Conductors by Electric Currents. 271 



electromotive force Y 2 —V! before it had time to be heated bj 

 the current, V*V t = 0, 



for c and k are constant since the temperature is uniform. 

 Thus F(3) = A + BV , where A and B have to be determined 

 to suit the boundary conditions. Now F(d) is by (6) a known 

 function of •&, since k and /($) are known functions of S, except 

 as regards an arbitrary constant. It is here convenient to 

 take this such that F(y) = 0, and to denote by — X and fi 

 respectively the values of F(S) at the two electrodes, which 

 are maintained finally at the temperatures $ x and 3 2 . Hence, 

 determining A and B, we have finally 



(V 2 -V 1 )F(3)=(X + rtV -(XV 2+/ ,V 1 ). . (7) 

 Thus the final temperature 3, and therefore also by (2) the 

 final potential, is determined in terms of the initial potential 

 in the same conductor uniformly heated, which may be found 

 by the ordinary methods. 



In the case § 2 = $iy which is really the fundamental one, 

 and from which the more general case could have been 

 derived, we see by (6) that X will be equal to fju, and therefore 

 (7) will become 



(V 2 -V 1 )F(d) = 2\!V -|(V 1 + V 2 )}. . . (7') 



Thus, comparing (7') and (4/), we see that the surface of 

 maximum temperature has initially and finally the same 

 potential, the mean of that of the electrodes, and being, there- 

 fore, independent of k and c, its form and position will be the 

 same for all homogeneous isotropic conductors which have 

 the same form of boundary and electrodes. 



In the general case, we see that surfaces which are equi- 

 potential initially are also so finally, but the values of their 

 potentials are in general different. It is not, however, to be 

 supposed that equipotentials remain so during the intermediate 

 state ; in fact, it is easy to show that they do not, except in 

 the particular cases where they are a set of planes, spheres, or 

 circular cylinders. It is also obvious that the initial and final 

 coincidence of equipotentials would in general be destroyed 

 by any emission from the bounding surface, the equipotentials 

 then also ceasing to be isothermal. 



7. Special Results for Cylindric Surfaces. 



The methods of Sections 4 to 6 may be slightly modified to 

 give in the case of cylindric surfaces a new set of theorems 

 analogous to those of these sections, but in which current takes 

 the place of potential, and resistance of conductivity. 



