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.

PERTURBATION 0F RADIUS VECTOR. 69
88. Before proceeding with the solution of our equation, we shall make one important remark. R, so far as
it depends on the place of m, is a function of r and 0.
Now r, when given in terms of no variable but t, is expressed entirely by constants and by cosines of multiples of
(nt + e - ): 0 is nt + e + a series depending on constants
and on sines of multiples of (nt + c - ). Consequently,
whether we suppose R completely expanded or not, if it
be expressed in terms of t, wherever we find nt we shall
find 6d. Suppose then
R = A + Bcos (nt+ + e - E) + Ccos (2nt + 2 - F) + &c.,
where A, B, C, &c. are any functions whatever of the coordinates of n': then taking d(R) witl the restriction
dt
imposed on it in (80) and (81).
- ) - n. B sin (nt + e - E)
dt
-2nC.sin (2nt+ e - ) - &c.
dR
But = - B. sin (nt + e - E)
- 2 C. sin (2nt + 2e - F) - &c.
d (R) dR
therefore d() n= d -
dt de
dR
where it is supposed that, in order to take —, both r and
0 have been expressed in terms of nt +e; which is in fact
the easiest method for the planets. We have now got rid
of the restriction with which we were before incumbered (80).
If we had developed A, B, C, &c. we should have had for
each a series of such terms as L cos q (n't + c'-E') which when
This is also true, if the inclinations of the orbits be taken into account.
t From this and other instances, the reader will observe the importance (in
giving the partial differential coefficient of a function which admits of being
expanded in terms of different quantities) of stating distinctly the supposition
made relative to the expansion.