1909.] LOVETT— PROBLEM OF THREE BODIES. 117 



-\v»i — I 



2v{ 



<40) /;. = ^- (f/ -f i)"^' sin {n, tan-^ ^, + /3,) ; 



■or finally the equations of the orbits become 



(41) Csin(/r^, + /3^) = 7,; 



the corresponding form of the force-function may be written down 



without difficulty. 



If we note that for three equal masses the relations (7) squared 

 give 



(42) Pi/ = 4(^i^ + ^/)+^fcS U^ = 123, 231,312, 



we see that the solution (41) for m = 3 is also a solution of the 

 problem of three equal masses under forces varying as the masses 

 and the cube of the distance. 



(c) Let (^i contain both ^i and Vi, while \pi is a function only of 

 riiZi; the condition (26) becomes 



(43) 2( I - ;/.)(^. + ;/.5,<^,,^ -f r.<^,. ,^ = o ; (/ = i , 2, 3) ; 

 whence it appears that <^i must have one of the forms 



(44) z, - ^\i:y r, ^\jk)^ 



in case Ui is unity an arbitrary additive constant may be appended 

 to each of these forms. Since $» and ^i are arbitrary functions we 

 have here an infinitude of problems. Considering the second of the 

 forms (32) a little further, the equations (25) become in this case 



<45) 



\ i / 



