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BELL SYSTEM TECHNICAL JOURNAL 



3. The characteristic impedance matrix Zo satisfies the relation 



Z = ZoFZo (Al.ll) 



4. The inverse matrices P~ and Q~ always exist if 71 , 72 , 73 are distinct. 



APPENDIX II 



Classical Solution of Symmetrical Section Line Equations — I 



The method of this section is very similar to that of Appendix I. The 

 equations to be solved are (2.10) or one of the sets 



v{n) = Zni(n) — Zniin + 1) 



v(n + 1) = Zuiin) — Zni(n + 1) 



i(n) = Ynv(n) + Ynvin + 1) 



-i(n-\- 1) = Ynv{n)-\- Ynv{n + 1) 



(A2.1) 



(A2.2) 



which are obtained from (2.5) and (2.6). We shall use the notation asso- 

 ciated with equation (2.19), /(f) being the characteristic matrix of A, F(f) 

 the adjoint of /(f), and ^1 , ^2 , ■ ■ - ^m the roots, assumed unequal, of the 

 characteristic equation | /(f) | = 0. The diagonal matrices A and 2 are 

 defined by 



A = 



Xi 

 

 







X2 











where 



2fr = \ 'T ^T , 



Vf?-i •• • 



\/f 1 - 1 



. Vtl- iJ 



Xr = f r - Vf? - 1 = 



(A2.3) 



r. + Vtl-i 



(A2.4) 



In general, electrical energy will be dissipated in the typical section and 

 from the physical significance of Xr , as seen from equations (A2.8) below, it 

 follows that the sign of the radical in (A2.4) may be chosen so that | Xr | < 1. 

 Since \/fr — 1 = fr — Xr = KX7 — X,) it follows that 



KA-^^ - A) 



(A2.5) 



Let the column matrix Sr be proportional to any non-zero column in 

 F(^r) where F(^) is the adjoint of /(f). (It follows from the theory of 

 matrices that the non-zero columns of F(f r) differ from each other only by a 



