PROFESSOR IN BERLIN 353 



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When a traveller wants to cross a mountain ridge, the 

 height of the pass is of course the maximum height to which 

 he must rise, while if he crossed at any other point he would 

 have to climb still higher. This is called in mathematics 

 a maximo-minimum of height, and all such values, as well as 

 the complete minima and maxima of variable magnitudes, are 

 known as limiting values.' 



So long as this principle was applied only to the obvious 

 motions of ponderable bodies, it seemed to have no other 

 real content than that contained in Newton's equations of 

 motion, but it soon acquired much greater significance, when 

 the investigation was extended to bodies within which per- 

 sistent concealed motions were proceeding. Helmholtz 

 ascribes a fundamental theoretical interest to the formulation 

 of the Law of Least Action, in relation to the course of all 

 natural processes, inasmuch as the energy components, with 

 which mechanics was originally concerned, entirely disappear 

 from the problem, and ' the question is now reduced to the two 

 chief forms of energy, the total value of which is unalterable 

 and eternal, but which fluctuate to and fro in the most 

 complex forms of manifestation in natural bodies. By this law 

 the ebb and flow of energy is brought under a brief but all- 

 embracing rule, whereby everything that happens in the world 

 is resolved simply and solely into a question of the distribution 

 of energy in time '. 



As the first and most striking example of the application of 

 the law of least action to the investigation of bodies, in the 

 interior of which concealed motion is proceeding, Helmholtz cites 

 the ' originally mysterious and incomprehensible laws of 

 the mechanical theory of heat' of Sadi Carnot, Clausius, 

 and Boltzmann ; he points out that F. E. Neumann expressed 

 the law of the electromagnetic action of closed galvanic 

 currents in the same form as results from the law of least 

 action, and remarks that all the hypotheses advanced by 

 W. Weber, Clerk Maxwell, Riemann, C. Neumann, and Clausius, 

 for the resolution of the reciprocal actions of many electrical 

 masses into elementary actions, have resulted in forms of calcula- 

 tion which correspond to the law of least action, although 

 what corresponds to vis viva and inertia in electricity is ex- 

 pressed in a different form from those used for ponderable 



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