HELMHOLTZ IN BERLIN 



moving system for a short interval of time, is usually 

 defined as the product of double the average kinetic 

 energy during that time, multiplied by the time ; or, 

 shortly, it is double the time-integral of the kinetic 

 energy. Discussions based on this definition give the 

 action in terms of the initial and final co-ordinates of 

 the system and the time prescribed for the motion. 

 Again, action may be expressed in another way, as 

 the sum of the products of the average momentum 

 for the spaces through which particles move, multi- 

 plied by the length of the spaces. For double the 

 kinetic energy during a short interval of time multi- 

 plied by the time, is equal to the average momentum 

 during that time multiplied by the space described. 

 Investigations on the second definition of action 

 give the action in terms of the initial and final 

 co-ordinates of the system, and the constant sum 

 of the potential and kinetic energies. 1 



It is not easy finding simple examples of the appli- 

 cation of the principle ; nor can it be said that the 

 fundamental dynamic significance of the principle has 

 been made clear. Quantities like velocity, momentum, 

 kinetic energy, potential energy, are what might be 

 called instantaneous properties of a system. Their 

 values are definite at each instant, and can be assigned 

 without reference to what has taken place previously. 

 But the Action of a system at any instant depends on 



1 Thomson and Tait, Natural Philosophy, vol. i., part i., sec. 326 to 

 368, pp. 337-439- 



245 



