Instantaneous Axis of a Cylinder rolling on a Cylinder.Let a cylinder **bbb**, whose axis of figure is B and angular velocity -y, roll on a fixed cylinder acm, whose axis of figure is A, either outside (as infig.

The line T on the surface **bbb** has for the instant no velocity it a direction perpendicular to AB; becau2e for the instant it touches, without sliding, the line T on the fixed surface aaa.

The line T on the surface **bbb** has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

The line of contact T, therefore, on the surface of the cylinder **bbb**, is for the instant at rest, and is the instantaneous axis FIG.

About which the cylinder **bbb** turns, together with any body rigidly attached to that cylinder.

To find, then, the direction and velocity at the given instant of any point P, either in or rigidly attached to the rolling cylinder T, draw the plane PT; the direction of motion of P will be perpendicular to that plane, and towards the right or left hand according to the direction of the rotation of **bbb**; and the velocity of P will be v~=y.PT, (3)

If P is in the circumference of **bbb**, that path becomes an epic ycloid.

The whole of the foregoing reasonings are applicable, not merely when acm and **bbb** are actual cylinders, but also when they are the osculating cylinders of a pair of cylindroidal surfaces of varying curvature, A and B being the axes of curvature of the parts of those surfaces which are in contact for the instant under consideration.