A marriage of first cousins or a bigamous marriage may be declared void.
Such a bigamous marriage is a true marriage in the sight of God (the necessity being proved), but it is not a true marriage in the eye of public law and custom.
His children therefore were bastards, the offspring of a bigamous union.
A bigamous marriage, by the ecclesiastical law of England, is simply void.
For example, if a woman marry B during the lifetime of her husband A, and after A's death marry C during the lifetime of B, her marriage with C is not bigamous, because her marriage with B was a nullity.