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.

104 PLANKEARY THEORY.
Multiplying cross-wise, and taking the coefficient of sin kw,
(a2 + a'2). k. C(k) aa'. (k - 1). C -'aa'. (k ) (k)
-aa' C(-1)- aa' C(k+l)
2 2k a~+ a~ 2k- i
whence Ck+) = C(k) - k C1
2k +1 aa 2Pk + 1
125. This formula includes (as will be seen on trial)
C(0), C<(), C(2). And as the two first are calculated, all the
others may be calculated from them by this formula.
126. In the same manner,
if ---- ---— _I. 1 D(0) + D(') cos w + &c.
(a - a a' cos w + a'2Y)
2 ]k1 aa' 2-k 1
*oJC CC2D- D( k - 1a
2k-i aa' 2/c0o -i
127. PRor. 50. To express D(<;) in terms of C(k-') and
C(;~)
CA.
Since
i I
1 - (_ -2aa'cosw +a'2) _,
(a_-2 aa cosw+aC2)i ) (a2- 2aa acos w + a'2)
or C() + C() cos w + &c.
= (a2- 2aacos W + a'2) 2.D(O)+ -D(1) cos w + &c.
we have, comparing the coefficients of coskw,
C(ka) = (CL2 + "') D(k) _aa'D-) - aa'D(k+l).
Putting for D('-1 the value in (126),
1 2
C(-) - (a+r a"2)D(C - + aa'D) +(k
k- 1 2k- 1
Again, comparing the coefficients of cos (k - 1)w
C(-) =_ (a2- a'2) D(-1) - a'D(-) - aa'D(k.