If an agreement under section 513.08 of the Revised Code provides for representation of a joint township hospital district, or of a county, on the board of hospital commissioners or board of governors of a municipal hospital, or both, the representatives of the district shall be appointed by the joint township district hospital board and shall be electors of the district; in the case of a county, the representatives of the county shall be appointed by the board of county commissioners and shall be electors of the county or of an adjacent county; and, in the case of membership on the board of governors, at least one elector, so appointed, shall be a doctor of medicine. In the case of a joint township hospital district, not more than one elector from any township included in the district shall be appointed to either board unless each township has first had an elector of that township appointed to that board.
District or county representatives appointed to a board of hospital commissioners shall serve until the duties of the board, pursuant to sections 749.06 to 749.14 of the Revised Code, have terminated.
District or county representatives on the board of governors of a municipal hospital shall be appointed in the following manner:
The joint township district hospital board or the board of county commissioners, by a majority vote of its members, may remove any appointee to either board, for good and sufficient cause, after a hearing upon written charges.
R.C. § 513.10