49 Lastly, when the succession derived through family descent from Aaron, contained rather heirs of the family than sharers in his righteousness, there came, after the likeness of that Melchisedech, of whom we read in the Old Testament, the true Melchisedech, the true King of peace, the true King of righteousness, for this is the interpretation of the Name, “without father, without mother, without genealogy, having neither beginning of days nor end of life,” which also refers to the Son of God, Who in His Divine Generation had no mother, was in His Birth of the Virgin Mary without a father; begotten before the ages of the Father alone, born in this age of the Virgin alone, and certainly could have no beginning of days seeing He “was in the beginning.” And how could He have any end of life, Who is the Author of life to all? He is “the Beginning and the Ending.” But this also is referred to Him as an example, that a priest ought to be without father and without mother, since in him it is not nobility of family, but holiness of character and pre-eminence in virtue which is elected.
Source: Letters (New Advent)