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.

LUNAR PRECESSION. 205
and for the motion perpendicular to the Moon's orbit,
S7r.B.sinl ' cos 2
T' (n+ 1) 2 '
I being now measured from the intersection of the equator
with the Moon's orbit.
34. The latter expression, and the second term of the
former, are periodical terms, going through all their changes
of value twice in a month: their magnitudes, besides, are
so small, that they are generally neglected.
Supposing I increased by 2 r, we find, for the motion
of the pole produced by the Moon's action in a sidereal
revolution,
67r2. B.sin '. cos '
T'. (n+ 1)
which motion is parallel to the Moon's orbit, or perpendicular to the great circle joining the pole of the Earth
with the pole of the Moon's orbit.
35. PitOP. 14. To investigate the precessional motion
produced by the Moon's action during a long period.
Let Q, (fig. 8), be the pole of the ecliptic; q that of
the Moon's orbit; P that of the earth: let them be joined
by arcs of great circles; then, by the last article, it appears
that by the action of the Moon, the pole is in the time
T' carried in the direction Pp, perpendicular to Pq, through
the arc
6wr'. B
T,'w(n+B sin q P. cos q P.;T'o (n + 1)
This may be represented by supposing the pole to have
the velocity
67r2. B
sin qP. cos qP,
T'2 ~ (n + 1)
in the direction Pp. Its velocity then in a direction perpendicular to QP, is
T -' " (n + l sin qP. cos q P.cos QPq
T'". w (n + 1)