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it rljcht angle* to one another are equivalent to a single dilatation or condensation equal lo all dfrecrlon*. The tingle atreaa equivalent to three equal tensions or pressure* In directions at right angles to one another la a negative or posture ITtsflu.ro equal In all directions.

(») If a certain stress or Infinitely small strain he defined (Chapter IIL Cor. 2, or Chapter IV.) by the ellipsoid

(1+A) X»+a+B)T»+a+C)Z»+DTZ+EZI+FST5=l, and another stress or Infinitely small strain by the ellipsoid;

a+A0X*+O+B')Y*+a+C')Z1+D'YZ+E-ZX+rXT=l.

where A, B, C, D. E, F, dec, are all Infinitely small, their resultant stress or strain la that represented by the ellipsoid

a+A+A^'+a+B+B^n+a+C+OZ^+CD+DOTI+CE+EOZX

Chapter VIII.— Specification of Strains and Stresses by their Components according lo chosen Types,

Prop. 8ix stresses or six strains of six distinct arbitrarily chosen types may be determined to fulfil the condition of having a given stress or a giren 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 fulfil ling six equations among the parameters which they involve; and therefore the magnitudes of six stresses or strains belonging to the six arbitarilv 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.

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

Eiempift —O.) Six strains In each of whleh one of the six edges of a tetrahedron of the solid Is elongated while the of hers remain onehanged, maybe used as types of reference for the specification of any kind of strain oratress. The elilpsolii representing any one of those six types will hare Its two circular sections parallel to the faces of the tetrahedron which do not contain the stretched side.

<2> Six strains consisting, anyone of them, of an Infinitely small alteration el' her of one of the three edges, or of one of the three angles between the faces, of a parallelepiped of the sol<d. while the other fire 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 might possibly be convenient to nse an oblique parallelepiped for such a system of types of reference; hut more frrqoently It will b* convenient to adopt a system of types related to the deformations of a cube of the solid.

Chapter IX.—Orthogonal Types of Refercnos, D*f. 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 elonents specify, ing, with reference to such a system, any stress or strain, will be called orthogonal components or orthogonal coordinates.

MrsmsiM <D The six types described In Example (7) of Chapter VTTI. are clearly orthogonal If the parallelepiped referred to Is rectangular. Three of these arc simple longitudinal extensions, parallel to the three sets of rectangular edges or the parallelepiped The remaining three are plana distortions parallel to the faces, their axes bisecting the ancles between the edges. They constitute the system of types of reference uniformly used hitherto by writers on the theory of elasticity.

(7) The six strains In which a spherical portion of the solid is changed into ellipsoids having the following equation*—

a+Axv+THz'-i

Xa-i (l+B)T*+Z»-l

x»4 Y*+a+c>z>=i

XJ+TM-ZJ+EZX = 1
X*+T*+Z*+FXTml.

are of the same kind as those considered In the preceding example, snd therefore constitute a normal syetem of types of reference. The resultant of the strains specified, sccording to those equations, by the elements A, B, C, D, E, F, Is a strain in which the sphere becomes an eUlpsold whose equation—eee above, Chapter VII. Ex IS)—ts

<l + A)X»+(l + B)Y*+0 + OZ7+DYZ+EZJE-rrXY»l.

f3)< A compression equal in all directions 11.), three simple distortions having their planes at right angles to one another and their axes'bisecting the angles between the lines of Intersection of the** planes (11.) (III.) (IV.), any simple or compound distortion consisting of a combination of longitudinal strains psrallel to taovj linos of intersections (V.), and the distortion (AT ), con■tltoted from the same elementa which is orthogonsl to the last, afford a system nf six mutually orthogonal typea which will be used aa types of reference below In sxpresflinir (he elasticity of cnbieajly isotropic solid*. (Compare Chapter X. Example 7 below.)

