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.

VARIATION OF PERIHELION. 91
the constant, in the integration of the term multiplying se,
being made =0, because, whatever be the value of e, we
always make the equation of the center = 0 when nt+ e-tr= O
(19), or when 0- ar = 0: and the same must therefore hold
for < e.
ad /(1 - e') sin (0 - -r) (2 + e cos 0 - -r) e
And t + - --- *
//(u) ' (1 $+ e cos 0 -.)2
Ja (1 - e2)e
^J ~/(/) (1 + e cos 0 - z')2
The second side will be some function of 0 which we
will call b (0). For convenience, put t' for the second
term on the first side, then
() (0) = t + t',
and consequently
(0o + 0') = t + t'+ >'(O).0',
O' being supposed small. Here the longitude 0 + ' corresponds to the time t + + '' (0). 0', on the same suppositions as those on which the investigation preceding has been
made, namely, that the eccentricity is e + Se. Make
t + (' (0). 0'=, or 0'=
then the longitude 0 - (O) corresponds to the time t; or
the longitude at the time t = longitude as usually expressed
a{XV/l-e2 sin(0-wr)(2+ecos0 —a ) <. /' (+ e cos0 - i)2
$ -]- ~ e~
V/ ~ (1 + e cos 0- ) )2 < a (1 - es) '
=longitude as usually expressed
sin (O - r) (2 + e cos 0 - zr),1 - e2
and hence by the definition of the differential coefficient,
dO sin (0 - ar) (2 + e cos 0 - r)
de 1 - e2