Of such ellipsoids, consisting of homogeneous matter, as are capable of having the resultant of the attraction of the mass upon a particle in the surface, and a centrifugal force caused by re­volving about one of the axes, made perpendicular to the surface

Abstract

Lagrange, who has considered the problem of the attractions of homogeneous ellipsoids in all its generality, and has given the true equations from which its solution must be derived, inferred from them that a homogeneous planet cannot be in equilibrium unless it has a figure of revolution. But M. Jacobi has proved that an equi­librium is possible in some ellipsoids of which the three axes are un­ equal and have a certain relation to one another. His transcend­ental equations, however, although adapted to numerical computa­tion on particular suppositions, still leave the most interesting points of the problem unexplored The author of the present paper points out the following property as being characteristic of all spheroids with which an equilibrium is possible on the supposition of a centrifugal force. From any point in the surface of the ellipsoid draw a perpendicular to the least axis, and likewise a line at right angles to the surface : if the plane pass­ing through these two lines contain the resultant of the attractions of all the particles of the spheroid upon the point in the surface, the equilibrium will be possible, otherwise it will not. For the re­sultant of the centrifugal force and the attraction of the mass must be a force perpendicular to the surface of the ellipsoid, which re­quires that the directions of the three forces shall be contained in one plane. This determination obviously comprehends all spheroids of revolution; but, on account of the complicated nature of the attractive force, it is difficult to deduce from it whether an equilibrium be possible or not in spheroids of three unequal axes, a problem which is unconnected with the physical conditions of equilibrium, and which is a purely geometrical question respecting a property of cer­tain ellipsoids.