carbon arc lamps use a solenoid-like device for mechanical feed of one of the rods. if the arc begins to weaken because of the electrode eroding, the voltage goes up across the electrodes. the coil is in parallel with the electrodes. as the voltage goes up, the solenoid pulls in, pulling the electrodes closer together. once the voltage begins to drop off again, the solenoid stops moving at a point where equilibrium is maintained.. this keeps the arc stable, while feeding the electrodes. once the electrodes erode past a certain point, the solenoid reaches a mechanical stop, and since it can't feed anymore, the arc goes out. i have the description and a drawing of this device in an old applied physics textbook, but my flatbed scanner bit the dust a couple of weeks ago. i also seem to remember that the carbon rods have some kind of mineral in the core that makes the white light even brighter than just the carbon alone (possibly some thorium salts or other metallic salt that emits a bright spectrum when heated). when i was a teenager i played around with pieces of graphite from pencils to make a small arc lamp. it kinda-sorta worked, except i didn't have the parts for the solenoid device, so it would work for a little bit and go out.
also, a safety note: be aware that large amounts of carbon dioxide AND carbon MONOXIDE are produced by an arc lamp. these devices need plenty of ventilation. also, the light from one of these (even using pencil graphite) is very intense and contains a lot of ultraviolet light. and you should have some kind of eye protection like welder's goggles. sunglasses are NOT sufficient.