i94 THE POPULAR SCIENCE MONTHLY 



such unusual phases of matter as radiation in rarefied gases, where the 

 system has no temperature at all, because its internal motions have 

 not settled down to a definite average. Helmholtz's dynamic proof of 

 the second law assumes the existence of cyclic systems with reversible 

 circular motions, like those of the gyroscope or the governor of a 

 steam engine, in other words it assumes matter to be made of rotatioual 

 or gyrostatic stresses in the ether. Gibbs's " Elementary Principles of 

 Statical Mechanics" (1903) 133 is based upon no assumptions whatever 

 except that the systems involved are mechanical, obeying the equations 

 of motion of Lagrange and Hamilton. " One is building on insecure 

 foundations," he says, "who rests his work on hypotheses concerning 

 the constitution of matter," and his statistics deal, not with tbe behavior 

 of gas molecules in isolated systems, but with large averages of vast 

 ensembles of systems of the same kind (solid, liquid or gas), " differing 

 in the configurations and velocities which they have at any given 

 instant, and differing not merely infinitesimally, but it may be so as to 

 embrace every conceivable combination of configuration and veloci- 

 ties." The problem is, given the distribution of these ensembles in 

 phase (i. e., in regard to configuration and velocities) at some one time, 

 to find their distribution at any required time. To solve this problem 

 Gibbs establishes a fundamental equation of statistical mechanics, which 

 gives the rate of change of the systems in regard to distribution in 

 phase. A particular case of this equation gives the condition for sta- 

 tistical equilibrium or permanent distribution in phase. Integration 

 of the equation in the general case gives certain constants relating to 

 the extent, density and probability of distribution of the systems in 

 phase, which Gibbs interprets as the principles of conservation of 

 " extension in phase," of " density in phase," and of " probability in 

 phase." Boltzmann found that when the gas molecules have more than 

 two degrees of freedom, the equations can not be integrated and further 

 progress is impossible. He got around this difficulty by using Jacobi's 

 " method of the last multiplier," which integrates the equations of mo- 

 tion. Gibbs found that the principle of " conservation of extension-in- 

 phase," supplies such a Jacobian multiplier, " if we have the skill or 

 good fortune (he says) to perceive that the multiplier will make tbe 

 first member of the equation an exact differential." Boltzmann's prob- 

 ability coefficient is used as the index of the canonical distribution of 

 ensembles, and when the exponent of this coefficient is zero, the latter 

 becomes unity, producing a distribution in phase called "micro- 

 canonical," in which all the systems in the ensemble have the same 

 energy, as in Maxwell's " state." After demonstrating the possibility 

 of irreversible phenomena in the various ensembles, and after a careful 

 study of their behavior when isolated, subjected to external forces or to 



133 « Yale Bicentennial Publications," 1903. Translated into German by 

 Ernst Zermelo, Leipzig, 1905. 



