Analysis of I he Mecanique Celeste of M. La Place, 41 



small orb around a fictitious planet, moved according to the 

 jiaws of elliptic motion, in an ellipsis, the elements of which 

 vary by insensible shades : hence are deduced with facility, 

 the differential c'quations which determine these variations. 

 By applving these results to tbe case of orbits eccentric in a 

 .small degree, and little inclined towards each other, and 

 neglecting the second powers of the perturbating masses, we 

 see in the first place that the transverse axes and the mean 

 motions are only subjected to periodical variations depending 

 upon the mutual configuration of the bodies of the system, 

 and thereby even not very extensive : hence it follows that 

 by neglecting these quantities, the major axes of the orbits are 

 constant, and the mean motions are uniform ; a result pre- 

 viously found by a different method. This property only takes 

 place when the mean motions of the bodies of the system are 

 incommensurable among each other, which is the case with 

 the planets : if, with respect to some of these bodies, very 

 iittle is required in order to fulfil this condition, the elliptic 

 elements, and particularly the n)ean longitude which de- 

 pends upon two integrations, acquire in certain terms very 

 jarge divisions, which introduce there some very perceptible 

 inequalities. The author gives the method of determining 

 those which affect the mean longitude, and he shows that 

 when there are inequalities of this kind, which the action of 

 one of the bodies of the system produces upon the mean 

 motion of another, it is easy to deduce those which the ac- 

 tion of the second body produces on the mean motion of the 

 first, and he proves that these inequalities are affected by 

 contrary signs, and reciprocal to the products of the masses 

 of bodies by the square roots of the major axes of their or- 

 bits. The illustrious autlior of this work has been the first 

 to demonstrate that to a similar cause is owing the accele- 

 ration of the mean motion of Jupiter, and the retardation of 

 that of Saturn. {Memoires de I' Academic, 1 784-8.5). 



'file author afterwards examines the case in which the 

 most perceptible iuequalilies of the mean motion are only to 

 be met with among the terms of the order of the square of 

 the peiturl'alrix masses : this singular circuuistance takes 

 place in the system of the sjitellites of Jupiter, and it depends 



UjlOI^ 



