Ivan wrote:
I have two adjacent ceiling fixtures holding two 40W 48" fluorescent
tubes, both controlled by a wall switch. Sometimes, they go on
properly; other times they'll go on if their glass tubes are touched;
still other times they need to be rotated slightly in their sockets.
And sometimes they won't go on at all.

What should I check?

The fixtures are at least 18 years old, but do not have the little
starter cans that older fixtures used to.

Flakey fixtures, such as you describe, are often traced to a problematic
ground connection. The ground for a florescent fixture, unlike its use for
most other electrical devices, is a necessary component of the device in
creating a flux this close to the galactic core - or something like that.