23^ CELESTIAL MECHANICS. I. vi. 24, 



ratel,.0..4.(fX)_Pd.J.0=..{d(|.:) 



—Qdt^, and 0= I71A d {^^•^) - -Rd# |, exactly in the 



same manner as similar equations have been deduced from 

 (P) in article 322; and as it was inferred in that case, that 

 the motion of the centre of gravity must be uniform, so 

 now, if the system be only subject to the mutual attraction 

 of the bodies comprehended in it, since SwiP, SwQ, and 

 YmR are evanescent, on account of the reciprocality of 



action and reaction, we have czi2m— ; c'z:2m-j^.~ 



at V dt V 



and c"=i2w-T-.--; but w-^.' — zn wi® -— , which is the 

 dt V dt V ds 



finite force of the body m, reduced to the direction of x: 



consequently the sum of the finite forces of the system, 



reduced to the direction of any given axis, is constant, 



whatever may be the relation of the force to the velocity; 



aud the state of rest is distinguished by the disappearance 



of that sum. This result is common to every hypothesis 



respecting the relation of force to motion, but it is only 



in the natural state of this relation that the motion of the 



centre of gravity becomes uniform and rectilinear. 



337- Theorem. The sum of the finite 

 forces, tending to turn the system round any- 

 given axis, is constant, and vanishes in the case 

 of equihbrium. 



We may make again, in the equation {S\ 



y y 



