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.

DIFFERENTIAL EQUATIONS FOR THE MOON S MOTION. 19
(E +M)s m's MG EG
p'(1 +-S)~ + /(1 +s1) y' + "
E M
And MG= E+ Mp/(1+ ); EG=-E M /( +s2).
29. We have now the values of the forces upon M in
three directions, considering E as fixed. We must, therefore, investigate the differential equations, for the motion
of a body about a fixed center, acted on by forces in
these directions.
30. PLtOP. 10. To find the differential equations for
the motion of M about the fixed center E.
Let E y, Fig. 3, be a straight line in the plane of the
ecliptic, drawn from E towards the first point of Aries:
Mb perpendicular to the ecliptic: then, r E b= Moon's longitude = 0. Also, if x, y, and z be rectangular co-ordinates
(z being perpendicular to the plane of the ecliptic, and
x measured on the line drawn towards the first point of
Aries),
x=pcosO, y=psinO, z=ps.
And X, or the force in direction of, = - P cos 0 - T sin 0,
Y........................ =.-Psin 0 + Tcos0,
Z.................................... S.
Hence the equations of motion
d2 X x 2y d2e d Z
-t-= x, - -=Y, z=
dt2 dt dt -
are changed into the following,
= p -- TY,
Y=_PP
dt2P P
dt p oS
dt2
"= — ' ~2S2
2_2