1909.] LOVETT— PROBLEM OF THREE BODIES. 113 



If in addition we have 



H^ - A^B^ = H^ - A^B^ = H^ - A^B.^ = some constant, 



(9) k\^ V 



say -2 1- ^>hCi ' 



the function Q may be written 



( I o) S = ^ E ^^Y- + c' '^^r^ — ^ ^Q, + h. 



Finally, no noting that the equations (7) lead to the relation 



(II) ( X) ^>h ) Z '^r- = '«1 W2P,/ + m^m^^ + W/gWj/Jg^' , 



we have Q^ in the well-known form 

 k 



which is thus made to appear as the unique case of conic section 

 orbits for all initial conditions under forces varying as the masses 

 and a function of the mutual distances. 



It may be observed here parenthetically that if a similar study 

 be made for the cubic a first condition will be found to demand that 

 the orbits be defined by equations of the form 



(13) aiXi^ -^^i^iXcyi — T^aiXiyi^ — hiyi^ — Ci = o, (i:=i,2, 3); 



the remaining analysis of the problem offers no difficulty. 

 3. Writing 



(H) |^'=«„ 1^.',, 



t I 



the function (2) becomes 



<'5) iS'<"'+3)/IS'H'^ 



