http://www.vistax64.com/
Bill,
I did a quick search at the link above. You are not alone. There are a few answers to your computer rebooting after shutting down. There are several things to check and I leave you to looking into those since, well, the computer is at your house
Anyway, the key words I used are "computer reboots after shutting down".
It is like a small quirk I had with Vista. It would not connect to the internet upon boot. I went into power management and unchecked "shut down this device when not in use"(internet network card). I guess MS is going green
I suspect you will find your cure in the power management area of Vista. Anyway, all my questions and problems have been resolved at this site. Plus there are a few real Vista Gurus that moderate and answer.