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The records of the Foundling Hospital are now in the public domain:
http://www.nationalarchives.gov.uk/nra/ ... ?ID=O57555 but there would be no record of Lady Hamilton having deposited a child there as the whole point of the Foundling Hospital was that babies were left there anonymously so that the mother could not be traced.
The story of the child Emma supposedly abandoned has never been more than a rumour, which the research of Flora Fraser, mentioned above, seems to cast even more doubt on.