(1)  If  such  vacancy  occurs at least seventy-five days prior to the
third Tuesday of the month preceding the end  of  the  current  official
year, for the balance of such year.
  (2)  If  such  vacancy occurs less than seventy-five days prior to the
third Tuesday of the month preceding the end  of  the  current  official
year,  for  the  balance of the year and, if the term of office does not
expire at the end of such year, by further appointment to the end of the
next official year.
  4. Elective offices, which have been filled by appointment for part of
the unexpired terms thereof shall further be filled  for  the  remaining
part  of such terms by election at the general village election, if any,
regularly scheduled to be held prior to the end of the  period  of  such
appointment.  In  the  event no general village election is scheduled, a
special village election shall be held on the third Tuesday of the month
preceding the end of  the  period  of  such  appointment.  Such  special
elections  shall  be noticed, held and conducted in the same manner as a
general village election.
  5. A village trustee shall be  eligible  for  appointment  to  fill  a
vacancy  in  the  office  of  mayor.  If  a  trustee  shall  accept such
appointment, his office as trustee shall be vacant.
  6. The village clerk shall, within three days after the appointment of
a village officer, notify each person so appointed  of  his  appointment
and  the  date thereof, and that, in order to qualify: he is required to
file his oath of office with such clerk together with proof by affidavit
of his eligibility to hold such office before entering upon  the  duties
thereof;  and  if an official undertaking be required by or in pursuance
of law, that he is also required to file the same with  such  clerk  and
that  upon  his failure to do so, he will be deemed to have declined the
office. In the case of a person appointed  to  fill  a  vacancy  in  the
position of village justice or acting village justice, such notice shall
further  state  that the filing of his oath with the county clerk of the
county and with the chief administrator of the courts is also  required.
If  an  undertaking is required of a village officer after entering upon
the duties of his office, the clerk of the village shall thereupon serve
upon such officer, personally, a written notice that he is  required  to
file  such  undertaking with the clerk within ten days after the service
of the notice and that upon his failure to do so his office will  become
vacant.
Structure New York Laws
Article 3 - Officers and Elections
3-300 - Eligibility for Election or Appointment To, and Continuance in Office.
3-302 - Official Year, Terms of Office, Extension or Reduction of Terms, Biennial Elections.
3-303 - Additional Village Justices in Certain Villages.
3-304 - Changing Number of Trustees.
3-306 - Official Undertakings.
3-308 - Separate Boards of Commissioners.
3-310 - Abolition of Separate or Municipal Boards.
3-312 - Filling of Offices and Vacancies by Election or Appointment.