II. On the Potential of Ellipsoidal Bodies, and the Figures of Equilibrium of 



Rotating Liquid Masses. 



By J. H. JEANS, M.A., F.R.S. 



Received May 29, Kead June 25, 1914. 



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/, = 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 = upwards, 

 and the final figure may be taken to be 



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. POINCARE,* 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. 



The theory of figures of equilibrium of rotating masses of liquid is at present 

 in an unsatisfactory state. It has been shown by Lord KELVIN that the Jacobian 

 ellipsoid is stable at the point at which it coalesces with the Maclaurin spheroid, 

 and it has been shown by POINCARE to remain stable up to the point at which 



* " Sur la StabiliW de FEquilibre des Figures Pyriformes affectees par une Masse Fluide en Rotation," 

 ' Phil. Trans.,' A, vol. 198, p. 333. 



(524.) E 2 [Published February 2, 1915. 



