If the city council shall refuse to confirm any nomination, the mayor shall then within ten (10) days thereafter, nominate another person to fill the office and he may continue to nominate until his nominee is confirmed. If the mayor fails to make another nomination for the same office within ten (10) days after the rejection of a nominee, the city council shall appoint a suitable person to fill the office during the term. The affirmative vote of one half (1/2) plus one (1) of the members of the full council shall be required to confirm any nomination made by the mayor. Whenever a vacancy shall occur in an appointive office, the vacancy for the unexpired term shall be filled by appointment in the same manner as the original appointment.