THE 11-DIMENSIONAL 5-BRANE

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