N.H. Rev. Stat. Ann. § 41:3

All town officers shall continue in office until the next annual meeting and until others shall be chosen or appointed and qualified in their stead, except in cases where the law otherwise directs. If a board of officers is balloted for and one or more of them are chosen the term of office of all the members of the preceding board shall cease when those who are chosen upon such balloting have qualified, as required by law.