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.

FORCE EXPRESSED BY PARTIAL DIFFERENTIALS. 61
in the directions of x and y. These however are not the
forces which,disturb the planet. To obtain these, we must,
as in (9) and (25), find the forces which m' exerts upon
the Sun, and apply them with sign changed to the disturbed planet: the expressions will then represent the forces
which disturb the relative motions of the Sun and planet,
and which are in fact the objects of this investigation. The
forces exerted by the planet on the Sun are; in the direction
mTX tfm y
of,' (,2 +y2) and in the direction of y, (x'2 +y)
HIence the disturbing forces on m are,
m' (ca - x) m't
',!(m )2 + (y'-/_ y)}- ('2 + y2),
in Y m'(y' -y) n'y'
" (-tlV)2 +(y'- y)2} ('2+y2)
mN 'm (m'w' + y'y)
Now let R — +
d i (w' - w) + (y' - y)2 } (w2 + y' )
t (w'- w)i + (y'- y)21 + (w'2 +y
dR - '(y - y) m'y'
dy = ('- )2 + (y' _ y)2i (I2 + yt2)i
Consequently the disturbing force in x =
dx
dR................................... in y = -
dy
in which expressions the differential coefficients are to be
taken as if x and y were independent of each other, and w'
and y' did not vary when x and y varied.
78. If we put j for the sum of the masses of the Sun
and the disturbed planet, the force arising from their mutual
attraction, as in (9), is - - in the direction of w, and
(x2 + yP)}
- - rY in the direction of y. Hence the whole force
(Xac + y2)
acting on m is,