352 HERMANN VON HELMHOLTZ 



is determined by the relative positions of the masses, but in 

 every case can only be ascertained by a knowledge of their 

 particular nature. The discussion of the different forms of 

 energy as well as the condition of its conversion from one form 

 into another represents, according to Hertz, the subject-matter 

 of the whole of physics and chemistry. 



From the Law of the Constancy of the Sum of Kinetic 

 and Potential Energy, the important consequence followed 

 immediately that, if a system of bodies is at rest in any position, 

 from which every movement compatible with the restrictions 

 of the system tends to a position with higher potential energy, 

 no vis viva, and therefore no motion of the bodies, can arise ; 

 there must accordingly be stable equilibrium in any position 

 at which the potential energy is at a minimum. The Law of the 

 Conservation of Energy, however, tells us nothing in the case 

 of motion, as to the succession of positions which the system 

 has to traverse, in order to get from a given initial to a given 

 final position : it is this which is elucidated by the principle of 

 least action. 



Leibniz had already asked himself what work can be done 

 by the inertia which distinguishes space filled by mass from 

 geometrical bodies ; he found that the work was the greater, 

 in proportion to the magnitude of the mass in motion, the 

 length of the path through which it moves, and the velocity 

 with which it is moving. The amount of the action was thus the 

 product of mass, distance, and velocity, or, what amounts to the 

 same, of vis viva and time. We thus arrive at a law that 

 completely embraces all possible motions of any given number 

 of material bodies under the influence of conservative forces, 

 in part exerted reciprocally, in part suffered from fixed centres, 

 as is summed up in the law of least action. According 

 to this, when such a material system passes with free and 

 undisturbed motion from a given initial to a given final 

 position, with a definite energy-value, the action has a limiting 

 value, and for short phases of the motion this is a minimum. 

 Accordingly, with given values of energy, the inertia must 

 always bring the masses in motion to their end by a path 

 which, at any rate for short distances, exacts the least amount of 

 work. To define the mathematical conception of the limiting 

 value Helmholtz says : 



