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THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



The slope at cut-off: 



("^ - l) coth 6 



•sgno-, 



and 



0(3' 



d0' 



^dp J a p coth 6 \da Jp ' 



Type 2. Intersections of 



(0„')^ - 0, at (T = (70 = a/^ + 1 - I 5 type 2/ 



(0„) r — 0/ at 0- = — (To ; type 2„ . 



jS' 5^ 



Define Xo as a root of 



(47) 



(48) 



o-o = 



-1 



I"x7l 



1 - (1 - Xo') 



ro 



F-'(\o)/ J 



(49) 



with the follomng convention: for 2„', Xo is negative and the 

 n^ branch of F~^(\o) is used; for 2n , Xq is positive and the n^ 

 branch of F~^(Xo) is used. Now 



= !A_I 



O"0 



Near cut-off 



^2 _ I Ao I ro / 1 + o-o- , A 



^ — + ir ^"i^^^^y 1 



|_X^ 

 o-o 



o"o — I 0- I 

 + I Xo I o-o 



(50) 



(51) 



with 



(1 - \o')A = 



2 Xo 



1 - 



ro 



1 + Xo I O-Q 



) ((70 + 2 1 Xo I + aoXo') 

 / 



1 + o'o I Xo 



+ 



Ao 



For 2o a special situation arises for 1 < ro < i/i where there are two 

 positive solutions for Xo . We write 2oi and 2o2 for the points corresponding 

 to the smaller and greater of these. A special cut-off point will be labeled 

 2ooo and arises from 0/ and (Oo)r as Xi goes to infinity. For this point 



