  A new local world volume supersymmetric Lagrangian for the bosonic membrane
is presented. The starting Lagrangian is the one constructed by Dolan and
Tchrakian with vanishing cosmological constant, with quadratic and quartic
derivative terms. Our Lagrangian differs from the one constructed by Lindstrom
and Rocek in the fact that it is polynomial in the fields facilitating the
quantization process. It is argued, rigorously, that if one wishes to construct
polynomial actions without a curvature scalar term and, where supersymmetry is
linearly realized in the space of physical fields, after the elimination of
auxiliary fields, one must relinquish $S$ supersymmetry, altogether, and
concentrate solely on the $Q$ supersymmetry associated with the superconformal
algebra in three dimensions. A full $''Q+S''$ supersymmetry cannot be
implemented in a linearly realized way satisfying all of the above-mentioned
requirements, unless a non-polynomial action is chosen.
  PACS:04.65.+e, 04.20.Fy.
