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See also: mathematics, the See also: curve assumed by a See also: uniform 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 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, Leibnitz, 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
.
' Logic, i
.
83 ; cf
.
Bain, Ded
.
Log., App
.
C . The See also: mechanical properties of the curves are treated in the article
See also: MEcHANIcs,where various forms are illustrated
.
The See also: simple See also: catenary
is shown in the figure
.
The cartesian 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 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 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 vertical axis See also: wear away equally (see MECHANICS: Applied)
.
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