NASA TT F-452 =1 H67~ Z7279 (ACCESSION NUMBER) 4^^ (NASA CR OH TMX OR AD NUMBER) 11/ (CODE) GPO PRICE $ CFSTI PRICE(S) $ Hard copy (HC) Microfiche IMF). fres3 Juiyes ^^^^^ 8 H ^ (CATEOORV] NASA TT F-452 NEW METHODS OF CELESTIAL MECHANICS VOLUME m. INTEGRAL INVARIANTS, PERIODIC SOLUTIONS OF THE SECOND TYPE, DOUBLY ASYMPTOTIC SOLUTIONS By H. Poincare Translation of "Les Methodes Nouvelles de la Mecanlque Celeste. Tome III. Invariants integraux. - Solutions pferiodiques du deuxigme genre. Solutions doublement asymptotiques." Dover Publications, New York, 1957 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Cleoringhouso for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 c PRECEDING PAGE BLANK NOT FILMED. TABLE OF CONTENTS VOLUME III CHAPTER XXII INTEGRAL INVARIANTS Page Steady Motion of a Fluid Definition of Integral Invariants Relationships Between the Invariants and the Integrals ' Relative Invariants • Relationship Between the Invariants and the Variational Equation ii Transformation of the Invariants • Other Relationships Between the Invariants and the Integrals ^1 Change in Variables „ , General Remarks CHAPTER XXIII FORMATION OF INVARIANTS 43 45 Use of the Last Multiplier Equations of Dynamics Integral Invariants and the Characteristic Exponents JU Use of Kepler Variables Remarks on the Invariant Given in No . 256 Case of the Reduced Problem CHAPTER XXIV USE OF INTEGRAL INVARIANTS 65 69 71 Test Procedures Relationship to a Jacob! Theorem . . . Application to the Two-Body Problem Application to Asymptotic Solutions 73 81 83 88 CHAPTER XXV INTEGRAL INVARIANTS AND ASYMPTOTIC SOLUTIONS Return to the Method of Bohlin Relationship with Integral Invariants 91 115 iii Page Another Discussion Method 120 Quadratic Invariants 130 Case of the Restricted Problem 134 CHAPTER XXVI POISSON STABILITY Different Definitions of Stability 142 Motion of a Liquid 143 Probabilitites 153 Extension of the Preceding Results 156 Application to the Restricted Problem 159 Application to the Three-Body Problem 167 CHAPTER XXVII THEORY OF CONSEQUENTS Theory of Consequents 176 Invariant Curves 179 Extension of the Preceding Results 188 Application to Equations of Dynamics 190 Application to the Restricted Problem 197 CHAPTER XXVIII PERIODIC SOLUTIONS OF THE SECOND TYPE Periodic Solutions of the Second Type 203 Case in Which Time Does Not Enter Explicitly 209 Application to the Equations of Dynamics 215 Solutions of the Second Type for Equations of Dynamics 228 Theorems Considering the Maxima 232 Existence of Solutions of the Second Type 241 Remarks , 245 Special Cases 246 CHAPTER XXIX DIFFERENT FORMS OF THE PRINCIPLE OF LEAST ACTION Different Forms of the Principle of Least Action , 250 Kinetic Focus 262 iv Page Maupertuis Focus 268 Application to Periodic Solutions 270 Case of Stable Solutions 271 Unstable Solutions 273 CHAPTER XXX FORMULATION OF SOLUTIONS OF THE SECOND TYPE Formulation of Solutions of the Second Type 294 Effective Formulation of the Solutions 296 Discussion 310 Discussion of Particular Cases 320 Application to Equations of No. 13 322 CHAPTER XXXI PROPERTIES OF SOLUTIONS OF THE SECOND TYPE Solutions of the Second Type and the Principle of Least Action 329 Stability and Instability 340 Application to the Orbits of Darwin 348 CHAPTER XXXII PERIODIC SOLUTIONS OF THE SECOND TYPE Periodic Solutions of the Second Type 357 CHAPTER XXXIII DOUBLY ASYMPTOTIC SOLUTIONS Different Methods of Geometric Representation 366 Homoclinous Solutions 377 Heteroclinous Solutions 383 Comparison with No. 225 386 Examples of Heteroclinous Solutions 388 FOLD LINE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D.C. 20546 POSTAGE AND FEES PAID NATIONAL AERONAUTICS & SPACE ADMINISTRATION OFFICIAL BUSINESS NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CODE USS-T WASHINGTON, D.C. 20546 NASA TTF No. 452 FOLD LINE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION TECHNICAL TRANSLATION EVALUATION TO: THE USERS OF THIS TRANSLATION . Budeel Bureau No. 104-R037 Approvml Expires: Sept. 30, 1969 NASA TTF NO. 452 w' MAINTAINING THE QUALITY OF NASA TRANSLATIONS REQUIRES A CONTINUOUS EVALUATION PROGRAM. PLEASE COMPLETE AND MAIL THIS FORM TO AID IN THE EVALUATION OF THE USEFULNESS AND QUALITY OF THE TRANSLATING SERVICE. THIS PUBLICATION (Check one or more) Q FURNISHED VALUABLE NEW DATA OR A NEW APPROACH TO RESEARCH. r"] VERIFIED INFORMATION AVAILABLE FROM OTHER SOURCES. I I FURNISHED INTERESTING BACKGROUND INFORMATION. I I OTHER (Exptain): — FOLD LINE fOLD L_1NE O z o u TRANSLATION TEXT (Check one; f" 1 IS TECHNICALLY ACCURATE. i I IS SUFFICIENTLY ACCURATE FOR OUR PURPOSE. CJ IS SATISFACTORY, BUT CONTAINS MINOR ERRORS. (31! IS UNSATISFACTORY BECAUSE OF (Check one or more): \ZJ POOR TERMINOLOGY. O NUMERICAL INACCURACIES. I I INCOMPLETE TRANSLATION. [7! ILLEGIBLE SYMBOLS. TABULATIONS, OR CURVES. n OTHER (Explain): FOLD LINE FOLD LINE REMARKS 3 o z o -I •< I- 3 U FROM NOTE: REMOVE THIS SHEET FROM THE PUBLICATION, FOLD AS INDICATED, STAPLE OR TAPE, ANDMAIL. NO POSTAGE NECESSARY. NASA Form 1332 AUG 66 GP 9 15-777 NEW METHODS OF CELESTIAL MECHANICS VOLUME III H. POINCARE ABSTRACT Integral invariants are introduced using the steady motion of the fluid as an example. The use- fulness of invariants in celestial mechanics is demon- strated. Various forms of the three-body problem are treated. Poisson stability is defined for the steady motion of a liquid, the general and restricted three- body problem. The theory of "consequents" is intro- duced in the discussion. The existence, stability, and properties of periodic solutions of the second type are treated. These are related to the principle of least action and the Darwin orbits. The concepts of kinetic focuses and Maupertius focuses are introduced in the discussion. Periodic solutions of the second type are treated, Homoclinous and heteroclinous doubly asymptotic solutions are discussed for the three-body problem. CHAPTER XXII INTEGRAL INVARIANTS Steady Motion of a Fluid 233. In order to clarify the origin and importance of the idea /l^ of integral invariants, it is useful to start with a study of a par- ticular example from the field of physics. Let us consider an arbitrary fluid, and let u, v, w be the three velocity components of the molecule which has the coordinates x, y, z at time t. * Numbers given in the margin indicate pagination in the original foreign text. We will consider u, v, w as functions of t, x, y, z, and we will assume that these functions are given. If u, V, w are independent of t and only depend on x, y, z, the motion of the fluid is said to be steady . We will assume that this con- dition is satisfied. The trajectory of an arbitrary molecule of the fluid is therefore a curve which is defined by the differential equation dx _ dy (1) If it were possible to integrate these equations, one would obtain _/2 ^ = ?l(', ^0, ro. «o), such that X, y and z would be expressed as a function of time t and their initial values xg , yg , Zq . If the initial position of a molecule were known, one could deduce the position of this same molecule at time t. Let us consider fluid molecules the group of which forms a certain figure Fq at the initial instant of time; when these molecules are dis- placed, their group will form a new figure which will move while being continuously deformed, and at the time t the group of molecules under consideration will form a new figure F. We will assume that the movement of the fluid is continuous, i.e., u, V, w are continuous functions of x, y, z; there are therefore certain relationships between the figures Fq and F which are obvious from the conditions of continuity. If the figure Fq is a curve or a continuous surface, the figure F will be a curve or a continuous surface. If the figure Fq is a simply connected volume, the figure F will be a simply connected volume. If the figure Fq is a curve or a closed surface, the same will hold true for the figure F. In particular, let us examine the case of liquids where the fluid is incompressible, i.e., where the volume of a mass of fluid is invari- able. Let us assume that the figure Fg is a volume. At time t the mass of fluid which fills out this volume will occupy a different volume which will be nothing else than the figure F. The volume of the mass of fluid did not change; thus, Fg and F have the same volume. Therefore, one can write ///"^■^ "^-^ ''' "///'^^° '^•^° '^•'° ' ^^^ The first integral is extended over the volume F and the other over the volume Fg . We will then say that the integral 11 fjjdxdyd. is an integral in variant . It is known that the condition of incompressibility can be expressed by the equation du dv dw _ dx dy dz ~ ' (3) The two equations (2) and (3) are thus equivalent. Let us again consider the case of a gas, i.e., the case where the volume of a mass of fluid is variable. Thus, the mass becomes invariable, such that if one calls p the density of the gas, one has / / \ dx dy dz =. j I /po dxo dy„ dz,. (4) The first integral is extended over the volume F, the second over the volume Fg. In other words, the integral / / I p dx dy dz is an integral invariant. In this case, where the motion is steady, the equation of continuity can be written as d(pu) d(pi>) , d(pw) ,c^ — J r ■ — :; h —3 =0. \J) dx dy dz The conditions (4) and (5) remain equivalent. 234. The theory of vortices of Helmholtz provides us with a second example. Let us assume that the figure Fq is a closed curve. The same will hold true for the figure F. Let us assume that the fluid, whether it is compressible or not, is at a constant temperature, and is only subjected to forces which have a potential. In order that the motion remains steady, it is necessary that u, v, w satisfy certain conditions. It is not useful to develop these conditions here. Let us assume that they are satisfied. /4 Under this assumption, let us consider the integral I (u dr -h V dj' -h w dz). As the theorem of Helmholtz shows, it has the same value along the curve F and along the curve Fq . In other words, this integral is an integral invariant. Definition of Integral Invariants 235. Due to the nature of the question, the examples which I have just presented readily lead one to the consideration of integral invari- ants. It is clear that these invariants can be used by generalizing their definition for cases which are much broader, in which it is not possible to give a simple physical meaning to the invariants. Let us consider differential equations of the form dx (fy dz X = 'Y- = T = '''' (1) X, Y, Z are given functions of x, y, z. If they could be integrated, one would obtain x, y, z as a function of t and their initial values xg , yg , zg. If we assume that the time is represented by t and x, y, z repre- sent the coordinates of a moving point M in space, equations (1) de- fine the laws of motion of this moving point. If these equations are integrated once, one can find the position of the point M at time t, if its initial position Mq, given by the coordinates xq , yo , zq' ^^ known. If one considers moving points which obey the same law and the group of vhich forms a figure F at the initial Instant of time, the group of these same points wiirform a different figure F at time t which will be a line, a surface, or a volume depending on whether the figure Fq is a line, a surface or a volxnne. Let us consider a simple integral -l— f{Adx + Bdy-hCdz), (2) where A, B, C are the known functions of x, y, and z. If Fq is a line, it may happen that this integral (2) extended over all of the elements of the line F is a constant which is independent of time, and xs conse- quently equal to the value of this same integral extended over all of the elements of the line Fg . Let us now assume that F and Fq are surfaces, and let us imagine the double integral f fiX'dfdz+B'dxdz + C'dxdj'), where A', B' , C are functions of x, y, and z. It may happen that this integral has the same value which is extended over all the elements of the surface F, or over all of those of the surface Fq. Let us now assume that F and Fq are volumes, and let us imagine the triple integral JJJndxdydz; ^^^ M is a function of x, y, z. It is possible that it may have the same value for F and Fq . In these different cases, we say that the integrals (2), (3) or (4) are integral invariants . It occasionally happens that the simple integral (2) will only have the same value for the lines F and Fg if these two curves are closed, or the double integral (3) will only have the same value for the surfaces F and Fg if these two surfaces are closed. We may thus say that (2) is an integral invariant with respect to the closed curves and that (3) is an integral Invariant with respect to the closed surfaces . 236. The geometric representation which we have employed plays no important role. We can thus lay it aside, and nothing prevents us from extending the preceding definitions to the case in which the number of variables is greater than three. Let us consider the /6 following equations XT" xr=---= x7='''' (1) where Xj, X2, ..., X are the given functions of x^, X2 x . If one could integrate them, one would find x^ , X2 , ..., x^^ as functions of t and of their initial values x^, x^ , ..., x^. In order to retain the same terminology, we may call point M the system of values xj , X2 , ..., x^, and the point Mq the system of values x^, x^, ..., x^. Let us consider a group of points Mg forming a subset Fg and the group of corresponding points M forming another subset F^ ■'. We shall assume that Fg and F are continuous subsets having p dimensions where p <^ n. Let us consider an Integral of the order p ^SArfu), (2) /^ where A is a function of xi, X2, ...» x^j, and where dco is the product of p differentials chosen among the n differentials dxi, dx,, ..., dx„. '^•^ The word subset is now commonly employed, so that I did not feel it was necessary to recall the definition. Every continuous group of points (or system of values) is named this way: In three- dimensional space, an arbitrary surface is a subset having two di- mensions, and an arbitrary line is a subset having one dimension. It is possible to give this integral the same value for the two subsets F and Fq. We may thus say that it is an integral invariant . It may also happen that this integral has the same value for the two subsets F and Fq , but only under the condition that these two sub- sets are closed . It is thus an integral invariant with respect to the closed subsets. Other types of integral invariants may be also assumed. For example, let us assume that p = 1 and that F and Fg may be reduced to lines. It is possible that the integral /7_ A A, <:^j,H- A, rfx, + . . . + A „ rfx„ ) = / S A, dr, has the same value for F and Fq , and is an integral invariant. This may also be the case for the following integral ( \lz B, cfrf -!- 2 s C, <■ dxi dxk, where B and C are like the A of the functions of xi, X2, ..., x^j. As I stated, it is possible that this Integral may have the same value for F and Fg , and other similar examples may be readily envisaged. The quantity p will be called the order of the integral invariant . Relationships Between the Invariants and the Integrals 237. Let us again consider the system rfr, rfr, dr,, , x7= xr=--=xr = '"- (1) If one could integrate it, all of its integral invariants could be formed. If integration were performed, the result could be presented in the following form 7i = Ci, z = t 1- G„, \^ Cj , Cj, ...J C are arbitrary constants, and the y's and z's are the given functions of the x's. Let us change the variables by taking y's and z for the new variables, instead of x's. Let us now consider an arbitrary integral invariant. Under the sign / (which will be repeated p times if the invariant is of the order p) , this invariant must include a certain expression, the function of the J8_ x's and of their differentials dx. After a change in the variables, this expression will become a function of the y's, z, and of their differentials dy and dz. Without changing the y's, in order to pass from one point of the figure Fq to a corresponding point in the figure F, it is necessary to change z into z + t. Therefore, when passing from an infinitely small arc of Fq to the corresponding arc of F, the differentials dy and dz do not change (the quantity t which is added to z is, in effect, the same for the two ends of the arc) . If one considers an infinitely small figure Fq having an arbitrary number of dimensions and the corresponding figure F, the product of a number (equalling that of the dimensions of Fg and F) of differentials dy or dz will not change either when one passes from one figure to the other. In short, in order that an expression may be an integral invariant, it is necessary and sufficient that z is not contained in it ; the y's, the dy's, and dz may be included in an arbitrary manner. Let us consider an expression having the same form as that which we discussed in the preceding section /= A ^03, (3) This expression represents an integral of the order p, A is a function of xi, X2, ..., x^, dco is a product of p differentials chosen from the n differentials rf.r,, dx„ ..., dx^. We would like to know whether this is an integral invariant. By carrying out a change in variables as indicated above, we find that expression (3) becomes ' £ B dm', P B is a function of the y's and of z, dco' is a product of p differen- tials chosen from the n differentials dyu dyt, .--. '0''.-" '^^• In order that expression (3) be an integral invariant, it is necessary and sufficient that all of the B's be independent of z and only depend on the y's. Just as in the preceding section, let us again consider the ex- /9 pression C\Ji B, dxf + 2 s C,-.* dxi dxk, W The B^'s and the C^j^'s are functions of the x's. After the change in the variables, this expression becomes f\'-Z B; dx'i^ -r- ■?. L C-.j. dx'i dx\ ; For greater symmetry in the notation, I have set the following 3:'i=yi\ (J = i, 2, ..., 71 — i); x'n= z, In order that expression (4) be an integral invariant, it is necessary and sufficient that all of the B'^'s and the C ^^y^ s be in- dependent of z, and depend only on y. Relative Invariants 238. We are now led to attempt to form the integral invariants relative to the closed subsets. Let us first assume that p - 1, and let us determine the condition by which the simple integral / {\,dx,-+- A,(fxj+. . .+ An(far«) is an integral invariant with respect to closed lines. Let us carry out the change in variables as indicated above, and our integral will become n B , :/>'i -\-Btd/, +...->- B„_, dy„.^ ■+■ B, ds), (1) which I can write again, taking the most symmetrical notation from the end of the preceding section / ^Bidx[. (1') This simple integral, extended over a closed, one-dimensional subset — i.e., over a closed line — may be transformed by the Stokes theorem /lO into a double integral extended over a non-closed, two-dimensional subset — i.e., over a non-closed surface. We then have /-'--/2(S-Sf)-'^'*- (» However, the integral of the second member of (2) must be an abso- lute integral invariant, and not only with respect to the closed subsets. We can therefore conclude the following: In order that (1) be an integral invariant with respect to the closed lines it is necessary and sufficient that the binomials dBj dBk dr't dr'i all be independent of z. Similarly, and more generally, let be an integral expression of the order p, having the same form as those which were discussed in the preceding sections. We would like to know whether this is an integral invariant with respect to the closed subsets of the order p. Let us assume that this integral is extended over an arbitrary closed subset of the order p. A theorem similar to that of Stokes states that it may be transformed into an integral of the order p + 1, extended over an arbitrary subset, which may be closed or not closed, of the order p + 1. The transformed integral may be written / ^^*-^'^-^^^'"- (4) One always takes the sign + if p is even, and the signs + and - al- ternately if p is odd. [For additional details, refer to my report on 10 the residuals of double integrals (Acta Mathematica, Volume VIII), and to my ^report contained in the Special Centenary Edition of the Journal de I'Ecole Polytechnique. ] The condition which is necessary and sufficient for (3) to be an integral invariant of the order p with respect to closed subsets is /ll that (4) be an absolute integral invariant of the order p + 1. 239. Let us again consider expression (1) of the preceding section, and let us assume that it is a relative invariant, that is, an integral invariant with respect to closed lines. Let us change it to the form (1') by our change in variables. Let Mq be a point of Fg and X\, yi, ■••. :yn-u ■= be its coordinates (with the new variables) . Let M be the corresponding point of F and ji, 7i, ■•-, yn-u z-^t be its coordinates. The B^^'s will be functions of the y's and of z, but I will make z appear, writing B^ in the following form B,(z). If the line Fq is closed, we will then have fz B/,(j -4- dx'i. = fzBk{z)dTi., that is, the expression Z[Bi(z+t)~Bi{z)]dx'„ (3) is an exact differential which I set equal to dV. The function V will depend not only on the y's and z, but also on t. In order that t - o, it must be reduced to a constant. If we assume that t is infinitely small and if we call B\(z) the derivative of \{z) with respect to z, expression (3) may be reduced to s[tB;(z}]dx',. 11 The expression SB',.(^)rf:ri (4) is then an exact differential which I set equal to dU. The function U which is thus defined will depend on the y's and z, but it will no longer depend on t. I shall again make z appear by writing U(z) . It then happens that 1^2 d\ r p f(t) is an arbitrary function of t. However, U(z) may be regarded as the derivative with respect to z of another function W(z) which is also dependent on the y's, and we will then have On the other hand, since V must be reduced to a constant for t = o, we may finally conclude that v = w(i + o-w(-) + (t) designates an arbitrary function of t only, and may be assumed to be zero without essentially restricting the conditions of generality. One then finds ^^^'^=5^^^(^) + C.- C^^ is independent of z, so that the expression (1') may be reduced to Jd\\+Jzc,dx\, and the first integral is that of an exact differential, and the second integral is an absolute integral invariant. 240. In a similar way let us discuss a relative invariant which is of a higher order than the first. Let us assume that 12 /= 2 A dw is this invariant which, after the change in variables, will become A B dm'. The integral must be zero, whatever may be the closed subset of order p over which it is extended. It must therefore satisfy certain "integrability conditions" which /13 are similar to those stating that a total differential of the first order is an exact differential. Let us now consider a subset V of p dimensions, which is not closed and limited by a subset v of p - 1 dimensions which will serve as the boundary for it. The integral (1), which is extended over the subset V, will not be zero. However, if it i s calculated for other similar subsets V , V", etc., having the same boundary v , one will obtain the same value — i.e., the value of the integral (1) only depends on the boundary v. It equals an integral of the order p - 1 which is extended over the subset v and where du" designates an arbitrary product of p - 1 differentials, while C is a function of the y's, z and t. This integral (2) is clearly a function of t, which depends in addi- tion on the subset v. Let us consider its derivative with respect to t. We will have As its last expression shows, this derivative does not change when one changes t into t - h or when, at the same time, one transforms V (or v) by changing z everywhere into z + h. 13 It can be concluded that J has the following form J= l'zi>{z+t)d^'~ rzD{s)dw', D(z) is a function of x, y, z. The integral JzD{z)d^-, + G dz). The B's and the C's must depend on the y's, but not on z. If this expression dU is an exact differential, the function U must therefore have the following form Uq and Ui are integrals of equation (1) . We will then have /29 If we return to the old variables x-j^, we will have rfU _ d{] rfU ., rfU „ -dl^dl.'^d^r'^'"^ 5^ "■ It therefore results that dXi dxx dx„ is an integral of equations (1). If this expression is zero, we have U, = o, U = Uo, and U is an integral of equations (1) . 252. We could cite numerous examples of this type. I shall only 30 present one example. Let us consider an invariant of the first order having the form Let A be the discriminant of the quadratic form $ • Let us make the change in variables according to No. 237, and our invariant will become ry/^T/:^' + 2 s c;* dx\ dV^ =y /*"• Let A' be the discriminant of the quadratic form $' . Let J be the Jacobian or the working determinant of the x's with respect to the x' 's. We will have A'r-. AJ5. The quantity A' will obviously be (like the B' 's and the C's) an integral of equations (1). Now let an invariant of the n£ll order be / .M dxx dxi . . . dxn^ After the change in the variables according to No. 237, it becomes IM / MJ dx\dT'j .. . dx',„ and MJ must be an integral of equations (1) . I may conclude from this that i' i.e. , ^ must be an integral of equations (1). 31 Change In Variables 253. Iflien the variables x-j^ are changed in an arbitrary manner, without affecting the variable t which represents time, it is only necessary to apply the customary rules for the variable change for single or multiple definite integrals to the integral invariants. This is the procedure we have already followed several times. However, when the variable t is changed, greater difficulty is encountered. It would even appear a priori that this transformation cannot lead to any result. Let us consider the system di='!^ = i'l^. ^r'-^jf. (1) -^1 X. ■■ X„ Let us introduce a new variable tj defined by the relationshi P dt, "■' Z is the given function of xj, X2, ..., x . System (1) must become zx, zx. --zx:;- ^^^ Let us assume that the initial values x°, x^, ..,, x^ represent the coordinates of a certain point Mg in space having n dimensions. If the motion of this point is defined by equations (1), with t /31 representing time, at the time t = t this point will move to M. On the other hand, if the motion is defined by equations (2), with t^ representing time, at the time ti = t the point Mq will move to M' . Let us now consider a figure Fq occupied at the time zero by different points Mg. If the motion and the deformation of this figure are defined by equations (1) , at the time t = t it will become a new figure F. If the motion is defined by equations (2), at the time ti - x the figure Fq will become a new figure F* which is different from F. 32 Not only will ^ be different from F, but in general it will no longer coincide with one of the positions occupied by F at a time which is different from the time t = x. It thus appears that we have profoundly changed the given quantities of the problem' and we must not expect that the invariants of (2) may be de- duced from the invariants of (1). However, this is what occurs for invariants of order n. Let us make the change in variables of No. 237. System (1) will become ~ O ~ O " ' O I and system (2) dvi dri dY„~\ _dz /oM at I = — = — = . . . — — - — — 7 ' ^ ' We must then assume that Z is expressed as functions of the y's and of z. Let us then set with integration being performed with respect to z (the y's are Assumed to be constants), and starting with an arbitrary origin which may depend on the y's. System (2) becomes „ dy, _dyr _ _ dy„-^ _ rffi (2") and will have the same form as (!')• Then let / M dxf dxt . . . dxn, be an invariant of the order n of equations (1). When the variables are changed according to No. 237, it becomes y M J dyt dyt... dyn-i dz; /32 33 J is the Jacobian of the x's with respect to the y's and z; MJ must be a function of the y's. And then y ,MJrf7.'„-,c/5, will be an invariant of equations (2") ; will be an invariant of equations (2'), and finally / ^ dxi dx,... dx„ will be an invariant of equations (2) . General Remarks 253'. Let us consider a system of differential equations dxi = Xidt, (1) and their variational equations d\i = Stdt. (2) Let us assume that equations (1) include an integral invariant of the first order Czkidxr Expression EA^^C^ will be an integral of equations (2) . On the other hand, these equations (2) will have the solution /33 with e being an arbitrary infinitesimal constant. Let ^1 = 9.(0 34 be an arbitrary solution of equations (1) . If e is a very small con- stant will still be a solution of equations (1) , and dx- 5,- = '^i(t + e)— 0,(0 = ^ -^ = ^^' will be a solution of equations (2) . It thus results that must be a constant. Therefore, EA^X^ is an integral of equations (1). Let us now assume that equations (1) include an integral invariant of the second order I jZAikd.Cidx;,. 11 be an integral of equations (2) and of equations (2'), which may be deduced by changing the C^'s into K\ • wi Let us set with e being a constant. This is permissible, because C'^ = eX^ is a solution of (2') . Then will be an integral of (2) . This shows that 35 is an integral invariant of the first order of equations (1) . This procedure makes it possible to obtain an invariant of the /34 order n - 1, when one knows an invariant of the order n. The procedure may sometimes be illusory, because the invariant which is thus obtained may be equal to zero. Let us now envisage an invariant having the following form f-S.{\i + tBt)dxt, where A^ and B^ are functions of the x's. We shall encounter invariants having this form below. Then will be an integral of equations (2). As a result, must be a constant. For purposes of brevity, let us set * = 2A,X,-; *,= vB.x,^ and the expression * + /*, must be a constant, which entails the condition — -f- < ^- + *, = o, or The X-l's, the A^'s, and the B^'s are functions of the x's. The same holds true for 36 The identity (3) can only occur if we also have identically IS"'- and The first relationship shows us that $i is an integral of equations (1). i35 253". Let •t" — consl. be an integral of equations (2). The function $ must be of a specific form, a whole and homogeneous polynomial with respect to the 5^ s, where the coefficients depend on the x^'s in an arbitrary manner. Let m be the degree of this polynomial. The expression (where *' is nothing else than $, where the ^'s were replaced by the differentials dx^^) will be an integral invariant of equations (1). Under this assumption, let I be an arbitrary invariant of the specific form $. Let us make the change in variables according to No. 237, and the equations (1) will become HT '^' di "'■ and, if one employs n^ and ? to designate the variations of y^ and z, the variational equations of (1') will be reduced to dt ~ dt ~ °" 37 With these new variables, $ will have the specific form $0' which is whole, homogeneous, and has the degree m with respect to the n-; 's and C The coefficients may be arbitrary functions of the yj^'s. However, according to the theory presented in No. 237, since we are dealing with an integral invariant, these coefficients cannot depend on z. The x^'s are functions of the y's and of z, and the following relationships between the variations may be deduced The Cs are therefore linear functions of the n's and of C, and the de- terminant of these linear equations (4) is nothing else than the Jaco- 736 bian of the x's with respect to y and to z. I have called the Jacobian J. One then passes from the form o, which corresponds to the invariant I of $. We will have I = IoJ/' with p being the degree of the invariant. However, Ig is a function of the coefficients of $o and, conse- quently, a function of the y's, which is independent of z. It is therefore an integral of equations (1). Let M be the last multiplier of equations (1) , in such a way that we have and that / M d:ri dx,... ilx„ is an integral invariant of the order n. We have seen in No. 252 that MJ will be an integral of equations (1). Therefore, 38 I„(M J )/---= IMP will be an integral of equations (1) . An integral of these equations therefore corresponds to each invariant of the form $. Now let C be a covariant of the form $, having the degree p with respect to the coefficients of 0, and the degree q with respect to the variables £,. If Cq is the corresponding covariant of $o» ^^ will have C =-- CoJf. The coefficients of Cq are functions of the coefficients of $o. and they are therefore independent of z. The same holds true for those of Co(MJ)/'=C.M''. Therefore, CMp is an integral of equations (2); therefore, is an integral invariant of equations (1), where C is none other than /37. C, where the ^^'s have been replaced by dxj^. We therefore have a method of forming a great number of integral invariants. The particular case in which p is zero (i.e., the case of the so-called absolute invariants or covariants) merits particular attention. If C, for example, is an absolute covariant of $ will be an integral invariant of equations (1) . One may therefore form a new integral invariant without knowing the last multiplier M. The same procedure may be applied to integral invariants of higher order. For example, let be an integral invariant of the second order. The bilinear form '1. = ZXa-di'k-Ul':) 39 \_/ which is an integral of equations (2) and (2') is connected with this integral invariant. Every invariant or covariant of this form, multiplied by one appropriate power of M, will be an integral of equations (2), (2") and will consequently produce a new integral invariant. In the same way, if one has a system of integral invariants, a system of forms which are similar to $ may be deduced from it, which will be integrals of equations (2), (2'). An integral of equations (1) will correspond to every invariant of this system of forms. An integral invariant of equations (1) will correspond to every covariant of this system of forms. For example, let F and Fj be two quadratic forms with respect to the C's. They become F' and 'E'l when the C^'s are replaced by the dif- ferentials dx-j^. Let us assume that F and F^ are integrals of (2) and that, consequently, are integral invariants of (1) . Let us consider the form F-XF, where X is an unknown. When stating that the discriminant of this form is zero, we shall obtain an algebraic equation of degree n in X, for which the n roots will obviously be absolute invariants of the system of forms F, F]^ . These will therefore be integrals of equa- tions (1) . However, this is not all. Let Xi, X2 ^n ^^ these roots, and F and F^ can be written in the form F --=X,A«+X,Ai + ...+ X„AJ, F,= A«+ AI+...-H AJ, with Ai , A2 , . . . , A^ being the linear forms which may be determined by purely algebraic operations. The quantities A^ , A2, ..., A^ may be regarded as the covariants /38 40 of zero degree of the F, F^ system, so that are the integral invariants of equations (1), if A'i designates what Ai becomes when the ?.'s are replaced by the differentials dx^. However, there would be an exception if the equation for X had multiple roots. For example, if Xi were equal to A2. it could no longer be stated that /a;, /a; are integral invariants, but only that is an integral invariant. Now let / S \ii,(lxidxk, I S Mikdxidxk be two integral invariants of the second order. The two bilinear /39 forms will be integrals of (2) and (2'). The most interesting case is that in which n is even; therefore, let n = 2m. Let us consider the form -f — X*, and let us make its determinant equal to 0. We shall have an algebraic equation for X of degree n = 2m. However, the first term in this 41 equation is a perfect square, so that it may be reduced to an equa- tion of order m. The m roots will be Integrals of equations (1), for the same reason as above. Now $ and $i can be written in the form '!• = Vx,(p,q;-q,p;) i — ;i .!>, = i(p,q;.-q,p;) and the P-^'s and the Qi's are 2m linear polynomials with respect to the C's. The P'i s and the Q'^'s are the same polynomials, where the ^^'s have been replaced by the ?' s . Then the expressions i'.q; -Q.f"., PjQi-Q.Pi, ..., p„,q;„-q„.p;„ will be covariants of the system $. $i, and consequently integrals of (2), (2') to which the integral invariants will correspond. There would be an exception to this if the equation for A had multiple roots. If we have, for example, X, = X. it could no longer be stated that the two expressions /40 p.q; -p;q.. P2Q;-PiQ, are integrals of (2), (2'), but only that their sum p.q;-p;q, + p,q;_pjq, is an Integral of (2), (2'). 42 CHAPTER XXIII. FORMATION OF INVARIANTS Use of the Last Multiplier 25A. There is an integral invariant which may be formed very 1^1 readily when the last multiplier of the differential equations is known. Let dxi dx^ _ dxa _ J x7=xT x7~ ' (1) be our differential equations. Let us assiime that there is a function M of xx, X2, ..., Xj^, so that we also have , identically rf(MX,) rf(MX,) rf(MX„) dxi . axt dx„ This function M is called the last multiplier . It may then be stated that the integral of the order n J = / M dx^ dxx . . . dxn is an integral invariant. Let us assume that equations (1) have been integrated; expressing xi, X2, •••, x^ as functions of t and of n integration constants the integral J will become J =: / i\I irfoti dli . . . di„. The quantity A is the Jacobian or the functional determinant of the x's with respect to the a's. We will then have - — rfj rrfiM A , , -=J -^d^,da,...d:r„ 43 However, On the other hand, c/Mi dt " dSl dt' d^\ dt ~ = dx, dtxx - . , • ■ • ) dx„ da I I may only write the first line of this determinant; the others may be deduced from It by changing a^ to 02, 03, ...» a^. Therefore, A + dt t— must be the Jacoblan of the dt 7 dj~, ^' '^ ~(IT ^ ^' ~^' • '^^ with respect to the a's. This will be the product of the Jacoblan of the x^'s with respect to the a's — I.e., of A, and the Jacoblan of the Xj^ + X^dt's with respect to the x-^'s, which I shall call D. I may write i>.-\-dt -r = A.D. dt However, the Jacoblan D may be readily formed. The elements of the principal diagonal are finite, and that belonging to the L^ll line and to the i£il column may be written i-+-rf« dXt dxi The other elements are Infinitely small; that belonging to the iS^ line and to the k£lL column (1 ^ k) may be written di . '. dxk It thus results that, neglecting terms on the order of dt^, we will have D=,..,2S; from which it follows that /43 44 dt Xi^ dxt One may conclude that rf.NTi ,V^ = i-X, rf.M (/^ " "' dxi from which we finally have + AiM4-=i cfx,- rf(MX,) ^2^ dxi 0, q.e.d. Equations of Dynamics 255. In the case of equations of dynamics, a great number of inte- gral invariants may be readily formed. From Sections 56 on, we learned how to form a certain number of integrals of the variational equations, and in the preceding chapter we learned how to deduce inte- gral invariants from them. The first integral (equations 3, Vol. I, p. 167) is as follows T,'i ?i — $i ^;i -H Vj 5. — $i Is + • • • = coosl. The integral invariant which may be deduced from it is as follows Ji = / [dXidyi-^-dx^dyt-h. ..+ dx„y„). It is of the second order and is of the greatest importance for the statements which will follow. A little farther on (still p. 167, Vol. I), I obtained a second integral which I may write = const. The integral invariant which I may deduce from it is of the fourth order and may be written as follows h ?i Vi 57 1* V/ r^i rrt 5* \'k r* n Ti* n'k T/* rTk Jj= jzdxidyidxkdyi,. 45 The siammation indicated by the sign E may be extended over the — ^ — - — ^ /44 combinations of the indices i and k. In the same way, the integral J , = I ^ dxi dyi dxk (If/, dxi dy,, where the sinranation is extended over — ^ ^-r-^ '- combinations of 6 the three indices i, k and 1, will still be an invariant, and so on. We thus obtain n integral invariants if we have n pairs of conjugate variables. One of these invariants Ji will be of the second order; another J2 will be of the fourth order; another J3 will be of the sixth order, ..., and the last 3^ will be of the order 2n. However, it is not necessary to assume that these invariants are all different. At the end of No. 247, I stated that from an invariant of the second order, one can always deduce an invariant of the fourth order, an invariant of the sixth order, and so on. The invariants J^ , J2, ..., Jji which I have just defined are none other than those which may be deduced from the first of them. These invariants may be considered in another way. At the beginning of page 169, Volume I, I demonstrated the manner in which one could de- duce the Poisson theorem from the integral (3) on page 157, or — which amounts to the same thing — from the integral invariant Ji . Following the same procedure with the invariant J2j one would obtain a theorem similar to that of Poisson. Let '?>, '^,, <^^, '\>„ be four integrals of the equations of djmamics. Let be the Jacobian of these four integrals with respect to ^n yi, ^k, yk- 46 The expression where the summation is extended over all combinations of the indices a,k, will still be an Integral. A similar theorem would be obtained by commencing with any of /45 the invariants J3, Jt, , ...» J^* However, according to the statements which I have just made, none of the theorems is different from that of Poisson in reality. However, from among all of these invariants, great importance may be attributed to the last of them J^ = / J.ri dy, dr^ dy^ . . . dx,, dy„. It could be obtained by the procedure given in the preceding section. It is known that the equations of dynamics have unity as the last multiplier. 256. I shall now assume that the x's designate the rectangular coordinates of n points in space, and I shall employ the notation given on page 169 of Vol. 1. On page 170, we obtained the following integral of the variational equations > -^ J — > -y- ^ = con^l. The corresponding integral invariant may be written J MKi\ m dx J In the same way, the invariant {dyn-h dyii+...^dy,u) corresponds to the integral -■'iit = const. The invariant / S (.z-,/ dy^j —yu dxii — xu dyu + yu dxu) 47 corresponds to the integral S/(T|,r,2,— /iijai— J^2i'i-,ii+/2j?ii) = consl. However, none of these invariants is of great interest, since they may be immediately deduced from the integrals of energy, center of gravity, and area. This does not hold for the following, which occurs when the /46 function V is homogeneous with respect to the x's. In No. 56, we learned that if V is homogeneous of degree -1, the variational equations have the integral K.x.,..- -..M = 3 ' [2 (■^- - ^. a.)] -^ con.., or, removing the indices, we have It may be stated more generally that if V is homogeneous of order p, the same procedure leads to the following integral S{ixr, -^py^)^(^2~p)c'^(-^ - ^ n+consl., from which we obtain the integral invariant an invariant which has a very special nature since It depends on time. The second integral may be written /'^V('Zl._v), J JLd\iin /' and is t|ierefore an integral of an exact differential. It may be readily seen that 48 IM is none other than the energy constant, which I shall call C. The invariant J is of the first order; it is therefore an inte- gral taken along an arc of an arbitrary curve. Let Cq and Ci be the values for the energy constant at the two ends of this arc. This arc is the figure which we have called Fq in the preceding chapter. When this figure is deformed to become F, Cq and Ci do not change, as I explained in the preceding chapter. As a result, we have J = Cz{ixdy — py dx) -\- {p — i)t{Ci — €■<,)■ The integral is therefore not constant when figure F (which is reduced to an arc of a curve here) is deformed; however, these variations are proportional to time. The integral is constant, if the two ends of the arc correspond to a single value for the energy constant. In particular, this is also true if the arc of the curve is closed. This integral is therefore a relative invariant, as I desig- nated it in the preceding chapter. However, if one assumes that the arc of the curve is closed, an arbitrary exact differential may be added under the / sign without changing the value of the integral. For example, we may add Z{xdy-k-ydx), with an arbitrary constant coefficient. Thus, the integrals jlydx, JT.xdy are also relative invariants. 49 We saw in No. 238 that an absolute invariant of the second order may always be deduced from a relative invariant of the first order. The invariant of the second order which is thus obtained is none other than which we studied above. This is the case in which the expression S(iT 4. Let us con- sider the n relationships ?,n,-,-H?.R,!-H?3B/3H-3in,i=.o (i- = i, 1, ..., n). (17) Let us form the Table T of the 4 n coefficients B. All of the determinants formed by means of the four columns in this table must be zero. If this is not the case, we shall obtain one or more relationships which must be satisfied by all the singular solutions, which will in- clude only the x's and which will not Include the indeterminate g's. If they are identically equal to zero, let us Consider three of the relationships (17) , and we may deduce the following from them 'h " h h "h ' The M's are minors of the first order of Table T. We will therefore have /53 MiH,-f-M,H,4- M3Hj + MtHt = o. (18) This relationship (18) must be identical, because the coefficient of K]f^ is one of the determinants of Table T, which I assume to be identically zero. We shall therefore have a relationship of the form (6") , which is opposed to our hypothesis, unless one only assumes that all of the M's are identically zero. If all of the minors of the first order of Table T are identically zero, let us form the minors of the second one. Let M' I, M'2 , 1^3 be three of these minors obtained by taking three arbitrary columns in the table and by cancelling the lines 1 and 4 for M'l , 2 and 4 for M'2 , 3 and 4 for ^3. It will become M',Hj-+-MiH. + MiH, = o. (19) This relationship must be identical, because the coefficient of K}^ in the first member is one of the minors of the first order of T which I have assumed to be identically zero. 55 This would still be a relationship of the form (6") , unless one only assumes that all of the minors of the second order M* are identi- cally zero. If this is the case, it will become identically B„n.-B,iHj = o, which is still a relationship of the form (6"). It may therefore only be the case that all of the determinants of Table T vanish identically. We shall therefore have at least one relationship (and, consequently, a system of invariant relationships) which must be satisfied by all the singular solutions of equations (1) . It may be immediately concluded that all of the solutions of equa- tions (1) cannot be singular. But this is not all; we may expand our definition of singular solutions. We have just defined the singular solutions with respect to q 754 integrals H^ of equations (2) which are linear with respect to the £,'s and which correspond to q invariants (linear and of the first order) of equations (1) . In the same way, we may provide a definite definition of the singu- lar solutions with respect to q arbitrary integrals H., H. If 1 of equations (2) and of equations (2') obtained by replacing the Vs by the V 's. These integrals must be homogeneous and of the same order, both with respect to the ^'s and with respect to the C' 's. They will be whole polynomials with respect to these variables, but they will not be necessarily linear with respect to the 5's. They may therefore corres- pond to integral invariants of a higher order, or to integral invariants of the first order, but which are not linear. In addition, these integrals must be different — i.e., they must not satisfy identically a relationship of the form (6), (6') or (6"). I may then state that a special solution S is singular if a rela- tionship (6) is satisfied for the values of x which correspond to this solution. 56 We shall then have The quantity \ is a monomial formed by the product of a certain number of factors 5i, C2. •••> ^n' ^'^' ^2' '••' ^n ^^^^sed to a suitable power, and the B^-j^'s are algebraic functions of the x's. We shall first set, as was done above, B/ = i3iB,M+ ?iB/.,-i-...+ |3,,B,-.y, and no changes need be made in the line of reasoning pursued above. We shall arrive at the same conclusion. Every singular solution with respect to the q integrals H^ satis- fies one and the same system of algebraic invariant relationships. These results are still valid if one envisages the integrals in the following form ni=B,.,«i + B,.,5,-(-...-*-B„.,f„+n„+i.,V?, + B„+,.,?$j-H...-i-Bj„.,7C„. The definition of the singular solutions, with respect to these /55 integrals, will still be the same, and these singular solutions will satisfy one and the same system of algebraic invariant relationships. The proof presented above need only be repeated, without any changes. The coefficients of the quantities Bj^ .^ — which will play the same role in this proof as the d's — ■ may be either the E,±'s, the products of Ci and of V^, or the products of the form tCi. 259. I do not wish to delve into the reasons for my belief that all periodic solutions cannot be singular solutions in the case of the three-body problem. This would take me too far afield from my subject; I shall return to this later. In the meantime, I shall provisionally assume that this proposition is correct, only observing that it is very unlikely that all of the periodic solutions of the three-body problem satisfy a system of invariant relationships, which would be necessary — according to the preceding section — in order that they may be singular. We shall again employ the notation and the numbering of equations in No. 257. If equations (1) and (2) include q different integrals which are linear with respect to the £,' s and algebraic with respect to the x's, these q integrals will still be different when the x's are replaced by the values corresponding to a non-singular periodic solution. 57 By stating that these q integrals are constants, and by replacing the x's by the values corresponding to a periodic solution in the equations thus obtained, one will obtain q equations of form (5), but where the exponent a\i^ will be zero. These q equations must therefore be included among equations (5) . Therefore, in order that equations (1) Include g different integral invariants which are linear with respect to the x's, it is necessary that q of the characteristic ex- ponents aj^ be zero for every non-singular periodic solution. Let us now try to determine the integral invariants of the form These invariants will correspond to the integrals of equations (1) and /56 (2) which are quadratic with respect to the C's. The integral F(t,)=con5l. will correspond to the invariant (7); this integral must be quadratic with respect to the 5's and algebraic with respect to the x's. In this equation, let us replace the x's by the values corresponding to a non- singular periodic solution. We shall have F*(5,)- const.. (8) where F* is a quadratic polynomial which is homogeneous with respect to the C's, whose coefficients are periodic functions of t. It must be possible to deduce all equations of the form (8) from equations (5) in the following manner. When dealing with a problem of dynamics — in particular, in the case of the three-body problem — we have seen that the characteristic exponents are pairwise equal and have the opposite sign. We can there- fore group equations (5) by pairs. Let us set A^c'i' =:c,f,o,/„ (5') Bitf -^I'^s.e.O'a.. (5") When multiplying equations (5') and (5") by each other, we will obtain an equation of the form (8) , and all equations of the form (8) must be linear combinations of the equations thus obtained. If we therefore assume that equations (1) have the canonical form of the equations of dynamics, and that they contain p pairs of conju- gate variables, we shall have p pairs of equations similar to (5') and (5"). Consequently, for each periodic solution, we shall have p 58 equations of the form (8) which are linearly independent. Let us choose one equation from these p equations and their linear combinations, for instance, F*(Ci) • Let us follow the same procedure for all of the other periodic solutions. We shall then have a certain poly- nomial F*(Ci) which is homogeneous and of the second degree with respect to the 5's, whose coefficients will be functions of the x's which are J5]_ only defined for values of x which correspond to a periodic solution. We must now determine whether the selection may be made in such a way that the coefficients of F* are algebraic functions of the x's, or even of the knovm functions of the x's. I shall simply pose this problem, without attempting to solve it at the present time. Let us now try to determine the invariants of the second order — i.e., those having the form of a double integral where F is a linear function of the products dXj^dx^^ (the coefficients of this linear function ^.re naturally functions of the x's). These invariants of the second order will have the following significance. Let us select equations (1) and (2) once again (we shall always retain the numbering given in No. 257), and let us form in addition the equations They will lead us to equations which are similar to (5) , and which I may write as follows Ai.c^i'=s$;6,vt. (5a) They only differ from equations (5) because the letters are accented. According to the preceding chapter, the invariants of the second order will then correspond to those of the integrals of (1) , (2) and (2a) , which are linear with respect to the determinants and algebraic with respect to the x's. Let F(^$i-bV.) 59 be one of these integrals. If the x's are replaced by the values corresponding to a periodic solution, we will obtain an equation having the form /^^ F'($<$i— $*$;) = consi., (9) where F* will be a linear function with respect to the determinants and whose coefficients will be periodic functions of t. We have now determined the manner in which all relationships of the form (9), relative to a given periodic solution, may be formed. In the case of equations of dynamics, equations (5a) may be divided into pairs like equations (5) . Let Ai.c'"' = 2$;o,v„ (5a') n;.e '■»' = i;?;o;v.. (5a") be one of these pairs. Let us multiply (5a') by (5"), (5a'') by (5'), and let us subtract. We shall obtain an equation having the form (9). Each pair of equations will give us one, and all other equations of the form (9) will only be linear combinations of those which thus may be formed. Let us choose one equation from among all equations of the form (9) thus obtained. Let us follow the same procedure for all other periodic solutions. We shall then have a relationship F*(5,$A-?t-5;) = const. whose first term will be a linear function of the determinants. The coefficients of this linear function will be functions of the x's which are only defined for values of the x's corresponding to a periodic solution. We must now determine whether the selection may be made so that these coefficients are algebraic functions or even the known functions of the x's. Let us now return to the linear invariants of the first order. According to No. 29, the form of equations (4), and consequently that of equations (5), is modified when two or more characteristic expo- nents become equal. If, for example, nine of these exponents equal zero, we may write 60 the corresponding equations (5) in the following form /59 The quantity Pj^ designates a whole polynomial with respect to t, having constants for coefficients. These polynomials are of the degree q - 1 at most. In order to specify this more precisely, the number of polynomials is q. The first may be reduced to a constant, the second is of degree one at most, the third is of degree two at most, and so on, and finally the last is of degree q - 1 at most. In the case in which the degree of this last polynomial reaches its maximum and is equal to q - 1, the polynomial before the last is a derivative of the last, the q - 2nd one the derivative of the q - 1st one, and so on. In every case, the q polynomials may be divided into several groups. In each group, the first pol3niomial may be reduced to a con- stant, and each of them is the derivative of the follovrlng. In order that there may be p linear Integral Invariants, it is not sufficient that p of the characteristic exponents are zero. It is necessary that p of the polynomials Vy^ be reduced to constants (or, which is the same thing, that these polynomials be at least divided into p groups) . From the point of view of our study, what is then the signifi- cance of equations (10) where Pj^ may not be reduced to a constant? In No. 216 we defined an integral invariant whose role is very important. This invariant has the form /'"-'A- where F and Fj are functions which are algebraic with respect to the x's, and linear with respect to the differentials dx. A similar invariant corresponds to an integral of equations (2) having the following form 1'' -T- M'l — consl., where F and F^ are functions which are algebraic with respect to the x's, and linear with respect to the ^'s. 61 In this integral, if I replace the x's by the values which corres- pond to a periodic solution, we shall have /60 F*-+-iFt= const., (11) where F* and F'^ are functions which are linear with respect to the C's, whose coefficients are periodic functions of t. We have now determined the manner in which we may obtain all rela- tionships of (11) starting with equations (10). Let us consider two poljmomials Pj^, the first being reduced to a constant, and the second being of the first degree; the first is the derivative of the second. The corresponding equations (10) may be written (10') (10") where the 9i's and the 6'^^ s are periodic in t. We may thus deduce 2^,0;— ii;j,0, = const., which is a relationship of the form (11) . We should note that equation (10'), raised to the square, provides us with a relationship of form (8) , and that a relationship of form (9) may be deduced from equations (10') and (10"), that is, (Z5,p,)(2riQ;)_(v5;0,^(2$,0;.)