272 BELL SYSTEM TECHNICAL JOURNAL 



potential at a point due to unit current entering the earth at a distance r 

 from the point; i.e. Q\_){r) is the mutual resistance between two points on the 

 earth's surface separated by the distance r. In the case of a two-layer 

 horizontally stratified earth the function (^oW may be approximated by 

 the following expression : 



<3oW = ^ [p, + (pi - p,)c-'''\ (47) 



where 



Pi = Resistivity of upper layer, meter-ohms 

 P2 = Resistivity of lower layer, meter-ohms 

 7o = k/2d 



d = Depth of upper layer, meters 



k = Constant depending on the ratio pi/p2 



When r is sufficienth' small compared to d, the above expression ap- 

 proaches the limit Qf,{r) = pi/livr and when r is sufiiciently large compared 

 to d the expression becomes ()(i(r) = pi/lirr. Expression (47) gives a fair 

 approximation to the function ()o(^) as given by curves calculated from 

 rather complicated integrals. Earth resistivity measurements by the 

 so-called "four-electrode measurements" are based on measurements of 

 Qo{r) and the results of such measurements may usually be approximated 

 to the same degree of accuracy by (47) as by the curves applying accurately 

 for two-layer earth, for the reason that the earth structure usually departs 

 considerably from an ideal two-layer earth. For various ratios pi/p2 the 

 constant k is about as follows : 



Inserting (47) in {2>3) and proceeding as before, the voltage between core 

 and sheath due to a stroke at the distance y may be written: 



F(0, y, t) = \ (0, a, t) — — (48) 



P2 A(a) -f (pi — P2)fj-[a) 



where V{0, a, () is the voltage for a direct stroke calculated for an equivalent 

 earth-resistivity: 



p, = p2X(a) + (pi - p2)M(a) (49) 



and where 



X(v) = log [(1 + ry)/ryl (50) 



