Next suppose that the curve is "rough"; and let F[delta]s be the tangential force of friction on [delta]s. We have [delta]T ± F[delta]s = 0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the coefficient of friction. If the string be on the point of slipping in the direction in which [psi] increases, the lower sign is to be taken; hence [delta]T = F[delta]s = [mu]T[delta][psi], whence T = T0 e^([mu][psi]), (2) if T0 be the tension corresponding to [psi] = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put [mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on. Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let [psi] denote the inclination to the horizontal, and w [delta]s the weight of an element [delta]s. The tangential and normal components of w[delta]s are -s sin [psi] and -w [delta]s cos [psi]. Hence [delta]T = w [delta]s sin [psi], T [delta][psi] = w [delta]s cos [psi] + R[delta]s. (3) If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T = w[delta]y. If the string be uniform, w is constant, and T = wy + const. = w(y - y0), (4) say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula d[psi] R = T ------ - w cos [psi]. (5) ds In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have T cos [psi] = T0, T sin [psi] = W, (6) where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant. [Illustration: FIG. 55.] If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes s = a tan [psi]. (7) This is the "intrinsic" equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive x = a log (sec [psi] + tan [psi]), y = a sec [psi]. (8) Eliminating [psi] we obtain the Cartesian equation x y = a cosh --- (9) a