transmission of a simple longitudinal Velocity of by the weight of the unit length. It is useful in many a tions of the theory of elasticity; as, for instance, in this p which will be proved later:-the velocity of transmissi longitudinal vibrations (as of sound) along a bar or cor equal to the velocity acquired by a body in falling fr height equal to half the length of the modulus.1 stress through a rod. Specific modulus of body. 688. The specific modulus of elasticity of an isotropies an isotropic stance, or, as it is most often called, simply the modaln elasticity of the substance, is the modulus of elasticity of of it having some definitely specified sectional area. If be such that the weight of unit length is unity, the mods, the substance will be the same as the length of the molaits any bar of it: a system of reckoning which, as we have s has some advantages in application. It is, however, more us to choose a common unit of area as the sectional area of bar referred to in the definition. There must also be a defia“ understanding as to the unit in terms of which the forb In terms of measured, which may be either the absolute unit (§ 223 :unit; or of the gravitation unit for a specified locality; that is § 226.1 weight in that locality of the unit of mass. Experiment mass in any hitherto have stated their results in terms of the gravitati. unit, each for his own locality; the accuracy hitherto attai being scarcely in any cases sufficient to require corrections for the different forces of gravity in the different places the absolute the force of gravity on the unit of particular locality. observation. " 689. The most useful and generally convenient specitor tion of the modulus of elasticity of a substance is in gramines weight per square centimetre. This has only to be divided the specific gravity of the substance to give the length of modulus. British measures, however, being still unhap dy sometimes used in practical and even in high scientific st..? 1 It is to be understood that the vibrations in question are so much sprea! through the length of the body, that inertia does not sensibly influence the tra verse contractions and dilatations which (unless the substance have in this res the peculiar character presented by cork, § 684) take place along with them. Aunder thermodynamics, we shall see that changes of temperature produced by t varying stresses cause changes of temperature which, in ordinary solids, render tor velocity of transmission of longitudinal vibrations sensibly greater than that lated by the rule stated in the text, if we use the static modulus as underst l ** the definition there given; and we shall learn to take into account the thermal ef by using a definite static modulus, or kinetic modulus, according to the care anska, v of any case that may occur. ents, we may have occasion to refer to reckonings of the odulus in pounds per square inch or per square foot, or to ength of the modulus in feet. 690. The reckoning most commonly adopted in British reatises on mechanics and practical statements is pounds per quare inch. The modulus thus stated must be divided by he weight of 12 cubic inches of the solid, or by the product f its specific gravity into 4337, to find the length of the nodulus, in feet. To reduce from pounds per square inch to grammes per quare centimetre, multiply by 70:31, or divide by 014223. French engineers generally state their results in kilogrammes er square millimetre, and so bring them to more convenient numbers, being 100000 of the inconveniently large numbers xpressing moduli in grammes weight per square centimetre. 691. 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 (§ 680) of an isotropic body; or, in general, to any one of the 21 co- Metrical efficients in the expressions [§ 673 (14)] for stresses in terms of tions of strains, or to the reciprocal of any one of the 21 coefficients of elasticity in the expressions [§ 673 (16)] for strains in terms of stresses, as well as to the modulus defined by Young. 1 This decimal being the weight in lbs. of 12 cubic inches of water. The one great advantage of the French metrical system is, that the mass of the unit volume (I cubic centimetre) of water at its temperature of maximum density (3945 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. denomina coefficients in general. Stress required to maintain a simple strain. 