182. A STRING is said to be perfectly flexible when any force, however small, which is applied otherwise than along the direction of the string will change its form. For shortness, we use the word flexible as equivalent to perfectly flexible.

If a flexible string be kept in equilibrium by two forces, one at each end, we assume as self-evident that those forces must be equal and act in opposite directions, so that the string assumes the form of a straight line in the direction of the forces. In this case the tension of the string is measured by the force applied at one end.

A

B C

Let ABC represent a string kept in equilibrium by a force T at one end A and an equal force T at the other end C acting in opposite directions along the line AC. Since any portion AB of the string is in equilibrium it follows that a force T must act on AB at B from B towards C in order to balance the force acting at A; and similarly, T' must act on BC from B towards A in order that BC may be in equilibrium. This result is expressed by saying that the tension of the string is the same throughout.

Unless the contrary be expressed, a string is supposed inextensible and a transverse section of it is supposed to be a curve every chord of which is indefinitely small.

183. When a flexible string is acted on by other forces besides one at each end it may in equilibrium assume a curvilinear form. If at any point of the curve we suppose a section made by a plane perpendicular to the tangent, the mutual action of the portions on opposite sides of this plane must be in the direction of the tangent, or else equilibrium would not hold, since the string is perfectly flexible.

184. PROP. A heavy string of uniform density and thickness is suspended from two given points; required to find the equation to the curve in which the string hangs when it is in equilibrium.

Let A, B be the fixed points to which the ends are attached; the string will rest in a vertical plane passing through A and B, because there is no reason why it should deviate to one side rather than the other of this ver- B tical plane. Let ACB be the form it assumes, C being the lowest point; take this as the origin of coordinates; let P be any point in

M.

T

the curve; CM, which is vertical, = y; MP, which is horizontal,= x; CP= 8.

The equilibrium of any portion CP will not be disturbed if we suppose it to become rigid. Let c and t be the lengths of portions of the chain of which the weights equal the tensions at C and P. Then CP is a rigid body acted on

by three forces which are proportional to c, s, and t, and act respectively, horizontally, vertically, and along the tangent at P. Draw PT the tangent at P meeting the axis of y in T; then the forces holding CP in equilibrium have their directions parallel to the sides of the triangle PMT, and therefore bear the same proportion one to another that these sides do (see Art. 19); therefore

the constant added being such that y=0 when s=0; therefore

the constant being chosen so that x and y vanish together.

Any one of these five equations may be taken as the equation to the curve. If in equation (4) we write y' for y+c, which amounts to moving the origin to a point vertically below the lowest point of the curve at a distance c from it, we have

When the chain is uniform in density and thickness, as in the present instance, the curve is called the common catenary.

185. PROP. To find the tension of the chain at any point. Let the tension at P be equal to the weight of a length t of the chain; then, as shewn in the last article,

But s2 = y2+2yc by equation (2) of Art. (184), therefore

This shews that the tension at any point is the weight of a portion of string whose length is the ordinate at that point, the origin being at a distance c below the lowest point.

Hence, if a uniform chain hang freely over any two points, the extremities of the chain will lie in the same horizontal line when the chain is in equilibrium.

186. To determine the constant c, the points of suspension and the length of the string

OM=a, ON=a', MA=b, NB=b, CA-1, CB=l'.

then h, k, λ are known quantities, since the length of the string and the positions of its ends are given. From Art. (184)

Equations (1) and (2) are theoretically sufficient to enable us to eliminate a, a, b, b', l, and l' and to determine c. We may deduce from them

This is the equation from which c is to be found, but on account of its transcendental form it could only be solved by approximation. If the exponents of e are small, we may expand by the exponential theorem and thus obtain the approximate value of c. In order that the exponents may be small, c must be large compared with h; since

dy

ds √(c2 + s2)