CHAPTER VIII. SECOND METHOD OF APPKOXIMATION TO THE MOTIONS OF THE HEAVENLY BODIES. 634
This method is founded on the variations which the elements of the motion, supposed to be elliptical, suffer by means of the periodical and secular inequalities. General method of finding these variations. The finite equations of the elliptical motion, and their first differentials, are the same in the variable as in the invariable ellipsis, [1167' — 1169^"] § 63
Expressions of the elements of the elliptical motion in the disturbed orbit, whatever be its excentricity and its inclination to the plane of the orbits of the disturbing masses, [1170—1194]. . . § 64
Development of these expressions, when the excentricities and the inclinations of the orbits are small. Considering, in the first place, the mean motions and transverse axes ; it is proved, that if we neglect the squares and the products of the disturbing forces, these two elements are subjected only to periodical inequalities, depending on the configuration of the bodies of the system. If the mean motions of the two planets are very nearly commensurable with each other, there may result, in the mean longitude, two very sensible inequalities, affected with contrary signs, and reciprocally proportional to the products of the masses of the bodies, by the square roots of the transverse axes of their orbits. The acceleration of the motion of Jupiter, and the retardation of the mean motion of Saturn, are produced by similar inequalities. Expressions of these inequalities, and of those which the same ratio of the mean motions may render sensible, in the terms depending on the second power of the disturbing masses, [1195—1214] §65
Examination of the case where the most sensible inequalities of the mean motion, occur only among terms of the order of the square of the disturbing masses. This very remarkable circumstance takes place in the system of the satellites of Jupiter, whence has been deduced the two following theorems, T%e mean motion of the first satellite, minus three times that of the second, plus tunce that of the third, is accurately and invariably equal to nothing, [1239'"]. The mean longitude of the first satellite, minu^ three times that of the second, plus ttoice thai of the ihird, is invariably equal to two right angles, [1239^]. These two theorems take place, notwithstanding the alterations which the mean motions of the satellites may suffer, either from a cause similar to that which alters the mean motion of the moon, or from the resistance of a very rare medium. These theorems give rise to an arbitrary inequality, which differs for each of the three satellites, only by its coefficient. This inequality is insensible by observation, [1240 — 1242^] • ... § 66
Diflerential equations which determine the variations of the excentricities and of the perihelia, [1243—1266] §67
Development of these equations. The values of these elements are composed of two parts, the one depending on the mutual configuration of the bodies of the system, which comprises the periodical inequalities ; the other independent of that configuration, which comprises the secular inequalities. This second part is given by the same differential equations as those which we have before considered, [1266', 1279] § 68
A very simple method of obtaining the variations of the excentricities and of the perihelia of the orbits, arising from the ratio of the mean motions being nearly commensurable ; these variations
