  The exact solution for the shape and gravitational field of a rotating
two-layer Maclaurin ellipsoid of revolution is compared with predictions of the
theory of figures up to third order in the small rotational parameter of the
theory of figures. An explicit formula is derived for the external
gravitational coefficient $J_2$ of the exact solution. A new approach to the
evaluation of the theory of figures based on numerical integration of ordinary
differential equations is presented. The classical Radau-Darwin formula is
found not to be valid for the rotational parameter \epsilon_2 = \Omega^2/(2\pi
G\rho_2) >= 0.17 since the formula then predicts a surface eccentricity that is
smaller than the eccentricity of the core-envelope boundary. Interface
eccentricity must be smaller than surface eccentricity. In the formula for
$\epsilon_2$, $\Omega$ is the angular velocity of the two-layer body, $\rho_2$
is the density of the outer layer, and G is the gravitational constant. For an
envelope density of 3000 kg m^(-3) the failure of the Radau-Darwin formula
corresponds to a rotation period of about 3 hr. Application of the exact
solution and the theory of figures is made to models of Earth, Mars, Uranus,
and Neptune. The two-layer model with constant densities in the layers can
provide realistic approximations to terrestrial planets and icy outer planet
satellites. The two-layer model needs to be generalized to allow for a
continuous envelope (outer layer) radial density profile in order to
realistically model a gas or ice giant planet.
