The cause of this progressive deterioration may be thus explained: Referring again to [fig. 1], we there see the base of the curve A B divided into the equal parts a b, b c, and c d; and observing the passage of the generating circle D, from the origin of the curve at d, to the first division c on the base, we shall find no more than the small portion d e, of the curve developed, whereas a second equal step of the generating circle c b, will extend the curve forward from e to f, a greater distance than the former; while a third equal step a b, will extend the curve from f to g, a distance greater than the last; and the successive increments of the curve will be still greater, as it approaches its summit; yet all these parts correspond to equal advances of the wheel, namely, to the equal parts a b, b c and c d of the base, and to equal ones of rotation of the generating circle. Surely then the parts s g, g f, of the epicycloidal tooth will be worn out sooner than those f e, e d, which are rubbed with so much less velocity than the other, even though the pressure were the same. But the pressure is not the same. For, the line a g is the direction in which the pressure of the curve acts at the point g, and the line p q, is the length of the lever-arm on which that pressure acts, to turn the generating circle on its axis (now supposed to be fixt;) but, as the turning force or rotatory effort of the wheels, is by hypothesis uniform, the pressure at g must be inversely as p q; that is, inversely as the cosine of half the angle of rotation of the generating circle; hence it would be infinite at s, the summit of the curve, when this circle has made a semi-revolution.
Thus it appears that independently of the effects of percussion, the end of an epicycloidal tooth must wear out sooner than any part nearer its base, (and if so, much more it may be supposed of a tooth of another form;) and that when its form is thus changed, the advantage it gave must cease, since nothing in the working of the wheel can afterwards restore the form, or remedy the growing evil.
Having now shewn one great defect in the common system of wheels, I shall proceed to develope the principles of the new system, which may be understood through the medium of the three following propositions.
1. The action of a wheel of the new kind on another with which it works or geers is the same at every moment of its revolution, so that the least possible motion of the circumference of one, generates an exactly equal and similar motion in that of the other.
2. There are but two points, one in each wheel, that necessarily touch each other at the same time, and their contact will always take place indefinitely near the plane that passes through the two axes of the wheels, if the diameters of the latter, at the useful or pressing points are in the exact ratio of their number of teeth respectively; in which case there will be no sensible friction between the points in contact.
3. In consequence of the properties above-mentioned, the epicycloidal or any other form of the teeth, is no longer indispensable; but many different forms may be used, without disturbing the principle of equable motion.
With regard to the demonstration of the first proposition, I must premise an observation of M. Camus on this subject, in his Mechanics, 3d. part, page 306, viz. “if all wheels could have teeth infinitely fine, their geering, which might then be considered as a simple contact, would have the property required, [that of acting uniformly] since we have seen that a wheel and a pinion have the same tangential force, when the motion of one is communicated to the other, by an infinitely small penetration of the particles of their respective circumferences.”
