Any of said officers that may be elected or appointed to fill vacancies may qualify and enter upon the duties of their office immediately thereafter, and when elected they may hold the same during the unexpired term for which they were elected, and until their successors are elected and qualified; but if appointed, unless otherwise provided by law, they shall hold the same only until their successors are elected and qualified. Appointment to fill a vacancy in an office which is to be abolished shall extend until the date of such abolition, and only until such date.
K.S.A. 25-314