Unplug the lamp and remove the socket.
Set a multitester to RX1 and clip a probe to one end of the cable. Place the other probe against one peg and then the other. The meter should indicate continuity on only one lead.
Repeat this step for the other end of the cable. If the cord and plug fail any of the tests, rewire the fixture.
Strip the ends of the new cable and twist the strands clockwise.
At the top of the light fixture, hook the ends of the old and new wires together.
Wrap electrical tape around the connection. If it doesn't fit through the fixture, unhook one pair of wires and reattach them along with the remaining hooked pair.
Using a paring knife, carefully peel back the felt cover from the base just enough to expose the wire in the lamp base.
Insert the connection into the central tube at the top of the lamp. From the base, pull the old cable until the connection appears. Next, pull the old wire at the edge of the lamp base to bring the new wire through the channel.
Undo the connection and discard the old cable. Install a new socket and plug.
Unplug the lamp and remove the shade. Remove plugs. Test the cables for each outlet.
Pull the wires out of the sockets with long-nose pliers.
Note that one wire to each plug is connected to each of the wires of the lamp cord. Remove the electrical tape, remove the caps from the old wires and disconnect the wires. Cut and fish the new cord..
For each socket, cut a piece of lamp cord the same length as the old socket cords, then separate and strip both ends of the socket cords, as well as the top end of a new lamp cord.
Connect the neutral wire, the one with the ridge on its insulation, to the silver terminal and the other wire to the brass terminal. Place the socket on the lamp, passing the cable through the socket cap.
Reinstall the insulating boot and socket housing, pressing the housing until it clicks into place.
Twist the neutral wire of the cable together with the neutral wires of each plug and the unconnected wire of the cable with the unconnected wires of each plug.
Secure each connection with a wire cap and electrical tape.
Refold the wires at the top of the light and replace the cover.
