Dear Carolyn, I am going from memory, and I cannot claim to have expertise on this family, but I am relatively sure of the following: Jaycox, Jeacocks, Giccox, etc. (with the second a common variation in North America) families of Dutchess Co. and other southern counties of New York, and of Long Island are descendants of a family unit consisting of (I believe) Thomas, William and Francis Jeacocks and their sister Catherine (called Catern) Jeacocks. They were children of Francis and Grace Jeacockes of Stratford-upon-Avon, Warwickshire, England (and locality.) If I remember rightly, Catern was baptized in nearby Bishopton in 1613. I don't believe that Francis Sr. and Grace came to America. Catern married Robert Ashman, who was first in Connecticut, possibly Massachusetts in the 1630's-1640', was one of the first settlers of Hempstead, Long Island, NY about 1655, and died in Jamaica, Long Island, NY 1683. The Betts refers to the family of Richard Betts, I believe also of Hempstead.