There is hereby constituted a board of law examiners consisting of 5 active members of the bar each of whom shall hold office for 5 years and 1 of whom shall be appointed by the governor on nomination by the supreme court on the first day of July in each year. Vacancies on the board shall be filled in like manner for the unexpired term. The president of the board is the member of the board whose term first expires. The board shall elect a secretary annually from its own membership. The clerk of the supreme court ex-officio is the assistant secretary and treasurer of the board. If a vacancy occurs in the office of president, the board may elect a president for the unexpired term from its own membership.