317-318] SPECIAL CASES. 593 



since the conditions are evidently satisfied by the superposition of the irrota- 

 tional motion which would be produced by the revolution of a rigid ellipsoidal 

 envelope with angular velocity n - o> on the uniform rotation o> (cf. Art. 107). 

 Hence 



2a 2 , 2& 2 W 



Substituting in (i), and integrating, we find 



-G + const 

 Hence the conditions for a free surface are 



(v). 



This includes a number of interesting cases. 

 1. If we put ?i = o>, we get the conditions of Jacobi's ellipsoid (Art. 315). 



2. If we put 7i = 0, so that the external boundary is stationary in space, 

 we get 



(vii), 



These are equivalent to 



and = . | - ................. 



2?rp 4a 2 6 2 a 2 - o 2 



It is evident, on comparison with Art. 315, that c must be the least axis 

 of the ellipsoid, and that the value (viii) of <n> 2 /27rp is positive. 



The paths of the particles are determined by 



26 2 



- 



whence x = ka cos (<rt + f), y = ^6sin(o-^-ff), z=0 ............... (x), 



lf * 



and ^, e are arbitrary constants. 



These results are due to Dedekind*. 



* 1. c. ante p. 589. See also Love, "On Dedekind's Theorem,...," Phil. Mag., 

 Jan. 1888. 



L. 38 