» This example, aa well as (7) of Chapter X. (5)"oi XL. and the example of Chspter XII, are Intended to prepare for tahe application of the theory of Principal Elasticities to cublcally and spherically Isotropic bodies, in Part IL Chspter XV. ,

» The "axes of a simple distortion" ave the lines of its two component longt

Chapteb 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 ono another in the same ratios as the normal components of die forces between the parts of the solid on the two sides of either plane dne 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, a, L *# C denote orthogonal components of a certain strain, and if r, Q, R, S, T, U denote components, of the same type respectively, of a stress applied to a body while acquiring that strain, toe work done upon it per unit of its volume will be

Pjr-rQr+B#+S£+Tn+Uj.

Cor. 2. The condition that two strains or stresses specified by (g, y, s, {, n> 0 *nd (af*, j/, */, f), in terms of a normal system of types of reference, may be orthogonal to one another is aV+rs'+u'+fr+fl'i'+jr-O.

Cor. 8. The rasgnitude 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 severs! components will be Txt Qy, kc, and therefore

Fr=Px + Qy+Ac

Now we have, to express the proportionality of the stresses and strains,

a* *• f Each member must be equal to

y + ty + flte.. P* + Or + Ac, *

and also equal to

Pj -f Qg + Ac.
7+P + Ac •

—which gives F» - P* + Q» + 4c

Hence and

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r jr» + r» + Ac..

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

Examptet,—(1) Rence 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 K to thelr>rlglna] 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 — */Z: a uniform compression In all directions of unity per unit of surface Is a negative stress equal to In absolute value.

(3) A unilorm dilatation In all directions. In which lineal dimensions are angmented in the ratio 1:1 + x, la a strain equal In magnitude to x •/ 3; or a uni

form " cable expansion" E ta a strain equal to ,

(4) A stress compounded of nnlt pressure In one direction and en equal tension In a direction at right angles to It, or which la the same thing, a stress compounded of two balancing couples of unit tangential tensiona In planes at angle* of 45* to the direction of those forces, and at right angle* to ono another amount* In magnitude to

(5) A strain compounded of a simple longitudinal extension i, and a simple longitudinal condensation of equal absolute value, in a direction perpendicular to it, is a strain of magnitude x*/ 2; or, which lathe same thing (It 9 = 3x), a 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

mentioned above 1* a motion c parallel to"

f«) If a strain be such that a sphere of nnlt radios Id the body becomes an ellipsoid whoso equation la

(1 - A)X« + (1 - B)T* + <1 - C)Z» - DTZ - EZX - PXT = 1, the values of the component strains corresponding, aa explained In Example (1 of Chap. IX.. to the different coefficients respectively ..are D E - F: sv»' sva" 17s •

For the components corresponding to A, B, C are simple longitudinal strains. In which diameter* of the sphere along the axes of coordtnatea become elongated from 2 to 2 + A, 2 + B. 2 + C respectively .Diss distortion in which diameter* in the plane TOZ. bisecting the angles TOZ and V 07, become respectively 'ami contracted from 2 to 2 + JD, and from 2 to 2 - TD; and an f«* if w* take x,», r, £, tj, J to denote the magnltades ot sit:

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component strains, according to tho orthogonal system of types described in six orthogonal types of reference, and , m', *', X', a, u those of Examples (1) and (2) of Chap. IX. the resultant strain equivalent to them will be one in whieb a sphere of radius 1 in the solld becomes an ellipsoid whose cu

the other. won is

Cor. 7. The most convenient specification of a type for strains or (1 - 22).X2+ (1 - 24)12+(1 – 22)Z2 - 3 V 2(EYZ+nZX+X -1, stresses, being in general a statement of the components, according

to the ty and its magnitude will be

ence, of a unit strain or stress of the type to ha v (x2+3+-++ m2 +.

specified, becomes a statemert of its concurrences with the types of

reference when these are orthogonal. (7) The specifications, according to the system of reference used in the pre

Erampics (1) The mutual concurrence of the simple longitudinal strains or ceding Example, ot the unit strains of the six orthogonal types defined in Example (3) of Chap. LX. are respectively as follows:

stresses, inclined to one another as an anglo e, is cos? 0.

(2) The mutual concurrence of two simple distortions in the sarre place whose axes are inclined at an angie @ to one another, is cos' 0 - sina, os 2 sin (45° - ) cos (45°-O).

