which can then be drawn through those points. This is the projection of one coil; and as many successive coils as may be required may be projected by simply repeating the same curve. In fig. 30 the projection, E H M F, of a second coil is shown; and it has been constructed by laying off an uniform distance parallel to the axis and equal to the pitch from each projected point of the first coil; for example, G H, L M, E F.

The half coils on the nearer side of the cylinder, viz., GLE and H M F, are drawn in thicker lines than the half coils, CG, E H, on the farther side of the cylinder. The screw is righthanded. Had it been left-handed, C G and EH would have been on the nearer side, and G E and HF on the farther side of the cylinder.

63. Development of a Lincar Screw.-The development of any figure drawn on a curved surface consists in supposing the surface to be a flexible sheet, and drawing the appearance which the figure would present if that sheet were spread out flat on a plane. Some curved surfaces only are capable of being developed, such as a cylinder, and a cone; others, such as a sphere, are not. To draw the development of the cylindrical surface in fig. 30, as bounded by the two circles whose projections are CD and FI, draw Cc perpendicular to A B, and equal in length to the circumference of the cylinder (see Article 51, page 27), and complete the rectangular parallelogram Cef F; this will be the required development of the cylindrical surface.

To draw the development of one coil, C G E, of the linear screw, take ce the pitch CE; draw the straight line Ce; this will be the required development. To draw the development of the second coil, EHF, take ef = the pitch, and draw the straight line Ef; and so on for any required number of coils.

The uniform angle of inclination of the linear screw to the axis is ECe FEƒ

=

One method of drawing a screw on a cylindrical surface is first to draw its development on a sheet of some flexible substance, and then to roll that sheet round a cylinder of the proper radius.

The several points in the development marked with small letters are the respective developments of the points marked with the corresponding capital letters in the projection.

To draw the development of the series of lines parallel to the axis which pass through the points of division of the circumference, divide Cc into twice as many equal parts as the semicircle C D is divided into, and draw straight lines parallel to C F through the points of division, such as di, qw, &c.

The length of one coil of the screw is

Ce (circumference 2+ pitch 2).

=

A.

64. The Radius of Curvature of a linear screw is found by the following construction:-In fig. 31, let A. C be the radius of the cylindrical surface in which the screw is situated. Draw A Y, making the angle CA Y equal to the angle which the screw makes with a plane perpendicular to its axis. Draw CY perpendicular to AC, cutting AY in Y, and Y Z perpendicular to A Y, cutting AC produced in Z. A Z will be the required radius of curvature. Its length may also be found by calculation, as follows:

65. Normal Pitch. (4. M., 472.)-By the normal pitch Fig. 31.

Y

of a linear screw is to be understood the distance from one coil to the next, measured, not parallel to the axis, but along the shortest line on the cylindrical surface between the two coils; that is to say, along another helix or linear screw which cuts all the coils of the original linear screw at right angles. The normal pitch is to be determined from the development of the screw, as follows:-In fig. 30, from c let fall en perpendicular to Ce; cn will be the normal pitch. The straight line c n is part of the development of the normal helix, as it may be called. When produced, it cuts Ef, the development of the next coil, in o, and no = cn is also the normal pitch. By finding the intersections, such as p, of the development of the normal helix with the series of straight lines parallel to the axis, any number of points, such as P, in the projection of the normal helix, C NOP, may be found if required. The normal pitch may be calculated by the following formula:circumference × pitch length of one coil

c n =

Сс се
Се

=

The pitch of the normal helix, if required, may be found by producing cp in fig. 30 till it cuts C F produced, and measuring the distance of the point of intersection from C; and then its radius of curvature may be found by a construction like that in fig. 31; but in general it is more convenient to find these quantities by calculation, as follows:

circumference 2

pitch of original helix

circumference 2

radius of curvature of normal helix = AC c. (1+ pitch of orig. helix?)

The sum of the reciprocals of the radii of curvature of the original helix and the normal helix is equal to the reciprocal of the radius of the cylinder; that is, to

AC

The pitch of a screw as measured parallel to the axis may be called the axial pitch, in order to distinguish it from the normal pitch; but when the word "pitch" is used without qualification, axial pitch is always to be understood.

The several linear screws which exist in the figure of an actual solid screw, or which are described by points in it or rigidly attached to it, have all the same axial pitch; but they have not the same normal pitch except when they are situated on the same cylindrical surface.

66. Divided Pitch.-A screw sometimes has more than one thread, in which case the distance between any coil of any one thread and the next coil of the same thread is divided by the other threads into as many parts as the total number of threads. In that case the distance from a point in one thread to the corresponding point in the next thread may be called the divided pitch, to distinguish it from the distance between two successive coils of the same thread, or pitch proper, which may, when required, be designated as the total pitch. The advance of the screw at each revolution depends on the total pitch only, in the manner already explained, and is wholly independent of the number of threads and of the divided pitch; so that division of the pitch does not affect the motion of a screw as a primary piece. Its use in combinations of pieces will be afterwards explained.

Division of the pitch of a linear screw is illustrated in fig. 30, where two additional linear screw threads, marked by dotted lines, are shown dividing the pitch of the original screw into three equal parts. To avoid confusion, one only of the additional screw-lines is lettered, viz., that marked QRSTUVW in the projection, and qr, Rstu, Uvw, in the development. The other is unlettered. The divided axial pitch is C R = CE, and the divided normal pitch cx = fcn.

The several linear screw threads in a case of this kind are all parallel and similar to each other; and in the development they are represented by parallel straight lines. They divide the circumference into as many equal parts as there are threads; and the length of one of those parts may be called the circular or circumferential pitch. In fig. 30, the circular pitch is represented by the arc C Q, and by its development e q.

43

CHAPTER III.

OF THE MOTIONS OF SECONDARY MOVING PIECES.

67. General Principles, (A. M., 383, 384, 503.)—In the present chapter the general principles only of the motions of secondary moving pieces in machines can be given, many of their most important applications being reserved for that chapter which will treat of "Aggregate Combinations in Mechanism," and some for the chapter on Elementary Combinations." The mechanism for producing the motions of secondary moving pieces belongs wholly to those later chapters.

Secondary moving pieces have already been defined (in Article 37, page 17) as those which are carried by other moving pieces, or which have their motions not wholly guided by their connection with the frame. Their motions, therefore, are not restricted, like those of primary pieces, to translation in a straight line, rotation about a fixed axis, and that combination of those two motions which constitutes the motion of a screw with a fixed axis; they comprehend translations along curved lines of various figures, rotations about shifting axes, and various combinations of translations and rotations. The paths of points, too, in secondary pieces are not restricted to three forms-the straight line, the circle, and the helix; they comprehend a great variety of curved lines, both plane and of double curvature. The comparative motions. of any two points in a primary piece are constant. The comparative motions of two points in a secondary piece very often vary from instant to instant as the piece changes its position.

In many cases the motions of secondary pieces are partially guided or restricted. For example, a secondary piece may be so guided that all its movements take place parallel to a fixed plane; in which case its motions are restricted to translations parallel to the fixed plane, and rotations about axes perpendicular to it; and the paths of its points are restricted to lines, straight or curved, in or parallel to that plane; and this restricted case is by far the most common in mechanism. Another kind of restriction on the movements of a secondary piece is when it turns about a ball and socket joint, or some equivalent contrivance, so that one point at the centre of the joint is kept fixed: in this case its motions are restricted to rotations about axes traversing that fixed point; and the motions of points in it are restricted to