484 EXPLORATION GEOPHYSICS 



Designate the potential values in the media 0, I, and II by Vo, Vi, and V2 re- 

 spectively. Let M2 be the electrical image of So in medium II, A''2 the electrical image 

 of M2 in medium 0, Mi the electrical image of N2 in medium II and so on ad infinitum, 

 all the images in medium II being denoted by M, those in medium by A'^ and the 

 subscript denoting the distance of the image from the plane bounding media and I 

 in terms of the thickness d of the layer I as unity ; i.e., each image is distant from So 

 according to 



2d + (the distance of the source, of which it is an image, from 5*0) 

 For example, 



distance from So of Mt =z 2d -{- (distance of So from So) 



= 2d -\- 

 distance from So of Nt = 2d 

 distance from So of M^ — 2d + distance of Ni from So 



= 2d + 2d 



= 4d 

 etc. 



Vo may be expressed as the sum of potentials due to 5'o and the images M in 

 medium II ; i.e., 



J, _ 5*0 M2 , Mi 



^0 ^^2^^2y/,^ [^2+^2d-sy]'^ [r'+ {Ad-zYY^ 



where r* = 4:^ -f y and So, Mt, Mi,, etc., are constants to be evaluated later, and z is 

 negative. 



Similarly, V2 is the sum of the potentials due to So and the images A'' in medium 

 0. Fi, however, is due to the source So and to the images M and A^ in medium II and 

 medium 0. Thus, the potentials in the three media are given by the equations : 



„ _ oSo , 0M2 , oMt 



[r^ + s"]^ [r" + (2d - sV]'^ [r'+(4d-zyV^ 

 00 



000 I \'' oM2k 



+ 



i^^l 



with 3 negative. *= 1 



J- T 2-JO J 2N2 , 2A' 4 



y 2— r 1 , JTTT I r o , — 7x~r~, — rrrrr > 



{r- + z-\ '/' [r= + (2d + zy\^ \r' + (4d + zy\ '^ 



1 20 , X* 2N2IC 



Z. [r 



[r' + 2="] '-^ Zj [r" + {2kd + zy] "''' 

 positive. * = l 



y _ iSo 1M2 , 1M4 



[r' + zT' [r'+(2d — sy]'^ [r^+iid-zyV' '''' 



, lA^ , 1A/4 , 



[r' + (2d + zy]'^ [r'+ (Ad + zyV' 



_ iSo . v' 1M21C , v' 1A/2* 



[r^ + 5^]^ Z, [r'+(2kd-zyr^ Zd [r' + (2kd + zy]''' 

 z positive. k — 1 k = i 



All of these potentials satisfy Laplace's equation (4). If it can be proved that 

 they also satisfy the boundary conditions, they will constitute the exact and unique 



