Maximum Effect of Machines* 279 



i = the quantity of matter, which if placed at the working 

 point, would oppose the same resistance as the inertia of 

 all the parts of the machinery. 



Since D and d are the radii of the wheel and axle, we shall 

 have D : d : : r : , a weight equal to that part of the power p 

 which is in equilibrium with the resistance. We have, therefore, 

 p -- as an expression for the effective force of the power ; and 



as D is the distance at which this force is applied, we have 



p D r d 



to represent the force which is employed in giving a rotatory 

 motion to the machine. The resistance which friction opposes 

 to this force will be/d ; the moment of inertia of the power p 

 will be as wo 2 ; the moment of inertia of the resistance as nd 2 , 

 and the moment of inertia of the machinery will be as id 2 . 

 Since the moving force is diminished by the resistance of friction, 

 we shall have JOD rd /d for the moving force; and since 

 the resistance arises from the moment of inertia of the resistance, 

 the moment of inertia of the power, and that of the machinery, it 

 will be as mo 2 -f nd 2 + *<? 2 - But the velocity is propor- 

 tional to the moving force directly and to the resitance inverse- 

 ly ; therefore 



the rotatory velocity will be 



pv r $ fd 



mo 2 -J-wd 2 -fid 2 ' 



Now, since the velocities of the impelled and working points are as 

 their distances from the centre of motion, or as D and d, we shall 

 obtain these velocities respectively by multiplying the rotatory 

 velocity by D and d ; and as the work performed is equal to 

 the resistance multiplied by the velocity of the working point: 



we shall have for the velocity of the impelled point 



D 2 rod 



mD 2 -|- n d 2 -f i d a 

 for the velocity of the working point 



