Looking good there Richard. Using the tailstock center to support the outboard end is always a good idea if you have the room after all it would be a shame to put a load of dents into it and possibly your good self, after all that hard polishing!
Couple of safety watch points for new players just in case and I am not implying there is anything wrong going on here but using hand held items on/against fast rotating parts carries a whole gamut of associated risks. Having said that we all do it and attending to the following will control the majority of them:
Never be tempted to wrap emery or sand paper etc. around a digit to aid hanging on to it. Hold the paper with finger tips such that if anything catches it will be whipped away harmlessly without pulling any body parts into the machine.
If you are using a file on the lathe be very careful, think about the forces and which way things will go if you slip, catch a chuck jaw etc. and make sure it has a handle on it!
Don't wear rings or have any loose clothing dangling anywhere near rotating parts
Don't whatever you do poke a finger inside a hole in the centre of a rotating workpiece! If you need to polish inside cut a piece of wooden dowel, put a slot across the end and slip your emery into the slot such that it tails around the dowel and fits easily inside the hole
I'll get down off my soapbox now!
