Forced Vibrations of Electromagnetic Systems. 499 



if J„„. or J, n (sr) is the ??ith Bessel • function, and G mr its 

 companion, whilst the ' means djdr. 

 The boundary condition is 



E! = E 2 -/ at r = a, .... (397) 



Ei being the inside, E 2 the outside value of the force of the 

 flux. Therefore, using (396) with y = inside, we obtain 



Hv ma[v ma J/^ma) /» 



a= /-f p./ f, — p — r cpj 



= C ^J ma (J ma --y& m a)f, .... (398) 



where x is a constant, being 7r/2 when m = 0, according to 

 (307), and always tt/2 if Gr m has the proper numerical factor 

 to fix its size. 



We see that when 



f=f cos m9 cos nt 



when / is constant, the boundary H, and with it the whole 

 external field, electric and magnetic, vanishes when 



J»ia = 0. 



If m=0, or there is no variation with 6, the impressed force 

 may be circular, outside the cylinder, and varying as r -1 . 



If m = l, the impressed force may be transverse, within the 

 cylinder, and of uniform intensity. 



77. Conducting tube, e circular, a function of 6 and t. — 

 This is merely chosen as the easiest extension of the last case. 

 In it let there be two cylindrical surfaces of f, infinitely 

 close together. They will cancel one another if equal and 

 opposite, but if we fill up the space between them with a tube 

 of c mductance K per unit area, we get the case of e circular 

 in tne tube, e varying with 6 and t, and produce a discon- 

 tinuity in H (which is still longitudinal, of course). Let E a 

 be the common value of E just outside and inside the tube ; 

 e + E a is then the force of the flux in the substance of the 

 tube, and 



H 1 -H 8 = 47rK(e + E.) .... (399) 



the discontinuity equation, leads, by the use of (396) and the 

 conjugate property of J m and G„ t as standardized in the last 

 paragraph, through 



d'-^-^KjE^TrK* 



