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



considering the case in which 



3 / ^2\ 3 



(-16) ^ ^ ^i-Z=_^=> 



\^^J f 3 12 — I 3 



we find immediately that 



(17) ^i = ~. ^/ = ^>.-t/; 



and on subjecting these values of Ui and Vi to the condition of in- 

 tegrability we have the following relations 



(18) ri-(f)i — i/'i" = some constant, say Ai", (i=i,2, 3), 



connecting the functions <j)i and xf/i. The construction of the func- 

 tions defined by the equations (17) and (18) is effected directly by 

 a simple integration which yields the result that under forces derived 

 from the potential function 



(.,. -?t..,(*^A')/||:.*|', 



three arbitrary masses nti describe the respective orbits 



(20) J tM dr, = ± v.. + ^„ {i=i, 2, 3) 



where the function \J/i is absolutely arbitrary, and the quantities At, ixi 

 are any two constants. 



In virtue of the relations (7) the function R contains only the 

 masses and mutual distances of the bodies, further, on writing the 

 function ij/r in the form 



(21) xj^i^^aiTi^ -{-<j)i(ri), 



where wi is an arbitrary function and ai any constant, it is evident 

 that R can be written in the form (4) ; whence it follows that the 

 three bodies under forces derived from (19) describe orbits of the 

 form (20) whatever be the initial conditions of the motion of the 

 system. 



