112 Lord Ray lei gh on the Law of 



The whole number of systems is to be found by integrating 

 (38), the integral being so taken as to give the variables all 

 values consistent with the condition that />i 2 +/?2 2 + « • 'V\-\ 1S 

 not greater than unity. Now 



J.'J W i- Pl *- ... -^_ l} - rci^Dj j 1 *> "** 



and . . (39) 



^ P ^- idPi= m^^, . . (40) 



in which T(J)= s/tt. Thus the whole number of systems is 



r(i«) ' 



or on restoration of 2T, equal to 2E — 2 V , 



ffffil-"i2E-2V}i»-' (41) 



1 \2 n ) 



To this we shall return later ; but for the present what we 

 require to ascertain is the distribution of one of the momenta, 

 say p u irrespectively of the values of the remaining momenta. 

 By (39), (40) the number of systems for which p Y lies between 

 pi and p\ + dp x in comparison with the whole number of 

 systems is 



r(i)T( 



This is substantially Maxwell's investigation, and (42) corre- 

 sponds with his equation (51). As was to be expected, the 

 law of distribution is the same for all the momenta. From 

 the manner of its formation, we note that the integral of (42), 

 taken between the limits p l = + \/(2T), is equal to unity. 



Maxwell next proceeds to the consideration of the special 

 form assumed by (42), when the number n of degrees of 

 freedom is extremely great *. This part of the work seems to 

 be very important ; but it has been much neglected, probably 

 because the result was not correctly stated. 



Dropping the suffix as unnecessary, we have to consider 

 the form of 



1 " 2TJ 



when n is very great, the mean value of p 2 becoming at the 

 * The particular cases where w=2, or n=3, are also worthy of notice. 



■(i«-i)V 2T/ V(2T)- • • {il > 



{ 



