and the method of drawing those arcs is as follows:-Let fig. 107 represent the trace of part of a wheel with its teeth, that wheel being the smallest wheel of a set that are to be capable of gearing together; because the smallest wheel of such a set requires the greatest addendum: let C be the centre, A A the addendum-circle,

B B the pitch-circle, D D the root-circle, and let E" I" J", EIJ, EI J', be the fronts of teeth designed according to the proper rules, and F", F, F', the pitch-points of the backs of those teeth. To any one of those back pitch-points, as F, draw the radius C F; bisect C F in G, and about G draw the semicircle F K C. Draw a straight line, H K, perpendicular to and bisecting the distance, EF, between the crest E and back pitch-point F; and let that straight line cut the semicircle in K. About the centre C, with the radius CK, draw the circle K" K K'; this will be the base-circle of the required involutes (see Article 131, page 121).

To draw the circular arcs approximating to those involutes, lay off, from the back pitch-points to the base-circle, the equal distances F" K" = F' K' = F K, &c; and about the respective

n

upper part of the figure is a projection of the rim of a wheel with stepped teeth on a plane parallel to the axis, and the lower part is a projection on a plane perpendicular to the axis. A wheel thus formed resembles in shape a series of equal and similar toothed discs placed side by side, with the teeth of each a little behind those of the preceding disc. In such a wheel, let Ρ be the circular pitch, and n the number of steps. Then the path of contact, the addendum, and the extent of sliding, are those due to the divided pitch 2, while the strength of the teeth is that due to the thickness corresponding to the total pitch p; so that the smooth action of small teeth and the strength of large teeth are combined. The action of small teeth is smoother and steadier than that of large teeth, because they can be made to approximate more closely to the exact theoretical figure; and also because the sliding motion of one tooth upon another is of less extent. In the example shown in fig. 108 there are four steps, so that the divided pitch is onefourth of the total pitch; and the path of contact (EI F, in the lower part of the figure) is of the length suited to the divided pitch, being only one-fourth of the length which would have been required had the fronts of the teeth not been stepped.

151. Helical Teeth, also invented by Dr. Hooke with the same object, are teeth whose fronts, instead of being parallel to the line of contact of the pitch-cylinders of a pair of spur-wheels, cross that line obliquely, so as to be of a screw-like or helical form: in other words, they are teeth of the figure of short portions of screw-threads (Article 58, page 36); the trace of each thread on a plane perpendicular to the axis being similar to that of a stepped tooth, as shown in the lower part of fig. 108. Fig. 108 A shows a projection of the rim of a wheel with helical teeth on a plane parallel to

the axis.

In order that a pair of wheels with parallel axes and helical teeth may gear correctly together, the teeth, besides being of the same circular pitch, must have the same transverse obliquity; and if in outside gearing, they must be right-handed on one wheel and left-handed on the other. If in inside gearing, they must be either right-handed or left-handed on both wheels. In fig. 108 A the teeth are left-handed. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical fronts, and terminates at the aftermost end; and the rims of the wheels are to be made of such a breadth that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.

Helical teeth are open to the objection that they exert a laterally oblique pressure, which tends to increase friction.

When, in designing a skew-bevel wheel, a portion of the tangent cylinder at the throat of the hyperboloïd (Article 106, page 87;

and Article 85, page 73) is used as an approximation to the true pitch-surface, the teeth of that wheel become screw-threads, having a transverse obliquity determined by the principles of Article 147, page 152; and, as has been already stated in the article referred to, they are either right-handed or left-handed in both wheels.

