whose hyperbolic logarithm is 2.07345. This number is 7-95; so that each additional coil increases the friction nearly eight times. Had the rope been dry, this proportion would have been much greater. If an additional half coil had been supposed continually to be put upon the rope instead of a whole coil, the friction would have been found in the same way to increase in geometrical progression, but the common ratio would in this case have been tan. instead of 2 tan.. In the above example the value of this ratio would for each half coil have been 2.82.

The enormous increase of friction which results from each additional turn of the cord upon a capstan or drum, may from these results be understood.

188. We may, from what has been stated above, readily explain the reason why a knot connecting the two extremities of a cord effectually resists the action of any force tending to separate them. If a wetted cord be wound round a

which R is attached be brought underneath the other string so as to be pressed by it against the surface of the cylinder, as at m, fig. 2.; then, provided the friction produced by this pressure be not less than one eighth of P, the string will not move even although the force R cease to act. And if both extremities of the string be thus made to pass between the coil and the cylinder, as in fig. 3., a still less pressure upon each will be requisite. Now, by diminishing the radius of the cylinder, this pressure can be increased to any extent, since, by a known property of funicular curves, it varies in

versely as the radius.* We may, therefore, so far diminish the radius of a cylinder, as that no force, however great, shall be able to pull away a rope coiled upon it, as represented in fig. 3., even although one extremity were loose, and acted upon by no force.

Fig. 4.

Let us suppose the rope to be doubled as in fig. 4., and coiled as before. Then it is apparent, from what has been said, that the cylinder may be made so small, that no forces P and P' applied to the extremities

of either of the double cords will be sufficient to pull them from it, in whatever directions these are applied.

Now let the cylinder be removed. The cord then being drawn tight, instead of being coiled round the cylinder, will be coiled round portions of itself, at the points m and n; and instead of being pressed at those points upon the cylinder, by a force acting on one portion of its circumference, it will be pressed by a greater force acting all round its circumference. All that has been proved before, with regard to the impossibility of pulling either of the cords away from the coil when the cylinder is inserted, will therefore now obtain in a greater degree; whence it follows that no forces P and P' acting to pull the extremities of the cords asunder, may be sufficient to separate the knot.

THE FRICTION BREAK.

189. There are certain machines whose motion tends, at certain stages, to a destructive acceleration; as, for instance, a crane, which, having raised a heavy weight in one position of its beam, allows it to descend by the action of gravity in another; or a railway train, which, on a certain portion of its line of transit, descends a gradient, having an inclination greater

* This property will be proved in that portion of the work which treats of the THEORY of ConstrucTION.

than the limiting angle of resistance. In each of these cases, the work done by gravity on the descending weight exceeds the work expended on the ordinary resistance due to the friction of the machine; and if some other resistance were not, under these circumstances, opposed to its motion, this excess (of the work done by gravity upon it over that expended upon the friction of its rubbing surfaces) would be accumulated in it (Art. 130.) under the form of vis viva, and be accompanied by a rapid acceleration and a destructive velocity of its moving parts. The extraordinary resistance required to take up this excess of work, and to prevent this accumulation, is sometimes supplied in the crane by the work of the labourer, who, to let the weight down gradually, exerts upon the revolving crank a pressure in a direction opposite to that which he used in raising it. It is more commonly supplied in the crane, and always in the railway train, without any work at all of the labourer, by a simple pressure of his hand or foot on the lever of the friction break, which useful instrument is represented in the accompanying figure under the form in which it is commonly applied to the crane,—a form of it which may serve to illustrate the principle of its application under every other. BC represents a wheel fixed commonly upon that axis of the machine to which the crank is attached, and which axis is carried round by it with greater velocity than any other. The periphery of this wheel, which is usually of cast iron, is embraced by a strong band* ABCE of wrought iron, fixed firmly by its extremity A to the frame of the machine, and by its extremity E to the short arm AE of a bent lever PAE, which turns upon a fixed axis or fulcrum at A, and whose arm PA, being prolonged, carries a counterpoise D just sufficient to overbalance the weight of the arm AP, and to relieve the point E of all tension, and loosen the strap from the periphery of the wheel,

* Blocks of wood are interposed between the band, the periphery of the break wheel. This case will be discussed in the Appendix.

when no force P is applied to the extremity of the arm AP, or when the break is out of action.

It is evident that a pressure P applied to the extremity of the lever will produce a pressure upon the point E, and a tension upon the band in the direction ABCE, and that being fixed at its extremity A, the band will thus be tightened upon the wheel, producing by its friction a certain resistance upon the circumference of the wheel.

Moreover, it is evident that this resistance of friction upon the circumference of the wheel is precisely equal to the tension upon the extremity A of the band, being, indeed, wholly borne by that tension; and that it is the same whether the wheel move, as in this case it does, under the band at rest, or whether the band move (under the same tensions upon its extremities, but in the opposite direction) over the wheel at rest. Let R and Q represent the tensions upon the extremities A and E of the band; then if we suppose the wheel to be at rest, and the band to be drawn over it in the direction ECB by the tension R, and to represent the angle subtended at the centre of the wheel by that part of its circumference which the band embraces, we have (equation 205)

R=Qetan..

Let a represent the length of the arm AP, and a, the length of the perpendicular let fall from A upon the direction of a tangent to that point in the circumference of the wheel where the end EC of the band leaves it.

Then, neglecting the friction of the axis A, we have (Art. 5.)

2

If P, represent any pressure applied to the circumference of the break wheel, and P, a pressure applied to the working point of the machine, whatever it may be, to which the break is applied, and if P,=aP, +b (Art. 152.) represent the relation between P, and P, in the inferior state bordering upon motion by the preponderance of P2; then, when P2 is taken

2

in this expression to represent the pressure W, whose action upon the working point of the machine the break is intended to control, P, will represent that value R of the friction upon the break which must be produced by the intervention of the lever to control the action of the pressure W upon the machine; so that taking R to represent the same quantity as in equation (207), we have

R=aW+b.

Eliminating R between this equation and equation (207), and solving in respect to P,

190. When the circular motion of any shaft in a machine, and the pressure which accompanies that motion, constituting together with it the work of the shaft, are to be communicated to any other distant shaft, this communication is usually

established by means of a band of leather, which passes round drums fixed upon the two shafts, and has its extremities drawn together with a certain pressure and united, so as to produce a tension, which should be just that necessary to prevent the band from slipping upon the drums, subject to the pressure under which the work is transferred. The facility with which this communication of rotatory motion may be established or broken at any distance and under almost every variety of circumstance, has brought the band so extensively into use in machinery, that it may be considered as a principal channel through which work is made to flow in its distribution to the successive stages of every process of mechanism, carried on in the same workshop or manufactory.

191. The sum of the tensions upon the two parts of a band is the same, whatever be the pressure under which the band is