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.

DETERMINED BY OBSERVATION. 183
The ellipticity, found by comparing the observed inequality
in longitude with the calculated inequality, differs little
from this.
85. The two latter methods, it will be observed, depend entirely upon the theory which we have laid down:
the first and second are quite independent of theory. The
near agreement of their results is one of the most convincing proofs that the principle of gravitation, and the
suppositions upon which our theory is founded, are true.
86. For the calculation of parallax, it is necessary to
know the distance of any point on the Earth's surface
from the Earth's center, and the angle ACp, (fig. 19),
which is called the corrected latitude: ALp being the true
latitude, which = the elevation of the pole. The difference
between the true and corrected latitude is called the angle
of the vertical.
87. PROP. 34. To find the distance of any point on
the Earth's surface from its center, in terms of the latitude of that point.
Cp2 = CN + Np LN2 + Ap2 = Lp2. (eos +sin2 =
(putting for Lp2 the value found in Prop. 32.)
a4 cos21 + c4 sin2 1 /a4 cos2 1 + c4 sin2 1
a cos2 + c2 sin and Cp a cos2 + cc sin2
If the ellipticity be small,
/(1 + 4e) cos21 + sin2 1
CP = C /(1 + 2e)cos21+ sin2 t
= e(l + ecos2l) nearly = c ( + e -e sin2 1).
88. PRoP. 35. To find the angle of the vertical.
Let ACp be the corrected latitude = 1'.
piN p N
Then tan= pN tan l'
NL t a CN;