822 LASTICITY component strains, according to the orthogonal system of types described in Examples (1) and (2) of Chap. IX., the resultant strain equivalent to them will be one in whieh a sphere of radius 1 in the solid beeomes an ellipsoid whose equa tion is and its magnitude will be v . (7) The specifications, according to the system of reference used in the pre ding Example, of the unit strains of the six orthogonal types defined in Example ) of Chap. IX. are respectively as follows: IS) of C X y z
1
(I.) i 1
-y O v " V3 (II.)
1
(III.)
1
(IV.)
1
(V.) I
m
n
(VI.) I m n
where?, m, n, I , m , n denote quantities fulfilling the following conditions: P+m* -f n 2 =1, / +m +n 0, IV + 4-7(71 = 0,
- /2 +; 2 +n - l,
I +m + =0. (8) If a-2eP)X 2 +(l-2eQ)Y 2 + (l-2eK)Z 2 -2c v /2(SYZ+TZX + UXY)=l be the equation of the ellipsoid representing a certain s!/ess, the amount of work done by this stress, if applied to a body while acquiring the strain represented by the equation in the preceding example, will be Cor. ITence, 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, referied to another system of rectangular axes, the coefficients .r, y, z, &.C., x t , y,, z lt &c., in two homogeneous quadratic functions of three variables, (1 - 2.r)X 2 + (1 - 2;/) Y 2 + (1 - 2;)Z 2 - 2 i/2( YZ + )|ZX -f XY) and (1 - 2.r,)X 2 + (1 - 2!/ / )Y 3 + (1 - 2z ; )Z 2 - 2 and the corresponding coefficients x f , y , z , Ac., r 1 ^ j/ ( , z t , <fec., in these func tions transformed to X 1 , j/, z 1 , will be so related that or the function ix l -- yy t 4- zz t + !;, + />), + <,<C, of the coefficients is an "invariant" for linear transformations fulfilling the conditions of transformation from one to another set of rectangular axes. Since x+y+z ami x,--y,+z, are clearly in variants also, it follows that AA,4-I5I5 / +CC,-t-2DD / 4-2EE / - r -2FF / is an invariant function of the coefficients of the two quadratics AX 2 +BY 2 +CZ 2 +2DYZ+2EZX + 2FXY and A,X 2 +B / Y 2 -t-C which it is easily proved to be by direct transformation. This is the simplest form of the algebraic theorem of invariance with which we are concerned. CHAPTER XI. O/i Imperfect Concurrences of two Stress or Strain Types. Dcf. 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 acquirirg 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 com ponent 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. Cur. 3. The concurrence of two orthogonal stresses and strains is zero. Cor. 4. The concurrence of two directly opposite stresses or strains is - 1. Cor. 5. If x, y, z, {, 77, , are orthogonal components of any strain or stress r, its concurrences with the types of reference are respectively where Cor. 6. The mutual concurrence of two stresses or strains is // -f mm -f nn + XX + u/u + vv , if I, m, n, A, w, v denote the concurrences of one of them with six orthogonal types of reference, and I , m , n , A/, //, v 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 two simple longitudinal strains or stresses, inclined to one another at an angle 0, is cos- 6. (. ) The mutual concurrence of two simple distortions in the sane plane, whose axes are inclined at an angle to one another, is cos" - sin 2 0, or 2 sin (45 D -0) cos (40 - t);. Hence the components of a simple distortion o along two rectangular axes in its plane, and two others bisecting the angle between these taken as axes of component simple distortions, are (cos 2 t) sin 2 0) and .2 sin cos respectively, if 6 be the angle between the axis of elongation in the given dis tortion and in the first component type. (3) The mutual concurrence of a simple longitudinal strain and a simple dis tortion is /2.cos a cos /3 , if a and /3 be the angles at which the direction of the longitudinal strain is inclined to the line bisecting the angles between the axes of the distortion ; it is also equal to (cos 2 <p - cos 2 i//) , if <f) and x// 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 dilatation is r: -v/3 (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 ( . ), Chap. IX. CHAPTER XII. On the Transformation of Types of Reference for Stresses or Strains. To transform the specification (x, y, ~, {, 77, ) of a stress or strain with reference to one system of types into (x ly x.-,, x 3 , x 4 , x 5 , x 6 ) with reference to another system of types. Let (,, b lt t 1; c x , f it g-^ be the components, according to the original system, of a unit strain of the first type of the new system ; let (a 2 , b. 2 , c. 2 , c, 2 , f. 2 , g, 2 ) be the corresponding specification of the second type of the new system ; and so on. Then we have, for the required formula; of transformation = i^i^iiy^ig^ a- Example. The transforming equations to pass from a specification (x, y, t, , ), ) in terms of the system of reference used in Examples (C) and (7), Chapter X., to a specification (<r, , );, , w, w) in terms of the new system escribed in Example (3) of Cluiptei IX., and specified in Example (7) of Chapter
X., are as follows : where, as before stated, I, m, re, I , m , n are by quantities fulfilling the conditions P+nP + 2 =1, I +m +TZ =0, 2+m 2 +n 2 = l +m +n =0, ll +mm +nn Q , 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, second half year, it was proved, from general principles in the theory of the Transformation of Energy, that the amount of work (w) re quired to reduce an elastic solid, kept at a constant temperature, from one stated condition of internal strain to another depend.* 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