Ed1960,
You can, but of course you have to use a CD with the same version and edition of Windows as your original, because the Product Key is matched to the version and edition.
If "someone else's disc" is a downloaded or burned copy, Watch Out! Microsoft's research has shown that about half of all downloaded copies of XP are rife with rootkits, spyware, trojans, viruses,a nd who knows what else!
If you bought a retail copy of XP, ask Microsoft for a replacement CD. http://support.microsoft.com/kb/326246
If you bought an OEM copy of XP, your systembuilder can get you a replacement CD for free from Microsoft.
For great advice on all topics XP, visit http://www.annoyances.org/exec/forum/winxp