We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium. It may be proved that the attraction of the ellipsoid x² + y² + z²(1 + ε²) = c²(1 + ε²); upon a particle P of its mass at x, y, z has for components
X = − Ax, Y = − Ay, Z = − Cz,
where
| A = 2πk²ρ( | 1 + ε² | tan−1ε − | 1 | ), |
| ε³ | ε² |
| C = 4πk²ρ( | 1 + ε² | − | 1 + ε² | tan−1ε), |
| ε² | ε³ |
and k² the constant of attraction. Besides the attraction of the mass of the ellipsoid, the centrifugal force at P has for components + xω², + yω², 0; then the condition of fluid equilibrium is
(A − ω²)xdx + (A − ω²)ydy + Czdz = 0,
which by integration gives
(A − ω²)(x² + y²) + Cz² = constant.
This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible. The equation coincides with that of the surface of the fluid mass if we make