by direct consideration of the series that there are no others expansible in any fractional powers of m. 188. Application of Jacobi's Integral. The differential equations admit Jacobi's integral, of which no use has yet been made, and it is the only integral not involving the independent variable. Upon transforming the integral to the variables used in this chapter, it is found without difficulty that its explicit form is (56) where C is the constant of integration. It will be shown that this integral can be used as a searching test on the accuracy of the computations of the solutions, or to replace the second differential equation of (7). Since the periodic solutions are developable as power series in m, the integral can be expanded as a power series in m and written in the form F +F l (p i ,p j ,w 3 ,w 1 ;r)m+ +F n (p 1 ,'p ] ,w 1 ,w j ;T}m n + =C. (57) In the F n the highest value of j is n. Since the integral converges for all \m\ sufficiently small, each F n separately is constant, and therefore F(PJ, pi, Wj, w } ; r} = C a . (58) It follows from the form of (55) and (56) that F n is a sum of cosines of integral multiples of r. In F n the sum of the products of the exponents and subscripts of the factors of any term not involving COSJT or sinjr can not exceed n; and in any term involving COSJT or sinjr the sum can not exceed ni. Therefore the highest multiple of r in F n is n, and (58) can be written in the form +7fcos;V+ +7 cos TIT = (7,,. (59) 1+P INFINITESIMAL SATELLITES AND INKKIUoK PLANETS. 373 Since this relation is an identity in T, it follows that T. W -C., 7-0 0-1, ---- n). (60) relations are functions of a, . . . , aj; /3f, . . . , 0*', and their fulfillment forj= 1, . . . , n serves as a thorough check on the expansion of t/ and on all of the computations. It will now be shown how the relations (60) can be used in place of the second equation of (7). If (56) is expanded as a power series in m, it is found that F. = 4 p.+2w.+4 pb.,+0. (p,, p,, tb,; T), (61) where G, is a polynomial in p h p,, w if and w, and involves r only in sines and cosines. Moreover, the greatest value of j in G, is n 2. Suppose that p,, . . . , p._,; u> lf . . . , w>._, are entirely known, and that p._, and to,_i are known except for the undetermined coefficients cj"~"; it will be shown that equations (60) and the third and fourth equations of (55) define the aj" and /3J" uniquely. It follows from the properties of F u and equation (61) that this func- tion can be written in the form Consequently equations (60) become (62) 7, w = 4ar+2^r+CT = It follows from ccjuations (55) that Upon comparing equations (62) and (63), it is found that (63) 2BJ*-/Cy (n = 2, . . . oo ; j-1, . . . , n). (64) Therefore the third to the seventh equations of (55) can be written _ - , -, _ _- . 1 - - (65) 374 PERIODIC ORBITS. These equations express the coefficients, which are determined at this step of the integration, uniquely in terms of constants which depend only upon the first equation of (7) and upon the integral. In practical computation it is more convenient to make the determination of the coefficients depend upon the A and C than upon the A and B, for the former have many terms in common, except for numerical multipliers, and both are coefficients of cosine series, which are easier to check than are the sine series on which the B depend. But the chances of error in lengthy computations are so great that if the developments are to be made to high powers of m, the only safe method is to use both the second equation of (7) and the integral, or, what is the same thing, to secure the fulfillment of equations (64). In order to illustrate the process the expression for F^ will be developed. It is found from equation (56) that (66) Upon developing the right member explicitly by means of the first four equations of (43), it is found that sin 2r+^M*r) [9+20 cos2r+ 35 cos4r] . cos 2r + 4a + 6#> + f It is found in the notation of (62), and by comparing with the right member of the second equation of (35), that (67) exactly fulfilling equations (64). INFINITESIMAL SATELLITES AND INFERIOR PLANETS. 375 189. The Solutions as Functions of the Jacobian Constant. It follows from equation (57) that when the periodic solution is given, the constant C is uniquely denned. The relation between C and the constant of the Jacobian integral, when it is expressed in terms of the variables in more ordinary use, will ho found. If the origin is taken at the center of gravity of the system, the differential equations of motion in rectangular coordinates are t _ dU f OU jr K Jill | " | ^A -= '- J fy ^S ~ ' " f dx aw n (68) These equations admit the integral x*+y'*-2N(xy'-yx') = 2U-C t , (69) where C is the constant of the Jacobian integral and Af is denned in (3). The relation between C and C of equation (56) is required. The variables x and y are expressed in terms of the polar coordinates, r and v of (1), by the equations z = rcosi; AcosNt, y = rs\nv- AsinM; from which it follows that z"+ y'* - 2N(xy' - yx') = r"+rV' - 2Nr*v' 2m 1 AJVrcoB(-M) - mJAW. (70) Upon making the transformations (6) and referring to (3) and (5), it is easily found that , , = r, r, w f (l+p) (71) Upon substituting (71) in (70) and (69) and comparing with (56), the relation between C 8 and C is found to be 376 PERIODIC ORBITS. It is seen from (56) that when C is expanded as a power series in m, or in m 1/3 if a/ A is eliminated by the last equation of (6), it starts off with a term which is independent of m. Therefore C , the Jacobian constant for the integral in the ordinary form, is expansible as a power series in m 1/3 and is infinite for m = 0. The three periodic orbits, of which two are complex for \m\ sufficiently small, corresponding to the three determinations of a/A in (6), coincide and branch at m = or C = oo . Since the coordinates in the periodic orbits are analytic functions of w 1/3 , and w 1/3 is an analytic function of C through the inversion of (72), it follows that the coordinates in the periodic orbits are analytic functions of C a . One branch-point is at C = oo . In the special problem treated by Darwin,* in which the ratio of the finite masses is 10 to 1, he found by computation in the case of the orbits around the smaller finite mass that there is another branch-point for a certain value of C , at which the complex orbits first become real and coincident, and then real and distinct. 190. Applications to the Lunar Theory. In the development of the Lunar Theory the differential equations have been so treated that the resulting expression for the distance, or its reciprocal, is a sum of terms which involve the time only under the cosine and sine functions. The longitude involves terms of the same type and the time multiplied by a constant factor. Considering the problem in the plane of the ecliptic, there are terms whose period is equal to one-half the synodic period of the moon. They are known as the variational terms. Now the period of the periodic orbits which have been found above is the synodic period of the revolution of the infinitesimal body, or twice that of the variational terms. The terms of the solutions which are of even degree in n have the period of half the synodical period of revolution. The variational terms in fact belong to the class of periodic orbits treated here. The detailed comparison, up to m 9 , with the work of Delaunay was made in the Transactions of the American Mathematical Society, vol. VII (1906), p. 562, and perfect agreement was found except in the coefficients of the higher powers of m, where errors are almost unavoid- able in Delaunay's complicated method. Hill wrote a remarkable series of papers on the Lunar Theory in the American Journal of Mathematics, vol. I (1878), in which he proposed to start from the variational orbit, instead of from an ellipse, as an intermediate orbit for the determination of the motion of the moon. The elliptic orbit as an intermediate orbit came down from Newton and his successors, and the inertia of the human mind is such that it was retained for over a century in spite of the fact that it has little to recommend it. Hill has the great honor of initiating a new movement which, it seems certain, will be of the highest importance. Ada Mathematica, vol. XXI (1897), pp. 99-242. 1M I\III>I\IAL SATELLITES AND INKKItloK PLANETS. 877 The results obtained by Hill are coextensive with those given here if \ve put M = and 77 = 1 in the latter series. The method employed by Hill was entirely different from that of this chapter. It was convenient in prac- tice, but its validity can not easily be established. The same method was extended by Broun to include terms which contain M as a factor* to the first, second, and third degrees. A comparison of the results obtained by the met hods of this paper with those of several writers on the Lunar Theory, especially in the coefficient of a/ A which converges most slowly, will be found in the Transactions of the American Mathematical Society, vol. VII (1906), p. 569. 191. Applications to Darwin's Periodic Orbits. In Darwin's compu- tations, f the ratio of the masses of the finite bodies was ten to one. It is found from the definition of j and the last equation of (6) that for the motion around the smaller of the finite bodies a_ _ / m l \ l/l /WV*... fjA 1 * ( m y A Vmi+Tnt/ \n/ \U/ \i+m/ Darwin defined his orbits by the value of the Jacobian constant, and their periods were found from the detailed computations. In comparing with his work it is simpler to take the periods which he obtained and to find the orbits from equation (43). The comparison will be made first with his "Satellite .A" for the Jacobian constant in his notation equal to 40.5, loc. cit., p. 199. The synodic period was found to be 61 23'= 61.383, where the period of the finite bodies is 360. Therefore ro - yp =0.17051, f = (^"(^ST)" -0-12449. (74) The m for this orbit is more than twice that occurring in the Lunar Theory. With these values of the constants substituted in the series of 186, it is found that r = 0.1 2427 +0.00652 cosr-0.00420 cos2T+0.00004 cos3r -0.00006 cos 4r+ (75) w= -0.12062 sin T+ 0.05079 sin2r-0.00184 sin3r + 0.00095 sin 4r+ The infinitesimal body is in a line with the finite bodies and between them when r = 0. The value of r at this time is found from (75) to be r(0) =0.12657. The corresponding value given by Darwin is 0.1265. The infinitesimal body is in opposition at r = ir, and it is found from (75) that r(*0 = 0.11345. Darwin's value is 0.1135. These agreements show that the "Satellites A" are of the class treated in this chapter. American Journal of MaUiemaKct, vol. XIV (1891), pp. 14O-160. \.\cta MaUumatica, vol. XXI (1887), pp. 99-242. 378 PERIODIC ORBITS. In a retrograde orbit having the same sidereal period, the expression for m is - N - N/n n+N l+N/n In this case N 613.83 n 61.383 + 360' therefore m= -0.12715. The value of a/ A is the same as before, and the series for r gives r = 0.12412 -0.00057 cos r- 0.00158 cos 2r- 0.00000 cos 3r -0.00001 cos 4r+ (77) w = 0.00820 sin r+0.01654 sin 2r+0.00047 sin 3r +0.00010 sin 4r+ which are seen to converge somewhat more rapidly than the series of (95). No retrograde orbits were computed by Darwin in his memoir in the Ada Mathematica. Comparison will also be made with one of Darwin's "Planets A." In this case m t = 10, m 2 =l. A = l, The orbit will be taken for which the Jacobian constant is 40.0. The period given by Darwin (loc. cit. p. 225) is 154 13'. Therefore m = = 0.42838, - = 0.43404. (78) ouU xL With these values of the parameters, the series for r gives r = 0.43373+0.00776 COST- 0.01286 cos2r- 0.00104 cos 3r+ ---- (79) From this series it is found that r(0) = 0.42759, r(ir) = 0.41415. Darwin's results in the respective cases were r(0) = 0.423 and r(r)= 0.4140. The agreement of these results shows the identity of his "Planets A" and the orbits covered by the analysis of this chapter. CHAPTER XII. PERIODIC ORBITS OF SUPERIOR PLANETS. 192. Introduction. The preceding chapter was devoted to the consid- ation of the motion of an infinitesimal body subject to the attraction of two Unite bodies which revolve in circles. The periodic orbits whose existence \\as there proved inclose only one of the finite bodies, and they are more nearly circular the smaller their dimensions and the shorter their periods. The present chapter also will be devoted to the consideration of the motion of an infinitesimal body subject to the attraction of two finite bodies which revolve in circles; but the periodic orbits now under discussion inclose lioth of the finite bodies and are more nearly circular the larger their dimensions and the longer their periods. There are three families of orbits of this class in which the motion is direct, and three in which it is retrograde. For small values of the parameter in terms of which the solutions are developed, only one family each of the direct and of the retrograde orbits is real. The mode of treatment of the problem of this chapter is similar to that of the preceding. A certain parameter n naturally enters the problem. When n is zero, the problem reduces to that of two bodies, which admits a circular orbit as a periodic solution. The existence of the analytic continu- ation of this orbit with respect to the parameter n is proved, and direct methods of constructing the solutions are developed. It is shown also how th- integral can be used as a check on the computations, or as a substitute for one of the differential equations in the construction of the solutions. The results of the preceding chapter were directly applicable to the Lunar Theory; those of this chapter have no direct bearing on the practical problems of the solar system, at least as they are at present treated. Their chief value at present is that they cover a part of the field of the problem of three bodies in which one is infinitesimal and in which the finite bodies revolve in circles. 193. The Differential Equations. Let the origin of coordinates be at the center of gravity of the finite bodies w, and m, , and take the xy-plane as the plane of their motion. Suppose the infinitesimal body moves in the ///-plane. Let the coordinates of w,, w,, and the infinitesimal body be ( x i, Vi), fe> l/i), and ( x > I/) respectively. Then the differential equations of motion for the infinitesimal body are (1) _dU -. Tr _i . de"~"dx" d? "~ dy' r, r, = V(x-x t ) t +(y-Vi) t , 379 380 PERIODIC ORBITS. L ^ r=V^W, R^V^W^^r^R, -y*Y, Rt=Vxl+yl = Then, in polar coordinates, equations (1) become (2) _ = -_ /o\ df " V " dr ' r df~* dt dt ~ r dv ' The potential function U will now be developed. From (1) and (2) it is found that ff {i [1+3cos2(0 _ 1)] (4) Then equations (3) become _ r + t = _ , l + 3 cos 2 (,-,, r L L4 cos - (5) d 2 ?; -j i 2j7jT - T df dtdt mi+rrh r 4 [ If the orbits of m l and m 2 are circles, which is assumed to be the case, equations (1) admit the Jacobian integral dy_ ,. dt y dt It follows that r^ is the mean motion of the finite bodies and that v t = n 1 (t . In polar coordinates the integral becomes r' 2 +/V 2 2n,r*v' = 2U C, (7) 1 7 \ / where the primes indicate derivatives with respect to t. When the right members of (5) are put equal to zero, the equations admit the particular solution where n is the angular velocity of the infinitesimal body in its orbit and t is an arbitrary constant. It will be supposed that n is given by the observations, or that its value is assumed, and that a is determined by the second equation of (8). The constant a has three values, only one of which is real. PKIUODIC OKHI!.- >1 >l I'KKIOR PLANETS. 381 New variables, p, 9, and T, and new constants, n and M , will be introduced by the equations Af. (9) I it It follows from (6), (8), and (9) that and equations (5) become "; (io) (ii) where the dots over the letters indicate derivatives with respect to T. These equations are valid for the determination of the motion of the infinitesimal body provided |M| being cosines and sines respectively of odd multiples of T. 194. Proof of the Existence of Periodic Solutions. Suppose p = /3, p = 0, = 0, = 7 at r = 0, and let the solution of (11) be written in the form P=/03,T;T), 6 = ^(0, y; T). (12) Now make the transformation P = Pl , 0=-0 lt r=-r l . (13) The resulting equations have precisely the form (11). Consequently their solutions with the initial conditions pi = /3, p, = 0, 6 = 0, 0, = 7 are Pi=/(0, 7;T,)=/G3, T;-T) = P, 6 l = (f3,y;-T) = -0. (14) Therefore, with these initial conditions, p is an even function of T, and 6 is an odd function of T. The orbit is symmetrical with respect to the p-axis both geometrically and in r. Such an orbit will be called symmetrical, whether it is periodic or not. 382 PERIODIC ORBITS. Now consider the conditions for a closed symmetrical orbit. Since the right members of (11) involve only sines and cosines of integral multiples of r, sufficient conditions that in symmetrical orbits p and 6 shall be periodic with the period 2jir are p = /03, 7 ;=(), = *>G8,7;./V) = 0; (15) and these conditions are necessary, provided they are distinct. In order to examine the solutions of (15), it is convenient to use par- ameters other than /3 and 7. Suppose that, at r = 0, r = a(l+p)=a(l-fa)(l e), f = ap = 0, M , A_ M Vi+e ~T(/ , 1-M I-/ (16) It follows that a(l+a) and e are the major semi-axis and eccentricity of the elliptic orbit which would be obtained if the right members of equations (11) were zero. Because of the well-known properties of the solutions of the two-body problem in terms of these elements, the properties of the general solutions, so far as they do not depend upon the right members of (11), are known. These properties will be important in solving the conditions for periodic solutions. Equations (11) are regular in the vicinity of M = 0, p = 0, p = 0, 6 = 0, 6 = for all values of T. It follows that the moduli of a, e, and /x 1/3 can be taken so small that the solutions will be regular while T runs through any finite preassigned range of values. We shall choose as the interval for T the range Q^.T^2jir and integrate (11) as power series in o, e, and M I/S , vanishing with a = e = /j. l/3 = 0. That is, the results will be the analytic continuation with respect to these parameters of the particular solution r = a, v = nt, which exists when ju, = 0. The results may be written in the form > = p 3 (a,e,M V3 ;r), ) where p l} . . . , p^ are power series in a, e, and M 1/3 > with T in the coefficients. The conditions for a periodic solution, (15), become p 3 (a, e, M 1/3 J = 0, p 3 (a, e, M 1/3 ; = 0. (18) It will be shown that these equations can be solved for a and e as power series in /z I/3 , vanishing with // /3 = 0, which converge if the modulus of /* l/3 is sufficiently small. Since the right members of (11) carry /i 10/3 as a factor, the part of the solution depending on the right members will be divisible by M 10/3 - If the right members of (11) were zero and if the solution were formed with the initial conditions (16), the mean angular motion of the infinitesimal body in its orbit would],be (19) PKUIODIC (IHBIT8 OF SUPERIOR PLANETS. 383 Consequent ly, from the solution of the two-body problem, it follows that equations (18) have the form = 0, (20) when- the unwritten parts in the brackets are sines of multiples of vjr, and carry e 1 as a factor. Upon referring to (19), it is observed that the first equation of (20) is divisible by M' and the second by n. After dividing by these factors the equations are still satisfied by a = e = n = 0; moreover, the determinant of their linear terms in a and e is , jw 3 V 9^ A 7H w J J (21) Therefore, besides the solution n = 0, equations (20) have a unique solution for a and e as power series in /x !/ *, vanishing with /n Vl = 0, which converge for the modulus of M V> sufficiently small. These power series carry n v * as a factor, and can be written in the form a = M 4 *P,(M*), e-|i*W*). (22) Upon substituting these series in the right members of (17), which vanish with a = e = n t/t = Q, the result is P = M V,Q I(M V, ; T)| 8- MM*', T). (23) The series Q, and Q t are periodic in r with the period 2jir because the con- ditions that the solutions shall have this period have been satisfied. Since (17) converge for all O^T^2jr if the moduli of a, e, and M'* are sufficiently small, and since the expressions for o and e given in (22) vanish for M = 0, it follows that the modulus of n :/t can be taken so small that the series (23) con- verge for all T in the interval; and since they are periodic with the period 2jr, the convergence holds for all finite values of r. The integer j has so far been undetermined. When j is unity, the periodic solutions exist uniquely and their period is 2*-. When .;' is greater than unity the periodic solutions also exist uniquely. Since the periodic orbits for j greater than unity include those for j equal to unity, and since in both cases there is precisely one periodic orbit for a given value of ^, it follows that all the symmetrical periodic orbits of the class under consideration have the period 2* in the independent variable T. 384 PERIODIC ORBITS. It follows from (6) and (9) that T+6 = vv 1 . Since in the periodic solution 6 is periodic with the period 2w, the period of the solution is the synodic period of the three bodies. Hence, if the motion of the infinitesimal body is referred to a set of axes having their origin at the center of gravity of the system and rotating in the direction of motion of the finite bodies at the angular rate at which they move, and if the :r-axis passes through the finite bodies, then the periodic orbit of the infinitesimal body, which has been proved to exist, will be symmetrical with respect to the z-axis. Since, by hypothesis, a > R, it follows from (6) and (8) that n : >n. Therefore, even if the motion of the infinitesimal body is forward with respect to fixed axes, it is retrograde with respect to the rotating axes. It is supposed that the period of the finite bodies, and therefore n l} is given in advance and remains fixed. The variation of the parameter p" corresponds to a variation of the period of the infinitesimal body defined by n. If the motion with respect to fixed axes is forward, n has the same sign as n l} and n l/3 has three values, one of which is real and positive while the other two are complex. If the motion is retrograde, n vs has three different values, one of which is real and negative while the other two are complex. Therefore, for a given period, there are six symmetrical orbits, three direct and three retrograde; and for small /x one direct orbit is real and one retrograde orbit is real, while in the others the coordinates are complex. This means, of course, that the corresponding solutions do not exist in the physical problem. The coordinates of the complex orbits are conjugate in pairs. For a certain value of /z 1/3 they may become equal, and therefore real, and, for larger values of / /3 , real and distinct. Upon transforming the integral (7) by (9), it is found that '* 111" _ . . ~l . 1/1,0. ill,. It. " I _ * . _*. if- . \ (24) +5cos3(r+0)]+ ]- for /i 1/3 = 0. Therefore the periodic orbits branch at C= oo , and there are two cycles of three each. PERIODIC OKBITS OF Kri'KHluK PLANETS. .'>S"> 195. Practical Construction of the Periodic Solutions. It has been proved that the symmetrical periodic solutions under discussion are expres- sible in the form (26) whore the p, and 8, arc functions of r. Since these series are periodic and converge for all IM'*] sufficiently small, it follows that each p, and 0, separately is periodic; that is, p,(r+2T)==p,(T), 6l (T+2ir)=6 l (r). (27) In every closed orbit there are points at which dp/dr = 0. Suppose / of (9) is so determined that this condition is satisfied at T = 0; it will follow from this and the convergence of (26) for all |M' /J | sufficiently small that P, = (t-4, ... ). (28) In the symmetrical periodic orbits the value of 8 is zero at r = 0. But this condition will not be imposed, because the general periodic orbits, whose initial conditions are not specialized, include those which are symmetrical; and in the construction it will appear that the conditions for symmetry are a consequence of those for periodicity. Hence all the periodic orbits of the class under consideration are symmetrical. Equations (26) are to be substituted in (11), arranged as power series in M' /J , and the coefficients of the several powers of M I/J set equal to zero. The coefficients of M V| set equal to zero give the equations p>0, 0>0. (29) The solutions of these equations satisfying (27) and (28) are P = a, 04 = &4, (30) where a, and & are so far undetermined constants. The coefficients of /i w , . . . , /** /l are the same as (29) except for their subscripts, and their solutions satisfying (27) and (28) are similarly Pj = aj, e, = b, a-5, . . . , 9), (31) where all the a, and b, are so far undetermined constants. The coefficients of M' O/I give the equations p w = 3a 4 -fA/[l+3cos2r], 0,. = - f M sin 2r. (32) In order that the solution of the first of these equations shall be periodic the condition a 4 =M (33) must be imposed, which uniquely determines the constant o 4 . Then the solution of (32) satisfying (27) and (28) is P,,=Oi.+ ^cos2T, 10 -& lo +fMsin2r, (34) where a, and & are as yet undetermined. 386 PERIODIC ORBITS. The coefficients of /x 11/3 are defined by the equations p n = 3a 6 , 0ii = 0, from which it follows that o 6 = 0, Pn = a n , 0ii = & n , (35) where a u and 6 n are as yet undetermined. The coefficients of M I2/3 in the solutions satisfy the equations where a 12 and 6 12 remain so far undetermined. In a similar way it is found from the coefficients of /x 13/3 that Upon imposing conditions (27) and integrating, it is found that a 6 = 0, and (36) L si ined. j coefficients of fj. n/3 that + ^Msin2r, (37) where a 13 and b l3 remain undetermined at this step. So far all the b } have remained arbitrary, and it is necessary to carry the integration one step further in order to see how they are determined. The coefficients of M H/3 are defined by (38) H = a 4 10 3M [6 4 cos2r 2a 4 sin2r]. Upon substituting the values of a 4 and 10 from (33) and (34), imposing the conditions (27) and integrating, it is found that a 8 = ^M 2 , and (39) The condition (28) for,;' = 14 gives the equations &4= 0, PU = OH ^If 2 cos2r, 14 = 6 14 ^M 2 sin2T. (40) ID Ow It is found in a similar way from the coefficients of /x u/3 that & 6 = 0, and (41) where a 15 and b K remain so far arbitrary. c.KHITS ()K ST I'M: IOR PLANETS. It will l>e observed that, so far as the computation has l>een carried, the coefficients of the p. are cosines of integral multiples of r, and that the coeffi- cients of the 8j , except for the undetermined additive constants, are sines of integral multiples of T. In the computation of p, the periodicity conditions have uniquely determined a/,,, and the condition p, = at T = has required that fe,_, = 0. It will now be shown that these properties are general. Suppose p 4 , . . . , p.; 4 , . . . , 0. have been computed and that the coefficients are all known except a,-!, . . . , a,, which enter additively in p,_,, . . . , p. respectively, and 6._,, . . . , b m , which enter additively in 0,_,, . . ., 0. respectively. The differential equations for the determination of p..,., and 0.+, are p. +l = 3a._,+ fM&._,sin2T+F. +1 (T), 0. +l =G, +1 (r), (42) where /'+, (T) and G n+l (r) are entirely known functions of T. It follows from the assumptions respecting p 4 , . . . , p.; 4 , . . . , 0. and the properties of equations (11) that F u+l (r) is a sum of cosines of integral multiples of T, and that G n+ i(r) is a sum of sines of integral multiples of T. Hence they may be written in the form FH.,(T) = 2 A**" COSJr, (?. + ,(T) = 2 B^ Bin jr. In order that the solution of the first equation of (42) shall be periodic the condition = (43) must be imposed, and this condition uniquely determines a,_,. After equation (43) has been satisfied, the solution of the first equation of (42) is I = o. +l - M6._,8in2r+2 a^" cos jr. <+''= -U^". (44) The condition p = at T = makes it necessary to take 6.-, = 0. (45) Then p. + i is completely determined except for the additive constant 0.+, , and it is a sum of cosines of integral multiples of T. The solution of the second equation of (42) is 0. +1 = 6. +1 +20r +1) n;V, ff* n - -^T 11 . (46) Hence 0,+, is a sum of sines of integral multiples of T, except for the unde- termined constant 6. +1 , which must be put equal to zero in order to satisfy the condition on p. +II . These results lead, by induction, to the conclusion that the p, and 9, 0'=4, . . . GO) are sums of cosines and sines respectively of integral multiples of T whose coefficients are uniquely determined. 388 PERIODIC ORBITS. From the properties of the solutions which have just been established, it follows that not only is p = at T = 0, but also 0(0) = 0. Therefore these periodic orbits are the symmetrical orbits whose existence was established in 194. In the construction it was not assumed that the orbits were symmetrical, and since this property is a necessary consequence of the periodicity conditions, it follows that all periodic solutions which are expan- sible as power series in ju 1/3 are symmetrical. It is easily shown, by direct consideration of the construction of periodic solutions, that they can not be expanded as power series in /* 1/3 except when j is a multiple of 3, and that then they reduce to those found above. 196. Application of the Integral. The differential equations admit the integral (24), which, for brevity, can be written in the form It follows from the form of (24) and the expansions (26) that the left member of this equation can be developed as a power series in ju 1/3 , giving + =0. (47) Since the p } and 0, are sums of cosines and sines respectively of integral multiples of T, and since p enters in (24) only in the second degree and 6 only in even degrees, it follows that the Fj are sums of cosines of integral multiples of T. Equation (47) is an identity in ;u 1/3 , whence F n =SC f fcos./T=0 (n=0, ... oo). Since these equations hold for all values of T, it follows that Cf = (n = 0, ... . ; j = 0, . . . oo). (48) The C* are functions of the < , . . . , af and # 0) , . . . , p . Hence equations (48) can be used as check formulas on the computation of the coefficients of the solutions. Equations (48) can be used in place of the second equation of (11) for the determination of the $ B) , the coefficients of the trigonometric terms in the expression for n . Suppose p 4 , . . . , p n - t and 4 , . . . , n _ 1 have been determined except for additive constants in p B _ 6 , . . . , p B _! . It follows from (24) that F n is F. = - 26 n + P a (P,, PI, e } , 0,) (j=4,, . . . , n- l), where P n is a polynomial in the arguments indicated. Consequently equations (48) are of the form which uniquely determine the $"'. CHAPTER XIII. A CLASS OF PERIODIC ORBITS OF A PARTICLE SUB- JECT TO THE ATTRACTION OF ;/ SPHERES HAVING PRESCRIBED MOTION. BY WILLIAM RAYMOND LONGLEY. 197. Introduction. The restricted problem of three bodies furnishes naturally the starting-point* for the consideration of the periodic orbits of an infinitesimal body, or particle, which is subject to the Newtonian attraction of certain finite spheres whose motion is supposed to be known. The two finite bodies are supposed to revolve in circles about their common center of mass, and the motion of the particle is restricted to the plane in which the finite bodies move. One class of orbits occurring in this problem is that in which the particle revolves about one of the finite bodies, and for the consideration of these orbits it is convenient to refer the motion of the particle to a plane rotating with the angular velocity of the finite bodies. All of the known periodic orbits of this type possess one and only one line of symmetry, namely, the line joining the finite bodies, and this property of symmetry plays an important part in the proof of their existence and the construction of series to represent them. The purpose of this chapter is to generalize the restricted problem by introducing into the plane of motion more than two finite bodies. The coordinates of the finite bodies (spheres) are supposed to be known functions of the time, that is, the motion of the spheres is prescribed. For the analysis which follows the nature of the forces producing this motion is unimportant. The spheres are supposed to attract the particle according to the Newtonian law. Besides involving additional terms in the disturbing function, this generalization modifies the original problem by introducing cases where the periodic orbits have no line of symmetry, and cases where there are more lines of symmetry than one. This modification necessitates some changes in the details of the analysis which must be worked out. In order to avoid cumbersome notation, the analysis will be developed for simple particular cases of the motion of spheres under their Newtonian attraction; with slight changes it is applicable to more general types of prescribed motion of the finite bodies, which are indicated in 207. See papers by Hill, American Journal of Malhematict, vol. 1 (1878), p. 245; Darwin, Ada Mathcmatica, vol. 21 (1897), p. 99; and Moulton, Transaction* of the American Mathematical Society, vol. 7 (1906), p. 537. 