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.

NEW SOLUTION OF EQUATION TO SECOND ORDER. 43
3k2 m2
And u therefore = a { 1 - - - + e cos (cO- a)
4 2
k'
- - cos (2go - ) 2m) 0 - 2/3
4
15 3
+ - me cos (2-2m-c) 0-2/3+a - - me'. cos (m0+S3-)].
8 2
56. PnoiP. 21. To form the differential equation for s.
First, s, as we have observed, will be approximately
represented by k sin (gO - y), where g differs little from 1;
the difference, which is caused entirely by the disturbing
force, being of the second order. Hence,
d2 s
d + s, or k (1 - g9) sin (gO - y),
will be small, of the third order: consequently, the terin
-d + s 2/ hU3
(d02 ) Jo h2 3
in equation (I) of Art. 41, will be of the fifth order, and is
not to be considered.
S- Ps u'(3 3
Second, or ms- + -cos2. -0
ma3 (4 +e'cosm+-)_ -— m 2 3 --
m'a"3 ( ~e' c~s mO+[3-). ksin gO-y. + - cos2.0-)
h2"a (+e cosc0-a)4
m a'3
Now h2 k or m2k, form a product of the third order:
hence, in the quantities which multiply then, all small terms
are to be rejected;
S-Ps (s.. u - =m2 k. sin (gO - y) + -cos (2 - 2m) 0- 23j
s
- mk sin (gO - y) + - sin (2 - 2m + g) 0 - 23 - 7
- sin (2 - 2m - g) 0 - 2/3 + y}.
4