ll mention
the long inequality of Jupiter and Saturn. Halley had found that Jupiter
was continually lagging behind its true place as given by the theory of
gravitation; and, on the other hand, that Saturn was being accelerated.
The lag on the part of Jupiter amounted to about 34-1/2 minutes in a
century. Overhauling ancient observations, however, Halley found signs
of the opposite state of things, for when he got far enough back Jupiter
was accelerated and Saturn was being retarded.

Here was evidently a case of planetary perturbation, and Laplace and
Lagrange undertook the working of it out. They attacked it as a case of
the problem of three bodies, viz. the sun, Jupiter, and Saturn; which
are so enormously the biggest of the known bodies in the system that
insignificant masses like the Earth, Mars, and the rest, may be wholly
neglected. They succeeded brilliantly, after a long and complex
investigation: succeeded, not in solving the problem of the three
bodies, but, by considering their mutual action as perturbations
superposed on each other, in explaining the most conspicuous of the
observed anomalies of their motion, and in laying the foundation of a
general planetary theory.

[Illustration: FIG. 79.--Shewing the three conjunction places in the
orbits of Jupiter and Saturn. The two planets are represented as leaving
one of the conjunctions where Jupiter was being pulled back and Saturn
being pulled forward by their mutual attraction.]

One of the facts that plays a large part in the result was known to
the old astrologers, viz. that Jupiter and Saturn come into
conjunction with a certain triangular symmetry; the whole scheme
being called a trigon, and being mentioned several times by Kepler.
It happens that five of Jupiter's years very nearly equal two of
Saturn's,[25] so that they get very nearly into conjunction three
times in every five Jupiter years, but not exactly. The result of
this close approach is that periodically one pulls the other on and
is itself pulled back; but since the three points progress, it is
not always the same planet which gets pulled back. The complete
theory shows that in the year 1560 there was no marked
perturbation: before that it was in one direction, while afterwards
it was in the other direction, and the period of the whole cycle of
disturbances is 929 of our years. The solution of this long
outstanding pu