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 puzzle by the theory of gravitation was hailed with the greatest enthusiasm by astronomers, and it established the fame of the two French mathematicians. Next they attacked the complicated problem of the motions of Jupiter's satellites. They succeeded in obtaining a theory of their motions which represented fact very nearly indeed, and they detected the following curious relationship between the satellites:--The speed of the first satellite + twice the speed of the second is equal to the speed of the third. They found this, not empirically, after the manner of Kepler, but as a deduction from the law of gravitation; for they go on to show that even if the satellites had not started with this relation they would sooner or later, by mutual perturbation, get themselves into it. One singular consequence of this, and of another quite similar connection between their positions, is that all three satellites can never be eclipsed at once.