200 Bh.LL SVSrF.M TECHNICAL JOURNAL 



the conductors, as shown, and $), $2 are the normal values of the 

 magnetic flux that cuts the surfaces bounded by the contours. The 

 term — 2E->" results from the symmetry of the system, which imposes 

 the condition that the electric and magnetic forces at corresponding 

 points in the outgoing and return conductors are equal and oppositely 

 directed. Also, it is unnecessary to write a third equation for the 

 field between the other wire and its sheath, because this equation 

 would be the same as (12). Therefore, the transmission is charac- 

 terized by only two modes of propagation. 



Since all the variables are propagated at the same rate, and since 

 sinusoidal currents are being considered, djdz may be replaced by — F 

 and didt by iw. Then 



£,' - E," + TV, = XJu (14) 



- lEo" + TFs = X.iU + /2), (15) 



where V is the propagation constant and 



A'l = i(joLi2 = reactance arising from the magnetic field between the 

 outer surface of the wire and the inner surface of 

 the sheath. 



Xo = ioiLoi = reactance arising from the magnetic field between the 

 two sheaths. 



The potential dilTerences Fi, Fo can be expressed in terms of the 

 currents by writing Maxwell's Law, curl II = 4tI, around contours in 

 the outside surfaces of wire and sheath. (Such a contour for the 

 wire is indicated by dotted lines in the sketch.) This gives 



2T:a^-^= - 47rFiFi, (16) 



dz 



27ra.^^= - 47rF2Fo. (17) 



az 



where 



Y\ = admittance across the insulation between wire and sheath. 

 F2 = admittance across the insulation between the two sheaths. 



Smce ill = — and //■> = > 



a 1 a 2 



r/i = \\Yu (18) 



r(/i + Li) = F2F2. (19) 



