Definition:Brachistochrone

Definition
Let a point $A$ be joined by a wire to a lower point $B$.

Let the wire be allowed to be bent into whatever shape is required.

Let a bead be released at $A$ to slide down without friction to $B$.

The shape of the wire so that the bead takes least time to descend from $A$ to $B$ is called the brachistochrone.

Also see

 * Brachistochrone is Cycloid

Linguistic Note
The word brachistochrone comes from the Greek βράχιστος χρόνος (brakhistos khrónos), which means shortest time.