Conductors by Electric Currents. 261 



and s the thermal capacity per unit volume at #, y, z,t ; then 

 the rate at which heat accumulates per unit volume per unit 

 time is 2 . 



and, by the law of Joule, the rate at which heat is generated 

 by the current per unit volume per unit time is 



. fdV 2 dY 2 dJP\ 

 W \dx 2 + dy 2 + dz 2 f 9 



J G 



where j is the factor which reduces work expressed in the 

 electrical units to its equivalent in the thermal units adopted ; 

 hence, finally, the equation to be satisfied by S will be 



d a A2 c • fW 2 , dY 2 dV 2 \ 



a* = W+J*{„ + W+&r • • (B) 



The equations (A) and (B) — which it will be noted must be 

 solved simultaneously in the general case, c being a function 

 of $, — in conjunction with the initial and boundary conditions, 

 fully determine V and 3. As the initial and boundary con- 

 ditions which may be imposed are almost entirely arbitrary, 

 they need not be specially formulated here ; but it is to be 

 noted that if ultimately they do not vary with the time, the 

 thermal and electrical distributions will tend to stationary 

 values, and ultimately the equation (B) to be satisfied by 3 

 may be replaced by 



When the conductor is homogeneous, and c, s, and k are 

 taken, as within certain limits they may be without material 

 error, independent of 3, the equations (A) and (B) reduce to 



V 2 V=0, (D) 



and 



& ;„,. ■ f d Y 2 dY 2 dY' 2 \ 



It is this case which will be discussed in the next four 

 sections. 



2. The Ultimate Steady Distribution : Invention of 

 Solvable Cases. 



As the direct problem of finding the solution for a body of 

 given forms involves great difficulties, indirect methods giving 

 us forms for which solutions may be directly obtained are of 



Phil. Mag. S. 5. Vol. 31. No. 190. Mar. 1891. U 



