GENERAL PRELIMINARY RESULTS. 35 



such a function as will satisfy the equation = 8 (p) : it is more- 

 over evident, that (p) can never become infinite when p is within 

 the surface. 



It now remains to prove that the formula 



shall always give V = F, for any point within the surface and 

 infinitely near it, whatever may be the assumed value of V. 



For this, suppose the point p to approach infinitely near the 

 surface ; then it is clear that the value of (p) , the density of the 

 electricity induced by p, will be insensible, except for those 

 parts infinitely near to p, and in these parts it is easy to see, 

 that the value of (p) will be independent of the form of the sur- 

 face, and depend only on the distance p, dcr. But, we shall after- 

 wards show (art. 10), that when this surface is a sphere of any 

 radius whatever, the value of (p) is 



a being the shortest distance between p and the surface, and/ 

 representing the distance p, da-. This expression will give an 

 idea of the rapidity with which (p) decreases, in passing from 

 the infinitely small portion of the surface in the immediate 

 vicinity of p, to any other part situate at a finite distance from 

 it, and when substituted in the above written value of F, gives, 

 by supposing a to vanish, 



F=F. 



It is also evident, that the function F, determined by the above 

 written formula, will have no singular values within the surface 

 under consideration. 



What was before proved, for the space within any closed 

 surface, may likewise be shown to hold good, for that exterior 

 to a number of closed surfaces, of any forms whatever, provided 

 we introduce the condition, that V shall be equal to zero at an 

 infinite distance from these surfaces. For, conceive a surface at 



32 



