526 BELL SYSTEM TECHNICAL JOURNAL 



where 



and 



Zjj = Zkk, 



the Z coefficients being the self and mutual impedances of the indi- 

 vidual conductors. The required self and mutual impedances, Zs and 

 Zm, respectively, however, are the impedances of the circuits 1-4 and 

 2-3. Owing to the relations, I\ = — h and h = — Is, of the cur- 

 rents, Zs and Z„i are given by 



Zs = 2{Z\i — Z41) 

 and (23) 



Zm = 2(Z21 — Z31). 



Thus, from equations (22) and (23), we have 



ZJi + Z„j2= 7(Fi - F4) = 2(£i+ F), n = a. (24) 



The problem is then reduced to the determination of E and F in terms 

 of /i and I2. 



The function F must satisfy Laplace's equation in two dimensions 

 and may be resolved into four waves centered respectively on the axes 

 of the four wires, each satisfying Laplace's equation. Thus, at any 

 point {rj, Qj) in the dielectric, F may be written 



F = Fi -f 7^2 + F3 + F,, (25) 



where 



Fj = Aoi log Tj -\- Z ( A ni ^ „ + Bnj ^ „ I , J = 1, 2. 3, 4. 



w = 1 \ ' J ' J / 



The arbitrary constants Aoj are determined by the relations 



But owing to the specific configuration and to the conditions 



U^ - U and /s = - I2, (27) 



the 8m arbitrary constants Ani, Bnj may be reduced to 4». Thus, we 

 have 



An\ = — Ani = ^n, B n\ = B ,,4 = B n, 



(28) 



An2 — ~ An^ = C „, Bn2 = -6„3 = /^rii 



and also 



Aui = — Aiu = — 2iuicx}Iu Ao2 = — Ai):i = — Ifjiicol-i. 



