  The relationships between the X-ray determined bolometric luminosity Lx, the
temperature T of the intracluster gas, and the optical measured velocity
dispersion sigma of the cluster galaxies are updated for galaxy clusters using
the largest sample of 256 clusters drawn from literature. The newly established
relationships, based on the doubly weighted orthogonal distance regression
(ODR) method, are justified by both their self-consistency and co-consistency,
which can then be used to test the theoretical models of cluster formation and
evolution. The observationally determined Lx-T and Lx-sigma relationships,
$L_x\propto T^{2.72\pm0.05}\propto \sigma^{5.24\pm0.29}$, are marginally
consistent with those predicted in the scenario that both intracluster gas and
galaxies are in isothermal and hydrostatic equilibrium with the underlying
gravitational potential of clusters. A comparison between these observed and
predicted Lx-T relationships also suggests that the mean cluster baryon
fraction fb remains approximately constant among different clusters,
$f_b\approx0.17$, which gives rise to a low-mass density universe of
$\Omega_m\approx0.3$.
