418 BELL SYSTEM TECHNICAL JOURNAL 



where the £'s are given by (31)-(33) for s < and by (34)-(36) for 

 < 2. We now subtract from this integral 47r times the current in 

 the earth which threads the rectangle, this quantity being found by the 

 appropriate integration of E^, as given by (31), over that portion of the 

 area of the rectangle which lies below the surface of the earth. As a 

 final result w^e obtain the expression 



(46) - r C-i- Vm'^ + V- + y-'F, + ^iF. - Vm'-^ + v'Pr - ixP.) 



'^ Jo Jo 



X COS xn sin r]v djxdv, 



after simplifying by means of the divergence conditions (37) and (38). 

 The net current threading the rectangle, after subtracting the current 

 in the earth, is / if the rectangle is situated between the two grounding 

 points, but is zero if it is outside them; that is, the above integral (46) 

 equals 4x7 if \x\< a, but equals zero if a < |:x;j , for any positive value 

 of Tj. It is readily verified that the Fourier integral 



167 r°° r" sin an cos xjj. sin 171' , , 

 ~^ Jo Jo ^'^ 



has the desired properties. Accordingly we must have 



(47) - Vm- + V- + tF, + ixF, - VmM^'/'x - iJ^Pz 



_ 87w/ sin a IX 



TV jX 



We can now solve equations (37)-(40), (43), and (47) for the six 

 arbitrary functions, obtaining 



(48) F. = P. = -^ 



_ \M- + V- + 7- 



_ mVm' + v'~ 



sm an, 



(49) Fy= Py= ^- . /^ ., sin a IX, 



21 . 

 (43) Fz= ^sin Gju, 



(50) P. = ^^^ ^^' + »-' + ^'^ sin an. 



Substituting these values in equations (31)-(33) and letting a 

 approach zero such that 2a = dS, we find, for the electric force in the 



