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.

6 LUNAR AND PLANETARY THEORIES.
10. PROP. 2. The orbit which the Sun appears to describe about a planet is a conic section.
Let M = mass of Sun (estimated by the accelerating force
which its attraction produces at distance 1), M' = that of the
planet: let their distance = r. The accelerating force on the
M'
Sun, according to the law of gravitation, = -: that on the
M
planet = -: if then we suppose this force applied in the opposite direction to the Sun, the whole accelerating force on the
M + M'
Sun, supposing the planet at rest, = = (M + M') u2,
if u =-. Substituting this for P in the equation above,
d2U M+ M'
d -,2 +u h = 0,
the solution of which, by (3), is
M+M'
u = h + A cos - B,
h2
or r=M+
+ A cos - B
which is the general polar equation to the conic sections, the
focus being the pole.
11. The conic section which a planet appears to describe
about the Sun, or the Sun about a planet, is found to be an
ellipse. Let a and e be the semi-axis-major and eccentricity,
B the longitude of the perihelion, then (Hamilton's Conic
Sections, Art. 115; Analytical Geometry, Art. 145),
1
I e
+ cosO - B
oa ( -e') (1i - e)cs
Comparing this with the expression above,