  This paper derives a set of general relativistic Cardinal Equations for the
equilibrium of an extended body in a uniform gravitational field. These
equations are essential for a proper understanding of the mechanics of
suspended relativistic systems. As an example, the prototypical case of a
suspended vessel filled with radiation is discussed. The mechanics of Casimir
apparatuses at rest in the gravitational field of the Earth is then considered.
Starting from an expression for the Casimir energy-momentum tensor in a weak
gravitational field recently derived by the authors, it is here shown that, in
the case of a rigid cavity supported by a stiff mount, the weight of the
Casimir energy $E_C$ stored in the cavity corresponds to a gravitational mass
$M=E_C/c^2$, in agreement with the covariant conservation law of the
regularized energy-momentum tensor. The case of a cavity consisting of two
disconnected plates supported by separate mounts, where the two measured forces
cannot be obtained by straightforward arguments, is also discussed.
