Under the council, council-supervisor and council-administrator forms of government provided for in this chapter the county treasurer and the county auditor shall be elected. Officials serving unexpired terms when a form of government provided for in this chapter is adopted by a particular county shall continue to serve until successors are elected and qualify. Under the council-manager form the county treasurer and county auditor shall serve out their unexpired terms but shall thereafter be elected or appointed as council shall by ordinance prescribe.