468 X. STATICS OF DEFORMABLE BODIES. 



which is the equation of an oblate ellipsoid of revolution. The 

 ellipticity is 



> + )- Sir' 



The difference between this and the value - given in 149 is due 



to the fact that we have neglected the attraction of the water for 

 itself and that the nucleus is not exactly centrobaric. 



18O. Gravitating, rotating Fluid. A problem of great 

 importance in connection with the figure of the earth and other 

 planets is the form of the bounding surface of a mass of homogeneous 

 rotating liquid under the action of its own gravitation. 



If V is the potential of the mass of fluid at any internal point, 

 and we take the X-axis for the axis of rotation, we have 



The form of the function V depends upon the shape of the bounding 

 surface of the liquid, which is to be determined by the problem 

 itself. The complete problem is thus one of very great difficulty 

 and has been only partially solved. 1 ) 



We will examine whether an ellipsoid is a possible figure of 

 equilibrium. 



We have found in 157, 37) for the potential of a homogeneous 

 ellipsoid 



QO 

 C '( T 2 it 2 ?"* \ fiii 



30) V=x<>al>c J |l--^- & ^--^| 



o 



= const. -^{Lx 2 + Mf + 

 where 



du 



L = 2itoal>c I - 



J ( a ^-\-u}-[/(a^-\-u)(l^-^ 



31) M=2it(>abc C = 



I (h% _j_ -jA I/Yd 2 -4- u 

 ^/ \ J r \ 







I - 

 J ( 





1) Poincare, "Sur Tequilibre (Tune masse fluide animee d'un mouvement 

 de rotation." Acta math., t. VII, 1885. Also, Figures d'equilibre d'une masse 

 fluide. Paris, 1903. 



