As with virtually any mathematical or scientific problem, a first good step towards finding the solution is to draw a diagram of the situation under consideration.  Consider the diagram below.  In this diagram, we assume that the wire AB lies in the vertical x-y plane.

The diagram shows a bead sliding without friction on a curve AB, relative to a suitably chosen coordinate system.  Now, in the article on Snell’s Law (see the link on the left hand side of the page), we derived an important rule that holds for a curve that minimises the travel time between any two points on the curve.  This is illustrated in the following diagram:

If the curve shown here is the curve that minimises the travel time between any two points on the curve, then the following condition must hold: 

To begin, we need to find the velocity v.  To this end, we note that the bead is sliding without friction on the wire.  Therefore, there are no energy losses during the motion of the bead, and energy is conserved.  In the first diagram above, we assume that the bead starts at point A.  At the position of the bead in the diagram, it has lost a potential energy mgy relative to its starting point A.  By conservation of energy, we must have 

Next, we need to express  in terms of the coordinates x and y.  This is done as shown in the light blue inset in the first diagram above, and is just application of simple geometry and the definition of the sine function.  Thus:

We now have all the information we need.  Combining these four equations and simplifying leads to the following differential equation: 

where c is the constant on the right-hand side of the first equation above. 

To solve this differential equation, we make a variable change: 

Substituting into the previous equation then yields 

This is easily integrated, noting that we wish our curve to pass through the origin for some value of ϕ, which we will take to be zero.  We find that the equation of our brachistochrone curve can be written in the following parametric form: 

This is the solution that we seek – the form of the curve that minimises the travel time between our two points A and B.  These expressions for x  and y define a curve that is known as a cycloid.  An example of a cycloid is shown in the following diagram, with the constant c = 1. 

The cycloid has another interesting property that is known as the tautochrone property.  Considering the diagram above, it can be shown that the time required for the bead to slide from any starting point to the bottom of the curve is independent of the starting point.  That is, wherever we start the bead from, it always takes the same amount of time to reach the bottom.  The time required to reach the bottom can be shown to be 

where g is the acceleration due to gravity.  To derive this, we note that the travel time for a small arc length ds along the curve is equal to 

We then express ds in terms of x and y by noting that (see first diagram) 

Now, we use the definition of the cycloid to express dt in terms of ϕ and integration of dt from any start point to the bottom gives the tautochrone result, namely that the travel time is independent of the start point and given by the expression for T above.