ON FARADAY'S LINES OF FORCE. 197 



If a, /?, y be the electro-motive forces, p the electric tension, and k the 

 coefficient of resistance, then the above equation is identical with the equation 

 of continuity 



da, ,db t dc, 



j -- r -j r -y- + 4;rp = ; 

 ax ay dz 



and the theorem shews that when the electro-motive forces and the rate of 

 production of electricity at every part of space are given, the value of the 

 electric tension is determinate. 



Since the mathematical laws of magnetism are identical with those of elec- 

 tricity, as far as we now consider them, we may regard a, /?, y as magnetizing 

 forces, p as magnetic tension, and p as real magnetic density, k being the 

 coefficient of resistance to magnetic induction. 



The proof of this theorem rests on the determination of/ the minimum value 

 of 



where V is got from the equation 



dfV 



and p has to be determined. 



The meaning of this integral in electrical language may be thus brought 

 out. If the presence of the media in which k has various values did not 

 affect the distribution of forces, then the "quantity" resolved in x would be 



dV dV 



simply -j and the intensity k -y . But the actual quantity and intensity are 



y- (a. -/-} and a -/-, and the parts due to the distribution of media alone 

 K \ ax/ ax 



are therefore 



k \ dx) dx ' dx dx ' 



Now the product of these represents the work done on account of this 

 distribution of media, the distribution of sources being determined, and taking 

 in the terms in y and z we get the expression Q for the total work done 



