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spaces. However, when in n aysiem the same molecule lijis bad a 

 great niimOer of collisions, the extension has had a projectioji on the 

 space determined by the axes of the coordinates of the h^ molecule 

 for each of these collisions. So when everv molccide has had a 

 great number of collisions tiie abo\e mentioned xc^ctor of vidocity 

 has passed roujul the sphere on which the poinis of velocilN lie, a 

 great many times in every direction. The [lomts of Nclocily which 

 originally covered an element of the s|)herical shell, will now occupy 

 the whole spherical surface many times. As, however, the points 

 of the sphere, where a point of velocity is after one, two etc. revo- 

 lutions in a certain direction, come to the same thing with I'espect 

 to the distribution of the velocities of the molecules, as in all the 

 preceding cases we may again assume that tinally the densily is the 

 same all over the spherical surface. 



For the rest of the elements of the same spherical shell originally 

 occupied the same reasoning holds. If there are also systems in the 

 ensemble with another kinetic energy, the |)oin(s disperse also here 

 homogeneously in spherical layers; as, however, one kind cannot 

 pass into another, the density may be ditferent for the layers. It is 

 the same as in the distribution of i)lace when the gas masses are in 

 different vessels. ^) 



Now the problem of the placing of the molecules. For this i)urpose we 

 consider a part of the phase-extension, originally determined with regard 

 to place by limits lying indelinitely near each other, but occupying 

 a fmite part of the space of xelocilies. Now we have to demonstrate 



1) Also without Borel's way of re|)resentalioii tlie above mentioned dispersion 

 may be imagined to a certain extent. In each of a number of systems the mole- 

 cules liave the same velocities, liul different places. Now it will entirely depcjid 

 on tiie mutual situation e.g. of the molecules 1 and 2 in connection with their 

 velocities, what the direction of the normal becomes in the collision, and so up 

 to a certain extent, what the final velocities will be. In any case we get an in- 

 linitely large number from a single pair of velocities. Wlicreas we bad before infinitely 

 narrow limits between which the components of velocity had lu lie, now we get 

 a finite region. If we have chosen a definite one from the pairs of final velocities 

 so also a definite velocity of the Isl molecule, this molecule may have all kinds 

 of positions with regard to the 3rd molecule, against which it will presently strike, 

 so also the normal of collision can have all kinds of directions, and so the limits 

 thought infinitely nr.rrow are again removed to a finite distance, etc. If we now 

 take as variables the angles of the general vector of velocity with the axes of 

 velocity instead of the components of velocity, the moving apart of the limits will 

 yield a larger amount of occupied angles, so thai finally lliere is occupied an amount 

 of a large number of times St. If we now take into account, that increase of such 

 an angle by an amount 2t has no effect, we arrive at considerations and results 

 of quite the same nature as in the problems ti'eated above. 



