Jackson doesn't join the Doctor, it's more like the other way around. The TARDIS materializes in 1851 with the Doctor but no companion. The Doctor has to ask a kid on the street what year it is, which makes you think either that the destination time was selected by the TARDIS rather than the Doctor, or that the Doctor has some kind of temporal dice-roller or coin-flipper he can use to pick a year at random. I suspect it's the TARDIS.
Anyway he soon meets Jackson. The interesting thing about Jackson is that he thinks he's the Doctor! This happened when a Cyberman data storage device containing information about all the real Doctors was accidently blasted into his head. So he knows he needs a TARDIS but he doesn't know what a TARDIS is. He knows he needs a sonic screwdriver and carries a regular screwdriver around with him. He also knows that he needs a companion and there he catches a break and finds Rosita Farisi played by Velile Tshabalala who is quite a good companion.
The Doctor, Jackson and Rosita proceed to wipe out a bunch of Cybermen who are, as usual, trying to take over the Earth, and the Doctor eventually gets Jackson straightened out. In return, Jackson does two things that no other companion has done, he gets the London crowd to applaud the Doctor as he lands Jackson's BLIMP, I mean his TARDIS, and he invites the Doctor to a Christmas dinner and the Doctor, for the first time in Dr Who history, accepts!Jackson Lake is played by David Morrisey (born June, 21 1964)