WAVE PROPAGATION OVER CONTINUOUSLY LOADED WIRES 201 

 Substituting (18) and (19) in (14) and (15), respectively, gives 



(20) 



(21) 



Xl — yT ] IX = £0' — El", 



X, - Y^ ) (/, + /2) = - 2E,", 



and substituting (1) in (20) and (21) gives the two equations of the 

 currents. In order that they shall be consistent, the determinant 

 of the coefficients must vanish. Therefore 



r2 



X\ — ^ — Z'ji -\- Zii 



J 1 



Xi — — + 2Zoi 

 ■^ 2 



A'o 





7 ' 



+ 2Z.J 



= 0. 



(22) 



The roots of this equation give the required solutions for the propaga- 

 tion constant. First, however, it is convenient to introduce two 

 known propagation constants. Let 



712 = propagation constant determining tran smissio n along one 

 wire with its sheath as the return = -ylYiZn- 



722 = propagation constant determining transmiss ion a long the 

 two sheaths if the wires were removed = VFoZoa- 



Then, from (1), (20) and (21), 



Z12 — Zii — Z21 ~r Z22 r A 1) 



Z22 ^^^ 2/^22 I -^2> 



substituting (23) in (22) and rearranging, 



712- - r2 



(23) 



Fi 

 - 2Z..0' 



— /v 22 



T22" ^ " 



Fo 



= 0. 



(24) 



Expanding 



r - P(722- + 7l2-) + T12-722- - 2Z22''FlF2 = 0. 



(25) 



The remaining impedance can be eliminated by introducing 7, the 

 propagation constant that would characterize transmission if the 



