356 HERMANN VON HELMHOLTZ 



affected by variable external influences, if we can regard them 

 as independent of any reaction of the moving system, as e. g. in 

 the case of the forces proceeding from fixed centres. 



'In any case the universality of the law of least action 

 appears to me so far assured that it may be assigned a high 

 value as a heuristic principle and guide in the attempt to 

 formulate laws for new classes of phenomena/ 



Helmholtz then, with a view to more exact investigation, sets 

 out from the law of least action in the form proposed by 

 Hamilton, according to which the negative mean value of the 

 difference in potential and kinetic energy (of the kinetic poten- 

 tial), calculated for each time-element, is minimal in the actual 

 path of the system, and has a limiting value over a longer portion 

 as compared with all adjacent paths leading in the same time 

 from the initial to the final position. Without separating kinetic 

 and potential energy, he develops the analytical expression for 

 this law with the utmost freedom for the nature of the kinetic 

 potential, and derives the form of Lagrange's equations of 

 motion from it, showing that already in the mechanics of ponder- 

 able masses (under special conditions, and with the elimination 

 of single parameters of the problem) these more general forms 

 may arise, in which the two energies are not separated. Even 

 under this most general assumption he deduces the law of the 

 conservation of energy, and finds that it is not true conversely 

 that in every case in which the conservation of energy obtains, 

 the law of least action holds good also. ' The last expresses 

 more than the first, and it is our task to find out what more it 

 expresses/ He shows, on the assumption of the validity of the 

 law of least action, how it is possible from the complete 

 knowledge of the dependence of energy upon the co-ordinates 

 and the velocities to find values for the kinetic potential, and 

 therewith for all the laws of motion of the system. In the 

 mechanics of ponderable bodies the kinetic potential is a homo- 

 geneous function of the second order of the velocities ; yet 

 under certain conditions, with elimination of co-ordinates, 

 Lagrange's equations of motion may stand for the remaining 

 co-ordinates in exactly the same form as for the kinetic potential, 

 in which case the velocities are also linear. In correspondence 

 with this analogy from the mechanics of ponderable bodies, 

 Helmholtz designates other cases of physical processes, in 



