Not at home, so can't find the exact place to help you:
Google: PC Rebooting with error messages
Will give you a BSOD so you can google the error message and fix the error
start button > Run > eventvwr
check it around the times the computer shut down, it might give you a clue in there. Also generally if your computer shut down because of temperatures the next time you turn it on it'll warn you of this before asking you to press a key to boot.
If it helps there's a PC at work doing the same thing (Dell Optiplex 745) I'll let you know if I find out what's causing it.
Doesnt "normaly" your pc-speaker beep a few times in advance? (before shutting down)?
(tho you must have plugged it in haha )
maybe try cleaning the cooling blocks (or what ever its called in english)
Best way to go is vacuum cleaner.. and perhaps a toothpick to clean between the ribs.
But i also had a problem like that a while back.
some error on my HD. everytime it read that sector it crashed :S
new HD fixed it tho
I currently work in a school ( handed my notice in on monday)
Yeah, probably best you leave before little suzy works out what "taste the snake" was really all about...
Hope you're leaving for something better tho.