In a single revolution the length of the hanging part of the rope is diminished by a length equal to the difference between the circumferences of the large and small parts of the axle. The weight is lifted by means of the pulley D, which is suspended by equal lengths of rope. The distance through which the pulley, and consequently the weight, will be raised by each revolution, is therefore half the length by which the rope is diminished.

Equilibrium will therefore be established when the power, multiplied by the length of the lever which turns the axle, is equal to the weight multiplied by half the difference of the radii of the two parts of the axle.

One advantage in this machine is, that there is no reaction of the weight when the power is withdrawn. On the other hand, a large amount of rope must be wound up to raise the weight even a small distance; though less than is requisite in any other machine of the same efficiency, in which the power is transmitted through ropes.

Fusee. When a power which varies in intensity is required to produce uniform motion in the weight, it is effected by adopting this principle of various diameters in the cylinder, or, in other words, by applying different leverages to the weight. This is well illustrated in the fuseel of a watch (fig. 46). A is a drum containing a

Fig. 46.

spiral spring of finely tempered steel, fixed at one end on an axis. The other end is attached to the inside of the drum A, which freely revolves round the axis. A chain, C, is coiled round the drum; one end being fixed

1 L., fusus, a spindle; F., fuseau.

to the drum, and the other to the lower part of the fusee B. This fusee is a conical figure turning on an axis, and on it a spiral groove is cut to receive the chain C.

Let us suppose the watch to be wound up. All the chain will then be wound off the drum A on to the fusee B, the last coil being at the small end of the fusee; and the spring in A will be stretched to its utmost intensity. Its tendency to recoil will be then at its greatest, and will diminish in proportion to the extent of its recoil. Now, if the chain were attached to a cylinder, the motion imparted to it by the spring would be a gradually diminishing one, the watch would consequently lose time. But the chain being wound on a cone, when the pulling force of the spring is at its greatest, the leverage on the axis of the fusee is at its least; and as the strength of the spring diminishes, so the leverage on the axis of the fusee increases. These are so adjusted as to result in a motion of the axis of the fusee at a uniform rate.

CHAPTER VIII.

THE MECHANICAL POWERS (continued).
2 (b). COMPOUND WHEEL WORK.

Relations with Levers.-As it is sometimes convenient to adopt a system of levers to raise a weight, so it is occasionally desirable to produce a certain effect by a system of wheels, called a train of wheels. The wheel and axle having been shown to be a modification of the lever, so a system of wheels is only another form of compound lever; and the conditions necessary to bring about equilibrium are the same in both.

As in the compound lever the power is applied to the longer arm of the lever, which acts through the short arm on the long arm of the next lever; so in complex wheelwork the power is applied to the circumference of the first wheel, which transmits its effect through its own

axle to the circumference of the next wheel. This is effected by various means.

Friction Wheels. Where the weight is very small, the mere friction of the wheels in contact is sufficient to impart motion, as in fig. 47, where the power is

00001.

Fig. 47.

P

applied to the circumference of the wheel A, and is transmitted through its axle to the circumference of the second wheel, B, and so on. Each separate wheel and axle being a lever, the effect of the above combination is the same as in a system of levers the longer arms of which are equal to the radii of the wheels, and the shorter arms equal to the radii of the axles. Equilibrium will therefore be produced when the product of the power by the radii of all the wheels is equal to the product of the weight by the radii all the axles.

Pulleys, or Drums.-Another method of transmitting the power of the axle to the succeeding circumference, is by straps or cords, GG, HH (fig. 48), passing over

and being

the circumference of the wheel and axle ; drawn tight, the tension,1 or strain, will generate sufficien

1 L., tendo, to stretch.

F

friction between the rope or band and the circumferences to transmit the power from one to the other. The wheel is called a drum when a strap is used, and a pulley when a cord is used.

When it is intended that both the driving wheel and the wheel to which it communicates motion should revolve in the same direction, the cord is merely carried round both circumferences, as in G G. But when it is intended that the wheels shall revolve in opposite directions, the strap is crossed, as HH. This latter method possesses the advantage of having more surface of the circumference to act upon, and consequently the friction, or bite, is greater.

Cog Wheels.-The most common method, however, of transmitting this force, is by means of cog-wheels, that

is, wheels with projections standing out from their circumferences, which are called webs. The projections on the webs of the wheel are called cogs, or teeth; and those on the web of the axle are termed leaves; and the axle itself is generally called a pinion.1

Several conditions must be fulfilled in each pair of wheels, to ensure their working together. The teeth of

each wheel must be equal and equidistant; and the teeth of one wheel must be equal to and equidistant from those of the other. The number of teeth in each wheel will therefore be proportional to their circumferences. Consequently, in speaking of the ratio of one wheel to another, we may either say that one is a wheel of 72 teeth, and the other one of 12, or that one is six times larger than the other. The power multiplied by the teeth in the wheels, equals the weight multiplied by the leaves in the pinions, when the machine is in equilibrium. The circle midway between the grooves and summits of the teeth, is called the pitch circle; and the motion transmitted by the contact of the teeth is the same as would be produced by the rolling contact of the pitch circles.1

Hunting Cog. In determining the number of teeth in a wheel and the pinion which works with it, it is desirable that the same pair of teeth should come in contact as seldom as possible, in order to lessen the friction, and consequent wear and tear. For example, if a wheel has 80 teeth, and its pinion 10, it is evident that every tenth tooth of the wheel would engage with the same leaf of the pinion, and that each leaf of the pinion would come in contact with the same eight teeth in every revolution of the wheel.

Were the teeth mathematically exact, this would be of little consequence. But as the best-cut wheels vary slightly in their teeth, these inequalities are compensated for by making teeth and leaves so work that each leaf

1Equation of equilibrium for a pair of toothed wheels.-Let the wheels be in equilibrium with a weight, A, at the axle of the larger, and a weight, B, at the axle of the smaller, and let R be the pitch circle of the larger wheel and S that of the smaller. Let the radii of the axles be equal and represented by r, and let the reaction of the wheels equal X. Then the equilibrium of the large wheel gives Axr = X× R, and of the smaller, B × r = X× S.