§ 13. Terms of office. The term of office of each elective officer, unless elected to fill a vacancy then existing, shall commence on the first day of January next succeeding his election. The term of office of each appointive officer shall commence on the day succeeding his appointment unless a different date is specified in the certificate of appointment. The term of office of the mayor, comptroller, treasurer and president of the common council shall be two years. The term of office of alderman and supervisor shall be two years. The term of office of the assessors shall be four years, except that at the city election first held in the city after the same shall have become a city of the second class and the provisions of this chapter relating to the election of its officers shall have become applicable thereto, two of the assessors shall be elected for a term of two years and two for a term of four years. The term of office of the corporation counsel, city engineer, commissioner of public works and sealer of weights and measures shall be two years, unless sooner removed by the mayor. The term of office of the commissioner of public welfare shall be five years. Where the term of office of an appointive officer is not specifically fixed by statute it shall be deemed to continue only during the pleasure of the officer, officers, board or body authorized to make the appointment.