I've found the documents holding the names of the convicts transported on board the ship Justitia in 1766: You need to request details for your ancestor and they then send them to you in the post.
If you go to http://www.nationalarchives.gov.uk
and key in the item reference T 1/450/96-101.
Or, try this link:http://www.nationalarchives.gov.uk/catalogue/displaycatalogu...
I hope that helps and that you find him on the register.