Thanks to the DIY and all its pictures, I removed the antenna unit from my '81 240D today and opened it up to find out if the mast is the smooth or toothed kind. It's smooth, so I ordered a new one. The mast and white plastic cable has been gone since I bought the car. The motor runs, so I hope a new mast will put me back in business.

I'm so thankful for the amazing amount of information on the forum. Without it I would never have the nerve to do work like this on the car. I'd end up paying someone to do a mediocre job. Thank you all for the amazing spirit of sharing.