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.

196 PRECESSION AND NUTATION.
in the direction parallel to RA. Now if a force f were
applied to every point of the spheroid, in the direction SA,
it would produce no effect in giving the Earth a motion
about A; without altering the motion, therefore, we may
suppose this force applied; that is, we may suppose the
only force acting at each point, to be the difference of the
force really acting there, and the force at A. In the figure,
the point A is evidently the projection of the axis, about
which these forces would make the Earth revolve. We
RA
must, therefore, find the momentum of the force f imSA
pressed on a particle 3m at P, about the center A. Let
SQ = SA; then, S being very distant, AQ is nearly perpendicular to SA; and
SQ3
SR ( p =SQ + 2 PQ, nearly;
RA
or RQ = 2 PQ. The force f. - Sm may now be resolved
SA
into
fRSQ QA 2PQ PN
f SAm and f.-A m or fm.n-a and fm. SA'
SA SA SA SA
acting at P in the directions QP, PN. Their momenta to
turn the Earth in the direction KB, are
2 PQ.PN PN. PQ
- fm. SA and - fm. SA
SA SA
(considering PN and QA as sensibly equal); the sum of
these
PN.PQ
= - sf.m. SA
If the absolute force of the attracting body S = S, and
SA = r, then f=, and the moment of the force on P
r3
Let AM=x; MP=y; BAS =.