  In a remark to Green's conjecture, Paranjape and Ramanan analyzed the vector
bundle $E$ which is the pullback by the canonical map of the universal quotient
bundle $T_{\Pp^{g-1}}(-1)$ on $\Pp^{g-1}$ and stated a more general conjecture
and proved it for the curves with Clifford Index $1$ (trigonal and plane
quintic). In this paper, we state the conjecture for general linear systems and
obtain the results for the case of hyperelliptic curves.
