nearly, then half a dozen cutters will be sufficient, and these must be made as nearly as possible to suit the wheels of 150, 50, 30, 21, 16, 13.

146. In the epicycloid abc (fig. 35, p. 65) join Tb, and let TOb = p, AT = R, and Tk 2r, then radius of

=

(Peacock's Examples,

p. 195), and this radius passes through T, for Tb is a

normal to abc at b. Now Tb = 2r. sin

which expression becomes identical with the value of D in

It appears therefore that if, in fig. 59, mn were an arc of an epicycloid whose base were the pitch circle, and

ΚΤ

diameter of the describing circle

=

sin

then would Pm

be its radius of curvature at m; and in like manner Q'm can be shewn to be the radius of curvature of the corresponding hypocycloid mp.

Consequently teeth described by this method approximate to epicycloidal teeth, and when described in sets by the Odontograph, approximate to those of the third solution (Art. 113). Hence the rules that have been given for the least numbers, and the length or addenda of all such teeth, may also be applied to these.

147. In all the figures of teeth hitherto given the teeth are symmetrical, so that they will act whether the wheels be turned one way or the other. If a machine be of such a nature that the wheels are only required to turn in one direction, the strength of the teeth may be doubled by an alteration of form, exhibited in fig. 64.

This represents a portion of the circumference of a pair

64

of wheels of which the lowest is the driver, and always moves in the direction of the arrow, consequently the right side of its teeth and the left side of the follower's teeth are the only portions that are ever called into action; and these sides are formed exactly as usual. But the back of each tooth, both in the driver and follower, is proposed to be bounded by an arc of an involute, as eg or cb.

The bases of these involutes being proportional to the pitch circles, they will during the motion be sure to clear each other, because, geometrically speaking, they would, if the wheels moved the reverse way, work together correctly; but the inclination of their common normal to the line of centers is too great for the transmission of pressure. The effect of this shape is to produce a very strong root, by taking away matter from the extremity of the tooth where the ordinary form has more than is required for strength, and adding it to the root.

148. In Hooke's system, under its second form (Art. 68), it has been shewn that the point of contact travels during the motion of the wheels from one side to the other; a fresh contact always beginning on the first side just before the last contact has quitted the other side. To ensure this, the teeth of the wheels in each section B (fig. 32) must be so formed that when the angular velocity ratio is constant the teeth may begin and end contact on the line of centers; otherwise, if the teeth were formed upon the principles of the previous Articles of this Chapter, it is evident that the sliding contact of the teeth before and after the line of centers would still remain. The simplest mode of effecting this object is to make the flanks of the teeth radial, as in the second solution, and their faces any arc of a circle that will lie within the epicycloidal face required by that solution. If, for example, the portion of tooth that lies beyond the pitch line be a complete semicircle whose center is upon that line, this condition will be complied with. I have described the teeth of B, fig. 32, in this manner. The figures A and C are nearly the same as Hooke's, but he has given no front view of his wheels, and has said nothing respecting the forms of the teeth.

To describe the teeth of wheels when their axes are not parallel.

149. To describe the teeth of bevil-wheels, let ACT, ATD, fig. 66, be the pitch cones of a pair of bevil-wheels described as in Arts. 43, 44; AT their line of contact. Let AET be any other cone also lying in contact with ATD along AT, and having its apex at A; therefore the axes of the three cones will be in the same plane ABF. Also the circumferences of their bases being at the same distance AT from A, will lie on the surface of a sphere whose center and radius are A and AT.

Let the three cones revolve round their axes with the

same relative velocity as would be produced by the rolling contact of their surfaces, then the line of contact will always be AT, and (calling the intermediate cone AET the describing cone) a line nm upon the surface of the describing cone directed to the common apex will generate one surface ompn on the outside of the cone ATD, and another surface smrn on the inside of the cone ACT.

Also, these surfaces will touch along the describing line nm, for since ponm is generated by the rolling of the describing cone upon the surface of the cone ADT, the motion of nm is at every instant perpendicular to the line of contact AT; and therefore, the normal plane at nm to the

surface generated by nm will pass through AT. And in like manner, the normal plane to the surface rsnm will pass through AT; therefore the surfaces touch along nm.

If these surfaces be employed as teeth, and the rotation of the cone ATD be communicated to the cone ACT by their contact action, the angular velocity ratio will, from the mode of their generation, be precisely the same as that produced by the rolling contact of the conical surfaces; for at the beginning of the motion op and rs coincide with AT, and in the position of the figure the arcs To, T's respectively described by the bases of the two cones are each equal to Tm, and therefore themselves equal.

150. The arc om is an arc of a spherical epicycloid* whose base is the cone ADT, and describing cone the cone AET; and in like manner sm is an arc of a spherical hypocycloid whose base is the cone ATC, and describing cone AET. But in practice, the portion of spherical surface occupied by these arcs, when employed for teeth, is a narrow belt extending to a small distance only from ToD

• DEFINITION. If a cone ABC be made to roll upon another fixed cone ADCE in such a manner that their summits A always coincide; then a tracing 65

point C in the circumference of the base of the rolling cone will trace a kind of epicycloid Ckm, which will plainly lie on the surface of a sphere whose center is A and radius AC, whence this curve is termed a spherical epicycloid. If the cone roll on the concave surface of the base, the curve becomes a spherical hypocycloid.