  A simplified nonlinear memory function is proposed in the ideal
time-convolutionless mode-coupling theory equation to study the dynamics of
glass-forming liquids. The numerical solutions are then compared with the
simulation results performed on fragile liquids and strong liquids. They are
shown to recover the simulation results in a supercooled state well within
error, except at a $\beta$-relaxation stage because of the ideal equation. A
temperature dependence of the nonlinearity $\mu$ in the memory function then
suggests that the supercooled state must be clearly separated into two
substates, a weakly supercooled state in which $\mu$ increases rapidly as $T$
decreases and a deeply supercooled state in which $\mu$ becomes constant up to
the glass transition as $T$ decreases. On the other hand, it is shown that in a
glass state $\mu$ increases rapidly as $T$ decreases, while it is constant in a
liquid state. Thus, it is emphasized that the new model for the simplified
memory function is much more reasonable than the conventional one proposed
earlier by the present author not only qualitatively but also quantitatively.
