644 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



^' = Xl = 1 + 



1 + 0-' 



and 



(59) 



1 \ 2 2 



1 — 0-Xi ro^ ' 



The identification scheme is again that shown above. 



For — o-Q < 0- < an isolated identifiable point arises from Ib- 

 {In)T which is expressible in inverse form by the relations 



2 



1 - 





+ -r-; iS" and 



J2 Jn' 



p = (^'- 1) 



1 - 



(60) 



for 



To" 



\^i- 



The identification of the modes with n proceeds as in the earlier parts of 

 this section. 



III. Small p. 



To order p" there exist the following expansions for jQ": for the TEi„- 

 mode 



3^ = /3o^+ ^' 



1 - a"" 



2 



Un^ — 1 



^0 



V + 



+ 



2/3o 



(1 - o^r 



5 + 5Un — 2Un 



(Un' -ly ' HUn' - 1) 2 .. , X 



(^0^ - 1) (1 - Uj) ^ • • • ' ^""'^ 



where 



/3o^ = 1 - ^ 



To' 



For the TMi„-modes 



^2 o 2 0" , 1 3^0 — 2 2 



/3 = /3o - :^ -, p + 7:, :;^, ^^, ^^ p 



1 - a-" 



(1 - (7^)2 2(1 -^0^) 



(62) 



