  We have calculated the first and second order corrections to several deep
inelastic sum rules which are due to heavy flavour contributions. A comparison
is made with the existing perturbation series which has been computed up to
third order for massless quarks only. In general it turns out that the effects
of heavy quarks are very small except when $Q \sim m$ or $Q \gg m$. Here $Q$
and $m$ denote the virtual mass of the vector boson and the mass of the heavy
quark, respectively. For $Q \gg m$ the radiative corrections reveal large
logarithms of the type $\ln Q^2/m^2$ which have to be absorbed in the running
coupling constant so that the number of light flavours $n_f$ is enhanced by one
unit. However this has to happen at much larger values of Q i.e. $Q \sim 6.5~
m$ than one usually assumes for the flavour thresholds which appear in the
running coupling constant. An alternative description for the heavy flavour
dependence of the running coupling constant in the context of the MOM-scheme is
discussed.
