618 Mr. J. H. Jeans on the 



in which f a is the law of distribution of the molecules of the 

 first kind, and so on. The conditions to which f a) f p .\. are 

 subject are, firstly, conditions of the form 



^§f a dudvdwdxdydz=l, . . . (42) 



and similar conditions for f p , &c, and, secondly, the single 

 energy condition 



SaffffiT \m a {u 2 + v 2 + vfi)fa du dv dw dx dy <fe = E. . (43) 



2' 



The equations giving fa, fp ... are found to be 



1 + log fa + X a + ifim a {u 2 + r + w 2 ) = 0, 



and this gives a solution of the form 



f a — Ke~ ^Mu 2 + v 2 + w 2 ). 



f^Be-^^+^+v 2 ), Ac. . . . (44) 



The constant h is the same throughout, whence the Boyle- 

 Oharles-Avogadro law can be deduced at once. 



III. — Molecules of Finite Size. 



§ 36, The analysis of § 27 applies without alteration to 

 this case, being independent of the size of the molecules. The 

 analysis of §§ 19-26 also applies if a correction is made for 

 the regions of the generalized space which are excluded by 

 equations (3). It will be found that this correction has no 

 effect on the ultimate result. 



We also see from § 13 that the " chance of two molecules 

 having assigned positions " is independent of the velocities of 

 these molecules — a result which is in direct opposition to the 

 views put forward by Burbury*. 



IV. — The Law of Equipartition. 



§ 37. The method of the present paper is in no way limited 

 to the simple type of molecule which has been discussed up to 

 the present. If each molecule has m degrees of freedom we 

 must suppose our generalized space to possess 2?nN dimen- 

 sions, and we can proceed exactly as before. If E now 

 denotes the energy of a single molecule, we find that no 

 matter what function of the 2m coordinates E may be, the 

 law of distribution is oi the form 



f =Ae -2h-E ...... (45) 



* ' Kinetic Theory of Gases,' chapters v. and vi. 



