2l8 MOTION OF A VARIABLE SYSTEM. [389. 



tions for all the particles. The left-hand member of the result- 

 ing equation is again dftymiP. In the right-hand member we 

 find, besides the term ^(Xdx+ Ydy + Zdz), such terms as 



The coefficients of X, //,, vanish only when the conditions 

 (i) are independent of the time, for then the differentiation of 

 these equations (i) gives 



In other words, in this case the constraining forces do no work 

 during the actual displacement of the system, as they are all 

 perpendicular to the paths of the particles, and we find equa- 

 tion (6). 



If, however, the conditional equations (i) contain the time, 

 their differentiation gives 



and we find in the place of equation (6) : 



Ydy + Zdz) -\h-pfy -. (7) 



389. If the conditional equations do not contain the time, and 

 if, moreover, there exists a force-function U for all the forces, 

 equation (6) can be put into the form 



which gives, by integration, 



^\m& - ^mv* = U- 7 , (8) 



or, by putting U= V, 



r+F=7- +F . (9) 



This equation expresses the principle of the conservation of 

 energy. 



It should be noticed that, even when there exists no force- 

 function, the elementary work ^(Xdx+ Ydy + Zdz) is a quantity 

 independent of the co-ordinate system, and the sum of these 