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

8(cos? 0 - sin?) and 0.2 sin cos 8 (IT.)

respectively, be tho angle between the axis of elongation in the giren die (UL)

tortion and in the first component type.

(3) The mutual concurrence of a simple longitudinal strain and a simple dis (IV.)

tortion is ()

V2.cos a cos B,
(1T P { min on

Il a and B bo the angles at which the direction of the longitudinal sfrala 19
Inclined to the Uno bisecting the angles between the azes of the distortion; it is

also equal to wherol, m, mer', m, n' denoto quantities fulfilling the following conditions:

Ja roosa p-cos? 4).
72 m2 + =1,
1 + + = 0,

Il d and vs denoto tbe angles at which the direction of the longitudinal train ir + mm' + nn'= 0,

is inclined to the axis of the distortion. 7 +m? +*% = 1,

(4) The mutual concurrence of a simple longitudinal strain and of a uniform 7 +m' + ' = 0.

dllatation is 23 (*) 11 (1 - 2P)X?+ (1 - 26QY2+(1-26R)Z.-2012(SYZ+TZX+UXY)=1

(5) Tho specifying elements exhibited In Example (7) of the preceding Se the equation of the ellipsoid representing a certain stress, the amount of work

Chapter are the concurrences of the new system of orthogonal types described in done by this stress, if applied to a body while acquiring the strain represented.

Examplo (8) of Chap. IX. with the ordinary by the equation in the preceding example, will be

stem, Examples (1) and (.

Chap. LX. Px+y+Rs+SE+In+ UY. Cor. Hence, 11 variables X, Y, Z be transformed to any other set (X', Y', 2) CHAPTER XII.-On the Transformation of Types of Refcrcrce for fulfilling the condition of being the coordinates of the same point, referred to

Stresses or Strains. another system of rectangular axes, the coefficients , y, &, &c., 14., $, &c., in two homogeneous quadratic functions of three variables,

To transform the specification (2, 4, 2, 6, 7, R) of a stress or strain (1 - 21)X+ (1 - 2y)Y+ (1 - 22)22 - 2V2EYZ+nZX + YXY)

with reference to one system of types into (2), Zg, xg, u Zg, as) and

with reference to another system of types. Let (a, b, c, cu fugi) (1 - 2x)X2 + (1 - 2y,)Yo + (1 - 22,923 - 2V2EYZ+nZx+ XY),

be the components, according to the original system, of a unit and the corresponding coefficients s', v', s', &c. ,0 ,2,, &c., in these func strain of the first type of the new system; let (ag, by Co, fy 9,) tions transformed to 3', 7, 8, will be so related that

be the corresponding specification of the second type of the ne 3/8',+y't,+8/,+EE,+9'',+8'8,-43,+, +,+EE,+999, +88; systein; and so on. Then we have, for the required formula of

transformation or the function xx,+ yy, +25, + EE + +X7, of the coeficients is an “Invariant" for linear transformations fulfiling the conditions of transformation from one to

IEQ4 +agtatagtstart.+dgtstarta, another set of rectangular axes. Since r+y+! and fi+y.+?, are clearly in

y=biti+bata+bata+ba+bststborg
Tariants also, It follows that A4,+BB,+CC,+ 2DD,+2EE,+2FF, is an inyeriant
function of the coeficients of the two quadratics

. . . . . . . . . . . . . .
AX+BY+CZ2+2DYZ+2EZX+2FXY
AX+BY+CZ2+2D,YZ+2E,ZX+2F,XY,

%= %17i+Izra+!3+x+fototysrst96703 which it is easily proved to be by direct transformation.

ErampleThe transforming equations to pass from a specifcation (x,y. This is the simplest form of the algebraic theorem of Invariance with which we & 1 ) in terms of the system of reference used in Examples (6) and (7) are concerned.

Chapter X., to a specification (o. E, M, 8. w, w) in terms of the new system

described in Example (3) of Chapter IX., and specified in Example (7) of Chapter CHAPTER XI.-On Imperfect concurrences of two Stress or Strain X., are as follows: Types.

