ON FARADAY'S LINES OF FORCE. 213 



Let the value of the potential undisturbed by the presence of the sphere be 



p = Ix. 

 Let the sphere produce an additional potential, which for external points is 



and let the potential within the sphere be 



p 1 = Ex. 



Let k' be the coefficient of resistance outside, and k inside the sphere, then 

 the conditions to be fulfilled are, that the interior and exterior potentials should 

 coincide at the surface, and that the induction through the surface should be the 

 same whether deduced from the external or the internal potential. Putting 

 x = r cos 6, we have for the external potential , 



and for the internal 



p 1 = Br cos 6, 



and these must be identical when r = a, or 



= B. 



The induction through the surface in the external medium is 



and that through the interior surface is 



and .-. (/-24) = 



k "\Tf 



A - T P r 



~ j 



These equations give 



The effect outside the sphere is equal to that of a little magnet whose 

 length is I and moment ml, provided 



