542 BELL SYSTEM TECHNICAL JOURNAL 



also 



f — f A 



dx dt ' 



f f 4- ^^ — ^* (1\ 



J IV J g ~r -JZ — TT » (^) 



where $ denotes the impressed magnetic flux threading the contour, 

 per unit length. Subtracting (2) from (1) and observing that 



V - F = ^Q, 



C (3) 



- $ = L/, 

 we get 



where, of course, Q is the charge, C the capacity to ground and L the 

 external inductance, per unit length of the wire. 



To eliminate Q from (4) we make use of the equation of current 

 continuity, namely 



-i = f + ^'' (^) 



where /' is the leakage current per unit length of the wire. If the 

 wire is embedded in a homogeneous leaky medium, then 



/' = |r"(2 = G\V - F), (6) 



where G^ is proportional to the conductivity of the medium.^" If, 

 furthermore, there is direct leakage admittance from the wire to 

 ground (as at poles and insulators) of amount Y' per unit length,^^ 

 when regarded as uniformly distributed, then 



I' =^Q+ Y'V = ^Q+ TF, (7) 



where 



G = GO + r. (8) 



On substituting the last value of /' into (5), setting d/dt — io), then 

 differentiating with respect to x, and finally substituting the resulting 

 value of dQ/dx into (4), we get 



f ^ ■ T^T 1 dU y dF 



G + loiC dx^ G + ii>iC dx 



'■'■^ A formula for G" is equation (18) derived below. 



^' While G" is merely a pure conductance, Y' is in general an admittance (leakage 

 admittance), because the insulators and poles have capacity as well as conductance. 

 Hence G, defined by (8), is an admittance. 



