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.

14 LUNAR THEORY.
hence, accelerating force on center of gravity in direction Gm'
m= EM2+ME2 r2 /3 9. -co + &c.
r {1+ (E+ M)3 r'2 4
m rME r2 /3 9 \
r {1 (E+ M)2r 2 +-cos2w +&c.}.
r - ) 2 4 " r \24 4
Now this differs from m- only by a quantity which is multiplied
r2 E M
by, and by E.-M E M and which, in the lunar
r 1 M i\
theory, (where = nearly, and E = -) is quite
'heor,. er' L~400 E + Ml 80
insensible. In the sane manner, we find the accelerating
force perpendicular to Gm'
m'3 EM2+ ME2 r2
2 2 (E+M): sin 2 w,
m 3 ME lr2
or - -;. sin
)r 2' (M + E)2 r 'sin
which, for the same reason, is too small to be perceptible in its
effects. Hence, the only accelerating force acting on the
center of gravityand and is in the direction of r; and
therefore the center of gravity moves, very nearly, in the
same manner in which a body would move placed at G.
Similarly it appears, that the force on the Sun, and the motion
of the Sun, are the same, as if a mass = E + M were collected
at G; therefore the relative motion of the center of gravity
about the Sun, is tle same as that of a mass = E + M; that is,
it will very nearly describe an ellipse in one plane, making the
areas proportional to the times.
23. COR. The Sun's apparent longitude, therefore, is
not that found by the elliptic theory, for that is his longitude as seen from G; but must be obtained by adding to
the longitude so found, the angle Em'G. Now sinEEn'G
or Em'G, (since it is a small angle, never exceeding 10")
EG. M r.
sin EGm' sinw very nearly: and, since
y E + M rsn