N.Y. High. Law § 237
The towns of this state, except as otherwise herein provided, shall be liable to pay the expenses for the construction and repair of its public or free bridges constructed over streams or other waters within their bounds, and their just and equitable share of such expenses when so constructed over streams or other waters upon their boundaries, except between the counties of Westchester and New York; and when such bridges are constructed over streams or other waters forming the boundary line of towns, either in the same or adjoining counties, such towns shall be jointly liable to pay such expenses. When such bridges are constructed over streams or other waters forming the boundary line between a city of the third class and a town, such city and town shall be liable each to pay its just and equitable share of the expenses for the construction, maintenance and repair of such bridges. Except as otherwise provided by law, a city of the third class shall be deemed a town for the purposes of this article. Each of the counties of this state shall also be liable to pay for the construction, care, maintenance, preservation and repair of public bridges lawfully constructed over streams or other waters forming its boundary line, not less than one-sixth part of the expense of construction, care, maintenance, preservation and repair, and, except in a county containing a portion of the Adirondack park, the whole of such expenses of public bridges lawfully constructed or to be constructed over streams, or waterways, intersecting county roads.
N.Y. High. Law § 237