TO THE THEORY OP ELECTRICITY. 53 



to the sphere, to which A' belongs. When r is finite, we have 

 evidently V = -7 . Therefore by equating 



r being made infinite. Having thus the value of $, the value 

 of p becomes known. 



To give an example of the application of the second equation 

 in p let us suppose a spherical conducting surface, whose radius 

 is a, in communication with the earth, to be acted upon by any 

 bodies situate within it, and B' to be the value of the potential 

 function arising from them, for a point p exterior to it. The 

 total potential function, arising from the interior bodies and 

 surface itself, will evidently be equal to zero at this surface, and 

 consequently (art. 5), at any point exterior to it. Hence 

 V + j?'=0; V being due to the surface. Thus the second of 

 the equations (9) becomes 



, 



4?rp = 2 -j-r + . 

 dr a 



We are therefore able, by means of this very simple equation, to 

 determine the density of the electricity induced on the surface in 

 question. 



Suppose now all the interior bodies to reduce themselves to 

 a single point P, in which a unit of electricity is concentrated, 

 and /to be the distance Pp : the potential function arising from 



P will be -j, , and hence 

 j 



TV l 

 =/' 



r being, as before, the distance between p and the centre of 

 the shell. Let now b represent the distance OP, and 6 the 

 angle POp' , then will f 2 = b 2 - 2Jr . cos + r\ From which 

 equation we deduce successively, 



r'-i cos e 



