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BELL SYSTEM TECHNICAL JOURNAL 



X being the positive horizontal distance from ds to S, and e the angle 

 between ds and S. 



This result is derived immediately from formula {B) upon assuming 

 S to be doubly infinite and then integrating over its entire length. 

 The first part of the expression comes from the .1/" function, the second 

 part from the Mz function. The other functions contribute nothing 

 to the leading term in the expansion. 



The expression enclosed in braces in (4) is the mutual impedance 

 gradient parallel to an infinite wire at a positive horizontal distance x 

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

 F. PoUaczek,^ J. R. Carson,'* and G. Haberland.^ Pollaczek has also 

 investigated the case of two grounded circuits of finite length, with 

 certain modifications.^ 



For purposes of computation, however, formula {A) is better, in 

 general, than formula {B). A distinct improvement is effected by 

 multiplying all distances which occur in {A) by the attenuation con- 

 stant, that is, by (aip/lpy^. The numerical value thus obtained for 

 any one distance is indicated by a prime accent on the corresponding 

 letter. We then find the mutual impedance expressed in the following 

 form: 



Zl2 — 



where 



(wvp 



F/ir- 



Qjr'jr ji') 

 dS'ds' 



+ cos e N{r',H' 



'jn ] 



dS'ds', (C) 



Q{r',II',h') = Q,{r') + Q,{r' ,11' + //') - Q,{r',\H' - h'\), 

 N{r',ir,h') = No{r') + Ni{r',ir + h') - N2{r',\H' - h'\), 



Qo(r') =p' 



Qiir'y) = i 







1 - r 





Jo{/ij.)diJi, 



Q,{r'4') =i[d' ^^^ ^r'-' + dy^ + d' _ ^^,, ^ ^,. ^,, ^ ^, 



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

 Einfachleitung," Elektrische Nachrichten-technik, 3, 339-359 (September, 1926). 



^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 W'echselstrom durch die Erde," 

 Zeitschrift fiir angewandte Mathematik und Mechanik, 6, 366-379 (October, 1926). 



* F. Pollaczek, "Gegenseitige Induktion zwischen Wechselstromfreileitungen von 

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

 assumptions regarding conditions at the ground connections seem to depart con- 

 siderably 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 Zn. 



