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in the filaments towards the corners, and so vitiate the solution. Unhappily mathematicians have not hitherto succeeded in solving, possibly not even tried to solve, the beautiful problem thus presented by the flexure of a broad very thin band (such as a watch spring) into a circle of radius comparable with a third proportional to its thickness and its breadth.

681. But, provided the radius of curvature of the flexure is not only a large multiple of the greatest diameter, but also of a third proportional to the diameters in and perpendicular to the plane of flexure; then however great may be the ratio of the greatest diameter to the least, the preceding solution is applicable ; and it is remarkable that the necessary distortion of the normal section (illustrated in the diagram of $ 679) does not sensibly impede the free lateral contractions and expansions in the filaments, even in the case of a broad thin lamina (whether of precisely rectangular section, or of unequal thicknesses in different parts).

682. In our sections on hydrostatics, the problem of finding the deformation produced in a spheroid of incompressible liquid by a given disturbing force will be solved; and then we shall consider the application of the preceding methods to an elastic solid sphere in their bearing on the theory of the tides and the rigidity of the earth. This proposed application, however, reminds us of a general remark of great practical importance, with which we shall leave elastic solids for the present. Considering different elastic solids of similar substance and similar shapes, we see that if by forces applied to them in any way they are similarly strained, the surface tractions in or across similarly situated elements of surface, whether of their boundaries or of surfaces imagined as cutting through their substances, must be equal, reckoned as usual per unit of area. Hence; the force across, or in, any such surface, being resolved into components parallel to any directions; the whole amounts of each such component for similar surfaces of the different bodies are in proportion to the squares of their lineal dimensions. Hence, if equilibrated similarly under the action of gravity, or of their kinetic reactions ($ 230) against equal accelerations (32), the greater body would be more strained than the less; as the amounts of gravity or of kinetic reaction of similar portions of them are as the cubes of their linear dimensions. Definitively, the strains at similarly situated points of the bodies will be in simple proportion to their linear dimensions, and the displacements will be as the squares of these lines, provided that there is no strain in any part of any of them too great to allow the principle of superposition to hold with sufficient exactness, and that no part is turned through more than a very small angle relatively to any other part. To illustrate by a single example, let us consider a uniform long, thin, round rod held horizontally by its middle. Let its substance be homogeneous, of density p, and Young's modulus, M; and let its length, l, be p times its diameter. Then (as the moment of inertia of a circular area of radius r round a diameter is dar4) the

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M 1 flexural rigidity of the rod will (8 679) be

This gives us

4 for the curvature at the middle of the rod the elongation and contraction where greatest, that is, at the highest and lowest points of the normal section through the middle point; and the droop of the ends; the following expressions

2pp Plp palap

and M M

8M

Thus, for a rod whose length is 200 times its diameter, if its substance be iron or steel, for which p=7.75, and M=194 x 107 grammes per square centimetre, the maximum elongation and contraction (being at the top and bottom of the middle section where it is held) are each equal to .8 x 10-6 xl, and the droop of its ends 2 x 10-5 x 22. Thus a steel or iron wire, ten centimetres long, and half a millimetre in diameter, held horizontally by its middle, would experience only •000008 of maximum elongation and contraction, and only .002 of a centimetre of droop in its ends: a round steel rod, of half a centimetre in diameter, and one metre long, would experience :00008 of maximum elongation and contraction, and 2 of a centimetre of droop: a round steel rod, of ten centimetres diameter, and twenty metres long, must be of remarkable temper (see Properties of Matter) to bear being held by the middle without taking a very sensible permanent set: and it is probable that no temper of steel is high enough in a round shaft forty metres long, if only two decimetres in diameter, to allow it to be held by its middle without either bending it to some great angle, and beyond all appearance of elasticity, or breaking it.

683. In passing from the dynamics of perfectly elastic solids to abstract hydrodynamics, or the dynamics of perfect fluids, it is convenient and instructive to anticipate slightly some of the views as to intermediate properties observed in real solids and fluids, which, according to the general plan proposed (§ 402) for our work, will be examined with more detail under Properties of Matter.

