which are involved in the preceding formulae are also noteworthy. It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a chain of given length hanging between fixed points must be stationary (§ 9). The length a is called the parameter of the catenary; it determines the scale of the curve, all catenaries being geometrically similar. If weights be suspended from various points of a hanging chain, the intervening portions will form arcs of equal catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain pass over a perfectly smooth peg, the catenaries in which it hangs on the two sides, though usually of different parameters, will have the same directrix, since by (10) y is the same for both at the peg.
As an example of the use of the formulae we may determine the maximum span for a wire of given material. The condition is that the tension must not exceed the weight of a certain length λ of the wire. At the ends we shall have y = λ, or
| λ = a cosh | x | , |
| a |
(11)
and the problem is to make x a maximum for variations of a. Differentiating (11) we find that, if dx/da = 0,
| x | tanh | x | = 1. |
| a | a |
(12)
It is easily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.200); the span is therefore
2x = 2au = 2λ/sinh u = 1.326 λ,
and the length of wire is