Although mounted as a tailstock the unit on the end of the bed is probably a "sliding-spindle" headstock - a mechanism used to manufacture batches of  precision components or, with a chase-screwcutting attachment, as a means of making very accurate threads

Stark No. 2 watchmakers' lathe showing the patent dates, town of manufacture and production date - though unfortunately without the year

A slightly different Stark No. 2 watchmakers' lathe. With a bed some 1/8" longer the T-Slot down the middle is slightly deeper, and the base of a different design - although the headstock and tailstock are interchangeable between the two lathes. The original finish looks to have been "blued"