388 MATHEMATICAL APPENDICES 18 * 



The different values of the absolute spaed are separated by this 

 most probable value W in such wise that other values of the 

 speed, whether greater or less, have the less probability the more 

 they differ in value from W. 



The law of this distribution is graphically represented in 26 

 on p. 52. The ordinate of the curve is such that 



y dx v/N, 



so that it simply denotes the probability of a value w : the abscissa 

 on the contrary is 



x = 



and is therefore equal to the ratio of the corresponding value w to 

 the most probable value W. 



19*. Mean Values of the Speed and Energy 



From the trend of the curve we easily see that the most 

 probable value of the speed is no average value in the usual sense 

 of the words. But the arithmetical mean value ft of all the 

 values of the speeds, when each is reckoned according to the 

 frequency of its occurrence, is given by 



ft = 4?i 



We may compare with this mean value of the molecular speed 

 that corresponding to the mean energy of a particle in motion 



E = 



o 



which agrees with the formula already found in 16*, this other 

 mean value G of the molecular speed being therefore given by 



/A_ 



V 2&m' 



the meaning of which is as simple and important as that of the 

 arithmetical mean value ft. We have in fact to understand by 

 G- that speed which all the particles would have if without 

 addition or subtraction of energy the speeds of all were made equal. 

 Then the simple relation in which these two mean values 

 stand to each other is given by the formula 



G = ON/ (37r/8) = 1-08540 



