of the two laws will assure the existence of an analogous relation between the mean longitudes for any instant of time whatever, whether past or future.  Laplace has shown, as the author has stated in the text, that if these relations had only been approximately true at the origin, the mutual attraction of the three satellites would have ultimately rendered them rigorously so; under such circumstances, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, would continually oscillate about 180 deg. as a mean value.  The three satellites would participate in this libratory movement, the extent of oscillation depending in each case on the mass of the satellite and its distance from the primary, but the period of libration is the same for all the satellites, amounting to 2,270 days 18 hours, or rather more than six years.  Observations of the eclipses of the satellites have not afforded any indications of the actual existence of such a libratory motion, so that the relations between the mean motions and mean longitudes may be presumed to be always rigorously true.—­Translator.

[39] Laplace has explained this theory in his Exposition du Systeme du Monde (liv. iv. note vii.).—­Translator.

APPENDIX.

(A.)

THE FOLLOWING IS A BRIEF NOTICE OF SOME OTHER INTERESTING RESULTS OF THE RESEARCHES OF LAPLACE WHICH HAVE NOT BEEN MENTIONED IN THE TEXT.

Method for determining the orbits of comets.—­Since comets are generally visible only during a few days or weeks at the utmost, the determination of their orbits is attended with peculiar difficulties.  The method devised by Newton for effecting this object was in every respect worthy of his genius.  Its practical value was illustrated by the brilliant researches of Halley on cometary orbits.  It necessitated, however, a long train of tedious calculations, and, in consequence, was not much used, astronomers generally preferring to attain the same end by a tentative process.  In the year 1780, Laplace communicated to the Academy of Sciences an analytical method for determining the elements of a comet’s orbit.  This method has been extensively employed in France.  Indeed, previously to the appearance of Olber’s method, about the close of the last century, it furnished the easiest and most expeditious process hitherto devised, for calculating the parabolic elements of a comet’s orbit.

Invariable plane of the solar system.—­In consequence of the mutual perturbations of the different bodies of the planetary system, the planes of the orbits in which they revolve are perpetually varying in position.  It becomes therefore desirable to ascertain some fixed plane to which the movements of the planets in all ages may be referred, so that the observations of one epoch might be rendered readily