The paddle-wheel of a steamboat acts on the same principle; the water, which offers a resistance to the motion of the paddle-boards, is the power.]

Wheels and axles are often made to act upon one another by the aid of cogs, as in clockwork and mill machinery. In these cases the cogs on the periphery of the wheel take the name of teeth, those on the axle the name of leaves, and the axle itself is called a pinion.

The law of equilibrium of such machines may be easily demonstrated to be, that the power multiplied by the product of the number of teeth in all

W

Fig. 167.

the wheels, is equal to the weight multiplied by the product of the number of leaves in all the pinions.

A system of wheel and pinion work is represented at Fig. 169. It is scarcely necessary to observe, that in it, as in all other cases, the law of virtual velocities holds good-the power multiplied by the velocity of the power is equal to the weight multiplied by the velocity of the weight.

In the construction of such machinery attention has to be paid to the form of the teeth, so that they may not scrape or jolt upon one another. Several of them should be in contact at once, to diminish the risk of fracture and the wear.

If the teeth of a wheel be in the direction of radii from its centre it is called a spur-wheel.

If the teeth are parallel to the axis of the wheel it is called a crown-wheel. If the teeth are oblique to the axis of the wheel it is called a bevelledwheel.

By combining these different forms of wheel suitably together, the resulting motion can be transferred to any required plane. Thus by a pair of bevelled-wheels motion round a vertical axis may be transferred to a horizontal one, or, indeed, one in any other direction.

When a pinion is made to work on a toothed bar, it constitutes a rack. This contrivance is under the same law as the wheel and axle.

CHAPTER XXIX.

THE INCLINED PLANE-THE WEDGE-THE SCREW.

Description of the Inclined Plane-Modes of applying the Power-Conditions of Equilibrium when the Power is Parallel to the Plane or Parallel to the Base-Position of Greatest Advantage-Description and Mode of using the Wedge-Formation of the Screw.

By the inclined plane we mean the unyielding plane surface inclined obliquely to the resistance to be overcome.

In Fig. 170, A C represents the inclined plane; the angle at A is the elevation of the plane; the line A C is the length, CB is the height, A B the base.

Fig. 170.

B

In the inclined plane the power may be applied in the following directions:-1. Parallel to the plane; 2. Parallel to its base; 3. Parallel to neither of these lines.

As in the former cases, so in this-the conditions of the equilibrium may be deduced from those of the lever.

Let us take the first instance, when the power is applied parallel to the inclined plane. Let Q, Fig. 170, be a body placed upon the plane, A C, the height of which is B C, and the base A B. The weight of this body acts in the vertical direction, a W; the body rests on the point, c, as on a fulcrum; and the power, P, under the supposition, acts on Q, in the direction a P. From the fulcrum, c, draw the perpendicular, c b, to the line of direction of the weight, a W; draw also ca. Then does be a represent a bent lever, the power being applied to the point, a, and the weight at the point, b; and, therefore, the power is to the weight as b c is to ac; but the triangles, a bc, A B C, are similar to each other. Therefore we arrive at the following law :

When the power acts in a direction parallel to the inclined plane, it will be in equilibrio with the weight when it is to the weight as the perpendicular of the plane is to its length.

In a similar manner it may be shown that when the power acts parallel to the base it will be in equilibrio with the weight, if it be to the weight as the perpendicular of the plane is to its base.

In different inclined planes the power increases as the height of the plane, compared with its length, diminishes, and the best direction of action is parallel to the inclined plane. This is very evident from the consideration that, if the power be directed above the plane, a portion of it is expended in lifting the weight off the plane, while the diminished residue draws it up. If it be directed downward a part is expended in pressing the weight upon the plane, and the diminished residue draws it up. Therefore, if the power acts parallel to the plane, it operates under the most advantageous condition. The laws of the inclined plane may be illustrated by an instrument such

Fig. 171,

as is represented in Fig. 171, in which A c Ac' is the plane, which may be set at any angle. It works upon an axis, A A'. Upon the plane a roller, e, moves. It has a string passing over a pulley, d, and terminating in a scale-pan, f, in which weights may be placed. The direction of the string may be varied, so as to be parallel to the plane, or the base, or any other direction.

The inclined plane is used for a variety of purposes-very frequently for facilitating the movements of heavy loads

THE WEDGE.

The wedge may be regarded as two inclined planes laid base to base- ·

BDA, Fig. 172, being one, and B C A being the other. The planes, CA and D A, constitute the sides or faces of the wedge; B is its back, and BA its length.

B

Fig. 172.

C'

The mode of employing the wedge is not by the agency of pressure, but of percussion. Its edge being inserted into a fissure, the wedge, as in Fig. 173, is driven in by blows upon its back.

After it has been. struck for some tine, the wedge enters further into the substance of the wood, as in Fig. 174, and when the wood cannot be compressed any more, the wedge splits it, as in Fig. 175. It is kept from recoiling by the friction of its sides against the surfaces past which it has been forced.

Fig. 175. Fig. 174.

Fig. 173.

This mode of application of the wedge prevents us from comparing its theory with that of the inclined plane-a power to which it has so much external resemblance.

The power of the wedge increases as the length of its back, compared with that of its sides, is diminished. As instances of its application, we may

B

Fig. 176.

mention the splitting of timber, the raising of heavy weights, such as ships. Different cutting instruments, as chisels, &c., act in consequence of their wedge-shaped form.

