Two years later (1775) Laplace, much struck by the method which Lagrange had used, applied it to the discussion of the secular variations of the eccentricity, and found that these were also of a periodic nature, so that the eccentricity also could not increase or decrease indefinitely.
In the next year Lagrange, in a remarkable paper of only 14 pages, proved that whether the eccentricities and inclinations were treated as small or not, and whatever the masses of the planets might be, the changes in the length of the axis of any planetary orbit were necessarily all periodic, so that for all time the length of the axis could only fluctuate between certain definite limits. This result was, however, still based on the assumption that the disturbing forces could be treated as small.
Next came a series of five papers published between 1781 and 1784 in which Lagrange summed up his earlier work, revised and improved his methods, and applied them to periodical inequalities and to various other problems.
Lastly in 1784 Laplace, in the same paper in which he explained the long inequality of Jupiter and Saturn, established by an extremely simple method two remarkable relations between the eccentricities and inclinations of the planets, or any similar set of bodies.
The first relation is:—
If the mass of each planet be multiplied by the square root of the axis of its orbit and by the square of the eccentricity, then the sum of these products for all the planets is invariable save for periodical inequalities.
The second is precisely similar, save that eccentricity is replaced by inclination.[150]
The first of these propositions establishes the existence of what may be called a stock or fund of eccentricity shared by the planets of the solar system. If the eccentricity of any one orbit increases, that of some other orbit must undergo a corresponding decrease. Also the fund can never be overdrawn. Moreover observation shews that the eccentricities of all the planetary orbits are small; consequently the whole fund is small, and the share owned at any time by any one planet must be small.[151] Consequently the eccentricity of the orbit of a planet of which the mass and distance from the sun are considerable can never increase much, and a similar conclusion holds for the inclinations of the various orbits.
One remarkable characteristic of the solar system is presupposed in these two propositions; namely, that all the planets revolve round the sun in the same direction, which to an observer supposed to be on the north side of the orbits appears to be contrary to that in which the hands of a clock move. If any planet moved in the opposite direction, the corresponding parts of the eccentricity and inclination funds would have to be subtracted instead of being added; and there would be nothing to prevent the fund from being overdrawn.
A somewhat similar restriction is involved in Laplace’s earlier results as to the impossibility of permanent changes in the eccentricities, though a system might exist in which his result would still be true if one or more of its members revolved in a different direction from the rest, but in this case there would have to be certain restrictions on the proportions of the orbits not required in the other case.