= const. 260. Let us apply this procedure to the three-body problem, and let us determine what may be the maximum number of integral invariants, of the several types studied in the preceding section, for this problem. That is: The first type: linear invariants with respect to the differen- tials dx; The second type: invariants where the function under the sign / is the square root of a second-degree polynomial with respect to the differentials of the x's; The third type: invariants of the second order, which are linear with respect to products of the differentials dxj[^dx{^; 62 The fourth type: invariants having the form considered at the end of the preceding section — i.e., having the form i^^ P'^-'f These different types of invariants correspond to different types of integrals of equations (2) and (2a), that is: The first type: linear integrals with respect to C's; The second type: quadratic integrals with respect to the C's; The third type: linear integrals with respect to the determinants ?l?'k- ^k^'i ' The fourth type: integrals having the form where F and F, are linear with respect to the 5's. We may assume that it is extremely probable that none of the periodic solutions of the three-body problem is singular. In the three-body problem, the number of degrees of freedom is six; the number of characteristic exponents is twelve. According to the ideas presented in No. 78, there are six, and six alone, which vanish; the six others are equal pairwise, and have the opposite sign. There are therefore six equations of form (10) and six polynomials P, , of which four are of degree zero and two are of degree one. Or, there are three pairs of equations having the form (5'), (5"), four equa- tions having the form (10'), and two equations having the form (10")- Let us therefore determine how many independent invariants of each type there will be. I shall state more precisely what I mean. I do not regard n in- variants of the first type as independent or n invariants of the second type or n invariants of the third type 63 //p., //p. .... //f. or n invariants of the fourth type /62 fP"fP^ /F" (Fr i'J+iF;), when there is an identical relationship between Fj , F2 , ...» F^j having the form where $1, $2> •••» *n ^'^^ integrals of equations (1). It is apparent that we cannot have more than four invariants of the first type, i.e., no more than the number of equations (10') already known. We cannot have more than thirteen invariants of the second type, of which three will come from the three pairs of equations having the form (5') and (5"), and the six others will be obtained by means of the squares of the four equations (10') and of their products by pairs. These last ten exist in actuality. However, they are not independent of the four invariants of the first type, since they may be deduced by the procedure given in No. 245. We may therefore have three new in- variants. We cannot have more than eleven invariants of the third type, of which three will come from the three pairs of equations having the form (5') (5"). Six will be obtained by combining the four equations (10') by pairs; two will be obtained by combining the two equations (10") with the corresponding equation (10'). Seven of these invariants are known. One is the invariant Jj of No. 255; the six others are those which may be deduced from the four equations (10'), but they may not be regarded as independent of the four invariants of the first type, since they may be deduced by the procedure given in No. 247. We may therefore have four new invariants of the third type. Finally, we may not have more than two invariants of the fourth type, i.e., no more than the number of equations (lO") , One of these invariants is known, that of No. 256; we may still 64 have a new Invariant. It is probable that these new invariants, the possibility of which was not excluded in the preceding discussion, do not exist. However, /63 in order to prove this, we must resort to other procedures — for example, procedures similar to those of the method advanced by Bruns. Use of Kepler Variables 261. The invariant of the fourth type in No. 256 may be written in still another form. Let us set an arbitrary system of canonical equations (faV _ £/F f^ ^ _ "'■'^ . (1) "di " djTi' dt dxi Let us consider the following integral taken along an arbitrary curve arc J = / (a-, dy, -f- a-j dyi + . . . + x„ dy„)- Let us assume that we are writing the equations of the curve arc along which integration is performed, expressing the x's and the y's as a function of the parameter ex, and that the values of this parameter which correspond to the ends of the arc are oq and ai. The integral J will equal Let us assume that we are considering our curve arc like the figure F in the preceding chapter, which varies with time and may be reduced to Fq for t - 0. Then the x's, the y's and the functions of the x's and the y's, such as F, H, "l^ will be functions of a and of t. We shall have dt J l^'rti dt rfaj J I dCdx] or 65 dt \-~imM'--sv-'imY when Integrating by parts However , /64 ^ rf^ da 2Lidx di~~ di ' and therefore dJ (2) If we assume that F is homogeneous and has the degree p with respect to the x's, it will become Let C be the energy constant, so that the equation of energy may be written F=G. Let Cq and Ci be the values of this constant which correspond to otQ and ai ; it will become "'J ^7=('-i=')(C,-c„). (3) Therefore, strictly speaking, J is not an invariant. However, its derivative, with respect to time, is constant and — to use the expres- sion defined in the preceding section, it is an invariant of the fourth type. 262. Let us now assume that F presents another type of homogeneity. Let us divide the pairs of conjugated variables into two classes, and let us use x^^, y^^ to designate the pairs of conjugated variables of the first class, and let us use x'^^, y'j^ to designate the pairs of con- jugated variables of the second class. 66 I shall assume that F is homogeneous of the order p with respect to the Xi's, to the (.x\)^ 's, and to the (y'i)2's, so that we have Let us then set or from which it follows that dv x^t ,d^ _^^.d^\rv^'^^ or or finally g=[F^;,F]-S; g=(,-/))(C,-C„), /65 which shows that J is still an invariant of the fourth type. 263. Let us apply the preceding statements to the three-body problem, and let us determine the change in the invariant of No. 256 67 with the different variables chosen. In No. 11, we used the following as variables 3L, pG, pe, p'L', p'G', p'e', I, g, 9, V, g\ 0'. F Is homogeneous of degree -2 with respect to the variables of the first series. Therefore, (\_^{h dl -t- ddg + 8 (i6)-h P'(LV/'+ Q'dg'-\- 3',/0'}] I- 3«(C, — Co) will be an Invariant. The same homogeneity remains If the following variables are chosen, /66 as In No. 12, A, n, z, A', H', Z', \ h, ?. X', h', $'. Therefore, J{Kd\ -hHdh-hZdii-i- X'dX'+ ll'dh'-hZ'.'^') -T- 3<(C,- Go) will be an Invariant. If the following are chosen as variables (see No. 12) A. A', \, ?', P, P', X, X', 7), 7)', q, q', the function F will be homogeneous of degree -2 with respect to the A's, to the C's, to the n's, to the p's, and to the q's. As a result, /2(aAtA-+-$rf7) — i)€^-i-^rfy_yrf^)-t-6ir(C, — Co) is an invariant. The sign Z indicates that the term which is deduced when the letters are accented must be added to each term. Thus, we have 68 If finally we select the variables of Nos. 131 and 137 A, A'; T,-, A, A ; T,-, we shall see that f[ ■>. A dl + 7. V H'i ~ n'it ^- Vj, and six other integration constants » I ' ^'0 ,.'0 ^'0 t' •^Oj Aj, x,", .( 2 , x^ , 3-4 , and we found that the equations of motion could be satisfied in the /72 following way. The quantities 73 \^ A, A', Xi— (Vi, X; — tvj, T,-, ■:/ may be developed in powers of v and of the x'j^° s . Each term is periodic with respect to the w's and the v/ 's, and depends in addition on the two integration constants Aq and A'q . The constants n^^ and n'^^ may be developed in powers of y and of the x'^^'s , and depend on Aq and A'q in addition. The Fj^'s and the oT'^^'s are six integration constants. Finally, Kcfk -H A'(A'i-t-S(T,6/-:( is an exact differential when the twelve variables A, X, a and t are replaced by their expansions, and when the w's and the w's are regarded as six independent variables and the quantities Aq, A'q, 3^^° are regarded as constants in these expansions. Our quantities x^ and y^ which I have just defined may be ex- pressed readily by means of the twelve variables A, X, and t. It may be concluded that xi and y^ may be developed in series in powers of y and of the x'^^ 's , as well as according to the cosines and the sines of multiples of the w's and the w" 's . In addition, each co- efficient depends on A and A'q . The expression S Xi dyi will be an exact differential, if the w's and the w' 's are regarded as six independent variables and Aq, A'q, x'^° are regarded as constants. We need barely point out that the series thus obtained are not convergent. They are only of value with respect to formal calculations, which gives them, however, a certain practical utility as I explained in Chapter VIII. Nevertheless, if we substitute these expansions for the x^^'s and the y^'s in the expression of an Integral invariant, the result of this substitution must, from the formal point of view, satisfy the conditions which must be satisfied by an integral invariant. This provides me with the verification procedure to which I wish to draw attention. 74 267. We saw above that HA fl{-ixdy-^ydx)—Zt{ci — ca) (1) is an integral invariant. In order that we may make use of this invariant, we are going to perform a change of variables which is similar to that given in No. 237. In order to have greater symmetry in the notation, let us set ip; = w.'+j ( i = 1 , 2, 3, 4), '^i = "'>!. ^'' - '''+'' We have seen that we may develop the x's and the y's in series depending on the w's, the w' s , the Aq, A'o . and the x'^O s — i.e., with our new notation, the w^'s and the ^i's (i = 1, 2, 3, 4, 5, 6). For new variables we may then take the ^^'s and the w-^'s, and then the differential equations of motion will take the form ^' == ^' = dt ^^^ [just as in No. 237, equations (1) become, after the change in variables, 1 as we have seen] • The n^'s are functions of the K±' s alone. However, it is more advantageous to select other variables. Due to the fact that the six n^'s are only functions of the six Ci's, nothing prevents us from taking the n^'s and the Wi's as variables, instead of the 5i's and the Wi's, so that the differential equations become ^ = ^=c/^ (3) o n-i An integral invariant of the first order will take the form where A and B are functions of the ni's and the w^'s. 75 I may assume that figure F Is a curve arc for which the equations, /74 which are variable with time, have the following form n/=/i{ct, I): i"/ =/;(»,/), where the variables n^ and w^ are expressed as functions of time t and of a parameter a which varies from ag to cxi when the arc F is en- tirely traversed. The equation of the arc Fq will then be /i/=--/,(a, o); ip,=/;(a, o). With these stipulations, I may then write '-rM^'-'<"-^h- from which it follows that dS dt ~ J ''^ Zi\-driii ^ ^ ^ii -"- '^' dTTa -"' ^' dfd^)- However, we have dRi ^ dB, ~dr=^'"'-dw from which it finally follows that dS If J is an absolute integral invariant, we must therefore have ..,u.^^ = -B. (5) Let us now determine what occurs in the case when the A's and the B's are periodic functions of the w's and may be, consequently, devel- oped in trigonometric series. Let us first consider equation (4) , and let us set 76 B,-.^S[/;cos(/rti«',-t-.,.-i-/,i,ip,5)_t. 6' sin(»!, IP, -;-...+ „,j^j)]_ where the b's and the b' s depend on the n^'s. Equation (4) becomes ZZi S(oti71i + . ..-t- /^i6"6)[— ^ sin (m, «', + ...+ niiWe) -I- Z)'cos(/7ii(Pi -t-. .-. + meivj)] = o, (6) which may only hold if «!i«i -)-. . .1- wis/ie = O, or if i = 6' = o. However, the m's are integer constants, and the n's are our inde- pendent variables between which no linear relationship may hold. Equa- tion (6) therefore entails the following m,--m2—...= nii — o. This means that the trigonometric expansion of Bj^ may be reduced to its known term — i.e., B^ is a function of the n^'s alone, and is independent of the w's. Let us now pass to equation (5) . Let us set A, — S(a cosio -4- a'sinto), writing o), for purposes of brevity, instead of m, ti-, +-. , .+ niiiVi. Equation (5) may then be written S(/n, «| -t-. . .+ mini){ — a sino) + a'cosoj) — — B,. Let us first consider a term which is dependent on the w's. I.e., such that mj, m2 mg are not zero at the same time. We shall then have In the second term, B-j^ does not depend on the w's. This second term contains neither a term for coso), nor a term for sinio. As a 77 result , we have a • a — o. Therefore, A^ does not depend on the w's, and may be reduced to the known term of its trigonometric expansion, a term which depends only on the n^^'s. However, equation (5) may then be reduced to /76 In general, every linear absolute, integral invariant of the first order, where the term under the sign / is algebraic with respect to the x's and the y's and, consequently, periodic with respect to the w's, must have the following form / ZKidni, where the A^'s depend only on the n^'s. In reality, this is what occurs for the absolute invariants which we know and which are obtained by differentiating the integrals of area, energy or motion of the center of gravity. However, the relative invariant i ~- I '^{nx dy k- y dx) deserves more attention. We have seen that J -3<(C,-C(,) (where Cq and Ci are the values of the energy constant at the two ends of the arc Fq) is an integral invariant. We shall therefore have If we set J = Cz{Sidni+V,idwi), equation (7) becomes 78 because the energy constant C is only a function of the n^'s. Equations (4) and (5) must therefore be replaced by the following equations 111 dwk dni ' (5') The A's and the B's must be periodic functions of the w's. If we treat equations (4') and (5') just the same as we treated equations (4) and (5), we find the following: 1. The B-j^'s are independent of the w's; 2. The Aj^'s are independent of the w's; 3. And that ^ dm - ^'■ We finally obtain ^{o.xdy ^^ydx) =-. lLkidni+ 3^ ~ dwi, where the Aj^'s depend only on the n^'s. In other words, expressions V ( <^yi dxi\ or ^'(^^'|S-*-^'£) (8) do not depend on the w's and are only functions of either the 5's or the n's, depending on whether everything is expressed as a function of the 5's and the w's, or as a function of the n's and the w's. In the same way, we shall have 79 hi I iXi -/■ — hVi -. — ) = 3 -, — \y) As I have already stated, the x^^'s, the y/s, and C are developed in powers of y and of the x'i" 's. Expressions (8) and the two terms in equations (9) may therefore also be developed in powers of these quanti- ties. All of the expansion terms of expressions (8) , which are expanded in powers of y and of the x'j^^ 's, must therefore be independent of the /78 w's. On the other hand, each expansion term of the first member of (9) must equal the corresponding term of the second member. We thus have numerous procedures for verifying our computations. 268. I have stated that S Xi dyi is an exact differential, if the Ci's are regraded as constants, and the w's are regarded as independent variables. We then obtain J-L{>x df +x dx) r= lj\ ^^ dw,, or, since the - — 's depend only on the ?'s, they must be consequently regarded as constants from which it follows that / -Zx dy -^ j Z{x dy + y dx) - 'i^'j^ f/. from which we finally have r dc (^°) Let us briefly return to the notation given in No. 162. In this 80 section, just as In No. 152, we chose the following as variables i ^' '^; ''■' (11) I '-I I ^1) ''II and we set On the other hand, the variables (11), just as the variables Xj^, yj^, are conjugate variables. As a result, just as I have explained several times , the expression Z.ri dyi - A .-Ai,X', — a»'T,. (12) Just as when the procedure outlined in Chapter XV is applied, one is led to formulate the function S, and equation (12) furnishes us with the desired verification in a new form. Relationship to a Jacobi Theorem 269. It is known that at the beginning of his Vorlesun^en aber Dynamik , Jacobi demonstrated the fact that, in the case of Newtonian attraction, the mean value of the kinetic energy equals, with the exception of a constant factor , the mean value of the potential energy, assuming that the coordinates may be expressed by the trigonometric series having the same form as those which we are presently studying. This Jacobi theorem is directly related to the preceding state- ments. The equations of motion may be written dxi dyt rfV from which it follows that I 2771/ vzL_v = G. 81 Then -V represents the potential energy, C the total energy, and the kinetic energy. On the other hand, due to the fact that V Is homogeneous of degree -1, we shall have The energy equation may therefore be written Xsii Let us take equations (9) from No. 217 and let us add them, after having multiplied them respectively by n^^. We shall have If we note that dx dx ^ "■'• J — = "77 divfc at (since ^ = nv) , we may conclude that dt / dVi dxi \ _ ^ dC -(-^•' W^y<-d )-'-''' dnl- When making a comparison with the energy equation, we find that 2 which shows that C must be homogeneous of degree y with respect to the ni.'s, which could be seen directly. The mean value of a function U, which I shall designate by the notation [U] , will be zero if U is the derivative of a periodic function. We shall therefore have 82 and, connecting this with the energy equation, we obtain from which we have I /:i ■>. nij I ^ _ I ■(-VJ -" 2' This is the Jacobi theorem. If the partial derivatives ^ are considered instead of the total /81 derivatives ^, similar results would be obtained. We would obtain dt and consequently 1* rfC Application to the T wo-Body Problem 270. In particular, the preceding considerations may be applied to the two-body problem. Let us consider a planet and the Sun. and let us refer the planet to axes having fixed directions and passing through the Sun. Consequently, let us consider the relative motion of the planet with respect to the Sun. Let XI X2, X3 be the three coordinates of the planet; let yi, y2, ya be the three components of angular momentum. Let £. n, ^ be the three coordinates of the planet with respect to particular axes, i.e.: The major axis of the orbit, a parallel line to the minor axis, and a perpendicular line to the orbital plane. We shall have 83 :ci -: /i,5 -I- /I'lT, -t- /i;^, where the h's are constants which are connected by the well-known rela- tionships which indicate that the transformation of coordinates is orthogonal. In the same way, we shall have where y is the mass of the planet. It is now evident that t, is zero, and that 5 and n are functions of one single argument w, which is the mean anomaly, and of two constants, which are the major axis a and the eccentricity e. In addition, the h's are the functions of the three Euler angles, or more generally, of three arbitrary functions coj , 0)2, 0)3 of these /82 three angles. Thus, the x's and the y's are functions of w, a, e, and of the (d's. If we designate C as the energy constant and n as the mean motion, we shall then have and, in addition, the expressions must be independent of w. Some of the statements were apparent beforehand, and provide us with no new verification. 84 In actuality, the — i's are linear functions of the x^'s whose coefficients depend on the oo's and are such that J.;.r,- - — ^ o. «(■>/(■ As a result, we may write the following identity dx, dXi dXi doit diiii: dint a, aj a, X, Xi X} -i --— S Vi/J ^'"/"i' +(' - e-')[T. r^.., cos/7ip]2 .= W, For purposes of brevity, I have written J__i everywhere, instead of Jp_l(pe). We must then have W-34^. an However , C = — '-'-- , n^a' — in, where m designates the mass of the Sun plus that of the planet. We therefore have 1 1 C= -I ,n''n' 2 and However, since we find W r^ jji a ViW, W =-. - 1 . When identifying the similar terms, we have a series of relation- /85 ships between the Bessel functions J. A study of expression E leads us to a series of relationships which are similar, in which the Bessel functions J and their first derivatives will be included this time. 271. Numerous examples of these particular applications could be provided. For example, after having treated the case of Keplerian motion as we have just done in the preceding section — i.e., after having taken into account terms of the degree zero with respect to the disturbed masses — one could apply the same principles to the entire 87 group of terms of degree 1. There is no doubt that this would lead to interesting results. Using the same procedure, we could also study the secular varia- tional equations which we discussed in Chapter X. In place of the integral invariant Cl.iixidyi-i-yidxi), we would have the advantage of employing similar invariants which we defined in Nos. 261, 262, 263. We shall put these questions aside. Application to Asymptotic Solutions 272. Let us apply these principles to asymptotic solutions. Let us take the coordinates Xj^ and the dxi as the variables. Let us consider the invariant J = / 2(2ar dy-\-y dx). We know that if C is the energy constant, and C^ and Cq are the values of this constant at the two ends of the integration line, we shall have J — 3<(G,— Go) = const. (1) If we consider a system of asymptotic solutions, it will have the following form: The x^'s and the yi's will be developed in powers of /86 where the coefficients are periodic in t + h, where A,, A„ ..., \k, h are k + 1 arbitrary constants. If these values of the x^'s and the y^'s are substituted in the energy equation, the first member is always developed in powers of 88 where the coefficients are periodic in t + h. Since it must be inde- pendent of t, it will also be independent of Ai , A2, . • • , A^^. and h. If the values of the x^'s and yi's are substituted in equation (1) , we shall have Ci = Go, and, consequently, J = const. In J, the expression under the sign /, is developed in powers of A,e».', Aje«.', .... A*c«i'; The coefficients are periodic in t + h; it depends linearly on the k + 1 differentials dXi, dKi, ..., dkk, dh. must therefore have 'lIix dXi dx\ : consl., 1.1 T.X dy dh dx\ : consl. (2) The first terms of equations (2) are developed in powers of the A-e^i^'s. All terms of this development must be zero, except for the known term. One thus obtains a multitude of relationships between ^^^^ the coefficients of the development of the x^'s in powers of the A^e s, By way of an example, I shall confine myself to considering the first term, and I shall write where X^ and Z^ are periodic in t + h. We may deduce where X'^ and Z\ designate the derivatives of X^^ and Z^. Neglecting all terms in e^"*^, etc., we then have 89 /87 ;4^ ,.y 'J;'\-:Z,ae-'[;\a''^'^'f')+ X'Z]. ; m(aX7/-+- 2aX7. ^;- X'Z) = o, which provides us with the first relationship between the coefficients X and Z-^. The relationship ^(■iJC dy d.v IT -^- y -,7 J "' const. dh ■' dhj furnishes us with another one which, in reality, would not differ from the first, since — when it is combined with the first relationship — an equation is obtained which is an immediate consequence of the energy principle. 90 CHAPTER XXV INTEGRAL INVARIANTS AND ASYMPTOTIC SOLUTIONS Return to the Method of Bohlln 273. Before proceeding any further, I must supplement some of /88 the results given in Chapters VII, XIX and XX. I would first like to sum up the results which I wish to compare and which will serve as my point of departure. We saw in Chapter VII that if a system cit has a periodic solution Xi - - ^i , (2) and if we set Zi -. X? -h ^i, the £;i's may be developed in increasing powers of (3) where the coefficients are periodic functions of t. The A^'s are inte- gration constants; the ai's are the characteristic exponents of the periodic solution (2) . The series always satisfy equations (1) formally. They are con- vergent under certain conditions, which we have discussed in No. 105. There is an exception in the case where we have a relationship having the following form between the exponents a y/ "l-^ Sa^ -z, = o (4) where the coefficients B are whole, positive, or zero, and the coeffi- ^89 cient Y is whole, positive, or negative. (See Volume I, page 338, line 5. When writing this relationship, I assvnned that the unit of 91 time was chosen so that the period of the solution (2) equalled 2it) . If there is a relationship having the form (4), the C's cannot be developed in powers of the quantities (3) , but in powers of these quantities (3) and of t. This is precisely what occurs if the equations (1) have the canonical form of the equations of dynamics. In actuality, in this case two of the exponents are zero, and the others are equal in pairs and have the opposite sign. In the case of equations of dynamics [or, more generally, when there is a relationship having the form (4)], we were still able to obtain a result. It is sufficient to give special values to the inte- gration constants A, so as to cancel the values of these constants corresponding to a zero exponent , and one of the two corresponding to each pair of equal exponents having opposite signs. [More generally, the constant A corresponding to one of the exponents included in the relationship having the form (4) would be cancelled, so that there would no longer be a relationship having this form between the ex- ponents corresponding to the constants A which are not zero,] For example, if 3,= «, =0, 1,= ~a„ a,^-a a„_,= -a„ (n even) , we would make Ai=Aj = 0| Aj = o, Ai = o, ..., An_i = o. The C's may then be developed in powers of those quantities (3) which are not zero. However, we shall no longer have the general solu- tion of equations (1) , but a special solution depending on the number of arbitrary constants which is less than n (i.e., ^ ~ -^ in the general 2 case of the equations of dynamics) . We have thus arrived at the asymptotic solutions: We have done this by cancelling a certain number of constants A, not only those which we have set equal to zero for the reason which I have just given, but also those which we had to cancel in order to satisfy the conver- gence conditions given in No. 105. /90 For the time being, I shall not deal with the development of the 5's in powers of y or of /\i. 92 In Chapter XIX I studied the method derived by M. Bohlin, which is basically only an application of the Jacob! method, since the problem is reduced to obtaining a function S which satisfies an equation with partial derivatives. Only this function S has a form which is particu- larly suitable for the case in which there is approximately a linear relationship having whole coefficients between the mean motions. The cases which are of greatest interest to us are those which are similar to that which I have designated as the limiting case (No. 207). In this section, we saw that the function S may be developed in powers of /y , in the following form and that So -1- V [iSi I- jaSj ^-. . . is periodic with the period 2tt with respect to ^'- Xl, ■... Xn (employing the notation in the section indicated above) . However, the results may be simplified by performing the change in variables which was discussed in No. 209 and 210. In section No. 206, I defined n + 1 functions which are periodic with respect to the variables r'-> 73, .... yn, and which I regarded as generalizations of periodic solutions. In No. 210, we set the following dye ' (iyi The equations retain the canonical form with the new variables x ' , y ' . Only the new equations have the following invariant relationships /91 93 which, with respect to the new canonical equations, may be regarded as generalizations of periodic solutions, Just as is the case for x, = i], y\=-Kj ^i-'Vi with respect to the old ones. Without limiting the conditions of generality , we may assume that our canonical equations imply the following invariant relationships .r, - Xi ^-yy = o. If this is the case, we saw in No. 210 that yi = is a simple zero for the derivatives 1^, and a double zero for the derivatives dyi f!E (i > 1). dyi Thus S, or rather S - Sq, may be developed in powers of Y, and the expansion will begin with a term of the second degree. We shall have S - So -I- >'-iy\ -I- >^37? + Sv7} + . . . (5) where the E's are series depending on ya, ys. •••. Yn ^"^ ^^^ developed in powers of /^. In addition, it may be seen that the Vs are periodic functions of y2, Ya* •••» yn" Unfortunately, this is not sufficient for our purposes. The function S, which is defined by equation (5), depends only on n - 1 arbitrary constants ^1, .-^S, ■■•- ^-s. whereas n would be required for the complete solution of the problem. In order to pursue the study in greater detail, we shall resort to the change in variables, given in No. 206. If we employ the nota- tion given in this section — i.e., if we set I r (fy , and if we define. Just as in the indicated section, the variables x'., 94 ^1 , vj, and the functions T and V, the derivatives of V with respect /92 to VI and to the z-j^'s will be periodic functions of the z^'s (see Volume II, p. 361). Let us examine in greater detail the equations which appear at the beginning of page 363 (Vol. II) and which are written 7i=- oC^i.r^.rs, ■■■,yn), Regarding 72, 73 7^ as constants, let us consider the following equations (alwa7s just as in the indicated section) y\ = OCvi), ar, = Ci(f,). When we var7 Vj, the point (xi , 7^) will describe a curve which I wish to stud7. Let us assume that we vary x'l , instead of varying the constants x'2, X3, ..., x'j^, and we shall obtain an infinit7 of curves corresponding to different values of x'l . We assumed above that the following invariant relationships hold a-, = xt -=yi =0 which are like a generalization of periodic solutions. The following point will correspond to these relationships a^i - 7i = o i.e., the origin of the coordinates. I would like to stud7 our curves in the vicinit7 of this point. Let us assign to x\ the value corresponding to the special function S defined b7 equation (5) , and we shall have ^,^i^,yii-3-S:,y\ + .... The corresponding curve passes through the origin. B7 changing vy into -/p, we would obtain a second curve passing through the origin. We therefore have two curves crossing at the origin. The center curves ma7 pass near the origin, without reaching it and without inter- secting each other, so that all of our curves together will look like (in terms of their general form in the immediate vicinit7 of the origin) the figure formed b7 a series of h7perbolas having the same as7mptotes /93 95 and formed by their asjnnp totes. 274. In order to study these curves and their corresponding functions S in greater detail, let us limit ourselves to the case in which there are only two degrees of freedom. Let us assume that the change in variables of No. 208 was performed in such a way that is a periodic solution, which amounts to stating that for .T| rz^ .Tj ~ J, r- O we have '/F dF dF f, ifi, ... with 98 respect to 72- In order that the canonical form of the equations is not changed, /96 it is sufficient to set •ill = 0,0'— 9'f', ; 2K :^- o,<;-'— o'>', ; 2L = 4/, <>'— <;"}',. It may be seen that H, K, L are periodic functions of 72, from which it follows that the form of the function F will not be changed either. However, if we set e = 0, our equations must have the solution from which it follows that a Ii = > A = t.=o. Without limiting the conditions of generality, we may assume that 11 = 1, A = C— o, B = consl. from which it follows (since we have removed the accents) We shall follow this procedure from this point on. Let us perform a change in variables, setting X\ Since is an exact differential, the canonical form will not be changed. We then thave The function F may then be developed in powers of e, xt, /il, e", e-", e'y<, e-'r.. 99 We have Fj, ~ X, -h 'i a. Let us write F in the following form and let us define a function S by the Jacobl equation ZIZ where C is a constant. Let us develop S and C in powers of e S •; So l--:S, + E^Sj H..., G ^. Co 4-:C,-i-£2Ci -+-.... In order to determine Sq, Si, S2, ••., by a recurrence method, we shall have the following equations ^-2l? ;? -:C„ d/i dv t-^'^S="-^" (2) dfi dv As I have already done previously, I shall designate every known function by $. In the second equation (2), I assume that Sq is known. In the third equation, I assume that Sq and Sj are known, and so on. Let us set with the condition Since Cg is arbitrary, the two constants o.q and 3o "i^Y be chosen arbitrarily. Nevertheless, it is important that we do not set Qq = 0. Following is the reason for this. Let us assume that it has been shown that dv ' dv " ' ' dv may be developed in powers of 100 t, f", e'O-.. We may conclude (if 60 is not zero) that the same holds true for l/^ <1^2 + ,'^' ^-... -+-.;. ^^'^ dv dv dj since the quantity under the radical may be reduced to go io^' e = 0. /98 This conclusion could not be reached if Sq were zero. It is important that this conclusion may be reached, due to the presence of the radical /u in F. Let us now consider the second equations (2) . The function $ which it includes depends on v and on y2, and has the following form The coefficients A are constants which may depend on oq and on Bq- The indices m and n may take all whole, positive, negative, or zero values. When removing it from the sign E, I have shown the term in which these two indices are zero. The second equation (2) then gives us the following with the condition Oi-i-aBPi= Aoo-t-C]. Except for this condition, the constants ai , 9>\ and C\ are arbitrary. I shall therefore assume that a, = p, = 0. I shall determine S2 by the third equation (2) . Due to the fact that this equation has exactly the same form as the second, it will be treated in the same manner, and so on. To sum up , the derivatives -r — and -r- may be developed in powers of If one compares this analysis with that given in No. 125, it may 101 be seen that there is an exact analogy between them. However, instead of having only imaginary exponentials we here have real exponentials 275. Once the function S has been determined, by applying the Jacobi method, we may arrive at series which are similar to those given in No. 127. The function S depends on v, on y2, and the two constants Oq and /99 60 . The energy constant C = Co+sCi-t-... is a function of ag and Bq- As a solution of our canonical differential equations, we then have the following equations dS dS dS dS '2; mt -^T:!t= -pr- =v. We have four equations from which we may obtain X2, u, y2 and v developed in powers of e, depending on oq , Qq, nj t + oji, n2 t + 0)2- By pursuing a line of reasoning exactly like that given in No. 127, we may see that 102 may be developed in powers of The same will hold true for i^u, x^ , and ji . I would like to add that all of these quantities may be developed in powers of E, Bo, e*'i'i,'+ra,), /3^e'".'+=='.', v'%>'~'"''"^°"'i and S — Sq may be developed in powers of If we set for the time being /lOO the two equations will take the form -,= £']/2, J3=e^„ (3) where ij;2 and \li^ may be developed in powers of I, ao, c="'(".'+'3.', /^e(".'+a.), ^/^g_!n,<-cj,i_ ^^^ jj^ [and, for example, we have and similar formulas for e±-'-y2, and /gQe""^] . In order to prove the postulate presented above, it is suffi- cient to apply the theorem given in No. 30 to equations (3). Let us now compare the results obtained with that given in Chapter VII, which I reiterated at the beginning of this chapter. We saw in Chapter VII that, in the vicinity of the periodic solu- tion ^1 = ri = ^2 = o, 103 the variables xi , yi, X2 , y2 niay be developed in powers of g±nii[i+c!,i^ Ad>";', X'e-"'.', el l- where A, A' are integration constants, n'^ and n'2 are absolute con- stants, depending only on the period of the periodic solution and the characteristic exponents. We have just seen that these same variables must be developed in powers of e^iin.,^c,i ^%e'".'^^.<, ^f,e-o:'-^.>. The two results clearly are in agreement. We may first set A = /B^e^^ A' = /07e"^'. In addition, n^ and n2 are constants, but constants which may be /lOl developed in powers of e, ag and Bq. and which may be reduced to u\ and n2 for e = ttQ = 60 = 0- We may then write, for example, r>ii't p{n,—fi')l and may then develop the second factor in powers of e, ag, Bq- 1" addition, the second factor will then be developed in powers of t. It is for this reason that we saw in Chapter VII the time t and its powers emerge from the exponential and trigonometric signs, which could have led to a certain amount of difficulty in certain cases. The pre- ceding analysis shows that this difficulty was entirely artificial. If I now wish to compare out result with those given in Chapter XIX, I shall consider the curves ri = o(''.), ^i = ?i(^-,) whose definition I presented at the end of No. 273. In order to obtain the equations for these curves, I need only take the expressions of X]^ and Yi and assign a constant value to ag , Bq » "1 t + iJiri . Then y^ and xi may be developed in powers of When n2t + '(JI2 is varied, it may be seen that the curves have the form which I described at the end of No. 273. In conclusion, I should point out that all of these results are only valid from the formal point of view. The series only converge in 104 /102 the case of asymptotic solutions, for which one obtains the equations by setting So = O, T32 = -i- »; I mean by this, setting or even setting ?o = o, nTj = — x; I mean by this, setting v/i^t"' = o, y/S^c-'^' = A', where A and A' designate the finite constants. 276. Let us proceed to the case in which there are more than two degrees of freedom. The preceding results may be generalized in two different manners. In order to explain this, it is sufficient to assume three degrees of freedom. It may happen that we may wish to study our equations in the vicinity of a system of invariant relationships xi = x\ = Xi =7i = o, which play the role of a generalization of the periodic solutions, in the sense of that given in No. 209. It may also happen that we wish to study them in the vicinity of a true periodic solution X, = a7j = sc3=xi —Xi = o. In the first case, there are four invariant relationships and one linear relationship between the mean motions, a relationship which we have represented in the following form, employing the change in variables of No. 202 if necessary In the second case, there are five invariant relationships and two linear relationships between the mean motions, which we have repre- sented in the following form Tii = o, nj = o. 105 We shall begin with the first case, and we shall set The equations remain canonical equations, and F' may be developed in powers of e , in the following form F'=Fi-+-£F',-H.... We then have F', = hi x\ -h hi x'j -1- A a; i> + 2 B a?', y\ ■+■ C^',« , or, removing the accents which have become useless, we have /103 Fo - hiXj -\- hiXi I- Kx] + -2 H ^•^y^ -i- C_yJ . The functions h2 , h3. A, B, C depend only on y2 and y3, and are periodic with the period Ztt with respect to these two variables. I am going to perform the change in variables of No. 274 again. Everything which I have stated remains valid, but only from the formal point of view . In order that I may apply the principles of formal calculation, it is necessary that there be a parameter with respect to the powers of which the expansions may be performed. This will be the parameter p. F and, consequently, h2, h3, A, B, C may be developed in whole powers of y. I should add that, for y = 0, B and C may be reduced to and that h^, h3, A may be reduced to constants which I designate as hO. h03 andA^. Let us try to integrate the following equations t-— ".. *--'- (1) 1 shall try to perform integration in such a manner that are periodic functions having the period Ztt of the two new variables y2 and y3 which must themselves have the following form The quantities n2 and n3 are constants which may be developed in powers of y; F2 and (03 are integration constants. 106 Equations (1) then take the form fly, tly, , c'/i , '(/i , (2) We shall set n,- - 11° -\- ['■■i'/' 'h il' n'r'' -t- . . . _ Ck^t^, „__ r-nnal-anta l-Viat the hV '"j and we shall assume that the n^ ^'s are constants, that the h^^ 's are periodic functions of yz and of 73 (the h°'s may be reduced to con- /104 stants, as we have seen), and finally that the y^ ^'s are periodic functions of y2 and y'3, except for the y^'s, which may be reduced to y'i- In equations (2) , we shall equate the equations having similar powers of y, and we shall have a series of equations which will enable us to determine the y^'^^'s and n('^)'s by a recurrence method. These equations may be written / «; ==-/.», 1 "1 ' ' <'y'. + = ■^. 1 «3 "♦- dy'z = *, (3) I shall designate every known function by $. In the second equation, I assumed that the y(°^'s and the n^°)'s are known; in the third equa- tion the y9's, the y^^'s, the n°'s, and the n^^'s, and so on. We then have J 'i J i' J 3 — y i ' /ij- /jj, «j — — ./(^, so that equations (3) may be reduced to .,«'^y:il+nl'i^+nr = 1>. (3') dy'i dy'z n{^> = 'l-. 107 to which we must add the following equations /.; ' - ,^. '-"V-""- (3") which may be deduced from the second equation (2), just as equations (3") are from the first equation (2). All of these equations may be Integrated in the same manner. Let us take, for example, the first equation (3'). The function $ which it contains, (like all the other functions $) is periodic in y\ and y'3 . We shall set n^ / equal to the mean value of this function, and by em- /105 ploying the procedure which we have already applied several times we shall be able to satisfy our equation by a function y^^^ which is periodic in y'2 and y'3. Having thus determined y2 and y3 as functions of y2 and y'3, I may set It is apparent that x\dy^^x\dy\~xt dyt — xj dy, , which is zero, is an exact differential and, consequently, that the canonical form of the equations is not changed when one takes x'2, x'3, y'2, y'3 for new variables, instead of X2, X3, y2, ya- The form of the function F is not changed either, but it may be seen that we have the identity — n,r', — njxi = hiXi-^- h,x,, which shows that the coefficients of x'2 and of x'3 may be reduced to constants. I may therefore assume that h2 and h3 are constants. I shall make this assumption from this point on . Let us now integrate the equations 108 "'^^=2(Bar, + C/,), ^ = -2(Aa:.4-B^,). or, which is the same thing . , dx-i , dxx ^"''"=''\ of e^^^''":'-"'.) , and of v/p^e'".'+'=''', v/?oe~<"''+°''- The constants ni , n'l and 1^2 may themselves be developed in powers of e, i/y, ccq u'q and Bp. 277. Let us proceed to the second generalization method, and let us assume that we wish to study the equations in the vicinity of a true periodic solution having the form ar, = ar, = i, = ^1 =^, = o. We shall set F = £> r, I, = ix\ , 7, = £/', , Tt = «a:',, Xt = E^;. from. which it follows that F = f;-+-sF', H-.... The equations remain canonical equations , and we have where $ is a homogeneous quadratic form in -^i, y*! , X2, ^2- The co- efficients of $ and h are periodic functions of ys = y'3. However, we shall remove the accents which have become useless, and we shall simply write F, = hx,-i-'P(xuy,, T,,^,). Just as in No. 274 and 276, it may be shown that we may always assume that h may be reduced to a constant. Let us now consider the equations 112 lU ~ ' dt dys. ' ^' ^'^i rfr, _ rf* £Xi— _^. ~dc ~ dyt' dt dxf They are linear and have periodic coefficients. Their general solution will have the form Xi = Aie"'?!.]-*- A,e-«',.,-i-y,B;j.,-(-a:',ij).,.,-)-y,9j.,, Xi = ar'i . and if), may be developed in series of sines and cosines of the multiples of 2tt (t + h) . T Let us consider the solutions which are near this periodic j^ili solution. According to the preceding statements, they may be written in the following form: x^ and y^ will be developed in powers of 2n - 2 quantities which are conjugate by pairs, and which I shall call A,e''.', A', e-^.' ■ Aje'.', A'.e-^.' A„_,e»-.', A'„_, £-«-.'. The A's and the A' 's are arbitrary integration constants. The ex- ponents a may themselves be developed in powers of A^Ai, A2A 2. •••> An-lVl- In addition, the expansion coefficients of x^ and of y^ are periodic functions of t + h, having period T. These coefficients (just like the exponents a) depend, in addition, on the energy constant C. We know that there is an Integral invariant Jy:dT,dy,; (2) from which it follows that, if g and y are two integration constants, we must have Zj\d? d-( d'l d<^ ) We could write this equation in another form. Let us assume that 3 is increased by 66, and that as a result for x-^, y^, A^e'*! have the following increases: 115 o.r,-, oj',-, oA/e'.' On the other hand, let us assume that y is increased by 6'y, and that .ng incr as a result we have the following increases for x , y.. • • t Our equation may be written - ( '-ri '.'vi — ?_/, f/.Ci) ^. consl. (3) The second number is a constant. By this I mean that it is a function of the integration constants multiplied by <56i5'y. We obviously have /113 oAe« = e^'(3A + tZi). On the other hand, we have S^, = __ oC + -^^^- ok +2 ^Z(A, .-^) '^^ '^' -^li ^A:,..--!'-) -^* -^ ="'' M = -.y, oC -I- 7,7,-; -r^v o(AkAk) (/(j .<- ia( Ak Ak) rf(AKAii) It can thus be seen that 6x and 6y have the following form ^yt = r^i + iM.i\ ^'yt -' ■>•,',• 4- t r/, f 3.r,- = 5,- 4- ; ^, ,. ; 3'^,. =^ ^'. 4_ ; J- ^^ where £.,£, ^.n.-n . are linear with respect to 6C, 5h, and to the 1 l.i 1 1.1 6Ae s and {A'e"'* s. In addition, they may be developed in powers of the Ae*^^ s and the A'e'Ct's and the sines and the cosines of the multiples of — (t + h) . The expressions of 6'x , i5'y. may be readily obtained. It is sufficient to change 6 into 6 ' in those of 6x and (5y . . It may be then seen that equation (3) may be written in the following form D + E« + F^! = const., from which it follows that F- S($,./l',..— ?'...■ ^, I..) are developed in powers of the Ae""^^ A'e"*^^ s and the sines and cosines of the multiples of — (t + h) , and they are bilinear with respect to the 116 5'Atf-^', 3'A'e «', S'G, o7). The first term must be independent of t, and we shall have which has already provided us with certain verification relationships which must be satisfied by the expansions of the Xj^'s and the y-^'s. Thus, D must be independent of t. It will therefore be linear /114 with respect to the following determinants aX^.o A'j. — o'AiAAi, A'a A;(oAj.o'Ay — oAy o'A/,), Ai.(cAA3'G-o'A;,i5C), (4) Ai(oAi o'/j -- o'Ax-o/i) (or with respect to similar determinants determined from the former by interchanging Aj^ with A'j^, or A^ with A' .). The coefficients will be developed in powers of the Aj^Aj^'s, and will depend in addition on C. The time must disappear. The exponentials must therefore disappear, which can only happen if each factor Ae'^t is multiplied by a factor A'e""^ or 6A'e-«t, or 6'A'e-"t. A new series of verification relationships may thus be deduced from this. 279. Among the ay^ exponents, some are imaginary, and others are real. Among the real exponents, some are positive, and others are nega- tive. However, since I may arbitrarily choose an exponent which I may call ot]^ from between two exponents which are equal and have opposite sign, I shall not limit the conditions of generality by assuming that Oj^ is positive if it is real. Let us now cancel the coefficients Aj^^ which correspond to an imaginary exponent, or to a positive exponent. We will then have the following, if aj^^ is real A;[. =0, A'^>o 117 ^ x and if a is imaginary A^-^Al. In addition, I shall set where C is the value of the energy constant which corresponds to the periodic solution under consideration. Our series will then be convergent, and will represent the asymp- totic solutions which we studied in Chapter VII. They include h and the A^'s, which correspond to negative exponents, as arbitrary constants. We shall therefore have 2n equalities which will express the x 's /llS and the y. 's as functions of t and of these constants h and AJ. If we eliminate t, h and the V's between these 2n equalities, we shall have a certain number of invariant relationships between the y^'s. If a group of values of the x 's and the y. 's is regarded as repre- senting a point in space having 2n dimensions, these invariant relation- ships will represent a certain subset V of this space. This is what I shall designate as the asymptotic subset . Let us reconsider the integral invariant X d.Ti d/i f^ and let us extend the integration over a portion of this asjnnptotic subset V. In other words, let us assume that every system of values of the x.'s and the y.'s, which form a part of the integration region, satisfies our invariant relationships. I may state that the integral invariant will be zero . It is sufficient for me to demonstrate the fact that and this is apparent , because we have A/c — o, G = Co, from which it follows that N 118 SA^ = o, 3C = o, S'Ai = 0, o'C = o, which shows that all of the expressions (A) are cancelled. We can also set C — Co, Ai^), A'<. = o (for real a^) , A4.