Section 85. The county commissioners shall provide for the maintenance, operation and repair of the institutions under section eighty-two provided that the expenditure of money for the purposes of this section shall be limited to such amounts as may be authorized by the various advisory boards on county expenditures. The commissioners shall, immediately after July tenth of each year, apportion the balance of the cost of maintenance, operation and repair of such hospitals, including the interest paid or due on temporary notes issued therefor, for the previous fiscal year, as defined by section sixteen of chapter thirty-five, to the cities and towns in the county in accordance with their evaluation as used in assessing the county taxes, and shall issue their warrants against such cities and towns for the amount for which they are assessed. Any balance of such cost not so assessed for such year by reason of an underestimate by the commissioners shall be assessed for the next subsequent fiscal year and any excess assessed in accordance with the foregoing provision by reason of an over-estimate by them shall be deducted in determining the amount to be assessed for the next subsequent fiscal year. The county may, thirty days after a written demand for payment, recover in contract against any city or town the amount for which it is liable under this section.