Sec. 10. (a) The appointments of members of the hospital governing board may not be made before the third Monday of January in the respective years and shall be made by the appointing board.
(b) If the county executive in any year fails to appoint one (1) of the board's members to the appointing board for the current year, the county executive who served on the appointing board in the preceding year, if the individual is still a member of the county executive, continues as a member of the appointing board until the county executive makes the board's annual appointment.