rigidity and resistance to compression in any class of solids. Thus clear elastic jellies, and India rubber, present familiar specimens of isotropic homogeneous solids, which, while differing very much from one another in rigidity ('stiffness '), are probably all of very nearly the same compressibility as water. This being 308000 per pound per square inch; the resistance to compression, measured by its reciprocal, or, as we may read it, 308000 lbs. per square inch,' is obviously many hundred times the absolute amount of the rigidity of the stiffest of those substances. A column of any of them, therefore, when pressed together or pulled out, within its limits of elasticity, by balancing forces applied to its ends (or an India rubber band when pulled out), experiences no sensible change of volume, though a very sensible change of length. Hence the proportionate extension or contraction of any transverse diameter must be sensibly equal to the longitudinal contraction or extension: and for all ordinary stresses, such substances may be practically regarded as incompressible elastic solids. Stokes gave reasons for believing that metals also have in general greater resistance to compression, in proportion to their rigidities, than according to the fallacious theory, although for them the discrepancy is very much less than for the gelatinous bodies. This probable conclusion was soon experimentally demonstrated by Werthiem, who found the ratio of lateral to longitudinal change of lineal dimensions, in columns acted on solely by longitudinal force, to be about for glass or brass; and by Kirchhoff, who, by a very well-devised experimental method, found 387 as the value of that ratio for brass, and 294 for iron. For copper we find that it probably lies between 226 and 441, by recent experiments1 of our own, measuring the torsional and longitudinal rigidities (§§ 609, 657) of a copper wire. 656. All these results indicate rigidity less in proportion to the compressibility than according to Navier's and Poisson's theory. And it has been supposed by many naturalists, who have seen the necessity of abandoning that theory as inapplicable to ordinary solids, that it may be regarded as the proper theory for an ideal perfect solid, and as indicating an amount of rigidity not quite reached in any real substance, but approached to in some of the most rigid of natural solids (as, for instance, iron). But it is scarcely possible to hold a piece of cork in the hand without preceiving the fallaciousness of this last attempt to maintain a theory which never had any good foundation. By careful measurements on columns of cork of various forms (among them, cylindrical pieces cut in the ordinary way for bottles) before and after compressing them longitudinally in a Brahmah's press, we have found that the change of lateral dimensions is insensible both with small longitudinal contractions and return dilatations, within the limits of elasticity, and with such enormous longitudinal contractions as to or of the original length. It is thus proved decisively that cork is much more rigid, while metals, 'On the Elasticity and Viscosity of Metals' (W. Thonison), Proc. R. S., May 1865. R glass, and gelatinous bodies are all less rigid, in proportion to resistance to compression than the supposed 'perfect solid'; and the utter worthlessness of the theory is experimentally demonstrated. 657. The modulus of elasticity of a bar, wire, fibre, thin filament, band, or cord of any material (of which the substance need not be isotropic, nor even homogeneous within one normal section), as a bar of glass or wood, a metal wire, a natural fibre, an India rubber band, or a common thread, cord, or tape, is a term introduced by Dr. Thomas Young to designate what we also sometimes call its longitudinal rigidity: that is, the quotient obtained by dividing the simple longitudinal force required to produce any infinitesimal elongation or contraction by the amount of this elongation or contraction reckoned as always per unit of length. 658. Instead of reckoning the modulus in units of weight, it is sometimes convenient to express it in terms of the weight of the unit length of the rod, wire, or thread. The modulus thus reckoned, or, as it is called by some writers, the length of the modulus, is of course found by dividing the weight-modulus by the weight of the unit length. It is useful in many applications of the theory of elasticity; as, for instance, in this result, which will be proved later:--the velocity of transmission of longitudinal vibrations (as of sound) along a bar or cord, is equal to the velocity acquired by a body in falling from a height equal to half the length of the modulus 1. 