  Theoretical study indicates that a contact binary system would merge into a
rapidly rotating single star due to tidal instability when the spin angular
momentum of the system is more than a third of its orbital angular momentum.
Assuming that W UMa contact binary systems rigorously comply with the Roche
geometry and the dynamical stability limit is at a contact degree of about 70%,
we obtain that W UMa systems might suffer Darwin's instability when their mass
ratios are in a region of about 0.076--0.078 and merge into the fast-rotating
stars. This suggests that the W UMa systems with mass ratio $q\leq0.076$ can
not be observed. Meanwhile, we find that the observed W UMa systems with a mass
ratio of about 0.077, corresponding to a contact degree of about 86% would
suffer tidal instability and merge into the single fast-rotating stars. This
suggests that the dynamical stability limit for the observed W UMa systems is
higher than the theoretical value, implying that the observed systems have
probably suffered the loss of angular momentum due to gravitational wave
radiation (GR) or magnetic stellar wind (MSW).
