Forced Vibrations of Electromagnetic Systems. 403 

 acting radially on the surface of the sphere. Thus, 

 (in) H = (^)V-^ e-^->cos{^-<a-r)-|}(197) 



when xa is very great. When the vorticity is confined to one 

 line of latitude, H in (197) vanishes everywhere except at the 

 vortex line. But a further approximation is required, or a 

 different form of solution, to show the disturbance round the 

 vortex line explicitly, i. e. when n is great, though not 

 infinitely great. 



29. A Conducting Dielectric. i»==l. — Here, if k is the con- 

 ductivity, c the permittivity, and fi the inductivity, let 



q=(4:irfi Q kp + fx, cp 2 )i = n l + n 2 i, . • . (198) 



when p = ni. Then n x and n 2 will be given by 





(199) 



Using this q in the general external H solution, but ignoring 

 the explicit connexion with the impressed force, we shall 

 arrive at 



(out) H=^[( 1+ -- ) 



~K^?) sin ] ( " 2 "- Bi) ' • ' (200) 



where C is an undetermined constant, depending upon the 

 magnitude of the disturbance at r=a. So far as the external 

 solution goes, however, the internal connexions are quite 

 arbitrary save in the periodicity and confinement to producing 

 magnetic force proportional in intensity to the cosine of the 

 latitude. The solution (200) may be continued unchanged as 

 near to the centre as we please. Stopping it anywhere, there 

 are various ways of constructing complementary distributions 

 in the rest of space, from which (200) is excluded. 



n x is zero when h=Q. We then have the dielectric solu- 

 tion, with n 2 = n/v. On the other hand, c = makes 



n 1 =n 2 = (27r/j, Jcn)i 

 in § 28. The value of n 1 2 + n 2 2 is 

 n? 



o,2 



Hs^+m)* • «j 



