266 BELL SYSTEM TECHNICAL JOURNAL 



where 



P{r,H,h) = Po(r) + P'{rJI,}i) + P,{rJI + h), 



M{r,H,h) = M\r,II,h) + il/aCr,// + h), 





p\r,n,h) ='-^[11 log ^ 



+ h log 



[r^ + (H + Z?)'']^^'' + /J^ + /? 

 r2 + (// - hyj'^ + II - h 



[^2 -f (77 + /;,)2]l/2 + 7/ + /? 



[r2 + (7/ - A)2]i/2 - II Jrh 

 + [r2 + (7/ - /z)2]i/2 - [r2 + (77 + hyj'^ | 



AP{r,H,h) ^^ I j-^2 + (77 _ /;)2]i/2 [;,2 ^ (7/ + /,)2-]i/2 1 



il/3('',5) = 2^ I ^..2 |_ •n2M/2 ^ __ Jo{riJi)dtJL. 



(M^ + P)l/2 + ;u 



The functions P and M are again divided into three parts: first, 

 Po, the term giving the direct-current mutual resistance; secondly, 

 P° and AP, terms giving the mutual impedance on the assumption of 

 a perfectly conducting earth; and thirdly, P3 and AI3, the correction 

 terms for the finite conductivity of the earth. The P" and AP terms 

 thus give icov/Sir times the mutual Neumann integral of the two com- 

 plete circuits formed from the actual wire circuits by adding to them 

 their reflections in the surface of the earth. 



For small values of T, the P3 and AI3 terms can be expanded as 

 follows: 



p^(r,s) = ^ { - 5 log r - 5 log [(r2 + ^2)1/2 + 5] - r 



-f {r' + s'yi' + [2 log 2 + '/'(I) + |> 



-f -i^^r - i^s{3r^ - 252) r2 log r + • • • 

 Ahir,s) = '^ { ^^,^\,^,,, -jr - i^n log r + . • 



(1) 



By means of these expansions, the complete P and AI functions, as 

 given by formula (B), can be put into the form: 