390 PERIODIC ORBITS. 198. Existence of Periodic Orbits Having no Line of Symmetry. It was shown by Lagrange* that an equilateral triangle is a possible configu- ration for three spheres revolving in circles about their common center of mass. This motion of three finite bodies will serve to illustrate the case when the periodic orbits of the particle about one of the bodies possess no line of symmetry. Let the masses of the three finite bodies moving accord- ing to the equilateral-triangle solution be denoted by M , M 1} M, , and sup- pose the particle P revolves about the mass M. Suppose also that the masses M l and M 2 are unequal.f With reference to M as origin and an axis having a fixed direction in space, let the polar coordinates of M lt M 2 , and P be respectively (R lf VJ, (R t , F 2 ), and (r, v). The coordinates of the bodies are expressed in terms of the time, t, as follows: R^R^A, V^V z -l=Nt, (1) where Here N denotes the angular velocity, A the length of a side of the triangle, and A; is a constant depending upon the units employed. The differential equations of motion of P are df T \dt) r 2 dr ' 1 'dt 2 H dt dt == r dv ' where _, 2 rMi , Ms _ MI, / _y\_Mt, , _y\\ (3) Let us define m and a by the relations mv = N, i/*a' = fc f M, (4) where v is a quantity to be assigned later. By the substitution v = w+V 1 = w + Nt the motion is referred to an axis rotating with the angular velocity of the finite bodies and passing always through M^, and factors depending upon the units employed are eliminated by the relations r = ap, vi T. On making these substitutions in equations (2) and dividing by v"a, the differential equations of relative motion become *Prize memoir, Essai sur le Probleme des Trois Corps, 1772; Coll. Works, vol. 6, p. 229. flf Mi =M, the periodic orbit of P about M has a line of symmetry, namely, the median of the triangle from the vertex M, which is the line joining M to the center of mass of the system. For the treatment of this special case it is convenient to make use of the property of symmetry and to employ analysis similar to that developed in 202 and 203. PARTICLE ATTRACTED BY SI'llKltKS. 391 We can expand <> as a power series in r/p/.l which is convergent for all values of w provided the distance Ml' = np is l.-s- than .-1 ; and in all that follows this condition is supposed to he satisfied. The expansion has the fiinu Let X, and X, he defined by the relations Af^X.M+A/,), J/.-X.CAf.+Af,). (6) From equations (1) we have Af.+Af. . A* Then, on setting Ml + Mt - it follows that the second members of equations (5) have the form (7) a 1 " li = - /i:wl p[|x I sin2u;-|-|x l (-|p)|sinu>+5sin3u>)H !)}+] It is convenient to introduce a parameter n into the differential equations (5) by the relations m=n, X, = XM, J =IWi ( 8 ) wherever the degree of a/A is higher than the first. The quantities X and ij are numerical constants. By relating X, and /i the existence proof is made to depend only upon general properties and certain terms of the differential equations which involve X, ; that is, upon terms in the disturbing function which are due to the body M , . We shall consider the solution of equations (5) as power series in the parameter jt. The differential equations, and consequently also the solution, do not represent the physical problem under consideration for any value of the parameter except the one satisfying the relations (8). But if the solution is valid when this particular numerical value of M is substituted, then it is a solution of the differential equations 392 PERIODIC ORBITS. representing the physical problem and therefore has a physical interpre- tation. The generalization of the parameter a/A is merely for convenience in having finite expressions in the equations which determine the coefficients at the various steps in the solution. On introducing the parameter n as indicated, equations (5) become d*p fdw dr 2 \ dr where v F fi 3 a 1 1 L I 2 8 -4 J J g--zA where F and G are functions of the indicated arguments. Equations (9) are periodic in w with the period 2w and do not involve T explicitly. Suppose that P = ^iO), W = fa(r) is a solution. Sufficient conditions that the solution shall be periodic with the period 2pir (where p is an integer) are 2T=tf(0), j =#(0), J where \f/[ and \]/' a denote derivatives of fa and \f/ 2 with respect to T. When n a periodic solution, which will be called the undisturbed orbit, is known, namely, P = l, W = T, (11) and the initial conditions are P = l, p' = 0, w = Q, w' = l. (12) It will be shown by the process of analytic continuation that, for values of fj. different from zero, but sufficiently small, there exists a periodic solution which, for n = 0, reduces to equations (11). For this purpose we consider the solution of equations (9) subject to the initial conditions P=l+]8 1 , p' = &, w = &, w' = l+p t , (13) where ft , j8j , ft , /3 4 , are to be determined as functions of /*, vanishing with H, so that the conditions of periodicity (10) shall hold. It follows from the differential equations that the solution is expressible as power series in ft , ft > ft > ft j an d M and that, for sufficiently small values of the parameters, the series are convergent for all values of T from to 2pir. We suppose that this condition on the moduli of the parameters is satisfied. For the determination of those terms in the series which involve the initial conditions but not (j?, it is possible to use the known solution of the two-body problem, since for ^ = equations (9) reduce to the equations of PARTICLE ATTRACTED BY Tt SPHERES. motion of a particle /' when subject to the attraction of M alone. Hence, instead of the additive increments /3, , t , 0, , /3 4 , it is convenient to introduce new parameters a, e, 9,

7ra 2pirfjL 6pirae-\- (d) (17) PARTICLE ATTRACTED BY n SIMI1 1 395 The conditions (17) involve the four quantities a, e, 0, tp, and, if independent . would determine them in terms of ^. Hut the differential equations (9) do not involve r explicitly and hence admit the integral of Jacobi. This furnishes a relation of the type F(a, e, 0, = 0. Consider the solution of equations (a), (6), and (c) for a, e, and 0. The equations have the following properties: (I) There are no terms independent of a and /*. This follows from the fact that , in the two-body problem, the period does not depend upon e and d. (II) There are no terms involving M to the first degree except the one term 2pirn, which occurs in (c). (III) There are no terms in d independent of e, since 6 does not enter the initial conditions independently of e. It follows from these properties and the particular form of the first terms of the equations that a, e, and 8 are determined uniquely as power series in n by the following steps: (1) From (c) we obtain a = M[-f+ + f unction (M, e, 0)]. (2) This value of a when substituted in (6) permits a factor M to be divided out. We can then solve the result for e as a power series in n and 6 which contains n as a factor, and obtain e = 4loi + ' ' ' + function (,*)]. (3) When the values of a and e are substituted in (a) a factor f can be divided out and obtained as a power series in /* alone, vanishing with p. (4) By the substitution of the value of thus found in the expressions for e and a, we obtain finally a=Mpi(/<0, e = np t (n), =/#,(/*). The preceding operations are known to be convergent for all values of a, e, 6, and n which are sufficiently small. Hence, for a given value of M sufficiently small, it is possible to determine the initial conditions (14) as power series in ^ such that the solution of the differential equations (9) shall be periodic in r with the period 2pr. See Poincar6, loe. eit., p. 87. tWben the finite bodies do not form a fixed configuration in the rotating plane the integral of Jacobi does not exist and the origin of time U not arbitrary. In thU case it u neceasary to determine the four parameters from the conditions of periodicity. The case of the triangular solution when the finite bodiea move in ellipses has been treated by Ingley in a paper in the Tran*tctunu of the American Mathematical Society, vol. 8 (1907), PP- 159-188. 396 PERIODIC ORBITS. When the values of a, e, 6 in terms of n are substituted in equations (16) the periodic solution is obtained. The period of the solution is 2pir in T, where p is an integer, and from the conditions of periodicity (10) it is apparent that the particle makes p revolutions in the rotating plane during a period. The process by which the periodic solution was obtained yields a unique result; therefore, for an assigned value of n, there exists one, and only one, orbit having the period 2pir. Since the orbits having the period 2pir, p>l, include those having the period 27r, it follows that all the orbits of this analytic type are closed after one synodic revolution* Since r = vt the period of the solution in t is 27r/, and the quantity v, which is so far arbitrary, can be determined by assigning the period of the solution. The parameter ^ is then determined by the relation fj, = m=N/v; that is, the numerical value of p is the ratio of the mean motions of the finite bodies and of the particle. If the direction of revolution of the par- ticle is the same as that of the finite bodies that is, if the orbit is direct v and N have the same sign and n is positive; if the orbit is retrograde, v and N have opposite signs and /z is negative. Since for an assigned value of M there exists one, and only one, periodic orbit, and since values of fj, which are numerically equal, but opposite in sign, give orbits having the same period in T, it follows that for a given period there exist two, and only two, real orbits of the type under consideration. In one the motion is direct, and in the other it is retrograde. We may now state the result as follows : The period 2ir/v of the solution may be assigned arbitrarily in advance, subject only to the condition that the ratio N/v is sufficiently small, where 2ir/N is the period of the motion of the finite bodies. Then there exist two, and only two, real periodic orbits of the particle having the required period. In one the motion is direct, and in the other it is retrograde. All the orbits of this type are closed after one synodic revolution. In deriving this conclusion no use was made of the explicit values of those terms in the disturbing function which are due to the body M 2 . The proof depends entirely upon the form of certain terms of the solution which involve \. Hence the analysis and conclusions are applicable without change to the case where n finite bodies revolve in circles in such a way as to form in the rotating plane a fixed configuration. 199. Construction of Periodic Orbits Having no Line of Symmetry. It is possible to construct the periodic solutions of the differential equations (9) by the method indicated in the existence proof, but the process is labo- rious. A method will now be given by which the solution to any desired number of terms can be conveniently constructed. It is not necessary to determine the initial conditions explicitly in advance, and the computation involves only algebraic processes. *Since no new orbits are obtained by taking p> 1 we will assume hereafter that p = 1. PARTICLE ATTRACTED BY n SPHERES. 397 It has been proved that the periodic solutions are expressible in the form - = 1 * ' ' ' * ' ; i I ('8) The series (18) satisfy the differential equations (9) uniformly over a finite interval in n, and hence, when the series are substituted in the differential equations, the coefficient of each power of p must vanish. Furthermore, the series are periodic with period 2* in r; and, because the periodicity holds for a continuous range of values of n, each coefficient p, and w t sepa- rately is {xriodic with the period 2r in r. It has been shown also that we can choose w = when r = Q, and because this holds identically in ft, it follows that w t (0) = for every t. Let the solution (18) be substituted in the differential equations (9) and arrange the results as power series in ft. The terms of the first members have the following forms, where the accents indicate derivatives with respect to r: + 2pX-.+ -+' +K+PX-.+ +P.-XV+ (19) 398 PERIODIC ORBITS. The second members have no terms independent of M 2 - Therefore, on equating to zero the coefficients of the first power of n, we have for the determination of p l and w l pi_ n x om ~ It follows from these equations that Pl = 2 ( 1 + O + c<" cos T + c sin r, 1 where cj", c^", c ( ", c ( " are constants of integration. Since Pl and w^ are periodic, the coefficient of T in w^ must vanish. This condition determines the constant cf, namely, c= 4/3. Since w\ = when r = 0, C4 = 2c"'. The constants c," and c"' are so far undetermined. On equating to zero the coefficients of the second power of /*, the fol- lowing set of equations is obtained : j 2) sin 3T. On integrating the second equation, we have -cos3r. (24) On eliminating -p 2 from the first of equations (23) by means of equation (24), there results s -*^ /\ I O xi^*' I I 2 /^ I yi ' ' ^_ O If I f*O^ T* - x /* ^1T1 T (25) - |Z))cos3r. PARTICLE ATTRACTED BY Tl SPHERES. 399 In order that the solution of equations (25) shall contain no non-periodic term, the coefficients of COST and sinr must vanish; hence ri" and c{" are determined by the conditions With these values of c and c"' the solution becomes P, = 4 where On substituting this value of p, in equation (24) and integrating, we obtain for u>, a solution of the form Wt = cf - (2.C+3O r-2cJ I> 8inT+2c > cl) cosT+; i) 8in2r+^ a> sin3r, where 6? = - 4 (D?+2jti?) 0'-2, 3). Since w t is periodic, c is determined by the condition 2 A? +3 , cj n , cj", ci", c) have been determined uniquely; ej" has been expressed uniquely in terms of cf ; while the remaining two (c?, O are still arbitrary. By equating to zero the coefficients of the third power of M the following set of equations is obtained: (96) where /, and ^^B^ + S (ay-" siiyV + 7?- The constants of integration have been uniquely determined except c""", c?~", and ci'~". The first two are so far arbitrary, while c""" is expressible in terms of c~ u by the relation PAUTICLK ATTRACTED BY n SI'lll 401 The equations for the determination of p, and w, have the form 1+1 + S CJS) Thr cuetlicients A, B, C, D arc known constants and equations (28) are solved \}\ the steps employed in the solution of equations (23). During the process four constants of integration are introduced, namely c{'\ cj, cj, c"', and four are uniquely determined by the conditions ?-"= -2ci'-- S T)''", 3_ 1 = - (0 - (n --- (n (31) The formulas (31) together with the conditions (29) are sufficient to construct the periodic solution of equations (9) to any desired degree of accuracy; the computation is entirely algebraic. In order to determine the constants of integration entering in the last (I th ) step, it is necessary to compute the coefficients ^| l+l) , fl^", CT", D? + " of the next following step. 402 PERIODIC ORBITS. 200. Numerical Example 1. For the purpose of illustration, we assign numbers to the constants involved in the preceding analysis and construct an orbit. In this and the other numerical examples which occur later, it has not been shown that the processes are valid for the numerical values which are employed and which have been selected for convenience in graphical representation. It is probable that the series are convergent, although it has not been found possible to determine the true radii of convergence. The differential equations of motion are equations (5). On putting ra = M and writing the second members explicitly as far as terms of the second degree in a/A, we have ; ~i )+35cos4(>-J)]+ (32) d*w We select M for the unit of mass and suppose M l = 10, M 2 = 5. For the unit of distance we take the distance between the finite bodies, that is, A = l; and the unit of time is selected so that N=l. The period of the solution is assigned so that v = 5, whence PAUTICLK ATTUACTKI) BY tl 8PHKHKS. The constant A" is detenniaed from the relation #'/!' = A'l.U + A/.+A/,), whence -0.06250, A- 1 A/, = 0.62500, ^,= 1.56250^. The constant a was defined by i*a' = fc*Af, whence \Yith these numerical values equations (32) become f - t = (0.31250+0.93750cos2?)pM' P + (0.78125-1. 17188 cos2u>+2.02977sin2tt>)pM' + (0.47714cosu>+0.79523cos3u>)pV + (0.59643 cos w>+ 1 .03308 sin w - 1 .98808 cos 3u>)pV + (0.16188+0.35974cos2u>+0.62954cos4u>)pV+- (4* +) = - (0.93750sin2w)p M 1 + ( 1 . 1 7 1 88 sin 2w + 2.02977 cos 2u>) p M ' - (0.159058inu?+0.79523sin3u>)pV + (-0.19881sinu>+0.34436cosu)+1.98808sin3tc)pV 4 - (0.1 7987 sin 2^+0.62954 sin 4 u;)pV+ The periodic solution of equations (33) is p= 1 --/x+ (0.45139+0.39762 COST -0.62500 cos2T)M J 3 + ( -0.47647+ 1.04542 cosT+0.86090sinr+ 0.46875 cos 2r -1.35318sin2T-0.16567cos3TV+ ', -0.79524sinT+0.85938sin2T)M* + (0.13882-3.25720sinT+1.72180cosT+0.27995sin2T -1.86062cos2T+0.19881sin3T) M '+ < (33) 404 PERIODIC ORBITS. Substituting the numerical value /x = 0.2, equations (34), if convergent, are the equations of motion of the particle P. The orbit is shown in Fig. 9. In this and the figures of the following numerical examples the comparison circles are not the circular orbits which have been called the undisturbed orbits. The undisturbed orbits are referred to fixed axes while the draw- ings are made with reference to rotating axes. The comparison circles represent orbits in which the particle would make a complete revolution with respect to the rotating axes during the period. The points which are numbered 1, 2, . . . ,8 represent positions of the particle in the periodic orbit at intervals of r = w/4. The corresponding positions in the comparison circle are indicated by the numbers 1', 2', . . . , 8'. 201. Some Particular Solutions of the Problem of n Bodies. The existence of symmetrical periodic orbits of the particle depends upon the masses and motion of the finite bodies. So far as the analysis is concerned, this motion may be arbitrarily periodic, without reference to the nature of the forces producing it. It is required only that the motion of the finite bodies shall be known and that they shall attract the particle according to the Newtonian law. It will be interesting, however, in developing the analysis, to prescribe motion for the finite bodies, which is possible under the law of the inverse square of the distance. For three finite bodies the two solutions of Lagrange are well known. In the case of the equilateral- triangle solution the periodic orbits of the particle about one of the bodies PARTICLE ATTRACTED BY n SPHERES. 405 have a line of symmetry if the other two masses are equal. In the case of the straight-line solution the periodic orbits of the particle about any one of the bodies are symmetrical with respect to the line joining the finite bodies. The particular solutions which Lagrange has given for three bodies have been extended to some cases of more than three bodies,* and we shall consider two examples: (1) in which there are five bodies, and (2) in which there are nine bodies. Let the masses of n finite bodies be represented by M lt M t , . . . ,M m . Suppose that the bodies lie always in the same plane, and that their coordi- nates with respect to their common center of mass as origin and a system of rectangular axes which rotate with the uniform angular velocity N are, respectively, (x lt y,), (x t , j/ t ), . . . , (x,,y n ). Supposing that the bodies attract each other according to the Newtonian law, the differential equations of motion are (35) + 2 N - JVfc = - ar at V(x l -Xj) t +(y t -y ) y (-l, 2, .... n; J If we assume that each body is revolving in a circle about the common center of mass of the system with the uniform angular velocity N, its coordi- nates with respect to the rotating axes are constants and the derivatives of the coordinates with respect to the time are zero. Equations (35) therefore reduce to the following system of algebraic equations =0 (36) It follows from these equations that M l x l +M s x t + -fA/,,x. = 0, Af.T/.-f-Af.t/.-r- +^. = 0, (37) which express the fact that the origin of coordinates is at the center of mass. These equations may be used instead of two of (36). See Hoppe, "Krweiterung der bekannlen Special loaungen dec Dnskorperprubleins;" Arekit.der Matk. tmd P*., vol. 64, pp. 218-223. Andoyer. "Sur l' (c), (e), (/), (0), and (t) are satisfied. Equations (6) and (d) become identical, yielding 1? = (2^7 + A'( Vl+K*)' + ^ ' and equations (h) and (j) become identical, yielding JV 2Af' 2Af" M, (41) When M ', M", M t , K, and A have been chosen or determined, these equa- tions insure a positive value for N*. On eliminating N*/k? between equations (40) and (41), we obtain for the relation between the masses, M' (42) The choice of the constant K, which is the ratio of the diagonals of the rhombus, is limited by the condition that the resulting ratio of the masses must be positive. To investigate this condition we set A/ t =l. Then, regarding K as a parameter, equation (42) represents a straight line in the 408 PERIODIC ORBITS. M'M" plane. Only those pairs of values (M',M") which represent a point in the first quadrant are admissible. This condition will certainly be satisfied if the slope is positive, that is, if the coefficient of M' is positive. This condition is easily found to be nding to equations (38) of the five-body problem, there is a set of equations, which, upon the assumptions (44), reduce to K 45) Eliminating N*A l /l from equations (45), there results the following condition on the masses A/', A/', A/,, and the ratio, K, of the radii of the circles, M , | 1 2Af V \W^4j^ (46) - - - M/ l^i~4~ The choice of the ratio K and two of the masses is limited by the condi- tions that the third mass, as determined by equation (46) and the square of the angular velocity, N', from (45), shall be positive. The limits within which the choice can be made have not been determined, but for the purpose of application in numerical example 3, the following set of values satisfying the conditions has been computed: A' = 2, A/' = l, A = l, Af. = l, A/' = 8.2526, #' = 1.6399**. 410 PERIODIC ORBITS. 202. Existence of Symmetrical Periodic Orbits. For the development of the type of analysis applicable to symmetrical orbits we shall use configu- ration (^4.) of the preceding section, the notation being unchanged except that the mass at the center will now be denoted by M instead of M 5 . With reference to M as origin and an axis having a fixed direction in space, let the polar coordinates of M lt M 2 , M 3 , M 4 , and P be, respectively, (/2 U F,), (# 2 ,F 2 ), (# 3 ,F 3 ), (# 4 ,F 4 ), and (r, v). The coordinates of the bodies are expressed in terms of the time as follows: where N is given by equation (40) or equation (41). The differential equations of motion of P are r dv (A ~ where T\ T% 7*3 7*4 v- F 2 ) - rcos(z>- F 3 ) - jjrcos(v- F 4 ) -2rA cos(v- FO, r 3 = Vi*+A'-2rAcos(v- F 3 ), r* = Vr t +A 2 -2rKAcos(v- F,), r,= Vr*+K*A*-2rKAcos(v- F 4 ). We now define m and a by the relations where, as in the preceding case, v denotes the mean angular velocity of P. The motion is referred to an axis rotating with the angular velocity N and passing always through M l by the substitution v = w-\-V l = w+Nt, and factors depending on the units employed are eliminated by the substitution r = ap, vt = T. We obtain then the differential equations of relative motion ^V^. (48) These equations have the same form as the set (9) and the analysis and results of that problem are applicable in this case. We know, then, that there exists one, and only one, orbit in which the particle P moves with direct motion and with a preassigned period. We shall see that there exists one, and only one, such orbit which is symmetrical to the line joining M and M l (also to the line joining M and M s ) ; hence it will follow that there are no unsymmetrical orbits of this type. Furthermore, all the periodic orbits which are given by this analysis are closed after one revolution in the rotating plane; hence in case symmetrical orbits exist, they are also closed after one revolution. PARTICLE ATTRACTED BY T SPHERES. 411 We can expand S2 as a power series in up/A which i< convergent fur all values of w so long as the distance, ap, of tin- panicle from M remains less than the distance from M to the nearest finite body; and in all that follow.- this condition is supposed to he satisfied. The expansion has the form i(jp)'|3cos(to-r)+5cos3(tr-r)}+ From the conditions of the configuration (A) we have Let X, and X, be defined by the relations \ = X 'V1" ""(V From equations (40) we have On setting the coefficient of N* in this equation equal to , it follows that the second members of equations (48) have the form -I- J*L = )+X i (l-3cos2uO + terms involving only cosines of even multiples of tc], (49) -- ^ = - K m t pr3X l sin2u)-3X t sin2u; f t a t o dw + terms involving only sines of even multiples of w\. 412 PERIODIC ORBITS. On introducing the parameter of j integration n by the relations m = n and a/ A = TJ/Z, where T\ is a numerical constant, equations (48) become /dw d?- p (d^ where /= K p[X 1 (l+3cos2w)+X 2 (l-3cos2w/)+ ], g= Kp[3(\ X 2 )sin2w+ ] Suppose P = *I(T), W = ^(T) (51) is a solution of equations (50) such that p' = w = at r = 0. Then it follows from the form of the differential equations that ^, is an even function, and that ^ 8 is an odd function of T. Hence, if the particle crosses the w-axis orthogonally, the orbit is symmetrical with respect to the line w = 0, and with respect to the time of crossing this line.* Suppose that when T = TT the particle crosses this line (or, what is the same thing, the line w = T) again orthogonally; the orbit will be symmetrical with respect to this line and the time of crossing, and the particle will have again its initial position and relative components of velocity at the end of the period T = 2-ir. Hence sufficient conditions that the solution (51) shall be periodic are p'(7r)=0, ;(*)- IT = 0. (52) For fj. = the equations have the form which occurs in the problem of two bodies, and a symmetrical solution having the required period is known, namely, p=l, W = T. The initial conditions for this solution are p = l, p' = 0, w = 0, w' = \. Consider the solution for values of n different from zero but sufficiently small, and let the initial conditions for T = be (53) *It may be remarked also, in this particular example, that if the solution (51) is subject to the initial conditions, p'=0, w=jr/2, (T = TI), then 1^1 is an even function, and 4/, jr/2 is an odd function of T T, and hence, if the particle crosses the line w r/2 orthogonally, the orbit is symmetrical to this line and the epoch T = TI. The sufficient conditions of periodicity (52) might be replaced by the conditions since the orbit would then have two lines of symmetry. For the purpose of covering more general cases it is better to base the existence proof on only one line of symmetry. PARTICLE ATTRACTED BY n SPHERES. 413 The solution i.s symmetrical with respect to tho epoch T = 0, and can be expressed as power series in o, r, and M. which arc convergent for an interval in T including the interval to *-, if the parameters are sufficiently small. If a and 6 ean l>e determined in terms of /u, vanishing with /i, so that the conditions (52) are satisfied, then the solution will be periodic with the period 2w. All terms of the solution which are independent of M* can be obtained from the two-body problem by making the substitution w = u pr. These terms are given in finite form by the expressions ( C08g ~ c j) = \l-ecosES = arccos = arc sn l where E is defined by the relation ( )n returning to the variable w, writing the terms in a and e as power series by Taylor's expansion, and applying the conditions (52), we obtain the equations 0=-fTae+ , 0=-fTa-T M + (54) It follows from the known properties of the series that there are no terms in e alone, and there are no terms involving ju to the first degree except the term TM- The equations are satisfied by a = e = n = 0, and in the second the coefficient of the first power of a is not zero; hence the second equation can be solved uniquely for a as a power series in e and n, which contains n as a factor. The result has the form (55) When this value of a is substituted in the first of equations (54), a factor M can be divided out, leaving = ire+ This equation is satisfied by e = M = 0, and since the coefficient of the first power of e is not zero, it furnishes a unique determination of e in terms of n, vanishing with n. When this value of e is substituted in equation (55), we have o expressed uniquely in terms of /*, vanishing with n. Hence for a given value of n sufficiently small it is possible to determine the initial conditions (53) as power series in n such that the solution in n is symmetrical with respect to the line w=* and the epoch T -K. Since it is sym- tn< Irical also with respect to the line w = and the epoch r = 0, it is periodic in T with period IT. The orbit is symmetrical with respect to the line joining the bodies M and M,. If we take for the initial line the line joining M and Af,, it follows from the same analysis that the orbit is symmetrical with respect to 414 PERIODIC ORBITS. this line and the time of crossing it. Since for a given value of /x there is only one periodic orbit of this type, it follows that it is symmetrical with respect to both the lines joining M with M l and with M t . For the con- figuration (A) of the finite bodies the periodic orbits of the particle have two lines of symmetry; there are four apses and the apsidal angle is ir/2. For the configuration (B) (see numerical example 3) the periodic orbits have four lines of symmetry, there are eight apses, and the apsidal angle is jr/4 In establishing the uniqueness of the periodic solution of equations (54) it is to be noted that no use was made of the explicit values of the terms from the second members of equations (50). Hence, if the second members have forms which permit symmetrical solutions, the preceding analysis is applicable without change to show the existence of symmetrical periodic orbits. // the masses and motions of the finite bodies are such that there can exist orbits of the particle about one of the bodies having a line of symmetry, we have established the existence of periodic orbits having this line of symmetry. As in the problem treated in 198, the period of the solution in t may be assigned arbitrarily (that is, when the finite bodies form a fixed configuration in the rotating plane). There exist then two, and only two, symmetrical closed orbits having the required period; in one the motion is direct, and in the other it is retrograde. 