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.

56 LUNAR THEORY.
a' E - M
multiples of mWa-. -.cos ( - m) 0- j3, of the fourth
a g
order; which, upon integration, rise to the third order (the
coefficient of 0 differing little from 1); and the corresponding
inequality in longitude is of the third order. The comparison of this observed inequality with its computed value, gives
us the means of determining - being known pretty exa \E.E-M E-M, \,
actly, and therefore, or being known) that
E+M
is, the ratio of the Sun's parallax to the Moon's. The latter
of these is very well known from observation: hence, the
former can be found. Its quantity thus determined agrees
very exactly with that determined by transits of Venus.
(2) The value of c, found by the second approximaS32
tion, is 1 -, which gives, for the progression of the
3m2
perigee in a revolution of the Moon,. 2 r 1~. 30'.
4
This is about half its true quantity, which is 3~. 2'. 22".
By continuing the approximation, we find that c is expressed
by a slowly converging series, of which the first terms are
3 225
1 —m - m,
4 32
and the progression of the perigee in one revolution is
(3 2 225
2 T. - m2+ — + m + &c.
4 32
On calculating a sufficient number of terms, the theoretical
value of the progression of the perigee is found to agree
most accurately with the observed value.
The series for g, on the contrary, converges fast; it is
3 9
1 + -m2 - - 3 + &c.
4 32
and the regression of the node in one revolution is
2 4 - 32 )