N.Y. Pub. Off. Law § 38

§ 38. Terms of officers chosen to fill vacancies. If an appointment of\na person to fill a vacancy in an appointive office be made by the\nofficer, or by the officers, body or board of officers, authorized to\nmake appointment to the office for the full term, the person so\nappointed to such vacancy shall hold office for the balance of the\nunexpired term. The term of office of an officer appointed to fill a\nvacancy in an elective office, shall be until the commencement of the\npolitical year next succeeding the first annual election after the\nhappening of the vacancy, if the office be made elective by the\nconstitution, or at which the vacancy can be filled by election, if the\noffice be otherwise made elective.\n