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at right angles to one another are equivalent to a single dilatation or condensation equal in all directions. The single stress equivalent to three equal tensions or pressures in directions at right angles to one another is a negative or positive pressure equal in all directions.

(5) If a certain stress or infinitely small strain be defined (Chapter III. Cor. 3, or Chapter IV.) by the ellipsoid

(1+A) X2+(1+B)Y2+(1+C)Z2+DYZ+EZX+FXY=1,

and another stress or infinitely small strain by the ellipsoid

(1+A)X2+(1+B')Y2+(1+C')Z2+D′YZ+E'ZX+F′XY=1. where A, B, C, D, E, F, &c., are all infinitely small, their resultant stress or strain is that represented by the ellipsoid

(1+A+A)X2+(1+B+B')Y2+(1+C÷C)Z2+(D+D)YZ+(E+E)ZX
+(F+F)XY=1.

CHAPTER VIII.—Specification of Strains and Stresses by their
Components according to chosen Types.

Prop. Six stresses or six strains of six distinct arbitrarily chosen types may be determined to fulfil the condition of having a given stress or a given strain for their resultant, provided those six types are so chosen that a strain belonging to any one of them cannot be the resultant of any strains whatever belonging to the others,

For, just six independent parameters being required to express any stress or strain whatever, the resultant of any set of stresses or strains may be made identical with a given stress or strain by fulfilling six equations among the parameters which they involve; and therefore the magnitudes of six stresses or strains belonging to the Six arbitarily chosen types may be determined, if their resultant be assumed to be identical with the given stress or strain.

Cor. Any stress or strain may be numerically specified in terms of numbers expressing the amounts of six stresses or strains of six arbitrarily chosen types which have it for their resultant.

The

Types arbitrarily chosen for this purpose will be called types of reference. The specifying elements of a stress or strain will be called its components according to types of reference. specifying elements of a strain may also be called its coordinates, with reference to the chosen types.

Examples-(1) Six strains in each of which one of the six edges of a tetrahedron of the solid is elongated while the others remain unchanged, may be used as types of reference for the specification of any kind of strain or stress. The ellipsoid representing any one of those six types will have its two circular sections parallel to the faces of the tetrahedron which do not contain the stretched side.

(2) Six strains consisting, any one of them, of an infinitely small alteration elther of one of the three edges, or of one of the three angles between the faces, of a parallelepiped of the solid. while the other five angles and edges remain unchanged, may be taken as types of reference, for the specification of either stresses or strains. In some cases, as for instance in expressing the probable elastic properties of a crystal of Iceland spar, it night possibly be convenient to use an oblique parallelepiped for such a system of types of reference; but more frequently it will be convenient to adopt a system of types related to the deformations of a cube of the solid.

CHAPTER IX.-Orthogonal Types of Reference. Def. A normal system of types of reference is one in which the strains or stresses of the different types are all six mutually orthogonal (fifteen conditions). A normal system of types of reference may also be called an orthogonal system. The elgrients specify ing, with reference to such a system, any stress or strain, will be called orthogonal components or orthogonal coordinates.

Examples(1) The six types described in Example (2) of Chapter VIII. are clearly orthogonal, if the parallelepiped referred to is rectangular. Three of these are simple longitudinal extensions, parallel to the three sets of rectangular edges of the parallelepiped The remaining three are plane distortions parallel to the faces, their axes bisecting the angles between the edges. They constitute the system of types of reference uniformly used hitherto by writers on the theory of elasticity.

(2) The six strains in which a spherical portion of the solid is changed into ellipsoids having the following equations

(1+A)X2+Y2+Z2=1

X2+ (1+B) Y2+72=1 X2+Y2+ (1+C)Z2=1 x2+Y+Z3+ DYZ=1 X2+Y2+22+ EZX=1 X2+Y2+Z2+FXY=1,

are of the same kind as those considered in the preceding_example, and therefore constitute a normal system of types of reference. The resultant of the strains specified, according to those equations, by the elements A, B, C, D, E, F, is a strain in which the sphere becomes an ellipsoid whose equation-see above, Chapter VII. Ex (5)-is

(1+A)X2+(1+B)Y2+(1+C)Z2+DYZ+EZX+FXY=1.

