so that it is greater the farther the point of contact is from the line of centres; and at the instant when that point passes the line of centres, and coincides with the pitch-point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact.
IV. The path of contact is the line traversing the various positions of the point T. If the line of connexion preserves always the same position, the path of contact coincides with it, and is straight; in other cases the path of contact is curved.
It is divided by the pitch-point I into two parts—the arc or line of approach described by T in approaching the line of centres, and the arc or line of recess described by T after having passed the line of centres.
During the approach, the flank D1B1 of the driving tooth drives the face D2B2 of the following tooth, and the teeth are sliding towards each other. During the recess (in which the position of the teeth is exemplified in the figure by curves marked with accented letters), the face B1′A1′ of the driving tooth drives the flank B2′A2′ of the following tooth, and the teeth are sliding from each other.
The path of contact is bounded where the approach commences by the addendum-circle of the follower, and where the recess terminates by the addendum-circle of the driver. The length of the path of contact should be such that there shall always be at least one pair of teeth in contact; and it is better still to make it so long that there shall always be at least two pairs of teeth in contact.
V. The obliquity of the action of the teeth is the angle EIT = IC1, P1 = IC2P2.
In practice it is found desirable that the mean value of the obliquity of action during the contact of teeth should not exceed 15°, nor the maximum value 30°.
It is unnecessary to give separate figures and demonstrations for inside gearing. The only modification required in the formulae is, that in equation (26) the difference of the angular velocities should be substituted for their sum.
§ 46. Involute Teeth.—The simplest form of tooth which fulfils the conditions of § 45 is obtained in the following manner (see fig. 102). Let C1, C2 be the centres of two wheels, B1IB1′, B2IB2′ their pitch-circles, I the pitch-point; let the obliquity of action of the teeth be constant, so that the same straight line P1IP2 shall represent at once the constant line of connexion of teeth and the path of contact. Draw C1P1, C2P2 perpendicular to P1IP2, and with those lines as radii describe about the centres of the wheels the circles D1D1′, D2D2′, called base-circles. It is evident that the radii of the base-circles bear to each other the same proportions as the radii of the pitch-circles, and also that
C1P1 = IC1 · cos obliquity
C2P2 = IC2 · cos obliquity.