73 .] ENERGY AND WORK. 4I 



Force can therefore be defined as the rate at which the kinetic 

 energy cJianges with the space. (Compare the end of Art. 60.) 



72. Integrating the last equation (10), we find 



Jv 

 Fds = ^mv' 2 ^mv 2 . (n) 



The product F (s r s) of a constant force F into the space s' s 

 described in the direction of the force, and in the case of a 

 variable force, the space-integral \ Fds, is called the work of 

 the force for this space. 



The equations (8) and (11) show that the work of a force is 

 equal to the corresponding change of tJie kinetic energy. 



We have here assumed that the force acts in the direction of 

 motion of the particle. A more general definition of work 

 including the above as a special case will be given later (Art. 

 232 sq.). 



The ideas of energy and work have attained the highest 

 importance in mechanics and mathematical physics within com- 

 paratively recent times. Their full discussion belongs to 

 Kinetics. 



73. According to their definitions, both momentum (Art. 56) 

 and force (Art. 60) may be regarded mathematically as mere 

 numerical multiples of velocity and acceleration, respectively. 

 They are therefore so-called vector-quantities ; i.e. a momentum 

 as well as a force can be represented geometrically by a segment 

 of a straight line of definite length, direction, and sense. 

 Moreover, as they are referred to a particular point, viz. to the 

 point whose mass is m, the line representing a momentum or a 

 force must be drawn through this point ; the line has therefore 

 not only direction, but also position ; i.e. a momentum as well 

 as a force is represented geometrically by a rotor (compare Kine- 

 matics, Arts. 57, 68, 291 sq.}. 



It follows that concurrent forces, for instance, can be com- 



