204 ON FARADAY'S LINES OF FORCE. 



Let us now consider the conditions of the conduction of the electric 

 currents within the medium during changes in the electro-tonic state. The 

 method which we shall adopt is an application of that given by Helmholtz in 

 his memoir on the Conservation of Force*. 



Let there be some external source of electric currents which would generate 

 in the conducting mass currents whose quantity is measured by a,, 6,, c, and 

 their intensity by a,, /J,, y r 



Then the amount of work due to this cause in the time dt is 



^ /J/( a J a > + && + c 7) dxdydz 

 in the form of resistance overcome, and 



tor dt I 1 1 



in the form of work done mechanically by the electro-magnetic action of these 

 currents. If there be no external cause producing currents, then the quantity 

 representing the whole work done by the external cause must vanish, and we 

 have 



dt 1 1 I (a,a, + 6^8, + c,y,) dxdydz + ^ I J I (a^ + b& + c,y ) dxdydz, 



where the integrals are taken through any arbitrary space. We must therefore 

 have 



a a o, + b& + c,y, = j t (a.0,, + &,& + c,y.) 



for every point of space ; and it must be remembered that the variation of 

 Q is supposed due to variations of a,, $,, y,, and not of a,, 6,, c,. We must 

 therefore treat a*, 6,, c, as constants, and the equation becomes 



In order that this equation may be independent of the values of a,, 6,, c,, 

 each of these coefficients must = ; and therefore we have the following 

 expressions for the electro-motive forces due to the action of magnets and 

 currents at a distance in terms of the electro-tonic functions, 



1 rfo, 1 dfr 1 dy 



"tor'dt' "tor dt' ~ 47T dt' 



* Translated in Taylor's New Scientific Memoirs, Part n. 



