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.

190 PRECESSION AND NUTATION.
Suppose 1" divided into n parts, as in the third Proposition, and suppose AB, AC, (fig. 1), to be at right
angles. If we suppose the angular velocity -, about the
n
axis AC, to be impressed instantaneously, and suppose AD
to be the new axis of revolution; then, by Prop. 1,
sin BAD a
sinCAD nw '
or, in the present case,
sin BAD a a
-- 7 = -, or tan BAD =-.
cos BAD n nnB
Suppose n very inuch increased; then tan BAD being diminished without limit, we may put the arc for the tangent;
hence, BAD =. And since, by the last Proposition, the
angular velocity remains unaltered, angles equal to BAD
will be added to BAD in each succeeding interval; and,
therefore, at the end of 1", the axis of revolution will be
inclined to the line which was the axis of revolution at the
beginning of that 1", by the angle -. Since the same is
(X)
true of every successive 1", the axis of revolution will move
from the position AB, towards the position AC, with the
angular velocity -.
c)
13. PiKO. 5. Under the same circumstances, if a spherical surface be described in the body about the point A,
at which the axes intersect each other; the points at which
the successive axes of revolution cut this surface will lie
in the circumference of a small circle, whose radius = radius
of the sphere x 4, nearly.
Let AB, (fig. 3), be the original axis, and, as before,
suppose 1" divided into n parts, and the angular velocity -,
about an ais perpendicular to th as f rotation, to be
about an axis perpendicular to the axis of rotation, to be