A^(r.) = rTT^Vfa) • (2-l«) 



TRANS^fJSSION LINE EQL'ATIONS 155 



where P(.tr) is not a square matrix but a scalar and N{^r) is the square 

 matrix 



l/(r.) 



F(f) is the adjoint of the characteristic matrix 



/(r) ^ n- A (2.19) 



and Ti, Ta, • • • fm are the roots, assumed to be unequal, of the character- 

 istic equation 



l/(r)l = 0. 



The denominator in the expression for N{^r) is the derivative of the char- 

 acteristic function: 



" d 



fi^r) 



,,,/(f), 



The formal application of Sylvester's theorem then gives 

 cosh r = ^ = S.V(rr)fr 



e-^ = A - U" - /)* = 2A^(f.)X. 



,-nT ^ 2A^(r.)x;' 



Zo = (sinh T)-'B = S J^^^Al^ B 

 (X7' - X.) 



(2.20) 



where N{^r) is given by (2.18), the summations run from r = I to r — m, 

 and Xr is related to fr through 



2f. = X. + X7' , X. = f. - VrT^ (2.21) 



where the sign of the square root is chosen so that | Xr | < 1. Xr is related 

 to e~ in the same way that fr is related to cosh T. 

 2.4 Results for Any Symmetrical Section Line — Passive 



The solutions of equations (2.10) which reduce to the given values v(o), 

 i(o) at n = are 



v{n) = cosh uF v(o) — sinh nF Zoi(o) 



(2.22) 

 i{n) — —sinh nT' Zo v{o) + cosh nV i{o) 



where e~ and Zo are the matrices of §2.3. These may be put in slightly 

 different form by using the relations 



sinh ;/r Zo — Z^ sinh nV' 

 sinh nV' Z^ — Z1 sinh nV 



