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.

COMPOSITION OF ROTATORY MOTION. 191
impressed at the end of each. At the end of the first
interval, the axis will be transported from AB to AD, the
angle BAD being = a; and at the end of the second and
nw
succeeding intervals, it will have the positions AD', AD'', &c.
in space, each of the angles DAD', D'AD", &c. being =
1/n
Now, in -, the body revolves through; hence, if on
n n
the surface of the sphere, the angle D'Dd' be made -,
10 n
and Dd'= DD', d' is the point which, at the end of the
second interval, coincides with D'; and Ad' is, therefore,
the line in the body, which, at the end of the second interval, is the axis of rotation. In the same manner, if d'd"
inake with Dd' produced the angle, and d'd"= D'D"
Ad" is the line, which, at the end of the third interval,
is the axis of rotation, &c. From the construction it is
evident, that BD, Dd', d'd", &c., are the sides of a regular polygon. Suppose now, n increased without limit,
or the number of the sides of the polygon increased without limit; the limit of the line traced on the spherical
surface by its intersections with the successive axes, is a
circle.
14. To find the radius of this circle, we observe, that
if the circle and polygon be small, the sum of all the
angles at D, d', &c. =27r; but, since each of them=-,
2nTr
their number, or the number of sides of the polygon =
And the length of each = AB. -; therefore the circumfer> r. AB. a
ence of the polygon, or ultimately of the circle =
therefore the radius of the circle = AB..
Co