2Q6 Mr. J. McCowan on the Heating of 



tained at potentials V x and V 2 and at temperatures S L and 3 2 

 respectively, and the remainder of the boundary is such that 

 no flux of heat or electricity takes place across it, then, when 

 the steady state supervenes, the equipotential surfaces will be 

 isothermal and the temperature at any point will be given in 

 terms of the initial (which is also the final) potential at^that 

 point by the formula 



^ == iy c (v 2 -V)(v-y 1 )+w 2 (^- y i)+^O r 2 -v)}/(v 2 --v 1 ).(i) 



Thus, if the problem of the electrical distribution can be 

 solved, the temperatures may be immediately obtained. From 

 this theorem, as remarkable in generality as in simplicity, an 

 extraordinary consequence results ; for (1) may be written 



2k\ 2 jc' Y 2 -rY x /' 



which shows that S has the maximum value 



*'=^i^-y^U^ ■ • (2) 



over the surface for which V has the value 



Y /_ VH-Vi . * **-», m 



Thus the maximum temperature attained is entirely inde- 

 pendent of the form or size of the body or its electrodes, 

 depending only on the applied electromotive force (V 2 — V t ), 

 the temperatures $$ and Sj at which the electrodes are main- 

 tained, and the physical character of the body as defined by k 

 and c. 



It must be noted, however, that as V must lie between V 2 

 and Y 1 in value, we have the condition that 2/ (3 2 - $i) must 

 lie between the limits ±jc(V 2 — Vi) 2 for a true maximum: 

 when this condition is not satisfied, 3 2 an d $± will be the 

 highest and lowest temperatures in the body. 



When $2 = ^ the condition for a maximum is always 

 satisfied, and this case is perhaps more instructive than the 

 more general one in which the natural gradient of temperature 

 between 3 2 an d $i is superposed on that due to the electrical 

 heating. Let then ^ 2 = ^, and for simplicity take this as the 

 zero from which the temperature is reckoned, then (1) 

 reduces to S> =\jc(V s -\)(y-Y l )lk; 



