The model consists in taking linearized approximations to the curvatures of the upper and lower surfaces, but the problem posed by the stable equilibrium of a suspended drop remains nonlinear because of the free boundary where the two surfaces join. A theory based on variational methods is developed on this basis, applying to drops suspended within non-circular frames. The mathematical problem is formulated in $\S$ 2, and the existence of solutions is proved in $\S$ 3. Conditions relating to the presence of a free boundary and to the uniqueness of solutions are discussed in $\S$ 4. To bear out a comment made in part I (Benjamin & Cocker, Proc. R. Soc. Lond. A 394, 19, 1984), the connection between the present topic and a familiar model for confined plasmas is made clear in $\S$ 5. Finally, in $\S$ 6, various unresolved aspects of this new topic are reviewed.