138 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



neither increased nor diminished by any internal action, but can 

 exist in any of the forms of which energy is susceptible. 



When we do not retain the conception of bodies as continuous, 

 but make a more complete analysis of the circumstances of any 

 motion, we are guided by the same principle in the form first 

 stated, viz. : when all motions are allowed for, the total energy of 

 the system is constant, and exists only in the forms of kinetic and 

 potential energy. 



154. Power. When work is done by the action of a system 

 S upon a system S' the forces exerted by the particles of S upon 

 the particles of S' do work in the displacements of the particles of 

 S'. In cases where the energy can be localised the energy of the 

 system $' is increased, and that of S diminished, by a quantity equal 

 to the amount of work so done. The number of units of work done 

 in any interval bears a definite ratio to the number of units of 

 time in the interval ; and, when the interval is indefinitely short, 

 this ratio has a limit, which is the rate at which work is being done 

 per unit of time. 



The power of a system acting on another system is the rate per 

 unit of time at which the first system does work upon the second. 



Corresponding to each force between particles of the two 

 systems there is a certain power measured by the product of the 

 magnitude of the force and the resolved part, in its direction, of 

 the velocity of the particle on which it acts, or by the product of 

 the magnitude of the velocity of the particle and the resolved part, 

 in its direction, of the force exerted upon it. The sum of all these 

 powers is the power of the first system acting on the second. 



The power can be measured equally by the rate at which work 

 is done upon the second system or by the rate at which the first 

 system does work. 



Thus in any machine performing mechanical work a certain 

 amount of energy is expended, and an equal amount of work done, 

 per unit of time ; and the machine is said to be " working up to a 

 power " measured by the rate at which the work is done. 



155. Kinetic Energy produced by Impulses. As in 



Article 113 let x, y, z be the resolved parts parallel to the axes of 

 the velocity of the particle of mass m just after an impulse, f , ?), 