659. The specific modulus of elasticity of an isotropic substance, or, as it is most often called, simply the modulus of elasticity of the substance, is the modulus of elasticity of a bar of it having some definitely specified sectional area. If this be such that the weight of unit length is unity, the modulus of the substance will be the same as the length of the modulus of any bar of it; a system of reckoning which, as we have seen, has some advantages in application. It is, however, more usual to choose a common unit of area as the sectional area of the bar referred to in the definition. There must also be a definite understanding as to the unit in terms of which the force is measured, which may be either the absolute unit (§ 188): or the gravitation unit for a specified locality; that is (§ 191), the weight in that locality of the unit of mass. Experimenters hitherto have stated their results in terms of the gravitation unit, each for his own locality; the accuracy hitherto attained being scarcely in any cases sufficient to 1 It is to be understood that the vibrations in question are so much spread out through the length of the body, that inertia does not sensibly influence the transverse contractions and dilatations which (unless the substance have in this respect the peculiar character presented by cork, § 656) take place along with them. Also, under thermodynamics, we shall see that changes of temperature produced by the varying strains cause changes of stress which, in ordinary solids, render the velocity of transmission of longitudinal vibrations sensibly greater than that calculated by the rule stated in the text, if we use the static modulus as understood from the definition there given; and we shall learn to take into account the thermal effect by using a definite static modulus, or kinetic modulus, according to the circumstances of any case that may occur. require corrections for the different forces of gravity in the different places of observation. 660. The most useful and generally convenient specification of the modulus of elasticity of a substance is in grammes-weight per square centimetre. This has only to be divided by the specific gravity of the substance to give the length of the modulus. British measures, however, being still unhappily sometimes used in practical and even in high scientific statements, we may have occasion to refer to reckonings of the modulus in pounds per square inch or per square foot, or to length of the modulus in feet. 661. The reckoning most commonly adopted in British treatises on mechanics and practical statements is pounds per square inch. The modulus thus stated must be divided by the weight of 12 cubic inches of the solid, or by the product of its specific gravity into 43371, to find the length of the modulus, in feet. To reduce from pounds per square inch to grammes per square centimetre, multiply by 70·31, or divide by 014223. French engineers generally state their results in kilogrammes per square millimetre, and so bring them to more convenient numbers, being 100000 of the inconveniently large numbers expressing moduli in grammes-weight per square centimetre.. 662. The same statements as to units, reducing factors, and nominal designations, are applicable to the resistance to compression of any elastic solid or fluid, and to the rigidity (§ 651). of an isotropic body; or, in general, to any one of the 21 co-efficients in the expressions for terms in stresses of strains, or to the reciprocal of any one of the 21 co-efficients in the expressions for strains in terms of stresses, as well as to the modulus defined by Young. 663. In §§ 652,653 we examined the effect of a simple longitudinal stress, in producing elongation in its own direction, and contraction 1 This decimal being the weight in pounds of 12 cubic inches of water. The one great advantage of the French metrical system is, that the mass of the unit volume (1 cubic centimetre) of water at its temperature of maximum density (3°.945 C.) is unity (1 gramme) to a sufficient degree of approximation for almost all practical purposes. Thus, according to this system, the density of a body and its specific gravity mean one and the same thing; whereas on the British no-system the density is expressed by a number found by multiplying the specific gravity by one number or another, according to the choice of a cubic inch, cubic foot, cubic yard, or cubic mile that is made for the unit of volume; and the grain, scruple, gunmaker's drachm, apothecary's drachm, ounce Troy, ounce avoirdupois, pound Troy, pound avoirdupois, stone (Imperial, Ayrshire, Lanarkshire, Dumbartonshire), stone for hay, stone for corn, quarter (of a hundredweight), quarter (of corn), hundredweight, or ton, that is chosen for unit of mass. It is a remarkable phenomenon, belonging rather to moral and social than to physical science, that a people tending naturally to be regulated by common sense should voluntarily condemn themselves, as the