HELMHOLTZ IN BERLIN 



dynamical system considered by him, the internal 

 forces are assumed to be conservative, but the external 

 forces depend on the time and work done by them and 

 can be specially calculated. 



Helmholtz based his investigations on two functions 

 of the co-ordinates and the velocities, namely, Hamil- 

 ton's Principal Function, and the total energy. The 

 total energy is the sum of the potential and kinetic 

 energies, while the Principal Function (called by 

 Helmholtz the Kinetic Potential] is the difference of 

 these two quantities. The principle of least action 

 was then defined in these words : If calculated for 

 equal short intervals of time, the negative mean value 

 of the kinetic potential, while the system passes by its 

 natural path from one configuration to another, is a 

 minimum as compared with its value by all other 

 contiguous paths described in the same time from the 

 initial to the final configuration. For cases of equili- 

 brium, the kinetic potential becomes the potential 

 energy ; and the principle just enunciated becomes 

 the well-known condition for stable equilibrium, 

 namely, the potential energy must be a minimum. 

 From the minimum theorem of the kinetic potential, 

 Helmholtz then deduced the principle of the constancy 

 of energy, and applied the principle to important 

 general problems in thermodynamics and electro- 

 dynamics. He considered that the truth of the 

 principle went far beyond the dynamics of ponderable 

 masses, and that it was the general law for all reversible 

 247 



