328 APPENDIX. 



Now if we consider a unit of positive electricity placed at P, 

 and if p B be the density on an element dS B at B, we shall have, 

 similarly, 



for all points outside the surface, or on it, since the potential 

 is zero on the surface. 



Let X be on the surface, say at A, this equation becomes 



Hence, substituting in equation (2) we get 



and as this is the same as we shall obtain for G p \ the property 

 is proved. 



Note to Art. 10, pp. 50, 51. 



The equation </> (r) =-^-J proved on p. 51, may be ex- 



pressed in words as follows. Let be the centre of a sphere of 

 radius a, and A, B, two points each of which is the electrical 

 image of the other with respect to the sphere (i.e. let 0, A, B 

 be in the same straight line, and OA . OB = a?), then, if electricity 

 be distributed in any manner over the surface of the sphere, 

 the potential at A is to the potential at B as a is to OA or 

 as OB is to a. 



For, if a point P move in such a manner that the ratio BP 

 to AP is constant (= X supposs) it will describe a sphere, and 

 if C, C' be the points in which this sphere cuts AB, 



AC _AB AB m 



- A( = 



