544 BELL SYSTEM TECHNICAL JOURNAL 



current /' flowing outward, per unit length of the wire, is given by 



r = f aErdS = ^ f eErdS, (15) 



the surface integral being taken over the unit length of the wire. 

 But, by Gauss' theorem (^ being a constant whose value depends 

 only on the units), 



feE4S = 4TQI^, (16) 



the resultant axial electric flux from the ends of the element being 

 negligible compared with the radial electric flux from the lateral 

 surface. Thus 



/' = ^e, (17) 



and comparison of this equation with (6) gives the result 



GVC = iwa/e^. (18) 



In this connection it may be noted that the leakage current repre- 

 sented by (15) does not directly depend on the impressed field, but 

 only on the field produced by the wire itself. This is because the 

 assumed medium is homogeneous and isotropic; hence a in (15) can 

 be taken outside the sign of integration, and then the conclusion 

 follows from (15) by noting that one of the constituents of E,- is the 

 impressed radial electric force/,, and that 



ff4S = fd[vf-dv = 0, 



since the divergence of the impressed electric force must be zero. 

 The conclusion would not follow, in general, if the medium were either 

 heterogeneous or aeolotropic. It may be noted that a homogeneous 

 isotropic medium surrounding a wire and containing direct leakage 

 admittance paths from the wire to ground may be regarded as a 

 heterogeneous aeolotropic medium. 



The value given for 8 in equation (9) of the text, namely 8 = 47ro-/e^, 

 is readily derivable by combining equations (4), (10), (5) of the text 

 with (17) of this Appendix. 



Equations (90) 

 If there were no impressed potential at the primary wires (F/, = 0), 

 the equations of continuity would be merely 



-^'= L F.,F,.., (A= 1, ••• n), (19) 



dx 



1=1 