(3) A compression equal in all directions (I.), three simple distortions hav Ing their planes at right angles to one another and their axes bisecting the angles between the lines of intersection of these planes (II.) (III.) (IV.), any simple or compound distortion consisting of a combination of longitudinal strains parallel to those lines of intersections (V.), and the distortion (VI), constituted from the same elements which is orthogonal to the last, afford a system of six mutually orthogonal types which will be used as types of reference below in expressing the elasticity of cubically isotropic solids. (Compare Chapter X. Example 7 below.)

This example, as well as (7) of Chapter X. (5)`o. XI., and the example of Chapter XII, are intended to prepare for the application of the theory of Principal Elasticities to cubically and spherically isotropic bodies, in Part II. Chapter XV.

2 The "axes of a simple distortion" are the lines of its two component longisudinal strains.

CHAPTER X.-On the Measurement of Strains and Stresses

Def. Strains of any types are said to be to one another in the same ratios as stresses of the same types respectively, when any particular plane of the solid acquires, relatively to another plane parallel to it, motions in virtue of those strains which are to one another in the same ratios as the normal components of the forces between the parts of the solid on the two sides of either plane due to the respective stresses.

Def. The magnitude of a stress and of a strain of the same type are quantities which, multiplied one by the other, give the work done on unity of volume of a body acted on by the stress while acquiring the strain.

Cor. 1. If x, y, z,, n, denote orthogonal components of a certain strain, and if P, Q, R, S, T, U denote components, of the same type respectively, of a stress applied to a body while acquiring that strain, the work done upon it per unit of its volume will be Pz+Qy+Re+SE+Tn+UZ.

Cor. 2. The condition that two strains or stresses specified by (x, y, z, E, n, () and (x', y', z, E', n', '), in terms of a normal system of types of reference, may be orthogonal to one another is

xx'+w+2+EE'+nn'+SS'=0.

Cor. 3. The magnitude of the resultant of two, three, four, five. or six mutually orthogonal strains or stresses is equal to the square root of the sum of their squares. For if P, Q, &c., denote several orthogonal stresses, and F the magnitude of their resultant; and x, y, &c., a set of proportional strains of the same types respectively, and r the magnitude of the single equivalent strain, the resultant stress and strain will be of one type, and therefore the work done by the resultant stress will be Fr. But the amounts done by the several components will be Pr, Qy, &c., and therefore

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F

Fr Fr

=

and

which gives F P2 + Q2 + &c.`

x2 + y2 + &c.. which gives

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➡ x2 + y2 + &c. Cor. 4. A definite stress of some particular type chosen arbitrarily may be called unity; and then the numerical reckoning of all strains and stresses becomes perfectly definite.

Def. A uniform pressure or tension in parallel lines, amounting be called a stress of unit magnitude, and will be reckoned as positive in intensity to the unit of force per unit of area normal to it, will when it is tension, and negative when pressure.

Examples.-(1) Hence the magnitude of a simple longitudinal strain, in which lines of the body parallel to a certain direction experience elongation to an extent bearing the ratio x to their original dimensions, must be called K.

(2) The magnitude of the single stress equivalent to three simple pressures in directions at right angles to one another each unity is 3; a uniform compression in all directions of unity per unit of surface is a negative stress equal to 3 in absolute value.

(3) A uniform dilatation in all directions, in which lineal dimensions are angmented in the ratio 1:1+x, is a strain equal in magnitude to z✅✔✅3; or a uniE form "cubic expansion " E is a strain equal to 3.

(4) A stress compounded of unit pressure in one direction and an equal tension In a direction at right angles to it, or which is the same thing, a stress compounded of two balancing couples of unit tangential tensions in planes at angles of 45° to the direction of those forces, and at right angles to one another amounts in magnitude to √2.

(5) A strain compounded of a simple longitudinal extension r, and a simple longitudinal condensation of equal absolute value, in a direction perpendicular to it, is a strain of magnitude 2; or, which is the same thing (if σ = 2x), & simple distortion such that the relative motion of two planes at unit distances parallel to either of the planes bisecting the angles between the two planes mentioned above is a motion a parallel to themselves, is a strain amounting in magnitude to

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(6) If a strain be such that a sphere of unit radius in the body becomes an ellipsoid whose equation is

(1 − A)X2 + (1 − B) Ya + (1 — C)Z3 — DYZ — EZX — FXY = 1, the values of the component strains corresponding, as explained in Example (3 of Chap. IX. to the different coefficients respectively, are

