317] VARYING ELLIPSOIDAL SURFACE. 591 



diffusion, the ellipsoid will oscillate regularly between the prolate 

 and oblate forms, with a period depending on the amplitude ; 

 whilst if the energy exceed this limit it will not oscillate, but will 

 tend to one or other of two extreme forms, viz. an infinite line of 

 matter coinciding with the axis of z, or an infinite film coincident 

 with the plane xy*. 



If, in the case of an ellipsoid of revolution, we superpose on the 

 irrotational motion given by (1) a uniform rotation o&amp;gt; about the axis of z t the 

 component angular velocities (relative to fixed axes) are 



d d c ... 



i(,-=. x w?/, v = y-\-G)X) w := - z \*J 



a a c 



The Eulerian equations then reduce to 



d . d 2 _ 1 dp dQ, ^ 



d d 2 1 dp do, 



-y + d&amp;gt;x + 2-G&amp;gt;x-o&amp;gt; 2 y= f- 5-, 



a* a pdy dy 



c I dp do. 



-z = f r i 



c p dz dz 



The first two equations give, by cross-differentiation, 



t +2 a = &amp;lt;**&amp;gt;&amp;gt; 



or o&amp;gt;a 2 = G&amp;gt; a 2 (iv), 



which is simply the expression of von Helmholtz theorem that the strength 

 of a vortex is constant (Art. 142). In virtue of (iii), the equations (ii) have 

 the integral 



/n /ri. \ ^&quot; 



,(V). 



Introducing the value of Q from Art. 313 (4), we find that the pressure will 

 be constant over the surface 



provided + 2irpa -a&amp;gt; 2 



In virtue of the relation (iii), and of the condition of constancy of volume 



2- + - 



a c 



* Dirichlet, /. c. When the amplitude of oscillation is small, the period 

 must coincide with that obtained by putting n 2 in the formula (10) of Art. 241. 

 This has been verified by Hicks, Proc. Camb. Phil. Soc., t. iv., p. 309 (1883). 



