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.

66 PLANETARY THEORY.
85. There are two methods of deriving a result from
these equations. One consists in obtaining immediately the
alterations in the radius vector and the longitude produced
by the disturbing force: this is the method commonly used
for the periodical terms of short period. The other consists
in obtaining the variation of the elements of the orbit: it
is best adapted to the discovery of secular terms, whether
periodical or permanent. We shall at present proceed with
the former.
PERTURBATIONS OF RADIUS VECTOR AND
LONGITUDE.
86. PROP. 31. To investigate very approximately the
equation for the perturbation of the radius vector.
Let r be the length of the radius vector, and 0 the
longitude, in the orbit which m would have described if
undisturbed: îr the perturbation of the radius vector: then
r+ ir= r. Substituting this in the last equation,
d_(r +_ _. r d(R) dR
-- t = 2C + rY 4J- -. (r + r) -.
df r+ 4r dt dri
Now Îr is supposed to be so small that we inay neglect
dR
its square. Also --- is itself a quantity depending on the
dr:
disturbing force, and therefore we may neglect the product
dR
of or and d. Again, as we have stated (76) the effect of
dr
the disturbing force in altering the body's place is so small
that in calculating R, (the function whose differential coefficients express its value), we may take the undisturbed
place of the body instead of the true one: that is, we may
now suppose
m'r cos '-) m__
Î~ Î ~r72 ""/{r2" - 2r cos (0'- 0) + r '2
dR dR
and for - may put -dr
dr dr