Turntable Hum


I've got an old Technics that is giving me fits...there's a slight hum present, that's there regardless of whethter the unit has a headshell attached or not, and regardless of whether it's AC power is plugged in. When I swing the arm across the platter, the hum varies from crackling and loud to nothing. It's not near a transformer or anything like that. Any ideas? I'm really perplexed by the behavior of the arm.
vanmeter
Not quite - I still need to connect a second wire to ground the tonearm, and am feeling a little nervous about getting inside to do it, so I'm hoping there'll be an obvious place to connect it once inside. I have a young daughter, so I have to do this stuff when the opportunity arises!
I got into the turntable night before last and discovered that the prior owner's installation of the Ultralink cable had resulted in some circuit board trace destruction. I can't really solder to the next point as they're too close together. It's a stand alone circuit board, meant to be a bridge between the tonearm and RCA cables, and there is a short length of wire between the two.

What would you do? I'm tempted to A) strip the wires going to the tonearm base and solder the RCAs to that or 2) remove that wire altogether and solder directly onto the tonearm base. Would option 2 be the no-brainer?