The regular election of members of the City Council to the positions on the City Council occupied by persons whose terms are expiring shall be held on the first Saturday in May of each year and at every such election each qualified voter shall vote for not more than one candidate for each City Council position to be filled. Said election shall be ordered by the Mayor, and in the event of the failure of the Mayor to order the same, the City Council shall make such order. In the event of the inability or failure of the Mayor and the City Council to act, the election may be called by the City Secretary, and in the event of the inability of the City Secretary to act, by the County Judge of Brazoria County, Texas, and in the event of the inability of the County Judge to act, by the Governor of the State of Texas. The City Secretary shall give notice of such election by causing said notice to be published at least thirty days prior to the date of such election in the official newspaper of the City.