2012 New York Consolidated Laws
PBO - Public Officers
38 - Terms of officers chosen to fill vacancies.

NY Pub Off L § 38 (2012) What's This?
    § 38. Terms of officers chosen to fill vacancies. If an appointment of
  a  person  to  fill  a  vacancy  in  an appointive office be made by the
  officer, or by the officers, body or board of  officers,  authorized  to
  make  appointment  to  the  office  for  the  full  term,  the person so
  appointed to such vacancy shall hold  office  for  the  balance  of  the
  unexpired  term.  The  term  of office of an officer appointed to fill a
  vacancy in an elective office, shall be until the  commencement  of  the
  political  year  next  succeeding  the  first  annual election after the
  happening of the  vacancy,  if  the  office  be  made  elective  by  the
  constitution,  or at which the vacancy can be filled by election, if the
  office be otherwise made elective.

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