166 



BELL SYSTEM TECHNICAL JOURNAL 



The ambiguous cases arise only in the limits X -^ co , w — > and X -^ 0, 

 CO— > 00, the product Xw appearing in the expressions then being 

 strictly indeterminate, until the order of the limits is defined. 



II 



Different problems are encountered in obtaining numerical results 

 for the two functions Q(A~B)[a-b) and N ss', Q(A~B)(a-b} is determined in 

 terms of four values, with proper sign, as given in equation (II), of 

 Q(r) ; while Q(r) apparently is not generally expressible in terms of 

 known functions it may always be evaluated by numerical integration. 

 The case of Nss is different because of the necessity of double integra- 

 tion over the wires; general numerical results involve carrying out at 

 least one of these integrations in addition to that required in evaluating 

 P{r), involving a considerable amount of labor and complexity of 

 results. 



However, without carrying out either of the evaluations completely, 

 the formulas for the limiting cases may be used to obtain results 

 approximating certain practical conditions. The important limiting 

 cases are (i) one wire infinite, and (ii) zero frequency. Curves for 

 these cases for wires on the surface of the ground and for special 

 values of the parameters are given in Figs. 1-6, as described below. 



0.0022 



0.0020 



,,, 0.0018 



j: 0.0016 

 <J 



a: 0.0014 

 ui 



Q. 



u 0.0012 



_j 



5 0.0010 



cc 



^ 0.0008 



5 Q0006 



° 0.0004 



0.0002 



0.05 



bp^ X 1000 



Fig. 1— Mutual impedance gradient at earth's surface parallel to an infinite 

 straight wire on the earth's surface; real component; two-layer earth Xi = IOX2; 

 b and y in feet, frequency in cycles per second, conductivities in abmhos per cm. 



Figures 1 and 2 show, respectively, the real and imaginary parts of 

 the mutual impedance gradient parallel to an infinite straight wire for 

 the conductivity ratio X2/X1 = 0.1; Figs. 3 and 4 show the same 



