202 BELL SYSTEM TECHNICAL JOURNAL 



wires were in contact with the sheaths. (In order not to disturb the 

 dimensions, it may be imagined that the insulation between wire and 

 sheath be replaced by an infinitely conducting material, which, how- 

 ever, is assumed to conduct no current axially. Then E^' — Ei" 

 — Xili.) To find 7, make Yi infinite and solve (25). Then 



r = Y.Z, 



and 



Z = Z,, - ^^ • (26) 



Zi2 



Therefore 



2Z22''J^ll^2 = T22^7]2' ~ 7"7l2"- (27) 



Finally, substituting (27) in (25) and solving the resulting equation 

 gives the two solutions for the propagation constant, 



2P = 712- + 722' ± V(7i2'' + 722'/' - 47-712-. (28) 



The arbitrary constants remain to be determined. The currents 

 are, in general, 



7i = ^iie-f-- + ^126-1- + ^iie^- + ^126^-% 



(29 



The condition of principal practical interest is that of a long cable 

 with connection made to the two wires and with the sheaths left free 

 at the sending end. For this case, the conditions are 



(1) At s = 0, /i = Jo and h = 0, 



(2) At s = 00 , /i = and A = 0, 



where Jo is the current delivered to the cable pair at the sending end. 

 From the second condition, 



Bii = B12 = B21 — B22 — 0. (30) 



From the first condition, 



Au +-4 12 = /o, 



(31) 

 A21 + ^22 = 0. 



But these constants must satisfy, for each of the two values of F, 

 the equations of the currents, whose coefficients are given in the 



