1909.] LOVETT— PROBLEM OF THREE BODIES. 115 



4. In order to generalize certain of the preceding results further, 

 let us write the equations of the orbits thus 



(22) Si^fi(xi,yi) — Ci = 0, (f=i,2, 3) 



and the potential function as follows : 



(23) ^ = 2 1^ '"'^^'^ ^ "^'^1 [S ''''^'^'^' ^ -^ •^'^]"' 



the axes being rectangular about the center of gravity of the system 

 as origin. 



Let us consider now the case in which we have 



, , \pi^^qi- = <f>ii-Vi,yi,Si;ri), . , , /• > 



(24) , , , ri^ = Xi-+yi^ (t=i,2,3), 



[■Vipi + yiQi = \]/i{.Vi,yi, Si'jTi) ; 



from these it at once follows that 



\ri-pi = XirPi dzyiVri^(l>i — if/i'^, 



(25) (f=I,2,3) 



I rrqi = ynj/i qp x^^/ri^4>i — ^p^-. 



The condition of integrability applied to (25) gives 



(26) ..... . 



(^=1,2,3) 



an equation whose integration determines ^i when \pi is given, and 

 conversely. 



(a) In case the functions ^i and \pi contain only ri the equation 

 (26) becomes 



that is to say, it takes the form (18). Accordingly the equations 

 (25) assume the simpler forms 



(28) rcpi = xnj/i ± Xiyi, rcqi = yixj/i zp XiXi, 



whence, by integration, the orbits (20) reappear. 



{b) Let ^i be a function only of ri and \pi a function of niZi, 



