This testing device purported came from Charles Brush's lab. These coils seemingly performed two functions: acting as a ballast for the arc lamp, and acting as a variable strength magnetic field to adjust the gap between the carbon rod electrodes. This device could be used two ways. With a known current it could test the coil strength (in millihenrys), or for a known coil strength it could measure the current (in amperes).
Since the hotter an arc gets, the less resistive it gets, any arc lamp needs some form of ballast to prevent it from quickly becoming a short circuit.