and we can find, as in the cylinder-problem, which agrees with equation (53) in the case of isotropy. .....(95), The cubical dilatation of the spherical cavity is the value of 3U/r when rr1, and this is P1r1" — Porn This result is of importance in the theory of piezometer experiments, for which a discrepancy appears to have been observed between the results obtained and the dilatation that would have place if the material were isotropic. The solution in (96) contains 3 independent constants and Saint-Venant' held that these could be adjusted so as to explain the experiments in question. 1 See Pearson's Elastical Researches of Barré de Saint-Venant, p. 82. CHAPTER VIII. GENERAL SOLUTIONS, 133. Statement of the Problem. The general problem of the Mathematical Theory of Elasticity consists in the discovery of functions u, v, w which satisfy the system of equations where P, Q, R, S, T, U are the partial differential coefficients of a quadratic function W of the six quantities at all points within a certain closed surface, the surface of the strained solid, and also fulfil certain conditions given at the boundary. We shall consider separately problems in which a solid is considered as held strained by the application of forces, and problems involving small motions, and shall proceed now to the consideration of the equilibrium of an isotropic solid body. Suppose then that a mass of homogeneous isotropic elastic matter is subject to bodily forces whose components at any point are X, Y, Z. The equations of equilibrium which hold at every In order to solve these equations we seek first any set of particular integrals in terms of X, Y, Z, and secondly the most general complementary solutions of the same equations with X, Y, Z all equal to zero. The first set of particular integrals obtained will not in general lead to values of the stresses or displacements which satisfy the boundary-conditions. In that case we have to determine the arbitrary functions or arbitrary constants, that occur in the complementary solutions, so that the complete solutions, consisting of particular integrals and complementary functions, may satisfy these conditions. 134. Formulæ for the Bodily Forces1. Let X, Y, Z be the components of the bodily force, per unit of mass, supposed finite continuous and one-valued functions of x, y, z throughout the body; we seek to throw X, Y, Z into the forms where U, V, W, and fare functions of x, y, z. .(3), By differentiating the above equations with respect to x, y, z and adding, we obtain 1 The subject-matter of this and the two following articles is due to Prof. BettiTeoria della Elasticita. Il Nuovo Cimento, 1872. Let be the value of Þat (x', y', z'), and r the distance of (x, y, z) from (x', y', z'), then a particular integral of the equation for ƒ is the potential of a distribution whose density at (x', y', z') p', so that we may write is 1 4.π 1 - 4+ say...... 4π r where the integration extends throughout the solid, and is a function which is finite continuous and one-valued within the body and satisfies the equation ▼2 = 0...... ..(6); we may complete the definition of by subjecting it to the condition - (lX + mY + nZ) = 0 ...................... .(7) at the boundary, (l, m, n) denoting the direction-cosines of the normal drawn outwards, and do the element of this normal. Thus the function f is completely determined. then G, H, K are completely determined. By differentiating these equations with respect to x, y, z, adding, and using (4), we find where, as in the case of ', G' is the value of G at (x', y', z'), and in We can now write G = V3A = дуду дх Əz əx and we have similar equations for H and K. X, Y, Z will be thrown into the forms (3), and all the functions f, U, V, W will be well defined. 135. Interpretation. Consider any surface σ drawn within the body. The surfaceintegral of the normal component of the system of forces depending P [[født, where dT is the element of the volume within the surface σ, and, when the surface is contracted to a point, we see that this system of forces tends to vary the volume of an element. The surface-integral of the normal component of the G, H, K system is |