GENERAL PRELIMINARY RESULTS. 33 



value V is known at this surface, we will now show, that what- 

 ever the value "of V may be, the general value of V deduced 

 from it by the formula just given shall satisfy the equation 



For, the value of V at any point p whose co-ordinates are x, y, z, 

 deduced from the assumed value of V, by the above written 

 formula, is 



U being the total potential function within the surface, arising 

 from a unit of electricity concentrated in the point p, and the 

 electricity induced on the surface itself by its action. Then, 

 since T is evidently independent of x, y, z, we immediately 

 deduce 



Now the general value of U will depend upon the position 

 of the point p producing it, and upon that of any other point p 

 whose co-ordinates are x, y ', z, to which it is referred, and will 

 consequently be a function of the six quantities, x, y^ z, x, y ', z. 



But we may conceive U to be divided into two parts, one = - 



(r being the distance pp r ) arising from the electricity in p, the 

 other, due to the electricity induced on the surface by the action 

 oi'p, and which we shall call U t . Then since U l has no singular 

 values within the surface, we may deduce its general value from 

 that at the surface, by a formula similar to the one just given. 

 Thus 



where U' is the total potential function, which would be pro- 

 duced by a unit of electricity inj/, and therefore, (-T- ) is inde- 

 pendent of the co-ordinates x 9 y, z, of p, to which 8 refers. 



