1174 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



along Avhich L2 = ao); crosses from one allowed region to the next 

 through the intersection of 0„ with {Ob')t and proceeds to its cut-off 

 point (0„ - (0^)r). 



Formulae giving ^ in the neighborhood of special points on the solu- 

 tion curves, or for special values of the parameters, are obtained exactly 

 as in Part I, by expansion of the L-equation. The results are tabulated 

 l)elow. They relate to antisymmetric modes only. 



Cut-off of the TEM-limit mode ((t = 1 - p, /3' = 0) 



Near cut-off 



d(3' 1 



P 



/ tan a\ 



For p > I, the TEM mode has no lower branch. 

 Behavior near resonance (a = 1): 



(37) 



TEM mode: /3' = :r-^p, 



n ' incipient mode: j8 = 



\ 2a2 /l - a 



(n = 1,2, •••) 

 Cut-off of the n incipient mode (parametric representation) : 



Xi = e ; a = e I 1 ^"2 I + -^-2 « 



V 



a — Xi; 



= 



7o2 2 2/ 2 2 



rf/3 mir f m T 



- 1 coth e 



2 2 

 m T 



(38) 



(39) 



, . _e , 2 tan a -e e 

 1 e + — z — e — e 



(The reader may note the similarities of these formulas to those 

 for the cylindrical waveguide). 



Spot-point for the n incipient mode 



_2 



1 - 



a 



r 





(40) 



p = (1 L- a)(/3' - 1\ 



