Forced Vibrations of Electromagnetic Systems. 439 



from which all the rest follows. Merely remarking concern- 

 ing k that the realization of (316) when k is finite requires 

 the splitting up of the Bessel functions into real and imaginary 

 parts, that the results are complex, and that there are no 

 striking peculiarities readily deducible; let us take A; = at 

 once, and keep to non-conducting dielectrics. Then, from 

 (316), follow the equations of E and H, in and out ; thus 



E (i „ )0 r (out) = J °^;f G ^ or J^-ytH , . . (317) 



xj _ c/)J lr (J a— yGroa) or J 0a (J lr — ?/Grir) /Q1Q\ 



-^(in) 0r (out) — T~ : 1 J * 'KOLOJ 



un, wuw s same denominator 



which we can now examine in detail. 



52. Vanishing of External Field. J 0a = 0. — The very first 

 thing to be observed is that J 0a =0 makes E and H and there- 

 fore also F vanish outside the tube, and that this property 

 is independent of y, or of the nature of the external medium. 

 "We require the impressed force to be sinusoidal or simply 

 periodic with respect to z and t, thus 



e = e sin (mz + u) sin (nt + fi), . . . (319) 

 so that ultimately 



s 2 = n 2 /v 2 -m 2 ; (320) 



and any one of the values of s given by J 0o =0 causes the 

 evanescence of the external field. The solutions just given 

 reduce to 



H=-47rK(J lr /J la > 



E = (*/m)4^K(J 0r /J,«>'* . . . (32 



F= - {cn)- l A7rK(J lr /J la )i(de/dz) 



which are fully realized, because i signifies pin, or involves 

 merely a time-differentiation performed on the e of (319). 



The electrification is solely upon the inner surface of the 

 tube. In its substance H falls from —AirKe inside to zero 

 outside, and E a being zero, the current in the tube is Ke per 

 unit surface. 



The independence of y raises suspicion at first that (321) 

 may not represent the state which is tended to after e is 

 started. But since the resistance of the tube itself is sufficient 

 to cause initial irregularities to subside to zero, even were 

 there a perfectly reflecting barrier outside the tube to prevent 

 dissipation of these irregularities in space, there seems no 

 reason to doubt that (321) do represent the state asymptoti- 

 cally tended to. Changing the form of y will only change 



J- 



