9* MAXWELL'S LAW 369 



justified if the mathematical formulae are differently interpreted, v 

 The differential calculus expresses the property of a function in 

 having a maximum in the same way as its behaviour iri maintain- 

 ing its value when its argument is varied. My formulae, which I 

 again give in 12* in unchanged form, need not therefore contain 

 the meaning that Maxwell's law is the most probable of all 

 conceiyable_Ia!W_s_; but they show, as Boltzmann had already 

 recognised before Pirogoff, that among a limited number of 

 molecules the values of the speeds may be distributed in different / 

 ways, and that all these different ways possess an ^qual degree y 

 of probability. On this theorem Pirogoff founded his proof, 

 which he carried out by the same process as I did mine. Since 

 this altered proof by Pirogoff was originally published in 

 Eussian only, and is therefore little known elsewhere, I will here 

 give his method at length. 



Pirogoff starts with the assumption that out of an unlimited 

 number of gaseous molecules, whose motions have already become 

 in accordance with Maxwell's law, a group of N particles is 

 so picked out that the choice is guided only by chance. He 

 then investigates the probability that given values of the velocity- 

 components u t v, w are to be found in this group. By these 

 values also the average state of the motion of the group, its 

 average speed and energy, are determined. If now a second 

 group of N particles is again picked out by chance, there will be 

 other values of the components in this second group ; but the 

 average values of the speed and energy may, in spite of this, be 

 the same as with the first group. The probability that each 

 group will have the same average value is the same for both. 



That we may arrive at the formulae of_my_jormer proof by 

 the stricter way suggested by Pirogoff was shown me on 

 February 5, 1882, by Gustav Liibeck, with whom I was then / 

 corresponding on the subject of my memoir and his. 1 I had 

 then, unfortunately, no opportunity of making use of this com- 

 munication. I will therefore now lay the foundation of this 

 proof in a way which will, I hope, be valid as a more comprehen- 

 sive improvement. 



1 Festschrift zur zweiten Sdcularfeier des Friedrichswerderschen 

 Gymnasiums in Berlin, 1881, p. 295 ; Ueber die Bewegung eines kugelfdrmigen 

 Atoms. 



B B 



