1144 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



Hence, assuming the pencil coaxial with the guide. 



2A. 



CO 



Jo L ^0^ \ ru/ 



, W r 2 / r 



+ — Jn I Unm - 



r- \ To 



r (h- 



+ -Ittk^J -2^/1 f " ./„ f //„„, -^ ./,/ (,(,„„ - \ dr, 

 rn Jo \ ?■()/ \ n,/ 



= 27r/i„ 



(m — mo) / 

 Jo 



xJ';{x) + 



/;(.r)" 



dx 



' «nm (ri/ro) 



+ 2m / Jn{x)Jn{x) dx 



Jo 



= 9 



27r/3„ 



J? r/2/ 



(m -Mo)(.r./„(.r)./„'(.r)+.^J::(x) 



+ — -^ ./«(.V) ) + K-7lJ\ I ?'« 



Making use of the \'alue of Am found in the preceeding paragraphs, we 

 have 



[UnnT — l)Jn{Unm)~ L\MO / \ 2 



+ ""-^ Jlix) 



+ n - J ,A Unm - 



jJ-Q 



ro 



(13) 



For TM modes. 



Ht,n = jo^eiV*\l/,„ , 



./«./« 



To 



In this case. 



a - 2 Jnm ^ 2 Jnm 



fi,n = o) ei/io — — ^ = ^1 — — r 



2x(a)ei) ( (/X — /io) / ~ Jn \ jnm - I 



\ Jo L^" V ^0/ 



+ H J'n ijnn, -)] T dv + 2^/1 f ^^ ^^ ./„ (jn,n "^ /,/ 6»« -^ ^r) . 



/•o^ \ ro/J Jo ro \ ro/ V ro/ / 



