ELASTICITY 823 the elasticity under any of the strains it has experienced. Thus for a homogeneous solid homogeneously strained, it appears that w is a function of six independent variables x, y, z, |, rj, C, ty 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 (a;, y, z, {, 17, to (x + dx, y + dy,z + dz, J + df, -n+dr,, C+C)> the work required to be done upon it is dx dy dz d% dii d The stress which must be applied to its surface to keep the body in equilibrium in the state (,i; y, z, |, , 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, d, dy, d ; that is, it must be the resultant of six stresses: one orthogonal to the five strains dy, dz, d, dn, d, and of such a magnitude as to do the work dw - dx when the body acquires the strain dx ; a second or thogonal to dx, dz, rf|, d-,], d, and of such a magnitude as to do the work -r- dy when the body acquires the strain dy ; and so on. y ,. 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, |, TJ, f the strains, the amounts of the six stresses which fulfil those conditions will (Chapter XL) be given by the equations b dy 1 dw -, c dz 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 dw " die . dw 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, , TJ, be chosen each zero for this condition, we shall have, by Maclaurin s theorem, w=H 2 (x, y, z, , )j, )-t-II 3 (.r, y, z, g, i), )+ &c., w^ere H 2 , H 3 , &c., denote homogeneous functions of the second order, third order, &c., respectively. Hence j-^> ~j> &c., will each be a linear function of the strain coordinates, together with func tions of higher orders derived from H 3 , &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 H 3 , &c., maybe neglected, and we have simply 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 The equations of equilibrium then become p=^, Q=? y , R=z, a b c o_-_ T *i TT C ~r 9 ~A I 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 transforms- 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 ex pressible 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 form are 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 conditions, 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 tor 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 strain- coordinates 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, c, f, g, h, is unity when the six principal strain-types are chosen for the coordinates. The equa tions of equilibrium of an elastic solid may therefore be expressed as follows : P AJ-, Q = Bj/, R = Cz, S-F, T = Gi/, U = H, where x, y, z, , 77, denote strains belonging to the six Principal Types, and P, Q, li, 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 Def. The coefficients A, B, C, F, G, H are called the six Prin cipal 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 Prin cipal 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 ov negative normal pressure all round its surface, experience a uniform condensation or dilation in all directions. Hence a uniform condensa tion 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 another, any strain whatever compounded of these three is a principal strain. T.neflv n rilnnp di-jtnrfinn whns<> MYCS cniiifide wifh nnv two f fV^ps (if th(? Clllwv ceding example, vi;h the farther condition B = C. Prop. Unless some of the six Principal Elasticities be equal to one another, the stress required to keep the body strained other wise 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 II the least of the six quantities A, B, C, F, G, H, the principal type to wiiich the first corresponds
is that of a strain requiring a greater stress to maintain U than any