# 1911 Encyclopædia Britannica/Catenary

The mechanical properties of the curves are treated in the article Mechanics, where various forms are illustrated. The simple catenary is shown in the figure. The cartesian equation referred to the axis and directrix is ${\displaystyle y=c\cosh(x/c)}$ or ${\displaystyle y={\tfrac {1}{2}}c(e^{x/c}+e^{-x/c})}$ ; other forms are ${\displaystyle s=c\sinh(x/c)}$ and ${\displaystyle y^{2}=c^{2}+s^{2}}$, ${\displaystyle s}$ being the arc measured from the vertex; the intrinsic equation is ${\displaystyle s=c\tan \psi }$. The radius of curvature and normal are each equal to ${\displaystyle c\sec ^{2}\psi }$.
${\displaystyle x=\surd (c^{2}-y^{2})+{\tfrac {1}{2}}c\log[\{c-\surd (c^{2}-y^{2})\}/\{c-\surd (c^{2}-y^{2})\}}$,