That gets it coaxial. --Jerry
Co-axial for the last few inches, which is what matters. Most bores are bananas, at least slightly. I thread mostly for suppressors, and I don't really care what the chamber end looks like. I use the same method as Alex, to ensure that the bullet leaves the bore in (coaxial) alignment with the threads.
With a long bed and long headstock (or small headstock bore), I chuck the receiver or chamber-end of the barrel in a 4-jaw, and run the muzzle end in a steady. If needed, a cat's head can be used in the steady (dialing this in is miserable). Or, a cleanup pass can be taken on the OD while the muzzle is held in a center, and the steady can run on that. I just did this on a personal PS90 project because the barrel is hidden when installed, and I think the direct contact gives the best alignment. However, I wouldn't ship back a customer gun with a band of finish removed, unless they expressly wanted that done!!!
With a nice short, large-diameter headstock, I just run the receiver end in a spider on outboard side, and the muzzle end in a 4-jaw or inboard spider (the IB spider takes up less real estate, for shorter barrels).
Note that a wrap of copper wire is used inside the 4-jaw if it's used at either end, to allow pivoting and protect the finish.
In either case, I have the ability to dial in from both end, achieving that coaxial alignment with the machine's spindle axis. I will dial in the chamber end off OD to get it close, but then all adjustments are done with the long-reach indicator.
It really makes life easier if the muzzle-end support is co-planar (in Z) with the inboard indication point reached with the long-stem indicator. That way you can dial in that point to run concentric first, and then achieve co-axial alignment with the other end only (or at least with only a minor tweak to the muzzle end to get it perfect). If you have to work both against each other, it's a slower process.
