bar, on the edge of which teeth are raised, which engage themselves in cor- 

 responding teeth on the wheel or axle. Such a bar is called a rack ; and an 

 instance of its use may be observed in the manner of working the pistons of an 

 air-pump. 



The power of the wheel and axle being expressed by the number of times 

 the diameter of the axle is contained in that of the wheel, there are obviously 

 only two ways by which this power may be increased, viz., either by dimin- 

 ishing the diameter of the axle, or increasing that of the wheel. In cases 

 where great power is required, each of these methods is attended with practi- 

 cal inconvenience and difficulty. If the diameter of the wheel be considerably 

 enlarged, the machine will become unwieldy, and the power will work through 

 an unmanageable space. If, on the other hand, the power of the machine be 

 increased by reducing the thickness of the axle, the strength of the axle will 

 become insufficient for the support of that weight, the magnitude of which had 

 rendered the increase of the power of the machine necessary. To combine 

 the requisite strength with moderate dimensions and great mechanical power 

 is, therefore, impracticable in the ordinary form of the wheel and axle. This 

 has, however, been accomplished by giving different thicknesses to different 

 parts of the axle, and carrying a rope, which is coiled on the thinner part, 

 through a wheel attached to the weight, and coiling it in the opposite direction 

 on the thicker part, as in fig. 18. To investigate the proportion of the power 



Fig. 18. 



Fig. 19. 



to the weight in this case, let fig. 19 represent a section of the apparatus at 

 right angles to the axis. The weight is equally suspended by the two parts 

 of the rope, S and S', and therefore each part is stretched by a force equal to 

 half the weight. The moment of the force which stretches the rope S is half 

 the weight multiplied by the radius of the thinner part of the axle. This force, 

 being at the same side of the centre with the power, co-operates with it in sup- 

 porting the force which stretches S', and which acts at the other side of the 

 centre. The moments of P and S are equal to that of S 7 ; and therefore, 'if P 

 be multiplied by the radius of the wheel, and added to half the weight multi- 

 plied by the radius of the thinner part of the axle, we must obtain a sum equal 

 to half the weight multiplied by the radius of the thicker part of the axle. 

 Hence it is easy to perceive, that the power multiplied by the radius of the 

 wheel is equal to half the weight multiplied by the difference of the radii of the 

 thicker and thinner parts of the axle ; or, what is the same, the power multi- 

 plied by the diameter of the wheel is equal to the weight multiplied by half the 

 difference of the diameters of the thinner and thicker parts of the axle. 



A wheel and axle constructed in this manner is equivalent to an ordinary 

 one, in which the wheel has the same diameter, and whose axle has a diame- 

 ter equal to half the difference of the diameters of the thicker and thinner 

 parts. The power of the machine is expressed by the proportion which the 

 diameter of the wheel bears to half the difference of these diameters ; and 



