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We find that the observed log N - log S relation of X-ray clusters can bereproduced remarkably well with a certain range of values for the fluctuationamplitude $\sigma_8$ and the cosmological density parameter $\Omega_0$ in colddark matter (CDM) universes. The $1\sigma$ confidence limits on $\sigma_8$ inthe 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 thedimensionless Hubble and cosmological constants. The errors quoted aboveindicate the statistical ones from the observed log N - log S only, and thesystematic uncertainty from our theoretical modelling of X-ray flux in thebest-fit value of $\sigma_8$ is about 15%. In the case of $n=1$, we find thatthe 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 yeardata. The derived values assume the observations are without systematic errors,and we discuss in details other theoretical uncertainties which may change thelimits on $\Omega_0$ and $\sigma_8$ from the log N - log S relation. We haveshown the power of this new approach which will become a strong tool as theobservations attain more precision.Comment: 15 pages. Substantial revision. Accepted for publication in  Astrophysical Journal, Part

Constraints on the Fluctuation Amplitude and Density Parameter from X‐Ray Cluster Number Counts