ON CERTAIN GENERALIZATIONS OF THE PROBLEM 

 OF THREE BODIES. 



By EDGAR ODELL LOVETT, 



The Rice Institute, Houston, Texas. 



(Read April 23, /pop.) 



The object of the following note is fourfold: first, to determine 

 all the problems of three bodies in which the bodies describe conic 

 sections, under central conservative forces, whatever be the initial 

 conditions of the motion; second, to specialize the preceding solu- 

 tions, so as to single out those in which the force-function contains 

 only the masses of the bodies and their mutual distances; third, 

 to generalize the latter group to the case in which the orbits are 

 the most arbitrary possible; fourth to generalize the last to the 

 case in which the functions defining the orbits appear in the poten- 

 tial function. 



I. If three given particles {ri,0^;mj, {r^,Bo;ni^), {rz,Oz',ni^) 

 describe, under central censervative forces, three given coplanar 

 curves whose equations in polar coordinates referred to the center 

 of gravity of the system are 



(i) /i(^,^i)=o, f,{r,,e,)=o, f,(r„e,)=o, 



the forces are derived from a potential function which may be 

 written in the form^ 



^ 4-.|(l:)'-(;,l:)' 



(2) ^=2 1^; — w, 



' On employing the usual substitutions the form given follows immediately 

 from Oppenheim's solution in rectangular coordinates. See his memoir in 

 the third volume of the Publications of the von Kuffner Observatory. 



Ill 



PROC. AMER. PHIL. SOC, XLVIII. I9I H, PRINTED JULY 6, I9O9. 



