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.

120 PLANETARY THEORY.
not very long. For these we either neglect or estimate
approximately the secular variations of the elements. Similarly to calculate accurately the secular variations, we may
neglect the variations of short period. Putting ecos' = u,
esin s = v, and e'cos = u', e' sin' = v', the equations of
(141) for the variations of the eccentricity and perihelion
are easily changed to
du rn' na2a'
dv m naa
d-u' nn a ( (D(")u - D(2)v ')
dt JU 4
v=. sin (- Dq + p2sin, D(2)),
dt f 4
du' m n'a' a
dt -A 4
Solving these as simultaneous equations, we find
u = pi cos (ait - qi) + p2 cos (a2t - q),
v = pi sin (a t - qi) + p2 sin (a2t - q2),
-- - cos (a, t - q,) + -- cos (a2t - q2),
X1'r2
Pi P2
v'= -sin (ait - q,) + -sin (at - q2),
where p, P2, p,i q2, are arbitrary constants, to be determined so as to make the expressions for u, v, &c. represent
the known values of u, v, &c. at any one time: the two
values of ai and a, are
a (a nr'ar + mtna) D()
8y
= %\/(m'na - mn'a' 12 D( )2+ 4nz'na. mn'a'. D(2)2)},
and the two values of x~ and %2
2 m n'a'D( { (m n'a - m'n a) D(
n /(mna - m na'li"D(l) + 4m'na. mn'a'. D(22)}.