= Ai = o (for imaginary a]^) . We shall have obtained a new series of asymptotic solutions and, conse- quently, a new asymptotic subset to which the same conclusions will apply. The procedure which we followed for the invariant (2) could be followed for an arbitrary bilinear invariant (invariant of the third type. No. 260), i.e., having the form /116 CCzudxidx^, (5) where B is a function of the x^'s and of the y^^'s and where one or two of the differentials dxj^, dx^ may be replaced by iy-^ or dyj^ under the sign 1. The expression will still be linear with respect to the quantities (4) . This would still apply to a quadratic invariant (invariant of the second type. No. 260) having the form J^/I^Td^^d^t, (6) where B is a function of the x^^'s and the y^j^'s, and where one or two of the differentials dx^, dxj^ may be replaced by dyj^, dyj^ under the sign Z, It may be seen that the expression SB IxiZxk must be linear with respect to the expressions / SA^oAi, ) Ai.A}5Ai8Ay, ^^1 J Ai8AiSC, SCSA 119 \ and to those which may be deduced from them when interchanging Ajj^ and A'l^., Aj and A' j . For every asymptotic subset, the invariant (5), like the invariant (6), must be cancelled. Another Discussion Method 280. This same study may be pursued farther, while presenting it in a different form. For example, we shall assume that we are dealing with a problem of dynamics, that the x^ s are the coordinates of different points of matter of the system, and that the conjugate variables y^ are the com- /117 ponents of their momentum. We plan to study the integral invariants which are algebraic with respect to the Xj^'s and to the y^^'s, and to determine whether one may exist in addition to the one which is known, and which is written // S dxi dyi. We have seen that, in the vicinity of a periodic solution, the x^'s and the y^'s may be developed in powers of the Ae^^'s, . . . . We are going to consider these expansions again, but we shall assume that the value of the energy constant corresponding to the periodic solution is zero, so that the expansions will not only proceed in powers of the Ae"*^'8, but even in powers of C. In addition, they will depend on t + h. By equating the Xj^'s and the y^'s to these expansions, we obtain 2n equations, which we shall solve with respect to the Ae^'^'s, C and t + h. We have G = '!>, We should point out that oq . like a.^, may be developed in powers of C and of the A^/iij^'s. It may be seen that fj^^, f^, $, cosG, sine are uniform functions of the x^'s and the yj_'s in the vicinity of the periodic solution. In addition, the Xj^'s and the y^^'s may be developed 120 V-^ in powers of the fj^^'s, the f^'s, and $, and according to the sines and cosines of the multiples of 0. On the other hand, the expression which corresponds to the invariant (2) , or the similar expressions which would correspond to another bilinear invariant of the form (5) , must be developed in powers of the f^^, fj^, $'s and be bilinear with respect to /118 Vk, s/i., S'J>, Se, «'/*. o7i-> S', o'e. In addition, when we replace fj^, f'j^, $, by their values (7), this expression must be independent of t. The time t may be introduced in three different ways: 1. In the exponential form; 2. In the form of the cosine or sine of the multiples of (t + h) ; 3. Outside of the exponential and trigonometric expressions (and, as we shall see, of the second degree and more). It must not enter in any of these three ways. 1. In order that it does not enter in the exponential form, it is necessary and sufficient that the expression be linear with respect to the following quantities which are similar to (4) ^/k ^''f'k — ^'fk ?/I, /U3Ao''l>-o'/^.5'^), (8) where the coefficients may be developed in powers of the f^^^'s, fj^^'s, and of $. 2. In order that t does not enter in the trigonometric form, it is necessary and sufficient that our expression does not depend on 0, but only on its variations 60, 6'0. 121 3. We must now determine the condition under which t does not enter outside the exponential and trigonometric expressions. We should point out that we have We may distinguish five types of terms in our expression, depending on whether they contain as a factor a quantity (8) included in the first, second, third, fourth, or fifth line of the table (8). Under this assumption, if we replace 6f , ... by their values (9), we shall see that the five types of terms include as a factor, respec- /119 tively, (3A*8'Ai— SA'*S'A*)H-<[8»«S'(A*.\'*)-S'a*o(A*Ai)], A'*A}(8A*o'Ay-8AyS'At) + AiA)<[A*(Sa*8'Ay— o'a*5Ay) — A/(Sa/S'A*-o'aySA*)] -1- AxrAiAyA;t«(Sat8'a; — SayS'.ai), ( A'*(SA*8'C — S'A*8C)H-A*AVtCS«*8'C-8'a*SC), ^q^ A'i(8A*S'Po-S'A*8fJo)-i-A*Ai«(8iA-8'Po-8'a*8Po) + Ai/(8A*3'oo — S'AA■8ao)-^- AiAi^Voiio'"*— o'"*^"*). (SCS'P,— 8'C8?o)+<(8GS'o,-S'C8I(,).• It may be seen that the time can enter in the second power. Let us first make the terms for t^ disappear. They may only begin with terms of the second type or of the fourth type. It may be stated that the coefficient of i'(Sat8'ay — 8ay8'at) must be zero. In actuality, due to the fact that the virtual displacements in the constants are arbitrary, we may asstraie that all the 6a. 's vanish, with the exception of 6a , and in the same way it may be assumed that all the iC 6'a. 's vanish with the exception of 6'a . All the terms in t^ cancel, with the exception of the term in «»(8a*8'ay — 8a/8'i^). There would be an exception if there were a relationship between the n - ] exponents o . . We could no longer assume that all the 6a 's 122 cancel except one, unless the last one itself cancels too. There are now four terms of the second type which result In terms For purposes of brevity, I may write them in the following form l};! (Ill -H i)*! "> "*" 't' J "» *+■ 4'k "* > The ijj'sare developed in powers of the f^, f'^ and of $. I have employed iMl to designate the expression which appears in the second line of the table (8): 11^ 0)2 may be deduced from ui by interchanging f^^- and f'^, 0)3 may be deduced from o)i by interchanging fj and f'j, 0)4 may be deduced from 0)^ by making these two permutations at the same time. In order that the terms in t^ disappear, it is necessary and suffi- cient that 'I'l — 'J's — "^i -+- ^i =0. If this condition is fulfilled, our four terms will provide us with the following terms in t (11) -i-{^, — 6t)t \j A;[ Ssy o'( Ai- A'i) — o' ay o( A/, A'^.)]. Let us now consider terms of the fourth type, which we shall group together by pairs. Let the following be one group of two terms where fi and i|j2 may be developed in powers of C and of the Aj^^Al^^' s, where o)^ is the expression included on the fourth line of the table (10) , and where 0)2 is that which is deduced by interchanging A^ and AlJ^ and changing a^^ to In order that the terms in t^ disappear, it is necessary that 4'i = ). becomes. We must also have identically -l-SO*[o(AxAi.)&'au-o'(AiA'i.)oj„l-f-D„(oG'!'ao-o'Coao) = o. For purposes of brevity, let us write Yj^ instead of \^^, yo instead of C and instead of We have or d(u, v) ouo'c — ZvZ u ; SSBt ^ d(v^, Yt.-)+ S2Di ^° d(Y,, Yy)-4- SDo ^ ^(>, y> ) = o. "7/ "Ty "".'y Under the sign H or EE, k may take on the values 1, 2, ..., n-1 and j may take on the values 0, 1, 2, ..., n-1. When setting the coefficient of 3(y4, Yfc) equal to zero, we obtain d-(j d-'^k a-zj d-^k 124 By setting the coefficient of 3(yo. Yj) equal to zero, we have "'To ' d-i, " d-^j (12') These equations indicate that is an exact differential. We must set yj = in equations (12) and (12'). The ^'s are there- fore constants. The a.'s are therefore linear functions of the y's. In actuality, as we have seen, the a's may be developed in powers of ^ /122 the y's. However, the result which we have just obtained is only valid if we neglect the squares of the y's, and if we stop the expansions of the a's at the terms of the first degree. In addition, the B's and D's are constants. The expression (13) is therefore the exact differential of a polynomial of the second degree. In order to carry this investigation further, let us express the of 7o, 7i, •■■> Ti-i. a 's not only as functions of k but also as functions of ^'o, Yi. •■•, Y«-i' In order to avoid any confusion, let us employ 3 to designate the deriva- tives chosen with respect to the new variables, and the d's to designate the derivatives chosen with respect to the old variables. It may then be seen that is an exact differential, which entails the following conditions „ d^^ „ rJsf, (14) 'h - - = L>/ , -• If one knew the relationships between the a's and the y's, these equations would allow us to determine the coefficients B^^. We can express ^D^Y-j as a function of the variables 'oi Ti' Ti' •••> T"-i while writing 125 \ The E, 's will be given by the equations and Eq may be chosen arbitrarily. It is necessary that equations (14) be compatible, which requires certain conditions in the case of n > 3 doi dxi doj _ Ja£ dij d^_ (15) 'o^i O'lj o^k " ''r* ''y' '^ij These conditions (15) will always be fulfilled, since there is always an integral invariant /123 /= tlxidyj. If there are several integral invariants which do not vanish iden- tically for the periodic solution under consideration, a system of values of the coefficients Bj and E. must correspond to each of these invariants . If equations (14) have q solutions which are linearly independent, we may calculate the corresponding values of the E^'s by means of equations (14'). Since Eq remains arbitrary, we shall have q + 1 systems of values, which are linearly independent, of the coefficients B. and E . We may therefore have q + 1 different integral invariants (if the periodic solution under consideration is not singular, with the meaning attributed to this word in No. 257), but we cannot have any more. 282. I stated above that conditions (15) were definitely fulfilled; there may still be some doubt on this point. If equations (14) have q different solutions, we may have q + 1 invariants. If there is only one invariant, we could assijme that q = 0. The presence of a single invariant / 'S.dxidyi would not enable us to state that equations (14) definitely have a solu- tion. This is the doubt which I wish to dispel. I would first like to note that in the case of the three-body 126 problem, there are not one, but two integral invariants. In Volume I, Chapter IV, we studied the variational equations of this problem. On pages 170 and 172 we obtained the following integrals yui —y ^ m Jmi dx In the same way, we could obtain d^ , , (1) 5 = consl. (2) 7124 ^yr/ yr^dV „ . (1') > <—^ — >—- f = const. ^ III ^.i dx Let us multiply (2') by (1), d') by (2), and let us subtract. We then have ^d\m dx I ^g^ ^^/Zl _ g t') 1(^3.^, _t-75) = consl. The first term is linear with respect to the determinants having the form We therefore have an integral of the variational equations, and we may deduce from it a new bilinear integral invariant. In the case of the three-body problem, we therefore have at least q=l, and it may be stated that conditions (15) are fulfilled. 283. Is this still true in the general case? Let us assume that it is not. Then all the coefficients which we have called B^ must be zero, as well as all of the Ej^'s, with the exception of Eq. Therefore, when we attribute the values corresponding to the periodic solution under consideration to the Xi's and the yi's, i.e., when we set C =A,--u e,c>,) However, this is impossible, since the first term is a bilinear form with determinant 1, and the second is a bilinear form with determinant 0. We must therefore conclude that conditions (15) are always fulfilled. 284. Let us now try to determine whether equations (14) may have several solutions. Let I^'.- BV B' be these two solutions and let us assume that we do not have B,- and then the two equations Hi — — = Ui -— , 128 will imply Then the indices I, ..., 2, n will be divided into a certain number of groups, as many groups as there Bi different values for the ratio rr". Two indices will belong to the same /126 group, if they correspond to the same value of the ratio -rf-. ^i In order that o.^ depends on Yi (or a^ on Y]^) > it is necessary that the indices i and k belong to the same group . In order to formulate these ideas clearly, let us assume that there are only two groups containing the indices, respectively, Then will depend only on and will depend only on I. ' P, /) -r- I , /) 4- 2, . . . , n — I . '0. vi. Te. ••■, r/'. 'o, T/..-1. V*-"-' ■•■. T"-!- It then appears that the characteristic exponents aj^ form several indepen- dent groups, in such a way that the aj^^'s of one group do not depend on the products AjA'. corresponding to another group. The periodic solutions for which this condition will be produced (or for which there would be one relationship between the a^'s) may be called particular solutions. We therefore arrive at the following conclusion: In order that there be another algebraic invariant, in addition to 129 % V those which we know, it would be necessary tha t all the periodic solu- tions be particular solutions, or thau they all b e singular solutions. with the meaning KJven in No. 257 . I shall not try to demonstrate the fact that this condition could not occur in the three-body problem, but this would seem to be very unlikely. Quadratic Invariants /1^7 285. Let us now study the quadratic invariants from the same point of view, i.e., the integral invariants having the form //p. where F is a quadratic form with respect to the differentials dx^, dy^. Let us set where the H's are functions of the x's and the y's, and where the product dx dx may be replaced in certain terms by the product dx^dy^^ or dy^dy^^. X K We may then write the following equation which is similar to equation (3) of No. 278 ZMoxi^ivj; — const. H) On the other hand, we find in No. 278 that oxi ^ ^, + l^,h 3/,- =--= Tj, + tr,,j. We may then write equation (1) in the form D ■(- lie + F r- -= con?t., where D, E, F may be developed in powers of the A^"*^ s , A^"""^ s and of the sines and cosines of the multiples of y^ (t + h) , and where D, E, F are quadratic with respect to the oXc-J-', o.S.'e-=", ZC, ?,h. We must therefore have E .--: V r. O, and, in addition, D must be independent of t, which shows that D must be /128 130 linear with respect to the following expressions A';. oAa- oC, oG o/i, or with respect to the expressions which may be deduced by interchanging k]^ and A!y., or Aj and A'^ . The coefficients will be developed in powers of the products k■^^\ and of C (if one assumes that the periodic solution corresponds to the zero value of the energy constant) . 286. Let us return to equations (7) given in No. 280, and let us pursue the same line of reasoning as given in No. 280. We shall find that the expression n .-^ I II or, or,, must satisfy the follow \ig conditions when the x^'s and yi's are replaced by their expansions as functions of the fj^, fj^^, $ and s; 1. It must be linear with respect to the following quantities: (8') fi/'j 5/a- 5/;. o'l. 00, o'l.v^e' where the coefficients are developed in powers of the f^ffe's and of $. 2. It will not depend on 0, but only on 60. 3. If these conditions are fulfilled, expression n will not include the time, neither in the exponential form nor in the trigonometric form. We must now determine the condition under which the time is not included outside either the exponential or trigonometric terms. Let us consider equations (9) again from Section No. 280. We shall find that the following terms will correspond to the different terms UAl given in the Table (8'): 131 N A'^.A^oAi o\j-hA'/,:Vjt(\),Zn. SA; •+- XjOxj oA*) A^. oAioC h X^.X'^to:ij,?jC, (10') A'i.oA^o3o + A:iV(oA^.oao + AiOaiO,3o)-f-Ai.A'^.;'5i^8i(,, Let us first make the terms in t^ vanish. The entire group of these terms is a quadratic form with respect to This quadratic form must be zero. The coefficient of Saj^Sa^t^ must therefore be zero. However, there are four terms which could introduce the product t^6aj^6a^ ; these are the terms in fkfj^^j^M,, /*/.?/* 5/;, fkfjZfWj, hfjinyy For purposes of brevity, let us designate these four expressions by loi, U2, "3, ui+. The entire group of our four terms may then be written where i^^, i|/2 , ^-t, and ^b, may be developed in powers of the fj^f'j<.'s and of $. In order that the coefficient of t^Saj^^Saj vanish, we must have identically 'l'l+ 'J-S + '5-3 + ■>» = O. In the same way, the coefficient of t^S^aj^ must vanish. It arises from terms in Va-/i-, si^n, nyi- For purposes of brevity, let us designate these three expressions by '*''l » ^li ^Zy s'^d the entire group of the three terms by where i|j\ , 1J/2, ip'a may be developed in powers of the fj^f, 's and of $. In order that the coefficient of t^S^Uj^ may vanish, we must have /130 (11) 132 For the periodic solution, we must have fi=f\ =/l =/*« = •■• =fn-\ =/'/.-( - o. All the terms including as a factor one of the expressions appearing on the 2nd, 3rd, or 4th lines of the Table (8') must then vanish, because each of these expressions includes fj^ or fj^ as a factor. The only terms of expression IT which do not vanish for the periodic solution are therefore the terms in Equation (11) shows that ^i contains fj^fj^. as a factor. Therefore, the term i|; j 6 f j^^S f j^ must also vanish. We then have only the terms in The first does not Include t, the second includes it in the first power , and the third includes it in the second power. Due to the fact that this third term is the only one which includes t^, it must be zero. If it is zero, the second term will also be zero, due to the fact that it is the only one which includes t. Finally, all the terms of IT vanish for the periodic solution, except the term in 6i|>^. In the general problem of dynamics, just as in the case of the three- body problem which we have designated as the restricted problem, the general reduced problem, and the planar reduced problem , we have a quadratic in- variant, but no more than one. I may write the energy equation in the following form F :-= const . This invariant is nothing else than and the term in 6$^ which does not vanish corresponds to this invariant. If there is a quadratic invariant, other than that which is known, /131 this invariant must vanish for all points of the periodic solution. In other words, this periodic solution must be singular in the sense of the meaning given in No. 257. 133 There would be an exception, if the n exponents '^a, o,, or,, ..., a„_i were not independent of each other, but if there were one relationship between them. In this case, the coefficient of t^, which is a quad- ratic form with respect to the n variables oil), Sli, ..., Olfl-i, could vanish without all of its coefficients being zero, since these n variables will no longer be independent. To sum up, in order that there may be other quadratic invariants, in addition to those which we are acquainted with, it is necessary that all periodic solutions be singular or particular . It is very unlikely that this will be the case for the three-body problem. Case of the Restricted Problem 287. We may conceive of another discussion method which we shall only apply to the case of the restricted problem. The discussion presented in No. 257 has presented the possibility of two quadratic invariants, of which one is known. Let us assume that these two quadratic invariants exist, and let II be the quadratic form corresponding to one of these invariants. According to the preceding statements , IT may include terms in ( y.5/,, /;2/.S'^> /.^/i^'i-. /,o/.«9, /.3/,59, (j^) On the other hand, IT is a quadratic form with respect to the quantities o.f,, Zxi, Zyi, o/j, whose coefficients are the algebraic functions of x^ , x^, y^ , y^- /132 Following are the variables x and y which we shall select. In this problem, which I have called the restricted problem, two of the bodies de- scribe concentric circumferences, and the third (whose mass is zero) moves in the plane of these circvraif erences . I shall refer this third body to moving axes turning uniformly around the center of gravity of the first two. One of these axes will constantly coincide with the line joining these two first bodies. I shall use x^ and x^ to designate the co- ordinates of the third body with respect to these moving axes, and y, and y to designate the projections of the absolute velocity on the 134 moving axes . Let us then set '!> r^ F -+- (0 G, where F and G designate the energy function and the area function in the absolute motion, and where o) designates the angular rotational velocity of the two first bodies around their common center of gravity. The equations take the canonical form (/.r,- _ fM> dfi _ c/•^ dt ~ dfi ' dt "' dxj The integral $ = const, is nothing else than "the Jacob! integral" (see Volume I, No. 9, page 23). Under this assumption, our expression n will be a quadratic form in o.r,, ox,, y',, ly~, for which the coefficients will be algebraic in x^ and y^. If we assume that the four variables x and y are related by the relationship •!' ..= const., which entails the following condition o'^ r^ o, our four variables Sx^ , 6y^ will no longer be independent. One of them could be eliminated, and n will become a ternary quadratic form. Let us consider one point of the periodic solution. For this point, we shall have All the expressions (1) will therefore vanish with the exception of /133 5/,?/;, Be', S'l"'*? and ''"'■ If we set 6$ = 0, they will all vanish with the exception of Vi'^f\ and 5e'. Therefore, for a point of the periodic solution, let us set n--=B5/,J/', + Coe'. The entire group of terms for t^ will therefore be reduced, for 135 this same point, to (see, supra. Table 10') and, since fi = i\ = 0, may be reduced to CrSaJ. The terms in t^ must vanish. The latter is the only one which does not vanish for the point under consideration; all the others are zero, even when the condition 6$ = is not imp osed , because &'^6Q and 6$ do not _ provide terms in t^. However, Solq is not also zero. For one point of the periodic solu- tion, we have 'J/\~"J/\'''d§~ °' dao but we cannot be sure of having = 0. This would assume that there d$ is a continuous infinity of periodic solutions having the same period, which does not occur. dao Nevertheless , it may be noted that -jT~ includes the small quantity which I may designate by y as a factor, i.e., the mass of the second body. Consequently, it may be noted that 6ao vanishes for y = 0, i.e., in Keplerian motion. The terms in t^ can only vanish if we have from which it follows 11 =B3/,3A. However, this latter equation would indicate that II may be reduced to a binary quadratic form and, consequently, that its discriminant is zero. Thus, the discriminant A of n must vanish for every point of /134 every periodic solution . 288. However, an algebraic relationship such as cannot be valid, unless it is reduced to an identity, for every point of every periodic solution. If the relationship a = o 136 is supplemented by two other relationships F-P, G-v (3) (where 3 and y are two arbitrary constants, and F and G are the two functions which were designated in the preceding section) and a fourth arbitrary algebraic relationship H-o, (4) the number of solutions of these four algebraic equations will be limited whatever the constants g and y may be. Let us now consider a periodic solution, and the variables x^^ and y^ will be developed in powers of y in the following form ( ,r/ — x'i -h iixl -h. . . In the same way, F will be developed in powers of y, and we shall have and G and H will be independent of y. The quantity A remains. It may be stated that this function, which is algebraic in x^ and y^ under the terms of the hypothesis, also depends algebraically on y. If we state that f^n is an integral invariant, we will be led to certain relationships which include the coefficinets of II , their derivatives, and the coefficients of the differential equations of motion. We assumed that n is an algebraic function of the Xj^'s and the y.'s. We may assume that this algebraic function is included as a special case /135 in a definite type, not containing y explicitly, but depending algebraically on a certain number of arbitrary parameters. The quantity / /jf will not be an integral invariant no matter what these parameters may be, but only when these parameters take on certain special values depending on y . When stating that / v^ is an Integral invariant, one Is led to certain algebraic equations between y and these parameters. These equa- tions must be compatible, and it is apparent that the parameters will be obtained as algebraic functions of y. 137 The coefficients of the form II and A will also be algebraic in y. The equation A = is therefore algebraic in y, and we may assume that it has undergone a transformation in such a way that the first term is a whole polynomial in y. We may therefore write In addition, Aq will not be identically zero, unless A is. If Aq would vanish, A would contain a factor y which could be made to vanish. The function A must vanish when the x^'s and y^'s are replaced by the expansions (5). It may then be developed in powers of y and, due to the fact that the term which is independent of y must vanish we shall have (2') We should now point out that we must have I G(:r°,/°)--Yo. ^^ ^ where Bq ^^^ Yo ^^^^ constants. In order that this may be the case, it is sufficient to recall that, for y = 0, the motion may be reduced to Keplerian motion. Now, for example, let us take /136_ and let us write the equation (x^l' + Ca-?)'^ r. (4') If we set y = 0, we may then observe that the third body will describe a Keplerian ellipse. Let 5 and n be the coordinates of this body, not with respect to the moving axes, but with respect to the axes of symmetry of this ellipse. The equations of the Keplerian ellipse will then be written I ,) = T,isinijj + T,jsin2tp + The coefficients E.^, r]^ will depend on two constants which are the major axis and the eccentricity of the ellipse and, consequently, on Bq 138 and Yo • We shall have o = 7i| ? 4- raj, where the mean motion ni depends on Bq and where a)^ is a new integration constant. The intersection of the ellipse (6) with the circle ^5 _1- T,» = I will occur at two points which will be given by the equations $ = cosO, T, = +5inO, ip = ±(po. We will then have (7) :f2 = ^ sin (to ; -i'W,)— r, cos(io? -1- ra,), (8) where (1)2 is a new integration constant. We shall obtain solutions of the equation (4') by combining equa- tions (7) and (8) , which yields (k is an arbitrary whole number) . In order that the solution be periodic, it is necessary and suf fi- /137 cient that the ratio -^ be commensurable. Let us write this ratio in the form of a fraction reduced to its most simple expression, and let D be its denominator. It may be seen that equation (4') has 2D different solutions. Equations (2'), (3'), and (4') must have only a limited number of solutions, no matter what the constants Bq and yo ™ay be. I may choose 60 in such a way that -^ has the value which I desire, and consequently that D may also be as large as I desire. This can only occur if Aq, and consequently if A, are identically zero. Consequently, the discriminant having the form n is identically zero, and this form must be reduced to a binary form. 139 It could be shown in the same way that, in the sense of No. 257, it is impossible that every periodic solution be a singular solution. This has only been proven in a very special case, but it is possi- ble that this proof may be extended to the general case. 289. The form 11 regarded as a binary form, must be reducible to for one point of a periodic solution. The binary form will therefore be definite (i.e., equal to the sum of two squares) if the periodic solution is stable — i.e., if the characteristic exponents are imaginary. It will be indefinite (i.e., equal to the difference of two squares) if the periodic solution is unstable — i.e., if the characteristic exponents are real. Let us assume that y is very small, and let us reconsider equation (4') . According to the principles outlined in Chapter III (No. 42), for a /138 given value of Bq, we shall have at least two periodic solutions, of which one is stable and one is unstable. Let be the corresponding values of the constants Hi and (^2. Let us set Oh (?o — ='i) + '^j^Y. 9-i- — (?o—ran + '^s = •!''. and equation (4') will give us, for the first periodic solution, :r; = COS I i)/ -\ 1 and for the second Without restricting the conditions of generality, we may assume that ^" > i|j' and that ijj' and \p" are contained between zero and il. Then the D form n will be 1)40 definite for x]---. cosf^ -h"^) indefinite for x\ r.-. cos U" i- ^^j definite for t\=. cos^^y -^- -^- indefinite for -!== cos(,/+ i^^: definite for :r;== cos(i>' + 2:1), indefinite for .rj -- coscl-' +2-); which shows that the discriminant of H, considered as a binary form, must vanish at least 2D times. Just as above, It may be concluded from this that it is identically zero. The form n may therefore be reduced to a square term. Therefore, since it must equal B 0/, o/'i for every point of a periodic solution, it must vanish for all of /139 these points. The same line of reasoning would show that it is identically zero. To sum up, there is no other quadratic invariant except the one which is known, at least for the special case of the restricted problem. 141 CHAPTER XXVI POISSON STABILITY Different Definitions of Stability 290. The word stability has been understood to have several dif- /140 ferent meanings, and the difference between these meanings is clearly apparent if we recall the history of science. Lagrange has shown that. If the squares of the masses are neglected, the major axes of the orbits are invariant. This means that, with this degree of approximation, the major axes may be developed in series whose terms have the following form where A, a and 6 are constants. If these series are uniformly convergent, this results in the fact that the major axes are contained between certain limits. The system of stars cannot therefore pass through every situation which is compatible with the Integrals of energy and area, and furthermore It will repass an infinite number of times as close as desired to the initial situation. This is complete stability. Carrying the approximation further, Poisson demonstrated that the stability continues to exist when one takes into accout the squares of the masses and when the cubes are neglected. However, this does not have the same meaning. He meant that the major axes may be developed in series, containing not only terms having the form A sin(a^ -h jB), but also terms having the form /141 \liin{at-i- P). The value of the major axis then undergoes continuous oscillations, but nothing indicates that the amplitude of these oscillations does not increase indefinitely with time. We may state that the system will always repass an infinite number 142 of times as close as desired to the initial situation. However, we may not state that it does not recede from it very much. The word stability does not therefore have the same meaning for Lagrange as for Poisson. It is advantageous to point out that the theorems of Lagrange and Poisson include one important exception: They are no longer valid if the ratio of the mean motion is commensurable. The two scientists concluded from it that stability exists, because the probability that they are commensurable is infinitely small . It is therefore advantageous to provide an exact definition of sta- bility. In order that there be complete stability In the three-body problem, the three folowing conditions are necessary: 1. None of the three bodies can recede Indefinitely; 2. Two of the bodies cannot collide with each other, and the dis- tance of these two bodies cannot desend below a certain limit; 3. The system repasses an infinite number of times as desired to the initial situation. If the third condition alone is fulfilled, without knowing whether the first two conditions are fulfilled, I would say that there is only Poisson stability . A case has been known to exist for a long time for which the first condition is fulfilled. We shall see that the third condition is ful- filled also. I can say nothing with respect to the second condition. This is the case given in the problem of Section No. 9, where I assumed that the three-bodies move in the same plane, that the mass of the third is zero, and that the first two describe concentric circumfer- ences around the common center of gravity. For purposes of brevity, I shall call this the restricted problem . Motion of a Liquid /142 291. In order to provide a better explanation of the principle un- derlying the proof, I am now going to present a simple example. 143 \ Let us consider a liquid which is enclosed in a vessel having an in- variable form and which is completely filled. Let x, y, z be the coor- dinates of a liquid molecule, u, v, w the velocity components, in such a way that the equations of motion may be written 1^ ^ ^ = f^ :. ,/,. (1) The components u, v, w are functions, which I assume to be given functions, of x, y, z and t. I shall assume that the motion is steady, in such a way that u, v, w depend only on x, y and z. Since the liquid is incompressible, we shall have t/u (!v d>v . — t- - -,- - - -„. ilx dy dz In other words, the volume is an integral invariant, / Lv dy dz Let us study the trajectory of an arbitrary molecule. I may state that this molecule will repass an infinite number of times as close as desired to its initial position. More precisely, let U be an arbitrary vol- ume inside of the vessel, which is as small as desired. It may be stated that there will be molecules crossing this volume an infinite number of times. Let U be an arbitrary volume inside of the vessel. The liquid molecules which fill this volume at the time will fill a certain volume U, at the time t, a certain volume U„, ..., at the time 2t , and a certain volume U at the time nx. n The incompressibility of the liquid or, which is the same thing, the existence of the integral invariant, indicates to us that all the volumes /143 Uo, U„ Uj u, are equal. Let V be the total volume of the vessel, and if V<(n-i-,)U<„ we shall have 144 V i) . Let M be a point in common to U^ and U^.. The molecule which is at the point M at the time ix is, at the time 0, at a point Mq belonging to Uq, since the point M belongs to U^. In the same way, the molecule which is at the point M at the time kx is, at the time (k-i)T, at the point Mq, since the motion is steady. On the other hand, it is at the time at a point Mi belonging to Uq, since M belongs to U^, and we must conclude from this that Mq belongs to Uk-i- Therefore, U^ . and Un have points in common. •^"J- q.e.d. There fore, it is possible to choose the number a in such a way that Uq and U^ have a part in common. Let U'o be the part in common, and let us form If i , U2, ■••. with I^q, as we formed Ui , U2 , . . • , with Uq . We may obtain a number g in such a way that U'o and U'g have a part in common. Let If' be this part in common. We may obtain a number y in such a way that ifj and IT^ have a part in common. This procedure may then be continued. As a result, U'q is part of Uq, U" of U'o, and U'q" of U',], ... In general, uJP"^^^ will be part of U^P^ When the number p increases in- definitely, the volume uj^^ must therefore become smaller and smaller. According to a well-known theorem, there will be at least one Point, perhaps several, or perhaps an infinity, which belong at the same time /144 to Uq, to U'o, to U'o' , ..., and to ujp) , however large p may be. 145 This group of points, which I shall call E, will be in a measure the limit toward which the volume u(p) tends, when p increases indefin- itely. ^ It may be composed of isolated points; however, it may be somewhat different. For example, it may happen that E is a region in space having a finite volume. A molecule which will be inside of U^, and, consequently, inside of Uq, at the time zero, will be inside of Uq at the time -ar. A molecule which will be inside of U'J and, consequently, inside of U' at the time zero, will be inside of l/g at the time -gx, and, consequently inside of Uq at the time -(a + B)t. A molecule which will be inside of U'J' at the time zero will be in- side of U'(j at the time -yx, inside of Ujj at the time -(g + y)t, and in- side of Uq at the time -(a + 6 + y)t. Since Uq" , Uq , U'q are part of Uq , this molecule will be inside of Uq at four different times (multiples of t) . In the same way, and more generally, a molecule which is inside of U^P-' at the time zero will be inside of Uq at p different previous times (which will equal the negative multiples of t) . Since E is part of U^P^ , however large p may be, as a result a mole- cule which, at the time zero, is part of E will cross Uq an infinite number of different times, which all equal a negative multiple of x. There are therefore molecules which cross the volume Uq an infinite number of times, however small this volume may be. q.e.d. The equations dx ily u'o>u;>..., and, if E has a finite volume which I may also call E, no matter what p may be, we have E "Zi since, without this stipulation, every molecule which was located in this common region at the time zero would cross Oq, and consequently Uq , more than k times between the times -nx and 0. We therefore have (n-\i). Let us now set n = k^ + x. The probability that our molecules does not cross Uq more than k times between the times -(k^ + x)t and 0, will be smaller than . bV ( ^3 + a? H- I ) Uo ' It tends toward zero when k increases indefinitely. The probability P that our molecule does not cross Uq an infinite number of times between the times — °° and is therefore infinitely small. In reality, this probability P is the svan of the probabilities that the molecule crosses Uq only once, that it crosses Uq twice and only twice, that it crosses Uq three times and only three times, etc. However, the probability that the molecule crosses Uq k times and k times only, between the times — «> and 0, is obviously smaller than the probability that it will cross Ug k times or less than k times between the times -(k^ + x)t and — it is consequently smaller than A-V The total probability P is therefore smaller than V 2V kV P< (a- i--2)Uo (^H-9)Uo "^•■■"^ (A-'-h.rH-i)Uo The series of the second term is uniformly convergent. Each of the terms tends to zero when x tends to infinity. Therefore the sum of the /154 series tends to zero, and P is infinitely small. In the same way, the probability is infinitely small that our mole- cule does not cross Ug an infinite number of times between the times and + «>. The same results are obtained when any other choice is made for the function (jj, instead of setting <|i = 1. Equation (1) must then be replaced by the following 155 where J(oo) and JCUq) designate the integral J extended over the regions Oq and Uq, respectively. I shall assume that the function i|) is continuous; consequently, it does not become infinite, and I may assign an upper limit y to it. We then have and since (/iM)(<^o)<''>V, we may deduce the following lUV No matter how small J(Uo) is, or how large k is, we may always take n large enough that the second term of this inequality is also as small as desired. We again obtain the same results which are therefor e indepen- dent of the choice of the function <^ , To sum up, the molecules which cross Uq only a finite number of times are unusual, in the same way as the commensurable numbers which are only an exception in the series of numbers, while the incommensurable numbers are the rule. Therefore, if Poisson could provide an affirmative answer to the sta- bility question which was posed, although he had excluded the cases in which the ratio of the mean motion is commensurable, we have the right to state that the stability which we have defined has been proven, although we are forced to exclude the unusual molecules which we have just dis- cussed. I would like to add that the existence of asymptotic solutions pro- /155 vides sufficient proof for the fact that these unusual molecules exist in reality. Extension of the Preceding Results 297. Up to the present time, we have limited ourselves to a very special case ~ that in which an incompressible liquid is enclosed in a vessel, i.e.j — to employ analytical language — the case of the 156 following equations dx __ dy dz where X, Y, Z are three functions which are interrelated by the followinc relationship d\ dX d\ ax tly dz and such that on every point of a closed surface (that of the vessel) we have where 1, m, n are the direction cosines of the normal to this closed surface. However, all of the preceding results are still valid even in the more extended cases without changing a thing, including the line of reasoning leading to these results. Let the n variables xi, xg, ..., x^, satisfy the differential equa- „, i'fi _ -A. (4) We are therefore led to compile the following curves V-)- "-(^'HTi^)-,. const. The first term in relationship (4) is necessarily positive, because we have V — — -1- — I where m and m are the masses of the two principal bodies, and r^ and v^ are their distances to the zero mass. The first term of (4) becomes in- finite for r =0, for r = 0, as well as at infinity. It must therefore 159 have at least a minimum, and two points where its two first derivatives vanish without there being a maximum or a minimina. More generally, if there are n relative minima or maxima, there will be n + 1 points where the two derivatives vanish without there being a maximum or a minimum. However, it is apparent that these points, where the two derivatives vanish, correspond to the special solutions of the three-body problem which Laplace studied in Chapter VI of Book X of his Mecanique Celeste (Celestial Mechanics) . Two of these points may be obtained by constructing an equilateral triangle on m]^m2, either above or below the line mim2 which we shall use for the axis of the 5's. The third apex of this triangle represents one of the solutions in question. All the other points satisfying the condition are located on the axis of the C's. It may be readily seen that the first term of (4) has three minima, and only three minima, when E, varies between — «> and +». The first minimum is located between infinity and the mass m^, the second is located between the two masses m^ and m2, and the third is located between infinity and the mass m2 . dV 9 The derivative — + n'^E, only vanishes (for n = 0) once in each of these intervals, since it is the sum of three terms which all increase. The equations /159 dV _ dV — TJ- -t- /2 - f = — h «' 7) — O indicating that the first derivatives of the first term of (4) are zero, have only five solutions, namely, the points Bj and B2 which are the apexes of the equilateral triangles, and the points Ai , A2 and A3 located on the axis of the 5's. We shall assume that these points occur in the following order — x), A|, m,, A,, m,, A3, -H ». We must now determine which of these points correspond to a minimum, and we know in advance that there are two. We should point out that if we vary the two masses mj and m2 continu- ously, any of the five points A and B will always correspond to a minimum, or will never correspond to one. One may only proceed from one case to another if tjie Hessian of the first term of (4) vanishes, i.e., if two of 160 the points A and B coincide, which will never occur. It is sufficient to examine a special case — for example, that in which m^ = m2. In this case, the S3nmnetry is sufficient for indi- cating to us that the two solutions A^ and A3 must have the same nature, just like the two solutions Bi and B2. It is therefore Ai and A3 alone, or Bx and B2 alone, which correspond to a minimum. Therefore, A does not correspond to a minimum. It can be seen that A^ does not correspond to a minimum. The two minima correspond therefore to Bj and B2. Let us now assume that mi is a great deal smaller than m2 , which is the case in nature. For sufficiently large values of -h, the curve will be composed of three closed branches Cj encircling mx , C2 encircling m2, and C3 encircling Ci and C2 . For smaller values, it will be composed of two closed branches, Cj encircling mx and m2 , and C2 encircling Cx- For values which are still smaller, we shall have only one closed branch leaving mx and m2 on the outside, and encircling Bx and B2. Finally, for even still smaller values, we shall have two closed /160 symmetrical curves, each of which encircles Bx and B2, respectively. The statements below will only apply to the two first cases; we shall therefore put the last two cases aside. In the first case, the group of points satisfying the inequality (4) may be divided into three partial groups: The group of points which are inside of Cx , the group of points which are inside of C2 , and the group of points which are outside of C3. In the second case, the group of points satisfying (4) may be divided into two partial groups : The group of points which are inside of Cx , and the group of points which are outside of C2. The statements below do not apply either In the first case to the group of points which are outside of C3, nor in the second case to the group of points which are outside of €2- On the contrary, in the first case this applies to the group of 161 points which are inside of Ci or to the group of points which are in- side of C.2 and, in the second case, to the group of points which are inside of Cj . In order to formulate these ideas more clearly, let us consider the first case and the group of points which are inside of C2. As the region V we shall take the region defined by the inequalities f- -t- t'' n' 2 X (5) We shall assume that e is small and that h has a value which we have employed in the first case. Finally, in order to conclude the definition of the region V, we shall impose the condition that the point (C, n) is located within the curve C2. It is then clear that, if the point (5, n, 5', n') is located in the region V at the initial instant of time, it will always remain there. In order to illustrate the fact that the results presented in the preceding paragraphs may be applied to the case which we are discussing, we must now show that the integral fdlclrid'-'dr/ (2) extended over the region V is finite. How may this integral become infinite? Due to the fact that the /161 curve C2 is closed, E, and n are limited. The integral can therefore only become infinite if C and n' are infinite. However, because of the inequalities (5) , ^ and n' may only become infinite if becomes infinite, or — since E, and n are limited — if V becomes infinite. However, V becomes infinite for rj = and for r2 = 0, Since the point mj is outside of C2, we need only examine the case of Let us therefore evaluate the portion of the integral which is in the vicinity of the point m2. If r2 is very small, £;^ + n^ is equal to (0 m2) , and. the term — is also constant, so that if we set 162 H will be regarded as a constant. If we then set (5 — 0//!j) = Tjcosu, 1!) = Tj sino); 5'=pcosp;_^>„_, (5.) and the integral (2) will become I pr^dp d/'i dt.0 df. \^ ) We shall add the inequality to the inequalities (5'), where a is very small, since it is the part of the integral which is close to m2 which must be evaluated, and since the other part is definitely finite. If we integrate with respect to co and (j), the integral (2') will be- come 4-^' / pr^ap dr,. (2") Let us integrate first with respect to p. We must calculate the /162 integral which is chosen between the limits /.(ll-..-^') and Py.(H^e-.^), which provides us with e. The integral (2") may therefore be reduced to 163 \^ /•j dri It is therefore finite. The theorems which were proven above may be applied to the case which we are discussing. The zero mass will repass its initial position as close as may be desired an infinite number of times , if one does not impose certain unusual, initial conditions for which the probability is infinitely small. In the restricted problem, if we assume that the initial conditions are such that the point E,, n must remain within a closed curve Ci or C2, the first of the stability conditions, which were defined in No. 290, is fulfilled. However, the third condition is also fulfilled; therefore, Poisson stability exists. 300. The result will clearly be the same whatever the law of attraction may be. If the motion of a material point 5, n is governed by the equa- tions d^^ _ dV J^ _ dV dO~ ~d\' ~dif " d^ or, in the case of relative motion, by the equations ^»? dn dC^ *" dt = rfV dV = d^' in such a way that the energy integral may be written /163 and if the function V and the constant h are such that the values of 5 and of n remain limited, we shall have Poisson stability. However, this is not all. The same holds true in the more extended case. Let xi, X2, . . . , Xjj be the coordinates of -r material points. 164 Let V be the force potential depending on these n variables. Let mj , m2 , . . . , nijj be the corresponding masses , in such a way that we may employ m^ , m2 or m3 at random to designate the mass of the material point whose coordinates are xi, X2 and X3. The equations may be written d^T-i _ dV "'' ^ii' - dFt and the energy integral may be written In virtue of this equation, if the function V and the constant h are such that the coordinates x^ are limited, there will be Poisson stability. What must be demonstrated is the fact that the integral invariant Jdx', dx\ . . . dx\ dxsdxt... dx„ (x'i =~) is finite when the integration is extended over the region I have called V, which is defined by the inequalities Let us call A the integral jdx\dx\...dx'„, extended over the region defined by the inequality The same integral extended over the region /164 will obviously be AR". 165 When extended over the region defined by the inequalities (1) , it will be or, since e is very small. Our integral invariant therefore equals n A ij( V + hy~'d.T, dx,... dx„ ^2) and the integration must be extended over every point, such that V + h is positive. According to my hypothesis, the region V + h > is limited. It may then be readily verified whether the integral (2) is finite or infinite. It will always be finite if n = 2, because the exponent of V + h is then zero. Let us now assume that n is > 2, and that V + h becomes infinitely large of the order p when the distance between the two points xj , X2, X3 and X[+, X5, xg becomes infinitely small of the first order. Then the quantity under the sign / in the integral (2) is of the order '(':-)■ The subset has n - 3 dimensions. The integral is of the order n; the condition under which the integral is finite may therefore be written /i-(n -3)>^(^^-,j, from which it follows that /165 ^<^-r 166 This is the condition under which there is Poisson stability. Application to the Three-Body Problem 301. The preceding considerations apply to the case in which the following equation /.I ■>. \ ill J results in the fact that the x^'s can only vary between finite limits. Unfortunately, this is not the case in the three-body problem. I shall employ the notation presented in No. 11. I shall use x^, X2, X3 to designate the coordinates of the second body with respect to the first, xi+, X5, xg to designate the coordinates of the third body with respect to the center of gravity of the first two, a, b, c to designate the dis- tances of the three bodies, and Mi, M2 , M3 to designate their masses. Finally, I shall employ mi — nil ~ '"3 ~ ?> in; — m-^ -- /Hi -- ^' to designate the quantities which I have called 6 and 3' in No. 11. We shall then have a ' b c Equation (1) entails the inequality V ; A>o. (2) The function V is essentially positive. Therefore, if the constant h is positive, the inequality will always be satisfied. However, the question is whether we may assign small enough negative values to h so that the inequality can only be satisfied for limited values of the coordinates x^. This amounts to inquiring whether the inequality /166 ^h^^^^^'!l^^.^h>o (3) a c with those which are imposed at the three sides of a triangle a-hb>c, b + c>a, rt + c>6 (^) can only be satisfied for finite values of a, b, c. Let us set a = c, and assume that it is very large; we shall assume that b is very small. 167 The inequalities (4) will be satisfied by them. With respect to inequality (3) which becomes a b no matter what h may be, it may be satisfied by arbitrarily large values of a. No matter how small h may be, or how large a may be, we may always assume that b is small enough that the first term may be positive. The existence of area integrals does not modify this conclusion. These integrals may be written I P(:r,37', — a-|a7'3)-H,3'(.ro^; — ar^x',)^ «,, (5) ' ^'^^\x\ — Xix\)-\-^'\x^x\ — XiX\')^. aj. In virtue of these equations , we have 2k' + x- + .-)+|'(x- + x- + ..v)> "L±|l±_"J, (6) where I is the moment of inertia of a system which is formed of two material points whose masses are B and 3' and the coordinates with respect to three fixed axes are xi , X2, X3; x^, X5, xg. I repeat, that I is the moment of inertia which this system would have with respect to the line serving as the instantaneous axis of rotation for a solid, which would coincide momentarily with this system and would rotate in such a way that the area constants are the same as for the system. Inequality (2) must then be replaced by the following V + A>°? + '^|_J1^. (2') /167 However, this equality, just like inequality (2) itself, may be satisfied by arbitrarily large values of the x^'s, because — for very large values of the x^'s — the moment of inertia I is very large, and, due to the fact that the second term is very close to zero, we return to inequality (2). We must therefore conclude that the considerations given in the preceding section are not applicable. In order to provide a better determination of this, let us calcu- late the integral invariant 168 / d.v\ dx\ . . dx'^ dxi cfxi . . . dXi, extending it over a region defined by the following inequalities k — e o Is infinite, although the denominator VI1I2I3 becomes infinite when one of the points XI, X2, X3 or x^, X5, xg recedes indefinitely. The inte- /168 gration field is then triply infinite, and the denominator only becomes doubly infinite. 302. Even if the considerations presented in the preceding sections are no longer applicable, we may nevertheless draw certain interesting conclusions from the existence of the integral invariant. Let us assume that the distance b of two of the bodies becomes small, and that the third body recedes indefinitely. Due to its great distance, the third body will no longer disturb the motion of the first two, which will become essentially elliptic. 