[SOLVED] building old computer. psu problem... or something

I will refer to my computers as #1 and #2.

I have got a couple of Pentium 4 computers at home as my secondary computer. I recently found a better motherboard so I decided to swap the CPU's around (moving the CPU from computer #2 to computer #1).

Funny thing is, when I turn on my computer (#1), it stays on for a couple of seconds, goes through the normal POST sequence, then just shuts down and loses power.

Even funnier is that I thought it was the power supply, so I took the CPU and put it back in the other computer (#2). Now both computers are doing the same thing.

I notice something else. If I turn the computers on right after they shut off, they stay on for about 4 seconds or so, whereas if I turn on the computer after a couple minutes of waiting, they stay on for a longer time.

I suspect a PSU problem, but since this is happening to both computers, I'm not sure anymore. Is it possible that it's a CPU problem or something else? They both POST fine. They just turn off for no reason. I've tried resetting the BIOS and everything. I've even tried using a new power outlet. If it really is the power supply, I would have a hard time understanding how both of my power supplies died.

Keep in mind that computer #2 was running this morning, and computer #1 has not been used for a couple of months now.

update: I'm running my computer with no RAM at all, and it's still doing it. This is very weird.