Chap. 7] 



GBAVITATIONAL METHODS 



93 



from circular shape and by assuming that the two equatorial moments of 

 inertia are equal. Thus, if ^ — B, the final expression for the total 

 potential is 



t/= ~ +^,(C-.1)(1 -3sin%) + 

 r 2r 



2 2 



0) r 



cos ip. 



(7-9c) 



From this expression, the gravity may be obtained with sufficient ap- 

 proximation, by differentiation with respect to r: 



g 



CtU fCIVl . die ,^y A \ /t n • 2 \ 2 ' 



— 2 + oi (^ — ^)(1 — 3 Sin <p) — cor cos" <p 



dr 



2r* 



or 



''=i^[i + 2i,^^<^-^^<'-^™'^^ 



2 3 



CD r 2 

 ^^ COS (p 



kM 



(7-lOa) 



The second and third terms in the first of the above equations are of the 

 second order and are small. Therefore, another simplification may be 

 made by letting r = a, that is, by replacing the radius of the earth with the 

 equatorial radius, a. For reasons which will be evident from what is to 

 follow, it is convenient to express g in terms of U. Eq. (7-96) may be 

 written : 



rj ^kM 



{C — A) , o • 2 \ , w r 2 



By substituting a for r in the brackets, 

 kM 



U 



^ . {C — A) f^ o . 2 \ , CO a 2 



•] 



Ushig the abbreviated notation o for (C — A)(l — 3 sin" ^)/2a^M and 

 p for (0 V cos^ (p/2kM, 



and 



g = ^f [1 + 30 - 2p] 



f/ = ^[l + o-f p], 

 r 



(7-106) 



The r may be eliminated from the last two equations so that 



g = 



U^ 1 + 3o - 2p 



kM {1 + + p)2' 



