In this note radiative corrections to the total hadronic decay rate of the $τ$-lepton are studied employing perturbative QCD and the operator product expansion. We calculate quadratic quark mass corrections to the decay rate ration $R_τ$ to the order ${\cal O}(α_s^2 m^2)$ and find that they contribute appreciably to the Cabbibo supressed decay modes of the $τ$-lepton. We also discuss corrections of mass dimension D=4, where we emphasize the need of a suitable choice of the renormalization scale of the quark and gluon condensates.

13 pages, LaTeX, no figures. This version fixes a typo in eq. (25) of the original paper (Z. Phys. C59 (1993) 525) and an errror in a numerical integration procedure which has resulted to a significant increase of the O(\alpha_s^2) coefficient in eq. (27). As a consequence also some tables in Section 4 have been modified