112 LOVETT— PROBLEM OF THREE BODIES. [April 23, 



where c is the constant in the integral of areas, that is, 



dt' 



(3) ^ = Z) ifh^i 



i=\ 



In case the orbits are described independently of the initial condi- 

 tions, Oppenheim has remarked that it must be possible to throw the 

 function P into the form 



(4) P = P,-^h, 



where /i is a constant independent of the parameters which enter 

 P^ ; if such a decomposition of P is impossible, the motion takes 

 place only for special values of the initial constants. 



When the orbits are conic sections the equations (i) become 



(5) /t(^'i> Oi)=r^i(Ai cos- Oi -\- 2Hi sin di cos Oi 



-{-Bi sin^ Oi)-\- 2ri{Gi cos di + Fi sin 9i) — Q = 0, (i= i, 2, 3.) 



If the corresponding functions 



dr.' ^'dr: r.dd. 

 t t t 1 



are constructed, and substituted in the form (2) the latter becomes 



(6) Q= 



E m^{{m - AByi + 2 \_{HK - Bfi^ cos B, 

 c' + {fffi-A,F;) sin '^,+/^^+ ^/-f {A, + B^) Q 



^ { i: ^«. [(G^ cos e, + R sin e,)r^ - Q^' 



this is the most general form of potential function giving rise to 

 conic section trajectories in the problem of three bodies under central 

 conservative forces. 



2. From the relations 



(7) mimjpij^ = ini(mi -f nij)ri- -f mj(mi -f mj)rj~ — nik-n", 



1/^=123,231.312, 



where pij is the distance between the bodies (vi, Bi] mi) and 

 {tj, Oj] fHj), it follows that if Q is to be a function of the masses 

 and mutual distances alone we must have 



(8) Fi = Gi = o, (i=i,2, 3). 



