504 Prof. Thomsou on the Uniform Motion of Heat 



are no sources without the sui-facc. The temperature of an ex- 

 ternal point is consequently the same as if all the sources were 

 distributed over this surface in such a manner as to produce the 

 given constant temperature. Hence we may consider the tem- 

 perature of any point without the isothermal surface as the sum 

 of the temperatures due to certain constant sources of heat distri- 

 buted over that surface. 



To find the temperature produced by a single source of heat, 

 let r be the distance of any point Irom it, and let v be the tem- 

 perature at that point. Then, since the temperature is the same 

 for all points situated at the same distance from the source, it is 

 readily shown that v is determined by the equation 



—r--r =A. 

 dr 



Dividing both members by ?'^, and integrating, we have 



r 



Now let us suppose that the natural temperature of the solid, 

 or the temperature at an infinite distance from the source, is 

 zero ; then we shall have C«=0, and consequently 



-7 (1) 



Hence that part of the temperature of a point without an iso- 

 thermal surface which is due to the sources of heat situated on 



7 2 



any element, dw^, of the surface, is "' ^, where i\ is the di- 



stance from the element to that point, and pj a quantity mea- 

 suring the intensity of the sources of heat at different parts of 

 the surface. Hence, the supposition being still made that there 

 are no sources of heat without the surface, if v be the tempera- 

 ture at the external point, we have 



^=fr^- (^) 



the integrals being extended over the whole surface. The quan- 

 tity p, must be determined by the condition 



«^=^i (3) 



for any point in the surface, v^ being a given constant tempe- 

 rature. 



Let us now consider what will be the temperature of a point 

 within the surface, supposing all the sources of heat by which 

 the surface is retained at the temperature i\ to be distributed 

 over it. Since there are no sources in the interior of the surface, 



