( 493 ) 



by Boltzmann and Jeans that Maxwell's distribution of velocities in 

 a gas should give the most probable state, and demonstrated that tin \ 

 wrongly assume an equality of' the probabilities a priori that the point 

 of velocity of an arbitrary molecule would fall into an arbitrary 



element of the space. 



The question, however, may be raised whether it would nol be 

 possible to interpret the analysis given by Boltzmann and Jeans in 

 a somewhat different way, so that avoiding the incorrect fundamental 

 assumption, the result could all the same be retained. And I hen this 

 proves really to be the case. When the most probable distribution 

 of' velocities is sought from the ensemble of equally possible combi- 

 nations of' velocities with equal total energy, we make only use 

 of the fact that the different combinations of velocities are equally 

 possible, how they have got to be so is after all of no consequence 

 Or else, it had not been necessary to occupy ourselves with the 

 separate velocities of the molecules and make an assumption as to them. 



This way of looking upon the matter is of exactly the same nature 

 as that constantly followed by Gibbs in his above-mentioned work. 

 Gibbs treats in his book all the time instead of a definite system, 

 an ensemble of systems of the same nature and determined mostly 

 by the same number of' general coordinates and momenta (p x . . . p n , 

 q l . . . q„), which he follows in their general course. Such an ensemble 

 will best illustrate the behaviour of a. system (e.g. a gas-mass), of 

 which only a few data are known and of which the others can assume 

 all kinds of values. He calls such an ensemble micro-canonical when 

 all systems, belonging to it, have an energy lying between E and 

 E-\-dE and for the rest the systems are uniformly distributed over 

 all possibilities of' phase or uniformly distributed over the whole 

 extension-in-phase the energy of which lies between E and E -\- dE. 

 When the energy of a gas-mass is given (naturally only up to a 

 certain degree of accuracy) we should have reason according to Gibbs 

 to study the microcanonical ensemble determined by this energy, and 

 to consider the gas-mass as taken at random from such an ensemble. 

 The extension-in-phase considered is thought to be determined by 



I dp, . . . dq n , but in the case of a gas-mass with simple equal 



molecules this is proportional to 



dx x dy 1 dz., . . . dx n dy n dz n , dx\ dy x dz x . . . dx n dy n dz n , 



■ 



so that we may say that every combination of velocities and con- 

 figuration is of equally frequent occurrence in the ensemble. 



It is now easy to see that when the energv is purely kinetic the 



33 



Proceedings Royal Acad. Amsterdam. Vol. IX. 



