Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family [Ascr ]=(Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash-Williams for any family [ufr ]. We also present a self-contained proof that if [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I, then our condition is necessary and sufficient for the family [ufr ] to have a transversal.