GROUND RETURN IMPEDANCE 97 



the harmonic terms representing secondary reflection at the surface 

 of the dielectric cylinder {p = a). If a is made sufficiently small, 

 however, the harmonic terms become negligible. In view of this 

 fact, the large amount of tedious additional analysis required, if the 

 harmonic terms are retained, is not believed to be justified by the 

 practical applications contemplated. E^ will therefore be taken as in 

 formula (4). 



The secondary electric intensity E' can always be written as the 

 Fourier integral 



E' = F(M)e^'^'''+*cos.%-V dix, 0^3'^ //, 



(6) 



where x' = xVa, y' = 3'Va, and the Fourier function F is to be 

 determined. For the formulation of the boundary conditions at 

 y = hw& also require the expansion of KQ{p'i-y[i) as a Fourier integral; 

 the required expansion is * 



K^ip'i^i) = ^==^_e-y'^'^'+'cosx'ix dfi, p > 0. (7) 

 Jo Vm^ + « 



In the dielectric, the magnetic forces Hx, Hy are taken as 



£00 

 (i){lj)e-y>' cos XIX dfjL 



{00 

 4>(li)e~^'^ sin Xfi dp. 



In the ground, on the other hand, we have 



iwHx = ——E, 

 dy 



iuHy = —-E. 

 ox 



y^h. (8) 



(9) 



In order to satisfy the boundary conditions at the plane y = //, we 

 equate Hx as given by (8) and (9), and Hy as given by (8) and (9). 

 The explicit formulas for Hx and Hy are derived from (9) by substi- 

 tuting the Fourier integral for Ko, as given by (7), in (4) and differ- 

 entiating as indicated. 



The two equations resulting from equating the two expressions for 

 Hx and the two expressions for Hy can be solved for the Fourier function 



* See note at end of this paper. 

 7 



