When you charge the plastic rod with the wool exciter and then charge the metal cylinder with it by stroking the rod on it or the ball electrode on top, the two pith balls resting on the outside of the cylinder fly outward, while the two pith balls resting inside the cylinder on the bottom remain there.
Shown in the photograph is a metal cylinder that is 3-1/2 inches long and two inches in diameter, mounted horizontally on an insulated stand. Protruding from the top of the cylinder is a wire with a ball electrode on top, which has two pith balls tied to it. These pith balls hang to either side of the wire and rest on the outside of the cylinder. Inside the cylinder, resting on the bottom, are two pith balls. (One is sitting exactly behind the other; the two strings to which they are tied are visible in the photograph.) When you charge the plastic rod with the wool exciter, the rod charges negative. When you now stroke the ball electrode on top of the wire, you charge the metal cylinder negative. The charge you deposit on the cylinder stays on the outer surface of the cylinder. The pith balls resting on the outside of the cylinder also become negatively charged, which results in their repulsion from the cylinder, and they fly outward and float above the cylinder. To charge the cylinder you can also stroke it directly, but if you bring the rod too close to one of the pith balls, it will fly around to the other side of the cylinder. (If you used a glass rod rubbed with silk, the rod would charge positive. You would then charge the cylinder positive, with the same result.) Since the charge on the cylinder stays on the outer surface, the two pith balls sitting on the bottom of the inside surface do not pick up any charge. There is thus no repulsion between them and the cylinder wall, and they just sit there.
This is a consequence of Gauss’ law, according to which ΦE = q/ε0, where ΦE is the total electric flux through a closed hypothetical surface (called a Gaussian surface), and q is the charge enclosed by the surface. The flux is the integral of the electric field over the entire surface, or ΦE = ∮E · dS. Putting these together, we have ∮E · dS = q/ε0. Only charge enclosed by the surface contributes to the electric field inside it. Charge outside the surface does not. If we charge the surface, all of the charge moves to the outside, and does not contribute to the electric field inside it. Since there is no charge inside the cylinder, and no electric field, the two pith balls sitting on the bottom inside the cylinder stay there. The charge distributed over the outer surface of the cylinder, however, produces an electric field outside the cylinder. The pith balls that had been resting on the outside of the cylinder became charged when you charged the cylinder, and now sit in the electric field produced by the charge on the outer surface of the cylinder. As a result, they are repelled by the cylinder and fly outward.
References:
1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, Inc., 1978), pp. 603-6.