CATENARY (from

See also:

• LAT

Lat. eaten¢, a

See also:

• CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain)  , in mathematics, the curve assumed by a uniform chain or string hanging freely between two supports . It was investigated by Galileo, who erroneously determined it to be a parabola; Jungius detected Galileo's error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Acta Erudilorum . Bernoulli also considered the cases when (1) the chain was of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre . These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others . Essais de critique generale (2nd ed.), La Logique, i. pp . 184, 190, 207-225 . Discussions, p . 577 . ' Logic, i . 83 ; cf . Bain, Ded . Log., App .

C . 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 y=c cosh (x/c) or y= 2c(ez/°+e-,/,) ; other forms are s=c sinh (x/c) and y2=c2+s2, s being the arc measured from the vertex; the intrinsic equation is s=c tan tt . The radius of curvature and normal are each equal to c sect ' . The surface formed by revolving the catenary about its directrix is named the alysseide . It is a minimal surface, i.e. the catenary solves the problem : to find a curve joining two given points, which when revolved about a line co-planar with the points traces a surface of minimum area (see VARIATIONS, CALCULUS OF) . x The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix . The cartesian equation is x=d (c'—y') +;clog[[c- d (c2_y2)}/(c+jl (c2+y2)}), and the curve has the geometrical property that the length of its tangent is constant . It is named the tractory, since a weight placed on the ground and drawn along by means of a flexible string by a person travelling in a straight line, the weight not being in this line, describes the curve in question . It is named the antifriction curve, since a pivot and step having the form of the surface generated by revolving the curve about its vertical axis wear away equally (see MECHANICS: Applied) .