  In the framework of spatially covariant gravity, it is natural to extend a
gravitational theory by putting the lapse function $N$ and the spatial metric
$h_{ij}$ on an equal footing. We find two sufficient and necessary conditions
for ensuring two physical degrees of freedom (DoF) for the theory with the
lapse function being dynamical by Hamiltonian analysis. A class of quadratic
actions with only two DoF is constructed. In the case that the coupling
functions depend on $N$ only, we find that the spatial curvature term cannot
enter the Lagrangian and thus this theory possesses no wave solution and cannot
recover general relativity (GR). In the case that the coupling functions depend
on the spatial derivatives of $N$, we perform a spatially conformal
transformation on a class of quadratic actions with nondynamical lapse function
to obtain a class of quadratic actions with $\dot{N}$. We confirm this theory
has two DoF by checking the two sufficient and necessary conditions. Besides,
we find that a class of quadratic actions with two DoF can be transformed from
GR by disformal transformation.
