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



The fastest rotation which admits of an ellipsoidal form of revolution, 

 for such a mass, has a period of 2 h. 25 m. 



If m be the total mass, h its angular momentum, we have 



m = 7rpa?c, h = |ma 2 w .................. (8), 



whence we find 



This gives the angular momentum of a given volume of given 

 fluid in terms of , and thence in terms of the excentricity e. 

 It appears from the discussion of an equivalent formula by 

 Laplace, or from the table given below, that the right-hand 

 side increases continually as f decreases from oo to 0. Hence 

 for a given volume of given fluid there is one, and only one, 

 form of Maclaurin's ellipsoid having any prescribed angular mo- 

 mentum. 



The following table, giving numerical details of a series of Maclaurin's 

 ellipsoids, is derived from Thomson and Tait*, with some modifications intro- 

 duced for the purpose of a more ready comparison with the corresponding 

 results for Jacobi's ellipsoids, obtained by Darwin (see Art. 315). The unit of 

 angular momentum is m^ a*. 



* Natural Philosophy, Art. 772. 



