If the votes cast favored the change, but no city or town received a majority vote, the board shall thereupon enter an order directing a special election to be held for the purpose of determining the location of the county seat, which shall be held not less than thirty nor more than sixty days from the date of the order. The election shall be for the purpose of determining which of the two places receiving the highest number of votes at the preceding election shall be the county seat, and only the names of such two places shall be placed upon the ballot. On the second Monday after the election the board shall meet and canvass the votes and thereupon enter an order directing that the county seat be moved to the city or town receiving the highest number of votes.