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.

116 PLANETARY THEORY,
The product of the two first cosines will produce one term
cosnt + e - - (n't + e'- ~r'),
and lience there will be in R the term
m a
-, -ee'. cos n t + E - - (n't + c'- 'r'). cos n t + e - (n't +
one part of the development of which is
ma eef
—,.. cos (i - r'i).
a1 2
Again, if we observe that, in the expansion of cos (0-0'),
(130), the term of the second order is
- cos nt + e - (n't + E).
which contains one term
cos nt + e - (n't + e'). 4e'. sinnt +e - r. sin n't + - '
we shall see that the product of the last factors will have
one term
ee. cos nt + e - v - (n't + e - r')
which multiplied by the first, and by -, will introduce
a
m'a ee'
the term + --. cos (r - z').
a2 2
This term destroys that which we have just found. There
are other terms involving higher powers of e and e', which
we shall not consider.
(2) If we examine the development of
V 2r' - 2r'"cos (O - ') + r' in (129)
we shall find in - these con/ 2strr cos(an- t) + ter
stant ternis