There is only one Adam Bell of the right age born Jedburgh in 1851 Census:
Lanton, Jedburgh
David Bell 71 House Proprietor and Grocer born Jedburgh, Roxburghshire
Adam Bell son 46 Tailor born Wilton, Roxb.
Isabella Bell dau. 34 born Ancrum, Roxb.
Sarah Bell dau. 31 born Eckford, Roxb.
George L. Bell son 23 born Crailing, Roxb.
Adam Bell grandson 2 born Jedburgh
Thomas Rutherford grandson 12 ditto
1841 - Lanton, Jedburgh
David Bell 60 Grocer
Margaret Bell 58
David Bell 19 Tailor
George Bell 14
Jannet Fairburn 5
Adam Hardie 2
Now, I don't know if you are aware, but it was quite common in Scotland for illegitimate children to grow up with their natural fathers surname and, indeed, often with their paternal family. In 1841 there is no Adam Bell in Jedburgh, and bearing in mind that in 1851 grandson Adam Bell is 12 and with paternal grandfather David Bell, I would think that Adam 'Hardie' 2 and Adam Bell are one and the same. This would indicate his parents were not married.
I cannot find a W. Bell married to an Agnes Hardie on familysearch so quite who 'W' is I don't know.
David Bell was married to Margaret Brown but few birth entries on familysearch (perhaps more revealed on Scotlands People) for their children - looks like they married ca.1805 and can only find 3 births on familysearch:
Thomas born 15/2/1808 Crailing, Roxburghshire son of David Bell and Margaret Brown
David born 13/4/1822 Crailing, ditto
George Lily born 18/12/1827 Crailing, ditto
plus - from census' we know there were also:
Adam bc.1805 Welton, Roxb.
Isabella bc.1817 Ancrum, Roxb.
Sarah bc.1820 Eckford, Roxb.
plus a 'W'?
plus a Janet bc.1811 Bowden, Roxb.
Is it definitely a 'W' shown for his father or could it have been a badly written 'A'? I can't see the Adam Bell bc.1805 on census after 1851 (but then he may have died which we can't confirm at the moment with SP site down) and wondered whether Adam was his father (shown as unmarried 1851) and both of them possibly went to Canada?? Perhaps I'm just clutching at straws!
The 1851 census also shows at Southdean, Roxburghshire:
Thomas Sword 37 Shepherd born Southdean
Janet Sword 40 born Bowden
Margaret Sword 5
John Sword 7mos
Janet Fairbairne 15 stepdaughter
Isabella Bell 5 niece
Clearly, the stepdaughter is the Jannet 5 with David Bell in 1841, so her mother Janet is another daughter of David - Thomas Sword married Janet Bell 11/12/1846 Jedburgh. This would suggest that dau. Janet Fairburn/Fairbaine bc.1836 was also illegitimate but raised with her natural fathers surname, like Adam. So, if Isabella Bell is Adam Bell's sister Bella (as seems likely) was her mother also Agnes Hardie or possibly someone else?? Such a nuisance that SP site is down right now.
This posting turned out longer than I thought!
Annette