Hi andy,
Welcome to Rootschat!
I couldn't find your William and his wife Elizabeth in the 1881 Cencus.
I found another William, living in Broxburn, who was born in Kirkliston who would be round about the right age for having a son born in 1855?
Census Place: Uphall, Linlithgow, Scotland
William RARITY, age 56, Birthplace: Kirkliston, Linlithgow, Scotland
Rel: Head, Occ: Surfaceman Ry
Elizabeth RARITY, age 69, Birthplace: Abercorn, Linlithgow, Scotland
Rel: Wife
George RARITY, age 7, Birthplace: Abercorn, Linlithgow, Scotland
Rel: Grandson, Occ: Scholar
Then there's this family:
Dwelling: 12 Back Street
Census Place: Burntisland, Fife, Scotland
George RARITY, age 25, Birthplace: Kirkliston, Linlithgow, Scotland
Rel: Head Occ: Carter (Railway)
Agnes RARITY, age 21, Birthplace: Kirkliston, Linlithgow, Scotland
Rel: Wife Occ: Carter (Railway) Wife
Agnes RARITY, age 11 m, Birthplace:Burntisland, Fife, Scotland
Rel: Daur
Could this George be William's younger brother? (Son of the above William?)
And then there was a Mary . . . .
Dwelling: 74 High Street
Census Place: Burntisland, Fife, Scotland
James MORISON, Widower, age 88, Birthplace: Burntisland, Fife, Scotland
Rel: Head Occ: Retrd Baker
Grace W. MORISON, Unmarried, age 55, Birthplace: Burntisland, Fife, Scotland
Rel: Dau Occ: Housekeeper
Mary RARITY, Unmarried, age 31 , Birthplace: Kirkliston, Linlithgow, Scotland
Rel: Serv Occ: Gen Serv
Could Mary be the elder sister of George and William?
In the first family above, George Rarity age 7 is listed as being the grandson of William and Elizabeth, so, who's son is he?
It's a bit unlikely that he's the son of family number 2 (George and Agnes) as Agnes is listed as only being 21.
Is there any possibility he could be Mary's illegitimate child?
I also think it would be a bit unlikely for him to be the missing William and Elizabeth unless they had him in 1874, 6 years before they married.
Is it possible your dates are mixed up and family number one, above, is who you are looking for?
Your best option is probably to check for dates of birth and marriages on Scotlands People
www.scotlandspeople.gov.uk. It's a pay per view site but you can view or download certificates for Births 1855-1905, Marriages 1855-1930 and Deaths 1855-1955.
Good Luck!
Anne