Chap. 7] GRAVITATIONAL METHODS 97 



problems in gravitational exploration involving calculations of normal 

 gravity, normal gravity gradient, and so on. 



Likewise, for many problems in geodesy and geophysical science it is 

 desirable to use the same reference surface (namely, an ellipsoid of revo- 

 lution) for both normal gravity and geodetic measurements. For this 

 reason the International Association of Geodesy adopted at the Stockholm 

 meeting of the International Geodetic and Geophysical Union in 1930 a 

 formula not including a longitude term, based on an ellipsoid of revolution 

 with a flattening of 1/297: 



g = 978,049 (1 + 0.0052884 sin' <p - 0.0000059 sin' 2<p). (7-1 5c) 



This international gravity formula is now used in all gravity reductions 

 by the U. S. Coast and Geodetic Survey.^^ 



V. PENDULUM AND GRAVIMETER METHODS 



A. Theory of the Pendulum on Fixed and Moving Support 



1. Pendulum on fixed support. A mathematical pendulum consists of a 

 particle of mass suspended from a point 'by means of a massless, flexible, 

 inextensible cord. In Fig. 7-6 let m bo the mass, I the length of the cord, 

 and 6 the angle of deflection from its rest position. In the state of motion 

 the inertia force m-l-d 6/dt balances the restoring force —jn^g-sm d for 

 sustained amplitudes; the weight component m-g- cos 6 and the centrifugal 

 force m-l-{d6/dtf are compensated by the tension of the suspension cord 

 and need not be considered. Hence, 



3-? + ?.sin^ = 0. (7-16a) 



dl- L 



An exact evaluation of this expression leads to an elliptical integral. 

 For small amplitudes, sin ^ = ^ and 



+ co'0 = (7-166) 



de , 2 



d<2 



where w = -s/g/l is the natural angular frequency or the number of oscilla- 

 tions in 27r sec, so that with / as frequency and T as period, w = 2-wf = 

 2Tr/T. 



For finite amplitudes, equation (7-166) does not apply. A solution of 

 (7-1 6a) is possible by decreasing the order of the differential equation and 

 considering the energy of motion, assuming again that no energy is con- 



*^ Personal communication, courtesy of Admiral L. O. Colbert, Director, U. S. 

 Coast and Geodetic Survey. 



