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. 109
And when this is multiplied by the series wlich in (129) stands
as its multiplier, the product will be a series to which the same
remarks may be applied, in which the coefficient of every term
is, by the methods above given, numerically calculable. The
multiples of (nt+e)-(n't+e'), nt+e-r, and n't+e'-'r',
will, as is plain, be arranged in every possible way of addition
and subtraction in the argument of each cosine.
131. The value of R (86) being
m r. cos (0 - 0') m
-~ {.~ - 2rr' cos (0 - 0') +.'2}
we are now able to expand it completely (as the first term
will be expanded by multiplying together the series for r,
cos( -'), and -). It is only necessary now to explain
dR
the expansion of r -d In the introduction of this exdr
pression {see (82) &c. it is to be remarked that R was to
be differentiated with respect to r as if r had no dependdR
ance on 0. Consequently r d means the quantity which
r 1
dR
a. -d becomes (R being the value of R when a is put for
r) on putting a +v for a, without altering 0. If then in
_dR
R the coefficient of cos k (0-0t) is C(, it will be in ad
da
d C(k) d R
a d -; and in r -,
da dr
dC~() d / d. C(k)
a- +- (a --.v
da da da da
da, d a &.
=a.- + [a 2 + d v
da da da
d2 C(k)
+ a - v', + &c.
dad a
and the coefficient of every term can be calculated.