The irregularity of ruled surfaces in $\textbf{P}^3$

Let $S$ be a ruled surface in $\textbf{P}^3$ with no multiple generators. Let $d$ and $q$ be nonnegative integers. In this paper we determine which pairs $d(d,q)$ correspond to the degree and irregularity of a ruled surface, by considering these surfaces as curves in a smooth quadric hypersurface in $\textbf{P}^5$.