in Conducting Sheets and Solid Bodies. 5 



j- (Pr), where — aP is the potential of a distribution of density 



<l> over the shell. The corresponding vector potential may be 

 represented, on the analogy of the case of plane strata (Maxwell, 



art. 657), by components 0, — . n 7 . , ^-ttt in the direc- 



J1 J r ' rsmudcf) rdd 



tion of dr, rd0, rsinOdcj). This is true, in fact, because 



Laplace's equation for P can be written in the form 



The given state of the external field we can express similarly 

 in terms of a function P . 



The equations giving P, Q, E the components of the electro- 

 motive force at a point are (Maxwell, art. 598) 



P = _^_<^ 

 dt dr 



-> dGr ^ dty 



^~~~~~dt riTtf 



P_ ^?- d ^ 



dt rsmdd0 l 



and therefore, by Ohm's law, 



R d® _ d rf(P + P ) __ dty 

 a sin dcf) "" dt sin d(j> a d0' 

 _ n d& d d(P + P ) #* 



where a is the radius of the shell. 

 These equations are satisfied by 



B<I>=-a|(P + P ), (1) 



without the intervention of any electrostatic potential i|r. 



To obtain a solution, let us suppose the external field to be 

 given by 



P =A ,-'(t)Y„ 



where Y n is a spherical harmonic of the rath order, and 



•=(-1)*. 



Let the corresponding values of P be 





P 1 =Ae«'(-) Y» inside; 



