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



where Hi is an arbitrary constant; then the condition (26) becomes 



(29) ^i I 2«,^A - ^ {^'<f'.) I = o, (z" = 1 , 2, 3), 

 from which we conclude that 



(30) <^.. = a/r5'-i), 



ai being any constant. The expressions (25) in this case assume the 

 form 



(31) 



The determination of the form of Si from the equations (31) can 

 be affected perhaps most simply in the following manner: That if/i 

 is a function only of tiiSi amounts to saying that 



(32) ^. = -^-/.(S): 



substituting the partial derivatives of this function in one or the 

 other of the expressions (31) we obtain the following equation: 



(33) db T/a,v»-^^-«,v = «,^/;-( I + p)//. 



whence we have the ordinary differential equation 



(34) /' = p^- { ^'lA ± v/^/d,^ + 1 r - n^/n , f. = f; • 



The integration of the latter equation may be facilitated by the 

 substitution 

 (35) 2Vi = ni\og($i^ + i), 



under which (34) takes the form 



e"i — I 



Putting now 

 (37) n.fjaf^ = six\u., 



(36) i; =/^ =^ — =i^r= AV- - nH^. 