169 This third body will essentially describe a hyperbola around the center of gravity of the first two. In order to elucidate this point, I shall present a simple example. I shall assume that we have a body describing a hyperbola around a fixed point. The hyperbola is composed of two branches. One of these branches is the analytical extension of the other, although the tra- jectory is only composed of one single branch for the engineer. We may then inquire whether the trajectory has an analyti- cal extension in the case of the three-body problem, and how it may be defined. The coordinates of the second body with respect to the first are xi , X2, X3; the coordinates of the third body with respect to the center of gravity of the first two are x^, X5, xg , so that we must envisage the motion of two imaginary points whose coordinates , with respect to three fixed axes, are xj , X2 , X3 for the first and x^, X5, X5 for the second. The first of these points will essentially describe an ellipse, the second essentially a hyperbola, and it will continue receding indefin- itely on one of the brances of this hyperbola. In order to obtain the desired analytical extension, let us construct the second branch of this hyperbola, and let us relate it to the ellipse described by the first point . Let us then consider two special trajectories of our system. For the first, the initial conditions of motion will be such that, if t is positive and very large, the point x^, X5, X5 will be very close to the first branch of the hyperbola and the point xi, X2, X3 will be very close to the /169 ellipse, in such a way that the distances of these tvro points — either to the hyperbola or to the ellipse — tend to zero when p increases indefinitely. Let us take the asymptote of the hyperbola as the axis of the xi+'s, and let V be the velocity of the point which describes this hyperbola, for a value of t which is positive and very large. Then Xi — yt will tend toward a finite and determinate limit X when t increases in- definitely. In the same way, let n be the mean motion on the ellipse and £ be the mean anomaly, and the difference I — nt 170 will tend toward a finite and determinate limit Iq. If we specify the ellipse and the hyperbola and, consequently, V and n, and in addition if we specify X and Iq, the initial conditions of motion corresponding to the first trajectory will be completely determined. Let us now consider the second trajectory, and let us assume that the initial conditions of motion are such that, for t which is negative and very large, the point xit, X5, xg Is very close to the second branch of the hyperbola, and the point xi , X2, X3 is very close to the ellipse, and that these two points come together indefinitely from these two curves when t tends toward — <». The differences n- V/, I — lit tend toward the finite and determinate limits X' and £'0 when t tends toward infinity. The initial conditions corresponding to the second trajectory are completely defined when we specify the ellipse, the hyperbola, and X' and If we have the two trajectories may be regarded as the analytical extension of each other . Let us now consider a system of differential equations /170 '1^1 = S (,■.= ,,,., ...,rt), (1) dt where the functions Xj^, which depend solely on xi, X2, ..•, x^, satisfy the relationship Y dXi _ ^ These equations will have the integral invariant r (2) Let us assume that we know arbitrarily that the point x^ , X2 Xn must remain within a certain region V, which is similar to the region V which was considered in the preceding sections, but extending indefinitely 171 so that the integral (2) extended over this region is infinite. The conclusions of Nos . 297 and 298 will no longer be applicable. However, let us replace equations (1) by the following where M is a given arbitrary function of xi, X2, .... x^^. The point xi , X2, ..., x^, whose motion is defined by equations (1'), will describe the same trajectories as that whose motion is defined by equations (1). The differential equations of these trajectories are in both cases dx\ __ dx, _ dj-^ X, " X. -• •= XT' However, if I employ P to designate the point whose motion is de- fined by equations (1) and P' to designate that whose motion is defined by equations (1'), we may see that these two points describe the same trajectory, but obey different laws. If I employ t to designate the time when P passes by a point of its trajectory, and t' to designate the time when P" passes by this same point, these two times will be related in the following way dt _ I dt' " ^i ' We have /171 which indicates that the equations a I have the integral invariant J^ldx,dTt...dx„. (2') Let us assume that the function M is always positive, and that it tends toward zero when the point x^, X2, . . . , Xn recedes indefinitely, and recedes rapidl y enough that the integral (2') extended over the region V is finite . The conclusions presented in Nos. 297 on may be applied to equations (1'). Tnese equations (1') therefore have Poisson stability. Since they define the same trajectories as equations (1), it may be stated in a 172 certain sense that the trajectories of the point P also have Poisson stability. I shall clarify this point. We have i^ f''^. (3) Since M is essentially positive, t increases with t' . However, since M may vanish, it may happen that the integral of the second term of (3) is infinite. For example, let us assume .that M vanishes for t' = T ; then t will be infinite for r ■- T or for «'>t. Let us consider the trajectory of the point P' . We may divide it into two parts, the first which P' traverses from the time t' = to the time t'= T; the second C" which P' traverses from the time t' = T to t' = CO . The point P will describe the same trajectory as P', but it will only describe the part C, because it can only reach the part C' after an in- finite time t. For the engineer, the trajectory of P would only be composed of C. For the analyst, it would be composed not only of C, but also of C , /172 which is the analytical continuation . Let us imagine a point Pj whose position is defined as follows: The point Pi will occupy at the time t^ the same position that the point P' occupies at the time t' . With respect to tj, it will be defined by the equality '.- r'^^' (where i'„>T). J,, M The motion of the point Pj will conform to equations (1) , and this point Pi will describe C , in such a way that the trajectories of the points P and P^ may be regarded as the analytical continuation of each other. Let us now assume that the point P is within a certain region Ug at the initial instant of time. If the initial conditions of motion are not unusual, in the sense attributed to this word in No. 296, the trajectory 173 of the point P and its successive analytical continuations will cut across the region U an infinite number of times, no matter how small it may be. However, it may happen that the point P never re-enters this region, because this region is not traversed by the trajectory, strictly speaking, of the point P, but by its analytical continuations, 303. This may be applied to the three-body problem. We saw above that we must consider the integral / rfrj . . . (Imc dx\ . . . dx\, which we have reduced to the sixfold integral /(v-'.--^!-^) a J \ ' dXi dr, . . . dx^ /'>I.I. However, we have seen that this integral, extended over the region V, is infinite, and this has prevented us from arriving at Poisson stability. Let us write the equations of motion in the form /173 dxi _ d.c'i "di -' ^'' 'Ji ^ ^'' where the Xj^'s and the Y^'s are functions of the xi's and the x'.'s. Then let us set M — ' '^ (x]-T-xl-^xl-i-.. .-hxl-i-i)* and let us write the new equations d.Vi ^ X,- dx'i _ Y,- dc' "" M ' d7 " M ■ The new equations will all have the following as the integral invariant / .M dXi . . . dXi dx\ . . . dx'g or However, this integral is finite . Therefore, if the initial situation of the system is such that the point P in space has 12 dimensions whose coordinates are 174 a;,, .r,, ..., .v^, j:,, x,, ..., a-;, and if this point P is within a certain region Uq at the initial in- stant of time, the trajectory of this point and its analytical continua- tions — such as we have defined at the end of No. 302 — will cut across this region Uq an infinite number of times unless the Initial situation of the system is not unusual, in the meaning attributed to this word in No. 296. 304. It may first appear that this result is only of interest for the analyst, and has no physical significance. However, this point of view is not entirely justified. It may be concluded that, if the system does not repass arbitrarily close to its initial position an infinite number of times, the integral /174 / Jt {.t] + x] +. . .H-.rJ-hl)' will be finite. This proposition is valid, if we overlook certain unusual trajec- tories whose probability is zero, in the meaning attributed to this word in No. 296. If this integral is finite, it may be concluded that the time during which the perimeter of the triangle formed by the three bodies remains less than a given quantity is always finite. 175 CHAPTER XXVII THEORY OF CONSEQUENTS 305. We may obtain other conclusions from the theory of inte- /175 gral invariants which will be of use to us below, although they will be presented in a somewhat different form. Let us commence by investigating a simple example. Let us assume a point whose coordinates in space are x, y and z and whose motion is defined by the equations >'■' ^ V ''y -y 'i'- . z. (1) -cu--^' di '^' di where X, Y and Z are the given, uniform functions of x, y, z. Let us assume that X and Y vanish all along the z axis, in such a way that X --. y -^ o is a solution of equations (1) . Let us then set a: = p coso), y ~ ? sinu), and equations (1) will become § = «. S-". S=^. <^' where R, Q, and Z are the functions of p , w and z which are periodic having the period Its with respect to oj. It is advantageous for us to assign only positive values to p, and we may do this with no difficulty since x = y = is a solution. I shall now assume in addition that n can never vanish and, for example, always remains positive. Then u will always increase with t. Let us assume that equations (2) have been integrated, and that we have the solution in the following form /176 The letters a and b represent integration constants. Let us set 176 Po--^/ifo, (i,b), Zt,rr-.f^{o, a, b), Pi --^/i(2-, 'T, b), z; =--f.i(?.r., a, b). Let Mq be the point whose coordinates are ■^ " ?<>• y - 1>. " .= -0. and Mj be the point whose coordinates are These two points both belong to the half-plane of the xz's located on the side of the positive x's. The point M^ will be the consequent of Mq. If we consider the bundle of curves which satisfy the differential equations (1), if we pass a curve through the point Mq , and if we extend it until it encounters the half-plane (y = 0, x > 0) again, the preceding definition is justified by the fact that this new encounter will occur at Ml. If an arbitrary figure Fq is drawn in this half-plane, the conse- quents of the different points of Fq will form a figure Fi which will be called the consequent of Fq. It is evident that pi and z^ are continuous functions of Po and Zq. Therefore, the consequent of a continuous curve will be a continuous curve, the consequent of a closed curve will be a closed curve, and the consequent of an area which is connected n times will be an area which is connected n times . Let us now assume that the three functions X, Y and Z are related as follows ^MX ^MY dm. _ dx dy dz ^ ' where M is a positive, uniform function of x, y, z. Equations (1) then have the integral invariant J Mdirdy dz m and equations (2) have the following invariant ,,^^ / M p dp (/w (/;;. Let us now consider the equations 'II ^ R ^'^ _ z Jz (see No. 253) . / MO.o dp o), and let Fi be its consequent. Let Jq be the Integral hlQpdpd::, (5) extended over the planar area Fg , and let Jj be the same integral ex- tended over the planar area Fj . Then let $o ^^ the volume produced by the area Fq when it is ro- tated around the z axis by an infinitely small angle e, and the in- tegral (4) extended over $ will be J e. ^ In the same way, let $i be the volume produced by the area F^ when it is turned around the z axis by an angle e, and the integral (4) ex- tended over $ will be J. e. 1 1 The integral invariant (4) must have the same value for $o as for $1, and we must have Jo ^ - Jl . Thus, the Integral (5) has the same value for an arbitrary area and its consequent . 178 This is a new form of the basic property of integral invariants. 306. Let us then assume a closed curve Cq located in the JJJA half-plane (y = 0, x > 0) and encompassing an area Fq. Let Ci be the consequent of Cq . This will also be a closed curve which will en- compass an area Fi , and this area Fi will be the consequent of Fq. If the integral (5) , extended over Fq and over Fi , has the value Jo and Ji , we shall have Jo-- J 1 I from which it follows that Fq cannot be a part of Y^ , and F^ cannot be a part of F., Four hypotheses may be formulated regarding the relative position of the two closed curves Cq and C^. 1. Ci is within Cq; 2. Co is within Ci ; 3. The two curves are outside of each other; 4. The two curves intersect. The equation Jq = Ji excludes the two first hypotheses. If the third is also excluded, for whatever reason, the two curves will definitely intersect. For example, let us assume that X, Y, Z depend on an arbitrary parameter y and that for p = 0, Co is its own consequent. For very small values of u, Co will differ very little from Ci. Therefore, it could not happen that the two curves Cq and Ci are outside of each other, and they must intersect. Invariant Curves 307. Any curve which will be its own consequent will be called an invariant curve . Invariant curves may be readily formed. Let Mq be an arbitrary point of the half-plane, and let M^ be its consequent. Let us connect Mo to Ml by an arc of an arbitrary curve Cq. Let Ci be the consequent of Co, C2 be the consequent of Cj, and so on. The entire group of arcs of the curve Cq, C^ , C2 , ... will obviously constitute an invariant curve . 179 But we may also consider Invariant curves whose formation will be more natural. Let us assume that equations (1) have a periodic solution. Let /179 ^ = ?i(0, r = ?!(0, - = =3(0 ^^^ be the equations of this periodic solution, in such a way that the func- tions (i>^ are periodic in t, having the period T. I shall assume that when t increases by T, oj increases by 2tt . Equations (6) represent a curve. Let Mq be the point where this curve intersects the half-plane; this point Mq will obviously be its own consequent . Let us now assume that there are asymptotic solutions which are very close to the periodic solution (6) . Let ^ = *.(0> 7 = *2(0. z^'P,{i) (7) be the equations of these solutions. The functions $£ may be developed in powers of Ae^'^, and the co- efficients are themselves periodic functions of t. In this expression, a is a characteristic exponent, and A is an integration constant. In equations (7) , the three coordinates x, y, z are therefore ex- pressed as a function of two parameters, A and t. These equations there- fore represent a surface which may be called the asymptotic surface . This asymptotic surface will pass through the curve (6) , since equations (7) may be reduced to equations (6) when we set A = 0. The asymptotic surface will intersect the half-plane along a certain curve which passes through the point Mq and which is obviously an invariant curve. 308. Let us consider an invariant curve K. I shall assume that X, Y, Z depend on the parameter y, as well as the curve K. I shall assume that for y = 0, the curve K is closed, but that it ceases to be closed for small values of y. Let Aq be a point of K. The position of this point will depend on y. For y = 0, the curve K is closed, so that, after having traversed this curve starting with Aq , one returns to the point Ag . If y is very small, this will no longer be the case, but one will pass very close to Aq. Therefore, on the curve K there will be a curve arc which is 180 different from that where Aq is located, but which will pass very close to Aq. Let Bo be the point of this curve arc which is closest to Aq. /180 I shall join AqBq. Let Ai and B^ be the consequents of Aq and Bq. These two points will be located on K. Let AiBj be the consequent curve of the small line AqBo- We must consider the closed curve Cq which is composed of the arc AqMBq of curve K, included between Aq and Bq, and of the small line AqEq. What will its consequent be? In order to define our ideas more precisely, let us assume that the four points Ai , Aq , B^, Bq follow each other on K in the order AiAqBiBq. The consequent Ci of Cq will be composed of the arc AiMBi of the curve K and of the small arc AiBi, the consequent of the small line AoBq. Several hypotheses may then be formulated: 1. The small curvilinear quadrilateral AqBoAiBi is convex, that is, none of these curvilinear sides have a double point, and the only points which the two sides have in common are the apexes. In this hypothesis, the form of the curve would be that indicated in one of the following figures Figure 1 Figure 2 This hypothesis must be rejected, because it is apparent that the integral J ig larger in the case of Figure 1 for Ci than for Cq, and smaller in the case of Figure 2. 181 2. The arc AqAi or BqBi has a double point. If this were the case for the invariant curve K, there would have to be a double point on the arc joining an arbitrary point on the curve to its first consequent; /181 we shall assume that this is not the case. Actually, this condition would not occur in any of the applications which I have in mind. It does not apply, in particular, in the case of the invariant curve produced by an asymptotic surface , as I explained at the end of the preceding section. It may be readily stated that the asymptotic surface does not have a double line if we limit ourselves to the portion of this surface corres- ponding to small values of the quantities which I have designated as Ae'^^ above. On the other hand, the line AqB does not have a double point, and the same must be true for its consequent A^Bx. To sum up, we shall assimie that the four sides of our quadrilateral do not have a double point. 3. The arc AqAj intersects the arc BoBj. (As a special case, this case includes that in which the curve K would be closed.) Our curves will then have the form shown in Figure 3. Figure 3 4. The arc AqEq intersects its consequent AjBi. Our curves will then have the form shown in Figure 4. There are cases in which this hypothesis must be rejected. For example, let us assume that X, Y, Z depend on one parameter y, and that for V = the curve K is closed and that each of its points is its own consequent, so that for y = the four apexes of the quadrilateral coincide. Then the four distances AqBq, A^Bi, AjAq, BiBq will be infinitely small quantities if y is the main infinitely small quantity. Let us /182 assume that AjAq is an infinitely small quantity of the order p, AqBq an infinitely small quantity of the order q, and that q is 182 Figure 4 larger than p. Since A^Bi is the consequent of AgBo, the length of the arc A^Bi must be of the order q. Then let C be one of the intersection points of AgBQ. In the mixtilinear triangle whose two sides are the lines AiAq and AqC, and whose third side is the arc of the curve A^C which is part of AiBi, the side AjC is larger than the difference between the two others, it should therefore be of the order p, and we have seen that it must be of the order q. The hypothesis must therefore be rejected. AiAg 5. Two adjacent sides of the quadrilateral intersect, for example , and AjBi. It is then necessary that AqBq, which is the antecedent of AiBi, intersect K itself. If A'q is the intersection of AqEq with K, and A'l is the intersection of A^Bi with the arc AqAj , A'l will be the con- sequent of A'o, and we shall obtain the following figure. Figure 5 It is apparent that A'q and A\ may play the same role as Aq and Ax, 183 and that we therefore return to the first case. This new hypothesis must therefore be rejected. J183 To sum up, the two arcs AqAj and BqBi will intersect every time that hypotheses 2 and 4 must be rejected, for one reason or another. We must now examine the case in which the points Ai , Aq, Bi, Bq follow one another in a different order on K. The orders BiBqAiAq, BoBiAqAi, AqAiBoBi do not differ essentially from that which we have just studied. Orders such as AiBiBqAo, AiBqBiAo, AiBqAqBi, ... will not appear in the applications which follow. We shall always assume that, if y is very small, the distances AqAi and BqBi are very small with respect to the length of the arcs AqMBq or A^MBi. The order AiAqBqBi, or the equivalent orders, remain, and we shall no longer discuss them. It is apparent that if they appear, on the arc AqMBq there will be a point which will be its own consequent. 309. For example, let us assume that equations (1) have a periodic solution ^ = ?.(0. 7 = ?5(0, •5 = ?j(0 (6) and asymptotic solutions a: = *,(0, 7 = *5(0, z = ^,{t). (7) Let us assume that equations (1) depend on a very small parameter y, and that X, Y, Z may be developed in powers of this parameter. For y = 0, let us assume that the asymptotic solutions (7) may be reduced to periodic solutions. This may be done as follows. We have stated that the $i's may be developed in powers of Ae'^ , with the coefficients themselves being periodic functions of p. However, the exponent a depends on y; let us assume that it vanishes for y = 0. Then for y = the functions ^^ will become periodic functions of t, and the solutions (7) may be reduced to periodic solutions. The asymptotic surface intersects the half-plane along a certain curve Co which passes through the point Mq, which is the intersection of the half-plane with the left curve (6) . 11^ The curve Cq is obviously invariant, as I stated at the end of No. 307. Foy y = 0, each of the points of Cq is its own consequent. 184 In addition, I shall assvime that the curve Cq is closed for p = 0. Let us refer back to Chapter VII, Volume I. We saw from Nos. 107 on that, in the case of dynamics, the characteristic exponents may be developed in powers of /y", and are equal pairwise and have the opposite sign. We shall assume that this is the case. In reality, we then have two asymptotic surfaces corresponding to the two equal exponents having opposite sign a and -a. We therefore have two curves Cq which will intersect at the point Mq. We may distinguish between four branches of the curve c c c r" all four of which end at the point Mq ; Cq and Cq will correspond to the exponent a, C'^ and C" to the exponent -a. Figure 6 These different branches of the curve are shown in Figure 6. The branch C'q is the branch MoPoPiAqAi, the branch Cq is the branch MqEqEi, the branch C^ is the branch MqQiQo and the branch C" is the branch MoRiRoBiBq. These four branches of the curve are obviously invariant. Now, for y = 0, C'q is identical to C^ , Cq is identical to C'j", and (if we assume that the curve Cq is closed for y = 0. which we shall /185 call cj,) these four branches of the curve will coincide on the closed curve Cq. It may be deduced from this that, for very small y, these branches 185 of the curve will differ very little from each other, that C'g will deviate very little from C\, CJ will deviate very little from C", and that, if Cg is sufficiently extended, it will pass very close to C", if it is sufficiently extended. I have indicated on the figure different points of these branches of the curve and their consequents. Thus, A^ , Bj, E^, P^ , Qi, Ri are, respectively, the consequents of Aq , Bq, Eq, Pq* Qo » ^0 • We would first like to note that the points Aj , Aq , Bj, Bq do follow each other (as we assvmied at the beginning of No. 308) in the order A^AqBiBo when the invariant curve formed of the two branches Cq and C" is traversed from A], to Bq. This invariant curve is not closed, but it differs very little from the closed curve Cq . In this connection, let us examine the five hypotheses of No. 308. As we have seen, the first must be rejected. The second will no longer occur. It could only occur if the asymptotic surface (7) had a double line. We have stated that the $j^'s may be developed in powers of Ae'^'^. Therefore let us set ,• ---. '[>? -)- Ae^''^,' 4- A'e'«''I>' -h. . . . If our surface had a double line, this double line would have to satisfy equations (1). Actually, the asymptotic surface is produced by an infinite number of lines satisfying these equations in such a way that, if two layers of this surface happen to intersect, the intersection could only be one of these lines . Since $-l depends on the time t and the parameter A at the same time, we may show this by writing 'I>,-=-f,(/, A). If there were a double line, we would have to have the three identities ,(<, A), *,(<, A)] and in the same way '^*4 ==X [*,(«', B), +,(;', B), *,(<', B;], from which it follows that rf*, _ d'Pi dt' _ ~dT ~ ~di' ' "di ~ ^' from which we have f=z t-hh, where h is a constant. We would thus obtain the following where G= Be^fi. The identity must be valid for t = — <», from which it follows that Ae»' = 065" = o, and we have *?(')= *?('-+- A), from which we have h = and or A,'(04-... = C'J>;(0-t-C»c«'*?(0-H.-. or, setting t = — «>, we have A = C = B. 187 Due to the fact that the two values A and B are equal, there is no double line. ^ The third hypothesis may be adopted. Let us pass on to the fourth hypothesis. In order to determine whether it must be rejected, we must try to determine the order of magnitude of the distances AjAq and AqBo- This is what we shall do in the different applications which follow. Finally, the fifth hypothesis is always reduced to the first one, as we have seen. Extension of the Preceding Results 310. We formulated very special hypotheses above concerning equa- tions (1), but all of them are not equally necessary. Let us consider a region D which is simply connected and which is part of the half-plane (y = 0, x > 0). Let us assume that we know arbi- trarily that, if the point (x, y, z) is located at a point Mq in this region at the initial instant of time, co will constantly increase from to 27T when t increases from to to, in such a way that the curve satis- fying equations (1) and passing through the point Mq ~ assuming that it is extended from this point Mq up to its new intersection with the half- plane — is never tangent to a plane passing through the z axis. Just as in No. 305, we may then define the consequent of the point Mq, and it is apparent that all the preceding statements will still be applicable to the figures which are located within the region D. It will not be necessary that the curves satisfying equations (1) and intersecting the half-plane outside of D be subjected to the condi- tion of never being tangent to a plane passing through the z axis. It will no longer be necessary that x = y = be a solution of equations (1). Then, if Cg is a closed curve inside of D and if Ci is its conse- quent, the two curves will be outside of each other or will intersect. The results given in No. 308 will be equally applicable to the invariant curves which do not leave the region D. If even one invariant curve leaves the region D when it is sufficiently extended, the results will still be applicable to the portion of this curve which is within /188 this region. 311. Let us now consider a curved surface S which is simply 188 connected, instead of a plane region D. Let us pass a curve y satis- fying equations (1) through a point Mq of this curved surface, and let us extend this curve until it again intersects S. The new point of intersection Mj may still be called the consequent of Mq . If we consider two points Mq and Mq which are very close to each other their consequents will be, in general, very close to each other. There would be an exception if the point Mj were located at the boundary of S, or if the curve y touched the surface at the point M^ or at the point Mq. Except for these exceptions, the coordinates of Mj are analy- tic functions of the coordinates of Mg. In order to avoid these exceptions, I shall consider a region D which is part of S and such that the curve y, proceeding from a point Mq inside of D, intersects S at a point M^ which is never located at the boundary of S — so that the curve y does not touch S either at Mq or at Mj. Finally, I shall assume that this region D is simply connected. Let us adopt a special system of coordinates which I shall call C, n and <;, for example, and for which I shall only assume the following: 1. When Id and | n | are smaller than 1, the rectangular coordinates X, y and z will be analytic and uniform functions of 5, n and ?, which are periodic with the period 27i with respect to ?. 2. No more than one system of values of 5, n, ^ can correspond to a point (x, y, z) in space, such that i{|(xi) is an odd function of x^ which increases from to 1 when xj increases from to + °o . We may set, for example /i + ^J If this hypothesis is adopted, the point M will always be within a torus of radius 1, which is tangent to the z axis. An infinite number of systems of values of x , y, and y will /191 correspond to each point M within this torus. However, these systems will not differ essentially from each other, since one passes from one to the other by increasing y^ or y^ by a multiple of 2ir, or by changing x, into -X and y into y, + tt. If xi, Yi and y2 are given, X2 may be deduced by means of equation (2) . Let us assume that the variables x and y vary in accordance with 191 equations (1) . and the corresponding point M will describe a certain curve which I shall call the trajectory. One and only one trajectory passes through each point inside the torus. The form of these trajectories for y = may be readily determined. For y = 0, the differential equations may be reduced to dxi _ dy£ ^^ _ 'di "" °' dt ^ dxi The x.'s are therefore constants, which Indicates that our trajec- tories arehocated on the tori, and the y.'s are linear functions of time, because rfF„ dTj depends only on the x^'s and is a constant. If the ratio ni :n2 is commensurable, the trajectories are closed curves. Conversely, they are not closed if this ratio is Incommensurable, Let mi, m2, Pi, P2 be four whole numbers, such that Let US set x\ = pl^i —pix„ The identity 3^'.7i -^- ^''.y'i = ^'^' "^ ^'-y' indicates that when one passes from the variables x^ y^ to the variables x'i, y\, the canonical form of the equations is not changed. We shall assume that nz does not vanish when xi remains less than a certain limit a. Then ^ will always retain the same sign, and we /192 shall have, for example rf,,. ^>'- 192 This inequality, which is valid for y = 0, will still be valid for small values of y. The relationships will then define a certain plane region D which will have the form of a circle. The trajectories starting from a point in this region will never be tangent to a plane passing through the z axis, at least before having cut across the half -plane y = again. Our region may therefore play the role of region D in No. 310. The equations (1) have the integral Invariant from which we may deduce the following by means of the Integral F = const. ' A5B5 are the successive consequents of AqEq. The arcs AiB^, A2B2, ..., A5B5 are no longer rectilinear in general, but are very small arcs of a curve. Figures 1 or 2 shown in No. 308 reproduce the part of the figure shown by the solid line. The entire group of our curves shown by the solid lines represents an invariant curve K. I have drawn the figure based on the first hypothesis, which — as we have seen — must be rejected along with the fifth hypothesis. Accord- ing to the statements I made in No. 309, this also holds true for the second hypothesis. We must examine the fourth hypothesis In greater detail. In order to do this, let us try to determine the equation of our asymptotic sur- faces. Based on the statements presented in No. 207, this equation may be obtained in the following way. A function S is formulated which may be developed in powers of /\i, in such a way that p S = SoH-/uS,-+-...-4- fi'Sp-i-.... Regarding Sp, it is a periodic funtion of the period 2Tr with respect to y2, and Att with respect to y\. We shall have dy\' -^'"^ ds ds W Equation (4) is the equation of the asympototic surface. If the series S were convergent, the periodicity of the S 's would entail the condition that our curves must be closed and that the two points Aq and Bq must coincide. However, this is not the case (see No. 225 , and the following) . What significance does equation (4) have? It may only be valid from the formal point of view, i.e., if E is the sum of the p + 1 first /196 terms of the series S, so that 196 Sp = So+ /jJiS, -H, . .+ p' Sp, the equation dllp dz„ £±L 2 will be valid up to quantities of the order y However, equation (4') represents a closed surface, and p is ar- bitrarily large. We must therefore conclude that the distance AqBo is an infinitely small quantity on the order of infinity (see Nosj^ 225 on) . In addition, the distance A0A5 (or B0B5) is on the order of /y, and is consequently infinitely small of the order of ^. The distance AqBq is therefore infinitely small with respect to AqA^ , which indicates that the fourth hypothesis must be rejected. The only possible hypothesis is therefore the third. Therefore the two arcs A0A5 and B0B5 intersect. Application to the Restricted Problem 313. I am going to apply the preceding principles to the problem presented in No. 9, and I shall employ the notation given in that section. Consequently, we shall have the canonical equations dx\ _dF^ dy\ _ dF' dt dy'i dt dx'i based on which we may set and, in addition. X, — L, x'^ = G, F' = R-t-G==Fo4-(jiF,4-. (5) IX Let us now set 197 Xf—L — G, xj = L -r- G, and the equations will retain the canonical form and will become /197 dxi _dr dyt ^ dr dl ~ dyi ' dt ~' dxi We will have ■?•; — ^1 {Xi+XtY ' 1 Fo = , ■_ — — ^ from which it follows that + 4 I -H 4 n, = ;--" r- -t- - , ^l = (Xi+.r5)' ■! {Xi-'r X.,Y 2 If we assume that the eccentricity is very small, L and G will differ very little in absolute value. Therefore, one of the two quanti- ties xi and X2 is very small. I would like to note in addition that the equations L = /n, G - ^a{i^ e') indicate that G is always smaller than L in absolute value. Therefore, XI and X2 are essentially positive. Let us assume that x^ is very small. The function F' will be a function of a and of £ + g - t which may be developed in powers of e cos g and of e sin g. Therefore, this will also be a function of X2 and of y2 which may be developed in powers of /^ cos/i and Ai sin^,. It will be periodic with the period 2ir both in yi and in y2. If, on the other hand, it is X2 which is very small, the function F' will be a function of xi and of yi , which may be developed in powers of s/xt cosj, and /■^^i sin/i. Let us now assxime that our four variables x and y are related by the equation of energy F = C . 198 This equation may be approximately reduced to Fo = C. Let us construct the curve Fq = C, taking x^ and X2 as the coordi- nates of a point in a plane. The equation may be written This curve has two asymptotes /198 Xl -H Tj — O, a"i — T| = a C and it is symmetrical with respect to the first of these two asymptotes. However, it should be noted that the only portion of the curve which is of use to us is that which is located in the first quadrant 3:1 >o, r, >o. Based on the values of C, the curve may have one of the forms shown in the two following figures Figure 8 The axes of the coordinates are represented by the dot-dash line, the asymptotes and the utilizable portions of the curve are shown by the solid line, and the portions of the curve which are of no use are shown by the dotted line. 199 Figure 9 We shall assume that a value Is assigned to C, so that the curve has the form shown in Figure 9 and so that it contains two utilizable arcs AB and CD. We shall no longer consider the arc AB. We should point out that when one traverses this arc AB, xi de- creases constantly from OA to zero, X2 increases constantly from zero to OB and — increases constantly from zero to +■» . If we now construct the curve F = C, assuming that yi and y2 are constants and x^ and X2 are the coordinates of a point in a plane, the /199 curve will differ very little from Fq = C and can still be represented by Figure 9. It will have a utilizable arc AB, and when one traverses X2 this arc the ratio — will increase constantly from zero to + " . We thus arrive at the following method of geometric representation. The location of the system will be represented by the point whose rec- tangular coordinates are /^J + 4^1 — 2 /^ cos^i /xj ~^- ix, — 2 /j7, COS^i 2 /t, sinji These three functions may be developed in powers of 1/xi cos yi and \/k^ sin yi, if xj is very small, and may be developed in powers of 200 yx2 cos 72 and I/X2" sin 72, if X2 Is very small. They only depend on the ratio — . Thus, one and only one point In space corresponds to each system of values of yi and of y2 and to each point on the utilizable arc AB. The functional determinant of the three coordinates with respect to yi, y2 . and with respect to ^_L , always retains the same sign. ▼ ^2 We may therefore apply the results obtained in the preceding sec- /200 tion within all of the region D where n2 does not vanish. However, n2 vanishes for xi + X2 = 2. But, if we have x^ + X2 = 2, x^ > 0, X2 > 0, we shall obviously have 2 Xi — a-i ^ 1 Ti + Xi _ 3 (ri-i-.r,)» ■ Z ■" (a-,-F j-,)» 2 ~4 However, the first term of this equation is Fq and, when compiling the curve Fq = C, we assumed that we were dealing with the case presented in Figure 9. However, the case shown in Figure 9 assumes that Since Fq differs very little from F, and consequently from C, we cannot have at the same time 4 4 3 (unless C is very close to its limit -t", which we have not assumed). Under the conditions with which we are now dealing, we shall not have n2 = . Thus, the results presented in the preceding section are applicable, and if we construct the asymptotic surfaces and if we consider the inter- section of these surfaces with the half-plane y2 = 0, the two arcs which are similar to those which we designated as AqAs and B0B5 above will intersect. 201 I would like to add one word to this . The coordinates of the third body, with respect to the major axis and the minor axis of the ellipse which it describes, are — according to the well-known formula L2(cos;-4-.. .), LG{sin/ H-...). It may thus be seen that, when G changes sign, the second of these coordinates changes sign. As a result, the perturbed planet turns in the same direction as the perturbing planet if G is positive, and it turns in the opposite direc- tion if G is negative. 202 CHAPTER XXVIII PERIODIC SOLUTIONS OF THE SECOND TYPE 314. Let us consider a system of equations /201 W " ^' ^'"='' '' ••■' ^>' (!■) where the X^'s are functions of x^, X2, ..., x^, and of t, which are periodic having the period T with respect to t. Let (2) be a periodic solution of period T of equations (1) . We shall try to determine whether equations (1) have other periodic solutions which are very close to (2) and whose period is a multiple of T. These solutions, if they exist, will be called periodic solutions of the second type . Let us consider a solution of equations (1) which is very close to (2) . Let 9,(0) +- ?/ be the value of x^ for t = 0, and let ?<(o)- - ?i + 'I/, - o,( A-T) + p,--(- <},,■ be the value of x. for t = kT (k is a whole number) . The B^'s and the tj^i'sj whose definition is the same as that given in Chapter III, will be very small. Just as in Chapter III, it will be found that the ^'s are ftinctions of the g's which may be developed in increasing powers of the 3's. In order that the solution may be periodic having the period kT, it /202 is necessary and sufficient that 203 Due to the fact that the (f)-j^(t)'s are periodic functions, the iJj's vanish with the 3's. We shall assimie that the functions X^ which appear in equations (1) depend on a certain parameter y. Then the functions (j).(t) will depend not only on t, but also on y. As regards t, they will be periodic of period T, with T being a constant which is independent of y. Under these conditions, the functions 4), whose definition remains the same, will depend not only on the 3's, but also on y. If we assume that p,, p., •••, Pi, }^ are coordinates of a point in space having n + 1 dimensions, equations (3) will represent a curve in this space. A periodic solution, of period kT, will correspond to each point on this curve. Since the ((/'s all vanish when the 3's all vanish at the same time, this curve will consist of the straight line p,= p,-.... .;5„^e. (4) The solution (2) will correspond to different points on this straight line. Due to the fact that this solution is a periodic solu- tion of period T, it is for that reason a periodic solution of period kT. But we must try to determine whether there are other periodic solu- tions which are very similar to the first or — in other words — if curve (3) includes, in addition to the straight line (4), other branches of the curve which are very close to the straight line(4). In other words, are there points on the straight line (4) through which branches of the curve (3) pass, other than this line? Let P, == p, rr^ . . . r^ P, = O, Ji == Jio be a point P of the line (4) . In order that, several branches of the curve may pass through the point P, it is necessary that at this point P the functional determinant , or the Jacobian, of the 4''s» with respect to the B's, vanishes . 204 This condition is not sufficient, as we shall see at a later /203 point, for several real branches of the curve to pass through the point P. Let us formulate the determinant of the i|;'s with respect to the e's, let us add -S to all the diagonal terms, and let us set the de- terminant thus obtained equal to zero. ¥e shall thus obtain the equa- tion which is known as the equation for S . The roots of this equation (see No. 80) are where a is one of the characteristic exponents of equation (1) . In order that the functional determinant may be zero, it is necessary and sufficient that one of the roots be zero. We must therefore have which means that kaT is a multiple of 2iir. Therefore, in order that several branches of the curve pass through the point P, it is necessary that one of the characteristic exponents be a multiple of t^. kT 315. This condition is not sufficient, and a more extensive dis- cussion is necessary. Let us set and let us try to develop the g's in whole or fractional powers of X. We shall assume that the Jacobian of the (j^'s, with respect to the 6's, is zero. This Jacobian vanishes for X = 0, but will not be identi- cally zero, in general. In order that this may be the case, it is necessary that one of the characteristic exponents be constant, indepen- dent of y, and equal to a multiple of ~-. We shall therefore assume that the Jacobian vanishes for X = Q, but that its derivative, with respect to X, does not vanish. In the same way, we shall assume that the minors of the first order of this Jacobian do not all vanish at the same time. In this case, based on the theorem in No. 30, from n - 1 of equa- tions (3) we may derive n - 1 of the quantities 6 in the form of series 205 developed in whole powers of X and of the n^ quantity 6, for example of 6n- ^^ Let us substitute the values of h, p.. ■••• P""" thus derived in the ntk equation (3). The first term of this r^ equation will be developed in powers of X and of $^. Let us write it in the following form. e(X, 3„) - o. I may first point out that must be divisible by &^, because the line (4) must be part of the curve (3). On the other hand, the derivative of with respect to Bn '""st vanish for X - 0, since the Jacobian vanishes. For X = 0, does not contain a term of the first degree. Let us assume that it no longer contains terms of the second degree. . . . . p - UiL degree, but that it does contain a term of degree p. Finally, since the derivative of the Jacobian with respect to X does not vanish, we shall have a term containing Xe^- I may therefore write 9.. A),3„-i-nii^f-C, where C is the total group of terms containing $P , Xg^. o^ ^ &n ^^ ^ factor. A and B are constant coefficients which are not zero. It may be seen that we may derive 6n from this in terms of a series which progresses according to the powers of 1^' , and the problem is to determine whether this series is real. If p is even or if, p being odd, A and B have opposite signs, the series is real, and periodic solutions of the second type exist. If p is odd, and if A and B have opposite signs, the series is imaginary, and there is no periodic solution of the second type. I shall now assume that not only the Jacobian vanishes for X = 0, but that the same holds true for all of its minors of the first, the second, etc., and p - iJ^ order. I shall nevertheless assume that the /205 minors of the p£^ order are not all zero at the time. 206 According to the statements presented In No. 57, under these con- ditions, there will be not one, but p, characteristic exponents which Ziir will be multiples of ttt". ^ kT From n - p of equations (3) , we may then derive n - p of the quantities g in the form of series developed in powers of X and of the p last quantities B. For purposes of brevity, I shall employ the B' 's to designate the n - p first quantities 6, and the g" »s to designate the p last quanti- ties 3. We shall therefore have the 3' 's developed in powers of X and of the 3" 's . Let us substitute these expansions in the place of the 3' 's in the p last equations (3) , and we shall obtain p equations e, ^ej.-^...-^ epr=o, (5) whose first terms will be developed in powers of A and of the 3" 'g Due to the fact that the Jacobian and its minors of the first p - 1 orders are zero, these first terms will not include terms of the first degree in 3" which are independent of X. We must now determine whether the first terms of equations (5) will contain terms of the first degree with respect to the 3" 's, and at the same time of the first degree with respect to X. Let 6-^ be the total group of terms of 0^ which are of the first degree with respect to the 3" 's- It is apparent that 6^ may be developed in powers of A. Let be this expansion. The e|^)'s will be homogeneous polynomials of the first degree with respect the the 3" 's. According to the preceding statements, e9 will be identically zero, but we must now determine whether the same holds true for q\ . The Jacobian of the ip's with respect to the 3's equals n(i --e'-^'f), The product indicated by the sign IT extends over n factors corresponding to the n characteristic exponents a. 207 Let ai , 02, ..., o^ be these n exponents, and let /206 (j> (a )-0 — £*'■•■) The Jacobian will equal the product ?(ai)?(aj)--.'f(i/i). In order that A = 0, the Jacobian vanishes as well as its minors of p - 1 first orders. As a result, p of the exponents are multiples of -j-^. Therefore, p of the factors ())(oi) vanish for X = and are, consequently, divisible by X. The product, i.e., the Jacobian, will therefore be divisible by XP. dct We shall assume that for X = none of the -r- vanishes, which is dX what we already assumed previously. Under these conditions, none of the (j)(a)'s are divisible by X^. Therefore, the product is not divisible by XP+1. Thus, the Jacobian is divisible by XP , but not by xP As a result, the determinant of the 6, 's is different from zero, and consequently none of the O.'s vanishes identically. The simplest case is that in which, for X = 0, the terms of the second degree do not vanish in the expressions for 0^, and in which these terms of the second degree cannot vanish at the same time, unless all the 3" 's vanish at the same time. Let us assume that m is the total group of terms of the second degree of 0-j^ for X = 0. It will be sufficient to consider the algebraic equations T„+X8,':---0, whose first terms are homogeneous polynomials of the second degree with respect to X and the g" 's. If these equations have real solutions, we shall have periodic solutions of the second type. I shall not extend the discussion to the other cases, but shall com- plete this discussion when treating the equations of dynamics. 208 Case In Which Time Does Not Enter Explicitly 1201 316. Let us assume that the functions X^ which appear in equations (1) do not depend on time t. As we have seen in No. 61, in this case one of the characteristic exponents is always zero. In addition, if is a periodic solution of period T, the same also holds for whatever the constant h may be. In the preceding section, we assumed that — no matter what y might be — there was a periodic solution and the period could only be T, since the Xj^'s were periodic functions of t, of period T. The period was therefore independent of \i. The same is not true in this case. We shall always assume that, no matter what y might be, equations (1) have a periodic solution However, the period will depend on y, in general. I shall call T the period, and Tg the value of T for y = yg, i.e., for X = 0. We shall then modify the definition of the quantities g and (j; to a certain extent . We shall always designate the value of x^ by ^±(0) + 6^ for t = 0, However, we shall designate the value of x^ by (|>j^(0) + g^ + tj^^^ for t = k (T + x) (and not for t = kT) . Then, the 4i-l's will be functions of the n + 2 variables 209 If we continue to assume that the g's and A's are the coordinates of a point in space having n + 1 dimensions, the equations /208 :• o (3) will no longer represent a curve, but will represent a surface, since we may vary the two parameters t and X independently and continuously. However, we should point out that curves are drawn on this surface whose different points correspond to periodic solutions which may not be regarded as being essentially different. If is a periodic solution, the same will hold true for no matter what the constant h may be, and this new solution will not differ from the first in reality. The following point corresponds to the first p,-.=/,(o) -0,(0), and the following point corresponds to the second ?<■ -/,(/') -?/(o). When h is varied continuously, the second point describes a curve whose different points do not correspond to solutions which are actually different. In particular, let us consider the solution The following point will correspond to this solution which belongs to the line (4) . The following point P,= rp,(A)-cp,(0), C'^') which belongs to a certain surface (4) making up the surface (3) , will cor- respond to the solution which is not actually different from the first. 210 We must now determine whether the surface (.3) Includes layers other than C^') approaching very close to C^'), i.e., whether there are points /2Q9 on the surface (4') through which other layers of the surface (3) pass in addition to the surface C^') itself. Without limiting the conditions of generality, we may assume that 6i = (or we may impose another arbitrary relationship between the g's). In actuality, the solutions are not different, and it is sufficient to take one of them into consid- eration. We may choose the constant- h p bitrarily, and we may take it in such a way that, for example, /.(A)-?.(o), from which we have q.e.d. If we impose this condition g^ = 0, the two surfaces (3) and (4') may be reduced to curves, and the surface (4') may be reduced to the line (4), in particular. We would like to again determine whether another branch of the curve (3) passes through a point of the line (4) . For this purpose, let us combine equation gj = with equations (3). These equations will represent the curve (3) , or a curve of which (3) is only a part. In the region under consideration, in order that this curve may not be reduced to the line (4) , it is necessary that the Jaco- bian ti > 'l'2» •■•> ^-a.* ^1 with respect to Bi, B2> •••> ^n* "^ ' ^^^ ^'^^'- °^ ^ly 'f'2. •••> ipn with respect to B2» ^3. •••» ^n» '^ > ^^ ^^^° ^°^ X = 0. Since nothing distinguishes 6i from other g's, the Jacobians of the ifj's with respect to t and with respect to n - 1 arbitrary B's must all vanish. That is, all the determinants included in the matrix of Nos. 38 and 63 must vanish at the same time. By pursuing a line of reasoning similar to that presented in No. 63, we may see that the equation for S must have two zero roots. As a result, two of the characteristic exponents must be multiples of -^. This is already true for the one of them which is zero. A second T J -1 fT .inr exponent must be a multiple of , . 211 If this condition is fulfilled, we shall formulate a system of /210 n + 1 equations including equations (3) and Bi = 0, We shall derive T and the B's in the form of a series developed in whole and fractional powers of X • If the series are real, we shall have periodic solutions of the second type; if the series are imaginary, this will not be the case. I shall not continue this discussion any further. 