234 CELESTIAL MECHANICS. I. vi. 24. 



constancy of the areas described by the revolving radii in 

 given times ; but this constancy is not observed in the case 

 of any other supposed relation. 



[Scholium. The definition of rotatory povrer ought 

 properly to have been premised to this demonstration, but 

 its nature is so purely speculative, that it was thought un- 

 necessary to anticipate any part of the important investi- 

 gation of rotation on this occasion; it is already as intelli- 

 gible as there is any reason to desire : especially consider- 

 ing the unavoidable confusion attending the idea of a 

 finite force as possessed by a moving body, which is almost 

 incompatible with the true conception of force, as a cause 

 of a change of motion.'] 



338. Theorem. The sum of the fluents 

 of the finite forces of a system, multiplied by 

 the fluxions of the paths described, is always 

 a minimum, and vanishes in the case of equi- 

 librium. 



By taking the variation of the function ^fhKpds, which is 

 here considered, we have ^^/m(pdszz'£fm(pdds + Xfmds^(p: 



ds V \dt dt 



now 



^ ddz) (265); consequently ^ifm^dszzX ^(^ 8a;+ ^ 



g, +gs.)-xy. { ,. d (^-|.£) +a, d (Ij)-^a. d (^; 



.t) \ +-LrmMs. since d \ ^(^8;c + ^Sy + ^8z) \ = 

 v/ J ^ I V \dt dt dt y J 



— .-T- d^j-f 8xd (-1^.-— ). .. Now the terminations of 

 V dt ^ V dt ^ 



the curves, described bv the different bodies of the 



