112 BELL SYSTEM TECHNICAL JOURNAL 



The transcendental equation defining the propagation constant F 

 may be solved by successive approximations; a convenient first 

 approximation is T = Ti = 4gZ{G), Z(0) being the earth-return self- 

 impedance of the conductor. Equations (3) and (4) are of the same 

 form as the solution for conductors of small leakance, except that the 

 propagation constant for the latter is taken equal to Fi above. The 

 effect of the earth potential appears in a first approximation as the 

 second term in the expression for G{T). 



For earth resistivities within the usual range and for electric railway 

 tracks or underground cables the two terms in the expression for 

 G{V) are frequently of the same order of magnitude. Appreciable 

 errors may therefore be obtained by neglecting the second term, and 

 in correlating the results of measurements this must be kept in mind. 



The second terms in the expansions are given below, but may be 

 neglected in the range of most practical applications. 



/2i(x) = /2i(- x) = - /(O)^^^ 1(1 + Vx)e^^Ei{Vx) 



47r 



- (1 - Fx)e-r^[£i(- Fx) + ^V]}, (5) 

 F2i(x) = - F2i(- x) = I{{i)^ [Txe^-Ei{Vx) 



- Vxe-^^'lEii- Vx) -f iV] - 2}, (6) 



I u 



where Ei{u) — | — du is the exponential integral. 



For sufficiently large values of Fx the bracket terms of expressions 

 (5) and (6) vanish as — 8/(Fx)=' and 4/(Fx)2, respectively, so that in 

 this case the second terms in the expansions predominate. 



