Unfortunately, emails accessed via Firefox (or any other web-browser, such as Safari) aren't actually stored on your computer - they're stored on the email provider's servers, and as such, Time Machine has no way to back them up.
Have you ever viewed these emails using an email client program, like the Mail app found in your Applications folder, or through an email feature built into a phone / tablet of some sort? They may still have a copy stored, even if the Hotmail servers do not.
Thanks for the answer. Last night when I was falling asleep, I got the idea to use TM to find that email, and was happy about that. But this morning as I was going into TM, I realized that it most probably wouldn't work.
I'm not sure what you mean by using Mail. I don't use it at all, and am not sure how it could be linked to hotmail.
The idea is that, although emails sent to you are initially stored on your email provider's servers, some "email client" programs (eg Mail) will download copies (so they can eg let you read old emails while you're not connected to the internet) - and just because you've delete emails from Hotmail's servers, doesn't mean they've been deleted from the stores of any email clients you might've used!
If you've never used Mail with your Hotmail account, then we can eliminate that as an option - if it's never had a chance to download these emails, then we can be sure it doesn't hold copies now. But have you ever looked at the emails outside of a web browser? For example, have you ever had a phone set up to view emails?
If you've never used a non-web-based email client, then I'm afraid you're likely out of luck. Contacting Hotmail's support staff may yield results but I wouldn't bet on it.