CHAPTER XII. 



EQUILIBRIUM OF ROTATING MASSES OF LIQUID. 



313. THIS subject had its origin in the investigations on the 

 theory of the Earth's Figure which so powerfully engaged the 

 attention of mathematicians near the end of the last and the 

 beginning of the present century. 



Considerations of space forbid our attempting more than a 

 rapid sketch, with references to the original memoirs, of the case 

 where the fluid is of uniform density, and the external boundary is 

 ellipsoidal. With this is incorporated a slight account of some 

 cognate investigations by Dirichlet and others, which claim notice 

 not only on grounds of physical interest, but also by reason of the 

 elegance of the analytical methods employed. 



We write down, in the first place, some formulae relating to 

 the attraction of ellipsoids. 



If the density p be expressed in ' astronomical ' measure, the 

 gravitation-potential (at internal points) of a uniform mass en- 

 closed by the surface 



s 



where A = {(a 2 + X) (fc 2 + X) (c 2 + \)}* ............... (3). 



This may be written 



ft = 7r/>(ac^ + /3o2/ 2 + 7o* 2 -%o) ............... (4), 



where 



(5), 



* For references see p. 534. The sign of 1) has been changed from the usual 

 reckoning. 





