344 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 accu- 

 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. 



