axes will be however changed, but they will not meet, and their common perpendicular will be equal to DE. Since DE is the locus of contact, it is clear that this motion will not disturb the angular position of either wheel in its own plane; and hence the angular velocity ratio of the wheels will remain constant and unaltered by the change of position. Involute wheels, therefore, may be employed to communicate a constant velocity ratio between axes that are inclined at any angle to each other, but which do

not meet.

But the demonstration supposes the wheels to be very thin, since they coincide with the planes that meet in the line DE, and the invariable points of contact are situated in this line. The edge of one of the wheels must be in practice rounded so that it may touch the other teeth in a point only.

ON CAMS AND SCREWS.

202. Having disposed of the teeth of wheels, we may now return to the remaining combinations in which sliding contact is employed to communicate a constant velocity ratio between two pieces.

If the motion of these pieces be limited to a not very considerable angle, or if one of them moves in a short rectilinear path in the manner of a rack, any of the pairs of curves in the first part of this chapter (in Arts. 116 to 127) may be employed in the single forms there shown, instead of being reduced to short arcs, and placed in successive order as teeth. To avoid unnecessary details, I shall confine myself to the examination of the cases in which one of these curves is reduced to a pin, as in the First Solution; for this method is generally preferred, and it has this advantage, that whereas greater friction is introduced when a long curved plate is substituted for a series of teeth,* the pin can be made into a roller, and thus the abrasion which would tend to destroy the form of the curved edge is transferred to the axis of the roller, which can be easily repaired when worn out.

203. In fig. 122, A is the center of motion of a revolving plate in which a slit ab is pierced, having parallel sides so as to embrace and nearly fit a pin m, which is carried by a bar CD fitted between guides so as to be capable of sliding in the direction of its length.

By carrying the point of contact farther from the line of centers (Art. 32).

If the plate revolve in the direction of the arrow the inner side of the slit presses against the pin and moves it further from the center A, but when the plate revolves in the opposite direction the outer edge of the slit acts against the pin and moves it in the opposite direction.

If the curved edges of the slit be involutes of the circle whose radius is Ac, where Ac is a perpendicular upon the path m c of the

bar, it appears from Art. 133 that the velocity ratio of plate and bar will be constant, and the linear velocity of the bar equal to that of the point c of the plate. But if any other velocity ratio be required, let Pc (fig. 123) be the path of the sliding bar, P the pin, A the center of the curve, aP the curve.

Let CAP=6, PAa=0, Ac=a, AP=r, then while a has moved from e to a, let P have moved from c to P; so that ca=mx cP; preserving a constant velocity ratio during the motion;

If the velocity of the circumference of the circle (radius Ac) equals the linear velocity of the bar,

which is the equation to the involute of the circle as it ought to be.*

If, however, the line Pc of the follower's path pass through the center A, then since equal angles described to the curve are to produce equal differences of radial distance in the pin, the curve becomes evidently the spiral of Archimedes; a curve which, although, as we see, capable of communicating velocity in a constant ratio between a circular and rectilinear path, cannot be employed for the teeth of racks, because the pitch line passes through the center of the wheel.

204. Sometimes the pin, instead of being mounted on a slide, is carried by an arm revolving round a center E, as mE, and therefore describes an arc of a circle. The curve is then derived from the first solution (Art. 129), the line of centers AE having been previously divided, in the ratio of the required angular velocities.

D

The angular motion of the curved plate which is the driver is Fig. 124. of course limited to the length of the slit a b, but this may be carried through several convolutions, as in fig. 124, where it is shown in the form of a spiral groove, excavated in the face of a revolving plate, and communicating rectilinear motion to the bar Dm by means of the pin at its extremity m, which lies always in the groove.

This may be termed a flut screw or plane

screw.

205. Combinations of this kind assume a great many different forms, the complete exhibition of which belongs rather to descriptive mechanism than to the plan of the present work. Thus, instead of employing the slit or groove, shown in these figures, the object of which is to produce action in both directions, a single curved edge may be employed, and the returning action produced by a weight or spring, which may be applied to the bar so as to keep the pin constantly in contact with it.

Curved plates of this kind are termed cams, or, when small, tappets, and they are more used to produce varying velocity ratios than constant ones. For which reason I shall refer to Chapters VI. and VII. for some other forms in which they appear. 206. If the path both of driver and follower be rectilinear, the slit will become straight.

* Peacock's Examples, p. 177.

Fig. 125.

Let a plane rectangle CD move in its own plane in a path parallel to its longest side, and have a straight slit cut in it making an angle with that side, and let a bar AB moving in the direction of its own length below this plane be provided with a projecting pin G which enters the slit, the slit making an angle with the path of this bar. Therefore the paths of the plane and bar make an angle + with each other. Ꮎ

G

B

A

=

If the plane move through a space Gf, draw gf parallel to the first position of the slit, then g will be the new position of the pin, and Gg the space described by the pin or bar;

velocity of plane_ Gf_sin Ggf_sin o
velocity of bar

a constant ratio.

=

=

=

Gg sin Gfg sin o'

If the bar move perpendicularly to the plane, 0+&=

207. To return to the revolving plate and bar; if the path of

Fig. 126.

B

the bar be not parallel to the plane of rotation of the plate, the latter must be formed into the cone or hyperboloid that would be generated by the rotation round its axis of the line which is the path of the pin, or other point of contact of the bar. Thus, in fig. 126, AB is the axis, A CD the sliding bar, e its pin, the path cd of whose acting extremity is in this case supposed to meet the axis. If this line cd generate a cone D by revolving round AB, the pin will always lie at the same depth in any groove excavated in the conical surface. Also, if this surface be developed, the groove ef will be the spiral of Archimedes. It is unnecessary to follow into detail all the forms, curves, and combinations, that arise in this manner. One case only requires more particular attention.

208. If the path of the bar CD be parallel to the axis of rotation AB, the conical surface upon which the groove is traced will become a cylinder; and to produce a constant velocity ratio

the spiral groove must be at every point equally inclined to a line drawn upon the surface parallel to the axis.

For it has been shown that a plane surface mh, fig. 127, moving perpendicularly to a sliding bar cd, will communicate motion to it in a constant ratio, by means of a

straight slit pr in which lies a pin fixed
to the bar, and that

velocity of plane =tan ;
velocity of bar

where is the angle rpd made by the
slit with the path of the bar.

Fig. 127.

D

m

d

с

B

If this plane be wrapped round the cylinder, keeping its axis parallel to the path of the bar, the groove will become a spiral, inclined at the angle & to a line drawn parallel to this axis. But the motion given to the bar by this spiral when the cylinder revolves will be exactly the same as if the plane had passed under it through the line kl and perpendicularly to the plane of the paper.

The velocity of the plane is now the velocity of rotation of the cylindrical surface, and therefore we have, if r be the radius of the cylinder, A its angular velocity, V the velocity of the bar,

rA
=tan &.
V

If the length of the plane be greater than the circumference of the cylinder, the spiral groove will encompass its surface through more than one revolution, and may, in this way, proceed in many convolutions from one extremity of the cylinder. to the other, its inclination to the axis of the cylinder remaining constant and equal to ; such a recurring spiral is termed a

screw.

Draw pq, qr respectively perpendicular and parallel to the path of the bar; if pq is equal to the circumference of the cylinder, qr will be the distance between two successive convolutions of the screw, and qr=. This is termed the pitch

2πr

tan o

of the screw, from its analogy to the pitch of a rack or toothed wheel. Every revolution of the screw carries the bar through a space equal to the pitch.

209. The screw is sometimes made in this elementary form,