the Kinetic Theory of Gases. 345 



(c) Maxwell proceeds as if every particle of one system 

 impinged upon one of the other system at each stage of the 

 process— i. e. he calculates the transference of energy as if 

 each pair of particles, one from each system, had simultaneously 

 a typical impact. This neglect of the immensely greater 

 number of particles which either had no impact or impinged 

 on others of their own group makes the calculated rate of 

 equalization far too rapid. 



(d) Attention is not called to the fact that impacts between 

 particles are numerous in proportion to their relative speed, 

 nor is this consideration introduced in the calculations. 



(e) Throughout the investigation each step of the process 

 of averaging is performed (as a rule) before the expressions 

 are ripe for it. 



2. In seeking for a proof of Maxwell's Theorem, I found it 

 absolutely essential to the application of the statistical method 

 to premise : — 



(A) That the particles of the two systems are thoroughly 

 mixed. 



(B) That, in any region containing a very large number of 

 particles, the particles of each kind acquire and maintain the 

 error-law distribution of velocities. This will be referred to 

 as the "special" state. The disturbances of this arrange- 

 ment produced in either system by impacts on members 

 of the other are regarded as being promptly repaired by 

 means of the internal collisions in the system itself. This 

 is the sole task assigned to these internal collisions. We 

 assume that they accomplish it, so we need not further allude 

 to them. 



[The warrant for these assumptions is sought in the fact 

 that only a small fraction of the whole particles are at any 

 instant in collision ; i. e. that each particle advances, on the 

 average, through a considerable multiple of its diameter before 

 it encounters another.] 



(C) That there is perfectly free access for collision between 

 each pair of particles, whether of the same or of different 

 systems ; and that, in the mixture, the number of particles of 

 one kind is not overwhelmingly greater than that of the other 

 kind. 



[This is one of the essential points which seem to be wholly 

 ignored by Boltzmann and his commentators. There is no 

 proof given by them that one system, while regulating by its 

 internal collisions the distribution of energy among its own 

 members, can also by impacts regulate the distribution of 

 energy among the members of another system, when these 

 are not free to collide with one another. In fact ; if (to take 



