TO THE THEORY OF ELECTRICITY. 77 



through the points P, P lf then will the value of V at any point 

 p, on L, be expressed by 



F.+0X; 



X being the distance Pp, measured along the line L, considered 

 as increasing in the direction PP, , and F , the given value of 

 V at P. For it is very easy to see that the value of V furnished 

 by this construction, satisfies the partial differential equation (a), 

 and is its general integral ; moreover the system of lines 

 L, L', L", &c. belonging to the points P, P', P", &c. on S, are 

 evidently those along which the electric fluid tends to move, and 

 will move during the following instant. 



Let now V+DV represent what V becomes at the end of 

 the time t + dt ; substituting this for V in (a) we obtain 



Q = dV dDV dV. dDV dV dDV 



dx' dx dy ' dy dz ' dz ' 



Then, if we designate by D' V, the augmentation of the potential 

 function, arising from the change which takes place in the 

 exterior forces during the element of time dt, 



DV-D'V 



will be the increment of the potential function, due to the cor- 

 responding alterations Dp and Dp in the densities of the electric 

 fluid at the surface of A and within it, which may be deter- 

 mined from DV D'Fby Art. 7. But, by the known theory of 

 partial differential equations, the most general value of DV satis- 

 fy ing (5), will be constant along every one of the lines L, L', L", 

 &c., and may vary arbitrarily in passing from one of them to 

 another : as it is also along these lines the electric fluid moves 

 during the instant dt, it is clear the total quantity of fluid in any 

 infinitely thin needle, formed by them, and terminating in the 

 opposite surfaces of A, will undergo no alteration during this 

 instant. Hence therefore 



(c); 



dv being an element of the volume of the needle, and da, d& l , 

 the two elements of A's surface by which it is terminated. This 



