1172 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



Thus for a < - the analysis may be confined to equation (34) which 



will give the course of the TEIM limit mode. The graphical analysis 

 follows closely that of the cylindrical waveguide (Part I, Section 4.1); 

 the part played by the G-function there is now played bj' the function 



^ == Vr^. '"^ ^l/T 



- X2 



- <tX 



A contour map is drawn of the function L in the upper half of the X-o- 

 plane. Onlj^ the upper half plane need be considered since the L-equation 

 (34) is unchanged by reflection in the origin. This, incidentally, means 

 that the present situation is reciprocal, so that a, p hereafter can be 

 considered positive. Pairs of values Xi(>0) and X2(<0) satisfying the 

 L-equation can then immediately be read off, but while they will give 

 ^ = X1X2 , they do not necessarily, for given o-, p, satisfy the Polder re- 

 lation, equation (35). To ensure this, the quadrant, X < 0, <r > 0, 

 and the L-contours therein are transformed on to X > 0, cr > by the 

 Polder relation for fixed p : 



X3 = '^ + P - ^' = rao. 



1 — crXi 



The surfaces L\ = L(Xi , a) and Li = L(T(\i), a) will intersect in a 

 number of curves, along whose base curves in the X-o- plane both the 

 Polder equation and the L equation are satisfied, and along which /3^ = 

 — X1X2 is known as a function of a, p. The zero and infinity curves of 

 L(X, a) are denoted by 0, / when X > 0, and by 0', /' when X < 0, The 

 transforms of 0', /' onto X > are denoted by (0') r , (I')t . The suflfix 

 n denotes the infinity (zero) curves corresponding to 



1 — X^ / l\^ ^ /1 — X" ^\ 



r^7x = ("+2J^' ir:^X = "'5^j n integral. 



The lines X = 0, +1, —1, are all zero curves denoted b}^ 0^ , 0^ , Ob', 

 respectively. The line Xo- = 1 is a conditional infinity curve, called Ic . 

 It is an / curve when viewed from crX > 1 for X < 1, and when ^'ie^^•ed 

 from o-X < 1 for X > 1. Otherwise (for aX < 1, X < 1 and o-X > 1, X > 1) 

 it is a limit curve of all possible curves L = const., where the constant 

 takes on any value indefinitely many times. (See Part I, Section 4, 

 where the curve o-X = 1 is a conditional zero curve, Oc .) Fig. 8, drawn 

 for a '^ 1, p '^ 0.5, shows the part of the first ({uadrant allowed b}' the 

 Polder relation divided into regions of like and unlike sign of L(X, a) 

 and L(T(\i , a)) by the various 0, / and (0')?- , (I')t curves. The un- 



