  A local world volume Q-supersymmetric Weyl invariant Lagrangian for the
membrane is presented. An analysis is provided which solves the problems raised
by some authors in the past concerning the algebraic elimination of the
auxiliary fields belonging to the coupling function supermultiplet. The
starting bosonic action is the one given by Dolan and Tchrakian with vanishing
cosmological constant and with quadratic, quartic derivative terms. Our
Lagrangian differs from the one of Lindstrom and Rocek in the fact that is
polynomial in the fields facilitating the quantization process. It is argued,
rigorously, that if one wishes to construct polynomial actions without
curvature terms and where supersymmetry is linearly realized, after the
elimination of auxiliary fields, one must relinquish S supersymmetry and
concentrate solely on the Q-supersymmetry associated with the superconformal
algebra in three dimensions. The role that this spinning membrane action may
have in the theory of D-branes, Skyrmions and BPS monopoles is also pointed
out.
