On or before the 5th day of January next, following each regular municipal election, the members of Council shall meet at the Council chambers for the purpose of organization. At such meeting, the Council shall elect one (1) of its members thereof as President of the Council to serve until a successor shall have been elected. In the event the President of the Council ceases to be a member thereof or succeeds to the office of Mayor, the Council shall elect another member to serve as President until another President of Council is elected in the manner hereinbefore provided.