The time required for a particle to slide from the cusp of a cycloid to the bottom is  \(t=\pi \sqrt{a / 2 g}\). Show that if the particle starts from rest at any point other than the cusp, it will take this same length of time to reach the bottom. The cycloid is therefore also the solution of the tautochrone, or "equal-time" problem. Hint: The energy equation for the particle speed in terms of \(y\) written must be modified to take into account the new starting condition. [The tautochrone result was known to the author Herman Melville. In the chapter called "The Try-Works" in Moby-Dick, the narrator Ishmael, on board the whaling ship Pequod, describes the great try-pots used for boiling whale blubber: "Sometimes they are polished with soapstone and sand, till they shine within like silver punchbowls. ... It was in the lefthand try-pot of the Pequod, with the soapstone diligently circling around me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time."]