946 Mr. G. EL Livens on some Farther Problems 



the conditions of the one cannot be determined from those 

 of the other. 



The case of a dielectric is, however, different. The charge 

 is here rigidly attached to the boundary and must move 

 round if the sphere is rotated. 



Adopting a similar notation to that employed in previous 

 papers and following Walker's method closely, we assume 

 the field to be of the form given by 



x = 4 2 , y = o, z = c - s ^(ry ,; +7/') 



2 cos e 



n6 



«=-^/'+/), /s-^^r+^+A 7=0, 



outside the sphere and inside 



X=Y=0, Z ='~" e [rW + fc"):M*i'-fc')]. 



/is a function of (ct— r), fa of (c r t—r) and fa of (c't + r), all 

 being supposed to be small quantities proportional to the 

 angular velocity o>, and squares of <o are neglected. 



Since the field must be finite at the origin we must have 



fa{c't) + faXc f t)=:0. 



. . . . (1) 



The normal component of the magnetic force is continuous 

 at r = a. Thus 



«/ f +/=«(*i , -*i0 + *i + *2» • 



(2) 



and this, according to "Walker, also secures the continuity of 

 the tangential component of electric force at r = a. 



The discontinuity of the tangential component of the 

 magnetic force determines the surface current which is due 



e 



\.ira< 



to the rotation of the uniform surface charge a = 



rotating with the sphere. Thus if tw is the angular velocity 

 of the sphere, supposed to be rotating round the polar axis, 

 we have 



ea'co 



= <*f'+af+f- [aW' + fc'0 + «(*i-fcO + *i + fc] 



