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.

DEVELOPMENT 0F R. 107
and similarly r'= a'+v', where the expression for v' is similar
to that for v, putting a', e', n', e', -', for a, e, n, e, r. The
expansion required will therefore be effected if we first expand
1
V/{a - aa'cos( (- O') + a'
and then in every part put a + v for a and a' + v' for a'. Let
w = 0 - ': then by Prop. 49 and 50
1
V/{2- 2aa'cos (0 - 0) + a'2}
may be expanded in the form
+ C(') cos (0 - 0') + C) cos 2 (0 - 0') + C cos 3 (0 0') + &c.,
where C), C(), &c. are functions of a and a'. If then in
each term we put a+ v for a and a'+ v' for a', the series
will become the sum of a multitude of series, in this forn
1
/ r2- 2rr cos ( - O') + r'2
- 1 C(O) +._} d C(~)
= 2 dan + d a2
d C(O) d2 C()
=C ) ~ + da2
+d-v + 2, d2,v
da 2 dada'
d2 C(O)
4 -a'22+&c.
c() 1 cdC(l) 2
+ cos (0 - 0') x + dC d v
d + dada &C.
C(1) + -- v+ dc.
" da2
+cos2(O-O') x d
i t~ d C(1)
-+ + dtfv + &c.
+ &c.