These, together with values of nt 2 N for cylindrical rods, and of N and m 2 N for ellipsoids of revolution, are given in the following useful table (loc. cit.
The magnetometric method, except when employed in connexion with ellipsoids, for which the demagnetizing factors are [[[Magnetic Measurements]] accurately known, is generally less satisfactory for the exact determination of induction or magnetization than the ballistic method.
Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids.
A system of confocal ellipsoids is taken y2 (3) a 2 +X b 2 +X c2 + A= I, and a velocity function of the form = x1 P, (4) where 4' is a function of X only, so that 4) is constant over an ellipsoid; and we seek to determine the motion set up, and the form of >G which will satisfy the equation of continuity.
The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.
- a of the two ellipsoids made by x=o; or if aU - a?
Well-known theorem in attractions that if a shell is made of gravitative matter whose inner and outer surfaces are similar ellipsoids, it exercises no attraction on a particle of matter in its interior.
The potential of such a shell at any internal point is constant, and the equi-potential surfaces for external space are ellipsoids confocal with the ellipsoidal shell.
Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.
During forty years the resources of analysis, even in the hands of d'Alembert, Lagrange and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached.
The ellipsoids (41) and (4~i) are reciprocal polars with respect to a sphere having 0 as centre.
The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, I) =const., V(x, y, 1) =const.
For example, all ellipsoids referred to co-ordinates parallel to any three conjugate diameters are parallel projections of each other and of a sphere referred to rectangular co-ordinates.
Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.
We have a memoir On the Attraction of Homogeneous Ellipsoids, and the already mentioned memoir Allgemeine LehrsÃtze, on the theory of forces attracting according to the inverse square of the distance.