TRANSIENTS IN GROUNDED WIRES 423 



the earth. The formula for straight parallel wires, wire 5" extending 

 along the s axis from — a to + a, and wire 5 from Si to z^ at distance 

 X from it, is obtained by double integration between these limits with 

 ^2 = x^ + {S - 5)2, cos e = 1. 



The result of integrating once, with respect to S, is: 



V.v2 + (a - 5)2 ^Jx' + (a + ^)' J 



+ 0(5 + a) - 0(5 - a) I (Is, (2 



where 



(t>{n) = " -erf ( V.v' + u~ J^ 



x-^{x^ + It-) \ ^ ^ 



1 / 7rX.r2\ ./ [^ 



where u is to be replaced by 5 + a and 5 — a in equation (2). 



Equation (2) is checked as follows. In the first term substitute 

 limits after removing differentiation and integration with respect to 

 S, which cancel each other. In the second term integrate by parts: 



/ ,,+ '-.,J. «fVf[.v'+(^-^)-]'^-^ 



■^"' ierfJ^[.v^ + (5--0=] 



The integral coming from this operation combines with the remaining 

 term to give : 



-2^/lexp{-^[..' + (5-.n}</5, 



which can be simplified in terms of the error function to the form in 

 equation (2). 



Integration from Si to z^ gives the result: 



Fi2(0 = tV ['/'(S2 + a) - '/'(S2 - a) - '/'(si + g) + H^i - «)], (^^) 



ZTTaX 



where 



^(«) = - w+i? + ""^ "' \ " + ^''^ 



7/. / ttX-V^ \ / /ttX 



