it was shown mathematically by Laplace that if the longitudes and mean motions were such that the angle U differed a little from 180°, there was a minute residual force arising from the mutual actions of the several bodies tending to bring this angle towards the value 180°. Consequently, if the mean motions were such that this angle increased only with great slowness, it would after a certain period tend back toward the value 180°, and then beyond it, exactly as a pendulum drawn out of the perpendicular oscillates towards and beyond it. Thus an oscillation would be engendered in virtue of which the angle would oscillate very slowly on each side of the central value. Computation of the mean longitude from observations has indicated that the angle does differ from 180°, but it is not certain whether this deviation is greater than the possible result of the errors of observation. However this may be, the existence of the libration, and its period if it does exist, are still unknown.
The following are the principal elements of the orbits of the five inner satellites, arranged in the order of distance from Jupiter. The mean longitudes are for 1891, 20th of October, G.M.T., and are referred to the equinox of the epoch, 1891, 2nd of October:—
| Satellite | V. | I. | II. | III. | IV. |
| Mean Long. | 264°.29 | 313°.7193 | 39°.1187 | 171°.2448 | 62°.2000 |
| Synodic Period | 11 h. 58 m. | 1 d. 18 h. .48 | 3d. 13h. .30 | 7d. 3h. .99 | 16d. 18m. .09 |
| Mean Distance | 106,400 m. | 260,000 m. | 414,000 m. | 661,000 m. | 1,162,000 m. |
| Mass ÷ Mass of Jup. | (?) | .00002831 | .00002324 | .00008125 | .00002149 |
| Stellar Mag. | 13 | 6.0 | 6.1 | 5.6 | 6.6 |
The following numbers relating to the planet itself have been supplied mostly by Professor Hermann Struve.
| Filar Mic. | Heliom. | |
| Equatorial diameter of Jupiter (Dist. 5.2028) | 38″.50 | 37″.50 |
| Polar diameter of Jupiter | 36″.02 | 35″.23 |
| Ellipticity | 1 ÷ 15.5 | 1 ÷ 16.5 |
| Theoretical ellipticity from motion of 900″ in the pericentreof Sat. V | 1 ÷ 15.3 | |
| Centrifugal force ÷ gravity at equator | 0.0900 | |
| Mass of Jupiter ÷ Mass of Sun, now used in tables | 1 ÷ 1047.34 | |
| Inclination of planet’s equator to ecliptic | 2° 9′.07 + 0.006t | |
| Inclination of planet’s equator to orbit | 3° 4′.80 | |
| Long. of Node of equator on ecliptic | 336° 21′.47 + 0′.762t | |
| Long. of Node of equator on orbit | 135°25′.81 + 0.729t | |
The longitudes are referred to the mean terrestrial equinox, and t is the time in years from 1900.0.
For the elements of Jupiter’s orbit, see [Solar System]; and for physical constants, see [Planet].
(S. N.)
