CriOKD-W AVK PU()l'A(iATl().\ 'nillorcJll (iYK().MA(ii\KTIC MKDIA (127 



d3- 

 curve along which -j— < 0, will tend smoothly towards the ff — a ciirxe 

 acr 



for ro just greater than Ui . The course of the TEn solution ('ur\-e remains 



qualitatively unchanged for all ro > iii . 



When ro passes through Ui , and the TEu solution curves escapes dis- 

 continuously to infinity, the solution curves below it disengage from 

 their former end points In+i — {Ia)t and instead end at the point 

 In — {I .a.)t ■ When ro exceeds ji , the curves /i and Oi escape intersection 

 with (X = \ — p simultaneously, for a > cto , and the curve (Ii)t makes 

 its first appearance. From the asymptotic formulae (App. II) the latter 

 runs to infinity between 7i and Oi , and now the solution curve which 

 ended for Ui < ro < ji at /i — (Ia)t is carried to infinity between 7] 

 and (/i)r • The asymptotic expression for /3 versus a, given in formula 

 (56) indicates that /3- tends to the isotropic value for the TMn mode. 

 No further qualitative changes w411 take place in behavior of this mode 

 as ro increases. 



As /"o increases through H2 (the value at which the isotropic medium 

 supports the TE12 mode), the (Oi)t curve makes its appearance, an 

 event accompanied by the escape of the uppermost incipient solution 

 curve (the one ending at (Ia)t — 1 2) to infinity. The escape takes place 

 in the same way as that of the TEu solution curve as ro passed through 

 Ui . The newl}^ escaped cur^^e, of course, represents the TEi2-limit mode. 

 The end points of the remaining incipient solution curves also jump 

 discontinuously to their next higher neighbors as they did at ro = ih . 

 The course of events as ro is increased further should now be abundantly 

 clear, and is summarized in Table I on page 642. 



We now turn to the region < o- < o-q and consider first the situation 

 < To < th . It is clear that in the area bounded by 0^, Oo , (/aOt and 

 {I' b)t both G functions are negative. There is no simple geometrical argu- 

 ment which determines the existence of a solution curve in this region. 

 It is therefore necessary to use a type of analytic argument, which is use- 

 ful in a number of other cases, although fully discussed only in the pre- 

 sent instance. 



We show that the least ^•alue attained by G(\i , a) in the admissible 

 region for p = (which contains all regions admissible for other p-values) 

 is greater than the maximum value of G(X2 , cr) in the range — 1 < X2 

 < 0, (T > 0. Consider the variation of G(ki , c) as the point Xi = 1, 

 (T = 1 is approached along a line of constant x iii the admissible region 

 f or p = (see Fig. 4). We have the relation 



G{\<y) = \ 



\ Firox) - 1 



A 



