752 For the condition of equilibrium of forces not parallel is that they shall be represented in direction and magnitude by the sides and diagonals of certain parallelograms, and of parallel forces that they shall divide certain straight lines in certain ratios ; and the parallel projection of a parallelogram is a parallelogram, and that of a straight line divided in a given ratio is a straight line divided in the same ratio. The resultant of a parallel projection of any system of forces is the projection of their resultant ; and the centre of gravity of a parallel projection of a solid is the projection of the centre of gravity of the first solid. 17. Principle of the Transformation of Structures. Here we have the following theorem :< If a structure of a given figure have stability of position under a system of forces represented by a given system of lines, then will any structure whose figure is a parallel projection of that of the first structure have stability of position under a system of forces represented by the corresponding projection of the first systein of lines. For in the second structure the weights, external pressures, and resistances will balance each other as in the first structure ; the weights of the pieces and all other parallel systems of forces will have the same ratios as in the first structure ; and the several centres of resistance will divide the depths of the joints in the same proportions as in the first structure. If the first structure have stability of friction, the second struc ture will have stability of friction also, so long as the effect of the projection is not to increase the obliquity of the resistance at any joint beyond the angle of repose. The lines representing the forces in the second figure show their relative directions and magnitudes. To find their absolute directions and magnitudes, a vertical line is to be drawn in the first figure, of s ich a length as to represent the weight of a particular portion of the structure. Then will the projection of that line in the projected figure indicate the vertical direction, and represent the weight of the part of the second structure corresponding to the before- mentioned portion of the first structure. The foregoing "principle of the transformation of structures" was first announced, though in a somewhat less comprehensive form, to the Royal Society on the 6th of March 1856. It is useful in practice, by enabling the engineer easily to deduce the conditions of equilibrium and stability of structures of complex and unsymmetrical figures from those of structures of simple and symmetrical figures. By its aid, for example, the whole of the properties of elliptical arches, whether square or skew, whether level or sloping in their span, are at once deduced by projection from those of symmetrical circular arches, and the properties of ellipsoidal and elliptic-conoidal domes from those of hemispherical and circu- lar-conoidal domes ; and the figures of arches fitted to resist the thrust of earth, which is less horizontally than vertically in a cer tain given ratio, can be deduced by a projection from those of arches fitted to resist the thrust of a liquid, which is of equal intensity, horizontally and vertically. 18. Conditions of Stiffness and Strength. After the arrangement of the pieces of a structure and the size and figure of their joints or surfaces of contact have been determined so as to fulfil the conditions of stability, conditions which depend mainly on the position and direction of the resultant or total load on each piece, and the revive magnitude of the loads on the different pieces, the dimen sions of each piece singly have to be adjusted so as to fulfil the con ditions of stiffness and strength, conditions which depend not only on the absolute magnitude of the load on each piece, and of the resistances by which it is balanced, but also on the mode of distri bution of the load over the piece, and of the resistances over the joints. The effect of the pressures applied to a piece, consisting of the load and the supporting resistances, is to force the piece into a state of strain or disfigurement, which increases until the elasticity, or resistance to strain, of the material causes it to exert a stress, or effort to recover its figure, equal and opposite to the system of applied pressures. The condition of stiffness is that the strain or disfigurement shall not be greater than is consistent with the pur poses of the structure ; and the condition of strength is that the stress shall be within the limits of that which the material can bear with safety against breaking. The ratio in which the utmost stress before breaking exceeds the safe working stress is called the factor of safety, and is determined empirically. It varies from three to twelve for various materials and structures. The STRENGTH OF MATERIALS forms the subject of a special article, to which the reader is referred. TART II. THEORY OF MACHINES. 19. Parts of a Machine Frame and Mechanism. The parts of a machine may be distinguished into two principal divisions, the frame, or fixed parts, and the mechanism, or moving parts. The frame is a structure which supports the pieces of the mechanism, and to a certain extent determines the nature of their motions. [APPLIED MECHANICS. The form and arrangement of the pieces of the frame depend upon the arrangement and the motions of the mechanism ; the dimen sions of the pieces of the frame required in order to give it stability and strength are determined from the pressures applied to it by means of the mechanism. It appears therefore that in general the mechanism is to be designed first and the frame afterwards, and that the designing of the frame is regulated by the principles of the stability of structures and of the strength and stiffness of materials, care being taken to adapt the frame to the most severe load which can be thrown upon it at any period of the action of the mechanism. Each independent piece of the mechanism also is a structure, and its dimensions are to be adapted, according to the principles of the strength and stiffness of materials, to the most severe load to which it can be subjected during the action of the machine. 20. Definition and Division of the Theory of Machines. From what has been said in the last section it appears that the depart ment of the art of designing machines which has reference to the stability of the frame and to the stiffness and strength of the frame and mechanism is a branch of the art of construction. It is there fore to be separated from the theory of machines, properly speaking, which has reference to the action of machines considered as moving. In the action of a machine the following three things take place : First, Some natural source of energy communicates motion and force to a piece or pieces of the mechanism, called the receiver of power or prime mover. Secondly, The motion and force are transmitted from the prime mover through the train of mechanism to the working piece or pieces, and during that transmission the motion and force are modi fied in amount and direction, so as to be rendered suitable for the purpose to which they are to be applied. Thirdly, The working piece or pieces by their motion, or by their motion and force combined, produce some useful effect. Such are the phenomena of the action of a machine, arranged in the order of causation. But in studying or treating of the theory of machines, the order of simplicity is the best ; and in this order the first branch of the subject is the modification of motion and force by the train of mechanism ; the next is the effect or purpose of the machine ; and the last, or most complex, is the action of the prime mover. The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and sim plicity by first considering alone the principles of the modifica tion of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the com bined modification of motion and force, which are founded both on geometry and on the laws of dynamics. The separation of kine matics from dynamics is due mainly to Monge, Ampere, and Willis. The theory of machines in the present article will be considered under the following four heads : I. PURE MECHANISM, or APPLIED KINEMATICS ; being the theory of machines considered simply as modifying motion. II. APPLIED DYNAMICS ; being the theory of machines con sidered as modifying both motion and force. III. PURPOSES AND EFFECTS OF MACHINES. IV, APPLIED ENERGETICS ; being the theory of prime movers and sources of power. CHAP. I. ON PURE MECHANISM. 21. Division of the Subject. Proceeding in the order of simplicity, the subject of Pure Mechanism, or Applied Kinematics, may be thus divided : Division 1. Motion of a point. Division 2. Motion of the surface of a fluid. Division 3. Motion of a rigid solid. Division 4. Motions of a pair of connected pieces, or of an " elementary combination " in mechanism. Division 5. Motions of trains of pieces of mechanism. Division 6. Motions of sets of more than two connected pieces, or of "aggregate combinations." A point is the boundary of a line, which is the boundary of a surface, which is the boundary of a volume. Points, lines, and surfaces have no independent existence, and consequently those divisions of this chapter which relate to their motions are only preliminary to the subsequent divisions, which relate to the motions of bodies. Division 1. Motion of a Point. 22. Path and Direction. See above, p. 679, 21. 23. Uniform Velocity. See p. 680, 25. 24. Varied Velocity. See p. 680, 25. 25. Direct Deviation, or Acceleration and Retardation. See pp.
680, 681, 27-29.