V 2 |V 2 = W 2 V+^-+iV + ' 



264 Mr. J. McCowan on the Heating of 



then $ = when -cr is equal to a or b, and this is therefore the 

 solution for a long tube or solenoid with current flowing 

 round it. 



The extension of Saint- Yenant's method to this class of 

 examples seems of some value, owing to the large part played 

 by ring-shaped conductors in practical applications of elec- 

 tricity. 



3. Transformation of the Equations and General Method of 

 Solution for the ease of Homogeneous Conductors with 

 Unvarying Conductivities fyc. 



For brevity, put 



U = y c Y 2 + k$, 



then since identically 



dY 2 dY 2 dY 2 

 dx l + dy 2 + dz 2 ' 



and V' 2 V = by (D), (E) may be reduced to 



^J=W 2 U, (F) 



and the general problem is thus reduced to the solution of 

 (D) and (F) subject to the boundary conditions. These equa- 

 tions are of the same form as those for the steady and varying 

 motion of heat in the ordinary case of no internal heating which 

 have been very fully discussed ; it follows, therefore, that the 

 more general problem of electrical heating is thus reduced to 

 the same state of completeness as the more particular one. 

 Thus, for example, if we can write down for a conductor of 

 specified form an expression for the temperature at any point 

 and time when every point of the surface is maintained at an 

 arbitrary temperature, then we can also write it down when 

 every point of the surface is maintained at an arbitrary tem- 

 perature and potential % and if there were no flux of heat over 

 a specified portion of the surface, this would afford the solution 

 for the case of no flux of heat or electricity over the same 

 portion in the more general case. 



General solutions may thus be obtained for the cuboid 

 (Fourier), the sphere (Poisson), the circular cylinder (Ma- 

 thieu), and in the case of steady motion for the ellipsoid 

 (Lame), various surfaces of revolution and cylindric surfaces. 

 We may take for example, as the results can be written in 

 very brief form, the 



General Case of Steady Distribution in a Sphere. 

 Taking polar coordinates, r, #, <j>, let every point of the 

 surface of the sphere r=a be maintained at the arbitrary 





