My Divisumma worked, somewhat at least, straight out of the carton. With some very minor attention, it briefly performed all four functions, despite sitting unused in a cupboard somewhere in Italy for probably three decades or more. Not everybody is as lucky. There's an excellent article here (in French) about the restoration of a Tetractys, the Divisumma's big brother (with practically identical internals). The author describes all the lubricants as having gummed up to the point where he had to clean and relubricate it extensively before it would even do basic addition.
I asked a couple of people how best to lubricate the machine, and they both recommended the same thing: AeroKroil. So I sent off for some, and a few other products from the same source while I was at it. Meanwhile, my Divisumma was not working very well at all, culminating of course in the catastrophic jam. There seemed to be no lubricant present, except some dark, dried-out grease in a few places.
When the AeroKroil showed up, I squirted tiny amounts of it into every bearing I could find. What a difference! It was now much easier to turn over by hand, and seemed to run smoothly under power as well, although after The Jam I had become very reluctant to use the motor anyway. I also came across an article in Italian which, after Google translation, said "Lubrication life is entrusted to 4 types of lubricants, each optimized for the kind of friction generated in the appropriate place: dense oil, fat, low viscosity, fat infusions and grease molybdenum disulfide Molykote.". This seemed like a mistranslation - it's hard to imagine them using lard - and indeed it turns out that "grasso" is Italian for both grease and fat. However there's still no indication what the "low viscosity fat" and "fat infusions" actually are. Molykote seemed easier to track down, but there's a bewilderingly huge variety of greases under this name. Also, since it's an industrial product, most suppliers want to sell it to you in 20kg drums or packs of 10 tubes. Luckily I found this supplier. Perusing the list, the BG-20 grease seemed about the best fit, so I sent off for some of that too.
At the same time I was dealing with the machine's various other caprices and dysfunctions, such as The Battle of the Springs. Finally, the Molykote showed up. I spent an hour or two applying tiny beads of it on the end of a nickel spatula (the extremely useful result of the two weeks I spent studying Chemistry at university) to everything that slides - all the cams, the various sliding bearings, and of course the motor and its gearing. That, and plenty of AeroKroil, have made a huge difference to the smoothness of things. One little indication: if you've ever used a Divisumma, you may have noticed that at the end of every operation it makes a gentle "clunk, clunk" noise. That's the one cam that turns directly off the motor output, not through the clutch, bouncing as the motor runs on. When I first got the machine, I barely got three clunks. Now I get eight or nine - an indication of how freely it is now running.
There have been plenty of other little adventures too. For example, for a while when it multiplied, it would sometimes add two to the value of the multiplier - so 9x1 was 11, and 7x1 was 9. I figured it must have been a feature for working with sales forecasts. It's fixed now - it turns out to have been an improbable side effect of The Battle of the Springs. I'm sure you can't wait to read about it.