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 eo on the uniform rotation o&amp;gt; (cf. Art. 107). 

 Hence 



Substituting in (i), and integrating, we find 



-fl + const 

 Hence the conditions for a free surface are 



a 2 6 2 26 2 



~ &quot; ~ 



= -7rpy c 2 (v). 



This includes a number of interesting cases. 



1. If we put ?i=o&amp;gt;, we get the conditions of Jacobi s ellipsoid (Art. 315). 



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

 we get 



f 2&amp;lt;* 2 6 2 



Woo- 



These are equivalent to 



and 



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

 of the ellipsoid, and that the value (viii) of a&amp;gt; 2 /2irp is positive. 



The paths of the particles are determined by 



26 2 



- 



whence x = ka cos (&amp;lt;rt + e), y = ^6sin(o-^ + e), ^=0 ............... (x), 



-&quot; ................................. 



and &, e are arbitrary constants. 



These results are due to Dedekind*. 



* I. c. ante p. 589. See also Love, &quot;On Dedekind s Theorem,...,&quot; Phil. Mag., 

 Jan. 1888. 



L. 38 



