N.Y. Gen. City Law § 41
The term of office of each member of such board shall be three years, from the first day of January following his appointment. Vacancies occurring by expiration of a term shall be filled by the mayor for a full term. Vacancies by death, removal, inability to act, resignation or removal from the city of any member shall be filled by him for the unexpired term. The chief inspector of plumbing and drainage and the engineer in charge of sewers or the officers holding equivalent positions or acting in like capacities designated or appointed by the mayor as herein provided, shall be ex officio members of such examining board, and when they shall cease to hold their offices by reason or on account of which they were so designated or appointed, their successors shall act on the examining board in their stead.
N.Y. Gen. City Law § 41