§ 38. Terms of officers chosen to fill vacancies. If an appointment of a person to fill a vacancy in an appointive office be made by the officer, or by the officers, body or board of officers, authorized to make appointment to the office for the full term, the person so appointed to such vacancy shall hold office for the balance of the unexpired term. The term of office of an officer appointed to fill a vacancy in an elective office, shall be until the commencement of the political year next succeeding the first annual election after the happening of the vacancy, if the office be made elective by the constitution, or at which the vacancy can be filled by election, if the office be otherwise made elective.