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.

188 PRECESSION AND NUTATION.
7. We have supposed, in the enunciations of the Propositions above, that an angular motion about one axis is
suddenly impressed upon a body which had previously an
angular motion about another axis. It is evident, that the
conclusions are the same, if we suppose both angular motions
to be impressed at once.
8. From these Propositions, compared with the usual
propositions on the composition of forces, it appears, that
if two forces in the directions AB, AC, be proportional
to w, w', their resultant will be in the direction AD, and
will be proportional to w". And hence, if several angular
motions were impressed upon a body at the same time, the
new axis of rotation and the angular velocity about that
axis would be found, by finding the direction and magnitude of the resultant of forces which are in the directions
of the several axes of rotation, and are proportional to the
angular velocities.
9. If then a body revolve about an axis, and angular
motions about two other axes be impressed upon it, it is
indifferent whether we first compound the two impressed
motions, and then compound their resultant with the original
motion, or compound the original motion with one of the
impressed motions, and then compound their resultant with
the other motion. For, in compounding the forces proportional to these angular velocities, the order in which they
are taken is indifferent.
10. If the angular motion about AC be not produced
instantaneously, but by the continued action of a finite force,
its effect may be found, by supposing the time divided into
a great number of small intervals, and supposing the angular velocity generated in each of those intervals to be
impressed at the end of each, and then finding the limit
to which we approach, by increasing indefinitely the number
of these intervals.
11. PROP. 3. If a uniform force act upon the body,
tending to give it a motion of rotation about an axis which
is always perpendicular to the axis about which it is at