190 GRAVITATIONAL METHODS [Chap. 7 



slowly rotated wire while a record is taken of the beam position with respect 

 to the instrument ease. As the gradient effects are proportional to the 

 single azimuth, and the curvature values proportional to the double 

 azimuth, one revolution yields an irregular curve whose positive and 

 negative portions have different amplitudes. Evaluation is bnspH on a 

 determination of two equal ordinates of the same sign having an interval 

 of, IT. These ordinates are proportional to the gradients in the two direc- 

 tions. Curvatures may be calculated from two pairs of equal but opposite 

 ordinates of the interval tt. The total period of observation was intended 

 to be 2 hours with a 40 minute wait period to allow the beams to come to 

 rest. Extensive experimentation with this balance did not show any 

 superiority over the standard instrument. 



5. Vertical gradiometers and similar instruments. The opinion has been 

 expressed in the literature that a determination of the vertical gradient 

 of gravity would be very desirable for a more complete interpretation of 

 gravitational data. However, the geologic importance of the vertical 

 gravity gradient has possibly been overstressed since for two-dimensional 

 geologic bodies it may be readily determined from the corrected curvature 

 values. Be that as it may, several attempts have been made to determine 

 the vertical gravity gradient directly. These date back to 1880 and were 

 continued in subsequent years in connection with measurements of the 

 gravitational constant and of the mean density of the earth. Assume that 

 in a sensitive balance the pans are replaced by a weight fixed to one end 

 of the beam and by an equal weight suspended at a lower level on the 

 other end (Fig. 7-64). Compared with the weight positions in the same 

 level, the beam is unbalancedjaecause of the increase in weight of the 

 suspended mass. If the addition of a weight. Am, is required to rebalance 

 the beam, if the masses are m, and their difference in elevation is h, 

 (m -f- Am)g = m(g 4- dg/dz-h), and therefore 



Am = ^.^-^ - (7-58a) 



9 dz 



In this manner Jolly found that with 5 kg (mercury) weights at a differ- 

 ence of elevation of 21 m, an addition of 31.69 mg was necessary to re- 

 balance the beam, which gave 3.01 X 10" for dg/dz. 



The normal value of the vertical gravity gradient may be obtained 

 (1) from Clairaut's theorem,- (2) from the curvature of the reference 

 elUpsoid. Since in Clairaut's theorem, gravity is expressed as a function 

 of the earth's radius, the vertical gravity gradient may be obtained by 

 differentiation with respect to the radius, so that 



" See also F. R. Helmert, Higher Geodesy, Part 2, pp. 94-98, and formulas (7-36o) 

 and (7-366). 



