98 BELL SYSTEM TECHNICAL JOURNAL 



minus sign being due to the fact that the current in the return is 

 in the negative direction of z. 



Inserting (23) and (25) in (22) gives 



Z — Zi -\- Z2 -{- ipLu. 



The quantity Z2 may be determined in the following manner. From 

 (7) we have 



E2' = Z21 /i + Z22 I2, (26) 



where I2 is the current in the iron sheath. The value of this current 

 can be found by applying the condition of continuity of electric 

 field intensity at the common surface of the iron and the sea water, 

 as in equation (9). This gives 



Z21 1 1 + Z22 1 2 — Zii 1 1 -\- Z32 1 2 + Z33 Iz, 



in which I3 is the current in the sea water. From (8) it can be seen 

 that Z33 = 0, since X3 = 00 , therefore 



■^2 = 777 77- -^1- (27) 



Substituting (27) in (26) gives 



Z'n 



E2' — Z21 + -1^, ^r Z'n /i, 



L ^22 ~ -^32 J 



and by comparison with (25) we have 



7' — 7" 



2 = — ^21 — ^77 ^7- ^l^^ V^o; 



^22 ~ ^n 



as the internal impedance of the return conductor. The resistance 

 and reactance per unit length of this portion of the circuit are then 

 represented by the real and imaginary parts of (28) respectively. 

 We may then determine R and L from the formula 



Z = i? + ii>L = Zi + Z2 + t>Li2, (29) 



where Zi and Z2 are calculated from (23) and (28) and 



L12 = 2 log—, 



62 and ai being the inner radius of the iron and the outer radius of 

 the core conductor, respectively. 



For purposes of comparison, the return impedance is calculated 

 for the case where the iron armoring is absent, the return current 



