Distribution e~ 2h x and van der Waals' Theorem. 417 



the law of equal partition, agreeing in substance with Maxwell 

 and Rayleigh. The law may for these modes of motion be 

 true, but I think it is Tiot proved, except by the great per- 

 sonal authority of the three eminent physicists. Ipsi dixerunt. 

 We have for the kinetic energy of any system 



E=a in 2 + a 2 r 2 2 + , . +0.1^, . . . (9) 



where r 1 r 2 &c. are Boltzmann , s et momentoids." Consider 

 those systems, out of an infinite number of systems, at a given 

 level of potential, for which the kinetic energy lies between 

 E and E+rfE. 



I will now make an assumption C. " All combinations of 

 the r'a which satisfy (9) are equally probable." That being 

 assumed, it follows that the mean value of any function, <£, 

 of the rs is given by 



£=jj. . . <f> dr x ...dr jj . . . dr x . . . dr n , . (10) 



the limits of integration being given by (9). 



Substituting, as Boltzmann does, E for i\ as one of the 

 n variables, we deduce 



**§->^-'-*>l§"^**--'** (n) 



and if <j> = a l r 1 2 , this leads, as Boltzmann proves, to 



ain 2 =l • • VE— a 2 r 2 l — . .-a n rn 2 dr 2 . . . drj 



dr 2 . . dr n , (12) 



-5 



E 



VE-arf-. ■-«V n a 



which proves the proposition. 



The mathematics are very interesting. But the logical 

 process is I think defective. Because the assumption C 

 is in no degree more self-evident than the proposition which 

 we propose to prove. What is really proved is, not that 

 every ar 2 is in fact equal to every other, but that, in the 

 present state of our knowledge, we have no reason for saying 

 that any of them, or if any which, is greater than any other. 



It may be said that Boltzmann does not make assumption 

 C. It is true he does not expressly make it. But if as- 

 sumption C is untrue, the whole reasoning falls to the ground. 

 The assumption then is made if not expressly, then by necessary 

 implication. It is an essential condition ; but it is not, I think, 

 an axiom. 



