27] EQUATION OF ENERGY. 67 



Integrating equation 20) with respect to t between the limits 



and ^, 



21) 



The square brackets with the affixes t Q , ^ denote that the value 

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

 value for t = t 1 . 



The integral on the right of 21), which may be written 



X r dx r -f Y r dy r -f 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 lf and the sum of such integrals 

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

 during the motion. The equation 21) thus becomes 



22) T tl - T to 



to 



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 

 represented by the partial derivatives of a single function of the 

 coordinates 



7 



r r 



In this case the expression 



2r{XrdXr + Y r dy r + Z r dg r ] =^ 

 is the exact differential of the function U, and the integral 



that is the work done in the motion, does not depend upon the 

 paths described by the various particles, but only on the initial and 

 final configurations of the system, since 



