370 MATHEMATICAL APPENDICES 



10*. Hypotheses Used in the Proof 



Maxwell's law of distribution refers to the state which a 

 group of gaseous molecules finally attains as its state of equi- 

 librium in consequence of their encounters. If this state once 

 >/occurs it is maintained to the last in unchanged fashion. But, 

 strictly speaking, it can only be reached when the number of 

 gaseous molecules is unlimited ; for by an encounter any value 

 whatever "of velocity may 'result, and only with an infinite 

 number of particles can all possible values of the speed be 

 actually existing at each moment. 



If the number of particles is limited, Maxwell's law must 

 " x be understood otherwise. Since the state of motion of the group 

 of particles is altered in a perceptible degree at every single 

 encounter between two particles, Maxwell's distribution cannot 

 exist at every moment, but will occur with exactness only when 

 the changing states which succeed each other in the course of a 

 sufficiently long period are all taken into account together. If all 

 these different distributions did not succeed each other, but 

 occurred simultaneously together in an unlimited number of 

 particles, the law would not thereby be changed ; but Maxwell's 

 law must be equally valid in both cases. 



After this remark we can proceed to investigate more closely 

 the function required which expresses the value of the probability. 

 For this purpose let us consider a large number Z, say 1,000 or 

 100,000, of the changing states succeeding each other, which a 

 group of N particles pass through. On the whole, then, NZ 

 different states of a single particle come into account. Among 

 these numerous cases it will often happen that a given particle 

 m } attains a velocity the components of which in three rectangular 

 directions are u^v^w^. The number of cases in which this occurs 

 we may represent by a function of the form 



for it must be proportional to the number NZ of states, and it 

 must further depend on the values u^ v lt w l of the components. 

 For Z = co the value of the function 



which expresses this law of dependence is the probability of 