D E F: JA, JB. JC,

For the components corresponding to A, B, C are simple longitudinal strains, in which diameters of the sphere along the axes of coordinates become elongated froin 2 to 2+ A, 2 + B. 2 + C respectively: D is a distortion in which diameters in the plane YOZ, bisecting the angles YOZ and Y'OZ, become respectively elongated and contracted from 2 to 2+D, and from 2 to 2- ¿D; and so for the others. Hence, if we take r, 3. r. E, n, to denote the magnitudes of six

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Cor. Hence, if variables X, Y, Z be transformed to any other set (X, Y, Z) fulfilling the condition of being the coordinates of the same point, referred to another system of rectangular axes, the coefficients x, y, z, &c., x, y, z,, &c., in two homogeneous quadratic functions of three variables.

and

(1 − 2x) X2 + (1 − 2y) Y2 + (1 − 2z) Z2 − 2 √/2{EYZ+nZX+{X})
(1−2x)X2+(1−2y,) Ya + (1 − 2x,)Z2 −2√2(E,YZ+n,ZX+Y,XY),

and the corresponding coefficients z', y', s', &c., x',,,,,, &c., in these functions transformed to x, y, z, will be so related that

x'x,+y'V,+z's,+E' E',+n'n',+S′S,−xx,+vy,+=z,+EE,+nn,+53,;

or the function xx,+wy,+21,+EE,+nn,+CS, of the coefficients is an "invariant" for linear transformations fulfilling the conditions of transformation from one to another set of rectangular axes. Since r+y+ and x,+y+, are clearly invariants also, it follows that AA,+BB,+CC,+2DD,+2EE,+2FF, is an invariant function of the coefficients of the two quadratics

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CHAPTER XI.-On Imperfect Concurrences of two Stress or Strain Types.

Def. The concurrence of any stresses or strains of two stated types is the proportion which the work done when a body of unit volume experiences a stress of either type, while acquiring a strain of the other, bears to the product of the numbers measuring the stress and strain respectively.

Cor. 1. In orthogonal resolution of a stress or strain, its component of any stated type is equal to its own amount multiplied by its concurrence with that type; or the stress or strain of a stated type which, along with another or others orthogonal to it, have a given stress or strain for their resultant, is equal to the amount of the given stress or strain reduced in the ratio of its concurrence with that stated type.

Cor. 2. The concurrence of two coincident stresses or strains is unity; or a perfect concurrence is numerically equal to unity.

Cor. 3. The concurrence of two orthogonal stresses and strains is

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six orthogonal types of reference, and l', m', n', x', u', ✔ those of the other.

Cor. 7. The most convenient specification of a type for strains Or stresses, being in general a statement of the components, according to the types of reference, of a unit strain or stress of the type to be specified, becomes a statement of its concurrences with the types of reference when these are orthogonal.

Examples-(1) The mutual concurrence of twe simple longitudinal strains or stresses, inclined to one another at an angle 6, is cos2 0.

(2) The mutual concurrence of two simple distortions in the same plane whose axes are inclined at an angie to one another, is cos3 ✪ – sin2 0, os 2 sin (45°-) cos (45° — 0).

Hence the components of a simple distortion d along two rectangular axes in its plane, and two others bisecting the angle between these taken as axes of component simple distortions, are

(cos-sin? 0) and 8.2 sin 0 cos 8

respectively, if be the angle between the axis of elongation in the gfren distortion and in the first component type.

(3) The mutual concurrence of a simple longitudinal strain and a simple distortion is √2.cos a cos ẞ,

if a and ẞ be the angles at which the direction of the longitudinal strain 19 inclined to the line bisecting the angles between the azes of the distortion; it is also equal to

1 (0052 - cos2),

if and denote the angles at which the direction of the longitudinal strain is inclined to the axis of the distortion.

(4) The mutual concurrence of a simple longitudinal strain and of a uniform 1 dilatation is

(5) The specifying elements exhibited in Example (7) of the preceding Chapter are the concurrences of the new system of orthogonal types described in Example (3) of Chap. IX. with the ordinary system, Examples (1) and (2). Chap. LX.

CHAPTER XII.-On the Transformation of Types of Reference for Stresses or Strains.

