270 Mr. J. McCowan on the Heating of 



From this, as in Section 4, we may deduce that : 

 yS. The temperature will have a maximum value d', 

 given by 



/(y)= /JM^ + i ( y 2 _ Vl)2+ ^/J^L)} 2 (3) 



over the surface for which the potential has the value 



V i +V 1 1/(^-/(30 

 " 2 + j %=T t • ■ ■ ■ U 



It should be noted that since k and c are essentially positive 

 and finite, /(£) must increase with 3 continually and without 

 limit. 



In the important case where $i = d 2 it is convenient to take 

 the arbitrary constant in /(d) such that 



/(»t) =/&) = <>, 

 and then the equations (2), (3), (4) become respectively 



/W=ii(v 2 -v)(V-v I ) .... (2') 



(/(y)=i;(V 2 -v 1 ) 2 (3') 



I v^m^+vo (4-) 



There is, however, a further problem to be solved ; the ulti- 

 mate relation between V and h has been obtained, but as the 

 distribution of V is affected by the heating, there remains to 

 be determined the final in terms of the initial distribution. 

 The means of doing this is afforded by the foregoing results: — 



Making use of (3) and (4), (2) may be written 



/(*)=/(») -1/(7- v') 2 • • (5; 



.-. V=V'±*/(2/iK{/(a')-/(S)}. 

 dY h=& 



••• b ^ (1 )' c ^ = vrw)-2./7W =F/w ' say ' (6) 



where the upper sign is to be taken in the space between the 

 electrode at potential Y 2 and the surface of maximum tem- 

 perature, and the lower sign from the latter surface to the 

 other electrode. 



By means of (6), (A) reduces to 



V 2 F(S) = 0. 



Now if V denote the potential at any point of the same con- 

 ductor when at the same temperature throughout, as it would 

 be, if initially so, immediately after the application of the 



