648 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



X-pairs determined from the if-diagram do not necessarily solve the 

 problem, since for a fixed q, they may not satisfy the plasma relation 

 (G7b). To take it into account, we interpret it as a transformation of the 

 whole of the second quadrant onto part of the first, and of the whole 

 fourth quadrant on part of the third. Writing (67b) in the form 



2 

 '^ .-1 



(T a i. — crA2 



we see that the curves X2 = const, transform into a bundle of hyperbola 

 passing through the intersection of a- =1 — 2 with a = 1/X, that is, 

 through 



Xio = _ rr~ ; > o'o — Vl — 



VT 



e 



These hyperbolae have vertical asymptotes Xi = — — , and cut the line 



X2 



0- = Oin — Xa.For a fixed positive <j < ao, Xi decreases from l/c to a/ 

 {l — q) as X2 increases from — =0 to 0, but, when cr > o-q , Xi increases 

 from l/o- to o-/(l — q). Thus the second quadrant transforms into the 

 region between a = X(l — g") and a = 1/X in the first quadrant. Simi- 

 larly, the inverse transformation X2 = T'(Xi) transforms the fourth 

 quadrant into the region between o- = X(l — g) and o- = 1/X in the 

 third quadrant. Points outside these regions cannot be site of acceptable 

 solutions of the H equation. In order to locate acceptable solutions, the 

 H = equation is now written in the form 



H(ki , a, n) = H(T{\i), a, To) 



when cr > 0, and in the form 



H{\2 , c, n) = H{T{\i), cr, n) 



when a- < 0. These equations represent the curves of intersection of 

 the //-surfaces. Along each such curve, both //-equation and plasma 

 relation are satisfied. Their projections onto the first (or third) quadrant 

 give Xi (or X2) as a function of a, and hence X2 (or Xi) from the plasma re- 

 lation. Thus jS" = — X1X2 is known along each solution curve. The rough 

 locating of the solution curves, and the estabhshment of precise analyti- 

 cal formulae near special points on them proceeds in complete analogy 

 with the ferrite case. Here we shall consider only the radius ro '^ 2.2, 

 as typical of radii large enough to permit propagation of the TEu mode 



