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.

COMPUTATION OF SUN S DISTURBING FORCE. 15
the orbits of the Earth and Moon are nearly circular,
this angle varies as sin w very nearly. And if the Moon
be above the plane of the ecliptic, the Earth will be below
it, and the Sun will appear to have a latitude, which can
be calculated from the latitude of the Moon.
PERTURBATION OF THE MOON'S MOTION.
24. If the Sun did not attract the Earth and Moon,
or if it attracted them equally, their relative motions would
not be disturbed, and the Moon would accurately describe
an ellipse about the Earth. But the Sun attracts them
unequally, and in different directions; so that not only is
the force altered in the direction of the radius vector, but
a force also acts perpendicular to it. And as the Moon's
orbit is inclined to the ecliptic, the disturbing force draws
the Moon from the plane in which she is moving, and thus
the plane of her orbit is perpetually changing. There
appears to be no better mode of estimating the disturbing
force, than by resolving it into three parts, one in the
direction of the projection of the radius vector on the
ecliptic, another in the plane of the ecliptic, perpendicular
to this projection, and a third perpendicular to the plane
of the ecliptic.
25. PROP. 6. To find the resolved part of the Sun's
disturbing force on the Moon, in the direction of the projection of the radius vector on the ecliptic.
Let E, M, m', (fig. 2.), be the Earth, Moon, and Sun:
G the center of gravity of the Earth and Moon, which by
Prop. 5., describes an ellipse in one plane about the Sun,
or about which the Sun appears to describe an ellipse in
one plane: draw MB, EiA, perpendicular to the plane of
the ecliptic; join m'M, m'B, BGA: let m'G = r', AB = p
EM = r, m'E = y, m'M = y, tanMGB = s. Then AB is
the projection of EM on the plane of the ecliptic. The
force of m' upon M is -, in the direction Mm' which,t!
m' MG m' Gm'
is equivalent to,. in direction MG, and -. — in
tn p l to
direction parallel to Gm'.