GUIDED-WAVE PKOPAGATION TIIKOUGII GYKOMAGNETIC MEDIA G55 



The equation may also be used to furnish an expansion near x = 0. 

 This is 



.2 ,4 



F(x) = 1 - ^ - ^ + higher terms. (81) 



Finally, putting x = jy, one finds from Eq. (79) for large y 



1 3 1 



FUy) = 2/-:^ + g-+ higher terms. (82) 



APPENDIX II. INFORMATION PERTAINING TO THE CONSTRUCTION OF 

 G^-DIAGRAMS 



The accurate construction of the contours G = const, is conveniently 

 based on the contours (l — X )/(l ~ o-X) = const, along any one of which 

 G is a function of X alone. These contours are shown in Fig. 4. Their 

 asymptotic properties are almost self-evident. 



The curves G = g = const, have various asymptotes. These, together 

 with their range of ^'alidity, and their Polder transforms where needed 

 are stated in Table IV. 



The formulas given in Table IV show that the curves G = const, 

 generally have two kinds of as3rmptotes; linear and hyperbolic. Formula 

 (83) shows the behavior of G along a line of constant finite slope unequal 

 to 7*0 /jn , the asymptotic slope of the /„ curves. Parallel to a line of 

 slope ro /jn all G contours must be found, not just the restricted range 

 given by the first formula. Writing o- = ( ro^/jn) X + x in the equation 

 G = g, and expanding F near its pole j„ , we find x in terms of g and ob- 

 tain (84) which holds for all g, from — cc to + =o . When g = it also 

 gives the linear asymptotes of 0„ , 0„' curves except Oo , as is readily 

 verified from the equation 



_ 1 _ ro\l - X') 



"" X x[F-Hx)? ' 



for the zero curves. 



Formula (85) shows how the G-contours tend towards o-X = 1 from 

 the side o-X > 1 as X -^ 0. 



Formula (86) relates the asymptotic behavior of the curves G = g 

 to the zero curves, g = 0, for small X. All G curves approach zero curves 

 arbitrarily closely as X — > 0. The only exceptions are the infinity curves 

 whose form near X = is 



