Tat the other, p the pressure of the rope on the cylinder per unit of length.

0

Then p.4B=2Tsin=Te approximately. Also μp.AB=T—T when the rope is just about to slip, i. e.

Hence, for equal small deflections, 0, of the rope, the tension increases in the geometrical ratio (1+μ0): 1; and thus by a common theorem (compound interest payable every instant) we have T= Тo, if T, To be the tensions at the ends of a cord wrapped on a cylinder, when the external angle between the directions of the free ends is a. [e is the base of Napier's Logarithms.] We thus obtain the singular result, that the dimensions of the cylinder have no influence on the increase of tension by friction, provided the cord is perfectly flexible.

593. Having thus briefly considered the equilibrium of a rigid body, we propose, before entering upon the subject of deformation of elastic solids, to consider certain intermediate cases, in each of which a particular assumption is made the basis of the investigation-thereby avoiding a very considerable amount of analytical difficulties.

594. Very excellent examples of this kind are furnished by the statics of a flexible and inextensible cord or chain, fixed at both ends, and subject to the action of any forces. The curve in which the chain hangs in any case may be called a Catenary, although the term is usually restricted to the case of a uniform chain acted on by gravity only.

595. We may consider separately the conditions of equilibrium of each element; or we may apply the general condition (§ 257) that the whole potential energy is a minimum, in the case of any conservative system of forces; or, especially when gravity is the only external force, we may consider the equilibrium of a finite portion of the chain treated for the time as a rigid body (§ 584).

596. The first of these methods gives immediately the three following equations of equilibrium, for the catenary in general:

(1) The rate of variation of the tension per unit of length along the cord is equal to the tangential component of the applied force, per unit of length.

(2) The plane of curvature of the cord contains the normal component of the applied force, and the centre of curvature is on the opposite side of the arc from that towards which this force acts.

(3) The amount of the curvature is equal to the normal component of the applied force per unit of length at any point divided by the tension of the cord at the same point.

The first of these is simply the equation of equilibrium of an infinitely small element of the cord relatively to tangential motion. The second and third express that the component of the resultant

of the tensions at the two ends of an infinitely small arc, along the normal through its middle point, is directly opposed and is equal to the normal applied force, and is equal to the whole amount of it on the arc. For the plane of the tangent lines in which those tensions act is (§ 12) the plane of curvature. And if be the angle between them (or the infinitely small angle by which the angle between their positive directions falls short of π), and T the arithmetical mean of their magnitudes, the component of their resultant along the line bisecting the angle between their positive directions is 27'sin 40, rigorously or Te, since is infinitely small. Hence 70-Nos if ds be the length of the arc, and Nds the whole amount of normal force applied to it. But (§ 9) 0= if P be the radius of curvature; and

therefore

δε

Ρ

which is the equation stated in words (3) above.

597. From (1) of § 596, we see that if the applied forces on any particle of the cord constitute a conservative system, and if any equal infinitely small lengths of the string experience the same force and in the same direction when brought into any one position by motion of the string, the difference of the tensions of the cord at any two points of it when hanging in equilibrium, is equal to the difference of the potential (§ 504) of the forces between the positions occupied by these points. Hence, whatever the position where the potential is réckoned zero, the tension of the string at any point is equal to the potential at the position occupied by it, with a constant added.

598. From § 596 it follows immediately that if a material particle of unit mass be carried along any catenary with a velocity, s, equal to T, the numerical measure of the tension at any point, the force upon it by which this is done is in the same direction as the resultant of the applied force on the catenary at this point, and is equal to the amount of this force per unit of length, multiplied by T. For denoting by S the tangential, and (as before) by N the normal component of the applied force per unit of length at any point P of the catenary, we have, by § 596 (1), S for the rate of variation of s per unit length, and therefore Ss for its variation per unit of time. That is to say,

=Ss=ST,

or (225) the tangential component force on the moving particle is equal to ST. Again, by § 596 (3),

T2 j2
NT=- =
P

or the centrifugal force of the moving particle in the circle of curvature of its path, that is to say, the normal component of the force on it, is equal to And lastly, by (2) this force is in the same direction as N. We see therefore that the direction of the

NT.

whole force on the moving particle is the same as that of the resultant of S and N; and its magnitude is T' times the magnitude of this resultant.

599. Thus we see how, from the more familiar problems of the kinetics of a particle, we may immediately derive curious cases of catenaries. For instance: a particle under the influence of a constant force in parallel lines moves in a parabola with its axis vertical, with velocity at each point equal to that generated by the force acting through a space equal to its distance from the directrix. Hence, if z denote this distance, and f the constant force, T= √2f%

in the allied parabolic catenary; and the force on the catenary is parallel to the axis, and is equal in amount per unit of length, to

f √2f%

Hence if the force on the catenary be that of gravity, it must have its axis vertical (its vertex downwards of course for stable equilibrium) and its mass per unit length at any point must be inversely as the square root of the distance of this point above the directrix. From this it follows that the whole weight of any arc of it is proportional to its horizontal projection.

600. Or, if the question be, to find what force towards a given fixed point, will cause a cord to hang in any given plane curve with this point in its plane; it may be answered immediately from the solution of the corresponding problem in 'central forces.'

