Abstract. For complex parameters a, c, we consider the Hénon mapping Ha,c : C 2 → C 2 , given by (x, y) → (x 2 + c − ay, x), and its Julia set, J. In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which DH (and DH −1 ) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.