Successor-invariance in the finite

A first-order sentence θ of vocabulary σ ∪ {S} is successor-invariant in the finite if for every finite σ-structure M and successor relations S₁ and S₂ on M, (M, S₁) ╞ θ ⇔ (M, S₂) ╞ θ. In this paper I give an example of a non-first-order definable class of finite structures, which is, however, defined by a successor-invariant first-order sentence. This strengthens a corresponding result for order-invariant in the finite, due to Y. Gurevich.