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. 1 87
the axis of rotation is the line of intersection of all the
planes perpendicular to the motion of every point, AD must
be the axis of rotation.
4. It is here supposed, that the angular motion about
AB tends to raise all the particles between AD and AC,
and that the angular motion about AC tends to depress
them. If, however, the angular motions about both AB
and AC, (fig. 2), tend to raise the particles between AB
and AC, produce CA to C': then the angular motion about
AC' tends to raise the particles between AC' and AB,
and the angular motion about AB tends to depress them.
Silence, the new axis of rotation will be the line AD,
which makes sinBAD: sin C'AD:: w':. The same is
true, if both angular motions tend to depress the particles
between AB and AC.
5. PlOP. 2. The angular velocity about the new axis
AD, will be to the original angular velocity about AB, as
sinBAC to sinDAC; and the angular velocity about AD
to the original angular velocity about AC, as sin BAC to
sin BAD.
6. Let o" be the angular velocity about AD;
then, PS = w"t. sin DP.
Now PQ: PS:: sinPSQ: sinPQS:: sinSPR: sin QPR:: sin DPC: sin BPC,
since BP, DP, CP, are perpendicular to PQ, PS, P/,
respectively. But
sin DC. sin DCP sin BC. sin BCP
sin DPC: sin BPC sin DP sin P
sin DP sin BP
putting then for PQ and PS their values,
sin DC sin BC
wt. sinBP: w"t. sinDP:: - D;
sin DP sin BP
whence, w: ":: sinDC: sin BC:: sinDAC: sin BAC.
And, since w'::: sinBAD: sinDAC by (2);. '. ow ):: sin BAD: sin BAC.