334 ELECTROKINETICS. [PT. II. CH. VIII. 



have /units entering at potential V A and emerging at potential V B . 

 The energy converted into heat in that portion of the conductor 

 will accordingly be 



(i) - 



But transforming the integral, 165 (i), by Green's theorem, 

 and taking the normal at A, B, and the surfaces of discontinuity 

 always in the direction of the current, 



J 



/*/*/* C ^ / O T7*> 

 / / / __ I n / Or 



- I Ma- X -5-) + :T l A '^-) + 5- 



JJJ (9#V oai J ty\3yj fa 



The volume integral vanishes by the equation 162 (5), and 

 in virtue of the surface conditions 164 (10) and (13), 



(3) J=-(E + 



The integral which is a minimum in the actual distribution of 

 current accordingly represents the heat generated in the conductor 

 in the unit of time. 



The equation (i), written 



EI= <W + H 

 dt 



is the equation of activity for steady currents. It may serve for 

 a definition of the magnitude of an impressed electromotive force, 

 as the rate at which energy is taken into the system per unit of 

 current in its direction. Combining with (i) the equation of 

 Ohm's Law, 



(5) Ri- 



we have 



(6) H 



