406 VIII. NEWTONIAN POTENTIAL FUNCTION. 



will be small, its cosine will differ from unity by a small quantity 

 of the second order, neglecting which we may put 



-i on\ 



139) # = - y ^ = 



. 9 B- A 



, 

 COS COS 



Determinations with the pendulum show that g varies very slightly 

 with the longitude, we may therefore put B=A, so that 



140) U- 



1A1\ 



On a level surface, such as the surface of the ocean will be shown 

 in 179 to be, 



U= const. = U . 



^M. 

 For such a surface, equation 140) gives, putting a = ^r and in the 



^0 



parenthesis substituting a for r, we obtain the equation of the surface, 

 142) r = al + (l 



The substitution of o, for r is permitted in the higher powers because 

 of the assumption that differs from unity only by a small quantity 77, 



where square is neglected, and thus r m = a m (l -\-mrf). Inserting the 

 value of r from 142) in 141), and approximating in like manner, 



143) . 9 = 



or 



4A\ r^ 



144) g= s r-r n 



145) g = g (1 + n sin 2 ^), 



AZ\ 2co 2 a 8 ZK 



146) ^ = ^ .r-^- 



ylf 2a 2 



The equation 142) is easily seen to be that of an ellipsoid of revo 

 lution, and putting ^ = 0, ^ = its semi -axes are found to be 



M7\ 



147) 



