N.Y. County Law § 274
A county district may be extended so as to include territory not previously included within its boundaries in the same manner as hereinbefore prescribed for the original establishment of the district. If the cost of establishment of the original district and of providing the improvement therein was assessed, levied and collected pursuant to the provisions of section two hundred seventy, then the cost of the extension and of the improvements therein shall be assessed, levied and collected pursuant to the provisions of said section. If the cost of establishment of the original district and of providing of the improvement therein was assessed, levied and collected pursuant to the provisions of section two hundred seventy-one, then the cost of the extension and of the improvement therein shall be assessed, levied and collected pursuant to the provisions of said section. The cost of the extension shall include such proportion of the cost of the system of the original district as the board of supervisors shall determine.
N.Y. County Law § 274