Section 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.