142 MEMOIR OF LEGENDRE. 



M. Leg-ciulre subsequently resumed the questions treated in these first and 

 memorabh.^ memoirs, purticuhxrly 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 contriltuted to his preceding labors on this subject by 

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

 by 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 M. Legendre's 

 attention, and the varied nature 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 surp»rising fecundity of his 

 genius. 



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

 indeterminate analysis, which includes numerous propositions on the theory of 

 numl)ers, and especially the celebrated tJieorem of reciprocity known nnder the 

 name of the kuv 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 aics of the ellipsis, and on the comparison of these 

 arcs, I memoirs which contain the first rudiments of his theory of eUiptical f mic- 

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

 dift'erences. By a simple change of variables, he arrives rigorously at the inte- 

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

 certain metaphysical principles about which there still existed some doubts. By 

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

 reptitation of the illustrious author of the apiilication of analysis to geometry, 

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

 school. In this same memoir he gives by his metliod the integrals of several 

 classes of equations with partial dift'erences of superior orders ; then, 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 \k\^ particidar inteyrals oi 

 difterential equations, of which he modestly says that the principle and demon- 

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

 M. de Lagrange had given in the Memoirs of the Academy 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 preparing \\\q> 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 which occnpied him at intervals for a great number of years, and in 

 which his labors were fertile in important results. 



In 1787, some donbts 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, MM. Cassini, Mechain, 

 and Legendre, the execution of this operation, in concert with Major (reneral 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 

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



"Mini, de V Academie des Sciences, vol. for 1785. 

 iM^m. de I' Academic des Sciences, vol. for 1786, p. 7. 

 XMim. de I' Academic des Sciences, for 1786, pp. 616-644. 



