13G LECTURE XV. 



the centres of the wheels. In calculating 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 resistance is not at all increased by in- 

 creasing the relative velocity ; but the effect 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 velocity 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 diminished in the proportion 

 of the difference of the diameters to their sum. If the face of the teeth, 

 where they are in contact, is too much inclined to the radius, their mutual 

 friction is not much affected, but a great pressure on their axes is pro- 

 duced ; and this occasions a strain on the machinery, as well as an increase 

 of the friction on the axes. 



If it is desired to produce a great angular velocity with the smallest pos- 

 sible quantity of wheelwork, the diameter of each wheel must be between 

 three and four times as great as that of the pinion on which it acts. Where 

 the pinion impels the wheel, it is sometimes made with three or four teeth 

 only, but it is much better in general to have at least six or eight ; and 

 considering the additional labour of increasing the number of wheels, it 

 may be advisable to allot more teeth to each of them than the number re- 

 sulting from the calculation ; so that we may allow 30 or 40 teeth to a 

 wheel acting on a pinion of 6 or 8. In works which do not require a great 

 degree of strength, the wheels have sometimes a much greater number of 

 teeth than this ; and on the other hand, an endless screw or a spiral acts as 

 a pinion of one tooth, since it propels the wheel through the breadth of one 

 tooth only in each revolution. For a pinion of six teeth, it would be 

 better to have a wheel of 35 or 37 than 36 ; for each tooth of the wheel 

 would thus act in turn upon each tooth of the pinion, and the work would 

 be more equally worn than if the same teeth continued to meet in each 

 revolution. The teeth of the pinion should also be somewhat stronger than 

 those of the wheel, in order to support the more frequent recurrence of 

 friction. It has been proposed for the coarser kinds of wheelwork, to di- 

 vide the distance between the middle points of two adjoining teeth into 

 30 parts, and to allot 16 to the tooth of the pinion, and 13 to that of the 

 wheel, allowing 1 for freedom of motion. 



The wheel and pinion may either be situated in the same plane, both 

 being commonly of the kind denominated spur wheels, or their planes may 

 form an angle : in this case one of them may be a crown or contrate wheel, 

 or both of them may be bevilled, the teeth being cut obliquely. According 

 to the relative magnitude of the wheels, the angle of the bevil must be dif- 

 ferent, so that the velocities of the wheels may be in the same proportion 

 at both ends of their oblique faces : for this purpose the faces of all the 

 teeth must be directed to the point where the axes would meet. (Plate XV. 

 Fig. 193, 194.) 



In cases where a motion not quite equable is required, as it sometimes 

 happens in the construction of clocks, but more frequently in orreries, the 

 wheels may either be divided a little unequally, or the axis may be placed 