152. Screw and Nut.-The figure of a true screw, external or internal, and the motion of a screw working in a corresponding screw-shaped bearing, have been described in Articles 57 to 66, pages 36 to 42. In the elementary combination of an external and internal screw, more commonly called a screw and nut, the two pieces have threads, one external and the other internal, of similar figures and equal dimensions, so as to fit each other truly; and one of them turns about their common axis without translation, while the other slides parallel to that axis without rotation. The best form of section for the threads is rectangular. The comparative motion is, that the sliding piece advances through a distance equal to the pitch (viz., the "total axial pitch") during each revolution of right-handed, the sliding the turning piece. If the threads are left-handed,

piece is made to move towards an observer at one end of the axis leftJ right-handed

by {

left-handed rotation, and to move from him by right{

handed

handed rotation, of the turning piece. The combination belongs to Mr. Willis's Class A, because the velocity-ratio is constant; and the extent of the motion is limited by the length of the screw.

153. Screw Wheel-Work in General.-Screw wheel-work consists of wheels with cylindrical pitch-surfaces, having screw-threads or helical teeth instead of ordinary teeth One case of screw-gearing has been described in Article 151, page 156-viz., that in which the axes are parallel. The cases to which this and the following articles relate are those in which the axes are not parallel; so that the pitch-surfaces in an elementary combination are a pair cylinders touching each other in one pitch-point, like those represented in Article 85, fig. 55, page 73. The pitch-point (O', fig. 55) is obviously in the common perpendicular of the two axes (FG', fig. 55); and there is one straight line traversing the pitch-point (O C', fig. 55), which is a tangent at once to the two pitchcylinders and to the acting surfaces or fronts of each pair of threads at the instant when those surfaces touch each other at the pitch-point: that straight line may be called the LINE OF CONTACT. The angles of inclination of the screw-threads to the two axes (see Article 63, page 40) are equal respectively to the angles made by the line of contact with those axes. The PITCH-CIRCLES of the two screws are the two circular sections of the pitch-cylinders which traverse the pitch-point. The PLANE OF CONNECTION, or PLANE OF

ACTION, is a plane traversing the pitch-point normal to the line of contact: that plane, of course, traverses the common perpendicular of the axes.

When the line of contact is found by the rule given in Article 84, page 71, the cylindrical pitch-surfaces represent the tangentcylinders at the throats of a pair of hyperboloïds; and the screwthreads are approximations to the skew-bevel teeth suited for that combination, as already stated in Article 151, page 156. But in many cases the line of contact has positions greatly differing from this; and then the comparative motion becomes different from that of a pair of skew-bevel wheels; the object of screw-gearing in such cases being to obtain, with a given pair of cylindrical pitchsurfaces, a velocity-ratio of rotation independent of the radii of those surfaces; and such is the difference between approximate skew-bevel gearing and screw gearing in general.

In every elementary combination in screw wheel-work, each of the two pieces is at once a screw and a wheel; but it is customary, when their diameters are very different, to call that which has the smaller diameter the ENDLESS SCREW, or WORM, and that which has the greater diameter the wORM-WHEEL. For example, in fig. 111 (farther on) a' is the worm, or endless screw, and A' the wormwheel. The word "endless" is used because of the extent of the motion being unlimited.

Screw wheel-work belongs to Mr. Willis's Class A, the velocityratio being constant.

The following are the general principles of elementary combina tions in screw wheel-work::

I. The angular velocities of the two screws are inversely, and their times of revolution directly, as the numbers of threads; whence it follows that the angular velocity-ratio must be expressible in whole numbers, as in the case of ordinary toothed wheels.

II. The divided normal pitch (see Article 66, page 42), as measured on the pitch-cylinders, must be the same in two screws that gear together.

III. The common component of the velocities of a pair of points in the two screws at the instant when those two points touch each other and pass the pitch-point, is perpendicular to the line of contact and to the common perpendicular of the axes; in other words, it coincides with the intersection of the plane of connection and the common tangent-plane of the two pitch-cylinders.

IV. The circular or circumferential pitches of the two screws (Article 42, page 66), as measured on their pitch-cylinders, are proportional to the total velocities of points (called the surface velocities) in those cylinders; and they bear the same proportion to the divided normal pitch that those total velocities bear to their common component.