98 Laffransre^s Memoirs 



<b' 



this body. M. Lagrange had already arrived at a result of about 

 the same kind, for the moon. We can doubt, however, that the 

 proposition was true in all its rigor. M. Lagrange had demon- 

 strated it directly, and without supposing the orbits nearly circular, 

 but with neglecting the squares, and the primary products of the 

 masses. M. Poisson has since extended the demonstration to quan- 

 tities of the second order. It is presumed that he will extend it to 

 products of all orders. As to the rest, what is already done, suffices 

 to show us that henceforth all fear in this respect, will be very fool- 

 ish and very chimerical. 



The common method of integrating equations of planetary motions, 

 had an inconvenience which rendered solutions almost illusory, that 

 of arcs of circles increasing indefinitely with the time. In cerlain 

 cases, the arcs could be expunged. M. Laplace had made upon 

 this kind very important remarks, but grounded on a metapkysique 

 too subtle to offer the clearness of a purely analytical demonstration. 

 Lagrange perceived that on making vary arbitrary constants, accord- 

 ing to the principles employed in the theory of particular integrals, 

 we can always avoid arcs of a circle in the calculation of perturba- 

 tions. 



The question of trajectories, or of families of curves, cutting at 

 given angles an infinity of other curves, all of the same kind, had 

 busied all geometers, from Leibnitz and Bernouilli, until Euler, who 

 seemed to have left nothing undone upon this question. Lagrange 

 made of it a new question, by carrying it from simple curves to sur- 

 faces. It leads to an equation of partial differences, integrable only 

 in the case where the angle of intersection is right. 



We have presented only a very imperfect idea of the immense 

 series of labors which have given so much value to the Memoirs of 

 the Academy of Berlin, while it had the inestimable advantage of 

 being directed by M. Lagrange. It is such of these memoirs as by 

 their extent and importance, can pass as a great work, and yet they 

 were only a part of what those twenty years had seen him produce. 

 He had therein composed his Mecanique Analytique, but he desired 

 that it should be printed at Paris, where he hoped that his formulas 

 would be given with more care and fidelity. It was moreover run- 

 ning too great hazard to trust such a manuscript in the hands of a 

 traveller, who could not feel sufficiently all its worth. Lagrange 

 made of it a copy, which M. Duchatelet took the trouble of remit- 

 ting to the Abbe Marie, with whom he was much connected. Marie 



