176 LECTURE XV. 



circle rolling on the wheel, of which the diameter must be half that of the 

 opposite wheel; and in this case it is demonstrable that the plane surface of 

 each 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, to 

 form all the surfaces into portions of the involutes of circles, or the curves 

 described 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 an 

 easier and simpler manner than the former. It may be experimentally de- 

 monstrated, that an equable motion is produced by the action of these curves 

 on each other: if we cut two boards into forms terminated by them, divide 

 the surfaces by lines into equal or proportional angular portions, and fix 

 them on any two centres, we shall find that as they revolve, whatever parts 

 of the surfaces may be in contact, the corresponding lines will always meet 

 each other. (Plate XV. Fig. 190 . . 192.) 



Both of these methods may be derived from the general principle, that the 

 teeth of the one wheel must be of such a form, that their outline may be 

 described by the revolution of a curve upon a given circle, while the outline 

 of the teeth of the other wheel is described by the same curve revolving 

 within the circle. It has been supposed by some of the best authors that the 

 epicycloidal tooth has also the advantage of completely avoiding friction; 

 this is however by no means true, and it is even impracticable to invent any 

 form for the teeth of a wheel, which will enable them to act on other teeth 

 without friction. In order to diminish it as much as possible, the teeth 

 must 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 the 

 point 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 slide 

 over each other has generally been taken for its measure: this is a slight 

 inaccuracy of conception, for, as we have already seen, the actual resist- 

 ance is not at all increased by increasing the relative velocity; but the 

 cflect 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 on 

 teeth 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. If 



