MUTUAL IMPEDANCE OF GROUNDED WIRES 165 



formula for arbitrary paths; then follow the d.-c. mutual resistance and 

 inductance, and in the last columns exact and approximate expressions 

 for the mutual impedance gradient parallel to a straight wire of infinite 

 length. 



The first entry in each group is the general case of two-layer earth. 

 In the first group, the next three entries are those in which one of the 

 conductivities is given the special value zero or infinity, one of the 

 four possible cases being trivial. The fifth and sixth entries involve 

 finite surface conductivity which is defined by cr = lim b\u in the first 



of these the surface conductivity differs from the conductivity of the 

 homogeneous earth below it, in the second the earth below is abolished. 

 The latter may serve as a convenient approximation to the case in 

 which the earth consists of a thin upper layer of high conductivity 

 relative to the layer, below. The final entry of this group is the case 

 of homogeneous ground. In the second group the second entry is 

 the limiting case for 6 = co , which places the wires at the plane of 

 separation of two semi-infinite media of conductivities Xi and X2; 

 the general formula for this case has been independently obtained by 

 R. M. Foster. With either conductivity zero this case reduces to the 

 case of homogeneous earth; with equal conductivities the case of an 

 infinite medium is obtained, which is the final entry of this group. 

 The third entry is the case of wires at depth b in homogeneous earth ; 

 for sufficiently large depths the formulas approach those of an infinite 

 medium. 



Further information regarding these special cases may be obtained 

 from the papers referred to in Table I. 



In case 1.4 where the conductivity X2 approaches an infinite limit, 

 an ambiguity arises concerning the d.-c. mutual inductance, two 

 cases appearing according as the approach of X2 to infinity is assumed 

 faster or slower, respectively, than the approach of the frequency to 

 zero, that is, according as the limits are taken X2 -> =0 , co — > or 

 0) — > 0, X2 — > °o . The entry in the table corresponds to the latter 

 limit and also to d.-c. distribution of earth current. The alternate 

 limit gives: 



L° Sa = N° Sa — N°Ss' + ^ (/l-ii)(a-6), 



where 



N°ss' — Mutual Neumann integral of one wire and the image of the 

 other wire, the image plane being the plane of separation 

 of the media. 



r , 1 - e-^'"' - 2ku Jo{u) - 1 , 



X (1 + e^ ) « 



(0 > N°{r) > - .2r). 



