Chap. 7] GRAVITATIONAL METHODS 95 



which may be written with the approximations used before : 



Since V is kM/r (the attraction potential), the total (attraction and 

 centrifugal) potential at the equator would be 



Hence, a is kM (1 + s + t/4)/t7, so that 



r^ = a(l — a' sin'^ cp) (7-12) 



where a' is t/4 + 3s. 

 Resubstituting the values of the coeflScients a' and b', 



a' ^ 3s + * = ^^^ - ^^ 4- ^ 

 ^4 2a2M ^2A;M' 



and 



Their sum is 



2a2M ^ fcM 



a' + b' = ^t= -.'^\ 

 ^ 4 2 kM' 



or, substituting c' for (/a/kM, 



a' + b' = 5 c' (7-13) 



This equation represents Clairaut's theorem. To determine the physical 

 significance of the three coefficients, a', b', and c', use eq. (7-12) thus: 

 r = a(l — a' sin ^). If ^ is 90°, then r is the polar radius, or the minor 

 axis, of the earth ellipsoid, which may be denoted by c. Hence, c = a 

 (l-a'),or 



a' = ^— -^ (7-14a) 



a 



The coefficient a' is the ratio of the difiference of the polar and equatorial 

 radii, divided by the equatorial radius. It is called the flattening (com- 

 pression). In eq. (7-11), which expresses the variation of gravity with 



