Consider the equilibrium configuration of an inhomogeneous selfgravitating celestial body, composed of incompressible fluid,
that is rotating steadily about some fixed axis passing through its center of mass. The analysis of Section 6.5
implies that the body's equation of hydrostatic equilibrium (in the corotating frame) can be written

(D.1) 
where

(D.2) 
and

(D.3) 
Here, , , are spherical coordinates whose symmetry axis coincides with the axis of rotation.
Moreover, is the fluid pressure, the mass density,
the gravitational potential, the centrifugal
potential, and
the angular rotation velocity.
Taking the curl of Equation (D.1), we
deduce that
, and, hence, that

(D.4) 
Let us assume that the contours of are spheroidal, so that

(D.5) 
where

(D.6) 
Here, and
are the mean radius and ellipticity of the density contour
, respectively. Furthermore, it is assumed that
.
(See Section 3.6).
It follows from Equations (D.4) and (D.5) that

(D.7) 
The aim of this appendix is to deduce
the relationship between the degree of flattening of the body, its rotational angular velocity, and its
moment of inertia about its rotation axis.