? 612 Mr. J. R. Carson on 



We are now prepared to take up the analysis of the 

 problem of wave propagation along parallel wires ; in the 

 course of this analysis the significance and utility of 

 the simplifying assumptions will become more apparent. 



From the general solution of the wave equation in 

 polar co-ordinates and the special conditions of symmetry 

 which obtain, the axial electric force in wire #1 is given 

 by the Fourier-Bessel expansion 



E, = 1 AJ n ( P] ) cos n0 u ..... (8) 



o 



and in wire £2 by 



E, = -5(-l)»A„J M (p s ) cos/i0 2 , ... (9) 







where 



In these equations J n (p) is the Bessel function of order n 

 and argument p, and the coefficients A . . . A n are to be 

 determined from the boundary conditions at the surfaces 

 of the wires. In either wire the magnetic force is then 

 derivable from 



flip H d == =^E,, 

 fiipK r = — ~^Q% 



. (10) 



where r, 6 denote either r 1? 6i or r 2 , 6 2 according as wire #1 

 or wire *2 is under consideration. From the symmetry of 

 the system, however, the satisfaction of the boundary con- 

 ditions imposed at the surface of one wire insures their 

 satisfaction at the surface of the other. 



In the dielectric the electric and magnetic forces 

 are expressible as Fourier-Bessel expansions, the Bessel 

 functions, however, being of the " external " or second 

 kind. In accordance with the assumption, however, that 

 y is a small quantity of: the order of magnitude of p/'v, 

 it follows that so long as pc/v (where c is the separation 

 between wires) is a small quantity compared with unity, 

 the Bessel functions in the neighbourhood of the wires 

 may be replaced by the limiting forms which they assume 

 for vanishing! v small arguments. In particular, the mag- 

 netic forces Hz and H y in the neighbourhood of the wires 



