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.

98 PLANETARY THEORY.
(which we shall call nt + e - -z + x)
SnaP
e = e -ea e' cos T.. i(l3n - 8,')
Substituting these in the series for 0, and observing that
sin (nt + e - ' r + x) = sin (nt + e - r) + xi cos (n t + e - r),
and neglecting products which produce higher powers than e3 e",
3
O, = nt + e + (2e -- + &c.) sin (nt + e - ar)
4
( —+ &c.) sin (2nt + 2e - 2r) + &c.
39 n2aaP 2na2 dP SnaP 1 -.
+-+ I- es sin T,u(13n —8n')2 4(13n-8n') da 2,(13n-8n')
+ -(i8 " ----e e2e'2. { sin T- (nt + e - Z) + sin T + (n t + e - r) }
ji(13nl- Sn')
3naP
+ ^3 - 8 ee'2. 5 sin T- (nt + e - wr) - sin T + (nt+ e- o).
The first two lines are the longitude calculated with invariable
elements. The additional terms are
39 na P 2na2 dP 3nc>aP
\u(13n - 8n)2,(13n - Sn') da 2,(13n - 8n')
x e' e' sin 13 (n t + e) - 8 (n't + c')- - 3- - r'
GnaP
4-+ -,) e e2 sin. 12(nt + e) - 8 (n't + e') - 2'r - ar,(l3n - s87)
constituting two distinct inequalities. The value of these
has been adverted to in (100) &c. supposing 13n - 8n' small.
We may remark that in this case the first inequality is
twice divided by 3n- 8n': the second, though only once
divided by 13n - 8n', is multiplied only by 2 ee instead of
3e'2: the comparative magnitude of the terms depends then
13n - 8n'
on the proportion of -- and e.