260 Mr. J. McCowan on the Heating of 



sources of heat; but he omits the consideration of such sources 

 in forming the differential equations of the motion. 



In the case of heating by an electric current there is not 

 only the thermal problem of finding the temperature at any 

 point, but also the further electrical problem of determining 

 the effect of the heating in changing the distribution of 

 electric potential by the alteration which it produces in the 

 conductivity. I am not aware that any solution of a problem 

 of this latter class has hitherto been given. 



In the following paper I have sought to discuss these 

 problems more or less fully. The steady state which in 

 general ultimately supervenes is considered in greater detail, 

 not only because the mathematical treatment is much simpler, 

 but also because it is of greater importance than the inter- 

 mediate varying state ; and for like reasons attention is chiefly 

 given to the case of homogeneous isotropic conductors, but I 

 have shown that many of the general results for these may be 

 at once extended to heterogeneous seolotropic conductors. 



It will be seen that the more general of the results at which 

 I have arrived depend on and, in fact, are an almost immediate 

 outcome of a remarkable transformation of which the thermal 

 equation is susceptible even in its most general case, and by 

 which the problem of electrical heating is immediately reduced 

 to the ordinary case where there is no such heating. 



It has been my object to develop the general theory rather 

 than to seek results specially suitable to practical applications ; 

 but it will be seen that many of the results suggest electrical 

 methods of experimentally examining the thermal character- 

 istics of substances, and that the maximum theorems may be 

 a useful guide where it is desired to produce high temperatures 

 by electric means. 



1. Formation of the General Equations for the case of 

 Heterogeneous Isotropic Conductors. 



Let V denote the electric potential and c the conductivity 

 at time t at a point %, y, z in a heterogeneous isotropic con- 

 ductor, then, there being no internal electrification, and for 

 brevity using the notation 



2= d d_ d^ d d d 

 yc ~~dx C dx + dy C dy + dz C Jz' 



V must satisfy the equation 



V*V = (A) 



Let <& denote the temperature, k the thermal conductivity, 