I-70+la+loc Def. The concurrence of any stresses or strains of two stated types is the proportion which the work done when a body of unit

tm'w volume experiences & stress of either type, while acquirirg a strain of ithe other, bears to the product of the numbers measuring the stress and strain respectively.

37 zotnataw, Cor. 1. Iu orthogonal resolution of a stress or strain, its com. ponent of any stated type is equal to its own amount multiplied by

E-6 n=14. -Si its concurrence with that type; or the stress or strain of a stated wbere, as before stated, l, m, n, l, m, n' are by quantities fuldblog the conditions type which, along with another or others orthogonal to it, have a

P in + el, given stress oc strain for their resultant, is equal to the amount of

1 +m + =0, the given stress or strain reduced in the rolio of its concurrence

1+m? +A%=1 with that stated type.

+m +x =0, Cor. 2. The concurrence of two coincident stresses or strains is

"+mm'tan'=0, unity; or a perfect concurrence is numerically equal to unity. Cor. 3. The concurrence of two orthogonal stresses and strains is

PART 11.-ON TRE DYNAMICAL RELATIONS BETWEEN STRESIES zero. Cor. 4. The concurrence of two directly opposite strssses or

AND STRAINS EXPERIENCED BY AN ELASTIC SOLID. strains is -1.

CHAPTER XIII.-Interpretation of the Differential Equation Cor. 5. If x, y, z, &; n, %, are orthogonal components of any

of Energy. strain or stress T, its concurrences with the types of reference are respectively

In a paper on the Thermo-elastic Properties of Matter, published in the first pinber of the Quarterly Aathematical Journal, April

1855, and republished in the Philosophical Manazine, 1877. second whera

half year, it was proved, from general principles in the theory of

the 'Transformation of Energy, that the amount of work (W) reço (+18+23+0+ m2 +8,5.

quired to reduce an elastic solid, kept at a constant temperature, Cor. 6. The mutual concurrence of two stresses or strains is from one stated condition of internal strain to another depende # + mm' + ax' +11+ uued tv,

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 IL m, n, d, the vdepote the concurrenoes of one of them with effecting the change, provided always there has been no failure in

and

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fni tfolUcity under any of the strains it has experienced. Thus fur a homogeneous solid homogeneously strained, ft appears that to is a function of six independent variables & y, ?, f, yf £ by which tbe condition of the solid as to strain is specified. Hence to strain the body to the infinitely small extent expressed by the variation Irom(:r,y, c, flto (x+dx. y + dytz+dz, { + ii + c?i?, C"H*0i the work required to be done upon it is

ix dp di rf£ rfl Tbe stress which must be applied to its surface to keep the body in rqmlibrium in the state {x, y, z, {, ij, f) mast 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, at, dti, d(; that is, it must be the resultant of six stresses:—one orthogonal to the five strains dy, dz, <£{, dr), d(, and of such a magnitude as to do tho dio

work -jr dx when the body acquires the strain dj:; a second orthogonal to dx, dz, d{, d% d{, and of such a magnitude as to do the work -j^ dy when the body acquires the strain dy j and so on.

If a, ft, e, f, g, h denote the respective concurrences of these six stresses, with the types of reference used in the specification (x, y, z, V, 0 of the strains, the amounts of tho six stresses which fulfil those conditions will (Chapter XL) be given by the equations

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«nd 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

_ dv _ dip <f#

r-£'

da _ rfif *»

Chapter XIV.—Reduction of the Potential Function, and of the Equation* of Equilibrium, of an Elastic Solid to their simplest Forms.

