N.Y. Alt. County Gov't Law § 50
A county which shall adopt the county administrator form of government shall have a county executive to be called "county administrator." The board of supervisors shall appoint the county administrator to serve during the term of office for which the members of such board then in office were elected. Successors shall be appointed for like terms. Vacancies shall be filled by appointment by the board of supervisors for the remainder of the term of office for which the members of such board then in office were elected. No member of the board of supervisors, other than the chairman shall, during the term for which he was elected, be eligible for appointment as county administrator. The board of supervisors may appoint the chairman of the board of supervisors as county administrator, who shall perform the duties of county administrator concurrently with those of chairman but without additional compensation. The county administrator may, when authorized by local law, serve as the head of one or more departments not administered by an elective official but without additional compensation. The board of supervisors shall appoint a county auditor who shall serve during the term of office for which the members of such board then in office were elected. Successors shall be appointed for like terms. In any county having an elective county auditor, the incumbent of the office shall continue in office, unless a vacancy otherwise occurs, until the expiration of the term of office for which he was elected. Vacancies in the office of county auditor shall be filled by appointment by the board of supervisors for the remainder of the term of office for which the members of such board then in office were elected. The county auditor shall audit all claims, accounts and demands which are made county charges by law and which otherwise would be audited by the board of supervisors. The county treasurer shall pay the amounts audited and certified by such auditor as if the same had been audited by the board of supervisors. The provisions of section three hundred sixty-nine of the county law shall apply to the audit and payment of claims against the county. The county administrator, subject to confirmation by the board of supervisors, shall appoint a county purchasing agent who shall serve during the term of office of the administrator. Successors shall be appointed for like terms. Vacancies in the office of county purchasing agent shall be filled by appointment in like manner for the remainder of the term. The county purchasing agent shall have and exercise all functions, powers and duties vested in or imposed upon the office under article nine of this chapter or other general law. The foregoing provisions of this section, together with other applicable provisions of this chapter, including those provisions which may be made applicable, shall constitute the structure of the county administrator form of government authorized to be adopted by this chapter.
N.Y. Alt. County Gov't Law § 50