162 BELL SYSTEM TECHNICAL JOURNAL 



the m modes of propagation are the same in the positive as in the negative 

 direction, as indicated by the appearance of A" and A~" in (2.45). Cor- 

 responding to any given propagation constant say \r , there are two modes of 

 propagation, one in a positive direction and the other in the negative 

 direction. The distribution of the voltages corresponding to these two 

 modes are given by the rth columns in P and P, respectively. The fact 

 that P and P differ shows that the distributions differ according to the 

 direction of propagation even though the propagation constant is the same. 

 A is still the diagonal matrix defined in (A2.3) but now the computation of 

 the elements Xr is more difficult than when the section is symmetrical. 

 They are defined as the roots of the equation 



I GX2 + //A + G' I = (2.46) 



which are less than unity in absolute value. The second of the equations 

 (A4.5) shows that the roots of (2.46) are the same whether the Z's or the 

 r''s are used in place of G and H. Of course, this is to be expected on 

 physical grounds. The third of the equations (A4.5) may be used to show 

 that the roots of (2.46) are also the roots of 



X^ - / \B 



= (2.47) 



XC \D - I 



From the form of (2.46) it follows that if X,. is a root so is X7 . This fact 

 may be used to simplify the determination of X,. . When the substitution 



2r = X + \-\ \ = t - VF^^l (A2.4) 



is made equation (2.46) may be written as 



= I (G + G')f ^ H -\- (C - G) Vr^^=l I 

 = I (G + G')f + H I 



+ (f" — 1) times the sum of determinants each ob- 

 tained by replacing two columns of | (G + G')^ -\- H \hy the cor- 

 responding columns of (G — G') 



, ,^^ . NO • , r ^(^ ~ \)(m — 2)(m — 3) , 

 + (r - 1)- times the sum of -^ ^^-— ^ determi- 

 nants each obtained by replacing four columns of |(G + G') f + 

 H [by the corresponding columns of (G — G') 



+ ••• 



The last equation is a polynomial of degree fn in f which is to be solved 

 for its roots ^1,^2, • • • Tm • For simplicity we assume that these roots are 

 distinct. Xr is then determined from fr by the relations (A2.4), the sign 



