We'll arrive at the same time
is a curve with a remarkable property: if you let some beads slide down it, they all reach the bottom at the same time! Ignoring friction, that is.
Even better, this curve is just an upside-down cycloid! The cycloid
is the curve you get by rolling a wheel on a flat road and tracing the motion of a point on the rim.
This was proved by Huyghens in 1659. He also showed that the time of descent equals the time it takes for a rock to fall a distance of π/2 times the diameter of the wheel you used to make the cycloid!
Amazingly, Huyghens did all this without calculus. Later, in 1690, Jakob Bernoulli solved this problem using calculus. This was the first published paper that contains the word "integral" in its modern calculus meaning!
In the picture, each bead slides down the tautochrone, with a little arrow showing its acceleration vector. At right we see a graph of the distance they travel as a function of time.
It's quite easy to show that the tautochrone is an upside-down cycloid if you use calculus and Lagrangian mechanics - the approach to classical mechanics that says roughly this: a system will move in a way that minimizes its total action. Its action
is its kinetic energy minus its potential energy, integrated over time.
For more see:http://en.wikipedia.org/wiki/Tautochrone_curve