Why can I play iTunes-purchased songs on another computer?
Well obviously you’re gona have to get computer #2 onto the internet some how so you can authorize it. There’s no real way around it. However if you are on windows, maybe give this a try http://hymn-project.org/ It allows you to remove the DRM protection from music that you own (I stress, music that *YOU* own, as otherwise you would be doing something illegal. It of course requires your itunes login so only owner could remove the protection anyways). By doing this you can play the the music on any media player that will support the file format (or convert it from AAC to mp3 to increase compatibility), even if its not itunes, nor an iPod. (good if you got a linux computer, or perhaps an mp3 player that isn’t an ipod). Its basically a way of exercising your fair-use rights as allowed by copyright laws. (Apple just will get pissed as they don’t want you doing that as per their iTunes store agreement).