Advanced book on Mathematics Olympiad

Geometry and Trigonometry 635

On the other hand, for two infinitesimally close points, the difference in the vertical
tension is given bydV=ρds, whereρis the density of the chain anddsis the length of
the arc between the two poins. Sinceds=

√

1 +(y′(x))^2 dx, it follows thatysatisfies
the differential equation

Hy′′=ρ

√

1 +(y′)^2.

If we setz(x)=y′(x), we obtain the separable first-order equation

Hz′=ρ

√

1 +z^2.

By integration, we obtainz=sinh(Hρx+C 1 ). The answer to the problem is therefore

y(x)=

H

ρ

cosh

(ρ

H

+C 1

)

+C 2.

Remark.Galileo claimed that the curve was a parabola, but this was later proved to be
false. The correct equation was derived by G.W. Leibniz, Ch. Huygens, and Johann
Bernoulli. The curve is called a “catenary’’ and plays an important role in the theory of
minimal surfaces.

626.An edge adjacent to the main diagonal describes a cone. For an edge not adjacent
to the main diagonal, consider an orthogonal system of coordinates such that the rotation
axis is thez-axis and, in its original position, the edge is parallel to they-plane (Figure 83).
In the appropriate scale, the line of support of the edge isy=1,z=

√

3 x.

x y

z

O

Figure 83

The locus of points on the surface of revolution is given in parametric form by

(x,y,z)=(tcosθ+sinθ,cosθ−tsinθ,

√

3 t), t∈R,θ∈[ 0 , 2 π).