  Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$.
In this paper we consider elementary transformations of linear pfaffian
representations of $C$. Elementary transformations can be interpreted as
actions on a rank 2 vector bundle on $C$ with canonical determinant and no
sections, which corresponds to the cokernel of a pfaffian representation. Every
two pfaffian representations of $C$ can be bridged by a finite sequence of
elementary transformations. Pfaffian representations and elementary
transformations are constructed explicitly. For a smooth quartic, applications
to Aronhold bundles and theta characteristics are given.
