N.H. Rev. Stat. Ann. § 659:85

If, upon the conclusion of the canvass of votes in any state general election, any person shall have received the largest number of votes for 2 or more incompatible offices, the secretary of state shall notify the candidate of that fact. Thereupon, the candidate shall, within 10 days, notify the secretary of state of which office he intends to accept. The secretary of state shall then declare that person elected to that office and declare vacant the other offices for which the person received a sufficient number of votes to win. Such a vacancy shall be filled as provided by RSA 661.