Modern Machine-Shop Practice, Volumes I and II - Joshua Rose

The method of using the table is as follows:—Suppose it is required to make a set of wheels, the smallest of which is to contain 50 teeth and the largest 150, and it is determined to use but one cutter, then that cutter should be made correct for a wheel containing 76; because in the table 76 is midway between 50 and 150.

But suppose it were determined to employ two cutters, then one of them should be made correct for a wheel having 60 teeth, and used on all the wheels having between 50 and 76 teeth, while the other should be made correct for a wheel containing 100 teeth, and used on all wheels containing between 76 and 150 teeth.

In the following table, also arranged by Professor Willis, the most desirable selection of cutters for different circumstances is given, it being supposed that the set of wheels contains from 12 teeth to a rack.

Number of cutters in the set.	Number of Teeth in Wheel for which the Cutter is to be made correct.
2	50	16
3	75	25	15
4	100	34	20	14
6	150	50	30	21	16	13
8	200	67	40	29	22	18	15	13
10	200	77	50	35	27	22	19	16	14	13
12	300	100	60	43	34	27	23	20	17	15	14	13
18	300	150	100	70	50	40	30	26	24	22	20	18	16	15	14	13	12
24	Rack	300	150	100	76	60	50	43	38	34	30	27	25	23	21	20	19	18	17	16	15	14	13	12

Suppose now we take the cutters, of a given pitch, necessary to cut all the wheels from 12 teeth to a rack, then the thickness of the teeth at the pitch line will for the purposes of designation be the thickness of the teeth of all the wheels, which thickness may be a certain proportion of the pitch.

But in involute teeth while the depth of tooth on the cutter may be taken as the standard for all the wheels in the range, and the actual depth for the wheel for which the cutter is correct, yet the depth of the teeth in the other wheels in the range may be varied sufficiently on each wheel to make the thickness of the teeth equal the width of the spaces (notwithstanding the variation between the arc and chord pitches), so that by a variation in the tooth depth the error induced by that variation may be corrected. The following table gives the proportions in the Brown and Sharpe system.

Arc Pitch.	Depth of Tooth.	Depth in terms of the arc pitch.
inches.	inches.	inches.
1.570	1.078	.686
1.394	.958	.687
1.256	.863	.686
1.140	.784	.697
1.046	.719	.687
.896	.616	.686
.786	.539	.685
.628	.431	.686
.524	.359	.685
.448	.307	.685
.392	.270	.686
.350	.240	.686
.314	.216	.687

To avoid the trouble of measuring, and to assist in obtaining accuracy of depth, a gauge is employed to mark on the wheel face a line denoting the depth to which the cutter should be entered.

Suppose now that it be required to make a set of cutters for a certain range of wheels, and it be determined that the cutters be so constructed that the greatest permissible amount of error in any wheel of the set be ¹⁄₁₀₀ inch. Then the curves for the smallest wheel, and those for the largest in the set, and the amount of difference between them ascertained, and assuming this difference to amount to ¹⁄₁₆ inch, which is about ⁶⁄₁₀₀, then it is evident that 6 cutters must be employed for the set.

It has been shown that on bevel-wheels the tooth curves vary at every point in the tooth breadth; hence it is obvious that the cutter being of a fixed curve will make the tooth to that curve. Again, the thickness of the teeth and breadth of the spaces vary at every point in the breadth, while with a cutter of fixed thickness the space cut will be parallel from end to end. To overcome these difficulties it is usual to give to the cutter a curve corresponding to the curve required at the middle of the wheel face and a thickness equal to the required width of space at its smallest end, which is at the smallest face diameter.