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.

ALTERATION OF MOON S MEAN MOTION. 57
the computed value of which also agrees well with the observed value. The ratio of the motion of the perigee to
the motion of the node, which is expressed by the fraction
225 75
-m2 + -- 3 + &c. 1 +-m + &c.
4 32 8,or
3 9 +
-m2 - - 3 + &c. 1 - - m +&c.
4 32 8
is much greater for the Moon, where m =-, than for one
of Jupiter's satellites,* where m is extremely small. This
is alluded to by Newton, Lib. III. Prop. 23.
73. Upon continuing the approximations, it appears
that p, the coefficient of t in the Moon's mean longitude,
depends upon e', and consequently, an alteration in e' produces an alteration in the Moon's mean motion. Now e',
the eccentricity of the Sun's or Earth's orbit, is slowly diminishing from the attraction of the other planets, and this
causes an increase in the Moon's mean motion. It is remarkable, that the indirect effect on the Moon is much
greater than the direct effect on the Earth.
74. The coefficients of inequalities of a high order
cannot easily be calculated from theory. Even Laplace
supposed it necessary, after finding the forms of the inequalities from theory, to discover the coefficientst from observation. Proceeding on this principle, he suggested that
there must be, in the expression for the time in terms of
the Moon's longitude, a term of the form
sin (3 -2g- c) 0 + a+ 2y - 3.
This could result only from the multiplication of sines or
cosines of these arcs; (3-3m)0 - 33, (3m0 + 3 - 3),
* That is, supposing the motions of the perigee and node of Jupiter's satellites to be caused by the disturbing force of the Sun. In reality, the principal
part of these motions is occasioned by the oblate form of Jupiter.
t In Damoiseau's tables, and in Plana's and Lubbock's treatises, the coefficients
are calculated from theory.