If no one place has a majority of all the votes polled, as provided by section 22-203, it shall be the duty of the county board within one month after the officers elected at the first election have qualified according to law, to order a special election and give ten days’ notice thereof by posting up three notices in each precinct in said county, at which election votes shall be taken by ballot between the three highest places voted for at the first election. If no choice is made at such election, notice of another election shall be given as above provided for, to decide between the two places having the highest number of votes in the last election; and the place having the highest number of votes shall be the county seat.