60 Distribution of Electricity on two Spherical Surfaces. 



a 2 



writing therein for x, becomes 



c ~~~ x 



and similarly that 



b 2 (c — x) — x(c 2 — a 2 — ex) , 



b 2 

 writing therein ■ for x, becomes 



c x 



b 2 

 = ( c _ x y2 {aXc-x)-<c 2 -b 2 -cx)\. 



More generally, the terms to be added are for <f>x a term as 

 above, where P denotes a function of x which remains unal- 



Q ' ( C ~~~ X) 



tered where x is changed into ' v 79 — '— , and for <£>x a term 

 to <r — b z — ex 



as above with P / instead of P, where P / denotes what P be- 



a 2 

 comes when x is changed into . But these additional 



& c — x 



terms vanish for the electrical problem, and the correct values 

 of cpx, &x are the particular values given above. 

 It is to be remarked that the function 

 a 2 (c—x) . 



is = 



c 2 —b 2 —cx "* b 2 



c — 



c—x 



viz., considering x as the distance of a point X from A, then 

 taking the image of X in regard to the sphere B, and again 

 the image of this image in regard to the sphere A, the function 

 in question is the distance of this second image from A. And 

 similarly the function 



b 2 (c-x) . b 2 



9 9 IS — 9 5 



c* — a z — cx a 2 

 c 



c — X 



viz., considering here x as the distance of the point X from B, 

 then taking the image of X in regard to the sphere A, and 

 again the image of this image in regard to the sphere B, the 

 function in question is the distance of this second image from 

 B. It thus appears that Poisson's solution depends upon the 

 successive images of X in regard to the spheres B and A alter- 

 nately, and also on the successive images of X in regard to 

 the spheres A and B alternately. This method of images is 

 in fact employed in Sir W. Thomson's paper " On the Mutual 

 Attraction or Repulsion between two Electrified Spherical 

 Conductors," Phil. Mag., April and August 1853. 



