POTENTIAL COEFFICIENTS FOR GROUND RETURN CIRCUITS 369 



and i{kl — k') = Airau ll-\-i Vy" ) ~ "^^ 



2'k r" e-"^'' cosy'v-dv 



V^ Jo (v + Vj^ 4- is^)(^ + tWv^ + is^) 



where, as in Q — iP, w' = w \/a and y' — y \/a. 

 ^ 



Noting next that — e~^ " = — y/ ave~^ " 

 oz 



we have 



ico ^ 2 ^ r * (1 — r^jg""'" cos y V'y»(/y 



1? dz ^' ^ ~^' ^ i {v+ V»^T1^)(^ + tW^^^~+1^) 



= -Qc — 2 / — — o ' e cos/r.(l ~ r)i'.^»'. 



Since (1 - r^)]/ = v + r^ -y/v^ + w^ - t2( VJM^^ + ^) 

 this is 



4 r*r r^w^ 1 



^ i^ Je L I' + r' Vv^ + w2 J ^ 



On adding (6) to (5) we have 



QPn = QcV log p'Vp' + 4(M 4- tW)] (7) 



where 



r e-^^'cosyv 

 Jo V y^ + ^2 + ^'/r 



Af + ^^^ vanishes as/— >0,/— > <», e— ^ oo oro--^ oo. 

 When kl — k is minute the leading terms in the approximation (9) 



1 



for K + iiV are ^ - ^ - ^ log (p" Vfe' - k")/!) - i ^. 



An Approximation for M + iN 



It is possible to get series expansions for M + iN but those which have 

 been obtained do not facilitate computation. A fairly good approximation 

 to M + iN is arrived at as follows. 



