[142] The establishment of the general equations of motion by a combination of virtual velocities and D’Alembert’s principle.
[143] Théorie des Fonctions Analytiques (1797); Resolution des Équations Numériques (1798); Leçons sur le Calcul des Fonctions (1805).
[144] Théorie Analytique des Probabilités.
[145] The fact that the post was then given by Napoleon to his brother Lucien suggests some doubts as to the unprejudiced character of the verdict of incompetence pronounced by Napoleon against Laplace.
[146] Outlines of Astronomy, § 656.
[147] Laplace, Système du Monde.
[148] If n, n′ are the mean motions of the two planets, the expression for the disturbing force contains terms of the type
| = | sin | (np ± n′p′) t, |
| cos |
where p, p′ are integers, and the coefficient is of the order p⁓p′ in the eccentricities and inclinations. If now p and p′ are such that np⁓n′p′ is small, the corresponding inequality has a period 2π∕(np⁓n′p′), and though its coefficient is of order p⁓p′, it has the small factor np⁓np′ (or its square) in the denominator and may therefore be considerable. In the case of Jupiter and Saturn, for example, n = 109,257 in seconds of arc per annum, n′ = 43,996; 5n′ - 2n = 1,466; there is therefore an inequality of the third order, with a period (in years) = 360°∕1,466″ = 900.
[149] This statement requires some qualification when perturbations are taken into account. But the point is not very important, and is too technical to be discussed.