598 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



pressed solely in terms of Xi,2 , ph and vh in the form 



Xl.2 = " 



1-1 



1 - '-^ X,.2 



9h 

 Again using Polder's formulae, this becomes 



2 1 ~ Xl,2 /QoN 



Xl.2 = :; ^— . (32) 



1 — (rA],2 



With these expressions for the x, the characteristic equation takes the 

 form 



G(Xi , cr, ro) = G{\, , a, To), (33) 



where 



G(X,,.„) = l^^ [In (,■./! 



- X2 



aX 



— n 



(34) 



Equations (31b) and (33) may now be considered for a fixed a and p 

 as determining associated pairs of values for Xi and X2 . Such a pair in 

 turn determines /3' = — X1X2 . Since /S' must be positive for propagation 

 Xi , Xo must have opposite signs. The convention will be adopted that Xi 

 is positive and Xo is negative. Equation (31b) will hereafter be called the 

 Polder relation and Equation (33) the G'-equation. 



An important fact of which frequent use will be made is that the 

 transformation 



Xi — > — X2 , X2 — ^ — Xi , <T ^> —a, V ~^ ~V 



leaves the Polder relation and /3 unchanged and converts n to — n in the 

 G-equation. It follows that it is necessary to consider positive n only, 

 provided we allow the pair cr, p to take on negative as well as positive 

 values. This corresponds to the physical fact that a right-circular wave 

 in a backward-directed magnetizing field behaves like a left-circular 

 wave in a forward field. 



The discussion in this paper is confined to the first azimuthal mode 

 number n = ±1. Accordingly the symbol F will replace Fi in what 

 follows. 



Before commencing the graphical analysis of the G function it is ad- 

 vantageous to consider briefly the function F{x) = xJi(x)/Ji(x), which 

 we require for real and for purely imaginary x. By logarithmic differentia- 



