  We construct explicit solutions to continuous motion of discrete plane curves
described by a semi-discrete potential modified KdV equation. Explicit formulas
in terms the $\tau$ function are presented. B\"acklund transformations of the
discrete curves are also discussed. We finally consider the continuous limit of
discrete motion of discrete plane curves described by the discrete potential
modified KdV equation to motion of smooth plane curves characterized by the
potential modified KdV equation.
