III. 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 surgace, and a centrifugal force caused by revolving about one of the axes, made perpendicular to the surface

Abstract

1. In the Conn. des Temps for 1837 it is announced that a homogeneous ellipsoid with three unequal axes, and consisting of particles that attract one another according to the law of nature, may be in equilibrium when it revolves with a proper velocity about the least axis. Lagrange has considered this problem in its utmost generality. The illustrious Geometer found the true equations from which the solution must be derived: but he inferred from them that a homogeneous planet cannot be in equilibrium unless it have a figure of revolution. Nevertheless M. Jacobi has proved that an equilibrium is possible in some ellipsoids of which the three axes have a certain relation to one another. The same thing is demonstrated my M. Liousville in 23rd cahir of the Journal de l'École Polytechnique . M. de Pontécoulant has also touched on the subject*. M. Jacobi has thus detected an inadvertence into which those had fallen who preceded him in this research. He has shown that the equations which, according to Lagrange, are capable of solution only in figures of revolution, may be solved in a certain class of ellipsoids with three unequal axes. But the transcendent equations of M. Jacobi, although fit for numerical computation on particular suppositions, leave unexplored the points of the problem which it is most interesting to know. It is easy to find a property characteristical 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 not. This will be evident, if it be considered that 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 with three unequal axes.