The mayor shall call together the heads of departments for consultation and advice upon the affairs of the city, at least once a month, and at such meetings he may call on the heads of departments for such reports as to the subject-matters under their control and management as he may deem proper, which it shall be their duty to prepare and submit at once to the mayor. Records shall be kept of such meetings; and rules and regulations shall be adopted thereat for the administration of the affairs of the city departments not inconsistent with any law or ordinance, which regulations shall prescribe a common and systematic method of ascertaining the comparative fitness of applicants for office, position and promotion, and of selecting, appointing and promoting those found to be the best fitted.