LECTURE XV.

176

circle rolling on the wheel, of which the diameter must be half that of theopposite wheel; and in this case it is demonstrable that the plane surface ofeach tooth will act on the curved surface of the opposite tooth so as to pro-duce an equable angular motion in both wheels: the other method is, toform all the surfaces into portions of the involutes of circles, or the curvesdescribed by a point of a thread which has been wound round the wheel,while it is uncoiled; and this method appears to answer the purpose in aneasier and simpler manner than the former. It may be experimentally de-monstrated, that an equable motion is produced by the action of these curveson each other: if we cut two boards into forms terminated by them, dividethe surfaces by lines into equal or proportional angular portions, and fixthem on any two centres, wc shall find that as they revolve, whatever partsof the surfaces may be in contact, the corresponding lines will always meeteach other. (Plate XV. Fig. 190 . . 192.)

Both of these methods may be derived from the general principle, that theteeth of the one wheel must be of such a form, that their outline may bedescribed by the revolution of a curve upon a given circle, while the outlineof the teeth of the other wheel is described by the same curve revolvingwithin the circle. It has been supposed by some of the best authors that theepicycloidal tooth has also the advantage of completely avoiding friction;this is however by no means true, and it is even impracticable to invent anyform for the teeth of a wheel, which will enable them to act on other teethwithout friction. In order to diminish it as much as possible, the teethmust be as small and as numerous as is consistent with strength and dura-bility ; for the effect of friction always increases with the distance of thepoint of contact from the line joining the centres of the wheels. In calcu-lating the quantity of the friction, the velocity with which the parts slideover each other has generally been taken for its measure: this is a slightinaccuracy of conception, for, as we have already seen, the actual resist-ance is not at all increased by increasing the relative velocity; but theeffect of that resistance, in retarding the motion of the wheels, may be shown,from the general laws of mechanics, to be proportional to the relative ve-locity thus ascertained. When it is possible to make one wheel act onteeth fixed in the concave surface of another, the friction may be thus dimi-nished in the proportion of the difference of the diameters to their sum. ^