To start with, IGI (
www.familysearch.org) lists the two sons-
1. William Thomas
Henry Derrington born 1 May 1870 Co.Armagh
2. John Joseph Derrington born 18 Dec.1871 Co.Armagh
Index to civil registrations is now online (free)-
http://pilot.familysearch.org/recordsearch/start.html#p=collectionDetails;t=searchable;c=1408347Did a search and found what seems to be William's death-
William DERRINGTON death 1873 Armagh registration district volume 1 page 26 (age 35)- so born c1838.
Certificate will list name, age, marital status, occupation, cause of death, name of informant. It will not list other details like wife (unless informant), children, parents, birthdate, birthplace, place of burial.
Details about ordering N.I. certificates here-
www.rootschat.com/forum/index.php/topic,383526.0.html