XXIV.— On the development of the disturbing function, upon which depend the inequalities of the motions of the planets, caused by their mutual attrac­tion

The perturbations of the planets caused by their mutual attraction depend chiefly upon one algebraic expression, from the development of which all the inequalities of their motions are derived. This function is very complicated, and requires much labour and many tedious operations to expand it in a series of parts which can be separately computed according to the occasions of the astronomer. The progress of physical astronomy has undoubtedly been re­tarded by the excessive length and irksomeness attending the arithmetical calculation of the inequalities. On this subject astronomers generally and continually complain; and that their complaints are well founded, is very aptly illustrated by a paper contained in the last year’s Transactions of this Society. The disturbing function is usually expanded in parts arranged according to the powers and products of the excentricities and the inclinations of the orbits to the ecliptic; and, as these elements are always small, the resulting series decreases in every case with great rapidity. No difficulty would therefore be found in this research, if an inequality depended solely on the quantity of the coefficient of its argument in the expanded function; because the terms of the series decrease so fast, that all of them, except those of the first order, or, at most, those of the first and second orders, might be safely neglected, as pro­ducing no sensible variation in the planet’s motion. But the magnitude of an inequality depends upon the length of its period, as well as upon the coefficient of its argument. When the former embraces a course of many years, the latter, although almost evanescent in the differential equation, acquires a great mul­tiplier in the process of integration, and thus comes to have a sensible effect on the place of the planet. Such is the origin of some of the most remarkable of the planetary irregularities, and in particular, of the great equations in the mean motions of Jupiter and Saturn, the discovery of which does so much honour to the sagacity of Laplace. It is not, therefore, enough to calculate the terms of the first order, or of the first and second orders, in the expansion of the disturbing function. This is already done in most of the books that treat of physical astronomy with all the care and fulness which the importance of the subject demands, leaving little room for further improvement. In the present state of the theory of the planetary motions, it is requisite that the astronomer have it in his power to compute any term in the expansion of the disturbing function below the sixth order; since it has been found that there are inequalities depending upon terms of the fifth order, which have a sensible effect on the motions of some of the planets.