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HISTOEY OF PHYSICAL ASTRONOMY.

We have given some account in a previous chapter of the researches ofLagrange and Laplace on the secular inequalities of the planetary motions,and have also mentioned the results to which they were conductedrelative to the stability of the system of the world. Laplace de-monstrated that neither the eccentricities nor the inclinations would in-crease indefinitely with the time, but he did not assign any means ofdetermining the exact limits between which they would perpetually oscil-late. It is conceivable, indeed, that both elements might vary to a con-siderable extent, without invalidating the famous theorems of that geometerrelative to their limits. Le Vender found that a small planet revolvinground the sun at twice the mean distance of the earth would be sodisturbed by Jupiter and Saturn that the inclinations of its orbit relativeto the orbits of those planets would attain considerable magnitudes,even although they were originally very small*. Now it is remarkablethat the small planets discovered between the orbits of Mars andJupiter , whose inclinations we know to be greater than those of theother planets, have all been found to revolve in the neighbourhood ofthis region. Le Vender also found a similar region between Mercury and Venus , in which, if a small planet revolved, its inclinations relativeto the orbits of the Earth and Venus would experience considerablevariations from the disturbing action of those planets.

The only definite conclusion which could be drawn from Laplace’s re-searches, was that the expressions for the eccentricities and inclinationswould consist of a series of sines and cosines of angles, increasing withthe time, but w'ould not contain any term involving the time, without thefunctional symbols. It still remained to compute the numerical valuesof the constants entering into these terms, in order to ascertain the valuesof the elements corresponding to any time, past or future, and to assignthe exact limits within which they would perpetually oscillate. Thisoperation, however, was one of extreme difficulty, for it involved the resolu-tion of an algebraic equation, equal in degree to the number of planetswhose mutual action was considered, and demanded also a most laboriousprocess of elimination. Lagrange made the first successful attack on thisproblem, by the aid of an ingenious simplification, which consisted ingrouping the planets into two systems, one composed of Jupiter andSaturn , to which he subsequently added Uranus ; and the other composedof Mercury , Venus , the Earth , and Mars , taking also into account theaction of the larger planets upon each of these bodies. By this processhe found that the eccentricities and inclinations would continually oscil-late between very narrow limits, and he assigned the numerical values ofthe limits for each planet. This investigation of Lagrange’s, althoughvaluable as a first attempt to establish an important point in the systemof the world, was considerably vitiated by the erroneous values he assignedto the masses of the smaller planets, especially Venus , the mass of whichhe estimated at a half more than its true value. In more recent times,when the masses of the planets and the elements of their motions cameto be ascertained with greater accuracy, a strong desire was felt that the

des Planetes, No. 1) he explained the ingenious process by which he eliminated the co-efficients of the disturbing function. A translation of this memoir is given in Taylor'sScientific Memoirs, part xviii. See also on this subject a paper by Sir John Lubbock , inthe Philosophical Magazine for August, 1848.

* Memoire sur les Mouvements des Inclinaisons et des Nceuds des trots Plandtes Ju -piter, Saturn et Uranus .