Finally in 1784 Laplace arrived at the true explanation. Lagrange had observed in 1776 that if the times of revolution of two planets are exactly proportional to two whole numbers, then part of the periodic disturbing force produces a secular change in their motions, acting continually in the same direction; though he pointed out that such a case did not occur in the solar system. If moreover the times of revolution are _nearly_ proportional to two whole numbers (neither of which is very large), then part of the periodic disturbing force produces an irregularity that is not strictly secular, but has a very long period; and a disturbing force so small as to be capable of being ordinarily overlooked may, if it is of this kind, be capable of producing a considerable effect.[148] Now Jupiter and Saturn revolve round the sun in about 4,333 days and 10,759 days respectively; five times the former number is 21,665, twice the latter is 21,518, which is very little less. Consequently the exceptional case occurs; and on working it out Laplace found an appreciable inequality with a period of about 900 years, which explained the observations satisfactorily. The inequalities of this class, of which several others have been discovered, are known as =long inequalities=, and may be regarded as connecting links between secular inequalities and periodical inequalities of the usual kind. 244. The discovery that the observed inequality of Jupiter and Saturn was not secular may be regarded as the first step in a remarkable series of investigations on secular inequalities carried out by Lagrange and Laplace, for the most part between 1773 and 1784, leading to some of the most interesting and general results in the whole of gravitational astronomy. The two astronomers, though living respectively in Berlin and Paris, were in constant communication, and scarcely any important advance was made by the one which was not at once utilised and developed by the other. The central problem was that of the secular alterations in the elements of a planet’s orbit regarded as a varying ellipse. Three of these elements, the axis of the ellipse, its eccentricity, and the inclination of its plane to a fixed plane (usually the ecliptic), are of much greater importance than the other three. The first two are the elements on which the size and shape of the orbit depend, and the first also determines (by Kepler’s Third Law) the period of revolution and average rate of motion of the planet;[149] the third has an important influence on the mutual relations of the two planets. The other three elements are chiefly of importance for periodical inequalities. It should be noted moreover that the eccentricities and inclinations were in all cases (except those specially mentioned) considered as small quantities; and thus all the investigations were approximate, these quantities and the disturbing forces themselves being treated as small.