Mich. Comp. Laws § 397.153
Upon the adoption of the regional library proposal, each board of supervisors shall name members to a library board, the members to be chosen from the citizens at large of each county with reference to their fitness for office. Not more than 1 member of the board of supervisors of each county shall be at any one time a member of said library board. Each county shall be entitled to 2 members on the regional library board, the members to be appointed for a term of 4 years each, except that the first members shall be appointed, 1 for 2 years and 1 for 4 years, or until their successors have been appointed. In the case of only 2 counties joining in the regional library, the library board shall consist of not more than 4 members from each county, each for a term of 4 years, except that the first members shall be appointed, 2 for 2 years and 2 for 4 years, or until their successors have been appointed. Vacancies in the board of trustees shall be filled in like manner as the original appointments. Members of the board of trustees shall receive no compensation except their actual and necessary expenses.
MCL 397.153