We derive a stability
criterion for a catenoidal liquid bridge making contact angles γ1 and γ2
with two parallel plates. We show that for the case of equal contact angles
γ1= γ2= γ the stability and instability sets are connected on the interval of
admissible γ. We also give an example showing that for unequal contact angles,
the family of stable catenoidal drops with one contact angle fixed can be
disconnected with respect to the other angle. At the end of the paper we give a
complete description of the stability and instability sets for various contact
angles.