N.Y. County Law § 361-A
The board of elections in each county, outside of the city of New York, on or before the fifteenth day of December and not earlier than the first day of October, in each year, shall certify to the clerk of the legislative body of the county, the total amount of the expenses of such board of elections, including salaries for the preceding year, and, if the legislative body of any county shall so direct, shall certify to such clerk the portions of such expenses which under provisions of law are to be borne by any city or cities in said county and the portion thereof which is to be borne by the rest of such county and such clerk shall thereupon notify the proper local official or officials, who, in spreading upon the assessment-rolls the taxes to be levied upon the taxable property in the city or any such cities, and in the rest of the county, shall include in the amount so spread the amounts certified by the board of elections to be borne by such city or cities, respectively, and in the amount spread upon the assessment-rolls of the taxable property in the several towns or other political subdivisions of the rest of the county the amount so certified by said board of elections to be borne by such towns or political subdivisions respectively. Whenever any additional expenses either for salaries or supplies in addition to the regular county-wide primary and election expenses are incurred by a board of elections incidental to any election in any city, town or village, such board of elections shall certify to the county legislative body a detailed statement of such expenses and said county legislative body may cause the amount thereof to be levied against such city, town or village or may certify the amount thereof to such city, town or village and such city, town or village shall upon such certification, include the amount so certified in the next budget and tax levy and shall pay the same to the county.
N.Y. CountyLaw § 361-A