Mr. Sturkey, thanks for your reply. According to the Rose Hill website, is an Andrew Smith buried in Soldier's Square. But it does not list his unit. According to the site Mr. Sanders posted, http://ftp.rootsweb.com/pub/usgenweb/ga/bibb/cemeteries/rosehill.txt, there is an A.J. Smith listed from company B of the 3rd MS who died July 24th, 1864.
An Andrew J. Smith is found on the muster roll of the 3rd, companies B and A. I don't know anything more about him. I live on the Gulf Coast Coast, about three hours from Jackson, so I'm not up that way too often to check the archives. I wish I had access to his service records without having to purchase them.
It's tempting to conclude that these are the same man. I'm also wondering if A.J and N.J. Smith are the same man since they died on the same day and in the same unit. Add that to the fact that I can't find anything on N.J. Smith's burial and I'm curious as to whether the records got messed up or something.
I was mistakenly sent N.J. Smith's, Co. B, 3rd MS infantry, service records for Nimrod Smith's, Co. H, 3rd Mississippi infantry. Nimrod, apparently, doesn't have a service record and doesn't appear on any muster rolls but does have pension and headstone applications.
It's a big, confusing, mystery I wish I could solve.