320] STABILITY OF MACLAURIN's ELLIPSOID. 597 



interval. It is easily seen from considerations of continuity that this value 

 must be always negative, and a minimum*. Hence the altitude at this point 

 of the surface is either a minimum, or a minimax. Moreover, since there 

 is a limit to the negative value of V, viz. when the ellipsoid becomes a 

 sphere, there is always at least one finite point of minimum (and negative) 

 altitude on the surface. 



Now it appears, on reference to the tables on pp. 584, 586, that when 

 h < '304 m a*> there is one and only one ellipsoidal form of equilibrium, 

 viz. one of revolution. The preceding considerations shew that this corre- 

 sponds to a point of minimum altitude, and is therefore secularly stable (for 

 symmetrical ellipsoidal disturbances). 



When h> "304 m^a^, there are three points of stationary altitude, viz. one 

 in the plane of symmetry, corresponding to a Maclaurin's ellipsoid, and two 

 others symmetrically situated on opposite sides of this plane, corresponding to 

 the Jacobian form. It is evident from topographical considerations that the 

 altitude must be a minimum at the two last-named points, and a minimax at 

 the former. Any other arrangement would involve the existence of additional 

 points of stationary altitude. 



The result of the investigation is that Maclaurin's ellipsoid is 

 secularly stable or unstable, for ellipsoidal disturbances, according 

 as e is less or greater than "8127, the eccentricity of the ellipsoid 

 of revolution which is the starting point of Jacobi's series *(. 



The further discussion of the stability of Maclaurin's ellipsoid, 

 though full of interest, would carry us too far. It appears that the 

 equilibrium is secularly stable for deformations of any type so long 

 as e falls below the above-mentioned limit. This is established by 

 shewing that there is no form of bifurcation (Art. 316) for any 

 Maclaurin's ellipsoid of smaller eccentricity. 



Poincare has also examined the stability of Jacobi's ellipsoids. 

 He finds that these are secularly stable provided the ratio a : b 

 (where a is the greater of the two equatorial axes) does not 

 exceed a certain limit. 



The secular stability of a rotating elliptic cylinder has been in- 

 vestigated directly from the equations of motion of a viscous fluid 

 by BryanJ. 



* It follows that Maclaurin's ellipsoid is always stable for a deformation such 

 that the surface remains an ellipsoid of revolution. Thomson and Tait, Natural 

 Philosophy (2nd ed.), Art. 778". 



t This result was stated, without proof, by Thomson and Tait, L c. 



Proc. Camb. Phil. Soc., t. vi. (1888). 



