If you have darker brown furniture and a white ring, a dab of mayonnaise on a soft cloth might fix it for you. Just rub it in and buff with a soft dry cloth. It may take a couple of times to get it done. It worked on my very old sewing machine cabinet.
P.S. Mayo works to clean green house plant leaves also. I use a soft paper towel (NOT for African Violets though).
By Vi Johnson from Moorpark, CA
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Removing white ring water marks on polished wood. The stain is caused by trapped water underneath the surface which needs to be dried out. Cover with a plain dish-towel and then point a hairdryer over the towel 15 cm from the stain for ten to fifteen minutes.