In Fig. 1, the small sphere and the hemisphere are shown in blue, and the embedded point mass in black. Two lines originating from the bottom center 0 of the hemisphere are drawn in green, and three lines passing through the center C of the small sphere are shown in red.

Figure 1 shows an instant at which the small sphere of radius r has rolled a distance s to the right of its starting point through an angle ϕ about C, so that s = rϕ. If we imagined the small sphere to be covered in paint, it would leave an arc of the same length s on the surface of the hemisphere of radius 5r which subtends an angle θ about 0, so that s = 5rθ. Equating these two expressions for s implies that
ϕ=5θ.
(1)...
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