Chap. 7] GRAVITATIONAL METHODS 191 



, ^ = 3.086 (1 + 7. MO"*. COS 2^) microgals . cm*"\ (7-586) 

 oz 



The second method^^ uses Laplace's equation (7-5) and the curvatures of 

 the reference ellipsoid in the meridian and the prime vertical, which are 

 given by 



i=_i^ and i=-i^ 



so that 



„ y(-^-i)+2co^ (7-58c) 



dg _i ^ , ^ \ , „ 2 



where 2a)^ = 10.52 E.U. For g, px, and p„, their values as function of 

 latitude must be used (see Fig. 7-73b). The vertical gravity gradients 

 calculated from eq. (7-58c) agree with those obtained from (7-586) to a 

 tenth Eotvos. 



Forty years after Jolly's experiments, Berroth^' proposed to use a 

 standard torsion balance for the measurement of vertical gravity gradients 

 by suspending it on nearly horizontal wires. Deflections were to be 

 measured in different azimuths and at different starting angles of the beam 

 against the horizontal. Another design proposed by Schmerwitz** aims 

 to increase the sensitivity of a regular balance by the addition of a hori- 

 zontal pendulum, that is, by astatization (see page 127). A horizontal 

 pendulum oscillating about a vertical axis is in labile equilibrium, but 

 when it is tilted forward by an angle <p, it will assume a definite rest posi- 

 tion. If the axis of revolution is then tilted sideways by the angle 6, 

 a deflection \j/ = d/(p from the rest position results. When a horizontal 

 pendulum is placed on a balance, as in Fig. 7-64, the deflection 6 is due 

 to an increase in weight of the suspended mass. The deflection ^ throws 

 additional weight over to the right, the moment being m'V sin \p. For 

 an ordinary balance with (equal) lever arms L, beam mass Mo , and vertical 

 distance of center of gravity from the axis of rotation d, the sensitivity 

 2 = L/Mod. With the horizontal pendulum, 



S- ^,^^ n> - (7-5Sd) 



Mod<p — ml 



With a horizontal pendulum balance {V = 14 cm, m' = 100 mg, sus- 

 pended by two 17/x wires [Zoellner suspension] at an angle of about 2°), 

 a sensitivitj"^ of 10~ mg per mm scale deflection could be obtained at a 



" R. V. Eotvos, op. cit., p. 362. 



" A. Berroth, Zeit. Instr., 40, 210-211 (1920). 



"G. Schraerwitz, Zeit. Geophys., 7(1/2), 104 (1931). 