203. Construction of Symmetrical Periodic Orbits. The method of constructing symmetrical periodic solutions is similar to that explained in 199. There is a slight difference in the conditions which determine the constants of integration, and the calculation is simpler because p contains only cosines of multiples of r, while w contains only sines. It has been proved that symmetrical periodic solutions of equations (50) exist, and that they are expressible in the form (18). Since the solution is periodic for a continuous range of values of /i, each coefficient p, and w t is periodic with period 2 w in T. Also, since the initial conditions p'(0) = 0, w(0) = hold identically in p., it follows that p^(0) =0, and w t (0) = for every i. The left members of equations (50) are the same as the left members of equations (9), and therefore, when the solution (18) is substituted in equations (50), the terms of the left members have the form (19). The right members have no terms independent of ju 2 > and the equations for the determination of the coefficients of the first power of n are of which the solution is +cj l) )+^ 1) cosr+^ ) sinr, i) )T-2c; i) sinr+2riodic the coefficient of T must be zero; whence rl" = -4/3. The constants c"' and f{" arc determined ly the conditions Pi(0)=0, ir,(0)-0. Therefore rj" = c? = 0. The constant ri" is determined in the following step of the integration [see equations (23)]. This process is applicable to all the succeeding steps. The differential equations (50) have a particular form which admits a symmetrical solution, and it can be established by complete induction that the equations for the determination of p, and ', have the form (fr 1 (56) P| indpt _ where c"~" is determined by the condition [compare equations (29)] The solution of equations (56) is "cos2T a"co8iY ( (57) w t = Oinr+6;"sin2T+ +OintY, where __ _> _ 3^.' a > - 1 / O 7Y> > (58) 204. Numerical Example 2. As a first example of a symmetrical periodic orbit we consider three finite bodies revolving in circles according to the straight-line solution of Lagrange. We suppose that the mass M , about which the particle revolves, is between M t and M l . Choosing M for tin- unit of mass, we select M, = 10, M t = 5. The unit of distance is MA/,; and it follows from the solution of the quintic equation of Lagrange* that the distance M,M is /?, = 0.77172 .... The unit of time is selected so that N = I, and the period of the solution is assigned so that v = 5; whence See Moulton, Celettial Mtctumic* (second edition), p. 312. 416 PERIODIC ORBITS. The differential equations of relative motion of the particle are where a is given by the relation i?a 3 = k 2 . The constant A; 2 is determined by i /,, . M 8 \ . R ^l* K ' whence A; 2 = 0.23763, A; 2 ^^ 2. 37630, ^^=2.58518. o= 1.05914/*, - = 1.37222 M. /Vj The diJGFerential equations of motion become ~ p +JU+ P = (2-48074+7.44222 - (1.15938 cos w?+ 1.93230 cos3w)pV + (4.23765+9.41700 cos2w;+16.47975cos4w)pV+ (60) + (0.38646 sinw+1.93230 sin3w)pV - (4.70850 sin2w+ 16.47975 sin 4w)pV+- The periodic solution of equations (60) is P = 1 - M- (0.27136+0.96615 cosr+4.96148 cos2r) M 2 + (0.62584- 19. 13740 cos r-2.47963 cos2r+0.40256 cos3r) M 3 + , (61) W = T+ (1.93230 sin r+ 6.82204 sin 2 TV + (41. 10885 sinT+10.74793sin2r-0.48307sin3TVH I'AKTICLK AI1KACTED BY It Sl'HI.i 417 On substituting the value n = 0.2, the orbit represented by eeriodic orbit we use the configuration (B), 201, of nine finite bodies, the numerical values l>eing those given. The unit of time is selected so that .V = 1, and the period of the solution is assigned so that V' = 5, whence M = 0.2. The differential equations [corresponding to equations (50)1 of relative motion of the particle are ' + ? - 2 ' '"[* . /t, ,'- ir,)+35cos4(ic- H' ( P j~ 418 PERIODIC ORBITS. On taking account of the relations (44) , and choosing the unit of distance so that -4 = 1, the preceding set of equations takes the form ~1 -J ISM" , fas/a ~~ From 201 we have the following values: K = 2, M'=l, M" = Since AT = 1, the last equation gives A; 2 = 0.60994. It follows that P M" / n \ 2 ~- =0.62920, a 2 = 2.10300M 2 , (|?) =0.52575 /A On substituting these numerical values, the differential equations become ~ P + y+ ~* = 2-47828 pM 2 +[3.63039+8.32914cos4io]pV+ , The periodic solution of these differential equations is P= l-M-0.27054 M *+1.70909M'-3.43127M 4 -0.8329lM 4 cos4r+ On substituting the value M = 0.2, the final result is found to be P = 0.86403-0.00133cos4r+ , w = r+0.00150sin4T+ The orbit has four axes of symmetry, namely, the lines connecting the central body with the others. It differs from the orbits of the other numer- ical examples in one respect that is, it lies entirely inside the comparison circle (see 200). In terms of p the radius of the comparison circle is about 0.8855. Fig. 13 is not drawn to scale, but the characteristic properties of the orbit, which are readily seen from the numerical values of p and w, are exaggerated to make them apparent in a small drawing. The inner circle is drawn merely to indicate the direction of the deviation of the orbit from a circle. PARTICLE ATTRACTED BY H SIMIKKi:-. 419 206. The Undisturbed Orbit Must be Circular. In tin- proofs of the exigence of periodic orbits (198-201) it was assumed that the undisturbed orl>it is circular. It remains to be shown that this assumption is necessary. The proof will be made for the case of symmetrical orbits [equations (50)] and is applicable also to the orbits of 198. The undisturbed orbit is given by the solution of equations '>()> when ^ = 0. For M = 0, the equations are the equations of motion of a particle subject to the attraction of a central force varying inversely as the square of the distance. The undisturbed orbit is therefore a conic ; and, since we are concerned only with closed orbits, must bean ellipse. Since the period in r is 2*, the major semi-axis of the ellipse must be unity (in p). The eccentricity, which will be denoted by e, is, however, arbitrary; that is, for M = the differential equations admit an infinite number* of symmetrical periodic solutions. Starting now with an ellipse for the undisturbed orbit, it will be shown that the eccentricity must be zero in order to fulfill the conditions of periodicity. For /u=0 the solution of equa- tions (50) representing an elliptic orbit of eccentricity e is *. FIGURE 13. p= 1 ecosE, C( /'.' . i-ecosE-> - ecosE where E is defined by the relation r = arc P = l-, P' = 0, Eem\E. The initial conditions for w' % i-,' 7 (T-e)* Consider the solution for values of n different from zero, but sufficiently small, and let the initial conditions be v'l -?.i... x'l-Ce+c)* /; ^s ~rP4 /. _\iW. ri i -\li M- If a and e can be determined in terms of n, vanishing with /*, so that the conditions (52) are satisfied, then the solution will he periodic with the The general CMC when the differential equation* (for M-0) admit a periodic solution containing an arbitrary parameter has been mentioned by Poincart, toe. <., vol. 1, p. 84. 420 PERIODIC ORBITS. period 2ir. All terms of the solution which are independent of ju* may be obtained from the two-body problem by making the substitution w = u /rr. These terms are given in finite form by the expressions u = are cos = M (je+e)cosE/ where E is defined by the relation ) , /t =E-(e+e)smE. On returning to the variable w, writing the terms in a and e as power series by Taylor's expansion, and applying the conditions (52), we obtain the equations It follows from the known properties of the series that there are no terms in e alone, and there are no terms involving /i to the first degree except the term PTTJU. Hence the second of equations (63) can be solved for a as a power series in e and /* in which ^ is contained as a factor; the result is a = MT T When this value of a is substituted in the first equation, a factor /j. can be divided out, leaving This equation can be solved for e as a power series in n, which vanishes with n, if and only ifH = Q. Since only those solutions are under consideration which are the analytic continuations with respect to /* of those for n = Q, the con- dition e = must be imposed. The condition e = Q means that the undisturbed orbit must be circular. 207. More General Types of Motion for the Finite Bodies. This sec- tion contains some remarks upon possible extensions of the analysis which will permit applications to practical problems of celestial mechanics, and is followed by an illustrative example. The particular problems treated in the preceding articles have no appli- cation in nature because the configurations assumed for the finite bodies do not exist. But a glance at the details shows that these configurations are not essential to the proofs. The possible generalizations of the motion of the finite bodies can be made in three ways: PAKTK'KE ATTKACTEU BY SPHKIiKS. 421 (1) In 198 the existence proof depends only upon certain terms of the disturbing function which are due to the body M l . If A/, retains the motion there prescribed, we may add other bodies to the fixed configuration in the rotating plane provided the operations with the power series are valid. This merely increases the number of terms in the second members of the equa- tions of motion ; the existence proof and method of construction are unchanged . (2) In the examples treated the finite bodies form a fixed configuration in a plane rotating with constant angular velocity. This is not necessary for the type of analysis used. If M, moves in a circle with uniform angular velocity, the other bodies can have any periodic motion, provided always that the convergence conditions hold. In this case the differential equations of motion of the particle involve r explicitly and are periodic in r. Two points of difference occur in the analysis: (a) Suppose the period in r of the differential equations is T; then the assigned period of the motion of the particle must be a multiple of T. (b) The differential equations do not admit the integral of Jacobi, and hence no use can be made of this in the existence proof. This is equivalent to saying that at r = we can not assume u; = 0, but must determine the initial longitude of the particle by the conditions of periodicity. The method of determining the constants of integration in the construction of the solutions is explained in a paper in the Transactions of the American Mathematical Society, vol. 8 (1907), pp. 177-181. (3) A further generalization of the motion of the finite bodies is pos- sible by permitting M, to move in a path which is not circular. It is possible to show that the analysis can be used if the motion of A/, is subject only to the mild restrictions that the expression for the radius vector shall contain only cosines of multiples of T while that for the longitude shall contain only sines. The case when the orbit of Af , is an ellipse is treated in the article referred to above. For this generalized motion of the finite bodies there may exist symmetrical orbits of the particle. In equations (50) the first contains only cosines of multiples of w, and the second only sines of multiples of u;. The periodic orbit of the particle may be symmetrical if the first equation contains also sines of multiples of w multiplied by odd functions of r, and cosines of multiples of to multiplied by even functions of T, and the second contains cosines of multiples of w multiplied by odd functions of r, and sines of multiples of to multiplied by even functions of T. From these remarks it is apparent that the treatment can be made sufficiently general to permit applications in the problems presented by the motions of the solar system. For example, suppose P is a satellite of one of the planets M, and that A/, is the sun . This implies that the disturbing effects of the satellite upon the other bodies are neglected, since we assume that its mass is infinitesimal. The conditions upon the motion of Af, are fulfilled if we neglect the perturbations of the other planets upon M; that is, if we suppose the orbit of A/, relative to M is an ellipse. If we neglect the incli- nations of the orbits of the other planets, and suppose that their motion is 422 PERIODIC ORBITS. periodic (that is, we assign a periodic motion which is approximately correct), it is possible by the methods given to treat the periodic motion of the satel- lite in the plane of the planetary orbit, when subject to the attraction of the sun and all the planets. The following numerical example is a simple illustration of the general idea. 208. Numerical Example 4. The mass of M is taken as the unit of mass and M lt of mass 10, is supposed to revolve about M in a circle of unit radius with uniform angular velocity N. A third mass, M t = M = 1, is supposed to revolve about M t in a circle of radius A t with uniform angular velocity N t . The unit of time is chosen so that N=l, and the period of the motion of the particle is assigned so that v = 5, whence N H = m= =0.2. v With reference to M as origin and an axis passing always through M l , the coordinates of M lt M t , and P are, respectively (1,0), (R t , W s ), and (r, w). The differential equations of relative motion of the particle [corresponding to equations (50)] are 16 - W t ) + f |- P {sin(;- WJ + 5sin3 (w- W t ) J (64) The constant A; 2 is given by the relation N* = k*(M-}-M l ), whence # = 0.09091, A;W! = 0.90909, k*M t = 0.09091. From the relation v*a 3 = k*M, it follows that The angular velocity of M t about M l will be selected so that its period with respect to the rotating axis MM l is one-half the period assigned for P. Hence N t -N = 2v, or #,= 11. The radius, A t , of the circular orbit of M t with respect to M l is determined by the relation N$A* t = ^(.M^+M,) . whence A 9 = 1 .01200ju. On assuming that PARTICLE ATTRACTED BY U SPHERES. 42.3 at T = 0the finite bodies are in conjunction in the order M, A/,, A/,, the coordi- nates (/(",, M',) of A/ f with respect to M are given by the expressions coslf = ^ 2ft On substituting the values of the constants and the coordinates ft, M , in equations (64), we obtain for the numerical differential equations of relative motion -*(4r +M )'+-, =(0.50000 + 1.50000 cos 2u>)pM vw.wvrw ~r~ * .tjwwv/vro ttw mi* p + (0.86523 cos w+ 1.44205 cos 3u>)pV + (0.33273 + 0.73940 cos 2u> + 1 .29395 cos 4u?)pV + (0.27600sin2rsin2t0-0. 13800 cos2T-0.41400co82rco82u>)pM' + (0.10474+0.17456 cos4T+0.31422cos2u>+0.10471 cos 4rcos2u> -0.55864 sin 4r sin 2u>)pM 4 + (0.07960sin2Tsinu>-0.31840cos2rcost0-0.53068cos2TC083u> +0.39801 sin2rsin3u>)pV+ , (65) +2^(^ + M ) = -(1.50000sin2u;)pM t - (0.28841 sin u>+ 1.44205 sin 3u>)pV - (0.36970sin2u>+ 1 .29395 sin 4u;)pV 4 + (0.27600 sin 2r cos 2w> +0.4 1400 cos 2r sin 2u>)pM* -(0.31422 sin 2u>+0.10741 cos 4r sin 2ic+0.55864 sin 4rcos2u;)pM 4 + (0.02653 sin2TCOSU>+0.10613cos2rsint/;+0.39801sin2TCos3u> +0.53068cos2Tsin3uj)pV+ The right member of the first equation contains only, (1) cosines of multiples of w, (2) cosines of multiples of w multiplied by cosines of multiples of T, and (3) sines of multiples of w multiplied by sines of multiples of T. The first equation is then unchanged if we replace w by - w, and T by - T. The right member of the second equation contains only, (1) sines of mul- tiples of w, (2) sines of multiples of w multiplied by cosines of multiples of T, and (3) cosines of multiples of w multiplied by sines of multiples of T. Hence the second equation is also unchanged if we replace w by w and T by T. Now let us suppose that is a solution of equations (65) satisfying the conditions p'(0) =u;(0) = 0. It follows from the form of the differential equations that ^, is an even function, and ^ t is an odd function of T. When r = the finite bodies are in conjunction in the order M, A/,, A/,. Therefore, if the particle P crosses the line A/ A/, orthogonally when the finite bodies are in conjunction in the order M, A/, , A/ t , the <>rl>H In the rotating plane is symmetrical with respect to this line and this epoch. 424 PERIODIC ORBITS. On constructing the solution of equations (65) by the formulas (58), we get P = l- .66667^+ (0.38889+0.72102 cosr-cos2r)M 2 + (-0.02616+2.09168 cos r- 0.45400 cos 2r- 0.30043 cos 3r +0.03450 cos 4rV+ . ((50) w = T + (- 1.44204 sin r+ 1.37500 sin 2r) M l + (-6.29838 sinT+2.12066sin2T+0.36051sin3T -0.03881 sin 4rV+ The orbit represented by equations (66) is shown in Fig. 14. The points which are num- bered 1, 2, . . . , 8 rep- resent the positions of the particle in the peri- odic orbit at intervals of T = 7T/4 . The corre- sponding positions in the comparison circle are indicated by the numbers 1', 2',..., 8'. With reference to the differential equa- tions (65) we can make the same statement concerning the unique- ness of the solution that was made in 198. These equations were written on the assump- tion that, at T = 0, the finite bodies are in conjunction in the order M, M l , M, . Without this assumption the expressions for R t and W t contain a parameter indicating the position of M t in its orbit at the origin of time. With reference to the physical problem, therefore, we can not affirm in this case that, for a preassigned period, there exists one and only one direct periodic orbit. It is necessary here to add a condition on the form of the configu- ration of the finite bodies and particle at the origin of time. For example, we might obtain another symmetrical periodic orbit having the preassigned period if the particle crosses the line M, M l when the finite bodies are in conjunction in the order M, M s , M l ; and we might have still other orbits with the preassigned period if P crosses this line when the finite bodies are not in conjunction. Fro. 14. CHAPTER XIV. CERTAIN PERIODIC ORBITS OF / FINITE BODIES REVOLVING ABOUT A RELATIVELY LARGE CENTRAL MASS. BY FRANK LOXLEY GRIFFIN. 209. The Problem. For a given system of k finite bodies, moving in a given plane relative to another given body, there is a 4fc-fold infinitude of possible orbits the variations which the configuration of the system under- goes and its orientation in the plane being determined jointly by the mutual attractions of the bodies according to the Newtonian law, and by the values at any instant of the 2k relative coordinates and their first derivatives with respect to the time. The differential equations admit no algebraic or uni- form transcendental integrals,* aside from the two fundamental integrals of energy and areas, even when the masses of all the bodies except one are very small; nevertheless, by restricting the initial values of the coordinates and their derivatives, in a manner to be shown below, it is possible to find an extensive class of periodic solutions. In fact, for arbitrary values of the masses (save that one of them, M , shall be large in comparison with the others, M,, M t , . . . , A/),f there exists a k-fold infinitude of distinct periodic orbits of the system, having an arbitrarily preassigned period T. In these orbits (which, for small finite values of M lt M t , . . . , M tl depart but little from a set of concentric circles about the planet) the k satellites come periodically into a "symmet- rical conjunction," that is, they are all momentarily in one straight line with the planet and moving at right angles to that line. These conjunctions may, or may not, always occur at the same absolute longitude; in the latter case the motion is periodic with reference to a uniformly rotating line. Besides the demonstration of the existence of such periodic orbits, this chapter contains : A method of constructing the solutions without integration, a single application of that process having provided formulas which reduce the problem to one of algebraic computation ; a numerical application to the case of Jupiter's satellites I, II, and III; a proof of the non-existence of cer- tain other types of orbits; and a brief consideration of some related questions. See memoirs by Bruns and Poincare, in Ada MaAematica, vols. 1 1 and 13. tin other words, the distribution of masses is such as is presented by the sun and any number of planets, or by a planet and any number of satellites. For convenience, in what follows, a single expression, planet and satellites, will be used with the understanding that it covers also sun and planets. 426 426 PERIODIC ORBITS. The problem may be formulated thus : Let quantities ju, ft , . . . , ft be defined by M^ = M i (i = i, ...,*), (1) where one of the j8's is to be selected arbitrarily. Let a system of positive or negative integers without common divisor, p t (i = l, . . , fc 1), and a number <7 t 5^0 be selected arbitrarily, save for the restriction mentioned below, and let v , n t , and a, be defined by _ 2?r _ n t _ n t n t .. , . (ey \ "~" ~ -! *- (t = l,...,*), (3) where K* denotes the gravitational constant, and where, of the three values of a, satisfying (3), that one is to be selected which is real. Also let the notation be so selected that a l} . . . , a t are in ascending order of magnitude, the PI being so selected that no two of the ck are equal and no n ( vanishes. If M were zero that is, if the satellites were "infinitesimal" possible orbits would be circles about the planet with c^, . . . , a t as radii; from (3) it follows that the angular velocities would be r^, . . . , n t . It is quite immaterial whether any of the n t are negative; the results obtained hold irrespective of retrograde motion of some of the bodies. The configuration of the infinitesimal system would undergo periodic variations with the period T; for, it foUows from (2) that 27T = T } or each synodic period is a sub-multiple of T. This condition being satisfied,* the motion of the infinitesimal system would be periodic with respect to a line through M, rotating with uniform angular velocity that of M t , or, indeed, that of any other bodyf M , though whether or not the system ever returns to the same position in space depends upon whether q t is rational or irrational. In describing the orbits mentioned, the infinitesimal satellites would be subject to certain initial conditions, the 2k coordinates and their derivatives with respect to the time having at the instant t = t certain values, say Co(t=l, . . . ,k;j=l, . . . ,4); but if the k finite satellites are subjected to these same initial conditions, their mutual disturbances in general destroy periodicity. The first problem is, then, to determine what, if any, incre- ments Ac w can be given to the former initial values di to preserve the periodicity when all the satellites are finite. *Poincar6, treating three satellites (Mtthodes Nouvelles de la Mtcanique C&este, vol. 1, pp. 154-6), states the condition thus: Integers o, ft, y, mutually prime, exist such that 0+^+7=0 and an l +ftn,+yn,=0. Evidently, in the case of three satellites, this condition ia equivalent to (2), since (n, n,)//J = (n, n,)/( o); but for a greater number it is not so. Thus, if n,=7, n, = 5, n,=3 v^7 n, = V2~ t the integers 5, 7, 1, 3 satisfy a condition similar to Poincar6's, but periodicity is impossible. For the general case a re-formulation such as (2) is necessary. fThe commensurability of n n t (i = l, . . . , k 1) evidently involves that of n n/ (f = l, . . . , j l,j+l, . . . , fc). For from n t nt = ptv and n ; n t = p ; K it follows that n i n l "(p i p 1 ). PROBLEM OF k SATELLITES. 427 210. The Differential Equations. Let the common plane of relative motion of the k bodies be selected as the //-plane, the origin being at M, :uid ME and M II being rectangular axes which rotate in the plane with the uniform angular velocity N. Let the coordinates of A/, referred to these axes be , and T;,; then the differential equations of motion are () *3- ^^-N>l l +t(M+M t )*--Z>SM l (S> .*)=0, (4) (6) 5J + 2N ^ - A-'ib+^V+M ,) J' ar at 77 where mid -' means 2'!* QVi)- Except in proving a certain symmetry theorem, t hese coordinates are less convenient than polar coordinates referred to rotat- ing reference lines. Besides (1), (2), and (3), let the following definitions be made: i tj = a Jo., , vq t = n, (-!,...,*), (5) where the X, are arbitrary constants, later to be taken as the longitudes of the Mi at the origin of time for n = 0. Let polar coordinates be introduced by the equations , = r,cosu,, ij, = r,smui, r t = a t x l , u t = w { -\-p i r-\-\, , (6) and N be taken equal to n, so that w, is the longitude of 3f, referred to a line rotating with uniform speed n, . The differential equations become (7) (6) -.) - =0, where a]a] J = a\^ i +a]x L J -2a t a i x t x J cos(4> Jl +w ) -w l ), and where the accents on the variables indicate derivatives with respect to T. 211. Symmetry Theorem. // a symmetrical conjunction occurs at any instant t = t,, then the orbit of each satellite before and after the conjunction is symmetrical, both with regard to geometric equality of figures and witii regard to intervals of lime. A proof will be given only for the case <, = 0, which cloc- not limit the generality since any other case is reduced to this one by the substitution < = 428 PERIODIC ORBITS. The differential equations (4) are invariant under the substitution ?< = &, if =-17., l=-t. (8) Consequently, every solution of (4) is transformed by (8) into some solution of (4). Moreover, the initial conditions & = *, * = 0, '=0, 4j*=b. (9) are transformed into ?< = ., * = 0, ^=0, g'=6,. (9') Therefore, that solution of (4) which satisfies the initial conditions (9) is transformed by (8) into itself. Hence, if that solution is , = *,(*), * = *,(*) (t = l, ...,*), (10) then *(0 =*(!) =*i(-0, *(0 = -*.() = -*,(-), whence also It will be noted that the proof holds, whatever the value of N. It is also geometrically evident that the symmetry, if present at all, is independent of the rate of rotation of the reference line. 212. Conditions for Periodic Solutions. Since the differential equations (7) are unchanged if r is replaced by T+2mr, or t by t+nT (n being an integer), it follows that if (t = l, ..>.,*) (11) is a solution, then so is x t = x t (T+2nir), w l = w,(T+2mr). (12) These two will be the same solution if the coordinates and their derivatives have the same values at r = r ; that is, if x' t (r l) +2mr}=x' t (T ), wfa^nir) =w' i (r ). ) If these conditions are satisfied, then, for all values of T, r) =XI(T), w t that is, (13) are sufficient conditions for the periodicity of the solutions. That they are also necessary is obvious, if the period is to be 2mr. I'HOBI-KM OK k S.MKI.1.IIE8. Special case. In the case of a symmetrical conjunction at r = 0, other sufficient conditions can he formulated. For, if x'(0) =u?,(0) =0 (t-l, . . . , fc), and if every X, is a multiple of T, it follows from the symmetry theorem that -T), 1 -,T). J But, by equations (13), if T, is put equal to -T, the conditions for period- icity of X,(T) and U,(T) are (a) *,(*)=*,(-*). (c) w t (ir) =>,( -T), J (b) *;(*)= *;(-*), (d) ,;(*) =*;(-*). } Of these conditions (a) and (d) are satisfied by virtue of (14), while (6) and (c) are also satisfied if z'(n-) = U?,(T) =0. It may then be stated that sufficient conditions for the periodicity of x, and w, (with period in t equal to T) are (a) *;>)=0, tr,(0) = 0, X,=OorT, J (6) *;=<), >,(ir)=0. J Moreover, after conditions (16a) have been imposed, conditions (6) are necessary as well as sufficient. 213. Nature of the Periodicity Conditions. For M = the differential ec (nations (7) admit the solution with period 2r (or T in giving the circular orbits r,=a, , w,=X,+p,T, in which at T-0, x,-l, if =tp,=t01-0. If these initial values are given increments Ac,j(t=l, . . . , k; j=l, 2, 3, 4), then the solutions of the differential equations (7) for M^O are developable as power series in n and the Ac,, , which converge throughout a preassigned in- terval of T for sufficiently small values of those parameters.* Such solutions are in general non-periodic; in fact, the periodicity conditions (13) or (16) impose the condition that 4Jk power series in these 4fc+l parameters shall vanish. In the cases to be considered these 4A; equations will determine the Ac,, as unique functions of n, holomorphic in the vicinity of n = Q and vanishing with n; so that, for sufficiently small values of M, there exist initial conditions (depending upon T, q u the p, , M, and the ft) such that the orbits described are periodic with the required period. Evidently for smaller and smaller values of /u, smaller and smaller devi- ations from the initial conditions of undisturbed motion are sufficient in order to get periodic orbits. These orbits for ^^0 may be said to "grow out of" the undisturbed circular orbits as M grows from zero. Of course, for any given masses, n and the 0, being fixed, the possible orbits of this sort can See fJU-16. 430 PERIODIC OUBITS. vary only with T, q t , or the p ( ; but to a range of values of /j, there corresponds a class of orbits. In what follows it will be inquired whether the conditions for periodicity can be satisfied by such values of the Ac,, as to prove the existence of a class of periodic orbits of each of the following types : Type I. The finite system has a symmetrical conjunction. Type II. The infinitesimal system has a symmetrical conjunction, but the finite system has none. Type III. Neither system has a symmetrical conjunction. 214. Integration of the Differential Equations as Power Series in Parameters. It will be necessary to obtain the first few terms of the devel- opments mentioned in the preceding article. Instead of increments Ac,, to the initial undisturbed values of the coordinates it will be more convenient, in finding the properties of the solutions, to employ parameters An, , e, , w, , T, , defined as follows. At T = let (17) the v t being equal to ti,+g t r, the true longitudes from a fixed reference lino. It is evident that the Ac w are holomorphic functions of the An,, e ( , w,, and r t for sufficiently small values of the latter quantities. Consequently, solutions of (7) exist also as power series in the new parameters. Further, since the real positive values of the radicals and the smallest values of the inverse cosines are to be taken in (17), the Ac, y are given uniquely in terms of the An, , e t , w< , and T< . From these two facts it follows that, if the latter quantities can be determined as unique power series in n, satisfying the con- ditions for periodicity, then also there exist for the Ac,, unique power series in ju, satisfying the conditions. Conversely, while the Jacobian of the Ac,, with respect to the new parameters is zero for An, =e t = o>, = T, =0, yet, in the only case where discussion will be necessary (viz., for Ac ( ,= Ac f3 = 0, whence u, =T, = 0), the solution for the An, and e t in terms of the Ac,, and Ac, 4 is unique; for the Jacobian of the Ac (l and Ac, 4 with respect to An, and e, is distinct from zero for An, = e, = 0. Hence, in this case, if the Ac, t and Ac, 4 exist as unique series in /*, satisfying the periodicity conditions, so also must the An, and e f exist as such series. In the developments of the coordinates as power series in ju and the new parameters all those terms independent of n may be obtained, together with a knowledge of theirjproperties, in the following simple manner: The terms 1'KOBLEM OF A' SA1 KI.I.I I B8. HI in question are those remaining when n is put equal to zero, and are there- fore the solution of the problem of A infinitesimal satellites when the initial conditions are (17) in other words, the solutions of k two-body problems. The dynamical meaning of the new parameters is then evident. The orbits of the infinitesimal system, subjected to the initial conditions (17), are ellipses in which the mean angular motions, major semi-axes, eccentricities, longitudes of pericenter, and times of pericenter passage are respectively n,(l+AnO, o,(l+AnO~ l , ft, X,+u ( , - If in the development of x, the coefficient of Af cj o>? r? be denoted by x, iAM then, by applying Taylor's theorem to the well-known developments of the coordinates in elliptic motion as power series in the eccentricity,* it is found that the following coefficients of first and second degree terms do not vanish : (18) ,=28019,7-, u>t,nn = 2q,coaq { r, From simple dynamical considerations the following important properties can be established. Let x,. /M be written zjj^ to indicate its dependence upon T. Then (-l *), (19) where m is any integer; for, the coefficients x^, etc., are those of the terms which do not involve An, , w, , and r t , these terms being obtained by putting An, = co, = T, = in the developments. But for these parameters equal to zero the initial positions are apses and the periods (in are 2r/n ( . Hence, at T = mTr/q,, M t is at an apse and x' l = w t = Q, whatever the value of f t . Since this is true for a range of values of e t , it follows that the coeffi- cient of each power of e< in z,' and in w t is zero at T = mr/g,. It is evident that in the terms independent of M only those parameters appear whose subscript is the same as that of the coordinate developed ; the terms involving /i introduce, however, the other 4(A 1) parameters. Terms involving p. The only terms involving M whose coefficients are needed in the sequel are n and pe, (j-1, . . . , k). Let the coefficient of n in the development of x, be z,(0; T), and that of ne, be z(j; T); let the coefficients of the same quantities in w t be respectively w,(0; T) and w,(j; T) (t=l, . . . ,k;j = l, . . . ,k). The process of finding these depends as fol- lows upon two properties of the solutions: Moulton, Introduction to CeUttial Mechanic* (new edition), p. 171. 