ELECTRICAL PROBING WITH AN APPARATUS IN WATER 



After substituting and simple conversions we have 



181 



QrM -JQO 



n = l ■■ J - 



2. / ■^(r-2^ %) 

 ^ 2 ^"i[r2 + 4V]'/» 



+ 2j^" 



r^[r^-2{nh+h^)^] r^[r-2{nh-hoy-] 



(22) 



Representing here 



= /j'" as the coefficient of re- 



7" 



2h 



3 r / \ 2 



2h 



r 



2h. 



5/2 



cession of the radial apparatus, ^ve obtain a formula similar to the for- 

 mula for the symmetrical AMNB apparatus: 



^ru ~ 2^^ 



^ + 2 ^ ^"^" + -^ ^0 ^P. + Z-J ^ri{ln-po + in+p,) 



. (23) 



It can be readily sho\\aa that for any other arrangement (quadrilateral, per- 

 pendicular, etc.) at the bottom of the water, the general form of the formula 

 for the apparent specific resistance will not differ from the expressions obtain- 

 ed, but only the coefficients of recession will change. Thus, in general we 

 have 





n = l 



in = l 



(24) 



§ 



Let us consider the asymptotic values of Qj^^. For this we suppose that 7- -> oo, 



flQ 



and since Iiq ^ h, then -7- -> 00, in this case (/„_p^ + ^n+p) ~^ 2Z„ , and Ip^ -> 1, 



00 

 and consequently, Qm~^Qo(^'^ ■^^^n^n) ~ Q^ ^•^- ^^^^h sufficiently large 



n = l 



operating distances r in comparison with the depth of immersion, the apparent 

 resistance measured with this apparatus will not differ from the value of the 

 apparent resistance which would be obtained if a similar apparatus of these 

 dimensions were placed at the water surface. 



Assuming that 'T'~^ and also t > 0, we find that all coefficients of 



