According to my 1953 "Editors and Engineers Handbook", the "disk" of the discone is 70% of a half-wave length of the lowest frequency you wish to monitor. It is made up of a sheet metal disk or wires horizontal to the terrain. There is NO "vertical" element. The ground radials stick out from the grounded insulator at a 45º angle. This "insulator" can be made from plexaglass or wood in a "doughnut" shape to mount in a piece of 1-3/8" fence top rail, with the top being solid and a small hole drilled to mount the horizontal disk. The center conductor of the coax connects here with a soldering lug to the underside of the disk and the bolt or screw on top with a generous supply of silicone.
The underside of this is drilled and as many as possible ground radials are added at a 45º angle. These are all bonded together with solder wick and soldered and "glued in" with silicone. The "shield" of the coax attaches here, soldered and insulated with silicone to prevent water from getting into the coax. ;-)
Even though the true discone, LQ^QKS like a horizontally-polarized antenna, it's a vertical...confusing, I know
Radio $hack's version of the "Discone", is more along the lines of a "Ground Plane" with the lower radials bent to a 45º to lower the antenna's impedance to 50 ohms. If the lower elements were left straight out, parallel to the terrain, the antenna impedance would be closer to 75 ohms.
A Discone and Ground Plane have "0" gain or less. If you live close by or don't want to pick up the pager systems, these are a wise choice. However, there ARE antennas you can build with wire and PVC that will do a better job of longer distance reception (and transmission, if you're a HAM).
http://www.hamuniverse.com/g2bcxslimjimantenna.html Be SURE to silicone the hot and ground-side leads on the coax for wx protection. ;-)