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 F, 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&amp;lt; 304m^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 &amp;gt; &quot;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 f. 



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 



* 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&quot;. 



t This result was stated, without proof, by Thomson and Tait, 1. c. 



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



