which is the equation of the parabola in question. The result might of course have been inferred from the theory of the parabolic funicular in § 2.
Finally, we may refer to the catenary of uniform strength, where the cross-section of the wire (or cable) is supposed to vary as the tension. Hence w, the weight per foot, varies as T, and we may write T = wλ, where λ is a constant length. Resolving along the normal the forces on an element δs, we find Tδψ = wδs cos ψ, whence
| ρ = | ds | = λ sec ψ. |
| dψ |
(15)
From this we derive
| x = λψ, y = λ log sec | x | , |
| λ |
(16)
where the directions of x and y are horizontal and vertical, and the origin is taken at the lowest point. The curve (fig. 58) has two vertical asymptotes x = ± 1⁄2πλ; this shows that however the thickness of a cable be adjusted there is a limit πλ to the horizontal span, where λ depends on the tensile strength of the material. For a uniform catenary the limit was found above to be 1.326λ.
| Fig. 58. |
For investigations relating to the equilibrium of a string in three dimensions we must refer to the textbooks. In the case of a string stretched over a smooth surface, but in other respects free from extraneous force, the tensions at the ends of a small element δs must be balanced by the normal reaction of the surface. It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.e. the curve must be a “geodesic,” and that the normal pressure per unit length must vary as the principal curvature of the curve.