[en] Necessary and sufficient variational criteria for the instability of two-dimensional, static, periodic equilibria of a system of gas, magnetic fields, and cosmic rays are derived; they emphasize the important role played by the curvature of the field lines. Application to periodic curved equilibrium states, related through flux freezing to an unstable, one-dimensional equilibrium of the Parker type, indicates that they are also unstable. We show that the growth rate of the instability, especially of cosmic rays are present, can be much faster than was previously thought