Let V be a finite-dimensional vector space over a finite field. The general and special linear groups, GL(V) and SL(V), act on the exterior algebras $\Lambda^*V$ and $\Lambda^*V^*$ of V and its dual $V^*$, and on the symmetric algebra $\S^*V$. The subring of SL(V)-invariants of $\Lambda^*V\otimes\S^*V$ was determined by Dickson and Mui. This paper describes the equivalent, but simpler, calculation of the invariant subring of $\Lambda^*V^*\otimes\S^*V$ as a representation of GL(V)/SL(V).