The Rosat all sky survey is used to investigate the X-ray properties of a sample of Abell clusters with measured redshifts and accurate positions. The sample comprises the 145 clusters within a 561 square degree region at high galactic latitude. The mean redshift is 0.17. The sample mean exposure time is higher than average and its mean galactic column density is very low. These together produce a flux limit of about 4.2 x 10(exp -13) erg/sq cm/s in the 0.5 to 2.5 keV energy band. By analyzing the excess of positive fluctuations of the X-ray flux at the cluster positions, compared with the fluctuations of randomly drawn background fields, it is possible to extend these results below the nominal flux limit. It is found that 80 of richness R equal to or greater than 0 and 86 of R equal to or greater than 1 clusters are X-ray emitters with fluxes above 1.10 to the power of minus 13 erg/sq cm/s. Nearly 90 of the clusters meeting the requirements to be in Abell's statistical sample emit above the same level. It is concluded that almost all Abell clusters are real clusters and the Abell catalogue is not strongly contaminated by projection effects. The Kaplan-Meier product limit estimator is used to calculate the cumulative X-ray luminosity function. It is shown that the shape of the luminosity functions are similar for different richness classes, but the characteristic luminosities of richness 2 clusters are about twice of those of richness 1 clusters which are in turn about twice of those of richness zero clusters. This result is another manifestation of the luminosity richness relation for Abell cluster.