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jointly. Hence, let a and a' denote the angular velocities of two rotating pieces, or a pair of numbers proportional to those angular velocities; r and, the perpendicular distances of a pair of points in those two pieces from their respective axes, or a pair of numbers proportional to those distances; and v and v', the respective velocities of those two points, or a pair of numbers proportional to those velocities; then the velocity-ratio of the points is,

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In order that a pair of points in a pair of rotating pieces may have equal velocities-that is, in order that

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SECTION IV.-Screw-like Motion of Primary Pieces.

57. Helical or Screw-like Motion (A. M., 382,) is compounded of rotation about a fixed axis, and of translation along that axis: the advance (as the translation in a given time is called) bearing a constant proportion to the rotation in the same time; in other words, the moving piece advances along the axis of rotation through an uniform length during each turn.

The subject of the resolution of screw-like motion into components in other and more complex ways will be considered in the next chapter.

58. General Figure of a Screw-Pitch. (A. M., 471.)—In order that a primary moving piece may have screw-like motion, its figure ought to be that of a true screw; and it ought to turn in a bearing of the same figure, fitting it accurately. The figure of a screw may be described in general terms as consisting of a projection of uniform cross-section called the thread, winding in successive coils round a circular cylinder. The best form of section for the thread of a screw that is to be used as a primary moving piece for producing helical motion only, and not as a fastening, nor in "screw gearing," is rectangular. The forms suited for other purposes will be considered later. There are two

sorts of screws, convex, or external, and concave, or internal; in the former the thread winds round the outside of a cylindrical spindle; in the latter it winds round the inside of a hollow cylinder. When the word "screw "is used without qualification, an external screw is usually meant; an internal screw is called a "nut.” When a primary moving piece is an external screw, its bearing is an internal screw; when the primary moving piece is an internal screw, the bearing is an external screw. The truth or accuracy of

figure of a screw depends mainly on the perfect uniformity of the pitch; that is to say, the distance, measured parallel to the axis, from any point in one coil of the thread to the corresponding point in the next coil. For example, the pitch of the screw R in fig. 29, so long as it is measured parallel to the axis, may be measured either from D to F, from E to G, from F to H, or from G to K, or between any pair of corresponding points in two successive coils; and it ought to be exactly the same wheresoever it is measured. The pitch is also the uniform distance through which the screw advances at each turn.

59. Right-Handed and Left-Handed Screws.-A screw is said to

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R

K

H

be right-handed or left-handed according as right-handed or left-handed rotation is required in order to make it advance; and this is a permanent distinction, and not dependent on the position of the spectator, as the distinction between right-handed and lefthanded rotation is (Article 48, page 25). For example, in fig. 29, L is a left-handed screw, and R a right-handed screw. Most screws used in the arts are right-handed; left-handed screws are made for special purposes only.

Fig.

29.

60. Comparative Motion of a Point in a Screw.-The principles of the present Article apply not only to any point in the thread or in the spindle of a screw, but to any point in a body that is rigidly attached to the screw, so as to move along with the screw as one piece; such as a wheel or a lever fixed to and turning with the screw. In fig. 29, let A B be the axis of the screw, and C a point rigidly attached to it at the perpendicular distance C A from the axis. Then, while the screw makes one turn the motion of the point C is the resultant of two components at right angles to each other: an advance, along with the whole screw, in a direction parallel to the axis, through a distance equal to the pitch of the screw; and a revolution, round a circle described about the axis with the radius A C, and having, therefore, the circumference 6.2832 A C. In most questions of comparative motion connected with screws, the quantity of most importance is the velocity-ratio of those two components of the motion of a given point, and it is expressed as follows:

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61. Path of a Point in a Screw-Linear Screw or Helix.-A point in, or rigidly attached to, a screw, traces a path which may be called a screw-shaped line or linear screw. By mathematicians it is called a helix. A helix winds round in successive similar coils upon a cylindrical surface described about the axis of rotation with a radius equal to the perpendicular distance of the tracing point from the axis. The distance, measured parallel to the axis, between any two successive coils is everywhere the same, and is identical with the pitch of the screw; and the angle of inclination of the inear screw to the axis is everywhere the same.

Points in, or rigidly attached to, a screw, at equal distances from the axis, trace by their motion equal and similar linear screws on one and the same cylindrical surface. Points at unequal distances from the axis trace different linear screws, inclined to the axis at different angles, and situated on cylindrical surfaces of unequal radii; but the pitch of all those linear screws is the same. All the edges, whether projecting or re-entering, of a screw-thread are linear screws.

A linear screw may be traced on a cylindrical surface by any mechanical contrivance which ensures that, while the cylinder rotates, the tracing point shall advance along a line parallel to the axis at a rate bearing a constant proportion to the rate of rotation. This will be further considered in that part of this treatise which relates to the construction of machinery.

A linear screw is the shortest line on the surface of a cylinder between two points that are neither in one plane traversing the axis nor in one plane perpendicular to the axis; and a cord or a flexible wire stretched on a cylindrical surface between two such points tends to assume of itself the figure of a linear screw.

62. Projection of a Linear Screw. The most useful projection of a linear screw is that upon a plane traversing the axis, and is drawn as follows:-In fig. 30, let A B represent the axis of the screw. Draw DAC perpendicular to A B, making AC = A D = the radius of the cylindrical surface in which the helix is to be situated. Draw DI and C F parallel to A B; those two lines will be the traces of the cylindrical surface. About A, with the radius A C, draw the semicircle C K D; this represents the trace of one-half of the cylindrical surface on a plane perpendicular to its axis, "rabatted" upon the plane of projection. Divide the semicircle into any convenient number of. equal arcs (Article 51, page 27); the greater the number of those divisions, the greater will be the accuracy of the projection. In fig. 29 the semicircle is divided into six equal arcs only; in practice a greater number will in general be required.

On C F, or any other line parallel to the axis, lay off CE = the intended pitch of the screw, and divide it into twice as many

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