§ 31. Scope of authority's jurisdiction. The territorial jurisdiction
of an authority established for a city or village shall be coterminous
with the territorial limits of such city or village, and the territorial
jurisdiction of an authority established for a town shall include all
such town, except that such territorial jurisdiction shall not include
any territory that lies within the boundaries of any village, whether
such village has or has not established an authority. The members of
such town authority shall if they consent and when authorized by
resolutions of the town board and the village board, act as the
authority in and for said village, the same as if it had been created
especially for said village.