1276 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



If x' » n" these expressions may be further simphfied : 



2 1 



Unix, px) = - —j=- sin (1 - p)x, 

 T \/px 



2 1 



Sn(x, px) = — -j^r- COS (1 — p)x. 



^ \/ pX 



Substituting these values in (36) we get : 



Wn{x, px) 



Unix, px) 



X cot (1 — p)x — n, 



i 



n 1 



Vnix, px) 

 Znix, px) 



= —px tan (1 — p)x ~ 



cot (1 — p)x 



1 tan (1 — p)x 



X 



With the restrictions (1 — p)x < 7r/2 and 7i <<C x we may write: 



Wnix, px) 



Unix, px) 

 Vnix, px) 



= px cot (1 — p)x, 



(46) 



n 



= - px tan (1 - p)x. 



Zjn[X, pX) 



With (46) the characteristic equation (1) is 

 1 1 



Xl .12 _ 



.x,'-x,' ri/»Vi)^i^^^5^^ 



P 2 2 



X-z — eXi 



_Xl J nipXl) X-i 



] 



1 Jn'ipXi) 1 I ^ 



^ , , — — tan 8x2 I = 0. 



X2 JnipXl) X2 



■]■ 



Solving this equation for cot 8x2 we get: 



F / F- 



cot 5x-2 = - + 4/ 1 4- J , 



9 



(47) 



in which 



F = 



Xl JnipXi) 

 X2 Jn ipXl) 



1 + 



2/2 2\ / 2 2\ ■ 



n [Xi — X2 j(a;2 — eXi ) 



fp" 



X^X^ 



.T2 JnipXl) 

 eXi JnipXl) 



To evaluate (47) we specify a value of A^/^nm and enter the right hand 

 side of (47) with 



2 2 0^^/32 2/1,1 A/3\ 



a;i = Pnm - 2 -— (S „,„a (1 + 9 ^— ) , 



X2 = (e — l)/?„^ a + Xl , 

 p = 1 - 5i 



