The ESO Nearby Abell Cluster Survey ( the ENACS ) has yielded 5634 redshifts for galaxies in the directions of 107 rich , Southern clusters selected from the ACO catalogue ( Abell et al .
1989 ) .
By combining these data with another 1000 redshifts from the literature , of galaxies in 37 clusters , we construct a volume-limited sample of 128 R _ { ACO } \geq 1 clusters in a solid angle of 2.55 sr centered on the South Galactic Pole , out to a redshift z = 0.1 .
For a subset of 80 of these clusters we can calculate a reliable velocity dispersion , based on at least 10 ( but very often between 30 and 150 ) redshifts .

We deal with the main observational problem that hampers an unambiguous interpretation of the distribution of cluster velocity dispersions , namely the contamination by fore- and background galaxies .
We also discuss in detail the completeness of the cluster samples for which we derive the distribution of cluster velocity dispersions .
We find that a cluster sample which is complete in terms of the field-corrected richness count given in the ACO catalogue gives a result that is essentially identical to that based on a smaller and more conservative sample which is complete in terms of an intrinsic richness count that has been corrected for superposition effects .

We find that the large apparent spread in the relation between velocity dispersion and richness count ( based either on visual inspection or on machine counts ) must be largely intrinsic ; i.e .
this spread is not primarily due to measurement uncertainties .
One of the consequences of the ( very ) broad relation between cluster richness and velocity dispersion is that all samples of clusters that are defined complete with respect to richness count are unavoidably biased against low- \sigma _ { V } clusters .
For the richness limit of our sample this bias operates only for velocity dispersions less than \approx 800 km/sec .

We obtain a statistically reliable distribution of global velocity dispersions which , for velocity dispersions \sigma _ { V } \ga 800 km/s , is free from systematic errors and biases .
Above this value of \sigma _ { V } our distribution agrees very well with the most recent determination of the distribution of cluster X-ray temperatures , from which we conclude that \beta = \sigma _ { V } ^ { 2 } \mu m _ { H } / kT _ { X } \approx 1 .

The observed distribution n ( > \sigma _ { V } ) , and especially its high- \sigma _ { V } tail above \approx 800 km/s , provides a reliable and discriminative constraint on cosmological scenarios for the formation of structure .
We stress the need for model predictions that produce exactly the same information as do the observations , namely dispersions of line-of-sight velocity of galaxies within the turn-around radius and inside a cylinder rather than a sphere , for a sample of model clusters with a richness limit that mimics that of the sample of observed clusters .