FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 123 



between thern requires that a certain function of the parameters — in 

 heat, the temperature — should have the same value in all the systems, 

 then the third property of heat expressed by the law of Carnot, namely, 

 the restricted capacity for transformation, is here in evidence. 



The three last memoirs of Helmholtz are in part in correction and in 

 part in amplification of these investigations, fundamental for the prin- 

 ciples of mechanics, on the principle of least action. They rest upon 

 the conceptions worked out by Faraday, Maxwell, and Hertz, accord- 

 ing to which the electrical oscillations in the ether are, in their velocity 

 of propagation, their nature as transverse vibrations, and the conse- 

 quent possibility of phenomena of polarization, refraction, and reflection, 

 exactly analogous to the oscillations of light and heat, and constitute 

 the method of performance of the apparent actions at a distance by 

 conveying the force through the intervening medium. 



In the memoir on "The principle of least action in electrodynamics" 

 (1892), Helmholtz investigated whether the empiric laws of electrody- 

 namics, as expressed in Maxwell's equations, may be brought into the 

 form of a law of minima. Upon the ground of considerations already 

 referred to, he was able to show that the ponderomotive forces could 

 in fact be deduced from the generalized Hamilton principle in a form 

 completely agreeing with Maxwell's theory. The energy was divided 

 into two parts which played the same roles toward one another as the 

 potential and actual energy for ponderable masses. The electrical 

 energy corresponded to the potential energy of masses at rest so long 

 as no changes occurred in the moments or electric currents, while the 

 magnetic energy corresponded to vis viva. 



He penetrated still further in the electromagnetic theory of light, 

 and proceeding from the consideration that the dispersion of light is 

 brought about only at the boundaries of spaces which contain ponder- 

 able masses besides the ether, he sought, in an article which appeared 

 in the year 1892, to explain the color dispersion with the aid of this 

 theory. The mathematical theory of Maxwell requires that pondero- 

 motive forces must exist along with the electric oscillations of the ether, 

 which are capable of setting in motion heavy atoms which lie in the 

 ether. Helmholtz showed that the material particles must also have 

 charges of electricity, so that in the equations of motion to be formed 

 the electric movements due to the electricity of each particle, since 

 they are of different magnitude and direction, and are also affected by 

 other than electrical forces — as, for example, inertia, friction, etc. — are 

 to be distinguished from those in the free ether and particularly inves- 

 tigated in order to deduce the laws of color dispersion. 



His general "Consequences of Maxwell's theory of the motions of 

 the pure ether" (1893) is of very great interest. Capacity for motion 

 is ascribed to the ether, and it is represented as receiving motions 

 imparted to it by ponderable masses which permeate it, and as being 

 thus moved along with them. Such mixtures occur in all substances 



