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.

MOTION OF TWO BODIES. 5
7. When m does not differ much from n, it appears
from (4), that the coefficient of cos m8 + D, in the expression
for u, will be much greater than that in the original equation.
This remark we shall find to be very important. The solution
in the 4th case assumes a form different from any of the others:
its peculiarity will materially affect our future operations.
MOTION OF TWO BODIES.
8. IF the Sun were supposed to be at rest, the motion
of a planet about it might be found by the formula for central
d2u P
forces. In the equation -d +u- -- = 0, where h is a coni dO
stant and = -. - (Whewell on the Free Motion of Points,
Art. 24; Earnshaw's Dynamies, Art. 87; or Art. 32 below,
if T=,) we must put for P the attraction of the Sun on
the planet, and by solving the equation, we should find u
in terms of 0, and the form of the orbit which the planet
describes would then be known.
9. In the actual case of the Sun and a planet, these
bodies move about their common center of gravity. But
their relative motion will be the same as if we suppose the
Sun to be at rest, provided we add to the accelerating forces
which really act on the planet, another force equal and opposite to that which acts on the Sun. For if the same accelerating force be supposed to act on both, since the absolute
motion which it communicates to both is the same, and in
the same direction, their relative motion will be the same
as if that force did not act: and if that force be equal and
opposite to the force really acting on the Sun, the Sun will
be at rest. Or, instead of this, if we add to the forces acting
on the Sun, a force equal and opposite to that acting on
the planet, the planet will be at rest, and the relative motions
will be unaltered. We shall generally make the latter supposition.