92 GRAVITATIONAL METHODS [Chap. 7 



moments of inertia, A, B, and C, about the three princiDal axes, h ,h , and 



A = I lldni ; -^ = / ^2^^ ; ^ ~ I ^^^^w ; 



when li = V?/^ + ? ; Za = a/^' + a;' ; ^3 = Vx^ + y^ ; hence, 



A^ j {y~ + z')dm; B = j (z'-h x')dm; C = f (x' -h y^)dm. (7-8) 



Substituting these values in (7-7), 



V=^-+-HB + C-2A) + ^HC + A-2B)+^AA+B- 2C). 

 n 2rl 2r\ 2r\ 



If we drop subscripts, the location of any surface point may be written 

 in geocentric coordinates: 



X = r cos ip cos X and x" = r cos" o-\{\ + cos 2\); 



y = r cos (f sin X and y' = r cos" ip-\{l ~ cos 2X); 



z — r sin v? and 2" = r" sin <p. 



Hence, after combining terms containing <p and X, 



y = — + o . ( C^ - -o~ - (i - 3 SHI -r) + -r^ cos <p COS 2X(fi - /I) 



r 2r^ \ 2 ) 4r 



(7-9a) 



This is the potential of the attraction only. The potential of the centrif- 

 ugal force must be added to it. Its three components are Cx = x-ut \ 

 Cy = y-<J]Cz = 0, when CO is the 'angular velocity. Thus, the resultant 



centrifugal force is o)'^\/x^ + y^ and its potential is V = "o*^^" "^ y^' 

 In polar coordinates, V = %. -r' cos" cp. Then the total gravity potential, 



\ + y = V, is 



J, kM , k I ^ A -\- 5V. Q • 2 X 

 (/ = — -f ( C — — ^r — • 1(1 - 3 sm if)) 



1" 



2 



+ 1^ cos' ^ cos 2X(5 - ^) + '*Q^ cos^' <p. (7-96) 



This expression may be further simplified by confining the derivation 

 to a iifo-axial ellipsoid, that is, by neglecting the deviation of the equator 



