372 MATHEMATICAL APPENDICES 10* 



the chance of the event that, among all the changing states of the 

 group, the first particle w t has the components u\, v { , io l , the 

 second m 2 the components ^ 2 V 2> W 2> ar| d so on > * ne l as t w# having 

 the components U N , V N , W N . 



In this nothing is assumed regarding the time at which these 

 values of the velocity for the individual particles of the group 

 occur. We may therefore apply the formula to the values of the 

 components which the N particles may have at any given moment 

 whatever. 



At another time the particles have different velocities, and 

 u'i, v'i, w\ may then be the components of m lt also u' 2 , v f , w f 2 

 those of m. 2 , &c. The probability of this changed state is then 

 given by the product 



^iJP,....^3-J^^ , 



which contains the same function F as the first, but with different 

 arguments. 



The two products are equal to each other in value, for each of 

 the states of distribution is as likely as the other, because, accord- 

 ing to our supposition, both form part of the state of equilibrium 

 which finally ensues. For equilibrium, therefore, it results that 

 the function F must satisfy the equation 



or that the product 



F(u lt v it w { } 



must always have one and the same value for all systems of the 

 values of the variables that occur. 1 



1 This theorem is proved differently by Pirogoff (Journ. d. russ. phys.- 

 chem. Ges. 1885, xvii.). P i r o g o f f~considers an infinite number of gaseous 

 particles which are in a state of equilibrium. From this infinite multitude 

 N particles are taken out. The probability of finding given values of the 

 components u, v, w among these N particles is expressed by the given 

 product. Of the same magnitude is the chance of taking a second group of 

 JV other particles which, though having different components of velocity from 

 those of the first group, have the same total kinetic energy and the 

 same motion of their centroid. Maxwell's law follows likewise from this 

 assumption. 



