The board of trustees of the Ohio state university shall elect, fix the compensation of, and remove, the president and such number of professors, teachers, and other employees as are necessary. Except as provided under division (C) of section 3335.02 of the Revised Code, no trustee, or relative of a trustee by blood or marriage, shall be eligible to a professorship or position in the university, the compensation for which is payable out of the state treasury or a university fund. The board shall fix and regulate the course of instruction and prescribe the extent and character of experiments to be made at the university.