I can't watch a rental???

[JLP.0]

I recently spent my first bit of money in iTunes renting a movie. I had downloaded it on to my computer, then realised I'd put it on the computer that can't properly work! I tried to go over to my other computer, which had a much more recent chipset, and it wouldn't download. It wouldn't even show up in the home sharing thing! Am I missing something, or is this by design? Either way I'm getting a very bad first impression, but hopefully somebody can help me so that I didn't waste $4 on absolutely nothing...