When the time within which to appeal to the Supervisors, as in this Article provided for, has expired, said warrant, assessment and diagram shall be delivered to the contractor, or his agent or assigns, on demand, provided no appeal has been filed as in this Article provided for, but not until after the payment to said Director of the incidental expenses not previously paid. If an appeal to the Supervisors be taken within the time and in the manner in this Article provided for, then, upon the confirmation of a warrant, assessment and diagram, in connection with the work done, by the Supervisors, the same shall be delivered to the contractor, or his agents or assigns, on demand, after payment of the incidental expenses as provided in this Article. By virtue of said warrant so delivered said contractor, or his agents or assigns, shall be authorized to demand and receive the amounts of the several assessments made to cover the sum due for the work specified in the contract and assessment.