AND OF THE SECOND LAW 59 



of its own elementary volume, the product of said probabilities 

 will likewise be proportional to the product 



dx-dy-dz-dZ-drj-d^^a ..... (12) 



of the two elementary volumes. Here a can be regarded as a sort 

 of fictitious volume or region, constructed, say, in a six-dimensional 

 space. 1 



The extent of such an elementary region is very minute in 

 comparison with the total space under consideration, but still 

 it must be conceived as sufficiently large to embrace many mole- 

 cules, otherwise its state would not be one of " elementary chaos." 

 On account of the equivalence here of probability and number 

 of concurring molecules, we may for the present say that the 

 number of the latter is proportional to the magnitude of this 

 elementary region a. But before we proceed further this last 

 statement must be subjected to a correction, for we temporarily 

 assumed above that there was an equal distribution of molecules 

 and velocities throughout the whole volume. Now at the start, 

 in defining the contemplated state, it was distinctly announced 

 that there was an unequal distribution of such elementary con- 

 ditions, the law of their distribution being given by the known 

 number of molecules in each elementary volume dV and in each 

 constructed elementary velocity volume d^-dy-d^. This correction 

 is effected by the introduction of a finite proportionality factor, 



(13) 



which can be any given function of the location and velocity 

 co-ordinates, so long as it fulfills the one condition (put in abbre- 

 viated form), 



Vf-'-N, ....... (14) 



where N = the total number of molecules of the gas. 



1 Such a fictitious space does not occur in the proof of MAXWELL'S distribution, 

 because there conditions are simpler. See foot-note to p. 49. 



