162 On a Certain Integral of Problem of Three Bodies. 



3. The Nature of the Integral. 



(i.) If the above argument is applied to the problem of 

 attraction to two fixed centres, motion being coplanar, it will 

 be found that a similar result is obtained. Here H is a 

 function of p x only, and <I> may exist, provided <3> involves ij 2 . 

 But we know that an integral <3> can be obtained in this case,, 

 and consequently the presumption is that it exists in the 

 problem of three bodies as here considered. 



(ii.) Poincare's original proof of his theorem was on some- 

 what different lines *. The existence of a one-valued integral 

 means that two out of the six characteristic exponents (which 

 are equal and opposite in pairs) vanish. In one plane, there 

 are four, and Poincare- shows that they cannot all vanish. 

 But there is no argument against four of the six in our 

 case vanishing. 



(iii.) The writer has made attempts to find <£> by direct 

 integration, but without success. It is unlikely that integra- 

 tion in finite terms will be effective. 



(iv.) The integral obviously vanishes identically in one 

 plane. 



(v.) There is a particular solution of this problem which 

 has been discussed by Pavaninif. Here the body P oscillates 

 along the z axis, the two. masses M and /j, being equal. The 

 integral <3> must either coincide with the Jacobian integral, 

 or vanish identically when at, y, #, y vanish and M = yu,. 



(vi.) It has been proved by Lindstedt J and others that the 

 problem may be solved by trigonometric series involving three 

 arguments. This is not incompatible with Poincare's theorem, 

 since the series cannot converge in all cases. But what has 

 been said above leads us to suppose that one particular series 

 may do so. 



Now we know, from Poincare's paper already referred to, 

 that the problem admits of an infinite number of periodic 

 solutions (which reduce to circles when the disturbing mass 

 vanishes). The Lindstedt series will then become Fourier 

 series. Assuming the solutions to be stable, we have, in the 

 neighbourhood of each, other solutions which may be re- 

 garded as oscillations about them, and thus a gradual 

 transition in the nature of the curve. Now these periodic 

 orbits must exist for all values of the inclination, and as we 



* " Sur le probleme de trois corps et les Equations de la dynaniique," 

 Acta Math. xiii. (1890). 



f " Sopra una nuova categoria di soluzioni periodiclie nel problema 

 dei tre corpi," Ami. di Mat. (1907). 



X (i Ueber die allgemeine Forme der Integrale der Dreikiirperproblems," 

 Astr. Nachr. cv. (1383). 



