Hello All,
Centre drills with small pilots will not tolerate any miss alignment between the lathe spindle centreline and the drill centreline Miss match may be due to ,wear on the bedways, wear on the underside of the tailstock body, inaccuricies in the drill chuck or its mounting or the tailstock being twisted sideways.
It is worth the trouble of making a holder for that size centre drill body.
Onto the end of a piece of free cutting mild steel bar machine a taper to match the tailstock barrel. Where it will be clear of the barrel machine a flat, say 8mm square, so you can put the taper back in the barrel in the same orientation every time. If you intend to lock the centre drill in place with a grub screw, rather than Loctite, drill and tap the hole for it now. Cut the taper off the bar leaving say 8mm clear of the tailstock barrel.
Mount the taper in the tailstock, retract the barrel as far as it will go, without ejecting the taper and clamp it. Using a boring head in the lathe spindle, or a slot drill or D bit machine a tight sliding fit hole for the centre drill body into the end of your holder. Fasten the centre drill in place.
Regards, John