MUTUAL IMPEDANCE OF GROUNDED WIRES 415 



which is satisfied by the electric force in the earth; 7 = {iAivX^Y''- is 

 the propagation constant for plane waves which vary with the time as 

 f*"'. The parameters /, m, n satisfy the relation 



(29) /- + ni- + 7/2 _ ^2 = Q^ 



In the air, the same equations hold, but with the propagation constant 

 7 equal to zero, and we note that the solution in the air must be chosen 

 to vanish at an infinite height, while in the earth the solution must 

 vanish at an infinite depth. 



For convenience in this method we start with a short straight wire of 

 length 2a lying along the x axis, later allowing a to approach zero. 

 Thus we suppose that the current /e*"' enters the earth at the point 

 (a, 0, 0) and leaves it at the point (-a, 0, 0). The factor e^"' will be 

 omitted, however, throughout the following work. The current flow 

 in this system is symmetrical with respect to the vertical plane through 

 the wire, the xz plane, and is also symmetrical, but with sign reversed, 

 with respect to the vertical plane normal to the wire at its midpoint, 

 the yz plane. Then if we replace the three parameters /, w, n of (27) 

 by two independent parameters m. J^. such that 



(30) / = ± /m, m = ± iv, n = ± Vm' + f' + 7', 



formula (29) is identically satisfied, and we can then replace the four 

 solutions e"^^'^'^'"" by their corresponding expressions in terms of sines 

 and cosines, namely, 



sin XjjL sin yp, sin Xfx cos yv, cos xii sin yv, cos .y/x cos yv. 



The above considerations of symmetry will eliminate, for each com- 

 ponent of the electric force, all but one of these forms. With the re- 

 maining solution as a basis we build up, by means of the Fourier in- 

 tegral, a general expression for any possible steady harmonic oscilla- 

 tion. Hence we may write down the general solutions for the total 

 electric force in the earth (s < 0), as follows. 



(31) £x = F,{iJL,v)e'^''"-+'"+''' cos Xfx cos yudfxdv, 



Ju Jo 



I fyil^, J/) 6^^"'+"-+^' sin xn sin yudndv, 

 - „ Jo 



I /^,(m, J/) 6^^"^+"=+^' sin xix cos yv dfxdu, 

 Jo 



(32) 

 (33) 