THE SCREW.

If we take a piece of paper cut into a long, right-angled triangle, Fig. 176, and wind it about a cylinder, Fig. 177, so that the height, C B, of the triangle is parallel to the axis, the length, A C, will trace a screw

Fig. 177.

line on the surface. The same results if we take a cylinder and wind upon it a flexible cord, so that the strands of the cord uniformly touch one another.

In any screw the line which is thus traced upon the cylinder goes under the name of the "worm," or "thread," and each complete turn that it makes is called a spire." The distance from one thread to another, which, of course, must be perfectly uniform throughout the screw, is called the breadth of the worm.

Fig. 178. Fig. 179.

[The thread of a screw may have a thin sharp edge, as in Fig. 178, or a square edge, as in Fig.

179. In either case the principle of its action is the same.

In most cases the screw requires a corresponding cavity in which it may work; this passes under the name of a "nut." Sometimes the nut is caused to move upon the screw, and sometimes the screw in the nut. In either case the movable part requires a lever to be attached, to the end of which the power is applied.

The law of equilibrium of the screw is, that "the power is to the weight as the breadth of the worm is to the circumference described by that point of the lever to which the power is attached."

When the end of a screw is advancing through a nut, this law evidently becomes that the power is to the weight as the circumference described by the power is to the space through which the end of the screw advances. It is obvious, therefore, that the force of the screw increases as its threads are finer, and as the lever by which it is urged is longer.

When the thread of a screw works in the teeth of a wheel, as shown in Fig. 180, it constitutes an endless screw. An important use of this con

Fig. 180.

trivance is in the engine for dividing graduated circles. This screw is also used to produce slow motions, or to measure, by the advance of its point, minute spaces. In the spherometer, represented in Fig. 5, page 6, we have an example of its use.

For all those purposes where slow motions have to be given, or minute spaces divided, the efficacy of the screw will increase with the closeness of its thread. But there is soon a practical limit attained; for, if the thread be too fine, it is liable to be torn off. To avoid this, and to attain those objects almost to an unlimited extent, Hunter's screw is often used. It may be

e

a

understood from Fig. 181. It consists of a screw, working in a nut, a b. To a movable piece, e, a second screw, c, is affixed. This screw works in the interior of a, which is hollow, and in which a corresponding thread is cut.

While, therefore, a is screwed downward, the threads of c pass upward, and the movable piece, e, advances through a space which is equal to the difference of the breadth of the two screws. Fig. 181. In this way very slow or minute motions may be obtained with a screw, the threads of which are very coarse.

[Sometimes the lever is inserted into, or passes through, the nut into which the screw is inserted, as in Fig. 182. In this case the nut forces the screw upwards or downwards, according to circumstances.]

Fig. 182.

CHAPTER XXX.

OF PASSIVE OR RESISTING FORCES.

Difference between the Theoretical and Actual Results of Machinery-Of Impediments to Motion-Friction-Sliding and Rolling Friction-Coefficient of Friction-Action of Unguents-Resistance of Media-General Phenomena of Resistance-Rigidity of Cordage.

Ir has already been stated, in the foregoing chapters, that the properties of machinery are described without taking into account any of those resisting agencies which so greatly complicate their action. The results of the theory of a machine in this respect differ very widely from its practical operation. There are resisting forces or impeding agencies which have thus far been kept out of view. We have described levers as being inflexible, the cords of pulleys as perfectly pliable, and machinery generally as experiencing no friction. In the case of one of the powers, it is true that this latter resisting force must necessarily be taken into account; for it is upon it that the efficacy of the wedge chiefly depends.

So, too, in speaking of the motion of projectiles, it has been stated that the parabolic theory is wholly departed from, by reason of the resistance of the air; and that not only is the path of such bodies changed, but their range becomes vastly less than what, upon that theory, it should be. Thus, a 24-pound shot, discharged at an elevation of 45° with a velocity of 2,000 feet per second, would range a horizontal distance of 125,000 feet were it not for the resistance of the air; but through that resistance its range is limited to about 7,300 feet.

Of these impediments to motion, or passive or resisting forces, three leading ones may be mentioned. They are, 1st, friction; 2nd, resistance of the media moved through; 3rd, rigidity of cordage.

OF FRICTION.

Friction arises from the adhesion of surfaces brought into contact, and is of different kinds- -as sliding friction, when one surface moves parallel to the other; rolling friction, when a round body turns upon the surface of another.

By the measure of friction we mean that part of the weight of the moving body which must be expended in overcoming the friction. The friction which expresses this is termed the co-efficient of friction. Thus, the co-efficient of sliding friction in the case of hard bodies, and when the weight is small, ranges from one-seventh to one-third.

[We are not to consider friction as a small force, slightly modifying the effects of other agencies. On the contrary, its amount is in most cases very great. When a body lies loose on the ground the friction is equal to onethird or one-half, or in some cases of bodies supported by oblique pressure the amount is far more enormous. In the arch of a bridge, the friction which is called into play between two of the vaulting stones may be equal to the whole weight of the bridge. In such cases this conservative force is so great, that the common theory, which neglects it, does not help us even to guess what will take place. According to the theory, certain forms of arches only will stand; but in practice almost any form will stand, and it is not easy to construct a model of a bridge which will fall.