where is the termed the

Now, according to Equation (889) and (890), the gravitational
potential generated *outside* an axially symmetric mass distribution
can be written

(902) |

Here, the integral is taken over the whole cross-section of the distribution in - space.

It follows that for a uniform spheroid

(904) |

(905) |

(906) |

since .

Thus, the gravitational potential outside a uniform spheroid of
total mass , mean radius , and ellipticity , is

(910) |

where use has been made of Equation (901).

Consider a self-gravitating spheroid of mass , mean radius , and ellipticity : *e.g.*, a star, or a planet. Assuming, for the sake of simplicity, that the
spheroid is composed of uniform density incompressible fluid, the gravitational potential on its surface is
given by Equation (911). However, the condition for an equilibrium
state is that the potential be *constant* over the surface. If this is not
the case then there will be gravitational forces acting *tangential* to the
surface. Such forces cannot be balanced by internal pressure, which only
acts *normal* to the surface. Hence, from (911), it is clear that the
condition for equilibrium is . In other words, the equilibrium
configuration of a self-gravitating mass is a *sphere*. Deviations
from this configuration can only be caused by forces in addition to self-gravity
and internal pressure: *e.g.*, centrifugal forces due to rotation, or tidal
forces due to orbiting masses.