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.

18 LUNAR THEORY.
Applying this to M in the opposite direction, we have, for
the whole disturbing force perpendicular to the plane of
the ecliptic,
1s (IMG EG\
m / ---Y >3 - +
'v/i +i Y13 y3.
28. PROP. 9. To find the whole force upon M in
these directions: or to find P, T, and S.
Besides the disturbing forces, we must also find the forces
resulting from the mutual attraction of E and M. The atE M
traction of E upon M= -: that of M upon E = -, in the
opposite direction: applying the latter to M with its direction
changed, we have, for the whole force on M, -Er acting in
the direction ME. The resolved part of this, in the direction
E+M
of the projection of the radius vector, is x cos MGB
E+ M
p2 (1 + s$){'
the resolved part in the plane of the ecliptic, perpendicular
to this projection, = 0:
the resolved part, perpendicular to the plane of the ecliptic,
(E + M)s
p (i + ~s")
If, then, we put P, T, and S', for the whole forces on M,
parallel to the projection of the radius, perpendicular to
the projection of the radius, and perpendicular to the
ecliptic, supposing E at rest, we have
P
MG E+M
P 2(1r + cs)
+. m/(1 + s2) y3 y+y —) -'. cos ( - 0) y 3 - ) j
T-=-m'.r' sin(0-0').(, - 3),