Every year after the election of the first board of supervisors, at such time and place in the county in which the district was organized as the board of supervisors may designate, and upon not less than fifteen days’ notice, unless waived in writing, such owners shall meet and elect one supervisor in the same manner as provided in section 31-306, who shall hold his office for five years and until his successor is elected and qualified. Notice may be given by personal service or by publication for two weeks in a newspaper in each county in the district and of general circulation therein, which notice shall be sufficient if it notifies the landowners of the district without naming them individually, of the time, place, and purpose of the meeting. The secretary of the board shall mail to the last-known post office address of each owner of land in the district a copy of a published notice. In case of a vacancy in any office of supervisor, the remaining supervisors may fill such vacancy until the next annual election when a successor shall be elected for the unexpired term.