1887.] Equilibrium of Rotating Masses of Fluid. 361 



towards answering which we at present offer no contribution."* This 

 paper is intended to be such a contribution, although an imperfect 

 one. 



M. Poincare has made an admirable investigation of the forms of 

 equilibrium of a single rotating mass of fluid, and has specially con- 

 sidered the stability of Jacobi's ellipsoid.f He has shown by a 

 difficult analytical process, that when the ellipsoid is moderately 

 elongated, instability sets in by a furrowing of the ellipsoid along a 

 line which lies in a plane perpendicular to the longest axis. It is, 

 however, extremely remarkable that the furrow is not symmetrical 

 with respect to the two ends, and there thus appears to be a tendency 

 to form a dumb-bell with unequal bulbs. 



M. Poincare's work seemed so important that, although the figures 

 above referred to were already drawn a year ago, this paper was kept 

 back in order that an endeavour might be made to apply the prin- 

 ciples enounced by him, concerning the stability of such systems. 

 The attempt, which proved abortive on account of the imperfection 

 of approximation of spherical harmonic analysis, is given in the 

 appendix to the paper, because, notwithstanding its failure, it presents 

 features of interest. 



The calculations in this paper being made by means of spherical 

 harmonic analysis, it is necessary to consider whether this approxi- 

 mate method has not been pushed too far in the computation of 

 figures of equilibrium which depart considerably from spheres. A 

 rough criterion of the applicability of the analysis is derived from a 

 comparison between the two values of the ellipticity of an isolated 

 revolutional ellipsoid of equilibrium as derived from the rigorous 

 formula and from spherical harmonic analysis. As judged by this 

 criterion, which is necessarily in some respects too severe, the figures 

 drawn appear to present a fair approximation to accuracy. 



Since, as above stated, the rigorous method of discussing the 

 stability of the system fails, certain considerations are adduced which 

 bear on the conditions under which there is a form of equilibrium 

 consisting of two fluid masses in close proximity, and it appears that 

 there cannot be such a form, unless the smaller of the two masses 

 exceeds about one-thirtieth of the larger. It seemed therefore worth 

 while to find to what results the analysis would lead when two masses, 

 one of which is 27 times as great as the other, are brought close 

 together. As judged by this criterion the computed result must be 

 very far from the truth, but as the criterion is too severe, it seemed 

 worth while to give the figure. The smaller mass is found to be 



* Thomson and Tait, 'Natural Philosophy,' (1883), §778 (i). He also remarks 

 elsewhere that by thinning a Jacobian ellipsoid in the middle, we shall get a figure 

 of the same moment of momentum and less kinetic energy. 



f ' Acta Math.,' 7, 3 and 4, 1885. 



