Hiya aghadowey,
You're not the only one to express surprise that the families named Adams mentioned in Danny's message are the only ones in Ulster. There were a few more in other towns in 1852. There may well have been many more not mentioned in either the street directory or Valuation.
1852 Directory of Belfast and the Province of Ulster
Armagh
John Clarke Adams, woollen and hat warehouse, English Street
William Adams, publican, Scotch Street
Ballymoney
Robert Adams, shuttle maker, Linenhall Street
Robert Adams, wheelwright, Linenhall Street
Note ... that may be the one and the same man.
County Cavan
C. J. Adams, magistrate, Shinan House, Shercock
Very Rev S. Adams, Dean of Cashel and magistrate, Northlands, Shercock
Monaghan
Francis Adams, woollen draper, Market Street
Francis Adams, miller, Mill Street
Note ... that may be the one and the same man.
Newtownards
William Adams, grocer, Little Francis Street
Randalstown
Hugh Adams, baker and grocer
You'll also need to have a look at Griffiths Valuations which took place 1848-64
The Valuations in Ulster took place on the following dates.....
Antrim 1861-2
Armagh 1864
Cavan 1856-7
Derry 1858-9
Donegal 1857
Down 1863-4
Fermanagh 1862
Monaghan 1858-60
Tyrone 1851
Christopher