The corresponding problem of displacement within a cylinder due to a rotation of its boundary is much simpler.

Consider a solution in which ▲ = 0, and is constant and equal to o, we have

v/h=abp, u/h = e−200 (a+b)2 + sin 26.

Thus the above solution satisfies all the conditions.

NOTES.

NOTE A. ON SHEAR AND SHEARING STRESS.

THE term "shear" was first used by engineers to denote tangential stress, and is so used in Rankine's Applied Mechanics. The usage of it for sliding strain in this work might be justified by reference to Sir W. Thomson, now Lord Kelvin, and many other eminent authorities, theoretical and practical. The kind of strain called shear has been considered in ch. 1, and the kind of stress called shearing stress has been considered in ch. II. The object of this note is to insist more fully than is done in those chapters on the twofold character of both shear and shearing stress as they occur in the mathematical expressions. For simplicity we shall limit our consideration to the case of infinitesimal displacements.

The shears are represented by such expressions as dw/dy + dv/dz. Now this expression is the sum of two simple shears, viz. : a simple shear dw/dy of the planes y=const. parallel to the axis z, and a simple shear dv/dz of the planes z=const. parallel to the axis y. In like manner if we define the (infinitesimal) shear of two initially rectangular lines (1) and (2) to be the cosine of the angle between them after strain—a definition which has been shewn to coincide with the definition in terms of sliding motion-then this shear will be made up of a simple shear parallel to (2) of the planes perpendicular to (1), and a simple shear parallel to (1) of the planes perpendicular to (2). The shears that occur in mathematical expressions are in fact generally the sums of two such simple shears which are not at first separated. Thus in the energy-function the terms in a for example are just the same whatever be the proportion in a of the simple shear parallel to y to that parallel to z.

Shearing stress also is of a twofold character, but the like ambiguity does not occur. Shearing stress consists of tangential stresses across two perpendicular planes, but these are always equal.

We know that a simple shear c is equivalent to equal extension and contraction each c, and conversely that equal extension and contraction each e are equivalent to a simple shear of amount 2e, and in the same way the extension and contraction might be taken to be equivalent to two simple shears each of amount e, which combine in the manner explained above;

or again the same extension and contraction will be the equivalents of two simple shears whose sum is 2e and whose ratio is anything whatever.

Equal pressure and tension each P are in like manner equivalent to a shearing stress, but the amount of the shearing stress is P. This shearing stress is really a stress-system consisting of equal tangential stresses P on two perpendicular planes.

The above remarks appear to contain the secret of the " discrepant reckonings of shear and shearing stress" to which Lord Kelvin has frequently called attention. (See e.g. Thomson and Tait's Nat. Phil. Part II. art. 681, and Lectures on Molecular Dynamics p. 176.) The discrepancy appears to arise from the combination in a shear of two simple shears whose ratio it is unnecessary to know, while the tangential stresses combined in a shearing stress are always equal. Writing the discrepant statements in parallel columns we have

Equal extension and contraction each e are equivalent to two simple shears of perpendicular planes; the sum of the shears is 2e and their ratio may be anything whatever.

Equal pressure and tension each Pare equivalent to tangential stresses on two perpendicular planes; each of these is of amount P.

Finally we may note that the values of the two simple shears will be equal if the strain be pure. It follows that, if we regard any small strain as analysed into a small rotation and a small pure strain, then the extensions and contractions to which the pure shears are equivalent are always obtained from the simple shears by precisely the same rule as that by which the pressures and tensions are obtained from the tangential stresses.

NOTE B. ON EOLOTROPIC BODIES.

Eolotropy has been defined in art. 24 as variability of the physical character of a body depending on directions fixed with reference to the body. Fibrous and luminated bodies as well as crystals exhibit such variability of elastic character, and in regard to other physical properties (optical, magnetic, thermal &c.) such variability is exhibited by many wellknown crystalline bodies. The theory of elastic crystals given in the text takes account of elastic properties only. This theory is not proved, and it is not here suggested that, even supposing it proved for elastic properties, it would hold for other physical properties. In other words it is not suggested that the solotropy of a body for the transmission of light waves (for example) is similar to its æolotropy for elastic reactions.

