414 BELL SYSTEM TECLINICAL JOURNAL 



The expression in square brackets in (24) is the mutual impedance 

 gradient parallel to an infinite wire at a positive distance x from the 

 wire. It agrees with the results published independently by F. 

 Pollaczek/- J. R. Carson,'^ and G. Haberland/^ eind has been employed 

 by us to obtain numerical results since 1917. Pollaczek has also in- 

 vestigated the case of two gounded circuits of finite length. ^^ 



The mutual impedance dZx^ between a short grounded circuit dS 

 and a counterclockwise small loop of area da, on the surface of the 

 earth, is given by the formula 



,^r-^ 7-7 dSda sin </) p- /-> i ^ i o on ^^,-1 



(25) dZ^o = -^^ • —^l^ - (3 + ^ir + 7-r)e ^'■], 



where (/> is the angle which dS makes with r, the line from da to dS. 

 This may be obtained from Sommerfeld's formulae for the horizontal 

 electric force due to a vertical magnetic antenna, or it may be obtained 

 by an application of Stokes's theorem to formula (20) above. 



By a further application of Stokes's theorem we may obtain the 

 mutual impedance between two counterclockwise small loops dA and 

 da, namely, 



(26) dZu = ^ • 1 [(9 + 97r + 47^'^ + yr')^-^'- - 9]. 



This result might also be derived from Sommerfeld's formula for the 

 vertical magnetic force due to a vertical magnetic antenna. 



We shall now indicate briefly how the same value of E as given in 

 (18) above may be obtained directly, though more laboriously, from 

 first principles. In this method we start from the fundamental 

 solution ^^ 



(27) u = e'-^+'»!/+"V' 

 of the wave equation 



^ ^ dx~ ay- oz- 



12 F. Pollaczek, " Uber das Feld einer unendlich langen wechselstromdurchflossenen 

 Einfachleitung," Elektnsche Nachrichten-techuik, 3, 339-359 (September 1926). 



'3 J. R. Carson, "Wave Propagation in Overhead Wires with Ground Return," 

 Bell System Technical Journal, 5, 539-554 (October 1926). 



"G. Haberland, "Theorie der Leitung von Wechselstrom durch die Erde," 

 Zeitschrift fur angeimndte Mathematik und Mechanik, 6, 366-379 (October 1926). 



'5 F". Pollaczek, "(iegenseitige Induktion zwischen Wechselstromfreiieitungen von 

 endlicher Lange," Annalen der Physik. (4), 87, 965-999 (December 1928). His as- 

 sumptions regarding conditions at the ground connections seem to depart considerably 

 from the conditions assumed in the present paper, and moreover his results are not 

 expressed in convenient form for direct comparison with the formula given above 

 for Z\i. 



"H. Bateman, op. cit.,% 4, pages 6, 7; § 11, page 26. 



