The gun was reproved when sleeved in 1983. J1B in a circle is the Birmingham date stamp for that year.

The original Proof Marks are Birmingham 1904 to 1925 but almost certainly before 1921 as it does not appear to bear their date stamp introduced that year.

Nigel Brown Volume 2 lists a number of Jacksons, but most are too early or too late.

The Nottingham Jackson, Samuel Jackson, trading up to 1912 seems likely.

The original top rib is often used when re-sleeving a gun. Does it bear any clue in the form of an initial or town?

I take the stamp along the bottom rib is to identify the importer into the United States.