40 KINETICS OF A PARTICLE. [72 



We find, therefore, 



dx 



dt 



or, multiplying by dt and integrating, 



dx+ Ydy + Zds), (3) 



where V Q is the initial velocity. 



The left-hand member represents the increase in the kinetic 

 energy of the particle ; the right-hand member represents the 

 work done by the resultant force ; and equation (3) expresses 

 the equality between the work done and the change in the 

 kinetic energy, that is, the principle of work or of kinetic 

 energy for the curvilinear motion of a particle (comp. Art. 

 47). Sometimes the name principle of vis viva is given to 

 this proposition, the term vis viva, or living force, meaning the 

 same, as kinetic energy, or, in older works, twice the kinetic 

 energy. 



72. The principle of work can be deduced still more directly 

 from the equations (i). Multiplying the former of these equa- 

 tions by vdt=ds, we find 



d( J mi?) = Fds cos 1^ ; 

 hence, integrating, 



= Fds cos ^, (4) 



where z/ is the velocity of the particle at the place specified by 

 s (comp. Part II., Art. 72). 



73. The principle of kinetic energy gives a first integral of 

 the equations of motion whenever the integration indicated in 

 the right-hand member of (3) or (4) can be performed. We 

 proceed to investigate under what conditions this integration 

 becomes possible. 



In the most general case the components X, Y, Z, in (3), as 

 well as the tangential force Fcos t/r in (4), are functions of the 