British have so long done, to unnecessary hard labour in every action of common business or scientific work related to measurement, from which all the other nations of Europe have emancipated themselves. We have been informed, through the kindness of Professor W. H. Miller, of Cambridge, that he concludes, from a very trustworthy comparison of standards by Kupffer, of St. Petersburgh, that the weight of a cubic decimetre of water at temperature of maximum density is 1000-013 grammes. in lines perpendicular to it. With stresses substituted for strains, and strains for stresses, we may apply the same process to investigate the longitudinal and lateral tractions required to produce a simple longitudinal strain (that is, an elongation in one direction, with no change of dimensions perpendicular to it) in a rod or solid of any shape. Thus a simple longitudinal strain e is equivalent to a cubic dilatation e without change of figure (or linear dilatatione equal in all directions), and two distortions consisting each of dilatatione in the given direction, and contraction e in each of two directions perpendicular to it and to one another. To produce the cubic dilatation, e, alone requires (§ 651) a normal traction ke equal in all directions. And, to produce either of the distortions simply, since the measure (§ 154) of each is e, requires a distorting stress equal to n xe, which consists of tangential tractions each equal to this amount, positive (or drawing outwards) in the line of the given elongation, and negative (or pressing inwards) in the perpendicular direction. Thus we have in all normal traction (k+n) e, in the direction of the given strain, and normal traction=(k—n) e, in (4) 664. If now we suppose any possible infinitely small strain (e, ƒ, g, a, b, c), according to the specification of § 640, to be given to a body, the stress (P, Q, R, S, T, U) required to maintain it will be expressed by the following formulae, obtained by successive applications of § 663 (4) to the components e, f, g separately, and of § 651 to a, b, c :— 665. Similarly, by § 651 and § 654 (3), we have (5) (6) as the formulae expressing the strain (e, f, g, a, b, c) in terms of the stress (P, Q, R, S, T, U). They are of course merely the algebraic inversions of (5); and they might have been found by solving these for e, f, g, a, b, c, regarded as the unknown quantities. M is here introduced to denote Young's modulus. 666. To express the equation of energy for an isotropic substance, we may take the general formula, w=1(Pe+Of+Rg+Sa+Tb+Uc), and eliminate from it P, Q, etc., by (5) of § 664, or, again, e, ƒ, etc., by (6) of § 665, we thus find 4n. 3 2n 2w = (k+)(e2 + ƒ2 + g2) + 2(k − −)(fg + ge+ ef) + n(a2 + b2 + c2) 1 1 -- 3 1 2n 3k) (QR+RP + PQ ) } + − (S2 + T2 — U2). = ↓ { ( } + {} } ) ( P2 + Q2 + R2) − 2(1)(Q) n 3 k n (7) 667. The mathematical theory of the equilibrium of an elastic solid presents the following general problems: A solid of any given shape, when undisturbed, is acted on in its substance by force distributed through it in any given manner, and displacements are arbitrarily produced, or forces arbitrarily applied, over its bounding surface. It is required to find the displacement of every point of its substance. This problem has been thoroughly solved for a shell of homogeneous isotropic substance bounded by surfaces which, when undisturbed, are spherical and concentric; but not hitherto for a body of any other shape. The limitations under which solutions have been obtained for other cases (thin plates and rods), leading, as we have seen, to important practical results, have been stated above (§ 605). To demonstrate the laws (§ 607) which were taken in anticipation will also be one of our applications of the general equations for interior equilibrium of an elastic solid, which we now proceed to investigate. 668. Any portion in the interior of an elastic solid may be regarded as becoming perfectly rigid (§ 584) without disturbing the equilibrium either of itself or of the matter round it. Hence the traction exerted by the matter all round it, regarded as a distribution of force applied to its surface, must, with the applied forces acting on the substance of the portion considered, fulfil the conditions of equilibrium of forces acting on a rigid body. This statement, applied to an infinitely small rectangular parallelepiped of the body, gives the general differential equations of internal equilibrium of an elastic solid. It is to be remarked that three equations suffice; the conditions of equilibrium for the couples being secured by the relation established above (§ 632) among the six pairs of tangential component tractions on the six faces of the figure. 669. One of the most beautiful applications of the general equations of internal equilibrium of an elastic solid hitherto made is |