NATURE 735 



tions of the parameters q, attainable by experiment. These equations, 

 as I have said, should conform to the principles of dynamics, and, in 

 particular, to the principle of the conservation of energy, and to that 

 of least action. 



The first of these two principles teaches us that the total energy is 

 constant, and may be divided into two parts : 



(1) Kinetic energy, or vis viva, which depends on the masses of 

 the hypothetical molecules m, and on their velocities. This I shall 

 call T. (2) The potential energy which depends only on the co- 

 ordinates of these molecules, and this I shall call U. It is the sum 

 of the energies T and U that is constant. 



Now what are we taught by the principle of least action? It 

 teaches us that to pass from the initial position occupied at the in- 

 stant t to the final position occupied at the instant t 1} the system 

 must describe such a path that in the interval of time between the 

 instant t and t 1} the mean value of the action i.e., the difference 

 between the two energies T and U, must be as small as possible. The 

 first of these two principles is, moreover, a consequence of the second. 

 If we know the functions T and U, this second principle is sufficient 

 to determine the equations of motion. 



Among the paths which enable us to pass from one position to 

 another, there is clearly one for which the mean value of the action 

 is smaller than for all the others. In addition, there is only such 

 path; and it follows from this, that the principle of least action is 

 sufficient to determine the path followed, and therefore the equations 

 of motion. We thus obtain what are called the equations of La- 

 grange. In these equations the independent variables are the co-ordi- 

 nates of the hypothetical molecules m; but I now assume that we take 

 for variables the parameters q, which are directly accessible to experi- 

 ment. 



The two parts of the energy should then be expressed as a function 

 of the parameters q and their derivatives; it is clear that it is under 

 this form that they will appear to the experimenter. The latter will 

 naturally endeavor to define kinetic and potential energy by the aid 

 of quantities he can directly observe. 1 If this be granted, the system 

 will always proceed from one position to another by such a path that 

 the mean value of the action is a minimum. It matters little that T 

 and U are now expressed by the aid of the parameters q and their 

 derivatives ; it matters little that it is also by the aid of these parame- 

 ters that we define the initial and final positions; the principle of 

 least action will always remain true. 



Now here again, of the whole of the paths which lead from one 



1 We may add that U will depend only on the q parameters, that T will 

 depend on them and their derivatives with respect to time, and will be 

 a homogeneous polynomial of the second degree with respect to these deriva- 

 tives. 



