OF THE MOTION OF A SYSTEM. 199 



. . . Thus, let mP, mQ, mR, be the motive forces which 

 impel the body m in directions parallel to the orthogonal 

 coordinates x, y, z; let 711 P\ 7n' Q, m' R', be the forces 

 belonging to m'; and let the time be t. The momentum 

 of m, reduced to the respective directions, will be m 



-7->wi-r^> and m -r- '. to this the force P, so far as it is not 

 at dt &t 



other <rise compensated, will add a momentum, which may 



be expressed by 7?^ * P' At, and which is obviously equal 



dx 

 to m A-Z-, since in the time At the momentum becomes 7n 

 at 



dx djz7 dx 



-T^ + mA -r- ; and wi ' P ' dt^md-rr : consequently the un- 



dr at dr ' "^ 



compensated force in the direction of x will he m P 



ddx ddx 



dt -r— [or more properly m P — -r-^; for it is nn- 



necessary to combine the idea of time with that of force in 

 estimating its comparative magnitude] ; and the same may 

 be shown with respect to the other forces concerned. We 

 have, therefore, from the principle of virtual velocities, 



that is 0= XmS^s (Z) (305), 0=mdx [~^p) + mdy 



From this general equation we may eliminate, by means 

 of the particular conditions of the system, as many of the 

 variations as there are of these conditions ; and then by 

 making the coefficients of the remaining variations vanish 

 separately, we shall obtain all the equations necessary for 

 determining the motion of the different bodies of the 

 system. 



