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N.Y. Senate Rules § R3S1, under The Temporary President.

N.Y. Senate Rules § R3S1

Section 1. The Senate shall be the judge of the elections, returns and qualifications of its own members and the Senate shall choose a Temporary President, by resolution adopted upon the vote of a majority of the members of the Senate elected, and who shall preside, or designate some other member to preside, in case of the absence from the Chamber or impeachment of the Lieutenant Governor, or when he or she shall refuse to act as President, or shall act as Governor. The Temporary President shall exercise such powers and duties as given by law or these rules during those periods as designated in the resolution and such resolution shall be effective for the 2013-2014 session. The Temporary President may not serve in such capacity longer than eight years.