766 



BELL SYSTEM TECHNICAL JOURNAL 



I am indebted to Dr. F. H. Murray of the American Telephone and 

 Telegraph Company for the following solution of Zi2{t) as given by (2) : 



ZM 



where 



and 



2h 



M\^y[\^t 



^i 



2\\Mi^ 

 e 



' + ' 



Mi" 



^^1^^ nr IT 



Ml'' 



erfc 



V/ 



+ 





Mo 



erfc 



i/sVx 



V/ 



(4) 



\M\ = \Mi\ = \M2\ 

 Ml = {h - jx)\^, 

 Mi = (h -j- jx)\ir, 

 h = hi + ho 



erfc Z = 1 - erf Z = 1 



_2_ r^ 



Vx Jo 



,-t2 



dx. 



Taking the limit of equation (4) as h approaches zero there results 



1 



^12(0 =-^,(1 -e-^^^'i^), 



ttXx 



(5) 



which formula is of fundamental importance in the present analysis.^ 

 This equation is also plotted on Fig. 4 for two different values of irXx^. 

 Assuming now I{t) = sin co/ formula (1) gives 



sm 



'r + e 



.-0t 



r e-^^M+Pr^r], 



(6) 



8 This formula can readily be checked in the following manner. The mutual 

 impedance between wires on the surface of the ground is 



Zi2(co) = 



J 7_ 



K.iyx), 



where y = yATrXjco and Ki is the Bessel function of the second kind with imaginary 

 argument defined by Watson, "Bessel Functions." 



Replace jco by p, and interpret the function of p so obtained according to opera- 

 tional methods. The first term is independent of p and therefore of t. The second 

 term is transformed according to the equivalent 



ayJpKiiayJp) = e-cV^', 



given as pair 922 in G. A. Campbell's paper "The Practical Application of the 

 Fourier Integral," Bell System Technical Journal, Oct. 1928. 

 Equation (5) is then immediately obtained. 



