Colt and UMC developed the .38 Long Centerfire in the early 1870s for use in Colt conversion revolvers and newly manufactured revolvers like the New Line, New Police, and New House models.
As you know, it was originally loaded with a heeled, outside lubricated lead bullet of approximately .375 diameter.
The .38 Long Centerfire is, for all intents and purposes, identical to the .38 Long Colt version that was adopted by the US Navy in 1889.
A few years later the US Army adopted the Colt revolver and cartridge, but with a modern style heeless bullet of .357 diameter, a style that was pretty quickly adopted by ammo manufacturers.
To allow the new version of the cartridge to be fired in the old guns, the round was loaded with a hollow base bullet that would expand into the rifling on firing. Accuracy was generally passable.
If you want to shoot your rifle today, I would suggest first doing a chamber cast to make sure that you have the chambering correct. Many of these rifles were also chambed for .38 Extra Long, and some may have been chambered for .38 Short, as well.
If you do have a .38 Long, you can trim .38 Special cases to fit.
Bullets are more difficult, as I don't know of anyone offering molds for a heeled .375 bullet. You would also need specialized loading dies.
The alternative is to use bullets such as a hollow base wadcutter. I've seen .38 Long Colt revolvers fired with HBWCs, and accuracy, while not great, was suitable.
With a gun this old, you should also stick to either black powder or pyrodex.
