Let t' be the tension at A; then, by the given conditions,

The equation (4) may be put in a more convenient form for use, in the following manner :

Let S be the length of a uniform chain of the thickness of the chain of suspension at A, of which the weight is equal to the weight of the portion AP of the suspended chain. Then evidently

Either of the equations (4) or (5) may be considered as the

equation of the catenary of equal strength.

In order to express the tension t and area of section at Pin terms of S and a, we have

Hence by (1), the area of the section at Pis

The equation (6) gives the tension at P in terms of S and a, and (7) gives the value of the section at P in terms of the same quantities.

Obs. In order to examine the nature of the curve called the Catenary and to determine some of its properties, we have assumed as origin of coordinates a point whose position is unknown. The parameter a, therefore, which is involved in each of the equations of the Catenary, being unknown, is to be determined experimentally by means of the tension at the lowest point A.

If a known point be made the origin, the equation will then contain three parameters, viz. a and the coordinates of A, to determine which we have three conditions-the curve passes through the two points of suspension, and the distance between these points is of given length.

VARIABLE DENSITY.

138. It is proposed in this article to generalize the equations of the catenary given in the preceding articles, so as to include chains of variable density.

PROP. A chain PAQ (fig. 64), of variable thickness and density, being suspended from two given points P and Q, it is required to determine the nature and properties of the curve in which the chain will hang.

Let p be the density of the chain at P, p' its density at A, and the other notation as in Art. 132. Then the weight of an element of the chain at P is

pg wds,

and therefore the weight of the portion AP of the chain is

Let t be the tension of the chain at P and a the length of a chain such that the tension t' at A of the suspended chain is given by the equation

ß being the area of the section at A of the suspended chain.

This is the differential equation of the curve in which the

chain hangs.

Differentiating this equation with respect to x, we obtain the

which gives the law of thickness corresponding to a given form. And by (3) and the value of t' in (2), we get

This gives the tension at any point.

[For some of the following examples, the author is indebted to Price's Infinitesimal Calculus, and Latham's Treatise on Wrought Iron Bridges.]

Examples for illustration.

1. Determine the law of variation of the thickness of a heavy homogeneous string suspended from two given points, in order that it may be in equilibrium in the form of a parabola with its vertex downwards and its axis vertical.

Let be the area of the section of the string at any point P in it, and ẞ the corresponding area at the lowest point; then referring the curve to rectangular coordinates which originate at the vertex, the axis of x being horizontal, the law of the thickness of the string is comprehended in the equation (Art. 132)

313

d2 y dx2

a

(1).

Differentiating the equation (2) with respect to x, we get

Wherefore the equation (1) becomes, by substitution,

which gives the law of variation of the thickness of the string.

2. Find the equation of the catenary when the weight of each element of the curve varies as the horizontal projection of that element.

Refer the curve to rectangular coordinate axes as in Art. 132, the lowest point of the curve being taken for the origin of coordinates, and a horizontal line for the axis of x.

Then if we denote AP (fig. 64), a portion of the curve measured from A, by s, and the area of the section at P by w, the element of the curve at P will be denoted by

w ds,

and the projection of this element on the axis of x, by

w dx,

the density of the chain being uniform.

Hence, μ being a constant, we have the condition

But a being the length of a uniform chain of the same thickness as the chain of suspension at A, and of a weight equal to the tension at A, we have by Art. 132

B being the area of the section at A.

Wherefore, integrating the right-hand member of this equation,

Hence, integrating the last equation, and determining the constant from the condition that when y = 0, x = 0, we get

which is the equation of a parabola, with its axis vertical and vertex downwards.

Obs. The case which has been considered in this problem is approximately that of Suspension Bridges, in which each element of the chain bears that part of the roadway or platform corresponding to the horizontal projection of it.

EE