432 PERIODIC ORBITS. (a) Since the solutions must satisfy the differential equations identi- cally in the parameters, the equating of coefficients of corresponding powers on both sides furnishes sets of differential equations for the successive co- efficients in the solutions. (6) The arbitrary constants which the successive coefficients carry are determined by the conditions that the solutions shall reduce identically to equations (17) at r = 0. For each pair of coefficients x f (f; r) and w t (f; r) (/=0, . . . , K), equations (7) give two simultaneous differential equations of the second order. The one from (7a) can be integrated once immediately, and its integral combined with the equation from (76) renders the latter a well-known type, m-O (20) Its solution, z ( (/; T), when substituted into the first integral, permits the final integration for w t (f; T). The initial conditions are x t (f;0)=x'(f;0)=w t (f;Q) = w' i (f;0}=0 (i=l, . . , fc;/=0, . . . , *), (21) for the conditions (17) do not involve /* at all. Now the form of the solution varies greatly according as a term COS$,T or sin^T is or is not present in (20). In the former case the solutions con- tain a so-called Poisson term, TCOsg,T or r sin q,r, and in the latter case they do not. In all the x t (f;r) and Wt(f;r) (/ = !,..., /c), a Poisson term is present; they are present in the Z,(O;T) if, and only if, for some pair of the HI , say n, and n, , there exists an integer J such that / (H,-n.)=n f . (22) The meaning and consequences of such a relation will be discussed in 219. In performing the integrations it is necessary to expand ( 1 - 2e w cos 4> fl + ,* ) ~5 ( = 3, 5) as a cosine series, where, for the sake of a uniform notation, the following definitions are made: e = a,, and 77y = o/,, if j i. (23) Then (24) 00 = 2 (?,((,) cos m ;,, m-O where the F m and G n are well-known power series in e u , beginning with e- PROBLEM OF k SATELLITES. 433 Finally, the desired coefficients of the functions x, and to, are, for r'=l ..... A and fnr/ = (). 1 ..... /,- , +K,,T*\n2q l T+ 2 (a^' (25) 2E, f TS\nq,T r +-K,,Tcos2q,T+ 2 (frT/al where /> = () and J (/ =K,, = for/^t, while if every ^ = or T, then y=o, c (/ =A / =, / =/> 1/ =c 1 7=C=o H: , it i*,tn integer. Here A, = (). Nevertheless the .lacohian of the with respect to the A/> ; , taken at An, = e, = p = 0, is A^T'M^O. (31) (The only possibility for q t = is p, = -q t which requires n, = 0, and such a -election for p, has been excluded.) Hence (29a) can be solved for the A;/,* as power series in the e, and n, converging for sufficiently small values of the latter (plant it ies. Now, by (19), every term in (29a, 6) has either n or some 11, as a factor: hence the solutions have the form A//,= M /' ( (C ; ,M) ('-! ..... t; j-l ..... k). (32) If the power >eries (32) are substituted in (296), the resulting series converge for sufficiently small values of M and e, and contain n as a factor. (This merely means that AM,=M = O satisfy the periodicity conditions, whatever may be the values of the e,). If p is divided out.f relations are obtained among the e, and n of the form ' At this point two questions of importance arise, viz., as to the vanishing of the x',(0; T) and as to the vanishing of the determinant of the coefficients of the linear terms in e,. By (25) and (27) every x',(Q; r) is zero unless the relation (22) holds for some pair of the n,. When such a relation does hold, the equations (33) are not satisfied by f, = /u = 0, so that solutions for the c,, vanishing with p, do not exist. Hence, periodic orbits of Type I, "growing out of the circular orbits," do not exist, if, for any n, and n,, J (n/ n,) = n, , where J is an integer. When no such relation exists, J the equations (33) are satisfied by e, = n = Q. It remains to examine the determinant A, of the first degree terms in the e,. This involves, in each of its elements, power series in the to, or (it/a,, a,/o; and it is unknown whether there are any sets of values of the e/ for which A, = 0. It will, how- ever, be shown that there is an infinite number of values for which the determinant is distinct from zero. Let P/ be any element of A,, the first subscript indicating the row and the second the column; then, by equations (33) and (25), P l , = x' l (f;T)=q l (-ir>*H, f (/*<), 1 P ll =x' l (i;*)-q l (-l)'l, where the Qv(a) (/= 1> > &) are power series in a, beginning with a constant term* In P under the sign S', the lowest power of a for j < i is lOt 4j 6k, this exponent having its smallest value 6i + 4 6/c, when j = il; while for j>i the lowest power is 6j 6/k, whose smallest value is 6t'+6 Gfc. Hence in the i* row of A, the lowest power of a in any of the P v is for/i, 6i+8-6/c, viz., for/=i+l. But, in the first row P u carries a 11 "" as against a 14 "'* in P u , which is the next lowest power. Evidently, then, in every row of A 3 the lowest power of a occurs in the element of the main diagonal; and if that lowest power of a is removed from the row as a factor of A 3 , a new determinant A 4 is obtained, all of whose main diagonal elements begin with a constant term, while the series in every other element carries a positive power of a as a factor. Thus the development of A 4 as a single series in a begins with a constant term (the product of those in the principal diagonal) and is distinct from zero both for a = and for all values of a up to some finite value. Hence A 3 also must be distinct from zero for all values of a sufficiently small, and vanishes, if at all, at a finite number of points. In case some -i;<7 main diagonal element than elsewhere in the same row, except in the first n>\\, where /' may now carry a a ~ u , as does P u . And even this exception is immaterial, since, when the rows have been factored as before, the constant term in the element Q u is to be multiplied by a minor whose first column contains a* as a factor, and hence can not destroy the constant term in the main diagonal product. Therefore, it is true without exception that Aj?* for all values of a sufficiently small. Therefore, equations (33) can be solved for the e, in terms of n, vanishing with n. The substitution of these solutions in (32) gives the An<, as well as t he e, , as holomorphic functions of /*, for ju sufficiently small. Hence there exist initial conditions giving periodic orbits of Type I even when q t is an integer, provided that no relation (22) holds and that a is sufficiently small. In drawing this last conclusion, however, a point of delicacy arises. The p, are functions of a in the foregoing argument, and obviously the p, are not integers (as the formulation of the problem requires them to be) for all values of a on any interval. The question arises as to whether there are, indeed, any values of a, "sufficiently small," for which the p, are integers. From (36) and p t = q t qt, it follows that l) 0-1 t-1). (37) The present discussion will be confined to exhibiting a selection of the l>, such that there is an infinite number of values of a less than any assigned quantity, for each of which the p t are integers without common divisor. Let the assigned value be a, and let the 6 be defined by (i " ..... *- |)! (38) and consider (37) for a = a 9 -\/l/n, where n is an integer. Evidently, since t is an integer, and since -q t 0-1 ..... fc-1), (39) '- the p, are integers. Consider the possibility of a common factor. If p,_, and p_, have a common factor, their difference has the same factor. Thus, if there is a factor common to (g*+l)n-g and (g*+2)n*-g, it is also a factor of n*+n(n !)(&+ 1). Hence if n is prime and greater than q t , such a factor must divide n-|-(n l)(g+l) and also the difference between this number and p_,, or (n 1). But, as n and n 1 are mutually prime, there is no factor of n 1 which divides n+(n 1) ( ( ,I(T) must satisfy (a) < 1 +2g < < 1 +S' w sin^l - tf, I F.(O*fe = 0, (6) 2 y L m F^e^cosnxb J =0. =o (43) Since every it is a multiple of T plus a multiple of IT, equations (43) are of the typo (a) >", + 2?X.+ SDftsramr-0, (44) where the D'' ari d ^iTi are linearly related to the F m , and can be ex|>rossed in terms of the latter as soon as the p t are chosen. The solutions are S A* cos WIT, where the c|* (j = 1, . . . , 4; t = 1, . . . , fc) are the constants of integration and (' -^D-4 M = #M - ^ DM, ^ 2 ^'"' = ^M - 2wi9, ^1 M. (46) A *6 Poisson terms do not appear in (45); for, since no relation (22) holds, no term in cosq,r or sin^T is present in (44). Now, by (41) and (42), ~<3> _ <<> _ A /- ( " - F l,l t/ (,l~ u C M OQ -"lill so that the A; constants c<" alone remain to be determined. And here arise two cases, just as in the existence proof: PROBLEM OF k SATELLITES. Case I. q t is not, an integer. Here, q, not being an integer, coeq, r does not have the period 2*; consequently, by (42), c" = (t 1, . . . , k). ('use II. q t is an integer. Here cosq t r has the period 2ir, and (42) is satisfied for the arbitrary c.^O. These k Constanta remain undetermined until the second-order terms are found, when the c* are uniquely determined in destroying Poisson terms. Terms of any order. Case I. Assume that for n-1, . . . , h\, the x,, n (T) and w,.. (T) have been found, the constants being determined, and have the form x lin (r)= 2 ^coawr. W,..(T)- 2 B^rinmr (-!,...,*). (47) An induction will show that X^(T) and u>(,(r) have the form (47); that, moreover, the differential equations for x,, and u\* are of the type (44); and that the constants of integration are determined just as in the preceding case for n = 1. The differential equations are [see (7)] (a) 2 .T, t u>",+2 2 .r;..tc;.,+2g,jr;.. (b) a-;;- 2 * l * t+l+m-k +q t l ti l (x7\- l +2'8 l ,>a lj 2 (48) + 2 Xj^cos^+Wj-w^xT'-ffu 1 }^ =0, *++--! where an expression in parenthesis, having a subscript / outside, denotes the sum of all those terms in the expression which involve /*' It is to be shown (a) that the variables in (48) whose second subscript is h enter in the same form as the x<,i and w,,i enter (44), and 03) that the remaining terms of (48) (a) and (6) reduce respectively to a sine series and a cosine series in multiples of T. Evidently in (48a) the only terms involving the x,. and i0,,(t'= 1, . . . , h) are w" Jt +2q t x' IJt ; and in (486), aside from (x,' 1 )., they are x' t ' JI -2q l w' IJt -q t t x^. Now it can be shown easily by induction that r (2;)'', * V M 'M-0 where the N, are positive numbers and the v, are jwsitive integers (or zero) satisfying the conditions Now x ljt enters only through (efz./d//)'*, and for this term N r =2, v t = 1, v k = 1, and f = 0(f= I, . . ,h-l); hence, in (xr'), x ljt appears with the coefficient - 2. Therefore, in (486), the terms involving x ljt and w tjt are x' t 'j t -2q t w f ljt -3q t l x tj> , and this establishes statement (a) above. 440 PERIODIC ORBITS. On using the notation F'(T) and F'(T) to designate respectively a cosine series and a sine series in multiples of T, it is evident that (F') n = F e , (FT = F e , (FT +l = F' . Hence, as every x,, n and w' tM (n=l, . . . , h-1) is a F e (r}, so also are all sums of products of these quantities, and also all polynomials in the x iM , e. g. Or x ), and parts of (o-- 3 ),. Similarly, the sums of all products x tti w" t and or'.X.i are F'(T). There remain in (48) only the terms (sin mv+Wj Wt). where w (> = < ; , in (48a) and m (J = <^, + 7r/2 in (486). Let, for the moment, z t j = m tJ +Wj w t . Then, since [d f z ij /dn r ] lt . = w j . f w tif , it can be shown by a simple induction that where the N, are numbers and the v f satisfy (49) after h is replaced by /. Now since every (w } ., w t , f ) (/ = !, . . . , h\) is a F'(T), the product ^'[(Wj.f w^ is a F'(r) or F(T) according as ^' t ~\v, is odd or even. But when this sum is odd, v t is odd; and when even, i> is even. The entire product no is then always a F'(T] if m t) = ]t , or a F C (T) if m u = 1( +7r/2. The differential equations (48) are, therefore, of the form (a) M& + 2q t x' (A + S D sin mr = 0, m = l (b) a-,',;-2 < 7X,-3^,,+^+ S ^ m-l (50) where no term in cos q t T or sin g,r occurs under the summation sign, since q, is not an integer. Obviously the integration of (50) is the same problem as that of (44), so that the solutions are 2 A + 2 m-l = !, where Crw 1 n 2 ^ 4 Lf t,h Also, by (41) and (42), r (1) - ff" o) /> (2) /.' /. (4) n ( ~ Cl r c c - PROBLEM OF k SATELLITES. 441 so that the induction is completely established. Thus the successive z (- and w tM reduce to * +24 cosmr, W IM (T) = 2 B% sin mr - 1 fc) , (5 1') and may be obtained without inhynitiim by applying (52). It is merely necessary to compute the D and E from equations (48) at each step. T> rms of any order. Case II. Assume, as in Case I, that for n= 1, . . . , (/-!) the Z,..(T) and W IM (T) have the form (47), all the constants of integration having been determined except the cJJ|_,. The differential equations for the x ljk and w ljt are again (50) (a) and (6), where, however, since (j t is an integer, cos q t T and sin q, T may occur under the summation sign. These terms arise from twosources: from the terms c*_, cos q, rand c^_, sing,T in the z,._, and u>,._i, and from similar terms in the earlier x, M and u' l-t as well as (usually) from combinations of the ,, in the coefficients. When (50a) are integrated and combined with (506), equations for the z (JI are obtained; to avoid Poisson terms in the x ljt , the coefficients of the terms in cos q, T in these last equations must be made to vanish. These co- efficients, E% 2D'tf, involve the c^_, and various known constants; e.g., the c (n = 1, . . . , h-2); and the vanishing of the E% - 2 D% will usually determine the c^_, . In the first place, the only terms of (48) in which the j-, ,_, and H' (Jk _, appear are* (a) *,,_, . I , . ^ x + 26,, { (2z M _,cos /( +if M _, Wut-iSmt,,) i I , sin ) > >( ) (aj/r,.,., +Z M _, - a ir r M _, +*,.._, cos ; , (54) For A -2 the first four U-niu in (a) and in (b) re to b divided by 2. 442 PERIODIC ORBITS. where (,> 9D"' 5 fl 'il.n+l *M.n+l- Remarks: (1) In constructing the solutions it has been tacitly assumed that the Fourier series representation of the x it and its multiples, and those which may appear because of relations among the various $. Terms of the latter sort can not be collected, unless the numerical values of the p t have been chosen; but their possible presence will be shown to be immaterial so far as the conclusions are concerned. In equations (48a), using h = 3, the terms which reduce identically to constants may be selected by fixing upon some one jt and expressing the coefficients of cos ra yi in x tA and w iA , and of sin m

ti . The resulting complicated constant, involving several series in the F n (c tJ ) and ?(), can be treated advantageously by expressing all the a,, e t} , q t , and 8 (J in terms of a single parameter a, just as in (35) and (36). In the H the coefficient of each cJ, say M ti> becomes then a power series in a; and it is found that M it = S' M (l , and i Mt] = a +-. Ntj - >t -), Af tf = a"- .N ti (j. 445 a in the various elements of any row of A, may be ascertained. Evidently the exponent 4i+2j 6k takes its smallest value 4i+2-6Jfc when; = l, and its largest value 6i2 6k when j = i-i- while QjQk takes its smallest value 61 +6 6A- when j = i+l. A> t his last is greater than the highest value of the former exponent, it is clear that the lowest power of a in any of the M,,(j=}, . . . ,t-l,7 + l, . . . . k - 1) is 4i+2 6k, which occurs in M (l and also in M tt . Thus, in the H* row of \ (where i = r+l), the lowest power is 4r+6 6A; and this appears for r=l, . . . , k 2 in two columns, the first and (r +!)<*. But, in the last row, where r = k 1, the lowest power can appear only in the first column. Hence if the factor a*' +4 ~** is removed from the elements of the r 1 * row (r= 1 k l),& new deter- minant \ is obtained in whose first column the series of each element beiiins with a constant term, as does also the series of one other element in each row except the last row. Now, if A, is developed by the minors of its last row, it is clear that a constant term can not be lacking when the development is rearranged as a single power series in o. For the only element of the last row which can con- tribute to the constant term is that in the first column; and in the minor of this element constant terms are present in all the elements of the main diagonal, and nowhere else. Thus A, ^0 at a = 0; and hence A, and likewise A, are distinct from zero for all values of a sufficiently small. Therefore the vanishing of the H determines the c| = 1 > k 1) uniquely and homogeneously in terms of rJ . But this latter constant is to IMJ put equal to zero by reason of the choice of the origin of time, as noted above in discussing the first-order terms. Hence every c% = 0. Thus the W IA (T) reduce to the values which they would have for orbits of Type I. Then also every P*' = Q*' = ; and the x tA (r) and H>,.(T) reduce to the values in Type I, except for the c^, which remain as yet undetermined for j=l, - . , k1. In the next step these c% will enter the H (by identity) in precisely the same way as the c% entered the //", and must likewise vanish. Simi- larly, the c in the terms of order n will remain undetermined until the //+, are set equal to zero, when they must vanish together with the P +1 and Q"Vu to which they will in the meantime have given rise. Thus the final determination of the constants arising at any step reduces the terms of that order to the values which they would have for Type I. If various relations among the p, give rise to other terms in the A/J5J than those which are present identically in some one )t , even if the-' terms introduce into various elements lower powers of a than have been treated in A, -- these new terms can not affect the argument in general; for, if they introduce into the development lower powers of a than were previously present, with non-vanishing coefficients, then this new development is equally as useful as the former value of A,; while, if they do not furni.-li 446 PERIODIC ORBITS. terms of lower order, neither can they in general destroy the terms treated in A 6 , since they must involve new /3's in their coefficients. Any cancellation could occur, then, only for a few special relations among the masses. The conclusion is not yet warranted that no orbits of Type II exist in Case I; but there can be none when a is below some ''sufficiently small" finite value, unless possibly for a few very special relations among the masses. Terms of any order. Case II. The differential equations for the second- order terms X IA (T), w,, 2 (r) are again (48); but because of the cf\ and c"J the equations reduce to (57), as in Case I above, where, however, the H and J now involve them (linearly) and vanish with these two sets of constants. In order that z,, 2 and w iit shall be periodic, it is necessary, just as for Type I, that E^ 2D ( ^, the coefficient of cos^r in the final differential equa- tion for x t , t (T), shall vanish. It is equally necessa^ that J$+2Hl'f, the coefficient of sin q,r in the same equation, shall vanish. Further, the "secular terms" Hf^r must vanish. The H^ are free from the c"J, just as in Case I above; but they now contain the cj\ and c^J, vanishing with the latter set. Now, from (54) and the fact that no relation (22) holds, it is evident that the E^-2D^ do not involve the c$ nor the c$; also the J$+2H involve neither the cf^ nor the c^ , and vanish with the cf^ . Further, since the x tssible exceptions as in Case I, it is impossible in Case II to determine the constants otherwise than as for Type I. !{< marks: (1) If either of the determinants A, or A s vanishes for some value of o, there may still be no new values for the constants of integration which would satisfy later conditions in subsequent differential equations and render the solutions periodic in form. Whether the series converge for this value of a would be unknown; so that the mere vanishing of A, or A, would not warrant the conclusion that orbits of Type II exist. (2) The foregoing conclusions extend beyond a denial of the possibility of obtaining equations of periodic orbits by a certain method of analysis, or solving the differential equations in a certain way: the non-existence of a class of physical orbits of a certain type is asserted, though possibly there exist numerous individual orbits of the type. All orbits, periodic or not, arising from any set of Ac,, and n [see (17) and (7)], can be represented by power series in these parameters converging through the interval 0^7-^2*-, provided that the parameters are sufficiently small. And from the exist- ence of a class of periodic orbits of the type sought would follow the existence of a range of values of the Ac,, and n (including zero) satisfying the periodicity conditions. These equations obviously could not be satisfied for a range of values of M by arbitrary values of the Ac,, , and therefore they would define the Ac,, as functions of n, holomorphic for n sufficiently small and van- ishing with At. The substitution of these values of the Ac,, into the original developments of the coordinates would render the latter series in M alone, having properties (40a, 6, d). If these series are impossible save for Type I, then there does not exist a class of physical orbits of Type II growing out of the circles named. 218. Concerning Orbits of Type IE. It may be inquired whether there r\ists a class of periodic orbits growing out of circular orbits of an infini- tesimal system which has no "grand conjunctions"; that is, whether then- are periodic solutions of (7) when some of the X, have other values than are possible in Type I. Let the initial conditions be (17), let some instant when A/, is at an apse be selected as the origin of time, and let the origin of longitude be the 448 PERIODIC ORBITS. apsidal position of M t , both for ^ = and for /z^O.* Then A h = 0, and w = T = identically in k and /x; and the conditions for periodicity are (13). But equations (7) admit two integrals, an examination of which shows that two equations of (13), namely x k (2ii) =x t (0) and x' t (2ir) = x' t (Q) , are a conse- quence of the other 4k 2 equations. Hence equations (13) become (a) (6) (c) (d) = e f (l - cos2q i ir) +M X li0 -\-e t r t ( - sin 2<7,ir) -cos2g i7 r) = e,(g ( sin = e t ( - 2q t T^ cos 2^) + M I F,', + 6,7,. (2^ sin 2 9i 7r /=! (t = l, ... . ,*-!), (63) where and the here. (j=l, 3, 4), etc., are constants whose values will not be needed *In treating orbits of Type I. the instant of a symmetrical conjunction was regarded as the beginning of a period and was taken as the origin of time; but this was merely for simplicity, since in periodic motion any other instant could be so regarded. By taking as T =0 the instant when M t of the infinitesimal system is at arbitrarily selected longitude, and choosing that longitude as a new origin of longitude (so that Xt = 0), it is clear that the longitudes which the other infinitesimal satellites have at that instant constitute a set of X's, not all of which are multiples of ir. Each family of orbits of Type I may thus be said to arise from any one of an infinitude of sets of the \ t other than multiples of w, though all the sets have X* = 0. By reason of the w t (T), which are in general distinct from zero, the absolute longitude t>< of any finite satellite at the new r = would vary with MJ (but, since families of orbits of Type I exist for every position of the line of conjunction, one may obtain a family in which the initial longitude of Jl/t is zero, identically as to n, by selecting orbits for different n's from different families, taking the conjunction line as needed for the M used.) For such an origin of tune M* would not in general be at an apse for T = 0. Hence if the origins of time and longitude be chosen at an apsidal position of Af t for all values of M, the sets of X, other than multiples of r, which can give rise to orbits of Type I, are largely excluded. (Whether any such sets remain, depends upon whether, in Type I, M t has any apses other than at the symmetrical conjunctions of the system; and this has not been ascertained.) riO>BI.KM (IK /, > \TKI.I. I IKS. While the determinant of the linear terms of tin- A;/ . - . u.-,, and T, in ii;:<) is zero. yet. when u- rv'(4, 8ln2g.y ur'Mnl -COS27.T) ** J H ' T /L A < / T 2(1 -C0827 ( ir) W '' J / +T '- ifi4) The constant term in (64a) vanishes, since for q t not an integer no relation (22) holds; the constant term in (646) reduces to -2q t C IJt . If any of the C tM are distinct from zero, (64) are not satisfied by w,=-T, = p-0; hence w t and T do not exist as holomor])hic functions of /* vanishing with n. or periodic orbits of the type sought do not exist. The necessary condition for periodicity, namely, that the C IJt vanish, is, by (28), 2'd,, 2 e.,(>inw(X,-X,)=0 (i-1 ..... k). (65) j m-l Of these A-ccjuations in the quantities X y O' 1, . . . , k 1), one is evidently a consequence of the others; for, before X t is put equal to zero, the Jacobian of the C tJlt with respect to the X ; (t = 1, . . . k; j= 1, . . . , k) is identically zero. Let the equation for t = 1 be suppressed. If particular values be assigned to all, save one, of the X ; , the last X can still be given an infinitude of values for which any one T M is distinct from zero: hence equations (65) imixw very special conditions upon the X,. Whether there are any sets of X's, other than those of Type I which satisfy (65), is unknown.* It will, however, be shown that there are no others "in the vicinity of" multiples of T, provided a is sufficiently small. TV limitation* upon any uch net* r*m fully a* evere here aa in Type I. 450 PERIODIC ORBITS. If the C 1|0 are developed as power series in the \ t J,ir (/, = (), 1), the coefficients of the linear terms are simply [dC,, /d\ f ] for \j = Jjir. Evidently, from (28), _ m-i (66) Denoting by S {/ the coefficient of X, in the i tt equation of (65), and intro- ducing a by (35), the S tl are obtained as power series in a. It is found that the lowest exponent of a present in S (f is (/ i) if f i. Hence, of all the S tf (f i), S ttt+1 carries the lowest, namely, the fifth. Consequently, if a' 1 "' 1 be removed as a factor from all the S u '(/=!, . . . , k 1; i = 2, . . . , k), a determinant A 7 is obtained (equal to the determinant of the coefficients S if multiplied by a power of a) , in whose r* A row all elements save those of the first and (r-\-l) th columns begin with a power of a. Therefore A 7 is of precisely the same type as A, , and the dis- cussion of the latter shows also that A 7 (and hence the determinant of the S t/ ) is distinct from zero for all values of a sufficiently small. Therefore quantities A / (/ = l, . . . , fc 1) exist such that equations (65) are not satisfied for any values of the X, for which \\ f J f ir\ .\ I KI.l.I I K.-. }.',i When k = 2. (22) always holds* if q t is an integer; for then ^=1 (other- wise T would not l.c the smallest synodic period of the infinitesimal and hence Jp { = q t is satisfied by giving J the integral value q t . Since /?,7' = 2\ T fl , since "Jr. = . 2r ' "/-" it follows that ,7' /f = 2ir./ when (22) holds; or the consecutive conjunc- tions of the infinitesimal pair M, and M, occur all in the same absolute longitude. Moreover, since all the u\ vanish at the beginning and end of each period, all the "grand conjunctions" of the finite system occur at the same longi- tudes as those of the infinitesimal system, and intermediate conjunctions of any finite pair occur at very nearly the same longitudes as those of the corresponding infinitesimal pair. These facts suggest a physical reason for the non-existence of periodic orbits under certain circumstances. The greater part of the mutual dis- turbances of two bodies occur while they are near conjunction; and, if the consecutive grand conjunctions occur at exactly the same longitude, the perturbations of the elements would tend to be cumulative. Nevertheless, if there are more than two bodies, the mutual disturbances may so balance each other as to yield periodic orbits, especially if the bodies are far apart (t. e., a sufficiently small) unless (22) holds. But if two bodies have con- junctions between the grand conjunctions, all occurring very near the same longitude, the other bodies can not counterbalance the large perturbations of the two. More exactly, there exists a range of values of the masses and the e t , including zero, for which periodic orbits are impossible; so that, unless the orbits for n = are eccentric rather than circular, there are for small values of /u no periodic plane orbits of k satellites when (22) holds. So far as this result extends, it would indicate that no asteroids having nearly circular orbits would be found, whose periods compared to Jupiter's are in the ratios J, f , f , etc. Those whose periods are nearly in any such ratio should be found subject to very great perturbations. It is well known that lacunary spaces of the sort just mentioned do occur among the asteroids. That there are such spaces also when the ratio of the periods is J, and other such values, is not surprising, as in any case the slightest deviation from the correct initial values destroys periodicity, there being Poisson terms in the solutions. In PoinrmvV .li*-UH.iMii of Ih.- |.rol,l.-in ..f thr.-.- I.., I,- i|,.-r.|..n <\,'- MM hw ,, ,- without such a relation aa (22) holding does not ari*e. 452 PERIODIC ORBITS. 220. Jupiter's Satellites I, II, and HI. Of Jupiter's longer known satel- lites, the innermost three move almost exactly in a plane, apparently in periodic orbits having symmetrical conjunctions; and their masses with respect to that of the planet are very small. Since for orbits of Type I the increase in the longitude of M t during a period is independent of //, being equal to n t T, the average angular velocity of each finite satellite for a period may be taken as the corresponding n. The unit of time being the sidereal day, and the unit of mass being the mass of Jupiter, the observational data are :* , = 0.000017, M t = 0.000023, M t = 0.000088, n, = 3. 551552261, n,= 1.769322711, n,= .878207937. (67) Since (w 1 -n,)/3 = . 891 114775, and w,-n, = . 891 114774, the w, of (67) satisfy, far beyond observational accuracy, the equations (2), where v = .891114774. Then the period T is 7.0509271 days, and Pi = 3, p t = l, q, = .985516077,1 (68) K = T, } so that satellite III advances 354?785788 during each period, while satel- lites I and II advance 1080 and 360 respectively more] than this. If /z is taken arbitrarily as .0001, then ft = .17, ft = .23, ft = .88. It seems desirable, however, to retain the /8's in the computations, inasmuch as a new determination of the masses may render it necessary to use other values than those given above. The D and E of (46) are obtained by writing equations (43) in the form (69) 1 m-0 where, rearranging according to multiples of T, *Tisserand, Traite de Mfcanique Celeste, vol. 4, p. 2. The n t Riven there (203?48895528, 101?37472396, and 50T31760833) are here reduced to circular measure. PROBLEM OF tc SATKU.1TE8. Evidently the U. V, and U enter respectively the D, D. and D%, etc. From (23), (5), and the relation (n,/n,Y=(n./n,Y. the (/ and S,,are found to be = = . 6284333, ,, = = . 3939606, - = . 6268932, *? =3.1366 A, *,*'' = 1.23875/3,, ^ =.19132/3,, u nil = ! 2326 ft , = .77464 /3, , = .30443 /3, . ,7(1 The F m (e,j), and also the G m (e tJ ), etc., encountered in the higher orders are readily computed by using the tables of coefficients given by LeVerrier.* In obtaining the successive A and B from (52), the smallest divisors introduced are 16-tf,4-gJ, and 1-gJ, or. 1156616, .05772591, and .02875806 respectively. These divisors decrease materially the effectiveness of the small value of /*; nevertheless the terms above those of the second order seem relatively unimportant and will not be computed. The coefficient^ of n in the X,(T) and w,(r) are found to be:f *U(T) = (.3/3, - .1/3,) - .8/3,cos2T - .l&cosSr - 203.70 s cos4r + (.7/3 1 -.2/3 1 )cos6T-.2/3 t cos8T+.lftcoslOr+ - (1.1/3,-. 