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.

28 LUNAR THEORY.
putting ^, -a; that of the second,
s = k. sin (0 - y).
The first shews that the Moon's orbit is an ellipse; the second,
that the tan latitude = k. sin longitude from node, and therefore, that she apparently inoves in a great circle. For, if
we conceive the earth to be in the center of a sphere, and
if upon the sphere we describe a spherical right-angled triangle, in which one of the sides including the right angle
(and representing the Moon's longitude on the ecliptic as
measured from a certain point) is 0 - y, and the other side
including the right angle (and representing the Moon's latitude) is p, where tan = k.sin0 - y; and if we draw a
great circle for the hypothenuse of the triangle, then the
angle k opposite to the side ( is determined by this equation
tan à
tan & =
sin 0 - y
The value found for s or tan ( reduces this equation to
the form
tan b/ = k.
Therefore, from whatever point of the Moon's path we draw
a great circle to a certain point on the ecliptic, the angle
at which it meets it is invariable: consequently the same
great circle must pass through every point of the Moon's
path, or the Moon moves apparently in a great circle.
40. PRop. 14. To integrate the differential equations,
second approximation.
As terms of the second order are to be included, we shall
here have the first terms of the disturbing force.
i therefore = 2x 1 - - s - r' - + - cos 2. (O - 0)
T 3 m'2 U'3 S-Ps
T 3 m 2. (O- S-Ps
— 'J -- --. 4 sin2.(9-?: — s = 0.
M 2 u4 u
We have just found for u the expression a {1 + e cos (9 - a)};
but it is evident that, in the substitution of this value in terms