In [Fig. 71] are represented a wheel and pinion, the pinion having but four teeth less than the wheel, and a tooth, j, being shown in position in which it has contact at two places. Thus at k it is in contact with the flank of a tooth on the annular wheel, while at l it is in contact with the face of the same tooth.
As the faces of the teeth on the wheel do not have contact higher than point t, it is obvious that instead of having them 3⁄10 of the pitch as at the bottom of the figure, we may cut off the portion x without diminishing the arc of contact, leaving them formed as at the top of the figure. These faces being thus reduced in height we may correspondingly reduce the depth of flank on the pinion by filling in the portion g, leaving the teeth formed as at the top of the pinion. The teeth faces of the wheel being thus reduced we may, by using a sufficiently large intermediate circle, obtain interior and exterior describing circles that will form teeth that will permit of the pinion having but one tooth less than the wheel, or that will form a wheel having but one tooth more than the pinion.
Fig. 72.
The limits to the diameter of the intermediate describing circle are as follows: in [Fig. 72] it is made equal in diameter to the pitch diameter of the pinion, hence b will represent the centre of the intermediate circle as well as of the pinion, and the pitch circle of the pinion will also represent the intermediate circle r. To obtain the radius for the interior describing circle we subtract the radius of the intermediate circle from the radius of the annular wheel, which gives a p, hence the pitch circle of the pinion also represents the interior circle r. But when we come to obtain the radius for the exterior describing circle (t), by subtracting the radius of the pinion from that of the intermediate circle, we find that the two being equal give o for the radius of (t), hence there could be no flanks on the pinion.
Now suppose that the intermediate circle be made equal in diameter to the pitch circle of the annular wheel, and we may obtain the radius for the exterior describing circle t; by subtracting the radius of the pinion from that of the intermediate circle, we shall obtain the radius a b; hence the radius of (t) will equal that of the pinion. But when we come to obtain the radius for the interior describing circle by subtracting the radius of the intermediate circle from that of the annular wheel, we find these two to be equal, hence there would be no interior describing circle, and, therefore, no faces to the pinion.
Fig. 73.