601. When a perfectly flexible string is stretched over a smooth surface, and acted on by no other force throughout its length than the resistance of this surface, it will, when in stable equilibrium, lie along a line of minimum length on the surface, between any two of its points. For ($ 584) its equilibrium can be neither disturbed nor rendered unstable by placing staples over it, through which it is free to slip, at any two points where it rests on the surface and for the intermediate part the energy criterion of stable equilibrium is that just stated.

The

There being no tangential force on the string in this case, and the normal force upon it being along the normal to the surface, its osculating plane (§ 596) must cut the surface everywhere at right angles. These considerations, easily translated into pure geometry, establish the fundamental property of the geodetic lines on any surface. analytical investigations of the question, when adapted to the case of a chain of not given length, stretched between two given points on a given smooth surface, constitute the direct analytical demonstration of this property.

In this case it is obvious that the tension of the string is the same at every point, and the pressure of the surface upon it is [§ 596 (3)] at each point proportional to the curvature of the string.

602. No real surface being perfectly smooth, a cord or chain may rest upon it when stretched over so great length of a geodetic on a convex rigid body as to be not of minimum length between its extreme points: but practically, as in tying a cord round a ball, for permanent security it is necessary, by staples or otherwise, to constrain it from lateral slipping at successive points near enough to one another to make each free portion a true minimum on the surface.

603. A very important practical case is supplied by the consideration of a rope wound round a rough cylinder. We may suppose it to lie in a plane perpendicular to the axis, as we thus simplify the question very considerably without sensibly injuring the utility of the solution. To simplify still further, we shall suppose that no forces act on the rope but tensions and the reaction of the cylinder. In practice this is equivalent to the supposition that the tensions and reactions are very large compared with the weight of the rope or chain; which, however, is inadmissible in some important cases, especially such as occur in the application of the principle to brakes for laying submarine cables, to dynamometers, and to windlasses (or capstans with horizontal axes).

By § 592 we have

T=TEMO,

showing that, for equal successive amounts of integral curvature (§ 14), the tension of the rope augments in geometrical progression. To give an idea of the magnitudes involved, suppose μ=5, 0=π, then

T=T5=4817, roughly.

Hence if the rope be wound three times round the post or cylinder the ratio of the tensions of its ends, when motion is about to commence, is

56: 1 or about 15,000: 1.

Thus we see how, by the aid of friction, one man may easily check the motion of the largest vessel, by the simple expedient of coiling a rope a few times round a post. This application of friction is of great importance in many other applications, especially to dynamometers (§§ 389, 390).

604. With the aid of the preceding investigations, the student may easily work out for himself the solution of the general problem of a cord under the action of any forces, and constrained by a rough surface; it is not of sufficient importance or interest to find a place here.

605. An elongated body of elastic material, which for brevity we shall generally call a wire, bent or twisted to any degree, subject only to the condition that the radius of curvature and the reciprocal of the twist are everywhere very great in comparison with the greatest transverse dimension, presents a case in which, as we shall see, the solution of the general equations for the equilibrium of an elastic solid is either obtainable in finite terms, or is reducible

to comparatively easy questions agreeing in mathematical conditions with some of the most elementary problems of hydrokinetics, electricity, and thermal conduction. And it is only for the determination of certain constants depending on the section of the wire and the elastic quality of its substance, which measure its flexural and torsional rigidity, that the solutions of these problems are required. When the constants of flexure and torsion are known, as we shall now suppose them to be, whether from theoretical calculation or experiment, the investigation of the form and twist of any length of the wire, under the influence of any forces which do not produce a violation of the condition stated above, becomes a subject of mathematical analysis involving only such principles and formulae as those that constitute the theory of curvature (§§ 9-15) and twist in geometry or kinematics.

606. Before entering on the general theory of elastic solids, we shall therefore, according to the plan proposed in § 593, examine the dynamic properties and investigate the conditions of equilibrium of a perfectly elastic wire, without admitting any other condition_or limitation of the circumstances than what is stated in § 605, and without assuming any special quality of isotropy, or of crystalline, fibrous or laminated structure in the substance.

607. Besides showing how the constants of flexural and torsional rigidity are to be determined theoretically from the form of the transverse section of the wire, and the proper data as to the elastic qualities of its substance, the complete theory simply indicates that, provided the conditional limit of deformation is not exceeded, the following laws will be obeyed by the wire under

Let the whole mutual action between the parts of the wire on the two sides of the cross section at any point (being of course the action of the matter infinitely near this plane on one side, upon the matter infinitely near it on the other side), be reduced to a single force through any point of the section and a single couple. Then

I. The twist and curvature of the wire in the neighbourhood of this section are independent of the force, and depend solely on the couple.

II. The curvatures and rates of twist producible by any several couples separately, constitute, if geometrically compounded, the curvature and rate of twist which are actually produced by a mutual action equal to the resultant of those couples.

608. It may be added, although not necessary for our present purpose, that there is one determinate point in the cross section such that if it be chosen as the point to which the forces are transferred, a higher order of approximation is obtained for the fulfilment of these laws than if any other point of the section be taken. That point, which in the case of a wire of substance uniform through its cross section is the centre of inertia of the area of the section, we