  We find that the observed log N - log S relation of X-ray clusters can be
reproduced remarkably well with a certain range of values for the fluctuation
amplitude $\sigma_8$ and the cosmological density parameter $\Omega_0$ in cold
dark matter (CDM) universes. The $1\sigma$ confidence limits on $\sigma_8$ in
the CDM models with $n=1$ and $h = 0.7$ are expressed as $(0.54 \pm 0.02)
\Omega_0^{-0.35-0.82\Omega_0+0.55\Omega_0^2}$ ($\lambda_0=1-\Omega_0$) and
$(0.54 \pm 0.02) \Omega_0^{-0.28-0.91\Omega_0+0.68\Omega_0^2}$ ($\lambda_0=0$),
where $n$ is the primordial spectral index, and $h$ and $\lambda_0$ are the
dimensionless Hubble and cosmological constants. The errors quoted above
indicate the statistical ones from the observed log N - log S only, and the
systematic uncertainty from our theoretical modelling of X-ray flux in the
best-fit value of $\sigma_8$ is about 15%. In the case of $n=1$, we find that
the CDM models with $(\Omega_0,\lambda_0,h,\sigma_8) \simeq (0.3,0.7,0.7,1)$
and $(0.45, 0, 0.7, 0.8)$ simultaneously account for the cluster log N -
log$S$, X-ray temperature functions, and the normalization from the COBE 4 year
data. The derived values assume the observations are without systematic errors,
and we discuss in details other theoretical uncertainties which may change the
limits on $\Omega_0$ and $\sigma_8$ from the log N - log S relation. We have
shown the power of this new approach which will become a strong tool as the
observations attain more precision.
