N.H. Rev. Stat. Ann. § 21:34

The term "vacancy shall be filled for the unexpired term", when used in connection with appointments to the state service by the governor and council shall be construed to mean that the successor shall be commissioned for a term equal in length to the unexpired balance of the term of his predecessor and where an appointee has held over beyond the expiration date of his commission, so far as the appointment of a successor is concerned, a vacancy shall be deemed to have been created.