54 MOLECULAR MOTION AND ITS ENERGY 27 



means, but the values of the speed which correspond to 

 the arithmetic means of the energy of the different particles. 

 (See 10.) 



By a simple mathematical consideration we may easily see 

 that the method by which Joule and Clausius calculated 

 the mean values of the molecular speed must in all cases 

 give numbers which are greater than the real arithmetic 

 means. Consider n particles moving respectively with the 

 speeds a, &, c . . . ; the mean value of these different speeds 

 is then 



H = (a + b + c + . . . ) /n. 



Calculating also the mean value of the molecular energy of 

 a particle, 



E = \mG\ 



wherein m, as before, represents the molecular mass, and G 

 the mean value of the speed, we obtain 



E = 



so that the mean value G of the speed introduced by Joule 

 and Clausius has the signification 



G 2 = (a 2 + Z> 2 + c 2 + ...)/* 

 Comparing this expression with 



H 2 = (a 2 + b* + c 2 + . . . + 26c + 2ca + Zab + . . . )/n 2 , 



which, as we see from the known relation 



a 2 + V > 2ab, 

 leads to 



or, since each square octfurs n times in the numerator, to 



H 2 < (a 2 + b 2 + c 2 + ...)/ 

 we find 



that is, the arithmetic mean value H of the speed is less than 

 the mean value G calculated by Joule and Clausius from 

 the mean kinetic energy. 



If Maxwell's law is true, this relation, which holds in 

 general between the two mean values, takes the following 



