1142 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



with variation e"'"^, n ^ 0. We shall have, 



Elm = -jo)HO^*^m , 

 Htm = -j^mV^m , 



where 



and 



^m = Jn I Unm - j fi'"'^, J n{Unm) = 



r, 2 2 Unm 



Pm = (^ eoMo — — r 



For the fields on the axis, one finds for n = dzl, 



nHr = -jH^ = -jn — — e' 

 Zro 



and 



nE^ = jEr = jo)on -^ e'"*'. 

 li \n \ 9^ I there is no first order perturbation. We now have 



2 



2A, 

 But we have 



Am 



Trri 



e — eo 2 2 Unm~ , IJ-fiQm'Unm" jJL -\- UK — /UO 



Co — — w Mo — r + ^ j 1 



e + €o r(/ To- fji -\- UK -\- Mo_ 



Jo 



= — 27r-w/xo/3« 



• ^<mr dr, 



Jo 



Jn'ixf + 



Jn{x)' 



X- 



X dx, 



r» a 'J n\.Unm) / 2 i > 



= —ZTW/JLoPm ?; \llnm — 1. 



and then, 



''7171 



1 /-f 



2/3„, 7-0- Jn{nnmy-(u„m- — 1) 



2 M + ^« — Mo I ^ 2 €l — Co 

 Pm 1 j -t- PC , 



H + riK -jr Ho ei -\- €n_ 



. (11) 



