561 
figure of the earth. 
(17.) Now since this is the resolved part of the whole 
gravity at any point, which is necessarily perpendicular to 
the surface, we shall find the whole gravity by dividing this 
expression by the cosine of the angle of the vertical. To the 
order which we are considering, the angle of the vertical is 
the same in the spheroid and in the ellipsoid with the same 
axes, and it is therefore = 2 e . sin lat. cos lat. = 2 e . ^ y' 1 — f*. 
Its cosine therefore = 1 
2 e 
l* 2 — 4 : consequently 
gravity at any point = ~~ { — - 2 -- J ( - a --- . i~V + 
. 2 e 2 . 
2 a . 1 — [S -f- - 
: 12 e . -J/ (a) 
’J' (a) 
3 n 
5 * a 4 
3 e 2 . p (a) 
. i 3 fj. z T 
2 Aip (a) 
b 2 — b* + 
b — b — 
3 ^ ii z ■ 
Pp.aea. 1— [^1 + 
v ( a ) 
5 
l-b*‘l —3b~ 
TjT “ T" ^ + “T ^ J } * But 
this is in terms of f* , the sine of the corrected latitude : it 
will be more convenient to have it in terms of X , the sine of 
the real latitude. Since the corr. lat. = real lat. — 2 e . sin. 
lat. cos. lat, it is easily found that (** — x 2 — 4e . X 2 — X 4 : 
substituting, gravity = -Tl J lJ?L (1 - n| 3 e 3 ) -f 
’Ha) 
— — - — e) 
5 5 ’ 
3 * 
2T 5 
(a) 
a (2 + 2 e) + — . + 
e - s e’)-^ if - f e ) + & a (,+ , e) + ? . 
( 6 A )- f ■ 4 P+&* « e + f ■ x’.r= 3 ?} • 
(18.) From the equations of (1 5), i-i! _j* 
J_M JJL a 1 Jiy. p] . (a) I 7 e 2 . 2 A \ 
21 ) 2T 2 | 3 • 21 /’a 6 a 2 \ 3 “T* 3 j 
. -j- a e . Substituting these, gravity 
= ± f^(^-e + e*+^ ]-#.a( 3 + f-e) 
+ ( 4 $- (— e + 2e'+ y A) a (S + -f e)).X» 
) . X*.TZ^} . 
_ jlW. (— — -f 3 A) + 5 ae 
