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.

PRECESSION OF THE EQUINOXES,
AND
NUTATION OF THE EARTH's AXIS.
ON THE COMPOSITION OF ROTATORY MOTION.
1. P1OP. 1. IF a body revolve about an axis AB,
(fig. 1) with an angular velocity w, and if a force be inpressed upon it which would make it revolve about the
axis AC, with an angular velocity w'; then the body will
not revolve about either of the axes AB, AC, but about
an axis AD, in the plane BAC, dividing the angle BAC
so that sin BAD: sin CAD:: w: w.
2. It is evident, that the new axis of rotation is
that line in the body, which, when the effect of both the
original motions is considered, remains at rest. If, then,
a line AD in the plane BAC be that axis, the angular
motion about AB would tend to raise any point in it, as
D, above the plane of the paper, as much as the angular
motion about AC would tend to depress it below the
plane of the paper. From D, draw DE, DF, perpendicular to AB, AC. In consequence of the angular motion
about AB, the point D would be raised above the paper,
with the velocity
w x ED = w x AD. sin BAD.
And, in consequence of the angular motion about AC, D
would be depressed below the paper, with the velocity
o' x FD = 'x AD.sin CAD.
Making these equal,. AD. sin BAD = w'. AD. sin CAD;. sin BAD: sin CAD:: w.