for the whole chain, but between any two links in the chain. With the crosshead in place, the connecting-rod can only swing about the crosshead pin, and the crosshead can only slide to and fro in the guides. With the higher pair introduced, the pin is part of the rod, and turns in the guides as well as moving to and fro; or, to express it differently, the pin not only slides in the guides, but rolls in them as well. When a higher pair does occur, it must be imagined replaced by two lower pairs and one element; or, in other words, when applying the formula, each higher pair must

FIG. 16.

be considered as equivalent to two lower pairs and one additional element.

§ 12. Minimum Number of Pairs in a Machine. Simple Machines. -In considering machines, it is clear that no machine can consist of only one link, but that it must have at least two. A bar of iron, for example, is not in itself a machine; neither is a pivot. But if the bar be rested on the pivot so that the latter acts the part of a fulcrum, the combination may be used as, and will fulfil all the functions of, a machine. A machine, therefore, must consist of at least two links forming one pair. The machines containing only one pair are usually known as the simple machines, and include the lever, wheel and axle, inclined plane, and screw. first two are examples of a turning pair, and the second two of a sliding pair. In each there is no train of mechanism, and the driver and follower are one and the same piece. They are briefly discussed in Chapter II.

The

§ 13. Machines consisting of Two Pairs.-Kinematic chains having two lower pairs do not exist. Chains having two higher pairs do exist; but generally these are equivalent, kinematically, to four lower pairs. For example, in Fig. 8, the blocks may be dispensed with, and pins, attached to the rod, allowed to slide in the two slots, as in Fig. 16. In such a case, the mechanism will

consist of two higher pairs and two elements; but the chain, kinematically, is equivalent to four elements and four lower pairs.

§ 14. Machines consisting of Three Pairs.-An example of a machine consisting of three lower pairs is the wedge, which consists of three sliding pairs; another is the hand press, consisting of one turning, one sliding, and one screw pair. By far the most common illustration of a chain having two lower pairs and one higher pair is friction rollers and toothed wheels. Here the two wheels form turning pairs with the frame, and the rollers or teeth, having line contact, constitute a higher pair. A further example of two lower pairs and one higher pair is in belting. These are all fully discussed in Chapter II.

§ 15. Machines consisting of Four Lower Pairs.-Examples of chains consisting of four lower pairs (and therefore of four elements) are exceedingly numerous, and generally occur in link work. They include such familiar mechanisms as the direct-acting engine, the crank and slotted lever, the donkey engine, Watt's parallel motion, toggle joints, Oldham's coupling, Hooke's joint, etc. These, and others, are fully discussed in Chapters III. and IV.

§ 16. Complex Machines.-When a mechanism consists of more than four lower pairs, the motions, as a rule, are more complicated than in the foregoing cases. The same general treatment may be extended to all whatever the number of pairs; and the methods are fully discussed in Chapters V. and VI. The general question of higher pairing is discussed in Chapters VII. and VIII.

§ 17. Plan of the Book.-As already indicated, the general plan of the book is, roughly, to consider machines according to the number of pairs of elements which they contain. The simple machines, consisting of one pair, are first discussed; then those consisting of three pairs, or a repetition of three pairs, the latter including those machines in which gearing (belting or toothed wheels) is the important factor, and under this head a considerable number of machine tools and other appliances are described and illustrated. Following this, the mechanisms consisting of four lower pairs are discussed at length: in the first place, when the mechanisms are used because of some geometrical property; and in the second place, when they are used for power

purposes, so that the modification of motion is the important item to consider. The general consideration of mechanisms consisting of any number of lower pairs is next discussed, the determination. of velocity and acceleration ratios being explained. The two closing chapters are occupied with problems of higher pairing, and include a full discussion of toothed wheels, gear-cutting machines, cams, etc.

CHAPTER II.

SIMPLE MACHINES AND MACHINE TOOLS, ETC.

WE shall consider, in the present chapter, the simpler kinds of machines.

§ 18. Machines consisting of One Lower Pair: Lever, Wheel and Axle. The lever consists of a link, AB, rotating about a pin, P, the combination forming a turning pair

(Fig. 17). When motion takes place, the driving and following points A and B describe circular arcs about the fulcrum P as centre, and the ratio of the

velocities of the points A and B is equal to the ratio of the lengths of the arms AP and PB.

In the wheel and axle (Fig. 18), the lever is replaced by two cylindrical pulleys, A and B, keyed to a spindle which rotates in bearings in the frame of the machine. If indefinitely thin belts be wrapped round the pulleys, the velocity ratio of any two points a and b in them will be equal to the ratio of the radii of the pulleys A and B.

§19. Inclined Plane.-In the inclined plane we usually require the ratio of the velocity in any given direction to that of the block

along the plane. If the displacement of the block along the plane in a given time is represented by AB (Fig. 19), the vertical displacement will be represented by AC, and the horizontal displacement by CB, and these may be taken to represent the velocities in the three directions. Along any direction whatever, such as AD, the velocity will be represented by AD, D being the foot of the perpendicular from B on the direction AD.

§ 20. Screw: Single and Multiple Threaded; Right and Left Handed. -Imagine the plane in the previous figure to be a thin thread, and let the thread be wrapped round a circular cylinder in such a way that its inclination with a transverse plane is always the same. The curve assumed by the initially straight thread is called a helix (Fig. 20), and the thread may be wrapped round the cylinder

any number of times in succession. The pitch of the helix is the distance, measured parallel to the axis of the cylinder, between any pair of corresponding points in two successive coils, and, if the thread be correctly wrapped on, will be the same wherever it is measured. Instead of defining the pitch directly as a length, the number of threads or coils in one-inch length of cylinder may be stated; thus eight threads to the inch represent a pitch of oneeighth of an inch. If a small particle be moved along the helix, it will advance a distance equal to the pitch along the axis in the same time that it makes one turn round the cylinder, and any other linear advance will be proportional to the angle turned through measured on a transverse plane.

In practice the thread must be of finite dimensions, and takes the form of a bearing surface as shown in Fig. 21. That sur

face is such that all cylinders co-axial with the initial cylinder