The terms of office of the mayor and other councilmen shall begin at 7:30 o'clock P.M. on the first Monday in May following their election. The council shall hold a meeting at that time, and those whose terms are beginning shall be inducted into office. If the mayor-elect or any other councilman-elect fails to qualify within one month thereafter, his office shall become vacant and the council shall fill the vacancy. The council shall meet regularly at least monthly at such times as it may prescribe by ordinance or otherwise. The mayor or any three councilmen may call special meetings. All meetings of the council shall be open to the public, the journal of its proceedings shall be open to public inspection.