Members of the Board shall hold office until their respective successors have been appointed and qualified. Any member may resign from his office to take effect when his successor has been appointed and has qualified. The appointing officer may remove any member of the Board appointed by him, in case of incompetency, neglect of duty, or malfeasance in office, after service on him, by registered United States mail, return requested, of a copy of the written charges against him and an opportunity to be publicly heard in person or by counsel in his own defense upon not less than 10 days notice. This same action may be taken against the City Manager of Rock Island and the City Administrator of Moline by their respective city councils. In case of failure to qualify within the time required, or of abandonment of his office, or in case of death, conviction of a felony or removal from office, his office shall become vacant. Each vacancy or position succession shall be filled for the unexpired term by appointment in like manner, as in case of expiration of the term of a member of the Board.