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How to rewire a dual/two socket lamp?

I have an old Aladdin Alacite lamp that I am rewiring and need instructions on how to rewire with TWO sockets, one in base of the lamp and the standard bulb on top. I have rewire many lamps with the standard one socket but have never tackled this type of rewire. Can anyone help me with this? Thanks so much for any help!

If you can do one socket, you can do two. The second socket is in parallel with the first, so the hots of the two sockets are connected and neutrals of the two sockets are connected. If there are 2 separate switches, then the power sides of the two switches are connected, each switch goes to the hot of its socket, and the socket neutrals are connected to the power side neutral.

The main thing is having a polarized plug, and make sure the switch is in the hot side and the screw shell of the socket is connected to the neutral. The lamp will work with either of these wrong, but it is a safety hazard. This is true for all lamps, not just with two sockets.

The correct term for a socket is a lampholder.

If you need more help, ask a more specific question.