ELECTRICAL METHODS 



483 



Mathematical Solution of the Tzvo-Layer Problem 

 Consider a uniform layer (I) of thickness d and resistivity px overlying 

 a semi-infinite homogeneous medivmi (II) of resistivity po- This geo- 

 logical two-layer structure corresponds to an electrical three-layer problem, 

 the third layer (0) being air. The "layer" of air lies above medium I and 

 has a resistivity of po = °o- Assume that a source So and a sink —5^0 are 

 located at the boundary of media and /. (Figure 292.) 



As in case III, the solution is facilitated by considering the effects of 

 the source and sink independently and then combining the effects. Further- 



t:^: 



5.1 ^ 



-■So 



V,p, 



I 

 I 



"^pz 



,eo 



,eo»^ 



Fig. 292. — The case of three layers. 



more the solution for the potential due to the sink — Sq at any point on 

 the line passing through the source and sink can be obtained from the 

 solution for the potential due to the source 6^0 by replacing x (the distance 

 of the point from ^o) by L — ;ir where L is the distance between the source 

 and sink. Thus the problem reduces to calculating the potential due to 

 the source ^o- This is most readily accomplished by assuming that in 

 addition to Sq there exists an infinite number of images of ^'o.f It will 

 be evident from the preceding problem that the relative magnitudes of 

 the current source and its electrical images will depend on the values 

 of the resistivities of the media in which the images are assumed to exist. 

 Thus the solution of the problem consists, essentially, in determining 

 the relative magnitudes of these images. 



t J. H. Jeans, loc. cit. 



