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



by Art. 21 (4); and substituting the value of O from Art. 313 

 we find 



- n = - \ (-** + 1 f- + ~ 



p \U/ U v 



.................. (4). 



The conditions that the pressure may be uniform over the 

 external surface 



/yi2 rt/2 /y^ 



J + t + J- 1 ........................ &amp;lt; 5 &amp;gt;&amp;gt; 



are therefore 



v 



g + 2*p,) a = fa +2,rp/3 ) b* = g + 2^-py.) c* . . . . (6). 



These equations, with (2), determine the variations of a, b, c. 

 If we multiply the three terms of (2) by the three equal magni 

 tudes in (6), we obtain 



ail + 1)b 4- cc + 2?r/) (a ad + fijbb + %cc) = ......... (7). 



If we substitute the values of , /3 , 70 from Art. 313, this has the 

 integral 



r /1\ 

 a 2 + 6 2 + c 2 - 4&amp;gt;7rpabc I -r- = const ............. (8). 



7 o ^ 



It has been already proved that the potential energy is 



/* /7&quot;X 

 F= const. - ft TrytfbW I -^ ............... (9), 



and it easily follows from (1) that the kinetic energy is 



+6 2 + c 2 ) .................. (10). 



Hence (8) is recognized as the equation of energy 



T+F=const ....................... (11). 



When the ellipsoid is of revolution (a = b), the equation (8), 

 with a 2 c = a 3 , is sufficient to determine the motion. We find 



(l + |0 c 2 + F= const (12). 



The character of the motion depends on the total energy. If 

 this be less than the potential energy in the state of infinite 