By induction from a great variety of observed phenomena, we are compelled to conclude that no change of volume or of shape can be produced in any kind of matter without dissipation of energy ($ 247); so that if in any case there is a return to the primitive configuration, some amount (however small) of work is always required to compensate the energy dissipated away, and restore the body to the same physical and the same palpably kinetic condition as that in which it was given. We have seen ($ 643), by anticipating something of thermodynamic principles, how such dissipation is inevitable, even in dealing with the absolutely perfect elasticity of volume presented by every fluid, and possibly by some solids, as, for instance, homogeneous crystals. But in metals, glass, porcelain, natural stones, wood, Indiarubber, homogeneous jelly, silk fibre, ivory, etc., a distinct frictional

S

resistance against every change of shape is, as we shall see later, under Properties of Matter, demonstrated by many experiments, and is found to depend on the speed with which the change of shape is made. A

very remarkable and obvious proof of frictional resistance to change of shape in ordinary solids, is afforded by the gradual, more or less rapid, subsidence of vibrations of elastic solids; marvellously rapid in India-rubber, and even in homogeneous jelly; less rapid in glass and metal springs, but still demonstrably, much more rapid than can be accounted for by the resistance of the air. This molecular friction in elastic solids may be properly called viscosity of solids, because, as being an internal resistance to change of shape depending on the rapidity of the change, it must be classed with fluid molecular friction, which by general consent is called viscosity of fluids. But, at the same time, we feel bound to remark that the word viscosity, as used hitherto by the best writers, when solids or heterogeneous semisolid-semifluid masses are referred to, has not been distinctly applied to molecular friction, especially not to the molecular friction of a highly elastic solid within its limits of high elasticity, but has rather been employed to designate a property of slow, continual yielding through very great, or altogether unlimited, extent of change of shape, under the action of continued stress. It is in this sense that Forbes, for instance, has used the word in stating that Viscous Theory of Glacial Motion’ which he demonstrated by his grand observations on glaciers. As, however, he, and many other writers after him, have used the words plasticity and plastic, both with reference to homogeneous solids (such as wax or pitch, even though also brittle; soft metals; etc.), and to heterogeneous semisolid-semifluid masses (as mud, moist earth, mortar, glacial ice, etc.), to designate the property?, common to all those cases, of experiencing, under continued stress either quite continued and unlimited change of shape, or gradually very great change at a diminishing (asymptotic) rate through infinite time; and as the use of the term plasticity implies no more than does viscosity, any physical theory or explanation of the property, the word viscosity is without inconvenience left available for the definition we have given of it above.

684. A perfect fluid, or (as we shall call it) a fluid, is an unrealizable conception, like a rigid, or a smooth, body : it is defined as a body incapable of resisting a change of shape and therefore incapable of experiencing distorting or tangential stress ($ 640). Hence its pressure on any surface, whether of a solid or of a contiguous portion of

1 See Proceedings of the Royal Society, May 1865, 'On the Viscosity and Elasticity of Metals' (W. Thomson).

2 Some confusion of ideas might have been avoided on the part of writers who have professedly objected to Forbes' theory while really objecting only (and we believe groundlessly) to his usage of the word viscosity, if they had paused to consider that no one physical explanation can hold for those several cases ; and that Forbes' theory is merely the proof by observation that glaciers have the property that mud (heterogeneous), mortar (heterogeneous), pitch (homogeneous), water (homogeneous), all have of changing shape indefinitely and continuously under the action of continued stress.

the fluid, is at every point perpendicular to the surface. In equilibrium, all common liquids and gaseous fluids fulfil the definition. But there is finite resistance, of the nature of friction, opposing change of shape at a finite rate; and, therefore, while a fluid is changing shape, it exerts tangential force on every surface other than normal planes of the stress (§ 635) required to keep this change of shape going on. Hence; although the hydrostatical results, to which we immediately proceed, are verified in practice; in treating of hydrokinetics, in a subsequent chapter, we shall be obliged to introduce the consideration of fluid friction, except in cases where the circumstances are such as to render its effects insensible.

