When they arived, the Father,
Elendil was 'high King' over both
Gondor &
Arnor. His sons
Isildur & Anarion ruled Gondor for him.
Onlt Isildur remained after the end of the 2nd Age.
He took
the Ring & went to Arnor to take up the high kingship, commiting rule in the South (Gondor) to Anarion's son Melendil (I think that's his name).
Of course, Isildur's company was ambushed by
Orcs & he was slain, &
the One Ring lost.
The line of Isildur & Anarion continued, but were basically split into 2
kingdoms from that point.
The line in Gondor failed, & was ruled by a Steward 'until the King returns'; requests at that time from the
King of Arnor to be named
King of Gondor were rejected in Third Age 1975 I believe.
Soon after, the
North Kingdom was destroyed.
Read Appendix A in ROTK for all the details.
I have to fly to work or I'd go into more detail; maybe somebody with more time can.....