MUTUAL IMPEDANCE OF GROUNDED WIRES 419 



earth. 



(51) £. = ^ r r r^^i= - <^^ + ^ + A .^v: 



^"^ Jo Jo L Vm' + i'' J 



jj2-)-v2-f--y2 



X COS x/i cos yv dfidp, 



(52) Ey = ^^ r r . '-"' eW/l^+i:i+^ sin x^ sin 3,^, ^^,/^^ 



^"-^ Jo Jo Vm- + i^' 



(53) E,= --^l i fxi^'^'^'+^'+y' sin xfi cos yudndu. 



These are precisely the values found by the former method, for the 

 integrals P and Q may be expressed as double integrals by substituting 

 for Jo{rp) the integral expression given by the formula ^"^ 



I 2 f^'^ 



(54) Jo{ryJ fx- + v-} — — | cos(r/i cos d) cos {rp sin ^)^^, 



^Jo 



and introducing rectangular coordinates in place of r, 6. These inte- 

 grals may, therefore, be written in the equivalent forms, 



• 00 ^00 g2VM2+l'2+72 



(55) P = — \ I , " COS XjjL cosyy dfxdp, 



""Jo Jo - ■ " ■ ^ 



2 /'•oo /-.oo ^Z-\Jn2+v2+yi 



(56) <2 = — I I cos xiJL cos ypdfjidp, 



^Jo Jo Vm' + J^'Vm' + J'' + t' 



and comparison with (51)-(53) again leads to the values 



(18) (E E E)=^(-^^-'^ -^^ ^) 



where P and Q are evaluated in (15) and (16). Thus the mutual im- 

 pedance formula presented in this paper may be derived directly from 

 first principles, without reference to the work of Sommerfeld. 



I am greatly indebted to my colleague. Dr. Marion C. Gray, for 

 putting into its present form the derivation of my formula from 

 Sommerfeld 's results. 



" G. N. Watson, op. cit., page 21, formula (1) of § 2.21. 



