Brand, Louis Formerly, Department of Mathematics, University of Houston, Houston, Texas.
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The curve formed by an ideal heavy uniform string hanging freely from two points of support. The lowest point A (Fig. 1) is the vertex. The portion AP is an equilibrium under the horizontal tension H at A, the tension F directed along the tangent at P, and the weight W of AP. If the weight of the string is w per unit length and s is the arc AP, W = ws; and from the force triangle, tan ψ = ws/H = s/c, where c = H/w, is called the parameter of the catenary. Thus the catenary has the differential equation
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