(1) If any member of an authority dies, resigns, is removed, or for any other reason ceases to be a member of the authority, the mayor or the county judge, as appropriate to the governmental unit which the member represented, shall appoint another person to fill the unexpired portion of the term of the member.
(2) A member of the authority once qualified shall not be removed during his appointment except for cause by the mayor or the county judge, as appropriate to the governmental unit which the member represents, or upon such other conditions as shall be set forth in the agreement establishing the authority.