38 GENERAL PRELIMINARY RESULTS. 



F, with that employed at the beginning of this article in proving 

 the equation = 8 F, which last, having performed its office, will 

 be no longer employed. 



The equation = 8' V gives in the same way 



V being what F becomes by bringing the point p to any other 

 element dv of the surface A. This substituted for V, in the ex- 

 pression before given, there arises 



V=+fj(p)( P ")dcd a 'V': 



in which double integral, the signs of integration, relative to each 

 of the independent elements da and dv, must extend over the 

 whole surface. 



If now, we represent by F', the value of the potential func- 

 tion at p arising from the surface A, when the electricity Q is 

 concentrated in p, we shall evidently have 



where the order of integrations alone is changed, the limits re- 



j~ 



maining unaltered : V t being what F, would become, by first 

 bringing the electrical point p to the surface, and afterward the 

 point p to which V t belongs. This being done, it is clear that ~V' 



T~ 



and F / represent but one and the same quantity, seeing that each 

 of them serves to express the value of the potential function, 

 at any point of the surface A, arising from the surface itself, 

 when the electricity is induced upon it by the action of an elec- 

 trified point, situate in any other point of the same surface, and 

 hence we have evidently 



F F 

 ~ */ 



as was asserted at the commencement of this article. 



