Neither the council nor any of its members shall direct or request the appointment of any person to, or his removal from, office or position by the mayor or by any of his subordinates, or in any manner take part in the appointment or removal of officers and employees in the administrative service of the city. Except for the purpose of inquiry, the council and its members shall deal with the administrative service solely through the mayor and neither the council nor any member thereof may give orders to any subordinates of the mayor, either publicly or privately.