This is the sequel to a previous paper on “Ellipsoidal Harmonic Analysis” (‘Phil. Trans..,’ A, vol. 197, pp. 461-557). I here make use of the methods of that paper, and for brevity shall refer to it as “Harmonics.” The sections 1 to 4 are preparatory, and might have been included in “Harmonics,” but seem more appropriate here. Section 5 is an independent investigation of so much of M. Poincaré’s celebrated memoir on rotating liquid as relates to the immediate object in view. It is not necessary to say more here, since I give a short summary in the last section.