27, 28] WORK OF AN IMPULSE. 73 



where / is the total impulse. From the equation of motion we have 



" dt ~ dt 

 so that we obtain for the work 



*i *i i 



W=jFds =JFvdt = lj[v, + s (v, - t, )] d* = * I( Vl + ). 



t t 



Thus we find as before for the work of an impulsive force the 

 product of the impulse by the average velocity at the beginning and 

 end of the action. It is evident that the same is true for the 

 infinitesimal work done by an ordinary, that is finite force, during 

 an infinitesimal interval. This conception of the impulse will be 

 useful to us hereafter, in connection with the following. For a 

 system of particles, we have for the kinetic energy, 



Now the kinetic energy is known when we know the velocities of 

 every particle of the system, as well as their masses, no matter what 

 their positions. If we consider T as a function of the velocities, 

 we have accordingly 



dT dT cT 



M xr = m r v xr = Q- V ; M yr = ni. r Vy r = Q-- > M zr = m r v zr = ^ ,; 



or the momentum components of any particle are the partial 

 derivatives of the kinetic energy of the system, considered as a 

 function of all the velocities of the particles, by the respective 

 velocity -components. Thus we may write 



$ T 



a-;; + 



which by the theorem of Euler is true for any homogeneous quadratic 

 function. 



28. Particular Case of Porce- function. The conditions 

 necessary for the existence of a force -function being 23), we must 

 have, since 



d*U_ d*U d*U d z U d*U d*U 



dxdy dydx oydz dzdy dzdx dxdz 



dY r _ dX r - 



== ' 



It will be shown below ( 31) that these conditions are also sufficient. 



