Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

180 FIGURE OF THE EARTH,
then, the numerator = sin?0. And the denominator
= 3 sin (l + I) sin (I'- 1).
Hence log e = 2 log sin 0 - log 3 - logsin (' + 1) - log sin (' - 1).
By calculating from the data above, e=
306,3
78. Attempts have also been made to determine the
ellipticity of the Earth by measuring the distance between
two places on the same parallel, and determining the difference of longitude, either by observation on Jupiter's satellites, or by observing the flash of gunpowder fired on a
conspicuous place between them. The difference of longitude may also be determined by mere observation of angles,
(see P/il. Trans. 1790).
79. PROP. 33. To express the distance of two places
on the sane parallel, in terms of their difference of longitude.
Let p, q, (fig. 19), be the places; L their difference of
longitude. Then, pq (which, when the arc is sinall, may be
measured as a great circle without sensible error,) = L. CN.
T Taf2 - a2 cos
Now CN = pL.cosl
c -\/(a cosl + c sin2 l)
pq= L. a cos si = L.ccos/(l + 2e - e cos2 )
-,,(a cos, + c sn 0
= L.ccosl(l + e + e sin21),
if the ellipticity be small.
80. If then one arc has been measured in the meridian, and another on a parallel, and if I be the latitude
of the middle of the meridional arc, 1' that of the parallel,
we shall have these equations:
PQ
D = c(1 - e + 3e. sin2 1),
Pq
L - c cos ( e + e sin el').
L