317. Let us now assume that the equations ^ = x. (1) where time enters explicitly have a uniform integral in such a way that we have We saw in No. 64 that in this case the Jacobian of the tjj's with re- spect to the g's vanishes, and that one of the characteristic exponents is zero. The equations ^1 ^ ■■■< i'"-!. F. with respect to the 3's vanish. This condition may be written in still another form. Let us assume that we have solved equation with respect to x^, and that this solution yields Let us substitute in place of x^ in Xj^, and let X^^ be the result of this substitution. Equations (1) are thus replaced by the following -^-- = x; (,-.,,. ;,-,). (1-) These equations (1') will have the following periodic solution a:,- = 9,(0- The number of characteristic exponents of this periodic solution, which is assumed to belong to equations (1'), will be n - 1. Let ai , a2, ...» Ojj.i be these n - 1 exponents. These will be the same as those for this periodic solution x^ = i(t), which are assumed to belong to equations (1), suppressing the n exponents which equal zero. In order that equations (1) have periodic solutions of the second /212 type in the vicinity of a point on the line (4) , it is necessary and sufficient that equations (1') have them, i.e., that one of the n - 1 characteristic exponents oij^, 012, ..., ex , is a multiple of -r— at a 213 point on the line (4) . Thus, the condition, which was presented above, that the Jacobian of tjji, 11^2. •••» V-l» ^ is ^^^° "'^y ^^ expressed in a completely differ- ent manner. In order that it may be fulfilled, it is necessary that two of the exponents be multiples of -^. This is always true for the one of them which is zero; this must be true for a second exponent. Let us assume that this condition is fulfilled. From equations (3') we shall derive the 3's in series which are ordered in whole and frac- tional powers of X. I shall not extend this discussion, to determine whether these series are real. 318. Let us now assume that the X±' s do not depend explicitly on time and that equations (1) have an integral F = C. In this case, according to No. 66, two of the characteristic expo- nents are zero. If the equations have a periodic solution for a system of values of y and of C, they will still have it for the adjacent values, so that we shall have a double infinity of periodic solutions which depend on the two parameters y and C. The period T will not be constant; it will be a function of y and of C. Let us then assign a fixed value Cq to C, and let be the values of x-^ for t = and for t = k (T + t) . We shall add equation F = Cq, and then an arbitrary relationship be- tween the e's ~ for example, Bj = ~ to the equations (3) Without limiting the conditions of generality, and for the same /213 reason as was given in No. 316, we may assume that 6i = 0. We shall thus obtain the system 4-v = o, F=^.Co, p, = o. O") These equations represent a curve. The number of equations equals n + 2, but the n equations (3) are not different, and may be replaced by n - 1 of them. This is justified by the same line of reasoning that was presented in the preceding section. System (3") may thus be reduced to 214 n + 1 equations. The number of variables is n + 2 — i.e.. This curve (3") includes the line P, = o. W Let Bj^ = 0, y = yo^^^ point on this line. In order that another branch of the curve may pass through this point, it is necessary that the Jacobian of the first terms of equations (3") be zero or — which amounts to the same thing — that the Jacobian of n - 1 of the ifi's and of F with respect to B2> ^3: •••> ^n ^"^^^ "^ ^^ zero. Finally, since nothing distinguishes Bi from the other 3's, it is necessary that the Jaco- blans of F and of n - 1 arbitrary il^'s with respect to t and to n - 1 ar- bitrary 6's all be zero. This condition may be expressed in another way. Just as in the preceding section, we shall derive the following from the equation F = Cg T„ ----- 0(3ri, .T2, .. ., .r„_,)i and we shall obtain the equations ''^^=Xi (t=.,,a,...,n-.). (1') tit According to No. 316, of the n - 1 characteristic exponents, it Is necessary that one of them be zero and that the other be a multiple of -r=r [if it is assumed that the periodic solution belongs to equations (1')]. In other words — which amounts to the same thing — it Is neces- sary that of the n characteristic exponents [if it Is assumed that the periodic solution belongs to equations (1)], two be zero, and a third /214 be a multiple of "tzT' Let us assume that this condition is fulfilled. We shall derive the 6's and the t from (3") In series which are ordered according to whole or fractional powers of X. I shall still forego a discussion of this point. Application to the Equations of Dynamics 319. I would like to discuss the equations of dynamics in greater detail. However, in order to do this I must first present an important property of these equations. 215 Let C^ and n^ be the values of x-j^ and y^ for t = 0. Let X^ and Y-j^ be the values of x^ and y^ for t = T. We know that is an integral invariant. We shall therefore have with the double integral extending over an arbitrary area A. This may be written as follows A(X, dXi - Y, rfX, - ^- dr^i H- T„ dli) -- o, where the simple integral is extended along the contour of the area A, i.e., an arbitrary closed contour. In other words, the expression S ( X| S] For T = 0, we have dS = and consequently S ~ const. We shall take this constant to be zero so that S will vanish identi- cally for T = 0. The function S is thus completely determined. 321. Let us determine the maxima and the minima of the function S. Let us first consider T as a constant. In order that the function S has a maximum or a minimum, assuming that this function S may be regarded as a uniform function of the variables Xi + gj and Yj^ + m in the region under consideration , it is necessary that its derivatives with respect to these variables are zero — i.e., that we have llld X, - 5,-, Y,- = 7),. The corresponding solution is therefore a periodic solution of period T . and this period T is one of the known quantities of the problem at hand. We shall no longer regard T as a known quantity. In order that S has a maximum or a minimum, it would be necessary that we first have X; = ?„ Y,= 7;,, and in addition ^g However, if X = 5, Y = n, we still have from which it follows that ^^=.(F_C), F = C. 217 The corresponding solution will still be a periodic solution of period T. However, the period T will no longer be a given quantity. The energy constant C, which did not enter the preceding case, will be a given quantity. The two methods for determining the maxima of S are related to the two methods of interpreting the principle of least action, that of Hamil- ton and that of Maupertuis. This will be clear to the reader after the following chapter has been read. 322. The definition of the function S may also be modified in the following way. In a large number of applications, F is a periodic function of period 2n with respect to the y^^'s. In this case, a solution may be assumed to be periodic when ^± = ^±, and when Y^ - m is a multiple of 2tt. It is then apparent that if we set where m^ , m2, . . . , 1% are arbitrary whole numbers, the expression dS will still be an exact differential. We shall thus obtain /217 dS rfT We shall set For T = 0, we have dS ==S4/;i,TCrff/. We shall set c , v ^ which concludes the determination of the function S. Assuming that T is a given quantity, the maxima and minima of S will be obtained by setting its derivatives equal to zero, which yields 218 '^^ aF-sFcx-S) )^| +(Y-r, — 2«ir)^'^^l+ arbitrary function of T. X,- = I/, Yf = T^i + 2;n,Tr. The corresponding solution is still a periodic solution, since Y^-Tii is a multiple of 2tt. The period T is given. If T is not given, it Is first necessary that and, in addition. >'', = ?i, Y,— r;,-{- 2/n/Tt rfS from which we have dT = "• Fr=C. 323. It is now necessary that we learn to distinguish between the real maxima and the real minima of S. Up to this point, we have only de- termined the condition for which the first derivatives of S are zero, but it is known that this condition is not sufficient for providing a maximum. It is still necessary that the second derivatives satisfy certain inequali- ties. Let us first assume that the conditions presented in No. 319 hold, and let us regard T as given. Let ^/ = ?-('). r<--?HO be a periodic solution of period T, so that J218 tp,(o) = 9;.f, o;,-, e;., will be imaginary and conjugate of 0<-.6 0/,.n QI.;, or.,-. Let us now assume that the x'. 'g and the y^^ 's are real. In order to calculate the constants A, B, C, D, we shall have 2n equations which we shall obtain, for example, by setting the following in the equation for x'. 1:^0, l = T, t=-.-2T, ..., / = (2n — i)T. These 2n equations are linear with respect to the 2n unknowns A, B, C, D. The second terms are real, and the coefficients are real or imaginary and conjugate pairwise. When we change \^ - 1 into -y-l: 1. Aj^ and Bj^ do not change when aj^ is real; 2. Aj^ and B, interchange when oj^ is purely imaginary; 3. Aj^ and Bj^ change into Aj and Bj when a]^ is complex and imaginary and conjugate of cxj . Therefore: 1. Aj^ and Bj^ are real when ajj^ is real; 2. Aj^ and B^^ are imaginary and conjugate when aj^ is purely imaginary; 3. Ajj^ and A., Bj^ and B. are Imaginary and conjugate when a,^ is /222 complex, and imaginary and conjugate of a ^ • Finally, C and D are real. These conditions are sufficient for x?j^ and y'j to be real. Let us assign values satisfying these conditions to the constants A{j^, B}^, C, D, as well as to the constants Aj^, Bj^^, C' , D' . Then the second term of (2) must be real, and in order that it may be real the following is necessary: 1. That Mj^ is real if ay^ is real; 223 2. That Mn is purely Imaginary if ay^ is purely imaginary; 3. That tiy^ and M. are imaginary and conjugate if a^ and a are complex, and imaginary and conjugate. Form (3) contains a term and does not contain another term depending on Aj^ or \. If the exponent a^ is real, the presence of a term containing Aj^B^ is sufficient for providing that the quadratic form (3) can be defined. Therefore, if only one of the exponents aj^ is real, the function S cannot have either a maximum or a minimvmi. Let us now assume that two exponents aj^ and a^ are complex, and imaginary and conjugate. Let us cancel all the constants except for Ai, B<-, Ay, Bj, aid the form (3) may be reduced to These two terms are imaginary and conjugate, so that form (3) is real. Let us assume that Aj^ does not change, and that B^ changes sign. A., which is imaginary and conjugate of Aj^., will change no longer, and B., which is imaginary and conjugate of B^, will change into -Bj . Therefore, form (3) will change sign; therefore, it cannot be defined. Therefore, if only one of the exponents aj^ is complex, the function S cannot have either a maximum or a minimum. Let us now assume that ay^ is purely imaginary. Then A^ and Bj^ are /223 imaginary and conjugate, and the product \By^ is the sum of two squares. In order that S have a maximum, it is necessary and sufficient that all of the quantities ^sin^^, -NT / - ■ I y/— I 224 be negative. In order that S have a mlnimian, it is necessary and suffi- cient that all these quantities be positive. It should be pointed out that all these quantities are real, because ^ and ^ are real. 325. How may these results be modified if it is assumed that the energy constant is one of the given quantities of this problem? We then have identically /dP , (IF ,\ dF dF where we assume that in — and -j— , x^ and y^^ have been replaced by the periodic functions ^^{.t) and ijj^Ct). The constant value of the function F must be the same for the periodic solution and for the infinitely close solution xi --: y,(0 + ^'h yi --= iM+y'i- This relationship is a linear equation between the constants Ai, Bi, C, D and the coefficients must be independent of t. It follows from this that Aj^ and B^ must not be included in the re- lationship, since these constants are always multiplied by e^^^k and since this exponential cannot vanish. In addition, C is no longer included, since the solution a^i = ?/(«j-HC ^^y, y,-. = 7,(0-1 „_,, 2r, o, ..., o, o, 2-, ..., 0, * • • . . . , o, o, ..., ■y.r., will play the role of periods in this line of reasoning. There would be an exception, if these periods were not different — i.e., if one of the quantities oj were connnensurable with Zir, or, more generally, if there were a linear combination of the z's which had only one single period — i.e., if there were a relationship having the form ^iWi+ 6,(0,-)-. . .4- 6„_,w„^i-t- 2-ii6„ = o, ^2) where the b's are whole numbers. Let us disregard the case of this exception. The quantities (1) will equal —^=z 5111 Zl. We may choose the whole nvraiber m in such a way that these quantities repre- sent a combination having a given sign — i.e., that there are numbers zj^ which satisfy inequalities having the form «1<-Il< rtl 1- 'T, fll < Jj < ai -t- Tt, ,.., fl„_,< 2„.-i< rt„_,^-1r, y-o\ where the aj^'s equal or tt. This results directly from the statements which we have just /228 made above . Let us move on to the case in which we have a relationship of the form (2) . We may always assume that the whole numbers b are primes among themselves. In this case, the expression has only the period 2Tr. In order that there may be no numbers z, satisfying the inequalities (3) , it is necessary and sufficient that the difference between the larg- est value and the smallest value which expression (4) takes — when all values which are compatible with the inequalities (3) are assigned to be z^'s — is smaller than 27r, i.e., smaller than a period of this expression (4). This difference is obviously as follows 229 and we must therefore have |i,l-t-lfcil-t-.---t-l*'.->l-^- (^5) The inequality can only hold if all of the b's are zero, except for one of them which must equal +1. In this case uiy, must equal a multiple of 2t7. This means that a^ must be zero, since a^^ is only determined up to a multiple of ---^ We have excluded the case in which one of the aj^-'s is zero. The equation can only be valid if all the b's are zero, except for two of them which must equal +1. Then the sum of the difference between two of the uj^-'s will be a multiple of 2tt. If we note that the a^^'s are only determined up to a multiple of HL/Z^, we may express this result in another way. Two of the characteristic exponents will be equal. This is the only exception which still exists, and it may be readily excluded. 329. Let us now assume that the equations of dynamics under consid- eration depend on an arbitrary parameter y, just as is the case for the /229 three-body problem, as we know. When we vary y continuously, the periodic solution will also vary continuously, as we may determine from the discussion in Chapter III. The quantities Mu will also vary continuously, but ~ as was ex- plained in No. 323 ~ they can never vanish. Therefore, they will always retain the same sign , and It is their sign alone in which we are interested. The energy constant will be regarded as one of the given quantities of the problem at hand, but this given quantity may depend on y, and we 230 shall choose it in such a way that the period T of the periodic solu- tion remains constant. The exponents a^^ will also vary continuously when we vary y con- tinuously. Let us clarify to a certain extent the manner in which this variation should be handled in the case of the three-body problem. For y = 0, all the exponents are zero. However, as soon as y ceases to be zero, the exponents cease to be zero also. One of these exponents can only vanish, or become equal to a multiple of ^^^^^ , or become equal to another characteristic exponent for certain special values of y. 330. Let us consider a periodic solution of period T, such that all the exponents aj^ are purely imaginary. This is what we designated above by a stable solution. In Chapters III and IV, we proved the existence of these solutions. Let us consider one of the exponents, cxi, for example. When y varies continuously, T^fj' — which is real — will become commensurable with 27T . ^. . — an infinity of times. Let us assign a value yQ to y, such that where k and p are the prime whole numbers among themselves. In add- /230 ft . ition, this value does not correspond to a maximum or a minimum of At a later point, in No. 334. we shall see why I have placed 2kTT in the numerator, and not kir . In any interval, no matter how small it may be, there is an infinite number of similar values. If m is an arbitrary whole number, for this value yg the expression /- . / "-7 is zero. In addition, since yg does not correspond to a maximum or a minimum of -y-'^-' this expression will change sign when y passes from yg — e to yg + e. For example, let us assume that it changes from being negative to being positive. 231 Pursuing the line of reasoning presented in No. 328, we will find that we may choose the whole number m in such a way that the expressions — -= tin ^r-^- (A— 2, 3, . . . , n — i) y/ - I v'-^ I have all possible combinations of signs, and that they are all negative. Under this assumption, for m =- Mq - c, our function S^^^p will have a maximum, since all our expressions will be negative. However, for ^jj = ^Q + e^ our periodic solution will no longer correspond to a maximum of S , since one of these expressions will have become positive. Theorems Considering the Maxima 331. In order to pursue this subject further, it is necessary to illustrate one property of the maxima. Let V be a function of the three variables xi , X2 and z, which may be developed in increasing powers of these three variables. I shall assume the following: , , . dV dV 1. For XI = X2 = 0, V vanishes as well as its derivatives ■^, ^, no matter what z may be; 2. For xi = X2 = 0, V has a maximum for z > and a minimum for /231 z < 0. It may be stated that the equations dxi dxx have other real solutions in addition to the solution Let us develop V in powers of z, and let The functions Vq , Vi , V2 , ... may themselves be developed in powers of xi and of X2. However, these expansions will contain neither terms of degree nor terms of degree 1, because — no matter what z may be ~ we must have V - i— - -- - dx^ dxf for xi = X2 = 0. In addition, Vq does not contain terms of the second degree either. 232 Without the second degree terms, it is impossible to pass from the case of the maximum to the case of the minimum, when going from z > to z < 0. Conversely, Vi will contain first degree terms, at least we shall assume this is the case. Let us then consider the equations dxi dxi clxt . . o = -J i- z -.— -t- j' --, H . . . rfr, dX} clXi which must be solved. Let Uq and Ui be the lowest degree terms of Vq and of Vi . According to the statements which we have discussed, Ui is of the second degree, and Uq is of the degree p ~ with p being larger than 2. Let us set {p~7.)iL^.i; x,=y,i, x,^-r,t, \ = ^ytp■, 3==±tp-' W may be developed in powers of t. Let us set ^v== w,-r- AV,i-i-Wj-i-.... We obviously have /£3£ U\ = Uit~P and Uq = Uot~P are two homogeneous polynomials in yi and y2 — one of degree 2 and the other of degree p. I shall employ the sign + or -, depending on how I have set z =+tP"2. The expression dV dV, dV dV, dXf dxx dxi dxx will also be developed in powers of t when xi and X2 are replaced by yit and yat. It will include a certain power of t as a factor. Let us divide by this factor, and let H be the quotient. This quotient developed in powers of t may be written 11 =no+HIi+<'H, + ...; Hq will be the first of the expressions dWj. dU'i _ dyVt dU\ dy, dyi dyt dyi which will not vanish. The equations ^y rfV dXi dXi 233 may be replaced by the following equations II = o -,— = o. I shall prove that we may derive the y's from these equations in the form of series which are ordered in fractional and whole powers of t, which vanish with t and which have real coefficients. In order to do this, according to statements presented in Nos. 32 and 33, it is sufficient to establish the fact that for t = 0, these equations have a real solution of odd order . For t = 0, these equations may be reduced to llo = o, -^=c, or /233 rfw< dv\ _ d\yj, cnj\_ __ and rfU', d\}' Equation (2) indicates that Wi^^ has a maximum or a minimum, if we assume that y\ and y2 are related by the relationship U'j = const. For the present, if we assume that yi and y2 are the coordinates of a point in a plane, the relationship U', = const will represent an ellipse, because the quadratic form Ui (and, consequently, the form up must be defined in order that V may have a maximum or a minimum. Due to the fact that an ellipse is a closed curve, the function W2 must have at least a maximum and a minimum when the point y^, y2 describes this closed curve. Therefore, whatever the constant value may be which is assigned to Uj^, equation (2) will have at least two roots, and two roots of odd order , because we have seen in No. 34 that a maximum or a minimum always corres- ponds to a root of odd order. At this point, where we have no more than one independent variable, the theorem presented in No. 34 is almost self- evident. Under this assumption, we may distinguish between two cases: First case . Uq is not a power of \^y In this case, we do not have identically ^^o d\]\ d\\\ d\}\ d^i dyi dyi dyi = o. 234 We shall therefore have W]^ = Wq , and dU' dV\ dU' dU'. Equation Hg = is then homogeneous in yj and yi . No matter what the constant value is which is assigned to U', , it will provide us with the yi same values for the ratio — . 72 We may derive — from equation (2) and, according to the preceding statements, we shall obtain at least two solutions of odd order. /234 vi °'i Let -i-i- = — be one of these solutions. Let us set vi = aiu, yo = = a2U and let us substitute in equation (3) . We shall have and equation (3) may be reduced to If p - 2 is odd, this equation will give us a real value for u. If p - 2 is even, we may distinguish between two cases. If A and B have the same sign, we shall take the lower sign A «/>-'— 15 =0. If A and B have opposite signs, we shall take the upper sign Ait/'-s ;-B :;-- o, and we shall have two real values for u. In every case, these real solutions are simple. Thus, equations (2) and (3) will always h.ave solutions of odd order. Second case . We have u; = A(u;)f. We shall begin by solving equation (3), which may be written as follows p ''-1 :^A(U',)' ±1 = 0. 2 235 This equation provides us with the value of U'^ . This value is real and simple, but this is not sufficient because U*^ is a negative definite form. In order that the solution may be suitable, it is necessary that the value found for U'^ be negative; as a consequence, we shall choose the sign +. The value of U^ having thus been determined, we may assign this constant value to U'^ , and in order to solve equation (3) we need only /235 determine the maxima and minima of Wj^. As we have seen, we shall derive at least two solutions of odd order. We have therefore established the fact that equations (2) and (3) always have real solutions of odd order. The theorem presented at the beginning of this section has thus been proven. 332. Now let V be a function of n + 1 variables T], 2-2J ••■> ^n and z. I shall assume the following: 1. V may be developed in powers of x and of z; 2. For a^i m 2*3 = . . . = Xfi =^ o, we have the following, no matter what z may be rfV _ £/ V _ _ ^v _ dx^ Jj-, ' ' ' d.T„ 3. Let us consider the group of terms of V which are second degree terms with respect to the x's. They represent a quadratic form which may be equated to the sum of n squares having positive or negative coefficients. When z changes from positive to negative, I shall assume that two of these n coefficients change from positive to negative, and that the n - 2 other coefficients do not vanish. Under these conditions, it may be stated that the equations ^ _ JV _ _ ^ _ (1) dXi dxt ' ' ' dx„ have real solutions which differ from .Ti — rj = . . . = ^n " o. 236 Let us develop V in powers of z and let us set Let Uq and U^ be the group of second degree terms of Vq and Vj . The group Uj is a quadratic form which may be decomposed into a sum of n - 2 squares, because we know that, for z = 0, two of the coefficients which were in question above vanish. /236 Therefore, if we consider the discriminant of Uq, i.e., the func- tional determinant of (/j-, ' c/.r, ' ' d.fn with respect to ■^'ii ^i, ■■-, ■i"/i, this determinant vanishes, as well as all of its minors of the first order. However, all of the second-order minors do not vanish, unless a third coefficient is zero, which we have not assumed. We may also assume that a linear change in the variables has been performed, so that Uq is restored to the form Consequently, the functional determinant of (Wo 0, V — regarded as a function of x^ and of X2, — has a maximum when these two variables are zero. In order to illustrate this, we must try to find the second-degree terms with respect to x^ and X2 in V. Let 238 Wo 4- 3W,-h-'-W,+... be these terms. In order to obtain W„-i- j\V, which are the only ones which interest me, I shall take the two /238 terms Uo-l-3U,, and I shall neglect the other terms of V which cannot influence Wq + zWi . I may derive the following from equations (2) in the form of series ordered in powers of xx and X2. In these series, I shall only retain the terms which are of degree 1 with respect to x^ and X2, and of degree with respect to z. The other terms may be neg- lected, because they do not influence Wo+aW,. Equations (2) may then be reduced to 1 A i, .Ti + z - - — -- o, . rfU, i.\„Xn'^ z - — — o. dXn If we substitute the values thus obtained in place of X3, xt^, ..., Xjj, in Uq, we shall find that Uq is divisible by z^. With respect to Uj, it may be reduced to where Uj is none other than the quantity which Uj becomes when we cancel X3, xi^, ..., Xjj, and where u] and U^ are two other quadratic forms with respect to the x's. We shall therefore have and Uo-f- ;:U, == 5 U}H- ;:2(U3 -H Ul)+ ;:3 UJ. 239 In order to calculate Wq + zWi , I may neglect the last two terms which may be divided by z^ and z^, and I shall simply have I shall demonstrate the fact that V has a maximum for xi = X2 = /239 and for z which is positive and which is very small. It is sufficient to illustrate this for Wq + zWi, i.e., for zU^ . Finally, we must prove that uj is a negative definite form. For this purpose, we shall write the quadratic form Ui as follows u; is a sum of two squares having coefficients whose sign I shall not predict. U" depends only on the n - 2 variables This is always possible, according to the general properties of quadratic forms. Let lis consider the form where z is assumed to be positive and very small. The form U^ + zU'^' , which depends only on the n - 2 variables X3, x^, ..., x^^, may be equated to a sum of n - 2 squares having coefficients whose signs must be the same as those for A3, A^, . . . , A^, since — due to the fact that z is very small — this form differs very little from Uq. Therefore, they do not change sign when z makes a transition from positi ve to negative. According to our hypotheses, when z makes the transition from posi- tive to negative, n - 2 of our coefficients do not vanish, and, on the con- trary, two coefficients make the transition from negative to positive. These last two coefficients can only be the coefficients of Ui. Therefore, V[ is the sum of two squares having negative co efficients. In order to have u", it is necessary to set the following in U'^ iCj = T^ = . , . =^ 37/1 O, Then U" vanishes, and U^ may be reduced to U'^ Therefore, U^ is a negative definite form. q.e.d. 240 Therefore, V, regarded as a function of x^ and X2, is maximtan for z which is positive and is very small, and for x^ = X2 = 0. /24Q One will find in the same way — or rather one will find at the same time — that V is minimum for z which is negative and very small, and for x^ = X2 = 0. As I have stated, we have thus returned to the conditions stipu- lated in the preceding section, and it may be assumed that the theorem presented at the beginning of this section has been substantiated. Existence of Solutions of the Second Type 333. Let us return to the hypotheses given in No. 330. We have defined the function S^jj- , which depends on y, of the 2n variables The Sj^'s and the ri-j's are the values of x^ and y-j^ for t = 0. The Xj^'s and the Y^'s are the values of Xj^ and y^ for t = mpT. We sould like to study the solutions of the equations According to Nos. 321 and 322, these solutions correspond to periodic solutions of period mpT. We already know one of them, since a periodic solution of period T is at the same time periodic having the period mpT. I propose to show that there are others in addition. First, however, I would like to illustrate the method which may be employed to regard S as being dependent only on p and on the 2n - 1 variables ( X| + ^,, Xj-H^j, ..., X„-i + ^„_,, , . ( Yi-t-T,,, Yj-T-T,,, ..., Y„_,4-T)„_,, Y„ + i]n. For this purpose, we shall assume that Let us now consider the equations dS„ J(X(H-$,-) OCii-i-r,i) (1') We shall employ the d's to represent the derivatives of S which is assumed to be a function of the variables (a), and shall employ the 3's 241 to represent the derivatives of this same function S which is assumed /241 to be a function of the variables (B) . I plan to show that equations (1) and (1') are equal. Section No. 322 has provided us with the following dS = Z[( \i - I) diY, + r,i) - ( Y; - T„- - 2 m,7!}d( X, -4- r„-)]. Equations (1) may therefore be written — ( Y, — -/,,■ — ■> »),-) -= X, — ^,- = o (' ^ 'i ' ")> and equations (1') may be written as follows — (Y/ — T,, — -imtr.) -. \i — u ^o {i ^ I, 1, ..., n ~i), X„ -f, — o. In view of the energy eqiiation, we have also According to equations (1'), all of the X^'s equal the C^'s, and all of the Y.'s (except one) equal n^^ + 2m-j^ir. The preceding identity may therefore be written as follows. For purposes of abbreviation, I shall write F(;i. ;!> ■•■. ?«; T,i-HQ7n,T7, rij-t-iOTjTT, ..., T,„_,-T--2m„^|-;t, Y„) = F(Y„). My identity may therefore be written in the following form F[r,„ u 9. m„ 77 + ( Y„ — T), — 2 /«„-)] - 1? (t], ■;- ini^Ti) -^ o, or, in view of the theorem of finite increases (Y„ — T,„ — a77i„T:)F'[Tj„-+-2m„^ -i-0(Y„ — T,„ — jm„-n:)] = o, (2) where 6 is included between and 1, and where F' is the derivative of F with respect to Y^^. Let i. and n? be the values of E,^ and n^ which correspond to the periodic solution of period T. The region under consideration only in- cludes the immediate vicinity of the point \i = \iq, E,^ = g_^, t], = n^. Therefore, 1,^ and X. will never deviate greatly from 5?, and n^ or Y. - 2m^TT will never deviate greatly from n^. Therefore, the second factor F' of relationship (2) will never deviate greatly from its value 242 ^ _ tor ^^ - K -^t '^1 ~ ^i' ^ ^^ general this value will not be zero. Therefore, the first factor of relationship (2) must vanish, and /242 we have Y„— T,,— im„TX =r o. In other words, equations (1') entail equations (1). We may therefore regard S as a function of the variables (B). When it is a maximum, considered as a function of the variables (B) , it will also be a maximum as a function of the variables (a) . I have employed 5? and n? to designate the values of C^ and of n^ which correspond to the periodic solution of period T. The corresponding values of X^^ + ^^ and Y^ + m will be 2^^ and 2n9 + 2m£mpTr (if the periodic solution of period T changes y^ into y^ + 2m^7T, in conformance with the hypotheses formulated in No. 322). Let Sq be the corresponding value of Sjjj . Let us set and let us consider V as a function of y' , of the 5' 's, and of the n' 's . The function V will be governed by the same conditions as the function V of the preceding section. No matter what y' may be, V and its first derivatives with respect to the C 's and to the n' 's will vanish when ?;-=T/i---o. If we consider the group of second degree terms of V with respect to the 5' 's and the n' 's , and if we regard it as one quadratic form which is decomposed into a svrai of square terms, it may be seen that two of these coefficients of these square terms both make a transition from negative to positive, or both make a transition from positive to negative, when y changes sign. The other coefficients do not vanish. The expression — _ —sini — . changes sign, and the other expressions : sini /ITT /: 243 do not vanish. The coefficient which I have designated as D in No. 323 no longer vanishes, and there is not another one because we have only /243 2n - 1 variables , the variables (6) . The conditions presented in the preceding section therefore hold, and we may state that the equations rfV d\ d\i drii have other real solutions in addition to E.'^ = n'^ = or, which means the same thing, equations i'^mp ^ ^ ( f^mp _ g (1) have other real solutions other than those corresponding to the periodic solution of period T. The maxima of the function Sjjjp, or more generally the solutions of equations (1), correspond to periodic solutions of period mpT. We must therefore conclude that our differential equations have periodic solutions of period mpT, which differ from the solution of period T, which is identical to that for y = yo. and which differ only slightly for y close to yg. If attention is drawn to the preceding line of reasoning, we shall find that the periodic solution of period T need not correspond to a maximum of S^j-. We shall therefore set m = 1. It is not necessary that the solution of period T be stable. It is sufficient that one of the characteristic exponents ai equals for y = yo. We therefore obtain the following result. If the equations of dynamics have a periodic solution of period T, such that one of the characteristic exponents is close to they will also have periodic solutions of period pT which differ very /244 244 little from the solution of period T, and which are identical to the latter when the characteristic exponent equals These are solutions of the second type. Remarks 334. This entire line of reasoning assumes that S^p is a uniform function of X^ + g^ , Y^ + ri-j^ . Under this condition alone may it be stated that all the maxima of S correspond to a periodic solution (see No. 321). This fact cannot be stressed enough. It is an obstacle which will be encountered frequently when we wish to derive the results of the theorem presented in No. 321. Let us determine whether S is a uniform function of these variables. We may assume that m = 1, which we have just illustrated. In addition, Sp is clearly a uniform function of the 5^'s and the Hi's. It will also be a uniform function of the X^ + ^^'s and the Y^ + n^-s. provided that the functional determinant of the X^ + 5^'s and the Y^ + ri-j^'s with respect to the C^'s and the n^'s does not vanish in the region under con- sideration. Due to the fact that this region may be reduced to the imme- diate vicinity of the values J'o, 5.= $?, n?. it will be sufficient that the functional determinant is not zero at this point. This functional determinant may be written as follows (assuming that n = 2 , to formulate our ideas more clearly) ^^' u , 'l^L dXi dX, /f, -"- ' (Ir^'t dU dM dY, 'dU dY, d'ii d'n dX, dU dX, dn. dY, dY, dY, dY, dU dra di. dn. It must therefore be verified that the equation in S /245 245 d\, dhi dX, 'du dX, dr„ 4. dn 1 d\\ 'dU dr^ d\i dni d\r ^ -df.-^ dX, dr,. d\. ,?Y, d\. <^Y, 'du dfj, d-t dr„ does not have a root which is equal to -1. According to the statements presented in No. 60, the roots of this equation equal where the a's are characteristic exponents. We must therefore verify the fact that we do not have /'I where k is an integer number. By hypothesis, the exponent a^ equals 2 f>~ v/-7 where k is an integer number, and the other exponents are not commensur- able with!L./:il. , in general. T The difficulty with which we are concerned will not therefore occur. In order to avoid this, in No. 330. I assumed that ,'iTi/^7 (.^ integer number) and not _ /,Trv/£_|_ ''' ■ pT (k integer number) Special Cases 335. Let us say a few words about the simplest cases, and let us assume only two degrees of freedom. Let us assume that the form which is similar to that which I have 246 designated as Uq , In the analysis of No. 331, is homogeneous of the /246 third degree only in xi and X2 . The equation dxl Jzj clxl Z^ ^- " (1) always has real r 0, a maximum for p = and two minima. For z < 0, a minimum for p = and two maxima. Employing the English term, I shall use the word minima to desig- nate a point for which the first derivatives vanish, and where there is neither a maximum or a minimim. The same will hold true for the function V, since — if z is very small — the terms Uq + zUi alone will have an influence. Therefore, no matter what z may be, the differential equations will have : 247 ^^ A solution of period T, of the first type, which is stable; A solution of period pT. of the second type, which is stable for z < and unstable for z > 0. Let us now assume that equation (1) has three real roots. The function f (<(.) will have three maxima and three minima which are equal pairwise and have opposite signs. In this case Uq + zUi , and consequently, V have: For z > 0, a maximum for p =0, and six minima; For z < 0, a minimum for p = 0, six maxima. No matter what z may be, the differential equations will therefore have: A solution of period T, of the first type, which is stable; Three solutions of period pT, of the second type. We shall see below that, from a certain point of view, none of these solutions are different. Let us proceed to a case which is a little more complicated, and let us assume that Uq is of the fourth degree. In this case, equation (1) is of the fourth degree, and, since it always has at least two real roots according to No. 331, it will have two or four. We then no longer have but rather /() will then have a maximum and a minimum when (t> varies from to tt, as well as when ^ varies from u to 2tt. A distinction may be drawn between two cases, depending on the signs of this maximum and this minimum. First case . The maximum and the minimum are positive. The functions Uq + zUi and V have: 248 For z > 0, a maximinn for p = 0, two minima and two maxima. For z < 0, a minimum for p = 0. In addition to the solution of the first type which always exists, the differential equations have two solutions of the second type for /248 z > 0, and do not have any for z < 0. Of these two solutions, one is stable and one is iinstable. Second case . The maximum is positive, and the minimum is negative. The constants Ug + zU^ and V have: For z > 0, a maximum for p = 0, two minima; For z < 0, a minimimia for p = 0, two minima. The differential equations always have an unstable solution of the second type, in addition to the solution of the first type which is stable. Third case . The maximtm itself is negative. The differential equations then have: For z > 0, a solution of the first type which is stable; For z < 0, a solution of the first type which is stable, and two solutions of the second type of which one is stable and one is unstable. We must now examine the case in which equation (1) has four real roots. The equations then have: For z > 0, a solution of the first type which is stable, h solutions of the second type which are unstable, and k solutions of the second type which are stable; For z < 0, a solution of the first type which is stable, 2 - h solu- tions of the second type which are stable, and 2 - k solutions of the second type which are unstable. The integer numbers h and k may take the following values, depending upon the signs of the maxima and the minima of f ((J>) : A = ^ r= 2 ; A = 2, /: — i; h = ■>., k = o\ A =- i , k —o; h = k =o; A = A- = 1 . 249 \-/ CHAPTER XXIX DIFFERENT FORMS OF THE PRINCIPLE OF LEAST ACTION 336. Let Ji, 72. ■■•. 7" be a double series of variables, and let F be an arbitrary function of /249 these variables. Let us consider the integral ,=jr"(-,M..„5),„. The variation of this integral may be written as follows. In order that this variation may vanish, it is necessary that we have jxi _ ,/v_ -^— -^^' -^^~iJ' ■■■' dx, dt J' (7) In the place of the y^'s, let us substitute their values (7) in J and in the equation /253 F =^0. dx, dx]j ^1 We shall derive -tT" as a function of the xj^^'s and the -r — 's from this 253 equation. We shall then substitute this value of -^ in expressions (7) and in J. This last integral will take the following form /^/'£:'^--/"'^^" dx]^ where $ is a function of the xj^'s and of the derivatives ■^—. This integral, which is thus written in a form independent of time, is still minimum. This is the principle of least action in its Maupertuis form. If h were not zero, we would only have to change F into F - h. 337. Let lis first examine the most important particular case. Let us assume that we have F =^ T — U, where T is homogeneous of the second degree with respect to the variables y., while U is independent of these variables. We then have Zj^^^-^^-T, II = T-i-U. According to the principle of Hamilton, the integral r '(th-u)(/« must be minimum. Let us determine what the principle of Maupertuis becomes. The energy equation may be written T U = h. The Maupertuis action then has the following expression ^(T4-U + /t)(/^ The equations ^^^^ ^ £{F = i^ di d/i dyi have their second terms which are linear and homogeneous with respect to 254 the y^ s. Therefore, T is homogeneous of the second degree with /254 respect to the "^ s . Let di"^ represent that which T becomes when dx^ -J— is replaced by dx^ ; we shall have T - - - and di will be a form which is linear and homogeneous with respect to the n differentials dx^. We may deduce the following from this di= ''' '^^ The Maupertuis action will then have the following expression ■>. fdz/uT-Jt. 338. For purposes of brevity, in order to be able to study other particular cases, let us set , d.T( and let us derive the y-j^'s of the equations ,_ dP so as to take the x^'s and the x^ 's for new variables. Let us employ the ordinary d's to designate the derivatives taken with respect to the Xj^'s and to the y^'s, and let us employ round 8's to designate the deriva- tives taken with respect to the Xj^'s and the x'^ 's. We may readily obtain the well-known relationships drj- O.Tj dxt and we will see that equations (1) are equivalent to the Lagrange equa- tions dt dx'i Oxi /255 Under this assumption, let us examine the case in which H has the 255 following form H = H,-HH,-t-lI,, where Hq , H^ , H2 are homogeneous, of degree 0, 1, 2, respectively, with respect to the variables x'^. We then have £ici:r-^ =2H,-t-n,, F = n,— Ho and the _ m^ dH, ^' " dx'i "^ d^ are linear functions, but they are not homogeneous with respect to the ■k\ 's. The Hamiltonian action retains the same form Al dt. Let vis determine what the Maupertuis action becomes. Let h be the energy constant. The Maupertuis action will have the following expression r{H-i-h)dt but it must be written in the form which is independent of time. For this purpose, let us set and , IT - " H2 is nothing other than energy, and dx^ is that which this energy be- comes when x^ is replaced by dx^. In the same way, do is that which Hi becomes when x'^ is replaced by dx^. It is therefore a form which is linear and homogeneous with respect to the differentials dx^. If we take the energy equation into account H,= Ho+A, 256 from which we have the Maupertuis action will become /256 The Maupertuis principle may therefore be applied to the case in which we are interested, as well as to that of absolute motion. However, there is one essential difference from the point of view of the following statements . In all the problems which will be encountered, the energy T or H2 is essentially positive; it is a quadratic, positive definite form. In the case of absolute motion (No. 337), the action is essentially positive. It does not change when the limits are inter- changed. On the contrary, in actuality, the action is composed of two terms. The first f'. d: v/Ij; h is always positive, and does not change when the limits are interchanged. The second J da changes sign when the limits are interchanged, and it may therefore be positive or negative. If we also note that in certain cases, the first term vanishes with- out the second term vanishing, we will find that the action is not always positive. This fact will cause a great deal of difficulty later on. 339. In order to show how the preceding considerations may be applied to relative motion, let us first consider the absolute motion of a system. Therefore, let H = T -H U and let us assume that the position of this system is defined by n + 1 variables X|, Tt, ..., X„, O), where xi , X2, ..., x^ are sufficient to find the relative position of different points of the system, and o) is the orientation of the system in space. 257 If the system is isolated, U will depend only on x^, X2, ..., x^^. T will be a form which is quadratic and homogeneous with respect to /257 x' , x' ..., x' , cj' whose coefficients depend only on Xj^ , X2, ..., x^^. We will then have the equation where p is a constant. This is the area integral. Under this assumption, let J be the Hamiltonian action J= f Udt; We shall have the following, if the equations of motion are satisfied 5J -_^\y ^?,r+~ ou)l'"'' The action will be minimum (or rather its first variation will be zero) if the initial and final values of the x^'s and of u are assumed to be given — i.e., if 6x^ = 6a) = for t = tg and for t = tj. Let us now assume that the initial and final values of these x^'s are given, but not those of w. We shall have Then let and r= Tll'dt, and we shall obviously have We may derive to', which is a linear, nonhomogeneous function of the x'.'s, from the equation x", = P- It may also be seen that H is a 1 dw quadratic function which is not homogeneous with respect to the x^ 's. H' therefore has the form Hq + H^ + H2 which was studied in No. 338. 258 The integral J' will thus be minimuni, even though the initial and final values of cj are not assumed to be given. We have /258 J' - J — jD('Ji — i)o)) where wg and o)]^ are the values of co for t = tg and t = tj. 340. Let us now assume that we have a system referred to moving axes and subjected to forces which depend only on the relative situa- tion of the system with respect to the moving axes. In addition, let us assume that the axes rotate uniformly with a constant angular velocity This problem may be directly related to the preceding one. We need only assign a very large moment of inertia to the moving axes, in such a way that its angular velocity remains constant. For the absolute motion, we then have II .^ T -4- U := T, -*- T, + U. The function of the forces U depends only on the variables x-j^ which define the position of the system with respect to the moving axes. Ti, which is the energy of the system, depends on the Xj^'s, and is a quad- ratic form with respect to the x^'s and to u' . T2, which is the energy of the moving axes, equals 2 and the moment of inertia I is very large. We then have and H'= H-j3u' = (T,-T-T,-i-U)- ^ to'-Iu'«" °^ dT, Ito'« H'= T,-hU-5^u>'-:^^. atij 2 However, 1 r "' 1 259 K^ dTi Since I and p are very large with respect to -— -, this equation d(jj gives us approximately the following X£2Z. and more exactly I , _ p I rfTj ,0 „ -- _ _ -^_. , In addition, we have Ia>'' _ /)' -P rfu7 I /rfT, y 2 2 I 1 2 1 \ (?lo' / rfT, We thus obtain In the second member, the term before the last is a constant. The last term is negligible, because I is very large. Since we may add an arbitrary constant to H' without changing the Hamiltonian principle, we may set H'=T,-^U and we know that the integral J"= Cwdt must be minimum (even though the initial and final values of u) are not given) . In the expression of H" , oi' must be regarded as a given constant. H" is then a quadratic function, which is not homogeneous with respect to the Xj^'s, having the form Hq + Hi + H2. For example, let a material point having the mass 1 move in a plane, whose coordinates with respect to the moving axes are 5 and n. We shall have We therefore have 260 The integral is then minimum, when we assume that the limits tg and t^ are given, /260 as well as the initial and the final values of 5 and n. The energy integral may then be written and we have seen that the integral is minimum even though we do not assume that tg and t^ are given. We then obtain i'==Ji2lh -.- II, ) dt ^f[ds /Ho -*- h + o. The first term of the quantity under the sign J which is ds v/Hj + h is essentially positive. This will "not be true for the second term, which changes sign when we reverse the direction in which the trajectory is assumed to be traversed. If the point 5, n is very close to the border of the region in which it is confined, and if, consequently, Hq + h is very small, the first term will be very small, and the second term is the one which will give the term its sign. J' is therefore not essentially positive. This can also be seen by means of the following equation 261 If h Is negative, the first term J is positive and the second is negative. Kinetic Focus /261 341. Up to the present, when I have stated that a certain integral is minimum . I was employing abridged terminology which was incorrect and could not deceive anyone. I should say the first variation of this inte- gral is zero ; this condition is necessary in order that there be a minimum, but it is not sufficient. We shall now try to determine the condition for which the integrals J and J' , which we studied in the preceding sections and whose first variations are zero, are effectively minimum. This investigation is related to the difficult question of second variations and the excellent theory of kinetic focus. Let us recall the principles of these theories. Let xi, X2, ..., x^ be the functions of t; let x\, i^2» • • • . x^ be their derivatives. Let us consider the integral J =y '/(xi, x[)dt, whose first variation 6 J is zero, assuming that the initial and final values of the x^'s are given. In order that this integral may be minimum, a condition which I shall call condition (A) is necessary, but not sufficient. The condition is that /(Xi, X'i -I- E,) — Se/ -^ , regarded as a function of the Cj^'s, is minimum . Condition (A) is not sufficient, unless the integration limits are not very close. Except for this case, it is necessary to add another condition which I shall call condition (B) . In order to explain this, I must first recall the definition of kinetic focus. In order that 262 ^:J ^ o, it is necessary and sufficient that the x^'s satisfy n differential /262 equations of the second order, which I shall call equations (C) . Let be a solution of these equations. Let us set the following for an infinitely close solution and let us formulate the variational equations, the linear equations of which satisfy the d's and which I shall call (D) . form The general solution of these equations (D) will have the following i = J/i ii=J^^khk (i = i, a, ..., rt). The Aj^'s are 2n integration constants, and the S^k'^ are 2n^ func- tions of t, which are determined perfectly and which correspond to 2n particular solutions of the linear equations (D) . Under this assumption, let us state that the ^^^'s all vanish for two given times t = t' , and t = t". We shall have 2n linear equations between which we may eliminate the 2n unknowns Aj^.. We shall thus obtain the equation where A is the determinant Sn.i td.i SI. lit ?n.t ?/»> ••• in.tn I The quantities C'^j^ and S^j^ represent that which the function E,^y^ becomes when t is replaced by t' and by t". 263 If the times t' and t" satisfy the equation A = 0, we may say that these are two conjugate times and that the two points M* and M" in space having n dimensions, which have J_lb3^ ■•?i(''). ojC.'') ?'.(f). respectively as coordinates, are two conjugate points . In addit is the close ttion, if t" is the time conjugate to t' after t', which 2St to' t' , we may state that M" is the focus of M' . We may now state the following condition (B) : There is no conju- gate time of to between to and t\. In order that J be a minimum, it is necessary and sufficient that the conditions (A) and (B) be fulfilled. A direct consequence may be inferred from this. Let to, ti, t2, t3 be four times. Let Mq, Ml, M2, M3 be the corresponding points of the curve Let us assume that M^ is the focus of Mq and M3 that of M2. If condition (A) is fulfilled, we may have or or But we cannot have Otherwise, the integral hh readily seen. A curve in space having n dimensions Xi -- 0,(0 representing a solution of the equations (c) can be called a trajectory, which I shall call (T) . The curve Xj — o,- ^- li will represent an infinitely close trajectory. If we draw one of these trajectories (T' ) which are infinitely close to (T) through the point M' , and if this trajectory again intersects the trajectory (T) at M" (more precisely, the distance from M" to this tra- jectory will be an infinitely small quantity of higher order), the points M' and M" will be conjugate if, in addition, the point which follows (T') passes through M' and infinitely close to M" at the times t' and t". 342. In the case of the Kamiltonian principle, condition (A) is always fulfilled. In effect, we have II = H(,+ H,-^H„ and H2 is a quadratic form which is homogeneous with respect to the xL 's. In all of these problems of dynamics, this quadratic form is definite and positive. If we change xj^ into x'^ + e^. Hi will change into H,(.;)-.s.g and H2 will change into 265 dUt and in addition we have Therefore , we have from which we finally have 7265 The first term corresponds to the function Since the quadratic form E^ie^ ^^ positive definite and we may see that the expression is minimum for Ei = - i.e., that condition (A) is fulfilled. 343 Let us proceed to the case of the Maupertuis principle in absolute motion. The integral to be examined may then be written where dx^ is a positive definite, quadratic form with respect to the differentials dx-j^. For the time being, let us select xi as the independent variable, The integral becomes J dx, dT\ „here l^^ is a polynomial of the second order P which is not homo- Y dx]^ / dx-j^ geneous (but essentially positive) with respect to the -j^'s. Therefore, let us set d-z dx \-\/^^' 266 We must determine whether \/"(i;*'.)--';^-'i^ Is minimum for e^ = 0. In other words, we must determine whether the second derivative, with respect to t, of the radical IS positive. dxj , No matter what the -5 — 's and the e^'s may be, we shall have /266 P[~- -1- £,«) = af^ + 2bt + c, \ - -1- e,n = at ■1 / where a, b, c are independent of t. The second derivative of the radical then equals ac — 6» («<' -h 2bt +- c)' Since the polynomial P is essentially positive, this expression is also always positive, and condition (A) is always fulfilled. 344. Let us proceed to the Maupertuis principle in relative motion. We must then consider the integral J [ds /U„ 4- h + a;'(J dr, — tj d'^)], or, choosing E, as the independent variable, we have J '( Jt/- 7)) is positive. This derivative is Condition (A) is therefore always fulfilled. Thus, condition (A) is itself fulfilled in every case which we shall 267 examine , Maupertuis Focus 345, The kinetic focuses are not always the same, depending on whether Hamiltonian action or Maupertuis action is being considered. In order to clarify this point, let us assume only two degrees of free- dom, and let x and y be the two variables which define the position of /267 the' system, and which we may regard as the coordinates of a point in a plane. Let ^-/.