teeth be applied to these frustra, in the same manner as in spur gearing they are attached to cylindrical surfaces, bevel gearing will be formed acting on the same principles of sliding contact which we have already discussed.

Let A B C, ACD (fig. 199) be two cones rolling in contact; take any other cone A E C also rolling in contact with A B C, in the line A C. As these cones roll together, the generating cone AEC will describe an epicycloidal surface pqrs on the outside of the cone a C D, and a hypocycloidal surface pt v s on the inside of the cone A C D. These surfaces will touch in the line ps, and will have a plane normal to their common tangent passing through A c. If, therefore, these surfaces be attached respectively to the cones A B C, A C D, and the motion of one cone be communicated to the other through the sliding contact of these surfaces, the motion will be uniform, as if the cones were driven by rolling contact at a c.

The curves pt, p q, lie in reality on the surface of a sphere of a radius equal to A c; but in practice, in bevel wheels, a small frustrum of a cone, tangential to the sphere at the circumference of the pitch line, is substituted for the spherical segment. Thus draw F C G (fig. 199) perpendicular to A C, cutting the axes of the cones in F and G. Let these lines revolve over the pitch lines of the cones and describe the narrow frustra. Then the epicycloidal surfaces may, without sensible error, be supposed to lie in these frustra, and to be generated there by the revolution of a generating circle c E.

Imagine the surface of these frustra to be unwrapped so as to lie in one plane; they will form parts of circular annuli. Thus let A B C, A C D (fig. 200) be two conical frustra; draw F C G as before, perpendicular to the line of contact A c. From G, with radii GH, G C, and G K, describe the circles K L, CM, HN; and from F, with radii F K, F C, F н, describe similar circles K P, CQ, HR; then the surfaces K P R H and K L N H will be developements of the frustra C D, C B. Let these be treated as spur wheels, and CQ, CM being treated as the pitch lines, let teeth be described by a describing circle in the method already explained for epicycloidal or other teeth. If, then, the plane on which these have been described, and which we suppose of drawing paper or other flexible material, be cut along the arcs K P, H R, KL, HN,

the circular annuli may be wrapped round the frustra C B, C D, and the forms of the teeth traced off upon them.

The axes of bevel wheels are generality of cases, at right angles. of bevels, with the frustra of the loped in the manner described.

in practice, in the great Fig. 200 shows such a pair extremity of the teeth deve

Skew Bevels.

When two axes or shafts, which have to be connected by bevel wheels, do not meet in direction, it is usual, as stated in the preliminary Chapter,* to introduce an intermediate bevel wheel with two frustra. But the same object can more easily be accomplished by adopting skew bevels.

Let в p q (fig. 201) be the place of one of the two frustra, a its centre, and a e the shortest distance between the axis of

* Mills and Mill-work, Vol. I., p. 47, §§ 70, 71.

Bp q, and the axis of the wheel to be connected with it. Divide a e in c, so that ac : ec mean radius of ABC: mean radius of frustrum working with A B C. Draw cpq perpendicular to a e; then cp or c q is the line of action of the teeth, according to the direction in which the teeth are laid out in the pinion.

Figure 202 shows two wheels laid out in this manner; a e, as before, is the eccentricity or shortest distance between the two shafts, and is divided in c proportionally to the mean radii of the wheels; with centre a and radius a c describe a circle, and draw ed perpendicular to a e. Take df = ce; then d will be the centre of the other wheel. From centre d, with radius fd, describe a circle. Then the directions of all the teeth in A B C will be tangents to the circle described about a, and the directions of all the teeth in D E F will be tangents to the circle described about f. Fig. 203 shows two such wheels in gear, the eccentricity permitting the shafts to pass each other.

Fig. 203.

The Worm and Wheel.

By this contrivance the motion of a screw is communicated with great smoothness to oblique teeth on a spur wheel.

The section of a screw through its axis is precisely similar to that of a double rack. Let A B be such a section, and for simplicity suppose that the form of the threads of the screw has been determined by one of the rules already given for racks.

Then the teeth of the wheel CDE may evidently be formed so as to work with the centre section of the screw. Now the effect of the revolution of the screw is precisely similar to that of the racks, and the sections of the threads of the screw will appear to travel from end to end, in the same way as a rack pushed forwards in the same direction. If, therefore, it is sufficient that the wheel teeth be in contact with the screw at one point only, the teeth of the wheel may be made oblique, but straight, the obliquity being equal to the pitch of the screw. This is the usual practice of millwrights. If, however, the teeth are required to be in contact with the entire breadth of the tooth, the outline of the tooth must vary in every section of the wheel, and the process of describing these teeth becomes very complex. Practically, the difficulty has been overcome by first making a pattern screw of steel, notched in the threads to convert it into a cutting instrument. The wheel is then roughly cut out, and, being fixed in a frame, the screw is used to cut out the spaces between the teeth to their true form.

Strength of the Teeth of Wheels.

The pressure on the teeth varies directly as the horse-power transmitted, and inversely as the velocity of revolution. Thus if one wheel transmit 5 horse-power and another 10 horse