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.

186 PRECESSION AND NUTATION.
3. Now, to shew that AD really is the axis of rotation, take any point P in the body: with center A, suppose
a spherical surface described, passing through P, and cutting
AB, AC, AD, in B, C, D; let BDC, BP, CP, be arcs
of great circles. Let PQ, drawn perpendicular to PB on
the surface of the sphere, be the motion of P, produced
by the rotation about AB only, in the very small time t;
let PR, drawn perpendicular to PC, on the surface of the
sphere, be the motion of P, produced by the rotation about
AC only, iln the same time; then, if the parallelogramn
QR be completed, PS, the diagonal, is the true motion
of P, in that time. Now CPR = 900 = BPQ; adding RPB
to both, CPB= RPQ. Also
sinSPQ: sin SPR:: sin SPQ: sinPSQ:: SQ: PQ:: PR: PQ.
But PR= w't.sinPC to radius AC;
PQ = wt. sin PB;.. sinSPQ: sinSPR:: '.sinPC: w.sinPB.
sin BD. sin BDP
And sin BPD: sin CPD:: sinBP
* -:: sin PC. sin PC.sinBD: sin PB. sin CD
sin CP::. sin PC:. sin PB,
(since sinBD: sin CD:: sinBAD: sin CAD:: w: o).
Hence, sin SPQ: sin SPR:: sin BPD: sin CPD.
Since, then, the two angles CPB, RPQ, are equal, and are
divided into parts whose sines are in the same ratio, it
follows that those parts are equal, or BPD = QPS. Adding
to each BPS,
DPS = BPQ = 900;
and, therefore, PS is perpendicular to the plane APD. In
the same manner it may be shewn, that the plane passing
through any other point of the body, and through AD,
is perpendicular to the motion of that point; and, since