  In this paper we consider the $T$- and $V$- versions, $T_{\tau}$ and
$V_{\tau}$ , of the irrational slope Thompson group $F_{\tau}$ considered in
[3]. We give infinite presentations for these groups and show how they can be
represented by tree-pair diagrams similar to those for $T$ and $V$. We also
show that $T_{\tau}$ and $V_{\tau}$ have index-2 normal subgroups, unlike their
original Thompson counterparts $T$ and $V$. These index-2 subgroups are shown
to be simple.
