If we multiply the number of teeth by their pitch to obtain the circumference of the pitch circle we shall obtain the circumference due to the radius of _g_ _h_, from the wheel axis, and so long as _g_ _h_ is parallel to the wheel axis we shall by this means obtain the same diameter of pitch circle, so long as we measure it on a line parallel to the line of centres _c_ _d_. The pitch of the worm is the same at whatever point in the tooth depth it may be measured, because the teeth curves are parallel one to the other, thus in Fig. 77 the pitch measures are equal at _m_, _n_, or _o_.
But the action of the worm and wheel will nevertheless not be correct unless the pitch line from which the curves were rolled coincides with the pitch line of the wheel on the line of centres, for although, if the pitch lines do not so coincide, the worm will at each revolution move the pitch line of the wheel through a distance equal to the pitch of the worm, yet the motion of the wheel will not be uniform because, supposing the two pitch lines not to meet, the faces of the pinion teeth will act against those of the wheel, as shown in Fig. 78, instead of against their flanks, and as the faces are not formed to work correctly together the motion will be irregular.
The diameter of the worm is usually made equal to four times the pitch of the teeth, and if the teeth are curved as in figure 76 they are made to envelop not more than 30° of the worm.
The number of teeth in the wheel should not be less than thirty, a double worm being employed when a quicker ratio of wheel to worm motion is required.
When the teeth of the wheel are curved to partly envelop the worm circumference it has been found, from experiments made by Robert Briggs, that the worm and the wheel will be more durable, and will work with greatly diminished friction, if the pitch line of the worm be located to increase the length of face and diminish that of the flank, which will decrease the length of face and increase the length of flank on the wheel, as is shown in Fig. 79; the location for the pitch line of the worm being determined as follows:--
The full radius of the worm is made equal to twice the pitch of its teeth, and the total depth of its teeth is made equal to .65 of its pitch. The pitch line is then drawn at a radius of 1.606 of the pitch from the worm axis. The pitch line is thus determined in Fig. 76, with the result that the area of tooth face and of worm surface is equalized on the two sides of the pitch line in the figure. In addition to this, however, it may be observed that by thus locating the pitch line the arcs both of approach and of recess are altered. Thus in Fig. 80 is represented the same worm and wheel as in Fig. 79, but the pitch lines are here laid down as in ordinary gearing. In the two figures the arcs of approach are marked by the thickened part of the generating circle, while the arcs of recess are denoted by the dotted arc on the generating circle, and it is shown that increasing the worm face, as in Fig. 79, increases the arc of recess, while diminishing the worm flank diminishes the arc of approach, and the action of the worm is smoother because the worm exerts more pulling than pushing action, it being noted that the action of the worm on the wheel is a pushing one before reaching, and a pulling one after passing, the line of centres.