Page:The Kinematics of Machinery.djvu/184

 162 of Euler, is shortly as follows. Let A and B (Fig. 117) be the centres of rotation of two bodies which can drive each other by means of the circular profiles touching in R, and drawn from the centres P and Q; then the intersection of the two lines of centres P Q and A B is the point of contact of the centroids corresponding to the relative motion of the figures A and B (see § 8), which therefore have for their angular velocity ratio O B: O A. In order that this may remain nearly constant for a short interval of time,  117. P Q must in its motion continue to pass as nearly as possible through O. The instantaneous centre, however, of P Q relatively to A B is the point C at the intersection of P A and B Q produced, and if C be a point upon a perpendicular to P Q at O, as C′ in our figure, then the instantaneous motion of P Q will be in fact through the point O. If therefore one of the centres, as P, be chosen, the position of the other must be the intersection Q′ of P Q and the line B C′. We thus obtain in the distance P Q, or rather P Q′, the sum of the required radii of curvature, but may take the point of contact R in any position, as R′ for example, as follows from what we have said in § 35.

In order to adapt this elegant method to set-wheels, Willis chose three constant magnitudes, the distances O C′and O R and the angle P O A; the latter he made 75°. If the teeth were to be profiled by one arc only he took O C′ = ∞, O R = 0, the circular arcs becoming approximations to involutes (see § 33). If two circular arcs were to be used, joining at the pitch line into an S-shaped figure,—as is usual,—the method was applied twice over, once for the portion of the tooth on each side of the pitch circle. Fig. 118 shows this;—O R′ and O R″ are each equal to half the pitch, and O C′ and O C″ are so taken that for a pinion