Cities of the first class be, and they are hereby authorized and directed to pay the amount of deficits, as hereinabove defined, which have arisen, and which may arise, on or before the thirty-first day of December, one thousand nine hundred and thirty-eight, in equal, annual installments, during the year one thousand nine hundred and thirty-nine and succeeding years; and to provide, by appropriation, for the payment thereof during the said years. The council of said cities, in levying and fixing the tax rate, and in making appropriations, and in framing and formulating their annual financial programs or budgets, during the said years, shall not be required to provide for or include therein the amount of such deficits, as hereinabove defined, except as to two million five hundred thousand dollars ($2,500,000) for each year; and the city controller, in furnishing data for the preparation of the financial programs or budgets for said years, as required by law, shall take into account as a current liability of said cities by reason of such deficits to be discharged during each of said years, only the said two million five hundred thousand dollars ($2,500,000); and in countersigning and approving contracts and warrants, he is hereby duly authorized to approve and countersign the same, notwithstanding that no provision has been made by appropriation for the payment of the balance of the amounts of such accrued deficits; and contracts may be entered into and warrants drawn and issued by municipal authorities of such cities and duly approved by the city controller, although no appropriations have been made to discharge such deficits, except as herein provided: Provided, however, That all indebtedness, arising from writs of mandamus executions, incurred in accordance with law during the year one thousand nine hundred and thirty-nine or any subsequent year and not paid during such year, shall be included and certified as a current liability by the city controller in furnishing data¹ for the preparation of the financial programs or budgets for the next succeeding year, and shall be provided for an the budget and tax levy for such next succeeding year.