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.

INFLUENCE OP A DISTANT BODY CALCULATED. 195
20. PROP. 8. The velocity of the Earth's rotation is
unaltered by the action of the Sun and Moon.
Since the Sun's action would give the Earth a motion
of rotation about an axis in the plane of the equator of
the terrestrial spheroid, and the Moon's action would give
it a rotation about another axis in the same plane, their
combined action would give the Earth a rotation about a
third axis in that plane, by (7).
Now to shew that the Earth's angular velocity is unaltered, we must shew, that this axis is always perpendicular to the axis of rotation. Let AC, (fig. 3), be this
axis, AB the axis of rotation; by (15), the points of intersection of this axis with a sphere described in the Earth
about A, lie nearly in a small circle, whose center is E.
The ellipticity of the Earth is produced by its rotation;
and since the axis of rotation passes successively through
all points of the circle BF in one revolution, the axis of
the spheroid will pass through E, the center of that circle.
AE, therefore, is perpendicular to AC; or if EC be joined
by an arc of a great circle, EC is a quadrant. And EBC
is a right angle; hence, BC is also a quadrant, or BAC
is always a right angle. Consequently, by Prop. 3, the
velocity of rotation is not altered.
21. PROP. 9, To calculate the value of a; the force
acting on the Earth being the attraction of a distant body;
and the Earth being a homogeneous spheroid.
Let A, (fig. 6), be the Earth's center; AB the axis of
the spheroid: S the attracting body; take P, the projection
of any point of the Earth, and draw PN perpendicular to
SA, and PM perpendicular to the projection of the equator.
Let f be the attraction of S upon A; then the attraction
SA' SA3 SR
of S upon P, is f,^p-, or, if SR = -5, it =f S.
If SA, then, be taken to represent the force f, SR will
represent the force on P, and RA the difference of forces
on P and; or that difference of forces and i
on P and A; or that difference of forces =f./ —, and is
13-2