The board of trustees shall consist of five members as follows: the chief of the fire department, the mayor of the city, or one member of council to be appointed by the mayor who will serve a two-year term, and three members elected in such manner and with such qualifications as council by ordinance shall prescribe and having terms of office of not more than three years. Any member is eligible to succeed himself. The board shall select from its members a president and secretary and the city treasurer is ex officio treasurer of the board and custodian of the pension fund, but has no vote as a member of the board.