Wheel and Axle. 115 



machine, we are able to raise a very considerable w< ight with 

 a moderate force. 



194. Toothed wheels serve several purposes. Sometimes we 

 employ them to augment a force, at others to increase a velocity, 

 often to change the direction of a motion, and still more frequently 

 to adapt motion to certain periods of time, or to render sensible 

 certain motions or spaces, that the eye cannot distinguish. 



Several toothed wheels W, X, Y, Z, being connected together Fig.lOS. 

 by the pinions w, a?, ?/, z, it is proposed to find the ratio of 

 the power </, applied to the first wheel, to the weight or effort p, 

 sustained by the last pinion. Let D, Z>, D", D"', be the greater 

 distances or radii of the wheels, d, d', d", d'", the less distances 

 or radii of the pinions. We shall consider the effort made by 

 the leaf of any one of the pinions upon the tooth of the neigh- 

 bouring wheel, as a power applied to this last ; then E, E', E", 

 being these efforts, we shall have 186 



q : E : : d : D, E : E' : : 6' : &, 



E' : E" : d" : D", E" : p : : d'" ' D"' ; 



whence, by taking the products of the corresponding terms, 

 q : p : : d d' d" d'" : D- D' D" D"' ; 



that is, the power is to the weight as the product of the radii of 

 all the pinions to the product of the radii of all the wheels. 



If, for example, the radius of each pinion is one tenth of that 

 of the corresponding wheel, we should have 



q : p :: 1 : 10 X 10 X 10 X 10, 

 1 : 10000; 



that is, a power of one pound would counterbalance a weight of 

 10000 pounds. 



What is gained, however, in point of force by the use of 

 wheels and pinions, is lost in respect to velocity. Indeed while 

 the wheel W turns once, the pinion w, turning in the same time, 

 causes to pass only as many teeth of the wheel A 7 , as it has leaves ri S- 106 - 

 in its own circumference, so that if the wheel X has 48 teeth, and 

 the pinion w six leaves, the wheel X would make only T 8 or | 

 part of a revolution, while W turns once round ; it will hence be 



