Chap. 7] GRAVITATIONAL METHODS 91 



Thus, 



- = (r^ + r! - 2rri cos 7)~^ (7-6a) 



e 



which may be written 



;-i[-(;;-;"')r. 



so that by series expansion and considering only terms up to the second 

 order : 



l = i[l+cos.I + ^:(-l + |cosS)]. (7-66) 



Substituting the value given above for cos 7, the potential by multiplica- 

 tion with k / dm becomes : 



y = -/ dm-| 3-/ xdm -{- -^ I ydm-\--^l zdm 



riJ rl J rl J rl 



h f /n 2 2 2\ 1 , kyi 



2r, •' r; J 



+ ^,f(2x' -i-z')dm + ^f (2y' - z' - x') dm 



2r; 



(7-7) 



The integrals have to be extended over the mass of the whole earth. 

 If we assume the latter to be concentrated in the center of gravity and make 

 it the zero point of the system of coordinates, 



/ dm = M, I xdm = / ydm = I zdm = 0; 



/ xydm = I yzdm = I zxdm = 0. 



The integrals involving the squares of the coordinates are not zero. 

 Assuming that the earth is a three-axial ellipsoid of rotation with three 



