EVALUATING THE ACCURACY OF A GRAVIMETRIC SURVEY 149 



it is necessary to derive an equation for the secant passing through the points 

 (%'^i) ^^^^ (^'2' ^2)' ^^^ to determine g^ from this equation. Then, using 

 the paraboHc equation (13), we substitute in the expression for e,- the 

 magnitudes gj^, g2, g^ by g^ and Iq. As a result we obtain 



£/ = ^0 [(a^i + -^1^) - (a + 2.Ti) x^ + x^^]. 



In this expression the variables are the arbitrarily selected points x-^ in the 

 range (0,2) and the point x^, in the range {x^^x^. In order to obtain the 

 value £,-^, it is necessary first of all to find the mean integral value e^ in the 

 range % < ^3 <rv:i+a. Dispensing with laborious calculations, we give the 

 final result 



e^m = ^g^a\ (15) 



where a should be measured in fractions of Iq, since in deriving the para- 

 bolic equation (13) it was assumed that Iq = 1. 



In view of the fact that the results of integration along x^ were not de- 

 pendent on x^, the relationship (15) will also determine the required va- 

 lue of £,-„,. 



2. Deriving the Relationship d{^ = ^i„^{g, a,p)—Let two neighbouring 

 isoanomalies of gravity have abscissae x^ and x^ ^ x^ + d (Fig. 4). We find 

 the expression 



^i = C?4-^3)-(^4-g'3)' 



where (gi—gs) is the cross-section of the isoanomalies; 



igi~gz) is the actual difference in the values of gravity at the points 

 with abscissae corresponding to the two adjacent isoanomalies. 



Here (as in the derivation of the value Sq^) there can be two cases ; both 

 points (^3 and x^ lie in the range (%, ^2)' the point x^ lies in the range 

 (xi, x^ and the point x^ outside this range. The values of 6^ will be different 

 for these cases. 



The position of the points x^ and x^ for the first case supposes that 

 < <i < a ; this position is represented in Fig. 4. For this case 



d,=^gQd[{d-a-2x,) + 2x,]. 



Squaring both sides of this expression and integrating for x^ in the limits 

 ^1 < ^3 < '^1 + ci—d, we obtain 



dfm = jgo'dHa-dr. (16) 



