Fig. 78.
Evidently, if this is to be so, the point d must be on a semicircle, whose diameter is f b, for in that case the angle f d b will always be a right angle.
Fig. 79.
The surface must then be so arranged that, whatever be the position of the cam and of the rod b d, the point of contact between them must always be on the semicircle f c d; that is to say, as the cam moves round the axis a its shape must be such that a line drawn from f to the point where it cuts the circle f d b is always perpendicular to the curve.
Now suppose that we move a circle whose centre is at a, and radius a f, so as to roll the circle f d b by simple surface friction round its centre o, then any point d on it would mark out a curve on a piece of paper attached to the moving circle whose centre is at a, and the direction of motion of the curve would always be such that the point d on it would at any instant be describing a circle round f, and the direction of the curve would thus at any point always be at right angles to the line d f for the time being.
Fig. 80.
This curve, caused by the rolling of one circle on another, is called an epicycloid. Hence, then, for a clock, if we make the pinion wheel with straight spokes and the driving wheel with its teeth cut in the form of epicycloids, caused by rolling a circle with a diameter equal to the radius of the pinion upon the driving wheel, we shall get a uniform ratio of leverage one upon the other.
The circles with radii a f, b f, are called the “pitch circles,” and these radii are in the ratio of the movement that is required for the wheels, usually six to one or eight to one, as the case may be. The sides of the teeth of the pinion wheels are straight lines radiating from the centre, and rounded off at the ends so as to avoid accidental jambing. The teeth of the cogwheel have epicycloidal sides. The tips are cut off so as to be out of the way, and spaces are left between them for the width of the leaves of the pinion wheel.