  We address the point of application A of the buoyancy force (also known as
the Archimedes force) by using two different definitions of the point of
application of a force, derived one from the work-energy relation and another
one from the equation of motion. We present a quantitative approach to this
issue based on the concept of the hydrostatic energy, considered for a general
shape of the immersed cross-section of the floating body. We show that the
location of A depends on the type of motion experienced by the body. In
particular, in vertical translation, from the work-energy viewpoint, this point
is fixed with respect to the centre of gravity G of the body. In contrast, in
rolling/pitching motion there is duality in the location of A ; indeed, the
work-energy relation implies A to be fixed with respect to the centre of
buoyancy C, while from considerations involving the rotational moment it
follows that A is located at the metacentre M. We obtain analytical expressions
of the location of M for a general shape of the immersed cross-section of the
floating body and for an arbitrary angle of heel. We show that three different
definitions of M viz., the ?geometrical? one, as the centre of curvature of the
buoyancy curve, the Bouguer's one, involving the moment of inertia of the plane
of flotation, and the ?dynamical? one, involving the second derivative of the
hydrostatic energy, refer to one and the same special point, and we demonstrate
a close relation between the height of M above the line of flotation and the
stability of the floating body. Finally, we provide analytical expressions and
graphs of the buoyancy, flotation and metacentric curves as functions of the
angle of heel, for some particular shapes of the floating bodies.
