Press Alt-e and choose Account Settings from the drop-down menu that appears. At the bottom of the left-hand column of the screen that opens is am "Account Actions" button. Click that then Add Mail Account.

I've tried that, and instead of seeing an add account screen it came up with Set Up Your Existing Email Address, which obviously is the email address that Thunderbird located when I first installed it. So, how do we add a second account?

If you want a second e-mail account you'll have to obtain that from an e-mail provider before you can use it in Thunderbird. HotMail/Outlook and GMail are well known a search for "Free e-mail account" will find plenty more.

I already have the second email account, I just want to add it my existing one on TB. In other words I need to have two email accounts using a single copy of TB.

In that case enter the details into the "Set Up Your Existing Email Address" screen.

Won't this override my existing account?

No. You can add as many as you wish to.

I entered the second account details, restarted TB, and saw that now the incoming account had changed to the new account. However, I could no longer see emails arriving for my first account. I changed back to my first account, restarted, and there was no sign of the second account. I checked this by creating a new mail and attempting to send from the second account, but it wasn't presented it as an option. I may have been spoilt by Outlook, that combines the two accounts in a seamless way, but I think it would be a good feature for TB.

I have two e-mail accounts connected to Thunderbird and set them up in the way I've detailed above.

I entered the second account details, restarted TB, and saw that now the incoming account had changed to the new account.

I don't honestly know what you mean by "the incoming account" as both are incoming and outgoing. Both your accounts should be showing in the folder tree to the left in a similar way.

Take my set-up as an example, screenshot below. If I want to look at my HotMail Inbox I click 'Inbox' under 'HotMail.' If I want to see my GMail one I click 'Inbox' under 'GMail.' If I want to look at a signal I've sent from Hotmail I click 'Sent' under 'Hotmail.' I can drag and drop things from folders on one account to the other as I wish.

If I'm in a Hotmail folder and click 'Write' the message will be sent from that account. If I'm in a Gmail one it'll be sent from that. I can switch to the other in the composition window by clicking the down-pointing arrow at the end of the 'From' field.

I've never used Outlook, though would still be using Outlook Express if I could.

I started again from the beginning and, as you said, I ended up with both accounts listed. I must have thought I'd created a second account when I hadn't. Thanks for your perseverence. You have been most helpful.

Glad you got there in the end.