TRANSMISSION LINE EQUATIONS 163 



of the radical being chosen so that | X, | < 1 as in the symmetrical case. In 

 his second paper Koizumi has given a procedure which amounts to an 

 alternative method of determining A. 



We shall first assume that the T's are known and that our equations are 



i(n) = Ynv(n) + Vuv{n + 1) 



(2.48) 



-i{n + 1) = y2iv{n) + Y22v(n + 1) 



As described above A may be computed from the determinantal equation 



l/(X)| = 



where /(X) represents the matrix 



/(X) = r^2X^' + (Vn + F22)X + I'2i (2.49) 



Let Pr be proportional to any non-zero column in F(Xr) where F(\) is the 

 adjoint of /(X) and let pr be proportional to any non-zero row of F(\r). 

 Then the matrices P and P in the expressions (2.45) for v(n) and i(n) are 

 given by 



P=iP^,p2,--- Pn,] 



(2.50) 



P = [Pl,p2, ••■ Pm] 



where pr is the column obtained by transposing the row pr . The matrices 

 Q and Q are obtained from P and P by means of the equations 



Q = Y,,P + I'i2PA = - F22P - I'2iPA-' 



(2.51) 

 Q = -YnP - YnPA-' = YnP + I'2iPA 



which are derived from (2.45) and (2.48). 



The properties of the individual columns of P and P lead to the relations 



F12PA2 -}- (I'll + Y,,)PA + Y,rP = 



_ , _ (2.52) 



F12PA-- + (I'll + I'22)PA'^' + I'2lP = 



and these guarantee that the difference equations (2.48) will be satisfied 

 when the expressions (2.45) for v{n) and i{n) are used. 



When the Z's are known instead of the I"s the procedure is much the 

 same. The difference equations are 



v(n) = Zni(n) — Zuiin + 1) 



(2.53) 

 v(n + 1) = Z2ii{n) — Z^iiin -\- 1) 



and the equation to determine the Xr's is 



|/(X)I =0 



