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



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

 we find 



p fti b c 



p 2 \d b c 



W- 



The conditions that the pressure may be uniform over the 

 external surface 



are therefore 



. c 2 .... (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 



ad + bb -f cc + 2?r/o (a aa + @ bb + y cc) = ......... (7). 



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

 integral 



d 2 + 6 2 + c 2 - fapabc ( ~ = const ............. (8). 



It has been already proved that the potential energy is 



f 30 d\. 

 F= const. - 7 8 57rya 2 6 2 c 2 ^- ............... (9), 



J o A 



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



(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 



-5-V+ F= const (12). 



2c 3 / 



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

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



