HAMILTON'S PRINCIPLE 323 



inch, then B invariably rises through two inches. The mechanism 

 may be a lever, an arrangement of pulleys, or clockwork. But 

 whether it is any one of these, or something entirely different from 

 any of them, it will be known that the motion of rope A down- 

 wards can be restrained by exerting on rope B a force equal to half 

 of that applied to A. This fact follows from the principle of virtual 

 work, quite apart from any conjecture as to the nature of the 

 hidden mechanism. Now the question before us is as follows: 

 Can we, without any knowledge of the hidden mechanism, discover 

 what motion of the ropes will ensue, if they are started in any 

 given way. And the answer is that we can, provided we know 

 the amount of energy involved in a motion of any kind, i.e. pro- 

 vided we know the kinetic energy of every motion, and also the 

 potential energy of every configuration. 



So also, to pass from analogies to realities, we can, without any 

 knowledge of the ultimate mechanism of the universe, discover 

 what motion will ensue from any initial conditions, provided that 

 we know the kinetic and potential energies of all configurations 

 of the portion of the universe with which we are dealing. 



HAMILTON'S PRINCIPLE 



264. Let us suppose that any single particle of a material sys- 

 tem has at any instant coordinates x lt y lt z lt its mass being ra^and 

 that it is acted upon by forces of which the resultant has compo- 

 nents X lt Y lt Z r Let the velocity of this particle have components 

 u lt v lt w lt so that dx 



u. = - > etc. 

 dt 



Then, if the motion of this particle is governed by Newton's laws, 

 we shall have , 



(149) 