692. In §§ 681, 682 we examined the effect of a E longitudinal stress, in producing elongation in its own i longitudinal tion, and contraction in lines perpendicular to it. With tsubstituted for strains, and strains for stresses, we may the same process to investigate the longitudinal and b tractions required to produce a simple longitudinal strain is, an elongation in one direction, with no change of dimens perpendicular to it) in a rod or solid of any shape. Stress components strain for isotropic body. Thus a simple longitudinal strain e is equivalent to a dilatation e without change of figure (or linear dilatatiz equal in all directions), and two distortions consisting el dilatation fe in the given direction, and contraction je in of two directions perpendicular to it and to one another produce the cubic dilatation, e, alone requires (§ 680) a n traction ke equal in all directions. And, to produce eitt the distortions simply, since the measure (§ 175) of each requires a distorting stress equal to n × e, which consists tangential tractions each equal to this amount, positive drawing outwards) in the line of the given elongation, s negative (or pressing inwards) in the perpendicular directi Thus we have in all normal traction = (k+n)e, in the direction of the given strain, and normal traction = (kn)e, in every direction perpen- 693. If now we suppose any possible infinitely small stru in terms of (e, f, g, a, b, c), according to the specification of § 669, to 'given to a body, the stress (P, Q, R, S, T, U) required: maintain it will be expressed by the following formula, tained by successive applications of § 692 (4) to the cct ponents e, f, g separately, and of § 680 to a, b, c : 694. Similarly, by § 680 and § 682 (3), we have Straincomponents in terms of stress for isotropic body. M= the formule expressing the strain (e, f, g, a, b, c) in terms of e stress (P, Q, R, S, T, U). They are of course merely the Igebraic inversions of (5); and (§ 673) they might have been ound by solving these for e, f, g, a, b, c, regarded as the unnown quantities. M is here introduced to denote Young's aodulus (§ 683). and m from § 698 (5). 695. To express the equation of energy for an isotropic Equation of ubstance, we may take the general formula, w=1(Pe+Qf+Rg+Sa+Tb+Uc)... [§ 673 (20)], and eliminate from it P, Q, etc., by (5) of § 693, or, again, e, f, tc., by (6) of § 694, we thus find energy for the same. 3k n 1 696. The mathematical theory of the equilibrium of an Fundaelastic solid presents the following general problems: mental problems of mathe theory, A solid of any given shape, when undisturbed, is acted on in matical 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 (§ 735); 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 (S$ 588, 632). To Scales of average ousness. Isotropic and aeolo tropic defined. under Properties of Matter; and in the meantime need o say that the definition of homogeneousness may be appli practically on a very large scale to masses of building or coarsegrained conglomerate rock, or on a more moderate scale blocks of common sandstone, or on a very small scale to seela ingly homogeneous metals; or on a scale of extreme, un Lcovered fineness, to vitreous bodies, continuous crystals, soliditel gums, as India rubber, gum-arabic, etc., and fluids. 676. The substance of a homogeneous solid is called is Substances tropic when a spherical portion of it, tested by any physic agency, exhibits no difference in quality however it is tur Or, which amounts to the same, a cubical portion cut from ar position in an isotropic body exhibits the same qualities rel tively to each pair of parallel faces. Or two equal and sin portions cut from any positions in the body, not subject to th condition of parallelism (§ 675), are undistinguishable from another. A substance which is not isotropic, but exhibits de ferences of quality in different directions, is called acolotrop Isotropy and aeolotropy of different sets of properties. 677. An individual body, or the substance of a hone geneous solid, may be isotropic in one quality or class of qualities, but aeolotropic in others. Thus in abstract dynamics a rigid body, or a group of beds rigidly connected, contained within and rigidly attached to a rigid spherical surface, is kinetically symmetrical (§ 285, if its centre of inertia is at the centre of the sphere, and if its momer's of inertia are equal round all diameters. It is also isotrop relatively to gravitation if it is centrobaric (§ 526), so that the centre of figure is not merely a centre of inertia, but a tr centre of gravity. Or a transparent substance may transit light at different velocities in different directions through it (that is, be doubly refracting), and yet a cube of it may and generally does in natural crystals) absorb the same part of a beam of white light transmitted across it perpendicularly t any of its three pairs of faces. Or (as a crystal which exhil s dichroism) it may be aeolotropic relatively to the latter, or to either, optic quality, and yet it may conduct heat equally all directions. Which, however, we know, as recently proved by Deville and Van Troest, tr porous enough at high temperatures to allow very free percolation of gases, |