X and Y; the perpendicular A E, from the touching surfaces in all situations, cuts the line of centres at the termination A of their proportional radii. But we saw p. 19, that when this was the case, the proportional circles must have equal velocities.

It is principally from this, that we shall deduce the best figure which can be given to the teeth of wheels and pinions, when one part of the wheel and pinion, or of both, ought to be a straight line tending to the centre of such wheel or pinion.

VI.

9. If in the same plane we have but two circles, R, Y, which touch in the point A, and if the movement of the one, communicate itself to the other, by this point of contact, any point E of the circumference of the circle Y, describes upon the plane of the moveable circleR, an epicycloid C.E.

A

B

F

Suppose this epicycloid attached to the circle R, it (the epicycloid) shall conduct the circle Y, pushing it round by the point E of its circumference, in the same manner as the circle R might conduct the same circle Y in communicating motion to it by the point of contact A.

And in like manner, the point E of the circumference of the circle Y, turns the

circle R, in pushing it by the epicycloid C E, supposed to be attached to R, in the same way that the circle Y would conduct the circle R in communicating its motion by the point of contact A *.

The same mode of proof applies here that did to the corollary immediately preceding.

This last corollary enables us to determine the best figure which can be given to the

*The experiment to prove this is similar to the former, but with this difference, that in the circumference of one of them, A, is fixed a fine needle, which is made to act against a piece of wood, formed into an epicycloid, fixed upon the other, B, which epicycloid is generated by A upon B as a base.

teeth of wheels, when the pinion shall be a trundle composed of staves.

We shall likewise determine from it the most advantageous figure which can be given to the teeth of a pinion, when the wheel shall have staves in place of teeth.

36

CHAP. II.

• OF THE APPLICATION OF THE PRINCI

PLES OF THE CONFIGURATION OF THE

TEETH OF WHEELS.

1. Having endeavoured to show, that an epicycloid is a curve, whereby two circles may conduct themselves as if put forward by the simple contact of their circumferences, I shall now attempt a practical explanation of this curve, in giving the best form to the teeth of wheels.

SECTION I.

Of Spur Geers.

2. By Spur Geers is understood wheels acting together, with their axes parallel and in the same plane; under this head