  A new criterion for pressure-driven interchange instabilities in cylindrical
geometry is derived, based on an alternate use of the Energy Principle. This
criterion is inequivalent to Suydam's criterion and does not contain the
magnetic shear. In fact, it is shown that Suydam's criterion relates to the
instability of the slow magnetosonic branch, while the present criterion
relates to the Alfv\'enic one, which is the most dangerous of the two. These
findings explain why pressure-driven modes nearly always exist even if Suydam's
criterion is satisfied by a large margin.
