Integral of the Problem of Three Bodies. 



159 



The Jacobian integral o£ the system is obtained by 

 multiplying the first equation o£ (1) by k, the second by if, 

 the third by i, adding, and integrating. This gives 



£» + 0» + P = 2Xl-C 



(2) 



where C is a constant of integration. 



Now these equations can be converted, either by a contact 

 transformation, or by the method of variation of constants 

 of integration, into the following Hamiltonian system : 



dq r 



£H dp r "dH f a . 



5 ^=~Wr (, " =1 ' 2 ' 3) 



(3) 



~dp r dt 

 where H = constant is the Jacobian integral, and 



^! = mean anomaly in the instantaneous ellipse, 



^ 2 -= angle between line of nodes and axis of #, 



^ 3 = angle between line of nodes and line of apsides. 



pi = square root of semi-major axis, 



p 2 = p3 cos i, where i = inclination of the orbit, 



p 3 = square root of semi-latus rectum. 



If /jl is not too great, H may be expanded in powers of it. 



H==H + / *H 1 + /* 2 H 8 + .... (4) 



and 



H =- ; 



n 



Now the Hessian 



a 2 H 



B 2 H 

 dpi Bpi 



*Pi 2 



a 2 H 



a a H 



P2 



(5) 



= 0. 



This happens to be inconvenient, but the difficulty may 

 be avoided by using H 2 in place of H, when the Hessian 

 will not vanish. 



2. Poincare's Theorem, 



The proof here attempted follows, in a condensed form,, 

 that given by Whit taker *. 



If possible, let <I> be an integral, one-valued, regular,, 

 expansible in powers of /jl. and periodic in q l9 q 2 , q s . 



Then 



<S> = <S> + A i<S> 1 + A i*<S> a + 



• Op. cit. 



