By the introduction into the figure of a segment of a spur-wheel also having 22 teeth and placed on the other side of the pinion, it is shown that the path of contact is greater, and therefore the angle of action is greater, in internal than in spur gearing. Thus suppose the pinion to drive in the direction of the arrows and the thickened arcs a b will be the arcs of approach, a measuring longer than b. The dotted arcs c d represent the arcs of receding contact and c is found longer than d, the angles of action being 66° for the spur-wheels and 72° for the annular wheel.
On referring again to [Fig. 62] it will be observed that it is the faces of the teeth on the two wheels that interfere and will prevent them from engaging, hence it will readily occur to the mind that it is possible to form the curves of the pinion faces correct to work with the faces of the wheel teeth as well as with the flanks; or it is possible to form the wheel faces with curves that will work correctly with the faces, as well as with the flanks of the pinion teeth, which will therefore increase the angle of action, and Professor McCord has shown in an article in the London Engineering how to accomplish this in a simple and yet exceedingly ingenious manner which may be described as follows:—
It is required to find a describing circle that will roll the curves for the flanks of the pinion and the faces of the wheels, and also a describing circle for the flanks of the wheel and the faces of the pinion; the curve for the wheel faces to work correctly with the faces as well as with the flanks of the pinion, and the curve for the pinion faces to work correctly with both the flanks and faces of the internal wheel.
Fig. 65.
Fig. 66.
In [Fig. 65] let p represent the pitch circle of an annular or internal wheel whose centre is at a, and q the pitch circle of a pinion whose centre is at b, and let r be a describing circle whose centre is at c, and which is to be used to roll all the curves for the teeth. For the flanks of the annular wheel we may roll r within p, while for the faces of the wheel we may roll r outside of p, but in the case of the pinion we cannot roll r within q, because r is larger than q, hence we must find some other rolling circle of less diameter than r, and that can be used in its stead (the radius of r always being greater than the radius of the axis of the wheel and pinion for reasons that will appear presently). Suppose then that in [Fig. 66] we have a ring whose bore r corresponds in diameter to the intermediate describing circle r, [Fig. 65] and that q represents the pinion. Then we may roll r around and in contact with the pinion q, and a tracing point in r will trace the curve m n o, giving a curve a portion of which may be used for the faces of the pinion. But suppose that instead of rolling the intermediate describing circle r around p, we roll the circle t around p, and it will trace precisely the same curve m n o; hence for the faces of the pinion we have found a rolling circle t which is a perfect substitute for the intermediate circle q, and which it will always be, no matter what the diameters of the pinion and of the intermediate describing circle may be, providing that the diameter of t is equal to the difference between the diameters of the pinion and that of the intermediate describing circle as in the figure. If now we use this describing circle to roll the flanks of the annular wheel as well as the faces of the pinion, these faces and flanks will obviously work correctly together. Since this describing circle is rolled on the outside of the pinion and on the outside of the annular wheel we may distinguish it as the exterior describing circle.