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 while combinations of pulleys or "sheaves," mounted in fixed or movable frames or "blocks," constitute mechanisms used to facilitate the raising of heavy weights. The word appears in Mid. Eng. as pulley or polley (late), also as poleyne (Prompt. Parvul). The first forms seem to be from the O. Fr. poulie, which itself is regarded as coming from the O. Eng. pullèan, to pull. The Low Lat. forms polea, polegia, whence Span. polea and Ital. poleggia, are apparently from the Fr. poulie. The earliest form, poleyne, is represented in Fr. by poulain, literally a colt, Low Lat. pullanus, pullus, the young of any animal, the root of which is seen in English "foal." Poulain was used of a rope to let casks down into a cellar or to raise heavy weights. The use of the name of an animal for a mechanical device is not uncommon, cf. "crane," or "easel," from Du. exel, literally "little ass."

Driving pulleys are usually constructed of cast iron, and are of circular form, having a central nave by which they are secured to the shaft by keys or other fastenings, and straight or curved arms connecting the nave to the rim, which latter is of a form adapted to the connector. Pulleys are usually cast in one piece, and the proportions of the various parts are designed to resist the unknown stresses due to contraction of the casting in cooling, in addition to the stresses to which pulleys are subjected in use. The rim is slightly wider than the belt, and is of such a section as will suiiice to resist the stress due to the pull of the belt, which is commonly taken as per inch of width for single belting and  lb per inch of width for double belting. The rim is also subject to a centrifugal tension of amount wv2/g pounds per square inch of section, where w is the weight in pounds of a length of one foot of the pulley rim one square inch in section, and 'v is the velocity of the rim in feet per second. This stress amounts to 1043 lb per square inch, if the velocity is 100 ft. per second. The combination of these stresses generally limits the rim velocity of cast-iron pulleys to 80 or 100 ft. per second. The dimensions of the nave depend to a large extent on the method of keying or otherwise securing the pulley to the shaft. The number of the arms is arbitrary, and they may be curved to diminish the liability to fracture from contraction in the cooling of the cast iron, but in other respects are preferably straight, since they are then lighter and stronger. The arms are elliptical in cross-section, diminishing from the nave to the rim, and are usually designed as equally loaded cantilevers, nxed at the nave and free at the rim. These assumptions are probably not nearly correct, and, as the stresses caused by the cooling of the casting are unknown, it is necessary to choose a low working stress of about one ton per square inch. The statical experiments of C. H. Benjamin (American Machinist, 1898) on cast-iron pulleys loaded by a belt to imitate the conditions in practice led him to the conclusion that the rim is usually not sufficiently rigid to load the arms equally, and that the ends of the arms are subjected to bending movements of opposite sign, that at the nave being almost invariably the greater.

Pulleys are also built up of wrought iron and steel, and can then be constructed entirely free from internal stress; they are thus much lighter and stronger, and are not liable to fly to pieces like cast iron if they break. Fig. 1 shows a built-up pulley having a cast-iron nave A, straight wrought-iron arms B, screwed therein and connected to a steel plate-rim C by riveted ends, and also by screwed flanges D riveted on each side to the rim. The pulley is in halves to facilitate fixing, and when in place the sections C are joined by plates E, bolted or riveted to the rim. The two halves of the nave are secured by bolts or rivets passing through the flanges F, and the pulley is connected to the

. 1.—Built-up Pulley.

shaft by a sunk key or by conical keys driven in between the shaft and the boss, which latter is bored to suit. A modified form of this arrangement of cone keys is shown in the figure, in which a screwed conical bush M, divided into several parts longitudinally, is clamped round the shaft, and screwed into the corresponding part of the nave until the grip is sufficient. The parts of the bush are glued to a sheet of emery paper, so that its rough side may give a better grip on the shaft. Pulleys are also made of paper, wood and other materials. Wooden pulleys are preferably made of maple, the rim being formed of small sections morticed, pinned and glued together, with the grain set in such directions that any warping of the material will leave the cylindrical form practically unaltered. Wooden pulleys are generally made in two halves, bolted together at the rim and nave, and are provided with wooden spokes dovetailed into the rim and secured by keys. The pulley is secured to the shaft by conical keys, to give a frictional grip on both the shaft and the pulley; these keys may have their exterior surfaces eccentric to the shaft, with corresponding recesses in the nave, so that the pulley and keys virtually form one piece.

If the centre of gravity of a pulley is on the axis of rotation, and the whole mass is distributed so that the axis of inertia coincides with the axis of rotation, there can be no 'unbalanced force or unbalanced couple as the pulley revolves. The magnitude of the unbalanced force, for a mass of w pounds at a radius of r feet and a velocity of v feet per second, is expressed by wv/; and, since the force varies as the square of the velocity, it is necessary carefully to balance a pulley running at a high speed to 'prevent injurious vibrations. This can be accomplished by attaching balance-weights to the pulley until it will remain stationary in all positions, when its shaft rests on two horizontal knife-edges in the same horizontal plane, or, preferably, the pulley and shaft may be supported on bearings resting on springs, and balanced by attached masses until there is no perceptible vibration of the springs at the highest speed of rotation.

The rims of pulleys, round which flat bands are wrapped, may be truly cylindrical, in which case the belt will run indifferently at any part of the pulley, or the rim may be swelled towards the centre, when the central line of the band will tend to run in the diametral plane of the pulley. This self-guiding property may be explained by the tendency which fa fiat band has, when running upon a conical pulley in a direction normal to its axis, to describe a spiral path as it wraps on to the surface because of the lateral stiffness of the material; the advancing side therefore tends to rise towards the highest part of the cone. If two cones are placed back to back the belt tends to rise to the ridge and stay there. In practice the pulley rim is curved to a radius of from three to five times its breadth, and this not only guides the belt, but allows the line of direction of the advancing side to deviate to a small extent, depending on the elasticity of the material.

Parallel shafts may be driven by flexible bands or connectors passing over pulleys, the central planes of which coincide, without any guiding arrangements for the belting. The shafts revolve in the same or opposite directions, according as the belt is open or crossed. Means of changing the relative speeds of rotation are furnished by pulleys of continuously varying diameter, or by speed cones (see : Applied). A common arrangement for driving a lathe spindle, in either direction at several definite speeds, is to provide a counter shaft on which are mounted two fixed pulleys and two loose pulleys to accommodate two driving belts from the main shaft, one of which is open and the other crossed. The belts are moved laterally by the forks of a striking gear pressing on the advancing sides of the belts, and the pulleys are arranged so that the belts either wrap round the loose pulleys, or can be shifted so that one wraps round a-fixed pulley, while the other still remains on its loose pulley. Motion in either direction is thereby obtained, and a considerable variation in the speed of rotation can be obtained by providing a cone pulley on the counter shaft, which drives the cone pulley secured to the lathe