Every bill duly passed by the board of aldermen and presented to the mayor and by him approved shall become an ordinance, and every bill presented as aforesaid, but returned with the mayor’s objections thereto, shall stand reconsidered. The board of aldermen shall cause the objections of the mayor to be entered at large upon the journal, and proceed at its convenience to consider the question pending, which shall be in this form: Shall the bill pass, the objections of the mayor thereto notwithstanding? The vote on this question shall be taken by ayes and nays and the names entered upon the journal, and if two-thirds of all the members-elect shall vote in the affirmative, the city clerk shall certify the fact on the roll, and the bill thus certified shall be deposited with the proper officer, and shall become an ordinance in the same manner and with like effect as if it had received the approval of the mayor. The mayor shall have power to sign or veto any ordinance passed by the board of aldermen; provided, that should he neglect or refuse to sign any ordinance and return the same with his objections, in writing, at the next regular meeting of the board of aldermen, the same shall become a law without his signature.