  We show that a capillary surface in a solid cone, that is, a surface that has
constant mean curvature and the boundary of surface meets the boundary of the
cone with a constant angle, is radially graphical if the mean curvature is
non-positive with respect to the Gauss map pointing toward the domain bounded
by the surface and the boundary of the cone. In the particular case that the
cone is circular, we prove that the surface is a spherical cap or a planar
disc. The proofs are based on an extension of the Alexandrov reflection method
by using inversions about spheres.
