212 MOTION OF A VARIABLE SYSTEM. [378. 



it expresses the equality of the sum of the virtual works of the 

 effective forces to the sum of the virtual works of the external 

 and internal forces, for any infinitesimal displacement of the 

 system. The internal forces do not enter into this equation if 

 they occur in pairs of equal and opposite forces, as will usually 

 be the case. 



378. As there are no conditions, we may select for Ss the 

 actual displacement ds of every particle, so that the equation 

 (3) becomes 



^m(xdx+ydy+zdz) = ^(Xdx+ Ydy+Zdz}. 



The left-hand member is the exact differential ^ 

 = ^^mz> 2 . Hence, integrating between the limits o and 

 and denoting by V Q the velocity of the particle m at the time o 

 we find 



Ydy + Zdz). (4 



This is the equation of kinetic energy. The right-hand mem 

 ber represents the work done by the forces during the time /. 



379. If there exists a force function or potential U for th< 

 forces X, Y y Z, i.e. if these forces are the partial derivative 

 with respect to x, y, z of one and the same function U> the sys 

 tern is said to be conservative. We have then 



and the integration of (4) gives 



where 7 is the value of U for t=o. 



Denoting as usual the kinetic energy by T, the potential 

 energy U by V, this equation can be written 



7 T 4- V rnn <5t (f\\ 



* flT^ * Vv**8ll , \\J 



it expresses the principle of the conservation of energy. (Comp. 

 Art. 79). 



