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.

4 LUNAR AND PLANETARY THEORIES.
5. Case 4. Let 0 =b.cosnO - D.
Instead of multiplying by cos n0, multiply by sinnO + D,
(which will do equally well).. sinnO 0 + D.-+n. sin nO + D.u + -.sin 2n0 + 2D = o.
dO2 2
Integrating by parts, as in (1),
du b
sin n + D. -- ncos nO + D.u - -.cos 2n0 + 2D + C o,
d9 4n
du b b
or sinz0+D. - -n.cosn0+ D.u+ - sin2nO+D+ C- - =0.
dO 2n 4n
Dividing by sin2nO + D, and integrating,
i-n + D + 0t- f- b cot0+ D + C'= 0,
sinnO + D 2 \n n 4ln
or u= ---O.sinn+D+ ( --- cosnO+D-C'sinnO+D,
2n nz 4nJ
which as before may be put under the form
u:= - - 0.sin n + D + A cosnO - B.
2n
6. Remarks. If 0 consisted of several terms, the expression for u would contain one term corresponding to each.
The part which depends upon the abitrary constants, is entirely independent of O. The process above having shewn
what is the form of the expression for u, we may son-etimes
solve the equation with greater ease, by assuming an expression with indeterminate coefficients. Thus, if we had the
equation
d + u + a b cosn + B + Q
df +nu + a + bcosm0 + B + pcosq0 + Q = o,
dO.
we might assume
u= - - + Acos(nO + C)+Dos + Dcos B + Ecosq+ Q,
n2
and, substituting this series in the equation, determine the
values of D and E.