274 Heating of Conductors by Electric Currents. 



and by means of the preceding this reduces to 



Hence, if we assume, as experiment to some extent at least 

 entitles us, that 



(k.r.s)(1.2.S) = (c.p.q)(1.2.B)f'($) 



respectively, this further reduces to 



arf-vta.af/w+t/v*}. . . . m 



In the case where /'(S) is a constant, m say, and s is also 

 independent of S, we have, putting U =/($) + ■£,; V 2 , 



*^F =m V? ,U, .... (F) 



which has the same form as the ordinary equation for the 

 movement of heat in a heterogeneous conductor when there 

 is no internal heating, and hence, as in Section 3, we see how 

 results for this case may be generalized to include electric 

 heating. 



In any case, however, we see, comparing (A') and (G') } 

 that a solution for the steady state is given by 



and thus the theorems (a) and (/3) of Section 6 are immedi- 

 ately extended to heterogeneous conductors. 



If we now assume, further, more or less on the warrant of 

 experiment, that 



.p . q)(l . 2 . 3) = (c . Po . (/0 )(l . 2 . 3) 0(d) 



respectively, where the zero subscripts refer to the values at 

 the temperature zero, and put 



where §' is the maximum temperature, and the signs are to be 

 taken as in (6), Section 6, we get 



Again, if V denote the potential with the same surface- 

 conditions, but $ = throughout the conductor, 



Therefore F($)=A + BV , or definitely, as in Section 6 (7'), 

 (V 2 -Y 1 )F(^) = 2X{V ~i(V 1 + V 2 )}, . 



