326 BELL SYSTEM TECHNICAL JOURNAL 



This assumption, which is equivalent to assuming that tanh cyp differs 

 appreciably from —1, does not appear to restrict our results since tanh 

 cyp is either purely imaginary or else real and positive. 



Substituting the appropriate values in (3.3-8), neglecting higher order 

 terms, and using the definition (3.3-6) for Cmp leads to 



Xp = a~p bpC ^[1 — (1 — bp/<J:^Vpp + Zl (o-p O'm)" {Cmp{<Tp — 8p){<Tm — 8m) 



m 



+ DmpiDpm — (Tmkpm) ~\~ \(Tp — dpjDpmVmp — 8mDmpVpm]] (3.3—10) 



A reduction similar to that used in going from the first to the second line 

 of (3.3-7) ^ives our final expression for Xp 



Xp = - I — jz I 7~\o I ^PP ^ -* '' pm rmp^m/yi- l *n 



8p + yptp (1 + ' '^ ' 



"^ ^ S: a a \ * \rx^ X^\ \VprabmFmp/tp (3.3-11) 



m OmOp\y- -T tm)\pm ~ Op) 



— VmpbpFpm + FpmFmp{dp + 8m/tp){8m "— ^p) ) 



The above expressions for Xi and Xp have been derived from the first of 

 equations (2.3-3). The second of equations (2.3-3) determines the column 

 matrix y in the same way that the first equation determines x except that 

 coth cT now replaces tanh cT. Therefore, we may obtain expressions for 

 the elements of y by replacing the /'s (where /» = tanh cyi) by their recipro- 

 cals in the expressions for the corresponding x's (i.e. in (3.3-7) and (3.3-11). 

 The values obtained in this way lead to, when i 9^ p, 



fT = c'^'ixi + y») 



= ^-T^e'"''^'--''-'-' sinh c(yi + yp)[-Vip8p - Fip/{8i + 8p)] 



(3.3-12) 

 ft = e''{xi - yi) 



= 57^e<=<^i+«p-^^-V sinh c(yi - yp)lVip8p - Fip/(8i - 8p)] 



where we have used the expressions (2.3-4) and (2.3-5) for/" and/+. 

 When i = p, w 



Fp = «'''(^p + yp) - ^'''' = -e''''-''\A^ (sinh 2cyp)/2 + ^2] (3.3-13) 

 where 



Ai — 2vpp -\- {yp — 8p)8p — JL, Vpm Vmp + 72 .2 



m L Om — Op J 



_ V' (cosh 2cyp - e-'^'y^) ^ ^ , , 2 



A2 ^^ 2Lj /)£ 5 /«2 52 \ I*' pm '^ inp i^m \ * mp i^ pm p "T~ l^ pm l\ip\ 



m lOmOpKPn ~ p) 



