Moving time machine backup to new computer

I just bought a new iMac to replace my older one. I did not use migration assistant to transfer my files since there was some junk I didn't want on the old computer. I have all my files moved over. I am having problems with the time machine backup. I have an external firewire drive for time machine. I want to just move it to the new machine and have access to the old backups and new backups like it was the same machine. But when I configure time machine to use it it seems to not show any of the old backups. Is there a way to get it to just continue using the drive as if it was still on the same computer?

Thank you in advance for any advice. This is my second Mac so I haven't had to do a Mac to Mac migration before.