For these reasons the cutters are limited to fewer wheels according as the number of teeth decreases, or, per contra, are allowed to be used over a greater range of wheels as the number of teeth in the wheels is increased.
Thus in the Brown and Sharpe system for involute teeth there are 8 cutters numbered numerically (for convenience in ordering) from 1 to 8, and in the following table the range of the respective cutters is shown, and the number of teeth for which the cutter is theoretically correct is also given.
BROWN AND SHARPE SYSTEM.
| No. of cutter. | Involute teeth. | Teeth. | |||||||||
| 1 | Used | upon all | wheels | having | from | 135 | teeth | to a rack correct for | 200 | ||
| 2 | „ | „ | „ | „ | „ | 55 | „ | to | 134 | teeth, | 68 |
| 3 | „ | „ | „ | „ | „ | 35 | „ | to | 54 | „ | 40 |
| 4 | „ | „ | „ | „ | „ | 26 | „ | to | 34 | „ | 29 |
| 5 | „ | „ | „ | „ | „ | 21 | „ | to | 25 | „ | 22 |
| 6 | „ | „ | „ | „ | „ | 17 | „ | to | 20 | „ | 18 |
| 7 | „ | „ | „ | „ | „ | 14 | „ | to | 16 | „ | 16 |
| 8 | „ | „ | „ | „ | „ | 12 | „ | to | 14 | „ | 13 |
Suppose that it was required that of a pair of wheels one make twice the revolutions of the other; then, knowing the particular number of teeth for which the cutters are made correct, we may obtain the nearest theoretically true results as follows: If we select cutters Nos. 8 and 4 and cut wheels having respectively 13 and 26 teeth, the 13 wheel will be theoretically correct, and the 26 will contain the minute error due to the fact that the cutter is used upon a wheel having three less teeth than the number it is theoretically correct for. But we may select the cutters that are correct for 16 and 29 teeth respectively, the 16th tooth being theoretically correct, and the 29th cutter (or cutter No. 4 in the table) being used to cut 32 teeth, this wheel will contain the error due to cutting 3 more teeth than the cutter was made correct for. This will be nearer correct, because the error is in a larger wheel, and, therefore, less in actual amount. The pitch of teeth may be selected so that with the given number of teeth the diameters of the wheels will be that required.
We may now examine the effect of the variation of curvature in combination with that of the thickness, upon a wheel having less and upon one having more teeth than the number in the wheel for which the cutter is correct.
First, then, suppose a cutter to be used upon a wheel having less teeth and it will cut the spaces too wide, because of the variation of thickness, and the curves too straight or insufficiently curved because of the error of curvature. Upon a wheel having more teeth it will cut the spaces too narrow, and the curvature of the teeth too great; but, as before stated, the number of wheels assigned to each cutter may be so apportioned that the error will be confined to practically unappreciable limits.
If, however, the teeth are epicycloidal, it is apparent that the spaces of one wheel must be wide enough to admit the teeth of the other to a depth sufficient to permit the pitch lines to coincide on the line of centres; hence it is necessary in small diameters, in which there is a sensible difference between the arc and the chord pitches, to confine the use of a cutter to the special wheel for which it is designed, that is, having the same number of teeth as the cutter is designed for.
Thus the Pratt and Whitney arrangement of cutters for epicycloidal teeth is as follows:—