478 EXPLORATION GEOPHYSICS 



From this equation the graph given in Figure 286 is readily obtained : 



VERTICAL DISTANCE FROM POINT M 



Fig. 286. — Current density versus depth in a plane 

 Q which is normal to and midway between the ener- 

 gizing electrodes. 



In order to calculate the fraction of the total current flowing below 

 plane C, consider an element dy dz oi plane Q. The current flowing through 

 this element is 



— dydz* — S— 



Hence, the total current flowing below plane C is 



h= - 



where 



/oo /+« 



/d / — CO 



- ^-— dy dz = 



/oo /+00 

 J d /— 00 



dy dz 



7^ 



ri^ = 



=[(*) 



2 -j3/2 



2 4- r,2 



+ y^ + z^ 



To carry out the integration, it is convenient to make the following sub- 

 stitutions. Set 



and set 

 Then 



y = a tan 6; dy — a sec^ 6 dO 



_ SLi \( asec'ede l _SL\ 

 -2SL\ 



dz = 1 -r dz 



{L/zy + i- p 



=¥[i— -'(f)] 



Or on substituting the value for 5 and dividing through by / 



t=f[y-- '(f)] 



