Whenever a vacancy for any cause whatever shall occur in the office of Mayor or Treasurer, the Council shall elect some qualified person to fill said vacancy until the next City election, and until his successor shall have been elected and qualified; and when such vacancy shall occur in the office of any appointive officer, his successor shall be appointed by the person making the original appointment, or his successor in office, as hereinbefore provided; and all elective and appointive officers of said City shall hold their respective offices until their successors are elected, or appointed, and qualified, unless sooner removed.