Section 237. When town or county expense

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.