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.

60 PLANETARY THEORY.
sions proceed by powers of P (35): now P in the lunar theory
r r
is about, but in the planetary theory it sometimes ap400
proaches to - And while two terms are sufficient in the
4'
former case, thirteen or more are necessary in the latter.
Here then the planetary theory is much more complicated
than the lunar theory.
In following for this investigation a process somewhat
different from that of the former, we shall introduce the
reader to some of the most remarkable methods of modern
analysts. To avoid complexity, we shall suppose that all
the planets move in the same plane.
77. PROP. 29. To express the disturbing force of one
planet upon another, by partial differential coefficients (taken
with respect to the co-ordinates of the disturbed planet) of a
function of their co-ordinates.
Let x and y be the co-ordinates of the disturbed planet
(which we shall call m,) the Sun being the origin, and x,
being measured towards the first point of Aries: w' and y'
those of the disturbing planet: n' the mass of the latter
(expressed by the acceleration which it would produce at
distance I in the unit of time). The difference of the coordinates of the two planets is x'- and y'-y in the directions of x and y: and their distance is
v(~ - )2 + (y'-y.)2
The accelerating force of m' upon m is
m
(' - r)2 + (y - y)2
in the direction of the line joining them: the resolved parts
of which are
m' ('- ) and '(y- y)
(X' _-)2 + (y'-y)2} { (, '._ )2 + (y'- y)2 t