II. On the potential of ellipsoidal bodies, and figures of equilibrium of rotating liquid masses

By an ellipsoidal body is meant, in the present paper, any homogeneous body which can be arrived at by continuous distortion of an ellipsoid. If ƒ 0 = 0 is the equation of the ellipsoid from which we start, and e is a parameter, the distortion of the ellipsoid may be supposed to proceed by e increasing from the value e = 0 upwards, and the final figure may be taken to be ƒ 0 + e ƒ 1 + e 2 ƒ 2 + e 3 ƒ 3 + ... = 0. For very small distortions the first two terms will adequately represent the distorted figure, and as we pass to higher orders the remaining terms will enter successively. The potential problem, to some extent interesting in itself, derives its chief importance from its application to the determination of the possible figures of equilibrium of a rotating mass of liquid. Poincaré, using his ingenious method of double layers, has shown how the potential of an ellipsoidal body can be carried as far as the second-order terms when the distortion is small, but gives no indication of how it is possible to carry it further, and indeed his method is one which hardly seems susceptible of being developed further than he himself has developed it. It is clear, however, that progress with the problem of rotating liquids can only be made when a method is available for writing down the potential of an ellipsoidal body distorted as far as we please. I believe the method explained in the present paper will be found capable of giving the potential of a body distorted to any extent, although (for reasons which will be explained later) I have not in the present paper carried the calculations further than second-order terms.