N.Y. High. Law § 240
If the town board of either of such towns, after notice in writing from the town board of any other of such towns, given by the town clerk thereof, shall not within twenty days give their consent in writing to build or repair any such bridge, and shall not within a reasonable time thereafter direct, by resolution, the same to be done, the town board giving such notice may direct the town superintendent to make or repair such bridge, and then maintain an action in the name of the town, against the town which neglects or refuses to join in such making or repairing, and in such action, the plaintiffs shall be entitled to recover so much from the defendant, as the town would be liable to contribute to the same, together with costs and interest.
N.Y. High. Law § 240