3/3,)sin6r + .3/3 1 sin8T-3/3 J sin9T-.l/3 1 sinlOr + T,.,(T) = (.5/3, + .3&- .1/3.) + .8/3, COST- (58.4/3, + 100.1^,) cos2T - .7/3,cos3r (.7/3,-.2/3 1 )cos4T-.l/3 J cos5T-.2/3,co86T+.l/J I cos8T + -(.8/3,-.2/3,)8in4r+.l/3 J sin5T+.2/3 1 sin6T-.l/3,sin8T (.4/3, + .5/3,+ .3&) +28.9/3.COST+ .6/3,cos2r - (.4/3, - .2ft) cos.'lr + .l/3,cos4T , For the second-order terms the >!!' and E are found from (54), where it is convenient first to rearrange the coefficients of formerly published (rranMe(ion4 all save the following terms fall below the limit of accu- racy in the x til and w l:l above. To facilitate comparison, the second-order terms are shown multiplied by n .0001. They are -(2.9/3 1 /3 2 -12.2$-9.2&/3 3 )cos4T-2.0/3 2 ! cos8T + .0001 W U (T) = .Sftftsinr- (.3/3 1 & + .2$+.3&&)sin2r + (5.8/3^ -24. 4ft 2 -18.3/3 2 /3 3 )sin4r+.l&&sin6T+5.1/3 2 2 sin8T , .0001z 2 . 2 (r) = (.2/3I+.6&&+.5/3D - (.!&&+. 1$) COST + (6.3$+12.6/3 1 /3 2 +12.6&/3 3 +3.1$)cos2T .0001 w M (T) = (.2 j 8 1 /3,+.2|8 1 /3 l + 1.3ft&+6.2/3 3 Osin2T + (.l/3 3 2 sin3r + .4# + . 1 ft &+ 1.40,0, + 1.30J)sin4r+ .0001 X M (T) = - (2.3ftft+2.5$+3.9ftft) COST-. I^cos2r -.Iftftcos7r+ , ) = (4.7ftft+4.9/3 2 ! +7.6ftft)sinT+.2ft 2 sin2H ---- . Since / ( = a ( (l+z ( .,M+z,,2M 2 + ) and v f = the radius vector and absolute longitude of each satellite are obtained by computing the a, from (3) and using the coefficients above; the deviations from their values in the undisturbed circular orbits are given to five significant figures, so far as the terms of the first two orders are concerned, How much these would be affected by terms of higher orders is unknown; in fact no proof has been given that the series converge for/z = .0001, although they have been proved convergent for all /x sufficiently small. To show the general shape of these orbits, the values of the v t and r ( /a t will be given to four decimals, using the values of the ft tabulated above: r ] /a 1 = l-.0044cos4r, w 1 = 7 r+3.9855T+.0089sin4r, r 2 /a,= l-.0093cos2T, ,= 1.9855 T-. 0003 sin r + . 0185 sin 2 T + . 0001 sin 3 r+. 0001 sin 4 T, r l /a,= l + . 0006 COST, ^ = .9855r-.0011sinT. Hence if these orbits are thought of as ellipses rotating in the plane, the major semi-axes would be the respective a,, the several eccentricities would be .0044, .0093, .0006, and the axes would rotate forward at rates whose average values are the n f . The three satellites are in line with Jupiter at the beginning and middle of each period, II and III being on the same side of the planet at T = 0, and I and III on the same side at T = TT. Whenever PROBLEM OF k SATELLITES. 455 II is in conjunction with I or III, the inner of the pair is near a perijove and the outer is near an apojove, which decreases the amount of their mutual perturbations. No radius vector or longitude differs very widely at any time from its value in a circular orbit (a, and \+qj, respectively). The largest departures arc for satellite II, as fj/o, reaches a minimum of .9907 at r = and a maximum of 1.0093 at about r = r/2 and every half-period thereafter, t', meanwhile ranging from 64' more to 64' less than the mean longitude of II. Similarly, satellites I and III get 30' and 4' respectively ahead of and behind their mean positions, and the r,/o, at such instants closely approximate their mean value, unity. For satellite I the maxima of r/o occur at intervals of a quarter-period, and for satellite III they occur at intervals of a period. Finally, it may be noted that, for this system of bodies, the increments Ac u (see 209) which have been given to the initial values of the coordinates and their time-derivatives to preserve periodicity when the bodies are finite are approximately r< < v, v, -.0044 a, .0310 (radians per day) -.00930, .0329 (radians per day) + .0006o, -.0010 (radians per day) 221. Orbits About an Oblate Central Body.* If the central body is an oblate spheroid and the satellites are spheres moving in its equatorial plane, periodic orbits of Type I still exist, the successive grand conjunctions falling at the same or a different longitude, according as q t is or is not an integer. The differential equations for this case are obtained from (4) by simply multiplying each 1/rJ (j = 1, . . . , k) by /(r,), where o being the equatorial radius and e the eccentricity of the spheroid.f Let oV/aJ = 7,M; then equations (7) are replaced by (a) xri'+2x[(w' l +q l )+2'6 l sr i 8m( ) ,+w j -w l ) ThwMction take* a firrt step in direction Mggerted by Professor K. Lave*, particularly with rafenoM to Jupiter's satellite*. tMoulton, Cele*ial Mechanic*, p. 122, where a'+r-rj, and 2fc'v'i-..-3t'Af/2a. 456 PERIODIC ORBITS. This substitution requires the flattening to vanish with the masses Af ,,..., M t , so that the central body becomes spherical if the others become infinitesimal; but the amount of flattening corresponding to any given set of finite masses remains quite arbitrary, even if the p t and q t are specified; for the values of the a, merely determine the ratios of the 7 , and one 7 may be taken at pleasure. In the solutions of (71) satisfying initial conditions (17), the terms independent of ju are the same as formerly. Hence in Case I where q t is not integral, the Aw, and e t still exist as convergent power series in n satisfying (29 a, 6), though of course their values in terms of n are now different because of the 7, . Thus periodic orbits exist. In case q t is an integer the j, enter the x f (0; T) and x,(i; T), but do not appear in the P tl or P (f of A 3 . Thus the argument in Case II is likewise unaltered, and periodic orbits of Type I exist under the same conditions as when the bodies are all spherical. The numerical results for Jupiter's satellites given above are affected but slightly in the first and second orders by including the flattening of Jupiter. The corrections to be added are in fact to o: M (T) add .17, cos 4r to .0001 a-,,, add 1.8 #,7, cos 4r to .0001 W M add -3.6 &7, sin 4r to z,,i( T ) add .l7 2 cos 2r to .0001 x ta add (.2/3,+. 4ft) 7 2 cos 2r to .0001 My, add -(.50 I + .8ft)7 1 sin 2r toar,,,(T) add .17, cos T to .0001.T,,, add - .1&7, cos r to .0001 w, iZ add -1&7, sni r And since 7, = .36, 7 2 = 14.3, and 7 3 = 5.7, the values of the r t /a t and v t given above are changed as follows: to r, a, add .0018 cos 4r. to r 2 /a 2 add .0007 cos 2r, to r, add -.0030 sin 4r, to v 2 add -.0011 sin 2r; then r, a, = 1 - .0026 cos 4r, t;, = Tr + 3.99855r+.0059 sin 4r, r,/a 2 = 1 - .0086 cos 2r, v,= 1.9855r- .0003 sinr + .0174 sin 2r + .0001 sin 3r+.0001 sin 4r, r 3 /a 3 --= 1 + .0006 cos T, v 3 = .9855r - .001 1 sin T. CHAPTER XV. CLOSED ORBITS OF EJECTION AND RELATED PERIODIC ORBITS. 222. Introduction. In the problem of two bodies there is in no sense ( ontinuity between circular orbits revolving in the forward and retrograde directions, except where their dimensions shrink to zero or become infinitely great. But in the restricted problem of three bodies the deviations from the circular forms of the orbits are such that there is geometrical continuity in some classes between those which revolve in the forward direction and those which are retrograde; and the limit between the two types is an orbit passing through one of the finite bodies. If the infinitesimal body leaves one of the finite bodies, its orbit is called an orbit of ejection; and if it strikes a finite mass, it is called an orbit of collision. In certain cases orbits of ejection are also orbits of collision, or closed orbits of ejection. When the direction of collision is exactly opposite to that of ejection, they are the limits of two classes of periodic orbits, in one of which the motion is direct and in the other of which it is retrograde. The closed orbits of ejection are not themselves periodic orbits, even if the physical impossibility be disregarded and the problem considered purely from the mathematical point of view; for, if the expressions for the coordi- nates are followed, in the sense of analytic continuity, beyond the values of t for which a collision occurs they become complex, and never become real again for increasing real values of /. Those orbits in which the ejection and collision are not in opposite directions are not the limits of periodic orbits, or at least of orbits which re-enter after a single revolution. The object of the investigations of this chapter is to determine the limiting types of certain classes of periodic orbits, and thus partially to prepare the way for the discussion of the evolution of the various classes of [>oriodic orbits with varying values of the parameters on which they depend, and to show the relations among these various classes. The existence of the closed orbits of ejection will be established, some of their properties will be derived, and it will be proved that each one in which the direction of ejec- tion and collision is opposite is the limit of two series of periodic orbits. 223. Ejectional Orbits in the Two-Body Problem. As preliminary to the general problem, the special case in which there is only one finite mass will first be treated. Let the mass of the finite body be 1 - n and let the units to chosen that the gravitational constant is unity. Then the motion of 457 458 PERIODIC ORBITS. the infinitesimal body projected along the fixed -axis satisfies the differential equation d? _ _ i- M df = " + T where the sign is or + according as the motion is in the positive or negative direction from the origin. Suppose f = f an d d/dt = ' = ' at T = T O . Then the first integral of equation (1) becomes #V _,./_ + 2(1-M) - 2 (l-/i) , r , 2 _ + 2(l- M ) , m di) 1 "IT ~F ' If Cj is negative, || has a finite maximum for which ' vanishes; if c^ is zero, % approaches zero as || becomes infinitely large; if c t is positive, % is finite for || infinite. It will be assumed that c, is negative in order to get orbits of ejection which are closed. Then, without loss of generality, it can be supposed that is the greatest value of for projection in the positive direc- tion, or the least for projection in the negative direction. Then 1-0, Cl =T^P^- (3) ?o With the initial values (3), the integral (2) becomes where j is an integer. Now consider as a function of (t t a ). Since the right members of (1) and (2) are regular for all values of t and all values of except = 0, it follows that is a regular function of t for all values of t except those for which vanishes. These values of t are easily determined from (4), and are found to be where j takes all integral values. The character of $ as a function of (t in the vicinity oit = t, is easily determined from (4). The left member is expansible as a power series in = 11, and the equation can be written in the form It is found that , , , . Therefore 17 is expansible as a power series in (t tj) lf3 , starting with a term of the first degree in (t ^) 1/3 . Since = r? 2 , it follows that is expansible as a power series in (t /) 1/3 , starting with a term of the second degree in (t t,) l/3 . It is easily seen from (4) that F(ry) is an odd function of r?. There- fore 17 is an odd series in (t ^) 1/3 , and is an even series in (t tj) l/3 . Since the only singularities are given by (5), the radius of the circle of conver- gence for the series for both T? and is 7r| U .~J . . CLOSED ORBITS OK KJK(TK>X AND HKI..V1 Kl> I'KltlnDIC ORBITS. 459 The form of the solution in the vicinity of t = t, beinn known, the co- efficients of the series can easily be found from (1) by the method of undetermined coefficients. It is convenient in the computation to let T-tf-W*, (6) nftor which (1) becomes * CD The solution of this equation with initial value of equal to zero has the form {-o,T+o 4 T 4 + +O^T"+ (8) By direct substitution and comparison of coefficients, it is found that (9) More convenient formulas for use can be developed by eliminating the term in "' from (7). After the transformation (6) the integral (2) becomes C-g -|l/l =-(l M), a = arbitrary constant. On using this equation to eliminate ~* from (7), the result is found to be T j ~ 2 ^ +^ ( -p) "=+9(1 M)T*. (10) Now it follows from equations (8) and (9) that J-t _ 2^ = - 4 has a 1 as a factor. This is a general property which will be needed in establishing the existence of the closed orbits of ejection in the problem of three bodies (230). It follows at once from (11) that the part of the co- efficient a 2J which comes from the first term on the right is opposite in sign to the coefficient of the preceding term. Since the sum of the subscripts of the product terms in the right member of (11) is 2j, their product has the same sign as the coefficient of Oj,_ 2 . Therefore, a 2J and a 2 ,_ 2 are opposite in sign for all j. It also follows from (11) that a tj contains a as a factor to one degree higher than it appears in a 2J _ 2 . The expression for has a branch-point at t = i h where three branches unite. If t t j = pe v ^ r ' ( ' f+ * nt) , the three distinct branches are (12) If t tj is real and negative,

i.K inCfKM \M) HKI.MKD 1'KUK l|)IC i.UHII- Mil 224. The Integral. -Equation (2) holds for all values of /. and therefore when the scries (9) is substituted in it the result is an identity in r . The conditions that the coefficients of tin- various powers of T shall he identical in the left and right members furnish severe tests of the accuracy of the com- putation of (9). The integral also gives the relation between the arbitrary a and the greatest distance . By direct substitution, it is found that The interval from ejection to collision is found from equation* (5), (9), and (13) to be _r\' (14) V2(1-M) V2(1-M) For a = 0, which corresponds to a parabolic orbit, P is infinite. 225. Orbits of Ejection in Rotating Axes. In the problem of three bodies the motion of the infinitesimal body will be referred to rotating axes. In the demonstration of the existence of closed orbits of ejection in the prob- lem of three bodies it will be necessary to use some of the properties of the orbits of ejection in that of two bodies. For this reason the orbits now under consideration will be referred to axes rotating uniformly with the period 2r. Suppose the ejection takes place at t = t t and along the x-axis, where the rectangular coordinates are denoted by x and y. Then x and y are given by the equations x=-Kcos(<-{,), y= -sin (<-/,). (15) Those orbits which re-enter in the direction opposite to that of ejection are of greatest interest in the present connection. The condition that they shall have this property is that their period in t from ejection to collision shah 1 be a multiple of 2*. This condition becomes, by virtue of (14), ( .) M -2;V2TI^), (16) where .; is a positive integer. Figs. 15, 16, and 17 show the curves for j equal to 1, 2, and 3, at least as to general form, in full lines for ejection along the x-axis in the positive direc- tion, and in dotted lines for ejection in the negative direction. These curves for the three values of.;' are not drawn to the same scale, for it follows from ( 16) that their linear dimensions are proportional iof 1 *. One of the important properties of all these curves is that they intersect the x-axis perpendicularly at their mid-points. 462 PERIODIC ORBITS. 226. Ejectional Orbits in the Problem of Three Bodies. The differential equations of motion for the infinitesimal body when the finite masses describe circular orbits are, in canonical units, (17) c. = df" dt dx' _ dt dy These equations have the integral (18) When /x is zero the problem reduces to that of two bodies, which was treated in 223. The singularity in the solution, whether /x is zero or not, comes from the fact that i\ tends toward zero as a limit as t tends toward t t . It is intuitionally clear that the mass /x will have only a slight influence on the motion of the infinitesimal body while r l is small, and it seems probable, therefore, that the nature of the singularity at t = t^ is the same whether /x is distinct from zero or not. This is, indeed, the case, as was first proved by Levi-Civita in a very important memoir.* It follows from (18) that x' 2 + 7/' 2 tends toward infinity as r l tends toward zero, but that the limit of TJ [x' 2 +y'*], for r, equal to zero, is the finite quantity 2(1 M). If M is zero, x and y are developable as power series in (t ,) 1/3 , and this suggests defining an independent variable a in terms of which x, y, and tti are expressible by series of the form 1 (19) Fia. 15. Since the solutions of analytic differential equations in the vicinity of points for which they are regular are themselves regular, while in the vicinity of singular points the solutions are regular or not, depending on supple- mentary circumstances, it is advantageous to choose such dependent vari- ables that the equations shall be regular for r t = 0, o- = 0. This Levi-Civita has done, preserving the canonical form, with rare skill and elegance. His dependent variables, which may be denoted here by p, q, u, and v, are related to the rectangular coordinates and their first derivatives by *Sur la resolution qualitative du probl&ne restraint dea trois corps, Ada Mathematica, vol. 30 (1906), pp. 305-327. ( 1.0SKI) oKHll.- OK KJKtTIuN AM) HKI.A1KI) PKIUODIC OHBIIiv u V \v I'd In these variables equations (17) da ~ du dff~ dv ff-J . 1 du da dp dv da _ dq (21) It follows from (20) that p = q = if x+M = y = 0, and that u and v are finite for r, = 0. Therefore the form of H shows that the differential equations (21) are regular in the vicinity of p = q = 0. Consequently, p and q are developable Fio. 16. i. 17. as power series in a, vanishing with a. It follows from (20) that x, y, and t (, , considered as functions of (r) = -r^(r,) = +r V (-T), j TY/(T)=-TMr t )=-TM-T), rV(r) = +T^(r 1 )=-r'V(-r).| It follows that x and dy/dr are even functions of T and that dx/dr and i/ are odd functions of r. The first equation of (22) contains only even powers of T, and the second contains only odd powers, starting with a term of the fifth degree as the lowest. With the initial conditions (24), the solution of (23) is found to be ' (27) ty/ | i,) -i 1/3 2 > a = arbitrary constant, where the positive or negative signs are to be used in the left members according as the initial projection is in the positive or negative direction. The constant of the integral (18) is given by the equation (28) (UKH OHHII.- n\ K.IK< IK.N .\M> KKI.AIKI) I'KHK'DH .|(|t||> It,.', It follows from (27) that the right memherof this equation can be developed as a scries of the form Since thi> equation must be an identity in T it follows that C = C,, ('-.= ('t = (\ = (\= =0. Those expressions which are zero constitute a check on thecompuiationof t lie coefficients of (27). By direct substitution of (27) in (28), it is found that (29) The force function is sometimes used in the symmetrical form instead of in the form given in (17). Then the constant C becomes (30) 228. Recursion Formulas for Solutions. The second terms in the right members of (23) give rise to a large part of the labor of constructing the feolutions. They can be eliminated by use of the integral, and relatively simple recursion formulas can be developed for the construction of the solu- tion after the terms of lowest order have been found. The integral (18) becomes in the notation of (23) (31) On multiplying the first equation of (23) by Z+M and the second by y and adding the results; and then multiplying the first by y and the second by (z+/u) and adding the results, it is found that (32) 466 PERIODIC ORBITS. On eliminating the second term of the first of these equations by means of the integral (31), the simplified equations become = 27 M r 6 The solution (27) may be written in the form (33) [9(1 )H 1/ ' 3 2~^ > Oj= 6 2 = a = arbitrary constant. (34) The next step is to form general expressions for the terms involved in (33). Nearly all of the terms written are of the second degree in Z+M and y; those which are of degree higher than the second are all multiplied by the factor M and will be in general of little importance. They will not be included in the general formula and must be added to it when it is used. Since these terms contain T U as a factor, they will not contribute much to the solution unless it is carried very far. The general expressions for the terms of the second degree in x-\-fi and y are found from (34) to be ;p =2cv{ *- l+ 1 = 1 J=2 *=1 = -4cv{l+ 1 1 = 1 }~2 t=l -22/ = -2cv5r-2 J (j+2)6 w - 6 T^+ S S J=4 ;=5 t=l {5r- * < I.(KI) incurs iiK K.IKI-IIUN AM) HKI.AIKI) I'KHK.Dlr oHHII.-v 467 /- t-i f (H -V>M.r+M)'- - f ^l+2 M )r'fr'-f-22^.,r-4- 2 J 1 ;-4 ,- ., C l- /-T y- *., /-I . /-, >- *-i (>+l)(2j+l)a,-6jr" -T(x-f n)%% = -2cVJ- 10- 2 " T I TM + 2 S (k+2)(2k+5)a lj . u b^, n dx vT/- iTiV^vi -2y^- = -4c 1 TN-l- >, 0+1)0^-6^ r w + X X 1 Q 7 ^*^^ L ^^^ ^*^ Gr 1 yg = -6r r'{5r'-2 2 0>2)6 w _,r u + JJ 2 * ;-4 ;- -i 2c I r'{-5- 2 [H, 1 /-i ^-1 *-! 468 PERIODIC ORBITS. On substituting these expressions in (33) and equating the coefficients of r* i+3 and T M+6 respectively, it is found that - +62 (2fc+5)6, A,-*-.+ f (1 +2 M ) 97 ^C^ - (1 IJL) ^b u b 2J _ 2k _ u -{- quantities coming from terms in (33) of the third and higher degrees, (35) - 12(./+2)6 2> _ 6 + ^-2'-6+ quantities *=i coming from terms in (33) of the third and higher degrees. These formulas are to be used when j is 3 or greater, and care must be taken in adding terms coming from the higher powers of (X+M) and y in the right members of equations (33). The results of applying (35) are 2,418,092 6 , 55,964,945 , 14 _ 38.481,084,886 , 16 , ~| 3,972,769 '67,540,473 32,937,237,333 J (36) ']' CLOSED ORBITS OF EJECTION AND HKI..VI Kl> PERIODIC ORBITS. 469 229. The Conditions for Existence of Closed Orbits of Ejection. The series (34 1 CM urn-rue for all |M|^/U and |o-o,|p provided \r\R, where R is a positive constant depending on ^>, a,, and p. The coefficients of the various powers of T are polynomials in a and M, so far as M occurs explicitly, and they also involve M implicitly through c. These coefficients are expansible as power series in a a,, and n which converge for all finite values of |a a,| and for |//| < 1. Therefore the expressions for Z+M and y are expansible as power sn-ie> in a-o = /3 and , and if |/3|p, |M|P the series converge for all \T\ ^R. They may be written X = PI (IJ,;T), ^=p,03,/i;T), y = p,(0,M;r), ^= P< (0,M;r), (37) where /;,,..., p, are power series in /3 and p. Now o, will be determined so that when n is zero the period from ejection to collision shall be 2 jr. From (5) and (13) it is found that a, satisfying this condition is . (38) Suppose T and let the values of the coordinates at r=T be (39) The conditions that these values of the coordinates shall belong to an orbit of ejection are T), y.+A-p.O, M; T), I (40) , M; T). Since the right members of these equations are expansible as converging power series in and M, it follows from the definitions of x t , z{, y,, and y' t that A = fc(/3,/0, A = ftO,M), 0, = 9,G3, M), h>~q<(fi,ri, (41) where 9, , . . . , ,- If the initial conditions are Zo+ft, x' a +(3 t , y 9 +0 3 , t/o+& and the parameter 5 has been introduced by (43), these equations become CLOSED ORBITS OF EJECTION AND I; D 1'KKIODIC ORBITS. 471 ^=F l (0 l ,...,0 t ,o,n;P/2) = 0, y-P l (A,...,A, l M;P/2)-0, (44) where P, and P, are converging power series in 0, , . . . , 4 , 5, and M- It follows from (42) that P, and P, vanish for 0,= =/3 4 =j=i M 0. If 0, , . . . , 4 are determined by (41) the orbit is an orbit of ejection. Therefore, upon substituting the series for these constants in (44), sufficient conditions for the existence of closed orbits of ejection become = Q,(0, 5, M ; P/2) = 0, y = Q f (0, 5, ; P/2) = 0, (45) where Q, and Q, are power series in 0, 5, and M, which vanish with = 6 = n = 0. The coordinates can, therefore, be developed as power series in 0, 3, and M and the moduli of these parameters can be taken so small that the series converge for |T| T the value of y is zero whatever a may be. Therefore dy/da is zero and the determinant becomes simply &=(dx/da)(dy/d&). In the case under consideration the value of t <, for which A is formed is _( 1 = T = T J(l + 5 )=-^(l+)'=;V(l+) 1 . (46) The second factor of A will be computed first. Upon putting a-o, = = 0, it is found from the second equation of (15) that i do L do -JJ- 472 PERIODIC ORBITS. Before computing the first factor of A the parameter 5 may be put equal to zero. Hence it follows, from (46), (15), and (9), that It was proved in 223 that the signs in this series alternate and that a is negative for those orbits which lie entirely in the finite part of the plane. Therefore dx/da is distinct from zero for all values of a under consideration. It follows from this discussion that A is distinct from zero for /3=a-a =5=0, T=JT, and consequently that the sufficient conditions for the existence of closed orbits of ejection can be uniquely satisfied for [/* suffi- ciently small. There is a closed orbit of the type in question for ejection in both the positive and the negative direction for all integral values of j upon which the , or a , of (16) depends. In the special case in which the finite masses are equal, a closed orbit of ejection for.;' = 2, with ejection in the positive direction,* was discovered from numerical experiments by Burrau in two interesting memoirs. f Since in his problem ju had the large value 0.5, it is not to be expected that the results of this analysis would agree very closely with the results of his computations. Hence the comparison will be made only for the constant of the Jacobian integral. Upon taking into account the difference in his units and those employed here, it is found that his Jacobian constant C B , equation (5) loc. cit., is related to C of (29) by the equation Burrau's computation gave-2C B = 2.2528; and for M=0.5, ?i? = it is found that C = 2.38, and the agreement is fully as close as would be expected. It follows from these numbers that a larger value of the con- stant a, corresponding to a smaller value of , belongs to the undisturbed orbit having the period 2ir than to that computed by Burrau. In the undis- turbed orbit the greatest distance to which the infinitesimal body recedes is, by (16), = 2; it has this value at t t t = w, and it is then on the negative half of the z-axis. The greatest distance found by Burrau in his computation was 1.9972, or a little less than that in the undisturbed motion. If the infinitesimal body is ejected toward or from the body ^ with a small value of | , it will be disturbed so that on its return it will revolve around 1 M in the positive direction. This can be seen when the motion is considered in fixed axes, for under the conditions postulated the disturb- ance is positive all the time that the infinitesimal body is going out and returning. If it is ejected farther, it will be accelerated by /z in the negative *Burrau's orbit of ejection was from the body called it here, but permuting 1 p and M and changing the positive directions of the axes, the statements are correct. fRecherches numeriques concernant des solutions periodiques d'un cas special du probleme des trois corps, Astronomische Nachrichten, vol. 135 (1894), No. 3230; and ibid, vol. 136 (1894), No. 3251. CI.(KI) ORBITS OF EJlriK'V AM) UKI. \ I Kl> PKKIODIC ORBITS. 47.'* direction part of the time. While in general the body will not collide with 1 At on its return, it may possibly do so under special conditions. Indeed, Sir George Darwin has discovered one such orbit by numerical ex|>eriment* having the period *-. The ejection was from /i in the direction of 1 n, and the body collided with n going in the same direction. t This orbit is one of a pair which together are the limit of certain periodic orbit*, though they are not periodic themselves, either pliy>ically or mathematically. The constant (' belonging to this orbit in the units employed here is 20/11 = 1.818. The values of the masses used by Darwin were 1 M= 10/11, /*= 1/11. It follows from (16) that { =2 I/> for this period, and from (30) that C= 1.716. In this case the belonging to the undisturbed orbit is larger than that belonging to Darwin's orbit. The value of is 2' '= 1.26; thegreatest distance in Darwin's orbit, according to his diagram, is 1.3. 231. Conditions at an Arbitrary Point for an Orbit of Ejection. Since the motion of the infinitesimal body is regular for all finite values of r and all finite values of the coordinates except those for which it collides with one of the finite masses,^ it becomes a matter of interest to determine in any special case whether the trajectory is one of ejection or collision for a finite value of t. It is sufficient, as Painleve" conjectured and as Levi-Civita proved, that the coordinates and velocities shall satisfy one analytic con- dition in order that the orbit shall pass through one of the finite masses for a finite value of t. This conclusion will be established here in a different way. Suppose M is zero and consider the problem of defining the initial con- ditions for an orbit of ejection so that it shall pass through the point in question, and so that the components of velocity at the point shall satisfy as many conditions as possible. The velocity in rotating axes at any distance from the finite mass l-n is the resultant of the velocity with respect to fixed axes and that due to the rotation of the axes. The velocity with respect to fixed axes at any finite distance can be made any finite quantity by a suitable determination of the constant a, or the equivalent constant & . Consequently, an arbitrary speed, or one of the components of velocity, with respect to rotating axes at any distance can be secured. Suppose the speed at a given distance has been assigned; then it is possible to determine the initial direction of ejection so that the orbit of the body will pass through any point having the given distance, for it is possible to do it in fixed axes and the rotation simply changes the direction of ejection by an angle which is proportional to the time required for the body to reach the distance in question. It is clear from this that when At = the conditions of ejection can be so determined that the infinitesimal On c^UuTf mUia of periodk orbita, MonlUg Notieu tf 0* Royal Agronomical SM**, vol. 70 tS< further remark* on thia orbit t the doe of |234. if'ainlrve, Lecont ntr la Tktorie AnalyHque dtt Equation* Dtff,r, nluUrt. p. S83. \Acta Mathrmatica, vol. 30 (1906), pp. 305-327. 474 PERIODIC ORBITS. body shall pass through any assigned point with any assigned speed. Of the four quantities required to define an orbit, viz., two coordinates and the speed and direction of motion, three can be taken arbitrarily and the fourth is determined by the condition that the orbit shall be one of ejection. The determination of the fourth quantity is double because the body has the same speed twice, once when it is receding from 1 M, and once when it is returning toward 1 /x. Suppose that, for /x = 0, an orbit of ejection passes through the point X T , y T with the speed V T = Vxy+y'r at (t t^ 1 ^ = T. If 1- T represents the speed with respect to fixed axes, then, since the component of velocity due to the rotation of the axes equals numerically the distance of the point from the origin, the relation between V T and ' T is (47) Equation (2) determines , the greatest distance to which the body recedes, and (13) gives the constant a c . Equation (4) gives the value of T, and the direction of ejection is T degrees in the negative direction from the line joining 1 /x and the point (X T , y T }. Let the angle of ejection be . Now suppose that /x is distinct from zero, but small. Let the initial values of a and 6 be a +/3 and +7- Let a new independent variable T I and a parameter 6 be introduced by (43). Then the solution can be written in the form X=PI(P, 7, 5, M; O, y=p 3 (P, 7, ,/*; T,), where Pi , , p t are power series in /3, 7, 5, and /x. The moduli of these parameters can be taken so small that the series converge for ^ TJ ^ T. The conditions that the body shall pass through the point (X T , y T ) with the velocity V T at T I = T are l-v T = 0. (48) Since these equations are satisfied by /3 = 7 = 6 = ju=0, they can be written as power series in ft, 7, 8, and /x, vanishing with /3 = 7 = 5 = /x = 0, of the form ^03,7,6^; D = 0, P a (/3,7,6,M; T) = 0, P,(/3, 7, 5, MJ T) = 0. (49) Equations (49) are not satisfied by M = unless also = 7 = 5 = 0. There- fore they have solutions for /3, 7, and 5 in terms of M which vanish for ju = 0. If the determinant of the linear terms in /3, 7, and 6 is distinct from zero the solution is unique. In treating the problem it is convenient to use equations derived from (49) rather than these equations themselves. Let (f> represent the angle between the positive end of the re-axis and the line from the origin to the point (x r ,y T ). Then let Q, , Q t , and Q 3 be defined by P 3 . (50) CM >SKI) ORBITS OF EJECTION AND RK1.AIK1) PKRIODIC ORBITS. 475 This transformation is equivalent to rotating the axes so that (x r , y r ) lies on the positive half of the new z-axis. The solutions of .tf, 7, ,M; 70=0 (51) are identical with those of (49), for the two sets of functions are linearly related with non-vanishing determinant. The determinant of the terms of the Q, which are linear in ft, y, and 6 is (83) Before forming this determinant M may be put equal to zero, and before computing the elements of each column the parameters with respect to which the derivations are taken in the other columns may be put equal to zero. When y=S=n=Q the value of Q, is zero for all values of /3; there- fore aQ,/ 3/3 = 0. Since, for /i = 0, the distance of the infinitesimal body from the origin is independent of y it follows that Q, is an even function of y; therefore dQ l /dy = for = y = d = n = Q. Also, since, for = 5 = ^ = 0, the velocity is independent of y, it follows that dQ^/dy = 0. Hence the determin- ant reduces to aft, aft, aft a/3 dy a aQ, aQ,, aQ, 30 ' dy 05 aQ,, aQ, aQ, a/3 dy d a& ~ a T aQ, 3/3 d/3 aQ, 06 d6 (58) When M = the values of x and y, which are the coordinates referred to rotating axes, are z = cos where has the value given in (9). Therefore P, and P t become , 0, If /3 = 5 = Q, the first terms of the expansions of these expressions as power series in 7 are Under the restrictions which have been imposed,

