  We consider the action, in arbitrary curved background, of the
eleven-dimensional five-brane to second order in the curvature of the
worldvolume tensor field. We show that this action gives upon double
dimensional reduction the action of the Dirichlet four-brane up to the same
order. We use this result as a starting point to discuss the structure of the
action including terms of higher order in the worldvolume curvature.
