Jim: Being from Chillicothe in Ross county, I went to school with Barnharts. I once had a rifle by William the elder. He signed with a script signature. I believe William the younger usually had a stamped signature. They built rifles in Green Township near Kingston, Ohio. The Ross County Historical Society museum used to have William the elder's tools and some Barnhart rifles on display. They were on loan from the Barnhart family, and I don't know if they are still there. I sold the rifle to someone in Ohio to, at least, return it to its home state. Once on a visit to Chilli. I went to the museum to look around and found the rifles. About seven of them. The lady let me take them out of the glass case and look at them. Then she brought me some oil, and I oiled them best I could. THEN she brought me some glass cleaner to clean the glass while the case was open. I had a fun afternoon.
Being a flintlock it is probably by the elder Barnhart. There was also a Nehemiah Barnhart who built later than the Williams.
Brice