ELECTRICAL PROBING WITH AN APPARATUS IN WATER 193 



The function — - has one extreme value for Hq = 0.25a'; for n^>-0 and 

 dn 



ri -> oo the function — - -> 0, i.e. Hm — - = 0. 

 dn n^o.oo dn 



Figure 1 shows graphs for the functions — ~ and . 



dn dn^ 



A study of these functions shows that in the summation sign in expression 

 (36) in straight brackets, the components can be positive for re > Wq, and 

 negative for n <^ n^. To evaluate the error, these terms should be summed 

 to the absolute value. We will consider two cases. 



(a) Let m> Hq. In this case, all the components in the summation sign 

 of expression (36) are positive and the error can be evaluated by the 

 inequality 



1 I °° 



"^ n = m + l 



We will consider the series: 



oo oo oo 



n=m+l n=m+l n=m+l 



^ V^m + l-p„~~^m + l)+ (^m + 2-p„~^m + 2) + (^m-|-3-po~"^m+3) + •" + 



"^ v^m + l~^m + l+po/ '^ v^m+2~^m+2+Po^ '^ (^m+3~^m+3+Po) "^ •" ~ 



~i^m + l~^/n + l+p„)~(^m + 2~^m + 2+p„)~(^m+3~^m+3+Po)~ •'• ~ 



^ (^m + l-p„+ ^m + l)+ (^m + 2-p„+ ^m + 2) + ••' + \''m~''m+p) ~ 



Po 



^^ 2j \^m-po+i~''m+i)' 

 1 = 1 



The magnitude of the error is therefore expressed in this case by the 



inequality 



1 I 1 ^° 



^5i <-^ I graax \ 2 (^m-p,+i -^m+z)- (37) 



^ 1 = 1 



(b) We will now put m <i n^. In this case, the values in the summation 

 sign of expression (36) in straight brackets are both positive and negative; 

 thus it can be written 



n=m + l n=/7o + l 



2 \K-p„'^h+p„ 2Z„|— 2 (^n-po+ ^n+p„~2/J — 

 n=/7o^-l 

 no 

 2 (^n-po'^~ ^n+Po~'^^n) 



n=m+l 

 Applied geophysics 13 