To transform the specification (x, y, z, E, n, C) of a stress or strain with reference to one system of types into (21, Z, X3, X4 X5, Xs) with reference to another system of types. Let (a, b, c, c12 $1,91) be the components, according to the original system, of a unit strain of the first type of the new system; let (ag bg ca cn f g 95) be the corresponding specification of the second type of the new systein; and so on. Then we have, for the required formula of transformation

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Example.-The transforming equations to pass from a specification (r. . E. n. in terms of the system of reference used in Examples (6) and (7), Chapter X., to a specification (o. En, 3., w) in terms of the new system described in Example (3) of Chapter IX., and specified in Example (7) of Chapter X., are as follows:

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PART II.-ON THE DYNAMICAL RELATIONS BETWEEN STRESSES
AND STRAINS EXPERIENCED BY AN ELASTIC SOLID.
CHAPTER XIII.-Interpretation of the Differential Equation
of Energy.

In a paper on the Thermo-elastic Properties of Matter, published in the first number of the Quarterly Mathematical Journal, April 1855, and republished in the Philosophical Magazine, 1877, secondl half year, it was proved, from general principles in the theory of the Transformation of Energy, that the amount of work (w) required to reduce an elastic solid, kept at a constant temperature, from one stated condition of internal strain to another depends solely on these two conditions, and not at all on the cycle of varied states through which the body may have been made to pass in effecting the change, provided always there has been no failure in

the elasticity under any of the strains it has experienced. Thus for a homogeneous solid homogeneously strained, it appears that 10 is a function of six independent variables x. y, z, ¿, n, S, by which the condition of the solid as to strain is specified. Hence to strain the body to the infinitely small extent expressed by the variation from (x, y, z, E, n, () to (x+dx. y+dy, z+dž, §+d§, n+dn, $+d$), the work required to be done upon it is

dw

'dy+

dw

du
dr
an+

dn વ

du dw dx+ dz dy dz The stress which must be applied to its surface to keep the body in equilibrium in the state (x, y, z, E, n, C) must therefore be such that it would do this amount of work if the body, under its action, were to acquire the arbitrary strain dx, dy, dz, de, dn, ds; that is, it must be the resultant of six stresses:-one orthogonal to the five strains dy, dz, dɛ, dŋ, d§, and of such a magnitude as to do the work dx de when the body acquires the strain de; a second orthogonal to dx, dz, d§, dn, d§, and of such a magnitude as to do the work dy when the body acquires the strain dy; and so on. dy If a, b, c, f, g, h denote the respective concurrences of these six stresses, with the types of reference used in the specification (x, y, z, En, S) of the strains, the amounts of the six stresses which fulfil those conditions will (Chapter XI.) be given by the equations

dw

P=

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Q=

1 d
'b dy'

R=
1 du

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and the types of these component stresses are determined by being orthogonal to the fives of the six strain-types, wanting the first, the second, &c., respectively.

Cor. If the types of reference used in expressing the strain of the body constitute an orthogonal system, the types of the component stresses will coincide with them, and each of the concurrences will be unity. Hence the equations of equilibrium of an elastic solid referred to six orthogonal types are simply

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dir

P=dz' Q= 'ẫy'

dw S=

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dro

R=dz

1 U=

CHAPTER XIV.-Reduction of the Potential Function, and of the
Equations of Equilibrium, of an Elastic Solid to their simplest
Forms.

If the condition of the body from which the work denoted by w is reckoned be that of equilibrium under no stress from without. and if x, y, z, E, n, be chosen each zero for this condition, we shall have, by Maclaurin's theorem,

w=H2(x, y, 2, E, n. Y)+H3(x, y, z, E, n, Y+ &c., where H, H3, &c., denote homogeneous functions of the second order, third order, &c., respectively. Hence dw dw dx dy &c., will each be a linear function of the strain coordinates, together with functions of higher orders derived from H,, &c. (section 37 above) that, within the elastic limits, the stresses are But experience shows very nearly if not quite proportional to the strains they are capable of producing; and therefore H,, &c., may be neglected, and we have simply

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the simplest possible form under which they can be presented. The interpretation can be expressed as follows.

Prop. An infinite number of systems of six types of strains or stresses exist in any given elastic solid such that, if a strain of any one of those types be impressed on the body, the elastic reaction is balanced by a stress orthogonal to the five others of the same system.

