We consider some of the complications that arise in attempting to generalize a version of Archimedes' principle concerning floating bodies to account for capillary effects. The main result provides a means to relate the floating position (depth in the liquid) of a symmetrically floating sphere in terms of other observable geometric quantities.A similar result is obtained for an idealized case corresponding to a symmetrically floating infinite cylinder.These results depend on a definition of equilibrium for capillary systems with floating objects which to our knowledge has not formally appeared in the literature. The definition, in turn, depends on a variational formula for floating bodies which was derived in a special case earlier (Pacific J. Math. 231:1 (2007), 167-191) and is here generalized to account for gravitational forces.A formal application of our results is made to the problem of a ball floating in an infinite bath asymptotic to a prescribed level. We obtain existence and nonuniqueness results.

The meniscus in a symmetric annular capillary tube is investigated. The contact angles on the inner and outer tube surface need not be the same. Existence and qualitative properties of solutions are obtained using an iteration similar to that used by Johnson and Perko in the case of a circular capillary tube. If the contact angles have the same sign ideas of Siegel are used to give asymptotic estimates using circular arcs as comparison curves.

The height of the surface of a fluid in an annular tube is explored using a shooting method to solve a boundary value problem where the radii and the contact angles are given. The contact angles on the inner and outer tube surface need not be the same. These surfaces are then extended so that they are no longer graphs. The extended surfaces are shown to solve a boundary value problem over an annular base domain where given inclination angles are achieved at given radii.

An existence theorem for floating drops due to Elcrat, Neel, and Siegel is generalized. The theorem applies to all radially symmetric domains, and to both light and heavy floating drops, and utilizes new results in annular capillary theory. Mathematics Subject Classification (2000). 76B45, 35A15, 35R35, 49J05.

The symmetry of floating drops is considered. Under conditions that the free boundary is flat it is shown that all three component interfaces are symmetric about a vertical line.