From Methods of series and fluxions:
"Hence (inverting the cycloids) at the cusp K of the upper cycloid support a weight hanging by a thread at the distance KA or 2EA. As the weight vibrates, the thread will wrap itself along the parts KF and KI of the cycloid: these on either side resist its straight-line extension, compelling it, as it departs from the vertical, gradually to mold itself to their shape in its upper portion while leaving its remaining part CD below the lowest point of contact extended in a straight line. Accordingly, the weight will move in the perimeter of the lower cycloid since the thread CD is always perpendicular to it" (Whiteside 163). (see demo)