96 GRAVITATIONAL METHODS [Chap. 7 



latitude, the gravity at the pole becomes Qc = Qail + ^^), if <p = 90°. 

 Therefore, the coefficient 



b' = ?i:iA» (7-146) 



represents the ratio between the diiference of polar and equatorial gravity 

 and equatorial gravity, or the gravitational flattening. Finally, the 

 coefficient 



2 3 2 2 2 



^ ~ icM " kMj^' ~ TjTa~ J^ (7-14CJ 



indicates the ratio of the centrifugal force at the equator to the gravity 

 at the equator. Therefore, the theorem of Clairaut may be stated as 

 follows : 



, .^ ^. ^ n :.^ ■ ^ -., equatorial centrifugal force 



geometric + gravitational nattening = ^ X -^^ r^ r-^-;;^ 



2 equatorial gravity force 



Since this relation involves only surface quantities, the figure of the 

 earth may be computed from a known surface distribution of gravity. 

 From a number of carefully selected stations, gravity as a function of 

 latitude, and thus the coefficient b', may be determined. The coefficient 

 c' is computed from the known velocity of revolution of the earth. Thus, 

 l)y applying Clairaut's theorem, the flattening may be calculated. With a 

 more rigorous derivation involving spherical harmonics of higher order in 

 (7-66) and all moments of inertia in (7-8), Clairaut's theorem may be 

 stated in more extended form. If the variation of gravity with longitude, 

 in addition to its change with latitude, is considered, 



g = go(l 4- b' sin^ (p + b" cos' v'-cos 2X + ••••). (7-15a) 



By a careful analysis of the distribution of gravity and by eliminating 

 stations with large topographic effects and local anomalies, Berroth has 

 computed the following values for the coefficients in (7-1 5a): 



g = 978.046 [1 + 0.005296 sin' <p 



±4.4 



+ 0.0000116 cos' q^ cos 2(X + 10°) - 0.000007 sin' 2^] (7-156) 



from which follows the flattening as a function of longitude (from Green- 

 wich): a' = 0.003358 + 0.000012 cos 2(X + 10°). The major axis of the 

 elliptical equator is 10° west of Greenwich. The flattening in this meridian 

 is 1/296.7, and at right angles thereto it is 1/298.9. The mean flattening 

 is 1/297.8. The difference of the equatorial radii is only 150 ± 58 meters. 

 Hence, the equator is practically a circle and is considered as such in all 



