The napkin rings of bone range from simple smooth bands, rather like a
giant wedding band, to those which have two concentric channels just in
from each outside edge to some rather intricate carvings for decorative
effect. None have a hole (apart from the center one which holds the
napkin of course for the literal minded). Some may in fact be ivory.
The edges tend to be rather sharp, but all have been smoothed to some
extent. The very plain ones will tend to have smoothed/rounded edges.

There are also sterling silver, plate silver and other varieties of
materials of which these things are made.

Also, the diameter of the hole through which the napkins fits is about
the same for all the specimens I've seen, but the width of the band
varies from the 0.5cm that Paul enquired about to about 4 cms.

I've seen them in antique shops in sets.

The curtain ring types should have had some wear lines from contact
with the rail, shouldn't they?

Lyle Browning