N.H. Rev. Stat. Ann. § 4:2

Whenever the trustee of any state institution shall have been absent from 3 consecutive meetings of the board of which he is a member, the governor and council may, after investigation of the cause of such absence, in their discretion declare the office of such trustee vacant, and thereupon his commission shall be automatically revoked. The vacancy shall then be filled for the unexpired term in the same manner as by law provided for an original appointment.