Conductors by Electric Currents. 2T6 



transformed equally directly. The case of constant conduc- 

 tivities is, of course, obtained directly from the above by putting 

 f (5) -=k$/p; and it should be noted that, owing to the con- 

 jugate properties of the current and potential functions, the 

 general method of Section 3 gives easily, on Fourier principles, 

 the solution for the case of arbitrary (two-dimensional) sur- 

 face-temperatures, for any strip for which these functions are 

 known. As an example, take a strip for which V and Q are 

 the potential and current functions, its boundaries being given 

 by +Q and + Y , these being maintained at zero tempera- 

 ture, and the remaining plane sides being impermeable. Pro- 

 ceeding as in Section 3, with the modification suggested in 

 Section 5, we take 



U = ^-i./c{V 2 -V 2 }, and .-. V 2 U = 0. 



Expanding next by Fourier's theorem, having the boundary- 

 conditions in view, we get 



iW-V*) _ ijcYo 2 . g 2 _^±_ 3 cos(2n + 1)7rV/2Vo _ 

 Hence we obtain U, and therefore finally 



ir 6 "(2n + l) 3 

 cosh(2n + l)7rpQ/2V 



cosh (2n + l)7r/)Q /2Vo 



cos(2n+l)7rV/2V, 



8. General Case of Heterogeneous jEolotropic Conductor. 

 The equation satisfied by V is, in its most general form, 



d c dV dY dV\ d r dV , dY , dY) 

 aTr\ C ^ + ^Ty + ^Tz J t + *l»^ + *^' + *TS } 



d ( dY ■ dV dY\ A 



^ V? c . p . 2) V=0 (A') 



but probably the p's are identical with the q's*. 

 The equation satisfied by S is 



d * n2 a^-f <iy* dY* dY*. , .^V^V 

 * Maxwell's ' Electricity and Magnetism,' Part ii. chap. viii. 



