M8 CELESTIAL MECHANICS. I. V. 23. 



OziXm^x (^- P) + . . . , we obtain 0=:Xm (Sa-d 1^ + 



gy d T^ + Sz d -^\—lmdtvdv. Now d5 being the fluxion 



of the path of m, let d*' be that of the path of 7n\ . . . , and 



vdi=:ds, v'dt=ds\...; ds being ^ {dx'' + df + dz^): and 



as it has already been shown (266) with respect to any par- 



^ , , , dxgx + d?/Si/ + dzgz , . 



ticular body, that S(t;d5)=:d -^ , we obtain 



by adding the results for the different bodies, 2m8 {vds):=i 



Imd ^^_±Mi±^^, and the fluent, which is taken in- 

 d^ 



dependently both of the variation, and of the integration ex- 



^^ ^ . ^_^ , ^,_ dx^x -\- d2/^i/ + dz^z 

 pressed by 2, gives us z^djmvds-iz C + 2.m- — ; 



the variations of the ordinates being those which belong to 

 the extreme points of the curves to be compared. Hence it 

 appears, that when these points are supposed invariable, the 

 equation becomes X^/mvdszzO, consequently the quantity 

 ^Jmvds is a minimum. And this is the law of the least ac- 

 tion, as applied to the motion of a system of bodies, a law 

 which is evidently derivable, by mathematical considera- 

 tions, from the fundamental principles of equilibrium and 

 of motion. 



Scholium. It is also apparent that this law, combined 

 with that of the preservation of impetus, would affbrd the 

 equation (P) (317), which includes all that is necessary to 

 the determination of the motions of the system ; and it ap- 

 pears from the preceding propositions, that the same prin- 

 ciples are applicable to the case of a moveable system of 

 bodies, provided that the motion of its centre of gravity be 

 uniform and rectilinear, and the system be detached from 

 the operation of all foreign forces. 



