  The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler
\cite{Ge} and independently by Y. Zhang \cite{Z} in order to describe an
integrable hierarchy which governs the Gromov--Witten invariants of $\C P^1$.
The {\em Lax type} presentation of the ETH was given in \cite{CDZ}. In this
paper we give a description of the ETH in terms of {\em tau-functions} and
Hirota Quadratic Equations (known also as Hirota Bilinear Equations). A new
feature here is that the Hirota equations are given in terms of vertex
operators taking values in the algebra of differential operators on the affine
line.