(O, JK=/,(0 be the equations of a trajectory (T) which will be a plane curve. Let us set and, neglecting the squares of C and of n, let us formulate the varia- tional equations. Since they are linear and of the fourth order, we shall have t; = a, j;i + rtjr,, 4- ajT,, + n^r^i, where the aj^'s are integration constants, and the ^i's and n^'s are functions of t. The equation given in No. 341 may then be written w r, r, w T.'i T.'j -''% li £" t' f* f -.1 ?j ?3 ■:» I'm T,I ^13 T,j (1) It is this equation which defines the Hamiltonian focus . It indicates that the point x, y, which describes the trajectory (T), and the point x + C, y + n, which describes the infinitely close trajectory (T'), occur at two different times, i.e., at the times t' and t", separated by an infinitely small distance of higher order. 268 However, these are not the conditions which the Maupertuis focuses must fulfill. Two points of the trajectory (T) — i.e., the two points M and M" which correspond to the times t' and t" — must be separated by^an infinitely small distance of higher order from the trajectory (T'). However, it is not necessary that the moving point which tra- verses (T') passes precisely at the time t" — for example, infinitely close to M". On the other hand, the energy constant must have the same value for (T) and for (T' ) . This last condition is not imposed on Hamiltonian focuses. One of the solutions of the variational equations is /268 We may therefore assvmie that The two functions C^ and m are thus defined. In addition, the difference between the energy constant relative to (T) and the energy constant relative to (T' ) is infinitely small. This is obviously a linear function of the four infinitely small constants Sl» ^2, 33, at^. Without limiting the conditions of generality, we may assume that this difference is precisely equal to a^. The condition stipulating that the value of the energy constant be the same for T and (T' ) is then ai+ = 0, or For t = t', C and n must be zero, from which we have equations «j y,\ + a, T,', + a, T,', = o. In addition, the value of x + 5, y + n for t = t" + e must be the same (up to quantities which are infinitely small of a higher degree) as that of x and y for t = t", which may be written (£ + a,)?; + a,fJ + a, tJ = o, (£ + a, )t,; -t- (7,T,; -4- a,Ti; = o, from which we have, by elimination. 269 i; k, r. o 'ii l', t/, o By developing the determinant, we obtain sj'ii ?:'ii ii'u — Cj^ii (2) and, setting 7269 equation (2) becomes = a'), ?(«') = !:('';• (3) Application to Periodic Solutions 346 If we are dealing with a periodic solution of period Ztt, the functions fi(t) and f2(t) of the preceding section will be periodic of the period 2tt. The same holds true for In addition, according to Chapter IV, the variational equations will have other particular solutions which will have the following form In these equations, g is a constant, a and -a are the characteristic exponents, and the t', it is necessary that k be positive. .11 4-1 By setting k = 1, we shall give the smallest value to t" - t'. We have T ^ T = — - and the point M" is then the focus of M' . One factor must be pointed out. In order that the preceding line of reasoning may be applicable, it is necessary that log G(t) be a periodic function. However, in general, all that we know is that G(t) is a periodic function, and as a result logGC/) is increased by a multiple of 2iTr, for example, of 2ki7r, when t increases by 2iT. Then logG(0 — I*' is a periodic function. Let us then set q.^,) ^ q^,)^-,*-/^ a = a H ; 2 we have ^(,) = g,„G(0 = e"''G'(0- We shall then no longer set I 21 272 but rather Since log G(t) will be periodic, the preceding conclusions remain valid, and equation (3) will be written T — X = — — (m is an integer number) and, in addition, M" will be the focus of M' if /212 7. 348. One of our three hypotheses stating that log G(t) must be periodic has thus been proven, I may now state that the function t must be constantly increasing, as we assumed. Let us assume that this function has a maximum tq for t = tg. We may then find two times t'^ and t'j* such that the corresponding values x\ and Tj of the function t are equal, and two other times t2 and t^^ such that Tj = T^ and such that the five times which are very close to one another satisfy the following inequalities Then t" will be the focus of t' , t" that of t\. We saw above that such inequalities are impossible when condition A is fulfilled. I may now state that G(t) cannot vanish. We have The numerator and the denominator of c(t) are imaginary and conju- gate. If one of them vanishes, the other also vanishes, so that the function ?(t) cannot become either zero or infinite. Thus, all of our hypotheses have been proven. Unstable Solutions 349. Let us now assume that the unstable solution and a^ are positive; in this case Sa. 12. ?3, na, ?, a, G are real. For the same reason as given above, the function t will be con- stantly increasing. However, two hypotheses are possible: 273 1. The quantity c(t) cannot vanish nor become Infinite , and Increases constantly from to +«> when t increases from -» to +°°, It then happens that no point of our periodic solution has a Maupertuis focus . 2. The quantity C(t) may vanish for t = to- It will also vanish /273 for t = to + 2Tr, and since It cannot have either a maximum or a minimum it must become infinite In the Interval. In the same way, if C(t) can become infinite, it must also be able to vanish. In order to clarify our thoughts, let us assume that c(t) becomes infinite for and for values which differ from these by a multiple of 2Tr, and vanishes for I shall assume that h. The closed trajectory (T) which represents our periodic solution will therefore be divided into two arcs, whose extremities will corres- pond to the following values of t Each of the points of one of the arcs will have Its first focus on the following arc. I may add that the points corresponding to the values of t 'oi 'o' 'l> 'l coincide with their two focuses. Let t" be a value of t corresponding to an arbitrary point of (T) , and let t^ be the value of t which corresponds to its niS. focus. We shall have lim = n 1 Ilk However, this is not all; we shall have If n is very large and if G(t") is not infinite, since t^ - t" is very large and since we assume that a is positive, G(t") will be very /274 small, so that if t" is, for example, included between tg and ti, the difference will strive toward t' when n increases indefinitely. If n strives toward -°°, this difference will strive toward tg or toward t^ , depending on whether t" will be Included between tg and t'o or between tVand t^. I should add that the difference f'jj, - 2nTr is either constantly increasing or constantly decreasing with n. The values t q , t' correspond to the points where but 51^2 ~ ?2^1 ^s ^ periodic function multiplied by e*^ . However, a periodic function must vanish an even number of times in one period. Consequently, the closed trajectory (T) will be divided by the points to, ti, tg + 2-n into a certain nxjmber of arcs, and this number will always be even . 350. From the point of view in which we are interested, the un- stable, periodic solutions may be divided into two categroies. However, it could be asked whether these two categories exist in actuality. It is therefore advantageous to cite some examples. Let p and w be the polar coordinates of a moving point in a plane. The equations of motion may be written For p = 1, let us assume that we have t/w dp ' dp' '' '^ • Equations (1) will have the solutions and this solution will correspond to a closed trajectory which will be a circtnnference. 275 K^ Let us set and let us formulate the variational equations. They may be written The second may be integrated immediately dt ^ but this constant must be zero if we want the energy constant to have the same value for the trajectory (T) and for the infinitely close tra- jectory. Therefore, if we replace -^ by -2c, the first variational equation will become ;5J| = a-f(0-3]. (2) Equation (2) which remains to be integrated is a linear equation having a periodic coefficient. These equations were discussed in Sections 29 and 189 (see, in addition. Chapter IV, in various places) . It is known that they have two solutions of the following form: where G and Gi are periodic functions. We are going to present examples for every case mentioned above. Let us first assume that ^ may be reduced to a constant A (case of central forces) . If A < 3, we shall have a stable, periodic solution. If A > 3, there will not be a Maupertuis focus on (T) , and we shall have an unstable, periodic solution of the first category. I must now show that we may also have periodic, unstable solutions of the second category. The solution will be unstable and of the second category if G vanishes in such a way that the ratio 276 1115 which corresponds to the ftinctlon ^(t) of the preceding sections can Jllh vanish, and consequently can become infinite. We may obviously formulate a periodic function G which satisfies the following conditions: 1. It has two simple zeros and only two; 2. These zeros will also vanish if G dC, dl'- dt As a result, every time that vanishes, its second derivative will also vanish in such a way that the ratio remains finite. One could obviously formulate a function G which satisfies these conditions. The periodic function ({> formulated by means of this func- tion G will correspond to an unstable, periodic solution of the second category. As an example of function G satisfying this condition, we may set G — sliif— - (cost — cosSO- 4 This fxmction vanishes for t = and t = ir, and it does not have another zero if /a For t = and for t = tt, we have di'-^'-^dT--''- Q In order that the ratio -r— may vanish, it Is not sufficient that G vanish;, it is still necessary that G^ does not vanish. 277 \^ However, this is what occurs, because if G and Gi vanished at the same time, the two solutions /277 'G(0, <,'-^'G,(0 could only differ by a constant factor (since they satisfy the same differential equation of the second order), and this is absurd. 351. One point to which I would like to draw attention is the fact that the unstable solutions of the first category and of the second category form two separate groups, so that we cannot pass from one to another continuously without passing through the intermediary of the stable solutions. Let us first confine ourselves to the particular case given in the preceding section, and let us reconsider the equation 3). (2) Let us vary the function <() continuously, and let us determine whether we can pass directly from an unstable solution of the first category to an unstable solution of the second category. For this purpose, it is neces- sary that the function G, which is real, be first incapable of vanishing, and then be capable of vanishing. We would thus pass from the case in which the equation G = has all its imaginary roots to the case in which it has real roots. At the time of passage, it would have a double root or, more generally, a multiple root on the order of 2m. This zero, which would be on the order of 2m for G, would be on the ■ of presslon order of 2m - 1 for ■^, on the order of 2m - 2 for -^, so that the ex- Tin"' 2 s -; \- r-G dl would be come infinite, which is impossible since it equals '-r/^'). (7) ... . d^U _^ d^U , I shall designate the sum of the two second derivatives — -y + — 2 ^y AU (or more briefly by A) . The following identity may be easily verified = 1{/ x' ~ y x')(\ x' ^ r^ y -\x' -- -ry), or, taking into account (5), (6), (7) and (3), we have (yj_,-_y'i)(6'- 6i)- 2(x'a:' + y7')0' -H 2(x"» + y )0 ^ o. (8) This is the differential equation which defines the unknown function 9. We shall set and our equation becomes 282 X — 2 ui r = — , y + 2 w a:' = -j— , ax ay (9) an equation having the same form as equation (2) of the preceding section. The conclusions of the preceding section therefore remain in force. One periodic unstable solution is of the second category, or of the first category, depending on whether the function i) can vanish or not. We can- not pass directly from an unstable solution of the first category to an unstable solution of the second category, but can only pass through stable solutions. 354. Do the same results still remain valid in the case of relative motion? The equations of motion then become d\i . , dM (1") dx' ^+---=^. where o) designates the speed of rotation of moving axes. The variational equations will be /283 Due to the fact that the energy equation is still valid, the same will hold true for Let us set and equations (5) and (6) will continue to hold. In addition, since x' and y' must satisfy equations (2'), we shall have , ■^'U , d^M , rf'U , rf'U , Taking these equations into account, as well as equations (2'), and also taking into account equation (3) , we may simplify the expression of 6", and we again obtain the equation 283 Since the identity given in the preceding section is always valid, we shall obtain equations (8) and (9) again. Therefore, nothing needs to be changed in the conclusions given in the preceding section. 355. However, one new question arises. The trajectory (T) is a closed curve. Up to the present, we have tried to determine whether an arc AB of this curve would correspond to an action which is smaller than any infinitely adjacent arc with the same end points. However, we may also question whether this entire closed curve corresponds to an action which is smaller than every infinitely small closed curve. Let us first assume that a point A of the curve (T) has its first focus B on the curve (T) , so that the arc AB is smaller than the entire /284 closed curve. This is what occurs for unstable solutions of the first category. We have seen that the curve (T) may be divided into a certain even number of arcs for these solutions, and that every point on one of these arcs has its first focus on the following arc, so that — starting from an arbitrary point — its first focus will be encountered before the entire curve (T) has been traversed. This also occurs for certain stable solutions. In the case of stable solutions, we have set (No. 347) and we have seen that the x of a point, and that of its first focus, differ by — . Therefore, if y is larger than y, the focus of a point will be encountered before (T) is completely traversed. If this is the case, the action cannot be less for the curve (T) than it is for any Infinitely adjacent curve. Let ABCDEA be the curve (T) , and let us assume that D is the focus of C. Since E is outside the focus of C, we may attach C to E by an arc CME which is very close to CDE, and which corresponds to a smaller action. If I represent the action corresponding to the arc CME by (CME), we shall have 284 and, consequently, (CME)<(CDE) (ABC.MEA)<(ABCDEA). Let us now consider a stable solution, such that It may be stated that the action will no longer be less for (T) than it is for any infinitely adjacent closed curve. In order to clarify these ideas, I have compiled a figure, assuming a 11 that — ranges between -r and 7-, in such a way that the focus of a point is encountered before traversing (T) three times, and after traversing /285 (T) twice. Let ABCDA be the curve (T) . The focus F will be located between AB, and it will be encountered after traversing (T) twice. Since B is located beyond this focus, we may attach A to B by an arc AEHNKHMEB, such that (Ai;iIMaiMl-:B)<(ABCABC\B). Since the focus of A is not encountered by describing the arc AB without Figure 10 traversing (T) , we shall have in addition (AE + EB)>(AB), 285 v-^ from which we have the following by subtraction (F,IL\KHME)<(ABC.\BCA) °^ (EII.ME)-+-(H.N'KH)<^.{ABCA). We must therefore have either (EII.MK)<(ACGA) °^ (IIMvIIXCABCA), In every case, there is a closed curve which differs little from (T) and corresponds to a smaller action. Therefore, in order that a closed curve may correspon d to an action which is less than any infinitlv adiacent closed cu rve, it is necessary /286 that this closed curve correspond to an unstable, peri odic solution of the first category . 356. Is this condition sufficient? In order to determine this, let us study the asymptotic solutions corresponding to a similar un- stable, periodic solution. Let a- = ?o(' , y = 'WiO be the equations of the periodic solution, and let X = 00(0 + Ae^'9i{/)H- A'e"''fj(0-t-- • •, ^.-=^„(04-Ae^'|,(0-+-A«c'-'.^,(0 + --- be the equations of the asymptotic solutions. The functions {A,B,). We therefore have (CB3)>(A,B0, and, consequently, action of (T') > action of (T) . q.e.d. 358. We must now determine whether the same result is still obtained for relative -motion. The irreversibility of the equations constitutes a great difference 289 from the preceding case. The action for an arbitrary arc AB is no longer the same as for the same arc traversed in a different direction. If an arbitrary curve satisfies differential equations, this will not hold true for the same curve traversed in a different direction. Finally, the orthogonal trajectories of the curves CD will no longer have the basic property which I disciissed in the preceding section. However, there are other curves which I shall define, and which have this property. This is sufficient for the result of the preceding section to remain valid. In No. 340, we obtained the following for the expression of the Z290 action J'= f[ds v/H7+A + lo'(? c/t, - T^ d';)]. For purposes of simplification, I shall set V/Hq + h = F. I shall no longer designate the coordinates by C and n, but rather by x and y, in order to approximate the notation employed in the preceding sections. And I shall no longer designate the angular velocity by u' , but rather by 0), removing the accent which has become useless. I shall then have J'=: f[F (,-i-e, -Hs'-tj-t- I would like to point out that ^y^ is homogeneous of degree k + 2, if we assume that Cp and u^ are of degree p + 2, t\ and v are of the 297 KJ degree p, X and v^ is of degree p. It is therefore a whole polynomial with respect to ^p. Up, T^P, <>. '^P< PP (/>>0)i and with respect to /;(„«"<, /" ofi" These last two quantities may be assumed to be on the order of 1. Finally, the coefficients of this polynomial are periodic functions of no whose period is 2v . In addition, we shall obtain where Hq and Bq are the values of ^ and ^ for X = y = 0. (We may assume that we have -^ = -^ = for A = y = 0.) In addition, 0^ depends only on Ip, -Tip, Up, Vp, Ip, Up ip^k — i). Our differential equations may then be written /299 'iT^'J^,' dt'^ di,' dt ch, dt du. For k = 0, they may be reduced to d\i, _ rff'j _ . lAo _ . dva _ . llT ~ ~dt ~ ° ' dt "'' ~dt "^ "*' They demonstrate the fact that Co and ^^O are constants, and that 7)0 — t-i vi= int + EI, where oT is a constant which must be determined. We may advantageously add other equations having a similar form to equations (4) and (5), which are only transformations of them. Let us develop xi and yi in powers of e, and let ^i-ro + ^ri +^'li +■■•- (4') The expansions (4') may be directly concluded from the two last ex- pansions (4) . We then find that ^-^^ is a whole polynomial with respect to the 298 quantities ?p. ^p, \'p, 1^. '^p, \^p (writing Hq separately) , (6) and that this polynomial is homogeneous of degree k + 2, if we assume that ?p is of degree p + ■»., ?p, r^'p is of degree p -hi, >■,/). ^p> \^p is of degree p. We then have the following equations 'l\\. ^ "+?'''^?='?, dt ' dt ' and we shall derive the following ^ ^ii^p-h qn) ^ _ ^ . r gf(p*^-?^<+JCTl_|- Y'e-'"', ' j^i(.p-\- qn-^ n) where y and y' are integration constants. Therefore, if X and Y are whole and homogeneous polynomials with respect to the same will hold true for 5 and n, unless it is assumed that the con- stants Y and y' are zero. If it is not assumed that these constants are zero, K and n will still be whole polynomials, but not homogeneous. Let us apply these principles to the quantities which we have just computed. Due to the fact that 302 dSt rfSi rfd| dOi rf;o ^1o rf;i rflj are polynomials which, according to the convention which we have em- ployed regarding degrees, are of the following degrees, respectively 1, 3, 1, 2, the same will hold true for III ?l. ')l> ?!• When we substitute the values of these quantities which are, re- spectively, of degrees 1, 3, 2, 2, instead of these quantities in'e2, it may be seen that 02 becomes a polynomial of the fourth degree, and /304 that -^ rfe, de, de, de, d^o' dTTio' d^',' rfT)J will be polynomials of the following degrees, respectively 2, 4, 3, 3. We may therefore formulate a generalization of this result. Equations (5) and (5') enable us to compute the unknowns 5, , t\,, C\c> n|^ from place to place. This would only be prevented if the mean value of the second term of one of the equations (5) were different from zero. Let us assume that this does not occur. It may be stated that b-< T)i, ^i, r/i will be polynomials of the following degrees X + 2, X-, k+l, k + l with respect to where the coefficients of these polynomials are themselves periodic func- tions of t of period 2Tr. Let us assume that this is valid for every value o,f the index which is less than k. We know that 0j^ is a whole polynomial of degree k + 2 with respect to ??• ';?> i'j, ^I'q (?<*) /■^2_) 303 \^ assuming that these quantities are of degree q + 2, q, q + 1, q + 1» respectively. If we substitute polynomials whose degree, wxth respect to the quantities (10), is precisely q + 2, q, q + 1, q + 1, in place of these quantities (11) , it is apparent that the result of the substi- tution will be a polynomial of degree k + 2 with respect the the quan- tities (10) . Therefore, 0], is a polynomial of degree k + 2 with respect to the quantities (10) , and for the same reason den dOk dQk dSj will be polynomials of the following degrees k, k-hi, k-\-l, k-hl with respect to the same quantities. The same holds true for the second terms of the first, second, fifth, and sixth equations (7). Consequently, by repeating the previous line of reasoning, we should readily see that the same holds true for /305 T,*, U■^ ^'kl I'll- q.e.d. The integration of equations (7) has introduced four new integration constants. They provide us with information concerning 5i, ni, ?i , ni, up to the following terms containing the four arbitrary constants Yi, Si. Yi' S'l- We shall retain only one of these constants and we shall set Y, = 8i = o, S', = — y',. Under this assumption, let us try to determine ?I, 1,, J',, T|',, by means of equations (5) and (5') and by setting k = 2. It is necessary that the second term of the first equation (5) has a mean value of zero. This mean value equals [S]' 304 and we always employ the brackets to represent the mean value of a function. We must therefore have m- (9') Let us assume that 02 is developed in Fourier series in the following form Since 62 is a polynomial of the fourth degree, q could not exceed 4 in absolute value. Consequently, if the denominator of n is larger than /306 4, [©2] will be identically zero, and the condition (9') will be ful- filled by itself. The constant u will remain undetermined. If the denominator of n equals 2 or 4, the condition (9') will de- termine TU. If the denominator of n equals 3, the constant "S has already been determined by condition (9), and condition (9') will enable us to deter- mine the constant y'l • Let us calculate the terms depending on this constant y\ in 02. We obviously will obtain Yi rfe, i.e. The mean value of this will be The condition (9') may therefore be written { if it is noted that _lL ^'fQ.I ^ H - „ where H depends on oT, but not on ■/, . If the denominator of n does not equal 3, [Gj] is zero and condition (9') is independent of y\. Therefore, if this denominator equals 2 or 4, 305 equation (9') will depend on w and not on fi and will determine o). If the denominator equals 3, condition C9') depends on y'l and will determine y' (it will provide us with y\ = 0) . In any case, having thus determined £.2, let us try to calculate n2 by means of the second equation (.5). We shall employ X2 in such a way that the second term has a mean value of zero. We should point out that X2 will not be zero in general, and will not be zero in general, because, due to the fact that 02 is a poly- nomial of degree 4, it will include a term containing Cg which is inde- pendent of the Cy.'s and n'^'s. The coefficient of this term will be a periodic function of t of period 2tt, and the mean value will not be zero in general. Let us proceed to equations (5') or, which is the same thing, to the last two equations (5). The second terms of these last two equations must have a mean value of zero. /3Q7 We must have [£]=--■'•• which determines v^' However, de, , f/B, M, Mj -3 — — ^lo "7JT -+- ?o ~J~r dua d^fi dr^^ is a polynomial of the fourth order. F2 therefore includes terms containing x?v?. and consequently U2 —^ includes a term containing ^^ dUQ III = (/^e''"*=')HV'«^e-'""^ra))'. The coefficient of this term is a periodic function of t, whose mean value is not zero in general. Therefore, in general L_i and, consequently. y2 are not zero. This is the same line of reasoning as is employed for '2 A2. We must then have f de,! [d^i~°^ (12) 306 However, it may be stated that this condition is fulfilled by itself. We have the energy integral F = const., from which we may deduce the series of equations 'J'o = const., , = consl., *j = const., Let us consider the third of these equations *i = 6j— ;»— inui + ^iIlo?o + 2B(i[jii«(i = const. This equation may replace the fourth equations (5) and, when X2, ^2. ^2 . /308 TI2 and v„ have been determined by means of the first three equations (5), it will determine U2 without any integration . We may therefore be assured that the determination of U2 is possible, and, consequently, that the con- dition (12) is fulfilled. We will have thus determined C2. ^2. C2 . n'2 up to the following terms depending on the four arbitrary constants. We shall retain only one of these constants, and we shall set 363. The calculation will be continued in the same way. The ability of equations (5) to be integrated requires the following conditions The last two conditions will determine X^ and yj^. The second will be a consequence of the first, according to what we have learned with respect to condition (12). We must then study the first. dGk Expression -^ is a polynomial of order k + 2. If it is developed in Fourier series the integer number q cannot exceed k + 2 in absolute value. If k + 2 is smaller than the denominator of n, we could not have p -i- gn =zo and the mean value of our expression will be zero. The condition L5^J=** (13) 307 will therefore be fulfilled by itself. We have introduced the following arbitrary constants: n, Y„ Y„ ... (1^) ^ J /309 and Oy^ may depend on -^ '^' Yi. Yi> ■■■• Y*-i- Let us determine the form of this dependence. Let us assume that we are considering the expansion and that in this expansion we replace the 5's, the n's, the C -s and the n' 's by their values. The different terms of the expansion will then de- pend on the constants (14) . In this expansion (15) , let us cancel all the constants y' , retaining only aJ, We will thus obtain a new expansion *;+£*', + £'*', -1- (16) In the expansion (16), let us now replace the constant IB by the expan- sion ra -t- EtTTi + e» Oj -V- . . . , where wi , W2 are new constants. Each term in the expansion (16) may be developed in its turn in powers of e. When this expansion is ordered anew in powers of e , we obtain a new expansion This expansion must be identical to the expansion (15) , under the condition that the constants u are replaced by the suitably chosen functions of the constants yl. It may be readily seen that $j^ depends only on O, TO,, ..-, ^k-i and that \ depends only on ra. Yi' •■■' If'*-'' We may conclude from this that To depends only on Yi. Yi' •••' f* and y'^ on ^,_ ^^^ ..., n,*. 308 It may be readily seen that /310 t>\ = SAD*;,, Ts^ raj- . . . Bjjt, where A is a numerical coefficient and where D$' is a derivative of $' _ mm with respect to w. The order of this derivative equals "i + if-h. ..-hat and we then have k = m + tit-h2Xt-h...+ /c-xt. Since m is at least equal to 1, and since "fo does not depend on ¥,' it may be seen that aj^ is zero, which we already knew. Let us consider an arbitrary term where Oj^,, a^^^i, ..., ct, ,, are zero, but where a^ is not zero. We must have rngfc-h. If the denominator of n is larger than k - h + 2, the mean value of D$ will be zero. This means that those terms of 4>|^ which depend on IS^ have a mean value of zero. An important result may be concluded from this concerning the mean value of $1^, and consequently the mean value of 0. . If the denominator of n equals k + 2, [©k ] will depend only on w. If the denominator of n equals k+1, [0j^] will depend on IS and oTi . If the denominator of n equals k, [0j^] will depend on To, ui and 702. _If the denominator of n equals k - 1, [6^^] will depend on IS, TSi , -©2 and 003. The statements which I have Just made concerning [Qi,] also apply to j-dG^I ^ UnoJ ■ Therefore, if the denominator of n equals k + 2, relationship (13), which will only include w, will determine oT. If the denominator equals k+1, relationship (13) will contain W and 732 . However, "io will have been previously determined by the relation- ship 309 ^-/ Relationship (13) will therefore deterniine ui and, consequently, Vi- If the denominator equals k, relationship (13) will contain w, a^i and 11^2. However, Hi and UJ, will have been previously determined by rela- /311 tionships having the same form as (13). Therefore, (13) wxll determine 1:^2 and consequently Vs- This process will then be continued. Discussion 364. The solution which we have obtained still includes the following arbitrary constants With respect to the parameters A and p, we have obtained them from their expansions in increasing powers of e, and we have successively cal- culated the coefficients of these expansions. These coefficients X^ and y,, depend on the two constants Co and uq ; these coefficients were calculated by means of the following equations where O, , ^ and uq 4^ are whole polynomials in k' dCo 6i-+- A, — 6, — A, > denominator of /2>2, which indicates that R is divisible by uq , since uq is included with the exponent - (bi + hi + ba + ha). There would only be an exception to this if we had b,-h A,= b,-hh^ but we would then have either in such a way that R would be divisible by uq , or b,— hi = bi = ht = o, from which we have A, -+■ A, = o. a However, the corresponding terms would not then appear in un — ^ . In the same way R will always be divisible by E,q , unless hs = 0, in which case the term would not be included in E,q \—^\ . /312 311 Therefore, to sum up we have \den [fife*-] \du,\' L'/uJ and, consequently, \y, and m^ are whole polynomials of Cq and \^. There- fore X and y are series which may be developed in powers of Z313 but these three constants do not enter arbitrarily. Let us recall the method which we employed to introduce the auxil- iary constant e, which only served to simplify the discussion. For this purpose, let us again consider the notation given in No. 274, and on page 95 . We have set X, = EX'j, >'i=Ey,, X, = E'xi, >'j=7i. Therefore, our equations do not cease to be satisfied when we change into e*-i, x\k, y\k, x',A» and when the parameters X and u retain their initial values. We then remove the accents which have become useless, and we develop x' , y', x', y', which we shall hereafter designate by the letters x^', y ^^ y^2* ^1^ ^" powers of e. We thus obtained the expansions ( 'ii + er/j + Eir, ',+.... e We shall not cease to satisfy the equations if we change e into j^, and if we multiply the four expansions (19) , respectively by *', I, *, k, or, which is the same thing, if we change 5pi ^P' ^p' ^'P ^^^° l,k^-p, npk-p, lp*•-^ -,;.*'-''• 312 By means of this change, we must again obtain expansions which are identical to the expansions (19), but with different values of the con- /314 stants Co ^^d un. However, it may be seen that Co and uq are changed into k Co and k^UQ by means of this change. Therefore !Pi ''>pi Ipt 1^ change into J/-*'-", ^pk-P, I'pk'-P, y^'^k'-P when Co and uq change into k^Co and k^ug . In other words , if the four expansions (19) are multiplied respec- tively by E^, 1, e, e, the four products thus obtained may be developed in powers of The same must be true of X and y, which did not have to change when e, Co, uq were changed into ^, k^Co , ^^^o- Therefore, let us assume that A and y are expressed as functions of e Co and gy/uq^ It is apparent that we shall thus have relationships from which we may derive e^Co and ey^ inversely as functions of X and y. 365, Let k + 2 be the denominator of n. The constant US will then be determined by the equation There is only an exception to this in the case of k + 2 = 2, where "cj is determined by dOk 10 The expression -^ — is a whole polynomial of degree k + 2 with respect to Therefore, each of these terms contains factors having the form Only terms which are independent of t will remain in the mean value LdnoJ 313 and we have seen that q must be divisible by the denominator of n, i.e., /315 by k + 2. Therefore, our expression has the following form I shall now show that the coefficient b is zero. For this purpose, I shall employ the following method. Let us cal- culate lo, ?ll •••' ?*■- 1' ■^0. ^i ^*-i' So I Tl; •••• ?*-l' To I III •••' '^*-l' by the procedure presented above. However, when computing Cj^, I shall retain an arbitrary value for C, instead of assigning a value which can- cels [^^1 to u5. Then the following equation LdnoJ dt d7\a Will allow me to compute ^k- However, instead of being a periodic func- tion of t, ?k will be a periodic function of t in addition to a non-periodic term We have another method of calculating 5o, 5ii ■ • ■'' b) lOl 1l> .•■■■ li— 1' • • • 1 • • ■) ' and, consequently, this term t[.^J. This method consists of again per- forming the calculation presented in No. 274. We shall determine Sq, Si, ..., by means of equations (2) on page 100. No difficulty will be entailed in calculating Sq, Si, ..., \.i, t)ut we shall encounter some difficulty when calculating S^ by the equation -r^ -4-2B -i— = *-t-G/. dyi dv In effect, the second term represents a group of terms having the ^^^^ following form 314 Ae'"'.r.+'"t'', where m^ and m2 are integer numbers. Nothing impedes us from performing integration, provided that we do not have I'm, + 2mjB =0. Since 2B equals in , where n is a commensurable number whose denominator equals k + 2, the second term of our equation will include terms satisfying this condition. As a result, Sj^ will not be a periodic function of y, and V, but may equal where Tj^ and Uj^ are periodic. Having thus determined the fpctlon S and having obtained the approxi- mation to terms of the order £^+ , we may employ the procedure given in No. 275 and may thus determine Xj , y^ , X2 , y^ . These two computational methods must lead to the same result. There- fore, let us set S = So-4-eSi-H... + E*S*. Let us compile the equations (see page 102) rfs dz ^ dz dz dyi dv d(Xo d^o dC dC and let us derive X2 from them as a function of t. The value of x, which is thus obtained must equal ?o-t-E?iH-...-He*^t up to terms of the order e^"^l . We are interested in calculating g^, particularly that of the sec- ular term m- This secular term can only come from the secular term of S, , which eouals We thus have the following, up to terms of the order £^+l (equating /317 the secular terms in the equation x„ = — ) 2 dy2 315 \^ In the first approximation — i.e., up to terms of the order e — we have (see page 102) n, = i; nt=in\ mt -^ w^ = i> = Va= i{nt -^w). We shall therefore commit an error of the order e^+'^ if, in the second term of (20) , we replace fo, Po, yi, p by 5o> Mo, ', ((nt-hTa). We shall therefore obtain h — J by making the same substxtution m ^^^ . However, % only includes terms containing dy2 inixyt -+■ rritv, Where im,^ ^m,B^-o. We therefore have —. = — 2 b — 7— • dy, dv dUk However, \i\^ is a periodic function of yz and iv. Therefore, ^^ r^Qki cannot contain a term which is independent of v. Therefore [^J does not contain a term which is independent of w. q . e . d . In order to clarify the preceding calculation, I would like to make one more remark. The mean motions nj and n^ are given by dC dC tl = — J— > Tli^ — -Jar' aio d^o In general, they depend on e, and they are only reduced to 1 and in for E = 0, However, we are here employing two parameters X and y which may be replaced by the arbitrary functions of e, .or, if it is preferred, we may ^318 316 employ an infinite number of constants Ai , A2, ..., yi, y2» ••• • We may then employ these constants in such a way that n^ and n2 remain equal to 1 and to in , no matter what e may be. 366. In order to determine "w, we therefore have an equation of the following form where a and c are conjugate and imaginary. In general, a and c are not zero, otherwise H could only be determined to the following approximation. ^The equation will provide us with the following series of real values for oj ■K III Stt It is apparent that we do not have two values which are actually different when we change oJ into u + Ztt, but we have more than this. It may be stated that the two values 211 k ~t- 1 do not correspond to two periodic solutions which are actually different. Since t is not explicitly included in our equations, by changing t into t + h we may transform an arbitrary periodic solution into another solution which is not essentially different. Therefore, let us change t into t + 2hTT, where h is an integer number. Then no changes into hq + 2hTr and vq = i(nt + w) into Since all of our functions are periodic, of the period Ztt, in Hq and iv, we shall not change our solution in any way by subtracting two multiples of 2tt from tiq and — , respectively, for example 2hiT and 2h'TT . Then no will again become no and vo will change into i(nt -}- 2/iAit + xa — iA't). In other words, we will have changed o) into /319 CI -+- iT.{nh — h'). However, we may always choose the integer numbers h and h' in such a way that 317 ^^ nh — A' = , , : • A: -+- 2 We therefore do not obtain a solution which ia actually new by - - 2Tr changing co into u + j^ ^ 2 ' q.e.d. We therefore have only two solutions which are actually different, corresponding to the two following values of o) TSa ■ /t^-2 We must now determine the constants z^E.^ and e^UQ. For this purpose, we shall employ equations which relate these two constants to X and p. In the questions which are customarily discussed, there is only one arbitrary parameter, and we have introduced two in order to facilitate the discussion. It is therefore convenient to assume that X and v are related by one rela- tionship — for example, X = y. The expansion of X and that of y in powers of £^^0 and ey^ begins in general with terms containing e^^o and e^uQ (if we disregard the case m which the denominator of n equals 3) . If we therefore assume that y = X, we shall derive z^^q and e\/^ from this which may be developed in powers of \/T. Either the coefficients of the expansion in powers of l/T will be real, or, on the contrary, the coefficients of the expansion in powers of V^ will be the ones which are real. In the first case, the problem will have two real solutions for X > and will not have any for X < 0. In the second case, the opposite will hold true. In order to determine which of these two cases is valid, let us examine the equation which relates y to uq, restricting ourselves to terms con- /320 taining e^. We shall have I may first observe that[^] and[^] are not only independent of t but also of oT. There is only one exception for k+2=2, 3 or 4. 318 \^ This is due to the fact that, for k + 2 > 4, terms having the following form ei'pl ^rjnl ^./a) which may be included in the second term in one of the equations (21) can only be independent of t if since |q| cannot exceed 4 and since qn must be an integer number. Thus, the second terms of equations (21) are linear and homogeneous functions of Cq and uq . The coefficients of these linear functions are absolute constants which are independent of oT. However, uo must be positive; otherwise Vuo"would be Imaginary. The equations (21) added to inequality uq > will determine the sign of A. .- ^rT^f.°^^^ P°^''^ °''^ ^"^^^ ^^^^ ^^8" ^°^^ ^°t depend on u, since equa- tions (21^ do not depend on it. We have seen that the equation which de- termines w has two solutions which are actually different ra = 73(1, w — T3o -I- A- -T- a In conformance with the preceding statements, a periodic solution which will be real if the sign of A is suitably chosen corresponds to each of them. The choice of this sign does not depend on 15. and these two solutions will both be real for A > and will both be imaginary for A < or the opposite will hold true. ' It first appears tha£ two periodic solutions correspond to each solu- tion of the equation for O), since two systems of values for the unknowns /321 e Co and z\/^ are obtained from the relationships between A, n, z^^^ and z^. This is not the case, however. Without restricting the conditions of generality, we may assume that ^/^^^ is positive, because we do not change our formulas in any way by changing y/^ into -y^. and ^ into (J + ^. _ Out of our two systems of values, there is only one for which V^ is positive. ' " Therefore, we have: Two real, periodic solutions of the second type for A > (or for A < 0) . No solution of the second type for A < (or for A > 0) . 319 .et us again employ the nctatio. given In Chapter XXVIII and. In particular, that given in No. 331. U, may be reduced to p^ and corresponds to the tern, containing x^yi which appears in 0q. U„ „ay be reduced to a constant factor multiplied hy P^ corresponding to terms coming fromj^J and |_^^J. The first term of W which may not be reduced to a power U, has the following form pi'-2[Acos(A- -2) 0; -ro solution for X < 0; one real solution of the second type for X > 0; one solution for 320 X < 0; Zero real solution of the second type for A > Q; two solutions for A < 0. The function Uq + zU^ given on page 247 becomes p'(A cos4f -r- B) — ^p». Let us now assume that the denominator of n equals 3. The expansion of \i in powers of e then begins with a term containing ^yuQ, so that if we set y = X, we shall obtain e^^q and ev/uo~ in series which may be developed in powers of X, and no longer ofyT". The sign of yTi^ will depend on m, and if it is positive f or ~ = 53- it 7T will be negative for u = oig + -r Therefore, if it is always convenient for us to assume thatyuQ is mainly positive, we shall readily find that we have: A real solution of the second type for X > and a real solution of the second type for X < 0. The function Uq + zUi given on page 247 becomes Ap'cos3^ — sp*. Finally, if the denominator of n equals 2, [©z^shn" . hr~^ > include terms containing e-'*-'-^, e~^^. /323 The equation for oT takes the form Acos(4nj-t- B)-f- A'cos(2ra -f-B') =0 and it has eight solutions •jr 3^ 2 a 3i: rai, CI, H — , cj, -t-TT, nr, H Of the two terms "aJQ and wj , at least one is real. The following hypotheses are possible: (4, 0), (3, 1), (2, 2), (1, 3), (0, 4), (2, 0), (1, 1), (0, 2). The first number between the parenthesis represents the number of periodic solutions for X > 0, and the second is the same number for X < 0. 321 The function given on page 247 becomes Ap»cos i9 H-Bpicos2'f -+- Cp'sin2^ and ip^ may be developed in increasing powers of y. /324 Let us consider one of these solutions (2) . Let T be the period, and let a be one of the characteristic exponents. There will be two of them, which are different from zero, which are equal and have opposite signs, where we may assume two degrees of freedom. We saw in Chapter IV that a depends on y, and may be developed in powers of ^T. When y varies continuously, the same will hold true for a. For y = yg, let us assume that aT is commensurable with 2iTT and equal to 2niir. We may conclude from this that, for y which is close to yQ, there are solutions of the second type, which are derived from (2) and whose period is (k + 2)T, where k + 2 designates the denominator of n. If we put aside the cases in which k + 2 equals 2, 3, or 4, we have seen that two of these solutions exist when X (here y - yo) has a certain sign, and that they do not exist when X (here y - yg) has the opposite sign. I have stated that the cases in which k + 2 = 2, 3, 4 have been dis- regarded, and I may do this without causing any inconvenience. The 322 following aT = n 2in may be developed in powers of \/^, and vanishes with yyT For small values of y, n is therefore very small, and its denominator is definitely larger than 4. We therefore have two hypotheses: Either the solutions of the second type occur only for y > yo, or they occur for y < yg. Which of these two hypotheses is valid? Everything depends on the sign of a certain term Q, which depends itself on the coefficients of ug and ^o i^ In order to determine this sign, we shall not need to formulate this term, and the following considerations will suffice. 369. Let us first take a simple case, which will be that presented /325 in No. 199. Let us set F = Xi -(- t| -H ji COS/j with the canonical equations which yields rfr, _ rfF dy,- _ dF dt ~ dy'i dl dxi' §=«. t = -' 'S-"'"'" *-— (1) The function S of Jacob! may be written S = xlyt-^ J y/C^ fi cos_^, dy, 2 and C (2) with two constants xS and C. We may derive the following from this J 2/0 — |Jl< where A and y^ are two new integration constants. 323 \J It may be seen that the elliptical integral is introduced J a v/C — li cosjKi This integral has a real period, which is the integral taken between and 2." il 1C| > lul, and two times the integral taken between if Ic! < |y|. Let us call u this real period. A periodic solution corresponds to each value of o) which is commensur- able with 2TT. However, we must distinguish between two cases. If Id > |y|, yi and ya increase by a multiple of Ztt during one perio" tL corUsponding periodic solutions are solutions of the fxrst type. If Icl < |y|, yi increases by a multiple of 27t during one period. ^326 and yf retims'to it2 original value. The corresponding solutxons are solutions of the second type. This discussion must be supplemented by two unusual Periodic solu- tions which must be regarded as solutions of the fxrst type. Let us set y > 0, and these solutions will then be I have stated that it must be assumed that these l^"er solutions are of the first type, and that the solutions corresponding to |C| |p1 must be regarded as solutions of the second type. Let us assign to C a value which is a little higher than -y . and let us set C = (e — i)!i, Where e is very small, yi will not be able to deviate very greatly from .. We shall approximately have C — jx cosjKi = H 1 5 '2 J ' and the period o) will be equal to ■It 324 from which we may draw the following conclusions. Let a be an arbitrary number which is commensurable with 2-w. There is a series of periodic solutions such that |c| < |y| and that u = a. If |/^is very close to TT , C will be very close to -]i, and for 2ct these periodic solutions will coincide with the second solution (4) which is of the first type. We may now recognize the characteristic property of solutions of the second type. It may be seen that the second solution (4) — i.e., that of the two solutions (4) which is stable — gives rise to solutions of the second type, as was explained in Chapter XXVIII. If the other solutions of the first type — those which are such that |c| > |y| — do not produce solutions of the second type, this is due to /327 the very particular form of the equations (1). (For these solutions, the characteristic exponents are always zero.) Let us first consider solutions of the first type, such that [c| > |y[. Let us set C = Cq + e. The period w, i.e., the integral (3) taken between and Ztt, may be developed in powers of e and of y, and the known terms may be reduced to Wo' Let us assign an arbitrary commensurable value to \/^. We shall have a periodic solution every time that we have 0) = . /Co The equation is satisfied for e = y = 0, and we may derive e and, consequently, C from this equation, in series which develop in powers of y. The equations (2) will then give us xj and yj developed in powers of y. These are the expansions of Chapter III. Let us pass to the second type, such that |c| < |y|. Let us set C = ey . We shall have ^^J '. dy\ 2 . Therefore, there is only a periodic solution corresponding to a given value of 0), which is commensurable with Ztt, if The coefficients of the Fourier expansion are therefore functions of y, which are real for (1) yi and imaginary for ,- n VP< — 7=- to y2 It is apparent that the same line of reasoning would lead to the same result if, instead of 326 \^ F = xt -h x] -h ij. cos_;'i, we had set F=.Fo-+-h[F,], where Fq depends only on xi and X2, and iFiJ depends only on xj , X2 and /329 yi. The solutions of the second type would still have been real for y > Pq. 370. In the general case, the quantity Q, which was in question at the end of No. 368 and whose sign we shall try to determine, obviously de- pends on y. If y is sufficiently small, the first term of the expansion will provide its sign. Let us determine the function S by the Bohlin method, and let us set S = So+ /jjiSi -+- fjLSi + . ... If y is small enough, it will obviously be the first two terms which will be the most important. If we set F = Fo+,uF,-+-(jlJF, + ..., we have seen in Chapter XIX that Sq and Sj depend neither on F2 or F^ - [Fil, but only on Fq and [fJ, where the mean value of Ft is designated by [Fil. Let us again take the quantity Q from No. 368. The first term of its expansion will only depend on Sq and Si, and consequently on Fq and [Fj]. The same would hold true if we had set F = F„+,4F,], which is, consequently, the same as in the preceding section. In the preceding section we found that solutions of the second type exist only for This conclusion still holds in the general case, provided that yg is suffi- ciently small. What is the value of yg for which this conclusion would no longer hold? Let us again consider the notation given in No. 361, which is that of No. 275. The exponent a which appears there may be developed in powers of 327 \^ the product AA' . It may be reduced to the characteristic exponent for AA' = 0. Since we assume that the solution of the first type is stable ^^^^330 a is imaginary. A and A' are imaginary and conjugate, and the product AA is positive. For small values of y, a decreases when AA' increases. If the re- verse were true, solutions of the second type would exist only for The desired value of Uo ±^ therefore that for which a ^^f ^^^°^^- _ crease when AA' increases. It is therefore that which cancels the deriva tive of a with respect to AA' . 328 CHAPTER XXXI PROPERTIES OF SOLUTIONS OF THE SECOND TYPE Solutions of the Second Type and the Principle of Least Action /331 371, I cannot pass over the relationships between the theory of solu- tions of the second type and the principle of least action in silence. I wrote Chapter XXIX just for these relationships. However, in order to understand them some preliminary remarks are still necessary. Let us assume two degrees of freedom. Let x^ and X2 be the two variables of the first series, which may be regarded as the coordinates of a point in a plane. The plane curves which satisfy our differential equations will comprise what I have designated as trajectories . Let M be an arbitrary point in the plane. Let us consider the group of trajectories emanating from the point M, and let E be their envelope. Let F be the niS- kinetic focus of M on the trajectory (T) . This trajec- tory will touch the envelope E at the point F, according to the definition of kinetic focuses. I would like to recall that the nt£ focus of M, or its focus of the order n, is the n^ point of intersection of T with the in- finitely adjacent trajectory passing through M. However, the conditions of this contact may vary. It may happen that F is not a singular point of the curve E, and that the contact is of the first order. This is the most general case. Let a-i = 0(2-1) be the equations of the trajectory (T) and of a trajectory (T' ) which is very close, emanating from the point M. Let zi and Z2 be the coordinates of the point M, and let ui and U2 /332 be the coordinates of F. Since (T) passes through M and F, and since (T' ) passes through M, we shall have Due to the fact that the trajectory (T') is very close to (T) , the function if) will be very small. I may call a the angle at which two tra- jectories Intersect the point M. It is this angle which will define the trajectory (T'), and the function ii will depend on the angle a. It will be very small if, as we have assumed, this angle ct is itself very small, and it will vanish with a. The value of i;;'(z2) (designating the derivative of i|j by i/;') will have 329 \^ sign if F is a focus of odd order. one characteristic of the case in which we -^/^^-^^f ,f,,^'^ '"^' that Hu2) is of the same order as a^. and always of the same sxgn. For example, let us assume that ^Cuz) is positive. If the sign of a is such that ^' (uz) is positive, the trajectory (T') w 1 inteSect (T) at a point F' which is close to the PO- F and -ir^ -^LferE^rfo^rF^fShiL^anrchirF ^^^i^: ^^. ^ra'w nU::; li^e'of reasoni;g. the action is larger (at least .n abso- lute motion) when we pass from M to F' proceeding along (T ) than xt when we pass from M to F' proceeding along (T) . If the sign of a is such that *' (u^) is negative. (T) intersects (T) . -nrv' which is farther away from M than F. In this case, (T ) ? u^h^faftef F-! i^dlx) touche ^^^^f •'' ' lof/ y^ '^'^ ^"^ " ^ F', the action is greater along (T) than it is along (T ). The results would be just the opposite if ^(u,) -f ^^^^ff J^^;^ l^' sect (T) close to F and just short of F. In this case, we may say that F is an ordinary focus . It cannot happen that F is an ordinary point of E, and that the con- tact is of a higher order than the first. Let us develop ij;(x^) in powers of a. and let us set /333 The condition under which there would be a contact of higher order But we already have i;>,(«,) = o ^e coefficient of the second derivative is reduced to unity. 