| Most white marks or rings on furniture are usually left by water or moisture penetrating through the protective finish (varnish, shellac, lacquer etc.) and then getting trapped below the finish. When this happens the finish in that area appears white and looses its transparency. This usually appears in the shape of a ring and happens on a surface like a tabletop. The ring shaped mark happens because it is usually left by a wet drinking glass or hot cup under which moisture condenses (builds up) on the protective finish that has been applied to the piece of furniture. This usually happens on older pieces where the finish has started to crack or craze. The moisture will seep through the small cracks in the finish.
There are a few methods you can use to remove water marks, and I will list them one by one.
Use A Lubricant
This first method is the easiest and less likely to damage the finish. Apply any type of lubricant oily substance (petroleum jelly, furniture wax, liquid furniture polish etc.) and then let it sit at least 8 hours. If the water ring was not too pronounced, the water (moisture) under the finish will have been replaced by the lubricant you applied.
Try Alcohol
Slightly dampen a clean cloth with some denatured alcohol and then pass it lightly over the water mark. Make sure you keep the cloth moving and just try to skim the surface. BE VERY CAREFUL. Too much alcohol will dissolve a shellac finish and could damage a lacquer or water based finish. Start with a very small amount of alcohol on the cloth and add more if needed. Make sure you watch closely what is happening. You may want to test this process on an unseen area of the piece first to make sure it will not harm the finish.
After about 30 seconds to 1 minute you should start to see the ring lighten up. Alcohol will remove most water rings, however there are exceptions.
Rubbing Out The Marks
In the worst cases you will have to physically remove the mark by using a lubricant and abrasive to rub it out. Apply some paraffin oil or mineral oil over the mark and use 0000 steel wool to rub the mark out. Be careful not to cut completely through the finish. Once the mark has been removed, the steel wool will have left the area dull. You will now have to even out the sheen in that area so it is even with the rest of the piece. To do this, once again use paraffin oil or mineral oil as a lubricant along with finer abrasives like pumice powder, or even finer, rottenstone. The finer the abrasive you use, the shiner the finish. Rubbing out the mark is the most likely to do damage to the finish and is the most time consuming. It should only be a last resort if the other methods have failed. | 
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