I replaced the bulb once again in a troublesome fixture again. With a 10,000 hour life, It shouldn't be a couple of times a year. It came on right after I turned it on, but went out again. A rap on the housing brought it on again, another and it went off. If jumpering the photo switch and checking all the connections doesn't do the job, should I replace the ballast?

Yes,sounds like there is a short in the ballast.

A short or an open? Extra cycles from an intermittent would shorten bulb life wouldn't it?

If the lamp just did not work the ballast windings would be opened ,and the lamp would be fine, just need a new ballast

If the lamp life is shorted, the internal windings of the ballast is shorted, and not delivering the proper voltage and increased the milliamps to the lamp, thus shortening the lamp life, and now need both lamp and ballast.

Not sure what kind of fixture you have, but look into a price of a new fixture and compare it to the price of the ballast alone.You may be surprised a new fixture is cheaper. Now you get a new ballasts, housing, socket and refractor lens.