x = a log (sec ψ + tan ψ), y = a sec ψ.
(8)
Eliminating ψ we obtain the Cartesian equation
| y = a cosh | x |
| a |
(9)
of the common catenary, as it is called (fig. 56). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin O of co-ordinates; viz. O is at a distance a vertically below the lowest point (ψ = 0) of the curve. The horizontal line through O is called the directrix. The relations
| s = a sinh | x | , y2 = a2 + s2, T = T0 sec ψ = wy, |
| a |
(10)
| Fig. 56. |