FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 115 



their formation only the differential coefficients of the so-called ergals, 

 completely determined as a function of the temperature. For processes 

 taking place at constant temperature the value of the potential func- 

 tion is thus determined, and this value he designated as the free energy. 

 So that, if we call the difference between the total inner energy and 

 the ergals the bound energy, this latter, divided by the temperature, 

 gives the quantity termed entropy, already in use. In order to clearly 

 distinguish what had in theoretical mechanics been called the vis viva 

 or kinetic energy from the mechanical equivalent of heat energy, 

 which was to be considered for the most part as invisible molecular 

 kinetic energy, he called the first the kinetic energy of organized motion, 

 defining organized motion — and this distinction is of fundamental signifi- 

 cance for later works of Helmholtz — as such that the components of 

 the velocity of the moving masses may be regarded as the differential 

 coefficients of the space coordinates. Unorganized motion, on the 

 other hand, is such that as in heat the motion of a single particle has 

 no necessary similarity to that of its neighbor, and, on account of the 

 relatively coarse means at our disposal, can not be directly transformed 

 into other forms of energy. In this sense Helmholtz designated the 

 value of the entropy as a measure of the disorganization. If a change 

 of condition proceeding with constant entropy be defined as adiabatic, 

 the entropy becomes the heat capacity for the heat generated during 

 adiabatic processes at the expense of the free energy. For all changes 

 of condition in which the temperature remains constant work is per- 

 formed only at the expense of the free energy, the bound energy chang- 

 ing in amount at the expense of the heat entering and proceeding away. 

 Assembling these results, it appears that all exterior work is done at 

 the expense of the free energy, all evolution of heat is at the expense of 

 the bound energy, and finally for each rise of temperature of the system 

 free energy goes over in definite quantity and becomes bound. From 

 this Helmholtz derived results on the emission and absorption of heat 

 in the formation and breaking up of chemical compounds, which were 

 substantiated by observations with galvanic elements. 



In the last ten years of his life, from 1884 to 1894, he was occupied 

 with his great researches on the principles of mechanics, in which he 

 verified all the theoretical conclusions which he had derived in the 

 course of his long and difficult investigations in the whole range of 

 physics and physiology. 



The general principles of mechanics, the principle of d'Alembert, 

 the law of the motion of a material point, the law of surfaces, the prin- 

 ciple of the conservation of kinetic energy, and the principle of least 

 action were all proved with the assumption of Newtonian forces and 

 rigid connection. It had later been found by observation that the laws 

 so derived were much more general in their application in nature than 

 followed from their proofs, and it had been on the one hand supposed 

 that certain general characteristics of the Newtonian conservative 



