In the event of a vacancy, from any cause, in the office of mayor, the council shall appoint a person to succeed to the office of mayor for the unexpired term. In the event of a vacancy in the office of the president of the council, the office shall be filled from the membership of the council. In the event of a vacancy, from any cause, in the office of councilman, the council shall appoint a person from the district in which such vacancy occurs to succeed to the office of councilman for the unexpired term.