a 
344 \e* LAPLACE. 
If we add to the mean motion of the first satellite 
twice the mean motion of the third, the sum is exactly 
equal to three times the mean motion of the second.* 
This numerical coincidence, which is perfectly aceu- 
rate, would be one of the most mysterious phenomena in 
the system of the universe if Laplace had not proved 
that the law need only have been approximate at the 
origin, and that the mutual action of the satellites has 
sufficed to render it rigorous. 
The illustrious geometer, who always pursued his 
researches to their most remote ramifications, arrived 
* 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°. 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° as a mean value. The three satel- 
lites 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 libra- 
tory motion, so that the relations between the mean motions and 
mean longitudes may be presumed to be always rigorously true.— 
Translator. 
