616 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



function through all possible values of the other in the region bounded by 

 /i , {Ib')t , (Oi')r we deduce that the solution curve just discussed con- 

 tinues into that region and persists as o- -^ =c . For, the asymptote of 



(0/)r is 0- = -^ X, and between it and X = 1, which is the asymptote of 



{Ib)t , G(T(\i), a) takes on all values between and — x ; in particular, 

 the limited range of values assumed by G(Xi , a) in this region. The be- 

 havior of the solution curve for large a may be deduced by using the 

 asymptotic formulae for curves G(Xi , a) = g and GiT{\i) a) = g which 

 are given in the appendix. These are 



and 



0- = — Xi — 



1 - ^ 



p+211-f?^)--^: 



* ro^/ 1 — V_ 



It is clear that ^ at a point of intersection is given by — ro /u{ plus terms 

 of order 1/Xi ; substituting this value in the second equation gives the 

 solution curve correctly to order 1/Xi in the form. 



2 



To . 



(T = — \l- V- 



When the solution curve has such a linear asymptote it is convenient 

 to calculate /3 from the formula 



j3^= 1 + - V terms of order higher than 1/cr 



a a 



which is readily obtained from (30). In the present case 



^2 = /i _ !^"U 1 -f ? j -f higher terms in l/cr. (38) 



As 0" — > °o, iS" tends to the value appropriate to the TEu-mode in an 

 isotropic medium (^t — > jUz = /xo , k ^ as a- ^ 30). Thereby the whole 

 solution curve is classified as specifying part of a TEn-limit mode. 



The remaining section of the TEn-limit mode in the upper half -plane 

 is again found in the region between (O/)?- and (7b') r foi" «■ < ao . Any 

 Ime 0- = constant < o-q cuts these two curves at two values of Xi . As \\ 

 varies between these values, G(T{\i), a) varies from to — 00 ; it is, thus, 



