"Her that had been the wife of Uriah"

Matthew 1.6: "David the king begat Solomon of her that had been the wife of Urias."

I was intrigued here by St. Matthew's genealogy of Christ. Why does he describe Bathsheba in this way? Why does he avoid using her name?

Matthew Henry's commentary on this passage gives a good explanation of this text:

"There are four women, and but four, named in this genealogy; two of them were originally strangers to the commonwealth of Israel, Rachab [sic; "Rahab"] a Canaanitess, and a harlot besides, and Ruth the Moabitess; for in Jesus Christ there is neither Greek nor Jew; those that are strangers and foreigners are welcome in Christ, to the citizenship of the saints. The other two were adulteresses, Tamar and Bathsheba; which was a further mark of humiliation put upon our Lord Jesus, that not only he descended from such, but that his descent from them is particularly remarked in his genealogy, and no veil drawn over it. He took upon him the likeness of sinful flesh (Rom. 8.3), and takes even great sinners, upon their repentance, into the nearest relationship to himself. Note, We ought not to upbraid people with the scandals of their ancestors; it is what they cannot help, and has been the lot of the best, even of our Master himself. David's begetting Solomon of her that had been the wife of Urias is taken notice of (says Dr. Whitby) to show that that crime of David, being repented of, was so far from hindering the promise made to him, that it pleased God by this very woman to fulfill it."