The result obtained in this process is precisely the same as that by the construction in [Fig. 93], as will be plainly seen, because there are marked on [Fig. 93] all the circles by which point f was arrived at in [Fig. 95]; and line 3, which in [Fig. 95] gives the centre wherefrom to strike curve o, is coincident with point f, as is shown in [Fig. 95]. By marking the graduated edge of c in quarter-inch divisions, as 1, 2, 3, &c., then every division will represent the distance from a for the centre for every inch of wheel radius. Suppose, for example, that a wheel has 3 inches radius, then with the scale c set to the radial line r, the centre therefrom to strike the curve o will be at 3; were the radius of the wheel 4 inches, then the scale being set the same as before (one edge coincident with r), the centre for the curve o would be at 4, and arc t would require to meet the edge of c at 4. Having found the radius from the centre of the wheel of point f for one tooth, we may mark circle t, cutting point f, and mark off all the teeth by setting one point of the compasses (set to radius a f) on one side of the tooth and marking on circle t the centre wherefrom to mark the curve (as o), continuing the process all around the wheel and on both sides of the tooth.
This operation of finding the location for the centre wherefrom to strike the tooth curves, must be performed separately for each wheel, because the distance or radius of the tooth curves varies with the radius of each wheel.
Fig. 96.
In [Fig. 96] this template is shown with all the lines necessary to set it, those shown in [Fig. 95] to show the identity of its results with those given in [Fig. 93] being omitted.
Fig. 97.
The principles involved in the construction of a rack to work correctly with a wheel or pinion, having involute teeth, are as in [Fig. 97], in which the pitch circle is shown by a dotted circle and the base circle by a full line circle. Now the diameter of the base circle has been shown to be arbitrary, but being assumed the radius b q will be determined (since it extends from the centre b to the point of contact of d q, with the base circle); b d is a straight line from the centre b of the pinion to the pitch line of the rack, and (whatever the angle of q d to b d) the sides of the rack teeth must be straight lines inclined to the pitch line of the rack at an angle equal to that of b d q.