166 BELL SYSTEM TECHNICAL JOURNAL 



where P, P, Q, Q are determined as before and ii(n) and j(n) depend upon 

 the generators. 



Here we shall consider only the physically important case in which the 

 voltages of the generators in the nth section are proportional to e~" where 6 

 is a constant. In this case g{n) may be expressed as 



giti) = ge""" (2.66) 



where g is a column matrix whose elements are independent of n. A 

 particular solution is obtained by assuming 



x{n) = ye 



Setting this in (2.42) gives 



(Ge"'' + He-' + G')y = g 



Hence a particular solution is 



x{:n) = {Ge-'' + He'' + G')"V"' (2.67) 



This method fails when 6 is equal to one of the roots Xi , • • • Xm , Xr , • • • Xm • 

 In this case, a particular integral may be obtained by a method similar to 

 one described in §5.11 of F.D.C. 



APPENDIX I 



Classical Solution of Uniform Transmission Line Equations 



For the sake of convenience we again assume that there are three circuits 

 in the transmission line. The equations to be solved are: 



^ = -Zi, f=-Yi (1.48) 



ax ax 



We adopt here the notation associated with equations (1.19) and (1.20), 

 Jiy") being the characteristic matrix of ZY , F(y~) the adjoint of /(7 ), and 

 7i , 72 , 73(w = 3) being the roots, supposed distinct, of \f{y ) | = 0. The 

 propagation constants 71,72, 73 are those square roots of 71 , 72 , 73 which in 

 physical systems have a positive real part. 



The solution may be constructed as follows: Let the column pr be 

 proportional (with any convenient constant of proportionality) to any non- 

 zero column of F{yr). The non-zero columns of Fiyr) are proportional to 

 each other according to a theorem in matrix algebra. " Construct the 

 square matrix P from the columns pi , p2 , pz '• 



P - [Pi,p2,pz\ (Al.l) 



" The method is that described in F.D.C. §5. 7(a) and §5.10 

 ^ F.D.C. §3.5 Theorem D. 



