Section 6. If, at or after the time for the mayor and aldermen to enter upon the performance of their duties, it appears that the mayor or the full number of aldermen has not been elected, such of said officers as have been elected shall issue warrants for the election of a mayor or of aldermen as the case may be. If none of said officers has been elected, the president of the common council shall issue such warrants. If no mayor has been elected, the president of the board of aldermen shall perform the duties of the office until a mayor is chosen and sworn. If there is a vacancy in the board of aldermen, in the common council, or in a city or ward office which is to be filled by an election of the people, the board of aldermen shall issue their warrant for an election to fill such vacancy at a time and place designated by them. A city officer shall, notwithstanding his removal from one ward of the city to another, continue to perform his official duties during his term of office. This section shall not be in effect in any city unless the city council accepts it or has accepted corresponding provisions of earlier laws.