57, 58] WORK AND ENERGY. 109 



and the equation may be written 



Integrating with respect to t between the limits t and t l} 

 T 1V \(dx r \ (dy r \* , [dz r \Y\^ 



(22) ^.|^) + ^j + ^jj^ 



d(C r,y fy r 7 r 



r r 



The square brackets with the affixes t , ti denote that the value 

 of the expression in brackets for t = t Q is to be subtracted from the 

 value for t = ti. 



The integral on the right of (22), which may be written 

 X r dx r + Y r dy r + Z r dz r > 



\ 



denotes the work done by the forces of the system on the particle 

 m r during the motion from t to t l} and the sum of such integrals 

 denotes the total work done by the forces acting on the system 

 during the motion. 



The expression 



/7'r \ 2 frlii \ 2 / r) y \1\ 



U + =M 



the half-sum of the products of the mass of each particle by the 

 square of its velocity, is called the Kinetic Energy of the system. 

 If we denote it by T, the equation (22) becomes 



(23) T tl - T to = 2 r j\X r dx r + Y r dy r -h Z r dz r ). 



This is called the equation of energy, and states that the gain of 

 kinetic energy is equal to the work done by the forces during the 

 motion. 



The equation of energy assumes an important form in the 

 particular case that the forces acting on the particles depend only 

 on the positions of the particles, and that the components may be 



