CHAP. II., 3.] 



PHYSICAL ASTRONOMY LEGENDRE IVORY. 



25 



(99.) 



Legendre's 

 researches 

 on the at- 

 traction of 

 Ellipsoids, 



(100.) 

 sometimes 

 in part at- 

 tributed to 

 Laplace. 



(101.) 

 History of 

 the subject 



(102.) 

 Other 

 works. 



(103.) 

 Ivory. 



him when alive. The French Academy, which had 

 buried his memoir intheirmostinaccessible"archives," 

 decreed too late the unprecedented honour of a post- 

 humous medal to his mother. Jacobi, the friendly 

 rival of Abel in his discoveries, died recently, at a 

 mature though not advanced age, at Kbnigsberg, 

 where he was professor. 



But to return to the labours of Legendre. The 

 theory of the figure and attraction of the earth 

 and of other planets naturally divides itself into two 

 parts (1.) the law of attraction of an ellipsoid on 

 a material point without or within it ; (2.) the figure 

 of equilibrium of a fluid subjected to no forces but 

 the mutual attractions of its particles, and the cen- 

 trifugal force due to its rotation. The latter of 

 these problems is still imperfectly solved, the former 

 completely so, and that mainly in consequence of the 

 labours of Legendre and Ivory. 1 



Though the services of Legendre are well known 

 and admitted, the superior address of Laplace in the 

 applications of mathematics has occasioned his re- 

 ceiving the credit of what in some instances belonged 

 to the former. 



Maclaurin, by an exercise of synthetic skill not 

 exceeded since the death of Newton, had demon- 

 strated the attraction of an ellipsoid of revolution 

 upon a material point anywhere within it or on its 

 surface, as well as for an exterior point in the 

 prolongation of its axis or in the plane of its 

 equator. The same problem was afterwards ana- 

 lytically solved by D'Alembert and Lagrange. In 

 1782, Legendre, by a profound and complicated 

 analysis, obtained an expression, by means of series, 

 for the attraction of an exterior particle generally, 

 and he was the first to imagine and employ those 

 artifices of calculation known usually by the name 

 of " Laplace's functions." Laplace made a step 

 towards the simplification of the expression of ex- 

 terior attractions, but the complete solution was 

 reserved for Mr Ivory, as I shall mention below. 



The other labours of Legendre need not be spe- 

 cified here. He co-operated in the trigonometrical 

 survey of France, and gave the formula, known by 

 his name, for approximately reducing a spherical to 

 a plane triangle. He also wrote on the orbits of 

 comets, and on the method of least squares. 



JAMES IVORY, the most considerable British mathe- 

 matician of his time, or that had appeared since Mac- 

 laurin, was born at Dundee in 1765, and studied at St 

 Andrews along with Sir John Leslie. The most ac- 

 tive period of his life was passed as mathematical pro- 

 fessor at the Military College of Marlow (afterwards 

 removed to Sandhurst). He was essentially a self- 

 taught mathematician, and spent much of his time in 

 retirement. He fathomed in private the profoundest 

 writings of the most learned continental mathemati- 



cians, and, at a period when but few Englishmen 

 were able to understand those difficult works, he 

 showed his capacity of adding to their value by ori- 

 ginal contributions, not unworthy of the first ana- 

 lysts. We pass over his earlier contributions con- 

 nected with mathematics and astronomy, several of 

 which are contained in the Transactions of the 

 Royal Society of Edinburgh, and proceed to his 

 most celebrated paper, published in the Philoso- 

 phical Transactions for 1809, in which he com- 

 pletely and definitely resolves the problem of attrac- 

 tions for every class of ellipsoidal bodies. After 

 what has been stated above as to the position of the 

 problem as treated by Legendre, a few words will 

 explain the precise import of Ivory" 1 s Theorem, one of 

 the most celebrated mathematical results of that time. 



We have seen that the attraction of an ellipsoid (104.) 

 on a point within or at its surface had been assigned His i? 01 ^ 

 by Maclaurin. The theorem in question enables *"* 



* "clU OH IDG 



us at once to reduce the case oi an exterior attracted attraction 

 point to that of a point on the surface of the ellipsoid, of Ellip- 

 Suppose an ellipsoid having the same excentricity, 80lds ' 

 and with the principal sections parallel to the first, 

 but whose surface passes through the given exterior 

 point. Let points on the surface whose co-ordinates 

 parallel to the three axes of the respective solids are 

 proportional to those axes be called corresponding 

 points; then the attraction parallel to each axis 

 which one of these bodies exerts on a point situated 

 on the surface of the other is to the attraction of the 

 latter on the " corresponding point" of the sur- 

 face of the former, as the product of the two other 

 axes of the first ellipsoid is to the product of the two 

 other axes of the second. By this means the attrac- 

 tion on an exterior point is accurately expressed in 

 terms of the attraction on an interior point, which 

 is known. It is fair to add that both Legendre and 

 Laplace had some glimpses of the principle, though 

 they failed to apply it to the direct solution of the 

 problem, and between the publication of their Me- 

 moirs and that of Mr Ivory there elapsed nearly a 

 quarter of a century. 



Besides this paper, Ivory contributed many others (105.) 

 on the subject of the attraction of spheroids and the His other 

 theory of the figure of the earth, during a period of pa 

 nearly thirty years; several of these were controversial, 

 and did not add materially to the progress of the sub- 

 ject ; others are considered as masterpieces of analyti- 

 cal skill. One of the last subjects which occupied his 

 attention was the possible equilibrium of a spheroid 

 with three unequal axes, which Jacobi had discovered. 



Between the labours of Ivory and those of Legen- (106.) 

 dre a great analogy subsists ; for the doctrine of ellip- 

 tic integrals also occupied the attention of the former. 



But next to the theory of attractions, that of At- (107.) 

 mospheric Refraction most seriously engaged Ivory's A ^" c Re _ 

 attention. Its great importance in astronomy, and f racfc i n. 



1 See the articles ATTRACTION and FIGURE OF THE EAKTH (the former by Ivory) in this Encyclopaedia. 



