The difference between a verbal participle and a gerund is
not always obvious, but note what is really said in each of
the following.
Do you
mind me asking a question?
Do you mind my asking a question?
In the first sentence, the queried objection is to me, as op-
posed to other members of the group, asking a question. In
the second example, the issue is whether a question may be
asked at all.