262 Mr. J. McCowan on the Heating of 



considerable importance. Saint- Venant obtained many in- 

 teresting and practically important solutions of problems on 

 the torsion of elastic prisms in this way, and his results have 

 been adapted by Thomson and Tait and others to problems in 

 fluid motion in two dimensions. In what immediately follows 

 I have applied, this method to an interesting set of heating- 

 problems, and have shown how the method, can be considerably 

 extended. 



a. To find the steady distribution of temperature in a long 

 cylindric bar carrying a current of strength q per unit area, 

 its surface being maintained at a constant temperature, taken 

 for convenience as the zero. 



It is evidently only necessary to consider equation (E), 

 which in this case reduces to 



*? + ** + #«() 



da? dy 2 ck 

 and therefore we may take 



*«:«— J(a*»+V)j 



where ar is any solution of 



dV dV _ 

 da; 2 + dtf ~ °' 



and (a -f b) =jq 2 /ck ; and thus we shall have the solution for 

 a bar whose surface is given by the equation 



2(7 = ax 2 + by 2 . 



One or two examples only need be given : — 

 (i) Let the cross section be an ellipse, equation 



b 2 x 2 + a 2 y 2 = a 2 b 2 . 

 Then clearly 



jq 2 W_ ( _a?_f I 

 2cka 2 + b 2 \ a 2 b 2 )> 



and we see that the isothermal surfaces are similar concentric 

 elliptic cylinders. 



In the case of circular cross section this reduces to 



$=jq 2 \a 2 -x 2 -f\/4ck. 



Again, when b is infinite, it reduces to the case of a flat 

 plate of thickness 2a, and 



S=jq 2 {a 2 -a; 2 \/2ck. 



(ii) Let the cross section be an equilateral triangle, and let 

 a be the radius of the inscribed circle and a, /?, y the perpen- 



