Forced Vibrations of Electromagnetic Systems. 379 



priate to the functions, viz. such as may be produced by 

 vorticity of e in spherical layers, proportional to v (or vQ\ in 

 general). But it is scarcely necessary to say that these solu- 

 tions in infinite series, of so very general a character, are more 

 ornamental than useful. On the other hand, the immediate 

 integration of the differential equations to show the develop- 

 ment of waves becomes excessively difficult, from the great 

 complexity, when there is a change of medium to produce 

 reflexion. 



46. Thin Metal Screens. — This case is sufficiently simple 

 to be useful. Let there be at r=a x a thin metal sheet inter- 

 posed between the inner and outer nonconducting dielectrics, 

 the latter extending to infinity. If made infinitely thin, E is 

 continuous, and H discontinuous to an amount equal to 47r 

 times the conduction current (tangential) in the sheet. Let 

 Ki be the conductance of the sheet (tangential) per unit area; 

 then 



(H/E) in -(H/E) 0Ut =47rK 1 at r = a v 



Therefore by (284), when the dielectric is the same on both 

 sides, 



\<-V u{-y x w{) 



where the functions w 1? &c. have the r = ax values. From 

 this, 



expresses y x for an outer thin conducting metal screen, to be 

 used in (286). If of no conductivity, it has no effect at all, 

 passing disturbances freely, and y^ — 1. At the other extreme 

 we have infinite conductivity, making y Y — Ui/ioi, with com- 

 plete stoppage of outward-going waves, and reflexion without 

 absorption, destroying the tangential electric disturbance. 



When the screen, on the other hand, is within the surface 

 of/, say, at r=a , of conductance K per unit area, we shall 

 find 



_ (±7rK /cpg)u /2 

 Jo ~ l + (±TrK /cprj)u o 'w » ' ' ' ^* 1} 



where w , &c. have the r = a values. The difference of form 

 from y, arises from the different nature of the r functions in 

 the region including the origin. As before, no conductivity 

 gives transparency (y =0), and infinite conductivity total 

 reflexion (y {) = u '/ii\i). When the inner screen is shifted up 

 to the origin, we make y =0 and so remove it. 



