ON MACHINERY. 135 



Where a broad strap runs on a wheel, it is usually confined to its situation, 

 not by causing the margin of the wheel to project, but, on the contrary, 

 by making the middle prominent : the reason of this may be understood 

 by examining the manner in which a tight strap running on a cone would 

 tend to run towards its thickest part. Sometimes, also, pins are fixed in 

 the wheels, and admitted into perforations in the straps ; a mode only 

 practicable where the motion is slow and steady. A smooth motion may 

 also be obtained, with considerable force, by forming the surfaces of the 

 wheels into brushes of hair. (Plate XV. Fig. 189.) 



More commonly, however, the circumferences of the contiguous wheels 

 are formed into teeth, impelling each other, as with the extremities of so 

 many levers, either exactly or nearly in the common direction of the cir- 

 cumferences ; and sometimes an endless screw is substituted for one of the 

 wheels. In forming the teeth of wheels, it is of consequence to deter- 

 mine the curvature which will procure an equable communication of 

 motion, with the least possible friction. For the equable communication 

 of motion, two methods have been recommended ; one, that the lower part 

 of the face of each tooth should be a straight line in the direction of the 

 radius, and the upper a portion of an epicycloid, that is, of a curve de- 

 scribed by a point of a 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 produce 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 demonstrated that an equable motion is pro- 

 duced 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 a second circle. It has been supposed by some of the 

 best authors that the epicycloidal tooth has also the advantage of com- 

 pletely 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 durability ; for the effect of friction always 

 increases with the distance of the point of contact from the line joining 



* For a demonstration of these propositions, see Airy on the Teeth of Wheels, 

 Trans, of the Camb. Phil. Soc. ii. 279. 



