I've always checked tail stock alignment by mounting a bar, turning two areas (one near the chuck and one near the tail stock) and comparing them,
I was aware of the method using a 'square cylinder' but, understanding these to be very costly, never obtained one.
While listening on the (amateur) radio this evening I heard another method which sounds quicker than the turning method. I should have 'called into the QSO (contact) but didn't and I'm now curious.
As I heard it, a bit of bar is mounted in the chuck. The bar has a bracket to hold a dial gauge (a 'clock' in conversation) so the tip can rest on either the quill or centre in the quill.
The dial gauge is 'zeroed' at a 12 o'clock point on, say, the centre. The chuck is rotated by hand through 180 and the run out calculated. Repeat at 3 o'clock and 9 o'clock. Adjust the tail stock and repeat.
I'm pretty sure my quill has a groove at 6 o'clock so I'd have to use a centre. I'm not sure if all quills have grooves.
I'm looked on the 'net and sure enough it is there, including on YouTube. However, that doesn't always mean it is proven.
This seems to be a beautifully simple technique. So much so, I'm wondering if I've missed something.