OP AN ELECTRIC DISCHARGE. 3 



in which the signs of summation S refer to all the masses of 

 the system, and to their coordinates, forces, and velocities. 



In this equation the left-hand side is immediately integrable, 

 and so is the right side, if we regard the quantities x, y, z; 

 a/, y' z', &c., not as independent variables, but as w hat they really 

 are, all of them functions of one and the same variable, namely, 

 of the time t. We thus obtain 



^-^mv^=fX{Xdx-\-^dy-\-Zdz)+con?^i.', 



or, if we call the velocities at any time of commencement what- 

 ever Vq, Vq', Vq, &c., and take this time as the lower limit of the 

 integral, 



\y,mv^-\Zmv\=r*L{Xdx^-Ydy + Zdz), . . (2) 



The quantities which appear in these equations are of very 

 common occurrence in mechanics, and have on this account re- 

 ceived particular names. With regard to that on the left-hand 

 side, it is known that the product mv^ is called the vis viva of 

 the mass m ; and according to this ^nvif^ would express the vis 

 viva of the entire system. Here, however, only the half of this 

 quantity makes its appearance ; and as this will always be the 

 case in the following pages, and likewise occurs so in most me- 

 chanical investigations, we propose to call, as other authors also 



have done, the quantity -'Xmv'^ the vis viva of the system. 



The quantity on the right-hand side can, in the first place, be 

 brought to a more simple form. Let ds represent the element 

 of its path which the mass m describes during the time dt, and 

 of which dx, dy, dz are the three projections on the three axes 

 of coordinates; further, let S be the component of the total 

 force acting upon m in this direction, we then have 



^ds^r^Xdx + Ydy + Zdz', 

 the quantity ^ds is that which in treatises on mechanics is called 

 the quantity of work produced by the force acting upon m during 

 the time dt, bearing in mind that this work must be regarded as 

 positive or negative, according as the element ds is in the same 

 direction as S or opposed to it. 



Applying these names, the meaning of the above equation may 



B 2 