electe point such that, for sufficiently small, the infinitesimal body will pass the point with the given velocity. The direction with which the body passes the point vanishing with M~MI> is unique. This is always true unless the solution becomes multiple. If the multiplicity is odd, there is one real solution for both positive and negative values of M~ Mi- There are three solutions altogether for |ju sufficiently small because , defined in (16), has three values for which the conditions of a closed orbit of ejection can be satisfied, but only one of them is real. Consequently the real solution can not disappear by uniting with one of the others unless they first unite and become real. Then, if two of the real solutions should unite and become complex, there would be one real one left. That is, there is one real closed orbit of ejection from 1 ^ for all values of /x from to 1, excluding the value unity. The argument has been made for the period 2,ir, but it is entirely similar for any multiple of 2ir. It has been tacitly assumed in the argument that none of the orbits of ejection under consideration passes through the position of M for any value of fj,; for it was only under this condition that the convergence of the series was assured. It has been proved that the closed orbits of ejection do not pass through /x for fj. sufficiently small. Since the coordinates in these orbits are regular analytic functions of /z, it follows that if any one of them passes through the position of ^ for any value of n, then for values near this one it will pass near ^- The motion of the infinitesimal body in the vicinity of a finite body when referred to rotating axes is always in the CLOSED ORBITS OF EJECTION AND RELATKD PERIODIC ORBITS. 479 retrograde direction, and the orbits in question are always symmetrical with respect to the x-axis. Consider the motion with respect to M in a closed orbit of ejection from I-M. Whether the ejection is toward or from ft the motion with respect ton in those parts of the orbit which are near to it is direct instead of retrograde. Therefore, the orbit can not pass near ft without first developing folds so that a line from n in certain directions will intersect it three times. It is extremely improbable, though not absolutely certain, that this sort of development could take place. It is probable that the real orbits of ejection which exist for ft sufficiently small continue in the analytic sense for all values of ft from to unity, and that they do not pass near ft. For M = the orbits in which the ejection is toward n are symmetrically opposite to those in which the ejection is away from ft, and these conditions \ / \ / \ 4 \ I \ i \ \ v?2 i >\**5 '>"* ///-t i i r \ Fio. 18. KM. 19. are approximately realized when n is small. When ft increases, the loops which surround 1 M diminish in size and preserve their approximate forms, while those which surround ft approach circles whose radii are f*, where 2jw is the period, and whose centers are at ft. For ft = 1 , where t is very small, the orbits have the form shown in Figs. 18 and 19. There are, of course, closed orbits of ejection from both of the finite bodies. 233. Periodic Orbits Related to Closed Orbits of Ejection. There are periodic orbits passing near one of the finite bodies of which the closed orbits of ejection are the limits. Suppose ft = and consider the motion of the infinitesimal body with respect to the finite body I/*. Let the infini- tesimal body cross the x-axis perpendicularly at t = near the body 1 it, and let the initial velocity be determined so that the period is 2r, or a mul- tiple of 2ir. Then the motion with respect to the rotating axes is periodic, 480 PERIODIC OKBITS. the orbit is symmetrical with respect to the x-axis, and crosses it perpen- dicularly at the half period. The limits of these orbits, as the nearest approach to 1 M becomes zero, are the closed orbits of ejection. The orbits under consideration exist for initial motion near the finite body in both the positive and the retrograde directions, but in both cases the motion in the remote parts of the orbits when referred to rotating axes is in the retrograde direction. Therefore, those in which the motion in the vicinity of the finite body is direct have loops, while the others do not. The character of the two classes of orbits for periods 2ir and 4?r are shown in Figs. 20 and 21. There are, of course, orbits of a similar character which are sym- metrically opposite with respect to the y-axis. Suppose the initial conditions for one of the periodic orbits in question when /x = 0, are x(0) = o, z'(0) = 0, 2/(0) = 0, y'(0) = b, and represent the coordinates for this solution by x , x' , y , and y' a . The distance a is small and the orbit does not pass through the point (1, 0) occupied by M- FIG. 20. FIG. 21. Now suppose n is distinct from zero and that the initial conditions are x(0) = a+a, a;'(0) = 0, ?/(0) = 0, y'(Q) = b+(3. Let the variable T be intro- duced by t = T(l + d), where 6 is an undetermined parameter. Then the solutions of the differential equations of motion, which are regular functions of the coordinates and 5 in the vicinity of x = x a , x' = x' <> , y = y , y' = y' a , d = for HHII> UK K.IKCIION \M KK.I..VI K.l> 1'KKIuDH' uKHI The problem is to show that o. 4. and 6 can IM- determined >o thai the-e (((nations shall he satisfied for an arbitrary n sufficiently small. In the problem for ^ = 0, either n or l> can he taken arbitrarily when the oth. determined in terms of / except for sign: one -ign belongs to the direct and the other to the retrograde orbit. It will be stip|>os<>d that n is the arbitrary. Therefore it may be supposed that it absorbs the undetermined a and l< only two parameters in (67) besides the arbitrary ^. Kquations .117 can be solved uniquely for tf and 5 as po\\er -erie* in ^, vanishing with n- provided dp, .'/' i< distinct from /.en>. Before this determinant is formed n can he put e<|ual to /em. and therefore A de|>ends only ujxm the two-body problem. Before the second column is formed can IM- put equal to zero. When /i = # = the period in t is 2jr; hence at the half j>eriod the infinitesimal body is on the x-axis and the value of j is an even function of t-jr. Now the parameter 5 serves only to vary the i>eriod in / (keeping it fixed in T), and i- therefore equivalent to varying t from the half jx-riod. When t is nearer, can be determined so that t-jir=jir(l+i)-jr consistently with the definition of r. Therefore ;>,(0, 0, , n\ jv) is an even function of 5, and consequently (")/>,'d5 = for = 5 = M = 0- Hence the determinant 6 become- The second factor is distinct from zero because, except for a constant factor, it is the derivative of y with respect to t at r =jr, and this derivative is distinct from zero. In considering the first factor of (69) the parameter 5 can be put equal to zero. If and > represent the coiirdinates referred to fixed axes, the expression for x becomes Therefore the value of x f at t =]* is The problem is reduced to finding whether or not the expression ^' a/3 r-iv -^'+- (70) } ^ is xero under the a>umed conditions. 482 PERIODIC ORBITS. The expressions for , ', and ij, as given by the two-body problem, are = a[cosE-e], ^'=- where E is the eccentric anomaly, a is the major semi-axis of the orbit, and co is the mean angular motion in the orbit. Since sin' = and cosE= 1 for t =jir and /3 = 0, it follows that d^_ coa dE ^_-A| 2^ dp~ ~l+e dp' dp~^ l ~ e ~d~p' Therefore (70) is not zero unless dE/dp is zero. But it is found from Kepler's equation that dE jir 3co dp = l-e ~d~P' It follows from the properties of the two-body problem that at t = From these equations it is found that Therefore neither factor of the right member of A is zero, and conse- quently the solution of (67) for /3 and 8 as power series in p, vanishing with M, is unique. When the results of the solution of (67) are substituted in (66), the expressions for x, x', y, and y' become power series in fj. alone (a having been taken equal to zero) and they are periodic with the period 2 jir. When /x = the limits, as a approaches zero, of the periodic orbits which are being considered are the closed orbits of ejection. There are two families, depending upon the sign of 6, which have the same limit. Now when n is distinct from zero the expressions for the coordinates are expan- sible as power series in /x, the parts independent of ^ are the periodic orbits for n 0, and the series converge, for |/z| sufficiently small, while t runs through at least half a period. Therefore the coordinates for any value of t are continuous functions of yu- Since the solutions were developed as power series in a they are continuous functions of a for any t and M if a is distinct from zero. But in the variables of Levi-Civita it is not necessary to make any restrictions on the initial conditions. The coordinates are in all cases uniformly continuous functions of the initial conditions for all n, and therefore the limits of the periodic orbits under discussion as a becomes zero are the closed orbits of ejection, and this holds not only for /i equal to zero but also for all n sufficiently small. n|{HII> "| K.IK. II..N \M) KK1.AIKD I'KltK >\>\< oKHIIS. 483 2.U. Periodic Orbits having Many Near Approaches. Tin- nrl>it> con- sidered in the preceding article an- characterized l>\ the fart tlial. at least for small values of p, after the infinitesimal Ixxly leaves the |M>int neare-t 1 n it continually recedes until the mid-|x'ri(xl, which is a multiple of r, and thru returns s\ mmetrically. Orbits \vill now lie considered in which the infinitesimal body recedes fn.ni and return- toward n many times Ix-fore they re-enter. Suppose /i is zero and that the infinitesimal body Ls started near 1 M on, and ix-rpendicularly to, the line joining 1 ^ and n; and suppose the initial conditions are Mich that its |x-riod is commensurable with 2* without being a multiple of '2w. Let the |x-riod be P = 2i/. (72) where /> and q are relatively prime integers. Then the motion with respect to the rotating axes is periodic with the period T = Pq = 2irp. (73) In thi> period the infinitesimal Ixxly runs through q of its jxriods with resjx'ct to fixexl axes and the movable axes make p rotations. Now suppose that n is distinct from zero and that the initial conditions are given slight variations, but of such a character that the infinitesimal body is started at right angles to the line joining the finite bodies. A new indejx'ndent variable T and a parameter 5 are introduced by the relation t 7(1+6). Tin- solutions can be developed as power series in ft, 6, and the increments to the initial condi- tions, and the moduli of these quantities can be taken so >mall that these series converge for T 7V'2. Then the conditions *^ that the solution shall be periodic are that the orbit shall cross the x-axis perpendicularly at T = T/2. These condi- tions have the form (67) and all of the properties of (67) which were used in proving the existence and character of their solution. Therefore, the j>eriodic orbits which are in question exist. Now suppose that the initial distance from 1 n, which was arbitrary, is made to approach /ero as a limit. During this approach to zero the distances to the other near apses vary, but there is no apparent reason why all these a|>sidal di-- t.-.nces should vanish at the same time. In fact, from the lack of symmetry it is doubtful whether any two of them are simultaneously zero. 484 PERIODIC ORBITS. The simplest orbits of the type under consideration are those for which p = l, <7 = 2. Their general form for retrograde motion in the vicinity of the finite body 1 /j. is shown in Fig. 22. If the distance from 1 /x to a becomes zero, the orbit of ejection discovered by Darwin is obtained. The question whether the distance from 1 ju to b becomes zero is one that is hard to answer. Certainly it can not be answered affirmatively with com- plete rigor by numerical experiments, though the existence of certain classes of periodic orbits can be proved in this way. If, for perpendicular projection from a given point on the z-axis with a certain speed, the next crossing of the x-axis is at an angle which is greater than ?r/2 ; and if, for a perpendicular projection from the same point with a different speed, the next crossing is at an angle less than ?r/2, then, from the analytic continuity, it can be inferred that there is an intermediate speed at which the crossing will be perpendicular. But in the present case these conditions are not present, and all that can be said is that when the distance from 1 ju to a vanishes, the distance from 1 p to 6 is small, and the approach to 1 n is almost exactly along the z-axis. This is, of course, to be expected from the nature of these orbits. CHAPTER XVI. SYNTHESIS OF PERIODIC ORBITS IN THE RESTRICTED PROBLEM OF THREE BODIES 235. Statement of Problem. In the problem of two bodies then- arc circular orbits whose dimensions range from infinitely great to infinitely small. They form a continuous scries geometrically and their coordinates are continuou> functions of the \an<.ii- parameters by which they may be defined. There are orbits in which the direction of motion is forward, and others in which it is retrograde. The two series are identical only when the orbits are infinitely great and when they are infinitely small. In the restricted problem of three bodies* the orbits which are analogous to the circular orbits in the problem of two bodies are those which revol\e around one or both of the finite bodies and which re-enter, when referred to rotating axes, after one >> nodical revolution. Those inclosing but a single finite body were treated in Chapter XII, and it was shown there that the delations from uniform circular motion are due to the attraction of the .second finite mass. Those orbits which revolve around both finite masses, and which are analogous to circular orbits, were treated in Chapter XIII, and it was shown there that the deviations from uniform circular motion are due to the fact that the finite masses are separated by a finite distance. The problem of the present chapter is to trace, so far as possible, a con- tinuous series of orbits from those inclosing both finite masses and having infinitely great dimensions to those revolving around the two finite mosses separately in orbits of infinitesimal dimensions. There are no difficulties for very great or for very small orbits, but since in some way the infinitely great are eventually divided into two series which become infinitely small, it is clear that there is a region in which the resemblance to the two-body problem is very remote. The difficulties arise in following the orbits through these critical forms. The method of treatment is that of analytic continuation of the solu- tions with respect to the parameters ujwn which they de|x-nd. The differ- ential equations which define the motion are, when referred to rotating a\e< and in canonical units, (1) r,' The restricted problem of three bodies n that in which there arc two finite bodie* ami one infini the finite bodie* revolving in riroles. 48eing retro- grade. For small values of the parameter in terms of which the solutions are develojK-d, corresponding to \t-ry small and very large orbit.- res|>ectively. I here are three classes in which the motion i> direct and three in which it i> retrograde. Only one class of the direct and one of the retrograde orbits is real for small values of the parameters. Darwin's computations show that at least in some cases the complex orbits may become real. One of the most important properties of the orbits in question is that they all cro-s the line joining the finite bodies perpendicularly at every half period. If the parameters upon which the periodic orbits depend are varied. these properties persist, for the solutions are expansible in integral or frac- t ioiial powers < if the parameters, and the property in question is a consequence of the character of their coefficients. Therefore, the whole series of orbits from infinite to infinitesimal dimensions will possess this property, unle-s indeed they branch into two series which arc symmetrical with resjM-ct to the line joining the finite bodies. If the coordinates of a real periodic orbit are analytic in a parameter. then the orbit can not disappear without becoming identical with another periodic orbit;* and a complex orbit can not become real without becoming identical with another complex orbit, the two becoming real as they become identical. Heal orbits disappear and appear in pairs with the variation of the parameters in terms of which their coordinates are analytic functions. 237. The Non-Existence of Isolated Periodic Orbits. Appeal will be made to the numerical computations in establishing the existence of j>eriodic orbits near certain critical forms. In order that this procedure may be justified, it is necessary to prove that the orbits which they have shown to exist arc not isolated examples which exist only for special values of the masses and the other parameter on which they depend. Suppose that for M = MO equations (1) admit the periodic solution (5) where the period is 2ir/no. The initial conditions are *(0)=*o, x'(0)=x' , y(0)=!/o, y'(0)=yV (6) The initial conditions determine the value of the constant of the Jacobian integral C,= -(xV+^+xo'+yo'+^+r (7) The orbit in question will not pass through one of the finite bodies, for then it would not be strictly periodic, as was explained in 222. It will not have any infinite branches, for then it could not have a finite period. LM Mlthodea Nouvelles tie la M3j a, a', ft, ft', X), *>((), 0, 0; a, a', ft, ft', X) = 0. (12) Therefore, if pi, p 2 , and p 3 are periodic with the period P, then by virtue of this relation p 4 is also periodic with the period P. Hence the fourth equa- tion of (11) is redundant and may be suppressed. M MIIKsl- OK I'KKIoDI. .,|(lill- |V.| rt consider the solution of the first three equations ,,f i 1 1 ) for a, a', and .i in terms ..f X. The parameter .-/ i- >ii|ierflui.us and m.-.x be taken e, ll:i l to xer... This amount- to a definite determination of the initial tune. Sine,- i.")j is a real periodic solution t he coefficients of p,, p,, and p,aren lU a- tions (11) are not satisfied by X = with a, a', and ft arbitrary, for then all orbits would be periodic with the same period when M = MO- Similarly, they an- not satisfied by x = () unless all three of the parameters a, a', and ft are /ero. They are not satisfied by a = a' = = and X arbitrary, for then fixed initial conditions would give a periodic orbit for all distribution of mass between the finite lx)dies. This is certainly not true for M "ear /en>. Tl fore, the three equations can be solved for o, a', and ,3 as po\\ integral or fractional powers of X. If the powers an- integral the solution will be unique and real for both positive and negative values of X. If the powers are odd fractional, there will be a single real soli it ion for both posit i vi- and negative values of X. If the powers are even fractional, there will be t\\o real solutions for X either positive or negative, de|H-nding on the M^IIS of the coefficients of certain terms, and all the solutions will Iw complex for X negative or positive, depending upon the signs of the same eoeffici. m>. Hence in all cases there are real periodic solutions for small values of X which reduce to (5) for X = 0. (B) Let C = C +y and take the initial conditions (8). Also let * = (l+i)T, (13) where 6 is an arbitrary parameter and r a new indejxjndent variable. The solutions in this case have the form (14) X =Fi (MO; Wor) + a -t-pi(a, a', ft, ft', y, i; T), x = FI'(JIQ', ^o 7 ") ~r"ft'~r"pi(o <*', ft, ft', y, ij T), y =F t (MO; nT)+0 +p,(a, a', ft, ft', y, S; T), y' = F t '(nt; n<>T}+ft'+p 4 (a, o', ft, ft', y, i; T), where p 1( . . . , p 4 are power series in a, o', ft, ft', y, and 5. The parameter y is determined in terms of a, a', ft, ft', and 6 by equation (2) at r = 0. There- fore it may be omitted from p !( . . . , p 4 . Or, if y ' is not zero, equation (2) determines ft' as a power series in o, a', ft, y, and 5, vanishing with these quantities. It will be supposed that ft' is eliminated. It follows from the integral that the last equation corresponding to (11) is redundant, and it will be suppressed. The constant o' is superfluous and may be taken equal to zero. Then the conditions that the solution (14) shall be periodic with the period P in T are p,(o, ft, y, 6; P) =0, p,(a, ft, y, 5; P) =0, p,(a, ft, y, *; P) -0. (15) Consider the solution of equations (15) for a, ft, and 6 in terms of y. They are not satisfied unless all four of these parameters are zero, and con- sequently solutions for a, ft, and 6 as power series in integral or fractional powers of y exist, and the circumstances under which they are real are strict ly analogous to those of Case (A). 490 PERIODIC ORBITS. 238. The Persistence of Double Orbits with Changing Mass-Ratio of the Finite Bodies. In Chapter XI, p. 359, it was stated that Darwin's com- putations show that the two orbits which are complex for small values of m, or for large values of the Jacobian constant C, unite and become real for a certain value of C. In this computation the ratio of the masses of the finite bodies was 10 to 1. The question naturally arises whether a corresponding double periodic orbit exists for other ratios of the finite masses. The conditions for a double periodic orbit will first be developed. Suppose *=/i(0, *'=//(), y=h(C), y'=f *'((), (16) is a periodic solution of equations (1) for M = MO, having the period P. All orbits except those around the equilateral triangular points, which will be given special consideration in 242, are symmetrical with respect to the x-axis. In them the origin of time can be so chosen that the initial con- ditions are z(0)=* , *'(0)=0, 2/(0)=0, 2/'(0) = ?/ , (17) and from (2), C = C . Now consider a solution with the initial conditions z'(0)=0, |/(0)=0, y'(0)=y'o+|8, C = C +y. (18) The constant /8 can be expressed in terms of a and 7 by means of (2), and it will be supposed that j3 is eliminated by this relation. Let a new inde- pendent variable T be defined by t=(l+d)r, (19) where 5 is a parameter as yet undetermined. Then the solution of (1) can be developed in the form x = pi(a, 7, 5; T), z' = p 2 (a, 7, 5; T), y = P^(a, y, 5; T), y' = p 4 (a, 7, 8; T), where pi, . . . , p^ are power series in a, 7, and 5. The conditions that (20) shall be a periodic solution with the period P in r are , 7, 6; P/2)=0, p t (a, 7, 8; P/2) = 0. (21) It will now be shown that 5 can be eliminated by means of the second of these equations. Suppose equations (1) with t as the independent variable and initial conditions (17) are integrated as power series in 5. The terms inde- pendent of 5 will be periodic with the period P. Non-periodic terms will enter only when the terms in the right members at some stage of the inte- gration contain terms with the period P. Then t times periodic terms will appear in the solution, and terms multiplied by t 2 and higher powers of t will not enter until later stages of the integration. Terms of this character 8YMHI-I- \\t\ will actually M1M, for otherwise tin- soluti.ui would IN- | N -ri.Nlir fur all *; that is. the coordinates would In- constants \\ith res|M-ct to /. Tl,. sion for // = /,, is an odd function of I with tin- initial conditi,,: fore the (mn in / is multiplied by an even function, that is, a cosine term which does not vanish at t = P/2. Therefore /,, cam,- ,, to tl,,. first Ic.gree and this parameter can Jx- eliminated, giving **(, 7)-0, (22) where /' is a power >eries in a and y vanishing for a="y-0. SupiH)se y is taken as the independent parameter and that (2'J i- solved for a in terms of y. A necessary and sufficient condition that |i, I).- a double periodic solution with respect to the Jacol.ian constant r is that -*<, T)-0 (23) for a = y = 0. Suppose this condition is satisfied. Now suppose /i = /tio + X and consider the (|iie.stion of the existence of a douMe periodic solution for this value of ji. The double periodic solution will exist provided the equation corresponding to (23) is satisfied. Suppose the initial value of x is (24) \\h( re a is & parameter which remains to be determined. The steps cor- responding to those leading up to (23) can be taken in this case, and the ((luatidn corresponding to (23) becomes P,((r, X)=0 (25) for a = y = 0, where Pj is a power series in a and X vanishing for t her. That is, for X = two real double periodic orbits unite and disappear 492 PERIODIC ORBITS. by becoming complex. And in general, double periodic solutions appear or disappear in pairs, which become identical for certain values of fj., a result analo- gous to Poincar6's theorem respecting the appearance and disappearance of real periodic orbits. Now consider the real periodic satellite orbits which are re-entrant after a single synodical revolution. Darwin's computation shows that for the mass ratio of 10 to 1 there is one, and but one, real double periodic orbit in which the motion is direct. Hence, there is no other double periodic orbit with which it could unite to disappear for any value of M- The same result also is a consequence of the analysis of Chapter XI, where it was shown that for M = 0, and therefore for all n, there are but three real and complex orbits of the type in question. From the fact that there are only three periodic orbits of the type in question it follows that there can not be more than one direct double periodic orbit; and from its existence for M = Vn> 1 M = I %I, it follows that there is one direct double periodic orbit for all values of n from zero to unity. The orbits of inferior planets differ from those of satellites only in the ratio of the masses. Therefore, there are also double periodic orbits of inferior planets in which the motion is in the forward direction. When one of the masses which revolve in circles becomes zero, those orbits around the other which are complex for small periods are complex for all periods from zero to infinity. In this case there are no double orbits except those of infinite and infinitesimal dimensions. The question arises as to the character of the double orbits for very small values of the second finite mass. Consider the totality of real circular orbits around one of the finite bodies when the mass of the other one is zero. As the second mass becomes finite, the periodic orbits are continuous deformations of the cir- cular orbits with the exception of that circular orbit whose period is 2ir. It passes through the point where the second mass becomes finite, and the force function for this orbit has a discontinuity. This means that the orbit itself has a discontinuity. It is conjectured that the complex orbits have corresponding discontinuities; that when the second finite mass is very small there are three real orbits about the larger mass in which the motion is in the forward direction and which have a mean distance near unity and a period near 2?r; that there are three corresponding real orbits of small dimen- sions about the smaller mass; and, finally, that there are three similar orbits of the nature of superior planets. For increasing values of the Jacobian constant two of the three in each case unite and form the double orbits. Consequently, if this conjecture is correct, and if the three types of double orbits are followed as one of the finite bodies approaches zero as a limit, the one around the larger finite mass and the one around both finite masses approach the unit circle, and the one around the smaller finite mass approaches zero dimensions. M \ I UK-MS cK I'KUIMIIH The question of direct double |M-riodic orbits was |)iit to numerical te>t for n = \. In all, ~M orbits wen- computed for vari>u> values of (', start- ing with C-8.58 and ending with 66. lor direct periodic orbits which were geometrically much alike, and which intersected the ,r-a\i> near .7v~>. For smaller values of C the correspond- ing periodic orbits were more nearly identical. For r = 3.0Si they were sensibly identical. The computation for this value of C gave the results set forth in the following table (Fig. 23): ;{.. r >K there wore two *-Hi C-3.086, Double IXTIO.II. orbit. M-M, C-3XW6, I)..ul.l.-|Tflic ..rl.,t ( X 1 i' .'/' J z V * / -7740 -I 193 .40 -5643 .:>. m - .317 (:, -.7693 -.(. IN: -1.168 BO -.4893 - :14 .738 .072 10 - 7556 -.1161 -1.097 60 -.-I17I-. - :ti7n aw + .159 II -.7 -.1684 I'.IS - .990 7ii -.MIO 1197 H M .20 -.7064 -.>\\^ tat - .862 Ml -.2917 -.2698 i.u -.6741 -..' ten - .724 .90 -.2429 - . 1974 .420 V. ; .30 -.6389 -.2872 ;j-, - .585 1 00 - 2097 -.1048 1 mfi -.6020 -.31.51 74B - .448 1 10 -.1978 + 0007 -.003 1 DT'i There are also three retrograde periodic orbits, only one of which is real for small values of the parameters in terms of which t heir coordinates are developed. The question arises as to whether the retrograde complex peri- odic orbits unite and become real. In order to test the question by numeri- cal experiment, 20 orbits were computed. For n = \ and C = 3.7. r >, five orbits were computed, starting with various values of X* and determining j/ ' so that C should be 3.75. The correspondence between x and the angle

e given for enough values of / to show its geometrical characteristics (Fig. 24). I:. Jl 494 PERIODIC ORBITS. M=J^, C = 3.75, Retrograde periodic orbit. t X y f V -.6500 2.052 .026 -.6420 .0504 .637 1.943 .050 -.6189 .0955 1 . 194 1.634 .O7. r > -.6835 .1308 1.611 1.173 .100 -.5398 . 1534 1.854 .625 . 125 -.4922 .1618 1.920 .048 .150 -.4451 .1561 1.821 - .501 .175 -.4024 .1374 1.582 - .982 .200 -.3670 .1078 1.233 -1.367 .225 -.3414 .0699 .807 -1.637 .250 -.3270 .0266 .336 -1.781 .275 -.3246 -.0184 - .151 -1.800 Eight retrograde orbits were computed for M i> (7=3.214. In this case also the existence of but one periodic orbit was indicated, as is shown M = /4, C 3.214, Motion retrograde. Xe -1.0000, - .9000, - .8000, - .7700, - .719, - .685, - .650, - .625 , near collision 80, 8321', 8456', 8741', 9115', 96 100 by the accompanying table of correspondences be- tween x and (7 = 2.95. In this case also the existence of only one periodic orbit was indicated. The correspond- ence between x and

