In Gallifreyan politics, Presidential Resignation Day was the day on which the incumbent Lord President resigned, and another Lord President was chosen. The alternative to a presidential resignation was an election. The outgoing Lord President usually named his successor, who was often Chancellor to the leaving Lord President.
On one occasion, with the intervention of the Master and the assistance of Chancellor Goth, the current Lord President was assassinated before naming his successor (who turned out to be current Prydonian Lord Cardinal Borusa). (TV: The Deadly Assassin)