In case two or more counties establish a hospital jointly, the thirteen members of the board of trustees shall be chosen as provided from the county in which the institution is located and each county legislative authority of the other county or counties which contributed to the establishment of the hospital shall appoint two additional members of the board of trustees. The regular term of each of the two additional members shall be four years and until their successors are appointed and qualified. Such additional members shall be residents of the respective counties from which they are appointed and shall otherwise possess the same qualifications as other trustees. The first term of office of the persons first appointed as additional members shall be fixed by the county legislative authority of the county in which said hospital or institution is located, but shall not be for more than four years.