If the condition of the body from which the work denoted by 10 U reckoned be that of equilibrium under no stress from without, and if x, y, %t £, 77, ( be chosen each zero for this condition! we shall hare, by ilaclanriu'a theorem,

If=h1u, y. t, £ „, p+rijO, f, f, £, „, Ac.,

trhcre Ha, H3, kc, denote homogeneous functions of the second

order, third order, fee., respectively. Hence -^g9 &a, will each

be a linear function of the strain coordinates, together with functions of higher orders derived from H., &c. But experience shows (section 37 above) that, within the elastic limits, the stresses are very nearly if not quite proportional to the strains they are capable of producing; and therefore Hp ice, may be neglected, ana wo bave simply

t^HiCr, y,t, £ tj, I). Now in general there will be twenty-one terms, with independent coefficients, in this function; but by a choice of types of reference, that is, by a linear transformation of the independent variables, we may, in an infinite variety of ways, reduce it to the form « - J (A** + By»+Ct*+rp+G«*+Hf*),

Tho equations of equilibrium then become

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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 ouly fifteen equations to satisfy ; while we have thirty disposable transforming coefficients* there being five independent elements to specify a type, and six types to Ds changed. Any further condition expressible by just fifteen independent equations may be satisfied, and makes the transformation deu-nuinatc. Xo\r the condition that six strains may bo mutually orthogonal is expressible by just c.3 many equations as there are dUlerent pair* of six tilings, that is, fifteen. The well-known algebraic theory of the linear transformation of quadratic functions shows for the case of six variables—(1) that tho six coefficients in the reduced form . if the roots of a "determinant" of tho sixth degree necessarily real: (2) that this multiplicity of roots leads detetminately to one, and only one system of six types fulfilling the prescribed conditions, unless two or more of tho roots are equal to ono another, when there will be an infinite number of solutions and definite degrees of isotropy among them; and (8) 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 tho sum ofthe products of the squares of tho components of the strain, according to those types, respectively multiplied by six determinate coefficients.

Dcf. 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 the quantities denoted above by a, b, e, /, g, A, is unity when the six principal strain-types are chosen for the coordinates. The equations of equilibrium ol an elastic solid may therefore be expressed as follows:—

P—Ax, Q—By, Ii — Cz,
S-F£, T-gij, U-nj,

where X, y, 2, {, 77, (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

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

The equations of equilibrium express the following propositions :—

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

ExampUir-Q) If a solid lie cubically Isotropic In Its elastic properties, as crystals of tho cubical class probably are, any portion of If will, when subject to a uniform positive cr negative normal pressure aU round Its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensation la one of Ita six principal strains. Three plane distortions with axes bisecting the angles between the edges of tho cube of symmetry are clearly also principal strains, and since the throe corresponding principal elasticities are equal to one anothor, any strain wnaterer compounded of these three is a principal strain. Lastly, a plane distortion whoso axes coincide with any two edges of the cube, 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 la also a principal distortion.

Hence the system of orthogonal types treated of In Examples (3) Clinp. TX, and (7) Chap. X., or any system in which, for (II.), (Ill), and (IV.), any three orthogonal strains compounded of them aro substituted, constitutes a system of six Principal Strains In a solid cubical!/ isotropic. There are only three distinct Principal Elasticities for snch a body, and these are—(A) its modulus of compressibility, (B) Its rigidity agninst 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).

l2) In a perfectly Isotropic solid, tho rigidity against all distortions la equal. Hcneo the rigidity (B) against diagonal distortion mustjie equal to the rieidity (C) against rectangular distortion, In a cube; and it Is easily seen that If this condition Is fulfilled for ono set of three rectangular planes for which a substance U isotiopic, the Isotropy must bo 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 tfniform condensation In all directions, and any system whatever of fire 1 • - —- of six Principal Strains in a sphert

simply Its M

s Uodulus of Compret

Prop. Unless some of the six Principal Elasticities be equal to one another, the streos 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 tho 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 foQr other principal strain-types have each the maximum •mini mum property in a determinate way.

Prop. If a body be strained in the direction of which the concurrences with the. principal strain-types are I, m, n, x, u, r, and tor an amount equal to r, the stress required to maintain it in this state will be equal to fir, where

Q - (A*fS+B>*>++F*X*+GV*+HV)t. and will be of a type of which tho concurrences with the principal types are respectively

A/ Bm Cfi FX On Th

U' U' U * W "0"* Q'

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 rotator)', 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 heterogeneousncss 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 plagiedrol faces discovered by Sir John Hersehel 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 tho 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.

