270 



GRAVITATIONAL METHODS 



[Chap. 7 



With one of the integrations carried out, the gradients and curvatures 

 as measured by this instrument are, therefore, for two-dimensional bodies: 



r^ r^ dr r^ dr 



Ux'z = 2k8 I / sin 2<p dtp — = k8 I — (cos 2(pi — cos 2^) 

 Jri J<pi r Jti r 



U^> = 2kb 



/ / cos 2<p dip — 



Jrx •'vi 1" 



kh 



I — (sin 2ip2 — sin 2(pi) 



(7-966) 



and for three-dimensional bodies 



«r2 



t/„ = Sk8 



cos a da (sin tp — sin (p) dip -— 



r 



/*"2 /•V2 t"' 

 Jai J<pi Jri 



= k8 I cos ada I — (-j-cos3<pi 4-|cos^i — |cos3<^ — f cos<p2) 



Uyz = same with sin a instead of cos a. 



•'■2 



-Ut 



m 



dr 



cos 2a da • cos ip dip- 

 r 



H7-96c) 



r"! r<fi /•'■2 

 ♦'oi J^i Jri 



cos 2a da / — (f sin^2+isin3v72 — f sin^i — isinSsPi) 



Uxy = same with sin 2a in place of cos 2a. 



The mechanism of this integraph is enclosed in a circular case and 

 pivots about a point near the supporting base, this being equivalent to 

 rotation about the angle ip. The variation in the r direction is brought 

 about by radial motion of the pointer touching the board, which by a 

 rack and pinion movement rotates a keyed shaft shown in the center of 

 the arm assembly. This instrument is probably the most advanced means 

 for a rapid determination of torsion balance anomafies and has an ac- 

 curac}'^ of 1 to 2 per cent. A simple integraph for the evaluation of two- 

 dimensional features has been constructed by Gamburzefif.^"^ 



G. Discussion of Torsion Balance Results 



1. Measurements on lakes. The logical approach to the verification of 

 the theory of subsurface effects discussed in the preceding section is a 

 measurement of the anomalies due to known subsurface mass distributions. 

 Most suitable for this purpose are frozen lakes, whose bottom contours 

 are usually well known from soundings. Measurements on the ice are 

 readily made. The instrument can be moved from one point to an- 

 other on skids, and terrain corrections are usually negligible. Eotvos, 



12" Gerl. Beitr., 24, 83-93 (1929). 



