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Fluorescent tube light fixtures with two ballasts are typically wired in the same manner. Individual wires of one ballast connect to the one side of the fixture, individual wires from the other ballast connect to the other side of the fixture. The common wires of each ballast are connected to the other side of the fixture from each of the individual wires.
As seen below on ballast 1, individual wires are connected to the lampholders of lamp 2 on the left side, and the common wires on the right.
On ballast 2, common wires are connected to the lampholders of lamps 1 and 3 on the left side, and the individual wires on the right.
A 3 lamp fixture with two series ballasts can be replaced with a 3 lamp parallel ballast. As seen below, connecting the 3 lamp parallel ballast to the three lamps can be a bit confusing. This is because the wire colors on the ballast and the lampholders will not match up.