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The police and fire departments shall, as to their membership and component parts, remain as now constituted until the same shall be changed by action of the common council. The common council has power at all times by ordinance to determine the number of officers and members of each of said departments and the classes and grades into which they shall be divided, except that it shall not have the power to diminish the number of the members of either of said departments as now fixed. The number of officers or members of either of said departments shall not be increased without the approval of the board of estimate and apportionment. The common council may pass ordinances not inconsistent with law for the government of the police and fire departments, and regulating the powers and duties of their officers and members. The commissioner shall appoint, as vacancies in said department occur, all officers and members thereof, and classify and apportion them into grades to conform to such ordinances.