where [delta]p is the displacement of any joint in the direction of the corresponding force P. If [Sigma](P·[delta]p) = 0, the stresses are merely indeterminate as before; but if [Sigma] (P·[delta]p) does not vanish, the equation cannot be satisfied by any finite value of S, since [delta]s = 0. This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces. With actual materials, the frame would yield elastically, until its configuration is no longer "critical." The stresses in the bars would then be comparatively very great, although finite. The use of frames which approximate to a critical form is of course to be avoided in practice. A brief reference must suffice to the theory of three dimensional frames. This is important from a technical point of view, since all structures are practically three-dimensional. We may note that a frame of n joints which is just rigid must have 3n - 6 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of "critical forms." § 10. _Statics of Inextensible Chains._--The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics. It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element [delta]s must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4). [Illustration: FIG. 54.] We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= [delta]s) in the directions of the tangent and normal respectively. If T, T + [delta]T be the tensions at P, Q, and [delta][psi] be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or [delta]T, ultimately; whilst for the component along the normal at P we have (T + [delta]T) sin [delta][psi], or T[delta][psi], or T[delta]s/[rho], where [rho] is the radius of curvature. Suppose, for example, that we have a light string stretched over a smooth curve; and let R[delta]s denote the normal pressure (outwards from the centre of curvature) on [delta]s. The two resolutions give [delta]T = 0, T[delta][psi] = R[delta]s, or T = const., R = T/[rho]. (1) The tension is constant, and the pressure per unit length varies as the curvature.