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.

IMPORTANT TERMS IN LONGITUDE. 77
found nearly in the proportion of the mean motions of n'
and m, there will be a large term in the perturbation of m
depending on the argument
p(nt + c) - q(n't + ') +Q.
There is no difficulty in finding, among tlie planetary motions, proportions nearly coinciding with those of two numbers. The most remarkable is that of Jupiter and Saturn,
where, the proportions of the mean motions being nearly
5: 2, and the masses being large, although the term in
question (as we shall shew liereafter) is a multiple of the
cube of the eccentricities, yet the inequality depending on
the term
cos 5(nt + ) - 2 (n't+ ') + Q
amounts to nearly fifty minutes. A singular instance has
also been lately discovered in the perturbation of the Earth
by Venus: eiglt times the mean motion of Venus is very
nearly equal to thirteen times the mean motion of the
Earth: and though (as we shall shew) this term
cos 13 (nt f+ ) - 8 (n't + E) + Q
must be multiplied by the fifth powers of the eccentricities, &c. which are very small, and though the disturbing
mass is very small, this inequality is sensible.*
96. There are, in the value of r3r, terms of the same
form (produced originally by the terms
cos (p ~ 1) (n t + e) - (n't + c') + Q)
which are once divided by
(pn l= n - qn')2 - n, or (pn = 2n - qn') (pn - qn').
These terms then may be considerable in Scr, but by no
means so conspicuous as in $0. There are others introduced by the terms
cos (p 2) (n t + C) - q (nt + e') + Q, &c.:
the same remark applies to them.
~ The coefficient of the term cos 13 ( + e) - 8 (n't + c') + Q in R is in this
instance multiplied in the integration by a number exceeding two millions.