If any vacancy occurs in the membership of the board of trustees between the annual elections, the remaining members of the board shall have power to fill such vacancies by appointment of persons having the same qualifications as themselves. The persons so appointed shall qualify in the same manner and hold office until the next annual election when their successors shall be elected. In the event that all places on the board become vacant, then a new board shall be appointed by the auditor, or if more than one county, then by the auditor of the county in which the greater acreage of the district is located. The persons so appointed shall hold office until the next annual election and until their successors are elected and qualify.