Every year in the same month after the time for the election of the first board of supervisors they shall call a meeting of the landowners in the district in the same manner as is provided for in § 49-19-510 and the owners of land in such district shall meet at the stated time and place and elect one supervisor therefor or, in case of their failure so to elect, the Director of the Department of Administration shall appoint such supervisor, in like manner as prescribed in § 49-19-530, who shall hold his office for three years or until his successor is elected and qualified.