Forced Vibrations of Electromagnetic Si/stems. 365 



If a direct proof be required, expand the exponential opera- 

 tor in (221) containing r in the way indicated in (216), and 

 let the result operate upon J (<Tti). The integrated result can 

 be simplified down to (221). 



37. Now use (221) in (220). Let/e^=/ e-<^ where / is 

 constant; and the square bracket in (220) becomes known, 

 being in fact the right member of (221) multiplied by / . 

 So, making use also of (228), we bring (220) to 



H yafi 



JH 1+ {(HW-=}?b] 



J. {J [('-«)'-»'«']'} 1(229) 



to which must be added the other part, beginning 2a/v later,, 

 got by negativing a, except the first one. The operation 

 (p 2 — c7 2 ) -1 may be replaced by two integrations with respect 

 to r. 



Let r and a be infinitely great, thus abolishing the curva- 

 ture. Let r— a=z, and fova/r } which is now constant, be 

 called e . Then we have simply 



H= 



J o{^(* 2 -^ 2 ) # }, • - • (230) 



showing the H produced in an infinite homogeneous conduct- 

 ing dielectric medium at time t after the introduction of a 

 plane sheet (at z=0), of vorticity of impressed electric force, 

 the surface density of vorticity being e €~ 2 P it . This corrobo- 

 rates the solution in § 8, equation (51) (vol. xxv. p. 140), 

 whilst somewhat extending its meaning. 



The condition to which / is subject may be written, by 

 (208), 



/=/»e- 2 "' ( , (231) 



where / is constant. If, then, we desire / to be constant, p 1 

 must vanish, which, by (208), requires £ = 0, whilst g may 

 be finite. 



But we can make the problem real thus. In (229) change 

 H to E and fiv to cv ; we have now the solution of the 

 problem of finding the electric field produced by suddenly 

 magnetizing uniformly a spherical portion of a conducting 

 dielectric ; i. e. the vorticity of the impressed magnetic force 

 is to be on the surface of the sphere r = a, parallel to its lines 

 of latitude, and of surface-density fv, such that/ve 2 **' is con- 

 stant. This makes / constant when g = and k finite, repre- 

 senting a real conducting dielectric. 