CHAPTER XV.-On the Six Principal Strains of an Elastic Solid. To reduce the twenty-one coefficients of the quadratic terms in the expression for the potential energy to six by a linear transforma

tion. we have only fifteen equations to satisfy; while we have thirty disposable transforming coefficients. there being five independent elements to specify a type, and six types to be changed. Any further condition expressible by just fifteen independent equations may be satisfied. and makes the transformation determinate. Now the condition that six strains may be mutually orthogonal is expressible by just as many equations as there are different pairs of six things, that is, fifteen. The well-known algebraic theory of the linear transformation of' quadratic functions shows for the case of Six variables-(1) that the six coefficients in the reduced forme the roots of a "determinant" of the sixth degree necessarily real; (2) that this multiplicity of roots leads determinately to one, and only one system of six types fulfilling the prescribed conditious, unless two or more of the roots are equal to one another, when there will be an infinite number of solutions and definite degrees of isotropy among them; and (3) that there is no equality between any of the six roots of the determinant in general, when there are twenty-one independent coefficients in the given quadratic.

Prop. Hence a single system of six mutually orthogonal types may be determined for any homogeneous elastic solid, so that its potential energy when homogeneously strained in any way is expressed by the sum of the products of the squares of the components of the strain, according to those types, respectively multiplied by six determinate coefficients.

Def. The six strain-types thus determined are called the Six Principal Strain-types of the body.

The concurrences of the stress-components used in interpreting the differential equation of energy with the types of the straincoordinates in terms of which the potential of elasticity is expressed, being perfect when these constitute an orthogonal system, each of six principal strain-types are chosen for the coordinates. The equa the quantities denoted above by a, b, c, f, g, h, is unity when the tions of equilibrium of an elastic solid may therefore be expressed

as follows:

P-A, Q-By, R-Cz,
S-FE, TG, U-HY,

where x, y, z, §, n, denote strains belonging to the six Principal Types, and P, Q, R, S, T, U the components according to the same types, of the stress required to hold the body in equilibrium when in the condition of having those strains. The amount of work that must be spent upon it per unit of its volume, to bring it to this state from an unconstrained condition, is given by the equation

w=}(Ax2+By2+Cz2+FF®+Gn®+HY3)

Def. The coefficients A, B, C, F, G, H are called the six Principal Elasticities of the body.

The equations of equilibrium express the following proposi tions:

Prop. If a body be strained according to any one of its six Principal Types, the stress required to hold it so is directly concurrent with the strain.

Examples.-(1) If a solid be cubically isotropic in its elastic properties, as crystals of the cubical class probably are, any portion of it will, when subject to a uniform positive or negative normal pressure all round its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensation is one of its six principal strains. Three plane distortions with axes bisecting the angles between the edges of the cube of symmetry are clearly also principal strains, and since the three corresponding principal elasticities are equal to one Lastly, a plane distortion whose axes coincide with any two edges of the cube, another, any strain whatever compounded of these three is a principal strain. being clearly a principal distortion, and the principal elasticities corresponding to the three distortions of this kind being equal to one another, any distortion compounded of them is also a principal distortion.

Hence the system of orthogonal types treated of in Examples (3) Chap. IX., and (7) Chap. X., or any system in which, for (II.), (III.), and (IV.), any three orthogonal strains compounded of them are substituted, constitutes a system of six Principal Strains in a solid cubically isotropic. There are only three distinct Principal Elasticities for such a body, and these are-(A) its modulus of compressibility, (B) its rigidity against diagonal distortion in any of its principal planes (three equal elasticities), and (C) its rigidity against rectangular distortions of a cube of symmetry (two equal elasticities).

(2) In a perfectly isotropic solid. the rigidity against all distortions is equal. Hence the rigidity (B) against diagonal distortion must be equal to the rigidity (C) against rectangular distortion, in a cube; and it is easily seen that if this condition is fulfilled for one set of three rectangular planes for which a substance is isotropic, the isotropy must be complete. The conditions of perfect or spherical isotropy are therefore expressed in terms of the conditions referred to in the preceding example, with the farther condition B=C.

A uniform condensation in all directions, and any system whatever of five orthogonal distortions, constitute a system of six Principal Strains in a spherically isotrople solid. Its Principal Elasticities are simply its Modulus of Compressibility and its Rigidity.

Prop. Unless some of the six Principal Elasticities be equal to one another, the stress required to keep the body strained otherwise than according to one or other of six distinct types is oblique to the strain.

