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 FI' E'; this will be the trace,

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 ie, 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, i, é; these points, marked with smail 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 ares 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, hgegh, 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 are through 2, draw the two straight lines A g, A g; these will cut that are 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 hyperboloïdal 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 E 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 normal 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 I N 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

IN will be the normal pitch at the middle pitch-circle. About S, with the radius S I, draw a circular arc, and lay off on that arc the distance, D D, equal to the normal pitch, one-half to each side of I. Lay off the intended middle thickness, G G, of a tooth, onehalf to each side of I. Then draw, by the rules for spur-wheels, the normal section, H G E G H, of a tooth, being its trace upon a surface which cuts it normally at the middle of the breadth of the rim of the wheel.

II. Trace of a Tooth on the Normal Cone.-Through A in fig. 104 draw A i parallel and equal to CI, and through I draw Ii parallel and equal to CA. About A, with the radius A i, draw the circular arc d d, equal in length to I P, the pitch on the pitch-circle, and having the middle of its length at the point i. This will be the are on the pitch-circle corresponding to the arc D D on the normal spiral.

Divide E F, the middle depth of the tooth, into any convenient number of intervals; and through E and F and the points of division draw straight lines parallel to I i, cutting A ie in a series of corresponding points. Through the points in E F draw circular. ares about S. Through the corresponding points of ef draw circular arcs about A. From the points where the arcs cut the trace EG H measure oblique half-thicknesses to the centre line, E F, of the tooth, along oblique lines drawn parallel to P I; and lay off those half-thicknesses at both sides of e ƒ, along the arcs which cross it.

Through the points thus found draw the curve hgegh; this will be the projection, on a plane perpendicular to the axis, of the trace of a tooth upon the normal cone of the pitch-surface at the middle of its breadth; that is, upon the cone whose trace is A. I in fig. 103. (If it be desired to draw the development of that trace, lay off the oblique half-thicknesses along arcs drawn about A, through the points of division of the radius A FIE. The result is the drawing of an outline outside of, and nearly parallel to, HGEGH. To prevent confusion, it is not shown in the figure.)

If the pitch-circle chosen is at the throat of the hyperboloïd, the normal cone becomes simply the plane of that circle; and in fig. 104, A fi e coincides with A FIE.

III. Projections of the Middle Lines of a Tooth.-In fig. 103, let FIE and fie, as before, represent the projections of the central depth of a tooth, being part of a normal (E IF A, eifa) to the pitch-surface at a point, I i, in the middle pitch-circle, whose radius is CI=ai; so that F, f, I, i, and E, e are the projections of the middle points of the tooth at the root, at the pitch-surface, and at the crest respectively; and let it be required to find the projections of the middle lines of that tooth at the root, pitch-surface, and crest respectively.

About a draw the circles ƒ ƒ', i ï', and e e; being the projections,