  By explicitly eliminating all gauge degrees of freedom in the $3+1$-gauge
description of a classical relativistic (open) membrane moving in $\Real^3$ we
derive a $2+1$-dimensional nonlinear wave equation of Born-Infeld type for the
graph $z(t,x,y)$ which is invariant under the Poincar\'e group in four
dimensions. Alternatively, we determine the world-volume of a membrane in a
covariant way by the zeroes of a scalar field $u(t,x,y,z)$ obeying a
homogeneous Poincar\'e-invariant nonlinear wave-equation. This approach also
gives a simple derivation of the nonlinear gas dynamic equation obtained in the
light-cone gauge.
