occur which result in such remarkable work being accomplished. The principle of the machine may be described in the following way. A spindle is supported near the centre in a ball-bearing

FIG. 157.

socket, so as to give it free play in every direction. At the top of the spindle is provided a roller, which is kept in contact with a template of any desired shape by means of springs. The lower end of the spindle carries the cutting tool, which is of peculiar shape, and so arranged that the cutting edge of the tool is on the centre line of the spindle. The spindle is rotated by means of gearing, and at the same time the roller is made to travel along the inside surface of the template. Whatever shape the template may be, the centre line of the spindle, and with it the cutting edge of the tool, will faithfully reproduce the shape to a reduced scale. The machine, like that just described, is therefore a copying machine.

A reference to Figs. 158 and 159, which represent a side section and a front elevation respectively, will make the component parts of the mechanism clear. A is the spindle turning in the ball-socket bearing B, formed in the sliding-head C, and, passing through the spur wheel D, is provided with the conical roller E, running free upon its upper end. The spindle is embraced in the spur wheel D by a sliding block, G, free to move in a slide formed in the wheel, and pulled off the centre outwards by a pair of strong springs, H, H. These springs cause the roller E to be kept in contact with the template ring F surrounding it. If F be square, it is clear that the centre of the roller will likewise describe a square; if it be

triangular, so likewise will the path described by the centre of the roller; if it be circular, so likewise will the path described by the centre of the roller; and so on. The template F is secured to projections, J, from the head by means of clips, K, so as to admit of easy replacement with other forms, and also to admit of rotation when desired. The set-screw L is provided in the wheel D, to set the sliding block in the centre position when required.

The roller E is of conical form, and is connected to the adjusting screw M above it, so that by adjusting the position of the

roller, a portion of the roller of lesser or greater diameter, as required, may be brought against the template, and consequently the tool is caused to describe a figure corresponding to the template, but varying in size according to the position of the roller. This is useful in adjusting the finishing cut to great nicety, and enables a tapered hole to be drilled. The screw is operated through a pair of spur wheels and the hand-wheel N.

The feed motion is obtained by moving the entire head downwards. The self-acting feed is operated in the ordinary manner, motion being given to the screw O. The nut P is made with a bevel wheel at its upper end, and this is in mesh with a wheel on a shaft having a hand-wheel, Q, secured to it. By this means it is possible to vary the feed during the progress of the self-acting motion.

CHAPTER IV.

MECHANISMS CONSISTING OF FOUR LOWER PAIRS.

§ 104. THE mechanisms considered in Chapter II. illustrate the uses to which a combination of ordinary gearing may be put. They are, in point of fact, made up of chains consisting of three pairs constantly repeated. The preceding chapter deals mostly with chains which consist of four lower pairs-the chain being used principally on account of some geometrical property rather than to transmit power.

Next, let us consider more fully the simple chains, which consist of four lower pairs, and therefore of four elements. Such a chain may consist of four turning pairs, in which case it is called the four-bar chain; of three turning and one sliding pair, in which case it is called the slider-crank chain; or of two turning and two sliding pairs, in which case it is called the double-slider crank chain. Diagrammatic sketches of these three chains have been already shown in Figs. 9, 1, and 8 respectively. Each chain consists of four elements, so that by fixing each link in turn, four mechanisms may be obtained from each chain. But although these mechanisms may differ greatly from each other in constructive details, and in the purposes to which they are put, it must be remembered that since they are the same kinematic chain, the relative motions of the different parts must always be the same. The different purposes for which they are used will appear as the subject proceeds.

§ 105. Determination of Velocity Ratio.-In mechanisms such as are about to be discussed, the different elements usually consist of links which, neglecting the slight changes due to change of temperature and elasticity, are of invariable length. The object

is to determine the velocity ratio of any two points, or of any two turning pairs, in the chain relative to one of the links; and in determining that velocity ratio, one or other of two methods may be adopted, namely

(1) The method of the instantaneous centre.

(2) By drawing the velocity diagram.

For simple linkwork chains, such as those consisting of four links, the first method is, on the whole, the more convenient, and will be discussed first; but for more complicated mechanisms, the second possesses considerable advantages over the first.

§ 106. Instantaneous Centre of Rotation.-Let us assume that all the links in the chain move parallel to one plane, in which case the displacement of any

link is the same as the displacement of any straight line in that link. The displacement of any link is then equivalent to a rotation about some finite or infinitely distant point. To prove this statement, let (Fig. 160) AB, A'B' represent any two successive

A

FIG. 160.

B

B'

positions of a link which moves parallel to the plane of the paper, and bisect AA', BB' at right angles by lines which intersect at the point O. Since AB is equal to A'B', the triangles OAB, OA'B' are equal in all respects, so that OAB may be imagined a rigid triangular frame rotating about the point O. By a finite rotation about O, the body can be moved from its first position, represented by AB, to its second position, represented by A'B'. For any intermediate position, the points A and B will lie on circular arcs having O as centre, and precisely the same motion. could be obtained by having circular slots, having the common centre O, in which pins at the ends of the link AB are allowed to slide (Fig. 161). The motion in such a case, whether for large or small displacements, would be precisely the same as if the link AB were rigidly attached to an arm rotating about the centre O.

Now, in the general case, the paths traced out by A and B will not be circular arcs, but may be curves of any shape whatever.

FIG. 161.

In that case, although the body may be brought from its initial to its final position by rotation about one point, yet the intermediate positions which it would then trace out would not coincide with the positions which it is constrained to successively occupy. But if, instead of attempting to bring the body from its initial to its final posi

tion by a finite rotation about one fixed point, it is taken from the first position to a second position, very near to the first, by rotation about some centre (obtained precisely as before), and from its second position to a third, very near to the second, by rotation about some other centre, and so on, until it reaches its final position; then, by successive rotations through small angles about different centres, the body will trace out the exact path required of it. The centres thus obtained are called virtual or instantaneous centres of rotation.

A second proposition is immediately obvious, namely, The direction of motion of every point in a body is perpendicular to the line joining that point to the instantaneous centre, and its velocity is proportional to the length of that line.

Hence, whenever the directions of motion of two points in a rigid link are known, the instantaneous centre is at once determined by drawing perpendiculars to those directions to intersect. That point of intersection is the point about which, for the moment, the body is rotating; and the body may be imagined to extend to that centre, and to actually turn about a pin attached to the frame of the machine. At the instant considered, the instantaneous centre is, therefore, that point in the moving link which is at rest, and its position may be obtained by methods other than that just described (see § 109).