Conditions are determined under which solid bodies will float on a liquid surface in stable equilibrium, under the influence
of gravity and of surface tension. These include configurations in which the density of the body exceeds the density of the
ambient liquid, so that for an infinitely deep liquid in a downward gravity field there is no absolute energy minimum. Of
notable interest are the results (a) that if a smooth body is held rigidly and translated downward into an infinite fluid
bath through a family of fluid equilibrium configurations in a downward gravity field, the transition is necessarily discontinuous,
and (b) a formal proof that there can be a free-floating locally energy minimizing configuration that does not globally minimize,
even if the density of the body exceeds that of the liquid. The present work is limited to the two dimensional case corresponding
to a long cylinder that is floating horizontally. The more physical three-dimensional case can be studied in a similar way,
although details of behavior can change significantly. That work will appear in an independent study written jointly with
T. I. Vogel.