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.

16 LUNAR THEORY.
m. MG
Resolving the force into one parallel to MB,
and one parallel to BG,
m'. n.MG m' MG
the latter - x cos MGB = ---
~y O'1 y3 Vl1 + s
and resolving the force into one parallel to BG, and
yt.3
another perpendicular to BG, in the plane of the ecliptic,
ml. Gm'
the former = -. cos n'GB.
yl"
Let 0 be the longitude of M, seen from G; 0' the longitude
'. Gm'
of m': then z m'GB = 0 - 0': and the part of m
yt3
parallel to BG
m'. Gm'
= -- -cos 0 -0.
y
Hence, the whole force on M in the direction BG, produced
by the Sun's attraction, is, { MG Gmi'
mt [ — -- -- cos O- CS0 -yn 3,1 / S y3C
W l0'V y
Similarly, the whole force on E, estimated in the same direction, is
EG Gm'
m (- /+ - y- cos 0 - ').
)
y3%/1 + y3
If, then, in the same manner as in (9), we suppose this
force applied to M in the opposite direction, we have, for
the whole disturbing force on M, in direction of the projection of the radius vector,
MG EG
m G-1 + EG - Gm'cosO - O'( -
y'3 2/ / y+i Y'3 +S
* It is always to be understood that the orbital motions of the Earth and the
Moon are in the direction opposite to that in which the hands of a watch revolve,
and the angles are therefore estimated positive in that direction.