Forced Vibrations of Electromagnetic Systems. 489 



within the cylinder, producing /= e, we have the same discon- 

 tinuity produced by/. 



H being circular, we use the operator (313). Applying it 

 to (356) we obtain 



/ -^jt.~j^5J h *' • • (357) 



from which, by the conjugate property (307), and the operator 

 (313), we derive 



ir as 



E(in) 01' (out) = — [J 0r (J, a — yGr la ) Or Jia(J r — ?/Gr r)]/, (358) 

 H ( in) Or (out )= 7I |^[J 1 r(Jla-?/G-la) 01' J la (J lr - ?/G lr ) ]/, (359) 



in which / is a function of t, and it may be also of z. If so, 

 then we have the radial component F of electric force given 



F ( in) or (out) = - J!? [J lr (J la -yG la ) or Ji«(Ji r -yG- lr )]^. (360) 



From these, by the use of Fourier's theorem, we can build 

 up the complete solutions for any distribution of/ with respect 

 to z ; for instance, the case of a single circular line of curl e. 



64. J la =0. Vanishing of external field. — Let / be simply 

 periodic with respect to t and z ; then J la = 0, or 



J 1 {a\/n 2 /v 2 -m 2 \=0, .... (361) 



produces evanescence of E and H outside the cylinder. The 

 independence of this property of y really requires an un- 

 bounded external medium, or else boundary resistance, to let 

 the initial effects escape or be dissipated, because no resistance 

 appears in our equations excepting. The case s = or n—mv 

 is to be excepted from (361) ; it is treated later. 



65. y=i. Unbounded medium. — When n/v>m y s is real, and 

 our equations give at once the fully realized solutions in the 

 case of no boundary, by taking y — i y 



H ( in) or ( out) = i7r«cn[J lr (J la — iGi fl ) or Ji a (J ]r — iG lr )]/, \ 



E (i n) or (out) = -i7ras[J 0r (G la + iJ la ) or J,a(G r + iJor)]/, T ( 362 ) 



F(i n ) Or (0 rt) = ira[J lr (Gia + *Jia) 0r J l«(^lr + & ir)] Wl dz ), 



in which i means pin. 



The instantaneous outward transfer of energy per unit 

 length of cylinder is (by Poynting's formula) 



EH . 



3 X 27T7% 



47T 



