150 Proceedings of Philosophical Societies. [Feb* 



inained so imperfectly known, as to be still considered as ons 

 natural order : for, since the accurate description and delineation 

 of upwards of 400 species, by Scheuchzer, enabled Linn^fius to 

 characterize the genera then known^ and the addition of others 

 by Konig and Thunberg, with the numerous species described 

 by Leers, Swartz, Schreber, and Brown, no subdivision of these 

 plants has been effected on the principles of the natural method, 

 even that of Jussieu being merely artificial. The present there- 

 fore is the first attempt to arrange a part of this extensive series 

 on the principle of natural affinity. 



IMPERIAL INSTiTUTE OF FRANCS. 



Analysis of the Lahours of the Class of Mathematical and Phy^ 

 siccd Scieikes of the Imperial Institute during the year 1812. 



Mathematical Part, By M, le Chevcdier Delamhre^ 

 Perpetual Secretary, 



analysis. 



Memoir on the Attraction of Homogeneous Spheroids, By 

 JSI, le Chevcdier Legendre, 



This is the third time that M. Legendre has returned to 

 this difficult problem, which for more than / O years has occupied 

 various mathematicians of great celebrity. Maclaurin, in his 

 Dissertation on the Flux and Reflux of the Tide, which shared 

 the prize of the Academy of Sciences for 17^0, resolved it in a 

 satisfactory manner in the case when the point attracted is within 

 the surface of the elipsoid. He attempted, likewise, the other 

 case when the point attracted is without the surface ; but he 

 could only give the solution of a particular case, namely, when 

 the external point is situated in the prolongation of one of the 

 three axes of the elipsoid. The theorem which he obtained 

 brought this particular case to that where the point attracted is 

 upon the surface of the elipsoid, and at the extremity of one of 

 its axes. M. Le Comte Lagrange had, by an elegant analysis, 

 confirmed all the results of Maclaurin; D'Alembert had demon- 

 strated them in the Berlin Memoirs :~M. Legendre, in Vol. x. 

 of the Memoirs presented, gave a new demonstration of them ; 

 and in considering generally the attraction of spheroids of revo- 

 lution upon an exterior point, he succeeded in reducing it to the 

 attraction on a point situated in the axis of the solid, wiiatever 

 was the figure of the meridian. It remained to be shown how 

 this solution could be extended to all elipsoids. This is what M. 

 Laplace did in his admirable Dissertation on the Figure of the 

 Planets, expressing the attraction of any spheroids whatever in 

 series. Excited by the desire of overcoming more directly a 

 great difficulty, M. Legendre resumed in 178S the consideration 

 of this subject. He demonstrated that the theorem of Maclaurin 



