ELECTRICAL PROBING WITH AN APPARATUS IN WATER l85 



For instruments at the boundary of separation of the first and second 

 layers, this formula becomes somewhat complex. 



Let us consider first a two -layered medium. In this case the problem of 

 the calculation of the theoretical curves is solved most simply. From equation 

 (27) we obtain 



n=l 



oo 



Having substituted the value for the series ^ ^n ^n "^ *^^ expression (25). 



n = l 



we obtain the following after simple conversions: 



M 



H [A, 



/< + 1 fjfi 



-1 \Qq I. 



(28) 



[1 — 1 

 bearing in mind that k — 



[i + l 



From this formula, which has the coordinates of the two -layer curves for 

 an apparatus at the surface, it is easy, without additional summation, to 

 calculate the coordinates of the corresponding two -layer curves for an 

 apparatus at the bottom of the -^vater. 



For a three-layer section, the formulae for the apparent resistance of 

 a bottom apparatus cannot be brought to a form suitable for simple conversion 

 and the calculation of the three-layer curves for a bottom apparatus requires 

 an additional summation of infinite series. 



A certain part of the curves, i.e. the curves with — =00 and — = 1/5. 



1/3, 1/2, 1, 2, 3, 5 were calculated by the method of splitting up the function B 

 into simple fractions. This method was proposed by S. Stefanesko and 

 has been further developed in a number of papers ^^' '' ^> ^\ 



The essence of this method is as follows. If the underlying medium has 

 a very high (^^ = 00) or a very low (^„ = 0) specific electrical resistance, 

 then the function 



Pn(e-^"^^) 

 ^ (?„(e-2mh) 



for such sections can be represented in the form of the sum of elementary 

 fractions. 



