The names given to the various lines of a tooth on a gear-wheel are as follows:

In Figure 233, A is the face and B the flank of a tooth, while C is the point, and D the root of the tooth; E is the height or depth, and F the breadth. P P is the pitch circle, and the space between the two teeth, as H, is termed a space.

Fig. 234.

Fig. 235.

It is obvious that the points of the teeth and the bottoms of the spaces, as well as the pitch circle, are concentric to the axis of the wheel bore. And to pencil in the teeth these circles must be fully drawn, as in Figure 234, in which P P is the pitch circle. This circle is divided into as many equal divisions as the wheel is to have teeth, these divisions being denoted by the radial lines, A, B, C, etc. Where these divisions intersect the pitch circle are the centres from which all the teeth curves may be drawn. The compasses are set to a radius equal to the pitch, less one-half the thickness of the tooth, and from a centre, as R, two face curves, as F G, may be marked; from the next centre, as at S, the curves D E may be marked, and so on for all the faces; that is, the tooth curves lying between the outer circle X and the pitch circle P. For the flank curves, that is, the curve from P to Y, the compasses are set to a radius equal to the pitch; and from the sides of the teeth the flank curves are drawn. Thus from J, as a centre flank, K is drawn; from V, as a centre flank, H is drawn, and so on.

The proportions of the teeth for cast gears generally accepted in this country are those given by Professor Willis, as average practice, and are as follows:

Instead, however, of calculating the dimensions these proportions give for any particular pitch, a diagram or scale may be made from which they may be taken for any pitch by a direct application of the compasses. A scale of this kind is given in Figure 235, in which the line A B is divided into inches and parts to represent the pitches; its total length representing the coarsest pitch within the capacity of the scale; and, the line B C (at a right-angle to A B) the whole depth of the tooth for the coarsest pitch, being 7/10 of the length of A B.

Fig. 236.

The other diagonal lines are for the proportion of the dimensions marked on the figure. Thus the depth of face, or distance from the pitch line to the extremity or tooth point for a 4 inch pitch, would be measured along the line B C, from the vertical line B to the first diagonal. The thickness of the tooth would be for a 4 inch pitch along line B C from B to the second diagonal, and so on. For a 3 inch pitch the measurement would be taken along the horizontal line, starting from the 3 on the line A B, and so on. On the left of the diagram or scale is marked the lbs. strain each pitch will safely transmit per inch width of wheel face, according to Professor Marks.

Fig. 237.

The application of the scale as follows: The pitch circles P P and P' P', Figure 236, for the respective wheels, are drawn, and the height of the teeth is obtained from the scale and marked beyond the pitch circles, when circles Q and Q' may be drawn. Similarly, the depths of the teeth within the pitch circles are obtained from the scale or diagram and marked within the respective pitch circles, and circles R and R' are marked in. The pitch circles are divided off into as many points of equal division, as at a, b, c, d, e, etc., as the respective wheels are to have teeth, and the thickness of tooth having been obtained from the scale, this thickness is marked from the points of division on the pitch circles, as at f in the figure, and the tooth curves may then be drawn in. It may be observed, however, that the tooth thicknesses will not be strictly correct, because the scale gives the same chord pitch for the teeth on both wheels which will give different arc pitches to the teeth on the two wheels; whereas, it is the arc pitches, and not the chord pitches, that should be correct. This error obviously increases as there is a greater amount of difference between the two wheels.

The curves given to the teeth in Figure 234 are not the proper ones to transmit uniform motion, but are curves merely used by draughtsmen to save the trouble of finding the true curves, which if it be required, may be drawn with a very near approach to accuracy, as follows, which is a construction given by Rankine:

Draw the rolling circle D, Figure 237, and draw A D, the line of centres. From the point of contact at C, mark on D, a point distant from C one-half the amount of the pitch, as at P, and draw the line P C of indefinite length beyond C. Draw the line P E passing through the line of centres at E, which is equidistant between C and A. Then increase the length of line P F to the right of C by an amount equal to the radius A C, and then diminish it to an amount equal to the radius E D, thus obtaining the point F and the latter will be the location of centre for compasses to strike the face curve.

Fig. 238.

Another method of finding the face curve, with compasses, is as follows: In Figure 238 let P P represent the pitch circle of the wheel to be marked, and B C the path of the centre of the generating or describing circle as it rolls outside of P P. Let the point B represent the centre of the generating circle when it is in contact with the pitch circle at A. Then from B mark off, on B C, any number of equidistant points, as D, E, F, G, H, and from A mark on the pitch circle, with the same radius, an equal number of points of division, as 1, 2, 3, 4, 5. With the compasses set to the radius of the generating circle, that is, A B, from B, as a centre, mark the arc I, from D, the arc J, from E, the arc K, from F, and so on, marking as many arcs as there are points of division on B C. With the compasses set to the radius of divisions 1, 2, etc., step off on arc M the five divisions, N, O, S, T, V, and at V will be a point on the epicycloidal curve. From point of division 4, step off on L four points of division, as a, b, c, d; and d will be another point on the epicycloidal curve. From point 3, set off three divisions, and so on, and through the points so obtained draw by hand, or with a scroll, the curve.