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.

DISTURBANCE IN TIIE MOON S MOTION. 173
terms which, as remarked in Art. 46. of the Lunar Theory,
will be much increased by integration. With this restriction, and observing that s is expressed nearly by k. sin g0-y,
where g differs little from 1, we have
4 7r u 2 (p (c) 24
p =- 1-; — ) —U. zi (c) sin c. cos w. sin 0. s},
47r (c).(
1= - - — U4. sin o. cos. cos 0. s,
3 5
47r f y2 )' (C). S 6u\ ()
S = -. < ().s + s --. sin o.cosw.sin 0.
3 (1 + 8 5
Let E be the nass of the Earth. The volume of a spheroid,
47rwhose semi-axes are c and c (1 + e), is -c3 (1 + 2e): hence,
3
the volume included between two spheroidal surfaces, is
ultimately
47r d{ce.(i +2 e)
3 dc
and if the density be p, the increment of the mass is
47r d\ ic.. (1 + 2e);
dc
therefore the mass
4,7r 4 {c. (1+ 2e) 47rr 4 7
- 3= U d dc =-: /(c); E (..
de 3 3
Also, by the equation of (61),
_ (p (c) 3 J (c)
2 c3 5' c5
whence
4.r.J(c)= (e — 2c ().- (- 2c. E.
3 5 27 3 f 2
Hence, we finally obtain
P = 1.s" + (e- - 8c2 4 u. sin. cos. sin. s,
(1 +~ s~Y 2/