N.H. Rev. Stat. § 24:14-a

The commissioners may apply to the county convention for an appropriation to be made subsequent to the adoption of the annual county budget, or the convention, on its own initiative, may consider and act upon a proposed supplemental appropriation. The clerk of the convention shall deliver or mail to each member of the county convention (who will be in office on the date of the convention vote on the proposed supplemental appropriation) and to the chairperson of the board of selectmen in each town and the mayor of each city within the county and to the secretary of state a statement including the amount of the proposed supplemental appropriation and the objects for which the money is required. The convention shall schedule a public hearing on such appropriation to be held within 30 days of the mailing or delivery of said statement. Notice of the date of said hearing and the date of the convention vote on the proposed appropriation shall accompany said statement. A supplemental county appropriation shall require a vote of the county convention as provided in RSA 24:13.