316 Astronomical and Nautical Collections. 
the chronometer being generally below deck, as one person might 
have his eye upon it, and another immediately above him on the 
upper deck might give a stamp with his foot the instant the lamp is 
darkened. 
The longitude of Cape Castle appears from eclipses of Jupiter’s 
satellites to be about 18° 21’ E. 
The height of Table Mountain above the sea was found, 
Entrance from the narrow passage on the top (5 obs.) 3430 F. 
Highest western point (13 obs.) .......... 3536 
Highest eastern point (ll obs.) .......... 3545 
ili. Easy APPROXIMATION ¢o the difference of LATITUDE on a 
SPHEROID. 
Supposine the excess of the equatorial semidiameter to be known 
and equal to e, while the semiaxis is = 1, and having the linear 
dimensions of a portion of a perpendicular to the meridian, we may 
compute the difference of latitude and of longitude of its two ex- 
tremities by considering the case of a sphere touching the surface 
of the spheroid in the given parallel of latitude, and having the same 
curvature with the perpendicular to the meridian at its origin, 
which must therefore be extremely near to the points that require 
to be compared with each other, so that they may be supposed to 
be in the surface of this sphere. 
Now the local semidiameter will always be 1 + ecos* L, Lbeing- 
the true latitude, whence, by taking the fluxion, we obtain for the 
tangent or the sine of the inclination of the surface, or the correc- 
tion of the latitude, 2e sin cos L, consequently the sine of the cor- 
rected or geocentric latitude will be sin L — 2e sin cos Leos L = 
sin L(1 — 2e cos? LL). Hence we find, by trigonometry, the normal 
to the equatorial plane (1 + e cos? L) sin L (1 — 2e cos* L).: sin L 
= (1 + ecos? L) (1 — 2e cos? L) = 1 — ecos? L, e being a small 
fraction ; and the normal to the axis, which is the radius of curva- 
ture of the perpendicular circle at its origin, = (1 + e cos? L) cos 
(L— 2e sin cos L) : cos L; but cos (L— 2e sincos L) = cos 
L + 2esin cos L sin L = cos L (1 + 2e sin? L) and this normal 
