Recherches sur la Theorie des Perturbations que les Comètes peuvent eprouver par L'Action des Planètes.

1785. (Paris, Moutard, 1785). 4to. Extracted from "Mémoires fe Mathematique et de Physique, Présentés à l'Academie des Sciences par divers Savans", Tome X. Pp. 65-160. Wide-margined, clean and fine. First appearance of this groundbreaking paper in which Lagrange presented his invention of the new method of solving differential equations as VARIATION OF PARAMETERS, and in which the method was completely developed for the first time. He applies the method to the determination of the orbit of a comet from three observations; this formed the basis of subsequent recherches on the subject." In the summer of 1779 Lagrange submitted "Recherches sur la théorie des perturbations que les cométes peuvent éprouver par l’action des planétes", which won the double prize of 4,000 livres. This was the last time that he participated in the competitions of the Paris Academy."(DSB)."Lagrange, whose contributions to celestial mechanics were of the most brilliant characther, wrote his first memoir in 1766 on the perturbations of Jupiter and Saturn. In this work he developed still further the method of the variation of parameters, leaving his final equations still incorrect....their true form being that of the long period terms, as was shown by Laplace in 1784...The method of variatiobn of parameters was completely developed for the first time in 1782 by Lagrange in his prize memoir on the perturbations...(the paper offered)."