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By aid of the methods of a paper on “ Ellipsoidal Harmonic Analysis ” (‘ Phil. Trans., A, vol. 197, pp. 461-557), I here resume the subject of a previous paper (‘ Phil. Trans.' A, vol. 198, pp. 301-331). These papers will be referred to hereafter by the abridged titles of " Harmonics ” and " The Pear-shaped Figure.” At the end of the latter of these it was stated that the stability of the figure could not he proved definitely without approximation of a higher order of accuracy. After some correspondence with M. Poincare during the course of my work, I made an attempt to carry out this further approximation, but found that the expression for a certain portion of the energy entirely foiled me. Meanwhile he had turned his attention to the subject, and he has shown (‘ Phil. Trans.,’A, vol. 198, pp. 333-373) by a method of the greatest ingenuity and skill how the problem may be solved. He has not, however, pursued the arduous task of converting his analytical results into numbers, so that he left the question as to the stability of the pear still unanswered.

VII. The stability of the pear-shaped figure of equilibrium of a rotating mass of liquid