Such civil service board shall, within ninety days after the initial appointment of such commissioners, classify all the officers and employees in the departments for which the merit system has been adopted, except elective officers, as provided in section 7-407; and the board may also, in its discretion, exempt from test and competition the officer responsible for the policy of any department, and one deputy, private secretary or other confidential employee of such officer. The officers and employees so classified by such board shall constitute the classified civil service, and no appointment of any such officer or employment of any such employee shall be made except in accordance with the rules hereinafter provided for.