23S French National Institute. 



have been attained bv ib^ power of analysis^ even without 

 knowinn the peculiar expressions of the quantities relative 

 to the ellipiic <irb\t. 



In this manner he demonstrates, in its whole possible 

 generality, and whatevtrr may be the inchnation ofXhe pri- 

 mitive orbit, that tfie var'wjio7i of the great axis cannot con- 

 tain am/ term n it periodical, either in i he first or the second 

 approximation, at least ii hile regard is had in the latter 

 only to the variations of the elements of the distiirhed orlif, 

 Tbi« prevfnts the s.une analv-is from btint^ also eTclended 

 to the terms proceudinii from the elements of the per- 

 turbing planets : it is becan •«• in thi? caso the function is not 

 symmetrical with respect lo the co-ordinates of all the 

 planets. 



But by carrying the planets, not to the centre of the sun, 

 but to the centre of gravity of the sun, and of the planets 

 around which the motion is more regular than around the 

 sun, M. L'igrange obtains a symmeirical function, which 

 is the same with respect to all the planets: the calculation 

 then becomes uniform, and is not subject to any exception; 

 and we demonstrate by one ana the same analysis, that the 

 great axis of each of the orbits cannot have, in the two first 

 approximation':, any inequality increasing with the time. 



It is afterwards easv to pass from the motion around the 

 common centre of gravity to the motion armmd the sun j 

 and we at length succeed in demonstrating the general pro- 

 position of the non-existence of the inequalities propor- 

 tional to the time in the great axes of the planets referred 

 to the sun. 



To return to the memoir of M. Lagrange: we there find 

 his new formulae for the variations of the elements of the 

 planets, as well as their application to the variations of f'ue 

 grand axes. His analysis is worthy of the attention of 

 geoiTictricians from its uniftirinitv, generality, and elegance, 

 and because it is indtpendtni of the elliptic figure ot the 

 orbits, and may be applied with the same success to every 

 other hypothesis of gravitation, in which the ovbi'.s would 

 no longer be conic sections. 



The whole of this analysis is preceded" by a historical de- 

 tail of this great problem, drawn up with all possible clear- 

 ness, in such a manner as to interest even thc»se who should 

 not have all the knowledge necessary for following the au- 

 thor into the whole details /)f his theory. 



In this memoir, read to 'he class on the e-l'd of August, 

 the generality of the analysis permitted M. Lagrange lo 



express 



