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It seems as though Thomas was the only one of the three to have survived for long. Emma Butler aged 8 was buried at Lt Stukeley on 5 Oct 1841, and James Butler aged 8 on 22 Mar 1844. The full entry might give parents, or the death certificate might do likewise.
There's also a burial of Mary Butler aged 43 on 4 Nov 1838 and I wonder if she might have been their mother. No sign of a burial of a William in Lt Stukeley though.
Huntingdon Archives online catalogue at
http://calm.cambridgeshire.gov.uk/ArchiveCatalogue/SearchArchives.htm shows admissions to Huntingdon Union Workhouse on 1 Jan 1841 of Emma Butler born 1833; James Butler born 1835, and Thomas Butler born 1837, all of Lt Stukeley.
They were also admitted on 15 Oct 1840. Possibly the full entry might contain more detail.
But there are also entries in 1840 of William Butler, farmer, 47 of Lt Stukeley who was accused of horse stealing and was sentenced to 15 years transportation on each of two convictions on 19th Oct 1840 (4 days after the children went into the workhouse) at Huntingdon County Sessions. If the mother of the three children was the Mary who died in 1838, and if their father was the William and he was transported in 1840-41, then it would explain why the children were in the workhouse. One of the entries states that William was a native of Gt Raveley, Hunts.
But the only marriage in Hunts of a William Butler and Mary was in 1813 in Abbots Ripton - William Butler of Ramsey and Mary Bond. Whilst it doesn't look too promising, there are also Butler burials in Lt Stukeley of George infant on 12 March 1828, Martha age 21 on 23 May 1837, and Elizabeth age 14 on 6 Apr 1838. If this was all one family then an 1813 marriage would be feasible.
There's a member submission on the IGI, which I never place much reliance on but this one looks better than most, which shows a burial of Elizabeth Butler on 6 Apr 1838, which is right, and a christening on 11 Dec 1825, which again looks right, where her parents are shown as William and Mary.
Little Stukeley parish register hasn't been extracted onto the IGI or BVRI other than baptisms 1803-22 which shows three children of William & Mary - Mary Furness Butler 4 Oct 1818, Ann 10 March 1822, and William also 10 March 1822. Hunts FHS has produced a transcript of Lt Stukeley parish register.
I now move on from speculation to pure fantasy!
Why did Mary baptised 1818 have the middle name Furness?
IF the 1813 marriage is the right one and Mary Bond was the one producing children in Lt Stukeley there's a marriage in 1785 in Alconbury between Robert Bond of Stukeley and Mary Furness which might be worth pursuing. Was Mary Bond the daughter of this couple? Said it was fantasy!
It would also be worth looking at transportation records to see if William went to Australia and what happened to him when he got there
David