The theory connects elastic quality with crystallographic form; and it leads, in the case of each crystal form, to a certain number of elastic constants. In the absence of definite experimental evidence the assumption that the maximum number of these constants for a given body, and the way they enter into the stress-strain relations, are correctly given appears to have considerable probability. I think it will be generally admitted that a spherical portion of a cubic crystal, for example, would exhibit identity of physical

properties after rotation through 90° about any one of the crystallographic axes. It may however be questioned whether the constants given by the theory are really independent. In other words I think it will be generally admitted that crystalline bodies are at least as nearly isotropic as the theory makes them, but it may be questioned whether they are not more nearly isotropic. Optical experiments appear in some cases to favour an affirmative answer to this question.

Taking again the case of cubic crystals, it is easy to shew that the rigidity (art. 42) for two directions in a principal plane of symmetry, making half right angles with the two principal axes of symmetry that lie in the plane, is (α11-12), while the rigidity for two principal axes of symmetry is ɑ·

This is the property which Lord Kelvin has noted as characteristic of "cubic asymmetry" or "cyboïd æolotropy", and he has, on optical grounds, questioned the existence of bodies possessing the property. (Lectures on Molecular Dynamics p. 158.) The experiments of Prof. Voigt (art. 45) appear to shew that (α11-α12) and a44 have, for some well-known cubic crystals, widely different values.

With regard to cubic crystals it may be as well to notice further two points :

(a) That if the luminiferous ether in any body were similar in elastic quality to the elastic cubic crystals discussed in art. 37 the body would be doubly refracting and would exhibit conical refraction, but the wavesurface would be much more complicated than Fresnel's.

(b) That although the three principal Young's moduluses, the three principal rigidities, and the three principal Poisson's ratios are equal, such , bodies are not "transversely isotropic".

With regard to "transverse isotropy" it may be noticed that a body cannot be transversely isotropic in the plane (x, y) unless its energy-function reduce to the form for hexagonal crystals, viz :

A (e+f)2+Cg2+2F (e+f)g+N (c2-4ef)+L (a2+b2).

For example a tetragonal crystal is not transversely isotropic although it has two principal Young's moduluses, two principal rigidities, and two principal Poisson's ratios equal.

NOTE C. ON BETTI'S METHOD OF INTEGRATION.

Mr Larmor suggests to me that the analysis in arts. 141, 142 admits of a physical interpretation.

Suppose a small spherical element of a solid whose centre is a given point is uniformly extended. If the solid be unlimited and under no bodily force, the displacements at any point can be shewn to be proportional to dr-1/dx, dr-1/dy, dr-1/dz. If the solid be limited by a free surface certain displacements will take place at the surface. If the surface be fixed certain tractions will have to be applied to the surface. The interpretation to be made involves the displacements that exist when the surface is free and the spherical element

about a given point is extended, and the surface-tractions that must be applied to hold the surface fixed when the same state of dilatation is produced in the spherical element.

Equation (40) on p. 244 shews that the dilatation produced at any point by a given system of surface-displacements is proportional to the work done by the tractions that must be applied to hold the surface fixed, when there is dilatation of the spherical element about the point, acting through the given surface-displacements; and equation (41) on the same page shews that the dilatation produced at any point by a given system of surface-tractions is proportional to the work done by these tractions acting through the displacements that take place when the surface is free and there is dilatation of the spherical element about the point.

There is a like interpretation of such equations as (48) and (46) on p. 246 for rotation about any given line in terms of the tractions that must be applied to hold the surface fixed when a spherical element about a given point is made to rotate about the line, and of the displacements that take place when the surface is free and a similar rotation is effected at the point. In fact in the above statements we have merely to read 'rotation about a given line' for 'dilatation'.