RIEMANN, Bernhard. Ein Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides.
Göttingen: Dieterischen Buchhandlung, 1861. First edition, offprint issue.

His classic paper on the rotating-liquid ellipsoid. "Continuing work of Dirichlet, in 1861 Riemann studied the motion of a liquid mass under its own gravity, within a varying ellipsoidal surface ..., a problem that had been the subject of many works. One of Riemann's classic results deals with the stability of an ellipsoid rotating around a principal axis under equatorial disturbances" (D.S.B.). "Investigations of the equilibrium shapes of homogeneous rotating fluids, which are of obvious interest because of the shapes of the celestial bodies, date back to Newton and were continues by MacLaurin (1742) and Lagrange (1811), besides others. In 1834 Jacobi made the surprising discovery that if the angular momentum exceeds a certain critical value, then ellipsoids without rotational symmetry can also appear as equilibrium shapes. Beginning in the winter of 1856/57 Dirichlet investigated what happens when the critical value is exceeded ... (Laugwitz: Bernhard Riemann, p.265) and posed the Dirichlet problem of determining the motion of a gravitating body that has the shape of an ellipsoid at each instant of time, when the coordinates of the particles are linear functions of their initial values. "Dirichlet himself considered the special case of a spheroid. Dedekind, who prepared Dirichlet's work for publication, found yet another class of solutions. A complete solution of Dirichlet's problem of stationary figures was given only by Riemann. He showed that, given a linear dependence of the velocity field on coordinates of the most general type of motion under which the ellipsoidal shape of the figure of equilibrium is preserved, there is a superposition of uniform rotation and internal movements with uniform vorticity of the liquid. Riemann's ellipsoids include the classical ellipsoids of Maclaurin, Jacobi, and Dedekind and, in addition, three new classes of figures. (Monastyrsky: Riemann, Topology, and Physics, p.71).

Offprint from: Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen Bd. 9. 4to: 258 x 200 mm. Bound in recent marbled paper. Last page with some soiling, otherwise fine. 36 pp.