of Equilibrium for Revolving Fluids. 123 



the differential equation for the surfaces de niveau in general, 

 according to equation (7), will be 



( p+ h(x*+f+z*?) {3cd3C+ydy+zdz)=0 - • (l2) 



This shows that the surfaces de niveau become all spherical. 

 If p denote the pressure on a point of the fluid with the coordi- 

 nates Xj y } z t we have 



z$y = ( p+ l (v /^+*y) (**+yfr+«fe)- ( 13 ) 



If the radius of the spherical surfaces of niveau is supposed to 

 be n, whereof xdx + ydy + zdz=ndn, this equation will be 



dp -r, 7 2r 3 7 



which, when integrated, gives 



p _Yn* 2r* 



2V 2 /~ 2 3rc 



If we now suppose that the equilibrium is only sustained by 

 the attraction and the centrifugal force acting on the particles, 

 the pressure on the inner surface of the fluid must be =0. If 

 we^ determine the arbitrary constant by this condition, we get, 

 lastly, 



In order to examine the circumstances under which equilibrium 

 can be sustained by attraction and centrifugal force, we shall sup- 

 pose an infinitely small canal in the fluid with a constant sec- 

 tional area going from the exterior surface of the fluid to its 

 inner surface. We may suppose this canal to be separated from 

 the other fluid by an infinitely thin tube ; and if now the acting 

 forces tend to conduct the fluid in the canal towards the cavity, 

 no equilibrium is possible under the supposed circumstances. 

 But if, on the contrary, the resultant of these forces, taken with 

 regard to all the particles of fluid that are in the canal, tends to 

 conduct the fluid towards the solid ellipsoid, equilibrium may 

 exist if at the same time the forces acting on the particles upon 

 the inner surface are also directed towards the solid ellipsoid. 

 As now the surfaces de niveau are spherical, it is sufficient to 

 examine a canal going in the direction of the axis of revolution. 

 If the forces acting on the particles in this canal are directed 

 outward, this must be still more the case with the other canals. 



If now c denote half the major axis of the inner spheroid, 



K3 



