49 S Lieut. Col. Lambton’s abstract of the results deduced 
: : (Cos. 2 7. a- + Sin. 7. b * ) —f : ( Cos. 9 /. d 2 + Sin. 1 /. 6 4 ) — | . 
Now to simplify this expression, if a — i, e the ellipticity ; 
and therefore b — l — e, and 6 2 = i — 2 e nearly, because 
e 3 must be very small. Then will ml : m : : (Cos. 2 7 (i < 2 e ). 
Sin.* 7) — i : ( Cos. * / ( 1 — s * ) . Sin. • /) —1. And if 
( i ■ — Sin. 8 7) and ( l — Sin. 8 /) be substituted for Cos.* 7 and 
Cos. 2 /, the expression will be transformed into (i — 2 e. 
Sin. 2 7) — I and (1 ze. Sin. 9 /) —I; or 1 ge. Sin. 2 7 and 
1 -j- ge. Sin. 2 / nearly, by developing the series, and leaving 
out all the quantities involving e 9 or its higher powers. 
Hence ml: m : : 1 -j- 3 e • Sin. 2 7: 1 + 3^- Sin. 2 / . ( 1 ) 
which reduced gives e = -7 — ■ . 2 A . ( 2 ) 
and when the degrees are contiguous, or very near to each 
other, this expression may be rendered still more simple by 
making ml = m in the denominator, which under these cir- 
cumstances will scarcely affect the result : whence 
___ m‘ — m , . 
e 3 m (Sin . 1 7 — Sin. 1 /)' w) 
By this expression it will be easy to estimate the incre- 
ments to degrees lying contiguous to each other : for if m, 
ml , in", &c. be contiguous degrees in latitudes / ; 7, '7, &c. 
that is /, / -f- T; / + 2 ° &c. Then we shall have 
' = which being made equal 
and reduced, we get ml — m : ml' — m : : Sin. 2 7 — Sin. 8 /: 
Sin. 1 '7 — Sin. * /. ; and in like manner m" — m : m'" — ■ m : : 
Sin. 2 '7 — Sin. * / : Sin. 8 "7 — Sin. 2 / ; and m "' — m : m"" — m : : 
Sin.* "7 — - Sin. 2 / : Sin. 2 "" / — Sin. 9 /. &c. from which it ap- 
pears that the increments to the degrees, beginning with the 
lowest latitude, will always be as the increments to the 
