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



this may be put in the form 



2da + cc + 2 (o> 2 ad + o>o>a 2 ) + 4irpa Q aa + 2irpy Q cc = ............ (ix), 



rd\ 

 - r = const ..... (x). 

 .( S +X)(<*+A)* 



This, again, may be identified as the equation of energy. 

 In terms of c as dependent variable, (x) may be written 



If the initial circumstances be favourable, the surface will oscillate regularly 

 between two extreme forms. Since, for a prolate ellipsoid, V increases with 

 c, it is evident that, whatever the initial conditions, there is a limit to the 

 elongation in the direction of the axis which the rotating ellipsoid can attain. 

 On the other hand, we may have an indefinite spreading out in the equatorial 

 plane *. 



318. For the further study of the motion of a fluid mass 

 bounded by a varying ellipsoidal surface we must refer to the 

 paper by Riemann already cited, and to the investigations of 

 Brioschif", LipschitzJ, Greenhill and Basset ||. We shall here 

 only pursue the case where the ellipsoidal boundary is invariable 

 in form, but rotates about a principal axis (z)*\\. 



If u, v, w denote the velocities relative to axes x, y rotating in their own 

 plane with constant angular velocity n, the equations of motion are, by 

 Art. 199, 



Du _ 9 I dp do. 



- 2nv -n z x=---f--j- 



Dt p dx dx 



Dv 1 dp do. 



7r + 2mi - n 2 y = - - J- - -=- 

 Dt p dy dy 



Dw _\dp do, 



~Di ~~p~dz~~dz 



If the fluid have an angular velocity &> about lines parallel to z, the actual 

 velocities parallel to the instantaneous positions of the axes will be 



* Dirichlet, 7. c. f Crelle, t. lix. (1861). 



I Crelle, t. Ixxviii. (1874). I.e. ante p. 589. 



|| "On the Motion of a Liquid Ellipsoid under the Influence of its own 

 Attraction," Proc. Lond. Math. Soc., t. xvii., p. 255 (1886) ; Hydrodynamics, c. xv. 



IT Greenhill, "On the Rotation of a Liquid Ellipsoid about its Mean Axis," 

 Proc. Camb. Phil. Soc., t. iii. (1879). 



