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CATENARY (from Lat. eaten¢, a chain)

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Originally appearing in Volume V05, Page 512 of the 1911 Encyclopedia Britannica.
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CATENARY (from See also:Lat. eaten¢, a See also:chain)  , in See also:mathematics, the See also:curve assumed by a See also:uniform See also:chain or See also:string See also:hanging freely between two supports . It was investigated by Galileo, who erroneously determined it to be a See also:parabola; Jungius detected Galileo's See also:error, but the true See also:form was not discovered until 1691, when See also:James See also:Bernoulli published it as a problem in the Acta Erudilorum . Bernoulli also considered the cases when (1) the chain was of variable See also:density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre . These curves attracted much See also:attention and were discussed by See also:John Bernoulli, See also:Leibnitz, See also:Huygens, See also:David See also:Gregory and others . Essais de critique generale (2nd ed.), La Logique, i. pp . 184, 190, 207-225 . Discussions, p . 577 . ' See also:Logic, i . 83 ; cf . See also:Bain, Ded . See also:Log., App .

C . The See also:

mechanical properties of the curves are treated in the See also:article See also:MEcHANIcs,where various forms are illustrated . The See also:simple See also:catenary is shown in the figure . The cartesian See also:equation referred to the See also: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 See also:intrinsic equation is s=c tan tt . The See also:radius of curvature and normal are each equal to c See also:sect ' . The See also: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 See also:line co-planar with the points traces a surface of minimum See also:area (see See also:VARIATIONS, CALCULUS OF) . x The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a See also: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 See also:property that the length of its tangent is See also:constant . It is named the tractory, since a See also:weight placed on the ground and See also:drawn along by means of a flexible string by a See also: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 See also:pivot and step having the form of the surface generated by revolving the curve about its See also:vertical axis See also:wear away equally (see MECHANICS: Applied) .

End of Article: CATENARY (from Lat. eaten¢, a chain)
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