eriodi<-. Its coordinates fur \;iri value.- c.f / an- niven in tin- following table (Kin- --'I'., page -196): *-K, C-2.W ,-,,rl.,l M-.'i, C-2.95, RetracrMle pprioli'- .rl.n Retrograde | t i y i- '< t X V *' 9" -.7190 .718 .200 -.6007 .2146 1 716 - .054 tat -.7140 IH.M .334 ..::, at -4581 .2088 1 i. - .408 .050 -.7088 .0832 IBB .-.77 m - 4171 I'M.i t.ira - .752 .075 -.68M 111 IS m lir, m -.3805 .1715 1 .Htt - 072 100 -.'. i:..:r. 1.21'J 100 MM Illl 1 140 - .:W2 1 .'.-. -.65 1.427 .034 ne -.3235 iota - in Ml 2000 i 587 at Ml IN, .'7 IT.", - * nie 1.685 m -.2987 nisj 1.-' - Ml Two orbits \\ere also computed for C = 2.75. The results were similar to those for = 2.95. All the results indicate that there is only one real retrograde |>eriodic orbit about each of the finite Ixxlies. 239. Cusps on Periodic Orbits. The orbits of ejection in a certain >en>e have cusps at the point of collision with a finite body. But they have been treated in Chapter XIV and require no further comments here. The coordinates of the infinitesimal body can be expressed as power M ! -ie,s in t ti, if for I = t\ it is at any point for which the differential equations are regular. A necessary condition that the orbit shall have a cusp at t = t t is that the expressions for both x and y shall have no linear terms in t 1\. It follows that if the orbit has a cusp at * = ti, then z'(f ,) = y'(t t ) = 0. That is, the body is on a curve of zero relative velocity at t = ti. Suppose its coordinates at t = ti are x t and y\. Now let x = Xi+, y=!/i+ij; (28) then equations (1) become M 496 PERIODIC ORBITS. The solution of equations (29) as power series in i- conditions = i\ = = i\' = is 24 B 0f . . r A 0f . f x-, , l = "^ t ~ tl )"~~\ t ~ t i) " ti with the initial (30) 24 The direction cosines of the normal to the curve of zero relative velocity at the point (z.i, ?/i) are proportional to A and B . Therefore the tangent to the cusp is perpendicular to the curve of zero relative velocity at the point (xi, y\}. Now take a new set of axes (u, v ) with origin at (xi, yi), with u having the direction of the tan- gent and v perpendicular to it. If the positive ends of the new axes are chosen so that the cosine and sine of the angle from the positive end of the -axis counted counter-clockwise to the positive end of the w-axis are proportional to A and B , and if the positive end of the u-axis is 90 forward counter-clockwise from the positive end of the tt-axis, then the equations of transformation are Fia. 26. !-K I'KHluIMc OKHII- |'I7 deed. all periodic orbits of superior planets revolve in tin- retrograde direction with respect to the n.tatiiisi axes. But the ,,rl)it.s with cusps, if they have DO Other double points. revolve in the forward direction with ri-|M-rt t.. ftXee, because at the cusps they have precisely tin- forward nioli..n ,,f the rotating; a \ 240. Periodic Orbits Having Loops Which Are Related to Cusps. Suppose the orbit defined by tin- initial conditions j(0) = . x'(0)=0, y(0)=0, y'(0)-y.', C-C., (32) i> periodic witli the |>eriod /'. Therefore x'(P/2)=0, y(P/2)0. Suppose it has a cusp at t = ti, or x(t t )= Xl , *'(,) -0 (,) -y,, y'(<,)0. (34) If the initial conditions are varied in such a way that the orbit remains periodic, its character in the vicinity of the cusp will be changed. The nature of these changes will now he considered. Supixwc the initial con- ditions are x(0)=zo+a, x'(0)=0, y(0)=0, y'(0)=y,'+/3, C-C,+ 7 ; (35) and that <-(l+)T, (36) \\here T is a new inde]M-ndent variable and 6 Ls an undetcnnined paramete periodic are z'[(l-H)P/2] = p,(a, ft, )-0, y[(l+*)P/2] = p,(, ft 3)=0, (37) here /;, and p 2 are power series in a, ft and 6, vanishing identically with a. 0, and i. Now consider the solution of (37) for and 6 in terms of o, vanish- ing with a. The solution is always possible either in integral or fractional powers unless the equations are identically satisfied by /3 = 5 = 0. But this means that all orbits in which the infinitesimal body crosses the x-axis near x with fixed velocity y' are periodic, and that they all have the same period. Since the results arc analytic in o, the orbits may be continued wit h re-|>ect to a, in the analytic sense, until the place of crossing, x,, is small. But then the methods of Chapter XIV apply and it is known that the period depends upon x . Therefore equations (37) are not identically satis- fied by (3 = 5 = 0, and they can be solved for and 6 in terms of o. The solu- tions will have the form ^ = a l/ 'P,(o 1 /'), i-a"^. 1 "), where p is unity if the Jacobism of PI and p* with respect to ft and f> is li>- tinct from zero for o = = 5 = 0. If p is not unity, it is some other |*.itivr integer. In general it will be unity. 498 PERIODIC ORBITS. In the computations of Darwin 7 was taken as the parameter which defines the orbits. The change can be made here because (2) is uniquely solvable for a as a power series in /3, 7, and 5 unless -O o- _ n * ~v~~ r s - m u, 2/0 -u. '1 ^2 But these equalities are satisfied only at one of the collinear solution points. The orbits in the vicinity of these points will be omitted from this discus- sion because they belong to quite another category. If a is eliminated by means of (2), and /8 and 5 by means of (38), the solutions will be expressed as power series in 7 1 fp , and the values of the coor- dinates at r = ti are ^' x'(r) r= ,, = 0+7 1/ ^ 2 ,-| y'(r)r. tt = 0+y 1/ "0J where 61, . . . , 64 are power series in 7 1/p . The expressions for the coordinates in the vicinity of t = t t , satisfying the relations (39), can be expanded as power series in t 1\. The solution is found from equations (29) to be (40) i) \ ,) where a 2 = - &3 = -f(l+6)a 2 +|(l+5) 2 [ J B 1 a 1 + J B 2 6 1 ], j i where A , AI, A 2 , B , B lt B 2 are given in (29) and P is a power series in 7' -'" The transformation (31) gives u = y l/p (0 2 A +6 4 Bo) (r ti) (42) It follows from (41) that for 7 = M = iUo ! +5o 2 )(T- 1 ) 2 + . . . , v= -|(4o 2 +5o 2 )(r-^) 3 + , (43) The equation v = determines the points at which the periodic orbit crosses the w-axis. It follows from the second of (43) that (t-ti)=Q is a triple but not a quadruple solution for 7 = 0. Therefore there are three solutions of v = Q for t t t as power series in integral or fractional powers of 7 1/p , vanishing with 7. One of them is simply T -<, = () (44) -1 \ I HK.-I- iiK l'Kl;l|)|( MHHIl-v The other.-. depend U|M)II till' \allle- of the coejfieielil- i .f the right of the >(<<, ii,l <>iiii:itioii of :}'_'.. I'lih-ss Mo-M*o = A" = n for ) = the two remaining solution-. ha\e tin- fr)rm - l " + vmK'-'' :r -' v/w 1 " If A" = (). which will he exceptionally if :it .-ill. the n.rre>|omling >..liition- <:\i>t hut may he in integral po\\er> of 7' '. It has hem remarked that p will in general be unity. When it is odd the periodic orhit with the cusp at (z,, y t ) is a multiple orhit. If / i> even. t woorhits which are real when y ha> <>m- -ign unite for y = and disap|x>ar hy becoming imaginary when y has the other sign. When {> is odd then' i- a -inule real orhit for 7 both positive and negative. It is clear that only exceptionally, if at all, will a double periodic orhit have a cusp. If it v so iii any particular case the value of ^ could be changed, when it would no longer he true. Therefore it will he sup]>osed that f> is unity. It follows from i4">) that the second and third intersections of the curve with the w-axis are real for y positive or negative according at< A' Is positive or negative, and that they are not real when 7 has the other sign. When the second and third intersections of the curve are real the curve consists of a small loop as is indicated in Fig. _'s; and when they are complex the curve has a point near which the curvature is sharp, as is indicated in Fig. 29. When there are three inter- -ections of the curve with the u-axi-. one occurs before t\ and one after /i. It follows from this discussion that if a periodic <>rl>it for a cirtnin mint' (\ of the Jacobian constant has a nis/,. then for n xliyhlly larger (or smaller) value it has a point, near a curve of zero relative velocity, in the vicinity of tchich there is very sharp curmlun; for diminishing (increasing) valws of (' the print of xhnrp cunatnn- upproachfs the cusp form on the fvnv.s-/Mm//m/ ninr <>/ -rr<> velocity, which it reaches for C = (\; andfartttill further diminishing (increasing) values of (' it ha* a small loop near a curve of zero velocity. In Darwin's computations examples of IH-riodic orbits with cusps were found. It follows, of course, from the sym- metry of the periodic orbits with respect to the ar-axis that if there is a ru*p at x = Ji, y = y\, then there is also a cusp at zZi, l/= -y\- 241. The Persistence of Cusps with Changing Mass-Ratio of the Finite Bodies. Suppose for M - Mo equations (1) have a periodic solution satisfying the initial conditions z(0) = x . x'(0)=0, y(0) =0, y'(0)-yV (46) and the cusp conditions at l = l\ 500 PERIODIC ORBITS. x(t 1 )=x l! z'(*i)=0, y(*0=J/i, y'(*i)=0. (47) If P represents the period of the solution, the expressions for x' and y satisfy the equations x'(P/2)=0, Z/(P/2)=(). (48) Now suppose M = Mo+X and consider the question of the existence of a periodic orbit having a cusp for this value of /z- Let t=(l+5)r, z(0)=z + a, s'(0)=0, J/(0)=0, ?/(0) = >/'o+0. (49) The solution can be expanded as power series in a, 0, 5, and X. The con- ditions that it shall be periodic in r with the period P are x'(r)^,, /2 = Pl (a, 0, 6, X) =0, 7/(r) r=/V2 = p 2 (a, 0, 5, X) =0, (50) where pi and p 2 are power series in a, 0, 5, and X, vanishing with a, 0, d, and X. Unless the initial conditions (46) define a double periodic orbit these equa- tions can be solved uniquely for and 5 as power series in a and X, vanishing with a and X. The results will have the form = 9l (a, X), = g S (a,X), (51) where qi and q 2 are power series in a and X, vanishing for a = X = 0. Suppose and 6 are eliminated from the solutions by means of equa- tions (51). The results will be expanded as power series in a and X. The values of the coordinates at r ti will be l (a, X), s'(T)r-, = 0+Pi'(a, X),' , X), y'(T),- h = 0+P,'(a, X) where PI, PI, P 2 , P 2 ' are power series in a and X which vanish with a and X. The values of the coordinates near r = t } can be expanded as power series in r ti satisfying equations (52). The results are s'(r)=0 +Pi'(a, X)+2a(r-< 1 ) 2/'(r)=0 +P 2 '(a, X)+26(T-<0 + . . . The conditions that the orbit shall have a cusp at T = t% are = Pi'(o,X)+2a( a -* 1 )+ . . . , = P 2 '(o,X)+26(< 2 -<,)+ .... (54) These equations are not satisfied by X = 0, because a and 6 contain terms which depend upon x v and yi alone. Neither are they satisfied by a = 0, t 2 ti = Q, unless orbits crossing the x-axis at x = x are periodic for all values of M and have a cusp for the same t ti. But it is known that the points at which the periodic orbits cross the z-axis depend upon the value of M- There- fore equations (54) can be solved for t 2 ti and a in integral or fractional powers of X. In general the solution will be in integral powers of X. If the BYMTHMH <>! i-Kitiooir OKHIIS. 501 solution is in intr^ntl cm. 1,1 fractional powers of X, it is real for both |o>itive and negative values of X. If the solution is in even fractional |x>were of X, there are t\\o real solutions \vlu-n X has on.- sign and only complex solutions when it has the othrr. It follows from this iliscu>-ion that if a real cusp exists for any value of n, it will exi>t for all other values of ^ unless two real cusps become identical and disappear by becoming complex. Since an orbit N uniquely defined by the condition- fora cusp, as well as by any other initial conditions, cusps disappear by becoming complex only when two orbits become identical. Darwin's computations >howed that in the case of one of the orbits which was complex for large values of the Jacobian constant ("satellites of ( 'lass C") there were j)eriodic orbits without loops near the cusp form, and >\ here for smaller values of the Jacobian constant having loops. It follows from the results of 240 that between the two orbits there exists one having two cusps which are symmetrically situated with respect to the z-axis; and it follows from the discussion of this article that the orbits with cusps e\i-t for all values of n unless a cusp develops on another orbit which lat-r becomes identical with this. 242. Some Properties of the Periodic Oscillating Satellites near the Equilateral Triangular Points. In Chapter IX, Dr. Buck has treated the periodic oscillating satellites which are near the equilateral triangular points, using in a general way the methods of Chapter V. It will be necessary for t he purposes of the latter part of this chapter to develop a few additional prop- erties of these orbits; and the most important of them can not be established by the methods of Chapter V, but follow from the methods of Chapter VI. The differential equations will be transformed by letting n = f For motion in the vicinity of the equilateral triangular points they are where X and Y are of the second and higher degrees in z, y, and X. In this article the character of the small oscillations will be discussed. In treating them X and Y may be provisionally put equal to zero. The characteristic equation on which the nature of the solutions depends is -^)-0. (56) 4' 4^ 01 9 The roots of this equation are all purely imaginary or complex in conjugate pairs according as l-27/io(l-w) is or is not greater than /en.. \Vh.-n 1 -27^(1 -/*>) is zero there are two pairs of equal purely imaginary wilut im> 502 PERIODIC ORBITS. of (56). It will be supposed for the present that MO has such a value that the roots of (56) are pure imaginaries and distinct, and that M has such a value that X = M MO is very small. Let the roots of (56) be +oV^l, Q at t = 0, the solution having the period 2ir/ 1 = "I >, f <'. = (', is 2 v ro. Hence the value of the .lacobian con-taut I- greater for the orl.it- who>e period is JIT a. and less for tho>e whose |M-riod is '2r p. than it is for the Lagrangiaii e(iuilateral triangular |M>int solution. No\\ consider the curves of zero relative velocity. They are known to he real only if the value of (' is greater than that which belong- to the equilateral triangular point solution. Therefore they are real only for the solution with the period '2r a. Their equal ion i< ^'.=i*'+^(i-2*)*j/+!i/'. This is the e< |iiation of an ellipse, the direction of \\ho-e major axis i> given by (64) The limit of the right member of this e\pre>- ion for Mo = i v/. Since a is small when MO is small, the approximate \alue of the risilit member of equation (59) is V3(l 2/n). Therefon- the orbit whose |>eriod i- JT a has its axes, for small MO. nearly coincident with the axes of the eorrrespondinK curves of zero relative velocity. The j-axis is in the line joining the finite body M with the equilateral triangular ix>int, and the other i> of course at right angles to it. Ix>t the coordinates in the orbit whose jieriod is 2v/a referred to its axes be $ and 77; its equation is then 66) A , = I[(a, cos 0-sin )+&,* cos 1 0], c, #, = ![((!, sin 0+cos 0)*+&, s sin 1 0]. The corresponding equations for the curves of zero relative velocity are cos 1 v+~(l -2Mo) sin ^ cos in ^ cos ^ + J cosVl- 66) It follows from (58) that when M is snuill the approximate values of a, and &i are v/3 3 and zero resect ively. Sinn- the limit of 9 for Mo = is -30, it is found from (65) that lira At lim cos* g+ 2V3 sin g coe 0+3 sin 1 6 _ Q (6?) ^ = 0#, =: M = 03c<* J 0-2v'38in0co80+8in t The limit of the ellipse for Mo = is a straight line through the origin and the position of M. and for small values of Mo the eccentricity i- near unity. 504 PERIODIC ORBITS. The approximate value of = &% Therefore, for these lo lo orbits also tan 26 = - v/3 when MO has the limit zero. It is found from the equations corresponding to (65) that Km At _ i Mo = 2 ~ 4 ' Therefore in these orbits the length of one axis is twice that of the other. The limit, for MO = 0, of the eccentricity of the orbits whose period is 2w/ff is unity and the limit of the periods is infinity; the corresponding limits for the orbits whose period is 2ir/p are \/3/2 and 2ir. 243. The Analytic Continuity of the Orbits about the Equilateral Triangular Points. The periodic solutions as developed by the methods of Chapter VI are power series in ^X*, and they involve MO- The coefficients of the power series are continuous functions of MO- The orbits are real when X has one sign and complex when it has the other. As X passes through zero from one sign to the other, two real solutions for =*=X- unite and disappear by becoming complex. They do not belong to the physical problem except when X = M MO, but since the orbits exist for every value of MO distinct from zero it is easy to get an understanding of the situation from the behavior of M M III -I- OK PEUloDK ..Kill IS the more general .-olutions when X dn>idered as varying \vitli X so that their sum is n. That is. th<- solutions and the |>eriod can lie expressed in terms of / and X by replacing /io by n X. The two real orbits which unite and disappear for X = are not geo- metrically distinct. This appears to be an exception to the theorem that real orl>it> appear or disappear only in pairs. It arises because in the analy.-is adopted the conditions that the orbit shall be periodic give a double determination of the same orbit. The two determination.-, coincide when the orbits shrink to zero dimensions for X = 0. Such a situation can arise only at the live equilibrium points. Moreover, the matter i.- quite different when the solutions are developed by the method of Chapter V. When the para- meter t' passes through zero the orbits do not disupj>ear. but t he same 861168 is obtained for both i>ositive and negative values of t', the origin of time belonging to a different place on the orbit. In the symmetrical orbits around the equilibrium points which are on the z-a\i> the origin of time is displaced by half a period. In the non-symmetrical orbit > about the equilateral triangular ]M>ints the origin is shifted from one |M>int where the arbitrary initial condition, e. g., x'(0)=0, is satisfied to the other point in the orbit w hoe it is also satisfied. The Jacobian integral exists when the right members of the differential equations are not limited to their linear terms. Hence, in place of (60), the right member is an infinite series in x and y. When the expressions for x and y as series in X 1 are substituted the constant C becomes a power serie- in X 1 , the term of the lowest degree in X being of the first degree. Conse- quently the series can be solved for X 1 as a power series in =*C', and the result substituted for the solution in powers of X' will give x and y expressed as power series in satisfies the equation 1-2W1-/*,)=0 the values of a- and p are equal to V2/2. In this case four solutions branch at X = 0. 244. The Existence of Periodic Orbits about the Equilateral Trian- gular Points for Large Values of it. The orbits heretofore discussed have been for values of n such that 1 -27n(l -M) s positive. If it is zero, t here i> a double solution of zero dimensions. Suppose now that n has such a value that this function of M is very little less than zero, and take *> so that l-2?Mo(l-Ao) is a little greater than zero. Then there are real periodic orbits with the periods 2r/) =0, and that the analytic continuation can be made with respect to X for all MO- The expressions for the coordinates in the two classes of orbits are the same except that a and p are interchanged. As MO approaches 0.0385 . . . a and p approach equality, and the correspond- ing orbits approach identity for X = /* /j, . A difficulty in attempting com- plete rigor arises from the fact that a certain determinant which is distinct from zero in the proof of the existence of the solutions approaches zero as /x approaches 0.0385 .... But if it is admitted that the analytic continua- tion with respect to X can be made starting with any ^o, it follows that even if ju is a little larger than 0.0385 . . . there is a double periodic orbit, and it surrounds a small real curve of zero relative velocity in the vicinity of one of the equilateral triangular solution points. As it is decreased toward the limit 0.0385 . . . , the dimensions of this double periodic orbit diminish toward zero as a limit. There is in this analysis a double determination of a double periodic orbit, just as of a single periodic orbit, and the two deter- minations coincide when it has zero dimensions. Consequently it can disap- pear at zero dimensions without uniting with another double periodic orbit. If ju increases the double periodic orbit persists, according to the prin- ciples of 238, unless it becomes identical with another double periodic orbit. If there were another double periodic orbit with which it could unite it would envelop neither of the finite masses and would have two distinct branches which are symmetrical with respect to the z-axis. It is improbable in the extreme that there is another such double periodic orbit, which would mean M \ 1111 -I- nh 1-1 KIUDIi ..Kill t*. the existence of four single periodic orhil- of tin- type under consideration. There an- only two periodic orhit> which >hrink mi tin- equilateral triangular points Miul others of the type could arise only from orhits. which originally had loo|is about a finite liody. pa--in thnniuh :tn ejectional form. The existence of a double periodic orhit f(r all values of ft iniplu- the existence or two single |>eriodic orhit- which liranch from it for value- of the parameter- which define the orhit, for example the .lacohiaii con-taut ('. il- linear dimen-ion. or its |>eriod. It should In- added, of cour-e. that the t\vo serie> of orhits may branch at the doulile orhit when considered with n-pect to one parameter, and form a continuoii- -.-rn-- when considered with respect to another. 245. Numerical Periodic Orbits about Equilateral Triangular Points. In accordance with the principles of 244. two periodic orliits about (In- equilateral triangular points should exist for nil values of ^ from to J, and for all values of <' near that belonging to the equilateral triangular ((luililirium points. The only way they could oe:ise to exist for at least .Minn- value of (' would he for all of them to pass through an ejectional form for every ('. These orhits have an axis of symmetry only when M = }- I' i- very difficult to establish by numerical processes the existence of a |>eriodic orhit when it has no axis of symmetry because, for a given initial point, there are two arbitrary comjwmcnts of velocity, and interpolations must he made from a two-parameter family. Then-fore the computations wen restricted to the case n = \. It follows from the differential equations that in this case the orbits in question have the line z = as a line of symmetry. Since /x = i is far from the values (0^/u^ 0.0385 . . .) for which the existence of the orbits in question was established by direct processes, those found by computation can not be expected to have much geometrical resemblance to those found by analysis. Since the surfaces of zero relative velocity expand with increasing (' and unite on the jr-axis. it follows that either the |>ericxlic orbits about the equilateral triangular jxunts unite in pairs and disappear with increasing values of C, or they pass through the collinear equilibrium points with infinite periods. Therefore it seemed best to start computations for values of (' not much greater than that belonging to the equilibrium point, viz, 3. In attempting to discover j>eriodic orhits about t he equilateral t riangular points 40 orhits were computed. In 17 of these C was taken equal to :\.(Y.\: in 10 it was taken equal to 3.20; in the remaining 7 it was taken equal to 3.284. The initial values of the coordinates and eom|>onent of velocity were x = 0, T/ arbitrarily chosen, j/ ' = 0, x ' determined so as to give tin- adopted value of C. The computation was continued until x became again equal to zero, and the approach to periodicity was determined by the approxi- mation of y ' to zero. 508 PERIODIC ORBITS. C = 3.03, Period =2X4.388 = 8.776 (Fig. 30). I X y x' V t X y x' y' 0.0 .1200 -1.060 1.2 -1.0287 - .5841 - .708 - .070 0.05 - .0533 .1216 -1.076 .067 1.4 -1.1768 - .5687 - .762 + .222 0.10 - .1083 .1266 -1.127 .131 1.6 -1.3285 - .4967 - .745 .495 0.15 - .1667 .1345 -1.219 .185 1.8 - .4701 - .3726 - .660 .740 0.20 - .2311 .1446 -1.366 .214 2.0 - .5890 - .2032 - .521 .947 0.25 - .3047 . 1549 -1.593 .182 2.2 - .6755 + .0030 - .338 1.107 0.30 - .3921 .1600 -1.919 - .022 2.4 - .7221 .2361 - .124 1.216 0.35 - .4966 . 1454 -2.217 - .661 2.6 - .7238 .4856 + .108 1.271 0.40 - .6014 .0865 -1.798 -1.661 2.8 - .6786 .7409 .344 1.274 0.45 - .6685 - .0060 - .927 -1.897 3.0 - .5865 .9917 .575 1.226 0.50 - .7019 - .0957 - .479 -1.674 3.2 - .4497 1.2285 .790 1.134 0.6 - .7353 - .2397 - .273 -1.233 3.4 - .2720 1.4429 .982 1.003 0.7 - .7638 - .3475 - .313 - .941 3.6 - .0586 1.6274 1.146 .838 0.8 - .7995 - .4305 - .403 - .725 3.8 - .8158 1.7762 1.277 .646 0.9 - .8445 - .4937 - .498 - .543 4.0 - .5503 1.8846 1.371 . 435 1.0 - .8986 - .5396 - .583 - .378 4.2 - .2699 1.9495 1.427 .212 1.1 - .9605 - .5695 - .654 - .221 4.4 + .0177 1.9689 1.443 - .018 C=3.03, Period =2X5.95 = 11.90 (Fig. 31). I X V of y' t X y x' y' .7428 .074 2.3 - .7940 - .5196 - .323 - .605 .1 .0072 .7406 .070 - .044 2.4 - .8320 - .5766 - .434 - .500 .2 .0137 .7340 .058 - .087 2.5 - .8804 - .6198 - .532 - .365 .3 .0185 .7232 .038 - .130 2.6 - .9380 - .6496 - .616 - .231 .4 .0210 .7082 .010 - .170 2.7 -1.0030 - .6660 - .684 - .097 .5 .0202 .6892 - .026 - .209 2.8 -1.0742 - .6689 - .735 + .039 .6 .0154 .6664 - .070 - .245 3.0 -1.2272 - .6340 - .784 .312 .7 .0060 .6402 - .121 - .278 3.2 -1.3828 - .5459 - .762 .570 .8 - .0090 .6109 - .180 - .308 3.4 -1.5275 - .4076 - .675 .808 .9 - .0303 .5788 - .247 - .332 3.6 -1.6490 - .2249 - .532 1.018 1.0 - .0588 .5446 - .323 - .352 3.8 -1.7222 - .0054 - .343 1.174 1.1 - .0954 .5086 - .411 - .367 4.0 -1.7841 + .2412 - .121 1.284 1.2 - .1414 .4713 - .514 - .378 4.2 -1.8095 . 5047 + .120 1.342 1.3 - .1988 .4329 - .638 - .390 4.4 -1.7358 .7742 .368 1.344 1.4 - .2702 .3925 - .793 - .416 4.6 -1.6376 1.0390 .612 1.296 1.5 - .3587 .3482 - .986 - .489 4.8 -1.4920 1.2892 .842 1.199 1.6 - .4683 .2906 -1.204 - .696 5.0 -1.3024 1.5157 .049 1.060 1.7 - .5951 .1988 -1.269 -1.199 5.2 -1.0744 1.7107 .227 .884 1.8 - .6994 .0492 - .710 -1.700 5.4 - .8140 1.8674 .371 .679 1.9 - .7384 - .1158 - .157 -1.522 5.6 - .5288 1.9806 .475 .450 2.0 - .7459 - .2519 - .042 -1.209 5.8 - .2268 2.0464 .538 .205 2.1 - .7522 - .3604 - .100 - .972 6.0 + .0834 2.0623 .557 - .024 2.2 - .7674 - .4483 - .207 - .792 Fia. 30. Fio. 31. HK sog It \v:is found that for r --;{.(>:{ there are t\\n jMTiodic .iit the equilateral triangular points din"erin; n>n-ideral>ly in dimensions and IM-riod.-: for (' :\:20 there an- al-<> t\v<> periodic orbits which differ C-3.20, IVriod -2X4.68-9.36 (Fi. 8). t i > f * t i v r 1 n n .2030 - .883 1.6 - 2334 .4760 - .683 H - 0449 aou - .904 1 x .8008 .3976 - .831 Ml Hi - .0908 10M - .936 OM - 4772 - .2780 - .537 U i:i'i -Mill - 001 - '. - .1146 - .408 -ii - 1"I7 2152 - 07., .104 -.' 1 _ ,. + ' - .346 014 - .'.M7J - I'M .-,x., INK) ;M) - .3KM - 661'J + i:tx - .4l.lx 2110 - - - .6136 Ml - - 17 1 -1 - H, ,.' n m - .71150 - .01 IJ - -1.470 34 _ i. 1741 Ml 7 - .1484 - :t7.' -1 199 - 2458 Ml 004 1 - !8181 - .2548 - .320 - .940 - 0588 H4I (M - .KM', - - - ,7:tl 40 - .8470 6736 111 010 1 II - .8901 4022 - llx - 1 J - 6154 -Kin l"x 1 1 - - !4498 - .484 - .399 4.4 - 3696 S537 i. a - .9864 - 1 - - .254 46 - .1150 MM .284 OHO i i -1.1036 - .6065 - .626 + .018 4.8 + 1426 Moa Ml - .098 C-3.20, Period-2X6.72-11.44 (Fig. 33). 1 z V z' f t z z' f o MM - .188 22 - 0165 - 5708 - - .129 M - .0003 .5549 - .1x7 - .020 2.4 - .1359 - ' - + .114 10 :,-,;( - .I'M - 040 26 - 2650 - .6266 - .15 - !0285 MOO - .199 - .060 2.8 - - 1 - .Ml .569 .20 - .0387 .5474 - .209 - .080 :j o - .6059 - .3997 - .520 .763 .25 BOO fft - .098 3.2 - 5970 - .1307 - .30 - 0609 - .238 - 117 3.4 - .6577 + 0660 - .218 OM I - .0867 6341 - .2NO - l.v.' 3.6 - .6827 .2S1S .029 111 - .117:! 1073 - - .184 38 - .6690 107V + .168 i ;> - .1542 .4874 - .406 - .214 4.0 - .6168 ....; u.; .7 - .1990 .4645 - .494 - 244 42 - .6243 on .549 OM .8 - .2537 IHS - .603 - .278 44 - .3974 1600 718 Ml 9 - 3205 11 IX- 1 - .737 - .331 4.6 - .2388 ua M Ml 1.0 - .4019 - .4989 1708 .3196 - .893 -1.050 - .428 - .618 4.8 5.0 - .0531 - .8453 oon MOl on .577 1 J 6079 .2419 -1.098 - .963 6.2 - .6208 7354 .165 414 i i - 7697 - .0152 - Ill -1.400 54 - .3861 xin:t 198 -Ml I r, 1 X - .8120 - .8495 - .2561 - .137 - 259 - .990 - .806 66 5.8 - .14.M + .0989 8338 .8301 m 205 on - nm -' o - .9180 - .5202 - .423 - .649 FIG. 32. Flo. 33. 510 PERIODIC ORBITS. in dimensions and periods; and for (7 = 3.3284 there is a very close approach to a periodic orbit, though one was not actually found. The computations indicate that the two series of periodic orbits unite and disappear for some value of C slightly smaller than 3.3284. But there is an orbit so nearly periodic for (7 = 3.3284 that it is included as being very nearly that double C = 3.3284, Approximate Double Periodic Orbit, Period =2X6. 14 = 12.28 (Fig. 34, page 511). 1 X y x' .'/' t X y x' .'/' .3500 - .567 .000 2.2 -1.3027 - .3950 - .497 . 434 .05 - .0284 .3499 - .571 - .003 2.4 -1.3953 - .2911 - .422 .(MM) .10 - .0572 .3496 - .583 - .006 2.6 - .4691 - .1504 - .310 .731 .18 - .0868 .3492 - .603 - .010 2.8 - .5178 + .0051 - .173 .821 .20 - .1176 .3486 - .632 - .016 3.0 - .5375 .1748 - .023 . S(>9 .25 - .1501 .3476 - .670 - .024 3.2 - .5270 .3498 + .128 .876 .30 - .1848 .3461 - .717 - .036 3.4 - .4869 .5226 .270 .846 .4 - .2624 .3405 - .840 - .081 3.6 - .4201 .6862 .395 .785 .5 - .3542 .3281 -1.000 - .180 3.8 - .3307 .8350 . 495 .701 .6 - .4628 .3009 -1.168 - .390 4.0 - .2239 .9655 .569 .603 .7 - .5842 .2444 -1.226 - .768 4.2 -1.1050 .075!) .615 .500 .8 - .6972 .1460 - .979 - 1 . 174 4.4 - .9795 . 165S .631 .400 .9 - .7742 .0206 - .568 -1.273 4.6 - .9517 .2364 .638 .308 1.0 - .8166 - .1004 - .315 -1.126 4.8 - .7254 .2900 .623 . 229 1.1 - .8428 - .2032 - .232 - .931 5.0 - .6033 .3292 .597 .166 1.2 - .8658 - .2871 - .237 - .752 5.2 - .4869 .3575 .567 .120 1.3 - .8915 - .3542 - .280 - .595 5.4 - .3763 .3782 . 540 .089 1.4 - .9223 - .4066 - .335 - .455 5.6 - .2705 .3939 .520 .070 1.5 - .9586 - .4455 - .390 - .326 5.8 - .1675 .4008 .511 .060 1.6 -1.0000 - .4720 - .439 - .204 6.0 - .0651 1.4182 .516 .055 1.8 -1.0954 - .4895 - .507 + .026 6.2 + .0397 1.4288 .536 .051 2.0 -1.1996 - .4627 - .527 .240 periodic orbit at which the two single periodic orbits unite and disappear. It is believed that in all cases the computations covered so wide a range of initial conditions that no periodic orbits of the type in question escaped detection. The results shown in the preceding tables (omitting intermediate steps) were obtained by the computations, the origin of coordinates being at the center of gravity of the finite bodies. 246. Closed Orbits of Ejection for Large Values of n- It was shown in Chapter XV that for small values of n there exist closed orbits of ejection from 1 ju for projections both toward and from 1 /z and that their periods reduce to 2jw (j=l, 2, . . . ) for /x = 0. It was also shown in 234 that these orbits can be continued, in the analytic sense, to any value of n unless two of them disappear by becoming identical and vanishing. In order to confirm this conclusion and to get an idea of the form of these orbits for large values of /*, 63 orbits of ejection were computed. It was also desired to discover orbits which are orbits of ejection from one finite mass and of collision with the other. The computations were all started by means of the series (36) of 228. After the values of x, y, x', and y' had been determined for a few small values of t, the computations were continued by the ordinary processes. M N I IIK>I> I'1.|;|"UH .-,11 In all cases the infinitesimal body \v;is ejected from the finite lunly 1 in the positive or negative r-ilirection. M -i 2 , r- 2.242. doMd Orbit of Ejection H ' i 'i f 1 * t " AIMHI + oo 1 1 M01 - ' - - - lo - - - 0266 - 1241 .' - .Ml '. U 1592 - O.MJ - 549 1 x .--.I - .759 - .703 20 - 0950 - .(Ml 1 .112 - 644 J u 1221 1940 - .979 - .560 2~, _ ( ] - .ll.Vi ! HM - - .0929 - 5884 163 - .367 -|- .Oli'.x l.'i3O -.7 - J 1 34O1 6388 - - ' - .1934 .'HI - _ ,. 6386 - + .139 III HIM - 901 - - H KOO - :. 1964 - - 3 - ' noo - '. 2767 .4070 77.. - MJ - 4203 - .2931 - 220 OOK 8406 4884 070 - 799 34 ivuvi - 0654 - BOO I 1112 - '.M.7I !CM - 7M S284 7910 - 172 1 I'-" > - IJI 7"7 - 9559 - 4804 - 4MS 821 1 II - : - 799 1 u -2 .0209 - .1171 - 162 ooo 1 J - .8878 oil - 42 -20201 + 1938 + 170 ooo -}$, C- 2.840, dosed Orbit of Ejection (Fig. 36). i X V z' y' t i V * i - .5000 + oo 3 - 9546 - 019 - 171 U) - -Mi:. 1 .Ml - .408 3 2 0000 - 9930 - 045 - 210 II - .170 - !0485 1 214 - .506 3 1 3710 - 0390 - .086 - .250 20 - .121.'