N.Y. High. Law § 83
If the estimated cost of a project in any county shall exceed the amount available in such county for such work to be provided by the state and federal government, the board of supervisors of the county, by resolution, act or ordinance, may approve of such project within such estimate and may provide that the county shall pay the amount of such excess. If the commissioner of transportation thereafter shall determine to proceed with the project, the board of supervisors shall appropriate sufficient moneys to pay the amount of such excess. Moneys to meet such appropriation may be provided from moneys raised by tax, from available moneys in the county road fund, from other available moneys or pursuant to the local finance law. Moneys so appropriated shall be paid out by the county treasurer on the written order of the commissioner of transportation. The form of any resolution, act or ordinance making such an appropriation, shall be approved by the commissioner of transportation, and immediately upon the adoption thereof the clerk of the board of supervisors shall transmit a certified copy to him. No resolution, act or ordinance thereafter adopted by such board shall amend or repeal any prior resolution, act or ordinance adopted pursuant to this section, either directly or indirectly, excepting under the advice and with the consent of the commissioner of transportation.
N.Y. High. Law § 83