  In a preceding paper, using Eggleton's evolution code we have discussed the
structure and evolution of low-mass W UMa type contact binaries without angular
momentum loss (AML). The models exhibit cyclic behavior about a state of
marginal contact on a thermal time-scale. Part of the time of each cycle is
spent in contact and part in a semi-detached state. According to observations,
W UMa systems suffer AML. We present the models of low-mass contact binaries
with AML due to gravitational wave radiation (GR) or magnetic stellar wind
(MSW) are presented. We find that gravitational radiation cannot prevent the
cyclic evolution of W UMa systems, and the effect of gravitational radiation on
the cyclic behavior of contact binary evolution is almost negligible. We also
find that the most likely AML mechanism for W UMa systems is magnetic braking,
and that magnetic braking effects can increase the period of the cyclic
evolution, and shorten the fraction of the time spent in the poor thermal
contact state. If W UMa stars do not undergo cyclic evolution, and their
angular momentum loss is caused simultaneously by MSW of both components, we
find that the value of the parameter, $\lambda$, should be taken a larger value
in comparison with those derived from observations of single stars. This
indicates that the AML efficiency in W UMa systems may be lowered in comparison
with non-contact stars because of the less mass contained in the convective
envelopes of the components in W UMa systems. If W UMa systems lose their
angular momentum at a constant rate. An angular momentum rate of $\frac{{\rm
dln}J}{{\rm d}t}\approx 1.6\times 10^{-9} {\rm yr^{-1}}$ can prevent the cyclic
behaviour of the model, and the model can keep in good contact with an
essentially constant depth of contact.
