  The excitation curve for the $\tau^+ \, \tau^-$ production in electron
positron annihilation near the threshold is revisited with the aim of updating
and extending a previous work. We find that the corrections of the relative
magnitude $O(\alpha)$ near the threshold are significantly contributed by the
radiative modification of the Coulomb interaction between the $\tau$ leptons.
The interpolation between the Coulomb effects at the threshold and the
relativistic effects at higher energies is considered and the resulting formula
is argued to have relative accuracy up to, but not including, terms of the
order of $\alpha^2$ at any velocity of the $\tau$ leptons.
