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.

T1ME OF DESCRIBING PART OF AN ELLIPSE. 9
we have
1 i= f1 e (1 + /- 2 - e')
( + ecos 0 - B)2 (1 - e2) 1 + -
e (1 + 2 \/V - e')
+ 2 ^(1+- Ve.)2 cos2.0-B
(1 + 2/1 - e2)2
el(1 + 3v/1 -e2)
-2 31- -- cos3. - B + &c.
(1+ 1-e )3
dt ai
15. Hence (12), x = --
_ 2e2 (1 + /i - e9
1 -2e cos 0 - B + cos 2.0 - B - &c..
(1 + V/ti-e )
Integrating and correcting, so as to make nt + e - B = 0,
when 0 - B = 0, e being a constant depending on the time
when the planet passed the perihelion, and putting
— n,
e (1+2 /1-e")
nt+e-B=0-B-2e sin -B+ +2sin 2. 0-B&c
(1 + p - e))
2 e.(1 +p - e') sinp. 0 -B =F &c.
p (1 + v/l - e')p
For a whole revolution, suppose 0 increased by 27r: this
change makes no alteration in the values of sin 0 - B,
sin 2.0 - B, &c.: then if T be the periodic time, t will be
increased by T, and
27r 2 7r. a1
nT=27r, or T - =n %/M + M'
16. The term anomaly is used generally to denote the
angular distance of that body which is supposed to move, from
its apse. The true anomaly, then, in the present case, is 0-B.