Investigation of the terms which can become sensible by the effect of small divisors, introduced by integration, although they may be multiplied by the excentricitics, which are very small [6245–6270] § 7
The sun's action produces also in the motion of the satellites some sensible inequalities, depending on the excentricities. Expression of these inequalities. That which affects the longitude is composed of two parts analogous to the evection and to the annual equation in the lunar theory [6271–6294] § 8
CHAPTER IV. ON THE INEQUALITIES OF THE SATELLITES IN LATITUDE 62
Analytical expressions of the latitude of a satellite and of the motion of its nodes [6295–6336] § 9
The part of this expression which depends on the displacements of the equator and of the orbit of Jupiter, represents the latitude which each satellite would have, if it should move in an intermediate plane which passes between the equator and orbit of Jupiter, through their common intersection. This effect is analogous to that which the earth produces upon the moon, as we have seen in [5352, &c.], but it is much more sensible. Determination of its value [6337–6431]; § 10
Investigation of the terms which acquire very small divisors by integration in the expression of the latitude, in consequence of the nearly commensurable values of the mean motions of the three inner satellites. Estimates of their values [6432–6486] § 11
CHAPTER V. INEQUALITIES DEPENDING ON THE SQUARES AND PRODUCTS OF THE EXCENTRICITIES AND INCLINATIONS OF THE ORBITS 102
Calculation of these inequalities. It is sufficient to notice those only which have a long period [6487–6524] § 12
The terms which become the most important in the secular equations of the satellites, are those which depend on the secular variations of the equator and of the orbit of Jupiter, and on the motion of the nodes of the fourth satellite. They are analogous to those which produce the moon's secular equation, and the equation of the moon's motion depending on the longitude of its nodes. Calculation of these terms [6525–6555'] § 13
CHAPTER VI. ON THE INEQUALITIES DEPENDING ON T[IE SQUARE OF THE DISTURBING FORCE. 126
The most remarkable of these inequalities has already been discussed under its general form, in Book II. [1214' — 1242y]. It depends on the circumstance, that, at the origin of the motion, the mean longitude of the first satellite, minus three times that of the second, plus twice that of the third, was very nearly equal to the semi-circumference π ; and subsequently, by means of the mutual action of the three bodies upon each other, it became accurately equal to π. Development of the theory of these motions by a different method from that which is used in [1214', &c.]. From this it follows, as in [1242v], that the mean motions of the three inner satellites are subjected to a species of libration, and it is of importance for astronomers to investigate and determine the limits of this libration by observations. Hitherto it has appeared to be insensible. From this it follows that the relation which now exists between the mean motions of the three inner satellites, will continue unchanged to future ages. Moreover, the two inequalities of the first satellite, arising from the attractions of the second and third
