H6 Statics. 



seen that the wheel X goes just so much slower than W, Y so 

 much slower than X, and so on. 



195. From what is above said, it will be perceived how, bj 

 means of toothed wheels, the velocity may be augmented in any 

 Fig.107. given ratio. Let there be, for example, the toothed wheel W, 

 acting upon the pinion w ; it is clear, that during one revolution 

 of W, the pinion zv will turn as many times as the number of leaves 

 in the pinion is contained in the number of teeth of the wheel ; 



that is, during one revolution of the wheel, the pinion will turn 



times, JV denoting the number of teeth in the wheel, and v the 

 number of leaves in the pinion. 



If therefore the axis of the pinion w carries a wheel, which 

 acts also on a pinion a:, we shall see that during one revolution 



,7V* 

 of the wheel X^ or of the pinion w, the pinion x will turn T 



times, JV denoting the number of teeth in the wheel X, and V* the 

 number of leaves in the pinion x. Therefore while the wheel X 



N 

 makes a number of turns expressed by , that is, during one 



revolution of the wheel W, the pinion x revolves a number of 



JV' jV N' 'V 



times expressed by X or -f . And by reasoning in 



this manner for a greater number of wheels and pinions, it will be 

 perceived that the number of times that the last pinion turns, 

 during one revolution of the first wheel, is expressed by a frac- 

 tion having for its numerator the product of the number of teeth 

 in the several wheels, and for a denominator the product of the 

 number of leaves in the several pinions. 



When it is asked, therefore, what must be the number of 

 teeth and leaves for a proposed number of wheels and pinions, 

 in order that the velocity of the last piece shall be to that of the 

 first in a given ratio, the question is indeterminate, that is, one 

 which admits of several answers. Two examples will suffice 

 to show how we ought to proceed in questions of this kind. 



We will suppose that it is required to find how many teeth 

 must be given to the two wheels W and Jf, and how many leaves 

 to the pinions w and , in order that the pinion x may make 50 

 revolutions while the wheel W makes one. We shall have 



