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



We multiply the x l equation by sb lt the y l equation by y lf and 

 so on, and add together all the equations thus obtained. This 

 process gives 



the summation extending to all the particles. 

 The left-hand member of this equation is 



where T is the kinetic energy of the system (Art. 104). 



If then we multiply by dt, and integrate both sides with 

 respect to t from t up to t, we obtain the equation 



T) dy + (Z + Z ) dz, 



where T Q is the kinetic energy at time t , and the integrals on the 

 right are line-integrals taken along the paths of the particles from 

 their positions at time t to their positions at time t. 



The right-hand member of the equation last written is the 

 work done by all the forces (internal and external) acting on all 

 the particles as they move from their positions at time t to their 

 positions at time t, and the equation can be stated in words : the 

 change of the kinetic energy of the system, as it passes from one 

 set of positions to another set of positions, is equal to the work 

 done by the forces in the same displacements. 



When the system is conservative, let V be the potential energy 

 in the position at time t, and V the potential energy in the 

 position at time t , then the right-hand member of the equation is 

 V V. The equation can therefore be written 



and this result can be expressed in words : The total energy of 

 the system, or the sum of the kinetic energy and the potential 

 energy, is a constant quantity. 



The equations of motion of a conservative system always possess 

 an integral which expresses the constancy of the total energy, and 

 this integral is, as we have seen, equivalent to the statement that 

 the change of kinetic energy between two positions is equal to the 

 work done by the forces on the particles as they pass from the first 

 position to the second position. The latter statement is true 



