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.

70 PLANETARY TIIEORY.
multiplied by one of the terms depending on nt + e -E would
have produced such terms as
P. cosp (nt + e- E) q (n't + '- E').
d(R) dR
The equation - =n would still have been found true,
dt de
as the term n't depends on t only because the place of m'
varies with t, and therefore for our restricted value of ()
dt
it must not be differentiated.
Should the reader find any difficulty in conceiving a
differential coefficient taken with regard to one of the quantities commonly considered constant, we beg him to observe
tIhat the variation of value, used in the definition at least
of a differential coefficient, is hypothetical only and not
necessarily true. We state that if we alter the value of
one quantity, and if we find the corresponding alteration
of the function, and if we divide the latter alteration by
the former, and if we find the limit to which the quotient
approaches on diminishing the former indefinitely, this limit
is the differential coefficient. The whole of this operation
can b)e performed equally well, whether the quantity be in
the nature of things variable, or we be compelled to conceive a variation where none really exists.
89. To return to our equation: we may now put it
under the form
d2(rr) rdR dR
O= -- +n2. r + 2n -+ --
dt'r 'de dr
3e2 9el
+n2. rrS3ecos(nt+e- 7r)+ -- - cos(2nt+2e-2'r) +&c.}.
a 2
This is most easily solved by approximation: neglecting first
the terms which are multiplied by e and its powers, which
reduces the equation to
d'(rtr) +. dR dR
O= dt-, + dr
ddt R d r