42 Analysis of the Mecanique Celeste of M. La Place. 



upon the relations which observations point out between 

 the mean motions of the three first among each other. The 

 author develops at proper length this important point of the 

 system of the world ; and from it he infers that tJie mean 

 motion of the first satellite, minus thrice that of the second, 

 plus tuice that of the third, is exactly and constantly equal 

 to nothing, and that the mean longitude of the first satellite 

 v}inus thrice that of the second, plus thrice that of the third, 

 is exactly and constantly equal to two right angles. These 

 elegant theorems, which would of themselves be sufficient 

 to immortalize their author, have been presented fur the 

 first time to geometricians and astronomers, in the Memoirs 

 above quoted. 



The author afterwards develops the differential equations 

 •w'hich determine the variations of the other elements of the 

 orbit : he shows that the values of the unknown quantities 

 which they contain are composed of two parts ; the one, de- 

 pending upon the mutual configuration of the bodies of the 

 system, contains the variations called periodic; the other, in- 

 dependent of this configuration, contains the variations called 

 secular. The author gives a very sinjple method of obtaining 

 the first part ; and as to the second, he proves that it is given 

 by the same differential equations deprived of their last terms ; 

 a circumstance which refers them to those which he had 

 formerly treated of at full length, by the first method of ap- 

 proximation which we have mentioned. 



We have said that the relations of the mean motions may 

 introduce, in the expression of the mean longitude, a sen- 

 sible inequality among the terras depending upon the second 

 power of the disturbing masses. The author examines the 

 influence of these same relations upon the other elements, 

 and he determines the inequalities resulting from them; be 

 establishes the very simple relations which connect these in- 

 equalities with those of the mean motion ; he discusses the 

 variations to which the same cause subjects the expressions 

 of the latitude above a fixed plane a little inclined to the or- 

 bit ; he shows how by means of these results wc obtain the 

 values of the latitude, longitude, and of the radius vector of 

 the disturbed orbit, variable, quantities which determine 



the 



