  We construct $W_3$ null vectors of a restricted class explicitly in two
different forms. The method we use is an extension of that of Bauer et al.~in
the Virasoro case. Our results are analogous to the formulae of Benoit and
St.~Aubin for the Virasoro null vectors. We derive in the Virasoro case some
alternative formulae for the same null vectors involving only the $L_{-1}$ and
$L_{-2}$ modes of the Virasoro algebra. }