. - .0758 - 580 3 r, - 0930 - Ml - 290 .25 - .0744 - 1002 074 - 636 3121 - 1546 - 220 - 325 .;n - .0335 - .1391 TOO - .677 4 - 2222 - 312 - .35 .40 + .0029 0800 - .1737 - .2092 ooo - 7i rj - 71.-, 1 J 44 1861 0007 - 292K 8021 - ll'i - - - OMO - .2806 .546 .706 46 - .0284 - 4251 - 666 - 280 1 I .VI - .34J2 IM, 663 I x - 1717 - 4761 - - 211 .7 1918 - 4126 - 604 .-, II .3877 - 5087 - - 100 2823 - .4698 - .540 5.2 - 5232 - 5165 - 969 + 033 9 1870 - .5208 821 - .481 54 - 7234 - 4938 -1 ins 2 4 6 B 2 II J 1 26 J'.Mi| 8403 8929 - .5662 - .6424 - .7028 - .7509 - .7894 - .8208 - .8474 - .8716 - .8959 174 102 on 022 006 ooo (KK) - .001 - .427 - 338 - .269 - .214 - 173 - 143 - .125 - 1 !'. - .126 56 6.0 6 4 7 u 7 2 - .9319 - 1408 - 3417 - 5249 - - - SX34 - 9164 - S976 - 4359 - 3395 - 2033 - - M7J - :>764 - 3130 -1 - .968 - .857 - - .Mil - - + 780 HO 188 ."7 -.' 2.8 3923 - .9228 - .005 - .11.-, FIG. 35. 512 PERIODIC ORBITS. Any orbit which intersects the x-axis perpendicularly is symmetrical with respect to the .r-axis. Hence, it follows that if one of these orbits of ejection intersects the .r-axis perpendicularly, then it is a closed orbit of ejec- tion of the type treated in Chapter XV. Computations were first made for /* = f to discover orbits of the typo characterized by j=l with ejection toward fj, and shown in Fig. 15. It was proved in Chapter XV that such an orbit exists for small values of n and that its period is approximately 2ir. Such an orbit was found for ju = |, but its period was about 8. Another orbit, also of a similar type, was discovered whose period was about 14. One of these orbits is undoubtedly the limit, for decreasing values of C, of the oscillating satellite about the collinear equilibrium point, as Burrau's calculations have indicated. The value of C corresponding to the equilibrium point for M = | is about 3.46, and the values of C for these orbits are 2.24 and 2.84. The question arises regarding the origin of the other orbit of this type. It is probably the limiting form of a periodic orbit about 1 n consisting of a double loop and having a double point on the x-axis. Such orbits were treated by Poincare in Les Methodes Nouvelles de la Mecanique Celeste, Chapter XXXI. The ana- lytic continuation of the former beyond the ejectional form for decreasing values of C is also a periodic orbit with two loops. For greater or smaller values of C the latter will have also the character of an oscillating satellite, but it can not reduce to the equilibrium point because there is only one orbit of this type. The results for ^ ^ are given in the tables of page 511. For M = | similar results were found. Since orbits of this type have not been computed heretofore, the results for the four orbits will be given for enough values of t to exhibit their properties. The corresponding results for /x = y> with ejections from 1 ju = |, are given in the following tables : M=4/5, C= 2.696, Closed Orbit of Ejection (Fig. 37, page 513) I X V x' V 1 X y x' V 1 - .8000 + co 2.0 .5307 -1.0537 - .130 - .605 .10 - .6021 - .0198 i.252 - .324 2.2 .4863 -1.1762 - .314 - .616 .15 - .5447 - .0383 1.062 - .413 2.4 .4051 -1.2978 - .497 - ..)< .20 - .4947 - .0609 .945 - .489 2.6 .2892 -1.4112 - .672 - .:,:!.-, .25 - .4495 - .0870 .870 - .554 2.8 .1384 -1.5094 - .834 - .440 .30 - .4073 - .1162 .822 - .610 3.0 - .0429 -1.5851 - .975 - .311 .35 - .3669 - .1478 .796 - .657 3.2 - .2498 -1.6316 -1.090 - .150 .40 - .3273 - .1816 .788 - .694 3.4 - .4766 -1.6430 -1.172 + .040 .5 - .2477 - .2535 .811 - .735 3.6 - .7158 -1.6142 -1.214 .250 .6 - .1642 - .3270 .861 - .725 3.8 - .9591 -1.5419 -1.212 .474 .7 - .0758 - .3969 .904 - .668 4.0 -1.1972 -1.4244 -1.162 .702 .8 + .0154 - .4596 .912 - .585 4.2 -1.4205 -1.2617 -1.063 .924 .9 .1052 - .5140 .878 - .506 4.4 -1.6189 -1.0561 - .915 .128 1.0 .1899 - .5615 .810 - .449 4.6 -1.7829 - .8125 - .720 .303 1.2 .3343 - .6464 .628 - .415 4.8 -1.9041 - .5384 - .487 .433 1.4 .4404 - .7321 .434 - .450 5.0 -1.9758 - .2429 - .226 .513 1.6 .5081 - .8282 .244 - .511 5.2 -1.9936 + .0627 + .049 .533 1.8 .5382 - .9361 .056 - .567 VI 1IK-1- (>K I'KUIODK MKIIII-. 513 ,-4/5, r-2965. Clonl KHtivc direction from 1 p. One periodic orbit of the type characterized by ./I, Fiji. 15, was discovered, and its coordinates are pven in the following table: _!.,, C- 1.8224, Clowcl Orbit of Kjrrti.m (Fig 39). < X 1 r- 1 i 9 z' < - Muni 00 .3 -1 0232 4027 902 .10 - 7 0873 -1.736 IIS I - .9251 .5296 050 \ 1 '>- . K. - .KMC, .0580 -1.457 .-.77 5 - .Klls .8337 208 1 IX, J - '.U '.K! "s 17 -1.264 J - .6843 7344 Ml - .97*4 1218 -1.111 7-1 .7 - 5439 Mfl Bl M - 0306 - .979 -7H - - .3919 ttl n .'K - .0765 - .Mil .947 2.0 - .0595 711 '4 - lir.s - 7.M Ol'i 'I + .2993 2 03X7 - .1816 - .547 189 24 MM J 0011 -1- - 'j. - 2265 1827 991 26 1 0331 1 ss7'l 77- - 7 - .ivji - !l64 Ml J - 1 .!717 1.7009 i in - 2593 :2 :i it 1 6777 1 UM :{93 -1 436 9 - 2478 907 3 2 1.9271 1 1303 -1 7(M 10 1 1 1 i' - .2180 - 17(Ct 1 OOWi 1 1 1 2780 7:!7 Ml MB 295 : 4 36 3.8 2 217tt 2 2445 7..S-, 3755 - 0315 :Ws .070 -1 901 -2 01.-, -2 039 9 Am T-A.i I 514 PERIODIC ORBITS. If n were zero and the infinitesimal body were ejected from 1 M either toward or from ^ in such a way that its period would be TT, the orbits described in rotating axes would consist of two parts symmetrical with respect to the x-axis, as shown in Figs. 40, a, and 41, a. These curves are the limits sepa- rating two types of periodic orbits (for M = 0) in the rotating plane, as is shown in Figs. 40, b, and 41, c. As n increases a dissymmetry develops with respect to the line through 1 M perpendicular to the x-axis. Suppose the orbits are followed as yu increases in such a way that they shall remain orbits of ejection in one way or the other. Then orbits of the type Fig. 40, a, will go into types having some of the characteristics of both types a and c of Fig. 41. That they partake of the characteristics of type c instead of those of type b was proved by computations for both n = % and n= i. The two following tables give the results for ejection from 1n toward fj., with n having the values \ and -|- The first orbit lacks a little of being closed, but an exactly closed orbit exists between this one and the one which was computed for (7 = 3.478. M = Ji = 3.489, Closed Orbit of Ejection (Fig. 42). t X V x' y' t X y x' .'/' -.5000 + CO .7 -.0225 -.2933 - .038 -.229 .1 -.2559 - .0244 1.353 -.380 .8 -.0290 -.3090 - .088 -.089 .15 -.1972 - .0455 1.021 -.456 .9 -.0393 -.31 OS - .114 .054 .20 -.1521 -.0697 .793 -.506 .0 -.0511 -.2980 - . 11<> .203 .25 -.1169 -.0958 .621 -.533 .1 -.0625 -.2703 - .107 .353 .30 -.0894 - . 1227 .486 -.543 .2 -.0721 -.2276 - .083 .499 .35 -.0680 - . 1498 .375 -.537 .3 -.0790 - . 1708 - .055 .634 .40 - .0516 - . 1762 .283 -.518 .4 - .0833 - . 1018 - .033 .741 .5 - .0308 -.2248 .141 -.448 .5 -.0860 - .0244 - .022 .796 .6 - .0222 -.2647 .037 -.346 1.55 - .0871 + .0154 - .022 .794 Six orbits of a similar type were computed for M = |- The following approximately closed orbit was obtained: -YMIIl -I- I'KIM ....... .1:1111-. 515 M-4/5, (' -3.5927, OOM-.I ori.it ,.: . ). ( z V -r- >/ i 1 / ' ii -.SHOO + 00 ii 7 - - i - 036 - 097 1 - 0177 '..:,7 s - 477'. - 1"'.7 - 056 + fllo u - _ n 7(0 - 1 .t - 4X34 - - 000 ISO Jii 5611 0497 _ . 1 II - 4892 - 17H. BO -.0077 in 1 - 4937 - 1433 - 03X :i - .MIMI -.368 .' -.4977 -.1051 - 094 - i - nut - - 4987 - 0.188 - 010 i - ; - r.Ms 1..7 - :u.i 1 -..VXK - m:t - 191 .."> 1711 - i .-,.-. 069 m - ! -1- - -'!> - rr. - I7i,7 mil -.195 Computation.- wen- made for /*= 1 in which the ejections were from 1 n in tin- direction o|)|>osite to p. One dosed orbit corresponding to j = | was t'ound, the results for which are contained in the following table. Corre- sponding computations were not made ,1-1. .1 .'7;. n.iHM>itof Kjivii,,n iKin II t X V z' ' t z * z- v' - .SOOO oo 4 - .7711 Ml .566 .162 -1 - 7 -1.509 .415 5 - .7IM u - - 0490 -1 -!!.-. :>i:< 1 - .6644 .8156 :,is - .067 .11 - .8800 .0766 - .996 592 7 - 6151 HMfl 465 - .163 M - .9254 1117s - .823 6M 1 - 5719 M - .383 - .9628 Ills - 1,77 71 W '.1 - 5367 307 - .367 - .9934 1799 - .549 7111 20 - .Mil 71J1 - III - Ills!) 21M - .434 767 .' 1 - .4987 am m - .633 .5 - 0.-.K) J'UH - .795 22 - .4951 MM - 052 - .615 .6 - .tNVtf _ II.-.N .793 2 :t - .5076 - an - .7 - .(Hi3Ti 1517 + out 766 J 4 - .5356 MM - :ii - SM 8 - (H7i. 233 717 - - - .9 - (ll'.'v :tti 651 26 - 6374 MM - 619 - JII 1 (i - .9821 .419 .-,7(1 27 - 6944 1214 - u:. - 1 1 - .9366 7081 .487 .478 J 72.'. - 7 intersects the y-axis perpendicularly is symmetrical with respect to the t/-axis. Hence it follows that if, for n = %, an orbit of ejection from 1 M intersects the 7/-axis perpendicularly, then it has a symmetrical col- lision with the second finite mass. After 14 computations had been made, an orbit of ejection from 1 /x and collision with /z (ju = |) was discovered in which the ejection was toward M and in which the collision took place without the infinitesimal having encircled, in the rotating plane, either 1 n or n. The following table gives the results at a considerable number of intervals from the time of ejection of the infinitesimal body from 1 n until it crossed the line x = : it= l A, = 3.4174, Orbit of Ejection and Collision (Fig. 45, page 517). t X y x' y' t x y x' y' -.5000 + 00 . 55 -.0082 -.2549 .138 -.440 .10 -.2540 -.0240 1.371 -.383 .(JO -.000.-) -.2751 .094 -.391 .15 - . 1949 -.0459 1.042 -.402 .65 + .0012 -.2933 .058 -.338 .20 -.1487 -.0704 .818 -.514 .70 .0033 -.3088 .028 -.281 .25 -.1122 -.0970 .650 -.545 .75 .0040 -.3214 .004 -.222 .30 -.0831 - . 1247 .518 -.559 .80 .0038 -.3310 - .014 -.161 .35 -.0600 - . 1527 .412 -.557 .85 .0028 -.3375 - .026 -.099 .40 -.0417 - . 1803 .324 -.542 .90 .0013 -.3409 - .032 -.037 .45 -.0274 -.2068 .251 -.516 .95 -.0003 -.3411 - .032 + .027 .50 -.0104 -.2318 .190 -.482 Another somewhat similar, but larger, orbit of ejection from 1 /x and collision with /x was found, after a number of computations, in which the ejection was toward n. Part of the results of the computation are given in the following table : M = K, C = 2.739, Orbit of Ejection and Collision. I x y x' y' t x y x' y' -.5000 + 1.4 .3941 - .7683 .032 -.379 .10 -.2428 - .0258 1.537 -.412 1.5 .3947 - .8051 - .018 -.357 .15 - . 1739 - .0490 1.242 -.514 1.6 .3906 - .8398 - .063 -.330 .20 -.1109 - .0707 1.048 -.592 1.7 .3822 - .8723 - .105 -.315 .25 -.0682 - .1079 .910 -.652 1.8 .3698 - .9028 - .142 -.295 .30 - .0254 - .1417 .808 -.096 1.9 .3539 - .9313 - .176 -.274 .35 + .0131 - .1773 .733 -.720 2.0 .3348 - .9577 - .206 -.253 .40 .0483 - .2140 .077 -.742 2.1 .3129 - .9819 - .233 -.232 .45 .0810 - .2512 .836 -.745 2.2 .2884 -1.0041 - .257 -.211 .50 .1119 - .2883 .000 -.737 2.3 .2616 -1.0241 - .278 -.189 .68 .1412 - .3248 .570 -.720 2.4 .2328 -1.0419 - .296 -.107 .6 . 1690 - .3602 .542 -.697 2.5 .2024 -1.0575 - .312 -.145 .7 .2202 - .4273 .482 -.042 2.6 .1705 -1.0709 - .326 -.122 .8 .2651 - .4880 .410 -.580 2.7 .1373 -1.0820 - .337 -.100 .9 .3033 - .5440 .347 -.530 2.8 .1031 -1.0908 - .346 -.077 1.0 .3345 - .5960 .277 -.493 2.9 .0682 -1.0973 - .353 - .053 1.1 .3588 - .0435 .210 -.458 3.0 .0326 -1.1014 - .357 -.030 1.2 .3706 - .6878 .146 -.428 3.1 -.0032 -1.1032 - .360 -.006 1.3 .3882 - .7293 .087 -.402 M \ I IU.-I- 'l Kl ...... iillHIIS. 517 This orbit has a luu|. about the equilateral triangular point It follou.- that then- an- twu families of |>eriodic orhit> <,f the type.- >ho\\n in I IM |i; Ten urhits were computed in an attempt to find one of tlu-in. The diffi- ciillics of making the calculations when the infinitesimal body \V:L- near one of the finite bodies were >o K reat that wide depart urcs from the orl.n ejection had to he attempted. Indications of such jM-riodic orbits \ obtained, but none \\a> actually found. Via. 45. Kin. 46. 248. Proof of the Existence of an Infinite Number of Closed Orbits of Ejection and of Orbits of Ejection and Collision when n = \. It was proved in Chapter XV that when n is sufficiently small there are infinitely many closed orbits of ejection, and reasons were given for believing that the-. orbits persist for all values of /u- The question of the existence of orbit > of ejection and collision was not considered. Fio. 47. Fio. 48. It will now be shown that there are infinitely many closed orbits of ejec- tion, and of ejection and collision, for M = i- The differential equations of motion in fixed rectangular axes with the origin at the center of gravity < f the system are r Vl), ,,s dP r,' r, dP where, if the finite bodies are on the r-axis at in*. (69) 518 PERIODIC ORBITS. Now let x = r cos 6, and y r sin 6, after which equations (68) become tfr /de\~ a ri 11 ifi 11 jJI-Mjrl 3|-lH il r 2 ~i -- i COS (0 Oi d- \d(). (72) always positive, it follows from the first of (70) that dV - (r+il &> (r-i)*' The integral of this inequality gives dr Suppose K>0 and that ^>0 at 1 = T. Then F(r) will always exceed A' in value and r will become infinite with t. Computations were made in which the infinitesimal body was ejected from 1 M both toward and from /j. with initial conditions corresponding to dr K>0, and they were both followed until r>2 with ^ positive. Hence in both cases the infinitesimal body would recede to infinity. Moreover, it follows from the second equation of (70) that, when referred to rotating axes, the infinitesimal body revolves infinitely many times about the finite bodies, and its distance from the origin continually increases. Now consider, for example, a closed orbit of ejection from 1 M, for M = 5, in which the infinitesimal body makes at least one circuit about the finite body ju. Its orbit therefore crosses the z-axis perpendicularly exactly once, but it does not cross the ?/-axis perpendicularly. Moreover, it follows from the symmetry of the orbit with respect to the x-axis, that if it crosses the i/-axis in the first half of the orbit at an angle T/2 + a, then in the second half it crosses the i/-axis at the angle ir/2 a. One or the other of these angles is less than ir/2. Suppose it is the latter. Now suppose the initial conditions of ejection are changed so as to increase K. The orbit will cease to be a closed orbit of ejection and will tend toward one which winds out to infinity with continually increasing r. The angles at which the orbit crosses the axes are continuous functions of the parameter defining the initial conditions, as K for example. Hence the intersection with the ^/-axis which was at the M viiiK-i- i.i i-i iMdim ORB1 .")!'. angle T 2 o and less than TT 2 for tin- c|o>ed orbit of ejection will \H- exactly perpendicular fur a certain value of A'. Tin- <>rl>it \\ill be therefore an orbit of ejection from 1 ^ and collision with /j. Now consider an orbit of ejection from 1 -M and collision with p. cross- ing the .r-axis at least once. From t he s\ ninietn of these orbits with reaped to the //-axis it follows that they cm the X-tOU an even number of time* and that if such an i rbit crosses the .r-a\i- once at an angle T '2+J, win positive quantity, then it also crosses it at an angle of T 2 ^. If the initial conditions are so changed as to increase the constant K, the orbit ceases to be an orbit of ejection and collision, the angle corresjxmding to TT '2 ,i eventually becomes greater than r 2, and therefore. Miice it i- ;i continuous function of A', there is at lea>t one value of A' for which it i- exactly TT I*. Such an orbit is a closed orbit of ejection. It follows from this discussion that if, for any value of A", there is a closed orbit of ejection, then for some larger value of A", oomspoodmg to a smaller value of the Jacohian constant (', there is an orbit of ejection and collision; and that if. for any value of A", there is an orbit of ejection and collision, then for some larger value of A. corresponding to some smaller value of the Jacobian constant (', there is a closed orbit of ejection. Hence, for 1 M* J, there are infinitely many closed orbits of ejection and collision. They are all distinct because they have distinct values of A'. And since it has i shown that, when K>0 for an orbit of ejection, r increases continuously to infinity, it follows that the infinite sets of values of A' corres|xmding to these classes of orbits are bounded. The orbits may be characterized by the num- ber of times they cross the //-axis. For ejections in both the positive and the negative direction there are closed orbits of ejection from each of the finite bodies, and also orbits of ejection from one and collision with the other, which eross the //-axis 2(2/-f 1) times, j = Q, 1, 2, ... 249. On the Evolution of Periodic Orbits about Equilibrium Points. The evolution of the jwriodic orbits about the equilibrium points un and (c) which are on the .r-axi> and not between 1 /* and n was traced, for decreasing values of C. by Hurrau's computations from small ovals to the ejectional form. For M = $ they are shown in Fig. 'M. and for /* = Fig. 37. Heyond these forms they have loops about 1 -p. The periodic orbits about the equilibrium ]>oint . It intersects the //-axis si\ times, twice ]>er|)cndicularly. Heyond this form it has loops about the finite bodies, and the motion in these loops is in the retrograde direction. With decreasing values of C these loops probably enlarge, the loop about each body eventually In-com- ing an orbit of collision with the other IMM!V. In this MM th- orbit of 520 PERIODIC ORBITS. ejection and collision intersects the t/-axis six times, two of the intersec- tions being perpendicular. As C decreases, the ejectional and colliskmal form passes into a loop about the second body, which in turn expands and becomes an ejectional and collisional form with respect to the first body. In this manner the loops of the periodic orbit, with decreasing values of d, pass through ejec- tional and collisional forms, first with one finite body and then with the other, in a never-ending series, the ejectional and collisional forms being those shown to exist, for M = i> i n 248, in which the ejections from each body are in the direction away from the other. They are characterized by the fact that they cross the ?/-axis 2(2/+l) times, j = Q, 1, 2, . . . If the finite masses are unequal the evolution of the periodic orbits is in a general way similar, except that the ejectional forms for the two masses do not occur for the same values of C. Periodic orbits about the equilateral triangle equilibrium points have been shown in Figs. 30 and 34. With decreasing values of C they probably increase in size and pass through ejectional forms, but ejectional forms in which the direction of ejection is not along the .r-axis. Consequently they can not be discovered by numerical processes. If this conjecture is cor- rect, for still smaller values of C they possess loops about the finite bodies, and for still smaller values of C the loop about each of the finite bodies may pass through an ejectional form with the other. There is, however, no evidence to guide conjectures. 250. On the Evolution of Direct Periodic Satellite Orbits. There are three direct periodic satellite orbits, two of which are complex for large values of the Jacobian constant, but all of which are real for smaller values of C. Suppose the orbits about the finite body n are under consideration. With decreasing value of C they can pass through ejectional forms. In fact, Darwin's computations showed that two of them were approaching such forms, one by approaching p. from the positive direction and the other by approaching it from the negative direction. The motion of the infinitesimal body when it is near collision is nearly the same as it would be if the mass of the second body were zero. Conse- quently its properties can be inferred from a consideration of the motion in the neighborhood of an ejection in the problem of two bodies referred to rotating axes. It is clear that if the ejection is in the positive direction, the curve near the point of ejection lies on the negative side of the x-axis, while if the ejection is in the negative direction the curve near the point of ejection lies on the positive side of the x-axis. When the ejection is in the positive direction, the two families of periodic orbits which are near the ejectional orbit both intersect the z-axis in the negative direction from the point of ejection, and the small complete loop about the point of ejection is then described in the retrograde direction, while the partial loop is described in the MMIH-.sIS oK l-KKIolMC oldUl-v .YJ 1 positive direction; while if tin- ejection is in the negatm- direction, the t\\.. families of periodic orltits which arc near the ejections! orbit both inleraact ilie .r-axis in the positive direction from the point of ejection, l.nt in this case the .small loops arc hoth described in the same direction- a- in the other case. The .r-a\i- roiisi>t- of three part-. \ i/. that extending from - -c to the 1-osiiion of 1 IJ . that extending from 1 -^ to n, inul that extending from n to -foo . If a periodic oriiit inter-ect- the j--axis |x-r|M-ndicularly in anyone of the-e three parts before it goes through an eject imial form. I hen it will also intersect the .r-a\is pcr|>eiidicul;irly in the same part after it p:isM8 through the ejectional form. Moreover, the branches of a closed orbit of ejection extend from the finite body with which there is collision in tin- direct ion opposite to that in which the neighboring jx-riodic orbit intersects the .c-axis perpendicularly. In the case of the direct periodic orbit about 1 n which enlarge- in t he po-itive direction and approaches 1 ^ from the negative direction, the ejection is in the positive direction, the collision is in the negative direction, and the orbit has the form shown in Fig. 42. The computation shows that it has two loops and. therefore, that it had two cusps symmetrical with respect to the x-axis before it arrived at the eject ional form. After this orbit passes beyond the ejectional form, with decreasing values of (', it acquires a loop about I n, which intersects the x-axis |>eri>cndicularly in the negative direction from ft and which has a double point on the x-axis between 1 M and n.. Consider the further evolution of the |>eriodic orbit. If the small loop al H nit 1 n should again pass to the ejectional form, the ejectional orbit would be exactly of the tyix- of that from which the loop developed. It is improba- ble that such an additional ejectional orbit exists for another value of ('. Now consider the possibility of that part of the orbit which crosses the x-axis perpendicularly in the positive direction between 1 p and n passing through an eject ional form. It can not pass to an ejectional form with n because, in accordance with the general conclusions respecting the motion near a ix>int of ejection, the branches of the curve near n would lie in the positive direction from it. and the partial loop about n just Ix-fore tin- eject ional form was reached would be described in the retrograde direction. Hut this branch of the curve could evolve to an ejectional form with 1 n, when the orbit would have the form shown in Fig. 47. With decreasing values of C this orbit acquires an additional loop about 1 M, which is described in the retrograde direction and which intersects the x-axis JHT- l>endicularly in the negative direction between 1 -M and n. This loop can expand and take an ejectional form with n, then acquire a loop about M. which can become an ejectional form with 1 - M, and so on, being an ejecti< >n.d form first with one of the finite masses and then with the other in a never- ending sequence. The ejections from 1 -// are all in th< % negative direction. 522 PERIODIC ORBITS. and from M they are all in the positive direction. It is probable, though not certain, that this is qualitatively the course of evolution of the direct satellite orbit from which the start was made. Now consider the direct satellite orbit about 1 M which enlarges in the negative direction and which approaches the ejectional form from the positive direction. The ejectional form was found by computation and is shown in Fig. 44. With decreasing C this orbit acquires a loop about 1 M, which may pass to the ejectional form with M, as shown in Fig. 48. The other branch which crosses the z-axis perpendicularly may pass to the ejectional form with 1 M. With decreasing values of C this orbit acquires a small loop about 1 n which never again passes through the ejectional form. But the loop about M enlarges and becomes an ejectional form with 1 n with the ejection in the negative direction. Then follows a loop which becomes an ejectional form with M> followed by a loop about M which be- comes an ejectional form with 1 M, and so on, first with one finite body and then with the other in a never-ending sequence. The ejections from 1 /x are in the negative direction, and from n they are in the positive direction. There is a third direct satellite orbit whose evolution has not been traced. Only a conjecture can be made in regard to it, and that conjecture is that it acquires cusps and then loops, probably about the region of the equilateral triangular points. 251. On the Evolution of Retrograde Periodic Satellite Orbits. Con- sider the retrograde periodic satellites about 1 M. There are three such orbits, only one of which is real for large values of C. The numerical experi- ments which were made, 238, indicate that only one of them is real for any value of C. For large values of C the retrograde periodic orbit about 1 M is small and nearly circular in form. As C diminishes the orbit increases in size and departs widely from a circle. Consider the question of its passing through an ejectional form with 1 /*. If the periodic orbit should approach the ejectional form by shrinking upon 1 /z from the positive or the negative direction, just before arriving at the ejectional form it would make a partial loop about 1 M in the retrograde direction, and just after passage through the ejectional form it would make a complete loop about 1 /* in the positive direction. But it was seen in 250, in connection with a consideration of an ejectional orbit in the problem of two bodies referred to rotating axes, that this is impossible. Hence the retrograde satellite orbit about 1 n can not become an ejectional orbit with 1 n, at least until after it has passed through an ejectional form with n- Now consider the possibility of the retrograde satellite orbit about 1 n passing through a collisional form with M- Since it intersects the z-axis between 1 /* and M such an orbit must be one in which the collision is in the negative direction, in which the ejection is in the positive direction, in which t \-i -.1:1 <>KHII- the partial loop just before collision is descril>ed in the |>iiti\c direction, and in which the complete loop just :ifter collision is described in the retrograde direction. Tin- i- precisely the way in which such :i limiting form can be p:i^>ed. and the periodic orbit pa-se- through this form. An orbit of tin- type, with the roles of 1 - M and ^ interchanged, was computed and i- -ho\\ n in I ig. :>'. After the retrograde |M-riodic orltit about 1 n passes through an ejec- tional form \\ith ^. i' acquire- a retrograde loop about n which cr..e< the ./-axis in the positive direction between 1 n and p. This loop enlarges and passes through an ejectional form with 1 M, after which it acquires u retro- urade loop about 1 - p, which, in turn, enlarges and passes through an ejcc- tional form with M- This process continues, the form becoming ejectional nr-t with one finite mass and then with the other in a never-ending sequence In all of these orbits the parts near the ejection points are on the negative side of 1 n or the {x>sitivc side of n, and never between 1 j* and M- ' I"' orbit- of these series \\hich are closed orbits of ejection with 1 M are a part of those which \\ere shown to exist in Chapter XV for sufficiently small values f M: ;"'d those which are closed orbits of ejection with M are the corresjwnding orbits for the other finite mass. ( 'oiisider first the orbits of the type under consideration which are orbits of collision and ejection with 1 M- All these orbits are orbits of ejection in the negative direction; they have double points on the .r-uxis in the posi- tive direction from n, and intersect the x-axis ixT|>endicuIarly only in tin- negative direction from I n. They are therefore only those orbits of 226 which are characterized by ejection in the negative direction and by even values of j; those characterized by odd values of j have a different origin. The orbits of the type under consideration which are orbits of ejection and collision with n also intersect the x-axis perpendicularly only on the negative side of 1 M- On interchanging the rdles of 1 M and n in 226, orbits of ejection from n in the |X)sitive direction were proved to exist for 1 -M sufficiently small. Those which are characterized by odd values of j intersect the x-axis |x-rpendicularly on the negative side of I-M. They are of the type of the part of those under consideration which are (.rl.il- of collision and ejection with p. To summarize: The retrograde |H-riodic satellite orbits about 1-ji, with decreasing values of (\ go through an infinite series of ejectional forms with 1 M. the ejections all being in the negative direction, and these orbits are those of the orbits treated in 22(3 which are ejected from I-/* in the negative direction and which are characterized by even values of j. The retrograde |x>riodic satellite orbits also go through an infinite series of ejec- tional forms with n. the ejections all being in the positive direction, and these orbits are those which can be shown to exist by the methods of 226, and which are characterized by ejection in the positive direction from n and l>\ odd values of j. There are similar retrograde periodic satellite orbits alnmt 524 PERIODIC ORBITS. /z, and they go through a similar series of critical ejectional forms with both 1 ju and M- The ejections from 1 M and M are also respectively in the nega- tive and positive directions, but those which are ejectional forms with 1 M are characterized by odd values of j, while those which are ejectional forms with // are characterized by even values of j. Therefore, the retrograde periodic satellites about the two finite bodies together pass through all the ejectional forms from both finite bodies, of the type treated in 226, in which the ejection from one body is in the opposite direction from the other. 252. On the Evolution of Periodic Orbits of Superior Planets. It was shown in Chapter XII that for large values of C there are two periodic orbits in which the infinitesimal body makes simple circuits about both of the finite bodies in the retrograde direction. When the system is referred to fixed axes, one of the orbits is direct and one is retrograde. There are also four orbits in which the coordinates are complex, two of them being direct when referred to fixed axes and two being retrograde. It is not known whether or not either pair of the complex orbits becomes real with decreasing values of C. Since none of these orbits was computed, very little is positively known about their geometrical characteristics or about their evolution. It was shown in 248 that there are two infinite sets of orbits which are orbits of ejection from one finite body and of collision with the other. One set is characterized by the fact that the ejection from each finite body is in the direction away from the other. Reasons were given in 249 for believing that they are limiting forms of the analytic continuations of the oscillating satellites about the equilibrium point b. The other set is characterized by the fact that the ejection from each finite body is toward the other finite body. These orbits are probably limiting forms of the analytic continua- tions of retrograde periodic planetary orbits. The probable series of changes to the first limiting form is shown qualitatively in Figs. 49 and 50. Beyond the limiting form the orbits acquire loops about each of the finite bodies. There are two retrograde periodic orbits of the types of superior planets. Probably they both undergo evolutions to limiting forms of the types de- scribed. This conjecture is supported by the fact that two closed orbits of ejection were found by computation, 247, for a related type of orbits. Fio. FIG. 50. UNIVERSITY OF CALIFORNIA LIBRARY *Y VeoK BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 2 1961 MAR 8 - I960 LD 21-100m-7,'52(A2528sl6)476 PCX for.. THE UNIVERSITY OF CALIFORNIA LIBRARY