Yes, 10.6 should be the bore diameter in mm (your measurement of .425 equals 10.79 mm)
and 60 mm the case length.
Bullet diameter between 11.15 mm (.43) and 11.5 mm (.45)

My guess is this is one of the pre-WWI blackpowder cartridges, and there were oh so many, see below.
My first guess would be a straight-walled 11,15x60R Express or LK (not the bottlenecked .43 Mauser) but I can be wrong .
I second Wayne59: proofmarks may help. And pictures of the gun are interesting, of course.
But: there were many very similar cartridges around, and proof laws back then did not require unambiguous caliber designation on the rifle.
I practical terms, you do not need to know the old name - take exact measurements from chamber cast and a lead slug, and start preparing cases from there.

Bertram in Australia and Horneber in Germany make cases for many old calibers. I do not know if they will have this particular one.
But maybe some available case can be adapted.
With your .479 base diameter in front of the rim (12.17 mm) I would check 9,3x74R, .30 R Blaser, .405 Winchester etc.
Regards, fuhrmann