N.H. Rev. Stat. § 38-A:4
The authority shall be composed of 5 members who shall be appointed by the board of aldermen, city council, or selectmen in such town or city as follows: One member to serve for a period of 5 years, one member to serve for a period of 4 years, one member to serve for a period of 3 years, one member to serve for a period of 2 years, and one member to serve for a period of one year. Each member shall serve for the term for which he is appointed and until his successor has been appointed and qualified. Members shall be eligible for reappointment. Beginning with the first appointment made due to expiration of term members shall be appointed to serve for a period of 5 years. Vacancies, whether occurring by expiration of term, by resignation, or for other cause shall be filled by the appropriate board of aldermen, city council, or board of selectmen. Appointment shall be for the unexpired term when made to fill vacancies created by cause other than expiration of term. If the officials who have the power and duty of appointment shall fail to fill any vacancy within 30 days following the creation thereof, then such vacancy may be filled by the remaining members of the authority.
RSA 38-A:4
1963, 278:1, eff. Sept. 1, 1963.