Max imitates Ronald Colman from The Prisoner of Zenda (1937) while posing as the King of Coronia. Don Adams was known for his impression of Colman in his stand-up act.
The scene in the train was filmed in the 'Varnish Car', the same set used on 'The Wild Wild West'. You can see Jim West's initials 'JTW' over the door.
Johnny Carson makes his second and final appearance on the series, this time as the servant who announces the King's visitors.
Traveling by train to the coronation, with Max impersonating King Charles, the Chief quizzes him on the king's immediate circle of relations. When the Chief asks him who the king's mother was, Max replies, "Queen Helena of Ruritania." Ruritania is the fictional setting for Anthony Hope's novel The Prisoner of Zenda (1937), upon which this episode is based, and whose best-known film version (cited here) stars Ronald Colman, whom Don Adams imitates throughout this episode.