Forced Vibrations of Electromagnetic Systems. 373 



Let / vary sinusoidally with the time, and observe the be- 

 haviour of $1 and (j) 2 as the frequency changes. The full 

 development which I have worked out is very complex. But 

 it is sufficient to consider the case in which k is big enough, 

 in concert with the radius a and frequency n/27r, to make the 

 disturbances in the sphere be practically confined to a spherical 

 shell whose depth is a small part of the radius. Let s= 

 (27r^ 1 A: 1 ??a 2 )^ ; then our assumption requires e~ s to be small. 

 This makes 



and, if further, s itself be a large number, this reduces to 



<j) 1 = (l+{){fju i n/S7rk l )^ (276) 



Adding on the other part of <£, similarly transformed by 

 p 2 = — n 2 , we obtain 



^ = \^ V l + (nalvy 2 + \&kj ) ~ l \{na/v) + (na/tO 8 V^,/ M 27?) 



where the terms containing k^ show the difference made by 

 its not being infinite. The real part is very materially 

 affected. Thus, copper, let 



fc l r=(1600)- 1 ,/*i=l, 2tj7i=1600, a = 10, .'. 5 = 10. 



These make 5 large enough. Now na/v is very small, but, on 

 the other hand, 



(^8^)^=130, 



so that the real part of </> depends almost entirely on the 

 sphere, whilst the other part is little affected. 



Now make n extremely great, say na/v = l ; else the same. 

 Then 



</> = (§ x 10 lo + 44x 10 4 )-2(| x 10 10 -44x 10 4 ), 



from which we see that the dissipation in space has become 

 relatively important. The ultimate form, at infinite fre- 

 quency, is 



^ = ^ + (/^ 1 n/8^ 1 )*(l + 0; .... (278) 



so that we come to a third state, in which the conductor puts 

 a stop to all disturbance. This is, however, because it has 

 been assumed not to be a dielectric also, so that inertia 

 ultimately controls matters. But if. as is infinitely more 

 probable, it is a dielectric, the case is quite changed. We 

 shall have 



*i = (4^i + /*i ptyvfei+qp)-*! . . . (279) 



