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NGC 7644. There is nothing at Swift's nominal position. The faint galaxy two arcmin following is too faint and too small to match Swift's description.
There are two reasonable candidates for Swift's object: NGC 7643 (found by Stephan) and NGC 7651, found by Swift himself just four weeks before N7644 (on 1 Sept 1886). N7643 is exactly two degrees south of Swift's nominal position, and 22 seconds of time preceding. The description pretty well matches the galaxy. N7651 has the same declination, and is 1 minute 15 seconds of time east of N7644's position. This is a fainter object -- Swift called it "extremely faint" rather than the "very faint" he used for N7644. In neither description does he refer to nearby stars, though N7651 is noted as being "in vacancy." This almost certainly rules out a third candidate, N7580 (5 min 32 sec west, but only 1.5 arcmin north) around which Swift noted four stars when he found it four nights earlier (on 25 Sept 1886).
The object that RNGC and Wolfgang choose as N7644 is an even fainter object of lower surface brightness just over an arcmin east-northeast of N7651. Had Swift seen this, he certainly would have noted the two as a close pair -- in his 32 arcmin field, they would appear to be almost on top of one another.
In the end, I suspect that N7643 is the most likely candidate, though N7651 has the advantage of only an error in RA. However, choosing either one is speculation, so I've sprinkled question marks liberally among the several galaxies in the table.
The RNGC (Sulentic and Tifft 1973) notes that this is a galaxy. Their coded description reads EL,BM.
The Messier objects
The Bennett objects
The Caldwell list
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