416 BELL SYSTEM TECHNICAL JOURNAL 



where the positive sign is chosen in the exponential term containing s 

 since the solution must vanish at an infinite depth, s being negative in 

 the earth ; and that value of the radical is taken which has a positive 

 real part. Fx, Fy, Fz are arbitrary functions of their arguments, to be 

 determined by the physical conditions of the problem. 



In the air (0 < z) we may formulate the corresponding solutions for 

 the total electric force as 



(34) 



(35) E 



(36) 



I I P^ifjL, p)e-^^'''+'''cosxiJLCOi^yvdiJ.(h, 



y = i I PyiiJi, v)e-^'''+'"s'in Xfi sin yvdjxdv, 



Jti Jo 



= I I F.iij., v)c-'^^-'^''^ s\n Xjx COS yvdixdv, 



Jo Jo 



where the propagation constant is zero in the air; the negative sign is 

 chosen in the exponential term containing z since the solution must 

 vanish at an infinite height, z being positive in the air; and Px, Py Pz 

 are arbitrary functions of their arguments. 



To determine these six arbitrary functions we need six independent 

 relations among them. Two of these relations are obtained by 

 utilizing the fact that the divergence of the electric force either in the 

 earth or in the air is equal to zero, that is, 



dEx dEy dE,^ 

 dx dy dz 



By means of this we obtain from (31)-(33), 



(37) - fxF, + vFy + Vm' + j'- + y'F, = 0, 



and from (34)-(36), 



(38) - fiP, + vPy - ^ix' ± z^-P. = 0. 



Since the horizontal components of the electric force are continuous 

 at the surface of the earth (s = 0) we see that we must also have, from 

 (31) and (34), 



(39) F, = P., 

 and from (32) and (35), 



(40) Fy - Fy. 



