I really think it is a hardware problem, too, William. But I do not have the tools to check every individual component.
It first occurred when I was changing around some hard drives. In order to ghost a drive I added an extra drive without screwing it into place and left it dangling. At first I thought it had shorted, but now I don't think it did because something would have failed immediately, not on reboot. The IDE cable looked a bit iffy so I changed it for a new one.
Having finished my work on the drives I installed the two that were going to remain in that computer in their proper place.
I was advised to check the tops of capacitors on the MB for bubbling, as this is a common cause of impending failure, but all was OK.
The case has been put back on and the machine has been running for a few days and seems fine. But I am concerned about turning it off as it might not switch on again!