A chain or a homogeneous flexible thin wire takes a form resembling an arc of a parabola when suspended at its ends. The arc is not from a parabola but from the graph of the hyperbolic cosine function in a suitable coordinate system.

Let’s derive the equation y = y(x) of this curve, called the catenary, in its plane with x-axis horizontal and y-axis vertical. We denote the line density of the weight of the wire by σ.

In any point (x, y) of the wire, the tangent line of the curve forms an angle φ with the positive direction of x-axis. Then,

In the point, a certain Tension T of the wire acts in the direction of the tangent; it has the horizontal component T cos φ which has apparently a constant value a. Hence we may write

whence the vertical component of T is

and its differential

But this differential is the amount of the supporting force acting on an infinitesimal portion of the wire having the projection dx on the x-axis. Because of the equilibrium, this force must be equal the weight σ dx (see the arc length). Thus we obtain the differential equation

σ dx = ady′,

(1)

which allows the separation of variables:

This may be solved by using the substitution

giving

i.e.

This leads to the final solution

of the equation (1). We have denoted the constants of integration by x₀ and y₀. They determine the position of the catenary in regard to the coordinate axes. By a suitable choice of the axes and the measure units one gets the simple equation

y = a cosh

(2)

of the catenary.

Some properties of catenary

• tan φ = sinh , sin φ = tanh
• The arc length of the catenary (2) from the apex (0, a) to the point (x, y) is a sinh = .
• The radius of curvature of the catenary (2) is a cosh ², which is the same as length of the normal line of the catenary between the curve and the x-axis.
• The catenary is the catacaustic of the exponential curve reflecting the vertical rays.
• If a parabola rolls on a straight line, the focus draws a catenary.
• The involute (or evolvent) of the catenary is the tractrix.