160 BELL SYSTEM TECHNICAL JOURNAL 



Hence, letting n -^ ^ and using condition (b) satisfied by e~^ , equation 

 (2.28) is obtained provided that (i) the terminal conditions at the far end 

 are such that v^n) remains finite, (ii) the sum in (2.28) converges, and (iii) 

 the expression in the last line in the equation just above approaches zero 

 as w -^ 00 . 



The results obtained by the formal application of Sylvester's theorem 

 may be verified by using the results of Appendix II and writing iV(fr) as 

 the product of a column matrix and a row matrix. They may also be 

 verified more directly. For example, setting w = in the expressions (2.23) 

 for v{n) and i{n) in any passive symmetrical section line and using 



m 



E N{^r) = T, (2.38) 



r=l 



which follows from Sylvester's theorem, we see that v(n) and i{n) reduce 

 to the appropriate values v(o) and i(o) at n = 0. Substituting v{n) and 

 i(n) into the difference equations (2.10) and using 



BC ^ A~ - I 



(I^r - A)N(^r) = Ni^rX^r - A) = 



(2.39) 

 BN'(^) = N(^)B 



CN(^) = 7V'(f)C, 



shows that they are solutions. The second of the relations (2.39) follows 

 from the fact that N(^r) is proportional to the adjoint F(^r) of /(j'r). In 

 the third and fourth relations 



iv(r)= ^''^^ 



\m 



(1) 



which is in agreement with the definition (2.18) of iV(fr)- To establish 

 the third relation we start from,^" 



(^i-A)Fir) = i\m\ ' 



(f/-^).v(r) = /|/(f)|/l/(f)r" 



Postmultiplication by B gives 



(^i-A)N(nB = B\m\/\m\''^ 



We also have 



i» F.D.C. §3.5. 



(f/-.4')FX0 = / I /(f) I 



i^i - A')N'(s) = i\m\/\M)\''' 



