Bingo. You were right first time, Mack:
1901 - Third Battalion the Manchester Regiment and Married Families, North Half of West Square, Farnborough, Hants.
Arthur Buckingham, aged 34, Sergeant, Instructor of Musketry, b.Lambeth
Teresa (wife), aged 29, b.Ireland
Olivia (dau), aged 7, b.Ireland
Rose (dau), aged 1, b.Manchester
Arthur was the son of Frederick Buckingham, painter, (b.Lambeth c.1831) and his wife Jane.