506 Prof. Thomson on the Uniform Motion of Heat 



flux of heat at any point in the first case will be proportional to 

 the attraction on an electrical point similarly situated in the 

 second ; and the direction of the flux will correspond to that of 

 the attraction. 



Let — T-i be the external value of — j- at the original sur- 

 flH, an 



face, or the attraction on a point without it, and indefinitely near 



it. Now this attraction is composed of two parts ; one, the 



attraction of the adjacent element of the surface, and the other 



the attraction of all the rest of the surface. Hence, calling the 



former of these «, and the latter b, we have 



dn^ 



Now, since the adjacent element of the surface may be taken as 

 infinitely larger, in its linear dimensions, than the distance from 

 it of the point attracted, its attraction will be the same as that of 

 an infinite plane, of the density pj. Hence a is independent of 

 the distance of the point from the sm-face, and is equal to ^irpy 

 Hence 



Now, for a point within the surface, the attraction of the adja- 

 cent element will be the same, but in a contrary direction ; and 

 the attraction of the rest of the surface will be the same, and in 

 the same direction. Hence the attraction on a point within the 

 surface and indefinitely near it is — 2Trp^ + b ; and consequently, 

 since this is equal to nothing, we must have b = 27rp^, and 

 therefore 



-1;=*-''. P) 



Hence pi is equal to the total flux of heat at any point of the 

 surface divided by 47r. 



It also follows, that if the attraction of matter spread over the 

 surface be nothing on an interior point, the attraction on an 

 exterior point indefinitely near the surface is perpendicular to 

 the surface, and equal to the density of the matter at the part of 

 the surface adjacent to that point multiplied by 47r. 



If V be the temperature at any isothermal surface, and p the 

 intensity of the sources at any point of this surface which would 

 be necessaiy to sustain the temperature v, we have, by (5), 



dv . 

 which equation holds, whatever be the manner in which the 



