nt to one twentieth of the
whole, whence it followed that the true value of the parallax could not
exceed 8".2. Laplace, by an analogous process, determined the parallax
to be 8".45. Encke, by a profound discussion of the observations of the
transits of Venus in 1761 and 1769, found the value of the same element
to be 8".5776.--_Translator_.

[37] The theoretical researches of Laplace formed the basis of
Burckhardt's Lunar Tables, which are chiefly employed in computing the
places of the moon for the Nautical Almanac and other Ephemerides. These
tables were defaced by an empiric equation, suggested for the purpose of
representing an inequality of long period which seemed to affect the
mean longitude of the moon. No satisfactory explanation of the origin of
this inequality could be discovered by any geometer, although it formed
the subject of much toilsome investigation throughout the present
century, until at length M. Hansen found it to arise from a combination
of two inequalities due to the disturbing action of Venus. The period of
one of these inequalities is 273 years, and that of the other is 239
years. The maximum value of the former is 27".4, and that of the latter
is 23".2.--_Translator_.

[38] This law is necessarily included in the law already enunciated by
the author relative to the mean longitudes. The following is the most
usual mode of expressing these curious relations: 1st, the mean motion
of the first satellite, plus twice the mean motion of the third, minus
three times the mean motion of the second, is rigorously equal to zero;
2d, the mean longitude of the first satellite, plus twice the mean
longitude of the third, minus three times the mean longitude of the
second, is equal to 180 deg.. It is plain that if we only consider the mean
longitude here to refer to a _given epoch_, the combination 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 sat