594 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII 



3. Let G>=0, so that the motion is irrotational. The conditions (v) 

 reduce to 



n* } f (6 2 -a 2 )(3a 2 + 6 2 ) n> \ 

 iBST " r ~ltf + &*?""" M 



These may be replaced by 



...... (xiii), 



_ pp 



4 22 4 ' 22 



The equation (xiii) determines c in terms of a, b. Let us suppose that 

 a>b. Then the left-hand side is easily seen to be positive for c = a, and nega- 

 tive for c = 6. Hence there is some real value of c, between a and 6, for which 

 the condition is satisfied ; and the value of n, given by (xiv) is then real, for 

 the same reason as in Art. 315. 



4. In the case of an elliptic cylinder rotating about its axis, the condition 



(v) takes the form 



4a 2 6 2 4irab . ... 



(xv) 



If we put n = a>, we get the case of Art. 315 (i). 



If n = 0, so that the external boundary is stationary, we have 



If o, = o, i. e. the motion is irrotational, we have 



319. The small disturbances of a rotating ellipsoidal mass 

 have been discussed by various writers. 



The simplest types of disturbance which we can consider are 

 those in which the surface remains ellipsoidal, with the axis of 

 revolution as a principal axis. In the case of Maclaurin's ellipsoid, 

 there are two distinct types of this character ; in one of these the 

 surface remains an ellipsoid of revolution, whilst in the other 

 the equatorial axes become unequal, one increasing and the other 

 decreasing, whilst the polar axis is unchanged. It was shewn by 

 Riemannf that the latter type is unstable when the eccentricity 

 (e) of the meridian section is greater than '9529. The periods of 



* Greenhill, I. c. ante p. 589. 



t I. c. ante p. 589. See also Basset, Hydrodynamics, Art. 367. Eiemann has 

 further shewn that Jacobi's ellipsoid is always stable for ellipsoidal disturbances. 



