Roche's problem is concerned with the equilibrium and the stability of rotating homogeneous masses which are, further, distorted by the constant tidal action of an attendant rigid spherical mass. This ancient problem is reconsidered in this paper with the principal oblect of determining the stability of the equilibrium configurations (the ellipsoids of Roche) by a direct evaluation of their characteristic frequencies of oscillation belonging to the second harmonics The result of the evaluation is the demonstration that the Roche ellipsoid becomes unstable at a point subsequent to the Roche limit where the angular velocity of rotation, consistent with equilibrium, attains its maximum value. This result requires a revision of the current common view regarding the meaning that is to be attached to the Roche limit Among related matters which are considered are the following: the relationships that exist between the sequences of Roche and those of Maclaurin, Jacobi, and Jeans; the exhibition and the isolation of the second neutral point (belonging to the third harmonics) along the Roche sequences; and the effect of compressibility on the stability of the Roche ellipsoids. A result which emerges from these considerations is the universal instability of the Jacobi ellipsoids under the least tidal action. The methods used in this paper are those derived from the virial theorem and its various extensions. The principal results are summarized in Section X and are exhibited in Figures 1, 2, and 3.