Whenever any election shall have been held for the purpose of determining whether bonds shall be issued for the purpose of acquiring a site, building and equipping such hospital and the electors of such city have declared by their votes in favor of the issuance of the same, the governing body shall proceed at once to appoint three trustees, chosen from the citizens of said city with reference to their fitness for such office, who shall constitute a board of trustees for said hospital. The said trustees shall hold their offices until the next following city election, when three trustees shall be elected and hold their offices, one for one year, one for two years and one for three years. At each subsequent city election, the trustee whose term is about to expire shall be filled by the nomination and election of hospital trustee in the same manner as other officers are elected but at no time shall more than one member of the board be a physician.