98 Lagrange’s Memoirs. 
this body. M. Lagrange had already arrived ata 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 certain 
cases, the arcs could be expunged. M. Laplace had made upon 
this kind very important remarks, but grounded on. a métaphysique 
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 = 
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 Mécanique Analytique, but he desired 
that it should be printed at Paris, where he hoped that bis 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 tothe Abbé Marie, with whom he was much connected. — 
