(b) Whenever any county, city, other than the city of New York, village or school district which is coterminous with, or partly within, or wholly within, a city having less than one hundred twenty-five thousand inhabitants according to the latest federal census provides by direct budgetary appropriation for any fiscal year for the payment in such fiscal year or in any future fiscal year or years of all or any part of the cost of an object or purpose for which a period of probable usefulness has been determined by law, the taxes required for such appropriation shall be excluded from the tax limitation prescribed by section ten of this article unless the legislature otherwise provides.