It's running, all right...exactly the same way as it had been! Yesterday I took the motor out and gave it a cleaning, removing scads of old grease, and relubricating with lithium/moly. I examined all the parts, and nothing
looks amiss, so I was hopeful that the problem would now be solved. No such luck.
Kirkwood, thanks for the suggestion above. I wondered if it might be that, too. But reviewing the various speculation in this thread two nights ago, it occurred to me that the machine runs the same way whether or not the rack is engaged (one of those "duh" moments!). So I figured that it had to be some general gumminess causing the speed to vary, if not a fault in one of the gears. While cleaning the motor, I found that the smooth ellipse under the motorboard, along which the friction spring moves, was pretty gummy with the remains of something like old Three-in-One oil, and I had a flash of hope that that could be the problem -- even though that part doesn't come into play unless the reproducer has been lowered.
Anyway, I put it back together, and whaddaya know -- the motor still runs with the same darn pattern of varying speed. Back to the ol' drawing board!
The Mechanical Service and Repairs manual that David Clark reproduced for the Forum, which has been a huge help, does offer this:
If all these inspections fail to restore the motor to uniform running so that the music is no longer subject to change of pitch, the cause is very likely in the spring case. A dry central shaft may be one of the causes. The caking of graphite on the spring coils will consume power, cause the motor to fluctuate, and very probably cause jumping or pounding of the spring while the motor is running. To eliminate these complaints it is necessary to cleanse and lubricate the main spring as per directions in paragraph (9), page 7.
I'm going to have to put this machine aside for awhile, I think, in order to finish the Credenza, which I had
hoped to have done by Christmas. When I do get that done, it'll obviously be time for...
spring cleaning! 
I'm actually pretty likely to send the springs out for that, as the Edison springs are well-known to be extra-strong and extra-perilous, having famously killed 23 children and a nun when one got loose in an orphanage in Belgium in 1924.
As a side-note, I experimented last night with playing a diamond disc on the Actuelle, since I was recently given the stylus required to do so. (Yes, I moved the cone to the "vertical" position.) It played the record perfectly well, though with less volume than the Actuelle usually gives. I watched it closely under a bright light, and it
did change the appearance of the record very subtly as it played. The point on the stylus is in good condition, so I don't think it was scraping the groove, as such, and it didn't remove the record's sheen at all, it just changed it very slightly. I'm going to guess that, however light the Actuelle tone arm may be, it's still several grams heavier than an Edison stylus, which would account for some of the change. But also, the Actuelle's needle chuck sits at a fixed 45-degree angle, and it seems to me that for Edison discs, which have a softer surface than the tough old Actuelle verticals, which are in turn played with the rounder sapphire tip, that angle
can't be good. The record I used for this experiment was a nice clean copy of
Whispering Hope, by Helen Clark and good ol' Walter Van Brunt, so, y'know, nothing ventured, nothing gained, and...nothing lost. But given the evidence, I won't be playing any more Edisons on the Actuelle.
)
Is that story true about the Nun and children? Sounds like a haunted phonograph if true truly sad but I think you are joking Too bad about not getting machine running right it is probably as someone said after all the obvious is eliminated what is left is probably the answer. It sounds the spring or a bearing directly r/t the spring or a bent shaft is the culprit not a project for average guy fingers, limbs and other body parts a more important. Maybe a previous owner gaffed something up. is it possible to remove the mechanism and set it that you watch it operate to see if something revolves not in a perfect circle as it should? Just a thought.