The side to side motion of the steering wheel without the front wheels turning is called "backlash". The easiest thing to do is to check out your 1931 Chevrolet Repair Manual since it give instructions on how to adjust the steering box to remove some of the "backlash". However, if your steering box is badly worn then most likely the adjustments will do little to correct the problem.
By the way, if you don't have a 1931 Chevrolet Repair Manual you can get one from the Filling Station in Lebanon, Oregon.
www.fillingstation.com
