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.

12 LUNAR THEORY.
LUNAR THEORY.
PERTURBATION OF THE EARTH'S MOTION.
20. IF the Earth, in its revolution round the Sun, were
unaccompanied by any other body, it would accurately describe an ellipse. By the attraction of the Moon, the orbit
will be altered: to assist us in the discovery of the orbit
really described, the following theorem will be useful.
21. PitoP. 5. The center of gravity of the Earth and
Moon describes about the Sun, very nearly, an ellipse in one
plane, and the area passed over by its radius vector is very
nearly proportional to the time.
Let E and M, (fig. 1.) be the Earth and Moon, m' the
Sun; G the center of gravity of the Earth and Moon;
join m'E, m'G, m'M; and draw EH, MIK, perpendicular
to m'G; let
m'G = r', EM = r, m'E = y, m'M = ', m'GM w.
Now the accelerating force of m' on E in the direction Em'
Mn' m'. E
is -; therefore tlie moving force in that direction =;
y- Y2
therefore the moving force in direction parallel to Gm' is
nm'. E m'I
y y
m'. E r'+ GE.cos w
2
q~ ^ y
nm'. E (r + GE. cos w)
y,
Similarly, the moving force on M in direction parallel to Gm'
m'. m'JC
= =
m'. M (r - Gi. cos w)
y/3
therefore the accelerating force on tlie center of gravity in
the direction Gm' is