ellites 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 comparable
with those of another. This object was accomplished by Laplace, who
discovered that notwithstanding the perpetual fluctuations of the
planetary orbits, there exists a fixed plane, to which the posi