  The form factors of the weak $\Xi_c\to \Xi(\Lambda)$ transitions are
calculated in the framework of the relativistic quark model based on the
quasipotential approach. All relativistic effects including transformations of
the baryon wave functions from the rest to moving reference frame and
contributions of the intermediate negative energy states are systematically
taken into account. The explicit analytic expressions which reliably
approximate the momentum transfer $q^2$ dependence of the form factors in the
whole accessible kinematical range are given. The calculated form factors are
applied for the evaluation of the semileptonic $\Xi_c\to \Xi \ell\nu_\ell$ and
$\Xi_c\to \Lambda \ell\nu_\ell$ ($\ell=e,\mu$) decay rates, different asymmetry
and polarization parameters within helicity formalism. The obtained results are
compared with available experimental data and previous calculations.
