  We extend the notion of pseudo-differential operators that are used to
represent the Gelfand-Dickey hierarchies, and obtain a similar representation
for the full Drinfeld-Sokolov hierarchies of $D_n$ type. By using such
pseudo-differential operators we introduce the tau functions of these
bi-Hamiltonian hierarchies, and prove that these hierarchies are equivalent to
the integrable hierarchies defined by Date-Jimbo-Kashiware-Miwa and
Kac-Wakimoto from the basic representation of the Kac-Moody algebra
$D_n^{(1)}$.
