It is true the official name was 18th Regiment of Foot until the reforms but it was frequently called the Royal Irish long before that as indicated in the name of the work mentioned in this thread:
Historical Record of the Eighteenth, the Royal Irish Regiment of Foot; Containing an Account of the Formation of the Regiment in 1684, and of its Subsequent Services to 1848, Compiled by Richard Cannon, Esq., Adjutant-general's Office, Horse Guards, London
Which was published in 1848. This book, with some transcription errors and minus plates, is available on line at :
http://www.archive.org/stream/recordofeighteen00canniala/recordofeighteen00canniala_djvu.txtThere is a 1922 history of the regiment available also called The Campaigns and History of the Royal Irish Regiment From 1684 to 1902 by Lieutenant-Colonel G. Le M. Gretton. It has recently been reprinted and is available through Chapters and Amazon.
Our 18th connection is much later than that being discussed here. My great-great-great-grandfather William Doheny and his son in law Michael Kelly were both in the Regiment as were family members in the First World War. These books helped me to reconnect with this part of our family heritage.
God bless
John Paul Bradford