94 GRAVITATIONAL METHODS [Chap. 7 



The division gives gr= U (l+o — 4p)/A;M or in the original notation 



.2 j._ 2 



Substituting 1 — sin ^ for cos <p, and using the abbreviations s, = 

 (C - ^)/2a'M and t = 2coV/fcMy 



r2 



for which 



g = ~ [l + s-t + sin'^(t-3s)], 



^ [(1 + s - t)(l + sinV(t - 3s))] 



approximately. The neglected term, sin^ (p (4st — 3s^ — t^), is very small, 

 since all terms in the brackets involve the square of the earth's mass in the 

 denominator. As g = V/r and V = kM/r, 1/r = V /kM) thus, g — V^/kM. 

 Therefore, the term before the bracket in eq. (7-lOc) is the gravity at the 

 equator (since a was previously substituted for r) or rather the portion 

 of gravity due to attraction only. Since the term (s — t) expresses the 

 effect of inertia and centrifugal force upon the attraction, V^/kM 

 • (1 + s — t) represents the total equatorial gravity, ga . Substituting 

 ga for F'' (1 4" s — t)/kM, and b' for ( t— 3s), we obtain a simple form 

 for the gravity at any point at the surface, thus: 



g^ == g.jl -^ h' sm\) (7-11) 



This equation represents gravity as a function of latitude. It will also 

 be convenient to express the earth's radius, r, as a function of latitude. 

 From (7-106) 



kM r^ . , , 



r = -^ [1 + + p] . 



Recalling the significance of the abbreviated notations o, p, s, and t, o 

 may be expressed in terms of s, and p in terms oft: o = s(l — 3 sin^ <p) 

 and p = (t/4) • cos (p. Thus, for r we have : 



U 



1 + s — 3s sin^ <p + T'cos'* tp , 



Again substituting (1 — sin^ (p) for cos^ tp 

 kM 



r = 



u 



[l + s + ^-sin^(3s + ^)] 



