656 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



When n = u„+i , the 0„ curve merges with the On' curve at 



-2 ro 



(w„+i2 — 1) Un+l^ ' 



Similarly all of the contours G = g oi the sheet to which 0„ belongs 

 merge with the corresponding G = g oi the sheet to which 0„' belongs, at 



_-^(1+4J^^. (88) 



These remarks apply to all 0„ , 0„' curves, Oo included. 

 The Oo-curve, for large X, behaves as 



<r = 



1 2 



1 — ro 



and so tends to a from above for n < I and from below when n > 1. 

 (In fact, for ro < 1, Oo lies wholly in the first quadrant; when Ui > ro > 

 1, Oo cuts cr = once.) 



The saddle points of G are most easily found by considering G in the 

 coordinate net formed by the curves x = const, and X = const. At a 

 saddle point 



dG 

 d: 

 and simultaneously 



^x ^x Lx^ \X 



^-2 = 



= 



The only saddle points that might be missed in this way are points at 

 which the two derivatives are not independent, that is points where the 

 X contours have vertical tangents, and it is easily verified that no saddle 

 points exist there. 



Proceeding with the differentiations, we find that — - = gives 



dX 



F(rox) = 

 or 



rox = Un (89) 



and so — =0 gives 

 dx 



- + ? F\u:) = 

 X X 



