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.

FORCES PRODUCING LARGE TERMS IN RADIUS. 79
suppose pn - qn-n to be small: ir will have a considerable terni of the form aN. cos (pnt -qn't + Q): and
adding this to the first variable term of r in (87), the
principal inequality of r is
a - ecos (nt + e - wr) + N. cos (pnt - qn't + Q)}
= a - ecos (nt + e - r)
+ N.cos (pnt - qn't - nt + Q - e + 5r) + (nt + e - r)}
= -cos(nt t+e-r){ae- aN. cos(pnt -qn't-nt +Q-e +aur)
- sin (nt + e - ar). aN. sin (pnt - qn't - n t + Q - + r).
As in (66) and (68) this may be put under the form
- aE. cos (nt + e - w - K),
where E = e - N.cos (pnt - qn't - nt+ Q -e + ~ )
N
K= - sin (pnt - qn't - nt + Q - e + r).
This is the same as the elliptic inequality in an orbit where
E is the eccentricity and '- + K the longitude of the perihelion. The alterations then both in the eccentricity and
in the longitude of the perihelion are periodical: and their
common period is -,- which is very long.
pn - qn - n
99. The sanie term of R will also produce considerable
2,7r
terms in r3r whose period is-: but they will
pn - qn- n
be multiplied by e (see Art. 90). Hence there will be in
ir no terms whose period is long that are comparable in
magnitude to those whose period is nearly the same as
the periodic time of the disturbed planet.
100. PROP. 37. To describe the nature of the forces,which produce the most striking inequalities in the longitude,
and to investigate their periodic times.
We have already remarked (94) that the terms of 0O
produced by the considerable terms of ràr, will have a