112 BELL SYSTEM TECHNICAL JOURNAL 



to a maximum value of F, there would still be a correlation of a similar 

 sort, though not so marked a one: the higher the velocity-component in the 

 x-direction, the lower would the y and the z components be likely to be. 

 One could imagine distributions for which the higher the velocity-component 

 in the x-direction, the higher would the y and the z components be likely 

 to be. 



The "assumption of independence" is, that in the most probable state 

 there is no correlation at all. Whether the x-component of the velocity of a 

 molecule is high or low is a detail which has no influence whatever on the 

 possible or the probable values of the y and the z components. Low values 

 of Vy go just as well and just as abundantly with low values of Vx as with high, 

 and reversely. 



The Maxwell-Boltzmann law, as I said, is the distribution-law which 

 conforms to both the assumption of isotropy and the assumption of inde- 

 pendence. So the question arises: do those two assumptions have the 

 quality of plausibility and of convincingness, which make the average per- 

 son say "Surely these must be the attributes of the most probable state of a 

 gas!" I do not know what result a referendum on this question would give, 

 but it is my guess that most physicists would feel more satisfied with these 

 than they would with the Maxwell-Boltzmann distribution-law if it were 

 tossed out to them with the bare afi&rmation "This is assumed to be the 

 attribute of the most probable state". Clearly this is how Maxwell felt, and 

 there is no better guide than the intuition of a Maxwell. 



The foregoing question is something else than the question whether the 

 assumptions, and the Maxwell-Boltzmann distribution-law which follows 

 from them, are truly the attributes of the most probable state. It is a strange 

 historical fact that not for many years after the promulgation of this famous 

 law, and not till after both of its sponsors were dead, was there any proper 

 test of it. The derivations of the law were exercises in abstract and un- 

 renumerated thought. Nevertheless experiment — applied to thermionic 

 electrons, to molecules of ordinary gases, to thermal neutrons — came at long 

 last to justify Maxwell. To any who may feel that the assumption of 

 independence is in itself too reasonable to require any proof, I disclose that 

 in other forms of statistics this assumption is declared to be false, except as 

 an approximation. 



The "Maxwell statistics" therefore consists in the main of the statement: 



The most probable state of a gas is that in which isotropy and independence 

 prevail among the velocity-vectors of the molecules. 



We now require some terminology and some notation. 



I take for granted an understanding of the terms 'Velocity-vector" and 

 "distribution-in-velocity", these being learned by physicists out of kinetic 

 theory if not out of S.M. A velocity-vector may be replaced by a point 



