128 



<p l and <p 2 denoting two arbitrary functions. Hence, according to 

 Fourier, we have, for the integrals of the equations (3), 



(6), 



and the solution of the problem is expressed in terms of these inte- 

 grals by (4). 



If now we suppose the cable to have one end at a finite distance 

 from the part considered, for instance at the point O from which x is 

 reckoned, and if at this end each wire is subjected to electric action 

 so as to make its potential vary arbitrarily with the time, there will 

 be the additional condition 



i=*i(0 and 



when 



(7), 



to be fulfilled. In the other conditions, (5), only positive values of 

 x have now to be considered, but they must be fulfilled in such a way 

 as not to interfere with the prescribed values of the potentials at the 

 ends of the wires ; which may be done according to the principle of 

 images, by still supposing the wires to extend indefinitely in both 

 directions, and in the beginning to be symmetrically electrified with 

 contrary electricities on the two sides of O. To express the new 

 condition (7), a form of integral, investigated in a communication to 

 the Royal Society ('Proceedings,' May 10, 1855, p. 385), may be 

 used ; and we thus have for the integrals of equations (3), 



(8). 



Lastly, instead of the cable extending indefinitely on one side of 



