On my 400W MH F-can ballast, it indicates on the label that the 120V input lead can be used to power an auxilary incandescent lamp of 250W maximum when running on 277V.  Well, when the ballast is running on 120V, there is 277V at the unused 277V input lead. Would this auxilary supply feature work the other way around, so as to use it as sort of a step up transformer? Could this drive an inductive load, or would it be restricted to resistive only?
 
Thanks.

For short time it may work, but for longer time it would overheat the primary winding.
The reason is, then with the extinct main lamp the ballast "consume" quite large reactive power (either from the PF compensation capacitor, usually connected on some higher voltage tap on the HX ballast, or by the main inductance on the CWA one).
This power need a current, what would be rather high when supplied on the 120V tap, while become lower (or nearly none, in case it s the tap with the PFC capacitor on HX) on the higher voltage taps.
As the primary (or it's large section) is wound by the same wire as the 120V section, the supply from the higher voltage taps cause less heat dissipated in the primary winding, so the primary get a thermal  margin to deliver some extra power when acting as the step-down to 120V transformer, usually allowing the same resistive load (e.g. used for the stand-by halogen incandescent backup to cover the main HID hot reignition time) as is the main HID rated power.

For low loads it would be possible to use, when these are so small, so may be neglected in the extra dissipation they cause (let say about 10..20% of the HID rated power).
All important here is the apparent power (so product of Vrms*Arms), not the real one, as we are principally talking about a load of the power distribution system, not generation, nor consumption.

If this ballast s connected without lamp, lets assum that the 120V primary can take 400VA (ie. for the load of the lamp that's missing) then  i think that this is about what the 277V tap can supply - when limited to 400VA by the 120V coil and 650VA by the core

Not 100/5 sure

I see what you're saying. So could it work if I wanted to use it to drive, say overseas 220-240V 35W SOX gear off of the 240V input of a 4 tap 400W MH ballast, since the power consumed by the SOX gear would be really low.  Would removing the MH lamp increase the available power as a step up transformer?

There is a step-up autotransformer in any CWA ballast, the unused lamp output, however, as Medved said, the capacitor could use more power, so it should work. However, the coil could overheat (if the SOX wattage is high).

You should use the apparent power.
The 35W SOX would consume about 0.4A, what correspond to about 110VA on the 277V (you would have to use the 277V tap to compensate for the higher frequency; assume the SOX ballast is series choke), what would be just on the edge to be still neglected.

You should use the apparent power.
The 35W SOX would consume about 0.4A, what correspond to about 110VA on the 277V (you would have to use the 277V tap to compensate for the higher frequency; assume the SOX ballast is series choke), what would be just on the edge to be still neglected.

So why should the 277V be used if it's 50Hz gear on 60Hz. Would it still run the SOX lamp at correct operating parameters, or would it be slightly overdriving the lamp and possibly overheating the SOX ballast? Does 277V at 60Hz have an approximate 50Hz equivalent apparent voltage as far as the driven ballast is concerned? What would happen if a ballast rated for 220V 50Hz is run on 240V 60Hz?

I have another ballast at home that's a 4 tap, so I have 120V, 208V, 240V, and 277V available on that one.

I guess the easy solution to that is to buy a step up transformer. They're pretty cheap, but it still runs at 60Hz.

So why should the 277V be used if it's 50Hz gear on 60Hz.

For this statement I assumed the gear is based on the inductor connected in series with the lamp. What the lamp care for is the arc circuit current.
This is given by the inductor reactance, what is 2*Pi()*Frequency*Inductance. That mean, then on the same voltage if you increase the frequency by factor of 6/5, so would do the ballast impedance, so the current would would decrease by the same ratio (5/6).
To compesae for this, you have to either change the inductance (I doubt this is feasible), or increase the input voltage, so there would be factor 6/5 higher voltage across the inductance. So the closest level to that is the 277V tap.

A bit different story would be with the electronic ballast. As the first thing these do is to rectify it, the exact mains frequency become irrelevant for the ballast functionality. So with these the 240V tap would be the correct one.

Worst situation for you would be with hybrid ballast. These use (beside the input power factor correction) not only inductances, but as well the capacitors. As their design count for given component impedances, you can not use these on any different frequency then they are rated for, the compensation by different voltage is not possible there.

Ok, thanks! 

The purpose of hooking up a lamp to the 120v input when running on 277v is as a backup lamp.  I believe it would only be powered when the main bulb is not running, such as a dead bulb or during the cool down period of a hot-restrike.  Then the main bulb lights, the aux lamp will go out.

To accomplish this you'd need some relay in the ballast to disconnect the 120V input

But then it'd go off as well when the ballast is powered by 120V, disconnecting the power and causing the ballast to cycle

This can be resolved by some more relay circuits in theballast, but why do all that ? It'd be easier to just add another disconnecting 120V output (separate from the 120V input) for this, but it is not done

So i think that it does not disconnect

The purpose of hooking up a lamp to the 120v input when running on 277v is as a backup lamp.  I believe it would only be powered when the main bulb is not running, such as a dead bulb or during the cool down period of a hot-restrike.  Then the main bulb lights, the aux lamp will go out.

All taps have their voltages present all the time the ballast is powered, the control should be provided by the separate box (as Ash mentioned).