Prop. The stress required to maintain a given amount of strain is a maximum or a maximum-minimum, or a minimum, if it is of one of the six Principal Types.

Cor. If A be the greatest and H the least of the six quantities A, B, C, F, G, H, the principal type to which the first corresponds is that of a strain requiring a greater stress to maintain it than any

other strain of equal amount; and the principal type to which the last corresponds is that of a strain which is maintained by a less stress than any other strain of equal amount in the same body. The stresses corresponding to the four other principal strain-types have each the maximum-minimum property in a determinate way.

Prop. If a body be strained in the direction of which the concur rences with the principal strain-types are l, m, n, λ, μ, v, and to an amount equal to r, the stress required to maintain it in this state will be equal to ♫r, where

Q = (A22 + B2m2 + C3n2 + F®)2+G3μ3 +H22)* ̧

periences simply a motion of translation-but a motion differing from the motions of particles in planes parallel to the same. Let OX, OY, OZ be three fixed rectangular axes; OX perpendicular to the wave front (as any of the parallel planes of moving particles referred to in the definition is called), and OY, OZ in the wave front. Let x+u, y+v, z+w be the coordinates at time t of a particle which, if the solid were free from strain, would be at (z, yi ;). The definition of wave motion amounts simply to this, that u, v, 30 are functions of x and t.

The strain of the sclid (Chap. VII. above) is the resultant of a du

and will be of a type of which the concurrences with the principal simple longitudinal strain in the direction OX, equal to dz types are respectively

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Prop. A homogeneous elastic solid, crystalline or non-crystalline, subject to magnetic force or free from magnetic force, has neither any right-handed or left-handed, nor any dipolar, properties dependent on elastic forces simply proportional to strains.

Cor. The elastic forces concerned in the luminiferous vibrations of a solid or fluid. medium possessing the right or left-handed property, whether axial or rotatory, such as quartz crystal, or tartaric acid, or solution of sugar, either depend on the heterogeneousness or on the magnitude of the strains experienced.

Hence as they do not depend on the magnitude of the strain, they do depend on its heterogeneousness through the portion of a medium containing a wave.

Cor. There cannot possibly be any characteristic of elastic forces simply proportional to the strains in a homogeneous body, corresponding to certain peculiarities of crystalline form which have been observed,-for instance corresponding to the plagiedral faces discovered by Sir John Herschel to indicate the optical character, whether right-handed or left-handed, in different specimens of quartz crystal, or corresponding to the distinguishing characteristics of the crystals of the right-handed and left-handed tartaric acids obtained by M. Pasteur from racemic acid, or corresponding to the dipolar characteristics of form said to have been discovered in electric crystals.

CHAPTER XVI.-Application of Conclusions to Natural Crystals. It is easy to demonstrate that a body, homogeneous when regarded on a large scale, may be constructed to have twenty-one arbitrarily prescribed values for the coefficients in the expression for its potential energy in terms of any prescribed system of strain coordinates. This proposition was first enunciated in the paper on the Thermo-elastic Properties of Solids, published April 1855, in the Quarterly Mathematical Journal alluded to above. We may infer the following.

dv, dw,
dx dx

and

two differential slips parallel to OY and OZ, constituting simple distortions of which the numerical magnitudes (Chap. X.)

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M (5). The three components of the whole force due to the tractions on the sides of an infinitely small parallelepiped dz, dy, &z of the solid are clearly

dar.8yde, dr.8yde, and droz.dyds

dr

1

dr

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(6),

and therefore, if p be its density, and consequently pdx dy dz its
mean, the equations of its motion are
dudp do dq diy dr
Pandr Pdr2 ̄dx' Pdd

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=

=M

(7).

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(3).

Prop. A solid may be constructed to have arbitrarily prescribed values for its six Principal Elasticities and an arbitrary orthogonal system of six strain-types, specified by fifteen independent eleinents, for its principal strains: for instance, five arbitrarily chosen systems of three rectangular axes, for the normal axes of five of the Principal Types; those of the sixth consequently in general These, putting for p, q, r their values by (5), become distinct from all the others, and determinate; and the six times two ratios between the three stresses or strains of each type, also determinate. The fifteen equations expressing (Chap. VI.) the mutual orthogonality of the six types determine the twelve ratios for the six types, and the three quantities specifying the axes of the sixth type in the particular case here suggested or generally the fifteen equations determino fifteen out of the thirty quantities (viz. twelve ratios and eighteen angular coordinates) specifying six Principal Types.

