FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 119 



iu addition to the potential and actual energy of weighable masses, 

 thermal, electrodynamic, and electromagnetic equivalents of work 

 appear. .For he had expressed the laws of reversible heat processes in 

 the form of Lagrange's equations of motion, and therefore through the 

 law of the minimum characteristic value of the kinetic potential. It 

 was found, however, that the temperature as a measure of the thermal 

 motions did not, like the velocities in the kinetic energy of a ponder- 

 able system, enter into the expression only in the second power. Hence 

 if it is desired to determine the general characteristics of systems which 

 are governed by the principle of least action, the assumption must be 

 abandoned that the velocities enter only in the values of the kinetic 

 energy, in the form of homogeneous functions of the second degree, and 

 the principle must be discussed under the supposition that the princi- 

 pal function is any function whatever of the coordinates and the veloci- 

 ties. The immediate occasion for these general considerations on the 

 part of Helmholtz was the investigation of the form of the kinetic poten- 

 tial demanded for Maxwell's theory of electrodynamics, in which the 

 velocities of electricity enter as a function of the second degree whose 

 coefficients are not constants as in the measure of the value of the 

 kinetic energy for ponderable systems, and where besides these appear 

 linear functions of the velocity, whenever the action of permanent 

 magnets comes into account. 



Since the phenomena of light may be in the main explained under 

 the hypothesis that the ether is a medium with properties similar to 

 those of ponderable elastic solids, the principle of least action must 

 be looked upon as applicable to the motion of light. Thus Helmholtz 

 regarded the proper domain of this principle as far ou treadling the 

 bounds within which is included the mechanics of weighable bodies, 

 and he held it as in the highest degree probable that it is the general 

 law of all reversible natural processes. It is, moreover, to be noted 

 that irreversibility rests not in the nature of things but in the limita- 

 tions of our means of investigation, which do not enable us to reor- 

 ganize unorganized atomic motions so as, for example, to reverse the 

 motion of all atoms affected by the motions characteristic of heat. 



The general validity of the principle of least action makes it of great 

 value in formulating the laws of new classes of phenomena, in that it 

 embraces in a single mathematical expression all the essential condi- 

 tions of these phenomena. All cases of physical processes in which 

 the kinetic potential contains the velocities in linear members were 

 called by Helmholtz instances of hidden motion. It was shown that 

 the principle of least action, as expressed in the above-mentioned gen- 

 eral form, embraces the principle of the conservation of energy, and 

 that the value of the energy may be determined from the values of the 

 kinetic potential. As it does not, on the other hand, appear that in 

 all cases where the constancy of energy is preserved the principle of 

 least action is obeyed, the latter asserts more than the former, and 



