Thank you once again everybody for the advice on how to adjust a derailleur. I am pleased to announce, that with your help, it is done.
To recap the problem I was having: When downshifting to the lowest gear, there was nothing stopping the chain from going past it and slipping off the cog. Here is how we corrected this:
The all-important tool
The derailleur. This is a rear SunTour Vx derailleur from the late 1970s. Note the two screws on the left.
A side view of the screws.
And here they are close up. Notice the letters "L" and "H" next to the screws. The "L" indicates low gear (the largest cog). The "H" indicates high gear (smallest cog). To stop the chain from going past the largest cog when downshifting into the lowest gear, tighten the "L" screw.
Here is the screw, being tightened. Conversely, if you find that the chain does not travel sufficiently to reach the largest cog when shifting to your lowest gear, you need to loosen this screw a bit. And if you are having this problem when upshifting to the highest gear, simply do the same thing to the "H" screw.
A close-up of the procedure. This takes very little time.
After the adjustment, test the derailleur: first by manually spinning the pedals as you shift, then by test-riding the bike.
Here I am, having happily shifted into the lowest gear without the chain coming off.
All done, and ready for the steepest hills. A big Thank You again to dukiebiddle, cyclemaniac, somervillain, and all the others who kindly offered advice and posted links. Your support is very much appreciated.
Some classic derailleur adjustment instructions, using more conventional tools:
. Sheldon Brown's thorough article on "derailler" adjustment
. The Bicycle Tutor's instructions and video