626 thp: bell system technical journal, may 1954 



if the n*** branch of F~ (X) is used. Thus, as ro increases, the successive 

 cur^'es either all pass through a fixed point (which can only be X = ± 1 , 

 or = ±1, 71 > 0) or move steadily up or down without further intersec- 

 tion. An 0„ curve starts from trX = 1 at ro = and falls for X < 1, rises 

 for X > 1, as ro increases. For large X, since a ^^ jn/fo (X — 2), n > 0, 

 the 0„ and /„ curves move together with a constant separation. Oo is 

 singular, since it does not pass through X, o- = 1 and falls steadily for all 

 X; it tends to o- = for large X. TheOn' curves rise from crX = 1 at ro = 

 for — 1 < X, fall for X < —1. They run parallel to /„+/ for —X very 

 large. For small X there is an expansion 



,= U-rLy J'i + o(x) 



holding for 0„ and for 0„'. This indicates that for ro < u„ , 0„ goes to 

 -f ^ and 0„' to — 2o for small X, but at ro = ?i„+i , 0„ and 0„' merge 

 momentarily at 



0, 



2ro' 



•U„+l2(w„+i2 — 1) 



< 0. 



For larger ro , 0„ goes to — » and 0„' to + =o . Since the union of 0„ and 

 0„' takes place at a negative o-, it is clear that 0„ curves, unlike /„ curves, 

 may cross the line <r = twice. Intersections of the 0„ , 0„' curves with 

 the Polder boundary are difficult to examine explicitly and this may lead 

 to some obscure situations for < | X | < 1. However, for a > o-o , 

 since 0„ and /„ have a fixed separation for large | X |, this pair escape 

 intersection with the boundary at the same value of ro , namely jn ■ 

 Similarly 0„' and /n+i escape together at ro = jn+i for a < — <ro . 



We shall now examine the effect of varying ro upon the sequence of 

 modes when a > ao . When ro is less than iii , a case in which the iso- 

 tropic medium would not propagate, no part of Oo' Ues in the upper 

 half plane and there is then no (Oo')r curve. The solution curve w'hich in 

 the previous discussion of Section 4.12 was assigned to TEn , after 

 passing the intersection [/i — (Ib) t] can no longer escape to infinity and 

 terminates on [/i — (Ia)t]- Thus, the TEn mode at this radius has 

 become an incipient mode with cut-off and other properties given by the 

 formulae already quoted for such modes. As ro approaches ?/i from 

 below, the j3' — a curve is double valued between o-cut-off and some larger 

 value. This is borne out by the fact that d/3 /da becomes positive at 

 cut-off, and by the observation that the solution curve bulges towards 

 large a- between /i — (Ib')t and its terminus. The part of the /3 — o- 



