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 NUTATION. 209
41. In the two last Propositions we have considered i
to be constant, and Q to be proportional to t. It appears,
however, from Art. 68, of the Lunar Theory, that the inclination is expressed, nearly, by
$m
i. 1+ -- cos 2. (long. node - long. Sun);
8
and that from the longitude of the node, found on the supposition of its uniform retrogradation, we must subtract
3m,
sin 2. (long. node - long. Sun).
2 r:-t
Now, the Sun's longitude = - + C, nearly: the longitude
27rt
of the node = 180 ---: hence, for i we ought to put
Ir
[1 + - cos{47rt +r + - c+C,
\T rL /
and for Q we should put
27rt 3m sin i 1\.
-^ 8 -sin{47rt +L+ +2 C
before performing the integrations: and sin 2 Q, &c. could
be expanded, as in Art. 49, of Lunar Theory. But the
additional terms thus introduced have small coefficients, and
upon integration receive large divisors, so that they become
quite insensible. The expressions which we have found are,
therefore, subject to no sensible error.
42. If now, in the terms of Nutation, which depend
on twice the longitude of the Moon's ascending node, we
used not the mean longitude of the node but the true,
we should add to the expressions terms which have small
coefficients, but which are not integrated, and, therefore, do
not receive large divisors. The values thus found for the
parts of Nutation at any given time, would, therefore, sen14