distance FG, draw DE perpendicular to AG. similar triangles ADE and AFG, we have, AE AG DE: FG; : :. FG = From the AG × DE AE But AG, DE = Psina, and AE = W + Pcosa, hence we have, FG = Pa sina And, since HG equals a, we have the following conditions for stability, indifference and instability, respectively, If we denote the distance FG by y, and the weight of a cubic foot of the material of the pillar by W, we shall have, since W = 4a2xw, y = Psina x 4a wxPcosa If, now, we suppose the intensity and direction of the force P to remain the same, whilst x is made to assume every possible value from 0 up to any assumed limit, the value of y will undergo corresponding changes. The suc cessive points thus determined make up a line which is called the line of resistance, and whose equation is that just deduced. If the pillar is made up of uncemented blocks, it will remain in equilibrium so long as each joint is pierced by the line of resistance, provided that the tangent to the line of resistance makes with the normal to the joint an angle less than that of least resistance (Art. 88). The highest degree of stability will be attained when the line of resistance is normal to every joint, and when it passes through the centre of gravity of each. 9. To determine the conditions of equilibrium and stability of an arch of uncemented stones. SOLUTION. D K EL Fig. 58. M V a Let MNLK represent half of an arch sustained in equilibrium by a horizontal force P, and by the weight of the archstones. Through the centre of gravity of the first arch-stone draw a vertical line, and on it lay off a distance to represent the weight of that stone. Prolong the direction of P, and lay off a distance equal to the horizontal pressure. Complete the parallelogram of forces, oabB, and draw the diagonal oB. This will be the resultant of the forces combined. Combine this resultant with the weight of the second arch-stone, and this with the weight of the third, and so on, till the last inclusive. The polygon oBCDE, thus found, is the line of resistance, and if this lies wholly within the solid part of the arch, the arch will be stable; but, if it does not lie within it, the arch will be unstable. A rupture will take place at the joint where the line of resistance passes without the solid part of the arch. This problem may be solved analytically, in accordance with the principles already illustrated. It is only intended to indicate the general method of proceeding. CHAPTER IV. ELEMENTARY МАСНINES. Definitions and General Principles. 75. A MACHINE is a contrivance by means of which a force applied at one point is made to produce an effect at some other point. The force applied is called the power, and the point at which it is applied, is called the point of application. The force to be overcome is called the resistance, and the point at which it is to be overcome is called the working point. The working of any machine requires a continued application of power. The source of this power is called the motor. Motors are exceedingly various. Some of the most important are muscular effort, as exhibited by man and beast in various kinds of work; the weight and living force of water, as exhibited in the various kinds of water-mills; the expansive force of vapors and gases, as displayed in steam and caloric engines; the force of air in motion, as exhi bited in the windmill, and in the propulsion of sailing vessels; the force of magnetic attraction and repulsion, as shown in the magnetic telegraph and various magnetic machines; the elastic force of springs, as shown in watches and various other machines. Of these motors, the most important ones are steam, air, and water power. To work, is to exert a certain pressure through a certain distance. The measure of the quantity of work performed by any force, is the product obtained by multiplying the effective pressure exerted, by the distance through which it is exerted. Machines serve simply to transmit and modify the action of forces. They add nothing to the work of the motor; on the contrary, they absorb and render inefficient much of the work that is impressed upon them. For example, in the case of a water-mill, only a small portion of the work expended by the motor is transmitted to the machine, on account of the imperfect manner of applying it, and of this portion a very large fraction is absorbed and rendered practically useless by the various resistances, so that, in reality, only a small fractional portion of the work expended by the motor becomes effective. Of the applied work, a part is expended in overcoming friction, stiffness of cords, bands, or chains, resistance of the air, adhesion of the parts, &c. This goes to wear out the machine. A second portion is expended in overcoming sudden impulses, or shocks, arising from the nature of the work to be accomplished, as well as from the imperfect connection of the parts, and from the want of hardness and elasticity in the connecting pieces. This also goes to strain and wear out the machine, and also to increase the sources of waste already mentioned. There is often a waste of work arising from a greater supply of motive power than is required to attain the desired result. Thus, in the movement of a train of cars on a railroad, the excess of the work of the steam, above what is just necessary to bring the train to the station, is wasted, and has to be consumed by the application of brakes, an operation which not only wears out the brakes, but also, by creating shocks, injures and ultimately destroys the cars themselves. Such are some of the sources of the loss of work. A part of these may, by judicious combinations and appliances, be greatly diminished; but, under the most favorable circumstances, there must be a continued loss of work, which requires a continued supply of power from the motor. In any machine, the quotient obtained by dividing the quantity of useful, or effective work, by the quantity of applied work, is called the modulus of the machine. As the resistances are diminished, the modulus increases, and the machine becomes more perfect. Could the modulus ever become equal to 1, the machine would be absolutely perfect. Once set in motion, it would continue to move forever, realizing the solution of the problem of perpetual motion. It is needless to state that, until the laws of nature are changed, no such realization need be looked for. In studying the principles of machines, we proceed by approximation. For a first result, it is usual to neglect the effect of hurtful resistances, such as friction, adhesion, stiffness of cords, &c. Having found the relations between the power and resistance under this hypothesis, these relations are afterwards modified, so as take into account the various resistances. We shall, therefore, in the first instance, regard cords as destitute of weight and thickness, perfectly flexible, and inextensible. We shall also regard bars and connecting pieces as destitute of weight and inertia, and perfectly rigid; that is, incapable of compression or extension by the forces to which they may be subjected. 76. Elementary Machines. The elementary machines are seven in number— viz., the cord; the lever; the inclined plane; the pulley, a combination of the cord and lever; the wheel and axle, also a combination of the cord and lever; the screw, a combination of two inclined planes twisted about an axis; and the wedge, a simple combination of two inclined planes. It may easily be seen that there are in reality but three elementary machines-the cord, the lever, and the inclined plane. It is, however, more convenient to consider the seven abovenamed as elementary. By a suitable combination of these seven elements, the most complicated pieces of mechanism are produced. The Cord. 77. Let AB represent a cord solicited by two forces, P and R, applied at its extremi ties, A and B. In order that the cord may be in equilibrium, A B Fig. 59. it is evident, in the first place, that two forces must act in the direction of the cord, and in |