The county officers to be appointed must be appointed by the board of county commissioners at the following times in the year in which the county consolidated office form of government goes into effect:
Thereafter, the appointments must be made within ten days prior to the expiration of the terms of office of the officers. The failure of the board of county commissioners to make any appointment within the time prescribed does not impair its power to make the appointment subsequently for the remainder of the term of office of the officer so appointed.
N.D.C.C. § 11-08-08