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.

TIME OF DESCRIBING PART OF AN ELLIPSE. 7
M+M'
h2 a (1 -e2'
or h= /a (1 - e) (M + M').
12. PROP. S. To find the time of describing any part
of the ellipse, or to express nt + e- B (the mean anomaly)
in terms of 0 - B, (the true anomaly).
By Whewell On the Free Motion, Art. 25, or Earndt r2
shaw's Dynamics, Art. 88, - = - = (in the present instance),
dO h
a. (1 - e2) 1
— ==M ==~ + M' (i +e -. The most convenient form
/M + M' (1 + e cos 0 - B)~2
into which this can be expanded, is a series of cosines of
multiple arcs, as
A + CcosO - B + D cos 2.0 - B + Ecos3.0 - B + &c.
To effect this, we shall first expand in a similar
1 + ecos 0- B
series.
i + -
13. If for cos 0 - B, we put - (where = e9x - -"1),
2
we have
1 1
+ ecos -B e (
2 k <J
1 e
which will =, if a2 + 3 = 1, a/ =-.
(a+3x) (a+-) 2
Fromn these equations a = - (/1 + e + /1 - e):
- = 1 ( +/1 + e-/ - e);
1 + x/1 -es 3 e
2 ' a i + /1 - e