330 If the integral tf^i (X2) vanishes, as well as its first derivative, for X2 = U2, it would be identically zero, which is absurd. Therefore, there is never a contact of higher order. However, it may happen that F is a cusp of the curve E. Either the cusp point is on the side of M, so that a moving point proceeding from M to F will encounter M with the cusp point directed at M, or the cusp point is turned the opposite way so that the moving point encounters M with the cusp point turned away from M. In the first case, I shall state that F is a pointed focus , and in the second case I shall state that F is a taloned focus . In one and the other case, 4'(u2) is on the order of a^. In this case, the pointed focus has the sign of a, if P is a focus of odd order, and it has the opposite sign of a if F is a focus of even order. The opposite is true iu the case of a taloned focus. In the case of a pointed focus, all the trajectories (T') intersect (T) at a point F' which is close to F and beyond F. Proceeding from M to F' , the action is greater along (T) than it is along (T'). In the case of a taloned focus, all the trajectories (T') intersect (T) at a point F' which is close to F and just short of F. Proceeding from M to F' , the action is greater along (T') than it is along (T) . Let F' be a point of (T) which is sufficiently close to F. In the case of a pointed focus, I may join M with F' by a trajectory (T'), if F' is beyond F. In the case of a taloned focus, I may join M with F' if F' is just short of F. It could finally be the case that F is a singular point of E which is /334 more complicated than an ordinary cusp. I would then state that it is a singular focus . I would only like to note that we cannot pass from a pointed focus to a taloned focus except through a singular focus, because at the time of passage iij(u2) must be of the order a"*. 372. Let us now consider an arbitrary periodic solution. It will correspond to a closed trajectory (T) . Let a be the characteristic ex- ponent and T be the period. In Chapter XXIX we saw how to determine successive kinetic focuses (No. 347). Let us assume that a equals — = — , where n is a commensurable number whose numerator is p. In this case, the application of the rule given in No. 347 shows that each point of (T) coincides with its 2p^ focus. 331 If, just as in No. 347, we take a unit of time such that the period T equals Ztt, we have a = in. If we designate the value of the function T at the point M by tq, and if xi, T2, ..., T2p are the values of this function T at the first, second up to the 2p^ focus of M, according to the rule given in No. 347, we shall have the following "0 = ) • ■ •) ij/j -0 - a n If p is the numerator of n, it can be seen that Xjp - tq is a multi- ple of 2tt, i.e., that M and its 2p^ focus coincide. The trajectory emanating from the point M which is infinitely close to (T) will therefore pass through the point M again after having gone around the closed trajectory (T) k + 2 times, if k + 2 is the denominator of n. The point M is therefore its 2p^ focus. However, we may wish to know what category of focuses it belongs to, from the point of view of the classification presented in the preceding section. Let us adopt a system of coordinates which are similar to the polar coordinates, so that the equation for the closed trajectory (T) is and so that w varies from to 2tt when one passes around this closed tra- jectory. The curves p = const, are then closed curves which form an en- /335 velope around each other in the same way as concentric circles. The curves cj = const, form a bundle of divergent curves which intersect all the curves p = const., in such a way that the curve u = a + 2it coincides with the curve a> = a. Then let coq be the value of w which corresponds to the point of de- parture M. The value of w which will correspond to this same point M, regarded as the 2p^ focus of the point of departure, will be Let 0|)-t- 2( A- -i- 2)7t. p — 1 -r ({/(w) be the equation of a trajectory (T') which is close to (T) and passes through M. The function i|j(u)) will correspond to the function ((j(x2) given in the preceding section. We shall have )|;(a)o) =0, and we must now discuss the sign of 332 We must therefore formulate the function !l;(a)), and for this purpose we need only apply the principles of Chapter VII, or the principles given m No. 274. For example, if we apply the latter principles, we shall ob- tain the following. The function i/j(a)) may be developed in powers of the two quantities The coefficients of the expansion are periodic functions of the period 27t; A and A' are two integration constants. With respect to a, it is a constant which may be developed in powers of the product AA' I -^ a„ -;- a,(AA') -,- cc,(A,Vp -i- The term qq equals the characteristic exponent of (T) , i.e. it equals in. ' If (T') differs very little from (T) , the two constants A and A' are very small. They are on the order of the angle which I called a in the /336 preceding section, and which must not be confused with the exponent which I have designated by the same symbol in the present section. If we take the approximation up to the third order inclusively with respect to A and A', ^(w) will be reduced to a polynomial of the third order with respect to these two constants, and I may then write ■{-(w) = Ae'wcr-t- A'e-!'w) = Acre'>>f».+a,AA,^.^'j'^_u(3.*j,AA'i_j_y(Agoi.o), A'e-".'-') or -)-st,(oAA'(Ae«.'Oj- A'e-='oWa') + /(Ae»o« A'e-»."). When 0) increases by (2k + 4)^, the coefficients of f , as well as o and a', do not change. The same holds true for e"0'^, since — = n has k + 2 for the denominator. Therefore, the same still holds true for Ae'."a, A'e-".'",', /(Ae=<.^) = (A- + 2)^^. AV(A e-."a - AV-^.-O- However, ^(wq) is zero. The term whose sign we must determine is therefore (2/.- + 2):ti, AA'(Ae»='"o!7j_ A'e-»«".!j;, ). I Shall employ Oq and c/q to designate the values of a and o' for o) = (.q • I should first point out that this term is of the third order which, i337 according to the preceding section, indicates to us that our focuses wxll m general be pointed focuses or taloned focuses. It may now be stated that this term always has the same sign, and that its coefficient cannot vanish. The two constants A and A' are related by the following relationship t;/( 4, we will only be finished with the calculation of the terms of higher order, which are not included in the preceding analysis. 374. For example, let us assume that all the focuses are pointed. Let M be an arbitrary point of (T) ; this point will be the 2p^ focus with respect to itself. Let M' be a point located a little beyond the point M in the direction in which the trajectory (T) and the trajectories close to (T') are traversed. I may draw a trajectory (T') emanating from point (M) , which will deviate very little from (T) , which will pass around (T) k + 2 times, which will finally end at the point M' , and which will have 2p + 1 points of intersection with (T) , counting the intersection points M and M' . Due to the fact that the focus is a pointed focus, the trajectories (T') which are close to (T) will all intersect (T) again beyond the focus. We may therefore draw the trajectory (T*) which satisfies the conditions I have just discussed, provided that the distance MM' is smaller than 6. It is apparent that the upper limit, which must not exceed the distance MM', depends upon the position of M on (T) . However, it never vanishes, since there is not a singular focus. It is therefore sufficient for me to set 6 equal to the smallest value which this upper limit can take on, and I shall assume that 5 is a constant. Therefore, if the distance MM' is smaller than 6, we may draw a tra- jectory (T') satisfying our conditions. We may even draw two of them, one intersecting (T) at M at a positive angle, and the other intersecting it at a negative angle. Under this assumption, let us assume that our differential canonical equations depend on the parameter X. For A = 0, the closed trajectory (T) has ag = in as the characteristic exponent. Let us assume that, for X > 0, the characteristic exponent divided by i is larger than n, and that for X < 0, on the other hand, it is smaller than n. For X ^ 0, the point M will no longer be its own 2p£ll focus. Its 2p^ focus will be located a little short of M for X > 0, and beyond M 335 K^ -' X < 0. -t . .e th- focus X.e distance » -n_^na.„aU. .epen.^^^ Slf ^is": e!" ? 13°; a ;„t\haf c 'llAe a co.tlnuou. function of H il i: rAl-^foJt :; aSaJ-f ?f?f -r^ r J.^-ac^oo^/fA. to thrrnncljlei given L No. 347. depending on the value of the charac- teristic exponent. The distance MF can never vanish. Let F- be a point located a little beyond F. We may connect M with F' bv a traiectory (T'), provided that the distance FF' Is less than a certlin J^^titH'. It L apparent that 6' is a continuous function of X, and that it may be reduced to 6 for X - 0. Let us set X > 0, In such a way that M is beyond F. We may have M play ihe "le of F' . and we may connect M to itself by a trajectory (T ), provided that the distance MF is smaller than 6 ' , or provided that e<8' For X = 0, e is zero, and 6 ' =6 > 0. Therefore, we may take X small enough so that'the inequality is satisfied. we may then connect the point M to itself through a trajectory (T') deviating a little from (+) , passing around (T) k + 2 txmes, and inter- secting (T) 2p + 1 times. Figure 12 In the figure, BA represents an arc of (T) on which M is located. MC is L arc ofcT') starting from M and DM is another arc of this same trajec'rj bordering upon M. The arrows indicate the direction xn which the trajectories are traversed. The noint M may also be connected to itself not through one trajec- tory.^ut' through So (T'). For one, as the figure i-^i-tes the angle ^340 ^"^Is positive! so that CM is above MA. For the other, the angle CMA 336 . would be negative. The trajectory (T') must not be regarded as a closed trajectory. It leaves the point M to return to the point M, but the direction of the tangent is not the same at the point of departure as it is at the point of arrival, so that the arcs MC and DM do not join each other. The trajectory (T'), thus proceeding from M to M with a hooked angle at M, will form what may be called a loop . If the same construction is followed for the points M of (T) , we shall obtain a series of loops . We shall obtain two of them, the first corresponding to the case in which the angle CMA is positive, and the second corresponding to the case in which this angle is negative. These two series are separated from each other, and the passage from one to another may only be made if the angle CMA is infinitely small. The trajectory (T'), which is infinitely close to (T) , would pass through the focus F, according to the definition of focuses. However, since it must end at the point M, the points M and F 5^ould coincide, and this cannot happen according to the principles presented in No. 347. Therefore, if all of the focuses are pointed, we have two series of loops for A > 0, and we have no more for A < 0. If all the focuses were taloned, the same line of reasoning could be repeated. We would find that there are two series of loops for X < 0, and that there are no more for X > 0. 375. Let us consider one of the series of loops defined in the pre- ceding section. The action calculated along one of these loops will vary with the position of the point M; it will have at least one maximum or one minimum. If the action is maximum or minimum, it may be stated that the two arcs MC and CD coincide, so that the trajectory (T') is closed and corres- ponds to a periodic solution of the second type. For example, let us assume that the trajectory (T') corresponds to the minimum of the action, and that the angle CMA is larger than the angle BMD, just as in the figure. Let us then take a point Mi to the left of M and infinitely close to M, and let us construct a loop (T?) which /341 differs by an infinitely small amount from the loop (T'), having its hooked point at Mi. Let MiCi and MiDi be two arcs of this loop. From M and from Mi I may draw two normals MP and MiQ on MiCi and MD. According to a well-known theorem, the action along (T') from the point M up to the point Q will equal the action along (Tj) from the point 337 \^ P to Ml. We shall therefore have action (T',)== action (T') -i- action (.M,P) — action (MQ) or action (T'j)=: action (T') -4- action (.\lM,)(cosCMA — cosBMQ), or finally action (T'j )< action (T'J, which is absurd, since (T') was assumed to correspond to the minimum of the action. If we set CM A < B.MD, we would arrive at the same absurd result placing Mi to the right of M. We must therefore assume that CM A = BMD, i.e., that the two arcs coincide. The same line of reasoning may be applied to the case of the maximum. Each series of loops therefore contains at least two closed trajec- tories. Each of these closed trajectories passes around (T) k + 2 times, and intersects (T) at 2p points. For p of these points, the angle similar to CMA is positive, and for the other p points, it is negative. Due to the fact that the curve (T') is closed, it must intersect (T) as many times m one direction as in the other direction. Therefore, it may be assumed that this closed trajectory consists of 2p types of loops . because we may regard any arbitrary one of our 2p points of intersection as the hooked point. For p of these types, the loop thus defined would belong to the first series, and for the other p types, it would belong to the second series. Among the loops of each series, there are therefore not two, but at least 2p of them, which may be reduced to closed trajectories. However, JJ^ one thus obtains not 4p , but only two different closed trajectories. The fact that there are not more of them is, in general, not the result of the preceding line of reasoning, but may be concluded from the principles presented in the preceding chapter. 338 The trajectory (T') thus defined will have yCk + l)p double points, if k is odd, and — (k + 2)p double points if k is even. This is valid for small values of A, and it remains valid no matter how large \ may be as long as (T') exists. The number of double points could only vary if two branches of the curve (T') were tangent to each, other. However, two tra- jectories cannot be tangent to each other without coinciding. For the same reason, no matter how large X may be, as long as the two trajectories (T) and (T') exist, they will intersect at 2p points. 376. The entire line of reasoning presented In the preceding section assumes that we are dealing with absolute motion . If this line of reasoning is extended to the case of relative motion, difficulties will be encountered which are not insurmountable, but which I shall not try to surmount at this point. To begin with, we must modify the construction employed in the preceding section. Instead of drawing MP and M^Q normal to MjCi and MD, we must pro- ceed as follows. In order to construct MP, for example, we should construct a circle which is infinitely small and which satisfies the following condi- tions. It intersects M^Ci at P and touches the line MP at this point. The line connecting M to the center must have a given direction, and the ratio of the line length to the radius must be given. The line MP thus constructed has the same properties as the normal in absolute motion. Unfortunately, in certain cases this construction entails certain difficulties. In addition, the action (MM^) is not always positive. If it became zero, this line of reasoning would still have a defect. The maximum or the minimum could be reached at the point M, so that the action (MMi) is zero, and this could occur without the necessity of the arcs MC and DM coinciding. /343 Our line of reasoning therefore only applies to the case of relative motion, if the action is positive along (T) . In any case, one of the conclusions is still valid. The closed trajec- tory (T') always exists, since — if the line of reasoning given in the pre- ceding section is lacking — the same does not hold true for the line of reasoning given in Chapters XXVIII and XXX. In addition, (T') intersects (T) at 2p points, and has ^(k + 1) or |-(.k + 2) double points. This is valid for small values of X, but it cannot be concluded any longer that this is valid no matter what A may be, because two trajectories may be tangent without coinciding, provided that they are traversed in the opposite direction. 339 stability and Instability 377. Let us assume that there are only two degrees of freedom, two of the characteristic exponents are zero, and the two others are equal and have opposite signs. The equation which has the following as roots is an equation of the second order whose coefficients are real (T represents the period and a represents one of the characteristic exponents) . Its roots are therefore real or imaginary and conjugate. If they are real and positive, the ot's are real, and the periodic solu- tion is unstable. If they are imaginary, the a's are imaginary and conjugate. Since the product equals +1, the a's are purely imaginary, and the periodic solution is stable. If they are real and negative, the a's are imaginary but complex , with the imaginary part equalling y". The periodic solution is still unstable. They cannot be real and have opposite signs, since the product equals +1. There are therefore two kinds of unstable solutions, corresponding to the following two hypotheses /344 The passage from stable solutions to unstable solutions of the first type occurs for the value The passage from stable solutions to unstable solutions of the second type occurs for the value "= T- 378. Let us first study the passage to unstable solutions of the first type. At the moment of passage, we have 340 Let us again consider the terms gj^ and i)^ defined in Chapter III, and let us consider the equation d?, d?, d?l d^i dfr dft d'W dh o-O/, d^, dh dh rf'l. d^. dh dh d^. ^ dh ^ dh d^, dh dh CD This equation has the following roots o, o, e«T__[^ e-iT_,. At the time of passage, the four roots become zero. Before studying the simple case in which we are dealing with equations of dynamics with two degrees of freedom, and in which we assume that the function F does not depend explicitly on time and that, consequently, the equations have the energy integral F = const., it is advantageous to consider for a moment a case which is even simpler. Let F be an arbitrary function of x, y and t, which is periodic of period T with respect to t. Let us consider the canonical equations 7345 dx dJ dF dt dF dx'' (2) These are the equations of dynamics with only one degree of freedom. How- ever, due to the fact that F depends on t, they do not have the energy equa- tion F = const. Let us assume that these equations (2) have a periodic solution of period T. The characteristic exponents will be provided by the following equation which is similar to (1) <^1 « dh~ dh dh ~ which has the following roots hi (3) e'T_, These roots all become zero at the moment of passage. Let us assume that F depends on a certain parameter y and that, for X = 0, the two roots of the equation (3) are zero. The functions ipi and ^pz will depend not only on Bj and 62. but also on y. We shall assume that F 341 may be developed in powers of y, and that consequently ^ and ^2 may be developed in powers of 61, B2 and y. The periodic solutions will be provided by the following equations (4) ij^j =0. For y = 0, Pi = B2 = 0, the functional determinant of the i|;'s with^_ respect to the P's is zero. However, in general the four derivatives ^ will not vanish at the same time. For example, let us assume and we shall derive Bi in series developed in powers of B2 and y from the ?irst equation (4), aid we shall substitute it in the second equation (1). i346 Let be the result of the substitution. Our functional determinant being zero, we shall have However, we may distinguish between two cases: 1. The derivative j^ is not zero, or, in other words, the functional determinant of i>^ and ^^ with respect to B^ and y is not zero. In this case if we assume that B2 and y are the coordinates of a point in a pLne' the c^rve represented by equation (5) will have an ordinary point at the origin, where the tangent will be the line y - 0. In general, the second derivative Will not be zero, i.e.. the origin will not be a point of inflection of the curve (5) . If we intersect the line y = yo, where yo is a rather small constant, we may have, two points of intersection for this line and the curve (5) in the vicinity of the origin, or we may not have any, depending on the sign of yg. 342 For example, if this curve is above its tangent, we shall have two intersections for uq > 0, and consequently two periodic solutions, and for Mo ^ we shall not have any. We have thus seen two periodic solutions approach each other, coincide, and then disappear. Let us consider the two points of intersection of the line y = yg with the curve (5) . They will correspond to two consecutive roots of the equation (5) and, consequently, to two values having opposite signs of the derivative ■^^, and therefore to two values of opposite signs of the func- tional determinant of the ifi^'s with respect to the B's, that is, of the /347 product i.e. , of a^ . Therefore, one of the two periodic solutions which coincides then to disappear is always stable, and the other is unstable . 2. The derivative ^ = 0, or in other words the functional determinant of ^i and 1^2 with respect to 6i and y, is zero. The curve (5) then has a singular point at the origin which, in general, will be an ordinary, double point. Two branches of the curve intersect at the origin, and the line y = Uq will always meet the curve at two points. We shall therefore have two periodic solutions, no matter what the sign of \iq may be. The two branches of the curve determine four regions in the vicinty of the origin. In two of these regions which are opposite the peak ¥ will be positive; in the other two regions, it will be negative. Let OPi , OP2, OP 3, OPi+ be the four half-branches which converge at the origin. OPi will be the extension of OP3 and OP2 will be the extension of OP^. OPi and OP2 will correspond to yg > 0; OP3 and OP4 will correspond to yo < 0. The function "V will be positive for the angles P1OP2, P3OP1+, and negative for the angles P2OP3, P1OP1+. We have just seen that the stability depends on the sign of the deriva- df ^ diT* ^°^ example, when we pass over OP2 , 4* will change from negative to positive. The derivative will be positive, and the solution will be stable, for example. It will also be stable when we pass over OPi^, and unstable when we pass over OP2 or OP3. 343 The periodic solutions corresponding to OPi are stable and they form an analytical sequence with respect to those which correspond to OP3 and which are unstable. Conversely, those which correspond to OP2 and which are unstable are the analytical sequence with respect to those which correspond to OP,, and which are stable. We thus have two analytical series of periodic solutions which coincide for y = 0, and at this instant of tiir.e the two series excha nge tneir stabil - We have just studied the two simplest cases, but there may be ^ multi- tude of other cases corresponding to different singularities which the IJh^ curve (5) may have at the origin. However, no matter what these singularities may be, we shall observe an even p + q number of half-branches emanating from the origin i.e. p forTT and q for u < 0. Let us assume that a small circle about the origin encounters them in the following order OP,, OP, 0?,,+^. Let OP,, 0I'„ ..., OP,, (6) be those which correspond to y > and let 0P;,-„ or,,^„ ..., ov,^, be those which correspond to y < 0. Then the half-branches (6) will correspond alternately to periodic stable solutions and to unstable solutions. For purposes of brevity, I may state that these half-branches are alternately stable or unstable. The same holds true for the half-branches (7). In addition, OPp and OPp^.^ are both stable or both unstable. Consequently, the same holds true for OPp+q and OP^. Therefore, let p' and p" be the number of stable half-branches and the number of unstable half-branches for y > 0, so that we have p' + p" = p- Let q' and q" be the corresponding numbers for y < 0, so that q' + q" = q- There are therefore only three possible hypotheses 344 P' = p' :-', 7'= ?'-'-'. p' = p"-u q'=q'—\- In any case, we have Let us assume that p does not equal q, and, for example, that p > q, /349 in such a way that a certain number of periodic solutions disappears when we pass from p > to y < 0. It may be seen that this number is always even, and in addition as many stable solutions as unstable solutions would always disappear , according to the preceding equation. Let us now assume that wq have an analytical series of periodic solu- tions and that, for u = 0, we pass from stability to instability, or vice- versa (in such a way that the exponent a vanishes). Then q' and p" (for example) are at least equal to 1. Therefore, p' + q" is at least equal to 2. It follows from this that we shall have at least another analytical series of real, periodic solutions which intersect the first for y = 0. Therefore, if. for a certain value of y. a periodic solution loses sta- bility or acquires it (in such a way that the exponent a is zero) it will coincide with anoth er periodic solution , with which it will have exchanged its stability. 379. Let us now return to the case which I was first going to discuss — that in which the time does not enter explicitly in the equations, where, consequently, we have the energy integral F = C, where finally there are two degrees of freedom. I shall pursue the same line of reasoning as was the case in No. 317, and I shall assume that the period of the periodic solution, which Is T for the solution which corresponds to y = 0, B^ = 0, equals T + t, and differs very little from T for adjacent periodic solutions. I shall write the following equations (1) A, = 0, -}', = o, -Ij^o, F = Go, which include the following variables P'' h, P), Pi, ^, T. According to our hypotheses, the functional determinant of the i|;'s with respect to the B's must vanish, as well as all its minors of the first order. However, the minors of the second order will not all be zero at the same time, in general. Therefore let us set 6i = in equations (1), and let us consider the 345 K^ functional determinant A of /35Q •h. ^" '>» F with respect to p„ p,, Pv, t. This determinant vanishes when the g's, y's and x's vanish, but in general the minors of the first order will not vanish. Let us consider the functional determinants of F and of two of the four functions i> with respect to x, and with respect to two of the four variables 6. Can they all be zero at the same time? According to the theory of determinants this could only happen if the following were true: 1. All the minors of the two first orders of the determinants of the IP's with respect to the x's were zero at the same time, which does not occur, in general, and which we shall not assume. 2. The derivatives of F were all zero at the same time. We saw in No. 64* that they must be zero all along the periodic solution. We shall no longer assume this. 3. The derivatives of the i|j's and of F with respect to x were all zero at the same time. The following values would not correspond to a periodic solution strictly speaking, but to a position of equilibrium (see No. 68). We shall no longer assvraie this . We may therefore always assume that all the minors of the first order of A are not zero. Let us then eliminate four of our unknowns B and x among the equations (1). For example, let us eliminate 6i, B3, 04, ^ ; we shall still have an equation of the following form Due to the fact that this equation has exactly the same form as equation (5) of the preceding section, it will be handled in the same way, and we shall arrive at the same results: 346 1. When periodic solutions disappear after having coincided, an even number, and as many stable as unstable solutions, always disappear. 2. When a periodic solution loses or acquires stability when we vary VI continuously (in such a way that a vanishes), we may always be certain /351 that at the moment of passage another real, periodic solution of the same period coincides with it. 380. Let us proceed to the second case, that in which Due to the fact that none of the characteristic exponents vanishes for H =0, except the two which are always zero, there is no periodic solution of period T which coincides with the first for fx = o. On the other hand, according to principles presented in Chapter XXVIII, there are periodic solutions of the second type, of period 2T, which coin- cide with the given solution whose period is T for y = 0. What may we say regarding their stability? For y > 0, for example, we shall have a stable solution of period T which will become unstable for y < 0. For y > 0, let p' and p" be the number of stable solutions and the num- ber of unstable solutions which have the period 2T, without having the period T. Let q' and q" be the corresponding numbers for y < 0. Let us then consider all the solutions of period 2T, whether they have have the period T or not. Applying the principles presented in No. 378 to them, I find that I may postulate the following three hypotheses regarding these four numbers : 7.+p' = p\ q=q'. However, if we refer to the principles given in Chapter XXVIII, we shall find that these four numbers cannot take all values which are compatible with the three hypotheses. The simplest and most frequent cases are investi- gated in No. 335. 347 Application to the Orbits of Darwin i}^ 381. In Volume XXI of Acta Mathematica , M. G. H. Darwin studied cer- tain periodic solutions in detail. He discusses the hypotheses given in No. 9, and considers a perturbing planet which he calls Jupiter, and to which he attributes a mass which is ten times smaller than that of the Sun. This fictitious planet describes a circular orbit around the Sun, and a small perturbed planet having zero mass moves in the plane of this orbit. He has thus acknowledged the existence of certain periodic solutions which are again Included in those which I have called solutions of the first type, and which he has studied in detail. These orbits are referred to moving axes, turning around the Sun with the same angular velocity as Jupiter. These orbits are closed curves, in relative motion with respect to these moving axes. M. Darwin has called the first class of periodic orbits the class of planets A. The orbit is a closed curve encircling the Sun, but not encircling Jupiter. The orbit is stable when the Jacob! constant is larger than 39, and unstable in the opposite case. The instability corresponds to a characteris- iiT tic exponent having — as the imaginary part . For values of the Jacobi constant which are close to 39, there are therefore periodic solutions of the second type whose period is double. The corresponding orbit will be a closed curve with a double point passing around the Sun twice. The two loops of this curve differ very little from each other, and both differ very little from a circle. We shall study these solutions of the second type in greater detail at a later point. M. Darwin also obtained oscillating satellites which he called a and b, and are those which we discussed in No. 52. They are always unstable. Finally, he obtained satellites which, strictly speaking, with respect to the system of moving axes under consideration, describe closed curves en- circling Jupiter, but not encircling the Sun. /353 For C = 40 (C is the Jacobi constant) , we have only one satellite A which is stable. For C = 39.5, the satellite A becomes unstable with a real exponent a. However, we have two new satellites B and C, the second of which is stable, and the first of which is unstable with a real exponent a. For C = 39. we obtain the same result. For C = 38.5, the satellite C becomes iiT unstable with a complex exponent a (whose imaginary part is — ). Finally, for C = 38, we obtain the same result. 348 We must therefore consider three passages: 1. The passage of satellite A from stability to instability; 2. The appearance of the satellites B and C; 3. The passage of satellite C from stability to instability; The last two passages do not entail any difficulties. Two periodic solutions B and C will appear simultaneously which differ very little from each other. One is stable and the other is unstable; the exponent a is real for the unstable solution. This conforms with the con- clusions reached in No. 378. The passage of the satellite C from stability to instability no longer presents any difficulties, because the exponent a is complex in the case of mstabxlity. The conditions presented in No. 380 therefore hold. We there- fore have periodic solutions of the second type corresponding to closed curves which pass around Jupiter twice. 382. On the other hand, the passage of satellite A from stability to instability entails great difficulty, because the exponent a is real in the case of instability. According to No. 378, we should therefore have exchange of stability , with other periodic solutions corresponding to closed curves passing around Jupiter only once. This would not seem to result from the calculations of Darwin. We are naturally led to think that the unstable satellites A discovered by Darwin do not represent the analytical extension of the stable satellites A A Other considerations lead to the same result. The stable satellites A have ordinary closed curves for orbits; the /354 unstable satellites A have orbits in the form of a figure eight. How may we pass from one case to another? This may only be done by a curve having a cusp, but the velocity must be zero at the cusp and. for reasons of symmetry, this cusp could only be located on the axis of the x's It could not be between the Sun and Jupiter. In Figure 1, Darwin gives the curves of zero velocity. For C > 40, 18, these curves intersect the axis of the X s between the Sun and Jupiter, but this no . longer holds for C < 40 18, and the passage occurs between C = 40 and C = 39.5. ' We are left with the hypothesis that the cusp is located beyond Jupiter but this is no longer satisfactory. Let us compare the two orbits corres- ponding to C = 40 and to C = 39.5. The first Intersects the axis of the x's 349 *- . ■.--,• «hi- an^le once beyond Jupiter and once just short of it. Stl ndVbe t^: -o'inte'LecIion poLts. In the sa.e way the second nlLt Hf we disregard the double point) intersects the axxs of the x s r^rl It a rieht Se once beyond Jupiter, and once just short of Jupxter. Ziyj ? bl Se t^o intersection points. Let us consider the inter- section point P or P' which is beyond Jupiter, and let us determine the sigt. of ^. We shall see that this sign is positive for one orbit or the other. However, ^ would have to change sign when passing through the cusp. The point P, the hypothetical cusp, and the point P' cannot therefore h. rPBarded as the analytical extension of each other. We must then assumption. Therefore, I may conclude that the unstable satellites A are not the analyticaiextension of the stable satellites A. But when do the satel- lites A become stable? /355 I can only formulate hypotheses on this point and, in order to do .u ■ T ?t would be necessary to reconsider the mechanical quadratures f S'^ia^^ii Soiever ?f we^amine the behavior of the curves, it appears that'a?Tc:;ta!n time 'the orbit of the satellite A must pass through Jupxter, Ind that it then becomes what M. Darwin has called an oscilMtxn^,satellxte. 383. Let us study the planets A in greater detail, and the passage of these planets from stability to instability. ThP orbits of these planets correspond to what we have designated as ^...^: ^olu^L^s ofthe^irst type (No. 40^. The orbit with a oubepoxnt, w hich passes around the S un twice and which differs very Ixttle from that ot Ihe planet 1 at the moment when the orbit of this planet has Just become unstable! corresponds to which we have designated as periodxc solutxons of the second type (47) . If we apply the procedure by which we deduced periodic solutions of the second t'eTrom those of the first type to solutions of the fxrst type, we shall obtain solutions of the second type exactly. Tr. eoi„rinn<, of the second type, the mean anomalistic motions, which differ verj little from the mean motions strictly speaking, are in a co:™nensur- tble rali^ We must therefore consider the case in which, for our solution of the second type (and. consequently, for the planet A at the time of 350 passing from stability to unstability) , the ratio of the mean motions is close to a simple commensurable number. Since the orbit must pass around the Sun twice, this ratio will be close to a multiple of i. 2 In other words, at the moment of passage, the term which M. Darwin has called nT must be close to a multiple of it. In effect, this is what occurs. The tables of M, Darwin provide us with the following C = ,{o A stable, nT -~ ii>4°, C=-^39,5 A stable, «T = i6>°, C " 3r) AunstablejrtT = 177", G = 38,5 Aunstablej,/iT = 191°. It can be seen that the passage must be made around nT = 170°, and /356 this number is close to 180°. The mean motion of the planet A is therefore almost three times that of Jupiter. We could consider applying the principles presented in Chapter XXX to a study of these solutions of the second type, but several difficulties would be encountered because we would be dealing with an exception. It would be better to resume this study directly. 384. Let us again consider the notation given in No. 313, and let us set the following, just as in this section j-i =- L - G, 3-, = L + G, F' = R-i-G.^ Fo+pF,-f-..., The term L must have the same sign as G (see page 201, in fine), and the eccentricity must be very small. Since xi is on the order of the square of the eccentricity, this variable will also be very small. Since we only wish to determine the number of periodic solutions and their stability, we shall be content with an approximation. We shall therefore neglect y^p^ and the following terms. In the term yFi , we only take into account secular terms and terms with a very long period, and we shall neglect the powers which are higher than x^. We shall have 351 ri^-a-^hx,-h cxi cosuj, Where a. b, c are functions only of .„ and where cxicos. is the very long period term which has been retained. The very long period terms are terms with I + 3g - 3t, i.e., terms with 2y2 - yi- We therefore have lo = iyi — iyi- We then have and we may apply the method of Delaunay. / 357 The canonical equations have the integral Xt -+- 2x1 = k, from which we have p., ? , ^ _ Ifil _H ,x{a i-6x,-HCJr,cos(d). With the approximation which has been adopted, we may replace a, b, c by aa — 2X,a'„, bo, Co, designating that which a, fj, h, c, become by ao , a'„. b„. c„ when we re- place X2 by k. Thus, designate the constants which depend on k, and we have p^ _ . ^ >. 1— iJ^ -t- ijL(a -^ ^Xi -4- YX, cosoj). (A- - - x, ;' ■ 2 -2 Let us assume that k is a constant, v'^cos'% v/^sln'^ are rectangular coordinates of a point in a plane, and let us compile the curve F' = G, where C designates a second constant. This curve also depends on the two constants k and C. If it has a doubirjoinrthis double point will correspond to a periodic solutxon. 352 which will be stable if the two tangents to the double point are imaginary, and unstable if the two tangents are real. We should note that the curve is symmetrical with respect to the two axes of the coordinates and that the two double points, which are symmetri- cal to each other with respect to the origin, do not correspond to two periodic solutions which are actually different. The double points may only be located on one of the axes of the coordin- ates, so that they will be obtained by setting If we set 1 k C == -7^ -♦-■-+ ^I, the curve F' = C passes through the origin and has a double point. The tan- gents to the double point are given by the equation /358 Therefore, if 4 3. , ;- ,u3 -t- uY cos 01 = o. 4 ^ „^ ^7-- M^?>I^Y (1) the tangents are imaginary. If FY> ^-- -i-F?;- -J^T.. (2) the tangents are real. Finally, if the tangents are again imaginary. The coefficient B is positive. I wrote the preceding inequalities also assuming that y is positive. If y were negative, we would only have to change o) into oi + tt. The double point at the origin corresponds to the solution of the first type, i.e., to planet A of M. Darwin. It may be seen that this solution is stable when the inequalities (1) or (3) hold, and is unstable when the in- equalities (2) hold. Let us now study the double points which may be located on the line u = 0. 353 If we set w = 0, the function F' becomes F'= -, i/>--^.;' -tjta -1- iJt2:i(P + y)= C. (4) Keeping k constant, if we vary xi from to k, we find that the maxima and minima of F' are given by the equation (5) (A--T,;» ■f(|3 + y) = o. Which has a solution if the inequality (3) holds, and does not have a solu- tion in the opposite case. Therefore, if the inequality (3) does not hold, the function F' is con- stantly decreasing if it holds. The function F' first increases, reaching a maximum, and then decreases. This maximum corresponds to a double~point located on the line u = 0./359 or rather to two double points which are symmetrical with respect to the origin. However we must determine how we may obtain these double points for a given value of the constant C. Equation (5) provides us with xi as a func- tion of k. We must deduce xi from it as a function of C. However, equations (4) and (5) may be written from which we have dC _dV^_^dl^^^^dlL, dFt " dXi "*" dk dxx dk dx, dx\ dk dxi dx, Neglecting terms containing y. we have dF^ rfP dk dx, from which we have dF' dk «"F' d^r dk dxi dx] = o; dx\ - {k-x,}' \8/ and 354 dx^ ' dxi It results from this tliat xj is a constantly decreasing function of C. For a value of C, we have only a maximum at the most, i.e., we have at the most two double points which are symmetrical to each other with respect to the origin on the line o) = 0. Let Cq be the value of C which satisfies the double equality „ 1 k ""o = ^ + -^- -t- I". We shall see that, for C > Cg, there will not be a double point on the line (0=0 and that, for C < Cq, there will be two of them. /360 The same discussion may be applied to the case of double points located on the line co = ir. The values of xj will be given by the equation (5') which has a solution if the inequalities (2) or (3) hold. If Ci is the value of C which satisfies the double equality „ 1 k X-j — ---■- H( ? — */) = o. the condition for which there are two double points on the line to = 7r, is C < Ci. We would like to point out that C^ > Cq , that Cq is the value of C for which one passes from inequality (2) to inequality (3) , and that Ci is the one for which we may pass from inequality (1) to inequality (2) . When compiling the curves, we would readily find that the tangents are real for the double points located on u = 0, and that they are imaginary for the double points located on u) = tt. 355 \^ we may therefore sum up our results as follows: First case c > c,. /361 The inequality (1) holds. The solution of the first type (planet A) is stable. There is no solution of the second type (orbit with double point). Second case The inequalities (2) hold. The solution of the first type becomes unstable. There is a solution of the second type which is stable. Third case Ce the points where the four collisions occur. We may specify these four points arbitrarily, provided that they are located on the same line passing through F. We must construct two ellipses E and Ei which intersect at Qi and Q^.two 360 ellipses E' and E'^ which intersect at Q2 and Q3, two others E" and EV which intersect at Q3 and Qi^, and finally two others E'" and E'V which intersect at Qi+and Qj . The orbit of P is composed of arcs pertaining to the four ellipses E, E' , E", S" , and the orbit of P^ is composed of arcs pertaining to the four ellipses E^ , E\, E'{, E'|' . We shall specify the energy and area constants arbitrarily. These /367 constants must be the same for the interval between the first two colli- sions (orbits E and E^) for the following interval, and for all the other intervals. According to the statements presented in the preceding section, this is the only condition which must be fulfilled. In order to compile E and E^ , we shall proceed as follows. Let us consider the motion of three bodies. Since we assume p = 0, this motion is Keplerian, and the central body may be regarded as being fixed at F. We know the total energy of the system. The two planets P and Pj must leave the point Qj simultaneously in order to arrive at the point Q2 simultane- ously. When P and Pi go from Qi to Q2, the true longitude of P increases by (2m + 1)tt, and that of P^ increases by (2mi + l)Tr. We may still specify the two integer numbers m and mi arbitrarily. The problem has then been completely determined. It should be pointed out that the inclination of the orbits does not intervene. In order to resolve this, we may assume planar motion. The problem can always be resolved. We need only apply the principle of Maupertuis, and Maupertuis action, which is essentially positive, always has a minimum. We must now determine the planes of the two ellipses. We know the area constants. We therefore know the invariable plane which passes through the line FQxQ2- The areal velocity of the system is represented by a vector perpendicular to the invariable plane, whose magnitude and direction we know. It is the geometric sum of the areal velocities of the two planets, represented by two vectors whose magnitude we know, since they equal, respectively, mp and mip, where m and mi are the masses of the two planets and p is the common parameter of the two ellipses E and Ei. We may therefore compile the directions of these two base vectors which are perpen- dicular to the plane of E and to the plane of Ei , respectively. The terms E' and E^ , E" and E'j", ..., may be determined in the same way. 389. Let us now assume that all of the successive collisions occur at the same point Q. The period will be divided into as many intervals as there will be collisions. Let us consider one of these intervals during which the two planets describe the two ellipses E and Ei. As in the pre- /368 ceding section, we will specify the energy constant and the area constant which must be the same for all the intervals. We must construct E and Ei. 361 K^ Let us assume that during the interval under consideration the planet P has performed m complete revolutions, and that the planet Pj has completed mi complete revolutions. We can arbitrarily specify the two whole numbers m and mi- Since we know these two whole numbers, we know the ratio of the major axes. Since we know, on the other hand, the energy constant, we also know the major axes themselves. On the other hand, we know the area constant. Consequently, we know the vector which represents the areal velocity of the system. This vector can be decomposed an infinite number of ways into two base vectors which represent the areal velocities of P and Pj. We shall arbitrarily specify this decomposition. If we know the two base vectors, we know the planes of the two ellipses and their parameters. The orientation of each of these ellipses in its plane remains to be determined. We will determine it by passing the ellipse through the point Q. Summarizing, we can arbitrarily specify: 1. The point Q and the number of intervals; 2. For all the intervals, the area constant and the energy constant; 3. For each interval, the whole numbers m and mi and the decomposition of the areolar vector. In order to make the problem tractable, these arbitrary numbers must satisfy certain inequalities which I will not describe. 390. Let us disregard the exceptional case where all the collisions take place along the same line or at the same point, and let us consider the case of motion in a plane. Let Qi, Qz, • • • , be the points where the succes- sive collisions take place. We will arbitrarily specify the energy constant and the area constant which must be the same for all the intervals. Let us consider one of the intervals, for example, the one where the two planets pass from Qi to Qz- We will arbitrarily specify the magnitude of the radius vectors FQi and FQ2, but not the angle between these two radius vectors, nor the duration of the interval. We know that in this interval the difference in longitude of the two /369 planets has increased by Zmir. Let us arbitrarily specify the whole number m. Since we know this whole number, the two lengths FQi and FQ2, as well as the two energy constants and the area constants, we have everything needed to determine the orbits E and Ej. This means that the principle of Maupertuis must be applied. However, the Hamiltonian action must be defined as was done in No. 339 and the Maupertuis action must be derived according 362 to the procedure of Nos. 336 and 337. Unfortunately, this Maupertuls action is not always positive and therefore one Is not certain that it always has a minimum. Summarizing, we can arbitrarily specify: 1. The number of intervals and the lengths FQj^ , FQ2, ...; 2. The area constants and the energy constants; 3. For each interval, the whole number m. The collision orbits obtained in this way are all planar . Among the periodic orbits of the second kind which reduce to these collision orbits for y = 0, there are certainly some which are planar. It is also possible that there are some which are not planar for p > 0, and only become so at the limit. 391. Let us now see how one may demonstrate the existence of periodic solutions of the second kind which, in the limit, reduce to the collision orbits which we constructed above. Let us now consider one of the collision orbits and let tg be a time before the first collision and t^ a time between the first and the second collisions. In the same way, let t2 be a time between the second and the third collisions. For the discussion I will assume that there are three collisions. I will call T the period in such a way that at the time tg + T the three bodies appear in the same configuration as was the case at the time tg. As the variables, I will take the major axes, the inclinations and the eccentricities, and the differences of the mean longitudes, the longitudes of the perihelia and the nodes. In all, there are eleven variables. The orbit is regarded as periodic if the three bodies have the same relative configuration at the end of the period. Let Xj , Xg , ..., X]j be the values of these variables at the instant tQ for the collision orbit under discussion and consequently for y = 0. /370 Let X. be the values of these variables at the time ti for this same colli- sion orbit, x. their values at the time t^, and x^ their values at the time to + T. One will have ■rf = x^ H- idht: where m^ is a whole number which must be zero for the major axes, the eccen- tricities and the inclinations. 363 \^ Let us now consider an orbit which is slightly different from the col- lision orbit. Let us assign a very small value to y, but different^ from zero. In this new orbit, our variables will have the values x^ + S^ at the time to. x^ + B^ at the time t, . x? + B^ at the time t^ and finally X? + b| at the time tQ + T + t. The condition for which the solution is periodic with period T + t is P? = ??• Assuming y = 0, in order that a collision occurs between the time to and the time ti, the variables bJ must satisfy two conditions. Let be these two conditions. Let us set it can be seen that the gO's are holomorphic functions of the yO's and of y. By applying the principles of Chapter II, i^ can be shown that the same holds for the B^'s. In order that there be a collision between the times ti and t2 (assuming that y = 0), two conditions are necessary, which I may write as follows /■(?!)=/.(?;)= °- (1) Replacing the b}'s in relationships (1) by their values as a function of the Y°'s and of y, and then setting y = 0, I obtain 0.(y?)-''-'(t?)-o- Let us then set I find that the BJ^'s and the B^'s are holomorphic functions of the y^'s ^371 and of y. The same holds true for the yj's, and consequently for the bJ's. Finally, in order that there be a collision between the times t2 and to + Tt, two conditions are necessary which I may write as follows 364 If the 32 rg 3j.g replaced by their values as a function of the y^'s i and of y, and if we then set y = 0, they become I may set ■')i(r,')=TiF. M{t\)-=ll\^, H=ll (^- = 3, .... II) and I then find that the bJ's, the e^'s, and the sj's are holomorphic func- tions of the Y?'s and of y. In the same way, the 6|'s are holomorphic func- tions of the Y?'s, of y, and of t. The relationships bJ = 3° are therefore equations whose two terms are holomorphic with respect to the yj's, y, and t. These equations could be discussed in the same manner as in Chapter III. The existence of solutions of the second type could then be demonstrated. I do not believe that this is necessary, because these solutions deviate too much from the orbits traversed in actuality by celestial bodies. 365 \^ CHAPTER XXXIII DOUBLY ASYMPTOTIC SOLUTIONS Different Methods of Geometric Representation 392 In order to study doubly asymptotic solutions, we shall confine ^372 ourselves to a very special case, that of Section No. 9: Zero mass of the perturbed planet; circular orbit of the perturbing planet; zero inclinations. The three-body problem then has the well-known integral called the Jacob i integral . Returning to No. 299 devoted to this problem from Eo. 9, we must dis- tinguish between several cases. We saw on page 159 that we must have the following inequality ■?« r, r, 2 (1) 2 We then distinguished between the case in which m^ is much smaller than m^, and in which -h is sufficiently large (page 160) . We saw that the following curve V .--(?'