French Nat'ional Institute, 233 



express certain values by the symbols of functions : it was 

 useless for him to give developments which rendered the 

 demonstration k-ss clear ynd more difficult : but in order to 

 apply his forniuiai to llie numerical calculation of the plane- 

 tary perluihations, these develcpmenls became indispensa- 

 ble. In a supplement read to the class on the 2d of Septem- 

 ber, M. Lagrange gave these calculations; but he i<ne\v 

 aduiirablv well how lo abridge them by the consideration 

 of the eccentric anomaly ; and in order to demonstrate the 

 exactitude of this new process, he showed that it led to the 

 same formulce which he had obtained by another way. 

 These substitutions, v\hich should stem to have been very 

 couipiicated, admit of astonishing simplifications, by means 

 of several equations of condition which M. Lagrange has 

 drawn from his theory. 



Bef )ri. reading this memoir to the Class of Sciences, 

 M. Lagrange had communicated it to the Board of Loneiiude 

 on the same day on which M. La Place detailed the method 

 by which he attained tiic same results. 



The object of M. La I'lace in iliis work, which he has 

 printed separately, was the perfection of the methods he 

 had given in the MecoTUcpie CeUste. On endeavouring to 

 give to the expressions of the elements of the orbits, the 

 simplest form of which they are susceptible, he succeeded 

 in maKint.'; ihem depend only on partial differentials of one 

 and itie same function ; and what is remarkable, the Cv^ef- 

 ficicnts of these differences are only the function of the elc- 

 ni;-ii'S themselves ; an advantage also enjoved by the for- 

 jiidlae of Ni. Lagrange, who had long ago given the example 

 in the expiLSsioii which he had found for the great axis, — an 

 expression which had led him to demonstrate, in a very for- 

 lunaic manner, the invanatsifuy of the mean motions, when 

 we have regard only to 'iu' first power of the perturbing 

 masses. iVl. La I'lace has subsequently given the same form 

 v6 the differential expressions of ihe eccentricity of the orbit, 

 the 11. liuation and the longitude of the node. It still rc- 

 jnained tu transform in the same way the differential ex- 



firessiou^' oi tlie lou-iitudes f)f the epoch and of the perlhe- 

 ion. This is what VL La Place executes in the supplement 

 of which we now give an account; and thereby ihe finite 

 varia'.ions of the differentials flow from the development of 

 a very simide function, which performs an important part 

 - ^n the MeiwKjue CHcslr. These new expressions lead very 

 naturally to the elei/iut theorcij) of iV] Poissou on the \\\- 

 variabdity of mean motions ; thev had also to the most 

 simple and gi'uera! solution of the se.ular variations of the 



ekments 