Cor. There is no reason for believing that natural crystals do not exist for which there are six unequal Principal Elasticities, and six distinct strain-types for which the three normal axes constitute six distinct sets of three principal rectangular axes of elasticity.

It is easy to give arbitrary illustrative examples regarding Principal Elasticities: also, to investigate the principal strain-types and the equations of elastic force referred to them or to other uatural types, for a body possessing the kind of symmetry as to elastic forces that is possessed by a crystal of Iceland spar, or by a crystal of the "tesseral class," or of the included "cubical class." Such illustrations and developments, though proper for a students' text book of the subject, are unnecessary here.

For applications of the Mathematical Theory of Elasticity to the question of the earth's rigidity and elasticity as a whole, and to the equilibrium of elastic solids in general, which are beyond the scope of the present article, the reader is referred to Thomson and Tait's Natural Philosophy, §§ 588, 740, 832, 849, and Appendix C.

CHAPTER XVII.-Plane Waves in a Homogeneous Eolotropic
Solid.

A plane wave in a homogeneous elastic solid is a motion in which every line of particles in a plane parallel to one fixed plane ex.

And by (4) and (1) we have

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Hence, instead of three different waves with different velocities, we
have just two,-a wave (like that of sound in air or other elastic
fluid) in which the motions are perpendicular to the wave front,
and the other (like the waves of light in an isotropic medium) in
which the motions are parallel to the wave front.

Waves in an Incompressible Solid (Eolotropic or Isotropic).-If
the solid be incompressible, we have A∞o, and u must be zero.
Hence
W=Bn®+CS®+2Dn

and by a determinantal quadratic. instead of cubic, we find two
wave-velocities and two wave-modes, in each of which the motion
is parallel to the wave front. In the case of isotropy the two
wave velocities are equal.

It is to be noticed that M,, M., M, in the preceding investigation corresponding to the particular plane chosen for the wave front. In are not generally true "principal moduluses." but special moduluses the particular case of isotropy, however, the equal moduluses modulus of rigidity, but M, is a mixed modulus of compressibility M., M. of (11) are principal moduluses, being each equal to the and rigidity-not a principal modulus. In the case of incompressibility, the two moduluses found from the determinantal quadratic rally, because the distortions by the differential motions of planes by the process indicated above are not principal moduluses genetangential stresses orthogonal to them, which do not influence the of particles parallel to the wave front must generally give rise to wave motion. (W.TH.)

ELATERIUM, a drug consisting of a sediment deposited by the juice of the fruit of Ecbalium Elaterium, the squirt ing cucumber (see vol. vi. p. 688.) To prepare it, the fruit is sliced lengthwise and slightly pressed; the greenish and slightly turbid juice thus obtained is strained and set aside; and the deposit of elaterium formed after a few hours is collected on a linen filter, rapidly drained, and dried on porous tiles at a gentle heat. Elaterium is met with in commerce in light, thin, friable, flat or slightly incurved opaque cakes, of a greyish-green colour, bitter taste, and tea-like smell. The best kind is the English, prepared at Hitchin Market Deeping, Mitcham, and elsewhere; the Maltese is generally very inferior. Elaterium is an exceedingly powerful hydragogue and drastic purgative, and not unfrequently produces vomiting. Its active principal is elaterin, a crystallizable body of the formula C20H2805

