64G THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



With these expressions for the x? the characteristic equation (63) takes 

 the form 



H(\i , a, n) = H(X2 , 0-, ro), 



where 



TO,,,,.).u^; 



^/i 



XF„ I ro i/ :---— 1 - n 



(69) 



For given a, and q, equations (67b) and (69) are simultaneous equa- 

 tions for Xi , X2 . When Xi , X2 have been found, ^^ = — X1X2 is kno^vn. 

 Since /3" must be positive, Xi , X2 must have opposite signs. As in the 

 ferrite, the convention Xi > 0, X2 < will be adopted. Equation (67b) 

 will hereafter be called the plasma relation. The transformation 



Xi — > — X2 J X2 — ^ — Xi , (T — > — (T 



leaves the plasma relation unchanged and changes n to —n in the 

 H-equation. As more fully explained in connection with the ferrite sec- 

 tion, it is therefore necessary to consider positive n only if a is allowed 

 to take on negative as well as positive values. As before, only the first 

 azimuthal mode number (n = ±1) is considered in this paper. 



The method of analysis is the same as that used for the ferrite. Here 

 we shall only sketch the most important steps; the reader will have no 

 difficulty in completing the analysis by referring to Section 4.11. For 

 fixed ro , a contour map of H is drawn in the X, a plane (see Fig. 16 

 drawn for ro '^ 2.2). The gross features of this map are determined 

 by the lines H = 0, H = ± =0 . For greater detail recourse is had to the 



1 — X^ 



lines = constant, along which values of H are readily generated . 



1 — ffX 



Further help is obtained from a knowledge of the location of the saddle 



point of H. The infinity curves are given by the same formulae as for 



the ferrite, except that the line X = is no longer an infinity line: along 



X = 0, H = —1. Zero curves are given by 



0" = — — 



1 _ ro^(l - X^) 

 X X[F-KX-0? 



The branches of aX = 1 are also zero curves in the same restricted 

 sense as for the G function. In the same notation as for the ferrite, all 

 In curves pass through a = 1, X = 1; all /„' curves through —1, —1. 

 The same is true for all 0„ , 0„' curves (n > 0) . The only exception is 

 denoted by Oq , it arises from that branch of F'^ along which F~^(l) = 0. 



