New York Consolidated Laws, Public Officers Law - PBO § 43. Filling other vacancies

If a vacancy shall occur, otherwise than by expiration of term, with no provision of law for filling the same, if the office be elective, the governor shall appoint a person to execute the duties thereof until the vacancy shall be filled by an election.  But if the term of such officer shall expire with the calendar year in which the appointment shall be made, or if the office be appointive, the appointee shall hold for the residue of the term.