I. Development of Teeth.-Let O, fig. 101, be the common apex of the pitch-cones, O B I, O B' I, of a pair of bevel-wheels; O C, O C', the axes of those cones; O I their line of contact. Perpendicular to OI draw A I A', cutting the axes in A, A'; make the

Б

Fig. 101.

D'

outer rims of the patterns and of B the wheels portions of the cones

A BI, A' B' I, of which the narrow zones occupied by the teeth A will be sufficiently near for practical purposes to a spherical surface described about O. As the cones, DABI, A'B' I, cut the pitch-cones at right angles in the outer pitchcircles, I B, I B', they may be called the normal cones. To find the traces of the teeth upon the normal cones, draw on a flat surface

circular arcs, I D, I D', with the radii A I, A' I; those arcs will be the developments of arcs of the pitch-circles, I B, I B', when the conical surfaces, A B I, A' B' I, are spread out flat. Describe the traces of teeth for the developed arcs as for a pair of spur-wheels, then wrap the developed arcs on the normal cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.

II. Traces and Projections of Teeth.-Fig. 102 illustrates the process of drawing the projection of a tooth of a bevel-wheel on a plane perpendicular to the axis. In the first place, let A C represent the common axis of the pitch-cone and normal cone; A being the apex of the normal cone. Let A I be the trace of the normal cone on a plane traversing the axis; and let I I', perpendicular to I A, be part of the trace of the pitch-cone on the same plane, of a length equal to the intended breadth of the toothed rim of the wheel. CI perpendicular to A C is the radius of the pitch-circle in which the pitch-cone and normal cone intersect each other. About A, with the radius A I, draw the circular arc DID, making DI=ID: half the pitch; DID will be the development of an arc of the pitch-circle of a length equal to the pitch. On the arc D I D lay off I G IG half the thickness of a tooth on the outer pitch-circle. Then, by the rules for spurwheels, draw the trace, H G EG H, of one tooth and a pair of half-spaces, with a suitable addendum-circle through E, and a suitable root-circle, H F H.

=

=

The straight line F I E will be the trace, upon a plane traversing the axis, of the outer side of a tooth; and E and F will be the traces, on that plane, of the outer addendum-circle and rootcircle respectively. From E and F draw straight lines, E E and

FF, converging towards the apex of the pitch-cone; these will be the traces of the addendum-cone and root-cone respectively. (For want of space, the apex of the pitch-cone is not shown in fig. 102.) Through I, parallel to F I E, draw F'I' E'; this will be the trace,

h

Fig. 102.

on a plane traversing the axis, of the inner side of a tooth; and the points E, I, and F' will be respectively the traces of the inner addendum-circle, inner pitch-circle, and inner root-circle.

Through A, parallel to C I, draw the straight line A i e, and conceive this line to be traversed by a plane perpendicular to the axis, as a new plane of projection. Through the points F, I, E, F, I', E, draw straight lines parallel to C A, cutting A e in f. i, e, f, e; these points, marked with small letters, will be the projections, on the new plane, of the points marked with the corresponding capital letters.

L

Divide the depth, F E, of the tooth at its outer side into any convenient number of intervals. Through the points of division draw straight lines parallel to CA; these will cut fe in a series of points, which will be the projections of the points of division of FE. Through the points of division of F E, and also through the projections of those points, draw circular arcs about A as a centre. Measure a series of thicknesses of the tooth on the arcs which cross F E, and lay off the same series of thicknesses on the corresponding arcs which cross fe; a curve, hge gh, drawn through the points thus found, will be the required projection, on a plane parallel to the axis, of the outer side of a tooth.

The projection, l' g' é g' h', of the inner side of a tooth is found by a similar process, except that the measuring and laying-off the thicknesses is rendered unnecessary by the fact that each pair of corresponding points in the projections of the outer and inner sides lie in one straight line with A. For example, having drawn about A a circular arc through i, draw the two straight lines A g, Ag; these will cut that arc in the points g', g', being the points in the projection of the inner side corresponding to g, g in the projection of the outer side; and thus it is unnecessary to lay off the thickness g' g.

145. Teeth of Skew-bevel Wheels — - General Conditions.-The surfaces of the teeth of a skew-bevel wheel belong, like its pitchsurface, to the hyperboloidal class, and may be conceived to be generated by the motion of a straight line which, in each of its successive positions, coincides with the line of contact of a tooth with the corresponding tooth of another wheel. Those surfaces may also be conceived to be traced by the rolling of a hyperboloïdal roller upon the hyperboloidal pitch-surface, in the manner described in Article 84, pages 70 to 73.

The conditions to be fulfilled by the traces of the fronts and backs of the teeth on the hyperboloidal pitch-surface are:-A. That each of those traces shall be one of the generating straight lines of the hyperboloid (Article 106, page 89); B. That the normal pitch, measured from front to front of the teeth along the normal spiral (Article 106, page 89), shall be the same in two wheels that gear together (this second condition is always fulfilled if the two pitch-surfaces are correctly designed, and the numbers of teeth made inversely proportional to the angular velocities); and C. That the teeth, if in outside gearing, shall be right-handed on both wheels, or left-handed on both wheels; and if in inside gearing, contrary-handed on the two wheels.

Skew-bevel teeth may be said to be RIGHT-HANDED or LEFTHANDED, according to the direction in which the generating lines of the teeth appear to deviate from the axis when looked at with the axis upright, as in fig. 103, page 147. For example, the wheel

in that figure has left-handed teeth; for the generating

line

I' I deviates to the left of the axis A' A. The same rule applies to the direction in which the crests of the teeth appear to deviate from the radii of the wheel, when looked at as in the upper part of fig. 105, page 150.

Right-handed

teeth have lefthanded normal

spirals, and left

right-handed nor- C mal spirals.

146. Skew-bevel Teeth-Rules.-I. Normal Section of a Tooth.-In fig. 103, let A a A' be the axis of a skew-bevel wheel: let a be the centre of the throat of its hyperboloidal pitch-surface; let the dotted curve through I be the trace of that surface on a plane traversing the axis; and let CI ai be the radius of the pitch-circle at the middle of the breadth of the rim of the intended

Fig. 103.

wheel, as found by Rule I. of Article 106, page 88. Draw by Rules II. and III. of that Article, pages 88, 89, the normal I A

and tangent I I" I', to the trace of the pitch-surface at I. Then find, by Rule V. of that Article, page 89, the radius of curvature of the normal spiral at the point I, and lay off that radius of curvature, I S, along the normal.

In fig. 104 (which is on a larger scale than fig. 103, for the sake of distinctness), let A C, as before, be the axis of the wheel, CI the radius of the middle pitch-circle, I A the normal, and I S the radius of curvature of the normal spiral; draw IN perpendicular to I S. Then, by Rule V. of Article 106, page 89, find the angle (=0g F in fig. 68, page 88) which a tangent to the normal spiral makes with a tangent to the pitch-circle, and draw I P, making that angle with IN. Lay off I P equal to the pitch as measured on the middle pitch-circle; let fall P N perpendicular to IN; then