But if the wheel w were the driver and v the driven, then these conditions would be exactly reversed. Thus, suppose this to be the case and the direction of motion be as denoted by arrow p, the contact would occur during the arc of approach, from h to x, ceasing at x.
Or if w were the driver, and the direction of motion was as denoted by q, then, again, the path of contact would be during the arc of approach only, beginning at b and ceasing at x, as denoted by the thickened arc b x.
Fig. 35.
The action of the teeth will in either case serve to give a theoretically perfect motion so far as uniformity of velocity is concerned, or, in other words, the motion of the driver will be transmitted with perfect uniformity to the driven wheel. It will be observed, however, that by the removal of the faces of the teeth, there are a less number of teeth in contact at each instant of time; thus, in [Fig. 33] there is driving contact at three points, c, f, and j, while in [Fig. 34] there is driving contact at two points only. From the fact that the faces of the teeth work with the flanks only, and that one side only of the teeth comes into action, it becomes apparent that each tooth may have curves formed by four different diameters of rolling or generating circles and yet work correctly, no matter which wheel be the driver, or which the driven wheel or follower, or in which direction motion occurs. Thus in [Fig. 35], suppose wheel v to be the driver, having motion in the direction of arrow p, then faces a on the teeth of v will work with flanks b of the teeth on w, and so long as the curves for these faces and flanks are obtained with the same diameter of rolling circle, the action of the teeth will be correct, no matter what the shapes of the other parts of the teeth. Now suppose that v still being the driver, motion occurs in the other direction as denoted by q, then the faces c of the teeth on v will drive the flanks c of the teeth on w, and the motion will again be correct, providing that the same diameter (whatever it may be) of rolling circle be used for these faces and flanks, irrespective, of course, of what diameter of rolling circle is used for any other of the teeth curves. Now suppose that w is the driver, motion occurring in the direction of p, then faces e will drive flanks f, and the motion will be correct as before if the curves e and f are produced with the same diameter of rolling circle. Finally, let w be the driving wheel and motion occur in the direction of q, and faces g will drive flanks h, and yet another diameter of rolling circle may be used for these faces and flanks. Here then it is shown that four different diameters of rolling circles may be used upon a pair of wheels, giving teeth-forms that will fill all the requirements so far as correctly transmitting motion is concerned. In the case of a pair of wheels having an equal number of teeth, so that each tooth on one wheel will always fall into gear with the same tooth on the other wheel, every tooth may have its individual curves differing from all the others, providing that the corresponding teeth on the other wheel are formed to match them by using the same size of rolling circle for each flank and face that work together.
It is obvious, however, that such teeth would involve a great deal of labor in their formation and would possess no advantage, hence they are not employed. It is not unusual, however, in a pair of wheels that are to gear together and that are not intended to interchange with other wheels, to use such sizes as will give to for the face of the teeth on the largest wheel of the pair and for the flanks of the teeth of the smallest wheel, a generating circle equal in diameter to the radius of the smallest wheel, and for the faces of the teeth of the small wheel and the flanks of the teeth of the large one, a generating circle whose diameter equals the radius of the large wheel.
Fig. 36.