The influence of nonvanishing quark masses on the total cross section
in electron positron collisions and on the $Z$ decay rate is calculated.
The corrections are expanded in $m^2/s$ and $\as$. Methods similar to
those applied for the quadratic mass terms allow to derive the
corrections of order $\as m^4/s^2$ and $\as^2m^4/s^2$. Coefficients
which depend logarithmically on $m^2/s$ and which
cannot be absorbed in a running quark mass arise in order $\as^2$.
The implications of these
results on electron positron annihilation cross sections
at LEP and at lower energies in particular between the
charm and the bottom threshold and for energies several GeV above the
$b\bar b$ threshold are discussed.