AND THE FIGURE OF THE EARTH. 
79 
From (4.), by subtraction. 
X — Y=V-llo gI 
1 + fX. . 
whence 
and 
.2 — 
4 e 
- (X - Y) V^l 
_ ( x - Y) V - 1 _ j 
: — (X — Y) a/"— T + ! 
1 ^ “ ( e -(X-Y)V-l + i) a ( £ -i(X-Y) V-l + £ i(X-Y)V-l ) 2 
consequently 
?Sy + - ( "7x-Y = °- Let» (»+!) = a. 
cos 2 
,X-Y 
Let 
4 cos 2 • 
d P K . ^ v 
jf = v, then + 
d Y 
^ 4 cos 2 ^ 
a P Y Y = an( l = — JY ~ cos2 
v — Y ’ » flX a 2 
d P„ 
</ 2 u 4 
v — dY ~ ~ dXd Y a 
— cos 2 
X-Y </v 4 X-Y . X-Y 
f/X « C0S 2 
sin 
oi- 
Let 
Then 
and 
whence 
d 2 v , dv ^ X-Y , 
dXdY dX tan 2 + 
dv , X-Y 
rfY + y tan ~ 2 — ** 
</ 2 v , f/u A X-Y , 
</X</Y + f/X tan * + 
a v 
4 cos 2 
X-Y 
= 0. 
<2 1 a — 2 
2 cos 2 
<// 
X-Y ~ dX 
dX 
4 2 
cos 2 
X-Y 
= 0 ; or v — — 
a — 2 
COS 2 
X-Y dt 
2 d X 
a — 2 
or 
Let 
d v 
dY ~ 
d?t 
dXdY “r c/X 
cos 2 
X-Y rf 2 / 
2 dXd Y a — 2 
COS 
X-Y . X-Y <2* 
sin 
2 f/X’ 
a — 2 
COS 2 
X-Y ffit 
8 
2 f/X</Y a-2 
i dt 
+ ,7y 2 tan 
X-Y 
+ * 
c — 2 
4 cos 2 
X-Y 
X-Y . X-Y dt 
COS 2 Sin- 2 dX ’ 
= 0. 
dt I q jl i X-Y 
dY + 2 2tan— 2 — = ?, 
