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 OF R. 103
cos 2 + a
/(1 + a cos 2+a2) + )
which gives
cos 2 / = - i +- a sin+ cos /(1 sin2),
this may be converted into
1i ir \1 ~, ca(2 sin24'- ) a
- -cos \L+ -.4- -.
a 2 4 (1- sin ) 4 / '(1- ~ sin2q)
1 a.( + a'){r cos2ff' I'=0o
a a 2 J '/(1 + 2 a'cos2 '+ a2) lrT
w- cC(0).
+ a. C'O>
Continuing the process we find
r cw a jr aa+ aaa?C(~) *C(~) + + + &c.
f4 "4 [2 2.2.2.2 j
CI)= c(0)sin (p sin sinq)'
or C(+ = C sin + &c.
2 2 2
which is very easily calculated.
124. PiloP. 49. Given the numerical values of two consecutive coefficients C(k-), C(k), to find the numerical value
of the next C(k+).
Since (_ i= - C(O) + C() cos w + &c.
(a2 - 2 aa'cos w + a"2) 2
+ C(- cos (k - 1)W) + C ) coskw + C(k+ ) cos(k + l)W + &c.;
differentiating the logarithm of each side with respect to o
a asin w
(a~ + a')) - 2aa'cos w
C(1 sinw + &c. + k-. C(-) sin (k- ) w + k. C() sin k o + k + 1.C(k+) sin (k + 1)+&c.
-C) + C')cosw+ &c.+C(k-)cos(k-1) w+ C(kcososkw+ C(k+) cos (k+ 1) w+ &c.