Chap. 7] GRAVITATIONAL METHODS 175 



SO that eq. (7-48^) becomes 



%2 rr ^2 



- g ( ^ - i) COS 2X = V^ - V^ . (7-480 



\Pl P2/ 





B. Theory of Torsion Balance 



The torsion balance measures horizontal and vertical gradients of 

 horizontal gravity components but does not react to vertical changes in 

 gravity. Its center of gravity is far enough below the point of suspension 

 so that tilting effects are not noticeable ; nor is it affected by gravity itself, 

 since its action on the center of gravity is compensated by the tension of 

 the suspension wire. The suspended lower weight will be deflected from 

 the vertical, but this is too small to be measured. Horizontal forces 

 alone are effective at both ends of the beam. For a given deflection the 

 torsional moment of the wire and the moment of the horizontal forces 

 are in equilibrium. 



1. The Eotvos curvature variometer is identical in form with the instru- 

 ments used by Cavendish, Heyl, and others for the determination of the 

 gravitational constant (see Fig. 7-3). Its action in a field with parabolic 

 lines of force was illustrated in Fig. 7-566. It will be shown in the next 

 paragraph that its deflections are proportional to the "curvature values" 

 9 U/dy — d U/dx and d U/dxdy. It has not been extensively used, 

 since curvature values are not too readily interpreted and are affected 

 more by masses near the horizon than underneath the instrument. Con- 

 sequently, terrain irregularities cause considerable interference. Eotvos 

 built three types, two with one beam and one with three beams 120° 

 apart. Separate curvature variometers are no longer used, since beams 

 designed to give gravity gradients furnish the curvature values at the same 

 time when they are set up in the requisite number of azimuths. Such 

 instruments are known as gravity variometers of the second type, com- 

 bined gradient and curvature variometers, or merely "gradient" vari- 

 ometers. 



2. The combined gradient and curvature variometer is illustrated in Fig. 

 2-4 of Chapter 2. If the equipotential surfaces passing through the 

 weights make an angle i with one another, a horizontal component H = 

 mgi is produced. Its moment Di =. Imgi, where I is one half of the beam 

 length and m the beam mass. Substituting eq. (7-476). 



Di = mhl^. (7-49a) 



as 



If the angle of the beam with north is a and that of the direction of 

 maximum gradient is (90 -|- a), 



dg d^U d'U . 



-^ = -— — cos a — sm a 



ds dydz dxdz 



