New York Consolidated Laws, Uniform City Court Act - UCT § 1303. Jury trial;  how obtained;  jury fee

(a) Either party after joinder of issue may demand a trial by jury.  The demand must be made in writing and must be filed with the clerk with the notice of trial set forth in § 1301. Any other party to the action within ten days after the service of a copy of the notice of trial upon him unaccompanied by a written notice demanding a trial by jury, may serve upon the attorneys for all the other parties to the action a written notice demanding a jury trial and file a copy of such notice with the clerk within three days after service thereof.  In a summary proceeding to recover possession of real property, the demand may be made by the tenant at the time of answering or by the landlord at any time before the day of trial.

(b) Unless a demand is made and the jury fee paid as provided in section nineteen hundred eleven of this act, a jury trial is waived.

(c) The court may relieve a party from the effect of failing to comply with this section if no undue prejudice to the rights of another party would result.