DIRICHLET, LEJEUNE (+) DEDEKIND.

Untersuchungen über ein Problem der Hydrodynamik [Dirichlet] (+) Zusatz zu der vorstehenden Abhandlung [Dedekind].

Berlin, Georg Reimer, 1861. 4to. As extracted from "Journal für die reine und angewandte Mathematik, 1861, band 58. Without backstrip. Fine and clean. Pp. 181-228.

First printing of these two important papers on gravitational interaction from bodies, such pseudo-rigid bodies, thereby "opened a new direction", a direction which later inspired Riemann to turn his attention to the problem

"Without the gravitational interaction from another body, such pseudo-rigid bodies have
received attention in many references. The interest was initiated by Newton in Principia, where he showed that a spinning axi-symmetric self-gravitating body of ?uid that is rotating slowly about the symmetry axis will be oblate. Jacobi in 1834 extended the work of Newton, but also work of Maclaurin, to show that a self-gravitating ?uid also can take on ellipsoidal shapes. The solutions of Jacobi, Maclaurin and Newton were, however, still all rigid. In a frame rotating with the body the ?uid is stationary.
Dirichlet and Dedekin, [in the present two papers], respectively, opened a new direction when they found a symmetry that applied to Jacobi's solution generated a new solution in which the body is
stationary in shape but the ?uid particles follow elliptical paths in planes orthogonal to a principle axis of the ellipsoid. Dirichlet's paper inspired Riemann to turn his attention to the problem. He gave a classi?cation of the solutions of Dirichlet's equations for which the ellipsoidal shape of the body remains constant. At the heart of this classi?cation lies what is now known as Riemann's theorem: the angular velocity and circulation (i) lie in the same principle plane and (ii) if the angular velocity is parallel to a principle axis then the circulation vector must also lie along that same principle axis." (Kristiansen, The two-body problem of a pseudo-rigid body and a rigid sphere)

Order-nr.: 49315