ELBA, the Aibalía of the Greeks, and Ilva of the Romans, is an island in the Mediterranean Sea, forming part of the Italian province of Livorno, and lying about 6 miles from the mainland of Italy, from which it is separated by the channel of Piombino, and about 34 miles E. of Corsica. It has a very irregular coast outline, is 18 miles long and 21 to 10 miles broad, and has a total area of nearly 90 square miles. It is throughout mountainous, and the highest point, Monte Capanne, is 2925 feet above sea-level. The western portion of the island is granitic, the eastern consists mainly of the sandstone locally known as verrucano, which in some places passes into a talc slate. In the vicinity of Porto Ferrajo the hills are cretaceous. The climate is mild, and, except at some spots on the coast, healthy. Springs are numerous, and the soil is not infertile; but agriculture and cattle-rearing are neglected, and there are no manufactures. Wine, wheat, aloes, dyer's lichen, and olives and other fruits are produced. The sardine and tunny fisheries, and the manufacture of sea-salt are of some importance; but the principal industry is mining. The iron mines are mostly in the vicinity of Rio Inferiore, and yield abundance of ore, chiefly hematite, of excellent quality. On account of the lack of fuel the ore is not smelted on the island, but is shipped direct to Follonica on the neighbouring coast of Italy, and to the ports of France and England. Marble, alabaster, sulphur, and ores of tin, lead, and silver are among the other mineral products The principal places in Elba are the chief town Porto Ferrajo, with about 5000 inhabitants, the residence of Napoleon from May 4, 1814, to February 26, 1815, Rio Ferrajo, San Pietro, Porto Longone, and the village of Capoliveri. The population of the island in 1871 was 21,755.

The Argonauts, in quest of Circo are said to have loaded at Portus Argous (Apy@os Auh), now Porto Ferrajo, in Elba. The island was early famous for the richness of its mines, alluded to by Virgil (En. x. 173). It was attacked by Phayllus with a Syracusan fleet, 458 B.C., and subsequently by Apelles, who is stated to have subjugated it. In the 10th century it became a possession of the Pisans, from who:n it was taken by the Geuouse in 1290. It fell

eventually into Spanish hands, came in 1786 under the jurisdiction of Naples, and in 1801 was ceded to the king of Etruria by the treaty of Lunéville. It was united to France in 1803, made over in the following year, and in 1860 annexed to Italy. to Napoleon by the Treaty of Paris in 1814, restored to Tuscany ELBE, the Albis of the Romans and the Labe of the Bohemians, a large river of Germany, with a total length of 705 miles, and a drainage area of about 55,000 square miles. It rises in Bohemia not far from the frontiers of Silesia, on the southern side of the Riesengebirge or Giants' Mountains, in 50° 46′ N. lat. and 15° 32′ E. long. Of the numerous small streams (Seifen or Flessen, as they are named in the district) whose confluent waters compose the infant river, the most important are the Weisswasser, or White Water, and the Elbseifen; the former rises to the S.W. of the Schneekuppe in the White Meadow, and the latter in a stone fountain in the Elb Meadow. Augmented successively by the Adler, the Iser, the Moldau, and the Eger, it cuts its way through the Mittelgebirge of Bohemia, traverses the sandstone mountains of Saxon Switzerland, and with a general N.W. direction continues to meander through Saxony, Anhalt, and Hanover, until at length it falls into the German Ocean about 53° 5′ N. lat. and 8° 50′ E. long. The principal towns on its banks are Leitmeritz, Pirna, Dresden, Meissen, Torgau, Wittenberg, Magdeburg, Wittenberge, Harburg, Hamburg, and Altona. A short distance above Hamburg the stream divides into a number of branches, but they all reunite before reaching the ocean.' At its source the Elbe is about 4600 feet above the level of the sea; after the first 40 miles of its course it is still 658 feet; but at Dresden it is only 279, and at Arneburg in Brandenburg only 176. At Königgratz the width is about 100 feet, at the mouth of the Moldau about 300, at Dresden 960, and at Magdeburg over 1000. The tide is perceptible as far up as Geesthacht. Of the fifty and more tributaries belonging to the system the most important are the Moldau, the Eger, the Mulde, and the Saale, the Moldau having a course of 267 miles, the Eger of 235, the Mulde of 185, and the Saale of 220. Though the channel in some places, and especially in the estuary, is encumbered with sandbanks and shallows, the Elbe is of great importance as a means of communication, steamboats being able to ascend the main stream as far as Melnick, and to reach Prague by means of the Moldau. Some idea of the extent of its traffic may be obtained from the statement that in 1870 at Schandau 489 passenger-steamers and 2658 vessels and barges of various kinds passed up the stream, and 489 passenger steamers, 2865 ships, and 1505 rafts down the stream. By one line of canal it communicates with Lübeck, by another with Bremen, and by others with the great network of Mecklenburg and Brandenburg; and several new lines are projected, by which a direct way will be opened up to Hanover, Leipsic, and various other important cities. For details see Dr Th. H. Schunke's "Die Schifffahrts-Kanäle im Deutschen Reiche," in Peter mann's Mittheil., 1877.

VII.

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