Partition of Kinetic Energy. 113 



same time small in comparison with 2T. If we write 



T = ??K = i«P 2 , (43) 



we have 



Limit {l-^} |W ~ f =^-^ 4K = ^^ 2p2 . • • (44) 



The limit of the fraction containing the Y functions may 

 be obtained by the formula 



T(m + 1) =e- m m m s/ (2m7r) ; 

 and the limiting form of (42) becomes 



£^ _i£_ or ^ &. . (45) 



v/(2tt) 4/(2K)' ° i/(27r) P l } 



It may be observed that the integral of (45) between the 

 limits + go is unity, and that this fact might have been used 

 to determine the numerical factor. 



Maxwell's result is given in terms of a quantity k, analo- 

 gous to K, and defined by 



\f=k (46) 



It is 



^W)h~ kdk (47) 



The corresponding form from (45) is 



i -j k 



~2Kdk (48) 



s/(2>ir)2s/(kK) 



In like manner if we inquire what proportion of the whole 

 number of systems have momenta lying within the limits 

 denoted by dp 1 dp 2 . . . dp r , where r is a number very small 

 relatively to n, we get 



e -<ft»+ft»+. . • +?v 2 )/4K dpi dpammm dpr 

 or, if we prefer it, 



e-<**+*r+>..+*f>i*** dpidp9mu , dpr 



{^(2tt)P P 1 * ' 



These results follow from the general expression (38), in 

 the same way as does (45), by stopping the multiple integra- 

 tion at an earlier stage. The remaining variables range over 

 values which may be considered in each case to be unlimited. 



Phil. Mag. S. 5. Vol. 49, No. 296. Jan. 1900. I 



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