42 Motions of a System of Bodies. 



directions of the axes of x, y, z, respectively, and by considering S 

 as the sign of integration. It may be remarked, that if the bodies 

 w, m^ &ic. receive finite changes in their motions in the indefinitely 

 small portion of time dt, by the actions of the forces P, Q, R, &:c., 



that (1), (2), (3) will be changed to D.^=P£?^, D.-^=Fc?^, kc. 



dv dy' dz „ dz' 



(10; D.J=Q*, Ji.±=Q.'dt, Uc., (2'); D.^=RA, D.^ = 



R'f?^, &LC. (3') ; D being the characteristic of finite differences. Hence 



we may find by the same reasoning as before used, D. 37= — yT j 



dY dt^mQ, dZ dtSmR , r ' 



D.-t7 = — ivj^) ^•'^f — — M — ' ("^0? which are mdependent oi any 



finite changes which the bodies receive from their reciprocal actions 

 in the instant dt; also if the bodies are subjected only to their mu- 



tual actions, SotP=0, SmQ=0, SmR=0; .'.as before D.-^=Oj 



dY dZ . d'X. ^^ dY ^^, 



D.--7T=0, D.-t: =0, whose finite integrals give -77= V, j7=» j 



dZ 



-t: =¥''; hence the same remarks concerning the motion of the cen- 

 ter of gravity apply as in the former case, when the bodies were only 

 subjected to their mutual actions. From what has been proved, it is 

 manifest that (4) are independent of any changes in the motions of 

 the bodies, and that whether they are gradual, or finite in an instant; 

 provided they arise from the mutual actions of the bodies on each 

 other. See Prin. cor. 4 to the laws of motion ; Mec. Anal. vol. i, 

 p. 259, Mec. Cel. vol. i, pp. 54, 70. 



II. When the bodies which compose the system are supposed to 

 revolve around a center of force situated at the origin of the co- 

 ordinates, and acted on by any other forces. . 



I shall consider all the forces except that which is directed towards 

 the origin of the coordinates as disturbing forces. Let each body, 

 (regarded as collected at its center of gravity,) and the forces which 

 affect it be reduced orthographically to the plane a?, 3/, (or fixed 

 plane,) as at p. 134, vol. xxli ; put r=the distance of m thus pro- 

 jected from the origin of the coordinates, t;=the angle made by r and 



c r"dt 

 the axis of x at the time ^, ^—1^ =the area described by r around 



the center of force in a unit of time, T=the intensity of the result- 