685. With reference to a fluid the pressure at any point in any direction is an expression used to denote the average pressure per unit of area on a plane surface imagined as containing the point, and perpendicular to the direction in question, when the area of that surface is indefinitely diminished.

686. At any point in a fluid at rest the pressure is the same in all directions: and, if no external forces act, the pressure is the same at every point. For the proof of these and most of the following propositions, we imagine, according to $ 584, a definite portion of the fluid to become solid, without changing its mass, form, or dimensions.

Suppose the fluid to be contained in a closed vessel, the pressure within depending on the pressure exerted on it by the vessel, and not on any external force such as gravity.

687. The resultant of the fluid pressures on the elements of any portion of a spherical surface must, like each of its components, pass through the centre of the sphere. Hence, if we suppose (S584) a portion of the fluid in the form of a plano-convex lens to be solidified, the resultant pressure on the plane side must pass through the centre of the sphere; and, therefore, being perpendicular to the plane, must pass through the centre of the circular area. From this it is obvious that the pressure is the same at all points of any plane in the fluid. Hence the resultant pressure on any plane surface passes through its centre of inertia.

Next, imagine a triangular prism of the fluid, with ends perpendicular to its faces, to be solidified. The resultant pressures on its ends act in the line joining the centres of inertia of their areas, and are equal since the resultant pressures on the sides are in directions perpendicular to this line. Hence the pressure is the same in all parallel planes.

But the centres of inertia of the three faces, and the resultant pressures applied there, lie in a triangular section parallel to the ends. The pressures act at the middle points of the sides of this triangle, and perpendicularly to them, so that their directions meet in a point. And, as they are in equilibrium, they must be proportional to the respective sides of the triangle; that is, to the breadths, or areas, of the faces of the prism. Thus the resultant pressures on the

faces must be proportional to the areas of the faces, and therefore the pressure is equal in any two planes which meet.

Collecting our results, we see that the pressure is the same at all points, and in all directions, throughout the fluid mass.

688. Hence if a force be applied at the centre of inertia of each face of a polyhedron, with magnitude proportional to the area of the face, the polyhedron will be in equilibrium. For we may suppose the polyhedron to be a solidified portion of the fluid. The resultant pressure on each face will then be proportional to its area, and will act at its centre of inertia ; which, in this case, is the Centre of Pressure.

689. Another proof of the equality of pressure throughout a mass of fluid, uninfluenced by other external force than the pressure of the containing vessel, is easily furnished by the energy criterion of equilibrium, § 254; but, to avoid complication, we will consider the fluid to be incompressible. Suppose a number of pistons fitted into cylinders inserted in the sides of the closed vessel containing the fluid. Then, if A be the area of one of these pistons, p the average pressure on it, x the distance through which it is pressed, in or out; the energy criterion is that no work shall be done on the whole, i. e. that

A,B,X, + A2P2X2 + ...={(Apx)=0, as much work being restored by the pistons which are forced out, as is done by those forced in. Also, since the fluid is incompressible, it must have gained as much space by forcing out some of the pistons as it lost by the intrusion of the others. This gives

44*, + Azxg+...=(Ax)=0. The last is the only condition to which x1, x2, etc., in the first equation, are subject; and therefore the first can only be satisfied if

D.=P.=Ps=etc., that is, if the pressure be the same on each piston. Upon this property depends the action of Bramah's Hydrostatic Press.

If the fluid be compressible, the work expended in compressing it from volume V to V-8V, at mean pressure p, is V.

If in this case we assume the pressure to be the same throughout, we obtain a result consistent with the energy criterion.

The work done on the fluid is 2 (Apx), that is, in consequence of the assumption, P (Ax). But this is equal to

p8V, for, evidently,

E(Ax)=8V. 690. When forces, such as gravity, act from external matter upon the substance of the fluid, either in proportion to the density of its own substance in its different parts, or in proportion to the density of electricity, or of magnetic polarity, or of any other conceivable accidental property of it, the pressure will still be the same in all directions at any one point, but will now vary continuously from point to point. For the preceding demonstration ( 687) may still

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