MEMOIR OF LEGENDRE. 141 



moment to tlie constitution and usages of the ancient Academy of Sciences of 

 Paris, from -which our own differ in some respects, though on many points they 

 have remained identically the same. 



I hasten to return to the labors of M. Legendre, which followed one another 

 at short intervals. He read to the Academy, July 4, 1784, Bcscarches on tJic 

 figure of planets, in which he again discussed in a felicitous manner a subject 

 treated by M. de Laplace. It had been ascertained by illustrious geometers 

 that when a planet, supposed to be fluid and homogeneous, revolves upon itself, 

 it arrives dctinitivel}' at an ellipsoidal figure, slightly flattened at the two poles 

 of rotation, and that among the figures which may be attributed to the meridian 

 curve, the ellipsis is one of those which satisfy the condition of equilibrium; but 

 no one had yet discovered that the ellipsis is the only curve which can satisfy the 

 condition. M. de Laplace, in his memoir of 1772, had said positively that ho 

 would not venture to assert that this figure was the only one which could do so ; 

 that it would be first necessary to know in finite terms the complete integral of 

 the diff'erential equation of the problem, and that he had not yet been able to 

 obtain it. This M. Legendre accomplished by availing himself of the ingenious 

 analysis of his memoir on the attraction of the spheroids, and he concludes that 

 if a planet in equilibrium be supposed to have the figure of a solid of revolution 

 little different from a sphere, and divided into two equal parts by its equator, the 

 meridian of that planet will necessarily be elliptical. 



^' The proposition which forms the object of this memoir," he observes in a 

 note, ^'having been demonstrated in a much more skilful and general manner in 

 a memoir which M. de Laplace has already published in the volume of 1782, 

 (printed later than its date,) I should draw attention to the fact that the date of 

 my own memoir is earlier, and that the proposition which appears here, as it was 

 read in June and July, 1784, gave occasion to M. de Laplace to investigate the 

 subject thoroughl}', and to present to geometers a complete theory thereof." 



Other great geometers also have added their discoveries to those of ISl. 

 Legendre,* but nothing has effaced the merit of his two memoirs drawn up in 

 1782. Hence, M. Poisson, in the learned and eloquent discourse which he pro- 

 nounced January 10, 1833, at the grave of Legendre, took occasion to say: 



The reduction into series of which he made use in the first memoir, gare rise to theorems 

 which have been since extended, but which are still the basis of the theory at which we have 

 subsequently arrived. In the second, he gave the only direct solution 3-et known of the 

 problem of the figure of a homogeneous planet, supposed to be fluid, and soon afterwards he 

 extended his researches to the general case of a planet, composed of heterogeneous strata. t 



In the course of his memoir, M. Legendre finds that the terrestrial spheroid, 

 which is in ecpiilibrium when the axes are in the ratio of 230 to 231, may still be 

 so if the axes be supposed in the ratio of 1 to 681, which affords quite a strange 

 figure, but one which recalls the ring of Saturn. He adds that d'Aleml^ert was 

 the first to remark that there might be several elliptical spheroids which would com- 

 port with equilibrium. We see by these different examples what emulation existed 

 between those fine intellects, d'Alembert, Lagrange, Laplace, Legendre ; with 

 what rapidity their labors succeeded, while they mutually completed one another. 

 It may further be remarked that M. Legendre supposes only in an imi)licit man- 

 ner that the spheroid is one of revolution. The equation found by him is that 

 of the meridian curve, and his analysis is in no respect contradicted b}' the dis- 

 covery, as curious as it was unexpected, made in our time almost simultaneously 

 by M. Liouville and M. Jacobi, that the planetary ellipsoid may have its three 

 axes unequal, and that the equator may itself be an ellipsis. 



* Since the death of M. Legendre, the question ot the attraction of an ellipsoid on an exter- 

 nal point has been completely resolved in an analytic manner by M. Poisson, ( Mf. moires dt 

 I' Acad, des Sciences dc l' Institute, t. xiii, p. 497, 1835 ;) ard in a synthetic muuiier by M. 

 Chasles, {Memoires des savants etrangcrs d V Acadeniie des Sciences, t. ix, p. G2'J, 1846.) 



t Discourse pronounced at the funeral of M. Legendre, January 10, 18'X>, by M. Poisson. 



