When a vacancy occurs in the City Council, the remaining members thereof, must, within ten (10) days, appoint a qualified person to fill the unexpired term of such vacancy; provided, however, the City Council shall not appoint more than one council member in any twelve-month period, and in the event of the occurrence of a vacancy in the City Council within a twelve-month period in which a vacancy in the City Council has been filled by the said City Council, or in the event that more than one vacancy shall occur at the same time; then in such event a special election shall be held within forty (40) days from the date the last vacancy occurred, for the purpose of electing the successor or successors. In the event that the Mayor and Mayor Pro Tem should resign at the same time, the remaining members of the City Council, or any of them, shall have the right and power to issue legal notice of such election to be so held, and to elect and appoint election officials. Should all positions on the City Council become vacant at one time, then in such event, the County Judge of Brazoria County, Texas, is hereby empowered and directed to issue notice of such special election for and on behalf of said City, and to appoint qualified persons as election officials.