Whenever any county, or any county and city jointly, or two or more counties jointly, establish a hospital for the care and treatment of the indigent, sick, injured, or infirm, under the provisions of this chapter, and such hospital is completed and ready for operation, the county legislative authority of the county in which the institution is located shall appoint thirteen persons as trustees for the institution. The thirteen trustees, together with the additional trustees required by RCW
36.62.130, if any, shall constitute a board of trustees for such hospital.