272 Mr. J. McCowan on the Heating of 



Let Q be the current-function, and p = 1/c the resistivity, 

 then 



P \dx 2 + dtf )~ G \ dx> + df f 



V p 2 Q = = V c 2 V. 



Thus the properties of Q and p are like those of V and c. 

 Put then 



*=^' (S) ' and ^W-W)l = TO ' 



where S' is the maximum value of 3 (see below), substitute 

 in the investigation of Section 6,/ for/, F for F, Q for V, 

 and p for c, adapt the language similarly, and the following 

 theorems are immediately obtained. 



Call the lamina enclosed between two cylindric surfaces a 

 cylindric sheet when unlimited, and a cylindric strip when it 

 is limited by two planes (its sides) cutting it orthogonally. 

 If the cylindric sheets are given by the values Q 2 and Qj of 

 the current function, then the whole current per unit breadth 

 between them, Q 2 — Qi, may be denoted by C, and the parts 

 flowing on each side of a point on the sheet Q, viz., Q 2 — Q 

 and Q — Qi by C 2 and C l respectively. 



a. If the surfaces of a cylindric sheet, or strip with im- 

 permeable sides, of a homogeneous conductor for which 

 Jc=pf '(§) 7 are maintained at temperature zero, and a current 

 C per unit breadth is maintained through it, then ultimately 

 the current-sheets will be isothermal, and the temperature at 

 any point will be given in terms of the parts C a and C 2 of the 

 whole current C which flow on each side of the point, by the 

 formula 



/o(*)=i/'CA, (i) 



and the temperature will have a maximum value 3' given by 

 / (3')=yC 2 /^ over the surface which equally divides the cur- 

 rent (the same initially and finally). 



Finally, if Q be the initial value of Q at any point, and 2\ 

 is the difference in the value of F (3) at the two boundaries, 

 then 



(Q 2 -Qi)F„(S)=2\{Q -i(Q. + Q 2 )}. • • (2) 



Thus the final temperature, and therefore also by (1) the final 

 current distribution, is given in terms of the initial current- 

 distribution which may be investigated by the ordinary 

 methods. 



For simplicity, the case of zero-bounding temperatures has 

 been taken above, but the general results of Section 6 may be 



