from the measurement of an arc of the meridian , &c. 499 
squares of the sines of the corresponding latitudes : and if 
m be at the equator where the Sin, / is 0, then we shall 
have m' — m : m" — m : : Sin. 2 7 : Sin. af 7 . 
Since by equation 1, m': m : : 1 -f* ge. Sin. 97 /: 
1 -j- ge. Sin . 2 1 
then m* = m 
and m = m! 
U) 
is) 
1 + 3e. Sin.* '/ \ 
1 + Sin.*/ ) 
i -f Sin.* 2 \ 
1 -jf- $e. Sin.*'// 
When m is at the equator, and therefore Sin. / = o; then 
m' = m ( 1 + 3 e - Sin. 2 7 ) . . . (6) 
If m' be at the pole, and therefore Sin. 2 7 = 1, then we 
have m‘ = m 
1 + 3* 
( 7 ) 
I 4. 3<r. Sin.* // 
If the degrees perpendicular to the meridian be made use 
of, let p‘ and p be the measures of those degrees in latitudes 
7 and /, then the radius of curvature of the perpendicular 
degree at 7 being as q r r r ~ S nr T TjI = i( , -“- Sin -’ V )7 = 
= ~ ( 1 -J- <?. Sin. 8 7 ) very nearly ; and for the same reason 
the radius of curvature of the perpendicular degree at / will be 
as £ ( 1 «£■ e. Sin. a /) very nearly ; so that we get p ( ; p : : 1 «f»^. 
Sin. 2 7 : 1 + Sin. 2 / 
and when reduced gives e 
P'-P 
( 8 ) 
( 9 ) 
p. Sin.* '2 — p‘ Sin.® 2 * 
From equation S,f = #(7^—7) . . 
and p=p\ l -±±J^L\ . . 
r r \i + e. Sin.* 7/ 
If p be on the equator where the Sin. I vanishes, then 
equation 10 becomes p* ^ p (i -f #. Sin.“ 7 ) . . . (12) 
If p ' be at the pole, and therefore the cosine of 7 unity , 
(to) 
(it) 
then equation 10 becomes />'==/> ( , +‘ e + sin */ ) 
MDCCCXVIIL 3 T 
(J 3 ) 
