32 GENERAL PRELIMINARY RESULTS. 



which shows, that V the value of F at the point p is given, 

 when V its value at the surface is known. 



To convince ourselves that there does exist such a function 

 as we have supposed Uto be; conceive the surface to be a per- 

 fect conductor put in communication with the earth, and a unit 

 of positive electricity to be concentrated in the point p' , then the 

 total potential function arising from p and from the electricity 

 it will induce upon the surface, will be the required value of U. 

 For, in consequence of the communication established between 

 the conducting surface and the earth, the total potential function 

 at this surface must be constant, and equal to that of the earth 

 itself, i. e. to zero (seeing that in this state they form but one 

 conducting body). Taking, therefore, this total potential func- 



_ 1 



tion for Z7, we have evidently = U 9 = 8 U 9 and U = - for 



those parts infinitely near to^>'. As moreover, this function has 

 no other singular points within the surface, it evidently possesses 

 all the properties assigned to U in the preceding proof. 



Again, since we have evidently Z7' = 0, for all the space 

 exterior to the surface, the equation (4) art. 4 gives 



where (p) is the density of the electricity induced on the surface, 

 by the action of a unit of electricity concentrated in the point p. 

 Thus, the equation (5) of this article becomes 



(6). 



This equation is remarkable on account of its simplicity and 

 singularity, seeing that it gives the value of the potential for 

 any point p, within the surface, when F, its value at the sur- 

 face itself is known, together with (p), the density that a unit 

 of electricity concentrated in p' would induce on this surface, 

 if it conducted electricity perfectly, and were put in communica- 

 tion with the earth. 



Having thus proved, that F' the value of the potential func- 

 tion F, at any point p within the surface is given, provided its 