+o«)=-A (2) 2 may be broken down into three closed branches which we have called Ci , Cg and C3. Therefore, in view of the inequality (1), the point 5, n must always remain inside of Ci , or always inside of C2, or always outside of C3 (?, n are the rectangular coordinates of the perturbed planet with respect to the moving axes) . We shall assume below that the value of the constant -h is large enough for curve (2) to be broken down into three closed branches, and that the point 5, n always remains inside of C2. In this way, the distance T2 ^''°'^ the perturbed planet to the central body may vanish, but this is not true 2J11 for the distance ri between the two planets. This hypothesis corresponds to the following hypothesis, which we formu- lated on pages 199 and 200 — i.e., the curve F = C has the form shown in Figure 9, and the point xi, X2 remains on the utilizable arc AB. We shall employ the notation given in No. 313, and we shall introduce the Keplerian variables L, G, 1, g. However, these Keplerian variables may be defined in two ways. Just as in No. 9, we could relate the perturbed body to the center of gravity of the perturbing body and of the central body. 366 and we could consider the oscillating ellipse described around this center of gravity. However, it is preferable to refer the perturbed body to the central body itself, and to consider the oscillating ellipse described around this central body. These two procedures are equally legitimate. We saw in No. 11 that the body B may be related to the body A, and the body C may be related to the center of gravity of A and of B. It is apparent that we could also refer C to A, and B to the center of gravity of A and C. If A represents the central body, B the perturbing body, and C the perturbed body, it can be seen that the first solution is that which was adopted in No. 9. It may also be seen that in the second solution, which we shall adopt from this point on, the two bodies B and C are both related to the central body since ~ due to the fact that the mass of C is zero — the center of gravity of A and C is at A. o j We then have ^^ '1 2 / 1 — JJl where y and 1 - p designate the masses of the perturbing body and of the central body, ri designates the distance between the two planets, 1 desig- nates the constant distance from the perturbing body to the central body, and r2 designates the distance of the perturbed body to the central body.' Just as in No. 313, we shall set r, ^-. L — G, a-, = L + G, ■iy-i^- l — S + i, ■iri=l-i-ff~r, ^ ' ' i /l — - fZ I would like to stress the following important point. It can be seen that /374 the function Fi always remains finite in the region from which the point 5 , n cannot leave. We shall employ the method of representation given on page 200 , and we shall represent the configuration of the system by the point in space whose rectangular coordinates are 7_ _ , I /. r, o. When xi, X2, yi, y2 vary in conformance with differential equations, the point X.'y, Z will describe a certain trajectory. When y2, which in- creases constantly, reaches the value 2it, the point X, Y, Z — which has moved to Ml ~ will again be located on the half-plane Y - 0, X > 0. The point Mi is then the consequent of M, according to the definition given in No. 305. Since y2 is always increasing, every point on the half- plane has a consequent and an antecedent. There is only an exception for 368 \^ very small X2 — i.e., for points on the half -plane which are very far from the origin, or very close to the Z axis. We shall have an integral invariant, in terms of the meaning attribu- ted to this word in No. 305. Let us try to formulate this invariant. Due to the fact that the equations are canonical equations , they have the following integral invariant I ilTidXidyicIy^. X2 Let us set z = — , and let us select F , z, y^, y2 as new variables. ^1 The invariant will become 'r; f/PV/if/yirfi-j _ (' x]dF'dzdy^dYt ■ ''' _ I ^1 "•' dzdy^d dV tfi' I ^1 /^i -t- x"5 /(. rf.r, dxi We may deduce the following triple invariant from this quadruple in- variant (due to the existence of the integral F' = C) x\dzdridYi rx\dzd y,dy^ In this triple integral, we assume that x,, ^2, «, = — ^— , «, = _ 5— are axi dx^ replaced as functions of z, yj^, y2 by means of the equations j?j = .r, ;, F'=C. Let us now take the variables X, Y, Z, and let us employ A to desig- /376 nate the Jacobian of X, Y, Z, with respect to z, yi , y2. The invariant will become r x\d\d \dl J (ar|/ii-+-^,/!,)i' Let us set R=^ ^ .^' — , Z = from which it follows that X =. R cos^», Y =^ R sin^,. Let US again set D ^-[(R -,)• + z*j[(R + ,)^4-Z»]- 369 \^ A simple calculation provides the following 8 /.-(; + 4) Our invariant may therefore be written r 8t> /^'(iT T) d\ d\ (17. The principles presented in No. 305 enable us to deduce the following invariant, in the sense of No. 305 8,r7v'^(--H4: r^iihS .r,/ii-+- T, n, d\ d'L. n2 and R play the role which fi and p played in the analysis of No. 305. The tern under the sign / is essentially positive, except for very small X2 — i.e., for points of the half -plane which are very far from the origin, or very close to the Z axis. 393. This fact (that a point will no longer have a consequent if it is too far, or if it is too close, to the Z axis) could cause some difficulty, and it would be advantageous to avoid this difficulty by whatever method. We could employ the statements presented in No. 311, and we could re- place our half-plane by a simply connected curve on a surface. We shall choose this curve on a surface in the following way. If X2 is very small, the eccentricity is very small, and the two planets turn in the opposite direction. The principles presented in No. 40 are applicable, and we may affirm the existence of a periodic solution of the /377 first type which will clearly satisfy the following conditions: The quanti- ties /z^cosj'2, /x^sin/,, xu cos/,, sinj, are periodic functions of the time t. These functions depend on y and on the energy constant C. They may be developed in powers of y; the period T also depends on y and on C. The angle yi increases by Zir when t increases by a period. Finally*, V^ cos ya and yx^ sin y2 are divisible by y, so that we have X2 = for y = 0. With our method of representation, this periodic solution, which I have called a, is represented by a closed curve K. Since X2 is very small when y is very small, this curve is displaced very little from the Z axis. 370 It may be stated that it is displaced from it very little, in the same way that a circle having a very large radius is displaced very little from a straight line. Every point on the K curve is either very far from the origin or very close to the Z axis. Under this assumption, our curve on a surface S would have the curve K for the perimeter, and it would be displaced very little from the half- plane Y = 0, X > 0, except in the immediate vicinity of the curve K. It would be very easy to conclude this determination in such a way that every point on this surface would have a consequent on this surface itself. For this purpose, if I designate an arbitrary trajectory by (T) — i.e., one of the curves defined in our method of representation by differential equa- tions — it would be sufficient that the surface S was not tangent at any point to any of the trajectories (T) . However, there is still another method, which does not basically differ from the first method. If we reflect on this a little, we will find that this difficulty is similar to that in Chapter XII. We must therefore perform the change in variables similar to that performed in No. 145. Let us first set and we then have S ---^•'■.i-h.r;^-, -hpS, where Si is a function of C'2. 12* x'l , yi. Let us then set /378 and finally I should first point out that the canonical form of the equations will not be changed when I pass from the variables xi , yi , X2, yz, to xi , yi, C2» TI2, then to x'l, y'l , ^\^ t\\, and finally to x\ , y'^ , x'^, y'g. I must now choose the function Si. I know that F' is a holomorphic function of v/^-^'i ♦'os^j^i, v'ax, sfnjKi, /axacosj^'j, y/axjsinjK- in the region under consideration. I would like it to 371 K^ remain a holomorphic function of the new variables v'T^cos/',, /^sin/;. For this purpose, I would like the old variables v/i^ gin yi ^° ^^ holomor- i COS phlc functions of the new variables V2x'^ 3^^ ^i ^^^ °^ ^' To do this, we need only assume that Si is a holomorphic function of and is divisible by x'^. For our periodic solution a, I would like to have tj = T/j — o, x', — xj = const. Therefore, let be the equations of the periodic solution. A, B, C are functions of yi which are periodic of the period 2tt and may be developed in powers of y . Then C - -^^ will also be a periodic function of yi- Let xj be its dyi mean value. We may obtain another periodic function a such that dy^. ilyx We shall no longer assume that, for x'l = xj, the function ySi may be re- /379. duced to «-B$;-+-Ar„. (2) This will be sufficient for the equations of the periodic solution to be reduced with the new variables to It is clearly possible to obtain a function ySi which may be developed in powers of V^i ^l^ Yl. which may be divisible by x'l, and which at the same time may be reduced to expression (2) for x'^ = x^ n Let VIS adopt the new variables x*i,y'i, x'2, y'z' 372 v_/ The function F', which was holomorphic with respect to l/2xi *^°^ vi ' ^ sin ' '■* V^x^j^^ 72* will also be holomorphic with respect to l/ix^ ^°^ y'l, I COS V2x'2g^^ 7 2 . In addition, since one of the solutions of the differential equations is we must have the following relationships for 5*2= n'2 = 0, x'l = xP ^F' _ d¥' _ rfF' _ t^^, - c/v; " ^ - "■ (3) For small values of C'2 and n'2 » F' may be developed in powers of 5*2 and n'2. In view of relationships (3), for x'l = xj , the terms of the first degree in this expansion will vanish, and the terms of zero degree will be reduced to a constant which is independent of y^. This constant can be nothing else than the energy constant C, so that the conditions 5'2 = n'2 = 0, x'l = xj may be replaced by the following con- ditions ?i = 7)', = 0, F'=C. Thus, for F' = C, the terms of the first degree in ^2 and r/2 will vanish in the expansion of F', The difficulty arises from the fact that F' and Fi include terms of the first degree in /380 and that, consequently, the derivative -~ includes terms — = — which be- come infinite for X2 = 0. This difficulty no longer exists now. We no longer have terms of the first degree in C'2, n'2- Therefore the derivative ^^ remains finite, dF' dFn even for X2 = 0,and — -, which differs very little from -ri; always retains CIX2 CIX2 the same sign. Therefore, with our new variables which only differ from the old variables by very small quantities on the order of m, we shall con- stantly have dx\ -dr>°- 373 \^ With our new variables, let us formulate a convention which is sindlar to th!t given in the preceding section, and let us -present the confxgura- tion of the system by the point in space whose coordinates are v/7; COS7J ^ Y ^ ___^/{:!.^'!lj:L_ , /r', -<-\iJ^'i-2 A; cosy, dx'2 Everything which we have stated still holds . However, since -^ can never vanish. PVPrv noint on the half -plane, without exception, will have a consequent . It may now be stated that the integral invariant is ^^^^y^ .P°^^^^^^- , There ca^ only be some question of doubt for the denominator which, wxth the same variables, was xjni + xzng and which now would be Which - assuming that F' is a function of the following four variables /381 may be written I A, dF' , dr „ d?' , dr\ -iK'^-du^'-d^r^'Wr^'-^WJ- In this form, it may be readily seen that the denominator ^^ ^°\f^°J^^ll with respect to the 5' 's, the n"s, and y. However, for y = 0, F may be reduced to (x\ -i-a?;)» and it may readily be shown that the denominator is always positive. It will still be positive for small values of y. 394. In the following statements, we shall adopt the variables defined in the preceding section. We shall remove the accents which have become useless, and we shall write F, Xi and y^ in place of F' , x . and yi- We then have the integral invariant (in the sense of No. 305) 374 "J D ■ —dV 7/F '^^ '^2 from which we have ' ''•'"s £>-[(.X-r)M-Z*J[(X-t-,}M-Z']. I would first like to note that this integral invariant, which is always positive, remains finite when it is extended over the entire half- plane. If l/(X - 1)2 + z2 is an infinitely small quantity of the first order, the numerator x2 Vz(z + 4) is an infinitely small quantit y of the second order, and the same holds true for D. If l/(X - 1)2 + z2 is an infinitely large quantity of the first order, the numerator remains finite, while D is very large of the fourth order. All of the other quantities remain finite. I shall call Jg the value of the invariant J extended over the entire half-plane. The periodic solutions and the trajectory curves which represent them are characterized by the fact that these curves intersect the half-plane at points whose successive consequents are finite in number. For example, let us refer to No. 312 and, in particular, to Figure 7 shown in page 195. In this figure, the closed trajectory which represents a periodic /382 solution intersects the half-plane at five points Mq, Mi. Mg, M3 , M^, each^ of which IS the consequent of the others. For purposes of brevity I shall call such a system a system of periodic points or a periodic 'system . Two systems of as^ptotic solutions correspond to each unstable, periodic r J^o^ ^"T^t solutions are represented by trajectories (in the sense of No. J12), and the total group of these trajectories forms what I have desig- nated as asymptotic surfaces. The intersection of an asymptotic surface with the half-plane will be called an asymptotic curve . Just as we saw in Figure 7, page 195, four branches of asymptotic curves (MA, MB MP MQ) ~ each two of which are located in the extension of the other ~ lead to each of the points M^ of an unstable periodic system. There is an infinite number of asymptotic curves, because there is an infinite number of unstable, periodic solutions and, consequently, an infinite number of systems of unstable periodic points, even if we confine ourselves to solutions of the first type which we defined in Nos. 42 and 44. A distinction may be drawn between asymptotic curves of the first 375 K^ family and of the second family, depending on whether ^^^ --"P^f^^^ characteristic exponent is positive or negative Curves "J^^^^ ^^^^^^^ family are characterized by the following P-P"^/' .^f ,f; Jf^ery of an arbitrary point is very close to a periodic P^^^^ xf^ eonleouent, laree For curves of the second family, it would be the n— consequent and n;t the rh antecedent, which would be very close to a periodic point. On the figure shown on page 195 . the curves MA and MP belong to the first fallly, Ld the curves MB and MQ belong to the second family. These asymptotic curves may be regarded as invariant curves in the Iti-nT'^u^:^ l^n^S^^^ ^^^^ --fit M3A3, M4A4, this total Sroup wii ' ^f 5 ^^d if we designate the 5p— we only consider the consequents m groups o^ ^' '^ ^ ,, ^hp nth conseauent, conseauent which it has been called up to the present, as the p— consequent , iriraprarent that only the curve MqAoA^ under consideration will be an in-^ variant curve. T wo curves of the same family can not intersect. These two curves will Pnd at the same periodic point - for example, the point Mo- These two curves will coincide (since MqAq with its extension MoPq is the only curve of the first family which passes through Mo), and we must determine whether an asymptotic curTe can have a' double point. The question has been answered in the negative (No . 309 , page 186) . Or these two curves will lead to two periodic points of the same periodic systL - for example, to the two points Mo ^^^ ^^ -^1',% 'ZT' which would then be MqAq and M.A^ , had a point m common Q the ^P-^nte cedent of Q would have to be very close to Mq for very ^^^S^P ' J^J^^"^^^^ Tould belong to MqAq, and it would have to be very close to M^ at the same time because Q would belong to MiAi . This is absurd. Or finally the two curves would lead to two points belonging to two different periodic systems. For example, let us assume that the two curves belong to ?he first family, and that Q is their point of intersection. For very large n. the n^l^ antecendent of Q would have to be very close to one of the points ;f the first periodic system and one of the points of the second system at the same time. This is also impossible. Conversely, ^h.rp i. no reason tha t two as ym ptotic curves of different families cannot intersect . Let S and S' be two unstable periodic solutions, let T and T' be the 376 corresponding closed trajectories, and let P and P' be the corresponding periodic systems. Let H and Z ' be two asymptotic surfaces wfiich pass through T and T' , respectively, and which intersect the half-plane along two asymptotic curves C and C' — one belonging to the first family, and the other belonging to the second family. What will happen if C and C' have a point in common Q? The two sur- faces E andE' will intersect along a trajectory x, which will correspond to a special solution 0. The trajectory x will belong to two asjmipototic surfaces, so that for t = -« it will closely approach T, and for t = +«> it will closely approach T' . For very large n, the n£ll antecedent of Q will be very close to one of the points of system P and its n£k consequent will be /384 very close to one of the points of system P'. The solution a is therefore doubly asymptotic . There is nothing absurd in any of these results. We must distinguish between two cases, however. The two solutions S and S' coincide, so that x first closely approaches T = T' , then recedes farther away from it, and again closely approaches this same tra- jectory T = T'. I could then state that the solution a is homoclinous . Or, S differs from S' , and T differs from T' ; I may then state that a is hetero- clinous . The existence of homoclinous solutions will be demonstrated very shortly. The existence of heteroclinous solutions remains doubtful, at least in the case of the three-body problem. Homoclinous Solutions 395. . At the end of No. 312, we found that "the arcs AqAs and B0B5 intersect". However, the arc A0A5 belongs to the curve MqAqAs which is an asymptotic curve of the first family, and the arc BQB5 is part of the curve M3B0 which belongs to the second family. The line of reasoning is general, and we must conclude that the two asymptotic surfaces which pass through the same closed trajectory must always intersect beyond this trajectory. The asymptotic curves of the first family which lead to the points of a periodic system always Intersect the curves of the second family, which lead to these same points. In other words, on each as3miptotlc surface there is at least one doubly asymptotic, homoclinous solution. We shall see very shortly that there is an infinite number of them, but we shall now show that there are at least two 377 of them. For this purpose, let us turn to the figure shown on page 195. Follow- ing the line of reasoning in Nos. 308 and 312, we find that tLe integral invariant J extended over the quadrilateral AqBqAsBs must be zero. It is for this reason that this curvilinear quadrilateral cannot be convex, and that the opposite sides A0A5 and B0B5 must intersect. Let Q be one of the /J«l intersection points of these two arcs. We should note that the point Bq was chosen arbitrarily on the asymptotic curve MAq . If we place the point An at the point Q itself, this point Aq will also be located on the curve _ M3B0 and will coincide with the point Bq . If the two points Aq and Bq coin- cide, the same will hold true for their five consequents A5 and B5. The quadrilateral A0B0A5B5 will therefore be reduced to the firgure formed by two arcs of a curve having the same end points. This figure can not be convex, since the integral invariant extended over the quadrilateral must be zero. Therefore the two arcs AqAj and B0B5 must have points in com- mon, other than their end points. There will therefore be at least two different intersection points (a point and an arbitrary consequent of it are not regarded as being differ- ent) . There will therefore always be at least two doubly asymptotic solutions. Let us assume that the points Aq and Bq coincide, and let us extend the arcs AqAj and B0B5 up to the first point at which they touch Cq. We will have thus determined an area which will be convex this time (from the point of view of Analysis situs) and which will be bounded by two arcs which are apart of the two arcs AqAj and B0B5, respectively, having the same end points — i.e., Aq = Bq and Cq. Let ao be this area, and let ct^ be its v^ consequent. The area a^ — like an ~ will obviously be convex and bounded by two arcs of a curve -- one belonging to the first family, and the other belonging to the second family. The integral J will have the same value for Oq and an- L et j be this value. Since the value Jq of the integral invariant for the entire half- plane is finite, following the line of reasoning presented in No. 291, we will find that, if ^ Jo J the area ao will have a part in common, at least with p of the areas M, ■'-1, ■ ■ •> 'n . 378 V-/ Since n cannot be taken arbitrarily large, I may stipulate the following result: Among the areas ot^, there is an infinite number o f them which have a part in common with ctp^ ' — How may it happen that qq has a part in common with 0^? /386 The area ao cannot be entirely within 0^, since the integral invariant has the same value for the two areas. For the same reason, the area an cannot be entirely within Oq. Neither can the two areas coincide. If one part of an asymptotic curve (for example, belonging to the first family) coincided with its n^ consequent, the same would hold true for its p£B. antecedent, no matter how large p may be. However, if p is large, this plk antecedent is very close to the periodic points, and the principles formu- lated in Chapter VII will demonstrate that this coincidence does not occur. We must therefore assume that the perimeter of uq intersects that of an. However, the perimeter of a^ is composed of an arc AqHoCq belonging to the curve MqAqAs of the first family, and of an arc belonging to the curve M3B5B0 of the second family. In the same way, the perimeter of a^ will be composed of the arc AnMnCn, the nta. consequent of AoHqCo, which will belong to the same asymptotic curve as AqHoCo — i.e., to a curve of the same family — and it will also be com- posed of the arc AnKnCn, the nth consequent of AoKqCo, which will belong to the same asymptotic curve as AqKoCq ~ i.e., to a curve of the second family. Due to the fact that two curves of the same family cannot intersect, it is necessary that AqHoCo intersect AnKnCn, or that AoKqCo intersects AnHnCn. However, if the two arcs AoKqCq and AnHnCn intersect, their nSll antecendents A_nK_nC_n and AqHoCq will equally intersect. It is therefore necessary that AqHoCo intersect the nth consequent, or the n£]l antecedent, of AoKqCq. However, the arc AoKqCq, all of its antecedents, and all of its conse- quents will belong to the same invariant curve of the second family, which was shown in the figure on page 195 by the total group of curves M3B0, M1B3, Mi^Bi , M2B4, M0B2. Ihe arc AoHqCq is therefore intersected an infinite n umber of times by this group of curves . " ' — The two surfaces Z and E' which passed through the closed trajectory T therefore have an infinite number of other intersection curves. 379 -n.prpfore. on the surface Y. there T^ an Infinite number of double asymptotic, homoclinous s olutions. • e.d. 396. Let AoHoCo be an arbitrary arc of our asymptotic curve of the first family, and Lt us assume that this arc intersects an asymptotic cirve ofthe second family at two end points A, and Co- It may be stated thT there will always be other points of intersection with the curve of the second family between these two points Aq and Cq. Let AqKoCo be the arc of the curve of the second family which unites these two points Aq and Cq- Either the two arcs AqHoCo and AoKqCq have points in common other than their end points, in which case the theorem has been proven. or these two arcs do not have a point in common other than their end joints An and Cn. The two arcs then bound an area ag which is similar to 'that whtch Se considered at the end of the preceding section Ihe same line of reasoning may then be applied, and we may conclude that the arc AoHqCq intersects the curve of the second family an infinite number of times. Therefore, there is an infinite number of other points on an asymptotic curve of the first family, between two arbitrary points of intersection with the curve of the second family. nn pn arbitrary asymptotic surface, between two doubly asympto.tl^_ar^ bitrary solutions, there is an inf inity of other solutions. We may not yet conclude that the doubly asymptotic solutions are ever^- where dense on the asymptotic surface, but this seems very likely. The points of intersection of two asymptotic curves may be divided into two categories. The asymptotic curve may be traversed in two ^PP^^^;^^/"^^: tions. We assume that this direction is positive, if we proceed from a point tolls consequent. L et A be a point of intersection of the two curves, and let BAB', CAC be two asymptotic curve arcs intersecting at A. Let us assume that BAB' belongs to the first family, and CAC belongs to the second family, and that -- when following the curves in the positive direction - one V^°- ceeds from A to B' . and from A to C . Depending upon whether the direc- /388 tlon AB- is to the right or the left of AC', the intersection point A will belong to the first or to the second category. Under this assumption, let AqHoCq be an arc of the first family, inter- sected at Ao and Co by an arc AqKoCq of the second family. No -^"er what category Aq and Cq belong to, the group of two arcs AoHoCoKqAo will form a closfd curve. If the two arcs have no other point in common except their end points, this closed curve does not have a double point and defines an 380 area Og. If the two arcs had points in common other than their end points, and if, for example, the two arcs AoHqDqH^Co, AqKoDoK^Cq inter- sect at Dq, we may replace the points Aq and Cq by the points Aq and Dq located between Ag and Cq , and the arcs AqHqCo , AqKoCq by the two arcs AoHqDo and AqKoDq. This may be continued until we arrive at two arcs which have no point in common other than their end points. Let us assume that the two arcs define an area clq. According to the statements we have just presented, the arc AqHoCo must intersect the asymp- totic curve of the second family an infinite number of times. Therefore, the curve of the second family must penetrate within qq an infinite number of times, and it must leave it an infinite number of times. It may pene- trate it or leave it only by intersecting AqHoCo, because it cannot inter- sect AoKqCq which also forms a part of the curve of the second family. It is apparent that points through which it will penetrate into the area, and the points through which it will leave the area, will not belong to the same category. Therefore, between two arbitrary intersection points of two curves. there i s an infinity of other points belonging to the first category, and an infinity of other points belonging to the second category . Let us employ (1), (2), (3), ..., to designate the successive points at which the curve of the second family and the arc AqHoCo meet, taken in the order in which they are encountered proceeding along the curve of the second family in the positive direction. They will belong to two categories in succession. Let us study the order in which they are encountered proceeding along the arc AqHoCo. This order cannot be completely arbitrary, and certain successions are excluded — for example, the following: /289 (7.ni), {xp)^ (rL?n H- I), (?./>-:- I) (iin+ I) (2/,), {■im), {,-).p .- \) ('•'«)> (2/J I- I}, {im+ i), (,,;,) (i"0. {r>.p), (2m -I), (2/?- I) as well as the same inverse successions, and the similar successions where 2m + 1 and 2p + 1 are replaced by 2m - 1 and 2 p - 1. 397. When we try to represent the figure formed by these two curves and their intersections in a finite number, each of which corresponds to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely serrated mesh. Neither of the two curves must ever cut across itself again, but it must bend back upon Itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times. 381 \^ The complexity of this figure will be striking and I shall ^ot ^^en try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three-body problem, and of all the Problems of dynamics in general, where there is no uniform integral and where the Bohlin series are divergent. Different hypotheses are possible. 1. We may assume that the group of points of two asymptotic _ curves Eq, or the group of points in the vicinity of which there is an infinite number of points belonging to Eq - i.e., the group E'o, the derivative of Eq" occupies the entire half -plane . We would then have to conclude that insta- bility of the solar system exists. 2. We may assume that the group E'o bas a finite area and occupies a finite region of the half-plane, but does not occupy it completely. Either one part of this half-plane remains outside of the meshes of our grid, or a "gap" remains within one of these meshes. For example, let Uq be one of _ these meshes bounded by two or more asymptotic curve arcs of the ^wo families Let us compile its successive consequents, and let us apply the procedure /390 presented in No. 291. Just as on page 145, let us formulate the following u„ u'„ u^, u;, u;, ..., k. If it is finite, the area E will represent one of the gaps which we just mentioned. It would appear that we may apply the line of reasoning employed in No. 294, and may conclude that this area must '^f ^J^'^^^^J^ °"^ of its consequents. However, this group E could be composed of a region of finite area and of a group located outside of this region whose total area^ would be zero. According to page 151, we may only conclude that E^ (the Xii^ consequent of E) includes E, and that the group Ex - E has — ^^^^J^^^^ same way, the groups E - E_x, E.^ - E_2X. •••. ^-n\ " E_(n+l)X will have area z^o Z area of a group, we meai the value of the integral J extended over this group), on thf other hand E_(,+0X is a part of E_,, . When n increases indefinitely. E_„x tends toward a group e including every point which is part of all the g;oups E_„, at the same time. The area of this group e is finite and equals that of E. Finally, e coincides with its A-^ consequent. 3. Finally, we may assume that the group E'o has area zero. It would then be similar to those "perfect groups which are not con- densed in any interval". 398 We may represent the different intersection points of the two curves in the following way. Let x be a variable which varies from -» to +", when the asymptotic curve of the first family MoAq is followed, from the point Mo up to infinity, and which increases by unity when we pass from one point to its fifth consequent ~ from Aq to A5, for example (to clarify this point, we shall assume that we are dealing with the conditions of the figure 382 shown on page 195). Let y be another variable which varies from +- to -» when the curve of the second family K^B^ is followed from the point M3 up to infinity, and which increases by unity when we pass from a point to its fifth consequent. The different intersection points of the two curves are characterized by two values of x and y, and each of them may be represented by the point on a plane whose rectangular coordinates are x and y. We shall thus have an infinite number of representative points of the /391 doubly asymptotic solutions in the plane. An infinite number of other pointl" may be deduced from each of these points. If the point x, y corresponds to an intersection of the two curves, the same will hold true for the points ^"i-i,^-hi; .r + 9., ;^' + 2; ...: .r-t-n, y -i- n, where n is a positive or negative whole number. In order to determine all the representative points, it is sufficient to know all those which are in- cluded in the region < x < 1, or in the region < y < 1. We would also like to note that the order in which the projections of these representative points will occur on the x axis will have no relation- ship with the order in which their projections will occur on the y axis ■nils results in the following. Let us consider several doubly asymptotic solutions. For t which Is negative and very large, they will all be very close to the periodic solu- tion, and they will appear in a certain order ~ some of them will be closer to, and others will be farther from, the periodic solution. All of them will then recede appreciably from the periodic solution, and -- for t which is positive and very large — they will all again be very close to It. However, th ey will then appear in an entirely different order , out of two solutions, if the first is closer than the second to the periodic solution for t = —, it may happen that for t = +- the first is farther away than the second from the periodic solution, but the opposite could also occur. We have pointed this out in order to illustrate the great complexity of the three-body problem, and to show how many different transcendents out of all those which we know must be considered in order to solve it. Heterocllnous Solutions 399. Do heterocllnous solutions exist? As far as we can determine, if there is one of them, there is an infinite number of them. 383 \^ Let Mo be a point belonging to a periodic system. Let MqAq and MqBo /392 be two asymptotic curves bordering upon this point Mq --one belonging to the first family, and the other belonging to the second family. We have just seen h^ tLse curves intersect, so that the doubly asymptotic, homo- clinous solutions may be determined. Now let Mb be a point belonging to another periodic solution. Let M'oA'o, M'uB'o be two asymptotic curves, M'A'q belongs to the first family, and M'B'o belongs to the second family. Let us assume that M'qA'o intersects MqBo at Qq. This intersection will correspond to a doubly asymptotic, heteroclinous solution. However, if these two curves intersect at Qo , they will also intersect at an infinite number of points Q^, the consequents of Qo- I shall state this precisely. For example, I shall assume that the periodic system of which Mq is a part is composed of five points Mq , Mi M2, M, Mu Then the fifth consequent of an arbitrary point of the curve MqEo will still be located on this curve, and in general — if Qo is on this curve ~ the same will hold true for its ntk consequent Q^, provided that n is a multiple of five. In the same way, let us assume that the periodic system of "hich M'q is a part is composed of seven points. Then, if Qo is on the curve M oAq, the same will hold true for its nth consequent Q^, provided that n is a multiple of 7. Therefore, if the two curves have an intersection at Q , they will still have an intersection at Q^, provided that n is a multiple of 35. Let QoHoQn be an arc of MoBq, and let QoKoQn be an arc of M'q A'o . Due to the fact that these two arcs have the same end points, together they will form a closed curve. We may pursue the same line of reasoning as in No. 396 for this closed curve. We shall find that, if the two arcs have no other point in common except their end points, this closed curve does not have a double point, and defines an area which is similar to the area ag given in Nos 395 and 396. If the two arcs have points in common other than their end points, we may obtain two other arcs which are part of the two arcs QoHoQn. QoKoQn which have only their end points in common and which define an area similar to ag. The same line of reasoning as was employed in Nos. 395 and 396 may be used for this area ao, and we will find that an infinite number of other points may be obtained on each of the two curves, between two arbitrary I39J points of intersection with the other curve. This line of reasoning shows that if there is one heteroclinous 384 ■ solution, there is an infinite number of them. 400. If there is a heteroclinous solution, the grid of which we spoke in No. 397 must be still more complicated. Instead of a single curve MqAq bending back upon itself without ever cutting across itself, and intersecting the other curve MqBq an infinite number of times, we shall have two curves MqAq , M'gA'g which must intersect MqEq an infinite number of times without ever cutting across each other. In No. 397, we defined the group E'g with respect to the point Mq and to the asjrmptotic curves MqAq, MqBo. We may also define a similar group with respect to the point M'g and to two asymptotic curves M'qA'o, M'q^O' If there is no heteroclinous solution, these two groups must be out- side of each other; therefore, they cannot occupy the half-plane. If, on the contrary, there is a heteroclinous solution, these two groups will coincide. It may be seen that the existence of such a solution — if it could be established — would provide an argument against stability. In Chapter XIII we studied the series of Newcomb and Lindstedt, and we showed in No. 149 that these series cannot converge for every value of the constants which they contain. However, one question remains in doubt. Could these series converge for certain values of these constants and, for example, could it happen that the convergence occurs when the ratio 111. is n2 the square root of a commensurable number which is not a perfect square (see Volume II, page 104, in fine). However, if a heteroclinous solution does exist, the answer to this ques- tion must be in the negative. Let us assume that for certain values of the ratio — the series of Newcomb and Lindstedt converge, and let us return to our method of representation. The solutions of the differential equations ni which would correspond to this value of — could be represented by certain trajectory curves. The group of these curves would form a surface, having the same connections as the torus, and this surface would intersect our /394 half-plane proceeding along a certain closed curve C. The group E'g which we just mentioned would have to be completely out- side of this curve, or completely inside of it. Let Mq and M'o be two points belonging to two different systems. If Mq is within the curve C and M'g is outside of this curve, the group E'q with respect to Mq would have to be entirely within it, while the group E'q with respect to M'q would have to be entirely outside of it. These two groups could not have any point in common, and no doubly 385 asymptotic, heteroclinous solution could exist, proceeding from Mq to M'q . If we admit the hypothesis advanced in Volume II, page 104, which X have just presented — i.e., if the convergence occurs for an infinite number of values of the ratio ^, for example, for those whose square is n2 commensurable ~ there would be an infinite number of curves C which would separate the points belonging to different periodic systems. This hypothesis is incompatible with the existence of heteroclinous solutions Cat least if the two points Mq and M'q which we are considering, or the corresponding periodic solutions, correspond to two different values of the number "_]_.) Comparison with No. 225 401. Before trying to present examples of heteroclinous solutions, we shall return to the example of No. 225, where the existence of doubly asymp- totic, homoclinous solutions may be illustrated. We set . F =/)-(-9'— ■Jiiisin*- — (iE^'e''] /396 In order to obtain the doubly asymptotic solutions, we must determine the Intersection of these two asymptotic surfaces. It will be sufficient for us to equate the two values of p and the two values of q. L et us set J„ ' + /» U :r^1 log/. We shall obtain ^ real part [J/e-''«+'.f] = o, ^- real part [Ue~a"*-i^]^-o, 387 K^ or, setting J = pe^", where K is a whole number. This is the equation of doubly asymptotic solutions. In reality, this equation provides us with two different solutions, one corresponding to even values of K, and the other corresponding to odd values of K. 402. We may be surprised at not obtaining more than two doubly asymp- totic solutions, when we know that there is an infinite number of them. The following approximations should provide us with no more than a finite number of doubly asymptotic solutions. How may this paradox be explained? In the preceding sections we saw that the different doubly asymptotic solutions correspond in an infinite number to different intersections of a certain arc AoHqCo with the different consequents of another arc AqKoCo. Let us assume that the first of its consequents which encounters AoHqCo is the consequent of order N. The number N will clearly depend on the con- ^397 stant e, and the smaller the constant is, the larger it will be. It will become infinite when e is zero . If we develop in powers of e and stop at an arbitrary term in the ex- pansion, it is as though we regarded e as being infinitely small. The arc AqHoCq no longer encounters the consequents of infinitely lar^e order of the other arc AqKoCo, and for this reason we have not analyzed the majority of the doubly asymptotic solutions. Examples of Heteroclinous Solutions 403. Let us try to generalize, and let us set F^-F<,-)-£F,. Fq is a function of p, q and y, and F^ is a function of p, q, x and y. These two functions are periodic, both in x and y. Let us consider the curves Fo = const. (1) 388 in which we regard p as a parameter, and q and y are regarded as the coor- dinates of a point. Out of these curves, those which must draw our attention are the ones having double points. These double points correspond to periodic solutions of the canonical equations when we assume that e is zero and that F may be reduced to Fq. We have a double infinity of curves (1) whose general equation is and which depend on two parameters p and h. I have just stated that the most interesting ones are those which have a double point, especially in the case in which some of these curves have two or more double points. It is in this case that we shall encounter /398 heteroclinous solutions. Just as in No. 225, let us try to formulate the function S of Jacobi, and let us set S =-- So + tSi-He'Sj-^- The function Sq may be formulated immediately. We shall have '/So rfSo where q is a function of y defined by equation (1) and depending on two para- meters p and h. We then obtain dp ~dx '^"dq -dy-^^''-"- ^2) dFg dFn We regard p as a constant in -r — ' -r— and Fi , and we replace q by its value obtained from equation (1). Equation (2) Is therefore a linear equa- tion with respect to the derivatives of Sj, whose coefficients are the given functions of x and y, which depend in addition on the parameters h and p. Since F^ ig periodic in x, I shall set F, ^: J:'I',„e'"'^, where ^^ only depends on y, just like the derivatives of Fq. In the same way, I shall set 389 and the function li-m will be given by the equation . ./P„ , di'o d.\,, (3) dp dij dy whose coefficients are the given functions of y. This equation may clearly be integrated by quadratures. Let us try to determine our asymptotic surfaces in this way. We must first choose the constants h and p so that the curve (1) has a double point. In addition, I shall assume that these constants are such that two real values of q correspond to each value of y (this is what occurs in the example presented in No. 225). These two values of q are periodic functions of y, which become equal /399 to each other at the double point ~ for example, for y = yo- Just as in No. 225, we may also assume that these two values of q are the analytical extension of each other. The function q then seems to us to be uniform in y and periodic of y period 4tt such as the function sin ^. This uniform function will take the same value for y = yo and y = yo + 27t. If we had several double points, instead of one, we could still regard q as a uniform function of y of period 4tt, if the number of double points were odd. On the other hand, if this number were even, we would have two values for q which would not be interchanged when y increased by 2tt and which could consequently be regarded as two different uniform functions of y, having 2tt for the period. In order to formulate our ideas more clearly, we shall assume that we have two double points corresponding to the values yo and yi of y. As a result, for y = yo and for y = yi, equation (1) must have a double root, since the two values of q coincide, and consequently -^ must vanish. Equation (3) is a linear equation with a second term, whose Integra- tion is similar to the integration of an equation without a second term, and consequently similar to the integration of the following equation 390 i^'Loo.,,f9.f =0 (4) dp dq dy from which we have The function 9 thus defined is a holomorphic function of y for all real values of this variable, except for the values y = yo, y = yi, which correspond to the double points. For these values, the function 6 — which plays a role similar to that of t = tan J in No. 226 — becomes zero or infinite. We then obtain /400 J iip where C^ is an integration constant, from which we have J dp In order to obtain equations of asymptotic surfaces, we may write assigning suitable values to the integration constants. Let us first neglect e. We shall set S = Sq, and we shall assign the values corresponding to the curve which has two double points to the con- stants h and p = pg. With this approximation, the differential equations have the following as periodic solutions (6) P'-P'>< q-iu r-=yu where yo , qo; yi , qi are the coordinates of the two double points. In order to represent our asymptotic surfaces, we may take a point in four-dimensional space, whose coordinates are 391 (/? -1^ a)co5.T, (p--- a)-' rro; Nj; p^^po, g^-q", y>yy\; Nti p-^po, q = q', y>y\\ It can be seen that the two surfaces N^ + N'^ and Ni^ + N^ coincide with this degree of approximation, just like the two surfaces N2 + N'2 and N3 + N'3 . Let us proceed to the following approximation, and let us set /402 O ^=- OQ -T- £ 5| . In order to define S^, we must choose the constants Cm. For the layers Nj and N'l, we must choose these constants so that the functions \^ have a regular behavior for q = q', y = yo- We need only refer to the analysis given on page 466, Volume II, in order to understand that this condition is sufficient for completely determining its constants. I shall call Si,i the function Si which is thus determined. For the layers N2 and N'2, we shall choose the Cm's so that the ^s are regular for q = q", y = yo, and we shall call Si 2 the ftmction Si which is thus determined. For the layers N2 and N'2, we shall choose the C^'s so that the i^^'s are regular for q - q", y - yi. For the layers Ni^ and N'^, the %^s must be regular for q = q', y = yi. We shall designate the two functions Si which are thus determined by Si ^3 and Si i^. The equations of our four surfaces thus become Ni-(-N;; /' dS,., p r/S,., ?-?^^ dy- p ■■ ^/'» + ^ dr ' rfS, 3 ^ = ^-^-^ dy' p- = ^» + ' dx ' 1-1^' dy- (8) However, we should note that the function Si,i, for example, has a regu- lar behavior for y = yo , and has an irregular behavior for y = yi . As a result, our equations cease to be valid, even as a first approximation, after the value yi is exceeded. In order to provide a better illustration of this, I shall confine 393 myself to the following remarks. Let y' and y" be two values of y such that Jo r»'o; N', et N', ; >'o> J >ri — »''• N, et N4; jo-t-it>;'>ri; N', etN;; }'i>r>yo- For example, if the surfaces Ni and N\ coincide, the intersection will correspond ^o a heteroclinous , doubly asymptotic solution -^i^^^^^i^J^^^, very close to the periodic solution (5) for t = -". and very close to the periodic solution (6) for t = +". In order to determine this intersection, let us compare the equations of Ni and N^ dS, 1 '^Sn and the intersection will clearly be given by rf(S,.i — Si^) _„ (9) dx Si.i - Sm is a function of x and y. which may be developed in positive and negative whole powers of ¥e". The fact that it is a periodic function of x is important to us. It there- ^re has at least a maximum and a minimum. Equation (9) therefore has at ieastl^o solutions, which means that there are at least two heteroclinous solutions . In the same way, it could be shown that there are two solutions corres- ponding to thrinter;ections of the surfaces N, and N'2 two corresponding ?o the surfaces N2 and N'a. and two corresponding to the surfaces N3 and N^. The preceding analysis does not yield the homoclinous solutions. 394 Vw^ 404. For example, let us set Fo=— p — o'-H2nsin'^^:^sin« -^~-^' , 1 a F,= ncosarsin(y— 7<,)sin(^— ^Xi). The periodic solutions (5) and (6) toward which the heteroclinous solutions /404 strive for t = -«> and t = +«= are then X ~^ t, p^-q^-0, y ya, X ---. C, p^- q -. o, y -.7,. It will be noted that, for y = 0, F may be reduced to -p - q^. Therefore, for y = 0, the function F depends only on variables of the first series p and q, and does not depend on variables of the second serjes x and y. The function F therefore has the form considered in Nos. 13, 125, etc. Nevertheless, we shall not be content with this example, which proves that the canonical equations having the form considered in No. 13 can have heteroclinous solutions. The two solutions (5) and (6) both correspond to the same value of the quantities -7- and -7^ — i.e. , dx ily (!c ' (It However, these quantities -rr» "TT are nothing else than the numbers which were called n^ and n2 above. Therefore, we find that doubly asjrmptotic solutions exist, which come infinitely close to two different periodic solutions for t = -<» and t = -H». However, these two periodic solutions correspond to the same values of the numbers n^ and vli. Therefore, I shall formulate another example, in which we shall deal with equations having the same form as those presented up to No. 13, and which have doubly asymptotic solutions coming arbitrarily close to two period- ic solutions which are not only different, but correspond to different values ni of the ratio — . n2 Unfortunately, I would like to show that these solutions exist for values of y which are close to 1, but I still am not able to establish the fact that they also exist for small values of y. 405. We shall take two pairs of conjugate variables 395 or /405 ^1. yi'y ^». y*< by setting This change in variables does not alter the canonical form of the equations. We shall set F = F<,(i — |Ji)-HftF,. We shall assume that Fq is a holomorphlc function of xi and X2, inde- „2 1 pendent of yi and ya, and that for xi = _, X2 = y, we have — = O, -J = M dx, axi 1 a^ , We shall also assvtme that for xg = y, xi = -y we have dxi ' dx, I shall assume that a < 1 holds for the quantity a. It follows from these hypotheses that, if we set y = 0, from which we have F = Fq, our equations will have two special periodic solutions. The first solution, which I shall call a, may be written a' > ^i = — > ^j= -' ri = '. ^» = o. Ji = ac05/; rn = as\ne, ?i=i, t,,=o. The second solution, which I shall call a', may be written ar, = - , ^« = — ' ri = o. n = '. $1 = 1. T,i = o, {j=acost, T), = a sin t. The first corresponds to ni = 1, n2 = 0, and the second corresponds to m = 0, n2 = 1. These two periodic solutions do not correspond to the same value of the ratio — . n2 In order to define Fj , I shall set 5i = I — rcosco, 5j = i — rsinio, 396 assigning a value which is essentially positive to the variable r. I shall then assume that (due to the fact that p is a positive, /406 very small quantity) we have the following for r > p F,= -^ ^ f-'-^l _ (r-i)' ^ 'H^ (1) r' where iJj(u)) is a function of to, which is regular for every real value of co, periodic with the period 2it, and finally which vanishes with its derivative for 0) = and for ui = ^. Since the function (1) would be infinite for r = — i.e., for 5i = E,2 = 1 — I shall assume that for r 4p, the function Fj takes on arbitrary values, in such a way that it nevertheless remains finite and continuous, as well as its derivatives of the two first orders. It may be readily verified that for y = 1 — i.e., for F = Fi — our equations still have two periodic solutions ct and a'. For the first of these solutions, we have o) = 0, and for the second we have o) = -S-. It may be iiranediately concluded that for every value of u . our equations will have these two periodic solutions. 406. We shall now integrate our equations in the case of y = 1 (assuming at least that r constantly remains > p) . If we first assumed that e = 0, we would be dealing with the problem of central forces, and the integration would be immediately possible, This is hardly true in the general case. The Jacobi method leads to the partial differential equation where h is a constant. Let us set where k is a second constant, and we shall have The general solution of our equations is therefore /407 397 /rdr C2) /a> ^ r 2J!L -_ =. k-, (3) where h' and k' are two new constants. We shall obtain our two periodic solutions a and a' , assigning the following particular values to the constants j,j , — ■i Let us assume that we would like to employ equation (2) to define r as a function of h' + t. If we assign values which are close to zero and 4 to the constants k and h. r will then be a periodic function of t + h'. We shall set u ^ n{t + A'), Where the number n is chosen in such a way that r is a P"^^^^^ J'^f '"J^f u with the period 2.. This number n. which is a type of mean motion, will naturally depend on the constants h and k. In the same way. ^ will be a periodic function of u. For k = 0, we simply have ,• r : I -': if ill COS u. L07 We therefore have two periodic solutions a and a' which will be represented by two closed curves, if we may regard the ^'s and the n's as tScoordinates of a point in four-dimensional ^P^^ J^/^^J^^ °^ ^"'^ faces pass through each of these curves -" °"%^^l°Sf ^^ J° f^l HH \^e /408 rui^^-frcerrin^efdrfa^^^^^^^^^ e is neglected. In order to obtain the equations of these surfaces, it is sufficient to 398 Ct2 assign the values zero and — to k and h. We thus have -■(?! -O'ii -+-(!.- Oil = /-/a»—(7--:r7)T,