The term of office of the members of the county hospital board shall be three (3) years and until a successor is appointed and qualified: Provided, that of the first board appointed one-third (1/3) of the membership, or as nearly one-third (1/3) as the number to be appointed shall permit, shall be appointed for the term of one (1) year; one-third (1/3) of the membership, or as nearly one-third (1/3) as the number to be appointed shall permit, for the term of two (2) years; and one-third (1/3) of the membership, or as nearly one-third (1/3) as the number to be appointed shall permit, for the term of three (3) years. All succeeding appointments shall be for the term of three (3) years, except that any vacancy occurring by reason of death, resignation or other cause shall be for the remainder of the term of the member whose office becomes vacant.