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.

68 PLANETARY THEORY.
Then, by (16) and (19), nt + e is the mean longitude of m
at the time t: nt+ - is its mean anomaly; also 0 - a
is the true anomaly; and the formula of (19) becomes
e3
0 = nt + e + (2e - - + &c.) sin (nt + e - r)
+ (-4 + &c. sin (2nt+ 2e - 2r) +&c.
\ 4 /
a (1 - e2)
Also r = - (
1 + e. cos (0 - ~r)
Putting in 0 - - the value found above for 0, and talking
its cosine, we get at length
e2
r = a - - + &c. - (e + ) cos (I + e - r)
e2&
(-+ &c.) cos ( t + 2e - 2) + &c.
which we shall call a + v. Similar expressions hold for the
place of m'.
l2 ai
Also - = n + n { Se cos (nt + e - r)
3Se2 e2
+- + - cos (2nt + 2e - 2r) + &c.1.
2 2
alfereaing d(r3r)
also by again differentiating -t and substituting in the equation
dc2(rr) Ul
d2 (ror) 9 8r + S = o,
d-~-) rr+S +
dx dM dy dN
dt dt dt '+dl ~
dy d dO d M I dYn 1x
From these (as x - y _ - d - A)h - - - y;
and as x = r cos 0, y = y sin 0,
we get r = - (cos 0ft r sin 0. S - sin 0ftr cos 0. S),
where the elliptical values are to be substituted for r and 0. It will easily be
seen that we might have put
ror = M ( cos a + ysin,) + N (ycosar - xsin ar),
and this would have given
or = - (cos -,rft r sin -. S- sin 0 - srt r cos 0 -, t. S),
which in actual computation would be rather more convenient. The method in
the tcxt is the most practicable.