OF THE MOTION OF A SYSTEM. 23S 



Sy = -f cy, , 6y = \- by ^ , . . . 



so that '^x may be made to disappear from the variations 

 of the mutual distances, f»f ,-* - , as in article 323, and 

 from the values of the forces depending on these distances. 

 We shall then have, if the system is free from foreign in- 

 terference, by making ^x =0, 0' i= 2/w. < xdi {— — j — yd 



^d^ ^ ^N 7 ^^(^py __ Q^) ^f . and by taking the fluent, 



ez= Swi ^ y~'^ ^ .— 4- S/m (Py—Qx) dt ; and in the same 

 dt V -' '' 



manner, taking c' and c" two other constant quantities, 



c'= S»i-^^5=^^. -^ + S/^ {Pz--Rx) dt, and 

 d^ ^; -^ 



If the system is only subject to the mutual actions of its 

 parts, we have Sz/i {Py—Qx)—0', Sm {Pz—Rx) — 0, and 

 ^m iQz — Ry)—0, as has been shown in article 323: and 



m {^ X ~~ — y — ) — is the rotatory power of the finite force 



of the body »j, reduced to the plane of x and y, and tend- 

 ing to turn the system round the axis parallel to z ; conse- 

 quently the integral 2m ( — ^-^^ — )— is the sum of the 



rotatory powers of all the finite forces of the bodies of the 

 system, with respect to the same axis ; and this sum is 

 shown to be constant: and in the state of equilibrium it 

 vanishes : so that there is here the same difference, be- 

 tween the conditions of motion and of rest, as \vith respect 

 to the forces parallel to any given axis. In the case of 

 the natural relation of the forces, this property implies tho 



