Integral of the Problem of Three Bodies. 161 



From equation (6) we have 



(Ho.tfJ + CH!, <J> )=0; 



^. e. 



3H 3$, JHo Wi^BHi^o 

 'dpi ' B^i "dp2 '■ d?2 ^Qi * dpi 



3Hi ago ag! ^ b^>o _ aHx ^ B3> 



d£ 2 " "bp* bqz * Bi>3 dps ' c^' 



whence, if <£ does not involve q z , we get, by substituting 

 for Hj and <E> l5 



Then, since this equation is true for all integral values of 

 m l5 ?n 2 , m 3 , we can find an infinite number of values of 

 wij, m 2 such that the left side vanishes. Then if B doesnot 

 vanish for these values, we have the bracket on the right 

 vanishing for all values of m 3 , and for the particular values 



of ?n l5 m 2 . Hence ^— ° = and we get 

 ops 



Hence the Jacobian ^, °' — ^- =0 for these values of 



0[J>i,Pi) 

 m>i, m 2 . But there are an infinite number of such values, 

 and the Jacobian is a continuous function. It therefore 

 vanishes for all values. Hence <E> is a function of H , which 

 we have proved not to be true : so that <I> cannot exist. 



Poincare shows that the case when B = does not affect 

 the result. 



Now we have assumed above that <I>o does not involve g s . 

 If it does, the whole argument is upset, and we cannot say 

 that <D> does not exist. In other words, if there are reasons for 

 supposing the existence of such an integral, Poincare's theorem 

 offers no objection. There cannot be two such integrals, 

 obtained independently, for in that case the problem could be 

 completely integrated, and therefore the problem in one plane: 

 which contravenes Poincare's theorem. On the other hand, 

 there may be two integrals, if one is the differential coefficient 

 of the other. 



Phil Mag. S. 6. Vol. 25. No. 145. Jan. 1913. M 



