ON FARADAY'S LINES OF FORCE. 211 



the pressure at the surface r = a will be reduced to zero. Now placing a source 



T?Ct 



47T^r- at the centre, the pressure at the surface will be uniform and equal to p. 



The whole amount of fluid emitted by the surface r = a may be fo^nd by 

 adding the rates of production of the sources within it. The result is 



fp P t P, , } 

 \ f - -j- - ~ --&c. [ . 

 (k b t b, 



To apply this result to the case of a conducting sphere, let us suppose 

 the external sources 4irP,, 4irP, to be small electrified bodies, containing e u e, 

 of positive electricity. Let us also suppose that the whole charge of the con- 

 ducting sphere is E previous to the action of the external points. Then all 

 that is required for the complete solution of the problem is, that the surface 

 of the sphere shall be a surface of equal potential, and that the total charge 

 of the surface shall be E. 



If by any distribution of imaginary sources within the spherical surface we 

 can effect this, the value of the corresponding potential outside the sphere is 

 the true and only one. The potential inside the sphere must really be constant 

 and equal to that at the surface. 



We must therefore find the images of the external electrified points, that 

 is, for every point at distance 6 from the centre we must find a point on the 



same radius at a distance j- , and at that point we must place a quantity 

 = e , - of imaginary electricity. 



At the centre we must put a quantity E' such that 



E' = E + e 1 ^ + e^ + &c. ; 



then if R be the distance from the centre, r t , r v &c. the distances from the 

 electrified points, and r\, r\, &c. the distances from their images at any point 

 outside the sphere, the potential at that point will be 



& , i 1 a 1\ i l a !\ , 

 P = TT + e, --- r }+eA -- r +&c. 



R V\r, 6, rj \r, b t rj 



272 





