Partition of Kinetic Energy. 115 



without reference to the question of configuration. We thus 

 arrive at the Maxwellian law of velocities in a single gas, as 

 well as the relation between the velocities in a mixture of 

 molecules of different kinds first laid down by Waterston. 



The assumptions which we have made as to the practical 

 regularity of statistics are those upon which the usual theory 

 of ideal gases is founded ; but the results are far more 

 general. Nothing whatever has been said as to the character 

 of the forces with which the molecules act upon one another, 

 or are acted upon by external agencies. Although for distinct- 

 ness a gas has been spoken of, the results apply equally to a 

 medium constituted as a liquid or a solid is supposed to be. 

 A kinetic theory of matter, as usually understood, appears to 

 require that in equilibrium the whole kinetic energy shall be 

 equally shared among all the degrees of freedom, and within 

 each degree of freedom be distributed according to the same 

 law. It is included in this statement that temperature is a 

 matter of kinetic energy only, e. g. that when a vertical column 

 of gas is in equilibrium, the mean velocity of a molecule is the 

 same at the top as at the bottom of the column. 



Reverting to (37), (41), in order to consider the distribution 

 of the systems as dependent upon the coordinates independ- 

 ently of the velocities, we have, omitting unnecessary factors, 



{-E-Y}^~id qi dq 2 . . . dq n . . . . (51) 



If n = 2, e.g. in the case already considered of a single 

 particle moving in two dimensions, or of two particles 

 moving in one dimension, or again whatever n may be, pro- 

 vided Y vanish, the first factor disappears, so that the 

 distribution is uniform with respect to the coordinates q x . . q n . 

 If n > 2 and V be finite, the distribution is such as to favour 

 those configurations for which V is least. 



" When the number of variables is very great, and when 

 the potential energy of the specified configuration is very 

 small compared with the total energy of the system, we may 

 obtain a useful approximation to the value of {E — Vp n_1 

 in an exponential form ; for if we write (as before) E = «K, 



\E-y}i n - l = & l - 1 e-^ 2K .... (52) 



nearly, provided n is very great and V is small compared 

 with E. The expression is no longer approximate when V 

 is nearly as great as E, and it does not vanish, as it ought 

 to do, when V = E » (Maxwell) . 



In the case of gas composed of molecules whose mutual 

 influence is limited to a small distance and which are not 

 subject to external forces, the distribution expressed by (51) 



