Previously the second author has constructed, for a uniform gravitational field, a container shape \({\mathcal C}\) such that there exists a continuum of rotational symmetric capillary surfaces which have the striking property that they all meet \({\mathcal C}\) in the same contact angle \(\gamma\), bound with \({\mathcal C}\) the same volume V of fluid, and yield identical mechanical energy [Manuscr. Math. 61, 347-372 (1988; Zbl 0659.53005)]. It was also shown that for any sufficiently weak gravitational field the energy of the configuration can be decreased by an asymmetric deformation. Here, an explicit asymmetric deformation is given for which the second variation of the energy can be computed in terms of hyperbolic functions and the hyperbolic Bessel functions \(I_ 0\) and \(I_ 1\). For all strengths of the gravitational field the second variation of the energy is seen (numerically) to be bounded uniformly below zero, and the asymptotic values are evaluated exactly.