618 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



0.5 



P>' 



0.2 



-t.O -0.8 >: -0.6 -0.4 -0.2 



'To=5.33 



0.4 0.6 



Fig. 10(b) — /32 versus p for small values of er — the TMu-liniit mode, 

 the negative root X20 (unique for the present radius) of 



Fin 



— (TfiK'i 



= \1. 



The associated ^^ is -X20X10 = -~ and is shown in Fig. 11(a). The way 



Co 



in which jS^ approaches this value as cr ^- o-q can be found and is one of 



the more subtle examples of behavior of a mode near a special point. 



Writing o- = (to — 5cr, Xi = Xio — 5Xi , we observe that, since cto + p 



0, the Polder relation in the form 



o-Q 



1 0- -f p - l/o- 



Xl = - -t- ' r 



a 1 — aK2 



d\i 



fully determines -— ^ ; any variation due to 6X2 vanishes at o- = ao . SXo can 

 a<T 



be determined from the G^-equation. Near o-X = 1 — (X > l),G(Xi,o-) is 

 given by 



ro 

 Xi 



1 - o-Xi 

 Xi^ - 1 ' 



which near ao , Xio may be written 



