INTRODUCTION

By aid of the methods of a paper on “Ellipsoidal Harmonic Analysis” [Paper 10] I here resume the subject of the immediately preceding paper [Paper 11]. 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 be proved definitely without approximation of a higher order of accuracy. After some correspondence with M. Poincaré 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.

M. Poincaré was so kind as to allow me to detain his manuscript on its way to the Royal Society for two or three days, and I devoted that time almost entirely to understanding the method of his attack on the key of the position—namely, the method of double layers, expounded in my own language in § 9 below.