Except when otherwise specially provided, all state and district officers shall qualify on the first day of January next succeeding their election and on said first day of January shall enter upon the discharge of the duties of their respective offices, provided that when a person is elected to fill an unexpired term in a district office then vacant or then held by an appointee, such person may qualify and enter upon the discharge of the duties of such office at any time after receiving a certificate of election to that office but not later than the first day of January next succeeding the date of election to the unexpired term of office.