152 LAPLACE. 



Let us cite two or three of the laws of Laplace : 



If we add to the mean longitude of the first satellite twice that of the 

 third, and subtract from the sum three times the meau longitude of the 

 secoiKl, the result will be exactly equal to 180°. Would it not be very 

 extraordinary if the three satellites had. been placed originally at the 

 distances from Jupiter, and in the positions, with respect to each other, 

 adapted for constantly and rigorously maintaining the foregoing rela- 

 tion ? Lai)lace has replied to this question by showing that it is not 

 necessary that this relation should have been rigorously true at the ori- 

 gin. The mutual action of the satellites would necessarily have reduced 

 it to its present mathematical condition, if once the distances and the 

 positions satisfied the law approximately. 



This first law is equally true when we employ the synodical elements. 

 It hence plainly results that the three first satellites of Jupiter can never 

 be all eclipsed at the same time. Bearing this in mind, we shall have 

 no difficulty in apprehending the import of a celebrated observation of 

 recent times, during which certain astronomers perceived the planet for 

 a short time without any of his four satellites. This would not by any 

 means authorize us in supposing the satellites to be eclipsed. A satel- 

 lite disappears when it is projected upon the central part of the lumi- 

 nous disk of Jui)iter, and also when it passes behind the opaque body 

 of the planet. 



The following is another very simple law to which the mean motions 

 of the same satellites of Jupiter are subject: 



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 co-incidence, which is perfectly accurate, would be one 

 of the most mysterious phenomena in the system of the universe if 



* This law is necessarily included in the law already enunciated by the autlior relative to 

 the mean longitudes. The lollowingis the most usual mode of expressing these curious rela- 

 tions : 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 )H0°. It is plain that if we only consider 

 the mean longitude liere to refer to n gicen epoch, the co.mbination of the two laws will as- 

 sure the existence of an analogous relation between the meau longitudes /or any ijistaiit of 

 time tcliatcver, 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 rigorous!}' 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 CLUtinually oscil- 

 late about J8(i° 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 'i,'27() days IH hours, or rather more than six years. Observations of the 

 eclipses of the satellites have not afi'orded 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. — TuANSL.VToit. 