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 elements, for its principal strains: for instance, five arbitrarily chosen systems of three rectangular axes, for the normal axes of fire of tne Principal Types; those of the sixth consequently in general 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 detennino 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 tho principal strain-types and the equations of elastic force referred to them or to other natural 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

Siestion of the earth's rigidity and elasticity as a whole, -and to e 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, U 668, 740, 832, 840, and A p. pendix C.

Chapter XVII.—Plane Waves in a Homogeneous JEMiropic 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

periences sirfiply a motion of translation—but a motion difiVrina> 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 fs called), and OY, OZ in tne ware front Let z + u, y + r, s+wbe the coordinates at time t of a particle which, if the solid were free from strain, would be at (z, y, ;). The definition of wave motion amounts simply to this, that u, v, «? are functions of * and L The strain of the solid (Chap. VII. above) is the resultant of a

simple longitudinal strain in the direction OX, equal to ^ and

two differential slips ~' parallel to OY and OZ, constituting

simple distortions of which the numerical magnitudes (Chap. X.) are

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

Watt* in an IncomprtssibU Solid (jSololropic or Isotropic).—If the solid be incompressible, we have A = oo, and u most be zero. Hence w-Bn'+Cp+JDnJ

and by a determinants! 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 iaotropy the two i wave velocities are equal

It is to be noticed that Mv if., M, in the preceding investigation are not generally true '* principal moduluses. but special moduluses corresponding to the particular plane chosen for the wave front. In the particular case of isotropy, however, the equal moduluses Mlf Ma of (11) are principal moduluses, being each equal to tht modulus of rigidity, but M, is a mixed modulus of compressibility and rigidity—not a principal modulus. In the case of iifcompres3ibility, the two moduluses found from the determinants! quadratic by the process indicated above are not principal moduluses generally, because the distortions by the differential motions of planes of particles parallel to the wave front must generally give rise to tangential stresses orthogonal to them, which do not influence the wave motion. (W. TH.)

ELATERItJM, a drug consisting of a sediment deposited by the juice of the fruit of Ecbalium Elaterium, the squirting 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 ox 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 pvoduces vomiting. Ita active principal is elaterin, a crystallizable body of the formula C^H^O,.

ELBA, the KWaXla of the Greeks, and Ilvaof the Romans, is an island in the Mediterranean Sea, forming part of the Italian province of Livomo, 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 2 J 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 I some importance; but the principal industry is mining. The iron mines are mostly in the vicinity of Rio Inferior©, 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 porta of .France and England. Marble, alabaster, sulpbur, 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 21,765.

The Argonauts, in quest of Clrcoi are said to have loaded at Tortus Argons (Apyiot Xim*», now Porto Ferrajo, is Elba. The island was early famous for the richness of its mines, alluded to by Vipgji (jBn. x. 178). Itwas attacked by Phayllua with a Svracusan fleet, 468 B.O., and subsequently by Apelles, who is stated to have subjugated it In the 10th century it became a possession of the Han, {pom who:u it was taken by tbe Oeuoese in 1200. It fell

eventually into Spanish hands, came in 1786 under tbe jurisdiction of Naples, and in 1801 was cjded to the king of Ktruria by the treaty of Luneville. It was united to France in 1803, made over to Napoleon by tbe Treaty of Paris in 1814. restored to Tuscsny in tbe following year, and in 1860 annexed to Italy.

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 Rieseugebirge 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 tbe 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 cute 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* fi' N. lat and 8* 50' E. long. The principal towns on its banks are Leitmeritz, Pirna, Dresden, Meissen, Torgan, Wittenberg, Magdeburg, Wittcnberge, Earburg, 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 tbe sea; after tbe first 40 miles of its course it is still 658 feet; but at Dresden it is only 279, and at Ameburg in Brandenburg only 176. At Koniggratz 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 anrt 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 Liibeck, 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-Kanale im Dcutschen Reiche," in Petermann's ilitthtti.. 1877.

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