α1 : α2 :: C2P2 : C1P1 :: IC2 :: IC1;
(24)
which is also the angular velocity ratio due to the rolling contact of the circles B1IB1′, B2IB2′.
This principle determines the forms of all teeth of spur-wheels. It also determines the forms of the teeth of straight racks, if one of the centres be removed, and a straight line EIE′, parallel to the direction of motion of the rack, and perpendicular to C1IC2, be substituted for a pitch-circle.
II. The component of the velocity of the point of contact of the teeth T along the line of connexion is
α1 · C1P1 = α2 · C2P2.
(25)
III. The relative velocity perpendicular to P1P2 of the teeth at their point of contact—that is, their velocity of sliding on each other—is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C1C2 to rotate backwards round C1 with the angular velocity α1, and the wheel 2 to rotate round C2 as before, with the angular velocity α2 relatively to the line of centres C1C2, so as to have the same motion as if its pitch-circle rolled on the pitch-circle of the first wheel. Thus the relative motion of the wheels is unchanged; but 1 is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity α1 + α2. Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is
(α1 + α2) · IT;
(26)