Vacancies occurring on any board of library trustees in a town shall be filled as provided in N.H. Rev. Stat. § 669:75. A vacancy occurring among the publicly elected members of the board of library trustees of a city library shall be filled by the city council or other appropriate appointing authority within 2 months of the notice by the remaining members of the board of trustees. The board of library trustees may recommend to the appointing authority names of persons for appointment to vacancies on expired terms. The board of library trustees may recommend to the appointing authority the names of no more than 3 persons who may serve as alternate members on the board when elected members of the board are unable to attend a board meeting. The alternate members shall be appointed to one-year terms.