Hello;

Our homes upstairs dormer bedroom has an overhead light fixture/fan that is turned on and off by a single pole switch at the top of the stairs. Two wires enter the switch box, one black and one white connected to the top and bottom terminals of the switch. The white wire is purely white , no black markings on it.
I added an additional light fixture via on wall track mounting next to the light switch as it is dark at the top of the stairs. When I connected the incoming white wire with the added light fixtures' white wire and then attached the incoming black wire and added light fixture black wire to the switch both light fixtures lit... but are blinking rapidly.
I then tried connecting the added light fixtures black and white wire to the switch along with the incoming black and white wire the 2 lights turned on and off in an opposite manner when the switch is flipped.
Can you advise

The white in the switch box is actually hot or switched hot(NO Neutra). See if you can get wiring from the existing light to the new light as it has a Neutral.
You can have three way switching so you can turn light on and off at the top and bottom of the Stairs(safer).
Just need to change switch out, and the new switch can be added anyhere you can get power to it, doesn't need wiring between the 2, you can also get a wireless switch and place anywhere. Can provide numbers if you want.

Thanks for your response Mr. Strat, There already is a separate 3 way switch for the stairs light fixture that is operable at top and bottom of the stairs. It has it's own B+W+R wiring.
The single pole switch switch is for the upstairs light fixture, but it's about 17 feet away from the switch and that's where I want the additional light in the upstairs bedroom area.
The existing upstairs light fixture switch is wall mounted and has Black and White wires entering the switch box and Black and White wires leading to the existing ceiling light; and .

Before trying to add the additional light fixture the existing lights' B+W wires are connected to the single pole switches' side terminals and the B+W wires entering the box are connected separately to the same switches 2 rear terminals

Just not sure how to wire up the additional lights' Black and White wire.

The common of one 3 way switch is always hot, the other switches common connects to the lights, the other 2 wires connected to each switch are "Travelers". You need to test off ground to see if that switch has the switched leg(to the light). And will need to connect to this and a neutral in that box, it is possible to have NO neutral in the box, in which case, connect to the existing light.

Just not sure how to wire up the additional lights' Black and White wire.
They have to be connected to the wiring in the light fixture, the one that is 17 ft away. The new fixture cannot be connected at the light switch.

They have to be connected to the wiring in the light fixture, the one that is 17 ft away. The new fixture cannot be connected at the light switch.

Okay , thanks for your post. Does this mean I can tie the B+W wire from both lights together and then to the switch with a pigtail ?

In the old fixture you should find at least two cables, each with a black and white wire. You should also find that a white wire of one cable is connected to the black wire of the other cable. The cable with the black wire connected is the cable that brings current to the ceiling box. The cable with the white wire goes to the switch. The white wire takes current to the switch and the black wire brings it back when the switch is on.
You should also find that the white wire of the lighting fixture itself is connected to the white wire of the cable bringing power to the ceiling box. That is your neutral. The black wire of the lighting will be connected to the black wire of the switch cable. Power comes to the ceiling box on the black wire of the power cable, it goes to the switch on the white wire of the switch cable. When the switch is on, power returns on the black wire to the lighting fixture. The power goes through the light bulb and comes out on the white wire of the fixture which is connected white wire of the power cable and returns to its source.
That is the basic principle of the wiring. You may have additional wires in the ceiling box but the wiring principle is the same.

To connect your new fixture, run a cable from the old fixture to the location of the new fixture. At the old fixture connect the wires of the same way the wires of the old lighting fixture are connected. White to the neutral and black to the black of the switch cable.
You should not have to make any changes at the switch.

Agreed, black to black, white to white and NO changes at the switch.
Also your ground.