so that this system does not tend to alter the volume of an element. Consider the line-integral of the tangential component of this latter system along any closed line s, and let dS be an element of a surface having the line s for an edge, then this line-integral is and, by the theorem for the transformation of line-integrals and surface-integrals, this is Thus if s be a very small closed curve in the plane (y, z), and S1 its area, the line-integral in question is S1VU, so that the system G, H, K tends to produce rotation of the elements. 136. Particular Integrals for the Bodily Forces. Now let u, v, w be the displacements at any point of the body, and suppose u, v, w expressed in the same way as X, Y, Z in the forms Then and u = Ə ƏN ƏM - бх ду + Əz + + ду д дх ӘФӘMӘL .(15). этоф дх + μ ( + dy дг af aw + дудг Hence we have a solution in the form + = + (x2+ 2μ) [] — da'dy'dz',) and similar forms for M and N, where as before f', U' are the values of ƒ, U at (x', y', z'). Hence we can write down u in the where cosa is the cosine of the angle between the axis x and the line r drawn from (x, y, z) to (x, y, z), and v and w can be written down by symmetry. These values of u, v, w are particular integrals of the equations of equilibrium. They will not however in general satisfy the boundary-conditions. We notice that in accordance with our interpretation of ƒ, U, V, W the cubical dilatation is - pf/(λ + 2μ). 137. Second form of Particular Integral. Another method of obtaining the particular integral will be given later (ch. IX. art. 150), where we shall shew that, if X', Y', Z' be the bodily forces, per unit mass, applied at the point (x', y', z'), the equations of equilibrium can be satisfied by the forms Solutions equivalent to these are given in Thomson and Tait's Natural Philosophy, Part II. art. 731. 138. Particular Integral for Forced Vibrations. Suppose the solid executes forced vibrations, under the action of periodic forces. Then we have to take X, Y, Z and consequently f, U, V, W all proportional to ept, where 2π/p is the period. In the forced vibrations u, v, w will also be proportional to ext, and thus the equations of small vibration may be written in such Now substituting from (15), and writing h2 = pp2/(x+2μ), x2=pp2/μ. we have three such equations as .(20), and thus all the equations can be satisfied by making a solution Now we know that a particular solution of (22) is (23). U=0... μ (see Lord Rayleigh's Theory of Sound, vol. II. art. 277), and in like manner for L, M, N we have such solutions as The values of u, v, w hence obtained are particular integrals of the equations of small motion (21), but they do not in general satisfy the boundary-conditions. 139. Particular Class of Cases. When the bodily forces have a potential ƒ which satisfies Laplace's equation, these particular solutions are very much simplified. For equilibrium we may take udx+vdy+wdz=do.. Then ▲ = V2, and we have three such equations as .(26). Now ƒ may be thrown into the form OF f=r + F. ..(28), where r2 = x2 + y2 + z2, and F satisfies Laplace's equation, and then constitute a set of particular integrals. For forced vibrations, taking the equations such as where f satisfies Vef=0, and has the time-factor ept, we may put Vv=0, V2w=0, A=0, and we have a set of particular integrals. 140. Description of Betti's Method of Integration. Prof. Betti has developed, by the aid of his theorem (art. 68), a general method of integrating the equations of elasticity, for an isotropic solid of any shape, with any given boundary-conditions, when the problem can be solved for the same solid with a certain set of boundary-conditions. In this method we seek in the first place to determine the cubical dilatation and the three component rotations, and from these we find the corresponding displacements. We have already shewn that it is always possible to find a particular integral for the bodily forces; so that we may divide the problem into two parts: (1) the determination of a system of particular displacements which satisfy the equations containing the bodily forces but do not satisfy the boundary-conditions; (2) the determination of a system of displacements which satisfy the equations when the bodily forces are null and which also satisfy arbitrary boundary-conditions. It is with the latter problem that we shall here occupy ourselves. We have to find a solution of the equations which hold at all points of the solid. We shall consider first the problem of determining the cubical dilatation ▲ and the three rotations w1, w2, w, so as to satisfy the differential equations, and so that it may be possible to satisfy the boundary-conditions; and we shall suppose that at the boundary of the solid either the surface-tractions or the displacements are given functions. When A, w1, w2, w, are known, we have Hence, if the surface-displacements be given, we have to find u, v, w to satisfy equations of the form Vua given function of x, y, z, and u = a given function at the boundary. If the surface-tractions F, G, H be given the boundary-conditions can be written, by (15) of art. 29, in the forms where (l, m, n) are the direction-cosines of the normal (dv) to the boundary drawn outwards from the space occupied by the solid. Thus we have to find u, v, w to satisfy equations of the form |