G28 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



For X negative, F(ri)x) is positive and thus, as X approaches unity from 

 above G decreases. Again for x^ positive and n < Ui , F(rox) is positive 

 and thus, as X approaches unity from below, G also decreases. Thus for 

 any x', G(\, a) takes on its least value at c = 1, X = 1 and this value is 



X^ 



The minimum value of this limit in this region is Fin) — 1 and is greater 

 than —1. In the region — 1 < Xo < 0, cr > 0, x is between and l,and 

 the X curves run from o- = to o- = cc . G' will clearly decrease as a 

 increases from zero on any one of these curves. Thus G attains its maxi- 

 mum on 0" = 0, where its value is 



' F{ro Vl - X^) ■ 



I X I ^ \ 



Since F(ro\/l — X-) is positive for n < Ui and | X | < I, G is clearly less 

 than — 1 . In passing we note that for ro = ?ii, both G functions may 

 attain the value — 1 . 



As ro passes through Ui , the (Oo')t curve appears in the region under 

 discussion and together with (Ib')t delimits the region carrying the TEu- 

 solution curve already discussed at length. No qualitative changes occur 

 in that curve as ro is increased indefinitely. When ro exceeds ji , the 

 (Ii)t curve appears between (Oo')?- and {Ia')t- Between {Ia)t and 

 (/i')r the G functions have a region of common sign, yet no solution 

 curve arises there for a given p until ro reaches Ji/a/I — V~* From then 

 on, the 7i curve cuts {Ia)t , see Fig. 9(c), and a solution curve exists 

 between (//)r and 7i . It is cut off at the intersection (Ia')t — h; there, 



pr = and a, -^— are given by the same parametric formulae (46-8) 

 da 



applying to the cut-off of incipient modes, the parameter 6 being nega- 

 tive. The curve begins at cr = 0, where it satisfies the usual ecjuation, 

 which for this radius has two solutions. The solution with the smaller 

 X, belonging to the present curve, tends to the isotropic TMn-limit as 

 p -^ 0. At a fixed ro , sufficiently below U2 , this mode does not exist at 



* There a re some exc eptions to this statement. When 4.82 < ro < id = 5.33 and 

 p exceeds ■\/T^^~j^^/r^, a double-valued /S^ — o- curve exists between two positive 

 a values. For values of n still closer to U2 further regions of common sign maj^ 

 arise as a result of the interplay of the {Oi')T and (/a')?" curves. We have not 

 examined these regions closely. Such dubious regions are confined to the immedi- 

 ate neighborhoods below the Un . 



