If any place, which under the Municipal Corporations Act, 1835, or any Act amending it, ceased to be included in a borough or county of a town or city, was before the passing of the Municipal Corporations Act, 1835, liable to contribute to any rate for satisfying any lawful debt to which the ratepayers of that borough or county were then liable, and if after the commencement of this Act any difference arises concerning the proportion of that debt to be contributed in respect of that place, the Secretary of State, on the application of the council, or of the chairman of a public meeting of the ratepayers of the place, may appoint by writing under his hand a barrister not having any interest in the question to arbitrate between the parties, and by his award under his hand and seal to assess the proportion aforesaid, if any; and the arbitrator shall assess the costs of the arbitration, and direct by whom and in what proportion and out of what fund they shall be paid; and the rate aforesaid shall continue to be levied by warrant of the council and to be paid by the place aforesaid to the treasurer of the borough, as if the Municipal Corporations Act, 1835, or any Act amending it, or this Act, had not been passed, until the proportion aforesaid is satisfied, and no longer.