Good find about Catherine FERGUSON/ADAMS/SMITH

Following you trail I see that in 1871 she has a POB as Ireland, the same as 1841, maybe she was an Army child as wasn't sure where she was born

I think her DOB is 1804 rather than 1824.
This looks like Martha's marriage which helps confirm that she is the daughter of James,
02 Apr 1854
St Mary, Lambeth
John STEVENSON
Bachelor
Baker
Father: Robert STEVENSON (decd) Forester
Martha ADAMS
Spinster
Father: James ADAMS (decd) Gunner
Both of full age, living at Brooks Street.
Witnesses: George ADAMS, Charle? ROBERTS
So who is George ADAMS? A brother, cousin, or just a coincidence?
You are trying to find out how James ADAMS connects to John ADAMS, is there a George in that family?
This link shows (amongst others) some possible children for James ADAMS and Catherine in Ireland,
Register of Marriages and Baptisms, [2 Battalion] Royal Artillery
http://discovery.nationalarchives.gov.uk/details/r/C11750956I think it means he was in the 2nd Btn at that time

If so it might help narrow down his records.
James, a child of James ADAMS of Royal Artillery and Catherine, was baptised on 5 December 1822 at Moy. Note: Born 20 November 1822.
Anne, a child of James ADAMS of Royal Artillery and Margaret, was baptised on 29 February 1824 at Charlemont. Note: Born 5 February 1824.
Mary, a child of James ADAMS of Royal Artillery and Catherine, was baptised on 28 November 1824 at Ballincollig. Note: Born 12 November 1824.
Moy and Charlemont are quite close together in County Tyrone, Northern Ireland.
Ballincollig is close to Carrigrohane in County Cork, Ireland, so I wonder if the above Mary b. 12 November 1824 belongs to this James and Catherine?
http://discovery.nationalarchives.gov.uk/details/r/C11750936James ADAMS, of Royal Artillery, and Catherine O'MEARA married on 23 November 1823 at Carrigrohane.
Daisy