148, 149] CLAIRAUT'S THEOREM. 405 



in terms of angular coordinates, let us introduce the geocentric 

 latitude ty = Q- and longitude qp, in terms of which 



x = r cos i[> cos qp, y = r cos ^ sin <p, = r sin iff. 

 The second term of 133) thus becomes 



~ [(B + C - 2A) cos 2 ^ cos 2 qp + (C + A - 2B) cos 2 $ sin 2 ? 



which, on putting 



1-1- c . 9 



cos 2 9 = - ; sin 2 = -- - -> cos 2 = 1 sin 



reduces to 



134) C-(l- 



1-1- cos 2 op . 9 1 cos2op 

 2 - 2 



In order to deal with the apparent gravity g, we have to add 

 to Vj the potential of the attraction that of the centrifugal force, as 

 in 123, 73), putting 



135) y V c = i co 2 (> 2 + y*) = { G3 2 r 2 sin 2 #. 



It is to he noticed that by writing 



136) 



F c is itself exhibited as o 2 r 2 plus a spherical harmonic. 

 If we now write 



^ 2 



K= -M~ 



we have the approximate expression for the potential of terrestrial 

 gravity 



137) U= 

 with 



If the surface of the earth is an ellipsoid whose radius vector 

 differs at every point from that of a sphere by a small quantity of 

 the first order, the angle between the normal and the radius vector 



