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We determine the scrollar invariants of the normalization $C$ of a nodalcurve $\Gamma$ of type $(k,a)$ on a smooth quadric $\mathbb{P}^1 \times\mathbb{P}^1$ associated to the $g^1_k$ defined by the pencil of lines of type$(0,1)$ in case all nodes are contained in at most $k-1$ lines of type $(1,0)$.This result is very much related to results obtained by E. Ballico, but in thispaper the proof follows directly from an easy lemma. Also a result of E.Ballico on the existence of curves with prescribed scrollar invariant is aconsequence of that lemma making the arguments much shorter.

The scrollar invariants of k-gonal curves having a nodal model on a smooth quadric having its nodes on few lines