The term of office of any such deputy sheriff shall expire with the term of the sheriff, unless sooner by him removed. And any such deputy shall be removed by the sheriff and another deputy appointed by him satisfactory to the president, treasurer or other executive officer of the park or place of amusement, whenever the president, treasurer or other executive officer shall make a statement in writing to the sheriff of the county that the deputy is not discharging his duties to the satisfaction of the president, treasurer or other executive officer and requesting a change of appointment.