The appointments hereafter to be made of the heads of all executive departments of said city, now by law authorized to be made by the mayor thereof, shall be subject to the approval of said council, to be evidenced by resolution, which shall receive the affirmative votes of a majority of all the members thereof; failing which approval, the mayor shall again designate another person or persons to fill the position or positions, as to which the former nominations have failed of approval, until the nominations are made which are approved by council in the manner above set forth.