642 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



Table I 



II. Asymptotes, genesis of the modes and spot poirits 

 For I cr I ^ CO there are asjnnptotic formulae: 

 For TEin-modes 



2N 



1 - 



ro' 



1+^ 



For TMi„-modes 



/3^ 





(55) 



(56) 



For 1 0- 1 > (To , all modes have their origui in the points cr = 1 , Xi = 1 

 oro-= — 1, X2=— 1, where /S" -^ co. The variation of jS with a in the 

 neighborhood of these points is described by the two expansions: 



for 0- ~' 1 



0n' = 



V 



an + 1 



J; , j l - ftn , 2a^ Z'? _i_ 1 ^ . ^ 



Xn"- \P 



+ 0((T - 1) 



(57) 



where 



and 



with the scheme 



an 



a„ + 1 2ro" 



F{Xn) = 1 



Xn > 0, 



