Thank you for your replies.
I'm not sure that there are not 2 Elizabeths involved but the evidence does stack up for Harriet Quinton Tapper in Dorset being the daughter of James Quinton who married Elizabeth Tapper in Salisbury. Harriet always maintained that she was born in Salisbury even though the IGI has her christened in Wyke Regis. Perhaps they moved there shortly after her birth. Her sister Caroline was, according to the IGI Christened at Fisherton Anger, (Salisbury).
Taking the ages given in the 1841 census Elizabeth would have been born in 1791 and Harriet in 1827 when Elizabeth was 36. For the dates given in 1851 Elizabeth would have been 39 when Harriet was born.
Assuming, and this seems very likely, that the families are the same, why did they move? One can speculate: Was James Quinton deported for an offence and when he returned the family thought it best to start a new life somewhere else? There does appear to be a large unexplained gap, 10 years, between Elizabeth's children, Caroline and Harriet.
Any further views?
Reg.