MEMORIAL TO CLASSICAL STATISTICS 111 



strange and sombre title "Memorial to the Classical Statistics". Yet even 

 as a past episode, it is worthy of remembrance; its didactic value may yet be 

 great; and perhaps the human mind may some day stretch its powers to the 

 point of conceiving the classical and the new statistics as aspects of a single 

 whole, as it has lately stretched itself to the extent of uniting the wave- 

 picture and the corpuscular picture of matter and of light. 



The Maxwell Statistics 



Since the main concern of S.M. is with the "most probable state", one 

 sees that its principal content must be made up of assertions about that most 

 probable state. Maxwell made such an assertion. He wrote down a for- 

 mula for the distribution-in-velocity of the molecules of a gas. It is the 

 formula now called "the Maxwell-Boltzmann distribution-law", which is so 

 well known to the readers of this journal that I will not bother to write it down 

 until there is actual need Jor having it on the page. Maxwell might have 

 said bluntly: "This is the distribution which I will assume for the most 

 probable state"; and having said so, left it at that. He did not leave it at 

 that, and presumably he would have been dissatisfied so to leave it, as most 

 of us would be. Instead, he postulated a pair of attributes for the most 

 probable state, and showed that if these are the attributes, then the distri- 

 bution is according to that formula. 



The attributes which Maxwell postulated are "isotropy" and "independ- 

 ence". 



The former is easy enough. One assumes that in the most probable state, 

 the distribution of velocities of the molecules is isotropic. Nothing can 

 usefully be added to this simple statement. 



The latter is a little harder to grasp. Perhaps it can best be exhibited by 

 describing a couple of imagined cases for which it would not be valid. Sup- 

 pose for instance that all of the molecules have the same speed — the same 

 magnitude of velocity, though their velocity-vectors be pointed in all 

 directions. Let this common value of speed be denoted by V, and let any 

 direction chosen at random be made the axis of x in an ordinary coordinate- 

 frame. If a molecule happened to be travelling with such a velocity that 

 the component thereof along the .T-axis, z^x let us call it, was just equal to V, 

 then it would be a certainty that Vy and v^ , the y and z components of the 

 velocity, were both of them zero. If a molecule happened to be travelling 

 in such a way that Vx was zero, then either Vy or v^ or both of them would have 

 to be different from zero, and the square root of the sum of the squares of 

 Vy and lis would have to be equal to V. There would consequently be a 

 correlation between the values of the three components, and the probable — 

 nay even the possible — values of any one of them would be affected by those 

 of the other two. If the molecules had a uniform distribution of speeds up 



