Fig. 25.
To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown in [Fig. 25] at d, e, &c., or the faces of notches cut in the wheel as at f, g, h, &c., the metal between f and g forming a tooth j, having flanks only. The wheel b has the curves of each tooth brought closer together to give room for the reception of the teeth upon a. We have here a pair of gears that possess sufficient strength and are capable of working correctly in either direction.
But the form of tooth on one wheel is conformed simply to suit those on the other, hence, neither two of the wheels a, nor would two of b, work correctly together.
Fig. 26.
They may be qualified to do so, however, by simply adding to the tops of the teeth on a, teeth of the form of those on b, and adding to those on b, and within the pitch circle, teeth corresponding to those on a, as in [Fig. 26], where at k′ and j′ teeth are provided on b corresponding to j and k on a, while on a there are added teeth o′, n′, corresponding to o, n, on b, with the result that two wheels such as a or two such as b would work correctly together, either being the driver or either the follower, and rotation may occur in either direction. In this operation we have simply added faces to the teeth on a, and flanks to those on b, the curves being generated or obtained by rolling the generating, or curve marking, circle c upon the pitch circles p and p′. Thus, for the flanks of the teeth of a, c is rolled upon, and within the pitch circle p of a; while for the face curves of the same teeth c is rolled upon, but without or outside of p. Similarly for the teeth of wheel b the generating circle c is rolled within p′ for the flanks and without for the faces. With the curves rolled or produced with the same diameter of generating circle the wheels will work correctly together, no matter what their relative diameter may be, as will be shown hereafter.
In this demonstration, however, the curves for the faces of the teeth being produced by an operation distinct from that employed to produce the flank curves, it is not clearly seen that the curves for the flanks of one wheel are the proper curves to insure a uniform velocity to the other. This, however, may be made clear as follows:—