  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.
