142 MEMOIR OF LEGENDRE. 



M. Le^C'iidre subsequently resumed the questions treated in these first and 

 memorable memoirs, particularly in 1790, in the sequel of his researches on the 

 figure of the planets; in 1789, in a memoir on double integrals, in which he 

 completes the analysis of his memoir on the attraction of spheroids ; and still 

 later, in a memoir read to the Academy in 1812. After having pointed out, in 

 this last, the improvements contributed to his preceding labors on this subject by 

 M. Biot, who had conceived the happy idea of applying thereto an integral given 

 ))y M. de Lagrange for another object, M. Legendre avails himself of the sub- 

 stitution discovered by M. Ivory to present the entire theory of the attraction 

 of homogeneous ellipsoids with all the simplicity of which it is susceptible. 



But these important labors were far from entirely absorbing JI. Legendre's 

 attention, and the varied natm'e of the memoirs which he presented in great fre- 

 quency to the Academy, to a mere enumeration of which I must here confine 

 myself, evinced the extent of his knowledge and the sm'j)rising fecundity of his 

 genius. 



In 1785, he read to the Academy a masterly memoir entitled Besearclies on 

 indeterminate analysis, which includes numerous propositions on the theory of 

 numbers, and especially the celebrated theorem of reciprocity kno-\\ir under the 

 name of the law of Legendre* In 1786, a memoir on the manner of distin- 

 guishing maxima from minima in the calculation of variations.t Also, two 

 memoirs on integrations by arcs of the ellipsis, and on the comparison of these 

 arcs, I memoirs which contain the first rudiments of his theory of elliptical func- 

 tions. In 1787, a memoir on the integration of certain equations with partial 

 differences. By a simple change of variables, he arrives rigorously at the inte- 

 gral of an equation which Monge had only integrated 1 )y a process depending on 

 certain metaphysical princi})les about which there still existed some doubts. By 

 proving that the integral was exact, M. Legendre contributed to corroborate the 

 reputation of the illustrious author of the application of analysis to geometry, 

 whose name also is one of the characteristic glories of the French mathematical 

 school. In this same memoh- he gives by his method the integrals of several 

 classes of equations with partial difierences of superior orders ; tlien, very hap- 

 pily extending an idea of Lagrange for the integration of non-linear equations 

 of the first order, he distinguishes therein six cases of integrability which they 

 may present. Again, in 1790 he read a memoir on the xmrticidar integrals of 

 difl'erential equations, of which he modestly says that the principle and demon- 

 stration are only consequences very easily to be deduced from the theory which 

 M. de Lagrange had given in the Men;ioirs of the Academj^ of Berlin for 1774. 

 He establishes that particular integrals are always comprised in a finite expres- 

 sion in which the number of arbitrary constants is less than in the complete inte- 

 gral, thus ])rcparing the way for the definitive labors which M. Poisson has since 

 made public on this subject. 



But at this epoch M. Legendre was already engaged in another series of 

 researches Avhich occupied him at intervals for a great number of years, and in 

 which his labors were fertile in important results. 



In 1787, some doubts having been raised upon the respective position of the 

 observatories of Paris and Greenwich, it was decided to connect the meridians 

 by a chain of triangles which should extend from one point to the other. The 

 Academy of Sciences confided to three of its members, ]\IM. Cassini, Mechain, 

 and Legendre, the execution of this operation, in concert with Major General Roy 

 and several other English savants. These important labors were accordingly 

 performed with all the exactness which the state of science then permitted — by 

 the help of an excellent quadrant prepared by the celebrated English artist 

 Kamsden, and the repeating circle constructed by Lenoir upon the principles of 



'Mem. de V Academie des Sciences, vol. for 1785. 

 jM^rn. de V Academie des Sciences, vol. for 1786, p. 7. 

 iMe7H. de I' Academie des Sciences, for 178(3, pp. (Ji-6-644. 



